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addMissionEventHandler addMPEventHandler addMusicEventHandler addOwnedMine addPlayerScores addPrimaryWeaponItem addPublicVariableEventHandler addRating addResources addScore addScoreSide addSecondaryWeaponItem addSwitchableUnit addTeamMember addToRemainsCollector addTorque addUniform addVehicle addVest addWaypoint addWeapon addWeaponCargo addWeaponCargoGlobal addWeaponGlobal addWeaponItem addWeaponPool addWeaponTurret admin agent agents AGLToASL aimedAtTarget aimPos airDensityRTD airplaneThrottle airportSide AISFinishHeal alive all3DENEntities allAirports allControls allCurators allCutLayers allDead allDeadMen allDisplays allGroups allMapMarkers allMines allMissionObjects allow3DMode allowCrewInImmobile allowCuratorLogicIgnoreAreas allowDamage allowDammage allowFileOperations allowFleeing allowGetIn allowSprint allPlayers allSimpleObjects allSites allTurrets allUnits allUnitsUAV allVariables ammo ammoOnPylon and animate animateBay animateDoor animatePylon animateSource animationNames animationPhase animationSourcePhase animationState append apply armoryPoints arrayIntersect asin ASLToAGL ASLToATL assert assignAsCargo assignAsCargoIndex assignAsCommander assignAsDriver assignAsGunner assignAsTurret assignCurator assignedCargo assignedCommander assignedDriver assignedGunner assignedItems assignedTarget assignedTeam assignedVehicle assignedVehicleRole assignItem assignTeam assignToAirport atan atan2 atg ATLToASL attachedObject attachedObjects attachedTo attachObject attachTo attackEnabled backpack backpackCargo backpackContainer backpackItems backpackMagazines backpackSpaceFor behaviour benchmark binocular boundingBox boundingBoxReal boundingCenter breakOut breakTo briefingName buildingExit buildingPos buttonAction buttonSetAction cadetMode call callExtension camCommand camCommit camCommitPrepared camCommitted camConstuctionSetParams camCreate camDestroy cameraEffect cameraEffectEnableHUD cameraInterest cameraOn cameraView campaignConfigFile camPreload camPreloaded camPrepareBank camPrepareDir camPrepareDive camPrepareFocus camPrepareFov camPrepareFovRange camPreparePos camPrepareRelPos camPrepareTarget camSetBank camSetDir camSetDive camSetFocus camSetFov camSetFovRange camSetPos camSetRelPos camSetTarget camTarget camUseNVG canAdd canAddItemToBackpack canAddItemToUniform canAddItemToVest cancelSimpleTaskDestination canFire canMove canSlingLoad canStand canSuspend canTriggerDynamicSimulation canUnloadInCombat canVehicleCargo captive captiveNum cbChecked cbSetChecked ceil channelEnabled cheatsEnabled checkAIFeature checkVisibility className clearAllItemsFromBackpack clearBackpackCargo clearBackpackCargoGlobal clearGroupIcons clearItemCargo clearItemCargoGlobal clearItemPool clearMagazineCargo clearMagazineCargoGlobal clearMagazinePool clearOverlay clearRadio clearWeaponCargo clearWeaponCargoGlobal clearWeaponPool clientOwner closeDialog closeDisplay closeOverlay collapseObjectTree collect3DENHistory collectiveRTD combatMode commandArtilleryFire commandChat commander commandFire commandFollow commandFSM commandGetOut commandingMenu commandMove commandRadio commandStop commandSuppressiveFire commandTarget commandWatch comment commitOverlay compile compileFinal completedFSM composeText configClasses configFile configHierarchy configName configProperties configSourceAddonList configSourceMod configSourceModList confirmSensorTarget connectTerminalToUAV controlsGroupCtrl copyFromClipboard copyToClipboard copyWaypoints cos count countEnemy countFriendly countSide countType countUnknown create3DENComposition create3DENEntity createAgent createCenter createDialog createDiaryLink createDiaryRecord createDiarySubject createDisplay createGearDialog createGroup createGuardedPoint createLocation createMarker createMarkerLocal createMenu createMine createMissionDisplay createMPCampaignDisplay createSimpleObject createSimpleTask createSite createSoundSource createTask createTeam createTrigger createUnit createVehicle createVehicleCrew createVehicleLocal crew ctAddHeader ctAddRow ctClear ctCurSel ctData ctFindHeaderRows ctFindRowHeader ctHeaderControls ctHeaderCount ctRemoveHeaders ctRemoveRows ctrlActivate ctrlAddEventHandler ctrlAngle ctrlAutoScrollDelay ctrlAutoScrollRewind ctrlAutoScrollSpeed ctrlChecked ctrlClassName ctrlCommit ctrlCommitted ctrlCreate ctrlDelete ctrlEnable ctrlEnabled ctrlFade ctrlHTMLLoaded ctrlIDC ctrlIDD ctrlMapAnimAdd ctrlMapAnimClear ctrlMapAnimCommit ctrlMapAnimDone ctrlMapCursor ctrlMapMouseOver ctrlMapScale ctrlMapScreenToWorld ctrlMapWorldToScreen ctrlModel ctrlModelDirAndUp ctrlModelScale ctrlParent ctrlParentControlsGroup ctrlPosition ctrlRemoveAllEventHandlers ctrlRemoveEventHandler ctrlScale ctrlSetActiveColor ctrlSetAngle ctrlSetAutoScrollDelay ctrlSetAutoScrollRewind ctrlSetAutoScrollSpeed ctrlSetBackgroundColor ctrlSetChecked ctrlSetEventHandler ctrlSetFade ctrlSetFocus ctrlSetFont ctrlSetFontH1 ctrlSetFontH1B ctrlSetFontH2 ctrlSetFontH2B ctrlSetFontH3 ctrlSetFontH3B ctrlSetFontH4 ctrlSetFontH4B ctrlSetFontH5 ctrlSetFontH5B ctrlSetFontH6 ctrlSetFontH6B ctrlSetFontHeight ctrlSetFontHeightH1 ctrlSetFontHeightH2 ctrlSetFontHeightH3 ctrlSetFontHeightH4 ctrlSetFontHeightH5 ctrlSetFontHeightH6 ctrlSetFontHeightSecondary ctrlSetFontP ctrlSetFontPB ctrlSetFontSecondary ctrlSetForegroundColor ctrlSetModel ctrlSetModelDirAndUp ctrlSetModelScale ctrlSetPixelPrecision ctrlSetPosition ctrlSetScale ctrlSetStructuredText ctrlSetText ctrlSetTextColor ctrlSetTooltip ctrlSetTooltipColorBox ctrlSetTooltipColorShade ctrlSetTooltipColorText ctrlShow ctrlShown ctrlText ctrlTextHeight ctrlTextWidth ctrlType ctrlVisible ctRowControls ctRowCount ctSetCurSel ctSetData ctSetHeaderTemplate ctSetRowTemplate ctSetValue ctValue curatorAddons curatorCamera curatorCameraArea curatorCameraAreaCeiling curatorCoef curatorEditableObjects curatorEditingArea curatorEditingAreaType curatorMouseOver curatorPoints curatorRegisteredObjects curatorSelected curatorWaypointCost current3DENOperation currentChannel currentCommand currentMagazine currentMagazineDetail currentMagazineDetailTurret currentMagazineTurret currentMuzzle currentNamespace currentTask currentTasks currentThrowable currentVisionMode currentWaypoint currentWeapon currentWeaponMode currentWeaponTurret currentZeroing cursorObject cursorTarget customChat customRadio cutFadeOut cutObj cutRsc cutText damage date dateToNumber daytime deActivateKey debriefingText debugFSM debugLog deg delete3DENEntities deleteAt deleteCenter deleteCollection deleteEditorObject deleteGroup deleteGroupWhenEmpty deleteIdentity deleteLocation deleteMarker deleteMarkerLocal deleteRange deleteResources deleteSite deleteStatus deleteTeam deleteVehicle deleteVehicleCrew deleteWaypoint detach detectedMines diag_activeMissionFSMs diag_activeScripts diag_activeSQFScripts diag_activeSQSScripts diag_captureFrame diag_captureFrameToFile diag_captureSlowFrame diag_codePerformance diag_drawMode diag_enable diag_enabled diag_fps diag_fpsMin diag_frameNo diag_lightNewLoad diag_list diag_log diag_logSlowFrame diag_mergeConfigFile diag_recordTurretLimits diag_setLightNew diag_tickTime diag_toggle dialog diarySubjectExists didJIP didJIPOwner difficulty difficultyEnabled difficultyEnabledRTD difficultyOption direction directSay disableAI disableCollisionWith disableConversation disableDebriefingStats disableMapIndicators disableNVGEquipment disableRemoteSensors disableSerialization disableTIEquipment disableUAVConnectability disableUserInput displayAddEventHandler displayCtrl displayParent displayRemoveAllEventHandlers displayRemoveEventHandler displaySetEventHandler dissolveTeam distance distance2D distanceSqr distributionRegion do3DENAction doArtilleryFire doFire doFollow doFSM doGetOut doMove doorPhase doStop doSuppressiveFire doTarget doWatch drawArrow drawEllipse drawIcon drawIcon3D drawLine drawLine3D drawLink drawLocation drawPolygon drawRectangle drawTriangle driver drop dynamicSimulationDistance dynamicSimulationDistanceCoef dynamicSimulationEnabled dynamicSimulationSystemEnabled echo edit3DENMissionAttributes editObject editorSetEventHandler effectiveCommander emptyPositions enableAI enableAIFeature enableAimPrecision enableAttack enableAudioFeature enableAutoStartUpRTD enableAutoTrimRTD enableCamShake enableCaustics enableChannel enableCollisionWith enableCopilot enableDebriefingStats enableDiagLegend enableDynamicSimulation enableDynamicSimulationSystem enableEndDialog enableEngineArtillery enableEnvironment enableFatigue enableGunLights enableInfoPanelComponent enableIRLasers enableMimics enablePersonTurret enableRadio enableReload enableRopeAttach enableSatNormalOnDetail enableSaving enableSentences enableSimulation enableSimulationGlobal enableStamina enableTeamSwitch enableTraffic enableUAVConnectability enableUAVWaypoints enableVehicleCargo enableVehicleSensor enableWeaponDisassembly endLoadingScreen endMission engineOn enginesIsOnRTD enginesRpmRTD enginesTorqueRTD entities environmentEnabled estimatedEndServerTime estimatedTimeLeft evalObjectArgument everyBackpack everyContainer exec execEditorScript execFSM execVM exp expectedDestination exportJIPMessages eyeDirection eyePos face faction fadeMusic fadeRadio fadeSound fadeSpeech failMission fillWeaponsFromPool find findCover findDisplay findEditorObject findEmptyPosition findEmptyPositionReady findIf findNearestEnemy finishMissionInit finite fire fireAtTarget firstBackpack flag flagAnimationPhase flagOwner flagSide flagTexture fleeing floor flyInHeight flyInHeightASL fog fogForecast fogParams forceAddUniform forcedMap forceEnd forceFlagTexture forceFollowRoad forceMap forceRespawn forceSpeed forceWalk forceWeaponFire forceWeatherChange forEachMember forEachMemberAgent forEachMemberTeam forgetTarget format formation formationDirection formationLeader formationMembers formationPosition formationTask formatText formLeader freeLook fromEditor fuel fullCrew gearIDCAmmoCount gearSlotAmmoCount gearSlotData get3DENActionState get3DENAttribute get3DENCamera get3DENConnections get3DENEntity get3DENEntityID get3DENGrid get3DENIconsVisible get3DENLayerEntities get3DENLinesVisible get3DENMissionAttribute get3DENMouseOver get3DENSelected getAimingCoef getAllEnvSoundControllers getAllHitPointsDamage getAllOwnedMines getAllSoundControllers getAmmoCargo getAnimAimPrecision getAnimSpeedCoef getArray getArtilleryAmmo getArtilleryComputerSettings getArtilleryETA getAssignedCuratorLogic getAssignedCuratorUnit getBackpackCargo getBleedingRemaining getBurningValue getCameraViewDirection getCargoIndex getCenterOfMass getClientState getClientStateNumber getCompatiblePylonMagazines getConnectedUAV getContainerMaxLoad getCursorObjectParams getCustomAimCoef getDammage getDescription getDir getDirVisual getDLCAssetsUsage getDLCAssetsUsageByName getDLCs getEditorCamera getEditorMode getEditorObjectScope getElevationOffset getEnvSoundController getFatigue getForcedFlagTexture getFriend getFSMVariable getFuelCargo getGroupIcon getGroupIconParams getGroupIcons getHideFrom getHit getHitIndex getHitPointDamage getItemCargo getMagazineCargo getMarkerColor getMarkerPos getMarkerSize getMarkerType getMass getMissionConfig getMissionConfigValue getMissionDLCs getMissionLayerEntities getModelInfo getMousePosition getMusicPlayedTime getNumber getObjectArgument getObjectChildren getObjectDLC getObjectMaterials getObjectProxy getObjectTextures getObjectType getObjectViewDistance getOxygenRemaining getPersonUsedDLCs getPilotCameraDirection getPilotCameraPosition getPilotCameraRotation getPilotCameraTarget getPlateNumber getPlayerChannel getPlayerScores getPlayerUID getPos getPosASL getPosASLVisual getPosASLW getPosATL getPosATLVisual getPosVisual getPosWorld getPylonMagazines getRelDir getRelPos getRemoteSensorsDisabled getRepairCargo getResolution getShadowDistance getShotParents getSlingLoad getSoundController getSoundControllerResult getSpeed getStamina getStatValue getSuppression getTerrainGrid getTerrainHeightASL getText getTotalDLCUsageTime getUnitLoadout getUnitTrait getUserMFDText getUserMFDvalue getVariable getVehicleCargo getWeaponCargo getWeaponSway getWingsOrientationRTD getWingsPositionRTD getWPPos glanceAt globalChat globalRadio goggles goto group groupChat groupFromNetId groupIconSelectable groupIconsVisible groupId groupOwner groupRadio groupSelectedUnits groupSelectUnit gunner gusts halt handgunItems handgunMagazine handgunWeapon handsHit hasInterface hasPilotCamera hasWeapon hcAllGroups hcGroupParams hcLeader hcRemoveAllGroups hcRemoveGroup hcSelected hcSelectGroup hcSetGroup hcShowBar hcShownBar headgear hideBody hideObject hideObjectGlobal hideSelection hint hintC hintCadet hintSilent hmd hostMission htmlLoad HUDMovementLevels humidity image importAllGroups importance in inArea inAreaArray incapacitatedState inflame inflamed infoPanel infoPanelComponentEnabled infoPanelComponents infoPanels inGameUISetEventHandler inheritsFrom initAmbientLife inPolygon inputAction inRangeOfArtillery insertEditorObject intersect is3DEN is3DENMultiplayer isAbleToBreathe isAgent isArray isAutoHoverOn isAutonomous isAutotest isBleeding isBurning isClass isCollisionLightOn isCopilotEnabled isDamageAllowed isDedicated isDLCAvailable isEngineOn isEqualTo isEqualType isEqualTypeAll isEqualTypeAny isEqualTypeArray isEqualTypeParams isFilePatchingEnabled isFlashlightOn isFlatEmpty isForcedWalk isFormationLeader isGroupDeletedWhenEmpty isHidden isInRemainsCollector isInstructorFigureEnabled isIRLaserOn isKeyActive isKindOf isLaserOn isLightOn isLocalized isManualFire isMarkedForCollection isMultiplayer isMultiplayerSolo isNil isNull isNumber isObjectHidden isObjectRTD isOnRoad isPipEnabled isPlayer isRealTime isRemoteExecuted isRemoteExecutedJIP isServer isShowing3DIcons isSimpleObject isSprintAllowed isStaminaEnabled isSteamMission isStreamFriendlyUIEnabled isText isTouchingGround isTurnedOut isTutHintsEnabled isUAVConnectable isUAVConnected isUIContext isUniformAllowed isVehicleCargo isVehicleRadarOn isVehicleSensorEnabled isWalking isWeaponDeployed isWeaponRested itemCargo items itemsWithMagazines join joinAs joinAsSilent joinSilent joinString kbAddDatabase kbAddDatabaseTargets kbAddTopic kbHasTopic kbReact kbRemoveTopic kbTell kbWasSaid keyImage keyName knowsAbout land landAt landResult language laserTarget lbAdd lbClear lbColor lbColorRight lbCurSel lbData lbDelete lbIsSelected lbPicture lbPictureRight lbSelection lbSetColor lbSetColorRight lbSetCurSel lbSetData lbSetPicture lbSetPictureColor lbSetPictureColorDisabled lbSetPictureColorSelected lbSetPictureRight lbSetPictureRightColor lbSetPictureRightColorDisabled lbSetPictureRightColorSelected lbSetSelectColor lbSetSelectColorRight lbSetSelected lbSetText lbSetTextRight lbSetTooltip lbSetValue lbSize lbSort lbSortByValue lbText lbTextRight lbValue leader leaderboardDeInit leaderboardGetRows leaderboardInit leaderboardRequestRowsFriends leaderboardsRequestUploadScore leaderboardsRequestUploadScoreKeepBest leaderboardState leaveVehicle libraryCredits libraryDisclaimers lifeState lightAttachObject lightDetachObject lightIsOn lightnings limitSpeed linearConversion lineIntersects lineIntersectsObjs lineIntersectsSurfaces lineIntersectsWith linkItem list listObjects listRemoteTargets listVehicleSensors ln lnbAddArray lnbAddColumn lnbAddRow lnbClear lnbColor lnbCurSelRow lnbData lnbDeleteColumn lnbDeleteRow lnbGetColumnsPosition lnbPicture lnbSetColor lnbSetColumnsPos lnbSetCurSelRow lnbSetData lnbSetPicture lnbSetText lnbSetValue lnbSize lnbSort lnbSortByValue lnbText lnbValue load loadAbs loadBackpack loadFile loadGame loadIdentity loadMagazine loadOverlay loadStatus loadUniform loadVest local localize locationPosition lock lockCameraTo lockCargo lockDriver locked lockedCargo lockedDriver lockedTurret lockIdentity lockTurret lockWP log logEntities logNetwork logNetworkTerminate lookAt lookAtPos magazineCargo magazines magazinesAllTurrets magazinesAmmo magazinesAmmoCargo magazinesAmmoFull magazinesDetail magazinesDetailBackpack magazinesDetailUniform magazinesDetailVest magazinesTurret magazineTurretAmmo mapAnimAdd mapAnimClear mapAnimCommit mapAnimDone mapCenterOnCamera mapGridPosition markAsFinishedOnSteam markerAlpha markerBrush markerColor markerDir markerPos markerShape markerSize markerText markerType max members menuAction menuAdd menuChecked menuClear menuCollapse menuData menuDelete menuEnable menuEnabled menuExpand menuHover menuPicture menuSetAction menuSetCheck menuSetData menuSetPicture menuSetValue menuShortcut menuShortcutText menuSize menuSort menuText menuURL menuValue min mineActive mineDetectedBy missionConfigFile missionDifficulty missionName missionNamespace missionStart missionVersion mod modelToWorld modelToWorldVisual modelToWorldVisualWorld modelToWorldWorld modParams moonIntensity moonPhase morale move move3DENCamera moveInAny moveInCargo moveInCommander moveInDriver moveInGunner moveInTurret moveObjectToEnd moveOut moveTime moveTo moveToCompleted moveToFailed musicVolume name nameSound nearEntities nearestBuilding nearestLocation nearestLocations nearestLocationWithDubbing nearestObject nearestObjects nearestTerrainObjects nearObjects nearObjectsReady nearRoads nearSupplies nearTargets needReload netId netObjNull newOverlay nextMenuItemIndex nextWeatherChange nMenuItems not numberOfEnginesRTD numberToDate objectCurators objectFromNetId objectParent objStatus onBriefingGroup onBriefingNotes onBriefingPlan onBriefingTeamSwitch onCommandModeChanged onDoubleClick onEachFrame onGroupIconClick onGroupIconOverEnter onGroupIconOverLeave onHCGroupSelectionChanged onMapSingleClick onPlayerConnected onPlayerDisconnected onPreloadFinished onPreloadStarted onShowNewObject onTeamSwitch openCuratorInterface openDLCPage openMap openSteamApp openYoutubeVideo or orderGetIn overcast overcastForecast owner param params parseNumber parseSimpleArray parseText parsingNamespace particlesQuality pickWeaponPool pitch pixelGrid pixelGridBase pixelGridNoUIScale pixelH pixelW playableSlotsNumber playableUnits playAction playActionNow player playerRespawnTime playerSide playersNumber playGesture playMission playMove playMoveNow playMusic playScriptedMission playSound playSound3D position positionCameraToWorld posScreenToWorld posWorldToScreen ppEffectAdjust ppEffectCommit ppEffectCommitted ppEffectCreate ppEffectDestroy ppEffectEnable ppEffectEnabled ppEffectForceInNVG precision preloadCamera preloadObject preloadSound preloadTitleObj preloadTitleRsc preprocessFile preprocessFileLineNumbers primaryWeapon primaryWeaponItems primaryWeaponMagazine priority processDiaryLink productVersion profileName profileNamespace profileNameSteam progressLoadingScreen progressPosition progressSetPosition publicVariable publicVariableClient publicVariableServer pushBack pushBackUnique putWeaponPool queryItemsPool queryMagazinePool queryWeaponPool rad radioChannelAdd radioChannelCreate radioChannelRemove radioChannelSetCallSign radioChannelSetLabel radioVolume rain rainbow random rank rankId rating rectangular registeredTasks registerTask reload reloadEnabled remoteControl remoteExec remoteExecCall remoteExecutedOwner remove3DENConnection remove3DENEventHandler remove3DENLayer removeAction removeAll3DENEventHandlers removeAllActions removeAllAssignedItems removeAllContainers removeAllCuratorAddons removeAllCuratorCameraAreas removeAllCuratorEditingAreas removeAllEventHandlers removeAllHandgunItems removeAllItems removeAllItemsWithMagazines removeAllMissionEventHandlers removeAllMPEventHandlers removeAllMusicEventHandlers removeAllOwnedMines removeAllPrimaryWeaponItems removeAllWeapons removeBackpack removeBackpackGlobal removeCuratorAddons removeCuratorCameraArea removeCuratorEditableObjects removeCuratorEditingArea removeDrawIcon removeDrawLinks removeEventHandler removeFromRemainsCollector removeGoggles removeGroupIcon removeHandgunItem removeHeadgear removeItem removeItemFromBackpack removeItemFromUniform removeItemFromVest removeItems removeMagazine removeMagazineGlobal removeMagazines removeMagazinesTurret removeMagazineTurret removeMenuItem removeMissionEventHandler removeMPEventHandler removeMusicEventHandler removeOwnedMine removePrimaryWeaponItem removeSecondaryWeaponItem removeSimpleTask removeSwitchableUnit removeTeamMember removeUniform removeVest removeWeapon removeWeaponAttachmentCargo removeWeaponCargo removeWeaponGlobal removeWeaponTurret reportRemoteTarget requiredVersion resetCamShake resetSubgroupDirection resize resources respawnVehicle restartEditorCamera reveal revealMine reverse reversedMouseY roadAt roadsConnectedTo roleDescription ropeAttachedObjects ropeAttachedTo ropeAttachEnabled ropeAttachTo ropeCreate ropeCut ropeDestroy ropeDetach ropeEndPosition ropeLength ropes ropeUnwind ropeUnwound rotorsForcesRTD rotorsRpmRTD round runInitScript safeZoneH safeZoneW safeZoneWAbs safeZoneX safeZoneXAbs safeZoneY save3DENInventory saveGame saveIdentity saveJoysticks saveOverlay saveProfileNamespace saveStatus saveVar savingEnabled say say2D say3D scopeName score scoreSide screenshot screenToWorld scriptDone scriptName scudState secondaryWeapon secondaryWeaponItems secondaryWeaponMagazine select selectBestPlaces selectDiarySubject selectedEditorObjects selectEditorObject selectionNames selectionPosition selectLeader selectMax selectMin selectNoPlayer selectPlayer selectRandom selectRandomWeighted selectWeapon selectWeaponTurret sendAUMessage sendSimpleCommand sendTask sendTaskResult sendUDPMessage serverCommand serverCommandAvailable serverCommandExecutable serverName serverTime set set3DENAttribute set3DENAttributes set3DENGrid set3DENIconsVisible set3DENLayer set3DENLinesVisible set3DENLogicType set3DENMissionAttribute set3DENMissionAttributes set3DENModelsVisible set3DENObjectType set3DENSelected setAccTime setActualCollectiveRTD setAirplaneThrottle setAirportSide setAmmo setAmmoCargo setAmmoOnPylon setAnimSpeedCoef setAperture setApertureNew setArmoryPoints setAttributes setAutonomous setBehaviour setBleedingRemaining setBrakesRTD setCameraInterest setCamShakeDefParams setCamShakeParams setCamUseTI setCaptive setCenterOfMass setCollisionLight setCombatMode setCompassOscillation setConvoySeparation setCuratorCameraAreaCeiling setCuratorCoef setCuratorEditingAreaType setCuratorWaypointCost setCurrentChannel setCurrentTask setCurrentWaypoint setCustomAimCoef setCustomWeightRTD setDamage setDammage setDate setDebriefingText setDefaultCamera setDestination setDetailMapBlendPars setDir setDirection setDrawIcon setDriveOnPath setDropInterval setDynamicSimulationDistance setDynamicSimulationDistanceCoef setEditorMode setEditorObjectScope setEffectCondition setEngineRPMRTD setFace setFaceAnimation setFatigue setFeatureType setFlagAnimationPhase setFlagOwner setFlagSide setFlagTexture setFog setFormation setFormationTask setFormDir setFriend setFromEditor setFSMVariable setFuel setFuelCargo setGroupIcon setGroupIconParams setGroupIconsSelectable setGroupIconsVisible setGroupId setGroupIdGlobal setGroupOwner setGusts setHideBehind setHit setHitIndex setHitPointDamage setHorizonParallaxCoef setHUDMovementLevels setIdentity setImportance setInfoPanel setLeader setLightAmbient setLightAttenuation setLightBrightness setLightColor setLightDayLight setLightFlareMaxDistance setLightFlareSize setLightIntensity setLightnings setLightUseFlare setLocalWindParams setMagazineTurretAmmo setMarkerAlpha setMarkerAlphaLocal setMarkerBrush setMarkerBrushLocal setMarkerColor setMarkerColorLocal setMarkerDir setMarkerDirLocal setMarkerPos setMarkerPosLocal setMarkerShape setMarkerShapeLocal setMarkerSize setMarkerSizeLocal setMarkerText setMarkerTextLocal setMarkerType setMarkerTypeLocal setMass setMimic setMousePosition setMusicEffect setMusicEventHandler setName setNameSound setObjectArguments setObjectMaterial setObjectMaterialGlobal setObjectProxy setObjectTexture setObjectTextureGlobal setObjectViewDistance setOvercast setOwner setOxygenRemaining setParticleCircle setParticleClass setParticleFire setParticleParams setParticleRandom setPilotCameraDirection setPilotCameraRotation setPilotCameraTarget setPilotLight setPiPEffect setPitch setPlateNumber setPlayable setPlayerRespawnTime setPos setPosASL setPosASL2 setPosASLW setPosATL setPosition setPosWorld setPylonLoadOut setPylonsPriority setRadioMsg setRain setRainbow setRandomLip setRank setRectangular setRepairCargo setRotorBrakeRTD setShadowDistance setShotParents setSide setSimpleTaskAlwaysVisible setSimpleTaskCustomData setSimpleTaskDescription setSimpleTaskDestination setSimpleTaskTarget setSimpleTaskType setSimulWeatherLayers setSize setSkill setSlingLoad setSoundEffect setSpeaker setSpeech setSpeedMode setStamina setStaminaScheme setStatValue setSuppression setSystemOfUnits setTargetAge setTaskMarkerOffset setTaskResult setTaskState setTerrainGrid setText 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e}}),s.helper.regexes={asteriskDashAndColon:/([*_:~])/g},s.helper.emojis={"+1":"👍","-1":"👎",100:"💯",1234:"🔢","1st_place_medal":"🥇","2nd_place_medal":"🥈","3rd_place_medal":"🥉","8ball":"🎱",a:"🅰️",ab:"🆎",abc:"🔤",abcd:"🔡",accept:"🉑",aerial_tramway:"🚡",airplane:"✈️",alarm_clock:"⏰",alembic:"⚗️",alien:"👽",ambulance:"🚑",amphora:"🏺",anchor:"⚓️",angel:"👼",anger:"💢",angry:"😠",anguished:"😧",ant:"🐜",apple:"🍎",aquarius:"♒️",aries:"♈️",arrow_backward:"◀️",arrow_double_down:"⏬",arrow_double_up:"⏫",arrow_down:"⬇️",arrow_down_small:"🔽",arrow_forward:"▶️",arrow_heading_down:"⤵️",arrow_heading_up:"⤴️",arrow_left:"⬅️",arrow_lower_left:"↙️",arrow_lower_right:"↘️",arrow_right:"➡️",arrow_right_hook:"↪️",arrow_up:"⬆️",arrow_up_down:"↕️",arrow_up_small:"🔼",arrow_upper_left:"↖️",arrow_upper_right:"↗️",arrows_clockwise:"🔃",arrows_counterclockwise:"🔄",art:"🎨",articulated_lorry:"🚛",artificial_satellite:"🛰",astonished:"😲",athletic_shoe:"👟",atm:"🏧",atom_symbol:"⚛️",avocado:"🥑",b:"🅱️",baby:"👶",baby_bottle:"🍼",baby_chick:"🐤",baby_symbol:"🚼",back:"🔙",bacon:"🥓",badminton:"🏸",baggage_claim:"🛄",baguette_bread:"🥖",balance_scale:"⚖️",balloon:"🎈",ballot_box:"🗳",ballot_box_with_check:"☑️",bamboo:"🎍",banana:"🍌",bangbang:"‼️",bank:"🏦",bar_chart:"📊",barber:"💈",baseball:"⚾️",basketball:"🏀",basketball_man:"⛹️",basketball_woman:"⛹️‍♀️",bat:"🦇",bath:"🛀",bathtub:"🛁",battery:"🔋",beach_umbrella:"🏖",bear:"🐻",bed:"🛏",bee:"🐝",beer:"🍺",beers:"🍻",beetle:"🐞",beginner:"🔰",bell:"🔔",bellhop_bell:"🛎",bento:"🍱",biking_man:"🚴",bike:"🚲",biking_woman:"🚴‍♀️",bikini:"👙",biohazard:"☣️",bird:"🐦",birthday:"🎂",black_circle:"⚫️",black_flag:"🏴",black_heart:"🖤",black_joker:"🃏",black_large_square:"⬛️",black_medium_small_square:"◾️",black_medium_square:"◼️",black_nib:"✒️",black_small_square:"▪️",black_square_button:"🔲",blonde_man:"👱",blonde_woman:"👱‍♀️",blossom:"🌼",blowfish:"🐡",blue_book:"📘",blue_car:"🚙",blue_heart:"💙",blush:"😊",boar:"🐗",boat:"⛵️",bomb:"💣",book:"📖",bookmark:"🔖",bookmark_tabs:"📑",books:"📚",boom:"💥",boot:"👢",bouquet:"💐",bowing_man:"🙇",bow_and_arrow:"🏹",bowing_woman:"🙇‍♀️",bowling:"🎳",boxing_glove:"🥊",boy:"👦",bread:"🍞",bride_with_veil:"👰",bridge_at_night:"🌉",briefcase:"💼",broken_heart:"💔",bug:"🐛",building_construction:"🏗",bulb:"💡",bullettrain_front:"🚅",bullettrain_side:"🚄",burrito:"🌯",bus:"🚌",business_suit_levitating:"🕴",busstop:"🚏",bust_in_silhouette:"👤",busts_in_silhouette:"👥",butterfly:"🦋",cactus:"🌵",cake:"🍰",calendar:"📆",call_me_hand:"🤙",calling:"📲",camel:"🐫",camera:"📷",camera_flash:"📸",camping:"🏕",cancer:"♋️",candle:"🕯",candy:"🍬",canoe:"🛶",capital_abcd:"🔠",capricorn:"♑️",car:"🚗",card_file_box:"🗃",card_index:"📇",card_index_dividers:"🗂",carousel_horse:"🎠",carrot:"🥕",cat:"🐱",cat2:"🐈",cd:"💿",chains:"⛓",champagne:"🍾",chart:"💹",chart_with_downwards_trend:"📉",chart_with_upwards_trend:"📈",checkered_flag:"🏁",cheese:"🧀",cherries:"🍒",cherry_blossom:"🌸",chestnut:"🌰",chicken:"🐔",children_crossing:"🚸",chipmunk:"🐿",chocolate_bar:"🍫",christmas_tree:"🎄",church:"⛪️",cinema:"🎦",circus_tent:"🎪",city_sunrise:"🌇",city_sunset:"🌆",cityscape:"🏙",cl:"🆑",clamp:"🗜",clap:"👏",clapper:"🎬",classical_building:"🏛",clinking_glasses:"🥂",clipboard:"📋",clock1:"🕐",clock10:"🕙",clock1030:"🕥",clock11:"🕚",clock1130:"🕦",clock12:"🕛",clock1230:"🕧",clock130:"🕜",clock2:"🕑",clock230:"🕝",clock3:"🕒",clock330:"🕞",clock4:"🕓",clock430:"🕟",clock5:"🕔",clock530:"🕠",clock6:"🕕",clock630:"🕡",clock7:"🕖",clock730:"🕢",clock8:"🕗",clock830:"🕣",clock9:"🕘",clock930:"🕤",closed_book:"📕",closed_lock_with_key:"🔐",closed_umbrella:"🌂",cloud:"☁️",cloud_with_lightning:"🌩",cloud_with_lightning_and_rain:"⛈",cloud_with_rain:"🌧",cloud_with_snow:"🌨",clown_face:"🤡",clubs:"♣️",cocktail:"🍸",coffee:"☕️",coffin:"⚰️",cold_sweat:"😰",comet:"☄️",computer:"💻",computer_mouse:"🖱",confetti_ball:"🎊",confounded:"😖",confused:"😕",congratulations:"㊗️",construction:"🚧",construction_worker_man:"👷",construction_worker_woman:"👷‍♀️",control_knobs:"🎛",convenience_store:"🏪",cookie:"🍪",cool:"🆒",policeman:"👮",copyright:"©️",corn:"🌽",couch_and_lamp:"🛋",couple:"👫",couple_with_heart_woman_man:"💑",couple_with_heart_man_man:"👨‍❤️‍👨",couple_with_heart_woman_woman:"👩‍❤️‍👩",couplekiss_man_man:"👨‍❤️‍💋‍👨",couplekiss_man_woman:"💏",couplekiss_woman_woman:"👩‍❤️‍💋‍👩",cow:"🐮",cow2:"🐄",cowboy_hat_face:"🤠",crab:"🦀",crayon:"🖍",credit_card:"💳",crescent_moon:"🌙",cricket:"🏏",crocodile:"🐊",croissant:"🥐",crossed_fingers:"🤞",crossed_flags:"🎌",crossed_swords:"⚔️",crown:"👑",cry:"😢",crying_cat_face:"😿",crystal_ball:"🔮",cucumber:"🥒",cupid:"💘",curly_loop:"➰",currency_exchange:"💱",curry:"🍛",custard:"🍮",customs:"🛃",cyclone:"🌀",dagger:"🗡",dancer:"💃",dancing_women:"👯",dancing_men:"👯‍♂️",dango:"🍡",dark_sunglasses:"🕶",dart:"🎯",dash:"💨",date:"📅",deciduous_tree:"🌳",deer:"🦌",department_store:"🏬",derelict_house:"🏚",desert:"🏜",desert_island:"🏝",desktop_computer:"🖥",male_detective:"🕵️",diamond_shape_with_a_dot_inside:"💠",diamonds:"♦️",disappointed:"😞",disappointed_relieved:"😥",dizzy:"💫",dizzy_face:"😵",do_not_litter:"🚯",dog:"🐶",dog2:"🐕",dollar:"💵",dolls:"🎎",dolphin:"🐬",door:"🚪",doughnut:"🍩",dove:"🕊",dragon:"🐉",dragon_face:"🐲",dress:"👗",dromedary_camel:"🐪",drooling_face:"🤤",droplet:"💧",drum:"🥁",duck:"🦆",dvd:"📀","e-mail":"📧",eagle:"🦅",ear:"👂",ear_of_rice:"🌾",earth_africa:"🌍",earth_americas:"🌎",earth_asia:"🌏",egg:"🥚",eggplant:"🍆",eight_pointed_black_star:"✴️",eight_spoked_asterisk:"✳️",electric_plug:"🔌",elephant:"🐘",email:"✉️",end:"🔚",envelope_with_arrow:"📩",euro:"💶",european_castle:"🏰",european_post_office:"🏤",evergreen_tree:"🌲",exclamation:"❗️",expressionless:"😑",eye:"👁",eye_speech_bubble:"👁‍🗨",eyeglasses:"👓",eyes:"👀",face_with_head_bandage:"🤕",face_with_thermometer:"🤒",fist_oncoming:"👊",factory:"🏭",fallen_leaf:"🍂",family_man_woman_boy:"👪",family_man_boy:"👨‍👦",family_man_boy_boy:"👨‍👦‍👦",family_man_girl:"👨‍👧",family_man_girl_boy:"👨‍👧‍👦",family_man_girl_girl:"👨‍👧‍👧",family_man_man_boy:"👨‍👨‍👦",family_man_man_boy_boy:"👨‍👨‍👦‍👦",family_man_man_girl:"👨‍👨‍👧",family_man_man_girl_boy:"👨‍👨‍👧‍👦",family_man_man_girl_girl:"👨‍👨‍👧‍👧",family_man_woman_boy_boy:"👨‍👩‍👦‍👦",family_man_woman_girl:"👨‍👩‍👧",family_man_woman_girl_boy:"👨‍👩‍👧‍👦",family_man_woman_girl_girl:"👨‍👩‍👧‍👧",family_woman_boy:"👩‍👦",family_woman_boy_boy:"👩‍👦‍👦",family_woman_girl:"👩‍👧",family_woman_girl_boy:"👩‍👧‍👦",family_woman_girl_girl:"👩‍👧‍👧",family_woman_woman_boy:"👩‍👩‍👦",family_woman_woman_boy_boy:"👩‍👩‍👦‍👦",family_woman_woman_girl:"👩‍👩‍👧",family_woman_woman_girl_boy:"👩‍👩‍👧‍👦",family_woman_woman_girl_girl:"👩‍👩‍👧‍👧",fast_forward:"⏩",fax:"📠",fearful:"😨",feet:"🐾",female_detective:"🕵️‍♀️",ferris_wheel:"🎡",ferry:"⛴",field_hockey:"🏑",file_cabinet:"🗄",file_folder:"📁",film_projector:"📽",film_strip:"🎞",fire:"🔥",fire_engine:"🚒",fireworks:"🎆",first_quarter_moon:"🌓",first_quarter_moon_with_face:"🌛",fish:"🐟",fish_cake:"🍥",fishing_pole_and_fish:"🎣",fist_raised:"✊",fist_left:"🤛",fist_right:"🤜",flags:"🎏",flashlight:"🔦",fleur_de_lis:"⚜️",flight_arrival:"🛬",flight_departure:"🛫",floppy_disk:"💾",flower_playing_cards:"🎴",flushed:"😳",fog:"🌫",foggy:"🌁",football:"🏈",footprints:"👣",fork_and_knife:"🍴",fountain:"⛲️",fountain_pen:"🖋",four_leaf_clover:"🍀",fox_face:"🦊",framed_picture:"🖼",free:"🆓",fried_egg:"🍳",fried_shrimp:"🍤",fries:"🍟",frog:"🐸",frowning:"😦",frowning_face:"☹️",frowning_man:"🙍‍♂️",frowning_woman:"🙍",middle_finger:"🖕",fuelpump:"⛽️",full_moon:"🌕",full_moon_with_face:"🌝",funeral_urn:"⚱️",game_die:"🎲",gear:"⚙️",gem:"💎",gemini:"♊️",ghost:"👻",gift:"🎁",gift_heart:"💝",girl:"👧",globe_with_meridians:"🌐",goal_net:"🥅",goat:"🐐",golf:"⛳️",golfing_man:"🏌️",golfing_woman:"🏌️‍♀️",gorilla:"🦍",grapes:"🍇",green_apple:"🍏",green_book:"📗",green_heart:"💚",green_salad:"🥗",grey_exclamation:"❕",grey_question:"❔",grimacing:"😬",grin:"😁",grinning:"😀",guardsman:"💂",guardswoman:"💂‍♀️",guitar:"🎸",gun:"🔫",haircut_woman:"💇",haircut_man:"💇‍♂️",hamburger:"🍔",hammer:"🔨",hammer_and_pick:"⚒",hammer_and_wrench:"🛠",hamster:"🐹",hand:"✋",handbag:"👜",handshake:"🤝",hankey:"💩",hatched_chick:"🐥",hatching_chick:"🐣",headphones:"🎧",hear_no_evil:"🙉",heart:"❤️",heart_decoration:"💟",heart_eyes:"😍",heart_eyes_cat:"😻",heartbeat:"💓",heartpulse:"💗",hearts:"♥️",heavy_check_mark:"✔️",heavy_division_sign:"➗",heavy_dollar_sign:"💲",heavy_heart_exclamation:"❣️",heavy_minus_sign:"➖",heavy_multiplication_x:"✖️",heavy_plus_sign:"➕",helicopter:"🚁",herb:"🌿",hibiscus:"🌺",high_brightness:"🔆",high_heel:"👠",hocho:"🔪",hole:"🕳",honey_pot:"🍯",horse:"🐴",horse_racing:"🏇",hospital:"🏥",hot_pepper:"🌶",hotdog:"🌭",hotel:"🏨",hotsprings:"♨️",hourglass:"⌛️",hourglass_flowing_sand:"⏳",house:"🏠",house_with_garden:"🏡",houses:"🏘",hugs:"🤗",hushed:"😯",ice_cream:"🍨",ice_hockey:"🏒",ice_skate:"⛸",icecream:"🍦",id:"🆔",ideograph_advantage:"🉐",imp:"👿",inbox_tray:"📥",incoming_envelope:"📨",tipping_hand_woman:"💁",information_source:"ℹ️",innocent:"😇",interrobang:"⁉️",iphone:"📱",izakaya_lantern:"🏮",jack_o_lantern:"🎃",japan:"🗾",japanese_castle:"🏯",japanese_goblin:"👺",japanese_ogre:"👹",jeans:"👖",joy:"😂",joy_cat:"😹",joystick:"🕹",kaaba:"🕋",key:"🔑",keyboard:"⌨️",keycap_ten:"🔟",kick_scooter:"🛴",kimono:"👘",kiss:"💋",kissing:"😗",kissing_cat:"😽",kissing_closed_eyes:"😚",kissing_heart:"😘",kissing_smiling_eyes:"😙",kiwi_fruit:"🥝",koala:"🐨",koko:"🈁",label:"🏷",large_blue_circle:"🔵",large_blue_diamond:"🔷",large_orange_diamond:"🔶",last_quarter_moon:"🌗",last_quarter_moon_with_face:"🌜",latin_cross:"✝️",laughing:"😆",leaves:"🍃",ledger:"📒",left_luggage:"🛅",left_right_arrow:"↔️",leftwards_arrow_with_hook:"↩️",lemon:"🍋",leo:"♌️",leopard:"🐆",level_slider:"🎚",libra:"♎️",light_rail:"🚈",link:"🔗",lion:"🦁",lips:"👄",lipstick:"💄",lizard:"🦎",lock:"🔒",lock_with_ink_pen:"🔏",lollipop:"🍭",loop:"➿",loud_sound:"🔊",loudspeaker:"📢",love_hotel:"🏩",love_letter:"💌",low_brightness:"🔅",lying_face:"🤥",m:"Ⓜ️",mag:"🔍",mag_right:"🔎",mahjong:"🀄️",mailbox:"📫",mailbox_closed:"📪",mailbox_with_mail:"📬",mailbox_with_no_mail:"📭",man:"👨",man_artist:"👨‍🎨",man_astronaut:"👨‍🚀",man_cartwheeling:"🤸‍♂️",man_cook:"👨‍🍳",man_dancing:"🕺",man_facepalming:"🤦‍♂️",man_factory_worker:"👨‍🏭",man_farmer:"👨‍🌾",man_firefighter:"👨‍🚒",man_health_worker:"👨‍⚕️",man_in_tuxedo:"🤵",man_judge:"👨‍⚖️",man_juggling:"🤹‍♂️",man_mechanic:"👨‍🔧",man_office_worker:"👨‍💼",man_pilot:"👨‍✈️",man_playing_handball:"🤾‍♂️",man_playing_water_polo:"🤽‍♂️",man_scientist:"👨‍🔬",man_shrugging:"🤷‍♂️",man_singer:"👨‍🎤",man_student:"👨‍🎓",man_teacher:"👨‍🏫",man_technologist:"👨‍💻",man_with_gua_pi_mao:"👲",man_with_turban:"👳",tangerine:"🍊",mans_shoe:"👞",mantelpiece_clock:"🕰",maple_leaf:"🍁",martial_arts_uniform:"🥋",mask:"😷",massage_woman:"💆",massage_man:"💆‍♂️",meat_on_bone:"🍖",medal_military:"🎖",medal_sports:"🏅",mega:"📣",melon:"🍈",memo:"📝",men_wrestling:"🤼‍♂️",menorah:"🕎",mens:"🚹",metal:"🤘",metro:"🚇",microphone:"🎤",microscope:"🔬",milk_glass:"🥛",milky_way:"🌌",minibus:"🚐",minidisc:"💽",mobile_phone_off:"📴",money_mouth_face:"🤑",money_with_wings:"💸",moneybag:"💰",monkey:"🐒",monkey_face:"🐵",monorail:"🚝",moon:"🌔",mortar_board:"🎓",mosque:"🕌",motor_boat:"🛥",motor_scooter:"🛵",motorcycle:"🏍",motorway:"🛣",mount_fuji:"🗻",mountain:"⛰",mountain_biking_man:"🚵",mountain_biking_woman:"🚵‍♀️",mountain_cableway:"🚠",mountain_railway:"🚞",mountain_snow:"🏔",mouse:"🐭",mouse2:"🐁",movie_camera:"🎥",moyai:"🗿",mrs_claus:"🤶",muscle:"💪",mushroom:"🍄",musical_keyboard:"🎹",musical_note:"🎵",musical_score:"🎼",mute:"🔇",nail_care:"💅",name_badge:"📛",national_park:"🏞",nauseated_face:"🤢",necktie:"👔",negative_squared_cross_mark:"❎",nerd_face:"🤓",neutral_face:"😐",new:"🆕",new_moon:"🌑",new_moon_with_face:"🌚",newspaper:"📰",newspaper_roll:"🗞",next_track_button:"⏭",ng:"🆖",no_good_man:"🙅‍♂️",no_good_woman:"🙅",night_with_stars:"🌃",no_bell:"🔕",no_bicycles:"🚳",no_entry:"⛔️",no_entry_sign:"🚫",no_mobile_phones:"📵",no_mouth:"😶",no_pedestrians:"🚷",no_smoking:"🚭","non-potable_water":"🚱",nose:"👃",notebook:"📓",notebook_with_decorative_cover:"📔",notes:"🎶",nut_and_bolt:"🔩",o:"⭕️",o2:"🅾️",ocean:"🌊",octopus:"🐙",oden:"🍢",office:"🏢",oil_drum:"🛢",ok:"🆗",ok_hand:"👌",ok_man:"🙆‍♂️",ok_woman:"🙆",old_key:"🗝",older_man:"👴",older_woman:"👵",om:"🕉",on:"🔛",oncoming_automobile:"🚘",oncoming_bus:"🚍",oncoming_police_car:"🚔",oncoming_taxi:"🚖",open_file_folder:"📂",open_hands:"👐",open_mouth:"😮",open_umbrella:"☂️",ophiuchus:"⛎",orange_book:"📙",orthodox_cross:"☦️",outbox_tray:"📤",owl:"🦉",ox:"🐂",package:"📦",page_facing_up:"📄",page_with_curl:"📃",pager:"📟",paintbrush:"🖌",palm_tree:"🌴",pancakes:"🥞",panda_face:"🐼",paperclip:"📎",paperclips:"🖇",parasol_on_ground:"⛱",parking:"🅿️",part_alternation_mark:"〽️",partly_sunny:"⛅️",passenger_ship:"🛳",passport_control:"🛂",pause_button:"⏸",peace_symbol:"☮️",peach:"🍑",peanuts:"🥜",pear:"🍐",pen:"🖊",pencil2:"✏️",penguin:"🐧",pensive:"😔",performing_arts:"🎭",persevere:"😣",person_fencing:"🤺",pouting_woman:"🙎",phone:"☎️",pick:"⛏",pig:"🐷",pig2:"🐖",pig_nose:"🐽",pill:"💊",pineapple:"🍍",ping_pong:"🏓",pisces:"♓️",pizza:"🍕",place_of_worship:"🛐",plate_with_cutlery:"🍽",play_or_pause_button:"⏯",point_down:"👇",point_left:"👈",point_right:"👉",point_up:"☝️",point_up_2:"👆",police_car:"🚓",policewoman:"👮‍♀️",poodle:"🐩",popcorn:"🍿",post_office:"🏣",postal_horn:"📯",postbox:"📮",potable_water:"🚰",potato:"🥔",pouch:"👝",poultry_leg:"🍗",pound:"💷",rage:"😡",pouting_cat:"😾",pouting_man:"🙎‍♂️",pray:"🙏",prayer_beads:"📿",pregnant_woman:"🤰",previous_track_button:"⏮",prince:"🤴",princess:"👸",printer:"🖨",purple_heart:"💜",purse:"👛",pushpin:"📌",put_litter_in_its_place:"🚮",question:"❓",rabbit:"🐰",rabbit2:"🐇",racehorse:"🐎",racing_car:"🏎",radio:"📻",radio_button:"🔘",radioactive:"☢️",railway_car:"🚃",railway_track:"🛤",rainbow:"🌈",rainbow_flag:"🏳️‍🌈",raised_back_of_hand:"🤚",raised_hand_with_fingers_splayed:"🖐",raised_hands:"🙌",raising_hand_woman:"🙋",raising_hand_man:"🙋‍♂️",ram:"🐏",ramen:"🍜",rat:"🐀",record_button:"⏺",recycle:"♻️",red_circle:"🔴",registered:"®️",relaxed:"☺️",relieved:"😌",reminder_ribbon:"🎗",repeat:"🔁",repeat_one:"🔂",rescue_worker_helmet:"⛑",restroom:"🚻",revolving_hearts:"💞",rewind:"⏪",rhinoceros:"🦏",ribbon:"🎀",rice:"🍚",rice_ball:"🍙",rice_cracker:"🍘",rice_scene:"🎑",right_anger_bubble:"🗯",ring:"💍",robot:"🤖",rocket:"🚀",rofl:"🤣",roll_eyes:"🙄",roller_coaster:"🎢",rooster:"🐓",rose:"🌹",rosette:"🏵",rotating_light:"🚨",round_pushpin:"📍",rowing_man:"🚣",rowing_woman:"🚣‍♀️",rugby_football:"🏉",running_man:"🏃",running_shirt_with_sash:"🎽",running_woman:"🏃‍♀️",sa:"🈂️",sagittarius:"♐️",sake:"🍶",sandal:"👡",santa:"🎅",satellite:"📡",saxophone:"🎷",school:"🏫",school_satchel:"🎒",scissors:"✂️",scorpion:"🦂",scorpius:"♏️",scream:"😱",scream_cat:"🙀",scroll:"📜",seat:"💺",secret:"㊙️",see_no_evil:"🙈",seedling:"🌱",selfie:"🤳",shallow_pan_of_food:"🥘",shamrock:"☘️",shark:"🦈",shaved_ice:"🍧",sheep:"🐑",shell:"🐚",shield:"🛡",shinto_shrine:"⛩",ship:"🚢",shirt:"👕",shopping:"🛍",shopping_cart:"🛒",shower:"🚿",shrimp:"🦐",signal_strength:"📶",six_pointed_star:"🔯",ski:"🎿",skier:"⛷",skull:"💀",skull_and_crossbones:"☠️",sleeping:"😴",sleeping_bed:"🛌",sleepy:"😪",slightly_frowning_face:"🙁",slightly_smiling_face:"🙂",slot_machine:"🎰",small_airplane:"🛩",small_blue_diamond:"🔹",small_orange_diamond:"🔸",small_red_triangle:"🔺",small_red_triangle_down:"🔻",smile:"😄",smile_cat:"😸",smiley:"😃",smiley_cat:"😺",smiling_imp:"😈",smirk:"😏",smirk_cat:"😼",smoking:"🚬",snail:"🐌",snake:"🐍",sneezing_face:"🤧",snowboarder:"🏂",snowflake:"❄️",snowman:"⛄️",snowman_with_snow:"☃️",sob:"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DefineExternal DefineInputStreamMethod DefineOutputStreamMethod DefineResourceFunction Definition Degree DegreeCentrality DegreeGraphDistribution DegreeLexicographic DegreeReverseLexicographic DEigensystem DEigenvalues Deinitialization Del DelaunayMesh Delayed Deletable Delete DeleteAnomalies DeleteBorderComponents DeleteCases DeleteChannel DeleteCloudExpression DeleteContents DeleteDirectory DeleteDuplicates DeleteDuplicatesBy DeleteFile DeleteMissing DeleteObject DeletePermissionsKey DeleteSearchIndex DeleteSmallComponents DeleteStopwords DeleteWithContents DeletionWarning DelimitedArray DelimitedSequence Delimiter DelimiterFlashTime DelimiterMatching Delimiters DeliveryFunction Dendrogram Denominator DensityGraphics DensityHistogram DensityPlot DensityPlot3D DependentVariables Deploy Deployed Depth DepthFirstScan Derivative DerivativeFilter DerivedKey DescriptorStateSpace DesignMatrix DestroyAfterEvaluation Det DeviceClose DeviceConfigure DeviceExecute DeviceExecuteAsynchronous DeviceObject DeviceOpen DeviceOpenQ DeviceRead DeviceReadBuffer DeviceReadLatest DeviceReadList DeviceReadTimeSeries Devices DeviceStreams DeviceWrite DeviceWriteBuffer DGaussianWavelet DiacriticalPositioning Diagonal DiagonalizableMatrixQ DiagonalMatrix DiagonalMatrixQ Dialog DialogIndent DialogInput DialogLevel DialogNotebook DialogProlog DialogReturn DialogSymbols Diamond DiamondMatrix DiceDissimilarity DictionaryLookup DictionaryWordQ DifferenceDelta DifferenceOrder DifferenceQuotient DifferenceRoot DifferenceRootReduce Differences DifferentialD DifferentialRoot DifferentialRootReduce DifferentiatorFilter DigitalSignature DigitBlock DigitBlockMinimum DigitCharacter DigitCount DigitQ DihedralAngle DihedralGroup Dilation DimensionalCombinations DimensionalMeshComponents DimensionReduce DimensionReducerFunction DimensionReduction Dimensions DiracComb DiracDelta DirectedEdge DirectedEdges DirectedGraph DirectedGraphQ DirectedInfinity Direction Directive Directory DirectoryName DirectoryQ DirectoryStack DirichletBeta DirichletCharacter DirichletCondition DirichletConvolve DirichletDistribution DirichletEta DirichletL DirichletLambda DirichletTransform DirichletWindow DisableConsolePrintPacket DisableFormatting DiscreteChirpZTransform DiscreteConvolve DiscreteDelta DiscreteHadamardTransform DiscreteIndicator DiscreteLimit DiscreteLQEstimatorGains DiscreteLQRegulatorGains DiscreteLyapunovSolve DiscreteMarkovProcess DiscreteMaxLimit DiscreteMinLimit DiscretePlot DiscretePlot3D DiscreteRatio DiscreteRiccatiSolve DiscreteShift DiscreteTimeModelQ DiscreteUniformDistribution DiscreteVariables DiscreteWaveletData DiscreteWaveletPacketTransform DiscreteWaveletTransform DiscretizeGraphics DiscretizeRegion Discriminant DisjointQ Disjunction Disk DiskBox DiskMatrix DiskSegment Dispatch DispatchQ DispersionEstimatorFunction Display DisplayAllSteps DisplayEndPacket DisplayFlushImagePacket DisplayForm DisplayFunction DisplayPacket DisplayRules DisplaySetSizePacket DisplayString DisplayTemporary DisplayWith DisplayWithRef DisplayWithVariable DistanceFunction DistanceMatrix DistanceTransform Distribute Distributed DistributedContexts DistributeDefinitions DistributionChart DistributionDomain DistributionFitTest DistributionParameterAssumptions DistributionParameterQ Dithering Div Divergence Divide DivideBy Dividers DivideSides Divisible Divisors DivisorSigma DivisorSum DMSList DMSString Do DockedCells DocumentGenerator DocumentGeneratorInformation DocumentGeneratorInformationData DocumentGenerators DocumentNotebook DocumentWeightingRules Dodecahedron DomainRegistrationInformation DominantColors DOSTextFormat Dot DotDashed DotEqual DotLayer DotPlusLayer Dotted DoubleBracketingBar DoubleContourIntegral DoubleDownArrow DoubleLeftArrow DoubleLeftRightArrow DoubleLeftTee DoubleLongLeftArrow DoubleLongLeftRightArrow DoubleLongRightArrow DoubleRightArrow DoubleRightTee DoubleUpArrow DoubleUpDownArrow DoubleVerticalBar DoublyInfinite Down DownArrow DownArrowBar DownArrowUpArrow DownLeftRightVector DownLeftTeeVector DownLeftVector DownLeftVectorBar DownRightTeeVector DownRightVector DownRightVectorBar Downsample DownTee DownTeeArrow DownValues DragAndDrop DrawEdges DrawFrontFaces DrawHighlighted Drop DropoutLayer DSolve DSolveValue Dt DualLinearProgramming DualPolyhedron DualSystemsModel DumpGet DumpSave DuplicateFreeQ Duration Dynamic DynamicBox DynamicBoxOptions DynamicEvaluationTimeout DynamicGeoGraphics DynamicImage DynamicLocation DynamicModule DynamicModuleBox DynamicModuleBoxOptions DynamicModuleParent DynamicModuleValues DynamicName DynamicNamespace DynamicReference DynamicSetting DynamicUpdating DynamicWrapper DynamicWrapperBox DynamicWrapperBoxOptionsE EarthImpactData EarthquakeData EccentricityCentrality Echo EchoFunction EclipseType EdgeAdd EdgeBetweennessCentrality EdgeCapacity EdgeCapForm EdgeColor EdgeConnectivity EdgeContract EdgeCost EdgeCount EdgeCoverQ EdgeCycleMatrix EdgeDashing EdgeDelete EdgeDetect EdgeForm EdgeIndex EdgeJoinForm EdgeLabeling EdgeLabels EdgeLabelStyle EdgeList EdgeOpacity EdgeQ EdgeRenderingFunction EdgeRules EdgeShapeFunction EdgeStyle EdgeThickness EdgeWeight EdgeWeightedGraphQ Editable EditButtonSettings EditCellTagsSettings EditDistance EffectiveInterest Eigensystem Eigenvalues EigenvectorCentrality Eigenvectors Element ElementData ElementwiseLayer ElidedForms Eliminate EliminationOrder Ellipsoid EllipticE EllipticExp EllipticExpPrime EllipticF EllipticFilterModel EllipticK EllipticLog EllipticNomeQ EllipticPi EllipticReducedHalfPeriods EllipticTheta EllipticThetaPrime EmbedCode EmbeddedHTML EmbeddedService EmbeddingLayer EmbeddingObject EmitSound EmphasizeSyntaxErrors EmpiricalDistribution Empty EmptyGraphQ EmptyRegion EnableConsolePrintPacket Enabled Encode Encrypt EncryptedObject EncryptFile End EndAdd EndDialogPacket EndFrontEndInteractionPacket EndOfBuffer EndOfFile EndOfLine EndOfString EndPackage EngineEnvironment EngineeringForm Enter EnterExpressionPacket EnterTextPacket Entity EntityClass EntityClassList EntityCopies EntityFunction EntityGroup EntityInstance EntityList EntityPrefetch EntityProperties EntityProperty EntityPropertyClass EntityRegister EntityStore EntityStores EntityTypeName EntityUnregister EntityValue Entropy EntropyFilter Environment Epilog EpilogFunction Equal EqualColumns EqualRows EqualTilde EqualTo EquatedTo Equilibrium EquirippleFilterKernel Equivalent Erf Erfc Erfi ErlangB ErlangC ErlangDistribution Erosion ErrorBox ErrorBoxOptions ErrorNorm ErrorPacket ErrorsDialogSettings EscapeRadius EstimatedBackground EstimatedDistribution EstimatedProcess EstimatorGains EstimatorRegulator EuclideanDistance EulerAngles EulerCharacteristic EulerE EulerGamma EulerianGraphQ EulerMatrix EulerPhi Evaluatable Evaluate Evaluated EvaluatePacket EvaluateScheduledTask EvaluationBox EvaluationCell EvaluationCompletionAction EvaluationData EvaluationElements EvaluationEnvironment EvaluationMode EvaluationMonitor EvaluationNotebook EvaluationObject EvaluationOrder Evaluator EvaluatorNames EvenQ EventData EventEvaluator EventHandler EventHandlerTag EventLabels EventSeries ExactBlackmanWindow ExactNumberQ ExactRootIsolation ExampleData Except ExcludedForms ExcludedLines ExcludedPhysicalQuantities ExcludePods Exclusions ExclusionsStyle Exists Exit ExitDialog ExoplanetData Exp Expand ExpandAll ExpandDenominator ExpandFileName ExpandNumerator Expectation ExpectationE ExpectedValue ExpGammaDistribution ExpIntegralE ExpIntegralEi ExpirationDate Exponent ExponentFunction ExponentialDistribution ExponentialFamily ExponentialGeneratingFunction ExponentialMovingAverage ExponentialPowerDistribution ExponentPosition ExponentStep Export ExportAutoReplacements ExportByteArray ExportForm ExportPacket ExportString Expression ExpressionCell ExpressionPacket ExpressionUUID ExpToTrig ExtendedEntityClass ExtendedGCD Extension ExtentElementFunction ExtentMarkers ExtentSize ExternalBundle ExternalCall ExternalDataCharacterEncoding ExternalEvaluate ExternalFunction ExternalFunctionName ExternalObject ExternalOptions ExternalSessionObject ExternalSessions ExternalTypeSignature ExternalValue Extract ExtractArchive ExtractLayer ExtremeValueDistributionFaceForm FaceGrids FaceGridsStyle FacialFeatures Factor FactorComplete Factorial Factorial2 FactorialMoment FactorialMomentGeneratingFunction FactorialPower FactorInteger FactorList FactorSquareFree FactorSquareFreeList FactorTerms FactorTermsList Fail Failure FailureAction FailureDistribution FailureQ False FareySequence FARIMAProcess FeatureDistance FeatureExtract FeatureExtraction FeatureExtractor FeatureExtractorFunction FeatureNames FeatureNearest FeatureSpacePlot FeatureSpacePlot3D FeatureTypes FEDisableConsolePrintPacket FeedbackLinearize FeedbackSector FeedbackSectorStyle FeedbackType FEEnableConsolePrintPacket FetalGrowthData Fibonacci Fibonorial FieldCompletionFunction FieldHint FieldHintStyle FieldMasked FieldSize File FileBaseName FileByteCount FileConvert FileDate FileExistsQ FileExtension FileFormat FileHandler FileHash FileInformation FileName FileNameDepth FileNameDialogSettings FileNameDrop FileNameForms FileNameJoin FileNames FileNameSetter FileNameSplit FileNameTake FilePrint FileSize FileSystemMap FileSystemScan FileTemplate FileTemplateApply FileType FilledCurve FilledCurveBox FilledCurveBoxOptions Filling FillingStyle FillingTransform FilteredEntityClass FilterRules FinancialBond FinancialData FinancialDerivative FinancialIndicator Find FindAnomalies FindArgMax FindArgMin FindChannels FindClique FindClusters FindCookies FindCurvePath FindCycle FindDevices FindDistribution FindDistributionParameters FindDivisions FindEdgeCover FindEdgeCut FindEdgeIndependentPaths FindEquationalProof FindEulerianCycle FindExternalEvaluators FindFaces FindFile FindFit FindFormula FindFundamentalCycles FindGeneratingFunction FindGeoLocation FindGeometricConjectures FindGeometricTransform FindGraphCommunities FindGraphIsomorphism FindGraphPartition FindHamiltonianCycle FindHamiltonianPath FindHiddenMarkovStates FindIndependentEdgeSet FindIndependentVertexSet FindInstance FindIntegerNullVector FindKClan FindKClique FindKClub FindKPlex FindLibrary FindLinearRecurrence FindList FindMatchingColor FindMaximum FindMaximumFlow FindMaxValue FindMeshDefects FindMinimum FindMinimumCostFlow FindMinimumCut FindMinValue FindMoleculeSubstructure FindPath FindPeaks FindPermutation FindPostmanTour FindProcessParameters FindRepeat FindRoot FindSequenceFunction FindSettings FindShortestPath FindShortestTour FindSpanningTree FindSystemModelEquilibrium FindTextualAnswer FindThreshold FindTransientRepeat FindVertexCover FindVertexCut FindVertexIndependentPaths Fine FinishDynamic FiniteAbelianGroupCount FiniteGroupCount FiniteGroupData First FirstCase FirstPassageTimeDistribution FirstPosition FischerGroupFi22 FischerGroupFi23 FischerGroupFi24Prime FisherHypergeometricDistribution FisherRatioTest FisherZDistribution Fit FitAll FitRegularization FittedModel FixedOrder FixedPoint FixedPointList FlashSelection Flat Flatten FlattenAt FlattenLayer FlatTopWindow FlipView Floor FlowPolynomial FlushPrintOutputPacket Fold FoldList FoldPair FoldPairList FollowRedirects Font FontColor FontFamily FontForm FontName FontOpacity FontPostScriptName FontProperties FontReencoding FontSize FontSlant FontSubstitutions FontTracking FontVariations FontWeight For ForAll Format FormatRules FormatType FormatTypeAutoConvert FormatValues FormBox FormBoxOptions FormControl FormFunction FormLayoutFunction FormObject FormPage FormTheme FormulaData FormulaLookup FortranForm Forward ForwardBackward Fourier FourierCoefficient FourierCosCoefficient FourierCosSeries FourierCosTransform FourierDCT FourierDCTFilter FourierDCTMatrix FourierDST FourierDSTMatrix FourierMatrix FourierParameters FourierSequenceTransform FourierSeries FourierSinCoefficient FourierSinSeries FourierSinTransform FourierTransform FourierTrigSeries FractionalBrownianMotionProcess FractionalGaussianNoiseProcess FractionalPart FractionBox FractionBoxOptions FractionLine Frame FrameBox FrameBoxOptions Framed FrameInset FrameLabel Frameless FrameMargins FrameRate FrameStyle FrameTicks FrameTicksStyle FRatioDistribution FrechetDistribution FreeQ FrenetSerretSystem FrequencySamplingFilterKernel FresnelC FresnelF FresnelG FresnelS Friday FrobeniusNumber FrobeniusSolve FromAbsoluteTime FromCharacterCode FromCoefficientRules FromContinuedFraction FromDate FromDigits FromDMS FromEntity FromJulianDate FromLetterNumber FromPolarCoordinates FromRomanNumeral FromSphericalCoordinates FromUnixTime Front FrontEndDynamicExpression FrontEndEventActions FrontEndExecute FrontEndObject FrontEndResource FrontEndResourceString FrontEndStackSize FrontEndToken FrontEndTokenExecute FrontEndValueCache FrontEndVersion FrontFaceColor FrontFaceOpacity Full FullAxes FullDefinition FullForm FullGraphics FullInformationOutputRegulator FullOptions FullRegion FullSimplify Function FunctionCompile FunctionCompileExport FunctionCompileExportByteArray FunctionCompileExportLibrary FunctionCompileExportString FunctionDomain FunctionExpand FunctionInterpolation FunctionPeriod FunctionRange FunctionSpace FussellVeselyImportanceGaborFilter GaborMatrix GaborWavelet GainMargins GainPhaseMargins GalaxyData GalleryView Gamma GammaDistribution GammaRegularized GapPenalty GARCHProcess GatedRecurrentLayer Gather GatherBy GaugeFaceElementFunction GaugeFaceStyle GaugeFrameElementFunction GaugeFrameSize GaugeFrameStyle GaugeLabels GaugeMarkers GaugeStyle GaussianFilter GaussianIntegers GaussianMatrix GaussianOrthogonalMatrixDistribution GaussianSymplecticMatrixDistribution GaussianUnitaryMatrixDistribution GaussianWindow GCD GegenbauerC General GeneralizedLinearModelFit GenerateAsymmetricKeyPair GenerateConditions GeneratedCell GeneratedDocumentBinding GenerateDerivedKey GenerateDigitalSignature GenerateDocument GeneratedParameters GeneratedQuantityMagnitudes GenerateHTTPResponse GenerateSecuredAuthenticationKey GenerateSymmetricKey GeneratingFunction GeneratorDescription GeneratorHistoryLength GeneratorOutputType Generic GenericCylindricalDecomposition GenomeData GenomeLookup GeoAntipode GeoArea GeoArraySize GeoBackground GeoBoundingBox GeoBounds GeoBoundsRegion GeoBubbleChart GeoCenter GeoCircle GeodesicClosing GeodesicDilation GeodesicErosion GeodesicOpening GeoDestination GeodesyData GeoDirection GeoDisk GeoDisplacement GeoDistance GeoDistanceList GeoElevationData GeoEntities GeoGraphics GeogravityModelData GeoGridDirectionDifference GeoGridLines GeoGridLinesStyle GeoGridPosition GeoGridRange GeoGridRangePadding GeoGridUnitArea GeoGridUnitDistance GeoGridVector GeoGroup GeoHemisphere GeoHemisphereBoundary GeoHistogram GeoIdentify GeoImage GeoLabels GeoLength GeoListPlot GeoLocation GeologicalPeriodData GeomagneticModelData GeoMarker GeometricAssertion GeometricBrownianMotionProcess GeometricDistribution GeometricMean GeometricMeanFilter GeometricScene GeometricTransformation GeometricTransformation3DBox GeometricTransformation3DBoxOptions GeometricTransformationBox GeometricTransformationBoxOptions GeoModel GeoNearest GeoPath GeoPosition GeoPositionENU GeoPositionXYZ GeoProjection GeoProjectionData GeoRange GeoRangePadding GeoRegionValuePlot GeoResolution GeoScaleBar GeoServer GeoSmoothHistogram GeoStreamPlot GeoStyling GeoStylingImageFunction GeoVariant GeoVector GeoVectorENU GeoVectorPlot GeoVectorXYZ GeoVisibleRegion GeoVisibleRegionBoundary GeoWithinQ GeoZoomLevel GestureHandler GestureHandlerTag Get GetBoundingBoxSizePacket GetContext GetEnvironment GetFileName GetFrontEndOptionsDataPacket GetLinebreakInformationPacket GetMenusPacket GetPageBreakInformationPacket Glaisher GlobalClusteringCoefficient GlobalPreferences GlobalSession Glow GoldenAngle GoldenRatio GompertzMakehamDistribution GoodmanKruskalGamma GoodmanKruskalGammaTest Goto Grad Gradient GradientFilter GradientOrientationFilter GrammarApply GrammarRules GrammarToken Graph Graph3D GraphAssortativity GraphAutomorphismGroup GraphCenter GraphComplement GraphData GraphDensity GraphDiameter GraphDifference GraphDisjointUnion GraphDistance GraphDistanceMatrix GraphElementData GraphEmbedding GraphHighlight GraphHighlightStyle GraphHub Graphics Graphics3D Graphics3DBox Graphics3DBoxOptions GraphicsArray GraphicsBaseline GraphicsBox GraphicsBoxOptions GraphicsColor GraphicsColumn GraphicsComplex GraphicsComplex3DBox GraphicsComplex3DBoxOptions GraphicsComplexBox GraphicsComplexBoxOptions GraphicsContents GraphicsData GraphicsGrid GraphicsGridBox GraphicsGroup GraphicsGroup3DBox GraphicsGroup3DBoxOptions GraphicsGroupBox GraphicsGroupBoxOptions GraphicsGrouping GraphicsHighlightColor GraphicsRow GraphicsSpacing GraphicsStyle GraphIntersection GraphLayout GraphLinkEfficiency GraphPeriphery GraphPlot GraphPlot3D GraphPower GraphPropertyDistribution GraphQ GraphRadius GraphReciprocity GraphRoot GraphStyle GraphUnion Gray GrayLevel Greater GreaterEqual GreaterEqualLess GreaterEqualThan GreaterFullEqual GreaterGreater GreaterLess GreaterSlantEqual GreaterThan GreaterTilde Green GreenFunction Grid GridBaseline GridBox GridBoxAlignment GridBoxBackground GridBoxDividers GridBoxFrame GridBoxItemSize GridBoxItemStyle GridBoxOptions GridBoxSpacings GridCreationSettings GridDefaultElement GridElementStyleOptions GridFrame GridFrameMargins GridGraph GridLines GridLinesStyle GroebnerBasis GroupActionBase GroupBy GroupCentralizer GroupElementFromWord GroupElementPosition GroupElementQ GroupElements GroupElementToWord GroupGenerators Groupings GroupMultiplicationTable GroupOrbits GroupOrder GroupPageBreakWithin GroupSetwiseStabilizer GroupStabilizer GroupStabilizerChain GroupTogetherGrouping GroupTogetherNestedGrouping GrowCutComponents Gudermannian GuidedFilter GumbelDistributionHaarWavelet HadamardMatrix HalfLine HalfNormalDistribution HalfPlane HalfSpace HamiltonianGraphQ HammingDistance HammingWindow HandlerFunctions HandlerFunctionsKeys HankelH1 HankelH2 HankelMatrix HankelTransform HannPoissonWindow HannWindow HaradaNortonGroupHN HararyGraph HarmonicMean HarmonicMeanFilter HarmonicNumber Hash Haversine HazardFunction Head HeadCompose HeaderLines Heads HeavisideLambda HeavisidePi HeavisideTheta HeldGroupHe HeldPart HelpBrowserLookup HelpBrowserNotebook HelpBrowserSettings Here HermiteDecomposition HermiteH HermitianMatrixQ HessenbergDecomposition Hessian HexadecimalCharacter Hexahedron HexahedronBox HexahedronBoxOptions HiddenMarkovProcess HiddenSurface Highlighted HighlightGraph HighlightImage HighlightMesh HighpassFilter HigmanSimsGroupHS HilbertCurve HilbertFilter HilbertMatrix Histogram Histogram3D HistogramDistribution HistogramList HistogramTransform HistogramTransformInterpolation HistoricalPeriodData HitMissTransform HITSCentrality HjorthDistribution HodgeDual HoeffdingD HoeffdingDTest Hold HoldAll HoldAllComplete HoldComplete HoldFirst HoldForm HoldPattern HoldRest HolidayCalendar HomeDirectory HomePage Horizontal HorizontalForm HorizontalGauge HorizontalScrollPosition HornerForm HostLookup HotellingTSquareDistribution HoytDistribution HTMLSave HTTPErrorResponse HTTPRedirect HTTPRequest HTTPRequestData HTTPResponse Hue HumanGrowthData HumpDownHump HumpEqual HurwitzLerchPhi HurwitzZeta HyperbolicDistribution HypercubeGraph HyperexponentialDistribution Hyperfactorial Hypergeometric0F1 Hypergeometric0F1Regularized Hypergeometric1F1 Hypergeometric1F1Regularized Hypergeometric2F1 Hypergeometric2F1Regularized HypergeometricDistribution HypergeometricPFQ HypergeometricPFQRegularized HypergeometricU Hyperlink HyperlinkCreationSettings Hyperplane Hyphenation HyphenationOptions HypoexponentialDistribution HypothesisTestDataI IconData Iconize IconizedObject IconRules Icosahedron Identity IdentityMatrix If IgnoreCase IgnoreDiacritics IgnorePunctuation IgnoreSpellCheck IgnoringInactive Im Image Image3D Image3DProjection Image3DSlices ImageAccumulate ImageAdd ImageAdjust ImageAlign ImageApply ImageApplyIndexed ImageAspectRatio ImageAssemble ImageAugmentationLayer ImageBoundingBoxes ImageCache ImageCacheValid ImageCapture ImageCaptureFunction ImageCases ImageChannels ImageClip ImageCollage ImageColorSpace ImageCompose ImageContainsQ ImageContents ImageConvolve ImageCooccurrence ImageCorners ImageCorrelate ImageCorrespondingPoints ImageCrop ImageData ImageDeconvolve ImageDemosaic ImageDifference ImageDimensions ImageDisplacements ImageDistance ImageEffect ImageExposureCombine ImageFeatureTrack ImageFileApply ImageFileFilter ImageFileScan ImageFilter ImageFocusCombine ImageForestingComponents ImageFormattingWidth ImageForwardTransformation ImageGraphics ImageHistogram ImageIdentify ImageInstanceQ ImageKeypoints ImageLevels ImageLines ImageMargins ImageMarker ImageMarkers ImageMeasurements ImageMesh ImageMultiply ImageOffset ImagePad ImagePadding ImagePartition ImagePeriodogram ImagePerspectiveTransformation ImagePosition ImagePreviewFunction ImagePyramid ImagePyramidApply ImageQ ImageRangeCache ImageRecolor ImageReflect ImageRegion ImageResize ImageResolution ImageRestyle ImageRotate ImageRotated ImageSaliencyFilter ImageScaled ImageScan ImageSize ImageSizeAction ImageSizeCache ImageSizeMultipliers ImageSizeRaw ImageSubtract ImageTake ImageTransformation ImageTrim ImageType ImageValue ImageValuePositions ImagingDevice ImplicitRegion Implies Import ImportAutoReplacements ImportByteArray ImportOptions ImportString ImprovementImportance In Inactivate Inactive IncidenceGraph IncidenceList IncidenceMatrix IncludeAromaticBonds IncludeConstantBasis IncludeDefinitions IncludeDirectories IncludeFileExtension IncludeGeneratorTasks IncludeHydrogens IncludeInflections IncludeMetaInformation IncludePods IncludeQuantities IncludeRelatedTables IncludeSingularTerm IncludeWindowTimes Increment IndefiniteMatrixQ Indent IndentingNewlineSpacings IndentMaxFraction IndependenceTest IndependentEdgeSetQ IndependentPhysicalQuantity IndependentUnit IndependentUnitDimension IndependentVertexSetQ Indeterminate IndeterminateThreshold IndexCreationOptions Indexed IndexGraph IndexTag Inequality InexactNumberQ InexactNumbers InfiniteLine InfinitePlane Infinity Infix InflationAdjust InflationMethod Information InformationData InformationDataGrid Inherited InheritScope InhomogeneousPoissonProcess InitialEvaluationHistory Initialization InitializationCell InitializationCellEvaluation InitializationCellWarning InitializationObjects InitializationValue Initialize InitialSeeding InlineCounterAssignments InlineCounterIncrements InlineRules Inner InnerPolygon InnerPolyhedron Inpaint Input InputAliases InputAssumptions InputAutoReplacements InputField InputFieldBox InputFieldBoxOptions InputForm InputGrouping InputNamePacket InputNotebook InputPacket InputSettings InputStream InputString InputStringPacket InputToBoxFormPacket Insert InsertionFunction InsertionPointObject InsertLinebreaks InsertResults Inset Inset3DBox Inset3DBoxOptions InsetBox InsetBoxOptions Insphere Install InstallService InstanceNormalizationLayer InString Integer IntegerDigits IntegerExponent IntegerLength IntegerName IntegerPart IntegerPartitions IntegerQ IntegerReverse Integers IntegerString Integral Integrate Interactive InteractiveTradingChart Interlaced Interleaving InternallyBalancedDecomposition InterpolatingFunction InterpolatingPolynomial Interpolation InterpolationOrder InterpolationPoints InterpolationPrecision Interpretation InterpretationBox InterpretationBoxOptions InterpretationFunction Interpreter InterpretTemplate InterquartileRange Interrupt InterruptSettings IntersectingQ Intersection Interval IntervalIntersection IntervalMarkers IntervalMarkersStyle IntervalMemberQ IntervalSlider IntervalUnion Into Inverse InverseBetaRegularized InverseCDF InverseChiSquareDistribution InverseContinuousWaveletTransform InverseDistanceTransform InverseEllipticNomeQ InverseErf InverseErfc InverseFourier InverseFourierCosTransform InverseFourierSequenceTransform InverseFourierSinTransform InverseFourierTransform InverseFunction InverseFunctions InverseGammaDistribution InverseGammaRegularized InverseGaussianDistribution InverseGudermannian InverseHankelTransform InverseHaversine InverseImagePyramid InverseJacobiCD InverseJacobiCN InverseJacobiCS InverseJacobiDC InverseJacobiDN InverseJacobiDS InverseJacobiNC InverseJacobiND InverseJacobiNS InverseJacobiSC InverseJacobiSD InverseJacobiSN InverseLaplaceTransform InverseMellinTransform InversePermutation InverseRadon InverseRadonTransform InverseSeries InverseShortTimeFourier InverseSpectrogram InverseSurvivalFunction InverseTransformedRegion InverseWaveletTransform InverseWeierstrassP InverseWishartMatrixDistribution InverseZTransform Invisible InvisibleApplication InvisibleTimes IPAddress IrreduciblePolynomialQ IslandData IsolatingInterval IsomorphicGraphQ IsotopeData Italic Item ItemAspectRatio ItemBox ItemBoxOptions ItemSize ItemStyle ItoProcessJaccardDissimilarity JacobiAmplitude Jacobian JacobiCD JacobiCN JacobiCS JacobiDC JacobiDN JacobiDS JacobiNC JacobiND JacobiNS JacobiP JacobiSC JacobiSD JacobiSN JacobiSymbol JacobiZeta JankoGroupJ1 JankoGroupJ2 JankoGroupJ3 JankoGroupJ4 JarqueBeraALMTest JohnsonDistribution Join JoinAcross Joined JoinedCurve JoinedCurveBox JoinedCurveBoxOptions JoinForm JordanDecomposition JordanModelDecomposition JulianDate JuliaSetBoettcher JuliaSetIterationCount JuliaSetPlot JuliaSetPointsK KagiChart KaiserBesselWindow KaiserWindow KalmanEstimator KalmanFilter KarhunenLoeveDecomposition KaryTree KatzCentrality KCoreComponents KDistribution KEdgeConnectedComponents KEdgeConnectedGraphQ KelvinBei KelvinBer KelvinKei KelvinKer KendallTau KendallTauTest KernelExecute KernelFunction KernelMixtureDistribution Kernels Ket Key KeyCollisionFunction KeyComplement KeyDrop KeyDropFrom KeyExistsQ KeyFreeQ KeyIntersection KeyMap KeyMemberQ KeypointStrength Keys KeySelect KeySort KeySortBy KeyTake KeyUnion KeyValueMap KeyValuePattern Khinchin KillProcess KirchhoffGraph KirchhoffMatrix KleinInvariantJ KnapsackSolve KnightTourGraph KnotData KnownUnitQ KochCurve KolmogorovSmirnovTest KroneckerDelta KroneckerModelDecomposition KroneckerProduct KroneckerSymbol KuiperTest KumaraswamyDistribution Kurtosis KuwaharaFilter KVertexConnectedComponents KVertexConnectedGraphQLABColor Label Labeled LabeledSlider LabelingFunction LabelingSize LabelStyle LabelVisibility LaguerreL LakeData LambdaComponents LambertW LaminaData LanczosWindow LandauDistribution Language LanguageCategory LanguageData LanguageIdentify LanguageOptions LaplaceDistribution LaplaceTransform Laplacian LaplacianFilter LaplacianGaussianFilter Large Larger Last Latitude LatitudeLongitude LatticeData LatticeReduce Launch LaunchKernels LayeredGraphPlot LayerSizeFunction LayoutInformation LCHColor LCM LeaderSize LeafCount LeapYearQ LearnDistribution LearnedDistribution LearningRate LearningRateMultipliers LeastSquares LeastSquaresFilterKernel Left LeftArrow LeftArrowBar LeftArrowRightArrow LeftDownTeeVector LeftDownVector LeftDownVectorBar LeftRightArrow LeftRightVector LeftTee LeftTeeArrow LeftTeeVector LeftTriangle LeftTriangleBar LeftTriangleEqual LeftUpDownVector LeftUpTeeVector LeftUpVector LeftUpVectorBar LeftVector LeftVectorBar LegendAppearance Legended LegendFunction LegendLabel LegendLayout LegendMargins LegendMarkers LegendMarkerSize LegendreP LegendreQ LegendreType Length LengthWhile LerchPhi Less LessEqual LessEqualGreater LessEqualThan LessFullEqual LessGreater LessLess LessSlantEqual LessThan LessTilde LetterCharacter LetterCounts LetterNumber LetterQ Level LeveneTest LeviCivitaTensor LevyDistribution Lexicographic LibraryDataType LibraryFunction LibraryFunctionError LibraryFunctionInformation LibraryFunctionLoad LibraryFunctionUnload LibraryLoad LibraryUnload LicenseID LiftingFilterData LiftingWaveletTransform LightBlue LightBrown LightCyan Lighter LightGray LightGreen Lighting LightingAngle LightMagenta LightOrange LightPink LightPurple LightRed LightSources LightYellow Likelihood Limit LimitsPositioning LimitsPositioningTokens LindleyDistribution Line Line3DBox Line3DBoxOptions LinearFilter LinearFractionalOptimization LinearFractionalTransform LinearGradientImage LinearizingTransformationData LinearLayer LinearModelFit LinearOffsetFunction LinearOptimization LinearProgramming LinearRecurrence LinearSolve LinearSolveFunction LineBox LineBoxOptions LineBreak LinebreakAdjustments LineBreakChart LinebreakSemicolonWeighting LineBreakWithin LineColor LineGraph LineIndent LineIndentMaxFraction LineIntegralConvolutionPlot LineIntegralConvolutionScale LineLegend LineOpacity LineSpacing LineWrapParts LinkActivate LinkClose LinkConnect LinkConnectedQ LinkCreate LinkError LinkFlush LinkFunction LinkHost LinkInterrupt LinkLaunch LinkMode LinkObject LinkOpen LinkOptions LinkPatterns LinkProtocol LinkRankCentrality LinkRead LinkReadHeld LinkReadyQ Links LinkService LinkWrite LinkWriteHeld LiouvilleLambda List Listable ListAnimate ListContourPlot ListContourPlot3D ListConvolve ListCorrelate ListCurvePathPlot ListDeconvolve ListDensityPlot ListDensityPlot3D Listen ListFormat ListFourierSequenceTransform ListInterpolation ListLineIntegralConvolutionPlot ListLinePlot ListLogLinearPlot ListLogLogPlot ListLogPlot ListPicker ListPickerBox ListPickerBoxBackground ListPickerBoxOptions ListPlay ListPlot ListPlot3D ListPointPlot3D ListPolarPlot ListQ ListSliceContourPlot3D ListSliceDensityPlot3D ListSliceVectorPlot3D ListStepPlot ListStreamDensityPlot ListStreamPlot ListSurfacePlot3D ListVectorDensityPlot ListVectorPlot ListVectorPlot3D ListZTransform Literal LiteralSearch LocalAdaptiveBinarize LocalCache LocalClusteringCoefficient LocalizeDefinitions LocalizeVariables LocalObject LocalObjects LocalResponseNormalizationLayer LocalSubmit LocalSymbol LocalTime LocalTimeZone LocationEquivalenceTest LocationTest Locator LocatorAutoCreate LocatorBox LocatorBoxOptions LocatorCentering LocatorPane LocatorPaneBox LocatorPaneBoxOptions LocatorRegion Locked Log Log10 Log2 LogBarnesG LogGamma LogGammaDistribution LogicalExpand LogIntegral LogisticDistribution LogisticSigmoid LogitModelFit LogLikelihood LogLinearPlot LogLogisticDistribution LogLogPlot LogMultinormalDistribution LogNormalDistribution LogPlot LogRankTest LogSeriesDistribution LongEqual Longest LongestCommonSequence LongestCommonSequencePositions LongestCommonSubsequence LongestCommonSubsequencePositions LongestMatch LongestOrderedSequence LongForm Longitude LongLeftArrow LongLeftRightArrow LongRightArrow LongShortTermMemoryLayer Lookup Loopback LoopFreeGraphQ LossFunction LowerCaseQ LowerLeftArrow LowerRightArrow LowerTriangularize LowerTriangularMatrixQ LowpassFilter LQEstimatorGains LQGRegulator LQOutputRegulatorGains LQRegulatorGains LUBackSubstitution LucasL LuccioSamiComponents LUDecomposition LunarEclipse LUVColor LyapunovSolve LyonsGroupLyMachineID MachineName MachineNumberQ MachinePrecision MacintoshSystemPageSetup Magenta Magnification Magnify MailAddressValidation MailExecute MailFolder MailItem MailReceiverFunction MailResponseFunction MailSearch MailServerConnect MailServerConnection MailSettings MainSolve MaintainDynamicCaches Majority MakeBoxes MakeExpression MakeRules ManagedLibraryExpressionID ManagedLibraryExpressionQ MandelbrotSetBoettcher MandelbrotSetDistance MandelbrotSetIterationCount MandelbrotSetMemberQ MandelbrotSetPlot MangoldtLambda ManhattanDistance Manipulate Manipulator MannedSpaceMissionData MannWhitneyTest MantissaExponent Manual Map MapAll MapAt MapIndexed MAProcess MapThread MarchenkoPasturDistribution MarcumQ MardiaCombinedTest MardiaKurtosisTest MardiaSkewnessTest MarginalDistribution MarkovProcessProperties Masking MatchingDissimilarity MatchLocalNameQ MatchLocalNames MatchQ Material MathematicalFunctionData MathematicaNotation MathieuC MathieuCharacteristicA MathieuCharacteristicB MathieuCharacteristicExponent MathieuCPrime MathieuGroupM11 MathieuGroupM12 MathieuGroupM22 MathieuGroupM23 MathieuGroupM24 MathieuS MathieuSPrime MathMLForm MathMLText Matrices MatrixExp MatrixForm MatrixFunction MatrixLog MatrixNormalDistribution MatrixPlot MatrixPower MatrixPropertyDistribution MatrixQ MatrixRank MatrixTDistribution Max MaxBend MaxCellMeasure MaxColorDistance MaxDetect MaxDuration MaxExtraBandwidths MaxExtraConditions MaxFeatureDisplacement MaxFeatures MaxFilter MaximalBy Maximize MaxItems MaxIterations MaxLimit MaxMemoryUsed MaxMixtureKernels MaxOverlapFraction MaxPlotPoints MaxPoints MaxRecursion MaxStableDistribution MaxStepFraction MaxSteps MaxStepSize MaxTrainingRounds MaxValue MaxwellDistribution MaxWordGap McLaughlinGroupMcL Mean MeanAbsoluteLossLayer MeanAround MeanClusteringCoefficient MeanDegreeConnectivity MeanDeviation MeanFilter MeanGraphDistance MeanNeighborDegree MeanShift MeanShiftFilter MeanSquaredLossLayer Median MedianDeviation MedianFilter MedicalTestData Medium MeijerG MeijerGReduce MeixnerDistribution MellinConvolve MellinTransform MemberQ MemoryAvailable MemoryConstrained MemoryConstraint MemoryInUse MengerMesh Menu MenuAppearance MenuCommandKey MenuEvaluator MenuItem MenuList MenuPacket MenuSortingValue MenuStyle MenuView Merge MergeDifferences MergingFunction MersennePrimeExponent MersennePrimeExponentQ Mesh MeshCellCentroid MeshCellCount MeshCellHighlight MeshCellIndex MeshCellLabel MeshCellMarker MeshCellMeasure MeshCellQuality MeshCells MeshCellShapeFunction MeshCellStyle MeshCoordinates MeshFunctions MeshPrimitives MeshQualityGoal MeshRange MeshRefinementFunction MeshRegion MeshRegionQ MeshShading MeshStyle Message MessageDialog MessageList MessageName MessageObject MessageOptions MessagePacket Messages MessagesNotebook MetaCharacters MetaInformation MeteorShowerData Method MethodOptions MexicanHatWavelet MeyerWavelet Midpoint Min MinColorDistance MinDetect MineralData MinFilter MinimalBy MinimalPolynomial MinimalStateSpaceModel Minimize MinimumTimeIncrement MinIntervalSize MinkowskiQuestionMark MinLimit MinMax MinorPlanetData Minors MinRecursion MinSize MinStableDistribution Minus MinusPlus MinValue Missing MissingBehavior MissingDataMethod MissingDataRules MissingQ MissingString MissingStyle MissingValuePattern MittagLefflerE MixedFractionParts MixedGraphQ MixedMagnitude MixedRadix MixedRadixQuantity MixedUnit MixtureDistribution Mod Modal Mode Modular ModularInverse ModularLambda Module Modulus MoebiusMu Molecule MoleculeContainsQ MoleculeEquivalentQ MoleculeGraph MoleculeModify MoleculePattern MoleculePlot MoleculePlot3D MoleculeProperty MoleculeQ MoleculeValue Moment Momentary MomentConvert MomentEvaluate MomentGeneratingFunction MomentOfInertia Monday Monitor MonomialList MonomialOrder MonsterGroupM MoonPhase MoonPosition MorletWavelet MorphologicalBinarize MorphologicalBranchPoints MorphologicalComponents MorphologicalEulerNumber MorphologicalGraph MorphologicalPerimeter MorphologicalTransform MortalityData Most MountainData MouseAnnotation MouseAppearance MouseAppearanceTag MouseButtons Mouseover MousePointerNote MousePosition MovieData MovingAverage MovingMap MovingMedian MoyalDistribution Multicolumn MultiedgeStyle MultigraphQ MultilaunchWarning MultiLetterItalics MultiLetterStyle MultilineFunction Multinomial MultinomialDistribution MultinormalDistribution MultiplicativeOrder Multiplicity MultiplySides Multiselection MultivariateHypergeometricDistribution MultivariatePoissonDistribution MultivariateTDistributionN NakagamiDistribution NameQ Names NamespaceBox NamespaceBoxOptions Nand NArgMax NArgMin NBernoulliB NBodySimulation NBodySimulationData NCache NDEigensystem NDEigenvalues NDSolve NDSolveValue Nearest NearestFunction NearestNeighborGraph NearestTo NebulaData NeedCurrentFrontEndPackagePacket NeedCurrentFrontEndSymbolsPacket NeedlemanWunschSimilarity Needs Negative NegativeBinomialDistribution NegativeDefiniteMatrixQ NegativeIntegers NegativeMultinomialDistribution NegativeRationals NegativeReals NegativeSemidefiniteMatrixQ NeighborhoodData NeighborhoodGraph Nest NestedGreaterGreater NestedLessLess NestedScriptRules NestGraph NestList NestWhile NestWhileList NetAppend NetBidirectionalOperator NetChain NetDecoder NetDelete NetDrop NetEncoder NetEvaluationMode NetExtract NetFlatten NetFoldOperator NetGraph NetInformation NetInitialize NetInsert NetInsertSharedArrays NetJoin NetMapOperator NetMapThreadOperator NetMeasurements NetModel NetNestOperator NetPairEmbeddingOperator NetPort NetPortGradient NetPrepend NetRename NetReplace NetReplacePart NetSharedArray NetStateObject NetTake NetTrain NetTrainResultsObject NetworkPacketCapture NetworkPacketRecording NetworkPacketRecordingDuring NetworkPacketTrace NeumannValue NevilleThetaC NevilleThetaD NevilleThetaN NevilleThetaS NewPrimitiveStyle NExpectation Next NextCell NextDate NextPrime NextScheduledTaskTime NHoldAll NHoldFirst NHoldRest NicholsGridLines NicholsPlot NightHemisphere NIntegrate NMaximize NMaxValue NMinimize NMinValue NominalVariables NonAssociative NoncentralBetaDistribution NoncentralChiSquareDistribution NoncentralFRatioDistribution NoncentralStudentTDistribution NonCommutativeMultiply NonConstants NondimensionalizationTransform None NoneTrue NonlinearModelFit NonlinearStateSpaceModel NonlocalMeansFilter NonNegative NonNegativeIntegers NonNegativeRationals NonNegativeReals NonPositive NonPositiveIntegers NonPositiveRationals NonPositiveReals Nor NorlundB Norm Normal NormalDistribution NormalGrouping NormalizationLayer Normalize Normalized NormalizedSquaredEuclideanDistance NormalMatrixQ NormalsFunction NormFunction Not NotCongruent NotCupCap NotDoubleVerticalBar Notebook NotebookApply NotebookAutoSave NotebookClose NotebookConvertSettings NotebookCreate NotebookCreateReturnObject NotebookDefault NotebookDelete NotebookDirectory NotebookDynamicExpression NotebookEvaluate NotebookEventActions NotebookFileName NotebookFind NotebookFindReturnObject NotebookGet NotebookGetLayoutInformationPacket NotebookGetMisspellingsPacket NotebookImport NotebookInformation NotebookInterfaceObject NotebookLocate NotebookObject NotebookOpen NotebookOpenReturnObject NotebookPath NotebookPrint NotebookPut NotebookPutReturnObject NotebookRead NotebookResetGeneratedCells Notebooks NotebookSave NotebookSaveAs NotebookSelection NotebookSetupLayoutInformationPacket NotebooksMenu NotebookTemplate NotebookWrite NotElement NotEqualTilde NotExists NotGreater NotGreaterEqual NotGreaterFullEqual NotGreaterGreater NotGreaterLess NotGreaterSlantEqual NotGreaterTilde Nothing NotHumpDownHump NotHumpEqual NotificationFunction NotLeftTriangle NotLeftTriangleBar NotLeftTriangleEqual NotLess NotLessEqual NotLessFullEqual NotLessGreater NotLessLess NotLessSlantEqual NotLessTilde NotNestedGreaterGreater NotNestedLessLess NotPrecedes NotPrecedesEqual NotPrecedesSlantEqual NotPrecedesTilde NotReverseElement NotRightTriangle NotRightTriangleBar NotRightTriangleEqual NotSquareSubset NotSquareSubsetEqual NotSquareSuperset NotSquareSupersetEqual NotSubset NotSubsetEqual NotSucceeds NotSucceedsEqual NotSucceedsSlantEqual NotSucceedsTilde NotSuperset NotSupersetEqual NotTilde NotTildeEqual NotTildeFullEqual NotTildeTilde NotVerticalBar Now NoWhitespace NProbability NProduct NProductFactors NRoots NSolve NSum NSumTerms NuclearExplosionData NuclearReactorData Null NullRecords NullSpace NullWords Number NumberCompose NumberDecompose NumberExpand NumberFieldClassNumber NumberFieldDiscriminant NumberFieldFundamentalUnits NumberFieldIntegralBasis NumberFieldNormRepresentatives NumberFieldRegulator NumberFieldRootsOfUnity NumberFieldSignature NumberForm NumberFormat NumberLinePlot NumberMarks NumberMultiplier NumberPadding NumberPoint NumberQ NumberSeparator NumberSigns NumberString Numerator NumeratorDenominator NumericalOrder NumericalSort NumericArray NumericArrayQ NumericArrayType NumericFunction NumericQ NuttallWindow NValues NyquistGridLines NyquistPlotO ObservabilityGramian ObservabilityMatrix ObservableDecomposition ObservableModelQ OceanData Octahedron OddQ Off Offset OLEData On ONanGroupON Once OneIdentity Opacity OpacityFunction OpacityFunctionScaling Open OpenAppend Opener OpenerBox OpenerBoxOptions OpenerView OpenFunctionInspectorPacket Opening OpenRead OpenSpecialOptions OpenTemporary OpenWrite Operate OperatingSystem OptimumFlowData Optional OptionalElement OptionInspectorSettings OptionQ Options OptionsPacket OptionsPattern OptionValue OptionValueBox OptionValueBoxOptions Or Orange Order OrderDistribution OrderedQ Ordering OrderingBy OrderingLayer Orderless OrderlessPatternSequence OrnsteinUhlenbeckProcess Orthogonalize OrthogonalMatrixQ Out Outer OuterPolygon OuterPolyhedron OutputAutoOverwrite OutputControllabilityMatrix OutputControllableModelQ OutputForm OutputFormData OutputGrouping OutputMathEditExpression OutputNamePacket OutputResponse OutputSizeLimit OutputStream Over OverBar OverDot Overflow OverHat Overlaps Overlay OverlayBox OverlayBoxOptions Overscript OverscriptBox OverscriptBoxOptions OverTilde OverVector OverwriteTarget OwenT OwnValuesPackage PackingMethod PaddedForm Padding PaddingLayer PaddingSize PadeApproximant PadLeft PadRight PageBreakAbove PageBreakBelow PageBreakWithin PageFooterLines PageFooters PageHeaderLines PageHeaders PageHeight PageRankCentrality PageTheme PageWidth Pagination PairedBarChart PairedHistogram PairedSmoothHistogram PairedTTest PairedZTest PaletteNotebook PalettePath PalindromeQ Pane PaneBox PaneBoxOptions Panel PanelBox PanelBoxOptions Paneled PaneSelector PaneSelectorBox PaneSelectorBoxOptions PaperWidth ParabolicCylinderD ParagraphIndent ParagraphSpacing ParallelArray ParallelCombine ParallelDo Parallelepiped ParallelEvaluate Parallelization Parallelize ParallelMap ParallelNeeds Parallelogram ParallelProduct ParallelSubmit ParallelSum ParallelTable ParallelTry Parameter ParameterEstimator ParameterMixtureDistribution ParameterVariables ParametricFunction ParametricNDSolve ParametricNDSolveValue ParametricPlot ParametricPlot3D ParametricRegion ParentBox ParentCell ParentConnect ParentDirectory ParentForm Parenthesize ParentList ParentNotebook ParetoDistribution ParetoPickandsDistribution ParkData Part PartBehavior PartialCorrelationFunction PartialD ParticleAcceleratorData ParticleData Partition PartitionGranularity PartitionsP PartitionsQ PartLayer PartOfSpeech PartProtection ParzenWindow PascalDistribution PassEventsDown PassEventsUp Paste PasteAutoQuoteCharacters PasteBoxFormInlineCells PasteButton Path PathGraph PathGraphQ Pattern PatternSequence PatternTest PauliMatrix PaulWavelet Pause PausedTime PDF PeakDetect PeanoCurve PearsonChiSquareTest PearsonCorrelationTest PearsonDistribution PercentForm PerfectNumber PerfectNumberQ PerformanceGoal Perimeter PeriodicBoundaryCondition PeriodicInterpolation Periodogram PeriodogramArray Permanent Permissions PermissionsGroup PermissionsGroupMemberQ PermissionsGroups PermissionsKey PermissionsKeys PermutationCycles PermutationCyclesQ PermutationGroup PermutationLength PermutationList PermutationListQ PermutationMax PermutationMin PermutationOrder PermutationPower PermutationProduct PermutationReplace Permutations PermutationSupport Permute PeronaMalikFilter Perpendicular PerpendicularBisector PersistenceLocation PersistenceTime PersistentObject PersistentObjects PersistentValue PersonData PERTDistribution PetersenGraph PhaseMargins PhaseRange PhysicalSystemData Pi Pick PIDData PIDDerivativeFilter PIDFeedforward PIDTune Piecewise PiecewiseExpand PieChart PieChart3D PillaiTrace PillaiTraceTest PingTime Pink PitchRecognize Pivoting PixelConstrained PixelValue PixelValuePositions Placed Placeholder PlaceholderReplace Plain PlanarAngle PlanarGraph PlanarGraphQ PlanckRadiationLaw PlaneCurveData PlanetaryMoonData PlanetData PlantData Play PlayRange Plot Plot3D Plot3Matrix PlotDivision PlotJoined PlotLabel PlotLabels PlotLayout PlotLegends PlotMarkers PlotPoints PlotRange PlotRangeClipping PlotRangeClipPlanesStyle PlotRangePadding PlotRegion PlotStyle PlotTheme Pluralize Plus PlusMinus Pochhammer PodStates PodWidth Point Point3DBox Point3DBoxOptions PointBox PointBoxOptions PointFigureChart PointLegend PointSize PoissonConsulDistribution PoissonDistribution PoissonProcess PoissonWindow PolarAxes PolarAxesOrigin PolarGridLines PolarPlot PolarTicks PoleZeroMarkers PolyaAeppliDistribution PolyGamma Polygon Polygon3DBox Polygon3DBoxOptions PolygonalNumber PolygonAngle PolygonBox PolygonBoxOptions PolygonCoordinates PolygonDecomposition PolygonHoleScale PolygonIntersections PolygonScale Polyhedron PolyhedronAngle PolyhedronCoordinates PolyhedronData PolyhedronDecomposition PolyhedronGenus PolyLog PolynomialExtendedGCD PolynomialForm PolynomialGCD PolynomialLCM PolynomialMod PolynomialQ PolynomialQuotient PolynomialQuotientRemainder PolynomialReduce PolynomialRemainder Polynomials PoolingLayer PopupMenu PopupMenuBox PopupMenuBoxOptions PopupView PopupWindow Position PositionIndex Positive PositiveDefiniteMatrixQ PositiveIntegers PositiveRationals PositiveReals PositiveSemidefiniteMatrixQ PossibleZeroQ Postfix PostScript Power PowerDistribution PowerExpand PowerMod PowerModList PowerRange PowerSpectralDensity PowersRepresentations PowerSymmetricPolynomial Precedence PrecedenceForm Precedes PrecedesEqual PrecedesSlantEqual PrecedesTilde Precision PrecisionGoal PreDecrement Predict PredictionRoot PredictorFunction PredictorInformation PredictorMeasurements PredictorMeasurementsObject PreemptProtect PreferencesPath Prefix PreIncrement Prepend PrependLayer PrependTo PreprocessingRules PreserveColor PreserveImageOptions Previous PreviousCell PreviousDate PriceGraphDistribution PrimaryPlaceholder Prime PrimeNu PrimeOmega PrimePi PrimePowerQ PrimeQ Primes PrimeZetaP PrimitivePolynomialQ PrimitiveRoot PrimitiveRootList PrincipalComponents PrincipalValue Print PrintableASCIIQ PrintAction PrintForm PrintingCopies PrintingOptions PrintingPageRange PrintingStartingPageNumber PrintingStyleEnvironment Printout3D Printout3DPreviewer PrintPrecision PrintTemporary Prism PrismBox PrismBoxOptions PrivateCellOptions PrivateEvaluationOptions PrivateFontOptions PrivateFrontEndOptions PrivateKey PrivateNotebookOptions PrivatePaths Probability ProbabilityDistribution ProbabilityPlot ProbabilityPr ProbabilityScalePlot ProbitModelFit ProcessConnection ProcessDirectory ProcessEnvironment Processes ProcessEstimator ProcessInformation ProcessObject ProcessParameterAssumptions ProcessParameterQ ProcessStateDomain ProcessStatus ProcessTimeDomain Product ProductDistribution ProductLog ProgressIndicator ProgressIndicatorBox ProgressIndicatorBoxOptions Projection Prolog PromptForm ProofObject Properties Property PropertyList PropertyValue Proportion Proportional Protect Protected ProteinData Pruning PseudoInverse PsychrometricPropertyData PublicKey PublisherID PulsarData PunctuationCharacter Purple Put PutAppend Pyramid PyramidBox PyramidBoxOptionsQBinomial QFactorial QGamma QHypergeometricPFQ QnDispersion QPochhammer QPolyGamma QRDecomposition QuadraticIrrationalQ QuadraticOptimization Quantile QuantilePlot Quantity QuantityArray QuantityDistribution QuantityForm QuantityMagnitude QuantityQ QuantityUnit QuantityVariable QuantityVariableCanonicalUnit QuantityVariableDimensions QuantityVariableIdentifier QuantityVariablePhysicalQuantity Quartics QuartileDeviation Quartiles QuartileSkewness Query QueueingNetworkProcess QueueingProcess QueueProperties Quiet Quit Quotient QuotientRemainderRadialGradientImage RadialityCentrality RadicalBox RadicalBoxOptions RadioButton RadioButtonBar RadioButtonBox RadioButtonBoxOptions Radon RadonTransform RamanujanTau RamanujanTauL RamanujanTauTheta RamanujanTauZ Ramp Random RandomChoice RandomColor RandomComplex RandomEntity RandomFunction RandomGeoPosition RandomGraph RandomImage RandomInstance RandomInteger RandomPermutation RandomPoint RandomPolygon RandomPolyhedron RandomPrime RandomReal RandomSample RandomSeed RandomSeeding RandomVariate RandomWalkProcess RandomWord Range RangeFilter RangeSpecification RankedMax RankedMin RarerProbability Raster Raster3D Raster3DBox Raster3DBoxOptions RasterArray RasterBox RasterBoxOptions Rasterize RasterSize Rational RationalFunctions Rationalize Rationals Ratios RawArray RawBoxes RawData RawMedium RayleighDistribution Re Read ReadByteArray ReadLine ReadList ReadProtected ReadString Real RealAbs RealBlockDiagonalForm RealDigits RealExponent Reals RealSign Reap RecognitionPrior RecognitionThreshold Record RecordLists RecordSeparators Rectangle RectangleBox RectangleBoxOptions RectangleChart RectangleChart3D RectangularRepeatingElement RecurrenceFilter RecurrenceTable RecurringDigitsForm Red Reduce RefBox ReferenceLineStyle ReferenceMarkers ReferenceMarkerStyle Refine ReflectionMatrix ReflectionTransform Refresh RefreshRate Region RegionBinarize RegionBoundary RegionBounds RegionCentroid RegionDifference RegionDimension RegionDisjoint RegionDistance RegionDistanceFunction RegionEmbeddingDimension RegionEqual RegionFunction RegionImage RegionIntersection RegionMeasure RegionMember RegionMemberFunction RegionMoment RegionNearest RegionNearestFunction RegionPlot RegionPlot3D RegionProduct RegionQ RegionResize RegionSize RegionSymmetricDifference RegionUnion RegionWithin RegisterExternalEvaluator RegularExpression Regularization RegularlySampledQ RegularPolygon ReIm ReImLabels ReImPlot ReImStyle Reinstall RelationalDatabase RelationGraph Release ReleaseHold ReliabilityDistribution ReliefImage ReliefPlot RemoteAuthorizationCaching RemoteConnect RemoteConnectionObject RemoteFile RemoteRun RemoteRunProcess Remove RemoveAlphaChannel RemoveAsynchronousTask RemoveAudioStream RemoveBackground RemoveChannelListener RemoveChannelSubscribers Removed RemoveDiacritics RemoveInputStreamMethod RemoveOutputStreamMethod RemoveProperty RemoveScheduledTask RemoveUsers RenameDirectory RenameFile RenderAll RenderingOptions RenewalProcess RenkoChart RepairMesh Repeated RepeatedNull RepeatedString RepeatedTiming RepeatingElement Replace ReplaceAll ReplaceHeldPart ReplaceImageValue ReplaceList ReplacePart ReplacePixelValue ReplaceRepeated ReplicateLayer RequiredPhysicalQuantities Resampling ResamplingAlgorithmData ResamplingMethod Rescale RescalingTransform ResetDirectory ResetMenusPacket ResetScheduledTask ReshapeLayer Residue ResizeLayer Resolve ResourceAcquire ResourceData ResourceFunction ResourceObject ResourceRegister ResourceRemove ResourceSearch ResourceSubmissionObject ResourceSubmit ResourceSystemBase ResourceUpdate ResponseForm Rest RestartInterval Restricted Resultant ResumePacket Return ReturnEntersInput ReturnExpressionPacket ReturnInputFormPacket ReturnPacket ReturnReceiptFunction ReturnTextPacket Reverse ReverseBiorthogonalSplineWavelet ReverseElement ReverseEquilibrium ReverseGraph ReverseSort ReverseSortBy ReverseUpEquilibrium RevolutionAxis RevolutionPlot3D RGBColor RiccatiSolve RiceDistribution RidgeFilter RiemannR RiemannSiegelTheta RiemannSiegelZ RiemannXi Riffle Right RightArrow RightArrowBar RightArrowLeftArrow RightComposition RightCosetRepresentative RightDownTeeVector RightDownVector RightDownVectorBar RightTee RightTeeArrow RightTeeVector RightTriangle RightTriangleBar RightTriangleEqual RightUpDownVector RightUpTeeVector RightUpVector RightUpVectorBar RightVector RightVectorBar RiskAchievementImportance RiskReductionImportance RogersTanimotoDissimilarity RollPitchYawAngles RollPitchYawMatrix RomanNumeral Root RootApproximant RootIntervals RootLocusPlot RootMeanSquare RootOfUnityQ RootReduce Roots RootSum Rotate RotateLabel RotateLeft RotateRight RotationAction RotationBox RotationBoxOptions RotationMatrix RotationTransform Round RoundImplies RoundingRadius Row RowAlignments RowBackgrounds RowBox RowHeights RowLines RowMinHeight RowReduce RowsEqual RowSpacings RSolve RSolveValue RudinShapiro RudvalisGroupRu Rule RuleCondition RuleDelayed RuleForm RulePlot RulerUnits Run RunProcess RunScheduledTask RunThrough RuntimeAttributes RuntimeOptions RussellRaoDissimilaritySameQ SameTest SampledEntityClass SampleDepth SampledSoundFunction SampledSoundList SampleRate SamplingPeriod SARIMAProcess SARMAProcess SASTriangle SatelliteData SatisfiabilityCount SatisfiabilityInstances SatisfiableQ Saturday Save Saveable SaveAutoDelete SaveConnection SaveDefinitions SavitzkyGolayMatrix SawtoothWave Scale Scaled ScaleDivisions ScaledMousePosition ScaleOrigin ScalePadding ScaleRanges ScaleRangeStyle ScalingFunctions ScalingMatrix ScalingTransform Scan ScheduledTask ScheduledTaskActiveQ ScheduledTaskInformation ScheduledTaskInformationData ScheduledTaskObject ScheduledTasks SchurDecomposition ScientificForm ScientificNotationThreshold ScorerGi ScorerGiPrime ScorerHi ScorerHiPrime ScreenRectangle ScreenStyleEnvironment ScriptBaselineShifts ScriptForm ScriptLevel ScriptMinSize ScriptRules ScriptSizeMultipliers Scrollbars ScrollingOptions ScrollPosition SearchAdjustment SearchIndexObject SearchIndices SearchQueryString SearchResultObject Sec Sech SechDistribution SecondOrderConeOptimization SectionGrouping SectorChart SectorChart3D SectorOrigin SectorSpacing SecuredAuthenticationKey SecuredAuthenticationKeys SeedRandom Select Selectable SelectComponents SelectedCells SelectedNotebook SelectFirst Selection SelectionAnimate SelectionCell SelectionCellCreateCell SelectionCellDefaultStyle SelectionCellParentStyle SelectionCreateCell SelectionDebuggerTag SelectionDuplicateCell SelectionEvaluate SelectionEvaluateCreateCell SelectionMove SelectionPlaceholder SelectionSetStyle SelectWithContents SelfLoops SelfLoopStyle SemanticImport SemanticImportString SemanticInterpretation SemialgebraicComponentInstances SemidefiniteOptimization SendMail SendMessage Sequence SequenceAlignment SequenceAttentionLayer SequenceCases SequenceCount SequenceFold SequenceFoldList SequenceForm SequenceHold SequenceLastLayer SequenceMostLayer SequencePosition SequencePredict SequencePredictorFunction SequenceReplace SequenceRestLayer SequenceReverseLayer SequenceSplit Series SeriesCoefficient SeriesData ServiceConnect ServiceDisconnect ServiceExecute ServiceObject ServiceRequest ServiceResponse ServiceSubmit SessionSubmit SessionTime Set SetAccuracy SetAlphaChannel SetAttributes Setbacks SetBoxFormNamesPacket SetCloudDirectory SetCookies SetDelayed SetDirectory SetEnvironment SetEvaluationNotebook SetFileDate SetFileLoadingContext SetNotebookStatusLine SetOptions SetOptionsPacket SetPermissions SetPrecision SetProperty SetSecuredAuthenticationKey SetSelectedNotebook SetSharedFunction SetSharedVariable SetSpeechParametersPacket SetStreamPosition SetSystemModel SetSystemOptions Setter SetterBar SetterBox SetterBoxOptions Setting SetUsers SetValue Shading Shallow ShannonWavelet ShapiroWilkTest Share SharingList Sharpen ShearingMatrix ShearingTransform ShellRegion ShenCastanMatrix ShiftedGompertzDistribution ShiftRegisterSequence Short ShortDownArrow Shortest ShortestMatch ShortestPathFunction ShortLeftArrow ShortRightArrow ShortTimeFourier ShortTimeFourierData ShortUpArrow Show ShowAutoConvert ShowAutoSpellCheck ShowAutoStyles ShowCellBracket ShowCellLabel ShowCellTags ShowClosedCellArea ShowCodeAssist ShowContents ShowControls ShowCursorTracker ShowGroupOpenCloseIcon ShowGroupOpener ShowInvisibleCharacters ShowPageBreaks ShowPredictiveInterface ShowSelection ShowShortBoxForm ShowSpecialCharacters ShowStringCharacters ShowSyntaxStyles ShrinkingDelay ShrinkWrapBoundingBox SiderealTime SiegelTheta SiegelTukeyTest SierpinskiCurve SierpinskiMesh Sign Signature SignedRankTest SignedRegionDistance SignificanceLevel SignPadding SignTest SimilarityRules SimpleGraph SimpleGraphQ SimplePolygonQ SimplePolyhedronQ Simplex Simplify Sin Sinc SinghMaddalaDistribution SingleEvaluation SingleLetterItalics SingleLetterStyle SingularValueDecomposition SingularValueList SingularValuePlot SingularValues Sinh SinhIntegral SinIntegral SixJSymbol Skeleton SkeletonTransform SkellamDistribution Skewness SkewNormalDistribution SkinStyle Skip SliceContourPlot3D SliceDensityPlot3D SliceDistribution SliceVectorPlot3D Slider Slider2D Slider2DBox Slider2DBoxOptions SliderBox SliderBoxOptions SlideView Slot SlotSequence Small SmallCircle Smaller SmithDecomposition SmithDelayCompensator SmithWatermanSimilarity SmoothDensityHistogram SmoothHistogram SmoothHistogram3D SmoothKernelDistribution SnDispersion Snippet SnubPolyhedron SocialMediaData Socket SocketConnect SocketListen SocketListener SocketObject SocketOpen SocketReadMessage SocketReadyQ Sockets SocketWaitAll SocketWaitNext SoftmaxLayer SokalSneathDissimilarity SolarEclipse SolarSystemFeatureData SolidAngle SolidData SolidRegionQ Solve SolveAlways SolveDelayed Sort SortBy SortedBy SortedEntityClass Sound SoundAndGraphics SoundNote SoundVolume SourceLink Sow Space SpaceCurveData SpaceForm Spacer Spacings Span SpanAdjustments SpanCharacterRounding SpanFromAbove SpanFromBoth SpanFromLeft SpanLineThickness SpanMaxSize SpanMinSize SpanningCharacters SpanSymmetric SparseArray SpatialGraphDistribution SpatialMedian SpatialTransformationLayer Speak SpeakTextPacket SpearmanRankTest SpearmanRho SpeciesData SpecificityGoal SpectralLineData Spectrogram SpectrogramArray Specularity SpeechRecognize SpeechSynthesize SpellingCorrection SpellingCorrectionList SpellingDictionaries SpellingDictionariesPath SpellingOptions SpellingSuggestionsPacket Sphere SphereBox SpherePoints SphericalBesselJ SphericalBesselY SphericalHankelH1 SphericalHankelH2 SphericalHarmonicY SphericalPlot3D SphericalRegion SphericalShell SpheroidalEigenvalue SpheroidalJoiningFactor SpheroidalPS SpheroidalPSPrime SpheroidalQS SpheroidalQSPrime SpheroidalRadialFactor SpheroidalS1 SpheroidalS1Prime SpheroidalS2 SpheroidalS2Prime Splice SplicedDistribution SplineClosed SplineDegree SplineKnots SplineWeights Split SplitBy SpokenString Sqrt SqrtBox SqrtBoxOptions Square SquaredEuclideanDistance SquareFreeQ SquareIntersection SquareMatrixQ SquareRepeatingElement SquaresR SquareSubset SquareSubsetEqual SquareSuperset SquareSupersetEqual SquareUnion SquareWave SSSTriangle StabilityMargins StabilityMarginsStyle StableDistribution Stack StackBegin StackComplete StackedDateListPlot StackedListPlot StackInhibit StadiumShape StandardAtmosphereData StandardDeviation StandardDeviationFilter StandardForm Standardize Standardized StandardOceanData StandbyDistribution Star StarClusterData StarData StarGraph StartAsynchronousTask StartExternalSession StartingStepSize StartOfLine StartOfString StartProcess StartScheduledTask StartupSound StartWebSession StateDimensions StateFeedbackGains StateOutputEstimator StateResponse StateSpaceModel StateSpaceRealization StateSpaceTransform StateTransformationLinearize StationaryDistribution StationaryWaveletPacketTransform StationaryWaveletTransform StatusArea StatusCentrality StepMonitor StereochemistryElements StieltjesGamma StirlingS1 StirlingS2 StopAsynchronousTask StoppingPowerData StopScheduledTask StrataVariables StratonovichProcess StreamColorFunction StreamColorFunctionScaling StreamDensityPlot StreamMarkers StreamPlot StreamPoints StreamPosition Streams StreamScale StreamStyle String StringBreak StringByteCount StringCases StringContainsQ StringCount StringDelete StringDrop StringEndsQ StringExpression StringExtract StringForm StringFormat StringFreeQ StringInsert StringJoin StringLength StringMatchQ StringPadLeft StringPadRight StringPart StringPartition StringPosition StringQ StringRepeat StringReplace StringReplaceList StringReplacePart StringReverse StringRiffle StringRotateLeft StringRotateRight StringSkeleton StringSplit StringStartsQ StringTake StringTemplate StringToByteArray StringToStream StringTrim StripBoxes StripOnInput StripWrapperBoxes StrokeForm StructuralImportance StructuredArray StructuredSelection StruveH StruveL Stub StudentTDistribution Style StyleBox StyleBoxAutoDelete StyleData StyleDefinitions StyleForm StyleHints StyleKeyMapping StyleMenuListing StyleNameDialogSettings StyleNames StylePrint StyleSheetPath Subdivide Subfactorial Subgraph SubMinus SubPlus SubresultantPolynomialRemainders SubresultantPolynomials Subresultants Subscript SubscriptBox SubscriptBoxOptions Subscripted Subsequences Subset SubsetEqual SubsetMap SubsetQ Subsets SubStar SubstitutionSystem Subsuperscript SubsuperscriptBox SubsuperscriptBoxOptions Subtract SubtractFrom SubtractSides SubValues Succeeds SucceedsEqual SucceedsSlantEqual SucceedsTilde Success SuchThat Sum SumConvergence SummationLayer Sunday SunPosition Sunrise Sunset SuperDagger SuperMinus SupernovaData SuperPlus Superscript SuperscriptBox SuperscriptBoxOptions Superset SupersetEqual SuperStar Surd SurdForm SurfaceArea SurfaceColor SurfaceData SurfaceGraphics SurvivalDistribution SurvivalFunction SurvivalModel SurvivalModelFit SuspendPacket SuzukiDistribution SuzukiGroupSuz SwatchLegend Switch Symbol SymbolName SymletWavelet Symmetric 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TouchPosition TouchscreenAutoZoom TouchscreenControlPlacement ToUpperCase Tr Trace TraceAbove TraceAction TraceBackward TraceDepth TraceDialog TraceForward TraceInternal TraceLevel TraceOff TraceOn TraceOriginal TracePrint TraceScan TrackedSymbols TrackingFunction TracyWidomDistribution TradingChart TraditionalForm TraditionalFunctionNotation TraditionalNotation TraditionalOrder TrainingProgressCheckpointing TrainingProgressFunction TrainingProgressMeasurements TrainingProgressReporting TrainingStoppingCriterion TransferFunctionCancel TransferFunctionExpand TransferFunctionFactor TransferFunctionModel TransferFunctionPoles TransferFunctionTransform TransferFunctionZeros TransformationClass TransformationFunction TransformationFunctions TransformationMatrix TransformedDistribution TransformedField TransformedProcess TransformedRegion TransitionDirection TransitionDuration TransitionEffect TransitiveClosureGraph TransitiveReductionGraph Translate TranslationOptions TranslationTransform 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ds_grid_clear ds_grid_set ds_grid_add ds_grid_multiply ds_grid_set_region ds_grid_add_region ds_grid_multiply_region ds_grid_set_disk ds_grid_add_disk ds_grid_multiply_disk ds_grid_set_grid_region ds_grid_add_grid_region ds_grid_multiply_grid_region ds_grid_get ds_grid_get_sum ds_grid_get_max ds_grid_get_min ds_grid_get_mean ds_grid_get_disk_sum ds_grid_get_disk_min ds_grid_get_disk_max ds_grid_get_disk_mean ds_grid_value_exists ds_grid_value_x ds_grid_value_y ds_grid_value_disk_exists ds_grid_value_disk_x ds_grid_value_disk_y ds_grid_shuffle ds_grid_write ds_grid_read ds_grid_sort ds_grid_set ds_grid_get effect_create_below effect_create_above effect_clear part_type_create part_type_destroy part_type_exists part_type_clear part_type_shape part_type_sprite part_type_size part_type_scale part_type_orientation part_type_life part_type_step part_type_death part_type_speed part_type_direction part_type_gravity part_type_colour1 part_type_colour2 part_type_colour3 part_type_colour_mix part_type_colour_rgb part_type_colour_hsv part_type_color1 part_type_color2 part_type_color3 part_type_color_mix part_type_color_rgb part_type_color_hsv part_type_alpha1 part_type_alpha2 part_type_alpha3 part_type_blend part_system_create part_system_create_layer part_system_destroy part_system_exists part_system_clear part_system_draw_order part_system_depth part_system_position part_system_automatic_update part_system_automatic_draw part_system_update part_system_drawit part_system_get_layer part_system_layer part_particles_create part_particles_create_colour part_particles_create_color part_particles_clear part_particles_count part_emitter_create part_emitter_destroy part_emitter_destroy_all part_emitter_exists part_emitter_clear part_emitter_region part_emitter_burst part_emitter_stream external_call external_define external_free window_handle window_device matrix_get matrix_set matrix_build_identity matrix_build matrix_build_lookat matrix_build_projection_ortho matrix_build_projection_perspective matrix_build_projection_perspective_fov matrix_multiply matrix_transform_vertex matrix_stack_push matrix_stack_pop matrix_stack_multiply matrix_stack_set matrix_stack_clear matrix_stack_top matrix_stack_is_empty browser_input_capture os_get_config os_get_info os_get_language os_get_region os_lock_orientation display_get_dpi_x display_get_dpi_y display_set_gui_size display_set_gui_maximise display_set_gui_maximize device_mouse_dbclick_enable display_set_timing_method display_get_timing_method display_set_sleep_margin display_get_sleep_margin virtual_key_add virtual_key_hide virtual_key_delete virtual_key_show draw_enable_drawevent draw_enable_swf_aa draw_set_swf_aa_level draw_get_swf_aa_level draw_texture_flush draw_flush gpu_set_blendenable gpu_set_ztestenable gpu_set_zfunc gpu_set_zwriteenable gpu_set_lightingenable gpu_set_fog gpu_set_cullmode gpu_set_blendmode gpu_set_blendmode_ext gpu_set_blendmode_ext_sepalpha gpu_set_colorwriteenable gpu_set_colourwriteenable gpu_set_alphatestenable gpu_set_alphatestref gpu_set_alphatestfunc gpu_set_texfilter gpu_set_texfilter_ext gpu_set_texrepeat gpu_set_texrepeat_ext gpu_set_tex_filter gpu_set_tex_filter_ext gpu_set_tex_repeat gpu_set_tex_repeat_ext gpu_set_tex_mip_filter gpu_set_tex_mip_filter_ext gpu_set_tex_mip_bias gpu_set_tex_mip_bias_ext gpu_set_tex_min_mip gpu_set_tex_min_mip_ext gpu_set_tex_max_mip gpu_set_tex_max_mip_ext gpu_set_tex_max_aniso gpu_set_tex_max_aniso_ext gpu_set_tex_mip_enable gpu_set_tex_mip_enable_ext gpu_get_blendenable gpu_get_ztestenable gpu_get_zfunc gpu_get_zwriteenable gpu_get_lightingenable gpu_get_fog gpu_get_cullmode gpu_get_blendmode gpu_get_blendmode_ext gpu_get_blendmode_ext_sepalpha gpu_get_blendmode_src gpu_get_blendmode_dest gpu_get_blendmode_srcalpha gpu_get_blendmode_destalpha gpu_get_colorwriteenable gpu_get_colourwriteenable gpu_get_alphatestenable gpu_get_alphatestref gpu_get_alphatestfunc gpu_get_texfilter gpu_get_texfilter_ext gpu_get_texrepeat gpu_get_texrepeat_ext gpu_get_tex_filter gpu_get_tex_filter_ext gpu_get_tex_repeat gpu_get_tex_repeat_ext gpu_get_tex_mip_filter gpu_get_tex_mip_filter_ext gpu_get_tex_mip_bias gpu_get_tex_mip_bias_ext gpu_get_tex_min_mip gpu_get_tex_min_mip_ext gpu_get_tex_max_mip gpu_get_tex_max_mip_ext gpu_get_tex_max_aniso gpu_get_tex_max_aniso_ext gpu_get_tex_mip_enable gpu_get_tex_mip_enable_ext gpu_push_state gpu_pop_state gpu_get_state gpu_set_state draw_light_define_ambient draw_light_define_direction draw_light_define_point draw_light_enable draw_set_lighting draw_light_get_ambient draw_light_get draw_get_lighting shop_leave_rating url_get_domain url_open url_open_ext url_open_full get_timer achievement_login achievement_logout achievement_post achievement_increment achievement_post_score achievement_available achievement_show_achievements achievement_show_leaderboards achievement_load_friends achievement_load_leaderboard achievement_send_challenge achievement_load_progress achievement_reset achievement_login_status achievement_get_pic achievement_show_challenge_notifications achievement_get_challenges achievement_event achievement_show achievement_get_info cloud_file_save cloud_string_save cloud_synchronise ads_enable ads_disable ads_setup ads_engagement_launch ads_engagement_available ads_engagement_active ads_event ads_event_preload ads_set_reward_callback ads_get_display_height ads_get_display_width ads_move ads_interstitial_available ads_interstitial_display device_get_tilt_x device_get_tilt_y device_get_tilt_z device_is_keypad_open device_mouse_check_button device_mouse_check_button_pressed device_mouse_check_button_released device_mouse_x device_mouse_y device_mouse_raw_x device_mouse_raw_y device_mouse_x_to_gui device_mouse_y_to_gui iap_activate iap_status iap_enumerate_products iap_restore_all iap_acquire iap_consume iap_product_details iap_purchase_details facebook_init facebook_login facebook_status facebook_graph_request facebook_dialog facebook_logout facebook_launch_offerwall facebook_post_message facebook_send_invite facebook_user_id facebook_accesstoken facebook_check_permission facebook_request_read_permissions facebook_request_publish_permissions gamepad_is_supported gamepad_get_device_count gamepad_is_connected gamepad_get_description gamepad_get_button_threshold gamepad_set_button_threshold gamepad_get_axis_deadzone gamepad_set_axis_deadzone gamepad_button_count gamepad_button_check gamepad_button_check_pressed gamepad_button_check_released gamepad_button_value gamepad_axis_count gamepad_axis_value gamepad_set_vibration gamepad_set_colour gamepad_set_color os_is_paused window_has_focus code_is_compiled http_get http_get_file http_post_string http_request json_encode json_decode zip_unzip load_csv base64_encode base64_decode md5_string_unicode md5_string_utf8 md5_file os_is_network_connected sha1_string_unicode sha1_string_utf8 sha1_file os_powersave_enable analytics_event analytics_event_ext win8_livetile_tile_notification win8_livetile_tile_clear win8_livetile_badge_notification win8_livetile_badge_clear win8_livetile_queue_enable win8_secondarytile_pin win8_secondarytile_badge_notification win8_secondarytile_delete win8_livetile_notification_begin win8_livetile_notification_secondary_begin win8_livetile_notification_expiry win8_livetile_notification_tag win8_livetile_notification_text_add win8_livetile_notification_image_add win8_livetile_notification_end win8_appbar_enable win8_appbar_add_element win8_appbar_remove_element win8_settingscharm_add_entry win8_settingscharm_add_html_entry win8_settingscharm_add_xaml_entry win8_settingscharm_set_xaml_property win8_settingscharm_get_xaml_property win8_settingscharm_remove_entry win8_share_image win8_share_screenshot win8_share_file win8_share_url win8_share_text win8_search_enable win8_search_disable win8_search_add_suggestions win8_device_touchscreen_available win8_license_initialize_sandbox win8_license_trial_version winphone_license_trial_version winphone_tile_title winphone_tile_count winphone_tile_back_title winphone_tile_back_content winphone_tile_back_content_wide winphone_tile_front_image winphone_tile_front_image_small winphone_tile_front_image_wide winphone_tile_back_image winphone_tile_back_image_wide winphone_tile_background_colour winphone_tile_background_color winphone_tile_icon_image winphone_tile_small_icon_image winphone_tile_wide_content winphone_tile_cycle_images winphone_tile_small_background_image physics_world_create physics_world_gravity physics_world_update_speed physics_world_update_iterations physics_world_draw_debug physics_pause_enable physics_fixture_create physics_fixture_set_kinematic physics_fixture_set_density physics_fixture_set_awake physics_fixture_set_restitution physics_fixture_set_friction physics_fixture_set_collision_group physics_fixture_set_sensor physics_fixture_set_linear_damping physics_fixture_set_angular_damping physics_fixture_set_circle_shape physics_fixture_set_box_shape physics_fixture_set_edge_shape physics_fixture_set_polygon_shape physics_fixture_set_chain_shape physics_fixture_add_point physics_fixture_bind physics_fixture_bind_ext physics_fixture_delete physics_apply_force physics_apply_impulse physics_apply_angular_impulse physics_apply_local_force physics_apply_local_impulse physics_apply_torque physics_mass_properties physics_draw_debug physics_test_overlap physics_remove_fixture physics_set_friction physics_set_density physics_set_restitution physics_get_friction physics_get_density physics_get_restitution physics_joint_distance_create physics_joint_rope_create physics_joint_revolute_create physics_joint_prismatic_create physics_joint_pulley_create physics_joint_wheel_create physics_joint_weld_create physics_joint_friction_create physics_joint_gear_create physics_joint_enable_motor physics_joint_get_value physics_joint_set_value physics_joint_delete physics_particle_create physics_particle_delete physics_particle_delete_region_circle physics_particle_delete_region_box physics_particle_delete_region_poly physics_particle_set_flags physics_particle_set_category_flags physics_particle_draw physics_particle_draw_ext physics_particle_count physics_particle_get_data physics_particle_get_data_particle physics_particle_group_begin physics_particle_group_circle physics_particle_group_box physics_particle_group_polygon physics_particle_group_add_point physics_particle_group_end physics_particle_group_join physics_particle_group_delete physics_particle_group_count physics_particle_group_get_data physics_particle_group_get_mass physics_particle_group_get_inertia physics_particle_group_get_centre_x physics_particle_group_get_centre_y physics_particle_group_get_vel_x physics_particle_group_get_vel_y physics_particle_group_get_ang_vel physics_particle_group_get_x physics_particle_group_get_y physics_particle_group_get_angle physics_particle_set_group_flags physics_particle_get_group_flags physics_particle_get_max_count physics_particle_get_radius physics_particle_get_density physics_particle_get_damping physics_particle_get_gravity_scale physics_particle_set_max_count physics_particle_set_radius physics_particle_set_density physics_particle_set_damping physics_particle_set_gravity_scale network_create_socket network_create_socket_ext network_create_server network_create_server_raw network_connect network_connect_raw network_send_packet network_send_raw network_send_broadcast network_send_udp network_send_udp_raw network_set_timeout network_set_config network_resolve network_destroy buffer_create buffer_write buffer_read buffer_seek buffer_get_surface buffer_set_surface buffer_delete buffer_exists buffer_get_type buffer_get_alignment buffer_poke buffer_peek buffer_save buffer_save_ext buffer_load buffer_load_ext buffer_load_partial buffer_copy buffer_fill buffer_get_size buffer_tell buffer_resize buffer_md5 buffer_sha1 buffer_base64_encode buffer_base64_decode buffer_base64_decode_ext buffer_sizeof buffer_get_address buffer_create_from_vertex_buffer buffer_create_from_vertex_buffer_ext buffer_copy_from_vertex_buffer buffer_async_group_begin buffer_async_group_option buffer_async_group_end buffer_load_async buffer_save_async gml_release_mode gml_pragma steam_activate_overlay steam_is_overlay_enabled steam_is_overlay_activated steam_get_persona_name steam_initialised steam_is_cloud_enabled_for_app steam_is_cloud_enabled_for_account steam_file_persisted steam_get_quota_total steam_get_quota_free steam_file_write steam_file_write_file steam_file_read steam_file_delete steam_file_exists steam_file_size steam_file_share steam_is_screenshot_requested steam_send_screenshot steam_is_user_logged_on steam_get_user_steam_id steam_user_owns_dlc steam_user_installed_dlc steam_set_achievement steam_get_achievement steam_clear_achievement steam_set_stat_int steam_set_stat_float steam_set_stat_avg_rate steam_get_stat_int steam_get_stat_float steam_get_stat_avg_rate steam_reset_all_stats steam_reset_all_stats_achievements steam_stats_ready steam_create_leaderboard steam_upload_score steam_upload_score_ext steam_download_scores_around_user steam_download_scores steam_download_friends_scores steam_upload_score_buffer steam_upload_score_buffer_ext steam_current_game_language steam_available_languages steam_activate_overlay_browser steam_activate_overlay_user steam_activate_overlay_store steam_get_user_persona_name steam_get_app_id steam_get_user_account_id steam_ugc_download steam_ugc_create_item steam_ugc_start_item_update steam_ugc_set_item_title steam_ugc_set_item_description steam_ugc_set_item_visibility steam_ugc_set_item_tags steam_ugc_set_item_content steam_ugc_set_item_preview steam_ugc_submit_item_update steam_ugc_get_item_update_progress steam_ugc_subscribe_item steam_ugc_unsubscribe_item steam_ugc_num_subscribed_items steam_ugc_get_subscribed_items steam_ugc_get_item_install_info steam_ugc_get_item_update_info steam_ugc_request_item_details steam_ugc_create_query_user steam_ugc_create_query_user_ex steam_ugc_create_query_all steam_ugc_create_query_all_ex steam_ugc_query_set_cloud_filename_filter steam_ugc_query_set_match_any_tag steam_ugc_query_set_search_text steam_ugc_query_set_ranked_by_trend_days steam_ugc_query_add_required_tag steam_ugc_query_add_excluded_tag steam_ugc_query_set_return_long_description steam_ugc_query_set_return_total_only steam_ugc_query_set_allow_cached_response steam_ugc_send_query shader_set shader_get_name shader_reset shader_current shader_is_compiled shader_get_sampler_index shader_get_uniform shader_set_uniform_i shader_set_uniform_i_array shader_set_uniform_f shader_set_uniform_f_array shader_set_uniform_matrix shader_set_uniform_matrix_array shader_enable_corner_id texture_set_stage texture_get_texel_width texture_get_texel_height shaders_are_supported vertex_format_begin vertex_format_end vertex_format_delete vertex_format_add_position vertex_format_add_position_3d vertex_format_add_colour vertex_format_add_color vertex_format_add_normal vertex_format_add_texcoord vertex_format_add_textcoord vertex_format_add_custom vertex_create_buffer vertex_create_buffer_ext vertex_delete_buffer vertex_begin vertex_end vertex_position vertex_position_3d vertex_colour vertex_color vertex_argb vertex_texcoord vertex_normal vertex_float1 vertex_float2 vertex_float3 vertex_float4 vertex_ubyte4 vertex_submit vertex_freeze vertex_get_number vertex_get_buffer_size vertex_create_buffer_from_buffer vertex_create_buffer_from_buffer_ext push_local_notification push_get_first_local_notification push_get_next_local_notification push_cancel_local_notification skeleton_animation_set skeleton_animation_get skeleton_animation_mix skeleton_animation_set_ext skeleton_animation_get_ext skeleton_animation_get_duration skeleton_animation_get_frames skeleton_animation_clear skeleton_skin_set skeleton_skin_get skeleton_attachment_set skeleton_attachment_get skeleton_attachment_create skeleton_collision_draw_set skeleton_bone_data_get skeleton_bone_data_set skeleton_bone_state_get skeleton_bone_state_set skeleton_get_minmax skeleton_get_num_bounds skeleton_get_bounds skeleton_animation_get_frame skeleton_animation_set_frame draw_skeleton draw_skeleton_time draw_skeleton_instance draw_skeleton_collision skeleton_animation_list skeleton_skin_list skeleton_slot_data layer_get_id layer_get_id_at_depth layer_get_depth layer_create layer_destroy layer_destroy_instances layer_add_instance layer_has_instance layer_set_visible layer_get_visible layer_exists layer_x layer_y layer_get_x layer_get_y layer_hspeed layer_vspeed layer_get_hspeed layer_get_vspeed layer_script_begin layer_script_end layer_shader layer_get_script_begin layer_get_script_end layer_get_shader layer_set_target_room layer_get_target_room layer_reset_target_room layer_get_all layer_get_all_elements layer_get_name layer_depth layer_get_element_layer layer_get_element_type layer_element_move layer_force_draw_depth layer_is_draw_depth_forced layer_get_forced_depth layer_background_get_id layer_background_exists layer_background_create layer_background_destroy layer_background_visible layer_background_change layer_background_sprite layer_background_htiled layer_background_vtiled layer_background_stretch layer_background_yscale layer_background_xscale layer_background_blend layer_background_alpha layer_background_index layer_background_speed layer_background_get_visible layer_background_get_sprite layer_background_get_htiled layer_background_get_vtiled layer_background_get_stretch layer_background_get_yscale layer_background_get_xscale layer_background_get_blend layer_background_get_alpha layer_background_get_index layer_background_get_speed layer_sprite_get_id layer_sprite_exists layer_sprite_create layer_sprite_destroy layer_sprite_change layer_sprite_index layer_sprite_speed layer_sprite_xscale layer_sprite_yscale layer_sprite_angle layer_sprite_blend layer_sprite_alpha layer_sprite_x layer_sprite_y layer_sprite_get_sprite layer_sprite_get_index layer_sprite_get_speed layer_sprite_get_xscale layer_sprite_get_yscale layer_sprite_get_angle layer_sprite_get_blend layer_sprite_get_alpha layer_sprite_get_x layer_sprite_get_y layer_tilemap_get_id layer_tilemap_exists layer_tilemap_create layer_tilemap_destroy tilemap_tileset tilemap_x tilemap_y tilemap_set tilemap_set_at_pixel tilemap_get_tileset tilemap_get_tile_width tilemap_get_tile_height tilemap_get_width tilemap_get_height tilemap_get_x tilemap_get_y tilemap_get tilemap_get_at_pixel tilemap_get_cell_x_at_pixel tilemap_get_cell_y_at_pixel tilemap_clear draw_tilemap draw_tile tilemap_set_global_mask tilemap_get_global_mask tilemap_set_mask tilemap_get_mask tilemap_get_frame tile_set_empty tile_set_index tile_set_flip tile_set_mirror tile_set_rotate tile_get_empty tile_get_index tile_get_flip tile_get_mirror tile_get_rotate layer_tile_exists layer_tile_create layer_tile_destroy layer_tile_change layer_tile_xscale layer_tile_yscale layer_tile_blend layer_tile_alpha layer_tile_x layer_tile_y layer_tile_region layer_tile_visible layer_tile_get_sprite layer_tile_get_xscale layer_tile_get_yscale layer_tile_get_blend layer_tile_get_alpha layer_tile_get_x layer_tile_get_y layer_tile_get_region layer_tile_get_visible layer_instance_get_instance instance_activate_layer instance_deactivate_layer camera_create camera_create_view camera_destroy camera_apply camera_get_active camera_get_default camera_set_default camera_set_view_mat camera_set_proj_mat camera_set_update_script camera_set_begin_script camera_set_end_script camera_set_view_pos camera_set_view_size camera_set_view_speed camera_set_view_border camera_set_view_angle camera_set_view_target camera_get_view_mat camera_get_proj_mat camera_get_update_script camera_get_begin_script camera_get_end_script camera_get_view_x camera_get_view_y camera_get_view_width camera_get_view_height camera_get_view_speed_x camera_get_view_speed_y camera_get_view_border_x camera_get_view_border_y camera_get_view_angle camera_get_view_target view_get_camera view_get_visible view_get_xport view_get_yport view_get_wport view_get_hport view_get_surface_id view_set_camera view_set_visible view_set_xport view_set_yport view_set_wport view_set_hport view_set_surface_id gesture_drag_time gesture_drag_distance gesture_flick_speed gesture_double_tap_time gesture_double_tap_distance gesture_pinch_distance gesture_pinch_angle_towards gesture_pinch_angle_away gesture_rotate_time gesture_rotate_angle gesture_tap_count gesture_get_drag_time gesture_get_drag_distance gesture_get_flick_speed gesture_get_double_tap_time gesture_get_double_tap_distance gesture_get_pinch_distance gesture_get_pinch_angle_towards gesture_get_pinch_angle_away gesture_get_rotate_time gesture_get_rotate_angle gesture_get_tap_count keyboard_virtual_show keyboard_virtual_hide keyboard_virtual_status 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Get-Date Get-Event Get-EventLog Get-EventSubscriber Get-ExecutionPolicy Get-FormatData Get-Host Get-HotFix Get-Help Get-History Get-IseSnippet Get-Item Get-ItemProperty Get-Job Get-JobTrigger Get-Location Get-Member Get-Module Get-PfxCertificate Get-Process Get-PSBreakpoint Get-PSCallStack Get-PSDrive Get-PSProvider Get-PSSession Get-PSSessionConfiguration Get-PSSnapin Get-Random Get-ScheduledJob Get-ScheduledJobOption Get-Service Get-TraceSource Get-Transaction Get-TypeData Get-UICulture Get-Unique Get-Variable Get-Verb Get-WinEvent Get-WmiObject Get-WSManCredSSP Get-WSManInstance Group-Object Import-Alias Import-Clixml Import-Counter Import-Csv Import-IseSnippet Import-LocalizedData Import-PSSession Import-Module Invoke-AsWorkflow Invoke-Command Invoke-Expression Invoke-History Invoke-Item Invoke-RestMethod Invoke-WebRequest Invoke-WmiMethod Invoke-WSManAction Join-Path Limit-EventLog Measure-Command Measure-Object Move-Item Move-ItemProperty New-Alias New-Event New-EventLog New-IseSnippet New-Item New-ItemProperty New-JobTrigger New-Object New-Module New-ModuleManifest New-PSDrive New-PSSession New-PSSessionConfigurationFile New-PSSessionOption New-PSTransportOption New-PSWorkflowExecutionOption New-PSWorkflowSession New-ScheduledJobOption New-Service New-TimeSpan New-Variable New-WebServiceProxy New-WinEvent New-WSManInstance New-WSManSessionOption Out-Default Out-File Out-GridView Out-Host Out-Null Out-Printer Out-String Pop-Location Push-Location Read-Host Receive-Job Register-EngineEvent Register-ObjectEvent Register-PSSessionConfiguration Register-ScheduledJob Register-WmiEvent Remove-Computer Remove-Event Remove-EventLog Remove-Item Remove-ItemProperty Remove-Job Remove-JobTrigger Remove-Module Remove-PSBreakpoint Remove-PSDrive Remove-PSSession Remove-PSSnapin Remove-TypeData Remove-Variable Remove-WmiObject Remove-WSManInstance Rename-Computer Rename-Item Rename-ItemProperty Reset-ComputerMachinePassword Resolve-Path Restart-Computer Restart-Service Restore-Computer Resume-Job Resume-Service Save-Help Select-Object Select-String Select-Xml Send-MailMessage Set-Acl Set-Alias Set-AuthenticodeSignature Set-Content Set-Date Set-ExecutionPolicy Set-Item Set-ItemProperty Set-JobTrigger Set-Location Set-PSBreakpoint Set-PSDebug Set-PSSessionConfiguration Set-ScheduledJob Set-ScheduledJobOption Set-Service Set-StrictMode Set-TraceSource Set-Variable Set-WmiInstance Set-WSManInstance Set-WSManQuickConfig Show-Command Show-ControlPanelItem Show-EventLog Sort-Object Split-Path Start-Job Start-Process Start-Service Start-Sleep Start-Transaction Start-Transcript Stop-Computer Stop-Job Stop-Process Stop-Service Stop-Transcript Suspend-Job Suspend-Service Tee-Object Test-ComputerSecureChannel Test-Connection Test-ModuleManifest Test-Path Test-PSSessionConfigurationFile Trace-Command Unblock-File Undo-Transaction Unregister-Event Unregister-PSSessionConfiguration Unregister-ScheduledJob Update-FormatData Update-Help Update-List Update-TypeData Use-Transaction Wait-Event Wait-Job Wait-Process Where-Object Write-Debug Write-Error Write-EventLog Write-Host Write-Output Write-Progress Write-Verbose Write-Warning Add-MDTPersistentDrive Disable-MDTMonitorService Enable-MDTMonitorService Get-MDTDeploymentShareStatistics Get-MDTMonitorData Get-MDTOperatingSystemCatalog Get-MDTPersistentDrive Import-MDTApplication Import-MDTDriver Import-MDTOperatingSystem Import-MDTPackage Import-MDTTaskSequence New-MDTDatabase Remove-MDTMonitorData Remove-MDTPersistentDrive Restore-MDTPersistentDrive Set-MDTMonitorData Test-MDTDeploymentShare Test-MDTMonitorData Update-MDTDatabaseSchema Update-MDTDeploymentShare Update-MDTLinkedDS Update-MDTMedia Add-VamtProductKey Export-VamtData Find-VamtManagedMachine Get-VamtConfirmationId Get-VamtProduct Get-VamtProductKey Import-VamtData Initialize-VamtData Install-VamtConfirmationId Install-VamtProductActivation Install-VamtProductKey Update-VamtProduct Add-CIDatastore Add-KeyManagementServer Add-NodeKeys Add-NsxDynamicCriteria Add-NsxDynamicMemberSet Add-NsxEdgeInterfaceAddress Add-NsxFirewallExclusionListMember Add-NsxFirewallRuleMember Add-NsxIpSetMember Add-NsxLicense Add-NsxLoadBalancerPoolMember Add-NsxLoadBalancerVip Add-NsxSecondaryManager Add-NsxSecurityGroupMember Add-NsxSecurityPolicyRule Add-NsxSecurityPolicyRuleGroup Add-NsxSecurityPolicyRuleService Add-NsxServiceGroupMember Add-NsxTransportZoneMember Add-PassthroughDevice Add-VDSwitchPhysicalNetworkAdapter Add-VDSwitchVMHost Add-VMHost Add-VMHostNtpServer Add-VirtualSwitchPhysicalNetworkAdapter Add-XmlElement Add-vRACustomForm Add-vRAPrincipalToTenantRole Add-vRAReservationNetwork Add-vRAReservationStorage Clear-NsxEdgeInterface Clear-NsxManagerTimeSettings Compress-Archive Connect-CIServer Connect-CisServer Connect-HCXServer Connect-NIServer Connect-NsxLogicalSwitch Connect-NsxServer Connect-NsxtServer Connect-SrmServer Connect-VIServer Connect-Vmc Connect-vRAServer Connect-vRNIServer ConvertFrom-Markdown ConvertTo-MOFInstance Copy-DatastoreItem Copy-HardDisk Copy-NsxEdge Copy-VDisk Copy-VMGuestFile Debug-Runspace Disable-NsxEdgeSsh Disable-RunspaceDebug Disable-vRNIDataSource Disconnect-CIServer Disconnect-CisServer Disconnect-HCXServer Disconnect-NsxLogicalSwitch Disconnect-NsxServer Disconnect-NsxtServer Disconnect-SrmServer Disconnect-VIServer Disconnect-Vmc Disconnect-vRAServer Disconnect-vRNIServer Dismount-Tools Enable-NsxEdgeSsh Enable-RunspaceDebug Enable-vRNIDataSource Expand-Archive Export-NsxObject Export-SpbmStoragePolicy Export-VApp Export-VDPortGroup Export-VDSwitch Export-VMHostProfile Export-vRAIcon Export-vRAPackage Find-Command Find-DscResource Find-Module Find-NsxWhereVMUsed Find-Package Find-PackageProvider Find-RoleCapability Find-Script Format-Hex Format-VMHostDiskPartition Format-XML Generate-VersionInfo Get-AdvancedSetting Get-AlarmAction Get-AlarmActionTrigger Get-AlarmDefinition Get-Annotation Get-CDDrive Get-CIAccessControlRule Get-CIDatastore Get-CINetworkAdapter Get-CIRole Get-CIUser Get-CIVApp Get-CIVAppNetwork Get-CIVAppStartRule Get-CIVAppTemplate Get-CIVM Get-CIVMTemplate Get-CIView Get-Catalog Get-CisCommand Get-CisService Get-CloudCommand Get-Cluster Get-CompatibleVersionAddtionaPropertiesStr Get-ComplexResourceQualifier Get-ConfigurationErrorCount Get-ContentLibraryItem Get-CustomAttribute Get-DSCResourceModules Get-Datacenter Get-Datastore Get-DatastoreCluster Get-DrsClusterGroup Get-DrsRecommendation Get-DrsRule Get-DrsVMHostRule Get-DscResource Get-EdgeGateway Get-EncryptedPassword Get-ErrorReport Get-EsxCli Get-EsxTop Get-ExternalNetwork Get-FileHash Get-FloppyDrive Get-Folder Get-HAPrimaryVMHost Get-HCXAppliance Get-HCXApplianceCompute Get-HCXApplianceDVS Get-HCXApplianceDatastore Get-HCXApplianceNetwork Get-HCXContainer Get-HCXDatastore Get-HCXGateway Get-HCXInterconnectStatus Get-HCXJob Get-HCXMigration Get-HCXNetwork Get-HCXNetworkExtension Get-HCXReplication Get-HCXReplicationSnapshot Get-HCXService Get-HCXSite Get-HCXSitePairing Get-HCXVM Get-HardDisk Get-IScsiHbaTarget Get-InnerMostErrorRecord Get-InstallPath Get-InstalledModule Get-InstalledScript Get-Inventory Get-ItemPropertyValue Get-KeyManagementServer Get-KmipClientCertificate Get-KmsCluster Get-Log Get-LogType Get-MarkdownOption Get-Media Get-MofInstanceName Get-MofInstanceText Get-NetworkAdapter Get-NetworkPool Get-NfsUser Get-NicTeamingPolicy Get-NsxApplicableMember Get-NsxApplicableSecurityAction Get-NsxBackingDVSwitch Get-NsxBackingPortGroup Get-NsxCliDfwAddrSet Get-NsxCliDfwFilter Get-NsxCliDfwRule Get-NsxClusterStatus Get-NsxController Get-NsxDynamicCriteria Get-NsxDynamicMemberSet Get-NsxEdge Get-NsxEdgeBgp Get-NsxEdgeBgpNeighbour Get-NsxEdgeCertificate Get-NsxEdgeCsr Get-NsxEdgeFirewall Get-NsxEdgeFirewallRule Get-NsxEdgeInterface Get-NsxEdgeInterfaceAddress Get-NsxEdgeNat Get-NsxEdgeNatRule Get-NsxEdgeOspf Get-NsxEdgeOspfArea Get-NsxEdgeOspfInterface Get-NsxEdgePrefix Get-NsxEdgeRedistributionRule Get-NsxEdgeRouting Get-NsxEdgeStaticRoute Get-NsxEdgeSubInterface Get-NsxFirewallExclusionListMember Get-NsxFirewallGlobalConfiguration Get-NsxFirewallPublishStatus Get-NsxFirewallRule Get-NsxFirewallRuleMember Get-NsxFirewallSavedConfiguration Get-NsxFirewallSection Get-NsxFirewallThreshold Get-NsxIpPool Get-NsxIpSet Get-NsxLicense Get-NsxLoadBalancer Get-NsxLoadBalancerApplicationProfile Get-NsxLoadBalancerApplicationRule Get-NsxLoadBalancerMonitor Get-NsxLoadBalancerPool Get-NsxLoadBalancerPoolMember Get-NsxLoadBalancerStats Get-NsxLoadBalancerVip Get-NsxLogicalRouter Get-NsxLogicalRouterBgp Get-NsxLogicalRouterBgpNeighbour Get-NsxLogicalRouterBridge Get-NsxLogicalRouterBridging Get-NsxLogicalRouterInterface Get-NsxLogicalRouterOspf Get-NsxLogicalRouterOspfArea Get-NsxLogicalRouterOspfInterface Get-NsxLogicalRouterPrefix Get-NsxLogicalRouterRedistributionRule Get-NsxLogicalRouterRouting Get-NsxLogicalRouterStaticRoute Get-NsxLogicalSwitch Get-NsxMacSet Get-NsxManagerBackup Get-NsxManagerCertificate Get-NsxManagerComponentSummary Get-NsxManagerNetwork Get-NsxManagerRole Get-NsxManagerSsoConfig Get-NsxManagerSyncStatus Get-NsxManagerSyslogServer Get-NsxManagerSystemSummary Get-NsxManagerTimeSettings Get-NsxManagerVcenterConfig Get-NsxSecondaryManager Get-NsxSecurityGroup Get-NsxSecurityGroupEffectiveIpAddress Get-NsxSecurityGroupEffectiveMacAddress Get-NsxSecurityGroupEffectiveMember Get-NsxSecurityGroupEffectiveVirtualMachine Get-NsxSecurityGroupEffectiveVnic Get-NsxSecurityGroupMemberTypes Get-NsxSecurityPolicy Get-NsxSecurityPolicyHighestUsedPrecedence Get-NsxSecurityPolicyRule Get-NsxSecurityTag Get-NsxSecurityTagAssignment Get-NsxSegmentIdRange Get-NsxService Get-NsxServiceDefinition Get-NsxServiceGroup Get-NsxServiceGroupMember Get-NsxServiceProfile Get-NsxSpoofguardNic Get-NsxSpoofguardPolicy Get-NsxSslVpn Get-NsxSslVpnAuthServer Get-NsxSslVpnClientInstallationPackage Get-NsxSslVpnIpPool Get-NsxSslVpnPrivateNetwork Get-NsxSslVpnUser Get-NsxTransportZone Get-NsxUserRole Get-NsxVdsContext Get-NsxtPolicyService Get-NsxtService Get-OSCustomizationNicMapping Get-OSCustomizationSpec Get-Org Get-OrgNetwork Get-OrgVdc Get-OrgVdcNetwork Get-OvfConfiguration Get-PSCurrentConfigurationNode Get-PSDefaultConfigurationDocument Get-PSMetaConfigDocumentInstVersionInfo Get-PSMetaConfigurationProcessed Get-PSReadLineKeyHandler Get-PSReadLineOption Get-PSRepository Get-PSTopConfigurationName Get-PSVersion Get-Package Get-PackageProvider Get-PackageSource Get-PassthroughDevice Get-PositionInfo Get-PowerCLICommunity Get-PowerCLIConfiguration Get-PowerCLIHelp Get-PowerCLIVersion Get-PowerNsxVersion Get-ProviderVdc Get-PublicKeyFromFile Get-PublicKeyFromStore Get-ResourcePool Get-Runspace Get-RunspaceDebug Get-ScsiController Get-ScsiLun Get-ScsiLunPath Get-SecurityInfo Get-SecurityPolicy Get-Snapshot Get-SpbmCapability Get-SpbmCompatibleStorage Get-SpbmEntityConfiguration Get-SpbmFaultDomain Get-SpbmPointInTimeReplica Get-SpbmReplicationGroup Get-SpbmReplicationPair Get-SpbmStoragePolicy Get-Stat Get-StatInterval Get-StatType Get-Tag Get-TagAssignment Get-TagCategory Get-Task Get-Template Get-TimeZone Get-Uptime Get-UsbDevice Get-VAIOFilter Get-VApp Get-VDBlockedPolicy Get-VDPort Get-VDPortgroup Get-VDPortgroupOverridePolicy Get-VDSecurityPolicy Get-VDSwitch Get-VDSwitchPrivateVlan Get-VDTrafficShapingPolicy Get-VDUplinkLacpPolicy Get-VDUplinkTeamingPolicy Get-VDisk Get-VIAccount Get-VICommand Get-VICredentialStoreItem Get-VIEvent Get-VIObjectByVIView Get-VIPermission Get-VIPrivilege Get-VIProperty Get-VIRole Get-VM Get-VMGuest Get-VMHost Get-VMHostAccount Get-VMHostAdvancedConfiguration Get-VMHostAuthentication Get-VMHostAvailableTimeZone Get-VMHostDiagnosticPartition Get-VMHostDisk Get-VMHostDiskPartition Get-VMHostFirewallDefaultPolicy Get-VMHostFirewallException Get-VMHostFirmware Get-VMHostHardware Get-VMHostHba Get-VMHostModule Get-VMHostNetwork Get-VMHostNetworkAdapter Get-VMHostNtpServer Get-VMHostPatch Get-VMHostPciDevice Get-VMHostProfile Get-VMHostProfileImageCacheConfiguration Get-VMHostProfileRequiredInput Get-VMHostProfileStorageDeviceConfiguration Get-VMHostProfileUserConfiguration Get-VMHostProfileVmPortGroupConfiguration Get-VMHostRoute Get-VMHostService Get-VMHostSnmp Get-VMHostStartPolicy Get-VMHostStorage Get-VMHostSysLogServer Get-VMQuestion Get-VMResourceConfiguration Get-VMStartPolicy Get-VTpm Get-VTpmCSR Get-VTpmCertificate Get-VasaProvider Get-VasaStorageArray Get-View Get-VirtualPortGroup Get-VirtualSwitch Get-VmcSddcNetworkService Get-VmcService Get-VsanClusterConfiguration Get-VsanComponent Get-VsanDisk Get-VsanDiskGroup Get-VsanEvacuationPlan Get-VsanFaultDomain Get-VsanIscsiInitiatorGroup Get-VsanIscsiInitiatorGroupTargetAssociation Get-VsanIscsiLun Get-VsanIscsiTarget Get-VsanObject Get-VsanResyncingComponent Get-VsanRuntimeInfo Get-VsanSpaceUsage Get-VsanStat Get-VsanView Get-vRAApplianceServiceStatus Get-vRAAuthorizationRole Get-vRABlueprint Get-vRABusinessGroup Get-vRACatalogItem Get-vRACatalogItemRequestTemplate Get-vRACatalogPrincipal Get-vRAComponentRegistryService Get-vRAComponentRegistryServiceEndpoint Get-vRAComponentRegistryServiceStatus Get-vRAContent Get-vRAContentData Get-vRAContentType Get-vRACustomForm Get-vRAEntitledCatalogItem Get-vRAEntitledService Get-vRAEntitlement Get-vRAExternalNetworkProfile Get-vRAGroupPrincipal Get-vRAIcon Get-vRANATNetworkProfile Get-vRANetworkProfileIPAddressList Get-vRANetworkProfileIPRangeSummary Get-vRAPackage Get-vRAPackageContent Get-vRAPropertyDefinition Get-vRAPropertyGroup Get-vRARequest Get-vRARequestDetail Get-vRAReservation Get-vRAReservationComputeResource Get-vRAReservationComputeResourceMemory Get-vRAReservationComputeResourceNetwork Get-vRAReservationComputeResourceResourcePool Get-vRAReservationComputeResourceStorage Get-vRAReservationPolicy Get-vRAReservationTemplate Get-vRAReservationType Get-vRAResource Get-vRAResourceAction Get-vRAResourceActionRequestTemplate Get-vRAResourceMetric Get-vRAResourceOperation Get-vRAResourceType Get-vRARoutedNetworkProfile Get-vRAService Get-vRAServiceBlueprint Get-vRASourceMachine Get-vRAStorageReservationPolicy Get-vRATenant Get-vRATenantDirectory Get-vRATenantDirectoryStatus Get-vRATenantRole Get-vRAUserPrincipal Get-vRAUserPrincipalGroupMembership Get-vRAVersion Get-vRNIAPIVersion Get-vRNIApplication Get-vRNIApplicationTier Get-vRNIDataSource Get-vRNIDataSourceSNMPConfig Get-vRNIDatastore Get-vRNIDistributedSwitch Get-vRNIDistributedSwitchPortGroup Get-vRNIEntity Get-vRNIEntityName Get-vRNIFirewallRule Get-vRNIFlow Get-vRNIHost Get-vRNIHostVMKNic Get-vRNIIPSet Get-vRNIL2Network Get-vRNINSXManager Get-vRNINodes Get-vRNIProblem Get-vRNIRecommendedRules Get-vRNIRecommendedRulesNsxBundle Get-vRNISecurityGroup Get-vRNISecurityTag Get-vRNIService Get-vRNIServiceGroup Get-vRNIVM Get-vRNIVMvNIC Get-vRNIvCenter Get-vRNIvCenterCluster Get-vRNIvCenterDatacenter Get-vRNIvCenterFolder Grant-NsxSpoofguardNicApproval Import-CIVApp Import-CIVAppTemplate Import-NsxObject Import-PackageProvider Import-PowerShellDataFile Import-SpbmStoragePolicy Import-VApp Import-VMHostProfile Import-vRAContentData Import-vRAIcon Import-vRAPackage Initialize-ConfigurationRuntimeState Install-Module Install-NsxCluster Install-Package Install-PackageProvider Install-Script Install-VMHostPatch Invoke-DrsRecommendation Invoke-NsxCli Invoke-NsxClusterResolveAll Invoke-NsxManagerSync Invoke-NsxRestMethod Invoke-NsxWebRequest Invoke-VMHostProfile Invoke-VMScript Invoke-XpathQuery Invoke-vRADataCollection Invoke-vRARestMethod Invoke-vRATenantDirectorySync Invoke-vRNIRestMethod Join-String Mount-Tools Move-Cluster Move-Datacenter Move-Datastore Move-Folder Move-HardDisk Move-Inventory Move-NsxSecurityPolicyRule Move-ResourcePool Move-Template Move-VApp Move-VDisk Move-VM Move-VMHost New-AdvancedSetting New-AlarmAction New-AlarmActionTrigger New-CDDrive New-CIAccessControlRule New-CIVApp New-CIVAppNetwork New-CIVAppTemplate New-CIVM New-Cluster New-CustomAttribute New-Datacenter New-Datastore New-DatastoreCluster New-DatastoreDrive New-DrsClusterGroup New-DrsRule New-DrsVMHostRule New-DscChecksum New-FloppyDrive New-Folder New-Guid New-HCXAppliance New-HCXMigration New-HCXNetworkExtension New-HCXNetworkMapping New-HCXReplication New-HCXSitePairing New-HCXStaticRoute New-HardDisk New-IScsiHbaTarget New-KmipClientCertificate New-NetworkAdapter New-NfsUser New-NsxAddressSpec New-NsxClusterVxlanConfig New-NsxController New-NsxDynamicCriteriaSpec New-NsxEdge New-NsxEdgeBgpNeighbour New-NsxEdgeCsr New-NsxEdgeFirewallRule New-NsxEdgeInterfaceSpec New-NsxEdgeNatRule New-NsxEdgeOspfArea New-NsxEdgeOspfInterface New-NsxEdgePrefix New-NsxEdgeRedistributionRule New-NsxEdgeSelfSignedCertificate New-NsxEdgeStaticRoute New-NsxEdgeSubInterface New-NsxEdgeSubInterfaceSpec New-NsxFirewallRule New-NsxFirewallSavedConfiguration New-NsxFirewallSection New-NsxIpPool New-NsxIpSet New-NsxLoadBalancerApplicationProfile New-NsxLoadBalancerApplicationRule New-NsxLoadBalancerMemberSpec New-NsxLoadBalancerMonitor New-NsxLoadBalancerPool New-NsxLogicalRouter New-NsxLogicalRouterBgpNeighbour New-NsxLogicalRouterBridge New-NsxLogicalRouterInterface New-NsxLogicalRouterInterfaceSpec New-NsxLogicalRouterOspfArea New-NsxLogicalRouterOspfInterface New-NsxLogicalRouterPrefix New-NsxLogicalRouterRedistributionRule New-NsxLogicalRouterStaticRoute New-NsxLogicalSwitch New-NsxMacSet New-NsxManager New-NsxSecurityGroup New-NsxSecurityPolicy New-NsxSecurityPolicyAssignment New-NsxSecurityPolicyFirewallRuleSpec New-NsxSecurityPolicyGuestIntrospectionSpec New-NsxSecurityPolicyNetworkIntrospectionSpec New-NsxSecurityTag New-NsxSecurityTagAssignment New-NsxSegmentIdRange New-NsxService New-NsxServiceGroup New-NsxSpoofguardPolicy New-NsxSslVpnAuthServer New-NsxSslVpnClientInstallationPackage New-NsxSslVpnIpPool New-NsxSslVpnPrivateNetwork New-NsxSslVpnUser New-NsxTransportZone New-NsxVdsContext New-OSCustomizationNicMapping New-OSCustomizationSpec New-Org New-OrgNetwork New-OrgVdc New-OrgVdcNetwork New-ResourcePool New-ScriptFileInfo New-ScsiController New-Snapshot New-SpbmRule New-SpbmRuleSet New-SpbmStoragePolicy New-StatInterval New-Tag New-TagAssignment New-TagCategory New-Template New-TemporaryFile New-VAIOFilter New-VApp New-VDPortgroup New-VDSwitch New-VDSwitchPrivateVlan New-VDisk New-VICredentialStoreItem New-VIInventoryDrive New-VIPermission New-VIProperty New-VIRole New-VISamlSecurityContext New-VM New-VMHostAccount New-VMHostNetworkAdapter New-VMHostProfile New-VMHostProfileVmPortGroupConfiguration New-VMHostRoute New-VTpm New-VasaProvider New-VcsOAuthSecurityContext New-VirtualPortGroup New-VirtualSwitch New-VsanDisk New-VsanDiskGroup New-VsanFaultDomain New-VsanIscsiInitiatorGroup New-VsanIscsiInitiatorGroupTargetAssociation New-VsanIscsiLun New-VsanIscsiTarget New-vRABusinessGroup New-vRAEntitlement New-vRAExternalNetworkProfile New-vRAGroupPrincipal New-vRANATNetworkProfile New-vRANetworkProfileIPRangeDefinition New-vRAPackage New-vRAPropertyDefinition New-vRAPropertyGroup New-vRAReservation New-vRAReservationNetworkDefinition New-vRAReservationPolicy New-vRAReservationStorageDefinition New-vRARoutedNetworkProfile New-vRAService New-vRAStorageReservationPolicy New-vRATenant New-vRATenantDirectory New-vRAUserPrincipal New-vRNIApplication New-vRNIApplicationTier New-vRNIDataSource Open-VMConsoleWindow Publish-Module Publish-NsxSpoofguardPolicy Publish-Script Register-PSRepository Register-PackageSource Remove-AdvancedSetting Remove-AlarmAction Remove-AlarmActionTrigger Remove-Alias Remove-CDDrive Remove-CIAccessControlRule Remove-CIVApp Remove-CIVAppNetwork Remove-CIVAppTemplate Remove-Cluster Remove-CustomAttribute Remove-Datacenter Remove-Datastore Remove-DatastoreCluster Remove-DrsClusterGroup Remove-DrsRule Remove-DrsVMHostRule Remove-FloppyDrive Remove-Folder Remove-HCXAppliance Remove-HCXNetworkExtension Remove-HCXReplication Remove-HCXSitePairing Remove-HardDisk Remove-IScsiHbaTarget Remove-Inventory Remove-KeyManagementServer Remove-NetworkAdapter Remove-NfsUser Remove-NsxCluster Remove-NsxClusterVxlanConfig Remove-NsxController Remove-NsxDynamicCriteria Remove-NsxDynamicMemberSet Remove-NsxEdge Remove-NsxEdgeBgpNeighbour Remove-NsxEdgeCertificate Remove-NsxEdgeCsr Remove-NsxEdgeFirewallRule Remove-NsxEdgeInterfaceAddress Remove-NsxEdgeNatRule Remove-NsxEdgeOspfArea Remove-NsxEdgeOspfInterface Remove-NsxEdgePrefix Remove-NsxEdgeRedistributionRule Remove-NsxEdgeStaticRoute Remove-NsxEdgeSubInterface Remove-NsxFirewallExclusionListMember Remove-NsxFirewallRule Remove-NsxFirewallRuleMember Remove-NsxFirewallSavedConfiguration Remove-NsxFirewallSection Remove-NsxIpPool Remove-NsxIpSet Remove-NsxIpSetMember Remove-NsxLoadBalancerApplicationProfile Remove-NsxLoadBalancerMonitor Remove-NsxLoadBalancerPool Remove-NsxLoadBalancerPoolMember Remove-NsxLoadBalancerVip Remove-NsxLogicalRouter Remove-NsxLogicalRouterBgpNeighbour Remove-NsxLogicalRouterBridge Remove-NsxLogicalRouterInterface Remove-NsxLogicalRouterOspfArea Remove-NsxLogicalRouterOspfInterface Remove-NsxLogicalRouterPrefix Remove-NsxLogicalRouterRedistributionRule Remove-NsxLogicalRouterStaticRoute Remove-NsxLogicalSwitch Remove-NsxMacSet Remove-NsxSecondaryManager Remove-NsxSecurityGroup Remove-NsxSecurityGroupMember Remove-NsxSecurityPolicy Remove-NsxSecurityPolicyAssignment Remove-NsxSecurityPolicyRule Remove-NsxSecurityPolicyRuleGroup Remove-NsxSecurityPolicyRuleService Remove-NsxSecurityTag Remove-NsxSecurityTagAssignment Remove-NsxSegmentIdRange Remove-NsxService Remove-NsxServiceGroup Remove-NsxSpoofguardPolicy Remove-NsxSslVpnClientInstallationPackage Remove-NsxSslVpnIpPool Remove-NsxSslVpnPrivateNetwork Remove-NsxSslVpnUser Remove-NsxTransportZone Remove-NsxTransportZoneMember Remove-NsxVdsContext Remove-OSCustomizationNicMapping Remove-OSCustomizationSpec Remove-Org Remove-OrgNetwork Remove-OrgVdc Remove-OrgVdcNetwork Remove-PSReadLineKeyHandler Remove-PassthroughDevice Remove-ResourcePool Remove-Snapshot Remove-SpbmStoragePolicy Remove-StatInterval Remove-Tag Remove-TagAssignment Remove-TagCategory Remove-Template Remove-UsbDevice Remove-VAIOFilter Remove-VApp Remove-VDPortGroup Remove-VDSwitch Remove-VDSwitchPhysicalNetworkAdapter Remove-VDSwitchPrivateVlan Remove-VDSwitchVMHost Remove-VDisk Remove-VICredentialStoreItem Remove-VIPermission Remove-VIProperty Remove-VIRole Remove-VM Remove-VMHost Remove-VMHostAccount Remove-VMHostNetworkAdapter Remove-VMHostNtpServer Remove-VMHostProfile Remove-VMHostProfileVmPortGroupConfiguration Remove-VMHostRoute Remove-VTpm Remove-VasaProvider Remove-VirtualPortGroup Remove-VirtualSwitch Remove-VirtualSwitchPhysicalNetworkAdapter Remove-VsanDisk Remove-VsanDiskGroup Remove-VsanFaultDomain Remove-VsanIscsiInitiatorGroup Remove-VsanIscsiInitiatorGroupTargetAssociation Remove-VsanIscsiLun Remove-VsanIscsiTarget Remove-vRABusinessGroup Remove-vRACustomForm Remove-vRAExternalNetworkProfile Remove-vRAGroupPrincipal Remove-vRAIcon Remove-vRANATNetworkProfile Remove-vRAPackage Remove-vRAPrincipalFromTenantRole Remove-vRAPropertyDefinition Remove-vRAPropertyGroup Remove-vRAReservation Remove-vRAReservationNetwork Remove-vRAReservationPolicy Remove-vRAReservationStorage Remove-vRARoutedNetworkProfile Remove-vRAService Remove-vRAStorageReservationPolicy Remove-vRATenant Remove-vRATenantDirectory Remove-vRAUserPrincipal Remove-vRNIApplication Remove-vRNIApplicationTier Remove-vRNIDataSource Repair-NsxEdge Repair-VsanObject Request-vRACatalogItem Request-vRAResourceAction Restart-CIVApp Restart-CIVAppGuest Restart-CIVM Restart-CIVMGuest Restart-VM Restart-VMGuest Restart-VMHost Restart-VMHostService Resume-HCXReplication Revoke-NsxSpoofguardNicApproval Save-Module Save-Package Save-Script Search-Cloud Set-AdvancedSetting Set-AlarmDefinition Set-Annotation Set-CDDrive Set-CIAccessControlRule Set-CINetworkAdapter Set-CIVApp Set-CIVAppNetwork Set-CIVAppStartRule Set-CIVAppTemplate Set-Cluster Set-CustomAttribute Set-Datacenter Set-Datastore Set-DatastoreCluster Set-DrsClusterGroup Set-DrsRule Set-DrsVMHostRule Set-FloppyDrive Set-Folder Set-HCXAppliance Set-HCXMigration Set-HCXReplication Set-HardDisk Set-IScsiHbaTarget Set-KeyManagementServer Set-KmsCluster Set-MarkdownOption Set-NetworkAdapter Set-NfsUser Set-NicTeamingPolicy Set-NodeExclusiveResources Set-NodeManager Set-NodeResourceSource Set-NodeResources Set-NsxEdge Set-NsxEdgeBgp Set-NsxEdgeFirewall Set-NsxEdgeInterface Set-NsxEdgeNat Set-NsxEdgeOspf Set-NsxEdgeRouting Set-NsxFirewallGlobalConfiguration Set-NsxFirewallRule Set-NsxFirewallSavedConfiguration Set-NsxFirewallThreshold Set-NsxLoadBalancer Set-NsxLoadBalancerPoolMember Set-NsxLogicalRouter Set-NsxLogicalRouterBgp Set-NsxLogicalRouterBridging Set-NsxLogicalRouterInterface Set-NsxLogicalRouterOspf Set-NsxLogicalRouterRouting Set-NsxManager Set-NsxManagerRole Set-NsxManagerTimeSettings 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current3DENOperation currentChannel currentCommand currentMagazine currentMagazineDetail currentMagazineDetailTurret currentMagazineTurret currentMuzzle currentNamespace currentTask currentTasks currentThrowable currentVisionMode currentWaypoint currentWeapon currentWeaponMode currentWeaponTurret currentZeroing cursorObject cursorTarget customChat customRadio cutFadeOut cutObj cutRsc cutText damage date dateToNumber daytime deActivateKey debriefingText debugFSM debugLog deg delete3DENEntities deleteAt deleteCenter deleteCollection deleteEditorObject deleteGroup deleteGroupWhenEmpty deleteIdentity deleteLocation deleteMarker deleteMarkerLocal deleteRange deleteResources deleteSite deleteStatus deleteTeam deleteVehicle deleteVehicleCrew deleteWaypoint detach detectedMines diag_activeMissionFSMs diag_activeScripts diag_activeSQFScripts diag_activeSQSScripts diag_captureFrame diag_captureFrameToFile diag_captureSlowFrame diag_codePerformance diag_drawMode diag_enable diag_enabled diag_fps diag_fpsMin diag_frameNo diag_lightNewLoad diag_list diag_log diag_logSlowFrame diag_mergeConfigFile diag_recordTurretLimits diag_setLightNew diag_tickTime diag_toggle dialog diarySubjectExists didJIP didJIPOwner difficulty difficultyEnabled difficultyEnabledRTD difficultyOption direction directSay disableAI disableCollisionWith disableConversation disableDebriefingStats disableMapIndicators disableNVGEquipment disableRemoteSensors disableSerialization disableTIEquipment disableUAVConnectability disableUserInput displayAddEventHandler displayCtrl displayParent displayRemoveAllEventHandlers displayRemoveEventHandler displaySetEventHandler dissolveTeam distance distance2D distanceSqr distributionRegion do3DENAction doArtilleryFire doFire doFollow doFSM doGetOut doMove doorPhase doStop doSuppressiveFire doTarget doWatch drawArrow drawEllipse drawIcon drawIcon3D drawLine drawLine3D drawLink drawLocation drawPolygon drawRectangle drawTriangle driver drop dynamicSimulationDistance dynamicSimulationDistanceCoef dynamicSimulationEnabled dynamicSimulationSystemEnabled echo edit3DENMissionAttributes editObject editorSetEventHandler effectiveCommander emptyPositions enableAI enableAIFeature enableAimPrecision enableAttack enableAudioFeature enableAutoStartUpRTD enableAutoTrimRTD enableCamShake enableCaustics enableChannel enableCollisionWith enableCopilot enableDebriefingStats enableDiagLegend enableDynamicSimulation enableDynamicSimulationSystem enableEndDialog enableEngineArtillery enableEnvironment enableFatigue enableGunLights enableInfoPanelComponent enableIRLasers enableMimics enablePersonTurret enableRadio enableReload enableRopeAttach enableSatNormalOnDetail enableSaving enableSentences enableSimulation enableSimulationGlobal enableStamina enableTeamSwitch enableTraffic enableUAVConnectability enableUAVWaypoints enableVehicleCargo enableVehicleSensor enableWeaponDisassembly endLoadingScreen endMission engineOn enginesIsOnRTD enginesRpmRTD enginesTorqueRTD entities environmentEnabled estimatedEndServerTime estimatedTimeLeft evalObjectArgument everyBackpack everyContainer exec execEditorScript execFSM execVM exp expectedDestination exportJIPMessages eyeDirection eyePos face faction fadeMusic fadeRadio fadeSound fadeSpeech failMission fillWeaponsFromPool find findCover findDisplay findEditorObject findEmptyPosition findEmptyPositionReady findIf findNearestEnemy finishMissionInit finite fire fireAtTarget firstBackpack flag flagAnimationPhase flagOwner flagSide flagTexture fleeing floor flyInHeight flyInHeightASL fog fogForecast fogParams forceAddUniform forcedMap forceEnd forceFlagTexture forceFollowRoad forceMap forceRespawn forceSpeed forceWalk forceWeaponFire forceWeatherChange forEachMember forEachMemberAgent forEachMemberTeam forgetTarget format formation formationDirection formationLeader formationMembers formationPosition formationTask formatText formLeader freeLook fromEditor fuel fullCrew gearIDCAmmoCount gearSlotAmmoCount gearSlotData get3DENActionState get3DENAttribute get3DENCamera get3DENConnections get3DENEntity get3DENEntityID get3DENGrid get3DENIconsVisible get3DENLayerEntities get3DENLinesVisible get3DENMissionAttribute get3DENMouseOver get3DENSelected getAimingCoef getAllEnvSoundControllers getAllHitPointsDamage getAllOwnedMines getAllSoundControllers getAmmoCargo getAnimAimPrecision getAnimSpeedCoef getArray getArtilleryAmmo getArtilleryComputerSettings getArtilleryETA getAssignedCuratorLogic getAssignedCuratorUnit getBackpackCargo getBleedingRemaining getBurningValue getCameraViewDirection getCargoIndex getCenterOfMass getClientState getClientStateNumber getCompatiblePylonMagazines getConnectedUAV getContainerMaxLoad getCursorObjectParams getCustomAimCoef getDammage getDescription getDir getDirVisual getDLCAssetsUsage getDLCAssetsUsageByName getDLCs getEditorCamera getEditorMode getEditorObjectScope getElevationOffset getEnvSoundController getFatigue getForcedFlagTexture getFriend getFSMVariable getFuelCargo getGroupIcon getGroupIconParams getGroupIcons getHideFrom getHit getHitIndex getHitPointDamage getItemCargo getMagazineCargo getMarkerColor getMarkerPos getMarkerSize getMarkerType getMass getMissionConfig getMissionConfigValue getMissionDLCs getMissionLayerEntities getModelInfo getMousePosition getMusicPlayedTime getNumber getObjectArgument getObjectChildren getObjectDLC getObjectMaterials getObjectProxy getObjectTextures getObjectType getObjectViewDistance getOxygenRemaining getPersonUsedDLCs getPilotCameraDirection getPilotCameraPosition getPilotCameraRotation getPilotCameraTarget getPlateNumber getPlayerChannel getPlayerScores getPlayerUID getPos getPosASL getPosASLVisual getPosASLW getPosATL getPosATLVisual getPosVisual getPosWorld getPylonMagazines getRelDir getRelPos getRemoteSensorsDisabled getRepairCargo getResolution getShadowDistance getShotParents getSlingLoad getSoundController getSoundControllerResult getSpeed getStamina getStatValue getSuppression getTerrainGrid getTerrainHeightASL getText getTotalDLCUsageTime getUnitLoadout getUnitTrait getUserMFDText getUserMFDvalue getVariable getVehicleCargo getWeaponCargo getWeaponSway getWingsOrientationRTD getWingsPositionRTD getWPPos glanceAt globalChat globalRadio goggles goto group groupChat groupFromNetId groupIconSelectable groupIconsVisible groupId groupOwner groupRadio groupSelectedUnits groupSelectUnit gunner gusts halt handgunItems handgunMagazine handgunWeapon handsHit hasInterface hasPilotCamera hasWeapon hcAllGroups hcGroupParams hcLeader hcRemoveAllGroups hcRemoveGroup hcSelected hcSelectGroup hcSetGroup hcShowBar hcShownBar headgear hideBody hideObject hideObjectGlobal hideSelection hint hintC hintCadet hintSilent hmd hostMission htmlLoad HUDMovementLevels humidity image importAllGroups importance in inArea inAreaArray incapacitatedState inflame inflamed infoPanel infoPanelComponentEnabled infoPanelComponents infoPanels inGameUISetEventHandler inheritsFrom initAmbientLife inPolygon inputAction inRangeOfArtillery insertEditorObject intersect is3DEN is3DENMultiplayer isAbleToBreathe isAgent isArray isAutoHoverOn isAutonomous isAutotest isBleeding isBurning isClass isCollisionLightOn isCopilotEnabled isDamageAllowed isDedicated isDLCAvailable isEngineOn isEqualTo isEqualType isEqualTypeAll isEqualTypeAny isEqualTypeArray isEqualTypeParams isFilePatchingEnabled isFlashlightOn isFlatEmpty isForcedWalk isFormationLeader isGroupDeletedWhenEmpty isHidden isInRemainsCollector isInstructorFigureEnabled isIRLaserOn isKeyActive isKindOf isLaserOn isLightOn isLocalized isManualFire isMarkedForCollection isMultiplayer isMultiplayerSolo isNil isNull isNumber isObjectHidden isObjectRTD isOnRoad isPipEnabled isPlayer isRealTime isRemoteExecuted isRemoteExecutedJIP isServer isShowing3DIcons isSimpleObject isSprintAllowed isStaminaEnabled isSteamMission isStreamFriendlyUIEnabled isText isTouchingGround isTurnedOut isTutHintsEnabled isUAVConnectable isUAVConnected isUIContext isUniformAllowed isVehicleCargo isVehicleRadarOn isVehicleSensorEnabled isWalking isWeaponDeployed isWeaponRested itemCargo items itemsWithMagazines join joinAs joinAsSilent joinSilent joinString kbAddDatabase kbAddDatabaseTargets kbAddTopic kbHasTopic kbReact kbRemoveTopic kbTell kbWasSaid keyImage keyName knowsAbout land landAt landResult language laserTarget lbAdd lbClear lbColor lbColorRight lbCurSel lbData lbDelete lbIsSelected lbPicture lbPictureRight lbSelection lbSetColor lbSetColorRight lbSetCurSel lbSetData lbSetPicture lbSetPictureColor lbSetPictureColorDisabled lbSetPictureColorSelected lbSetPictureRight lbSetPictureRightColor lbSetPictureRightColorDisabled lbSetPictureRightColorSelected lbSetSelectColor lbSetSelectColorRight lbSetSelected lbSetText lbSetTextRight lbSetTooltip lbSetValue lbSize lbSort lbSortByValue lbText lbTextRight lbValue leader leaderboardDeInit leaderboardGetRows leaderboardInit leaderboardRequestRowsFriends leaderboardsRequestUploadScore leaderboardsRequestUploadScoreKeepBest leaderboardState leaveVehicle libraryCredits libraryDisclaimers lifeState lightAttachObject lightDetachObject lightIsOn lightnings limitSpeed linearConversion lineIntersects lineIntersectsObjs lineIntersectsSurfaces lineIntersectsWith linkItem list listObjects listRemoteTargets listVehicleSensors ln lnbAddArray lnbAddColumn lnbAddRow lnbClear lnbColor lnbCurSelRow lnbData lnbDeleteColumn lnbDeleteRow lnbGetColumnsPosition lnbPicture lnbSetColor lnbSetColumnsPos lnbSetCurSelRow lnbSetData lnbSetPicture lnbSetText lnbSetValue lnbSize lnbSort lnbSortByValue lnbText lnbValue load loadAbs loadBackpack loadFile loadGame loadIdentity loadMagazine loadOverlay loadStatus loadUniform loadVest local localize locationPosition lock lockCameraTo lockCargo lockDriver locked lockedCargo lockedDriver lockedTurret lockIdentity lockTurret lockWP log logEntities logNetwork logNetworkTerminate lookAt lookAtPos magazineCargo magazines magazinesAllTurrets magazinesAmmo magazinesAmmoCargo magazinesAmmoFull magazinesDetail magazinesDetailBackpack magazinesDetailUniform magazinesDetailVest magazinesTurret magazineTurretAmmo mapAnimAdd mapAnimClear mapAnimCommit mapAnimDone mapCenterOnCamera mapGridPosition markAsFinishedOnSteam markerAlpha markerBrush markerColor markerDir markerPos markerShape markerSize markerText markerType max members menuAction menuAdd menuChecked menuClear menuCollapse menuData menuDelete menuEnable menuEnabled menuExpand menuHover menuPicture menuSetAction menuSetCheck menuSetData menuSetPicture menuSetValue menuShortcut menuShortcutText menuSize menuSort menuText menuURL menuValue min mineActive mineDetectedBy missionConfigFile missionDifficulty missionName missionNamespace missionStart missionVersion mod modelToWorld modelToWorldVisual modelToWorldVisualWorld modelToWorldWorld modParams moonIntensity moonPhase morale move move3DENCamera moveInAny moveInCargo moveInCommander moveInDriver moveInGunner moveInTurret moveObjectToEnd moveOut moveTime moveTo moveToCompleted moveToFailed musicVolume name nameSound nearEntities nearestBuilding nearestLocation nearestLocations nearestLocationWithDubbing nearestObject nearestObjects nearestTerrainObjects nearObjects nearObjectsReady nearRoads nearSupplies nearTargets needReload netId netObjNull newOverlay nextMenuItemIndex nextWeatherChange nMenuItems not numberOfEnginesRTD numberToDate objectCurators objectFromNetId objectParent objStatus onBriefingGroup onBriefingNotes onBriefingPlan onBriefingTeamSwitch onCommandModeChanged onDoubleClick onEachFrame onGroupIconClick onGroupIconOverEnter onGroupIconOverLeave onHCGroupSelectionChanged onMapSingleClick onPlayerConnected onPlayerDisconnected onPreloadFinished onPreloadStarted onShowNewObject onTeamSwitch openCuratorInterface openDLCPage openMap openSteamApp openYoutubeVideo or orderGetIn overcast overcastForecast owner param params parseNumber parseSimpleArray parseText parsingNamespace particlesQuality pickWeaponPool pitch pixelGrid pixelGridBase pixelGridNoUIScale pixelH pixelW playableSlotsNumber playableUnits playAction playActionNow player playerRespawnTime playerSide playersNumber playGesture playMission playMove playMoveNow playMusic playScriptedMission playSound playSound3D position positionCameraToWorld posScreenToWorld posWorldToScreen ppEffectAdjust ppEffectCommit ppEffectCommitted ppEffectCreate ppEffectDestroy ppEffectEnable ppEffectEnabled ppEffectForceInNVG precision preloadCamera preloadObject preloadSound preloadTitleObj preloadTitleRsc preprocessFile preprocessFileLineNumbers primaryWeapon primaryWeaponItems primaryWeaponMagazine priority processDiaryLink productVersion profileName profileNamespace profileNameSteam progressLoadingScreen progressPosition progressSetPosition publicVariable publicVariableClient publicVariableServer pushBack pushBackUnique putWeaponPool queryItemsPool queryMagazinePool queryWeaponPool rad radioChannelAdd radioChannelCreate radioChannelRemove radioChannelSetCallSign radioChannelSetLabel radioVolume rain rainbow random rank rankId rating rectangular registeredTasks registerTask reload reloadEnabled remoteControl remoteExec remoteExecCall remoteExecutedOwner remove3DENConnection remove3DENEventHandler remove3DENLayer removeAction removeAll3DENEventHandlers removeAllActions removeAllAssignedItems removeAllContainers removeAllCuratorAddons removeAllCuratorCameraAreas removeAllCuratorEditingAreas removeAllEventHandlers removeAllHandgunItems removeAllItems removeAllItemsWithMagazines removeAllMissionEventHandlers removeAllMPEventHandlers removeAllMusicEventHandlers removeAllOwnedMines removeAllPrimaryWeaponItems removeAllWeapons removeBackpack removeBackpackGlobal removeCuratorAddons removeCuratorCameraArea removeCuratorEditableObjects removeCuratorEditingArea removeDrawIcon removeDrawLinks removeEventHandler removeFromRemainsCollector removeGoggles removeGroupIcon removeHandgunItem removeHeadgear removeItem removeItemFromBackpack removeItemFromUniform removeItemFromVest removeItems removeMagazine removeMagazineGlobal removeMagazines removeMagazinesTurret removeMagazineTurret removeMenuItem removeMissionEventHandler removeMPEventHandler removeMusicEventHandler removeOwnedMine removePrimaryWeaponItem removeSecondaryWeaponItem removeSimpleTask removeSwitchableUnit removeTeamMember removeUniform removeVest removeWeapon removeWeaponAttachmentCargo removeWeaponCargo removeWeaponGlobal removeWeaponTurret reportRemoteTarget requiredVersion resetCamShake resetSubgroupDirection resize resources respawnVehicle restartEditorCamera reveal revealMine reverse reversedMouseY roadAt roadsConnectedTo roleDescription ropeAttachedObjects ropeAttachedTo ropeAttachEnabled ropeAttachTo ropeCreate ropeCut ropeDestroy ropeDetach ropeEndPosition ropeLength ropes ropeUnwind ropeUnwound rotorsForcesRTD rotorsRpmRTD round runInitScript safeZoneH safeZoneW safeZoneWAbs safeZoneX safeZoneXAbs safeZoneY save3DENInventory saveGame saveIdentity saveJoysticks saveOverlay saveProfileNamespace saveStatus saveVar savingEnabled say say2D say3D scopeName score scoreSide screenshot screenToWorld scriptDone scriptName scudState secondaryWeapon secondaryWeaponItems secondaryWeaponMagazine select selectBestPlaces selectDiarySubject selectedEditorObjects selectEditorObject selectionNames selectionPosition selectLeader selectMax selectMin selectNoPlayer selectPlayer selectRandom selectRandomWeighted selectWeapon selectWeaponTurret sendAUMessage sendSimpleCommand sendTask sendTaskResult sendUDPMessage serverCommand serverCommandAvailable serverCommandExecutable serverName serverTime set set3DENAttribute set3DENAttributes set3DENGrid set3DENIconsVisible set3DENLayer set3DENLinesVisible set3DENLogicType set3DENMissionAttribute set3DENMissionAttributes set3DENModelsVisible set3DENObjectType set3DENSelected setAccTime setActualCollectiveRTD setAirplaneThrottle setAirportSide setAmmo setAmmoCargo setAmmoOnPylon setAnimSpeedCoef setAperture setApertureNew setArmoryPoints setAttributes setAutonomous setBehaviour setBleedingRemaining setBrakesRTD setCameraInterest setCamShakeDefParams setCamShakeParams setCamUseTI setCaptive setCenterOfMass setCollisionLight setCombatMode setCompassOscillation setConvoySeparation setCuratorCameraAreaCeiling setCuratorCoef setCuratorEditingAreaType setCuratorWaypointCost setCurrentChannel setCurrentTask setCurrentWaypoint setCustomAimCoef setCustomWeightRTD setDamage setDammage setDate setDebriefingText setDefaultCamera setDestination setDetailMapBlendPars setDir setDirection setDrawIcon setDriveOnPath setDropInterval setDynamicSimulationDistance setDynamicSimulationDistanceCoef setEditorMode setEditorObjectScope setEffectCondition setEngineRPMRTD setFace setFaceAnimation setFatigue setFeatureType setFlagAnimationPhase setFlagOwner setFlagSide setFlagTexture setFog setFormation setFormationTask setFormDir setFriend setFromEditor setFSMVariable setFuel setFuelCargo setGroupIcon setGroupIconParams setGroupIconsSelectable setGroupIconsVisible setGroupId setGroupIdGlobal setGroupOwner setGusts setHideBehind setHit setHitIndex setHitPointDamage setHorizonParallaxCoef setHUDMovementLevels setIdentity setImportance setInfoPanel setLeader setLightAmbient setLightAttenuation setLightBrightness setLightColor setLightDayLight setLightFlareMaxDistance setLightFlareSize setLightIntensity setLightnings setLightUseFlare setLocalWindParams setMagazineTurretAmmo setMarkerAlpha setMarkerAlphaLocal setMarkerBrush setMarkerBrushLocal setMarkerColor setMarkerColorLocal setMarkerDir setMarkerDirLocal setMarkerPos setMarkerPosLocal setMarkerShape setMarkerShapeLocal setMarkerSize setMarkerSizeLocal setMarkerText setMarkerTextLocal setMarkerType setMarkerTypeLocal setMass setMimic setMousePosition setMusicEffect setMusicEventHandler setName setNameSound setObjectArguments setObjectMaterial setObjectMaterialGlobal setObjectProxy setObjectTexture setObjectTextureGlobal setObjectViewDistance setOvercast setOwner setOxygenRemaining setParticleCircle setParticleClass setParticleFire setParticleParams setParticleRandom setPilotCameraDirection setPilotCameraRotation setPilotCameraTarget setPilotLight setPiPEffect setPitch setPlateNumber setPlayable setPlayerRespawnTime setPos setPosASL setPosASL2 setPosASLW setPosATL setPosition setPosWorld setPylonLoadOut setPylonsPriority setRadioMsg setRain setRainbow setRandomLip setRank setRectangular setRepairCargo setRotorBrakeRTD setShadowDistance setShotParents setSide setSimpleTaskAlwaysVisible setSimpleTaskCustomData setSimpleTaskDescription setSimpleTaskDestination setSimpleTaskTarget setSimpleTaskType setSimulWeatherLayers setSize setSkill setSlingLoad setSoundEffect setSpeaker setSpeech setSpeedMode setStamina setStaminaScheme setStatValue setSuppression setSystemOfUnits setTargetAge setTaskMarkerOffset setTaskResult setTaskState setTerrainGrid setText setTimeMultiplier setTitleEffect setTrafficDensity setTrafficDistance setTrafficGap setTrafficSpeed setTriggerActivation setTriggerArea setTriggerStatements setTriggerText setTriggerTimeout setTriggerType setType setUnconscious setUnitAbility setUnitLoadout setUnitPos setUnitPosWeak setUnitRank setUnitRecoilCoefficient setUnitTrait setUnloadInCombat setUserActionText setUserMFDText setUserMFDvalue setVariable setVectorDir setVectorDirAndUp setVectorUp setVehicleAmmo setVehicleAmmoDef setVehicleArmor setVehicleCargo setVehicleId setVehicleLock setVehiclePosition setVehicleRadar setVehicleReceiveRemoteTargets setVehicleReportOwnPosition setVehicleReportRemoteTargets setVehicleTIPars setVehicleVarName setVelocity setVelocityModelSpace setVelocityTransformation setViewDistance setVisibleIfTreeCollapsed setWantedRPMRTD setWaves setWaypointBehaviour setWaypointCombatMode setWaypointCompletionRadius setWaypointDescription setWaypointForceBehaviour setWaypointFormation setWaypointHousePosition setWaypointLoiterRadius setWaypointLoiterType setWaypointName setWaypointPosition setWaypointScript setWaypointSpeed setWaypointStatements setWaypointTimeout setWaypointType setWaypointVisible setWeaponReloadingTime setWind setWindDir setWindForce setWindStr setWingForceScaleRTD setWPPos show3DIcons showChat showCinemaBorder showCommandingMenu showCompass showCuratorCompass showGPS showHUD showLegend showMap shownArtilleryComputer shownChat shownCompass shownCuratorCompass showNewEditorObject shownGPS shownHUD shownMap shownPad shownRadio shownScoretable shownUAVFeed shownWarrant shownWatch showPad showRadio showScoretable showSubtitles showUAVFeed showWarrant showWatch showWaypoint showWaypoints side sideChat sideEnemy sideFriendly sideRadio simpleTasks simulationEnabled simulCloudDensity simulCloudOcclusion simulInClouds simulWeatherSync sin size sizeOf skill skillFinal skipTime sleep sliderPosition sliderRange sliderSetPosition sliderSetRange sliderSetSpeed sliderSpeed slingLoadAssistantShown soldierMagazines someAmmo sort soundVolume spawn speaker speed speedMode splitString sqrt squadParams stance startLoadingScreen step stop stopEngineRTD stopped str sunOrMoon supportInfo suppressFor surfaceIsWater surfaceNormal surfaceType swimInDepth switchableUnits switchAction switchCamera switchGesture switchLight switchMove synchronizedObjects synchronizedTriggers synchronizedWaypoints synchronizeObjectsAdd synchronizeObjectsRemove synchronizeTrigger synchronizeWaypoint systemChat systemOfUnits tan targetKnowledge targets targetsAggregate targetsQuery taskAlwaysVisible taskChildren taskCompleted taskCustomData taskDescription taskDestination taskHint taskMarkerOffset taskParent taskResult taskState taskType teamMember teamName teams teamSwitch teamSwitchEnabled teamType terminate terrainIntersect terrainIntersectASL terrainIntersectAtASL text textLog textLogFormat tg time timeMultiplier titleCut titleFadeOut titleObj titleRsc titleText toArray toFixed toLower toString toUpper triggerActivated triggerActivation triggerArea triggerAttachedVehicle triggerAttachObject triggerAttachVehicle triggerDynamicSimulation triggerStatements triggerText triggerTimeout triggerTimeoutCurrent triggerType turretLocal turretOwner turretUnit tvAdd tvClear tvCollapse tvCollapseAll tvCount tvCurSel tvData tvDelete tvExpand tvExpandAll tvPicture tvSetColor tvSetCurSel tvSetData tvSetPicture tvSetPictureColor tvSetPictureColorDisabled tvSetPictureColorSelected tvSetPictureRight tvSetPictureRightColor tvSetPictureRightColorDisabled tvSetPictureRightColorSelected tvSetText tvSetTooltip tvSetValue tvSort tvSortByValue tvText tvTooltip tvValue type typeName typeOf UAVControl uiNamespace uiSleep unassignCurator unassignItem unassignTeam unassignVehicle underwater uniform uniformContainer uniformItems uniformMagazines unitAddons unitAimPosition unitAimPositionVisual unitBackpack unitIsUAV unitPos unitReady unitRecoilCoefficient units unitsBelowHeight unlinkItem 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",dragon_face:"🐲",dress:"👗",dromedary_camel:"🐪",drooling_face:"🤤",droplet:"💧",drum:"🥁",duck:"🦆",dvd:"📀","e-mail":"📧",eagle:"🦅",ear:"👂",ear_of_rice:"🌾",earth_africa:"🌍",earth_americas:"🌎",earth_asia:"🌏",egg:"🥚",eggplant:"🍆",eight_pointed_black_star:"✴️",eight_spoked_asterisk:"✳️",electric_plug:"🔌",elephant:"🐘",email:"✉️",end:"🔚",envelope_with_arrow:"📩",euro:"💶",european_castle:"🏰",european_post_office:"🏤",evergreen_tree:"🌲",exclamation:"❗️",expressionless:"😑",eye:"👁",eye_speech_bubble:"👁‍🗨",eyeglasses:"👓",eyes:"👀",face_with_head_bandage:"🤕",face_with_thermometer:"🤒",fist_oncoming:"👊",factory:"🏭",fallen_leaf:"🍂",family_man_woman_boy:"👪",family_man_boy:"👨‍👦",family_man_boy_boy:"👨‍👦‍👦",family_man_girl:"👨‍👧",family_man_girl_boy:"👨‍👧‍👦",family_man_girl_girl:"👨‍👧‍👧",family_man_man_boy:"👨‍👨‍👦",family_man_man_boy_boy:"👨‍👨‍👦‍👦",family_man_man_girl:"👨‍👨‍👧",family_man_man_girl_boy:"👨‍👨‍👧‍👦",family_man_man_girl_girl:"👨‍👨‍👧‍👧",family_man_woman_boy_boy:"👨‍👩‍👦‍👦",family_man_woman_girl:"👨‍👩‍👧",family_man_woman_girl_boy:"👨‍👩‍👧‍👦",family_man_woman_girl_girl:"👨‍👩‍👧‍👧",family_woman_boy:"👩‍👦",family_woman_boy_boy:"👩‍👦‍👦",family_woman_girl:"👩‍👧",family_woman_girl_boy:"👩‍👧‍👦",family_woman_girl_girl:"👩‍👧‍👧",family_woman_woman_boy:"👩‍👩‍👦",family_woman_woman_boy_boy:"👩‍👩‍👦‍👦",family_woman_woman_girl:"👩‍👩‍👧",family_woman_woman_girl_boy:"👩‍👩‍👧‍👦",family_woman_woman_girl_girl:"👩‍👩‍👧‍👧",fast_forward:"⏩",fax:"📠",fearful:"😨",feet:"🐾",female_detective:"🕵️‍♀️",ferris_wheel:"🎡",ferry:"⛴",field_hockey:"🏑",file_cabinet:"🗄",file_folder:"📁",film_projector:"📽",film_strip:"🎞",fire:"🔥",fire_engine:"🚒",fireworks:"🎆",first_quarter_moon:"🌓",first_quarter_moon_with_face:"🌛",fish:"🐟",fish_cake:"🍥",fishing_pole_and_fish:"🎣",fist_raised:"✊",fist_left:"🤛",fist_right:"🤜",flags:"🎏",flashlight:"🔦",fleur_de_lis:"⚜️",flight_arrival:"🛬",flight_departure:"🛫",floppy_disk:"💾",flower_playing_cards:"🎴",flushed:"😳",fog:"🌫",foggy:"🌁",football:"🏈",footprints:"👣",fork_and_knife:"🍴",fountain:"⛲️",fountain_pen:"🖋",four_leaf_clover:"🍀",fox_face:"🦊",framed_picture:"🖼",free:"🆓",fried_egg:"🍳",fried_shrimp:"🍤",fries:"🍟",frog:"🐸",frowning:"😦",frowning_face:"☹️",frowning_man:"🙍‍♂️",frowning_woman:"🙍",middle_finger:"🖕",fuelpump:"⛽️",full_moon:"🌕",full_moon_with_face:"🌝",funeral_urn:"⚱️",game_die:"🎲",gear:"⚙️",gem:"💎",gemini:"♊️",ghost:"👻",gift:"🎁",gift_heart:"💝",girl:"👧",globe_with_meridians:"🌐",goal_net:"🥅",goat:"🐐",golf:"⛳️",golfing_man:"🏌️",golfing_woman:"🏌️‍♀️",gorilla:"🦍",grapes:"🍇",green_apple:"🍏",green_book:"📗",green_heart:"💚",green_salad:"🥗",grey_exclamation:"❕",grey_question:"❔",grimacing:"😬",grin:"😁",grinning:"😀",guardsman:"💂",guardswoman:"💂‍♀️",guitar:"🎸",gun:"🔫",haircut_woman:"💇",haircut_man:"💇‍♂️",hamburger:"🍔",hammer:"🔨",hammer_and_pick:"⚒",hammer_and_wrench:"🛠",hamster:"🐹",hand:"✋",handbag:"👜",handshake:"🤝",hankey:"💩",hatched_chick:"🐥",hatching_chick:"🐣",headphones:"🎧",hear_no_evil:"🙉",heart:"❤️",heart_decoration:"💟",heart_eyes:"😍",heart_eyes_cat:"😻",heartbeat:"💓",heartpulse:"💗",hearts:"♥️",heavy_check_mark:"✔️",heavy_division_sign:"➗",heavy_dollar_sign:"💲",heavy_heart_exclamation:"❣️",heavy_minus_sign:"➖",heavy_multiplication_x:"✖️",heavy_plus_sign:"➕",helicopter:"🚁",herb:"🌿",hibiscus:"🌺",high_brightness:"🔆",high_heel:"👠",hocho:"🔪",hole:"🕳",honey_pot:"🍯",horse:"🐴",horse_racing:"🏇",hospital:"🏥",hot_pepper:"🌶",hotdog:"🌭",hotel:"🏨",hotsprings:"♨️",hourglass:"⌛️",hourglass_flowing_sand:"⏳",house:"🏠",house_with_garden:"🏡",houses:"🏘",hugs:"🤗",hushed:"😯",ice_cream:"🍨",ice_hockey:"🏒",ice_skate:"⛸",icecream:"🍦",id:"🆔",ideograph_advantage:"🉐",imp:"👿",inbox_tray:"📥",incoming_envelope:"📨",tipping_hand_woman:"💁",information_source:"ℹ️",innocent:"😇",interrobang:"⁉️",iphone:"📱",izakaya_lantern:"🏮",jack_o_lantern:"🎃",japan:"🗾",japanese_castle:"🏯",japanese_goblin:"👺",japanese_ogre:"👹",jeans:"👖",joy:"😂",joy_cat:"😹",joystick:"🕹",kaaba:"🕋",key:"🔑",keyboard:"⌨️",keycap_ten:"🔟",kick_scooter:"🛴",kimono:"👘",kiss:"💋",kissing:"😗",kissing_cat:"😽",kissing_closed_eyes:"😚",kissing_heart:"😘",kissing_smiling_eyes:"😙",kiwi_fruit:"🥝",koala:"🐨",koko:"🈁",label:"🏷",large_blue_circle:"🔵",large_blue_diamond:"🔷",large_orange_diamond:"🔶",last_quarter_moon:"🌗",last_quarter_moon_with_face:"🌜",latin_cross:"✝️",laughing:"😆",leaves:"🍃",ledger:"📒",left_luggage:"🛅",left_right_arrow:"↔️",leftwards_arrow_with_hook:"↩️",lemon:"🍋",leo:"♌️",leopard:"🐆",level_slider:"🎚",libra:"♎️",light_rail:"🚈",link:"🔗",lion:"🦁",lips:"👄",lipstick:"💄",lizard:"🦎",lock:"🔒",lock_with_ink_pen:"🔏",lollipop:"🍭",loop:"➿",loud_sound:"🔊",loudspeaker:"📢",love_hotel:"🏩",love_letter:"💌",low_brightness:"🔅",lying_face:"🤥",m:"Ⓜ️",mag:"🔍",mag_right:"🔎",mahjong:"🀄️",mailbox:"📫",mailbox_closed:"📪",mailbox_with_mail:"📬",mailbox_with_no_mail:"📭",man:"👨",man_artist:"👨‍🎨",man_astronaut:"👨‍🚀",man_cartwheeling:"🤸‍♂️",man_cook:"👨‍🍳",man_dancing:"🕺",man_facepalming:"🤦‍♂️",man_factory_worker:"👨‍🏭",man_farmer:"👨‍🌾",man_firefighter:"👨‍🚒",man_health_worker:"👨‍⚕️",man_in_tuxedo:"🤵",man_judge:"👨‍⚖️",man_juggling:"🤹‍♂️",man_mechanic:"👨‍🔧",man_office_worker:"👨‍💼",man_pilot:"👨‍✈️",man_playing_handball:"🤾‍♂️",man_playing_water_polo:"🤽‍♂️",man_scientist:"👨‍🔬",man_shrugging:"🤷‍♂️",man_singer:"👨‍🎤",man_student:"👨‍🎓",man_teacher:"👨‍🏫",man_technologist:"👨‍💻",man_with_gua_pi_mao:"👲",man_with_turban:"👳",tangerine:"🍊",mans_shoe:"👞",mantelpiece_clock:"🕰",maple_leaf:"🍁",martial_arts_uniform:"🥋",mask:"😷",massage_woman:"💆",massage_man:"💆‍♂️",meat_on_bone:"🍖",medal_military:"🎖",medal_sports:"🏅",mega:"📣",melon:"🍈",memo:"📝",men_wrestling:"🤼‍♂️",menorah:"🕎",mens:"🚹",metal:"🤘",metro:"🚇",microphone:"🎤",microscope:"🔬",milk_glass:"🥛",milky_way:"🌌",minibus:"🚐",minidisc:"💽",mobile_phone_off:"📴",money_mouth_face:"🤑",money_with_wings:"💸",moneybag:"💰",monkey:"🐒",monkey_face:"🐵",monorail:"🚝",moon:"🌔",mortar_board:"🎓",mosque:"🕌",motor_boat:"🛥",motor_scooter:"🛵",motorcycle:"🏍",motorway:"🛣",mount_fuji:"🗻",mountain:"⛰",mountain_biking_man:"🚵",mountain_biking_woman:"🚵‍♀️",mountain_cableway:"🚠",mountain_railway:"🚞",mountain_snow:"🏔",mouse:"🐭",mouse2:"🐁",movie_camera:"🎥",moyai:"🗿",mrs_claus:"🤶",muscle:"💪",mushroom:"🍄",musical_keyboard:"🎹",musical_note:"🎵",musical_score:"🎼",mute:"🔇",nail_care:"💅",name_badge:"📛",national_park:"🏞",nauseated_face:"🤢",necktie:"👔",negative_squared_cross_mark:"❎",nerd_face:"🤓",neutral_face:"😐",new:"🆕",new_moon:"🌑",new_moon_with_face:"🌚",newspaper:"📰",newspaper_roll:"🗞",next_track_button:"⏭",ng:"🆖",no_good_man:"🙅‍♂️",no_good_woman:"🙅",night_with_stars:"🌃",no_bell:"🔕",no_bicycles:"🚳",no_entry:"⛔️",no_entry_sign:"🚫",no_mobile_phones:"📵",no_mouth:"😶",no_pedestrians:"🚷",no_smoking:"🚭","non-potable_water":"🚱",nose:"👃",notebook:"📓",notebook_with_decorative_cover:"📔",notes:"🎶",nut_and_bolt:"🔩",o:"⭕️",o2:"🅾️",ocean:"🌊",octopus:"🐙",oden:"🍢",office:"🏢",oil_drum:"🛢",ok:"🆗",ok_hand:"👌",ok_man:"🙆‍♂️",ok_woman:"🙆",old_key:"🗝",older_man:"👴",older_woman:"👵",om:"🕉",on:"🔛",oncoming_automobile:"🚘",oncoming_bus:"🚍",oncoming_police_car:"🚔",oncoming_taxi:"🚖",open_file_folder:"📂",open_hands:"👐",open_mouth:"😮",open_umbrella:"☂️",ophiuchus:"⛎",orange_book:"📙",orthodox_cross:"☦️",outbox_tray:"📤",owl:"🦉",ox:"🐂",package:"📦",page_facing_up:"📄",page_with_curl:"📃",pager:"📟",paintbrush:"🖌",palm_tree:"🌴",pancakes:"🥞",panda_face:"🐼",paperclip:"📎",paperclips:"🖇",parasol_on_ground:"⛱",parking:"🅿️",part_alternation_mark:"〽️",partly_sunny:"⛅️",passenger_ship:"🛳",passport_control:"🛂",pause_button:"⏸",peace_symbol:"☮️",peach:"🍑",peanuts:"🥜",pear:"🍐",pen:"🖊",pencil2:"✏️",penguin:"🐧",pensive:"😔",performing_arts:"🎭",persevere:"😣",person_fencing:"🤺",pouting_woman:"🙎",phone:"☎️",pick:"⛏",pig:"🐷",pig2:"🐖",pig_nose:"🐽",pill:"💊",pineapple:"🍍",ping_pong:"🏓",pisces:"♓️",pizza:"🍕",place_of_worship:"🛐",plate_with_cutlery:"🍽",play_or_pause_button:"⏯",point_down:"👇",point_left:"👈",point_right:"👉",point_up:"☝️",point_up_2:"👆",police_car:"🚓",policewoman:"👮‍♀️",poodle:"🐩",popcorn:"🍿",post_office:"🏣",postal_horn:"📯",postbox:"📮",potable_water:"🚰",potato:"🥔",pouch:"👝",poultry_leg:"🍗",pound:"💷",rage:"😡",pouting_cat:"😾",pouting_man:"🙎‍♂️",pray:"🙏",prayer_beads:"📿",pregnant_woman:"🤰",previous_track_button:"⏮",prince:"🤴",princess:"👸",printer:"🖨",purple_heart:"💜",purse:"👛",pushpin:"📌",put_litter_in_its_place:"🚮",question:"❓",rabbit:"🐰",rabbit2:"🐇",racehorse:"🐎",racing_car:"🏎",radio:"📻",radio_button:"🔘",radioactive:"☢️",railway_car:"🚃",railway_track:"🛤",rainbow:"🌈",rainbow_flag:"🏳️‍🌈",raised_back_of_hand:"🤚",raised_hand_with_fingers_splayed:"🖐",raised_hands:"🙌",raising_hand_woman:"🙋",raising_hand_man:"🙋‍♂️",ram:"🐏",ramen:"🍜",rat:"🐀",record_button:"⏺",recycle:"♻️",red_circle:"🔴",registered:"®️",relaxed:"☺️",relieved:"😌",reminder_ribbon:"🎗",repeat:"🔁",repeat_one:"🔂",rescue_worker_helmet:"⛑",restroom:"🚻",revolving_hearts:"💞",rewind:"⏪",rhinoceros:"🦏",ribbon:"🎀",rice:"🍚",rice_ball:"🍙",rice_cracker:"🍘",rice_scene:"🎑",right_anger_bubble:"🗯",ring:"💍",robot:"🤖",rocket:"🚀",rofl:"🤣",roll_eyes:"🙄",roller_coaster:"🎢",rooster:"🐓",rose:"🌹",rosette:"🏵",rotating_light:"🚨",round_pushpin:"📍",rowing_man:"🚣",rowing_woman:"🚣‍♀️",rugby_football:"🏉",running_man:"🏃",running_shirt_with_sash:"🎽",running_woman:"🏃‍♀️",sa:"🈂️",sagittarius:"♐️",sake:"🍶",sandal:"👡",santa:"🎅",satellite:"📡",saxophone:"🎷",school:"🏫",school_satchel:"🎒",scissors:"✂️",scorpion:"🦂",scorpius:"♏️",scream:"😱",scream_cat:"🙀",scroll:"📜",seat:"💺",secret:"㊙️",see_no_evil:"🙈",seedling:"🌱",selfie:"🤳",shallow_pan_of_food:"🥘",shamrock:"☘️",shark:"🦈",shaved_ice:"🍧",sheep:"🐑",shell:"🐚",shield:"🛡",shinto_shrine:"⛩",ship:"🚢",shirt:"👕",shopping:"🛍",shopping_cart:"🛒",shower:"🚿",shrimp:"🦐",signal_strength:"📶",six_pointed_star:"🔯",ski:"🎿",skier:"⛷",skull:"💀",skull_and_crossbones:"☠️",sleeping:"😴",sleeping_bed:"🛌",sleepy:"😪",slightly_frowning_face:"🙁",slightly_smiling_face:"🙂",slot_machine:"🎰",small_airplane:"🛩",small_blue_diamond:"🔹",small_orange_diamond:"🔸",small_red_triangle:"🔺",small_red_triangle_down:"🔻",smile:"😄",smile_cat:"😸",smiley:"😃",smiley_cat:"😺",smiling_imp:"😈",smirk:"😏",smirk_cat:"😼",smoking:"🚬",snail:"🐌",snake:"🐍",sneezing_face:"🤧",snowboarder:"🏂",snowflake:"❄️",snowman:"⛄️",snowman_with_snow:"☃️",sob:"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DefineExternal DefineInputStreamMethod DefineOutputStreamMethod DefineResourceFunction Definition Degree DegreeCentrality DegreeGraphDistribution DegreeLexicographic DegreeReverseLexicographic DEigensystem DEigenvalues Deinitialization Del DelaunayMesh Delayed Deletable Delete DeleteAnomalies DeleteBorderComponents DeleteCases DeleteChannel DeleteCloudExpression DeleteContents DeleteDirectory DeleteDuplicates DeleteDuplicatesBy DeleteFile DeleteMissing DeleteObject DeletePermissionsKey DeleteSearchIndex DeleteSmallComponents DeleteStopwords DeleteWithContents DeletionWarning DelimitedArray DelimitedSequence Delimiter DelimiterFlashTime DelimiterMatching Delimiters DeliveryFunction Dendrogram Denominator DensityGraphics DensityHistogram DensityPlot DensityPlot3D DependentVariables Deploy Deployed Depth DepthFirstScan Derivative DerivativeFilter DerivedKey DescriptorStateSpace DesignMatrix DestroyAfterEvaluation Det DeviceClose DeviceConfigure DeviceExecute DeviceExecuteAsynchronous DeviceObject DeviceOpen DeviceOpenQ DeviceRead DeviceReadBuffer DeviceReadLatest DeviceReadList DeviceReadTimeSeries Devices DeviceStreams DeviceWrite DeviceWriteBuffer DGaussianWavelet DiacriticalPositioning Diagonal DiagonalizableMatrixQ DiagonalMatrix DiagonalMatrixQ Dialog DialogIndent DialogInput DialogLevel DialogNotebook DialogProlog DialogReturn DialogSymbols Diamond DiamondMatrix DiceDissimilarity DictionaryLookup DictionaryWordQ DifferenceDelta DifferenceOrder DifferenceQuotient DifferenceRoot DifferenceRootReduce Differences DifferentialD DifferentialRoot DifferentialRootReduce DifferentiatorFilter DigitalSignature DigitBlock DigitBlockMinimum DigitCharacter DigitCount DigitQ DihedralAngle DihedralGroup Dilation DimensionalCombinations DimensionalMeshComponents DimensionReduce DimensionReducerFunction DimensionReduction Dimensions DiracComb DiracDelta DirectedEdge DirectedEdges DirectedGraph DirectedGraphQ DirectedInfinity Direction Directive Directory DirectoryName DirectoryQ DirectoryStack DirichletBeta DirichletCharacter DirichletCondition DirichletConvolve DirichletDistribution DirichletEta DirichletL DirichletLambda DirichletTransform DirichletWindow DisableConsolePrintPacket DisableFormatting DiscreteChirpZTransform DiscreteConvolve DiscreteDelta DiscreteHadamardTransform DiscreteIndicator DiscreteLimit DiscreteLQEstimatorGains DiscreteLQRegulatorGains DiscreteLyapunovSolve DiscreteMarkovProcess DiscreteMaxLimit DiscreteMinLimit DiscretePlot DiscretePlot3D DiscreteRatio DiscreteRiccatiSolve DiscreteShift DiscreteTimeModelQ DiscreteUniformDistribution DiscreteVariables DiscreteWaveletData DiscreteWaveletPacketTransform DiscreteWaveletTransform DiscretizeGraphics DiscretizeRegion Discriminant DisjointQ Disjunction Disk DiskBox DiskMatrix DiskSegment Dispatch DispatchQ DispersionEstimatorFunction Display DisplayAllSteps DisplayEndPacket DisplayFlushImagePacket DisplayForm DisplayFunction DisplayPacket DisplayRules DisplaySetSizePacket DisplayString DisplayTemporary DisplayWith DisplayWithRef DisplayWithVariable DistanceFunction DistanceMatrix DistanceTransform Distribute Distributed DistributedContexts DistributeDefinitions DistributionChart DistributionDomain DistributionFitTest DistributionParameterAssumptions DistributionParameterQ Dithering Div Divergence Divide DivideBy Dividers DivideSides Divisible Divisors DivisorSigma DivisorSum DMSList DMSString Do DockedCells DocumentGenerator DocumentGeneratorInformation DocumentGeneratorInformationData DocumentGenerators DocumentNotebook DocumentWeightingRules Dodecahedron DomainRegistrationInformation DominantColors DOSTextFormat Dot DotDashed DotEqual DotLayer DotPlusLayer Dotted DoubleBracketingBar DoubleContourIntegral DoubleDownArrow DoubleLeftArrow DoubleLeftRightArrow DoubleLeftTee DoubleLongLeftArrow DoubleLongLeftRightArrow DoubleLongRightArrow DoubleRightArrow DoubleRightTee DoubleUpArrow DoubleUpDownArrow DoubleVerticalBar DoublyInfinite Down DownArrow DownArrowBar DownArrowUpArrow DownLeftRightVector DownLeftTeeVector DownLeftVector DownLeftVectorBar DownRightTeeVector DownRightVector DownRightVectorBar Downsample DownTee DownTeeArrow DownValues DragAndDrop DrawEdges DrawFrontFaces DrawHighlighted Drop DropoutLayer DSolve DSolveValue Dt DualLinearProgramming DualPolyhedron DualSystemsModel DumpGet DumpSave DuplicateFreeQ Duration Dynamic DynamicBox DynamicBoxOptions DynamicEvaluationTimeout DynamicGeoGraphics DynamicImage DynamicLocation DynamicModule DynamicModuleBox DynamicModuleBoxOptions DynamicModuleParent DynamicModuleValues DynamicName DynamicNamespace DynamicReference DynamicSetting DynamicUpdating DynamicWrapper DynamicWrapperBox DynamicWrapperBoxOptionsE EarthImpactData EarthquakeData EccentricityCentrality Echo EchoFunction EclipseType EdgeAdd EdgeBetweennessCentrality EdgeCapacity EdgeCapForm EdgeColor EdgeConnectivity EdgeContract EdgeCost EdgeCount EdgeCoverQ EdgeCycleMatrix EdgeDashing EdgeDelete EdgeDetect EdgeForm EdgeIndex EdgeJoinForm EdgeLabeling EdgeLabels EdgeLabelStyle EdgeList EdgeOpacity EdgeQ EdgeRenderingFunction EdgeRules EdgeShapeFunction EdgeStyle EdgeThickness EdgeWeight EdgeWeightedGraphQ Editable EditButtonSettings EditCellTagsSettings EditDistance EffectiveInterest Eigensystem Eigenvalues EigenvectorCentrality Eigenvectors Element ElementData ElementwiseLayer ElidedForms Eliminate EliminationOrder Ellipsoid EllipticE EllipticExp EllipticExpPrime EllipticF EllipticFilterModel EllipticK EllipticLog EllipticNomeQ EllipticPi EllipticReducedHalfPeriods EllipticTheta EllipticThetaPrime EmbedCode EmbeddedHTML EmbeddedService EmbeddingLayer EmbeddingObject EmitSound EmphasizeSyntaxErrors EmpiricalDistribution Empty EmptyGraphQ EmptyRegion EnableConsolePrintPacket Enabled Encode Encrypt EncryptedObject EncryptFile End EndAdd EndDialogPacket EndFrontEndInteractionPacket EndOfBuffer EndOfFile EndOfLine EndOfString EndPackage EngineEnvironment EngineeringForm Enter EnterExpressionPacket EnterTextPacket Entity EntityClass EntityClassList EntityCopies EntityFunction EntityGroup EntityInstance EntityList EntityPrefetch EntityProperties EntityProperty EntityPropertyClass EntityRegister EntityStore EntityStores EntityTypeName EntityUnregister EntityValue Entropy EntropyFilter Environment Epilog EpilogFunction Equal EqualColumns EqualRows EqualTilde EqualTo EquatedTo Equilibrium EquirippleFilterKernel Equivalent Erf Erfc Erfi ErlangB ErlangC ErlangDistribution Erosion ErrorBox ErrorBoxOptions ErrorNorm ErrorPacket ErrorsDialogSettings EscapeRadius EstimatedBackground EstimatedDistribution EstimatedProcess EstimatorGains EstimatorRegulator EuclideanDistance EulerAngles EulerCharacteristic EulerE EulerGamma EulerianGraphQ EulerMatrix EulerPhi Evaluatable Evaluate Evaluated EvaluatePacket EvaluateScheduledTask EvaluationBox EvaluationCell EvaluationCompletionAction EvaluationData EvaluationElements EvaluationEnvironment EvaluationMode EvaluationMonitor EvaluationNotebook EvaluationObject EvaluationOrder Evaluator EvaluatorNames EvenQ EventData EventEvaluator EventHandler EventHandlerTag EventLabels EventSeries ExactBlackmanWindow ExactNumberQ ExactRootIsolation ExampleData Except ExcludedForms ExcludedLines ExcludedPhysicalQuantities ExcludePods Exclusions ExclusionsStyle Exists Exit ExitDialog ExoplanetData Exp Expand ExpandAll ExpandDenominator ExpandFileName ExpandNumerator Expectation ExpectationE ExpectedValue ExpGammaDistribution ExpIntegralE ExpIntegralEi ExpirationDate Exponent ExponentFunction ExponentialDistribution ExponentialFamily ExponentialGeneratingFunction ExponentialMovingAverage ExponentialPowerDistribution ExponentPosition ExponentStep Export ExportAutoReplacements ExportByteArray ExportForm ExportPacket ExportString Expression ExpressionCell ExpressionPacket ExpressionUUID ExpToTrig ExtendedEntityClass ExtendedGCD Extension ExtentElementFunction ExtentMarkers ExtentSize ExternalBundle ExternalCall ExternalDataCharacterEncoding ExternalEvaluate ExternalFunction ExternalFunctionName ExternalObject ExternalOptions ExternalSessionObject ExternalSessions ExternalTypeSignature ExternalValue Extract ExtractArchive ExtractLayer ExtremeValueDistributionFaceForm FaceGrids FaceGridsStyle FacialFeatures Factor FactorComplete Factorial Factorial2 FactorialMoment FactorialMomentGeneratingFunction FactorialPower FactorInteger FactorList FactorSquareFree FactorSquareFreeList FactorTerms FactorTermsList Fail Failure FailureAction FailureDistribution FailureQ False FareySequence FARIMAProcess FeatureDistance FeatureExtract FeatureExtraction FeatureExtractor FeatureExtractorFunction FeatureNames FeatureNearest FeatureSpacePlot FeatureSpacePlot3D FeatureTypes FEDisableConsolePrintPacket FeedbackLinearize FeedbackSector FeedbackSectorStyle FeedbackType FEEnableConsolePrintPacket FetalGrowthData Fibonacci Fibonorial FieldCompletionFunction FieldHint FieldHintStyle FieldMasked FieldSize File FileBaseName FileByteCount FileConvert FileDate FileExistsQ FileExtension FileFormat FileHandler FileHash FileInformation FileName FileNameDepth FileNameDialogSettings FileNameDrop FileNameForms FileNameJoin FileNames FileNameSetter FileNameSplit FileNameTake FilePrint FileSize FileSystemMap FileSystemScan FileTemplate FileTemplateApply FileType FilledCurve FilledCurveBox FilledCurveBoxOptions Filling FillingStyle FillingTransform FilteredEntityClass FilterRules FinancialBond FinancialData FinancialDerivative FinancialIndicator Find FindAnomalies FindArgMax FindArgMin FindChannels FindClique FindClusters FindCookies FindCurvePath FindCycle FindDevices FindDistribution FindDistributionParameters FindDivisions FindEdgeCover FindEdgeCut FindEdgeIndependentPaths FindEquationalProof FindEulerianCycle FindExternalEvaluators FindFaces FindFile FindFit FindFormula FindFundamentalCycles FindGeneratingFunction FindGeoLocation FindGeometricConjectures FindGeometricTransform FindGraphCommunities FindGraphIsomorphism FindGraphPartition FindHamiltonianCycle FindHamiltonianPath FindHiddenMarkovStates FindIndependentEdgeSet FindIndependentVertexSet FindInstance FindIntegerNullVector FindKClan FindKClique FindKClub FindKPlex FindLibrary FindLinearRecurrence FindList FindMatchingColor FindMaximum FindMaximumFlow FindMaxValue FindMeshDefects FindMinimum FindMinimumCostFlow FindMinimumCut FindMinValue FindMoleculeSubstructure FindPath FindPeaks FindPermutation FindPostmanTour FindProcessParameters FindRepeat FindRoot FindSequenceFunction FindSettings FindShortestPath FindShortestTour FindSpanningTree FindSystemModelEquilibrium FindTextualAnswer FindThreshold FindTransientRepeat FindVertexCover FindVertexCut FindVertexIndependentPaths Fine FinishDynamic FiniteAbelianGroupCount FiniteGroupCount FiniteGroupData First FirstCase FirstPassageTimeDistribution FirstPosition FischerGroupFi22 FischerGroupFi23 FischerGroupFi24Prime FisherHypergeometricDistribution FisherRatioTest FisherZDistribution Fit FitAll FitRegularization FittedModel FixedOrder FixedPoint FixedPointList FlashSelection Flat Flatten FlattenAt FlattenLayer FlatTopWindow FlipView Floor FlowPolynomial FlushPrintOutputPacket Fold FoldList FoldPair FoldPairList FollowRedirects Font FontColor FontFamily FontForm FontName FontOpacity FontPostScriptName FontProperties FontReencoding FontSize FontSlant FontSubstitutions FontTracking FontVariations FontWeight For ForAll Format FormatRules FormatType FormatTypeAutoConvert FormatValues FormBox FormBoxOptions FormControl FormFunction FormLayoutFunction FormObject FormPage FormTheme FormulaData FormulaLookup FortranForm Forward ForwardBackward Fourier FourierCoefficient FourierCosCoefficient FourierCosSeries FourierCosTransform FourierDCT FourierDCTFilter FourierDCTMatrix FourierDST FourierDSTMatrix FourierMatrix FourierParameters FourierSequenceTransform FourierSeries FourierSinCoefficient FourierSinSeries FourierSinTransform FourierTransform FourierTrigSeries FractionalBrownianMotionProcess FractionalGaussianNoiseProcess FractionalPart FractionBox FractionBoxOptions FractionLine Frame FrameBox FrameBoxOptions Framed FrameInset FrameLabel Frameless FrameMargins FrameRate FrameStyle FrameTicks FrameTicksStyle FRatioDistribution FrechetDistribution FreeQ FrenetSerretSystem FrequencySamplingFilterKernel FresnelC FresnelF FresnelG FresnelS Friday FrobeniusNumber FrobeniusSolve FromAbsoluteTime FromCharacterCode FromCoefficientRules FromContinuedFraction FromDate FromDigits FromDMS FromEntity FromJulianDate FromLetterNumber FromPolarCoordinates FromRomanNumeral FromSphericalCoordinates FromUnixTime Front FrontEndDynamicExpression FrontEndEventActions FrontEndExecute FrontEndObject FrontEndResource FrontEndResourceString FrontEndStackSize FrontEndToken FrontEndTokenExecute FrontEndValueCache FrontEndVersion FrontFaceColor FrontFaceOpacity Full FullAxes FullDefinition FullForm FullGraphics FullInformationOutputRegulator FullOptions FullRegion FullSimplify Function FunctionCompile FunctionCompileExport FunctionCompileExportByteArray FunctionCompileExportLibrary FunctionCompileExportString FunctionDomain FunctionExpand FunctionInterpolation FunctionPeriod FunctionRange FunctionSpace FussellVeselyImportanceGaborFilter GaborMatrix GaborWavelet GainMargins GainPhaseMargins GalaxyData GalleryView Gamma GammaDistribution GammaRegularized GapPenalty GARCHProcess GatedRecurrentLayer Gather GatherBy GaugeFaceElementFunction GaugeFaceStyle GaugeFrameElementFunction GaugeFrameSize GaugeFrameStyle GaugeLabels GaugeMarkers GaugeStyle GaussianFilter GaussianIntegers GaussianMatrix GaussianOrthogonalMatrixDistribution GaussianSymplecticMatrixDistribution GaussianUnitaryMatrixDistribution GaussianWindow GCD GegenbauerC General GeneralizedLinearModelFit GenerateAsymmetricKeyPair GenerateConditions GeneratedCell GeneratedDocumentBinding GenerateDerivedKey GenerateDigitalSignature GenerateDocument GeneratedParameters GeneratedQuantityMagnitudes GenerateHTTPResponse GenerateSecuredAuthenticationKey GenerateSymmetricKey GeneratingFunction GeneratorDescription GeneratorHistoryLength GeneratorOutputType Generic GenericCylindricalDecomposition GenomeData GenomeLookup GeoAntipode GeoArea GeoArraySize GeoBackground GeoBoundingBox GeoBounds GeoBoundsRegion GeoBubbleChart GeoCenter GeoCircle GeodesicClosing GeodesicDilation GeodesicErosion GeodesicOpening GeoDestination GeodesyData GeoDirection GeoDisk GeoDisplacement GeoDistance GeoDistanceList GeoElevationData GeoEntities GeoGraphics GeogravityModelData GeoGridDirectionDifference GeoGridLines GeoGridLinesStyle GeoGridPosition GeoGridRange GeoGridRangePadding GeoGridUnitArea GeoGridUnitDistance GeoGridVector GeoGroup GeoHemisphere GeoHemisphereBoundary GeoHistogram GeoIdentify GeoImage GeoLabels GeoLength GeoListPlot GeoLocation GeologicalPeriodData GeomagneticModelData GeoMarker GeometricAssertion GeometricBrownianMotionProcess GeometricDistribution GeometricMean GeometricMeanFilter GeometricScene GeometricTransformation GeometricTransformation3DBox GeometricTransformation3DBoxOptions GeometricTransformationBox GeometricTransformationBoxOptions GeoModel GeoNearest GeoPath GeoPosition GeoPositionENU GeoPositionXYZ GeoProjection GeoProjectionData GeoRange GeoRangePadding GeoRegionValuePlot GeoResolution GeoScaleBar GeoServer GeoSmoothHistogram GeoStreamPlot GeoStyling GeoStylingImageFunction GeoVariant GeoVector GeoVectorENU GeoVectorPlot GeoVectorXYZ GeoVisibleRegion GeoVisibleRegionBoundary GeoWithinQ GeoZoomLevel GestureHandler GestureHandlerTag Get GetBoundingBoxSizePacket GetContext GetEnvironment GetFileName GetFrontEndOptionsDataPacket GetLinebreakInformationPacket GetMenusPacket GetPageBreakInformationPacket Glaisher GlobalClusteringCoefficient GlobalPreferences GlobalSession Glow GoldenAngle GoldenRatio GompertzMakehamDistribution GoodmanKruskalGamma GoodmanKruskalGammaTest Goto Grad Gradient GradientFilter GradientOrientationFilter GrammarApply GrammarRules GrammarToken Graph Graph3D GraphAssortativity GraphAutomorphismGroup GraphCenter GraphComplement GraphData GraphDensity GraphDiameter GraphDifference GraphDisjointUnion GraphDistance GraphDistanceMatrix GraphElementData GraphEmbedding GraphHighlight GraphHighlightStyle GraphHub Graphics Graphics3D Graphics3DBox Graphics3DBoxOptions GraphicsArray GraphicsBaseline GraphicsBox GraphicsBoxOptions GraphicsColor GraphicsColumn GraphicsComplex GraphicsComplex3DBox GraphicsComplex3DBoxOptions GraphicsComplexBox GraphicsComplexBoxOptions GraphicsContents GraphicsData GraphicsGrid GraphicsGridBox GraphicsGroup GraphicsGroup3DBox GraphicsGroup3DBoxOptions GraphicsGroupBox GraphicsGroupBoxOptions GraphicsGrouping GraphicsHighlightColor GraphicsRow GraphicsSpacing GraphicsStyle GraphIntersection GraphLayout GraphLinkEfficiency GraphPeriphery GraphPlot GraphPlot3D GraphPower GraphPropertyDistribution GraphQ GraphRadius GraphReciprocity GraphRoot GraphStyle GraphUnion Gray GrayLevel Greater GreaterEqual GreaterEqualLess GreaterEqualThan GreaterFullEqual GreaterGreater GreaterLess GreaterSlantEqual GreaterThan GreaterTilde Green GreenFunction Grid GridBaseline GridBox GridBoxAlignment GridBoxBackground GridBoxDividers GridBoxFrame GridBoxItemSize GridBoxItemStyle GridBoxOptions GridBoxSpacings GridCreationSettings GridDefaultElement GridElementStyleOptions GridFrame GridFrameMargins GridGraph GridLines GridLinesStyle GroebnerBasis GroupActionBase GroupBy GroupCentralizer GroupElementFromWord GroupElementPosition GroupElementQ GroupElements GroupElementToWord GroupGenerators Groupings GroupMultiplicationTable GroupOrbits GroupOrder GroupPageBreakWithin GroupSetwiseStabilizer GroupStabilizer GroupStabilizerChain GroupTogetherGrouping GroupTogetherNestedGrouping GrowCutComponents Gudermannian GuidedFilter GumbelDistributionHaarWavelet HadamardMatrix HalfLine HalfNormalDistribution HalfPlane HalfSpace HamiltonianGraphQ HammingDistance HammingWindow HandlerFunctions HandlerFunctionsKeys HankelH1 HankelH2 HankelMatrix HankelTransform HannPoissonWindow HannWindow HaradaNortonGroupHN HararyGraph HarmonicMean HarmonicMeanFilter HarmonicNumber Hash Haversine HazardFunction Head HeadCompose HeaderLines Heads HeavisideLambda HeavisidePi HeavisideTheta HeldGroupHe HeldPart HelpBrowserLookup HelpBrowserNotebook HelpBrowserSettings Here HermiteDecomposition HermiteH HermitianMatrixQ HessenbergDecomposition Hessian HexadecimalCharacter Hexahedron HexahedronBox HexahedronBoxOptions HiddenMarkovProcess HiddenSurface Highlighted HighlightGraph HighlightImage HighlightMesh HighpassFilter HigmanSimsGroupHS HilbertCurve HilbertFilter HilbertMatrix Histogram Histogram3D HistogramDistribution HistogramList HistogramTransform HistogramTransformInterpolation HistoricalPeriodData HitMissTransform HITSCentrality HjorthDistribution HodgeDual HoeffdingD HoeffdingDTest Hold HoldAll HoldAllComplete HoldComplete HoldFirst HoldForm HoldPattern HoldRest HolidayCalendar HomeDirectory HomePage Horizontal HorizontalForm HorizontalGauge HorizontalScrollPosition HornerForm HostLookup HotellingTSquareDistribution HoytDistribution HTMLSave HTTPErrorResponse HTTPRedirect HTTPRequest HTTPRequestData HTTPResponse Hue HumanGrowthData HumpDownHump HumpEqual HurwitzLerchPhi HurwitzZeta HyperbolicDistribution HypercubeGraph HyperexponentialDistribution Hyperfactorial Hypergeometric0F1 Hypergeometric0F1Regularized Hypergeometric1F1 Hypergeometric1F1Regularized Hypergeometric2F1 Hypergeometric2F1Regularized HypergeometricDistribution HypergeometricPFQ HypergeometricPFQRegularized HypergeometricU Hyperlink HyperlinkCreationSettings Hyperplane Hyphenation HyphenationOptions HypoexponentialDistribution HypothesisTestDataI IconData Iconize IconizedObject IconRules Icosahedron Identity IdentityMatrix If IgnoreCase IgnoreDiacritics IgnorePunctuation IgnoreSpellCheck IgnoringInactive Im Image Image3D Image3DProjection Image3DSlices ImageAccumulate ImageAdd ImageAdjust ImageAlign ImageApply ImageApplyIndexed ImageAspectRatio ImageAssemble ImageAugmentationLayer ImageBoundingBoxes ImageCache ImageCacheValid ImageCapture ImageCaptureFunction ImageCases ImageChannels ImageClip ImageCollage ImageColorSpace ImageCompose ImageContainsQ ImageContents ImageConvolve ImageCooccurrence ImageCorners ImageCorrelate ImageCorrespondingPoints ImageCrop ImageData ImageDeconvolve ImageDemosaic ImageDifference ImageDimensions ImageDisplacements ImageDistance ImageEffect ImageExposureCombine ImageFeatureTrack ImageFileApply ImageFileFilter ImageFileScan ImageFilter ImageFocusCombine ImageForestingComponents ImageFormattingWidth ImageForwardTransformation ImageGraphics ImageHistogram ImageIdentify ImageInstanceQ ImageKeypoints ImageLevels ImageLines ImageMargins ImageMarker ImageMarkers ImageMeasurements ImageMesh ImageMultiply ImageOffset ImagePad ImagePadding ImagePartition ImagePeriodogram ImagePerspectiveTransformation ImagePosition ImagePreviewFunction ImagePyramid ImagePyramidApply ImageQ ImageRangeCache ImageRecolor ImageReflect ImageRegion ImageResize ImageResolution ImageRestyle ImageRotate ImageRotated ImageSaliencyFilter ImageScaled ImageScan ImageSize ImageSizeAction ImageSizeCache ImageSizeMultipliers ImageSizeRaw ImageSubtract ImageTake ImageTransformation ImageTrim ImageType ImageValue ImageValuePositions ImagingDevice ImplicitRegion Implies Import ImportAutoReplacements ImportByteArray ImportOptions ImportString ImprovementImportance In Inactivate Inactive IncidenceGraph IncidenceList IncidenceMatrix IncludeAromaticBonds IncludeConstantBasis IncludeDefinitions IncludeDirectories IncludeFileExtension IncludeGeneratorTasks IncludeHydrogens IncludeInflections IncludeMetaInformation IncludePods IncludeQuantities IncludeRelatedTables IncludeSingularTerm IncludeWindowTimes Increment IndefiniteMatrixQ Indent IndentingNewlineSpacings IndentMaxFraction IndependenceTest IndependentEdgeSetQ IndependentPhysicalQuantity IndependentUnit IndependentUnitDimension IndependentVertexSetQ Indeterminate IndeterminateThreshold IndexCreationOptions Indexed IndexGraph IndexTag Inequality InexactNumberQ InexactNumbers InfiniteLine InfinitePlane Infinity Infix InflationAdjust InflationMethod Information InformationData InformationDataGrid Inherited InheritScope InhomogeneousPoissonProcess InitialEvaluationHistory Initialization InitializationCell InitializationCellEvaluation InitializationCellWarning InitializationObjects InitializationValue Initialize InitialSeeding InlineCounterAssignments InlineCounterIncrements InlineRules Inner InnerPolygon InnerPolyhedron Inpaint Input InputAliases InputAssumptions InputAutoReplacements InputField InputFieldBox InputFieldBoxOptions InputForm InputGrouping InputNamePacket InputNotebook InputPacket InputSettings InputStream InputString InputStringPacket InputToBoxFormPacket Insert InsertionFunction InsertionPointObject InsertLinebreaks InsertResults Inset Inset3DBox Inset3DBoxOptions InsetBox InsetBoxOptions Insphere Install InstallService InstanceNormalizationLayer InString Integer IntegerDigits IntegerExponent IntegerLength IntegerName IntegerPart IntegerPartitions IntegerQ IntegerReverse Integers IntegerString Integral Integrate Interactive InteractiveTradingChart Interlaced Interleaving InternallyBalancedDecomposition InterpolatingFunction InterpolatingPolynomial Interpolation InterpolationOrder InterpolationPoints InterpolationPrecision Interpretation InterpretationBox InterpretationBoxOptions InterpretationFunction Interpreter InterpretTemplate InterquartileRange Interrupt InterruptSettings IntersectingQ Intersection Interval IntervalIntersection IntervalMarkers IntervalMarkersStyle IntervalMemberQ IntervalSlider IntervalUnion Into Inverse InverseBetaRegularized InverseCDF InverseChiSquareDistribution InverseContinuousWaveletTransform InverseDistanceTransform InverseEllipticNomeQ InverseErf InverseErfc InverseFourier InverseFourierCosTransform InverseFourierSequenceTransform InverseFourierSinTransform InverseFourierTransform InverseFunction InverseFunctions InverseGammaDistribution InverseGammaRegularized InverseGaussianDistribution InverseGudermannian InverseHankelTransform InverseHaversine InverseImagePyramid InverseJacobiCD InverseJacobiCN InverseJacobiCS InverseJacobiDC InverseJacobiDN InverseJacobiDS InverseJacobiNC InverseJacobiND InverseJacobiNS InverseJacobiSC InverseJacobiSD InverseJacobiSN InverseLaplaceTransform InverseMellinTransform InversePermutation InverseRadon InverseRadonTransform InverseSeries InverseShortTimeFourier InverseSpectrogram InverseSurvivalFunction InverseTransformedRegion InverseWaveletTransform InverseWeierstrassP InverseWishartMatrixDistribution InverseZTransform Invisible InvisibleApplication InvisibleTimes IPAddress IrreduciblePolynomialQ IslandData IsolatingInterval IsomorphicGraphQ IsotopeData Italic Item ItemAspectRatio ItemBox ItemBoxOptions ItemSize ItemStyle ItoProcessJaccardDissimilarity JacobiAmplitude Jacobian JacobiCD JacobiCN JacobiCS JacobiDC JacobiDN JacobiDS JacobiNC JacobiND JacobiNS JacobiP JacobiSC JacobiSD JacobiSN JacobiSymbol JacobiZeta JankoGroupJ1 JankoGroupJ2 JankoGroupJ3 JankoGroupJ4 JarqueBeraALMTest JohnsonDistribution Join JoinAcross Joined JoinedCurve JoinedCurveBox JoinedCurveBoxOptions JoinForm JordanDecomposition JordanModelDecomposition JulianDate JuliaSetBoettcher JuliaSetIterationCount JuliaSetPlot JuliaSetPointsK KagiChart KaiserBesselWindow KaiserWindow KalmanEstimator KalmanFilter KarhunenLoeveDecomposition KaryTree KatzCentrality KCoreComponents KDistribution KEdgeConnectedComponents KEdgeConnectedGraphQ KelvinBei KelvinBer KelvinKei KelvinKer KendallTau KendallTauTest KernelExecute KernelFunction KernelMixtureDistribution Kernels Ket Key KeyCollisionFunction KeyComplement KeyDrop KeyDropFrom KeyExistsQ KeyFreeQ KeyIntersection KeyMap KeyMemberQ KeypointStrength Keys KeySelect KeySort KeySortBy KeyTake KeyUnion KeyValueMap KeyValuePattern Khinchin KillProcess KirchhoffGraph KirchhoffMatrix KleinInvariantJ KnapsackSolve KnightTourGraph KnotData KnownUnitQ KochCurve KolmogorovSmirnovTest KroneckerDelta KroneckerModelDecomposition KroneckerProduct KroneckerSymbol KuiperTest KumaraswamyDistribution Kurtosis KuwaharaFilter KVertexConnectedComponents KVertexConnectedGraphQLABColor Label Labeled LabeledSlider LabelingFunction LabelingSize LabelStyle LabelVisibility LaguerreL LakeData LambdaComponents LambertW LaminaData LanczosWindow LandauDistribution Language LanguageCategory LanguageData LanguageIdentify LanguageOptions LaplaceDistribution LaplaceTransform Laplacian LaplacianFilter LaplacianGaussianFilter Large Larger Last Latitude LatitudeLongitude LatticeData LatticeReduce Launch LaunchKernels LayeredGraphPlot LayerSizeFunction LayoutInformation LCHColor LCM LeaderSize LeafCount LeapYearQ LearnDistribution LearnedDistribution LearningRate LearningRateMultipliers LeastSquares LeastSquaresFilterKernel Left LeftArrow LeftArrowBar LeftArrowRightArrow LeftDownTeeVector LeftDownVector LeftDownVectorBar LeftRightArrow LeftRightVector LeftTee LeftTeeArrow LeftTeeVector LeftTriangle LeftTriangleBar LeftTriangleEqual LeftUpDownVector LeftUpTeeVector LeftUpVector LeftUpVectorBar LeftVector LeftVectorBar LegendAppearance Legended LegendFunction LegendLabel LegendLayout LegendMargins LegendMarkers LegendMarkerSize LegendreP LegendreQ LegendreType Length LengthWhile LerchPhi Less LessEqual LessEqualGreater LessEqualThan LessFullEqual LessGreater LessLess LessSlantEqual LessThan LessTilde LetterCharacter LetterCounts LetterNumber LetterQ Level LeveneTest LeviCivitaTensor LevyDistribution Lexicographic LibraryDataType LibraryFunction LibraryFunctionError LibraryFunctionInformation LibraryFunctionLoad LibraryFunctionUnload LibraryLoad LibraryUnload LicenseID LiftingFilterData LiftingWaveletTransform LightBlue LightBrown LightCyan Lighter LightGray LightGreen Lighting LightingAngle LightMagenta LightOrange LightPink LightPurple LightRed LightSources LightYellow Likelihood Limit LimitsPositioning LimitsPositioningTokens LindleyDistribution Line Line3DBox Line3DBoxOptions LinearFilter LinearFractionalOptimization LinearFractionalTransform LinearGradientImage LinearizingTransformationData LinearLayer LinearModelFit LinearOffsetFunction LinearOptimization LinearProgramming LinearRecurrence LinearSolve LinearSolveFunction LineBox LineBoxOptions LineBreak LinebreakAdjustments LineBreakChart LinebreakSemicolonWeighting LineBreakWithin LineColor LineGraph LineIndent LineIndentMaxFraction LineIntegralConvolutionPlot LineIntegralConvolutionScale LineLegend LineOpacity LineSpacing LineWrapParts LinkActivate LinkClose LinkConnect LinkConnectedQ LinkCreate LinkError LinkFlush LinkFunction LinkHost LinkInterrupt LinkLaunch LinkMode LinkObject LinkOpen LinkOptions LinkPatterns LinkProtocol LinkRankCentrality LinkRead LinkReadHeld LinkReadyQ Links LinkService LinkWrite LinkWriteHeld LiouvilleLambda List Listable ListAnimate ListContourPlot ListContourPlot3D ListConvolve ListCorrelate ListCurvePathPlot ListDeconvolve ListDensityPlot ListDensityPlot3D Listen ListFormat ListFourierSequenceTransform ListInterpolation ListLineIntegralConvolutionPlot ListLinePlot ListLogLinearPlot ListLogLogPlot ListLogPlot ListPicker ListPickerBox ListPickerBoxBackground ListPickerBoxOptions ListPlay ListPlot ListPlot3D ListPointPlot3D ListPolarPlot ListQ ListSliceContourPlot3D ListSliceDensityPlot3D ListSliceVectorPlot3D ListStepPlot ListStreamDensityPlot ListStreamPlot ListSurfacePlot3D ListVectorDensityPlot ListVectorPlot ListVectorPlot3D ListZTransform Literal LiteralSearch LocalAdaptiveBinarize LocalCache LocalClusteringCoefficient LocalizeDefinitions LocalizeVariables LocalObject LocalObjects LocalResponseNormalizationLayer LocalSubmit LocalSymbol LocalTime LocalTimeZone LocationEquivalenceTest LocationTest Locator LocatorAutoCreate LocatorBox LocatorBoxOptions LocatorCentering LocatorPane LocatorPaneBox LocatorPaneBoxOptions LocatorRegion Locked Log Log10 Log2 LogBarnesG LogGamma LogGammaDistribution LogicalExpand LogIntegral LogisticDistribution LogisticSigmoid LogitModelFit LogLikelihood LogLinearPlot LogLogisticDistribution LogLogPlot LogMultinormalDistribution LogNormalDistribution LogPlot LogRankTest LogSeriesDistribution LongEqual Longest LongestCommonSequence LongestCommonSequencePositions LongestCommonSubsequence LongestCommonSubsequencePositions LongestMatch LongestOrderedSequence LongForm Longitude LongLeftArrow LongLeftRightArrow LongRightArrow LongShortTermMemoryLayer Lookup Loopback LoopFreeGraphQ LossFunction LowerCaseQ LowerLeftArrow LowerRightArrow LowerTriangularize LowerTriangularMatrixQ LowpassFilter LQEstimatorGains LQGRegulator LQOutputRegulatorGains LQRegulatorGains LUBackSubstitution LucasL LuccioSamiComponents LUDecomposition LunarEclipse LUVColor LyapunovSolve LyonsGroupLyMachineID MachineName MachineNumberQ MachinePrecision MacintoshSystemPageSetup Magenta Magnification Magnify MailAddressValidation MailExecute MailFolder MailItem MailReceiverFunction MailResponseFunction MailSearch MailServerConnect MailServerConnection MailSettings MainSolve MaintainDynamicCaches Majority MakeBoxes MakeExpression MakeRules ManagedLibraryExpressionID ManagedLibraryExpressionQ MandelbrotSetBoettcher MandelbrotSetDistance MandelbrotSetIterationCount MandelbrotSetMemberQ MandelbrotSetPlot MangoldtLambda ManhattanDistance Manipulate Manipulator MannedSpaceMissionData MannWhitneyTest MantissaExponent Manual Map MapAll MapAt MapIndexed MAProcess MapThread MarchenkoPasturDistribution MarcumQ MardiaCombinedTest MardiaKurtosisTest MardiaSkewnessTest MarginalDistribution MarkovProcessProperties Masking MatchingDissimilarity MatchLocalNameQ MatchLocalNames MatchQ Material MathematicalFunctionData MathematicaNotation MathieuC MathieuCharacteristicA MathieuCharacteristicB MathieuCharacteristicExponent MathieuCPrime MathieuGroupM11 MathieuGroupM12 MathieuGroupM22 MathieuGroupM23 MathieuGroupM24 MathieuS MathieuSPrime MathMLForm MathMLText Matrices MatrixExp MatrixForm MatrixFunction MatrixLog MatrixNormalDistribution MatrixPlot MatrixPower MatrixPropertyDistribution MatrixQ MatrixRank MatrixTDistribution Max MaxBend MaxCellMeasure MaxColorDistance MaxDetect MaxDuration MaxExtraBandwidths MaxExtraConditions MaxFeatureDisplacement MaxFeatures MaxFilter MaximalBy Maximize MaxItems MaxIterations MaxLimit MaxMemoryUsed MaxMixtureKernels MaxOverlapFraction MaxPlotPoints MaxPoints MaxRecursion MaxStableDistribution MaxStepFraction MaxSteps MaxStepSize MaxTrainingRounds MaxValue MaxwellDistribution MaxWordGap McLaughlinGroupMcL Mean MeanAbsoluteLossLayer MeanAround MeanClusteringCoefficient MeanDegreeConnectivity MeanDeviation MeanFilter MeanGraphDistance MeanNeighborDegree MeanShift MeanShiftFilter MeanSquaredLossLayer Median MedianDeviation MedianFilter MedicalTestData Medium MeijerG MeijerGReduce MeixnerDistribution MellinConvolve MellinTransform MemberQ MemoryAvailable MemoryConstrained MemoryConstraint MemoryInUse MengerMesh Menu MenuAppearance MenuCommandKey MenuEvaluator MenuItem MenuList MenuPacket MenuSortingValue MenuStyle MenuView Merge MergeDifferences MergingFunction MersennePrimeExponent MersennePrimeExponentQ Mesh MeshCellCentroid MeshCellCount MeshCellHighlight MeshCellIndex MeshCellLabel MeshCellMarker MeshCellMeasure MeshCellQuality MeshCells MeshCellShapeFunction MeshCellStyle MeshCoordinates MeshFunctions MeshPrimitives MeshQualityGoal MeshRange MeshRefinementFunction MeshRegion MeshRegionQ MeshShading MeshStyle Message MessageDialog MessageList MessageName MessageObject MessageOptions MessagePacket Messages MessagesNotebook MetaCharacters MetaInformation MeteorShowerData Method MethodOptions MexicanHatWavelet MeyerWavelet Midpoint Min MinColorDistance MinDetect MineralData MinFilter MinimalBy MinimalPolynomial MinimalStateSpaceModel Minimize MinimumTimeIncrement MinIntervalSize MinkowskiQuestionMark MinLimit MinMax MinorPlanetData Minors MinRecursion MinSize MinStableDistribution Minus MinusPlus MinValue Missing MissingBehavior MissingDataMethod MissingDataRules MissingQ MissingString MissingStyle MissingValuePattern MittagLefflerE MixedFractionParts MixedGraphQ MixedMagnitude MixedRadix MixedRadixQuantity MixedUnit MixtureDistribution Mod Modal Mode Modular ModularInverse ModularLambda Module Modulus MoebiusMu Molecule MoleculeContainsQ MoleculeEquivalentQ MoleculeGraph MoleculeModify MoleculePattern MoleculePlot MoleculePlot3D MoleculeProperty MoleculeQ MoleculeValue Moment Momentary MomentConvert MomentEvaluate MomentGeneratingFunction MomentOfInertia Monday Monitor MonomialList MonomialOrder MonsterGroupM MoonPhase MoonPosition MorletWavelet MorphologicalBinarize MorphologicalBranchPoints MorphologicalComponents MorphologicalEulerNumber MorphologicalGraph MorphologicalPerimeter MorphologicalTransform MortalityData Most MountainData MouseAnnotation MouseAppearance MouseAppearanceTag MouseButtons Mouseover MousePointerNote MousePosition MovieData MovingAverage MovingMap MovingMedian MoyalDistribution Multicolumn MultiedgeStyle MultigraphQ MultilaunchWarning MultiLetterItalics MultiLetterStyle MultilineFunction Multinomial MultinomialDistribution MultinormalDistribution MultiplicativeOrder Multiplicity MultiplySides Multiselection MultivariateHypergeometricDistribution MultivariatePoissonDistribution MultivariateTDistributionN NakagamiDistribution NameQ Names NamespaceBox NamespaceBoxOptions Nand NArgMax NArgMin NBernoulliB NBodySimulation NBodySimulationData NCache NDEigensystem NDEigenvalues NDSolve NDSolveValue Nearest NearestFunction NearestNeighborGraph NearestTo NebulaData NeedCurrentFrontEndPackagePacket NeedCurrentFrontEndSymbolsPacket NeedlemanWunschSimilarity Needs Negative NegativeBinomialDistribution NegativeDefiniteMatrixQ NegativeIntegers NegativeMultinomialDistribution NegativeRationals NegativeReals NegativeSemidefiniteMatrixQ NeighborhoodData NeighborhoodGraph Nest NestedGreaterGreater NestedLessLess NestedScriptRules NestGraph NestList NestWhile NestWhileList NetAppend NetBidirectionalOperator NetChain NetDecoder NetDelete NetDrop NetEncoder NetEvaluationMode NetExtract NetFlatten NetFoldOperator NetGraph NetInformation NetInitialize NetInsert NetInsertSharedArrays NetJoin NetMapOperator NetMapThreadOperator NetMeasurements NetModel NetNestOperator NetPairEmbeddingOperator NetPort NetPortGradient NetPrepend NetRename NetReplace NetReplacePart NetSharedArray NetStateObject NetTake NetTrain NetTrainResultsObject NetworkPacketCapture NetworkPacketRecording NetworkPacketRecordingDuring NetworkPacketTrace NeumannValue NevilleThetaC NevilleThetaD NevilleThetaN NevilleThetaS NewPrimitiveStyle NExpectation Next NextCell NextDate NextPrime NextScheduledTaskTime NHoldAll NHoldFirst NHoldRest NicholsGridLines NicholsPlot NightHemisphere NIntegrate NMaximize NMaxValue NMinimize NMinValue NominalVariables NonAssociative NoncentralBetaDistribution NoncentralChiSquareDistribution NoncentralFRatioDistribution NoncentralStudentTDistribution NonCommutativeMultiply NonConstants NondimensionalizationTransform None NoneTrue NonlinearModelFit NonlinearStateSpaceModel NonlocalMeansFilter NonNegative NonNegativeIntegers NonNegativeRationals NonNegativeReals NonPositive NonPositiveIntegers NonPositiveRationals NonPositiveReals Nor NorlundB Norm Normal NormalDistribution NormalGrouping NormalizationLayer Normalize Normalized NormalizedSquaredEuclideanDistance NormalMatrixQ NormalsFunction NormFunction Not NotCongruent NotCupCap NotDoubleVerticalBar Notebook NotebookApply NotebookAutoSave NotebookClose NotebookConvertSettings NotebookCreate NotebookCreateReturnObject NotebookDefault NotebookDelete NotebookDirectory NotebookDynamicExpression NotebookEvaluate NotebookEventActions NotebookFileName NotebookFind NotebookFindReturnObject NotebookGet NotebookGetLayoutInformationPacket NotebookGetMisspellingsPacket NotebookImport NotebookInformation NotebookInterfaceObject NotebookLocate NotebookObject NotebookOpen NotebookOpenReturnObject NotebookPath NotebookPrint NotebookPut NotebookPutReturnObject NotebookRead NotebookResetGeneratedCells Notebooks NotebookSave NotebookSaveAs NotebookSelection NotebookSetupLayoutInformationPacket NotebooksMenu NotebookTemplate NotebookWrite NotElement NotEqualTilde NotExists NotGreater NotGreaterEqual NotGreaterFullEqual NotGreaterGreater NotGreaterLess NotGreaterSlantEqual NotGreaterTilde Nothing NotHumpDownHump NotHumpEqual NotificationFunction NotLeftTriangle NotLeftTriangleBar NotLeftTriangleEqual NotLess NotLessEqual NotLessFullEqual NotLessGreater NotLessLess NotLessSlantEqual NotLessTilde NotNestedGreaterGreater NotNestedLessLess NotPrecedes NotPrecedesEqual NotPrecedesSlantEqual NotPrecedesTilde NotReverseElement NotRightTriangle NotRightTriangleBar NotRightTriangleEqual NotSquareSubset NotSquareSubsetEqual NotSquareSuperset NotSquareSupersetEqual NotSubset NotSubsetEqual NotSucceeds NotSucceedsEqual NotSucceedsSlantEqual NotSucceedsTilde NotSuperset NotSupersetEqual NotTilde NotTildeEqual NotTildeFullEqual NotTildeTilde NotVerticalBar Now NoWhitespace NProbability NProduct NProductFactors NRoots NSolve NSum NSumTerms NuclearExplosionData NuclearReactorData Null NullRecords NullSpace NullWords Number NumberCompose NumberDecompose NumberExpand NumberFieldClassNumber NumberFieldDiscriminant NumberFieldFundamentalUnits NumberFieldIntegralBasis NumberFieldNormRepresentatives NumberFieldRegulator NumberFieldRootsOfUnity NumberFieldSignature NumberForm NumberFormat NumberLinePlot NumberMarks NumberMultiplier NumberPadding NumberPoint NumberQ NumberSeparator NumberSigns NumberString Numerator NumeratorDenominator NumericalOrder NumericalSort NumericArray NumericArrayQ NumericArrayType NumericFunction NumericQ NuttallWindow NValues NyquistGridLines NyquistPlotO ObservabilityGramian ObservabilityMatrix ObservableDecomposition ObservableModelQ OceanData Octahedron OddQ Off Offset OLEData On ONanGroupON Once OneIdentity Opacity OpacityFunction OpacityFunctionScaling Open OpenAppend Opener OpenerBox OpenerBoxOptions OpenerView OpenFunctionInspectorPacket Opening OpenRead OpenSpecialOptions OpenTemporary OpenWrite Operate OperatingSystem OptimumFlowData Optional OptionalElement OptionInspectorSettings OptionQ Options OptionsPacket OptionsPattern OptionValue OptionValueBox OptionValueBoxOptions Or Orange Order OrderDistribution OrderedQ Ordering OrderingBy OrderingLayer Orderless OrderlessPatternSequence OrnsteinUhlenbeckProcess Orthogonalize OrthogonalMatrixQ Out Outer OuterPolygon OuterPolyhedron OutputAutoOverwrite OutputControllabilityMatrix OutputControllableModelQ OutputForm OutputFormData OutputGrouping OutputMathEditExpression OutputNamePacket OutputResponse OutputSizeLimit OutputStream Over OverBar OverDot Overflow OverHat Overlaps Overlay OverlayBox OverlayBoxOptions Overscript OverscriptBox OverscriptBoxOptions OverTilde OverVector OverwriteTarget OwenT OwnValuesPackage PackingMethod PaddedForm Padding PaddingLayer PaddingSize PadeApproximant PadLeft PadRight PageBreakAbove PageBreakBelow PageBreakWithin PageFooterLines PageFooters PageHeaderLines PageHeaders PageHeight PageRankCentrality PageTheme PageWidth Pagination PairedBarChart PairedHistogram PairedSmoothHistogram PairedTTest PairedZTest PaletteNotebook PalettePath PalindromeQ Pane PaneBox PaneBoxOptions Panel PanelBox PanelBoxOptions Paneled PaneSelector PaneSelectorBox PaneSelectorBoxOptions PaperWidth ParabolicCylinderD ParagraphIndent ParagraphSpacing ParallelArray ParallelCombine ParallelDo Parallelepiped ParallelEvaluate Parallelization Parallelize ParallelMap ParallelNeeds Parallelogram ParallelProduct ParallelSubmit ParallelSum ParallelTable ParallelTry Parameter ParameterEstimator ParameterMixtureDistribution ParameterVariables ParametricFunction ParametricNDSolve ParametricNDSolveValue ParametricPlot ParametricPlot3D ParametricRegion ParentBox ParentCell ParentConnect ParentDirectory ParentForm Parenthesize ParentList ParentNotebook ParetoDistribution ParetoPickandsDistribution ParkData Part PartBehavior PartialCorrelationFunction PartialD ParticleAcceleratorData ParticleData Partition PartitionGranularity PartitionsP PartitionsQ PartLayer PartOfSpeech PartProtection ParzenWindow PascalDistribution PassEventsDown PassEventsUp Paste PasteAutoQuoteCharacters PasteBoxFormInlineCells PasteButton Path PathGraph PathGraphQ Pattern PatternSequence PatternTest PauliMatrix PaulWavelet Pause PausedTime PDF PeakDetect PeanoCurve PearsonChiSquareTest PearsonCorrelationTest PearsonDistribution PercentForm PerfectNumber PerfectNumberQ PerformanceGoal Perimeter PeriodicBoundaryCondition PeriodicInterpolation Periodogram PeriodogramArray Permanent Permissions PermissionsGroup PermissionsGroupMemberQ PermissionsGroups PermissionsKey PermissionsKeys PermutationCycles PermutationCyclesQ PermutationGroup PermutationLength PermutationList PermutationListQ PermutationMax PermutationMin PermutationOrder PermutationPower PermutationProduct PermutationReplace Permutations PermutationSupport Permute PeronaMalikFilter Perpendicular PerpendicularBisector PersistenceLocation PersistenceTime PersistentObject PersistentObjects PersistentValue PersonData PERTDistribution PetersenGraph PhaseMargins PhaseRange PhysicalSystemData Pi Pick PIDData PIDDerivativeFilter PIDFeedforward PIDTune Piecewise PiecewiseExpand PieChart PieChart3D PillaiTrace PillaiTraceTest PingTime Pink PitchRecognize Pivoting PixelConstrained PixelValue PixelValuePositions Placed Placeholder PlaceholderReplace Plain PlanarAngle PlanarGraph PlanarGraphQ PlanckRadiationLaw PlaneCurveData PlanetaryMoonData PlanetData PlantData Play PlayRange Plot Plot3D Plot3Matrix PlotDivision PlotJoined PlotLabel PlotLabels PlotLayout PlotLegends PlotMarkers PlotPoints PlotRange PlotRangeClipping PlotRangeClipPlanesStyle PlotRangePadding PlotRegion PlotStyle PlotTheme Pluralize Plus PlusMinus Pochhammer PodStates PodWidth Point Point3DBox Point3DBoxOptions PointBox PointBoxOptions PointFigureChart PointLegend PointSize PoissonConsulDistribution PoissonDistribution PoissonProcess PoissonWindow PolarAxes PolarAxesOrigin PolarGridLines PolarPlot PolarTicks PoleZeroMarkers PolyaAeppliDistribution PolyGamma Polygon Polygon3DBox Polygon3DBoxOptions PolygonalNumber PolygonAngle PolygonBox PolygonBoxOptions PolygonCoordinates PolygonDecomposition PolygonHoleScale PolygonIntersections PolygonScale Polyhedron PolyhedronAngle PolyhedronCoordinates PolyhedronData PolyhedronDecomposition PolyhedronGenus PolyLog PolynomialExtendedGCD PolynomialForm PolynomialGCD PolynomialLCM PolynomialMod PolynomialQ PolynomialQuotient PolynomialQuotientRemainder PolynomialReduce PolynomialRemainder Polynomials PoolingLayer PopupMenu PopupMenuBox PopupMenuBoxOptions PopupView PopupWindow Position PositionIndex Positive PositiveDefiniteMatrixQ PositiveIntegers PositiveRationals PositiveReals PositiveSemidefiniteMatrixQ PossibleZeroQ Postfix PostScript Power PowerDistribution PowerExpand PowerMod PowerModList PowerRange PowerSpectralDensity PowersRepresentations PowerSymmetricPolynomial Precedence PrecedenceForm Precedes PrecedesEqual PrecedesSlantEqual PrecedesTilde Precision PrecisionGoal PreDecrement Predict PredictionRoot PredictorFunction PredictorInformation PredictorMeasurements PredictorMeasurementsObject PreemptProtect PreferencesPath Prefix PreIncrement Prepend PrependLayer PrependTo PreprocessingRules PreserveColor PreserveImageOptions Previous PreviousCell PreviousDate PriceGraphDistribution PrimaryPlaceholder Prime PrimeNu PrimeOmega PrimePi PrimePowerQ PrimeQ Primes PrimeZetaP PrimitivePolynomialQ PrimitiveRoot PrimitiveRootList PrincipalComponents PrincipalValue Print PrintableASCIIQ PrintAction PrintForm PrintingCopies PrintingOptions PrintingPageRange PrintingStartingPageNumber PrintingStyleEnvironment Printout3D Printout3DPreviewer PrintPrecision PrintTemporary Prism PrismBox PrismBoxOptions PrivateCellOptions PrivateEvaluationOptions PrivateFontOptions PrivateFrontEndOptions PrivateKey PrivateNotebookOptions PrivatePaths Probability ProbabilityDistribution ProbabilityPlot ProbabilityPr ProbabilityScalePlot ProbitModelFit ProcessConnection ProcessDirectory ProcessEnvironment Processes ProcessEstimator ProcessInformation ProcessObject ProcessParameterAssumptions ProcessParameterQ ProcessStateDomain ProcessStatus ProcessTimeDomain Product ProductDistribution ProductLog ProgressIndicator ProgressIndicatorBox ProgressIndicatorBoxOptions Projection Prolog PromptForm ProofObject Properties Property PropertyList PropertyValue Proportion Proportional Protect Protected ProteinData Pruning PseudoInverse PsychrometricPropertyData PublicKey PublisherID PulsarData PunctuationCharacter Purple Put PutAppend Pyramid PyramidBox PyramidBoxOptionsQBinomial QFactorial QGamma QHypergeometricPFQ QnDispersion QPochhammer QPolyGamma QRDecomposition QuadraticIrrationalQ QuadraticOptimization Quantile QuantilePlot Quantity QuantityArray QuantityDistribution QuantityForm QuantityMagnitude QuantityQ QuantityUnit QuantityVariable QuantityVariableCanonicalUnit QuantityVariableDimensions QuantityVariableIdentifier QuantityVariablePhysicalQuantity Quartics QuartileDeviation Quartiles QuartileSkewness Query QueueingNetworkProcess QueueingProcess QueueProperties Quiet Quit Quotient QuotientRemainderRadialGradientImage RadialityCentrality RadicalBox RadicalBoxOptions RadioButton RadioButtonBar RadioButtonBox RadioButtonBoxOptions Radon RadonTransform RamanujanTau RamanujanTauL RamanujanTauTheta RamanujanTauZ Ramp Random RandomChoice RandomColor RandomComplex RandomEntity RandomFunction RandomGeoPosition RandomGraph RandomImage RandomInstance RandomInteger RandomPermutation RandomPoint RandomPolygon RandomPolyhedron RandomPrime RandomReal RandomSample RandomSeed RandomSeeding RandomVariate RandomWalkProcess RandomWord Range RangeFilter RangeSpecification RankedMax RankedMin RarerProbability Raster Raster3D Raster3DBox Raster3DBoxOptions RasterArray RasterBox RasterBoxOptions Rasterize RasterSize Rational RationalFunctions Rationalize Rationals Ratios RawArray RawBoxes RawData RawMedium RayleighDistribution Re Read ReadByteArray ReadLine ReadList ReadProtected ReadString Real RealAbs RealBlockDiagonalForm RealDigits RealExponent Reals RealSign Reap RecognitionPrior RecognitionThreshold Record RecordLists RecordSeparators Rectangle RectangleBox RectangleBoxOptions RectangleChart RectangleChart3D RectangularRepeatingElement RecurrenceFilter RecurrenceTable RecurringDigitsForm Red Reduce RefBox ReferenceLineStyle ReferenceMarkers ReferenceMarkerStyle Refine ReflectionMatrix ReflectionTransform Refresh RefreshRate Region RegionBinarize RegionBoundary RegionBounds RegionCentroid RegionDifference RegionDimension RegionDisjoint RegionDistance RegionDistanceFunction RegionEmbeddingDimension RegionEqual RegionFunction RegionImage RegionIntersection RegionMeasure RegionMember RegionMemberFunction RegionMoment RegionNearest RegionNearestFunction RegionPlot RegionPlot3D RegionProduct RegionQ RegionResize RegionSize RegionSymmetricDifference RegionUnion RegionWithin RegisterExternalEvaluator RegularExpression Regularization RegularlySampledQ RegularPolygon ReIm ReImLabels ReImPlot ReImStyle Reinstall RelationalDatabase RelationGraph Release ReleaseHold ReliabilityDistribution ReliefImage ReliefPlot RemoteAuthorizationCaching RemoteConnect RemoteConnectionObject RemoteFile RemoteRun RemoteRunProcess Remove RemoveAlphaChannel RemoveAsynchronousTask RemoveAudioStream RemoveBackground RemoveChannelListener RemoveChannelSubscribers Removed RemoveDiacritics RemoveInputStreamMethod RemoveOutputStreamMethod RemoveProperty RemoveScheduledTask RemoveUsers RenameDirectory RenameFile RenderAll RenderingOptions RenewalProcess RenkoChart RepairMesh Repeated RepeatedNull RepeatedString RepeatedTiming RepeatingElement Replace ReplaceAll ReplaceHeldPart ReplaceImageValue ReplaceList ReplacePart ReplacePixelValue ReplaceRepeated ReplicateLayer RequiredPhysicalQuantities Resampling ResamplingAlgorithmData ResamplingMethod Rescale RescalingTransform ResetDirectory ResetMenusPacket ResetScheduledTask ReshapeLayer Residue ResizeLayer Resolve ResourceAcquire ResourceData ResourceFunction ResourceObject ResourceRegister ResourceRemove ResourceSearch ResourceSubmissionObject ResourceSubmit ResourceSystemBase ResourceUpdate ResponseForm Rest RestartInterval Restricted Resultant ResumePacket Return ReturnEntersInput ReturnExpressionPacket ReturnInputFormPacket ReturnPacket ReturnReceiptFunction ReturnTextPacket Reverse ReverseBiorthogonalSplineWavelet ReverseElement ReverseEquilibrium ReverseGraph ReverseSort ReverseSortBy ReverseUpEquilibrium RevolutionAxis RevolutionPlot3D RGBColor RiccatiSolve RiceDistribution RidgeFilter RiemannR RiemannSiegelTheta RiemannSiegelZ RiemannXi Riffle Right RightArrow RightArrowBar RightArrowLeftArrow RightComposition RightCosetRepresentative RightDownTeeVector RightDownVector RightDownVectorBar RightTee RightTeeArrow RightTeeVector RightTriangle RightTriangleBar RightTriangleEqual RightUpDownVector RightUpTeeVector RightUpVector RightUpVectorBar RightVector RightVectorBar RiskAchievementImportance RiskReductionImportance RogersTanimotoDissimilarity RollPitchYawAngles RollPitchYawMatrix RomanNumeral Root RootApproximant RootIntervals RootLocusPlot RootMeanSquare RootOfUnityQ RootReduce Roots RootSum Rotate RotateLabel RotateLeft RotateRight RotationAction RotationBox RotationBoxOptions RotationMatrix RotationTransform Round RoundImplies RoundingRadius Row RowAlignments RowBackgrounds RowBox RowHeights RowLines RowMinHeight RowReduce RowsEqual RowSpacings RSolve RSolveValue RudinShapiro RudvalisGroupRu Rule RuleCondition RuleDelayed RuleForm RulePlot RulerUnits Run RunProcess RunScheduledTask RunThrough RuntimeAttributes RuntimeOptions RussellRaoDissimilaritySameQ SameTest SampledEntityClass SampleDepth SampledSoundFunction SampledSoundList SampleRate SamplingPeriod SARIMAProcess SARMAProcess SASTriangle SatelliteData SatisfiabilityCount SatisfiabilityInstances SatisfiableQ Saturday Save Saveable SaveAutoDelete SaveConnection SaveDefinitions SavitzkyGolayMatrix SawtoothWave Scale Scaled ScaleDivisions ScaledMousePosition ScaleOrigin ScalePadding ScaleRanges ScaleRangeStyle ScalingFunctions ScalingMatrix ScalingTransform Scan ScheduledTask ScheduledTaskActiveQ ScheduledTaskInformation ScheduledTaskInformationData ScheduledTaskObject ScheduledTasks SchurDecomposition ScientificForm ScientificNotationThreshold ScorerGi ScorerGiPrime ScorerHi ScorerHiPrime ScreenRectangle ScreenStyleEnvironment ScriptBaselineShifts ScriptForm ScriptLevel ScriptMinSize ScriptRules ScriptSizeMultipliers Scrollbars ScrollingOptions ScrollPosition SearchAdjustment SearchIndexObject SearchIndices SearchQueryString SearchResultObject Sec Sech SechDistribution SecondOrderConeOptimization SectionGrouping SectorChart SectorChart3D SectorOrigin SectorSpacing SecuredAuthenticationKey SecuredAuthenticationKeys SeedRandom Select Selectable SelectComponents SelectedCells SelectedNotebook SelectFirst Selection SelectionAnimate SelectionCell SelectionCellCreateCell SelectionCellDefaultStyle SelectionCellParentStyle SelectionCreateCell SelectionDebuggerTag SelectionDuplicateCell SelectionEvaluate SelectionEvaluateCreateCell SelectionMove SelectionPlaceholder SelectionSetStyle SelectWithContents SelfLoops SelfLoopStyle SemanticImport SemanticImportString SemanticInterpretation SemialgebraicComponentInstances SemidefiniteOptimization SendMail SendMessage Sequence SequenceAlignment SequenceAttentionLayer SequenceCases SequenceCount SequenceFold SequenceFoldList SequenceForm SequenceHold SequenceLastLayer SequenceMostLayer SequencePosition SequencePredict SequencePredictorFunction SequenceReplace SequenceRestLayer SequenceReverseLayer SequenceSplit Series SeriesCoefficient SeriesData ServiceConnect ServiceDisconnect ServiceExecute ServiceObject ServiceRequest ServiceResponse ServiceSubmit SessionSubmit SessionTime Set SetAccuracy SetAlphaChannel SetAttributes Setbacks SetBoxFormNamesPacket SetCloudDirectory SetCookies SetDelayed SetDirectory SetEnvironment SetEvaluationNotebook SetFileDate SetFileLoadingContext SetNotebookStatusLine SetOptions SetOptionsPacket SetPermissions SetPrecision SetProperty SetSecuredAuthenticationKey SetSelectedNotebook SetSharedFunction SetSharedVariable SetSpeechParametersPacket SetStreamPosition SetSystemModel SetSystemOptions Setter SetterBar SetterBox SetterBoxOptions Setting SetUsers SetValue Shading Shallow ShannonWavelet ShapiroWilkTest Share SharingList Sharpen ShearingMatrix ShearingTransform ShellRegion ShenCastanMatrix ShiftedGompertzDistribution ShiftRegisterSequence Short ShortDownArrow Shortest ShortestMatch ShortestPathFunction ShortLeftArrow ShortRightArrow ShortTimeFourier ShortTimeFourierData ShortUpArrow Show ShowAutoConvert ShowAutoSpellCheck ShowAutoStyles ShowCellBracket ShowCellLabel ShowCellTags ShowClosedCellArea ShowCodeAssist ShowContents ShowControls ShowCursorTracker ShowGroupOpenCloseIcon ShowGroupOpener ShowInvisibleCharacters ShowPageBreaks ShowPredictiveInterface ShowSelection ShowShortBoxForm ShowSpecialCharacters ShowStringCharacters ShowSyntaxStyles ShrinkingDelay ShrinkWrapBoundingBox SiderealTime SiegelTheta SiegelTukeyTest SierpinskiCurve SierpinskiMesh Sign Signature SignedRankTest SignedRegionDistance SignificanceLevel SignPadding SignTest SimilarityRules SimpleGraph SimpleGraphQ SimplePolygonQ SimplePolyhedronQ Simplex Simplify Sin Sinc SinghMaddalaDistribution SingleEvaluation SingleLetterItalics SingleLetterStyle SingularValueDecomposition SingularValueList SingularValuePlot SingularValues Sinh SinhIntegral SinIntegral SixJSymbol Skeleton SkeletonTransform SkellamDistribution Skewness SkewNormalDistribution SkinStyle Skip SliceContourPlot3D SliceDensityPlot3D SliceDistribution SliceVectorPlot3D Slider Slider2D Slider2DBox Slider2DBoxOptions SliderBox SliderBoxOptions SlideView Slot SlotSequence Small SmallCircle Smaller SmithDecomposition SmithDelayCompensator SmithWatermanSimilarity SmoothDensityHistogram SmoothHistogram SmoothHistogram3D SmoothKernelDistribution SnDispersion Snippet SnubPolyhedron SocialMediaData Socket SocketConnect SocketListen SocketListener SocketObject SocketOpen SocketReadMessage SocketReadyQ Sockets SocketWaitAll SocketWaitNext SoftmaxLayer SokalSneathDissimilarity SolarEclipse SolarSystemFeatureData SolidAngle SolidData SolidRegionQ Solve SolveAlways SolveDelayed Sort SortBy SortedBy SortedEntityClass Sound SoundAndGraphics SoundNote SoundVolume SourceLink Sow Space SpaceCurveData SpaceForm Spacer Spacings Span SpanAdjustments SpanCharacterRounding SpanFromAbove SpanFromBoth SpanFromLeft SpanLineThickness SpanMaxSize SpanMinSize SpanningCharacters SpanSymmetric SparseArray SpatialGraphDistribution SpatialMedian SpatialTransformationLayer Speak SpeakTextPacket SpearmanRankTest SpearmanRho SpeciesData SpecificityGoal SpectralLineData Spectrogram SpectrogramArray Specularity SpeechRecognize SpeechSynthesize SpellingCorrection SpellingCorrectionList SpellingDictionaries SpellingDictionariesPath SpellingOptions SpellingSuggestionsPacket Sphere SphereBox SpherePoints SphericalBesselJ SphericalBesselY SphericalHankelH1 SphericalHankelH2 SphericalHarmonicY SphericalPlot3D SphericalRegion SphericalShell SpheroidalEigenvalue SpheroidalJoiningFactor SpheroidalPS SpheroidalPSPrime SpheroidalQS SpheroidalQSPrime SpheroidalRadialFactor SpheroidalS1 SpheroidalS1Prime SpheroidalS2 SpheroidalS2Prime Splice SplicedDistribution SplineClosed SplineDegree SplineKnots SplineWeights Split SplitBy SpokenString Sqrt SqrtBox SqrtBoxOptions Square SquaredEuclideanDistance SquareFreeQ SquareIntersection SquareMatrixQ SquareRepeatingElement SquaresR SquareSubset SquareSubsetEqual SquareSuperset SquareSupersetEqual SquareUnion SquareWave SSSTriangle StabilityMargins StabilityMarginsStyle StableDistribution Stack StackBegin StackComplete StackedDateListPlot StackedListPlot StackInhibit StadiumShape StandardAtmosphereData StandardDeviation StandardDeviationFilter StandardForm Standardize Standardized StandardOceanData StandbyDistribution Star StarClusterData StarData StarGraph StartAsynchronousTask StartExternalSession StartingStepSize StartOfLine StartOfString StartProcess StartScheduledTask StartupSound StartWebSession StateDimensions StateFeedbackGains StateOutputEstimator StateResponse StateSpaceModel StateSpaceRealization StateSpaceTransform StateTransformationLinearize StationaryDistribution StationaryWaveletPacketTransform StationaryWaveletTransform StatusArea StatusCentrality StepMonitor StereochemistryElements StieltjesGamma StirlingS1 StirlingS2 StopAsynchronousTask StoppingPowerData StopScheduledTask StrataVariables StratonovichProcess StreamColorFunction StreamColorFunctionScaling StreamDensityPlot StreamMarkers StreamPlot StreamPoints StreamPosition Streams StreamScale StreamStyle String StringBreak StringByteCount StringCases StringContainsQ StringCount StringDelete StringDrop StringEndsQ StringExpression StringExtract StringForm StringFormat StringFreeQ StringInsert StringJoin StringLength StringMatchQ StringPadLeft StringPadRight StringPart StringPartition StringPosition StringQ StringRepeat StringReplace StringReplaceList StringReplacePart StringReverse StringRiffle StringRotateLeft StringRotateRight StringSkeleton StringSplit StringStartsQ StringTake StringTemplate StringToByteArray StringToStream StringTrim StripBoxes StripOnInput StripWrapperBoxes StrokeForm StructuralImportance StructuredArray StructuredSelection StruveH StruveL Stub StudentTDistribution Style StyleBox StyleBoxAutoDelete StyleData StyleDefinitions StyleForm StyleHints StyleKeyMapping StyleMenuListing StyleNameDialogSettings StyleNames StylePrint StyleSheetPath Subdivide Subfactorial Subgraph SubMinus SubPlus SubresultantPolynomialRemainders SubresultantPolynomials Subresultants Subscript SubscriptBox SubscriptBoxOptions Subscripted Subsequences Subset SubsetEqual SubsetMap SubsetQ Subsets SubStar SubstitutionSystem Subsuperscript SubsuperscriptBox SubsuperscriptBoxOptions Subtract SubtractFrom SubtractSides SubValues Succeeds SucceedsEqual SucceedsSlantEqual SucceedsTilde Success SuchThat Sum SumConvergence SummationLayer Sunday SunPosition Sunrise Sunset SuperDagger SuperMinus SupernovaData SuperPlus Superscript SuperscriptBox SuperscriptBoxOptions Superset SupersetEqual SuperStar Surd SurdForm SurfaceArea SurfaceColor SurfaceData SurfaceGraphics SurvivalDistribution SurvivalFunction SurvivalModel SurvivalModelFit SuspendPacket SuzukiDistribution SuzukiGroupSuz SwatchLegend Switch Symbol SymbolName SymletWavelet Symmetric 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TouchPosition TouchscreenAutoZoom TouchscreenControlPlacement ToUpperCase Tr Trace TraceAbove TraceAction TraceBackward TraceDepth TraceDialog TraceForward TraceInternal TraceLevel TraceOff TraceOn TraceOriginal TracePrint TraceScan TrackedSymbols TrackingFunction TracyWidomDistribution TradingChart TraditionalForm TraditionalFunctionNotation TraditionalNotation TraditionalOrder TrainingProgressCheckpointing TrainingProgressFunction TrainingProgressMeasurements TrainingProgressReporting TrainingStoppingCriterion TransferFunctionCancel TransferFunctionExpand TransferFunctionFactor TransferFunctionModel TransferFunctionPoles TransferFunctionTransform TransferFunctionZeros TransformationClass TransformationFunction TransformationFunctions TransformationMatrix TransformedDistribution TransformedField TransformedProcess TransformedRegion TransitionDirection TransitionDuration TransitionEffect TransitiveClosureGraph TransitiveReductionGraph Translate TranslationOptions TranslationTransform 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ds_grid_clear ds_grid_set ds_grid_add ds_grid_multiply ds_grid_set_region ds_grid_add_region ds_grid_multiply_region ds_grid_set_disk ds_grid_add_disk ds_grid_multiply_disk ds_grid_set_grid_region ds_grid_add_grid_region ds_grid_multiply_grid_region ds_grid_get ds_grid_get_sum ds_grid_get_max ds_grid_get_min ds_grid_get_mean ds_grid_get_disk_sum ds_grid_get_disk_min ds_grid_get_disk_max ds_grid_get_disk_mean ds_grid_value_exists ds_grid_value_x ds_grid_value_y ds_grid_value_disk_exists ds_grid_value_disk_x ds_grid_value_disk_y ds_grid_shuffle ds_grid_write ds_grid_read ds_grid_sort ds_grid_set ds_grid_get effect_create_below effect_create_above effect_clear part_type_create part_type_destroy part_type_exists part_type_clear part_type_shape part_type_sprite part_type_size part_type_scale part_type_orientation part_type_life part_type_step part_type_death part_type_speed part_type_direction part_type_gravity part_type_colour1 part_type_colour2 part_type_colour3 part_type_colour_mix part_type_colour_rgb part_type_colour_hsv part_type_color1 part_type_color2 part_type_color3 part_type_color_mix part_type_color_rgb part_type_color_hsv part_type_alpha1 part_type_alpha2 part_type_alpha3 part_type_blend part_system_create part_system_create_layer part_system_destroy part_system_exists part_system_clear part_system_draw_order part_system_depth part_system_position part_system_automatic_update part_system_automatic_draw part_system_update part_system_drawit part_system_get_layer part_system_layer part_particles_create part_particles_create_colour part_particles_create_color part_particles_clear part_particles_count part_emitter_create part_emitter_destroy part_emitter_destroy_all part_emitter_exists part_emitter_clear part_emitter_region part_emitter_burst part_emitter_stream external_call external_define external_free window_handle window_device matrix_get matrix_set matrix_build_identity matrix_build matrix_build_lookat matrix_build_projection_ortho matrix_build_projection_perspective matrix_build_projection_perspective_fov matrix_multiply matrix_transform_vertex matrix_stack_push matrix_stack_pop matrix_stack_multiply matrix_stack_set matrix_stack_clear matrix_stack_top matrix_stack_is_empty browser_input_capture os_get_config os_get_info os_get_language os_get_region os_lock_orientation display_get_dpi_x display_get_dpi_y display_set_gui_size display_set_gui_maximise display_set_gui_maximize device_mouse_dbclick_enable display_set_timing_method display_get_timing_method display_set_sleep_margin display_get_sleep_margin virtual_key_add virtual_key_hide virtual_key_delete virtual_key_show draw_enable_drawevent draw_enable_swf_aa draw_set_swf_aa_level draw_get_swf_aa_level draw_texture_flush draw_flush gpu_set_blendenable gpu_set_ztestenable gpu_set_zfunc gpu_set_zwriteenable gpu_set_lightingenable gpu_set_fog gpu_set_cullmode gpu_set_blendmode gpu_set_blendmode_ext gpu_set_blendmode_ext_sepalpha gpu_set_colorwriteenable gpu_set_colourwriteenable gpu_set_alphatestenable gpu_set_alphatestref gpu_set_alphatestfunc gpu_set_texfilter gpu_set_texfilter_ext gpu_set_texrepeat gpu_set_texrepeat_ext gpu_set_tex_filter gpu_set_tex_filter_ext gpu_set_tex_repeat gpu_set_tex_repeat_ext gpu_set_tex_mip_filter gpu_set_tex_mip_filter_ext gpu_set_tex_mip_bias gpu_set_tex_mip_bias_ext gpu_set_tex_min_mip gpu_set_tex_min_mip_ext gpu_set_tex_max_mip gpu_set_tex_max_mip_ext gpu_set_tex_max_aniso gpu_set_tex_max_aniso_ext gpu_set_tex_mip_enable gpu_set_tex_mip_enable_ext gpu_get_blendenable gpu_get_ztestenable gpu_get_zfunc gpu_get_zwriteenable gpu_get_lightingenable gpu_get_fog gpu_get_cullmode gpu_get_blendmode gpu_get_blendmode_ext gpu_get_blendmode_ext_sepalpha gpu_get_blendmode_src gpu_get_blendmode_dest gpu_get_blendmode_srcalpha gpu_get_blendmode_destalpha gpu_get_colorwriteenable gpu_get_colourwriteenable gpu_get_alphatestenable gpu_get_alphatestref gpu_get_alphatestfunc gpu_get_texfilter gpu_get_texfilter_ext gpu_get_texrepeat gpu_get_texrepeat_ext gpu_get_tex_filter gpu_get_tex_filter_ext gpu_get_tex_repeat gpu_get_tex_repeat_ext gpu_get_tex_mip_filter gpu_get_tex_mip_filter_ext gpu_get_tex_mip_bias gpu_get_tex_mip_bias_ext gpu_get_tex_min_mip gpu_get_tex_min_mip_ext gpu_get_tex_max_mip gpu_get_tex_max_mip_ext gpu_get_tex_max_aniso gpu_get_tex_max_aniso_ext gpu_get_tex_mip_enable gpu_get_tex_mip_enable_ext gpu_push_state gpu_pop_state gpu_get_state gpu_set_state draw_light_define_ambient draw_light_define_direction draw_light_define_point draw_light_enable draw_set_lighting draw_light_get_ambient draw_light_get draw_get_lighting shop_leave_rating url_get_domain url_open url_open_ext url_open_full get_timer achievement_login achievement_logout achievement_post achievement_increment achievement_post_score achievement_available achievement_show_achievements achievement_show_leaderboards achievement_load_friends achievement_load_leaderboard achievement_send_challenge achievement_load_progress achievement_reset achievement_login_status achievement_get_pic achievement_show_challenge_notifications achievement_get_challenges achievement_event achievement_show achievement_get_info cloud_file_save cloud_string_save cloud_synchronise ads_enable ads_disable ads_setup ads_engagement_launch ads_engagement_available ads_engagement_active ads_event ads_event_preload ads_set_reward_callback ads_get_display_height ads_get_display_width ads_move ads_interstitial_available ads_interstitial_display device_get_tilt_x device_get_tilt_y device_get_tilt_z device_is_keypad_open device_mouse_check_button device_mouse_check_button_pressed device_mouse_check_button_released device_mouse_x device_mouse_y device_mouse_raw_x device_mouse_raw_y device_mouse_x_to_gui device_mouse_y_to_gui iap_activate iap_status iap_enumerate_products iap_restore_all iap_acquire iap_consume iap_product_details iap_purchase_details facebook_init facebook_login facebook_status facebook_graph_request facebook_dialog facebook_logout facebook_launch_offerwall facebook_post_message facebook_send_invite facebook_user_id facebook_accesstoken facebook_check_permission facebook_request_read_permissions facebook_request_publish_permissions gamepad_is_supported gamepad_get_device_count gamepad_is_connected gamepad_get_description gamepad_get_button_threshold gamepad_set_button_threshold gamepad_get_axis_deadzone gamepad_set_axis_deadzone gamepad_button_count gamepad_button_check gamepad_button_check_pressed gamepad_button_check_released gamepad_button_value gamepad_axis_count gamepad_axis_value gamepad_set_vibration gamepad_set_colour gamepad_set_color os_is_paused window_has_focus code_is_compiled http_get http_get_file http_post_string http_request json_encode json_decode zip_unzip load_csv base64_encode base64_decode md5_string_unicode md5_string_utf8 md5_file os_is_network_connected sha1_string_unicode sha1_string_utf8 sha1_file os_powersave_enable analytics_event analytics_event_ext win8_livetile_tile_notification win8_livetile_tile_clear win8_livetile_badge_notification win8_livetile_badge_clear win8_livetile_queue_enable win8_secondarytile_pin win8_secondarytile_badge_notification win8_secondarytile_delete win8_livetile_notification_begin win8_livetile_notification_secondary_begin win8_livetile_notification_expiry win8_livetile_notification_tag win8_livetile_notification_text_add win8_livetile_notification_image_add win8_livetile_notification_end win8_appbar_enable win8_appbar_add_element win8_appbar_remove_element win8_settingscharm_add_entry win8_settingscharm_add_html_entry win8_settingscharm_add_xaml_entry win8_settingscharm_set_xaml_property win8_settingscharm_get_xaml_property win8_settingscharm_remove_entry win8_share_image win8_share_screenshot win8_share_file win8_share_url win8_share_text win8_search_enable win8_search_disable win8_search_add_suggestions win8_device_touchscreen_available win8_license_initialize_sandbox win8_license_trial_version winphone_license_trial_version winphone_tile_title winphone_tile_count winphone_tile_back_title winphone_tile_back_content winphone_tile_back_content_wide winphone_tile_front_image winphone_tile_front_image_small winphone_tile_front_image_wide winphone_tile_back_image winphone_tile_back_image_wide winphone_tile_background_colour winphone_tile_background_color winphone_tile_icon_image winphone_tile_small_icon_image winphone_tile_wide_content winphone_tile_cycle_images winphone_tile_small_background_image physics_world_create physics_world_gravity physics_world_update_speed physics_world_update_iterations physics_world_draw_debug physics_pause_enable physics_fixture_create physics_fixture_set_kinematic physics_fixture_set_density physics_fixture_set_awake physics_fixture_set_restitution physics_fixture_set_friction physics_fixture_set_collision_group physics_fixture_set_sensor physics_fixture_set_linear_damping physics_fixture_set_angular_damping physics_fixture_set_circle_shape physics_fixture_set_box_shape physics_fixture_set_edge_shape physics_fixture_set_polygon_shape physics_fixture_set_chain_shape physics_fixture_add_point physics_fixture_bind physics_fixture_bind_ext physics_fixture_delete physics_apply_force physics_apply_impulse physics_apply_angular_impulse physics_apply_local_force physics_apply_local_impulse physics_apply_torque physics_mass_properties physics_draw_debug physics_test_overlap physics_remove_fixture physics_set_friction physics_set_density physics_set_restitution physics_get_friction physics_get_density physics_get_restitution physics_joint_distance_create physics_joint_rope_create physics_joint_revolute_create physics_joint_prismatic_create physics_joint_pulley_create physics_joint_wheel_create physics_joint_weld_create physics_joint_friction_create physics_joint_gear_create physics_joint_enable_motor physics_joint_get_value physics_joint_set_value physics_joint_delete physics_particle_create physics_particle_delete physics_particle_delete_region_circle physics_particle_delete_region_box physics_particle_delete_region_poly physics_particle_set_flags physics_particle_set_category_flags physics_particle_draw physics_particle_draw_ext physics_particle_count physics_particle_get_data physics_particle_get_data_particle physics_particle_group_begin physics_particle_group_circle physics_particle_group_box physics_particle_group_polygon physics_particle_group_add_point physics_particle_group_end physics_particle_group_join physics_particle_group_delete physics_particle_group_count physics_particle_group_get_data physics_particle_group_get_mass physics_particle_group_get_inertia physics_particle_group_get_centre_x physics_particle_group_get_centre_y physics_particle_group_get_vel_x physics_particle_group_get_vel_y physics_particle_group_get_ang_vel physics_particle_group_get_x physics_particle_group_get_y physics_particle_group_get_angle physics_particle_set_group_flags physics_particle_get_group_flags physics_particle_get_max_count physics_particle_get_radius physics_particle_get_density physics_particle_get_damping physics_particle_get_gravity_scale physics_particle_set_max_count physics_particle_set_radius physics_particle_set_density physics_particle_set_damping physics_particle_set_gravity_scale network_create_socket network_create_socket_ext network_create_server network_create_server_raw network_connect network_connect_raw network_send_packet network_send_raw network_send_broadcast network_send_udp network_send_udp_raw network_set_timeout network_set_config network_resolve network_destroy buffer_create buffer_write buffer_read buffer_seek buffer_get_surface buffer_set_surface buffer_delete buffer_exists buffer_get_type buffer_get_alignment buffer_poke buffer_peek buffer_save buffer_save_ext buffer_load buffer_load_ext buffer_load_partial buffer_copy buffer_fill buffer_get_size buffer_tell buffer_resize buffer_md5 buffer_sha1 buffer_base64_encode buffer_base64_decode buffer_base64_decode_ext buffer_sizeof buffer_get_address buffer_create_from_vertex_buffer buffer_create_from_vertex_buffer_ext buffer_copy_from_vertex_buffer buffer_async_group_begin buffer_async_group_option buffer_async_group_end buffer_load_async buffer_save_async gml_release_mode gml_pragma steam_activate_overlay steam_is_overlay_enabled steam_is_overlay_activated steam_get_persona_name steam_initialised steam_is_cloud_enabled_for_app steam_is_cloud_enabled_for_account steam_file_persisted steam_get_quota_total steam_get_quota_free steam_file_write steam_file_write_file steam_file_read steam_file_delete steam_file_exists steam_file_size steam_file_share steam_is_screenshot_requested steam_send_screenshot steam_is_user_logged_on steam_get_user_steam_id steam_user_owns_dlc steam_user_installed_dlc steam_set_achievement steam_get_achievement steam_clear_achievement steam_set_stat_int steam_set_stat_float steam_set_stat_avg_rate steam_get_stat_int steam_get_stat_float steam_get_stat_avg_rate steam_reset_all_stats steam_reset_all_stats_achievements steam_stats_ready steam_create_leaderboard steam_upload_score steam_upload_score_ext steam_download_scores_around_user steam_download_scores steam_download_friends_scores steam_upload_score_buffer steam_upload_score_buffer_ext steam_current_game_language steam_available_languages steam_activate_overlay_browser steam_activate_overlay_user steam_activate_overlay_store steam_get_user_persona_name steam_get_app_id steam_get_user_account_id steam_ugc_download steam_ugc_create_item steam_ugc_start_item_update steam_ugc_set_item_title steam_ugc_set_item_description steam_ugc_set_item_visibility steam_ugc_set_item_tags steam_ugc_set_item_content steam_ugc_set_item_preview steam_ugc_submit_item_update steam_ugc_get_item_update_progress steam_ugc_subscribe_item steam_ugc_unsubscribe_item steam_ugc_num_subscribed_items steam_ugc_get_subscribed_items steam_ugc_get_item_install_info steam_ugc_get_item_update_info steam_ugc_request_item_details steam_ugc_create_query_user steam_ugc_create_query_user_ex steam_ugc_create_query_all steam_ugc_create_query_all_ex steam_ugc_query_set_cloud_filename_filter steam_ugc_query_set_match_any_tag steam_ugc_query_set_search_text steam_ugc_query_set_ranked_by_trend_days steam_ugc_query_add_required_tag steam_ugc_query_add_excluded_tag steam_ugc_query_set_return_long_description steam_ugc_query_set_return_total_only steam_ugc_query_set_allow_cached_response steam_ugc_send_query shader_set shader_get_name shader_reset shader_current shader_is_compiled shader_get_sampler_index shader_get_uniform shader_set_uniform_i shader_set_uniform_i_array shader_set_uniform_f shader_set_uniform_f_array shader_set_uniform_matrix shader_set_uniform_matrix_array shader_enable_corner_id texture_set_stage texture_get_texel_width texture_get_texel_height shaders_are_supported vertex_format_begin vertex_format_end vertex_format_delete vertex_format_add_position vertex_format_add_position_3d vertex_format_add_colour vertex_format_add_color vertex_format_add_normal vertex_format_add_texcoord vertex_format_add_textcoord vertex_format_add_custom vertex_create_buffer vertex_create_buffer_ext vertex_delete_buffer vertex_begin vertex_end vertex_position vertex_position_3d vertex_colour vertex_color vertex_argb vertex_texcoord vertex_normal vertex_float1 vertex_float2 vertex_float3 vertex_float4 vertex_ubyte4 vertex_submit vertex_freeze vertex_get_number vertex_get_buffer_size vertex_create_buffer_from_buffer vertex_create_buffer_from_buffer_ext push_local_notification push_get_first_local_notification push_get_next_local_notification push_cancel_local_notification skeleton_animation_set skeleton_animation_get skeleton_animation_mix skeleton_animation_set_ext skeleton_animation_get_ext skeleton_animation_get_duration skeleton_animation_get_frames skeleton_animation_clear skeleton_skin_set skeleton_skin_get skeleton_attachment_set skeleton_attachment_get skeleton_attachment_create skeleton_collision_draw_set skeleton_bone_data_get skeleton_bone_data_set skeleton_bone_state_get skeleton_bone_state_set skeleton_get_minmax skeleton_get_num_bounds skeleton_get_bounds skeleton_animation_get_frame skeleton_animation_set_frame draw_skeleton draw_skeleton_time draw_skeleton_instance draw_skeleton_collision skeleton_animation_list skeleton_skin_list skeleton_slot_data layer_get_id layer_get_id_at_depth layer_get_depth layer_create layer_destroy layer_destroy_instances layer_add_instance layer_has_instance layer_set_visible layer_get_visible layer_exists layer_x layer_y layer_get_x layer_get_y layer_hspeed layer_vspeed layer_get_hspeed layer_get_vspeed layer_script_begin layer_script_end layer_shader layer_get_script_begin layer_get_script_end layer_get_shader layer_set_target_room layer_get_target_room layer_reset_target_room layer_get_all layer_get_all_elements layer_get_name layer_depth layer_get_element_layer layer_get_element_type layer_element_move layer_force_draw_depth layer_is_draw_depth_forced layer_get_forced_depth layer_background_get_id layer_background_exists layer_background_create layer_background_destroy layer_background_visible layer_background_change layer_background_sprite layer_background_htiled layer_background_vtiled layer_background_stretch layer_background_yscale layer_background_xscale layer_background_blend layer_background_alpha layer_background_index layer_background_speed layer_background_get_visible layer_background_get_sprite layer_background_get_htiled layer_background_get_vtiled layer_background_get_stretch layer_background_get_yscale layer_background_get_xscale layer_background_get_blend layer_background_get_alpha layer_background_get_index layer_background_get_speed layer_sprite_get_id layer_sprite_exists layer_sprite_create layer_sprite_destroy layer_sprite_change layer_sprite_index layer_sprite_speed layer_sprite_xscale layer_sprite_yscale layer_sprite_angle layer_sprite_blend layer_sprite_alpha layer_sprite_x layer_sprite_y layer_sprite_get_sprite layer_sprite_get_index layer_sprite_get_speed layer_sprite_get_xscale layer_sprite_get_yscale layer_sprite_get_angle layer_sprite_get_blend layer_sprite_get_alpha layer_sprite_get_x layer_sprite_get_y layer_tilemap_get_id layer_tilemap_exists layer_tilemap_create layer_tilemap_destroy tilemap_tileset tilemap_x tilemap_y tilemap_set tilemap_set_at_pixel tilemap_get_tileset tilemap_get_tile_width tilemap_get_tile_height tilemap_get_width tilemap_get_height tilemap_get_x tilemap_get_y tilemap_get tilemap_get_at_pixel tilemap_get_cell_x_at_pixel tilemap_get_cell_y_at_pixel tilemap_clear draw_tilemap draw_tile tilemap_set_global_mask tilemap_get_global_mask tilemap_set_mask tilemap_get_mask tilemap_get_frame tile_set_empty tile_set_index tile_set_flip tile_set_mirror tile_set_rotate tile_get_empty tile_get_index tile_get_flip tile_get_mirror tile_get_rotate layer_tile_exists layer_tile_create layer_tile_destroy layer_tile_change layer_tile_xscale layer_tile_yscale layer_tile_blend layer_tile_alpha layer_tile_x layer_tile_y layer_tile_region layer_tile_visible layer_tile_get_sprite layer_tile_get_xscale layer_tile_get_yscale layer_tile_get_blend layer_tile_get_alpha layer_tile_get_x layer_tile_get_y layer_tile_get_region layer_tile_get_visible layer_instance_get_instance instance_activate_layer instance_deactivate_layer camera_create camera_create_view camera_destroy camera_apply camera_get_active camera_get_default camera_set_default camera_set_view_mat camera_set_proj_mat camera_set_update_script camera_set_begin_script camera_set_end_script camera_set_view_pos camera_set_view_size camera_set_view_speed camera_set_view_border camera_set_view_angle camera_set_view_target camera_get_view_mat camera_get_proj_mat camera_get_update_script camera_get_begin_script camera_get_end_script camera_get_view_x camera_get_view_y camera_get_view_width camera_get_view_height camera_get_view_speed_x camera_get_view_speed_y camera_get_view_border_x camera_get_view_border_y camera_get_view_angle camera_get_view_target view_get_camera view_get_visible view_get_xport view_get_yport view_get_wport view_get_hport view_get_surface_id view_set_camera view_set_visible view_set_xport view_set_yport view_set_wport view_set_hport view_set_surface_id gesture_drag_time gesture_drag_distance gesture_flick_speed gesture_double_tap_time gesture_double_tap_distance gesture_pinch_distance gesture_pinch_angle_towards gesture_pinch_angle_away gesture_rotate_time gesture_rotate_angle gesture_tap_count gesture_get_drag_time gesture_get_drag_distance gesture_get_flick_speed gesture_get_double_tap_time gesture_get_double_tap_distance gesture_get_pinch_distance gesture_get_pinch_angle_towards gesture_get_pinch_angle_away gesture_get_rotate_time gesture_get_rotate_angle gesture_get_tap_count keyboard_virtual_show keyboard_virtual_hide keyboard_virtual_status 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getSuppression getTerrainGrid getTerrainHeightASL getText getTotalDLCUsageTime getUnitLoadout getUnitTrait getUserMFDText getUserMFDvalue getVariable getVehicleCargo getWeaponCargo getWeaponSway getWingsOrientationRTD getWingsPositionRTD getWPPos glanceAt globalChat globalRadio goggles goto group groupChat groupFromNetId groupIconSelectable groupIconsVisible groupId groupOwner groupRadio groupSelectedUnits groupSelectUnit gunner gusts halt handgunItems handgunMagazine handgunWeapon handsHit hasInterface hasPilotCamera hasWeapon hcAllGroups hcGroupParams hcLeader hcRemoveAllGroups hcRemoveGroup hcSelected hcSelectGroup hcSetGroup hcShowBar hcShownBar headgear hideBody hideObject hideObjectGlobal hideSelection hint hintC hintCadet hintSilent hmd hostMission htmlLoad HUDMovementLevels humidity image importAllGroups importance in inArea inAreaArray incapacitatedState inflame inflamed infoPanel infoPanelComponentEnabled infoPanelComponents infoPanels inGameUISetEventHandler inheritsFrom 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isUAVConnected isUIContext isUniformAllowed isVehicleCargo isVehicleRadarOn isVehicleSensorEnabled isWalking isWeaponDeployed isWeaponRested itemCargo items itemsWithMagazines join joinAs joinAsSilent joinSilent joinString kbAddDatabase kbAddDatabaseTargets kbAddTopic kbHasTopic kbReact kbRemoveTopic kbTell kbWasSaid keyImage keyName knowsAbout land landAt landResult language laserTarget lbAdd lbClear lbColor lbColorRight lbCurSel lbData lbDelete lbIsSelected lbPicture lbPictureRight lbSelection lbSetColor lbSetColorRight lbSetCurSel lbSetData lbSetPicture lbSetPictureColor lbSetPictureColorDisabled lbSetPictureColorSelected lbSetPictureRight lbSetPictureRightColor lbSetPictureRightColorDisabled lbSetPictureRightColorSelected lbSetSelectColor lbSetSelectColorRight lbSetSelected lbSetText lbSetTextRight lbSetTooltip lbSetValue lbSize lbSort lbSortByValue lbText lbTextRight lbValue leader leaderboardDeInit leaderboardGetRows leaderboardInit leaderboardRequestRowsFriends leaderboardsRequestUploadScore leaderboardsRequestUploadScoreKeepBest leaderboardState leaveVehicle libraryCredits libraryDisclaimers lifeState lightAttachObject lightDetachObject lightIsOn lightnings limitSpeed linearConversion lineIntersects lineIntersectsObjs lineIntersectsSurfaces lineIntersectsWith linkItem list listObjects listRemoteTargets listVehicleSensors ln lnbAddArray lnbAddColumn lnbAddRow lnbClear lnbColor lnbCurSelRow lnbData lnbDeleteColumn lnbDeleteRow lnbGetColumnsPosition lnbPicture lnbSetColor lnbSetColumnsPos lnbSetCurSelRow lnbSetData lnbSetPicture lnbSetText lnbSetValue lnbSize lnbSort lnbSortByValue lnbText lnbValue load loadAbs loadBackpack loadFile loadGame loadIdentity loadMagazine loadOverlay loadStatus loadUniform loadVest local localize locationPosition lock lockCameraTo lockCargo lockDriver locked lockedCargo lockedDriver lockedTurret lockIdentity lockTurret lockWP log logEntities logNetwork logNetworkTerminate lookAt lookAtPos magazineCargo magazines magazinesAllTurrets magazinesAmmo magazinesAmmoCargo magazinesAmmoFull magazinesDetail magazinesDetailBackpack magazinesDetailUniform magazinesDetailVest magazinesTurret magazineTurretAmmo mapAnimAdd mapAnimClear mapAnimCommit mapAnimDone mapCenterOnCamera mapGridPosition markAsFinishedOnSteam markerAlpha markerBrush markerColor markerDir markerPos markerShape markerSize markerText markerType max members menuAction menuAdd menuChecked menuClear menuCollapse menuData menuDelete menuEnable menuEnabled menuExpand menuHover menuPicture menuSetAction menuSetCheck menuSetData menuSetPicture menuSetValue menuShortcut menuShortcutText menuSize menuSort menuText menuURL menuValue min mineActive mineDetectedBy missionConfigFile missionDifficulty missionName missionNamespace missionStart missionVersion mod modelToWorld modelToWorldVisual modelToWorldVisualWorld modelToWorldWorld modParams moonIntensity moonPhase morale move move3DENCamera moveInAny moveInCargo moveInCommander moveInDriver moveInGunner moveInTurret moveObjectToEnd moveOut moveTime moveTo moveToCompleted moveToFailed musicVolume name nameSound nearEntities nearestBuilding nearestLocation nearestLocations nearestLocationWithDubbing nearestObject nearestObjects nearestTerrainObjects nearObjects nearObjectsReady nearRoads nearSupplies nearTargets needReload netId netObjNull newOverlay nextMenuItemIndex nextWeatherChange nMenuItems not numberOfEnginesRTD numberToDate objectCurators objectFromNetId objectParent objStatus onBriefingGroup onBriefingNotes onBriefingPlan onBriefingTeamSwitch onCommandModeChanged onDoubleClick onEachFrame onGroupIconClick onGroupIconOverEnter onGroupIconOverLeave onHCGroupSelectionChanged onMapSingleClick onPlayerConnected onPlayerDisconnected onPreloadFinished onPreloadStarted onShowNewObject onTeamSwitch openCuratorInterface openDLCPage openMap openSteamApp openYoutubeVideo or orderGetIn overcast overcastForecast owner param params parseNumber parseSimpleArray parseText parsingNamespace particlesQuality pickWeaponPool pitch pixelGrid pixelGridBase pixelGridNoUIScale pixelH pixelW playableSlotsNumber playableUnits playAction playActionNow player playerRespawnTime playerSide playersNumber playGesture playMission playMove playMoveNow playMusic playScriptedMission playSound playSound3D position positionCameraToWorld posScreenToWorld posWorldToScreen ppEffectAdjust ppEffectCommit ppEffectCommitted ppEffectCreate ppEffectDestroy ppEffectEnable ppEffectEnabled ppEffectForceInNVG precision preloadCamera preloadObject preloadSound preloadTitleObj preloadTitleRsc preprocessFile preprocessFileLineNumbers primaryWeapon primaryWeaponItems primaryWeaponMagazine priority processDiaryLink productVersion profileName profileNamespace profileNameSteam progressLoadingScreen progressPosition progressSetPosition publicVariable publicVariableClient publicVariableServer pushBack pushBackUnique putWeaponPool queryItemsPool queryMagazinePool queryWeaponPool rad radioChannelAdd radioChannelCreate radioChannelRemove radioChannelSetCallSign radioChannelSetLabel radioVolume rain rainbow random rank rankId rating rectangular registeredTasks registerTask reload reloadEnabled remoteControl remoteExec remoteExecCall remoteExecutedOwner remove3DENConnection remove3DENEventHandler remove3DENLayer removeAction removeAll3DENEventHandlers removeAllActions removeAllAssignedItems removeAllContainers removeAllCuratorAddons removeAllCuratorCameraAreas removeAllCuratorEditingAreas removeAllEventHandlers removeAllHandgunItems removeAllItems removeAllItemsWithMagazines removeAllMissionEventHandlers removeAllMPEventHandlers removeAllMusicEventHandlers removeAllOwnedMines removeAllPrimaryWeaponItems removeAllWeapons removeBackpack removeBackpackGlobal removeCuratorAddons removeCuratorCameraArea removeCuratorEditableObjects removeCuratorEditingArea removeDrawIcon removeDrawLinks removeEventHandler removeFromRemainsCollector removeGoggles removeGroupIcon removeHandgunItem removeHeadgear removeItem removeItemFromBackpack removeItemFromUniform removeItemFromVest removeItems removeMagazine removeMagazineGlobal removeMagazines removeMagazinesTurret removeMagazineTurret removeMenuItem removeMissionEventHandler removeMPEventHandler removeMusicEventHandler removeOwnedMine removePrimaryWeaponItem removeSecondaryWeaponItem removeSimpleTask removeSwitchableUnit removeTeamMember removeUniform removeVest removeWeapon removeWeaponAttachmentCargo removeWeaponCargo removeWeaponGlobal removeWeaponTurret reportRemoteTarget requiredVersion resetCamShake resetSubgroupDirection resize resources respawnVehicle restartEditorCamera reveal revealMine reverse reversedMouseY roadAt roadsConnectedTo roleDescription ropeAttachedObjects ropeAttachedTo ropeAttachEnabled ropeAttachTo ropeCreate ropeCut ropeDestroy ropeDetach ropeEndPosition ropeLength ropes ropeUnwind ropeUnwound rotorsForcesRTD rotorsRpmRTD round runInitScript safeZoneH safeZoneW safeZoneWAbs safeZoneX safeZoneXAbs safeZoneY save3DENInventory saveGame saveIdentity saveJoysticks saveOverlay saveProfileNamespace saveStatus saveVar savingEnabled say say2D say3D scopeName score scoreSide screenshot screenToWorld scriptDone scriptName scudState secondaryWeapon secondaryWeaponItems secondaryWeaponMagazine select selectBestPlaces selectDiarySubject selectedEditorObjects selectEditorObject selectionNames selectionPosition selectLeader selectMax selectMin selectNoPlayer selectPlayer selectRandom selectRandomWeighted selectWeapon selectWeaponTurret sendAUMessage sendSimpleCommand sendTask sendTaskResult sendUDPMessage serverCommand serverCommandAvailable serverCommandExecutable serverName serverTime set set3DENAttribute set3DENAttributes set3DENGrid set3DENIconsVisible set3DENLayer set3DENLinesVisible set3DENLogicType set3DENMissionAttribute set3DENMissionAttributes set3DENModelsVisible set3DENObjectType set3DENSelected setAccTime setActualCollectiveRTD setAirplaneThrottle setAirportSide setAmmo 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setFSMVariable setFuel setFuelCargo setGroupIcon setGroupIconParams setGroupIconsSelectable setGroupIconsVisible setGroupId setGroupIdGlobal setGroupOwner setGusts setHideBehind setHit setHitIndex setHitPointDamage setHorizonParallaxCoef setHUDMovementLevels setIdentity setImportance setInfoPanel setLeader setLightAmbient setLightAttenuation setLightBrightness setLightColor setLightDayLight setLightFlareMaxDistance setLightFlareSize setLightIntensity setLightnings setLightUseFlare setLocalWindParams setMagazineTurretAmmo setMarkerAlpha setMarkerAlphaLocal setMarkerBrush setMarkerBrushLocal setMarkerColor setMarkerColorLocal setMarkerDir setMarkerDirLocal setMarkerPos setMarkerPosLocal setMarkerShape setMarkerShapeLocal setMarkerSize setMarkerSizeLocal setMarkerText setMarkerTextLocal setMarkerType setMarkerTypeLocal setMass setMimic setMousePosition setMusicEffect setMusicEventHandler setName setNameSound setObjectArguments setObjectMaterial setObjectMaterialGlobal setObjectProxy setObjectTexture setObjectTextureGlobal setObjectViewDistance setOvercast setOwner setOxygenRemaining setParticleCircle setParticleClass setParticleFire setParticleParams setParticleRandom setPilotCameraDirection setPilotCameraRotation setPilotCameraTarget setPilotLight setPiPEffect setPitch setPlateNumber setPlayable setPlayerRespawnTime setPos setPosASL setPosASL2 setPosASLW setPosATL setPosition setPosWorld setPylonLoadOut setPylonsPriority setRadioMsg setRain setRainbow setRandomLip setRank setRectangular setRepairCargo setRotorBrakeRTD setShadowDistance setShotParents setSide setSimpleTaskAlwaysVisible setSimpleTaskCustomData setSimpleTaskDescription setSimpleTaskDestination setSimpleTaskTarget setSimpleTaskType setSimulWeatherLayers setSize setSkill setSlingLoad setSoundEffect setSpeaker setSpeech setSpeedMode setStamina setStaminaScheme setStatValue setSuppression setSystemOfUnits setTargetAge setTaskMarkerOffset setTaskResult setTaskState setTerrainGrid setText setTimeMultiplier setTitleEffect setTrafficDensity setTrafficDistance setTrafficGap setTrafficSpeed setTriggerActivation setTriggerArea setTriggerStatements setTriggerText setTriggerTimeout setTriggerType setType setUnconscious setUnitAbility setUnitLoadout setUnitPos setUnitPosWeak setUnitRank setUnitRecoilCoefficient setUnitTrait setUnloadInCombat setUserActionText setUserMFDText setUserMFDvalue setVariable setVectorDir setVectorDirAndUp setVectorUp setVehicleAmmo setVehicleAmmoDef setVehicleArmor setVehicleCargo setVehicleId setVehicleLock setVehiclePosition setVehicleRadar setVehicleReceiveRemoteTargets setVehicleReportOwnPosition setVehicleReportRemoteTargets setVehicleTIPars setVehicleVarName setVelocity setVelocityModelSpace setVelocityTransformation setViewDistance setVisibleIfTreeCollapsed setWantedRPMRTD setWaves setWaypointBehaviour setWaypointCombatMode setWaypointCompletionRadius setWaypointDescription setWaypointForceBehaviour setWaypointFormation setWaypointHousePosition setWaypointLoiterRadius setWaypointLoiterType setWaypointName setWaypointPosition setWaypointScript setWaypointSpeed setWaypointStatements setWaypointTimeout setWaypointType setWaypointVisible setWeaponReloadingTime setWind setWindDir setWindForce setWindStr setWingForceScaleRTD setWPPos show3DIcons showChat showCinemaBorder showCommandingMenu showCompass showCuratorCompass showGPS showHUD showLegend showMap shownArtilleryComputer shownChat shownCompass shownCuratorCompass showNewEditorObject shownGPS shownHUD shownMap shownPad shownRadio shownScoretable shownUAVFeed shownWarrant shownWatch showPad showRadio showScoretable showSubtitles showUAVFeed showWarrant showWatch showWaypoint showWaypoints side sideChat sideEnemy sideFriendly sideRadio simpleTasks simulationEnabled simulCloudDensity simulCloudOcclusion simulInClouds simulWeatherSync sin size sizeOf skill skillFinal skipTime sleep sliderPosition sliderRange sliderSetPosition sliderSetRange sliderSetSpeed sliderSpeed slingLoadAssistantShown soldierMagazines someAmmo sort soundVolume spawn speaker speed speedMode splitString sqrt squadParams stance startLoadingScreen step stop stopEngineRTD stopped str sunOrMoon supportInfo suppressFor surfaceIsWater surfaceNormal surfaceType swimInDepth switchableUnits switchAction switchCamera switchGesture switchLight switchMove synchronizedObjects synchronizedTriggers synchronizedWaypoints synchronizeObjectsAdd synchronizeObjectsRemove synchronizeTrigger synchronizeWaypoint systemChat systemOfUnits tan targetKnowledge targets targetsAggregate targetsQuery taskAlwaysVisible taskChildren taskCompleted taskCustomData taskDescription taskDestination taskHint taskMarkerOffset taskParent taskResult taskState taskType teamMember teamName teams teamSwitch teamSwitchEnabled teamType terminate terrainIntersect terrainIntersectASL terrainIntersectAtASL text textLog textLogFormat tg time timeMultiplier titleCut titleFadeOut titleObj titleRsc titleText toArray toFixed toLower toString toUpper triggerActivated triggerActivation triggerArea triggerAttachedVehicle triggerAttachObject triggerAttachVehicle triggerDynamicSimulation triggerStatements triggerText triggerTimeout triggerTimeoutCurrent triggerType turretLocal turretOwner turretUnit tvAdd tvClear tvCollapse tvCollapseAll tvCount tvCurSel tvData tvDelete tvExpand tvExpandAll tvPicture tvSetColor tvSetCurSel tvSetData tvSetPicture tvSetPictureColor tvSetPictureColorDisabled tvSetPictureColorSelected tvSetPictureRight tvSetPictureRightColor tvSetPictureRightColorDisabled tvSetPictureRightColorSelected tvSetText tvSetTooltip tvSetValue tvSort tvSortByValue tvText tvTooltip tvValue type typeName typeOf UAVControl uiNamespace uiSleep unassignCurator unassignItem unassignTeam unassignVehicle underwater uniform uniformContainer uniformItems uniformMagazines unitAddons unitAimPosition unitAimPositionVisual unitBackpack unitIsUAV unitPos unitReady unitRecoilCoefficient units unitsBelowHeight unlinkItem 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",dragon_face:"🐲",dress:"👗",dromedary_camel:"🐪",drooling_face:"🤤",droplet:"💧",drum:"🥁",duck:"🦆",dvd:"📀","e-mail":"📧",eagle:"🦅",ear:"👂",ear_of_rice:"🌾",earth_africa:"🌍",earth_americas:"🌎",earth_asia:"🌏",egg:"🥚",eggplant:"🍆",eight_pointed_black_star:"✴️",eight_spoked_asterisk:"✳️",electric_plug:"🔌",elephant:"🐘",email:"✉️",end:"🔚",envelope_with_arrow:"📩",euro:"💶",european_castle:"🏰",european_post_office:"🏤",evergreen_tree:"🌲",exclamation:"❗️",expressionless:"😑",eye:"👁",eye_speech_bubble:"👁‍🗨",eyeglasses:"👓",eyes:"👀",face_with_head_bandage:"🤕",face_with_thermometer:"🤒",fist_oncoming:"👊",factory:"🏭",fallen_leaf:"🍂",family_man_woman_boy:"👪",family_man_boy:"👨‍👦",family_man_boy_boy:"👨‍👦‍👦",family_man_girl:"👨‍👧",family_man_girl_boy:"👨‍👧‍👦",family_man_girl_girl:"👨‍👧‍👧",family_man_man_boy:"👨‍👨‍👦",family_man_man_boy_boy:"👨‍👨‍👦‍👦",family_man_man_girl:"👨‍👨‍👧",family_man_man_girl_boy:"👨‍👨‍👧‍👦",family_man_man_girl_girl:"👨‍👨‍👧‍👧",family_man_woman_boy_boy:"👨‍👩‍👦‍👦",family_man_woman_girl:"👨‍👩‍👧",family_man_woman_girl_boy:"👨‍👩‍👧‍👦",family_man_woman_girl_girl:"👨‍👩‍👧‍👧",family_woman_boy:"👩‍👦",family_woman_boy_boy:"👩‍👦‍👦",family_woman_girl:"👩‍👧",family_woman_girl_boy:"👩‍👧‍👦",family_woman_girl_girl:"👩‍👧‍👧",family_woman_woman_boy:"👩‍👩‍👦",family_woman_woman_boy_boy:"👩‍👩‍👦‍👦",family_woman_woman_girl:"👩‍👩‍👧",family_woman_woman_girl_boy:"👩‍👩‍👧‍👦",family_woman_woman_girl_girl:"👩‍👩‍👧‍👧",fast_forward:"⏩",fax:"📠",fearful:"😨",feet:"🐾",female_detective:"🕵️‍♀️",ferris_wheel:"🎡",ferry:"⛴",field_hockey:"🏑",file_cabinet:"🗄",file_folder:"📁",film_projector:"📽",film_strip:"🎞",fire:"🔥",fire_engine:"🚒",fireworks:"🎆",first_quarter_moon:"🌓",first_quarter_moon_with_face:"🌛",fish:"🐟",fish_cake:"🍥",fishing_pole_and_fish:"🎣",fist_raised:"✊",fist_left:"🤛",fist_right:"🤜",flags:"🎏",flashlight:"🔦",fleur_de_lis:"⚜️",flight_arrival:"🛬",flight_departure:"🛫",floppy_disk:"💾",flower_playing_cards:"🎴",flushed:"😳",fog:"🌫",foggy:"🌁",football:"🏈",footprints:"👣",fork_and_knife:"🍴",fountain:"⛲️",fountain_pen:"🖋",four_leaf_clover:"🍀",fox_face:"🦊",framed_picture:"🖼",free:"🆓",fried_egg:"🍳",fried_shrimp:"🍤",fries:"🍟",frog:"🐸",frowning:"😦",frowning_face:"☹️",frowning_man:"🙍‍♂️",frowning_woman:"🙍",middle_finger:"🖕",fuelpump:"⛽️",full_moon:"🌕",full_moon_with_face:"🌝",funeral_urn:"⚱️",game_die:"🎲",gear:"⚙️",gem:"💎",gemini:"♊️",ghost:"👻",gift:"🎁",gift_heart:"💝",girl:"👧",globe_with_meridians:"🌐",goal_net:"🥅",goat:"🐐",golf:"⛳️",golfing_man:"🏌️",golfing_woman:"🏌️‍♀️",gorilla:"🦍",grapes:"🍇",green_apple:"🍏",green_book:"📗",green_heart:"💚",green_salad:"🥗",grey_exclamation:"❕",grey_question:"❔",grimacing:"😬",grin:"😁",grinning:"😀",guardsman:"💂",guardswoman:"💂‍♀️",guitar:"🎸",gun:"🔫",haircut_woman:"💇",haircut_man:"💇‍♂️",hamburger:"🍔",hammer:"🔨",hammer_and_pick:"⚒",hammer_and_wrench:"🛠",hamster:"🐹",hand:"✋",handbag:"👜",handshake:"🤝",hankey:"💩",hatched_chick:"🐥",hatching_chick:"🐣",headphones:"🎧",hear_no_evil:"🙉",heart:"❤️",heart_decoration:"💟",heart_eyes:"😍",heart_eyes_cat:"😻",heartbeat:"💓",heartpulse:"💗",hearts:"♥️",heavy_check_mark:"✔️",heavy_division_sign:"➗",heavy_dollar_sign:"💲",heavy_heart_exclamation:"❣️",heavy_minus_sign:"➖",heavy_multiplication_x:"✖️",heavy_plus_sign:"➕",helicopter:"🚁",herb:"🌿",hibiscus:"🌺",high_brightness:"🔆",high_heel:"👠",hocho:"🔪",hole:"🕳",honey_pot:"🍯",horse:"🐴",horse_racing:"🏇",hospital:"🏥",hot_pepper:"🌶",hotdog:"🌭",hotel:"🏨",hotsprings:"♨️",hourglass:"⌛️",hourglass_flowing_sand:"⏳",house:"🏠",house_with_garden:"🏡",houses:"🏘",hugs:"🤗",hushed:"😯",ice_cream:"🍨",ice_hockey:"🏒",ice_skate:"⛸",icecream:"🍦",id:"🆔",ideograph_advantage:"🉐",imp:"👿",inbox_tray:"📥",incoming_envelope:"📨",tipping_hand_woman:"💁",information_source:"ℹ️",innocent:"😇",interrobang:"⁉️",iphone:"📱",izakaya_lantern:"🏮",jack_o_lantern:"🎃",japan:"🗾",japanese_castle:"🏯",japanese_goblin:"👺",japanese_ogre:"👹",jeans:"👖",joy:"😂",joy_cat:"😹",joystick:"🕹",kaaba:"🕋",key:"🔑",keyboard:"⌨️",keycap_ten:"🔟",kick_scooter:"🛴",kimono:"👘",kiss:"💋",kissing:"😗",kissing_cat:"😽",kissing_closed_eyes:"😚",kissing_heart:"😘",kissing_smiling_eyes:"😙",kiwi_fruit:"🥝",koala:"🐨",koko:"🈁",label:"🏷",large_blue_circle:"🔵",large_blue_diamond:"🔷",large_orange_diamond:"🔶",last_quarter_moon:"🌗",last_quarter_moon_with_face:"🌜",latin_cross:"✝️",laughing:"😆",leaves:"🍃",ledger:"📒",left_luggage:"🛅",left_right_arrow:"↔️",leftwards_arrow_with_hook:"↩️",lemon:"🍋",leo:"♌️",leopard:"🐆",level_slider:"🎚",libra:"♎️",light_rail:"🚈",link:"🔗",lion:"🦁",lips:"👄",lipstick:"💄",lizard:"🦎",lock:"🔒",lock_with_ink_pen:"🔏",lollipop:"🍭",loop:"➿",loud_sound:"🔊",loudspeaker:"📢",love_hotel:"🏩",love_letter:"💌",low_brightness:"🔅",lying_face:"🤥",m:"Ⓜ️",mag:"🔍",mag_right:"🔎",mahjong:"🀄️",mailbox:"📫",mailbox_closed:"📪",mailbox_with_mail:"📬",mailbox_with_no_mail:"📭",man:"👨",man_artist:"👨‍🎨",man_astronaut:"👨‍🚀",man_cartwheeling:"🤸‍♂️",man_cook:"👨‍🍳",man_dancing:"🕺",man_facepalming:"🤦‍♂️",man_factory_worker:"👨‍🏭",man_farmer:"👨‍🌾",man_firefighter:"👨‍🚒",man_health_worker:"👨‍⚕️",man_in_tuxedo:"🤵",man_judge:"👨‍⚖️",man_juggling:"🤹‍♂️",man_mechanic:"👨‍🔧",man_office_worker:"👨‍💼",man_pilot:"👨‍✈️",man_playing_handball:"🤾‍♂️",man_playing_water_polo:"🤽‍♂️",man_scientist:"👨‍🔬",man_shrugging:"🤷‍♂️",man_singer:"👨‍🎤",man_student:"👨‍🎓",man_teacher:"👨‍🏫",man_technologist:"👨‍💻",man_with_gua_pi_mao:"👲",man_with_turban:"👳",tangerine:"🍊",mans_shoe:"👞",mantelpiece_clock:"🕰",maple_leaf:"🍁",martial_arts_uniform:"🥋",mask:"😷",massage_woman:"💆",massage_man:"💆‍♂️",meat_on_bone:"🍖",medal_military:"🎖",medal_sports:"🏅",mega:"📣",melon:"🍈",memo:"📝",men_wrestling:"🤼‍♂️",menorah:"🕎",mens:"🚹",metal:"🤘",metro:"🚇",microphone:"🎤",microscope:"🔬",milk_glass:"🥛",milky_way:"🌌",minibus:"🚐",minidisc:"💽",mobile_phone_off:"📴",money_mouth_face:"🤑",money_with_wings:"💸",moneybag:"💰",monkey:"🐒",monkey_face:"🐵",monorail:"🚝",moon:"🌔",mortar_board:"🎓",mosque:"🕌",motor_boat:"🛥",motor_scooter:"🛵",motorcycle:"🏍",motorway:"🛣",mount_fuji:"🗻",mountain:"⛰",mountain_biking_man:"🚵",mountain_biking_woman:"🚵‍♀️",mountain_cableway:"🚠",mountain_railway:"🚞",mountain_snow:"🏔",mouse:"🐭",mouse2:"🐁",movie_camera:"🎥",moyai:"🗿",mrs_claus:"🤶",muscle:"💪",mushroom:"🍄",musical_keyboard:"🎹",musical_note:"🎵",musical_score:"🎼",mute:"🔇",nail_care:"💅",name_badge:"📛",national_park:"🏞",nauseated_face:"🤢",necktie:"👔",negative_squared_cross_mark:"❎",nerd_face:"🤓",neutral_face:"😐",new:"🆕",new_moon:"🌑",new_moon_with_face:"🌚",newspaper:"📰",newspaper_roll:"🗞",next_track_button:"⏭",ng:"🆖",no_good_man:"🙅‍♂️",no_good_woman:"🙅",night_with_stars:"🌃",no_bell:"🔕",no_bicycles:"🚳",no_entry:"⛔️",no_entry_sign:"🚫",no_mobile_phones:"📵",no_mouth:"😶",no_pedestrians:"🚷",no_smoking:"🚭","non-potable_water":"🚱",nose:"👃",notebook:"📓",notebook_with_decorative_cover:"📔",notes:"🎶",nut_and_bolt:"🔩",o:"⭕️",o2:"🅾️",ocean:"🌊",octopus:"🐙",oden:"🍢",office:"🏢",oil_drum:"🛢",ok:"🆗",ok_hand:"👌",ok_man:"🙆‍♂️",ok_woman:"🙆",old_key:"🗝",older_man:"👴",older_woman:"👵",om:"🕉",on:"🔛",oncoming_automobile:"🚘",oncoming_bus:"🚍",oncoming_police_car:"🚔",oncoming_taxi:"🚖",open_file_folder:"📂",open_hands:"👐",open_mouth:"😮",open_umbrella:"☂️",ophiuchus:"⛎",orange_book:"📙",orthodox_cross:"☦️",outbox_tray:"📤",owl:"🦉",ox:"🐂",package:"📦",page_facing_up:"📄",page_with_curl:"📃",pager:"📟",paintbrush:"🖌",palm_tree:"🌴",pancakes:"🥞",panda_face:"🐼",paperclip:"📎",paperclips:"🖇",parasol_on_ground:"⛱",parking:"🅿️",part_alternation_mark:"〽️",partly_sunny:"⛅️",passenger_ship:"🛳",passport_control:"🛂",pause_button:"⏸",peace_symbol:"☮️",peach:"🍑",peanuts:"🥜",pear:"🍐",pen:"🖊",pencil2:"✏️",penguin:"🐧",pensive:"😔",performing_arts:"🎭",persevere:"😣",person_fencing:"🤺",pouting_woman:"🙎",phone:"☎️",pick:"⛏",pig:"🐷",pig2:"🐖",pig_nose:"🐽",pill:"💊",pineapple:"🍍",ping_pong:"🏓",pisces:"♓️",pizza:"🍕",place_of_worship:"🛐",plate_with_cutlery:"🍽",play_or_pause_button:"⏯",point_down:"👇",point_left:"👈",point_right:"👉",point_up:"☝️",point_up_2:"👆",police_car:"🚓",policewoman:"👮‍♀️",poodle:"🐩",popcorn:"🍿",post_office:"🏣",postal_horn:"📯",postbox:"📮",potable_water:"🚰",potato:"🥔",pouch:"👝",poultry_leg:"🍗",pound:"💷",rage:"😡",pouting_cat:"😾",pouting_man:"🙎‍♂️",pray:"🙏",prayer_beads:"📿",pregnant_woman:"🤰",previous_track_button:"⏮",prince:"🤴",princess:"👸",printer:"🖨",purple_heart:"💜",purse:"👛",pushpin:"📌",put_litter_in_its_place:"🚮",question:"❓",rabbit:"🐰",rabbit2:"🐇",racehorse:"🐎",racing_car:"🏎",radio:"📻",radio_button:"🔘",radioactive:"☢️",railway_car:"🚃",railway_track:"🛤",rainbow:"🌈",rainbow_flag:"🏳️‍🌈",raised_back_of_hand:"🤚",raised_hand_with_fingers_splayed:"🖐",raised_hands:"🙌",raising_hand_woman:"🙋",raising_hand_man:"🙋‍♂️",ram:"🐏",ramen:"🍜",rat:"🐀",record_button:"⏺",recycle:"♻️",red_circle:"🔴",registered:"®️",relaxed:"☺️",relieved:"😌",reminder_ribbon:"🎗",repeat:"🔁",repeat_one:"🔂",rescue_worker_helmet:"⛑",restroom:"🚻",revolving_hearts:"💞",rewind:"⏪",rhinoceros:"🦏",ribbon:"🎀",rice:"🍚",rice_ball:"🍙",rice_cracker:"🍘",rice_scene:"🎑",right_anger_bubble:"🗯",ring:"💍",robot:"🤖",rocket:"🚀",rofl:"🤣",roll_eyes:"🙄",roller_coaster:"🎢",rooster:"🐓",rose:"🌹",rosette:"🏵",rotating_light:"🚨",round_pushpin:"📍",rowing_man:"🚣",rowing_woman:"🚣‍♀️",rugby_football:"🏉",running_man:"🏃",running_shirt_with_sash:"🎽",running_woman:"🏃‍♀️",sa:"🈂️",sagittarius:"♐️",sake:"🍶",sandal:"👡",santa:"🎅",satellite:"📡",saxophone:"🎷",school:"🏫",school_satchel:"🎒",scissors:"✂️",scorpion:"🦂",scorpius:"♏️",scream:"😱",scream_cat:"🙀",scroll:"📜",seat:"💺",secret:"㊙️",see_no_evil:"🙈",seedling:"🌱",selfie:"🤳",shallow_pan_of_food:"🥘",shamrock:"☘️",shark:"🦈",shaved_ice:"🍧",sheep:"🐑",shell:"🐚",shield:"🛡",shinto_shrine:"⛩",ship:"🚢",shirt:"👕",shopping:"🛍",shopping_cart:"🛒",shower:"🚿",shrimp:"🦐",signal_strength:"📶",six_pointed_star:"🔯",ski:"🎿",skier:"⛷",skull:"💀",skull_and_crossbones:"☠️",sleeping:"😴",sleeping_bed:"🛌",sleepy:"😪",slightly_frowning_face:"🙁",slightly_smiling_face:"🙂",slot_machine:"🎰",small_airplane:"🛩",small_blue_diamond:"🔹",small_orange_diamond:"🔸",small_red_triangle:"🔺",small_red_triangle_down:"🔻",smile:"😄",smile_cat:"😸",smiley:"😃",smiley_cat:"😺",smiling_imp:"😈",smirk:"😏",smirk_cat:"😼",smoking:"🚬",snail:"🐌",snake:"🐍",sneezing_face:"🤧",snowboarder:"🏂",snowflake:"❄️",snowman:"⛄️",snowman_with_snow:"☃️",sob:"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DefineExternal DefineInputStreamMethod DefineOutputStreamMethod DefineResourceFunction Definition Degree DegreeCentrality DegreeGraphDistribution DegreeLexicographic DegreeReverseLexicographic DEigensystem DEigenvalues Deinitialization Del DelaunayMesh Delayed Deletable Delete DeleteAnomalies DeleteBorderComponents DeleteCases DeleteChannel DeleteCloudExpression DeleteContents DeleteDirectory DeleteDuplicates DeleteDuplicatesBy DeleteFile DeleteMissing DeleteObject DeletePermissionsKey DeleteSearchIndex DeleteSmallComponents DeleteStopwords DeleteWithContents DeletionWarning DelimitedArray DelimitedSequence Delimiter DelimiterFlashTime DelimiterMatching Delimiters DeliveryFunction Dendrogram Denominator DensityGraphics DensityHistogram DensityPlot DensityPlot3D DependentVariables Deploy Deployed Depth DepthFirstScan Derivative DerivativeFilter DerivedKey DescriptorStateSpace DesignMatrix DestroyAfterEvaluation Det DeviceClose DeviceConfigure DeviceExecute DeviceExecuteAsynchronous DeviceObject DeviceOpen DeviceOpenQ DeviceRead DeviceReadBuffer DeviceReadLatest DeviceReadList DeviceReadTimeSeries Devices DeviceStreams DeviceWrite DeviceWriteBuffer DGaussianWavelet DiacriticalPositioning Diagonal DiagonalizableMatrixQ DiagonalMatrix DiagonalMatrixQ Dialog DialogIndent DialogInput DialogLevel DialogNotebook DialogProlog DialogReturn DialogSymbols Diamond DiamondMatrix DiceDissimilarity DictionaryLookup DictionaryWordQ DifferenceDelta DifferenceOrder DifferenceQuotient DifferenceRoot DifferenceRootReduce Differences DifferentialD DifferentialRoot DifferentialRootReduce DifferentiatorFilter DigitalSignature DigitBlock DigitBlockMinimum DigitCharacter DigitCount DigitQ DihedralAngle DihedralGroup Dilation DimensionalCombinations DimensionalMeshComponents DimensionReduce DimensionReducerFunction DimensionReduction Dimensions DiracComb DiracDelta DirectedEdge DirectedEdges DirectedGraph DirectedGraphQ DirectedInfinity Direction Directive Directory DirectoryName DirectoryQ DirectoryStack DirichletBeta DirichletCharacter DirichletCondition DirichletConvolve DirichletDistribution DirichletEta DirichletL DirichletLambda DirichletTransform DirichletWindow DisableConsolePrintPacket DisableFormatting DiscreteChirpZTransform DiscreteConvolve DiscreteDelta DiscreteHadamardTransform DiscreteIndicator DiscreteLimit DiscreteLQEstimatorGains DiscreteLQRegulatorGains DiscreteLyapunovSolve DiscreteMarkovProcess DiscreteMaxLimit DiscreteMinLimit DiscretePlot DiscretePlot3D DiscreteRatio DiscreteRiccatiSolve DiscreteShift DiscreteTimeModelQ DiscreteUniformDistribution DiscreteVariables DiscreteWaveletData DiscreteWaveletPacketTransform DiscreteWaveletTransform DiscretizeGraphics DiscretizeRegion Discriminant DisjointQ Disjunction Disk DiskBox DiskMatrix DiskSegment Dispatch DispatchQ DispersionEstimatorFunction Display DisplayAllSteps DisplayEndPacket DisplayFlushImagePacket DisplayForm DisplayFunction DisplayPacket DisplayRules DisplaySetSizePacket DisplayString DisplayTemporary DisplayWith DisplayWithRef DisplayWithVariable DistanceFunction DistanceMatrix DistanceTransform Distribute Distributed DistributedContexts DistributeDefinitions DistributionChart DistributionDomain DistributionFitTest DistributionParameterAssumptions DistributionParameterQ Dithering Div Divergence Divide DivideBy Dividers DivideSides Divisible Divisors DivisorSigma DivisorSum DMSList DMSString Do DockedCells DocumentGenerator DocumentGeneratorInformation DocumentGeneratorInformationData DocumentGenerators DocumentNotebook DocumentWeightingRules Dodecahedron DomainRegistrationInformation DominantColors DOSTextFormat Dot DotDashed DotEqual DotLayer DotPlusLayer Dotted DoubleBracketingBar DoubleContourIntegral DoubleDownArrow DoubleLeftArrow DoubleLeftRightArrow DoubleLeftTee DoubleLongLeftArrow DoubleLongLeftRightArrow DoubleLongRightArrow DoubleRightArrow DoubleRightTee DoubleUpArrow DoubleUpDownArrow DoubleVerticalBar DoublyInfinite Down DownArrow DownArrowBar DownArrowUpArrow DownLeftRightVector DownLeftTeeVector DownLeftVector DownLeftVectorBar DownRightTeeVector DownRightVector DownRightVectorBar Downsample DownTee DownTeeArrow DownValues DragAndDrop DrawEdges DrawFrontFaces DrawHighlighted Drop DropoutLayer DSolve DSolveValue Dt DualLinearProgramming DualPolyhedron DualSystemsModel DumpGet DumpSave DuplicateFreeQ Duration Dynamic DynamicBox DynamicBoxOptions DynamicEvaluationTimeout DynamicGeoGraphics DynamicImage DynamicLocation DynamicModule DynamicModuleBox DynamicModuleBoxOptions DynamicModuleParent DynamicModuleValues DynamicName DynamicNamespace DynamicReference DynamicSetting DynamicUpdating DynamicWrapper DynamicWrapperBox DynamicWrapperBoxOptionsE EarthImpactData EarthquakeData EccentricityCentrality Echo EchoFunction EclipseType EdgeAdd EdgeBetweennessCentrality EdgeCapacity EdgeCapForm EdgeColor EdgeConnectivity EdgeContract EdgeCost EdgeCount EdgeCoverQ EdgeCycleMatrix EdgeDashing EdgeDelete EdgeDetect EdgeForm EdgeIndex EdgeJoinForm EdgeLabeling EdgeLabels EdgeLabelStyle EdgeList EdgeOpacity EdgeQ EdgeRenderingFunction EdgeRules EdgeShapeFunction EdgeStyle EdgeThickness EdgeWeight EdgeWeightedGraphQ Editable EditButtonSettings EditCellTagsSettings EditDistance EffectiveInterest Eigensystem Eigenvalues EigenvectorCentrality Eigenvectors Element ElementData ElementwiseLayer ElidedForms Eliminate EliminationOrder Ellipsoid EllipticE EllipticExp EllipticExpPrime EllipticF EllipticFilterModel EllipticK EllipticLog EllipticNomeQ EllipticPi EllipticReducedHalfPeriods EllipticTheta EllipticThetaPrime EmbedCode EmbeddedHTML EmbeddedService EmbeddingLayer EmbeddingObject EmitSound EmphasizeSyntaxErrors EmpiricalDistribution Empty EmptyGraphQ EmptyRegion EnableConsolePrintPacket Enabled Encode Encrypt EncryptedObject EncryptFile End EndAdd EndDialogPacket EndFrontEndInteractionPacket EndOfBuffer EndOfFile EndOfLine EndOfString EndPackage EngineEnvironment EngineeringForm Enter EnterExpressionPacket EnterTextPacket Entity EntityClass EntityClassList EntityCopies EntityFunction EntityGroup EntityInstance EntityList EntityPrefetch EntityProperties EntityProperty EntityPropertyClass EntityRegister EntityStore EntityStores EntityTypeName EntityUnregister EntityValue Entropy EntropyFilter Environment Epilog EpilogFunction Equal EqualColumns EqualRows EqualTilde EqualTo EquatedTo Equilibrium EquirippleFilterKernel Equivalent Erf Erfc Erfi ErlangB ErlangC ErlangDistribution Erosion ErrorBox ErrorBoxOptions ErrorNorm ErrorPacket ErrorsDialogSettings EscapeRadius EstimatedBackground EstimatedDistribution EstimatedProcess EstimatorGains EstimatorRegulator EuclideanDistance EulerAngles EulerCharacteristic EulerE EulerGamma EulerianGraphQ EulerMatrix EulerPhi Evaluatable Evaluate Evaluated EvaluatePacket EvaluateScheduledTask EvaluationBox EvaluationCell EvaluationCompletionAction EvaluationData EvaluationElements EvaluationEnvironment EvaluationMode EvaluationMonitor EvaluationNotebook EvaluationObject EvaluationOrder Evaluator EvaluatorNames EvenQ EventData EventEvaluator EventHandler EventHandlerTag EventLabels EventSeries ExactBlackmanWindow ExactNumberQ ExactRootIsolation ExampleData Except ExcludedForms ExcludedLines ExcludedPhysicalQuantities ExcludePods Exclusions ExclusionsStyle Exists Exit ExitDialog ExoplanetData Exp Expand ExpandAll ExpandDenominator ExpandFileName ExpandNumerator Expectation ExpectationE ExpectedValue ExpGammaDistribution ExpIntegralE ExpIntegralEi ExpirationDate Exponent ExponentFunction ExponentialDistribution ExponentialFamily ExponentialGeneratingFunction ExponentialMovingAverage ExponentialPowerDistribution ExponentPosition ExponentStep Export ExportAutoReplacements ExportByteArray ExportForm ExportPacket ExportString Expression ExpressionCell ExpressionPacket ExpressionUUID ExpToTrig ExtendedEntityClass ExtendedGCD Extension ExtentElementFunction ExtentMarkers ExtentSize ExternalBundle ExternalCall ExternalDataCharacterEncoding ExternalEvaluate ExternalFunction ExternalFunctionName ExternalObject ExternalOptions ExternalSessionObject ExternalSessions ExternalTypeSignature ExternalValue Extract ExtractArchive ExtractLayer ExtremeValueDistributionFaceForm FaceGrids FaceGridsStyle FacialFeatures Factor FactorComplete Factorial Factorial2 FactorialMoment FactorialMomentGeneratingFunction FactorialPower FactorInteger FactorList FactorSquareFree FactorSquareFreeList FactorTerms FactorTermsList Fail Failure FailureAction FailureDistribution FailureQ False FareySequence FARIMAProcess FeatureDistance FeatureExtract FeatureExtraction FeatureExtractor FeatureExtractorFunction FeatureNames FeatureNearest FeatureSpacePlot FeatureSpacePlot3D FeatureTypes FEDisableConsolePrintPacket FeedbackLinearize FeedbackSector FeedbackSectorStyle FeedbackType FEEnableConsolePrintPacket FetalGrowthData Fibonacci Fibonorial FieldCompletionFunction FieldHint FieldHintStyle FieldMasked FieldSize File FileBaseName FileByteCount FileConvert FileDate FileExistsQ FileExtension FileFormat FileHandler FileHash FileInformation FileName FileNameDepth FileNameDialogSettings FileNameDrop FileNameForms FileNameJoin FileNames FileNameSetter FileNameSplit FileNameTake FilePrint FileSize FileSystemMap FileSystemScan FileTemplate FileTemplateApply FileType FilledCurve FilledCurveBox FilledCurveBoxOptions Filling FillingStyle FillingTransform FilteredEntityClass FilterRules FinancialBond FinancialData FinancialDerivative FinancialIndicator Find FindAnomalies FindArgMax FindArgMin FindChannels FindClique FindClusters FindCookies FindCurvePath FindCycle FindDevices FindDistribution FindDistributionParameters FindDivisions FindEdgeCover FindEdgeCut FindEdgeIndependentPaths FindEquationalProof FindEulerianCycle FindExternalEvaluators FindFaces FindFile FindFit FindFormula FindFundamentalCycles FindGeneratingFunction FindGeoLocation FindGeometricConjectures FindGeometricTransform FindGraphCommunities FindGraphIsomorphism FindGraphPartition FindHamiltonianCycle FindHamiltonianPath FindHiddenMarkovStates FindIndependentEdgeSet FindIndependentVertexSet FindInstance FindIntegerNullVector FindKClan FindKClique FindKClub FindKPlex FindLibrary FindLinearRecurrence FindList FindMatchingColor FindMaximum FindMaximumFlow FindMaxValue FindMeshDefects FindMinimum FindMinimumCostFlow FindMinimumCut FindMinValue FindMoleculeSubstructure FindPath FindPeaks FindPermutation FindPostmanTour FindProcessParameters FindRepeat FindRoot FindSequenceFunction FindSettings FindShortestPath FindShortestTour FindSpanningTree FindSystemModelEquilibrium FindTextualAnswer FindThreshold FindTransientRepeat FindVertexCover FindVertexCut FindVertexIndependentPaths Fine FinishDynamic FiniteAbelianGroupCount FiniteGroupCount FiniteGroupData First FirstCase FirstPassageTimeDistribution FirstPosition FischerGroupFi22 FischerGroupFi23 FischerGroupFi24Prime FisherHypergeometricDistribution FisherRatioTest FisherZDistribution Fit FitAll FitRegularization FittedModel FixedOrder FixedPoint FixedPointList FlashSelection Flat Flatten FlattenAt FlattenLayer FlatTopWindow FlipView Floor FlowPolynomial FlushPrintOutputPacket Fold FoldList FoldPair FoldPairList FollowRedirects Font FontColor FontFamily FontForm FontName FontOpacity FontPostScriptName FontProperties FontReencoding FontSize FontSlant FontSubstitutions FontTracking FontVariations FontWeight For ForAll Format FormatRules FormatType FormatTypeAutoConvert FormatValues FormBox FormBoxOptions FormControl FormFunction FormLayoutFunction FormObject FormPage FormTheme FormulaData FormulaLookup FortranForm Forward ForwardBackward Fourier FourierCoefficient FourierCosCoefficient FourierCosSeries FourierCosTransform FourierDCT FourierDCTFilter FourierDCTMatrix FourierDST FourierDSTMatrix FourierMatrix FourierParameters FourierSequenceTransform FourierSeries FourierSinCoefficient FourierSinSeries FourierSinTransform FourierTransform FourierTrigSeries FractionalBrownianMotionProcess FractionalGaussianNoiseProcess FractionalPart FractionBox FractionBoxOptions FractionLine Frame FrameBox FrameBoxOptions Framed FrameInset FrameLabel Frameless FrameMargins FrameRate FrameStyle FrameTicks FrameTicksStyle FRatioDistribution FrechetDistribution FreeQ FrenetSerretSystem FrequencySamplingFilterKernel FresnelC FresnelF FresnelG FresnelS Friday FrobeniusNumber FrobeniusSolve FromAbsoluteTime FromCharacterCode FromCoefficientRules FromContinuedFraction FromDate FromDigits FromDMS FromEntity FromJulianDate FromLetterNumber FromPolarCoordinates FromRomanNumeral FromSphericalCoordinates FromUnixTime Front FrontEndDynamicExpression FrontEndEventActions FrontEndExecute FrontEndObject FrontEndResource FrontEndResourceString FrontEndStackSize FrontEndToken FrontEndTokenExecute FrontEndValueCache FrontEndVersion FrontFaceColor FrontFaceOpacity Full FullAxes FullDefinition FullForm FullGraphics FullInformationOutputRegulator FullOptions FullRegion FullSimplify Function FunctionCompile FunctionCompileExport FunctionCompileExportByteArray FunctionCompileExportLibrary FunctionCompileExportString FunctionDomain FunctionExpand FunctionInterpolation FunctionPeriod FunctionRange FunctionSpace FussellVeselyImportanceGaborFilter GaborMatrix GaborWavelet GainMargins GainPhaseMargins GalaxyData GalleryView Gamma GammaDistribution GammaRegularized GapPenalty GARCHProcess GatedRecurrentLayer Gather GatherBy GaugeFaceElementFunction GaugeFaceStyle GaugeFrameElementFunction GaugeFrameSize GaugeFrameStyle GaugeLabels GaugeMarkers GaugeStyle GaussianFilter GaussianIntegers GaussianMatrix GaussianOrthogonalMatrixDistribution GaussianSymplecticMatrixDistribution GaussianUnitaryMatrixDistribution GaussianWindow GCD GegenbauerC General GeneralizedLinearModelFit GenerateAsymmetricKeyPair GenerateConditions GeneratedCell GeneratedDocumentBinding GenerateDerivedKey GenerateDigitalSignature GenerateDocument GeneratedParameters GeneratedQuantityMagnitudes GenerateHTTPResponse GenerateSecuredAuthenticationKey GenerateSymmetricKey GeneratingFunction GeneratorDescription GeneratorHistoryLength GeneratorOutputType Generic GenericCylindricalDecomposition GenomeData GenomeLookup GeoAntipode GeoArea GeoArraySize GeoBackground GeoBoundingBox GeoBounds GeoBoundsRegion GeoBubbleChart GeoCenter GeoCircle GeodesicClosing GeodesicDilation GeodesicErosion GeodesicOpening GeoDestination GeodesyData GeoDirection GeoDisk GeoDisplacement GeoDistance GeoDistanceList GeoElevationData GeoEntities GeoGraphics GeogravityModelData GeoGridDirectionDifference GeoGridLines GeoGridLinesStyle GeoGridPosition GeoGridRange GeoGridRangePadding GeoGridUnitArea GeoGridUnitDistance GeoGridVector GeoGroup GeoHemisphere GeoHemisphereBoundary GeoHistogram GeoIdentify GeoImage GeoLabels GeoLength GeoListPlot GeoLocation GeologicalPeriodData GeomagneticModelData GeoMarker GeometricAssertion GeometricBrownianMotionProcess GeometricDistribution GeometricMean GeometricMeanFilter GeometricScene GeometricTransformation GeometricTransformation3DBox GeometricTransformation3DBoxOptions GeometricTransformationBox GeometricTransformationBoxOptions GeoModel GeoNearest GeoPath GeoPosition GeoPositionENU GeoPositionXYZ GeoProjection GeoProjectionData GeoRange GeoRangePadding GeoRegionValuePlot GeoResolution GeoScaleBar GeoServer GeoSmoothHistogram GeoStreamPlot GeoStyling GeoStylingImageFunction GeoVariant GeoVector GeoVectorENU GeoVectorPlot GeoVectorXYZ GeoVisibleRegion GeoVisibleRegionBoundary GeoWithinQ GeoZoomLevel GestureHandler GestureHandlerTag Get GetBoundingBoxSizePacket GetContext GetEnvironment GetFileName GetFrontEndOptionsDataPacket GetLinebreakInformationPacket GetMenusPacket GetPageBreakInformationPacket Glaisher GlobalClusteringCoefficient GlobalPreferences GlobalSession Glow GoldenAngle GoldenRatio GompertzMakehamDistribution GoodmanKruskalGamma GoodmanKruskalGammaTest Goto Grad Gradient GradientFilter GradientOrientationFilter GrammarApply GrammarRules GrammarToken Graph Graph3D GraphAssortativity GraphAutomorphismGroup GraphCenter GraphComplement GraphData GraphDensity GraphDiameter GraphDifference GraphDisjointUnion GraphDistance GraphDistanceMatrix GraphElementData GraphEmbedding GraphHighlight GraphHighlightStyle GraphHub Graphics Graphics3D Graphics3DBox Graphics3DBoxOptions GraphicsArray GraphicsBaseline GraphicsBox GraphicsBoxOptions GraphicsColor GraphicsColumn GraphicsComplex GraphicsComplex3DBox GraphicsComplex3DBoxOptions GraphicsComplexBox GraphicsComplexBoxOptions GraphicsContents GraphicsData GraphicsGrid GraphicsGridBox GraphicsGroup GraphicsGroup3DBox GraphicsGroup3DBoxOptions GraphicsGroupBox GraphicsGroupBoxOptions GraphicsGrouping GraphicsHighlightColor GraphicsRow GraphicsSpacing GraphicsStyle GraphIntersection GraphLayout GraphLinkEfficiency GraphPeriphery GraphPlot GraphPlot3D GraphPower GraphPropertyDistribution GraphQ GraphRadius GraphReciprocity GraphRoot GraphStyle GraphUnion Gray GrayLevel Greater GreaterEqual GreaterEqualLess GreaterEqualThan GreaterFullEqual GreaterGreater GreaterLess GreaterSlantEqual GreaterThan GreaterTilde Green GreenFunction Grid GridBaseline GridBox GridBoxAlignment GridBoxBackground GridBoxDividers GridBoxFrame GridBoxItemSize GridBoxItemStyle GridBoxOptions GridBoxSpacings GridCreationSettings GridDefaultElement GridElementStyleOptions GridFrame GridFrameMargins GridGraph GridLines GridLinesStyle GroebnerBasis GroupActionBase GroupBy GroupCentralizer GroupElementFromWord GroupElementPosition GroupElementQ GroupElements GroupElementToWord GroupGenerators Groupings GroupMultiplicationTable GroupOrbits GroupOrder GroupPageBreakWithin GroupSetwiseStabilizer GroupStabilizer GroupStabilizerChain GroupTogetherGrouping GroupTogetherNestedGrouping GrowCutComponents Gudermannian GuidedFilter GumbelDistributionHaarWavelet HadamardMatrix HalfLine HalfNormalDistribution HalfPlane HalfSpace HamiltonianGraphQ HammingDistance HammingWindow HandlerFunctions HandlerFunctionsKeys HankelH1 HankelH2 HankelMatrix HankelTransform HannPoissonWindow HannWindow HaradaNortonGroupHN HararyGraph HarmonicMean HarmonicMeanFilter HarmonicNumber Hash Haversine HazardFunction Head HeadCompose HeaderLines Heads HeavisideLambda HeavisidePi HeavisideTheta HeldGroupHe HeldPart HelpBrowserLookup HelpBrowserNotebook HelpBrowserSettings Here HermiteDecomposition HermiteH HermitianMatrixQ HessenbergDecomposition Hessian HexadecimalCharacter Hexahedron HexahedronBox HexahedronBoxOptions HiddenMarkovProcess HiddenSurface Highlighted HighlightGraph HighlightImage HighlightMesh HighpassFilter HigmanSimsGroupHS HilbertCurve HilbertFilter HilbertMatrix Histogram Histogram3D HistogramDistribution HistogramList HistogramTransform HistogramTransformInterpolation HistoricalPeriodData HitMissTransform HITSCentrality HjorthDistribution HodgeDual HoeffdingD HoeffdingDTest Hold HoldAll HoldAllComplete HoldComplete HoldFirst HoldForm HoldPattern HoldRest HolidayCalendar HomeDirectory HomePage Horizontal HorizontalForm HorizontalGauge HorizontalScrollPosition HornerForm HostLookup HotellingTSquareDistribution HoytDistribution HTMLSave HTTPErrorResponse HTTPRedirect HTTPRequest HTTPRequestData HTTPResponse Hue HumanGrowthData HumpDownHump HumpEqual HurwitzLerchPhi HurwitzZeta HyperbolicDistribution HypercubeGraph HyperexponentialDistribution Hyperfactorial Hypergeometric0F1 Hypergeometric0F1Regularized Hypergeometric1F1 Hypergeometric1F1Regularized Hypergeometric2F1 Hypergeometric2F1Regularized HypergeometricDistribution HypergeometricPFQ HypergeometricPFQRegularized HypergeometricU Hyperlink HyperlinkCreationSettings Hyperplane Hyphenation HyphenationOptions HypoexponentialDistribution HypothesisTestDataI IconData Iconize IconizedObject IconRules Icosahedron Identity IdentityMatrix If IgnoreCase IgnoreDiacritics IgnorePunctuation IgnoreSpellCheck IgnoringInactive Im Image Image3D Image3DProjection Image3DSlices ImageAccumulate ImageAdd ImageAdjust ImageAlign ImageApply ImageApplyIndexed ImageAspectRatio ImageAssemble ImageAugmentationLayer ImageBoundingBoxes ImageCache ImageCacheValid ImageCapture ImageCaptureFunction ImageCases ImageChannels ImageClip ImageCollage ImageColorSpace ImageCompose ImageContainsQ ImageContents ImageConvolve ImageCooccurrence ImageCorners ImageCorrelate ImageCorrespondingPoints ImageCrop ImageData ImageDeconvolve ImageDemosaic ImageDifference ImageDimensions ImageDisplacements ImageDistance ImageEffect ImageExposureCombine ImageFeatureTrack ImageFileApply ImageFileFilter ImageFileScan ImageFilter ImageFocusCombine ImageForestingComponents ImageFormattingWidth ImageForwardTransformation ImageGraphics ImageHistogram ImageIdentify ImageInstanceQ ImageKeypoints ImageLevels ImageLines ImageMargins ImageMarker ImageMarkers ImageMeasurements ImageMesh ImageMultiply ImageOffset ImagePad ImagePadding ImagePartition ImagePeriodogram ImagePerspectiveTransformation ImagePosition ImagePreviewFunction ImagePyramid ImagePyramidApply ImageQ ImageRangeCache ImageRecolor ImageReflect ImageRegion ImageResize ImageResolution ImageRestyle ImageRotate ImageRotated ImageSaliencyFilter ImageScaled ImageScan ImageSize ImageSizeAction ImageSizeCache ImageSizeMultipliers ImageSizeRaw ImageSubtract ImageTake ImageTransformation ImageTrim ImageType ImageValue ImageValuePositions ImagingDevice ImplicitRegion Implies Import ImportAutoReplacements ImportByteArray ImportOptions ImportString ImprovementImportance In Inactivate Inactive IncidenceGraph IncidenceList IncidenceMatrix IncludeAromaticBonds IncludeConstantBasis IncludeDefinitions IncludeDirectories IncludeFileExtension IncludeGeneratorTasks IncludeHydrogens IncludeInflections IncludeMetaInformation IncludePods IncludeQuantities IncludeRelatedTables IncludeSingularTerm IncludeWindowTimes Increment IndefiniteMatrixQ Indent IndentingNewlineSpacings IndentMaxFraction IndependenceTest IndependentEdgeSetQ IndependentPhysicalQuantity IndependentUnit IndependentUnitDimension IndependentVertexSetQ Indeterminate IndeterminateThreshold IndexCreationOptions Indexed IndexGraph IndexTag Inequality InexactNumberQ InexactNumbers InfiniteLine InfinitePlane Infinity Infix InflationAdjust InflationMethod Information InformationData InformationDataGrid Inherited InheritScope InhomogeneousPoissonProcess InitialEvaluationHistory Initialization InitializationCell InitializationCellEvaluation InitializationCellWarning InitializationObjects InitializationValue Initialize InitialSeeding InlineCounterAssignments InlineCounterIncrements InlineRules Inner InnerPolygon InnerPolyhedron Inpaint Input InputAliases InputAssumptions InputAutoReplacements InputField InputFieldBox InputFieldBoxOptions InputForm InputGrouping InputNamePacket InputNotebook InputPacket InputSettings InputStream InputString InputStringPacket InputToBoxFormPacket Insert InsertionFunction InsertionPointObject InsertLinebreaks InsertResults Inset Inset3DBox Inset3DBoxOptions InsetBox InsetBoxOptions Insphere Install InstallService InstanceNormalizationLayer InString Integer IntegerDigits IntegerExponent IntegerLength IntegerName IntegerPart IntegerPartitions IntegerQ IntegerReverse Integers IntegerString Integral Integrate Interactive InteractiveTradingChart Interlaced Interleaving InternallyBalancedDecomposition InterpolatingFunction InterpolatingPolynomial Interpolation InterpolationOrder InterpolationPoints InterpolationPrecision Interpretation InterpretationBox InterpretationBoxOptions InterpretationFunction Interpreter InterpretTemplate InterquartileRange Interrupt InterruptSettings IntersectingQ Intersection Interval IntervalIntersection IntervalMarkers IntervalMarkersStyle IntervalMemberQ IntervalSlider IntervalUnion Into Inverse InverseBetaRegularized InverseCDF InverseChiSquareDistribution InverseContinuousWaveletTransform InverseDistanceTransform InverseEllipticNomeQ InverseErf InverseErfc InverseFourier InverseFourierCosTransform InverseFourierSequenceTransform InverseFourierSinTransform InverseFourierTransform InverseFunction InverseFunctions InverseGammaDistribution InverseGammaRegularized InverseGaussianDistribution InverseGudermannian InverseHankelTransform InverseHaversine InverseImagePyramid InverseJacobiCD InverseJacobiCN InverseJacobiCS InverseJacobiDC InverseJacobiDN InverseJacobiDS InverseJacobiNC InverseJacobiND InverseJacobiNS InverseJacobiSC InverseJacobiSD InverseJacobiSN InverseLaplaceTransform InverseMellinTransform InversePermutation InverseRadon InverseRadonTransform InverseSeries InverseShortTimeFourier InverseSpectrogram InverseSurvivalFunction InverseTransformedRegion InverseWaveletTransform InverseWeierstrassP InverseWishartMatrixDistribution InverseZTransform Invisible InvisibleApplication InvisibleTimes IPAddress IrreduciblePolynomialQ IslandData IsolatingInterval IsomorphicGraphQ IsotopeData Italic Item ItemAspectRatio ItemBox ItemBoxOptions ItemSize ItemStyle ItoProcessJaccardDissimilarity JacobiAmplitude Jacobian JacobiCD JacobiCN JacobiCS JacobiDC JacobiDN JacobiDS JacobiNC JacobiND JacobiNS JacobiP JacobiSC JacobiSD JacobiSN JacobiSymbol JacobiZeta JankoGroupJ1 JankoGroupJ2 JankoGroupJ3 JankoGroupJ4 JarqueBeraALMTest JohnsonDistribution Join JoinAcross Joined JoinedCurve JoinedCurveBox JoinedCurveBoxOptions JoinForm JordanDecomposition JordanModelDecomposition JulianDate JuliaSetBoettcher JuliaSetIterationCount JuliaSetPlot JuliaSetPointsK KagiChart KaiserBesselWindow KaiserWindow KalmanEstimator KalmanFilter KarhunenLoeveDecomposition KaryTree KatzCentrality KCoreComponents KDistribution KEdgeConnectedComponents KEdgeConnectedGraphQ KelvinBei KelvinBer KelvinKei KelvinKer KendallTau KendallTauTest KernelExecute KernelFunction KernelMixtureDistribution Kernels Ket Key KeyCollisionFunction KeyComplement KeyDrop KeyDropFrom KeyExistsQ KeyFreeQ KeyIntersection KeyMap KeyMemberQ KeypointStrength Keys KeySelect KeySort KeySortBy KeyTake KeyUnion KeyValueMap KeyValuePattern Khinchin KillProcess KirchhoffGraph KirchhoffMatrix KleinInvariantJ KnapsackSolve KnightTourGraph KnotData KnownUnitQ KochCurve KolmogorovSmirnovTest KroneckerDelta KroneckerModelDecomposition KroneckerProduct KroneckerSymbol KuiperTest KumaraswamyDistribution Kurtosis KuwaharaFilter KVertexConnectedComponents KVertexConnectedGraphQLABColor Label Labeled LabeledSlider LabelingFunction LabelingSize LabelStyle LabelVisibility LaguerreL LakeData LambdaComponents LambertW LaminaData LanczosWindow LandauDistribution Language LanguageCategory LanguageData LanguageIdentify LanguageOptions LaplaceDistribution LaplaceTransform Laplacian LaplacianFilter LaplacianGaussianFilter Large Larger Last Latitude LatitudeLongitude LatticeData LatticeReduce Launch LaunchKernels LayeredGraphPlot LayerSizeFunction LayoutInformation LCHColor LCM LeaderSize LeafCount LeapYearQ LearnDistribution LearnedDistribution LearningRate LearningRateMultipliers LeastSquares LeastSquaresFilterKernel Left LeftArrow LeftArrowBar LeftArrowRightArrow LeftDownTeeVector LeftDownVector LeftDownVectorBar LeftRightArrow LeftRightVector LeftTee LeftTeeArrow LeftTeeVector LeftTriangle LeftTriangleBar LeftTriangleEqual LeftUpDownVector LeftUpTeeVector LeftUpVector LeftUpVectorBar LeftVector LeftVectorBar LegendAppearance Legended LegendFunction LegendLabel LegendLayout LegendMargins LegendMarkers LegendMarkerSize LegendreP LegendreQ LegendreType Length LengthWhile LerchPhi Less LessEqual LessEqualGreater LessEqualThan LessFullEqual LessGreater LessLess LessSlantEqual LessThan LessTilde LetterCharacter LetterCounts LetterNumber LetterQ Level LeveneTest LeviCivitaTensor LevyDistribution Lexicographic LibraryDataType LibraryFunction LibraryFunctionError LibraryFunctionInformation LibraryFunctionLoad LibraryFunctionUnload LibraryLoad LibraryUnload LicenseID LiftingFilterData LiftingWaveletTransform LightBlue LightBrown LightCyan Lighter LightGray LightGreen Lighting LightingAngle LightMagenta LightOrange LightPink LightPurple LightRed LightSources LightYellow Likelihood Limit LimitsPositioning LimitsPositioningTokens LindleyDistribution Line Line3DBox Line3DBoxOptions LinearFilter LinearFractionalOptimization LinearFractionalTransform LinearGradientImage LinearizingTransformationData LinearLayer LinearModelFit LinearOffsetFunction LinearOptimization LinearProgramming LinearRecurrence LinearSolve LinearSolveFunction LineBox LineBoxOptions LineBreak LinebreakAdjustments LineBreakChart LinebreakSemicolonWeighting LineBreakWithin LineColor LineGraph LineIndent LineIndentMaxFraction LineIntegralConvolutionPlot LineIntegralConvolutionScale LineLegend LineOpacity LineSpacing LineWrapParts LinkActivate LinkClose LinkConnect LinkConnectedQ LinkCreate LinkError LinkFlush LinkFunction LinkHost LinkInterrupt LinkLaunch LinkMode LinkObject LinkOpen LinkOptions LinkPatterns LinkProtocol LinkRankCentrality LinkRead LinkReadHeld LinkReadyQ Links LinkService LinkWrite LinkWriteHeld LiouvilleLambda List Listable ListAnimate ListContourPlot ListContourPlot3D ListConvolve ListCorrelate ListCurvePathPlot ListDeconvolve ListDensityPlot ListDensityPlot3D Listen ListFormat ListFourierSequenceTransform ListInterpolation ListLineIntegralConvolutionPlot ListLinePlot ListLogLinearPlot ListLogLogPlot ListLogPlot ListPicker ListPickerBox ListPickerBoxBackground ListPickerBoxOptions ListPlay ListPlot ListPlot3D ListPointPlot3D ListPolarPlot ListQ ListSliceContourPlot3D ListSliceDensityPlot3D ListSliceVectorPlot3D ListStepPlot ListStreamDensityPlot ListStreamPlot ListSurfacePlot3D ListVectorDensityPlot ListVectorPlot ListVectorPlot3D ListZTransform Literal LiteralSearch LocalAdaptiveBinarize LocalCache LocalClusteringCoefficient LocalizeDefinitions LocalizeVariables LocalObject LocalObjects LocalResponseNormalizationLayer LocalSubmit LocalSymbol LocalTime LocalTimeZone LocationEquivalenceTest LocationTest Locator LocatorAutoCreate LocatorBox LocatorBoxOptions LocatorCentering LocatorPane LocatorPaneBox LocatorPaneBoxOptions LocatorRegion Locked Log Log10 Log2 LogBarnesG LogGamma LogGammaDistribution LogicalExpand LogIntegral LogisticDistribution LogisticSigmoid LogitModelFit LogLikelihood LogLinearPlot LogLogisticDistribution LogLogPlot LogMultinormalDistribution LogNormalDistribution LogPlot LogRankTest LogSeriesDistribution LongEqual Longest LongestCommonSequence LongestCommonSequencePositions LongestCommonSubsequence LongestCommonSubsequencePositions LongestMatch LongestOrderedSequence LongForm Longitude LongLeftArrow LongLeftRightArrow LongRightArrow LongShortTermMemoryLayer Lookup Loopback LoopFreeGraphQ LossFunction LowerCaseQ LowerLeftArrow LowerRightArrow LowerTriangularize LowerTriangularMatrixQ LowpassFilter LQEstimatorGains LQGRegulator LQOutputRegulatorGains LQRegulatorGains LUBackSubstitution LucasL LuccioSamiComponents LUDecomposition LunarEclipse LUVColor LyapunovSolve LyonsGroupLyMachineID MachineName MachineNumberQ MachinePrecision MacintoshSystemPageSetup Magenta Magnification Magnify MailAddressValidation MailExecute MailFolder MailItem MailReceiverFunction MailResponseFunction MailSearch MailServerConnect MailServerConnection MailSettings MainSolve MaintainDynamicCaches Majority MakeBoxes MakeExpression MakeRules ManagedLibraryExpressionID ManagedLibraryExpressionQ MandelbrotSetBoettcher MandelbrotSetDistance MandelbrotSetIterationCount MandelbrotSetMemberQ MandelbrotSetPlot MangoldtLambda ManhattanDistance Manipulate Manipulator MannedSpaceMissionData MannWhitneyTest MantissaExponent Manual Map MapAll MapAt MapIndexed MAProcess MapThread MarchenkoPasturDistribution MarcumQ MardiaCombinedTest MardiaKurtosisTest MardiaSkewnessTest MarginalDistribution MarkovProcessProperties Masking MatchingDissimilarity MatchLocalNameQ MatchLocalNames MatchQ Material MathematicalFunctionData MathematicaNotation MathieuC MathieuCharacteristicA MathieuCharacteristicB MathieuCharacteristicExponent MathieuCPrime MathieuGroupM11 MathieuGroupM12 MathieuGroupM22 MathieuGroupM23 MathieuGroupM24 MathieuS MathieuSPrime MathMLForm MathMLText Matrices MatrixExp MatrixForm MatrixFunction MatrixLog MatrixNormalDistribution MatrixPlot MatrixPower MatrixPropertyDistribution MatrixQ MatrixRank MatrixTDistribution Max MaxBend MaxCellMeasure MaxColorDistance MaxDetect MaxDuration MaxExtraBandwidths MaxExtraConditions MaxFeatureDisplacement MaxFeatures MaxFilter MaximalBy Maximize MaxItems MaxIterations MaxLimit MaxMemoryUsed MaxMixtureKernels MaxOverlapFraction MaxPlotPoints MaxPoints MaxRecursion MaxStableDistribution MaxStepFraction MaxSteps MaxStepSize MaxTrainingRounds MaxValue MaxwellDistribution MaxWordGap McLaughlinGroupMcL Mean MeanAbsoluteLossLayer MeanAround MeanClusteringCoefficient MeanDegreeConnectivity MeanDeviation MeanFilter MeanGraphDistance MeanNeighborDegree MeanShift MeanShiftFilter MeanSquaredLossLayer Median MedianDeviation MedianFilter MedicalTestData Medium MeijerG MeijerGReduce MeixnerDistribution MellinConvolve MellinTransform MemberQ MemoryAvailable MemoryConstrained MemoryConstraint MemoryInUse MengerMesh Menu MenuAppearance MenuCommandKey MenuEvaluator MenuItem MenuList MenuPacket MenuSortingValue MenuStyle MenuView Merge MergeDifferences MergingFunction MersennePrimeExponent MersennePrimeExponentQ Mesh MeshCellCentroid MeshCellCount MeshCellHighlight MeshCellIndex MeshCellLabel MeshCellMarker MeshCellMeasure MeshCellQuality MeshCells MeshCellShapeFunction MeshCellStyle MeshCoordinates MeshFunctions MeshPrimitives MeshQualityGoal MeshRange MeshRefinementFunction MeshRegion MeshRegionQ MeshShading MeshStyle Message MessageDialog MessageList MessageName MessageObject MessageOptions MessagePacket Messages MessagesNotebook MetaCharacters MetaInformation MeteorShowerData Method MethodOptions MexicanHatWavelet MeyerWavelet Midpoint Min MinColorDistance MinDetect MineralData MinFilter MinimalBy MinimalPolynomial MinimalStateSpaceModel Minimize MinimumTimeIncrement MinIntervalSize MinkowskiQuestionMark MinLimit MinMax MinorPlanetData Minors MinRecursion MinSize MinStableDistribution Minus MinusPlus MinValue Missing MissingBehavior MissingDataMethod MissingDataRules MissingQ MissingString MissingStyle MissingValuePattern MittagLefflerE MixedFractionParts MixedGraphQ MixedMagnitude MixedRadix MixedRadixQuantity MixedUnit MixtureDistribution Mod Modal Mode Modular ModularInverse ModularLambda Module Modulus MoebiusMu Molecule MoleculeContainsQ MoleculeEquivalentQ MoleculeGraph MoleculeModify MoleculePattern MoleculePlot MoleculePlot3D MoleculeProperty MoleculeQ MoleculeValue Moment Momentary MomentConvert MomentEvaluate MomentGeneratingFunction MomentOfInertia Monday Monitor MonomialList MonomialOrder MonsterGroupM MoonPhase MoonPosition MorletWavelet MorphologicalBinarize MorphologicalBranchPoints MorphologicalComponents MorphologicalEulerNumber MorphologicalGraph MorphologicalPerimeter MorphologicalTransform MortalityData Most MountainData MouseAnnotation MouseAppearance MouseAppearanceTag MouseButtons Mouseover MousePointerNote MousePosition MovieData MovingAverage MovingMap MovingMedian MoyalDistribution Multicolumn MultiedgeStyle MultigraphQ MultilaunchWarning MultiLetterItalics MultiLetterStyle MultilineFunction Multinomial MultinomialDistribution MultinormalDistribution MultiplicativeOrder Multiplicity MultiplySides Multiselection MultivariateHypergeometricDistribution MultivariatePoissonDistribution MultivariateTDistributionN NakagamiDistribution NameQ Names NamespaceBox NamespaceBoxOptions Nand NArgMax NArgMin NBernoulliB NBodySimulation NBodySimulationData NCache NDEigensystem NDEigenvalues NDSolve NDSolveValue Nearest NearestFunction NearestNeighborGraph NearestTo NebulaData NeedCurrentFrontEndPackagePacket NeedCurrentFrontEndSymbolsPacket NeedlemanWunschSimilarity Needs Negative NegativeBinomialDistribution NegativeDefiniteMatrixQ NegativeIntegers NegativeMultinomialDistribution NegativeRationals NegativeReals NegativeSemidefiniteMatrixQ NeighborhoodData NeighborhoodGraph Nest NestedGreaterGreater NestedLessLess NestedScriptRules NestGraph NestList NestWhile NestWhileList NetAppend NetBidirectionalOperator NetChain NetDecoder NetDelete NetDrop NetEncoder NetEvaluationMode NetExtract NetFlatten NetFoldOperator NetGraph NetInformation NetInitialize NetInsert NetInsertSharedArrays NetJoin NetMapOperator NetMapThreadOperator NetMeasurements NetModel NetNestOperator NetPairEmbeddingOperator NetPort NetPortGradient NetPrepend NetRename NetReplace NetReplacePart NetSharedArray NetStateObject NetTake NetTrain NetTrainResultsObject NetworkPacketCapture NetworkPacketRecording NetworkPacketRecordingDuring NetworkPacketTrace NeumannValue NevilleThetaC NevilleThetaD NevilleThetaN NevilleThetaS NewPrimitiveStyle NExpectation Next NextCell NextDate NextPrime NextScheduledTaskTime NHoldAll NHoldFirst NHoldRest NicholsGridLines NicholsPlot NightHemisphere NIntegrate NMaximize NMaxValue NMinimize NMinValue NominalVariables NonAssociative NoncentralBetaDistribution NoncentralChiSquareDistribution NoncentralFRatioDistribution NoncentralStudentTDistribution NonCommutativeMultiply NonConstants NondimensionalizationTransform None NoneTrue NonlinearModelFit NonlinearStateSpaceModel NonlocalMeansFilter NonNegative NonNegativeIntegers NonNegativeRationals NonNegativeReals NonPositive NonPositiveIntegers NonPositiveRationals NonPositiveReals Nor NorlundB Norm Normal NormalDistribution NormalGrouping NormalizationLayer Normalize Normalized NormalizedSquaredEuclideanDistance NormalMatrixQ NormalsFunction NormFunction Not NotCongruent NotCupCap NotDoubleVerticalBar Notebook NotebookApply NotebookAutoSave NotebookClose NotebookConvertSettings NotebookCreate NotebookCreateReturnObject NotebookDefault NotebookDelete NotebookDirectory NotebookDynamicExpression NotebookEvaluate NotebookEventActions NotebookFileName NotebookFind NotebookFindReturnObject NotebookGet NotebookGetLayoutInformationPacket NotebookGetMisspellingsPacket NotebookImport NotebookInformation NotebookInterfaceObject NotebookLocate NotebookObject NotebookOpen NotebookOpenReturnObject NotebookPath NotebookPrint NotebookPut NotebookPutReturnObject NotebookRead NotebookResetGeneratedCells Notebooks NotebookSave NotebookSaveAs NotebookSelection NotebookSetupLayoutInformationPacket NotebooksMenu NotebookTemplate NotebookWrite NotElement NotEqualTilde NotExists NotGreater NotGreaterEqual NotGreaterFullEqual NotGreaterGreater NotGreaterLess NotGreaterSlantEqual NotGreaterTilde Nothing NotHumpDownHump NotHumpEqual NotificationFunction NotLeftTriangle NotLeftTriangleBar NotLeftTriangleEqual NotLess NotLessEqual NotLessFullEqual NotLessGreater NotLessLess NotLessSlantEqual NotLessTilde NotNestedGreaterGreater NotNestedLessLess NotPrecedes NotPrecedesEqual NotPrecedesSlantEqual NotPrecedesTilde NotReverseElement NotRightTriangle NotRightTriangleBar NotRightTriangleEqual NotSquareSubset NotSquareSubsetEqual NotSquareSuperset NotSquareSupersetEqual NotSubset NotSubsetEqual NotSucceeds NotSucceedsEqual NotSucceedsSlantEqual NotSucceedsTilde NotSuperset NotSupersetEqual NotTilde NotTildeEqual NotTildeFullEqual NotTildeTilde NotVerticalBar Now NoWhitespace NProbability NProduct NProductFactors NRoots NSolve NSum NSumTerms NuclearExplosionData NuclearReactorData Null NullRecords NullSpace NullWords Number NumberCompose NumberDecompose NumberExpand NumberFieldClassNumber NumberFieldDiscriminant NumberFieldFundamentalUnits NumberFieldIntegralBasis NumberFieldNormRepresentatives NumberFieldRegulator NumberFieldRootsOfUnity NumberFieldSignature NumberForm NumberFormat NumberLinePlot NumberMarks NumberMultiplier NumberPadding NumberPoint NumberQ NumberSeparator NumberSigns NumberString Numerator NumeratorDenominator NumericalOrder NumericalSort NumericArray NumericArrayQ NumericArrayType NumericFunction NumericQ NuttallWindow NValues NyquistGridLines NyquistPlotO ObservabilityGramian ObservabilityMatrix ObservableDecomposition ObservableModelQ OceanData Octahedron OddQ Off Offset OLEData On ONanGroupON Once OneIdentity Opacity OpacityFunction OpacityFunctionScaling Open OpenAppend Opener OpenerBox OpenerBoxOptions OpenerView OpenFunctionInspectorPacket Opening OpenRead OpenSpecialOptions OpenTemporary OpenWrite Operate OperatingSystem OptimumFlowData Optional OptionalElement OptionInspectorSettings OptionQ Options OptionsPacket OptionsPattern OptionValue OptionValueBox OptionValueBoxOptions Or Orange Order OrderDistribution OrderedQ Ordering OrderingBy OrderingLayer Orderless OrderlessPatternSequence OrnsteinUhlenbeckProcess Orthogonalize OrthogonalMatrixQ Out Outer OuterPolygon OuterPolyhedron OutputAutoOverwrite OutputControllabilityMatrix OutputControllableModelQ OutputForm OutputFormData OutputGrouping OutputMathEditExpression OutputNamePacket OutputResponse OutputSizeLimit OutputStream Over OverBar OverDot Overflow OverHat Overlaps Overlay OverlayBox OverlayBoxOptions Overscript OverscriptBox OverscriptBoxOptions OverTilde OverVector OverwriteTarget OwenT OwnValuesPackage PackingMethod PaddedForm Padding PaddingLayer PaddingSize PadeApproximant PadLeft PadRight PageBreakAbove PageBreakBelow PageBreakWithin PageFooterLines PageFooters PageHeaderLines PageHeaders PageHeight PageRankCentrality PageTheme PageWidth Pagination PairedBarChart PairedHistogram PairedSmoothHistogram PairedTTest PairedZTest PaletteNotebook PalettePath PalindromeQ Pane PaneBox PaneBoxOptions Panel PanelBox PanelBoxOptions Paneled PaneSelector PaneSelectorBox PaneSelectorBoxOptions PaperWidth ParabolicCylinderD ParagraphIndent ParagraphSpacing ParallelArray ParallelCombine ParallelDo Parallelepiped ParallelEvaluate Parallelization Parallelize ParallelMap ParallelNeeds Parallelogram ParallelProduct ParallelSubmit ParallelSum ParallelTable ParallelTry Parameter ParameterEstimator ParameterMixtureDistribution ParameterVariables ParametricFunction ParametricNDSolve ParametricNDSolveValue ParametricPlot ParametricPlot3D ParametricRegion ParentBox ParentCell ParentConnect ParentDirectory ParentForm Parenthesize ParentList ParentNotebook ParetoDistribution ParetoPickandsDistribution ParkData Part PartBehavior PartialCorrelationFunction PartialD ParticleAcceleratorData ParticleData Partition PartitionGranularity PartitionsP PartitionsQ PartLayer PartOfSpeech PartProtection ParzenWindow PascalDistribution PassEventsDown PassEventsUp Paste PasteAutoQuoteCharacters PasteBoxFormInlineCells PasteButton Path PathGraph PathGraphQ Pattern PatternSequence PatternTest PauliMatrix PaulWavelet Pause PausedTime PDF PeakDetect PeanoCurve PearsonChiSquareTest PearsonCorrelationTest PearsonDistribution PercentForm PerfectNumber PerfectNumberQ PerformanceGoal Perimeter PeriodicBoundaryCondition PeriodicInterpolation Periodogram PeriodogramArray Permanent Permissions PermissionsGroup PermissionsGroupMemberQ PermissionsGroups PermissionsKey PermissionsKeys PermutationCycles PermutationCyclesQ PermutationGroup PermutationLength PermutationList PermutationListQ PermutationMax PermutationMin PermutationOrder PermutationPower PermutationProduct PermutationReplace Permutations PermutationSupport Permute PeronaMalikFilter Perpendicular PerpendicularBisector PersistenceLocation PersistenceTime PersistentObject PersistentObjects PersistentValue PersonData PERTDistribution PetersenGraph PhaseMargins PhaseRange PhysicalSystemData Pi Pick PIDData PIDDerivativeFilter PIDFeedforward PIDTune Piecewise PiecewiseExpand PieChart PieChart3D PillaiTrace PillaiTraceTest PingTime Pink PitchRecognize Pivoting PixelConstrained PixelValue PixelValuePositions Placed Placeholder PlaceholderReplace Plain PlanarAngle PlanarGraph PlanarGraphQ PlanckRadiationLaw PlaneCurveData PlanetaryMoonData PlanetData PlantData Play PlayRange Plot Plot3D Plot3Matrix PlotDivision PlotJoined PlotLabel PlotLabels PlotLayout PlotLegends PlotMarkers PlotPoints PlotRange PlotRangeClipping PlotRangeClipPlanesStyle PlotRangePadding PlotRegion PlotStyle PlotTheme Pluralize Plus PlusMinus Pochhammer PodStates PodWidth Point Point3DBox Point3DBoxOptions PointBox PointBoxOptions PointFigureChart PointLegend PointSize PoissonConsulDistribution PoissonDistribution PoissonProcess PoissonWindow PolarAxes PolarAxesOrigin PolarGridLines PolarPlot PolarTicks PoleZeroMarkers PolyaAeppliDistribution PolyGamma Polygon Polygon3DBox Polygon3DBoxOptions PolygonalNumber PolygonAngle PolygonBox PolygonBoxOptions PolygonCoordinates PolygonDecomposition PolygonHoleScale PolygonIntersections PolygonScale Polyhedron PolyhedronAngle PolyhedronCoordinates PolyhedronData PolyhedronDecomposition PolyhedronGenus PolyLog PolynomialExtendedGCD PolynomialForm PolynomialGCD PolynomialLCM PolynomialMod PolynomialQ PolynomialQuotient PolynomialQuotientRemainder PolynomialReduce PolynomialRemainder Polynomials PoolingLayer PopupMenu PopupMenuBox PopupMenuBoxOptions PopupView PopupWindow Position PositionIndex Positive PositiveDefiniteMatrixQ PositiveIntegers PositiveRationals PositiveReals PositiveSemidefiniteMatrixQ PossibleZeroQ Postfix PostScript Power PowerDistribution PowerExpand PowerMod PowerModList PowerRange PowerSpectralDensity PowersRepresentations PowerSymmetricPolynomial Precedence PrecedenceForm Precedes PrecedesEqual PrecedesSlantEqual PrecedesTilde Precision PrecisionGoal PreDecrement Predict PredictionRoot PredictorFunction PredictorInformation PredictorMeasurements PredictorMeasurementsObject PreemptProtect PreferencesPath Prefix PreIncrement Prepend PrependLayer PrependTo PreprocessingRules PreserveColor PreserveImageOptions Previous PreviousCell PreviousDate PriceGraphDistribution PrimaryPlaceholder Prime PrimeNu PrimeOmega PrimePi PrimePowerQ PrimeQ Primes PrimeZetaP PrimitivePolynomialQ PrimitiveRoot PrimitiveRootList PrincipalComponents PrincipalValue Print PrintableASCIIQ PrintAction PrintForm PrintingCopies PrintingOptions PrintingPageRange PrintingStartingPageNumber PrintingStyleEnvironment Printout3D Printout3DPreviewer PrintPrecision PrintTemporary Prism PrismBox PrismBoxOptions PrivateCellOptions PrivateEvaluationOptions PrivateFontOptions PrivateFrontEndOptions PrivateKey PrivateNotebookOptions PrivatePaths Probability ProbabilityDistribution ProbabilityPlot ProbabilityPr ProbabilityScalePlot ProbitModelFit ProcessConnection ProcessDirectory ProcessEnvironment Processes ProcessEstimator ProcessInformation ProcessObject ProcessParameterAssumptions ProcessParameterQ ProcessStateDomain ProcessStatus ProcessTimeDomain Product ProductDistribution ProductLog ProgressIndicator ProgressIndicatorBox ProgressIndicatorBoxOptions Projection Prolog PromptForm ProofObject Properties Property PropertyList PropertyValue Proportion Proportional Protect Protected ProteinData Pruning PseudoInverse PsychrometricPropertyData PublicKey PublisherID PulsarData PunctuationCharacter Purple Put PutAppend Pyramid PyramidBox PyramidBoxOptionsQBinomial QFactorial QGamma QHypergeometricPFQ QnDispersion QPochhammer QPolyGamma QRDecomposition QuadraticIrrationalQ QuadraticOptimization Quantile QuantilePlot Quantity QuantityArray QuantityDistribution QuantityForm QuantityMagnitude QuantityQ QuantityUnit QuantityVariable QuantityVariableCanonicalUnit QuantityVariableDimensions QuantityVariableIdentifier QuantityVariablePhysicalQuantity Quartics QuartileDeviation Quartiles QuartileSkewness Query QueueingNetworkProcess QueueingProcess QueueProperties Quiet Quit Quotient QuotientRemainderRadialGradientImage RadialityCentrality RadicalBox RadicalBoxOptions RadioButton RadioButtonBar RadioButtonBox RadioButtonBoxOptions Radon RadonTransform RamanujanTau RamanujanTauL RamanujanTauTheta RamanujanTauZ Ramp Random RandomChoice RandomColor RandomComplex RandomEntity RandomFunction RandomGeoPosition RandomGraph RandomImage RandomInstance RandomInteger RandomPermutation RandomPoint RandomPolygon RandomPolyhedron RandomPrime RandomReal RandomSample RandomSeed RandomSeeding RandomVariate RandomWalkProcess RandomWord Range RangeFilter RangeSpecification RankedMax RankedMin RarerProbability Raster Raster3D Raster3DBox Raster3DBoxOptions RasterArray RasterBox RasterBoxOptions Rasterize RasterSize Rational RationalFunctions Rationalize Rationals Ratios RawArray RawBoxes RawData RawMedium RayleighDistribution Re Read ReadByteArray ReadLine ReadList ReadProtected ReadString Real RealAbs RealBlockDiagonalForm RealDigits RealExponent Reals RealSign Reap RecognitionPrior RecognitionThreshold Record RecordLists RecordSeparators Rectangle RectangleBox RectangleBoxOptions RectangleChart RectangleChart3D RectangularRepeatingElement RecurrenceFilter RecurrenceTable RecurringDigitsForm Red Reduce RefBox ReferenceLineStyle ReferenceMarkers ReferenceMarkerStyle Refine ReflectionMatrix ReflectionTransform Refresh RefreshRate Region RegionBinarize RegionBoundary RegionBounds RegionCentroid RegionDifference RegionDimension RegionDisjoint RegionDistance RegionDistanceFunction RegionEmbeddingDimension RegionEqual RegionFunction RegionImage RegionIntersection RegionMeasure RegionMember RegionMemberFunction RegionMoment RegionNearest RegionNearestFunction RegionPlot RegionPlot3D RegionProduct RegionQ RegionResize RegionSize RegionSymmetricDifference RegionUnion RegionWithin RegisterExternalEvaluator RegularExpression Regularization RegularlySampledQ RegularPolygon ReIm ReImLabels ReImPlot ReImStyle Reinstall RelationalDatabase RelationGraph Release ReleaseHold ReliabilityDistribution ReliefImage ReliefPlot RemoteAuthorizationCaching RemoteConnect RemoteConnectionObject RemoteFile RemoteRun RemoteRunProcess Remove RemoveAlphaChannel RemoveAsynchronousTask RemoveAudioStream RemoveBackground RemoveChannelListener RemoveChannelSubscribers Removed RemoveDiacritics RemoveInputStreamMethod RemoveOutputStreamMethod RemoveProperty RemoveScheduledTask RemoveUsers RenameDirectory RenameFile RenderAll RenderingOptions RenewalProcess RenkoChart RepairMesh Repeated RepeatedNull RepeatedString RepeatedTiming RepeatingElement Replace ReplaceAll ReplaceHeldPart ReplaceImageValue ReplaceList ReplacePart ReplacePixelValue ReplaceRepeated ReplicateLayer RequiredPhysicalQuantities Resampling ResamplingAlgorithmData ResamplingMethod Rescale RescalingTransform ResetDirectory ResetMenusPacket ResetScheduledTask ReshapeLayer Residue ResizeLayer Resolve ResourceAcquire ResourceData ResourceFunction ResourceObject ResourceRegister ResourceRemove ResourceSearch ResourceSubmissionObject ResourceSubmit ResourceSystemBase ResourceUpdate ResponseForm Rest RestartInterval Restricted Resultant ResumePacket Return ReturnEntersInput ReturnExpressionPacket ReturnInputFormPacket ReturnPacket ReturnReceiptFunction ReturnTextPacket Reverse ReverseBiorthogonalSplineWavelet ReverseElement ReverseEquilibrium ReverseGraph ReverseSort ReverseSortBy ReverseUpEquilibrium RevolutionAxis RevolutionPlot3D RGBColor RiccatiSolve RiceDistribution RidgeFilter RiemannR RiemannSiegelTheta RiemannSiegelZ RiemannXi Riffle Right RightArrow RightArrowBar RightArrowLeftArrow RightComposition RightCosetRepresentative RightDownTeeVector RightDownVector RightDownVectorBar RightTee RightTeeArrow RightTeeVector RightTriangle RightTriangleBar RightTriangleEqual RightUpDownVector RightUpTeeVector RightUpVector RightUpVectorBar RightVector RightVectorBar RiskAchievementImportance RiskReductionImportance RogersTanimotoDissimilarity RollPitchYawAngles RollPitchYawMatrix RomanNumeral Root RootApproximant RootIntervals RootLocusPlot RootMeanSquare RootOfUnityQ RootReduce Roots RootSum Rotate RotateLabel RotateLeft RotateRight RotationAction RotationBox RotationBoxOptions RotationMatrix RotationTransform Round RoundImplies RoundingRadius Row RowAlignments RowBackgrounds RowBox RowHeights RowLines RowMinHeight RowReduce RowsEqual RowSpacings RSolve RSolveValue RudinShapiro RudvalisGroupRu Rule RuleCondition RuleDelayed RuleForm RulePlot RulerUnits Run RunProcess RunScheduledTask RunThrough RuntimeAttributes RuntimeOptions RussellRaoDissimilaritySameQ SameTest SampledEntityClass SampleDepth SampledSoundFunction SampledSoundList SampleRate SamplingPeriod SARIMAProcess SARMAProcess SASTriangle SatelliteData SatisfiabilityCount SatisfiabilityInstances SatisfiableQ Saturday Save Saveable SaveAutoDelete SaveConnection SaveDefinitions SavitzkyGolayMatrix SawtoothWave Scale Scaled ScaleDivisions ScaledMousePosition ScaleOrigin ScalePadding ScaleRanges ScaleRangeStyle ScalingFunctions ScalingMatrix ScalingTransform Scan ScheduledTask ScheduledTaskActiveQ ScheduledTaskInformation ScheduledTaskInformationData ScheduledTaskObject ScheduledTasks SchurDecomposition ScientificForm ScientificNotationThreshold ScorerGi ScorerGiPrime ScorerHi ScorerHiPrime ScreenRectangle ScreenStyleEnvironment ScriptBaselineShifts ScriptForm ScriptLevel ScriptMinSize ScriptRules ScriptSizeMultipliers Scrollbars ScrollingOptions ScrollPosition SearchAdjustment SearchIndexObject SearchIndices SearchQueryString SearchResultObject Sec Sech SechDistribution SecondOrderConeOptimization SectionGrouping SectorChart SectorChart3D SectorOrigin SectorSpacing SecuredAuthenticationKey SecuredAuthenticationKeys SeedRandom Select Selectable SelectComponents SelectedCells SelectedNotebook SelectFirst Selection SelectionAnimate SelectionCell SelectionCellCreateCell SelectionCellDefaultStyle SelectionCellParentStyle SelectionCreateCell SelectionDebuggerTag SelectionDuplicateCell SelectionEvaluate SelectionEvaluateCreateCell SelectionMove SelectionPlaceholder SelectionSetStyle SelectWithContents SelfLoops SelfLoopStyle SemanticImport SemanticImportString SemanticInterpretation SemialgebraicComponentInstances SemidefiniteOptimization SendMail SendMessage Sequence SequenceAlignment SequenceAttentionLayer SequenceCases SequenceCount SequenceFold SequenceFoldList SequenceForm SequenceHold SequenceLastLayer SequenceMostLayer SequencePosition SequencePredict SequencePredictorFunction SequenceReplace SequenceRestLayer SequenceReverseLayer SequenceSplit Series SeriesCoefficient SeriesData ServiceConnect ServiceDisconnect ServiceExecute ServiceObject ServiceRequest ServiceResponse ServiceSubmit SessionSubmit SessionTime Set SetAccuracy SetAlphaChannel SetAttributes Setbacks SetBoxFormNamesPacket SetCloudDirectory SetCookies SetDelayed SetDirectory SetEnvironment SetEvaluationNotebook SetFileDate SetFileLoadingContext SetNotebookStatusLine SetOptions SetOptionsPacket SetPermissions SetPrecision SetProperty SetSecuredAuthenticationKey SetSelectedNotebook SetSharedFunction SetSharedVariable SetSpeechParametersPacket SetStreamPosition SetSystemModel SetSystemOptions Setter SetterBar SetterBox SetterBoxOptions Setting SetUsers SetValue Shading Shallow ShannonWavelet ShapiroWilkTest Share SharingList Sharpen ShearingMatrix ShearingTransform ShellRegion ShenCastanMatrix ShiftedGompertzDistribution ShiftRegisterSequence Short ShortDownArrow Shortest ShortestMatch ShortestPathFunction ShortLeftArrow ShortRightArrow ShortTimeFourier ShortTimeFourierData ShortUpArrow Show ShowAutoConvert ShowAutoSpellCheck ShowAutoStyles ShowCellBracket ShowCellLabel ShowCellTags ShowClosedCellArea ShowCodeAssist ShowContents ShowControls ShowCursorTracker ShowGroupOpenCloseIcon ShowGroupOpener ShowInvisibleCharacters ShowPageBreaks ShowPredictiveInterface ShowSelection ShowShortBoxForm ShowSpecialCharacters ShowStringCharacters ShowSyntaxStyles ShrinkingDelay ShrinkWrapBoundingBox SiderealTime SiegelTheta SiegelTukeyTest SierpinskiCurve SierpinskiMesh Sign Signature SignedRankTest SignedRegionDistance SignificanceLevel SignPadding SignTest SimilarityRules SimpleGraph SimpleGraphQ SimplePolygonQ SimplePolyhedronQ Simplex Simplify Sin Sinc SinghMaddalaDistribution SingleEvaluation SingleLetterItalics SingleLetterStyle SingularValueDecomposition SingularValueList SingularValuePlot SingularValues Sinh SinhIntegral SinIntegral SixJSymbol Skeleton SkeletonTransform SkellamDistribution Skewness SkewNormalDistribution SkinStyle Skip SliceContourPlot3D SliceDensityPlot3D SliceDistribution SliceVectorPlot3D Slider Slider2D Slider2DBox Slider2DBoxOptions SliderBox SliderBoxOptions SlideView Slot SlotSequence Small SmallCircle Smaller SmithDecomposition SmithDelayCompensator SmithWatermanSimilarity SmoothDensityHistogram SmoothHistogram SmoothHistogram3D SmoothKernelDistribution SnDispersion Snippet SnubPolyhedron SocialMediaData Socket SocketConnect SocketListen SocketListener SocketObject SocketOpen SocketReadMessage SocketReadyQ Sockets SocketWaitAll SocketWaitNext SoftmaxLayer SokalSneathDissimilarity SolarEclipse SolarSystemFeatureData SolidAngle SolidData SolidRegionQ Solve SolveAlways SolveDelayed Sort SortBy SortedBy SortedEntityClass Sound SoundAndGraphics SoundNote SoundVolume SourceLink Sow Space SpaceCurveData SpaceForm Spacer Spacings Span SpanAdjustments SpanCharacterRounding SpanFromAbove SpanFromBoth SpanFromLeft SpanLineThickness SpanMaxSize SpanMinSize SpanningCharacters SpanSymmetric SparseArray SpatialGraphDistribution SpatialMedian SpatialTransformationLayer Speak SpeakTextPacket SpearmanRankTest SpearmanRho SpeciesData SpecificityGoal SpectralLineData Spectrogram SpectrogramArray Specularity SpeechRecognize SpeechSynthesize SpellingCorrection SpellingCorrectionList SpellingDictionaries SpellingDictionariesPath SpellingOptions SpellingSuggestionsPacket Sphere SphereBox SpherePoints SphericalBesselJ SphericalBesselY SphericalHankelH1 SphericalHankelH2 SphericalHarmonicY SphericalPlot3D SphericalRegion SphericalShell SpheroidalEigenvalue SpheroidalJoiningFactor SpheroidalPS SpheroidalPSPrime SpheroidalQS SpheroidalQSPrime SpheroidalRadialFactor SpheroidalS1 SpheroidalS1Prime SpheroidalS2 SpheroidalS2Prime Splice SplicedDistribution SplineClosed SplineDegree SplineKnots SplineWeights Split SplitBy SpokenString Sqrt SqrtBox SqrtBoxOptions Square SquaredEuclideanDistance SquareFreeQ SquareIntersection SquareMatrixQ SquareRepeatingElement SquaresR SquareSubset SquareSubsetEqual SquareSuperset SquareSupersetEqual SquareUnion SquareWave SSSTriangle StabilityMargins StabilityMarginsStyle StableDistribution Stack StackBegin StackComplete StackedDateListPlot StackedListPlot StackInhibit StadiumShape StandardAtmosphereData StandardDeviation StandardDeviationFilter StandardForm Standardize Standardized StandardOceanData StandbyDistribution Star StarClusterData StarData StarGraph StartAsynchronousTask StartExternalSession StartingStepSize StartOfLine StartOfString StartProcess StartScheduledTask StartupSound StartWebSession StateDimensions StateFeedbackGains StateOutputEstimator StateResponse StateSpaceModel StateSpaceRealization StateSpaceTransform StateTransformationLinearize StationaryDistribution StationaryWaveletPacketTransform StationaryWaveletTransform StatusArea StatusCentrality StepMonitor StereochemistryElements StieltjesGamma StirlingS1 StirlingS2 StopAsynchronousTask StoppingPowerData StopScheduledTask StrataVariables StratonovichProcess StreamColorFunction StreamColorFunctionScaling StreamDensityPlot StreamMarkers StreamPlot StreamPoints StreamPosition Streams StreamScale StreamStyle String StringBreak StringByteCount StringCases StringContainsQ StringCount StringDelete StringDrop StringEndsQ StringExpression StringExtract StringForm StringFormat StringFreeQ StringInsert StringJoin StringLength StringMatchQ StringPadLeft StringPadRight StringPart StringPartition StringPosition StringQ StringRepeat StringReplace StringReplaceList StringReplacePart StringReverse StringRiffle StringRotateLeft StringRotateRight StringSkeleton StringSplit StringStartsQ StringTake StringTemplate StringToByteArray StringToStream StringTrim StripBoxes StripOnInput StripWrapperBoxes StrokeForm StructuralImportance StructuredArray StructuredSelection StruveH StruveL Stub StudentTDistribution Style StyleBox StyleBoxAutoDelete StyleData StyleDefinitions StyleForm StyleHints StyleKeyMapping StyleMenuListing StyleNameDialogSettings StyleNames StylePrint StyleSheetPath Subdivide Subfactorial Subgraph SubMinus SubPlus SubresultantPolynomialRemainders SubresultantPolynomials Subresultants Subscript SubscriptBox SubscriptBoxOptions Subscripted Subsequences Subset SubsetEqual SubsetMap SubsetQ Subsets SubStar SubstitutionSystem Subsuperscript SubsuperscriptBox SubsuperscriptBoxOptions Subtract SubtractFrom SubtractSides SubValues Succeeds SucceedsEqual SucceedsSlantEqual SucceedsTilde Success SuchThat Sum SumConvergence SummationLayer Sunday SunPosition Sunrise Sunset SuperDagger SuperMinus SupernovaData SuperPlus Superscript SuperscriptBox SuperscriptBoxOptions Superset SupersetEqual SuperStar Surd SurdForm SurfaceArea SurfaceColor SurfaceData SurfaceGraphics SurvivalDistribution SurvivalFunction SurvivalModel SurvivalModelFit SuspendPacket SuzukiDistribution SuzukiGroupSuz SwatchLegend Switch Symbol SymbolName SymletWavelet Symmetric 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TouchPosition TouchscreenAutoZoom TouchscreenControlPlacement ToUpperCase Tr Trace TraceAbove TraceAction TraceBackward TraceDepth TraceDialog TraceForward TraceInternal TraceLevel TraceOff TraceOn TraceOriginal TracePrint TraceScan TrackedSymbols TrackingFunction TracyWidomDistribution TradingChart TraditionalForm TraditionalFunctionNotation TraditionalNotation TraditionalOrder TrainingProgressCheckpointing TrainingProgressFunction TrainingProgressMeasurements TrainingProgressReporting TrainingStoppingCriterion TransferFunctionCancel TransferFunctionExpand TransferFunctionFactor TransferFunctionModel TransferFunctionPoles TransferFunctionTransform TransferFunctionZeros TransformationClass TransformationFunction TransformationFunctions TransformationMatrix TransformedDistribution TransformedField TransformedProcess TransformedRegion TransitionDirection TransitionDuration TransitionEffect TransitiveClosureGraph TransitiveReductionGraph Translate TranslationOptions TranslationTransform 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ds_grid_clear ds_grid_set ds_grid_add ds_grid_multiply ds_grid_set_region ds_grid_add_region ds_grid_multiply_region ds_grid_set_disk ds_grid_add_disk ds_grid_multiply_disk ds_grid_set_grid_region ds_grid_add_grid_region ds_grid_multiply_grid_region ds_grid_get ds_grid_get_sum ds_grid_get_max ds_grid_get_min ds_grid_get_mean ds_grid_get_disk_sum ds_grid_get_disk_min ds_grid_get_disk_max ds_grid_get_disk_mean ds_grid_value_exists ds_grid_value_x ds_grid_value_y ds_grid_value_disk_exists ds_grid_value_disk_x ds_grid_value_disk_y ds_grid_shuffle ds_grid_write ds_grid_read ds_grid_sort ds_grid_set ds_grid_get effect_create_below effect_create_above effect_clear part_type_create part_type_destroy part_type_exists part_type_clear part_type_shape part_type_sprite part_type_size part_type_scale part_type_orientation part_type_life part_type_step part_type_death part_type_speed part_type_direction part_type_gravity part_type_colour1 part_type_colour2 part_type_colour3 part_type_colour_mix part_type_colour_rgb part_type_colour_hsv part_type_color1 part_type_color2 part_type_color3 part_type_color_mix part_type_color_rgb part_type_color_hsv part_type_alpha1 part_type_alpha2 part_type_alpha3 part_type_blend part_system_create part_system_create_layer part_system_destroy part_system_exists part_system_clear part_system_draw_order part_system_depth part_system_position part_system_automatic_update part_system_automatic_draw part_system_update part_system_drawit part_system_get_layer part_system_layer part_particles_create part_particles_create_colour part_particles_create_color part_particles_clear part_particles_count part_emitter_create part_emitter_destroy part_emitter_destroy_all part_emitter_exists part_emitter_clear part_emitter_region part_emitter_burst part_emitter_stream external_call external_define external_free window_handle window_device matrix_get matrix_set matrix_build_identity matrix_build matrix_build_lookat matrix_build_projection_ortho matrix_build_projection_perspective matrix_build_projection_perspective_fov matrix_multiply matrix_transform_vertex matrix_stack_push matrix_stack_pop matrix_stack_multiply matrix_stack_set matrix_stack_clear matrix_stack_top matrix_stack_is_empty browser_input_capture os_get_config os_get_info os_get_language os_get_region os_lock_orientation display_get_dpi_x display_get_dpi_y display_set_gui_size display_set_gui_maximise display_set_gui_maximize device_mouse_dbclick_enable display_set_timing_method display_get_timing_method display_set_sleep_margin display_get_sleep_margin virtual_key_add virtual_key_hide virtual_key_delete virtual_key_show draw_enable_drawevent draw_enable_swf_aa draw_set_swf_aa_level draw_get_swf_aa_level draw_texture_flush draw_flush gpu_set_blendenable gpu_set_ztestenable gpu_set_zfunc gpu_set_zwriteenable gpu_set_lightingenable gpu_set_fog gpu_set_cullmode gpu_set_blendmode gpu_set_blendmode_ext gpu_set_blendmode_ext_sepalpha gpu_set_colorwriteenable gpu_set_colourwriteenable gpu_set_alphatestenable gpu_set_alphatestref gpu_set_alphatestfunc gpu_set_texfilter gpu_set_texfilter_ext gpu_set_texrepeat gpu_set_texrepeat_ext gpu_set_tex_filter gpu_set_tex_filter_ext gpu_set_tex_repeat gpu_set_tex_repeat_ext gpu_set_tex_mip_filter gpu_set_tex_mip_filter_ext gpu_set_tex_mip_bias gpu_set_tex_mip_bias_ext gpu_set_tex_min_mip gpu_set_tex_min_mip_ext gpu_set_tex_max_mip gpu_set_tex_max_mip_ext gpu_set_tex_max_aniso gpu_set_tex_max_aniso_ext gpu_set_tex_mip_enable gpu_set_tex_mip_enable_ext gpu_get_blendenable gpu_get_ztestenable gpu_get_zfunc gpu_get_zwriteenable gpu_get_lightingenable gpu_get_fog gpu_get_cullmode gpu_get_blendmode gpu_get_blendmode_ext gpu_get_blendmode_ext_sepalpha gpu_get_blendmode_src gpu_get_blendmode_dest gpu_get_blendmode_srcalpha gpu_get_blendmode_destalpha gpu_get_colorwriteenable gpu_get_colourwriteenable gpu_get_alphatestenable gpu_get_alphatestref gpu_get_alphatestfunc gpu_get_texfilter gpu_get_texfilter_ext gpu_get_texrepeat gpu_get_texrepeat_ext gpu_get_tex_filter gpu_get_tex_filter_ext gpu_get_tex_repeat gpu_get_tex_repeat_ext gpu_get_tex_mip_filter gpu_get_tex_mip_filter_ext gpu_get_tex_mip_bias gpu_get_tex_mip_bias_ext gpu_get_tex_min_mip gpu_get_tex_min_mip_ext gpu_get_tex_max_mip gpu_get_tex_max_mip_ext gpu_get_tex_max_aniso gpu_get_tex_max_aniso_ext gpu_get_tex_mip_enable gpu_get_tex_mip_enable_ext gpu_push_state gpu_pop_state gpu_get_state gpu_set_state draw_light_define_ambient draw_light_define_direction draw_light_define_point draw_light_enable draw_set_lighting draw_light_get_ambient draw_light_get draw_get_lighting shop_leave_rating url_get_domain url_open url_open_ext url_open_full get_timer achievement_login achievement_logout achievement_post achievement_increment achievement_post_score achievement_available achievement_show_achievements achievement_show_leaderboards achievement_load_friends achievement_load_leaderboard achievement_send_challenge achievement_load_progress achievement_reset achievement_login_status achievement_get_pic achievement_show_challenge_notifications achievement_get_challenges achievement_event achievement_show achievement_get_info cloud_file_save cloud_string_save cloud_synchronise ads_enable ads_disable ads_setup ads_engagement_launch ads_engagement_available ads_engagement_active ads_event ads_event_preload ads_set_reward_callback ads_get_display_height ads_get_display_width ads_move ads_interstitial_available ads_interstitial_display device_get_tilt_x device_get_tilt_y device_get_tilt_z device_is_keypad_open device_mouse_check_button device_mouse_check_button_pressed device_mouse_check_button_released device_mouse_x device_mouse_y device_mouse_raw_x device_mouse_raw_y device_mouse_x_to_gui device_mouse_y_to_gui iap_activate iap_status iap_enumerate_products iap_restore_all iap_acquire iap_consume iap_product_details iap_purchase_details facebook_init facebook_login facebook_status facebook_graph_request facebook_dialog facebook_logout facebook_launch_offerwall facebook_post_message facebook_send_invite facebook_user_id facebook_accesstoken facebook_check_permission facebook_request_read_permissions facebook_request_publish_permissions gamepad_is_supported gamepad_get_device_count gamepad_is_connected gamepad_get_description gamepad_get_button_threshold gamepad_set_button_threshold gamepad_get_axis_deadzone gamepad_set_axis_deadzone gamepad_button_count gamepad_button_check gamepad_button_check_pressed gamepad_button_check_released gamepad_button_value gamepad_axis_count gamepad_axis_value gamepad_set_vibration gamepad_set_colour gamepad_set_color os_is_paused window_has_focus code_is_compiled http_get http_get_file http_post_string http_request json_encode json_decode zip_unzip load_csv base64_encode base64_decode md5_string_unicode md5_string_utf8 md5_file os_is_network_connected sha1_string_unicode sha1_string_utf8 sha1_file os_powersave_enable analytics_event analytics_event_ext win8_livetile_tile_notification win8_livetile_tile_clear win8_livetile_badge_notification win8_livetile_badge_clear win8_livetile_queue_enable win8_secondarytile_pin win8_secondarytile_badge_notification win8_secondarytile_delete win8_livetile_notification_begin win8_livetile_notification_secondary_begin win8_livetile_notification_expiry win8_livetile_notification_tag win8_livetile_notification_text_add win8_livetile_notification_image_add win8_livetile_notification_end win8_appbar_enable win8_appbar_add_element win8_appbar_remove_element win8_settingscharm_add_entry win8_settingscharm_add_html_entry win8_settingscharm_add_xaml_entry win8_settingscharm_set_xaml_property win8_settingscharm_get_xaml_property win8_settingscharm_remove_entry win8_share_image win8_share_screenshot win8_share_file win8_share_url win8_share_text win8_search_enable win8_search_disable win8_search_add_suggestions win8_device_touchscreen_available win8_license_initialize_sandbox win8_license_trial_version winphone_license_trial_version winphone_tile_title winphone_tile_count winphone_tile_back_title winphone_tile_back_content winphone_tile_back_content_wide winphone_tile_front_image winphone_tile_front_image_small winphone_tile_front_image_wide winphone_tile_back_image winphone_tile_back_image_wide winphone_tile_background_colour winphone_tile_background_color winphone_tile_icon_image winphone_tile_small_icon_image winphone_tile_wide_content winphone_tile_cycle_images winphone_tile_small_background_image physics_world_create physics_world_gravity physics_world_update_speed physics_world_update_iterations physics_world_draw_debug physics_pause_enable physics_fixture_create physics_fixture_set_kinematic physics_fixture_set_density physics_fixture_set_awake physics_fixture_set_restitution physics_fixture_set_friction physics_fixture_set_collision_group physics_fixture_set_sensor physics_fixture_set_linear_damping physics_fixture_set_angular_damping physics_fixture_set_circle_shape physics_fixture_set_box_shape physics_fixture_set_edge_shape physics_fixture_set_polygon_shape physics_fixture_set_chain_shape physics_fixture_add_point physics_fixture_bind physics_fixture_bind_ext physics_fixture_delete physics_apply_force physics_apply_impulse physics_apply_angular_impulse physics_apply_local_force physics_apply_local_impulse physics_apply_torque physics_mass_properties physics_draw_debug physics_test_overlap physics_remove_fixture physics_set_friction physics_set_density physics_set_restitution physics_get_friction physics_get_density physics_get_restitution physics_joint_distance_create physics_joint_rope_create physics_joint_revolute_create physics_joint_prismatic_create physics_joint_pulley_create physics_joint_wheel_create physics_joint_weld_create physics_joint_friction_create physics_joint_gear_create physics_joint_enable_motor physics_joint_get_value physics_joint_set_value physics_joint_delete physics_particle_create physics_particle_delete physics_particle_delete_region_circle physics_particle_delete_region_box physics_particle_delete_region_poly physics_particle_set_flags physics_particle_set_category_flags physics_particle_draw physics_particle_draw_ext physics_particle_count physics_particle_get_data physics_particle_get_data_particle physics_particle_group_begin physics_particle_group_circle physics_particle_group_box physics_particle_group_polygon physics_particle_group_add_point physics_particle_group_end physics_particle_group_join physics_particle_group_delete physics_particle_group_count physics_particle_group_get_data physics_particle_group_get_mass physics_particle_group_get_inertia physics_particle_group_get_centre_x physics_particle_group_get_centre_y physics_particle_group_get_vel_x physics_particle_group_get_vel_y physics_particle_group_get_ang_vel physics_particle_group_get_x physics_particle_group_get_y physics_particle_group_get_angle physics_particle_set_group_flags physics_particle_get_group_flags physics_particle_get_max_count physics_particle_get_radius physics_particle_get_density physics_particle_get_damping physics_particle_get_gravity_scale physics_particle_set_max_count physics_particle_set_radius physics_particle_set_density physics_particle_set_damping physics_particle_set_gravity_scale network_create_socket network_create_socket_ext network_create_server network_create_server_raw network_connect network_connect_raw network_send_packet network_send_raw network_send_broadcast network_send_udp network_send_udp_raw network_set_timeout network_set_config network_resolve network_destroy buffer_create buffer_write buffer_read buffer_seek buffer_get_surface buffer_set_surface buffer_delete buffer_exists buffer_get_type buffer_get_alignment buffer_poke buffer_peek buffer_save buffer_save_ext buffer_load buffer_load_ext buffer_load_partial buffer_copy buffer_fill buffer_get_size buffer_tell buffer_resize buffer_md5 buffer_sha1 buffer_base64_encode buffer_base64_decode buffer_base64_decode_ext buffer_sizeof buffer_get_address buffer_create_from_vertex_buffer buffer_create_from_vertex_buffer_ext buffer_copy_from_vertex_buffer buffer_async_group_begin buffer_async_group_option buffer_async_group_end buffer_load_async buffer_save_async gml_release_mode gml_pragma steam_activate_overlay steam_is_overlay_enabled steam_is_overlay_activated steam_get_persona_name steam_initialised steam_is_cloud_enabled_for_app steam_is_cloud_enabled_for_account steam_file_persisted steam_get_quota_total steam_get_quota_free steam_file_write steam_file_write_file steam_file_read steam_file_delete steam_file_exists steam_file_size steam_file_share steam_is_screenshot_requested steam_send_screenshot steam_is_user_logged_on steam_get_user_steam_id steam_user_owns_dlc steam_user_installed_dlc steam_set_achievement steam_get_achievement steam_clear_achievement steam_set_stat_int steam_set_stat_float steam_set_stat_avg_rate steam_get_stat_int steam_get_stat_float steam_get_stat_avg_rate steam_reset_all_stats steam_reset_all_stats_achievements steam_stats_ready steam_create_leaderboard steam_upload_score steam_upload_score_ext steam_download_scores_around_user steam_download_scores steam_download_friends_scores steam_upload_score_buffer steam_upload_score_buffer_ext steam_current_game_language steam_available_languages steam_activate_overlay_browser steam_activate_overlay_user steam_activate_overlay_store steam_get_user_persona_name steam_get_app_id steam_get_user_account_id steam_ugc_download steam_ugc_create_item steam_ugc_start_item_update steam_ugc_set_item_title steam_ugc_set_item_description steam_ugc_set_item_visibility steam_ugc_set_item_tags steam_ugc_set_item_content steam_ugc_set_item_preview steam_ugc_submit_item_update steam_ugc_get_item_update_progress steam_ugc_subscribe_item steam_ugc_unsubscribe_item steam_ugc_num_subscribed_items steam_ugc_get_subscribed_items steam_ugc_get_item_install_info steam_ugc_get_item_update_info steam_ugc_request_item_details steam_ugc_create_query_user steam_ugc_create_query_user_ex steam_ugc_create_query_all steam_ugc_create_query_all_ex steam_ugc_query_set_cloud_filename_filter steam_ugc_query_set_match_any_tag steam_ugc_query_set_search_text steam_ugc_query_set_ranked_by_trend_days steam_ugc_query_add_required_tag steam_ugc_query_add_excluded_tag steam_ugc_query_set_return_long_description steam_ugc_query_set_return_total_only steam_ugc_query_set_allow_cached_response steam_ugc_send_query shader_set shader_get_name shader_reset shader_current shader_is_compiled shader_get_sampler_index shader_get_uniform shader_set_uniform_i shader_set_uniform_i_array shader_set_uniform_f shader_set_uniform_f_array shader_set_uniform_matrix shader_set_uniform_matrix_array shader_enable_corner_id texture_set_stage texture_get_texel_width texture_get_texel_height shaders_are_supported vertex_format_begin vertex_format_end vertex_format_delete vertex_format_add_position vertex_format_add_position_3d vertex_format_add_colour vertex_format_add_color vertex_format_add_normal vertex_format_add_texcoord vertex_format_add_textcoord vertex_format_add_custom vertex_create_buffer vertex_create_buffer_ext vertex_delete_buffer vertex_begin vertex_end vertex_position vertex_position_3d vertex_colour vertex_color vertex_argb vertex_texcoord vertex_normal vertex_float1 vertex_float2 vertex_float3 vertex_float4 vertex_ubyte4 vertex_submit vertex_freeze vertex_get_number vertex_get_buffer_size vertex_create_buffer_from_buffer vertex_create_buffer_from_buffer_ext push_local_notification push_get_first_local_notification push_get_next_local_notification push_cancel_local_notification skeleton_animation_set skeleton_animation_get skeleton_animation_mix skeleton_animation_set_ext skeleton_animation_get_ext skeleton_animation_get_duration skeleton_animation_get_frames skeleton_animation_clear skeleton_skin_set skeleton_skin_get skeleton_attachment_set skeleton_attachment_get skeleton_attachment_create skeleton_collision_draw_set skeleton_bone_data_get skeleton_bone_data_set skeleton_bone_state_get skeleton_bone_state_set skeleton_get_minmax skeleton_get_num_bounds skeleton_get_bounds skeleton_animation_get_frame skeleton_animation_set_frame draw_skeleton draw_skeleton_time draw_skeleton_instance draw_skeleton_collision skeleton_animation_list skeleton_skin_list skeleton_slot_data layer_get_id layer_get_id_at_depth layer_get_depth layer_create layer_destroy layer_destroy_instances layer_add_instance layer_has_instance layer_set_visible layer_get_visible layer_exists layer_x layer_y layer_get_x layer_get_y layer_hspeed layer_vspeed layer_get_hspeed layer_get_vspeed layer_script_begin layer_script_end layer_shader layer_get_script_begin layer_get_script_end layer_get_shader layer_set_target_room layer_get_target_room layer_reset_target_room layer_get_all layer_get_all_elements layer_get_name layer_depth layer_get_element_layer layer_get_element_type layer_element_move layer_force_draw_depth layer_is_draw_depth_forced layer_get_forced_depth layer_background_get_id layer_background_exists layer_background_create layer_background_destroy layer_background_visible layer_background_change layer_background_sprite layer_background_htiled layer_background_vtiled layer_background_stretch layer_background_yscale layer_background_xscale layer_background_blend layer_background_alpha layer_background_index layer_background_speed layer_background_get_visible layer_background_get_sprite layer_background_get_htiled layer_background_get_vtiled layer_background_get_stretch layer_background_get_yscale layer_background_get_xscale layer_background_get_blend layer_background_get_alpha layer_background_get_index layer_background_get_speed layer_sprite_get_id layer_sprite_exists layer_sprite_create layer_sprite_destroy layer_sprite_change layer_sprite_index layer_sprite_speed layer_sprite_xscale layer_sprite_yscale layer_sprite_angle layer_sprite_blend layer_sprite_alpha layer_sprite_x layer_sprite_y layer_sprite_get_sprite layer_sprite_get_index layer_sprite_get_speed layer_sprite_get_xscale layer_sprite_get_yscale layer_sprite_get_angle layer_sprite_get_blend layer_sprite_get_alpha layer_sprite_get_x layer_sprite_get_y layer_tilemap_get_id layer_tilemap_exists layer_tilemap_create layer_tilemap_destroy tilemap_tileset tilemap_x tilemap_y tilemap_set tilemap_set_at_pixel tilemap_get_tileset tilemap_get_tile_width tilemap_get_tile_height tilemap_get_width tilemap_get_height tilemap_get_x tilemap_get_y tilemap_get tilemap_get_at_pixel tilemap_get_cell_x_at_pixel tilemap_get_cell_y_at_pixel tilemap_clear draw_tilemap draw_tile tilemap_set_global_mask tilemap_get_global_mask tilemap_set_mask tilemap_get_mask tilemap_get_frame tile_set_empty tile_set_index tile_set_flip tile_set_mirror tile_set_rotate tile_get_empty tile_get_index tile_get_flip tile_get_mirror tile_get_rotate layer_tile_exists layer_tile_create layer_tile_destroy layer_tile_change layer_tile_xscale layer_tile_yscale layer_tile_blend layer_tile_alpha layer_tile_x layer_tile_y layer_tile_region layer_tile_visible layer_tile_get_sprite layer_tile_get_xscale layer_tile_get_yscale layer_tile_get_blend layer_tile_get_alpha layer_tile_get_x layer_tile_get_y layer_tile_get_region layer_tile_get_visible layer_instance_get_instance instance_activate_layer instance_deactivate_layer camera_create camera_create_view camera_destroy camera_apply camera_get_active camera_get_default camera_set_default camera_set_view_mat camera_set_proj_mat camera_set_update_script camera_set_begin_script camera_set_end_script camera_set_view_pos camera_set_view_size camera_set_view_speed camera_set_view_border camera_set_view_angle camera_set_view_target camera_get_view_mat camera_get_proj_mat camera_get_update_script camera_get_begin_script camera_get_end_script camera_get_view_x camera_get_view_y camera_get_view_width camera_get_view_height camera_get_view_speed_x camera_get_view_speed_y camera_get_view_border_x camera_get_view_border_y camera_get_view_angle camera_get_view_target view_get_camera view_get_visible view_get_xport view_get_yport view_get_wport view_get_hport view_get_surface_id view_set_camera view_set_visible view_set_xport view_set_yport view_set_wport view_set_hport view_set_surface_id gesture_drag_time gesture_drag_distance gesture_flick_speed gesture_double_tap_time gesture_double_tap_distance gesture_pinch_distance gesture_pinch_angle_towards gesture_pinch_angle_away gesture_rotate_time gesture_rotate_angle gesture_tap_count gesture_get_drag_time gesture_get_drag_distance gesture_get_flick_speed gesture_get_double_tap_time gesture_get_double_tap_distance gesture_get_pinch_distance gesture_get_pinch_angle_towards gesture_get_pinch_angle_away gesture_get_rotate_time gesture_get_rotate_angle gesture_get_tap_count keyboard_virtual_show keyboard_virtual_hide keyboard_virtual_status 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nearest pack present product ' +\n 'radix random_number random_seed range repeat reshape rrspacing scale scan selected_int_kind selected_real_kind ' +\n 'set_exponent shape size spacing spread sum system_clock tiny transpose trim ubound unpack verify achar iachar transfer ' +\n 'dble entry dprod cpu_time command_argument_count get_command get_command_argument get_environment_variable is_iostat_end ' +\n 'ieee_arithmetic ieee_support_underflow_control ieee_get_underflow_mode ieee_set_underflow_mode ' +\n 'is_iostat_eor move_alloc new_line selected_char_kind same_type_as extends_type_of' +\n 'acosh asinh atanh bessel_j0 bessel_j1 bessel_jn bessel_y0 bessel_y1 bessel_yn erf erfc erfc_scaled gamma log_gamma hypot norm2 ' +\n 'atomic_define atomic_ref execute_command_line leadz trailz storage_size merge_bits ' +\n 'bge bgt ble blt dshiftl dshiftr findloc iall iany iparity image_index lcobound ucobound maskl maskr ' +\n 'num_images parity popcnt poppar shifta shiftl shiftr this_image ' +\n 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manipScaleLimitsCtx marker match max memory menu menuBarLayout menuEditor menuItem menuItemToShelf menuSet menuSetPref messageLine min minimizeApp mirrorJoint modelCurrentTimeCtx modelEditor modelPanel mouse movIn movOut move moveIKtoFK moveKeyCtx moveVertexAlongDirection multiProfileBirailSurface mute nParticle nameCommand nameField namespace namespaceInfo newPanelItems newton nodeCast nodeIconButton nodeOutliner nodePreset nodeType noise nonLinear normalConstraint normalize nurbsBoolean nurbsCopyUVSet nurbsCube nurbsEditUV nurbsPlane nurbsSelect nurbsSquare nurbsToPoly nurbsToPolygonsPref nurbsToSubdiv nurbsToSubdivPref nurbsUVSet nurbsViewDirectionVector objExists objectCenter objectLayer objectType objectTypeUI obsoleteProc oceanNurbsPreviewPlane offsetCurve offsetCurveOnSurface offsetSurface openGLExtension openMayaPref optionMenu optionMenuGrp optionVar orbit orbitCtx orientConstraint outlinerEditor outlinerPanel overrideModifier paintEffectsDisplay pairBlend palettePort paneLayout panel panelConfiguration panelHistory paramDimContext paramDimension paramLocator parent parentConstraint particle particleExists particleInstancer particleRenderInfo partition pasteKey pathAnimation pause pclose percent performanceOptions pfxstrokes pickWalk picture pixelMove planarSrf plane play playbackOptions playblast plugAttr plugNode pluginInfo pluginResourceUtil pointConstraint pointCurveConstraint pointLight pointMatrixMult pointOnCurve pointOnSurface pointPosition poleVectorConstraint polyAppend polyAppendFacetCtx polyAppendVertex polyAutoProjection polyAverageNormal polyAverageVertex polyBevel polyBlendColor polyBlindData polyBoolOp polyBridgeEdge polyCacheMonitor polyCheck polyChipOff polyClipboard polyCloseBorder polyCollapseEdge polyCollapseFacet polyColorBlindData polyColorDel polyColorPerVertex polyColorSet polyCompare polyCone polyCopyUV polyCrease polyCreaseCtx polyCreateFacet polyCreateFacetCtx polyCube polyCut polyCutCtx polyCylinder 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Get-Date Get-Event Get-EventLog Get-EventSubscriber Get-ExecutionPolicy Get-FormatData Get-Host Get-HotFix Get-Help Get-History Get-IseSnippet Get-Item Get-ItemProperty Get-Job Get-JobTrigger Get-Location Get-Member Get-Module Get-PfxCertificate Get-Process Get-PSBreakpoint Get-PSCallStack Get-PSDrive Get-PSProvider Get-PSSession Get-PSSessionConfiguration Get-PSSnapin Get-Random Get-ScheduledJob Get-ScheduledJobOption Get-Service Get-TraceSource Get-Transaction Get-TypeData Get-UICulture Get-Unique Get-Variable Get-Verb Get-WinEvent Get-WmiObject Get-WSManCredSSP Get-WSManInstance Group-Object Import-Alias Import-Clixml Import-Counter Import-Csv Import-IseSnippet Import-LocalizedData Import-PSSession Import-Module Invoke-AsWorkflow Invoke-Command Invoke-Expression Invoke-History Invoke-Item Invoke-RestMethod Invoke-WebRequest Invoke-WmiMethod Invoke-WSManAction Join-Path Limit-EventLog Measure-Command Measure-Object Move-Item Move-ItemProperty New-Alias New-Event New-EventLog New-IseSnippet New-Item New-ItemProperty New-JobTrigger New-Object New-Module New-ModuleManifest New-PSDrive New-PSSession New-PSSessionConfigurationFile New-PSSessionOption New-PSTransportOption New-PSWorkflowExecutionOption New-PSWorkflowSession New-ScheduledJobOption New-Service New-TimeSpan New-Variable New-WebServiceProxy New-WinEvent New-WSManInstance New-WSManSessionOption Out-Default Out-File Out-GridView Out-Host Out-Null Out-Printer Out-String Pop-Location Push-Location Read-Host Receive-Job Register-EngineEvent Register-ObjectEvent Register-PSSessionConfiguration Register-ScheduledJob Register-WmiEvent Remove-Computer Remove-Event Remove-EventLog Remove-Item Remove-ItemProperty Remove-Job Remove-JobTrigger Remove-Module Remove-PSBreakpoint Remove-PSDrive Remove-PSSession Remove-PSSnapin Remove-TypeData Remove-Variable Remove-WmiObject Remove-WSManInstance Rename-Computer Rename-Item Rename-ItemProperty Reset-ComputerMachinePassword Resolve-Path Restart-Computer Restart-Service Restore-Computer Resume-Job Resume-Service Save-Help Select-Object Select-String Select-Xml Send-MailMessage Set-Acl Set-Alias Set-AuthenticodeSignature Set-Content Set-Date Set-ExecutionPolicy Set-Item Set-ItemProperty Set-JobTrigger Set-Location Set-PSBreakpoint Set-PSDebug Set-PSSessionConfiguration Set-ScheduledJob Set-ScheduledJobOption Set-Service Set-StrictMode Set-TraceSource Set-Variable Set-WmiInstance Set-WSManInstance Set-WSManQuickConfig Show-Command Show-ControlPanelItem Show-EventLog Sort-Object Split-Path Start-Job Start-Process Start-Service Start-Sleep Start-Transaction Start-Transcript Stop-Computer Stop-Job Stop-Process Stop-Service Stop-Transcript Suspend-Job Suspend-Service Tee-Object Test-ComputerSecureChannel Test-Connection Test-ModuleManifest Test-Path Test-PSSessionConfigurationFile Trace-Command Unblock-File Undo-Transaction Unregister-Event Unregister-PSSessionConfiguration Unregister-ScheduledJob Update-FormatData Update-Help Update-List Update-TypeData Use-Transaction Wait-Event Wait-Job Wait-Process Where-Object Write-Debug Write-Error Write-EventLog Write-Host Write-Output Write-Progress Write-Verbose Write-Warning Add-MDTPersistentDrive Disable-MDTMonitorService Enable-MDTMonitorService Get-MDTDeploymentShareStatistics Get-MDTMonitorData Get-MDTOperatingSystemCatalog Get-MDTPersistentDrive Import-MDTApplication Import-MDTDriver Import-MDTOperatingSystem Import-MDTPackage Import-MDTTaskSequence New-MDTDatabase Remove-MDTMonitorData Remove-MDTPersistentDrive Restore-MDTPersistentDrive Set-MDTMonitorData Test-MDTDeploymentShare Test-MDTMonitorData Update-MDTDatabaseSchema Update-MDTDeploymentShare Update-MDTLinkedDS Update-MDTMedia Add-VamtProductKey Export-VamtData Find-VamtManagedMachine Get-VamtConfirmationId Get-VamtProduct Get-VamtProductKey Import-VamtData Initialize-VamtData Install-VamtConfirmationId Install-VamtProductActivation Install-VamtProductKey Update-VamtProduct Add-CIDatastore Add-KeyManagementServer Add-NodeKeys Add-NsxDynamicCriteria Add-NsxDynamicMemberSet Add-NsxEdgeInterfaceAddress Add-NsxFirewallExclusionListMember Add-NsxFirewallRuleMember Add-NsxIpSetMember Add-NsxLicense Add-NsxLoadBalancerPoolMember Add-NsxLoadBalancerVip Add-NsxSecondaryManager Add-NsxSecurityGroupMember Add-NsxSecurityPolicyRule Add-NsxSecurityPolicyRuleGroup Add-NsxSecurityPolicyRuleService Add-NsxServiceGroupMember Add-NsxTransportZoneMember Add-PassthroughDevice Add-VDSwitchPhysicalNetworkAdapter Add-VDSwitchVMHost Add-VMHost Add-VMHostNtpServer Add-VirtualSwitchPhysicalNetworkAdapter Add-XmlElement Add-vRACustomForm Add-vRAPrincipalToTenantRole Add-vRAReservationNetwork Add-vRAReservationStorage Clear-NsxEdgeInterface Clear-NsxManagerTimeSettings Compress-Archive Connect-CIServer Connect-CisServer Connect-HCXServer Connect-NIServer Connect-NsxLogicalSwitch Connect-NsxServer Connect-NsxtServer Connect-SrmServer Connect-VIServer Connect-Vmc Connect-vRAServer Connect-vRNIServer ConvertFrom-Markdown ConvertTo-MOFInstance Copy-DatastoreItem Copy-HardDisk Copy-NsxEdge Copy-VDisk Copy-VMGuestFile Debug-Runspace Disable-NsxEdgeSsh Disable-RunspaceDebug Disable-vRNIDataSource Disconnect-CIServer Disconnect-CisServer Disconnect-HCXServer Disconnect-NsxLogicalSwitch Disconnect-NsxServer Disconnect-NsxtServer Disconnect-SrmServer Disconnect-VIServer Disconnect-Vmc Disconnect-vRAServer Disconnect-vRNIServer Dismount-Tools Enable-NsxEdgeSsh Enable-RunspaceDebug Enable-vRNIDataSource Expand-Archive Export-NsxObject Export-SpbmStoragePolicy Export-VApp Export-VDPortGroup Export-VDSwitch Export-VMHostProfile Export-vRAIcon Export-vRAPackage Find-Command Find-DscResource Find-Module Find-NsxWhereVMUsed Find-Package Find-PackageProvider Find-RoleCapability Find-Script Format-Hex Format-VMHostDiskPartition Format-XML Generate-VersionInfo Get-AdvancedSetting Get-AlarmAction Get-AlarmActionTrigger Get-AlarmDefinition Get-Annotation Get-CDDrive Get-CIAccessControlRule Get-CIDatastore Get-CINetworkAdapter Get-CIRole Get-CIUser Get-CIVApp Get-CIVAppNetwork Get-CIVAppStartRule Get-CIVAppTemplate Get-CIVM Get-CIVMTemplate Get-CIView Get-Catalog Get-CisCommand Get-CisService Get-CloudCommand Get-Cluster Get-CompatibleVersionAddtionaPropertiesStr Get-ComplexResourceQualifier Get-ConfigurationErrorCount Get-ContentLibraryItem Get-CustomAttribute Get-DSCResourceModules Get-Datacenter Get-Datastore Get-DatastoreCluster Get-DrsClusterGroup Get-DrsRecommendation Get-DrsRule Get-DrsVMHostRule Get-DscResource Get-EdgeGateway Get-EncryptedPassword Get-ErrorReport Get-EsxCli Get-EsxTop Get-ExternalNetwork Get-FileHash Get-FloppyDrive Get-Folder Get-HAPrimaryVMHost Get-HCXAppliance Get-HCXApplianceCompute Get-HCXApplianceDVS Get-HCXApplianceDatastore Get-HCXApplianceNetwork Get-HCXContainer Get-HCXDatastore Get-HCXGateway Get-HCXInterconnectStatus Get-HCXJob Get-HCXMigration Get-HCXNetwork Get-HCXNetworkExtension Get-HCXReplication Get-HCXReplicationSnapshot Get-HCXService Get-HCXSite Get-HCXSitePairing Get-HCXVM Get-HardDisk Get-IScsiHbaTarget Get-InnerMostErrorRecord Get-InstallPath Get-InstalledModule Get-InstalledScript Get-Inventory Get-ItemPropertyValue Get-KeyManagementServer Get-KmipClientCertificate Get-KmsCluster Get-Log Get-LogType Get-MarkdownOption Get-Media Get-MofInstanceName Get-MofInstanceText Get-NetworkAdapter Get-NetworkPool Get-NfsUser Get-NicTeamingPolicy Get-NsxApplicableMember Get-NsxApplicableSecurityAction Get-NsxBackingDVSwitch Get-NsxBackingPortGroup Get-NsxCliDfwAddrSet Get-NsxCliDfwFilter Get-NsxCliDfwRule Get-NsxClusterStatus Get-NsxController Get-NsxDynamicCriteria Get-NsxDynamicMemberSet Get-NsxEdge Get-NsxEdgeBgp Get-NsxEdgeBgpNeighbour Get-NsxEdgeCertificate Get-NsxEdgeCsr Get-NsxEdgeFirewall Get-NsxEdgeFirewallRule Get-NsxEdgeInterface Get-NsxEdgeInterfaceAddress Get-NsxEdgeNat Get-NsxEdgeNatRule Get-NsxEdgeOspf Get-NsxEdgeOspfArea Get-NsxEdgeOspfInterface Get-NsxEdgePrefix Get-NsxEdgeRedistributionRule Get-NsxEdgeRouting Get-NsxEdgeStaticRoute Get-NsxEdgeSubInterface Get-NsxFirewallExclusionListMember Get-NsxFirewallGlobalConfiguration Get-NsxFirewallPublishStatus Get-NsxFirewallRule Get-NsxFirewallRuleMember Get-NsxFirewallSavedConfiguration Get-NsxFirewallSection Get-NsxFirewallThreshold Get-NsxIpPool Get-NsxIpSet Get-NsxLicense Get-NsxLoadBalancer Get-NsxLoadBalancerApplicationProfile Get-NsxLoadBalancerApplicationRule Get-NsxLoadBalancerMonitor Get-NsxLoadBalancerPool Get-NsxLoadBalancerPoolMember Get-NsxLoadBalancerStats Get-NsxLoadBalancerVip Get-NsxLogicalRouter Get-NsxLogicalRouterBgp Get-NsxLogicalRouterBgpNeighbour Get-NsxLogicalRouterBridge Get-NsxLogicalRouterBridging Get-NsxLogicalRouterInterface Get-NsxLogicalRouterOspf Get-NsxLogicalRouterOspfArea Get-NsxLogicalRouterOspfInterface Get-NsxLogicalRouterPrefix Get-NsxLogicalRouterRedistributionRule Get-NsxLogicalRouterRouting Get-NsxLogicalRouterStaticRoute Get-NsxLogicalSwitch Get-NsxMacSet Get-NsxManagerBackup Get-NsxManagerCertificate Get-NsxManagerComponentSummary Get-NsxManagerNetwork Get-NsxManagerRole Get-NsxManagerSsoConfig Get-NsxManagerSyncStatus Get-NsxManagerSyslogServer Get-NsxManagerSystemSummary Get-NsxManagerTimeSettings Get-NsxManagerVcenterConfig Get-NsxSecondaryManager Get-NsxSecurityGroup Get-NsxSecurityGroupEffectiveIpAddress Get-NsxSecurityGroupEffectiveMacAddress Get-NsxSecurityGroupEffectiveMember Get-NsxSecurityGroupEffectiveVirtualMachine Get-NsxSecurityGroupEffectiveVnic Get-NsxSecurityGroupMemberTypes Get-NsxSecurityPolicy Get-NsxSecurityPolicyHighestUsedPrecedence Get-NsxSecurityPolicyRule Get-NsxSecurityTag Get-NsxSecurityTagAssignment Get-NsxSegmentIdRange Get-NsxService Get-NsxServiceDefinition Get-NsxServiceGroup Get-NsxServiceGroupMember Get-NsxServiceProfile Get-NsxSpoofguardNic Get-NsxSpoofguardPolicy Get-NsxSslVpn Get-NsxSslVpnAuthServer Get-NsxSslVpnClientInstallationPackage Get-NsxSslVpnIpPool Get-NsxSslVpnPrivateNetwork Get-NsxSslVpnUser Get-NsxTransportZone Get-NsxUserRole Get-NsxVdsContext Get-NsxtPolicyService Get-NsxtService Get-OSCustomizationNicMapping Get-OSCustomizationSpec Get-Org Get-OrgNetwork Get-OrgVdc Get-OrgVdcNetwork Get-OvfConfiguration Get-PSCurrentConfigurationNode Get-PSDefaultConfigurationDocument Get-PSMetaConfigDocumentInstVersionInfo Get-PSMetaConfigurationProcessed Get-PSReadLineKeyHandler Get-PSReadLineOption Get-PSRepository Get-PSTopConfigurationName Get-PSVersion Get-Package Get-PackageProvider Get-PackageSource Get-PassthroughDevice Get-PositionInfo Get-PowerCLICommunity Get-PowerCLIConfiguration Get-PowerCLIHelp Get-PowerCLIVersion Get-PowerNsxVersion Get-ProviderVdc Get-PublicKeyFromFile Get-PublicKeyFromStore Get-ResourcePool Get-Runspace Get-RunspaceDebug Get-ScsiController Get-ScsiLun Get-ScsiLunPath Get-SecurityInfo Get-SecurityPolicy Get-Snapshot Get-SpbmCapability Get-SpbmCompatibleStorage Get-SpbmEntityConfiguration Get-SpbmFaultDomain Get-SpbmPointInTimeReplica Get-SpbmReplicationGroup Get-SpbmReplicationPair Get-SpbmStoragePolicy Get-Stat Get-StatInterval Get-StatType Get-Tag Get-TagAssignment Get-TagCategory Get-Task Get-Template Get-TimeZone Get-Uptime Get-UsbDevice Get-VAIOFilter Get-VApp Get-VDBlockedPolicy Get-VDPort Get-VDPortgroup Get-VDPortgroupOverridePolicy Get-VDSecurityPolicy Get-VDSwitch Get-VDSwitchPrivateVlan Get-VDTrafficShapingPolicy Get-VDUplinkLacpPolicy Get-VDUplinkTeamingPolicy Get-VDisk Get-VIAccount Get-VICommand Get-VICredentialStoreItem Get-VIEvent Get-VIObjectByVIView Get-VIPermission Get-VIPrivilege Get-VIProperty Get-VIRole Get-VM Get-VMGuest Get-VMHost Get-VMHostAccount Get-VMHostAdvancedConfiguration Get-VMHostAuthentication Get-VMHostAvailableTimeZone Get-VMHostDiagnosticPartition Get-VMHostDisk Get-VMHostDiskPartition Get-VMHostFirewallDefaultPolicy Get-VMHostFirewallException Get-VMHostFirmware Get-VMHostHardware Get-VMHostHba Get-VMHostModule Get-VMHostNetwork Get-VMHostNetworkAdapter Get-VMHostNtpServer Get-VMHostPatch Get-VMHostPciDevice Get-VMHostProfile Get-VMHostProfileImageCacheConfiguration Get-VMHostProfileRequiredInput Get-VMHostProfileStorageDeviceConfiguration Get-VMHostProfileUserConfiguration Get-VMHostProfileVmPortGroupConfiguration Get-VMHostRoute Get-VMHostService Get-VMHostSnmp Get-VMHostStartPolicy Get-VMHostStorage Get-VMHostSysLogServer Get-VMQuestion Get-VMResourceConfiguration Get-VMStartPolicy Get-VTpm Get-VTpmCSR Get-VTpmCertificate Get-VasaProvider Get-VasaStorageArray Get-View Get-VirtualPortGroup Get-VirtualSwitch Get-VmcSddcNetworkService Get-VmcService Get-VsanClusterConfiguration Get-VsanComponent Get-VsanDisk Get-VsanDiskGroup Get-VsanEvacuationPlan Get-VsanFaultDomain Get-VsanIscsiInitiatorGroup Get-VsanIscsiInitiatorGroupTargetAssociation Get-VsanIscsiLun Get-VsanIscsiTarget Get-VsanObject Get-VsanResyncingComponent Get-VsanRuntimeInfo Get-VsanSpaceUsage Get-VsanStat Get-VsanView Get-vRAApplianceServiceStatus Get-vRAAuthorizationRole Get-vRABlueprint Get-vRABusinessGroup Get-vRACatalogItem Get-vRACatalogItemRequestTemplate Get-vRACatalogPrincipal Get-vRAComponentRegistryService Get-vRAComponentRegistryServiceEndpoint Get-vRAComponentRegistryServiceStatus Get-vRAContent Get-vRAContentData Get-vRAContentType Get-vRACustomForm Get-vRAEntitledCatalogItem Get-vRAEntitledService Get-vRAEntitlement Get-vRAExternalNetworkProfile Get-vRAGroupPrincipal Get-vRAIcon Get-vRANATNetworkProfile Get-vRANetworkProfileIPAddressList Get-vRANetworkProfileIPRangeSummary Get-vRAPackage Get-vRAPackageContent Get-vRAPropertyDefinition Get-vRAPropertyGroup Get-vRARequest Get-vRARequestDetail Get-vRAReservation Get-vRAReservationComputeResource Get-vRAReservationComputeResourceMemory Get-vRAReservationComputeResourceNetwork Get-vRAReservationComputeResourceResourcePool Get-vRAReservationComputeResourceStorage Get-vRAReservationPolicy Get-vRAReservationTemplate Get-vRAReservationType Get-vRAResource Get-vRAResourceAction Get-vRAResourceActionRequestTemplate Get-vRAResourceMetric Get-vRAResourceOperation Get-vRAResourceType Get-vRARoutedNetworkProfile Get-vRAService Get-vRAServiceBlueprint Get-vRASourceMachine Get-vRAStorageReservationPolicy Get-vRATenant Get-vRATenantDirectory Get-vRATenantDirectoryStatus Get-vRATenantRole Get-vRAUserPrincipal Get-vRAUserPrincipalGroupMembership Get-vRAVersion Get-vRNIAPIVersion Get-vRNIApplication Get-vRNIApplicationTier Get-vRNIDataSource Get-vRNIDataSourceSNMPConfig Get-vRNIDatastore Get-vRNIDistributedSwitch Get-vRNIDistributedSwitchPortGroup Get-vRNIEntity Get-vRNIEntityName Get-vRNIFirewallRule Get-vRNIFlow Get-vRNIHost Get-vRNIHostVMKNic Get-vRNIIPSet Get-vRNIL2Network Get-vRNINSXManager Get-vRNINodes Get-vRNIProblem Get-vRNIRecommendedRules Get-vRNIRecommendedRulesNsxBundle Get-vRNISecurityGroup Get-vRNISecurityTag Get-vRNIService Get-vRNIServiceGroup Get-vRNIVM Get-vRNIVMvNIC Get-vRNIvCenter Get-vRNIvCenterCluster Get-vRNIvCenterDatacenter Get-vRNIvCenterFolder Grant-NsxSpoofguardNicApproval Import-CIVApp Import-CIVAppTemplate Import-NsxObject Import-PackageProvider Import-PowerShellDataFile Import-SpbmStoragePolicy Import-VApp Import-VMHostProfile Import-vRAContentData Import-vRAIcon Import-vRAPackage Initialize-ConfigurationRuntimeState Install-Module Install-NsxCluster Install-Package Install-PackageProvider Install-Script Install-VMHostPatch Invoke-DrsRecommendation Invoke-NsxCli Invoke-NsxClusterResolveAll Invoke-NsxManagerSync Invoke-NsxRestMethod Invoke-NsxWebRequest Invoke-VMHostProfile Invoke-VMScript Invoke-XpathQuery Invoke-vRADataCollection Invoke-vRARestMethod Invoke-vRATenantDirectorySync Invoke-vRNIRestMethod Join-String Mount-Tools Move-Cluster Move-Datacenter Move-Datastore Move-Folder Move-HardDisk Move-Inventory Move-NsxSecurityPolicyRule Move-ResourcePool Move-Template Move-VApp Move-VDisk Move-VM Move-VMHost New-AdvancedSetting New-AlarmAction New-AlarmActionTrigger New-CDDrive New-CIAccessControlRule New-CIVApp New-CIVAppNetwork New-CIVAppTemplate New-CIVM New-Cluster New-CustomAttribute New-Datacenter New-Datastore New-DatastoreCluster New-DatastoreDrive New-DrsClusterGroup New-DrsRule New-DrsVMHostRule New-DscChecksum New-FloppyDrive New-Folder New-Guid New-HCXAppliance New-HCXMigration New-HCXNetworkExtension New-HCXNetworkMapping New-HCXReplication New-HCXSitePairing New-HCXStaticRoute New-HardDisk New-IScsiHbaTarget New-KmipClientCertificate New-NetworkAdapter New-NfsUser New-NsxAddressSpec New-NsxClusterVxlanConfig New-NsxController New-NsxDynamicCriteriaSpec New-NsxEdge New-NsxEdgeBgpNeighbour New-NsxEdgeCsr New-NsxEdgeFirewallRule New-NsxEdgeInterfaceSpec New-NsxEdgeNatRule New-NsxEdgeOspfArea New-NsxEdgeOspfInterface New-NsxEdgePrefix New-NsxEdgeRedistributionRule New-NsxEdgeSelfSignedCertificate New-NsxEdgeStaticRoute New-NsxEdgeSubInterface New-NsxEdgeSubInterfaceSpec New-NsxFirewallRule New-NsxFirewallSavedConfiguration New-NsxFirewallSection New-NsxIpPool New-NsxIpSet New-NsxLoadBalancerApplicationProfile New-NsxLoadBalancerApplicationRule New-NsxLoadBalancerMemberSpec New-NsxLoadBalancerMonitor New-NsxLoadBalancerPool New-NsxLogicalRouter New-NsxLogicalRouterBgpNeighbour New-NsxLogicalRouterBridge New-NsxLogicalRouterInterface New-NsxLogicalRouterInterfaceSpec New-NsxLogicalRouterOspfArea New-NsxLogicalRouterOspfInterface New-NsxLogicalRouterPrefix New-NsxLogicalRouterRedistributionRule New-NsxLogicalRouterStaticRoute New-NsxLogicalSwitch New-NsxMacSet New-NsxManager New-NsxSecurityGroup New-NsxSecurityPolicy New-NsxSecurityPolicyAssignment New-NsxSecurityPolicyFirewallRuleSpec New-NsxSecurityPolicyGuestIntrospectionSpec New-NsxSecurityPolicyNetworkIntrospectionSpec New-NsxSecurityTag New-NsxSecurityTagAssignment New-NsxSegmentIdRange New-NsxService New-NsxServiceGroup New-NsxSpoofguardPolicy New-NsxSslVpnAuthServer New-NsxSslVpnClientInstallationPackage New-NsxSslVpnIpPool New-NsxSslVpnPrivateNetwork New-NsxSslVpnUser New-NsxTransportZone New-NsxVdsContext New-OSCustomizationNicMapping New-OSCustomizationSpec New-Org New-OrgNetwork New-OrgVdc New-OrgVdcNetwork New-ResourcePool New-ScriptFileInfo New-ScsiController New-Snapshot New-SpbmRule New-SpbmRuleSet New-SpbmStoragePolicy New-StatInterval New-Tag New-TagAssignment New-TagCategory New-Template New-TemporaryFile New-VAIOFilter New-VApp New-VDPortgroup New-VDSwitch New-VDSwitchPrivateVlan New-VDisk New-VICredentialStoreItem New-VIInventoryDrive New-VIPermission New-VIProperty New-VIRole New-VISamlSecurityContext New-VM New-VMHostAccount New-VMHostNetworkAdapter New-VMHostProfile New-VMHostProfileVmPortGroupConfiguration New-VMHostRoute New-VTpm New-VasaProvider New-VcsOAuthSecurityContext New-VirtualPortGroup New-VirtualSwitch New-VsanDisk New-VsanDiskGroup New-VsanFaultDomain New-VsanIscsiInitiatorGroup New-VsanIscsiInitiatorGroupTargetAssociation New-VsanIscsiLun New-VsanIscsiTarget New-vRABusinessGroup New-vRAEntitlement New-vRAExternalNetworkProfile New-vRAGroupPrincipal New-vRANATNetworkProfile New-vRANetworkProfileIPRangeDefinition New-vRAPackage New-vRAPropertyDefinition New-vRAPropertyGroup New-vRAReservation New-vRAReservationNetworkDefinition New-vRAReservationPolicy New-vRAReservationStorageDefinition New-vRARoutedNetworkProfile New-vRAService New-vRAStorageReservationPolicy New-vRATenant New-vRATenantDirectory New-vRAUserPrincipal New-vRNIApplication New-vRNIApplicationTier New-vRNIDataSource Open-VMConsoleWindow Publish-Module Publish-NsxSpoofguardPolicy Publish-Script Register-PSRepository Register-PackageSource Remove-AdvancedSetting Remove-AlarmAction Remove-AlarmActionTrigger Remove-Alias Remove-CDDrive Remove-CIAccessControlRule Remove-CIVApp Remove-CIVAppNetwork Remove-CIVAppTemplate Remove-Cluster Remove-CustomAttribute Remove-Datacenter Remove-Datastore Remove-DatastoreCluster Remove-DrsClusterGroup Remove-DrsRule Remove-DrsVMHostRule Remove-FloppyDrive Remove-Folder Remove-HCXAppliance Remove-HCXNetworkExtension Remove-HCXReplication Remove-HCXSitePairing Remove-HardDisk Remove-IScsiHbaTarget Remove-Inventory Remove-KeyManagementServer Remove-NetworkAdapter Remove-NfsUser Remove-NsxCluster Remove-NsxClusterVxlanConfig Remove-NsxController Remove-NsxDynamicCriteria Remove-NsxDynamicMemberSet Remove-NsxEdge Remove-NsxEdgeBgpNeighbour Remove-NsxEdgeCertificate Remove-NsxEdgeCsr Remove-NsxEdgeFirewallRule Remove-NsxEdgeInterfaceAddress Remove-NsxEdgeNatRule Remove-NsxEdgeOspfArea Remove-NsxEdgeOspfInterface Remove-NsxEdgePrefix Remove-NsxEdgeRedistributionRule Remove-NsxEdgeStaticRoute Remove-NsxEdgeSubInterface Remove-NsxFirewallExclusionListMember Remove-NsxFirewallRule Remove-NsxFirewallRuleMember Remove-NsxFirewallSavedConfiguration Remove-NsxFirewallSection Remove-NsxIpPool Remove-NsxIpSet Remove-NsxIpSetMember Remove-NsxLoadBalancerApplicationProfile Remove-NsxLoadBalancerMonitor Remove-NsxLoadBalancerPool Remove-NsxLoadBalancerPoolMember Remove-NsxLoadBalancerVip Remove-NsxLogicalRouter Remove-NsxLogicalRouterBgpNeighbour Remove-NsxLogicalRouterBridge Remove-NsxLogicalRouterInterface Remove-NsxLogicalRouterOspfArea Remove-NsxLogicalRouterOspfInterface Remove-NsxLogicalRouterPrefix Remove-NsxLogicalRouterRedistributionRule Remove-NsxLogicalRouterStaticRoute Remove-NsxLogicalSwitch Remove-NsxMacSet Remove-NsxSecondaryManager Remove-NsxSecurityGroup Remove-NsxSecurityGroupMember Remove-NsxSecurityPolicy Remove-NsxSecurityPolicyAssignment Remove-NsxSecurityPolicyRule Remove-NsxSecurityPolicyRuleGroup Remove-NsxSecurityPolicyRuleService Remove-NsxSecurityTag Remove-NsxSecurityTagAssignment Remove-NsxSegmentIdRange Remove-NsxService Remove-NsxServiceGroup Remove-NsxSpoofguardPolicy Remove-NsxSslVpnClientInstallationPackage Remove-NsxSslVpnIpPool Remove-NsxSslVpnPrivateNetwork Remove-NsxSslVpnUser Remove-NsxTransportZone Remove-NsxTransportZoneMember Remove-NsxVdsContext Remove-OSCustomizationNicMapping Remove-OSCustomizationSpec Remove-Org Remove-OrgNetwork Remove-OrgVdc Remove-OrgVdcNetwork Remove-PSReadLineKeyHandler Remove-PassthroughDevice Remove-ResourcePool Remove-Snapshot Remove-SpbmStoragePolicy Remove-StatInterval Remove-Tag Remove-TagAssignment Remove-TagCategory Remove-Template Remove-UsbDevice Remove-VAIOFilter Remove-VApp Remove-VDPortGroup Remove-VDSwitch Remove-VDSwitchPhysicalNetworkAdapter Remove-VDSwitchPrivateVlan Remove-VDSwitchVMHost Remove-VDisk Remove-VICredentialStoreItem Remove-VIPermission Remove-VIProperty Remove-VIRole Remove-VM Remove-VMHost Remove-VMHostAccount Remove-VMHostNetworkAdapter Remove-VMHostNtpServer Remove-VMHostProfile Remove-VMHostProfileVmPortGroupConfiguration Remove-VMHostRoute Remove-VTpm Remove-VasaProvider Remove-VirtualPortGroup Remove-VirtualSwitch Remove-VirtualSwitchPhysicalNetworkAdapter Remove-VsanDisk Remove-VsanDiskGroup Remove-VsanFaultDomain Remove-VsanIscsiInitiatorGroup Remove-VsanIscsiInitiatorGroupTargetAssociation Remove-VsanIscsiLun Remove-VsanIscsiTarget Remove-vRABusinessGroup Remove-vRACustomForm Remove-vRAExternalNetworkProfile Remove-vRAGroupPrincipal Remove-vRAIcon Remove-vRANATNetworkProfile Remove-vRAPackage Remove-vRAPrincipalFromTenantRole Remove-vRAPropertyDefinition Remove-vRAPropertyGroup Remove-vRAReservation Remove-vRAReservationNetwork Remove-vRAReservationPolicy Remove-vRAReservationStorage Remove-vRARoutedNetworkProfile Remove-vRAService Remove-vRAStorageReservationPolicy Remove-vRATenant Remove-vRATenantDirectory Remove-vRAUserPrincipal Remove-vRNIApplication Remove-vRNIApplicationTier Remove-vRNIDataSource Repair-NsxEdge Repair-VsanObject Request-vRACatalogItem Request-vRAResourceAction Restart-CIVApp Restart-CIVAppGuest Restart-CIVM Restart-CIVMGuest Restart-VM Restart-VMGuest Restart-VMHost Restart-VMHostService Resume-HCXReplication Revoke-NsxSpoofguardNicApproval Save-Module Save-Package Save-Script Search-Cloud Set-AdvancedSetting Set-AlarmDefinition Set-Annotation Set-CDDrive Set-CIAccessControlRule Set-CINetworkAdapter Set-CIVApp Set-CIVAppNetwork Set-CIVAppStartRule Set-CIVAppTemplate Set-Cluster Set-CustomAttribute Set-Datacenter Set-Datastore Set-DatastoreCluster Set-DrsClusterGroup Set-DrsRule Set-DrsVMHostRule Set-FloppyDrive Set-Folder Set-HCXAppliance Set-HCXMigration Set-HCXReplication Set-HardDisk Set-IScsiHbaTarget Set-KeyManagementServer Set-KmsCluster Set-MarkdownOption Set-NetworkAdapter Set-NfsUser Set-NicTeamingPolicy Set-NodeExclusiveResources Set-NodeManager Set-NodeResourceSource Set-NodeResources Set-NsxEdge Set-NsxEdgeBgp Set-NsxEdgeFirewall Set-NsxEdgeInterface Set-NsxEdgeNat Set-NsxEdgeOspf Set-NsxEdgeRouting Set-NsxFirewallGlobalConfiguration Set-NsxFirewallRule Set-NsxFirewallSavedConfiguration Set-NsxFirewallThreshold Set-NsxLoadBalancer Set-NsxLoadBalancerPoolMember Set-NsxLogicalRouter Set-NsxLogicalRouterBgp Set-NsxLogicalRouterBridging Set-NsxLogicalRouterInterface Set-NsxLogicalRouterOspf Set-NsxLogicalRouterRouting Set-NsxManager Set-NsxManagerRole Set-NsxManagerTimeSettings 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current3DENOperation currentChannel currentCommand currentMagazine currentMagazineDetail currentMagazineDetailTurret currentMagazineTurret currentMuzzle currentNamespace currentTask currentTasks currentThrowable currentVisionMode currentWaypoint currentWeapon currentWeaponMode currentWeaponTurret currentZeroing cursorObject cursorTarget customChat customRadio cutFadeOut cutObj cutRsc cutText damage date dateToNumber daytime deActivateKey debriefingText debugFSM debugLog deg delete3DENEntities deleteAt deleteCenter deleteCollection deleteEditorObject deleteGroup deleteGroupWhenEmpty deleteIdentity deleteLocation deleteMarker deleteMarkerLocal deleteRange deleteResources deleteSite deleteStatus deleteTeam deleteVehicle deleteVehicleCrew deleteWaypoint detach detectedMines diag_activeMissionFSMs diag_activeScripts diag_activeSQFScripts diag_activeSQSScripts diag_captureFrame diag_captureFrameToFile diag_captureSlowFrame diag_codePerformance diag_drawMode diag_enable diag_enabled diag_fps diag_fpsMin diag_frameNo diag_lightNewLoad diag_list diag_log diag_logSlowFrame diag_mergeConfigFile diag_recordTurretLimits diag_setLightNew diag_tickTime diag_toggle dialog diarySubjectExists didJIP didJIPOwner difficulty difficultyEnabled difficultyEnabledRTD difficultyOption direction directSay disableAI disableCollisionWith disableConversation disableDebriefingStats disableMapIndicators disableNVGEquipment disableRemoteSensors disableSerialization disableTIEquipment disableUAVConnectability disableUserInput displayAddEventHandler displayCtrl displayParent displayRemoveAllEventHandlers displayRemoveEventHandler displaySetEventHandler dissolveTeam distance distance2D distanceSqr distributionRegion do3DENAction doArtilleryFire doFire doFollow doFSM doGetOut doMove doorPhase doStop doSuppressiveFire doTarget doWatch drawArrow drawEllipse drawIcon drawIcon3D drawLine drawLine3D drawLink drawLocation drawPolygon drawRectangle drawTriangle driver drop dynamicSimulationDistance dynamicSimulationDistanceCoef dynamicSimulationEnabled dynamicSimulationSystemEnabled echo edit3DENMissionAttributes editObject editorSetEventHandler effectiveCommander emptyPositions enableAI enableAIFeature enableAimPrecision enableAttack enableAudioFeature enableAutoStartUpRTD enableAutoTrimRTD enableCamShake enableCaustics enableChannel enableCollisionWith enableCopilot enableDebriefingStats enableDiagLegend enableDynamicSimulation enableDynamicSimulationSystem enableEndDialog enableEngineArtillery enableEnvironment enableFatigue enableGunLights enableInfoPanelComponent enableIRLasers enableMimics enablePersonTurret enableRadio enableReload enableRopeAttach enableSatNormalOnDetail enableSaving enableSentences enableSimulation enableSimulationGlobal enableStamina enableTeamSwitch enableTraffic enableUAVConnectability enableUAVWaypoints enableVehicleCargo enableVehicleSensor enableWeaponDisassembly endLoadingScreen endMission engineOn enginesIsOnRTD enginesRpmRTD enginesTorqueRTD entities environmentEnabled estimatedEndServerTime estimatedTimeLeft evalObjectArgument everyBackpack everyContainer exec execEditorScript execFSM execVM exp expectedDestination exportJIPMessages eyeDirection eyePos face faction fadeMusic fadeRadio fadeSound fadeSpeech failMission fillWeaponsFromPool find findCover findDisplay findEditorObject findEmptyPosition findEmptyPositionReady findIf findNearestEnemy finishMissionInit finite fire fireAtTarget firstBackpack flag flagAnimationPhase flagOwner flagSide flagTexture fleeing floor flyInHeight flyInHeightASL fog fogForecast fogParams forceAddUniform forcedMap forceEnd forceFlagTexture forceFollowRoad forceMap forceRespawn forceSpeed forceWalk forceWeaponFire forceWeatherChange forEachMember forEachMemberAgent forEachMemberTeam forgetTarget format formation formationDirection formationLeader formationMembers formationPosition formationTask formatText formLeader freeLook fromEditor fuel fullCrew gearIDCAmmoCount gearSlotAmmoCount gearSlotData get3DENActionState get3DENAttribute get3DENCamera get3DENConnections get3DENEntity get3DENEntityID get3DENGrid get3DENIconsVisible get3DENLayerEntities get3DENLinesVisible get3DENMissionAttribute get3DENMouseOver get3DENSelected getAimingCoef getAllEnvSoundControllers getAllHitPointsDamage getAllOwnedMines getAllSoundControllers getAmmoCargo getAnimAimPrecision getAnimSpeedCoef getArray getArtilleryAmmo getArtilleryComputerSettings getArtilleryETA getAssignedCuratorLogic getAssignedCuratorUnit getBackpackCargo getBleedingRemaining getBurningValue getCameraViewDirection getCargoIndex getCenterOfMass getClientState getClientStateNumber getCompatiblePylonMagazines getConnectedUAV getContainerMaxLoad getCursorObjectParams getCustomAimCoef getDammage getDescription getDir getDirVisual getDLCAssetsUsage getDLCAssetsUsageByName getDLCs getEditorCamera getEditorMode getEditorObjectScope getElevationOffset getEnvSoundController getFatigue getForcedFlagTexture getFriend getFSMVariable getFuelCargo getGroupIcon getGroupIconParams getGroupIcons getHideFrom getHit getHitIndex getHitPointDamage getItemCargo getMagazineCargo getMarkerColor getMarkerPos getMarkerSize getMarkerType getMass getMissionConfig getMissionConfigValue getMissionDLCs getMissionLayerEntities getModelInfo getMousePosition getMusicPlayedTime getNumber getObjectArgument getObjectChildren getObjectDLC getObjectMaterials getObjectProxy getObjectTextures getObjectType getObjectViewDistance getOxygenRemaining getPersonUsedDLCs getPilotCameraDirection getPilotCameraPosition getPilotCameraRotation getPilotCameraTarget getPlateNumber getPlayerChannel getPlayerScores getPlayerUID getPos getPosASL getPosASLVisual getPosASLW getPosATL getPosATLVisual getPosVisual getPosWorld getPylonMagazines getRelDir getRelPos getRemoteSensorsDisabled getRepairCargo getResolution getShadowDistance getShotParents getSlingLoad getSoundController getSoundControllerResult getSpeed getStamina getStatValue getSuppression getTerrainGrid getTerrainHeightASL getText getTotalDLCUsageTime getUnitLoadout getUnitTrait getUserMFDText getUserMFDvalue getVariable getVehicleCargo getWeaponCargo getWeaponSway getWingsOrientationRTD getWingsPositionRTD getWPPos glanceAt globalChat globalRadio goggles goto group groupChat groupFromNetId groupIconSelectable groupIconsVisible groupId groupOwner groupRadio groupSelectedUnits groupSelectUnit gunner gusts halt handgunItems handgunMagazine handgunWeapon handsHit hasInterface hasPilotCamera hasWeapon hcAllGroups hcGroupParams hcLeader hcRemoveAllGroups hcRemoveGroup hcSelected hcSelectGroup hcSetGroup hcShowBar hcShownBar headgear hideBody hideObject hideObjectGlobal hideSelection hint hintC hintCadet hintSilent hmd hostMission htmlLoad HUDMovementLevels humidity image importAllGroups importance in inArea inAreaArray incapacitatedState inflame inflamed infoPanel infoPanelComponentEnabled infoPanelComponents infoPanels inGameUISetEventHandler inheritsFrom initAmbientLife inPolygon inputAction inRangeOfArtillery insertEditorObject intersect is3DEN is3DENMultiplayer isAbleToBreathe isAgent isArray isAutoHoverOn isAutonomous isAutotest isBleeding isBurning isClass isCollisionLightOn isCopilotEnabled isDamageAllowed isDedicated isDLCAvailable isEngineOn isEqualTo isEqualType isEqualTypeAll isEqualTypeAny isEqualTypeArray isEqualTypeParams isFilePatchingEnabled isFlashlightOn isFlatEmpty isForcedWalk isFormationLeader isGroupDeletedWhenEmpty isHidden isInRemainsCollector isInstructorFigureEnabled isIRLaserOn isKeyActive isKindOf isLaserOn isLightOn isLocalized isManualFire isMarkedForCollection isMultiplayer isMultiplayerSolo isNil isNull isNumber isObjectHidden isObjectRTD isOnRoad isPipEnabled isPlayer isRealTime isRemoteExecuted isRemoteExecutedJIP isServer isShowing3DIcons isSimpleObject isSprintAllowed isStaminaEnabled isSteamMission isStreamFriendlyUIEnabled isText isTouchingGround isTurnedOut isTutHintsEnabled isUAVConnectable isUAVConnected isUIContext isUniformAllowed isVehicleCargo isVehicleRadarOn isVehicleSensorEnabled isWalking isWeaponDeployed isWeaponRested itemCargo items itemsWithMagazines join joinAs joinAsSilent joinSilent joinString kbAddDatabase kbAddDatabaseTargets kbAddTopic kbHasTopic kbReact kbRemoveTopic kbTell kbWasSaid keyImage keyName knowsAbout land landAt landResult language laserTarget lbAdd lbClear lbColor lbColorRight lbCurSel lbData lbDelete lbIsSelected lbPicture lbPictureRight lbSelection lbSetColor lbSetColorRight lbSetCurSel lbSetData lbSetPicture lbSetPictureColor lbSetPictureColorDisabled lbSetPictureColorSelected lbSetPictureRight lbSetPictureRightColor lbSetPictureRightColorDisabled lbSetPictureRightColorSelected lbSetSelectColor lbSetSelectColorRight lbSetSelected lbSetText lbSetTextRight lbSetTooltip lbSetValue lbSize lbSort lbSortByValue lbText lbTextRight lbValue leader leaderboardDeInit leaderboardGetRows leaderboardInit leaderboardRequestRowsFriends leaderboardsRequestUploadScore leaderboardsRequestUploadScoreKeepBest leaderboardState leaveVehicle libraryCredits libraryDisclaimers lifeState lightAttachObject lightDetachObject lightIsOn lightnings limitSpeed linearConversion lineIntersects lineIntersectsObjs lineIntersectsSurfaces lineIntersectsWith linkItem list listObjects listRemoteTargets listVehicleSensors ln lnbAddArray lnbAddColumn lnbAddRow lnbClear lnbColor lnbCurSelRow lnbData lnbDeleteColumn lnbDeleteRow lnbGetColumnsPosition lnbPicture lnbSetColor lnbSetColumnsPos lnbSetCurSelRow lnbSetData lnbSetPicture lnbSetText lnbSetValue lnbSize lnbSort lnbSortByValue lnbText lnbValue load loadAbs loadBackpack loadFile loadGame loadIdentity loadMagazine loadOverlay loadStatus loadUniform loadVest local localize locationPosition lock lockCameraTo lockCargo lockDriver locked lockedCargo lockedDriver lockedTurret lockIdentity lockTurret lockWP log logEntities logNetwork logNetworkTerminate lookAt lookAtPos magazineCargo magazines magazinesAllTurrets magazinesAmmo magazinesAmmoCargo magazinesAmmoFull magazinesDetail magazinesDetailBackpack magazinesDetailUniform magazinesDetailVest magazinesTurret magazineTurretAmmo mapAnimAdd mapAnimClear mapAnimCommit mapAnimDone mapCenterOnCamera mapGridPosition markAsFinishedOnSteam markerAlpha markerBrush markerColor markerDir markerPos markerShape markerSize markerText markerType max members menuAction menuAdd menuChecked menuClear menuCollapse menuData menuDelete menuEnable menuEnabled menuExpand menuHover menuPicture menuSetAction menuSetCheck menuSetData menuSetPicture menuSetValue menuShortcut menuShortcutText menuSize menuSort menuText menuURL menuValue min mineActive mineDetectedBy missionConfigFile missionDifficulty missionName missionNamespace missionStart missionVersion mod modelToWorld modelToWorldVisual modelToWorldVisualWorld modelToWorldWorld modParams moonIntensity moonPhase morale move move3DENCamera moveInAny moveInCargo moveInCommander moveInDriver moveInGunner moveInTurret moveObjectToEnd moveOut moveTime moveTo moveToCompleted moveToFailed musicVolume name nameSound nearEntities nearestBuilding nearestLocation nearestLocations nearestLocationWithDubbing nearestObject nearestObjects nearestTerrainObjects nearObjects nearObjectsReady nearRoads nearSupplies nearTargets needReload netId netObjNull newOverlay nextMenuItemIndex nextWeatherChange nMenuItems not numberOfEnginesRTD numberToDate objectCurators objectFromNetId objectParent objStatus onBriefingGroup onBriefingNotes onBriefingPlan onBriefingTeamSwitch onCommandModeChanged onDoubleClick onEachFrame onGroupIconClick onGroupIconOverEnter onGroupIconOverLeave onHCGroupSelectionChanged onMapSingleClick onPlayerConnected onPlayerDisconnected onPreloadFinished onPreloadStarted onShowNewObject onTeamSwitch openCuratorInterface openDLCPage openMap openSteamApp openYoutubeVideo or orderGetIn overcast overcastForecast owner param params parseNumber parseSimpleArray parseText parsingNamespace particlesQuality pickWeaponPool pitch pixelGrid pixelGridBase pixelGridNoUIScale pixelH pixelW playableSlotsNumber playableUnits playAction playActionNow player playerRespawnTime playerSide playersNumber playGesture playMission playMove playMoveNow playMusic playScriptedMission playSound playSound3D position positionCameraToWorld posScreenToWorld posWorldToScreen ppEffectAdjust ppEffectCommit ppEffectCommitted ppEffectCreate ppEffectDestroy ppEffectEnable ppEffectEnabled ppEffectForceInNVG precision preloadCamera preloadObject preloadSound preloadTitleObj preloadTitleRsc preprocessFile preprocessFileLineNumbers primaryWeapon primaryWeaponItems primaryWeaponMagazine priority processDiaryLink productVersion profileName profileNamespace profileNameSteam progressLoadingScreen progressPosition progressSetPosition publicVariable publicVariableClient publicVariableServer pushBack pushBackUnique putWeaponPool queryItemsPool queryMagazinePool queryWeaponPool rad radioChannelAdd radioChannelCreate radioChannelRemove radioChannelSetCallSign radioChannelSetLabel radioVolume rain rainbow random rank rankId rating rectangular registeredTasks registerTask reload reloadEnabled remoteControl remoteExec remoteExecCall remoteExecutedOwner remove3DENConnection remove3DENEventHandler remove3DENLayer removeAction removeAll3DENEventHandlers removeAllActions removeAllAssignedItems removeAllContainers removeAllCuratorAddons removeAllCuratorCameraAreas removeAllCuratorEditingAreas removeAllEventHandlers removeAllHandgunItems removeAllItems removeAllItemsWithMagazines removeAllMissionEventHandlers removeAllMPEventHandlers removeAllMusicEventHandlers removeAllOwnedMines removeAllPrimaryWeaponItems removeAllWeapons removeBackpack removeBackpackGlobal removeCuratorAddons removeCuratorCameraArea removeCuratorEditableObjects removeCuratorEditingArea removeDrawIcon removeDrawLinks removeEventHandler removeFromRemainsCollector removeGoggles removeGroupIcon removeHandgunItem removeHeadgear removeItem removeItemFromBackpack removeItemFromUniform removeItemFromVest removeItems removeMagazine removeMagazineGlobal removeMagazines removeMagazinesTurret removeMagazineTurret removeMenuItem removeMissionEventHandler removeMPEventHandler removeMusicEventHandler removeOwnedMine removePrimaryWeaponItem removeSecondaryWeaponItem removeSimpleTask removeSwitchableUnit removeTeamMember removeUniform removeVest removeWeapon removeWeaponAttachmentCargo removeWeaponCargo removeWeaponGlobal removeWeaponTurret reportRemoteTarget requiredVersion resetCamShake resetSubgroupDirection resize resources respawnVehicle restartEditorCamera reveal revealMine reverse reversedMouseY roadAt roadsConnectedTo roleDescription ropeAttachedObjects ropeAttachedTo ropeAttachEnabled ropeAttachTo ropeCreate ropeCut ropeDestroy ropeDetach ropeEndPosition ropeLength ropes ropeUnwind ropeUnwound rotorsForcesRTD rotorsRpmRTD round runInitScript safeZoneH safeZoneW safeZoneWAbs safeZoneX safeZoneXAbs safeZoneY save3DENInventory saveGame saveIdentity saveJoysticks saveOverlay saveProfileNamespace saveStatus saveVar savingEnabled say say2D say3D scopeName score scoreSide screenshot screenToWorld scriptDone scriptName scudState secondaryWeapon secondaryWeaponItems secondaryWeaponMagazine select selectBestPlaces selectDiarySubject selectedEditorObjects selectEditorObject selectionNames selectionPosition selectLeader selectMax selectMin selectNoPlayer selectPlayer selectRandom selectRandomWeighted selectWeapon selectWeaponTurret sendAUMessage sendSimpleCommand sendTask sendTaskResult sendUDPMessage serverCommand serverCommandAvailable serverCommandExecutable serverName serverTime set set3DENAttribute set3DENAttributes set3DENGrid set3DENIconsVisible set3DENLayer set3DENLinesVisible set3DENLogicType set3DENMissionAttribute set3DENMissionAttributes set3DENModelsVisible set3DENObjectType set3DENSelected setAccTime setActualCollectiveRTD setAirplaneThrottle setAirportSide setAmmo setAmmoCargo setAmmoOnPylon setAnimSpeedCoef setAperture setApertureNew setArmoryPoints setAttributes setAutonomous setBehaviour setBleedingRemaining setBrakesRTD setCameraInterest setCamShakeDefParams setCamShakeParams setCamUseTI setCaptive setCenterOfMass setCollisionLight setCombatMode setCompassOscillation setConvoySeparation setCuratorCameraAreaCeiling setCuratorCoef setCuratorEditingAreaType setCuratorWaypointCost setCurrentChannel setCurrentTask setCurrentWaypoint setCustomAimCoef setCustomWeightRTD setDamage setDammage setDate setDebriefingText setDefaultCamera setDestination setDetailMapBlendPars setDir setDirection setDrawIcon setDriveOnPath setDropInterval setDynamicSimulationDistance setDynamicSimulationDistanceCoef setEditorMode setEditorObjectScope setEffectCondition setEngineRPMRTD setFace setFaceAnimation setFatigue setFeatureType setFlagAnimationPhase setFlagOwner setFlagSide setFlagTexture setFog setFormation setFormationTask setFormDir setFriend setFromEditor setFSMVariable setFuel setFuelCargo setGroupIcon setGroupIconParams setGroupIconsSelectable setGroupIconsVisible setGroupId setGroupIdGlobal setGroupOwner setGusts setHideBehind setHit setHitIndex setHitPointDamage setHorizonParallaxCoef setHUDMovementLevels setIdentity setImportance setInfoPanel setLeader setLightAmbient setLightAttenuation setLightBrightness setLightColor setLightDayLight setLightFlareMaxDistance setLightFlareSize setLightIntensity setLightnings setLightUseFlare setLocalWindParams setMagazineTurretAmmo setMarkerAlpha setMarkerAlphaLocal setMarkerBrush setMarkerBrushLocal setMarkerColor setMarkerColorLocal setMarkerDir setMarkerDirLocal setMarkerPos setMarkerPosLocal setMarkerShape setMarkerShapeLocal setMarkerSize setMarkerSizeLocal setMarkerText setMarkerTextLocal setMarkerType setMarkerTypeLocal setMass setMimic setMousePosition setMusicEffect setMusicEventHandler setName setNameSound setObjectArguments setObjectMaterial setObjectMaterialGlobal setObjectProxy setObjectTexture setObjectTextureGlobal setObjectViewDistance setOvercast setOwner setOxygenRemaining setParticleCircle setParticleClass setParticleFire setParticleParams setParticleRandom setPilotCameraDirection setPilotCameraRotation setPilotCameraTarget setPilotLight setPiPEffect setPitch setPlateNumber setPlayable setPlayerRespawnTime setPos setPosASL setPosASL2 setPosASLW setPosATL setPosition setPosWorld setPylonLoadOut setPylonsPriority setRadioMsg setRain setRainbow setRandomLip setRank setRectangular setRepairCargo setRotorBrakeRTD setShadowDistance setShotParents setSide setSimpleTaskAlwaysVisible setSimpleTaskCustomData setSimpleTaskDescription setSimpleTaskDestination setSimpleTaskTarget setSimpleTaskType setSimulWeatherLayers setSize setSkill setSlingLoad setSoundEffect setSpeaker setSpeech setSpeedMode setStamina setStaminaScheme setStatValue setSuppression setSystemOfUnits setTargetAge setTaskMarkerOffset setTaskResult setTaskState setTerrainGrid setText setTimeMultiplier setTitleEffect setTrafficDensity setTrafficDistance setTrafficGap setTrafficSpeed setTriggerActivation setTriggerArea setTriggerStatements setTriggerText setTriggerTimeout setTriggerType setType setUnconscious setUnitAbility setUnitLoadout setUnitPos setUnitPosWeak setUnitRank setUnitRecoilCoefficient setUnitTrait setUnloadInCombat setUserActionText setUserMFDText setUserMFDvalue setVariable setVectorDir setVectorDirAndUp setVectorUp setVehicleAmmo setVehicleAmmoDef setVehicleArmor setVehicleCargo setVehicleId setVehicleLock setVehiclePosition setVehicleRadar setVehicleReceiveRemoteTargets setVehicleReportOwnPosition setVehicleReportRemoteTargets setVehicleTIPars setVehicleVarName setVelocity setVelocityModelSpace setVelocityTransformation setViewDistance setVisibleIfTreeCollapsed setWantedRPMRTD setWaves setWaypointBehaviour setWaypointCombatMode setWaypointCompletionRadius setWaypointDescription setWaypointForceBehaviour setWaypointFormation setWaypointHousePosition setWaypointLoiterRadius setWaypointLoiterType setWaypointName setWaypointPosition setWaypointScript setWaypointSpeed setWaypointStatements setWaypointTimeout setWaypointType setWaypointVisible setWeaponReloadingTime setWind setWindDir setWindForce setWindStr setWingForceScaleRTD setWPPos show3DIcons showChat showCinemaBorder showCommandingMenu showCompass showCuratorCompass showGPS showHUD showLegend showMap shownArtilleryComputer shownChat shownCompass shownCuratorCompass showNewEditorObject shownGPS shownHUD shownMap shownPad shownRadio shownScoretable shownUAVFeed shownWarrant shownWatch showPad showRadio showScoretable showSubtitles showUAVFeed showWarrant showWatch showWaypoint showWaypoints side sideChat sideEnemy sideFriendly sideRadio simpleTasks simulationEnabled simulCloudDensity simulCloudOcclusion simulInClouds simulWeatherSync sin size sizeOf skill skillFinal skipTime sleep sliderPosition sliderRange sliderSetPosition sliderSetRange sliderSetSpeed sliderSpeed slingLoadAssistantShown soldierMagazines someAmmo sort soundVolume spawn speaker speed speedMode splitString sqrt squadParams stance startLoadingScreen step stop stopEngineRTD stopped str sunOrMoon supportInfo suppressFor surfaceIsWater surfaceNormal surfaceType swimInDepth switchableUnits switchAction switchCamera switchGesture switchLight switchMove synchronizedObjects synchronizedTriggers synchronizedWaypoints synchronizeObjectsAdd synchronizeObjectsRemove synchronizeTrigger synchronizeWaypoint systemChat systemOfUnits tan targetKnowledge targets targetsAggregate targetsQuery taskAlwaysVisible taskChildren taskCompleted taskCustomData taskDescription taskDestination taskHint taskMarkerOffset taskParent taskResult taskState taskType teamMember teamName teams teamSwitch teamSwitchEnabled teamType terminate terrainIntersect terrainIntersectASL terrainIntersectAtASL text textLog textLogFormat tg time timeMultiplier titleCut titleFadeOut titleObj titleRsc titleText toArray toFixed toLower toString toUpper triggerActivated triggerActivation triggerArea triggerAttachedVehicle triggerAttachObject triggerAttachVehicle triggerDynamicSimulation triggerStatements triggerText triggerTimeout triggerTimeoutCurrent triggerType turretLocal turretOwner turretUnit tvAdd tvClear tvCollapse tvCollapseAll tvCount tvCurSel tvData tvDelete tvExpand tvExpandAll tvPicture tvSetColor tvSetCurSel tvSetData tvSetPicture tvSetPictureColor tvSetPictureColorDisabled tvSetPictureColorSelected tvSetPictureRight tvSetPictureRightColor tvSetPictureRightColorDisabled tvSetPictureRightColorSelected tvSetText tvSetTooltip tvSetValue tvSort tvSortByValue tvText tvTooltip tvValue type typeName typeOf UAVControl uiNamespace uiSleep unassignCurator unassignItem unassignTeam unassignVehicle underwater uniform uniformContainer uniformItems uniformMagazines unitAddons unitAimPosition unitAimPositionVisual unitBackpack unitIsUAV unitPos unitReady unitRecoilCoefficient units unitsBelowHeight unlinkItem 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",dragon_face:"🐲",dress:"👗",dromedary_camel:"🐪",drooling_face:"🤤",droplet:"💧",drum:"🥁",duck:"🦆",dvd:"📀","e-mail":"📧",eagle:"🦅",ear:"👂",ear_of_rice:"🌾",earth_africa:"🌍",earth_americas:"🌎",earth_asia:"🌏",egg:"🥚",eggplant:"🍆",eight_pointed_black_star:"✴️",eight_spoked_asterisk:"✳️",electric_plug:"🔌",elephant:"🐘",email:"✉️",end:"🔚",envelope_with_arrow:"📩",euro:"💶",european_castle:"🏰",european_post_office:"🏤",evergreen_tree:"🌲",exclamation:"❗️",expressionless:"😑",eye:"👁",eye_speech_bubble:"👁‍🗨",eyeglasses:"👓",eyes:"👀",face_with_head_bandage:"🤕",face_with_thermometer:"🤒",fist_oncoming:"👊",factory:"🏭",fallen_leaf:"🍂",family_man_woman_boy:"👪",family_man_boy:"👨‍👦",family_man_boy_boy:"👨‍👦‍👦",family_man_girl:"👨‍👧",family_man_girl_boy:"👨‍👧‍👦",family_man_girl_girl:"👨‍👧‍👧",family_man_man_boy:"👨‍👨‍👦",family_man_man_boy_boy:"👨‍👨‍👦‍👦",family_man_man_girl:"👨‍👨‍👧",family_man_man_girl_boy:"👨‍👨‍👧‍👦",family_man_man_girl_girl:"👨‍👨‍👧‍👧",family_man_woman_boy_boy:"👨‍👩‍👦‍👦",family_man_woman_girl:"👨‍👩‍👧",family_man_woman_girl_boy:"👨‍👩‍👧‍👦",family_man_woman_girl_girl:"👨‍👩‍👧‍👧",family_woman_boy:"👩‍👦",family_woman_boy_boy:"👩‍👦‍👦",family_woman_girl:"👩‍👧",family_woman_girl_boy:"👩‍👧‍👦",family_woman_girl_girl:"👩‍👧‍👧",family_woman_woman_boy:"👩‍👩‍👦",family_woman_woman_boy_boy:"👩‍👩‍👦‍👦",family_woman_woman_girl:"👩‍👩‍👧",family_woman_woman_girl_boy:"👩‍👩‍👧‍👦",family_woman_woman_girl_girl:"👩‍👩‍👧‍👧",fast_forward:"⏩",fax:"📠",fearful:"😨",feet:"🐾",female_detective:"🕵️‍♀️",ferris_wheel:"🎡",ferry:"⛴",field_hockey:"🏑",file_cabinet:"🗄",file_folder:"📁",film_projector:"📽",film_strip:"🎞",fire:"🔥",fire_engine:"🚒",fireworks:"🎆",first_quarter_moon:"🌓",first_quarter_moon_with_face:"🌛",fish:"🐟",fish_cake:"🍥",fishing_pole_and_fish:"🎣",fist_raised:"✊",fist_left:"🤛",fist_right:"🤜",flags:"🎏",flashlight:"🔦",fleur_de_lis:"⚜️",flight_arrival:"🛬",flight_departure:"🛫",floppy_disk:"💾",flower_playing_cards:"🎴",flushed:"😳",fog:"🌫",foggy:"🌁",football:"🏈",footprints:"👣",fork_and_knife:"🍴",fountain:"⛲️",fountain_pen:"🖋",four_leaf_clover:"🍀",fox_face:"🦊",framed_picture:"🖼",free:"🆓",fried_egg:"🍳",fried_shrimp:"🍤",fries:"🍟",frog:"🐸",frowning:"😦",frowning_face:"☹️",frowning_man:"🙍‍♂️",frowning_woman:"🙍",middle_finger:"🖕",fuelpump:"⛽️",full_moon:"🌕",full_moon_with_face:"🌝",funeral_urn:"⚱️",game_die:"🎲",gear:"⚙️",gem:"💎",gemini:"♊️",ghost:"👻",gift:"🎁",gift_heart:"💝",girl:"👧",globe_with_meridians:"🌐",goal_net:"🥅",goat:"🐐",golf:"⛳️",golfing_man:"🏌️",golfing_woman:"🏌️‍♀️",gorilla:"🦍",grapes:"🍇",green_apple:"🍏",green_book:"📗",green_heart:"💚",green_salad:"🥗",grey_exclamation:"❕",grey_question:"❔",grimacing:"😬",grin:"😁",grinning:"😀",guardsman:"💂",guardswoman:"💂‍♀️",guitar:"🎸",gun:"🔫",haircut_woman:"💇",haircut_man:"💇‍♂️",hamburger:"🍔",hammer:"🔨",hammer_and_pick:"⚒",hammer_and_wrench:"🛠",hamster:"🐹",hand:"✋",handbag:"👜",handshake:"🤝",hankey:"💩",hatched_chick:"🐥",hatching_chick:"🐣",headphones:"🎧",hear_no_evil:"🙉",heart:"❤️",heart_decoration:"💟",heart_eyes:"😍",heart_eyes_cat:"😻",heartbeat:"💓",heartpulse:"💗",hearts:"♥️",heavy_check_mark:"✔️",heavy_division_sign:"➗",heavy_dollar_sign:"💲",heavy_heart_exclamation:"❣️",heavy_minus_sign:"➖",heavy_multiplication_x:"✖️",heavy_plus_sign:"➕",helicopter:"🚁",herb:"🌿",hibiscus:"🌺",high_brightness:"🔆",high_heel:"👠",hocho:"🔪",hole:"🕳",honey_pot:"🍯",horse:"🐴",horse_racing:"🏇",hospital:"🏥",hot_pepper:"🌶",hotdog:"🌭",hotel:"🏨",hotsprings:"♨️",hourglass:"⌛️",hourglass_flowing_sand:"⏳",house:"🏠",house_with_garden:"🏡",houses:"🏘",hugs:"🤗",hushed:"😯",ice_cream:"🍨",ice_hockey:"🏒",ice_skate:"⛸",icecream:"🍦",id:"🆔",ideograph_advantage:"🉐",imp:"👿",inbox_tray:"📥",incoming_envelope:"📨",tipping_hand_woman:"💁",information_source:"ℹ️",innocent:"😇",interrobang:"⁉️",iphone:"📱",izakaya_lantern:"🏮",jack_o_lantern:"🎃",japan:"🗾",japanese_castle:"🏯",japanese_goblin:"👺",japanese_ogre:"👹",jeans:"👖",joy:"😂",joy_cat:"😹",joystick:"🕹",kaaba:"🕋",key:"🔑",keyboard:"⌨️",keycap_ten:"🔟",kick_scooter:"🛴",kimono:"👘",kiss:"💋",kissing:"😗",kissing_cat:"😽",kissing_closed_eyes:"😚",kissing_heart:"😘",kissing_smiling_eyes:"😙",kiwi_fruit:"🥝",koala:"🐨",koko:"🈁",label:"🏷",large_blue_circle:"🔵",large_blue_diamond:"🔷",large_orange_diamond:"🔶",last_quarter_moon:"🌗",last_quarter_moon_with_face:"🌜",latin_cross:"✝️",laughing:"😆",leaves:"🍃",ledger:"📒",left_luggage:"🛅",left_right_arrow:"↔️",leftwards_arrow_with_hook:"↩️",lemon:"🍋",leo:"♌️",leopard:"🐆",level_slider:"🎚",libra:"♎️",light_rail:"🚈",link:"🔗",lion:"🦁",lips:"👄",lipstick:"💄",lizard:"🦎",lock:"🔒",lock_with_ink_pen:"🔏",lollipop:"🍭",loop:"➿",loud_sound:"🔊",loudspeaker:"📢",love_hotel:"🏩",love_letter:"💌",low_brightness:"🔅",lying_face:"🤥",m:"Ⓜ️",mag:"🔍",mag_right:"🔎",mahjong:"🀄️",mailbox:"📫",mailbox_closed:"📪",mailbox_with_mail:"📬",mailbox_with_no_mail:"📭",man:"👨",man_artist:"👨‍🎨",man_astronaut:"👨‍🚀",man_cartwheeling:"🤸‍♂️",man_cook:"👨‍🍳",man_dancing:"🕺",man_facepalming:"🤦‍♂️",man_factory_worker:"👨‍🏭",man_farmer:"👨‍🌾",man_firefighter:"👨‍🚒",man_health_worker:"👨‍⚕️",man_in_tuxedo:"🤵",man_judge:"👨‍⚖️",man_juggling:"🤹‍♂️",man_mechanic:"👨‍🔧",man_office_worker:"👨‍💼",man_pilot:"👨‍✈️",man_playing_handball:"🤾‍♂️",man_playing_water_polo:"🤽‍♂️",man_scientist:"👨‍🔬",man_shrugging:"🤷‍♂️",man_singer:"👨‍🎤",man_student:"👨‍🎓",man_teacher:"👨‍🏫",man_technologist:"👨‍💻",man_with_gua_pi_mao:"👲",man_with_turban:"👳",tangerine:"🍊",mans_shoe:"👞",mantelpiece_clock:"🕰",maple_leaf:"🍁",martial_arts_uniform:"🥋",mask:"😷",massage_woman:"💆",massage_man:"💆‍♂️",meat_on_bone:"🍖",medal_military:"🎖",medal_sports:"🏅",mega:"📣",melon:"🍈",memo:"📝",men_wrestling:"🤼‍♂️",menorah:"🕎",mens:"🚹",metal:"🤘",metro:"🚇",microphone:"🎤",microscope:"🔬",milk_glass:"🥛",milky_way:"🌌",minibus:"🚐",minidisc:"💽",mobile_phone_off:"📴",money_mouth_face:"🤑",money_with_wings:"💸",moneybag:"💰",monkey:"🐒",monkey_face:"🐵",monorail:"🚝",moon:"🌔",mortar_board:"🎓",mosque:"🕌",motor_boat:"🛥",motor_scooter:"🛵",motorcycle:"🏍",motorway:"🛣",mount_fuji:"🗻",mountain:"⛰",mountain_biking_man:"🚵",mountain_biking_woman:"🚵‍♀️",mountain_cableway:"🚠",mountain_railway:"🚞",mountain_snow:"🏔",mouse:"🐭",mouse2:"🐁",movie_camera:"🎥",moyai:"🗿",mrs_claus:"🤶",muscle:"💪",mushroom:"🍄",musical_keyboard:"🎹",musical_note:"🎵",musical_score:"🎼",mute:"🔇",nail_care:"💅",name_badge:"📛",national_park:"🏞",nauseated_face:"🤢",necktie:"👔",negative_squared_cross_mark:"❎",nerd_face:"🤓",neutral_face:"😐",new:"🆕",new_moon:"🌑",new_moon_with_face:"🌚",newspaper:"📰",newspaper_roll:"🗞",next_track_button:"⏭",ng:"🆖",no_good_man:"🙅‍♂️",no_good_woman:"🙅",night_with_stars:"🌃",no_bell:"🔕",no_bicycles:"🚳",no_entry:"⛔️",no_entry_sign:"🚫",no_mobile_phones:"📵",no_mouth:"😶",no_pedestrians:"🚷",no_smoking:"🚭","non-potable_water":"🚱",nose:"👃",notebook:"📓",notebook_with_decorative_cover:"📔",notes:"🎶",nut_and_bolt:"🔩",o:"⭕️",o2:"🅾️",ocean:"🌊",octopus:"🐙",oden:"🍢",office:"🏢",oil_drum:"🛢",ok:"🆗",ok_hand:"👌",ok_man:"🙆‍♂️",ok_woman:"🙆",old_key:"🗝",older_man:"👴",older_woman:"👵",om:"🕉",on:"🔛",oncoming_automobile:"🚘",oncoming_bus:"🚍",oncoming_police_car:"🚔",oncoming_taxi:"🚖",open_file_folder:"📂",open_hands:"👐",open_mouth:"😮",open_umbrella:"☂️",ophiuchus:"⛎",orange_book:"📙",orthodox_cross:"☦️",outbox_tray:"📤",owl:"🦉",ox:"🐂",package:"📦",page_facing_up:"📄",page_with_curl:"📃",pager:"📟",paintbrush:"🖌",palm_tree:"🌴",pancakes:"🥞",panda_face:"🐼",paperclip:"📎",paperclips:"🖇",parasol_on_ground:"⛱",parking:"🅿️",part_alternation_mark:"〽️",partly_sunny:"⛅️",passenger_ship:"🛳",passport_control:"🛂",pause_button:"⏸",peace_symbol:"☮️",peach:"🍑",peanuts:"🥜",pear:"🍐",pen:"🖊",pencil2:"✏️",penguin:"🐧",pensive:"😔",performing_arts:"🎭",persevere:"😣",person_fencing:"🤺",pouting_woman:"🙎",phone:"☎️",pick:"⛏",pig:"🐷",pig2:"🐖",pig_nose:"🐽",pill:"💊",pineapple:"🍍",ping_pong:"🏓",pisces:"♓️",pizza:"🍕",place_of_worship:"🛐",plate_with_cutlery:"🍽",play_or_pause_button:"⏯",point_down:"👇",point_left:"👈",point_right:"👉",point_up:"☝️",point_up_2:"👆",police_car:"🚓",policewoman:"👮‍♀️",poodle:"🐩",popcorn:"🍿",post_office:"🏣",postal_horn:"📯",postbox:"📮",potable_water:"🚰",potato:"🥔",pouch:"👝",poultry_leg:"🍗",pound:"💷",rage:"😡",pouting_cat:"😾",pouting_man:"🙎‍♂️",pray:"🙏",prayer_beads:"📿",pregnant_woman:"🤰",previous_track_button:"⏮",prince:"🤴",princess:"👸",printer:"🖨",purple_heart:"💜",purse:"👛",pushpin:"📌",put_litter_in_its_place:"🚮",question:"❓",rabbit:"🐰",rabbit2:"🐇",racehorse:"🐎",racing_car:"🏎",radio:"📻",radio_button:"🔘",radioactive:"☢️",railway_car:"🚃",railway_track:"🛤",rainbow:"🌈",rainbow_flag:"🏳️‍🌈",raised_back_of_hand:"🤚",raised_hand_with_fingers_splayed:"🖐",raised_hands:"🙌",raising_hand_woman:"🙋",raising_hand_man:"🙋‍♂️",ram:"🐏",ramen:"🍜",rat:"🐀",record_button:"⏺",recycle:"♻️",red_circle:"🔴",registered:"®️",relaxed:"☺️",relieved:"😌",reminder_ribbon:"🎗",repeat:"🔁",repeat_one:"🔂",rescue_worker_helmet:"⛑",restroom:"🚻",revolving_hearts:"💞",rewind:"⏪",rhinoceros:"🦏",ribbon:"🎀",rice:"🍚",rice_ball:"🍙",rice_cracker:"🍘",rice_scene:"🎑",right_anger_bubble:"🗯",ring:"💍",robot:"🤖",rocket:"🚀",rofl:"🤣",roll_eyes:"🙄",roller_coaster:"🎢",rooster:"🐓",rose:"🌹",rosette:"🏵",rotating_light:"🚨",round_pushpin:"📍",rowing_man:"🚣",rowing_woman:"🚣‍♀️",rugby_football:"🏉",running_man:"🏃",running_shirt_with_sash:"🎽",running_woman:"🏃‍♀️",sa:"🈂️",sagittarius:"♐️",sake:"🍶",sandal:"👡",santa:"🎅",satellite:"📡",saxophone:"🎷",school:"🏫",school_satchel:"🎒",scissors:"✂️",scorpion:"🦂",scorpius:"♏️",scream:"😱",scream_cat:"🙀",scroll:"📜",seat:"💺",secret:"㊙️",see_no_evil:"🙈",seedling:"🌱",selfie:"🤳",shallow_pan_of_food:"🥘",shamrock:"☘️",shark:"🦈",shaved_ice:"🍧",sheep:"🐑",shell:"🐚",shield:"🛡",shinto_shrine:"⛩",ship:"🚢",shirt:"👕",shopping:"🛍",shopping_cart:"🛒",shower:"🚿",shrimp:"🦐",signal_strength:"📶",six_pointed_star:"🔯",ski:"🎿",skier:"⛷",skull:"💀",skull_and_crossbones:"☠️",sleeping:"😴",sleeping_bed:"🛌",sleepy:"😪",slightly_frowning_face:"🙁",slightly_smiling_face:"🙂",slot_machine:"🎰",small_airplane:"🛩",small_blue_diamond:"🔹",small_orange_diamond:"🔸",small_red_triangle:"🔺",small_red_triangle_down:"🔻",smile:"😄",smile_cat:"😸",smiley:"😃",smiley_cat:"😺",smiling_imp:"😈",smirk:"😏",smirk_cat:"😼",smoking:"🚬",snail:"🐌",snake:"🐍",sneezing_face:"🤧",snowboarder:"🏂",snowflake:"❄️",snowman:"⛄️",snowman_with_snow:"☃️",sob:"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DefineExternal DefineInputStreamMethod DefineOutputStreamMethod DefineResourceFunction Definition Degree DegreeCentrality DegreeGraphDistribution DegreeLexicographic DegreeReverseLexicographic DEigensystem DEigenvalues Deinitialization Del DelaunayMesh Delayed Deletable Delete DeleteAnomalies DeleteBorderComponents DeleteCases DeleteChannel DeleteCloudExpression DeleteContents DeleteDirectory DeleteDuplicates DeleteDuplicatesBy DeleteFile DeleteMissing DeleteObject DeletePermissionsKey DeleteSearchIndex DeleteSmallComponents DeleteStopwords DeleteWithContents DeletionWarning DelimitedArray DelimitedSequence Delimiter DelimiterFlashTime DelimiterMatching Delimiters DeliveryFunction Dendrogram Denominator DensityGraphics DensityHistogram DensityPlot DensityPlot3D DependentVariables Deploy Deployed Depth DepthFirstScan Derivative DerivativeFilter DerivedKey DescriptorStateSpace DesignMatrix DestroyAfterEvaluation Det DeviceClose DeviceConfigure DeviceExecute DeviceExecuteAsynchronous DeviceObject DeviceOpen DeviceOpenQ DeviceRead DeviceReadBuffer DeviceReadLatest DeviceReadList DeviceReadTimeSeries Devices DeviceStreams DeviceWrite DeviceWriteBuffer DGaussianWavelet DiacriticalPositioning Diagonal DiagonalizableMatrixQ DiagonalMatrix DiagonalMatrixQ Dialog DialogIndent DialogInput DialogLevel DialogNotebook DialogProlog DialogReturn DialogSymbols Diamond DiamondMatrix DiceDissimilarity DictionaryLookup DictionaryWordQ DifferenceDelta DifferenceOrder DifferenceQuotient DifferenceRoot DifferenceRootReduce Differences DifferentialD DifferentialRoot DifferentialRootReduce DifferentiatorFilter DigitalSignature DigitBlock DigitBlockMinimum DigitCharacter DigitCount DigitQ DihedralAngle DihedralGroup Dilation DimensionalCombinations DimensionalMeshComponents DimensionReduce DimensionReducerFunction DimensionReduction Dimensions DiracComb DiracDelta DirectedEdge DirectedEdges DirectedGraph DirectedGraphQ DirectedInfinity Direction Directive Directory DirectoryName DirectoryQ DirectoryStack DirichletBeta DirichletCharacter DirichletCondition DirichletConvolve DirichletDistribution DirichletEta DirichletL DirichletLambda DirichletTransform DirichletWindow DisableConsolePrintPacket DisableFormatting DiscreteChirpZTransform DiscreteConvolve DiscreteDelta DiscreteHadamardTransform DiscreteIndicator DiscreteLimit DiscreteLQEstimatorGains DiscreteLQRegulatorGains DiscreteLyapunovSolve DiscreteMarkovProcess DiscreteMaxLimit DiscreteMinLimit DiscretePlot DiscretePlot3D DiscreteRatio DiscreteRiccatiSolve DiscreteShift DiscreteTimeModelQ DiscreteUniformDistribution DiscreteVariables DiscreteWaveletData DiscreteWaveletPacketTransform DiscreteWaveletTransform DiscretizeGraphics DiscretizeRegion Discriminant DisjointQ Disjunction Disk DiskBox DiskMatrix DiskSegment Dispatch DispatchQ DispersionEstimatorFunction Display DisplayAllSteps DisplayEndPacket DisplayFlushImagePacket DisplayForm DisplayFunction DisplayPacket DisplayRules DisplaySetSizePacket DisplayString DisplayTemporary DisplayWith DisplayWithRef DisplayWithVariable DistanceFunction DistanceMatrix DistanceTransform Distribute Distributed DistributedContexts DistributeDefinitions DistributionChart DistributionDomain DistributionFitTest DistributionParameterAssumptions DistributionParameterQ Dithering Div Divergence Divide DivideBy Dividers DivideSides Divisible Divisors DivisorSigma DivisorSum DMSList DMSString Do DockedCells DocumentGenerator DocumentGeneratorInformation DocumentGeneratorInformationData DocumentGenerators DocumentNotebook DocumentWeightingRules Dodecahedron DomainRegistrationInformation DominantColors DOSTextFormat Dot DotDashed DotEqual DotLayer DotPlusLayer Dotted DoubleBracketingBar DoubleContourIntegral DoubleDownArrow DoubleLeftArrow DoubleLeftRightArrow DoubleLeftTee DoubleLongLeftArrow DoubleLongLeftRightArrow DoubleLongRightArrow DoubleRightArrow DoubleRightTee DoubleUpArrow DoubleUpDownArrow DoubleVerticalBar DoublyInfinite Down DownArrow DownArrowBar DownArrowUpArrow DownLeftRightVector DownLeftTeeVector DownLeftVector DownLeftVectorBar DownRightTeeVector DownRightVector DownRightVectorBar Downsample DownTee DownTeeArrow DownValues DragAndDrop DrawEdges DrawFrontFaces DrawHighlighted Drop DropoutLayer DSolve DSolveValue Dt DualLinearProgramming DualPolyhedron DualSystemsModel DumpGet DumpSave DuplicateFreeQ Duration Dynamic DynamicBox DynamicBoxOptions DynamicEvaluationTimeout DynamicGeoGraphics DynamicImage DynamicLocation DynamicModule DynamicModuleBox DynamicModuleBoxOptions DynamicModuleParent DynamicModuleValues DynamicName DynamicNamespace DynamicReference DynamicSetting DynamicUpdating DynamicWrapper DynamicWrapperBox DynamicWrapperBoxOptionsE EarthImpactData EarthquakeData EccentricityCentrality Echo EchoFunction EclipseType EdgeAdd EdgeBetweennessCentrality EdgeCapacity EdgeCapForm EdgeColor EdgeConnectivity EdgeContract EdgeCost EdgeCount EdgeCoverQ EdgeCycleMatrix EdgeDashing EdgeDelete EdgeDetect EdgeForm EdgeIndex EdgeJoinForm EdgeLabeling EdgeLabels EdgeLabelStyle EdgeList EdgeOpacity EdgeQ EdgeRenderingFunction EdgeRules EdgeShapeFunction EdgeStyle EdgeThickness EdgeWeight EdgeWeightedGraphQ Editable EditButtonSettings EditCellTagsSettings EditDistance EffectiveInterest Eigensystem Eigenvalues EigenvectorCentrality Eigenvectors Element ElementData ElementwiseLayer ElidedForms Eliminate EliminationOrder Ellipsoid EllipticE EllipticExp EllipticExpPrime EllipticF EllipticFilterModel EllipticK EllipticLog EllipticNomeQ EllipticPi EllipticReducedHalfPeriods EllipticTheta EllipticThetaPrime EmbedCode EmbeddedHTML EmbeddedService EmbeddingLayer EmbeddingObject EmitSound EmphasizeSyntaxErrors EmpiricalDistribution Empty EmptyGraphQ EmptyRegion EnableConsolePrintPacket Enabled Encode Encrypt EncryptedObject EncryptFile End EndAdd EndDialogPacket EndFrontEndInteractionPacket EndOfBuffer EndOfFile EndOfLine EndOfString EndPackage EngineEnvironment EngineeringForm Enter EnterExpressionPacket EnterTextPacket Entity EntityClass EntityClassList EntityCopies EntityFunction EntityGroup EntityInstance EntityList EntityPrefetch EntityProperties EntityProperty EntityPropertyClass EntityRegister EntityStore EntityStores EntityTypeName EntityUnregister EntityValue Entropy EntropyFilter Environment Epilog EpilogFunction Equal EqualColumns EqualRows EqualTilde EqualTo EquatedTo Equilibrium EquirippleFilterKernel Equivalent Erf Erfc Erfi ErlangB ErlangC ErlangDistribution Erosion ErrorBox ErrorBoxOptions ErrorNorm ErrorPacket ErrorsDialogSettings EscapeRadius EstimatedBackground EstimatedDistribution EstimatedProcess EstimatorGains EstimatorRegulator EuclideanDistance EulerAngles EulerCharacteristic EulerE EulerGamma EulerianGraphQ EulerMatrix EulerPhi Evaluatable Evaluate Evaluated EvaluatePacket EvaluateScheduledTask EvaluationBox EvaluationCell EvaluationCompletionAction EvaluationData EvaluationElements EvaluationEnvironment EvaluationMode EvaluationMonitor EvaluationNotebook EvaluationObject EvaluationOrder Evaluator EvaluatorNames EvenQ EventData EventEvaluator EventHandler EventHandlerTag EventLabels EventSeries ExactBlackmanWindow ExactNumberQ ExactRootIsolation ExampleData Except ExcludedForms ExcludedLines ExcludedPhysicalQuantities ExcludePods Exclusions ExclusionsStyle Exists Exit ExitDialog ExoplanetData Exp Expand ExpandAll ExpandDenominator ExpandFileName ExpandNumerator Expectation ExpectationE ExpectedValue ExpGammaDistribution ExpIntegralE ExpIntegralEi ExpirationDate Exponent ExponentFunction ExponentialDistribution ExponentialFamily ExponentialGeneratingFunction ExponentialMovingAverage ExponentialPowerDistribution ExponentPosition ExponentStep Export ExportAutoReplacements ExportByteArray ExportForm ExportPacket ExportString Expression ExpressionCell ExpressionPacket ExpressionUUID ExpToTrig ExtendedEntityClass ExtendedGCD Extension ExtentElementFunction ExtentMarkers ExtentSize ExternalBundle ExternalCall ExternalDataCharacterEncoding ExternalEvaluate ExternalFunction ExternalFunctionName ExternalObject ExternalOptions ExternalSessionObject ExternalSessions ExternalTypeSignature ExternalValue Extract ExtractArchive ExtractLayer ExtremeValueDistributionFaceForm FaceGrids FaceGridsStyle FacialFeatures Factor FactorComplete Factorial Factorial2 FactorialMoment FactorialMomentGeneratingFunction FactorialPower FactorInteger FactorList FactorSquareFree FactorSquareFreeList FactorTerms FactorTermsList Fail Failure FailureAction FailureDistribution FailureQ False FareySequence FARIMAProcess FeatureDistance FeatureExtract FeatureExtraction FeatureExtractor FeatureExtractorFunction FeatureNames FeatureNearest FeatureSpacePlot FeatureSpacePlot3D FeatureTypes FEDisableConsolePrintPacket FeedbackLinearize FeedbackSector FeedbackSectorStyle FeedbackType FEEnableConsolePrintPacket FetalGrowthData Fibonacci Fibonorial FieldCompletionFunction FieldHint FieldHintStyle FieldMasked FieldSize File FileBaseName FileByteCount FileConvert FileDate FileExistsQ FileExtension FileFormat FileHandler FileHash FileInformation FileName FileNameDepth FileNameDialogSettings FileNameDrop FileNameForms FileNameJoin FileNames FileNameSetter FileNameSplit FileNameTake FilePrint FileSize FileSystemMap FileSystemScan FileTemplate FileTemplateApply FileType FilledCurve FilledCurveBox FilledCurveBoxOptions Filling FillingStyle FillingTransform FilteredEntityClass FilterRules FinancialBond FinancialData FinancialDerivative FinancialIndicator Find FindAnomalies FindArgMax FindArgMin FindChannels FindClique FindClusters FindCookies FindCurvePath FindCycle FindDevices FindDistribution FindDistributionParameters FindDivisions FindEdgeCover FindEdgeCut FindEdgeIndependentPaths FindEquationalProof FindEulerianCycle FindExternalEvaluators FindFaces FindFile FindFit FindFormula FindFundamentalCycles FindGeneratingFunction FindGeoLocation FindGeometricConjectures FindGeometricTransform FindGraphCommunities FindGraphIsomorphism FindGraphPartition FindHamiltonianCycle FindHamiltonianPath FindHiddenMarkovStates FindIndependentEdgeSet FindIndependentVertexSet FindInstance FindIntegerNullVector FindKClan FindKClique FindKClub FindKPlex FindLibrary FindLinearRecurrence FindList FindMatchingColor FindMaximum FindMaximumFlow FindMaxValue FindMeshDefects FindMinimum FindMinimumCostFlow FindMinimumCut FindMinValue FindMoleculeSubstructure FindPath FindPeaks FindPermutation FindPostmanTour FindProcessParameters FindRepeat FindRoot FindSequenceFunction FindSettings FindShortestPath FindShortestTour FindSpanningTree FindSystemModelEquilibrium FindTextualAnswer FindThreshold FindTransientRepeat FindVertexCover FindVertexCut FindVertexIndependentPaths Fine FinishDynamic FiniteAbelianGroupCount FiniteGroupCount FiniteGroupData First FirstCase FirstPassageTimeDistribution FirstPosition FischerGroupFi22 FischerGroupFi23 FischerGroupFi24Prime FisherHypergeometricDistribution FisherRatioTest FisherZDistribution Fit FitAll FitRegularization FittedModel FixedOrder FixedPoint FixedPointList FlashSelection Flat Flatten FlattenAt FlattenLayer FlatTopWindow FlipView Floor FlowPolynomial FlushPrintOutputPacket Fold FoldList FoldPair FoldPairList FollowRedirects Font FontColor FontFamily FontForm FontName FontOpacity FontPostScriptName FontProperties FontReencoding FontSize FontSlant FontSubstitutions FontTracking FontVariations FontWeight For ForAll Format FormatRules FormatType FormatTypeAutoConvert FormatValues FormBox FormBoxOptions FormControl FormFunction FormLayoutFunction FormObject FormPage FormTheme FormulaData FormulaLookup FortranForm Forward ForwardBackward Fourier FourierCoefficient FourierCosCoefficient FourierCosSeries FourierCosTransform FourierDCT FourierDCTFilter FourierDCTMatrix FourierDST FourierDSTMatrix FourierMatrix FourierParameters FourierSequenceTransform FourierSeries FourierSinCoefficient FourierSinSeries FourierSinTransform FourierTransform FourierTrigSeries FractionalBrownianMotionProcess FractionalGaussianNoiseProcess FractionalPart FractionBox FractionBoxOptions FractionLine Frame FrameBox FrameBoxOptions Framed FrameInset FrameLabel Frameless FrameMargins FrameRate FrameStyle FrameTicks FrameTicksStyle FRatioDistribution FrechetDistribution FreeQ FrenetSerretSystem FrequencySamplingFilterKernel FresnelC FresnelF FresnelG FresnelS Friday FrobeniusNumber FrobeniusSolve FromAbsoluteTime FromCharacterCode FromCoefficientRules FromContinuedFraction FromDate FromDigits FromDMS FromEntity FromJulianDate FromLetterNumber FromPolarCoordinates FromRomanNumeral FromSphericalCoordinates FromUnixTime Front FrontEndDynamicExpression FrontEndEventActions FrontEndExecute FrontEndObject FrontEndResource FrontEndResourceString FrontEndStackSize FrontEndToken FrontEndTokenExecute FrontEndValueCache FrontEndVersion FrontFaceColor FrontFaceOpacity Full FullAxes FullDefinition FullForm FullGraphics FullInformationOutputRegulator FullOptions FullRegion FullSimplify Function FunctionCompile FunctionCompileExport FunctionCompileExportByteArray FunctionCompileExportLibrary FunctionCompileExportString FunctionDomain FunctionExpand FunctionInterpolation FunctionPeriod FunctionRange FunctionSpace FussellVeselyImportanceGaborFilter GaborMatrix GaborWavelet GainMargins GainPhaseMargins GalaxyData GalleryView Gamma GammaDistribution GammaRegularized GapPenalty GARCHProcess GatedRecurrentLayer Gather GatherBy GaugeFaceElementFunction GaugeFaceStyle GaugeFrameElementFunction GaugeFrameSize GaugeFrameStyle GaugeLabels GaugeMarkers GaugeStyle GaussianFilter GaussianIntegers GaussianMatrix GaussianOrthogonalMatrixDistribution GaussianSymplecticMatrixDistribution GaussianUnitaryMatrixDistribution GaussianWindow GCD GegenbauerC General GeneralizedLinearModelFit GenerateAsymmetricKeyPair GenerateConditions GeneratedCell GeneratedDocumentBinding GenerateDerivedKey GenerateDigitalSignature GenerateDocument GeneratedParameters GeneratedQuantityMagnitudes GenerateHTTPResponse GenerateSecuredAuthenticationKey GenerateSymmetricKey GeneratingFunction GeneratorDescription GeneratorHistoryLength GeneratorOutputType Generic GenericCylindricalDecomposition GenomeData GenomeLookup GeoAntipode GeoArea GeoArraySize GeoBackground GeoBoundingBox GeoBounds GeoBoundsRegion GeoBubbleChart GeoCenter GeoCircle GeodesicClosing GeodesicDilation GeodesicErosion GeodesicOpening GeoDestination GeodesyData GeoDirection GeoDisk GeoDisplacement GeoDistance GeoDistanceList GeoElevationData GeoEntities GeoGraphics GeogravityModelData GeoGridDirectionDifference GeoGridLines GeoGridLinesStyle GeoGridPosition GeoGridRange GeoGridRangePadding GeoGridUnitArea GeoGridUnitDistance GeoGridVector GeoGroup GeoHemisphere GeoHemisphereBoundary GeoHistogram GeoIdentify GeoImage GeoLabels GeoLength GeoListPlot GeoLocation GeologicalPeriodData GeomagneticModelData GeoMarker GeometricAssertion GeometricBrownianMotionProcess GeometricDistribution GeometricMean GeometricMeanFilter GeometricScene GeometricTransformation GeometricTransformation3DBox GeometricTransformation3DBoxOptions GeometricTransformationBox GeometricTransformationBoxOptions GeoModel GeoNearest GeoPath GeoPosition GeoPositionENU GeoPositionXYZ GeoProjection GeoProjectionData GeoRange GeoRangePadding GeoRegionValuePlot GeoResolution GeoScaleBar GeoServer GeoSmoothHistogram GeoStreamPlot GeoStyling GeoStylingImageFunction GeoVariant GeoVector GeoVectorENU GeoVectorPlot GeoVectorXYZ GeoVisibleRegion GeoVisibleRegionBoundary GeoWithinQ GeoZoomLevel GestureHandler GestureHandlerTag Get GetBoundingBoxSizePacket GetContext GetEnvironment GetFileName GetFrontEndOptionsDataPacket GetLinebreakInformationPacket GetMenusPacket GetPageBreakInformationPacket Glaisher GlobalClusteringCoefficient GlobalPreferences GlobalSession Glow GoldenAngle GoldenRatio GompertzMakehamDistribution GoodmanKruskalGamma GoodmanKruskalGammaTest Goto Grad Gradient GradientFilter GradientOrientationFilter GrammarApply GrammarRules GrammarToken Graph Graph3D GraphAssortativity GraphAutomorphismGroup GraphCenter GraphComplement GraphData GraphDensity GraphDiameter GraphDifference GraphDisjointUnion GraphDistance GraphDistanceMatrix GraphElementData GraphEmbedding GraphHighlight GraphHighlightStyle GraphHub Graphics Graphics3D Graphics3DBox Graphics3DBoxOptions GraphicsArray GraphicsBaseline GraphicsBox GraphicsBoxOptions GraphicsColor GraphicsColumn GraphicsComplex GraphicsComplex3DBox GraphicsComplex3DBoxOptions GraphicsComplexBox GraphicsComplexBoxOptions GraphicsContents GraphicsData GraphicsGrid GraphicsGridBox GraphicsGroup GraphicsGroup3DBox GraphicsGroup3DBoxOptions GraphicsGroupBox GraphicsGroupBoxOptions GraphicsGrouping GraphicsHighlightColor GraphicsRow GraphicsSpacing GraphicsStyle GraphIntersection GraphLayout GraphLinkEfficiency GraphPeriphery GraphPlot GraphPlot3D GraphPower GraphPropertyDistribution GraphQ GraphRadius GraphReciprocity GraphRoot GraphStyle GraphUnion Gray GrayLevel Greater GreaterEqual GreaterEqualLess GreaterEqualThan GreaterFullEqual GreaterGreater GreaterLess GreaterSlantEqual GreaterThan GreaterTilde Green GreenFunction Grid GridBaseline GridBox GridBoxAlignment GridBoxBackground GridBoxDividers GridBoxFrame GridBoxItemSize GridBoxItemStyle GridBoxOptions GridBoxSpacings GridCreationSettings GridDefaultElement GridElementStyleOptions GridFrame GridFrameMargins GridGraph GridLines GridLinesStyle GroebnerBasis GroupActionBase GroupBy GroupCentralizer GroupElementFromWord GroupElementPosition GroupElementQ GroupElements GroupElementToWord GroupGenerators Groupings GroupMultiplicationTable GroupOrbits GroupOrder GroupPageBreakWithin GroupSetwiseStabilizer GroupStabilizer GroupStabilizerChain GroupTogetherGrouping GroupTogetherNestedGrouping GrowCutComponents Gudermannian GuidedFilter GumbelDistributionHaarWavelet HadamardMatrix HalfLine HalfNormalDistribution HalfPlane HalfSpace HamiltonianGraphQ HammingDistance HammingWindow HandlerFunctions HandlerFunctionsKeys HankelH1 HankelH2 HankelMatrix HankelTransform HannPoissonWindow HannWindow HaradaNortonGroupHN HararyGraph HarmonicMean HarmonicMeanFilter HarmonicNumber Hash Haversine HazardFunction Head HeadCompose HeaderLines Heads HeavisideLambda HeavisidePi HeavisideTheta HeldGroupHe HeldPart HelpBrowserLookup HelpBrowserNotebook HelpBrowserSettings Here HermiteDecomposition HermiteH HermitianMatrixQ HessenbergDecomposition Hessian HexadecimalCharacter Hexahedron HexahedronBox HexahedronBoxOptions HiddenMarkovProcess HiddenSurface Highlighted HighlightGraph HighlightImage HighlightMesh HighpassFilter HigmanSimsGroupHS HilbertCurve HilbertFilter HilbertMatrix Histogram Histogram3D HistogramDistribution HistogramList HistogramTransform HistogramTransformInterpolation HistoricalPeriodData HitMissTransform HITSCentrality HjorthDistribution HodgeDual HoeffdingD HoeffdingDTest Hold HoldAll HoldAllComplete HoldComplete HoldFirst HoldForm HoldPattern HoldRest HolidayCalendar HomeDirectory HomePage Horizontal HorizontalForm HorizontalGauge HorizontalScrollPosition HornerForm HostLookup HotellingTSquareDistribution HoytDistribution HTMLSave HTTPErrorResponse HTTPRedirect HTTPRequest HTTPRequestData HTTPResponse Hue HumanGrowthData HumpDownHump HumpEqual HurwitzLerchPhi HurwitzZeta HyperbolicDistribution HypercubeGraph HyperexponentialDistribution Hyperfactorial Hypergeometric0F1 Hypergeometric0F1Regularized Hypergeometric1F1 Hypergeometric1F1Regularized Hypergeometric2F1 Hypergeometric2F1Regularized HypergeometricDistribution HypergeometricPFQ HypergeometricPFQRegularized HypergeometricU Hyperlink HyperlinkCreationSettings Hyperplane Hyphenation HyphenationOptions HypoexponentialDistribution HypothesisTestDataI IconData Iconize IconizedObject IconRules Icosahedron Identity IdentityMatrix If IgnoreCase IgnoreDiacritics IgnorePunctuation IgnoreSpellCheck IgnoringInactive Im Image Image3D Image3DProjection Image3DSlices ImageAccumulate ImageAdd ImageAdjust ImageAlign ImageApply ImageApplyIndexed ImageAspectRatio ImageAssemble ImageAugmentationLayer ImageBoundingBoxes ImageCache ImageCacheValid ImageCapture ImageCaptureFunction ImageCases ImageChannels ImageClip ImageCollage ImageColorSpace ImageCompose ImageContainsQ ImageContents ImageConvolve ImageCooccurrence ImageCorners ImageCorrelate ImageCorrespondingPoints ImageCrop ImageData ImageDeconvolve ImageDemosaic ImageDifference ImageDimensions ImageDisplacements ImageDistance ImageEffect ImageExposureCombine ImageFeatureTrack ImageFileApply ImageFileFilter ImageFileScan ImageFilter ImageFocusCombine ImageForestingComponents ImageFormattingWidth ImageForwardTransformation ImageGraphics ImageHistogram ImageIdentify ImageInstanceQ ImageKeypoints ImageLevels ImageLines ImageMargins ImageMarker ImageMarkers ImageMeasurements ImageMesh ImageMultiply ImageOffset ImagePad ImagePadding ImagePartition ImagePeriodogram ImagePerspectiveTransformation ImagePosition ImagePreviewFunction ImagePyramid ImagePyramidApply ImageQ ImageRangeCache ImageRecolor ImageReflect ImageRegion ImageResize ImageResolution ImageRestyle ImageRotate ImageRotated ImageSaliencyFilter ImageScaled ImageScan ImageSize ImageSizeAction ImageSizeCache ImageSizeMultipliers ImageSizeRaw ImageSubtract ImageTake ImageTransformation ImageTrim ImageType ImageValue ImageValuePositions ImagingDevice ImplicitRegion Implies Import ImportAutoReplacements ImportByteArray ImportOptions ImportString ImprovementImportance In Inactivate Inactive IncidenceGraph IncidenceList IncidenceMatrix IncludeAromaticBonds IncludeConstantBasis IncludeDefinitions IncludeDirectories IncludeFileExtension IncludeGeneratorTasks IncludeHydrogens IncludeInflections IncludeMetaInformation IncludePods IncludeQuantities IncludeRelatedTables IncludeSingularTerm IncludeWindowTimes Increment IndefiniteMatrixQ Indent IndentingNewlineSpacings IndentMaxFraction IndependenceTest IndependentEdgeSetQ IndependentPhysicalQuantity IndependentUnit IndependentUnitDimension IndependentVertexSetQ Indeterminate IndeterminateThreshold IndexCreationOptions Indexed IndexGraph IndexTag Inequality InexactNumberQ InexactNumbers InfiniteLine InfinitePlane Infinity Infix InflationAdjust InflationMethod Information InformationData InformationDataGrid Inherited InheritScope InhomogeneousPoissonProcess InitialEvaluationHistory Initialization InitializationCell InitializationCellEvaluation InitializationCellWarning InitializationObjects InitializationValue Initialize InitialSeeding InlineCounterAssignments InlineCounterIncrements InlineRules Inner InnerPolygon InnerPolyhedron Inpaint Input InputAliases InputAssumptions InputAutoReplacements InputField InputFieldBox InputFieldBoxOptions InputForm InputGrouping InputNamePacket InputNotebook InputPacket InputSettings InputStream InputString InputStringPacket InputToBoxFormPacket Insert InsertionFunction InsertionPointObject InsertLinebreaks InsertResults Inset Inset3DBox Inset3DBoxOptions InsetBox InsetBoxOptions Insphere Install InstallService InstanceNormalizationLayer InString Integer IntegerDigits IntegerExponent IntegerLength IntegerName IntegerPart IntegerPartitions IntegerQ IntegerReverse Integers IntegerString Integral Integrate Interactive InteractiveTradingChart Interlaced Interleaving InternallyBalancedDecomposition InterpolatingFunction InterpolatingPolynomial Interpolation InterpolationOrder InterpolationPoints InterpolationPrecision Interpretation InterpretationBox InterpretationBoxOptions InterpretationFunction Interpreter InterpretTemplate InterquartileRange Interrupt InterruptSettings IntersectingQ Intersection Interval IntervalIntersection IntervalMarkers IntervalMarkersStyle IntervalMemberQ IntervalSlider IntervalUnion Into Inverse InverseBetaRegularized InverseCDF InverseChiSquareDistribution InverseContinuousWaveletTransform InverseDistanceTransform InverseEllipticNomeQ InverseErf InverseErfc InverseFourier InverseFourierCosTransform InverseFourierSequenceTransform InverseFourierSinTransform InverseFourierTransform InverseFunction InverseFunctions InverseGammaDistribution InverseGammaRegularized InverseGaussianDistribution InverseGudermannian InverseHankelTransform InverseHaversine InverseImagePyramid InverseJacobiCD InverseJacobiCN InverseJacobiCS InverseJacobiDC InverseJacobiDN InverseJacobiDS InverseJacobiNC InverseJacobiND InverseJacobiNS InverseJacobiSC InverseJacobiSD InverseJacobiSN InverseLaplaceTransform InverseMellinTransform InversePermutation InverseRadon InverseRadonTransform InverseSeries InverseShortTimeFourier InverseSpectrogram InverseSurvivalFunction InverseTransformedRegion InverseWaveletTransform InverseWeierstrassP InverseWishartMatrixDistribution InverseZTransform Invisible InvisibleApplication InvisibleTimes IPAddress IrreduciblePolynomialQ IslandData IsolatingInterval IsomorphicGraphQ IsotopeData Italic Item ItemAspectRatio ItemBox ItemBoxOptions ItemSize ItemStyle ItoProcessJaccardDissimilarity JacobiAmplitude Jacobian JacobiCD JacobiCN JacobiCS JacobiDC JacobiDN JacobiDS JacobiNC JacobiND JacobiNS JacobiP JacobiSC JacobiSD JacobiSN JacobiSymbol JacobiZeta JankoGroupJ1 JankoGroupJ2 JankoGroupJ3 JankoGroupJ4 JarqueBeraALMTest JohnsonDistribution Join JoinAcross Joined JoinedCurve JoinedCurveBox JoinedCurveBoxOptions JoinForm JordanDecomposition JordanModelDecomposition JulianDate JuliaSetBoettcher JuliaSetIterationCount JuliaSetPlot JuliaSetPointsK KagiChart KaiserBesselWindow KaiserWindow KalmanEstimator KalmanFilter KarhunenLoeveDecomposition KaryTree KatzCentrality KCoreComponents KDistribution KEdgeConnectedComponents KEdgeConnectedGraphQ KelvinBei KelvinBer KelvinKei KelvinKer KendallTau KendallTauTest KernelExecute KernelFunction KernelMixtureDistribution Kernels Ket Key KeyCollisionFunction KeyComplement KeyDrop KeyDropFrom KeyExistsQ KeyFreeQ KeyIntersection KeyMap KeyMemberQ KeypointStrength Keys KeySelect KeySort KeySortBy KeyTake KeyUnion KeyValueMap KeyValuePattern Khinchin KillProcess KirchhoffGraph KirchhoffMatrix KleinInvariantJ KnapsackSolve KnightTourGraph KnotData KnownUnitQ KochCurve KolmogorovSmirnovTest KroneckerDelta KroneckerModelDecomposition KroneckerProduct KroneckerSymbol KuiperTest KumaraswamyDistribution Kurtosis KuwaharaFilter KVertexConnectedComponents KVertexConnectedGraphQLABColor Label Labeled LabeledSlider LabelingFunction LabelingSize LabelStyle LabelVisibility LaguerreL LakeData LambdaComponents LambertW LaminaData LanczosWindow LandauDistribution Language LanguageCategory LanguageData LanguageIdentify LanguageOptions LaplaceDistribution LaplaceTransform Laplacian LaplacianFilter LaplacianGaussianFilter Large Larger Last Latitude LatitudeLongitude LatticeData LatticeReduce Launch LaunchKernels LayeredGraphPlot LayerSizeFunction LayoutInformation LCHColor LCM LeaderSize LeafCount LeapYearQ LearnDistribution LearnedDistribution LearningRate LearningRateMultipliers LeastSquares LeastSquaresFilterKernel Left LeftArrow LeftArrowBar LeftArrowRightArrow LeftDownTeeVector LeftDownVector LeftDownVectorBar LeftRightArrow LeftRightVector LeftTee LeftTeeArrow LeftTeeVector LeftTriangle LeftTriangleBar LeftTriangleEqual LeftUpDownVector LeftUpTeeVector LeftUpVector LeftUpVectorBar LeftVector LeftVectorBar LegendAppearance Legended LegendFunction LegendLabel LegendLayout LegendMargins LegendMarkers LegendMarkerSize LegendreP LegendreQ LegendreType Length LengthWhile LerchPhi Less LessEqual LessEqualGreater LessEqualThan LessFullEqual LessGreater LessLess LessSlantEqual LessThan LessTilde LetterCharacter LetterCounts LetterNumber LetterQ Level LeveneTest LeviCivitaTensor LevyDistribution Lexicographic LibraryDataType LibraryFunction LibraryFunctionError LibraryFunctionInformation LibraryFunctionLoad LibraryFunctionUnload LibraryLoad LibraryUnload LicenseID LiftingFilterData LiftingWaveletTransform LightBlue LightBrown LightCyan Lighter LightGray LightGreen Lighting LightingAngle LightMagenta LightOrange LightPink LightPurple LightRed LightSources LightYellow Likelihood Limit LimitsPositioning LimitsPositioningTokens LindleyDistribution Line Line3DBox Line3DBoxOptions LinearFilter LinearFractionalOptimization LinearFractionalTransform LinearGradientImage LinearizingTransformationData LinearLayer LinearModelFit LinearOffsetFunction LinearOptimization LinearProgramming LinearRecurrence LinearSolve LinearSolveFunction LineBox LineBoxOptions LineBreak LinebreakAdjustments LineBreakChart LinebreakSemicolonWeighting LineBreakWithin LineColor LineGraph LineIndent LineIndentMaxFraction LineIntegralConvolutionPlot LineIntegralConvolutionScale LineLegend LineOpacity LineSpacing LineWrapParts LinkActivate LinkClose LinkConnect LinkConnectedQ LinkCreate LinkError LinkFlush LinkFunction LinkHost LinkInterrupt LinkLaunch LinkMode LinkObject LinkOpen LinkOptions LinkPatterns LinkProtocol LinkRankCentrality LinkRead LinkReadHeld LinkReadyQ Links LinkService LinkWrite LinkWriteHeld LiouvilleLambda List Listable ListAnimate ListContourPlot ListContourPlot3D ListConvolve ListCorrelate ListCurvePathPlot ListDeconvolve ListDensityPlot ListDensityPlot3D Listen ListFormat ListFourierSequenceTransform ListInterpolation ListLineIntegralConvolutionPlot ListLinePlot ListLogLinearPlot ListLogLogPlot ListLogPlot ListPicker ListPickerBox ListPickerBoxBackground ListPickerBoxOptions ListPlay ListPlot ListPlot3D ListPointPlot3D ListPolarPlot ListQ ListSliceContourPlot3D ListSliceDensityPlot3D ListSliceVectorPlot3D ListStepPlot ListStreamDensityPlot ListStreamPlot ListSurfacePlot3D ListVectorDensityPlot ListVectorPlot ListVectorPlot3D ListZTransform Literal LiteralSearch LocalAdaptiveBinarize LocalCache LocalClusteringCoefficient LocalizeDefinitions LocalizeVariables LocalObject LocalObjects LocalResponseNormalizationLayer LocalSubmit LocalSymbol LocalTime LocalTimeZone LocationEquivalenceTest LocationTest Locator LocatorAutoCreate LocatorBox LocatorBoxOptions LocatorCentering LocatorPane LocatorPaneBox LocatorPaneBoxOptions LocatorRegion Locked Log Log10 Log2 LogBarnesG LogGamma LogGammaDistribution LogicalExpand LogIntegral LogisticDistribution LogisticSigmoid LogitModelFit LogLikelihood LogLinearPlot LogLogisticDistribution LogLogPlot LogMultinormalDistribution LogNormalDistribution LogPlot LogRankTest LogSeriesDistribution LongEqual Longest LongestCommonSequence LongestCommonSequencePositions LongestCommonSubsequence LongestCommonSubsequencePositions LongestMatch LongestOrderedSequence LongForm Longitude LongLeftArrow LongLeftRightArrow LongRightArrow LongShortTermMemoryLayer Lookup Loopback LoopFreeGraphQ LossFunction LowerCaseQ LowerLeftArrow LowerRightArrow LowerTriangularize LowerTriangularMatrixQ LowpassFilter LQEstimatorGains LQGRegulator LQOutputRegulatorGains LQRegulatorGains LUBackSubstitution LucasL LuccioSamiComponents LUDecomposition LunarEclipse LUVColor LyapunovSolve LyonsGroupLyMachineID MachineName MachineNumberQ MachinePrecision MacintoshSystemPageSetup Magenta Magnification Magnify MailAddressValidation MailExecute MailFolder MailItem MailReceiverFunction MailResponseFunction MailSearch MailServerConnect MailServerConnection MailSettings MainSolve MaintainDynamicCaches Majority MakeBoxes MakeExpression MakeRules ManagedLibraryExpressionID ManagedLibraryExpressionQ MandelbrotSetBoettcher MandelbrotSetDistance MandelbrotSetIterationCount MandelbrotSetMemberQ MandelbrotSetPlot MangoldtLambda ManhattanDistance Manipulate Manipulator MannedSpaceMissionData MannWhitneyTest MantissaExponent Manual Map MapAll MapAt MapIndexed MAProcess MapThread MarchenkoPasturDistribution MarcumQ MardiaCombinedTest MardiaKurtosisTest MardiaSkewnessTest MarginalDistribution MarkovProcessProperties Masking MatchingDissimilarity MatchLocalNameQ MatchLocalNames MatchQ Material MathematicalFunctionData MathematicaNotation MathieuC MathieuCharacteristicA MathieuCharacteristicB MathieuCharacteristicExponent MathieuCPrime MathieuGroupM11 MathieuGroupM12 MathieuGroupM22 MathieuGroupM23 MathieuGroupM24 MathieuS MathieuSPrime MathMLForm MathMLText Matrices MatrixExp MatrixForm MatrixFunction MatrixLog MatrixNormalDistribution MatrixPlot MatrixPower MatrixPropertyDistribution MatrixQ MatrixRank MatrixTDistribution Max MaxBend MaxCellMeasure MaxColorDistance MaxDetect MaxDuration MaxExtraBandwidths MaxExtraConditions MaxFeatureDisplacement MaxFeatures MaxFilter MaximalBy Maximize MaxItems MaxIterations MaxLimit MaxMemoryUsed MaxMixtureKernels MaxOverlapFraction MaxPlotPoints MaxPoints MaxRecursion MaxStableDistribution MaxStepFraction MaxSteps MaxStepSize MaxTrainingRounds MaxValue MaxwellDistribution MaxWordGap McLaughlinGroupMcL Mean MeanAbsoluteLossLayer MeanAround MeanClusteringCoefficient MeanDegreeConnectivity MeanDeviation MeanFilter MeanGraphDistance MeanNeighborDegree MeanShift MeanShiftFilter MeanSquaredLossLayer Median MedianDeviation MedianFilter MedicalTestData Medium MeijerG MeijerGReduce MeixnerDistribution MellinConvolve MellinTransform MemberQ MemoryAvailable MemoryConstrained MemoryConstraint MemoryInUse MengerMesh Menu MenuAppearance MenuCommandKey MenuEvaluator MenuItem MenuList MenuPacket MenuSortingValue MenuStyle MenuView Merge MergeDifferences MergingFunction MersennePrimeExponent MersennePrimeExponentQ Mesh MeshCellCentroid MeshCellCount MeshCellHighlight MeshCellIndex MeshCellLabel MeshCellMarker MeshCellMeasure MeshCellQuality MeshCells MeshCellShapeFunction MeshCellStyle MeshCoordinates MeshFunctions MeshPrimitives MeshQualityGoal MeshRange MeshRefinementFunction MeshRegion MeshRegionQ MeshShading MeshStyle Message MessageDialog MessageList MessageName MessageObject MessageOptions MessagePacket Messages MessagesNotebook MetaCharacters MetaInformation MeteorShowerData Method MethodOptions MexicanHatWavelet MeyerWavelet Midpoint Min MinColorDistance MinDetect MineralData MinFilter MinimalBy MinimalPolynomial MinimalStateSpaceModel Minimize MinimumTimeIncrement MinIntervalSize MinkowskiQuestionMark MinLimit MinMax MinorPlanetData Minors MinRecursion MinSize MinStableDistribution Minus MinusPlus MinValue Missing MissingBehavior MissingDataMethod MissingDataRules MissingQ MissingString MissingStyle MissingValuePattern MittagLefflerE MixedFractionParts MixedGraphQ MixedMagnitude MixedRadix MixedRadixQuantity MixedUnit MixtureDistribution Mod Modal Mode Modular ModularInverse ModularLambda Module Modulus MoebiusMu Molecule MoleculeContainsQ MoleculeEquivalentQ MoleculeGraph MoleculeModify MoleculePattern MoleculePlot MoleculePlot3D MoleculeProperty MoleculeQ MoleculeValue Moment Momentary MomentConvert MomentEvaluate MomentGeneratingFunction MomentOfInertia Monday Monitor MonomialList MonomialOrder MonsterGroupM MoonPhase MoonPosition MorletWavelet MorphologicalBinarize MorphologicalBranchPoints MorphologicalComponents MorphologicalEulerNumber MorphologicalGraph MorphologicalPerimeter MorphologicalTransform MortalityData Most MountainData MouseAnnotation MouseAppearance MouseAppearanceTag MouseButtons Mouseover MousePointerNote MousePosition MovieData MovingAverage MovingMap MovingMedian MoyalDistribution Multicolumn MultiedgeStyle MultigraphQ MultilaunchWarning MultiLetterItalics MultiLetterStyle MultilineFunction Multinomial MultinomialDistribution MultinormalDistribution MultiplicativeOrder Multiplicity MultiplySides Multiselection MultivariateHypergeometricDistribution MultivariatePoissonDistribution MultivariateTDistributionN NakagamiDistribution NameQ Names NamespaceBox NamespaceBoxOptions Nand NArgMax NArgMin NBernoulliB NBodySimulation NBodySimulationData NCache NDEigensystem NDEigenvalues NDSolve NDSolveValue Nearest NearestFunction NearestNeighborGraph NearestTo NebulaData NeedCurrentFrontEndPackagePacket NeedCurrentFrontEndSymbolsPacket NeedlemanWunschSimilarity Needs Negative NegativeBinomialDistribution NegativeDefiniteMatrixQ NegativeIntegers NegativeMultinomialDistribution NegativeRationals NegativeReals NegativeSemidefiniteMatrixQ NeighborhoodData NeighborhoodGraph Nest NestedGreaterGreater NestedLessLess NestedScriptRules NestGraph NestList NestWhile NestWhileList NetAppend NetBidirectionalOperator NetChain NetDecoder NetDelete NetDrop NetEncoder NetEvaluationMode NetExtract NetFlatten NetFoldOperator NetGraph NetInformation NetInitialize NetInsert NetInsertSharedArrays NetJoin NetMapOperator NetMapThreadOperator NetMeasurements NetModel NetNestOperator NetPairEmbeddingOperator NetPort NetPortGradient NetPrepend NetRename NetReplace NetReplacePart NetSharedArray NetStateObject NetTake NetTrain NetTrainResultsObject NetworkPacketCapture NetworkPacketRecording NetworkPacketRecordingDuring NetworkPacketTrace NeumannValue NevilleThetaC NevilleThetaD NevilleThetaN NevilleThetaS NewPrimitiveStyle NExpectation Next NextCell NextDate NextPrime NextScheduledTaskTime NHoldAll NHoldFirst NHoldRest NicholsGridLines NicholsPlot NightHemisphere NIntegrate NMaximize NMaxValue NMinimize NMinValue NominalVariables NonAssociative NoncentralBetaDistribution NoncentralChiSquareDistribution NoncentralFRatioDistribution NoncentralStudentTDistribution NonCommutativeMultiply NonConstants NondimensionalizationTransform None NoneTrue NonlinearModelFit NonlinearStateSpaceModel NonlocalMeansFilter NonNegative NonNegativeIntegers NonNegativeRationals NonNegativeReals NonPositive NonPositiveIntegers NonPositiveRationals NonPositiveReals Nor NorlundB Norm Normal NormalDistribution NormalGrouping NormalizationLayer Normalize Normalized NormalizedSquaredEuclideanDistance NormalMatrixQ NormalsFunction NormFunction Not NotCongruent NotCupCap NotDoubleVerticalBar Notebook NotebookApply NotebookAutoSave NotebookClose NotebookConvertSettings NotebookCreate NotebookCreateReturnObject NotebookDefault NotebookDelete NotebookDirectory NotebookDynamicExpression NotebookEvaluate NotebookEventActions NotebookFileName NotebookFind NotebookFindReturnObject NotebookGet NotebookGetLayoutInformationPacket NotebookGetMisspellingsPacket NotebookImport NotebookInformation NotebookInterfaceObject NotebookLocate NotebookObject NotebookOpen NotebookOpenReturnObject NotebookPath NotebookPrint NotebookPut NotebookPutReturnObject NotebookRead NotebookResetGeneratedCells Notebooks NotebookSave NotebookSaveAs NotebookSelection NotebookSetupLayoutInformationPacket NotebooksMenu NotebookTemplate NotebookWrite NotElement NotEqualTilde NotExists NotGreater NotGreaterEqual NotGreaterFullEqual NotGreaterGreater NotGreaterLess NotGreaterSlantEqual NotGreaterTilde Nothing NotHumpDownHump NotHumpEqual NotificationFunction NotLeftTriangle NotLeftTriangleBar NotLeftTriangleEqual NotLess NotLessEqual NotLessFullEqual NotLessGreater NotLessLess NotLessSlantEqual NotLessTilde NotNestedGreaterGreater NotNestedLessLess NotPrecedes NotPrecedesEqual NotPrecedesSlantEqual NotPrecedesTilde NotReverseElement NotRightTriangle NotRightTriangleBar NotRightTriangleEqual NotSquareSubset NotSquareSubsetEqual NotSquareSuperset NotSquareSupersetEqual NotSubset NotSubsetEqual NotSucceeds NotSucceedsEqual NotSucceedsSlantEqual NotSucceedsTilde NotSuperset NotSupersetEqual NotTilde NotTildeEqual NotTildeFullEqual NotTildeTilde NotVerticalBar Now NoWhitespace NProbability NProduct NProductFactors NRoots NSolve NSum NSumTerms NuclearExplosionData NuclearReactorData Null NullRecords NullSpace NullWords Number NumberCompose NumberDecompose NumberExpand NumberFieldClassNumber NumberFieldDiscriminant NumberFieldFundamentalUnits NumberFieldIntegralBasis NumberFieldNormRepresentatives NumberFieldRegulator NumberFieldRootsOfUnity NumberFieldSignature NumberForm NumberFormat NumberLinePlot NumberMarks NumberMultiplier NumberPadding NumberPoint NumberQ NumberSeparator NumberSigns NumberString Numerator NumeratorDenominator NumericalOrder NumericalSort NumericArray NumericArrayQ NumericArrayType NumericFunction NumericQ NuttallWindow NValues NyquistGridLines NyquistPlotO ObservabilityGramian ObservabilityMatrix ObservableDecomposition ObservableModelQ OceanData Octahedron OddQ Off Offset OLEData On ONanGroupON Once OneIdentity Opacity OpacityFunction OpacityFunctionScaling Open OpenAppend Opener OpenerBox OpenerBoxOptions OpenerView OpenFunctionInspectorPacket Opening OpenRead OpenSpecialOptions OpenTemporary OpenWrite Operate OperatingSystem OptimumFlowData Optional OptionalElement OptionInspectorSettings OptionQ Options OptionsPacket OptionsPattern OptionValue OptionValueBox OptionValueBoxOptions Or Orange Order OrderDistribution OrderedQ Ordering OrderingBy OrderingLayer Orderless OrderlessPatternSequence OrnsteinUhlenbeckProcess Orthogonalize OrthogonalMatrixQ Out Outer OuterPolygon OuterPolyhedron OutputAutoOverwrite OutputControllabilityMatrix OutputControllableModelQ OutputForm OutputFormData OutputGrouping OutputMathEditExpression OutputNamePacket OutputResponse OutputSizeLimit OutputStream Over OverBar OverDot Overflow OverHat Overlaps Overlay OverlayBox OverlayBoxOptions Overscript OverscriptBox OverscriptBoxOptions OverTilde OverVector OverwriteTarget OwenT OwnValuesPackage PackingMethod PaddedForm Padding PaddingLayer PaddingSize PadeApproximant PadLeft PadRight PageBreakAbove PageBreakBelow PageBreakWithin PageFooterLines PageFooters PageHeaderLines PageHeaders PageHeight PageRankCentrality PageTheme PageWidth Pagination PairedBarChart PairedHistogram PairedSmoothHistogram PairedTTest PairedZTest PaletteNotebook PalettePath PalindromeQ Pane PaneBox PaneBoxOptions Panel PanelBox PanelBoxOptions Paneled PaneSelector PaneSelectorBox PaneSelectorBoxOptions PaperWidth ParabolicCylinderD ParagraphIndent ParagraphSpacing ParallelArray ParallelCombine ParallelDo Parallelepiped ParallelEvaluate Parallelization Parallelize ParallelMap ParallelNeeds Parallelogram ParallelProduct ParallelSubmit ParallelSum ParallelTable ParallelTry Parameter ParameterEstimator ParameterMixtureDistribution ParameterVariables ParametricFunction ParametricNDSolve ParametricNDSolveValue ParametricPlot ParametricPlot3D ParametricRegion ParentBox ParentCell ParentConnect ParentDirectory ParentForm Parenthesize ParentList ParentNotebook ParetoDistribution ParetoPickandsDistribution ParkData Part PartBehavior PartialCorrelationFunction PartialD ParticleAcceleratorData ParticleData Partition PartitionGranularity PartitionsP PartitionsQ PartLayer PartOfSpeech PartProtection ParzenWindow PascalDistribution PassEventsDown PassEventsUp Paste PasteAutoQuoteCharacters PasteBoxFormInlineCells PasteButton Path PathGraph PathGraphQ Pattern PatternSequence PatternTest PauliMatrix PaulWavelet Pause PausedTime PDF PeakDetect PeanoCurve PearsonChiSquareTest PearsonCorrelationTest PearsonDistribution PercentForm PerfectNumber PerfectNumberQ PerformanceGoal Perimeter PeriodicBoundaryCondition PeriodicInterpolation Periodogram PeriodogramArray Permanent Permissions PermissionsGroup PermissionsGroupMemberQ PermissionsGroups PermissionsKey PermissionsKeys PermutationCycles PermutationCyclesQ PermutationGroup PermutationLength PermutationList PermutationListQ PermutationMax PermutationMin PermutationOrder PermutationPower PermutationProduct PermutationReplace Permutations PermutationSupport Permute PeronaMalikFilter Perpendicular PerpendicularBisector PersistenceLocation PersistenceTime PersistentObject PersistentObjects PersistentValue PersonData PERTDistribution PetersenGraph PhaseMargins PhaseRange PhysicalSystemData Pi Pick PIDData PIDDerivativeFilter PIDFeedforward PIDTune Piecewise PiecewiseExpand PieChart PieChart3D PillaiTrace PillaiTraceTest PingTime Pink PitchRecognize Pivoting PixelConstrained PixelValue PixelValuePositions Placed Placeholder PlaceholderReplace Plain PlanarAngle PlanarGraph PlanarGraphQ PlanckRadiationLaw PlaneCurveData PlanetaryMoonData PlanetData PlantData Play PlayRange Plot Plot3D Plot3Matrix PlotDivision PlotJoined PlotLabel PlotLabels PlotLayout PlotLegends PlotMarkers PlotPoints PlotRange PlotRangeClipping PlotRangeClipPlanesStyle PlotRangePadding PlotRegion PlotStyle PlotTheme Pluralize Plus PlusMinus Pochhammer PodStates PodWidth Point Point3DBox Point3DBoxOptions PointBox PointBoxOptions PointFigureChart PointLegend PointSize PoissonConsulDistribution PoissonDistribution PoissonProcess PoissonWindow PolarAxes PolarAxesOrigin PolarGridLines PolarPlot PolarTicks PoleZeroMarkers PolyaAeppliDistribution PolyGamma Polygon Polygon3DBox Polygon3DBoxOptions PolygonalNumber PolygonAngle PolygonBox PolygonBoxOptions PolygonCoordinates PolygonDecomposition PolygonHoleScale PolygonIntersections PolygonScale Polyhedron PolyhedronAngle PolyhedronCoordinates PolyhedronData PolyhedronDecomposition PolyhedronGenus PolyLog PolynomialExtendedGCD PolynomialForm PolynomialGCD PolynomialLCM PolynomialMod PolynomialQ PolynomialQuotient PolynomialQuotientRemainder PolynomialReduce PolynomialRemainder Polynomials PoolingLayer PopupMenu PopupMenuBox PopupMenuBoxOptions PopupView PopupWindow Position PositionIndex Positive PositiveDefiniteMatrixQ PositiveIntegers PositiveRationals PositiveReals PositiveSemidefiniteMatrixQ PossibleZeroQ Postfix PostScript Power PowerDistribution PowerExpand PowerMod PowerModList PowerRange PowerSpectralDensity PowersRepresentations PowerSymmetricPolynomial Precedence PrecedenceForm Precedes PrecedesEqual PrecedesSlantEqual PrecedesTilde Precision PrecisionGoal PreDecrement Predict PredictionRoot PredictorFunction PredictorInformation PredictorMeasurements PredictorMeasurementsObject PreemptProtect PreferencesPath Prefix PreIncrement Prepend PrependLayer PrependTo PreprocessingRules PreserveColor PreserveImageOptions Previous PreviousCell PreviousDate PriceGraphDistribution PrimaryPlaceholder Prime PrimeNu PrimeOmega PrimePi PrimePowerQ PrimeQ Primes PrimeZetaP PrimitivePolynomialQ PrimitiveRoot PrimitiveRootList PrincipalComponents PrincipalValue Print PrintableASCIIQ PrintAction PrintForm PrintingCopies PrintingOptions PrintingPageRange PrintingStartingPageNumber PrintingStyleEnvironment Printout3D Printout3DPreviewer PrintPrecision PrintTemporary Prism PrismBox PrismBoxOptions PrivateCellOptions PrivateEvaluationOptions PrivateFontOptions PrivateFrontEndOptions PrivateKey PrivateNotebookOptions PrivatePaths Probability ProbabilityDistribution ProbabilityPlot ProbabilityPr ProbabilityScalePlot ProbitModelFit ProcessConnection ProcessDirectory ProcessEnvironment Processes ProcessEstimator ProcessInformation ProcessObject ProcessParameterAssumptions ProcessParameterQ ProcessStateDomain ProcessStatus ProcessTimeDomain Product ProductDistribution ProductLog ProgressIndicator ProgressIndicatorBox ProgressIndicatorBoxOptions Projection Prolog PromptForm ProofObject Properties Property PropertyList PropertyValue Proportion Proportional Protect Protected ProteinData Pruning PseudoInverse PsychrometricPropertyData PublicKey PublisherID PulsarData PunctuationCharacter Purple Put PutAppend Pyramid PyramidBox PyramidBoxOptionsQBinomial QFactorial QGamma QHypergeometricPFQ QnDispersion QPochhammer QPolyGamma QRDecomposition QuadraticIrrationalQ QuadraticOptimization Quantile QuantilePlot Quantity QuantityArray QuantityDistribution QuantityForm QuantityMagnitude QuantityQ QuantityUnit QuantityVariable QuantityVariableCanonicalUnit QuantityVariableDimensions QuantityVariableIdentifier QuantityVariablePhysicalQuantity Quartics QuartileDeviation Quartiles QuartileSkewness Query QueueingNetworkProcess QueueingProcess QueueProperties Quiet Quit Quotient QuotientRemainderRadialGradientImage RadialityCentrality RadicalBox RadicalBoxOptions RadioButton RadioButtonBar RadioButtonBox RadioButtonBoxOptions Radon RadonTransform RamanujanTau RamanujanTauL RamanujanTauTheta RamanujanTauZ Ramp Random RandomChoice RandomColor RandomComplex RandomEntity RandomFunction RandomGeoPosition RandomGraph RandomImage RandomInstance RandomInteger RandomPermutation RandomPoint RandomPolygon RandomPolyhedron RandomPrime RandomReal RandomSample RandomSeed RandomSeeding RandomVariate RandomWalkProcess RandomWord Range RangeFilter RangeSpecification RankedMax RankedMin RarerProbability Raster Raster3D Raster3DBox Raster3DBoxOptions RasterArray RasterBox RasterBoxOptions Rasterize RasterSize Rational RationalFunctions Rationalize Rationals Ratios RawArray RawBoxes RawData RawMedium RayleighDistribution Re Read ReadByteArray ReadLine ReadList ReadProtected ReadString Real RealAbs RealBlockDiagonalForm RealDigits RealExponent Reals RealSign Reap RecognitionPrior RecognitionThreshold Record RecordLists RecordSeparators Rectangle RectangleBox RectangleBoxOptions RectangleChart RectangleChart3D RectangularRepeatingElement RecurrenceFilter RecurrenceTable RecurringDigitsForm Red Reduce RefBox ReferenceLineStyle ReferenceMarkers ReferenceMarkerStyle Refine ReflectionMatrix ReflectionTransform Refresh RefreshRate Region RegionBinarize RegionBoundary RegionBounds RegionCentroid RegionDifference RegionDimension RegionDisjoint RegionDistance RegionDistanceFunction RegionEmbeddingDimension RegionEqual RegionFunction RegionImage RegionIntersection RegionMeasure RegionMember RegionMemberFunction RegionMoment RegionNearest RegionNearestFunction RegionPlot RegionPlot3D RegionProduct RegionQ RegionResize RegionSize RegionSymmetricDifference RegionUnion RegionWithin RegisterExternalEvaluator RegularExpression Regularization RegularlySampledQ RegularPolygon ReIm ReImLabels ReImPlot ReImStyle Reinstall RelationalDatabase RelationGraph Release ReleaseHold ReliabilityDistribution ReliefImage ReliefPlot RemoteAuthorizationCaching RemoteConnect RemoteConnectionObject RemoteFile RemoteRun RemoteRunProcess Remove RemoveAlphaChannel RemoveAsynchronousTask RemoveAudioStream RemoveBackground RemoveChannelListener RemoveChannelSubscribers Removed RemoveDiacritics RemoveInputStreamMethod RemoveOutputStreamMethod RemoveProperty RemoveScheduledTask RemoveUsers RenameDirectory RenameFile RenderAll RenderingOptions RenewalProcess RenkoChart RepairMesh Repeated RepeatedNull RepeatedString RepeatedTiming RepeatingElement Replace ReplaceAll ReplaceHeldPart ReplaceImageValue ReplaceList ReplacePart ReplacePixelValue ReplaceRepeated ReplicateLayer RequiredPhysicalQuantities Resampling ResamplingAlgorithmData ResamplingMethod Rescale RescalingTransform ResetDirectory ResetMenusPacket ResetScheduledTask ReshapeLayer Residue ResizeLayer Resolve ResourceAcquire ResourceData ResourceFunction ResourceObject ResourceRegister ResourceRemove ResourceSearch ResourceSubmissionObject ResourceSubmit ResourceSystemBase ResourceUpdate ResponseForm Rest RestartInterval Restricted Resultant ResumePacket Return ReturnEntersInput ReturnExpressionPacket ReturnInputFormPacket ReturnPacket ReturnReceiptFunction ReturnTextPacket Reverse ReverseBiorthogonalSplineWavelet ReverseElement ReverseEquilibrium ReverseGraph ReverseSort ReverseSortBy ReverseUpEquilibrium RevolutionAxis RevolutionPlot3D RGBColor RiccatiSolve RiceDistribution RidgeFilter RiemannR RiemannSiegelTheta RiemannSiegelZ RiemannXi Riffle Right RightArrow RightArrowBar RightArrowLeftArrow RightComposition RightCosetRepresentative RightDownTeeVector RightDownVector RightDownVectorBar RightTee RightTeeArrow RightTeeVector RightTriangle RightTriangleBar RightTriangleEqual RightUpDownVector RightUpTeeVector RightUpVector RightUpVectorBar RightVector RightVectorBar RiskAchievementImportance RiskReductionImportance RogersTanimotoDissimilarity RollPitchYawAngles RollPitchYawMatrix RomanNumeral Root RootApproximant RootIntervals RootLocusPlot RootMeanSquare RootOfUnityQ RootReduce Roots RootSum Rotate RotateLabel RotateLeft RotateRight RotationAction RotationBox RotationBoxOptions RotationMatrix RotationTransform Round RoundImplies RoundingRadius Row RowAlignments RowBackgrounds RowBox RowHeights RowLines RowMinHeight RowReduce RowsEqual RowSpacings RSolve RSolveValue RudinShapiro RudvalisGroupRu Rule RuleCondition RuleDelayed RuleForm RulePlot RulerUnits Run RunProcess RunScheduledTask RunThrough RuntimeAttributes RuntimeOptions RussellRaoDissimilaritySameQ SameTest SampledEntityClass SampleDepth SampledSoundFunction SampledSoundList SampleRate SamplingPeriod SARIMAProcess SARMAProcess SASTriangle SatelliteData SatisfiabilityCount SatisfiabilityInstances SatisfiableQ Saturday Save Saveable SaveAutoDelete SaveConnection SaveDefinitions SavitzkyGolayMatrix SawtoothWave Scale Scaled ScaleDivisions ScaledMousePosition ScaleOrigin ScalePadding ScaleRanges ScaleRangeStyle ScalingFunctions ScalingMatrix ScalingTransform Scan ScheduledTask ScheduledTaskActiveQ ScheduledTaskInformation ScheduledTaskInformationData ScheduledTaskObject ScheduledTasks SchurDecomposition ScientificForm ScientificNotationThreshold ScorerGi ScorerGiPrime ScorerHi ScorerHiPrime ScreenRectangle ScreenStyleEnvironment ScriptBaselineShifts ScriptForm ScriptLevel ScriptMinSize ScriptRules ScriptSizeMultipliers Scrollbars ScrollingOptions ScrollPosition SearchAdjustment SearchIndexObject SearchIndices SearchQueryString SearchResultObject Sec Sech SechDistribution SecondOrderConeOptimization SectionGrouping SectorChart SectorChart3D SectorOrigin SectorSpacing SecuredAuthenticationKey SecuredAuthenticationKeys SeedRandom Select Selectable SelectComponents SelectedCells SelectedNotebook SelectFirst Selection SelectionAnimate SelectionCell SelectionCellCreateCell SelectionCellDefaultStyle SelectionCellParentStyle SelectionCreateCell SelectionDebuggerTag SelectionDuplicateCell SelectionEvaluate SelectionEvaluateCreateCell SelectionMove SelectionPlaceholder SelectionSetStyle SelectWithContents SelfLoops SelfLoopStyle SemanticImport SemanticImportString SemanticInterpretation SemialgebraicComponentInstances SemidefiniteOptimization SendMail SendMessage Sequence SequenceAlignment SequenceAttentionLayer SequenceCases SequenceCount SequenceFold SequenceFoldList SequenceForm SequenceHold SequenceLastLayer SequenceMostLayer SequencePosition SequencePredict SequencePredictorFunction SequenceReplace SequenceRestLayer SequenceReverseLayer SequenceSplit Series SeriesCoefficient SeriesData ServiceConnect ServiceDisconnect ServiceExecute ServiceObject ServiceRequest ServiceResponse ServiceSubmit SessionSubmit SessionTime Set SetAccuracy SetAlphaChannel SetAttributes Setbacks SetBoxFormNamesPacket SetCloudDirectory SetCookies SetDelayed SetDirectory SetEnvironment SetEvaluationNotebook SetFileDate SetFileLoadingContext SetNotebookStatusLine SetOptions SetOptionsPacket SetPermissions SetPrecision SetProperty SetSecuredAuthenticationKey SetSelectedNotebook SetSharedFunction SetSharedVariable SetSpeechParametersPacket SetStreamPosition SetSystemModel SetSystemOptions Setter SetterBar SetterBox SetterBoxOptions Setting SetUsers SetValue Shading Shallow ShannonWavelet ShapiroWilkTest Share SharingList Sharpen ShearingMatrix ShearingTransform ShellRegion ShenCastanMatrix ShiftedGompertzDistribution ShiftRegisterSequence Short ShortDownArrow Shortest ShortestMatch ShortestPathFunction ShortLeftArrow ShortRightArrow ShortTimeFourier ShortTimeFourierData ShortUpArrow Show ShowAutoConvert ShowAutoSpellCheck ShowAutoStyles ShowCellBracket ShowCellLabel ShowCellTags ShowClosedCellArea ShowCodeAssist ShowContents ShowControls ShowCursorTracker ShowGroupOpenCloseIcon ShowGroupOpener ShowInvisibleCharacters ShowPageBreaks ShowPredictiveInterface ShowSelection ShowShortBoxForm ShowSpecialCharacters ShowStringCharacters ShowSyntaxStyles ShrinkingDelay ShrinkWrapBoundingBox SiderealTime SiegelTheta SiegelTukeyTest SierpinskiCurve SierpinskiMesh Sign Signature SignedRankTest SignedRegionDistance SignificanceLevel SignPadding SignTest SimilarityRules SimpleGraph SimpleGraphQ SimplePolygonQ SimplePolyhedronQ Simplex Simplify Sin Sinc SinghMaddalaDistribution SingleEvaluation SingleLetterItalics SingleLetterStyle SingularValueDecomposition SingularValueList SingularValuePlot SingularValues Sinh SinhIntegral SinIntegral SixJSymbol Skeleton SkeletonTransform SkellamDistribution Skewness SkewNormalDistribution SkinStyle Skip SliceContourPlot3D SliceDensityPlot3D SliceDistribution SliceVectorPlot3D Slider Slider2D Slider2DBox Slider2DBoxOptions SliderBox SliderBoxOptions SlideView Slot SlotSequence Small SmallCircle Smaller SmithDecomposition SmithDelayCompensator SmithWatermanSimilarity SmoothDensityHistogram SmoothHistogram SmoothHistogram3D SmoothKernelDistribution SnDispersion Snippet SnubPolyhedron SocialMediaData Socket SocketConnect SocketListen SocketListener SocketObject SocketOpen SocketReadMessage SocketReadyQ Sockets SocketWaitAll SocketWaitNext SoftmaxLayer SokalSneathDissimilarity SolarEclipse SolarSystemFeatureData SolidAngle SolidData SolidRegionQ Solve SolveAlways SolveDelayed Sort SortBy SortedBy SortedEntityClass Sound SoundAndGraphics SoundNote SoundVolume SourceLink Sow Space SpaceCurveData SpaceForm Spacer Spacings Span SpanAdjustments SpanCharacterRounding SpanFromAbove SpanFromBoth SpanFromLeft SpanLineThickness SpanMaxSize SpanMinSize SpanningCharacters SpanSymmetric SparseArray SpatialGraphDistribution SpatialMedian SpatialTransformationLayer Speak SpeakTextPacket SpearmanRankTest SpearmanRho SpeciesData SpecificityGoal SpectralLineData Spectrogram SpectrogramArray Specularity SpeechRecognize SpeechSynthesize SpellingCorrection SpellingCorrectionList SpellingDictionaries SpellingDictionariesPath SpellingOptions SpellingSuggestionsPacket Sphere SphereBox SpherePoints SphericalBesselJ SphericalBesselY SphericalHankelH1 SphericalHankelH2 SphericalHarmonicY SphericalPlot3D SphericalRegion SphericalShell SpheroidalEigenvalue SpheroidalJoiningFactor SpheroidalPS SpheroidalPSPrime SpheroidalQS SpheroidalQSPrime SpheroidalRadialFactor SpheroidalS1 SpheroidalS1Prime SpheroidalS2 SpheroidalS2Prime Splice SplicedDistribution SplineClosed SplineDegree SplineKnots SplineWeights Split SplitBy SpokenString Sqrt SqrtBox SqrtBoxOptions Square SquaredEuclideanDistance SquareFreeQ SquareIntersection SquareMatrixQ SquareRepeatingElement SquaresR SquareSubset SquareSubsetEqual SquareSuperset SquareSupersetEqual SquareUnion SquareWave SSSTriangle StabilityMargins StabilityMarginsStyle StableDistribution Stack StackBegin StackComplete StackedDateListPlot StackedListPlot StackInhibit StadiumShape StandardAtmosphereData StandardDeviation StandardDeviationFilter StandardForm Standardize Standardized StandardOceanData StandbyDistribution Star StarClusterData StarData StarGraph StartAsynchronousTask StartExternalSession StartingStepSize StartOfLine StartOfString StartProcess StartScheduledTask StartupSound StartWebSession StateDimensions StateFeedbackGains StateOutputEstimator StateResponse StateSpaceModel StateSpaceRealization StateSpaceTransform StateTransformationLinearize StationaryDistribution StationaryWaveletPacketTransform StationaryWaveletTransform StatusArea StatusCentrality StepMonitor StereochemistryElements StieltjesGamma StirlingS1 StirlingS2 StopAsynchronousTask StoppingPowerData StopScheduledTask StrataVariables StratonovichProcess StreamColorFunction StreamColorFunctionScaling StreamDensityPlot StreamMarkers StreamPlot StreamPoints StreamPosition Streams StreamScale StreamStyle String StringBreak StringByteCount StringCases StringContainsQ StringCount StringDelete StringDrop StringEndsQ StringExpression StringExtract StringForm StringFormat StringFreeQ StringInsert StringJoin StringLength StringMatchQ StringPadLeft StringPadRight StringPart StringPartition StringPosition StringQ StringRepeat StringReplace StringReplaceList StringReplacePart StringReverse StringRiffle StringRotateLeft StringRotateRight StringSkeleton StringSplit StringStartsQ StringTake StringTemplate StringToByteArray StringToStream StringTrim StripBoxes StripOnInput StripWrapperBoxes StrokeForm StructuralImportance StructuredArray StructuredSelection StruveH StruveL Stub StudentTDistribution Style StyleBox StyleBoxAutoDelete StyleData StyleDefinitions StyleForm StyleHints StyleKeyMapping StyleMenuListing StyleNameDialogSettings StyleNames StylePrint StyleSheetPath Subdivide Subfactorial Subgraph SubMinus SubPlus SubresultantPolynomialRemainders SubresultantPolynomials Subresultants Subscript SubscriptBox SubscriptBoxOptions Subscripted Subsequences Subset SubsetEqual SubsetMap SubsetQ Subsets SubStar SubstitutionSystem Subsuperscript SubsuperscriptBox SubsuperscriptBoxOptions Subtract SubtractFrom SubtractSides SubValues Succeeds SucceedsEqual SucceedsSlantEqual SucceedsTilde Success SuchThat Sum SumConvergence SummationLayer Sunday SunPosition Sunrise Sunset SuperDagger SuperMinus SupernovaData SuperPlus Superscript SuperscriptBox SuperscriptBoxOptions Superset SupersetEqual SuperStar Surd SurdForm SurfaceArea SurfaceColor SurfaceData SurfaceGraphics SurvivalDistribution SurvivalFunction SurvivalModel SurvivalModelFit SuspendPacket SuzukiDistribution SuzukiGroupSuz SwatchLegend Switch Symbol SymbolName SymletWavelet Symmetric 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TouchPosition TouchscreenAutoZoom TouchscreenControlPlacement ToUpperCase Tr Trace TraceAbove TraceAction TraceBackward TraceDepth TraceDialog TraceForward TraceInternal TraceLevel TraceOff TraceOn TraceOriginal TracePrint TraceScan TrackedSymbols TrackingFunction TracyWidomDistribution TradingChart TraditionalForm TraditionalFunctionNotation TraditionalNotation TraditionalOrder TrainingProgressCheckpointing TrainingProgressFunction TrainingProgressMeasurements TrainingProgressReporting TrainingStoppingCriterion TransferFunctionCancel TransferFunctionExpand TransferFunctionFactor TransferFunctionModel TransferFunctionPoles TransferFunctionTransform TransferFunctionZeros TransformationClass TransformationFunction TransformationFunctions TransformationMatrix TransformedDistribution TransformedField TransformedProcess TransformedRegion TransitionDirection TransitionDuration TransitionEffect TransitiveClosureGraph TransitiveReductionGraph Translate TranslationOptions TranslationTransform 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ds_grid_clear ds_grid_set ds_grid_add ds_grid_multiply ds_grid_set_region ds_grid_add_region ds_grid_multiply_region ds_grid_set_disk ds_grid_add_disk ds_grid_multiply_disk ds_grid_set_grid_region ds_grid_add_grid_region ds_grid_multiply_grid_region ds_grid_get ds_grid_get_sum ds_grid_get_max ds_grid_get_min ds_grid_get_mean ds_grid_get_disk_sum ds_grid_get_disk_min ds_grid_get_disk_max ds_grid_get_disk_mean ds_grid_value_exists ds_grid_value_x ds_grid_value_y ds_grid_value_disk_exists ds_grid_value_disk_x ds_grid_value_disk_y ds_grid_shuffle ds_grid_write ds_grid_read ds_grid_sort ds_grid_set ds_grid_get effect_create_below effect_create_above effect_clear part_type_create part_type_destroy part_type_exists part_type_clear part_type_shape part_type_sprite part_type_size part_type_scale part_type_orientation part_type_life part_type_step part_type_death part_type_speed part_type_direction part_type_gravity part_type_colour1 part_type_colour2 part_type_colour3 part_type_colour_mix part_type_colour_rgb part_type_colour_hsv part_type_color1 part_type_color2 part_type_color3 part_type_color_mix part_type_color_rgb part_type_color_hsv part_type_alpha1 part_type_alpha2 part_type_alpha3 part_type_blend part_system_create part_system_create_layer part_system_destroy part_system_exists part_system_clear part_system_draw_order part_system_depth part_system_position part_system_automatic_update part_system_automatic_draw part_system_update part_system_drawit part_system_get_layer part_system_layer part_particles_create part_particles_create_colour part_particles_create_color part_particles_clear part_particles_count part_emitter_create part_emitter_destroy part_emitter_destroy_all part_emitter_exists part_emitter_clear part_emitter_region part_emitter_burst part_emitter_stream external_call external_define external_free window_handle window_device matrix_get matrix_set matrix_build_identity matrix_build matrix_build_lookat matrix_build_projection_ortho matrix_build_projection_perspective matrix_build_projection_perspective_fov matrix_multiply matrix_transform_vertex matrix_stack_push matrix_stack_pop matrix_stack_multiply matrix_stack_set matrix_stack_clear matrix_stack_top matrix_stack_is_empty browser_input_capture os_get_config os_get_info os_get_language os_get_region os_lock_orientation display_get_dpi_x display_get_dpi_y display_set_gui_size display_set_gui_maximise display_set_gui_maximize device_mouse_dbclick_enable display_set_timing_method display_get_timing_method display_set_sleep_margin display_get_sleep_margin virtual_key_add virtual_key_hide virtual_key_delete virtual_key_show draw_enable_drawevent draw_enable_swf_aa draw_set_swf_aa_level draw_get_swf_aa_level draw_texture_flush draw_flush gpu_set_blendenable gpu_set_ztestenable gpu_set_zfunc gpu_set_zwriteenable gpu_set_lightingenable gpu_set_fog gpu_set_cullmode gpu_set_blendmode gpu_set_blendmode_ext gpu_set_blendmode_ext_sepalpha gpu_set_colorwriteenable gpu_set_colourwriteenable gpu_set_alphatestenable gpu_set_alphatestref gpu_set_alphatestfunc gpu_set_texfilter gpu_set_texfilter_ext gpu_set_texrepeat gpu_set_texrepeat_ext gpu_set_tex_filter gpu_set_tex_filter_ext gpu_set_tex_repeat gpu_set_tex_repeat_ext gpu_set_tex_mip_filter gpu_set_tex_mip_filter_ext gpu_set_tex_mip_bias gpu_set_tex_mip_bias_ext gpu_set_tex_min_mip gpu_set_tex_min_mip_ext gpu_set_tex_max_mip gpu_set_tex_max_mip_ext gpu_set_tex_max_aniso gpu_set_tex_max_aniso_ext gpu_set_tex_mip_enable gpu_set_tex_mip_enable_ext gpu_get_blendenable gpu_get_ztestenable gpu_get_zfunc gpu_get_zwriteenable gpu_get_lightingenable gpu_get_fog gpu_get_cullmode gpu_get_blendmode gpu_get_blendmode_ext gpu_get_blendmode_ext_sepalpha gpu_get_blendmode_src gpu_get_blendmode_dest gpu_get_blendmode_srcalpha gpu_get_blendmode_destalpha gpu_get_colorwriteenable gpu_get_colourwriteenable gpu_get_alphatestenable gpu_get_alphatestref gpu_get_alphatestfunc gpu_get_texfilter gpu_get_texfilter_ext gpu_get_texrepeat gpu_get_texrepeat_ext gpu_get_tex_filter gpu_get_tex_filter_ext gpu_get_tex_repeat gpu_get_tex_repeat_ext gpu_get_tex_mip_filter gpu_get_tex_mip_filter_ext gpu_get_tex_mip_bias gpu_get_tex_mip_bias_ext gpu_get_tex_min_mip gpu_get_tex_min_mip_ext gpu_get_tex_max_mip gpu_get_tex_max_mip_ext gpu_get_tex_max_aniso gpu_get_tex_max_aniso_ext gpu_get_tex_mip_enable gpu_get_tex_mip_enable_ext gpu_push_state gpu_pop_state gpu_get_state gpu_set_state draw_light_define_ambient draw_light_define_direction draw_light_define_point draw_light_enable draw_set_lighting draw_light_get_ambient draw_light_get draw_get_lighting shop_leave_rating url_get_domain url_open url_open_ext url_open_full get_timer achievement_login achievement_logout achievement_post achievement_increment achievement_post_score achievement_available achievement_show_achievements achievement_show_leaderboards achievement_load_friends achievement_load_leaderboard achievement_send_challenge achievement_load_progress achievement_reset achievement_login_status achievement_get_pic achievement_show_challenge_notifications achievement_get_challenges achievement_event achievement_show achievement_get_info cloud_file_save cloud_string_save cloud_synchronise ads_enable ads_disable ads_setup ads_engagement_launch ads_engagement_available ads_engagement_active ads_event ads_event_preload ads_set_reward_callback ads_get_display_height ads_get_display_width ads_move ads_interstitial_available ads_interstitial_display device_get_tilt_x device_get_tilt_y device_get_tilt_z device_is_keypad_open device_mouse_check_button device_mouse_check_button_pressed device_mouse_check_button_released device_mouse_x device_mouse_y device_mouse_raw_x device_mouse_raw_y device_mouse_x_to_gui device_mouse_y_to_gui iap_activate iap_status iap_enumerate_products iap_restore_all iap_acquire iap_consume iap_product_details iap_purchase_details facebook_init facebook_login facebook_status facebook_graph_request facebook_dialog facebook_logout facebook_launch_offerwall facebook_post_message facebook_send_invite facebook_user_id facebook_accesstoken facebook_check_permission facebook_request_read_permissions facebook_request_publish_permissions gamepad_is_supported gamepad_get_device_count gamepad_is_connected gamepad_get_description gamepad_get_button_threshold gamepad_set_button_threshold gamepad_get_axis_deadzone gamepad_set_axis_deadzone gamepad_button_count gamepad_button_check gamepad_button_check_pressed gamepad_button_check_released gamepad_button_value gamepad_axis_count gamepad_axis_value gamepad_set_vibration gamepad_set_colour gamepad_set_color os_is_paused window_has_focus code_is_compiled http_get http_get_file http_post_string http_request json_encode json_decode zip_unzip load_csv base64_encode base64_decode md5_string_unicode md5_string_utf8 md5_file os_is_network_connected sha1_string_unicode sha1_string_utf8 sha1_file os_powersave_enable analytics_event analytics_event_ext win8_livetile_tile_notification win8_livetile_tile_clear win8_livetile_badge_notification win8_livetile_badge_clear win8_livetile_queue_enable win8_secondarytile_pin win8_secondarytile_badge_notification win8_secondarytile_delete win8_livetile_notification_begin win8_livetile_notification_secondary_begin win8_livetile_notification_expiry win8_livetile_notification_tag win8_livetile_notification_text_add win8_livetile_notification_image_add win8_livetile_notification_end win8_appbar_enable win8_appbar_add_element win8_appbar_remove_element win8_settingscharm_add_entry win8_settingscharm_add_html_entry win8_settingscharm_add_xaml_entry win8_settingscharm_set_xaml_property win8_settingscharm_get_xaml_property win8_settingscharm_remove_entry win8_share_image win8_share_screenshot win8_share_file win8_share_url win8_share_text win8_search_enable win8_search_disable win8_search_add_suggestions win8_device_touchscreen_available win8_license_initialize_sandbox win8_license_trial_version winphone_license_trial_version winphone_tile_title winphone_tile_count winphone_tile_back_title winphone_tile_back_content winphone_tile_back_content_wide winphone_tile_front_image winphone_tile_front_image_small winphone_tile_front_image_wide winphone_tile_back_image winphone_tile_back_image_wide winphone_tile_background_colour winphone_tile_background_color winphone_tile_icon_image winphone_tile_small_icon_image winphone_tile_wide_content winphone_tile_cycle_images winphone_tile_small_background_image physics_world_create physics_world_gravity physics_world_update_speed physics_world_update_iterations physics_world_draw_debug physics_pause_enable physics_fixture_create physics_fixture_set_kinematic physics_fixture_set_density physics_fixture_set_awake physics_fixture_set_restitution physics_fixture_set_friction physics_fixture_set_collision_group physics_fixture_set_sensor physics_fixture_set_linear_damping physics_fixture_set_angular_damping physics_fixture_set_circle_shape physics_fixture_set_box_shape physics_fixture_set_edge_shape physics_fixture_set_polygon_shape physics_fixture_set_chain_shape physics_fixture_add_point physics_fixture_bind physics_fixture_bind_ext physics_fixture_delete physics_apply_force physics_apply_impulse physics_apply_angular_impulse physics_apply_local_force physics_apply_local_impulse physics_apply_torque physics_mass_properties physics_draw_debug physics_test_overlap physics_remove_fixture physics_set_friction physics_set_density physics_set_restitution physics_get_friction physics_get_density physics_get_restitution physics_joint_distance_create physics_joint_rope_create physics_joint_revolute_create physics_joint_prismatic_create physics_joint_pulley_create physics_joint_wheel_create physics_joint_weld_create physics_joint_friction_create physics_joint_gear_create physics_joint_enable_motor physics_joint_get_value physics_joint_set_value physics_joint_delete physics_particle_create physics_particle_delete physics_particle_delete_region_circle physics_particle_delete_region_box physics_particle_delete_region_poly physics_particle_set_flags physics_particle_set_category_flags physics_particle_draw physics_particle_draw_ext physics_particle_count physics_particle_get_data physics_particle_get_data_particle physics_particle_group_begin physics_particle_group_circle physics_particle_group_box physics_particle_group_polygon physics_particle_group_add_point physics_particle_group_end physics_particle_group_join physics_particle_group_delete physics_particle_group_count physics_particle_group_get_data physics_particle_group_get_mass physics_particle_group_get_inertia physics_particle_group_get_centre_x physics_particle_group_get_centre_y physics_particle_group_get_vel_x physics_particle_group_get_vel_y physics_particle_group_get_ang_vel physics_particle_group_get_x physics_particle_group_get_y physics_particle_group_get_angle physics_particle_set_group_flags physics_particle_get_group_flags physics_particle_get_max_count physics_particle_get_radius physics_particle_get_density physics_particle_get_damping physics_particle_get_gravity_scale physics_particle_set_max_count physics_particle_set_radius physics_particle_set_density physics_particle_set_damping physics_particle_set_gravity_scale network_create_socket network_create_socket_ext network_create_server network_create_server_raw network_connect network_connect_raw network_send_packet network_send_raw network_send_broadcast network_send_udp network_send_udp_raw network_set_timeout network_set_config network_resolve network_destroy buffer_create buffer_write buffer_read buffer_seek buffer_get_surface buffer_set_surface buffer_delete buffer_exists buffer_get_type buffer_get_alignment buffer_poke buffer_peek buffer_save buffer_save_ext buffer_load buffer_load_ext buffer_load_partial buffer_copy buffer_fill buffer_get_size buffer_tell buffer_resize buffer_md5 buffer_sha1 buffer_base64_encode buffer_base64_decode buffer_base64_decode_ext buffer_sizeof buffer_get_address buffer_create_from_vertex_buffer buffer_create_from_vertex_buffer_ext buffer_copy_from_vertex_buffer buffer_async_group_begin buffer_async_group_option buffer_async_group_end buffer_load_async buffer_save_async gml_release_mode gml_pragma steam_activate_overlay steam_is_overlay_enabled steam_is_overlay_activated steam_get_persona_name steam_initialised steam_is_cloud_enabled_for_app steam_is_cloud_enabled_for_account steam_file_persisted steam_get_quota_total steam_get_quota_free steam_file_write steam_file_write_file steam_file_read steam_file_delete steam_file_exists steam_file_size steam_file_share steam_is_screenshot_requested steam_send_screenshot steam_is_user_logged_on steam_get_user_steam_id steam_user_owns_dlc steam_user_installed_dlc steam_set_achievement steam_get_achievement steam_clear_achievement steam_set_stat_int steam_set_stat_float steam_set_stat_avg_rate steam_get_stat_int steam_get_stat_float steam_get_stat_avg_rate steam_reset_all_stats steam_reset_all_stats_achievements steam_stats_ready steam_create_leaderboard steam_upload_score steam_upload_score_ext steam_download_scores_around_user steam_download_scores steam_download_friends_scores steam_upload_score_buffer steam_upload_score_buffer_ext steam_current_game_language steam_available_languages steam_activate_overlay_browser steam_activate_overlay_user steam_activate_overlay_store steam_get_user_persona_name steam_get_app_id steam_get_user_account_id steam_ugc_download steam_ugc_create_item steam_ugc_start_item_update steam_ugc_set_item_title steam_ugc_set_item_description steam_ugc_set_item_visibility steam_ugc_set_item_tags steam_ugc_set_item_content steam_ugc_set_item_preview steam_ugc_submit_item_update steam_ugc_get_item_update_progress steam_ugc_subscribe_item steam_ugc_unsubscribe_item steam_ugc_num_subscribed_items steam_ugc_get_subscribed_items steam_ugc_get_item_install_info steam_ugc_get_item_update_info steam_ugc_request_item_details steam_ugc_create_query_user steam_ugc_create_query_user_ex steam_ugc_create_query_all steam_ugc_create_query_all_ex steam_ugc_query_set_cloud_filename_filter steam_ugc_query_set_match_any_tag steam_ugc_query_set_search_text steam_ugc_query_set_ranked_by_trend_days steam_ugc_query_add_required_tag steam_ugc_query_add_excluded_tag steam_ugc_query_set_return_long_description steam_ugc_query_set_return_total_only steam_ugc_query_set_allow_cached_response steam_ugc_send_query shader_set shader_get_name shader_reset shader_current shader_is_compiled shader_get_sampler_index shader_get_uniform shader_set_uniform_i shader_set_uniform_i_array shader_set_uniform_f shader_set_uniform_f_array shader_set_uniform_matrix shader_set_uniform_matrix_array shader_enable_corner_id texture_set_stage texture_get_texel_width texture_get_texel_height shaders_are_supported vertex_format_begin vertex_format_end vertex_format_delete vertex_format_add_position vertex_format_add_position_3d vertex_format_add_colour vertex_format_add_color vertex_format_add_normal vertex_format_add_texcoord vertex_format_add_textcoord vertex_format_add_custom vertex_create_buffer vertex_create_buffer_ext vertex_delete_buffer vertex_begin vertex_end vertex_position vertex_position_3d vertex_colour vertex_color vertex_argb vertex_texcoord vertex_normal vertex_float1 vertex_float2 vertex_float3 vertex_float4 vertex_ubyte4 vertex_submit vertex_freeze vertex_get_number vertex_get_buffer_size vertex_create_buffer_from_buffer vertex_create_buffer_from_buffer_ext push_local_notification push_get_first_local_notification push_get_next_local_notification push_cancel_local_notification skeleton_animation_set skeleton_animation_get skeleton_animation_mix skeleton_animation_set_ext skeleton_animation_get_ext skeleton_animation_get_duration skeleton_animation_get_frames skeleton_animation_clear skeleton_skin_set skeleton_skin_get skeleton_attachment_set skeleton_attachment_get skeleton_attachment_create skeleton_collision_draw_set skeleton_bone_data_get skeleton_bone_data_set skeleton_bone_state_get skeleton_bone_state_set skeleton_get_minmax skeleton_get_num_bounds skeleton_get_bounds skeleton_animation_get_frame skeleton_animation_set_frame draw_skeleton draw_skeleton_time draw_skeleton_instance draw_skeleton_collision skeleton_animation_list skeleton_skin_list skeleton_slot_data layer_get_id layer_get_id_at_depth layer_get_depth layer_create layer_destroy layer_destroy_instances layer_add_instance layer_has_instance layer_set_visible layer_get_visible layer_exists layer_x layer_y layer_get_x layer_get_y layer_hspeed layer_vspeed layer_get_hspeed layer_get_vspeed layer_script_begin layer_script_end layer_shader layer_get_script_begin layer_get_script_end layer_get_shader layer_set_target_room layer_get_target_room layer_reset_target_room layer_get_all layer_get_all_elements layer_get_name layer_depth layer_get_element_layer layer_get_element_type layer_element_move layer_force_draw_depth layer_is_draw_depth_forced layer_get_forced_depth layer_background_get_id layer_background_exists layer_background_create layer_background_destroy layer_background_visible layer_background_change layer_background_sprite layer_background_htiled layer_background_vtiled layer_background_stretch layer_background_yscale layer_background_xscale layer_background_blend layer_background_alpha layer_background_index layer_background_speed layer_background_get_visible layer_background_get_sprite layer_background_get_htiled layer_background_get_vtiled layer_background_get_stretch layer_background_get_yscale layer_background_get_xscale layer_background_get_blend layer_background_get_alpha layer_background_get_index layer_background_get_speed layer_sprite_get_id layer_sprite_exists layer_sprite_create layer_sprite_destroy layer_sprite_change layer_sprite_index layer_sprite_speed layer_sprite_xscale layer_sprite_yscale layer_sprite_angle layer_sprite_blend layer_sprite_alpha layer_sprite_x layer_sprite_y layer_sprite_get_sprite layer_sprite_get_index layer_sprite_get_speed layer_sprite_get_xscale layer_sprite_get_yscale layer_sprite_get_angle layer_sprite_get_blend layer_sprite_get_alpha layer_sprite_get_x layer_sprite_get_y layer_tilemap_get_id layer_tilemap_exists layer_tilemap_create layer_tilemap_destroy tilemap_tileset tilemap_x tilemap_y tilemap_set tilemap_set_at_pixel tilemap_get_tileset tilemap_get_tile_width tilemap_get_tile_height tilemap_get_width tilemap_get_height tilemap_get_x tilemap_get_y tilemap_get tilemap_get_at_pixel tilemap_get_cell_x_at_pixel tilemap_get_cell_y_at_pixel tilemap_clear draw_tilemap draw_tile tilemap_set_global_mask tilemap_get_global_mask tilemap_set_mask tilemap_get_mask tilemap_get_frame tile_set_empty tile_set_index tile_set_flip tile_set_mirror tile_set_rotate tile_get_empty tile_get_index tile_get_flip tile_get_mirror tile_get_rotate layer_tile_exists layer_tile_create layer_tile_destroy layer_tile_change layer_tile_xscale layer_tile_yscale layer_tile_blend layer_tile_alpha layer_tile_x layer_tile_y layer_tile_region layer_tile_visible layer_tile_get_sprite layer_tile_get_xscale layer_tile_get_yscale layer_tile_get_blend layer_tile_get_alpha layer_tile_get_x layer_tile_get_y layer_tile_get_region layer_tile_get_visible layer_instance_get_instance instance_activate_layer instance_deactivate_layer camera_create camera_create_view camera_destroy camera_apply camera_get_active camera_get_default camera_set_default camera_set_view_mat camera_set_proj_mat camera_set_update_script camera_set_begin_script camera_set_end_script camera_set_view_pos camera_set_view_size camera_set_view_speed camera_set_view_border camera_set_view_angle camera_set_view_target camera_get_view_mat camera_get_proj_mat camera_get_update_script camera_get_begin_script camera_get_end_script camera_get_view_x camera_get_view_y camera_get_view_width camera_get_view_height camera_get_view_speed_x camera_get_view_speed_y camera_get_view_border_x camera_get_view_border_y camera_get_view_angle camera_get_view_target view_get_camera view_get_visible view_get_xport view_get_yport view_get_wport view_get_hport view_get_surface_id view_set_camera view_set_visible view_set_xport view_set_yport view_set_wport view_set_hport view_set_surface_id gesture_drag_time gesture_drag_distance gesture_flick_speed gesture_double_tap_time gesture_double_tap_distance gesture_pinch_distance gesture_pinch_angle_towards gesture_pinch_angle_away gesture_rotate_time gesture_rotate_angle gesture_tap_count gesture_get_drag_time gesture_get_drag_distance gesture_get_flick_speed gesture_get_double_tap_time gesture_get_double_tap_distance gesture_get_pinch_distance gesture_get_pinch_angle_towards gesture_get_pinch_angle_away gesture_get_rotate_time gesture_get_rotate_angle gesture_get_tap_count keyboard_virtual_show keyboard_virtual_hide keyboard_virtual_status 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' +\n 'goto save else use module select case ' +\n 'access blank direct exist file fmt form formatted iostat name named nextrec number opened rec recl sequential status unformatted unit ' +\n 'continue format pause cycle exit ' +\n 'c_null_char c_alert c_backspace c_form_feed flush wait decimal round iomsg ' +\n 'synchronous nopass non_overridable pass protected volatile abstract extends import ' +\n 'non_intrinsic value deferred generic final enumerator class associate bind enum ' +\n 'c_int c_short c_long c_long_long c_signed_char c_size_t c_int8_t c_int16_t c_int32_t c_int64_t c_int_least8_t c_int_least16_t ' +\n 'c_int_least32_t c_int_least64_t c_int_fast8_t c_int_fast16_t c_int_fast32_t c_int_fast64_t c_intmax_t C_intptr_t c_float c_double ' +\n 'c_long_double c_float_complex c_double_complex c_long_double_complex c_bool c_char c_null_ptr c_null_funptr ' +\n 'c_new_line c_carriage_return c_horizontal_tab c_vertical_tab iso_c_binding c_loc c_funloc c_associated c_f_pointer ' +\n 'c_ptr c_funptr iso_fortran_env character_storage_size error_unit file_storage_size input_unit iostat_end iostat_eor ' +\n 'numeric_storage_size output_unit c_f_procpointer ieee_arithmetic ieee_support_underflow_control ' +\n 'ieee_get_underflow_mode ieee_set_underflow_mode newunit contiguous recursive ' +\n 'pad position action delim readwrite eor advance nml interface procedure namelist include sequence elemental pure ' +\n 'integer real character complex logical dimension allocatable|10 parameter ' +\n 'external implicit|10 none double precision assign intent optional pointer ' +\n 'target in out common equivalence data ' +\n // IRPF90 special keywords\n 'begin_provider &begin_provider end_provider begin_shell end_shell begin_template end_template subst assert touch ' +\n 'soft_touch provide no_dep free irp_if irp_else irp_endif irp_write irp_read',\n built_in: 'alog alog10 amax0 amax1 amin0 amin1 amod cabs ccos cexp clog csin csqrt dabs dacos dasin datan datan2 dcos dcosh ddim dexp dint ' +\n 'dlog dlog10 dmax1 dmin1 dmod dnint dsign dsin dsinh dsqrt dtan dtanh float iabs idim idint idnint ifix isign max0 max1 min0 min1 sngl ' +\n 'algama cdabs cdcos cdexp cdlog cdsin cdsqrt cqabs cqcos cqexp cqlog cqsin cqsqrt dcmplx dconjg derf derfc dfloat dgamma dimag dlgama ' +\n 'iqint qabs qacos qasin qatan qatan2 qcmplx qconjg qcos qcosh qdim qerf qerfc qexp qgamma qimag qlgama qlog qlog10 qmax1 qmin1 qmod ' +\n 'qnint qsign qsin qsinh qsqrt qtan qtanh abs acos aimag aint anint asin atan atan2 char cmplx conjg cos cosh exp ichar index int log ' +\n 'log10 max min nint sign sin sinh sqrt tan tanh print write dim lge lgt lle llt mod nullify allocate deallocate ' +\n 'adjustl adjustr all allocated any associated bit_size btest ceiling count cshift date_and_time digits dot_product ' +\n 'eoshift epsilon exponent floor fraction huge iand ibclr ibits ibset ieor ior ishft ishftc lbound len_trim matmul ' +\n 'maxexponent maxloc maxval merge minexponent minloc minval modulo mvbits nearest pack present product ' +\n 'radix random_number random_seed range repeat reshape rrspacing scale scan selected_int_kind selected_real_kind ' +\n 'set_exponent shape size spacing spread sum system_clock tiny transpose trim ubound unpack verify achar iachar transfer ' +\n 'dble entry dprod cpu_time command_argument_count get_command get_command_argument get_environment_variable is_iostat_end ' +\n 'ieee_arithmetic ieee_support_underflow_control ieee_get_underflow_mode ieee_set_underflow_mode ' +\n 'is_iostat_eor move_alloc new_line selected_char_kind same_type_as extends_type_of' +\n 'acosh asinh atanh bessel_j0 bessel_j1 bessel_jn bessel_y0 bessel_y1 bessel_yn erf erfc erfc_scaled gamma log_gamma hypot norm2 ' +\n 'atomic_define atomic_ref execute_command_line leadz trailz storage_size merge_bits ' +\n 'bge bgt ble blt dshiftl dshiftr findloc iall iany iparity image_index lcobound ucobound maskl maskr ' +\n 'num_images parity popcnt poppar shifta shiftl shiftr this_image ' +\n 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\ No newline at end of file diff --git a/docs/route-AlgoritmiEStruttureDati.chunk.d1ce1.js b/docs/route-AlgoritmiEStruttureDati.chunk.56b0d.js similarity index 97% rename from docs/route-AlgoritmiEStruttureDati.chunk.d1ce1.js rename to docs/route-AlgoritmiEStruttureDati.chunk.56b0d.js index a3e1c45..ebdf4d9 100644 --- a/docs/route-AlgoritmiEStruttureDati.chunk.d1ce1.js +++ b/docs/route-AlgoritmiEStruttureDati.chunk.56b0d.js @@ -1,2 +1,2 @@ -(window.webpackJsonp=window.webpackJsonp||[]).push([[2],{w9qZ:function(n,e,i){"use strict";i.r(e),function(n){function o(){var n=function(n,e){e||(e=n.slice(0));return n.raw=e,n}(["\n # Algoritmi e Strutture Dati\n\n Docente: [**Manuela Montangero**](mailto:manuela.montangero@unimore.it)\n\n Crediti: **9 CFU** (72 ore di lezione)\n\n Orario di ricevimento: **Giovedì dalle 14:30 alle 16:30**\n _Mandare una mail prima, altrimenti potrebbe andarsene_\n\n\n #### Regole particolari per le email\n\n - Oggetto \"ASD\"\n - Mail **firmata** con **nome e cognome**\n - Mail spedita dall'**account Unimore**\n\n ### Materiale\n\n Libri:\n - **Introduzione agli Algoritmi e Strutture Dati** di _T.H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein_\n\n **[Dolly (FIM)](https://dolly.fim.unimore.it/2018/course/view.php?id=26)**\n\n ### Tutorato\n\n Tutor: [**Gianluca d'Addese**](mailto:tutoratoalgoritmi@gmail.com)\n\n Cosa: **Esercizi sugli argomenti visti a lezione** e **preparatori all'esame**\n\n Quando: **Mercoledì dalle 09:00 alle 11:00**...?\n\n ### Esame\n\n Per iscriversi all'esame, bisogna aver passato:\n\n - Analisi matematica (propedeutica)\n - Programmazione I\n\n Formato:\n\n - Prima prova scritta\n - Risolvere problemi con algoritmi proposti a lezione\n - E' un test sulla preparazione\n - Dura 1h30m\n - Non si può usare nessun tipo di materiale\n - Seconda prova scritta\n - Proponi soluzioni per nuovi problemi non studiati a lezione\n - Domande teoriche sugli argomenti studiati\n - Dura 2h\n - E' il giorno dopo la prima prova\n - Si può utilizzare qualsiasi materiale, ma non deve permettere di comunicare\n - Orale facoltativo\n - Solo per chi supera entrambi gli scritti\n - Domande su quello che abbiamo visto a lezione (\"perchè gli algoritmi funzionano?\")\n - Informare via email entro 3 giorni dalla pubblicazione degli esiti della seconda prova\n - L'orale potrebbe migliorare o peggiorare il voto (anche \"molto\")\n\n **Attenzione:** Vengono verbalizzati anche i voti insufficienti; ricordarsi di rifiutarli!\n\n Ci sono i seguenti appelli:\n\n - 3 appelli tra Giugno e Luglio\n - 1 appello a Settembre\n - 2 appelli a Gennaio e Febbraio\n\n Vale il salto di appello se:\n\n - Uno studente regolarmente iscritto **non si presenta** all'appello e non ha avvisato via email almeno il giorno prima.\n - Uno studente ha riportato una **grave insufficienza** in uno dei due scritti, e l'appello successivo è nella stessa sessione\n\n\n ### Note\n\n Il corso è ben collegato con quello di Programmazione 2.\n\n Dormire non fa bene!\n "]);return o=function(){return n},n}var a=i("mbOI"),r=String.raw;e.default=function(){return n("div",null,n(a.o,{title:"Disperazione di Steffo"},"aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa tutte le formule in latex si sono rotteeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee"),n(a.o,null,n(a.m,null,r(o()))),n(a.o,null,n(a.m,null,"\n # Il nome del corso\n\n ## Cosa sono gli algoritmi?\n\n Gli algoritmi sono modi sistematici per risolvere problemi.\n\n Sono fondamentali per sviluppare software, in quanto i computer sono eccellenti esecutori di algoritmi.\n\n ## Come si sviluppa un algoritmo?\n\n Innanzitutto, bisogna conoscere gli _input_ e gli _output_ del problema, rispettivamente i dati di partenza e i dati di arrivo di esso; si ha quindi una fase di **ricerca**.\n\n Poi, si deve trovare un procedimento che ci faccia risolvere il nostro problema: è quello che faremo in questa materia!\n\n Infine, bisogna scrivere la soluzione in un modo che possa essere eseguita da un computer: questa è la **programmazione**.\n\n ## Che tipo di problemi possiamo risolvere?\n\n Un algoritmo risolve problemi di tipo generale, non ci interessa sapere _il risultato di 123+456_, ma vogliamo sapere _il risultato di x+y_, dove x e y sono due numeri naturali qualsiasi.\n\n Un problema può essere quindi considerato circa come una **funzione matematica**, che connette ogni input a un output corrispondente.\n\n ## Che caratteristiche ha un algoritmo?\n\n Per prima cosa, ripetendo l'algoritmo più volte con lo stesso ingresso deve dare sempre la stessa uscita come **risultato**, finendo in un **tempo finito**.\n\n Deve essere **ben ordinato**: cambiando l'ordine in cui vengono effettuate le operazioni, è probabile che anche il risultato cambi!\n\n Le sue istruzioni devono essere **non ambigue**, cioè che non possano essere interpretate in più modi, e **effettivamente realizzabili**, cioè realizzabili con l'esecutore che vogliamo usare per eseguire l'algoritmo.\n\n ## Esistono algoritmi equivalenti?\n\n **Sì!** Possono esserci due algoritmi che dati gli stessi input, hanno gli stessi output, e quindi risolvono lo stesso problema.\n\n In compenso, possono avere un numero di operazioni diverse, e quindi essere **uno più veloce** (da eseguire) dell'altro.\n\n ## Come si verifica la correttezza di un algoritmo?\n\n L'algoritmo deve essere **valido per tutti gli input**, anche se questi sono infiniti.\n\n Possiamo effettuare prove matematiche per verificarne la correttezza; il **principio di induzione** è dunque una dei teoremi fondamentali dell'algoritmica.\n\n Possiamo però verificare la _non correttezza_ di un algoritmo trovando un singolo controesempio.\n ")),n(a.o,null,n(a.m,null,"\n # Efficienza degli algoritmi\n \n Un buon algoritmo deve essere **efficiente**, ovvero deve usare il minimo delle risorse necessarie, come _usare il minimo di tempo possibile_.\n \n ## Come misuriamo il tempo necessario?\n \n Cerchiamo di astrarre il tempo dal particolare esecutore, e andiamo a contare il numero di operazioni elementari richieste per eseguire il nostro algoritmo nel caso peggiore.\n \n Un algoritmo efficiente, infatti, all'aumentare dei dati in ingresso, diventerà sempre più veloce rispetto a uno non efficiente, anche su computer più lenti!\n \n > Il [Bubble Sort](https://en.wikipedia.org/wiki/Bubble_sort) è sempre più lento di un [Tree Sort](https://en.wikipedia.org/wiki/Tree_sort), anche su computer più lenti, perchè, dovendo ordinare liste sempre più lunghe, prima o poi si raggiunge un punto in cui il primo è più veloce (in termini di tempo) dell'altro.\n \n Dobbiamo andare a vedere, quindi, _il numero di operazioni richieste per ottenere il risultato nel caso peggiore_.\n \n Consideriamo operazioni sia operazioni aritmetiche sia operazioni logiche, e diciamo che ciascuna costa 1.\n \n > L'[algoritmo di Euclide](https://it.wikipedia.org/wiki/Algoritmo_di_Euclide) per l'MCD costa `3` per ogni iterazione (un giro del ciclo `while`). Diciamo, allora, che costa `3n`, dove `n` è il numero più alto dei due, perchè nel caso peggiore (uno dei due numeri è 1) l'algoritmo compie `n` iterazioni.\n \n ## Altri parametri ottimizzabili\n \n È possibile che alcuni algoritmi per vari motivi cerchino di ottimizzare altri parametri diversi dal tempo, come ad esempio _la dimensione dell'input_ o la _memoria utilizzata_. \n \n > Generalmente, questo viene fatto sui dispositivi embedded, con memoria molto limitata.\n \n ### Criteri di costo di memoria\n \n Ci sono vari criteri con cui stimare la memoria richiesta da un dato: è possibile che il costo risultante vari in base al criterio scelto!\n \n #### Criterio di costo logaritmico\n \n Un dato costa il numero di bit necessari per rappresentarlo.\n \n > Un int che contiene il numero `n` costa `log_2(n)`. \n > Un array di `[n]` numeri tutti uguali costa `n log_2(n)`.\n \n #### Criterio di costo uniforme\n \n Un dato costa il numero di elementi che lo costituiscono.\n \n > Un int che contiene il numero `n` costa `1`. \n > Un array di `[n]` numeri costa `n`. \n > Una matrice `[m][n]` costa `m*n`.\n ")),n(a.o,null,n(a.m,null,"\n # Modelli algoritmici\n\n Per progettare un algoritmo, abbiamo bisogno di sapere le proprietà del nostro esecutore, ovvero il suo _modello algoritmico_.\n \n > Ad esempio, dobbiamo sapere quali istruzioni è in grado di eseguire, e quanto tempo queste istruzioni richiedono.\n \n ## Il modello RAM\n \n Il modello in uso su tutti i computer attuali è il _modello RAM_: \n \n - In ogni cella di memoria può essere archiviato un dato.\n - Il **tempo di accesso** alle celle è **costante** per tutte le celle.\n - La **memoria** principale è **infinita**.\n - Si ha **un solo processore**.\n ")),n(a.o,null,n(a.m,null,'\n # Notazione asintotica\n \n La _notazione asintotica_ è un sistema per **stimare** velocemente il costo di un algoritmo complesso.\n \n Ci permette di **confrontare velocemente il caso peggiore** degli algoritmi.\n \n In particolare, consideriamo il _rapporto tra il numero di operazioni nel caso peggiore e la dimensione dell\'input_.\n \n ## Limiti\n \n Possiamo dare a questa stima dei limiti, superiore e inferiore, che rappresenteranno rispettivamente un costo che non sarà **mai superato** e un costo che verrà **sempre superato**.\n \n Chiameremo questi limiti _upper bound_ e _lower bound_; la loro combinazione darà un _tight bound_.\n \n L\'obiettivo sarà di _ricavare i bound più precisi possibile_ per un dato algoritmo, ovvero l\'**upper bound più basso** e il **lower bound più alto**.\n \n ### O grande\n \n > "O grande" \n > O di g(n)\n > "big-O"\n \n Per rappresentare la stima, useremo una notazione particolare, detta _O grande_, con la seguente proprietà:\n - Date due funzioni `f(n) : N -> R` e `g(n) : N -> R`, diremo che `f(n) ∈ O(g(n))` se e soltanto se `∃ c > 0, n ≥ n_0` tali che `∀ n ≥ 0, f(n) ≤ c * g(n)`\n \n Quando una funzione è O grande di un altra, significa che **asintoticamente, la funzione in O grande è sempre maggiore di quella che sta venendo stimata**.\n \n > **Ipotesi** \n > - `f(n) = 2n² + 3n + 6` \n > - `g(n) = n²`\n >\n > **Tesi** \n > - `f(n) ∈ O(n²)`.\n >\n > **Svolgimento** \n > Scrivo una disequazione, lasciando intatto il termine noto: \n > 1. `2n² + 3n + 6 ≤ 2n² + 3n² + 6` \n > 2. `n² ≤ 2n² + 3n² + n² = 6n²` per `n ≥ 3`\n >\n > Sappiamo, allora, che `2n² + 3n + 6 ≤ 6n²`.\n \n #### Espressioni di O grande\n \n Questa tabella rappresenta le espressioni di O grande più comunemente utilizzate, in ordine **dalla più forte alla più debole**.\n \n > Più forte significa che, per ogni riga della tabella, tutte le righe sottostanti sono contenute nell\'espressione.\n >\n > Ad esempio, `O(n) ∈ O(1)`.\n \n | Espressione O() | Nome |\n |-----------------|------|\n | `O(1)` | Costante |\n | `O(log log n)` | loglog |\n | `O(log n)` | Logaritmica |\n | `Ω(n^{1/c})` (per c ≥ 1) | Sublineare |\n | `O(n)` | Lineare |\n | `O(n log n)` | nlogn |\n | `O(n²)` | Quadratica |\n | `O(n³)` | Cubica |\n | `O(n^k)` (per k ≥ 1) | Polinomiale |\n | `O(a^n)` (per a ≥ 1) | Esponenziale |\n | `O(n!)` | Fattoriale |\n \n #### Polinomiale\n \n Molto spesso, noteremo che il tempo richiesto da una funzione è O grande di un polinomio di grado K, ovvero `f(n) ∈ O(n^k)`.\n \n Notiamo che in questi casi, possiamo semplificare l\'O grande al grado massimo del polinomio.\n \n > Ad esempio, `O(n² + n + 1) = O(n²)`.\n \n ##### Dimostrazione\n \n Per `n > 0 \\and 0 ≤ i ≤ k`: \n ![LaTeX](https://latex.codecogs.com/png.latex?a_k%20n^k%20+%20a_{k-1}%20n^{k-1}%20+%20%E2%80%A6%20+%20a_1%20n%20+%20a_0%20%E2%89%A4%20|a_k|%20n^k%20+%20|a_{k-1}|%20n^k%20+%20%E2%80%A6%20+%20|a_1|%20n^k%20+%20|a_0|%20n^k%20=%20(|a_k|%20+%20|a_{k-1}|%20+%20%E2%80%A6%20+%20|a_1|%20+%20|a_0|)%20n^k)\n \n #### Proprietà di O grande\n \n 1. `f(n) ∈ O(g(n)) -> ∀ a > 0, a * f(n) ∈ O(g(n))`.\n 2. `f(n) ∈ O(g(n)), d(n) ∈ O(h(n)) -> f(n) + d(n) ∈ O(g(n) + h(n)) -> O(max\\{g(n), h(n)\\})`\n 3. `f(n) ∈ O(g(n)), d(n) ∈ O(h(n)) -> f(n) * d(n) ∈ O(g(n) * h(n))`\n \n In pratica, se una funzione è la _somma di più termini_, basta guardare l\'**`O()` più grande** tra tutti i suoi termini; se invece una funzione è un _prodotto di più termini_, si possono **omettere le costanti**, e l\'`O()` finale sarà dato dal **prodotto degli `O()`** dei termini.\n \n ## Lower bound\n \n Possiamo anche stimare il _lower bound_, il limite inferiore: il **numero minimo di operazioni** che viene effettuato **nel caso migliore** con la **massima dimensione dell\'ingresso**.\n \n ### Ω()\n \n > "Omega" \n > Omega di g(n)\n > "big-Omega"\n \n Esiste un equivalente di O grande per il lower bound: è detto _Omega grande_, o più semplicemente _Omega_, e funziona nello stesso identico modo, solo... al contrario.\n \n Diremo che `f(n) ∈ Ω(g(n))` se e solo se `∃ c > 0, n_0 ≥ 0 : ∀ n ≥ n_0 f(n) ≥ c * g(n)`.\n \n #### Espressioni di Ω()\n \n Anche in questa tabella le espressioni sono **dalla più forte alla più debole**.\n \n | Espressione Ω() | Nome |\n |-----------------|------|\n | `Ω(n!)` | Fattoriale |\n | `Ω(a^n)` (per a ≥ 1) | Esponenziale |\n | `Ω(n^k)` (per k ≥ 1) | Polinomiale |\n | `Ω(n³)` | Cubica |\n | `Ω(n²)` | Quadratica |\n | `Ω(n log n)` | nlogn |\n | `Ω(n)` | Lineare |\n | `Ω(n^{1/c})` (per c ≥ 1) | Sublineare |\n | `Ω(log n)` | Logaritmica |\n | `Ω(log log n)` | loglog |\n | `Ω(1)` | Costante |\n \n ## Tight bound\n \n Quando **upper e lower bound coincidono**, allora otteniamo un _tight bound_.\n \n ### θ()\n \n > "Theta" \n > Theta di g(n)\n > "big-Theta"\n \n Anche per il tight bound abbiamo una notazione equivalente a O grande e Omega grande: _Theta grande_!\n \n Diciamo che `f(n) ∈ θ(g(n))` se e solo se `∃ c_1, c_2 > 0, n_0 ≥ 0 : ∀ n ≥ n_0, c_1 * g(n) ≤ f(n) ≤ c_2 * g(n)`.\n \n Ha la particolarità che non valgono tutte le proprietà degli altri due: va usata quindi con cautela!\n \n ## Risorse utili\n \n [khanacademy.org](https://www.khanacademy.org/computing/computer-science/algorithms/asymptotic-notation/a/big-big-theta-notation)\n ')),n(a.o,null,n(a.m,null,"\n # Problemi algoritmici\n \n Un _problema algoritmico_ è un problema matematico che si vuole provare a risolvere con un algoritmo.\n \n > Dati 10 numeri, voglio sapere se sono in ordine crescente oppure no.\n \n ## Caterigorizzazione\n \n I problemi algoritmici si dividono in tre categorie: problemi _trattabili_, problemi _intrattabili_ e problemi _irrisolvibili_.\n \n ### Problema trattabile\n \n Perchè un problema algoritmico sia _trattabile_, deve avere **almeno un algoritmo con upper bound polinomiale**.\n \n > Questo significa che il tempo impiegato da un computer per risolvere il problema rimane ragionevole, e che quindi può essere utilizzato in maniera efficiente.\n \n La trattabilità è un campo ancora parecchio aperto: esistono anche tanti problemi di cui non si è ancora dimostrata la trattabilità o intrattabilità.\n \n > La [fattorizzazione](https://it.wikipedia.org/wiki/Fattorizzazione) è uno di questi problemi: l'assenza di una dimostrazione è ciò che la rende uno dei pilastri della sicurezza informatica moderna.\n \n ### Problema intrattabile\n \n Se **un problema non ha nessun algoritmo con upper bound polinomiale**, allora si dice che è **intrattabile**.\n \n ### Problema irrisolvibile\n \n Se **non esistono algoritmi per risolvere un problema**, allora questo si dice **irrisolvibile**.\n \n > [Dato un algoritmo con certi input, riusciamo a capire con un algoritmo se la sua esecuzione termina o no?](https://en.wikipedia.org/wiki/Halting_problem)\n \n ### Upper e lower bound di problemi\n \n Si può anche trovare un _upper bound_ e un _lower bound_ per un problema, ma bisogna generalizzare di più.\n \n L'**upper bound di un problema** è il minimo upper bound di tutti gli algoritmi che lo risolvono; deve esistere almeno un algoritmo che lo risolva che abbia lo stesso _upper bound_. E' praticamente il tempo migliore per risolvere il problema dato.\n \n Il **lower bound di un problema** è il minimo lower bound di tutti gli algoritmi che lo risolvono; non deve esistere nemmeno un algoritmo che abbia un lower bound migliore. E' il numero assolutamente minimo di operazioni richieste, non si può fare meglio di così.\n \n In particolare, abbiamo che l'_upper bound di un algoritmo_ -> l'_upper bound del suo problema_, \n e il _lower bound di un problema_ -> il _lower bound di un suo algoritmo_.\n \n Generalmente, il _lower bound di un problema_ è una rappresentazione abbastanza accurata della sua difficoltà.\n ")),n(a.o,null,n(a.m,null,"\n # Ricerca binaria\n \n Non credo di aver bisogno di studiare la ricerca binaria, quindi non ho preso appunti a riguardo.\n \n Se non siete me, e state cercando informazioni a riguardo, andate a vedere su Wikipedia!\n ")),n(a.o,null,n(a.m,null,"\n # Divide et impera\n \n Un modo efficace per risolvere un problema è di usare il metodo _divide et impera (et combina)_.\n \n - _Divide_: Divido il problema in **tanti sottoproblemi**.\n - _Impera_: **Risolvo indipendentemente dal resto** ciascuno dei sottoproblemi.\n - _Combina_: **Combino** i risultati dei sottoproblemi per **risolvere il problema principale**.\n \n ## Ricorsione\n \n Un algoritmo (o funzione) si dice _ricorsivo_ quando durante l'esecuzione **richiama sè stesso**.\n \n Dato che tutti gli algoritmi devono avere termine entro un tempo finito, se scriviamo una funzione ricorsiva è fondamentale finire con un **caso base**, che non chiami ulteriormente la ricorsione.\n \n Se esiste una funzione ricorsiva, allora esiste _sempre_ una _funzione iterativa_ che darà lo stesso risultato.\n \n > Sul pratico, una funzione ricorsiva tipicamente è **più costosa** del suo equivalente iterativo: se possibile, quindi, la ricorsione andrebbe evitata.\n \n #### Pseudocodice di esempio\n \n ```python\n def fattoriale(n):\n # Caso base; la ricorsione finisce e dà un risultato fisso\n if n <= 1:\n return 1\n # Caso ricorsivo; la funzione restituisce il risultato di sè stessa (ma con parametri diversi)\n else:\n return n * fattoriale(n - 1)\n ```\n ")),n(a.o,null,n(a.m,null,"\n # Master Theorem\n \n Il _Master Theorem_ è uno dei teoremi più importanti dell'algoritmica.\n \n Esso permette di **calcolare l'upper bound di un algoritmo ricorsivo** in modo piuttosto semplice.\n \n ## Ipotesi\n \n Dato un algoritmo:\n - Con uno o più casi base\n - Che richiama la funzione ricorsiva un numero n di volte\n \n ## Tesi\n \n Allora, il suo upper bound avrà la formula:\n \n ![](https://quicklatex.com/cache3/89/ql_08d29a7e55d561a900570bc83b93ff89_l3.png)\n ")),n(a.o,null,n(a.m,null,"\n # Caso particolare del Master Theorem\n \n ## Ipotesi\n \n Se: \n ![](https://quicklatex.com/cache3/57/ql_e34dc27b42831d3c3ff671b0f3861257_l3.png)\n \n Ovvero, se la dimensione dell'input viene divisa ad ogni ciclo da una costante b, è polinomiale e il caso base è costante...\n \n ## Tesi\n \n Allora: \n ![](https://quicklatex.com/cache3/ca/ql_26e3557a6ca2d6ac4b8481e7c5263fca_l3.png)\n \n > In pratica, se il costo dominante è quello della parte \"fissa\" dell'algoritmo, esso sarà `O(n^d)`, mentre se il costo dominante è quello delle chiamate ricorsive, esso sarà `O(n^{log_b e})`. \n > Se nessuno dei due è dominante... si dividono circa in parti uguali, creando un costo di `O(n^d log n)`.\n ")),n(a.o,null,n(a.m,null,"\n # Ordinamento\n \n Un problema molto frequente nell'informatica consiste nell'**ordinare efficientemente grandi quantità di elementi**.\n \n Esistono [tantissimi](https://it.wikipedia.org/wiki/Algoritmo_di_ordinamento) algoritmi per effettuare l'ordinamento. \n L'**efficienza** di ciascuno **varia** di caso in caso: alcuni sono estremamente efficienti se quasi tutti i numeri sono già nell'ordine giusto; altri, invece, potrebbero impiegare tantissimo tempo.\n \n In termini matematici, abbiamo:\n \n - **Input:** A[n]\n - **Output:** B, ∀ i < n, A[i] ≤ A[i+1]\n \n ## Ordinamento tramite confronto\n \n L'ordinamento \"tradizionale\" è detto _ordinamento tramite confronto_: funziona sempre, e **non ha altri modi di ottenere informazioni** se non con l'operazione logica di confronto tra i dati.\n \n ### Limiti\n \n E' un problema risolto: è dimostrabile che il suo **lower bound** è **`Ω(n log n)`**; possiamo quindi dire che qualsiasi algoritmo di ordinamento è in `Ω(n log n)`, e se riusciamo a trovare un algoritmo di ordinamento in `O(n log n)` siamo riusciti a raggiungere il massimo dell'efficienza.\n \n #### Dimostrazione\n \n Consideriamo **tutte le possibili permutazioni** della sequenza da ordinare: sono `n!`.\n \n Per ogni confronto che effettuiamo, **riduciamo la quantità di permutazioni** correttamente ordinate; prima o poi, rimarrà **una sola possibilità**.\n \n **TODO, non trovo la spiegazione corretta!**\n \n ### Esempi\n \n Algoritmi che effettuano l'ordinamento tramite confronto sono:\n - _Bubble sort_\n - _Merge sort_\n - _Insertion sort_\n - _Quick sort_\n - _Heap sort_\n - E tanti, tanti altri!\n \n ## Ordinamento con altri mezzi\n \n Esistono algoritmi che ricavano informazioni in altri modi, diversi dal confronto.\n \n Essi possono avere un lower bound più basso di `O(n log n)`, però hanno spesso limitazioni sul loro utilizzo.\n \n ### Esempi\n \n - _Counting sort_, indicizza i valori da ordinare\n - _Radix sort_, guarda singolarmente le cifre dei valori\n - _Sleep sort_, sfrutta i thread e la funzione sleep per ordinare valori\n - E altri ancora!\n ")),n(a.o,null,n(a.m,null,"\n # Insertion sort\n \n L'_insertion sort_ è una soluzione **iterativa** all'ordinamento per confronto.\n \n ## Funzionamento\n \n Considero la sequenza divisa in **due parti**: una parte **ordinata** e una parte **non ordinata**.\n \n Parto dal primo elemento della lista: è sempre ordinato con sè stesso. \n Poi, aggiungo uno alla volta i numeri della parte non ordinata a quella ordinata; prima trovo in quale posizione dovrò andare a mettere il numero, poi **faccio slittare tutti i numeri dopo quella posizione** avanti di 1, in modo da **creare lo spazio** in cui infine **inserirò** il numero.\n \n ## Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | `O(n²)` | `Ω(n)` | - |\n \n Nel _caso migliore_ (**lista già ordinata**), il numero da inserire è già nella posizione giusta, quindi non devo effettuare altri confronti oltre il primo, rendendo il lower bound dell'algoritmo `Ω(n)`.\n \n Nel _caso peggiore_ (**lista nell'ordine inverso**), dobbiamo confrontare il numero da inserire con tutti gli altri nella parte ordinata: dobbiamo allora eseguire `1+2+3+4+5+… = \\frac{(n-1)(n)}{2}` confronti; ciò significa che l'upper bound è `O(n²)`!\n \n ## Pseudocodice\n \n ```python\n def insertion_sorted(lista):\n # Itero su tutti i numeri della lista, dal primo all'ultimo.\n for divisore_ord in range(len(lista)):\n # Partendo dalla posizione attuale, creo l'indice di divisione numeri ordinati maggiori e minori\n divisore_magg = divisore_ord\n # Faccio slittare avanti i numeri maggiori di quello che stiamo inserendo\n # Se l'indice divisore_magg raggiunge 0, vuol dire che tutti i numeri della lista sono maggiori del numero attuale \n while divisore_magg >= 0 and lista[divisore_magg-1] > lista[divisore_magg]:\n # Scambio la posizione dei due elementi con gli indici specificati\n # Funzione inventata \n lista[divisore_magg], lista[divisore_magg-1] = lista[divisore_magg-1], lista[divisore_magg]\n # Diminuisco il separatore di 1\n divisore_magg -= 1\n ```\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/bn/sorting)\n ")),n(a.o,null,n(a.m,null,"\n # Merge sort\n \n Il _merge sort_ è una soluzione **ricorsiva** all'ordinamento per confronto.\n \n ## Funzionamento\n \n Per questo algoritmo, utilizziamo la tecnica del **divide et impera**.\n \n 1. _Divide_: Divido A in **due parti**.\n 2. _Impera_: Metto **separatamente in ordine** le parti.\n 3. _Unisci_: **Unisco** le due parti.\n \n Consideriamo come **caso base** della ricorsione una parte composta da un numero, che ovviamente è già ordinata con sè stessa.\n \n ### Merge\n \n Per **unire le due parti** usiamo una funzione detta `merge()`.\n \n Costruiamo una nuova sequenza uguale alla sequenza 1, ma **aggiungiamo alla fine un valore sentinella** sempre maggiore di tutti gli elementi contenuti.\n \n > ```\n > | 1 | 3 | 7 | 8 | ∞ |\n > ```\n \n Facciamo **la stessa cosa** per la sequenza due.\n \n > ```\n > | 2 | 4 | 5 | 6 | ∞ |\n > ```\n \n Prendo i primi numeri delle due sequenze e **metto il più piccolo nella sequenza iniziale**.\n \n > ```\n > | 1 | 2 | 3 | | | | | | \n > | | | 7 | 8 | ∞ | \n > | | 4 | 5 | 6 | ∞ | \n > ```\n \n **Continuo** finchè non ho messo tutti i numeri; **grazie alla sentinella non usciremo mai dalla sequenza**, in quanto essa è sempre maggiore di tutti gli altri numeri.\n \n > ```\n > | 1 | 2 | 3 | 4 | 5 | 6 | | | \n > | | | 7 | 8 | ∞ | \n > | | | | | ∞ |\n > ```\n \n Quando **rimangono solo le sentinelle** significa che abbiamo aggiunto tutti gli elementi, e quindi abbiamo finito.\n \n > ```\n > | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | \n > | | | | | ∞ | \n > | | | | | ∞ | \n > ```\n \n \n ## Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | `O(n log n)` | `Ω(n log n)` | **`θ(n log n)`** |\n \n Il merge sort è un algoritmo ricorsivo con un **caso base in tempo costante** e che **richiama sè stesso 2 volte**.\n \n ```latex\n T(n) =\\\\\n \\\\\n Θ(1) \\qquad n=1\\\\\n 2 T(\\frac{n}{2}) + Θ(n) \\qquad n \\neq 1\n ```\n \n Applicando il **caso particolare del Master Theorem**, otteniamo:\n \n ```latex\n T(n) =\\\\\n \\\\\n Θ(1) \\qquad n=1\\\\\n Θ(n log n) \\qquad n \\neq 1\n ```\n \n ## Pseudocodice\n \n ```python\n def merge_sorted(part):\n # Caso base\n if len(part) == 1:\n return part\n # Divide\n middle = len(part) // 2\n part_a = part[:middle]\n part_b = part[middle:]\n # Impera\n sort_a = merge_sorted(part_a)\n sort_b = merge_sorted(part_b)\n # Combina\n return merge(sort_a, sort_b)\n ```\n \n ## Visualizzazione\n \n [hackerearth.com](https://www.hackerearth.com/practice/algorithms/sorting/merge-sort/visualize/)\n \n [visualgo.net](https://visualgo.net/bn/sorting) (Nota: visualizza solo la fase _Unisci_ del sort)\n ")),n(a.o,null,n(a.m,null,"\n # Quick sort\n \n Il _quick sort_ è un altro approccio **ricorsivo** all'ordinamento per confronto.\n \n ## Funzionamento\n \n Anche qui, applichiamo il **divide et impera**.\n \n 1. _Divide_: Scelgo un **pivot** qualsiasi all'interno della lista. Metto alla sua **sinistra tutti i numeri minori** e alla sua **destra tutti i numeri maggiori**.\n 2. _Impera_: Eseguo un **quick sort su entrambe le sottoliste**.\n \n ### Esempi\n \n #### Iterazione con partizioni bilanciate\n \n Osserviamo come si formi una partizione con tre elementi e una con quattro. \n \n ```\n |¦ [2] 8 7 1 3 5 6 {4}\n 2 |¦ [8] 7 1 3 5 6 {4}\n 2 |8 ¦ [7] 1 3 5 6 {4}\n 2 |8 ¦ [7] 1 3 5 6 {4}\n 2 |8 7 ¦ [1] 3 5 6 {4}\n 2 1 |7 8 ¦ [3] 5 6 {4}\n 2 1 3 |8 7 ¦ [5] 6 {4}\n 2 1 3 |8 7 5 ¦ [6] {4}\n 2 1 3 |8 7 5 6 ¦ [{4}]\n [2 1 3] {4} [8 5 6 7]\n ```\n \n #### Iterazione con partizioni sbilanciate\n \n Osserviamo come si formi una partizione con **zero elementi** e una con tre.\n \n ```\n |¦ [4] 7 3 {1}\n |4 ¦ [7] 3 {1}\n |4 7 |¦ [3] {1}\n |4 7 3 |¦ [{1}]\n [] {1} [4 7 3]\n ```\n \n ## Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | `O(n²)` | `Ω(n log n)` | - |\n \n Il costo della funzione è dato dalla somma del costo per **dividere in due partizioni** con il costo per realizzare il **Quick sort delle due sottopartizioni**\n \n Possiamo applicare allora il **Master Theorem generale**:\n \n ```latex\n T(n)\\\\\n =\\\\\n Θ(1) \\qquad per\\ n = 1\\\\\n T(q) + T(dim_lista - pivot - 1) + Θ(n) \\qquad per\\ n > 1\n ```\n \n ### Il caso migliore\n \n Se il pivot `q` è la **mediana della partizione** che stiamo ordinando, si vengono a creare due _sottopartizioni bilanciate_, e sostituendo otteniamo:\n \n ```latex\n T(n)\\\\\n =\\\\\n Θ(1) \\qquad per\\ n = 1\\\\\n 2 T(\\frac{n}{2}) + Θ(n) \\qquad per\\ n > 1\n ``` \n \n Possiamo allora applicare il **Master Theorem particolare**:\n \n ```latex\n T(n)\\\\\n =\\\\\n Θ(1) \\qquad per\\ n = 1\\\\\n Θ(n log n) \\qquad per\\ n > 1\n ```\n \n ### Il caso peggiore\n \n Se il pivot è uno degli **estremi dell'array**, si creano due _partizioni sbilanciate_: una delle due sottoliste è sempre vuota! \n Allora:\n \n ```latex\n T(n) = T(n-1) + Θ(n)\\\\\n = T(n-2) + Θ(n-1) + Θ(n)\\\\\n = T(n-3) + Θ(n-2) + Θ(n-1) + Θ(n)\\\\\n = …\n ∈ Θ(n^2)\n ```\n \n > \"Non date da mangiare sequenze ordinate al Quicksort, gli sono indigeste.\"\n \n ## Pseudocodice\n \n ```python\n def partition(partizione, inizio, fine):\n \"\"\"Dividi una partizione in due, usando l'ultimo elemento come pivot.\n \n Note utili:\n partizione[fine] è il pivot\n partizione[maggiori] è il primo numero dei maggiori\n partizione[non_iterati] è l'elemento su cui si sta iterando al momento\"\"\"\n # Crea il primo separatore (la | pipe nell'esempio)\n maggiori = inizio\n # Crea il secondo separatore (la ¦ broken pipe nell'esempio)\n non_iterati = inizio\n # Itera su ogni numero tra inizio e fine (escluso!)\n while non_iterati < fine:\n # Se l'elemento su cui stiamo iterando è minore del pivot\n if partizione[non_iterati] <= partizione[fine]:\n # Mettilo nell'insieme dei minori, scambiandolo con il primo numero dei maggiori e incrementando il primo separatore\n partizione[maggiori], partizione[non_iterati] = partizione[non_iterati], partizione[maggiori]\n maggiori += 1\n # Incrementa sempre il secondo separatore\n non_iterati += 1\n # Inserisci il pivot tra le due sottopartizioni create, \n partizione[fine], partizione[non_iterati] = partizione[non_iterati], partizione[fine]\n return maggiori\n ```\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/bn/sorting) (Nota: invece che prendere l'ultimo numero come pivot prende il primo, cambiando leggermente l'algoritmo.)\n \n ## Note per l'esame\n \n > La domanda che fa sempre è **\"Qual è la sequenza di pivot utilizzata?\"**\n \n > Elementi da soli _non_ vengono presi come pivot!\n ")),n(a.o,null,n(a.m,null,'\n # Counting sort\n\n Il _counting sort_ è un approccio diverso all\'ordinamento: **non usa il confronto**!\n \n ## Requisiti\n \n Il counting sort può essere utilizzato solo su **sequenze di numeri interi**, e solo se **siamo a conoscenza del minimo e del massimo** dei numeri contenuti nell\'array, ed essi non sono troppo distanti uno dall\'altro. \n (La memoria occupata dal counting sort aumenta linearmente con la differenza tra minimo e massimo!)\n \n Per semplicità, consideriamo il **minimo `0`**. \n L\'input allora sarà una sequenza di interi `A`, e il valore del **massimo `k`**, tale che `∀ n ∈ A, 0 \\leq n \\leq K`.\n \n ## Funzionamento\n \n Il counting sort **conta le ripetizioni** delle chiavi nella sequenza originale e in seguito **sovrascrive i valori** della sequenza con i valori ordinati ripetuti il numero di volte che sono stati individuati nella sequenza.\n \n > ```\n > 1 4 5 3 4 1 4 2 5 1\n > ```\n >\n > L\'`1` appare 3 volte, il `2` 1 volta, il `3` 1 volta, il `4` tre volte e il `5` due volte.\n >\n > La sequenza viene quindi così sovrascritta:\n > ```\n > 1 1 1 3 4 1 4 2 5 1 # Sovrascriviamo la sequenza con 1 ripetuto 3 volte\n > 1 1 1 2 3 4 4 4 5 1 # Sovrascriviamo la sequenza con 2, 3, 4 ripetuti rispettivamente 1 1 e 3 volte\n > 1 1 1 2 3 4 4 4 5 5 # Sovrascriviamo la sequenza con 5 ripetuto 2 volte: abbiamo finito!\n > ``` \n \n Esiste anche una **versione stabile** del counting sort che, invece che sovrascrivere, **sposta i valori**, mantenendo le informazioni aggiuntive nel caso invece che interi fossero altri tipi di dati.\n \n ## Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | `O(k + n)` | `Ω(k + n)` | **`θ(k + n)`** |\n \n L\'algoritmo è composto da quattro parti:\n - Ricerca del minimo e massimo (in `θ(n)`)\n - Inizializzazione dell\'indice (in `θ(k)`)\n - Conteggio dei numeri (in `θ(n)`)\n - Sovrascrittura dei numeri (in `θ(k + n)`)\n \n `2 + O(k) + O(n) + O(k + n) -> O(k + n)`\n \n Notiamo che `k` è costante, l\'algoritmo è `O(n)`, estremamente efficiente.\n \n ## Pseudocodice\n \n ```python\n def counting_sort(lista: typing.List[int]):\n """Ordina in-place una lista con il counting sort."""\n # Trovo la dimensione della lista\n dim = len(lista)\n # Trovo il massimo e il minimo all\'interno della lista\n minimo = min(lista)\n massimo = max(lista)\n # Creo l\'indice dei numeri, in modo che sia lungo k e pieno di 0\n indice = [0 for _ in range(minimo, massimo+1)]\n # Conto i numeri presenti, scorrendo su lista e aggiungendo 1 al numero corrispondente\n for i in range(dim):\n indice[lista[i]] += 1\n # Sovrascrivo i numeri nella lista\n count = 0\n for pos, val in enumerate(indice):\n for _ in range(val):\n indice[count] = pos\n count += 1\n \n def stable_counting_sorted(lista: typing.List[int], k: int) -> typing.List[int]:\n """Ordina stabilmente una lista con il counting sort stabile, e restituiscila."""\n # Trovo la dimensione della lista\n dim = len(lista)\n # Trovo il massimo e il minimo all\'interno della lista\n minimo = min(lista)\n massimo = max(lista)\n # Creo l\'indice dei numeri, in modo che sia lungo k e pieno di 0\n indice = [0 for _ in range(minimo, massimo+1)]\n # Conto i numeri presenti, scorrendo su lista e aggiungendo 1 al numero corrispondente\n for i in range(dim):\n indice[lista[i]] += 1\n # Faccio diventare l\'indice "il numero di numeri \\leq i"\n for i in range(len(indice)):\n if i == 0:\n continue\n indice[i] += indice[i-1]\n assert indice[-1] == dim\n # Creo una nuova lista, che sarà quella che verrà restituita\n nuova = [None for _ in range(dim)]\n # Inizio a posizionare i numeri, al contrario\n for i in range(0, dim, -1):\n nuova[indice[lista[i]]] = lista[i]\n indice[lista[i]] -= 1\n return nuova\n ```\n ')),n(a.o,null,n(a.m,null,"\n ## Introduzione alle Strutture Dati\n\n Una _struttura dati_ è un modo in cui si possono organizzare i dati di un programma.\n \n Si possono definire in due modi: **elementari** e **astratte**.\n \n ### Strutture dati elementari\n \n Le strutture _elementari_ dipendono strettamente dal modo in cui vengono memorizzati i dati.\n \n > **Array** e **liste** sono strutture dati elementari: sono definite dicendo come sono memorizzati i dati, rispettivamente, in celle contigue di memoria e da una serie di nodi con un valore e che puntano al successivo.\n \n ### Strutture dati astratte\n \n Le strutture _astratte_ sono separate dal modo in cui vengono memorizzati i dati, sono più ad alto livello, e si definiscono descrivendo le **proprietà** della struttura e i **metodi** che su di essa possono essere effettuate.\n \n > Una _classe_ in un qualsiasi linguaggio di programmazione è una struttura dati astratta.\n \n > Una _pila_ astratta:\n > - memorizza dati tutti dello stesso tipo\n > - `pop()`, estrae dalla pila l'ultimo valore inserito\n > - `push(val)`, aggiunge alla pila un valore\n > - `top()`, permette di vedere l'ultimo valore inserito nella pila\n > - `vuota()`, dice se la pila è vuota oppure no.\n ")),n(a.o,null,n(a.m,null,'\n # Array\n\n Un _array_ è sequenza di dati di **lunghezza conosciuta**, tutti dello **stesso tipo** e di una **dimensione fissa**, immagazzinata in **blocchi di memoria contigui**.\n \n ## Proprietà\n \n - E\' possibile accedere a tutti i blocchi di memoria conoscendo la loro **posizione relativa al primo** elemento.\n \n ## Metodi\n \n ```python\n class Array:\n def __init__(self, size, type=int): "Crea un array di dimensione size di elementi di tipo int."\n def __getitem__(self, index): "Restituisci il valore alla posizione index."\n def __setitem__(self, index, value): "Cambia il valore alla posizione index."\n ```\n \n ### Costo computazionale\n \n Tutte le operazioni su un array sono in **accesso immediato `O(1)`**!\n ')),n(a.o,null,n(a.m,null,'\n # Lista\n \n Una _lista_ è una sequenza di dati immagazzinata in **blocchi di memoria qualsiasi**.\n \n ## Proprietà\n \n Ogni dato ha un **riferimento** alla collocazione di memoria **successiva** (un puntatore): insieme, sono detti un _nodo_.\n \n E\' di **natura ricorsiva**: qualsiasi nodo di una lista può essere visto come inizio della lista con sè stesso e i suoi successivi.\n \n ## Metodi\n \n ```python\n class LinkedList:\n def __init__(self, value, next: typing.Optional[Node] = None):\n self.value = value\n self.next: typing.Optional[Node] = next\n \n def is_empty(self) -> bool: "Restituisce se la lista è vuota o no."\n def is_full(self) -> bool: "Restituisce se la lista è piena o no.\n def append(self, value): "Aggiunge un nuovo elemento in testa alla lista."\n def insert(self, value, index): "Inserisce un elemento dopo il nodo in posizione index."\n def insert_node(self, value, node): "Inserisce un nuovo elemento subito dopo un dato nodo."\n def find(self, node) -> int: "Trova l\'indice del nodo."\n def delete(self, value): "Elimina il primo nodo con quel valore dalla lista."\n def delete_node(self, node): "Elimina il nodo dalla lista."\n def forward(self, index) -> Node: "Restituisce il nodo in posizione index."\n ```\n \n ### Costo computazionale\n \n #### `List.forward(index)`\n \n Per raggiungere l\'`n`-esimo elemento, bisogna _scorrere tutti gli elementi prima di esso_: è dunque un **accesso sequenziale** in **`O(n)`**.\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/list)\n ')),n(a.o,null,n(a.m,null,'\n # Coda (Queue)\n\n Una _coda_ è come una pila, ma segue la strategia **First In, First Out** (il primo inserito sarà il primo a essere estratto).\n \n ## Proprietà\n \n - I dati vi possono essere aggiunti solo tramite il **metodo `enqueue()`**\n - I dati possono essere estratti solo tramite il **metodo `dequeue()`**\n - Verranno restituiti i valori inseriti secondo la strategia **First In, First Out** (il primo inserito sarà il primo a essere estratto).\n \n > Hai presente quando fai la fila per pagare al supermercato? Beh, è quello, però non si possono superare le altre persone in nessun modo.\n \n ## Metodi\n \n ```python\n class Queue:\n def __init__(self): "Crea una nuova coda."\n def is_empty(self) -> bool: "Restituisce vero se la coda è vuota."\n def enqueue(self, data): "Aggiunge un dato alla coda."\n def first(self) -> ...: "Restituisce il primo dato della coda."\n def dequeue(self) -> ...: "Restituisce il primo dato della coda e lo rimuove."\n ```\n \n ## Implementazione tramite lista\n \n Posso implementare la coda con una lista, ma per realizzare l\'implementazione più efficiente devo tenere anche un puntatore all\'ultimo elemento della coda, in modo da non doverla scorrere ogni volta che voglio effettuare un\'operazione.\n \n Chiamiamo i due puntatori `head` e `tail` rispettivamente.\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/list)\n ')),n(a.o,null,n(a.m,null,'\n # Pila (Stack)\n\n Una _pila_ è una struttura dati contenente **valori omogenei**.\n \n ## Proprietà\n \n - I dati vi possono essere aggiunti solo tramite il **metodo `push()`**\n - I dati possono essere estratti solo tramite il **metodo `pop()`**\n - Verranno restituiti i valori inseriti secondo la strategia **Last In, First Out** (l\'ultimo inserito sarà il primo a essere estratto).\n \n > Ci si può immaginare una pila di libri, da cui si può solo prendere un libro alla volta, quello più in alto.\n \n ## Metodi\n \n ```python\n class Stack:\n def __init__(self): "Crea una nuova pila."\n def is_empty(self) -> bool: "Restituisce vero se la pila è vuota."\n def push(self, data): "Aggiunge un dato alla pila."\n def top(self) -> ...: "Restituisce il primo dato della pila."\n def pop(self) -> ...: "Restituisce il primo dato della pila e lo rimuove."\n ```\n \n ## Implementazione tramite lista\n \n Utilizzando una lista possiamo realizzare una pila!\n \n La direzione dei puntatori sarà dall\'ultimo al primo, che non punterà più a nessuno.\n \n Il costo di tutti i metodi è `Θ(1)`!\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/list)\n ')),n(a.o,null,n(a.m,null,"\n # Albero radicato\n \n Un _albero radicato_ è una struttura dati di **natura ricorsiva** che organizza i dati in maniera **non-lineare**.\n \n ## Proprietà\n \n - Ogni nodo dell'albero ha un **unico genitore**: `∀ (padre, figlio), (padre' figlio) ∈ E \\implies padre = padre'`\n - Ogni nodo dell'albero può avere **un numero qualsiasi di figli**.\n \x3c!----\x3e\n - I **nodi superiori al padre** vengono chiamati _antenati_.\n - I **nodi inferiori ai figli** vengono chiamati _discendenti_.\n \x3c!----\x3e\n - Nodi **senza padre** sono detti _radice_: `\\notexists (padre, radice) ∈ E`\n - Nodi **con padre e figli** sono detti _rami_ o interni.\n - Nodi **senza figli** sono detti _foglie_.\n \x3c!----\x3e\n - La **distanza** tra il nodo radice e i suoi discendenti è detta _livello_:\n - I figli immediati sono di livello 1.\n - I \"nipoti\" (figli dei figli) sono di livello 2.\n - I figli dei nipoti sono livello 3.\n - E così via.\n - Il **livello massimo** all'interno di un albero è detto _altezza_, _profondità_ oppure _h_, ed è sempre `1 ≤ h ≤ n-1`.\n \x3c!----\x3e\n - Un albero ha sempre `n-1` archi.\n \n ## Alberi particolari\n \n ### Alberi `d`-ari\n \n Un albero _`d`-ario_ è un particolare tipo di albero che **limita il numero massimo di figli di un nodo** a `d`.\n \n > Un albero _binario_ può avere **massimo 2 figli** per ogni nodo; un albero _ternario_ ne può avere **3**; un albero _`17`-ario_ ne potrà avere **17**\n \n #### Alberi completi\n \n Un albero `d`-ario si dice _completo_ se **tutti i nodi hanno 0 o `d` figli**, e mai una numero in mezzo.\n \n #### Alberi bilanciati\n \n Un albero `d`-ario si dice _bilanciato_ se **tutti i livelli eccetto l'ultimo** hanno il numero massimo di figli.\n \n #### Alberi perfettamente bilanciati\n \n Un albero `d`-ario si dice _perfettamente bilanciato_ se **tutti i livelli incluso l'ultimo** hanno il numero massimo di figli.\n \n ##### Particolarità degli alberi binari perfettamente bilanciati\n \n Si può dimostrare per induzione che:\n - Hanno sempre `2^h` foglie.\n - Hanno sempre `2^{h+1}-1` (`\\sum_i=0^n 2^i`) nodi.\n - L'altezza è in `Θ(log n)`.\n \n ## Implementazione degli alberi\n \n Possiamo scegliere se usare una rappresentazione con array o con nodi e puntatori: ognuna ha vantaggi e svantaggi diversi.\n \n ### Implementazione tramite array\n \n E' suggerito se l'albero è regolare; più è simile a un albero d-ario completo, meglio è.\n \n ### Implementazione tramite nodi e puntatori\n \n Più adatta ad alberi irregolari.\n \n Se l'albero è regolare, creiamo il numero esatto di campi:\n \n - Valore\n - Figlio1\n - Figlio2\n - _Opzionale:_ Padre\n \n Se un albero è irregolare, creiamo una specie di lista:\n \n - Valore\n - Primo figlio\n - Prossimo fratello\n - _Opzionale:_ Padre\n ")),n(a.o,null,n(a.m,null,"\n #### Breadth-first search (BFS)\n\n La _breadth-first search_ è un algoritmo che visita **ogni livello** dell'albero in ordine, dal più basso al più alto.\n \n ## Funzionamento\n \n > 1. __Visita radice__\n > 2. __Visita figli__\n > 3. __Visita nipoti__\n > 4. __Visita pronipoti__\n \n Si può implementare con una coda, in cui verranno inseriti i figli del nodo visitato da visitare ed estratti dopo avere completato la visita del livello attuale.\n \n ## Pseudocodice \n \n ```python\n def bfs(radice):\n c = Queue()\n c.enqueue(radice)\n while not c.is_empty():\n nodo = c.dequeue()\n print(nodo)\n for figlio in nodo.figli:\n nodo.enqueue(figlio)\n ```\n ")),n(a.o,null,n(a.m,null,"\n # Depth-First Search\n\n La _depth-first search_ è un algoritmo che visita **tutti i sottoalberi di un figlio** prima di passare ad un altro sfruttando la natura ricorsiva degli alberi.\n \n ## Funzionamento\n \n Ci sono diverse versioni della depth-first search: ognuna visita la radice in un momento diverso.\n \n ### DFS previsita (pre-order)\n \n La _DFS pre-visita_ visita la **radice per prima**, poi tutti i sottoalberi formati dai figli uno dopo l'altro.\n \n > 1. __Visita radice__\n > 2. dfs_pre_order(figlio1)\n > 3. dfs_pre_order(figlio2)\n \n ### Postvisita (post-order)\n \n La _DFS postvisita_ visita prima tutti i sottoalberi dei figli, e **alla fine la radice**.\n \n > 1. dfs_post_order(figlio1)\n > 2. dfs_post_order(figlio2)\n > 3. __Visita radice__\n \n ### Invisita (in-order)\n \n La _DFS invisita_ visita **un certo numero di figli**, poi la radice, e infine i figli restanti.\n \n > 1. dfs_in_order(figlio1, 1)\n > 2. __Visita radice__\n > 3. dfs_in_order(figlio2, 1)\n ")),n(a.o,null,n(a.m,null,"\n # Albero binario di ricerca\n\n ## Proprietà\n \n - Albero **binario**\n - Chiavi appartenenti ad un **insieme totalmente ordinato** (N, Q, R, ma non C)\n \x3c!----\x3e\n - Per ogni nodo con valore `x`, se un valore `v` è nel sottoalbero di sinistra allora `v ≤ x`, mentre se è nel sottoalbero di destra allora `v > x`.\n \n ## Costo computazionale\n \n - Trovare un valore: `O(h)`\n - Ordinare i valori: `O(n)`\n - Trovare il minimo: `O(h)`\n - Trovare il massimo: `O(h)`\n - Inserire un elemento: `O(h)`\n - Cancellare un elemento: `O(h)`\n \n `h` vale `log n` in un albero perfettamente bilanciato, e più l'albero diventa sbilanciato, più si avvicina a `n`, raggiungendola nel caso l'albero sia una lista.\n \n ## Pseudocodice\n \n ### Cancellazione ricorsiva\n \n ```python\n def delete(tree, key):\n if tree is not None:\n # Se ho trovato il nodo che cercavo...\n if tree.key == key:\n # E c'è una sola diramazione...\n # Semplicemente stacca il nodo come in una lista.\n if tree.left is None:\n return tree.right\n if tree.right is None:\n return tree.left\n # Altrimenti, diventa il minimo dell'albero di destra\n tree.key = tree.right.min()\n # Ed eliminalo dal sottoalbero\n tree.right = delete(tree.right, tree.key)\n # Se la chiave attuale è diversa da quella che cerchiamo, continuo a navigare l'albero\n elif tree.key < key:\n tree.left = delete(tree.left, key)\n else:\n tree.right = delete(tree.right, key)\n return tree\n ```\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/bst)\n \n ## Approfondimenti\n \n Esistono alberi più avanzati che mantengono le proprietà degli alberi binari di ricerca, ma che si autobilanciano, come il [Red Black Tree](https://it.wikipedia.org/wiki/RB-Albero).\n ")),n(a.o,null,n(a.m,null,'\n # Heap binario\n \n L\'_heap binario_ è un **albero binario bilanciato a sinistra**.\n \n ## Proprietà\n \n - _Proprietà strutturale_:\n - L\'albero è **perfettamente bilanciato** in tutti i livelli tranne l\'ultimo\n - Nell\'ultimo livello, le foglie occupano le **posizioni più a sinistra** possibili\n - _Proprietà di ordinamento_:\n - La chiave di un qualsiasi nodo è **più piccola** di tutte quelle dei nodi **del suo sottoalbero**\n \n ## Metodi\n \n ```python\n class Heap:\n def __init__(self, H): ...\n def _heapify_ancestors(self, i): "Ripristina le proprietà dell\'heap per il nodo all\'indice specificato e i suoi genitori."\n def minimum(self): "Restituisce la chiave con il valore minimo in H."\n def decrease_value(self, index, new_value): "Diminuisce il valore della chiave all\'indice index a new_value."\n def insert(self, value): "Inserisci un nuovo valore nell\'albero."\n def _heapify_children(self, i): "Ripristina le proprietà dell\'heap per il nodo all\'indice specificato e i suoi figli."\n def pop(self): "Restituisce la chiave con il valore minimo, e la elimina."\n def from_list(l): "Crea un heap da una lista."\n ```\n \n ## Implementazione con un array\n \n Possiamo implementare l\'albero utilizzando un array con le chiavi dell\'albero memorizzate nell\'ordine _breadth-first_.\n \n L\'indice del figlio sinistro può essere trovato a `2i+1`, mentre l\'indice del figlio sinistro può essere trovato a `2i+2`; il genitore è a `i//2-1`.\n \n ### Pseudocodice\n \n ```python\n class Heap:\n def __init__(self, size):\n self.array = Array(size) # Il tipo Array non esiste; consideriamolo pseudocodice\n self.next_value = 0\n \n def _heapify_ancestors(self, index)\n """Ripristina le proprietà dell\'heap per il nodo all\'indice specificato e i suoi genitori.\n \n Costo:\n O(log n)"""\n # Trovo l\'indice del genitore\n parent = index // 2 - 1\n # Controllo se viene mantenuta la proprietà di ordinamento dell\'heap\n if self.array[index] < self.array[parent]:\n # Scambio i valori dei due nodi\n self.array[index], self.array[parent] = self.array[parent], self.array[index]\n # Faccio la stessa cosa con il genitore\n self._heapify(parent)\n \n \n def minimum(self):\n """Restituisce la chiave con il valore minimo in H.\n \n Costo:\n O(1)"""\n return self.array[0]\n \n def decrease_value(self, index, new_value):\n """Diminuisce il valore della chiave all\'indice index a new_value.\n \n Costo:\n O(log n)"""\n # Diminuisco il valore del nodo\n self.array[index] = new_value\n # Aggiorno l\'heap \n self._heapify(index)\n \n def insert(self, value):\n """Inserisci un nuovo valore nell\'albero.\n \n Costo:\n O(log n)"""\n # Trovo l\'indice in cui inserire il valore\n index = self.next_index \n # Aggiungo il valore in fondo\n self.array[index] = value\n # Aggiorno l\'heap\n self._heapify(index)\n \n def _heapify_children(self, index):\n """Ripristina le proprietà dell\'heap per il nodo all\'indice specificato e i suoi figli.\n \n Costo:\n O(log n)"""\n # Trovo l\'indice dei figli\n left = index * 2 + 1\n right = index * 2 + 2\n # Mi assicuro che i figli esistano\n try:\n # Guardo quale dei figli è maggiore\n if self.array[left] > self.array[right]:\n # Scambio i valori\n self.array[left], self.array[index] = self.array[index], self.array[left]\n # Ripeto la procedura sul figlio modificato\n self._heapify_children(left)\n else:\n # Scambio i valori\n self.array[right], self.array[index] = self.array[index], self.array[right]\n # Ripeto la procedura sul figlio modificato\n self._heapify_children(left)\n except IndexError:\n # La foglia non ha figli: ho finito!\n return\n \n def pop(self):\n """Restituisce la chiave con il valore minimo, e la elimina.\n \n Costo:\n O(log n)"""\n # Mi salvo il valore della radice\n value = self.array[0]\n # Sostituisco la radice con l\'ultima foglia a destra\n self.array[0], self.array[self.next_value] = self.array[self.next_value], self.array[0]\n ### Non bisognerebbe eliminare la foglia...?\n # Riordino l\'heap\n self._heapify_children(0)\n return value\n \n @staticmethod\n def from_list(l):\n """Crea un heap da una lista.\n \n Costo:\n O(n log n), ma si può abbassare"""\n heap = Heap(len(l)))\n heap.array = Array.from_list(l) # Pseudocodice\n heap.next_value = len(l)\n # Cominciamo a riordinare l\'heap dalla fine, in modo che rispetti le proprietà\n for index in range(heap.next_value, 0, -1):\n heap._heapify_children(index)\n ```\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/heap)\n ')),n(a.o,null,n(a.m,null,"\n # Coda con priorità\n \n La _coda con priorità_ è una struttura dati dal funzionamento molto simile a quello di una coda, ma invece che restituire il primo elemento inserito, essa restituisce l'**elemento con il valore di priorità minore**.\n \n ## Proprietà\n \n - Ogni elemento è una coppia costituita da **valore** e **priorità** (un numero intero).\n - Nuovi elementi possono essere aggiunti solo tramite il **metodo `insert()`**\n - Gli elementi possono essere estratti solo tramite il **metodo `pop()`**\n - Verranno restituiti i valori inseriti secondo la strategia **Lower Priority** (l'elemento con la priorità minore sarà il primo ad essere restituito).\n - E' possibile diminuire la priorità di un elemento (e quindi anticipare la sua estrazione)\n \n ## Metodi\n \n ```python\n class PriorityQueue:\n def __init__(self): ...\n def insert(self, new_elem: Element): ...\n def minimum(self) -> Element: ...\n def pop(self) -> Element: ...\n def decrease_priority_for(self, elem: Element, priority: int): ...\n ```\n \n ## Implementazione con lista\n \n E' possibile implementare la coda con priorità tramite una **lista**: l'inserimento di nuovi valori diventerà molto efficiente, ma tutte le altre operazioni saranno linearmente lente.\n \n ### Costo computazionale\n \n - *insert()*: `O(1)`\n - *minimum()*: `O(n)`\n - *pop()*: `O(n)`\n - *decrease_priority_for()*: `O(n)`\n \n ## Implementazione con lista ordinata\n \n Implementando la coda con priorità con una **lista ordinata** si avrà un costo di ordinamento elevato negli inserimenti e modifiche alla priorità, ma costi costanti nell'estrazione di un elemento.\n \n ### Costo computazionale\n \n - *insert()*: `O(n)`\n - *minimum()*: `O(1)`\n - *pop()*: `O(1)`\n - *decrease_priority_for()*: `O(n)`\n \n ## Implementazione con heap\n \n La **soluzione migliore** è quella di implementare la coda con priorità tramite un **heap**: tutti i costi saranno logaritmici, eccetto l'inserimento che sarà costante.\n \n - `insert()` costa `O(1)`\n - `minimum()` costa `O(log n)`\n - `pop()` costa `O(log n)`\n - `decrease_priority_for()` costa `O(log n)`\n \n ## Approfondimenti\n \n Esistono code con priorità che restituiscono **l'elemento con priorità maggiore**, invece che quello minore.\n ")),n(a.o,null,n(a.m,null,"\n # Heap sort\n\n L'_heap sort_ è un algoritmo di ordinamento per confronto **iterativo**.\n \n ## Funzionamento\n \n Per effettuare un heap sort, creiamo un **heap massimo** in cui inseriamo tutti i valori che vogliamo ordinare.\n \n Una volta applicate le proprietà dell'heap, chiamiamo una versione particolare di `heap.pop()` che invece che rimuovere dall'array i valori estratti li posiziona nello spazio creatosi in fondo.\n \n ## Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | `O(n log n)` | `Ω(n log n)` | **`θ(n log n)`** |\n ")),n(a.o,null,n(a.m,null,"\n # Grafo\n \n Un _grafo_ è una struttura dati che rappresenta elementi interconnessi tra loro.\n \n Esistono due tipi di grafi: _orientati_ e _non orientati_.\n \n Per semplicità, consideriamo i nostri nodi numerati da 1 a `n`.\n \n ## Proprietà\n \n - Gli elementi sono rappresentati tramite _nodi_.\n - Il loro _grado_ è dato dal **numero degli archi** che vi incidono.\n - Se il grafo è orientato, hanno anche un _in-degree_ (**numero di archi entranti**) e un _out-degree_ (**numero di archi uscenti**).\n - Le connessioni tra elementi sono rappresentate tramite _archi_.\n - Un arco _incide_ esattamente su **due nodi**.\n - Se il grafo è orientato, sono _uscenti_ da uno dei due nodi ed _entranti_ nell'altro.\n - Sono matematicamente meno del **quadrato dei nodi**.\n \n ## Grafi particolari\n \n ### Catena\n \n Una _catena_ è un **grafo non orientato** composto da una **sequenza di nodi** aventi un **grado massimo di 2** tutti collegati tra loro.\n \n ### Cammino\n \n Un _cammino_ è un **grafo orientato** composto da una **sequenza di nodi** aventi un **in-degree** e un **out-degree** **massimo di 1**, collegati tra loro in modo che partendo dal primo e seguendo gli archi sia possibile arrivare all'ultimo.\n \n ### Cricca\n \n Una _cricca_ è un grafo in cui **tutti i nodi sono collegati tra loro**.\n \n Se il grafo è **non orientato**, la cricca ha `((n-1)n)/2` archi.\n \n Se il grafo è **orientato**, ha per ogni coppia un arco in entrambe le direzioni, quindi ha `(n-1)n` archi.\n \n ### Direct Acyclic Graph\n \n Un _DAG_ è un grafo diretto che non contiene nessun ciclo.\n \n Su di esso possiamo effettuare un ordinamento, detto _linearizzazione_, tra i nodi: otteniamo l'_ordine topologico_.\n \n I primi elementi dei DAG sono detti _Source_ (_Sorgente_), mentre gli ultimi sono detti _Sink_ (_Pozzo_).\n \n #### Albero\n \n Un **albero** può essere considerato un DAG con una **sorgente singola** e le **foglie come pozzi**.\n \n ### Grafo fortemente connesso\n \n Un insieme di nodi `V` di un **grafo diretto** `G` si dice una _componente fortemente connessa_ se:\n \n 1. Per ogni coppia di nodi `∀ u, v ∈ V' : ∃ un cammino u->v in G'`\n 2. Massimale (non può diventare più grande)\n \n > Praticamente una componente fortemente connessa è un gruppo di nodi tra i quali si può viaggiare liberamente da e a qualsiasi nodo al suo interno.\n \n Un grafo si dice _fortemente connesso_ se l'insieme `V` coincide con l'insieme dei nodi del grafo `G`.\n \n > Se partendo da qualsiasi nodo di un grafo riesco ad arrivare a qualsiasi altro nodo, allora il grafo è fortemente connesso.\n \n Inoltre, se creiamo un nuovo grafo, in cui **ogni nodo rappresenta una componente fortemente connessa** del nostro grafo iniziale, **otteniamo un DAG**, perchè tutti i cicli sono stati integrati nella componente.\n \n ### Trasposto di un grafo\n \n Il _trasposto_ di un **grafo diretto** `G` è il grafo stesso con gli archi che però vanno nella **direzione opposta**.\n \n ### Grafo pesato\n \n Un _grafo pesato_ è un particolare grafo che associa a ciascun arco un **costo** per attraversarlo.\n \n #### Costi negativi\n \n I costi possono anche essere negativi: rappresenteranno allora un **guadagno** ottenuto attraversando il nodo.\n \n ### Minimum spanning tree\n \n Un _minimum spanning tree_ è il **sottoinsieme degli archi** di un **grafo non diretto** che **connettono tutti i nodi** con il **minor costo possibile**.\n \n I MST hanno [molte proprietà](https://en.wikipedia.org/wiki/Minimum_spanning_tree#Properties); sono troppe da scrivere qui, e probabilmente non ci interesseranno nemmeno.\n \n ## Implementazione tramite matrice di adiacenza\n \n Possiamo implementare un grafo creando una **matrice di `bool` di dimensione `n * n`** in cui le **caselle collegate sono vere** e le caselle non collegate sono false.\n \n > Ad esempio, possiamo implementare un grafo non orientato in questo modo (`█` indica l'esistenza di un collegamento e ` ` indica la sua assenza):\n >\n > | |1|2|3|\n > |-|-|-|-|\n > |1|░|░|░|\n > |2|█|░|░|\n > |3|█| |░|\n >\n > Esistono gli archi `1-2` e `1-3`, ma non esiste un collegamento `2-3`.\n \n > Un grafo orientato invece si può implementare così:\n >\n > | |1|2|3|\n > |-|-|-|-|\n > |1|░|█| |\n > |2|█|░| |\n > |3|█| |░|\n >\n > Esistono gli archi `1->2`, `2->1` e `3->1`, ma non ci sono collegamenti `2->3`, `1->3` e `3->2`.\n \n ### Costo computazionale\n \n #### Tempo\n \n Le matrici di adiacenza portano alla realizzazione di algoritmi molto veloci: verificare l'esistenza di un arco è in `O(1)`!\n \n Abbiamo però penalità significative quando vogliamo effettuare operazioni sugli archi: ad esempio, trovare il trasposto di un grafo implementato con una matrice di adiacenza è in `O(nodi²)`.\n \n #### Memoria\n \n E' poco efficiente in quanto a memoria: l'upper bound è in `O(n^2)`. \n \n ## Implementazione tramite liste di adiacenza\n \n Un'alternativa alla matrice di adiacenza è quella di creare un'**array di liste**, le quali contengono i **vicini di ciascun nodo**.\n \n > |Posizione|Lista|\n > |-|-|\n > |1|[2, 3]|\n > |2|[]|\n > |3|[1]|\n >\n > Esistono gli archi `1->2`, `1->3`, e `3->1`, ma non esistono `2->1`, `2->3` e `3->2`.\n \n ### Costo computazionale\n \n #### Tempo\n \n Utilizzando le liste di adiacenza, il tempo richiesto per verificare l'esistenza di un arco sale a `O(max-out-degree)`.\n \n E' efficace però quando il problema che vogliamo risolvere riguarda operazioni su archi: trovare la trasposta è in `O(archi)`.\n \n #### Memoria \n \n La memoria richiesta dalle liste di adiacenza è minore di quella delle matrici: l'upper bound è in `O(nodi + archi)`.\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/graphds)\n ")),n(a.o,null,n(a.m,null,"\n # Visitare un grafo\n \n Come per gli alberi radicati, esistono due modi per visitare un grafo: _depth-first search_ e _breadth-first search_.\n \n In entrambi i casi, non visito mai due volte lo stesso nodo, e come risultato ottengo **molteplici alberi**, il cui insieme viene detto _foresta di copertura_.\n \n Se il grafo che vogliamo visitare è diretto, allora dobbiamo **considerare come vicini solo gli archi uscenti**\n \n ## Depth-first search\n \n La DFS ci può risultare utile per **identificare le componenti connesse** di un grafo e **identificare eventuali cicli**.\n \n ### Funzionamento\n \n Posso utilizzare la DFS per classificare gli archi di un grafo in quattro categorie:\n \n - _Tree_, archi che ci fanno **scoprire un nuovo nodo**\n - _Forward_, archi che ci portano a un **discendente**\n - _Back_, archi che ci portano ad un **antenato**\n - _Cross_, archi che **connettono due sottoalberi** diversi\n \n Usiamo due array inizializzati a 0 chiamati `pre` e `post`, grandi quanto il numero di archi del grafo, che ci indicano rispettivamente quando un nodo è stato scoperto e quando è terminata la visita. \n Inoltre, creiamo una variabile `clock` che avanza ad ogni evento. \n Alla scoperta di un nuovo nodo, mettiamo il valore attuale di `clock` all'interno di `pre[n]`. \n Alla fine della visita di un nodo invece mettiamo il valore di `clock` in `post[n]`.\n \n **Durante la visita**, gli archi avranno i seguenti valori:\n - _Tree_: `pre[dst] == 0`\n - _Forward_: `pre[src] < pre[dst] && post[dst] > 0`\n - _Back_: `pre[dst] < pre[src] && post[dst] == 0`\n - _Cross_: Tutti gli altri (`post[dst] < pre[src]`)\n \n **A fine visita**, gli archi avranno i seguenti valori:\n - _Tree_: `pre[dst] < pre[dst] < post[dst] < pre[src]`\n - _Forward_: `pre[dst] < pre[dst] < post[dst] < pre[src]`\n - _Back_: `pre[src] < pre[dst] < post[dst] < post[src]`\n - _Cross_: `pre[dst] < post[dst] < pre[src] < post[src]`\n \n Se un **grafo non diretto** contiene degli **archi Back**, allora esso **conterrà un ciclo**.\n \n #### DFS nel grafo trasposto\n \n Se effettuo una DFS sul trasposto di un grafo, posso **scoprire i nodi che hanno un cammino verso l'origine**.\n \n #### DFS nella componente fortemente connessa\n \n Se effettuo una DFS in una componente fortemente connessa e nella sua trasposta, il **`post` della trasposta sarà sempre minore** del `post` della componente originale.\n \n ### Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | `O(nodi + archi)` | `Ω(nodi + archi)` | **`θ(nodi + archi)`** |\n | Memoria | `O(nodi)` | `Ω(nodi)` | **`θ(nodi)`** |\n \n ### Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/dfsbfs)\n \n ## Breadth-first search\n \n La BFS ci può risultare utile per **trovare tutti i nodi a una certa distanza** da un'origine.\n \n ### Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | `O(nodi + archi)` | `Ω(nodi + archi)` | **`θ(nodi + archi)`** |\n | Memoria | `O(nodi + archi)` | `Ω(nodi + archi)` | **`θ(nodi + archi)`** |\n \n ### Pseudocodice\n \n Come per gli alberi, la implementiamo in modo **iterativo**:\n \n ```python\n queue = [starting_node]\n parents = [None for node in graph.nodes]\n distance = [-1 for node in graph.nodes]\n \n # TODO: controllami quando sei più sveglio\n \n while queue:\n node, source, distance = queue.pop(0)\n parents[node.number] = source\n distance[node.number] = distance\n for neighbour in node.neighbours:\n queue.append((neighbour, node, distance+1))\n ```\n \n > Nella coda, la distanza massima tra un nodo e l'altro è 1.\n \n ### Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/dfsbfs)\n ")),n(a.o,null,n(a.m,null,"\n # Algoritmi greedy\n\n Un modo per risolvere problemi algoritmici può essere usare una **tecnica** _greedy_.\n \n Le tecniche greedy consistono nel effettuare tanti piccoli passi, ed effettuare una **scelta** in base ai dati **locali al passo attuale**.\n \n > Scegli il numero di monete più piccole possibili per comporre €1.12.\n >\n > L'algoritmo cerca di scegliere sempre la moneta più grande possibile compatibile con il prezzo in quel momento, quindi:\n > | Moneta scelta | Rimanente |\n > |---------------|-----------|\n > | € 1.00 | € 0.12 |\n > | € 0.10 | € 0.02 |\n > | € 0.02 | € 0.00 |\n \n ## Esempi\n \n Sono algoritmi greedy:\n \n - L'_Algoritmo di Dijkstra_\n - L'_Algoritmo di Kruskal_\n - L'_Algoritmo di Prim_\n ")),n(a.o,null,n(a.m,null,"\n # Percorso più breve\n\n Trovare il _percorso più breve_ (o _cammino minimo_) tra due nodi di un **grafo pesato** è un problema frequente nell'informatica; per questo, sono stati sviluppati [numerosi algoritmi](https://en.wikipedia.org/wiki/Shortest_path_problem) per risolverlo.\n \n > Ad esempio, il pathfinding dei nemici nei videogiochi, oppure Google Maps!\n \n ## Percorso più breve da una sorgente singola\n \n Una sottocategoria del problema del percorso più breve è il caso in cui ci interessa sapere i percorsi più brevi che **partono da uno specifico nodo del grafo**: è detto problema del _percorso più breve da una sorgente singola_, o _single-source shortest path_.\n \n Si può notare che se il grafo contiene **costi negativi** allora è possibile che il percorso più breve non esista, in quanto diventa possibile la comparsa di **cicli di costo infinitamente negativo**.\n \n Possiamo notare che, se il percorso più breve tra `A` e `D` è `A-B-C-D`, allora il cammino minimo tra `B` e `D` passerà obbligatoriamente per `C` (`B-C-D`).\n Diremo più avanti che il percorso più breve ha una **sottostruttura ottimale**.\n \n ### Esempi\n \n Alcuni algoritmi che trovano il percorso più breve sono:\n - L'_Algoritmo di Dijkstra_\n - L'_Algoritmo di Bellman-Ford_\n - La [_ricerca A*_](https://en.wikipedia.org/wiki/A*_search_algorithm)\n ")),n(a.o,null,n(a.m,null,"\n # Algoritmo di Dijkstra\n \n L'_Algoritmo di [Dijkstra](https://upload.wikimedia.org/wikipedia/commons/8/85/Dijkstra.ogg)_ è un algoritmo che risolve il problema del **percorso più breve da una sorgente singola** per grafi con pesi **reali positivi** `\\mathbb{R}^+`.\n \n L'algoritmo trova tutti i percorsi più brevi per raggiungere qualsiasi nodo del grafo partendo da un dato nodo, assieme al costo richiesto per farlo.\n \n ## Funzionamento\n \n 1. Separiamo tutti i nodi del grafo in due gruppi: **visitati** e **non visitati**. \n - Tutti i nodi partono da **non visitati**.\n 2. Per ogni nodo, manteniamo un valore \"**costo richiesto per raggiungerlo**\", che verrà cambiato man mano che l'algoritmo avanza.\n - Il costo di partenza è `+∞`.\n - Il costo sarà **definitivo per i nodi visitati**, e **provvisorio per i non visitati**.\n 3. Creiamo un insieme detto _frontiera_ che conterrà tutti i **nodi non visitati adiacenti** a quelli visitati.\n 4. Prendiamo il nodo iniziale, che avrà un **costo di `0`**, e definiamolo il nodo _attuale_.\n 5. Finchè ci sono dei nodi non sono stati visitati, ripetiamo il seguente ciclo:\n 1. Aggiungiamo i nodi adiacenti al nodo attuale alla frontiera.\n - Il costo per raggiungerli sarà il **costo per il nodo attuale sommato al costo dell'arco** che li connette al nodo attuale. \n Se questo **costo** risulta essere **minore del costo provvisorio** precedente, esso **diventerà il nuovo costo**.\n - Questa operazione è detta _rilassamento dell'arco_.\n 2. Facciamo diventare **visitato** il nodo attuale.\n - Il percorso che abbiamo fatto per raggiungerlo è obbligatoriamente il più breve.\n 3. Il prossimo nodo attuale sarà il nodo di frontiera con un costo più basso.\n - Per questo, è possibile definire l'algoritmo di Dijkstra come un **algoritmo greedy**.\n \n ### Non funziona se...\n \n L'algoritmo smette di funzionare nel caso in cui siano presenti **costi negativi** e il grafo non sia **aciclico**, in quanto non saremmo mai in grado di rendere visitato un nodo.\n \n ## Costo computazionale\n \n | Categoria | Upper bound |\n |-----------|-------------|\n | Tempo | `O(nodi + archi) log nodi)` |\n \n ### Scomposizione\n \n - Inizializzazione: `O(nodi)`\n - Creazione coda priorità: `O(nodi log nodi)`\n - Ciclo: `O((nodi + archi) log nodi)`\n \n ## Pseudocodice\n \n ```python\n import math\n \n class Info:\n def __init__(self, distance=math.inf, previous=None):\n self.distance = distance\n self.previous = previous\n \n def dijkstra(graph, start):\n data = [Info() for node in graph.nodes]\n queue = PriorityQueue([start])\n while queue:\n node = queue.pop()\n for arc in node.connections:\n other = arc.other(node)\n if data[node.number].distance + arc.cost < data[other].distance:\n data[other].distance = data[node.number].distance + arc.cost\n queue.decrease_priority_for(other, data[other].distance)\n data[v].previous = node\n return data\n ```\n \n ## Visualizzazione\n \n [Visualgo](https://visualgo.net/en/sssp)\n ")),n(a.o,null,n(a.m,null,"\n # Algoritmo di Bellman-Ford\n\n L'_Algoritmo di Bellman-Ford_ è un algoritmo che, come l'Algoritmo di Dijkstra, risolve il problema del **percorso più breve da una sorgente singola**, però, a differenza da quest'ultimo, l'Algoritmo di Bellman-Ford accetta in input anche grafi con pesi **reali** `\\mathbb{R}` (sia positivi, sia negativi).\n \n ## Funzionamento\n \n L'approccio dell'algoritmo è simile a quello di Dijkstra: entrambi usano il **rilassamento** degli archi per ottenere un costo provvisorio per il raggiungimento di un nodo, ma invece che rilassare solo l'arco con costo inferiore, questo algoritmo **rilassa tutti gli archi** ripetutamente, eliminando la frontiera e il problema dei nodi negativi.\n \n L'operazione di rilassamento è ripetuta `nodi - 1` volte, ovvero la **lunghezza massima** di un cammino aciclico all'interno di un grafo.\n \n Possiamo individuare dopo i rilassamenti se è presente un nodo con un **ciclo negativo**: ci basta controllare se esiste un arco che connette due nodi con una distanza incompatibile: se `a.distanza + arco.costo < b.distanza`, allora è presente un ciclo negativo.\n \n ## Costo computazionale\n \n | Categoria | Upper bound |\n |-----------|-------------|\n | Tempo | `O(nodi * archi)` |\n \n ## Pseudocodice\n \n > TODO\n \n ## Visualizzazione\n \n [Visualgo](https://visualgo.net/en/sssp)\n ")),n(a.o,null,n(a.m,null,'\n # Disjoint set\n \n Il _disjoint set_ è una struttura dati che rappresenta elementi di un insieme raggruppati in **sottoinsiemi disgiunti**.\n \n ## Metodi\n \n ```python\n class DisjointNode:\n def __init__(self): ...\n def find_set(self): "Trova il rappresentante dell\'elemento."\n def union(self, other): "Unisce i sottoinsiemi che contengono questo nodo e `other`."\n ```\n \n ## Implementazione tramite array\n \n Possiamo implementare il disjoint set con due array: uno per l\'**indice del rappresentante** e uno per il **rango dell\'insieme**.\n \n Un singoletto avrà **sè stesso come rappresentante** e **rango `0`**.\n \n ### Costo computazionale\n \n - *create_set()*: `O(1)`\n - *find_set()*: `O(h)`\n - *union()*: `O(h)`\n \n ### Pseudocodice\n \n ```python\n class DisjointNode:\n def __init__(self):\n self.parent = self\n self.rank = 0\n \n def find_set(self):\n element = self\n while self.parent != element:\n element = self.parent\n return element\n \n def union(self, other):\n repres_self = self.find_set()\n repres_other = other.find_set()\n if repres_self == repres_other:\n return\n if repres_self.rank < repres_other.rank:\n repres_greater = repres_other\n repres_lesser = repres_self\n else:\n repres_greater = repres_self\n repres_lesser = repres_other\n repres_lesser.parent = repres_greater\n if repres_greater.rank == repres_lesser.rank:\n repres_greater.rank += 1\n ```\n \n ### Visualizzazione\n \n [cs.usfca.edu](https://www.cs.usfca.edu/~galles/JavascriptVisual/DisjointSets.html)\n ')),n(a.o,null,n(a.m,null,"\n # Trovare il minimum spanning tree\n\n Un altro problema ricorrente riguardante i grafi è trovare il _minimum spanning tree_ di un dato grafo non diretto.\n \n > E' utile per trovare il modo più efficiente per connettere le cose: ad esempio, per decidere la struttura di una rete internet!\n \n ## Esempi\n \n Gli algoritmi principali che risolvono il problema sono due, ed entrambi sono **algoritmi greedy**:\n \n - L'_Algoritmo di Kruskal_\n - L'_Algoritmo di Prim_\n ")),n(a.o,null,n(a.m,null,"\n # Algoritmo di Kruskal\n \n L'_Algoritmo di Kruskal_ è un algoritmo **greedy** che **trova il minimum spanning tree** di un grafo.\n \n ## Funzionamento\n \n 1. Ripetiamo questa procedura finchè tutti i nodi non sono connessi:\n 1. Prendiamo ad ogni passo **l'arco meno costoso** del grafo non ancora aggiunto all'insieme.\n 2. Assicuriamoci che **non si creino cicli**: se non se ne verrebbero a creare, possiamo **aggiungere l'arco all'insieme**.\n - Gli archi devono quindi connettere nodi in **componenti connesse diverse**.\n - Possiamo rappresentare le componenti connesse con un **Disjoint Set**.\n \n ## Costo computazionale\n \n | Categoria | Upper bound |\n |-----------|-------------|\n | Tempo | `O(archi²)` |\n \n ### Scomposizione\n \n - DisjointSet.__init__(): `O(archi)`\n - Per ogni ciclo: `O(archi²)`\n - DisjointSet.find_set(): `O(1)`\n - DisjointSet.union(): `O(archi)`\n \n ## Pseudocodice\n \n ```python\n def minimum_spanning_tree_kruskal(graph):\n ds = DisjointSet()\n for node in graph.nodes:\n ds.create_set(node)\n arcs = []\n sorted_arcs = sorted(graph.arcs, key=lambda arc: arc.cost)\n for arc in sorted_arcs:\n if ds.find_set(node.start) != ds.find_set(node.end):\n arcs.append(arc)\n ds.union(node.start, node.end)\n return arcs\n ```\n ")),n(a.o,null,n(a.m,null,"\n # Algoritmo di Prim\n \n L'_Algoritmo di Prim_ è un altro algoritmo **greedy** che **trova il minimum spanning tree** di un grafo.\n \n ## Funzionamento\n \n Creo una **coda con priorità** in cui inserisco tutti gli archi visibili dal mio albero, in cui la chiave è il **costo dell'arco**.\n \n Per trovare l'arco con costo più piccolo posso **estrarre un arco** dalla coda: la priorità ci garantisce che esso è l'**arco meno costoso**.\n \n Aggiungo allora un nuovo nodo all'albero, e con esso, **aggiungo alla coda** tutti gli **archi che scoprono un nuovo nodo**.\n \n ## Costo computazionale\n \n | Categoria | Upper bound |\n |-----------|-------------|\n | Tempo | `O(archi + nodi log nodi)` |\n \n ## Pseudocodice\n \n ```python\n import math\n \n def minimum_spanning_tree_prim(graph, cost_array, start_node):\n # E' un Array di bool: se l'indice corrispondente al nodo è uguale a true, vuol dire che il (nodo è contenuto nell'albero.\n contains = [False for _ in range(len(graph))]\n # Contiene il precedente di ogni nodo \n prev = [None for _ in range(len(graph))]\n # Contiene il costo per arrivare a quel nodo\n cost = [math.inf for _ in range(len(graph))]\n # Creo la priority queue\n pq = PriorityQueue(graph.arcs, key=lambda arc: arc.cost)\n # Parto dal nodo `start_index`\n # Il costo dell'origine è 0.\n cost[start_node.index] = 0\n contains[start_node.index] = True\n \n while not pq.is_empty():\n new_node = pq.pop()\n contains[new_node.index] = True\n for arc in new_node.connections:\n other_node = arc.other(new_node)\n if not contains[other_node.index] and cost[other_node.index] > arc.cost:\n cost[other_node.index] = arc.cost\n prev[other_node.index] = new_node\n pq.decrease_priority_for(other_node, arc.cost)\n # L'array di prev rappresenta un albero.\n return prev\n ```_\n ")),n(a.o,null,n(a.m,null,"\n # Compressione\n\n _Comprimere_ un file significa **ridurne le dimensioni** senza modificarne il significato.\n \n ## Categorie\n \n ### Compressione lossless\n \n Nella _compressione lossless_, i dati possono essere decompressi riottenendo una copia identica dell'originale.\n \n > Immaginiamo un file che contiene solo le lettere `a, b, c, d, e, f`.\n >\n > Le lettere compaiono con questa frequenza:\n > |a|b|c|d|e|f|\n > |45%|13%|12%|16%|9%|5%|\n >\n > Possiamo codificare le lettere nel seguente modo:\n > |a|b|c|d|e|f|\n > |`0b0`|`0b100`|`0b101`|`0b111`|`0b1100`|`0b1101`|\n >\n > Scrivere `abacadae` richiederebbe 64 bits con la codifica ASCII estesa, ma in questo modo riusciamo a scriverlo con soli 17 bits!\n \n Le codifiche di un file compresso devono rispettare la proprietà del _Codice a prefisso_, che dice che **nessun codice deve essere il prefisso di un altro codice**; altrimenti, si avrebbero ambiguità nella decodifica.\n \n > a = `1`\n > b = `11`\n >\n > `111` è `ab`, `ba` oppure `aaa`?\n \n Creiamo allora un _albero di decodifica_: un **albero binario** che, leggendo uno ad uno i bit codificati, ci permette di arrivare al **valore del codice presente sulle foglie** dell'albero.\n \n Gli alberi di decodifica sono sempre **completi**.\n \n > Un albero di decodifica incompleto sarebbe non ottimizzato!\n \n #### Esempi\n \n - .png\n - .flac\n - .zip\n - ...\n \n ### Compressione lossy\n \n Nella _compressione lossy_, alcuni dati [solitamente](http://needsmorejpeg.com/) irrilevanti vengono perduti: non si può, dunque, ricostruire l'originale.\n \n #### Esempi\n \n - .jpeg\n - .mp3\n - ...\n ")),n(a.o,null,n(a.m,null,"\n # Algoritmo di Huffman\n\n L'_Algoritmo di Huffman_ è un **algoritmo greedy** per la **costruzione di un albero di decodifica**.\n \n ## Funzionamento\n \n 1. Costruisco **un albero** (con un solo nodo) **per ogni elemento dell'alfabeto**.\n 2. Associo ad **ogni albero la frequenza dell'elemento** da cui è stato creato, per poi inserire tutti gli elementi in una coda con priorità.\n 3. Finchè non ho **un albero solo**:\n 1. Estraggo dalla coda i **due alberi con frequenza minore**.\n 2. **Li rendo fratelli**, creando un nuovo nodo in cui sono uno figlio destro e uno figlio sinistro.\n 3. Associo al nuovo nodo la **somma delle frequenze dei due alberi**, e inserisco il nuovo albero nella coda.\n \n > È molto raro che venga un albero \"dritto\"; se succede, probabilmente c'è qualcosa che non va.\n ")),n(a.o,null,n(a.m,null,"\n # Dizionario\n \n Un _dizionario_ è una struttura dati che **associa dei valori a delle chiavi**.\n \n ## Proprietà\n \n - Ogni elemento del dizionario è un **valore** che è stato **associato a una chiave**.\n - Possiamo aggiungere nuovi elementi con il **metodo `add(chiave, valore)`**.\n - Posiamo estrarre elementi con il **metodo `get(chiave)`**, che restituirà il valore associato a `chiave`.\n - E' possibile rimuovere elementi con il **metodo `delete(chiave)`**.\n - Due elementi con **chiavi diverse** non devono **mai restituire lo stesso valore**.\n \n ## Metodi\n \n Beh... Non ha molto senso in questo caso...\n \n ```python\n dict()\n ```\n \n ## Implementazione con tabella hash\n \n Una _tabella hash_ è un'array di coppie **chiave-valore**, che formano l'insieme _universo_.\n \n Per determinare l'**indice dell'array** in cui inserire una coppia, usiamo una [funzione _hash_](https://it.wikipedia.org/wiki/Hash#Algoritmo_di_hash) sulla chiave, che restituirà **numeri da `0` a `len(hash_table)`**.\n \n ### Risoluzione collisioni\n \n Potrebbe capitare però che **due chiavi diverse abbiano lo stesso indice**. Dobbiamo allora usare un metodo di _risoluzione collisioni_, che mi permetta di distinguere tra chiavi diverse.\n \n #### Lista di trabocco\n \n Possiamo salvare nell'array **liste di coppie** chiave-valore; in caso di collisione, **aggiungo un nuovo elemento alla lista**.\n \n In media, ciascuna di queste liste conterrà `elementi_inseriti / dimensione_tabella` elementi.\n \n #### Indirizzamento aperto\n \n Possiamo decidere di mettere le coppie che non trovano posto nel loro indice in **un altro indirizzo vuoto** dell'array.\n \n Ci sono diversi modi in cui decidere il nuovo indirizzo, ognuno con vantaggi e svantaggi: si può scegliere quello successivo, oppure il primo vuoto dell'array, oppure un indirizzo casuale.\n \n > Python, nei `dict`, usa indirizzamento aperto pseudorandom.\n \n ### Costo computazionale\n \n - Aggiungere una chiave: `O(n)`\n - Trovare una chiave: `O(n)`\n - Eliminare una chiave: `O(n)`\n ")),n(a.o,null,n(a.m,null,"\n # Programmazione dinamica\n \n La _programmazione dinamica_ è una **tecnica** di programmazione che prevede l'**estensione di una soluzione ottima precedente**.\n \n Tutti i problemi in cui si può applicare si possono risolvere anche con la **ricorsione**, ma a differenza della ricorsione, questa tecnica riesce ad evitare di ricalcolare la soluzione per ogni chiamata ricorsiva, ottenendo quindi tempi molto migliori.\n \n Si può applicare solo se un problema ha una **sottostruttura ottimale**, ovvero se la soluzione ottima di un sottoproblema è inclusa nella soluzione ottima del problema. \n \n ## Esempi\n \n - _Problema dello zaino_\n - ...\n \n > Il cammino minimo per raggiungere un nodo in un DAG è dato da `arco.costo + arco.primo_nodo.costo_cammino_minimo()`.\n >\n > ```python\n > def SPD_PD(graph, start):\n > distance = [float(inf) for node in graph.nodes]:\n > distance[start] = 0\n > # I nodi devono essere in ordine di linearizzazione\n > for node in graph.nodes:\n > distance[node] = min([(arc.cost + distance[arc.other(node)] for arc in node.connections])\n > ```\n \n > Ho una sequenza di interi da `a_1` a `a_n`. Voglio trovare la sottosequenza crescente più lunga.\n >\n > 5 2 3 4 7 3 6 3 1 6\n >\n > Trovo tutte le sequenze lunghe 1, e le rendo nodi di un grafo diretto.\n >\n > Da ogni nodo, creo una connessione verso i suoi maggiori.\n >\n > Infine, cerco i cammini massimi del grafo. \n > \n > Essi saranno la soluzione del problema.\n \n > Trova la lunghezza della sottosequenza più lunga che termina con `j`.\n >\n > `L[j]` = lunghezza della sottosequenza più lunga che termina in `j`\n > \n > ```python\n > L[j] = max([1 + L[node] for arc in node.connections)]\n > ```\n >\n > Esempio:\n > ```python\n > L[9] = max([1+L[8], 1+L[3], 1+L[6]])\n > ```\n ")),n(a.o,null,n(a.m,null,"\n # Problema dello zaino\n \n Il problema dello zaino è un problema _pseudo-trattabile_: non abbiamo dimostrazioni di se sia trattabile o intrattabile.\n \n ## Descrizione\n \n > Sei un ladro, e devi mettere **più refurtiva possibile** nello zaino per scappare. \n > Lo zaino può portare **al massimo `dim` kili**.\n >\n > **Quali** (e quanti) oggetti scegli?\n \n | Input | Output |\n |-------|--------|\n | `dim`ensione_zaino, `n`umero_oggetti, `oggetto.peso`, `oggetto.valore` | `profitto_massimo` |\n \n ## Categorie\n \n ### Problema con ripetizione\n \n Puoi prendere **tutte le copie che vuoi** di un oggetto.\n \n #### Soluzione\n \n `K(dim)` è il valore massimo ottenibile con uno zaino di capacità `dim`.\n \n > Se `i` appartenesse alla soluzione ottima, allora `K(dim) = i.valore + K(dim - i.peso)`...\n \n Possiamo dire che `K(dim) = max(i.valore + K(dim - i.peso))`.\n \n Inoltre, `K(0) = 0`.\n \n Ci salviamo tutte le soluzioni da `K(0)` a `K(dim)`, e le usiamo per calcolare il massimo in seguito.\n \n Calcolare `K(dim)` avrà allora un costo di `O(n * dim)`:\n - `n`, perchè trovare il massimo è un'operazione lineare\n - `dim`, perchè `dim` sono tutti i casi tra i quali devo andare a provare\n \n Il costo computazionale, allora, è in `O(n * dim)`. \n \n Però, il **tempo richiesto** dal nostro algoritmo dipende non dalla lunghezza dell'input, bensì dal **valore numerico** di `dim`, che corrisponde alla dimensione dell'array delle soluzioni. \n Allora, si dice che l'algoritmo è in **tempo _pseudo-polinomiale_**.\n \n ### Problema senza ripetizione\n \n Si può prendere **ogni oggetto una volta sola**.\n \n #### Soluzione _bruteforce_\n \n Scelgo se prendere o no l'item 1.\n \n Si creano due percorsi:\n - Non prendo l'oggetto: `valore = 0, peso = 0`\n - Prendo l'oggetto: `valore = oggetto.valore, peso = oggetto.peso`\n \n Continuo a creare percorsi, creando una specie di albero binario.\n \n Se a un certo punto vedo che `valore = x, peso = K` e `valore < x, peso = K`, allora posso escludere automaticamente tutto il sottoalbero destro, perchè non può essere migliore del sinistro: allora, sarò riuscito a ridurre il numero dei casi rispetto alla ricorsione.\n \n ### Problema in due variabili\n \n `K(j, w)` = massimo valore ottenibile con uno zaino di capacità `w` scegliendo gli item da `1` a `j`.\n \n Non possiamo più applicare la soluzione bruteforce, perchè abbiamo due variabili, `j` e `w`.\n \n Allora, prendo l'elemento `j`. Esso può essere o non essere nella soluzione: mi calcolo entrambe le alternative, e mi tengo l'alternativa dal valore più alto.\n \n Se `j` non è nella soluzione, il risultato diventerà `K(j-1, w)`; se invece è nella soluzione, il risultato sarà `j.valore + K(j-1, w-j.peso)`.\n \n In pratica, prendiamo\n \n ```latex\n K(j, w) = max\n \\begin{cases}\n V_j + K(j-1, w-w_j)\n K(j-1, w)\n \\end{cases}\n ```\n \n Costruisco allora una matrice con `j` su un asse e `w` sull'altro. \n Riempio le caselle con il valore di `K(j, w)`. \n Nella casella con `K(j, w)` avremo la soluzione ottima.\n \n Il tempo necessario per riempire tutte le caselle è nuovamente `O(n * w)`, ancora **pseudopolinomiale**.\n \n Per sapere che oggetti ho messo o no devo tenere traccia in qualche modo della catena del calcolo, usando, ad esempio, una pila.\n ")),n(a.o,null,n(a.m,null,"\n # Problemi intrattabili\n \n ## Problema di Set-Cover\n \n ### Input\n \n `U`niverso di `e`lementi\n \n `S`ottoinsieme di `s`ottoinsiemi di elementi di `U`\n \n ### Output\n \n Il minimo `S'`ottoinsieme di `s`ottoinsiemi che copra completamente `U`.\n \n ### Soluzione in `O(n^d)`\n \n Non c'è.\n \n ### Non-soluzione alternativa\n \n Faccio una scelta greedy, ma non posso dimostrare in alcun modo che la soluzione ottenuta sia quella ottima. \n Infatti, l'algoritmo non dà sempre la soluzione ottima, ma dà una soluzione accettabile in tempo polinomiale.\n \n Seleziono sempre il sottoinsieme che copre più elementi mancanti possibili.\n \n #### Costo computazionale\n \n `Costo greedy <= log(numero_elementi) * Costo ottimo`\n ")))}}.call(this,i("hosL").h)}}]); -//# sourceMappingURL=route-AlgoritmiEStruttureDati.chunk.d1ce1.js.map \ No newline at end of file +(window.webpackJsonp=window.webpackJsonp||[]).push([[2],{w9qZ:function(n,e,i){"use strict";i.r(e),function(n){function o(){var n=function(n,e){e||(e=n.slice(0));return n.raw=e,n}(["\n # Algoritmi e Strutture Dati\n\n Docente: [**Manuela Montangero**](mailto:manuela.montangero@unimore.it)\n\n Crediti: **9 CFU** (72 ore di lezione)\n\n Orario di ricevimento: **Giovedì dalle 14:30 alle 16:30**\n _Mandare una mail prima, altrimenti potrebbe andarsene_\n\n\n #### Regole particolari per le email\n\n - Oggetto \"ASD\"\n - Mail **firmata** con **nome e cognome**\n - Mail spedita dall'**account Unimore**\n\n ### Materiale\n\n Libri:\n - **Introduzione agli Algoritmi e Strutture Dati** di _T.H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein_\n\n **[Dolly (FIM)](https://dolly.fim.unimore.it/2018/course/view.php?id=26)**\n\n ### Tutorato\n\n Tutor: [**Gianluca d'Addese**](mailto:tutoratoalgoritmi@gmail.com)\n\n Cosa: **Esercizi sugli argomenti visti a lezione** e **preparatori all'esame**\n\n Quando: **Mercoledì dalle 09:00 alle 11:00**...?\n\n ### Esame\n\n Per iscriversi all'esame, bisogna aver passato:\n\n - Analisi matematica (propedeutica)\n - Programmazione I\n\n Formato:\n\n - Prima prova scritta\n - Risolvere problemi con algoritmi proposti a lezione\n - E' un test sulla preparazione\n - Dura 1h30m\n - Non si può usare nessun tipo di materiale\n - Seconda prova scritta\n - Proponi soluzioni per nuovi problemi non studiati a lezione\n - Domande teoriche sugli argomenti studiati\n - Dura 2h\n - E' il giorno dopo la prima prova\n - Si può utilizzare qualsiasi materiale, ma non deve permettere di comunicare\n - Orale facoltativo\n - Solo per chi supera entrambi gli scritti\n - Domande su quello che abbiamo visto a lezione (\"perchè gli algoritmi funzionano?\")\n - Informare via email entro 3 giorni dalla pubblicazione degli esiti della seconda prova\n - L'orale potrebbe migliorare o peggiorare il voto (anche \"molto\")\n\n **Attenzione:** Vengono verbalizzati anche i voti insufficienti; ricordarsi di rifiutarli!\n\n Ci sono i seguenti appelli:\n\n - 3 appelli tra Giugno e Luglio\n - 1 appello a Settembre\n - 2 appelli a Gennaio e Febbraio\n\n Vale il salto di appello se:\n\n - Uno studente regolarmente iscritto **non si presenta** all'appello e non ha avvisato via email almeno il giorno prima.\n - Uno studente ha riportato una **grave insufficienza** in uno dei due scritti, e l'appello successivo è nella stessa sessione\n\n\n ### Note\n\n Il corso è ben collegato con quello di Programmazione 2.\n\n Dormire non fa bene!\n "]);return o=function(){return n},n}var a=i("mbOI"),r=String.raw;e.default=function(){return n("div",null,n(a.p,{title:"Disperazione di Steffo"},"aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa tutte le formule in latex si sono rotteeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee"),n(a.p,null,n(a.n,null,r(o()))),n(a.p,null,n(a.n,null,"\n # Il nome del corso\n\n ## Cosa sono gli algoritmi?\n\n Gli algoritmi sono modi sistematici per risolvere problemi.\n\n Sono fondamentali per sviluppare software, in quanto i computer sono eccellenti esecutori di algoritmi.\n\n ## Come si sviluppa un algoritmo?\n\n Innanzitutto, bisogna conoscere gli _input_ e gli _output_ del problema, rispettivamente i dati di partenza e i dati di arrivo di esso; si ha quindi una fase di **ricerca**.\n\n Poi, si deve trovare un procedimento che ci faccia risolvere il nostro problema: è quello che faremo in questa materia!\n\n Infine, bisogna scrivere la soluzione in un modo che possa essere eseguita da un computer: questa è la **programmazione**.\n\n ## Che tipo di problemi possiamo risolvere?\n\n Un algoritmo risolve problemi di tipo generale, non ci interessa sapere _il risultato di 123+456_, ma vogliamo sapere _il risultato di x+y_, dove x e y sono due numeri naturali qualsiasi.\n\n Un problema può essere quindi considerato circa come una **funzione matematica**, che connette ogni input a un output corrispondente.\n\n ## Che caratteristiche ha un algoritmo?\n\n Per prima cosa, ripetendo l'algoritmo più volte con lo stesso ingresso deve dare sempre la stessa uscita come **risultato**, finendo in un **tempo finito**.\n\n Deve essere **ben ordinato**: cambiando l'ordine in cui vengono effettuate le operazioni, è probabile che anche il risultato cambi!\n\n Le sue istruzioni devono essere **non ambigue**, cioè che non possano essere interpretate in più modi, e **effettivamente realizzabili**, cioè realizzabili con l'esecutore che vogliamo usare per eseguire l'algoritmo.\n\n ## Esistono algoritmi equivalenti?\n\n **Sì!** Possono esserci due algoritmi che dati gli stessi input, hanno gli stessi output, e quindi risolvono lo stesso problema.\n\n In compenso, possono avere un numero di operazioni diverse, e quindi essere **uno più veloce** (da eseguire) dell'altro.\n\n ## Come si verifica la correttezza di un algoritmo?\n\n L'algoritmo deve essere **valido per tutti gli input**, anche se questi sono infiniti.\n\n Possiamo effettuare prove matematiche per verificarne la correttezza; il **principio di induzione** è dunque una dei teoremi fondamentali dell'algoritmica.\n\n Possiamo però verificare la _non correttezza_ di un algoritmo trovando un singolo controesempio.\n ")),n(a.p,null,n(a.n,null,"\n # Efficienza degli algoritmi\n \n Un buon algoritmo deve essere **efficiente**, ovvero deve usare il minimo delle risorse necessarie, come _usare il minimo di tempo possibile_.\n \n ## Come misuriamo il tempo necessario?\n \n Cerchiamo di astrarre il tempo dal particolare esecutore, e andiamo a contare il numero di operazioni elementari richieste per eseguire il nostro algoritmo nel caso peggiore.\n \n Un algoritmo efficiente, infatti, all'aumentare dei dati in ingresso, diventerà sempre più veloce rispetto a uno non efficiente, anche su computer più lenti!\n \n > Il [Bubble Sort](https://en.wikipedia.org/wiki/Bubble_sort) è sempre più lento di un [Tree Sort](https://en.wikipedia.org/wiki/Tree_sort), anche su computer più lenti, perchè, dovendo ordinare liste sempre più lunghe, prima o poi si raggiunge un punto in cui il primo è più veloce (in termini di tempo) dell'altro.\n \n Dobbiamo andare a vedere, quindi, _il numero di operazioni richieste per ottenere il risultato nel caso peggiore_.\n \n Consideriamo operazioni sia operazioni aritmetiche sia operazioni logiche, e diciamo che ciascuna costa 1.\n \n > L'[algoritmo di Euclide](https://it.wikipedia.org/wiki/Algoritmo_di_Euclide) per l'MCD costa `3` per ogni iterazione (un giro del ciclo `while`). Diciamo, allora, che costa `3n`, dove `n` è il numero più alto dei due, perchè nel caso peggiore (uno dei due numeri è 1) l'algoritmo compie `n` iterazioni.\n \n ## Altri parametri ottimizzabili\n \n È possibile che alcuni algoritmi per vari motivi cerchino di ottimizzare altri parametri diversi dal tempo, come ad esempio _la dimensione dell'input_ o la _memoria utilizzata_. \n \n > Generalmente, questo viene fatto sui dispositivi embedded, con memoria molto limitata.\n \n ### Criteri di costo di memoria\n \n Ci sono vari criteri con cui stimare la memoria richiesta da un dato: è possibile che il costo risultante vari in base al criterio scelto!\n \n #### Criterio di costo logaritmico\n \n Un dato costa il numero di bit necessari per rappresentarlo.\n \n > Un int che contiene il numero `n` costa `log_2(n)`. \n > Un array di `[n]` numeri tutti uguali costa `n log_2(n)`.\n \n #### Criterio di costo uniforme\n \n Un dato costa il numero di elementi che lo costituiscono.\n \n > Un int che contiene il numero `n` costa `1`. \n > Un array di `[n]` numeri costa `n`. \n > Una matrice `[m][n]` costa `m*n`.\n ")),n(a.p,null,n(a.n,null,"\n # Modelli algoritmici\n\n Per progettare un algoritmo, abbiamo bisogno di sapere le proprietà del nostro esecutore, ovvero il suo _modello algoritmico_.\n \n > Ad esempio, dobbiamo sapere quali istruzioni è in grado di eseguire, e quanto tempo queste istruzioni richiedono.\n \n ## Il modello RAM\n \n Il modello in uso su tutti i computer attuali è il _modello RAM_: \n \n - In ogni cella di memoria può essere archiviato un dato.\n - Il **tempo di accesso** alle celle è **costante** per tutte le celle.\n - La **memoria** principale è **infinita**.\n - Si ha **un solo processore**.\n ")),n(a.p,null,n(a.n,null,'\n # Notazione asintotica\n \n La _notazione asintotica_ è un sistema per **stimare** velocemente il costo di un algoritmo complesso.\n \n Ci permette di **confrontare velocemente il caso peggiore** degli algoritmi.\n \n In particolare, consideriamo il _rapporto tra il numero di operazioni nel caso peggiore e la dimensione dell\'input_.\n \n ## Limiti\n \n Possiamo dare a questa stima dei limiti, superiore e inferiore, che rappresenteranno rispettivamente un costo che non sarà **mai superato** e un costo che verrà **sempre superato**.\n \n Chiameremo questi limiti _upper bound_ e _lower bound_; la loro combinazione darà un _tight bound_.\n \n L\'obiettivo sarà di _ricavare i bound più precisi possibile_ per un dato algoritmo, ovvero l\'**upper bound più basso** e il **lower bound più alto**.\n \n ### O grande\n \n > "O grande" \n > O di g(n)\n > "big-O"\n \n Per rappresentare la stima, useremo una notazione particolare, detta _O grande_, con la seguente proprietà:\n - Date due funzioni `f(n) : N -> R` e `g(n) : N -> R`, diremo che `f(n) ∈ O(g(n))` se e soltanto se `∃ c > 0, n ≥ n_0` tali che `∀ n ≥ 0, f(n) ≤ c * g(n)`\n \n Quando una funzione è O grande di un altra, significa che **asintoticamente, la funzione in O grande è sempre maggiore di quella che sta venendo stimata**.\n \n > **Ipotesi** \n > - `f(n) = 2n² + 3n + 6` \n > - `g(n) = n²`\n >\n > **Tesi** \n > - `f(n) ∈ O(n²)`.\n >\n > **Svolgimento** \n > Scrivo una disequazione, lasciando intatto il termine noto: \n > 1. `2n² + 3n + 6 ≤ 2n² + 3n² + 6` \n > 2. `n² ≤ 2n² + 3n² + n² = 6n²` per `n ≥ 3`\n >\n > Sappiamo, allora, che `2n² + 3n + 6 ≤ 6n²`.\n \n #### Espressioni di O grande\n \n Questa tabella rappresenta le espressioni di O grande più comunemente utilizzate, in ordine **dalla più forte alla più debole**.\n \n > Più forte significa che, per ogni riga della tabella, tutte le righe sottostanti sono contenute nell\'espressione.\n >\n > Ad esempio, `O(n) ∈ O(1)`.\n \n | Espressione O() | Nome |\n |-----------------|------|\n | `O(1)` | Costante |\n | `O(log log n)` | loglog |\n | `O(log n)` | Logaritmica |\n | `Ω(n^{1/c})` (per c ≥ 1) | Sublineare |\n | `O(n)` | Lineare |\n | `O(n log n)` | nlogn |\n | `O(n²)` | Quadratica |\n | `O(n³)` | Cubica |\n | `O(n^k)` (per k ≥ 1) | Polinomiale |\n | `O(a^n)` (per a ≥ 1) | Esponenziale |\n | `O(n!)` | Fattoriale |\n \n #### Polinomiale\n \n Molto spesso, noteremo che il tempo richiesto da una funzione è O grande di un polinomio di grado K, ovvero `f(n) ∈ O(n^k)`.\n \n Notiamo che in questi casi, possiamo semplificare l\'O grande al grado massimo del polinomio.\n \n > Ad esempio, `O(n² + n + 1) = O(n²)`.\n \n ##### Dimostrazione\n \n Per `n > 0 \\and 0 ≤ i ≤ k`: \n ![LaTeX](https://latex.codecogs.com/png.latex?a_k%20n^k%20+%20a_{k-1}%20n^{k-1}%20+%20%E2%80%A6%20+%20a_1%20n%20+%20a_0%20%E2%89%A4%20|a_k|%20n^k%20+%20|a_{k-1}|%20n^k%20+%20%E2%80%A6%20+%20|a_1|%20n^k%20+%20|a_0|%20n^k%20=%20(|a_k|%20+%20|a_{k-1}|%20+%20%E2%80%A6%20+%20|a_1|%20+%20|a_0|)%20n^k)\n \n #### Proprietà di O grande\n \n 1. `f(n) ∈ O(g(n)) -> ∀ a > 0, a * f(n) ∈ O(g(n))`.\n 2. `f(n) ∈ O(g(n)), d(n) ∈ O(h(n)) -> f(n) + d(n) ∈ O(g(n) + h(n)) -> O(max\\{g(n), h(n)\\})`\n 3. `f(n) ∈ O(g(n)), d(n) ∈ O(h(n)) -> f(n) * d(n) ∈ O(g(n) * h(n))`\n \n In pratica, se una funzione è la _somma di più termini_, basta guardare l\'**`O()` più grande** tra tutti i suoi termini; se invece una funzione è un _prodotto di più termini_, si possono **omettere le costanti**, e l\'`O()` finale sarà dato dal **prodotto degli `O()`** dei termini.\n \n ## Lower bound\n \n Possiamo anche stimare il _lower bound_, il limite inferiore: il **numero minimo di operazioni** che viene effettuato **nel caso migliore** con la **massima dimensione dell\'ingresso**.\n \n ### Ω()\n \n > "Omega" \n > Omega di g(n)\n > "big-Omega"\n \n Esiste un equivalente di O grande per il lower bound: è detto _Omega grande_, o più semplicemente _Omega_, e funziona nello stesso identico modo, solo... al contrario.\n \n Diremo che `f(n) ∈ Ω(g(n))` se e solo se `∃ c > 0, n_0 ≥ 0 : ∀ n ≥ n_0 f(n) ≥ c * g(n)`.\n \n #### Espressioni di Ω()\n \n Anche in questa tabella le espressioni sono **dalla più forte alla più debole**.\n \n | Espressione Ω() | Nome |\n |-----------------|------|\n | `Ω(n!)` | Fattoriale |\n | `Ω(a^n)` (per a ≥ 1) | Esponenziale |\n | `Ω(n^k)` (per k ≥ 1) | Polinomiale |\n | `Ω(n³)` | Cubica |\n | `Ω(n²)` | Quadratica |\n | `Ω(n log n)` | nlogn |\n | `Ω(n)` | Lineare |\n | `Ω(n^{1/c})` (per c ≥ 1) | Sublineare |\n | `Ω(log n)` | Logaritmica |\n | `Ω(log log n)` | loglog |\n | `Ω(1)` | Costante |\n \n ## Tight bound\n \n Quando **upper e lower bound coincidono**, allora otteniamo un _tight bound_.\n \n ### θ()\n \n > "Theta" \n > Theta di g(n)\n > "big-Theta"\n \n Anche per il tight bound abbiamo una notazione equivalente a O grande e Omega grande: _Theta grande_!\n \n Diciamo che `f(n) ∈ θ(g(n))` se e solo se `∃ c_1, c_2 > 0, n_0 ≥ 0 : ∀ n ≥ n_0, c_1 * g(n) ≤ f(n) ≤ c_2 * g(n)`.\n \n Ha la particolarità che non valgono tutte le proprietà degli altri due: va usata quindi con cautela!\n \n ## Risorse utili\n \n [khanacademy.org](https://www.khanacademy.org/computing/computer-science/algorithms/asymptotic-notation/a/big-big-theta-notation)\n ')),n(a.p,null,n(a.n,null,"\n # Problemi algoritmici\n \n Un _problema algoritmico_ è un problema matematico che si vuole provare a risolvere con un algoritmo.\n \n > Dati 10 numeri, voglio sapere se sono in ordine crescente oppure no.\n \n ## Caterigorizzazione\n \n I problemi algoritmici si dividono in tre categorie: problemi _trattabili_, problemi _intrattabili_ e problemi _irrisolvibili_.\n \n ### Problema trattabile\n \n Perchè un problema algoritmico sia _trattabile_, deve avere **almeno un algoritmo con upper bound polinomiale**.\n \n > Questo significa che il tempo impiegato da un computer per risolvere il problema rimane ragionevole, e che quindi può essere utilizzato in maniera efficiente.\n \n La trattabilità è un campo ancora parecchio aperto: esistono anche tanti problemi di cui non si è ancora dimostrata la trattabilità o intrattabilità.\n \n > La [fattorizzazione](https://it.wikipedia.org/wiki/Fattorizzazione) è uno di questi problemi: l'assenza di una dimostrazione è ciò che la rende uno dei pilastri della sicurezza informatica moderna.\n \n ### Problema intrattabile\n \n Se **un problema non ha nessun algoritmo con upper bound polinomiale**, allora si dice che è **intrattabile**.\n \n ### Problema irrisolvibile\n \n Se **non esistono algoritmi per risolvere un problema**, allora questo si dice **irrisolvibile**.\n \n > [Dato un algoritmo con certi input, riusciamo a capire con un algoritmo se la sua esecuzione termina o no?](https://en.wikipedia.org/wiki/Halting_problem)\n \n ### Upper e lower bound di problemi\n \n Si può anche trovare un _upper bound_ e un _lower bound_ per un problema, ma bisogna generalizzare di più.\n \n L'**upper bound di un problema** è il minimo upper bound di tutti gli algoritmi che lo risolvono; deve esistere almeno un algoritmo che lo risolva che abbia lo stesso _upper bound_. E' praticamente il tempo migliore per risolvere il problema dato.\n \n Il **lower bound di un problema** è il minimo lower bound di tutti gli algoritmi che lo risolvono; non deve esistere nemmeno un algoritmo che abbia un lower bound migliore. E' il numero assolutamente minimo di operazioni richieste, non si può fare meglio di così.\n \n In particolare, abbiamo che l'_upper bound di un algoritmo_ -> l'_upper bound del suo problema_, \n e il _lower bound di un problema_ -> il _lower bound di un suo algoritmo_.\n \n Generalmente, il _lower bound di un problema_ è una rappresentazione abbastanza accurata della sua difficoltà.\n ")),n(a.p,null,n(a.n,null,"\n # Ricerca binaria\n \n Non credo di aver bisogno di studiare la ricerca binaria, quindi non ho preso appunti a riguardo.\n \n Se non siete me, e state cercando informazioni a riguardo, andate a vedere su Wikipedia!\n ")),n(a.p,null,n(a.n,null,"\n # Divide et impera\n \n Un modo efficace per risolvere un problema è di usare il metodo _divide et impera (et combina)_.\n \n - _Divide_: Divido il problema in **tanti sottoproblemi**.\n - _Impera_: **Risolvo indipendentemente dal resto** ciascuno dei sottoproblemi.\n - _Combina_: **Combino** i risultati dei sottoproblemi per **risolvere il problema principale**.\n \n ## Ricorsione\n \n Un algoritmo (o funzione) si dice _ricorsivo_ quando durante l'esecuzione **richiama sè stesso**.\n \n Dato che tutti gli algoritmi devono avere termine entro un tempo finito, se scriviamo una funzione ricorsiva è fondamentale finire con un **caso base**, che non chiami ulteriormente la ricorsione.\n \n Se esiste una funzione ricorsiva, allora esiste _sempre_ una _funzione iterativa_ che darà lo stesso risultato.\n \n > Sul pratico, una funzione ricorsiva tipicamente è **più costosa** del suo equivalente iterativo: se possibile, quindi, la ricorsione andrebbe evitata.\n \n #### Pseudocodice di esempio\n \n ```python\n def fattoriale(n):\n # Caso base; la ricorsione finisce e dà un risultato fisso\n if n <= 1:\n return 1\n # Caso ricorsivo; la funzione restituisce il risultato di sè stessa (ma con parametri diversi)\n else:\n return n * fattoriale(n - 1)\n ```\n ")),n(a.p,null,n(a.n,null,"\n # Master Theorem\n \n Il _Master Theorem_ è uno dei teoremi più importanti dell'algoritmica.\n \n Esso permette di **calcolare l'upper bound di un algoritmo ricorsivo** in modo piuttosto semplice.\n \n ## Ipotesi\n \n Dato un algoritmo:\n - Con uno o più casi base\n - Che richiama la funzione ricorsiva un numero n di volte\n \n ## Tesi\n \n Allora, il suo upper bound avrà la formula:\n \n ![](https://quicklatex.com/cache3/89/ql_08d29a7e55d561a900570bc83b93ff89_l3.png)\n ")),n(a.p,null,n(a.n,null,"\n # Caso particolare del Master Theorem\n \n ## Ipotesi\n \n Se: \n ![](https://quicklatex.com/cache3/57/ql_e34dc27b42831d3c3ff671b0f3861257_l3.png)\n \n Ovvero, se la dimensione dell'input viene divisa ad ogni ciclo da una costante b, è polinomiale e il caso base è costante...\n \n ## Tesi\n \n Allora: \n ![](https://quicklatex.com/cache3/ca/ql_26e3557a6ca2d6ac4b8481e7c5263fca_l3.png)\n \n > In pratica, se il costo dominante è quello della parte \"fissa\" dell'algoritmo, esso sarà `O(n^d)`, mentre se il costo dominante è quello delle chiamate ricorsive, esso sarà `O(n^{log_b e})`. \n > Se nessuno dei due è dominante... si dividono circa in parti uguali, creando un costo di `O(n^d log n)`.\n ")),n(a.p,null,n(a.n,null,"\n # Ordinamento\n \n Un problema molto frequente nell'informatica consiste nell'**ordinare efficientemente grandi quantità di elementi**.\n \n Esistono [tantissimi](https://it.wikipedia.org/wiki/Algoritmo_di_ordinamento) algoritmi per effettuare l'ordinamento. \n L'**efficienza** di ciascuno **varia** di caso in caso: alcuni sono estremamente efficienti se quasi tutti i numeri sono già nell'ordine giusto; altri, invece, potrebbero impiegare tantissimo tempo.\n \n In termini matematici, abbiamo:\n \n - **Input:** A[n]\n - **Output:** B, ∀ i < n, A[i] ≤ A[i+1]\n \n ## Ordinamento tramite confronto\n \n L'ordinamento \"tradizionale\" è detto _ordinamento tramite confronto_: funziona sempre, e **non ha altri modi di ottenere informazioni** se non con l'operazione logica di confronto tra i dati.\n \n ### Limiti\n \n E' un problema risolto: è dimostrabile che il suo **lower bound** è **`Ω(n log n)`**; possiamo quindi dire che qualsiasi algoritmo di ordinamento è in `Ω(n log n)`, e se riusciamo a trovare un algoritmo di ordinamento in `O(n log n)` siamo riusciti a raggiungere il massimo dell'efficienza.\n \n #### Dimostrazione\n \n Consideriamo **tutte le possibili permutazioni** della sequenza da ordinare: sono `n!`.\n \n Per ogni confronto che effettuiamo, **riduciamo la quantità di permutazioni** correttamente ordinate; prima o poi, rimarrà **una sola possibilità**.\n \n **TODO, non trovo la spiegazione corretta!**\n \n ### Esempi\n \n Algoritmi che effettuano l'ordinamento tramite confronto sono:\n - _Bubble sort_\n - _Merge sort_\n - _Insertion sort_\n - _Quick sort_\n - _Heap sort_\n - E tanti, tanti altri!\n \n ## Ordinamento con altri mezzi\n \n Esistono algoritmi che ricavano informazioni in altri modi, diversi dal confronto.\n \n Essi possono avere un lower bound più basso di `O(n log n)`, però hanno spesso limitazioni sul loro utilizzo.\n \n ### Esempi\n \n - _Counting sort_, indicizza i valori da ordinare\n - _Radix sort_, guarda singolarmente le cifre dei valori\n - _Sleep sort_, sfrutta i thread e la funzione sleep per ordinare valori\n - E altri ancora!\n ")),n(a.p,null,n(a.n,null,"\n # Insertion sort\n \n L'_insertion sort_ è una soluzione **iterativa** all'ordinamento per confronto.\n \n ## Funzionamento\n \n Considero la sequenza divisa in **due parti**: una parte **ordinata** e una parte **non ordinata**.\n \n Parto dal primo elemento della lista: è sempre ordinato con sè stesso. \n Poi, aggiungo uno alla volta i numeri della parte non ordinata a quella ordinata; prima trovo in quale posizione dovrò andare a mettere il numero, poi **faccio slittare tutti i numeri dopo quella posizione** avanti di 1, in modo da **creare lo spazio** in cui infine **inserirò** il numero.\n \n ## Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | `O(n²)` | `Ω(n)` | - |\n \n Nel _caso migliore_ (**lista già ordinata**), il numero da inserire è già nella posizione giusta, quindi non devo effettuare altri confronti oltre il primo, rendendo il lower bound dell'algoritmo `Ω(n)`.\n \n Nel _caso peggiore_ (**lista nell'ordine inverso**), dobbiamo confrontare il numero da inserire con tutti gli altri nella parte ordinata: dobbiamo allora eseguire `1+2+3+4+5+… = \\frac{(n-1)(n)}{2}` confronti; ciò significa che l'upper bound è `O(n²)`!\n \n ## Pseudocodice\n \n ```python\n def insertion_sorted(lista):\n # Itero su tutti i numeri della lista, dal primo all'ultimo.\n for divisore_ord in range(len(lista)):\n # Partendo dalla posizione attuale, creo l'indice di divisione numeri ordinati maggiori e minori\n divisore_magg = divisore_ord\n # Faccio slittare avanti i numeri maggiori di quello che stiamo inserendo\n # Se l'indice divisore_magg raggiunge 0, vuol dire che tutti i numeri della lista sono maggiori del numero attuale \n while divisore_magg >= 0 and lista[divisore_magg-1] > lista[divisore_magg]:\n # Scambio la posizione dei due elementi con gli indici specificati\n # Funzione inventata \n lista[divisore_magg], lista[divisore_magg-1] = lista[divisore_magg-1], lista[divisore_magg]\n # Diminuisco il separatore di 1\n divisore_magg -= 1\n ```\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/bn/sorting)\n ")),n(a.p,null,n(a.n,null,"\n # Merge sort\n \n Il _merge sort_ è una soluzione **ricorsiva** all'ordinamento per confronto.\n \n ## Funzionamento\n \n Per questo algoritmo, utilizziamo la tecnica del **divide et impera**.\n \n 1. _Divide_: Divido A in **due parti**.\n 2. _Impera_: Metto **separatamente in ordine** le parti.\n 3. _Unisci_: **Unisco** le due parti.\n \n Consideriamo come **caso base** della ricorsione una parte composta da un numero, che ovviamente è già ordinata con sè stessa.\n \n ### Merge\n \n Per **unire le due parti** usiamo una funzione detta `merge()`.\n \n Costruiamo una nuova sequenza uguale alla sequenza 1, ma **aggiungiamo alla fine un valore sentinella** sempre maggiore di tutti gli elementi contenuti.\n \n > ```\n > | 1 | 3 | 7 | 8 | ∞ |\n > ```\n \n Facciamo **la stessa cosa** per la sequenza due.\n \n > ```\n > | 2 | 4 | 5 | 6 | ∞ |\n > ```\n \n Prendo i primi numeri delle due sequenze e **metto il più piccolo nella sequenza iniziale**.\n \n > ```\n > | 1 | 2 | 3 | | | | | | \n > | | | 7 | 8 | ∞ | \n > | | 4 | 5 | 6 | ∞ | \n > ```\n \n **Continuo** finchè non ho messo tutti i numeri; **grazie alla sentinella non usciremo mai dalla sequenza**, in quanto essa è sempre maggiore di tutti gli altri numeri.\n \n > ```\n > | 1 | 2 | 3 | 4 | 5 | 6 | | | \n > | | | 7 | 8 | ∞ | \n > | | | | | ∞ |\n > ```\n \n Quando **rimangono solo le sentinelle** significa che abbiamo aggiunto tutti gli elementi, e quindi abbiamo finito.\n \n > ```\n > | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | \n > | | | | | ∞ | \n > | | | | | ∞ | \n > ```\n \n \n ## Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | `O(n log n)` | `Ω(n log n)` | **`θ(n log n)`** |\n \n Il merge sort è un algoritmo ricorsivo con un **caso base in tempo costante** e che **richiama sè stesso 2 volte**.\n \n ```latex\n T(n) =\\\\\n \\\\\n Θ(1) \\qquad n=1\\\\\n 2 T(\\frac{n}{2}) + Θ(n) \\qquad n \\neq 1\n ```\n \n Applicando il **caso particolare del Master Theorem**, otteniamo:\n \n ```latex\n T(n) =\\\\\n \\\\\n Θ(1) \\qquad n=1\\\\\n Θ(n log n) \\qquad n \\neq 1\n ```\n \n ## Pseudocodice\n \n ```python\n def merge_sorted(part):\n # Caso base\n if len(part) == 1:\n return part\n # Divide\n middle = len(part) // 2\n part_a = part[:middle]\n part_b = part[middle:]\n # Impera\n sort_a = merge_sorted(part_a)\n sort_b = merge_sorted(part_b)\n # Combina\n return merge(sort_a, sort_b)\n ```\n \n ## Visualizzazione\n \n [hackerearth.com](https://www.hackerearth.com/practice/algorithms/sorting/merge-sort/visualize/)\n \n [visualgo.net](https://visualgo.net/bn/sorting) (Nota: visualizza solo la fase _Unisci_ del sort)\n ")),n(a.p,null,n(a.n,null,"\n # Quick sort\n \n Il _quick sort_ è un altro approccio **ricorsivo** all'ordinamento per confronto.\n \n ## Funzionamento\n \n Anche qui, applichiamo il **divide et impera**.\n \n 1. _Divide_: Scelgo un **pivot** qualsiasi all'interno della lista. Metto alla sua **sinistra tutti i numeri minori** e alla sua **destra tutti i numeri maggiori**.\n 2. _Impera_: Eseguo un **quick sort su entrambe le sottoliste**.\n \n ### Esempi\n \n #### Iterazione con partizioni bilanciate\n \n Osserviamo come si formi una partizione con tre elementi e una con quattro. \n \n ```\n |¦ [2] 8 7 1 3 5 6 {4}\n 2 |¦ [8] 7 1 3 5 6 {4}\n 2 |8 ¦ [7] 1 3 5 6 {4}\n 2 |8 ¦ [7] 1 3 5 6 {4}\n 2 |8 7 ¦ [1] 3 5 6 {4}\n 2 1 |7 8 ¦ [3] 5 6 {4}\n 2 1 3 |8 7 ¦ [5] 6 {4}\n 2 1 3 |8 7 5 ¦ [6] {4}\n 2 1 3 |8 7 5 6 ¦ [{4}]\n [2 1 3] {4} [8 5 6 7]\n ```\n \n #### Iterazione con partizioni sbilanciate\n \n Osserviamo come si formi una partizione con **zero elementi** e una con tre.\n \n ```\n |¦ [4] 7 3 {1}\n |4 ¦ [7] 3 {1}\n |4 7 |¦ [3] {1}\n |4 7 3 |¦ [{1}]\n [] {1} [4 7 3]\n ```\n \n ## Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | `O(n²)` | `Ω(n log n)` | - |\n \n Il costo della funzione è dato dalla somma del costo per **dividere in due partizioni** con il costo per realizzare il **Quick sort delle due sottopartizioni**\n \n Possiamo applicare allora il **Master Theorem generale**:\n \n ```latex\n T(n)\\\\\n =\\\\\n Θ(1) \\qquad per\\ n = 1\\\\\n T(q) + T(dim_lista - pivot - 1) + Θ(n) \\qquad per\\ n > 1\n ```\n \n ### Il caso migliore\n \n Se il pivot `q` è la **mediana della partizione** che stiamo ordinando, si vengono a creare due _sottopartizioni bilanciate_, e sostituendo otteniamo:\n \n ```latex\n T(n)\\\\\n =\\\\\n Θ(1) \\qquad per\\ n = 1\\\\\n 2 T(\\frac{n}{2}) + Θ(n) \\qquad per\\ n > 1\n ``` \n \n Possiamo allora applicare il **Master Theorem particolare**:\n \n ```latex\n T(n)\\\\\n =\\\\\n Θ(1) \\qquad per\\ n = 1\\\\\n Θ(n log n) \\qquad per\\ n > 1\n ```\n \n ### Il caso peggiore\n \n Se il pivot è uno degli **estremi dell'array**, si creano due _partizioni sbilanciate_: una delle due sottoliste è sempre vuota! \n Allora:\n \n ```latex\n T(n) = T(n-1) + Θ(n)\\\\\n = T(n-2) + Θ(n-1) + Θ(n)\\\\\n = T(n-3) + Θ(n-2) + Θ(n-1) + Θ(n)\\\\\n = …\n ∈ Θ(n^2)\n ```\n \n > \"Non date da mangiare sequenze ordinate al Quicksort, gli sono indigeste.\"\n \n ## Pseudocodice\n \n ```python\n def partition(partizione, inizio, fine):\n \"\"\"Dividi una partizione in due, usando l'ultimo elemento come pivot.\n \n Note utili:\n partizione[fine] è il pivot\n partizione[maggiori] è il primo numero dei maggiori\n partizione[non_iterati] è l'elemento su cui si sta iterando al momento\"\"\"\n # Crea il primo separatore (la | pipe nell'esempio)\n maggiori = inizio\n # Crea il secondo separatore (la ¦ broken pipe nell'esempio)\n non_iterati = inizio\n # Itera su ogni numero tra inizio e fine (escluso!)\n while non_iterati < fine:\n # Se l'elemento su cui stiamo iterando è minore del pivot\n if partizione[non_iterati] <= partizione[fine]:\n # Mettilo nell'insieme dei minori, scambiandolo con il primo numero dei maggiori e incrementando il primo separatore\n partizione[maggiori], partizione[non_iterati] = partizione[non_iterati], partizione[maggiori]\n maggiori += 1\n # Incrementa sempre il secondo separatore\n non_iterati += 1\n # Inserisci il pivot tra le due sottopartizioni create, \n partizione[fine], partizione[non_iterati] = partizione[non_iterati], partizione[fine]\n return maggiori\n ```\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/bn/sorting) (Nota: invece che prendere l'ultimo numero come pivot prende il primo, cambiando leggermente l'algoritmo.)\n \n ## Note per l'esame\n \n > La domanda che fa sempre è **\"Qual è la sequenza di pivot utilizzata?\"**\n \n > Elementi da soli _non_ vengono presi come pivot!\n ")),n(a.p,null,n(a.n,null,'\n # Counting sort\n\n Il _counting sort_ è un approccio diverso all\'ordinamento: **non usa il confronto**!\n \n ## Requisiti\n \n Il counting sort può essere utilizzato solo su **sequenze di numeri interi**, e solo se **siamo a conoscenza del minimo e del massimo** dei numeri contenuti nell\'array, ed essi non sono troppo distanti uno dall\'altro. \n (La memoria occupata dal counting sort aumenta linearmente con la differenza tra minimo e massimo!)\n \n Per semplicità, consideriamo il **minimo `0`**. \n L\'input allora sarà una sequenza di interi `A`, e il valore del **massimo `k`**, tale che `∀ n ∈ A, 0 \\leq n \\leq K`.\n \n ## Funzionamento\n \n Il counting sort **conta le ripetizioni** delle chiavi nella sequenza originale e in seguito **sovrascrive i valori** della sequenza con i valori ordinati ripetuti il numero di volte che sono stati individuati nella sequenza.\n \n > ```\n > 1 4 5 3 4 1 4 2 5 1\n > ```\n >\n > L\'`1` appare 3 volte, il `2` 1 volta, il `3` 1 volta, il `4` tre volte e il `5` due volte.\n >\n > La sequenza viene quindi così sovrascritta:\n > ```\n > 1 1 1 3 4 1 4 2 5 1 # Sovrascriviamo la sequenza con 1 ripetuto 3 volte\n > 1 1 1 2 3 4 4 4 5 1 # Sovrascriviamo la sequenza con 2, 3, 4 ripetuti rispettivamente 1 1 e 3 volte\n > 1 1 1 2 3 4 4 4 5 5 # Sovrascriviamo la sequenza con 5 ripetuto 2 volte: abbiamo finito!\n > ``` \n \n Esiste anche una **versione stabile** del counting sort che, invece che sovrascrivere, **sposta i valori**, mantenendo le informazioni aggiuntive nel caso invece che interi fossero altri tipi di dati.\n \n ## Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | `O(k + n)` | `Ω(k + n)` | **`θ(k + n)`** |\n \n L\'algoritmo è composto da quattro parti:\n - Ricerca del minimo e massimo (in `θ(n)`)\n - Inizializzazione dell\'indice (in `θ(k)`)\n - Conteggio dei numeri (in `θ(n)`)\n - Sovrascrittura dei numeri (in `θ(k + n)`)\n \n `2 + O(k) + O(n) + O(k + n) -> O(k + n)`\n \n Notiamo che `k` è costante, l\'algoritmo è `O(n)`, estremamente efficiente.\n \n ## Pseudocodice\n \n ```python\n def counting_sort(lista: typing.List[int]):\n """Ordina in-place una lista con il counting sort."""\n # Trovo la dimensione della lista\n dim = len(lista)\n # Trovo il massimo e il minimo all\'interno della lista\n minimo = min(lista)\n massimo = max(lista)\n # Creo l\'indice dei numeri, in modo che sia lungo k e pieno di 0\n indice = [0 for _ in range(minimo, massimo+1)]\n # Conto i numeri presenti, scorrendo su lista e aggiungendo 1 al numero corrispondente\n for i in range(dim):\n indice[lista[i]] += 1\n # Sovrascrivo i numeri nella lista\n count = 0\n for pos, val in enumerate(indice):\n for _ in range(val):\n indice[count] = pos\n count += 1\n \n def stable_counting_sorted(lista: typing.List[int], k: int) -> typing.List[int]:\n """Ordina stabilmente una lista con il counting sort stabile, e restituiscila."""\n # Trovo la dimensione della lista\n dim = len(lista)\n # Trovo il massimo e il minimo all\'interno della lista\n minimo = min(lista)\n massimo = max(lista)\n # Creo l\'indice dei numeri, in modo che sia lungo k e pieno di 0\n indice = [0 for _ in range(minimo, massimo+1)]\n # Conto i numeri presenti, scorrendo su lista e aggiungendo 1 al numero corrispondente\n for i in range(dim):\n indice[lista[i]] += 1\n # Faccio diventare l\'indice "il numero di numeri \\leq i"\n for i in range(len(indice)):\n if i == 0:\n continue\n indice[i] += indice[i-1]\n assert indice[-1] == dim\n # Creo una nuova lista, che sarà quella che verrà restituita\n nuova = [None for _ in range(dim)]\n # Inizio a posizionare i numeri, al contrario\n for i in range(0, dim, -1):\n nuova[indice[lista[i]]] = lista[i]\n indice[lista[i]] -= 1\n return nuova\n ```\n ')),n(a.p,null,n(a.n,null,"\n ## Introduzione alle Strutture Dati\n\n Una _struttura dati_ è un modo in cui si possono organizzare i dati di un programma.\n \n Si possono definire in due modi: **elementari** e **astratte**.\n \n ### Strutture dati elementari\n \n Le strutture _elementari_ dipendono strettamente dal modo in cui vengono memorizzati i dati.\n \n > **Array** e **liste** sono strutture dati elementari: sono definite dicendo come sono memorizzati i dati, rispettivamente, in celle contigue di memoria e da una serie di nodi con un valore e che puntano al successivo.\n \n ### Strutture dati astratte\n \n Le strutture _astratte_ sono separate dal modo in cui vengono memorizzati i dati, sono più ad alto livello, e si definiscono descrivendo le **proprietà** della struttura e i **metodi** che su di essa possono essere effettuate.\n \n > Una _classe_ in un qualsiasi linguaggio di programmazione è una struttura dati astratta.\n \n > Una _pila_ astratta:\n > - memorizza dati tutti dello stesso tipo\n > - `pop()`, estrae dalla pila l'ultimo valore inserito\n > - `push(val)`, aggiunge alla pila un valore\n > - `top()`, permette di vedere l'ultimo valore inserito nella pila\n > - `vuota()`, dice se la pila è vuota oppure no.\n ")),n(a.p,null,n(a.n,null,'\n # Array\n\n Un _array_ è sequenza di dati di **lunghezza conosciuta**, tutti dello **stesso tipo** e di una **dimensione fissa**, immagazzinata in **blocchi di memoria contigui**.\n \n ## Proprietà\n \n - E\' possibile accedere a tutti i blocchi di memoria conoscendo la loro **posizione relativa al primo** elemento.\n \n ## Metodi\n \n ```python\n class Array:\n def __init__(self, size, type=int): "Crea un array di dimensione size di elementi di tipo int."\n def __getitem__(self, index): "Restituisci il valore alla posizione index."\n def __setitem__(self, index, value): "Cambia il valore alla posizione index."\n ```\n \n ### Costo computazionale\n \n Tutte le operazioni su un array sono in **accesso immediato `O(1)`**!\n ')),n(a.p,null,n(a.n,null,'\n # Lista\n \n Una _lista_ è una sequenza di dati immagazzinata in **blocchi di memoria qualsiasi**.\n \n ## Proprietà\n \n Ogni dato ha un **riferimento** alla collocazione di memoria **successiva** (un puntatore): insieme, sono detti un _nodo_.\n \n E\' di **natura ricorsiva**: qualsiasi nodo di una lista può essere visto come inizio della lista con sè stesso e i suoi successivi.\n \n ## Metodi\n \n ```python\n class LinkedList:\n def __init__(self, value, next: typing.Optional[Node] = None):\n self.value = value\n self.next: typing.Optional[Node] = next\n \n def is_empty(self) -> bool: "Restituisce se la lista è vuota o no."\n def is_full(self) -> bool: "Restituisce se la lista è piena o no.\n def append(self, value): "Aggiunge un nuovo elemento in testa alla lista."\n def insert(self, value, index): "Inserisce un elemento dopo il nodo in posizione index."\n def insert_node(self, value, node): "Inserisce un nuovo elemento subito dopo un dato nodo."\n def find(self, node) -> int: "Trova l\'indice del nodo."\n def delete(self, value): "Elimina il primo nodo con quel valore dalla lista."\n def delete_node(self, node): "Elimina il nodo dalla lista."\n def forward(self, index) -> Node: "Restituisce il nodo in posizione index."\n ```\n \n ### Costo computazionale\n \n #### `List.forward(index)`\n \n Per raggiungere l\'`n`-esimo elemento, bisogna _scorrere tutti gli elementi prima di esso_: è dunque un **accesso sequenziale** in **`O(n)`**.\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/list)\n ')),n(a.p,null,n(a.n,null,'\n # Coda (Queue)\n\n Una _coda_ è come una pila, ma segue la strategia **First In, First Out** (il primo inserito sarà il primo a essere estratto).\n \n ## Proprietà\n \n - I dati vi possono essere aggiunti solo tramite il **metodo `enqueue()`**\n - I dati possono essere estratti solo tramite il **metodo `dequeue()`**\n - Verranno restituiti i valori inseriti secondo la strategia **First In, First Out** (il primo inserito sarà il primo a essere estratto).\n \n > Hai presente quando fai la fila per pagare al supermercato? Beh, è quello, però non si possono superare le altre persone in nessun modo.\n \n ## Metodi\n \n ```python\n class Queue:\n def __init__(self): "Crea una nuova coda."\n def is_empty(self) -> bool: "Restituisce vero se la coda è vuota."\n def enqueue(self, data): "Aggiunge un dato alla coda."\n def first(self) -> ...: "Restituisce il primo dato della coda."\n def dequeue(self) -> ...: "Restituisce il primo dato della coda e lo rimuove."\n ```\n \n ## Implementazione tramite lista\n \n Posso implementare la coda con una lista, ma per realizzare l\'implementazione più efficiente devo tenere anche un puntatore all\'ultimo elemento della coda, in modo da non doverla scorrere ogni volta che voglio effettuare un\'operazione.\n \n Chiamiamo i due puntatori `head` e `tail` rispettivamente.\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/list)\n ')),n(a.p,null,n(a.n,null,'\n # Pila (Stack)\n\n Una _pila_ è una struttura dati contenente **valori omogenei**.\n \n ## Proprietà\n \n - I dati vi possono essere aggiunti solo tramite il **metodo `push()`**\n - I dati possono essere estratti solo tramite il **metodo `pop()`**\n - Verranno restituiti i valori inseriti secondo la strategia **Last In, First Out** (l\'ultimo inserito sarà il primo a essere estratto).\n \n > Ci si può immaginare una pila di libri, da cui si può solo prendere un libro alla volta, quello più in alto.\n \n ## Metodi\n \n ```python\n class Stack:\n def __init__(self): "Crea una nuova pila."\n def is_empty(self) -> bool: "Restituisce vero se la pila è vuota."\n def push(self, data): "Aggiunge un dato alla pila."\n def top(self) -> ...: "Restituisce il primo dato della pila."\n def pop(self) -> ...: "Restituisce il primo dato della pila e lo rimuove."\n ```\n \n ## Implementazione tramite lista\n \n Utilizzando una lista possiamo realizzare una pila!\n \n La direzione dei puntatori sarà dall\'ultimo al primo, che non punterà più a nessuno.\n \n Il costo di tutti i metodi è `Θ(1)`!\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/list)\n ')),n(a.p,null,n(a.n,null,"\n # Albero radicato\n \n Un _albero radicato_ è una struttura dati di **natura ricorsiva** che organizza i dati in maniera **non-lineare**.\n \n ## Proprietà\n \n - Ogni nodo dell'albero ha un **unico genitore**: `∀ (padre, figlio), (padre' figlio) ∈ E \\implies padre = padre'`\n - Ogni nodo dell'albero può avere **un numero qualsiasi di figli**.\n \x3c!----\x3e\n - I **nodi superiori al padre** vengono chiamati _antenati_.\n - I **nodi inferiori ai figli** vengono chiamati _discendenti_.\n \x3c!----\x3e\n - Nodi **senza padre** sono detti _radice_: `\\notexists (padre, radice) ∈ E`\n - Nodi **con padre e figli** sono detti _rami_ o interni.\n - Nodi **senza figli** sono detti _foglie_.\n \x3c!----\x3e\n - La **distanza** tra il nodo radice e i suoi discendenti è detta _livello_:\n - I figli immediati sono di livello 1.\n - I \"nipoti\" (figli dei figli) sono di livello 2.\n - I figli dei nipoti sono livello 3.\n - E così via.\n - Il **livello massimo** all'interno di un albero è detto _altezza_, _profondità_ oppure _h_, ed è sempre `1 ≤ h ≤ n-1`.\n \x3c!----\x3e\n - Un albero ha sempre `n-1` archi.\n \n ## Alberi particolari\n \n ### Alberi `d`-ari\n \n Un albero _`d`-ario_ è un particolare tipo di albero che **limita il numero massimo di figli di un nodo** a `d`.\n \n > Un albero _binario_ può avere **massimo 2 figli** per ogni nodo; un albero _ternario_ ne può avere **3**; un albero _`17`-ario_ ne potrà avere **17**\n \n #### Alberi completi\n \n Un albero `d`-ario si dice _completo_ se **tutti i nodi hanno 0 o `d` figli**, e mai una numero in mezzo.\n \n #### Alberi bilanciati\n \n Un albero `d`-ario si dice _bilanciato_ se **tutti i livelli eccetto l'ultimo** hanno il numero massimo di figli.\n \n #### Alberi perfettamente bilanciati\n \n Un albero `d`-ario si dice _perfettamente bilanciato_ se **tutti i livelli incluso l'ultimo** hanno il numero massimo di figli.\n \n ##### Particolarità degli alberi binari perfettamente bilanciati\n \n Si può dimostrare per induzione che:\n - Hanno sempre `2^h` foglie.\n - Hanno sempre `2^{h+1}-1` (`\\sum_i=0^n 2^i`) nodi.\n - L'altezza è in `Θ(log n)`.\n \n ## Implementazione degli alberi\n \n Possiamo scegliere se usare una rappresentazione con array o con nodi e puntatori: ognuna ha vantaggi e svantaggi diversi.\n \n ### Implementazione tramite array\n \n E' suggerito se l'albero è regolare; più è simile a un albero d-ario completo, meglio è.\n \n ### Implementazione tramite nodi e puntatori\n \n Più adatta ad alberi irregolari.\n \n Se l'albero è regolare, creiamo il numero esatto di campi:\n \n - Valore\n - Figlio1\n - Figlio2\n - _Opzionale:_ Padre\n \n Se un albero è irregolare, creiamo una specie di lista:\n \n - Valore\n - Primo figlio\n - Prossimo fratello\n - _Opzionale:_ Padre\n ")),n(a.p,null,n(a.n,null,"\n #### Breadth-first search (BFS)\n\n La _breadth-first search_ è un algoritmo che visita **ogni livello** dell'albero in ordine, dal più basso al più alto.\n \n ## Funzionamento\n \n > 1. __Visita radice__\n > 2. __Visita figli__\n > 3. __Visita nipoti__\n > 4. __Visita pronipoti__\n \n Si può implementare con una coda, in cui verranno inseriti i figli del nodo visitato da visitare ed estratti dopo avere completato la visita del livello attuale.\n \n ## Pseudocodice \n \n ```python\n def bfs(radice):\n c = Queue()\n c.enqueue(radice)\n while not c.is_empty():\n nodo = c.dequeue()\n print(nodo)\n for figlio in nodo.figli:\n nodo.enqueue(figlio)\n ```\n ")),n(a.p,null,n(a.n,null,"\n # Depth-First Search\n\n La _depth-first search_ è un algoritmo che visita **tutti i sottoalberi di un figlio** prima di passare ad un altro sfruttando la natura ricorsiva degli alberi.\n \n ## Funzionamento\n \n Ci sono diverse versioni della depth-first search: ognuna visita la radice in un momento diverso.\n \n ### DFS previsita (pre-order)\n \n La _DFS pre-visita_ visita la **radice per prima**, poi tutti i sottoalberi formati dai figli uno dopo l'altro.\n \n > 1. __Visita radice__\n > 2. dfs_pre_order(figlio1)\n > 3. dfs_pre_order(figlio2)\n \n ### Postvisita (post-order)\n \n La _DFS postvisita_ visita prima tutti i sottoalberi dei figli, e **alla fine la radice**.\n \n > 1. dfs_post_order(figlio1)\n > 2. dfs_post_order(figlio2)\n > 3. __Visita radice__\n \n ### Invisita (in-order)\n \n La _DFS invisita_ visita **un certo numero di figli**, poi la radice, e infine i figli restanti.\n \n > 1. dfs_in_order(figlio1, 1)\n > 2. __Visita radice__\n > 3. dfs_in_order(figlio2, 1)\n ")),n(a.p,null,n(a.n,null,"\n # Albero binario di ricerca\n\n ## Proprietà\n \n - Albero **binario**\n - Chiavi appartenenti ad un **insieme totalmente ordinato** (N, Q, R, ma non C)\n \x3c!----\x3e\n - Per ogni nodo con valore `x`, se un valore `v` è nel sottoalbero di sinistra allora `v ≤ x`, mentre se è nel sottoalbero di destra allora `v > x`.\n \n ## Costo computazionale\n \n - Trovare un valore: `O(h)`\n - Ordinare i valori: `O(n)`\n - Trovare il minimo: `O(h)`\n - Trovare il massimo: `O(h)`\n - Inserire un elemento: `O(h)`\n - Cancellare un elemento: `O(h)`\n \n `h` vale `log n` in un albero perfettamente bilanciato, e più l'albero diventa sbilanciato, più si avvicina a `n`, raggiungendola nel caso l'albero sia una lista.\n \n ## Pseudocodice\n \n ### Cancellazione ricorsiva\n \n ```python\n def delete(tree, key):\n if tree is not None:\n # Se ho trovato il nodo che cercavo...\n if tree.key == key:\n # E c'è una sola diramazione...\n # Semplicemente stacca il nodo come in una lista.\n if tree.left is None:\n return tree.right\n if tree.right is None:\n return tree.left\n # Altrimenti, diventa il minimo dell'albero di destra\n tree.key = tree.right.min()\n # Ed eliminalo dal sottoalbero\n tree.right = delete(tree.right, tree.key)\n # Se la chiave attuale è diversa da quella che cerchiamo, continuo a navigare l'albero\n elif tree.key < key:\n tree.left = delete(tree.left, key)\n else:\n tree.right = delete(tree.right, key)\n return tree\n ```\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/bst)\n \n ## Approfondimenti\n \n Esistono alberi più avanzati che mantengono le proprietà degli alberi binari di ricerca, ma che si autobilanciano, come il [Red Black Tree](https://it.wikipedia.org/wiki/RB-Albero).\n ")),n(a.p,null,n(a.n,null,'\n # Heap binario\n \n L\'_heap binario_ è un **albero binario bilanciato a sinistra**.\n \n ## Proprietà\n \n - _Proprietà strutturale_:\n - L\'albero è **perfettamente bilanciato** in tutti i livelli tranne l\'ultimo\n - Nell\'ultimo livello, le foglie occupano le **posizioni più a sinistra** possibili\n - _Proprietà di ordinamento_:\n - La chiave di un qualsiasi nodo è **più piccola** di tutte quelle dei nodi **del suo sottoalbero**\n \n ## Metodi\n \n ```python\n class Heap:\n def __init__(self, H): ...\n def _heapify_ancestors(self, i): "Ripristina le proprietà dell\'heap per il nodo all\'indice specificato e i suoi genitori."\n def minimum(self): "Restituisce la chiave con il valore minimo in H."\n def decrease_value(self, index, new_value): "Diminuisce il valore della chiave all\'indice index a new_value."\n def insert(self, value): "Inserisci un nuovo valore nell\'albero."\n def _heapify_children(self, i): "Ripristina le proprietà dell\'heap per il nodo all\'indice specificato e i suoi figli."\n def pop(self): "Restituisce la chiave con il valore minimo, e la elimina."\n def from_list(l): "Crea un heap da una lista."\n ```\n \n ## Implementazione con un array\n \n Possiamo implementare l\'albero utilizzando un array con le chiavi dell\'albero memorizzate nell\'ordine _breadth-first_.\n \n L\'indice del figlio sinistro può essere trovato a `2i+1`, mentre l\'indice del figlio sinistro può essere trovato a `2i+2`; il genitore è a `i//2-1`.\n \n ### Pseudocodice\n \n ```python\n class Heap:\n def __init__(self, size):\n self.array = Array(size) # Il tipo Array non esiste; consideriamolo pseudocodice\n self.next_value = 0\n \n def _heapify_ancestors(self, index)\n """Ripristina le proprietà dell\'heap per il nodo all\'indice specificato e i suoi genitori.\n \n Costo:\n O(log n)"""\n # Trovo l\'indice del genitore\n parent = index // 2 - 1\n # Controllo se viene mantenuta la proprietà di ordinamento dell\'heap\n if self.array[index] < self.array[parent]:\n # Scambio i valori dei due nodi\n self.array[index], self.array[parent] = self.array[parent], self.array[index]\n # Faccio la stessa cosa con il genitore\n self._heapify(parent)\n \n \n def minimum(self):\n """Restituisce la chiave con il valore minimo in H.\n \n Costo:\n O(1)"""\n return self.array[0]\n \n def decrease_value(self, index, new_value):\n """Diminuisce il valore della chiave all\'indice index a new_value.\n \n Costo:\n O(log n)"""\n # Diminuisco il valore del nodo\n self.array[index] = new_value\n # Aggiorno l\'heap \n self._heapify(index)\n \n def insert(self, value):\n """Inserisci un nuovo valore nell\'albero.\n \n Costo:\n O(log n)"""\n # Trovo l\'indice in cui inserire il valore\n index = self.next_index \n # Aggiungo il valore in fondo\n self.array[index] = value\n # Aggiorno l\'heap\n self._heapify(index)\n \n def _heapify_children(self, index):\n """Ripristina le proprietà dell\'heap per il nodo all\'indice specificato e i suoi figli.\n \n Costo:\n O(log n)"""\n # Trovo l\'indice dei figli\n left = index * 2 + 1\n right = index * 2 + 2\n # Mi assicuro che i figli esistano\n try:\n # Guardo quale dei figli è maggiore\n if self.array[left] > self.array[right]:\n # Scambio i valori\n self.array[left], self.array[index] = self.array[index], self.array[left]\n # Ripeto la procedura sul figlio modificato\n self._heapify_children(left)\n else:\n # Scambio i valori\n self.array[right], self.array[index] = self.array[index], self.array[right]\n # Ripeto la procedura sul figlio modificato\n self._heapify_children(left)\n except IndexError:\n # La foglia non ha figli: ho finito!\n return\n \n def pop(self):\n """Restituisce la chiave con il valore minimo, e la elimina.\n \n Costo:\n O(log n)"""\n # Mi salvo il valore della radice\n value = self.array[0]\n # Sostituisco la radice con l\'ultima foglia a destra\n self.array[0], self.array[self.next_value] = self.array[self.next_value], self.array[0]\n ### Non bisognerebbe eliminare la foglia...?\n # Riordino l\'heap\n self._heapify_children(0)\n return value\n \n @staticmethod\n def from_list(l):\n """Crea un heap da una lista.\n \n Costo:\n O(n log n), ma si può abbassare"""\n heap = Heap(len(l)))\n heap.array = Array.from_list(l) # Pseudocodice\n heap.next_value = len(l)\n # Cominciamo a riordinare l\'heap dalla fine, in modo che rispetti le proprietà\n for index in range(heap.next_value, 0, -1):\n heap._heapify_children(index)\n ```\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/heap)\n ')),n(a.p,null,n(a.n,null,"\n # Coda con priorità\n \n La _coda con priorità_ è una struttura dati dal funzionamento molto simile a quello di una coda, ma invece che restituire il primo elemento inserito, essa restituisce l'**elemento con il valore di priorità minore**.\n \n ## Proprietà\n \n - Ogni elemento è una coppia costituita da **valore** e **priorità** (un numero intero).\n - Nuovi elementi possono essere aggiunti solo tramite il **metodo `insert()`**\n - Gli elementi possono essere estratti solo tramite il **metodo `pop()`**\n - Verranno restituiti i valori inseriti secondo la strategia **Lower Priority** (l'elemento con la priorità minore sarà il primo ad essere restituito).\n - E' possibile diminuire la priorità di un elemento (e quindi anticipare la sua estrazione)\n \n ## Metodi\n \n ```python\n class PriorityQueue:\n def __init__(self): ...\n def insert(self, new_elem: Element): ...\n def minimum(self) -> Element: ...\n def pop(self) -> Element: ...\n def decrease_priority_for(self, elem: Element, priority: int): ...\n ```\n \n ## Implementazione con lista\n \n E' possibile implementare la coda con priorità tramite una **lista**: l'inserimento di nuovi valori diventerà molto efficiente, ma tutte le altre operazioni saranno linearmente lente.\n \n ### Costo computazionale\n \n - *insert()*: `O(1)`\n - *minimum()*: `O(n)`\n - *pop()*: `O(n)`\n - *decrease_priority_for()*: `O(n)`\n \n ## Implementazione con lista ordinata\n \n Implementando la coda con priorità con una **lista ordinata** si avrà un costo di ordinamento elevato negli inserimenti e modifiche alla priorità, ma costi costanti nell'estrazione di un elemento.\n \n ### Costo computazionale\n \n - *insert()*: `O(n)`\n - *minimum()*: `O(1)`\n - *pop()*: `O(1)`\n - *decrease_priority_for()*: `O(n)`\n \n ## Implementazione con heap\n \n La **soluzione migliore** è quella di implementare la coda con priorità tramite un **heap**: tutti i costi saranno logaritmici, eccetto l'inserimento che sarà costante.\n \n - `insert()` costa `O(1)`\n - `minimum()` costa `O(log n)`\n - `pop()` costa `O(log n)`\n - `decrease_priority_for()` costa `O(log n)`\n \n ## Approfondimenti\n \n Esistono code con priorità che restituiscono **l'elemento con priorità maggiore**, invece che quello minore.\n ")),n(a.p,null,n(a.n,null,"\n # Heap sort\n\n L'_heap sort_ è un algoritmo di ordinamento per confronto **iterativo**.\n \n ## Funzionamento\n \n Per effettuare un heap sort, creiamo un **heap massimo** in cui inseriamo tutti i valori che vogliamo ordinare.\n \n Una volta applicate le proprietà dell'heap, chiamiamo una versione particolare di `heap.pop()` che invece che rimuovere dall'array i valori estratti li posiziona nello spazio creatosi in fondo.\n \n ## Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | `O(n log n)` | `Ω(n log n)` | **`θ(n log n)`** |\n ")),n(a.p,null,n(a.n,null,"\n # Grafo\n \n Un _grafo_ è una struttura dati che rappresenta elementi interconnessi tra loro.\n \n Esistono due tipi di grafi: _orientati_ e _non orientati_.\n \n Per semplicità, consideriamo i nostri nodi numerati da 1 a `n`.\n \n ## Proprietà\n \n - Gli elementi sono rappresentati tramite _nodi_.\n - Il loro _grado_ è dato dal **numero degli archi** che vi incidono.\n - Se il grafo è orientato, hanno anche un _in-degree_ (**numero di archi entranti**) e un _out-degree_ (**numero di archi uscenti**).\n - Le connessioni tra elementi sono rappresentate tramite _archi_.\n - Un arco _incide_ esattamente su **due nodi**.\n - Se il grafo è orientato, sono _uscenti_ da uno dei due nodi ed _entranti_ nell'altro.\n - Sono matematicamente meno del **quadrato dei nodi**.\n \n ## Grafi particolari\n \n ### Catena\n \n Una _catena_ è un **grafo non orientato** composto da una **sequenza di nodi** aventi un **grado massimo di 2** tutti collegati tra loro.\n \n ### Cammino\n \n Un _cammino_ è un **grafo orientato** composto da una **sequenza di nodi** aventi un **in-degree** e un **out-degree** **massimo di 1**, collegati tra loro in modo che partendo dal primo e seguendo gli archi sia possibile arrivare all'ultimo.\n \n ### Cricca\n \n Una _cricca_ è un grafo in cui **tutti i nodi sono collegati tra loro**.\n \n Se il grafo è **non orientato**, la cricca ha `((n-1)n)/2` archi.\n \n Se il grafo è **orientato**, ha per ogni coppia un arco in entrambe le direzioni, quindi ha `(n-1)n` archi.\n \n ### Direct Acyclic Graph\n \n Un _DAG_ è un grafo diretto che non contiene nessun ciclo.\n \n Su di esso possiamo effettuare un ordinamento, detto _linearizzazione_, tra i nodi: otteniamo l'_ordine topologico_.\n \n I primi elementi dei DAG sono detti _Source_ (_Sorgente_), mentre gli ultimi sono detti _Sink_ (_Pozzo_).\n \n #### Albero\n \n Un **albero** può essere considerato un DAG con una **sorgente singola** e le **foglie come pozzi**.\n \n ### Grafo fortemente connesso\n \n Un insieme di nodi `V` di un **grafo diretto** `G` si dice una _componente fortemente connessa_ se:\n \n 1. Per ogni coppia di nodi `∀ u, v ∈ V' : ∃ un cammino u->v in G'`\n 2. Massimale (non può diventare più grande)\n \n > Praticamente una componente fortemente connessa è un gruppo di nodi tra i quali si può viaggiare liberamente da e a qualsiasi nodo al suo interno.\n \n Un grafo si dice _fortemente connesso_ se l'insieme `V` coincide con l'insieme dei nodi del grafo `G`.\n \n > Se partendo da qualsiasi nodo di un grafo riesco ad arrivare a qualsiasi altro nodo, allora il grafo è fortemente connesso.\n \n Inoltre, se creiamo un nuovo grafo, in cui **ogni nodo rappresenta una componente fortemente connessa** del nostro grafo iniziale, **otteniamo un DAG**, perchè tutti i cicli sono stati integrati nella componente.\n \n ### Trasposto di un grafo\n \n Il _trasposto_ di un **grafo diretto** `G` è il grafo stesso con gli archi che però vanno nella **direzione opposta**.\n \n ### Grafo pesato\n \n Un _grafo pesato_ è un particolare grafo che associa a ciascun arco un **costo** per attraversarlo.\n \n #### Costi negativi\n \n I costi possono anche essere negativi: rappresenteranno allora un **guadagno** ottenuto attraversando il nodo.\n \n ### Minimum spanning tree\n \n Un _minimum spanning tree_ è il **sottoinsieme degli archi** di un **grafo non diretto** che **connettono tutti i nodi** con il **minor costo possibile**.\n \n I MST hanno [molte proprietà](https://en.wikipedia.org/wiki/Minimum_spanning_tree#Properties); sono troppe da scrivere qui, e probabilmente non ci interesseranno nemmeno.\n \n ## Implementazione tramite matrice di adiacenza\n \n Possiamo implementare un grafo creando una **matrice di `bool` di dimensione `n * n`** in cui le **caselle collegate sono vere** e le caselle non collegate sono false.\n \n > Ad esempio, possiamo implementare un grafo non orientato in questo modo (`█` indica l'esistenza di un collegamento e ` ` indica la sua assenza):\n >\n > | |1|2|3|\n > |-|-|-|-|\n > |1|░|░|░|\n > |2|█|░|░|\n > |3|█| |░|\n >\n > Esistono gli archi `1-2` e `1-3`, ma non esiste un collegamento `2-3`.\n \n > Un grafo orientato invece si può implementare così:\n >\n > | |1|2|3|\n > |-|-|-|-|\n > |1|░|█| |\n > |2|█|░| |\n > |3|█| |░|\n >\n > Esistono gli archi `1->2`, `2->1` e `3->1`, ma non ci sono collegamenti `2->3`, `1->3` e `3->2`.\n \n ### Costo computazionale\n \n #### Tempo\n \n Le matrici di adiacenza portano alla realizzazione di algoritmi molto veloci: verificare l'esistenza di un arco è in `O(1)`!\n \n Abbiamo però penalità significative quando vogliamo effettuare operazioni sugli archi: ad esempio, trovare il trasposto di un grafo implementato con una matrice di adiacenza è in `O(nodi²)`.\n \n #### Memoria\n \n E' poco efficiente in quanto a memoria: l'upper bound è in `O(n^2)`. \n \n ## Implementazione tramite liste di adiacenza\n \n Un'alternativa alla matrice di adiacenza è quella di creare un'**array di liste**, le quali contengono i **vicini di ciascun nodo**.\n \n > |Posizione|Lista|\n > |-|-|\n > |1|[2, 3]|\n > |2|[]|\n > |3|[1]|\n >\n > Esistono gli archi `1->2`, `1->3`, e `3->1`, ma non esistono `2->1`, `2->3` e `3->2`.\n \n ### Costo computazionale\n \n #### Tempo\n \n Utilizzando le liste di adiacenza, il tempo richiesto per verificare l'esistenza di un arco sale a `O(max-out-degree)`.\n \n E' efficace però quando il problema che vogliamo risolvere riguarda operazioni su archi: trovare la trasposta è in `O(archi)`.\n \n #### Memoria \n \n La memoria richiesta dalle liste di adiacenza è minore di quella delle matrici: l'upper bound è in `O(nodi + archi)`.\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/graphds)\n ")),n(a.p,null,n(a.n,null,"\n # Visitare un grafo\n \n Come per gli alberi radicati, esistono due modi per visitare un grafo: _depth-first search_ e _breadth-first search_.\n \n In entrambi i casi, non visito mai due volte lo stesso nodo, e come risultato ottengo **molteplici alberi**, il cui insieme viene detto _foresta di copertura_.\n \n Se il grafo che vogliamo visitare è diretto, allora dobbiamo **considerare come vicini solo gli archi uscenti**\n \n ## Depth-first search\n \n La DFS ci può risultare utile per **identificare le componenti connesse** di un grafo e **identificare eventuali cicli**.\n \n ### Funzionamento\n \n Posso utilizzare la DFS per classificare gli archi di un grafo in quattro categorie:\n \n - _Tree_, archi che ci fanno **scoprire un nuovo nodo**\n - _Forward_, archi che ci portano a un **discendente**\n - _Back_, archi che ci portano ad un **antenato**\n - _Cross_, archi che **connettono due sottoalberi** diversi\n \n Usiamo due array inizializzati a 0 chiamati `pre` e `post`, grandi quanto il numero di archi del grafo, che ci indicano rispettivamente quando un nodo è stato scoperto e quando è terminata la visita. \n Inoltre, creiamo una variabile `clock` che avanza ad ogni evento. \n Alla scoperta di un nuovo nodo, mettiamo il valore attuale di `clock` all'interno di `pre[n]`. \n Alla fine della visita di un nodo invece mettiamo il valore di `clock` in `post[n]`.\n \n **Durante la visita**, gli archi avranno i seguenti valori:\n - _Tree_: `pre[dst] == 0`\n - _Forward_: `pre[src] < pre[dst] && post[dst] > 0`\n - _Back_: `pre[dst] < pre[src] && post[dst] == 0`\n - _Cross_: Tutti gli altri (`post[dst] < pre[src]`)\n \n **A fine visita**, gli archi avranno i seguenti valori:\n - _Tree_: `pre[dst] < pre[dst] < post[dst] < pre[src]`\n - _Forward_: `pre[dst] < pre[dst] < post[dst] < pre[src]`\n - _Back_: `pre[src] < pre[dst] < post[dst] < post[src]`\n - _Cross_: `pre[dst] < post[dst] < pre[src] < post[src]`\n \n Se un **grafo non diretto** contiene degli **archi Back**, allora esso **conterrà un ciclo**.\n \n #### DFS nel grafo trasposto\n \n Se effettuo una DFS sul trasposto di un grafo, posso **scoprire i nodi che hanno un cammino verso l'origine**.\n \n #### DFS nella componente fortemente connessa\n \n Se effettuo una DFS in una componente fortemente connessa e nella sua trasposta, il **`post` della trasposta sarà sempre minore** del `post` della componente originale.\n \n ### Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | `O(nodi + archi)` | `Ω(nodi + archi)` | **`θ(nodi + archi)`** |\n | Memoria | `O(nodi)` | `Ω(nodi)` | **`θ(nodi)`** |\n \n ### Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/dfsbfs)\n \n ## Breadth-first search\n \n La BFS ci può risultare utile per **trovare tutti i nodi a una certa distanza** da un'origine.\n \n ### Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | `O(nodi + archi)` | `Ω(nodi + archi)` | **`θ(nodi + archi)`** |\n | Memoria | `O(nodi + archi)` | `Ω(nodi + archi)` | **`θ(nodi + archi)`** |\n \n ### Pseudocodice\n \n Come per gli alberi, la implementiamo in modo **iterativo**:\n \n ```python\n queue = [starting_node]\n parents = [None for node in graph.nodes]\n distance = [-1 for node in graph.nodes]\n \n # TODO: controllami quando sei più sveglio\n \n while queue:\n node, source, distance = queue.pop(0)\n parents[node.number] = source\n distance[node.number] = distance\n for neighbour in node.neighbours:\n queue.append((neighbour, node, distance+1))\n ```\n \n > Nella coda, la distanza massima tra un nodo e l'altro è 1.\n \n ### Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/dfsbfs)\n ")),n(a.p,null,n(a.n,null,"\n # Algoritmi greedy\n\n Un modo per risolvere problemi algoritmici può essere usare una **tecnica** _greedy_.\n \n Le tecniche greedy consistono nel effettuare tanti piccoli passi, ed effettuare una **scelta** in base ai dati **locali al passo attuale**.\n \n > Scegli il numero di monete più piccole possibili per comporre €1.12.\n >\n > L'algoritmo cerca di scegliere sempre la moneta più grande possibile compatibile con il prezzo in quel momento, quindi:\n > | Moneta scelta | Rimanente |\n > |---------------|-----------|\n > | € 1.00 | € 0.12 |\n > | € 0.10 | € 0.02 |\n > | € 0.02 | € 0.00 |\n \n ## Esempi\n \n Sono algoritmi greedy:\n \n - L'_Algoritmo di Dijkstra_\n - L'_Algoritmo di Kruskal_\n - L'_Algoritmo di Prim_\n ")),n(a.p,null,n(a.n,null,"\n # Percorso più breve\n\n Trovare il _percorso più breve_ (o _cammino minimo_) tra due nodi di un **grafo pesato** è un problema frequente nell'informatica; per questo, sono stati sviluppati [numerosi algoritmi](https://en.wikipedia.org/wiki/Shortest_path_problem) per risolverlo.\n \n > Ad esempio, il pathfinding dei nemici nei videogiochi, oppure Google Maps!\n \n ## Percorso più breve da una sorgente singola\n \n Una sottocategoria del problema del percorso più breve è il caso in cui ci interessa sapere i percorsi più brevi che **partono da uno specifico nodo del grafo**: è detto problema del _percorso più breve da una sorgente singola_, o _single-source shortest path_.\n \n Si può notare che se il grafo contiene **costi negativi** allora è possibile che il percorso più breve non esista, in quanto diventa possibile la comparsa di **cicli di costo infinitamente negativo**.\n \n Possiamo notare che, se il percorso più breve tra `A` e `D` è `A-B-C-D`, allora il cammino minimo tra `B` e `D` passerà obbligatoriamente per `C` (`B-C-D`).\n Diremo più avanti che il percorso più breve ha una **sottostruttura ottimale**.\n \n ### Esempi\n \n Alcuni algoritmi che trovano il percorso più breve sono:\n - L'_Algoritmo di Dijkstra_\n - L'_Algoritmo di Bellman-Ford_\n - La [_ricerca A*_](https://en.wikipedia.org/wiki/A*_search_algorithm)\n ")),n(a.p,null,n(a.n,null,"\n # Algoritmo di Dijkstra\n \n L'_Algoritmo di [Dijkstra](https://upload.wikimedia.org/wikipedia/commons/8/85/Dijkstra.ogg)_ è un algoritmo che risolve il problema del **percorso più breve da una sorgente singola** per grafi con pesi **reali positivi** `\\mathbb{R}^+`.\n \n L'algoritmo trova tutti i percorsi più brevi per raggiungere qualsiasi nodo del grafo partendo da un dato nodo, assieme al costo richiesto per farlo.\n \n ## Funzionamento\n \n 1. Separiamo tutti i nodi del grafo in due gruppi: **visitati** e **non visitati**. \n - Tutti i nodi partono da **non visitati**.\n 2. Per ogni nodo, manteniamo un valore \"**costo richiesto per raggiungerlo**\", che verrà cambiato man mano che l'algoritmo avanza.\n - Il costo di partenza è `+∞`.\n - Il costo sarà **definitivo per i nodi visitati**, e **provvisorio per i non visitati**.\n 3. Creiamo un insieme detto _frontiera_ che conterrà tutti i **nodi non visitati adiacenti** a quelli visitati.\n 4. Prendiamo il nodo iniziale, che avrà un **costo di `0`**, e definiamolo il nodo _attuale_.\n 5. Finchè ci sono dei nodi non sono stati visitati, ripetiamo il seguente ciclo:\n 1. Aggiungiamo i nodi adiacenti al nodo attuale alla frontiera.\n - Il costo per raggiungerli sarà il **costo per il nodo attuale sommato al costo dell'arco** che li connette al nodo attuale. \n Se questo **costo** risulta essere **minore del costo provvisorio** precedente, esso **diventerà il nuovo costo**.\n - Questa operazione è detta _rilassamento dell'arco_.\n 2. Facciamo diventare **visitato** il nodo attuale.\n - Il percorso che abbiamo fatto per raggiungerlo è obbligatoriamente il più breve.\n 3. Il prossimo nodo attuale sarà il nodo di frontiera con un costo più basso.\n - Per questo, è possibile definire l'algoritmo di Dijkstra come un **algoritmo greedy**.\n \n ### Non funziona se...\n \n L'algoritmo smette di funzionare nel caso in cui siano presenti **costi negativi** e il grafo non sia **aciclico**, in quanto non saremmo mai in grado di rendere visitato un nodo.\n \n ## Costo computazionale\n \n | Categoria | Upper bound |\n |-----------|-------------|\n | Tempo | `O(nodi + archi) log nodi)` |\n \n ### Scomposizione\n \n - Inizializzazione: `O(nodi)`\n - Creazione coda priorità: `O(nodi log nodi)`\n - Ciclo: `O((nodi + archi) log nodi)`\n \n ## Pseudocodice\n \n ```python\n import math\n \n class Info:\n def __init__(self, distance=math.inf, previous=None):\n self.distance = distance\n self.previous = previous\n \n def dijkstra(graph, start):\n data = [Info() for node in graph.nodes]\n queue = PriorityQueue([start])\n while queue:\n node = queue.pop()\n for arc in node.connections:\n other = arc.other(node)\n if data[node.number].distance + arc.cost < data[other].distance:\n data[other].distance = data[node.number].distance + arc.cost\n queue.decrease_priority_for(other, data[other].distance)\n data[v].previous = node\n return data\n ```\n \n ## Visualizzazione\n \n [Visualgo](https://visualgo.net/en/sssp)\n ")),n(a.p,null,n(a.n,null,"\n # Algoritmo di Bellman-Ford\n\n L'_Algoritmo di Bellman-Ford_ è un algoritmo che, come l'Algoritmo di Dijkstra, risolve il problema del **percorso più breve da una sorgente singola**, però, a differenza da quest'ultimo, l'Algoritmo di Bellman-Ford accetta in input anche grafi con pesi **reali** `\\mathbb{R}` (sia positivi, sia negativi).\n \n ## Funzionamento\n \n L'approccio dell'algoritmo è simile a quello di Dijkstra: entrambi usano il **rilassamento** degli archi per ottenere un costo provvisorio per il raggiungimento di un nodo, ma invece che rilassare solo l'arco con costo inferiore, questo algoritmo **rilassa tutti gli archi** ripetutamente, eliminando la frontiera e il problema dei nodi negativi.\n \n L'operazione di rilassamento è ripetuta `nodi - 1` volte, ovvero la **lunghezza massima** di un cammino aciclico all'interno di un grafo.\n \n Possiamo individuare dopo i rilassamenti se è presente un nodo con un **ciclo negativo**: ci basta controllare se esiste un arco che connette due nodi con una distanza incompatibile: se `a.distanza + arco.costo < b.distanza`, allora è presente un ciclo negativo.\n \n ## Costo computazionale\n \n | Categoria | Upper bound |\n |-----------|-------------|\n | Tempo | `O(nodi * archi)` |\n \n ## Pseudocodice\n \n > TODO\n \n ## Visualizzazione\n \n [Visualgo](https://visualgo.net/en/sssp)\n ")),n(a.p,null,n(a.n,null,'\n # Disjoint set\n \n Il _disjoint set_ è una struttura dati che rappresenta elementi di un insieme raggruppati in **sottoinsiemi disgiunti**.\n \n ## Metodi\n \n ```python\n class DisjointNode:\n def __init__(self): ...\n def find_set(self): "Trova il rappresentante dell\'elemento."\n def union(self, other): "Unisce i sottoinsiemi che contengono questo nodo e `other`."\n ```\n \n ## Implementazione tramite array\n \n Possiamo implementare il disjoint set con due array: uno per l\'**indice del rappresentante** e uno per il **rango dell\'insieme**.\n \n Un singoletto avrà **sè stesso come rappresentante** e **rango `0`**.\n \n ### Costo computazionale\n \n - *create_set()*: `O(1)`\n - *find_set()*: `O(h)`\n - *union()*: `O(h)`\n \n ### Pseudocodice\n \n ```python\n class DisjointNode:\n def __init__(self):\n self.parent = self\n self.rank = 0\n \n def find_set(self):\n element = self\n while self.parent != element:\n element = self.parent\n return element\n \n def union(self, other):\n repres_self = self.find_set()\n repres_other = other.find_set()\n if repres_self == repres_other:\n return\n if repres_self.rank < repres_other.rank:\n repres_greater = repres_other\n repres_lesser = repres_self\n else:\n repres_greater = repres_self\n repres_lesser = repres_other\n repres_lesser.parent = repres_greater\n if repres_greater.rank == repres_lesser.rank:\n repres_greater.rank += 1\n ```\n \n ### Visualizzazione\n \n [cs.usfca.edu](https://www.cs.usfca.edu/~galles/JavascriptVisual/DisjointSets.html)\n ')),n(a.p,null,n(a.n,null,"\n # Trovare il minimum spanning tree\n\n Un altro problema ricorrente riguardante i grafi è trovare il _minimum spanning tree_ di un dato grafo non diretto.\n \n > E' utile per trovare il modo più efficiente per connettere le cose: ad esempio, per decidere la struttura di una rete internet!\n \n ## Esempi\n \n Gli algoritmi principali che risolvono il problema sono due, ed entrambi sono **algoritmi greedy**:\n \n - L'_Algoritmo di Kruskal_\n - L'_Algoritmo di Prim_\n ")),n(a.p,null,n(a.n,null,"\n # Algoritmo di Kruskal\n \n L'_Algoritmo di Kruskal_ è un algoritmo **greedy** che **trova il minimum spanning tree** di un grafo.\n \n ## Funzionamento\n \n 1. Ripetiamo questa procedura finchè tutti i nodi non sono connessi:\n 1. Prendiamo ad ogni passo **l'arco meno costoso** del grafo non ancora aggiunto all'insieme.\n 2. Assicuriamoci che **non si creino cicli**: se non se ne verrebbero a creare, possiamo **aggiungere l'arco all'insieme**.\n - Gli archi devono quindi connettere nodi in **componenti connesse diverse**.\n - Possiamo rappresentare le componenti connesse con un **Disjoint Set**.\n \n ## Costo computazionale\n \n | Categoria | Upper bound |\n |-----------|-------------|\n | Tempo | `O(archi²)` |\n \n ### Scomposizione\n \n - DisjointSet.__init__(): `O(archi)`\n - Per ogni ciclo: `O(archi²)`\n - DisjointSet.find_set(): `O(1)`\n - DisjointSet.union(): `O(archi)`\n \n ## Pseudocodice\n \n ```python\n def minimum_spanning_tree_kruskal(graph):\n ds = DisjointSet()\n for node in graph.nodes:\n ds.create_set(node)\n arcs = []\n sorted_arcs = sorted(graph.arcs, key=lambda arc: arc.cost)\n for arc in sorted_arcs:\n if ds.find_set(node.start) != ds.find_set(node.end):\n arcs.append(arc)\n ds.union(node.start, node.end)\n return arcs\n ```\n ")),n(a.p,null,n(a.n,null,"\n # Algoritmo di Prim\n \n L'_Algoritmo di Prim_ è un altro algoritmo **greedy** che **trova il minimum spanning tree** di un grafo.\n \n ## Funzionamento\n \n Creo una **coda con priorità** in cui inserisco tutti gli archi visibili dal mio albero, in cui la chiave è il **costo dell'arco**.\n \n Per trovare l'arco con costo più piccolo posso **estrarre un arco** dalla coda: la priorità ci garantisce che esso è l'**arco meno costoso**.\n \n Aggiungo allora un nuovo nodo all'albero, e con esso, **aggiungo alla coda** tutti gli **archi che scoprono un nuovo nodo**.\n \n ## Costo computazionale\n \n | Categoria | Upper bound |\n |-----------|-------------|\n | Tempo | `O(archi + nodi log nodi)` |\n \n ## Pseudocodice\n \n ```python\n import math\n \n def minimum_spanning_tree_prim(graph, cost_array, start_node):\n # E' un Array di bool: se l'indice corrispondente al nodo è uguale a true, vuol dire che il (nodo è contenuto nell'albero.\n contains = [False for _ in range(len(graph))]\n # Contiene il precedente di ogni nodo \n prev = [None for _ in range(len(graph))]\n # Contiene il costo per arrivare a quel nodo\n cost = [math.inf for _ in range(len(graph))]\n # Creo la priority queue\n pq = PriorityQueue(graph.arcs, key=lambda arc: arc.cost)\n # Parto dal nodo `start_index`\n # Il costo dell'origine è 0.\n cost[start_node.index] = 0\n contains[start_node.index] = True\n \n while not pq.is_empty():\n new_node = pq.pop()\n contains[new_node.index] = True\n for arc in new_node.connections:\n other_node = arc.other(new_node)\n if not contains[other_node.index] and cost[other_node.index] > arc.cost:\n cost[other_node.index] = arc.cost\n prev[other_node.index] = new_node\n pq.decrease_priority_for(other_node, arc.cost)\n # L'array di prev rappresenta un albero.\n return prev\n ```_\n ")),n(a.p,null,n(a.n,null,"\n # Compressione\n\n _Comprimere_ un file significa **ridurne le dimensioni** senza modificarne il significato.\n \n ## Categorie\n \n ### Compressione lossless\n \n Nella _compressione lossless_, i dati possono essere decompressi riottenendo una copia identica dell'originale.\n \n > Immaginiamo un file che contiene solo le lettere `a, b, c, d, e, f`.\n >\n > Le lettere compaiono con questa frequenza:\n > |a|b|c|d|e|f|\n > |45%|13%|12%|16%|9%|5%|\n >\n > Possiamo codificare le lettere nel seguente modo:\n > |a|b|c|d|e|f|\n > |`0b0`|`0b100`|`0b101`|`0b111`|`0b1100`|`0b1101`|\n >\n > Scrivere `abacadae` richiederebbe 64 bits con la codifica ASCII estesa, ma in questo modo riusciamo a scriverlo con soli 17 bits!\n \n Le codifiche di un file compresso devono rispettare la proprietà del _Codice a prefisso_, che dice che **nessun codice deve essere il prefisso di un altro codice**; altrimenti, si avrebbero ambiguità nella decodifica.\n \n > a = `1`\n > b = `11`\n >\n > `111` è `ab`, `ba` oppure `aaa`?\n \n Creiamo allora un _albero di decodifica_: un **albero binario** che, leggendo uno ad uno i bit codificati, ci permette di arrivare al **valore del codice presente sulle foglie** dell'albero.\n \n Gli alberi di decodifica sono sempre **completi**.\n \n > Un albero di decodifica incompleto sarebbe non ottimizzato!\n \n #### Esempi\n \n - .png\n - .flac\n - .zip\n - ...\n \n ### Compressione lossy\n \n Nella _compressione lossy_, alcuni dati [solitamente](http://needsmorejpeg.com/) irrilevanti vengono perduti: non si può, dunque, ricostruire l'originale.\n \n #### Esempi\n \n - .jpeg\n - .mp3\n - ...\n ")),n(a.p,null,n(a.n,null,"\n # Algoritmo di Huffman\n\n L'_Algoritmo di Huffman_ è un **algoritmo greedy** per la **costruzione di un albero di decodifica**.\n \n ## Funzionamento\n \n 1. Costruisco **un albero** (con un solo nodo) **per ogni elemento dell'alfabeto**.\n 2. Associo ad **ogni albero la frequenza dell'elemento** da cui è stato creato, per poi inserire tutti gli elementi in una coda con priorità.\n 3. Finchè non ho **un albero solo**:\n 1. Estraggo dalla coda i **due alberi con frequenza minore**.\n 2. **Li rendo fratelli**, creando un nuovo nodo in cui sono uno figlio destro e uno figlio sinistro.\n 3. Associo al nuovo nodo la **somma delle frequenze dei due alberi**, e inserisco il nuovo albero nella coda.\n \n > È molto raro che venga un albero \"dritto\"; se succede, probabilmente c'è qualcosa che non va.\n ")),n(a.p,null,n(a.n,null,"\n # Dizionario\n \n Un _dizionario_ è una struttura dati che **associa dei valori a delle chiavi**.\n \n ## Proprietà\n \n - Ogni elemento del dizionario è un **valore** che è stato **associato a una chiave**.\n - Possiamo aggiungere nuovi elementi con il **metodo `add(chiave, valore)`**.\n - Posiamo estrarre elementi con il **metodo `get(chiave)`**, che restituirà il valore associato a `chiave`.\n - E' possibile rimuovere elementi con il **metodo `delete(chiave)`**.\n - Due elementi con **chiavi diverse** non devono **mai restituire lo stesso valore**.\n \n ## Metodi\n \n Beh... Non ha molto senso in questo caso...\n \n ```python\n dict()\n ```\n \n ## Implementazione con tabella hash\n \n Una _tabella hash_ è un'array di coppie **chiave-valore**, che formano l'insieme _universo_.\n \n Per determinare l'**indice dell'array** in cui inserire una coppia, usiamo una [funzione _hash_](https://it.wikipedia.org/wiki/Hash#Algoritmo_di_hash) sulla chiave, che restituirà **numeri da `0` a `len(hash_table)`**.\n \n ### Risoluzione collisioni\n \n Potrebbe capitare però che **due chiavi diverse abbiano lo stesso indice**. Dobbiamo allora usare un metodo di _risoluzione collisioni_, che mi permetta di distinguere tra chiavi diverse.\n \n #### Lista di trabocco\n \n Possiamo salvare nell'array **liste di coppie** chiave-valore; in caso di collisione, **aggiungo un nuovo elemento alla lista**.\n \n In media, ciascuna di queste liste conterrà `elementi_inseriti / dimensione_tabella` elementi.\n \n #### Indirizzamento aperto\n \n Possiamo decidere di mettere le coppie che non trovano posto nel loro indice in **un altro indirizzo vuoto** dell'array.\n \n Ci sono diversi modi in cui decidere il nuovo indirizzo, ognuno con vantaggi e svantaggi: si può scegliere quello successivo, oppure il primo vuoto dell'array, oppure un indirizzo casuale.\n \n > Python, nei `dict`, usa indirizzamento aperto pseudorandom.\n \n ### Costo computazionale\n \n - Aggiungere una chiave: `O(n)`\n - Trovare una chiave: `O(n)`\n - Eliminare una chiave: `O(n)`\n ")),n(a.p,null,n(a.n,null,"\n # Programmazione dinamica\n \n La _programmazione dinamica_ è una **tecnica** di programmazione che prevede l'**estensione di una soluzione ottima precedente**.\n \n Tutti i problemi in cui si può applicare si possono risolvere anche con la **ricorsione**, ma a differenza della ricorsione, questa tecnica riesce ad evitare di ricalcolare la soluzione per ogni chiamata ricorsiva, ottenendo quindi tempi molto migliori.\n \n Si può applicare solo se un problema ha una **sottostruttura ottimale**, ovvero se la soluzione ottima di un sottoproblema è inclusa nella soluzione ottima del problema. \n \n ## Esempi\n \n - _Problema dello zaino_\n - ...\n \n > Il cammino minimo per raggiungere un nodo in un DAG è dato da `arco.costo + arco.primo_nodo.costo_cammino_minimo()`.\n >\n > ```python\n > def SPD_PD(graph, start):\n > distance = [float(inf) for node in graph.nodes]:\n > distance[start] = 0\n > # I nodi devono essere in ordine di linearizzazione\n > for node in graph.nodes:\n > distance[node] = min([(arc.cost + distance[arc.other(node)] for arc in node.connections])\n > ```\n \n > Ho una sequenza di interi da `a_1` a `a_n`. Voglio trovare la sottosequenza crescente più lunga.\n >\n > 5 2 3 4 7 3 6 3 1 6\n >\n > Trovo tutte le sequenze lunghe 1, e le rendo nodi di un grafo diretto.\n >\n > Da ogni nodo, creo una connessione verso i suoi maggiori.\n >\n > Infine, cerco i cammini massimi del grafo. \n > \n > Essi saranno la soluzione del problema.\n \n > Trova la lunghezza della sottosequenza più lunga che termina con `j`.\n >\n > `L[j]` = lunghezza della sottosequenza più lunga che termina in `j`\n > \n > ```python\n > L[j] = max([1 + L[node] for arc in node.connections)]\n > ```\n >\n > Esempio:\n > ```python\n > L[9] = max([1+L[8], 1+L[3], 1+L[6]])\n > ```\n ")),n(a.p,null,n(a.n,null,"\n # Problema dello zaino\n \n Il problema dello zaino è un problema _pseudo-trattabile_: non abbiamo dimostrazioni di se sia trattabile o intrattabile.\n \n ## Descrizione\n \n > Sei un ladro, e devi mettere **più refurtiva possibile** nello zaino per scappare. \n > Lo zaino può portare **al massimo `dim` kili**.\n >\n > **Quali** (e quanti) oggetti scegli?\n \n | Input | Output |\n |-------|--------|\n | `dim`ensione_zaino, `n`umero_oggetti, `oggetto.peso`, `oggetto.valore` | `profitto_massimo` |\n \n ## Categorie\n \n ### Problema con ripetizione\n \n Puoi prendere **tutte le copie che vuoi** di un oggetto.\n \n #### Soluzione\n \n `K(dim)` è il valore massimo ottenibile con uno zaino di capacità `dim`.\n \n > Se `i` appartenesse alla soluzione ottima, allora `K(dim) = i.valore + K(dim - i.peso)`...\n \n Possiamo dire che `K(dim) = max(i.valore + K(dim - i.peso))`.\n \n Inoltre, `K(0) = 0`.\n \n Ci salviamo tutte le soluzioni da `K(0)` a `K(dim)`, e le usiamo per calcolare il massimo in seguito.\n \n Calcolare `K(dim)` avrà allora un costo di `O(n * dim)`:\n - `n`, perchè trovare il massimo è un'operazione lineare\n - `dim`, perchè `dim` sono tutti i casi tra i quali devo andare a provare\n \n Il costo computazionale, allora, è in `O(n * dim)`. \n \n Però, il **tempo richiesto** dal nostro algoritmo dipende non dalla lunghezza dell'input, bensì dal **valore numerico** di `dim`, che corrisponde alla dimensione dell'array delle soluzioni. \n Allora, si dice che l'algoritmo è in **tempo _pseudo-polinomiale_**.\n \n ### Problema senza ripetizione\n \n Si può prendere **ogni oggetto una volta sola**.\n \n #### Soluzione _bruteforce_\n \n Scelgo se prendere o no l'item 1.\n \n Si creano due percorsi:\n - Non prendo l'oggetto: `valore = 0, peso = 0`\n - Prendo l'oggetto: `valore = oggetto.valore, peso = oggetto.peso`\n \n Continuo a creare percorsi, creando una specie di albero binario.\n \n Se a un certo punto vedo che `valore = x, peso = K` e `valore < x, peso = K`, allora posso escludere automaticamente tutto il sottoalbero destro, perchè non può essere migliore del sinistro: allora, sarò riuscito a ridurre il numero dei casi rispetto alla ricorsione.\n \n ### Problema in due variabili\n \n `K(j, w)` = massimo valore ottenibile con uno zaino di capacità `w` scegliendo gli item da `1` a `j`.\n \n Non possiamo più applicare la soluzione bruteforce, perchè abbiamo due variabili, `j` e `w`.\n \n Allora, prendo l'elemento `j`. Esso può essere o non essere nella soluzione: mi calcolo entrambe le alternative, e mi tengo l'alternativa dal valore più alto.\n \n Se `j` non è nella soluzione, il risultato diventerà `K(j-1, w)`; se invece è nella soluzione, il risultato sarà `j.valore + K(j-1, w-j.peso)`.\n \n In pratica, prendiamo\n \n ```latex\n K(j, w) = max\n \\begin{cases}\n V_j + K(j-1, w-w_j)\n K(j-1, w)\n \\end{cases}\n ```\n \n Costruisco allora una matrice con `j` su un asse e `w` sull'altro. \n Riempio le caselle con il valore di `K(j, w)`. \n Nella casella con `K(j, w)` avremo la soluzione ottima.\n \n Il tempo necessario per riempire tutte le caselle è nuovamente `O(n * w)`, ancora **pseudopolinomiale**.\n \n Per sapere che oggetti ho messo o no devo tenere traccia in qualche modo della catena del calcolo, usando, ad esempio, una pila.\n ")),n(a.p,null,n(a.n,null,"\n # Problemi intrattabili\n \n ## Problema di Set-Cover\n \n ### Input\n \n `U`niverso di `e`lementi\n \n `S`ottoinsieme di `s`ottoinsiemi di elementi di `U`\n \n ### Output\n \n Il minimo `S'`ottoinsieme di `s`ottoinsiemi che copra completamente `U`.\n \n ### Soluzione in `O(n^d)`\n \n Non c'è.\n \n ### Non-soluzione alternativa\n \n Faccio una scelta greedy, ma non posso dimostrare in alcun modo che la soluzione ottenuta sia quella ottima. \n Infatti, l'algoritmo non dà sempre la soluzione ottima, ma dà una soluzione accettabile in tempo polinomiale.\n \n Seleziono sempre il sottoinsieme che copre più elementi mancanti possibili.\n \n #### Costo computazionale\n \n `Costo greedy <= log(numero_elementi) * Costo ottimo`\n ")))}}.call(this,i("hosL").h)}}]); +//# sourceMappingURL=route-AlgoritmiEStruttureDati.chunk.56b0d.js.map \ No newline at end of file diff --git a/docs/route-AlgoritmiEStruttureDati.chunk.d1ce1.js.map b/docs/route-AlgoritmiEStruttureDati.chunk.56b0d.js.map similarity index 99% rename from docs/route-AlgoritmiEStruttureDati.chunk.d1ce1.js.map rename to docs/route-AlgoritmiEStruttureDati.chunk.56b0d.js.map index a90caad..74e4735 100644 --- a/docs/route-AlgoritmiEStruttureDati.chunk.d1ce1.js.map +++ b/docs/route-AlgoritmiEStruttureDati.chunk.56b0d.js.map @@ -1 +1 @@ -{"version":3,"sources":["webpack:///./routes/AlgoritmiEStruttureDati.js"],"names":["r","String","raw","title"],"mappings":"s+GAEMA,EAAIC,OAAOC,IAGF,qBACX,OACI,aACI,EAAC,IAAD,CAAOC,MAAO,0BAAd,uIAGA,EAAC,IAAD,KACI,EAAC,IAAD,KAAWH,EAAX,OA+EJ,EAAC,IAAD,KACI,EAAC,IAAD,guFA8CJ,EAAC,IAAD,KACI,EAAC,IAAD,ipGA6CJ,EAAC,IAAD,KACI,EAAC,IAAD,+1BAiBJ,EAAC,IAAD,KACI,EAAC,IAAD,2rPA0IJ,EAAC,IAAD,KACI,EAAC,IAAD,mtGA6CJ,EAAC,IAAD,KACI,EAAC,IAAD,oUAQJ,EAAC,IAAD,KACI,EAAC,IAAD,o1DAgCJ,EAAC,IAAD,KACI,EAAC,IAAD,0yBAoBJ,EAAC,IAAD,KACI,EAAC,IAAD,0/BAmBJ,EAAC,IAAD,KACI,EAAC,IAAD,qgGAqDJ,EAAC,IAAD,KACI,EAAC,IAAD,4gGA6CJ,EAAC,IAAD,KACI,EAAC,IAAD,+pJAyGJ,EAAC,IAAD,KACI,EAAC,IAAD,29MAsIJ,EAAC,IAAD,KACI,EAAC,IAAD,0uLAkGJ,EAAC,IAAD,KACI,EAAC,IAAD,kpDA2BJ,EAAC,IAAD,KACI,EAAC,IAAD,6mCAuBJ,EAAC,IAAD,KACI,EAAC,IAAD,qyEAyCJ,EAAC,IAAD,KACI,EAAC,IAAD,g/DAmCJ,EAAC,IAAD,KACI,EAAC,IAAD,uzDAqCJ,EAAC,IAAD,KACI,EAAC,IAAD,8rIAgFJ,EAAC,IAAD,KACI,EAAC,IAAD,uqCA4BJ,EAAC,IAAD,KACI,EAAC,IAAD,yjDAkCJ,EAAC,IAAD,KACI,EAAC,IAAD,m5FAyDJ,EAAC,IAAD,KACI,EAAC,IAAD,q3PA+IJ,EAAC,IAAD,KACI,EAAC,IAAD,6zGA4DJ,EAAC,IAAD,KACI,EAAC,IAAD,o6BAkBJ,EAAC,IAAD,KACI,EAAC,IAAD,m0QAmJJ,EAAC,IAAD,KACI,EAAC,IAAD,uhLAiGJ,EAAC,IAAD,KACI,EAAC,IAAD,soCAyBJ,EAAC,IAAD,KACI,EAAC,IAAD,2tDAwBJ,EAAC,IAAD,KACI,EAAC,IAAD,s5IAuEJ,EAAC,IAAD,KACI,EAAC,IAAD,8yDA4BJ,EAAC,IAAD,KACI,EAAC,IAAD,myFA6DJ,EAAC,IAAD,KACI,EAAC,IAAD,msBAeJ,EAAC,IAAD,KACI,EAAC,IAAD,4mEA2CJ,EAAC,IAAD,KACI,EAAC,IAAD,y2FAoDJ,EAAC,IAAD,KACI,EAAC,IAAD,uiFAsDJ,EAAC,IAAD,KACI,EAAC,IAAD,skCAiBJ,EAAC,IAAD,KACI,EAAC,IAAD,q8FAoDJ,EAAC,IAAD,KACI,EAAC,IAAD,2rFAmDJ,EAAC,IAAD,KACI,EAAC,IAAD,s1JAwFJ,EAAC,IAAD,KACI,EAAC,IAAD,8wC","file":"route-AlgoritmiEStruttureDati.chunk.d1ce1.js","sourcesContent":["import {Panel, Markdown} from \"bluelib\";\n\nconst r = String.raw;\n\n\nexport default function (props) {\n return (\n
\n \n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa tutte le formule in latex si sono rotteeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee\n \n \n {r`\n # Algoritmi e Strutture Dati\n\n Docente: [**Manuela Montangero**](mailto:manuela.montangero@unimore.it)\n\n Crediti: **9 CFU** (72 ore di lezione)\n\n Orario di ricevimento: **Giovedì dalle 14:30 alle 16:30**\n _Mandare una mail prima, altrimenti potrebbe andarsene_\n\n\n #### Regole particolari per le email\n\n - Oggetto \"ASD\"\n - Mail **firmata** con **nome e cognome**\n - Mail spedita dall'**account Unimore**\n\n ### Materiale\n\n Libri:\n - **Introduzione agli Algoritmi e Strutture Dati** di _T.H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein_\n\n **[Dolly (FIM)](https://dolly.fim.unimore.it/2018/course/view.php?id=26)**\n\n ### Tutorato\n\n Tutor: [**Gianluca d'Addese**](mailto:tutoratoalgoritmi@gmail.com)\n\n Cosa: **Esercizi sugli argomenti visti a lezione** e **preparatori all'esame**\n\n Quando: **Mercoledì dalle 09:00 alle 11:00**...?\n\n ### Esame\n\n Per iscriversi all'esame, bisogna aver passato:\n\n - Analisi matematica (propedeutica)\n - Programmazione I\n\n Formato:\n\n - Prima prova scritta\n - Risolvere problemi con algoritmi proposti a lezione\n - E' un test sulla preparazione\n - Dura 1h30m\n - Non si può usare nessun tipo di materiale\n - Seconda prova scritta\n - Proponi soluzioni per nuovi problemi non studiati a lezione\n - Domande teoriche sugli argomenti studiati\n - Dura 2h\n - E' il giorno dopo la prima prova\n - Si può utilizzare qualsiasi materiale, ma non deve permettere di comunicare\n - Orale facoltativo\n - Solo per chi supera entrambi gli scritti\n - Domande su quello che abbiamo visto a lezione (\"perchè gli algoritmi funzionano?\")\n - Informare via email entro 3 giorni dalla pubblicazione degli esiti della seconda prova\n - L'orale potrebbe migliorare o peggiorare il voto (anche \"molto\")\n\n **Attenzione:** Vengono verbalizzati anche i voti insufficienti; ricordarsi di rifiutarli!\n\n Ci sono i seguenti appelli:\n\n - 3 appelli tra Giugno e Luglio\n - 1 appello a Settembre\n - 2 appelli a Gennaio e Febbraio\n\n Vale il salto di appello se:\n\n - Uno studente regolarmente iscritto **non si presenta** all'appello e non ha avvisato via email almeno il giorno prima.\n - Uno studente ha riportato una **grave insufficienza** in uno dei due scritti, e l'appello successivo è nella stessa sessione\n\n\n ### Note\n\n Il corso è ben collegato con quello di Programmazione 2.\n\n Dormire non fa bene!\n `}\n \n \n {`\n # Il nome del corso\n\n ## Cosa sono gli algoritmi?\n\n Gli algoritmi sono modi sistematici per risolvere problemi.\n\n Sono fondamentali per sviluppare software, in quanto i computer sono eccellenti esecutori di algoritmi.\n\n ## Come si sviluppa un algoritmo?\n\n Innanzitutto, bisogna conoscere gli _input_ e gli _output_ del problema, rispettivamente i dati di partenza e i dati di arrivo di esso; si ha quindi una fase di **ricerca**.\n\n Poi, si deve trovare un procedimento che ci faccia risolvere il nostro problema: è quello che faremo in questa materia!\n\n Infine, bisogna scrivere la soluzione in un modo che possa essere eseguita da un computer: questa è la **programmazione**.\n\n ## Che tipo di problemi possiamo risolvere?\n\n Un algoritmo risolve problemi di tipo generale, non ci interessa sapere _il risultato di 123+456_, ma vogliamo sapere _il risultato di x+y_, dove x e y sono due numeri naturali qualsiasi.\n\n Un problema può essere quindi considerato circa come una **funzione matematica**, che connette ogni input a un output corrispondente.\n\n ## Che caratteristiche ha un algoritmo?\n\n Per prima cosa, ripetendo l'algoritmo più volte con lo stesso ingresso deve dare sempre la stessa uscita come **risultato**, finendo in un **tempo finito**.\n\n Deve essere **ben ordinato**: cambiando l'ordine in cui vengono effettuate le operazioni, è probabile che anche il risultato cambi!\n\n Le sue istruzioni devono essere **non ambigue**, cioè che non possano essere interpretate in più modi, e **effettivamente realizzabili**, cioè realizzabili con l'esecutore che vogliamo usare per eseguire l'algoritmo.\n\n ## Esistono algoritmi equivalenti?\n\n **Sì!** Possono esserci due algoritmi che dati gli stessi input, hanno gli stessi output, e quindi risolvono lo stesso problema.\n\n In compenso, possono avere un numero di operazioni diverse, e quindi essere **uno più veloce** (da eseguire) dell'altro.\n\n ## Come si verifica la correttezza di un algoritmo?\n\n L'algoritmo deve essere **valido per tutti gli input**, anche se questi sono infiniti.\n\n Possiamo effettuare prove matematiche per verificarne la correttezza; il **principio di induzione** è dunque una dei teoremi fondamentali dell'algoritmica.\n\n Possiamo però verificare la _non correttezza_ di un algoritmo trovando un singolo controesempio.\n `}\n \n \n {`\n # Efficienza degli algoritmi\n \n Un buon algoritmo deve essere **efficiente**, ovvero deve usare il minimo delle risorse necessarie, come _usare il minimo di tempo possibile_.\n \n ## Come misuriamo il tempo necessario?\n \n Cerchiamo di astrarre il tempo dal particolare esecutore, e andiamo a contare il numero di operazioni elementari richieste per eseguire il nostro algoritmo nel caso peggiore.\n \n Un algoritmo efficiente, infatti, all'aumentare dei dati in ingresso, diventerà sempre più veloce rispetto a uno non efficiente, anche su computer più lenti!\n \n > Il [Bubble Sort](https://en.wikipedia.org/wiki/Bubble_sort) è sempre più lento di un [Tree Sort](https://en.wikipedia.org/wiki/Tree_sort), anche su computer più lenti, perchè, dovendo ordinare liste sempre più lunghe, prima o poi si raggiunge un punto in cui il primo è più veloce (in termini di tempo) dell'altro.\n \n Dobbiamo andare a vedere, quindi, _il numero di operazioni richieste per ottenere il risultato nel caso peggiore_.\n \n Consideriamo operazioni sia operazioni aritmetiche sia operazioni logiche, e diciamo che ciascuna costa 1.\n \n > L'[algoritmo di Euclide](https://it.wikipedia.org/wiki/Algoritmo_di_Euclide) per l'MCD costa \\`3\\` per ogni iterazione (un giro del ciclo \\`while\\`). Diciamo, allora, che costa \\`3n\\`, dove \\`n\\` è il numero più alto dei due, perchè nel caso peggiore (uno dei due numeri è 1) l'algoritmo compie \\`n\\` iterazioni.\n \n ## Altri parametri ottimizzabili\n \n È possibile che alcuni algoritmi per vari motivi cerchino di ottimizzare altri parametri diversi dal tempo, come ad esempio _la dimensione dell'input_ o la _memoria utilizzata_. \n \n > Generalmente, questo viene fatto sui dispositivi embedded, con memoria molto limitata.\n \n ### Criteri di costo di memoria\n \n Ci sono vari criteri con cui stimare la memoria richiesta da un dato: è possibile che il costo risultante vari in base al criterio scelto!\n \n #### Criterio di costo logaritmico\n \n Un dato costa il numero di bit necessari per rappresentarlo.\n \n > Un int che contiene il numero \\`n\\` costa \\`log_2(n)\\`. \n > Un array di \\`[n]\\` numeri tutti uguali costa \\`n log_2(n)\\`.\n \n #### Criterio di costo uniforme\n \n Un dato costa il numero di elementi che lo costituiscono.\n \n > Un int che contiene il numero \\`n\\` costa \\`1\\`. \n > Un array di \\`[n]\\` numeri costa \\`n\\`. \n > Una matrice \\`[m][n]\\` costa \\`m*n\\`.\n `}\n \n \n {`\n # Modelli algoritmici\n\n Per progettare un algoritmo, abbiamo bisogno di sapere le proprietà del nostro esecutore, ovvero il suo _modello algoritmico_.\n \n > Ad esempio, dobbiamo sapere quali istruzioni è in grado di eseguire, e quanto tempo queste istruzioni richiedono.\n \n ## Il modello RAM\n \n Il modello in uso su tutti i computer attuali è il _modello RAM_: \n \n - In ogni cella di memoria può essere archiviato un dato.\n - Il **tempo di accesso** alle celle è **costante** per tutte le celle.\n - La **memoria** principale è **infinita**.\n - Si ha **un solo processore**.\n `}\n \n \n {`\n # Notazione asintotica\n \n La _notazione asintotica_ è un sistema per **stimare** velocemente il costo di un algoritmo complesso.\n \n Ci permette di **confrontare velocemente il caso peggiore** degli algoritmi.\n \n In particolare, consideriamo il _rapporto tra il numero di operazioni nel caso peggiore e la dimensione dell'input_.\n \n ## Limiti\n \n Possiamo dare a questa stima dei limiti, superiore e inferiore, che rappresenteranno rispettivamente un costo che non sarà **mai superato** e un costo che verrà **sempre superato**.\n \n Chiameremo questi limiti _upper bound_ e _lower bound_; la loro combinazione darà un _tight bound_.\n \n L'obiettivo sarà di _ricavare i bound più precisi possibile_ per un dato algoritmo, ovvero l'**upper bound più basso** e il **lower bound più alto**.\n \n ### O grande\n \n > \"O grande\" \n > O di g(n)\n > \"big-O\"\n \n Per rappresentare la stima, useremo una notazione particolare, detta _O grande_, con la seguente proprietà:\n - Date due funzioni \\`f(n) : N -> R\\` e \\`g(n) : N -> R\\`, diremo che \\`f(n) ∈ O(g(n))\\` se e soltanto se \\`∃ c > 0, n ≥ n_0\\` tali che \\`∀ n ≥ 0, f(n) ≤ c * g(n)\\`\n \n Quando una funzione è O grande di un altra, significa che **asintoticamente, la funzione in O grande è sempre maggiore di quella che sta venendo stimata**.\n \n > **Ipotesi** \n > - \\`f(n) = 2n² + 3n + 6\\` \n > - \\`g(n) = n²\\`\n >\n > **Tesi** \n > - \\`f(n) ∈ O(n²)\\`.\n >\n > **Svolgimento** \n > Scrivo una disequazione, lasciando intatto il termine noto: \n > 1. \\`2n² + 3n + 6 ≤ 2n² + 3n² + 6\\` \n > 2. \\`n² ≤ 2n² + 3n² + n² = 6n²\\` per \\`n ≥ 3\\`\n >\n > Sappiamo, allora, che \\`2n² + 3n + 6 ≤ 6n²\\`.\n \n #### Espressioni di O grande\n \n Questa tabella rappresenta le espressioni di O grande più comunemente utilizzate, in ordine **dalla più forte alla più debole**.\n \n > Più forte significa che, per ogni riga della tabella, tutte le righe sottostanti sono contenute nell'espressione.\n >\n > Ad esempio, \\`O(n) ∈ O(1)\\`.\n \n | Espressione O() | Nome |\n |-----------------|------|\n | \\`O(1)\\` | Costante |\n | \\`O(log log n)\\` | loglog |\n | \\`O(log n)\\` | Logaritmica |\n | \\`Ω(n^{1/c})\\` (per c ≥ 1) | Sublineare |\n | \\`O(n)\\` | Lineare |\n | \\`O(n log n)\\` | nlogn |\n | \\`O(n²)\\` | Quadratica |\n | \\`O(n³)\\` | Cubica |\n | \\`O(n^k)\\` (per k ≥ 1) | Polinomiale |\n | \\`O(a^n)\\` (per a ≥ 1) | Esponenziale |\n | \\`O(n!)\\` | Fattoriale |\n \n #### Polinomiale\n \n Molto spesso, noteremo che il tempo richiesto da una funzione è O grande di un polinomio di grado K, ovvero \\`f(n) ∈ O(n^k)\\`.\n \n Notiamo che in questi casi, possiamo semplificare l'O grande al grado massimo del polinomio.\n \n > Ad esempio, \\`O(n² + n + 1) = O(n²)\\`.\n \n ##### Dimostrazione\n \n Per \\`n > 0 \\\\and 0 ≤ i ≤ k\\`: \n ![LaTeX](https://latex.codecogs.com/png.latex?a_k%20n^k%20+%20a_{k-1}%20n^{k-1}%20+%20%E2%80%A6%20+%20a_1%20n%20+%20a_0%20%E2%89%A4%20|a_k|%20n^k%20+%20|a_{k-1}|%20n^k%20+%20%E2%80%A6%20+%20|a_1|%20n^k%20+%20|a_0|%20n^k%20=%20(|a_k|%20+%20|a_{k-1}|%20+%20%E2%80%A6%20+%20|a_1|%20+%20|a_0|)%20n^k)\n \n #### Proprietà di O grande\n \n 1. \\`f(n) ∈ O(g(n)) -> ∀ a > 0, a * f(n) ∈ O(g(n))\\`.\n 2. \\`f(n) ∈ O(g(n)), d(n) ∈ O(h(n)) -> f(n) + d(n) ∈ O(g(n) + h(n)) -> O(max\\\\{g(n), h(n)\\\\})\\`\n 3. \\`f(n) ∈ O(g(n)), d(n) ∈ O(h(n)) -> f(n) * d(n) ∈ O(g(n) * h(n))\\`\n \n In pratica, se una funzione è la _somma di più termini_, basta guardare l'**\\`O()\\` più grande** tra tutti i suoi termini; se invece una funzione è un _prodotto di più termini_, si possono **omettere le costanti**, e l'\\`O()\\` finale sarà dato dal **prodotto degli \\`O()\\`** dei termini.\n \n ## Lower bound\n \n Possiamo anche stimare il _lower bound_, il limite inferiore: il **numero minimo di operazioni** che viene effettuato **nel caso migliore** con la **massima dimensione dell'ingresso**.\n \n ### Ω()\n \n > \"Omega\" \n > Omega di g(n)\n > \"big-Omega\"\n \n Esiste un equivalente di O grande per il lower bound: è detto _Omega grande_, o più semplicemente _Omega_, e funziona nello stesso identico modo, solo... al contrario.\n \n Diremo che \\`f(n) ∈ Ω(g(n))\\` se e solo se \\`∃ c > 0, n_0 ≥ 0 : ∀ n ≥ n_0 f(n) ≥ c * g(n)\\`.\n \n #### Espressioni di Ω()\n \n Anche in questa tabella le espressioni sono **dalla più forte alla più debole**.\n \n | Espressione Ω() | Nome |\n |-----------------|------|\n | \\`Ω(n!)\\` | Fattoriale |\n | \\`Ω(a^n)\\` (per a ≥ 1) | Esponenziale |\n | \\`Ω(n^k)\\` (per k ≥ 1) | Polinomiale |\n | \\`Ω(n³)\\` | Cubica |\n | \\`Ω(n²)\\` | Quadratica |\n | \\`Ω(n log n)\\` | nlogn |\n | \\`Ω(n)\\` | Lineare |\n | \\`Ω(n^{1/c})\\` (per c ≥ 1) | Sublineare |\n | \\`Ω(log n)\\` | Logaritmica |\n | \\`Ω(log log n)\\` | loglog |\n | \\`Ω(1)\\` | Costante |\n \n ## Tight bound\n \n Quando **upper e lower bound coincidono**, allora otteniamo un _tight bound_.\n \n ### θ()\n \n > \"Theta\" \n > Theta di g(n)\n > \"big-Theta\"\n \n Anche per il tight bound abbiamo una notazione equivalente a O grande e Omega grande: _Theta grande_!\n \n Diciamo che \\`f(n) ∈ θ(g(n))\\` se e solo se \\`∃ c_1, c_2 > 0, n_0 ≥ 0 : ∀ n ≥ n_0, c_1 * g(n) ≤ f(n) ≤ c_2 * g(n)\\`.\n \n Ha la particolarità che non valgono tutte le proprietà degli altri due: va usata quindi con cautela!\n \n ## Risorse utili\n \n [khanacademy.org](https://www.khanacademy.org/computing/computer-science/algorithms/asymptotic-notation/a/big-big-theta-notation)\n `}\n \n \n {`\n # Problemi algoritmici\n \n Un _problema algoritmico_ è un problema matematico che si vuole provare a risolvere con un algoritmo.\n \n > Dati 10 numeri, voglio sapere se sono in ordine crescente oppure no.\n \n ## Caterigorizzazione\n \n I problemi algoritmici si dividono in tre categorie: problemi _trattabili_, problemi _intrattabili_ e problemi _irrisolvibili_.\n \n ### Problema trattabile\n \n Perchè un problema algoritmico sia _trattabile_, deve avere **almeno un algoritmo con upper bound polinomiale**.\n \n > Questo significa che il tempo impiegato da un computer per risolvere il problema rimane ragionevole, e che quindi può essere utilizzato in maniera efficiente.\n \n La trattabilità è un campo ancora parecchio aperto: esistono anche tanti problemi di cui non si è ancora dimostrata la trattabilità o intrattabilità.\n \n > La [fattorizzazione](https://it.wikipedia.org/wiki/Fattorizzazione) è uno di questi problemi: l'assenza di una dimostrazione è ciò che la rende uno dei pilastri della sicurezza informatica moderna.\n \n ### Problema intrattabile\n \n Se **un problema non ha nessun algoritmo con upper bound polinomiale**, allora si dice che è **intrattabile**.\n \n ### Problema irrisolvibile\n \n Se **non esistono algoritmi per risolvere un problema**, allora questo si dice **irrisolvibile**.\n \n > [Dato un algoritmo con certi input, riusciamo a capire con un algoritmo se la sua esecuzione termina o no?](https://en.wikipedia.org/wiki/Halting_problem)\n \n ### Upper e lower bound di problemi\n \n Si può anche trovare un _upper bound_ e un _lower bound_ per un problema, ma bisogna generalizzare di più.\n \n L'**upper bound di un problema** è il minimo upper bound di tutti gli algoritmi che lo risolvono; deve esistere almeno un algoritmo che lo risolva che abbia lo stesso _upper bound_. E' praticamente il tempo migliore per risolvere il problema dato.\n \n Il **lower bound di un problema** è il minimo lower bound di tutti gli algoritmi che lo risolvono; non deve esistere nemmeno un algoritmo che abbia un lower bound migliore. E' il numero assolutamente minimo di operazioni richieste, non si può fare meglio di così.\n \n In particolare, abbiamo che l'_upper bound di un algoritmo_ -> l'_upper bound del suo problema_, \n e il _lower bound di un problema_ -> il _lower bound di un suo algoritmo_.\n \n Generalmente, il _lower bound di un problema_ è una rappresentazione abbastanza accurata della sua difficoltà.\n `}\n \n \n {`\n # Ricerca binaria\n \n Non credo di aver bisogno di studiare la ricerca binaria, quindi non ho preso appunti a riguardo.\n \n Se non siete me, e state cercando informazioni a riguardo, andate a vedere su Wikipedia!\n `}\n \n \n {`\n # Divide et impera\n \n Un modo efficace per risolvere un problema è di usare il metodo _divide et impera (et combina)_.\n \n - _Divide_: Divido il problema in **tanti sottoproblemi**.\n - _Impera_: **Risolvo indipendentemente dal resto** ciascuno dei sottoproblemi.\n - _Combina_: **Combino** i risultati dei sottoproblemi per **risolvere il problema principale**.\n \n ## Ricorsione\n \n Un algoritmo (o funzione) si dice _ricorsivo_ quando durante l'esecuzione **richiama sè stesso**.\n \n Dato che tutti gli algoritmi devono avere termine entro un tempo finito, se scriviamo una funzione ricorsiva è fondamentale finire con un **caso base**, che non chiami ulteriormente la ricorsione.\n \n Se esiste una funzione ricorsiva, allora esiste _sempre_ una _funzione iterativa_ che darà lo stesso risultato.\n \n > Sul pratico, una funzione ricorsiva tipicamente è **più costosa** del suo equivalente iterativo: se possibile, quindi, la ricorsione andrebbe evitata.\n \n #### Pseudocodice di esempio\n \n \\`\\`\\`python\n def fattoriale(n):\n # Caso base; la ricorsione finisce e dà un risultato fisso\n if n <= 1:\n return 1\n # Caso ricorsivo; la funzione restituisce il risultato di sè stessa (ma con parametri diversi)\n else:\n return n * fattoriale(n - 1)\n \\`\\`\\`\n `}\n \n \n {`\n # Master Theorem\n \n Il _Master Theorem_ è uno dei teoremi più importanti dell'algoritmica.\n \n Esso permette di **calcolare l'upper bound di un algoritmo ricorsivo** in modo piuttosto semplice.\n \n ## Ipotesi\n \n Dato un algoritmo:\n - Con uno o più casi base\n - Che richiama la funzione ricorsiva un numero n di volte\n \n ## Tesi\n \n Allora, il suo upper bound avrà la formula:\n \n ![](https://quicklatex.com/cache3/89/ql_08d29a7e55d561a900570bc83b93ff89_l3.png)\n `}\n \n \n {`\n # Caso particolare del Master Theorem\n \n ## Ipotesi\n \n Se: \n ![](https://quicklatex.com/cache3/57/ql_e34dc27b42831d3c3ff671b0f3861257_l3.png)\n \n Ovvero, se la dimensione dell'input viene divisa ad ogni ciclo da una costante b, è polinomiale e il caso base è costante...\n \n ## Tesi\n \n Allora: \n ![](https://quicklatex.com/cache3/ca/ql_26e3557a6ca2d6ac4b8481e7c5263fca_l3.png)\n \n > In pratica, se il costo dominante è quello della parte \"fissa\" dell'algoritmo, esso sarà \\`O(n^d)\\`, mentre se il costo dominante è quello delle chiamate ricorsive, esso sarà \\`O(n^{log_b e})\\`. \n > Se nessuno dei due è dominante... si dividono circa in parti uguali, creando un costo di \\`O(n^d log n)\\`.\n `}\n \n \n {`\n # Ordinamento\n \n Un problema molto frequente nell'informatica consiste nell'**ordinare efficientemente grandi quantità di elementi**.\n \n Esistono [tantissimi](https://it.wikipedia.org/wiki/Algoritmo_di_ordinamento) algoritmi per effettuare l'ordinamento. \n L'**efficienza** di ciascuno **varia** di caso in caso: alcuni sono estremamente efficienti se quasi tutti i numeri sono già nell'ordine giusto; altri, invece, potrebbero impiegare tantissimo tempo.\n \n In termini matematici, abbiamo:\n \n - **Input:** A[n]\n - **Output:** B, ∀ i < n, A[i] ≤ A[i+1]\n \n ## Ordinamento tramite confronto\n \n L'ordinamento \"tradizionale\" è detto _ordinamento tramite confronto_: funziona sempre, e **non ha altri modi di ottenere informazioni** se non con l'operazione logica di confronto tra i dati.\n \n ### Limiti\n \n E' un problema risolto: è dimostrabile che il suo **lower bound** è **\\`Ω(n log n)\\`**; possiamo quindi dire che qualsiasi algoritmo di ordinamento è in \\`Ω(n log n)\\`, e se riusciamo a trovare un algoritmo di ordinamento in \\`O(n log n)\\` siamo riusciti a raggiungere il massimo dell'efficienza.\n \n #### Dimostrazione\n \n Consideriamo **tutte le possibili permutazioni** della sequenza da ordinare: sono \\`n!\\`.\n \n Per ogni confronto che effettuiamo, **riduciamo la quantità di permutazioni** correttamente ordinate; prima o poi, rimarrà **una sola possibilità**.\n \n **TODO, non trovo la spiegazione corretta!**\n \n ### Esempi\n \n Algoritmi che effettuano l'ordinamento tramite confronto sono:\n - _Bubble sort_\n - _Merge sort_\n - _Insertion sort_\n - _Quick sort_\n - _Heap sort_\n - E tanti, tanti altri!\n \n ## Ordinamento con altri mezzi\n \n Esistono algoritmi che ricavano informazioni in altri modi, diversi dal confronto.\n \n Essi possono avere un lower bound più basso di \\`O(n log n)\\`, però hanno spesso limitazioni sul loro utilizzo.\n \n ### Esempi\n \n - _Counting sort_, indicizza i valori da ordinare\n - _Radix sort_, guarda singolarmente le cifre dei valori\n - _Sleep sort_, sfrutta i thread e la funzione sleep per ordinare valori\n - E altri ancora!\n `}\n \n \n {`\n # Insertion sort\n \n L'_insertion sort_ è una soluzione **iterativa** all'ordinamento per confronto.\n \n ## Funzionamento\n \n Considero la sequenza divisa in **due parti**: una parte **ordinata** e una parte **non ordinata**.\n \n Parto dal primo elemento della lista: è sempre ordinato con sè stesso. \n Poi, aggiungo uno alla volta i numeri della parte non ordinata a quella ordinata; prima trovo in quale posizione dovrò andare a mettere il numero, poi **faccio slittare tutti i numeri dopo quella posizione** avanti di 1, in modo da **creare lo spazio** in cui infine **inserirò** il numero.\n \n ## Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | \\`O(n²)\\` | \\`Ω(n)\\` | - |\n \n Nel _caso migliore_ (**lista già ordinata**), il numero da inserire è già nella posizione giusta, quindi non devo effettuare altri confronti oltre il primo, rendendo il lower bound dell'algoritmo \\`Ω(n)\\`.\n \n Nel _caso peggiore_ (**lista nell'ordine inverso**), dobbiamo confrontare il numero da inserire con tutti gli altri nella parte ordinata: dobbiamo allora eseguire \\`1+2+3+4+5+… = \\\\frac{(n-1)(n)}{2}\\` confronti; ciò significa che l'upper bound è \\`O(n²)\\`!\n \n ## Pseudocodice\n \n \\`\\`\\`python\n def insertion_sorted(lista):\n # Itero su tutti i numeri della lista, dal primo all'ultimo.\n for divisore_ord in range(len(lista)):\n # Partendo dalla posizione attuale, creo l'indice di divisione numeri ordinati maggiori e minori\n divisore_magg = divisore_ord\n # Faccio slittare avanti i numeri maggiori di quello che stiamo inserendo\n # Se l'indice divisore_magg raggiunge 0, vuol dire che tutti i numeri della lista sono maggiori del numero attuale \n while divisore_magg >= 0 and lista[divisore_magg-1] > lista[divisore_magg]:\n # Scambio la posizione dei due elementi con gli indici specificati\n # Funzione inventata \n lista[divisore_magg], lista[divisore_magg-1] = lista[divisore_magg-1], lista[divisore_magg]\n # Diminuisco il separatore di 1\n divisore_magg -= 1\n \\`\\`\\`\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/bn/sorting)\n `}\n \n \n {`\n # Merge sort\n \n Il _merge sort_ è una soluzione **ricorsiva** all'ordinamento per confronto.\n \n ## Funzionamento\n \n Per questo algoritmo, utilizziamo la tecnica del **divide et impera**.\n \n 1. _Divide_: Divido A in **due parti**.\n 2. _Impera_: Metto **separatamente in ordine** le parti.\n 3. _Unisci_: **Unisco** le due parti.\n \n Consideriamo come **caso base** della ricorsione una parte composta da un numero, che ovviamente è già ordinata con sè stessa.\n \n ### Merge\n \n Per **unire le due parti** usiamo una funzione detta \\`merge()\\`.\n \n Costruiamo una nuova sequenza uguale alla sequenza 1, ma **aggiungiamo alla fine un valore sentinella** sempre maggiore di tutti gli elementi contenuti.\n \n > \\`\\`\\`\n > | 1 | 3 | 7 | 8 | ∞ |\n > \\`\\`\\`\n \n Facciamo **la stessa cosa** per la sequenza due.\n \n > \\`\\`\\`\n > | 2 | 4 | 5 | 6 | ∞ |\n > \\`\\`\\`\n \n Prendo i primi numeri delle due sequenze e **metto il più piccolo nella sequenza iniziale**.\n \n > \\`\\`\\`\n > | 1 | 2 | 3 | | | | | | \n > | | | 7 | 8 | ∞ | \n > | | 4 | 5 | 6 | ∞ | \n > \\`\\`\\`\n \n **Continuo** finchè non ho messo tutti i numeri; **grazie alla sentinella non usciremo mai dalla sequenza**, in quanto essa è sempre maggiore di tutti gli altri numeri.\n \n > \\`\\`\\`\n > | 1 | 2 | 3 | 4 | 5 | 6 | | | \n > | | | 7 | 8 | ∞ | \n > | | | | | ∞ |\n > \\`\\`\\`\n \n Quando **rimangono solo le sentinelle** significa che abbiamo aggiunto tutti gli elementi, e quindi abbiamo finito.\n \n > \\`\\`\\`\n > | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | \n > | | | | | ∞ | \n > | | | | | ∞ | \n > \\`\\`\\`\n \n \n ## Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | \\`O(n log n)\\` | \\`Ω(n log n)\\` | **\\`θ(n log n)\\`** |\n \n Il merge sort è un algoritmo ricorsivo con un **caso base in tempo costante** e che **richiama sè stesso 2 volte**.\n \n \\`\\`\\`latex\n T(n) =\\\\\\\\\n \\\\\\\\\n Θ(1) \\\\qquad n=1\\\\\\\\\n 2 T(\\\\frac{n}{2}) + Θ(n) \\\\qquad n \\\\neq 1\n \\`\\`\\`\n \n Applicando il **caso particolare del Master Theorem**, otteniamo:\n \n \\`\\`\\`latex\n T(n) =\\\\\\\\\n \\\\\\\\\n Θ(1) \\\\qquad n=1\\\\\\\\\n Θ(n log n) \\\\qquad n \\\\neq 1\n \\`\\`\\`\n \n ## Pseudocodice\n \n \\`\\`\\`python\n def merge_sorted(part):\n # Caso base\n if len(part) == 1:\n return part\n # Divide\n middle = len(part) // 2\n part_a = part[:middle]\n part_b = part[middle:]\n # Impera\n sort_a = merge_sorted(part_a)\n sort_b = merge_sorted(part_b)\n # Combina\n return merge(sort_a, sort_b)\n \\`\\`\\`\n \n ## Visualizzazione\n \n [hackerearth.com](https://www.hackerearth.com/practice/algorithms/sorting/merge-sort/visualize/)\n \n [visualgo.net](https://visualgo.net/bn/sorting) (Nota: visualizza solo la fase _Unisci_ del sort)\n `}\n \n \n {`\n # Quick sort\n \n Il _quick sort_ è un altro approccio **ricorsivo** all'ordinamento per confronto.\n \n ## Funzionamento\n \n Anche qui, applichiamo il **divide et impera**.\n \n 1. _Divide_: Scelgo un **pivot** qualsiasi all'interno della lista. Metto alla sua **sinistra tutti i numeri minori** e alla sua **destra tutti i numeri maggiori**.\n 2. _Impera_: Eseguo un **quick sort su entrambe le sottoliste**.\n \n ### Esempi\n \n #### Iterazione con partizioni bilanciate\n \n Osserviamo come si formi una partizione con tre elementi e una con quattro. \n \n \\`\\`\\`\n |¦ [2] 8 7 1 3 5 6 {4}\n 2 |¦ [8] 7 1 3 5 6 {4}\n 2 |8 ¦ [7] 1 3 5 6 {4}\n 2 |8 ¦ [7] 1 3 5 6 {4}\n 2 |8 7 ¦ [1] 3 5 6 {4}\n 2 1 |7 8 ¦ [3] 5 6 {4}\n 2 1 3 |8 7 ¦ [5] 6 {4}\n 2 1 3 |8 7 5 ¦ [6] {4}\n 2 1 3 |8 7 5 6 ¦ [{4}]\n [2 1 3] {4} [8 5 6 7]\n \\`\\`\\`\n \n #### Iterazione con partizioni sbilanciate\n \n Osserviamo come si formi una partizione con **zero elementi** e una con tre.\n \n \\`\\`\\`\n |¦ [4] 7 3 {1}\n |4 ¦ [7] 3 {1}\n |4 7 |¦ [3] {1}\n |4 7 3 |¦ [{1}]\n [] {1} [4 7 3]\n \\`\\`\\`\n \n ## Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | \\`O(n²)\\` | \\`Ω(n log n)\\` | - |\n \n Il costo della funzione è dato dalla somma del costo per **dividere in due partizioni** con il costo per realizzare il **Quick sort delle due sottopartizioni**\n \n Possiamo applicare allora il **Master Theorem generale**:\n \n \\`\\`\\`latex\n T(n)\\\\\\\\\n =\\\\\\\\\n Θ(1) \\\\qquad per\\\\ n = 1\\\\\\\\\n T(q) + T(dim_lista - pivot - 1) + Θ(n) \\\\qquad per\\\\ n > 1\n \\`\\`\\`\n \n ### Il caso migliore\n \n Se il pivot \\`q\\` è la **mediana della partizione** che stiamo ordinando, si vengono a creare due _sottopartizioni bilanciate_, e sostituendo otteniamo:\n \n \\`\\`\\`latex\n T(n)\\\\\\\\\n =\\\\\\\\\n Θ(1) \\\\qquad per\\\\ n = 1\\\\\\\\\n 2 T(\\\\frac{n}{2}) + Θ(n) \\\\qquad per\\\\ n > 1\n \\`\\`\\` \n \n Possiamo allora applicare il **Master Theorem particolare**:\n \n \\`\\`\\`latex\n T(n)\\\\\\\\\n =\\\\\\\\\n Θ(1) \\\\qquad per\\\\ n = 1\\\\\\\\\n Θ(n log n) \\\\qquad per\\\\ n > 1\n \\`\\`\\`\n \n ### Il caso peggiore\n \n Se il pivot è uno degli **estremi dell'array**, si creano due _partizioni sbilanciate_: una delle due sottoliste è sempre vuota! \n Allora:\n \n \\`\\`\\`latex\n T(n) = T(n-1) + Θ(n)\\\\\\\\\n = T(n-2) + Θ(n-1) + Θ(n)\\\\\\\\\n = T(n-3) + Θ(n-2) + Θ(n-1) + Θ(n)\\\\\\\\\n = …\n ∈ Θ(n^2)\n \\`\\`\\`\n \n > \"Non date da mangiare sequenze ordinate al Quicksort, gli sono indigeste.\"\n \n ## Pseudocodice\n \n \\`\\`\\`python\n def partition(partizione, inizio, fine):\n \"\"\"Dividi una partizione in due, usando l'ultimo elemento come pivot.\n \n Note utili:\n partizione[fine] è il pivot\n partizione[maggiori] è il primo numero dei maggiori\n partizione[non_iterati] è l'elemento su cui si sta iterando al momento\"\"\"\n # Crea il primo separatore (la | pipe nell'esempio)\n maggiori = inizio\n # Crea il secondo separatore (la ¦ broken pipe nell'esempio)\n non_iterati = inizio\n # Itera su ogni numero tra inizio e fine (escluso!)\n while non_iterati < fine:\n # Se l'elemento su cui stiamo iterando è minore del pivot\n if partizione[non_iterati] <= partizione[fine]:\n # Mettilo nell'insieme dei minori, scambiandolo con il primo numero dei maggiori e incrementando il primo separatore\n partizione[maggiori], partizione[non_iterati] = partizione[non_iterati], partizione[maggiori]\n maggiori += 1\n # Incrementa sempre il secondo separatore\n non_iterati += 1\n # Inserisci il pivot tra le due sottopartizioni create, \n partizione[fine], partizione[non_iterati] = partizione[non_iterati], partizione[fine]\n return maggiori\n \\`\\`\\`\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/bn/sorting) (Nota: invece che prendere l'ultimo numero come pivot prende il primo, cambiando leggermente l'algoritmo.)\n \n ## Note per l'esame\n \n > La domanda che fa sempre è **\"Qual è la sequenza di pivot utilizzata?\"**\n \n > Elementi da soli _non_ vengono presi come pivot!\n `}\n \n \n {`\n # Counting sort\n\n Il _counting sort_ è un approccio diverso all'ordinamento: **non usa il confronto**!\n \n ## Requisiti\n \n Il counting sort può essere utilizzato solo su **sequenze di numeri interi**, e solo se **siamo a conoscenza del minimo e del massimo** dei numeri contenuti nell'array, ed essi non sono troppo distanti uno dall'altro. \n (La memoria occupata dal counting sort aumenta linearmente con la differenza tra minimo e massimo!)\n \n Per semplicità, consideriamo il **minimo \\`0\\`**. \n L'input allora sarà una sequenza di interi \\`A\\`, e il valore del **massimo \\`k\\`**, tale che \\`∀ n ∈ A, 0 \\\\leq n \\\\leq K\\`.\n \n ## Funzionamento\n \n Il counting sort **conta le ripetizioni** delle chiavi nella sequenza originale e in seguito **sovrascrive i valori** della sequenza con i valori ordinati ripetuti il numero di volte che sono stati individuati nella sequenza.\n \n > \\`\\`\\`\n > 1 4 5 3 4 1 4 2 5 1\n > \\`\\`\\`\n >\n > L'\\`1\\` appare 3 volte, il \\`2\\` 1 volta, il \\`3\\` 1 volta, il \\`4\\` tre volte e il \\`5\\` due volte.\n >\n > La sequenza viene quindi così sovrascritta:\n > \\`\\`\\`\n > 1 1 1 3 4 1 4 2 5 1 # Sovrascriviamo la sequenza con 1 ripetuto 3 volte\n > 1 1 1 2 3 4 4 4 5 1 # Sovrascriviamo la sequenza con 2, 3, 4 ripetuti rispettivamente 1 1 e 3 volte\n > 1 1 1 2 3 4 4 4 5 5 # Sovrascriviamo la sequenza con 5 ripetuto 2 volte: abbiamo finito!\n > \\`\\`\\` \n \n Esiste anche una **versione stabile** del counting sort che, invece che sovrascrivere, **sposta i valori**, mantenendo le informazioni aggiuntive nel caso invece che interi fossero altri tipi di dati.\n \n ## Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | \\`O(k + n)\\` | \\`Ω(k + n)\\` | **\\`θ(k + n)\\`** |\n \n L'algoritmo è composto da quattro parti:\n - Ricerca del minimo e massimo (in \\`θ(n)\\`)\n - Inizializzazione dell'indice (in \\`θ(k)\\`)\n - Conteggio dei numeri (in \\`θ(n)\\`)\n - Sovrascrittura dei numeri (in \\`θ(k + n)\\`)\n \n \\`2 + O(k) + O(n) + O(k + n) -> O(k + n)\\`\n \n Notiamo che \\`k\\` è costante, l'algoritmo è \\`O(n)\\`, estremamente efficiente.\n \n ## Pseudocodice\n \n \\`\\`\\`python\n def counting_sort(lista: typing.List[int]):\n \"\"\"Ordina in-place una lista con il counting sort.\"\"\"\n # Trovo la dimensione della lista\n dim = len(lista)\n # Trovo il massimo e il minimo all'interno della lista\n minimo = min(lista)\n massimo = max(lista)\n # Creo l'indice dei numeri, in modo che sia lungo k e pieno di 0\n indice = [0 for _ in range(minimo, massimo+1)]\n # Conto i numeri presenti, scorrendo su lista e aggiungendo 1 al numero corrispondente\n for i in range(dim):\n indice[lista[i]] += 1\n # Sovrascrivo i numeri nella lista\n count = 0\n for pos, val in enumerate(indice):\n for _ in range(val):\n indice[count] = pos\n count += 1\n \n def stable_counting_sorted(lista: typing.List[int], k: int) -> typing.List[int]:\n \"\"\"Ordina stabilmente una lista con il counting sort stabile, e restituiscila.\"\"\"\n # Trovo la dimensione della lista\n dim = len(lista)\n # Trovo il massimo e il minimo all'interno della lista\n minimo = min(lista)\n massimo = max(lista)\n # Creo l'indice dei numeri, in modo che sia lungo k e pieno di 0\n indice = [0 for _ in range(minimo, massimo+1)]\n # Conto i numeri presenti, scorrendo su lista e aggiungendo 1 al numero corrispondente\n for i in range(dim):\n indice[lista[i]] += 1\n # Faccio diventare l'indice \"il numero di numeri \\\\leq i\"\n for i in range(len(indice)):\n if i == 0:\n continue\n indice[i] += indice[i-1]\n assert indice[-1] == dim\n # Creo una nuova lista, che sarà quella che verrà restituita\n nuova = [None for _ in range(dim)]\n # Inizio a posizionare i numeri, al contrario\n for i in range(0, dim, -1):\n nuova[indice[lista[i]]] = lista[i]\n indice[lista[i]] -= 1\n return nuova\n \\`\\`\\`\n `}\n \n \n {`\n ## Introduzione alle Strutture Dati\n\n Una _struttura dati_ è un modo in cui si possono organizzare i dati di un programma.\n \n Si possono definire in due modi: **elementari** e **astratte**.\n \n ### Strutture dati elementari\n \n Le strutture _elementari_ dipendono strettamente dal modo in cui vengono memorizzati i dati.\n \n > **Array** e **liste** sono strutture dati elementari: sono definite dicendo come sono memorizzati i dati, rispettivamente, in celle contigue di memoria e da una serie di nodi con un valore e che puntano al successivo.\n \n ### Strutture dati astratte\n \n Le strutture _astratte_ sono separate dal modo in cui vengono memorizzati i dati, sono più ad alto livello, e si definiscono descrivendo le **proprietà** della struttura e i **metodi** che su di essa possono essere effettuate.\n \n > Una _classe_ in un qualsiasi linguaggio di programmazione è una struttura dati astratta.\n \n > Una _pila_ astratta:\n > - memorizza dati tutti dello stesso tipo\n > - \\`pop()\\`, estrae dalla pila l'ultimo valore inserito\n > - \\`push(val)\\`, aggiunge alla pila un valore\n > - \\`top()\\`, permette di vedere l'ultimo valore inserito nella pila\n > - \\`vuota()\\`, dice se la pila è vuota oppure no.\n `}\n \n \n {`\n # Array\n\n Un _array_ è sequenza di dati di **lunghezza conosciuta**, tutti dello **stesso tipo** e di una **dimensione fissa**, immagazzinata in **blocchi di memoria contigui**.\n \n ## Proprietà\n \n - E' possibile accedere a tutti i blocchi di memoria conoscendo la loro **posizione relativa al primo** elemento.\n \n ## Metodi\n \n \\`\\`\\`python\n class Array:\n def __init__(self, size, type=int): \"Crea un array di dimensione size di elementi di tipo int.\"\n def __getitem__(self, index): \"Restituisci il valore alla posizione index.\"\n def __setitem__(self, index, value): \"Cambia il valore alla posizione index.\"\n \\`\\`\\`\n \n ### Costo computazionale\n \n Tutte le operazioni su un array sono in **accesso immediato \\`O(1)\\`**!\n `}\n \n \n {`\n # Lista\n \n Una _lista_ è una sequenza di dati immagazzinata in **blocchi di memoria qualsiasi**.\n \n ## Proprietà\n \n Ogni dato ha un **riferimento** alla collocazione di memoria **successiva** (un puntatore): insieme, sono detti un _nodo_.\n \n E' di **natura ricorsiva**: qualsiasi nodo di una lista può essere visto come inizio della lista con sè stesso e i suoi successivi.\n \n ## Metodi\n \n \\`\\`\\`python\n class LinkedList:\n def __init__(self, value, next: typing.Optional[Node] = None):\n self.value = value\n self.next: typing.Optional[Node] = next\n \n def is_empty(self) -> bool: \"Restituisce se la lista è vuota o no.\"\n def is_full(self) -> bool: \"Restituisce se la lista è piena o no.\n def append(self, value): \"Aggiunge un nuovo elemento in testa alla lista.\"\n def insert(self, value, index): \"Inserisce un elemento dopo il nodo in posizione index.\"\n def insert_node(self, value, node): \"Inserisce un nuovo elemento subito dopo un dato nodo.\"\n def find(self, node) -> int: \"Trova l'indice del nodo.\"\n def delete(self, value): \"Elimina il primo nodo con quel valore dalla lista.\"\n def delete_node(self, node): \"Elimina il nodo dalla lista.\"\n def forward(self, index) -> Node: \"Restituisce il nodo in posizione index.\"\n \\`\\`\\`\n \n ### Costo computazionale\n \n #### \\`List.forward(index)\\`\n \n Per raggiungere l'\\`n\\`-esimo elemento, bisogna _scorrere tutti gli elementi prima di esso_: è dunque un **accesso sequenziale** in **\\`O(n)\\`**.\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/list)\n `}\n \n \n {`\n # Coda (Queue)\n\n Una _coda_ è come una pila, ma segue la strategia **First In, First Out** (il primo inserito sarà il primo a essere estratto).\n \n ## Proprietà\n \n - I dati vi possono essere aggiunti solo tramite il **metodo \\`enqueue()\\`**\n - I dati possono essere estratti solo tramite il **metodo \\`dequeue()\\`**\n - Verranno restituiti i valori inseriti secondo la strategia **First In, First Out** (il primo inserito sarà il primo a essere estratto).\n \n > Hai presente quando fai la fila per pagare al supermercato? Beh, è quello, però non si possono superare le altre persone in nessun modo.\n \n ## Metodi\n \n \\`\\`\\`python\n class Queue:\n def __init__(self): \"Crea una nuova coda.\"\n def is_empty(self) -> bool: \"Restituisce vero se la coda è vuota.\"\n def enqueue(self, data): \"Aggiunge un dato alla coda.\"\n def first(self) -> ...: \"Restituisce il primo dato della coda.\"\n def dequeue(self) -> ...: \"Restituisce il primo dato della coda e lo rimuove.\"\n \\`\\`\\`\n \n ## Implementazione tramite lista\n \n Posso implementare la coda con una lista, ma per realizzare l'implementazione più efficiente devo tenere anche un puntatore all'ultimo elemento della coda, in modo da non doverla scorrere ogni volta che voglio effettuare un'operazione.\n \n Chiamiamo i due puntatori \\`head\\` e \\`tail\\` rispettivamente.\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/list)\n `}\n \n \n {`\n # Pila (Stack)\n\n Una _pila_ è una struttura dati contenente **valori omogenei**.\n \n ## Proprietà\n \n - I dati vi possono essere aggiunti solo tramite il **metodo \\`push()\\`**\n - I dati possono essere estratti solo tramite il **metodo \\`pop()\\`**\n - Verranno restituiti i valori inseriti secondo la strategia **Last In, First Out** (l'ultimo inserito sarà il primo a essere estratto).\n \n > Ci si può immaginare una pila di libri, da cui si può solo prendere un libro alla volta, quello più in alto.\n \n ## Metodi\n \n \\`\\`\\`python\n class Stack:\n def __init__(self): \"Crea una nuova pila.\"\n def is_empty(self) -> bool: \"Restituisce vero se la pila è vuota.\"\n def push(self, data): \"Aggiunge un dato alla pila.\"\n def top(self) -> ...: \"Restituisce il primo dato della pila.\"\n def pop(self) -> ...: \"Restituisce il primo dato della pila e lo rimuove.\"\n \\`\\`\\`\n \n ## Implementazione tramite lista\n \n Utilizzando una lista possiamo realizzare una pila!\n \n La direzione dei puntatori sarà dall'ultimo al primo, che non punterà più a nessuno.\n \n Il costo di tutti i metodi è \\`Θ(1)\\`!\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/list)\n `}\n \n \n {`\n # Albero radicato\n \n Un _albero radicato_ è una struttura dati di **natura ricorsiva** che organizza i dati in maniera **non-lineare**.\n \n ## Proprietà\n \n - Ogni nodo dell'albero ha un **unico genitore**: \\`∀ (padre, figlio), (padre' figlio) ∈ E \\\\implies padre = padre'\\`\n - Ogni nodo dell'albero può avere **un numero qualsiasi di figli**.\n \n - I **nodi superiori al padre** vengono chiamati _antenati_.\n - I **nodi inferiori ai figli** vengono chiamati _discendenti_.\n \n - Nodi **senza padre** sono detti _radice_: \\`\\\\notexists (padre, radice) ∈ E\\`\n - Nodi **con padre e figli** sono detti _rami_ o interni.\n - Nodi **senza figli** sono detti _foglie_.\n \n - La **distanza** tra il nodo radice e i suoi discendenti è detta _livello_:\n - I figli immediati sono di livello 1.\n - I \"nipoti\" (figli dei figli) sono di livello 2.\n - I figli dei nipoti sono livello 3.\n - E così via.\n - Il **livello massimo** all'interno di un albero è detto _altezza_, _profondità_ oppure _h_, ed è sempre \\`1 ≤ h ≤ n-1\\`.\n \n - Un albero ha sempre \\`n-1\\` archi.\n \n ## Alberi particolari\n \n ### Alberi \\`d\\`-ari\n \n Un albero _\\`d\\`-ario_ è un particolare tipo di albero che **limita il numero massimo di figli di un nodo** a \\`d\\`.\n \n > Un albero _binario_ può avere **massimo 2 figli** per ogni nodo; un albero _ternario_ ne può avere **3**; un albero _\\`17\\`-ario_ ne potrà avere **17**\n \n #### Alberi completi\n \n Un albero \\`d\\`-ario si dice _completo_ se **tutti i nodi hanno 0 o \\`d\\` figli**, e mai una numero in mezzo.\n \n #### Alberi bilanciati\n \n Un albero \\`d\\`-ario si dice _bilanciato_ se **tutti i livelli eccetto l'ultimo** hanno il numero massimo di figli.\n \n #### Alberi perfettamente bilanciati\n \n Un albero \\`d\\`-ario si dice _perfettamente bilanciato_ se **tutti i livelli incluso l'ultimo** hanno il numero massimo di figli.\n \n ##### Particolarità degli alberi binari perfettamente bilanciati\n \n Si può dimostrare per induzione che:\n - Hanno sempre \\`2^h\\` foglie.\n - Hanno sempre \\`2^{h+1}-1\\` (\\`\\\\sum_i=0^n 2^i\\`) nodi.\n - L'altezza è in \\`Θ(log n)\\`.\n \n ## Implementazione degli alberi\n \n Possiamo scegliere se usare una rappresentazione con array o con nodi e puntatori: ognuna ha vantaggi e svantaggi diversi.\n \n ### Implementazione tramite array\n \n E' suggerito se l'albero è regolare; più è simile a un albero d-ario completo, meglio è.\n \n ### Implementazione tramite nodi e puntatori\n \n Più adatta ad alberi irregolari.\n \n Se l'albero è regolare, creiamo il numero esatto di campi:\n \n - Valore\n - Figlio1\n - Figlio2\n - _Opzionale:_ Padre\n \n Se un albero è irregolare, creiamo una specie di lista:\n \n - Valore\n - Primo figlio\n - Prossimo fratello\n - _Opzionale:_ Padre\n `}\n \n \n {`\n #### Breadth-first search (BFS)\n\n La _breadth-first search_ è un algoritmo che visita **ogni livello** dell'albero in ordine, dal più basso al più alto.\n \n ## Funzionamento\n \n > 1. __Visita radice__\n > 2. __Visita figli__\n > 3. __Visita nipoti__\n > 4. __Visita pronipoti__\n \n Si può implementare con una coda, in cui verranno inseriti i figli del nodo visitato da visitare ed estratti dopo avere completato la visita del livello attuale.\n \n ## Pseudocodice \n \n \\`\\`\\`python\n def bfs(radice):\n c = Queue()\n c.enqueue(radice)\n while not c.is_empty():\n nodo = c.dequeue()\n print(nodo)\n for figlio in nodo.figli:\n nodo.enqueue(figlio)\n \\`\\`\\`\n `}\n \n \n {`\n # Depth-First Search\n\n La _depth-first search_ è un algoritmo che visita **tutti i sottoalberi di un figlio** prima di passare ad un altro sfruttando la natura ricorsiva degli alberi.\n \n ## Funzionamento\n \n Ci sono diverse versioni della depth-first search: ognuna visita la radice in un momento diverso.\n \n ### DFS previsita (pre-order)\n \n La _DFS pre-visita_ visita la **radice per prima**, poi tutti i sottoalberi formati dai figli uno dopo l'altro.\n \n > 1. __Visita radice__\n > 2. dfs_pre_order(figlio1)\n > 3. dfs_pre_order(figlio2)\n \n ### Postvisita (post-order)\n \n La _DFS postvisita_ visita prima tutti i sottoalberi dei figli, e **alla fine la radice**.\n \n > 1. dfs_post_order(figlio1)\n > 2. dfs_post_order(figlio2)\n > 3. __Visita radice__\n \n ### Invisita (in-order)\n \n La _DFS invisita_ visita **un certo numero di figli**, poi la radice, e infine i figli restanti.\n \n > 1. dfs_in_order(figlio1, 1)\n > 2. __Visita radice__\n > 3. dfs_in_order(figlio2, 1)\n `}\n \n \n {`\n # Albero binario di ricerca\n\n ## Proprietà\n \n - Albero **binario**\n - Chiavi appartenenti ad un **insieme totalmente ordinato** (N, Q, R, ma non C)\n \n - Per ogni nodo con valore \\`x\\`, se un valore \\`v\\` è nel sottoalbero di sinistra allora \\`v ≤ x\\`, mentre se è nel sottoalbero di destra allora \\`v > x\\`.\n \n ## Costo computazionale\n \n - Trovare un valore: \\`O(h)\\`\n - Ordinare i valori: \\`O(n)\\`\n - Trovare il minimo: \\`O(h)\\`\n - Trovare il massimo: \\`O(h)\\`\n - Inserire un elemento: \\`O(h)\\`\n - Cancellare un elemento: \\`O(h)\\`\n \n \\`h\\` vale \\`log n\\` in un albero perfettamente bilanciato, e più l'albero diventa sbilanciato, più si avvicina a \\`n\\`, raggiungendola nel caso l'albero sia una lista.\n \n ## Pseudocodice\n \n ### Cancellazione ricorsiva\n \n \\`\\`\\`python\n def delete(tree, key):\n if tree is not None:\n # Se ho trovato il nodo che cercavo...\n if tree.key == key:\n # E c'è una sola diramazione...\n # Semplicemente stacca il nodo come in una lista.\n if tree.left is None:\n return tree.right\n if tree.right is None:\n return tree.left\n # Altrimenti, diventa il minimo dell'albero di destra\n tree.key = tree.right.min()\n # Ed eliminalo dal sottoalbero\n tree.right = delete(tree.right, tree.key)\n # Se la chiave attuale è diversa da quella che cerchiamo, continuo a navigare l'albero\n elif tree.key < key:\n tree.left = delete(tree.left, key)\n else:\n tree.right = delete(tree.right, key)\n return tree\n \\`\\`\\`\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/bst)\n \n ## Approfondimenti\n \n Esistono alberi più avanzati che mantengono le proprietà degli alberi binari di ricerca, ma che si autobilanciano, come il [Red Black Tree](https://it.wikipedia.org/wiki/RB-Albero).\n `}\n \n \n {`\n # Heap binario\n \n L'_heap binario_ è un **albero binario bilanciato a sinistra**.\n \n ## Proprietà\n \n - _Proprietà strutturale_:\n - L'albero è **perfettamente bilanciato** in tutti i livelli tranne l'ultimo\n - Nell'ultimo livello, le foglie occupano le **posizioni più a sinistra** possibili\n - _Proprietà di ordinamento_:\n - La chiave di un qualsiasi nodo è **più piccola** di tutte quelle dei nodi **del suo sottoalbero**\n \n ## Metodi\n \n \\`\\`\\`python\n class Heap:\n def __init__(self, H): ...\n def _heapify_ancestors(self, i): \"Ripristina le proprietà dell'heap per il nodo all'indice specificato e i suoi genitori.\"\n def minimum(self): \"Restituisce la chiave con il valore minimo in H.\"\n def decrease_value(self, index, new_value): \"Diminuisce il valore della chiave all'indice index a new_value.\"\n def insert(self, value): \"Inserisci un nuovo valore nell'albero.\"\n def _heapify_children(self, i): \"Ripristina le proprietà dell'heap per il nodo all'indice specificato e i suoi figli.\"\n def pop(self): \"Restituisce la chiave con il valore minimo, e la elimina.\"\n def from_list(l): \"Crea un heap da una lista.\"\n \\`\\`\\`\n \n ## Implementazione con un array\n \n Possiamo implementare l'albero utilizzando un array con le chiavi dell'albero memorizzate nell'ordine _breadth-first_.\n \n L'indice del figlio sinistro può essere trovato a \\`2i+1\\`, mentre l'indice del figlio sinistro può essere trovato a \\`2i+2\\`; il genitore è a \\`i//2-1\\`.\n \n ### Pseudocodice\n \n \\`\\`\\`python\n class Heap:\n def __init__(self, size):\n self.array = Array(size) # Il tipo Array non esiste; consideriamolo pseudocodice\n self.next_value = 0\n \n def _heapify_ancestors(self, index)\n \"\"\"Ripristina le proprietà dell'heap per il nodo all'indice specificato e i suoi genitori.\n \n Costo:\n O(log n)\"\"\"\n # Trovo l'indice del genitore\n parent = index // 2 - 1\n # Controllo se viene mantenuta la proprietà di ordinamento dell'heap\n if self.array[index] < self.array[parent]:\n # Scambio i valori dei due nodi\n self.array[index], self.array[parent] = self.array[parent], self.array[index]\n # Faccio la stessa cosa con il genitore\n self._heapify(parent)\n \n \n def minimum(self):\n \"\"\"Restituisce la chiave con il valore minimo in H.\n \n Costo:\n O(1)\"\"\"\n return self.array[0]\n \n def decrease_value(self, index, new_value):\n \"\"\"Diminuisce il valore della chiave all'indice index a new_value.\n \n Costo:\n O(log n)\"\"\"\n # Diminuisco il valore del nodo\n self.array[index] = new_value\n # Aggiorno l'heap \n self._heapify(index)\n \n def insert(self, value):\n \"\"\"Inserisci un nuovo valore nell'albero.\n \n Costo:\n O(log n)\"\"\"\n # Trovo l'indice in cui inserire il valore\n index = self.next_index \n # Aggiungo il valore in fondo\n self.array[index] = value\n # Aggiorno l'heap\n self._heapify(index)\n \n def _heapify_children(self, index):\n \"\"\"Ripristina le proprietà dell'heap per il nodo all'indice specificato e i suoi figli.\n \n Costo:\n O(log n)\"\"\"\n # Trovo l'indice dei figli\n left = index * 2 + 1\n right = index * 2 + 2\n # Mi assicuro che i figli esistano\n try:\n # Guardo quale dei figli è maggiore\n if self.array[left] > self.array[right]:\n # Scambio i valori\n self.array[left], self.array[index] = self.array[index], self.array[left]\n # Ripeto la procedura sul figlio modificato\n self._heapify_children(left)\n else:\n # Scambio i valori\n self.array[right], self.array[index] = self.array[index], self.array[right]\n # Ripeto la procedura sul figlio modificato\n self._heapify_children(left)\n except IndexError:\n # La foglia non ha figli: ho finito!\n return\n \n def pop(self):\n \"\"\"Restituisce la chiave con il valore minimo, e la elimina.\n \n Costo:\n O(log n)\"\"\"\n # Mi salvo il valore della radice\n value = self.array[0]\n # Sostituisco la radice con l'ultima foglia a destra\n self.array[0], self.array[self.next_value] = self.array[self.next_value], self.array[0]\n ### Non bisognerebbe eliminare la foglia...?\n # Riordino l'heap\n self._heapify_children(0)\n return value\n \n @staticmethod\n def from_list(l):\n \"\"\"Crea un heap da una lista.\n \n Costo:\n O(n log n), ma si può abbassare\"\"\"\n heap = Heap(len(l)))\n heap.array = Array.from_list(l) # Pseudocodice\n heap.next_value = len(l)\n # Cominciamo a riordinare l'heap dalla fine, in modo che rispetti le proprietà\n for index in range(heap.next_value, 0, -1):\n heap._heapify_children(index)\n \\`\\`\\`\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/heap)\n `}\n \n \n {`\n # Coda con priorità\n \n La _coda con priorità_ è una struttura dati dal funzionamento molto simile a quello di una coda, ma invece che restituire il primo elemento inserito, essa restituisce l'**elemento con il valore di priorità minore**.\n \n ## Proprietà\n \n - Ogni elemento è una coppia costituita da **valore** e **priorità** (un numero intero).\n - Nuovi elementi possono essere aggiunti solo tramite il **metodo \\`insert()\\`**\n - Gli elementi possono essere estratti solo tramite il **metodo \\`pop()\\`**\n - Verranno restituiti i valori inseriti secondo la strategia **Lower Priority** (l'elemento con la priorità minore sarà il primo ad essere restituito).\n - E' possibile diminuire la priorità di un elemento (e quindi anticipare la sua estrazione)\n \n ## Metodi\n \n \\`\\`\\`python\n class PriorityQueue:\n def __init__(self): ...\n def insert(self, new_elem: Element): ...\n def minimum(self) -> Element: ...\n def pop(self) -> Element: ...\n def decrease_priority_for(self, elem: Element, priority: int): ...\n \\`\\`\\`\n \n ## Implementazione con lista\n \n E' possibile implementare la coda con priorità tramite una **lista**: l'inserimento di nuovi valori diventerà molto efficiente, ma tutte le altre operazioni saranno linearmente lente.\n \n ### Costo computazionale\n \n - *insert()*: \\`O(1)\\`\n - *minimum()*: \\`O(n)\\`\n - *pop()*: \\`O(n)\\`\n - *decrease_priority_for()*: \\`O(n)\\`\n \n ## Implementazione con lista ordinata\n \n Implementando la coda con priorità con una **lista ordinata** si avrà un costo di ordinamento elevato negli inserimenti e modifiche alla priorità, ma costi costanti nell'estrazione di un elemento.\n \n ### Costo computazionale\n \n - *insert()*: \\`O(n)\\`\n - *minimum()*: \\`O(1)\\`\n - *pop()*: \\`O(1)\\`\n - *decrease_priority_for()*: \\`O(n)\\`\n \n ## Implementazione con heap\n \n La **soluzione migliore** è quella di implementare la coda con priorità tramite un **heap**: tutti i costi saranno logaritmici, eccetto l'inserimento che sarà costante.\n \n - \\`insert()\\` costa \\`O(1)\\`\n - \\`minimum()\\` costa \\`O(log n)\\`\n - \\`pop()\\` costa \\`O(log n)\\`\n - \\`decrease_priority_for()\\` costa \\`O(log n)\\`\n \n ## Approfondimenti\n \n Esistono code con priorità che restituiscono **l'elemento con priorità maggiore**, invece che quello minore.\n `}\n \n \n {`\n # Heap sort\n\n L'_heap sort_ è un algoritmo di ordinamento per confronto **iterativo**.\n \n ## Funzionamento\n \n Per effettuare un heap sort, creiamo un **heap massimo** in cui inseriamo tutti i valori che vogliamo ordinare.\n \n Una volta applicate le proprietà dell'heap, chiamiamo una versione particolare di \\`heap.pop()\\` che invece che rimuovere dall'array i valori estratti li posiziona nello spazio creatosi in fondo.\n \n ## Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | \\`O(n log n)\\` | \\`Ω(n log n)\\` | **\\`θ(n log n)\\`** |\n `}\n \n \n {`\n # Grafo\n \n Un _grafo_ è una struttura dati che rappresenta elementi interconnessi tra loro.\n \n Esistono due tipi di grafi: _orientati_ e _non orientati_.\n \n Per semplicità, consideriamo i nostri nodi numerati da 1 a \\`n\\`.\n \n ## Proprietà\n \n - Gli elementi sono rappresentati tramite _nodi_.\n - Il loro _grado_ è dato dal **numero degli archi** che vi incidono.\n - Se il grafo è orientato, hanno anche un _in-degree_ (**numero di archi entranti**) e un _out-degree_ (**numero di archi uscenti**).\n - Le connessioni tra elementi sono rappresentate tramite _archi_.\n - Un arco _incide_ esattamente su **due nodi**.\n - Se il grafo è orientato, sono _uscenti_ da uno dei due nodi ed _entranti_ nell'altro.\n - Sono matematicamente meno del **quadrato dei nodi**.\n \n ## Grafi particolari\n \n ### Catena\n \n Una _catena_ è un **grafo non orientato** composto da una **sequenza di nodi** aventi un **grado massimo di 2** tutti collegati tra loro.\n \n ### Cammino\n \n Un _cammino_ è un **grafo orientato** composto da una **sequenza di nodi** aventi un **in-degree** e un **out-degree** **massimo di 1**, collegati tra loro in modo che partendo dal primo e seguendo gli archi sia possibile arrivare all'ultimo.\n \n ### Cricca\n \n Una _cricca_ è un grafo in cui **tutti i nodi sono collegati tra loro**.\n \n Se il grafo è **non orientato**, la cricca ha \\`((n-1)n)/2\\` archi.\n \n Se il grafo è **orientato**, ha per ogni coppia un arco in entrambe le direzioni, quindi ha \\`(n-1)n\\` archi.\n \n ### Direct Acyclic Graph\n \n Un _DAG_ è un grafo diretto che non contiene nessun ciclo.\n \n Su di esso possiamo effettuare un ordinamento, detto _linearizzazione_, tra i nodi: otteniamo l'_ordine topologico_.\n \n I primi elementi dei DAG sono detti _Source_ (_Sorgente_), mentre gli ultimi sono detti _Sink_ (_Pozzo_).\n \n #### Albero\n \n Un **albero** può essere considerato un DAG con una **sorgente singola** e le **foglie come pozzi**.\n \n ### Grafo fortemente connesso\n \n Un insieme di nodi \\`V\\` di un **grafo diretto** \\`G\\` si dice una _componente fortemente connessa_ se:\n \n 1. Per ogni coppia di nodi \\`∀ u, v ∈ V' : ∃ un cammino u->v in G'\\`\n 2. Massimale (non può diventare più grande)\n \n > Praticamente una componente fortemente connessa è un gruppo di nodi tra i quali si può viaggiare liberamente da e a qualsiasi nodo al suo interno.\n \n Un grafo si dice _fortemente connesso_ se l'insieme \\`V\\` coincide con l'insieme dei nodi del grafo \\`G\\`.\n \n > Se partendo da qualsiasi nodo di un grafo riesco ad arrivare a qualsiasi altro nodo, allora il grafo è fortemente connesso.\n \n Inoltre, se creiamo un nuovo grafo, in cui **ogni nodo rappresenta una componente fortemente connessa** del nostro grafo iniziale, **otteniamo un DAG**, perchè tutti i cicli sono stati integrati nella componente.\n \n ### Trasposto di un grafo\n \n Il _trasposto_ di un **grafo diretto** \\`G\\` è il grafo stesso con gli archi che però vanno nella **direzione opposta**.\n \n ### Grafo pesato\n \n Un _grafo pesato_ è un particolare grafo che associa a ciascun arco un **costo** per attraversarlo.\n \n #### Costi negativi\n \n I costi possono anche essere negativi: rappresenteranno allora un **guadagno** ottenuto attraversando il nodo.\n \n ### Minimum spanning tree\n \n Un _minimum spanning tree_ è il **sottoinsieme degli archi** di un **grafo non diretto** che **connettono tutti i nodi** con il **minor costo possibile**.\n \n I MST hanno [molte proprietà](https://en.wikipedia.org/wiki/Minimum_spanning_tree#Properties); sono troppe da scrivere qui, e probabilmente non ci interesseranno nemmeno.\n \n ## Implementazione tramite matrice di adiacenza\n \n Possiamo implementare un grafo creando una **matrice di \\`bool\\` di dimensione \\`n * n\\`** in cui le **caselle collegate sono vere** e le caselle non collegate sono false.\n \n > Ad esempio, possiamo implementare un grafo non orientato in questo modo (\\`█\\` indica l'esistenza di un collegamento e \\` \\` indica la sua assenza):\n >\n > | |1|2|3|\n > |-|-|-|-|\n > |1|░|░|░|\n > |2|█|░|░|\n > |3|█| |░|\n >\n > Esistono gli archi \\`1-2\\` e \\`1-3\\`, ma non esiste un collegamento \\`2-3\\`.\n \n > Un grafo orientato invece si può implementare così:\n >\n > | |1|2|3|\n > |-|-|-|-|\n > |1|░|█| |\n > |2|█|░| |\n > |3|█| |░|\n >\n > Esistono gli archi \\`1->2\\`, \\`2->1\\` e \\`3->1\\`, ma non ci sono collegamenti \\`2->3\\`, \\`1->3\\` e \\`3->2\\`.\n \n ### Costo computazionale\n \n #### Tempo\n \n Le matrici di adiacenza portano alla realizzazione di algoritmi molto veloci: verificare l'esistenza di un arco è in \\`O(1)\\`!\n \n Abbiamo però penalità significative quando vogliamo effettuare operazioni sugli archi: ad esempio, trovare il trasposto di un grafo implementato con una matrice di adiacenza è in \\`O(nodi²)\\`.\n \n #### Memoria\n \n E' poco efficiente in quanto a memoria: l'upper bound è in \\`O(n^2)\\`. \n \n ## Implementazione tramite liste di adiacenza\n \n Un'alternativa alla matrice di adiacenza è quella di creare un'**array di liste**, le quali contengono i **vicini di ciascun nodo**.\n \n > |Posizione|Lista|\n > |-|-|\n > |1|[2, 3]|\n > |2|[]|\n > |3|[1]|\n >\n > Esistono gli archi \\`1->2\\`, \\`1->3\\`, e \\`3->1\\`, ma non esistono \\`2->1\\`, \\`2->3\\` e \\`3->2\\`.\n \n ### Costo computazionale\n \n #### Tempo\n \n Utilizzando le liste di adiacenza, il tempo richiesto per verificare l'esistenza di un arco sale a \\`O(max-out-degree)\\`.\n \n E' efficace però quando il problema che vogliamo risolvere riguarda operazioni su archi: trovare la trasposta è in \\`O(archi)\\`.\n \n #### Memoria \n \n La memoria richiesta dalle liste di adiacenza è minore di quella delle matrici: l'upper bound è in \\`O(nodi + archi)\\`.\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/graphds)\n `}\n \n \n {`\n # Visitare un grafo\n \n Come per gli alberi radicati, esistono due modi per visitare un grafo: _depth-first search_ e _breadth-first search_.\n \n In entrambi i casi, non visito mai due volte lo stesso nodo, e come risultato ottengo **molteplici alberi**, il cui insieme viene detto _foresta di copertura_.\n \n Se il grafo che vogliamo visitare è diretto, allora dobbiamo **considerare come vicini solo gli archi uscenti**\n \n ## Depth-first search\n \n La DFS ci può risultare utile per **identificare le componenti connesse** di un grafo e **identificare eventuali cicli**.\n \n ### Funzionamento\n \n Posso utilizzare la DFS per classificare gli archi di un grafo in quattro categorie:\n \n - _Tree_, archi che ci fanno **scoprire un nuovo nodo**\n - _Forward_, archi che ci portano a un **discendente**\n - _Back_, archi che ci portano ad un **antenato**\n - _Cross_, archi che **connettono due sottoalberi** diversi\n \n Usiamo due array inizializzati a 0 chiamati \\`pre\\` e \\`post\\`, grandi quanto il numero di archi del grafo, che ci indicano rispettivamente quando un nodo è stato scoperto e quando è terminata la visita. \n Inoltre, creiamo una variabile \\`clock\\` che avanza ad ogni evento. \n Alla scoperta di un nuovo nodo, mettiamo il valore attuale di \\`clock\\` all'interno di \\`pre[n]\\`. \n Alla fine della visita di un nodo invece mettiamo il valore di \\`clock\\` in \\`post[n]\\`.\n \n **Durante la visita**, gli archi avranno i seguenti valori:\n - _Tree_: \\`pre[dst] == 0\\`\n - _Forward_: \\`pre[src] < pre[dst] && post[dst] > 0\\`\n - _Back_: \\`pre[dst] < pre[src] && post[dst] == 0\\`\n - _Cross_: Tutti gli altri (\\`post[dst] < pre[src]\\`)\n \n **A fine visita**, gli archi avranno i seguenti valori:\n - _Tree_: \\`pre[dst] < pre[dst] < post[dst] < pre[src]\\`\n - _Forward_: \\`pre[dst] < pre[dst] < post[dst] < pre[src]\\`\n - _Back_: \\`pre[src] < pre[dst] < post[dst] < post[src]\\`\n - _Cross_: \\`pre[dst] < post[dst] < pre[src] < post[src]\\`\n \n Se un **grafo non diretto** contiene degli **archi Back**, allora esso **conterrà un ciclo**.\n \n #### DFS nel grafo trasposto\n \n Se effettuo una DFS sul trasposto di un grafo, posso **scoprire i nodi che hanno un cammino verso l'origine**.\n \n #### DFS nella componente fortemente connessa\n \n Se effettuo una DFS in una componente fortemente connessa e nella sua trasposta, il **\\`post\\` della trasposta sarà sempre minore** del \\`post\\` della componente originale.\n \n ### Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | \\`O(nodi + archi)\\` | \\`Ω(nodi + archi)\\` | **\\`θ(nodi + archi)\\`** |\n | Memoria | \\`O(nodi)\\` | \\`Ω(nodi)\\` | **\\`θ(nodi)\\`** |\n \n ### Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/dfsbfs)\n \n ## Breadth-first search\n \n La BFS ci può risultare utile per **trovare tutti i nodi a una certa distanza** da un'origine.\n \n ### Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | \\`O(nodi + archi)\\` | \\`Ω(nodi + archi)\\` | **\\`θ(nodi + archi)\\`** |\n | Memoria | \\`O(nodi + archi)\\` | \\`Ω(nodi + archi)\\` | **\\`θ(nodi + archi)\\`** |\n \n ### Pseudocodice\n \n Come per gli alberi, la implementiamo in modo **iterativo**:\n \n \\`\\`\\`python\n queue = [starting_node]\n parents = [None for node in graph.nodes]\n distance = [-1 for node in graph.nodes]\n \n # TODO: controllami quando sei più sveglio\n \n while queue:\n node, source, distance = queue.pop(0)\n parents[node.number] = source\n distance[node.number] = distance\n for neighbour in node.neighbours:\n queue.append((neighbour, node, distance+1))\n \\`\\`\\`\n \n > Nella coda, la distanza massima tra un nodo e l'altro è 1.\n \n ### Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/dfsbfs)\n `}\n \n \n {`\n # Algoritmi greedy\n\n Un modo per risolvere problemi algoritmici può essere usare una **tecnica** _greedy_.\n \n Le tecniche greedy consistono nel effettuare tanti piccoli passi, ed effettuare una **scelta** in base ai dati **locali al passo attuale**.\n \n > Scegli il numero di monete più piccole possibili per comporre €1.12.\n >\n > L'algoritmo cerca di scegliere sempre la moneta più grande possibile compatibile con il prezzo in quel momento, quindi:\n > | Moneta scelta | Rimanente |\n > |---------------|-----------|\n > | € 1.00 | € 0.12 |\n > | € 0.10 | € 0.02 |\n > | € 0.02 | € 0.00 |\n \n ## Esempi\n \n Sono algoritmi greedy:\n \n - L'_Algoritmo di Dijkstra_\n - L'_Algoritmo di Kruskal_\n - L'_Algoritmo di Prim_\n `}\n \n \n {`\n # Percorso più breve\n\n Trovare il _percorso più breve_ (o _cammino minimo_) tra due nodi di un **grafo pesato** è un problema frequente nell'informatica; per questo, sono stati sviluppati [numerosi algoritmi](https://en.wikipedia.org/wiki/Shortest_path_problem) per risolverlo.\n \n > Ad esempio, il pathfinding dei nemici nei videogiochi, oppure Google Maps!\n \n ## Percorso più breve da una sorgente singola\n \n Una sottocategoria del problema del percorso più breve è il caso in cui ci interessa sapere i percorsi più brevi che **partono da uno specifico nodo del grafo**: è detto problema del _percorso più breve da una sorgente singola_, o _single-source shortest path_.\n \n Si può notare che se il grafo contiene **costi negativi** allora è possibile che il percorso più breve non esista, in quanto diventa possibile la comparsa di **cicli di costo infinitamente negativo**.\n \n Possiamo notare che, se il percorso più breve tra \\`A\\` e \\`D\\` è \\`A-B-C-D\\`, allora il cammino minimo tra \\`B\\` e \\`D\\` passerà obbligatoriamente per \\`C\\` (\\`B-C-D\\`).\n Diremo più avanti che il percorso più breve ha una **sottostruttura ottimale**.\n \n ### Esempi\n \n Alcuni algoritmi che trovano il percorso più breve sono:\n - L'_Algoritmo di Dijkstra_\n - L'_Algoritmo di Bellman-Ford_\n - La [_ricerca A*_](https://en.wikipedia.org/wiki/A*_search_algorithm)\n `}\n \n \n {`\n # Algoritmo di Dijkstra\n \n L'_Algoritmo di [Dijkstra](https://upload.wikimedia.org/wikipedia/commons/8/85/Dijkstra.ogg)_ è un algoritmo che risolve il problema del **percorso più breve da una sorgente singola** per grafi con pesi **reali positivi** \\`\\\\mathbb{R}^+\\`.\n \n L'algoritmo trova tutti i percorsi più brevi per raggiungere qualsiasi nodo del grafo partendo da un dato nodo, assieme al costo richiesto per farlo.\n \n ## Funzionamento\n \n 1. Separiamo tutti i nodi del grafo in due gruppi: **visitati** e **non visitati**. \n - Tutti i nodi partono da **non visitati**.\n 2. Per ogni nodo, manteniamo un valore \"**costo richiesto per raggiungerlo**\", che verrà cambiato man mano che l'algoritmo avanza.\n - Il costo di partenza è \\`+∞\\`.\n - Il costo sarà **definitivo per i nodi visitati**, e **provvisorio per i non visitati**.\n 3. Creiamo un insieme detto _frontiera_ che conterrà tutti i **nodi non visitati adiacenti** a quelli visitati.\n 4. Prendiamo il nodo iniziale, che avrà un **costo di \\`0\\`**, e definiamolo il nodo _attuale_.\n 5. Finchè ci sono dei nodi non sono stati visitati, ripetiamo il seguente ciclo:\n 1. Aggiungiamo i nodi adiacenti al nodo attuale alla frontiera.\n - Il costo per raggiungerli sarà il **costo per il nodo attuale sommato al costo dell'arco** che li connette al nodo attuale. \n Se questo **costo** risulta essere **minore del costo provvisorio** precedente, esso **diventerà il nuovo costo**.\n - Questa operazione è detta _rilassamento dell'arco_.\n 2. Facciamo diventare **visitato** il nodo attuale.\n - Il percorso che abbiamo fatto per raggiungerlo è obbligatoriamente il più breve.\n 3. Il prossimo nodo attuale sarà il nodo di frontiera con un costo più basso.\n - Per questo, è possibile definire l'algoritmo di Dijkstra come un **algoritmo greedy**.\n \n ### Non funziona se...\n \n L'algoritmo smette di funzionare nel caso in cui siano presenti **costi negativi** e il grafo non sia **aciclico**, in quanto non saremmo mai in grado di rendere visitato un nodo.\n \n ## Costo computazionale\n \n | Categoria | Upper bound |\n |-----------|-------------|\n | Tempo | \\`O(nodi + archi) log nodi)\\` |\n \n ### Scomposizione\n \n - Inizializzazione: \\`O(nodi)\\`\n - Creazione coda priorità: \\`O(nodi log nodi)\\`\n - Ciclo: \\`O((nodi + archi) log nodi)\\`\n \n ## Pseudocodice\n \n \\`\\`\\`python\n import math\n \n class Info:\n def __init__(self, distance=math.inf, previous=None):\n self.distance = distance\n self.previous = previous\n \n def dijkstra(graph, start):\n data = [Info() for node in graph.nodes]\n queue = PriorityQueue([start])\n while queue:\n node = queue.pop()\n for arc in node.connections:\n other = arc.other(node)\n if data[node.number].distance + arc.cost < data[other].distance:\n data[other].distance = data[node.number].distance + arc.cost\n queue.decrease_priority_for(other, data[other].distance)\n data[v].previous = node\n return data\n \\`\\`\\`\n \n ## Visualizzazione\n \n [Visualgo](https://visualgo.net/en/sssp)\n `}\n \n \n {`\n # Algoritmo di Bellman-Ford\n\n L'_Algoritmo di Bellman-Ford_ è un algoritmo che, come l'Algoritmo di Dijkstra, risolve il problema del **percorso più breve da una sorgente singola**, però, a differenza da quest'ultimo, l'Algoritmo di Bellman-Ford accetta in input anche grafi con pesi **reali** \\`\\\\mathbb{R}\\` (sia positivi, sia negativi).\n \n ## Funzionamento\n \n L'approccio dell'algoritmo è simile a quello di Dijkstra: entrambi usano il **rilassamento** degli archi per ottenere un costo provvisorio per il raggiungimento di un nodo, ma invece che rilassare solo l'arco con costo inferiore, questo algoritmo **rilassa tutti gli archi** ripetutamente, eliminando la frontiera e il problema dei nodi negativi.\n \n L'operazione di rilassamento è ripetuta \\`nodi - 1\\` volte, ovvero la **lunghezza massima** di un cammino aciclico all'interno di un grafo.\n \n Possiamo individuare dopo i rilassamenti se è presente un nodo con un **ciclo negativo**: ci basta controllare se esiste un arco che connette due nodi con una distanza incompatibile: se \\`a.distanza + arco.costo < b.distanza\\`, allora è presente un ciclo negativo.\n \n ## Costo computazionale\n \n | Categoria | Upper bound |\n |-----------|-------------|\n | Tempo | \\`O(nodi * archi)\\` |\n \n ## Pseudocodice\n \n > TODO\n \n ## Visualizzazione\n \n [Visualgo](https://visualgo.net/en/sssp)\n `}\n \n \n {`\n # Disjoint set\n \n Il _disjoint set_ è una struttura dati che rappresenta elementi di un insieme raggruppati in **sottoinsiemi disgiunti**.\n \n ## Metodi\n \n \\`\\`\\`python\n class DisjointNode:\n def __init__(self): ...\n def find_set(self): \"Trova il rappresentante dell'elemento.\"\n def union(self, other): \"Unisce i sottoinsiemi che contengono questo nodo e \\`other\\`.\"\n \\`\\`\\`\n \n ## Implementazione tramite array\n \n Possiamo implementare il disjoint set con due array: uno per l'**indice del rappresentante** e uno per il **rango dell'insieme**.\n \n Un singoletto avrà **sè stesso come rappresentante** e **rango \\`0\\`**.\n \n ### Costo computazionale\n \n - *create_set()*: \\`O(1)\\`\n - *find_set()*: \\`O(h)\\`\n - *union()*: \\`O(h)\\`\n \n ### Pseudocodice\n \n \\`\\`\\`python\n class DisjointNode:\n def __init__(self):\n self.parent = self\n self.rank = 0\n \n def find_set(self):\n element = self\n while self.parent != element:\n element = self.parent\n return element\n \n def union(self, other):\n repres_self = self.find_set()\n repres_other = other.find_set()\n if repres_self == repres_other:\n return\n if repres_self.rank < repres_other.rank:\n repres_greater = repres_other\n repres_lesser = repres_self\n else:\n repres_greater = repres_self\n repres_lesser = repres_other\n repres_lesser.parent = repres_greater\n if repres_greater.rank == repres_lesser.rank:\n repres_greater.rank += 1\n \\`\\`\\`\n \n ### Visualizzazione\n \n [cs.usfca.edu](https://www.cs.usfca.edu/~galles/JavascriptVisual/DisjointSets.html)\n `}\n \n \n {`\n # Trovare il minimum spanning tree\n\n Un altro problema ricorrente riguardante i grafi è trovare il _minimum spanning tree_ di un dato grafo non diretto.\n \n > E' utile per trovare il modo più efficiente per connettere le cose: ad esempio, per decidere la struttura di una rete internet!\n \n ## Esempi\n \n Gli algoritmi principali che risolvono il problema sono due, ed entrambi sono **algoritmi greedy**:\n \n - L'_Algoritmo di Kruskal_\n - L'_Algoritmo di Prim_\n `}\n \n \n {`\n # Algoritmo di Kruskal\n \n L'_Algoritmo di Kruskal_ è un algoritmo **greedy** che **trova il minimum spanning tree** di un grafo.\n \n ## Funzionamento\n \n 1. Ripetiamo questa procedura finchè tutti i nodi non sono connessi:\n 1. Prendiamo ad ogni passo **l'arco meno costoso** del grafo non ancora aggiunto all'insieme.\n 2. Assicuriamoci che **non si creino cicli**: se non se ne verrebbero a creare, possiamo **aggiungere l'arco all'insieme**.\n - Gli archi devono quindi connettere nodi in **componenti connesse diverse**.\n - Possiamo rappresentare le componenti connesse con un **Disjoint Set**.\n \n ## Costo computazionale\n \n | Categoria | Upper bound |\n |-----------|-------------|\n | Tempo | \\`O(archi²)\\` |\n \n ### Scomposizione\n \n - DisjointSet.__init__(): \\`O(archi)\\`\n - Per ogni ciclo: \\`O(archi²)\\`\n - DisjointSet.find_set(): \\`O(1)\\`\n - DisjointSet.union(): \\`O(archi)\\`\n \n ## Pseudocodice\n \n \\`\\`\\`python\n def minimum_spanning_tree_kruskal(graph):\n ds = DisjointSet()\n for node in graph.nodes:\n ds.create_set(node)\n arcs = []\n sorted_arcs = sorted(graph.arcs, key=lambda arc: arc.cost)\n for arc in sorted_arcs:\n if ds.find_set(node.start) != ds.find_set(node.end):\n arcs.append(arc)\n ds.union(node.start, node.end)\n return arcs\n \\`\\`\\`\n `}\n \n \n {`\n # Algoritmo di Prim\n \n L'_Algoritmo di Prim_ è un altro algoritmo **greedy** che **trova il minimum spanning tree** di un grafo.\n \n ## Funzionamento\n \n Creo una **coda con priorità** in cui inserisco tutti gli archi visibili dal mio albero, in cui la chiave è il **costo dell'arco**.\n \n Per trovare l'arco con costo più piccolo posso **estrarre un arco** dalla coda: la priorità ci garantisce che esso è l'**arco meno costoso**.\n \n Aggiungo allora un nuovo nodo all'albero, e con esso, **aggiungo alla coda** tutti gli **archi che scoprono un nuovo nodo**.\n \n ## Costo computazionale\n \n | Categoria | Upper bound |\n |-----------|-------------|\n | Tempo | \\`O(archi + nodi log nodi)\\` |\n \n ## Pseudocodice\n \n \\`\\`\\`python\n import math\n \n def minimum_spanning_tree_prim(graph, cost_array, start_node):\n # E' un Array di bool: se l'indice corrispondente al nodo è uguale a true, vuol dire che il (nodo è contenuto nell'albero.\n contains = [False for _ in range(len(graph))]\n # Contiene il precedente di ogni nodo \n prev = [None for _ in range(len(graph))]\n # Contiene il costo per arrivare a quel nodo\n cost = [math.inf for _ in range(len(graph))]\n # Creo la priority queue\n pq = PriorityQueue(graph.arcs, key=lambda arc: arc.cost)\n # Parto dal nodo \\`start_index\\`\n # Il costo dell'origine è 0.\n cost[start_node.index] = 0\n contains[start_node.index] = True\n \n while not pq.is_empty():\n new_node = pq.pop()\n contains[new_node.index] = True\n for arc in new_node.connections:\n other_node = arc.other(new_node)\n if not contains[other_node.index] and cost[other_node.index] > arc.cost:\n cost[other_node.index] = arc.cost\n prev[other_node.index] = new_node\n pq.decrease_priority_for(other_node, arc.cost)\n # L'array di prev rappresenta un albero.\n return prev\n \\`\\`\\`_\n `}\n \n \n {`\n # Compressione\n\n _Comprimere_ un file significa **ridurne le dimensioni** senza modificarne il significato.\n \n ## Categorie\n \n ### Compressione lossless\n \n Nella _compressione lossless_, i dati possono essere decompressi riottenendo una copia identica dell'originale.\n \n > Immaginiamo un file che contiene solo le lettere \\`a, b, c, d, e, f\\`.\n >\n > Le lettere compaiono con questa frequenza:\n > |a|b|c|d|e|f|\n > |45%|13%|12%|16%|9%|5%|\n >\n > Possiamo codificare le lettere nel seguente modo:\n > |a|b|c|d|e|f|\n > |\\`0b0\\`|\\`0b100\\`|\\`0b101\\`|\\`0b111\\`|\\`0b1100\\`|\\`0b1101\\`|\n >\n > Scrivere \\`abacadae\\` richiederebbe 64 bits con la codifica ASCII estesa, ma in questo modo riusciamo a scriverlo con soli 17 bits!\n \n Le codifiche di un file compresso devono rispettare la proprietà del _Codice a prefisso_, che dice che **nessun codice deve essere il prefisso di un altro codice**; altrimenti, si avrebbero ambiguità nella decodifica.\n \n > a = \\`1\\`\n > b = \\`11\\`\n >\n > \\`111\\` è \\`ab\\`, \\`ba\\` oppure \\`aaa\\`?\n \n Creiamo allora un _albero di decodifica_: un **albero binario** che, leggendo uno ad uno i bit codificati, ci permette di arrivare al **valore del codice presente sulle foglie** dell'albero.\n \n Gli alberi di decodifica sono sempre **completi**.\n \n > Un albero di decodifica incompleto sarebbe non ottimizzato!\n \n #### Esempi\n \n - .png\n - .flac\n - .zip\n - ...\n \n ### Compressione lossy\n \n Nella _compressione lossy_, alcuni dati [solitamente](http://needsmorejpeg.com/) irrilevanti vengono perduti: non si può, dunque, ricostruire l'originale.\n \n #### Esempi\n \n - .jpeg\n - .mp3\n - ...\n `}\n \n \n {`\n # Algoritmo di Huffman\n\n L'_Algoritmo di Huffman_ è un **algoritmo greedy** per la **costruzione di un albero di decodifica**.\n \n ## Funzionamento\n \n 1. Costruisco **un albero** (con un solo nodo) **per ogni elemento dell'alfabeto**.\n 2. Associo ad **ogni albero la frequenza dell'elemento** da cui è stato creato, per poi inserire tutti gli elementi in una coda con priorità.\n 3. Finchè non ho **un albero solo**:\n 1. Estraggo dalla coda i **due alberi con frequenza minore**.\n 2. **Li rendo fratelli**, creando un nuovo nodo in cui sono uno figlio destro e uno figlio sinistro.\n 3. Associo al nuovo nodo la **somma delle frequenze dei due alberi**, e inserisco il nuovo albero nella coda.\n \n > È molto raro che venga un albero \"dritto\"; se succede, probabilmente c'è qualcosa che non va.\n `}\n \n \n {`\n # Dizionario\n \n Un _dizionario_ è una struttura dati che **associa dei valori a delle chiavi**.\n \n ## Proprietà\n \n - Ogni elemento del dizionario è un **valore** che è stato **associato a una chiave**.\n - Possiamo aggiungere nuovi elementi con il **metodo \\`add(chiave, valore)\\`**.\n - Posiamo estrarre elementi con il **metodo \\`get(chiave)\\`**, che restituirà il valore associato a \\`chiave\\`.\n - E' possibile rimuovere elementi con il **metodo \\`delete(chiave)\\`**.\n - Due elementi con **chiavi diverse** non devono **mai restituire lo stesso valore**.\n \n ## Metodi\n \n Beh... Non ha molto senso in questo caso...\n \n \\`\\`\\`python\n dict()\n \\`\\`\\`\n \n ## Implementazione con tabella hash\n \n Una _tabella hash_ è un'array di coppie **chiave-valore**, che formano l'insieme _universo_.\n \n Per determinare l'**indice dell'array** in cui inserire una coppia, usiamo una [funzione _hash_](https://it.wikipedia.org/wiki/Hash#Algoritmo_di_hash) sulla chiave, che restituirà **numeri da \\`0\\` a \\`len(hash_table)\\`**.\n \n ### Risoluzione collisioni\n \n Potrebbe capitare però che **due chiavi diverse abbiano lo stesso indice**. Dobbiamo allora usare un metodo di _risoluzione collisioni_, che mi permetta di distinguere tra chiavi diverse.\n \n #### Lista di trabocco\n \n Possiamo salvare nell'array **liste di coppie** chiave-valore; in caso di collisione, **aggiungo un nuovo elemento alla lista**.\n \n In media, ciascuna di queste liste conterrà \\`elementi_inseriti / dimensione_tabella\\` elementi.\n \n #### Indirizzamento aperto\n \n Possiamo decidere di mettere le coppie che non trovano posto nel loro indice in **un altro indirizzo vuoto** dell'array.\n \n Ci sono diversi modi in cui decidere il nuovo indirizzo, ognuno con vantaggi e svantaggi: si può scegliere quello successivo, oppure il primo vuoto dell'array, oppure un indirizzo casuale.\n \n > Python, nei \\`dict\\`, usa indirizzamento aperto pseudorandom.\n \n ### Costo computazionale\n \n - Aggiungere una chiave: \\`O(n)\\`\n - Trovare una chiave: \\`O(n)\\`\n - Eliminare una chiave: \\`O(n)\\`\n `}\n \n \n {`\n # Programmazione dinamica\n \n La _programmazione dinamica_ è una **tecnica** di programmazione che prevede l'**estensione di una soluzione ottima precedente**.\n \n Tutti i problemi in cui si può applicare si possono risolvere anche con la **ricorsione**, ma a differenza della ricorsione, questa tecnica riesce ad evitare di ricalcolare la soluzione per ogni chiamata ricorsiva, ottenendo quindi tempi molto migliori.\n \n Si può applicare solo se un problema ha una **sottostruttura ottimale**, ovvero se la soluzione ottima di un sottoproblema è inclusa nella soluzione ottima del problema. \n \n ## Esempi\n \n - _Problema dello zaino_\n - ...\n \n > Il cammino minimo per raggiungere un nodo in un DAG è dato da \\`arco.costo + arco.primo_nodo.costo_cammino_minimo()\\`.\n >\n > \\`\\`\\`python\n > def SPD_PD(graph, start):\n > distance = [float(inf) for node in graph.nodes]:\n > distance[start] = 0\n > # I nodi devono essere in ordine di linearizzazione\n > for node in graph.nodes:\n > distance[node] = min([(arc.cost + distance[arc.other(node)] for arc in node.connections])\n > \\`\\`\\`\n \n > Ho una sequenza di interi da \\`a_1\\` a \\`a_n\\`. Voglio trovare la sottosequenza crescente più lunga.\n >\n > 5 2 3 4 7 3 6 3 1 6\n >\n > Trovo tutte le sequenze lunghe 1, e le rendo nodi di un grafo diretto.\n >\n > Da ogni nodo, creo una connessione verso i suoi maggiori.\n >\n > Infine, cerco i cammini massimi del grafo. \n > \n > Essi saranno la soluzione del problema.\n \n > Trova la lunghezza della sottosequenza più lunga che termina con \\`j\\`.\n >\n > \\`L[j]\\` = lunghezza della sottosequenza più lunga che termina in \\`j\\`\n > \n > \\`\\`\\`python\n > L[j] = max([1 + L[node] for arc in node.connections)]\n > \\`\\`\\`\n >\n > Esempio:\n > \\`\\`\\`python\n > L[9] = max([1+L[8], 1+L[3], 1+L[6]])\n > \\`\\`\\`\n `}\n \n \n {`\n # Problema dello zaino\n \n Il problema dello zaino è un problema _pseudo-trattabile_: non abbiamo dimostrazioni di se sia trattabile o intrattabile.\n \n ## Descrizione\n \n > Sei un ladro, e devi mettere **più refurtiva possibile** nello zaino per scappare. \n > Lo zaino può portare **al massimo \\`dim\\` kili**.\n >\n > **Quali** (e quanti) oggetti scegli?\n \n | Input | Output |\n |-------|--------|\n | \\`dim\\`ensione_zaino, \\`n\\`umero_oggetti, \\`oggetto.peso\\`, \\`oggetto.valore\\` | \\`profitto_massimo\\` |\n \n ## Categorie\n \n ### Problema con ripetizione\n \n Puoi prendere **tutte le copie che vuoi** di un oggetto.\n \n #### Soluzione\n \n \\`K(dim)\\` è il valore massimo ottenibile con uno zaino di capacità \\`dim\\`.\n \n > Se \\`i\\` appartenesse alla soluzione ottima, allora \\`K(dim) = i.valore + K(dim - i.peso)\\`...\n \n Possiamo dire che \\`K(dim) = max(i.valore + K(dim - i.peso))\\`.\n \n Inoltre, \\`K(0) = 0\\`.\n \n Ci salviamo tutte le soluzioni da \\`K(0)\\` a \\`K(dim)\\`, e le usiamo per calcolare il massimo in seguito.\n \n Calcolare \\`K(dim)\\` avrà allora un costo di \\`O(n * dim)\\`:\n - \\`n\\`, perchè trovare il massimo è un'operazione lineare\n - \\`dim\\`, perchè \\`dim\\` sono tutti i casi tra i quali devo andare a provare\n \n Il costo computazionale, allora, è in \\`O(n * dim)\\`. \n \n Però, il **tempo richiesto** dal nostro algoritmo dipende non dalla lunghezza dell'input, bensì dal **valore numerico** di \\`dim\\`, che corrisponde alla dimensione dell'array delle soluzioni. \n Allora, si dice che l'algoritmo è in **tempo _pseudo-polinomiale_**.\n \n ### Problema senza ripetizione\n \n Si può prendere **ogni oggetto una volta sola**.\n \n #### Soluzione _bruteforce_\n \n Scelgo se prendere o no l'item 1.\n \n Si creano due percorsi:\n - Non prendo l'oggetto: \\`valore = 0, peso = 0\\`\n - Prendo l'oggetto: \\`valore = oggetto.valore, peso = oggetto.peso\\`\n \n Continuo a creare percorsi, creando una specie di albero binario.\n \n Se a un certo punto vedo che \\`valore = x, peso = K\\` e \\`valore < x, peso = K\\`, allora posso escludere automaticamente tutto il sottoalbero destro, perchè non può essere migliore del sinistro: allora, sarò riuscito a ridurre il numero dei casi rispetto alla ricorsione.\n \n ### Problema in due variabili\n \n \\`K(j, w)\\` = massimo valore ottenibile con uno zaino di capacità \\`w\\` scegliendo gli item da \\`1\\` a \\`j\\`.\n \n Non possiamo più applicare la soluzione bruteforce, perchè abbiamo due variabili, \\`j\\` e \\`w\\`.\n \n Allora, prendo l'elemento \\`j\\`. Esso può essere o non essere nella soluzione: mi calcolo entrambe le alternative, e mi tengo l'alternativa dal valore più alto.\n \n Se \\`j\\` non è nella soluzione, il risultato diventerà \\`K(j-1, w)\\`; se invece è nella soluzione, il risultato sarà \\`j.valore + K(j-1, w-j.peso)\\`.\n \n In pratica, prendiamo\n \n \\`\\`\\`latex\n K(j, w) = max\n \\\\begin{cases}\n V_j + K(j-1, w-w_j)\n K(j-1, w)\n \\\\end{cases}\n \\`\\`\\`\n \n Costruisco allora una matrice con \\`j\\` su un asse e \\`w\\` sull'altro. \n Riempio le caselle con il valore di \\`K(j, w)\\`. \n Nella casella con \\`K(j, w)\\` avremo la soluzione ottima.\n \n Il tempo necessario per riempire tutte le caselle è nuovamente \\`O(n * w)\\`, ancora **pseudopolinomiale**.\n \n Per sapere che oggetti ho messo o no devo tenere traccia in qualche modo della catena del calcolo, usando, ad esempio, una pila.\n `}\n \n \n {`\n # Problemi intrattabili\n \n ## Problema di Set-Cover\n \n ### Input\n \n \\`U\\`niverso di \\`e\\`lementi\n \n \\`S\\`ottoinsieme di \\`s\\`ottoinsiemi di elementi di \\`U\\`\n \n ### Output\n \n Il minimo \\`S'\\`ottoinsieme di \\`s\\`ottoinsiemi che copra completamente \\`U\\`.\n \n ### Soluzione in \\`O(n^d)\\`\n \n Non c'è.\n \n ### Non-soluzione alternativa\n \n Faccio una scelta greedy, ma non posso dimostrare in alcun modo che la soluzione ottenuta sia quella ottima. \n Infatti, l'algoritmo non dà sempre la soluzione ottima, ma dà una soluzione accettabile in tempo polinomiale.\n \n Seleziono sempre il sottoinsieme che copre più elementi mancanti possibili.\n \n #### Costo computazionale\n \n \\`Costo greedy <= log(numero_elementi) * Costo ottimo\\`\n `}\n \n
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\n \n aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa tutte le formule in latex si sono rotteeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee\n \n \n {r`\n # Algoritmi e Strutture Dati\n\n Docente: [**Manuela Montangero**](mailto:manuela.montangero@unimore.it)\n\n Crediti: **9 CFU** (72 ore di lezione)\n\n Orario di ricevimento: **Giovedì dalle 14:30 alle 16:30**\n _Mandare una mail prima, altrimenti potrebbe andarsene_\n\n\n #### Regole particolari per le email\n\n - Oggetto \"ASD\"\n - Mail **firmata** con **nome e cognome**\n - Mail spedita dall'**account Unimore**\n\n ### Materiale\n\n Libri:\n - **Introduzione agli Algoritmi e Strutture Dati** di _T.H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein_\n\n **[Dolly (FIM)](https://dolly.fim.unimore.it/2018/course/view.php?id=26)**\n\n ### Tutorato\n\n Tutor: [**Gianluca d'Addese**](mailto:tutoratoalgoritmi@gmail.com)\n\n Cosa: **Esercizi sugli argomenti visti a lezione** e **preparatori all'esame**\n\n Quando: **Mercoledì dalle 09:00 alle 11:00**...?\n\n ### Esame\n\n Per iscriversi all'esame, bisogna aver passato:\n\n - Analisi matematica (propedeutica)\n - Programmazione I\n\n Formato:\n\n - Prima prova scritta\n - Risolvere problemi con algoritmi proposti a lezione\n - E' un test sulla preparazione\n - Dura 1h30m\n - Non si può usare nessun tipo di materiale\n - Seconda prova scritta\n - Proponi soluzioni per nuovi problemi non studiati a lezione\n - Domande teoriche sugli argomenti studiati\n - Dura 2h\n - E' il giorno dopo la prima prova\n - Si può utilizzare qualsiasi materiale, ma non deve permettere di comunicare\n - Orale facoltativo\n - Solo per chi supera entrambi gli scritti\n - Domande su quello che abbiamo visto a lezione (\"perchè gli algoritmi funzionano?\")\n - Informare via email entro 3 giorni dalla pubblicazione degli esiti della seconda prova\n - L'orale potrebbe migliorare o peggiorare il voto (anche \"molto\")\n\n **Attenzione:** Vengono verbalizzati anche i voti insufficienti; ricordarsi di rifiutarli!\n\n Ci sono i seguenti appelli:\n\n - 3 appelli tra Giugno e Luglio\n - 1 appello a Settembre\n - 2 appelli a Gennaio e Febbraio\n\n Vale il salto di appello se:\n\n - Uno studente regolarmente iscritto **non si presenta** all'appello e non ha avvisato via email almeno il giorno prima.\n - Uno studente ha riportato una **grave insufficienza** in uno dei due scritti, e l'appello successivo è nella stessa sessione\n\n\n ### Note\n\n Il corso è ben collegato con quello di Programmazione 2.\n\n Dormire non fa bene!\n `}\n \n \n {`\n # Il nome del corso\n\n ## Cosa sono gli algoritmi?\n\n Gli algoritmi sono modi sistematici per risolvere problemi.\n\n Sono fondamentali per sviluppare software, in quanto i computer sono eccellenti esecutori di algoritmi.\n\n ## Come si sviluppa un algoritmo?\n\n Innanzitutto, bisogna conoscere gli _input_ e gli _output_ del problema, rispettivamente i dati di partenza e i dati di arrivo di esso; si ha quindi una fase di **ricerca**.\n\n Poi, si deve trovare un procedimento che ci faccia risolvere il nostro problema: è quello che faremo in questa materia!\n\n Infine, bisogna scrivere la soluzione in un modo che possa essere eseguita da un computer: questa è la **programmazione**.\n\n ## Che tipo di problemi possiamo risolvere?\n\n Un algoritmo risolve problemi di tipo generale, non ci interessa sapere _il risultato di 123+456_, ma vogliamo sapere _il risultato di x+y_, dove x e y sono due numeri naturali qualsiasi.\n\n Un problema può essere quindi considerato circa come una **funzione matematica**, che connette ogni input a un output corrispondente.\n\n ## Che caratteristiche ha un algoritmo?\n\n Per prima cosa, ripetendo l'algoritmo più volte con lo stesso ingresso deve dare sempre la stessa uscita come **risultato**, finendo in un **tempo finito**.\n\n Deve essere **ben ordinato**: cambiando l'ordine in cui vengono effettuate le operazioni, è probabile che anche il risultato cambi!\n\n Le sue istruzioni devono essere **non ambigue**, cioè che non possano essere interpretate in più modi, e **effettivamente realizzabili**, cioè realizzabili con l'esecutore che vogliamo usare per eseguire l'algoritmo.\n\n ## Esistono algoritmi equivalenti?\n\n **Sì!** Possono esserci due algoritmi che dati gli stessi input, hanno gli stessi output, e quindi risolvono lo stesso problema.\n\n In compenso, possono avere un numero di operazioni diverse, e quindi essere **uno più veloce** (da eseguire) dell'altro.\n\n ## Come si verifica la correttezza di un algoritmo?\n\n L'algoritmo deve essere **valido per tutti gli input**, anche se questi sono infiniti.\n\n Possiamo effettuare prove matematiche per verificarne la correttezza; il **principio di induzione** è dunque una dei teoremi fondamentali dell'algoritmica.\n\n Possiamo però verificare la _non correttezza_ di un algoritmo trovando un singolo controesempio.\n `}\n \n \n {`\n # Efficienza degli algoritmi\n \n Un buon algoritmo deve essere **efficiente**, ovvero deve usare il minimo delle risorse necessarie, come _usare il minimo di tempo possibile_.\n \n ## Come misuriamo il tempo necessario?\n \n Cerchiamo di astrarre il tempo dal particolare esecutore, e andiamo a contare il numero di operazioni elementari richieste per eseguire il nostro algoritmo nel caso peggiore.\n \n Un algoritmo efficiente, infatti, all'aumentare dei dati in ingresso, diventerà sempre più veloce rispetto a uno non efficiente, anche su computer più lenti!\n \n > Il [Bubble Sort](https://en.wikipedia.org/wiki/Bubble_sort) è sempre più lento di un [Tree Sort](https://en.wikipedia.org/wiki/Tree_sort), anche su computer più lenti, perchè, dovendo ordinare liste sempre più lunghe, prima o poi si raggiunge un punto in cui il primo è più veloce (in termini di tempo) dell'altro.\n \n Dobbiamo andare a vedere, quindi, _il numero di operazioni richieste per ottenere il risultato nel caso peggiore_.\n \n Consideriamo operazioni sia operazioni aritmetiche sia operazioni logiche, e diciamo che ciascuna costa 1.\n \n > L'[algoritmo di Euclide](https://it.wikipedia.org/wiki/Algoritmo_di_Euclide) per l'MCD costa \\`3\\` per ogni iterazione (un giro del ciclo \\`while\\`). Diciamo, allora, che costa \\`3n\\`, dove \\`n\\` è il numero più alto dei due, perchè nel caso peggiore (uno dei due numeri è 1) l'algoritmo compie \\`n\\` iterazioni.\n \n ## Altri parametri ottimizzabili\n \n È possibile che alcuni algoritmi per vari motivi cerchino di ottimizzare altri parametri diversi dal tempo, come ad esempio _la dimensione dell'input_ o la _memoria utilizzata_. \n \n > Generalmente, questo viene fatto sui dispositivi embedded, con memoria molto limitata.\n \n ### Criteri di costo di memoria\n \n Ci sono vari criteri con cui stimare la memoria richiesta da un dato: è possibile che il costo risultante vari in base al criterio scelto!\n \n #### Criterio di costo logaritmico\n \n Un dato costa il numero di bit necessari per rappresentarlo.\n \n > Un int che contiene il numero \\`n\\` costa \\`log_2(n)\\`. \n > Un array di \\`[n]\\` numeri tutti uguali costa \\`n log_2(n)\\`.\n \n #### Criterio di costo uniforme\n \n Un dato costa il numero di elementi che lo costituiscono.\n \n > Un int che contiene il numero \\`n\\` costa \\`1\\`. \n > Un array di \\`[n]\\` numeri costa \\`n\\`. \n > Una matrice \\`[m][n]\\` costa \\`m*n\\`.\n `}\n \n \n {`\n # Modelli algoritmici\n\n Per progettare un algoritmo, abbiamo bisogno di sapere le proprietà del nostro esecutore, ovvero il suo _modello algoritmico_.\n \n > Ad esempio, dobbiamo sapere quali istruzioni è in grado di eseguire, e quanto tempo queste istruzioni richiedono.\n \n ## Il modello RAM\n \n Il modello in uso su tutti i computer attuali è il _modello RAM_: \n \n - In ogni cella di memoria può essere archiviato un dato.\n - Il **tempo di accesso** alle celle è **costante** per tutte le celle.\n - La **memoria** principale è **infinita**.\n - Si ha **un solo processore**.\n `}\n \n \n {`\n # Notazione asintotica\n \n La _notazione asintotica_ è un sistema per **stimare** velocemente il costo di un algoritmo complesso.\n \n Ci permette di **confrontare velocemente il caso peggiore** degli algoritmi.\n \n In particolare, consideriamo il _rapporto tra il numero di operazioni nel caso peggiore e la dimensione dell'input_.\n \n ## Limiti\n \n Possiamo dare a questa stima dei limiti, superiore e inferiore, che rappresenteranno rispettivamente un costo che non sarà **mai superato** e un costo che verrà **sempre superato**.\n \n Chiameremo questi limiti _upper bound_ e _lower bound_; la loro combinazione darà un _tight bound_.\n \n L'obiettivo sarà di _ricavare i bound più precisi possibile_ per un dato algoritmo, ovvero l'**upper bound più basso** e il **lower bound più alto**.\n \n ### O grande\n \n > \"O grande\" \n > O di g(n)\n > \"big-O\"\n \n Per rappresentare la stima, useremo una notazione particolare, detta _O grande_, con la seguente proprietà:\n - Date due funzioni \\`f(n) : N -> R\\` e \\`g(n) : N -> R\\`, diremo che \\`f(n) ∈ O(g(n))\\` se e soltanto se \\`∃ c > 0, n ≥ n_0\\` tali che \\`∀ n ≥ 0, f(n) ≤ c * g(n)\\`\n \n Quando una funzione è O grande di un altra, significa che **asintoticamente, la funzione in O grande è sempre maggiore di quella che sta venendo stimata**.\n \n > **Ipotesi** \n > - \\`f(n) = 2n² + 3n + 6\\` \n > - \\`g(n) = n²\\`\n >\n > **Tesi** \n > - \\`f(n) ∈ O(n²)\\`.\n >\n > **Svolgimento** \n > Scrivo una disequazione, lasciando intatto il termine noto: \n > 1. \\`2n² + 3n + 6 ≤ 2n² + 3n² + 6\\` \n > 2. \\`n² ≤ 2n² + 3n² + n² = 6n²\\` per \\`n ≥ 3\\`\n >\n > Sappiamo, allora, che \\`2n² + 3n + 6 ≤ 6n²\\`.\n \n #### Espressioni di O grande\n \n Questa tabella rappresenta le espressioni di O grande più comunemente utilizzate, in ordine **dalla più forte alla più debole**.\n \n > Più forte significa che, per ogni riga della tabella, tutte le righe sottostanti sono contenute nell'espressione.\n >\n > Ad esempio, \\`O(n) ∈ O(1)\\`.\n \n | Espressione O() | Nome |\n |-----------------|------|\n | \\`O(1)\\` | Costante |\n | \\`O(log log n)\\` | loglog |\n | \\`O(log n)\\` | Logaritmica |\n | \\`Ω(n^{1/c})\\` (per c ≥ 1) | Sublineare |\n | \\`O(n)\\` | Lineare |\n | \\`O(n log n)\\` | nlogn |\n | \\`O(n²)\\` | Quadratica |\n | \\`O(n³)\\` | Cubica |\n | \\`O(n^k)\\` (per k ≥ 1) | Polinomiale |\n | \\`O(a^n)\\` (per a ≥ 1) | Esponenziale |\n | \\`O(n!)\\` | Fattoriale |\n \n #### Polinomiale\n \n Molto spesso, noteremo che il tempo richiesto da una funzione è O grande di un polinomio di grado K, ovvero \\`f(n) ∈ O(n^k)\\`.\n \n Notiamo che in questi casi, possiamo semplificare l'O grande al grado massimo del polinomio.\n \n > Ad esempio, \\`O(n² + n + 1) = O(n²)\\`.\n \n ##### Dimostrazione\n \n Per \\`n > 0 \\\\and 0 ≤ i ≤ k\\`: \n ![LaTeX](https://latex.codecogs.com/png.latex?a_k%20n^k%20+%20a_{k-1}%20n^{k-1}%20+%20%E2%80%A6%20+%20a_1%20n%20+%20a_0%20%E2%89%A4%20|a_k|%20n^k%20+%20|a_{k-1}|%20n^k%20+%20%E2%80%A6%20+%20|a_1|%20n^k%20+%20|a_0|%20n^k%20=%20(|a_k|%20+%20|a_{k-1}|%20+%20%E2%80%A6%20+%20|a_1|%20+%20|a_0|)%20n^k)\n \n #### Proprietà di O grande\n \n 1. \\`f(n) ∈ O(g(n)) -> ∀ a > 0, a * f(n) ∈ O(g(n))\\`.\n 2. \\`f(n) ∈ O(g(n)), d(n) ∈ O(h(n)) -> f(n) + d(n) ∈ O(g(n) + h(n)) -> O(max\\\\{g(n), h(n)\\\\})\\`\n 3. \\`f(n) ∈ O(g(n)), d(n) ∈ O(h(n)) -> f(n) * d(n) ∈ O(g(n) * h(n))\\`\n \n In pratica, se una funzione è la _somma di più termini_, basta guardare l'**\\`O()\\` più grande** tra tutti i suoi termini; se invece una funzione è un _prodotto di più termini_, si possono **omettere le costanti**, e l'\\`O()\\` finale sarà dato dal **prodotto degli \\`O()\\`** dei termini.\n \n ## Lower bound\n \n Possiamo anche stimare il _lower bound_, il limite inferiore: il **numero minimo di operazioni** che viene effettuato **nel caso migliore** con la **massima dimensione dell'ingresso**.\n \n ### Ω()\n \n > \"Omega\" \n > Omega di g(n)\n > \"big-Omega\"\n \n Esiste un equivalente di O grande per il lower bound: è detto _Omega grande_, o più semplicemente _Omega_, e funziona nello stesso identico modo, solo... al contrario.\n \n Diremo che \\`f(n) ∈ Ω(g(n))\\` se e solo se \\`∃ c > 0, n_0 ≥ 0 : ∀ n ≥ n_0 f(n) ≥ c * g(n)\\`.\n \n #### Espressioni di Ω()\n \n Anche in questa tabella le espressioni sono **dalla più forte alla più debole**.\n \n | Espressione Ω() | Nome |\n |-----------------|------|\n | \\`Ω(n!)\\` | Fattoriale |\n | \\`Ω(a^n)\\` (per a ≥ 1) | Esponenziale |\n | \\`Ω(n^k)\\` (per k ≥ 1) | Polinomiale |\n | \\`Ω(n³)\\` | Cubica |\n | \\`Ω(n²)\\` | Quadratica |\n | \\`Ω(n log n)\\` | nlogn |\n | \\`Ω(n)\\` | Lineare |\n | \\`Ω(n^{1/c})\\` (per c ≥ 1) | Sublineare |\n | \\`Ω(log n)\\` | Logaritmica |\n | \\`Ω(log log n)\\` | loglog |\n | \\`Ω(1)\\` | Costante |\n \n ## Tight bound\n \n Quando **upper e lower bound coincidono**, allora otteniamo un _tight bound_.\n \n ### θ()\n \n > \"Theta\" \n > Theta di g(n)\n > \"big-Theta\"\n \n Anche per il tight bound abbiamo una notazione equivalente a O grande e Omega grande: _Theta grande_!\n \n Diciamo che \\`f(n) ∈ θ(g(n))\\` se e solo se \\`∃ c_1, c_2 > 0, n_0 ≥ 0 : ∀ n ≥ n_0, c_1 * g(n) ≤ f(n) ≤ c_2 * g(n)\\`.\n \n Ha la particolarità che non valgono tutte le proprietà degli altri due: va usata quindi con cautela!\n \n ## Risorse utili\n \n [khanacademy.org](https://www.khanacademy.org/computing/computer-science/algorithms/asymptotic-notation/a/big-big-theta-notation)\n `}\n \n \n {`\n # Problemi algoritmici\n \n Un _problema algoritmico_ è un problema matematico che si vuole provare a risolvere con un algoritmo.\n \n > Dati 10 numeri, voglio sapere se sono in ordine crescente oppure no.\n \n ## Caterigorizzazione\n \n I problemi algoritmici si dividono in tre categorie: problemi _trattabili_, problemi _intrattabili_ e problemi _irrisolvibili_.\n \n ### Problema trattabile\n \n Perchè un problema algoritmico sia _trattabile_, deve avere **almeno un algoritmo con upper bound polinomiale**.\n \n > Questo significa che il tempo impiegato da un computer per risolvere il problema rimane ragionevole, e che quindi può essere utilizzato in maniera efficiente.\n \n La trattabilità è un campo ancora parecchio aperto: esistono anche tanti problemi di cui non si è ancora dimostrata la trattabilità o intrattabilità.\n \n > La [fattorizzazione](https://it.wikipedia.org/wiki/Fattorizzazione) è uno di questi problemi: l'assenza di una dimostrazione è ciò che la rende uno dei pilastri della sicurezza informatica moderna.\n \n ### Problema intrattabile\n \n Se **un problema non ha nessun algoritmo con upper bound polinomiale**, allora si dice che è **intrattabile**.\n \n ### Problema irrisolvibile\n \n Se **non esistono algoritmi per risolvere un problema**, allora questo si dice **irrisolvibile**.\n \n > [Dato un algoritmo con certi input, riusciamo a capire con un algoritmo se la sua esecuzione termina o no?](https://en.wikipedia.org/wiki/Halting_problem)\n \n ### Upper e lower bound di problemi\n \n Si può anche trovare un _upper bound_ e un _lower bound_ per un problema, ma bisogna generalizzare di più.\n \n L'**upper bound di un problema** è il minimo upper bound di tutti gli algoritmi che lo risolvono; deve esistere almeno un algoritmo che lo risolva che abbia lo stesso _upper bound_. E' praticamente il tempo migliore per risolvere il problema dato.\n \n Il **lower bound di un problema** è il minimo lower bound di tutti gli algoritmi che lo risolvono; non deve esistere nemmeno un algoritmo che abbia un lower bound migliore. E' il numero assolutamente minimo di operazioni richieste, non si può fare meglio di così.\n \n In particolare, abbiamo che l'_upper bound di un algoritmo_ -> l'_upper bound del suo problema_, \n e il _lower bound di un problema_ -> il _lower bound di un suo algoritmo_.\n \n Generalmente, il _lower bound di un problema_ è una rappresentazione abbastanza accurata della sua difficoltà.\n `}\n \n \n {`\n # Ricerca binaria\n \n Non credo di aver bisogno di studiare la ricerca binaria, quindi non ho preso appunti a riguardo.\n \n Se non siete me, e state cercando informazioni a riguardo, andate a vedere su Wikipedia!\n `}\n \n \n {`\n # Divide et impera\n \n Un modo efficace per risolvere un problema è di usare il metodo _divide et impera (et combina)_.\n \n - _Divide_: Divido il problema in **tanti sottoproblemi**.\n - _Impera_: **Risolvo indipendentemente dal resto** ciascuno dei sottoproblemi.\n - _Combina_: **Combino** i risultati dei sottoproblemi per **risolvere il problema principale**.\n \n ## Ricorsione\n \n Un algoritmo (o funzione) si dice _ricorsivo_ quando durante l'esecuzione **richiama sè stesso**.\n \n Dato che tutti gli algoritmi devono avere termine entro un tempo finito, se scriviamo una funzione ricorsiva è fondamentale finire con un **caso base**, che non chiami ulteriormente la ricorsione.\n \n Se esiste una funzione ricorsiva, allora esiste _sempre_ una _funzione iterativa_ che darà lo stesso risultato.\n \n > Sul pratico, una funzione ricorsiva tipicamente è **più costosa** del suo equivalente iterativo: se possibile, quindi, la ricorsione andrebbe evitata.\n \n #### Pseudocodice di esempio\n \n \\`\\`\\`python\n def fattoriale(n):\n # Caso base; la ricorsione finisce e dà un risultato fisso\n if n <= 1:\n return 1\n # Caso ricorsivo; la funzione restituisce il risultato di sè stessa (ma con parametri diversi)\n else:\n return n * fattoriale(n - 1)\n \\`\\`\\`\n `}\n \n \n {`\n # Master Theorem\n \n Il _Master Theorem_ è uno dei teoremi più importanti dell'algoritmica.\n \n Esso permette di **calcolare l'upper bound di un algoritmo ricorsivo** in modo piuttosto semplice.\n \n ## Ipotesi\n \n Dato un algoritmo:\n - Con uno o più casi base\n - Che richiama la funzione ricorsiva un numero n di volte\n \n ## Tesi\n \n Allora, il suo upper bound avrà la formula:\n \n ![](https://quicklatex.com/cache3/89/ql_08d29a7e55d561a900570bc83b93ff89_l3.png)\n `}\n \n \n {`\n # Caso particolare del Master Theorem\n \n ## Ipotesi\n \n Se: \n ![](https://quicklatex.com/cache3/57/ql_e34dc27b42831d3c3ff671b0f3861257_l3.png)\n \n Ovvero, se la dimensione dell'input viene divisa ad ogni ciclo da una costante b, è polinomiale e il caso base è costante...\n \n ## Tesi\n \n Allora: \n ![](https://quicklatex.com/cache3/ca/ql_26e3557a6ca2d6ac4b8481e7c5263fca_l3.png)\n \n > In pratica, se il costo dominante è quello della parte \"fissa\" dell'algoritmo, esso sarà \\`O(n^d)\\`, mentre se il costo dominante è quello delle chiamate ricorsive, esso sarà \\`O(n^{log_b e})\\`. \n > Se nessuno dei due è dominante... si dividono circa in parti uguali, creando un costo di \\`O(n^d log n)\\`.\n `}\n \n \n {`\n # Ordinamento\n \n Un problema molto frequente nell'informatica consiste nell'**ordinare efficientemente grandi quantità di elementi**.\n \n Esistono [tantissimi](https://it.wikipedia.org/wiki/Algoritmo_di_ordinamento) algoritmi per effettuare l'ordinamento. \n L'**efficienza** di ciascuno **varia** di caso in caso: alcuni sono estremamente efficienti se quasi tutti i numeri sono già nell'ordine giusto; altri, invece, potrebbero impiegare tantissimo tempo.\n \n In termini matematici, abbiamo:\n \n - **Input:** A[n]\n - **Output:** B, ∀ i < n, A[i] ≤ A[i+1]\n \n ## Ordinamento tramite confronto\n \n L'ordinamento \"tradizionale\" è detto _ordinamento tramite confronto_: funziona sempre, e **non ha altri modi di ottenere informazioni** se non con l'operazione logica di confronto tra i dati.\n \n ### Limiti\n \n E' un problema risolto: è dimostrabile che il suo **lower bound** è **\\`Ω(n log n)\\`**; possiamo quindi dire che qualsiasi algoritmo di ordinamento è in \\`Ω(n log n)\\`, e se riusciamo a trovare un algoritmo di ordinamento in \\`O(n log n)\\` siamo riusciti a raggiungere il massimo dell'efficienza.\n \n #### Dimostrazione\n \n Consideriamo **tutte le possibili permutazioni** della sequenza da ordinare: sono \\`n!\\`.\n \n Per ogni confronto che effettuiamo, **riduciamo la quantità di permutazioni** correttamente ordinate; prima o poi, rimarrà **una sola possibilità**.\n \n **TODO, non trovo la spiegazione corretta!**\n \n ### Esempi\n \n Algoritmi che effettuano l'ordinamento tramite confronto sono:\n - _Bubble sort_\n - _Merge sort_\n - _Insertion sort_\n - _Quick sort_\n - _Heap sort_\n - E tanti, tanti altri!\n \n ## Ordinamento con altri mezzi\n \n Esistono algoritmi che ricavano informazioni in altri modi, diversi dal confronto.\n \n Essi possono avere un lower bound più basso di \\`O(n log n)\\`, però hanno spesso limitazioni sul loro utilizzo.\n \n ### Esempi\n \n - _Counting sort_, indicizza i valori da ordinare\n - _Radix sort_, guarda singolarmente le cifre dei valori\n - _Sleep sort_, sfrutta i thread e la funzione sleep per ordinare valori\n - E altri ancora!\n `}\n \n \n {`\n # Insertion sort\n \n L'_insertion sort_ è una soluzione **iterativa** all'ordinamento per confronto.\n \n ## Funzionamento\n \n Considero la sequenza divisa in **due parti**: una parte **ordinata** e una parte **non ordinata**.\n \n Parto dal primo elemento della lista: è sempre ordinato con sè stesso. \n Poi, aggiungo uno alla volta i numeri della parte non ordinata a quella ordinata; prima trovo in quale posizione dovrò andare a mettere il numero, poi **faccio slittare tutti i numeri dopo quella posizione** avanti di 1, in modo da **creare lo spazio** in cui infine **inserirò** il numero.\n \n ## Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | \\`O(n²)\\` | \\`Ω(n)\\` | - |\n \n Nel _caso migliore_ (**lista già ordinata**), il numero da inserire è già nella posizione giusta, quindi non devo effettuare altri confronti oltre il primo, rendendo il lower bound dell'algoritmo \\`Ω(n)\\`.\n \n Nel _caso peggiore_ (**lista nell'ordine inverso**), dobbiamo confrontare il numero da inserire con tutti gli altri nella parte ordinata: dobbiamo allora eseguire \\`1+2+3+4+5+… = \\\\frac{(n-1)(n)}{2}\\` confronti; ciò significa che l'upper bound è \\`O(n²)\\`!\n \n ## Pseudocodice\n \n \\`\\`\\`python\n def insertion_sorted(lista):\n # Itero su tutti i numeri della lista, dal primo all'ultimo.\n for divisore_ord in range(len(lista)):\n # Partendo dalla posizione attuale, creo l'indice di divisione numeri ordinati maggiori e minori\n divisore_magg = divisore_ord\n # Faccio slittare avanti i numeri maggiori di quello che stiamo inserendo\n # Se l'indice divisore_magg raggiunge 0, vuol dire che tutti i numeri della lista sono maggiori del numero attuale \n while divisore_magg >= 0 and lista[divisore_magg-1] > lista[divisore_magg]:\n # Scambio la posizione dei due elementi con gli indici specificati\n # Funzione inventata \n lista[divisore_magg], lista[divisore_magg-1] = lista[divisore_magg-1], lista[divisore_magg]\n # Diminuisco il separatore di 1\n divisore_magg -= 1\n \\`\\`\\`\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/bn/sorting)\n `}\n \n \n {`\n # Merge sort\n \n Il _merge sort_ è una soluzione **ricorsiva** all'ordinamento per confronto.\n \n ## Funzionamento\n \n Per questo algoritmo, utilizziamo la tecnica del **divide et impera**.\n \n 1. _Divide_: Divido A in **due parti**.\n 2. _Impera_: Metto **separatamente in ordine** le parti.\n 3. _Unisci_: **Unisco** le due parti.\n \n Consideriamo come **caso base** della ricorsione una parte composta da un numero, che ovviamente è già ordinata con sè stessa.\n \n ### Merge\n \n Per **unire le due parti** usiamo una funzione detta \\`merge()\\`.\n \n Costruiamo una nuova sequenza uguale alla sequenza 1, ma **aggiungiamo alla fine un valore sentinella** sempre maggiore di tutti gli elementi contenuti.\n \n > \\`\\`\\`\n > | 1 | 3 | 7 | 8 | ∞ |\n > \\`\\`\\`\n \n Facciamo **la stessa cosa** per la sequenza due.\n \n > \\`\\`\\`\n > | 2 | 4 | 5 | 6 | ∞ |\n > \\`\\`\\`\n \n Prendo i primi numeri delle due sequenze e **metto il più piccolo nella sequenza iniziale**.\n \n > \\`\\`\\`\n > | 1 | 2 | 3 | | | | | | \n > | | | 7 | 8 | ∞ | \n > | | 4 | 5 | 6 | ∞ | \n > \\`\\`\\`\n \n **Continuo** finchè non ho messo tutti i numeri; **grazie alla sentinella non usciremo mai dalla sequenza**, in quanto essa è sempre maggiore di tutti gli altri numeri.\n \n > \\`\\`\\`\n > | 1 | 2 | 3 | 4 | 5 | 6 | | | \n > | | | 7 | 8 | ∞ | \n > | | | | | ∞ |\n > \\`\\`\\`\n \n Quando **rimangono solo le sentinelle** significa che abbiamo aggiunto tutti gli elementi, e quindi abbiamo finito.\n \n > \\`\\`\\`\n > | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | \n > | | | | | ∞ | \n > | | | | | ∞ | \n > \\`\\`\\`\n \n \n ## Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | \\`O(n log n)\\` | \\`Ω(n log n)\\` | **\\`θ(n log n)\\`** |\n \n Il merge sort è un algoritmo ricorsivo con un **caso base in tempo costante** e che **richiama sè stesso 2 volte**.\n \n \\`\\`\\`latex\n T(n) =\\\\\\\\\n \\\\\\\\\n Θ(1) \\\\qquad n=1\\\\\\\\\n 2 T(\\\\frac{n}{2}) + Θ(n) \\\\qquad n \\\\neq 1\n \\`\\`\\`\n \n Applicando il **caso particolare del Master Theorem**, otteniamo:\n \n \\`\\`\\`latex\n T(n) =\\\\\\\\\n \\\\\\\\\n Θ(1) \\\\qquad n=1\\\\\\\\\n Θ(n log n) \\\\qquad n \\\\neq 1\n \\`\\`\\`\n \n ## Pseudocodice\n \n \\`\\`\\`python\n def merge_sorted(part):\n # Caso base\n if len(part) == 1:\n return part\n # Divide\n middle = len(part) // 2\n part_a = part[:middle]\n part_b = part[middle:]\n # Impera\n sort_a = merge_sorted(part_a)\n sort_b = merge_sorted(part_b)\n # Combina\n return merge(sort_a, sort_b)\n \\`\\`\\`\n \n ## Visualizzazione\n \n [hackerearth.com](https://www.hackerearth.com/practice/algorithms/sorting/merge-sort/visualize/)\n \n [visualgo.net](https://visualgo.net/bn/sorting) (Nota: visualizza solo la fase _Unisci_ del sort)\n `}\n \n \n {`\n # Quick sort\n \n Il _quick sort_ è un altro approccio **ricorsivo** all'ordinamento per confronto.\n \n ## Funzionamento\n \n Anche qui, applichiamo il **divide et impera**.\n \n 1. _Divide_: Scelgo un **pivot** qualsiasi all'interno della lista. Metto alla sua **sinistra tutti i numeri minori** e alla sua **destra tutti i numeri maggiori**.\n 2. _Impera_: Eseguo un **quick sort su entrambe le sottoliste**.\n \n ### Esempi\n \n #### Iterazione con partizioni bilanciate\n \n Osserviamo come si formi una partizione con tre elementi e una con quattro. \n \n \\`\\`\\`\n |¦ [2] 8 7 1 3 5 6 {4}\n 2 |¦ [8] 7 1 3 5 6 {4}\n 2 |8 ¦ [7] 1 3 5 6 {4}\n 2 |8 ¦ [7] 1 3 5 6 {4}\n 2 |8 7 ¦ [1] 3 5 6 {4}\n 2 1 |7 8 ¦ [3] 5 6 {4}\n 2 1 3 |8 7 ¦ [5] 6 {4}\n 2 1 3 |8 7 5 ¦ [6] {4}\n 2 1 3 |8 7 5 6 ¦ [{4}]\n [2 1 3] {4} [8 5 6 7]\n \\`\\`\\`\n \n #### Iterazione con partizioni sbilanciate\n \n Osserviamo come si formi una partizione con **zero elementi** e una con tre.\n \n \\`\\`\\`\n |¦ [4] 7 3 {1}\n |4 ¦ [7] 3 {1}\n |4 7 |¦ [3] {1}\n |4 7 3 |¦ [{1}]\n [] {1} [4 7 3]\n \\`\\`\\`\n \n ## Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | \\`O(n²)\\` | \\`Ω(n log n)\\` | - |\n \n Il costo della funzione è dato dalla somma del costo per **dividere in due partizioni** con il costo per realizzare il **Quick sort delle due sottopartizioni**\n \n Possiamo applicare allora il **Master Theorem generale**:\n \n \\`\\`\\`latex\n T(n)\\\\\\\\\n =\\\\\\\\\n Θ(1) \\\\qquad per\\\\ n = 1\\\\\\\\\n T(q) + T(dim_lista - pivot - 1) + Θ(n) \\\\qquad per\\\\ n > 1\n \\`\\`\\`\n \n ### Il caso migliore\n \n Se il pivot \\`q\\` è la **mediana della partizione** che stiamo ordinando, si vengono a creare due _sottopartizioni bilanciate_, e sostituendo otteniamo:\n \n \\`\\`\\`latex\n T(n)\\\\\\\\\n =\\\\\\\\\n Θ(1) \\\\qquad per\\\\ n = 1\\\\\\\\\n 2 T(\\\\frac{n}{2}) + Θ(n) \\\\qquad per\\\\ n > 1\n \\`\\`\\` \n \n Possiamo allora applicare il **Master Theorem particolare**:\n \n \\`\\`\\`latex\n T(n)\\\\\\\\\n =\\\\\\\\\n Θ(1) \\\\qquad per\\\\ n = 1\\\\\\\\\n Θ(n log n) \\\\qquad per\\\\ n > 1\n \\`\\`\\`\n \n ### Il caso peggiore\n \n Se il pivot è uno degli **estremi dell'array**, si creano due _partizioni sbilanciate_: una delle due sottoliste è sempre vuota! \n Allora:\n \n \\`\\`\\`latex\n T(n) = T(n-1) + Θ(n)\\\\\\\\\n = T(n-2) + Θ(n-1) + Θ(n)\\\\\\\\\n = T(n-3) + Θ(n-2) + Θ(n-1) + Θ(n)\\\\\\\\\n = …\n ∈ Θ(n^2)\n \\`\\`\\`\n \n > \"Non date da mangiare sequenze ordinate al Quicksort, gli sono indigeste.\"\n \n ## Pseudocodice\n \n \\`\\`\\`python\n def partition(partizione, inizio, fine):\n \"\"\"Dividi una partizione in due, usando l'ultimo elemento come pivot.\n \n Note utili:\n partizione[fine] è il pivot\n partizione[maggiori] è il primo numero dei maggiori\n partizione[non_iterati] è l'elemento su cui si sta iterando al momento\"\"\"\n # Crea il primo separatore (la | pipe nell'esempio)\n maggiori = inizio\n # Crea il secondo separatore (la ¦ broken pipe nell'esempio)\n non_iterati = inizio\n # Itera su ogni numero tra inizio e fine (escluso!)\n while non_iterati < fine:\n # Se l'elemento su cui stiamo iterando è minore del pivot\n if partizione[non_iterati] <= partizione[fine]:\n # Mettilo nell'insieme dei minori, scambiandolo con il primo numero dei maggiori e incrementando il primo separatore\n partizione[maggiori], partizione[non_iterati] = partizione[non_iterati], partizione[maggiori]\n maggiori += 1\n # Incrementa sempre il secondo separatore\n non_iterati += 1\n # Inserisci il pivot tra le due sottopartizioni create, \n partizione[fine], partizione[non_iterati] = partizione[non_iterati], partizione[fine]\n return maggiori\n \\`\\`\\`\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/bn/sorting) (Nota: invece che prendere l'ultimo numero come pivot prende il primo, cambiando leggermente l'algoritmo.)\n \n ## Note per l'esame\n \n > La domanda che fa sempre è **\"Qual è la sequenza di pivot utilizzata?\"**\n \n > Elementi da soli _non_ vengono presi come pivot!\n `}\n \n \n {`\n # Counting sort\n\n Il _counting sort_ è un approccio diverso all'ordinamento: **non usa il confronto**!\n \n ## Requisiti\n \n Il counting sort può essere utilizzato solo su **sequenze di numeri interi**, e solo se **siamo a conoscenza del minimo e del massimo** dei numeri contenuti nell'array, ed essi non sono troppo distanti uno dall'altro. \n (La memoria occupata dal counting sort aumenta linearmente con la differenza tra minimo e massimo!)\n \n Per semplicità, consideriamo il **minimo \\`0\\`**. \n L'input allora sarà una sequenza di interi \\`A\\`, e il valore del **massimo \\`k\\`**, tale che \\`∀ n ∈ A, 0 \\\\leq n \\\\leq K\\`.\n \n ## Funzionamento\n \n Il counting sort **conta le ripetizioni** delle chiavi nella sequenza originale e in seguito **sovrascrive i valori** della sequenza con i valori ordinati ripetuti il numero di volte che sono stati individuati nella sequenza.\n \n > \\`\\`\\`\n > 1 4 5 3 4 1 4 2 5 1\n > \\`\\`\\`\n >\n > L'\\`1\\` appare 3 volte, il \\`2\\` 1 volta, il \\`3\\` 1 volta, il \\`4\\` tre volte e il \\`5\\` due volte.\n >\n > La sequenza viene quindi così sovrascritta:\n > \\`\\`\\`\n > 1 1 1 3 4 1 4 2 5 1 # Sovrascriviamo la sequenza con 1 ripetuto 3 volte\n > 1 1 1 2 3 4 4 4 5 1 # Sovrascriviamo la sequenza con 2, 3, 4 ripetuti rispettivamente 1 1 e 3 volte\n > 1 1 1 2 3 4 4 4 5 5 # Sovrascriviamo la sequenza con 5 ripetuto 2 volte: abbiamo finito!\n > \\`\\`\\` \n \n Esiste anche una **versione stabile** del counting sort che, invece che sovrascrivere, **sposta i valori**, mantenendo le informazioni aggiuntive nel caso invece che interi fossero altri tipi di dati.\n \n ## Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | \\`O(k + n)\\` | \\`Ω(k + n)\\` | **\\`θ(k + n)\\`** |\n \n L'algoritmo è composto da quattro parti:\n - Ricerca del minimo e massimo (in \\`θ(n)\\`)\n - Inizializzazione dell'indice (in \\`θ(k)\\`)\n - Conteggio dei numeri (in \\`θ(n)\\`)\n - Sovrascrittura dei numeri (in \\`θ(k + n)\\`)\n \n \\`2 + O(k) + O(n) + O(k + n) -> O(k + n)\\`\n \n Notiamo che \\`k\\` è costante, l'algoritmo è \\`O(n)\\`, estremamente efficiente.\n \n ## Pseudocodice\n \n \\`\\`\\`python\n def counting_sort(lista: typing.List[int]):\n \"\"\"Ordina in-place una lista con il counting sort.\"\"\"\n # Trovo la dimensione della lista\n dim = len(lista)\n # Trovo il massimo e il minimo all'interno della lista\n minimo = min(lista)\n massimo = max(lista)\n # Creo l'indice dei numeri, in modo che sia lungo k e pieno di 0\n indice = [0 for _ in range(minimo, massimo+1)]\n # Conto i numeri presenti, scorrendo su lista e aggiungendo 1 al numero corrispondente\n for i in range(dim):\n indice[lista[i]] += 1\n # Sovrascrivo i numeri nella lista\n count = 0\n for pos, val in enumerate(indice):\n for _ in range(val):\n indice[count] = pos\n count += 1\n \n def stable_counting_sorted(lista: typing.List[int], k: int) -> typing.List[int]:\n \"\"\"Ordina stabilmente una lista con il counting sort stabile, e restituiscila.\"\"\"\n # Trovo la dimensione della lista\n dim = len(lista)\n # Trovo il massimo e il minimo all'interno della lista\n minimo = min(lista)\n massimo = max(lista)\n # Creo l'indice dei numeri, in modo che sia lungo k e pieno di 0\n indice = [0 for _ in range(minimo, massimo+1)]\n # Conto i numeri presenti, scorrendo su lista e aggiungendo 1 al numero corrispondente\n for i in range(dim):\n indice[lista[i]] += 1\n # Faccio diventare l'indice \"il numero di numeri \\\\leq i\"\n for i in range(len(indice)):\n if i == 0:\n continue\n indice[i] += indice[i-1]\n assert indice[-1] == dim\n # Creo una nuova lista, che sarà quella che verrà restituita\n nuova = [None for _ in range(dim)]\n # Inizio a posizionare i numeri, al contrario\n for i in range(0, dim, -1):\n nuova[indice[lista[i]]] = lista[i]\n indice[lista[i]] -= 1\n return nuova\n \\`\\`\\`\n `}\n \n \n {`\n ## Introduzione alle Strutture Dati\n\n Una _struttura dati_ è un modo in cui si possono organizzare i dati di un programma.\n \n Si possono definire in due modi: **elementari** e **astratte**.\n \n ### Strutture dati elementari\n \n Le strutture _elementari_ dipendono strettamente dal modo in cui vengono memorizzati i dati.\n \n > **Array** e **liste** sono strutture dati elementari: sono definite dicendo come sono memorizzati i dati, rispettivamente, in celle contigue di memoria e da una serie di nodi con un valore e che puntano al successivo.\n \n ### Strutture dati astratte\n \n Le strutture _astratte_ sono separate dal modo in cui vengono memorizzati i dati, sono più ad alto livello, e si definiscono descrivendo le **proprietà** della struttura e i **metodi** che su di essa possono essere effettuate.\n \n > Una _classe_ in un qualsiasi linguaggio di programmazione è una struttura dati astratta.\n \n > Una _pila_ astratta:\n > - memorizza dati tutti dello stesso tipo\n > - \\`pop()\\`, estrae dalla pila l'ultimo valore inserito\n > - \\`push(val)\\`, aggiunge alla pila un valore\n > - \\`top()\\`, permette di vedere l'ultimo valore inserito nella pila\n > - \\`vuota()\\`, dice se la pila è vuota oppure no.\n `}\n \n \n {`\n # Array\n\n Un _array_ è sequenza di dati di **lunghezza conosciuta**, tutti dello **stesso tipo** e di una **dimensione fissa**, immagazzinata in **blocchi di memoria contigui**.\n \n ## Proprietà\n \n - E' possibile accedere a tutti i blocchi di memoria conoscendo la loro **posizione relativa al primo** elemento.\n \n ## Metodi\n \n \\`\\`\\`python\n class Array:\n def __init__(self, size, type=int): \"Crea un array di dimensione size di elementi di tipo int.\"\n def __getitem__(self, index): \"Restituisci il valore alla posizione index.\"\n def __setitem__(self, index, value): \"Cambia il valore alla posizione index.\"\n \\`\\`\\`\n \n ### Costo computazionale\n \n Tutte le operazioni su un array sono in **accesso immediato \\`O(1)\\`**!\n `}\n \n \n {`\n # Lista\n \n Una _lista_ è una sequenza di dati immagazzinata in **blocchi di memoria qualsiasi**.\n \n ## Proprietà\n \n Ogni dato ha un **riferimento** alla collocazione di memoria **successiva** (un puntatore): insieme, sono detti un _nodo_.\n \n E' di **natura ricorsiva**: qualsiasi nodo di una lista può essere visto come inizio della lista con sè stesso e i suoi successivi.\n \n ## Metodi\n \n \\`\\`\\`python\n class LinkedList:\n def __init__(self, value, next: typing.Optional[Node] = None):\n self.value = value\n self.next: typing.Optional[Node] = next\n \n def is_empty(self) -> bool: \"Restituisce se la lista è vuota o no.\"\n def is_full(self) -> bool: \"Restituisce se la lista è piena o no.\n def append(self, value): \"Aggiunge un nuovo elemento in testa alla lista.\"\n def insert(self, value, index): \"Inserisce un elemento dopo il nodo in posizione index.\"\n def insert_node(self, value, node): \"Inserisce un nuovo elemento subito dopo un dato nodo.\"\n def find(self, node) -> int: \"Trova l'indice del nodo.\"\n def delete(self, value): \"Elimina il primo nodo con quel valore dalla lista.\"\n def delete_node(self, node): \"Elimina il nodo dalla lista.\"\n def forward(self, index) -> Node: \"Restituisce il nodo in posizione index.\"\n \\`\\`\\`\n \n ### Costo computazionale\n \n #### \\`List.forward(index)\\`\n \n Per raggiungere l'\\`n\\`-esimo elemento, bisogna _scorrere tutti gli elementi prima di esso_: è dunque un **accesso sequenziale** in **\\`O(n)\\`**.\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/list)\n `}\n \n \n {`\n # Coda (Queue)\n\n Una _coda_ è come una pila, ma segue la strategia **First In, First Out** (il primo inserito sarà il primo a essere estratto).\n \n ## Proprietà\n \n - I dati vi possono essere aggiunti solo tramite il **metodo \\`enqueue()\\`**\n - I dati possono essere estratti solo tramite il **metodo \\`dequeue()\\`**\n - Verranno restituiti i valori inseriti secondo la strategia **First In, First Out** (il primo inserito sarà il primo a essere estratto).\n \n > Hai presente quando fai la fila per pagare al supermercato? Beh, è quello, però non si possono superare le altre persone in nessun modo.\n \n ## Metodi\n \n \\`\\`\\`python\n class Queue:\n def __init__(self): \"Crea una nuova coda.\"\n def is_empty(self) -> bool: \"Restituisce vero se la coda è vuota.\"\n def enqueue(self, data): \"Aggiunge un dato alla coda.\"\n def first(self) -> ...: \"Restituisce il primo dato della coda.\"\n def dequeue(self) -> ...: \"Restituisce il primo dato della coda e lo rimuove.\"\n \\`\\`\\`\n \n ## Implementazione tramite lista\n \n Posso implementare la coda con una lista, ma per realizzare l'implementazione più efficiente devo tenere anche un puntatore all'ultimo elemento della coda, in modo da non doverla scorrere ogni volta che voglio effettuare un'operazione.\n \n Chiamiamo i due puntatori \\`head\\` e \\`tail\\` rispettivamente.\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/list)\n `}\n \n \n {`\n # Pila (Stack)\n\n Una _pila_ è una struttura dati contenente **valori omogenei**.\n \n ## Proprietà\n \n - I dati vi possono essere aggiunti solo tramite il **metodo \\`push()\\`**\n - I dati possono essere estratti solo tramite il **metodo \\`pop()\\`**\n - Verranno restituiti i valori inseriti secondo la strategia **Last In, First Out** (l'ultimo inserito sarà il primo a essere estratto).\n \n > Ci si può immaginare una pila di libri, da cui si può solo prendere un libro alla volta, quello più in alto.\n \n ## Metodi\n \n \\`\\`\\`python\n class Stack:\n def __init__(self): \"Crea una nuova pila.\"\n def is_empty(self) -> bool: \"Restituisce vero se la pila è vuota.\"\n def push(self, data): \"Aggiunge un dato alla pila.\"\n def top(self) -> ...: \"Restituisce il primo dato della pila.\"\n def pop(self) -> ...: \"Restituisce il primo dato della pila e lo rimuove.\"\n \\`\\`\\`\n \n ## Implementazione tramite lista\n \n Utilizzando una lista possiamo realizzare una pila!\n \n La direzione dei puntatori sarà dall'ultimo al primo, che non punterà più a nessuno.\n \n Il costo di tutti i metodi è \\`Θ(1)\\`!\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/list)\n `}\n \n \n {`\n # Albero radicato\n \n Un _albero radicato_ è una struttura dati di **natura ricorsiva** che organizza i dati in maniera **non-lineare**.\n \n ## Proprietà\n \n - Ogni nodo dell'albero ha un **unico genitore**: \\`∀ (padre, figlio), (padre' figlio) ∈ E \\\\implies padre = padre'\\`\n - Ogni nodo dell'albero può avere **un numero qualsiasi di figli**.\n \n - I **nodi superiori al padre** vengono chiamati _antenati_.\n - I **nodi inferiori ai figli** vengono chiamati _discendenti_.\n \n - Nodi **senza padre** sono detti _radice_: \\`\\\\notexists (padre, radice) ∈ E\\`\n - Nodi **con padre e figli** sono detti _rami_ o interni.\n - Nodi **senza figli** sono detti _foglie_.\n \n - La **distanza** tra il nodo radice e i suoi discendenti è detta _livello_:\n - I figli immediati sono di livello 1.\n - I \"nipoti\" (figli dei figli) sono di livello 2.\n - I figli dei nipoti sono livello 3.\n - E così via.\n - Il **livello massimo** all'interno di un albero è detto _altezza_, _profondità_ oppure _h_, ed è sempre \\`1 ≤ h ≤ n-1\\`.\n \n - Un albero ha sempre \\`n-1\\` archi.\n \n ## Alberi particolari\n \n ### Alberi \\`d\\`-ari\n \n Un albero _\\`d\\`-ario_ è un particolare tipo di albero che **limita il numero massimo di figli di un nodo** a \\`d\\`.\n \n > Un albero _binario_ può avere **massimo 2 figli** per ogni nodo; un albero _ternario_ ne può avere **3**; un albero _\\`17\\`-ario_ ne potrà avere **17**\n \n #### Alberi completi\n \n Un albero \\`d\\`-ario si dice _completo_ se **tutti i nodi hanno 0 o \\`d\\` figli**, e mai una numero in mezzo.\n \n #### Alberi bilanciati\n \n Un albero \\`d\\`-ario si dice _bilanciato_ se **tutti i livelli eccetto l'ultimo** hanno il numero massimo di figli.\n \n #### Alberi perfettamente bilanciati\n \n Un albero \\`d\\`-ario si dice _perfettamente bilanciato_ se **tutti i livelli incluso l'ultimo** hanno il numero massimo di figli.\n \n ##### Particolarità degli alberi binari perfettamente bilanciati\n \n Si può dimostrare per induzione che:\n - Hanno sempre \\`2^h\\` foglie.\n - Hanno sempre \\`2^{h+1}-1\\` (\\`\\\\sum_i=0^n 2^i\\`) nodi.\n - L'altezza è in \\`Θ(log n)\\`.\n \n ## Implementazione degli alberi\n \n Possiamo scegliere se usare una rappresentazione con array o con nodi e puntatori: ognuna ha vantaggi e svantaggi diversi.\n \n ### Implementazione tramite array\n \n E' suggerito se l'albero è regolare; più è simile a un albero d-ario completo, meglio è.\n \n ### Implementazione tramite nodi e puntatori\n \n Più adatta ad alberi irregolari.\n \n Se l'albero è regolare, creiamo il numero esatto di campi:\n \n - Valore\n - Figlio1\n - Figlio2\n - _Opzionale:_ Padre\n \n Se un albero è irregolare, creiamo una specie di lista:\n \n - Valore\n - Primo figlio\n - Prossimo fratello\n - _Opzionale:_ Padre\n `}\n \n \n {`\n #### Breadth-first search (BFS)\n\n La _breadth-first search_ è un algoritmo che visita **ogni livello** dell'albero in ordine, dal più basso al più alto.\n \n ## Funzionamento\n \n > 1. __Visita radice__\n > 2. __Visita figli__\n > 3. __Visita nipoti__\n > 4. __Visita pronipoti__\n \n Si può implementare con una coda, in cui verranno inseriti i figli del nodo visitato da visitare ed estratti dopo avere completato la visita del livello attuale.\n \n ## Pseudocodice \n \n \\`\\`\\`python\n def bfs(radice):\n c = Queue()\n c.enqueue(radice)\n while not c.is_empty():\n nodo = c.dequeue()\n print(nodo)\n for figlio in nodo.figli:\n nodo.enqueue(figlio)\n \\`\\`\\`\n `}\n \n \n {`\n # Depth-First Search\n\n La _depth-first search_ è un algoritmo che visita **tutti i sottoalberi di un figlio** prima di passare ad un altro sfruttando la natura ricorsiva degli alberi.\n \n ## Funzionamento\n \n Ci sono diverse versioni della depth-first search: ognuna visita la radice in un momento diverso.\n \n ### DFS previsita (pre-order)\n \n La _DFS pre-visita_ visita la **radice per prima**, poi tutti i sottoalberi formati dai figli uno dopo l'altro.\n \n > 1. __Visita radice__\n > 2. dfs_pre_order(figlio1)\n > 3. dfs_pre_order(figlio2)\n \n ### Postvisita (post-order)\n \n La _DFS postvisita_ visita prima tutti i sottoalberi dei figli, e **alla fine la radice**.\n \n > 1. dfs_post_order(figlio1)\n > 2. dfs_post_order(figlio2)\n > 3. __Visita radice__\n \n ### Invisita (in-order)\n \n La _DFS invisita_ visita **un certo numero di figli**, poi la radice, e infine i figli restanti.\n \n > 1. dfs_in_order(figlio1, 1)\n > 2. __Visita radice__\n > 3. dfs_in_order(figlio2, 1)\n `}\n \n \n {`\n # Albero binario di ricerca\n\n ## Proprietà\n \n - Albero **binario**\n - Chiavi appartenenti ad un **insieme totalmente ordinato** (N, Q, R, ma non C)\n \n - Per ogni nodo con valore \\`x\\`, se un valore \\`v\\` è nel sottoalbero di sinistra allora \\`v ≤ x\\`, mentre se è nel sottoalbero di destra allora \\`v > x\\`.\n \n ## Costo computazionale\n \n - Trovare un valore: \\`O(h)\\`\n - Ordinare i valori: \\`O(n)\\`\n - Trovare il minimo: \\`O(h)\\`\n - Trovare il massimo: \\`O(h)\\`\n - Inserire un elemento: \\`O(h)\\`\n - Cancellare un elemento: \\`O(h)\\`\n \n \\`h\\` vale \\`log n\\` in un albero perfettamente bilanciato, e più l'albero diventa sbilanciato, più si avvicina a \\`n\\`, raggiungendola nel caso l'albero sia una lista.\n \n ## Pseudocodice\n \n ### Cancellazione ricorsiva\n \n \\`\\`\\`python\n def delete(tree, key):\n if tree is not None:\n # Se ho trovato il nodo che cercavo...\n if tree.key == key:\n # E c'è una sola diramazione...\n # Semplicemente stacca il nodo come in una lista.\n if tree.left is None:\n return tree.right\n if tree.right is None:\n return tree.left\n # Altrimenti, diventa il minimo dell'albero di destra\n tree.key = tree.right.min()\n # Ed eliminalo dal sottoalbero\n tree.right = delete(tree.right, tree.key)\n # Se la chiave attuale è diversa da quella che cerchiamo, continuo a navigare l'albero\n elif tree.key < key:\n tree.left = delete(tree.left, key)\n else:\n tree.right = delete(tree.right, key)\n return tree\n \\`\\`\\`\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/bst)\n \n ## Approfondimenti\n \n Esistono alberi più avanzati che mantengono le proprietà degli alberi binari di ricerca, ma che si autobilanciano, come il [Red Black Tree](https://it.wikipedia.org/wiki/RB-Albero).\n `}\n \n \n {`\n # Heap binario\n \n L'_heap binario_ è un **albero binario bilanciato a sinistra**.\n \n ## Proprietà\n \n - _Proprietà strutturale_:\n - L'albero è **perfettamente bilanciato** in tutti i livelli tranne l'ultimo\n - Nell'ultimo livello, le foglie occupano le **posizioni più a sinistra** possibili\n - _Proprietà di ordinamento_:\n - La chiave di un qualsiasi nodo è **più piccola** di tutte quelle dei nodi **del suo sottoalbero**\n \n ## Metodi\n \n \\`\\`\\`python\n class Heap:\n def __init__(self, H): ...\n def _heapify_ancestors(self, i): \"Ripristina le proprietà dell'heap per il nodo all'indice specificato e i suoi genitori.\"\n def minimum(self): \"Restituisce la chiave con il valore minimo in H.\"\n def decrease_value(self, index, new_value): \"Diminuisce il valore della chiave all'indice index a new_value.\"\n def insert(self, value): \"Inserisci un nuovo valore nell'albero.\"\n def _heapify_children(self, i): \"Ripristina le proprietà dell'heap per il nodo all'indice specificato e i suoi figli.\"\n def pop(self): \"Restituisce la chiave con il valore minimo, e la elimina.\"\n def from_list(l): \"Crea un heap da una lista.\"\n \\`\\`\\`\n \n ## Implementazione con un array\n \n Possiamo implementare l'albero utilizzando un array con le chiavi dell'albero memorizzate nell'ordine _breadth-first_.\n \n L'indice del figlio sinistro può essere trovato a \\`2i+1\\`, mentre l'indice del figlio sinistro può essere trovato a \\`2i+2\\`; il genitore è a \\`i//2-1\\`.\n \n ### Pseudocodice\n \n \\`\\`\\`python\n class Heap:\n def __init__(self, size):\n self.array = Array(size) # Il tipo Array non esiste; consideriamolo pseudocodice\n self.next_value = 0\n \n def _heapify_ancestors(self, index)\n \"\"\"Ripristina le proprietà dell'heap per il nodo all'indice specificato e i suoi genitori.\n \n Costo:\n O(log n)\"\"\"\n # Trovo l'indice del genitore\n parent = index // 2 - 1\n # Controllo se viene mantenuta la proprietà di ordinamento dell'heap\n if self.array[index] < self.array[parent]:\n # Scambio i valori dei due nodi\n self.array[index], self.array[parent] = self.array[parent], self.array[index]\n # Faccio la stessa cosa con il genitore\n self._heapify(parent)\n \n \n def minimum(self):\n \"\"\"Restituisce la chiave con il valore minimo in H.\n \n Costo:\n O(1)\"\"\"\n return self.array[0]\n \n def decrease_value(self, index, new_value):\n \"\"\"Diminuisce il valore della chiave all'indice index a new_value.\n \n Costo:\n O(log n)\"\"\"\n # Diminuisco il valore del nodo\n self.array[index] = new_value\n # Aggiorno l'heap \n self._heapify(index)\n \n def insert(self, value):\n \"\"\"Inserisci un nuovo valore nell'albero.\n \n Costo:\n O(log n)\"\"\"\n # Trovo l'indice in cui inserire il valore\n index = self.next_index \n # Aggiungo il valore in fondo\n self.array[index] = value\n # Aggiorno l'heap\n self._heapify(index)\n \n def _heapify_children(self, index):\n \"\"\"Ripristina le proprietà dell'heap per il nodo all'indice specificato e i suoi figli.\n \n Costo:\n O(log n)\"\"\"\n # Trovo l'indice dei figli\n left = index * 2 + 1\n right = index * 2 + 2\n # Mi assicuro che i figli esistano\n try:\n # Guardo quale dei figli è maggiore\n if self.array[left] > self.array[right]:\n # Scambio i valori\n self.array[left], self.array[index] = self.array[index], self.array[left]\n # Ripeto la procedura sul figlio modificato\n self._heapify_children(left)\n else:\n # Scambio i valori\n self.array[right], self.array[index] = self.array[index], self.array[right]\n # Ripeto la procedura sul figlio modificato\n self._heapify_children(left)\n except IndexError:\n # La foglia non ha figli: ho finito!\n return\n \n def pop(self):\n \"\"\"Restituisce la chiave con il valore minimo, e la elimina.\n \n Costo:\n O(log n)\"\"\"\n # Mi salvo il valore della radice\n value = self.array[0]\n # Sostituisco la radice con l'ultima foglia a destra\n self.array[0], self.array[self.next_value] = self.array[self.next_value], self.array[0]\n ### Non bisognerebbe eliminare la foglia...?\n # Riordino l'heap\n self._heapify_children(0)\n return value\n \n @staticmethod\n def from_list(l):\n \"\"\"Crea un heap da una lista.\n \n Costo:\n O(n log n), ma si può abbassare\"\"\"\n heap = Heap(len(l)))\n heap.array = Array.from_list(l) # Pseudocodice\n heap.next_value = len(l)\n # Cominciamo a riordinare l'heap dalla fine, in modo che rispetti le proprietà\n for index in range(heap.next_value, 0, -1):\n heap._heapify_children(index)\n \\`\\`\\`\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/heap)\n `}\n \n \n {`\n # Coda con priorità\n \n La _coda con priorità_ è una struttura dati dal funzionamento molto simile a quello di una coda, ma invece che restituire il primo elemento inserito, essa restituisce l'**elemento con il valore di priorità minore**.\n \n ## Proprietà\n \n - Ogni elemento è una coppia costituita da **valore** e **priorità** (un numero intero).\n - Nuovi elementi possono essere aggiunti solo tramite il **metodo \\`insert()\\`**\n - Gli elementi possono essere estratti solo tramite il **metodo \\`pop()\\`**\n - Verranno restituiti i valori inseriti secondo la strategia **Lower Priority** (l'elemento con la priorità minore sarà il primo ad essere restituito).\n - E' possibile diminuire la priorità di un elemento (e quindi anticipare la sua estrazione)\n \n ## Metodi\n \n \\`\\`\\`python\n class PriorityQueue:\n def __init__(self): ...\n def insert(self, new_elem: Element): ...\n def minimum(self) -> Element: ...\n def pop(self) -> Element: ...\n def decrease_priority_for(self, elem: Element, priority: int): ...\n \\`\\`\\`\n \n ## Implementazione con lista\n \n E' possibile implementare la coda con priorità tramite una **lista**: l'inserimento di nuovi valori diventerà molto efficiente, ma tutte le altre operazioni saranno linearmente lente.\n \n ### Costo computazionale\n \n - *insert()*: \\`O(1)\\`\n - *minimum()*: \\`O(n)\\`\n - *pop()*: \\`O(n)\\`\n - *decrease_priority_for()*: \\`O(n)\\`\n \n ## Implementazione con lista ordinata\n \n Implementando la coda con priorità con una **lista ordinata** si avrà un costo di ordinamento elevato negli inserimenti e modifiche alla priorità, ma costi costanti nell'estrazione di un elemento.\n \n ### Costo computazionale\n \n - *insert()*: \\`O(n)\\`\n - *minimum()*: \\`O(1)\\`\n - *pop()*: \\`O(1)\\`\n - *decrease_priority_for()*: \\`O(n)\\`\n \n ## Implementazione con heap\n \n La **soluzione migliore** è quella di implementare la coda con priorità tramite un **heap**: tutti i costi saranno logaritmici, eccetto l'inserimento che sarà costante.\n \n - \\`insert()\\` costa \\`O(1)\\`\n - \\`minimum()\\` costa \\`O(log n)\\`\n - \\`pop()\\` costa \\`O(log n)\\`\n - \\`decrease_priority_for()\\` costa \\`O(log n)\\`\n \n ## Approfondimenti\n \n Esistono code con priorità che restituiscono **l'elemento con priorità maggiore**, invece che quello minore.\n `}\n \n \n {`\n # Heap sort\n\n L'_heap sort_ è un algoritmo di ordinamento per confronto **iterativo**.\n \n ## Funzionamento\n \n Per effettuare un heap sort, creiamo un **heap massimo** in cui inseriamo tutti i valori che vogliamo ordinare.\n \n Una volta applicate le proprietà dell'heap, chiamiamo una versione particolare di \\`heap.pop()\\` che invece che rimuovere dall'array i valori estratti li posiziona nello spazio creatosi in fondo.\n \n ## Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | \\`O(n log n)\\` | \\`Ω(n log n)\\` | **\\`θ(n log n)\\`** |\n `}\n \n \n {`\n # Grafo\n \n Un _grafo_ è una struttura dati che rappresenta elementi interconnessi tra loro.\n \n Esistono due tipi di grafi: _orientati_ e _non orientati_.\n \n Per semplicità, consideriamo i nostri nodi numerati da 1 a \\`n\\`.\n \n ## Proprietà\n \n - Gli elementi sono rappresentati tramite _nodi_.\n - Il loro _grado_ è dato dal **numero degli archi** che vi incidono.\n - Se il grafo è orientato, hanno anche un _in-degree_ (**numero di archi entranti**) e un _out-degree_ (**numero di archi uscenti**).\n - Le connessioni tra elementi sono rappresentate tramite _archi_.\n - Un arco _incide_ esattamente su **due nodi**.\n - Se il grafo è orientato, sono _uscenti_ da uno dei due nodi ed _entranti_ nell'altro.\n - Sono matematicamente meno del **quadrato dei nodi**.\n \n ## Grafi particolari\n \n ### Catena\n \n Una _catena_ è un **grafo non orientato** composto da una **sequenza di nodi** aventi un **grado massimo di 2** tutti collegati tra loro.\n \n ### Cammino\n \n Un _cammino_ è un **grafo orientato** composto da una **sequenza di nodi** aventi un **in-degree** e un **out-degree** **massimo di 1**, collegati tra loro in modo che partendo dal primo e seguendo gli archi sia possibile arrivare all'ultimo.\n \n ### Cricca\n \n Una _cricca_ è un grafo in cui **tutti i nodi sono collegati tra loro**.\n \n Se il grafo è **non orientato**, la cricca ha \\`((n-1)n)/2\\` archi.\n \n Se il grafo è **orientato**, ha per ogni coppia un arco in entrambe le direzioni, quindi ha \\`(n-1)n\\` archi.\n \n ### Direct Acyclic Graph\n \n Un _DAG_ è un grafo diretto che non contiene nessun ciclo.\n \n Su di esso possiamo effettuare un ordinamento, detto _linearizzazione_, tra i nodi: otteniamo l'_ordine topologico_.\n \n I primi elementi dei DAG sono detti _Source_ (_Sorgente_), mentre gli ultimi sono detti _Sink_ (_Pozzo_).\n \n #### Albero\n \n Un **albero** può essere considerato un DAG con una **sorgente singola** e le **foglie come pozzi**.\n \n ### Grafo fortemente connesso\n \n Un insieme di nodi \\`V\\` di un **grafo diretto** \\`G\\` si dice una _componente fortemente connessa_ se:\n \n 1. Per ogni coppia di nodi \\`∀ u, v ∈ V' : ∃ un cammino u->v in G'\\`\n 2. Massimale (non può diventare più grande)\n \n > Praticamente una componente fortemente connessa è un gruppo di nodi tra i quali si può viaggiare liberamente da e a qualsiasi nodo al suo interno.\n \n Un grafo si dice _fortemente connesso_ se l'insieme \\`V\\` coincide con l'insieme dei nodi del grafo \\`G\\`.\n \n > Se partendo da qualsiasi nodo di un grafo riesco ad arrivare a qualsiasi altro nodo, allora il grafo è fortemente connesso.\n \n Inoltre, se creiamo un nuovo grafo, in cui **ogni nodo rappresenta una componente fortemente connessa** del nostro grafo iniziale, **otteniamo un DAG**, perchè tutti i cicli sono stati integrati nella componente.\n \n ### Trasposto di un grafo\n \n Il _trasposto_ di un **grafo diretto** \\`G\\` è il grafo stesso con gli archi che però vanno nella **direzione opposta**.\n \n ### Grafo pesato\n \n Un _grafo pesato_ è un particolare grafo che associa a ciascun arco un **costo** per attraversarlo.\n \n #### Costi negativi\n \n I costi possono anche essere negativi: rappresenteranno allora un **guadagno** ottenuto attraversando il nodo.\n \n ### Minimum spanning tree\n \n Un _minimum spanning tree_ è il **sottoinsieme degli archi** di un **grafo non diretto** che **connettono tutti i nodi** con il **minor costo possibile**.\n \n I MST hanno [molte proprietà](https://en.wikipedia.org/wiki/Minimum_spanning_tree#Properties); sono troppe da scrivere qui, e probabilmente non ci interesseranno nemmeno.\n \n ## Implementazione tramite matrice di adiacenza\n \n Possiamo implementare un grafo creando una **matrice di \\`bool\\` di dimensione \\`n * n\\`** in cui le **caselle collegate sono vere** e le caselle non collegate sono false.\n \n > Ad esempio, possiamo implementare un grafo non orientato in questo modo (\\`█\\` indica l'esistenza di un collegamento e \\` \\` indica la sua assenza):\n >\n > | |1|2|3|\n > |-|-|-|-|\n > |1|░|░|░|\n > |2|█|░|░|\n > |3|█| |░|\n >\n > Esistono gli archi \\`1-2\\` e \\`1-3\\`, ma non esiste un collegamento \\`2-3\\`.\n \n > Un grafo orientato invece si può implementare così:\n >\n > | |1|2|3|\n > |-|-|-|-|\n > |1|░|█| |\n > |2|█|░| |\n > |3|█| |░|\n >\n > Esistono gli archi \\`1->2\\`, \\`2->1\\` e \\`3->1\\`, ma non ci sono collegamenti \\`2->3\\`, \\`1->3\\` e \\`3->2\\`.\n \n ### Costo computazionale\n \n #### Tempo\n \n Le matrici di adiacenza portano alla realizzazione di algoritmi molto veloci: verificare l'esistenza di un arco è in \\`O(1)\\`!\n \n Abbiamo però penalità significative quando vogliamo effettuare operazioni sugli archi: ad esempio, trovare il trasposto di un grafo implementato con una matrice di adiacenza è in \\`O(nodi²)\\`.\n \n #### Memoria\n \n E' poco efficiente in quanto a memoria: l'upper bound è in \\`O(n^2)\\`. \n \n ## Implementazione tramite liste di adiacenza\n \n Un'alternativa alla matrice di adiacenza è quella di creare un'**array di liste**, le quali contengono i **vicini di ciascun nodo**.\n \n > |Posizione|Lista|\n > |-|-|\n > |1|[2, 3]|\n > |2|[]|\n > |3|[1]|\n >\n > Esistono gli archi \\`1->2\\`, \\`1->3\\`, e \\`3->1\\`, ma non esistono \\`2->1\\`, \\`2->3\\` e \\`3->2\\`.\n \n ### Costo computazionale\n \n #### Tempo\n \n Utilizzando le liste di adiacenza, il tempo richiesto per verificare l'esistenza di un arco sale a \\`O(max-out-degree)\\`.\n \n E' efficace però quando il problema che vogliamo risolvere riguarda operazioni su archi: trovare la trasposta è in \\`O(archi)\\`.\n \n #### Memoria \n \n La memoria richiesta dalle liste di adiacenza è minore di quella delle matrici: l'upper bound è in \\`O(nodi + archi)\\`.\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/graphds)\n `}\n \n \n {`\n # Visitare un grafo\n \n Come per gli alberi radicati, esistono due modi per visitare un grafo: _depth-first search_ e _breadth-first search_.\n \n In entrambi i casi, non visito mai due volte lo stesso nodo, e come risultato ottengo **molteplici alberi**, il cui insieme viene detto _foresta di copertura_.\n \n Se il grafo che vogliamo visitare è diretto, allora dobbiamo **considerare come vicini solo gli archi uscenti**\n \n ## Depth-first search\n \n La DFS ci può risultare utile per **identificare le componenti connesse** di un grafo e **identificare eventuali cicli**.\n \n ### Funzionamento\n \n Posso utilizzare la DFS per classificare gli archi di un grafo in quattro categorie:\n \n - _Tree_, archi che ci fanno **scoprire un nuovo nodo**\n - _Forward_, archi che ci portano a un **discendente**\n - _Back_, archi che ci portano ad un **antenato**\n - _Cross_, archi che **connettono due sottoalberi** diversi\n \n Usiamo due array inizializzati a 0 chiamati \\`pre\\` e \\`post\\`, grandi quanto il numero di archi del grafo, che ci indicano rispettivamente quando un nodo è stato scoperto e quando è terminata la visita. \n Inoltre, creiamo una variabile \\`clock\\` che avanza ad ogni evento. \n Alla scoperta di un nuovo nodo, mettiamo il valore attuale di \\`clock\\` all'interno di \\`pre[n]\\`. \n Alla fine della visita di un nodo invece mettiamo il valore di \\`clock\\` in \\`post[n]\\`.\n \n **Durante la visita**, gli archi avranno i seguenti valori:\n - _Tree_: \\`pre[dst] == 0\\`\n - _Forward_: \\`pre[src] < pre[dst] && post[dst] > 0\\`\n - _Back_: \\`pre[dst] < pre[src] && post[dst] == 0\\`\n - _Cross_: Tutti gli altri (\\`post[dst] < pre[src]\\`)\n \n **A fine visita**, gli archi avranno i seguenti valori:\n - _Tree_: \\`pre[dst] < pre[dst] < post[dst] < pre[src]\\`\n - _Forward_: \\`pre[dst] < pre[dst] < post[dst] < pre[src]\\`\n - _Back_: \\`pre[src] < pre[dst] < post[dst] < post[src]\\`\n - _Cross_: \\`pre[dst] < post[dst] < pre[src] < post[src]\\`\n \n Se un **grafo non diretto** contiene degli **archi Back**, allora esso **conterrà un ciclo**.\n \n #### DFS nel grafo trasposto\n \n Se effettuo una DFS sul trasposto di un grafo, posso **scoprire i nodi che hanno un cammino verso l'origine**.\n \n #### DFS nella componente fortemente connessa\n \n Se effettuo una DFS in una componente fortemente connessa e nella sua trasposta, il **\\`post\\` della trasposta sarà sempre minore** del \\`post\\` della componente originale.\n \n ### Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | \\`O(nodi + archi)\\` | \\`Ω(nodi + archi)\\` | **\\`θ(nodi + archi)\\`** |\n | Memoria | \\`O(nodi)\\` | \\`Ω(nodi)\\` | **\\`θ(nodi)\\`** |\n \n ### Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/dfsbfs)\n \n ## Breadth-first search\n \n La BFS ci può risultare utile per **trovare tutti i nodi a una certa distanza** da un'origine.\n \n ### Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | \\`O(nodi + archi)\\` | \\`Ω(nodi + archi)\\` | **\\`θ(nodi + archi)\\`** |\n | Memoria | \\`O(nodi + archi)\\` | \\`Ω(nodi + archi)\\` | **\\`θ(nodi + archi)\\`** |\n \n ### Pseudocodice\n \n Come per gli alberi, la implementiamo in modo **iterativo**:\n \n \\`\\`\\`python\n queue = [starting_node]\n parents = [None for node in graph.nodes]\n distance = [-1 for node in graph.nodes]\n \n # TODO: controllami quando sei più sveglio\n \n while queue:\n node, source, distance = queue.pop(0)\n parents[node.number] = source\n distance[node.number] = distance\n for neighbour in node.neighbours:\n queue.append((neighbour, node, distance+1))\n \\`\\`\\`\n \n > Nella coda, la distanza massima tra un nodo e l'altro è 1.\n \n ### Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/dfsbfs)\n `}\n \n \n {`\n # Algoritmi greedy\n\n Un modo per risolvere problemi algoritmici può essere usare una **tecnica** _greedy_.\n \n Le tecniche greedy consistono nel effettuare tanti piccoli passi, ed effettuare una **scelta** in base ai dati **locali al passo attuale**.\n \n > Scegli il numero di monete più piccole possibili per comporre €1.12.\n >\n > L'algoritmo cerca di scegliere sempre la moneta più grande possibile compatibile con il prezzo in quel momento, quindi:\n > | Moneta scelta | Rimanente |\n > |---------------|-----------|\n > | € 1.00 | € 0.12 |\n > | € 0.10 | € 0.02 |\n > | € 0.02 | € 0.00 |\n \n ## Esempi\n \n Sono algoritmi greedy:\n \n - L'_Algoritmo di Dijkstra_\n - L'_Algoritmo di Kruskal_\n - L'_Algoritmo di Prim_\n `}\n \n \n {`\n # Percorso più breve\n\n Trovare il _percorso più breve_ (o _cammino minimo_) tra due nodi di un **grafo pesato** è un problema frequente nell'informatica; per questo, sono stati sviluppati [numerosi algoritmi](https://en.wikipedia.org/wiki/Shortest_path_problem) per risolverlo.\n \n > Ad esempio, il pathfinding dei nemici nei videogiochi, oppure Google Maps!\n \n ## Percorso più breve da una sorgente singola\n \n Una sottocategoria del problema del percorso più breve è il caso in cui ci interessa sapere i percorsi più brevi che **partono da uno specifico nodo del grafo**: è detto problema del _percorso più breve da una sorgente singola_, o _single-source shortest path_.\n \n Si può notare che se il grafo contiene **costi negativi** allora è possibile che il percorso più breve non esista, in quanto diventa possibile la comparsa di **cicli di costo infinitamente negativo**.\n \n Possiamo notare che, se il percorso più breve tra \\`A\\` e \\`D\\` è \\`A-B-C-D\\`, allora il cammino minimo tra \\`B\\` e \\`D\\` passerà obbligatoriamente per \\`C\\` (\\`B-C-D\\`).\n Diremo più avanti che il percorso più breve ha una **sottostruttura ottimale**.\n \n ### Esempi\n \n Alcuni algoritmi che trovano il percorso più breve sono:\n - L'_Algoritmo di Dijkstra_\n - L'_Algoritmo di Bellman-Ford_\n - La [_ricerca A*_](https://en.wikipedia.org/wiki/A*_search_algorithm)\n `}\n \n \n {`\n # Algoritmo di Dijkstra\n \n L'_Algoritmo di [Dijkstra](https://upload.wikimedia.org/wikipedia/commons/8/85/Dijkstra.ogg)_ è un algoritmo che risolve il problema del **percorso più breve da una sorgente singola** per grafi con pesi **reali positivi** \\`\\\\mathbb{R}^+\\`.\n \n L'algoritmo trova tutti i percorsi più brevi per raggiungere qualsiasi nodo del grafo partendo da un dato nodo, assieme al costo richiesto per farlo.\n \n ## Funzionamento\n \n 1. Separiamo tutti i nodi del grafo in due gruppi: **visitati** e **non visitati**. \n - Tutti i nodi partono da **non visitati**.\n 2. Per ogni nodo, manteniamo un valore \"**costo richiesto per raggiungerlo**\", che verrà cambiato man mano che l'algoritmo avanza.\n - Il costo di partenza è \\`+∞\\`.\n - Il costo sarà **definitivo per i nodi visitati**, e **provvisorio per i non visitati**.\n 3. Creiamo un insieme detto _frontiera_ che conterrà tutti i **nodi non visitati adiacenti** a quelli visitati.\n 4. Prendiamo il nodo iniziale, che avrà un **costo di \\`0\\`**, e definiamolo il nodo _attuale_.\n 5. Finchè ci sono dei nodi non sono stati visitati, ripetiamo il seguente ciclo:\n 1. Aggiungiamo i nodi adiacenti al nodo attuale alla frontiera.\n - Il costo per raggiungerli sarà il **costo per il nodo attuale sommato al costo dell'arco** che li connette al nodo attuale. \n Se questo **costo** risulta essere **minore del costo provvisorio** precedente, esso **diventerà il nuovo costo**.\n - Questa operazione è detta _rilassamento dell'arco_.\n 2. Facciamo diventare **visitato** il nodo attuale.\n - Il percorso che abbiamo fatto per raggiungerlo è obbligatoriamente il più breve.\n 3. Il prossimo nodo attuale sarà il nodo di frontiera con un costo più basso.\n - Per questo, è possibile definire l'algoritmo di Dijkstra come un **algoritmo greedy**.\n \n ### Non funziona se...\n \n L'algoritmo smette di funzionare nel caso in cui siano presenti **costi negativi** e il grafo non sia **aciclico**, in quanto non saremmo mai in grado di rendere visitato un nodo.\n \n ## Costo computazionale\n \n | Categoria | Upper bound |\n |-----------|-------------|\n | Tempo | \\`O(nodi + archi) log nodi)\\` |\n \n ### Scomposizione\n \n - Inizializzazione: \\`O(nodi)\\`\n - Creazione coda priorità: \\`O(nodi log nodi)\\`\n - Ciclo: \\`O((nodi + archi) log nodi)\\`\n \n ## Pseudocodice\n \n \\`\\`\\`python\n import math\n \n class Info:\n def __init__(self, distance=math.inf, previous=None):\n self.distance = distance\n self.previous = previous\n \n def dijkstra(graph, start):\n data = [Info() for node in graph.nodes]\n queue = PriorityQueue([start])\n while queue:\n node = queue.pop()\n for arc in node.connections:\n other = arc.other(node)\n if data[node.number].distance + arc.cost < data[other].distance:\n data[other].distance = data[node.number].distance + arc.cost\n queue.decrease_priority_for(other, data[other].distance)\n data[v].previous = node\n return data\n \\`\\`\\`\n \n ## Visualizzazione\n \n [Visualgo](https://visualgo.net/en/sssp)\n `}\n \n \n {`\n # Algoritmo di Bellman-Ford\n\n L'_Algoritmo di Bellman-Ford_ è un algoritmo che, come l'Algoritmo di Dijkstra, risolve il problema del **percorso più breve da una sorgente singola**, però, a differenza da quest'ultimo, l'Algoritmo di Bellman-Ford accetta in input anche grafi con pesi **reali** \\`\\\\mathbb{R}\\` (sia positivi, sia negativi).\n \n ## Funzionamento\n \n L'approccio dell'algoritmo è simile a quello di Dijkstra: entrambi usano il **rilassamento** degli archi per ottenere un costo provvisorio per il raggiungimento di un nodo, ma invece che rilassare solo l'arco con costo inferiore, questo algoritmo **rilassa tutti gli archi** ripetutamente, eliminando la frontiera e il problema dei nodi negativi.\n \n L'operazione di rilassamento è ripetuta \\`nodi - 1\\` volte, ovvero la **lunghezza massima** di un cammino aciclico all'interno di un grafo.\n \n Possiamo individuare dopo i rilassamenti se è presente un nodo con un **ciclo negativo**: ci basta controllare se esiste un arco che connette due nodi con una distanza incompatibile: se \\`a.distanza + arco.costo < b.distanza\\`, allora è presente un ciclo negativo.\n \n ## Costo computazionale\n \n | Categoria | Upper bound |\n |-----------|-------------|\n | Tempo | \\`O(nodi * archi)\\` |\n \n ## Pseudocodice\n \n > TODO\n \n ## Visualizzazione\n \n [Visualgo](https://visualgo.net/en/sssp)\n `}\n \n \n {`\n # Disjoint set\n \n Il _disjoint set_ è una struttura dati che rappresenta elementi di un insieme raggruppati in **sottoinsiemi disgiunti**.\n \n ## Metodi\n \n \\`\\`\\`python\n class DisjointNode:\n def __init__(self): ...\n def find_set(self): \"Trova il rappresentante dell'elemento.\"\n def union(self, other): \"Unisce i sottoinsiemi che contengono questo nodo e \\`other\\`.\"\n \\`\\`\\`\n \n ## Implementazione tramite array\n \n Possiamo implementare il disjoint set con due array: uno per l'**indice del rappresentante** e uno per il **rango dell'insieme**.\n \n Un singoletto avrà **sè stesso come rappresentante** e **rango \\`0\\`**.\n \n ### Costo computazionale\n \n - *create_set()*: \\`O(1)\\`\n - *find_set()*: \\`O(h)\\`\n - *union()*: \\`O(h)\\`\n \n ### Pseudocodice\n \n \\`\\`\\`python\n class DisjointNode:\n def __init__(self):\n self.parent = self\n self.rank = 0\n \n def find_set(self):\n element = self\n while self.parent != element:\n element = self.parent\n return element\n \n def union(self, other):\n repres_self = self.find_set()\n repres_other = other.find_set()\n if repres_self == repres_other:\n return\n if repres_self.rank < repres_other.rank:\n repres_greater = repres_other\n repres_lesser = repres_self\n else:\n repres_greater = repres_self\n repres_lesser = repres_other\n repres_lesser.parent = repres_greater\n if repres_greater.rank == repres_lesser.rank:\n repres_greater.rank += 1\n \\`\\`\\`\n \n ### Visualizzazione\n \n [cs.usfca.edu](https://www.cs.usfca.edu/~galles/JavascriptVisual/DisjointSets.html)\n `}\n \n \n {`\n # Trovare il minimum spanning tree\n\n Un altro problema ricorrente riguardante i grafi è trovare il _minimum spanning tree_ di un dato grafo non diretto.\n \n > E' utile per trovare il modo più efficiente per connettere le cose: ad esempio, per decidere la struttura di una rete internet!\n \n ## Esempi\n \n Gli algoritmi principali che risolvono il problema sono due, ed entrambi sono **algoritmi greedy**:\n \n - L'_Algoritmo di Kruskal_\n - L'_Algoritmo di Prim_\n `}\n \n \n {`\n # Algoritmo di Kruskal\n \n L'_Algoritmo di Kruskal_ è un algoritmo **greedy** che **trova il minimum spanning tree** di un grafo.\n \n ## Funzionamento\n \n 1. Ripetiamo questa procedura finchè tutti i nodi non sono connessi:\n 1. Prendiamo ad ogni passo **l'arco meno costoso** del grafo non ancora aggiunto all'insieme.\n 2. Assicuriamoci che **non si creino cicli**: se non se ne verrebbero a creare, possiamo **aggiungere l'arco all'insieme**.\n - Gli archi devono quindi connettere nodi in **componenti connesse diverse**.\n - Possiamo rappresentare le componenti connesse con un **Disjoint Set**.\n \n ## Costo computazionale\n \n | Categoria | Upper bound |\n |-----------|-------------|\n | Tempo | \\`O(archi²)\\` |\n \n ### Scomposizione\n \n - DisjointSet.__init__(): \\`O(archi)\\`\n - Per ogni ciclo: \\`O(archi²)\\`\n - DisjointSet.find_set(): \\`O(1)\\`\n - DisjointSet.union(): \\`O(archi)\\`\n \n ## Pseudocodice\n \n \\`\\`\\`python\n def minimum_spanning_tree_kruskal(graph):\n ds = DisjointSet()\n for node in graph.nodes:\n ds.create_set(node)\n arcs = []\n sorted_arcs = sorted(graph.arcs, key=lambda arc: arc.cost)\n for arc in sorted_arcs:\n if ds.find_set(node.start) != ds.find_set(node.end):\n arcs.append(arc)\n ds.union(node.start, node.end)\n return arcs\n \\`\\`\\`\n `}\n \n \n {`\n # Algoritmo di Prim\n \n L'_Algoritmo di Prim_ è un altro algoritmo **greedy** che **trova il minimum spanning tree** di un grafo.\n \n ## Funzionamento\n \n Creo una **coda con priorità** in cui inserisco tutti gli archi visibili dal mio albero, in cui la chiave è il **costo dell'arco**.\n \n Per trovare l'arco con costo più piccolo posso **estrarre un arco** dalla coda: la priorità ci garantisce che esso è l'**arco meno costoso**.\n \n Aggiungo allora un nuovo nodo all'albero, e con esso, **aggiungo alla coda** tutti gli **archi che scoprono un nuovo nodo**.\n \n ## Costo computazionale\n \n | Categoria | Upper bound |\n |-----------|-------------|\n | Tempo | \\`O(archi + nodi log nodi)\\` |\n \n ## Pseudocodice\n \n \\`\\`\\`python\n import math\n \n def minimum_spanning_tree_prim(graph, cost_array, start_node):\n # E' un Array di bool: se l'indice corrispondente al nodo è uguale a true, vuol dire che il (nodo è contenuto nell'albero.\n contains = [False for _ in range(len(graph))]\n # Contiene il precedente di ogni nodo \n prev = [None for _ in range(len(graph))]\n # Contiene il costo per arrivare a quel nodo\n cost = [math.inf for _ in range(len(graph))]\n # Creo la priority queue\n pq = PriorityQueue(graph.arcs, key=lambda arc: arc.cost)\n # Parto dal nodo \\`start_index\\`\n # Il costo dell'origine è 0.\n cost[start_node.index] = 0\n contains[start_node.index] = True\n \n while not pq.is_empty():\n new_node = pq.pop()\n contains[new_node.index] = True\n for arc in new_node.connections:\n other_node = arc.other(new_node)\n if not contains[other_node.index] and cost[other_node.index] > arc.cost:\n cost[other_node.index] = arc.cost\n prev[other_node.index] = new_node\n pq.decrease_priority_for(other_node, arc.cost)\n # L'array di prev rappresenta un albero.\n return prev\n \\`\\`\\`_\n `}\n \n \n {`\n # Compressione\n\n _Comprimere_ un file significa **ridurne le dimensioni** senza modificarne il significato.\n \n ## Categorie\n \n ### Compressione lossless\n \n Nella _compressione lossless_, i dati possono essere decompressi riottenendo una copia identica dell'originale.\n \n > Immaginiamo un file che contiene solo le lettere \\`a, b, c, d, e, f\\`.\n >\n > Le lettere compaiono con questa frequenza:\n > |a|b|c|d|e|f|\n > |45%|13%|12%|16%|9%|5%|\n >\n > Possiamo codificare le lettere nel seguente modo:\n > |a|b|c|d|e|f|\n > |\\`0b0\\`|\\`0b100\\`|\\`0b101\\`|\\`0b111\\`|\\`0b1100\\`|\\`0b1101\\`|\n >\n > Scrivere \\`abacadae\\` richiederebbe 64 bits con la codifica ASCII estesa, ma in questo modo riusciamo a scriverlo con soli 17 bits!\n \n Le codifiche di un file compresso devono rispettare la proprietà del _Codice a prefisso_, che dice che **nessun codice deve essere il prefisso di un altro codice**; altrimenti, si avrebbero ambiguità nella decodifica.\n \n > a = \\`1\\`\n > b = \\`11\\`\n >\n > \\`111\\` è \\`ab\\`, \\`ba\\` oppure \\`aaa\\`?\n \n Creiamo allora un _albero di decodifica_: un **albero binario** che, leggendo uno ad uno i bit codificati, ci permette di arrivare al **valore del codice presente sulle foglie** dell'albero.\n \n Gli alberi di decodifica sono sempre **completi**.\n \n > Un albero di decodifica incompleto sarebbe non ottimizzato!\n \n #### Esempi\n \n - .png\n - .flac\n - .zip\n - ...\n \n ### Compressione lossy\n \n Nella _compressione lossy_, alcuni dati [solitamente](http://needsmorejpeg.com/) irrilevanti vengono perduti: non si può, dunque, ricostruire l'originale.\n \n #### Esempi\n \n - .jpeg\n - .mp3\n - ...\n `}\n \n \n {`\n # Algoritmo di Huffman\n\n L'_Algoritmo di Huffman_ è un **algoritmo greedy** per la **costruzione di un albero di decodifica**.\n \n ## Funzionamento\n \n 1. Costruisco **un albero** (con un solo nodo) **per ogni elemento dell'alfabeto**.\n 2. Associo ad **ogni albero la frequenza dell'elemento** da cui è stato creato, per poi inserire tutti gli elementi in una coda con priorità.\n 3. Finchè non ho **un albero solo**:\n 1. Estraggo dalla coda i **due alberi con frequenza minore**.\n 2. **Li rendo fratelli**, creando un nuovo nodo in cui sono uno figlio destro e uno figlio sinistro.\n 3. Associo al nuovo nodo la **somma delle frequenze dei due alberi**, e inserisco il nuovo albero nella coda.\n \n > È molto raro che venga un albero \"dritto\"; se succede, probabilmente c'è qualcosa che non va.\n `}\n \n \n {`\n # Dizionario\n \n Un _dizionario_ è una struttura dati che **associa dei valori a delle chiavi**.\n \n ## Proprietà\n \n - Ogni elemento del dizionario è un **valore** che è stato **associato a una chiave**.\n - Possiamo aggiungere nuovi elementi con il **metodo \\`add(chiave, valore)\\`**.\n - Posiamo estrarre elementi con il **metodo \\`get(chiave)\\`**, che restituirà il valore associato a \\`chiave\\`.\n - E' possibile rimuovere elementi con il **metodo \\`delete(chiave)\\`**.\n - Due elementi con **chiavi diverse** non devono **mai restituire lo stesso valore**.\n \n ## Metodi\n \n Beh... Non ha molto senso in questo caso...\n \n \\`\\`\\`python\n dict()\n \\`\\`\\`\n \n ## Implementazione con tabella hash\n \n Una _tabella hash_ è un'array di coppie **chiave-valore**, che formano l'insieme _universo_.\n \n Per determinare l'**indice dell'array** in cui inserire una coppia, usiamo una [funzione _hash_](https://it.wikipedia.org/wiki/Hash#Algoritmo_di_hash) sulla chiave, che restituirà **numeri da \\`0\\` a \\`len(hash_table)\\`**.\n \n ### Risoluzione collisioni\n \n Potrebbe capitare però che **due chiavi diverse abbiano lo stesso indice**. Dobbiamo allora usare un metodo di _risoluzione collisioni_, che mi permetta di distinguere tra chiavi diverse.\n \n #### Lista di trabocco\n \n Possiamo salvare nell'array **liste di coppie** chiave-valore; in caso di collisione, **aggiungo un nuovo elemento alla lista**.\n \n In media, ciascuna di queste liste conterrà \\`elementi_inseriti / dimensione_tabella\\` elementi.\n \n #### Indirizzamento aperto\n \n Possiamo decidere di mettere le coppie che non trovano posto nel loro indice in **un altro indirizzo vuoto** dell'array.\n \n Ci sono diversi modi in cui decidere il nuovo indirizzo, ognuno con vantaggi e svantaggi: si può scegliere quello successivo, oppure il primo vuoto dell'array, oppure un indirizzo casuale.\n \n > Python, nei \\`dict\\`, usa indirizzamento aperto pseudorandom.\n \n ### Costo computazionale\n \n - Aggiungere una chiave: \\`O(n)\\`\n - Trovare una chiave: \\`O(n)\\`\n - Eliminare una chiave: \\`O(n)\\`\n `}\n \n \n {`\n # Programmazione dinamica\n \n La _programmazione dinamica_ è una **tecnica** di programmazione che prevede l'**estensione di una soluzione ottima precedente**.\n \n Tutti i problemi in cui si può applicare si possono risolvere anche con la **ricorsione**, ma a differenza della ricorsione, questa tecnica riesce ad evitare di ricalcolare la soluzione per ogni chiamata ricorsiva, ottenendo quindi tempi molto migliori.\n \n Si può applicare solo se un problema ha una **sottostruttura ottimale**, ovvero se la soluzione ottima di un sottoproblema è inclusa nella soluzione ottima del problema. \n \n ## Esempi\n \n - _Problema dello zaino_\n - ...\n \n > Il cammino minimo per raggiungere un nodo in un DAG è dato da \\`arco.costo + arco.primo_nodo.costo_cammino_minimo()\\`.\n >\n > \\`\\`\\`python\n > def SPD_PD(graph, start):\n > distance = [float(inf) for node in graph.nodes]:\n > distance[start] = 0\n > # I nodi devono essere in ordine di linearizzazione\n > for node in graph.nodes:\n > distance[node] = min([(arc.cost + distance[arc.other(node)] for arc in node.connections])\n > \\`\\`\\`\n \n > Ho una sequenza di interi da \\`a_1\\` a \\`a_n\\`. Voglio trovare la sottosequenza crescente più lunga.\n >\n > 5 2 3 4 7 3 6 3 1 6\n >\n > Trovo tutte le sequenze lunghe 1, e le rendo nodi di un grafo diretto.\n >\n > Da ogni nodo, creo una connessione verso i suoi maggiori.\n >\n > Infine, cerco i cammini massimi del grafo. \n > \n > Essi saranno la soluzione del problema.\n \n > Trova la lunghezza della sottosequenza più lunga che termina con \\`j\\`.\n >\n > \\`L[j]\\` = lunghezza della sottosequenza più lunga che termina in \\`j\\`\n > \n > \\`\\`\\`python\n > L[j] = max([1 + L[node] for arc in node.connections)]\n > \\`\\`\\`\n >\n > Esempio:\n > \\`\\`\\`python\n > L[9] = max([1+L[8], 1+L[3], 1+L[6]])\n > \\`\\`\\`\n `}\n \n \n {`\n # Problema dello zaino\n \n Il problema dello zaino è un problema _pseudo-trattabile_: non abbiamo dimostrazioni di se sia trattabile o intrattabile.\n \n ## Descrizione\n \n > Sei un ladro, e devi mettere **più refurtiva possibile** nello zaino per scappare. \n > Lo zaino può portare **al massimo \\`dim\\` kili**.\n >\n > **Quali** (e quanti) oggetti scegli?\n \n | Input | Output |\n |-------|--------|\n | \\`dim\\`ensione_zaino, \\`n\\`umero_oggetti, \\`oggetto.peso\\`, \\`oggetto.valore\\` | \\`profitto_massimo\\` |\n \n ## Categorie\n \n ### Problema con ripetizione\n \n Puoi prendere **tutte le copie che vuoi** di un oggetto.\n \n #### Soluzione\n \n \\`K(dim)\\` è il valore massimo ottenibile con uno zaino di capacità \\`dim\\`.\n \n > Se \\`i\\` appartenesse alla soluzione ottima, allora \\`K(dim) = i.valore + K(dim - i.peso)\\`...\n \n Possiamo dire che \\`K(dim) = max(i.valore + K(dim - i.peso))\\`.\n \n Inoltre, \\`K(0) = 0\\`.\n \n Ci salviamo tutte le soluzioni da \\`K(0)\\` a \\`K(dim)\\`, e le usiamo per calcolare il massimo in seguito.\n \n Calcolare \\`K(dim)\\` avrà allora un costo di \\`O(n * dim)\\`:\n - \\`n\\`, perchè trovare il massimo è un'operazione lineare\n - \\`dim\\`, perchè \\`dim\\` sono tutti i casi tra i quali devo andare a provare\n \n Il costo computazionale, allora, è in \\`O(n * dim)\\`. \n \n Però, il **tempo richiesto** dal nostro algoritmo dipende non dalla lunghezza dell'input, bensì dal **valore numerico** di \\`dim\\`, che corrisponde alla dimensione dell'array delle soluzioni. \n Allora, si dice che l'algoritmo è in **tempo _pseudo-polinomiale_**.\n \n ### Problema senza ripetizione\n \n Si può prendere **ogni oggetto una volta sola**.\n \n #### Soluzione _bruteforce_\n \n Scelgo se prendere o no l'item 1.\n \n Si creano due percorsi:\n - Non prendo l'oggetto: \\`valore = 0, peso = 0\\`\n - Prendo l'oggetto: \\`valore = oggetto.valore, peso = oggetto.peso\\`\n \n Continuo a creare percorsi, creando una specie di albero binario.\n \n Se a un certo punto vedo che \\`valore = x, peso = K\\` e \\`valore < x, peso = K\\`, allora posso escludere automaticamente tutto il sottoalbero destro, perchè non può essere migliore del sinistro: allora, sarò riuscito a ridurre il numero dei casi rispetto alla ricorsione.\n \n ### Problema in due variabili\n \n \\`K(j, w)\\` = massimo valore ottenibile con uno zaino di capacità \\`w\\` scegliendo gli item da \\`1\\` a \\`j\\`.\n \n Non possiamo più applicare la soluzione bruteforce, perchè abbiamo due variabili, \\`j\\` e \\`w\\`.\n \n Allora, prendo l'elemento \\`j\\`. Esso può essere o non essere nella soluzione: mi calcolo entrambe le alternative, e mi tengo l'alternativa dal valore più alto.\n \n Se \\`j\\` non è nella soluzione, il risultato diventerà \\`K(j-1, w)\\`; se invece è nella soluzione, il risultato sarà \\`j.valore + K(j-1, w-j.peso)\\`.\n \n In pratica, prendiamo\n \n \\`\\`\\`latex\n K(j, w) = max\n \\\\begin{cases}\n V_j + K(j-1, w-w_j)\n K(j-1, w)\n \\\\end{cases}\n \\`\\`\\`\n \n Costruisco allora una matrice con \\`j\\` su un asse e \\`w\\` sull'altro. \n Riempio le caselle con il valore di \\`K(j, w)\\`. \n Nella casella con \\`K(j, w)\\` avremo la soluzione ottima.\n \n Il tempo necessario per riempire tutte le caselle è nuovamente \\`O(n * w)\\`, ancora **pseudopolinomiale**.\n \n Per sapere che oggetti ho messo o no devo tenere traccia in qualche modo della catena del calcolo, usando, ad esempio, una pila.\n `}\n \n \n {`\n # Problemi intrattabili\n \n ## Problema di Set-Cover\n \n ### Input\n \n \\`U\\`niverso di \\`e\\`lementi\n \n \\`S\\`ottoinsieme di \\`s\\`ottoinsiemi di elementi di \\`U\\`\n \n ### Output\n \n Il minimo \\`S'\\`ottoinsieme di \\`s\\`ottoinsiemi che copra completamente \\`U\\`.\n \n ### Soluzione in \\`O(n^d)\\`\n \n Non c'è.\n \n ### Non-soluzione alternativa\n \n Faccio una scelta greedy, ma non posso dimostrare in alcun modo che la soluzione ottenuta sia quella ottima. \n Infatti, l'algoritmo non dà sempre la soluzione ottima, ma dà una soluzione accettabile in tempo polinomiale.\n \n Seleziono sempre il sottoinsieme che copre più elementi mancanti possibili.\n \n #### Costo computazionale\n \n \\`Costo greedy <= log(numero_elementi) * Costo ottimo\\`\n `}\n \n
\n );\n}\n"],"sourceRoot":""} \ No newline at end of file diff --git a/docs/route-AlgoritmiEStruttureDati.chunk.1a105.esm.js b/docs/route-AlgoritmiEStruttureDati.chunk.92023.esm.js similarity index 97% rename from docs/route-AlgoritmiEStruttureDati.chunk.1a105.esm.js rename to docs/route-AlgoritmiEStruttureDati.chunk.92023.esm.js index 9649885..ad174d9 100644 --- a/docs/route-AlgoritmiEStruttureDati.chunk.1a105.esm.js +++ b/docs/route-AlgoritmiEStruttureDati.chunk.92023.esm.js @@ -1,4 +1,4 @@ -(window.webpackJsonp=window.webpackJsonp||[]).push([[2],{w9qZ:function(n,e,i){"use strict";i.r(e),function(n){var o=i("mbOI");let a,r=n=>n;const t=String.raw;e.default=function(){return n("div",null,n(o.o,{title:"Disperazione di Steffo"},"aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa tutte le formule in latex si sono rotteeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee"),n(o.o,null,n(o.m,null,t(a||(a=r` +(window.webpackJsonp=window.webpackJsonp||[]).push([[2],{w9qZ:function(n,e,i){"use strict";i.r(e),function(n){var o=i("mbOI");let a,r=n=>n;const t=String.raw;e.default=function(){return n("div",null,n(o.p,{title:"Disperazione di Steffo"},"aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa tutte le formule in latex si sono rotteeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee"),n(o.p,null,n(o.n,null,t(a||(a=r` # Algoritmi e Strutture Dati Docente: [**Manuela Montangero**](mailto:manuela.montangero@unimore.it) @@ -75,5 +75,5 @@ Il corso è ben collegato con quello di Programmazione 2. Dormire non fa bene! - `)))),n(o.o,null,n(o.m,null,"\n # Il nome del corso\n\n ## Cosa sono gli algoritmi?\n\n Gli algoritmi sono modi sistematici per risolvere problemi.\n\n Sono fondamentali per sviluppare software, in quanto i computer sono eccellenti esecutori di algoritmi.\n\n ## Come si sviluppa un algoritmo?\n\n Innanzitutto, bisogna conoscere gli _input_ e gli _output_ del problema, rispettivamente i dati di partenza e i dati di arrivo di esso; si ha quindi una fase di **ricerca**.\n\n Poi, si deve trovare un procedimento che ci faccia risolvere il nostro problema: è quello che faremo in questa materia!\n\n Infine, bisogna scrivere la soluzione in un modo che possa essere eseguita da un computer: questa è la **programmazione**.\n\n ## Che tipo di problemi possiamo risolvere?\n\n Un algoritmo risolve problemi di tipo generale, non ci interessa sapere _il risultato di 123+456_, ma vogliamo sapere _il risultato di x+y_, dove x e y sono due numeri naturali qualsiasi.\n\n Un problema può essere quindi considerato circa come una **funzione matematica**, che connette ogni input a un output corrispondente.\n\n ## Che caratteristiche ha un algoritmo?\n\n Per prima cosa, ripetendo l'algoritmo più volte con lo stesso ingresso deve dare sempre la stessa uscita come **risultato**, finendo in un **tempo finito**.\n\n Deve essere **ben ordinato**: cambiando l'ordine in cui vengono effettuate le operazioni, è probabile che anche il risultato cambi!\n\n Le sue istruzioni devono essere **non ambigue**, cioè che non possano essere interpretate in più modi, e **effettivamente realizzabili**, cioè realizzabili con l'esecutore che vogliamo usare per eseguire l'algoritmo.\n\n ## Esistono algoritmi equivalenti?\n\n **Sì!** Possono esserci due algoritmi che dati gli stessi input, hanno gli stessi output, e quindi risolvono lo stesso problema.\n\n In compenso, possono avere un numero di operazioni diverse, e quindi essere **uno più veloce** (da eseguire) dell'altro.\n\n ## Come si verifica la correttezza di un algoritmo?\n\n L'algoritmo deve essere **valido per tutti gli input**, anche se questi sono infiniti.\n\n Possiamo effettuare prove matematiche per verificarne la correttezza; il **principio di induzione** è dunque una dei teoremi fondamentali dell'algoritmica.\n\n Possiamo però verificare la _non correttezza_ di un algoritmo trovando un singolo controesempio.\n ")),n(o.o,null,n(o.m,null,"\n # Efficienza degli algoritmi\n \n Un buon algoritmo deve essere **efficiente**, ovvero deve usare il minimo delle risorse necessarie, come _usare il minimo di tempo possibile_.\n \n ## Come misuriamo il tempo necessario?\n \n Cerchiamo di astrarre il tempo dal particolare esecutore, e andiamo a contare il numero di operazioni elementari richieste per eseguire il nostro algoritmo nel caso peggiore.\n \n Un algoritmo efficiente, infatti, all'aumentare dei dati in ingresso, diventerà sempre più veloce rispetto a uno non efficiente, anche su computer più lenti!\n \n > Il [Bubble Sort](https://en.wikipedia.org/wiki/Bubble_sort) è sempre più lento di un [Tree Sort](https://en.wikipedia.org/wiki/Tree_sort), anche su computer più lenti, perchè, dovendo ordinare liste sempre più lunghe, prima o poi si raggiunge un punto in cui il primo è più veloce (in termini di tempo) dell'altro.\n \n Dobbiamo andare a vedere, quindi, _il numero di operazioni richieste per ottenere il risultato nel caso peggiore_.\n \n Consideriamo operazioni sia operazioni aritmetiche sia operazioni logiche, e diciamo che ciascuna costa 1.\n \n > L'[algoritmo di Euclide](https://it.wikipedia.org/wiki/Algoritmo_di_Euclide) per l'MCD costa `3` per ogni iterazione (un giro del ciclo `while`). Diciamo, allora, che costa `3n`, dove `n` è il numero più alto dei due, perchè nel caso peggiore (uno dei due numeri è 1) l'algoritmo compie `n` iterazioni.\n \n ## Altri parametri ottimizzabili\n \n È possibile che alcuni algoritmi per vari motivi cerchino di ottimizzare altri parametri diversi dal tempo, come ad esempio _la dimensione dell'input_ o la _memoria utilizzata_. \n \n > Generalmente, questo viene fatto sui dispositivi embedded, con memoria molto limitata.\n \n ### Criteri di costo di memoria\n \n Ci sono vari criteri con cui stimare la memoria richiesta da un dato: è possibile che il costo risultante vari in base al criterio scelto!\n \n #### Criterio di costo logaritmico\n \n Un dato costa il numero di bit necessari per rappresentarlo.\n \n > Un int che contiene il numero `n` costa `log_2(n)`. \n > Un array di `[n]` numeri tutti uguali costa `n log_2(n)`.\n \n #### Criterio di costo uniforme\n \n Un dato costa il numero di elementi che lo costituiscono.\n \n > Un int che contiene il numero `n` costa `1`. \n > Un array di `[n]` numeri costa `n`. \n > Una matrice `[m][n]` costa `m*n`.\n ")),n(o.o,null,n(o.m,null,"\n # Modelli algoritmici\n\n Per progettare un algoritmo, abbiamo bisogno di sapere le proprietà del nostro esecutore, ovvero il suo _modello algoritmico_.\n \n > Ad esempio, dobbiamo sapere quali istruzioni è in grado di eseguire, e quanto tempo queste istruzioni richiedono.\n \n ## Il modello RAM\n \n Il modello in uso su tutti i computer attuali è il _modello RAM_: \n \n - In ogni cella di memoria può essere archiviato un dato.\n - Il **tempo di accesso** alle celle è **costante** per tutte le celle.\n - La **memoria** principale è **infinita**.\n - Si ha **un solo processore**.\n ")),n(o.o,null,n(o.m,null,'\n # Notazione asintotica\n \n La _notazione asintotica_ è un sistema per **stimare** velocemente il costo di un algoritmo complesso.\n \n Ci permette di **confrontare velocemente il caso peggiore** degli algoritmi.\n \n In particolare, consideriamo il _rapporto tra il numero di operazioni nel caso peggiore e la dimensione dell\'input_.\n \n ## Limiti\n \n Possiamo dare a questa stima dei limiti, superiore e inferiore, che rappresenteranno rispettivamente un costo che non sarà **mai superato** e un costo che verrà **sempre superato**.\n \n Chiameremo questi limiti _upper bound_ e _lower bound_; la loro combinazione darà un _tight bound_.\n \n L\'obiettivo sarà di _ricavare i bound più precisi possibile_ per un dato algoritmo, ovvero l\'**upper bound più basso** e il **lower bound più alto**.\n \n ### O grande\n \n > "O grande" \n > O di g(n)\n > "big-O"\n \n Per rappresentare la stima, useremo una notazione particolare, detta _O grande_, con la seguente proprietà:\n - Date due funzioni `f(n) : N -> R` e `g(n) : N -> R`, diremo che `f(n) ∈ O(g(n))` se e soltanto se `∃ c > 0, n ≥ n_0` tali che `∀ n ≥ 0, f(n) ≤ c * g(n)`\n \n Quando una funzione è O grande di un altra, significa che **asintoticamente, la funzione in O grande è sempre maggiore di quella che sta venendo stimata**.\n \n > **Ipotesi** \n > - `f(n) = 2n² + 3n + 6` \n > - `g(n) = n²`\n >\n > **Tesi** \n > - `f(n) ∈ O(n²)`.\n >\n > **Svolgimento** \n > Scrivo una disequazione, lasciando intatto il termine noto: \n > 1. `2n² + 3n + 6 ≤ 2n² + 3n² + 6` \n > 2. `n² ≤ 2n² + 3n² + n² = 6n²` per `n ≥ 3`\n >\n > Sappiamo, allora, che `2n² + 3n + 6 ≤ 6n²`.\n \n #### Espressioni di O grande\n \n Questa tabella rappresenta le espressioni di O grande più comunemente utilizzate, in ordine **dalla più forte alla più debole**.\n \n > Più forte significa che, per ogni riga della tabella, tutte le righe sottostanti sono contenute nell\'espressione.\n >\n > Ad esempio, `O(n) ∈ O(1)`.\n \n | Espressione O() | Nome |\n |-----------------|------|\n | `O(1)` | Costante |\n | `O(log log n)` | loglog |\n | `O(log n)` | Logaritmica |\n | `Ω(n^{1/c})` (per c ≥ 1) | Sublineare |\n | `O(n)` | Lineare |\n | `O(n log n)` | nlogn |\n | `O(n²)` | Quadratica |\n | `O(n³)` | Cubica |\n | `O(n^k)` (per k ≥ 1) | Polinomiale |\n | `O(a^n)` (per a ≥ 1) | Esponenziale |\n | `O(n!)` | Fattoriale |\n \n #### Polinomiale\n \n Molto spesso, noteremo che il tempo richiesto da una funzione è O grande di un polinomio di grado K, ovvero `f(n) ∈ O(n^k)`.\n \n Notiamo che in questi casi, possiamo semplificare l\'O grande al grado massimo del polinomio.\n \n > Ad esempio, `O(n² + n + 1) = O(n²)`.\n \n ##### Dimostrazione\n \n Per `n > 0 \\and 0 ≤ i ≤ k`: \n ![LaTeX](https://latex.codecogs.com/png.latex?a_k%20n^k%20+%20a_{k-1}%20n^{k-1}%20+%20%E2%80%A6%20+%20a_1%20n%20+%20a_0%20%E2%89%A4%20|a_k|%20n^k%20+%20|a_{k-1}|%20n^k%20+%20%E2%80%A6%20+%20|a_1|%20n^k%20+%20|a_0|%20n^k%20=%20(|a_k|%20+%20|a_{k-1}|%20+%20%E2%80%A6%20+%20|a_1|%20+%20|a_0|)%20n^k)\n \n #### Proprietà di O grande\n \n 1. `f(n) ∈ O(g(n)) -> ∀ a > 0, a * f(n) ∈ O(g(n))`.\n 2. `f(n) ∈ O(g(n)), d(n) ∈ O(h(n)) -> f(n) + d(n) ∈ O(g(n) + h(n)) -> O(max\\{g(n), h(n)\\})`\n 3. `f(n) ∈ O(g(n)), d(n) ∈ O(h(n)) -> f(n) * d(n) ∈ O(g(n) * h(n))`\n \n In pratica, se una funzione è la _somma di più termini_, basta guardare l\'**`O()` più grande** tra tutti i suoi termini; se invece una funzione è un _prodotto di più termini_, si possono **omettere le costanti**, e l\'`O()` finale sarà dato dal **prodotto degli `O()`** dei termini.\n \n ## Lower bound\n \n Possiamo anche stimare il _lower bound_, il limite inferiore: il **numero minimo di operazioni** che viene effettuato **nel caso migliore** con la **massima dimensione dell\'ingresso**.\n \n ### Ω()\n \n > "Omega" \n > Omega di g(n)\n > "big-Omega"\n \n Esiste un equivalente di O grande per il lower bound: è detto _Omega grande_, o più semplicemente _Omega_, e funziona nello stesso identico modo, solo... al contrario.\n \n Diremo che `f(n) ∈ Ω(g(n))` se e solo se `∃ c > 0, n_0 ≥ 0 : ∀ n ≥ n_0 f(n) ≥ c * g(n)`.\n \n #### Espressioni di Ω()\n \n Anche in questa tabella le espressioni sono **dalla più forte alla più debole**.\n \n | Espressione Ω() | Nome |\n |-----------------|------|\n | `Ω(n!)` | Fattoriale |\n | `Ω(a^n)` (per a ≥ 1) | Esponenziale |\n | `Ω(n^k)` (per k ≥ 1) | Polinomiale |\n | `Ω(n³)` | Cubica |\n | `Ω(n²)` | Quadratica |\n | `Ω(n log n)` | nlogn |\n | `Ω(n)` | Lineare |\n | `Ω(n^{1/c})` (per c ≥ 1) | Sublineare |\n | `Ω(log n)` | Logaritmica |\n | `Ω(log log n)` | loglog |\n | `Ω(1)` | Costante |\n \n ## Tight bound\n \n Quando **upper e lower bound coincidono**, allora otteniamo un _tight bound_.\n \n ### θ()\n \n > "Theta" \n > Theta di g(n)\n > "big-Theta"\n \n Anche per il tight bound abbiamo una notazione equivalente a O grande e Omega grande: _Theta grande_!\n \n Diciamo che `f(n) ∈ θ(g(n))` se e solo se `∃ c_1, c_2 > 0, n_0 ≥ 0 : ∀ n ≥ n_0, c_1 * g(n) ≤ f(n) ≤ c_2 * g(n)`.\n \n Ha la particolarità che non valgono tutte le proprietà degli altri due: va usata quindi con cautela!\n \n ## Risorse utili\n \n [khanacademy.org](https://www.khanacademy.org/computing/computer-science/algorithms/asymptotic-notation/a/big-big-theta-notation)\n ')),n(o.o,null,n(o.m,null,"\n # Problemi algoritmici\n \n Un _problema algoritmico_ è un problema matematico che si vuole provare a risolvere con un algoritmo.\n \n > Dati 10 numeri, voglio sapere se sono in ordine crescente oppure no.\n \n ## Caterigorizzazione\n \n I problemi algoritmici si dividono in tre categorie: problemi _trattabili_, problemi _intrattabili_ e problemi _irrisolvibili_.\n \n ### Problema trattabile\n \n Perchè un problema algoritmico sia _trattabile_, deve avere **almeno un algoritmo con upper bound polinomiale**.\n \n > Questo significa che il tempo impiegato da un computer per risolvere il problema rimane ragionevole, e che quindi può essere utilizzato in maniera efficiente.\n \n La trattabilità è un campo ancora parecchio aperto: esistono anche tanti problemi di cui non si è ancora dimostrata la trattabilità o intrattabilità.\n \n > La [fattorizzazione](https://it.wikipedia.org/wiki/Fattorizzazione) è uno di questi problemi: l'assenza di una dimostrazione è ciò che la rende uno dei pilastri della sicurezza informatica moderna.\n \n ### Problema intrattabile\n \n Se **un problema non ha nessun algoritmo con upper bound polinomiale**, allora si dice che è **intrattabile**.\n \n ### Problema irrisolvibile\n \n Se **non esistono algoritmi per risolvere un problema**, allora questo si dice **irrisolvibile**.\n \n > [Dato un algoritmo con certi input, riusciamo a capire con un algoritmo se la sua esecuzione termina o no?](https://en.wikipedia.org/wiki/Halting_problem)\n \n ### Upper e lower bound di problemi\n \n Si può anche trovare un _upper bound_ e un _lower bound_ per un problema, ma bisogna generalizzare di più.\n \n L'**upper bound di un problema** è il minimo upper bound di tutti gli algoritmi che lo risolvono; deve esistere almeno un algoritmo che lo risolva che abbia lo stesso _upper bound_. E' praticamente il tempo migliore per risolvere il problema dato.\n \n Il **lower bound di un problema** è il minimo lower bound di tutti gli algoritmi che lo risolvono; non deve esistere nemmeno un algoritmo che abbia un lower bound migliore. E' il numero assolutamente minimo di operazioni richieste, non si può fare meglio di così.\n \n In particolare, abbiamo che l'_upper bound di un algoritmo_ -> l'_upper bound del suo problema_, \n e il _lower bound di un problema_ -> il _lower bound di un suo algoritmo_.\n \n Generalmente, il _lower bound di un problema_ è una rappresentazione abbastanza accurata della sua difficoltà.\n ")),n(o.o,null,n(o.m,null,"\n # Ricerca binaria\n \n Non credo di aver bisogno di studiare la ricerca binaria, quindi non ho preso appunti a riguardo.\n \n Se non siete me, e state cercando informazioni a riguardo, andate a vedere su Wikipedia!\n ")),n(o.o,null,n(o.m,null,"\n # Divide et impera\n \n Un modo efficace per risolvere un problema è di usare il metodo _divide et impera (et combina)_.\n \n - _Divide_: Divido il problema in **tanti sottoproblemi**.\n - _Impera_: **Risolvo indipendentemente dal resto** ciascuno dei sottoproblemi.\n - _Combina_: **Combino** i risultati dei sottoproblemi per **risolvere il problema principale**.\n \n ## Ricorsione\n \n Un algoritmo (o funzione) si dice _ricorsivo_ quando durante l'esecuzione **richiama sè stesso**.\n \n Dato che tutti gli algoritmi devono avere termine entro un tempo finito, se scriviamo una funzione ricorsiva è fondamentale finire con un **caso base**, che non chiami ulteriormente la ricorsione.\n \n Se esiste una funzione ricorsiva, allora esiste _sempre_ una _funzione iterativa_ che darà lo stesso risultato.\n \n > Sul pratico, una funzione ricorsiva tipicamente è **più costosa** del suo equivalente iterativo: se possibile, quindi, la ricorsione andrebbe evitata.\n \n #### Pseudocodice di esempio\n \n ```python\n def fattoriale(n):\n # Caso base; la ricorsione finisce e dà un risultato fisso\n if n <= 1:\n return 1\n # Caso ricorsivo; la funzione restituisce il risultato di sè stessa (ma con parametri diversi)\n else:\n return n * fattoriale(n - 1)\n ```\n ")),n(o.o,null,n(o.m,null,"\n # Master Theorem\n \n Il _Master Theorem_ è uno dei teoremi più importanti dell'algoritmica.\n \n Esso permette di **calcolare l'upper bound di un algoritmo ricorsivo** in modo piuttosto semplice.\n \n ## Ipotesi\n \n Dato un algoritmo:\n - Con uno o più casi base\n - Che richiama la funzione ricorsiva un numero n di volte\n \n ## Tesi\n \n Allora, il suo upper bound avrà la formula:\n \n ![](https://quicklatex.com/cache3/89/ql_08d29a7e55d561a900570bc83b93ff89_l3.png)\n ")),n(o.o,null,n(o.m,null,"\n # Caso particolare del Master Theorem\n \n ## Ipotesi\n \n Se: \n ![](https://quicklatex.com/cache3/57/ql_e34dc27b42831d3c3ff671b0f3861257_l3.png)\n \n Ovvero, se la dimensione dell'input viene divisa ad ogni ciclo da una costante b, è polinomiale e il caso base è costante...\n \n ## Tesi\n \n Allora: \n ![](https://quicklatex.com/cache3/ca/ql_26e3557a6ca2d6ac4b8481e7c5263fca_l3.png)\n \n > In pratica, se il costo dominante è quello della parte \"fissa\" dell'algoritmo, esso sarà `O(n^d)`, mentre se il costo dominante è quello delle chiamate ricorsive, esso sarà `O(n^{log_b e})`. \n > Se nessuno dei due è dominante... si dividono circa in parti uguali, creando un costo di `O(n^d log n)`.\n ")),n(o.o,null,n(o.m,null,"\n # Ordinamento\n \n Un problema molto frequente nell'informatica consiste nell'**ordinare efficientemente grandi quantità di elementi**.\n \n Esistono [tantissimi](https://it.wikipedia.org/wiki/Algoritmo_di_ordinamento) algoritmi per effettuare l'ordinamento. \n L'**efficienza** di ciascuno **varia** di caso in caso: alcuni sono estremamente efficienti se quasi tutti i numeri sono già nell'ordine giusto; altri, invece, potrebbero impiegare tantissimo tempo.\n \n In termini matematici, abbiamo:\n \n - **Input:** A[n]\n - **Output:** B, ∀ i < n, A[i] ≤ A[i+1]\n \n ## Ordinamento tramite confronto\n \n L'ordinamento \"tradizionale\" è detto _ordinamento tramite confronto_: funziona sempre, e **non ha altri modi di ottenere informazioni** se non con l'operazione logica di confronto tra i dati.\n \n ### Limiti\n \n E' un problema risolto: è dimostrabile che il suo **lower bound** è **`Ω(n log n)`**; possiamo quindi dire che qualsiasi algoritmo di ordinamento è in `Ω(n log n)`, e se riusciamo a trovare un algoritmo di ordinamento in `O(n log n)` siamo riusciti a raggiungere il massimo dell'efficienza.\n \n #### Dimostrazione\n \n Consideriamo **tutte le possibili permutazioni** della sequenza da ordinare: sono `n!`.\n \n Per ogni confronto che effettuiamo, **riduciamo la quantità di permutazioni** correttamente ordinate; prima o poi, rimarrà **una sola possibilità**.\n \n **TODO, non trovo la spiegazione corretta!**\n \n ### Esempi\n \n Algoritmi che effettuano l'ordinamento tramite confronto sono:\n - _Bubble sort_\n - _Merge sort_\n - _Insertion sort_\n - _Quick sort_\n - _Heap sort_\n - E tanti, tanti altri!\n \n ## Ordinamento con altri mezzi\n \n Esistono algoritmi che ricavano informazioni in altri modi, diversi dal confronto.\n \n Essi possono avere un lower bound più basso di `O(n log n)`, però hanno spesso limitazioni sul loro utilizzo.\n \n ### Esempi\n \n - _Counting sort_, indicizza i valori da ordinare\n - _Radix sort_, guarda singolarmente le cifre dei valori\n - _Sleep sort_, sfrutta i thread e la funzione sleep per ordinare valori\n - E altri ancora!\n ")),n(o.o,null,n(o.m,null,"\n # Insertion sort\n \n L'_insertion sort_ è una soluzione **iterativa** all'ordinamento per confronto.\n \n ## Funzionamento\n \n Considero la sequenza divisa in **due parti**: una parte **ordinata** e una parte **non ordinata**.\n \n Parto dal primo elemento della lista: è sempre ordinato con sè stesso. \n Poi, aggiungo uno alla volta i numeri della parte non ordinata a quella ordinata; prima trovo in quale posizione dovrò andare a mettere il numero, poi **faccio slittare tutti i numeri dopo quella posizione** avanti di 1, in modo da **creare lo spazio** in cui infine **inserirò** il numero.\n \n ## Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | `O(n²)` | `Ω(n)` | - |\n \n Nel _caso migliore_ (**lista già ordinata**), il numero da inserire è già nella posizione giusta, quindi non devo effettuare altri confronti oltre il primo, rendendo il lower bound dell'algoritmo `Ω(n)`.\n \n Nel _caso peggiore_ (**lista nell'ordine inverso**), dobbiamo confrontare il numero da inserire con tutti gli altri nella parte ordinata: dobbiamo allora eseguire `1+2+3+4+5+… = \\frac{(n-1)(n)}{2}` confronti; ciò significa che l'upper bound è `O(n²)`!\n \n ## Pseudocodice\n \n ```python\n def insertion_sorted(lista):\n # Itero su tutti i numeri della lista, dal primo all'ultimo.\n for divisore_ord in range(len(lista)):\n # Partendo dalla posizione attuale, creo l'indice di divisione numeri ordinati maggiori e minori\n divisore_magg = divisore_ord\n # Faccio slittare avanti i numeri maggiori di quello che stiamo inserendo\n # Se l'indice divisore_magg raggiunge 0, vuol dire che tutti i numeri della lista sono maggiori del numero attuale \n while divisore_magg >= 0 and lista[divisore_magg-1] > lista[divisore_magg]:\n # Scambio la posizione dei due elementi con gli indici specificati\n # Funzione inventata \n lista[divisore_magg], lista[divisore_magg-1] = lista[divisore_magg-1], lista[divisore_magg]\n # Diminuisco il separatore di 1\n divisore_magg -= 1\n ```\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/bn/sorting)\n ")),n(o.o,null,n(o.m,null,"\n # Merge sort\n \n Il _merge sort_ è una soluzione **ricorsiva** all'ordinamento per confronto.\n \n ## Funzionamento\n \n Per questo algoritmo, utilizziamo la tecnica del **divide et impera**.\n \n 1. _Divide_: Divido A in **due parti**.\n 2. _Impera_: Metto **separatamente in ordine** le parti.\n 3. _Unisci_: **Unisco** le due parti.\n \n Consideriamo come **caso base** della ricorsione una parte composta da un numero, che ovviamente è già ordinata con sè stessa.\n \n ### Merge\n \n Per **unire le due parti** usiamo una funzione detta `merge()`.\n \n Costruiamo una nuova sequenza uguale alla sequenza 1, ma **aggiungiamo alla fine un valore sentinella** sempre maggiore di tutti gli elementi contenuti.\n \n > ```\n > | 1 | 3 | 7 | 8 | ∞ |\n > ```\n \n Facciamo **la stessa cosa** per la sequenza due.\n \n > ```\n > | 2 | 4 | 5 | 6 | ∞ |\n > ```\n \n Prendo i primi numeri delle due sequenze e **metto il più piccolo nella sequenza iniziale**.\n \n > ```\n > | 1 | 2 | 3 | | | | | | \n > | | | 7 | 8 | ∞ | \n > | | 4 | 5 | 6 | ∞ | \n > ```\n \n **Continuo** finchè non ho messo tutti i numeri; **grazie alla sentinella non usciremo mai dalla sequenza**, in quanto essa è sempre maggiore di tutti gli altri numeri.\n \n > ```\n > | 1 | 2 | 3 | 4 | 5 | 6 | | | \n > | | | 7 | 8 | ∞ | \n > | | | | | ∞ |\n > ```\n \n Quando **rimangono solo le sentinelle** significa che abbiamo aggiunto tutti gli elementi, e quindi abbiamo finito.\n \n > ```\n > | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | \n > | | | | | ∞ | \n > | | | | | ∞ | \n > ```\n \n \n ## Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | `O(n log n)` | `Ω(n log n)` | **`θ(n log n)`** |\n \n Il merge sort è un algoritmo ricorsivo con un **caso base in tempo costante** e che **richiama sè stesso 2 volte**.\n \n ```latex\n T(n) =\\\\\n \\\\\n Θ(1) \\qquad n=1\\\\\n 2 T(\\frac{n}{2}) + Θ(n) \\qquad n \\neq 1\n ```\n \n Applicando il **caso particolare del Master Theorem**, otteniamo:\n \n ```latex\n T(n) =\\\\\n \\\\\n Θ(1) \\qquad n=1\\\\\n Θ(n log n) \\qquad n \\neq 1\n ```\n \n ## Pseudocodice\n \n ```python\n def merge_sorted(part):\n # Caso base\n if len(part) == 1:\n return part\n # Divide\n middle = len(part) // 2\n part_a = part[:middle]\n part_b = part[middle:]\n # Impera\n sort_a = merge_sorted(part_a)\n sort_b = merge_sorted(part_b)\n # Combina\n return merge(sort_a, sort_b)\n ```\n \n ## Visualizzazione\n \n [hackerearth.com](https://www.hackerearth.com/practice/algorithms/sorting/merge-sort/visualize/)\n \n [visualgo.net](https://visualgo.net/bn/sorting) (Nota: visualizza solo la fase _Unisci_ del sort)\n ")),n(o.o,null,n(o.m,null,"\n # Quick sort\n \n Il _quick sort_ è un altro approccio **ricorsivo** all'ordinamento per confronto.\n \n ## Funzionamento\n \n Anche qui, applichiamo il **divide et impera**.\n \n 1. _Divide_: Scelgo un **pivot** qualsiasi all'interno della lista. Metto alla sua **sinistra tutti i numeri minori** e alla sua **destra tutti i numeri maggiori**.\n 2. _Impera_: Eseguo un **quick sort su entrambe le sottoliste**.\n \n ### Esempi\n \n #### Iterazione con partizioni bilanciate\n \n Osserviamo come si formi una partizione con tre elementi e una con quattro. \n \n ```\n |¦ [2] 8 7 1 3 5 6 {4}\n 2 |¦ [8] 7 1 3 5 6 {4}\n 2 |8 ¦ [7] 1 3 5 6 {4}\n 2 |8 ¦ [7] 1 3 5 6 {4}\n 2 |8 7 ¦ [1] 3 5 6 {4}\n 2 1 |7 8 ¦ [3] 5 6 {4}\n 2 1 3 |8 7 ¦ [5] 6 {4}\n 2 1 3 |8 7 5 ¦ [6] {4}\n 2 1 3 |8 7 5 6 ¦ [{4}]\n [2 1 3] {4} [8 5 6 7]\n ```\n \n #### Iterazione con partizioni sbilanciate\n \n Osserviamo come si formi una partizione con **zero elementi** e una con tre.\n \n ```\n |¦ [4] 7 3 {1}\n |4 ¦ [7] 3 {1}\n |4 7 |¦ [3] {1}\n |4 7 3 |¦ [{1}]\n [] {1} [4 7 3]\n ```\n \n ## Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | `O(n²)` | `Ω(n log n)` | - |\n \n Il costo della funzione è dato dalla somma del costo per **dividere in due partizioni** con il costo per realizzare il **Quick sort delle due sottopartizioni**\n \n Possiamo applicare allora il **Master Theorem generale**:\n \n ```latex\n T(n)\\\\\n =\\\\\n Θ(1) \\qquad per\\ n = 1\\\\\n T(q) + T(dim_lista - pivot - 1) + Θ(n) \\qquad per\\ n > 1\n ```\n \n ### Il caso migliore\n \n Se il pivot `q` è la **mediana della partizione** che stiamo ordinando, si vengono a creare due _sottopartizioni bilanciate_, e sostituendo otteniamo:\n \n ```latex\n T(n)\\\\\n =\\\\\n Θ(1) \\qquad per\\ n = 1\\\\\n 2 T(\\frac{n}{2}) + Θ(n) \\qquad per\\ n > 1\n ``` \n \n Possiamo allora applicare il **Master Theorem particolare**:\n \n ```latex\n T(n)\\\\\n =\\\\\n Θ(1) \\qquad per\\ n = 1\\\\\n Θ(n log n) \\qquad per\\ n > 1\n ```\n \n ### Il caso peggiore\n \n Se il pivot è uno degli **estremi dell'array**, si creano due _partizioni sbilanciate_: una delle due sottoliste è sempre vuota! \n Allora:\n \n ```latex\n T(n) = T(n-1) + Θ(n)\\\\\n = T(n-2) + Θ(n-1) + Θ(n)\\\\\n = T(n-3) + Θ(n-2) + Θ(n-1) + Θ(n)\\\\\n = …\n ∈ Θ(n^2)\n ```\n \n > \"Non date da mangiare sequenze ordinate al Quicksort, gli sono indigeste.\"\n \n ## Pseudocodice\n \n ```python\n def partition(partizione, inizio, fine):\n \"\"\"Dividi una partizione in due, usando l'ultimo elemento come pivot.\n \n Note utili:\n partizione[fine] è il pivot\n partizione[maggiori] è il primo numero dei maggiori\n partizione[non_iterati] è l'elemento su cui si sta iterando al momento\"\"\"\n # Crea il primo separatore (la | pipe nell'esempio)\n maggiori = inizio\n # Crea il secondo separatore (la ¦ broken pipe nell'esempio)\n non_iterati = inizio\n # Itera su ogni numero tra inizio e fine (escluso!)\n while non_iterati < fine:\n # Se l'elemento su cui stiamo iterando è minore del pivot\n if partizione[non_iterati] <= partizione[fine]:\n # Mettilo nell'insieme dei minori, scambiandolo con il primo numero dei maggiori e incrementando il primo separatore\n partizione[maggiori], partizione[non_iterati] = partizione[non_iterati], partizione[maggiori]\n maggiori += 1\n # Incrementa sempre il secondo separatore\n non_iterati += 1\n # Inserisci il pivot tra le due sottopartizioni create, \n partizione[fine], partizione[non_iterati] = partizione[non_iterati], partizione[fine]\n return maggiori\n ```\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/bn/sorting) (Nota: invece che prendere l'ultimo numero come pivot prende il primo, cambiando leggermente l'algoritmo.)\n \n ## Note per l'esame\n \n > La domanda che fa sempre è **\"Qual è la sequenza di pivot utilizzata?\"**\n \n > Elementi da soli _non_ vengono presi come pivot!\n ")),n(o.o,null,n(o.m,null,'\n # Counting sort\n\n Il _counting sort_ è un approccio diverso all\'ordinamento: **non usa il confronto**!\n \n ## Requisiti\n \n Il counting sort può essere utilizzato solo su **sequenze di numeri interi**, e solo se **siamo a conoscenza del minimo e del massimo** dei numeri contenuti nell\'array, ed essi non sono troppo distanti uno dall\'altro. \n (La memoria occupata dal counting sort aumenta linearmente con la differenza tra minimo e massimo!)\n \n Per semplicità, consideriamo il **minimo `0`**. \n L\'input allora sarà una sequenza di interi `A`, e il valore del **massimo `k`**, tale che `∀ n ∈ A, 0 \\leq n \\leq K`.\n \n ## Funzionamento\n \n Il counting sort **conta le ripetizioni** delle chiavi nella sequenza originale e in seguito **sovrascrive i valori** della sequenza con i valori ordinati ripetuti il numero di volte che sono stati individuati nella sequenza.\n \n > ```\n > 1 4 5 3 4 1 4 2 5 1\n > ```\n >\n > L\'`1` appare 3 volte, il `2` 1 volta, il `3` 1 volta, il `4` tre volte e il `5` due volte.\n >\n > La sequenza viene quindi così sovrascritta:\n > ```\n > 1 1 1 3 4 1 4 2 5 1 # Sovrascriviamo la sequenza con 1 ripetuto 3 volte\n > 1 1 1 2 3 4 4 4 5 1 # Sovrascriviamo la sequenza con 2, 3, 4 ripetuti rispettivamente 1 1 e 3 volte\n > 1 1 1 2 3 4 4 4 5 5 # Sovrascriviamo la sequenza con 5 ripetuto 2 volte: abbiamo finito!\n > ``` \n \n Esiste anche una **versione stabile** del counting sort che, invece che sovrascrivere, **sposta i valori**, mantenendo le informazioni aggiuntive nel caso invece che interi fossero altri tipi di dati.\n \n ## Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | `O(k + n)` | `Ω(k + n)` | **`θ(k + n)`** |\n \n L\'algoritmo è composto da quattro parti:\n - Ricerca del minimo e massimo (in `θ(n)`)\n - Inizializzazione dell\'indice (in `θ(k)`)\n - Conteggio dei numeri (in `θ(n)`)\n - Sovrascrittura dei numeri (in `θ(k + n)`)\n \n `2 + O(k) + O(n) + O(k + n) -> O(k + n)`\n \n Notiamo che `k` è costante, l\'algoritmo è `O(n)`, estremamente efficiente.\n \n ## Pseudocodice\n \n ```python\n def counting_sort(lista: typing.List[int]):\n """Ordina in-place una lista con il counting sort."""\n # Trovo la dimensione della lista\n dim = len(lista)\n # Trovo il massimo e il minimo all\'interno della lista\n minimo = min(lista)\n massimo = max(lista)\n # Creo l\'indice dei numeri, in modo che sia lungo k e pieno di 0\n indice = [0 for _ in range(minimo, massimo+1)]\n # Conto i numeri presenti, scorrendo su lista e aggiungendo 1 al numero corrispondente\n for i in range(dim):\n indice[lista[i]] += 1\n # Sovrascrivo i numeri nella lista\n count = 0\n for pos, val in enumerate(indice):\n for _ in range(val):\n indice[count] = pos\n count += 1\n \n def stable_counting_sorted(lista: typing.List[int], k: int) -> typing.List[int]:\n """Ordina stabilmente una lista con il counting sort stabile, e restituiscila."""\n # Trovo la dimensione della lista\n dim = len(lista)\n # Trovo il massimo e il minimo all\'interno della lista\n minimo = min(lista)\n massimo = max(lista)\n # Creo l\'indice dei numeri, in modo che sia lungo k e pieno di 0\n indice = [0 for _ in range(minimo, massimo+1)]\n # Conto i numeri presenti, scorrendo su lista e aggiungendo 1 al numero corrispondente\n for i in range(dim):\n indice[lista[i]] += 1\n # Faccio diventare l\'indice "il numero di numeri \\leq i"\n for i in range(len(indice)):\n if i == 0:\n continue\n indice[i] += indice[i-1]\n assert indice[-1] == dim\n # Creo una nuova lista, che sarà quella che verrà restituita\n nuova = [None for _ in range(dim)]\n # Inizio a posizionare i numeri, al contrario\n for i in range(0, dim, -1):\n nuova[indice[lista[i]]] = lista[i]\n indice[lista[i]] -= 1\n return nuova\n ```\n ')),n(o.o,null,n(o.m,null,"\n ## Introduzione alle Strutture Dati\n\n Una _struttura dati_ è un modo in cui si possono organizzare i dati di un programma.\n \n Si possono definire in due modi: **elementari** e **astratte**.\n \n ### Strutture dati elementari\n \n Le strutture _elementari_ dipendono strettamente dal modo in cui vengono memorizzati i dati.\n \n > **Array** e **liste** sono strutture dati elementari: sono definite dicendo come sono memorizzati i dati, rispettivamente, in celle contigue di memoria e da una serie di nodi con un valore e che puntano al successivo.\n \n ### Strutture dati astratte\n \n Le strutture _astratte_ sono separate dal modo in cui vengono memorizzati i dati, sono più ad alto livello, e si definiscono descrivendo le **proprietà** della struttura e i **metodi** che su di essa possono essere effettuate.\n \n > Una _classe_ in un qualsiasi linguaggio di programmazione è una struttura dati astratta.\n \n > Una _pila_ astratta:\n > - memorizza dati tutti dello stesso tipo\n > - `pop()`, estrae dalla pila l'ultimo valore inserito\n > - `push(val)`, aggiunge alla pila un valore\n > - `top()`, permette di vedere l'ultimo valore inserito nella pila\n > - `vuota()`, dice se la pila è vuota oppure no.\n ")),n(o.o,null,n(o.m,null,'\n # Array\n\n Un _array_ è sequenza di dati di **lunghezza conosciuta**, tutti dello **stesso tipo** e di una **dimensione fissa**, immagazzinata in **blocchi di memoria contigui**.\n \n ## Proprietà\n \n - E\' possibile accedere a tutti i blocchi di memoria conoscendo la loro **posizione relativa al primo** elemento.\n \n ## Metodi\n \n ```python\n class Array:\n def __init__(self, size, type=int): "Crea un array di dimensione size di elementi di tipo int."\n def __getitem__(self, index): "Restituisci il valore alla posizione index."\n def __setitem__(self, index, value): "Cambia il valore alla posizione index."\n ```\n \n ### Costo computazionale\n \n Tutte le operazioni su un array sono in **accesso immediato `O(1)`**!\n ')),n(o.o,null,n(o.m,null,'\n # Lista\n \n Una _lista_ è una sequenza di dati immagazzinata in **blocchi di memoria qualsiasi**.\n \n ## Proprietà\n \n Ogni dato ha un **riferimento** alla collocazione di memoria **successiva** (un puntatore): insieme, sono detti un _nodo_.\n \n E\' di **natura ricorsiva**: qualsiasi nodo di una lista può essere visto come inizio della lista con sè stesso e i suoi successivi.\n \n ## Metodi\n \n ```python\n class LinkedList:\n def __init__(self, value, next: typing.Optional[Node] = None):\n self.value = value\n self.next: typing.Optional[Node] = next\n \n def is_empty(self) -> bool: "Restituisce se la lista è vuota o no."\n def is_full(self) -> bool: "Restituisce se la lista è piena o no.\n def append(self, value): "Aggiunge un nuovo elemento in testa alla lista."\n def insert(self, value, index): "Inserisce un elemento dopo il nodo in posizione index."\n def insert_node(self, value, node): "Inserisce un nuovo elemento subito dopo un dato nodo."\n def find(self, node) -> int: "Trova l\'indice del nodo."\n def delete(self, value): "Elimina il primo nodo con quel valore dalla lista."\n def delete_node(self, node): "Elimina il nodo dalla lista."\n def forward(self, index) -> Node: "Restituisce il nodo in posizione index."\n ```\n \n ### Costo computazionale\n \n #### `List.forward(index)`\n \n Per raggiungere l\'`n`-esimo elemento, bisogna _scorrere tutti gli elementi prima di esso_: è dunque un **accesso sequenziale** in **`O(n)`**.\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/list)\n ')),n(o.o,null,n(o.m,null,'\n # Coda (Queue)\n\n Una _coda_ è come una pila, ma segue la strategia **First In, First Out** (il primo inserito sarà il primo a essere estratto).\n \n ## Proprietà\n \n - I dati vi possono essere aggiunti solo tramite il **metodo `enqueue()`**\n - I dati possono essere estratti solo tramite il **metodo `dequeue()`**\n - Verranno restituiti i valori inseriti secondo la strategia **First In, First Out** (il primo inserito sarà il primo a essere estratto).\n \n > Hai presente quando fai la fila per pagare al supermercato? Beh, è quello, però non si possono superare le altre persone in nessun modo.\n \n ## Metodi\n \n ```python\n class Queue:\n def __init__(self): "Crea una nuova coda."\n def is_empty(self) -> bool: "Restituisce vero se la coda è vuota."\n def enqueue(self, data): "Aggiunge un dato alla coda."\n def first(self) -> ...: "Restituisce il primo dato della coda."\n def dequeue(self) -> ...: "Restituisce il primo dato della coda e lo rimuove."\n ```\n \n ## Implementazione tramite lista\n \n Posso implementare la coda con una lista, ma per realizzare l\'implementazione più efficiente devo tenere anche un puntatore all\'ultimo elemento della coda, in modo da non doverla scorrere ogni volta che voglio effettuare un\'operazione.\n \n Chiamiamo i due puntatori `head` e `tail` rispettivamente.\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/list)\n ')),n(o.o,null,n(o.m,null,'\n # Pila (Stack)\n\n Una _pila_ è una struttura dati contenente **valori omogenei**.\n \n ## Proprietà\n \n - I dati vi possono essere aggiunti solo tramite il **metodo `push()`**\n - I dati possono essere estratti solo tramite il **metodo `pop()`**\n - Verranno restituiti i valori inseriti secondo la strategia **Last In, First Out** (l\'ultimo inserito sarà il primo a essere estratto).\n \n > Ci si può immaginare una pila di libri, da cui si può solo prendere un libro alla volta, quello più in alto.\n \n ## Metodi\n \n ```python\n class Stack:\n def __init__(self): "Crea una nuova pila."\n def is_empty(self) -> bool: "Restituisce vero se la pila è vuota."\n def push(self, data): "Aggiunge un dato alla pila."\n def top(self) -> ...: "Restituisce il primo dato della pila."\n def pop(self) -> ...: "Restituisce il primo dato della pila e lo rimuove."\n ```\n \n ## Implementazione tramite lista\n \n Utilizzando una lista possiamo realizzare una pila!\n \n La direzione dei puntatori sarà dall\'ultimo al primo, che non punterà più a nessuno.\n \n Il costo di tutti i metodi è `Θ(1)`!\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/list)\n ')),n(o.o,null,n(o.m,null,"\n # Albero radicato\n \n Un _albero radicato_ è una struttura dati di **natura ricorsiva** che organizza i dati in maniera **non-lineare**.\n \n ## Proprietà\n \n - Ogni nodo dell'albero ha un **unico genitore**: `∀ (padre, figlio), (padre' figlio) ∈ E \\implies padre = padre'`\n - Ogni nodo dell'albero può avere **un numero qualsiasi di figli**.\n \x3c!----\x3e\n - I **nodi superiori al padre** vengono chiamati _antenati_.\n - I **nodi inferiori ai figli** vengono chiamati _discendenti_.\n \x3c!----\x3e\n - Nodi **senza padre** sono detti _radice_: `\\notexists (padre, radice) ∈ E`\n - Nodi **con padre e figli** sono detti _rami_ o interni.\n - Nodi **senza figli** sono detti _foglie_.\n \x3c!----\x3e\n - La **distanza** tra il nodo radice e i suoi discendenti è detta _livello_:\n - I figli immediati sono di livello 1.\n - I \"nipoti\" (figli dei figli) sono di livello 2.\n - I figli dei nipoti sono livello 3.\n - E così via.\n - Il **livello massimo** all'interno di un albero è detto _altezza_, _profondità_ oppure _h_, ed è sempre `1 ≤ h ≤ n-1`.\n \x3c!----\x3e\n - Un albero ha sempre `n-1` archi.\n \n ## Alberi particolari\n \n ### Alberi `d`-ari\n \n Un albero _`d`-ario_ è un particolare tipo di albero che **limita il numero massimo di figli di un nodo** a `d`.\n \n > Un albero _binario_ può avere **massimo 2 figli** per ogni nodo; un albero _ternario_ ne può avere **3**; un albero _`17`-ario_ ne potrà avere **17**\n \n #### Alberi completi\n \n Un albero `d`-ario si dice _completo_ se **tutti i nodi hanno 0 o `d` figli**, e mai una numero in mezzo.\n \n #### Alberi bilanciati\n \n Un albero `d`-ario si dice _bilanciato_ se **tutti i livelli eccetto l'ultimo** hanno il numero massimo di figli.\n \n #### Alberi perfettamente bilanciati\n \n Un albero `d`-ario si dice _perfettamente bilanciato_ se **tutti i livelli incluso l'ultimo** hanno il numero massimo di figli.\n \n ##### Particolarità degli alberi binari perfettamente bilanciati\n \n Si può dimostrare per induzione che:\n - Hanno sempre `2^h` foglie.\n - Hanno sempre `2^{h+1}-1` (`\\sum_i=0^n 2^i`) nodi.\n - L'altezza è in `Θ(log n)`.\n \n ## Implementazione degli alberi\n \n Possiamo scegliere se usare una rappresentazione con array o con nodi e puntatori: ognuna ha vantaggi e svantaggi diversi.\n \n ### Implementazione tramite array\n \n E' suggerito se l'albero è regolare; più è simile a un albero d-ario completo, meglio è.\n \n ### Implementazione tramite nodi e puntatori\n \n Più adatta ad alberi irregolari.\n \n Se l'albero è regolare, creiamo il numero esatto di campi:\n \n - Valore\n - Figlio1\n - Figlio2\n - _Opzionale:_ Padre\n \n Se un albero è irregolare, creiamo una specie di lista:\n \n - Valore\n - Primo figlio\n - Prossimo fratello\n - _Opzionale:_ Padre\n ")),n(o.o,null,n(o.m,null,"\n #### Breadth-first search (BFS)\n\n La _breadth-first search_ è un algoritmo che visita **ogni livello** dell'albero in ordine, dal più basso al più alto.\n \n ## Funzionamento\n \n > 1. __Visita radice__\n > 2. __Visita figli__\n > 3. __Visita nipoti__\n > 4. __Visita pronipoti__\n \n Si può implementare con una coda, in cui verranno inseriti i figli del nodo visitato da visitare ed estratti dopo avere completato la visita del livello attuale.\n \n ## Pseudocodice \n \n ```python\n def bfs(radice):\n c = Queue()\n c.enqueue(radice)\n while not c.is_empty():\n nodo = c.dequeue()\n print(nodo)\n for figlio in nodo.figli:\n nodo.enqueue(figlio)\n ```\n ")),n(o.o,null,n(o.m,null,"\n # Depth-First Search\n\n La _depth-first search_ è un algoritmo che visita **tutti i sottoalberi di un figlio** prima di passare ad un altro sfruttando la natura ricorsiva degli alberi.\n \n ## Funzionamento\n \n Ci sono diverse versioni della depth-first search: ognuna visita la radice in un momento diverso.\n \n ### DFS previsita (pre-order)\n \n La _DFS pre-visita_ visita la **radice per prima**, poi tutti i sottoalberi formati dai figli uno dopo l'altro.\n \n > 1. __Visita radice__\n > 2. dfs_pre_order(figlio1)\n > 3. dfs_pre_order(figlio2)\n \n ### Postvisita (post-order)\n \n La _DFS postvisita_ visita prima tutti i sottoalberi dei figli, e **alla fine la radice**.\n \n > 1. dfs_post_order(figlio1)\n > 2. dfs_post_order(figlio2)\n > 3. __Visita radice__\n \n ### Invisita (in-order)\n \n La _DFS invisita_ visita **un certo numero di figli**, poi la radice, e infine i figli restanti.\n \n > 1. dfs_in_order(figlio1, 1)\n > 2. __Visita radice__\n > 3. dfs_in_order(figlio2, 1)\n ")),n(o.o,null,n(o.m,null,"\n # Albero binario di ricerca\n\n ## Proprietà\n \n - Albero **binario**\n - Chiavi appartenenti ad un **insieme totalmente ordinato** (N, Q, R, ma non C)\n \x3c!----\x3e\n - Per ogni nodo con valore `x`, se un valore `v` è nel sottoalbero di sinistra allora `v ≤ x`, mentre se è nel sottoalbero di destra allora `v > x`.\n \n ## Costo computazionale\n \n - Trovare un valore: `O(h)`\n - Ordinare i valori: `O(n)`\n - Trovare il minimo: `O(h)`\n - Trovare il massimo: `O(h)`\n - Inserire un elemento: `O(h)`\n - Cancellare un elemento: `O(h)`\n \n `h` vale `log n` in un albero perfettamente bilanciato, e più l'albero diventa sbilanciato, più si avvicina a `n`, raggiungendola nel caso l'albero sia una lista.\n \n ## Pseudocodice\n \n ### Cancellazione ricorsiva\n \n ```python\n def delete(tree, key):\n if tree is not None:\n # Se ho trovato il nodo che cercavo...\n if tree.key == key:\n # E c'è una sola diramazione...\n # Semplicemente stacca il nodo come in una lista.\n if tree.left is None:\n return tree.right\n if tree.right is None:\n return tree.left\n # Altrimenti, diventa il minimo dell'albero di destra\n tree.key = tree.right.min()\n # Ed eliminalo dal sottoalbero\n tree.right = delete(tree.right, tree.key)\n # Se la chiave attuale è diversa da quella che cerchiamo, continuo a navigare l'albero\n elif tree.key < key:\n tree.left = delete(tree.left, key)\n else:\n tree.right = delete(tree.right, key)\n return tree\n ```\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/bst)\n \n ## Approfondimenti\n \n Esistono alberi più avanzati che mantengono le proprietà degli alberi binari di ricerca, ma che si autobilanciano, come il [Red Black Tree](https://it.wikipedia.org/wiki/RB-Albero).\n ")),n(o.o,null,n(o.m,null,'\n # Heap binario\n \n L\'_heap binario_ è un **albero binario bilanciato a sinistra**.\n \n ## Proprietà\n \n - _Proprietà strutturale_:\n - L\'albero è **perfettamente bilanciato** in tutti i livelli tranne l\'ultimo\n - Nell\'ultimo livello, le foglie occupano le **posizioni più a sinistra** possibili\n - _Proprietà di ordinamento_:\n - La chiave di un qualsiasi nodo è **più piccola** di tutte quelle dei nodi **del suo sottoalbero**\n \n ## Metodi\n \n ```python\n class Heap:\n def __init__(self, H): ...\n def _heapify_ancestors(self, i): "Ripristina le proprietà dell\'heap per il nodo all\'indice specificato e i suoi genitori."\n def minimum(self): "Restituisce la chiave con il valore minimo in H."\n def decrease_value(self, index, new_value): "Diminuisce il valore della chiave all\'indice index a new_value."\n def insert(self, value): "Inserisci un nuovo valore nell\'albero."\n def _heapify_children(self, i): "Ripristina le proprietà dell\'heap per il nodo all\'indice specificato e i suoi figli."\n def pop(self): "Restituisce la chiave con il valore minimo, e la elimina."\n def from_list(l): "Crea un heap da una lista."\n ```\n \n ## Implementazione con un array\n \n Possiamo implementare l\'albero utilizzando un array con le chiavi dell\'albero memorizzate nell\'ordine _breadth-first_.\n \n L\'indice del figlio sinistro può essere trovato a `2i+1`, mentre l\'indice del figlio sinistro può essere trovato a `2i+2`; il genitore è a `i//2-1`.\n \n ### Pseudocodice\n \n ```python\n class Heap:\n def __init__(self, size):\n self.array = Array(size) # Il tipo Array non esiste; consideriamolo pseudocodice\n self.next_value = 0\n \n def _heapify_ancestors(self, index)\n """Ripristina le proprietà dell\'heap per il nodo all\'indice specificato e i suoi genitori.\n \n Costo:\n O(log n)"""\n # Trovo l\'indice del genitore\n parent = index // 2 - 1\n # Controllo se viene mantenuta la proprietà di ordinamento dell\'heap\n if self.array[index] < self.array[parent]:\n # Scambio i valori dei due nodi\n self.array[index], self.array[parent] = self.array[parent], self.array[index]\n # Faccio la stessa cosa con il genitore\n self._heapify(parent)\n \n \n def minimum(self):\n """Restituisce la chiave con il valore minimo in H.\n \n Costo:\n O(1)"""\n return self.array[0]\n \n def decrease_value(self, index, new_value):\n """Diminuisce il valore della chiave all\'indice index a new_value.\n \n Costo:\n O(log n)"""\n # Diminuisco il valore del nodo\n self.array[index] = new_value\n # Aggiorno l\'heap \n self._heapify(index)\n \n def insert(self, value):\n """Inserisci un nuovo valore nell\'albero.\n \n Costo:\n O(log n)"""\n # Trovo l\'indice in cui inserire il valore\n index = self.next_index \n # Aggiungo il valore in fondo\n self.array[index] = value\n # Aggiorno l\'heap\n self._heapify(index)\n \n def _heapify_children(self, index):\n """Ripristina le proprietà dell\'heap per il nodo all\'indice specificato e i suoi figli.\n \n Costo:\n O(log n)"""\n # Trovo l\'indice dei figli\n left = index * 2 + 1\n right = index * 2 + 2\n # Mi assicuro che i figli esistano\n try:\n # Guardo quale dei figli è maggiore\n if self.array[left] > self.array[right]:\n # Scambio i valori\n self.array[left], self.array[index] = self.array[index], self.array[left]\n # Ripeto la procedura sul figlio modificato\n self._heapify_children(left)\n else:\n # Scambio i valori\n self.array[right], self.array[index] = self.array[index], self.array[right]\n # Ripeto la procedura sul figlio modificato\n self._heapify_children(left)\n except IndexError:\n # La foglia non ha figli: ho finito!\n return\n \n def pop(self):\n """Restituisce la chiave con il valore minimo, e la elimina.\n \n Costo:\n O(log n)"""\n # Mi salvo il valore della radice\n value = self.array[0]\n # Sostituisco la radice con l\'ultima foglia a destra\n self.array[0], self.array[self.next_value] = self.array[self.next_value], self.array[0]\n ### Non bisognerebbe eliminare la foglia...?\n # Riordino l\'heap\n self._heapify_children(0)\n return value\n \n @staticmethod\n def from_list(l):\n """Crea un heap da una lista.\n \n Costo:\n O(n log n), ma si può abbassare"""\n heap = Heap(len(l)))\n heap.array = Array.from_list(l) # Pseudocodice\n heap.next_value = len(l)\n # Cominciamo a riordinare l\'heap dalla fine, in modo che rispetti le proprietà\n for index in range(heap.next_value, 0, -1):\n heap._heapify_children(index)\n ```\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/heap)\n ')),n(o.o,null,n(o.m,null,"\n # Coda con priorità\n \n La _coda con priorità_ è una struttura dati dal funzionamento molto simile a quello di una coda, ma invece che restituire il primo elemento inserito, essa restituisce l'**elemento con il valore di priorità minore**.\n \n ## Proprietà\n \n - Ogni elemento è una coppia costituita da **valore** e **priorità** (un numero intero).\n - Nuovi elementi possono essere aggiunti solo tramite il **metodo `insert()`**\n - Gli elementi possono essere estratti solo tramite il **metodo `pop()`**\n - Verranno restituiti i valori inseriti secondo la strategia **Lower Priority** (l'elemento con la priorità minore sarà il primo ad essere restituito).\n - E' possibile diminuire la priorità di un elemento (e quindi anticipare la sua estrazione)\n \n ## Metodi\n \n ```python\n class PriorityQueue:\n def __init__(self): ...\n def insert(self, new_elem: Element): ...\n def minimum(self) -> Element: ...\n def pop(self) -> Element: ...\n def decrease_priority_for(self, elem: Element, priority: int): ...\n ```\n \n ## Implementazione con lista\n \n E' possibile implementare la coda con priorità tramite una **lista**: l'inserimento di nuovi valori diventerà molto efficiente, ma tutte le altre operazioni saranno linearmente lente.\n \n ### Costo computazionale\n \n - *insert()*: `O(1)`\n - *minimum()*: `O(n)`\n - *pop()*: `O(n)`\n - *decrease_priority_for()*: `O(n)`\n \n ## Implementazione con lista ordinata\n \n Implementando la coda con priorità con una **lista ordinata** si avrà un costo di ordinamento elevato negli inserimenti e modifiche alla priorità, ma costi costanti nell'estrazione di un elemento.\n \n ### Costo computazionale\n \n - *insert()*: `O(n)`\n - *minimum()*: `O(1)`\n - *pop()*: `O(1)`\n - *decrease_priority_for()*: `O(n)`\n \n ## Implementazione con heap\n \n La **soluzione migliore** è quella di implementare la coda con priorità tramite un **heap**: tutti i costi saranno logaritmici, eccetto l'inserimento che sarà costante.\n \n - `insert()` costa `O(1)`\n - `minimum()` costa `O(log n)`\n - `pop()` costa `O(log n)`\n - `decrease_priority_for()` costa `O(log n)`\n \n ## Approfondimenti\n \n Esistono code con priorità che restituiscono **l'elemento con priorità maggiore**, invece che quello minore.\n ")),n(o.o,null,n(o.m,null,"\n # Heap sort\n\n L'_heap sort_ è un algoritmo di ordinamento per confronto **iterativo**.\n \n ## Funzionamento\n \n Per effettuare un heap sort, creiamo un **heap massimo** in cui inseriamo tutti i valori che vogliamo ordinare.\n \n Una volta applicate le proprietà dell'heap, chiamiamo una versione particolare di `heap.pop()` che invece che rimuovere dall'array i valori estratti li posiziona nello spazio creatosi in fondo.\n \n ## Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | `O(n log n)` | `Ω(n log n)` | **`θ(n log n)`** |\n ")),n(o.o,null,n(o.m,null,"\n # Grafo\n \n Un _grafo_ è una struttura dati che rappresenta elementi interconnessi tra loro.\n \n Esistono due tipi di grafi: _orientati_ e _non orientati_.\n \n Per semplicità, consideriamo i nostri nodi numerati da 1 a `n`.\n \n ## Proprietà\n \n - Gli elementi sono rappresentati tramite _nodi_.\n - Il loro _grado_ è dato dal **numero degli archi** che vi incidono.\n - Se il grafo è orientato, hanno anche un _in-degree_ (**numero di archi entranti**) e un _out-degree_ (**numero di archi uscenti**).\n - Le connessioni tra elementi sono rappresentate tramite _archi_.\n - Un arco _incide_ esattamente su **due nodi**.\n - Se il grafo è orientato, sono _uscenti_ da uno dei due nodi ed _entranti_ nell'altro.\n - Sono matematicamente meno del **quadrato dei nodi**.\n \n ## Grafi particolari\n \n ### Catena\n \n Una _catena_ è un **grafo non orientato** composto da una **sequenza di nodi** aventi un **grado massimo di 2** tutti collegati tra loro.\n \n ### Cammino\n \n Un _cammino_ è un **grafo orientato** composto da una **sequenza di nodi** aventi un **in-degree** e un **out-degree** **massimo di 1**, collegati tra loro in modo che partendo dal primo e seguendo gli archi sia possibile arrivare all'ultimo.\n \n ### Cricca\n \n Una _cricca_ è un grafo in cui **tutti i nodi sono collegati tra loro**.\n \n Se il grafo è **non orientato**, la cricca ha `((n-1)n)/2` archi.\n \n Se il grafo è **orientato**, ha per ogni coppia un arco in entrambe le direzioni, quindi ha `(n-1)n` archi.\n \n ### Direct Acyclic Graph\n \n Un _DAG_ è un grafo diretto che non contiene nessun ciclo.\n \n Su di esso possiamo effettuare un ordinamento, detto _linearizzazione_, tra i nodi: otteniamo l'_ordine topologico_.\n \n I primi elementi dei DAG sono detti _Source_ (_Sorgente_), mentre gli ultimi sono detti _Sink_ (_Pozzo_).\n \n #### Albero\n \n Un **albero** può essere considerato un DAG con una **sorgente singola** e le **foglie come pozzi**.\n \n ### Grafo fortemente connesso\n \n Un insieme di nodi `V` di un **grafo diretto** `G` si dice una _componente fortemente connessa_ se:\n \n 1. Per ogni coppia di nodi `∀ u, v ∈ V' : ∃ un cammino u->v in G'`\n 2. Massimale (non può diventare più grande)\n \n > Praticamente una componente fortemente connessa è un gruppo di nodi tra i quali si può viaggiare liberamente da e a qualsiasi nodo al suo interno.\n \n Un grafo si dice _fortemente connesso_ se l'insieme `V` coincide con l'insieme dei nodi del grafo `G`.\n \n > Se partendo da qualsiasi nodo di un grafo riesco ad arrivare a qualsiasi altro nodo, allora il grafo è fortemente connesso.\n \n Inoltre, se creiamo un nuovo grafo, in cui **ogni nodo rappresenta una componente fortemente connessa** del nostro grafo iniziale, **otteniamo un DAG**, perchè tutti i cicli sono stati integrati nella componente.\n \n ### Trasposto di un grafo\n \n Il _trasposto_ di un **grafo diretto** `G` è il grafo stesso con gli archi che però vanno nella **direzione opposta**.\n \n ### Grafo pesato\n \n Un _grafo pesato_ è un particolare grafo che associa a ciascun arco un **costo** per attraversarlo.\n \n #### Costi negativi\n \n I costi possono anche essere negativi: rappresenteranno allora un **guadagno** ottenuto attraversando il nodo.\n \n ### Minimum spanning tree\n \n Un _minimum spanning tree_ è il **sottoinsieme degli archi** di un **grafo non diretto** che **connettono tutti i nodi** con il **minor costo possibile**.\n \n I MST hanno [molte proprietà](https://en.wikipedia.org/wiki/Minimum_spanning_tree#Properties); sono troppe da scrivere qui, e probabilmente non ci interesseranno nemmeno.\n \n ## Implementazione tramite matrice di adiacenza\n \n Possiamo implementare un grafo creando una **matrice di `bool` di dimensione `n * n`** in cui le **caselle collegate sono vere** e le caselle non collegate sono false.\n \n > Ad esempio, possiamo implementare un grafo non orientato in questo modo (`█` indica l'esistenza di un collegamento e ` ` indica la sua assenza):\n >\n > | |1|2|3|\n > |-|-|-|-|\n > |1|░|░|░|\n > |2|█|░|░|\n > |3|█| |░|\n >\n > Esistono gli archi `1-2` e `1-3`, ma non esiste un collegamento `2-3`.\n \n > Un grafo orientato invece si può implementare così:\n >\n > | |1|2|3|\n > |-|-|-|-|\n > |1|░|█| |\n > |2|█|░| |\n > |3|█| |░|\n >\n > Esistono gli archi `1->2`, `2->1` e `3->1`, ma non ci sono collegamenti `2->3`, `1->3` e `3->2`.\n \n ### Costo computazionale\n \n #### Tempo\n \n Le matrici di adiacenza portano alla realizzazione di algoritmi molto veloci: verificare l'esistenza di un arco è in `O(1)`!\n \n Abbiamo però penalità significative quando vogliamo effettuare operazioni sugli archi: ad esempio, trovare il trasposto di un grafo implementato con una matrice di adiacenza è in `O(nodi²)`.\n \n #### Memoria\n \n E' poco efficiente in quanto a memoria: l'upper bound è in `O(n^2)`. \n \n ## Implementazione tramite liste di adiacenza\n \n Un'alternativa alla matrice di adiacenza è quella di creare un'**array di liste**, le quali contengono i **vicini di ciascun nodo**.\n \n > |Posizione|Lista|\n > |-|-|\n > |1|[2, 3]|\n > |2|[]|\n > |3|[1]|\n >\n > Esistono gli archi `1->2`, `1->3`, e `3->1`, ma non esistono `2->1`, `2->3` e `3->2`.\n \n ### Costo computazionale\n \n #### Tempo\n \n Utilizzando le liste di adiacenza, il tempo richiesto per verificare l'esistenza di un arco sale a `O(max-out-degree)`.\n \n E' efficace però quando il problema che vogliamo risolvere riguarda operazioni su archi: trovare la trasposta è in `O(archi)`.\n \n #### Memoria \n \n La memoria richiesta dalle liste di adiacenza è minore di quella delle matrici: l'upper bound è in `O(nodi + archi)`.\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/graphds)\n ")),n(o.o,null,n(o.m,null,"\n # Visitare un grafo\n \n Come per gli alberi radicati, esistono due modi per visitare un grafo: _depth-first search_ e _breadth-first search_.\n \n In entrambi i casi, non visito mai due volte lo stesso nodo, e come risultato ottengo **molteplici alberi**, il cui insieme viene detto _foresta di copertura_.\n \n Se il grafo che vogliamo visitare è diretto, allora dobbiamo **considerare come vicini solo gli archi uscenti**\n \n ## Depth-first search\n \n La DFS ci può risultare utile per **identificare le componenti connesse** di un grafo e **identificare eventuali cicli**.\n \n ### Funzionamento\n \n Posso utilizzare la DFS per classificare gli archi di un grafo in quattro categorie:\n \n - _Tree_, archi che ci fanno **scoprire un nuovo nodo**\n - _Forward_, archi che ci portano a un **discendente**\n - _Back_, archi che ci portano ad un **antenato**\n - _Cross_, archi che **connettono due sottoalberi** diversi\n \n Usiamo due array inizializzati a 0 chiamati `pre` e `post`, grandi quanto il numero di archi del grafo, che ci indicano rispettivamente quando un nodo è stato scoperto e quando è terminata la visita. \n Inoltre, creiamo una variabile `clock` che avanza ad ogni evento. \n Alla scoperta di un nuovo nodo, mettiamo il valore attuale di `clock` all'interno di `pre[n]`. \n Alla fine della visita di un nodo invece mettiamo il valore di `clock` in `post[n]`.\n \n **Durante la visita**, gli archi avranno i seguenti valori:\n - _Tree_: `pre[dst] == 0`\n - _Forward_: `pre[src] < pre[dst] && post[dst] > 0`\n - _Back_: `pre[dst] < pre[src] && post[dst] == 0`\n - _Cross_: Tutti gli altri (`post[dst] < pre[src]`)\n \n **A fine visita**, gli archi avranno i seguenti valori:\n - _Tree_: `pre[dst] < pre[dst] < post[dst] < pre[src]`\n - _Forward_: `pre[dst] < pre[dst] < post[dst] < pre[src]`\n - _Back_: `pre[src] < pre[dst] < post[dst] < post[src]`\n - _Cross_: `pre[dst] < post[dst] < pre[src] < post[src]`\n \n Se un **grafo non diretto** contiene degli **archi Back**, allora esso **conterrà un ciclo**.\n \n #### DFS nel grafo trasposto\n \n Se effettuo una DFS sul trasposto di un grafo, posso **scoprire i nodi che hanno un cammino verso l'origine**.\n \n #### DFS nella componente fortemente connessa\n \n Se effettuo una DFS in una componente fortemente connessa e nella sua trasposta, il **`post` della trasposta sarà sempre minore** del `post` della componente originale.\n \n ### Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | `O(nodi + archi)` | `Ω(nodi + archi)` | **`θ(nodi + archi)`** |\n | Memoria | `O(nodi)` | `Ω(nodi)` | **`θ(nodi)`** |\n \n ### Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/dfsbfs)\n \n ## Breadth-first search\n \n La BFS ci può risultare utile per **trovare tutti i nodi a una certa distanza** da un'origine.\n \n ### Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | `O(nodi + archi)` | `Ω(nodi + archi)` | **`θ(nodi + archi)`** |\n | Memoria | `O(nodi + archi)` | `Ω(nodi + archi)` | **`θ(nodi + archi)`** |\n \n ### Pseudocodice\n \n Come per gli alberi, la implementiamo in modo **iterativo**:\n \n ```python\n queue = [starting_node]\n parents = [None for node in graph.nodes]\n distance = [-1 for node in graph.nodes]\n \n # TODO: controllami quando sei più sveglio\n \n while queue:\n node, source, distance = queue.pop(0)\n parents[node.number] = source\n distance[node.number] = distance\n for neighbour in node.neighbours:\n queue.append((neighbour, node, distance+1))\n ```\n \n > Nella coda, la distanza massima tra un nodo e l'altro è 1.\n \n ### Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/dfsbfs)\n ")),n(o.o,null,n(o.m,null,"\n # Algoritmi greedy\n\n Un modo per risolvere problemi algoritmici può essere usare una **tecnica** _greedy_.\n \n Le tecniche greedy consistono nel effettuare tanti piccoli passi, ed effettuare una **scelta** in base ai dati **locali al passo attuale**.\n \n > Scegli il numero di monete più piccole possibili per comporre €1.12.\n >\n > L'algoritmo cerca di scegliere sempre la moneta più grande possibile compatibile con il prezzo in quel momento, quindi:\n > | Moneta scelta | Rimanente |\n > |---------------|-----------|\n > | € 1.00 | € 0.12 |\n > | € 0.10 | € 0.02 |\n > | € 0.02 | € 0.00 |\n \n ## Esempi\n \n Sono algoritmi greedy:\n \n - L'_Algoritmo di Dijkstra_\n - L'_Algoritmo di Kruskal_\n - L'_Algoritmo di Prim_\n ")),n(o.o,null,n(o.m,null,"\n # Percorso più breve\n\n Trovare il _percorso più breve_ (o _cammino minimo_) tra due nodi di un **grafo pesato** è un problema frequente nell'informatica; per questo, sono stati sviluppati [numerosi algoritmi](https://en.wikipedia.org/wiki/Shortest_path_problem) per risolverlo.\n \n > Ad esempio, il pathfinding dei nemici nei videogiochi, oppure Google Maps!\n \n ## Percorso più breve da una sorgente singola\n \n Una sottocategoria del problema del percorso più breve è il caso in cui ci interessa sapere i percorsi più brevi che **partono da uno specifico nodo del grafo**: è detto problema del _percorso più breve da una sorgente singola_, o _single-source shortest path_.\n \n Si può notare che se il grafo contiene **costi negativi** allora è possibile che il percorso più breve non esista, in quanto diventa possibile la comparsa di **cicli di costo infinitamente negativo**.\n \n Possiamo notare che, se il percorso più breve tra `A` e `D` è `A-B-C-D`, allora il cammino minimo tra `B` e `D` passerà obbligatoriamente per `C` (`B-C-D`).\n Diremo più avanti che il percorso più breve ha una **sottostruttura ottimale**.\n \n ### Esempi\n \n Alcuni algoritmi che trovano il percorso più breve sono:\n - L'_Algoritmo di Dijkstra_\n - L'_Algoritmo di Bellman-Ford_\n - La [_ricerca A*_](https://en.wikipedia.org/wiki/A*_search_algorithm)\n ")),n(o.o,null,n(o.m,null,"\n # Algoritmo di Dijkstra\n \n L'_Algoritmo di [Dijkstra](https://upload.wikimedia.org/wikipedia/commons/8/85/Dijkstra.ogg)_ è un algoritmo che risolve il problema del **percorso più breve da una sorgente singola** per grafi con pesi **reali positivi** `\\mathbb{R}^+`.\n \n L'algoritmo trova tutti i percorsi più brevi per raggiungere qualsiasi nodo del grafo partendo da un dato nodo, assieme al costo richiesto per farlo.\n \n ## Funzionamento\n \n 1. Separiamo tutti i nodi del grafo in due gruppi: **visitati** e **non visitati**. \n - Tutti i nodi partono da **non visitati**.\n 2. Per ogni nodo, manteniamo un valore \"**costo richiesto per raggiungerlo**\", che verrà cambiato man mano che l'algoritmo avanza.\n - Il costo di partenza è `+∞`.\n - Il costo sarà **definitivo per i nodi visitati**, e **provvisorio per i non visitati**.\n 3. Creiamo un insieme detto _frontiera_ che conterrà tutti i **nodi non visitati adiacenti** a quelli visitati.\n 4. Prendiamo il nodo iniziale, che avrà un **costo di `0`**, e definiamolo il nodo _attuale_.\n 5. Finchè ci sono dei nodi non sono stati visitati, ripetiamo il seguente ciclo:\n 1. Aggiungiamo i nodi adiacenti al nodo attuale alla frontiera.\n - Il costo per raggiungerli sarà il **costo per il nodo attuale sommato al costo dell'arco** che li connette al nodo attuale. \n Se questo **costo** risulta essere **minore del costo provvisorio** precedente, esso **diventerà il nuovo costo**.\n - Questa operazione è detta _rilassamento dell'arco_.\n 2. Facciamo diventare **visitato** il nodo attuale.\n - Il percorso che abbiamo fatto per raggiungerlo è obbligatoriamente il più breve.\n 3. Il prossimo nodo attuale sarà il nodo di frontiera con un costo più basso.\n - Per questo, è possibile definire l'algoritmo di Dijkstra come un **algoritmo greedy**.\n \n ### Non funziona se...\n \n L'algoritmo smette di funzionare nel caso in cui siano presenti **costi negativi** e il grafo non sia **aciclico**, in quanto non saremmo mai in grado di rendere visitato un nodo.\n \n ## Costo computazionale\n \n | Categoria | Upper bound |\n |-----------|-------------|\n | Tempo | `O(nodi + archi) log nodi)` |\n \n ### Scomposizione\n \n - Inizializzazione: `O(nodi)`\n - Creazione coda priorità: `O(nodi log nodi)`\n - Ciclo: `O((nodi + archi) log nodi)`\n \n ## Pseudocodice\n \n ```python\n import math\n \n class Info:\n def __init__(self, distance=math.inf, previous=None):\n self.distance = distance\n self.previous = previous\n \n def dijkstra(graph, start):\n data = [Info() for node in graph.nodes]\n queue = PriorityQueue([start])\n while queue:\n node = queue.pop()\n for arc in node.connections:\n other = arc.other(node)\n if data[node.number].distance + arc.cost < data[other].distance:\n data[other].distance = data[node.number].distance + arc.cost\n queue.decrease_priority_for(other, data[other].distance)\n data[v].previous = node\n return data\n ```\n \n ## Visualizzazione\n \n [Visualgo](https://visualgo.net/en/sssp)\n ")),n(o.o,null,n(o.m,null,"\n # Algoritmo di Bellman-Ford\n\n L'_Algoritmo di Bellman-Ford_ è un algoritmo che, come l'Algoritmo di Dijkstra, risolve il problema del **percorso più breve da una sorgente singola**, però, a differenza da quest'ultimo, l'Algoritmo di Bellman-Ford accetta in input anche grafi con pesi **reali** `\\mathbb{R}` (sia positivi, sia negativi).\n \n ## Funzionamento\n \n L'approccio dell'algoritmo è simile a quello di Dijkstra: entrambi usano il **rilassamento** degli archi per ottenere un costo provvisorio per il raggiungimento di un nodo, ma invece che rilassare solo l'arco con costo inferiore, questo algoritmo **rilassa tutti gli archi** ripetutamente, eliminando la frontiera e il problema dei nodi negativi.\n \n L'operazione di rilassamento è ripetuta `nodi - 1` volte, ovvero la **lunghezza massima** di un cammino aciclico all'interno di un grafo.\n \n Possiamo individuare dopo i rilassamenti se è presente un nodo con un **ciclo negativo**: ci basta controllare se esiste un arco che connette due nodi con una distanza incompatibile: se `a.distanza + arco.costo < b.distanza`, allora è presente un ciclo negativo.\n \n ## Costo computazionale\n \n | Categoria | Upper bound |\n |-----------|-------------|\n | Tempo | `O(nodi * archi)` |\n \n ## Pseudocodice\n \n > TODO\n \n ## Visualizzazione\n \n [Visualgo](https://visualgo.net/en/sssp)\n ")),n(o.o,null,n(o.m,null,'\n # Disjoint set\n \n Il _disjoint set_ è una struttura dati che rappresenta elementi di un insieme raggruppati in **sottoinsiemi disgiunti**.\n \n ## Metodi\n \n ```python\n class DisjointNode:\n def __init__(self): ...\n def find_set(self): "Trova il rappresentante dell\'elemento."\n def union(self, other): "Unisce i sottoinsiemi che contengono questo nodo e `other`."\n ```\n \n ## Implementazione tramite array\n \n Possiamo implementare il disjoint set con due array: uno per l\'**indice del rappresentante** e uno per il **rango dell\'insieme**.\n \n Un singoletto avrà **sè stesso come rappresentante** e **rango `0`**.\n \n ### Costo computazionale\n \n - *create_set()*: `O(1)`\n - *find_set()*: `O(h)`\n - *union()*: `O(h)`\n \n ### Pseudocodice\n \n ```python\n class DisjointNode:\n def __init__(self):\n self.parent = self\n self.rank = 0\n \n def find_set(self):\n element = self\n while self.parent != element:\n element = self.parent\n return element\n \n def union(self, other):\n repres_self = self.find_set()\n repres_other = other.find_set()\n if repres_self == repres_other:\n return\n if repres_self.rank < repres_other.rank:\n repres_greater = repres_other\n repres_lesser = repres_self\n else:\n repres_greater = repres_self\n repres_lesser = repres_other\n repres_lesser.parent = repres_greater\n if repres_greater.rank == repres_lesser.rank:\n repres_greater.rank += 1\n ```\n \n ### Visualizzazione\n \n [cs.usfca.edu](https://www.cs.usfca.edu/~galles/JavascriptVisual/DisjointSets.html)\n ')),n(o.o,null,n(o.m,null,"\n # Trovare il minimum spanning tree\n\n Un altro problema ricorrente riguardante i grafi è trovare il _minimum spanning tree_ di un dato grafo non diretto.\n \n > E' utile per trovare il modo più efficiente per connettere le cose: ad esempio, per decidere la struttura di una rete internet!\n \n ## Esempi\n \n Gli algoritmi principali che risolvono il problema sono due, ed entrambi sono **algoritmi greedy**:\n \n - L'_Algoritmo di Kruskal_\n - L'_Algoritmo di Prim_\n ")),n(o.o,null,n(o.m,null,"\n # Algoritmo di Kruskal\n \n L'_Algoritmo di Kruskal_ è un algoritmo **greedy** che **trova il minimum spanning tree** di un grafo.\n \n ## Funzionamento\n \n 1. Ripetiamo questa procedura finchè tutti i nodi non sono connessi:\n 1. Prendiamo ad ogni passo **l'arco meno costoso** del grafo non ancora aggiunto all'insieme.\n 2. Assicuriamoci che **non si creino cicli**: se non se ne verrebbero a creare, possiamo **aggiungere l'arco all'insieme**.\n - Gli archi devono quindi connettere nodi in **componenti connesse diverse**.\n - Possiamo rappresentare le componenti connesse con un **Disjoint Set**.\n \n ## Costo computazionale\n \n | Categoria | Upper bound |\n |-----------|-------------|\n | Tempo | `O(archi²)` |\n \n ### Scomposizione\n \n - DisjointSet.__init__(): `O(archi)`\n - Per ogni ciclo: `O(archi²)`\n - DisjointSet.find_set(): `O(1)`\n - DisjointSet.union(): `O(archi)`\n \n ## Pseudocodice\n \n ```python\n def minimum_spanning_tree_kruskal(graph):\n ds = DisjointSet()\n for node in graph.nodes:\n ds.create_set(node)\n arcs = []\n sorted_arcs = sorted(graph.arcs, key=lambda arc: arc.cost)\n for arc in sorted_arcs:\n if ds.find_set(node.start) != ds.find_set(node.end):\n arcs.append(arc)\n ds.union(node.start, node.end)\n return arcs\n ```\n ")),n(o.o,null,n(o.m,null,"\n # Algoritmo di Prim\n \n L'_Algoritmo di Prim_ è un altro algoritmo **greedy** che **trova il minimum spanning tree** di un grafo.\n \n ## Funzionamento\n \n Creo una **coda con priorità** in cui inserisco tutti gli archi visibili dal mio albero, in cui la chiave è il **costo dell'arco**.\n \n Per trovare l'arco con costo più piccolo posso **estrarre un arco** dalla coda: la priorità ci garantisce che esso è l'**arco meno costoso**.\n \n Aggiungo allora un nuovo nodo all'albero, e con esso, **aggiungo alla coda** tutti gli **archi che scoprono un nuovo nodo**.\n \n ## Costo computazionale\n \n | Categoria | Upper bound |\n |-----------|-------------|\n | Tempo | `O(archi + nodi log nodi)` |\n \n ## Pseudocodice\n \n ```python\n import math\n \n def minimum_spanning_tree_prim(graph, cost_array, start_node):\n # E' un Array di bool: se l'indice corrispondente al nodo è uguale a true, vuol dire che il (nodo è contenuto nell'albero.\n contains = [False for _ in range(len(graph))]\n # Contiene il precedente di ogni nodo \n prev = [None for _ in range(len(graph))]\n # Contiene il costo per arrivare a quel nodo\n cost = [math.inf for _ in range(len(graph))]\n # Creo la priority queue\n pq = PriorityQueue(graph.arcs, key=lambda arc: arc.cost)\n # Parto dal nodo `start_index`\n # Il costo dell'origine è 0.\n cost[start_node.index] = 0\n contains[start_node.index] = True\n \n while not pq.is_empty():\n new_node = pq.pop()\n contains[new_node.index] = True\n for arc in new_node.connections:\n other_node = arc.other(new_node)\n if not contains[other_node.index] and cost[other_node.index] > arc.cost:\n cost[other_node.index] = arc.cost\n prev[other_node.index] = new_node\n pq.decrease_priority_for(other_node, arc.cost)\n # L'array di prev rappresenta un albero.\n return prev\n ```_\n ")),n(o.o,null,n(o.m,null,"\n # Compressione\n\n _Comprimere_ un file significa **ridurne le dimensioni** senza modificarne il significato.\n \n ## Categorie\n \n ### Compressione lossless\n \n Nella _compressione lossless_, i dati possono essere decompressi riottenendo una copia identica dell'originale.\n \n > Immaginiamo un file che contiene solo le lettere `a, b, c, d, e, f`.\n >\n > Le lettere compaiono con questa frequenza:\n > |a|b|c|d|e|f|\n > |45%|13%|12%|16%|9%|5%|\n >\n > Possiamo codificare le lettere nel seguente modo:\n > |a|b|c|d|e|f|\n > |`0b0`|`0b100`|`0b101`|`0b111`|`0b1100`|`0b1101`|\n >\n > Scrivere `abacadae` richiederebbe 64 bits con la codifica ASCII estesa, ma in questo modo riusciamo a scriverlo con soli 17 bits!\n \n Le codifiche di un file compresso devono rispettare la proprietà del _Codice a prefisso_, che dice che **nessun codice deve essere il prefisso di un altro codice**; altrimenti, si avrebbero ambiguità nella decodifica.\n \n > a = `1`\n > b = `11`\n >\n > `111` è `ab`, `ba` oppure `aaa`?\n \n Creiamo allora un _albero di decodifica_: un **albero binario** che, leggendo uno ad uno i bit codificati, ci permette di arrivare al **valore del codice presente sulle foglie** dell'albero.\n \n Gli alberi di decodifica sono sempre **completi**.\n \n > Un albero di decodifica incompleto sarebbe non ottimizzato!\n \n #### Esempi\n \n - .png\n - .flac\n - .zip\n - ...\n \n ### Compressione lossy\n \n Nella _compressione lossy_, alcuni dati [solitamente](http://needsmorejpeg.com/) irrilevanti vengono perduti: non si può, dunque, ricostruire l'originale.\n \n #### Esempi\n \n - .jpeg\n - .mp3\n - ...\n ")),n(o.o,null,n(o.m,null,"\n # Algoritmo di Huffman\n\n L'_Algoritmo di Huffman_ è un **algoritmo greedy** per la **costruzione di un albero di decodifica**.\n \n ## Funzionamento\n \n 1. Costruisco **un albero** (con un solo nodo) **per ogni elemento dell'alfabeto**.\n 2. Associo ad **ogni albero la frequenza dell'elemento** da cui è stato creato, per poi inserire tutti gli elementi in una coda con priorità.\n 3. Finchè non ho **un albero solo**:\n 1. Estraggo dalla coda i **due alberi con frequenza minore**.\n 2. **Li rendo fratelli**, creando un nuovo nodo in cui sono uno figlio destro e uno figlio sinistro.\n 3. Associo al nuovo nodo la **somma delle frequenze dei due alberi**, e inserisco il nuovo albero nella coda.\n \n > È molto raro che venga un albero \"dritto\"; se succede, probabilmente c'è qualcosa che non va.\n ")),n(o.o,null,n(o.m,null,"\n # Dizionario\n \n Un _dizionario_ è una struttura dati che **associa dei valori a delle chiavi**.\n \n ## Proprietà\n \n - Ogni elemento del dizionario è un **valore** che è stato **associato a una chiave**.\n - Possiamo aggiungere nuovi elementi con il **metodo `add(chiave, valore)`**.\n - Posiamo estrarre elementi con il **metodo `get(chiave)`**, che restituirà il valore associato a `chiave`.\n - E' possibile rimuovere elementi con il **metodo `delete(chiave)`**.\n - Due elementi con **chiavi diverse** non devono **mai restituire lo stesso valore**.\n \n ## Metodi\n \n Beh... Non ha molto senso in questo caso...\n \n ```python\n dict()\n ```\n \n ## Implementazione con tabella hash\n \n Una _tabella hash_ è un'array di coppie **chiave-valore**, che formano l'insieme _universo_.\n \n Per determinare l'**indice dell'array** in cui inserire una coppia, usiamo una [funzione _hash_](https://it.wikipedia.org/wiki/Hash#Algoritmo_di_hash) sulla chiave, che restituirà **numeri da `0` a `len(hash_table)`**.\n \n ### Risoluzione collisioni\n \n Potrebbe capitare però che **due chiavi diverse abbiano lo stesso indice**. Dobbiamo allora usare un metodo di _risoluzione collisioni_, che mi permetta di distinguere tra chiavi diverse.\n \n #### Lista di trabocco\n \n Possiamo salvare nell'array **liste di coppie** chiave-valore; in caso di collisione, **aggiungo un nuovo elemento alla lista**.\n \n In media, ciascuna di queste liste conterrà `elementi_inseriti / dimensione_tabella` elementi.\n \n #### Indirizzamento aperto\n \n Possiamo decidere di mettere le coppie che non trovano posto nel loro indice in **un altro indirizzo vuoto** dell'array.\n \n Ci sono diversi modi in cui decidere il nuovo indirizzo, ognuno con vantaggi e svantaggi: si può scegliere quello successivo, oppure il primo vuoto dell'array, oppure un indirizzo casuale.\n \n > Python, nei `dict`, usa indirizzamento aperto pseudorandom.\n \n ### Costo computazionale\n \n - Aggiungere una chiave: `O(n)`\n - Trovare una chiave: `O(n)`\n - Eliminare una chiave: `O(n)`\n ")),n(o.o,null,n(o.m,null,"\n # Programmazione dinamica\n \n La _programmazione dinamica_ è una **tecnica** di programmazione che prevede l'**estensione di una soluzione ottima precedente**.\n \n Tutti i problemi in cui si può applicare si possono risolvere anche con la **ricorsione**, ma a differenza della ricorsione, questa tecnica riesce ad evitare di ricalcolare la soluzione per ogni chiamata ricorsiva, ottenendo quindi tempi molto migliori.\n \n Si può applicare solo se un problema ha una **sottostruttura ottimale**, ovvero se la soluzione ottima di un sottoproblema è inclusa nella soluzione ottima del problema. \n \n ## Esempi\n \n - _Problema dello zaino_\n - ...\n \n > Il cammino minimo per raggiungere un nodo in un DAG è dato da `arco.costo + arco.primo_nodo.costo_cammino_minimo()`.\n >\n > ```python\n > def SPD_PD(graph, start):\n > distance = [float(inf) for node in graph.nodes]:\n > distance[start] = 0\n > # I nodi devono essere in ordine di linearizzazione\n > for node in graph.nodes:\n > distance[node] = min([(arc.cost + distance[arc.other(node)] for arc in node.connections])\n > ```\n \n > Ho una sequenza di interi da `a_1` a `a_n`. Voglio trovare la sottosequenza crescente più lunga.\n >\n > 5 2 3 4 7 3 6 3 1 6\n >\n > Trovo tutte le sequenze lunghe 1, e le rendo nodi di un grafo diretto.\n >\n > Da ogni nodo, creo una connessione verso i suoi maggiori.\n >\n > Infine, cerco i cammini massimi del grafo. \n > \n > Essi saranno la soluzione del problema.\n \n > Trova la lunghezza della sottosequenza più lunga che termina con `j`.\n >\n > `L[j]` = lunghezza della sottosequenza più lunga che termina in `j`\n > \n > ```python\n > L[j] = max([1 + L[node] for arc in node.connections)]\n > ```\n >\n > Esempio:\n > ```python\n > L[9] = max([1+L[8], 1+L[3], 1+L[6]])\n > ```\n ")),n(o.o,null,n(o.m,null,"\n # Problema dello zaino\n \n Il problema dello zaino è un problema _pseudo-trattabile_: non abbiamo dimostrazioni di se sia trattabile o intrattabile.\n \n ## Descrizione\n \n > Sei un ladro, e devi mettere **più refurtiva possibile** nello zaino per scappare. \n > Lo zaino può portare **al massimo `dim` kili**.\n >\n > **Quali** (e quanti) oggetti scegli?\n \n | Input | Output |\n |-------|--------|\n | `dim`ensione_zaino, `n`umero_oggetti, `oggetto.peso`, `oggetto.valore` | `profitto_massimo` |\n \n ## Categorie\n \n ### Problema con ripetizione\n \n Puoi prendere **tutte le copie che vuoi** di un oggetto.\n \n #### Soluzione\n \n `K(dim)` è il valore massimo ottenibile con uno zaino di capacità `dim`.\n \n > Se `i` appartenesse alla soluzione ottima, allora `K(dim) = i.valore + K(dim - i.peso)`...\n \n Possiamo dire che `K(dim) = max(i.valore + K(dim - i.peso))`.\n \n Inoltre, `K(0) = 0`.\n \n Ci salviamo tutte le soluzioni da `K(0)` a `K(dim)`, e le usiamo per calcolare il massimo in seguito.\n \n Calcolare `K(dim)` avrà allora un costo di `O(n * dim)`:\n - `n`, perchè trovare il massimo è un'operazione lineare\n - `dim`, perchè `dim` sono tutti i casi tra i quali devo andare a provare\n \n Il costo computazionale, allora, è in `O(n * dim)`. \n \n Però, il **tempo richiesto** dal nostro algoritmo dipende non dalla lunghezza dell'input, bensì dal **valore numerico** di `dim`, che corrisponde alla dimensione dell'array delle soluzioni. \n Allora, si dice che l'algoritmo è in **tempo _pseudo-polinomiale_**.\n \n ### Problema senza ripetizione\n \n Si può prendere **ogni oggetto una volta sola**.\n \n #### Soluzione _bruteforce_\n \n Scelgo se prendere o no l'item 1.\n \n Si creano due percorsi:\n - Non prendo l'oggetto: `valore = 0, peso = 0`\n - Prendo l'oggetto: `valore = oggetto.valore, peso = oggetto.peso`\n \n Continuo a creare percorsi, creando una specie di albero binario.\n \n Se a un certo punto vedo che `valore = x, peso = K` e `valore < x, peso = K`, allora posso escludere automaticamente tutto il sottoalbero destro, perchè non può essere migliore del sinistro: allora, sarò riuscito a ridurre il numero dei casi rispetto alla ricorsione.\n \n ### Problema in due variabili\n \n `K(j, w)` = massimo valore ottenibile con uno zaino di capacità `w` scegliendo gli item da `1` a `j`.\n \n Non possiamo più applicare la soluzione bruteforce, perchè abbiamo due variabili, `j` e `w`.\n \n Allora, prendo l'elemento `j`. Esso può essere o non essere nella soluzione: mi calcolo entrambe le alternative, e mi tengo l'alternativa dal valore più alto.\n \n Se `j` non è nella soluzione, il risultato diventerà `K(j-1, w)`; se invece è nella soluzione, il risultato sarà `j.valore + K(j-1, w-j.peso)`.\n \n In pratica, prendiamo\n \n ```latex\n K(j, w) = max\n \\begin{cases}\n V_j + K(j-1, w-w_j)\n K(j-1, w)\n \\end{cases}\n ```\n \n Costruisco allora una matrice con `j` su un asse e `w` sull'altro. \n Riempio le caselle con il valore di `K(j, w)`. \n Nella casella con `K(j, w)` avremo la soluzione ottima.\n \n Il tempo necessario per riempire tutte le caselle è nuovamente `O(n * w)`, ancora **pseudopolinomiale**.\n \n Per sapere che oggetti ho messo o no devo tenere traccia in qualche modo della catena del calcolo, usando, ad esempio, una pila.\n ")),n(o.o,null,n(o.m,null,"\n # Problemi intrattabili\n \n ## Problema di Set-Cover\n \n ### Input\n \n `U`niverso di `e`lementi\n \n `S`ottoinsieme di `s`ottoinsiemi di elementi di `U`\n \n ### Output\n \n Il minimo `S'`ottoinsieme di `s`ottoinsiemi che copra completamente `U`.\n \n ### Soluzione in `O(n^d)`\n \n Non c'è.\n \n ### Non-soluzione alternativa\n \n Faccio una scelta greedy, ma non posso dimostrare in alcun modo che la soluzione ottenuta sia quella ottima. \n Infatti, l'algoritmo non dà sempre la soluzione ottima, ma dà una soluzione accettabile in tempo polinomiale.\n \n Seleziono sempre il sottoinsieme che copre più elementi mancanti possibili.\n \n #### Costo computazionale\n \n `Costo greedy <= log(numero_elementi) * Costo ottimo`\n ")))}}.call(this,i("hosL").h)}}]); -//# sourceMappingURL=route-AlgoritmiEStruttureDati.chunk.1a105.esm.js.map \ No newline at end of file + `)))),n(o.p,null,n(o.n,null,"\n # Il nome del corso\n\n ## Cosa sono gli algoritmi?\n\n Gli algoritmi sono modi sistematici per risolvere problemi.\n\n Sono fondamentali per sviluppare software, in quanto i computer sono eccellenti esecutori di algoritmi.\n\n ## Come si sviluppa un algoritmo?\n\n Innanzitutto, bisogna conoscere gli _input_ e gli _output_ del problema, rispettivamente i dati di partenza e i dati di arrivo di esso; si ha quindi una fase di **ricerca**.\n\n Poi, si deve trovare un procedimento che ci faccia risolvere il nostro problema: è quello che faremo in questa materia!\n\n Infine, bisogna scrivere la soluzione in un modo che possa essere eseguita da un computer: questa è la **programmazione**.\n\n ## Che tipo di problemi possiamo risolvere?\n\n Un algoritmo risolve problemi di tipo generale, non ci interessa sapere _il risultato di 123+456_, ma vogliamo sapere _il risultato di x+y_, dove x e y sono due numeri naturali qualsiasi.\n\n Un problema può essere quindi considerato circa come una **funzione matematica**, che connette ogni input a un output corrispondente.\n\n ## Che caratteristiche ha un algoritmo?\n\n Per prima cosa, ripetendo l'algoritmo più volte con lo stesso ingresso deve dare sempre la stessa uscita come **risultato**, finendo in un **tempo finito**.\n\n Deve essere **ben ordinato**: cambiando l'ordine in cui vengono effettuate le operazioni, è probabile che anche il risultato cambi!\n\n Le sue istruzioni devono essere **non ambigue**, cioè che non possano essere interpretate in più modi, e **effettivamente realizzabili**, cioè realizzabili con l'esecutore che vogliamo usare per eseguire l'algoritmo.\n\n ## Esistono algoritmi equivalenti?\n\n **Sì!** Possono esserci due algoritmi che dati gli stessi input, hanno gli stessi output, e quindi risolvono lo stesso problema.\n\n In compenso, possono avere un numero di operazioni diverse, e quindi essere **uno più veloce** (da eseguire) dell'altro.\n\n ## Come si verifica la correttezza di un algoritmo?\n\n L'algoritmo deve essere **valido per tutti gli input**, anche se questi sono infiniti.\n\n Possiamo effettuare prove matematiche per verificarne la correttezza; il **principio di induzione** è dunque una dei teoremi fondamentali dell'algoritmica.\n\n Possiamo però verificare la _non correttezza_ di un algoritmo trovando un singolo controesempio.\n ")),n(o.p,null,n(o.n,null,"\n # Efficienza degli algoritmi\n \n Un buon algoritmo deve essere **efficiente**, ovvero deve usare il minimo delle risorse necessarie, come _usare il minimo di tempo possibile_.\n \n ## Come misuriamo il tempo necessario?\n \n Cerchiamo di astrarre il tempo dal particolare esecutore, e andiamo a contare il numero di operazioni elementari richieste per eseguire il nostro algoritmo nel caso peggiore.\n \n Un algoritmo efficiente, infatti, all'aumentare dei dati in ingresso, diventerà sempre più veloce rispetto a uno non efficiente, anche su computer più lenti!\n \n > Il [Bubble Sort](https://en.wikipedia.org/wiki/Bubble_sort) è sempre più lento di un [Tree Sort](https://en.wikipedia.org/wiki/Tree_sort), anche su computer più lenti, perchè, dovendo ordinare liste sempre più lunghe, prima o poi si raggiunge un punto in cui il primo è più veloce (in termini di tempo) dell'altro.\n \n Dobbiamo andare a vedere, quindi, _il numero di operazioni richieste per ottenere il risultato nel caso peggiore_.\n \n Consideriamo operazioni sia operazioni aritmetiche sia operazioni logiche, e diciamo che ciascuna costa 1.\n \n > L'[algoritmo di Euclide](https://it.wikipedia.org/wiki/Algoritmo_di_Euclide) per l'MCD costa `3` per ogni iterazione (un giro del ciclo `while`). Diciamo, allora, che costa `3n`, dove `n` è il numero più alto dei due, perchè nel caso peggiore (uno dei due numeri è 1) l'algoritmo compie `n` iterazioni.\n \n ## Altri parametri ottimizzabili\n \n È possibile che alcuni algoritmi per vari motivi cerchino di ottimizzare altri parametri diversi dal tempo, come ad esempio _la dimensione dell'input_ o la _memoria utilizzata_. \n \n > Generalmente, questo viene fatto sui dispositivi embedded, con memoria molto limitata.\n \n ### Criteri di costo di memoria\n \n Ci sono vari criteri con cui stimare la memoria richiesta da un dato: è possibile che il costo risultante vari in base al criterio scelto!\n \n #### Criterio di costo logaritmico\n \n Un dato costa il numero di bit necessari per rappresentarlo.\n \n > Un int che contiene il numero `n` costa `log_2(n)`. \n > Un array di `[n]` numeri tutti uguali costa `n log_2(n)`.\n \n #### Criterio di costo uniforme\n \n Un dato costa il numero di elementi che lo costituiscono.\n \n > Un int che contiene il numero `n` costa `1`. \n > Un array di `[n]` numeri costa `n`. \n > Una matrice `[m][n]` costa `m*n`.\n ")),n(o.p,null,n(o.n,null,"\n # Modelli algoritmici\n\n Per progettare un algoritmo, abbiamo bisogno di sapere le proprietà del nostro esecutore, ovvero il suo _modello algoritmico_.\n \n > Ad esempio, dobbiamo sapere quali istruzioni è in grado di eseguire, e quanto tempo queste istruzioni richiedono.\n \n ## Il modello RAM\n \n Il modello in uso su tutti i computer attuali è il _modello RAM_: \n \n - In ogni cella di memoria può essere archiviato un dato.\n - Il **tempo di accesso** alle celle è **costante** per tutte le celle.\n - La **memoria** principale è **infinita**.\n - Si ha **un solo processore**.\n ")),n(o.p,null,n(o.n,null,'\n # Notazione asintotica\n \n La _notazione asintotica_ è un sistema per **stimare** velocemente il costo di un algoritmo complesso.\n \n Ci permette di **confrontare velocemente il caso peggiore** degli algoritmi.\n \n In particolare, consideriamo il _rapporto tra il numero di operazioni nel caso peggiore e la dimensione dell\'input_.\n \n ## Limiti\n \n Possiamo dare a questa stima dei limiti, superiore e inferiore, che rappresenteranno rispettivamente un costo che non sarà **mai superato** e un costo che verrà **sempre superato**.\n \n Chiameremo questi limiti _upper bound_ e _lower bound_; la loro combinazione darà un _tight bound_.\n \n L\'obiettivo sarà di _ricavare i bound più precisi possibile_ per un dato algoritmo, ovvero l\'**upper bound più basso** e il **lower bound più alto**.\n \n ### O grande\n \n > "O grande" \n > O di g(n)\n > "big-O"\n \n Per rappresentare la stima, useremo una notazione particolare, detta _O grande_, con la seguente proprietà:\n - Date due funzioni `f(n) : N -> R` e `g(n) : N -> R`, diremo che `f(n) ∈ O(g(n))` se e soltanto se `∃ c > 0, n ≥ n_0` tali che `∀ n ≥ 0, f(n) ≤ c * g(n)`\n \n Quando una funzione è O grande di un altra, significa che **asintoticamente, la funzione in O grande è sempre maggiore di quella che sta venendo stimata**.\n \n > **Ipotesi** \n > - `f(n) = 2n² + 3n + 6` \n > - `g(n) = n²`\n >\n > **Tesi** \n > - `f(n) ∈ O(n²)`.\n >\n > **Svolgimento** \n > Scrivo una disequazione, lasciando intatto il termine noto: \n > 1. `2n² + 3n + 6 ≤ 2n² + 3n² + 6` \n > 2. `n² ≤ 2n² + 3n² + n² = 6n²` per `n ≥ 3`\n >\n > Sappiamo, allora, che `2n² + 3n + 6 ≤ 6n²`.\n \n #### Espressioni di O grande\n \n Questa tabella rappresenta le espressioni di O grande più comunemente utilizzate, in ordine **dalla più forte alla più debole**.\n \n > Più forte significa che, per ogni riga della tabella, tutte le righe sottostanti sono contenute nell\'espressione.\n >\n > Ad esempio, `O(n) ∈ O(1)`.\n \n | Espressione O() | Nome |\n |-----------------|------|\n | `O(1)` | Costante |\n | `O(log log n)` | loglog |\n | `O(log n)` | Logaritmica |\n | `Ω(n^{1/c})` (per c ≥ 1) | Sublineare |\n | `O(n)` | Lineare |\n | `O(n log n)` | nlogn |\n | `O(n²)` | Quadratica |\n | `O(n³)` | Cubica |\n | `O(n^k)` (per k ≥ 1) | Polinomiale |\n | `O(a^n)` (per a ≥ 1) | Esponenziale |\n | `O(n!)` | Fattoriale |\n \n #### Polinomiale\n \n Molto spesso, noteremo che il tempo richiesto da una funzione è O grande di un polinomio di grado K, ovvero `f(n) ∈ O(n^k)`.\n \n Notiamo che in questi casi, possiamo semplificare l\'O grande al grado massimo del polinomio.\n \n > Ad esempio, `O(n² + n + 1) = O(n²)`.\n \n ##### Dimostrazione\n \n Per `n > 0 \\and 0 ≤ i ≤ k`: \n ![LaTeX](https://latex.codecogs.com/png.latex?a_k%20n^k%20+%20a_{k-1}%20n^{k-1}%20+%20%E2%80%A6%20+%20a_1%20n%20+%20a_0%20%E2%89%A4%20|a_k|%20n^k%20+%20|a_{k-1}|%20n^k%20+%20%E2%80%A6%20+%20|a_1|%20n^k%20+%20|a_0|%20n^k%20=%20(|a_k|%20+%20|a_{k-1}|%20+%20%E2%80%A6%20+%20|a_1|%20+%20|a_0|)%20n^k)\n \n #### Proprietà di O grande\n \n 1. `f(n) ∈ O(g(n)) -> ∀ a > 0, a * f(n) ∈ O(g(n))`.\n 2. `f(n) ∈ O(g(n)), d(n) ∈ O(h(n)) -> f(n) + d(n) ∈ O(g(n) + h(n)) -> O(max\\{g(n), h(n)\\})`\n 3. `f(n) ∈ O(g(n)), d(n) ∈ O(h(n)) -> f(n) * d(n) ∈ O(g(n) * h(n))`\n \n In pratica, se una funzione è la _somma di più termini_, basta guardare l\'**`O()` più grande** tra tutti i suoi termini; se invece una funzione è un _prodotto di più termini_, si possono **omettere le costanti**, e l\'`O()` finale sarà dato dal **prodotto degli `O()`** dei termini.\n \n ## Lower bound\n \n Possiamo anche stimare il _lower bound_, il limite inferiore: il **numero minimo di operazioni** che viene effettuato **nel caso migliore** con la **massima dimensione dell\'ingresso**.\n \n ### Ω()\n \n > "Omega" \n > Omega di g(n)\n > "big-Omega"\n \n Esiste un equivalente di O grande per il lower bound: è detto _Omega grande_, o più semplicemente _Omega_, e funziona nello stesso identico modo, solo... al contrario.\n \n Diremo che `f(n) ∈ Ω(g(n))` se e solo se `∃ c > 0, n_0 ≥ 0 : ∀ n ≥ n_0 f(n) ≥ c * g(n)`.\n \n #### Espressioni di Ω()\n \n Anche in questa tabella le espressioni sono **dalla più forte alla più debole**.\n \n | Espressione Ω() | Nome |\n |-----------------|------|\n | `Ω(n!)` | Fattoriale |\n | `Ω(a^n)` (per a ≥ 1) | Esponenziale |\n | `Ω(n^k)` (per k ≥ 1) | Polinomiale |\n | `Ω(n³)` | Cubica |\n | `Ω(n²)` | Quadratica |\n | `Ω(n log n)` | nlogn |\n | `Ω(n)` | Lineare |\n | `Ω(n^{1/c})` (per c ≥ 1) | Sublineare |\n | `Ω(log n)` | Logaritmica |\n | `Ω(log log n)` | loglog |\n | `Ω(1)` | Costante |\n \n ## Tight bound\n \n Quando **upper e lower bound coincidono**, allora otteniamo un _tight bound_.\n \n ### θ()\n \n > "Theta" \n > Theta di g(n)\n > "big-Theta"\n \n Anche per il tight bound abbiamo una notazione equivalente a O grande e Omega grande: _Theta grande_!\n \n Diciamo che `f(n) ∈ θ(g(n))` se e solo se `∃ c_1, c_2 > 0, n_0 ≥ 0 : ∀ n ≥ n_0, c_1 * g(n) ≤ f(n) ≤ c_2 * g(n)`.\n \n Ha la particolarità che non valgono tutte le proprietà degli altri due: va usata quindi con cautela!\n \n ## Risorse utili\n \n [khanacademy.org](https://www.khanacademy.org/computing/computer-science/algorithms/asymptotic-notation/a/big-big-theta-notation)\n ')),n(o.p,null,n(o.n,null,"\n # Problemi algoritmici\n \n Un _problema algoritmico_ è un problema matematico che si vuole provare a risolvere con un algoritmo.\n \n > Dati 10 numeri, voglio sapere se sono in ordine crescente oppure no.\n \n ## Caterigorizzazione\n \n I problemi algoritmici si dividono in tre categorie: problemi _trattabili_, problemi _intrattabili_ e problemi _irrisolvibili_.\n \n ### Problema trattabile\n \n Perchè un problema algoritmico sia _trattabile_, deve avere **almeno un algoritmo con upper bound polinomiale**.\n \n > Questo significa che il tempo impiegato da un computer per risolvere il problema rimane ragionevole, e che quindi può essere utilizzato in maniera efficiente.\n \n La trattabilità è un campo ancora parecchio aperto: esistono anche tanti problemi di cui non si è ancora dimostrata la trattabilità o intrattabilità.\n \n > La [fattorizzazione](https://it.wikipedia.org/wiki/Fattorizzazione) è uno di questi problemi: l'assenza di una dimostrazione è ciò che la rende uno dei pilastri della sicurezza informatica moderna.\n \n ### Problema intrattabile\n \n Se **un problema non ha nessun algoritmo con upper bound polinomiale**, allora si dice che è **intrattabile**.\n \n ### Problema irrisolvibile\n \n Se **non esistono algoritmi per risolvere un problema**, allora questo si dice **irrisolvibile**.\n \n > [Dato un algoritmo con certi input, riusciamo a capire con un algoritmo se la sua esecuzione termina o no?](https://en.wikipedia.org/wiki/Halting_problem)\n \n ### Upper e lower bound di problemi\n \n Si può anche trovare un _upper bound_ e un _lower bound_ per un problema, ma bisogna generalizzare di più.\n \n L'**upper bound di un problema** è il minimo upper bound di tutti gli algoritmi che lo risolvono; deve esistere almeno un algoritmo che lo risolva che abbia lo stesso _upper bound_. E' praticamente il tempo migliore per risolvere il problema dato.\n \n Il **lower bound di un problema** è il minimo lower bound di tutti gli algoritmi che lo risolvono; non deve esistere nemmeno un algoritmo che abbia un lower bound migliore. E' il numero assolutamente minimo di operazioni richieste, non si può fare meglio di così.\n \n In particolare, abbiamo che l'_upper bound di un algoritmo_ -> l'_upper bound del suo problema_, \n e il _lower bound di un problema_ -> il _lower bound di un suo algoritmo_.\n \n Generalmente, il _lower bound di un problema_ è una rappresentazione abbastanza accurata della sua difficoltà.\n ")),n(o.p,null,n(o.n,null,"\n # Ricerca binaria\n \n Non credo di aver bisogno di studiare la ricerca binaria, quindi non ho preso appunti a riguardo.\n \n Se non siete me, e state cercando informazioni a riguardo, andate a vedere su Wikipedia!\n ")),n(o.p,null,n(o.n,null,"\n # Divide et impera\n \n Un modo efficace per risolvere un problema è di usare il metodo _divide et impera (et combina)_.\n \n - _Divide_: Divido il problema in **tanti sottoproblemi**.\n - _Impera_: **Risolvo indipendentemente dal resto** ciascuno dei sottoproblemi.\n - _Combina_: **Combino** i risultati dei sottoproblemi per **risolvere il problema principale**.\n \n ## Ricorsione\n \n Un algoritmo (o funzione) si dice _ricorsivo_ quando durante l'esecuzione **richiama sè stesso**.\n \n Dato che tutti gli algoritmi devono avere termine entro un tempo finito, se scriviamo una funzione ricorsiva è fondamentale finire con un **caso base**, che non chiami ulteriormente la ricorsione.\n \n Se esiste una funzione ricorsiva, allora esiste _sempre_ una _funzione iterativa_ che darà lo stesso risultato.\n \n > Sul pratico, una funzione ricorsiva tipicamente è **più costosa** del suo equivalente iterativo: se possibile, quindi, la ricorsione andrebbe evitata.\n \n #### Pseudocodice di esempio\n \n ```python\n def fattoriale(n):\n # Caso base; la ricorsione finisce e dà un risultato fisso\n if n <= 1:\n return 1\n # Caso ricorsivo; la funzione restituisce il risultato di sè stessa (ma con parametri diversi)\n else:\n return n * fattoriale(n - 1)\n ```\n ")),n(o.p,null,n(o.n,null,"\n # Master Theorem\n \n Il _Master Theorem_ è uno dei teoremi più importanti dell'algoritmica.\n \n Esso permette di **calcolare l'upper bound di un algoritmo ricorsivo** in modo piuttosto semplice.\n \n ## Ipotesi\n \n Dato un algoritmo:\n - Con uno o più casi base\n - Che richiama la funzione ricorsiva un numero n di volte\n \n ## Tesi\n \n Allora, il suo upper bound avrà la formula:\n \n ![](https://quicklatex.com/cache3/89/ql_08d29a7e55d561a900570bc83b93ff89_l3.png)\n ")),n(o.p,null,n(o.n,null,"\n # Caso particolare del Master Theorem\n \n ## Ipotesi\n \n Se: \n ![](https://quicklatex.com/cache3/57/ql_e34dc27b42831d3c3ff671b0f3861257_l3.png)\n \n Ovvero, se la dimensione dell'input viene divisa ad ogni ciclo da una costante b, è polinomiale e il caso base è costante...\n \n ## Tesi\n \n Allora: \n ![](https://quicklatex.com/cache3/ca/ql_26e3557a6ca2d6ac4b8481e7c5263fca_l3.png)\n \n > In pratica, se il costo dominante è quello della parte \"fissa\" dell'algoritmo, esso sarà `O(n^d)`, mentre se il costo dominante è quello delle chiamate ricorsive, esso sarà `O(n^{log_b e})`. \n > Se nessuno dei due è dominante... si dividono circa in parti uguali, creando un costo di `O(n^d log n)`.\n ")),n(o.p,null,n(o.n,null,"\n # Ordinamento\n \n Un problema molto frequente nell'informatica consiste nell'**ordinare efficientemente grandi quantità di elementi**.\n \n Esistono [tantissimi](https://it.wikipedia.org/wiki/Algoritmo_di_ordinamento) algoritmi per effettuare l'ordinamento. \n L'**efficienza** di ciascuno **varia** di caso in caso: alcuni sono estremamente efficienti se quasi tutti i numeri sono già nell'ordine giusto; altri, invece, potrebbero impiegare tantissimo tempo.\n \n In termini matematici, abbiamo:\n \n - **Input:** A[n]\n - **Output:** B, ∀ i < n, A[i] ≤ A[i+1]\n \n ## Ordinamento tramite confronto\n \n L'ordinamento \"tradizionale\" è detto _ordinamento tramite confronto_: funziona sempre, e **non ha altri modi di ottenere informazioni** se non con l'operazione logica di confronto tra i dati.\n \n ### Limiti\n \n E' un problema risolto: è dimostrabile che il suo **lower bound** è **`Ω(n log n)`**; possiamo quindi dire che qualsiasi algoritmo di ordinamento è in `Ω(n log n)`, e se riusciamo a trovare un algoritmo di ordinamento in `O(n log n)` siamo riusciti a raggiungere il massimo dell'efficienza.\n \n #### Dimostrazione\n \n Consideriamo **tutte le possibili permutazioni** della sequenza da ordinare: sono `n!`.\n \n Per ogni confronto che effettuiamo, **riduciamo la quantità di permutazioni** correttamente ordinate; prima o poi, rimarrà **una sola possibilità**.\n \n **TODO, non trovo la spiegazione corretta!**\n \n ### Esempi\n \n Algoritmi che effettuano l'ordinamento tramite confronto sono:\n - _Bubble sort_\n - _Merge sort_\n - _Insertion sort_\n - _Quick sort_\n - _Heap sort_\n - E tanti, tanti altri!\n \n ## Ordinamento con altri mezzi\n \n Esistono algoritmi che ricavano informazioni in altri modi, diversi dal confronto.\n \n Essi possono avere un lower bound più basso di `O(n log n)`, però hanno spesso limitazioni sul loro utilizzo.\n \n ### Esempi\n \n - _Counting sort_, indicizza i valori da ordinare\n - _Radix sort_, guarda singolarmente le cifre dei valori\n - _Sleep sort_, sfrutta i thread e la funzione sleep per ordinare valori\n - E altri ancora!\n ")),n(o.p,null,n(o.n,null,"\n # Insertion sort\n \n L'_insertion sort_ è una soluzione **iterativa** all'ordinamento per confronto.\n \n ## Funzionamento\n \n Considero la sequenza divisa in **due parti**: una parte **ordinata** e una parte **non ordinata**.\n \n Parto dal primo elemento della lista: è sempre ordinato con sè stesso. \n Poi, aggiungo uno alla volta i numeri della parte non ordinata a quella ordinata; prima trovo in quale posizione dovrò andare a mettere il numero, poi **faccio slittare tutti i numeri dopo quella posizione** avanti di 1, in modo da **creare lo spazio** in cui infine **inserirò** il numero.\n \n ## Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | `O(n²)` | `Ω(n)` | - |\n \n Nel _caso migliore_ (**lista già ordinata**), il numero da inserire è già nella posizione giusta, quindi non devo effettuare altri confronti oltre il primo, rendendo il lower bound dell'algoritmo `Ω(n)`.\n \n Nel _caso peggiore_ (**lista nell'ordine inverso**), dobbiamo confrontare il numero da inserire con tutti gli altri nella parte ordinata: dobbiamo allora eseguire `1+2+3+4+5+… = \\frac{(n-1)(n)}{2}` confronti; ciò significa che l'upper bound è `O(n²)`!\n \n ## Pseudocodice\n \n ```python\n def insertion_sorted(lista):\n # Itero su tutti i numeri della lista, dal primo all'ultimo.\n for divisore_ord in range(len(lista)):\n # Partendo dalla posizione attuale, creo l'indice di divisione numeri ordinati maggiori e minori\n divisore_magg = divisore_ord\n # Faccio slittare avanti i numeri maggiori di quello che stiamo inserendo\n # Se l'indice divisore_magg raggiunge 0, vuol dire che tutti i numeri della lista sono maggiori del numero attuale \n while divisore_magg >= 0 and lista[divisore_magg-1] > lista[divisore_magg]:\n # Scambio la posizione dei due elementi con gli indici specificati\n # Funzione inventata \n lista[divisore_magg], lista[divisore_magg-1] = lista[divisore_magg-1], lista[divisore_magg]\n # Diminuisco il separatore di 1\n divisore_magg -= 1\n ```\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/bn/sorting)\n ")),n(o.p,null,n(o.n,null,"\n # Merge sort\n \n Il _merge sort_ è una soluzione **ricorsiva** all'ordinamento per confronto.\n \n ## Funzionamento\n \n Per questo algoritmo, utilizziamo la tecnica del **divide et impera**.\n \n 1. _Divide_: Divido A in **due parti**.\n 2. _Impera_: Metto **separatamente in ordine** le parti.\n 3. _Unisci_: **Unisco** le due parti.\n \n Consideriamo come **caso base** della ricorsione una parte composta da un numero, che ovviamente è già ordinata con sè stessa.\n \n ### Merge\n \n Per **unire le due parti** usiamo una funzione detta `merge()`.\n \n Costruiamo una nuova sequenza uguale alla sequenza 1, ma **aggiungiamo alla fine un valore sentinella** sempre maggiore di tutti gli elementi contenuti.\n \n > ```\n > | 1 | 3 | 7 | 8 | ∞ |\n > ```\n \n Facciamo **la stessa cosa** per la sequenza due.\n \n > ```\n > | 2 | 4 | 5 | 6 | ∞ |\n > ```\n \n Prendo i primi numeri delle due sequenze e **metto il più piccolo nella sequenza iniziale**.\n \n > ```\n > | 1 | 2 | 3 | | | | | | \n > | | | 7 | 8 | ∞ | \n > | | 4 | 5 | 6 | ∞ | \n > ```\n \n **Continuo** finchè non ho messo tutti i numeri; **grazie alla sentinella non usciremo mai dalla sequenza**, in quanto essa è sempre maggiore di tutti gli altri numeri.\n \n > ```\n > | 1 | 2 | 3 | 4 | 5 | 6 | | | \n > | | | 7 | 8 | ∞ | \n > | | | | | ∞ |\n > ```\n \n Quando **rimangono solo le sentinelle** significa che abbiamo aggiunto tutti gli elementi, e quindi abbiamo finito.\n \n > ```\n > | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | \n > | | | | | ∞ | \n > | | | | | ∞ | \n > ```\n \n \n ## Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | `O(n log n)` | `Ω(n log n)` | **`θ(n log n)`** |\n \n Il merge sort è un algoritmo ricorsivo con un **caso base in tempo costante** e che **richiama sè stesso 2 volte**.\n \n ```latex\n T(n) =\\\\\n \\\\\n Θ(1) \\qquad n=1\\\\\n 2 T(\\frac{n}{2}) + Θ(n) \\qquad n \\neq 1\n ```\n \n Applicando il **caso particolare del Master Theorem**, otteniamo:\n \n ```latex\n T(n) =\\\\\n \\\\\n Θ(1) \\qquad n=1\\\\\n Θ(n log n) \\qquad n \\neq 1\n ```\n \n ## Pseudocodice\n \n ```python\n def merge_sorted(part):\n # Caso base\n if len(part) == 1:\n return part\n # Divide\n middle = len(part) // 2\n part_a = part[:middle]\n part_b = part[middle:]\n # Impera\n sort_a = merge_sorted(part_a)\n sort_b = merge_sorted(part_b)\n # Combina\n return merge(sort_a, sort_b)\n ```\n \n ## Visualizzazione\n \n [hackerearth.com](https://www.hackerearth.com/practice/algorithms/sorting/merge-sort/visualize/)\n \n [visualgo.net](https://visualgo.net/bn/sorting) (Nota: visualizza solo la fase _Unisci_ del sort)\n ")),n(o.p,null,n(o.n,null,"\n # Quick sort\n \n Il _quick sort_ è un altro approccio **ricorsivo** all'ordinamento per confronto.\n \n ## Funzionamento\n \n Anche qui, applichiamo il **divide et impera**.\n \n 1. _Divide_: Scelgo un **pivot** qualsiasi all'interno della lista. Metto alla sua **sinistra tutti i numeri minori** e alla sua **destra tutti i numeri maggiori**.\n 2. _Impera_: Eseguo un **quick sort su entrambe le sottoliste**.\n \n ### Esempi\n \n #### Iterazione con partizioni bilanciate\n \n Osserviamo come si formi una partizione con tre elementi e una con quattro. \n \n ```\n |¦ [2] 8 7 1 3 5 6 {4}\n 2 |¦ [8] 7 1 3 5 6 {4}\n 2 |8 ¦ [7] 1 3 5 6 {4}\n 2 |8 ¦ [7] 1 3 5 6 {4}\n 2 |8 7 ¦ [1] 3 5 6 {4}\n 2 1 |7 8 ¦ [3] 5 6 {4}\n 2 1 3 |8 7 ¦ [5] 6 {4}\n 2 1 3 |8 7 5 ¦ [6] {4}\n 2 1 3 |8 7 5 6 ¦ [{4}]\n [2 1 3] {4} [8 5 6 7]\n ```\n \n #### Iterazione con partizioni sbilanciate\n \n Osserviamo come si formi una partizione con **zero elementi** e una con tre.\n \n ```\n |¦ [4] 7 3 {1}\n |4 ¦ [7] 3 {1}\n |4 7 |¦ [3] {1}\n |4 7 3 |¦ [{1}]\n [] {1} [4 7 3]\n ```\n \n ## Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | `O(n²)` | `Ω(n log n)` | - |\n \n Il costo della funzione è dato dalla somma del costo per **dividere in due partizioni** con il costo per realizzare il **Quick sort delle due sottopartizioni**\n \n Possiamo applicare allora il **Master Theorem generale**:\n \n ```latex\n T(n)\\\\\n =\\\\\n Θ(1) \\qquad per\\ n = 1\\\\\n T(q) + T(dim_lista - pivot - 1) + Θ(n) \\qquad per\\ n > 1\n ```\n \n ### Il caso migliore\n \n Se il pivot `q` è la **mediana della partizione** che stiamo ordinando, si vengono a creare due _sottopartizioni bilanciate_, e sostituendo otteniamo:\n \n ```latex\n T(n)\\\\\n =\\\\\n Θ(1) \\qquad per\\ n = 1\\\\\n 2 T(\\frac{n}{2}) + Θ(n) \\qquad per\\ n > 1\n ``` \n \n Possiamo allora applicare il **Master Theorem particolare**:\n \n ```latex\n T(n)\\\\\n =\\\\\n Θ(1) \\qquad per\\ n = 1\\\\\n Θ(n log n) \\qquad per\\ n > 1\n ```\n \n ### Il caso peggiore\n \n Se il pivot è uno degli **estremi dell'array**, si creano due _partizioni sbilanciate_: una delle due sottoliste è sempre vuota! \n Allora:\n \n ```latex\n T(n) = T(n-1) + Θ(n)\\\\\n = T(n-2) + Θ(n-1) + Θ(n)\\\\\n = T(n-3) + Θ(n-2) + Θ(n-1) + Θ(n)\\\\\n = …\n ∈ Θ(n^2)\n ```\n \n > \"Non date da mangiare sequenze ordinate al Quicksort, gli sono indigeste.\"\n \n ## Pseudocodice\n \n ```python\n def partition(partizione, inizio, fine):\n \"\"\"Dividi una partizione in due, usando l'ultimo elemento come pivot.\n \n Note utili:\n partizione[fine] è il pivot\n partizione[maggiori] è il primo numero dei maggiori\n partizione[non_iterati] è l'elemento su cui si sta iterando al momento\"\"\"\n # Crea il primo separatore (la | pipe nell'esempio)\n maggiori = inizio\n # Crea il secondo separatore (la ¦ broken pipe nell'esempio)\n non_iterati = inizio\n # Itera su ogni numero tra inizio e fine (escluso!)\n while non_iterati < fine:\n # Se l'elemento su cui stiamo iterando è minore del pivot\n if partizione[non_iterati] <= partizione[fine]:\n # Mettilo nell'insieme dei minori, scambiandolo con il primo numero dei maggiori e incrementando il primo separatore\n partizione[maggiori], partizione[non_iterati] = partizione[non_iterati], partizione[maggiori]\n maggiori += 1\n # Incrementa sempre il secondo separatore\n non_iterati += 1\n # Inserisci il pivot tra le due sottopartizioni create, \n partizione[fine], partizione[non_iterati] = partizione[non_iterati], partizione[fine]\n return maggiori\n ```\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/bn/sorting) (Nota: invece che prendere l'ultimo numero come pivot prende il primo, cambiando leggermente l'algoritmo.)\n \n ## Note per l'esame\n \n > La domanda che fa sempre è **\"Qual è la sequenza di pivot utilizzata?\"**\n \n > Elementi da soli _non_ vengono presi come pivot!\n ")),n(o.p,null,n(o.n,null,'\n # Counting sort\n\n Il _counting sort_ è un approccio diverso all\'ordinamento: **non usa il confronto**!\n \n ## Requisiti\n \n Il counting sort può essere utilizzato solo su **sequenze di numeri interi**, e solo se **siamo a conoscenza del minimo e del massimo** dei numeri contenuti nell\'array, ed essi non sono troppo distanti uno dall\'altro. \n (La memoria occupata dal counting sort aumenta linearmente con la differenza tra minimo e massimo!)\n \n Per semplicità, consideriamo il **minimo `0`**. \n L\'input allora sarà una sequenza di interi `A`, e il valore del **massimo `k`**, tale che `∀ n ∈ A, 0 \\leq n \\leq K`.\n \n ## Funzionamento\n \n Il counting sort **conta le ripetizioni** delle chiavi nella sequenza originale e in seguito **sovrascrive i valori** della sequenza con i valori ordinati ripetuti il numero di volte che sono stati individuati nella sequenza.\n \n > ```\n > 1 4 5 3 4 1 4 2 5 1\n > ```\n >\n > L\'`1` appare 3 volte, il `2` 1 volta, il `3` 1 volta, il `4` tre volte e il `5` due volte.\n >\n > La sequenza viene quindi così sovrascritta:\n > ```\n > 1 1 1 3 4 1 4 2 5 1 # Sovrascriviamo la sequenza con 1 ripetuto 3 volte\n > 1 1 1 2 3 4 4 4 5 1 # Sovrascriviamo la sequenza con 2, 3, 4 ripetuti rispettivamente 1 1 e 3 volte\n > 1 1 1 2 3 4 4 4 5 5 # Sovrascriviamo la sequenza con 5 ripetuto 2 volte: abbiamo finito!\n > ``` \n \n Esiste anche una **versione stabile** del counting sort che, invece che sovrascrivere, **sposta i valori**, mantenendo le informazioni aggiuntive nel caso invece che interi fossero altri tipi di dati.\n \n ## Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | `O(k + n)` | `Ω(k + n)` | **`θ(k + n)`** |\n \n L\'algoritmo è composto da quattro parti:\n - Ricerca del minimo e massimo (in `θ(n)`)\n - Inizializzazione dell\'indice (in `θ(k)`)\n - Conteggio dei numeri (in `θ(n)`)\n - Sovrascrittura dei numeri (in `θ(k + n)`)\n \n `2 + O(k) + O(n) + O(k + n) -> O(k + n)`\n \n Notiamo che `k` è costante, l\'algoritmo è `O(n)`, estremamente efficiente.\n \n ## Pseudocodice\n \n ```python\n def counting_sort(lista: typing.List[int]):\n """Ordina in-place una lista con il counting sort."""\n # Trovo la dimensione della lista\n dim = len(lista)\n # Trovo il massimo e il minimo all\'interno della lista\n minimo = min(lista)\n massimo = max(lista)\n # Creo l\'indice dei numeri, in modo che sia lungo k e pieno di 0\n indice = [0 for _ in range(minimo, massimo+1)]\n # Conto i numeri presenti, scorrendo su lista e aggiungendo 1 al numero corrispondente\n for i in range(dim):\n indice[lista[i]] += 1\n # Sovrascrivo i numeri nella lista\n count = 0\n for pos, val in enumerate(indice):\n for _ in range(val):\n indice[count] = pos\n count += 1\n \n def stable_counting_sorted(lista: typing.List[int], k: int) -> typing.List[int]:\n """Ordina stabilmente una lista con il counting sort stabile, e restituiscila."""\n # Trovo la dimensione della lista\n dim = len(lista)\n # Trovo il massimo e il minimo all\'interno della lista\n minimo = min(lista)\n massimo = max(lista)\n # Creo l\'indice dei numeri, in modo che sia lungo k e pieno di 0\n indice = [0 for _ in range(minimo, massimo+1)]\n # Conto i numeri presenti, scorrendo su lista e aggiungendo 1 al numero corrispondente\n for i in range(dim):\n indice[lista[i]] += 1\n # Faccio diventare l\'indice "il numero di numeri \\leq i"\n for i in range(len(indice)):\n if i == 0:\n continue\n indice[i] += indice[i-1]\n assert indice[-1] == dim\n # Creo una nuova lista, che sarà quella che verrà restituita\n nuova = [None for _ in range(dim)]\n # Inizio a posizionare i numeri, al contrario\n for i in range(0, dim, -1):\n nuova[indice[lista[i]]] = lista[i]\n indice[lista[i]] -= 1\n return nuova\n ```\n ')),n(o.p,null,n(o.n,null,"\n ## Introduzione alle Strutture Dati\n\n Una _struttura dati_ è un modo in cui si possono organizzare i dati di un programma.\n \n Si possono definire in due modi: **elementari** e **astratte**.\n \n ### Strutture dati elementari\n \n Le strutture _elementari_ dipendono strettamente dal modo in cui vengono memorizzati i dati.\n \n > **Array** e **liste** sono strutture dati elementari: sono definite dicendo come sono memorizzati i dati, rispettivamente, in celle contigue di memoria e da una serie di nodi con un valore e che puntano al successivo.\n \n ### Strutture dati astratte\n \n Le strutture _astratte_ sono separate dal modo in cui vengono memorizzati i dati, sono più ad alto livello, e si definiscono descrivendo le **proprietà** della struttura e i **metodi** che su di essa possono essere effettuate.\n \n > Una _classe_ in un qualsiasi linguaggio di programmazione è una struttura dati astratta.\n \n > Una _pila_ astratta:\n > - memorizza dati tutti dello stesso tipo\n > - `pop()`, estrae dalla pila l'ultimo valore inserito\n > - `push(val)`, aggiunge alla pila un valore\n > - `top()`, permette di vedere l'ultimo valore inserito nella pila\n > - `vuota()`, dice se la pila è vuota oppure no.\n ")),n(o.p,null,n(o.n,null,'\n # Array\n\n Un _array_ è sequenza di dati di **lunghezza conosciuta**, tutti dello **stesso tipo** e di una **dimensione fissa**, immagazzinata in **blocchi di memoria contigui**.\n \n ## Proprietà\n \n - E\' possibile accedere a tutti i blocchi di memoria conoscendo la loro **posizione relativa al primo** elemento.\n \n ## Metodi\n \n ```python\n class Array:\n def __init__(self, size, type=int): "Crea un array di dimensione size di elementi di tipo int."\n def __getitem__(self, index): "Restituisci il valore alla posizione index."\n def __setitem__(self, index, value): "Cambia il valore alla posizione index."\n ```\n \n ### Costo computazionale\n \n Tutte le operazioni su un array sono in **accesso immediato `O(1)`**!\n ')),n(o.p,null,n(o.n,null,'\n # Lista\n \n Una _lista_ è una sequenza di dati immagazzinata in **blocchi di memoria qualsiasi**.\n \n ## Proprietà\n \n Ogni dato ha un **riferimento** alla collocazione di memoria **successiva** (un puntatore): insieme, sono detti un _nodo_.\n \n E\' di **natura ricorsiva**: qualsiasi nodo di una lista può essere visto come inizio della lista con sè stesso e i suoi successivi.\n \n ## Metodi\n \n ```python\n class LinkedList:\n def __init__(self, value, next: typing.Optional[Node] = None):\n self.value = value\n self.next: typing.Optional[Node] = next\n \n def is_empty(self) -> bool: "Restituisce se la lista è vuota o no."\n def is_full(self) -> bool: "Restituisce se la lista è piena o no.\n def append(self, value): "Aggiunge un nuovo elemento in testa alla lista."\n def insert(self, value, index): "Inserisce un elemento dopo il nodo in posizione index."\n def insert_node(self, value, node): "Inserisce un nuovo elemento subito dopo un dato nodo."\n def find(self, node) -> int: "Trova l\'indice del nodo."\n def delete(self, value): "Elimina il primo nodo con quel valore dalla lista."\n def delete_node(self, node): "Elimina il nodo dalla lista."\n def forward(self, index) -> Node: "Restituisce il nodo in posizione index."\n ```\n \n ### Costo computazionale\n \n #### `List.forward(index)`\n \n Per raggiungere l\'`n`-esimo elemento, bisogna _scorrere tutti gli elementi prima di esso_: è dunque un **accesso sequenziale** in **`O(n)`**.\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/list)\n ')),n(o.p,null,n(o.n,null,'\n # Coda (Queue)\n\n Una _coda_ è come una pila, ma segue la strategia **First In, First Out** (il primo inserito sarà il primo a essere estratto).\n \n ## Proprietà\n \n - I dati vi possono essere aggiunti solo tramite il **metodo `enqueue()`**\n - I dati possono essere estratti solo tramite il **metodo `dequeue()`**\n - Verranno restituiti i valori inseriti secondo la strategia **First In, First Out** (il primo inserito sarà il primo a essere estratto).\n \n > Hai presente quando fai la fila per pagare al supermercato? Beh, è quello, però non si possono superare le altre persone in nessun modo.\n \n ## Metodi\n \n ```python\n class Queue:\n def __init__(self): "Crea una nuova coda."\n def is_empty(self) -> bool: "Restituisce vero se la coda è vuota."\n def enqueue(self, data): "Aggiunge un dato alla coda."\n def first(self) -> ...: "Restituisce il primo dato della coda."\n def dequeue(self) -> ...: "Restituisce il primo dato della coda e lo rimuove."\n ```\n \n ## Implementazione tramite lista\n \n Posso implementare la coda con una lista, ma per realizzare l\'implementazione più efficiente devo tenere anche un puntatore all\'ultimo elemento della coda, in modo da non doverla scorrere ogni volta che voglio effettuare un\'operazione.\n \n Chiamiamo i due puntatori `head` e `tail` rispettivamente.\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/list)\n ')),n(o.p,null,n(o.n,null,'\n # Pila (Stack)\n\n Una _pila_ è una struttura dati contenente **valori omogenei**.\n \n ## Proprietà\n \n - I dati vi possono essere aggiunti solo tramite il **metodo `push()`**\n - I dati possono essere estratti solo tramite il **metodo `pop()`**\n - Verranno restituiti i valori inseriti secondo la strategia **Last In, First Out** (l\'ultimo inserito sarà il primo a essere estratto).\n \n > Ci si può immaginare una pila di libri, da cui si può solo prendere un libro alla volta, quello più in alto.\n \n ## Metodi\n \n ```python\n class Stack:\n def __init__(self): "Crea una nuova pila."\n def is_empty(self) -> bool: "Restituisce vero se la pila è vuota."\n def push(self, data): "Aggiunge un dato alla pila."\n def top(self) -> ...: "Restituisce il primo dato della pila."\n def pop(self) -> ...: "Restituisce il primo dato della pila e lo rimuove."\n ```\n \n ## Implementazione tramite lista\n \n Utilizzando una lista possiamo realizzare una pila!\n \n La direzione dei puntatori sarà dall\'ultimo al primo, che non punterà più a nessuno.\n \n Il costo di tutti i metodi è `Θ(1)`!\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/list)\n ')),n(o.p,null,n(o.n,null,"\n # Albero radicato\n \n Un _albero radicato_ è una struttura dati di **natura ricorsiva** che organizza i dati in maniera **non-lineare**.\n \n ## Proprietà\n \n - Ogni nodo dell'albero ha un **unico genitore**: `∀ (padre, figlio), (padre' figlio) ∈ E \\implies padre = padre'`\n - Ogni nodo dell'albero può avere **un numero qualsiasi di figli**.\n \x3c!----\x3e\n - I **nodi superiori al padre** vengono chiamati _antenati_.\n - I **nodi inferiori ai figli** vengono chiamati _discendenti_.\n \x3c!----\x3e\n - Nodi **senza padre** sono detti _radice_: `\\notexists (padre, radice) ∈ E`\n - Nodi **con padre e figli** sono detti _rami_ o interni.\n - Nodi **senza figli** sono detti _foglie_.\n \x3c!----\x3e\n - La **distanza** tra il nodo radice e i suoi discendenti è detta _livello_:\n - I figli immediati sono di livello 1.\n - I \"nipoti\" (figli dei figli) sono di livello 2.\n - I figli dei nipoti sono livello 3.\n - E così via.\n - Il **livello massimo** all'interno di un albero è detto _altezza_, _profondità_ oppure _h_, ed è sempre `1 ≤ h ≤ n-1`.\n \x3c!----\x3e\n - Un albero ha sempre `n-1` archi.\n \n ## Alberi particolari\n \n ### Alberi `d`-ari\n \n Un albero _`d`-ario_ è un particolare tipo di albero che **limita il numero massimo di figli di un nodo** a `d`.\n \n > Un albero _binario_ può avere **massimo 2 figli** per ogni nodo; un albero _ternario_ ne può avere **3**; un albero _`17`-ario_ ne potrà avere **17**\n \n #### Alberi completi\n \n Un albero `d`-ario si dice _completo_ se **tutti i nodi hanno 0 o `d` figli**, e mai una numero in mezzo.\n \n #### Alberi bilanciati\n \n Un albero `d`-ario si dice _bilanciato_ se **tutti i livelli eccetto l'ultimo** hanno il numero massimo di figli.\n \n #### Alberi perfettamente bilanciati\n \n Un albero `d`-ario si dice _perfettamente bilanciato_ se **tutti i livelli incluso l'ultimo** hanno il numero massimo di figli.\n \n ##### Particolarità degli alberi binari perfettamente bilanciati\n \n Si può dimostrare per induzione che:\n - Hanno sempre `2^h` foglie.\n - Hanno sempre `2^{h+1}-1` (`\\sum_i=0^n 2^i`) nodi.\n - L'altezza è in `Θ(log n)`.\n \n ## Implementazione degli alberi\n \n Possiamo scegliere se usare una rappresentazione con array o con nodi e puntatori: ognuna ha vantaggi e svantaggi diversi.\n \n ### Implementazione tramite array\n \n E' suggerito se l'albero è regolare; più è simile a un albero d-ario completo, meglio è.\n \n ### Implementazione tramite nodi e puntatori\n \n Più adatta ad alberi irregolari.\n \n Se l'albero è regolare, creiamo il numero esatto di campi:\n \n - Valore\n - Figlio1\n - Figlio2\n - _Opzionale:_ Padre\n \n Se un albero è irregolare, creiamo una specie di lista:\n \n - Valore\n - Primo figlio\n - Prossimo fratello\n - _Opzionale:_ Padre\n ")),n(o.p,null,n(o.n,null,"\n #### Breadth-first search (BFS)\n\n La _breadth-first search_ è un algoritmo che visita **ogni livello** dell'albero in ordine, dal più basso al più alto.\n \n ## Funzionamento\n \n > 1. __Visita radice__\n > 2. __Visita figli__\n > 3. __Visita nipoti__\n > 4. __Visita pronipoti__\n \n Si può implementare con una coda, in cui verranno inseriti i figli del nodo visitato da visitare ed estratti dopo avere completato la visita del livello attuale.\n \n ## Pseudocodice \n \n ```python\n def bfs(radice):\n c = Queue()\n c.enqueue(radice)\n while not c.is_empty():\n nodo = c.dequeue()\n print(nodo)\n for figlio in nodo.figli:\n nodo.enqueue(figlio)\n ```\n ")),n(o.p,null,n(o.n,null,"\n # Depth-First Search\n\n La _depth-first search_ è un algoritmo che visita **tutti i sottoalberi di un figlio** prima di passare ad un altro sfruttando la natura ricorsiva degli alberi.\n \n ## Funzionamento\n \n Ci sono diverse versioni della depth-first search: ognuna visita la radice in un momento diverso.\n \n ### DFS previsita (pre-order)\n \n La _DFS pre-visita_ visita la **radice per prima**, poi tutti i sottoalberi formati dai figli uno dopo l'altro.\n \n > 1. __Visita radice__\n > 2. dfs_pre_order(figlio1)\n > 3. dfs_pre_order(figlio2)\n \n ### Postvisita (post-order)\n \n La _DFS postvisita_ visita prima tutti i sottoalberi dei figli, e **alla fine la radice**.\n \n > 1. dfs_post_order(figlio1)\n > 2. dfs_post_order(figlio2)\n > 3. __Visita radice__\n \n ### Invisita (in-order)\n \n La _DFS invisita_ visita **un certo numero di figli**, poi la radice, e infine i figli restanti.\n \n > 1. dfs_in_order(figlio1, 1)\n > 2. __Visita radice__\n > 3. dfs_in_order(figlio2, 1)\n ")),n(o.p,null,n(o.n,null,"\n # Albero binario di ricerca\n\n ## Proprietà\n \n - Albero **binario**\n - Chiavi appartenenti ad un **insieme totalmente ordinato** (N, Q, R, ma non C)\n \x3c!----\x3e\n - Per ogni nodo con valore `x`, se un valore `v` è nel sottoalbero di sinistra allora `v ≤ x`, mentre se è nel sottoalbero di destra allora `v > x`.\n \n ## Costo computazionale\n \n - Trovare un valore: `O(h)`\n - Ordinare i valori: `O(n)`\n - Trovare il minimo: `O(h)`\n - Trovare il massimo: `O(h)`\n - Inserire un elemento: `O(h)`\n - Cancellare un elemento: `O(h)`\n \n `h` vale `log n` in un albero perfettamente bilanciato, e più l'albero diventa sbilanciato, più si avvicina a `n`, raggiungendola nel caso l'albero sia una lista.\n \n ## Pseudocodice\n \n ### Cancellazione ricorsiva\n \n ```python\n def delete(tree, key):\n if tree is not None:\n # Se ho trovato il nodo che cercavo...\n if tree.key == key:\n # E c'è una sola diramazione...\n # Semplicemente stacca il nodo come in una lista.\n if tree.left is None:\n return tree.right\n if tree.right is None:\n return tree.left\n # Altrimenti, diventa il minimo dell'albero di destra\n tree.key = tree.right.min()\n # Ed eliminalo dal sottoalbero\n tree.right = delete(tree.right, tree.key)\n # Se la chiave attuale è diversa da quella che cerchiamo, continuo a navigare l'albero\n elif tree.key < key:\n tree.left = delete(tree.left, key)\n else:\n tree.right = delete(tree.right, key)\n return tree\n ```\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/bst)\n \n ## Approfondimenti\n \n Esistono alberi più avanzati che mantengono le proprietà degli alberi binari di ricerca, ma che si autobilanciano, come il [Red Black Tree](https://it.wikipedia.org/wiki/RB-Albero).\n ")),n(o.p,null,n(o.n,null,'\n # Heap binario\n \n L\'_heap binario_ è un **albero binario bilanciato a sinistra**.\n \n ## Proprietà\n \n - _Proprietà strutturale_:\n - L\'albero è **perfettamente bilanciato** in tutti i livelli tranne l\'ultimo\n - Nell\'ultimo livello, le foglie occupano le **posizioni più a sinistra** possibili\n - _Proprietà di ordinamento_:\n - La chiave di un qualsiasi nodo è **più piccola** di tutte quelle dei nodi **del suo sottoalbero**\n \n ## Metodi\n \n ```python\n class Heap:\n def __init__(self, H): ...\n def _heapify_ancestors(self, i): "Ripristina le proprietà dell\'heap per il nodo all\'indice specificato e i suoi genitori."\n def minimum(self): "Restituisce la chiave con il valore minimo in H."\n def decrease_value(self, index, new_value): "Diminuisce il valore della chiave all\'indice index a new_value."\n def insert(self, value): "Inserisci un nuovo valore nell\'albero."\n def _heapify_children(self, i): "Ripristina le proprietà dell\'heap per il nodo all\'indice specificato e i suoi figli."\n def pop(self): "Restituisce la chiave con il valore minimo, e la elimina."\n def from_list(l): "Crea un heap da una lista."\n ```\n \n ## Implementazione con un array\n \n Possiamo implementare l\'albero utilizzando un array con le chiavi dell\'albero memorizzate nell\'ordine _breadth-first_.\n \n L\'indice del figlio sinistro può essere trovato a `2i+1`, mentre l\'indice del figlio sinistro può essere trovato a `2i+2`; il genitore è a `i//2-1`.\n \n ### Pseudocodice\n \n ```python\n class Heap:\n def __init__(self, size):\n self.array = Array(size) # Il tipo Array non esiste; consideriamolo pseudocodice\n self.next_value = 0\n \n def _heapify_ancestors(self, index)\n """Ripristina le proprietà dell\'heap per il nodo all\'indice specificato e i suoi genitori.\n \n Costo:\n O(log n)"""\n # Trovo l\'indice del genitore\n parent = index // 2 - 1\n # Controllo se viene mantenuta la proprietà di ordinamento dell\'heap\n if self.array[index] < self.array[parent]:\n # Scambio i valori dei due nodi\n self.array[index], self.array[parent] = self.array[parent], self.array[index]\n # Faccio la stessa cosa con il genitore\n self._heapify(parent)\n \n \n def minimum(self):\n """Restituisce la chiave con il valore minimo in H.\n \n Costo:\n O(1)"""\n return self.array[0]\n \n def decrease_value(self, index, new_value):\n """Diminuisce il valore della chiave all\'indice index a new_value.\n \n Costo:\n O(log n)"""\n # Diminuisco il valore del nodo\n self.array[index] = new_value\n # Aggiorno l\'heap \n self._heapify(index)\n \n def insert(self, value):\n """Inserisci un nuovo valore nell\'albero.\n \n Costo:\n O(log n)"""\n # Trovo l\'indice in cui inserire il valore\n index = self.next_index \n # Aggiungo il valore in fondo\n self.array[index] = value\n # Aggiorno l\'heap\n self._heapify(index)\n \n def _heapify_children(self, index):\n """Ripristina le proprietà dell\'heap per il nodo all\'indice specificato e i suoi figli.\n \n Costo:\n O(log n)"""\n # Trovo l\'indice dei figli\n left = index * 2 + 1\n right = index * 2 + 2\n # Mi assicuro che i figli esistano\n try:\n # Guardo quale dei figli è maggiore\n if self.array[left] > self.array[right]:\n # Scambio i valori\n self.array[left], self.array[index] = self.array[index], self.array[left]\n # Ripeto la procedura sul figlio modificato\n self._heapify_children(left)\n else:\n # Scambio i valori\n self.array[right], self.array[index] = self.array[index], self.array[right]\n # Ripeto la procedura sul figlio modificato\n self._heapify_children(left)\n except IndexError:\n # La foglia non ha figli: ho finito!\n return\n \n def pop(self):\n """Restituisce la chiave con il valore minimo, e la elimina.\n \n Costo:\n O(log n)"""\n # Mi salvo il valore della radice\n value = self.array[0]\n # Sostituisco la radice con l\'ultima foglia a destra\n self.array[0], self.array[self.next_value] = self.array[self.next_value], self.array[0]\n ### Non bisognerebbe eliminare la foglia...?\n # Riordino l\'heap\n self._heapify_children(0)\n return value\n \n @staticmethod\n def from_list(l):\n """Crea un heap da una lista.\n \n Costo:\n O(n log n), ma si può abbassare"""\n heap = Heap(len(l)))\n heap.array = Array.from_list(l) # Pseudocodice\n heap.next_value = len(l)\n # Cominciamo a riordinare l\'heap dalla fine, in modo che rispetti le proprietà\n for index in range(heap.next_value, 0, -1):\n heap._heapify_children(index)\n ```\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/heap)\n ')),n(o.p,null,n(o.n,null,"\n # Coda con priorità\n \n La _coda con priorità_ è una struttura dati dal funzionamento molto simile a quello di una coda, ma invece che restituire il primo elemento inserito, essa restituisce l'**elemento con il valore di priorità minore**.\n \n ## Proprietà\n \n - Ogni elemento è una coppia costituita da **valore** e **priorità** (un numero intero).\n - Nuovi elementi possono essere aggiunti solo tramite il **metodo `insert()`**\n - Gli elementi possono essere estratti solo tramite il **metodo `pop()`**\n - Verranno restituiti i valori inseriti secondo la strategia **Lower Priority** (l'elemento con la priorità minore sarà il primo ad essere restituito).\n - E' possibile diminuire la priorità di un elemento (e quindi anticipare la sua estrazione)\n \n ## Metodi\n \n ```python\n class PriorityQueue:\n def __init__(self): ...\n def insert(self, new_elem: Element): ...\n def minimum(self) -> Element: ...\n def pop(self) -> Element: ...\n def decrease_priority_for(self, elem: Element, priority: int): ...\n ```\n \n ## Implementazione con lista\n \n E' possibile implementare la coda con priorità tramite una **lista**: l'inserimento di nuovi valori diventerà molto efficiente, ma tutte le altre operazioni saranno linearmente lente.\n \n ### Costo computazionale\n \n - *insert()*: `O(1)`\n - *minimum()*: `O(n)`\n - *pop()*: `O(n)`\n - *decrease_priority_for()*: `O(n)`\n \n ## Implementazione con lista ordinata\n \n Implementando la coda con priorità con una **lista ordinata** si avrà un costo di ordinamento elevato negli inserimenti e modifiche alla priorità, ma costi costanti nell'estrazione di un elemento.\n \n ### Costo computazionale\n \n - *insert()*: `O(n)`\n - *minimum()*: `O(1)`\n - *pop()*: `O(1)`\n - *decrease_priority_for()*: `O(n)`\n \n ## Implementazione con heap\n \n La **soluzione migliore** è quella di implementare la coda con priorità tramite un **heap**: tutti i costi saranno logaritmici, eccetto l'inserimento che sarà costante.\n \n - `insert()` costa `O(1)`\n - `minimum()` costa `O(log n)`\n - `pop()` costa `O(log n)`\n - `decrease_priority_for()` costa `O(log n)`\n \n ## Approfondimenti\n \n Esistono code con priorità che restituiscono **l'elemento con priorità maggiore**, invece che quello minore.\n ")),n(o.p,null,n(o.n,null,"\n # Heap sort\n\n L'_heap sort_ è un algoritmo di ordinamento per confronto **iterativo**.\n \n ## Funzionamento\n \n Per effettuare un heap sort, creiamo un **heap massimo** in cui inseriamo tutti i valori che vogliamo ordinare.\n \n Una volta applicate le proprietà dell'heap, chiamiamo una versione particolare di `heap.pop()` che invece che rimuovere dall'array i valori estratti li posiziona nello spazio creatosi in fondo.\n \n ## Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | `O(n log n)` | `Ω(n log n)` | **`θ(n log n)`** |\n ")),n(o.p,null,n(o.n,null,"\n # Grafo\n \n Un _grafo_ è una struttura dati che rappresenta elementi interconnessi tra loro.\n \n Esistono due tipi di grafi: _orientati_ e _non orientati_.\n \n Per semplicità, consideriamo i nostri nodi numerati da 1 a `n`.\n \n ## Proprietà\n \n - Gli elementi sono rappresentati tramite _nodi_.\n - Il loro _grado_ è dato dal **numero degli archi** che vi incidono.\n - Se il grafo è orientato, hanno anche un _in-degree_ (**numero di archi entranti**) e un _out-degree_ (**numero di archi uscenti**).\n - Le connessioni tra elementi sono rappresentate tramite _archi_.\n - Un arco _incide_ esattamente su **due nodi**.\n - Se il grafo è orientato, sono _uscenti_ da uno dei due nodi ed _entranti_ nell'altro.\n - Sono matematicamente meno del **quadrato dei nodi**.\n \n ## Grafi particolari\n \n ### Catena\n \n Una _catena_ è un **grafo non orientato** composto da una **sequenza di nodi** aventi un **grado massimo di 2** tutti collegati tra loro.\n \n ### Cammino\n \n Un _cammino_ è un **grafo orientato** composto da una **sequenza di nodi** aventi un **in-degree** e un **out-degree** **massimo di 1**, collegati tra loro in modo che partendo dal primo e seguendo gli archi sia possibile arrivare all'ultimo.\n \n ### Cricca\n \n Una _cricca_ è un grafo in cui **tutti i nodi sono collegati tra loro**.\n \n Se il grafo è **non orientato**, la cricca ha `((n-1)n)/2` archi.\n \n Se il grafo è **orientato**, ha per ogni coppia un arco in entrambe le direzioni, quindi ha `(n-1)n` archi.\n \n ### Direct Acyclic Graph\n \n Un _DAG_ è un grafo diretto che non contiene nessun ciclo.\n \n Su di esso possiamo effettuare un ordinamento, detto _linearizzazione_, tra i nodi: otteniamo l'_ordine topologico_.\n \n I primi elementi dei DAG sono detti _Source_ (_Sorgente_), mentre gli ultimi sono detti _Sink_ (_Pozzo_).\n \n #### Albero\n \n Un **albero** può essere considerato un DAG con una **sorgente singola** e le **foglie come pozzi**.\n \n ### Grafo fortemente connesso\n \n Un insieme di nodi `V` di un **grafo diretto** `G` si dice una _componente fortemente connessa_ se:\n \n 1. Per ogni coppia di nodi `∀ u, v ∈ V' : ∃ un cammino u->v in G'`\n 2. Massimale (non può diventare più grande)\n \n > Praticamente una componente fortemente connessa è un gruppo di nodi tra i quali si può viaggiare liberamente da e a qualsiasi nodo al suo interno.\n \n Un grafo si dice _fortemente connesso_ se l'insieme `V` coincide con l'insieme dei nodi del grafo `G`.\n \n > Se partendo da qualsiasi nodo di un grafo riesco ad arrivare a qualsiasi altro nodo, allora il grafo è fortemente connesso.\n \n Inoltre, se creiamo un nuovo grafo, in cui **ogni nodo rappresenta una componente fortemente connessa** del nostro grafo iniziale, **otteniamo un DAG**, perchè tutti i cicli sono stati integrati nella componente.\n \n ### Trasposto di un grafo\n \n Il _trasposto_ di un **grafo diretto** `G` è il grafo stesso con gli archi che però vanno nella **direzione opposta**.\n \n ### Grafo pesato\n \n Un _grafo pesato_ è un particolare grafo che associa a ciascun arco un **costo** per attraversarlo.\n \n #### Costi negativi\n \n I costi possono anche essere negativi: rappresenteranno allora un **guadagno** ottenuto attraversando il nodo.\n \n ### Minimum spanning tree\n \n Un _minimum spanning tree_ è il **sottoinsieme degli archi** di un **grafo non diretto** che **connettono tutti i nodi** con il **minor costo possibile**.\n \n I MST hanno [molte proprietà](https://en.wikipedia.org/wiki/Minimum_spanning_tree#Properties); sono troppe da scrivere qui, e probabilmente non ci interesseranno nemmeno.\n \n ## Implementazione tramite matrice di adiacenza\n \n Possiamo implementare un grafo creando una **matrice di `bool` di dimensione `n * n`** in cui le **caselle collegate sono vere** e le caselle non collegate sono false.\n \n > Ad esempio, possiamo implementare un grafo non orientato in questo modo (`█` indica l'esistenza di un collegamento e ` ` indica la sua assenza):\n >\n > | |1|2|3|\n > |-|-|-|-|\n > |1|░|░|░|\n > |2|█|░|░|\n > |3|█| |░|\n >\n > Esistono gli archi `1-2` e `1-3`, ma non esiste un collegamento `2-3`.\n \n > Un grafo orientato invece si può implementare così:\n >\n > | |1|2|3|\n > |-|-|-|-|\n > |1|░|█| |\n > |2|█|░| |\n > |3|█| |░|\n >\n > Esistono gli archi `1->2`, `2->1` e `3->1`, ma non ci sono collegamenti `2->3`, `1->3` e `3->2`.\n \n ### Costo computazionale\n \n #### Tempo\n \n Le matrici di adiacenza portano alla realizzazione di algoritmi molto veloci: verificare l'esistenza di un arco è in `O(1)`!\n \n Abbiamo però penalità significative quando vogliamo effettuare operazioni sugli archi: ad esempio, trovare il trasposto di un grafo implementato con una matrice di adiacenza è in `O(nodi²)`.\n \n #### Memoria\n \n E' poco efficiente in quanto a memoria: l'upper bound è in `O(n^2)`. \n \n ## Implementazione tramite liste di adiacenza\n \n Un'alternativa alla matrice di adiacenza è quella di creare un'**array di liste**, le quali contengono i **vicini di ciascun nodo**.\n \n > |Posizione|Lista|\n > |-|-|\n > |1|[2, 3]|\n > |2|[]|\n > |3|[1]|\n >\n > Esistono gli archi `1->2`, `1->3`, e `3->1`, ma non esistono `2->1`, `2->3` e `3->2`.\n \n ### Costo computazionale\n \n #### Tempo\n \n Utilizzando le liste di adiacenza, il tempo richiesto per verificare l'esistenza di un arco sale a `O(max-out-degree)`.\n \n E' efficace però quando il problema che vogliamo risolvere riguarda operazioni su archi: trovare la trasposta è in `O(archi)`.\n \n #### Memoria \n \n La memoria richiesta dalle liste di adiacenza è minore di quella delle matrici: l'upper bound è in `O(nodi + archi)`.\n \n ## Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/graphds)\n ")),n(o.p,null,n(o.n,null,"\n # Visitare un grafo\n \n Come per gli alberi radicati, esistono due modi per visitare un grafo: _depth-first search_ e _breadth-first search_.\n \n In entrambi i casi, non visito mai due volte lo stesso nodo, e come risultato ottengo **molteplici alberi**, il cui insieme viene detto _foresta di copertura_.\n \n Se il grafo che vogliamo visitare è diretto, allora dobbiamo **considerare come vicini solo gli archi uscenti**\n \n ## Depth-first search\n \n La DFS ci può risultare utile per **identificare le componenti connesse** di un grafo e **identificare eventuali cicli**.\n \n ### Funzionamento\n \n Posso utilizzare la DFS per classificare gli archi di un grafo in quattro categorie:\n \n - _Tree_, archi che ci fanno **scoprire un nuovo nodo**\n - _Forward_, archi che ci portano a un **discendente**\n - _Back_, archi che ci portano ad un **antenato**\n - _Cross_, archi che **connettono due sottoalberi** diversi\n \n Usiamo due array inizializzati a 0 chiamati `pre` e `post`, grandi quanto il numero di archi del grafo, che ci indicano rispettivamente quando un nodo è stato scoperto e quando è terminata la visita. \n Inoltre, creiamo una variabile `clock` che avanza ad ogni evento. \n Alla scoperta di un nuovo nodo, mettiamo il valore attuale di `clock` all'interno di `pre[n]`. \n Alla fine della visita di un nodo invece mettiamo il valore di `clock` in `post[n]`.\n \n **Durante la visita**, gli archi avranno i seguenti valori:\n - _Tree_: `pre[dst] == 0`\n - _Forward_: `pre[src] < pre[dst] && post[dst] > 0`\n - _Back_: `pre[dst] < pre[src] && post[dst] == 0`\n - _Cross_: Tutti gli altri (`post[dst] < pre[src]`)\n \n **A fine visita**, gli archi avranno i seguenti valori:\n - _Tree_: `pre[dst] < pre[dst] < post[dst] < pre[src]`\n - _Forward_: `pre[dst] < pre[dst] < post[dst] < pre[src]`\n - _Back_: `pre[src] < pre[dst] < post[dst] < post[src]`\n - _Cross_: `pre[dst] < post[dst] < pre[src] < post[src]`\n \n Se un **grafo non diretto** contiene degli **archi Back**, allora esso **conterrà un ciclo**.\n \n #### DFS nel grafo trasposto\n \n Se effettuo una DFS sul trasposto di un grafo, posso **scoprire i nodi che hanno un cammino verso l'origine**.\n \n #### DFS nella componente fortemente connessa\n \n Se effettuo una DFS in una componente fortemente connessa e nella sua trasposta, il **`post` della trasposta sarà sempre minore** del `post` della componente originale.\n \n ### Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | `O(nodi + archi)` | `Ω(nodi + archi)` | **`θ(nodi + archi)`** |\n | Memoria | `O(nodi)` | `Ω(nodi)` | **`θ(nodi)`** |\n \n ### Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/dfsbfs)\n \n ## Breadth-first search\n \n La BFS ci può risultare utile per **trovare tutti i nodi a una certa distanza** da un'origine.\n \n ### Costo computazionale\n \n | Categoria | Upper bound | Lower bound | Tight bound |\n |-----------|-------------|-------------|-------------|\n | Tempo | `O(nodi + archi)` | `Ω(nodi + archi)` | **`θ(nodi + archi)`** |\n | Memoria | `O(nodi + archi)` | `Ω(nodi + archi)` | **`θ(nodi + archi)`** |\n \n ### Pseudocodice\n \n Come per gli alberi, la implementiamo in modo **iterativo**:\n \n ```python\n queue = [starting_node]\n parents = [None for node in graph.nodes]\n distance = [-1 for node in graph.nodes]\n \n # TODO: controllami quando sei più sveglio\n \n while queue:\n node, source, distance = queue.pop(0)\n parents[node.number] = source\n distance[node.number] = distance\n for neighbour in node.neighbours:\n queue.append((neighbour, node, distance+1))\n ```\n \n > Nella coda, la distanza massima tra un nodo e l'altro è 1.\n \n ### Visualizzazione\n \n [visualgo.net](https://visualgo.net/en/dfsbfs)\n ")),n(o.p,null,n(o.n,null,"\n # Algoritmi greedy\n\n Un modo per risolvere problemi algoritmici può essere usare una **tecnica** _greedy_.\n \n Le tecniche greedy consistono nel effettuare tanti piccoli passi, ed effettuare una **scelta** in base ai dati **locali al passo attuale**.\n \n > Scegli il numero di monete più piccole possibili per comporre €1.12.\n >\n > L'algoritmo cerca di scegliere sempre la moneta più grande possibile compatibile con il prezzo in quel momento, quindi:\n > | Moneta scelta | Rimanente |\n > |---------------|-----------|\n > | € 1.00 | € 0.12 |\n > | € 0.10 | € 0.02 |\n > | € 0.02 | € 0.00 |\n \n ## Esempi\n \n Sono algoritmi greedy:\n \n - L'_Algoritmo di Dijkstra_\n - L'_Algoritmo di Kruskal_\n - L'_Algoritmo di Prim_\n ")),n(o.p,null,n(o.n,null,"\n # Percorso più breve\n\n Trovare il _percorso più breve_ (o _cammino minimo_) tra due nodi di un **grafo pesato** è un problema frequente nell'informatica; per questo, sono stati sviluppati [numerosi algoritmi](https://en.wikipedia.org/wiki/Shortest_path_problem) per risolverlo.\n \n > Ad esempio, il pathfinding dei nemici nei videogiochi, oppure Google Maps!\n \n ## Percorso più breve da una sorgente singola\n \n Una sottocategoria del problema del percorso più breve è il caso in cui ci interessa sapere i percorsi più brevi che **partono da uno specifico nodo del grafo**: è detto problema del _percorso più breve da una sorgente singola_, o _single-source shortest path_.\n \n Si può notare che se il grafo contiene **costi negativi** allora è possibile che il percorso più breve non esista, in quanto diventa possibile la comparsa di **cicli di costo infinitamente negativo**.\n \n Possiamo notare che, se il percorso più breve tra `A` e `D` è `A-B-C-D`, allora il cammino minimo tra `B` e `D` passerà obbligatoriamente per `C` (`B-C-D`).\n Diremo più avanti che il percorso più breve ha una **sottostruttura ottimale**.\n \n ### Esempi\n \n Alcuni algoritmi che trovano il percorso più breve sono:\n - L'_Algoritmo di Dijkstra_\n - L'_Algoritmo di Bellman-Ford_\n - La [_ricerca A*_](https://en.wikipedia.org/wiki/A*_search_algorithm)\n ")),n(o.p,null,n(o.n,null,"\n # Algoritmo di Dijkstra\n \n L'_Algoritmo di [Dijkstra](https://upload.wikimedia.org/wikipedia/commons/8/85/Dijkstra.ogg)_ è un algoritmo che risolve il problema del **percorso più breve da una sorgente singola** per grafi con pesi **reali positivi** `\\mathbb{R}^+`.\n \n L'algoritmo trova tutti i percorsi più brevi per raggiungere qualsiasi nodo del grafo partendo da un dato nodo, assieme al costo richiesto per farlo.\n \n ## Funzionamento\n \n 1. Separiamo tutti i nodi del grafo in due gruppi: **visitati** e **non visitati**. \n - Tutti i nodi partono da **non visitati**.\n 2. Per ogni nodo, manteniamo un valore \"**costo richiesto per raggiungerlo**\", che verrà cambiato man mano che l'algoritmo avanza.\n - Il costo di partenza è `+∞`.\n - Il costo sarà **definitivo per i nodi visitati**, e **provvisorio per i non visitati**.\n 3. Creiamo un insieme detto _frontiera_ che conterrà tutti i **nodi non visitati adiacenti** a quelli visitati.\n 4. Prendiamo il nodo iniziale, che avrà un **costo di `0`**, e definiamolo il nodo _attuale_.\n 5. Finchè ci sono dei nodi non sono stati visitati, ripetiamo il seguente ciclo:\n 1. Aggiungiamo i nodi adiacenti al nodo attuale alla frontiera.\n - Il costo per raggiungerli sarà il **costo per il nodo attuale sommato al costo dell'arco** che li connette al nodo attuale. \n Se questo **costo** risulta essere **minore del costo provvisorio** precedente, esso **diventerà il nuovo costo**.\n - Questa operazione è detta _rilassamento dell'arco_.\n 2. Facciamo diventare **visitato** il nodo attuale.\n - Il percorso che abbiamo fatto per raggiungerlo è obbligatoriamente il più breve.\n 3. Il prossimo nodo attuale sarà il nodo di frontiera con un costo più basso.\n - Per questo, è possibile definire l'algoritmo di Dijkstra come un **algoritmo greedy**.\n \n ### Non funziona se...\n \n L'algoritmo smette di funzionare nel caso in cui siano presenti **costi negativi** e il grafo non sia **aciclico**, in quanto non saremmo mai in grado di rendere visitato un nodo.\n \n ## Costo computazionale\n \n | Categoria | Upper bound |\n |-----------|-------------|\n | Tempo | `O(nodi + archi) log nodi)` |\n \n ### Scomposizione\n \n - Inizializzazione: `O(nodi)`\n - Creazione coda priorità: `O(nodi log nodi)`\n - Ciclo: `O((nodi + archi) log nodi)`\n \n ## Pseudocodice\n \n ```python\n import math\n \n class Info:\n def __init__(self, distance=math.inf, previous=None):\n self.distance = distance\n self.previous = previous\n \n def dijkstra(graph, start):\n data = [Info() for node in graph.nodes]\n queue = PriorityQueue([start])\n while queue:\n node = queue.pop()\n for arc in node.connections:\n other = arc.other(node)\n if data[node.number].distance + arc.cost < data[other].distance:\n data[other].distance = data[node.number].distance + arc.cost\n queue.decrease_priority_for(other, data[other].distance)\n data[v].previous = node\n return data\n ```\n \n ## Visualizzazione\n \n [Visualgo](https://visualgo.net/en/sssp)\n ")),n(o.p,null,n(o.n,null,"\n # Algoritmo di Bellman-Ford\n\n L'_Algoritmo di Bellman-Ford_ è un algoritmo che, come l'Algoritmo di Dijkstra, risolve il problema del **percorso più breve da una sorgente singola**, però, a differenza da quest'ultimo, l'Algoritmo di Bellman-Ford accetta in input anche grafi con pesi **reali** `\\mathbb{R}` (sia positivi, sia negativi).\n \n ## Funzionamento\n \n L'approccio dell'algoritmo è simile a quello di Dijkstra: entrambi usano il **rilassamento** degli archi per ottenere un costo provvisorio per il raggiungimento di un nodo, ma invece che rilassare solo l'arco con costo inferiore, questo algoritmo **rilassa tutti gli archi** ripetutamente, eliminando la frontiera e il problema dei nodi negativi.\n \n L'operazione di rilassamento è ripetuta `nodi - 1` volte, ovvero la **lunghezza massima** di un cammino aciclico all'interno di un grafo.\n \n Possiamo individuare dopo i rilassamenti se è presente un nodo con un **ciclo negativo**: ci basta controllare se esiste un arco che connette due nodi con una distanza incompatibile: se `a.distanza + arco.costo < b.distanza`, allora è presente un ciclo negativo.\n \n ## Costo computazionale\n \n | Categoria | Upper bound |\n |-----------|-------------|\n | Tempo | `O(nodi * archi)` |\n \n ## Pseudocodice\n \n > TODO\n \n ## Visualizzazione\n \n [Visualgo](https://visualgo.net/en/sssp)\n ")),n(o.p,null,n(o.n,null,'\n # Disjoint set\n \n Il _disjoint set_ è una struttura dati che rappresenta elementi di un insieme raggruppati in **sottoinsiemi disgiunti**.\n \n ## Metodi\n \n ```python\n class DisjointNode:\n def __init__(self): ...\n def find_set(self): "Trova il rappresentante dell\'elemento."\n def union(self, other): "Unisce i sottoinsiemi che contengono questo nodo e `other`."\n ```\n \n ## Implementazione tramite array\n \n Possiamo implementare il disjoint set con due array: uno per l\'**indice del rappresentante** e uno per il **rango dell\'insieme**.\n \n Un singoletto avrà **sè stesso come rappresentante** e **rango `0`**.\n \n ### Costo computazionale\n \n - *create_set()*: `O(1)`\n - *find_set()*: `O(h)`\n - *union()*: `O(h)`\n \n ### Pseudocodice\n \n ```python\n class DisjointNode:\n def __init__(self):\n self.parent = self\n self.rank = 0\n \n def find_set(self):\n element = self\n while self.parent != element:\n element = self.parent\n return element\n \n def union(self, other):\n repres_self = self.find_set()\n repres_other = other.find_set()\n if repres_self == repres_other:\n return\n if repres_self.rank < repres_other.rank:\n repres_greater = repres_other\n repres_lesser = repres_self\n else:\n repres_greater = repres_self\n repres_lesser = repres_other\n repres_lesser.parent = repres_greater\n if repres_greater.rank == repres_lesser.rank:\n repres_greater.rank += 1\n ```\n \n ### Visualizzazione\n \n [cs.usfca.edu](https://www.cs.usfca.edu/~galles/JavascriptVisual/DisjointSets.html)\n ')),n(o.p,null,n(o.n,null,"\n # Trovare il minimum spanning tree\n\n Un altro problema ricorrente riguardante i grafi è trovare il _minimum spanning tree_ di un dato grafo non diretto.\n \n > E' utile per trovare il modo più efficiente per connettere le cose: ad esempio, per decidere la struttura di una rete internet!\n \n ## Esempi\n \n Gli algoritmi principali che risolvono il problema sono due, ed entrambi sono **algoritmi greedy**:\n \n - L'_Algoritmo di Kruskal_\n - L'_Algoritmo di Prim_\n ")),n(o.p,null,n(o.n,null,"\n # Algoritmo di Kruskal\n \n L'_Algoritmo di Kruskal_ è un algoritmo **greedy** che **trova il minimum spanning tree** di un grafo.\n \n ## Funzionamento\n \n 1. Ripetiamo questa procedura finchè tutti i nodi non sono connessi:\n 1. Prendiamo ad ogni passo **l'arco meno costoso** del grafo non ancora aggiunto all'insieme.\n 2. Assicuriamoci che **non si creino cicli**: se non se ne verrebbero a creare, possiamo **aggiungere l'arco all'insieme**.\n - Gli archi devono quindi connettere nodi in **componenti connesse diverse**.\n - Possiamo rappresentare le componenti connesse con un **Disjoint Set**.\n \n ## Costo computazionale\n \n | Categoria | Upper bound |\n |-----------|-------------|\n | Tempo | `O(archi²)` |\n \n ### Scomposizione\n \n - DisjointSet.__init__(): `O(archi)`\n - Per ogni ciclo: `O(archi²)`\n - DisjointSet.find_set(): `O(1)`\n - DisjointSet.union(): `O(archi)`\n \n ## Pseudocodice\n \n ```python\n def minimum_spanning_tree_kruskal(graph):\n ds = DisjointSet()\n for node in graph.nodes:\n ds.create_set(node)\n arcs = []\n sorted_arcs = sorted(graph.arcs, key=lambda arc: arc.cost)\n for arc in sorted_arcs:\n if ds.find_set(node.start) != ds.find_set(node.end):\n arcs.append(arc)\n ds.union(node.start, node.end)\n return arcs\n ```\n ")),n(o.p,null,n(o.n,null,"\n # Algoritmo di Prim\n \n L'_Algoritmo di Prim_ è un altro algoritmo **greedy** che **trova il minimum spanning tree** di un grafo.\n \n ## Funzionamento\n \n Creo una **coda con priorità** in cui inserisco tutti gli archi visibili dal mio albero, in cui la chiave è il **costo dell'arco**.\n \n Per trovare l'arco con costo più piccolo posso **estrarre un arco** dalla coda: la priorità ci garantisce che esso è l'**arco meno costoso**.\n \n Aggiungo allora un nuovo nodo all'albero, e con esso, **aggiungo alla coda** tutti gli **archi che scoprono un nuovo nodo**.\n \n ## Costo computazionale\n \n | Categoria | Upper bound |\n |-----------|-------------|\n | Tempo | `O(archi + nodi log nodi)` |\n \n ## Pseudocodice\n \n ```python\n import math\n \n def minimum_spanning_tree_prim(graph, cost_array, start_node):\n # E' un Array di bool: se l'indice corrispondente al nodo è uguale a true, vuol dire che il (nodo è contenuto nell'albero.\n contains = [False for _ in range(len(graph))]\n # Contiene il precedente di ogni nodo \n prev = [None for _ in range(len(graph))]\n # Contiene il costo per arrivare a quel nodo\n cost = [math.inf for _ in range(len(graph))]\n # Creo la priority queue\n pq = PriorityQueue(graph.arcs, key=lambda arc: arc.cost)\n # Parto dal nodo `start_index`\n # Il costo dell'origine è 0.\n cost[start_node.index] = 0\n contains[start_node.index] = True\n \n while not pq.is_empty():\n new_node = pq.pop()\n contains[new_node.index] = True\n for arc in new_node.connections:\n other_node = arc.other(new_node)\n if not contains[other_node.index] and cost[other_node.index] > arc.cost:\n cost[other_node.index] = arc.cost\n prev[other_node.index] = new_node\n pq.decrease_priority_for(other_node, arc.cost)\n # L'array di prev rappresenta un albero.\n return prev\n ```_\n ")),n(o.p,null,n(o.n,null,"\n # Compressione\n\n _Comprimere_ un file significa **ridurne le dimensioni** senza modificarne il significato.\n \n ## Categorie\n \n ### Compressione lossless\n \n Nella _compressione lossless_, i dati possono essere decompressi riottenendo una copia identica dell'originale.\n \n > Immaginiamo un file che contiene solo le lettere `a, b, c, d, e, f`.\n >\n > Le lettere compaiono con questa frequenza:\n > |a|b|c|d|e|f|\n > |45%|13%|12%|16%|9%|5%|\n >\n > Possiamo codificare le lettere nel seguente modo:\n > |a|b|c|d|e|f|\n > |`0b0`|`0b100`|`0b101`|`0b111`|`0b1100`|`0b1101`|\n >\n > Scrivere `abacadae` richiederebbe 64 bits con la codifica ASCII estesa, ma in questo modo riusciamo a scriverlo con soli 17 bits!\n \n Le codifiche di un file compresso devono rispettare la proprietà del _Codice a prefisso_, che dice che **nessun codice deve essere il prefisso di un altro codice**; altrimenti, si avrebbero ambiguità nella decodifica.\n \n > a = `1`\n > b = `11`\n >\n > `111` è `ab`, `ba` oppure `aaa`?\n \n Creiamo allora un _albero di decodifica_: un **albero binario** che, leggendo uno ad uno i bit codificati, ci permette di arrivare al **valore del codice presente sulle foglie** dell'albero.\n \n Gli alberi di decodifica sono sempre **completi**.\n \n > Un albero di decodifica incompleto sarebbe non ottimizzato!\n \n #### Esempi\n \n - .png\n - .flac\n - .zip\n - ...\n \n ### Compressione lossy\n \n Nella _compressione lossy_, alcuni dati [solitamente](http://needsmorejpeg.com/) irrilevanti vengono perduti: non si può, dunque, ricostruire l'originale.\n \n #### Esempi\n \n - .jpeg\n - .mp3\n - ...\n ")),n(o.p,null,n(o.n,null,"\n # Algoritmo di Huffman\n\n L'_Algoritmo di Huffman_ è un **algoritmo greedy** per la **costruzione di un albero di decodifica**.\n \n ## Funzionamento\n \n 1. Costruisco **un albero** (con un solo nodo) **per ogni elemento dell'alfabeto**.\n 2. Associo ad **ogni albero la frequenza dell'elemento** da cui è stato creato, per poi inserire tutti gli elementi in una coda con priorità.\n 3. Finchè non ho **un albero solo**:\n 1. Estraggo dalla coda i **due alberi con frequenza minore**.\n 2. **Li rendo fratelli**, creando un nuovo nodo in cui sono uno figlio destro e uno figlio sinistro.\n 3. Associo al nuovo nodo la **somma delle frequenze dei due alberi**, e inserisco il nuovo albero nella coda.\n \n > È molto raro che venga un albero \"dritto\"; se succede, probabilmente c'è qualcosa che non va.\n ")),n(o.p,null,n(o.n,null,"\n # Dizionario\n \n Un _dizionario_ è una struttura dati che **associa dei valori a delle chiavi**.\n \n ## Proprietà\n \n - Ogni elemento del dizionario è un **valore** che è stato **associato a una chiave**.\n - Possiamo aggiungere nuovi elementi con il **metodo `add(chiave, valore)`**.\n - Posiamo estrarre elementi con il **metodo `get(chiave)`**, che restituirà il valore associato a `chiave`.\n - E' possibile rimuovere elementi con il **metodo `delete(chiave)`**.\n - Due elementi con **chiavi diverse** non devono **mai restituire lo stesso valore**.\n \n ## Metodi\n \n Beh... Non ha molto senso in questo caso...\n \n ```python\n dict()\n ```\n \n ## Implementazione con tabella hash\n \n Una _tabella hash_ è un'array di coppie **chiave-valore**, che formano l'insieme _universo_.\n \n Per determinare l'**indice dell'array** in cui inserire una coppia, usiamo una [funzione _hash_](https://it.wikipedia.org/wiki/Hash#Algoritmo_di_hash) sulla chiave, che restituirà **numeri da `0` a `len(hash_table)`**.\n \n ### Risoluzione collisioni\n \n Potrebbe capitare però che **due chiavi diverse abbiano lo stesso indice**. Dobbiamo allora usare un metodo di _risoluzione collisioni_, che mi permetta di distinguere tra chiavi diverse.\n \n #### Lista di trabocco\n \n Possiamo salvare nell'array **liste di coppie** chiave-valore; in caso di collisione, **aggiungo un nuovo elemento alla lista**.\n \n In media, ciascuna di queste liste conterrà `elementi_inseriti / dimensione_tabella` elementi.\n \n #### Indirizzamento aperto\n \n Possiamo decidere di mettere le coppie che non trovano posto nel loro indice in **un altro indirizzo vuoto** dell'array.\n \n Ci sono diversi modi in cui decidere il nuovo indirizzo, ognuno con vantaggi e svantaggi: si può scegliere quello successivo, oppure il primo vuoto dell'array, oppure un indirizzo casuale.\n \n > Python, nei `dict`, usa indirizzamento aperto pseudorandom.\n \n ### Costo computazionale\n \n - Aggiungere una chiave: `O(n)`\n - Trovare una chiave: `O(n)`\n - Eliminare una chiave: `O(n)`\n ")),n(o.p,null,n(o.n,null,"\n # Programmazione dinamica\n \n La _programmazione dinamica_ è una **tecnica** di programmazione che prevede l'**estensione di una soluzione ottima precedente**.\n \n Tutti i problemi in cui si può applicare si possono risolvere anche con la **ricorsione**, ma a differenza della ricorsione, questa tecnica riesce ad evitare di ricalcolare la soluzione per ogni chiamata ricorsiva, ottenendo quindi tempi molto migliori.\n \n Si può applicare solo se un problema ha una **sottostruttura ottimale**, ovvero se la soluzione ottima di un sottoproblema è inclusa nella soluzione ottima del problema. \n \n ## Esempi\n \n - _Problema dello zaino_\n - ...\n \n > Il cammino minimo per raggiungere un nodo in un DAG è dato da `arco.costo + arco.primo_nodo.costo_cammino_minimo()`.\n >\n > ```python\n > def SPD_PD(graph, start):\n > distance = [float(inf) for node in graph.nodes]:\n > distance[start] = 0\n > # I nodi devono essere in ordine di linearizzazione\n > for node in graph.nodes:\n > distance[node] = min([(arc.cost + distance[arc.other(node)] for arc in node.connections])\n > ```\n \n > Ho una sequenza di interi da `a_1` a `a_n`. Voglio trovare la sottosequenza crescente più lunga.\n >\n > 5 2 3 4 7 3 6 3 1 6\n >\n > Trovo tutte le sequenze lunghe 1, e le rendo nodi di un grafo diretto.\n >\n > Da ogni nodo, creo una connessione verso i suoi maggiori.\n >\n > Infine, cerco i cammini massimi del grafo. \n > \n > Essi saranno la soluzione del problema.\n \n > Trova la lunghezza della sottosequenza più lunga che termina con `j`.\n >\n > `L[j]` = lunghezza della sottosequenza più lunga che termina in `j`\n > \n > ```python\n > L[j] = max([1 + L[node] for arc in node.connections)]\n > ```\n >\n > Esempio:\n > ```python\n > L[9] = max([1+L[8], 1+L[3], 1+L[6]])\n > ```\n ")),n(o.p,null,n(o.n,null,"\n # Problema dello zaino\n \n Il problema dello zaino è un problema _pseudo-trattabile_: non abbiamo dimostrazioni di se sia trattabile o intrattabile.\n \n ## Descrizione\n \n > Sei un ladro, e devi mettere **più refurtiva possibile** nello zaino per scappare. \n > Lo zaino può portare **al massimo `dim` kili**.\n >\n > **Quali** (e quanti) oggetti scegli?\n \n | Input | Output |\n |-------|--------|\n | `dim`ensione_zaino, `n`umero_oggetti, `oggetto.peso`, `oggetto.valore` | `profitto_massimo` |\n \n ## Categorie\n \n ### Problema con ripetizione\n \n Puoi prendere **tutte le copie che vuoi** di un oggetto.\n \n #### Soluzione\n \n `K(dim)` è il valore massimo ottenibile con uno zaino di capacità `dim`.\n \n > Se `i` appartenesse alla soluzione ottima, allora `K(dim) = i.valore + K(dim - i.peso)`...\n \n Possiamo dire che `K(dim) = max(i.valore + K(dim - i.peso))`.\n \n Inoltre, `K(0) = 0`.\n \n Ci salviamo tutte le soluzioni da `K(0)` a `K(dim)`, e le usiamo per calcolare il massimo in seguito.\n \n Calcolare `K(dim)` avrà allora un costo di `O(n * dim)`:\n - `n`, perchè trovare il massimo è un'operazione lineare\n - `dim`, perchè `dim` sono tutti i casi tra i quali devo andare a provare\n \n Il costo computazionale, allora, è in `O(n * dim)`. \n \n Però, il **tempo richiesto** dal nostro algoritmo dipende non dalla lunghezza dell'input, bensì dal **valore numerico** di `dim`, che corrisponde alla dimensione dell'array delle soluzioni. \n Allora, si dice che l'algoritmo è in **tempo _pseudo-polinomiale_**.\n \n ### Problema senza ripetizione\n \n Si può prendere **ogni oggetto una volta sola**.\n \n #### Soluzione _bruteforce_\n \n Scelgo se prendere o no l'item 1.\n \n Si creano due percorsi:\n - Non prendo l'oggetto: `valore = 0, peso = 0`\n - Prendo l'oggetto: `valore = oggetto.valore, peso = oggetto.peso`\n \n Continuo a creare percorsi, creando una specie di albero binario.\n \n Se a un certo punto vedo che `valore = x, peso = K` e `valore < x, peso = K`, allora posso escludere automaticamente tutto il sottoalbero destro, perchè non può essere migliore del sinistro: allora, sarò riuscito a ridurre il numero dei casi rispetto alla ricorsione.\n \n ### Problema in due variabili\n \n `K(j, w)` = massimo valore ottenibile con uno zaino di capacità `w` scegliendo gli item da `1` a `j`.\n \n Non possiamo più applicare la soluzione bruteforce, perchè abbiamo due variabili, `j` e `w`.\n \n Allora, prendo l'elemento `j`. Esso può essere o non essere nella soluzione: mi calcolo entrambe le alternative, e mi tengo l'alternativa dal valore più alto.\n \n Se `j` non è nella soluzione, il risultato diventerà `K(j-1, w)`; se invece è nella soluzione, il risultato sarà `j.valore + K(j-1, w-j.peso)`.\n \n In pratica, prendiamo\n \n ```latex\n K(j, w) = max\n \\begin{cases}\n V_j + K(j-1, w-w_j)\n K(j-1, w)\n \\end{cases}\n ```\n \n Costruisco allora una matrice con `j` su un asse e `w` sull'altro. \n Riempio le caselle con il valore di `K(j, w)`. \n Nella casella con `K(j, w)` avremo la soluzione ottima.\n \n Il tempo necessario per riempire tutte le caselle è nuovamente `O(n * w)`, ancora **pseudopolinomiale**.\n \n Per sapere che oggetti ho messo o no devo tenere traccia in qualche modo della catena del calcolo, usando, ad esempio, una pila.\n ")),n(o.p,null,n(o.n,null,"\n # Problemi intrattabili\n \n ## Problema di Set-Cover\n \n ### Input\n \n `U`niverso di `e`lementi\n \n `S`ottoinsieme di `s`ottoinsiemi di elementi di `U`\n \n ### Output\n \n Il minimo `S'`ottoinsieme di `s`ottoinsiemi che copra completamente `U`.\n \n ### Soluzione in `O(n^d)`\n \n Non c'è.\n \n ### Non-soluzione alternativa\n \n Faccio una scelta greedy, ma non posso dimostrare in alcun modo che la soluzione ottenuta sia quella ottima. \n Infatti, l'algoritmo non dà sempre la soluzione ottima, ma dà una soluzione accettabile in tempo polinomiale.\n \n Seleziono sempre il sottoinsieme che copre più elementi mancanti possibili.\n \n #### Costo computazionale\n \n `Costo greedy <= log(numero_elementi) * Costo ottimo`\n ")))}}.call(this,i("hosL").h)}}]); 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estraggo una pallina da una urna casuale ad ogni passo e la sposto nell'altra; con tante palline il sistema appare irreversibile."))),i(n.o,{title:"Fasi"},i("p",null,i("b",null,"Stati")," in cui si può trovare un sistema dinamico."),i("p",null,"Tutte insieme formano lo ",i("i",null,"(iper)",i("b",null,"spazio delle fasi")),".")),i(n.o,{title:"Attrattore"},i("p",null,"Lo ",i("b",null,"stato finale")," di un sistema dinamico."),i("p",null,"Tutte le fasi tendono a uno specifico attrattore.")),i(n.o,{title:"Elaborazione di informazione"},i("p",null,"I sistemi dinamici ",i("b",null,"elaborano informazione")," attraversando fasi e raggiungendo un attrattore."),i(o.a,null,"L'evoluzione biologica crea nuove specie partendo da quelle precedenti di maggiore successo fino a quando non si raggiunge la specie perfetta."),i(o.a,null,"Si può vedere l'universo come un gigantesco sistema dinamico. ",i("a",{href:"https://it.wikipedia.org/wiki/Ipotesi_della_simulazione"},"Che sia artificiale?")," ",i("a",{href:"https://it.wikipedia.org/wiki/Morte_termica_dell%27universo"},"Qual è il suo attrattore?")))),i(n.p,{title:"Sistemi lineari"},i(n.o,{title:"Cosa sono?"},i("p",null,"Sistemi dinamici i cui cambiamenti sono descritti da ",i("b",null,"funzioni lineari"),".")),i(n.o,{title:"Nodi"},i("p",null,i("b",null,"Situazioni iniziali")," di un sistema lineare."),i("p",null,"Possono essere:"),i("ul",null,i("li",null,i("u",null,"Stabili"),": ",i("b",null,"convergono")," ad un punto fisso"),i("li",null,i("u",null,"Instabili"),": ",i("b",null,"divergono")," da un punto fisso"),i("li",null,i("a",{href:"https://it.wikipedia.org/wiki/Punto_di_sella"},i("u",null,"Di sella")))),i("p",null,"Nell'insieme dei ",i(n.g,null,K(a||(a=X`\mathbb{C}`)))," possono anche dare origine a:"),i("ul",null,i("li",null,i("u",null,"Spirali stabili"),": spirali che ",i("b",null,"convergono")),i("li",null,i("u",null,"Spirali instabili"),": spirali che ",i("b",null,"divergono")),i("li",null,i("u",null,"Cicli"),": il sistema forma un ciclo diverso in base alla posizione del nodo"),i("li",null,i("u",null,"Cicli limite"),": il sistema evolve fino a formare un ciclo specifico")),i("p",null,"Infine, in sistemi dissipativi può anche comparire:"),i("ul",null,i("li",null,i("u",null,"Caos"),": il sistema evolve in maniera pseudo-casuale")),i(o.a,null,"Mai sentito parlare di ",i("a",{href:"https://en.wikipedia.org/wiki/Mersenne_Twister"},"Mersenne Twister"),"?")),i(n.o,{title:"Potenziale"},i("p",null,"Funzione che rappresenta lo stato attuale del sistema."),i("p",null,"Gli attrattori coincidono con i suoi ",i("b",null,"punti di minimo"),", detti ",i("i",null,"punti fissi"),"."),i("p",null,"Il suo complementare è la ",i("b",null,"funzione energia"),"."))),i(n.p,{title:"Agenti"},i(n.o,{title:"Secondo il paradigma debole"},i("p",null,"Sono ",i("b",null,"sistemi")," con le seguenti caratteristiche:"),i("ul",null,i("li",null,i("b",null,"Autonomia"),": agiscono gli uni indipendentemente dagli altri"),i("li",null,i("b",null,"Reattività"),": percepiscono ciò che sta nel loro ambiente e ",i("i",null,"reagiscono")," ai cambiamenti di quest'ultimo"),i("li",null,i("b",null,"Proattività"),": agiscono in maniera tale da portare a termine i loro obiettivi"),i("li",null,i("b",null,"Socialità"),": comunicano con gli altri agenti, scambiando informazioni"))),i(n.o,{title:"Secondo il paradigma forte"},i("p",null,"Hanno anche caratteristiche di ",i("b",null,"livello più alto")," derivate dalle quattro precedenti:"),i("ul",null,i("li",null,"Conoscenza"),i("li",null,"Intenzioni"),i("li",null,"Emozioni"),i("li",null,"Obblighi"),i("li",null,"Obiettivi"),i("li",null,"etc...")),i(o.a,null,"Gli umani possono benissimo essere considerati agenti del sistema universo.")),i(n.o,{title:"Caratteristiche aggiuntive"},i("p",null,"Gli agenti si distinguono anche in:"),i("ul",null,i("li",null,i("b",null,"Mobilità"),": quanto e come possono muoversi nell'ambiente"),i("li",null,i("b",null,"Veridicità"),": quanto producono informazioni corrette",i(o.a,null,"È possibile effettuare un attacco a un sistema introducendovi agenti maliziosi che producono intenzionalmente informazioni sbagliate!")),i("li",null,i("b",null,"Benevolenza"),": quanto beneficiano gli altri delle loro azioni",i(o.a,null,i("u",null,"Agenti malevoli"),": ad esempio, troll in siti web",i("br",null),i("u",null,"Agenti benevoli"),": ad esempio, filtri che bannano i troll")),i("li",null,i("b",null,"Razionalità"),": quanto le loro azioni sono coerenti con i loro obiettivi e lo stato dell'ambiente",i(o.a,null,i("u",null,"Razionalità limitata"),": gli agenti non conoscono completamente l'ambiente, e compiono le azioni che suppongono essere giuste"))))),i(n.p,{title:"Benefici degli agenti"},i(n.o,{title:"Emergenza"},i("p",null,"Lo sviluppo negli agenti di nuove capacità per cui non erano stati programmati."),i(o.a,null,"Ad esempio, la Swarm Intelligence, descritta dopo!"))),i(n.p,{title:"Architetture di agente"},i(n.o,{title:"Classe"},i("p",null,"Classificazione in base a ",i("b",null,"come prende le decisioni")," un agente:"),i("ul",null,i("li",null,i("b",null,"Logic-based"),": prende le decisioni attraverso deduzioni ",i("i",null,"logiche")),i("li",null,i("b",null,"Reactive"),": mappa una ",i("i",null,"reazione")," a ogni situazione dell'ambiente"),i("li",null,i("b",null,"Belief-desire-intention"),": per decidere, considera le proprie ",i("i",null,"assunzioni sul mondo")," (",i("i",null,"belief"),"), i propri ",i("i",null,"desideri")," (",i("i",null,"desire"),") e le sue ",i("i",null,"intenzioni")," correnti (",i("i",null,"intention"),")"),i("li",null,i("b",null,"Layered"),": utilizza diversi ",i("i",null,"strati")," di capacità cognitive per giungere a una decisione"))),i(n.o,{title:"Comportamento"},i("p",null,"Classificazione in base a ",i("b",null,"come sono definiti gli obiettivi")," di un agente:"),i("ul",null,i("li",null,i("b",null,"Teleonomico"),": gli obiettivi sono predefiniti ed espliciti"),i("li",null,i("b",null,"Riflessivo"),": l'agente è libero di scegliere il suo obiettivo in base alle proprie percezioni interne"))),i(n.o,{title:"Conoscenze"},i("p",null,"Classificazione in base a ",i("b",null,"quanto conosce dell'ambiente")," un agente:"),i("ul",null,i("li",null,i("b",null,"Cognitivo"),": l'agente è immediatamente a conoscenza di ",i("b",null,"tutto l'ambiente")),i("li",null,i("b",null,"Reattivo"),": l'agente deve scoprire l'ambiente con le sue capacità sensoriali")))),i(n.p,{title:"Sistemi multi-agente"},i(n.o,{title:"Vantaggi"},i("ul",null,i("li",null,i("b",null,"Distribuzione"),": più agenti possono coprire aree di ambiente più vaste, o elaborare più in fretta zone più dense di informazione"),i("li",null,i("b",null,"Rappresentazione"),": i sistemi multi-agente modellano più accuratamente il mondo reale"))),i(n.o,{title:"Feedback"},i("p",null,"Influenza esercitata dal sistema sugli agenti per guidarli verso il loro obiettivo."),i("p",null,"Può essere:"),i("ul",null,i("li",null,i("b",{class:"lime"},"Positivo"),": incentiva gli agenti ad avere un dato comportamento"),i("li",null,i("b",{class:"red"},"Negativo"),": disincentiva gli agenti ad avere un dato comportamento"))),i(n.o,{title:"Swarm intelligence"},i("p",null,"Comportamento ",i("b",null,"emergente")," che si manifesta nei sistemi multiagente con tantissimi agenti."),i("p",null,"Indica la capacità di risoluzione di problemi complessi attraverso la collaborazione di più agenti semplici."))),i(n.p,{title:"Evoluzione"},i(n.o,{title:"Algoritmi genetici"},i("p",null,"Meccanismi simili a quelli evolutivi umani che permettono ai tratti degli agenti di ",i("b",null,"convergere")," verso un valore.")),i(n.o,{title:"Fitness"},i("p",null,"Inizialmente definita come ",i("b",null,"numero di discendenti fertili"),", solitamente indica quanto è probabile che i tratti di un individuo siano passati alla generazione successiva.")),i(n.o,{title:"Cromosoma"},i("p",null,i("b",null,"Sequenza di valori")," che definisce uno o più tratti di un individuo.")),i(n.o,{title:"Popolazione"},i("p",null,"Un ",i("b",null,"insieme di individui")," aventi tutti gli stessi cromosomi.")),i(n.o,{title:"Mutazione"},i("p",null,"Fenomeno che causa una ",i("b",null,"piccola variazione casuale nei cromosomi")," dei figli."),i("p",null,"Previene la ",i("b",null,"convergenza prematura")," in un sistema.")),i(n.o,{title:"Crossover"},i("p",null,"Meccanismo di ",i("b",null,"costruzione dei cromosomi")," in un figlio: i cromosomi dei genitori vengono tagliati nello stesso punto scelto a caso, e per costruire quelli del figlio viene presa una parte dal padre e l'altra parte dalla madre."),i("p",null,"Può portare al miglioramento di un individuo e allo sviluppo di nuovi tratti, ma solo nelle parti di cromosoma che sono diverse tra i due genitori.")),i(n.o,{title:"Schema"},i("p",null,i("b",null,"Sequenza di valori")," all'interno di un cromosoma, che può includere anche sezioni in cui il valore è ",i("b",null,"irrilevante"),"."),i("p",null,"Gli algoritmi genetici permettono di trovare gli schemi con la ",i("b",null,"fitness più alta in assoluto")," in un tempo relativamente breve: il sistema ",i("i",null,"generalmente")," favorisce gli schemi corti con fitness alta.")),i(n.o,{title:"Convergenza prematura"},i("p",null,"Situazione in cui si è raggiunta una soluzione non-ottimale a causa dell'assenza di novità nel sistema."),i("p",null,"Si può impedire con vari metodi: con la ",i("b",null,"mutazione"),", introducendo ",i("b",null,"requisiti di località")," per l'accoppiamento, scegliendo diversamente i genitori, etc..."))),i(n.p,{title:"Sistema a classificatori"},i(n.o,{title:"Cosa sono?"},i("p",null,"Programmi che dati tanti esempi sono in grado di classificare un elemento in una o più categorie."),i("p",null,"Sono formati da ",i("i",null,"classificatori"),", liste di ",i("i",null,"messaggi"),", ",i("i",null,"detettori")," e ",i("i",null,"effettori"),"."))),i(n.p,null,i(n.o,{title:"Classificatori"},i("p",null,"Strutture logiche che ",i("b",null,"elaborano")," i messaggi."),i("p",null,"Valutano una espressione logica (",i("i",null,"condizione"),") sui messaggi in arrivo, e se questa risulta essere vera, emettono un nuovo messaggio in risposta (",i("i",null,"azione"),")."),i(o.a,null,"Condizione e azione possono essere considerati come due cromosomi di un algoritmo genetico!")),i(n.o,{title:"Messaggi"},i("p",null,i("b",null,"Unità di informazione")," di un sistema a classificatori: sono ",i("b",null,"generati")," da ",i("i",null,"detettori")," e ",i("i",null,"classificatori"),", e ",i("b",null,"consumati")," da ",i("i",null,"classificatori")," ed ",i("i",null,"effettori"),".")),i(n.o,{title:"Detettori"},i("p",null,i("b",null,"Sensori")," che percepiscono lo stato dell'ambiente esterno e lo riportano sotto forma di ",i("i",null,"messaggi"),".")),i(n.o,{title:"Effettori"},i("p",null,i("b",null,"Motori")," che rispondono ai ",i("i",null,"messaggi")," effettuando una qualche azione nell'ambiente."))),i(n.p,null,i(n.o,{title:"Forza"},i("p",null,"Un ",i("b",null,"punteggio")," associato ad ogni classificatore."),i("p",null,"Più un classificatore viene attivato, più la sua forza crescerà."),i(o.a,null,"I classificatori più deboli vengono lentamente eliminati!")),i(n.o,{title:"Specificità"},i("p",null,"Il ",i("b",null,"numero di condizioni che devono essere soddisfatte")," perchè il classificatore si attivi."),i("p",null)),i(n.o,{title:"Bid"},i("p",null,i("b",null,"Prodotto")," di specificità e forza di un classificatore."),i("p",null,"Rappresenta ",i("b",null,"quanto è probabile che venga utilizzato")," un dato classificatore nel caso che le condizioni di più di uno vengano soddisfatte."),i(o.a,null,"È la fitness degli algoritmi genetici applicata ai classificatori."))),i(n.p,null,i(n.o,{title:"Cover Detector"},i("p",null,"Se l'input non soddisfa nessun classificatore esistente, se ne crea uno nuovo soddisfatto dall'input attuale con una azione casuale.")),i(n.o,{title:"Cover Effector"},i("p",null,"Se i classificatori emettono in output un messaggio non valido, si crea un nuovo classificatore che trasforma quel messaggio in un output valido."))),i(n.p,{title:"Reti neurali"},i(n.o,{title:"Neuroni"},i("p",null,"Agenti che possono ",i("b",null,"collegarsi tra loro")," tramite ",i("i",null,"sinapsi")," (dirette) e ",i("b",null,"ricevere")," ed ",i("b",null,"emettere")," ",i("i",null,"impulsi")," lungo di esse."),i("p",null,"Gli impulsi ricevuti vengono temporaneamente ",i("b",null,"memorizzati")," dal neurone attraverso valori che decadono nel tempo."),i("p",null,"Se la somma dei valori di tutti gli impulsi ricevuti è ",i("b",null,"maggiore di una certa soglia"),", allora il neurone ",i("b",null,"emetterà")," un impulso."))),i(n.p,{title:"Modello booleano"},i(n.o,{title:"Cos'è?"},i("p",null,"Un modello semplificato di rete neurale in cui vengono considerati ",i("b",null,"solo tempi discreti")," (",i(t.a,null,"ticks"),"), e non è presente la memorizzazione degli impulsi nel tempo."),i("p",null,"È stato sviluppato da ",i("a",{href:"https://it.wikipedia.org/wiki/Warren_McCulloch"},"Warren McCulloch")," (un neurofisiologo) e ",i("a",{href:"https://it.wikipedia.org/wiki/Walter_Pitts"},"Walter Pitts")," (un matematico)."),i(o.a,null,"È importante perchè dimostra che le reti neurali ",i("b",null,"possono elaborare qualsiasi cosa"),", ma incompleto perchè non descrive nessun metodo per la loro creazione automatica.")),i(n.o,{title:"Neuroni"},i("p",null,"I neuroni ",i("b",null,"si attivano")," in un dato ",i(t.a,null)," se la ",i("b",null,"somma dei loro impulsi")," nel ",i(t.a,null)," precedente è ",i("b",null,"maggiore o uguale a 1"),".")),i(n.o,{title:"Intensità sinaptica"},i("p",null,"Le sinapsi hanno una ",i("i",null,"intensità"),": è un ",i("b",null,"moltiplicatore")," che viene applicato a tutti gli impulsi transitanti la sinapsi."))),i(n.p,{title:"Funzioni logiche nel modello booleano"},i(n.o,{title:"NOT"},i("p",null,"Un neurone con una sinapsi entrante con intensità ",i(n.g,null,K(u||(u=X`-1`))),".")),i(n.o,{title:"OR"},i("p",null,"Un neurone con due o più sinapsi entranti con intensità ",i(n.g,null,K(s||(s=X`1`))),".")),i(n.o,{title:"AND"},i("p",null,"Un neurone con due o più sinapsi entranti con intensità ",i(n.g,null,K(r||(r=X`\frac{1}{numero\ sinapsi}`))),"."))),i(n.p,{title:"Modello di Hopfield"},i(n.o,{title:"Cos'è?"},i("p",null,"Un'estensione del modello booleano per permettere l'apprendimento automatico delle configurazioni giuste di neuroni."),i("p",null,"È stato sviluppato da ",i("a",{href:"https://en.wikipedia.org/wiki/John_Hopfield"},"John Hopfield")," (uno scienziato)."),i(o.a,null,"Non è molto avanzato, ma ha portato a ulteriori studi nel campo delle reti neurali.")),i(n.q,null,i("thead",null,i("tr",null,i("th",null,i("abbr",{title:"Vettore / matrice"},"v")),i("th",null,i("abbr",{title:"Elemento singolo / scalare"},"s")),i("th",null,"Glossario"))),i("tbody",null,i("tr",null,i("td",null),i("td",null,i(n.a,null,K(c||(c=X`t`)))),i("td",null,i(t.a,null,"Tick")," attuale")),i("tr",null,i("td",null),i("td",null,i(n.g,null,K(p||(p=X`n`))),", ",i(n.g,null,K(d||(d=X`m`)))),i("td",null,"Identificatore di un neurone specifico")),i("tr",null,i("td",null,i(n.a,null,K(m||(m=X`\mathbf{W}`)))),i("td",null,i(n.a,null,K(g||(g=X`w_{nm}`)))),i("td",null,"Intensità della sinapsi diretta da ",i(n.g,null,K(b||(b=X`n`)))," verso ",i(n.g,null,K(f||(f=X`m`))))),i("tr",null,i("td",null,i(n.a,null,K(v||(v=X`\mathbf{\Theta}`)))),i("td",null,i(n.a,null,K(h||(h=X`\theta_n`)))),i("td",null,"Soglia di attivazione di un neurone")),i("tr",null,i("td",null,i(n.a,null,K(z||(z=X`\mathbf{X}(t)`)))),i("td",null,i(n.a,null,K(_||(_=X`x_n(t)`)))),i("td",null,"Emissione di un neurone")),i("tr",null,i("td",null,i(n.a,null,K(w||(w=X`\mathbf{I}(t)`)))),i("td",null,i(n.a,null,K(S||(S=X`i_n(t)`)))),i("td",null,"Somma degli ingressi di un neurone")),i("tr",null,i("td",null),i("td",null,i(n.a,null,K(k||(k=X`E`)))),i("td",null,"Energia del sistema")),i("tr",null,i("td",null,i(n.a,null,K(q||(q=X`\mathbf{A}`)))),i("td",null,i(n.a,null,K(C||(C=X`a_i`)))),i("td",null,"Stato di un neurone in un pattern")),i("tr",null,i("td",null),i("td",null,i(n.a,null,K(A||(A=X`Q(\mathbf{A}, \mathbf{B})`)))),i("td",null,"Sovrapposizione tra due pattern"))))),i(n.p,null,i(n.o,{title:"Emissione"},i("p",null,"In ogni ",i(t.a,null),", i neuroni:"),i("ul",null,i("li",null,"Emettono ",i(n.g,null,K(I||(I=X`0`)))," se gli input nel ",i(t.a,null)," precedente ",i("b",null,"erano inferiori")," alla soglia di attivazione"),i("li",null,"Emettono ",i(n.g,null,K(N||(N=X`1`)))," se gli input nel ",i(t.a,null)," precedente ",i("b",null,"superavano")," la soglia di attivazione"),i("li",null,"Non cambiano stato se gli input nel ",i(t.a,null)," precedente ",i("b",null,"erano uguali")," alla soglia di attivazione"))),i(n.o,{title:"Sinapsi"},i("p",null,i("b",null,"Tutti")," i neuroni del modello sono intercollegati tra loro da sinapsi."),i("p",null,"I neuroni non possono essere collegati a loro stessi."),i("p",null,"Questo porta il ",i("b",null,"costo computazionale")," del modello ad essere ",i(n.g,null,K(E||(E=X`O(n^2)`))),".")),i(n.o,{title:"Energia"},i("p",null,"Una funzione dell'intero sistema che rappresenta il totale degli stati di tutti i neuroni e tutte le connessioni."),i(n.n,null,K(P||(P=X` +(window.webpackJsonp=window.webpackJsonp||[]).push([[3],{"2w3n":function(i){i.exports={red:"red__2y1B_",orange:"orange__dD2kx",yellow:"yellow__OEpwl",lime:"lime__CVe41",cyan:"cyan__26ZAg",blue:"blue__LO7Xm",magenta:"magenta__1Akee",example:"example__2PzAa"}},"5rUc":function(i,l,e){"use strict";(function(i){var n=e("mDT8"),t=e.n(n);l.a=function(l){return 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minuti"))),i(n.p,{title:"Appelli"},i("ol",null,i("li",null,i(n.s,{to:"2020-06-23"})),i("li",null,i(n.s,{to:"2020-07-08"})),i("li",null,i(n.s,{to:"2020-07-22"}))))),i(n.q,{title:"NetLogo"},i(n.p,{title:"Cos'è?"},i("p",null,"NetLogo è un software di modellazione sistemi multiagente, da noi usato per le lezioni di laboratorio."),i("p",null,"Si può ",i("a",{href:"https://ccl.northwestern.edu/netlogo/download.shtml"},"scaricare")," o ",i("a",{href:"https://www.netlogoweb.org/launch"},"usare da browser"),"."),i("p",null,"Il suo codice sorgente è disponibile su ",i("a",{href:"https://github.com/NetLogo/NetLogo"},"GitHub"),", e ha una pagina di ",i("a",{href:"https://ccl.northwestern.edu/netlogo/docs/dictionary.html"},"documentazione"),"."),i("p",null,i("b",{style:"font-size: x-large;"},i("a",{href:"/apprendimento/netlogo"},"Consulta i miei appunti su NetLogo qui!"))))),i(n.q,{title:"Sistemi dinamici"},i(n.p,{title:"Cosa sono?"},i("p",null,"Sistemi naturali o artificiali che si basano su ",i("b",null,"leggi reversibili e deterministiche"),"."),i("p",null,"In natura, alcuni leggi possono sembrare irreversibili a livello macroscopico, ma sono in realtà reversibili a livello microscopico."),i(t.a,null,i("p",null,i("a",{href:"https://it.wikipedia.org/wiki/Modello_di_Ehrenfest"},i("u",null,"Urne di Ehrenfest")),": due urne con N palline; estraggo una pallina da una urna casuale ad ogni passo e la sposto nell'altra; con tante palline il sistema appare irreversibile."))),i(n.p,{title:"Fasi"},i("p",null,i("b",null,"Stati")," in cui si può trovare un sistema dinamico."),i("p",null,"Tutte insieme formano lo ",i("i",null,"(iper)",i("b",null,"spazio delle fasi")),".")),i(n.p,{title:"Attrattore"},i("p",null,"Lo ",i("b",null,"stato finale")," di un sistema dinamico."),i("p",null,"Tutte le fasi tendono a uno specifico attrattore.")),i(n.p,{title:"Elaborazione di informazione"},i("p",null,"I sistemi dinamici ",i("b",null,"elaborano informazione")," attraversando fasi e raggiungendo un attrattore."),i(t.a,null,"L'evoluzione biologica crea nuove specie partendo da quelle precedenti di maggiore successo fino a quando non si raggiunge la specie perfetta."),i(t.a,null,"Si può vedere l'universo come un gigantesco sistema dinamico. ",i("a",{href:"https://it.wikipedia.org/wiki/Ipotesi_della_simulazione"},"Che sia artificiale?")," ",i("a",{href:"https://it.wikipedia.org/wiki/Morte_termica_dell%27universo"},"Qual è il suo attrattore?")))),i(n.q,{title:"Sistemi lineari"},i(n.p,{title:"Cosa sono?"},i("p",null,"Sistemi dinamici i cui cambiamenti sono descritti da ",i("b",null,"funzioni lineari"),".")),i(n.p,{title:"Nodi"},i("p",null,i("b",null,"Situazioni iniziali")," di un sistema lineare."),i("p",null,"Possono essere:"),i("ul",null,i("li",null,i("u",null,"Stabili"),": ",i("b",null,"convergono")," ad un punto fisso"),i("li",null,i("u",null,"Instabili"),": ",i("b",null,"divergono")," da un punto fisso"),i("li",null,i("a",{href:"https://it.wikipedia.org/wiki/Punto_di_sella"},i("u",null,"Di sella")))),i("p",null,"Nell'insieme dei ",i(n.h,null,K(a||(a=X`\mathbb{C}`)))," possono anche dare origine a:"),i("ul",null,i("li",null,i("u",null,"Spirali stabili"),": spirali che ",i("b",null,"convergono")),i("li",null,i("u",null,"Spirali instabili"),": spirali che ",i("b",null,"divergono")),i("li",null,i("u",null,"Cicli"),": il sistema forma un ciclo diverso in base alla posizione del nodo"),i("li",null,i("u",null,"Cicli limite"),": il sistema evolve fino a formare un ciclo specifico")),i("p",null,"Infine, in sistemi dissipativi può anche comparire:"),i("ul",null,i("li",null,i("u",null,"Caos"),": il sistema evolve in maniera pseudo-casuale")),i(t.a,null,"Mai sentito parlare di ",i("a",{href:"https://en.wikipedia.org/wiki/Mersenne_Twister"},"Mersenne Twister"),"?")),i(n.p,{title:"Potenziale"},i("p",null,"Funzione che rappresenta lo stato attuale del sistema."),i("p",null,"Gli attrattori coincidono con i suoi ",i("b",null,"punti di minimo"),", detti ",i("i",null,"punti fissi"),"."),i("p",null,"Il suo complementare è la ",i("b",null,"funzione energia"),"."))),i(n.q,{title:"Agenti"},i(n.p,{title:"Secondo il paradigma debole"},i("p",null,"Sono ",i("b",null,"sistemi")," con le seguenti caratteristiche:"),i("ul",null,i("li",null,i("b",null,"Autonomia"),": agiscono gli uni indipendentemente dagli altri"),i("li",null,i("b",null,"Reattività"),": percepiscono ciò che sta nel loro ambiente e ",i("i",null,"reagiscono")," ai cambiamenti di quest'ultimo"),i("li",null,i("b",null,"Proattività"),": agiscono in maniera tale da portare a termine i loro obiettivi"),i("li",null,i("b",null,"Socialità"),": comunicano con gli altri agenti, scambiando informazioni"))),i(n.p,{title:"Secondo il paradigma forte"},i("p",null,"Hanno anche caratteristiche di ",i("b",null,"livello più alto")," derivate dalle quattro precedenti:"),i("ul",null,i("li",null,"Conoscenza"),i("li",null,"Intenzioni"),i("li",null,"Emozioni"),i("li",null,"Obblighi"),i("li",null,"Obiettivi"),i("li",null,"etc...")),i(t.a,null,"Gli umani possono benissimo essere considerati agenti del sistema universo.")),i(n.p,{title:"Caratteristiche aggiuntive"},i("p",null,"Gli agenti si distinguono anche in:"),i("ul",null,i("li",null,i("b",null,"Mobilità"),": quanto e come possono muoversi nell'ambiente"),i("li",null,i("b",null,"Veridicità"),": quanto producono informazioni corrette",i(t.a,null,"È possibile effettuare un attacco a un sistema introducendovi agenti maliziosi che producono intenzionalmente informazioni sbagliate!")),i("li",null,i("b",null,"Benevolenza"),": quanto beneficiano gli altri delle loro azioni",i(t.a,null,i("u",null,"Agenti malevoli"),": ad esempio, troll in siti web",i("br",null),i("u",null,"Agenti benevoli"),": ad esempio, filtri che bannano i troll")),i("li",null,i("b",null,"Razionalità"),": quanto le loro azioni sono coerenti con i loro obiettivi e lo stato dell'ambiente",i(t.a,null,i("u",null,"Razionalità limitata"),": gli agenti non conoscono completamente l'ambiente, e compiono le azioni che suppongono essere giuste"))))),i(n.q,{title:"Benefici degli agenti"},i(n.p,{title:"Emergenza"},i("p",null,"Lo sviluppo negli agenti di nuove capacità per cui non erano stati programmati."),i(t.a,null,"Ad esempio, la Swarm Intelligence, descritta dopo!"))),i(n.q,{title:"Architetture di agente"},i(n.p,{title:"Classe"},i("p",null,"Classificazione in base a ",i("b",null,"come prende le decisioni")," un agente:"),i("ul",null,i("li",null,i("b",null,"Logic-based"),": prende le decisioni attraverso deduzioni ",i("i",null,"logiche")),i("li",null,i("b",null,"Reactive"),": mappa una ",i("i",null,"reazione")," a ogni situazione dell'ambiente"),i("li",null,i("b",null,"Belief-desire-intention"),": per decidere, considera le proprie ",i("i",null,"assunzioni sul mondo")," (",i("i",null,"belief"),"), i propri ",i("i",null,"desideri")," (",i("i",null,"desire"),") e le sue ",i("i",null,"intenzioni")," correnti (",i("i",null,"intention"),")"),i("li",null,i("b",null,"Layered"),": utilizza diversi ",i("i",null,"strati")," di capacità cognitive per giungere a una decisione"))),i(n.p,{title:"Comportamento"},i("p",null,"Classificazione in base a ",i("b",null,"come sono definiti gli obiettivi")," di un agente:"),i("ul",null,i("li",null,i("b",null,"Teleonomico"),": gli obiettivi sono predefiniti ed espliciti"),i("li",null,i("b",null,"Riflessivo"),": l'agente è libero di scegliere il suo obiettivo in base alle proprie percezioni interne"))),i(n.p,{title:"Conoscenze"},i("p",null,"Classificazione in base a ",i("b",null,"quanto conosce dell'ambiente")," un agente:"),i("ul",null,i("li",null,i("b",null,"Cognitivo"),": l'agente è immediatamente a conoscenza di ",i("b",null,"tutto l'ambiente")),i("li",null,i("b",null,"Reattivo"),": l'agente deve scoprire l'ambiente con le sue capacità sensoriali")))),i(n.q,{title:"Sistemi multi-agente"},i(n.p,{title:"Vantaggi"},i("ul",null,i("li",null,i("b",null,"Distribuzione"),": più agenti possono coprire aree di ambiente più vaste, o elaborare più in fretta zone più dense di informazione"),i("li",null,i("b",null,"Rappresentazione"),": i sistemi multi-agente modellano più accuratamente il mondo reale"))),i(n.p,{title:"Feedback"},i("p",null,"Influenza esercitata dal sistema sugli agenti per guidarli verso il loro obiettivo."),i("p",null,"Può essere:"),i("ul",null,i("li",null,i("b",{class:"lime"},"Positivo"),": incentiva gli agenti ad avere un dato comportamento"),i("li",null,i("b",{class:"red"},"Negativo"),": disincentiva gli agenti ad avere un dato comportamento"))),i(n.p,{title:"Swarm intelligence"},i("p",null,"Comportamento ",i("b",null,"emergente")," che si manifesta nei sistemi multiagente con tantissimi agenti."),i("p",null,"Indica la capacità di risoluzione di problemi complessi attraverso la collaborazione di più agenti semplici."))),i(n.q,{title:"Evoluzione"},i(n.p,{title:"Algoritmi genetici"},i("p",null,"Meccanismi simili a quelli evolutivi umani che permettono ai tratti degli agenti di ",i("b",null,"convergere")," verso un valore.")),i(n.p,{title:"Fitness"},i("p",null,"Inizialmente definita come ",i("b",null,"numero di discendenti fertili"),", solitamente indica quanto è probabile che i tratti di un individuo siano passati alla generazione successiva.")),i(n.p,{title:"Cromosoma"},i("p",null,i("b",null,"Sequenza di valori")," che definisce uno o più tratti di un individuo.")),i(n.p,{title:"Popolazione"},i("p",null,"Un ",i("b",null,"insieme di individui")," aventi tutti gli stessi cromosomi.")),i(n.p,{title:"Mutazione"},i("p",null,"Fenomeno che causa una ",i("b",null,"piccola variazione casuale nei cromosomi")," dei figli."),i("p",null,"Previene la ",i("b",null,"convergenza prematura")," in un sistema.")),i(n.p,{title:"Crossover"},i("p",null,"Meccanismo di ",i("b",null,"costruzione dei cromosomi")," in un figlio: i cromosomi dei genitori vengono tagliati nello stesso punto scelto a caso, e per costruire quelli del figlio viene presa una parte dal padre e l'altra parte dalla madre."),i("p",null,"Può portare al miglioramento di un individuo e allo sviluppo di nuovi tratti, ma solo nelle parti di cromosoma che sono diverse tra i due genitori.")),i(n.p,{title:"Schema"},i("p",null,i("b",null,"Sequenza di valori")," all'interno di un cromosoma, che può includere anche sezioni in cui il valore è ",i("b",null,"irrilevante"),"."),i("p",null,"Gli algoritmi genetici permettono di trovare gli schemi con la ",i("b",null,"fitness più alta in assoluto")," in un tempo relativamente breve: il sistema ",i("i",null,"generalmente")," favorisce gli schemi corti con fitness alta.")),i(n.p,{title:"Convergenza prematura"},i("p",null,"Situazione in cui si è raggiunta una soluzione non-ottimale a causa dell'assenza di novità nel sistema."),i("p",null,"Si può impedire con vari metodi: con la ",i("b",null,"mutazione"),", introducendo ",i("b",null,"requisiti di località")," per l'accoppiamento, scegliendo diversamente i genitori, etc..."))),i(n.q,{title:"Sistema a classificatori"},i(n.p,{title:"Cosa sono?"},i("p",null,"Programmi che dati tanti esempi sono in grado di classificare un elemento in una o più categorie."),i("p",null,"Sono formati da ",i("i",null,"classificatori"),", liste di ",i("i",null,"messaggi"),", ",i("i",null,"detettori")," e ",i("i",null,"effettori"),"."))),i(n.q,null,i(n.p,{title:"Classificatori"},i("p",null,"Strutture logiche che ",i("b",null,"elaborano")," i messaggi."),i("p",null,"Valutano una espressione logica (",i("i",null,"condizione"),") sui messaggi in arrivo, e se questa risulta essere vera, emettono un nuovo messaggio in risposta (",i("i",null,"azione"),")."),i(t.a,null,"Condizione e azione possono essere considerati come due cromosomi di un algoritmo genetico!")),i(n.p,{title:"Messaggi"},i("p",null,i("b",null,"Unità di informazione")," di un sistema a classificatori: sono ",i("b",null,"generati")," da ",i("i",null,"detettori")," e ",i("i",null,"classificatori"),", e ",i("b",null,"consumati")," da ",i("i",null,"classificatori")," ed ",i("i",null,"effettori"),".")),i(n.p,{title:"Detettori"},i("p",null,i("b",null,"Sensori")," che percepiscono lo stato dell'ambiente esterno e lo riportano sotto forma di ",i("i",null,"messaggi"),".")),i(n.p,{title:"Effettori"},i("p",null,i("b",null,"Motori")," che rispondono ai ",i("i",null,"messaggi")," effettuando una qualche azione nell'ambiente."))),i(n.q,null,i(n.p,{title:"Forza"},i("p",null,"Un ",i("b",null,"punteggio")," associato ad ogni classificatore."),i("p",null,"Più un classificatore viene attivato, più la sua forza crescerà."),i(t.a,null,"I classificatori più deboli vengono lentamente eliminati!")),i(n.p,{title:"Specificità"},i("p",null,"Il ",i("b",null,"numero di condizioni che devono essere soddisfatte")," perchè il classificatore si attivi."),i("p",null)),i(n.p,{title:"Bid"},i("p",null,i("b",null,"Prodotto")," di specificità e forza di un classificatore."),i("p",null,"Rappresenta ",i("b",null,"quanto è probabile che venga utilizzato")," un dato classificatore nel caso che le condizioni di più di uno vengano soddisfatte."),i(t.a,null,"È la fitness degli algoritmi genetici applicata ai classificatori."))),i(n.q,null,i(n.p,{title:"Cover Detector"},i("p",null,"Se l'input non soddisfa nessun classificatore esistente, se ne crea uno nuovo soddisfatto dall'input attuale con una azione casuale.")),i(n.p,{title:"Cover Effector"},i("p",null,"Se i classificatori emettono in output un messaggio non valido, si crea un nuovo classificatore che trasforma quel messaggio in un output valido."))),i(n.q,{title:"Reti neurali"},i(n.p,{title:"Neuroni"},i("p",null,"Agenti che possono ",i("b",null,"collegarsi tra loro")," tramite ",i("i",null,"sinapsi")," (dirette) e ",i("b",null,"ricevere")," ed ",i("b",null,"emettere")," ",i("i",null,"impulsi")," lungo di esse."),i("p",null,"Gli impulsi ricevuti vengono temporaneamente ",i("b",null,"memorizzati")," dal neurone attraverso valori che decadono nel tempo."),i("p",null,"Se la somma dei valori di tutti gli impulsi ricevuti è ",i("b",null,"maggiore di una certa soglia"),", allora il neurone ",i("b",null,"emetterà")," un impulso."))),i(n.q,{title:"Modello booleano"},i(n.p,{title:"Cos'è?"},i("p",null,"Un modello semplificato di rete neurale in cui vengono considerati ",i("b",null,"solo tempi discreti")," (",i(o.a,null,"ticks"),"), e non è presente la memorizzazione degli impulsi nel tempo."),i("p",null,"È stato sviluppato da ",i("a",{href:"https://it.wikipedia.org/wiki/Warren_McCulloch"},"Warren McCulloch")," (un neurofisiologo) e ",i("a",{href:"https://it.wikipedia.org/wiki/Walter_Pitts"},"Walter Pitts")," (un matematico)."),i(t.a,null,"È importante perchè dimostra che le reti neurali ",i("b",null,"possono elaborare qualsiasi cosa"),", ma incompleto perchè non descrive nessun metodo per la loro creazione automatica.")),i(n.p,{title:"Neuroni"},i("p",null,"I neuroni ",i("b",null,"si attivano")," in un dato ",i(o.a,null)," se la ",i("b",null,"somma dei loro impulsi")," nel ",i(o.a,null)," precedente è ",i("b",null,"maggiore o uguale a 1"),".")),i(n.p,{title:"Intensità sinaptica"},i("p",null,"Le sinapsi hanno una ",i("i",null,"intensità"),": è un ",i("b",null,"moltiplicatore")," che viene applicato a tutti gli impulsi transitanti la sinapsi."))),i(n.q,{title:"Funzioni logiche nel modello booleano"},i(n.p,{title:"NOT"},i("p",null,"Un neurone con una sinapsi entrante con intensità ",i(n.h,null,K(u||(u=X`-1`))),".")),i(n.p,{title:"OR"},i("p",null,"Un neurone con due o più sinapsi entranti con intensità ",i(n.h,null,K(s||(s=X`1`))),".")),i(n.p,{title:"AND"},i("p",null,"Un neurone con due o più sinapsi entranti con intensità ",i(n.h,null,K(r||(r=X`\frac{1}{numero\ sinapsi}`))),"."))),i(n.q,{title:"Modello di Hopfield"},i(n.p,{title:"Cos'è?"},i("p",null,"Un'estensione del modello booleano per permettere l'apprendimento automatico delle configurazioni giuste di neuroni."),i("p",null,"È stato sviluppato da ",i("a",{href:"https://en.wikipedia.org/wiki/John_Hopfield"},"John Hopfield")," (uno scienziato)."),i(t.a,null,"Non è molto avanzato, ma ha portato a ulteriori studi nel campo delle reti neurali.")),i(n.r,null,i("thead",null,i("tr",null,i("th",null,i("abbr",{title:"Vettore / matrice"},"v")),i("th",null,i("abbr",{title:"Elemento singolo / scalare"},"s")),i("th",null,"Glossario"))),i("tbody",null,i("tr",null,i("td",null),i("td",null,i(n.a,null,K(p||(p=X`t`)))),i("td",null,i(o.a,null,"Tick")," attuale")),i("tr",null,i("td",null),i("td",null,i(n.h,null,K(c||(c=X`n`))),", ",i(n.h,null,K(d||(d=X`m`)))),i("td",null,"Identificatore di un neurone specifico")),i("tr",null,i("td",null,i(n.a,null,K(m||(m=X`\mathbf{W}`)))),i("td",null,i(n.a,null,K(g||(g=X`w_{nm}`)))),i("td",null,"Intensità della sinapsi diretta da ",i(n.h,null,K(b||(b=X`n`)))," verso ",i(n.h,null,K(f||(f=X`m`))))),i("tr",null,i("td",null,i(n.a,null,K(h||(h=X`\mathbf{\Theta}`)))),i("td",null,i(n.a,null,K(v||(v=X`\theta_n`)))),i("td",null,"Soglia di attivazione di un neurone")),i("tr",null,i("td",null,i(n.a,null,K(z||(z=X`\mathbf{X}(t)`)))),i("td",null,i(n.a,null,K(_||(_=X`x_n(t)`)))),i("td",null,"Emissione di un neurone")),i("tr",null,i("td",null,i(n.a,null,K(q||(q=X`\mathbf{I}(t)`)))),i("td",null,i(n.a,null,K(w||(w=X`i_n(t)`)))),i("td",null,"Somma degli ingressi di un neurone")),i("tr",null,i("td",null),i("td",null,i(n.a,null,K(S||(S=X`E`)))),i("td",null,"Energia del sistema")),i("tr",null,i("td",null,i(n.a,null,K(k||(k=X`\mathbf{A}`)))),i("td",null,i(n.a,null,K(C||(C=X`a_i`)))),i("td",null,"Stato di un neurone in un pattern")),i("tr",null,i("td",null),i("td",null,i(n.a,null,K(A||(A=X`Q(\mathbf{A}, \mathbf{B})`)))),i("td",null,"Sovrapposizione tra due pattern"))))),i(n.q,null,i(n.p,{title:"Emissione"},i("p",null,"In ogni ",i(o.a,null),", i neuroni:"),i("ul",null,i("li",null,"Emettono ",i(n.h,null,K(I||(I=X`0`)))," se gli input nel ",i(o.a,null)," precedente ",i("b",null,"erano inferiori")," alla soglia di attivazione"),i("li",null,"Emettono ",i(n.h,null,K(N||(N=X`1`)))," se gli input nel ",i(o.a,null)," precedente ",i("b",null,"superavano")," la soglia di attivazione"),i("li",null,"Non cambiano stato se gli input nel ",i(o.a,null)," precedente ",i("b",null,"erano uguali")," alla soglia di attivazione"))),i(n.p,{title:"Sinapsi"},i("p",null,i("b",null,"Tutti")," i neuroni del modello sono intercollegati tra loro da sinapsi."),i("p",null,"I neuroni non possono essere collegati a loro stessi."),i("p",null,"Questo porta il ",i("b",null,"costo computazionale")," del modello ad essere ",i(n.h,null,K(E||(E=X`O(n^2)`))),".")),i(n.p,{title:"Energia"},i("p",null,"Una funzione dell'intero sistema che rappresenta il totale degli stati di tutti i neuroni e tutte le connessioni."),i(n.o,null,K(P||(P=X` E = - \frac{1}{2} \sum_{n, m} ( w_{nm} \cdot x_n \cdot x_m ) + \sum_n ( \theta_n \cdot x_n ) - `))))),i(n.p,null,i(n.o,{title:"Apprendimento hebbiano"},i("p",null,"Un metodo per realizzare l'apprendimento nel modello di Hopfield."),i("p",null,"Si incrementa l'intensità delle sinapsi che connettono neuroni nello stesso stato, e invece si decrementa l'intensità di quelle che connettono neuroni in stati opposti."),i("p",null,"Considerando i neuroni spenti e quelli accesi come ",i(n.g,null,K(U||(U=X`0`)))," e ",i(n.g,null,K(x||(x=X`1`)))," rispettivamente, si ha che per ogni pattern:"),i(n.n,null,K(M||(M=X` + `))))),i(n.q,null,i(n.p,{title:"Apprendimento hebbiano"},i("p",null,"Un metodo per realizzare l'apprendimento nel modello di Hopfield."),i("p",null,"Si incrementa l'intensità delle sinapsi che connettono neuroni nello stesso stato, e invece si decrementa l'intensità di quelle che connettono neuroni in stati opposti."),i("p",null,"Considerando i neuroni spenti e quelli accesi come ",i(n.h,null,K(U||(U=X`0`)))," e ",i(n.h,null,K(x||(x=X`1`)))," rispettivamente, si ha che per ogni pattern:"),i(n.o,null,K(M||(M=X` \Delta w_{ik} = (2 \cdot A_i - 1)(2 \cdot A_k - 1) - `))),i(o.a,null,"Così facendo, si insegna sia il pattern normale sia il suo complementare!")),i(n.o,{title:"Simmetria"},i("p",null,"Applicando l'apprendimento hebbiano al modello di Hopfield si ottengono sinapsi simmetriche."),i("p",null,"Se è valida questa proprietà, si può dimostrare che l'",i("b",null,"energia del sistema è sempre decrescente"),", e che quindi che tenderà a un punto fisso!"),i("p",null,i(n.s,null,"TODO: Dopo il prof dimostra la relazione tra input netto e overlap, e che il sistema converge al pattern più simile."))),i(n.o,{title:"Overlap di due pattern"},i("p",null,"Il numero di neuroni attivati in entrambi i pattern."),i(n.n,null,K(D||(D=X` + `))),i(t.a,null,"Così facendo, si insegna sia il pattern normale sia il suo complementare!")),i(n.p,{title:"Simmetria"},i("p",null,"Applicando l'apprendimento hebbiano al modello di Hopfield si ottengono sinapsi simmetriche."),i("p",null,"Se è valida questa proprietà, si può dimostrare che l'",i("b",null,"energia del sistema è sempre decrescente"),", e che quindi che tenderà a un punto fisso!"),i("p",null,i(n.t,null,"TODO: Dopo il prof dimostra la relazione tra input netto e overlap, e che il sistema converge al pattern più simile."))),i(n.p,{title:"Overlap di due pattern"},i("p",null,"Il numero di neuroni attivati in entrambi i pattern."),i(n.o,null,K(D||(D=X` Q(A, B) = \sum_{i = 1}^n A_i B_i - `)))),i(n.o,{title:"Interferenza"},i("p",null,"Più pattern vengono imparati da un modello, più è facile che essi interferiscano tra loro."),i("p",null,"In caso di pattern completamente scorrelati tra loro, il limite di pattern imparabili è circa:"),i(n.n,null,K(L||(L=X`0.14 \cdot N`)))),i(n.o,{title:"Archetipi"},i("p",null,"Per minimizzare l'interferenza tra pattern, è possibile insegnare al modello un ",i("i",null,"archetipo"),": si insegna più volte il pattern originale applicandoci una minima quantità di interferenza casuale."))),i(n.p,{title:"Modello a percettroni"},i(n.o,{title:"Cos'è?"},i("p",null,"Un modello di rete neurale che supporta l'apprendimento e che presenta ",i("b",null,"più strati di neuroni"),"."),i("p",null,"Ha costi computazionali molto più bassi del modello di Hopfield.")),i(n.q,null,i("thead",null,i("tr",null,i("th",null,"Simbolo"),i("th",null,"Descrizione"))),i("tbody",null,i("tr",null,i("td",null,i(n.a,null,K(y||(y=X`N`)))),i("td",null,"Numero totale di neuroni nel sistema")),i("tr",null,i("td",null,i(n.a,null,K(T||(T=X`n`)))),i("td",null,"Numero di un neurone specifico")),i("tr",null,i("td",null,i(n.a,null,K(H||(H=X`w_{nm}`)))),i("td",null,"Intensità della sinapsi diretta da ",i(n.g,null,K(R||(R=X`n`)))," verso ",i(n.g,null,K(B||(B=X`m`))))),i("tr",null,i("td",null,i(n.a,null,K(G||(G=X`x_n`)))),i("td",null,"Emissione del neurone ",i(n.g,null,K(O||(O=X`n`))))),i("tr",null,i("td",null,i(n.a,null,K(F||(F=X`H(v)`)))),i("td",null,"Funzione che restituisce lo stato di un neurone dato un valore di input")),i("tr",null,i("td",null,i(n.a,null,K(V||(V=X`\sum_1^N ( w_n \cdot x_n )`)))),i("td",null,"Somma degli input di un neurone")),i("tr",null,i("td",null,i(n.a,null,K(W||(W=X`b`)))),i("td",null,"Bias di un neurone"))))),i(n.p,null,i(n.o,{title:"Percettrone"},i("p",null,"Una ",i("b",null,"rete neurale")," che viene incapsulata all'interno di un singolo neurone."),i("p",null,"La sua emissione è determinata dalla sua funzione di emissione ",i(n.g,null,K(Q||(Q=X`H`))),":"),i(n.n,null,K(j||(j=X` + `)))),i(n.p,{title:"Interferenza"},i("p",null,"Più pattern vengono imparati da un modello, più è facile che essi interferiscano tra loro."),i("p",null,"In caso di pattern completamente scorrelati tra loro, il limite di pattern imparabili è circa:"),i(n.o,null,K(L||(L=X`0.14 \cdot N`)))),i(n.p,{title:"Archetipi"},i("p",null,"Per minimizzare l'interferenza tra pattern, è possibile insegnare al modello un ",i("i",null,"archetipo"),": si insegna più volte il pattern originale applicandoci una minima quantità di interferenza casuale."))),i(n.q,{title:"Modello a percettroni"},i(n.p,{title:"Cos'è?"},i("p",null,"Un modello di rete neurale che supporta l'apprendimento e che presenta ",i("b",null,"più strati di neuroni"),"."),i("p",null,"Ha costi computazionali molto più bassi del modello di Hopfield.")),i(n.r,null,i("thead",null,i("tr",null,i("th",null,"Simbolo"),i("th",null,"Descrizione"))),i("tbody",null,i("tr",null,i("td",null,i(n.a,null,K(y||(y=X`N`)))),i("td",null,"Numero totale di neuroni nel sistema")),i("tr",null,i("td",null,i(n.a,null,K(T||(T=X`n`)))),i("td",null,"Numero di un neurone specifico")),i("tr",null,i("td",null,i(n.a,null,K(H||(H=X`w_{nm}`)))),i("td",null,"Intensità della sinapsi diretta da ",i(n.h,null,K(R||(R=X`n`)))," verso ",i(n.h,null,K(B||(B=X`m`))))),i("tr",null,i("td",null,i(n.a,null,K(G||(G=X`x_n`)))),i("td",null,"Emissione del neurone ",i(n.h,null,K(O||(O=X`n`))))),i("tr",null,i("td",null,i(n.a,null,K(F||(F=X`H(v)`)))),i("td",null,"Funzione che restituisce lo stato di un neurone dato un valore di input")),i("tr",null,i("td",null,i(n.a,null,K(V||(V=X`\sum_1^N ( w_n \cdot x_n )`)))),i("td",null,"Somma degli input di un neurone")),i("tr",null,i("td",null,i(n.a,null,K(W||(W=X`b`)))),i("td",null,"Bias di un neurone"))))),i(n.q,null,i(n.p,{title:"Percettrone"},i("p",null,"Una ",i("b",null,"rete neurale")," che viene incapsulata all'interno di un singolo neurone."),i("p",null,"La sua emissione è determinata dalla sua funzione di emissione ",i(n.h,null,K(Q||(Q=X`H`))),":"),i(n.o,null,K(j||(j=X` x_n = H \left( \sum_1^N ( w_n \cdot x_n + b) \right) - `))),i("p",null,i(n.g,null,K(J||(J=X`b`)))," è una costante configurabile, detta ",i("i",null,"bias"),", che rappresenta il valore di partenza della somma degli input.")),i(n.o,{title:"Percettrone booleano"},i("p",null,"Un percettrone la cui funzione di emissione è:"),i(n.n,null,K(Z||(Z=X` + `))),i("p",null,i(n.h,null,K(J||(J=X`b`)))," è una costante configurabile, detta ",i("i",null,"bias"),", che rappresenta il valore di partenza della somma degli input.")),i(n.p,{title:"Percettrone booleano"},i("p",null,"Un percettrone la cui funzione di emissione è:"),i(n.o,null,K(Z||(Z=X` \begin{cases} 1 \qquad se\ v > 0\\ 0 \qquad se\ v = 0\\ -1 \qquad se\ v < 0 \end{cases} - `)))),i(n.o,{title:"Apprendimento"},i("p",null,"Si parte da intensità casuali delle sinapsi."),i("p",null,"Si prova a classificare degli esempi pre-classificati: se un esempio viene classificato nel modo sbagliato, si alterano le intensità delle sinapsi in direzione della sua classificazione corretta."),i("p",null,"Nel caso che vi siano più strati di neuroni, allora sarà necessario ricorrere alla ",i("a",{href:"https://en.wikipedia.org/wiki/Backpropagation"},"backpropagation"),", che stima l'errore di classificazione di ogni singolo neurone e li corregge di conseguenza."))),i(n.p,{title:"Rete feed-forward"},i(n.o,{title:"Cos'è?"},i("p",null,"Un modello a percettroni in cui ",i("b",null,"non si presentano cicli"),"."),i("p",null,"Alcuni dei neuroni che vi sono all'interno saranno dunque dei ",i("b",null,"neuroni sorgente")," e dei ",i("b",null,"neuroni pozzo"),"."))))}}.call(this,e("hosL").h)},ke5e:function(i,l,e){"use strict";(function(i){var n=e("2w3n"),o=e.n(n);l.a=function(l){return i("div",{class:o.a.example},l.children)}}).call(this,e("hosL").h)},mDT8:function(i){i.exports={red:"red__1DkTW",orange:"orange__3-zNZ",yellow:"yellow__2ln-k",lime:"lime__3VExD",cyan:"cyan__YxBjD",blue:"blue__2bc0p",magenta:"magenta__KGEa7",tick:"tick__2T7VF"}}}]); -//# sourceMappingURL=route-ApprendimentoSistemiArtificiali.chunk.f7eaf.esm.js.map \ No newline at end of file + `)))),i(n.p,{title:"Apprendimento"},i("p",null,"Si parte da intensità casuali delle sinapsi."),i("p",null,"Si prova a classificare degli esempi pre-classificati: se un esempio viene classificato nel modo sbagliato, si alterano le intensità delle sinapsi in direzione della sua classificazione corretta."),i("p",null,"Nel caso che vi siano più strati di neuroni, allora sarà necessario ricorrere alla ",i("a",{href:"https://en.wikipedia.org/wiki/Backpropagation"},"backpropagation"),", che stima l'errore di classificazione di ogni singolo neurone e li corregge di conseguenza."))),i(n.q,{title:"Rete feed-forward"},i(n.p,{title:"Cos'è?"},i("p",null,"Un modello a percettroni in cui ",i("b",null,"non si presentano cicli"),"."),i("p",null,"Alcuni dei neuroni che vi sono all'interno saranno dunque dei ",i("b",null,"neuroni sorgente")," e dei ",i("b",null,"neuroni pozzo"),"."))))}}.call(this,e("hosL").h)},ke5e:function(i,l,e){"use strict";(function(i){var n=e("2w3n"),t=e.n(n);l.a=function(l){return i("div",{class:t.a.example},l.children)}}).call(this,e("hosL").h)},mDT8:function(i){i.exports={red:"red__1DkTW",orange:"orange__3-zNZ",yellow:"yellow__2ln-k",lime:"lime__3VExD",cyan:"cyan__YxBjD",blue:"blue__2bc0p",magenta:"magenta__KGEa7",tick:"tick__2T7VF"}}}]); +//# sourceMappingURL=route-ApprendimentoSistemiArtificiali.chunk.0fe89.esm.js.map \ No newline at end of file diff --git a/docs/route-ApprendimentoSistemiArtificiali.chunk.f7eaf.esm.js.map 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",n("b",null,"studente dell'Unimore"),", puoi accedere all'",n("b",null,n("a",{href:"https://drive.google.com/drive/folders/1GcjP1Z5UtRjyAag7qjFQ-kx3s06G8ZoP"},"archivio del corso su Google Drive")),"."))),n(J.p,{title:"Esame"},n(J.o,{title:"Scritto"},n("ul",null,n("li",null,"Scegli 2 tra 6 domande possibili"),n("li",null,"90 minuti"))),n(J.o,{title:"Appelli"},n("ol",null,n("li",null,n(J.r,{to:"2020-06-23"})),n("li",null,n(J.r,{to:"2020-07-08"})),n("li",null,n(J.r,{to:"2020-07-22"}))))),n(J.p,{title:"NetLogo"},n(J.o,{title:"Cos'è?"},n("p",null,"NetLogo è un software di modellazione sistemi multiagente, da noi usato per le lezioni di laboratorio."),n("p",null,"Si può ",n("a",{href:"https://ccl.northwestern.edu/netlogo/download.shtml"},"scaricare")," o ",n("a",{href:"https://www.netlogoweb.org/launch"},"usare da browser"),"."),n("p",null,"Il suo codice sorgente è disponibile su ",n("a",{href:"https://github.com/NetLogo/NetLogo"},"GitHub"),", e ha una pagina di 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impulso."))),n(J.p,{title:"Modello booleano"},n(J.o,{title:"Cos'è?"},n("p",null,"Un modello semplificato di rete neurale in cui vengono considerati ",n("b",null,"solo tempi discreti")," (",n(X.a,null,"ticks"),"), e non è presente la memorizzazione degli impulsi nel tempo."),n("p",null,"È stato sviluppato da ",n("a",{href:"https://it.wikipedia.org/wiki/Warren_McCulloch"},"Warren McCulloch")," (un neurofisiologo) e ",n("a",{href:"https://it.wikipedia.org/wiki/Walter_Pitts"},"Walter Pitts")," (un matematico)."),n(Z.a,null,"È importante perchè dimostra che le reti neurali ",n("b",null,"possono elaborare qualsiasi cosa"),", ma incompleto perchè non descrive nessun metodo per la loro creazione automatica.")),n(J.o,{title:"Neuroni"},n("p",null,"I neuroni ",n("b",null,"si attivano")," in un dato ",n(X.a,null)," se la ",n("b",null,"somma dei loro impulsi")," nel ",n(X.a,null)," precedente è ",n("b",null,"maggiore o uguale a 1"),".")),n(J.o,{title:"Intensità sinaptica"},n("p",null,"Le sinapsi 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",n("a",{href:"https://ccl.northwestern.edu/netlogo/docs/dictionary.html"},"documentazione"),"."),n("p",null,n("b",{style:"font-size: x-large;"},n("a",{href:"/apprendimento/netlogo"},"Consulta i miei appunti su NetLogo qui!"))))),n(J.q,{title:"Sistemi dinamici"},n(J.p,{title:"Cosa sono?"},n("p",null,"Sistemi naturali o artificiali che si basano su ",n("b",null,"leggi reversibili e deterministiche"),"."),n("p",null,"In natura, alcuni leggi possono sembrare irreversibili a livello macroscopico, ma sono in realtà reversibili a livello microscopico."),n(Z.a,null,n("p",null,n("a",{href:"https://it.wikipedia.org/wiki/Modello_di_Ehrenfest"},n("u",null,"Urne di Ehrenfest")),": due urne con N palline; estraggo una pallina da una urna casuale ad ogni passo e la sposto nell'altra; con tante palline il sistema appare irreversibile."))),n(J.p,{title:"Fasi"},n("p",null,n("b",null,"Stati")," in cui si può trovare un sistema dinamico."),n("p",null,"Tutte insieme formano lo ",n("i",null,"(iper)",n("b",null,"spazio delle fasi")),".")),n(J.p,{title:"Attrattore"},n("p",null,"Lo ",n("b",null,"stato finale")," di un sistema dinamico."),n("p",null,"Tutte le fasi tendono a uno specifico attrattore.")),n(J.p,{title:"Elaborazione di informazione"},n("p",null,"I sistemi dinamici ",n("b",null,"elaborano informazione")," attraversando fasi e raggiungendo un attrattore."),n(Z.a,null,"L'evoluzione biologica crea nuove specie partendo da quelle precedenti di maggiore successo fino a quando non si raggiunge la specie perfetta."),n(Z.a,null,"Si può vedere l'universo come un gigantesco sistema dinamico. ",n("a",{href:"https://it.wikipedia.org/wiki/Ipotesi_della_simulazione"},"Che sia artificiale?")," ",n("a",{href:"https://it.wikipedia.org/wiki/Morte_termica_dell%27universo"},"Qual è il suo attrattore?")))),n(J.q,{title:"Sistemi lineari"},n(J.p,{title:"Cosa sono?"},n("p",null,"Sistemi dinamici i cui cambiamenti sono descritti da ",n("b",null,"funzioni lineari"),".")),n(J.p,{title:"Nodi"},n("p",null,n("b",null,"Situazioni iniziali")," di un sistema lineare."),n("p",null,"Possono essere:"),n("ul",null,n("li",null,n("u",null,"Stabili"),": ",n("b",null,"convergono")," ad un punto fisso"),n("li",null,n("u",null,"Instabili"),": ",n("b",null,"divergono")," da un punto fisso"),n("li",null,n("a",{href:"https://it.wikipedia.org/wiki/Punto_di_sella"},n("u",null,"Di sella")))),n("p",null,"Nell'insieme dei ",n(J.h,null,K(W()))," possono anche dare origine a:"),n("ul",null,n("li",null,n("u",null,"Spirali stabili"),": spirali che ",n("b",null,"convergono")),n("li",null,n("u",null,"Spirali instabili"),": spirali che ",n("b",null,"divergono")),n("li",null,n("u",null,"Cicli"),": il sistema forma un ciclo diverso in base alla posizione del nodo"),n("li",null,n("u",null,"Cicli limite"),": il sistema evolve fino a formare un ciclo specifico")),n("p",null,"Infine, in sistemi dissipativi può anche comparire:"),n("ul",null,n("li",null,n("u",null,"Caos"),": il sistema evolve in maniera pseudo-casuale")),n(Z.a,null,"Mai sentito parlare di ",n("a",{href:"https://en.wikipedia.org/wiki/Mersenne_Twister"},"Mersenne Twister"),"?")),n(J.p,{title:"Potenziale"},n("p",null,"Funzione che rappresenta lo stato attuale del sistema."),n("p",null,"Gli attrattori coincidono con i suoi ",n("b",null,"punti di minimo"),", detti ",n("i",null,"punti fissi"),"."),n("p",null,"Il suo complementare è la ",n("b",null,"funzione energia"),"."))),n(J.q,{title:"Agenti"},n(J.p,{title:"Secondo il paradigma debole"},n("p",null,"Sono ",n("b",null,"sistemi")," con le seguenti caratteristiche:"),n("ul",null,n("li",null,n("b",null,"Autonomia"),": agiscono gli uni indipendentemente dagli altri"),n("li",null,n("b",null,"Reattività"),": percepiscono ciò che sta nel loro ambiente e ",n("i",null,"reagiscono")," ai cambiamenti di quest'ultimo"),n("li",null,n("b",null,"Proattività"),": agiscono in maniera tale da portare a termine i loro obiettivi"),n("li",null,n("b",null,"Socialità"),": comunicano con gli altri agenti, scambiando informazioni"))),n(J.p,{title:"Secondo il paradigma forte"},n("p",null,"Hanno anche caratteristiche di ",n("b",null,"livello più alto")," derivate dalle quattro precedenti:"),n("ul",null,n("li",null,"Conoscenza"),n("li",null,"Intenzioni"),n("li",null,"Emozioni"),n("li",null,"Obblighi"),n("li",null,"Obiettivi"),n("li",null,"etc...")),n(Z.a,null,"Gli umani possono benissimo essere considerati agenti del sistema universo.")),n(J.p,{title:"Caratteristiche aggiuntive"},n("p",null,"Gli agenti si distinguono anche in:"),n("ul",null,n("li",null,n("b",null,"Mobilità"),": quanto e come possono muoversi nell'ambiente"),n("li",null,n("b",null,"Veridicità"),": quanto producono informazioni corrette",n(Z.a,null,"È possibile effettuare un attacco a un sistema introducendovi agenti maliziosi che producono intenzionalmente informazioni sbagliate!")),n("li",null,n("b",null,"Benevolenza"),": quanto beneficiano gli altri delle loro azioni",n(Z.a,null,n("u",null,"Agenti malevoli"),": ad esempio, troll in siti web",n("br",null),n("u",null,"Agenti benevoli"),": ad esempio, filtri che bannano i troll")),n("li",null,n("b",null,"Razionalità"),": quanto le loro azioni sono coerenti con i loro obiettivi e lo stato dell'ambiente",n(Z.a,null,n("u",null,"Razionalità limitata"),": gli agenti non conoscono completamente l'ambiente, e compiono le azioni che suppongono essere giuste"))))),n(J.q,{title:"Benefici degli agenti"},n(J.p,{title:"Emergenza"},n("p",null,"Lo sviluppo negli agenti di nuove capacità per cui non erano stati programmati."),n(Z.a,null,"Ad esempio, la Swarm Intelligence, descritta dopo!"))),n(J.q,{title:"Architetture di agente"},n(J.p,{title:"Classe"},n("p",null,"Classificazione in base a ",n("b",null,"come prende le decisioni")," un agente:"),n("ul",null,n("li",null,n("b",null,"Logic-based"),": prende le decisioni attraverso deduzioni ",n("i",null,"logiche")),n("li",null,n("b",null,"Reactive"),": mappa una ",n("i",null,"reazione")," a ogni situazione dell'ambiente"),n("li",null,n("b",null,"Belief-desire-intention"),": per decidere, considera le proprie ",n("i",null,"assunzioni sul mondo")," (",n("i",null,"belief"),"), i propri ",n("i",null,"desideri")," (",n("i",null,"desire"),") e le sue ",n("i",null,"intenzioni")," correnti (",n("i",null,"intention"),")"),n("li",null,n("b",null,"Layered"),": utilizza diversi ",n("i",null,"strati")," di capacità cognitive per giungere a una decisione"))),n(J.p,{title:"Comportamento"},n("p",null,"Classificazione in base a ",n("b",null,"come sono definiti gli obiettivi")," di un agente:"),n("ul",null,n("li",null,n("b",null,"Teleonomico"),": gli obiettivi sono predefiniti ed espliciti"),n("li",null,n("b",null,"Riflessivo"),": l'agente è libero di scegliere il suo obiettivo in base alle proprie percezioni interne"))),n(J.p,{title:"Conoscenze"},n("p",null,"Classificazione in base a ",n("b",null,"quanto conosce dell'ambiente")," un agente:"),n("ul",null,n("li",null,n("b",null,"Cognitivo"),": l'agente è immediatamente a conoscenza di ",n("b",null,"tutto l'ambiente")),n("li",null,n("b",null,"Reattivo"),": l'agente deve scoprire l'ambiente con le sue capacità sensoriali")))),n(J.q,{title:"Sistemi multi-agente"},n(J.p,{title:"Vantaggi"},n("ul",null,n("li",null,n("b",null,"Distribuzione"),": più agenti possono coprire aree di ambiente più vaste, o elaborare più in fretta zone più dense di informazione"),n("li",null,n("b",null,"Rappresentazione"),": i sistemi multi-agente modellano più accuratamente il mondo reale"))),n(J.p,{title:"Feedback"},n("p",null,"Influenza esercitata dal sistema sugli agenti per guidarli verso il loro obiettivo."),n("p",null,"Può essere:"),n("ul",null,n("li",null,n("b",{class:"lime"},"Positivo"),": incentiva gli agenti ad avere un dato comportamento"),n("li",null,n("b",{class:"red"},"Negativo"),": disincentiva gli agenti ad avere un dato comportamento"))),n(J.p,{title:"Swarm intelligence"},n("p",null,"Comportamento ",n("b",null,"emergente")," che si manifesta nei sistemi multiagente con tantissimi agenti."),n("p",null,"Indica la capacità di risoluzione di problemi complessi attraverso la collaborazione di più agenti semplici."))),n(J.q,{title:"Evoluzione"},n(J.p,{title:"Algoritmi genetici"},n("p",null,"Meccanismi simili a quelli evolutivi umani che permettono ai tratti degli agenti di ",n("b",null,"convergere")," verso un valore.")),n(J.p,{title:"Fitness"},n("p",null,"Inizialmente definita come ",n("b",null,"numero di discendenti fertili"),", solitamente indica quanto è probabile che i tratti di un individuo siano passati alla generazione successiva.")),n(J.p,{title:"Cromosoma"},n("p",null,n("b",null,"Sequenza di valori")," che definisce uno o più tratti di un individuo.")),n(J.p,{title:"Popolazione"},n("p",null,"Un ",n("b",null,"insieme di individui")," aventi tutti gli stessi cromosomi.")),n(J.p,{title:"Mutazione"},n("p",null,"Fenomeno che causa una ",n("b",null,"piccola variazione casuale nei cromosomi")," dei figli."),n("p",null,"Previene la ",n("b",null,"convergenza prematura")," in un sistema.")),n(J.p,{title:"Crossover"},n("p",null,"Meccanismo di ",n("b",null,"costruzione dei cromosomi")," in un figlio: i cromosomi dei genitori vengono tagliati nello stesso punto scelto a caso, e per costruire quelli del figlio viene presa una parte dal padre e l'altra parte dalla madre."),n("p",null,"Può portare al miglioramento di un individuo e allo sviluppo di nuovi tratti, ma solo nelle parti di cromosoma che sono diverse tra i due genitori.")),n(J.p,{title:"Schema"},n("p",null,n("b",null,"Sequenza di valori")," all'interno di un cromosoma, che può includere anche sezioni in cui il valore è ",n("b",null,"irrilevante"),"."),n("p",null,"Gli algoritmi genetici permettono di trovare gli schemi con la ",n("b",null,"fitness più alta in assoluto")," in un tempo relativamente breve: il sistema ",n("i",null,"generalmente")," favorisce gli schemi corti con fitness alta.")),n(J.p,{title:"Convergenza prematura"},n("p",null,"Situazione in cui si è raggiunta una soluzione non-ottimale a causa dell'assenza di novità nel sistema."),n("p",null,"Si può impedire con vari metodi: con la ",n("b",null,"mutazione"),", introducendo ",n("b",null,"requisiti di località")," per l'accoppiamento, scegliendo diversamente i genitori, etc..."))),n(J.q,{title:"Sistema a classificatori"},n(J.p,{title:"Cosa sono?"},n("p",null,"Programmi che dati tanti esempi sono in grado di classificare un elemento in una o più categorie."),n("p",null,"Sono formati da ",n("i",null,"classificatori"),", liste di ",n("i",null,"messaggi"),", ",n("i",null,"detettori")," e ",n("i",null,"effettori"),"."))),n(J.q,null,n(J.p,{title:"Classificatori"},n("p",null,"Strutture logiche che ",n("b",null,"elaborano")," i messaggi."),n("p",null,"Valutano una espressione logica (",n("i",null,"condizione"),") sui messaggi in arrivo, e se questa risulta essere vera, emettono un nuovo messaggio in risposta (",n("i",null,"azione"),")."),n(Z.a,null,"Condizione e azione possono essere considerati come due cromosomi di un algoritmo genetico!")),n(J.p,{title:"Messaggi"},n("p",null,n("b",null,"Unità di informazione")," di un sistema a classificatori: sono ",n("b",null,"generati")," da ",n("i",null,"detettori")," e ",n("i",null,"classificatori"),", e ",n("b",null,"consumati")," da ",n("i",null,"classificatori")," ed ",n("i",null,"effettori"),".")),n(J.p,{title:"Detettori"},n("p",null,n("b",null,"Sensori")," che percepiscono lo stato dell'ambiente esterno e lo riportano sotto forma di ",n("i",null,"messaggi"),".")),n(J.p,{title:"Effettori"},n("p",null,n("b",null,"Motori")," che rispondono ai ",n("i",null,"messaggi")," effettuando una qualche azione nell'ambiente."))),n(J.q,null,n(J.p,{title:"Forza"},n("p",null,"Un ",n("b",null,"punteggio")," associato ad ogni classificatore."),n("p",null,"Più un classificatore viene attivato, più la sua forza crescerà."),n(Z.a,null,"I classificatori più deboli vengono lentamente eliminati!")),n(J.p,{title:"Specificità"},n("p",null,"Il ",n("b",null,"numero di condizioni che devono essere soddisfatte")," perchè il classificatore si attivi."),n("p",null)),n(J.p,{title:"Bid"},n("p",null,n("b",null,"Prodotto")," di specificità e forza di un classificatore."),n("p",null,"Rappresenta ",n("b",null,"quanto è probabile che venga utilizzato")," un dato classificatore nel caso che le condizioni di più di uno vengano soddisfatte."),n(Z.a,null,"È la fitness degli algoritmi genetici applicata ai classificatori."))),n(J.q,null,n(J.p,{title:"Cover Detector"},n("p",null,"Se l'input non soddisfa nessun classificatore esistente, se ne crea uno nuovo soddisfatto dall'input attuale con una azione casuale.")),n(J.p,{title:"Cover Effector"},n("p",null,"Se i classificatori emettono in output un messaggio non valido, si crea un nuovo classificatore che trasforma quel messaggio in un output valido."))),n(J.q,{title:"Reti neurali"},n(J.p,{title:"Neuroni"},n("p",null,"Agenti che possono ",n("b",null,"collegarsi tra loro")," tramite ",n("i",null,"sinapsi")," (dirette) e ",n("b",null,"ricevere")," ed ",n("b",null,"emettere")," ",n("i",null,"impulsi")," lungo di esse."),n("p",null,"Gli impulsi ricevuti vengono temporaneamente ",n("b",null,"memorizzati")," dal neurone attraverso valori che decadono nel tempo."),n("p",null,"Se la somma dei valori di tutti gli impulsi ricevuti è ",n("b",null,"maggiore di una certa soglia"),", allora il neurone ",n("b",null,"emetterà")," un impulso."))),n(J.q,{title:"Modello booleano"},n(J.p,{title:"Cos'è?"},n("p",null,"Un modello semplificato di rete neurale in cui vengono considerati ",n("b",null,"solo tempi discreti")," (",n(X.a,null,"ticks"),"), e non è presente la memorizzazione degli impulsi nel tempo."),n("p",null,"È stato sviluppato da ",n("a",{href:"https://it.wikipedia.org/wiki/Warren_McCulloch"},"Warren McCulloch")," (un neurofisiologo) e ",n("a",{href:"https://it.wikipedia.org/wiki/Walter_Pitts"},"Walter Pitts")," (un matematico)."),n(Z.a,null,"È importante perchè dimostra che le reti neurali ",n("b",null,"possono elaborare qualsiasi cosa"),", ma incompleto perchè non descrive nessun metodo per la loro creazione automatica.")),n(J.p,{title:"Neuroni"},n("p",null,"I neuroni ",n("b",null,"si attivano")," in un dato ",n(X.a,null)," se la ",n("b",null,"somma dei loro impulsi")," nel ",n(X.a,null)," precedente è ",n("b",null,"maggiore o uguale a 1"),".")),n(J.p,{title:"Intensità sinaptica"},n("p",null,"Le sinapsi hanno una ",n("i",null,"intensità"),": è un ",n("b",null,"moltiplicatore")," che viene applicato a tutti gli impulsi transitanti la sinapsi."))),n(J.q,{title:"Funzioni logiche nel modello booleano"},n(J.p,{title:"NOT"},n("p",null,"Un neurone con una sinapsi entrante con intensità ",n(J.h,null,K(Q())),".")),n(J.p,{title:"OR"},n("p",null,"Un neurone con due o più sinapsi entranti con intensità ",n(J.h,null,K(V())),".")),n(J.p,{title:"AND"},n("p",null,"Un neurone con due o più sinapsi entranti con intensità ",n(J.h,null,K(F())),"."))),n(J.q,{title:"Modello di Hopfield"},n(J.p,{title:"Cos'è?"},n("p",null,"Un'estensione del modello booleano per permettere l'apprendimento automatico delle configurazioni giuste di neuroni."),n("p",null,"È stato sviluppato da ",n("a",{href:"https://en.wikipedia.org/wiki/John_Hopfield"},"John Hopfield")," (uno scienziato)."),n(Z.a,null,"Non è molto avanzato, ma ha portato a ulteriori studi nel campo delle reti neurali.")),n(J.r,null,n("thead",null,n("tr",null,n("th",null,n("abbr",{title:"Vettore / matrice"},"v")),n("th",null,n("abbr",{title:"Elemento singolo / scalare"},"s")),n("th",null,"Glossario"))),n("tbody",null,n("tr",null,n("td",null),n("td",null,n(J.a,null,K(O()))),n("td",null,n(X.a,null,"Tick")," attuale")),n("tr",null,n("td",null),n("td",null,n(J.h,null,K(G())),", ",n(J.h,null,K(R()))),n("td",null,"Identificatore di un neurone specifico")),n("tr",null,n("td",null,n(J.a,null,K(H()))),n("td",null,n(J.a,null,K(B()))),n("td",null,"Intensità della sinapsi diretta da ",n(J.h,null,K(T()))," verso ",n(J.h,null,K(y())))),n("tr",null,n("td",null,n(J.a,null,K(L()))),n("td",null,n(J.a,null,K(M()))),n("td",null,"Soglia di attivazione di un neurone")),n("tr",null,n("td",null,n(J.a,null,K(D()))),n("td",null,n(J.a,null,K(U()))),n("td",null,"Emissione di un neurone")),n("tr",null,n("td",null,n(J.a,null,K(P()))),n("td",null,n(J.a,null,K(E()))),n("td",null,"Somma degli ingressi di un neurone")),n("tr",null,n("td",null),n("td",null,n(J.a,null,K(x()))),n("td",null,"Energia del sistema")),n("tr",null,n("td",null,n(J.a,null,K(N()))),n("td",null,n(J.a,null,K(I()))),n("td",null,"Stato di un neurone in un pattern")),n("tr",null,n("td",null),n("td",null,n(J.a,null,K(A()))),n("td",null,"Sovrapposizione tra due pattern"))))),n(J.q,null,n(J.p,{title:"Emissione"},n("p",null,"In ogni ",n(X.a,null),", i neuroni:"),n("ul",null,n("li",null,"Emettono ",n(J.h,null,K(C()))," se gli input nel ",n(X.a,null)," precedente ",n("b",null,"erano inferiori")," alla soglia di attivazione"),n("li",null,"Emettono ",n(J.h,null,K(k()))," se gli input nel ",n(X.a,null)," precedente ",n("b",null,"superavano")," la soglia di attivazione"),n("li",null,"Non cambiano stato se gli input nel ",n(X.a,null)," precedente ",n("b",null,"erano uguali")," alla soglia di attivazione"))),n(J.p,{title:"Sinapsi"},n("p",null,n("b",null,"Tutti")," i neuroni del modello sono intercollegati tra loro da sinapsi."),n("p",null,"I neuroni non possono essere collegati a loro stessi."),n("p",null,"Questo porta il ",n("b",null,"costo computazionale")," del modello ad essere ",n(J.h,null,K(S())),".")),n(J.p,{title:"Energia"},n("p",null,"Una funzione dell'intero sistema che rappresenta il totale degli stati di tutti i neuroni e tutte le connessioni."),n(J.o,null,K(w())))),n(J.q,null,n(J.p,{title:"Apprendimento hebbiano"},n("p",null,"Un metodo per realizzare l'apprendimento nel modello di Hopfield."),n("p",null,"Si incrementa l'intensità delle sinapsi che connettono neuroni nello stesso stato, e invece si decrementa l'intensità di quelle che connettono neuroni in stati opposti."),n("p",null,"Considerando i neuroni spenti e quelli accesi come ",n(J.h,null,K(q()))," e ",n(J.h,null,K(_()))," rispettivamente, si ha che per ogni pattern:"),n(J.o,null,K(z())),n(Z.a,null,"Così facendo, si insegna sia il pattern normale sia il suo complementare!")),n(J.p,{title:"Simmetria"},n("p",null,"Applicando l'apprendimento hebbiano al modello di Hopfield si ottengono sinapsi simmetriche."),n("p",null,"Se è valida questa proprietà, si può dimostrare che l'",n("b",null,"energia del sistema è sempre decrescente"),", e che quindi che tenderà a un punto fisso!"),n("p",null,n(J.t,null,"TODO: Dopo il prof dimostra la relazione tra input netto e overlap, e che il sistema converge al pattern più simile."))),n(J.p,{title:"Overlap di due pattern"},n("p",null,"Il numero di neuroni attivati in entrambi i pattern."),n(J.o,null,K(h()))),n(J.p,{title:"Interferenza"},n("p",null,"Più pattern vengono imparati da un modello, più è facile che essi interferiscano tra loro."),n("p",null,"In caso di pattern completamente scorrelati tra loro, il limite di pattern imparabili è circa:"),n(J.o,null,K(v()))),n(J.p,{title:"Archetipi"},n("p",null,"Per minimizzare l'interferenza tra pattern, è possibile insegnare al modello un ",n("i",null,"archetipo"),": si insegna più volte il pattern originale applicandoci una minima quantità di interferenza casuale."))),n(J.q,{title:"Modello a percettroni"},n(J.p,{title:"Cos'è?"},n("p",null,"Un modello di rete neurale che supporta l'apprendimento e che presenta ",n("b",null,"più strati di neuroni"),"."),n("p",null,"Ha costi computazionali molto più bassi del modello di Hopfield.")),n(J.r,null,n("thead",null,n("tr",null,n("th",null,"Simbolo"),n("th",null,"Descrizione"))),n("tbody",null,n("tr",null,n("td",null,n(J.a,null,K(b()))),n("td",null,"Numero totale di neuroni nel sistema")),n("tr",null,n("td",null,n(J.a,null,K(g()))),n("td",null,"Numero di un neurone specifico")),n("tr",null,n("td",null,n(J.a,null,K(f()))),n("td",null,"Intensità della sinapsi diretta da ",n(J.h,null,K(m()))," verso ",n(J.h,null,K(d())))),n("tr",null,n("td",null,n(J.a,null,K(p()))),n("td",null,"Emissione del neurone ",n(J.h,null,K(c())))),n("tr",null,n("td",null,n(J.a,null,K(s()))),n("td",null,"Funzione che restituisce lo stato di un neurone dato un valore di input")),n("tr",null,n("td",null,n(J.a,null,K(r()))),n("td",null,"Somma degli input di un neurone")),n("tr",null,n("td",null,n(J.a,null,K(u()))),n("td",null,"Bias di un neurone"))))),n(J.q,null,n(J.p,{title:"Percettrone"},n("p",null,"Una ",n("b",null,"rete neurale")," che viene incapsulata all'interno di un singolo neurone."),n("p",null,"La sua emissione è determinata dalla sua funzione di emissione ",n(J.h,null,K(a())),":"),n(J.o,null,K(o())),n("p",null,n(J.h,null,K(t()))," è una costante configurabile, detta ",n("i",null,"bias"),", che rappresenta il valore di partenza della somma degli input.")),n(J.p,{title:"Percettrone booleano"},n("p",null,"Un percettrone la cui funzione di emissione è:"),n(J.o,null,K(l()))),n(J.p,{title:"Apprendimento"},n("p",null,"Si parte da intensità casuali delle sinapsi."),n("p",null,"Si prova a classificare degli esempi pre-classificati: se un esempio viene classificato nel modo sbagliato, si alterano le intensità delle sinapsi in direzione della sua classificazione corretta."),n("p",null,"Nel caso che vi siano più strati di neuroni, allora sarà necessario ricorrere alla ",n("a",{href:"https://en.wikipedia.org/wiki/Backpropagation"},"backpropagation"),", che stima l'errore di classificazione di ogni singolo neurone e li corregge di conseguenza."))),n(J.q,{title:"Rete feed-forward"},n(J.p,{title:"Cos'è?"},n("p",null,"Un modello a percettroni in cui ",n("b",null,"non si presentano cicli"),"."),n("p",null,"Alcuni dei neuroni che vi sono all'interno saranno dunque dei ",n("b",null,"neuroni sorgente")," e dei ",n("b",null,"neuroni pozzo"),"."))))}}.call(this,e("hosL").h)},ke5e:function(n,i,e){"use strict";(function(n){var l=e("2w3n"),t=e.n(l);i.a=function(i){return n("div",{class:t.a.example},i.children)}}).call(this,e("hosL").h)},mDT8:function(n){n.exports={red:"red__1DkTW",orange:"orange__3-zNZ",yellow:"yellow__2ln-k",lime:"lime__3VExD",cyan:"cyan__YxBjD",blue:"blue__2bc0p",magenta:"magenta__KGEa7",tick:"tick__2T7VF"}}}]); +//# 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\n

Apprendimento ed evoluzione in sistemi artificiali

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\n Se sei uno studente dell'Unimore, puoi accedere all'archivio del corso su Google Drive.\n

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  • Scegli 2 tra 6 domande possibili
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  • 90 minuti
    • \n
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    2. \n
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\n \n

\n NetLogo è un software di modellazione sistemi multiagente, da noi usato per le lezioni di laboratorio.\n

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\n Si può scaricare o usare da browser.\n

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\n Il suo codice sorgente è disponibile su GitHub, e ha una pagina di documentazione.\n

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\n Consulta i miei appunti su NetLogo qui!\n

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\n \n

\n Sistemi naturali o artificiali che si basano su leggi reversibili e deterministiche.\n

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\n In natura, alcuni leggi possono sembrare irreversibili a livello macroscopico, ma sono in realtà reversibili a livello microscopico.\n

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\n Urne di Ehrenfest: due urne con N palline; estraggo una pallina da una urna casuale ad ogni passo e la sposto nell'altra; con tante palline il sistema appare irreversibile.\n

\n \n \n \n

\n Stati in cui si può trovare un sistema dinamico.\n

\n

\n Tutte insieme formano lo (iper)spazio delle fasi.\n

\n \n \n

\n Lo stato finale di un sistema dinamico.\n

\n

\n Tutte le fasi tendono a uno specifico attrattore.\n

\n \n \n

\n I sistemi dinamici elaborano informazione attraversando fasi e raggiungendo un attrattore.\n

\n \n L'evoluzione biologica crea nuove specie partendo da quelle precedenti di maggiore successo fino a quando non si raggiunge la specie perfetta.\n \n \n Si può vedere l'universo come un gigantesco sistema dinamico. Che sia artificiale? Qual è il suo attrattore?\n \n \n
\n
\n \n

\n Sistemi dinamici i cui cambiamenti sono descritti da funzioni lineari.\n

\n \n \n

\n Situazioni iniziali di un sistema lineare.\n

\n

\n Possono essere:\n

\n
    \n
  • Stabili: convergono ad un punto fisso
    • \n
  • Instabili: divergono da un punto fisso
    • \n
  • Di sella
    • \n
\n

\n Nell'insieme dei {r`\\mathbb{C}`} possono anche dare origine a:\n

\n
    \n
  • Spirali stabili: spirali che convergono
    • \n
  • Spirali instabili: spirali che divergono
    • \n
  • Cicli: il sistema forma un ciclo diverso in base alla posizione del nodo
    • \n
  • Cicli limite: il sistema evolve fino a formare un ciclo specifico
    • \n
\n

\n Infine, in sistemi dissipativi può anche comparire:\n

\n
    \n
  • Caos: il sistema evolve in maniera pseudo-casuale
    • \n
\n \n Mai sentito parlare di Mersenne Twister?\n \n \n \n

\n Funzione che rappresenta lo stato attuale del sistema.\n

\n

\n Gli attrattori coincidono con i suoi punti di minimo, detti punti fissi.\n

\n

\n Il suo complementare è la funzione energia.\n

\n \n
\n
\n \n

\n Sono sistemi con le seguenti caratteristiche:\n

\n
    \n
  • Autonomia: agiscono gli uni indipendentemente dagli altri
    • \n
  • Reattività: percepiscono ciò che sta nel loro ambiente e reagiscono ai cambiamenti di quest'ultimo
    • \n
  • Proattività: agiscono in maniera tale da portare a termine i loro obiettivi
    • \n
  • Socialità: comunicano con gli altri agenti, scambiando informazioni
    • \n
\n \n \n

\n Hanno anche caratteristiche di livello più alto derivate dalle quattro precedenti:\n

\n
    \n
  • Conoscenza
    • \n
  • Intenzioni
    • \n
  • Emozioni
    • \n
  • Obblighi
    • \n
  • Obiettivi
    • \n
  • etc...
    • \n
\n \n Gli umani possono benissimo essere considerati agenti del sistema universo.\n \n \n \n

\n Gli agenti si distinguono anche in:\n

\n
    \n
  • Mobilità: quanto e come possono muoversi nell'ambiente
    • \n
  • \n Veridicità: quanto producono informazioni corrette\n \n È possibile effettuare un attacco a un sistema introducendovi agenti maliziosi che producono intenzionalmente informazioni sbagliate!\n \n
    • \n
  • \n Benevolenza: quanto beneficiano gli altri delle loro azioni\n \n Agenti malevoli: ad esempio, troll in siti web
    \n Agenti benevoli: ad esempio, filtri che bannano i troll\n     \n
    • \n
  • \n Razionalità: quanto le loro azioni sono coerenti con i loro obiettivi e lo stato dell'ambiente\n \n Razionalità limitata: gli agenti non conoscono completamente l'ambiente, e compiono le azioni che suppongono essere giuste\n \n
    • \n
\n \n
\n
\n \n

\n Lo sviluppo negli agenti di nuove capacità per cui non erano stati programmati.\n

\n \n Ad esempio, la Swarm Intelligence, descritta dopo!\n \n \n
\n
\n \n

\n Classificazione in base a come prende le decisioni un agente:\n

\n
    \n
  • Logic-based: prende le decisioni attraverso deduzioni logiche
    • \n
  • Reactive: mappa una reazione a ogni situazione dell'ambiente
    • \n
  • Belief-desire-intention: per decidere, considera le proprie assunzioni sul mondo (belief), i propri desideri (desire) e le sue intenzioni correnti (intention)
    • \n
  • Layered: utilizza diversi strati di capacità cognitive per giungere a una decisione
    • \n
\n \n \n

\n Classificazione in base a come sono definiti gli obiettivi di un agente:\n

\n
    \n
  • Teleonomico: gli obiettivi sono predefiniti ed espliciti
    • \n
  • Riflessivo: l'agente è libero di scegliere il suo obiettivo in base alle proprie percezioni interne
    • \n
\n \n \n

\n Classificazione in base a quanto conosce dell'ambiente un agente:\n

\n
    \n
  • Cognitivo: l'agente è immediatamente a conoscenza di tutto l'ambiente
    • \n
  • Reattivo: l'agente deve scoprire l'ambiente con le sue capacità sensoriali
    • \n
\n \n
\n
\n \n
    \n
  • Distribuzione: più agenti possono coprire aree di ambiente più vaste, o elaborare più in fretta zone più dense di informazione
    • \n
  • Rappresentazione: i sistemi multi-agente modellano più accuratamente il mondo reale
    • \n
\n \n \n

\n Influenza esercitata dal sistema sugli agenti per guidarli verso il loro obiettivo.\n

\n

\n Può essere:\n

\n
    \n
  • Positivo: incentiva gli agenti ad avere un dato comportamento
    • \n
  • Negativo: disincentiva gli agenti ad avere un dato comportamento
    • \n
\n \n \n

\n Comportamento emergente che si manifesta nei sistemi multiagente con tantissimi agenti.\n

\n

\n Indica la capacità di risoluzione di problemi complessi attraverso la collaborazione di più agenti semplici.\n

\n \n
\n
\n \n

\n Meccanismi simili a quelli evolutivi umani che permettono ai tratti degli agenti di convergere verso un valore.\n

\n \n \n

\n Inizialmente definita come numero di discendenti fertili, solitamente indica quanto è probabile che i tratti di un individuo siano passati alla generazione successiva.\n

\n \n \n

\n Sequenza di valori che definisce uno o più tratti di un individuo.\n

\n \n \n

\n Un insieme di individui aventi tutti gli stessi cromosomi.\n

\n \n \n

\n Fenomeno che causa una piccola variazione casuale nei cromosomi dei figli.\n

\n

\n Previene la convergenza prematura in un sistema.\n

\n \n \n

\n Meccanismo di costruzione dei cromosomi in un figlio: i cromosomi dei genitori vengono tagliati nello stesso punto scelto a caso, e per costruire quelli del figlio viene presa una parte dal padre e l'altra parte dalla madre.\n

\n

\n Può portare al miglioramento di un individuo e allo sviluppo di nuovi tratti, ma solo nelle parti di cromosoma che sono diverse tra i due genitori.\n

\n \n \n

\n Sequenza di valori all'interno di un cromosoma, che può includere anche sezioni in cui il valore è irrilevante.\n

\n

\n Gli algoritmi genetici permettono di trovare gli schemi con la fitness più alta in assoluto in un tempo relativamente breve: il sistema generalmente favorisce gli schemi corti con fitness alta.\n

\n \n \n

\n Situazione in cui si è raggiunta una soluzione non-ottimale a causa dell'assenza di novità nel sistema.\n

\n

\n Si può impedire con vari metodi: con la mutazione, introducendo requisiti di località per l'accoppiamento, scegliendo diversamente i genitori, etc...\n

\n \n
\n
\n \n

\n Programmi che dati tanti esempi sono in grado di classificare un elemento in una o più categorie.\n

\n

\n Sono formati da classificatori, liste di messaggi, detettori e effettori.\n

\n \n
\n
\n \n

\n Strutture logiche che elaborano i messaggi.\n

\n

\n Valutano una espressione logica (condizione) sui messaggi in arrivo, e se questa risulta essere vera, emettono un nuovo messaggio in risposta (azione).\n

\n \n Condizione e azione possono essere considerati come due cromosomi di un algoritmo genetico!\n \n \n \n

\n Unità di informazione di un sistema a classificatori: sono generati da detettori e classificatori, e consumati da classificatori ed effettori.\n

\n \n \n

\n Sensori che percepiscono lo stato dell'ambiente esterno e lo riportano sotto forma di messaggi.\n

\n \n \n

\n Motori che rispondono ai messaggi effettuando una qualche azione nell'ambiente.\n

\n \n
\n
\n \n

\n Un punteggio associato ad ogni classificatore.\n

\n

\n Più un classificatore viene attivato, più la sua forza crescerà.\n

\n \n I classificatori più deboli vengono lentamente eliminati!\n \n \n \n

\n Il numero di condizioni che devono essere soddisfatte perchè il classificatore si attivi.\n

\n

\n\n

\n \n \n

\n Prodotto di specificità e forza di un classificatore.\n

\n

\n Rappresenta quanto è probabile che venga utilizzato un dato classificatore nel caso che le condizioni di più di uno vengano soddisfatte.\n

\n \n È la fitness degli algoritmi genetici applicata ai classificatori.\n \n \n
\n
\n \n

\n Se l'input non soddisfa nessun classificatore esistente, se ne crea uno nuovo soddisfatto dall'input attuale con una azione casuale.\n

\n \n \n

\n Se i classificatori emettono in output un messaggio non valido, si crea un nuovo classificatore che trasforma quel messaggio in un output valido.\n

\n \n
\n
\n \n

\n Agenti che possono collegarsi tra loro tramite sinapsi (dirette) e ricevere ed emettere impulsi lungo di esse.\n

\n

\n Gli impulsi ricevuti vengono temporaneamente memorizzati dal neurone attraverso valori che decadono nel tempo.\n

\n

\n Se la somma dei valori di tutti gli impulsi ricevuti è maggiore di una certa soglia, allora il neurone emetterà un impulso.\n

\n \n
\n
\n \n

\n Un modello semplificato di rete neurale in cui vengono considerati solo tempi discreti (ticks), e non è presente la memorizzazione degli impulsi nel tempo.\n

\n

\n È stato sviluppato da Warren McCulloch (un neurofisiologo) e Walter Pitts (un matematico).\n

\n \n È importante perchè dimostra che le reti neurali possono elaborare qualsiasi cosa, ma incompleto perchè non descrive nessun metodo per la loro creazione automatica.\n \n \n \n

\n I neuroni si attivano in un dato se la somma dei loro impulsi nel precedente è maggiore o uguale a 1.\n

\n \n \n

\n Le sinapsi hanno una intensità: è un moltiplicatore che viene applicato a tutti gli impulsi transitanti la sinapsi.\n

\n \n
\n
\n \n

\n Un neurone con una sinapsi entrante con intensità {r`-1`}.\n

\n \n \n

\n Un neurone con due o più sinapsi entranti con intensità {r`1`}.\n

\n \n \n

\n Un neurone con due o più sinapsi entranti con intensità {r`\\frac{1}{numero\\ sinapsi}`}.\n

\n \n
\n
\n \n

\n Un'estensione del modello booleano per permettere l'apprendimento automatico delle configurazioni giuste di neuroni.\n

\n

\n È stato sviluppato da John Hopfield (uno scienziato).\n

\n \n Non è molto avanzato, ma ha portato a ulteriori studi nel campo delle reti neurali.\n \n \n \n \n \n v\n s\n Glossario\n \n \n \n \n \n {r`t`}\n Tick attuale\n \n \n \n {r`n`}, {r`m`}\n Identificatore di un neurone specifico\n \n \n {r`\\mathbf{W}`}\n {r`w_{nm}`}\n Intensità della sinapsi diretta da {r`n`} verso {r`m`}\n \n \n {r`\\mathbf{\\Theta}`}\n {r`\\theta_n`}\n Soglia di attivazione di un neurone\n \n \n {r`\\mathbf{X}(t)`}\n {r`x_n(t)`}\n Emissione di un neurone\n \n \n {r`\\mathbf{I}(t)`}\n {r`i_n(t)`}\n Somma degli ingressi di un neurone\n \n \n \n {r`E`}\n Energia del sistema\n \n \n {r`\\mathbf{A}`}\n {r`a_i`}\n Stato di un neurone in un pattern\n \n \n \n {r`Q(\\mathbf{A}, \\mathbf{B})`}\n Sovrapposizione tra due pattern\n \n \n \n
\n
\n \n

\n In ogni , i neuroni:\n

\n
    \n
  • Emettono {r`0`} se gli input nel precedente erano inferiori alla soglia di attivazione
    • \n
  • Emettono {r`1`} se gli input nel precedente superavano la soglia di attivazione
    • \n
  • Non cambiano stato se gli input nel precedente erano uguali alla soglia di attivazione
    • \n
\n \n \n

\n Tutti i neuroni del modello sono intercollegati tra loro da sinapsi.\n

\n

\n I neuroni non possono essere collegati a loro stessi.\n

\n

\n Questo porta il costo computazionale del modello ad essere {r`O(n^2)`}.\n

\n \n \n

\n Una funzione dell'intero sistema che rappresenta il totale degli stati di tutti i neuroni e tutte le connessioni.\n

\n {r`\n E = - \\frac{1}{2} \\sum_{n, m} ( w_{nm} \\cdot x_n \\cdot x_m ) + \\sum_n ( \\theta_n \\cdot x_n )\n `}\n \n
\n
\n \n

\n Un metodo per realizzare l'apprendimento nel modello di Hopfield.\n

\n

\n Si incrementa l'intensità delle sinapsi che connettono neuroni nello stesso stato, e invece si decrementa l'intensità di quelle che connettono neuroni in stati opposti.\n

\n

\n Considerando i neuroni spenti e quelli accesi come {r`0`} e {r`1`} rispettivamente, si ha che per ogni pattern:\n

\n {r`\n \\Delta w_{ik} = (2 \\cdot A_i - 1)(2 \\cdot A_k - 1)\n `}\n \n Così facendo, si insegna sia il pattern normale sia il suo complementare!\n \n \n \n

\n Applicando l'apprendimento hebbiano al modello di Hopfield si ottengono sinapsi simmetriche.\n

\n

\n Se è valida questa proprietà, si può dimostrare che l'energia del sistema è sempre decrescente, e che quindi che tenderà a un punto fisso!\n

\n

\n TODO: Dopo il prof dimostra la relazione tra input netto e overlap, e che il sistema converge al pattern più simile.\n

\n \n \n

\n Il numero di neuroni attivati in entrambi i pattern.\n

\n {r`\n Q(A, B) = \\sum_{i = 1}^n A_i B_i\n `}\n \n \n

\n Più pattern vengono imparati da un modello, più è facile che essi interferiscano tra loro.\n

\n

\n In caso di pattern completamente scorrelati tra loro, il limite di pattern imparabili è circa:\n

\n \n {r`0.14 \\cdot N`}\n \n \n \n

\n Per minimizzare l'interferenza tra pattern, è possibile insegnare al modello un archetipo: si insegna più volte il pattern originale applicandoci una minima quantità di interferenza casuale.\n

\n \n
\n
\n \n

\n Un modello di rete neurale che supporta l'apprendimento e che presenta più strati di neuroni.\n

\n

\n Ha costi computazionali molto più bassi del modello di Hopfield.\n

\n \n \n \n \n Simbolo\n Descrizione\n \n \n \n \n {r`N`}\n Numero totale di neuroni nel sistema\n \n \n {r`n`}\n Numero di un neurone specifico\n \n \n {r`w_{nm}`}\n Intensità della sinapsi diretta da {r`n`} verso {r`m`}\n \n \n {r`x_n`}\n Emissione del neurone {r`n`}\n \n \n {r`H(v)`}\n Funzione che restituisce lo stato di un neurone dato un valore di input\n \n \n {r`\\sum_1^N ( w_n \\cdot x_n )`}\n Somma degli input di un neurone\n \n \n {r`b`}\n Bias di un neurone\n \n \n \n
\n
\n \n

\n Una rete neurale che viene incapsulata all'interno di un singolo neurone.\n

\n

\n La sua emissione è determinata dalla sua funzione di emissione {r`H`}:\n

\n {r`\n x_n = H \\left( \\sum_1^N ( w_n \\cdot x_n + b) \\right)\n `}\n

\n {r`b`} è una costante configurabile, detta bias, che rappresenta il valore di partenza della somma degli input.\n

\n \n \n

\n Un percettrone la cui funzione di emissione è:\n

\n {r`\n \\begin{cases}\n 1 \\qquad se\\ v > 0\\\\\n 0 \\qquad se\\ v = 0\\\\\n -1 \\qquad se\\ v < 0\n \\end{cases}\n `}\n \n \n

\n Si parte da intensità casuali delle sinapsi.\n

\n

\n Si prova a classificare degli esempi pre-classificati: se un esempio viene classificato nel modo sbagliato, si alterano le intensità delle sinapsi in direzione della sua classificazione corretta.\n

\n

\n Nel caso che vi siano più strati di neuroni, allora sarà necessario ricorrere alla backpropagation, che stima l'errore di classificazione di ogni singolo neurone e li corregge di conseguenza.\n

\n \n
\n
\n \n

\n Un modello a percettroni in cui non si presentano cicli.\n

\n

\n Alcuni dei neuroni che vi sono all'interno saranno dunque dei neuroni sorgente e dei neuroni pozzo.\n

\n \n
\n
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\n

Apprendimento ed evoluzione in sistemi artificiali

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\n \n \n \n \n

\n Se sei uno studente dell'Unimore, puoi accedere all'archivio del corso su Google Drive.\n

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  • Scegli 2 tra 6 domande possibili
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  • 90 minuti
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\n NetLogo è un software di modellazione sistemi multiagente, da noi usato per le lezioni di laboratorio.\n

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\n Si può scaricare o usare da browser.\n

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\n Il suo codice sorgente è disponibile su GitHub, e ha una pagina di documentazione.\n

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\n Consulta i miei appunti su NetLogo qui!\n

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\n Sistemi naturali o artificiali che si basano su leggi reversibili e deterministiche.\n

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\n In natura, alcuni leggi possono sembrare irreversibili a livello macroscopico, ma sono in realtà reversibili a livello microscopico.\n

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\n Urne di Ehrenfest: due urne con N palline; estraggo una pallina da una urna casuale ad ogni passo e la sposto nell'altra; con tante palline il sistema appare irreversibile.\n

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\n Stati in cui si può trovare un sistema dinamico.\n

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\n Tutte insieme formano lo (iper)spazio delle fasi.\n

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\n Lo stato finale di un sistema dinamico.\n

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\n Tutte le fasi tendono a uno specifico attrattore.\n

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\n I sistemi dinamici elaborano informazione attraversando fasi e raggiungendo un attrattore.\n

\n \n L'evoluzione biologica crea nuove specie partendo da quelle precedenti di maggiore successo fino a quando non si raggiunge la specie perfetta.\n \n \n Si può vedere l'universo come un gigantesco sistema dinamico. Che sia artificiale? Qual è il suo attrattore?\n \n \n
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\n Sistemi dinamici i cui cambiamenti sono descritti da funzioni lineari.\n

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\n Situazioni iniziali di un sistema lineare.\n

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\n Possono essere:\n

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  • Stabili: convergono ad un punto fisso
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  • Instabili: divergono da un punto fisso
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  • Di sella
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\n Nell'insieme dei {r`\\mathbb{C}`} possono anche dare origine a:\n

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  • Spirali stabili: spirali che convergono
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  • Spirali instabili: spirali che divergono
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  • Cicli: il sistema forma un ciclo diverso in base alla posizione del nodo
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  • Cicli limite: il sistema evolve fino a formare un ciclo specifico
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\n Infine, in sistemi dissipativi può anche comparire:\n

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  • Caos: il sistema evolve in maniera pseudo-casuale
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\n \n Mai sentito parlare di Mersenne Twister?\n \n \n \n

\n Funzione che rappresenta lo stato attuale del sistema.\n

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\n Gli attrattori coincidono con i suoi punti di minimo, detti punti fissi.\n

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\n Il suo complementare è la funzione energia.\n

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\n Sono sistemi con le seguenti caratteristiche:\n

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  • Autonomia: agiscono gli uni indipendentemente dagli altri
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  • Reattività: percepiscono ciò che sta nel loro ambiente e reagiscono ai cambiamenti di quest'ultimo
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  • Proattività: agiscono in maniera tale da portare a termine i loro obiettivi
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  • Socialità: comunicano con gli altri agenti, scambiando informazioni
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\n Hanno anche caratteristiche di livello più alto derivate dalle quattro precedenti:\n

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  • Conoscenza
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  • Intenzioni
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  • Emozioni
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  • Obblighi
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  • Obiettivi
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  • etc...
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\n \n Gli umani possono benissimo essere considerati agenti del sistema universo.\n \n \n \n

\n Gli agenti si distinguono anche in:\n

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  • Mobilità: quanto e come possono muoversi nell'ambiente
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  • \n Veridicità: quanto producono informazioni corrette\n \n È possibile effettuare un attacco a un sistema introducendovi agenti maliziosi che producono intenzionalmente informazioni sbagliate!\n \n
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  • \n Benevolenza: quanto beneficiano gli altri delle loro azioni\n \n Agenti malevoli: ad esempio, troll in siti web
    \n Agenti benevoli: ad esempio, filtri che bannano i troll\n     \n
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  • \n Razionalità: quanto le loro azioni sono coerenti con i loro obiettivi e lo stato dell'ambiente\n \n Razionalità limitata: gli agenti non conoscono completamente l'ambiente, e compiono le azioni che suppongono essere giuste\n \n
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\n Lo sviluppo negli agenti di nuove capacità per cui non erano stati programmati.\n

\n \n Ad esempio, la Swarm Intelligence, descritta dopo!\n \n \n
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\n Classificazione in base a come prende le decisioni un agente:\n

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  • Logic-based: prende le decisioni attraverso deduzioni logiche
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  • Reactive: mappa una reazione a ogni situazione dell'ambiente
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  • Belief-desire-intention: per decidere, considera le proprie assunzioni sul mondo (belief), i propri desideri (desire) e le sue intenzioni correnti (intention)
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  • Layered: utilizza diversi strati di capacità cognitive per giungere a una decisione
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\n Classificazione in base a come sono definiti gli obiettivi di un agente:\n

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    \n
  • Teleonomico: gli obiettivi sono predefiniti ed espliciti
    • \n
  • Riflessivo: l'agente è libero di scegliere il suo obiettivo in base alle proprie percezioni interne
    • \n
\n \n \n

\n Classificazione in base a quanto conosce dell'ambiente un agente:\n

\n
    \n
  • Cognitivo: l'agente è immediatamente a conoscenza di tutto l'ambiente
    • \n
  • Reattivo: l'agente deve scoprire l'ambiente con le sue capacità sensoriali
    • \n
\n \n
\n
\n \n
    \n
  • Distribuzione: più agenti possono coprire aree di ambiente più vaste, o elaborare più in fretta zone più dense di informazione
    • \n
  • Rappresentazione: i sistemi multi-agente modellano più accuratamente il mondo reale
    • \n
\n \n \n

\n Influenza esercitata dal sistema sugli agenti per guidarli verso il loro obiettivo.\n

\n

\n Può essere:\n

\n
    \n
  • Positivo: incentiva gli agenti ad avere un dato comportamento
    • \n
  • Negativo: disincentiva gli agenti ad avere un dato comportamento
    • \n
\n \n \n

\n Comportamento emergente che si manifesta nei sistemi multiagente con tantissimi agenti.\n

\n

\n Indica la capacità di risoluzione di problemi complessi attraverso la collaborazione di più agenti semplici.\n

\n \n
\n
\n \n

\n Meccanismi simili a quelli evolutivi umani che permettono ai tratti degli agenti di convergere verso un valore.\n

\n \n \n

\n Inizialmente definita come numero di discendenti fertili, solitamente indica quanto è probabile che i tratti di un individuo siano passati alla generazione successiva.\n

\n \n \n

\n Sequenza di valori che definisce uno o più tratti di un individuo.\n

\n \n \n

\n Un insieme di individui aventi tutti gli stessi cromosomi.\n

\n \n \n

\n Fenomeno che causa una piccola variazione casuale nei cromosomi dei figli.\n

\n

\n Previene la convergenza prematura in un sistema.\n

\n \n \n

\n Meccanismo di costruzione dei cromosomi in un figlio: i cromosomi dei genitori vengono tagliati nello stesso punto scelto a caso, e per costruire quelli del figlio viene presa una parte dal padre e l'altra parte dalla madre.\n

\n

\n Può portare al miglioramento di un individuo e allo sviluppo di nuovi tratti, ma solo nelle parti di cromosoma che sono diverse tra i due genitori.\n

\n \n \n

\n Sequenza di valori all'interno di un cromosoma, che può includere anche sezioni in cui il valore è irrilevante.\n

\n

\n Gli algoritmi genetici permettono di trovare gli schemi con la fitness più alta in assoluto in un tempo relativamente breve: il sistema generalmente favorisce gli schemi corti con fitness alta.\n

\n \n \n

\n Situazione in cui si è raggiunta una soluzione non-ottimale a causa dell'assenza di novità nel sistema.\n

\n

\n Si può impedire con vari metodi: con la mutazione, introducendo requisiti di località per l'accoppiamento, scegliendo diversamente i genitori, etc...\n

\n \n
\n
\n \n

\n Programmi che dati tanti esempi sono in grado di classificare un elemento in una o più categorie.\n

\n

\n Sono formati da classificatori, liste di messaggi, detettori e effettori.\n

\n \n
\n
\n \n

\n Strutture logiche che elaborano i messaggi.\n

\n

\n Valutano una espressione logica (condizione) sui messaggi in arrivo, e se questa risulta essere vera, emettono un nuovo messaggio in risposta (azione).\n

\n \n Condizione e azione possono essere considerati come due cromosomi di un algoritmo genetico!\n \n \n \n

\n Unità di informazione di un sistema a classificatori: sono generati da detettori e classificatori, e consumati da classificatori ed effettori.\n

\n \n \n

\n Sensori che percepiscono lo stato dell'ambiente esterno e lo riportano sotto forma di messaggi.\n

\n \n \n

\n Motori che rispondono ai messaggi effettuando una qualche azione nell'ambiente.\n

\n \n
\n
\n \n

\n Un punteggio associato ad ogni classificatore.\n

\n

\n Più un classificatore viene attivato, più la sua forza crescerà.\n

\n \n I classificatori più deboli vengono lentamente eliminati!\n \n \n \n

\n Il numero di condizioni che devono essere soddisfatte perchè il classificatore si attivi.\n

\n

\n\n

\n \n \n

\n Prodotto di specificità e forza di un classificatore.\n

\n

\n Rappresenta quanto è probabile che venga utilizzato un dato classificatore nel caso che le condizioni di più di uno vengano soddisfatte.\n

\n \n È la fitness degli algoritmi genetici applicata ai classificatori.\n \n \n
\n
\n \n

\n Se l'input non soddisfa nessun classificatore esistente, se ne crea uno nuovo soddisfatto dall'input attuale con una azione casuale.\n

\n \n \n

\n Se i classificatori emettono in output un messaggio non valido, si crea un nuovo classificatore che trasforma quel messaggio in un output valido.\n

\n \n
\n
\n \n

\n Agenti che possono collegarsi tra loro tramite sinapsi (dirette) e ricevere ed emettere impulsi lungo di esse.\n

\n

\n Gli impulsi ricevuti vengono temporaneamente memorizzati dal neurone attraverso valori che decadono nel tempo.\n

\n

\n Se la somma dei valori di tutti gli impulsi ricevuti è maggiore di una certa soglia, allora il neurone emetterà un impulso.\n

\n \n
\n
\n \n

\n Un modello semplificato di rete neurale in cui vengono considerati solo tempi discreti (ticks), e non è presente la memorizzazione degli impulsi nel tempo.\n

\n

\n È stato sviluppato da Warren McCulloch (un neurofisiologo) e Walter Pitts (un matematico).\n

\n \n È importante perchè dimostra che le reti neurali possono elaborare qualsiasi cosa, ma incompleto perchè non descrive nessun metodo per la loro creazione automatica.\n \n \n \n

\n I neuroni si attivano in un dato se la somma dei loro impulsi nel precedente è maggiore o uguale a 1.\n

\n \n \n

\n Le sinapsi hanno una intensità: è un moltiplicatore che viene applicato a tutti gli impulsi transitanti la sinapsi.\n

\n \n
\n
\n \n

\n Un neurone con una sinapsi entrante con intensità {r`-1`}.\n

\n \n \n

\n Un neurone con due o più sinapsi entranti con intensità {r`1`}.\n

\n \n \n

\n Un neurone con due o più sinapsi entranti con intensità {r`\\frac{1}{numero\\ sinapsi}`}.\n

\n \n
\n
\n \n

\n Un'estensione del modello booleano per permettere l'apprendimento automatico delle configurazioni giuste di neuroni.\n

\n

\n È stato sviluppato da John Hopfield (uno scienziato).\n

\n \n Non è molto avanzato, ma ha portato a ulteriori studi nel campo delle reti neurali.\n \n \n \n \n \n v\n s\n Glossario\n \n \n \n \n \n {r`t`}\n Tick attuale\n \n \n \n {r`n`}, {r`m`}\n Identificatore di un neurone specifico\n \n \n {r`\\mathbf{W}`}\n {r`w_{nm}`}\n Intensità della sinapsi diretta da {r`n`} verso {r`m`}\n \n \n {r`\\mathbf{\\Theta}`}\n {r`\\theta_n`}\n Soglia di attivazione di un neurone\n \n \n {r`\\mathbf{X}(t)`}\n {r`x_n(t)`}\n Emissione di un neurone\n \n \n {r`\\mathbf{I}(t)`}\n {r`i_n(t)`}\n Somma degli ingressi di un neurone\n \n \n \n {r`E`}\n Energia del sistema\n \n \n {r`\\mathbf{A}`}\n {r`a_i`}\n Stato di un neurone in un pattern\n \n \n \n {r`Q(\\mathbf{A}, \\mathbf{B})`}\n Sovrapposizione tra due pattern\n \n \n \n
\n
\n \n

\n In ogni , i neuroni:\n

\n
    \n
  • Emettono {r`0`} se gli input nel precedente erano inferiori alla soglia di attivazione
    • \n
  • Emettono {r`1`} se gli input nel precedente superavano la soglia di attivazione
    • \n
  • Non cambiano stato se gli input nel precedente erano uguali alla soglia di attivazione
    • \n
\n \n \n

\n Tutti i neuroni del modello sono intercollegati tra loro da sinapsi.\n

\n

\n I neuroni non possono essere collegati a loro stessi.\n

\n

\n Questo porta il costo computazionale del modello ad essere {r`O(n^2)`}.\n

\n \n \n

\n Una funzione dell'intero sistema che rappresenta il totale degli stati di tutti i neuroni e tutte le connessioni.\n

\n {r`\n E = - \\frac{1}{2} \\sum_{n, m} ( w_{nm} \\cdot x_n \\cdot x_m ) + \\sum_n ( \\theta_n \\cdot x_n )\n `}\n \n
\n
\n \n

\n Un metodo per realizzare l'apprendimento nel modello di Hopfield.\n

\n

\n Si incrementa l'intensità delle sinapsi che connettono neuroni nello stesso stato, e invece si decrementa l'intensità di quelle che connettono neuroni in stati opposti.\n

\n

\n Considerando i neuroni spenti e quelli accesi come {r`0`} e {r`1`} rispettivamente, si ha che per ogni pattern:\n

\n {r`\n \\Delta w_{ik} = (2 \\cdot A_i - 1)(2 \\cdot A_k - 1)\n `}\n \n Così facendo, si insegna sia il pattern normale sia il suo complementare!\n \n \n \n

\n Applicando l'apprendimento hebbiano al modello di Hopfield si ottengono sinapsi simmetriche.\n

\n

\n Se è valida questa proprietà, si può dimostrare che l'energia del sistema è sempre decrescente, e che quindi che tenderà a un punto fisso!\n

\n

\n TODO: Dopo il prof dimostra la relazione tra input netto e overlap, e che il sistema converge al pattern più simile.\n

\n \n \n

\n Il numero di neuroni attivati in entrambi i pattern.\n

\n {r`\n Q(A, B) = \\sum_{i = 1}^n A_i B_i\n `}\n \n \n

\n Più pattern vengono imparati da un modello, più è facile che essi interferiscano tra loro.\n

\n

\n In caso di pattern completamente scorrelati tra loro, il limite di pattern imparabili è circa:\n

\n \n {r`0.14 \\cdot N`}\n \n \n \n

\n Per minimizzare l'interferenza tra pattern, è possibile insegnare al modello un archetipo: si insegna più volte il pattern originale applicandoci una minima quantità di interferenza casuale.\n

\n \n
\n
\n \n

\n Un modello di rete neurale che supporta l'apprendimento e che presenta più strati di neuroni.\n

\n

\n Ha costi computazionali molto più bassi del modello di Hopfield.\n

\n \n \n \n \n Simbolo\n Descrizione\n \n \n \n \n {r`N`}\n Numero totale di neuroni nel sistema\n \n \n {r`n`}\n Numero di un neurone specifico\n \n \n {r`w_{nm}`}\n Intensità della sinapsi diretta da {r`n`} verso {r`m`}\n \n \n {r`x_n`}\n Emissione del neurone {r`n`}\n \n \n {r`H(v)`}\n Funzione che restituisce lo stato di un neurone dato un valore di input\n \n \n {r`\\sum_1^N ( w_n \\cdot x_n )`}\n Somma degli input di un neurone\n \n \n {r`b`}\n Bias di un neurone\n \n \n \n
\n
\n \n

\n Una rete neurale che viene incapsulata all'interno di un singolo neurone.\n

\n

\n La sua emissione è determinata dalla sua funzione di emissione {r`H`}:\n

\n {r`\n x_n = H \\left( \\sum_1^N ( w_n \\cdot x_n + b) \\right)\n `}\n

\n {r`b`} è una costante configurabile, detta bias, che rappresenta il valore di partenza della somma degli input.\n

\n \n \n

\n Un percettrone la cui funzione di emissione è:\n

\n {r`\n \\begin{cases}\n 1 \\qquad se\\ v > 0\\\\\n 0 \\qquad se\\ v = 0\\\\\n -1 \\qquad se\\ v < 0\n \\end{cases}\n `}\n \n \n

\n Si parte da intensità casuali delle sinapsi.\n

\n

\n Si prova a classificare degli esempi pre-classificati: se un esempio viene classificato nel modo sbagliato, si alterano le intensità delle sinapsi in direzione della sua classificazione corretta.\n

\n

\n Nel caso che vi siano più strati di neuroni, allora sarà necessario ricorrere alla backpropagation, che stima l'errore di classificazione di ogni singolo neurone e li corregge di conseguenza.\n

\n \n
\n
\n \n

\n Un modello a percettroni in cui non si presentano cicli.\n

\n

\n Alcuni dei neuroni che vi sono all'interno saranno dunque dei neuroni sorgente e dei neuroni pozzo.\n

\n \n
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extracted by mini-css-extract-plugin\nmodule.exports = {\"red\":\"red__2y1B_\",\"orange\":\"orange__dD2kx\",\"yellow\":\"yellow__OEpwl\",\"lime\":\"lime__CVe41\",\"cyan\":\"cyan__26ZAg\",\"blue\":\"blue__LO7Xm\",\"magenta\":\"magenta__1Akee\",\"example\":\"example__2PzAa\"};","import {Section, Panel, Timer, ILatex, TablePanel, BLatex, PLatex} from \"bluelib\";\nimport Example from \"../components/Example\";\n\nconst r = String.raw;\n\nexport default function (props) {\n return (\n
\n

Basi di dati

\n
\n \n \n \n \n

\n Se sei uno studente dell'Unimore, puoi accedere all'archivio del corso su Google Drive.\n

\n \n
\n
\n \n

\n Uno scritto con tre domande:\n

\n
    \n
  • Progettazione concettuale e logica
    • \n
  • Formulazione interrogazione
    • \n
  • Una domanda tra:\n
      \n
    • Studio dato derivato
      • \n
    • Progettazione fisica
      • \n
    • Tecnologia database
      • \n
    \n
    • \n
\n \n \n
    \n
    2. \n
    3. \n
    4. \n
\n \n
\n
\n \n

\n In gruppo, progettare un database secondo queste specifiche.\n

\n

\n Va consegnato via mail almeno 10 giorni prima dello scritto.\n

\n \n \n
    \n
    2. \n
    3. \n
    4. \n
\n \n
\n
\n \n

\n Link\n

\n \n
\n
\n \n \n \n Simbolo\n Descrizione\n \n \n \n \n {r`N_{tuple}`}\n Numero di tuple di una tabella\n \n \n {r`N_{blocchi}`}\n Numero di blocchi di una tabella\n \n \n {r`N_{foglie}`}\n Numero di foglie di un indice\n \n \n {r`N_{indici}`}\n Numero di indici presenti in una tabella\n \n \n {r`N_{unici}`}\n Numero di valori unici presenti in una colonna\n \n \n {r`F`}\n Fattore di selettività di una condizione\n \n \n {r`C`}\n Costo di accesso a una o più tuple\n \n \n {r`N_{selezionati}`}\n Numero di tuple del risultato\n \n \n \n
\n
\n \n

\n Espressione formata da una o più condizioni moltiplicate tra loro.\n

\n \n \n

\n Condizioni che soddisfano i seguenti requisiti:\n

\n
    \n
  • \n Usa uno dei seguenti operatori:\n
      \n
    • {r`<`}
      • \n
    • {r`\\leq`}
      • \n
    • {r`=`}
      • \n
    • {r`\\geq`}
      • \n
    • {r`>`}
      • \n
    • {r`BETWEEN`}
      • \n
    • {r`OR`} (solo in alcuni DBMS)
      • \n
    \n
    • \n
  • Uno dei due termini del confronto è una colonna della tabella
    • \n
  • L'altro termine del confronto è una costante numerica
    • \n
\n \n \n

\n È possibile costruire indici sulle colonne di una tabella per velocizzare le query che riguardano quelle colonne.\n

\n

\n In particolare, ogni tabella può avere un indice clustered e infiniti indici unclustered.\n

\n

\n Generalmente, l'indice clustered è costruito sulla colonna della primary key, ma non è sempre quello il caso.\n

\n

\n Gli indici vanno tenuti aggiornati, e ciò ha un costo di manutenzione:\n

\n
    \n
  • DELETE: {r`N_{indici} \\cdot 2 \\cdot N_{selezionati}`}
    • \n
  • UPDATE: {r`N_{indici} \\cdot 4 \\cdot N_{selezionati}`}
    • \n
\n

\n È possibile usare gli indici nelle query solo per gli argomenti di ricerca attraverso indice.\n

\n \n
\n
\n \n

\n La percentuale di tuple di una tabella che soddisfano una condizione.\n

\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
CondizioneFattore di selettività
col = $costante{r`F = \\frac{1}{N_{unici}`}
col != $costante{r`F = 1 - \\frac{1}{N_{unici}`}
col1 != col2{r`F = \\frac{1}{max(N_{unici_A}, N_{unici_B})`}
IN{r`F = \\frac{n_{IN}}{N_{unici}}`}
col > $costante{r`F = \\frac{ max(col) - \\$costante }{ max(col) - min(col) }`}
{r`col < $costante`}{r`F = \\frac{ \\$costante - min(col) }{ max(col) - min(col) }`}
col BETWEEN $lower AND $upper{r`F = \\frac{ \\$upper - \\$lower }{ max(col) - min(col) }`}
cond1 OR cond2{r`F = F_{cond1} + F_{cond2} - ( F_{cond1} \\cdot F_{cond2} )`}
\n \n \n

\n Quante unità di lavoro costa accedere a una specifica relazione:\n

\n {r`\n C = {\\color{yellow} {C_{indice}}} + {\\color{Orange} C_{relazione}}\n `}\n

\n Per accedere attraverso un indice a una specifica tupla della relazione, si spenderà:\n

\n {r`\n C = {\\color{yellow} 1} + {\\color{Orange} 1}\n `}\n

\n Se le tuple a cui si vuole accedere sono più di una, allora, il costo dipenderà da se l'indice è clustered o non-clustered.\n

\n \n Nel calcolo del costo di una query, si considerano tutti i cambi di pagina come page fault.\n \n

\n Per gli indici clustered, visto che basta caricare in memoria i blocchi in sequenza, il costo sarà:\n

\n {r`\n C = {\\color{yellow} ( F \\cdot N_{foglie} ) } + {\\color{Orange} ( F \\cdot N_{blocchi} ) }\n `}\n

\n Per gli indici unclustered, per i quali potremmo dover caricare e rimuovere lo stesso blocco dalla memoria più volte, il costo sarà:\n

\n {r`\n C = {\\color{yellow} ( F \\cdot N_{foglie} ) } + {\\color{Orange} ( F \\cdot N_{tuple} ) }\n `}\n

\n Se si stanno effettuando query su più indici unclustered, il costo sarà:\n

\n {r`\n C = {\\color{yellow}\\sum_k ( F_k \\cdot N_{foglie_k} )} + {\\color{Orange} \\left( \\prod_k F_k \\cdot N_{tuple} \\right)}\n `}\n \n \n

\n Quante unità di lavoro costa effettuare un join tra due relazioni.\n

\n

\n Varia in base al metodo di join utilizzato.\n

\n

\n Utilizzando il metodo del nested loop:\n

\n {r`\n C_{1 \\times 2} = C_1 + ( N_{selezionati_1} \\cdot C_2 )\n `}\n

\n La scelta della colonna su cui iterare è quindi importante!\n

\n \n
\n
\n )\n}\n","import style from \"./Example.less\";\r\n\r\nexport default function(props) {\r\n return (\r\n
\r\n {props.children}\r\n
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\n

Basi di dati

\n
\n \n \n \n \n

\n Se sei uno studente dell'Unimore, puoi accedere all'archivio del corso su Google Drive.\n

\n \n
\n
\n \n

\n Uno scritto con tre domande:\n

\n
    \n
  • Progettazione concettuale e logica
    • \n
  • Formulazione interrogazione
    • \n
  • Una domanda tra:\n
      \n
    • Studio dato derivato
      • \n
    • Progettazione fisica
      • \n
    • Tecnologia database
      • \n
    \n
    • \n
\n \n \n
    \n
    2. \n
    3. \n
    4. \n
\n \n
\n
\n \n

\n In gruppo, progettare un database secondo queste specifiche.\n

\n

\n Va consegnato via mail almeno 10 giorni prima dello scritto.\n

\n \n \n
    \n
    2. \n
    3. \n
    4. \n
\n \n
\n
\n \n

\n Link\n

\n \n
\n
\n \n \n \n Simbolo\n Descrizione\n \n \n \n \n {r`N_{tuple}`}\n Numero di tuple di una tabella\n \n \n {r`N_{blocchi}`}\n Numero di blocchi di una tabella\n \n \n {r`N_{foglie}`}\n Numero di foglie di un indice\n \n \n {r`N_{indici}`}\n Numero di indici presenti in una tabella\n \n \n {r`N_{unici}`}\n Numero di valori unici presenti in una colonna\n \n \n {r`F`}\n Fattore di selettività di una condizione\n \n \n {r`C`}\n Costo di accesso a una o più tuple\n \n \n {r`N_{selezionati}`}\n Numero di tuple del risultato\n \n \n \n
\n
\n \n

\n Espressione formata da una o più condizioni moltiplicate tra loro.\n

\n \n \n

\n Condizioni che soddisfano i seguenti requisiti:\n

\n
    \n
  • \n Usa uno dei seguenti operatori:\n
      \n
    • {r`<`}
      • \n
    • {r`\\leq`}
      • \n
    • {r`=`}
      • \n
    • {r`\\geq`}
      • \n
    • {r`>`}
      • \n
    • {r`BETWEEN`}
      • \n
    • {r`OR`} (solo in alcuni DBMS)
      • \n
    \n
    • \n
  • Uno dei due termini del confronto è una colonna della tabella
    • \n
  • L'altro termine del confronto è una costante numerica
    • \n
\n \n \n

\n È possibile costruire indici sulle colonne di una tabella per velocizzare le query che riguardano quelle colonne.\n

\n

\n In particolare, ogni tabella può avere un indice clustered e infiniti indici unclustered.\n

\n

\n Generalmente, l'indice clustered è costruito sulla colonna della primary key, ma non è sempre quello il caso.\n

\n

\n Gli indici vanno tenuti aggiornati, e ciò ha un costo di manutenzione:\n

\n
    \n
  • DELETE: {r`N_{indici} \\cdot 2 \\cdot N_{selezionati}`}
    • \n
  • UPDATE: {r`N_{indici} \\cdot 4 \\cdot N_{selezionati}`}
    • \n
\n

\n È possibile usare gli indici nelle query solo per gli argomenti di ricerca attraverso indice.\n

\n \n
\n
\n \n

\n La percentuale di tuple di una tabella che soddisfano una condizione.\n

\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
CondizioneFattore di selettività
col = $costante{r`F = \\frac{1}{N_{unici}`}
col != $costante{r`F = 1 - \\frac{1}{N_{unici}`}
col1 != col2{r`F = \\frac{1}{max(N_{unici_A}, N_{unici_B})`}
IN{r`F = \\frac{n_{IN}}{N_{unici}}`}
col > $costante{r`F = \\frac{ max(col) - \\$costante }{ max(col) - min(col) }`}
{r`col < $costante`}{r`F = \\frac{ \\$costante - min(col) }{ max(col) - min(col) }`}
col BETWEEN $lower AND $upper{r`F = \\frac{ \\$upper - \\$lower }{ max(col) - min(col) }`}
cond1 OR cond2{r`F = F_{cond1} + F_{cond2} - ( F_{cond1} \\cdot F_{cond2} )`}
\n \n \n

\n Quante unità di lavoro costa accedere a una specifica relazione:\n

\n {r`\n C = {\\color{yellow} {C_{indice}}} + {\\color{Orange} C_{relazione}}\n `}\n

\n Per accedere attraverso un indice a una specifica tupla della relazione, si spenderà:\n

\n {r`\n C = {\\color{yellow} 1} + {\\color{Orange} 1}\n `}\n

\n Se le tuple a cui si vuole accedere sono più di una, allora, il costo dipenderà da se l'indice è clustered o non-clustered.\n

\n \n Nel calcolo del costo di una query, si considerano tutti i cambi di pagina come page fault.\n \n

\n Per gli indici clustered, visto che basta caricare in memoria i blocchi in sequenza, il costo sarà:\n

\n {r`\n C = {\\color{yellow} ( F \\cdot N_{foglie} ) } + {\\color{Orange} ( F \\cdot N_{blocchi} ) }\n `}\n

\n Per gli indici unclustered, per i quali potremmo dover caricare e rimuovere lo stesso blocco dalla memoria più volte, il costo sarà:\n

\n {r`\n C = {\\color{yellow} ( F \\cdot N_{foglie} ) } + {\\color{Orange} ( F \\cdot N_{tuple} ) }\n `}\n

\n Se si stanno effettuando query su più indici unclustered, il costo sarà:\n

\n {r`\n C = {\\color{yellow}\\sum_k ( F_k \\cdot N_{foglie_k} )} + {\\color{Orange} \\left( \\prod_k F_k \\cdot N_{tuple} \\right)}\n `}\n \n \n

\n Quante unità di lavoro costa effettuare un join tra due relazioni.\n

\n

\n Varia in base al metodo di join utilizzato.\n

\n

\n Utilizzando il metodo del nested loop:\n

\n {r`\n C_{1 \\times 2} = C_1 + ( N_{selezionati_1} \\cdot C_2 )\n `}\n

\n La scelta della colonna su cui iterare è quindi importante!\n

\n \n
\n
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E' molto simile al massimo: ricordi le prime lezioni di Analisi?")),n(S.p,{title:"Norma a infinito"},n("p",null,"Massimo delle somme dei valori assoluti di tutti gli elementi di ogni riga di una matrice."),n("p",null,n(S.h,null,N(a())))),n(S.p,{title:"Norma a 1"},n("p",null,"Massimo delle somme dei valori assoluti di tutti gli elementi di ogni colonna di una matrice."),n("p",null,n(S.h,null,N(o())))),n(S.p,{title:"Norma a 2"},n("p",null,"Radice quadrata del rango del prodotto tra una matrice e la sua trasposta."),n("p",null,n(S.h,null,N(t()))))),n(S.q,{title:"Sistemi lineari"},n(S.p,{title:"Condizionamento"},n(S.o,null,N(r())),n("p",null,"In particolare, le ",n("b",null,"numero di condizionamento"),":"),n(S.o,null,N(e())))),n(S.q,{title:"Fattorizzazione"},n(S.p,{title:"Cos'è?"},n("p",null,"La fattorizzazione è il processo che permette di risolvere sistemi di equazioni lineari rappresentati in forma di matrice."),n("p",null,"Esistono molteplici algoritmi in grado di realizzarla: mentre tutti portano alla stessa soluzione, possono avere ",n("b",null,"velocità")," e ",n("b",null,"indici algoritmici")," diversi."),n("p",null,"Il sistema lineare da risolvere viene diviso in due parti: la ",n("i",null,"matrice dei coefficienti")," e il ",n("i",null,"vettore termine noto"),".")),n(S.p,{title:"Teorema di Rouché-Capélli"},n("p",null))))}}.call(this,i("hosL").h)}}]); 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\n

Calcolo Numerico

\n
\n \n \n \n \n

\n Se sei uno studente dell'Unimore, puoi accedere all'archivio del corso su Google Drive..\n

\n \n \n

\n Prima di studiare Calcolo Numerico, guardati i prerequisiti di Algebra Lineare!\n

\n \n
\n
\n \n

\n E' composto da:\n

\n
    \n
  • 2 domande sugli argomenti teorici
    • \n
  • 1 domanda di implementazione algoritmo in MATLAB
    • \n
\n \n \n
    \n
    2. \n
\n \n
\n
\n \n

\n Algoritmi che hanno:\n

\n
    \n
  • numeri reali in input e output
    • \n
  • successioni delle quattro operazioni aritmetiche fondamentali come passi
    • \n
\n \n
\n
\n \n

\n Con i numeri floating point può capitare che un certo numero {r`\\alpha`} non sia rappresentato correttamente.\n

\n

\n In tal caso, il numero si indica con {r`\\alpha^*`}.\n

\n \n
\n
\n \n

\n È la differenza tra il numero desiderato e il numero rappresentato:\n

\n {r`E_a = \\left | \\alpha - \\alpha^* \\right |`}\n \n \n

\n Indica quanto il numero rappresentato differisce dal numero desiderato:\n

\n {r`\\forall \\alpha \\neq 0, E_r = \\frac{E_a}{\\left | \\alpha \\right |}`}\n \n
\n
\n \n

\n Metodo con cui gestire gli underflow floating point: le cifre meno significative vengono rimosse.\n

\n \n 
\n                            1.00  →  1.0
\n                            1.01  →  1.0
\n                            1.10  →  1.1
\n                            1.11  →  1.1\n
\n \n \n \n

\n Metodo con cui gestire gli underflow floating point: se la cifra più significativa di quelle che devono essere rimosse è 1, allora aumenta di 1 anche quella meno signficativa che viene tenuta.\n

\n \n 
\n                            1.00  →  1.0
\n                            1.01  →  1.0
\n                            1.10  →  1.1
\n                            1.11  → 10.\n
\n \n \n
\n
\n \n

\n L'errore relativo di un numero reale rappresentato in virgola mobile è minore o uguale alla precisione di macchina:\n

\n

\n {r`E_r \\leq k \\cdot \\beta^{1-t}`}\n

\n
    \n
  • \n \\beta è uguale alla base utilizzata (solitamente 2).\n
    • \n
  • \n t è uguale al numero di cifre della mantissa.\n
    • \n
  • \n k è uguale a 1 se il numero viene rappresentato per troncamento oppure a {r`\\frac{1}{2}`} se viene rappresentato per arrotondamento.\n
    • \n
\n \n \n

\n Associa un valore reale al suo corrispondente valore floating point, utilizzando uno dei due metodi di gestione dell'undeflow.\n

\n {r`fl(x) = (x)(1 + \\epsilon_x)`}\n \n Indica che un valore è soggetto alla precisione di macchina.\n {r`fl(1.11) = 1.1`}\n \n \n
\n
\n \n

\n L'insieme {r`\\mathbb{F}`} è il sottoinsieme dei numeri reali rappresentabili in floating point dalla macchina che stiamo usando.\n

\n

\n Operazioni tra elementi di {r`\\mathbb{F}`} producono risultati in {r`\\mathbb{R}`}, che però decaderanno nuovamente a elementi di {r`\\mathbb{F}`}, perdendo informazioni.\n

\n

\n Il teorema della precisione di macchina si applica quindi anche ai risultati delle operazioni.\n

\n \n \n
    \n
  • Hanno più di un elemento neutro.
    • \n
  • Un numero ha più opposti.
    • \n
  • Non sono associative.
    • \n
  • Non sono distributive.
    • \n
  • Non vale la legge di annullamento del prodotto.
    • \n
\n \n
\n
\n \n

\n Errore derivato da underflow sui dati.\n

\n

\n Si indica con {r`\\epsilon_{variabile}`}.\n

\n \n L'errore sulla variabile x si indica con {r`\\epsilon_{x}`}.\n \n \n \n

\n Errore derivato da underflow durante l'esecuzione dell'algoritmo.\n

\n

\n Si indica con {r`\\epsilon_{passo}`}.\n

\n \n L'errore al primo passo dell'algoritmo si indica con {r`\\epsilon_{1}`}.\n \n \n
\n
\n \n

\n Sensibilità di un problema all'errore inerente.\n

\n \n {r`y = \\frac{1}{x}`} è mal condizionato intorno allo 0 e ben condizionato lontano dallo 0.\n \n \n \n

\n Sensibilità di un problema all'errore algoritmico.\n

\n \n

\n Cerchiamo un algoritmo che risolva {r`2x^* = 4`}.\n

\n

\n Calcolare prima {r`t = fl \\left( \\frac{1}{4} \\right)`} e poi {r`x = fl ( 2 \\cdot t )`} porta a una perdita di precisione.\n

\n

\n Calcolare direttamente {r`x = fl \\left( \\frac{2}{4} \\right)`} non ha alcuna perdita di precisione e rende l'algoritmo più stabile del precedente.\n

\n \n \n
\n
\n \n

\n È il coefficiente di proporzionalità tra i dati e l'errore inerente.\n

\n

\n Essendo sempre maggiore di uno, si può dire che sia un coefficiente di amplificazione.\n

\n

\n Minore è l'indice di condizionamento, meglio condizionato è un problema.\n

\n \n \n

\n È il coefficiente di proporzionalità tra i dati e l'errore algoritmico.\n

\n

\n Essendo sempre maggiore di uno, si può dire che sia un coefficiente di amplificazione.\n

\n \n
\n\n
\n \n

\n Funzione che associa un valore positivo a ogni matrice diversa da 0, e 0 alla matrice zero.\n

\n

\n Si ricavano dalle norme vettoriali:\n

\n

\n {r`\\Vert A \\Vert = sup_{x \\in \\mathbb{R}, x \\neq 0} \\frac{\\Vert A \\cdot x \\Vert}{\\Vert x \\Vert}`}\n

\n \n sup è l'estremo superiore di un insieme. E' molto simile al massimo: ricordi le prime lezioni di Analisi?\n \n \n \n

\n Massimo delle somme dei valori assoluti di tutti gli elementi di ogni riga di una matrice.\n

\n

\n {r`\\Vert A \\Vert_\\infty = max_{i = 1..n} \\sum_{j = 1}^n | a_{ij} |`}\n

\n \n \n

\n Massimo delle somme dei valori assoluti di tutti gli elementi di ogni colonna di una matrice.\n

\n

\n {r`\\Vert A \\Vert_1 = max_{j = 1..n} \\sum_{i = 1}^n | a_{ij} |`}\n

\n \n \n

\n Radice quadrata del rango del prodotto tra una matrice e la sua trasposta.\n

\n

\n {r`\\Vert A \\Vert_2 = \\sqrt{\\rho ( A^T \\times A ) }`}\n

\n \n
\n
\n \n\n {r`\\frac{{\\color{yellow} \\|A\\| \\cdot \\|A^{-1}\\|} \\cdot \\| \\Delta b \\|}{\\| b \\|}`}\n

\n In particolare, le numero di condizionamento:\n

\n \n {r`k(A) = `}\n \n \n
\n
\n \n

\n La fattorizzazione è il processo che permette di risolvere sistemi di equazioni lineari rappresentati in forma di matrice.\n

\n

\n Esistono molteplici algoritmi in grado di realizzarla: mentre tutti portano alla stessa soluzione, possono avere velocità e indici algoritmici diversi.\n

\n

\n Il sistema lineare da risolvere viene diviso in due parti: la matrice dei coefficienti e il vettore termine noto.\n

\n \n \n

\n\n

\n \n
\n
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\n

Calcolo Numerico

\n
\n \n \n \n \n

\n Se sei uno studente dell'Unimore, puoi accedere all'archivio del corso su Google Drive..\n

\n \n
\n
\n \n

\n E' composto da:\n

\n
    \n
  • 2 domande sugli argomenti teorici
    • \n
  • 1 domanda di implementazione algoritmo in MATLAB
    • \n
\n \n \n
    \n
    2. \n
    3. \n
    4. \n
    5. \n
    6. \n
    7. \n
\n \n
\n
\n \n

\n Algoritmi che hanno:\n

\n
    \n
  • numeri reali in input e output
    • \n
  • successioni delle quattro operazioni aritmetiche fondamentali come passi
    • \n
\n \n
\n
\n \n

\n Con i numeri floating point può capitare che un certo numero {r`\\alpha`} non sia rappresentato correttamente.\n

\n

\n In tal caso, il numero si indica con {r`\\alpha^*`}.\n

\n \n \n

\n È la differenza tra il numero desiderato e il numero rappresentato:\n

\n

\n {r`E_a = \\left | \\alpha - \\alpha^* \\right |`}\n

\n \n Vale sempre 0, tranne in caso di underflow o overflow.\n \n \n \n

\n Indica quanto il numero rappresentato differisce dal numero desiderato:\n

\n

\n {r`\\forall \\alpha \\neq 0, E_r = \\frac{E_a}{\\left | \\alpha \\right |}`}\n

\n \n
\n
\n \n

\n L'errore relativo di un numero reale rappresentato in virgola mobile è minore o uguale alla precisione di macchina:\n

\n

\n {r`E_r \\leq k \\cdot \\beta^{1-t}`}\n

\n
    \n
  • \n \\beta è uguale alla base utilizzata (solitamente 2).\n
    • \n
  • \n t è uguale al numero di cifre della mantissa.\n
    • \n
  • \n k è uguale a 1 se il numero viene rappresentato per troncamento oppure a {r`\\frac{1}{2}`} se viene rappresentato per arrotondamento.\n
    • \n
\n \n
\n
\n \n

\n L'insieme F è il sottoinsieme dei numeri reali rappresentabili in floating point dalla macchina che stiamo usando.\n

\n

\n Operazioni tra elementi di F producono risultati in {r`\\mathbb{R}`}, che però decaderanno nuovamente a elementi di F, perdendo informazioni.\n

\n

\n Il teorema della precisione di macchina si applica quindi anche ai risultati delle operazioni.\n

\n \n \n
    \n
  • Hanno più di un elemento neutro.
    • \n
  • Un numero ha più opposti.
    • \n
  • Non sono associative.
    • \n
  • Non sono distributive.
    • \n
  • Non vale la legge di annullamento del prodotto.
    • \n
\n \n \n

\n Indica che un valore è soggetto alla precisione di macchina:\n

\n

\n {r`fl(x) = (x)(1 + \\epsilon_x)`}\n

\n \n
\n
\n \n

\n Errore dipendente solo dai dati.\n

\n

\n TODO: qual è la formula?\n

\n

\n È l'errore che si presenterebbe se {r`\\epsilon_1 = \\epsilon_2 = \\dots = 0`}.\n

\n \n \n

\n Errore dipendente solo dalle operazioni effettuate.\n

\n

\n TODO: qual è la formula?\n

\n

\n È l'errore che si presenterebbe se {r`\\epsilon_x = \\epsilon_y = \\dots = 0`}.\n

\n \n
\n
\n \n

\n Un problema poco sensibile all'errore inerente.\n

\n \n {r`y = \\frac{1}{x}`} è mal condizionato intorno allo 0 e ben condizionato lontano dallo 0.\n \n \n \n

\n Un algoritmo poco sensibile all'errore algoritmico.\n

\n \n

\n Cerchiamo un algoritmo che risolva {r`2x = 4`}.\n

\n

\n Calcolare prima {r`t = fl \\left( \\frac{1}{4} \\right)`} e poi {r`x^* = fl ( 2 \\cdot t )`} porta a una perdita di precisione.\n

\n

\n Calcolare direttamente {r`x^* = fl \\left( \\frac{2}{4} \\right)`} non ha alcuna perdita di precisione e rende l'algoritmo più stabile del precedente.\n

\n \n \n
\n
\n \n

\n È il coefficiente di proporzionalità tra i dati e l'errore inerente.\n

\n

\n Essendo sempre maggiore di uno, si può dire che sia un coefficiente di amplificazione.\n

\n

\n Minore è l'indice di condizionamento, meglio condizionato è un problema.\n

\n \n \n

\n È il coefficiente di proporzionalità tra i dati e l'errore algoritmico.\n

\n

\n Essendo sempre maggiore di uno, si può dire che sia un coefficiente di amplificazione.\n

\n \n
\n
\n \n

\n Matrice con determinante 0.\n

\n

\n Le matrici singolari non sono invertibili.\n

\n \n
\n
\n \n

\n Funzione che associa un valore positivo a ogni vettore diverso da 0, e 0 al vettore zero.\n

\n \n Esempi su Wikipedia\n \n \n \n

\n Massimo dei valori assoluti di tutti gli elementi del vettore.\n

\n

\n {r`\\Vert x \\Vert_\\infty = max_{i = 1..n} | x_i |`}\n

\n \n \n

\n Somma dei valori assoluti di tutti gli elementi del vettore.\n

\n

\n {r`\\Vert x \\Vert_1 = \\sum_{i = 1}^n | x_i |`}\n

\n \n \n

\n Radice quadrata della somma dei quadrati di tutti gli elementi del vettore.\n

\n

\n {r`\\Vert x \\Vert_2 = \\sqrt{\\sum_{i = 1}^n x_i^2}`}\n

\n \n
\n
\n \n

\n Funzione che associa un valore positivo a ogni matrice diversa da 0, e 0 alla matrice zero.\n

\n

\n Si ricavano dalle norme vettoriali:\n

\n

\n {r`\\Vert A \\Vert = sup_{x \\in \\mathbb{R}, x \\neq 0} \\frac{\\Vert A \\cdot x \\Vert}{\\Vert x \\Vert}`}\n

\n \n sup è l'estremo superiore di un insieme. E' molto simile al massimo: ricordi le prime lezioni di Analisi?\n \n \n \n

\n Massimo delle somme dei valori assoluti di tutti gli elementi di ogni riga di una matrice.\n

\n

\n {r`\\Vert A \\Vert_\\infty = max_{i = 1..n} \\sum_{j = 1}^n | a_{ij} |`}\n

\n \n \n

\n Massimo delle somme dei valori assoluti di tutti gli elementi di ogni colonna di una matrice.\n

\n

\n {r`\\Vert A \\Vert_1 = max_{j = 1..n} \\sum_{i = 1}^n | a_{ij} |`}\n

\n \n \n

\n Radice quadrata del rango del prodotto tra una matrice e la sua trasposta.\n

\n

\n {r`\\Vert A \\Vert_2 = \\sqrt{\\rho ( A^T \\times A ) }`}\n

\n \n
\n
\n \n

\n Le norme sono usate per calcolare l'errore relativo tra due vettori o matrici:\n

\n

\n {r`\\frac{\\Vert x - y \\Vert}{\\Vert x \\Vert}`}\n

\n \n
\n
\n \n

\n La fattorizzazione è il processo che permette di risolvere sistemi di equazioni lineari rappresentati in forma di matrice.\n

\n

\n Esistono molteplici algoritmi in grado di realizzarla: mentre tutti portano alla stessa soluzione, possono avere velocità e indici algoritmici diversi.\n

\n

\n Il sistema lineare da risolvere viene diviso in due parti: la matrice dei coefficienti e il vettore termine noto.\n

\n \n \n

\n\n

\n \n
\n
\n \n )\n}\n"],"sourceRoot":""} \ No newline at end of file diff --git a/docs/route-Fisica.chunk.00106.esm.js b/docs/route-Fisica.chunk.00106.esm.js deleted file mode 100644 index 5cf5ab3..0000000 --- a/docs/route-Fisica.chunk.00106.esm.js +++ /dev/null @@ -1,2 +0,0 @@ -(window.webpackJsonp=window.webpackJsonp||[]).push([[6],{"2QOB":function(l,e,i){"use strict";i.r(e),function(l){var a=i("mbOI"),n=i("o8Jj"),o=i("ZHMS");let t,r,u,c,p,s,d,m,v,g,f,z,h,_,b,q,L,D,F,S,E,P,C,I,A,w,k,B,N,V,R,T,y,M,U,x,O,G,Q,j,J,W,H,Z,X,K,Y,$,ll,el,il,al,nl,ol,tl,rl,ul,cl,pl,sl,dl,ml,vl,gl,fl,zl,hl,_l,bl,ql,Ll,Dl,Fl,Sl,El,Pl,Cl,Il,Al,wl,kl,Bl,Nl,Vl,Rl,Tl,yl,Ml,Ul,xl,Ol,Gl,Ql,jl,Jl,Wl,Hl,Zl,Xl,Kl,Yl,$l,le,ee,ie,ae,ne,oe,te,re,ue,ce,pe,se,de,me,ve,ge,fe,ze,he,_e,be,qe,Le,De,Fe,Se,Ee,Pe,Ce,Ie,Ae,we,ke,Be=l=>l;const Ne=String.raw;e.default=function(){return l(a.k.Provider,{value:!1},l(a.j.Provider,{value:a.l.INLINE},l("div",null,l("h1",null,"Fisica"),l(a.p,{title:"Vettori"},l(a.o,{title:"Componenti cartesiane"},l("p",null,"Usa le regole base della trigonometria:"),l("p",null,l(a.i,null,Ne(t||(t=Be`\vec{v} = \vec{v}_x + \vec{v}_y`)))),l("p",null,l(a.i,null,Ne(r||(r=Be`\left | \vec{v}_x \right | = \left | \vec{v} \right | \sin \alpha`)))),l("p",null,l(a.i,null,Ne(u||(u=Be`\left | \vec{v}_y \right | = \left | \vec{v} \right | \cos \alpha`))))),l(a.o,{title:"Somma"},l("p",null,"Scomponi in componenti, poi sommali:"),l("p",null,l(a.i,null,Ne(c||(c=Be`\vec{v} + \vec{w} = (\vec{v}_x + \vec{w}_x) + (\vec{v}_y + \vec{w}_y)`)))),l("p",null,"Produce il vettore risultante dall'applicazione della regola del parallelogramma.")),l(a.o,{title:"Differenza"},l("p",null,"Alla fine è sempre una somma:"),l("p",null,l(a.i,null,Ne(p||(p=Be`\vec{v} - \vec{w} = (\vec{v}_x - \vec{w}_x) + (\vec{v}_y - \vec{w}_y)`)))),l("p",null,"Produce il vettore che parte da ",l(a.i,null,"w")," e arriva a ",l(a.i,null,"v"),".")),l(a.o,{title:"Prodotto scalare"},l("p",null,"Si chiama scalare perchè il risultato è uno scalare, non un vettore."),l("p",null,l(a.i,null,Ne(s||(s=Be`\vec{v} \cdot \vec{w} = \left | \vec{v} \right | \left | \vec{w} \right | \cos \alpha`)))),l("p",null,"Produce il modulo della proiezione di ",l(a.i,null,Ne(d||(d=Be`\vec{a}`)))," su ",l(a.i,null,Ne(m||(m=Be`\vec{b}`))),".")),l(a.o,{title:"Prodotto vettoriale"},l("p",null,"Si chiama vettoriale perchè il risultato è un altro vettore."),l("ul",null,l("li",null,l(a.i,null,Ne(v||(v=Be`\vec{c} = \vec{a} \times \vec{b}`)))),l("li",null,l(a.i,null,Ne(g||(g=Be`\left | \vec{c} \right | = \left | \vec{a} \right | \cdot \left | \vec{b} \right | \cdot \sin(\alpha)`)))),l("li",null,l("a",{href:"https://it.wikipedia.org/wiki/Regola_della_mano_destra"},"Regola della mano destra"))),l("p",null,"Non è commutativo!"))),l(a.p,{title:"Leggi di Newton"},l(a.o,{title:"1ᵃ: Inerzia"},l("p",null,"Se un corpo puntiforme ha forza risultante nulla, allora la sua velocità non cambia."),l("p",null,l(a.i,null,Ne(f||(f=Be`\Sigma \vec{F} = 0 \Longleftrightarrow \Delta v = 0`))))),l(a.o,{title:"2ᵃ: Proporzionalità"},l("p",null,"La forza risultante di un corpo è direttamente proporzionale alla sua accelerazione, e la costante di proporzionalità è la ",l("i",null,"massa"),"."),l("p",null,l(a.i,null,Ne(z||(z=Be`\Sigma \vec{F} = m \vec{a}`))))),l(a.o,{title:"3ᵃ: Azione e reazione"},l("p",null,"Due corpi esercitano forze uguali e opposte uno sull'altro."),l("p",null,l(a.i,null,Ne(h||(h=Be`\vec{F}_{21} = -\vec{F}_{12}`)))))),l(a.p,{title:"Forza di gravità"},l(a.o,{title:"Tra due corpi"},l("p",null,"Due corpi puntiformi si attirano uno verso l'altro con forza:"),l("p",null,l(a.i,null,Ne(_||(_=Be`\left | \vec{F} \right | = G \frac{m_1 m_2}{s^2}`)))),l("p",null,l(a.i,null,"G")," è la ",l("i",null,"costante di gravitazione universale")," e vale:"),l("p",null,l(a.i,null,Ne(b||(b=Be`G = 6.67 \cdot 10^{-11} \frac{N m^2}{{kg}^2}`))))),l(a.o,{title:"Verso la Terra"},l("p",null,"Se nel sistema di riferimento consideriamo la Terra ferma, allora un corpo è attratto verso la Terra con forza ",l("i",null,"peso")," uguale a:"),l("p",null,l(a.i,null,Ne(q||(q=Be`\left | \vec{F} \right | = g m`)))),l("p",null,l(a.i,null,"g")," è la ",l("i",null,"costante di gravità")," della Terra, e vale:"),l("p",null,l(a.i,null,Ne(L||(L=Be`g = 9.81 \frac{m}{s^2}`))))),l(a.o,{title:"Su pianeti diversi"},l("p",null,"Per pianeti diversi dalla Terra vale la stessa regola:"),l("p",null,l(a.i,null,Ne(D||(D=Be`\left | \vec{F} \right | = g m`)))),l("p",null,"L'unica differenza è che cambia la ",l("i",null,"costante di gravità"),":"),l("p",null,l(a.i,null,Ne(F||(F=Be`g_{luna} = 1.62 \frac{m}{s^2}`)))),l("p",null,l(a.i,null,Ne(S||(S=Be`g_{marte} = 3.71 \frac{m}{s^2}`)))))),l(a.p,{title:"Forze di contatto"},l(a.o,{title:"Normale"},l("p",null,"Si oppone alle forze applicate alla superficie di contatto."),l("p",null,"Un libro appoggiato su un tavolo ha la ",l("b",null,"forza di gravità")," che lo attira verso il terreno e la ",l("b",null,"forza normale")," che lo trattiene dal cadere.")),l(a.o,{title:"Attrito statico"},l("p",null,"Impedisce a un corpo di muoversi se non viene spinto da una forza che supera una certa soglia:"),l("p",null,l(a.i,null,Ne(E||(E=Be`\left | \vec{F} \right | \leq \mu_{s} \left | \vec{F}_{normale} \right |`))))),l(a.o,{title:"Attrito dinamico"},l("p",null,"Rallenta i corpi che si stanno muovendo finchè essi non si fermano:"),l("p",null,l(a.i,null,Ne(P||(P=Be`\left | \vec{F} \right | \leq \mu_{d} \left | \vec{F}_{normale} \right |`))))),l(a.o,{title:"Tensione"},l("p",null,"E' forza trasmessa tra due estremi di una fune."),l("p",null,"Può essere redirezionata per mezzo di carrucole.")),l(a.o,{title:"Elastica"},l("p",null,"Una molla cerca sempre di tornare alla sua posizione indeformata con forza:"),l("p",null,l(a.i,null,Ne(C||(C=Be`F = -k x`)))),l("p",null,"(E' negativa perchè la forza è opposta a quella applicata per deformarla.)"))),l(a.p,{title:"Cinematica"},l(a.o,{title:"Spostamento"},l("p",null,"È un vettore che indica la posizione di un corpo rispetto a un'origine."),l("p",null,l(a.i,null,Ne(I||(I=Be`\Delta \vec{s} = \vec{s}(fine) - \vec{s}(inizio)`))))),l(a.o,{title:"Velocità"},l("p",null,"È un vettore che misura la variazione di posizione nel tempo."),l("p",null,l(a.i,null,Ne(A||(A=Be`\vec{v} = \frac{\Delta \vec{s}}{\Delta t}`)))),l("p",null,"Se si considera un intervallo di tempo infinitesimale si dice ",l("i",null,"velocità istantanea"),":"),l("p",null,l(a.i,null,Ne(w||(w=Be`\vec{v} = \lim_{\Delta t \to 0} \frac{\Delta \vec{s}}{\Delta t} = \frac{d \vec{s}}{dt}`))))),l(a.o,{title:"Accelerazione"},l("p",null,"È un vettore che misura la variazione di velocità nel tempo."),l("p",null,l(a.i,null,Ne(k||(k=Be`\vec{a} = \frac{\Delta \vec{v}}{\Delta t}`)))),l("p",null,"Se si considera un intervallo di tempo infinitesimale si dice ",l("i",null,"accelerazione istantanea"),":"),l("p",null,l(a.i,null,Ne(B||(B=Be`\vec{a} = \lim_{\Delta v \to 0} \frac{\Delta \vec{v}}{\Delta t} = \frac{d \vec{v}}{d t} = \frac{d^2 \vec{s}}{d t^2}`))))),l(a.o,{title:l("span",null,"Quantità di moto ",l("small",null,"(momento lineare)"))},l("p",null,"La quantità di moto è una proprietà vettoriale dei corpi:"),l("p",null,l(a.i,null,Ne(N||(N=Be`\vec{p} = m \vec{v}`)))),l("p",null,"Se la forza risultante è nulla, la quantità di moto non cambia."),l("p",null,l(a.i,null,Ne(V||(V=Be`\Sigma \vec{F} = 0 \Longleftrightarrow \Delta \vec{p} = 0`)))))),l(a.p,{title:"Moto rettilineo uniforme"},l(a.o,{title:"Spostamento"},l("p",null,"La ",l("i",null,"legge oraria")," è:"),l("p",null,l(a.i,null,Ne(R||(R=Be`s(t) = v \cdot \Delta t + s(0)`))))),l(a.o,{title:"Velocità"},l("p",null,"È costante:"),l("p",null,l(a.i,null,Ne(T||(T=Be`v(t) = k`))))),l(a.o,{title:"Accelerazione"},l("p",null,"La velocità non varia:"),l("p",null,l(a.i,null,Ne(y||(y=Be`a(t) = 0`))))),l(a.o,{title:"Forze"},l("p",null,"Si applica la prima legge di Newton:"),l("p",null,l(a.i,null,"f(t) = 0")))),l(a.p,{title:"Moto rettilineo uniformemente accelerato"},l(a.o,{title:"Spostamento"},l("p",null,"La ",l("i",null,"legge oraria")," è:"),l("p",null,l(a.i,null,Ne(M||(M=Be`s(t) = \frac{1}{2} a \cdot (\Delta t)^2 + v(0) \cdot (\Delta t) + s(0)`))))),l(a.o,{title:"Velocità"},l("p",null,"È una retta:"),l("p",null,l(a.i,null,Ne(U||(U=Be`v(t) = a \Delta t + v(0)`))))),l(a.o,{title:"Accelerazione"},l("p",null,"È costante:"),l("p",null,l(a.i,null,Ne(x||(x=Be`a(t) = k`))))),l(a.o,{title:"Forze"},l("p",null,"Si applica la prima legge di Newton:"),l("p",null,l(a.i,null,"f(t) = m a")))),l(a.p,{title:"Moto armonico semplice"},l(a.o,{title:"Ampiezza"},l("p",null,"E' la distanza dal centro massima che raggiunge il corpo."),l("p",null,"(L'ampiezza di una sinusoide.)")),l(a.o,{title:"Velocità angolare"},l("p",null,"Indica quanto in fretta cambia la posizione del corpo."),l("p",null,"Dipende dal periodo:"),l("p",null,l(a.i,null,Ne(O||(O=Be`\omega = \frac{2 \pi}{T}`))))),l(a.o,{title:"Spostamento"},l("p",null,"E' una sinusoide:"),l("p",null,l(a.i,null,Ne(G||(G=Be`s(t) = A \sin (\omega \cdot t + \phi)`))))),l(a.o,{title:"Velocità"},l("p",null,"E' la sinusoide dello spostamento, sfasata di ",l(a.i,null,Ne(Q||(Q=Be`\frac{\pi}{2}`))),":"),l("p",null,l(a.i,null,Ne(j||(j=Be`v(t) = A \sin (\omega \cdot t + \phi + \frac{\pi}{2})`))))),l(a.o,{title:"Accelerazione"},l("p",null,"E' la sinusoide della velocità, sfasata di ",l(a.i,null,Ne(J||(J=Be`\pi`))),":"),l("p",null,l(a.i,null,Ne(W||(W=Be`a(t) = A \sin (\omega \cdot t + \phi + \pi)`))))),l(a.o,{title:"Forze"},l("p",null,"Si applica la prima legge di Newton:"),l("p",null,l(a.i,null,"f(t) = m a")))),l(a.p,{title:"Moti composti"},l(a.o,{title:"Moto parabolico"},l("p",null,"Il moto parabolico è dato sommando un moto rettilineo uniforme sull'asse orizzontale e un moto rettilineo uniformemente accelerato sull'asse verticale.")),l(a.o,{title:"Moto circolare uniforme"},l("p",null,"Il moto parabolico è dato sommando due moti armonici semplici: uno sull'asse X, e l'altro, sfasato di ",l(a.i,null,Ne(H||(H=Be`\frac{\pi}{2}`))),", sull'asse Y."))),l(a.p,{title:"Moto circolare uniforme"},l(a.o,null,l("h3",null,"Velocità angolare"),l("p",null,"Quanto cambia la fase nel tempo."),l("p",null,l(a.i,null,Ne(Z||(Z=Be`\omega = \frac{2 \pi}{T}`))))),l(a.o,{title:"Fase"},l("p",null,"E' l'angolo percorso dal corpo rispetto alla posizione iniziale."),l("p",null,"Si indica con ",l(a.i,null,Ne(X||(X=Be`\phi`))),", e generalmente si usa in radianti.")),l(a.o,{title:"Velocità"},l("p",null,"Si applicano le formule per la circonferenza:"),l("p",null,l(a.i,null,Ne(K||(K=Be`v = \frac{\Delta s}{t} = \frac{2 \pi \cdot r}{T} = \omega r`))))),l(a.o,{title:"Accelerazione"},l("p",null,"Il corpo ha sempre un accelerazione verso il centro che gli impedisce di abbandonare il moto:"),l("p",null,l(a.i,null,Ne(Y||(Y=Be`a = \frac{v^2}{r} = r \cdot \omega^2 = v \cdot \omega`))))),l(a.o,{title:"Forza centripeta"},l("p",null,"È verso il centro e si calcola con:"),l("p",null,l(a.i,null,Ne($||($=Be`F = m \cdot a`)))))),l(a.p,{title:"Lavoro ed energia"},l(a.o,{title:"Lavoro"},l("p",null,"E' compiuto da una forza che sposta un corpo."),l("p",null,l(a.i,null,Ne(ll||(ll=Be`W = \vec{F} \cdot \vec{s} = F \cdot \Delta s \cdot cos(\alpha )`)))),l("p",null,"(Se la forza non è parallela allo spostamento, il prodotto scalare ci fa considerare solo la componente parallela.)")),l(a.o,{title:"Energia cinetica"},l("p",null,"Un corpo ha energia cinetica in ogni momento uguale a:"),l("p",null,l(a.i,null,Ne(el||(el=Be`E_c = \frac{1}{2} m v^2`)))),l("p",null,"Se una forza effettua lavoro su un corpo, cambia la sua energia cinetica pari al lavoro effettuato:"),l("p",null,l(a.i,null,Ne(il||(il=Be`\Delta E_c = W`))))),l(a.o,{title:"Energia potenziale gravitazionale"},l("p",null,"Un corpo ha energia potenziale in ogni momento pari a:"),l("p",null,l(a.i,null,Ne(al||(al=Be`E_{p_g} = m \cdot g \cdot h`)))),l("p",null,"(Con ",l(a.i,null,"h")," uguale a un altezza scelta come punto di riferimento.)")),l(a.o,{title:"Energia potenziale elastica"},l("p",null,"Una molla ha sempre energia potenziale elastica pari a:"),l("p",null,l(a.i,null,Ne(nl||(nl=Be`E_{p_e} = \frac{1}{2} k x^2`))))),l(a.o,{title:"Forze conservative"},l("p",null,"Sono conservative le forze per le quali il lavoro compiuto non dipende dal percorso seguito per andare dalla partenza all'arrivo."),l("p",null,"Ad esempio, è conservativa la ",l("i",null,"forza di gravità"),", ma ",l("b",null,"non")," è conservativa la forza di attrito."),l("p",null,"Se in un sistema ci sono solo forze conservative, allora l'energia meccanica totale si conserva:"),l("p",null,l(a.i,null,Ne(ol||(ol=Be`E = E_k + E_p`))))),l(a.o,{title:"Potenza"},l("p",null,"È la velocità di trasferimento di energia:"),l("p",null,l(a.i,null,Ne(tl||(tl=Be`P = \frac{\Delta E}{\Delta t}`)))))),l(a.p,{title:"Elettrostatica"},l(a.o,{title:"Carica elettrica"},l("p",null,"È una proprietà dei corpi che può essere ",l(n.a,null,"positiva")," o ",l(o.a,null,"negativa"),"."),l("p",null,"Si conserva: in un sistema chiuso la carica totale è costante."),l("p",null,"Esiste un'unità elementare: ",l(a.i,null,Ne(rl||(rl=Be`C_{elettrone} = 1.602 \cdot 10^{-19}`))),"."),l("p",null,"Cariche ",l(n.a,null,"opp"),l(o.a,null,"oste")," si attraggono; cariche ",l(n.a,null,"uguali")," si respingono.")),l(a.o,{title:"Conduttori e isolanti"},l("p",null,"Più ",l("a",{href:"https://it.wikipedia.org/wiki/Ione"},"ioni")," ha un corpo, meglio la carica si muove attraverso di esso."),l("p",null,"I corpi in cui la carica si muove bene sono ",l("i",null,"conduttori"),", mentre quelli in cui si muove difficilmente sono ",l("i",null,"isolanti"),"."),l("p",null,l("i",null,"Il corpo umano è un buon conduttore.")))),l(a.p,{title:"Polarizzazione"},l(a.o,{title:"Polarizzazione"},l("p",null,"E' possibile polarizzare un corpo per accumulare la carica di un segno in una certa zona."))),l(a.p,null,l(a.o,{title:"Messa a terra"},l("p",null,"Se un corpo conduttore è in contatto con la Terra, le cariche su di esso saranno ",l("i",null,"equilibrate")," e il corpo diventerà elettricamente neutro (con stesso numero di ",l(n.a,null,"cariche positive")," e ",l(o.a,null,"negative")," all'interno)."))),l(a.p,null,l(a.o,{title:"Polarizzazione per strofinio"},l("p",null,"Strofinando tra loro due corpi isolanti, essi si ",l("i",null,"polarizzeranno per strofinio"),".")),l(a.o,{title:"Polarizzazione per contatto"},l("p",null,"Toccando un conduttore con un corpo carico, il conduttore potrà ",l("i",null,"polarizzarsi per contatto"),".")),l(a.o,{title:"Polarizzazione per induzione"},l("p",null,'Se un corpo conduttore ha cariche "esterne" di un ',l(n.a,null,"certo segno")," vicino, esso avrà tutte le cariche del ",l(o.a,null,"segno opposto")," in equilibrio vicino alle cariche esterne, e tutte le cariche dello ",l(n.a,null,"stesso segno")," più lontano possibile da esse."),l("p",null,"Mettendo a terra il conduttore, nuove cariche del ",l(o.a,null,"segno opposto")," saranno attratte all'interno del corpo per equilibrare le cariche che si sono allontanate."),l("p",null,"Staccando il conduttore da terra e rimuovendo le cariche esterne, esso si ritroverà ",l(o.a,null,"caricato del segno opposto")," rispetto alle cariche esterne."))),l(a.p,{title:"Forza elettrica"},l(a.o,{title:"Legge di Coulomb"},l("p",null,"Due corpi carichi si attraggono tra loro con forza:"),l("p",null,l(a.i,null,Ne(ul||(ul=Be`\left | \vec{F}_{elettrica} \right | = \frac{-k \cdot q_1 \cdot q_2}{s^2}`)))),l("p",null,l(a.i,null,Ne(cl||(cl=Be`k`)))," è la ",l("i",null,"costante di Coulomb"),", e vale ",l(a.i,null,Ne(pl||(pl=Be`k = 8.99 \cdot 10^9 \frac{N \cdot m^2}{C^2}`))),".")),l(a.o,{title:"Permeabilità dello spazio vuoto"},l("p",null,"La costante ",l(a.i,null,Ne(sl||(sl=Be`k`)))," è in realtà dipendente da un altra costante, ",l(a.i,null,Ne(dl||(dl=Be`\epsilon_0`))),", la ",l("i",null,"permeabilità del vuoto"),"."),l("p",null,l(a.i,null,Ne(ml||(ml=Be`k = \frac{1}{4 \pi \cdot \epsilon_0}`)))),l("p",null,l(a.i,null,Ne(vl||(vl=Be`\left | \vec{F}_{elettrica} \right | = \frac{q_1 \cdot q_2}{4 \pi \cdot \epsilon_0 \cdot s^2}`))))),l(a.o,{title:"Campo elettrico"},l("p",null,"Misura che forza viene applicata in ogni punto su una carica unitaria:"),l("p",null,l(a.i,null,Ne(gl||(gl=Be`\vec{E} = \frac{\vec{F}_{elettrica}}{q} = \frac{-k \cdot q}{s^2}`))))),l(a.o,{title:"Flusso elettrico"},l("p",null,'È la differenza tra "quanto" campo elettrico ',l(n.a,null,"entra")," e quanto campo elettrico ",l(o.a,null,"esce")," da una certa area."),l("p",null,"In qualsiasi superficie chiusa, il flusso elettrico è uguale alla componente perpendicolare del campo elettrico moltiplicato per l'area."),l("p",null,l(a.i,null,Ne(fl||(fl=Be`\Phi_E = \vec{E} \cdot \vec{A}`)))),l("p",null,"Se il campo elettrico è uniforme, se ne può calcolare facilmente il valore:"),l("p",null,l(a.i,null,Ne(zl||(zl=Be`\Phi_E = \vec{E} \cdot \vec{A} = E_\perp \cdot A \cdot \cos(\alpha)`)))),l("p",null,l(a.s,null,"Circa. E' una specie di integrale..."))),l(a.o,{title:"Legge di Gauss per i campi elettrostatici"},l("p",null,"Il flusso elettrico è direttamente proporzionale alla carica presente all'interno della superficie."),l("p",null,l(a.i,null,Ne(hl||(hl=Be`\Phi_E = 4 \pi \cdot k \cdot q = \frac{q}{\epsilon_0}`)))),l("p",null,"Ovvero, i campi elettrostatici sono generati dalle cariche elettriche."))),l(a.p,{title:"Energia elettrica"},l(a.o,{title:"Energia potenziale elettrica"},l("p",null,"Un corpo carico vicino ad altre cariche possiede un'",l("i",null,"energia potenziale elettrica")," ",l(a.i,null,Ne(_l||(_l=Be`U_e`))),"."))),l(a.p,{title:"Circuiti elettrici"},l(a.o,{title:l("span",null,"Potenziale elettrico ",l("small",null,"(tensione)"))},l("p",null,"È il valore dell'energia potenziale elettrica per una carica unitaria."),l("p",null,l(a.i,null,Ne(bl||(bl=Be`V = \frac{U_e}{q}`)))),l("p",null,"La sua unità di misura è il Volt (",l(a.i,null,Ne(ql||(ql=Be`V`))),")."),l("p",null,"In una batteria è detto ",l("i",null,"forza elettromotrice"),", e corrisponde al lavoro compiuto da una batteria ideale per spostare una carica unitaria tra i due poli.")),l(a.o,{title:l("span",null,"Corrente elettrica ",l("small",null,"(intensità)"))},l("p",null,"Quanta carica passa attraverso un'area (perpendicolare al flusso) nel tempo."),l("p",null,l(a.i,null,Ne(Ll||(Ll=Be`I = \frac{\Delta q}{\Delta t}`)))),l("p",null,"Fintanto che c'è differenza di potenziale, ci sarà anche intensità non nulla."),l("p",null,"La sua unità di misura è l'Ampere (",l(a.i,null,Ne(Dl||(Dl=Be`A`))),").")),l(a.o,{title:l("span",null,"Corrente continua ",l("small",null,"(",l("abbr",{title:"Direct Current"},"DC"),")"))},l("p",null,"Quando in un circuito la direzione della corrente è costante.")),l(a.o,{title:l("span",null,"Corrente alternata ",l("small",null,"(",l("abbr",{title:"Alternate Current"},"AC"),")"))},l("p",null,"Quando in un circuito la direzione della corrente si alterna periodicamente.")),l(a.o,{title:"Potenza elettrica"},l("p",null,"Possiamo calcolare la potenza di un circuito:"),l("p",null,l(a.i,null,Ne(Fl||(Fl=Be`P = \frac{\Delta U_e}{\Delta t} = I \cdot \Delta V = I^2 \cdot R = \frac{(\Delta V)^2}{R}`)))))),l(a.p,{title:"Elementi di un circuito"},l(a.o,{title:"Resistore"},l("p",null,"Riduce l'intensità di corrente, e converte parte del potenziale in calore."),l("p",null,"Il potenziale utilizzato è pari a:"),l("p",null,l(a.i,null,Ne(Sl||(Sl=Be`V = R \cdot I`)))),l("p",null,"Dove ",l(a.i,null,Ne(El||(El=Be`R`)))," è una costante detta ",l("i",null,"resistenza")," con unità di misura Ohm (",l(a.i,null,Ne(Pl||(Pl=Be`\Omega`))),")."),l("p",null,"La resistenza di un conduttore vale:"),l("p",null,l(a.i,null,Ne(Cl||(Cl=Be`R = \rho \frac{L_{unghezza}}{A_{rea}}`)))),l("p",null,l(a.i,null,Ne(Il||(Il=Be`\rho`)))," è la ",l("i",null,"resistività")," del materiale, e varia in base alla temperatura:"),l("p",null,l(a.i,null,Ne(Al||(Al=Be`\rho = \rho_0 (1 + \alpha(T - T_0))`))))),l(a.o,{title:"Condensatore"},l("p",null,"Immagazzina potenziale elettrico, permettendo di riutilizzarla in seguito."),l("p",null,"Per farlo, cattura cariche ",l(n.a,null,"positive")," e ",l(o.a,null,"negative")," sulle sue due armature; perchè questo avvenga, deve essere compiuto lavoro."),l("p",null,"Ha una ",l("b",null,"capacità")," caratteristica, che in un condensatore a facce piane parallele è:"),l("p",null,l(a.i,null,Ne(wl||(wl=Be`C = \frac{q_{massima}}{\Delta V}`)))),l("p",null,"Condensatori di capacità maggiore immagazzinano più potenziale con meno carica."),l("p",null,"La capacità aumenta se viene messo qualcosa tra le armature:"),l("p",null,l(a.i,null,Ne(kl||(kl=Be`C_{nuova} = \kappa \cdot \frac{\epsilon_0 \cdot A}{s}`)))),l("p",null,"Dove ",l(a.i,null,Ne(Bl||(Bl=Be`\kappa`)))," è la ",l("i",null,"costante dielettrica relativa")," del materiale inserito, ",l(a.i,null,Ne(Nl||(Nl=Be`A`)))," l'area di una armatura e ",l(a.i,null,Ne(Vl||(Vl=Be`s`)))," la distanza tra le due armature."),l("p",null,"Se il campo elettrico creatosi tra le due armature supera la ",l("i",null,"rigidità dielettrica")," del condensatore, la carica immagazzinata viene persa e ha luogo un ",l("i",null,"breakdown"),"."),l("p",null,"La sua unità di misura è il Farad (",l(a.i,null,Ne(Rl||(Rl=Be`Fa`))),")")),l(a.o,{title:"Amperometro"},l("p",null,"Misura la corrente elettrica se messo in serie."),l("p",null,"(Funzionamento: ha una resistenza interna bassisima in modo da non influire significativamente sulla corrente.)")),l(a.o,{title:"Voltmetro"},l("p",null,"Misura la differenza di potenziale se messo in parallelo."),l("p",null,"(Funzionamento: ha una resistenza altissima in modo da non influire significativamente sulla tensione.)"))),l(a.p,{title:"Principi di Kirchhoff"},l(a.o,{title:"Legge dei nodi"},l("p",null,"Per nodo si intende un qualsiasi punto del circuito."),l("p",null,"Da un nodo entra ed esce la stessa corrente.")),l(a.o,{title:"Legge delle maglie"},l("p",null,"Per maglia si intende un qualsiasi percorso chiuso all'interno del circuito."),l("p",null,"In una maglia chiusa, la somma delle differenze di potenziale è 0."))),l(a.p,{title:"Serie e Parallelo"},l(a.o,{title:"Circuito in serie"},l("p",null,"Più parti di circuito sono ",l("i",null,"in serie")," se sono consecutive e senza biforcazioni."),l("p",null,"Parti di circuito in serie sono attraversate dalla stessa corrente.")),l(a.o,{title:"Circuito in parallelo"},l("p",null,"Più parti di circuito sono ",l("i",null,"in parallelo")," tra loro se hanno lo stesso punto di partenza e lo stesso punto di arrivo."),l("p",null,"Parti di circuito in parallelo hanno la stessa differenza di potenziale."))),l(a.p,{title:"Resistenze equivalenti"},l(a.o,{title:"Circuiti in serie"},l("p",null,"Nei circuiti in serie, tutte le resistenze possono essere sostituite con una equivalente dalla resistenza della somma di tutte le quelle sostituite:"),l("p",null,l(a.i,null,Ne(Tl||(Tl=Be`R_{serie} = \sum_{i=1}^{n} R_i`))))),l(a.o,{title:"Circuiti in parallelo"},l("p",null,"Nei circuiti in parallelo, tutte le resistenze possono essere sostituite con una equivalente dalla resistenza di:"),l("p",null,l(a.i,null,Ne(yl||(yl=Be`R_{parallelo} = \frac{1}{\sum_{i=1}^{n} \frac{1}{R_i}}`)))))),l(a.p,{title:"Condensatori equivalenti"},l(a.o,{title:"Circuiti in serie"},l("p",null,"Nei circuiti in serie, tutti i condensatori possono essere sostituiti con uno equivalente dalla capacità di:"),l("p",null,l(a.i,null,Ne(Ml||(Ml=Be`C_{serie} = \frac{1}{\sum_{i=1}^{n} \frac{1}{C_i}}`))))),l(a.o,{title:"Circuiti in parallelo"},l("p",null,"Nei circuiti in parallelo, tutte i condensatori possono essere sostituite con uno equivalente dalla capacità della somma di tutti quelli sostituiti:"),l("p",null,l(a.i,null,Ne(Ul||(Ul=Be`C_{parallelo} = \sum_{i=1}^{n} C_n`)))))),l(a.p,{title:"Magnetismo"},l(a.o,{title:"Permeabilità magnetica dello spazio vuoto"},l("p",null,"E' una costante fisica fondamentale che rappresenta quanto un materiale si magnetizza facilmente."),l("p",null,l(a.i,null,Ne(xl||(xl=Be`\mu_0 = 4 \pi \cdot 10^{-7} \frac{H}{m}`)))," (",l(a.i,null,Ne(Ol||(Ol=Be`\frac{N}{A^2}`))),")")),l(a.o,{title:"Campo magnetico"},l("p",null,"Come un campo elettrico, ma per i magneti."),l("p",null,"Il suo simbolo è ",l(a.i,null,Ne(Gl||(Gl=Be`B`))),", e la sua unità di misura è il Tesla (",l(a.i,null,"T"),").")),l(a.o,{title:"Flusso magnetico"},l("p",null,'È "quanto" campo magnetico ',l("b",null,"attraversa")," un percorso chiuso."),l("p",null,'Per qualsiasi percorso chiuso, il flusso magnetico è uguale alla somma di tutti i "sottoflussi" magnetici calcolati sui suoi lati.'),l("p",null,l(a.i,null,Ne(Ql||(Ql=Be`\Phi_{B_{i}} = \vec{B} \cdot \vec{L}_n = B \cdot L_i \cdot \sin(\alpha) = B_\parallel \cdot L_i`)))),l("p",null,l(a.i,null,Ne(jl||(jl=Be`\Phi_{B} = \sum_{i=0}^{n_{lati}} \Phi_{Bn}`)))),l("p",null,"La sua unità di misura è il Weber (",l(a.i,null,Ne(Jl||(Jl=Be`Wb = T \cdot m^2`))),").")),l(a.o,{title:"Legge di Gauss per i campi magnetici"},l("p",null,"Il flusso magnetico attraverso qualsiasi superficie chiusa è sempre nullo."),l("p",null,"Ovvero, non esistono monopoli magnetici.")),l(a.o,{title:"Legge di Ampère"},l("p",null,"L'intensità di corrente che attraversa un percorso chiuso è direttamente proporzionale al flusso magnetico dello stesso percorso."),l("p",null,l(a.i,null,Ne(Wl||(Wl=Be`\Phi_B = \mu_0 \cdot I`)))))),l(a.p,{title:"Forze magnetiche"},l(a.o,{title:l("span",null,"Forza magnetica su carica puntiforme ",l("small",null,"(Forza di Lorentz)"))},l("p",null,"I campi magnetici applicano una forza sulle cariche vicine:"),l("p",null,l(a.i,null,Ne(Hl||(Hl=Be`\vec{F}_{B} = q \cdot (\vec{v} \times \vec{B})`)))),l("p",null,"Dove ",l(a.i,null,Ne(Zl||(Zl=Be`\vec{B}`)))," è l'intensità del campo magnetico e ",l(a.i,null,Ne(Xl||(Xl=Be`\vec{v}`)))," la velocità della carica considerata."),l("p",null,"Si ha una forza massima se la velocità è perpendicolare al campo magnetico."),l("p",null,"In un campo magnetico uniforme, una velocità perpendicolare al campo porta alla creazione di un moto circolare uniforme.")),l(a.o,{title:"Forza magnetica in un filo"},l("p",null,"I campi magnetici influenzano ovviamente anche le cariche presenti in un conduttore:"),l("p",null,l(a.i,null,Ne(Kl||(Kl=Be`\vec{F}_{magnetica} = I \cdot (\vec{L} \times \vec{B})`)))," ",l("a",{href:"https://it.openprof.com/wb/forza_di_lorentz_su_un_filo_percorso_da_corrente?ch=360"},"[1]")),l("p",null,"Dove ",l(a.i,null,Ne(Yl||(Yl=Be`I`)))," è la corrente elettrica, ",l(a.i,null,Ne($l||($l=Be`\vec{L}`)))," è un vettore che punta nella direzione di scorrimento della corrente e ha come modulo la lunghezza del conduttore."))),l(a.p,{title:"Campi magnetici"},l(a.o,{title:"Campo magnetico in una spira"},l("p",null,"Una spira in cui passa corrente produce un campo magnetico perpendicolare al piano creato dalla spira.")),l(a.o,{title:"Campo magnetico di un solenoide"},l("p",null,"Un solenoide sono tante spire avvolte in modo da formare una specie di cilindro."),l("p",null,"All'interno del solenoide si crea un campo (quasi) uniforme:"),l("p",null,l(a.i,null,Ne(le||(le=Be`\left | \vec{B} \right | = \mu_0 \cdot I \cdot \frac{A_{vvolgimenti}}{L_{unghezzafilo}}`))))),l(a.o,{title:"Legge di Oersted"},l("p",null,l("i",null,"Caso particolare della ",l("a",{href:"https://it.wikipedia.org/wiki/Legge_di_Amp%C3%A8re"},"Legge di Ampère"),".")),l("p",null,"Il modulo del campo magnetico ",l(a.i,null,"B")," prodotto da un filo in cui passa una corrente continua ",l(a.i,null,"I")," alla distanza ",l(a.i,null,"s")," è:"),l("p",null,l(a.i,null,Ne(ee||(ee=Be`\left | \vec{B} \right | = \frac{\mu \cdot I}{2 \pi r}`)))),l("p",null,"Il campo magnetico così creato gira attorno al filo in senso antiorario."),l("p",null,"Due fili attraversati dalla ",l(n.a,null,"stessa corrente")," si attraggono, due fili attraversati da ",l(n.a,null,"corr"),l(o.a,null,"enti")," ",l(n.a,null,"opp"),l(o.a,null,"oste")," si respingono."))),l(a.p,{title:"Induzione elettromagnetica"},l(a.o,{title:"Forza elettromotrice indotta"},l("p",null,"Un conduttore perpendicolare ad un campo magnetico può ottenere una differenza di potenziale se messo in movimento in un direzione perpendicolare alla direzione del conduttore e del campo."),l("p",null,"La differenza di potenziale si crea a causa della forza magnetica, che fa spostare tutti gli elettroni verso un capo del conduttore."),l("p",null,"Essa vale:"),l("p",null,l(a.i,null,Ne(ie||(ie=Be`\Delta V_{indotta} = v \cdot B \cdot L`)))),l("p",null,"Dove ",l(a.i,null,"v")," è la velocità del conduttore, ",l(a.i,null,"B")," è l'intensità del campo magnetico ed ",l(a.i,null,"L")," è la lunghezza del conduttore.")),l(a.o,{title:"Flusso magnetico in una spira"},l("p",null,"In un campo magnetico ",l(a.i,null,Ne(ae||(ae=Be`B`)))," uniforme e perpendicolare al piano di una spira di area ",l(a.i,null,Ne(ne||(ne=Be`A`))),", il flusso magnetico si può determinare con la ",l("i",null,"Legge di Faraday-Neumann-Lenz"),":"),l("p",null,l(a.i,null,Ne(oe||(oe=Be`\Phi_B = \vec{B} \cdot \vec{A} = B \cdot A \cdot \cos(\alpha)`)))))),l(a.p,null,l(a.o,{title:"Legge di Faraday-Neumann-Lenz"},l("p",null,"Dice che la forza elettromotrice media indotta in un percorso dipende dalla variazione nel tempo del flusso magnetico nello stesso percorso."),l("p",null,l(a.i,null,Ne(te||(te=Be`\Delta V_{indotta} = - \frac{\Delta \Phi_B}{\Delta t}`)))),l("p",null,"Il meno è dovuto alla ",l("a",{href:"https://it.wikipedia.org/wiki/Legge_di_Lenz"},"Legge di Lenz"),", che specifica qualitativamente il verso della forza elettromotrice indotta.")),l(a.o,{title:"Faraday in un solenoide"},l("p",null,"In un solenoide, la forza elettromotrice indotta è uguale a:"),l("p",null,l(a.i,null,Ne(re||(re=Be`\Delta V_{indotta} = - \frac{N \cdot \Delta \Phi_{B_{spira}}}{\Delta t} = - \frac{N \cdot B \cdot A \cdot cos(\alpha)}{\Delta t}`)))),l("p",null,"Dove ",l(a.i,null,Ne(ue||(ue=Be`N`)))," è il numero delle spire del solenoide.")),l(a.o,{title:"Legge di Ampère-Maxwell"},l("p",null,"Correnti o campi elettrici variabili creano un campo magnetico."))),l(a.p,{title:"Elettromagnetismo"},l(a.o,{title:"Onde elettromagnetiche"},l("p",null,"Nel vuoto, il campo elettrico ",l(a.i,null,Ne(ce||(ce=Be`E`)))," e il campo magnetico ",l(a.i,null,Ne(pe||(pe=Be`B`)))," sono perpendicolari tra loro e la direzione di propagazione, e sono entrambe funzioni del tempo."),l("p",null,"Si dice quindi che sono ",l("i",null,"onde elettromagnetiche"),"."),l("p",null,"Esse sono legate dalla relazione:"),l("p",null,l(a.i,null,Ne(se||(se=Be`E = c \cdot B`)))),l("p",null,"Dove ",l(a.i,null,Ne(de||(de=Be`c`)))," è la velocità delle onde (luce) nel vuoto, e a sua volta è uguale a:"),l("p",null,l(a.i,null,Ne(me||(me=Be`c = \frac{1}{\sqrt{\epsilon_0 \cdot \mu_0}} = 3.00 \cdot 10^8 \frac{m}{s}`))))),l(a.o,{title:"Formula delle onde"},l("p",null,l(a.i,null,Ne(ve||(ve=Be`A(t) = A_{max} \cdot \sin \left ( \frac{2 \pi}{\lambda} - \omega t + \phi \right )`)))),l("p",null,"Dove ",l(a.i,null,Ne(ge||(ge=Be`A_{max}`)))," è l'ampiezza massima che può avere l'onda, ",l(a.i,null,Ne(fe||(fe=Be`\frac{2 \pi}{\lambda} = \left | \vec{k} \right |`)))," è il vettore d'onda, ",l(a.i,null,Ne(ze||(ze=Be`\omega`)))," la frequenza angolare e ",l(a.i,null,Ne(he||(he=Be`\phi`)))," la fase."))),l(a.p,{title:"Spettroscopia"},l(a.o,{title:"Emissione"},l("p",null,"I solidi, se portati ad alta temperatura, emettono luce con uno ",l("a",{href:"https://it.wikipedia.org/wiki/Spettro_continuo"},"spettro continuo"),"."),l("p",null,"I gas, invece, ad alta temperatura emettono luce solo con particolari lunghezze d'onda."),l("p",null,"In un gas di idrogeno, le lunghezze d'onda emesse sono ricavabili con:"),l("p",null,l(a.i,null,Ne(_e||(_e=Be`\frac{1}{\lambda} = R \left ( \frac{1}{4} - \frac{1}{n^2} \right )`)))),l("p",null,"Con ",l(a.i,null,Ne(be||(be=Be`R = 1.097 \cdot 10^7 \frac{1}{m}`))),", detta costante di Rydberg, e ",l(a.i,null,Ne(qe||(qe=Be`n`)))," un numero intero.")),l(a.o,{title:"Grandezza quantizzata"},l("p",null,"Una grandezza si dice quantizzata (o discreta) se può assumere solo determinati valori."),l("p",null,"Una grandezza si dice continua se può assumere qualsiasi valore e quindi se non è quantizzata."),l("p",null,"Energia, momento angolare e raggio sono quantizzati."),l("p",null,"Nota costante quantica è ",l(a.i,null,Ne(Le||(Le=Be`h`))),", la costante di Planck, ovvero il valore minimo possibile per la carica (talvolta espressa come ",l(a.i,null,Ne(De||(De=Be`\hbar = \left ( \frac{h}{2 \pi} \right )`))),"."))),l(a.p,null,l(a.o,{title:"Modello di Bohr"},l("p",null,"L'energia degli elettroni è quantizzata."),l("p",null,"Inoltre, per essi è valido che:"),l("p",null,l(a.i,null,Ne(Fe||(Fe=Be`m \cdot v_n \cdot 2 \pi \cdot r = n \cdot h`)))),l("p",null,"Ancora, il raggio delle orbite è uguale a:"),l("p",null,l(a.i,null,Ne(Se||(Se=Be`r_n = n^2 \cdot a_0 = n^2 \cdot \frac{\hbar}{m_{elettrone} \cdot k \cdot e^2} `)))),l("p",null,"Con ",l(a.i,null,Ne(Ee||(Ee=Be`a_0 = \left ( \frac{h}{2 \pi} \right )^2 \cdot \frac{1}{m_{elettrone} \cdot k \cdot e^2} = 5.29 \cdot 10^{-11} m`))),"."),l("p",null,"Infine, in ogni stato, l'energia è pari a:"),l("p",null,l(a.i,null,Ne(Pe||(Pe=Be`E_n = \frac{1}{n^2} \cdot E_1 = - \frac{1}{n^2} \cdot \frac{a_0^2}{2 \cdot m \cdot \hbar^4} = - \frac{1}{n^2} \cdot \frac{m_{elettrone} \cdot k^2 \cdot e^4}{2 \cdot \hbar^2}`)))),l("p",null,"Due elettroni non possono occupare lo stesso stato."),l("p",null,"Questo modello funziona solo per atomi con numero atomico basso. Atomi con molti elettroni hanno comportamenti diversi, descritti dal modello di"))),l(a.p,null,l(a.o,{title:"Nei solidi"},l("p",null,"Nei solidi, le lunghezze d'onda sono talmente tanto vicine da poter essere considerate una banda."),l("p",null,"Possono però comunque avere dei gap dovuti agli intervalli di energia non ammessi."))),l(a.p,{title:"Semiconduttori"},l(a.o,{title:"Semiconduttori"},l("p",null,l(a.s,null,"Refactor this")),l("p",null,"Se la banda di emissione con energia più alta di un corpo è assente o è separata da un gap dell'ordine di grandezza maggiore di ",l(a.i,null,Ne(Ce||(Ce=Be`10^1 eV`))),", allora il corpo è un isolante."),l("p",null,"Se invece la banda di emissione si sovrappone a un altra, allora il corpo è un conduttore."),l("p",null,"Se il gap è invece dell'ordine di grandezza di ",l(a.i,null,Ne(Ie||(Ie=Be`1 eV`))),", allora il corpo è un semiconduttore.")),l(a.o,{title:"Lacune"},l("p",null,"Legami in cui ",l(n.a,null,"mancano elettroni"),"."),l("p",null,l(o.a,null,"Elettroni")," di altri legami possono spostarsi per colmare le ",l(n.a,null,"lacune"),", creandone altre, e spostandole in direzione opposta a quella della corrente.")),l(a.o,{title:"Accettori e donori"},l("p",null,"Se si inserisce in un cristallo semiconduttore si inserisce un atomo con numero atomico diverso, si otterrà:"),l("ul",null,l("li",null,"Con numero atomico maggiore, un semiconduttore di ",l(o.a,null,"tipo N")," con ",l(o.a,null,"elettroni in eccesso")," liberi di scorrere."),l("li",null,"Con numero atomico minore, un semiconduttore di ",l(n.a,null,"tipo P")," con ",l(n.a,null,"lacune in eccesso")," libere di catturare elettroni da altri legami.")),l("p",null,"Maggiore impurezza porta a maggiore conduttività.")),l(a.o,{title:"Temperatura"},l("p",null,"Aumentando la temperatura di un semiconduttore si aumenta la conduttività, perchè eccita le particelle e favorisce il movimento di ",l(o.a,null,"elettroni")," e ",l(n.a,null,"lacune"),"."))),l(a.p,{title:l("span",null,"Ottica ",l("small",null,"(non l'abbiamo fatta)"))},l(a.o,{title:"Assorbimento e riflessione"},l("p",null,"I corpi possono assorbire o riflettere le onde elettromagnetiche che li colpiscono.")),l(a.o,{title:"Corpo nero"},l("p",null,"Un corpo nero è un corpo che assorbe tutte le onde elettromagnetiche che riceve senza rifletterne nessuna."),l("p",null,"Le onde assorbite vengono poi riemesse sotto forma di un onda di ",l(a.i,null,Ne(Ae||(Ae=Be`\lambda`)))," variabile in base alla temperatura."),l("p",null,l(a.i,null,Ne(we||(we=Be`\lambda_{max} \cdot T`)))," è costante.")),l(a.o,{title:"Teoria di Planck per il corpo nero"},l("p",null,"L'energia assorbita e emessa dai corpi neri è quantizzata.")),l(a.o,{title:"Fotone"},l("p",null,"Un onda magnetica con un quanto di energia è detta ",l("i",null,"fotone"),":"),l("p",null,l(a.i,null,Ne(ke||(ke=Be`E_{fotone} = h \cdot f`))))),l(a.o,{title:"Effetto fotoelettrico"},l("p",null,"A volte, i fotoni che colpiscono un metallo possono estrarvi degli elettroni e creare una differenza di potenziale."),l("p",null,"Perchè avvenga, la frequenza deve essere maggiore di una certa soglia."),l("p",null,"Il numero di elettroni estratti dipende dall'intensità dell'onda, mentre l'energia cinetica degli elettroni dipende dalla frequenza."),l("p",null,"Non c'è nessun ritardo tra l'assorbimento del fotone e l'estrazione di elettroni."))))))}}.call(this,i("hosL").h)},ZHMS:function(l,e,i){"use strict";(function(l){var a=i("gJgL"),n=i.n(a);e.a=function(e){return 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W"]);return ke=function(){return e},e}function Be(){var e=Sn(["E_c = \frac{1}{2} m v^2"],["E_c = \\frac{1}{2} m v^2"]);return Be=function(){return e},e}function Ne(){var e=Sn(["W = \vec{F} cdot \vec{s} = F cdot Delta s cdot cos(alpha )"],["W = \\vec{F} \\cdot \\vec{s} = F \\cdot \\Delta s \\cdot cos(\\alpha )"]);return Ne=function(){return e},e}function Ve(){var e=Sn(["F = m cdot a"],["F = m \\cdot a"]);return Ve=function(){return e},e}function Re(){var e=Sn(["a = \frac{v^2}{r} = r cdot omega^2 = v cdot omega"],["a = \\frac{v^2}{r} = r \\cdot \\omega^2 = v \\cdot \\omega"]);return Re=function(){return e},e}function Te(){var e=Sn(["v = \frac{Delta s}{t} = \frac{2 pi cdot r}{T} = omega r"],["v = \\frac{\\Delta s}{t} = \\frac{2 \\pi \\cdot r}{T} = \\omega r"]);return Te=function(){return e},e}function xe(){var e=Sn(["phi"],["\\phi"]);return xe=function(){return e},e}function ye(){var e=Sn(["omega = \frac{2 pi}{T}"],["\\omega = \\frac{2 \\pi}{T}"]);return ye=function(){return e},e}function Ue(){var e=Sn(["\frac{pi}{2}"],["\\frac{\\pi}{2}"]);return Ue=function(){return e},e}function Me(){var e=Sn(["a(t) = A sin (omega cdot t + phi + pi)"],["a(t) = A \\sin (\\omega \\cdot t + \\phi + \\pi)"]);return Me=function(){return e},e}function Oe(){var e=Sn(["pi"],["\\pi"]);return Oe=function(){return e},e}function Ge(){var e=Sn(["v(t) = A sin (omega cdot t + phi + \frac{pi}{2})"],["v(t) = A \\sin (\\omega \\cdot t + \\phi + \\frac{\\pi}{2})"]);return Ge=function(){return e},e}function We(){var e=Sn(["\frac{pi}{2}"],["\\frac{\\pi}{2}"]);return We=function(){return e},e}function Qe(){var e=Sn(["s(t) = A sin (omega cdot t + phi)"],["s(t) = A \\sin (\\omega \\cdot t + \\phi)"]);return Qe=function(){return e},e}function Je(){var e=Sn(["omega = \frac{2 pi}{T}"],["\\omega = \\frac{2 \\pi}{T}"]);return Je=function(){return e},e}function He(){var e=Sn(["a(t) = k"]);return He=function(){return e},e}function Ze(){var e=Sn(["v(t) = a Delta t + v(0)"],["v(t) = a \\Delta t + v(0)"]);return Ze=function(){return e},e}function Xe(){var e=Sn(["s(t) = \frac{1}{2} a cdot (Delta t)^2 + v(0) cdot (Delta t) + s(0)"],["s(t) = \\frac{1}{2} a \\cdot (\\Delta t)^2 + v(0) \\cdot (\\Delta t) + s(0)"]);return Xe=function(){return e},e}function Ke(){var e=Sn(["a(t) = 0"]);return Ke=function(){return e},e}function Ye(){var e=Sn(["v(t) = k"]);return Ye=function(){return e},e}function $e(){var e=Sn(["s(t) = v cdot Delta t + s(0)"],["s(t) = v \\cdot \\Delta t + s(0)"]);return $e=function(){return e},e}function en(){var e=Sn(["Sigma \vec{F} = 0 Longleftrightarrow Delta \vec{p} = 0"],["\\Sigma \\vec{F} = 0 \\Longleftrightarrow \\Delta \\vec{p} = 0"]);return en=function(){return e},e}function nn(){var e=Sn(["\vec{p} = m \vec{v}"],["\\vec{p} = m \\vec{v}"]);return nn=function(){return e},e}function tn(){var e=Sn(["\vec{a} = lim_{Delta v \to 0} \frac{Delta \vec{v}}{Delta t} = \frac{d \vec{v}}{d t} = \frac{d^2 \vec{s}}{d t^2}"],["\\vec{a} = \\lim_{\\Delta v \\to 0} \\frac{\\Delta \\vec{v}}{\\Delta t} = \\frac{d \\vec{v}}{d t} = \\frac{d^2 \\vec{s}}{d t^2}"]);return tn=function(){return e},e}function ln(){var e=Sn(["\vec{a} = \frac{Delta \vec{v}}{Delta t}"],["\\vec{a} = \\frac{\\Delta \\vec{v}}{\\Delta t}"]);return ln=function(){return e},e}function an(){var e=Sn(["\vec{v} = lim_{Delta t \to 0} \frac{Delta \vec{s}}{Delta t} = \frac{d \vec{s}}{dt}"],["\\vec{v} = \\lim_{\\Delta t \\to 0} \\frac{\\Delta \\vec{s}}{\\Delta t} = \\frac{d \\vec{s}}{dt}"]);return an=function(){return e},e}function rn(){var e=Sn(["\vec{v} = \frac{Delta \vec{s}}{Delta t}"],["\\vec{v} = \\frac{\\Delta \\vec{s}}{\\Delta t}"]);return rn=function(){return e},e}function on(){var e=Sn(["Delta \vec{s} = \vec{s}(fine) - \vec{s}(inizio)"],["\\Delta \\vec{s} = \\vec{s}(fine) - \\vec{s}(inizio)"]);return on=function(){return e},e}function un(){var e=Sn(["F = -k x"]);return un=function(){return e},e}function cn(){var e=Sn(["left | \vec{F} \right | leq mu_{d} left | \vec{F}_{normale} \right |"],["\\left | \\vec{F} \\right | \\leq \\mu_{d} \\left | \\vec{F}_{normale} \\right |"]);return cn=function(){return e},e}function pn(){var e=Sn(["left | \vec{F} \right | leq mu_{s} left | \vec{F}_{normale} \right |"],["\\left | \\vec{F} \\right | \\leq \\mu_{s} \\left | \\vec{F}_{normale} \\right |"]);return pn=function(){return e},e}function sn(){var e=Sn(["g_{marte} = 3.71 \frac{m}{s^2}"],["g_{marte} = 3.71 \\frac{m}{s^2}"]);return sn=function(){return e},e}function dn(){var e=Sn(["g_{luna} = 1.62 \frac{m}{s^2}"],["g_{luna} = 1.62 \\frac{m}{s^2}"]);return dn=function(){return e},e}function fn(){var e=Sn(["left | \vec{F} \right | = g m"],["\\left | \\vec{F} \\right | = g m"]);return fn=function(){return e},e}function vn(){var e=Sn(["g = 9.81 \frac{m}{s^2}"],["g = 9.81 \\frac{m}{s^2}"]);return vn=function(){return e},e}function mn(){var e=Sn(["left | \vec{F} \right | = g m"],["\\left | \\vec{F} \\right | = g m"]);return mn=function(){return e},e}function gn(){var e=Sn(["G = 6.67 cdot 10^{-11} \frac{N m^2}{{kg}^2}"],["G = 6.67 \\cdot 10^{-11} \\frac{N m^2}{{kg}^2}"]);return gn=function(){return e},e}function _n(){var e=Sn(["left | \vec{F} \right | = G \frac{m_1 m_2}{s^2}"],["\\left | \\vec{F} \\right | = G \\frac{m_1 m_2}{s^2}"]);return _n=function(){return e},e}function hn(){var e=Sn(["\vec{F}_{21} = -\vec{F}_{12}"],["\\vec{F}_{21} = -\\vec{F}_{12}"]);return hn=function(){return e},e}function zn(){var e=Sn(["Sigma \vec{F} = m \vec{a}"],["\\Sigma \\vec{F} = m \\vec{a}"]);return zn=function(){return e},e}function jn(){var e=Sn(["Sigma \vec{F} = 0 Longleftrightarrow Delta v = 0"],["\\Sigma \\vec{F} = 0 \\Longleftrightarrow \\Delta v = 0"]);return jn=function(){return e},e}function qn(){var e=Sn(["left | \vec{c} \right | = left | \vec{a} \right | cdot left | \vec{b} \right | cdot sin(alpha)"],["\\left | \\vec{c} \\right | = \\left | \\vec{a} \\right | \\cdot \\left | \\vec{b} \\right | \\cdot \\sin(\\alpha)"]);return qn=function(){return e},e}function bn(){var e=Sn(["\vec{c} = \vec{a} \times \vec{b}"],["\\vec{c} = \\vec{a} \\times \\vec{b}"]);return bn=function(){return e},e}function Dn(){var e=Sn(["\vec{b}"],["\\vec{b}"]);return Dn=function(){return e},e}function Ln(){var e=Sn(["\vec{a}"],["\\vec{a}"]);return Ln=function(){return e},e}function Fn(){var e=Sn(["\vec{v} cdot \vec{w} = left | \vec{v} \right | left | \vec{w} \right | cos alpha"],["\\vec{v} \\cdot \\vec{w} = \\left | \\vec{v} \\right | \\left | \\vec{w} \\right | \\cos \\alpha"]);return Fn=function(){return e},e}function En(){var e=Sn(["\vec{v} - \vec{w} = (\vec{v}_x - \vec{w}_x) + (\vec{v}_y - \vec{w}_y)"],["\\vec{v} - \\vec{w} = (\\vec{v}_x - \\vec{w}_x) + (\\vec{v}_y - \\vec{w}_y)"]);return En=function(){return e},e}function An(){var e=Sn(["\vec{v} + \vec{w} = (\vec{v}_x + \vec{w}_x) + (\vec{v}_y + \vec{w}_y)"],["\\vec{v} + \\vec{w} = (\\vec{v}_x + \\vec{w}_x) + (\\vec{v}_y + \\vec{w}_y)"]);return An=function(){return e},e}function Pn(){var e=Sn(["left | \vec{v}_y \right | = left | \vec{v} \right | cos alpha"],["\\left | \\vec{v}_y \\right | = \\left | \\vec{v} \\right | \\cos \\alpha"]);return Pn=function(){return e},e}function Cn(){var e=Sn(["left | \vec{v}_x \right | = left | \vec{v} \right | sin alpha"],["\\left | \\vec{v}_x \\right | = \\left | \\vec{v} \\right | \\sin \\alpha"]);return Cn=function(){return e},e}function In(){var e=Sn(["\vec{v} = \vec{v}_x + \vec{v}_y"],["\\vec{v} = \\vec{v}_x + \\vec{v}_y"]);return In=function(){return e},e}function Sn(e,n){return n||(n=e.slice(0)),e.raw=n,e}var wn=t("mbOI"),kn=t("o8Jj"),Bn=t("ZHMS"),Nn=String.raw;n.default=function(){return e(wn.l.Provider,{value:!1},e(wn.k.Provider,{value:wn.m.INLINE},e("div",null,e("h1",null,"Fisica"),e(wn.q,{title:"Vettori"},e(wn.p,{title:"Componenti cartesiane"},e("p",null,"Usa le regole base della trigonometria:"),e("p",null,e(wn.j,null,Nn(In()))),e("p",null,e(wn.j,null,Nn(Cn()))),e("p",null,e(wn.j,null,Nn(Pn())))),e(wn.p,{title:"Somma"},e("p",null,"Scomponi in componenti, poi sommali:"),e("p",null,e(wn.j,null,Nn(An()))),e("p",null,"Produce il vettore risultante dall'applicazione della regola del parallelogramma.")),e(wn.p,{title:"Differenza"},e("p",null,"Alla fine è sempre una somma:"),e("p",null,e(wn.j,null,Nn(En()))),e("p",null,"Produce il vettore che parte da ",e(wn.j,null,"w")," e arriva a ",e(wn.j,null,"v"),".")),e(wn.p,{title:"Prodotto scalare"},e("p",null,"Si chiama scalare perchè il risultato è uno scalare, non un vettore."),e("p",null,e(wn.j,null,Nn(Fn()))),e("p",null,"Produce il modulo della proiezione di ",e(wn.j,null,Nn(Ln()))," su ",e(wn.j,null,Nn(Dn())),".")),e(wn.p,{title:"Prodotto vettoriale"},e("p",null,"Si chiama vettoriale perchè il risultato è un altro vettore."),e("ul",null,e("li",null,e(wn.j,null,Nn(bn()))),e("li",null,e(wn.j,null,Nn(qn()))),e("li",null,e("a",{href:"https://it.wikipedia.org/wiki/Regola_della_mano_destra"},"Regola della mano destra"))),e("p",null,"Non è commutativo!"))),e(wn.q,{title:"Leggi di Newton"},e(wn.p,{title:"1ᵃ: Inerzia"},e("p",null,"Se un corpo puntiforme ha forza risultante nulla, allora la sua velocità non cambia."),e("p",null,e(wn.j,null,Nn(jn())))),e(wn.p,{title:"2ᵃ: Proporzionalità"},e("p",null,"La forza risultante di un corpo è direttamente proporzionale alla sua accelerazione, e la costante di proporzionalità è la ",e("i",null,"massa"),"."),e("p",null,e(wn.j,null,Nn(zn())))),e(wn.p,{title:"3ᵃ: Azione e reazione"},e("p",null,"Due corpi esercitano forze uguali e opposte uno sull'altro."),e("p",null,e(wn.j,null,Nn(hn()))))),e(wn.q,{title:"Forza di gravità"},e(wn.p,{title:"Tra due corpi"},e("p",null,"Due corpi puntiformi si attirano uno verso l'altro con forza:"),e("p",null,e(wn.j,null,Nn(_n()))),e("p",null,e(wn.j,null,"G")," è la ",e("i",null,"costante di gravitazione universale")," e vale:"),e("p",null,e(wn.j,null,Nn(gn())))),e(wn.p,{title:"Verso la Terra"},e("p",null,"Se nel sistema di riferimento consideriamo la Terra ferma, allora un corpo è attratto verso la Terra con forza ",e("i",null,"peso")," uguale a:"),e("p",null,e(wn.j,null,Nn(mn()))),e("p",null,e(wn.j,null,"g")," è la ",e("i",null,"costante di gravità")," della Terra, e vale:"),e("p",null,e(wn.j,null,Nn(vn())))),e(wn.p,{title:"Su pianeti diversi"},e("p",null,"Per pianeti diversi dalla Terra vale la stessa regola:"),e("p",null,e(wn.j,null,Nn(fn()))),e("p",null,"L'unica differenza è che cambia la ",e("i",null,"costante di gravità"),":"),e("p",null,e(wn.j,null,Nn(dn()))),e("p",null,e(wn.j,null,Nn(sn()))))),e(wn.q,{title:"Forze di contatto"},e(wn.p,{title:"Normale"},e("p",null,"Si oppone alle forze applicate alla superficie di contatto."),e("p",null,"Un libro appoggiato su un tavolo ha la ",e("b",null,"forza di gravità")," che lo attira verso il terreno e la ",e("b",null,"forza normale")," che lo trattiene dal cadere.")),e(wn.p,{title:"Attrito statico"},e("p",null,"Impedisce a un corpo di muoversi se non viene spinto da una forza che supera una certa soglia:"),e("p",null,e(wn.j,null,Nn(pn())))),e(wn.p,{title:"Attrito dinamico"},e("p",null,"Rallenta i corpi che si stanno muovendo finchè essi non si fermano:"),e("p",null,e(wn.j,null,Nn(cn())))),e(wn.p,{title:"Tensione"},e("p",null,"E' forza trasmessa tra due estremi di una fune."),e("p",null,"Può essere redirezionata per mezzo di carrucole.")),e(wn.p,{title:"Elastica"},e("p",null,"Una molla cerca sempre di tornare alla sua posizione indeformata con forza:"),e("p",null,e(wn.j,null,Nn(un()))),e("p",null,"(E' negativa perchè la forza è opposta a quella applicata per deformarla.)"))),e(wn.q,{title:"Cinematica"},e(wn.p,{title:"Spostamento"},e("p",null,"È un vettore che indica la posizione di un corpo rispetto a un'origine."),e("p",null,e(wn.j,null,Nn(on())))),e(wn.p,{title:"Velocità"},e("p",null,"È un vettore che misura la variazione di posizione nel tempo."),e("p",null,e(wn.j,null,Nn(rn()))),e("p",null,"Se si considera un intervallo di tempo infinitesimale si dice ",e("i",null,"velocità istantanea"),":"),e("p",null,e(wn.j,null,Nn(an())))),e(wn.p,{title:"Accelerazione"},e("p",null,"È un vettore che misura la variazione di velocità nel tempo."),e("p",null,e(wn.j,null,Nn(ln()))),e("p",null,"Se si considera un intervallo di tempo infinitesimale si dice ",e("i",null,"accelerazione istantanea"),":"),e("p",null,e(wn.j,null,Nn(tn())))),e(wn.p,{title:e("span",null,"Quantità di moto ",e("small",null,"(momento lineare)"))},e("p",null,"La quantità di moto è una proprietà vettoriale dei corpi:"),e("p",null,e(wn.j,null,Nn(nn()))),e("p",null,"Se la forza risultante è nulla, la quantità di moto non cambia."),e("p",null,e(wn.j,null,Nn(en()))))),e(wn.q,{title:"Moto rettilineo uniforme"},e(wn.p,{title:"Spostamento"},e("p",null,"La ",e("i",null,"legge oraria")," è:"),e("p",null,e(wn.j,null,Nn($e())))),e(wn.p,{title:"Velocità"},e("p",null,"È costante:"),e("p",null,e(wn.j,null,Nn(Ye())))),e(wn.p,{title:"Accelerazione"},e("p",null,"La velocità non varia:"),e("p",null,e(wn.j,null,Nn(Ke())))),e(wn.p,{title:"Forze"},e("p",null,"Si applica la prima legge di Newton:"),e("p",null,e(wn.j,null,"f(t) = 0")))),e(wn.q,{title:"Moto rettilineo uniformemente accelerato"},e(wn.p,{title:"Spostamento"},e("p",null,"La ",e("i",null,"legge oraria")," è:"),e("p",null,e(wn.j,null,Nn(Xe())))),e(wn.p,{title:"Velocità"},e("p",null,"È una retta:"),e("p",null,e(wn.j,null,Nn(Ze())))),e(wn.p,{title:"Accelerazione"},e("p",null,"È costante:"),e("p",null,e(wn.j,null,Nn(He())))),e(wn.p,{title:"Forze"},e("p",null,"Si applica la prima legge di Newton:"),e("p",null,e(wn.j,null,"f(t) = m a")))),e(wn.q,{title:"Moto armonico semplice"},e(wn.p,{title:"Ampiezza"},e("p",null,"E' la distanza dal centro massima che raggiunge il corpo."),e("p",null,"(L'ampiezza di una sinusoide.)")),e(wn.p,{title:"Velocità angolare"},e("p",null,"Indica quanto in fretta cambia la posizione del corpo."),e("p",null,"Dipende dal periodo:"),e("p",null,e(wn.j,null,Nn(Je())))),e(wn.p,{title:"Spostamento"},e("p",null,"E' una sinusoide:"),e("p",null,e(wn.j,null,Nn(Qe())))),e(wn.p,{title:"Velocità"},e("p",null,"E' la sinusoide dello spostamento, sfasata di ",e(wn.j,null,Nn(We())),":"),e("p",null,e(wn.j,null,Nn(Ge())))),e(wn.p,{title:"Accelerazione"},e("p",null,"E' la sinusoide della velocità, sfasata di ",e(wn.j,null,Nn(Oe())),":"),e("p",null,e(wn.j,null,Nn(Me())))),e(wn.p,{title:"Forze"},e("p",null,"Si applica la prima legge di Newton:"),e("p",null,e(wn.j,null,"f(t) = m a")))),e(wn.q,{title:"Moti composti"},e(wn.p,{title:"Moto parabolico"},e("p",null,"Il moto parabolico è dato sommando un moto rettilineo uniforme sull'asse orizzontale e un moto rettilineo uniformemente accelerato sull'asse verticale.")),e(wn.p,{title:"Moto circolare uniforme"},e("p",null,"Il moto parabolico è dato sommando due moti armonici semplici: uno sull'asse X, e l'altro, sfasato di ",e(wn.j,null,Nn(Ue())),", sull'asse Y."))),e(wn.q,{title:"Moto circolare uniforme"},e(wn.p,null,e("h3",null,"Velocità angolare"),e("p",null,"Quanto cambia la fase nel tempo."),e("p",null,e(wn.j,null,Nn(ye())))),e(wn.p,{title:"Fase"},e("p",null,"E' l'angolo percorso dal corpo rispetto alla posizione iniziale."),e("p",null,"Si indica con ",e(wn.j,null,Nn(xe())),", e generalmente si usa in radianti.")),e(wn.p,{title:"Velocità"},e("p",null,"Si applicano le formule per la circonferenza:"),e("p",null,e(wn.j,null,Nn(Te())))),e(wn.p,{title:"Accelerazione"},e("p",null,"Il corpo ha sempre un accelerazione verso il centro che gli impedisce di abbandonare il moto:"),e("p",null,e(wn.j,null,Nn(Re())))),e(wn.p,{title:"Forza centripeta"},e("p",null,"È verso il centro e si calcola con:"),e("p",null,e(wn.j,null,Nn(Ve()))))),e(wn.q,{title:"Lavoro ed energia"},e(wn.p,{title:"Lavoro"},e("p",null,"E' compiuto da una forza che sposta un corpo."),e("p",null,e(wn.j,null,Nn(Ne()))),e("p",null,"(Se la forza non è parallela allo spostamento, il prodotto scalare ci fa considerare solo la componente parallela.)")),e(wn.p,{title:"Energia cinetica"},e("p",null,"Un corpo ha energia cinetica in ogni momento uguale a:"),e("p",null,e(wn.j,null,Nn(Be()))),e("p",null,"Se una forza effettua lavoro su un corpo, cambia la sua energia cinetica pari al lavoro effettuato:"),e("p",null,e(wn.j,null,Nn(ke())))),e(wn.p,{title:"Energia potenziale gravitazionale"},e("p",null,"Un corpo ha energia potenziale in ogni momento pari a:"),e("p",null,e(wn.j,null,Nn(we()))),e("p",null,"(Con ",e(wn.j,null,"h")," uguale a un altezza scelta come punto di riferimento.)")),e(wn.p,{title:"Energia potenziale elastica"},e("p",null,"Una molla ha sempre energia potenziale elastica pari a:"),e("p",null,e(wn.j,null,Nn(Se())))),e(wn.p,{title:"Forze conservative"},e("p",null,"Sono conservative le forze per le quali il lavoro compiuto non dipende dal percorso seguito per andare dalla partenza all'arrivo."),e("p",null,"Ad esempio, è conservativa la ",e("i",null,"forza di gravità"),", ma ",e("b",null,"non")," è conservativa la forza di attrito."),e("p",null,"Se in un sistema ci sono solo forze conservative, allora l'energia meccanica totale si conserva:"),e("p",null,e(wn.j,null,Nn(Ie())))),e(wn.p,{title:"Potenza"},e("p",null,"È la velocità di trasferimento di energia:"),e("p",null,e(wn.j,null,Nn(Ce()))))),e(wn.q,{title:"Elettrostatica"},e(wn.p,{title:"Carica elettrica"},e("p",null,"È una proprietà dei corpi che può essere ",e(kn.a,null,"positiva")," o ",e(Bn.a,null,"negativa"),"."),e("p",null,"Si conserva: in un sistema chiuso la carica totale è costante."),e("p",null,"Esiste un'unità elementare: ",e(wn.j,null,Nn(Pe())),"."),e("p",null,"Cariche ",e(kn.a,null,"opp"),e(Bn.a,null,"oste")," si attraggono; cariche ",e(kn.a,null,"uguali")," si respingono.")),e(wn.p,{title:"Conduttori e isolanti"},e("p",null,"Più ",e("a",{href:"https://it.wikipedia.org/wiki/Ione"},"ioni")," ha un corpo, meglio la carica si muove attraverso di esso."),e("p",null,"I corpi in cui la carica si muove bene sono ",e("i",null,"conduttori"),", mentre quelli in cui si muove difficilmente sono ",e("i",null,"isolanti"),"."),e("p",null,e("i",null,"Il corpo umano è un buon conduttore.")))),e(wn.q,{title:"Polarizzazione"},e(wn.p,{title:"Polarizzazione"},e("p",null,"E' possibile polarizzare un corpo per accumulare la carica di un segno in una certa zona."))),e(wn.q,null,e(wn.p,{title:"Messa a terra"},e("p",null,"Se un corpo conduttore è in contatto con la Terra, le cariche su di esso saranno ",e("i",null,"equilibrate")," e il corpo diventerà elettricamente neutro (con stesso numero di ",e(kn.a,null,"cariche positive")," e ",e(Bn.a,null,"negative")," all'interno)."))),e(wn.q,null,e(wn.p,{title:"Polarizzazione per strofinio"},e("p",null,"Strofinando tra loro due corpi isolanti, essi si ",e("i",null,"polarizzeranno per strofinio"),".")),e(wn.p,{title:"Polarizzazione per contatto"},e("p",null,"Toccando un conduttore con un corpo carico, il conduttore potrà ",e("i",null,"polarizzarsi per contatto"),".")),e(wn.p,{title:"Polarizzazione per induzione"},e("p",null,'Se un corpo conduttore ha cariche "esterne" di un ',e(kn.a,null,"certo segno")," vicino, esso avrà tutte le cariche del ",e(Bn.a,null,"segno opposto")," in equilibrio vicino alle cariche esterne, e tutte le cariche dello ",e(kn.a,null,"stesso segno")," più lontano possibile da esse."),e("p",null,"Mettendo a terra il conduttore, nuove cariche del ",e(Bn.a,null,"segno opposto")," saranno attratte all'interno del corpo per equilibrare le cariche che si sono allontanate."),e("p",null,"Staccando il conduttore da terra e rimuovendo le cariche esterne, esso si ritroverà ",e(Bn.a,null,"caricato del segno opposto")," rispetto alle cariche esterne."))),e(wn.q,{title:"Forza elettrica"},e(wn.p,{title:"Legge di Coulomb"},e("p",null,"Due corpi carichi si attraggono tra loro con forza:"),e("p",null,e(wn.j,null,Nn(Ae()))),e("p",null,e(wn.j,null,Nn(Ee()))," è la ",e("i",null,"costante di Coulomb"),", e vale ",e(wn.j,null,Nn(Fe())),".")),e(wn.p,{title:"Permeabilità dello spazio vuoto"},e("p",null,"La costante ",e(wn.j,null,Nn(Le()))," è in realtà dipendente da un altra costante, ",e(wn.j,null,Nn(De())),", la ",e("i",null,"permeabilità del vuoto"),"."),e("p",null,e(wn.j,null,Nn(be()))),e("p",null,e(wn.j,null,Nn(qe())))),e(wn.p,{title:"Campo elettrico"},e("p",null,"Misura che forza viene applicata in ogni punto su una carica unitaria:"),e("p",null,e(wn.j,null,Nn(je())))),e(wn.p,{title:"Flusso elettrico"},e("p",null,'È la differenza tra "quanto" campo elettrico ',e(kn.a,null,"entra")," e quanto campo elettrico ",e(Bn.a,null,"esce")," da una certa area."),e("p",null,"In qualsiasi superficie chiusa, il flusso elettrico è uguale alla componente perpendicolare del campo elettrico moltiplicato per l'area."),e("p",null,e(wn.j,null,Nn(ze()))),e("p",null,"Se il campo elettrico è uniforme, se ne può calcolare facilmente il valore:"),e("p",null,e(wn.j,null,Nn(he()))),e("p",null,e(wn.t,null,"Circa. E' una specie di integrale..."))),e(wn.p,{title:"Legge di Gauss per i campi elettrostatici"},e("p",null,"Il flusso elettrico è direttamente proporzionale alla carica presente all'interno della superficie."),e("p",null,e(wn.j,null,Nn(_e()))),e("p",null,"Ovvero, i campi elettrostatici sono generati dalle cariche elettriche."))),e(wn.q,{title:"Energia elettrica"},e(wn.p,{title:"Energia potenziale elettrica"},e("p",null,"Un corpo carico vicino ad altre cariche possiede un'",e("i",null,"energia potenziale elettrica")," ",e(wn.j,null,Nn(ge())),"."))),e(wn.q,{title:"Circuiti elettrici"},e(wn.p,{title:e("span",null,"Potenziale elettrico ",e("small",null,"(tensione)"))},e("p",null,"È il valore dell'energia potenziale elettrica per una carica unitaria."),e("p",null,e(wn.j,null,Nn(me()))),e("p",null,"La sua unità di misura è il Volt (",e(wn.j,null,Nn(ve())),")."),e("p",null,"In una batteria è detto ",e("i",null,"forza elettromotrice"),", e corrisponde al lavoro compiuto da una batteria ideale per spostare una carica unitaria tra i due poli.")),e(wn.p,{title:e("span",null,"Corrente elettrica ",e("small",null,"(intensità)"))},e("p",null,"Quanta carica passa attraverso un'area (perpendicolare al flusso) nel tempo."),e("p",null,e(wn.j,null,Nn(fe()))),e("p",null,"Fintanto che c'è differenza di potenziale, ci sarà anche intensità non nulla."),e("p",null,"La sua unità di misura è l'Ampere (",e(wn.j,null,Nn(de())),").")),e(wn.p,{title:e("span",null,"Corrente continua ",e("small",null,"(",e("abbr",{title:"Direct Current"},"DC"),")"))},e("p",null,"Quando in un circuito la direzione della corrente è costante.")),e(wn.p,{title:e("span",null,"Corrente alternata ",e("small",null,"(",e("abbr",{title:"Alternate Current"},"AC"),")"))},e("p",null,"Quando in un circuito la direzione della corrente si alterna periodicamente.")),e(wn.p,{title:"Potenza elettrica"},e("p",null,"Possiamo calcolare la potenza di un circuito:"),e("p",null,e(wn.j,null,Nn(se()))))),e(wn.q,{title:"Elementi di un circuito"},e(wn.p,{title:"Resistore"},e("p",null,"Riduce l'intensità di corrente, e converte parte del potenziale in calore."),e("p",null,"Il potenziale utilizzato è pari a:"),e("p",null,e(wn.j,null,Nn(pe()))),e("p",null,"Dove ",e(wn.j,null,Nn(ce()))," è una costante detta ",e("i",null,"resistenza")," con unità di misura Ohm (",e(wn.j,null,Nn(ue())),")."),e("p",null,"La resistenza di un conduttore vale:"),e("p",null,e(wn.j,null,Nn(oe()))),e("p",null,e(wn.j,null,Nn(re()))," è la ",e("i",null,"resistività")," del materiale, e varia in base alla temperatura:"),e("p",null,e(wn.j,null,Nn(ie())))),e(wn.p,{title:"Condensatore"},e("p",null,"Immagazzina potenziale elettrico, permettendo di riutilizzarla in seguito."),e("p",null,"Per farlo, cattura cariche ",e(kn.a,null,"positive")," e ",e(Bn.a,null,"negative")," sulle sue due armature; perchè questo avvenga, deve essere compiuto lavoro."),e("p",null,"Ha una ",e("b",null,"capacità")," caratteristica, che in un condensatore a facce piane parallele è:"),e("p",null,e(wn.j,null,Nn(ae()))),e("p",null,"Condensatori di capacità maggiore immagazzinano più potenziale con meno carica."),e("p",null,"La capacità aumenta se viene messo qualcosa tra le armature:"),e("p",null,e(wn.j,null,Nn(le()))),e("p",null,"Dove ",e(wn.j,null,Nn(te()))," è la ",e("i",null,"costante dielettrica relativa")," del materiale inserito, ",e(wn.j,null,Nn(ne()))," l'area di una armatura e ",e(wn.j,null,Nn(ee()))," la distanza tra le due armature."),e("p",null,"Se il campo elettrico creatosi tra le due armature supera la ",e("i",null,"rigidità dielettrica")," del condensatore, la carica immagazzinata viene persa e ha luogo un ",e("i",null,"breakdown"),"."),e("p",null,"La sua unità di misura è il Farad (",e(wn.j,null,Nn($())),")")),e(wn.p,{title:"Amperometro"},e("p",null,"Misura la corrente elettrica se messo in serie."),e("p",null,"(Funzionamento: ha una resistenza interna bassisima in modo da non influire significativamente sulla corrente.)")),e(wn.p,{title:"Voltmetro"},e("p",null,"Misura la differenza di potenziale se messo in parallelo."),e("p",null,"(Funzionamento: ha una resistenza altissima in modo da non influire significativamente sulla tensione.)"))),e(wn.q,{title:"Principi di Kirchhoff"},e(wn.p,{title:"Legge dei nodi"},e("p",null,"Per nodo si intende un qualsiasi punto del circuito."),e("p",null,"Da un nodo entra ed esce la stessa corrente.")),e(wn.p,{title:"Legge delle maglie"},e("p",null,"Per maglia si intende un qualsiasi percorso chiuso all'interno del circuito."),e("p",null,"In una maglia chiusa, la somma delle differenze di potenziale è 0."))),e(wn.q,{title:"Serie e Parallelo"},e(wn.p,{title:"Circuito in serie"},e("p",null,"Più parti di circuito sono ",e("i",null,"in serie")," se sono consecutive e senza biforcazioni."),e("p",null,"Parti di circuito in serie sono attraversate dalla stessa corrente.")),e(wn.p,{title:"Circuito in parallelo"},e("p",null,"Più parti di circuito sono ",e("i",null,"in parallelo")," tra loro se hanno lo stesso punto di partenza e lo stesso punto di arrivo."),e("p",null,"Parti di circuito in parallelo hanno la stessa differenza di potenziale."))),e(wn.q,{title:"Resistenze equivalenti"},e(wn.p,{title:"Circuiti in serie"},e("p",null,"Nei circuiti in serie, tutte le resistenze possono essere sostituite con una equivalente dalla resistenza della somma di tutte le quelle sostituite:"),e("p",null,e(wn.j,null,Nn(Y())))),e(wn.p,{title:"Circuiti in parallelo"},e("p",null,"Nei circuiti in parallelo, tutte le resistenze possono essere sostituite con una equivalente dalla resistenza di:"),e("p",null,e(wn.j,null,Nn(K()))))),e(wn.q,{title:"Condensatori equivalenti"},e(wn.p,{title:"Circuiti in serie"},e("p",null,"Nei circuiti in serie, tutti i condensatori possono essere sostituiti con uno equivalente dalla capacità di:"),e("p",null,e(wn.j,null,Nn(X())))),e(wn.p,{title:"Circuiti in parallelo"},e("p",null,"Nei circuiti in parallelo, tutte i condensatori possono essere sostituite con uno equivalente dalla capacità della somma di tutti quelli sostituiti:"),e("p",null,e(wn.j,null,Nn(Z()))))),e(wn.q,{title:"Magnetismo"},e(wn.p,{title:"Permeabilità magnetica dello spazio vuoto"},e("p",null,"E' una costante fisica fondamentale che rappresenta quanto un materiale si magnetizza facilmente."),e("p",null,e(wn.j,null,Nn(H()))," (",e(wn.j,null,Nn(J())),")")),e(wn.p,{title:"Campo magnetico"},e("p",null,"Come un campo elettrico, ma per i magneti."),e("p",null,"Il suo simbolo è ",e(wn.j,null,Nn(Q())),", e la sua unità di misura è il Tesla (",e(wn.j,null,"T"),").")),e(wn.p,{title:"Flusso magnetico"},e("p",null,'È "quanto" campo magnetico ',e("b",null,"attraversa")," un percorso chiuso."),e("p",null,'Per qualsiasi percorso chiuso, il flusso magnetico è uguale alla somma di tutti i "sottoflussi" magnetici calcolati sui suoi lati.'),e("p",null,e(wn.j,null,Nn(W()))),e("p",null,e(wn.j,null,Nn(G()))),e("p",null,"La sua unità di misura è il Weber (",e(wn.j,null,Nn(O())),").")),e(wn.p,{title:"Legge di Gauss per i campi magnetici"},e("p",null,"Il flusso magnetico attraverso qualsiasi superficie chiusa è sempre nullo."),e("p",null,"Ovvero, non esistono monopoli magnetici.")),e(wn.p,{title:"Legge di Ampère"},e("p",null,"L'intensità di corrente che attraversa un percorso chiuso è direttamente proporzionale al flusso magnetico dello stesso percorso."),e("p",null,e(wn.j,null,Nn(M()))))),e(wn.q,{title:"Forze magnetiche"},e(wn.p,{title:e("span",null,"Forza magnetica su carica puntiforme ",e("small",null,"(Forza di Lorentz)"))},e("p",null,"I campi magnetici applicano una forza sulle cariche vicine:"),e("p",null,e(wn.j,null,Nn(U()))),e("p",null,"Dove ",e(wn.j,null,Nn(y()))," è l'intensità del campo magnetico e ",e(wn.j,null,Nn(x()))," la velocità della carica considerata."),e("p",null,"Si ha una forza massima se la velocità è perpendicolare al campo magnetico."),e("p",null,"In un campo magnetico uniforme, una velocità perpendicolare al campo porta alla creazione di un moto circolare uniforme.")),e(wn.p,{title:"Forza magnetica in un filo"},e("p",null,"I campi magnetici influenzano ovviamente anche le cariche presenti in un conduttore:"),e("p",null,e(wn.j,null,Nn(T()))," ",e("a",{href:"https://it.openprof.com/wb/forza_di_lorentz_su_un_filo_percorso_da_corrente?ch=360"},"[1]")),e("p",null,"Dove ",e(wn.j,null,Nn(R()))," è la corrente elettrica, ",e(wn.j,null,Nn(V()))," è un vettore che punta nella direzione di scorrimento della corrente e ha come modulo la lunghezza del conduttore."))),e(wn.q,{title:"Campi magnetici"},e(wn.p,{title:"Campo magnetico in una spira"},e("p",null,"Una spira in cui passa corrente produce un campo magnetico perpendicolare al piano creato dalla spira.")),e(wn.p,{title:"Campo magnetico di un solenoide"},e("p",null,"Un solenoide sono tante spire avvolte in modo da formare una specie di cilindro."),e("p",null,"All'interno del solenoide si crea un campo (quasi) uniforme:"),e("p",null,e(wn.j,null,Nn(N())))),e(wn.p,{title:"Legge di Oersted"},e("p",null,e("i",null,"Caso particolare della ",e("a",{href:"https://it.wikipedia.org/wiki/Legge_di_Amp%C3%A8re"},"Legge di Ampère"),".")),e("p",null,"Il modulo del campo magnetico ",e(wn.j,null,"B")," prodotto da un filo in cui passa una corrente continua ",e(wn.j,null,"I")," alla distanza ",e(wn.j,null,"s")," è:"),e("p",null,e(wn.j,null,Nn(B()))),e("p",null,"Il campo magnetico così creato gira attorno al filo in senso antiorario."),e("p",null,"Due fili attraversati dalla ",e(kn.a,null,"stessa corrente")," si attraggono, due fili attraversati da ",e(kn.a,null,"corr"),e(Bn.a,null,"enti")," ",e(kn.a,null,"opp"),e(Bn.a,null,"oste")," si respingono."))),e(wn.q,{title:"Induzione elettromagnetica"},e(wn.p,{title:"Forza elettromotrice indotta"},e("p",null,"Un conduttore perpendicolare ad un campo magnetico può ottenere una differenza di potenziale se messo in movimento in un direzione perpendicolare alla direzione del conduttore e del campo."),e("p",null,"La differenza di potenziale si crea a causa della forza magnetica, che fa spostare tutti gli elettroni verso un capo del conduttore."),e("p",null,"Essa vale:"),e("p",null,e(wn.j,null,Nn(k()))),e("p",null,"Dove ",e(wn.j,null,"v")," è la velocità del conduttore, ",e(wn.j,null,"B")," è l'intensità del campo magnetico ed ",e(wn.j,null,"L")," è la lunghezza del conduttore.")),e(wn.p,{title:"Flusso magnetico in una spira"},e("p",null,"In un campo magnetico ",e(wn.j,null,Nn(w()))," uniforme e perpendicolare al piano di una spira di area ",e(wn.j,null,Nn(S())),", il flusso magnetico si può determinare con la ",e("i",null,"Legge di Faraday-Neumann-Lenz"),":"),e("p",null,e(wn.j,null,Nn(I()))))),e(wn.q,null,e(wn.p,{title:"Legge di Faraday-Neumann-Lenz"},e("p",null,"Dice che la forza elettromotrice media indotta in un percorso dipende dalla variazione nel tempo del flusso magnetico nello stesso percorso."),e("p",null,e(wn.j,null,Nn(C()))),e("p",null,"Il meno è dovuto alla ",e("a",{href:"https://it.wikipedia.org/wiki/Legge_di_Lenz"},"Legge di Lenz"),", che specifica qualitativamente il verso della forza elettromotrice indotta.")),e(wn.p,{title:"Faraday in un solenoide"},e("p",null,"In un solenoide, la forza elettromotrice indotta è uguale a:"),e("p",null,e(wn.j,null,Nn(P()))),e("p",null,"Dove ",e(wn.j,null,Nn(A()))," è il numero delle spire del solenoide.")),e(wn.p,{title:"Legge di Ampère-Maxwell"},e("p",null,"Correnti o campi elettrici variabili creano un campo magnetico."))),e(wn.q,{title:"Elettromagnetismo"},e(wn.p,{title:"Onde elettromagnetiche"},e("p",null,"Nel vuoto, il campo elettrico ",e(wn.j,null,Nn(E()))," e il campo magnetico ",e(wn.j,null,Nn(F()))," sono perpendicolari tra loro e la direzione di propagazione, e sono entrambe funzioni del tempo."),e("p",null,"Si dice quindi che sono ",e("i",null,"onde elettromagnetiche"),"."),e("p",null,"Esse sono legate dalla relazione:"),e("p",null,e(wn.j,null,Nn(L()))),e("p",null,"Dove ",e(wn.j,null,Nn(D()))," è la velocità delle onde (luce) nel vuoto, e a sua volta è uguale a:"),e("p",null,e(wn.j,null,Nn(b())))),e(wn.p,{title:"Formula delle onde"},e("p",null,e(wn.j,null,Nn(q()))),e("p",null,"Dove ",e(wn.j,null,Nn(j()))," è l'ampiezza massima che può avere l'onda, ",e(wn.j,null,Nn(z()))," è il vettore d'onda, ",e(wn.j,null,Nn(h()))," la frequenza angolare e ",e(wn.j,null,Nn(_()))," la fase."))),e(wn.q,{title:"Spettroscopia"},e(wn.p,{title:"Emissione"},e("p",null,"I solidi, se portati ad alta temperatura, emettono luce con uno ",e("a",{href:"https://it.wikipedia.org/wiki/Spettro_continuo"},"spettro continuo"),"."),e("p",null,"I gas, invece, ad alta temperatura emettono luce solo con particolari lunghezze d'onda."),e("p",null,"In un gas di idrogeno, le lunghezze d'onda emesse sono ricavabili con:"),e("p",null,e(wn.j,null,Nn(g()))),e("p",null,"Con ",e(wn.j,null,Nn(m())),", detta costante di Rydberg, e ",e(wn.j,null,Nn(v()))," un numero intero.")),e(wn.p,{title:"Grandezza quantizzata"},e("p",null,"Una grandezza si dice quantizzata (o discreta) se può assumere solo determinati valori."),e("p",null,"Una grandezza si dice continua se può assumere qualsiasi valore e quindi se non è quantizzata."),e("p",null,"Energia, momento angolare e raggio sono quantizzati."),e("p",null,"Nota costante quantica è ",e(wn.j,null,Nn(f())),", la costante di Planck, ovvero il valore minimo possibile per la carica (talvolta espressa come ",e(wn.j,null,Nn(d())),"."))),e(wn.q,null,e(wn.p,{title:"Modello di Bohr"},e("p",null,"L'energia degli elettroni è quantizzata."),e("p",null,"Inoltre, per essi è valido che:"),e("p",null,e(wn.j,null,Nn(s()))),e("p",null,"Ancora, il raggio delle orbite è uguale a:"),e("p",null,e(wn.j,null,Nn(p()))),e("p",null,"Con ",e(wn.j,null,Nn(c())),"."),e("p",null,"Infine, in ogni stato, l'energia è pari a:"),e("p",null,e(wn.j,null,Nn(u()))),e("p",null,"Due elettroni non possono occupare lo stesso stato."),e("p",null,"Questo modello funziona solo per atomi con numero atomico basso. Atomi con molti elettroni hanno comportamenti diversi, descritti dal modello di"))),e(wn.q,null,e(wn.p,{title:"Nei solidi"},e("p",null,"Nei solidi, le lunghezze d'onda sono talmente tanto vicine da poter essere considerate una banda."),e("p",null,"Possono però comunque avere dei gap dovuti agli intervalli di energia non ammessi."))),e(wn.q,{title:"Semiconduttori"},e(wn.p,{title:"Semiconduttori"},e("p",null,e(wn.t,null,"Refactor this")),e("p",null,"Se la banda di emissione con energia più alta di un corpo è assente o è separata da un gap dell'ordine di grandezza maggiore di ",e(wn.j,null,Nn(o())),", allora il corpo è un isolante."),e("p",null,"Se invece la banda di emissione si sovrappone a un altra, allora il corpo è un conduttore."),e("p",null,"Se il gap è invece dell'ordine di grandezza di ",e(wn.j,null,Nn(r())),", allora il corpo è un semiconduttore.")),e(wn.p,{title:"Lacune"},e("p",null,"Legami in cui ",e(kn.a,null,"mancano elettroni"),"."),e("p",null,e(Bn.a,null,"Elettroni")," di altri legami possono spostarsi per colmare le ",e(kn.a,null,"lacune"),", creandone altre, e spostandole in direzione opposta a quella della corrente.")),e(wn.p,{title:"Accettori e donori"},e("p",null,"Se si inserisce in un cristallo semiconduttore si inserisce un atomo con numero atomico diverso, si otterrà:"),e("ul",null,e("li",null,"Con numero atomico maggiore, un semiconduttore di ",e(Bn.a,null,"tipo N")," con ",e(Bn.a,null,"elettroni in eccesso")," liberi di scorrere."),e("li",null,"Con numero atomico minore, un semiconduttore di ",e(kn.a,null,"tipo P")," con ",e(kn.a,null,"lacune in eccesso")," libere di catturare elettroni da altri legami.")),e("p",null,"Maggiore impurezza porta a maggiore conduttività.")),e(wn.p,{title:"Temperatura"},e("p",null,"Aumentando la temperatura di un semiconduttore si aumenta la conduttività, perchè eccita le particelle e favorisce il movimento di ",e(Bn.a,null,"elettroni")," e ",e(kn.a,null,"lacune"),"."))),e(wn.q,{title:e("span",null,"Ottica ",e("small",null,"(non l'abbiamo fatta)"))},e(wn.p,{title:"Assorbimento e riflessione"},e("p",null,"I corpi possono assorbire o riflettere le onde elettromagnetiche che li colpiscono.")),e(wn.p,{title:"Corpo nero"},e("p",null,"Un corpo nero è un corpo che assorbe tutte le onde elettromagnetiche che riceve senza rifletterne nessuna."),e("p",null,"Le onde assorbite vengono poi riemesse sotto forma di un onda di ",e(wn.j,null,Nn(i()))," variabile in base alla temperatura."),e("p",null,e(wn.j,null,Nn(a()))," è costante.")),e(wn.p,{title:"Teoria di Planck per il corpo nero"},e("p",null,"L'energia assorbita e emessa dai corpi neri è quantizzata.")),e(wn.p,{title:"Fotone"},e("p",null,"Un onda magnetica con un quanto di energia è detta ",e("i",null,"fotone"),":"),e("p",null,e(wn.j,null,Nn(l())))),e(wn.p,{title:"Effetto fotoelettrico"},e("p",null,"A volte, i fotoni che colpiscono un metallo possono estrarvi degli elettroni e creare una differenza di potenziale."),e("p",null,"Perchè avvenga, la frequenza deve essere maggiore di una certa soglia."),e("p",null,"Il numero di elettroni estratti dipende dall'intensità dell'onda, mentre l'energia cinetica degli elettroni dipende dalla frequenza."),e("p",null,"Non c'è nessun ritardo tra l'assorbimento del fotone e l'estrazione di elettroni."))))))}}.call(this,t("hosL").h)},ZHMS:function(e,n,t){"use strict";(function(e){var l=t("gJgL"),a=t.n(l);n.a=function(n){return e("span",{class:a.a.minus},n.children)}}).call(this,t("hosL").h)},gJgL:function(e){e.exports={red:"red__3phX0",orange:"orange__1OSkI",yellow:"yellow__2pQOr",lime:"lime__3vRUg",cyan:"cyan__1cBtG",blue:"blue__1G1wb",magenta:"magenta__3Wagv",minus:"minus__2qLyv"}},jZax:function(e){e.exports={red:"red__ipSoq",orange:"orange__1nmkj",yellow:"yellow__353ge",lime:"lime__3FW1E",cyan:"cyan__2xhu7",blue:"blue__3MvDe",magenta:"magenta__i03ie",plus:"plus__1eV4_"}},o8Jj:function(e,n,t){"use strict";(function(e){var l=t("jZax"),a=t.n(l);n.a=function(n){return 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{Latex, LatexDisplay, Panel, Section, Todo, LatexDefaultInline, LatexDefaultDisplay} from \"bluelib\";\n\nimport Plus from \"../components/Fisica/Plus\";\nimport Minus from \"../components/Fisica/Minus\";\n\nconst r = String.raw;\n\n\nexport default function(props) {\n return (\n \n \n
\n

Fisica

\n
\n \n

\n Usa le regole base della trigonometria:\n

\n

\n {r`\\vec{v} = \\vec{v}_x + \\vec{v}_y`}\n

\n

\n {r`\\left | \\vec{v}_x \\right | = \\left | \\vec{v} \\right | \\sin \\alpha`}\n

\n

\n {r`\\left | \\vec{v}_y \\right | = \\left | \\vec{v} \\right | \\cos \\alpha`}\n

\n \n \n

\n Scomponi in componenti, poi sommali:\n

\n

\n {r`\\vec{v} + \\vec{w} = (\\vec{v}_x + \\vec{w}_x) + (\\vec{v}_y + \\vec{w}_y)`}\n

\n

\n Produce il vettore risultante dall'applicazione della regola del parallelogramma.\n

\n \n \n

\n Alla fine è sempre una somma:\n

\n

\n {r`\\vec{v} - \\vec{w} = (\\vec{v}_x - \\vec{w}_x) + (\\vec{v}_y - \\vec{w}_y)`}\n

\n

\n Produce il vettore che parte da w e arriva a v.\n

\n \n \n

\n Si chiama scalare perchè il risultato è uno scalare, non un vettore.\n

\n

\n {r`\\vec{v} \\cdot \\vec{w} = \\left | \\vec{v} \\right | \\left | \\vec{w} \\right | \\cos \\alpha`}\n

\n

\n Produce il modulo della proiezione di {r`\\vec{a}`} su {r`\\vec{b}`}.\n

\n \n \n

\n Si chiama vettoriale perchè il risultato è un altro vettore.\n

\n
    \n
  • {r`\\vec{c} = \\vec{a} \\times \\vec{b}`}
    • \n
  • {r`\\left | \\vec{c} \\right | = \\left | \\vec{a} \\right | \\cdot \\left | \\vec{b} \\right | \\cdot \\sin(\\alpha)`}
    • \n
  • Regola della mano destra
    • \n
\n

\n Non è commutativo!\n

\n \n
\n
\n \n

\n Se un corpo puntiforme ha forza risultante nulla, allora la sua velocità non cambia.\n

\n

\n {r`\\Sigma \\vec{F} = 0 \\Longleftrightarrow \\Delta v = 0`}\n

\n \n \n

\n La forza risultante di un corpo è direttamente proporzionale alla sua accelerazione, e la costante di proporzionalità è la massa.\n

\n

\n {r`\\Sigma \\vec{F} = m \\vec{a}`}\n

\n \n \n

\n Due corpi esercitano forze uguali e opposte uno sull'altro.\n

\n

\n {r`\\vec{F}_{21} = -\\vec{F}_{12}`}\n

\n \n
\n
\n \n

\n Due corpi puntiformi si attirano uno verso l'altro con forza:\n

\n

\n {r`\\left | \\vec{F} \\right | = G \\frac{m_1 m_2}{s^2}`}\n

\n

\n G è la costante di gravitazione universale e vale:\n

\n

\n {r`G = 6.67 \\cdot 10^{-11} \\frac{N m^2}{{kg}^2}`}\n

\n \n \n

\n Se nel sistema di riferimento consideriamo la Terra ferma, allora un corpo è attratto verso la Terra con forza peso uguale a:\n

\n

\n {r`\\left | \\vec{F} \\right | = g m`}\n

\n

\n g è la costante di gravità della Terra, e vale:\n

\n

\n {r`g = 9.81 \\frac{m}{s^2}`}\n

\n \n \n

\n Per pianeti diversi dalla Terra vale la stessa regola:\n

\n

\n {r`\\left | \\vec{F} \\right | = g m`}\n

\n

\n L'unica differenza è che cambia la costante di gravità:\n

\n

\n {r`g_{luna} = 1.62 \\frac{m}{s^2}`}\n

\n

\n {r`g_{marte} = 3.71 \\frac{m}{s^2}`}\n

\n \n
\n
\n \n

\n Si oppone alle forze applicate alla superficie di contatto.\n

\n

\n Un libro appoggiato su un tavolo ha la forza di gravità che lo attira verso il terreno e la forza normale che lo trattiene dal cadere.\n

\n \n \n

\n Impedisce a un corpo di muoversi se non viene spinto da una forza che supera una certa soglia:\n

\n

\n {r`\\left | \\vec{F} \\right | \\leq \\mu_{s} \\left | \\vec{F}_{normale} \\right |`}\n

\n \n \n

\n Rallenta i corpi che si stanno muovendo finchè essi non si fermano:\n

\n

\n {r`\\left | \\vec{F} \\right | \\leq \\mu_{d} \\left | \\vec{F}_{normale} \\right |`}\n

\n \n \n

\n E' forza trasmessa tra due estremi di una fune.\n

\n

\n Può essere redirezionata per mezzo di carrucole.\n

\n \n \n

\n Una molla cerca sempre di tornare alla sua posizione indeformata con forza:\n

\n

\n {r`F = -k x`}\n

\n

\n (E' negativa perchè la forza è opposta a quella applicata per deformarla.)\n

\n \n
\n
\n \n

\n È un vettore che indica la posizione di un corpo rispetto a un'origine.\n

\n

\n {r`\\Delta \\vec{s} = \\vec{s}(fine) - \\vec{s}(inizio)`}\n

\n \n \n

\n È un vettore che misura la variazione di posizione nel tempo.\n

\n

\n {r`\\vec{v} = \\frac{\\Delta \\vec{s}}{\\Delta t}`}\n

\n

\n Se si considera un intervallo di tempo infinitesimale si dice velocità istantanea:\n

\n

\n {r`\\vec{v} = \\lim_{\\Delta t \\to 0} \\frac{\\Delta \\vec{s}}{\\Delta t} = \\frac{d \\vec{s}}{dt}`}\n

\n \n \n

\n È un vettore che misura la variazione di velocità nel tempo.\n

\n

\n {r`\\vec{a} = \\frac{\\Delta \\vec{v}}{\\Delta t}`}\n

\n

\n Se si considera un intervallo di tempo infinitesimale si dice accelerazione istantanea:\n

\n

\n {r`\\vec{a} = \\lim_{\\Delta v \\to 0} \\frac{\\Delta \\vec{v}}{\\Delta t} = \\frac{d \\vec{v}}{d t} = \\frac{d^2 \\vec{s}}{d t^2}`}\n

\n \n Quantità di moto (momento lineare)}>\n

\n La quantità di moto è una proprietà vettoriale dei corpi:\n

\n

\n {r`\\vec{p} = m \\vec{v}`}\n

\n

\n Se la forza risultante è nulla, la quantità di moto non cambia.\n

\n

\n {r`\\Sigma \\vec{F} = 0 \\Longleftrightarrow \\Delta \\vec{p} = 0`}\n

\n \n
\n
\n \n

\n La legge oraria è:\n

\n

\n {r`s(t) = v \\cdot \\Delta t + s(0)`}\n

\n \n \n

\n È costante:\n

\n

\n {r`v(t) = k`}\n

\n \n \n

\n La velocità non varia:\n

\n

\n {r`a(t) = 0`}\n

\n \n \n

\n Si applica la prima legge di Newton:\n

\n

\n f(t) = 0\n

\n \n
\n
\n \n

\n La legge oraria è:\n

\n

\n {r`s(t) = \\frac{1}{2} a \\cdot (\\Delta t)^2 + v(0) \\cdot (\\Delta t) + s(0)`}\n

\n \n \n

\n È una retta:\n

\n

\n {r`v(t) = a \\Delta t + v(0)`}\n

\n \n \n

\n È costante:\n

\n

\n {r`a(t) = k`}\n

\n \n \n

\n Si applica la prima legge di Newton:\n

\n

\n f(t) = m a\n

\n \n
\n
\n \n

\n E' la distanza dal centro massima che raggiunge il corpo.\n

\n

\n (L'ampiezza di una sinusoide.)\n

\n \n \n

\n Indica quanto in fretta cambia la posizione del corpo.\n

\n

\n Dipende dal periodo:\n

\n

\n {r`\\omega = \\frac{2 \\pi}{T}`}\n

\n \n \n

\n E' una sinusoide:\n

\n

\n {r`s(t) = A \\sin (\\omega \\cdot t + \\phi)`}\n

\n \n \n

\n E' la sinusoide dello spostamento, sfasata di {r`\\frac{\\pi}{2}`}:\n

\n

\n {r`v(t) = A \\sin (\\omega \\cdot t + \\phi + \\frac{\\pi}{2})`}\n

\n \n \n

\n E' la sinusoide della velocità, sfasata di {r`\\pi`}:\n

\n

\n {r`a(t) = A \\sin (\\omega \\cdot t + \\phi + \\pi)`}\n

\n \n \n

\n Si applica la prima legge di Newton:\n

\n

\n f(t) = m a\n

\n \n
\n
\n \n

\n Il moto parabolico è dato sommando un moto rettilineo uniforme sull'asse orizzontale e un moto rettilineo uniformemente accelerato sull'asse verticale.\n

\n \n \n

\n Il moto parabolico è dato sommando due moti armonici semplici: uno sull'asse X, e l'altro, sfasato di {r`\\frac{\\pi}{2}`}, sull'asse Y.\n

\n \n
\n
\n \n

\n Velocità angolare\n

\n

\n Quanto cambia la fase nel tempo.\n

\n

\n {r`\\omega = \\frac{2 \\pi}{T}`}\n

\n \n \n

\n E' l'angolo percorso dal corpo rispetto alla posizione iniziale.\n

\n

\n Si indica con {r`\\phi`}, e generalmente si usa in radianti.\n

\n \n \n

\n Si applicano le formule per la circonferenza:\n

\n

\n {r`v = \\frac{\\Delta s}{t} = \\frac{2 \\pi \\cdot r}{T} = \\omega r`}\n

\n \n \n

\n Il corpo ha sempre un accelerazione verso il centro che gli impedisce di abbandonare il moto:\n

\n

\n {r`a = \\frac{v^2}{r} = r \\cdot \\omega^2 = v \\cdot \\omega`}\n

\n \n \n

\n È verso il centro e si calcola con:\n

\n

\n {r`F = m \\cdot a`}\n

\n \n
\n
\n \n

\n E' compiuto da una forza che sposta un corpo.\n

\n

\n {r`W = \\vec{F} \\cdot \\vec{s} = F \\cdot \\Delta s \\cdot cos(\\alpha )`}\n

\n

\n (Se la forza non è parallela allo spostamento, il prodotto scalare ci fa considerare solo la componente parallela.)\n

\n \n \n

\n Un corpo ha energia cinetica in ogni momento uguale a:\n

\n

\n {r`E_c = \\frac{1}{2} m v^2`}\n

\n

\n Se una forza effettua lavoro su un corpo, cambia la sua energia cinetica pari al lavoro effettuato:\n

\n

\n {r`\\Delta E_c = W`}\n

\n \n \n

\n Un corpo ha energia potenziale in ogni momento pari a:\n

\n

\n {r`E_{p_g} = m \\cdot g \\cdot h`}\n

\n

\n (Con h uguale a un altezza scelta come punto di riferimento.)\n

\n \n \n

\n Una molla ha sempre energia potenziale elastica pari a:\n

\n

\n {r`E_{p_e} = \\frac{1}{2} k x^2`}\n

\n \n \n

\n Sono conservative le forze per le quali il lavoro compiuto non dipende dal percorso seguito per andare dalla partenza all'arrivo.\n

\n

\n Ad esempio, è conservativa la forza di gravità, ma non è conservativa la forza di attrito.\n

\n

\n Se in un sistema ci sono solo forze conservative, allora l'energia meccanica totale si conserva:\n

\n

\n {r`E = E_k + E_p`}\n

\n \n \n

\n È la velocità di trasferimento di energia:\n

\n

\n {r`P = \\frac{\\Delta E}{\\Delta t}`}\n

\n \n
\n
\n \n

\n È una proprietà dei corpi che può essere positiva o negativa.\n

\n

\n Si conserva: in un sistema chiuso la carica totale è costante.\n

\n

\n Esiste un'unità elementare: {r`C_{elettrone} = 1.602 \\cdot 10^{-19}`}.\n

\n

\n Cariche opposte si attraggono; cariche uguali si respingono.\n

\n \n \n

\n Più ioni ha un corpo, meglio la carica si muove attraverso di esso.\n

\n

\n I corpi in cui la carica si muove bene sono conduttori, mentre quelli in cui si muove difficilmente sono isolanti.\n

\n

\n Il corpo umano è un buon conduttore.\n

\n \n
\n
\n \n

\n E' possibile polarizzare un corpo per accumulare la carica di un segno in una certa zona.\n

\n \n
\n
\n \n

\n Se un corpo conduttore è in contatto con la Terra, le cariche su di esso saranno equilibrate e il corpo diventerà elettricamente neutro (con stesso numero di cariche positive e negative all'interno).\n

\n \n
\n
\n \n

\n Strofinando tra loro due corpi isolanti, essi si polarizzeranno per strofinio.\n

\n \n \n

\n Toccando un conduttore con un corpo carico, il conduttore potrà polarizzarsi per contatto.\n

\n \n \n

\n Se un corpo conduttore ha cariche \"esterne\" di un certo segno vicino, esso avrà tutte le cariche del segno opposto in equilibrio vicino alle cariche esterne, e tutte le cariche dello stesso segno più lontano possibile da esse.\n

\n

\n Mettendo a terra il conduttore, nuove cariche del segno opposto saranno attratte all'interno del corpo per equilibrare le cariche che si sono allontanate.\n

\n

\n Staccando il conduttore da terra e rimuovendo le cariche esterne, esso si ritroverà caricato del segno opposto rispetto alle cariche esterne.\n

\n \n
\n
\n \n

\n Due corpi carichi si attraggono tra loro con forza:\n

\n

\n {r`\\left | \\vec{F}_{elettrica} \\right | = \\frac{-k \\cdot q_1 \\cdot q_2}{s^2}`}\n

\n

\n {r`k`} è la costante di Coulomb, e vale {r`k = 8.99 \\cdot 10^9 \\frac{N \\cdot m^2}{C^2}`}.\n

\n \n \n

\n La costante {r`k`} è in realtà dipendente da un altra costante, {r`\\epsilon_0`}, la permeabilità del vuoto.\n

\n

\n {r`k = \\frac{1}{4 \\pi \\cdot \\epsilon_0}`}\n

\n

\n {r`\\left | \\vec{F}_{elettrica} \\right | = \\frac{q_1 \\cdot q_2}{4 \\pi \\cdot \\epsilon_0 \\cdot s^2}`}\n

\n \n \n

\n Misura che forza viene applicata in ogni punto su una carica unitaria:\n

\n

\n {r`\\vec{E} = \\frac{\\vec{F}_{elettrica}}{q} = \\frac{-k \\cdot q}{s^2}`}\n

\n \n \n

\n È la differenza tra \"quanto\" campo elettrico entra e quanto campo elettrico esce da una certa area.\n

\n

\n In qualsiasi superficie chiusa, il flusso elettrico è uguale alla componente perpendicolare del campo elettrico moltiplicato per l'area.\n

\n

\n {r`\\Phi_E = \\vec{E} \\cdot \\vec{A}`}\n

\n

\n Se il campo elettrico è uniforme, se ne può calcolare facilmente il valore:\n

\n

\n {r`\\Phi_E = \\vec{E} \\cdot \\vec{A} = E_\\perp \\cdot A \\cdot \\cos(\\alpha)`}\n

\n

\n Circa. E' una specie di integrale...\n

\n \n \n

\n Il flusso elettrico è direttamente proporzionale alla carica presente all'interno della superficie.\n

\n

\n {r`\\Phi_E = 4 \\pi \\cdot k \\cdot q = \\frac{q}{\\epsilon_0}`}\n

\n

\n Ovvero, i campi elettrostatici sono generati dalle cariche elettriche.\n

\n \n
\n
\n \n

\n Un corpo carico vicino ad altre cariche possiede un'energia potenziale elettrica {r`U_e`}.\n

\n \n
\n
\n Potenziale elettrico (tensione)}>\n

\n È il valore dell'energia potenziale elettrica per una carica unitaria.\n

\n

\n {r`V = \\frac{U_e}{q}`}\n

\n

\n La sua unità di misura è il Volt ({r`V`}).\n

\n

\n In una batteria è detto forza elettromotrice, e corrisponde al lavoro compiuto da una batteria ideale per spostare una carica unitaria tra i due poli.\n

\n \n Corrente elettrica (intensità)}>\n

\n Quanta carica passa attraverso un'area (perpendicolare al flusso) nel tempo.\n

\n

\n {r`I = \\frac{\\Delta q}{\\Delta t}`}\n

\n

\n Fintanto che c'è differenza di potenziale, ci sarà anche intensità non nulla.\n

\n

\n La sua unità di misura è l'Ampere ({r`A`}).\n

\n \n Corrente continua (DC)}>\n

\n Quando in un circuito la direzione della corrente è costante.\n

\n \n Corrente alternata (AC)}>\n

\n Quando in un circuito la direzione della corrente si alterna periodicamente.\n

\n \n \n

\n Possiamo calcolare la potenza di un circuito:\n

\n

\n {r`P = \\frac{\\Delta U_e}{\\Delta t} = I \\cdot \\Delta V = I^2 \\cdot R = \\frac{(\\Delta V)^2}{R}`}\n

\n \n
\n
\n \n

\n Riduce l'intensità di corrente, e converte parte del potenziale in calore.\n

\n

\n Il potenziale utilizzato è pari a:\n

\n

\n {r`V = R \\cdot I`}\n

\n

\n Dove {r`R`} è una costante detta resistenza con unità di misura Ohm ({r`\\Omega`}).\n

\n

\n La resistenza di un conduttore vale:\n

\n

\n {r`R = \\rho \\frac{L_{unghezza}}{A_{rea}}`}\n

\n

\n {r`\\rho`} è la resistività del materiale, e varia in base alla temperatura:\n

\n

\n {r`\\rho = \\rho_0 (1 + \\alpha(T - T_0))`}\n

\n \n \n

\n Immagazzina potenziale elettrico, permettendo di riutilizzarla in seguito.\n

\n

\n Per farlo, cattura cariche positive e negative sulle sue due armature; perchè questo avvenga, deve essere compiuto lavoro.\n

\n

\n Ha una capacità caratteristica, che in un condensatore a facce piane parallele è:\n

\n

\n {r`C = \\frac{q_{massima}}{\\Delta V}`}\n

\n

\n Condensatori di capacità maggiore immagazzinano più potenziale con meno carica.\n

\n

\n La capacità aumenta se viene messo qualcosa tra le armature:\n

\n

\n {r`C_{nuova} = \\kappa \\cdot \\frac{\\epsilon_0 \\cdot A}{s}`}\n

\n

\n Dove {r`\\kappa`} è la costante dielettrica relativa del materiale inserito, {r`A`} l'area di una armatura e {r`s`} la distanza tra le due armature.\n

\n

\n Se il campo elettrico creatosi tra le due armature supera la rigidità dielettrica del condensatore, la carica immagazzinata viene persa e ha luogo un breakdown.\n

\n

\n La sua unità di misura è il Farad ({r`Fa`})\n

\n \n \n

\n Misura la corrente elettrica se messo in serie.\n

\n

\n (Funzionamento: ha una resistenza interna bassisima in modo da non influire significativamente sulla corrente.)\n

\n \n \n

\n Misura la differenza di potenziale se messo in parallelo.\n

\n

\n (Funzionamento: ha una resistenza altissima in modo da non influire significativamente sulla tensione.)\n

\n \n
\n
\n \n

\n Per nodo si intende un qualsiasi punto del circuito.\n

\n

\n Da un nodo entra ed esce la stessa corrente.\n

\n \n \n

\n Per maglia si intende un qualsiasi percorso chiuso all'interno del circuito.\n

\n

\n In una maglia chiusa, la somma delle differenze di potenziale è 0.\n

\n \n
\n
\n \n

\n Più parti di circuito sono in serie se sono consecutive e senza biforcazioni.\n

\n

\n Parti di circuito in serie sono attraversate dalla stessa corrente.\n

\n \n \n

\n Più parti di circuito sono in parallelo tra loro se hanno lo stesso punto di partenza e lo stesso punto di arrivo.\n

\n

\n Parti di circuito in parallelo hanno la stessa differenza di potenziale.\n

\n \n
\n
\n \n

\n Nei circuiti in serie, tutte le resistenze possono essere sostituite con una equivalente dalla resistenza della somma di tutte le quelle sostituite:\n

\n

\n {r`R_{serie} = \\sum_{i=1}^{n} R_i`}\n

\n \n \n

\n Nei circuiti in parallelo, tutte le resistenze possono essere sostituite con una equivalente dalla resistenza di:\n

\n

\n {r`R_{parallelo} = \\frac{1}{\\sum_{i=1}^{n} \\frac{1}{R_i}}`}\n

\n \n
\n
\n \n

\n Nei circuiti in serie, tutti i condensatori possono essere sostituiti con uno equivalente dalla capacità di:\n

\n

\n {r`C_{serie} = \\frac{1}{\\sum_{i=1}^{n} \\frac{1}{C_i}}`}\n

\n \n \n

\n Nei circuiti in parallelo, tutte i condensatori possono essere sostituite con uno equivalente dalla capacità della somma di tutti quelli sostituiti:\n

\n

\n {r`C_{parallelo} = \\sum_{i=1}^{n} C_n`}\n

\n \n
\n
\n \n

\n E' una costante fisica fondamentale che rappresenta quanto un materiale si magnetizza facilmente.\n

\n

\n {r`\\mu_0 = 4 \\pi \\cdot 10^{-7} \\frac{H}{m}`} ({r`\\frac{N}{A^2}`})\n

\n \n \n

\n Come un campo elettrico, ma per i magneti.\n

\n

\n Il suo simbolo è {r`B`}, e la sua unità di misura è il Tesla (T).\n

\n \n \n

\n È \"quanto\" campo magnetico attraversa un percorso chiuso.\n

\n

\n Per qualsiasi percorso chiuso, il flusso magnetico è uguale alla somma di tutti i \"sottoflussi\" magnetici calcolati sui suoi lati.\n

\n

\n {r`\\Phi_{B_{i}} = \\vec{B} \\cdot \\vec{L}_n = B \\cdot L_i \\cdot \\sin(\\alpha) = B_\\parallel \\cdot L_i`}\n

\n

\n {r`\\Phi_{B} = \\sum_{i=0}^{n_{lati}} \\Phi_{Bn}`}\n

\n

\n La sua unità di misura è il Weber ({r`Wb = T \\cdot m^2`}).\n

\n \n \n

\n Il flusso magnetico attraverso qualsiasi superficie chiusa è sempre nullo.\n

\n

\n Ovvero, non esistono monopoli magnetici.\n

\n \n \n

\n L'intensità di corrente che attraversa un percorso chiuso è direttamente proporzionale al flusso magnetico dello stesso percorso.\n

\n

\n {r`\\Phi_B = \\mu_0 \\cdot I`}\n

\n \n
\n
\n Forza magnetica su carica puntiforme (Forza di Lorentz)}>\n

\n I campi magnetici applicano una forza sulle cariche vicine:\n

\n

\n {r`\\vec{F}_{B} = q \\cdot (\\vec{v} \\times \\vec{B})`}\n

\n

\n Dove {r`\\vec{B}`} è l'intensità del campo magnetico e {r`\\vec{v}`} la velocità della carica considerata.\n

\n

\n Si ha una forza massima se la velocità è perpendicolare al campo magnetico.\n

\n

\n In un campo magnetico uniforme, una velocità perpendicolare al campo porta alla creazione di un moto circolare uniforme.\n

\n \n \n

\n I campi magnetici influenzano ovviamente anche le cariche presenti in un conduttore:\n

\n

\n {r`\\vec{F}_{magnetica} = I \\cdot (\\vec{L} \\times \\vec{B})`} [1]\n

\n

\n Dove {r`I`} è la corrente elettrica, {r`\\vec{L}`} è un vettore che punta nella direzione di scorrimento della corrente e ha come modulo la lunghezza del conduttore.\n

\n \n
\n
\n \n

\n Una spira in cui passa corrente produce un campo magnetico perpendicolare al piano creato dalla spira.\n

\n \n \n

\n Un solenoide sono tante spire avvolte in modo da formare una specie di cilindro.\n

\n

\n All'interno del solenoide si crea un campo (quasi) uniforme:\n

\n

\n {r`\\left | \\vec{B} \\right | = \\mu_0 \\cdot I \\cdot \\frac{A_{vvolgimenti}}{L_{unghezzafilo}}`}\n

\n \n \n

\n Caso particolare della Legge di Ampère.\n

\n

\n Il modulo del campo magnetico B prodotto da un filo in cui passa una corrente continua I alla distanza s è:\n

\n

\n {r`\\left | \\vec{B} \\right | = \\frac{\\mu \\cdot I}{2 \\pi r}`}\n

\n

\n Il campo magnetico così creato gira attorno al filo in senso antiorario.\n

\n

\n Due fili attraversati dalla stessa corrente si attraggono, due fili attraversati da correnti opposte si respingono.\n

\n \n
\n
\n \n

\n Un conduttore perpendicolare ad un campo magnetico può ottenere una differenza di potenziale se messo in movimento in un direzione perpendicolare alla direzione del conduttore e del campo.\n

\n

\n La differenza di potenziale si crea a causa della forza magnetica, che fa spostare tutti gli elettroni verso un capo del conduttore.\n

\n

\n Essa vale:\n

\n

\n {r`\\Delta V_{indotta} = v \\cdot B \\cdot L`}\n

\n

\n Dove v è la velocità del conduttore, B è l'intensità del campo magnetico ed L è la lunghezza del conduttore.\n

\n \n \n

\n In un campo magnetico {r`B`} uniforme e perpendicolare al piano di una spira di area {r`A`}, il flusso magnetico si può determinare con la Legge di Faraday-Neumann-Lenz:\n

\n

\n {r`\\Phi_B = \\vec{B} \\cdot \\vec{A} = B \\cdot A \\cdot \\cos(\\alpha)`}\n

\n \n
\n
\n \n

\n Dice che la forza elettromotrice media indotta in un percorso dipende dalla variazione nel tempo del flusso magnetico nello stesso percorso.\n

\n

\n {r`\\Delta V_{indotta} = - \\frac{\\Delta \\Phi_B}{\\Delta t}`}\n

\n

\n Il meno è dovuto alla Legge di Lenz, che specifica qualitativamente il verso della forza elettromotrice indotta.\n

\n \n \n

\n In un solenoide, la forza elettromotrice indotta è uguale a:\n

\n

\n {r`\\Delta V_{indotta} = - \\frac{N \\cdot \\Delta \\Phi_{B_{spira}}}{\\Delta t} = - \\frac{N \\cdot B \\cdot A \\cdot cos(\\alpha)}{\\Delta t}`}\n

\n

\n Dove {r`N`} è il numero delle spire del solenoide.\n

\n \n \n

\n Correnti o campi elettrici variabili creano un campo magnetico.\n

\n \n
\n
\n \n

\n Nel vuoto, il campo elettrico {r`E`} e il campo magnetico {r`B`} sono perpendicolari tra loro e la direzione di propagazione, e sono entrambe funzioni del tempo.\n

\n

\n Si dice quindi che sono onde elettromagnetiche.\n

\n

\n Esse sono legate dalla relazione:\n

\n

\n {r`E = c \\cdot B`}\n

\n

\n Dove {r`c`} è la velocità delle onde (luce) nel vuoto, e a sua volta è uguale a:\n

\n

\n {r`c = \\frac{1}{\\sqrt{\\epsilon_0 \\cdot \\mu_0}} = 3.00 \\cdot 10^8 \\frac{m}{s}`}\n

\n \n \n

\n {r`A(t) = A_{max} \\cdot \\sin \\left ( \\frac{2 \\pi}{\\lambda} - \\omega t + \\phi \\right )`}\n

\n

\n Dove {r`A_{max}`} è l'ampiezza massima che può avere l'onda, {r`\\frac{2 \\pi}{\\lambda} = \\left | \\vec{k} \\right |`} è il vettore d'onda, {r`\\omega`} la frequenza angolare e {r`\\phi`} la fase.\n

\n \n
\n
\n \n

\n I solidi, se portati ad alta temperatura, emettono luce con uno spettro continuo.\n

\n

\n I gas, invece, ad alta temperatura emettono luce solo con particolari lunghezze d'onda.\n

\n

\n In un gas di idrogeno, le lunghezze d'onda emesse sono ricavabili con:\n

\n

\n {r`\\frac{1}{\\lambda} = R \\left ( \\frac{1}{4} - \\frac{1}{n^2} \\right )`}\n

\n

\n Con {r`R = 1.097 \\cdot 10^7 \\frac{1}{m}`}, detta costante di Rydberg, e {r`n`} un numero intero.\n

\n \n \n

\n Una grandezza si dice quantizzata (o discreta) se può assumere solo determinati valori.\n

\n

\n Una grandezza si dice continua se può assumere qualsiasi valore e quindi se non è quantizzata.\n

\n

\n Energia, momento angolare e raggio sono quantizzati.\n

\n

\n Nota costante quantica è {r`h`}, la costante di Planck, ovvero il valore minimo possibile per la carica (talvolta espressa come {r`\\hbar = \\left ( \\frac{h}{2 \\pi} \\right )`}.\n

\n \n
\n
\n \n

\n L'energia degli elettroni è quantizzata.\n

\n

\n Inoltre, per essi è valido che:\n

\n

\n {r`m \\cdot v_n \\cdot 2 \\pi \\cdot r = n \\cdot h`}\n

\n

\n Ancora, il raggio delle orbite è uguale a:\n

\n

\n {r`r_n = n^2 \\cdot a_0 = n^2 \\cdot \\frac{\\hbar}{m_{elettrone} \\cdot k \\cdot e^2} `}\n

\n

\n Con {r`a_0 = \\left ( \\frac{h}{2 \\pi} \\right )^2 \\cdot \\frac{1}{m_{elettrone} \\cdot k \\cdot e^2} = 5.29 \\cdot 10^{-11} m`}.\n

\n

\n Infine, in ogni stato, l'energia è pari a:\n

\n

\n {r`E_n = \\frac{1}{n^2} \\cdot E_1 = - \\frac{1}{n^2} \\cdot \\frac{a_0^2}{2 \\cdot m \\cdot \\hbar^4} = - \\frac{1}{n^2} \\cdot \\frac{m_{elettrone} \\cdot k^2 \\cdot e^4}{2 \\cdot \\hbar^2}`}\n

\n

\n Due elettroni non possono occupare lo stesso stato.\n

\n

\n Questo modello funziona solo per atomi con numero atomico basso. Atomi con molti elettroni hanno comportamenti diversi, descritti dal modello di\n

\n \n
\n
\n \n

\n Nei solidi, le lunghezze d'onda sono talmente tanto vicine da poter essere considerate una banda.\n

\n

\n Possono però comunque avere dei gap dovuti agli intervalli di energia non ammessi.\n

\n \n
\n
\n \n

\n Refactor this\n

\n

\n Se la banda di emissione con energia più alta di un corpo è assente o è separata da un gap dell'ordine di grandezza maggiore di {r`10^1 eV`}, allora il corpo è un isolante.\n

\n

\n Se invece la banda di emissione si sovrappone a un altra, allora il corpo è un conduttore.\n

\n

\n Se il gap è invece dell'ordine di grandezza di {r`1 eV`}, allora il corpo è un semiconduttore.\n

\n \n \n

\n Legami in cui mancano elettroni.\n

\n

\n Elettroni di altri legami possono spostarsi per colmare le lacune, creandone altre, e spostandole in direzione opposta a quella della corrente.\n

\n \n \n

\n Se si inserisce in un cristallo semiconduttore si inserisce un atomo con numero atomico diverso, si otterrà:\n

\n
    \n
  • Con numero atomico maggiore, un semiconduttore di tipo N con elettroni in eccesso liberi di scorrere.
    • \n
  • Con numero atomico minore, un semiconduttore di tipo P con lacune in eccesso libere di catturare elettroni da altri legami.
    • \n
\n

\n Maggiore impurezza porta a maggiore conduttività.\n

\n \n \n

\n Aumentando la temperatura di un semiconduttore si aumenta la conduttività, perchè eccita le particelle e favorisce il movimento di elettroni e lacune.\n

\n \n
\n
Ottica (non l'abbiamo fatta)}>\n \n

\n I corpi possono assorbire o riflettere le onde elettromagnetiche che li colpiscono.\n

\n \n \n

\n Un corpo nero è un corpo che assorbe tutte le onde elettromagnetiche che riceve senza rifletterne nessuna.\n

\n

\n Le onde assorbite vengono poi riemesse sotto forma di un onda di {r`\\lambda`} variabile in base alla temperatura.\n

\n

\n {r`\\lambda_{max} \\cdot T`} è costante.\n

\n \n \n

\n L'energia assorbita e emessa dai corpi neri è quantizzata.\n

\n \n \n

\n Un onda magnetica con un quanto di energia è detta fotone:\n

\n

\n {r`E_{fotone} = h \\cdot f`}\n

\n \n \n

\n A volte, i fotoni che colpiscono un metallo possono estrarvi degli elettroni e creare una differenza di potenziale.\n

\n

\n Perchè avvenga, la frequenza deve essere maggiore di una certa soglia.\n

\n

\n Il numero di elettroni estratti dipende dall'intensità dell'onda, mentre l'energia cinetica degli elettroni dipende dalla frequenza.\n

\n

\n Non c'è nessun ritardo tra l'assorbimento del fotone e l'estrazione di elettroni.\n

\n \n
\n
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{Latex, LatexDisplay, Panel, Section, Todo, LatexDefaultInline, LatexDefaultDisplay} from \"bluelib\";\n\nimport Plus from \"../components/Fisica/Plus\";\nimport Minus from \"../components/Fisica/Minus\";\n\nconst r = String.raw;\n\n\nexport default function(props) {\n return (\n \n \n
\n

Fisica

\n
\n \n

\n Usa le regole base della trigonometria:\n

\n

\n {r`\\vec{v} = \\vec{v}_x + \\vec{v}_y`}\n

\n

\n {r`\\left | \\vec{v}_x \\right | = \\left | \\vec{v} \\right | \\sin \\alpha`}\n

\n

\n {r`\\left | \\vec{v}_y \\right | = \\left | \\vec{v} \\right | \\cos \\alpha`}\n

\n \n \n

\n Scomponi in componenti, poi sommali:\n

\n

\n {r`\\vec{v} + \\vec{w} = (\\vec{v}_x + \\vec{w}_x) + (\\vec{v}_y + \\vec{w}_y)`}\n

\n

\n Produce il vettore risultante dall'applicazione della regola del parallelogramma.\n

\n \n \n

\n Alla fine è sempre una somma:\n

\n

\n {r`\\vec{v} - \\vec{w} = (\\vec{v}_x - \\vec{w}_x) + (\\vec{v}_y - \\vec{w}_y)`}\n

\n

\n Produce il vettore che parte da w e arriva a v.\n

\n \n \n

\n Si chiama scalare perchè il risultato è uno scalare, non un vettore.\n

\n

\n {r`\\vec{v} \\cdot \\vec{w} = \\left | \\vec{v} \\right | \\left | \\vec{w} \\right | \\cos \\alpha`}\n

\n

\n Produce il modulo della proiezione di {r`\\vec{a}`} su {r`\\vec{b}`}.\n

\n \n \n

\n Si chiama vettoriale perchè il risultato è un altro vettore.\n

\n
    \n
  • {r`\\vec{c} = \\vec{a} \\times \\vec{b}`}
    • \n
  • {r`\\left | \\vec{c} \\right | = \\left | \\vec{a} \\right | \\cdot \\left | \\vec{b} \\right | \\cdot \\sin(\\alpha)`}
    • \n
  • Regola della mano destra
    • \n
\n

\n Non è commutativo!\n

\n \n
\n
\n \n

\n Se un corpo puntiforme ha forza risultante nulla, allora la sua velocità non cambia.\n

\n

\n {r`\\Sigma \\vec{F} = 0 \\Longleftrightarrow \\Delta v = 0`}\n

\n \n \n

\n La forza risultante di un corpo è direttamente proporzionale alla sua accelerazione, e la costante di proporzionalità è la massa.\n

\n

\n {r`\\Sigma \\vec{F} = m \\vec{a}`}\n

\n \n \n

\n Due corpi esercitano forze uguali e opposte uno sull'altro.\n

\n

\n {r`\\vec{F}_{21} = -\\vec{F}_{12}`}\n

\n \n
\n
\n \n

\n Due corpi puntiformi si attirano uno verso l'altro con forza:\n

\n

\n {r`\\left | \\vec{F} \\right | = G \\frac{m_1 m_2}{s^2}`}\n

\n

\n G è la costante di gravitazione universale e vale:\n

\n

\n {r`G = 6.67 \\cdot 10^{-11} \\frac{N m^2}{{kg}^2}`}\n

\n \n \n

\n Se nel sistema di riferimento consideriamo la Terra ferma, allora un corpo è attratto verso la Terra con forza peso uguale a:\n

\n

\n {r`\\left | \\vec{F} \\right | = g m`}\n

\n

\n g è la costante di gravità della Terra, e vale:\n

\n

\n {r`g = 9.81 \\frac{m}{s^2}`}\n

\n \n \n

\n Per pianeti diversi dalla Terra vale la stessa regola:\n

\n

\n {r`\\left | \\vec{F} \\right | = g m`}\n

\n

\n L'unica differenza è che cambia la costante di gravità:\n

\n

\n {r`g_{luna} = 1.62 \\frac{m}{s^2}`}\n

\n

\n {r`g_{marte} = 3.71 \\frac{m}{s^2}`}\n

\n \n
\n
\n \n

\n Si oppone alle forze applicate alla superficie di contatto.\n

\n

\n Un libro appoggiato su un tavolo ha la forza di gravità che lo attira verso il terreno e la forza normale che lo trattiene dal cadere.\n

\n \n \n

\n Impedisce a un corpo di muoversi se non viene spinto da una forza che supera una certa soglia:\n

\n

\n {r`\\left | \\vec{F} \\right | \\leq \\mu_{s} \\left | \\vec{F}_{normale} \\right |`}\n

\n \n \n

\n Rallenta i corpi che si stanno muovendo finchè essi non si fermano:\n

\n

\n {r`\\left | \\vec{F} \\right | \\leq \\mu_{d} \\left | \\vec{F}_{normale} \\right |`}\n

\n \n \n

\n E' forza trasmessa tra due estremi di una fune.\n

\n

\n Può essere redirezionata per mezzo di carrucole.\n

\n \n \n

\n Una molla cerca sempre di tornare alla sua posizione indeformata con forza:\n

\n

\n {r`F = -k x`}\n

\n

\n (E' negativa perchè la forza è opposta a quella applicata per deformarla.)\n

\n \n
\n
\n \n

\n È un vettore che indica la posizione di un corpo rispetto a un'origine.\n

\n

\n {r`\\Delta \\vec{s} = \\vec{s}(fine) - \\vec{s}(inizio)`}\n

\n \n \n

\n È un vettore che misura la variazione di posizione nel tempo.\n

\n

\n {r`\\vec{v} = \\frac{\\Delta \\vec{s}}{\\Delta t}`}\n

\n

\n Se si considera un intervallo di tempo infinitesimale si dice velocità istantanea:\n

\n

\n {r`\\vec{v} = \\lim_{\\Delta t \\to 0} \\frac{\\Delta \\vec{s}}{\\Delta t} = \\frac{d \\vec{s}}{dt}`}\n

\n \n \n

\n È un vettore che misura la variazione di velocità nel tempo.\n

\n

\n {r`\\vec{a} = \\frac{\\Delta \\vec{v}}{\\Delta t}`}\n

\n

\n Se si considera un intervallo di tempo infinitesimale si dice accelerazione istantanea:\n

\n

\n {r`\\vec{a} = \\lim_{\\Delta v \\to 0} \\frac{\\Delta \\vec{v}}{\\Delta t} = \\frac{d \\vec{v}}{d t} = \\frac{d^2 \\vec{s}}{d t^2}`}\n

\n \n Quantità di moto (momento lineare)}>\n

\n La quantità di moto è una proprietà vettoriale dei corpi:\n

\n

\n {r`\\vec{p} = m \\vec{v}`}\n

\n

\n Se la forza risultante è nulla, la quantità di moto non cambia.\n

\n

\n {r`\\Sigma \\vec{F} = 0 \\Longleftrightarrow \\Delta \\vec{p} = 0`}\n

\n \n
\n
\n \n

\n La legge oraria è:\n

\n

\n {r`s(t) = v \\cdot \\Delta t + s(0)`}\n

\n \n \n

\n È costante:\n

\n

\n {r`v(t) = k`}\n

\n \n \n

\n La velocità non varia:\n

\n

\n {r`a(t) = 0`}\n

\n \n \n

\n Si applica la prima legge di Newton:\n

\n

\n f(t) = 0\n

\n \n
\n
\n \n

\n La legge oraria è:\n

\n

\n {r`s(t) = \\frac{1}{2} a \\cdot (\\Delta t)^2 + v(0) \\cdot (\\Delta t) + s(0)`}\n

\n \n \n

\n È una retta:\n

\n

\n {r`v(t) = a \\Delta t + v(0)`}\n

\n \n \n

\n È costante:\n

\n

\n {r`a(t) = k`}\n

\n \n \n

\n Si applica la prima legge di Newton:\n

\n

\n f(t) = m a\n

\n \n
\n
\n \n

\n E' la distanza dal centro massima che raggiunge il corpo.\n

\n

\n (L'ampiezza di una sinusoide.)\n

\n \n \n

\n Indica quanto in fretta cambia la posizione del corpo.\n

\n

\n Dipende dal periodo:\n

\n

\n {r`\\omega = \\frac{2 \\pi}{T}`}\n

\n \n \n

\n E' una sinusoide:\n

\n

\n {r`s(t) = A \\sin (\\omega \\cdot t + \\phi)`}\n

\n \n \n

\n E' la sinusoide dello spostamento, sfasata di {r`\\frac{\\pi}{2}`}:\n

\n

\n {r`v(t) = A \\sin (\\omega \\cdot t + \\phi + \\frac{\\pi}{2})`}\n

\n \n \n

\n E' la sinusoide della velocità, sfasata di {r`\\pi`}:\n

\n

\n {r`a(t) = A \\sin (\\omega \\cdot t + \\phi + \\pi)`}\n

\n \n \n

\n Si applica la prima legge di Newton:\n

\n

\n f(t) = m a\n

\n \n
\n
\n \n

\n Il moto parabolico è dato sommando un moto rettilineo uniforme sull'asse orizzontale e un moto rettilineo uniformemente accelerato sull'asse verticale.\n

\n \n \n

\n Il moto parabolico è dato sommando due moti armonici semplici: uno sull'asse X, e l'altro, sfasato di {r`\\frac{\\pi}{2}`}, sull'asse Y.\n

\n \n
\n
\n \n

\n Velocità angolare\n

\n

\n Quanto cambia la fase nel tempo.\n

\n

\n {r`\\omega = \\frac{2 \\pi}{T}`}\n

\n \n \n

\n E' l'angolo percorso dal corpo rispetto alla posizione iniziale.\n

\n

\n Si indica con {r`\\phi`}, e generalmente si usa in radianti.\n

\n \n \n

\n Si applicano le formule per la circonferenza:\n

\n

\n {r`v = \\frac{\\Delta s}{t} = \\frac{2 \\pi \\cdot r}{T} = \\omega r`}\n

\n \n \n

\n Il corpo ha sempre un accelerazione verso il centro che gli impedisce di abbandonare il moto:\n

\n

\n {r`a = \\frac{v^2}{r} = r \\cdot \\omega^2 = v \\cdot \\omega`}\n

\n \n \n

\n È verso il centro e si calcola con:\n

\n

\n {r`F = m \\cdot a`}\n

\n \n
\n
\n \n

\n E' compiuto da una forza che sposta un corpo.\n

\n

\n {r`W = \\vec{F} \\cdot \\vec{s} = F \\cdot \\Delta s \\cdot cos(\\alpha )`}\n

\n

\n (Se la forza non è parallela allo spostamento, il prodotto scalare ci fa considerare solo la componente parallela.)\n

\n \n \n

\n Un corpo ha energia cinetica in ogni momento uguale a:\n

\n

\n {r`E_c = \\frac{1}{2} m v^2`}\n

\n

\n Se una forza effettua lavoro su un corpo, cambia la sua energia cinetica pari al lavoro effettuato:\n

\n

\n {r`\\Delta E_c = W`}\n

\n \n \n

\n Un corpo ha energia potenziale in ogni momento pari a:\n

\n

\n {r`E_{p_g} = m \\cdot g \\cdot h`}\n

\n

\n (Con h uguale a un altezza scelta come punto di riferimento.)\n

\n \n \n

\n Una molla ha sempre energia potenziale elastica pari a:\n

\n

\n {r`E_{p_e} = \\frac{1}{2} k x^2`}\n

\n \n \n

\n Sono conservative le forze per le quali il lavoro compiuto non dipende dal percorso seguito per andare dalla partenza all'arrivo.\n

\n

\n Ad esempio, è conservativa la forza di gravità, ma non è conservativa la forza di attrito.\n

\n

\n Se in un sistema ci sono solo forze conservative, allora l'energia meccanica totale si conserva:\n

\n

\n {r`E = E_k + E_p`}\n

\n \n \n

\n È la velocità di trasferimento di energia:\n

\n

\n {r`P = \\frac{\\Delta E}{\\Delta t}`}\n

\n \n
\n
\n \n

\n È una proprietà dei corpi che può essere positiva o negativa.\n

\n

\n Si conserva: in un sistema chiuso la carica totale è costante.\n

\n

\n Esiste un'unità elementare: {r`C_{elettrone} = 1.602 \\cdot 10^{-19}`}.\n

\n

\n Cariche opposte si attraggono; cariche uguali si respingono.\n

\n \n \n

\n Più ioni ha un corpo, meglio la carica si muove attraverso di esso.\n

\n

\n I corpi in cui la carica si muove bene sono conduttori, mentre quelli in cui si muove difficilmente sono isolanti.\n

\n

\n Il corpo umano è un buon conduttore.\n

\n \n
\n
\n \n

\n E' possibile polarizzare un corpo per accumulare la carica di un segno in una certa zona.\n

\n \n
\n
\n \n

\n Se un corpo conduttore è in contatto con la Terra, le cariche su di esso saranno equilibrate e il corpo diventerà elettricamente neutro (con stesso numero di cariche positive e negative all'interno).\n

\n \n
\n
\n \n

\n Strofinando tra loro due corpi isolanti, essi si polarizzeranno per strofinio.\n

\n \n \n

\n Toccando un conduttore con un corpo carico, il conduttore potrà polarizzarsi per contatto.\n

\n \n \n

\n Se un corpo conduttore ha cariche \"esterne\" di un certo segno vicino, esso avrà tutte le cariche del segno opposto in equilibrio vicino alle cariche esterne, e tutte le cariche dello stesso segno più lontano possibile da esse.\n

\n

\n Mettendo a terra il conduttore, nuove cariche del segno opposto saranno attratte all'interno del corpo per equilibrare le cariche che si sono allontanate.\n

\n

\n Staccando il conduttore da terra e rimuovendo le cariche esterne, esso si ritroverà caricato del segno opposto rispetto alle cariche esterne.\n

\n \n
\n
\n \n

\n Due corpi carichi si attraggono tra loro con forza:\n

\n

\n {r`\\left | \\vec{F}_{elettrica} \\right | = \\frac{-k \\cdot q_1 \\cdot q_2}{s^2}`}\n

\n

\n {r`k`} è la costante di Coulomb, e vale {r`k = 8.99 \\cdot 10^9 \\frac{N \\cdot m^2}{C^2}`}.\n

\n \n \n

\n La costante {r`k`} è in realtà dipendente da un altra costante, {r`\\epsilon_0`}, la permeabilità del vuoto.\n

\n

\n {r`k = \\frac{1}{4 \\pi \\cdot \\epsilon_0}`}\n

\n

\n {r`\\left | \\vec{F}_{elettrica} \\right | = \\frac{q_1 \\cdot q_2}{4 \\pi \\cdot \\epsilon_0 \\cdot s^2}`}\n

\n \n \n

\n Misura che forza viene applicata in ogni punto su una carica unitaria:\n

\n

\n {r`\\vec{E} = \\frac{\\vec{F}_{elettrica}}{q} = \\frac{-k \\cdot q}{s^2}`}\n

\n \n \n

\n È la differenza tra \"quanto\" campo elettrico entra e quanto campo elettrico esce da una certa area.\n

\n

\n In qualsiasi superficie chiusa, il flusso elettrico è uguale alla componente perpendicolare del campo elettrico moltiplicato per l'area.\n

\n

\n {r`\\Phi_E = \\vec{E} \\cdot \\vec{A}`}\n

\n

\n Se il campo elettrico è uniforme, se ne può calcolare facilmente il valore:\n

\n

\n {r`\\Phi_E = \\vec{E} \\cdot \\vec{A} = E_\\perp \\cdot A \\cdot \\cos(\\alpha)`}\n

\n

\n Circa. E' una specie di integrale...\n

\n \n \n

\n Il flusso elettrico è direttamente proporzionale alla carica presente all'interno della superficie.\n

\n

\n {r`\\Phi_E = 4 \\pi \\cdot k \\cdot q = \\frac{q}{\\epsilon_0}`}\n

\n

\n Ovvero, i campi elettrostatici sono generati dalle cariche elettriche.\n

\n \n
\n
\n \n

\n Un corpo carico vicino ad altre cariche possiede un'energia potenziale elettrica {r`U_e`}.\n

\n \n
\n
\n Potenziale elettrico (tensione)}>\n

\n È il valore dell'energia potenziale elettrica per una carica unitaria.\n

\n

\n {r`V = \\frac{U_e}{q}`}\n

\n

\n La sua unità di misura è il Volt ({r`V`}).\n

\n

\n In una batteria è detto forza elettromotrice, e corrisponde al lavoro compiuto da una batteria ideale per spostare una carica unitaria tra i due poli.\n

\n \n Corrente elettrica (intensità)}>\n

\n Quanta carica passa attraverso un'area (perpendicolare al flusso) nel tempo.\n

\n

\n {r`I = \\frac{\\Delta q}{\\Delta t}`}\n

\n

\n Fintanto che c'è differenza di potenziale, ci sarà anche intensità non nulla.\n

\n

\n La sua unità di misura è l'Ampere ({r`A`}).\n

\n \n Corrente continua (DC)}>\n

\n Quando in un circuito la direzione della corrente è costante.\n

\n \n Corrente alternata (AC)}>\n

\n Quando in un circuito la direzione della corrente si alterna periodicamente.\n

\n \n \n

\n Possiamo calcolare la potenza di un circuito:\n

\n

\n {r`P = \\frac{\\Delta U_e}{\\Delta t} = I \\cdot \\Delta V = I^2 \\cdot R = \\frac{(\\Delta V)^2}{R}`}\n

\n \n
\n
\n \n

\n Riduce l'intensità di corrente, e converte parte del potenziale in calore.\n

\n

\n Il potenziale utilizzato è pari a:\n

\n

\n {r`V = R \\cdot I`}\n

\n

\n Dove {r`R`} è una costante detta resistenza con unità di misura Ohm ({r`\\Omega`}).\n

\n

\n La resistenza di un conduttore vale:\n

\n

\n {r`R = \\rho \\frac{L_{unghezza}}{A_{rea}}`}\n

\n

\n {r`\\rho`} è la resistività del materiale, e varia in base alla temperatura:\n

\n

\n {r`\\rho = \\rho_0 (1 + \\alpha(T - T_0))`}\n

\n \n \n

\n Immagazzina potenziale elettrico, permettendo di riutilizzarla in seguito.\n

\n

\n Per farlo, cattura cariche positive e negative sulle sue due armature; perchè questo avvenga, deve essere compiuto lavoro.\n

\n

\n Ha una capacità caratteristica, che in un condensatore a facce piane parallele è:\n

\n

\n {r`C = \\frac{q_{massima}}{\\Delta V}`}\n

\n

\n Condensatori di capacità maggiore immagazzinano più potenziale con meno carica.\n

\n

\n La capacità aumenta se viene messo qualcosa tra le armature:\n

\n

\n {r`C_{nuova} = \\kappa \\cdot \\frac{\\epsilon_0 \\cdot A}{s}`}\n

\n

\n Dove {r`\\kappa`} è la costante dielettrica relativa del materiale inserito, {r`A`} l'area di una armatura e {r`s`} la distanza tra le due armature.\n

\n

\n Se il campo elettrico creatosi tra le due armature supera la rigidità dielettrica del condensatore, la carica immagazzinata viene persa e ha luogo un breakdown.\n

\n

\n La sua unità di misura è il Farad ({r`Fa`})\n

\n \n \n

\n Misura la corrente elettrica se messo in serie.\n

\n

\n (Funzionamento: ha una resistenza interna bassisima in modo da non influire significativamente sulla corrente.)\n

\n \n \n

\n Misura la differenza di potenziale se messo in parallelo.\n

\n

\n (Funzionamento: ha una resistenza altissima in modo da non influire significativamente sulla tensione.)\n

\n \n
\n
\n \n

\n Per nodo si intende un qualsiasi punto del circuito.\n

\n

\n Da un nodo entra ed esce la stessa corrente.\n

\n \n \n

\n Per maglia si intende un qualsiasi percorso chiuso all'interno del circuito.\n

\n

\n In una maglia chiusa, la somma delle differenze di potenziale è 0.\n

\n \n
\n
\n \n

\n Più parti di circuito sono in serie se sono consecutive e senza biforcazioni.\n

\n

\n Parti di circuito in serie sono attraversate dalla stessa corrente.\n

\n \n \n

\n Più parti di circuito sono in parallelo tra loro se hanno lo stesso punto di partenza e lo stesso punto di arrivo.\n

\n

\n Parti di circuito in parallelo hanno la stessa differenza di potenziale.\n

\n \n
\n
\n \n

\n Nei circuiti in serie, tutte le resistenze possono essere sostituite con una equivalente dalla resistenza della somma di tutte le quelle sostituite:\n

\n

\n {r`R_{serie} = \\sum_{i=1}^{n} R_i`}\n

\n \n \n

\n Nei circuiti in parallelo, tutte le resistenze possono essere sostituite con una equivalente dalla resistenza di:\n

\n

\n {r`R_{parallelo} = \\frac{1}{\\sum_{i=1}^{n} \\frac{1}{R_i}}`}\n

\n \n
\n
\n \n

\n Nei circuiti in serie, tutti i condensatori possono essere sostituiti con uno equivalente dalla capacità di:\n

\n

\n {r`C_{serie} = \\frac{1}{\\sum_{i=1}^{n} \\frac{1}{C_i}}`}\n

\n \n \n

\n Nei circuiti in parallelo, tutte i condensatori possono essere sostituite con uno equivalente dalla capacità della somma di tutti quelli sostituiti:\n

\n

\n {r`C_{parallelo} = \\sum_{i=1}^{n} C_n`}\n

\n \n
\n
\n \n

\n E' una costante fisica fondamentale che rappresenta quanto un materiale si magnetizza facilmente.\n

\n

\n {r`\\mu_0 = 4 \\pi \\cdot 10^{-7} \\frac{H}{m}`} ({r`\\frac{N}{A^2}`})\n

\n \n \n

\n Come un campo elettrico, ma per i magneti.\n

\n

\n Il suo simbolo è {r`B`}, e la sua unità di misura è il Tesla (T).\n

\n \n \n

\n È \"quanto\" campo magnetico attraversa un percorso chiuso.\n

\n

\n Per qualsiasi percorso chiuso, il flusso magnetico è uguale alla somma di tutti i \"sottoflussi\" magnetici calcolati sui suoi lati.\n

\n

\n {r`\\Phi_{B_{i}} = \\vec{B} \\cdot \\vec{L}_n = B \\cdot L_i \\cdot \\sin(\\alpha) = B_\\parallel \\cdot L_i`}\n

\n

\n {r`\\Phi_{B} = \\sum_{i=0}^{n_{lati}} \\Phi_{Bn}`}\n

\n

\n La sua unità di misura è il Weber ({r`Wb = T \\cdot m^2`}).\n

\n \n \n

\n Il flusso magnetico attraverso qualsiasi superficie chiusa è sempre nullo.\n

\n

\n Ovvero, non esistono monopoli magnetici.\n

\n \n \n

\n L'intensità di corrente che attraversa un percorso chiuso è direttamente proporzionale al flusso magnetico dello stesso percorso.\n

\n

\n {r`\\Phi_B = \\mu_0 \\cdot I`}\n

\n \n
\n
\n Forza magnetica su carica puntiforme (Forza di Lorentz)}>\n

\n I campi magnetici applicano una forza sulle cariche vicine:\n

\n

\n {r`\\vec{F}_{B} = q \\cdot (\\vec{v} \\times \\vec{B})`}\n

\n

\n Dove {r`\\vec{B}`} è l'intensità del campo magnetico e {r`\\vec{v}`} la velocità della carica considerata.\n

\n

\n Si ha una forza massima se la velocità è perpendicolare al campo magnetico.\n

\n

\n In un campo magnetico uniforme, una velocità perpendicolare al campo porta alla creazione di un moto circolare uniforme.\n

\n \n \n

\n I campi magnetici influenzano ovviamente anche le cariche presenti in un conduttore:\n

\n

\n {r`\\vec{F}_{magnetica} = I \\cdot (\\vec{L} \\times \\vec{B})`} [1]\n

\n

\n Dove {r`I`} è la corrente elettrica, {r`\\vec{L}`} è un vettore che punta nella direzione di scorrimento della corrente e ha come modulo la lunghezza del conduttore.\n

\n \n
\n
\n \n

\n Una spira in cui passa corrente produce un campo magnetico perpendicolare al piano creato dalla spira.\n

\n \n \n

\n Un solenoide sono tante spire avvolte in modo da formare una specie di cilindro.\n

\n

\n All'interno del solenoide si crea un campo (quasi) uniforme:\n

\n

\n {r`\\left | \\vec{B} \\right | = \\mu_0 \\cdot I \\cdot \\frac{A_{vvolgimenti}}{L_{unghezzafilo}}`}\n

\n \n \n

\n Caso particolare della Legge di Ampère.\n

\n

\n Il modulo del campo magnetico B prodotto da un filo in cui passa una corrente continua I alla distanza s è:\n

\n

\n {r`\\left | \\vec{B} \\right | = \\frac{\\mu \\cdot I}{2 \\pi r}`}\n

\n

\n Il campo magnetico così creato gira attorno al filo in senso antiorario.\n

\n

\n Due fili attraversati dalla stessa corrente si attraggono, due fili attraversati da correnti opposte si respingono.\n

\n \n
\n
\n \n

\n Un conduttore perpendicolare ad un campo magnetico può ottenere una differenza di potenziale se messo in movimento in un direzione perpendicolare alla direzione del conduttore e del campo.\n

\n

\n La differenza di potenziale si crea a causa della forza magnetica, che fa spostare tutti gli elettroni verso un capo del conduttore.\n

\n

\n Essa vale:\n

\n

\n {r`\\Delta V_{indotta} = v \\cdot B \\cdot L`}\n

\n

\n Dove v è la velocità del conduttore, B è l'intensità del campo magnetico ed L è la lunghezza del conduttore.\n

\n \n \n

\n In un campo magnetico {r`B`} uniforme e perpendicolare al piano di una spira di area {r`A`}, il flusso magnetico si può determinare con la Legge di Faraday-Neumann-Lenz:\n

\n

\n {r`\\Phi_B = \\vec{B} \\cdot \\vec{A} = B \\cdot A \\cdot \\cos(\\alpha)`}\n

\n \n
\n
\n \n

\n Dice che la forza elettromotrice media indotta in un percorso dipende dalla variazione nel tempo del flusso magnetico nello stesso percorso.\n

\n

\n {r`\\Delta V_{indotta} = - \\frac{\\Delta \\Phi_B}{\\Delta t}`}\n

\n

\n Il meno è dovuto alla Legge di Lenz, che specifica qualitativamente il verso della forza elettromotrice indotta.\n

\n \n \n

\n In un solenoide, la forza elettromotrice indotta è uguale a:\n

\n

\n {r`\\Delta V_{indotta} = - \\frac{N \\cdot \\Delta \\Phi_{B_{spira}}}{\\Delta t} = - \\frac{N \\cdot B \\cdot A \\cdot cos(\\alpha)}{\\Delta t}`}\n

\n

\n Dove {r`N`} è il numero delle spire del solenoide.\n

\n \n \n

\n Correnti o campi elettrici variabili creano un campo magnetico.\n

\n \n
\n
\n \n

\n Nel vuoto, il campo elettrico {r`E`} e il campo magnetico {r`B`} sono perpendicolari tra loro e la direzione di propagazione, e sono entrambe funzioni del tempo.\n

\n

\n Si dice quindi che sono onde elettromagnetiche.\n

\n

\n Esse sono legate dalla relazione:\n

\n

\n {r`E = c \\cdot B`}\n

\n

\n Dove {r`c`} è la velocità delle onde (luce) nel vuoto, e a sua volta è uguale a:\n

\n

\n {r`c = \\frac{1}{\\sqrt{\\epsilon_0 \\cdot \\mu_0}} = 3.00 \\cdot 10^8 \\frac{m}{s}`}\n

\n \n \n

\n {r`A(t) = A_{max} \\cdot \\sin \\left ( \\frac{2 \\pi}{\\lambda} - \\omega t + \\phi \\right )`}\n

\n

\n Dove {r`A_{max}`} è l'ampiezza massima che può avere l'onda, {r`\\frac{2 \\pi}{\\lambda} = \\left | \\vec{k} \\right |`} è il vettore d'onda, {r`\\omega`} la frequenza angolare e {r`\\phi`} la fase.\n

\n \n
\n
\n \n

\n I solidi, se portati ad alta temperatura, emettono luce con uno spettro continuo.\n

\n

\n I gas, invece, ad alta temperatura emettono luce solo con particolari lunghezze d'onda.\n

\n

\n In un gas di idrogeno, le lunghezze d'onda emesse sono ricavabili con:\n

\n

\n {r`\\frac{1}{\\lambda} = R \\left ( \\frac{1}{4} - \\frac{1}{n^2} \\right )`}\n

\n

\n Con {r`R = 1.097 \\cdot 10^7 \\frac{1}{m}`}, detta costante di Rydberg, e {r`n`} un numero intero.\n

\n \n \n

\n Una grandezza si dice quantizzata (o discreta) se può assumere solo determinati valori.\n

\n

\n Una grandezza si dice continua se può assumere qualsiasi valore e quindi se non è quantizzata.\n

\n

\n Energia, momento angolare e raggio sono quantizzati.\n

\n

\n Nota costante quantica è {r`h`}, la costante di Planck, ovvero il valore minimo possibile per la carica (talvolta espressa come {r`\\hbar = \\left ( \\frac{h}{2 \\pi} \\right )`}.\n

\n \n
\n
\n \n

\n L'energia degli elettroni è quantizzata.\n

\n

\n Inoltre, per essi è valido che:\n

\n

\n {r`m \\cdot v_n \\cdot 2 \\pi \\cdot r = n \\cdot h`}\n

\n

\n Ancora, il raggio delle orbite è uguale a:\n

\n

\n {r`r_n = n^2 \\cdot a_0 = n^2 \\cdot \\frac{\\hbar}{m_{elettrone} \\cdot k \\cdot e^2} `}\n

\n

\n Con {r`a_0 = \\left ( \\frac{h}{2 \\pi} \\right )^2 \\cdot \\frac{1}{m_{elettrone} \\cdot k \\cdot e^2} = 5.29 \\cdot 10^{-11} m`}.\n

\n

\n Infine, in ogni stato, l'energia è pari a:\n

\n

\n {r`E_n = \\frac{1}{n^2} \\cdot E_1 = - \\frac{1}{n^2} \\cdot \\frac{a_0^2}{2 \\cdot m \\cdot \\hbar^4} = - \\frac{1}{n^2} \\cdot \\frac{m_{elettrone} \\cdot k^2 \\cdot e^4}{2 \\cdot \\hbar^2}`}\n

\n

\n Due elettroni non possono occupare lo stesso stato.\n

\n

\n Questo modello funziona solo per atomi con numero atomico basso. Atomi con molti elettroni hanno comportamenti diversi, descritti dal modello di\n

\n \n
\n
\n \n

\n Nei solidi, le lunghezze d'onda sono talmente tanto vicine da poter essere considerate una banda.\n

\n

\n Possono però comunque avere dei gap dovuti agli intervalli di energia non ammessi.\n

\n \n
\n
\n \n

\n Refactor this\n

\n

\n Se la banda di emissione con energia più alta di un corpo è assente o è separata da un gap dell'ordine di grandezza maggiore di {r`10^1 eV`}, allora il corpo è un isolante.\n

\n

\n Se invece la banda di emissione si sovrappone a un altra, allora il corpo è un conduttore.\n

\n

\n Se il gap è invece dell'ordine di grandezza di {r`1 eV`}, allora il corpo è un semiconduttore.\n

\n \n \n

\n Legami in cui mancano elettroni.\n

\n

\n Elettroni di altri legami possono spostarsi per colmare le lacune, creandone altre, e spostandole in direzione opposta a quella della corrente.\n

\n \n \n

\n Se si inserisce in un cristallo semiconduttore si inserisce un atomo con numero atomico diverso, si otterrà:\n

\n
    \n
  • Con numero atomico maggiore, un semiconduttore di tipo N con elettroni in eccesso liberi di scorrere.
    • \n
  • Con numero atomico minore, un semiconduttore di tipo P con lacune in eccesso libere di catturare elettroni da altri legami.
    • \n
\n

\n Maggiore impurezza porta a maggiore conduttività.\n

\n \n \n

\n Aumentando la temperatura di un semiconduttore si aumenta la conduttività, perchè eccita le particelle e favorisce il movimento di elettroni e lacune.\n

\n \n
\n
Ottica (non l'abbiamo fatta)}>\n \n

\n I corpi possono assorbire o riflettere le onde elettromagnetiche che li colpiscono.\n

\n \n \n

\n Un corpo nero è un corpo che assorbe tutte le onde elettromagnetiche che riceve senza rifletterne nessuna.\n

\n

\n Le onde assorbite vengono poi riemesse sotto forma di un onda di {r`\\lambda`} variabile in base alla temperatura.\n

\n

\n {r`\\lambda_{max} \\cdot T`} è costante.\n

\n \n \n

\n L'energia assorbita e emessa dai corpi neri è quantizzata.\n

\n \n \n

\n Un onda magnetica con un quanto di energia è detta fotone:\n

\n

\n {r`E_{fotone} = h \\cdot f`}\n

\n \n \n

\n A volte, i fotoni che colpiscono un metallo possono estrarvi degli elettroni e creare una differenza di potenziale.\n

\n

\n Perchè avvenga, la frequenza deve essere maggiore di una certa soglia.\n

\n

\n Il numero di elettroni estratti dipende dall'intensità dell'onda, mentre l'energia cinetica degli elettroni dipende dalla frequenza.\n

\n

\n Non c'è nessun ritardo tra l'assorbimento del fotone e l'estrazione di elettroni.\n

\n \n
\n
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cdot \vec{A} = E_perp cdot A cdot cos(alpha)"],["\\Phi_E = \\vec{E} \\cdot \\vec{A} = E_\\perp \\cdot A \\cdot \\cos(\\alpha)"]);return he=function(){return e},e}function ze(){var e=wn(["Phi_E = \vec{E} cdot \vec{A}"],["\\Phi_E = \\vec{E} \\cdot \\vec{A}"]);return ze=function(){return e},e}function be(){var e=wn(["\vec{E} = \frac{\vec{F}_{elettrica}}{q} = \frac{-k cdot q}{s^2}"],["\\vec{E} = \\frac{\\vec{F}_{elettrica}}{q} = \\frac{-k \\cdot q}{s^2}"]);return be=function(){return e},e}function De(){var e=wn(["left | \vec{F}_{elettrica} \right | = \frac{q_1 cdot q_2}{4 pi cdot epsilon_0 cdot s^2}"],["\\left | \\vec{F}_{elettrica} \\right | = \\frac{q_1 \\cdot q_2}{4 \\pi \\cdot \\epsilon_0 \\cdot s^2}"]);return De=function(){return e},e}function qe(){var e=wn(["k = \frac{1}{4 pi cdot epsilon_0}"],["k = \\frac{1}{4 \\pi \\cdot \\epsilon_0}"]);return qe=function(){return e},e}function Le(){var e=wn(["epsilon_0"],["\\epsilon_0"]);return Le=function(){return e},e}function Fe(){var e=wn(["k"]);return Fe=function(){return e},e}function Ee(){var e=wn(["k = 8.99 cdot 10^9 \frac{N cdot m^2}{C^2}"],["k = 8.99 \\cdot 10^9 \\frac{N \\cdot m^2}{C^2}"]);return Ee=function(){return e},e}function Ae(){var e=wn(["k"]);return Ae=function(){return e},e}function Pe(){var e=wn(["left | \vec{F}_{elettrica} \right | = \frac{-k cdot q_1 cdot q_2}{s^2}"],["\\left | \\vec{F}_{elettrica} \\right | = \\frac{-k \\cdot q_1 \\cdot q_2}{s^2}"]);return Pe=function(){return e},e}function Ce(){var e=wn(["C_{elettrone} = 1.602 cdot 10^{-19}"],["C_{elettrone} = 1.602 \\cdot 10^{-19}"]);return Ce=function(){return e},e}function Ie(){var e=wn(["P = \frac{Delta E}{Delta t}"],["P = \\frac{\\Delta E}{\\Delta t}"]);return Ie=function(){return e},e}function Se(){var e=wn(["E = E_k + E_p"]);return Se=function(){return e},e}function we(){var e=wn(["E_{p_e} = \frac{1}{2} k x^2"],["E_{p_e} = \\frac{1}{2} k x^2"]);return we=function(){return e},e}function ke(){var e=wn(["E_{p_g} = m cdot g cdot h"],["E_{p_g} = m \\cdot g \\cdot h"]);return ke=function(){return e},e}function Be(){var e=wn(["Delta E_c = W"],["\\Delta E_c = W"]);return Be=function(){return e},e}function Ne(){var e=wn(["E_c = \frac{1}{2} m v^2"],["E_c = \\frac{1}{2} m v^2"]);return Ne=function(){return e},e}function Ve(){var e=wn(["W = \vec{F} cdot \vec{s} = F cdot Delta s cdot cos(alpha )"],["W = \\vec{F} \\cdot \\vec{s} = F \\cdot \\Delta s \\cdot cos(\\alpha )"]);return Ve=function(){return e},e}function Re(){var e=wn(["F = m cdot a"],["F = m \\cdot a"]);return Re=function(){return e},e}function Te(){var e=wn(["a = \frac{v^2}{r} = r cdot omega^2 = v cdot omega"],["a = \\frac{v^2}{r} = r \\cdot \\omega^2 = v \\cdot \\omega"]);return Te=function(){return e},e}function xe(){var e=wn(["v = \frac{Delta s}{t} = \frac{2 pi cdot r}{T} = omega r"],["v = \\frac{\\Delta s}{t} = \\frac{2 \\pi \\cdot r}{T} = \\omega r"]);return xe=function(){return e},e}function ye(){var e=wn(["phi"],["\\phi"]);return ye=function(){return e},e}function Ue(){var e=wn(["omega = \frac{2 pi}{T}"],["\\omega = \\frac{2 \\pi}{T}"]);return Ue=function(){return e},e}function Me(){var e=wn(["\frac{pi}{2}"],["\\frac{\\pi}{2}"]);return Me=function(){return e},e}function Oe(){var e=wn(["a(t) = A sin (omega cdot t + phi + pi)"],["a(t) = A \\sin (\\omega \\cdot t + \\phi + \\pi)"]);return Oe=function(){return e},e}function Ge(){var e=wn(["pi"],["\\pi"]);return Ge=function(){return e},e}function We(){var e=wn(["v(t) = A sin (omega cdot t + phi + \frac{pi}{2})"],["v(t) = A \\sin (\\omega \\cdot t + \\phi + \\frac{\\pi}{2})"]);return We=function(){return e},e}function Qe(){var e=wn(["\frac{pi}{2}"],["\\frac{\\pi}{2}"]);return Qe=function(){return e},e}function je(){var e=wn(["s(t) = A sin (omega cdot t + phi)"],["s(t) = A \\sin (\\omega \\cdot t + \\phi)"]);return je=function(){return e},e}function Je(){var e=wn(["omega = \frac{2 pi}{T}"],["\\omega = \\frac{2 \\pi}{T}"]);return Je=function(){return e},e}function He(){var e=wn(["a(t) = k"]);return He=function(){return e},e}function Ze(){var e=wn(["v(t) = a Delta t + v(0)"],["v(t) = a \\Delta t + v(0)"]);return Ze=function(){return e},e}function Xe(){var e=wn(["s(t) = \frac{1}{2} a cdot (Delta t)^2 + v(0) cdot (Delta t) + s(0)"],["s(t) = \\frac{1}{2} a \\cdot (\\Delta t)^2 + v(0) \\cdot (\\Delta t) + s(0)"]);return Xe=function(){return e},e}function Ke(){var e=wn(["a(t) = 0"]);return Ke=function(){return e},e}function Ye(){var e=wn(["v(t) = k"]);return Ye=function(){return e},e}function $e(){var e=wn(["s(t) = v cdot Delta t + s(0)"],["s(t) = v \\cdot \\Delta t + s(0)"]);return $e=function(){return e},e}function en(){var e=wn(["Sigma \vec{F} = 0 Longleftrightarrow Delta \vec{p} = 0"],["\\Sigma \\vec{F} = 0 \\Longleftrightarrow \\Delta \\vec{p} = 0"]);return en=function(){return e},e}function nn(){var e=wn(["\vec{p} = m \vec{v}"],["\\vec{p} = m \\vec{v}"]);return nn=function(){return e},e}function tn(){var e=wn(["\vec{a} = lim_{Delta v \to 0} \frac{Delta \vec{v}}{Delta t} = \frac{d \vec{v}}{d t} = \frac{d^2 \vec{s}}{d t^2}"],["\\vec{a} = \\lim_{\\Delta v \\to 0} \\frac{\\Delta \\vec{v}}{\\Delta t} = \\frac{d \\vec{v}}{d t} = \\frac{d^2 \\vec{s}}{d t^2}"]);return tn=function(){return e},e}function ln(){var e=wn(["\vec{a} = \frac{Delta \vec{v}}{Delta t}"],["\\vec{a} = \\frac{\\Delta \\vec{v}}{\\Delta t}"]);return ln=function(){return e},e}function an(){var e=wn(["\vec{v} = lim_{Delta t \to 0} \frac{Delta \vec{s}}{Delta t} = \frac{d \vec{s}}{dt}"],["\\vec{v} = \\lim_{\\Delta t \\to 0} \\frac{\\Delta \\vec{s}}{\\Delta t} = \\frac{d \\vec{s}}{dt}"]);return an=function(){return e},e}function on(){var e=wn(["\vec{v} = \frac{Delta \vec{s}}{Delta t}"],["\\vec{v} = \\frac{\\Delta \\vec{s}}{\\Delta t}"]);return on=function(){return e},e}function rn(){var e=wn(["Delta \vec{s} = \vec{s}(fine) - \vec{s}(inizio)"],["\\Delta \\vec{s} = \\vec{s}(fine) - \\vec{s}(inizio)"]);return rn=function(){return e},e}function un(){var e=wn(["F = -k x"]);return un=function(){return e},e}function cn(){var e=wn(["left | \vec{F} \right | leq mu_{d} left | \vec{F}_{normale} \right |"],["\\left | \\vec{F} \\right | \\leq \\mu_{d} \\left | \\vec{F}_{normale} \\right |"]);return cn=function(){return e},e}function pn(){var e=wn(["left | \vec{F} \right | leq mu_{s} left | \vec{F}_{normale} \right |"],["\\left | \\vec{F} \\right | \\leq \\mu_{s} \\left | \\vec{F}_{normale} \\right |"]);return pn=function(){return e},e}function sn(){var e=wn(["g_{marte} = 3.71 \frac{m}{s^2}"],["g_{marte} = 3.71 \\frac{m}{s^2}"]);return sn=function(){return e},e}function dn(){var e=wn(["g_{luna} = 1.62 \frac{m}{s^2}"],["g_{luna} = 1.62 \\frac{m}{s^2}"]);return dn=function(){return e},e}function fn(){var e=wn(["left | \vec{F} \right | = g m"],["\\left | \\vec{F} \\right | = g m"]);return fn=function(){return e},e}function vn(){var e=wn(["g = 9.81 \frac{m}{s^2}"],["g = 9.81 \\frac{m}{s^2}"]);return vn=function(){return e},e}function mn(){var e=wn(["left | \vec{F} \right | = g m"],["\\left | \\vec{F} \\right | = g m"]);return mn=function(){return e},e}function gn(){var e=wn(["G = 6.67 cdot 10^{-11} \frac{N m^2}{{kg}^2}"],["G = 6.67 \\cdot 10^{-11} \\frac{N m^2}{{kg}^2}"]);return gn=function(){return e},e}function _n(){var e=wn(["left | \vec{F} \right | = G \frac{m_1 m_2}{s^2}"],["\\left | \\vec{F} \\right | = G \\frac{m_1 m_2}{s^2}"]);return _n=function(){return e},e}function hn(){var e=wn(["\vec{F}_{21} = -\vec{F}_{12}"],["\\vec{F}_{21} = -\\vec{F}_{12}"]);return hn=function(){return e},e}function zn(){var e=wn(["Sigma \vec{F} = m \vec{a}"],["\\Sigma \\vec{F} = m \\vec{a}"]);return zn=function(){return e},e}function bn(){var e=wn(["Sigma \vec{F} = 0 Longleftrightarrow Delta v = 0"],["\\Sigma \\vec{F} = 0 \\Longleftrightarrow \\Delta v = 0"]);return bn=function(){return e},e}function Dn(){var e=wn(["left | \vec{c} \right | = left | \vec{a} \right | cdot left | \vec{b} \right | cdot sin(alpha)"],["\\left | \\vec{c} \\right | = \\left | \\vec{a} \\right | \\cdot \\left | \\vec{b} \\right | \\cdot \\sin(\\alpha)"]);return Dn=function(){return e},e}function qn(){var e=wn(["\vec{c} = \vec{a} \times \vec{b}"],["\\vec{c} = \\vec{a} \\times \\vec{b}"]);return qn=function(){return e},e}function Ln(){var e=wn(["\vec{b}"],["\\vec{b}"]);return Ln=function(){return e},e}function Fn(){var e=wn(["\vec{a}"],["\\vec{a}"]);return Fn=function(){return e},e}function En(){var e=wn(["\vec{v} cdot \vec{w} = left | \vec{v} \right | left | \vec{w} \right | cos alpha"],["\\vec{v} \\cdot \\vec{w} = \\left | \\vec{v} \\right | \\left | \\vec{w} \\right | \\cos \\alpha"]);return En=function(){return e},e}function An(){var e=wn(["\vec{v} - \vec{w} = (\vec{v}_x - \vec{w}_x) + (\vec{v}_y - \vec{w}_y)"],["\\vec{v} - \\vec{w} = (\\vec{v}_x - \\vec{w}_x) + (\\vec{v}_y - \\vec{w}_y)"]);return An=function(){return e},e}function Pn(){var e=wn(["\vec{v} + \vec{w} = (\vec{v}_x + \vec{w}_x) + (\vec{v}_y + \vec{w}_y)"],["\\vec{v} + \\vec{w} = (\\vec{v}_x + \\vec{w}_x) + (\\vec{v}_y + \\vec{w}_y)"]);return Pn=function(){return e},e}function Cn(){var e=wn(["left | \vec{v}_y \right | = left | \vec{v} \right | cos alpha"],["\\left | \\vec{v}_y \\right | = \\left | \\vec{v} \\right | \\cos \\alpha"]);return Cn=function(){return e},e}function In(){var e=wn(["left | \vec{v}_x \right | = left | \vec{v} \right | sin alpha"],["\\left | \\vec{v}_x \\right | = \\left | \\vec{v} \\right | \\sin \\alpha"]);return In=function(){return e},e}function Sn(){var e=wn(["\vec{v} = \vec{v}_x + \vec{v}_y"],["\\vec{v} = \\vec{v}_x + \\vec{v}_y"]);return Sn=function(){return e},e}function wn(e,n){return n||(n=e.slice(0)),e.raw=n,e}var kn=t("mbOI"),Bn=t("o8Jj"),Nn=t("ZHMS"),Vn=String.raw;n.default=function(){return e(kn.k.Provider,{value:!1},e(kn.j.Provider,{value:kn.l.INLINE},e("div",null,e("h1",null,"Fisica"),e(kn.p,{title:"Vettori"},e(kn.o,{title:"Componenti cartesiane"},e("p",null,"Usa le regole base della trigonometria:"),e("p",null,e(kn.i,null,Vn(Sn()))),e("p",null,e(kn.i,null,Vn(In()))),e("p",null,e(kn.i,null,Vn(Cn())))),e(kn.o,{title:"Somma"},e("p",null,"Scomponi in componenti, poi sommali:"),e("p",null,e(kn.i,null,Vn(Pn()))),e("p",null,"Produce il vettore risultante dall'applicazione della regola del parallelogramma.")),e(kn.o,{title:"Differenza"},e("p",null,"Alla fine è sempre una somma:"),e("p",null,e(kn.i,null,Vn(An()))),e("p",null,"Produce il vettore che parte da ",e(kn.i,null,"w")," e arriva a ",e(kn.i,null,"v"),".")),e(kn.o,{title:"Prodotto scalare"},e("p",null,"Si chiama scalare perchè il risultato è uno scalare, non un vettore."),e("p",null,e(kn.i,null,Vn(En()))),e("p",null,"Produce il modulo della proiezione di ",e(kn.i,null,Vn(Fn()))," su ",e(kn.i,null,Vn(Ln())),".")),e(kn.o,{title:"Prodotto vettoriale"},e("p",null,"Si chiama vettoriale perchè il risultato è un altro vettore."),e("ul",null,e("li",null,e(kn.i,null,Vn(qn()))),e("li",null,e(kn.i,null,Vn(Dn()))),e("li",null,e("a",{href:"https://it.wikipedia.org/wiki/Regola_della_mano_destra"},"Regola della mano destra"))),e("p",null,"Non è commutativo!"))),e(kn.p,{title:"Leggi di Newton"},e(kn.o,{title:"1ᵃ: Inerzia"},e("p",null,"Se un corpo puntiforme ha forza risultante nulla, allora la sua velocità non cambia."),e("p",null,e(kn.i,null,Vn(bn())))),e(kn.o,{title:"2ᵃ: Proporzionalità"},e("p",null,"La forza risultante di un corpo è direttamente proporzionale alla sua accelerazione, e la costante di proporzionalità è la ",e("i",null,"massa"),"."),e("p",null,e(kn.i,null,Vn(zn())))),e(kn.o,{title:"3ᵃ: Azione e reazione"},e("p",null,"Due corpi esercitano forze uguali e opposte uno sull'altro."),e("p",null,e(kn.i,null,Vn(hn()))))),e(kn.p,{title:"Forza di gravità"},e(kn.o,{title:"Tra due corpi"},e("p",null,"Due corpi puntiformi si attirano uno verso l'altro con forza:"),e("p",null,e(kn.i,null,Vn(_n()))),e("p",null,e(kn.i,null,"G")," è la ",e("i",null,"costante di gravitazione universale")," e vale:"),e("p",null,e(kn.i,null,Vn(gn())))),e(kn.o,{title:"Verso la Terra"},e("p",null,"Se nel sistema di riferimento consideriamo la Terra ferma, allora un corpo è attratto verso la Terra con forza ",e("i",null,"peso")," uguale a:"),e("p",null,e(kn.i,null,Vn(mn()))),e("p",null,e(kn.i,null,"g")," è la ",e("i",null,"costante di gravità")," della Terra, e vale:"),e("p",null,e(kn.i,null,Vn(vn())))),e(kn.o,{title:"Su pianeti diversi"},e("p",null,"Per pianeti diversi dalla Terra vale la stessa regola:"),e("p",null,e(kn.i,null,Vn(fn()))),e("p",null,"L'unica differenza è che cambia la ",e("i",null,"costante di gravità"),":"),e("p",null,e(kn.i,null,Vn(dn()))),e("p",null,e(kn.i,null,Vn(sn()))))),e(kn.p,{title:"Forze di contatto"},e(kn.o,{title:"Normale"},e("p",null,"Si oppone alle forze applicate alla superficie di contatto."),e("p",null,"Un libro appoggiato su un tavolo ha la ",e("b",null,"forza di gravità")," che lo attira verso il terreno e la ",e("b",null,"forza normale")," che lo trattiene dal cadere.")),e(kn.o,{title:"Attrito statico"},e("p",null,"Impedisce a un corpo di muoversi se non viene spinto da una forza che supera una certa soglia:"),e("p",null,e(kn.i,null,Vn(pn())))),e(kn.o,{title:"Attrito dinamico"},e("p",null,"Rallenta i corpi che si stanno muovendo finchè essi non si fermano:"),e("p",null,e(kn.i,null,Vn(cn())))),e(kn.o,{title:"Tensione"},e("p",null,"E' forza trasmessa tra due estremi di una fune."),e("p",null,"Può essere redirezionata per mezzo di carrucole.")),e(kn.o,{title:"Elastica"},e("p",null,"Una molla cerca sempre di tornare alla sua posizione indeformata con forza:"),e("p",null,e(kn.i,null,Vn(un()))),e("p",null,"(E' negativa perchè la forza è opposta a quella applicata per deformarla.)"))),e(kn.p,{title:"Cinematica"},e(kn.o,{title:"Spostamento"},e("p",null,"È un vettore che indica la posizione di un corpo rispetto a un'origine."),e("p",null,e(kn.i,null,Vn(rn())))),e(kn.o,{title:"Velocità"},e("p",null,"È un vettore che misura la variazione di posizione nel tempo."),e("p",null,e(kn.i,null,Vn(on()))),e("p",null,"Se si considera un intervallo di tempo infinitesimale si dice ",e("i",null,"velocità istantanea"),":"),e("p",null,e(kn.i,null,Vn(an())))),e(kn.o,{title:"Accelerazione"},e("p",null,"È un vettore che misura la variazione di velocità nel tempo."),e("p",null,e(kn.i,null,Vn(ln()))),e("p",null,"Se si considera un intervallo di tempo infinitesimale si dice ",e("i",null,"accelerazione istantanea"),":"),e("p",null,e(kn.i,null,Vn(tn())))),e(kn.o,{title:e("span",null,"Quantità di moto ",e("small",null,"(momento lineare)"))},e("p",null,"La quantità di moto è una proprietà vettoriale dei corpi:"),e("p",null,e(kn.i,null,Vn(nn()))),e("p",null,"Se la forza risultante è nulla, la quantità di moto non cambia."),e("p",null,e(kn.i,null,Vn(en()))))),e(kn.p,{title:"Moto rettilineo uniforme"},e(kn.o,{title:"Spostamento"},e("p",null,"La ",e("i",null,"legge oraria")," è:"),e("p",null,e(kn.i,null,Vn($e())))),e(kn.o,{title:"Velocità"},e("p",null,"È costante:"),e("p",null,e(kn.i,null,Vn(Ye())))),e(kn.o,{title:"Accelerazione"},e("p",null,"La velocità non varia:"),e("p",null,e(kn.i,null,Vn(Ke())))),e(kn.o,{title:"Forze"},e("p",null,"Si applica la prima legge di Newton:"),e("p",null,e(kn.i,null,"f(t) = 0")))),e(kn.p,{title:"Moto rettilineo uniformemente accelerato"},e(kn.o,{title:"Spostamento"},e("p",null,"La ",e("i",null,"legge oraria")," è:"),e("p",null,e(kn.i,null,Vn(Xe())))),e(kn.o,{title:"Velocità"},e("p",null,"È una retta:"),e("p",null,e(kn.i,null,Vn(Ze())))),e(kn.o,{title:"Accelerazione"},e("p",null,"È costante:"),e("p",null,e(kn.i,null,Vn(He())))),e(kn.o,{title:"Forze"},e("p",null,"Si applica la prima legge di Newton:"),e("p",null,e(kn.i,null,"f(t) = m a")))),e(kn.p,{title:"Moto armonico semplice"},e(kn.o,{title:"Ampiezza"},e("p",null,"E' la distanza dal centro massima che raggiunge il corpo."),e("p",null,"(L'ampiezza di una sinusoide.)")),e(kn.o,{title:"Velocità angolare"},e("p",null,"Indica quanto in fretta cambia la posizione del corpo."),e("p",null,"Dipende dal periodo:"),e("p",null,e(kn.i,null,Vn(Je())))),e(kn.o,{title:"Spostamento"},e("p",null,"E' una sinusoide:"),e("p",null,e(kn.i,null,Vn(je())))),e(kn.o,{title:"Velocità"},e("p",null,"E' la sinusoide dello spostamento, sfasata di ",e(kn.i,null,Vn(Qe())),":"),e("p",null,e(kn.i,null,Vn(We())))),e(kn.o,{title:"Accelerazione"},e("p",null,"E' la sinusoide della velocità, sfasata di ",e(kn.i,null,Vn(Ge())),":"),e("p",null,e(kn.i,null,Vn(Oe())))),e(kn.o,{title:"Forze"},e("p",null,"Si applica la prima legge di Newton:"),e("p",null,e(kn.i,null,"f(t) = m a")))),e(kn.p,{title:"Moti composti"},e(kn.o,{title:"Moto parabolico"},e("p",null,"Il moto parabolico è dato sommando un moto rettilineo uniforme sull'asse orizzontale e un moto rettilineo uniformemente accelerato sull'asse verticale.")),e(kn.o,{title:"Moto circolare uniforme"},e("p",null,"Il moto parabolico è dato sommando due moti armonici semplici: uno sull'asse X, e l'altro, sfasato di ",e(kn.i,null,Vn(Me())),", sull'asse Y."))),e(kn.p,{title:"Moto circolare uniforme"},e(kn.o,null,e("h3",null,"Velocità angolare"),e("p",null,"Quanto cambia la fase nel tempo."),e("p",null,e(kn.i,null,Vn(Ue())))),e(kn.o,{title:"Fase"},e("p",null,"E' l'angolo percorso dal corpo rispetto alla posizione iniziale."),e("p",null,"Si indica con ",e(kn.i,null,Vn(ye())),", e generalmente si usa in radianti.")),e(kn.o,{title:"Velocità"},e("p",null,"Si applicano le formule per la circonferenza:"),e("p",null,e(kn.i,null,Vn(xe())))),e(kn.o,{title:"Accelerazione"},e("p",null,"Il corpo ha sempre un accelerazione verso il centro che gli impedisce di abbandonare il moto:"),e("p",null,e(kn.i,null,Vn(Te())))),e(kn.o,{title:"Forza centripeta"},e("p",null,"È verso il centro e si calcola con:"),e("p",null,e(kn.i,null,Vn(Re()))))),e(kn.p,{title:"Lavoro ed energia"},e(kn.o,{title:"Lavoro"},e("p",null,"E' compiuto da una forza che sposta un corpo."),e("p",null,e(kn.i,null,Vn(Ve()))),e("p",null,"(Se la forza non è parallela allo spostamento, il prodotto scalare ci fa considerare solo la componente parallela.)")),e(kn.o,{title:"Energia cinetica"},e("p",null,"Un corpo ha energia cinetica in ogni momento uguale a:"),e("p",null,e(kn.i,null,Vn(Ne()))),e("p",null,"Se una forza effettua lavoro su un corpo, cambia la sua energia cinetica pari al lavoro effettuato:"),e("p",null,e(kn.i,null,Vn(Be())))),e(kn.o,{title:"Energia potenziale gravitazionale"},e("p",null,"Un corpo ha energia potenziale in ogni momento pari a:"),e("p",null,e(kn.i,null,Vn(ke()))),e("p",null,"(Con ",e(kn.i,null,"h")," uguale a un altezza scelta come punto di riferimento.)")),e(kn.o,{title:"Energia potenziale elastica"},e("p",null,"Una molla ha sempre energia potenziale elastica pari a:"),e("p",null,e(kn.i,null,Vn(we())))),e(kn.o,{title:"Forze conservative"},e("p",null,"Sono conservative le forze per le quali il lavoro compiuto non dipende dal percorso seguito per andare dalla partenza all'arrivo."),e("p",null,"Ad esempio, è conservativa la ",e("i",null,"forza di gravità"),", ma ",e("b",null,"non")," è conservativa la forza di attrito."),e("p",null,"Se in un sistema ci sono solo forze conservative, allora l'energia meccanica totale si conserva:"),e("p",null,e(kn.i,null,Vn(Se())))),e(kn.o,{title:"Potenza"},e("p",null,"È la velocità di trasferimento di energia:"),e("p",null,e(kn.i,null,Vn(Ie()))))),e(kn.p,{title:"Elettrostatica"},e(kn.o,{title:"Carica elettrica"},e("p",null,"È una proprietà dei corpi che può essere ",e(Bn.a,null,"positiva")," o ",e(Nn.a,null,"negativa"),"."),e("p",null,"Si conserva: in un sistema chiuso la carica totale è costante."),e("p",null,"Esiste un'unità elementare: ",e(kn.i,null,Vn(Ce())),"."),e("p",null,"Cariche ",e(Bn.a,null,"opp"),e(Nn.a,null,"oste")," si attraggono; cariche ",e(Bn.a,null,"uguali")," si respingono.")),e(kn.o,{title:"Conduttori e isolanti"},e("p",null,"Più ",e("a",{href:"https://it.wikipedia.org/wiki/Ione"},"ioni")," ha un corpo, meglio la carica si muove attraverso di esso."),e("p",null,"I corpi in cui la carica si muove bene sono ",e("i",null,"conduttori"),", mentre quelli in cui si muove difficilmente sono ",e("i",null,"isolanti"),"."),e("p",null,e("i",null,"Il corpo umano è un buon conduttore.")))),e(kn.p,{title:"Polarizzazione"},e(kn.o,{title:"Polarizzazione"},e("p",null,"E' possibile polarizzare un corpo per accumulare la carica di un segno in una certa zona."))),e(kn.p,null,e(kn.o,{title:"Messa a terra"},e("p",null,"Se un corpo conduttore è in contatto con la Terra, le cariche su di esso saranno ",e("i",null,"equilibrate")," e il corpo diventerà elettricamente neutro (con stesso numero di ",e(Bn.a,null,"cariche positive")," e ",e(Nn.a,null,"negative")," all'interno)."))),e(kn.p,null,e(kn.o,{title:"Polarizzazione per strofinio"},e("p",null,"Strofinando tra loro due corpi isolanti, essi si ",e("i",null,"polarizzeranno per strofinio"),".")),e(kn.o,{title:"Polarizzazione per contatto"},e("p",null,"Toccando un conduttore con un corpo carico, il conduttore potrà ",e("i",null,"polarizzarsi per contatto"),".")),e(kn.o,{title:"Polarizzazione per induzione"},e("p",null,'Se un corpo conduttore ha cariche "esterne" di un ',e(Bn.a,null,"certo segno")," vicino, esso avrà tutte le cariche del ",e(Nn.a,null,"segno opposto")," in equilibrio vicino alle cariche esterne, e tutte le cariche dello ",e(Bn.a,null,"stesso segno")," più lontano possibile da esse."),e("p",null,"Mettendo a terra il conduttore, nuove cariche del ",e(Nn.a,null,"segno opposto")," saranno attratte all'interno del corpo per equilibrare le cariche che si sono allontanate."),e("p",null,"Staccando il conduttore da terra e rimuovendo le cariche esterne, esso si ritroverà ",e(Nn.a,null,"caricato del segno opposto")," rispetto alle cariche esterne."))),e(kn.p,{title:"Forza elettrica"},e(kn.o,{title:"Legge di Coulomb"},e("p",null,"Due corpi carichi si attraggono tra loro con forza:"),e("p",null,e(kn.i,null,Vn(Pe()))),e("p",null,e(kn.i,null,Vn(Ae()))," è la ",e("i",null,"costante di Coulomb"),", e vale ",e(kn.i,null,Vn(Ee())),".")),e(kn.o,{title:"Permeabilità dello spazio vuoto"},e("p",null,"La costante ",e(kn.i,null,Vn(Fe()))," è in realtà dipendente da un altra costante, ",e(kn.i,null,Vn(Le())),", la ",e("i",null,"permeabilità del vuoto"),"."),e("p",null,e(kn.i,null,Vn(qe()))),e("p",null,e(kn.i,null,Vn(De())))),e(kn.o,{title:"Campo elettrico"},e("p",null,"Misura che forza viene applicata in ogni punto su una carica unitaria:"),e("p",null,e(kn.i,null,Vn(be())))),e(kn.o,{title:"Flusso elettrico"},e("p",null,'È la differenza tra "quanto" campo elettrico ',e(Bn.a,null,"entra")," e quanto campo elettrico ",e(Nn.a,null,"esce")," da una certa area."),e("p",null,"In qualsiasi superficie chiusa, il flusso elettrico è uguale alla componente perpendicolare del campo elettrico moltiplicato per l'area."),e("p",null,e(kn.i,null,Vn(ze()))),e("p",null,"Se il campo elettrico è uniforme, se ne può calcolare facilmente il valore:"),e("p",null,e(kn.i,null,Vn(he()))),e("p",null,e(kn.s,null,"Circa. E' una specie di integrale..."))),e(kn.o,{title:"Legge di Gauss per i campi elettrostatici"},e("p",null,"Il flusso elettrico è direttamente proporzionale alla carica presente all'interno della superficie."),e("p",null,e(kn.i,null,Vn(_e()))),e("p",null,"Ovvero, i campi elettrostatici sono generati dalle cariche elettriche."))),e(kn.p,{title:"Energia elettrica"},e(kn.o,{title:"Energia potenziale elettrica"},e("p",null,"Un corpo carico vicino ad altre cariche possiede un'",e("i",null,"energia potenziale elettrica")," ",e(kn.i,null,Vn(ge())),"."))),e(kn.p,{title:"Circuiti elettrici"},e(kn.o,{title:e("span",null,"Potenziale elettrico ",e("small",null,"(tensione)"))},e("p",null,"È il valore dell'energia potenziale elettrica per una carica unitaria."),e("p",null,e(kn.i,null,Vn(me()))),e("p",null,"La sua unità di misura è il Volt (",e(kn.i,null,Vn(ve())),")."),e("p",null,"In una batteria è detto ",e("i",null,"forza elettromotrice"),", e corrisponde al lavoro compiuto da una batteria ideale per spostare una carica unitaria tra i due poli.")),e(kn.o,{title:e("span",null,"Corrente elettrica ",e("small",null,"(intensità)"))},e("p",null,"Quanta carica passa attraverso un'area (perpendicolare al flusso) nel tempo."),e("p",null,e(kn.i,null,Vn(fe()))),e("p",null,"Fintanto che c'è differenza di potenziale, ci sarà anche intensità non nulla."),e("p",null,"La sua unità di misura è l'Ampere (",e(kn.i,null,Vn(de())),").")),e(kn.o,{title:e("span",null,"Corrente continua ",e("small",null,"(",e("abbr",{title:"Direct Current"},"DC"),")"))},e("p",null,"Quando in un circuito la direzione della corrente è costante.")),e(kn.o,{title:e("span",null,"Corrente alternata ",e("small",null,"(",e("abbr",{title:"Alternate Current"},"AC"),")"))},e("p",null,"Quando in un circuito la direzione della corrente si alterna periodicamente.")),e(kn.o,{title:"Potenza elettrica"},e("p",null,"Possiamo calcolare la potenza di un circuito:"),e("p",null,e(kn.i,null,Vn(se()))))),e(kn.p,{title:"Elementi di un circuito"},e(kn.o,{title:"Resistore"},e("p",null,"Riduce l'intensità di corrente, e converte parte del potenziale in calore."),e("p",null,"Il potenziale utilizzato è pari a:"),e("p",null,e(kn.i,null,Vn(pe()))),e("p",null,"Dove ",e(kn.i,null,Vn(ce()))," è una costante detta ",e("i",null,"resistenza")," con unità di misura Ohm (",e(kn.i,null,Vn(ue())),")."),e("p",null,"La resistenza di un conduttore vale:"),e("p",null,e(kn.i,null,Vn(re()))),e("p",null,e(kn.i,null,Vn(oe()))," è la ",e("i",null,"resistività")," del materiale, e varia in base alla temperatura:"),e("p",null,e(kn.i,null,Vn(ae())))),e(kn.o,{title:"Condensatore"},e("p",null,"Immagazzina potenziale elettrico, permettendo di riutilizzarla in seguito."),e("p",null,"Per farlo, cattura cariche ",e(Bn.a,null,"positive")," e ",e(Nn.a,null,"negative")," sulle sue due armature; perchè questo avvenga, deve essere compiuto lavoro."),e("p",null,"Ha una ",e("b",null,"capacità")," caratteristica, che in un condensatore a facce piane parallele è:"),e("p",null,e(kn.i,null,Vn(ie()))),e("p",null,"Condensatori di capacità maggiore immagazzinano più potenziale con meno carica."),e("p",null,"La capacità aumenta se viene messo qualcosa tra le armature:"),e("p",null,e(kn.i,null,Vn(le()))),e("p",null,"Dove ",e(kn.i,null,Vn(te()))," è la ",e("i",null,"costante dielettrica relativa")," del materiale inserito, ",e(kn.i,null,Vn(ne()))," l'area di una armatura e ",e(kn.i,null,Vn(ee()))," la distanza tra le due armature."),e("p",null,"Se il campo elettrico creatosi tra le due armature supera la ",e("i",null,"rigidità dielettrica")," del condensatore, la carica immagazzinata viene persa e ha luogo un ",e("i",null,"breakdown"),"."),e("p",null,"La sua unità di misura è il Farad (",e(kn.i,null,Vn($())),")")),e(kn.o,{title:"Amperometro"},e("p",null,"Misura la corrente elettrica se messo in serie."),e("p",null,"(Funzionamento: ha una resistenza interna bassisima in modo da non influire significativamente sulla corrente.)")),e(kn.o,{title:"Voltmetro"},e("p",null,"Misura la differenza di potenziale se messo in parallelo."),e("p",null,"(Funzionamento: ha una resistenza altissima in modo da non influire significativamente sulla tensione.)"))),e(kn.p,{title:"Principi di Kirchhoff"},e(kn.o,{title:"Legge dei nodi"},e("p",null,"Per nodo si intende un qualsiasi punto del circuito."),e("p",null,"Da un nodo entra ed esce la stessa corrente.")),e(kn.o,{title:"Legge delle maglie"},e("p",null,"Per maglia si intende un qualsiasi percorso chiuso all'interno del circuito."),e("p",null,"In una maglia chiusa, la somma delle differenze di potenziale è 0."))),e(kn.p,{title:"Serie e Parallelo"},e(kn.o,{title:"Circuito in serie"},e("p",null,"Più parti di circuito sono ",e("i",null,"in serie")," se sono consecutive e senza biforcazioni."),e("p",null,"Parti di circuito in serie sono attraversate dalla stessa corrente.")),e(kn.o,{title:"Circuito in parallelo"},e("p",null,"Più parti di circuito sono ",e("i",null,"in parallelo")," tra loro se hanno lo stesso punto di partenza e lo stesso punto di arrivo."),e("p",null,"Parti di circuito in parallelo hanno la stessa differenza di potenziale."))),e(kn.p,{title:"Resistenze equivalenti"},e(kn.o,{title:"Circuiti in serie"},e("p",null,"Nei circuiti in serie, tutte le resistenze possono essere sostituite con una equivalente dalla resistenza della somma di tutte le quelle sostituite:"),e("p",null,e(kn.i,null,Vn(Y())))),e(kn.o,{title:"Circuiti in parallelo"},e("p",null,"Nei circuiti in parallelo, tutte le resistenze possono essere sostituite con una equivalente dalla resistenza di:"),e("p",null,e(kn.i,null,Vn(K()))))),e(kn.p,{title:"Condensatori equivalenti"},e(kn.o,{title:"Circuiti in serie"},e("p",null,"Nei circuiti in serie, tutti i condensatori possono essere sostituiti con uno equivalente dalla capacità di:"),e("p",null,e(kn.i,null,Vn(X())))),e(kn.o,{title:"Circuiti in parallelo"},e("p",null,"Nei circuiti in parallelo, tutte i condensatori possono essere sostituite con uno equivalente dalla capacità della somma di tutti quelli sostituiti:"),e("p",null,e(kn.i,null,Vn(Z()))))),e(kn.p,{title:"Magnetismo"},e(kn.o,{title:"Permeabilità magnetica dello spazio vuoto"},e("p",null,"E' una costante fisica fondamentale che rappresenta quanto un materiale si magnetizza facilmente."),e("p",null,e(kn.i,null,Vn(H()))," (",e(kn.i,null,Vn(J())),")")),e(kn.o,{title:"Campo magnetico"},e("p",null,"Come un campo elettrico, ma per i magneti."),e("p",null,"Il suo simbolo è ",e(kn.i,null,Vn(j())),", e la sua unità di misura è il Tesla (",e(kn.i,null,"T"),").")),e(kn.o,{title:"Flusso magnetico"},e("p",null,'È "quanto" campo magnetico ',e("b",null,"attraversa")," un percorso chiuso."),e("p",null,'Per qualsiasi percorso chiuso, il flusso magnetico è uguale alla somma di tutti i "sottoflussi" magnetici calcolati sui suoi lati.'),e("p",null,e(kn.i,null,Vn(Q()))),e("p",null,e(kn.i,null,Vn(W()))),e("p",null,"La sua unità di misura è il Weber (",e(kn.i,null,Vn(G())),").")),e(kn.o,{title:"Legge di Gauss per i campi magnetici"},e("p",null,"Il flusso magnetico attraverso qualsiasi superficie chiusa è sempre nullo."),e("p",null,"Ovvero, non esistono monopoli magnetici.")),e(kn.o,{title:"Legge di Ampère"},e("p",null,"L'intensità di corrente che attraversa un percorso chiuso è direttamente proporzionale al flusso magnetico dello stesso percorso."),e("p",null,e(kn.i,null,Vn(O()))))),e(kn.p,{title:"Forze magnetiche"},e(kn.o,{title:e("span",null,"Forza magnetica su carica puntiforme ",e("small",null,"(Forza di Lorentz)"))},e("p",null,"I campi magnetici applicano una forza sulle cariche vicine:"),e("p",null,e(kn.i,null,Vn(M()))),e("p",null,"Dove ",e(kn.i,null,Vn(U()))," è l'intensità del campo magnetico e ",e(kn.i,null,Vn(y()))," la velocità della carica considerata."),e("p",null,"Si ha una forza massima se la velocità è perpendicolare al campo magnetico."),e("p",null,"In un campo magnetico uniforme, una velocità perpendicolare al campo porta alla creazione di un moto circolare uniforme.")),e(kn.o,{title:"Forza magnetica in un filo"},e("p",null,"I campi magnetici influenzano ovviamente anche le cariche presenti in un conduttore:"),e("p",null,e(kn.i,null,Vn(x()))," ",e("a",{href:"https://it.openprof.com/wb/forza_di_lorentz_su_un_filo_percorso_da_corrente?ch=360"},"[1]")),e("p",null,"Dove ",e(kn.i,null,Vn(T()))," è la corrente elettrica, ",e(kn.i,null,Vn(R()))," è un vettore che punta nella direzione di scorrimento della corrente e ha come modulo la lunghezza del conduttore."))),e(kn.p,{title:"Campi magnetici"},e(kn.o,{title:"Campo magnetico in una spira"},e("p",null,"Una spira in cui passa corrente produce un campo magnetico perpendicolare al piano creato dalla spira.")),e(kn.o,{title:"Campo magnetico di un solenoide"},e("p",null,"Un solenoide sono tante spire avvolte in modo da formare una specie di cilindro."),e("p",null,"All'interno del solenoide si crea un campo (quasi) uniforme:"),e("p",null,e(kn.i,null,Vn(V())))),e(kn.o,{title:"Legge di Oersted"},e("p",null,e("i",null,"Caso particolare della ",e("a",{href:"https://it.wikipedia.org/wiki/Legge_di_Amp%C3%A8re"},"Legge di Ampère"),".")),e("p",null,"Il modulo del campo magnetico ",e(kn.i,null,"B")," prodotto da un filo in cui passa una corrente continua ",e(kn.i,null,"I")," alla distanza ",e(kn.i,null,"s")," è:"),e("p",null,e(kn.i,null,Vn(N()))),e("p",null,"Il campo magnetico così creato gira attorno al filo in senso antiorario."),e("p",null,"Due fili attraversati dalla ",e(Bn.a,null,"stessa corrente")," si attraggono, due fili attraversati da ",e(Bn.a,null,"corr"),e(Nn.a,null,"enti")," ",e(Bn.a,null,"opp"),e(Nn.a,null,"oste")," si respingono."))),e(kn.p,{title:"Induzione elettromagnetica"},e(kn.o,{title:"Forza elettromotrice indotta"},e("p",null,"Un conduttore perpendicolare ad un campo magnetico può ottenere una differenza di potenziale se messo in movimento in un direzione perpendicolare alla direzione del conduttore e del campo."),e("p",null,"La differenza di potenziale si crea a causa della forza magnetica, che fa spostare tutti gli elettroni verso un capo del conduttore."),e("p",null,"Essa vale:"),e("p",null,e(kn.i,null,Vn(B()))),e("p",null,"Dove ",e(kn.i,null,"v")," è la velocità del conduttore, ",e(kn.i,null,"B")," è l'intensità del campo magnetico ed ",e(kn.i,null,"L")," è la lunghezza del conduttore.")),e(kn.o,{title:"Flusso magnetico in una spira"},e("p",null,"In un campo magnetico ",e(kn.i,null,Vn(k()))," uniforme e perpendicolare al piano di una spira di area ",e(kn.i,null,Vn(w())),", il flusso magnetico si può determinare con la ",e("i",null,"Legge di Faraday-Neumann-Lenz"),":"),e("p",null,e(kn.i,null,Vn(S()))))),e(kn.p,null,e(kn.o,{title:"Legge di Faraday-Neumann-Lenz"},e("p",null,"Dice che la forza elettromotrice media indotta in un percorso dipende dalla variazione nel tempo del flusso magnetico nello stesso percorso."),e("p",null,e(kn.i,null,Vn(I()))),e("p",null,"Il meno è dovuto alla ",e("a",{href:"https://it.wikipedia.org/wiki/Legge_di_Lenz"},"Legge di Lenz"),", che specifica qualitativamente il verso della forza elettromotrice indotta.")),e(kn.o,{title:"Faraday in un solenoide"},e("p",null,"In un solenoide, la forza elettromotrice indotta è uguale a:"),e("p",null,e(kn.i,null,Vn(C()))),e("p",null,"Dove ",e(kn.i,null,Vn(P()))," è il numero delle spire del solenoide.")),e(kn.o,{title:"Legge di Ampère-Maxwell"},e("p",null,"Correnti o campi elettrici variabili creano un campo magnetico."))),e(kn.p,{title:"Elettromagnetismo"},e(kn.o,{title:"Onde elettromagnetiche"},e("p",null,"Nel vuoto, il campo elettrico ",e(kn.i,null,Vn(A()))," e il campo magnetico ",e(kn.i,null,Vn(E()))," sono perpendicolari tra loro e la direzione di propagazione, e sono entrambe funzioni del tempo."),e("p",null,"Si dice quindi che sono ",e("i",null,"onde elettromagnetiche"),"."),e("p",null,"Esse sono legate dalla relazione:"),e("p",null,e(kn.i,null,Vn(F()))),e("p",null,"Dove ",e(kn.i,null,Vn(L()))," è la velocità delle onde (luce) nel vuoto, e a sua volta è uguale a:"),e("p",null,e(kn.i,null,Vn(q())))),e(kn.o,{title:"Formula delle onde"},e("p",null,e(kn.i,null,Vn(D()))),e("p",null,"Dove ",e(kn.i,null,Vn(b()))," è l'ampiezza massima che può avere l'onda, ",e(kn.i,null,Vn(z()))," è il vettore d'onda, ",e(kn.i,null,Vn(h()))," la frequenza angolare e ",e(kn.i,null,Vn(_()))," la fase."))),e(kn.p,{title:"Spettroscopia"},e(kn.o,{title:"Emissione"},e("p",null,"I solidi, se portati ad alta temperatura, emettono luce con uno ",e("a",{href:"https://it.wikipedia.org/wiki/Spettro_continuo"},"spettro continuo"),"."),e("p",null,"I gas, invece, ad alta temperatura emettono luce solo con particolari lunghezze d'onda."),e("p",null,"In un gas di idrogeno, le lunghezze d'onda emesse sono ricavabili con:"),e("p",null,e(kn.i,null,Vn(g()))),e("p",null,"Con ",e(kn.i,null,Vn(m())),", detta costante di Rydberg, e ",e(kn.i,null,Vn(v()))," un numero intero.")),e(kn.o,{title:"Grandezza quantizzata"},e("p",null,"Una grandezza si dice quantizzata (o discreta) se può assumere solo determinati valori."),e("p",null,"Una grandezza si dice continua se può assumere qualsiasi valore e quindi se non è quantizzata."),e("p",null,"Energia, momento angolare e raggio sono quantizzati."),e("p",null,"Nota costante quantica è ",e(kn.i,null,Vn(f())),", la costante di Planck, ovvero il valore minimo possibile per la carica (talvolta espressa come ",e(kn.i,null,Vn(d())),"."))),e(kn.p,null,e(kn.o,{title:"Modello di Bohr"},e("p",null,"L'energia degli elettroni è quantizzata."),e("p",null,"Inoltre, per essi è valido che:"),e("p",null,e(kn.i,null,Vn(s()))),e("p",null,"Ancora, il raggio delle orbite è uguale a:"),e("p",null,e(kn.i,null,Vn(p()))),e("p",null,"Con ",e(kn.i,null,Vn(c())),"."),e("p",null,"Infine, in ogni stato, l'energia è pari a:"),e("p",null,e(kn.i,null,Vn(u()))),e("p",null,"Due elettroni non possono occupare lo stesso stato."),e("p",null,"Questo modello funziona solo per atomi con numero atomico basso. Atomi con molti elettroni hanno comportamenti diversi, descritti dal modello di"))),e(kn.p,null,e(kn.o,{title:"Nei solidi"},e("p",null,"Nei solidi, le lunghezze d'onda sono talmente tanto vicine da poter essere considerate una banda."),e("p",null,"Possono però comunque avere dei gap dovuti agli intervalli di energia non ammessi."))),e(kn.p,{title:"Semiconduttori"},e(kn.o,{title:"Semiconduttori"},e("p",null,e(kn.s,null,"Refactor this")),e("p",null,"Se la banda di emissione con energia più alta di un corpo è assente o è separata da un gap dell'ordine di grandezza maggiore di ",e(kn.i,null,Vn(r())),", allora il corpo è un isolante."),e("p",null,"Se invece la banda di emissione si sovrappone a un altra, allora il corpo è un conduttore."),e("p",null,"Se il gap è invece dell'ordine di grandezza di ",e(kn.i,null,Vn(o())),", allora il corpo è un semiconduttore.")),e(kn.o,{title:"Lacune"},e("p",null,"Legami in cui ",e(Bn.a,null,"mancano elettroni"),"."),e("p",null,e(Nn.a,null,"Elettroni")," di altri legami possono spostarsi per colmare le ",e(Bn.a,null,"lacune"),", creandone altre, e spostandole in direzione opposta a quella della corrente.")),e(kn.o,{title:"Accettori e donori"},e("p",null,"Se si inserisce in un cristallo semiconduttore si inserisce un atomo con numero atomico diverso, si otterrà:"),e("ul",null,e("li",null,"Con numero atomico maggiore, un semiconduttore di ",e(Nn.a,null,"tipo N")," con ",e(Nn.a,null,"elettroni in eccesso")," liberi di scorrere."),e("li",null,"Con numero atomico minore, un semiconduttore di ",e(Bn.a,null,"tipo P")," con ",e(Bn.a,null,"lacune in eccesso")," libere di catturare elettroni da altri legami.")),e("p",null,"Maggiore impurezza porta a maggiore conduttività.")),e(kn.o,{title:"Temperatura"},e("p",null,"Aumentando la temperatura di un semiconduttore si aumenta la conduttività, perchè eccita le particelle e favorisce il movimento di ",e(Nn.a,null,"elettroni")," e ",e(Bn.a,null,"lacune"),"."))),e(kn.p,{title:e("span",null,"Ottica ",e("small",null,"(non l'abbiamo fatta)"))},e(kn.o,{title:"Assorbimento e riflessione"},e("p",null,"I corpi possono assorbire o riflettere le onde elettromagnetiche che li colpiscono.")),e(kn.o,{title:"Corpo nero"},e("p",null,"Un corpo nero è un corpo che assorbe tutte le onde elettromagnetiche che riceve senza rifletterne nessuna."),e("p",null,"Le onde assorbite vengono poi riemesse sotto forma di un onda di ",e(kn.i,null,Vn(a()))," variabile in base alla temperatura."),e("p",null,e(kn.i,null,Vn(i()))," è costante.")),e(kn.o,{title:"Teoria di Planck per il corpo nero"},e("p",null,"L'energia assorbita e emessa dai corpi neri è quantizzata.")),e(kn.o,{title:"Fotone"},e("p",null,"Un onda magnetica con un quanto di energia è detta ",e("i",null,"fotone"),":"),e("p",null,e(kn.i,null,Vn(l())))),e(kn.o,{title:"Effetto fotoelettrico"},e("p",null,"A volte, i fotoni che colpiscono un metallo possono estrarvi degli elettroni e creare una differenza di potenziale."),e("p",null,"Perchè avvenga, la frequenza deve essere maggiore di una certa soglia."),e("p",null,"Il numero di elettroni estratti dipende dall'intensità dell'onda, mentre l'energia cinetica degli elettroni dipende dalla frequenza."),e("p",null,"Non c'è nessun ritardo tra l'assorbimento del fotone e l'estrazione di elettroni."))))))}}.call(this,t("hosL").h)},ZHMS:function(e,n,t){"use strict";(function(e){var l=t("gJgL"),i=t.n(l);n.a=function(n){return e("span",{class:i.a.minus},n.children)}}).call(this,t("hosL").h)},gJgL:function(e){e.exports={red:"red__3phX0",orange:"orange__1OSkI",yellow:"yellow__2pQOr",lime:"lime__3vRUg",cyan:"cyan__1cBtG",blue:"blue__1G1wb",magenta:"magenta__3Wagv",minus:"minus__2qLyv"}},jZax:function(e){e.exports={red:"red__ipSoq",orange:"orange__1nmkj",yellow:"yellow__353ge",lime:"lime__3FW1E",cyan:"cyan__2xhu7",blue:"blue__3MvDe",magenta:"magenta__i03ie",plus:"plus__1eV4_"}},o8Jj:function(e,n,t){"use strict";(function(e){var l=t("jZax"),i=t.n(l);n.a=function(n){return 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l(i.l.Provider,{value:!1},l(i.k.Provider,{value:i.m.INLINE},l("div",null,l("h1",null,"Fisica"),l(i.q,{title:"Vettori"},l(i.p,{title:"Componenti cartesiane"},l("p",null,"Usa le regole base della trigonometria:"),l("p",null,l(i.j,null,Be(o||(o=ke`\vec{v} = \vec{v}_x + \vec{v}_y`)))),l("p",null,l(i.j,null,Be(r||(r=ke`\left | \vec{v}_x \right | = \left | \vec{v} \right | \sin \alpha`)))),l("p",null,l(i.j,null,Be(u||(u=ke`\left | \vec{v}_y \right | = \left | \vec{v} \right | \cos \alpha`))))),l(i.p,{title:"Somma"},l("p",null,"Scomponi in componenti, poi sommali:"),l("p",null,l(i.j,null,Be(c||(c=ke`\vec{v} + \vec{w} = (\vec{v}_x + \vec{w}_x) + (\vec{v}_y + \vec{w}_y)`)))),l("p",null,"Produce il vettore risultante dall'applicazione della regola del parallelogramma.")),l(i.p,{title:"Differenza"},l("p",null,"Alla fine è sempre una somma:"),l("p",null,l(i.j,null,Be(p||(p=ke`\vec{v} - \vec{w} = (\vec{v}_x - \vec{w}_x) + (\vec{v}_y - \vec{w}_y)`)))),l("p",null,"Produce il vettore che parte da ",l(i.j,null,"w")," e arriva a ",l(i.j,null,"v"),".")),l(i.p,{title:"Prodotto scalare"},l("p",null,"Si chiama scalare perchè il risultato è uno scalare, non un vettore."),l("p",null,l(i.j,null,Be(s||(s=ke`\vec{v} \cdot \vec{w} = \left | \vec{v} \right | \left | \vec{w} \right | \cos \alpha`)))),l("p",null,"Produce il modulo della proiezione di ",l(i.j,null,Be(d||(d=ke`\vec{a}`)))," su ",l(i.j,null,Be(m||(m=ke`\vec{b}`))),".")),l(i.p,{title:"Prodotto vettoriale"},l("p",null,"Si chiama vettoriale perchè il risultato è un altro vettore."),l("ul",null,l("li",null,l(i.j,null,Be(v||(v=ke`\vec{c} = \vec{a} \times \vec{b}`)))),l("li",null,l(i.j,null,Be(g||(g=ke`\left | \vec{c} \right | = \left | \vec{a} \right | \cdot \left | \vec{b} \right | \cdot \sin(\alpha)`)))),l("li",null,l("a",{href:"https://it.wikipedia.org/wiki/Regola_della_mano_destra"},"Regola della mano destra"))),l("p",null,"Non è commutativo!"))),l(i.q,{title:"Leggi di Newton"},l(i.p,{title:"1ᵃ: Inerzia"},l("p",null,"Se un corpo puntiforme ha forza risultante nulla, allora la sua velocità non cambia."),l("p",null,l(i.j,null,Be(f||(f=ke`\Sigma \vec{F} = 0 \Longleftrightarrow \Delta v = 0`))))),l(i.p,{title:"2ᵃ: Proporzionalità"},l("p",null,"La forza risultante di un corpo è direttamente proporzionale alla sua accelerazione, e la costante di proporzionalità è la ",l("i",null,"massa"),"."),l("p",null,l(i.j,null,Be(z||(z=ke`\Sigma \vec{F} = m \vec{a}`))))),l(i.p,{title:"3ᵃ: Azione e reazione"},l("p",null,"Due corpi esercitano forze uguali e opposte uno sull'altro."),l("p",null,l(i.j,null,Be(h||(h=ke`\vec{F}_{21} = -\vec{F}_{12}`)))))),l(i.q,{title:"Forza di gravità"},l(i.p,{title:"Tra due corpi"},l("p",null,"Due corpi puntiformi si attirano uno verso l'altro con forza:"),l("p",null,l(i.j,null,Be(_||(_=ke`\left | \vec{F} \right | = G \frac{m_1 m_2}{s^2}`)))),l("p",null,l(i.j,null,"G")," è la ",l("i",null,"costante di gravitazione universale")," e vale:"),l("p",null,l(i.j,null,Be(j||(j=ke`G = 6.67 \cdot 10^{-11} \frac{N m^2}{{kg}^2}`))))),l(i.p,{title:"Verso la Terra"},l("p",null,"Se nel sistema di riferimento consideriamo la Terra ferma, allora un corpo è attratto verso la Terra con forza ",l("i",null,"peso")," uguale a:"),l("p",null,l(i.j,null,Be(q||(q=ke`\left | \vec{F} \right | = g m`)))),l("p",null,l(i.j,null,"g")," è la ",l("i",null,"costante di gravità")," della Terra, e vale:"),l("p",null,l(i.j,null,Be(b||(b=ke`g = 9.81 \frac{m}{s^2}`))))),l(i.p,{title:"Su pianeti diversi"},l("p",null,"Per pianeti diversi dalla Terra vale la stessa regola:"),l("p",null,l(i.j,null,Be(L||(L=ke`\left | \vec{F} \right | = g m`)))),l("p",null,"L'unica differenza è che cambia la ",l("i",null,"costante di gravità"),":"),l("p",null,l(i.j,null,Be(D||(D=ke`g_{luna} = 1.62 \frac{m}{s^2}`)))),l("p",null,l(i.j,null,Be(F||(F=ke`g_{marte} = 3.71 \frac{m}{s^2}`)))))),l(i.q,{title:"Forze di contatto"},l(i.p,{title:"Normale"},l("p",null,"Si oppone alle forze applicate alla superficie di contatto."),l("p",null,"Un libro appoggiato su un tavolo ha la ",l("b",null,"forza di gravità")," che lo attira verso il terreno e la ",l("b",null,"forza normale")," che lo trattiene dal cadere.")),l(i.p,{title:"Attrito statico"},l("p",null,"Impedisce a un corpo di muoversi se non viene spinto da una forza che supera una certa soglia:"),l("p",null,l(i.j,null,Be(S||(S=ke`\left | \vec{F} \right | \leq \mu_{s} \left | \vec{F}_{normale} \right |`))))),l(i.p,{title:"Attrito dinamico"},l("p",null,"Rallenta i corpi che si stanno muovendo finchè essi non si fermano:"),l("p",null,l(i.j,null,Be(E||(E=ke`\left | \vec{F} \right | \leq \mu_{d} \left | \vec{F}_{normale} \right |`))))),l(i.p,{title:"Tensione"},l("p",null,"E' forza trasmessa tra due estremi di una fune."),l("p",null,"Può essere redirezionata per mezzo di carrucole.")),l(i.p,{title:"Elastica"},l("p",null,"Una molla cerca sempre di tornare alla sua posizione indeformata con forza:"),l("p",null,l(i.j,null,Be(P||(P=ke`F = -k x`)))),l("p",null,"(E' negativa perchè la forza è opposta a quella applicata per deformarla.)"))),l(i.q,{title:"Cinematica"},l(i.p,{title:"Spostamento"},l("p",null,"È un vettore che indica la posizione di un corpo rispetto a un'origine."),l("p",null,l(i.j,null,Be(C||(C=ke`\Delta \vec{s} = \vec{s}(fine) - \vec{s}(inizio)`))))),l(i.p,{title:"Velocità"},l("p",null,"È un vettore che misura la variazione di posizione nel tempo."),l("p",null,l(i.j,null,Be(I||(I=ke`\vec{v} = \frac{\Delta \vec{s}}{\Delta t}`)))),l("p",null,"Se si considera un intervallo di tempo infinitesimale si dice ",l("i",null,"velocità istantanea"),":"),l("p",null,l(i.j,null,Be(A||(A=ke`\vec{v} = \lim_{\Delta t \to 0} \frac{\Delta \vec{s}}{\Delta t} = \frac{d \vec{s}}{dt}`))))),l(i.p,{title:"Accelerazione"},l("p",null,"È un vettore che misura la variazione di velocità nel tempo."),l("p",null,l(i.j,null,Be(w||(w=ke`\vec{a} = \frac{\Delta \vec{v}}{\Delta t}`)))),l("p",null,"Se si considera un intervallo di tempo infinitesimale si dice ",l("i",null,"accelerazione istantanea"),":"),l("p",null,l(i.j,null,Be(k||(k=ke`\vec{a} = \lim_{\Delta v \to 0} \frac{\Delta \vec{v}}{\Delta t} = \frac{d \vec{v}}{d t} = \frac{d^2 \vec{s}}{d t^2}`))))),l(i.p,{title:l("span",null,"Quantità di moto ",l("small",null,"(momento lineare)"))},l("p",null,"La quantità di moto è una proprietà vettoriale dei corpi:"),l("p",null,l(i.j,null,Be(B||(B=ke`\vec{p} = m \vec{v}`)))),l("p",null,"Se la forza risultante è nulla, la quantità di moto non cambia."),l("p",null,l(i.j,null,Be(N||(N=ke`\Sigma \vec{F} = 0 \Longleftrightarrow \Delta \vec{p} = 0`)))))),l(i.q,{title:"Moto rettilineo uniforme"},l(i.p,{title:"Spostamento"},l("p",null,"La ",l("i",null,"legge oraria")," è:"),l("p",null,l(i.j,null,Be(V||(V=ke`s(t) = v \cdot \Delta t + s(0)`))))),l(i.p,{title:"Velocità"},l("p",null,"È costante:"),l("p",null,l(i.j,null,Be(R||(R=ke`v(t) = k`))))),l(i.p,{title:"Accelerazione"},l("p",null,"La velocità non varia:"),l("p",null,l(i.j,null,Be(T||(T=ke`a(t) = 0`))))),l(i.p,{title:"Forze"},l("p",null,"Si applica la prima legge di Newton:"),l("p",null,l(i.j,null,"f(t) = 0")))),l(i.q,{title:"Moto rettilineo uniformemente accelerato"},l(i.p,{title:"Spostamento"},l("p",null,"La ",l("i",null,"legge oraria")," è:"),l("p",null,l(i.j,null,Be(y||(y=ke`s(t) = \frac{1}{2} a \cdot (\Delta t)^2 + v(0) \cdot (\Delta t) + s(0)`))))),l(i.p,{title:"Velocità"},l("p",null,"È una retta:"),l("p",null,l(i.j,null,Be(M||(M=ke`v(t) = a \Delta t + v(0)`))))),l(i.p,{title:"Accelerazione"},l("p",null,"È costante:"),l("p",null,l(i.j,null,Be(U||(U=ke`a(t) = k`))))),l(i.p,{title:"Forze"},l("p",null,"Si applica la prima legge di Newton:"),l("p",null,l(i.j,null,"f(t) = m a")))),l(i.q,{title:"Moto armonico semplice"},l(i.p,{title:"Ampiezza"},l("p",null,"E' la distanza dal centro massima che raggiunge il corpo."),l("p",null,"(L'ampiezza di una sinusoide.)")),l(i.p,{title:"Velocità angolare"},l("p",null,"Indica quanto in fretta cambia la posizione del corpo."),l("p",null,"Dipende dal periodo:"),l("p",null,l(i.j,null,Be(x||(x=ke`\omega = \frac{2 \pi}{T}`))))),l(i.p,{title:"Spostamento"},l("p",null,"E' una sinusoide:"),l("p",null,l(i.j,null,Be(O||(O=ke`s(t) = A \sin (\omega \cdot t + \phi)`))))),l(i.p,{title:"Velocità"},l("p",null,"E' la sinusoide dello spostamento, sfasata di ",l(i.j,null,Be(G||(G=ke`\frac{\pi}{2}`))),":"),l("p",null,l(i.j,null,Be(Q||(Q=ke`v(t) = A \sin (\omega \cdot t + \phi + \frac{\pi}{2})`))))),l(i.p,{title:"Accelerazione"},l("p",null,"E' la sinusoide della velocità, sfasata di ",l(i.j,null,Be(J||(J=ke`\pi`))),":"),l("p",null,l(i.j,null,Be(W||(W=ke`a(t) = A \sin (\omega \cdot t + \phi + \pi)`))))),l(i.p,{title:"Forze"},l("p",null,"Si applica la prima legge di Newton:"),l("p",null,l(i.j,null,"f(t) = m a")))),l(i.q,{title:"Moti composti"},l(i.p,{title:"Moto parabolico"},l("p",null,"Il moto parabolico è dato sommando un moto rettilineo uniforme sull'asse orizzontale e un moto rettilineo uniformemente accelerato sull'asse verticale.")),l(i.p,{title:"Moto circolare uniforme"},l("p",null,"Il moto parabolico è dato sommando due moti armonici semplici: uno sull'asse X, e l'altro, sfasato di ",l(i.j,null,Be(H||(H=ke`\frac{\pi}{2}`))),", sull'asse Y."))),l(i.q,{title:"Moto circolare uniforme"},l(i.p,null,l("h3",null,"Velocità angolare"),l("p",null,"Quanto cambia la fase nel tempo."),l("p",null,l(i.j,null,Be(Z||(Z=ke`\omega = \frac{2 \pi}{T}`))))),l(i.p,{title:"Fase"},l("p",null,"E' l'angolo percorso dal corpo rispetto alla posizione iniziale."),l("p",null,"Si indica con ",l(i.j,null,Be(X||(X=ke`\phi`))),", e generalmente si usa in radianti.")),l(i.p,{title:"Velocità"},l("p",null,"Si applicano le formule per la circonferenza:"),l("p",null,l(i.j,null,Be(K||(K=ke`v = \frac{\Delta s}{t} = \frac{2 \pi \cdot r}{T} = \omega r`))))),l(i.p,{title:"Accelerazione"},l("p",null,"Il corpo ha sempre un accelerazione verso il centro che gli impedisce di abbandonare il moto:"),l("p",null,l(i.j,null,Be(Y||(Y=ke`a = \frac{v^2}{r} = r \cdot \omega^2 = v \cdot \omega`))))),l(i.p,{title:"Forza centripeta"},l("p",null,"È verso il centro e si calcola con:"),l("p",null,l(i.j,null,Be($||($=ke`F = m \cdot a`)))))),l(i.q,{title:"Lavoro ed energia"},l(i.p,{title:"Lavoro"},l("p",null,"E' compiuto da una forza che sposta un corpo."),l("p",null,l(i.j,null,Be(ll||(ll=ke`W = \vec{F} \cdot \vec{s} = F \cdot \Delta s \cdot cos(\alpha )`)))),l("p",null,"(Se la forza non è parallela allo spostamento, il prodotto scalare ci fa considerare solo la componente parallela.)")),l(i.p,{title:"Energia cinetica"},l("p",null,"Un corpo ha energia cinetica in ogni momento uguale a:"),l("p",null,l(i.j,null,Be(el||(el=ke`E_c = \frac{1}{2} m v^2`)))),l("p",null,"Se una forza effettua lavoro su un corpo, cambia la sua energia cinetica pari al lavoro effettuato:"),l("p",null,l(i.j,null,Be(al||(al=ke`\Delta E_c = W`))))),l(i.p,{title:"Energia potenziale gravitazionale"},l("p",null,"Un corpo ha energia potenziale in ogni momento pari a:"),l("p",null,l(i.j,null,Be(il||(il=ke`E_{p_g} = m \cdot g \cdot h`)))),l("p",null,"(Con ",l(i.j,null,"h")," uguale a un altezza scelta come punto di riferimento.)")),l(i.p,{title:"Energia potenziale elastica"},l("p",null,"Una molla ha sempre energia potenziale elastica pari a:"),l("p",null,l(i.j,null,Be(nl||(nl=ke`E_{p_e} = \frac{1}{2} k x^2`))))),l(i.p,{title:"Forze conservative"},l("p",null,"Sono conservative le forze per le quali il lavoro compiuto non dipende dal percorso seguito per andare dalla partenza all'arrivo."),l("p",null,"Ad esempio, è conservativa la ",l("i",null,"forza di gravità"),", ma ",l("b",null,"non")," è conservativa la forza di attrito."),l("p",null,"Se in un sistema ci sono solo forze conservative, allora l'energia meccanica totale si conserva:"),l("p",null,l(i.j,null,Be(tl||(tl=ke`E = E_k + E_p`))))),l(i.p,{title:"Potenza"},l("p",null,"È la velocità di trasferimento di energia:"),l("p",null,l(i.j,null,Be(ol||(ol=ke`P = \frac{\Delta E}{\Delta t}`)))))),l(i.q,{title:"Elettrostatica"},l(i.p,{title:"Carica elettrica"},l("p",null,"È una proprietà dei corpi che può essere ",l(n.a,null,"positiva")," o ",l(t.a,null,"negativa"),"."),l("p",null,"Si conserva: in un sistema chiuso la carica totale è costante."),l("p",null,"Esiste un'unità elementare: ",l(i.j,null,Be(rl||(rl=ke`C_{elettrone} = 1.602 \cdot 10^{-19}`))),"."),l("p",null,"Cariche ",l(n.a,null,"opp"),l(t.a,null,"oste")," si attraggono; cariche ",l(n.a,null,"uguali")," si respingono.")),l(i.p,{title:"Conduttori e isolanti"},l("p",null,"Più ",l("a",{href:"https://it.wikipedia.org/wiki/Ione"},"ioni")," ha un corpo, meglio la carica si muove attraverso di esso."),l("p",null,"I corpi in cui la carica si muove bene sono ",l("i",null,"conduttori"),", mentre quelli in cui si muove difficilmente sono ",l("i",null,"isolanti"),"."),l("p",null,l("i",null,"Il corpo umano è un buon conduttore.")))),l(i.q,{title:"Polarizzazione"},l(i.p,{title:"Polarizzazione"},l("p",null,"E' possibile polarizzare un corpo per accumulare la carica di un segno in una certa zona."))),l(i.q,null,l(i.p,{title:"Messa a terra"},l("p",null,"Se un corpo conduttore è in contatto con la Terra, le cariche su di esso saranno ",l("i",null,"equilibrate")," e il corpo diventerà elettricamente neutro (con stesso numero di ",l(n.a,null,"cariche positive")," e ",l(t.a,null,"negative")," all'interno)."))),l(i.q,null,l(i.p,{title:"Polarizzazione per strofinio"},l("p",null,"Strofinando tra loro due corpi isolanti, essi si ",l("i",null,"polarizzeranno per strofinio"),".")),l(i.p,{title:"Polarizzazione per contatto"},l("p",null,"Toccando un conduttore con un corpo carico, il conduttore potrà ",l("i",null,"polarizzarsi per contatto"),".")),l(i.p,{title:"Polarizzazione per induzione"},l("p",null,'Se un corpo conduttore ha cariche "esterne" di un ',l(n.a,null,"certo segno")," vicino, esso avrà tutte le cariche del ",l(t.a,null,"segno opposto")," in equilibrio vicino alle cariche esterne, e tutte le cariche dello ",l(n.a,null,"stesso segno")," più lontano possibile da esse."),l("p",null,"Mettendo a terra il conduttore, nuove cariche del ",l(t.a,null,"segno opposto")," saranno attratte all'interno del corpo per equilibrare le cariche che si sono allontanate."),l("p",null,"Staccando il conduttore da terra e rimuovendo le cariche esterne, esso si ritroverà ",l(t.a,null,"caricato del segno opposto")," rispetto alle cariche esterne."))),l(i.q,{title:"Forza elettrica"},l(i.p,{title:"Legge di Coulomb"},l("p",null,"Due corpi carichi si attraggono tra loro con forza:"),l("p",null,l(i.j,null,Be(ul||(ul=ke`\left | \vec{F}_{elettrica} \right | = \frac{-k \cdot q_1 \cdot q_2}{s^2}`)))),l("p",null,l(i.j,null,Be(cl||(cl=ke`k`)))," è la ",l("i",null,"costante di Coulomb"),", e vale ",l(i.j,null,Be(pl||(pl=ke`k = 8.99 \cdot 10^9 \frac{N \cdot m^2}{C^2}`))),".")),l(i.p,{title:"Permeabilità dello spazio vuoto"},l("p",null,"La costante ",l(i.j,null,Be(sl||(sl=ke`k`)))," è in realtà dipendente da un altra costante, ",l(i.j,null,Be(dl||(dl=ke`\epsilon_0`))),", la ",l("i",null,"permeabilità del vuoto"),"."),l("p",null,l(i.j,null,Be(ml||(ml=ke`k = \frac{1}{4 \pi \cdot \epsilon_0}`)))),l("p",null,l(i.j,null,Be(vl||(vl=ke`\left | \vec{F}_{elettrica} \right | = \frac{q_1 \cdot q_2}{4 \pi \cdot \epsilon_0 \cdot s^2}`))))),l(i.p,{title:"Campo elettrico"},l("p",null,"Misura che forza viene applicata in ogni punto su una carica unitaria:"),l("p",null,l(i.j,null,Be(gl||(gl=ke`\vec{E} = \frac{\vec{F}_{elettrica}}{q} = \frac{-k \cdot q}{s^2}`))))),l(i.p,{title:"Flusso elettrico"},l("p",null,'È la differenza tra "quanto" campo elettrico ',l(n.a,null,"entra")," e quanto campo elettrico ",l(t.a,null,"esce")," da una certa area."),l("p",null,"In qualsiasi superficie chiusa, il flusso elettrico è uguale alla componente perpendicolare del campo elettrico moltiplicato per l'area."),l("p",null,l(i.j,null,Be(fl||(fl=ke`\Phi_E = \vec{E} \cdot \vec{A}`)))),l("p",null,"Se il campo elettrico è uniforme, se ne può calcolare facilmente il valore:"),l("p",null,l(i.j,null,Be(zl||(zl=ke`\Phi_E = \vec{E} \cdot \vec{A} = E_\perp \cdot A \cdot \cos(\alpha)`)))),l("p",null,l(i.t,null,"Circa. E' una specie di integrale..."))),l(i.p,{title:"Legge di Gauss per i campi elettrostatici"},l("p",null,"Il flusso elettrico è direttamente proporzionale alla carica presente all'interno della superficie."),l("p",null,l(i.j,null,Be(hl||(hl=ke`\Phi_E = 4 \pi \cdot k \cdot q = \frac{q}{\epsilon_0}`)))),l("p",null,"Ovvero, i campi elettrostatici sono generati dalle cariche elettriche."))),l(i.q,{title:"Energia elettrica"},l(i.p,{title:"Energia potenziale elettrica"},l("p",null,"Un corpo carico vicino ad altre cariche possiede un'",l("i",null,"energia potenziale elettrica")," ",l(i.j,null,Be(_l||(_l=ke`U_e`))),"."))),l(i.q,{title:"Circuiti elettrici"},l(i.p,{title:l("span",null,"Potenziale elettrico ",l("small",null,"(tensione)"))},l("p",null,"È il valore dell'energia potenziale elettrica per una carica unitaria."),l("p",null,l(i.j,null,Be(jl||(jl=ke`V = \frac{U_e}{q}`)))),l("p",null,"La sua unità di misura è il Volt (",l(i.j,null,Be(ql||(ql=ke`V`))),")."),l("p",null,"In una batteria è detto ",l("i",null,"forza elettromotrice"),", e corrisponde al lavoro compiuto da una batteria ideale per spostare una carica unitaria tra i due poli.")),l(i.p,{title:l("span",null,"Corrente elettrica ",l("small",null,"(intensità)"))},l("p",null,"Quanta carica passa attraverso un'area (perpendicolare al flusso) nel tempo."),l("p",null,l(i.j,null,Be(bl||(bl=ke`I = \frac{\Delta q}{\Delta t}`)))),l("p",null,"Fintanto che c'è differenza di potenziale, ci sarà anche intensità non nulla."),l("p",null,"La sua unità di misura è l'Ampere (",l(i.j,null,Be(Ll||(Ll=ke`A`))),").")),l(i.p,{title:l("span",null,"Corrente continua ",l("small",null,"(",l("abbr",{title:"Direct Current"},"DC"),")"))},l("p",null,"Quando in un circuito la direzione della corrente è costante.")),l(i.p,{title:l("span",null,"Corrente alternata ",l("small",null,"(",l("abbr",{title:"Alternate Current"},"AC"),")"))},l("p",null,"Quando in un circuito la direzione della corrente si alterna periodicamente.")),l(i.p,{title:"Potenza elettrica"},l("p",null,"Possiamo calcolare la potenza di un circuito:"),l("p",null,l(i.j,null,Be(Dl||(Dl=ke`P = \frac{\Delta U_e}{\Delta t} = I \cdot \Delta V = I^2 \cdot R = \frac{(\Delta V)^2}{R}`)))))),l(i.q,{title:"Elementi di un circuito"},l(i.p,{title:"Resistore"},l("p",null,"Riduce l'intensità di corrente, e converte parte del potenziale in calore."),l("p",null,"Il potenziale utilizzato è pari a:"),l("p",null,l(i.j,null,Be(Fl||(Fl=ke`V = R \cdot I`)))),l("p",null,"Dove ",l(i.j,null,Be(Sl||(Sl=ke`R`)))," è una costante detta ",l("i",null,"resistenza")," con unità di misura Ohm (",l(i.j,null,Be(El||(El=ke`\Omega`))),")."),l("p",null,"La resistenza di un conduttore vale:"),l("p",null,l(i.j,null,Be(Pl||(Pl=ke`R = \rho \frac{L_{unghezza}}{A_{rea}}`)))),l("p",null,l(i.j,null,Be(Cl||(Cl=ke`\rho`)))," è la ",l("i",null,"resistività")," del materiale, e varia in base alla temperatura:"),l("p",null,l(i.j,null,Be(Il||(Il=ke`\rho = \rho_0 (1 + \alpha(T - T_0))`))))),l(i.p,{title:"Condensatore"},l("p",null,"Immagazzina potenziale elettrico, permettendo di riutilizzarla in seguito."),l("p",null,"Per farlo, cattura cariche ",l(n.a,null,"positive")," e ",l(t.a,null,"negative")," sulle sue due armature; perchè questo avvenga, deve essere compiuto lavoro."),l("p",null,"Ha una ",l("b",null,"capacità")," caratteristica, che in un condensatore a facce piane parallele è:"),l("p",null,l(i.j,null,Be(Al||(Al=ke`C = \frac{q_{massima}}{\Delta V}`)))),l("p",null,"Condensatori di capacità maggiore immagazzinano più potenziale con meno carica."),l("p",null,"La capacità aumenta se viene messo qualcosa tra le armature:"),l("p",null,l(i.j,null,Be(wl||(wl=ke`C_{nuova} = \kappa \cdot \frac{\epsilon_0 \cdot A}{s}`)))),l("p",null,"Dove ",l(i.j,null,Be(kl||(kl=ke`\kappa`)))," è la ",l("i",null,"costante dielettrica relativa")," del materiale inserito, ",l(i.j,null,Be(Bl||(Bl=ke`A`)))," l'area di una armatura e ",l(i.j,null,Be(Nl||(Nl=ke`s`)))," la distanza tra le due armature."),l("p",null,"Se il campo elettrico creatosi tra le due armature supera la ",l("i",null,"rigidità dielettrica")," del condensatore, la carica immagazzinata viene persa e ha luogo un ",l("i",null,"breakdown"),"."),l("p",null,"La sua unità di misura è il Farad (",l(i.j,null,Be(Vl||(Vl=ke`Fa`))),")")),l(i.p,{title:"Amperometro"},l("p",null,"Misura la corrente elettrica se messo in serie."),l("p",null,"(Funzionamento: ha una resistenza interna bassisima in modo da non influire significativamente sulla corrente.)")),l(i.p,{title:"Voltmetro"},l("p",null,"Misura la differenza di potenziale se messo in parallelo."),l("p",null,"(Funzionamento: ha una resistenza altissima in modo da non influire significativamente sulla tensione.)"))),l(i.q,{title:"Principi di Kirchhoff"},l(i.p,{title:"Legge dei nodi"},l("p",null,"Per nodo si intende un qualsiasi punto del circuito."),l("p",null,"Da un nodo entra ed esce la stessa corrente.")),l(i.p,{title:"Legge delle maglie"},l("p",null,"Per maglia si intende un qualsiasi percorso chiuso all'interno del circuito."),l("p",null,"In una maglia chiusa, la somma delle differenze di potenziale è 0."))),l(i.q,{title:"Serie e Parallelo"},l(i.p,{title:"Circuito in serie"},l("p",null,"Più parti di circuito sono ",l("i",null,"in serie")," se sono consecutive e senza biforcazioni."),l("p",null,"Parti di circuito in serie sono attraversate dalla stessa corrente.")),l(i.p,{title:"Circuito in parallelo"},l("p",null,"Più parti di circuito sono ",l("i",null,"in parallelo")," tra loro se hanno lo stesso punto di partenza e lo stesso punto di arrivo."),l("p",null,"Parti di circuito in parallelo hanno la stessa differenza di potenziale."))),l(i.q,{title:"Resistenze equivalenti"},l(i.p,{title:"Circuiti in serie"},l("p",null,"Nei circuiti in serie, tutte le resistenze possono essere sostituite con una equivalente dalla resistenza della somma di tutte le quelle sostituite:"),l("p",null,l(i.j,null,Be(Rl||(Rl=ke`R_{serie} = \sum_{i=1}^{n} R_i`))))),l(i.p,{title:"Circuiti in parallelo"},l("p",null,"Nei circuiti in parallelo, tutte le resistenze possono essere sostituite con una equivalente dalla resistenza di:"),l("p",null,l(i.j,null,Be(Tl||(Tl=ke`R_{parallelo} = \frac{1}{\sum_{i=1}^{n} \frac{1}{R_i}}`)))))),l(i.q,{title:"Condensatori equivalenti"},l(i.p,{title:"Circuiti in serie"},l("p",null,"Nei circuiti in serie, tutti i condensatori possono essere sostituiti con uno equivalente dalla capacità di:"),l("p",null,l(i.j,null,Be(yl||(yl=ke`C_{serie} = \frac{1}{\sum_{i=1}^{n} \frac{1}{C_i}}`))))),l(i.p,{title:"Circuiti in parallelo"},l("p",null,"Nei circuiti in parallelo, tutte i condensatori possono essere sostituite con uno equivalente dalla capacità della somma di tutti quelli sostituiti:"),l("p",null,l(i.j,null,Be(Ml||(Ml=ke`C_{parallelo} = \sum_{i=1}^{n} C_n`)))))),l(i.q,{title:"Magnetismo"},l(i.p,{title:"Permeabilità magnetica dello spazio vuoto"},l("p",null,"E' una costante fisica fondamentale che rappresenta quanto un materiale si magnetizza facilmente."),l("p",null,l(i.j,null,Be(Ul||(Ul=ke`\mu_0 = 4 \pi \cdot 10^{-7} \frac{H}{m}`)))," (",l(i.j,null,Be(xl||(xl=ke`\frac{N}{A^2}`))),")")),l(i.p,{title:"Campo magnetico"},l("p",null,"Come un campo elettrico, ma per i magneti."),l("p",null,"Il suo simbolo è ",l(i.j,null,Be(Ol||(Ol=ke`B`))),", e la sua unità di misura è il Tesla (",l(i.j,null,"T"),").")),l(i.p,{title:"Flusso magnetico"},l("p",null,'È "quanto" campo magnetico ',l("b",null,"attraversa")," un percorso chiuso."),l("p",null,'Per qualsiasi percorso chiuso, il flusso magnetico è uguale alla somma di tutti i "sottoflussi" magnetici calcolati sui suoi lati.'),l("p",null,l(i.j,null,Be(Gl||(Gl=ke`\Phi_{B_{i}} = \vec{B} \cdot \vec{L}_n = B \cdot L_i \cdot \sin(\alpha) = B_\parallel \cdot L_i`)))),l("p",null,l(i.j,null,Be(Ql||(Ql=ke`\Phi_{B} = \sum_{i=0}^{n_{lati}} \Phi_{Bn}`)))),l("p",null,"La sua unità di misura è il Weber (",l(i.j,null,Be(Jl||(Jl=ke`Wb = T \cdot m^2`))),").")),l(i.p,{title:"Legge di Gauss per i campi magnetici"},l("p",null,"Il flusso magnetico attraverso qualsiasi superficie chiusa è sempre nullo."),l("p",null,"Ovvero, non esistono monopoli magnetici.")),l(i.p,{title:"Legge di Ampère"},l("p",null,"L'intensità di corrente che attraversa un percorso chiuso è direttamente proporzionale al flusso magnetico dello stesso percorso."),l("p",null,l(i.j,null,Be(Wl||(Wl=ke`\Phi_B = \mu_0 \cdot I`)))))),l(i.q,{title:"Forze magnetiche"},l(i.p,{title:l("span",null,"Forza magnetica su carica puntiforme ",l("small",null,"(Forza di Lorentz)"))},l("p",null,"I campi magnetici applicano una forza sulle cariche vicine:"),l("p",null,l(i.j,null,Be(Hl||(Hl=ke`\vec{F}_{B} = q \cdot (\vec{v} \times \vec{B})`)))),l("p",null,"Dove ",l(i.j,null,Be(Zl||(Zl=ke`\vec{B}`)))," è l'intensità del campo magnetico e ",l(i.j,null,Be(Xl||(Xl=ke`\vec{v}`)))," la velocità della carica considerata."),l("p",null,"Si ha una forza massima se la velocità è perpendicolare al campo magnetico."),l("p",null,"In un campo magnetico uniforme, una velocità perpendicolare al campo porta alla creazione di un moto circolare uniforme.")),l(i.p,{title:"Forza magnetica in un filo"},l("p",null,"I campi magnetici influenzano ovviamente anche le cariche presenti in un conduttore:"),l("p",null,l(i.j,null,Be(Kl||(Kl=ke`\vec{F}_{magnetica} = I \cdot (\vec{L} \times \vec{B})`)))," ",l("a",{href:"https://it.openprof.com/wb/forza_di_lorentz_su_un_filo_percorso_da_corrente?ch=360"},"[1]")),l("p",null,"Dove ",l(i.j,null,Be(Yl||(Yl=ke`I`)))," è la corrente elettrica, ",l(i.j,null,Be($l||($l=ke`\vec{L}`)))," è un vettore che punta nella direzione di scorrimento della corrente e ha come modulo la lunghezza del conduttore."))),l(i.q,{title:"Campi magnetici"},l(i.p,{title:"Campo magnetico in una spira"},l("p",null,"Una spira in cui passa corrente produce un campo magnetico perpendicolare al piano creato dalla spira.")),l(i.p,{title:"Campo magnetico di un solenoide"},l("p",null,"Un solenoide sono tante spire avvolte in modo da formare una specie di cilindro."),l("p",null,"All'interno del solenoide si crea un campo (quasi) uniforme:"),l("p",null,l(i.j,null,Be(le||(le=ke`\left | \vec{B} \right | = \mu_0 \cdot I \cdot \frac{A_{vvolgimenti}}{L_{unghezzafilo}}`))))),l(i.p,{title:"Legge di Oersted"},l("p",null,l("i",null,"Caso particolare della ",l("a",{href:"https://it.wikipedia.org/wiki/Legge_di_Amp%C3%A8re"},"Legge di Ampère"),".")),l("p",null,"Il modulo del campo magnetico ",l(i.j,null,"B")," prodotto da un filo in cui passa una corrente continua ",l(i.j,null,"I")," alla distanza ",l(i.j,null,"s")," è:"),l("p",null,l(i.j,null,Be(ee||(ee=ke`\left | \vec{B} \right | = \frac{\mu \cdot I}{2 \pi r}`)))),l("p",null,"Il campo magnetico così creato gira attorno al filo in senso antiorario."),l("p",null,"Due fili attraversati dalla ",l(n.a,null,"stessa corrente")," si attraggono, due fili attraversati da ",l(n.a,null,"corr"),l(t.a,null,"enti")," ",l(n.a,null,"opp"),l(t.a,null,"oste")," si respingono."))),l(i.q,{title:"Induzione elettromagnetica"},l(i.p,{title:"Forza elettromotrice indotta"},l("p",null,"Un conduttore perpendicolare ad un campo magnetico può ottenere una differenza di potenziale se messo in movimento in un direzione perpendicolare alla direzione del conduttore e del campo."),l("p",null,"La differenza di potenziale si crea a causa della forza magnetica, che fa spostare tutti gli elettroni verso un capo del conduttore."),l("p",null,"Essa vale:"),l("p",null,l(i.j,null,Be(ae||(ae=ke`\Delta V_{indotta} = v \cdot B \cdot L`)))),l("p",null,"Dove ",l(i.j,null,"v")," è la velocità del conduttore, ",l(i.j,null,"B")," è l'intensità del campo magnetico ed ",l(i.j,null,"L")," è la lunghezza del conduttore.")),l(i.p,{title:"Flusso magnetico in una spira"},l("p",null,"In un campo magnetico ",l(i.j,null,Be(ie||(ie=ke`B`)))," uniforme e perpendicolare al piano di una spira di area ",l(i.j,null,Be(ne||(ne=ke`A`))),", il flusso magnetico si può determinare con la ",l("i",null,"Legge di Faraday-Neumann-Lenz"),":"),l("p",null,l(i.j,null,Be(te||(te=ke`\Phi_B = \vec{B} \cdot \vec{A} = B \cdot A \cdot \cos(\alpha)`)))))),l(i.q,null,l(i.p,{title:"Legge di Faraday-Neumann-Lenz"},l("p",null,"Dice che la forza elettromotrice media indotta in un percorso dipende dalla variazione nel tempo del flusso magnetico nello stesso percorso."),l("p",null,l(i.j,null,Be(oe||(oe=ke`\Delta V_{indotta} = - \frac{\Delta \Phi_B}{\Delta t}`)))),l("p",null,"Il meno è dovuto alla ",l("a",{href:"https://it.wikipedia.org/wiki/Legge_di_Lenz"},"Legge di Lenz"),", che specifica qualitativamente il verso della forza elettromotrice indotta.")),l(i.p,{title:"Faraday in un solenoide"},l("p",null,"In un solenoide, la forza elettromotrice indotta è uguale a:"),l("p",null,l(i.j,null,Be(re||(re=ke`\Delta V_{indotta} = - \frac{N \cdot \Delta \Phi_{B_{spira}}}{\Delta t} = - \frac{N \cdot B \cdot A \cdot cos(\alpha)}{\Delta t}`)))),l("p",null,"Dove ",l(i.j,null,Be(ue||(ue=ke`N`)))," è il numero delle spire del solenoide.")),l(i.p,{title:"Legge di Ampère-Maxwell"},l("p",null,"Correnti o campi elettrici variabili creano un campo magnetico."))),l(i.q,{title:"Elettromagnetismo"},l(i.p,{title:"Onde elettromagnetiche"},l("p",null,"Nel vuoto, il campo elettrico ",l(i.j,null,Be(ce||(ce=ke`E`)))," e il campo magnetico ",l(i.j,null,Be(pe||(pe=ke`B`)))," sono perpendicolari tra loro e la direzione di propagazione, e sono entrambe funzioni del tempo."),l("p",null,"Si dice quindi che sono ",l("i",null,"onde elettromagnetiche"),"."),l("p",null,"Esse sono legate dalla relazione:"),l("p",null,l(i.j,null,Be(se||(se=ke`E = c \cdot B`)))),l("p",null,"Dove ",l(i.j,null,Be(de||(de=ke`c`)))," è la velocità delle onde (luce) nel vuoto, e a sua volta è uguale a:"),l("p",null,l(i.j,null,Be(me||(me=ke`c = \frac{1}{\sqrt{\epsilon_0 \cdot \mu_0}} = 3.00 \cdot 10^8 \frac{m}{s}`))))),l(i.p,{title:"Formula delle onde"},l("p",null,l(i.j,null,Be(ve||(ve=ke`A(t) = A_{max} \cdot \sin \left ( \frac{2 \pi}{\lambda} - \omega t + \phi \right )`)))),l("p",null,"Dove ",l(i.j,null,Be(ge||(ge=ke`A_{max}`)))," è l'ampiezza massima che può avere l'onda, ",l(i.j,null,Be(fe||(fe=ke`\frac{2 \pi}{\lambda} = \left | \vec{k} \right |`)))," è il vettore d'onda, ",l(i.j,null,Be(ze||(ze=ke`\omega`)))," la frequenza angolare e ",l(i.j,null,Be(he||(he=ke`\phi`)))," la fase."))),l(i.q,{title:"Spettroscopia"},l(i.p,{title:"Emissione"},l("p",null,"I solidi, se portati ad alta temperatura, emettono luce con uno ",l("a",{href:"https://it.wikipedia.org/wiki/Spettro_continuo"},"spettro continuo"),"."),l("p",null,"I gas, invece, ad alta temperatura emettono luce solo con particolari lunghezze d'onda."),l("p",null,"In un gas di idrogeno, le lunghezze d'onda emesse sono ricavabili con:"),l("p",null,l(i.j,null,Be(_e||(_e=ke`\frac{1}{\lambda} = R \left ( \frac{1}{4} - \frac{1}{n^2} \right )`)))),l("p",null,"Con ",l(i.j,null,Be(je||(je=ke`R = 1.097 \cdot 10^7 \frac{1}{m}`))),", detta costante di Rydberg, e ",l(i.j,null,Be(qe||(qe=ke`n`)))," un numero intero.")),l(i.p,{title:"Grandezza quantizzata"},l("p",null,"Una grandezza si dice quantizzata (o discreta) se può assumere solo determinati valori."),l("p",null,"Una grandezza si dice continua se può assumere qualsiasi valore e quindi se non è quantizzata."),l("p",null,"Energia, momento angolare e raggio sono quantizzati."),l("p",null,"Nota costante quantica è ",l(i.j,null,Be(be||(be=ke`h`))),", la costante di Planck, ovvero il valore minimo possibile per la carica (talvolta espressa come ",l(i.j,null,Be(Le||(Le=ke`\hbar = \left ( \frac{h}{2 \pi} \right )`))),"."))),l(i.q,null,l(i.p,{title:"Modello di Bohr"},l("p",null,"L'energia degli elettroni è quantizzata."),l("p",null,"Inoltre, per essi è valido che:"),l("p",null,l(i.j,null,Be(De||(De=ke`m \cdot v_n \cdot 2 \pi \cdot r = n \cdot h`)))),l("p",null,"Ancora, il raggio delle orbite è uguale a:"),l("p",null,l(i.j,null,Be(Fe||(Fe=ke`r_n = n^2 \cdot a_0 = n^2 \cdot \frac{\hbar}{m_{elettrone} \cdot k \cdot e^2} `)))),l("p",null,"Con ",l(i.j,null,Be(Se||(Se=ke`a_0 = \left ( \frac{h}{2 \pi} \right )^2 \cdot \frac{1}{m_{elettrone} \cdot k \cdot e^2} = 5.29 \cdot 10^{-11} m`))),"."),l("p",null,"Infine, in ogni stato, l'energia è pari a:"),l("p",null,l(i.j,null,Be(Ee||(Ee=ke`E_n = \frac{1}{n^2} \cdot E_1 = - \frac{1}{n^2} \cdot \frac{a_0^2}{2 \cdot m \cdot \hbar^4} = - \frac{1}{n^2} \cdot \frac{m_{elettrone} \cdot k^2 \cdot e^4}{2 \cdot \hbar^2}`)))),l("p",null,"Due elettroni non possono occupare lo stesso stato."),l("p",null,"Questo modello funziona solo per atomi con numero atomico basso. Atomi con molti elettroni hanno comportamenti diversi, descritti dal modello di"))),l(i.q,null,l(i.p,{title:"Nei solidi"},l("p",null,"Nei solidi, le lunghezze d'onda sono talmente tanto vicine da poter essere considerate una banda."),l("p",null,"Possono però comunque avere dei gap dovuti agli intervalli di energia non ammessi."))),l(i.q,{title:"Semiconduttori"},l(i.p,{title:"Semiconduttori"},l("p",null,l(i.t,null,"Refactor this")),l("p",null,"Se la banda di emissione con energia più alta di un corpo è assente o è separata da un gap dell'ordine di grandezza maggiore di ",l(i.j,null,Be(Pe||(Pe=ke`10^1 eV`))),", allora il corpo è un isolante."),l("p",null,"Se invece la banda di emissione si sovrappone a un altra, allora il corpo è un conduttore."),l("p",null,"Se il gap è invece dell'ordine di grandezza di ",l(i.j,null,Be(Ce||(Ce=ke`1 eV`))),", allora il corpo è un semiconduttore.")),l(i.p,{title:"Lacune"},l("p",null,"Legami in cui ",l(n.a,null,"mancano elettroni"),"."),l("p",null,l(t.a,null,"Elettroni")," di altri legami possono spostarsi per colmare le ",l(n.a,null,"lacune"),", creandone altre, e spostandole in direzione opposta a quella della corrente.")),l(i.p,{title:"Accettori e donori"},l("p",null,"Se si inserisce in un cristallo semiconduttore si inserisce un atomo con numero atomico diverso, si otterrà:"),l("ul",null,l("li",null,"Con numero atomico maggiore, un semiconduttore di ",l(t.a,null,"tipo N")," con ",l(t.a,null,"elettroni in eccesso")," liberi di scorrere."),l("li",null,"Con numero atomico minore, un semiconduttore di ",l(n.a,null,"tipo P")," con ",l(n.a,null,"lacune in eccesso")," libere di catturare elettroni da altri legami.")),l("p",null,"Maggiore impurezza porta a maggiore conduttività.")),l(i.p,{title:"Temperatura"},l("p",null,"Aumentando la temperatura di un semiconduttore si aumenta la conduttività, perchè eccita le particelle e favorisce il movimento di ",l(t.a,null,"elettroni")," e ",l(n.a,null,"lacune"),"."))),l(i.q,{title:l("span",null,"Ottica ",l("small",null,"(non l'abbiamo fatta)"))},l(i.p,{title:"Assorbimento e riflessione"},l("p",null,"I corpi possono assorbire o riflettere le onde elettromagnetiche che li colpiscono.")),l(i.p,{title:"Corpo nero"},l("p",null,"Un corpo nero è un corpo che assorbe tutte le onde elettromagnetiche che riceve senza rifletterne nessuna."),l("p",null,"Le onde assorbite vengono poi riemesse sotto forma di un onda di ",l(i.j,null,Be(Ie||(Ie=ke`\lambda`)))," variabile in base alla temperatura."),l("p",null,l(i.j,null,Be(Ae||(Ae=ke`\lambda_{max} \cdot T`)))," è costante.")),l(i.p,{title:"Teoria di Planck per il corpo nero"},l("p",null,"L'energia assorbita e emessa dai corpi neri è quantizzata.")),l(i.p,{title:"Fotone"},l("p",null,"Un onda magnetica con un quanto di energia è detta ",l("i",null,"fotone"),":"),l("p",null,l(i.j,null,Be(we||(we=ke`E_{fotone} = h \cdot f`))))),l(i.p,{title:"Effetto fotoelettrico"},l("p",null,"A volte, i fotoni che colpiscono un metallo possono estrarvi degli elettroni e creare una differenza di potenziale."),l("p",null,"Perchè avvenga, la frequenza deve essere maggiore di una certa soglia."),l("p",null,"Il numero di elettroni estratti dipende dall'intensità dell'onda, mentre l'energia cinetica degli elettroni dipende dalla frequenza."),l("p",null,"Non c'è nessun ritardo tra l'assorbimento del fotone e l'estrazione di elettroni."))))))}}.call(this,a("hosL").h)},ZHMS:function(l,e,a){"use strict";(function(l){var i=a("gJgL"),n=a.n(i);e.a=function(e){return 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dell'",i(r.b,{href:"https://www.unimore.it/"},"Unimore"),".")),i(r.p,{title:i(a.a,{href:"/mingwinstall"},"Come installare MinGW")},i("p",null,"Un breve tutorial con immagini su come installare e configurare ",i(r.b,{href:"https://it.wikipedia.org/wiki/MinGW"},"MinGW")," per compilare programmi C e C++ su Windows."))),i(r.q,{title:"Collegamenti utili"},i(r.p,{title:i(a.a,{href:"https://github.com/topics/unimore-informatica"},"unimore-informatica")},i("p",null,"Un topic di GitHub che raccoglie tutti i repository realizzati dagli studenti di Informatica dell'Unimore.")),i(r.p,{title:i(a.a,{href:"https://t.me/unimoreinfo"},"Unimore Informatica")},i("p",null,"Il gruppo ",i(r.b,{href:"https://telegram.org/"},"Telegram")," del corso di Informatica dell'Unimore!")),i(r.p,{title:i(a.a,{href:"https://calendar.google.com/calendar?cid=MmYza2o2M3VuZWQ1cmZqaGpmOGY0MWFrNmdAZ3JvdXAuY2FsZW5kYXIuZ29vZ2xlLmNvbQ"},"Calendario Lezioni")},i("p",null,"Calendario Google ",i("small",null,"quasi")," sempre aggiornato delle lezioni e degli esami del secondo anno dell'",i(r.b,{href:"https://www.unimore.it/"},"Unimore")," durante l'Anno Accademico 2019/2020.")),i(r.p,{title:i(a.a,{href:"http://erre2.fermitech.info/dashboard"},"Erre2")},i("p",null,"Portale contenente appunti e riassunti mantenuto da ",i(r.b,{href:"https://github.com/LBindustries"},"Lorenzo Balugani"),"."))))}}.call(this,t("hosL").h)}}]); +//# sourceMappingURL=route-Home.chunk.005a2.js.map \ No newline at end of file diff --git a/docs/route-Home.chunk.f25c8.js.map b/docs/route-Home.chunk.005a2.js.map similarity index 99% rename from docs/route-Home.chunk.f25c8.js.map rename to docs/route-Home.chunk.005a2.js.map index 1d1a795..c967cc2 100644 --- a/docs/route-Home.chunk.f25c8.js.map +++ b/docs/route-Home.chunk.005a2.js.map @@ -1 +1 @@ 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\n Appunti scritti mentre studiavo per l'esame di Calcolo Numerico del corso triennale di Informatica all'Unimore.\n

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\n Appunti scritti mentre studiavo per l'esame di Ottimizzazione lineare intera del corso triennale di Informatica all'Unimore.\n

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\n
\n Basi di dati}>\n

\n Appunti scritti mentre studiavo per l'esame di Basi di dati del corso triennale di Informatica all'Unimore.\n

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\n Appunti scritti mentre studiavo per l'esame di Apprendimento ed evoluzione in sistemi artificiali del corso triennale di Informatica all'Unimore.\n

\n \n Statistica ed elementi di probabilità}>\n

\n Appunti scritti mentre studiavo per l'esame di Statistica ed elementi di probabilità del corso triennale di Informatica all'Unimore del Prof. Luca La Rocca.\n

\n \n Cleaver}>\n

\n Progetto in Java sviluppato per l'esame di Programmazione ad Oggetti del corso triennale di Informatica all'Unimore, tenuto dai Prof. Giacomo Cabri e Nicola Capodieci.\n

\n \n Fisica}>\n

\n Appunti delle lezioni di Fisica del corso triennale di Informatica all'Unimore, tenute dalla Prof.ssa Rossella Brunetti nel primo semestre dell'Anno Accademico 2019/2020.\n

\n \n Sistemi Operativi}>\n

\n Soluzioni agli Arzigogoli proposti dal Prof. Mauro Andreolini durante le lezioni di Sistemi Operativi del corso triennale di Informatica all'Unimore tenutesi nel primo semestre dell'Anno Accademico 2019/2020.\n

\n \n Algoritmi e Strutture Dati}>\n

\n Appunti delle lezioni di Algoritmi e Strutture Dati del corso triennale di Informatica all'Unimore, tenute dalla Prof.ssa Manuela Montangero nel secondo semestre dell'Anno Accademico 2018/2019.\n

\n

\n TODO: finisci di migrare da GitHub!\n

\n \n Videolezioni di Geometria}>\n

\n Ottime videolezioni di Geometria con licenza CC BY-NC-SA 4.0 che ho trovato sul portale Dolly 2018 dell'Unimore.\n

\n \n Come installare MinGW}>\n

\n Un breve tutorial con immagini su come installare e configurare MinGW per compilare programmi C e C++ su Windows.\n

\n \n
\n
\n unimore-informatica}>\n

\n Un topic di GitHub che raccoglie tutti i repository realizzati dagli studenti di Informatica dell'Unimore.\n

\n \n Unimore Informatica}>\n

\n Il gruppo Telegram del corso di Informatica dell'Unimore!\n

\n \n Calendario Lezioni}>\n

\n Calendario Google quasi sempre aggiornato delle lezioni e degli esami del secondo anno dell'Unimore durante l'Anno Accademico 2019/2020.\n

\n \n Erre2}>\n

\n Portale contenente appunti e riassunti mantenuto da Lorenzo Balugani.\n

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\n Appunti scritti mentre studiavo per l'esame di Ottimizzazione lineare intera del corso triennale di Informatica all'Unimore.\n

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\n
\n Basi di dati}>\n

\n Appunti scritti mentre studiavo per l'esame di Basi di dati del corso triennale di Informatica all'Unimore.\n

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\n Appunti scritti mentre studiavo per l'esame di Apprendimento ed evoluzione in sistemi artificiali del corso triennale di Informatica all'Unimore.\n

\n \n Statistica ed elementi di probabilità}>\n

\n Appunti scritti mentre studiavo per l'esame di Statistica ed elementi di probabilità del corso triennale di Informatica all'Unimore del Prof. Luca La Rocca.\n

\n \n Cleaver}>\n

\n Progetto in Java sviluppato per l'esame di Programmazione ad Oggetti del corso triennale di Informatica all'Unimore, tenuto dai Prof. Giacomo Cabri e Nicola Capodieci.\n

\n \n Fisica}>\n

\n Appunti delle lezioni di Fisica del corso triennale di Informatica all'Unimore, tenute dalla Prof.ssa Rossella Brunetti nel primo semestre dell'Anno Accademico 2019/2020.\n

\n \n Sistemi Operativi}>\n

\n Soluzioni agli Arzigogoli proposti dal Prof. Mauro Andreolini durante le lezioni di Sistemi Operativi del corso triennale di Informatica all'Unimore tenutesi nel primo semestre dell'Anno Accademico 2019/2020.\n

\n \n Algoritmi e Strutture Dati}>\n

\n Appunti delle lezioni di Algoritmi e Strutture Dati del corso triennale di Informatica all'Unimore, tenute dalla Prof.ssa Manuela Montangero nel secondo semestre dell'Anno Accademico 2018/2019.\n

\n

\n TODO: finisci di migrare da GitHub!\n

\n \n Videolezioni di Geometria}>\n

\n Ottime videolezioni di Geometria con licenza CC BY-NC-SA 4.0 che ho trovato sul portale Dolly 2018 dell'Unimore.\n

\n \n Come installare MinGW}>\n

\n Un breve tutorial con immagini su come installare e configurare MinGW per compilare programmi C e C++ su Windows.\n

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\n
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\n Un topic di GitHub che raccoglie tutti i repository realizzati dagli studenti di Informatica dell'Unimore.\n

\n \n Unimore Informatica}>\n

\n Il gruppo Telegram del corso di Informatica dell'Unimore!\n

\n \n Calendario Lezioni}>\n

\n Calendario Google quasi sempre aggiornato delle lezioni e degli esami del secondo anno dell'Unimore durante l'Anno Accademico 2019/2020.\n

\n \n Erre2}>\n

\n Portale contenente appunti e riassunti mantenuto da Lorenzo Balugani.\n

\n \n
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\n\t\t\t

Come installare MinGW

\n\t\t\t\n\t\t\t\t

Scaricate l'installer ufficiale,\n\t\t\t\t\ted eseguitelo.

\"\"/\n\t\t\t\t

Dovrebbe comparire questa schermata. Cliccate su Install, poi scegliete una cartella di installazione\n\t\t\t\t\t(ricordatevela!) e poi Continue. Lasciate stare le altre opzioni, dovrebbero essere tutte spuntate,\n\t\t\t\t\ttranne For all users, che dovrebbe essere disattivato.

\"\"/\n\t\t\t\t

Aspettate che finisca il download. Pochi secondi dopo, dovrebbe finire e dovrebbe apparire un tasto\n\t\t\t\t\tContinue. Premetelo.

\"\"/\n\t\t\t\t

Dovrebbe apparirvi questa finestra. L'installer di MinGW è una specie di gestore pacchetti (tipo apt su\n\t\t\t\t\tUbuntu); potete scegliere quali pacchetti installare, e quindi quali funzionalità.

\"\"/\n\t\t\t\t

Nel nostro caso, dovrebbero servirci mingw32-base-bin (per il C e alcune librerie C++) e\n\t\t\t\t\tmingw32-gcc-g++-bin (per il C++). Cliccate, quindi, sui due quadratini corrispondenti, e premete\n\t\t\t\t\tMark for Installation. Dovrebbe comparire una freccia gialla sul quadratino.

\"\"/\n\t\t\t\t

Ora, è il momento di installare i pacchetti. Aprite il menù Installation, poi premete\n\t\t\t\t\tApply Changes, e di nuovo Apply.

\"\"/\n\t\t\t\t

Lasciate che scarichi, ci vorrà un po'. Guardatevi un video nel frattempo, fatevi una partitina a qualcosa, tornate\n\t\t\t\t\tdopo circa 10 minuti.

\"\"/\n\t\t\t\t

Una volta installato, dobbiamo aggiungere g++ ai programmi eseguibili da Prompt dei Comandi: premete il\n\t\t\t\t\ttasto Windows, e scrivete PATH. Windows dovrebbe trovarvi automaticamente quell'opzione.

\n\t\t\t\t\"\"/\n\t\t\t\t

Dentro la finestra di Proprietà del Sistema, premete Variabili d'ambiente.

\"\"/\n\t\t\t\t

Trovate la variabile d'ambiente globale Path, e fateci doppio click per modificarla.

\"\"/\n\t\t\t\t

Ora dovreste vedere l'elenco di tutte le cartelle contenenti programmi eseguibili da terminale: dobbiamo aggiungere\n\t\t\t\t\tquella di MinGW! Premete Sfoglia.

\"\"/\n\t\t\t\t

Trovate la cartella in cui avete installato MinGW (vi avevo detto di ricordarvela!); entrateci, poi selezionate la\n\t\t\t\t\tsottocartella bin e premete OK su tutte le finestre che avete aperto fino ad ora,\n\t\t\t\t\tchiudendole.

\n\t\t\t\t

Complimenti! Avete installato MinGW e potete compilare programmi C e C++ da Windows! Avete a disposizione\n\t\t\t\t\tgcc e g++ sul Prompt dei Comandi, e potete finalmente creare dei file .exe!

\n\t\t\t\n\t\t
\n\t);\n}\n\n"],"sourceRoot":""} \ No newline at end of file +{"version":3,"sources":["webpack:///./routes/MingwInstall.js"],"names":["href","src","alt"],"mappings":"2FAAA,mCAGe,qBACd,OACC,aACC,qCACA,EAAC,IAAD,KACC,yBAAc,OAAGA,KAAK,wEAAR,yBAAd,oBACmB,SAAKC,IAAI,kCAAkCC,IAAI,KAClE,gEAAqD,yBAArD,wEACwB,0BADxB,+EAEQ,+BAFR,sCAEwE,SAAKD,IAAI,kCAAkCC,IAAI,KACvH,kHACC,0BADD,gBACsC,SAAKD,IAAI,kCAAkCC,IAAI,KACrF,gHAAqG,qBAArG,0FACuF,SAAKD,IAAI,kCAAkCC,IAAI,KACtI,oDAAyC,kCAAzC,sCACC,qCADD,gFAEC,uCAFD,2DAE8F,SAAKD,IAAI,kCAAkCC,IAAI,KAC7I,2EAAgE,8BAAhE,gBACC,+BADD,gBACwC,uBADxC,KAC+D,SAAKD,IAAI,kCAAkCC,IAAI,KAC9G,yJAC0B,SAAKD,IAAI,kCAAkCC,IAAI,KACzE,yDAA8C,qBAA9C,oEACO,wBADP,gBACsC,sBADtC,8DAEA,SAAKD,IAAI,kCAAkCC,IAAI,KAC/C,qCAA0B,oCAA1B,aAAgE,sCAAhE,KAAsG,SAAKD,IAAI,kCAAkCC,IAAI,KACrJ,uDAA4C,sBAA5C,4CAAyG,SAAKD,IAAI,kCAAkCC,IAAI,KACxJ,4JAC0B,yBAD1B,KACmD,SAAKD,IAAI,kCAAkCC,IAAI,KAClG,gJACe,qBADf,cAC0C,oBAD1C,oEAGA,wHACC,qBADD,MACoB,qBADpB,2E","file":"route-MingwInstall.chunk.cb2f5.js","sourcesContent":["import {Panel} from \"bluelib\";\n\n\nexport default function(props) {\n\treturn (\n\t\t
\n\t\t\t

Come installare MinGW

\n\t\t\t\n\t\t\t\t

Scaricate l'installer ufficiale,\n\t\t\t\t\ted eseguitelo.

\"\"/\n\t\t\t\t

Dovrebbe comparire questa schermata. Cliccate su Install, poi scegliete una cartella di installazione\n\t\t\t\t\t(ricordatevela!) e poi Continue. Lasciate stare le altre opzioni, dovrebbero essere tutte spuntate,\n\t\t\t\t\ttranne For all users, che dovrebbe essere disattivato.

\"\"/\n\t\t\t\t

Aspettate che finisca il download. Pochi secondi dopo, dovrebbe finire e dovrebbe apparire un tasto\n\t\t\t\t\tContinue. Premetelo.

\"\"/\n\t\t\t\t

Dovrebbe apparirvi questa finestra. L'installer di MinGW è una specie di gestore pacchetti (tipo apt su\n\t\t\t\t\tUbuntu); potete scegliere quali pacchetti installare, e quindi quali funzionalità.

\"\"/\n\t\t\t\t

Nel nostro caso, dovrebbero servirci mingw32-base-bin (per il C e alcune librerie C++) e\n\t\t\t\t\tmingw32-gcc-g++-bin (per il C++). Cliccate, quindi, sui due quadratini corrispondenti, e premete\n\t\t\t\t\tMark for Installation. Dovrebbe comparire una freccia gialla sul quadratino.

\"\"/\n\t\t\t\t

Ora, è il momento di installare i pacchetti. Aprite il menù Installation, poi premete\n\t\t\t\t\tApply Changes, e di nuovo Apply.

\"\"/\n\t\t\t\t

Lasciate che scarichi, ci vorrà un po'. Guardatevi un video nel frattempo, fatevi una partitina a qualcosa, tornate\n\t\t\t\t\tdopo circa 10 minuti.

\"\"/\n\t\t\t\t

Una volta installato, dobbiamo aggiungere g++ ai programmi eseguibili da Prompt dei Comandi: premete il\n\t\t\t\t\ttasto Windows, e scrivete PATH. Windows dovrebbe trovarvi automaticamente quell'opzione.

\n\t\t\t\t\"\"/\n\t\t\t\t

Dentro la finestra di Proprietà del Sistema, premete Variabili d'ambiente.

\"\"/\n\t\t\t\t

Trovate la variabile d'ambiente globale Path, e fateci doppio click per modificarla.

\"\"/\n\t\t\t\t

Ora dovreste vedere l'elenco di tutte le cartelle contenenti programmi eseguibili da terminale: dobbiamo aggiungere\n\t\t\t\t\tquella di MinGW! Premete Sfoglia.

\"\"/\n\t\t\t\t

Trovate la cartella in cui avete installato MinGW (vi avevo detto di ricordarvela!); entrateci, poi selezionate la\n\t\t\t\t\tsottocartella bin e premete OK su tutte le finestre che avete aperto fino ad ora,\n\t\t\t\t\tchiudendole.

\n\t\t\t\t

Complimenti! Avete installato MinGW e potete compilare programmi C e C++ da Windows! Avete a disposizione\n\t\t\t\t\tgcc e g++ sul Prompt dei Comandi, e potete finalmente creare dei file .exe!

\n\t\t\t\n\t\t
\n\t);\n}\n\n"],"sourceRoot":""} \ No newline at end of file diff --git a/docs/route-NetLogo.chunk.41701.esm.js b/docs/route-NetLogo.chunk.661e6.esm.js similarity index 50% rename from docs/route-NetLogo.chunk.41701.esm.js rename to docs/route-NetLogo.chunk.661e6.esm.js index 47f1559..e2245a5 100644 --- a/docs/route-NetLogo.chunk.41701.esm.js +++ b/docs/route-NetLogo.chunk.661e6.esm.js @@ -1,25 +1,25 @@ -(window.webpackJsonp=window.webpackJsonp||[]).push([[9],{"2w3n":function(e){e.exports={red:"red__2y1B_",orange:"orange__dD2kx",yellow:"yellow__OEpwl",lime:"lime__CVe41",cyan:"cyan__26ZAg",blue:"blue__LO7Xm",magenta:"magenta__1Akee",example:"example__2PzAa"}},hkqw:function(e,l,i){"use strict";i.r(l),function(e){var t=i("mbOI"),n=i("ke5e");let a,o,r,u,s,p,c,g,d,v,b,m,h,f,w,_=e=>e;const k=String.raw;l.default=function(){return e("div",null,e("h1",null,e("a",{href:"/apprendimento"},"Apprendimento"),": NetLogo"),e(t.p,{title:"Agenti di NetLogo"},e(t.o,{title:"Observer"},e("p",null,"L'",e("b",null,"utente")," di NetLogo, che vive nel contesto globale e dà ordini agli altri agenti.")),e(t.o,{title:"Turtle"},e("p",null,"Entità che possono ",e("b",null,"muoversi")," per il mondo."),e(n.a,null,"Sono più o meno come le ",e("a",{href:"https://www.computercraft.info/wiki/Turtle"},"Turtle di Computercraft"),"!"),e("p",null,"Possono essere di una ",e("i",null,"breed")," (razza) specifica, definite con:"),e(t.e,{language:"lisp"},k(a||(a=_`breed [ ]`))),e(n.a,null,"Le breed sono praticamente sottoclassi di turtle.")),e(t.o,{title:"Patch"},e("p",null,"I ",e("b",null,"quadratini")," del mondo. 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\n

\n Apprendimento: NetLogo\n

\n
\n \n

\n L'utente di NetLogo, che vive nel contesto globale e dà ordini agli altri agenti.\n

\n \n \n

\n Entità che possono muoversi per il mondo.\n

\n \n Sono più o meno come le Turtle di Computercraft!\n \n

\n Possono essere di una breed (razza) specifica, definite con:\n

\n {r`breed [ ]`}\n \n Le breed sono praticamente sottoclassi di turtle.\n \n \n \n

\n I quadratini del mondo. Possono essere di vari colori e avere varie proprietà.\n

\n \n ...possiamo dire che questi siano i blocchi di Minecraft?\n \n \n \n

\n Collegamenti tra due turtle.\n

\n \n
\n
\n \n

\n L'observer viene creato automaticamente all'apertura della simulazione.\n

\n \n \n

\n Nuove turtle possono essere create con il comando:\n

\n {r`- `}\n

\n Dove prefisso varia in base al contesto attuale:\n

\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
ContestoPrefisso
Observer{r`create`}
Turtle{r`hatch`}
Patch{r`sprout`}
\n \n
\n
\n \n

\n All'inizio del codice:\n

\n {`globals [var1 var2 var3]`}\n \n Sono a tutti gli effetti variabili globali.\n \n \n \n

\n Aggiungi proprietà a tutte le turtle:\n

\n {`turtles-own [var1 var2 var3]`}\n

\n Aggiungi proprietà alle turtle di un breed:\n

\n {`-own [var1 var2 var3]`}\n \n \n

\n Aggiungi proprietà alle patch:\n

\n {r`patches-own [var1 var2 var3]`}\n \n \n

\n Aggiungi proprietà ai link:\n

\n {r`links-own [var1 var2 var3]`}\n \n
\n
\n \n

\n Si possono creare con:\n

\n {r`let `}\n \n \n

\n Si può impostare il valore di una variabile con:\n

\n {r`set `}\n \n
\n
\n \n

\n Per cambiare contesto:\n

\n {r`\n ask [\n \n ]\n `}\n

\n Nel caso ci siano più bersagli, verranno eseguiti uno\n

\n \n \n

\n Funzioni che non restituiscono nulla:\n

\n {r`\n to \n \n end\n `}\n

\n Con argomenti:\n

\n {r`\n to [var1 var2 var3]\n \n end\n `}\n \n \n

\n Funzioni che restituiscono un valore:\n

\n {r`\n to-report [var1 var2 var3]\n \n report \n end\n `}\n \n
\n
\n \n

\n Muoviti di 10 unità:\n

\n {r`\n forward 10\n back 10\n `}\n \n \n

\n Ruota di 10 gradi:\n

\n {r`\n left 10\n right 10\n `}\n \n
\n
\n )\n}\n","import style from \"./Example.less\";\r\n\r\nexport default function(props) {\r\n return (\r\n
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\n

\n Apprendimento: NetLogo\n

\n
\n \n

\n L'utente di NetLogo, che vive nel contesto globale e dà ordini agli altri agenti.\n

\n \n \n

\n Entità che possono muoversi per il mondo.\n

\n \n Sono più o meno come le Turtle di Computercraft!\n \n

\n Possono essere di una breed (razza) specifica, definite con:\n

\n {r`breed [ ]`}\n \n Le breed sono praticamente sottoclassi di turtle.\n \n \n \n

\n I quadratini del mondo. Possono essere di vari colori e avere varie proprietà.\n

\n \n ...possiamo dire che questi siano i blocchi di Minecraft?\n \n \n \n

\n Collegamenti tra due turtle.\n

\n \n
\n
\n \n

\n L'observer viene creato automaticamente all'apertura della simulazione.\n

\n \n \n

\n Nuove turtle possono essere create con il comando:\n

\n {r`- `}\n

\n Dove prefisso varia in base al contesto attuale:\n

\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
ContestoPrefisso
Observer{r`create`}
Turtle{r`hatch`}
Patch{r`sprout`}
\n \n
\n
\n \n

\n All'inizio del codice:\n

\n {`globals [var1 var2 var3]`}\n \n Sono a tutti gli effetti variabili globali.\n \n \n \n

\n Aggiungi proprietà a tutte le turtle:\n

\n {`turtles-own [var1 var2 var3]`}\n

\n Aggiungi proprietà alle turtle di un breed:\n

\n {`-own [var1 var2 var3]`}\n \n \n

\n Aggiungi proprietà alle patch:\n

\n {r`patches-own [var1 var2 var3]`}\n \n \n

\n Aggiungi proprietà ai link:\n

\n {r`links-own [var1 var2 var3]`}\n \n
\n
\n \n

\n Si possono creare con:\n

\n {r`let `}\n \n \n

\n Si può impostare il valore di una variabile con:\n

\n {r`set `}\n \n
\n
\n \n

\n Per cambiare contesto:\n

\n {r`\n ask [\n \n ]\n `}\n

\n Nel caso ci siano più bersagli, verranno eseguiti uno\n

\n \n \n

\n Funzioni che non restituiscono nulla:\n

\n {r`\n to \n \n end\n `}\n

\n Con argomenti:\n

\n {r`\n to [var1 var2 var3]\n \n end\n `}\n \n \n

\n Funzioni che restituiscono un valore:\n

\n {r`\n to-report [var1 var2 var3]\n \n report \n end\n `}\n \n
\n
\n \n

\n Muoviti di 10 unità:\n

\n {r`\n forward 10\n back 10\n `}\n \n \n

\n Ruota di 10 gradi:\n

\n {r`\n left 10\n right 10\n `}\n \n
\n
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Sì, lo abbiamo fatto anche in Algebra Lineare.")),n(hl.o,{title:"Variabili nella base"},n("p",null,"Variabili che hanno ",n("b",null,"tutti 0 e un solo 1")," nella loro colonna del tableau."),n("p",null,"La loro controparte sono le ",n("i",null,"variabili fuori base"),", che hanno qualsiasi altro valore.")),n(hl.o,{title:"Valore attuale"},n("p",null,"Il valore della funzione obiettivo che si otterrebbe se ",n("b",null,"tutte le variabili fuori base valessero 0"),"."),n("p",null,"Procedendo nella risoluzione (descritta in seguito) del tableau, questo valore aumenterà, fino a raggiungere il valore ottimo quando la risoluzione sarà completata."))),n(hl.p,null,n(hl.o,{color:yl.a,title:"Un esempio"},n("p",null,"Il sistema:"),n(hl.n,null,Al(qn())),n("p",null,"Diventa il tableau:"),n("table",{className:"right"},n("thead",null,n("tr",null,n("th",null,n(hl.i,null,"x_1")),n("th",null,n(hl.i,null,"x_2")),n("th",null,n(hl.i,null,"s_1")),n("th",null,n(hl.i,null,"s_2")),n("th",null,n(hl.i,null,"s_3")),n("th",null,n("abbr",{title:"Termine noto"},"TN")))),n("tbody",null,n("tr",null,n("td",null,n(hl.i,null,"1")),n("td",null,n(hl.i,null,"0")),n("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},n(hl.i,null,"1")),n("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},n(hl.i,null,"0")),n("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},n(hl.i,null,"0")),n("td",{style:"background-color: rgba(255, 0, 0, 0.1);"},n(hl.i,null,"3"))),n("tr",null,n("td",null,n(hl.i,null,"0")),n("td",null,n(hl.i,null,"1")),n("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},n(hl.i,null,"0")),n("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},n(hl.i,null,"1")),n("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},n(hl.i,null,"0")),n("td",{style:"background-color: rgba(255, 0, 0, 0.1);"},n(hl.i,null,"3"))),n("tr",null,n("td",null,n(hl.i,null,"2")),n("td",null,n(hl.i,null,"2")),n("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},n(hl.i,null,"0")),n("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},n(hl.i,null,"0")),n("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},n(hl.i,null,"1")),n("td",{style:"background-color: rgba(255, 0, 0, 0.1);"},n(hl.i,null,"7"))),n("tr",{style:"background-color: rgba(0, 255, 0, 0.1);"},n("td",null,n(hl.i,null,"2000")),n("td",null,n(hl.i,null,"1000")),n("td",null,n(hl.i,null,"0")),n("td",null,n(hl.i,null,"0")),n("td",null,n(hl.i,null,"0")),n("td",{style:"background-color: rgba(255, 255, 0, 0.2);"},n(hl.i,null,"0"))))),n("ul",null,n("li",null,n("u",{style:"color: #7dff7d;"},"Verde"),": funzione obiettivo"),n("li",null,n("u",{style:"color: #ffff7d;"},"Giallo"),": valore attuale della funzione"),n("li",null,n("u",{style:"color: #ff7d7d;"},"Rosso"),": termini noti"),n("li",null,n("u",{style:"color: #7d7dff;"},"Blu"),": Variabili slack")))),n(hl.p,{title:"Simplex primale"},n(hl.o,{title:"Cos'è?"},n("p",null,"Un algoritmo per trovare efficientemente il ",n("b",null,"valore ottimo")," e le coordinate di un ",n("b",null,"vertice ottimo")," in problemi di ottimizzazione lineare."),n(vl.a,null,"Ricordi ",n(hl.b,{href:"/vldigeometria"},"Gauss-Jordan"),"? 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",n(hl.i,null,Al(zn())),".")),n(hl.o,{title:"Procedimento"},n("ol",null,n("li",null,"Crea un nuovo tableau, ",n("b",null,"aggiungendo variabili artificiali")," in modo da avere una base ammissibile."),n("li",null,"Sostituisci la vecchia funzione obiettivo con una nuova che ",n("b",null,"minimizzi la somma")," di tutte le variabili artificiali."),n("li",null,n("u",null,"Fase 1"),": ",n("b",null,"Risolvi")," il nuovo problema con il Simplex primale."),n("li",null,"Se il Simplex termina quando ci sono ancora ",n("b",null,"variabili artificiali nella base"),", allora il poliedro è ",n("b",null,n(gl.a,null)),"."),n("li",null,"Una volta che le variabili artificiali sono fuori base, ",n("b",null,"elimina")," le loro colonne e la nuova funzione obiettivo.",n("br",null)),n("li",null,"Riporta il tableau in forma base compiendo operazioni per ",n("b",null,"azzerare i coefficienti")," delle variabili di base nella funzione obiettivo."),n("li",null,n("u",null,"Fase 2"),": ",n("b",null,"Risolvi")," il tableau con il Simplex primale.")))),n(hl.p,{title:"Rilassamento"},n(hl.o,{title:"Cos'è?"},n("p",null,"Una versione semplificata di un problema nella quale si ",n("b",null,"ignora la violazione")," di uno o più vincoli.")),n(hl.o,{title:"Rilassamento di Lagrange"},n("p",null,"Un rilassamento che permette di misurare ",n("b",null,"di quanto i vincoli vengono violati"),"."),n("p",null,"I vincoli, moltiplicati per ",n("b",null,"coefficienti di rilassamento"),", vengono inseriti nella funzione obiettivo."),n("p",null,"Il vettore dei coefficienti di rilassamento solitamente è indicato con ",n(hl.i,null,Al(_n())),"."),n(vl.a,null,n("p",null,"Il sistema:"),n(hl.i,{inline:!1},Al(gn())),n("p",null,"diventa:"),n(hl.i,{inline:!1},Al(vn()))))),n(hl.p,{title:"Dualità"},n(hl.o,{title:"Duale"},n("p",null,"Il sistema che ",n("b",null,n(xl.a,null,"massimizza"),"/",n(ql.a,null,"minimizza")," i moltiplicatori di rilassamento")," di un problema detto ",n("i",null,"primale"),".")),n(hl.o,{title:"In termini matriciali"},n("p",null,"Possiamo ",n("b",null,"trasporre")," il tableau e sostituire le variabili ",n(hl.i,null,Al(hn()))," con variabili ",n(hl.i,null,Al(pn()))," per ottenere il sistema duale!"),n("p",null,"I maggiori e minori dei vincoli diventeranno maggiori e minori delle variabili e viceversa.")),n(hl.o,{title:"Feasibility del duale"},n("ul",null,n("li",null,"Se un problema ha una ",n("b",null,"soluzione finita"),", allora anche il suo duale la avrà."),n("li",null,"Se un problema è ",n("b",null,n(gl.a,null)),", allora il suo duale potrà essere ",n(gl.a,null)," oppure ",n(_l.a,null),"."),n("li",null,"Se un problema è ",n("b",null,n(_l.a,null)),", allora il suo duale sarà certamente ",n(gl.a,null),"."))),n(hl.o,{title:"Variabili e vincoli"},n("p",null,"Variabili e vincoli del duale corrispondono rispettivamente a vincoli e variabili del primale."),n("p",null,"In particolare:"),n("table",null,n("thead",null,n("tr",null,n("th",null,n(xl.a,null,"Min")),n("th",null,n(ql.a,null,"Max")))),n("tbody",null,n("tr",null,n("td",null,"Vincolo ",n(hl.g,null,"\\leq")),n("td",null,"Variabile ",n(hl.g,null,"\\leq"))),n("tr",null,n("td",null,"Vincolo ",n(hl.g,null,"\\geq")),n("td",null,"Variabile ",n(hl.g,null,"\\geq"))),n("tr",null,n("td",null,"Vincolo ",n(hl.g,null,"=")),n("td",null,"Variabile ",n("b",null,"libera"))),n("tr",null,n("td",null,"Variabile ",n(hl.g,null,"\\leq")),n("td",null,"Vincolo ",n(hl.g,null,"\\geq"))),n("tr",null,n("td",null,"Variabile ",n(hl.g,null,"\\geq")),n("td",null,"Vincolo ",n(hl.g,null,"\\leq"))),n("tr",null,n("td",null,"Variabile ",n("b",null,"libera")),n("td",null,"Vincolo ",n(hl.g,null,"="))))))),n(hl.p,{title:"Un po' di teoria"},n(hl.o,{title:"Lemma di Farkas"},n("p",null,"Una disuguaglianza lineare ",n(hl.i,null,Al(mn()))," è verificata da tutti i punti di un poliedro non-",n(gl.a,null)," se e solo se esiste un vettore ",n(hl.i,null,Al(bn()))," tale che:"),n(hl.n,null,Al(dn())),n(hl.n,null,Al(sn()))),n(hl.o,{title:"Dualità forte"},n("p",null,"Il teorema che dimostra l'equivalenza tra primale e duale."),n("p",null,"Se uno dei due problemi è finito, la soluzione di uno coincide con la soluzione dell'altro."),n("p",null,n(hl.i,null,Al(fn()))),n("p",null,n(hl.s,null,"TODO: Anche qui c'è una lunga dimostrazione..."))),n(hl.o,{title:"Dualità debole"},n("p",null,"Il teorema che dimostra che il valore della funzione obiettivo del duale (di un qualsiasi tableau) è sempre ",n(xl.a,null,"minore o uguale"),"/",n(ql.a,null,"maggiore o uguale")," alla soluzione del corrispettivo primale."),n("p",null,n(hl.s,null,"TODO: Dimostrazione cortina, ma sembra complicata."))),n(hl.o,{title:"Condizioni di ottimalità"},n("p",null,"Il teorema che ci permette di passare dalla soluzione del duale alla soluzione del primale. ",n(hl.s,null,"TODO: credo?")),n("p",null,"Si deriva combinando le seguenti condizioni:"),n("ul",null,n("li",null,"Ammissibilità del primale: ",n(hl.i,null,Al(cn()))),n("li",null,"Ammissibilità del duale: ",n(hl.i,null,Al(rn()))),n("li",null,"Teorema della dualità forte: ",n(hl.i,null,Al(on()))," (alla soluzione ottima)")),n("p",null,"Ne risulta che una soluzione è ottima se e solo se:"),n(hl.n,null,Al(un())),n(hl.n,null,Al(an())))),n(hl.p,{title:"Simplex duale"},n(hl.o,{title:"Cos'è?"},n("p",null,"Un'estensione al Simplex primale che opera sul problema duale.")),n(hl.o,{title:"Come funziona?"},n("p",null,"Funziona esattamente come il Simplex primale, ma opera sul duale."))),n(hl.p,{title:"Analisi di sensibilità"},n(hl.o,{title:"Cos'è?"},n("p",null,"Un procedimento che misura di ",n("b",null,"quanto può variare")," il termine noto di un vincolo ",n(hl.i,null,Al(tn()))," o il coefficiente della funzione obiettivo ",n(hl.i,null,Al(en()))," prima che la base degeneri."))),n(hl.p,{title:"Ottimizzazione lineare intera"},n(hl.o,{title:"Cos'è?"},n("p",null,"Particolari problemi di ottimizzazione lineare in cui le ",n("b",null,"variabili sono vincolate ad essere numeri interi"),"."),n(hl.n,null,Al(ln())),n("p",null,"Spesso detti anche ",n("i",null,"problemi di ",n("abbr",{title:"Integer Linear Programming"},"ILP")),".")),n(hl.o,{title:"Rilassamento lineare"},n("p",null,"Un rilassamento che rimuove il ",n("b",null,"vincolo di integrità")," a un problema, trovando la sua ",n("b",null,"soluzione continua"),"."))),n(hl.p,{title:"Dal rilassamento alla soluzione"},n(hl.o,{title:"Enumerazione totale"},n("p",null,"Un ",n("b",null,"modo")," per passare dalla soluzione del rilassamento alla soluzione intera di un problema di ILP."),n("p",null,"Consiste nel calcolare la soluzione di ogni singolo punto incluso nel poliedro, e selezionare la ",n(xl.a,null,"minore"),"/",n(ql.a,null,"maggiore"),"."),n("p",null,"Trova ",n("b",null,"sicuramente")," la soluzione giusta, ma il costo computazionale è esponenziale ",n(hl.g,null,"O(n^k)"),"!")),n(hl.o,{title:"Arrotondamento"},n("p",null,"Un altro ",n("b",null,"modo")," per passare dalla soluzione del rilassamento alla soluzione intera di un problema di ILP."),n("p",null,"Consiste nell'",n("b",null,"arrotondare tutte le variabili al loro valore intero più vicino"),", e calcolarne il valore ottimo."),n("p",null,"Funziona bene per valori grandi, ma più essi si avvicinano allo 0 più l'",n("b",null,"errore diventa grande"),".")),n(hl.o,{title:"Piani secanti"},n("p",null,"Un altro ",n("b",null,"modo")," ancora per passare dalla soluzione del rilassamento alla soluzione intera di un problema di ILP."),n("p",null,"Consiste nel tagliare il poliedro con nuovi vincoli (",n("i",null,"piani secanti"),") che ",n("b",null,"riducono le possibili soluzioni continue")," ma non quelle intere."),n("p",null,"Per selezionare i vincoli, si usano i ",n("b",null,"tagli di Gomory"),":"),n(hl.n,null,Al(nn())),n("p",null,"Per ogni valore noto frazionario si viene quindi a creare ",n("b",null,"una nuova variabile in base")," e un nuovo vincolo formato dall'opposto di tutti i valori frazionari dei coefficienti fuori base."),n(vl.a,null,n("p",null,"Il tableau:",n("table",null,n("thead",null,n("tr",null,n("th",null,n(hl.a,null,Al($()))),n("th",null,n(hl.a,null,Al(K()))),n("th",null,n(hl.a,null,Al(Y()))),n("th",null,n(hl.a,null,Al(Q()))),n("th",null,n("abbr",{title:"Termine 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esplorazione")," dell'albero:",n("ul",null,n("li",null,n("b",null,"depth-first"),": permette di raggiungere immediatamente a una soluzione accettabile (ma non ottimale)"),n("li",null,n("b",null,"best-first"),": permette di raggiungere più velocemente alla soluzione corretta")))),n(hl.o,{title:"Seca et impera"},n("p",null,"È possibile combinare il metodo dei ",n("b",null,"tagli secanti")," con la tecnica ",n("b",null,"divide et impera")," per raggiungere ancora più velocemente a una soluzione."),n("p",null,"Si effettuano ",n("b",null,"poche iterazioni")," del metodo dei tagli secanti, e sul risultato di quelle iterazioni si applica il ",n("b",null,"divide et impera"),"."))),n(hl.p,{title:n("span",null,"Terminologia dei grafi ",n(hl.s,null,"TODO: migliorare"))},n(hl.o,{title:"Grafo"},n("p",null,"Insieme di ",n("b",null,"nodi")," ",n(hl.g,null,Al(a()))," e ",n("b",null,"archi")," ",n(hl.g,null,Al(t()))," che li connettono."),n("p",null,"Può essere ",n("b",null,"diretto")," se gli archi hanno una direzione.")),n(hl.o,{title:"Nodi adiacenti"},n("p",null,"Nodi ",n("b",null,"connessi da un arco"),".")),n(hl.o,{title:"Arco incidente"},n("p",null,"Arco ",n("b",null,"connesso a un dato nodo"),".")),n(hl.o,{title:"Arco entrante o uscente"},n("p",null,"Un arco diretto che ",n("b",null,"termina")," o ",n("b",null,"inizia")," da un dato nodo.")),n(hl.o,{title:"Grado"},n("p",null,n("b",null,"Conteggio")," degli archi incidenti di un nodo."),n("p",null,"Si può calcolare anche relativamente agli archi entranti o agli archi uscenti.")),n(hl.o,{title:"Percorso"},n("p",null,"Sequenza di ",n("b",null,"archi consecutivi"),".")),n(hl.o,{title:"Connessione"},n("p",null,"Due nodi sono connessi se tra loro esiste ",n("b",null,"almeno un percorso"),"."),n("p",null,"Un grafo è connesso se tutti i suoi nodi sono connessi.")),n(hl.o,{title:"Cicli e circuiti"},n("p",null,"Percorsi rispettivamente indiretti e diretti in cui l'inizio coincide con la fine.")),n(hl.o,{title:"Grafo completo"},n("p",null,"Grafo in cui ogni nodo è connesso con ogni altro."),n("p",null,"Se diretto, contiene ",n(hl.g,null,Al(e()))," archi; altrimenti, ne contiene la metà.")),n(hl.o,{title:"Matrice di adiacenza"},n("blockquote",null,"Vedi ",n(hl.b,{href:"/algoritmiestrutturedati"},"Algoritmi"),".")),n(hl.o,{title:"Lista di adiacenza"},n("blockquote",null,"Vedi ",n(hl.b,{href:"/algoritmiestrutturedati"},"Algoritmi"),".")),n(hl.o,{title:"Taglio"},n("p",null,"Sottoinsieme di archi che connettono due sottoinsiemi di nodi."),n("p",null,"Può essere anche uscente o entrante; in tal caso include solo gli archi entranti o uscenti dal sottoinsieme.")),n(hl.o,{title:"Sottografo"},n("p",null,"Sottoinsieme di nodi e archi di un grafo."),n("p",null,"Tutti gli archi di un sottografo possono connettere solo nodi all'interno di esso.")),n(hl.o,{title:"Albero"},n("p",null,"Sottografo connesso e aciclico.")),n(hl.o,{title:"Spanning tree"},n("p",null,"Albero che include tutti i nodi di un grafo."))),n(hl.p,{title:"Algoritmi con i grafi"},n(hl.o,{title:"Prim"},n("p",null,"Crea uno spanning tree."),n("ol",null,n("li",null,"Aggiungi l'arco di costo minimo all'albero."),n("li",null,"Finchè mancano ancora archi:",n("ol",null,n("li",null,"Trova tutti gli archi che aggiungerebbero un nuovo nodo all'albero."),n("li",null,"Seleziona l'arco di costo minore."))))),n(hl.o,{title:"Ordine topologico"},n("p",null,"Trova l'ordine topologico di un albero."),n("ol",null,n("li",null,"Ripeti finchè ci sono nodi nel grafo:",n("ol",null,n("li",null,"Assegna un numero sequenziale a un nodo senza archi entranti."),n("li",null,"Elimina il nodo a cui hai assegnato il numero."),n("li",null,"Elimina tutti gli archi incidenti sul nodo che hai eliminato."))))),n(hl.o,{title:"Percorsi minimi in grafo diretto"},n("p",null,"Trova i percorsi di costo minimo in un albero."),n("ol",null,n("li",null,"Trova l'ordine topologico dell'albero."),n("li",null,"Invece che provare ogni singola combinazione di nodi, 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x")),n("tr",null,n("td",null,n(hl.j,null,yl(En()))),n("td",null,"Arrotondamento per eccesso di x")),n("tr",null,n("td",null,n(hl.j,null,yl(Gn()))),n("td",null,"Parte frazionaria di x (se non è negativo)")))))),n(hl.q,{title:"Problemi di ottimizzazione lineare"},n(hl.p,{title:"Cosa sono?"},n("p",null,"Problemi che cercano di ",n(xl.a,null,"minimizzare"),"/",n(ql.a,null,"massimizzare")," il valore di una ",n("i",null,"funzione obiettivo")," le cui incognite sono sottoposte a un ",n("b",null,"sistema di ",n("i",null,"vincoli")),"."),n("p",null,"Spesso sono detti anche ",n("i",null,"problemi di ",n("abbr",{title:"Linear Programming"},"LP")),".")),n(hl.p,{title:"Funzione obiettivo"},n("p",null,"La funzione da ",n(xl.a,null,"minimizzare"),"/",n(ql.a,null,"massimizzare"),"."),n("p",null,"Il vettore dei suoi coefficienti è detto ",n(hl.j,null,yl(Rn())),", mentre quello delle sue incognite ",n(hl.j,null,yl(Dn())),"."),n("p",null,"In genere, la funzione obiettivo è scritta in questa 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",n("i",null,n(_l.a,null)),"."),n(vl.a,null,"Si chiama così perchè se si disegna su un piano cartesiano, esso forma una figura geometrica a più lati, ovvero un ",n("a",{href:"https://it.wikipedia.org/wiki/Poliedro"},"poliedro"),".")),n(hl.p,{title:"Valore ottimo"},n("p",null,"La ",n("b",null,"soluzione")," di un problema, ricavabile dal prodotto ",n(hl.j,null,yl(Ln())),"."),n("p",null,"In particolare, il valore ottimo è un ",n("b",null,"vertice")," del poliedro, detto ",n("i",null,"vertice ottimo"),"."))),n(hl.q,{title:"Forme di un problema di ottimizzazione"},n(hl.p,{title:"Forma generale"},n("p",null,"Un problema con:"),n("ul",null,n("li",null,n(jl.a,null,"Equazioni e disequazioni")),n("li",null,n(jl.a,null,"Variabili non vincolate"))),n(hl.o,null,yl(Pn()))),n(hl.p,{title:"Forma canonica"},n("p",null,"Un problema con:"),n("ul",null,n("li",null,n(jl.a,null,"Solo disequazioni")),n("li",null,n(Tl.a,null,"Vincoli di non-negatività sulle incognite"))),n(hl.o,null,yl(kn()))),n(hl.p,{title:"Forma standard"},n("p",null,"Un problema con:"),n("ul",null,n("li",null,n(Tl.a,null,"Solo equazioni")),n("li",null,n(Tl.a,null,"Vincoli di non-negatività sulle incognite"))),n(hl.o,null,yl(In())))),n(hl.q,{title:"Conversioni tra le forme"},n(hl.p,{title:"Standard e generale"},n("p",null,"Applica questa conversione a ogni equazione nel sistema:"),n("p",null,n(hl.j,{inline:!1},yl(An()))),n(vl.a,null,"Serve solo nella teoria per dimostrare che le forme sono equivalenti.")),n(hl.p,{title:"Canonica e standard"},n("p",null,"Aggiungi una ",n("i",null,"variabile slack")," ",n(hl.j,null,yl(yn()))," ",n("b",null,"non-vincolata")," a ogni disequazione nel sistema:"),n("p",null,n(hl.j,{inline:!1},yl(Sn()))),n("p",null,n(hl.j,{inline:!1},yl(Tn())))),n(hl.p,{title:"Generale e canonica"},n("p",null,"Sdoppia ogni variabile non-vincolata in due variabili con vincolo di non-negatività:"),n("p",null,n(hl.j,{inline:!1},yl(jn()))))),n(hl.q,null,"Valore attuale"),n(hl.q,{title:"Tableau"},n(hl.p,{title:"Cos'è?"},n("p",null,"Un modo per rappresentare sistemi in forma standard, anche noto come ",n("b",null,"matrice equivalente completa")," del sistema.")),n(hl.p,{title:"Trasformazioni"},n("p",null,"Un tableau è un sistema di equazioni in ",n("b",null,"forma matriciale completa"),"."),n("p",null,"È possibile effettuare senza che cambi il risultato finale le seguenti trasformazioni:"),n("ul",null,n("li",null,n("b",null,"Moltiplicare")," un'intera riga per una costante."),n("li",null,n("b",null,"Sommare")," una riga a un'altra")),n(vl.a,null,"Suona familiare? Sì, lo abbiamo fatto anche in Algebra Lineare.")),n(hl.p,{title:"Variabili nella base"},n("p",null,"Variabili che hanno ",n("b",null,"tutti 0 e un solo 1")," nella loro colonna del tableau."),n("p",null,"La loro controparte sono le ",n("i",null,"variabili fuori base"),", che hanno qualsiasi altro valore.")),n(hl.p,{title:"Valore attuale"},n("p",null,"Il valore della funzione obiettivo che si otterrebbe se ",n("b",null,"tutte le variabili fuori base valessero 0"),"."),n("p",null,"Procedendo nella risoluzione (descritta in seguito) del tableau, questo valore aumenterà, fino a raggiungere il valore ottimo quando la risoluzione sarà completata."))),n(hl.q,null,n(hl.p,{color:Sl.a,title:"Un esempio"},n("p",null,"Il sistema:"),n(hl.o,null,yl(qn())),n("p",null,"Diventa il tableau:"),n("table",{className:"right"},n("thead",null,n("tr",null,n("th",null,n(hl.j,null,"x_1")),n("th",null,n(hl.j,null,"x_2")),n("th",null,n(hl.j,null,"s_1")),n("th",null,n(hl.j,null,"s_2")),n("th",null,n(hl.j,null,"s_3")),n("th",null,n("abbr",{title:"Termine noto"},"TN")))),n("tbody",null,n("tr",null,n("td",null,n(hl.j,null,"1")),n("td",null,n(hl.j,null,"0")),n("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},n(hl.j,null,"1")),n("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},n(hl.j,null,"0")),n("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},n(hl.j,null,"0")),n("td",{style:"background-color: rgba(255, 0, 0, 0.1);"},n(hl.j,null,"3"))),n("tr",null,n("td",null,n(hl.j,null,"0")),n("td",null,n(hl.j,null,"1")),n("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},n(hl.j,null,"0")),n("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},n(hl.j,null,"1")),n("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},n(hl.j,null,"0")),n("td",{style:"background-color: rgba(255, 0, 0, 0.1);"},n(hl.j,null,"3"))),n("tr",null,n("td",null,n(hl.j,null,"2")),n("td",null,n(hl.j,null,"2")),n("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},n(hl.j,null,"0")),n("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},n(hl.j,null,"0")),n("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},n(hl.j,null,"1")),n("td",{style:"background-color: rgba(255, 0, 0, 0.1);"},n(hl.j,null,"7"))),n("tr",{style:"background-color: rgba(0, 255, 0, 0.1);"},n("td",null,n(hl.j,null,"2000")),n("td",null,n(hl.j,null,"1000")),n("td",null,n(hl.j,null,"0")),n("td",null,n(hl.j,null,"0")),n("td",null,n(hl.j,null,"0")),n("td",{style:"background-color: rgba(255, 255, 0, 0.2);"},n(hl.j,null,"0"))))),n("ul",null,n("li",null,n("u",{style:"color: #7dff7d;"},"Verde"),": funzione obiettivo"),n("li",null,n("u",{style:"color: #ffff7d;"},"Giallo"),": valore attuale della funzione"),n("li",null,n("u",{style:"color: #ff7d7d;"},"Rosso"),": termini noti"),n("li",null,n("u",{style:"color: #7d7dff;"},"Blu"),": Variabili slack")))),n(hl.q,{title:"Simplex primale"},n(hl.p,{title:"Cos'è?"},n("p",null,"Un algoritmo per trovare efficientemente il ",n("b",null,"valore ottimo")," e le coordinate di un ",n("b",null,"vertice ottimo")," in problemi di ottimizzazione lineare."),n(vl.a,null,"Ricordi ",n(hl.b,{href:"/vldigeometria"},"Gauss-Jordan"),"? Il Simplex è la stessa cosa, aggiungendo criteri per la selezione del pivot."),n(vl.a,{title:"Esempio"},n("a",{href:"https://i.imgur.com/1r405Mb.jpg"},"Questa")," è la soluzione passo per passo del problema 3 del file ",n("a",{href:"https://dolly.fim.unimore.it/2019/mod/resource/view.php?id=2716"},n("code",null,"Ex_LP_testo")),"."),n("p",null,"Perchè sia possibile effettuare il Simplex è necessario che l'",n("b",null,"origine sia nel poliedro"),": pertanto, ",n("b",null,"non")," è possibile che un problema risolto con il Simplex sia ",n(gl.a,null),".")),n(hl.p,{title:"I passi"},n("ol",null,n("li",null,"Trasforma il sistema in ",n("b",null,"forma standard"),"."),n("li",null,"Trova tante variabili ",n("b",null,"linearmente indipendenti")," quante siano le righe: esse saranno la ",n("i",null,"base iniziale"),"."),n("li",null,"Finchè ci sono variabili con coefficienti ",n(xl.a,null,"positivi"),"/",n(ql.a,null,"negativi")," nella funzione obiettivo:",n("ol",null,n("li",null,n("b",null,"Scegli")," la prima variabile fuori base con coefficiente ",n(xl.a,null,"positivo"),"/",n(ql.a,null,"negativo")," nella funzione obiettivo: essa è la ",n("i",null,"variabile entrante"),".",n("br",null),n("aside",null,n("u",null,"Regola di Bland"),": Si potrebbe scegliere qualsiasi variabile come entrante, ma scegliendo sempre la prima ammissibile ci si assicura che l'algoritmo termini.")),n("li",null,n("b",null,"Scegli")," la variabile in base con il minor rapporto positivo ",n(hl.j,null,yl(xn())),".",n("aside",null,"Se non sei riuscito a trovare nessuna variabile con un rapporto positivo, significa che il poliedro è ",n(_l.a,null),".")),n("li",null,n("u",null,"Pivot"),": ",n("b",null,"trasforma")," tutte le funzioni del sistema in modo che abbiano 0 nella colonna della variabile entrante, tranne nella riga della variabile uscente, in cui avrà 1."))),n("li",null,"Il poliedro è ",n(zl.a,null),": i ",n("b",null,"termini noti dei vincoli")," sono le coordinate del suo vertice ottimo, mentre il ",n("b",null,"termine noto della funzione obiettivo")," è il valore ottimo."))),n(hl.p,{title:"Soluzioni di base degenerata"},n("p",null,"Una soluzione con almeno una variabile di valore ",n(hl.j,null,"0"),", dovuta a uno o più ",n("b",null,"vincoli ridondanti"),"."),n("p",null,"Senza ",n("b",null,"Regola di Bland")," e in presenza di vincoli ridondanti si rischia di trovarsi a fare pivot infiniti."))),n(hl.q,{title:"Metodo delle due fasi"},n(hl.p,{title:"Metodo delle due fasi"},n("p",null,"Un ",n("b",null,"estensione del Simplex")," per permettere la risoluzione di problemi la cui ",n("b",null,"origine non è una soluzione ammissibile"),"."),n("p",null,"Prevede l'introduzione di un ",n("i",null,"problema ausiliario"),", le cui incognite sono dette ",n("i",null,"artificiali"),"."),n("p",null,"Il vettore delle incognite artificiali è solitamente chiamato ",n(hl.j,null,yl(zn())),".")),n(hl.p,{title:"Procedimento"},n("ol",null,n("li",null,"Crea un nuovo tableau, ",n("b",null,"aggiungendo variabili artificiali")," in modo da avere una base ammissibile."),n("li",null,"Sostituisci la vecchia funzione obiettivo con una nuova che ",n("b",null,"minimizzi la somma")," di tutte le variabili artificiali."),n("li",null,n("u",null,"Fase 1"),": ",n("b",null,"Risolvi")," il nuovo problema con il Simplex primale."),n("li",null,"Se il Simplex termina quando ci sono ancora ",n("b",null,"variabili artificiali nella base"),", allora il poliedro è ",n("b",null,n(gl.a,null)),"."),n("li",null,"Una volta che le variabili artificiali sono fuori base, ",n("b",null,"elimina")," le loro colonne e la nuova funzione obiettivo.",n("br",null)),n("li",null,"Riporta il tableau in forma base compiendo operazioni per ",n("b",null,"azzerare i coefficienti")," delle variabili di base nella funzione obiettivo."),n("li",null,n("u",null,"Fase 2"),": ",n("b",null,"Risolvi")," il tableau con il Simplex primale.")))),n(hl.q,{title:"Rilassamento"},n(hl.p,{title:"Cos'è?"},n("p",null,"Una versione semplificata di un problema nella quale si ",n("b",null,"ignora la violazione")," di uno o più vincoli.")),n(hl.p,{title:"Rilassamento di Lagrange"},n("p",null,"Un rilassamento che permette di misurare ",n("b",null,"di quanto i vincoli vengono violati"),"."),n("p",null,"I vincoli, moltiplicati per ",n("b",null,"coefficienti di rilassamento"),", vengono inseriti nella funzione obiettivo."),n("p",null,"Il vettore dei coefficienti di rilassamento solitamente è indicato con ",n(hl.j,null,yl(_n())),"."),n(vl.a,null,n("p",null,"Il sistema:"),n(hl.j,{inline:!1},yl(gn())),n("p",null,"diventa:"),n(hl.j,{inline:!1},yl(vn()))))),n(hl.q,{title:"Dualità"},n(hl.p,{title:"Duale"},n("p",null,"Il sistema che ",n("b",null,n(xl.a,null,"massimizza"),"/",n(ql.a,null,"minimizza")," i moltiplicatori di rilassamento")," di un problema detto ",n("i",null,"primale"),".")),n(hl.p,{title:"In termini matriciali"},n("p",null,"Possiamo ",n("b",null,"trasporre")," il tableau e sostituire le variabili ",n(hl.j,null,yl(hn()))," con variabili ",n(hl.j,null,yl(pn()))," per ottenere il sistema duale!"),n("p",null,"I maggiori e minori dei vincoli diventeranno maggiori e minori delle variabili e viceversa.")),n(hl.p,{title:"Feasibility del duale"},n("ul",null,n("li",null,"Se un problema ha una ",n("b",null,"soluzione finita"),", allora anche il suo duale la avrà."),n("li",null,"Se un problema è ",n("b",null,n(gl.a,null)),", allora il suo duale potrà essere ",n(gl.a,null)," oppure ",n(_l.a,null),"."),n("li",null,"Se un problema è ",n("b",null,n(_l.a,null)),", allora il suo duale sarà certamente ",n(gl.a,null),"."))),n(hl.p,{title:"Variabili e vincoli"},n("p",null,"Variabili e vincoli del duale corrispondono rispettivamente a vincoli e variabili del primale."),n("p",null,"In particolare:"),n("table",null,n("thead",null,n("tr",null,n("th",null,n(xl.a,null,"Min")),n("th",null,n(ql.a,null,"Max")))),n("tbody",null,n("tr",null,n("td",null,"Vincolo ",n(hl.h,null,"\\leq")),n("td",null,"Variabile ",n(hl.h,null,"\\leq"))),n("tr",null,n("td",null,"Vincolo ",n(hl.h,null,"\\geq")),n("td",null,"Variabile ",n(hl.h,null,"\\geq"))),n("tr",null,n("td",null,"Vincolo ",n(hl.h,null,"=")),n("td",null,"Variabile ",n("b",null,"libera"))),n("tr",null,n("td",null,"Variabile ",n(hl.h,null,"\\leq")),n("td",null,"Vincolo ",n(hl.h,null,"\\geq"))),n("tr",null,n("td",null,"Variabile ",n(hl.h,null,"\\geq")),n("td",null,"Vincolo ",n(hl.h,null,"\\leq"))),n("tr",null,n("td",null,"Variabile ",n("b",null,"libera")),n("td",null,"Vincolo ",n(hl.h,null,"="))))))),n(hl.q,{title:"Un po' di teoria"},n(hl.p,{title:"Lemma di Farkas"},n("p",null,"Una disuguaglianza lineare ",n(hl.j,null,yl(mn()))," è verificata da tutti i punti di un poliedro non-",n(gl.a,null)," se e solo se esiste un vettore ",n(hl.j,null,yl(bn()))," tale che:"),n(hl.o,null,yl(dn())),n(hl.o,null,yl(sn()))),n(hl.p,{title:"Dualità forte"},n("p",null,"Il teorema che dimostra l'equivalenza tra primale e duale."),n("p",null,"Se uno dei due problemi è finito, la soluzione di uno coincide con la soluzione dell'altro."),n("p",null,n(hl.j,null,yl(fn()))),n("p",null,n(hl.t,null,"TODO: Anche qui c'è una lunga dimostrazione..."))),n(hl.p,{title:"Dualità debole"},n("p",null,"Il teorema che dimostra che il valore della funzione obiettivo del duale (di un qualsiasi tableau) è sempre ",n(xl.a,null,"minore o uguale"),"/",n(ql.a,null,"maggiore o uguale")," alla soluzione del corrispettivo primale."),n("p",null,n(hl.t,null,"TODO: Dimostrazione cortina, ma sembra complicata."))),n(hl.p,{title:"Condizioni di ottimalità"},n("p",null,"Il teorema che ci permette di passare dalla soluzione del duale alla soluzione del primale. ",n(hl.t,null,"TODO: credo?")),n("p",null,"Si deriva combinando le seguenti condizioni:"),n("ul",null,n("li",null,"Ammissibilità del primale: ",n(hl.j,null,yl(cn()))),n("li",null,"Ammissibilità del duale: ",n(hl.j,null,yl(rn()))),n("li",null,"Teorema della dualità forte: ",n(hl.j,null,yl(on()))," (alla soluzione ottima)")),n("p",null,"Ne risulta che una soluzione è ottima se e solo se:"),n(hl.o,null,yl(un())),n(hl.o,null,yl(an())))),n(hl.q,{title:"Simplex duale"},n(hl.p,{title:"Cos'è?"},n("p",null,"Un'estensione al Simplex primale che opera sul problema duale.")),n(hl.p,{title:"Come funziona?"},n("p",null,"Funziona esattamente come il Simplex primale, ma opera sul duale."))),n(hl.q,{title:"Analisi di sensibilità"},n(hl.p,{title:"Cos'è?"},n("p",null,"Un procedimento che misura di ",n("b",null,"quanto può variare")," il termine noto di un vincolo ",n(hl.j,null,yl(tn()))," o il coefficiente della funzione obiettivo ",n(hl.j,null,yl(en()))," prima che la base degeneri."))),n(hl.q,{title:"Ottimizzazione lineare intera"},n(hl.p,{title:"Cos'è?"},n("p",null,"Particolari problemi di ottimizzazione lineare in cui le ",n("b",null,"variabili sono vincolate ad essere numeri interi"),"."),n(hl.o,null,yl(ln())),n("p",null,"Spesso detti anche ",n("i",null,"problemi di ",n("abbr",{title:"Integer Linear Programming"},"ILP")),".")),n(hl.p,{title:"Rilassamento lineare"},n("p",null,"Un rilassamento che rimuove il ",n("b",null,"vincolo di integrità")," a un problema, trovando la sua ",n("b",null,"soluzione continua"),"."))),n(hl.q,{title:"Dal rilassamento alla soluzione"},n(hl.p,{title:"Enumerazione totale"},n("p",null,"Un ",n("b",null,"modo")," per passare dalla soluzione del rilassamento alla soluzione intera di un problema di ILP."),n("p",null,"Consiste nel calcolare la soluzione di ogni singolo punto incluso nel poliedro, e selezionare la ",n(xl.a,null,"minore"),"/",n(ql.a,null,"maggiore"),"."),n("p",null,"Trova ",n("b",null,"sicuramente")," la soluzione giusta, ma il costo computazionale è esponenziale ",n(hl.h,null,"O(n^k)"),"!")),n(hl.p,{title:"Arrotondamento"},n("p",null,"Un altro ",n("b",null,"modo")," per passare dalla soluzione del rilassamento alla soluzione intera di un problema di ILP."),n("p",null,"Consiste nell'",n("b",null,"arrotondare tutte le variabili al loro valore intero più vicino"),", e calcolarne il valore ottimo."),n("p",null,"Funziona bene per valori grandi, ma più essi si avvicinano allo 0 più l'",n("b",null,"errore diventa grande"),".")),n(hl.p,{title:"Piani secanti"},n("p",null,"Un altro ",n("b",null,"modo")," ancora per passare dalla soluzione del rilassamento alla soluzione intera di un problema di ILP."),n("p",null,"Consiste nel tagliare il poliedro con nuovi vincoli (",n("i",null,"piani secanti"),") che ",n("b",null,"riducono le possibili soluzioni continue")," ma non quelle intere."),n("p",null,"Per selezionare i vincoli, si usano i ",n("b",null,"tagli di Gomory"),":"),n(hl.o,null,yl(nn())),n("p",null,"Per ogni valore noto frazionario si viene quindi a creare ",n("b",null,"una nuova variabile in base")," e un nuovo vincolo formato dall'opposto di tutti i valori frazionari dei coefficienti fuori base."),n(vl.a,null,n("p",null,"Il tableau:",n("table",null,n("thead",null,n("tr",null,n("th",null,n(hl.a,null,yl($()))),n("th",null,n(hl.a,null,yl(K()))),n("th",null,n(hl.a,null,yl(Y()))),n("th",null,n(hl.a,null,yl(Q()))),n("th",null,n("abbr",{title:"Termine 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esplorazione")," dell'albero:",n("ul",null,n("li",null,n("b",null,"depth-first"),": permette di raggiungere immediatamente a una soluzione accettabile (ma non ottimale)"),n("li",null,n("b",null,"best-first"),": permette di raggiungere più velocemente alla soluzione corretta")))),n(hl.p,{title:"Seca et impera"},n("p",null,"È possibile combinare il metodo dei ",n("b",null,"tagli secanti")," con la tecnica ",n("b",null,"divide et impera")," per raggiungere ancora più velocemente a una soluzione."),n("p",null,"Si effettuano ",n("b",null,"poche iterazioni")," del metodo dei tagli secanti, e sul risultato di quelle iterazioni si applica il ",n("b",null,"divide et impera"),"."))),n(hl.q,{title:n("span",null,"Terminologia dei grafi ",n(hl.t,null,"TODO: migliorare"))},n(hl.p,{title:"Grafo"},n("p",null,"Insieme di ",n("b",null,"nodi")," ",n(hl.h,null,yl(a()))," e ",n("b",null,"archi")," ",n(hl.h,null,yl(t()))," che li connettono."),n("p",null,"Può essere ",n("b",null,"diretto")," se gli archi hanno una direzione.")),n(hl.p,{title:"Nodi adiacenti"},n("p",null,"Nodi ",n("b",null,"connessi da un arco"),".")),n(hl.p,{title:"Arco incidente"},n("p",null,"Arco ",n("b",null,"connesso a un dato nodo"),".")),n(hl.p,{title:"Arco entrante o uscente"},n("p",null,"Un arco diretto che ",n("b",null,"termina")," o ",n("b",null,"inizia")," da un dato nodo.")),n(hl.p,{title:"Grado"},n("p",null,n("b",null,"Conteggio")," degli archi incidenti di un nodo."),n("p",null,"Si può calcolare anche relativamente agli archi entranti o agli archi uscenti.")),n(hl.p,{title:"Percorso"},n("p",null,"Sequenza di ",n("b",null,"archi consecutivi"),".")),n(hl.p,{title:"Connessione"},n("p",null,"Due nodi sono connessi se tra loro esiste ",n("b",null,"almeno un percorso"),"."),n("p",null,"Un grafo è connesso se tutti i suoi nodi sono connessi.")),n(hl.p,{title:"Cicli e circuiti"},n("p",null,"Percorsi rispettivamente indiretti e diretti in cui l'inizio coincide con la fine.")),n(hl.p,{title:"Grafo completo"},n("p",null,"Grafo in cui ogni nodo è connesso con ogni altro."),n("p",null,"Se diretto, contiene ",n(hl.h,null,yl(e()))," archi; altrimenti, ne contiene la metà.")),n(hl.p,{title:"Matrice di adiacenza"},n("blockquote",null,"Vedi ",n(hl.b,{href:"/algoritmiestrutturedati"},"Algoritmi"),".")),n(hl.p,{title:"Lista di adiacenza"},n("blockquote",null,"Vedi ",n(hl.b,{href:"/algoritmiestrutturedati"},"Algoritmi"),".")),n(hl.p,{title:"Taglio"},n("p",null,"Sottoinsieme di archi che connettono due sottoinsiemi di nodi."),n("p",null,"Può essere anche uscente o entrante; in tal caso include solo gli archi entranti o uscenti dal sottoinsieme.")),n(hl.p,{title:"Sottografo"},n("p",null,"Sottoinsieme di nodi e archi di un grafo."),n("p",null,"Tutti gli archi di un sottografo possono connettere solo nodi all'interno di esso.")),n(hl.p,{title:"Albero"},n("p",null,"Sottografo connesso e aciclico.")),n(hl.p,{title:"Spanning tree"},n("p",null,"Albero che include tutti i nodi di un grafo."))),n(hl.q,{title:"Algoritmi con i grafi"},n(hl.p,{title:"Prim"},n("p",null,"Crea uno spanning tree."),n("ol",null,n("li",null,"Aggiungi l'arco di costo minimo all'albero."),n("li",null,"Finchè mancano ancora archi:",n("ol",null,n("li",null,"Trova tutti gli archi che aggiungerebbero un nuovo nodo all'albero."),n("li",null,"Seleziona l'arco di costo minore."))))),n(hl.p,{title:"Ordine topologico"},n("p",null,"Trova l'ordine topologico di un albero."),n("ol",null,n("li",null,"Ripeti finchè ci sono nodi nel grafo:",n("ol",null,n("li",null,"Assegna un numero sequenziale a un nodo senza archi entranti."),n("li",null,"Elimina il nodo a cui hai assegnato il numero."),n("li",null,"Elimina tutti gli archi incidenti sul nodo che hai eliminato."))))),n(hl.p,{title:"Percorsi minimi in grafo diretto"},n("p",null,"Trova i percorsi di costo minimo in un albero."),n("ol",null,n("li",null,"Trova l'ordine topologico dell'albero."),n("li",null,"Invece che provare ogni singola combinazione di nodi, prova solo i nodi che hanno un numero topologico maggiore di quello del nodo attuale.")),n("p",null,n(hl.t,null,"TODO: forse spiegarlo meglio non farebbe male"))),n(hl.p,{title:"Algoritmo di Dijkstra"},n("blockquote",null,"Vedi ",n(hl.b,{href:"/algoritmiestrutturedati"},"Algoritmi"),".")),n(hl.p,{title:"Algoritmo di Ford-Fulkerson"},n(vl.a,null,"Trova il volume massimo di acqua che è possibile fare scorrere attraverso tubature con una data capacità."),n("p",null,"Costruisci il grafo residuo e vedi se c'è un percorso che va dalla sorgente alla destinazione."),n("p",null,n(hl.i,{src:"https://i.imgur.com/FJk44q0.png"})),n("p",null,n(hl.i,{src:"https://i.imgur.com/fzb6xz2.png"})))))}}.call(this,i("hosL").h)},jZax:function(n){n.exports={red:"red__ipSoq",orange:"orange__1nmkj",yellow:"yellow__353ge",lime:"lime__3FW1E",cyan:"cyan__2xhu7",blue:"blue__3MvDe",magenta:"magenta__i03ie",plus:"plus__1eV4_"}},ke5e:function(n,l,i){"use strict";(function(n){var 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extracted by mini-css-extract-plugin\nmodule.exports = {\"red\":\"red__2y1B_\",\"orange\":\"orange__dD2kx\",\"yellow\":\"yellow__OEpwl\",\"lime\":\"lime__CVe41\",\"cyan\":\"cyan__26ZAg\",\"blue\":\"blue__LO7Xm\",\"magenta\":\"magenta__1Akee\",\"example\":\"example__2PzAa\"};","// extracted by mini-css-extract-plugin\nmodule.exports = {\"red\":\"red__UmOgB\",\"orange\":\"orange__1FTS8\",\"yellow\":\"yellow__3dJAV\",\"lime\":\"lime__3kPc-\",\"cyan\":\"cyan__11RpC\",\"blue\":\"blue__320AC\",\"magenta\":\"magenta__VPUPe\",\"unbounded\":\"unbounded__25UN8\",\"unfeasible\":\"unfeasible__2MuIF\",\"finite\":\"finite__2EP_f\",\"min\":\"min__3lPpn\",\"max\":\"max__1hxKl\"};","import style from \"./Styles.less\";\n\nexport default function (props) {\n return (\n {props.children ? props.children : \"max\"}\n );\n}\n","import style from \"./Styles.less\";\n\nexport default function (props) {\n return (\n {props.children ? props.children : \"finito\"}\n );\n}\n","import style from \"./Styles.less\";\n\nexport default function (props) {\n return (\n {props.children ? props.children : \"vuoto\"}\n );\n}\n","import style from \"./Minus.less\";\n\nexport default function(props) {\n\treturn {props.children};\n}\n","import style from \"./Styles.less\";\n\nexport default function (props) {\n return (\n {props.children ? props.children : \"illimitato\"}\n );\n}\n","import style from \"./Styles.less\";\n\nexport default function (props) {\n return (\n {props.children ? props.children : \"min\"}\n );\n}\n","// extracted by mini-css-extract-plugin\nmodule.exports = {\"red\":\"red__3phX0\",\"orange\":\"orange__1OSkI\",\"yellow\":\"yellow__2pQOr\",\"lime\":\"lime__3vRUg\",\"cyan\":\"cyan__1cBtG\",\"blue\":\"blue__1G1wb\",\"magenta\":\"magenta__3Wagv\",\"minus\":\"minus__2qLyv\"};","import {\n Section,\n Latex,\n Panel,\n Todo,\n Timer,\n PLatex,\n TablePanel,\n LatexDefaultInline,\n ILatex,\n BLatex,\n BaseLink, Image\n} from \"bluelib\";\nimport Example from \"../components/Example\";\nimport Empty from \"../components/OttimizzazioneLineare/Empty\";\nimport Unbounded from \"../components/OttimizzazioneLineare/Unbounded\";\nimport Finite from \"../components/OttimizzazioneLineare/Finite\";\nimport Min from \"../components/OttimizzazioneLineare/Min\";\nimport Max from \"../components/OttimizzazioneLineare/Max\";\nimport Plus from \"../components/Fisica/Plus\";\nimport Minus from \"../components/Fisica/Minus\";\nimport ExampleBoxColor from \"../components/ExampleBoxColor\";\nimport Link from \"../components/Link\";\n\nconst r = String.raw;\n\n\nexport default function(props) {\n return (\n
\n

Ottimizzazione lineare intera

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  • Progettazione concettuale e logica
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\n \n
\n \n
\n \n \n \n v\n s\n Significato\n \n \n \n \n {r`\\mathbf{x}`}\n {r`x_i`}\n Incognite\n \n \n {r`\\mathbf{s}`}\n {r`s_i`}\n Variabili slack\n \n \n {r`\\mathbf{c}`}\n {r`c_i`}\n Coefficienti della funzione obiettivo\n \n \n {r`\\mathbf{A}`}\n {r`a_{ij}`}\n Coefficienti dei vincoli\n \n \n {r`\\mathbf{b}`}\n {r`b_i`}\n Termini noti dei vincoli\n \n \n {r`\\mathbf{y}`}\n {r`y_i`}\n Incognite artificiali\n \n \n {r`\\mathbf{u}`}\n {r`u_i`}\n Coefficienti di rilassamento\n \n \n \n {r`c_0`}\n Valore ottimo di un problema\n \n \n {r`\\mathbf{x}_B`}\n \n Incognite in base\n \n \n {r`\\mathbf{c}_B`}\n \n Coefficienti della funzione obiettivo delle variabili in base\n \n \n {r`\\mathbf{B}`}\n \n Coefficienti dei vincoli delle variabili in base\n \n \n {r`\\mathbf{x}_F`}\n \n Incognite fuori base\n \n \n {r`\\mathbf{c}_F`}\n \n Coefficienti della funzione obiettivo delle variabili fuori base\n \n \n {r`\\mathbf{F}`}\n \n Coefficienti dei vincoli delle variabili fuori base\n \n \n \n \n \n \n Simboli\n Significato\n \n \n \n \n {r`\\mathbf{c}^T \\mathbf{x}`}\n Soluzione del problema\n \n \n {r`\\mathbf{A} \\mathbf{x} = \\mathbf{b}`}\n Vincoli in forma standard\n \n \n {r`z(\\dots)`}\n Funzione obiettivo\n \n \n {r`\\mathbf{u}^T \\mathbf{b}`}\n Soluzione del problema duale\n \n \n {r`\\mathbf{u}^T \\mathbf{A} = \\mathbf{c}^T`}\n Vincoli del problema duale in forma standard\n \n \n {r`\\lfloor x \\rfloor`}\n Arrotondamento per difetto di x\n \n \n {r`\\lceil x \\rceil`}\n Arrotondamento per eccesso di x\n \n \n {r`x - \\lfloor x \\rfloor`}\n Parte frazionaria di x (se non è negativo)\n \n \n \n
\n \n
\n \n

\n Problemi che cercano di minimizzare/massimizzare il valore di una funzione obiettivo le cui incognite sono sottoposte a un sistema di vincoli.\n

\n

\n Spesso sono detti anche problemi di LP.\n

\n \n \n

\n La funzione da minimizzare/massimizzare.\n

\n

\n Il vettore dei suoi coefficienti è detto {r`\\mathbf{c}`}, mentre quello delle sue incognite {r`\\mathbf{x}`}.\n

\n

\n In genere, la funzione obiettivo è scritta in questa forma:\n

\n

\n {r`z(\\mathbf{x}) = c_1 \\cdot x_1 + c_2 \\cdot x_2 + \\dots + c_n \\cdot x_n`}\n

\n \n \n

\n Funzione della funzione obiettivo che restituisce la direzione del suo aumento più veloce.\n

\n

\n {r`\\nabla (f) = \\frac{d f}{d x_1} I_1 + \\frac{d f}{d x_2} I_2 + \\frac{d f}{d x_n} I_n`}\n

\n \n La matrice {r`\\mathbf{I}`} è la matrice identità.\n \n \n Se la funzione obiettivo è z = 2w + 3x + 4y, il suo gradiente è {r`\\nabla z = (2, 3, 4)`}.\n \n \n \n

\n Equazioni e disequazioni a cui devono sottostare le incognite perchè esse formino una soluzione valida.\n

\n

\n I loro coefficienti sono contenuti nella matrice {r`\\mathbf{A}`}, mentre i loro termini noti nel vettore {r`\\mathbf{b}`}.\n

\n \n \n

\n L'insieme che racchiunde tutte le soluzioni ammissibili di un problema.\n

\n

\n Può essere , oppure .\n

\n \n Si chiama così perchè se si disegna su un piano cartesiano, esso forma una figura geometrica a più lati, ovvero un poliedro.\n \n \n \n

\n La soluzione di un problema, ricavabile dal prodotto {r`\\mathbf{c}^T \\mathbf{x}`}.\n

\n

\n In particolare, il valore ottimo è un vertice del poliedro, detto vertice ottimo.\n

\n \n
\n
\n \n

\n Un problema con:\n

\n
    \n
  • Equazioni e disequazioni
    • \n
  • Variabili non vincolate
    • \n
\n {r`min \\left\\{ \\mathbf{c}^T \\mathbf{x} : \\mathbf{A} \\mathbf{x} = b,\\quad \\mathbf{A'} \\mathbf{x} \\geq \\mathbf{b'} \\quad x_j \\geq 0,\\quad j = 1 \\dots n \\right\\}`}\n \n \n

\n Un problema con:\n

\n
    \n
  • Solo disequazioni
    • \n
  • Vincoli di non-negatività sulle incognite
    • \n
\n {r`min \\left\\{ \\mathbf{c}^T \\mathbf{x} : \\mathbf{A} \\mathbf{x} \\geq b,\\quad x_j \\geq 0,\\quad j = 1 \\dots n \\right\\}`}\n \n \n

\n Un problema con:\n

\n
    \n
  • Solo equazioni
    • \n
  • Vincoli di non-negatività sulle incognite
    • \n
\n {r`min \\left\\{ \\mathbf{c}^T \\mathbf{x} : \\mathbf{A} \\mathbf{x} = b,\\quad x_j \\geq 0,\\quad j = 1 \\dots n \\right\\}`}\n \n
\n
\n \n

\n Applica questa conversione a ogni equazione nel sistema:\n

\n

\n {r`a = b \\Leftrightarrow\n \\begin{cases}\n a \\leq b\\\\\n a \\geq b\n \\end{cases}\n `}\n

\n Serve solo nella teoria per dimostrare che le forme sono equivalenti.\n \n \n

\n Aggiungi una variabile slack {r`s`} non-vincolata a ogni disequazione nel sistema:\n

\n

\n {r`\n a \\leq b \\Leftrightarrow a + s = b\n `}\n

\n

\n {r`\n a \\geq b \\Leftrightarrow a - s = b\n `}\n

\n \n \n

\n Sdoppia ogni variabile non-vincolata in due variabili con vincolo di non-negatività:\n

\n

\n {r`\\begin{cases}\n a = a^+ - a^-\\\\\n a^+ \\geq 0\\\\\n a^- \\geq 0\n \\end{cases}`}\n

\n \n
\n
\n Valore attuale\n
\n
\n \n

\n Un modo per rappresentare sistemi in forma standard, anche noto come matrice equivalente completa del sistema.\n

\n \n \n

\n Un tableau è un sistema di equazioni in forma matriciale completa.\n

\n

\n È possibile effettuare senza che cambi il risultato finale le seguenti trasformazioni:\n

\n
    \n
  • Moltiplicare un'intera riga per una costante.
    • \n
  • Sommare una riga a un'altra
    • \n
\n \n Suona familiare? Sì, lo abbiamo fatto anche in Algebra Lineare.\n \n \n \n

\n Variabili che hanno tutti 0 e un solo 1 nella loro colonna del tableau.\n

\n

\n La loro controparte sono le variabili fuori base, che hanno qualsiasi altro valore.\n

\n \n \n

\n Il valore della funzione obiettivo che si otterrebbe se tutte le variabili fuori base valessero 0.\n

\n

\n Procedendo nella risoluzione (descritta in seguito) del tableau, questo valore aumenterà, fino a raggiungere il valore ottimo quando la risoluzione sarà completata.\n

\n \n
\n
\n \n

\n Il sistema:\n

\n {r`\n \\begin{cases}\n 2000x_1 + 1000x_2 = z\\\\\n 1x_1 \\leq 3\\\\\n 1x_2 \\leq 3\\\\\n 2x_1 + 2x_2 \\leq 7\n \\end{cases}\n `}\n

\n Diventa il tableau:\n

\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
x_1x_2s_1s_2s_3TN
101003
010103
220017
200010000000
\n
    \n
  • \n Verde: funzione obiettivo\n
    • \n
  • \n Giallo: valore attuale della funzione\n
    • \n
  • \n Rosso: termini noti\n
    • \n
  • \n Blu: Variabili slack\n
    • \n
\n \n
\n
\n \n

\n Un algoritmo per trovare efficientemente il valore ottimo e le coordinate di un vertice ottimo in problemi di ottimizzazione lineare.\n

\n \n Ricordi Gauss-Jordan? Il Simplex è la stessa cosa, aggiungendo criteri per la selezione del pivot.\n \n \n Questa è la soluzione passo per passo del problema 3 del file Ex_LP_testo.\n \n

\n Perchè sia possibile effettuare il Simplex è necessario che l'origine sia nel poliedro: pertanto, non è possibile che un problema risolto con il Simplex sia .\n

\n \n \n
    \n
  1. Trasforma il sistema in forma standard.
    2. \n
  2. Trova tante variabili linearmente indipendenti quante siano le righe: esse saranno la base iniziale.
    3. \n
  3. Finchè ci sono variabili con coefficienti positivi/negativi nella funzione obiettivo:\n
      \n
    1. \n Scegli la prima variabile fuori base con coefficiente positivo/negativo nella funzione obiettivo: essa è la variabile entrante.
      \n \n
      2. \n
    2. \n Scegli la variabile in base con il minor rapporto positivo {r`\\frac{termine\\ noto}{coeff.\\ variabile\\ entrante}`}.\n \n
      3. \n
    3. Pivot: trasforma tutte le funzioni del sistema in modo che abbiano 0 nella colonna della variabile entrante, tranne nella riga della variabile uscente, in cui avrà 1.
      4. \n
    \n
    4. \n
  4. Il poliedro è : i termini noti dei vincoli sono le coordinate del suo vertice ottimo, mentre il termine noto della funzione obiettivo è il valore ottimo.
    5. \n
\n \n \n

\n Una soluzione con almeno una variabile di valore 0, dovuta a uno o più vincoli ridondanti.\n

\n

\n Senza Regola di Bland e in presenza di vincoli ridondanti si rischia di trovarsi a fare pivot infiniti.\n

\n \n
\n
\n \n

\n Un estensione del Simplex per permettere la risoluzione di problemi la cui origine non è una soluzione ammissibile.\n

\n

\n Prevede l'introduzione di un problema ausiliario, le cui incognite sono dette artificiali.\n

\n

\n Il vettore delle incognite artificiali è solitamente chiamato {r`\\mathbf{y}`}.\n

\n \n \n
    \n
  1. Crea un nuovo tableau, aggiungendo variabili artificiali in modo da avere una base ammissibile.
    2. \n
  2. Sostituisci la vecchia funzione obiettivo con una nuova che minimizzi la somma di tutte le variabili artificiali.
    3. \n
  3. Fase 1: Risolvi il nuovo problema con il Simplex primale.
    4. \n
  4. Se il Simplex termina quando ci sono ancora variabili artificiali nella base, allora il poliedro è .
    5. \n
  5. Una volta che le variabili artificiali sono fuori base, elimina le loro colonne e la nuova funzione obiettivo.
    6. \n
  6. Riporta il tableau in forma base compiendo operazioni per azzerare i coefficienti delle variabili di base nella funzione obiettivo.
    7. \n
  7. Fase 2: Risolvi il tableau con il Simplex primale.
    8. \n
\n \n
\n
\n \n

\n Una versione semplificata di un problema nella quale si ignora la violazione di uno o più vincoli.\n

\n \n \n

\n Un rilassamento che permette di misurare di quanto i vincoli vengono violati.\n

\n

\n I vincoli, moltiplicati per coefficienti di rilassamento, vengono inseriti nella funzione obiettivo.\n

\n

\n Il vettore dei coefficienti di rilassamento solitamente è indicato con {r`\\mathbf{u}`}.\n

\n \n

\n Il sistema:\n

\n {r`\n \\begin{cases}\n z = 3 x_1 + 5 x_2\\\\\n 2 x_1 + 3 x_2 \\geq 12\\\\\n - x_1 + 3 x_2 \\geq 3\\\\\n x_1 \\geq 0\\\\\n x_2 \\geq 0\n \\end{cases}\n `}\n

\n diventa:\n

\n {r`\n \\begin{cases}\n z = 3 x_1 + 5 x_2 + u_1 ( 12 - 2 x_1 - 3 x_2 ) + u_2 ( 3 + x_1 - 3 x_2 )\\\\\n x_1 \\geq 0\\\\\n x_2 \\geq 0\n \\end{cases}\n `}\n \n \n
\n
\n \n

\n Il sistema che massimizza/minimizza i moltiplicatori di rilassamento di un problema detto primale.\n

\n \n \n

\n Possiamo trasporre il tableau e sostituire le variabili {r`x_n`} con variabili {r`u_n`} per ottenere il sistema duale!\n

\n

\n I maggiori e minori dei vincoli diventeranno maggiori e minori delle variabili e viceversa.\n

\n \n \n
    \n
  • Se un problema ha una soluzione finita, allora anche il suo duale la avrà.
    • \n
  • Se un problema è , allora il suo duale potrà essere oppure .
    • \n
  • Se un problema è , allora il suo duale sarà certamente .
    • \n
\n \n \n

\n Variabili e vincoli del duale corrispondono rispettivamente a vincoli e variabili del primale.\n

\n

\n In particolare:\n

\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
MinMax
Vincolo \\leqVariabile \\leq
Vincolo \\geqVariabile \\geq
Vincolo =Variabile libera
Variabile \\leqVincolo \\geq
Variabile \\geqVincolo \\leq
Variabile liberaVincolo =
\n \n
\n
\n \n

\n Una disuguaglianza lineare {r`c_0 \\leq \\mathbf{c}^T \\mathbf{x}`} è verificata da tutti i punti di un poliedro non- se e solo se esiste un vettore {r`u \\in \\mathfrak{R}^m`} tale che:\n

\n {r`\\mathbf{c}^T \\geq \\mathbf{u}^T \\mathbf{A}`}\n {r`c_0 \\leq \\mathbf{u}^T \\mathbf{b}`}\n \n \n

\n Il teorema che dimostra l'equivalenza tra primale e duale.\n

\n

\n Se uno dei due problemi è finito, la soluzione di uno coincide con la soluzione dell'altro.\n

\n

\n {r`\\mathbf{c}^T \\mathbf{x} = \\mathbf{u}^T \\mathbf{b}`}\n

\n

\n TODO: Anche qui c'è una lunga dimostrazione...\n

\n \n \n

\n Il teorema che dimostra che il valore della funzione obiettivo del duale (di un qualsiasi tableau) è sempre minore o uguale/maggiore o uguale alla soluzione del corrispettivo primale.\n

\n

\n TODO: Dimostrazione cortina, ma sembra complicata.\n

\n \n \n

\n Il teorema che ci permette di passare dalla soluzione del duale alla soluzione del primale. TODO: credo?\n

\n

\n Si deriva combinando le seguenti condizioni:\n

\n
    \n
  • Ammissibilità del primale: {r`\\mathbf{A} \\mathbf{X} \\geq \\mathbf{b}, \\quad \\mathbf{x} \\geq 0`}
    • \n
  • Ammissibilità del duale: {r`\\mathbf{u}^T \\mathbf{A} \\leq \\mathbf{c}^T, \\quad \\mathbf{u} \\geq 0`}
    • \n
  • Teorema della dualità forte: {r`\\mathbf{c}^T \\mathbf{x} = \\mathbf{u}^T \\mathbf{b}`} (alla soluzione ottima)
    • \n
\n

\n Ne risulta che una soluzione è ottima se e solo se:\n

\n {r`\\left( \\mathbf{c}^T - \\mathbf{u}^T \\mathbf{A} \\right) \\mathbf{x} = 0`}\n {r`\\mathbf{u}^T \\left( \\mathbf{A} \\mathbf{x} - \\mathbf{b} \\right) = 0`}\n \n
\n
\n \n

\n Un'estensione al Simplex primale che opera sul problema duale.\n

\n \n \n

\n Funziona esattamente come il Simplex primale, ma opera sul duale.\n

\n \n
\n
\n \n

\n Un procedimento che misura di quanto può variare il termine noto di un vincolo {r`b_i`} o il coefficiente della funzione obiettivo {r`c_i`} prima che la base degeneri.\n

\n \n
\n
\n \n

\n Particolari problemi di ottimizzazione lineare in cui le variabili sono vincolate ad essere numeri interi.\n

\n {r`\n \\mathbf{x} \\in \\mathbb{Z}^n\n `}\n

\n Spesso detti anche problemi di ILP.\n

\n \n \n

\n Un rilassamento che rimuove il vincolo di integrità a un problema, trovando la sua soluzione continua.\n

\n \n
\n
\n \n

\n Un modo per passare dalla soluzione del rilassamento alla soluzione intera di un problema di ILP.\n

\n

\n Consiste nel calcolare la soluzione di ogni singolo punto incluso nel poliedro, e selezionare la minore/maggiore.\n

\n

\n Trova sicuramente la soluzione giusta, ma il costo computazionale è esponenziale O(n^k)!\n

\n \n \n

\n Un altro modo per passare dalla soluzione del rilassamento alla soluzione intera di un problema di ILP.\n

\n

\n Consiste nell'arrotondare tutte le variabili al loro valore intero più vicino, e calcolarne il valore ottimo.\n

\n

\n Funziona bene per valori grandi, ma più essi si avvicinano allo 0 più l'errore diventa grande.\n

\n \n \n

\n Un altro modo ancora per passare dalla soluzione del rilassamento alla soluzione intera di un problema di ILP.\n

\n

\n Consiste nel tagliare il poliedro con nuovi vincoli (piani secanti) che riducono le possibili soluzioni continue ma non quelle intere.\n

\n

\n Per selezionare i vincoli, si usano i tagli di Gomory:\n

\n {r`\n \\sum_{j \\in F} \\left( \\left( a_{tj} - \\lfloor a_{tj} \\rfloor \\right) \\cdot x_j \\right) \\geq (b_t - \\lfloor b_t \\rfloor)\n `}\n

\n Per ogni valore noto frazionario si viene quindi a creare una nuova variabile in base e un nuovo vincolo formato dall'opposto di tutti i valori frazionari dei coefficienti fuori base.\n

\n \n

\n Il tableau:\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
{r`x_1`}{r`x_2`}{r`s_1`}{r`s_2`}TN
{r`1`}{r`1`}{r`0`}{r`0`}{r`0`}
{r`1`}{r`0`}{r`1`}{r`0`}{r`3`}
{r`\\frac{3}{2}`}{r`\\frac{1}{2}`}{r`0`}{r`1`}{r`\\frac{6}{5}`}
\n

\n

\n Diventa:\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
{r`x_1`}{r`x_2`}{r`s_1`}{r`s_2`}{r`s_3`}TN
{r`1`}{r`1`}{r`0`}{r`0`}{r`0`}{r`0`}
{r`1`}{r`0`}{r`1`}{r`0`}{r`0`}{r`3`}
{r`\\frac{3}{2}`}{r`\\frac{1}{2}`}{r`0`}{r`1`}{r`0`}{r`\\frac{6}{5}`}
{r`-\\frac{1}{2}`}{r`-\\frac{1}{2}`}{r`0`}{r`0`}{r`1`}{r`-\\frac{1}{5}`}
\n

\n \n \n \n

\n È possibile usare la tecnica divide et impera per rendere più efficiente l'enumerazione totale.\n

\n

\n Si divide il problema principale (trovare il valore ottimo di un problema di ILP) in più sottoproblemi (trovare il valore ottimo di un problema di ILP con una variabile impostata a un valore fisso).\n

\n

\n Si crea così un albero.\n

\n

\n È possibile chiudere in anticipo alcuni nodi dell'albero se il loro miglior possibile valore ottimo è inferiore a uno precedentemente trovato o se il loro poliedro è .\n

\n

\n È possibile utilizzare diverse strategie di esplorazione dell'albero:\n

    \n
  • depth-first: permette di raggiungere immediatamente a una soluzione accettabile (ma non ottimale)
    • \n
  • best-first: permette di raggiungere più velocemente alla soluzione corretta
    • \n
\n

\n \n \n

\n È possibile combinare il metodo dei tagli secanti con la tecnica divide et impera per raggiungere ancora più velocemente a una soluzione.\n

\n

\n Si effettuano poche iterazioni del metodo dei tagli secanti, e sul risultato di quelle iterazioni si applica il divide et impera.\n

\n \n
\n
Terminologia dei grafi TODO: migliorare}>\n \n

\n Insieme di nodi {r`N`} e archi {r`E`} che li connettono.\n

\n

\n Può essere diretto se gli archi hanno una direzione.\n

\n \n \n

\n Nodi connessi da un arco.\n

\n \n \n

\n Arco connesso a un dato nodo.\n

\n \n \n

\n Un arco diretto che termina o inizia da un dato nodo.\n

\n \n \n

\n Conteggio degli archi incidenti di un nodo.\n

\n

\n Si può calcolare anche relativamente agli archi entranti o agli archi uscenti.\n

\n \n \n

\n Sequenza di archi consecutivi.\n

\n \n \n

\n Due nodi sono connessi se tra loro esiste almeno un percorso.\n

\n

\n Un grafo è connesso se tutti i suoi nodi sono connessi.\n

\n \n \n

\n Percorsi rispettivamente indiretti e diretti in cui l'inizio coincide con la fine.\n

\n \n \n

\n Grafo in cui ogni nodo è connesso con ogni altro.\n

\n

\n Se diretto, contiene {r`n \\cdot (n - 1)`} archi; altrimenti, ne contiene la metà.\n

\n \n \n
\n Vedi Algoritmi.\n
\n \n \n
\n Vedi Algoritmi.\n
\n \n \n

\n Sottoinsieme di archi che connettono due sottoinsiemi di nodi.\n

\n

\n Può essere anche uscente o entrante; in tal caso include solo gli archi entranti o uscenti dal sottoinsieme.\n

\n \n \n

\n Sottoinsieme di nodi e archi di un grafo.\n

\n

\n Tutti gli archi di un sottografo possono connettere solo nodi all'interno di esso.\n

\n \n \n

\n Sottografo connesso e aciclico.\n

\n \n \n

\n Albero che include tutti i nodi di un grafo.\n

\n \n
\n
\n \n

\n Crea uno spanning tree.\n

\n
    \n
  1. Aggiungi l'arco di costo minimo all'albero.
    2. \n
  2. Finchè mancano ancora archi:\n
      \n
    1. Trova tutti gli archi che aggiungerebbero un nuovo nodo all'albero.
      2. \n
    2. Seleziona l'arco di costo minore.
      3. \n
    \n
    3. \n
\n \n \n

\n Trova l'ordine topologico di un albero.\n

\n
    \n
  1. Ripeti finchè ci sono nodi nel grafo:\n
      \n
    1. Assegna un numero sequenziale a un nodo senza archi entranti.
      2. \n
    2. Elimina il nodo a cui hai assegnato il numero.
      3. \n
    3. Elimina tutti gli archi incidenti sul nodo che hai eliminato.
      4. \n
    \n
    2. \n
\n \n \n

\n Trova i percorsi di costo minimo in un albero.\n

\n
    \n
  1. Trova l'ordine topologico dell'albero.
    2. \n
  2. Invece che provare ogni singola combinazione di nodi, prova solo i nodi che hanno un numero topologico maggiore di quello del nodo attuale.
    3. \n
\n

\n TODO: forse spiegarlo meglio non farebbe male\n

\n \n \n
\n Vedi Algoritmi.\n
\n \n \n \n Trova il volume massimo di acqua che è possibile fare scorrere attraverso tubature con una data capacità.\n \n

\n Costruisci il grafo residuo e vedi se c'è un percorso che va dalla sorgente alla destinazione.\n

\n

\n \n

\n

\n \n

\n \n
\n
\n )\n}\n","// extracted by mini-css-extract-plugin\nmodule.exports = {\"red\":\"red__ipSoq\",\"orange\":\"orange__1nmkj\",\"yellow\":\"yellow__353ge\",\"lime\":\"lime__3FW1E\",\"cyan\":\"cyan__2xhu7\",\"blue\":\"blue__3MvDe\",\"magenta\":\"magenta__i03ie\",\"plus\":\"plus__1eV4_\"};","import style from \"./Example.less\";\r\n\r\nexport default function(props) {\r\n return (\r\n
\r\n {props.children}\r\n
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extracted by mini-css-extract-plugin\nmodule.exports = {\"red\":\"red__2y1B_\",\"orange\":\"orange__dD2kx\",\"yellow\":\"yellow__OEpwl\",\"lime\":\"lime__CVe41\",\"cyan\":\"cyan__26ZAg\",\"blue\":\"blue__LO7Xm\",\"magenta\":\"magenta__1Akee\",\"example\":\"example__2PzAa\"};","// extracted by mini-css-extract-plugin\nmodule.exports = {\"red\":\"red__UmOgB\",\"orange\":\"orange__1FTS8\",\"yellow\":\"yellow__3dJAV\",\"lime\":\"lime__3kPc-\",\"cyan\":\"cyan__11RpC\",\"blue\":\"blue__320AC\",\"magenta\":\"magenta__VPUPe\",\"unbounded\":\"unbounded__25UN8\",\"unfeasible\":\"unfeasible__2MuIF\",\"finite\":\"finite__2EP_f\",\"min\":\"min__3lPpn\",\"max\":\"max__1hxKl\"};","import style from \"./Styles.less\";\n\nexport default function (props) {\n return (\n {props.children ? props.children : \"max\"}\n );\n}\n","import style from \"./Styles.less\";\n\nexport default function (props) {\n return (\n {props.children ? props.children : \"finito\"}\n );\n}\n","import style from \"./Styles.less\";\n\nexport default function (props) {\n return (\n {props.children ? props.children : \"vuoto\"}\n );\n}\n","import style from \"./Minus.less\";\n\nexport default function(props) {\n\treturn {props.children};\n}\n","import style from \"./Styles.less\";\n\nexport default function (props) {\n return (\n {props.children ? props.children : \"illimitato\"}\n );\n}\n","import style from \"./Styles.less\";\n\nexport default function (props) {\n return (\n {props.children ? props.children : \"min\"}\n );\n}\n","// extracted by mini-css-extract-plugin\nmodule.exports = {\"red\":\"red__3phX0\",\"orange\":\"orange__1OSkI\",\"yellow\":\"yellow__2pQOr\",\"lime\":\"lime__3vRUg\",\"cyan\":\"cyan__1cBtG\",\"blue\":\"blue__1G1wb\",\"magenta\":\"magenta__3Wagv\",\"minus\":\"minus__2qLyv\"};","import {\n Section,\n Latex,\n Panel,\n Todo,\n Timer,\n PLatex,\n TablePanel,\n LatexDefaultInline,\n ILatex,\n BLatex,\n BaseLink, Image\n} from \"bluelib\";\nimport Example from \"../components/Example\";\nimport Empty from \"../components/OttimizzazioneLineare/Empty\";\nimport Unbounded from \"../components/OttimizzazioneLineare/Unbounded\";\nimport Finite from \"../components/OttimizzazioneLineare/Finite\";\nimport Min from \"../components/OttimizzazioneLineare/Min\";\nimport Max from \"../components/OttimizzazioneLineare/Max\";\nimport Plus from \"../components/Fisica/Plus\";\nimport Minus from \"../components/Fisica/Minus\";\nimport ExampleBoxColor from \"../components/ExampleBoxColor\";\nimport Link from \"../components/Link\";\n\nconst r = String.raw;\n\n\nexport default function(props) {\n return (\n
\n

Ottimizzazione lineare intera

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\n \n
\n \n
\n \n \n \n v\n s\n Significato\n \n \n \n \n {r`\\mathbf{x}`}\n {r`x_i`}\n Incognite\n \n \n {r`\\mathbf{s}`}\n {r`s_i`}\n Variabili slack\n \n \n {r`\\mathbf{c}`}\n {r`c_i`}\n Coefficienti della funzione obiettivo\n \n \n {r`\\mathbf{A}`}\n {r`a_{ij}`}\n Coefficienti dei vincoli\n \n \n {r`\\mathbf{b}`}\n {r`b_i`}\n Termini noti dei vincoli\n \n \n {r`\\mathbf{y}`}\n {r`y_i`}\n Incognite artificiali\n \n \n {r`\\mathbf{u}`}\n {r`u_i`}\n Coefficienti di rilassamento\n \n \n \n {r`c_0`}\n Valore ottimo di un problema\n \n \n {r`\\mathbf{x}_B`}\n \n Incognite in base\n \n \n {r`\\mathbf{c}_B`}\n \n Coefficienti della funzione obiettivo delle variabili in base\n \n \n {r`\\mathbf{B}`}\n \n Coefficienti dei vincoli delle variabili in base\n \n \n {r`\\mathbf{x}_F`}\n \n Incognite fuori base\n \n \n {r`\\mathbf{c}_F`}\n \n Coefficienti della funzione obiettivo delle variabili fuori base\n \n \n {r`\\mathbf{F}`}\n \n Coefficienti dei vincoli delle variabili fuori base\n \n \n \n \n \n \n Simboli\n Significato\n \n \n \n \n {r`\\mathbf{c}^T \\mathbf{x}`}\n Soluzione del problema\n \n \n {r`\\mathbf{A} \\mathbf{x} = \\mathbf{b}`}\n Vincoli in forma standard\n \n \n {r`z(\\dots)`}\n Funzione obiettivo\n \n \n {r`\\mathbf{u}^T \\mathbf{b}`}\n Soluzione del problema duale\n \n \n {r`\\mathbf{u}^T \\mathbf{A} = \\mathbf{c}^T`}\n Vincoli del problema duale in forma standard\n \n \n {r`\\lfloor x \\rfloor`}\n Arrotondamento per difetto di x\n \n \n {r`\\lceil x \\rceil`}\n Arrotondamento per eccesso di x\n \n \n {r`x - \\lfloor x \\rfloor`}\n Parte frazionaria di x (se non è negativo)\n \n \n \n
\n \n
\n \n

\n Problemi che cercano di minimizzare/massimizzare il valore di una funzione obiettivo le cui incognite sono sottoposte a un sistema di vincoli.\n

\n

\n Spesso sono detti anche problemi di LP.\n

\n \n \n

\n La funzione da minimizzare/massimizzare.\n

\n

\n Il vettore dei suoi coefficienti è detto {r`\\mathbf{c}`}, mentre quello delle sue incognite {r`\\mathbf{x}`}.\n

\n

\n In genere, la funzione obiettivo è scritta in questa forma:\n

\n

\n {r`z(\\mathbf{x}) = c_1 \\cdot x_1 + c_2 \\cdot x_2 + \\dots + c_n \\cdot x_n`}\n

\n \n \n

\n Funzione della funzione obiettivo che restituisce la direzione del suo aumento più veloce.\n

\n

\n {r`\\nabla (f) = \\frac{d f}{d x_1} I_1 + \\frac{d f}{d x_2} I_2 + \\frac{d f}{d x_n} I_n`}\n

\n \n La matrice {r`\\mathbf{I}`} è la matrice identità.\n \n \n Se la funzione obiettivo è z = 2w + 3x + 4y, il suo gradiente è {r`\\nabla z = (2, 3, 4)`}.\n \n \n \n

\n Equazioni e disequazioni a cui devono sottostare le incognite perchè esse formino una soluzione valida.\n

\n

\n I loro coefficienti sono contenuti nella matrice {r`\\mathbf{A}`}, mentre i loro termini noti nel vettore {r`\\mathbf{b}`}.\n

\n \n \n

\n L'insieme che racchiunde tutte le soluzioni ammissibili di un problema.\n

\n

\n Può essere , oppure .\n

\n \n Si chiama così perchè se si disegna su un piano cartesiano, esso forma una figura geometrica a più lati, ovvero un poliedro.\n \n \n \n

\n La soluzione di un problema, ricavabile dal prodotto {r`\\mathbf{c}^T \\mathbf{x}`}.\n

\n

\n In particolare, il valore ottimo è un vertice del poliedro, detto vertice ottimo.\n

\n \n
\n
\n \n

\n Un problema con:\n

\n
    \n
  • Equazioni e disequazioni
    • \n
  • Variabili non vincolate
    • \n
\n {r`min \\left\\{ \\mathbf{c}^T \\mathbf{x} : \\mathbf{A} \\mathbf{x} = b,\\quad \\mathbf{A'} \\mathbf{x} \\geq \\mathbf{b'} \\quad x_j \\geq 0,\\quad j = 1 \\dots n \\right\\}`}\n \n \n

\n Un problema con:\n

\n
    \n
  • Solo disequazioni
    • \n
  • Vincoli di non-negatività sulle incognite
    • \n
\n {r`min \\left\\{ \\mathbf{c}^T \\mathbf{x} : \\mathbf{A} \\mathbf{x} \\geq b,\\quad x_j \\geq 0,\\quad j = 1 \\dots n \\right\\}`}\n \n \n

\n Un problema con:\n

\n
    \n
  • Solo equazioni
    • \n
  • Vincoli di non-negatività sulle incognite
    • \n
\n {r`min \\left\\{ \\mathbf{c}^T \\mathbf{x} : \\mathbf{A} \\mathbf{x} = b,\\quad x_j \\geq 0,\\quad j = 1 \\dots n \\right\\}`}\n \n
\n
\n \n

\n Applica questa conversione a ogni equazione nel sistema:\n

\n

\n {r`a = b \\Leftrightarrow\n \\begin{cases}\n a \\leq b\\\\\n a \\geq b\n \\end{cases}\n `}\n

\n Serve solo nella teoria per dimostrare che le forme sono equivalenti.\n \n \n

\n Aggiungi una variabile slack {r`s`} non-vincolata a ogni disequazione nel sistema:\n

\n

\n {r`\n a \\leq b \\Leftrightarrow a + s = b\n `}\n

\n

\n {r`\n a \\geq b \\Leftrightarrow a - s = b\n `}\n

\n \n \n

\n Sdoppia ogni variabile non-vincolata in due variabili con vincolo di non-negatività:\n

\n

\n {r`\\begin{cases}\n a = a^+ - a^-\\\\\n a^+ \\geq 0\\\\\n a^- \\geq 0\n \\end{cases}`}\n

\n \n
\n
\n Valore attuale\n
\n
\n \n

\n Un modo per rappresentare sistemi in forma standard, anche noto come matrice equivalente completa del sistema.\n

\n \n \n

\n Un tableau è un sistema di equazioni in forma matriciale completa.\n

\n

\n È possibile effettuare senza che cambi il risultato finale le seguenti trasformazioni:\n

\n
    \n
  • Moltiplicare un'intera riga per una costante.
    • \n
  • Sommare una riga a un'altra
    • \n
\n \n Suona familiare? Sì, lo abbiamo fatto anche in Algebra Lineare.\n \n \n \n

\n Variabili che hanno tutti 0 e un solo 1 nella loro colonna del tableau.\n

\n

\n La loro controparte sono le variabili fuori base, che hanno qualsiasi altro valore.\n

\n \n \n

\n Il valore della funzione obiettivo che si otterrebbe se tutte le variabili fuori base valessero 0.\n

\n

\n Procedendo nella risoluzione (descritta in seguito) del tableau, questo valore aumenterà, fino a raggiungere il valore ottimo quando la risoluzione sarà completata.\n

\n \n
\n
\n \n

\n Il sistema:\n

\n {r`\n \\begin{cases}\n 2000x_1 + 1000x_2 = z\\\\\n 1x_1 \\leq 3\\\\\n 1x_2 \\leq 3\\\\\n 2x_1 + 2x_2 \\leq 7\n \\end{cases}\n `}\n

\n Diventa il tableau:\n

\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
x_1x_2s_1s_2s_3TN
101003
010103
220017
200010000000
\n
    \n
  • \n Verde: funzione obiettivo\n
    • \n
  • \n Giallo: valore attuale della funzione\n
    • \n
  • \n Rosso: termini noti\n
    • \n
  • \n Blu: Variabili slack\n
    • \n
\n \n
\n
\n \n

\n Un algoritmo per trovare efficientemente il valore ottimo e le coordinate di un vertice ottimo in problemi di ottimizzazione lineare.\n

\n \n Ricordi Gauss-Jordan? Il Simplex è la stessa cosa, aggiungendo criteri per la selezione del pivot.\n \n \n Questa è la soluzione passo per passo del problema 3 del file Ex_LP_testo.\n \n

\n Perchè sia possibile effettuare il Simplex è necessario che l'origine sia nel poliedro: pertanto, non è possibile che un problema risolto con il Simplex sia .\n

\n \n \n
    \n
  1. Trasforma il sistema in forma standard.
    2. \n
  2. Trova tante variabili linearmente indipendenti quante siano le righe: esse saranno la base iniziale.
    3. \n
  3. Finchè ci sono variabili con coefficienti positivi/negativi nella funzione obiettivo:\n
      \n
    1. \n Scegli la prima variabile fuori base con coefficiente positivo/negativo nella funzione obiettivo: essa è la variabile entrante.
      \n \n
      2. \n
    2. \n Scegli la variabile in base con il minor rapporto positivo {r`\\frac{termine\\ noto}{coeff.\\ variabile\\ entrante}`}.\n \n
      3. \n
    3. Pivot: trasforma tutte le funzioni del sistema in modo che abbiano 0 nella colonna della variabile entrante, tranne nella riga della variabile uscente, in cui avrà 1.
      4. \n
    \n
    4. \n
  4. Il poliedro è : i termini noti dei vincoli sono le coordinate del suo vertice ottimo, mentre il termine noto della funzione obiettivo è il valore ottimo.
    5. \n
\n \n \n

\n Una soluzione con almeno una variabile di valore 0, dovuta a uno o più vincoli ridondanti.\n

\n

\n Senza Regola di Bland e in presenza di vincoli ridondanti si rischia di trovarsi a fare pivot infiniti.\n

\n \n
\n
\n \n

\n Un estensione del Simplex per permettere la risoluzione di problemi la cui origine non è una soluzione ammissibile.\n

\n

\n Prevede l'introduzione di un problema ausiliario, le cui incognite sono dette artificiali.\n

\n

\n Il vettore delle incognite artificiali è solitamente chiamato {r`\\mathbf{y}`}.\n

\n \n \n
    \n
  1. Crea un nuovo tableau, aggiungendo variabili artificiali in modo da avere una base ammissibile.
    2. \n
  2. Sostituisci la vecchia funzione obiettivo con una nuova che minimizzi la somma di tutte le variabili artificiali.
    3. \n
  3. Fase 1: Risolvi il nuovo problema con il Simplex primale.
    4. \n
  4. Se il Simplex termina quando ci sono ancora variabili artificiali nella base, allora il poliedro è .
    5. \n
  5. Una volta che le variabili artificiali sono fuori base, elimina le loro colonne e la nuova funzione obiettivo.
    6. \n
  6. Riporta il tableau in forma base compiendo operazioni per azzerare i coefficienti delle variabili di base nella funzione obiettivo.
    7. \n
  7. Fase 2: Risolvi il tableau con il Simplex primale.
    8. \n
\n \n
\n
\n \n

\n Una versione semplificata di un problema nella quale si ignora la violazione di uno o più vincoli.\n

\n \n \n

\n Un rilassamento che permette di misurare di quanto i vincoli vengono violati.\n

\n

\n I vincoli, moltiplicati per coefficienti di rilassamento, vengono inseriti nella funzione obiettivo.\n

\n

\n Il vettore dei coefficienti di rilassamento solitamente è indicato con {r`\\mathbf{u}`}.\n

\n \n

\n Il sistema:\n

\n {r`\n \\begin{cases}\n z = 3 x_1 + 5 x_2\\\\\n 2 x_1 + 3 x_2 \\geq 12\\\\\n - x_1 + 3 x_2 \\geq 3\\\\\n x_1 \\geq 0\\\\\n x_2 \\geq 0\n \\end{cases}\n `}\n

\n diventa:\n

\n {r`\n \\begin{cases}\n z = 3 x_1 + 5 x_2 + u_1 ( 12 - 2 x_1 - 3 x_2 ) + u_2 ( 3 + x_1 - 3 x_2 )\\\\\n x_1 \\geq 0\\\\\n x_2 \\geq 0\n \\end{cases}\n `}\n \n \n
\n
\n \n

\n Il sistema che massimizza/minimizza i moltiplicatori di rilassamento di un problema detto primale.\n

\n \n \n

\n Possiamo trasporre il tableau e sostituire le variabili {r`x_n`} con variabili {r`u_n`} per ottenere il sistema duale!\n

\n

\n I maggiori e minori dei vincoli diventeranno maggiori e minori delle variabili e viceversa.\n

\n \n \n
    \n
  • Se un problema ha una soluzione finita, allora anche il suo duale la avrà.
    • \n
  • Se un problema è , allora il suo duale potrà essere oppure .
    • \n
  • Se un problema è , allora il suo duale sarà certamente .
    • \n
\n \n \n

\n Variabili e vincoli del duale corrispondono rispettivamente a vincoli e variabili del primale.\n

\n

\n In particolare:\n

\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
MinMax
Vincolo \\leqVariabile \\leq
Vincolo \\geqVariabile \\geq
Vincolo =Variabile libera
Variabile \\leqVincolo \\geq
Variabile \\geqVincolo \\leq
Variabile liberaVincolo =
\n \n
\n
\n \n

\n Una disuguaglianza lineare {r`c_0 \\leq \\mathbf{c}^T \\mathbf{x}`} è verificata da tutti i punti di un poliedro non- se e solo se esiste un vettore {r`u \\in \\mathfrak{R}^m`} tale che:\n

\n {r`\\mathbf{c}^T \\geq \\mathbf{u}^T \\mathbf{A}`}\n {r`c_0 \\leq \\mathbf{u}^T \\mathbf{b}`}\n \n \n

\n Il teorema che dimostra l'equivalenza tra primale e duale.\n

\n

\n Se uno dei due problemi è finito, la soluzione di uno coincide con la soluzione dell'altro.\n

\n

\n {r`\\mathbf{c}^T \\mathbf{x} = \\mathbf{u}^T \\mathbf{b}`}\n

\n

\n TODO: Anche qui c'è una lunga dimostrazione...\n

\n \n \n

\n Il teorema che dimostra che il valore della funzione obiettivo del duale (di un qualsiasi tableau) è sempre minore o uguale/maggiore o uguale alla soluzione del corrispettivo primale.\n

\n

\n TODO: Dimostrazione cortina, ma sembra complicata.\n

\n \n \n

\n Il teorema che ci permette di passare dalla soluzione del duale alla soluzione del primale. TODO: credo?\n

\n

\n Si deriva combinando le seguenti condizioni:\n

\n
    \n
  • Ammissibilità del primale: {r`\\mathbf{A} \\mathbf{X} \\geq \\mathbf{b}, \\quad \\mathbf{x} \\geq 0`}
    • \n
  • Ammissibilità del duale: {r`\\mathbf{u}^T \\mathbf{A} \\leq \\mathbf{c}^T, \\quad \\mathbf{u} \\geq 0`}
    • \n
  • Teorema della dualità forte: {r`\\mathbf{c}^T \\mathbf{x} = \\mathbf{u}^T \\mathbf{b}`} (alla soluzione ottima)
    • \n
\n

\n Ne risulta che una soluzione è ottima se e solo se:\n

\n {r`\\left( \\mathbf{c}^T - \\mathbf{u}^T \\mathbf{A} \\right) \\mathbf{x} = 0`}\n {r`\\mathbf{u}^T \\left( \\mathbf{A} \\mathbf{x} - \\mathbf{b} \\right) = 0`}\n \n
\n
\n \n

\n Un'estensione al Simplex primale che opera sul problema duale.\n

\n \n \n

\n Funziona esattamente come il Simplex primale, ma opera sul duale.\n

\n \n
\n
\n \n

\n Un procedimento che misura di quanto può variare il termine noto di un vincolo {r`b_i`} o il coefficiente della funzione obiettivo {r`c_i`} prima che la base degeneri.\n

\n \n
\n
\n \n

\n Particolari problemi di ottimizzazione lineare in cui le variabili sono vincolate ad essere numeri interi.\n

\n {r`\n \\mathbf{x} \\in \\mathbb{Z}^n\n `}\n

\n Spesso detti anche problemi di ILP.\n

\n \n \n

\n Un rilassamento che rimuove il vincolo di integrità a un problema, trovando la sua soluzione continua.\n

\n \n
\n
\n \n

\n Un modo per passare dalla soluzione del rilassamento alla soluzione intera di un problema di ILP.\n

\n

\n Consiste nel calcolare la soluzione di ogni singolo punto incluso nel poliedro, e selezionare la minore/maggiore.\n

\n

\n Trova sicuramente la soluzione giusta, ma il costo computazionale è esponenziale O(n^k)!\n

\n \n \n

\n Un altro modo per passare dalla soluzione del rilassamento alla soluzione intera di un problema di ILP.\n

\n

\n Consiste nell'arrotondare tutte le variabili al loro valore intero più vicino, e calcolarne il valore ottimo.\n

\n

\n Funziona bene per valori grandi, ma più essi si avvicinano allo 0 più l'errore diventa grande.\n

\n \n \n

\n Un altro modo ancora per passare dalla soluzione del rilassamento alla soluzione intera di un problema di ILP.\n

\n

\n Consiste nel tagliare il poliedro con nuovi vincoli (piani secanti) che riducono le possibili soluzioni continue ma non quelle intere.\n

\n

\n Per selezionare i vincoli, si usano i tagli di Gomory:\n

\n {r`\n \\sum_{j \\in F} \\left( \\left( a_{tj} - \\lfloor a_{tj} \\rfloor \\right) \\cdot x_j \\right) \\geq (b_t - \\lfloor b_t \\rfloor)\n `}\n

\n Per ogni valore noto frazionario si viene quindi a creare una nuova variabile in base e un nuovo vincolo formato dall'opposto di tutti i valori frazionari dei coefficienti fuori base.\n

\n \n

\n Il tableau:\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
{r`x_1`}{r`x_2`}{r`s_1`}{r`s_2`}TN
{r`1`}{r`1`}{r`0`}{r`0`}{r`0`}
{r`1`}{r`0`}{r`1`}{r`0`}{r`3`}
{r`\\frac{3}{2}`}{r`\\frac{1}{2}`}{r`0`}{r`1`}{r`\\frac{6}{5}`}
\n

\n

\n Diventa:\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
{r`x_1`}{r`x_2`}{r`s_1`}{r`s_2`}{r`s_3`}TN
{r`1`}{r`1`}{r`0`}{r`0`}{r`0`}{r`0`}
{r`1`}{r`0`}{r`1`}{r`0`}{r`0`}{r`3`}
{r`\\frac{3}{2}`}{r`\\frac{1}{2}`}{r`0`}{r`1`}{r`0`}{r`\\frac{6}{5}`}
{r`-\\frac{1}{2}`}{r`-\\frac{1}{2}`}{r`0`}{r`0`}{r`1`}{r`-\\frac{1}{5}`}
\n

\n \n \n \n

\n È possibile usare la tecnica divide et impera per rendere più efficiente l'enumerazione totale.\n

\n

\n Si divide il problema principale (trovare il valore ottimo di un problema di ILP) in più sottoproblemi (trovare il valore ottimo di un problema di ILP con una variabile impostata a un valore fisso).\n

\n

\n Si crea così un albero.\n

\n

\n È possibile chiudere in anticipo alcuni nodi dell'albero se il loro miglior possibile valore ottimo è inferiore a uno precedentemente trovato o se il loro poliedro è .\n

\n

\n È possibile utilizzare diverse strategie di esplorazione dell'albero:\n

    \n
  • depth-first: permette di raggiungere immediatamente a una soluzione accettabile (ma non ottimale)
    • \n
  • best-first: permette di raggiungere più velocemente alla soluzione corretta
    • \n
\n

\n \n \n

\n È possibile combinare il metodo dei tagli secanti con la tecnica divide et impera per raggiungere ancora più velocemente a una soluzione.\n

\n

\n Si effettuano poche iterazioni del metodo dei tagli secanti, e sul risultato di quelle iterazioni si applica il divide et impera.\n

\n \n
\n
Terminologia dei grafi TODO: migliorare}>\n \n

\n Insieme di nodi {r`N`} e archi {r`E`} che li connettono.\n

\n

\n Può essere diretto se gli archi hanno una direzione.\n

\n \n \n

\n Nodi connessi da un arco.\n

\n \n \n

\n Arco connesso a un dato nodo.\n

\n \n \n

\n Un arco diretto che termina o inizia da un dato nodo.\n

\n \n \n

\n Conteggio degli archi incidenti di un nodo.\n

\n

\n Si può calcolare anche relativamente agli archi entranti o agli archi uscenti.\n

\n \n \n

\n Sequenza di archi consecutivi.\n

\n \n \n

\n Due nodi sono connessi se tra loro esiste almeno un percorso.\n

\n

\n Un grafo è connesso se tutti i suoi nodi sono connessi.\n

\n \n \n

\n Percorsi rispettivamente indiretti e diretti in cui l'inizio coincide con la fine.\n

\n \n \n

\n Grafo in cui ogni nodo è connesso con ogni altro.\n

\n

\n Se diretto, contiene {r`n \\cdot (n - 1)`} archi; altrimenti, ne contiene la metà.\n

\n \n \n
\n Vedi Algoritmi.\n
\n \n \n
\n Vedi Algoritmi.\n
\n \n \n

\n Sottoinsieme di archi che connettono due sottoinsiemi di nodi.\n

\n

\n Può essere anche uscente o entrante; in tal caso include solo gli archi entranti o uscenti dal sottoinsieme.\n

\n \n \n

\n Sottoinsieme di nodi e archi di un grafo.\n

\n

\n Tutti gli archi di un sottografo possono connettere solo nodi all'interno di esso.\n

\n \n \n

\n Sottografo connesso e aciclico.\n

\n \n \n

\n Albero che include tutti i nodi di un grafo.\n

\n \n
\n
\n \n

\n Crea uno spanning tree.\n

\n
    \n
  1. Aggiungi l'arco di costo minimo all'albero.
    2. \n
  2. Finchè mancano ancora archi:\n
      \n
    1. Trova tutti gli archi che aggiungerebbero un nuovo nodo all'albero.
      2. \n
    2. Seleziona l'arco di costo minore.
      3. \n
    \n
    3. \n
\n \n \n

\n Trova l'ordine topologico di un albero.\n

\n
    \n
  1. Ripeti finchè ci sono nodi nel grafo:\n
      \n
    1. Assegna un numero sequenziale a un nodo senza archi entranti.
      2. \n
    2. Elimina il nodo a cui hai assegnato il numero.
      3. \n
    3. Elimina tutti gli archi incidenti sul nodo che hai eliminato.
      4. \n
    \n
    2. \n
\n \n \n

\n Trova i percorsi di costo minimo in un albero.\n

\n
    \n
  1. Trova l'ordine topologico dell'albero.
    2. \n
  2. Invece che provare ogni singola combinazione di nodi, prova solo i nodi che hanno un numero topologico maggiore di quello del nodo attuale.
    3. \n
\n

\n TODO: forse spiegarlo meglio non farebbe male\n

\n \n \n
\n Vedi Algoritmi.\n
\n \n \n \n Trova il volume massimo di acqua che è possibile fare scorrere attraverso tubature con una data capacità.\n \n

\n Costruisci il grafo residuo e vedi se c'è un percorso che va dalla sorgente alla destinazione.\n

\n

\n \n

\n

\n \n

\n \n
\n
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suoi coefficienti è detto ",l(e.j,null,yi(J||(J=Si`\mathbf{c}`))),", mentre quello delle sue incognite ",l(e.j,null,yi(M||(M=Si`\mathbf{x}`))),"."),l("p",null,"In genere, la funzione obiettivo è scritta in questa forma:"),l("p",null,l(e.j,null,yi(N||(N=Si`z(\mathbf{x}) = c_1 \cdot x_1 + c_2 \cdot x_2 + \dots + c_n \cdot x_n`))))),l(e.p,{title:"Gradiente"},l("p",null,l("b",null,"Funzione")," della funzione obiettivo che restituisce la direzione del suo aumento più veloce."),l("p",null,l(e.j,null,yi(B||(B=Si`\nabla (f) = \frac{d f}{d x_1} I_1 + \frac{d f}{d x_2} I_2 + \frac{d f}{d x_n} I_n`)))),l(a.a,null,"La matrice ",l(e.j,null,yi(Z||(Z=Si`\mathbf{I}`)))," è la matrice identità."),l(a.a,null,"Se la funzione obiettivo è ",l(e.j,null,"z = 2w + 3x + 4y"),", il suo gradiente è ",l(e.j,null,yi(H||(H=Si`\nabla z = (2, 3, 4)`))),".")),l(e.p,{title:"Vincoli"},l("p",null,"Equazioni e disequazioni a cui devono sottostare le incognite perchè esse formino una soluzione valida."),l("p",null,"I loro coefficienti sono contenuti nella matrice ",l(e.j,null,yi(W||(W=Si`\mathbf{A}`))),", mentre i loro termini noti nel vettore ",l(e.j,null,yi(X||(X=Si`\mathbf{b}`))),".")),l(e.p,{title:"Poliedro"},l("p",null,"L'",l("b",null,"insieme")," che racchiunde tutte le ",l("b",null,"soluzioni ammissibili")," di un problema."),l("p",null,"Può essere ",l("i",null,l(u.a,null)),", ",l("i",null,l(o.a,null))," oppure ",l("i",null,l(t.a,null)),"."),l(a.a,null,"Si chiama così perchè se si disegna su un piano cartesiano, esso forma una figura geometrica a più lati, ovvero un ",l("a",{href:"https://it.wikipedia.org/wiki/Poliedro"},"poliedro"),".")),l(e.p,{title:"Valore ottimo"},l("p",null,"La ",l("b",null,"soluzione")," di un problema, ricavabile dal prodotto ",l(e.j,null,yi(Q||(Q=Si`\mathbf{c}^T \mathbf{x}`))),"."),l("p",null,"In particolare, il valore ottimo è un ",l("b",null,"vertice")," del poliedro, detto ",l("i",null,"vertice ottimo"),"."))),l(e.q,{title:"Forme di un problema di ottimizzazione"},l(e.p,{title:"Forma generale"},l("p",null,"Un problema con:"),l("ul",null,l("li",null,l(d.a,null,"Equazioni e disequazioni")),l("li",null,l(d.a,null,"Variabili non vincolate"))),l(e.o,null,yi(Y||(Y=Si`min \left\{ \mathbf{c}^T \mathbf{x} : \mathbf{A} \mathbf{x} = b,\quad \mathbf{A'} \mathbf{x} \geq \mathbf{b'} \quad x_j \geq 0,\quad j = 1 \dots n \right\}`)))),l(e.p,{title:"Forma canonica"},l("p",null,"Un problema con:"),l("ul",null,l("li",null,l(d.a,null,"Solo disequazioni")),l("li",null,l(c.a,null,"Vincoli di non-negatività sulle incognite"))),l(e.o,null,yi(K||(K=Si`min \left\{ \mathbf{c}^T \mathbf{x} : \mathbf{A} \mathbf{x} \geq b,\quad x_j \geq 0,\quad j = 1 \dots n \right\}`)))),l(e.p,{title:"Forma standard"},l("p",null,"Un problema con:"),l("ul",null,l("li",null,l(c.a,null,"Solo equazioni")),l("li",null,l(c.a,null,"Vincoli di non-negatività sulle incognite"))),l(e.o,null,yi($||($=Si`min \left\{ \mathbf{c}^T \mathbf{x} : \mathbf{A} \mathbf{x} = b,\quad x_j \geq 0,\quad j = 1 \dots n \right\}`))))),l(e.q,{title:"Conversioni tra le forme"},l(e.p,{title:"Standard e generale"},l("p",null,"Applica questa conversione a ogni equazione nel sistema:"),l("p",null,l(e.j,{inline:!1},yi(ll||(ll=Si`a = b \Leftrightarrow \begin{cases} a \leq b\\ a \geq b \end{cases} - `)))),l(a.a,null,"Serve solo nella teoria per dimostrare che le forme sono equivalenti.")),l(e.o,{title:"Canonica e standard"},l("p",null,"Aggiungi una ",l("i",null,"variabile slack")," ",l(e.i,null,Ti(il||(il=yi`s`)))," ",l("b",null,"non-vincolata")," a ogni disequazione nel sistema:"),l("p",null,l(e.i,{inline:!1},Ti(nl||(nl=yi` + `)))),l(a.a,null,"Serve solo nella teoria per dimostrare che le forme sono equivalenti.")),l(e.p,{title:"Canonica e standard"},l("p",null,"Aggiungi una ",l("i",null,"variabile slack")," ",l(e.j,null,yi(il||(il=Si`s`)))," ",l("b",null,"non-vincolata")," a ogni disequazione nel sistema:"),l("p",null,l(e.j,{inline:!1},yi(nl||(nl=Si` a \leq b \Leftrightarrow a + s = b - 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il risultato finale le seguenti trasformazioni:"),l("ul",null,l("li",null,l("b",null,"Moltiplicare")," un'intera riga per una costante."),l("li",null,l("b",null,"Sommare")," una riga a un'altra")),l(a.a,null,"Suona familiare? Sì, lo abbiamo fatto anche in Algebra Lineare.")),l(e.o,{title:"Variabili nella base"},l("p",null,"Variabili che hanno ",l("b",null,"tutti 0 e un solo 1")," nella loro colonna del tableau."),l("p",null,"La loro controparte sono le ",l("i",null,"variabili fuori base"),", che hanno qualsiasi altro valore.")),l(e.o,{title:"Valore attuale"},l("p",null,"Il valore della funzione obiettivo che si otterrebbe se ",l("b",null,"tutte le variabili fuori base valessero 0"),"."),l("p",null,"Procedendo nella risoluzione (descritta in seguito) del tableau, questo valore aumenterà, fino a raggiungere il valore ottimo quando la risoluzione sarà completata."))),l(e.p,null,l(e.o,{color:m.a,title:"Un esempio"},l("p",null,"Il sistema:"),l(e.n,null,Ti(ol||(ol=yi` + \end{cases}`)))))),l(e.q,null,"Valore attuale"),l(e.q,{title:"Tableau"},l(e.p,{title:"Cos'è?"},l("p",null,"Un modo per rappresentare sistemi in forma standard, anche noto come ",l("b",null,"matrice equivalente completa")," del sistema.")),l(e.p,{title:"Trasformazioni"},l("p",null,"Un tableau è un sistema di equazioni in ",l("b",null,"forma matriciale completa"),"."),l("p",null,"È possibile effettuare senza che cambi il risultato finale le seguenti trasformazioni:"),l("ul",null,l("li",null,l("b",null,"Moltiplicare")," un'intera riga per una costante."),l("li",null,l("b",null,"Sommare")," una riga a un'altra")),l(a.a,null,"Suona familiare? Sì, lo abbiamo fatto anche in Algebra Lineare.")),l(e.p,{title:"Variabili nella base"},l("p",null,"Variabili che hanno ",l("b",null,"tutti 0 e un solo 1")," nella loro colonna del tableau."),l("p",null,"La loro controparte sono le ",l("i",null,"variabili fuori base"),", che hanno qualsiasi altro valore.")),l(e.p,{title:"Valore attuale"},l("p",null,"Il valore della funzione obiettivo che si otterrebbe se ",l("b",null,"tutte le variabili fuori base valessero 0"),"."),l("p",null,"Procedendo nella risoluzione (descritta in seguito) del tableau, questo valore aumenterà, fino a raggiungere il valore ottimo quando la risoluzione sarà completata."))),l(e.q,null,l(e.p,{color:m.a,title:"Un esempio"},l("p",null,"Il sistema:"),l(e.o,null,yi(ol||(ol=Si` \begin{cases} 2000x_1 + 1000x_2 = z\\ 1x_1 \leq 3\\ 1x_2 \leq 3\\ 2x_1 + 2x_2 \leq 7 \end{cases} - `))),l("p",null,"Diventa il tableau:"),l("table",{className:"right"},l("thead",null,l("tr",null,l("th",null,l(e.i,null,"x_1")),l("th",null,l(e.i,null,"x_2")),l("th",null,l(e.i,null,"s_1")),l("th",null,l(e.i,null,"s_2")),l("th",null,l(e.i,null,"s_3")),l("th",null,l("abbr",{title:"Termine noto"},"TN")))),l("tbody",null,l("tr",null,l("td",null,l(e.i,null,"1")),l("td",null,l(e.i,null,"0")),l("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},l(e.i,null,"1")),l("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},l(e.i,null,"0")),l("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},l(e.i,null,"0")),l("td",{style:"background-color: rgba(255, 0, 0, 0.1);"},l(e.i,null,"3"))),l("tr",null,l("td",null,l(e.i,null,"0")),l("td",null,l(e.i,null,"1")),l("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},l(e.i,null,"0")),l("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},l(e.i,null,"1")),l("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},l(e.i,null,"0")),l("td",{style:"background-color: rgba(255, 0, 0, 0.1);"},l(e.i,null,"3"))),l("tr",null,l("td",null,l(e.i,null,"2")),l("td",null,l(e.i,null,"2")),l("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},l(e.i,null,"0")),l("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},l(e.i,null,"0")),l("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},l(e.i,null,"1")),l("td",{style:"background-color: rgba(255, 0, 0, 0.1);"},l(e.i,null,"7"))),l("tr",{style:"background-color: rgba(0, 255, 0, 0.1);"},l("td",null,l(e.i,null,"2000")),l("td",null,l(e.i,null,"1000")),l("td",null,l(e.i,null,"0")),l("td",null,l(e.i,null,"0")),l("td",null,l(e.i,null,"0")),l("td",{style:"background-color: rgba(255, 255, 0, 0.2);"},l(e.i,null,"0"))))),l("ul",null,l("li",null,l("u",{style:"color: #7dff7d;"},"Verde"),": funzione obiettivo"),l("li",null,l("u",{style:"color: #ffff7d;"},"Giallo"),": valore attuale della funzione"),l("li",null,l("u",{style:"color: #ff7d7d;"},"Rosso"),": termini noti"),l("li",null,l("u",{style:"color: #7d7dff;"},"Blu"),": Variabili slack")))),l(e.p,{title:"Simplex primale"},l(e.o,{title:"Cos'è?"},l("p",null,"Un algoritmo per trovare efficientemente il ",l("b",null,"valore ottimo")," e le coordinate di un ",l("b",null,"vertice ottimo")," in problemi di ottimizzazione lineare."),l(a.a,null,"Ricordi ",l(e.b,{href:"/vldigeometria"},"Gauss-Jordan"),"? Il Simplex è la stessa cosa, aggiungendo criteri per la selezione del pivot."),l(a.a,{title:"Esempio"},l("a",{href:"https://i.imgur.com/1r405Mb.jpg"},"Questa")," è la soluzione passo per passo del problema 3 del file ",l("a",{href:"https://dolly.fim.unimore.it/2019/mod/resource/view.php?id=2716"},l("code",null,"Ex_LP_testo")),"."),l("p",null,"Perchè sia possibile effettuare il Simplex è necessario che l'",l("b",null,"origine sia nel poliedro"),": pertanto, ",l("b",null,"non")," è possibile che un problema risolto con il Simplex sia ",l(o.a,null),".")),l(e.o,{title:"I passi"},l("ol",null,l("li",null,"Trasforma il sistema in ",l("b",null,"forma standard"),"."),l("li",null,"Trova tante variabili ",l("b",null,"linearmente indipendenti")," quante siano le righe: esse saranno la ",l("i",null,"base iniziale"),"."),l("li",null,"Finchè ci sono variabili con coefficienti ",l(r.a,null,"positivi"),"/",l(s.a,null,"negativi")," nella funzione obiettivo:",l("ol",null,l("li",null,l("b",null,"Scegli")," la prima variabile fuori base con coefficiente ",l(r.a,null,"positivo"),"/",l(s.a,null,"negativo")," nella funzione obiettivo: essa è la ",l("i",null,"variabile entrante"),".",l("br",null),l("aside",null,l("u",null,"Regola di Bland"),": Si potrebbe scegliere qualsiasi variabile come entrante, ma scegliendo sempre la prima ammissibile ci si assicura che l'algoritmo termini.")),l("li",null,l("b",null,"Scegli")," la variabile in base con il minor rapporto positivo ",l(e.i,null,Ti(tl||(tl=yi`\frac{termine\ noto}{coeff.\ variabile\ entrante}`))),".",l("aside",null,"Se non sei riuscito a trovare nessuna variabile con un rapporto positivo, significa che il poliedro è ",l(t.a,null),".")),l("li",null,l("u",null,"Pivot"),": ",l("b",null,"trasforma")," tutte le funzioni del sistema in modo che abbiano 0 nella colonna della variabile entrante, tranne nella riga della variabile uscente, in cui avrà 1."))),l("li",null,"Il poliedro è ",l(u.a,null),": i ",l("b",null,"termini noti dei vincoli")," sono le coordinate del suo vertice ottimo, mentre il ",l("b",null,"termine noto della funzione obiettivo")," è il valore ottimo."))),l(e.o,{title:"Soluzioni di base degenerata"},l("p",null,"Una soluzione con almeno una variabile di valore ",l(e.i,null,"0"),", dovuta a uno o più ",l("b",null,"vincoli ridondanti"),"."),l("p",null,"Senza ",l("b",null,"Regola di Bland")," e in presenza di vincoli ridondanti si rischia di trovarsi a fare pivot infiniti."))),l(e.p,{title:"Metodo delle due fasi"},l(e.o,{title:"Metodo delle due fasi"},l("p",null,"Un ",l("b",null,"estensione del Simplex")," per permettere la risoluzione di problemi la cui ",l("b",null,"origine non è una soluzione ammissibile"),"."),l("p",null,"Prevede l'introduzione di un ",l("i",null,"problema ausiliario"),", le cui incognite sono dette ",l("i",null,"artificiali"),"."),l("p",null,"Il vettore delle incognite artificiali è solitamente chiamato ",l(e.i,null,Ti(ul||(ul=yi`\mathbf{y}`))),".")),l(e.o,{title:"Procedimento"},l("ol",null,l("li",null,"Crea un nuovo tableau, ",l("b",null,"aggiungendo variabili artificiali")," in modo da avere una base ammissibile."),l("li",null,"Sostituisci la vecchia funzione obiettivo con una nuova che ",l("b",null,"minimizzi la somma")," di tutte le variabili artificiali."),l("li",null,l("u",null,"Fase 1"),": ",l("b",null,"Risolvi")," il nuovo problema con il Simplex primale."),l("li",null,"Se il Simplex termina quando ci sono ancora ",l("b",null,"variabili artificiali nella base"),", allora il poliedro è ",l("b",null,l(o.a,null)),"."),l("li",null,"Una volta che le variabili artificiali sono fuori base, ",l("b",null,"elimina")," le loro colonne e la nuova funzione obiettivo.",l("br",null)),l("li",null,"Riporta il tableau in forma base compiendo operazioni per ",l("b",null,"azzerare i coefficienti")," delle variabili di base nella funzione obiettivo."),l("li",null,l("u",null,"Fase 2"),": ",l("b",null,"Risolvi")," il tableau con il Simplex primale.")))),l(e.p,{title:"Rilassamento"},l(e.o,{title:"Cos'è?"},l("p",null,"Una versione semplificata di un problema nella quale si ",l("b",null,"ignora la violazione")," di uno o più vincoli.")),l(e.o,{title:"Rilassamento di Lagrange"},l("p",null,"Un rilassamento che permette di misurare ",l("b",null,"di quanto i vincoli vengono violati"),"."),l("p",null,"I vincoli, moltiplicati per ",l("b",null,"coefficienti di rilassamento"),", vengono inseriti nella funzione obiettivo."),l("p",null,"Il vettore dei coefficienti di rilassamento solitamente è indicato con ",l(e.i,null,Ti(rl||(rl=yi`\mathbf{u}`))),"."),l(a.a,null,l("p",null,"Il sistema:"),l(e.i,{inline:!1},Ti(sl||(sl=yi` + `))),l("p",null,"Diventa il tableau:"),l("table",{className:"right"},l("thead",null,l("tr",null,l("th",null,l(e.j,null,"x_1")),l("th",null,l(e.j,null,"x_2")),l("th",null,l(e.j,null,"s_1")),l("th",null,l(e.j,null,"s_2")),l("th",null,l(e.j,null,"s_3")),l("th",null,l("abbr",{title:"Termine noto"},"TN")))),l("tbody",null,l("tr",null,l("td",null,l(e.j,null,"1")),l("td",null,l(e.j,null,"0")),l("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},l(e.j,null,"1")),l("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},l(e.j,null,"0")),l("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},l(e.j,null,"0")),l("td",{style:"background-color: rgba(255, 0, 0, 0.1);"},l(e.j,null,"3"))),l("tr",null,l("td",null,l(e.j,null,"0")),l("td",null,l(e.j,null,"1")),l("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},l(e.j,null,"0")),l("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},l(e.j,null,"1")),l("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},l(e.j,null,"0")),l("td",{style:"background-color: rgba(255, 0, 0, 0.1);"},l(e.j,null,"3"))),l("tr",null,l("td",null,l(e.j,null,"2")),l("td",null,l(e.j,null,"2")),l("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},l(e.j,null,"0")),l("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},l(e.j,null,"0")),l("td",{style:"background-color: rgba(0, 0, 255, 0.1);"},l(e.j,null,"1")),l("td",{style:"background-color: rgba(255, 0, 0, 0.1);"},l(e.j,null,"7"))),l("tr",{style:"background-color: rgba(0, 255, 0, 0.1);"},l("td",null,l(e.j,null,"2000")),l("td",null,l(e.j,null,"1000")),l("td",null,l(e.j,null,"0")),l("td",null,l(e.j,null,"0")),l("td",null,l(e.j,null,"0")),l("td",{style:"background-color: rgba(255, 255, 0, 0.2);"},l(e.j,null,"0"))))),l("ul",null,l("li",null,l("u",{style:"color: #7dff7d;"},"Verde"),": funzione obiettivo"),l("li",null,l("u",{style:"color: #ffff7d;"},"Giallo"),": valore attuale della funzione"),l("li",null,l("u",{style:"color: #ff7d7d;"},"Rosso"),": termini noti"),l("li",null,l("u",{style:"color: #7d7dff;"},"Blu"),": Variabili slack")))),l(e.q,{title:"Simplex primale"},l(e.p,{title:"Cos'è?"},l("p",null,"Un algoritmo per trovare efficientemente il ",l("b",null,"valore ottimo")," e le coordinate di un ",l("b",null,"vertice ottimo")," in problemi di ottimizzazione lineare."),l(a.a,null,"Ricordi ",l(e.b,{href:"/vldigeometria"},"Gauss-Jordan"),"? Il Simplex è la stessa cosa, aggiungendo criteri per la selezione del pivot."),l(a.a,{title:"Esempio"},l("a",{href:"https://i.imgur.com/1r405Mb.jpg"},"Questa")," è la soluzione passo per passo del problema 3 del file ",l("a",{href:"https://dolly.fim.unimore.it/2019/mod/resource/view.php?id=2716"},l("code",null,"Ex_LP_testo")),"."),l("p",null,"Perchè sia possibile effettuare il Simplex è necessario che l'",l("b",null,"origine sia nel poliedro"),": pertanto, ",l("b",null,"non")," è possibile che un problema risolto con il Simplex sia ",l(o.a,null),".")),l(e.p,{title:"I passi"},l("ol",null,l("li",null,"Trasforma il sistema in ",l("b",null,"forma standard"),"."),l("li",null,"Trova tante variabili ",l("b",null,"linearmente indipendenti")," quante siano le righe: esse saranno la ",l("i",null,"base iniziale"),"."),l("li",null,"Finchè ci sono variabili con coefficienti ",l(r.a,null,"positivi"),"/",l(s.a,null,"negativi")," nella funzione obiettivo:",l("ol",null,l("li",null,l("b",null,"Scegli")," la prima variabile fuori base con coefficiente ",l(r.a,null,"positivo"),"/",l(s.a,null,"negativo")," nella funzione obiettivo: essa è la ",l("i",null,"variabile entrante"),".",l("br",null),l("aside",null,l("u",null,"Regola di Bland"),": Si potrebbe scegliere qualsiasi variabile come entrante, ma scegliendo sempre la prima ammissibile ci si assicura che l'algoritmo termini.")),l("li",null,l("b",null,"Scegli")," la variabile in base con il minor rapporto positivo ",l(e.j,null,yi(tl||(tl=Si`\frac{termine\ noto}{coeff.\ variabile\ entrante}`))),".",l("aside",null,"Se non sei riuscito a trovare nessuna variabile con un rapporto positivo, significa che il poliedro è ",l(t.a,null),".")),l("li",null,l("u",null,"Pivot"),": ",l("b",null,"trasforma")," tutte le funzioni del sistema in modo che abbiano 0 nella colonna della variabile entrante, tranne nella riga della variabile uscente, in cui avrà 1."))),l("li",null,"Il poliedro è ",l(u.a,null),": i ",l("b",null,"termini noti dei vincoli")," sono le coordinate del suo vertice ottimo, mentre il ",l("b",null,"termine noto della funzione obiettivo")," è il valore ottimo."))),l(e.p,{title:"Soluzioni di base degenerata"},l("p",null,"Una soluzione con almeno una variabile di valore ",l(e.j,null,"0"),", dovuta a uno o più ",l("b",null,"vincoli ridondanti"),"."),l("p",null,"Senza ",l("b",null,"Regola di Bland")," e in presenza di vincoli ridondanti si rischia di trovarsi a fare pivot infiniti."))),l(e.q,{title:"Metodo delle due fasi"},l(e.p,{title:"Metodo delle due fasi"},l("p",null,"Un ",l("b",null,"estensione del Simplex")," per permettere la risoluzione di problemi la cui ",l("b",null,"origine non è una soluzione ammissibile"),"."),l("p",null,"Prevede l'introduzione di un ",l("i",null,"problema ausiliario"),", le cui incognite sono dette ",l("i",null,"artificiali"),"."),l("p",null,"Il vettore delle incognite artificiali è solitamente chiamato ",l(e.j,null,yi(ul||(ul=Si`\mathbf{y}`))),".")),l(e.p,{title:"Procedimento"},l("ol",null,l("li",null,"Crea un nuovo tableau, ",l("b",null,"aggiungendo variabili artificiali")," in modo da avere una base ammissibile."),l("li",null,"Sostituisci la vecchia funzione obiettivo con una nuova che ",l("b",null,"minimizzi la somma")," di tutte le variabili artificiali."),l("li",null,l("u",null,"Fase 1"),": ",l("b",null,"Risolvi")," il nuovo problema con il Simplex primale."),l("li",null,"Se il Simplex termina quando ci sono ancora ",l("b",null,"variabili artificiali nella base"),", allora il poliedro è ",l("b",null,l(o.a,null)),"."),l("li",null,"Una volta che le variabili artificiali sono fuori base, ",l("b",null,"elimina")," le loro colonne e la nuova funzione obiettivo.",l("br",null)),l("li",null,"Riporta il tableau in forma base compiendo operazioni per ",l("b",null,"azzerare i coefficienti")," delle variabili di base nella funzione obiettivo."),l("li",null,l("u",null,"Fase 2"),": ",l("b",null,"Risolvi")," il tableau con il Simplex primale.")))),l(e.q,{title:"Rilassamento"},l(e.p,{title:"Cos'è?"},l("p",null,"Una versione semplificata di un problema nella quale si ",l("b",null,"ignora la violazione")," di uno o più vincoli.")),l(e.p,{title:"Rilassamento di Lagrange"},l("p",null,"Un rilassamento che permette di misurare ",l("b",null,"di quanto i vincoli vengono violati"),"."),l("p",null,"I vincoli, moltiplicati per ",l("b",null,"coefficienti di rilassamento"),", vengono inseriti nella funzione obiettivo."),l("p",null,"Il vettore dei coefficienti di rilassamento solitamente è indicato con ",l(e.j,null,yi(rl||(rl=Si`\mathbf{u}`))),"."),l(a.a,null,l("p",null,"Il sistema:"),l(e.j,{inline:!1},yi(sl||(sl=Si` \begin{cases} z = 3 x_1 + 5 x_2\\ 2 x_1 + 3 x_2 \geq 12\\ @@ -26,15 +26,15 @@ x_1 \geq 0\\ x_2 \geq 0 \end{cases} - `))),l("p",null,"diventa:"),l(e.i,{inline:!1},Ti(dl||(dl=yi` + `))),l("p",null,"diventa:"),l(e.j,{inline:!1},yi(dl||(dl=Si` \begin{cases} z = 3 x_1 + 5 x_2 + u_1 ( 12 - 2 x_1 - 3 x_2 ) + u_2 ( 3 + x_1 - 3 x_2 )\\ x_1 \geq 0\\ x_2 \geq 0 \end{cases} - `)))))),l(e.p,{title:"Dualità"},l(e.o,{title:"Duale"},l("p",null,"Il sistema che ",l("b",null,l(r.a,null,"massimizza"),"/",l(s.a,null,"minimizza")," i moltiplicatori di rilassamento")," di un problema detto ",l("i",null,"primale"),".")),l(e.o,{title:"In termini matriciali"},l("p",null,"Possiamo ",l("b",null,"trasporre")," il tableau e sostituire le variabili ",l(e.i,null,Ti(cl||(cl=yi`x_n`)))," con variabili ",l(e.i,null,Ti(ml||(ml=yi`u_n`)))," per ottenere il sistema duale!"),l("p",null,"I maggiori e minori dei vincoli diventeranno maggiori e minori delle variabili e viceversa.")),l(e.o,{title:"Feasibility del duale"},l("ul",null,l("li",null,"Se un problema ha una ",l("b",null,"soluzione finita"),", allora anche il suo duale la avrà."),l("li",null,"Se un problema è ",l("b",null,l(o.a,null)),", allora il suo duale potrà essere ",l(o.a,null)," oppure ",l(t.a,null),"."),l("li",null,"Se un problema è ",l("b",null,l(t.a,null)),", allora il suo duale sarà certamente ",l(o.a,null),"."))),l(e.o,{title:"Variabili e vincoli"},l("p",null,"Variabili e vincoli del duale corrispondono rispettivamente a vincoli e variabili del primale."),l("p",null,"In particolare:"),l("table",null,l("thead",null,l("tr",null,l("th",null,l(r.a,null,"Min")),l("th",null,l(s.a,null,"Max")))),l("tbody",null,l("tr",null,l("td",null,"Vincolo ",l(e.g,null,"\\leq")),l("td",null,"Variabile ",l(e.g,null,"\\leq"))),l("tr",null,l("td",null,"Vincolo ",l(e.g,null,"\\geq")),l("td",null,"Variabile ",l(e.g,null,"\\geq"))),l("tr",null,l("td",null,"Vincolo ",l(e.g,null,"=")),l("td",null,"Variabile ",l("b",null,"libera"))),l("tr",null,l("td",null,"Variabile ",l(e.g,null,"\\leq")),l("td",null,"Vincolo ",l(e.g,null,"\\geq"))),l("tr",null,l("td",null,"Variabile ",l(e.g,null,"\\geq")),l("td",null,"Vincolo ",l(e.g,null,"\\leq"))),l("tr",null,l("td",null,"Variabile ",l("b",null,"libera")),l("td",null,"Vincolo ",l(e.g,null,"="))))))),l(e.p,{title:"Un po' di teoria"},l(e.o,{title:"Lemma di Farkas"},l("p",null,"Una disuguaglianza lineare ",l(e.i,null,Ti(bl||(bl=yi`c_0 \leq \mathbf{c}^T \mathbf{x}`)))," è verificata da tutti i punti di un poliedro non-",l(o.a,null)," se e solo se esiste un vettore ",l(e.i,null,Ti(pl||(pl=yi`u \in \mathfrak{R}^m`)))," tale che:"),l(e.n,null,Ti(fl||(fl=yi`\mathbf{c}^T \geq \mathbf{u}^T \mathbf{A}`))),l(e.n,null,Ti(gl||(gl=yi`c_0 \leq \mathbf{u}^T \mathbf{b}`)))),l(e.o,{title:"Dualità forte"},l("p",null,"Il teorema che dimostra l'equivalenza tra primale e duale."),l("p",null,"Se uno dei due problemi è finito, la soluzione di uno coincide con la soluzione dell'altro."),l("p",null,l(e.i,null,Ti(hl||(hl=yi`\mathbf{c}^T \mathbf{x} = \mathbf{u}^T \mathbf{b}`)))),l("p",null,l(e.s,null,"TODO: Anche qui c'è una lunga dimostrazione..."))),l(e.o,{title:"Dualità debole"},l("p",null,"Il teorema che dimostra che il valore della funzione obiettivo del duale (di un qualsiasi tableau) è sempre ",l(r.a,null,"minore o uguale"),"/",l(s.a,null,"maggiore o uguale")," alla soluzione del corrispettivo primale."),l("p",null,l(e.s,null,"TODO: Dimostrazione cortina, ma sembra complicata."))),l(e.o,{title:"Condizioni di ottimalità"},l("p",null,"Il teorema che ci permette di passare dalla soluzione del duale alla soluzione del primale. ",l(e.s,null,"TODO: credo?")),l("p",null,"Si deriva combinando le seguenti condizioni:"),l("ul",null,l("li",null,"Ammissibilità del primale: ",l(e.i,null,Ti(vl||(vl=yi`\mathbf{A} \mathbf{X} \geq \mathbf{b}, \quad \mathbf{x} \geq 0`)))),l("li",null,"Ammissibilità del duale: ",l(e.i,null,Ti(_l||(_l=yi`\mathbf{u}^T \mathbf{A} \leq \mathbf{c}^T, \quad \mathbf{u} \geq 0`)))),l("li",null,"Teorema della dualità forte: ",l(e.i,null,Ti(zl||(zl=yi`\mathbf{c}^T \mathbf{x} = \mathbf{u}^T \mathbf{b}`)))," (alla soluzione ottima)")),l("p",null,"Ne risulta che una soluzione è ottima se e solo se:"),l(e.n,null,Ti(xl||(xl=yi`\left( \mathbf{c}^T - \mathbf{u}^T \mathbf{A} \right) \mathbf{x} = 0`))),l(e.n,null,Ti(ql||(ql=yi`\mathbf{u}^T \left( \mathbf{A} \mathbf{x} - \mathbf{b} \right) = 0`))))),l(e.p,{title:"Simplex duale"},l(e.o,{title:"Cos'è?"},l("p",null,"Un'estensione al Simplex primale che opera sul problema duale.")),l(e.o,{title:"Come funziona?"},l("p",null,"Funziona esattamente come il Simplex primale, ma opera sul duale."))),l(e.p,{title:"Analisi di sensibilità"},l(e.o,{title:"Cos'è?"},l("p",null,"Un procedimento che misura di ",l("b",null,"quanto può variare")," il termine noto di un vincolo ",l(e.i,null,Ti(Sl||(Sl=yi`b_i`)))," o il coefficiente della funzione obiettivo ",l(e.i,null,Ti(yl||(yl=yi`c_i`)))," prima che la base degeneri."))),l(e.p,{title:"Ottimizzazione lineare intera"},l(e.o,{title:"Cos'è?"},l("p",null,"Particolari problemi di ottimizzazione lineare in cui le ",l("b",null,"variabili sono vincolate ad essere numeri interi"),"."),l(e.n,null,Ti(Tl||(Tl=yi` + `)))))),l(e.q,{title:"Dualità"},l(e.p,{title:"Duale"},l("p",null,"Il sistema che ",l("b",null,l(r.a,null,"massimizza"),"/",l(s.a,null,"minimizza")," i moltiplicatori di rilassamento")," di un problema detto ",l("i",null,"primale"),".")),l(e.p,{title:"In termini matriciali"},l("p",null,"Possiamo ",l("b",null,"trasporre")," il tableau e sostituire le variabili ",l(e.j,null,yi(cl||(cl=Si`x_n`)))," con variabili ",l(e.j,null,yi(ml||(ml=Si`u_n`)))," per ottenere il sistema duale!"),l("p",null,"I maggiori e minori dei vincoli diventeranno maggiori e minori delle variabili e viceversa.")),l(e.p,{title:"Feasibility del duale"},l("ul",null,l("li",null,"Se un problema ha una ",l("b",null,"soluzione finita"),", allora anche il suo duale la avrà."),l("li",null,"Se un problema è ",l("b",null,l(o.a,null)),", allora il suo duale potrà essere ",l(o.a,null)," oppure ",l(t.a,null),"."),l("li",null,"Se un problema è ",l("b",null,l(t.a,null)),", allora il suo duale sarà certamente ",l(o.a,null),"."))),l(e.p,{title:"Variabili e vincoli"},l("p",null,"Variabili e vincoli del duale corrispondono rispettivamente a vincoli e variabili del primale."),l("p",null,"In particolare:"),l("table",null,l("thead",null,l("tr",null,l("th",null,l(r.a,null,"Min")),l("th",null,l(s.a,null,"Max")))),l("tbody",null,l("tr",null,l("td",null,"Vincolo ",l(e.h,null,"\\leq")),l("td",null,"Variabile ",l(e.h,null,"\\leq"))),l("tr",null,l("td",null,"Vincolo ",l(e.h,null,"\\geq")),l("td",null,"Variabile ",l(e.h,null,"\\geq"))),l("tr",null,l("td",null,"Vincolo ",l(e.h,null,"=")),l("td",null,"Variabile ",l("b",null,"libera"))),l("tr",null,l("td",null,"Variabile ",l(e.h,null,"\\leq")),l("td",null,"Vincolo ",l(e.h,null,"\\geq"))),l("tr",null,l("td",null,"Variabile ",l(e.h,null,"\\geq")),l("td",null,"Vincolo ",l(e.h,null,"\\leq"))),l("tr",null,l("td",null,"Variabile ",l("b",null,"libera")),l("td",null,"Vincolo ",l(e.h,null,"="))))))),l(e.q,{title:"Un po' di teoria"},l(e.p,{title:"Lemma di Farkas"},l("p",null,"Una disuguaglianza lineare ",l(e.j,null,yi(bl||(bl=Si`c_0 \leq \mathbf{c}^T \mathbf{x}`)))," è verificata da tutti i punti di un poliedro non-",l(o.a,null)," se e solo se esiste un vettore ",l(e.j,null,yi(pl||(pl=Si`u \in \mathfrak{R}^m`)))," tale che:"),l(e.o,null,yi(fl||(fl=Si`\mathbf{c}^T \geq \mathbf{u}^T \mathbf{A}`))),l(e.o,null,yi(gl||(gl=Si`c_0 \leq \mathbf{u}^T \mathbf{b}`)))),l(e.p,{title:"Dualità forte"},l("p",null,"Il teorema che dimostra l'equivalenza tra primale e duale."),l("p",null,"Se uno dei due problemi è finito, la soluzione di uno coincide con la soluzione dell'altro."),l("p",null,l(e.j,null,yi(hl||(hl=Si`\mathbf{c}^T \mathbf{x} = \mathbf{u}^T \mathbf{b}`)))),l("p",null,l(e.t,null,"TODO: Anche qui c'è una lunga dimostrazione..."))),l(e.p,{title:"Dualità debole"},l("p",null,"Il teorema che dimostra che il valore della funzione obiettivo del duale (di un qualsiasi tableau) è sempre ",l(r.a,null,"minore o uguale"),"/",l(s.a,null,"maggiore o uguale")," alla soluzione del corrispettivo primale."),l("p",null,l(e.t,null,"TODO: Dimostrazione cortina, ma sembra complicata."))),l(e.p,{title:"Condizioni di ottimalità"},l("p",null,"Il teorema che ci permette di passare dalla soluzione del duale alla soluzione del primale. ",l(e.t,null,"TODO: credo?")),l("p",null,"Si deriva combinando le seguenti condizioni:"),l("ul",null,l("li",null,"Ammissibilità del primale: ",l(e.j,null,yi(vl||(vl=Si`\mathbf{A} \mathbf{X} \geq \mathbf{b}, \quad \mathbf{x} \geq 0`)))),l("li",null,"Ammissibilità del duale: ",l(e.j,null,yi(_l||(_l=Si`\mathbf{u}^T \mathbf{A} \leq \mathbf{c}^T, \quad \mathbf{u} \geq 0`)))),l("li",null,"Teorema della dualità forte: ",l(e.j,null,yi(zl||(zl=Si`\mathbf{c}^T \mathbf{x} = \mathbf{u}^T \mathbf{b}`)))," (alla soluzione ottima)")),l("p",null,"Ne risulta che una soluzione è ottima se e solo se:"),l(e.o,null,yi(xl||(xl=Si`\left( \mathbf{c}^T - \mathbf{u}^T \mathbf{A} \right) \mathbf{x} = 0`))),l(e.o,null,yi(jl||(jl=Si`\mathbf{u}^T \left( \mathbf{A} \mathbf{x} - \mathbf{b} \right) = 0`))))),l(e.q,{title:"Simplex duale"},l(e.p,{title:"Cos'è?"},l("p",null,"Un'estensione al Simplex primale che opera sul problema duale.")),l(e.p,{title:"Come funziona?"},l("p",null,"Funziona esattamente come il Simplex primale, ma opera sul duale."))),l(e.q,{title:"Analisi di sensibilità"},l(e.p,{title:"Cos'è?"},l("p",null,"Un procedimento che misura di ",l("b",null,"quanto può variare")," il termine noto di un vincolo ",l(e.j,null,yi(ql||(ql=Si`b_i`)))," o il coefficiente della funzione obiettivo ",l(e.j,null,yi(Sl||(Sl=Si`c_i`)))," prima che la base degeneri."))),l(e.q,{title:"Ottimizzazione lineare intera"},l(e.p,{title:"Cos'è?"},l("p",null,"Particolari problemi di ottimizzazione lineare in cui le ",l("b",null,"variabili sono vincolate ad essere numeri interi"),"."),l(e.o,null,yi(yl||(yl=Si` \mathbf{x} \in \mathbb{Z}^n - `))),l("p",null,"Spesso detti anche ",l("i",null,"problemi di ",l("abbr",{title:"Integer Linear Programming"},"ILP")),".")),l(e.o,{title:"Rilassamento lineare"},l("p",null,"Un rilassamento che rimuove il ",l("b",null,"vincolo di integrità")," a un problema, trovando la sua ",l("b",null,"soluzione continua"),"."))),l(e.p,{title:"Dal rilassamento alla soluzione"},l(e.o,{title:"Enumerazione totale"},l("p",null,"Un ",l("b",null,"modo")," per passare dalla soluzione del rilassamento alla soluzione intera di un problema di ILP."),l("p",null,"Consiste nel calcolare la soluzione di ogni singolo punto incluso nel poliedro, e selezionare la ",l(r.a,null,"minore"),"/",l(s.a,null,"maggiore"),"."),l("p",null,"Trova ",l("b",null,"sicuramente")," la soluzione giusta, ma il costo computazionale è esponenziale ",l(e.g,null,"O(n^k)"),"!")),l(e.o,{title:"Arrotondamento"},l("p",null,"Un altro ",l("b",null,"modo")," per passare dalla soluzione del rilassamento alla soluzione intera di un problema di ILP."),l("p",null,"Consiste nell'",l("b",null,"arrotondare tutte le variabili al loro valore intero più vicino"),", e calcolarne il valore ottimo."),l("p",null,"Funziona bene per valori grandi, ma più essi si avvicinano allo 0 più l'",l("b",null,"errore diventa grande"),".")),l(e.o,{title:"Piani secanti"},l("p",null,"Un altro ",l("b",null,"modo")," ancora per passare dalla soluzione del rilassamento alla soluzione intera di un problema di ILP."),l("p",null,"Consiste nel tagliare il poliedro con nuovi vincoli (",l("i",null,"piani secanti"),") che ",l("b",null,"riducono le possibili soluzioni continue")," ma non quelle intere."),l("p",null,"Per selezionare i vincoli, si usano i ",l("b",null,"tagli di Gomory"),":"),l(e.n,null,Ti(Il||(Il=yi` + `))),l("p",null,"Spesso detti anche ",l("i",null,"problemi di ",l("abbr",{title:"Integer Linear Programming"},"ILP")),".")),l(e.p,{title:"Rilassamento lineare"},l("p",null,"Un rilassamento che rimuove il ",l("b",null,"vincolo di integrità")," a un problema, trovando la sua ",l("b",null,"soluzione continua"),"."))),l(e.q,{title:"Dal rilassamento alla soluzione"},l(e.p,{title:"Enumerazione totale"},l("p",null,"Un ",l("b",null,"modo")," per passare dalla soluzione del rilassamento alla soluzione intera di un problema di ILP."),l("p",null,"Consiste nel calcolare la soluzione di ogni singolo punto incluso nel poliedro, e selezionare la ",l(r.a,null,"minore"),"/",l(s.a,null,"maggiore"),"."),l("p",null,"Trova ",l("b",null,"sicuramente")," la soluzione giusta, ma il costo computazionale è esponenziale ",l(e.h,null,"O(n^k)"),"!")),l(e.p,{title:"Arrotondamento"},l("p",null,"Un altro ",l("b",null,"modo")," per passare dalla soluzione del rilassamento alla soluzione intera di un problema di ILP."),l("p",null,"Consiste nell'",l("b",null,"arrotondare tutte le variabili al loro valore intero più vicino"),", e calcolarne il valore ottimo."),l("p",null,"Funziona bene per valori grandi, ma più essi si avvicinano allo 0 più l'",l("b",null,"errore diventa grande"),".")),l(e.p,{title:"Piani secanti"},l("p",null,"Un altro ",l("b",null,"modo")," ancora per passare dalla soluzione del rilassamento alla soluzione intera di un problema di ILP."),l("p",null,"Consiste nel tagliare il poliedro con nuovi vincoli (",l("i",null,"piani secanti"),") che ",l("b",null,"riducono le possibili soluzioni continue")," ma non quelle intere."),l("p",null,"Per selezionare i vincoli, si usano i ",l("b",null,"tagli di Gomory"),":"),l(e.o,null,yi(Tl||(Tl=Si` \sum_{j \in F} \left( \left( a_{tj} - \lfloor a_{tj} \rfloor \right) \cdot x_j \right) \geq (b_t - \lfloor b_t \rfloor) - `))),l("p",null,"Per ogni valore noto frazionario si viene quindi a creare ",l("b",null,"una nuova variabile in base")," e un nuovo vincolo formato dall'opposto di tutti i valori frazionari dei coefficienti fuori base."),l(a.a,null,l("p",null,"Il tableau:",l("table",null,l("thead",null,l("tr",null,l("th",null,l(e.a,null,Ti(kl||(kl=yi`x_1`)))),l("th",null,l(e.a,null,Ti(Pl||(Pl=yi`x_2`)))),l("th",null,l(e.a,null,Ti(Al||(Al=yi`s_1`)))),l("th",null,l(e.a,null,Ti(Ll||(Ll=yi`s_2`)))),l("th",null,l("abbr",{title:"Termine 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l'",l("b",null,"enumerazione totale"),"."),l("p",null,"Si divide il problema principale (trovare il valore ottimo di un problema di ILP) in più sottoproblemi (trovare il valore ottimo di un problema di ILP con una variabile impostata a un valore fisso)."),l("p",null,"Si crea così un ",l("b",null,"albero"),"."),l("p",null,"È possibile ",l("b",null,"chiudere in anticipo")," alcuni nodi dell'albero se il loro miglior possibile valore ottimo è inferiore a uno precedentemente trovato o se il loro poliedro è ",l(o.a,null),"."),l("p",null,"È possibile utilizzare diverse ",l("b",null,"strategie di esplorazione")," dell'albero:",l("ul",null,l("li",null,l("b",null,"depth-first"),": permette di raggiungere immediatamente a una soluzione accettabile (ma non ottimale)"),l("li",null,l("b",null,"best-first"),": permette di raggiungere più velocemente alla soluzione corretta")))),l(e.o,{title:"Seca et impera"},l("p",null,"È possibile combinare il metodo dei ",l("b",null,"tagli secanti")," con la tecnica ",l("b",null,"divide et impera")," per raggiungere ancora più velocemente a una soluzione."),l("p",null,"Si effettuano ",l("b",null,"poche iterazioni")," del metodo dei tagli secanti, e sul risultato di quelle iterazioni si applica il ",l("b",null,"divide et impera"),"."))),l(e.p,{title:l("span",null,"Terminologia dei grafi ",l(e.s,null,"TODO: migliorare"))},l(e.o,{title:"Grafo"},l("p",null,"Insieme di ",l("b",null,"nodi")," ",l(e.g,null,Ti(xi||(xi=yi`N`)))," e ",l("b",null,"archi")," ",l(e.g,null,Ti(qi||(qi=yi`E`)))," che li connettono."),l("p",null,"Può essere ",l("b",null,"diretto")," se gli archi hanno una direzione.")),l(e.o,{title:"Nodi adiacenti"},l("p",null,"Nodi ",l("b",null,"connessi da un arco"),".")),l(e.o,{title:"Arco incidente"},l("p",null,"Arco ",l("b",null,"connesso a un dato nodo"),".")),l(e.o,{title:"Arco entrante o uscente"},l("p",null,"Un arco diretto che ",l("b",null,"termina")," o ",l("b",null,"inizia")," da un dato nodo.")),l(e.o,{title:"Grado"},l("p",null,l("b",null,"Conteggio")," degli archi incidenti di un nodo."),l("p",null,"Si può calcolare anche relativamente agli archi entranti o agli archi uscenti.")),l(e.o,{title:"Percorso"},l("p",null,"Sequenza di ",l("b",null,"archi consecutivi"),".")),l(e.o,{title:"Connessione"},l("p",null,"Due nodi sono connessi se tra loro esiste ",l("b",null,"almeno un percorso"),"."),l("p",null,"Un grafo è connesso se tutti i suoi nodi sono connessi.")),l(e.o,{title:"Cicli e circuiti"},l("p",null,"Percorsi rispettivamente indiretti e diretti in cui l'inizio coincide con la fine.")),l(e.o,{title:"Grafo completo"},l("p",null,"Grafo in cui ogni nodo è connesso con ogni altro."),l("p",null,"Se diretto, contiene ",l(e.g,null,Ti(Si||(Si=yi`n \cdot (n - 1)`)))," archi; altrimenti, ne contiene la metà.")),l(e.o,{title:"Matrice di adiacenza"},l("blockquote",null,"Vedi ",l(e.b,{href:"/algoritmiestrutturedati"},"Algoritmi"),".")),l(e.o,{title:"Lista di adiacenza"},l("blockquote",null,"Vedi ",l(e.b,{href:"/algoritmiestrutturedati"},"Algoritmi"),".")),l(e.o,{title:"Taglio"},l("p",null,"Sottoinsieme di archi che connettono due sottoinsiemi di nodi."),l("p",null,"Può essere anche uscente o entrante; in tal caso include solo gli archi entranti o uscenti dal sottoinsieme.")),l(e.o,{title:"Sottografo"},l("p",null,"Sottoinsieme di nodi e archi di un grafo."),l("p",null,"Tutti gli archi di un sottografo possono connettere solo nodi all'interno di esso.")),l(e.o,{title:"Albero"},l("p",null,"Sottografo connesso e aciclico.")),l(e.o,{title:"Spanning tree"},l("p",null,"Albero che include tutti i nodi di un grafo."))),l(e.p,{title:"Algoritmi con i grafi"},l(e.o,{title:"Prim"},l("p",null,"Crea uno spanning tree."),l("ol",null,l("li",null,"Aggiungi l'arco di costo minimo all'albero."),l("li",null,"Finchè mancano ancora archi:",l("ol",null,l("li",null,"Trova tutti gli archi che aggiungerebbero un nuovo nodo all'albero."),l("li",null,"Seleziona l'arco di costo minore."))))),l(e.o,{title:"Ordine topologico"},l("p",null,"Trova l'ordine topologico di un albero."),l("ol",null,l("li",null,"Ripeti finchè ci sono nodi nel grafo:",l("ol",null,l("li",null,"Assegna un numero sequenziale a un nodo senza archi entranti."),l("li",null,"Elimina il nodo a cui hai assegnato il numero."),l("li",null,"Elimina tutti gli archi incidenti sul nodo che hai eliminato."))))),l(e.o,{title:"Percorsi minimi in grafo diretto"},l("p",null,"Trova i percorsi di costo minimo in un albero."),l("ol",null,l("li",null,"Trova l'ordine topologico dell'albero."),l("li",null,"Invece che provare ogni singola combinazione di nodi, prova solo i nodi che hanno un numero topologico maggiore di quello del nodo attuale.")),l("p",null,l(e.s,null,"TODO: forse spiegarlo meglio non farebbe male"))),l(e.o,{title:"Algoritmo di Dijkstra"},l("blockquote",null,"Vedi ",l(e.b,{href:"/algoritmiestrutturedati"},"Algoritmi"),".")),l(e.o,{title:"Algoritmo di Ford-Fulkerson"},l(a.a,null,"Trova il volume massimo di acqua che è possibile fare scorrere attraverso tubature con una data capacità."),l("p",null,"Costruisci il grafo residuo e vedi se c'è un percorso che va dalla sorgente alla destinazione."),l("p",null,l(e.h,{src:"https://i.imgur.com/FJk44q0.png"})),l("p",null,l(e.h,{src:"https://i.imgur.com/fzb6xz2.png"})))))}}.call(this,n("hosL").h)},jZax:function(l){l.exports={red:"red__ipSoq",orange:"orange__1nmkj",yellow:"yellow__353ge",lime:"lime__3FW1E",cyan:"cyan__2xhu7",blue:"blue__3MvDe",magenta:"magenta__i03ie",plus:"plus__1eV4_"}},ke5e:function(l,i,n){"use strict";(function(l){var e=n("2w3n"),a=n.n(e);i.a=function(i){return l("div",{class:a.a.example},i.children)}}).call(this,n("hosL").h)},mGtH:function(l,i,n){"use strict";var e=n("uzfk");i.a=n.n(e).a.example},o8Jj:function(l,i,n){"use strict";(function(l){var e=n("jZax"),a=n.n(e);i.a=function(i){return 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l'",l("b",null,"enumerazione totale"),"."),l("p",null,"Si divide il problema principale (trovare il valore ottimo di un problema di ILP) in più sottoproblemi (trovare il valore ottimo di un problema di ILP con una variabile impostata a un valore fisso)."),l("p",null,"Si crea così un ",l("b",null,"albero"),"."),l("p",null,"È possibile ",l("b",null,"chiudere in anticipo")," alcuni nodi dell'albero se il loro miglior possibile valore ottimo è inferiore a uno precedentemente trovato o se il loro poliedro è ",l(o.a,null),"."),l("p",null,"È possibile utilizzare diverse ",l("b",null,"strategie di esplorazione")," dell'albero:",l("ul",null,l("li",null,l("b",null,"depth-first"),": permette di raggiungere immediatamente a una soluzione accettabile (ma non ottimale)"),l("li",null,l("b",null,"best-first"),": permette di raggiungere più velocemente alla soluzione corretta")))),l(e.p,{title:"Seca et impera"},l("p",null,"È possibile combinare il metodo dei ",l("b",null,"tagli secanti")," con la 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di costo minore."))))),l(e.p,{title:"Ordine topologico"},l("p",null,"Trova l'ordine topologico di un albero."),l("ol",null,l("li",null,"Ripeti finchè ci sono nodi nel grafo:",l("ol",null,l("li",null,"Assegna un numero sequenziale a un nodo senza archi entranti."),l("li",null,"Elimina il nodo a cui hai assegnato il numero."),l("li",null,"Elimina tutti gli archi incidenti sul nodo che hai eliminato."))))),l(e.p,{title:"Percorsi minimi in grafo diretto"},l("p",null,"Trova i percorsi di costo minimo in un albero."),l("ol",null,l("li",null,"Trova l'ordine topologico dell'albero."),l("li",null,"Invece che provare ogni singola combinazione di nodi, prova solo i nodi che hanno un numero topologico maggiore di quello del nodo attuale.")),l("p",null,l(e.t,null,"TODO: forse spiegarlo meglio non farebbe male"))),l(e.p,{title:"Algoritmo di Dijkstra"},l("blockquote",null,"Vedi ",l(e.b,{href:"/algoritmiestrutturedati"},"Algoritmi"),".")),l(e.p,{title:"Algoritmo di Ford-Fulkerson"},l(a.a,null,"Trova 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extracted by mini-css-extract-plugin\nmodule.exports = {\"red\":\"red__2y1B_\",\"orange\":\"orange__dD2kx\",\"yellow\":\"yellow__OEpwl\",\"lime\":\"lime__CVe41\",\"cyan\":\"cyan__26ZAg\",\"blue\":\"blue__LO7Xm\",\"magenta\":\"magenta__1Akee\",\"example\":\"example__2PzAa\"};","import {ILatex, Panel, PLatex, Section} from \"bluelib\";\nimport Example from \"../components/Example\";\n\nconst r = String.raw;\n\n\nexport default function(params) {\n return (\n
\n

Ripasso di Algebra Lineare per Calcolo Numerico

\n
\n \n

\n Elemento neutro della moltiplicazione matriciale.\n

\n \n {r`\n \\begin{pmatrix}\n {\\color{Yellow} 1} & {\\color{Yellow} 0} & {\\color{Yellow} 0}\\\\\n {\\color{Yellow} 0} & {\\color{Yellow} 1} & {\\color{Yellow} 0}\\\\\n {\\color{Yellow} 0} & {\\color{Yellow} 0} & {\\color{Yellow} 1}\n \\end{pmatrix}\n `}\n \n \n \n

\n Matrice con elementi diversi da 0 solo sulla diagonale.\n

\n \n {r`\n \\begin{pmatrix}\n {\\color{Yellow} 3} & {\\color{Gray} 0} & {\\color{Gray} 0}\\\\\n {\\color{Gray} 0} & {\\color{Yellow} 4} & {\\color{Gray} 0}\\\\\n {\\color{Gray} 0} & {\\color{Gray} 0} & {\\color{Yellow} 5}\n \\end{pmatrix}\n `}\n \n \n \n

\n Matrice con elementi diversi da 0 sopra la diagonale.\n

\n \n {r`\n \\begin{pmatrix}\n {\\color{Yellow} 3} & {\\color{Gray} 0} & {\\color{Gray} 0}\\\\\n {\\color{Orange} 4} & {\\color{Yellow} 4} & {\\color{Gray} 0}\\\\\n {\\color{Orange} 5} & {\\color{Orange} 5} & {\\color{Yellow} 5}\n \\end{pmatrix}\n `}\n \n \n \n

\n Matrice con elementi diversi da 0 sotto la diagonale.\n

\n \n {r`\n \\begin{pmatrix}\n {\\color{Yellow} 3} & {\\color{Orange} 3} & {\\color{Orange} 3}\\\\\n {\\color{Gray} 0} & {\\color{Yellow} 4} & {\\color{Orange} 4}\\\\\n {\\color{Gray} 0} & {\\color{Gray} 0} & {\\color{Yellow} 5}\n \\end{pmatrix}\n `}\n \n \n \n

\n Matrice con determinante diverso da 0.\n

\n {r`det(A) \\neq 0`}\n

\n Sono anche dette matrici linearmente indipendenti o matrici invertibili.\n

\n \n {r`\n \\begin{pmatrix}\n {\\color{Yellow} 1} & {\\color{Yellow} 1} & {\\color{Yellow} 2}\\\\\n {\\color{Orange} 2} & {\\color{Orange} 1} & {\\color{Orange} 1}\\\\\n {\\color{Red} 1} & {\\color{Red} 2} & {\\color{Red} 1}\n \\end{pmatrix}\n `}\n \n \n \n

\n Matrice con un asse di simmetria lungo la diagonale.\n

\n {r`A = A^T`}\n \n {r`\n \\begin{pmatrix}\n 1 & {\\color{Yellow} 2} & {\\color{Orange} 4}\\\\ \n {\\color{Yellow} 2} & 3 & {\\color{Red} 5}\\\\ \n {\\color{Orange} 4} & {\\color{Red} 5} & 6\n \\end{pmatrix}\n `}\n \n \n \n

\n Matrice con un asse di simmetria lungo la diagonale; gli elementi nel triangolo superiore sono però l'opposto di quelli del triangolo inferiore.\n

\n

\n Ha sempre degli 0 lungo la diagonale.\n

\n {r`A = -A^T`}\n \n {r`\n \\begin{pmatrix}\n {\\color{Gray} 0} & {\\color{Yellow} -2} & {\\color{Orange} -4}\\\\ \n {\\color{Yellow} 2} & {\\color{Gray} 0} & {\\color{Red} -5}\\\\ \n {\\color{Orange} 4} & {\\color{Red} 5} & {\\color{Gray} 0}\n \\end{pmatrix}\n `}\n \n \n \n

\n Matrice in cui i valori della diagonale sono maggiori della somma di tutti gli altri nella riga/colonna.\n

\n \n {r`\n \\begin{pmatrix}\n {\\color{Orange} 9} & 1 & 2\\\\\n 1 & {\\color{Orange} 8} & 1\\\\\n 1 & 2 & {\\color{Orange} 7}\n \\end{pmatrix}\n `}\n \n \n \n

\n Matrice che se moltiplicata per la sua trasposta dà come risultato la matrice identità.\n

\n {r`A^T \\cdot A = I`}\n \n {r`\n \\begin{pmatrix}\n \\frac{1}{3} & \\frac{2}{3} & -\\frac{2}{3}\\\\\n \\frac{2}{3} & \\frac{1}{3} & \\frac{2}{3}\\\\\n \\frac{2}{3} & -\\frac{2}{3} & -\\frac{1}{3}\\\\\n \\end{pmatrix}\n `}\n \n \n \n

\n Matrice tale che:\n

\n {r`A^{-1} \\cdot A = I`}\n \n \n

\n Matrice con pochissimi valori diversi da 0.\n

\n \n {r`\n \\begin{pmatrix}\n {\\color{Gray} 0} & 1 & {\\color{Gray} 0}\\\\\n 1 & 1 & {\\color{Gray} 0}\\\\\n {\\color{Gray} 0} & {\\color{Gray} 0} & 1\n \\end{pmatrix}\n `}\n \n \n
\n
\n \n

\n Funzione che associa un valore positivo a ogni vettore diverso da 0, e 0 al vettore zero.\n

\n \n Esempi su Wikipedia\n \n \n \n

\n Massimo dei valori assoluti di tutti gli elementi del vettore.\n

\n

\n {r`\\Vert x \\Vert_\\infty = max_{i = 1..n} | x_i |`}\n

\n \n \n

\n Somma dei valori assoluti di tutti gli elementi del vettore.\n

\n

\n {r`\\Vert x \\Vert_1 = \\sum_{i = 1}^n | x_i |`}\n

\n \n \n

\n Radice quadrata della somma dei quadrati di tutti gli elementi del vettore.\n

\n

\n {r`\\Vert x \\Vert_2 = \\sqrt{\\sum_{i = 1}^n x_i^2}`}\n

\n \n \n

\n Le norme sono usate per calcolare l'errore relativo tra due vettori o matrici:\n

\n

\n {r`\\frac{\\Vert x - y \\Vert}{\\Vert x \\Vert}`}\n

\n \n
\n
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probabilità"},l(e.p,{title:"Classica"},l("p",null,l(e.j,null,ia(u||(u=la`P(E) = \frac{casi\ favorevoli}{casi\ possibili}`))))),l(e.p,{title:"Frequentista"},l("p",null,l(e.j,null,ia(r||(r=la`P(E) = \frac{successi}{prove\ totali}`))))),l(e.p,{title:"Soggettiva"},l("p",null,"Il prezzo che un individuo coerente riterrebbe equo per ricevere ",l("b",null,"1")," nel caso l'evento si verificasse e ",l("b",null,"0")," nel caso l'evento non si verificasse."))),l(e.q,{title:"Linguaggio matematico"},l(e.p,{title:"Spazio campionario"},l("blockquote",null,'"omegone"'),l("p",null,"L'",l("b",null,"insieme")," di tutti gli esiti possibili di un esperimento."),l("p",null,l(e.j,null,ia(p||(p=la`\Omega = \left \{ 1, 2, 3, 4, 5, 6 \right \}`))))),l(e.p,{title:"Esito"},l("blockquote",null,'"omeghino"'),l("p",null,"Un ",l("b",null,"elemento")," dello spazio campionario."),l("p",null,l(e.j,null,ia(s||(s=la`\omega = 1`))))),l(e.p,{title:"Evento"},l("blockquote",null,'"e"'),l("p",null,"Un ",l("b",null,"sottoinsieme")," dello spazio campionario."),l("p",null,l(e.j,null,ia(d||(d=la`E = \left \{ 1, 2 \right \}`)))),l("p",null,"Lo spazio campionario stesso è un ",l("b",null,"evento certo"),".")),l(e.p,{title:"Not"},l("blockquote",null,'"not e"'),l("p",null,"Il ",l("b",null,"complementare")," di un sottoinsieme."),l("p",null,l(e.j,null,ia(m||(m=la`\bar{E} = \left \{ 3, 4, 5, 6 \right \}`))))),l(e.p,{title:"And"},l("blockquote",null,'"e intersecato effe"'),l("p",null,"L'",l("b",null,"intersezione")," di più sottoinsiemi."),l("p",null,l(e.j,null,ia(c||(c=la`E \cap F = \left \{ 1 \right \}`))))),l(e.p,{title:"Or"},l("blockquote",null,'"e unito a effe"'),l("p",null,"L'",l("b",null,"unione")," di più sottoinsiemi."),l("p",null,l(e.j,null,ia(b||(b=la`E \cup F = \left \{ 1, 2, 3, 4 \right \}`))))),l(e.p,{title:"Differenza"},l("blockquote",null,'"e meno effe"'),l("p",null,l(e.j,null,ia(v||(v=la`E \setminus F = E \cap \bar{F}`))))),l(e.p,{title:"Implicazione"},l("blockquote",null,'"e contenuto in effe"'),l("p",null,"L'",l("b",null,"inclusione")," del primo insieme in un altro."),l("p",null,l(e.j,null,ia(g||(g=la`E \subseteq F`)))),l("p",null,"Se si verifica ",l(e.j,null,"E"),", allora si verifica anche ",l(e.j,null,"F"),".")),l(e.p,{title:"Evento impossibile"},l("blockquote",null,'"e è impossibile"'),l("p",null,"Un sottoinsieme ",l("b",null,"vuoto"),"."),l("p",null,l(e.j,null,ia(f||(f=la`E = \emptyset`))))),l(e.p,{title:"Mutua esclusione"},l("blockquote",null,'"e ed effe si escludono mutualmente"'),l("p",null,"La ",l("b",null,"disgiunzione")," di due insiemi."),l("p",null,l(e.j,null,ia(j||(j=la`E \cap F = \emptyset`)))))),l(e.q,null,l(e.p,{title:"Famiglia degli eventi"},l("blockquote",null,'"famiglia effe"'),l("p",null,"I sottoinsiemi dello spazio campionario formano una ",l("b",null,"famiglia")," di sottoinsiemi detta ",l("i",null,"famiglia degli eventi"),"."),l("p",null,l(e.j,null,ia(_||(_=la`\mathcal{F}`)))),l("p",null,"Qualsiasi sottoinsieme appartenente a ",l(e.j,null,ia(h||(h=la`\mathcal{F}`)))," è considerato un evento.")),l(e.p,{title:l("span",null,l(e.j,null,ia(z||(z=la`\sigma`))),"-algebra")},l("blockquote",null,'"sigma algebra"'),l("p",null,"Se la famiglia degli eventi soddisfa questi tre requisiti, allora viene detta ",l("i",null,l(e.j,null,ia(X||(X=la`\sigma`))),"-algebra"),":"),l("ol",null,l("li",null,"Lo spazio campionario è un evento: ",l(e.j,null,ia(q||(q=la`\Omega \in \mathcal{F}`)))),l("li",null,"Se un sottoinsieme è un evento, allora anche il suo complementare lo è: ",l(e.j,null,ia(E||(E=la`E \in \mathcal{F} \implies \bar{E} \in \mathcal{F}`)))),l("li",null,"Se due sottoinsiemi sono eventi, allora lo sono anche la loro unione e intersezione: ",l(e.j,null,ia(P||(P=la`(E, F) \in \mathcal{F} \implies (E \cup F, E \cap F) \in \mathcal{F}`))))),l("p",null,"Un esempio: ",l(e.j,null,ia(L||(L=la`E \in \mathcal{F} \implies \mathcal{F} = \{ \emptyset, E, \bar{E}, \Omega \}`)))))),l(e.q,null,l(e.p,{title:"Partizione"},l("blockquote",null,'"la partizione e composta da e uno, e due, e tre..."'),l("p",null,"Un insieme di esiti e eventi:"),l("ul",null,l("li",null,l("b",null,"Finito"),"."),l("li",null,"In cui tutti gli eventi hanno ",l("b",null,"probabilità diversa da 0"),"."),l("li",null,"In cui tutti gli eventi sono ",l("b",null,"mutualmente esclusivi"),"."),l("li",null,"In cui l'unione di tutti i suoi elementi ",l("b",null,"copre lo spazio campionario"),".")),l("p",null,"La partizione ",l(e.j,null,ia(x||(x=la`E_i`)))," è composta dagli eventi ",l(e.j,null,ia(k||(k=la`E_1`))),", ",l(e.j,null,ia(y||(y=la`E_2`))),", ",l(e.j,null,ia(F||(F=la`E_3`))),", fino a ",l(e.j,null,ia(S||(S=la`E_n`))),"."),l(a.a,null,"Se lo spazio campionario fosse una torta, una sua partizione sarebbe l'insieme delle fette di uno dei modi in cui si potrebbe tagliare."))),l(e.q,{title:"Assiomi della probabilità"},l(e.p,{title:"Primo assioma della probabilità"},l("p",null,"La probabilità di un evento è un numero tra 0 e 1."),l("p",null,l(e.j,null,ia(I||(I=la`\forall E \in \mathcal{F}, 0 \leq P(E) \leq 1`))))),l(e.p,{title:"Secondo assioma della probabilità"},l("p",null,"La probabilità dello spazio campionario è sempre 1."),l("p",null,l(e.j,null,ia(C||(C=la`P(\Omega) = 1`))))),l(e.p,{title:"Terzo assioma della probabilità"},l("p",null,"La probabilità dell'unione di eventi indipendenti è uguale alla somma delle loro probabilità."),l("p",null,l(e.j,null,ia(N||(N=la`P \left ( \bigcup_i E_i \right ) = \sum_i P ( E_i )`)))))),l(e.q,{title:"Conseguenze degli assiomi"},l(e.p,{title:"Probabilità di un evento negato"},l("p",null,"La probabilità di un evento negato è uguale a 1 meno la probabilità dell'evento non negato."),l("p",null,l(e.j,null,ia(V||(V=la`P(\bar{E}) = 1 - P({E})`))))),l(e.p,{title:"Probabilità di un evento incluso"},l("p",null,"La probabilità di un evento incluso in un altro è sempre minore o uguale alla probabilità dell'evento in cui è incluso."),l("p",null,l(e.j,null,ia(w||(w=la`F \subseteq E \implies P(F) \leq P(E)`))))),l(e.p,{title:"Unione"},l("p",null,"La probabilità di un evento unito a un altro è uguale alla somma delle probabilità dei due eventi meno la probabilità della loro intersezione."),l("p",null,l(e.j,null,ia(D||(D=la`P(E \cup F) = P(E) + P(F) - P(E \cap F)`)))),l(a.a,null,"Sommando le probabilità dei due eventi, l'intersezione viene contata due volte, e va quindi rimossa!"))),l(e.q,{title:"Spazi equiprobabili"},l(e.p,{title:"Cosa sono?"},l("p",null,"Spazi campionari in cui ci sono un numero finito di esiti e ogni esito ha la stessa probabilità di verificarsi."),l("p",null,l(e.j,null,ia(Y||(Y=la`P(E) = \frac{len(E)}{len(\Omega)}`))))),l(e.p,{title:"Spazi equiprobabili geometrici"},l("p",null,"Gli spazi campionari possono avere un numero infinito di esiti: sono ",l("i",null,"equiprobabili geometrici")," se nessun esito è privilegiato rispetto agli altri."))),l(e.q,{title:"Calcolo combinatorio"},l(e.p,{title:"Disposizioni"},l("p",null,"Estraggo un numero, da un sacchetto con ",l(e.j,null,"n")," numeri, mi segno che numero ho estratto e lo ",l("b",null,"tengo fuori dal sacchetto"),". Ripeto per ",l(e.j,null,"k")," volte."),l("p",null,l("b",null,"Tengo conto")," dell'ordine in cui ho estratto i numeri."),l("p",null,l(e.j,null,ia(U||(U=la`\boldsymbol{D}_{n, k} = \frac{n!}{(n - k)!}`))))),l(e.p,{title:"Disposizioni con ripetizione"},l("p",null,"Estraggo un numero, da un sacchetto con ",l(e.j,null,"n")," numeri, mi segno che numero ho estratto e lo ",l("b",null,"rimetto nel sacchetto"),". Ripeto per ",l(e.j,null,"k")," volte."),l("p",null,l("b",null,"Tengo conto")," dell'ordine in cui ho estratto i numeri."),l("p",null,l(e.j,null,ia(M||(M=la`\boldsymbol{D}^{r}_{n, k} = n^k`))))),l(e.p,{title:"Combinazioni"},l("p",null,"Estraggo un numero, da un sacchetto con ",l(e.j,null,"n")," numeri, mi segno che numero ho estratto e lo ",l("b",null,"tengo fuori dal sacchetto"),". Ripeto per ",l(e.j,null,"k")," volte."),l("p",null,l("b",null,"Non mi interessa")," l'ordine in cui ho estratto i numeri."),l("p",null,l(e.j,null,ia(B||(B=la`\boldsymbol{C}_{n, k} = \binom{n}{k} = \frac{n!}{(k)! \cdot (n - k)!}`))))),l(e.p,{title:"Combinazioni con ripetizione"},l("p",null,"Estraggo un numero, da un sacchetto con ",l(e.j,null,"n")," numeri, mi segno che numero ho estratto e lo ",l("b",null,"rimetto nel sacchetto"),". Ripeto per ",l(e.j,null,"k")," volte."),l("p",null,l("b",null,"Non mi interessa")," l'ordine in cui ho estratto i numeri."),l("p",null,l(e.j,null,ia(A||(A=la`\boldsymbol{C}^{r}_{n, k} = \binom{n + k - 1}{k} = \frac{(n + k - 1)!}{(k)! \cdot (n - 1)!}`))))),l(e.p,{title:"Permutazioni"},l("p",null,"Estraggo ",l(e.j,null,"n")," numeri e guardo in quanti ordini diversi li posso mettere."),l("p",null,l(e.j,null,ia(T||(T=la`\boldsymbol{P}_n = n!`)))))),l(e.q,{title:"Probabilità condizionata"},l(e.p,{title:"Eventi condizionati"},l("blockquote",null,'"E dato F"'),l("p",null,"La probabilità che si verifichi ",l(e.j,null,"E")," sapendo che ",l("b",null,"si è già verificato")," ",l(e.j,null,"F"),"."),l("p",null,l(e.j,null,ia(G||(G=la`P(E|F) = \frac{P(E \cap F)}{P(F)}`)))),l(a.a,null,"Ricorda vagamente le pipe di ",l("code",null,"bash"),", però al contrario...")),l(e.p,{title:"Eventi mutualmente esclusivi"},l("p",null,"Se due eventi sono mutualmente esclusivi, entrambe le loro probabilità condizionate saranno uguali a 0."),l("p",null,l(e.j,null,ia(O||(O=la`E \cap F = \emptyset \Longleftrightarrow P(E|F) = P(F|E) = 0`)))))),l(e.q,null,l(e.p,{title:"Regola della catena"},l("p",null,"Si può sfruttare la formula inversa della probabilità condizionata per calcolare catene di intersezioni:"),l("p",null,l(e.j,null,ia(R||(R=la`P(E_1 \cap \times \cap E_n) = P(E_1) \times P(E_2 | E_1) \times \dots \times P(E_n | E_1 \cap E_2 \cap \dots \cap E_{n-1})`)))))),l(e.q,{title:"Le alternative"},l(e.p,{title:"Legge delle alternative"},l("p",null,"La probabilità che si verifichi un evento è pari alla somma delle probabilità dell'evento stesso dati tutti gli eventi di una partizione."),l("p",null,l(e.j,null,ia(K||(K=la`P(F) = \sum_{i} P(F|E_i) \cdot P(E_i)`))))),l(e.p,{title:"Legge condizionata delle alternative"},l("p",null,"La legge delle alternative funziona anche quando ad essere partizionato è un ",l("b",null,"evento"),":"),l("p",null,l(e.j,null,ia(Q||(Q=la`P(F|G) = \sum_i P(F|E_i \cap G) \cdot P(E_i | G)`))))),l(e.p,{title:"Formula di Bayes"},l("p",null,"Tramite la ",l("i",null,"formula di Bayes")," possiamo risalire alla probabilità di un evento condizionato a un altro partendo dalla probabilità di quest'ultimo condizionato al primo:"),l("p",null,l(e.j,null,ia(Z||(Z=la`P(E_h | F) = \frac{P(F | E_h) \cdot P(E_h)}{P(F)}`)))),l(a.a,null,"In pratica, invertiamo gli eventi."))),l(e.q,{title:"Eventi indipendenti"},l(e.p,{title:"Due eventi indipendenti"},l("blockquote",null,'"eventi indipendenti a due a due"'),l("p",null,"Se due eventi sono indipendenti, sapere che uno dei due si è verificato non influisce sulle probabilità che si sia verificato l'altro."),l("p",null,l(e.j,null,ia(W||(W=la`P(E \cap F) = P(E) \cdot P(F) \Longleftrightarrow P(E|F) = P(E) \Longleftrightarrow P(F|E) = P(F)`))))),l(e.p,{title:"Più eventi indipendenti"},l("blockquote",null,'"eventi indipendenti a tre a tre, a quattro a quattro, a cinque a cinque..."'),l("p",null,"Si può verificare l'indipendenza di più eventi alla volta:"),l("p",null,l(e.j,null,ia(J||(J=la`P(E \cap F \cap G) = P(E) \cdot P(F) \cdot P(G)`)))),l("p",null,"Eventi indipendenti a due a due non sono per forza indipendenti a tre a tre, e viceversa.")),l(e.p,{title:"Famiglia di eventi indipendenti"},l("p",null,"Un insieme di ",l(e.j,null,"n")," eventi è una ",l("i",null,"famiglia di eventi indipendenti")," se, preso un qualsiasi numero di eventi da essa, essi risulteranno indipendenti."),l(a.a,null,"Tutti gli eventi provenienti da essa saranno indipendenti sia a due a due, sia a tre a tre, sia a quattro a quattro, e così via!"))),l(e.q,{title:"Variabili aleatorie"},l(e.p,{title:"Variabile aleatoria"},l("p",null,"Una funzione che fa corrispondere un numero reale a ogni possibile esito dello spazio campionario. ",l(e.j,null,ia(H||(H=la`X(\omega) : \Omega \to \mathbb{R}`))),".")),l(e.p,{title:l("abbr",{title:"Nome artigianale dato da Steffo."},"Insieme di ripartizione")},l("p",null,"Ad ogni variabile aleatoria sono associati gli eventi ",l(e.j,null,ia($||($=la`A_t = \{ \omega | X(\omega) \leq t \}`))),", che contengono tutti gli esiti a cui la variabile aleatoria associa un valore minore o uguale a ",l(e.j,null,"t"),"."),l("p",null,"Per definizione, tutte le variabili aleatorie devono rispettare questa condizione:"),l("p",null,l(e.j,null,ia(ll||(ll=la`\forall t \in \mathbb{R}, A_t \in \mathcal{F}`)))),l(a.a,null,"All'aumentare di t, l'insieme conterrà sempre più elementi.")),l(e.p,{title:"Supporto"},l("blockquote",null,'"supporto di X"'),l("p",null,"Il ",l("b",null,"codominio")," della variabile aleatoria è il suo ",l("i",null,"supporto"),"."),l("p",null,"Per indicare che un valore ",l(e.j,null,"x_0")," appartiene al supporto di ",l(e.j,null,"X"),", si usa la notazione ",l(e.j,null,"X \\mapsto x_0"),"."))),l(e.q,{title:"Densità"},l(e.p,{title:"Funzione probabilità"},l("p",null,"La ",l("i",null,"funzione probabilità")," ",l(e.j,null,ia(il||(il=la`p_X : X \to [0, 1]`)))," di una variabile aleatoria ",l("b",null,"discreta")," ",l(e.j,null,"X")," è la funzione che associa ad ogni esito la sua probabilità:"),l("p",null,l(e.j,null,ia(nl||(nl=la` + p_X (x) = \begin{cases} + P([X = x]) \quad se\ X \mapsto x \\ + 0 \qquad \qquad \quad se\ X \not\mapsto x + \end{cases} + `))))),l(e.p,{title:"Funzione densità"},l("p",null,"La ",l("i",null,"funzione densità")," ",l(e.j,null,ia(el||(el=la`f_X : X \to [0, 1]`)))," di una variabile aleatoria ",l("b",null,"continua")," ",l(e.j,null,"X")," è l'equivalente continuo della funzione probabilità:"),l("p",null,l(e.j,null,ia(al||(al=la`P([a < X \leq b]) = \int_a^b f_X (x) dx`)))),l("p",null,"A differenza della funzione probabilità, è possibile che la funzione densità ",l("b",null,"non esista")," per una certa variabile aleatoria."),l(a.a,null,"Rappresenta \"quanta\" probabilità c'è in un'unità di x!"))),l(e.q,{title:"Funzione di ripartizione"},l(e.p,{title:"Definizione"},l("p",null,"Ogni variabile aleatoria ha una ",l("i",null,"funzione di ripartizione")," ",l(e.j,null,ia(ol||(ol=la`F_X : \mathbb{R} \to [0, 1]`)))," associata, che rappresenta la probabilità che la variabile aleatoria assuma un valore minore o uguale a ",l(e.j,null,"t"),":"),l("p",null,"Si può dire che essa rappresenti la probabilità dell'evento ",l(e.j,null,ia(tl||(tl=la`A_t`))),":"),l("p",null,l(e.j,null,ia(ul||(ul=la` + F_X (t) = P(A_t) = \begin{cases} + \sum_{i = 0}^{t} p_X (x_i) \quad nel\ discreto\\ + \\ + \int_{-\infty}^t f_X (x) dx \quad nel\ continuo + \end{cases} + `))))),l(e.p,{title:"Proprietà della funzione"},l("ul",null,l("li",null,"È sempre ",l("b",null,"monotona crescente")," (non strettamente)."),l("br",null),l("li",null,"Vale ",l("b",null,"0")," a ",l(e.j,null,"-\\infty")," e ",l("b",null,"1")," a ",l(e.j,null,"+\\infty"),"."),l("br",null),l("li",null,"È ",l("b",null,"continua da destra"),": ",l(e.j,null,ia(rl||(rl=la`\forall x_0 \in \mathbb{R}, F_X (x_0) = \lim_{t \to x^+_0} F_X (t)`)))))),l(e.p,{title:"Probabilità di un valore"},l("p",null,"Possiamo usare la funzione di ripartizione per calcolare la probabilità di un certo valore reale:"),l("p",null,l(e.j,null,ia(pl||(pl=la`P([X = x_0]) = \lim_{t \to x^+_0} F_X (t) - \lim_{t \to x^-_0} F_X (t)`)))))),l(e.q,{title:"Trasformazioni di variabili aleatorie"},l(e.p,{title:"Nel discreto"},l("p",null,"Nel discreto basta abbinare un nuovo valore a ogni valore della variabile originale.")),l(e.p,{title:"Nel continuo (invertibile)"},l("p",null,"Nel continuo applichiamo la formula dell'integrazione per sostituzione:"),l("p",null,l(e.j,null,ia(sl||(sl=la`f_Y (y) = \int_{g(a)}^{g(b)} f_X ( g^{-1} (x) ) g^{-2} (x)`))))),l(e.p,{title:"Nel... digitale"},l("p",null,"Trasformare variabili aleatorie è molto utile nell'informatica per creare distribuzioni partendo da una funzione ",l("a",{href:"https://docs.python.org/3/library/random.html#random.random"},l("code",null,"random()"))," che restituisce numeri da 0 a 1 con una distribuzione lineare."))),l(e.q,{title:"Informazioni delle variabili aleatorie"},l(e.p,{title:"Media"},l("p",null,"Ogni variabile aleatoria che ha una ",l("b",null,"funzione di ripartizione")," e un ",l("b",null,"supporto finito")," ha anche una ",l("i",null,"media")," (o ",l("i",null,"valore medio")," o ",l("i",null,"atteso"),"):"),l("p",null,l(e.j,null,ia(dl||(dl=la`E(X) = \int_0^{+infty} (1 - F_X (t)) dt - \int_{-\infty}^{0} F_X (t) dt`)))),l("p",null,"Nel discreto, si può calcolare con:"),l("p",null,l(e.j,null,ia(ml||(ml=la`E(X) = \sum_i P(X = x_i) \cdot x_i`)))),l("p",null,"Nel continuo, si può calcolare con:"),l("p",null,l(e.j,null,ia(cl||(cl=la`E(X) = \int_{-\infty}^{+\infty} f_X (x) \cdot x \cdot dx`)))))),l(e.q,null,l(e.p,{title:"Moda"},l("p",null,"Valore per cui la ",l("b",null,"funzione probabilità")," o ",l("b",null,"funzione densità")," è ",l("b",null,"massima"),".")),l(e.p,{title:"Quantili"},l("p",null,"Il ",l("i",null,"quantile")," ",l(e.j,null,ia(bl||(bl=la`x_{\alpha}`)))," di ordine ",l(e.j,null,ia(vl||(vl=la`0 \leq \alpha \leq 1`)))," della variabile aleatoria ",l(e.j,null,"X")," è il più piccolo numero tale che:"),l("p",null,l(e.j,null,ia(gl||(gl=la`P([X < x_{\alpha}]) \leq \alpha \leq P([X \leq x_{\alpha}])`)))),l("p",null),l("p",null,"Il quantile di ordine 0.5 ",l(e.j,null,ia(fl||(fl=la`x_{0.5}`)))," è detto ",l("i",null,"mediana"),"."),l("p",null,"I quantili di ordine 0.25 ",l(e.j,null,ia(jl||(jl=la`x_{0.25}`)))," e 0.75 ",l(e.j,null,ia(_l||(_l=la`x_{0.75}`)))," sono detti ",l("i",null,"quartili"),"."),l("p",null,"I quantili di ordine ",l(e.j,null,ia(hl||(hl=la`\frac{n}{100}`)))," sono detti ",l("i",null,l(e.j,null,"n"),"-esima percentile"),".")),l(e.p,{title:"Varianza"},l("p",null,"È un valore che indica quanto la variabile aleatoria si discosta generalmente dalla media:"),l("p",null,l(e.j,null,ia(zl||(zl=la`Var(X) = E( (X - E(X) )^2 ) = E ( X^2 ) - (E(X))^2`)))))),l(e.q,{title:"Disuguaglianze notevoli"},l(e.p,{title:"Disuguaglianza di Markov"},l("p",null,"Data una variabile aleatoria non-negativa:"),l("p",null,l(e.j,null,ia(Xl||(Xl=la`\forall k > 0, P([X \geq k]) \leq \frac{E(X)}{k}`)))),l("p",null,"Divide in due parti (",l(e.j,null,ia(ql||(ql=la`P(X < k)`)))," e ",l(e.j,null,ia(El||(El=la`P(X \geq k)`))),") la funzione X, la cui media risulterà uguale a:"),l("p",null,l(e.j,null,ia(Pl||(Pl=la`E(X) = \overline{k} \cdot P(X < k) + k \cdot P(X \geq k)`))))),l(e.p,{title:"Disuguaglianza di Čebyšëv"},l("blockquote",null,'"disuguaglianza di cebicev"'),l("p",null,"Se la variabile aleatoria ",l(e.j,null,"X")," ha media e varianza, allora la probabilità che essa abbia un valore a più di ",l(e.j,null,ia(Ll||(Ll=la`\epsilon`)))," di distanza dal valore medio è minore o uguale a ",l(e.j,null,ia(xl||(xl=la`\frac{Var(X)}{\epsilon^2}`))),"."),l("p",null,l(e.j,null,ia(kl||(kl=la`\forall \epsilon > 0, P([ \left| X - E(X) \right| \geq \epsilon]) \leq \frac{Var(X)}{\epsilon^2}`)))),l("p",null,"E anche:"),l("p",null,l(e.j,null,ia(yl||(yl=la`\forall \epsilon > 0, P([ \left| X - E(X) \right| < \epsilon]) \geq 1 - \frac{Var(X)}{\epsilon^2}`)))),l(a.a,null,"Serve per semplificare i calcoli quando la funzione di ripartizione è difficile da calcolare!"))),l(e.q,{title:"Un momento...!"},l(e.p,{title:"Momento"},l("p",null,"Il ",l("i",null,"momento")," ",l(e.j,null,"k"),"-esimo di una variabile aleatoria è:"),l("p",null,l(e.j,null,ia(Fl||(Fl=la` + \mu_k = E ( X^k ) = \begin{cases} + \sum_i x_i^k p_X (x_i) \qquad nel\ discreto\\ + \\ + \int_{-\infty}^{+\infty} x^k f_X (x) dx \qquad nel\ continuo + \end{cases}`)))),l(a.a,null,"La media di una variabile aleatoria è anche il suo primo momento.")),l(e.p,{title:"Funzione generatrice dei momenti"},l("p",null,"La ",l("i",null,"funzione generatrice dei momenti")," è:"),l("p",null,l(e.j,null,ia(Sl||(Sl=la`m_X (t) = E( e^{t \cdot X} )`)))),l("p",null,"Se due variabile aleatorie hanno la stessa funzione generatrice dei momenti, allora esse hanno la ",l("b",null,"stessa distribuzione"),"."),l("p",null,"E' la ",l("b",null,"trasformata di Laplace")," della variabile aleatoria di X.")),l(e.p,{title:"Funzione caratteristica"},l("p",null,"La ",l("i",null,"funzione caratteristica")," è:"),l("p",null,l(e.j,null,ia(Il||(Il=la`H_X (t) = E ( e^{i \cdot t \cdot X} )`)))),l("p",null,"Se due variabile aleatorie hanno la stessa funzione caratteristica, allora esse hanno la ",l("b",null,"stessa distribuzione"),"."),l("p",null,"E' la ",l("b",null,"trasformata di Fourier")," della variabile aleatoria di X."))),l(e.q,{title:"Prove e schemi"},l(e.p,{title:"Variabile con distribuzione"},l("p",null,"Per dire che una variabile ha una certa distribuzione, si usa la notazione:"),l("p",null,l(e.j,null,ia(Cl||(Cl=la`X \sim Distribuzione()`))))),l(e.p,{title:"Prova di Bernoulli"},l("p",null,"Una prova con solo due possibili esiti: ",l(o.a,null,"successo")," e ",l(t.a,null,"insuccesso"),".")),l(e.p,{title:"Schema di Bernoulli"},l("p",null,"Una sequenza di prove di Bernoulli per le quali le probabilità di successo e fallimento rimangono invariate."))),l(e.q,{title:"Bernoulliana"},l(e.p,{title:"Distribuzione bernoulliana"},l("p",null,"Una variabile aleatoria che rappresenta una prova di Bernoulli:"),l("ul",null,l("li",null,"vale ",l(o.a,null,"1")," in caso di ",l(o.a,null,"successo"),"."),l("li",null,"vale ",l(t.a,null,"0")," in caso di ",l(t.a,null,"insuccesso"),".")),l("p",null,"Il suo simbolo è ",l(e.j,null,ia(Nl||(Nl=la`Ber(p)`))))),l(e.p,{title:"Densità della bernoulliana"},l("p",null,"La distribuzione bernoulliana ha come densità:"),l("p",null,l(e.j,null,ia(Vl||(Vl=la` + f_X (k) : \{0, 1\} = \begin{cases} + p \quad se\ k = 1\\ + q \quad se\ k = 0\\ + 0 \quad altrimenti + \end{cases} = p^x \cdot q^{1 - k}`)))))),l(e.q,{title:"Binomiale"},l(e.p,{title:"Distribuzione binomiale"},l("p",null,"Una variabile aleatoria che conta il numero di successi di ",l(e.j,null,"n")," prove di uno schema di Bernoulli."),l("p",null,"Il suo simbolo è ",l(e.j,null,ia(wl||(wl=la`Bin(n, p)`))),".")),l(e.p,{title:"Densità della binomiale"},l("p",null,"La binomiale ha come densità:"),l("p",null,l(e.j,null,ia(Dl||(Dl=la`f_X (k) : \{0..n\} = \binom{n}{k} \cdot p^k \cdot q^{n - k}`))))),l(e.p,{title:"Momenti della binomiale"},l("p",null,"La ",l("b",null,"funzione generatrice dei momenti")," della binomiale è:"),l("p",null,l(e.j,null,ia(Yl||(Yl=la`m_X (t) = (q + p \cdot e^t) ^ n`)))),l("p",null,"La ",l("b",null,"media")," di una binomiale è:"),l("p",null,l(e.j,null,ia(Ul||(Ul=la`E(X) = n \cdot p`)))),l("p",null,"La ",l("b",null,"varianza")," di una binomiale è:"),l("p",null,l(e.j,null,ia(Ml||(Ml=la`Var(X) = n \cdot p \cdot q`)))))),l(e.q,{title:"Geometrica"},l(e.p,{title:"Distribuzione geometrica"},l("p",null,"Una variabile aleatoria che conta il numero di prove in uno schema di Bernoulli fino alla comparsa del primo successo."),l("p",null,"Il suo simbolo è ",l(e.j,null,"Geo(p)"),".")),l(e.p,{title:"Densità della geometrica"},l("p",null,"La geometrica ha come densità:"),l("p",null,l(e.j,null,ia(Bl||(Bl=la`f_X (k) : \mathbb{N} = q^{k - 1} p`))))),l(e.p,{title:"Momenti della geometrica"},l("p",null,"La ",l("b",null,"funzione generatrice dei momenti")," della geometrica è:"),l("p",null,l(e.j,null,ia(Al||(Al=la`m_X (t) = \frac{p \cdot e^t}{1 - q \cdot e^t}`)))),l("p",null,"La ",l("b",null,"media")," della geometrica è:"),l("p",null,l(e.j,null,ia(Tl||(Tl=la`E(X) = \frac{1}{p}`)))),l("p",null,"La ",l("b",null,"varianza")," della geometrica è:"),l("p",null,l(e.j,null,ia(Gl||(Gl=la`Var(X) = \frac{q}{p^2}`))))),l(e.p,{title:"Assenza di memoria della geometrica"},l("p",null,"La geometrica non tiene conto degli eventi avvenuti in passato: ha la proprietà dell'assenza di memoria:"),l("p",null,l(e.j,null,ia(Ol||(Ol=la`P([X = i + j | X > i ]) = P([X = j])`)))),l(a.a,null,"Ovvero, riscalando opportunamente l'asse Y posso prendere come 0 qualsiasi punto dell'asse X."))),l(e.q,{title:"Binomiale negativa"},l(e.p,{title:"Distribuzione binomiale negativa"},l("p",null,"Una variabile aleatoria che conta il numero di prove in uno schema di Bernoulli necessarie perchè si verifichi l'",l(e.j,null,"n"),"-esimo successo."),l("p",null,"Il suo simbolo è ",l(e.j,null,ia(Rl||(Rl=la`\overline{Bin}(n, p)`))),".")),l(e.p,{title:"Densità della binomiale negativa"},l("p",null,"La binomiale negativa ha come densità:"),l("p",null,l(e.j,null,ia(Kl||(Kl=la`f_X (k) : \{ n .. +\infty \} \in \mathbb{N} = \binom{k - 1}{n - 1} \cdot p^n \cdot q^{k - n} `))))),l(e.p,{title:"Momenti della binomiale negativa"},l("p",null,l("p",null,"La ",l("b",null,"funzione generatrice dei momenti")," della binomiale negativa è:"),l("p",null,l(e.j,null,ia(Ql||(Ql=la`m_X (t) : \{ t < ln(\frac{1}{q}) \} = \left( \frac{p \cdot e^t}{1 - q \cdot e^t} \right) ^n`)))),l("p",null,"La ",l("b",null,"media")," della binomiale negativa è:"),l("p",null,l(e.j,null,ia(Zl||(Zl=la`E(X) = \frac{n}{p}`)))),l("p",null,"La ",l("b",null,"varianza")," della binomiale negativa è:"),l("p",null,l(e.j,null,ia(Wl||(Wl=la`Var(X) = \frac{n \cdot q}{p^2}`))))))),l(e.q,{title:"Geometrica traslata"},l(e.p,{title:"Distribuzione geometrica traslata"},l("p",null,"Una variabile aleatoria che conta il numero ",l(e.j,null,"k")," di insuccessi consecutivi in uno schema di Bernoulli:"),l("p",null,"Il suo simbolo rimane ",l(e.j,null,ia(Jl||(Jl=la`Geo(p)`))),".")),l(e.p,{title:"Densità della geometrica tralsata"},l("p",null,"La geometrica traslata ha come densità:"),l("p",null,l(e.j,null,ia(Hl||(Hl=la`f_X (k) : \mathbb{N} = p \cdot q^k `))))),l(e.p,{title:"Momenti della geometrica traslata"},l("p",null,"La ",l("b",null,"funzione generatrice dei momenti")," della geometrica traslata è:"),l("p",null,l(e.j,null,ia($l||($l=la`m_X (t) : \left\{ t < ln \left( \frac{1}{q} \right) \right\} = \frac{p}{1 - q \cdot e^t}`)))),l("p",null,"La ",l("b",null,"media")," della geometrica traslata è:"),l("p",null,l(e.j,null,ia(li||(li=la`E(X) = \frac{q}{p}`)))),l("p",null,"La ",l("b",null,"varianza")," della geometrica è:"),l("p",null,l(e.j,null,ia(ii||(ii=la`Var(X) = \frac{q}{p^2}`))))),l(e.p,{title:"Assenza di memoria della geometrica traslata"},l("p",null,"La geometrica traslata non tiene conto degli eventi avvenuti in passato: ha la proprietà dell'assenza di memoria:"),l("p",null,l(e.j,null,ia(ni||(ni=la`P([X = i + j | X > i ]) = P([X = j])`)))),l(a.a,null,"Ovvero, riscalando opportunamente l'asse Y posso prendere come 0 qualsiasi punto dell'asse X."))),l(e.q,{title:"Binomiale negativa traslata"},l(e.p,{title:"Distribuzione binomiale negativa traslata"},l("p",null,"Una variabile aleatoria che conta il numero di insuccessi in uno schema di Bernoulli prima che si verifichi l'",l(e.j,null,"n"),"-esimo successo."),l("p",null,"Il suo simbolo rimane ",l(e.j,null,ia(ei||(ei=la`\overline{Bin}(n, p)`))),".")),l(e.p,{title:"Densità della binomiale negativa traslata"},l("p",null,"La binomiale negativa traslata ha come densità:"),l("p",null,l(e.j,null,ia(ai||(ai=la`f_X (k) : \mathbb{N} = \binom{k + n - 1}{n - 1} \cdot p^n \cdot q^k `))))),l(e.p,{title:"Momenti della binomiale negativa traslata"},l("p",null,l("p",null,"La ",l("b",null,"funzione generatrice dei momenti")," della binomiale negativa traslata è:"),l("p",null,l(e.j,null,ia(oi||(oi=la`m_X (t) : \left\{ t < ln \left( \frac{1}{q} \right) \right\} = \left( \frac{p \cdot e^t}{1 - q \cdot e^t} \right) ^n`)))),l("p",null,"La ",l("b",null,"media")," della binomiale negativa traslata è:"),l("p",null,l(e.j,null,ia(ti||(ti=la`E(X) = \frac{n \cdot q}{p}`)))),l("p",null,"La ",l("b",null,"varianza")," della binomiale negativa traslata è:"),l("p",null,l(e.j,null,ia(ui||(ui=la`Var(X) = \frac{n \cdot q}{p^2}`))))))),l(e.q,{title:"Ipergeometrica"},l(e.p,{title:"Distribuzione ipergeometrica"},l("p",null,"Una variabile aleatoria che, sapendo il numero di successi ",l(e.j,null,"K")," e di insuccessi ",l(e.j,null,"N-K"),", conta quanti successi si otterrebbero se se ne estraessero ",l(e.j,null,"n")," in blocco."),l("p",null,"Il suo simbolo è ",l(e.j,null,"Ipe(N, K, n)"),".")),l(e.p,{title:"Densità della ipergeometrica"},l("p",null,"La ipergeometrica ha come densità:"),l("p",null,l(e.j,null,ia(ri||(ri=la`f_X (k) : \{0..n\} \in \mathbb{N} = \frac{\binom{K}{k} \cdot \binom{N - K}{n - k}}{\binom{N}{n}}`))))),l(e.p,{title:"Momenti della ipergeometrica"},l("p",null,l("p",null,"La ",l("b",null,"funzione generatrice dei momenti")," della ipergeometrica è trascurabile."),l("p",null,"La ",l("b",null,"media")," della ipergeometrica è:"),l("p",null,l(e.j,null,ia(pi||(pi=la`E(X) = n \cdot \frac{K}{N}`)))),l("p",null,"La ",l("b",null,"varianza")," della ipergeometrica è:"),l("p",null,l(e.j,null,ia(si||(si=la`Var(X) = n \cdot \frac{K}{N} \cdot \frac{N - K}{N} \cdot \frac{N - n}{N - 1}`))))))),l(e.q,{title:"Poissoniana"},l(e.p,{title:"Distribuzione poissoniana"},l("p",null,"Una variabile aleatoria che soddisfa tutte le seguenti caratteristiche:"),l("ul",null,l("li",null,"Binomiale: ",l(e.j,null,ia(di||(di=la`X \sim Bin(n, p)`)))),l("li",null,"Il numero di prove tende a infinito: ",l(e.j,null,ia(mi||(mi=la`n \to +\infty`)))),l("li",null,"La probabilità di successo tende a 0: ",l(e.j,null,ia(ci||(ci=la`p \to 0`)))),l("li",null,"La media è finita: ",l(e.j,null,ia(bi||(bi=la`E(X) = n \cdot p \to \mu \neq 0`))))),l("p",null,"Il suo simbolo è ",l(e.j,null,ia(vi||(vi=la`Poi(\mu)`))))),l(e.p,{title:"Densità della poissoniana"},l("p",null,"La poissoniana ha come densità:"),l("p",null,l(e.j,null,ia(gi||(gi=la`f_X (k) : \mathbb{N} = \frac{e^{-\mu} \cdot \mu^k}{k!}`))))),l(e.p,{title:"Momenti della poissoniana"},l("p",null,l("p",null,"La ",l("b",null,"funzione generatrice dei momenti")," della poissoniana è:"),l("p",null,l(e.j,null,ia(fi||(fi=la`m_X (t) = e^{\mu \cdot (e^t - 1)}`)))),l("p",null,"La ",l("b",null,"media")," della poissoniana è:"),l("p",null,l(e.j,null,ia(ji||(ji=la`E(X) = \mu`)))),l("p",null,"La ",l("b",null,"varianza")," della poissoniana è:"),l("p",null,l(e.j,null,ia(_i||(_i=la`Var(X) = \mu`)))),l("p",null,"Gli altri momenti della poissoniana sono:"),l("ol",{start:2},l("li",null,l(e.j,null,ia(hi||(hi=la`E(X^2) = \mu^2 + \mu`)))))))),l(e.q,{title:"Un altro schema"},l(e.p,{title:"Schema di Poisson"},l("p",null,"Una successione di ",l("b",null,"arrivi")," avvenuti in un certo arco temporale che:"),l("ul",null,l("li",null,"non sono sovrapposti."),l("li",null,"hanno intensità ",l(e.j,null,ia(zi||(zi=la`\lambda`)))," costante."),l("li",null,"avvengono indipendentemente gli uni dagli altri."))),l(e.p,{title:"Processo di Poisson"},l("p",null,"Una variabile aleatoria ",l(e.j,null,"N_t")," che conta il numero di arrivi di uno schema di Poisson di intensità ",l(e.j,null,ia(Xi||(Xi=la`\lambda`)))," in un intervallo di tempo di durata ",l(e.j,null,"t"),"."),l("p",null,"E' una distribuzione poissoniana con ",l(e.j,null,ia(qi||(qi=la`\mu = t \cdot \lambda`))),": ",l(e.j,null,ia(Ei||(Ei=la`Poi(t \cdot \lambda)`)))),l(a.a,null,"E' paragonabile a una bernoulliana: ogni successo corrisponde a un arrivo, mentre il tempo è il numero di prove effettuate (ma nel continuo)."))),l(e.q,{title:"Esponenziale"},l(e.p,{title:"Distribuzione esponenziale"},l("p",null,"Una variabile aleatoria che conta il tempo diwidehattesa prima del primo arrivo di un processo di Poisson di intensità ",l(e.j,null,ia(Pi||(Pi=la`\lambda`))),"."),l("p",null,"Il suo simbolo è ",l(e.j,null,ia(Li||(Li=la`Esp(\lambda)`))),".")),l(e.p,{title:"Densità dell'esponenziale"},l("p",null,"L'esponenziale ha come ",l("b",null,"densità"),":"),l("p",null,l(e.j,null,ia(xi||(xi=la` + f_X (x) = \begin{cases} + 0 \qquad \qquad x < 0\\ + \lambda \cdot e^{-\lambda \cdot x} \quad x > 0 + \end{cases}`)))),l("p",null,"L'esponenziale ha come ",l("b",null,"funzione di ripartizione"),":"),l("p",null,l(e.j,null,ia(ki||(ki=la` + F_X (t) = \begin{cases} + 0 \qquad \qquad t < 0\\ + 1 - e^{-\lambda \cdot t} \quad t \geq 0 + \end{cases}`))))),l(e.p,{title:"Momenti dell'esponenziale"},l("p",null,"La ",l("b",null,"funzione generatrice dei momenti")," dell'esponenziale è:"),l("p",null,l(e.j,null,ia(yi||(yi=la`m_X (t) : \{ t | t < \lambda \} \in \mathbb{R} = \frac{\lambda}{\lambda - t}`)))),l("p",null,"La ",l("b",null,"media")," dell'esponenziale è:"),l("p",null,l(e.j,null,ia(Fi||(Fi=la`E(X) = \frac{1}{\lambda}`)))),l("p",null,"La ",l("b",null,"varianza")," dell'esponenziale è:"),l("p",null,l(e.j,null,ia(Si||(Si=la`Var(X) = \frac{1}{\lambda^2}`))))),l(e.p,{title:"Assenza di memoria della esponenziale"},l("p",null,"L'esponenziale non tiene conto degli eventi avvenuti in passato: ha la proprietà dell'assenza di memoria:"),l("p",null,l(e.j,null,ia(Ii||(Ii=la`P([X > s + t | X > s]) = P([X > t])`)))),l(a.a,null,"Ovvero, riscalando opportunamente l'asse Y posso prendere come 0 qualsiasi punto dell'asse X."))),l(e.q,{title:"Legge gamma"},l(e.p,{title:"Distribuzione gamma"},l("p",null,"Una variabile aleatoria che conta il tempo diwidehattesa prima dell'",l(e.j,null,"n"),"-esimo arrivo di un processo di Poisson di intensità ",l(e.j,null,ia(Ci||(Ci=la`\lambda`))),"."),l("p",null,"Il suo simbolo è ",l(e.j,null,ia(Ni||(Ni=la`\Gamma(n, \lambda)`))),".")),l(e.p,{title:"Densità della legge gamma"},l("p",null,"La legge gamma ha come densità:"),l("p",null,l(e.j,null,ia(Vi||(Vi=la` + f_X (x) = \begin{cases} + 0 \qquad \qquad \qquad \qquad \qquad x < 0\\ + \frac{1}{(n-1)!} \cdot \lambda^n \cdot x^{n-1} \cdot e^{-\lambda \cdot x} \quad k > 0 + \end{cases}`))))),l(e.p,{title:"Momenti della legge gamma"},l("p",null,l("p",null,"La ",l("b",null,"funzione generatrice dei momenti")," della legge gamma è:"),l("p",null,l(e.j,null,ia(wi||(wi=la`m_X (t) : ( t < \lambda ) \in \mathbb{R} = \left( \frac{\lambda}{\lambda - t} \right) ^\alpha`)))),l("p",null,"La ",l("b",null,"media")," della legge gamma è:"),l("p",null,l(e.j,null,ia(Di||(Di=la`E(X) = \frac{\alpha}{\lambda}`)))),l("p",null,"La ",l("b",null,"varianza")," della legge gamma è:"),l("p",null,l(e.j,null,ia(Yi||(Yi=la`Var(X) = \frac{\alpha}{\lambda^2}`))))))),l(e.q,{title:"Uniforme"},l(e.p,{title:"Distribuzione uniforme"},l("p",null,"Una variabile aleatoria che può assumere qualsiasi valore in un intervallo ",l(e.j,null,ia(Ui||(Ui=la`[a, b]`)))," in modo equiprobabile."),l("p",null,"Il suo simbolo è ",l(e.j,null,ia(Mi||(Mi=la`Uni(a, b)`)))),l("p",null,"Su di essa vale la seguente proprietà:"),l("p",null,l(e.j,null,ia(Bi||(Bi=la`P(X \in (c, d)) = \frac{d - c}{b - a}`))))),l(e.p,{title:"Densità della distribuzione uniforme"},l("p",null,"La distribuzione uniforme ha come ",l("b",null,"densità"),":"),l("p",null,l(e.j,null,ia(Ai||(Ai=la` + f_X (x) = \begin{cases} + \frac{1}{b - a} \qquad a \leq x \leq b\\ + 0 \qquad \quad altrimenti + \end{cases} + `)))),l("p",null,"La distribuzione uniforme ha come ",l("b",null,"funzione di ripartizione"),":"),l("p",null,l(e.j,null,ia(Ti||(Ti=la` + f_X (x) = \begin{cases} + 0 \qquad \quad x < a + \frac{1}{b - a} \qquad a \leq x \leq b\\ + 1 \qquad \quad x > b + \end{cases}`))))),l(e.p,{title:"Momenti della distribuzione uniforme"},l("p",null,l("p",null,"La ",l("b",null,"funzione generatrice dei momenti")," della distribuzione uniforme è:"),l("p",null,l(e.j,null,ia(Gi||(Gi=la`m_X (t) = \frac{e^{b \cdot t} - e^{a \cdot t}}{(b - a) \cdot t}`)))),l("p",null,"La ",l("b",null,"media")," della distribuzione uniforme è:"),l("p",null,l(e.j,null,ia(Oi||(Oi=la`E(X) = \frac{a + b}{2}`)))),l("p",null,"La ",l("b",null,"varianza")," della distribuzione uniforme è:"),l("p",null,l(e.j,null,ia(Ri||(Ri=la`Var(X) = \frac{(b - a)^2}{12}`))))))),l(e.q,{title:"Normale o Gaussiana"},l(e.p,{title:"Distribuzione normale"},l("p",null,"Una variabile aleatoria con una specifica distribuzione."),l("p",null,"Il suo simbolo è ",l(e.j,null,ia(Ki||(Ki=la`Nor(\mu, \sigma^2)`))),"."),l(a.a,null,l(e.j,null,"\\mu")," e ",l(e.j,null,"\\sigma^2")," sono rispettivamente la media e la varianza della distribuzione!")),l(e.p,{title:"Densità della distribuzione normale"},l("p",null,"La distribuzione normale ha come densità:"),l("p",null,l(e.j,null,ia(Qi||(Qi=la`f_X (x) = \frac{e^{-\frac{(x - \mu)^2}{2 \sigma^2}}}{\sqrt{2 \pi \cdot \sigma^2}}`))))),l(e.p,{title:"Momenti della distribuzione normale"},l("p",null,l("p",null,"La ",l("b",null,"funzione generatrice dei momenti")," della distribuzione normale è:"),l("p",null,l(e.j,null,ia(Zi||(Zi=la`m_X (t) = e^{\mu \cdot t + \frac{\sigma^2 \cdot t^2}{2}}`)))),l("p",null,"La ",l("b",null,"media")," della distribuzione normale è:"),l("p",null,l(e.j,null,ia(Wi||(Wi=la`E(X) = \mu`)))),l("p",null,"La ",l("b",null,"varianza")," della distribuzione normale è:"),l("p",null,l(e.j,null,ia(Ji||(Ji=la`Var(X) = \sigma^2`))))))),l(e.q,null,l(e.p,{title:"Trasformazione della normale"},l("p",null,"Qualsiasi normale può essere trasformata in qualsiasi altra normale:"),l("p",null,l(e.j,null,ia(Hi||(Hi=la`X \sim Nor(m, v^2) \implies \alpha X + \beta \sim Nor(\alpha m + \beta, (\alpha v)^2)`))))),l(e.p,{title:"Normale standard"},l("p",null,"La distribuzione normale standard ",l(e.j,null,"Z")," è:"),l("p",null,l(e.j,null,"Z \\sim Nor(0, 1)")),l("p",null,"La sua funzione di ripartizione è detta ",l(e.j,null,ia($i||($i=la`\phi(z)`)))," e vale:"),l("p",null,l(e.j,null,ia(ln||(ln=la`F_Z(z) = \phi(z) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{z} e^{-\frac{x^2}{2}} dx`))))),l(e.p,{title:"Quantili normali"},l("p",null,"Da un quantile ",l(e.j,null,ia(nn||(nn=la`z_\alpha`)))," della normale standard è possibile risalire allo stesso quantile di qualsiasi altra normale:"),l("p",null,l(e.j,null,ia(en||(en=la`x_\alpha = \mu + z_\alpha \cdot \sqrt{\sigma^2}`)))))),l(e.q,null,l(e.p,{title:"Gamma e normale"},l("p",null,"La distribuzione normale ha una particolare relazione con la distribuzione Gamma:"),l("p",null,l(e.j,null,ia(an||(an=la`Z^2 \sim \chi^2 (v = 1)`))))),l(e.p,{title:"La funzione Chi"},l("blockquote",null,'"chi-quadro a un grado di libertà"'),l("p",null,"Esiste una distribuzione Gamma particolare:"),l("p",null,l(e.j,null,ia(on||(on=la`\Gamma \left( \frac{1}{2}, \frac{1}{2} \right) = \chi^2 (v = 1)`)))),l("p",null,"Più chi-quadro possono essere sommate per aumentare i loro gradi di libertà:"),l("p",null,l(e.j,null,ia(tn||(tn=la`\chi^2 (n) + \chi^2 (m) = \chi^2 (n + m)`))))),l(e.p,{title:"T di Student"},l("p",null,"Un'altra funzione particolare è la funzione T di Student:"),l("p",null,l(e.j,null,ia(un||(un=la`T(v) = \frac{Nor(0, 1)}{\sqrt{\frac{\chi^2(v)}{v}}}`)))))),l(e.q,{title:"Approssimazioni notevoli"},l(e.p,{title:"Ipergeometrica e binomiale"},l("p",null,"La binomiale è come una ipergeometrica ma con ripetizioni, quindi per valori molto grandi di ",l(e.j,null,"N")," rispetto a ",l(e.j,null,"n"),", si può dire che:"),l("p",null,l(e.j,null,ia(rn||(rn=la`Ipe(N, K, n) \approx Bin(n, \frac{K}{N})`))))),l(e.p,{title:"Binomiale e poissoniana"},l("p",null,"La binomiale non è altro che una poissoniana a tempo discreto, quindi, se ",l(e.j,null,"n")," è grande e ",l(e.j,null,"n \\cdot p")," è nell'ordine di grandezza delle unità, allora:"),l("p",null,l(e.j,null,ia(pn||(pn=la`Bin(n, p) \approx Poi(n \cdot p)`))))),l(e.p,{title:"Binomiale e normale"},l("p",null,"Per il Teorema di De Moivre-Laplace, se una binomiale ha una ",l(e.j,null,"n")," grande e ",l(e.j,null,"p")," non vicina a 0 o 1, si può approssimare con:"),l("p",null,l(e.j,null,ia(sn||(sn=la`Bin(n, p) \approx Nor(n \cdot p, n \cdot p \cdot q)`))))),l(e.p,{title:"Correzione di Yates"},l("p",null,"Passando da una variabile discreta ",l(e.j,null,"X")," a una continua ",l(e.j,null,"Y"),", per ogni valore discreto ",l(e.j,null,"k"),' la probabilità viene "spalmata" su tutto l\'intervallo ',l(e.j,null,ia(dn||(dn=la`(k - \frac{1}{2}, k + \frac{1}{2})`))),":"),l("ul",null,l("li",null,l(e.j,null,ia(mn||(mn=la`P(X < k) \simeq P(Y \leq k - \frac{1}{2})`)))),l("li",null,l(e.j,null,ia(cn||(cn=la`P(X \leq k) \simeq P(Y \leq k + \frac{1}{2})`)))),l("li",null,l(e.j,null,ia(bn||(bn=la`P(X \geq k) \simeq P(Y \geq k - \frac{1}{2})`)))),l("li",null,l(e.j,null,ia(vn||(vn=la`P(X > k) \simeq P(Y \geq k + \frac{1}{2})`))))))),l(e.q,{title:"Vettori aleatori"},l(e.p,{title:"Vettore aleatorio"},l("p",null,"Un vettore ",l("b",null,"composto da variabili aleatorie"),"."),l("p",null,"Il suo simbolo generalmente è ",l(e.j,null,ia(gn||(gn=la`\boldsymbol{X}`)))," oppure ",l(e.j,null,ia(fn||(fn=la`X, Y`))),".")),l(e.p,{title:"Funzioni di ripartizione"},l("p",null,"I vettori aleatori hanno più funzioni di ripartizione che si differenziano in base al numero di parametri."),l("p",null,"Se il numero di parametri coincide con la dimensione del vettore aleatorio, allora la funzione sarà una ",l("i",null,"funzione di ripartizione congiunta"),":"),l("p",null,l(e.j,null,ia(jn||(jn=la`F_{X, Y} (x, y) = P(X \leq x, Y \leq y)`)))),l("p",null,"Se il numero di parametri è minore della dimensione del vettore aleatorio, allora la funzione sarà una ",l("i",null,"funzione di ripartizione marginale"),":"),l("p",null,l(e.j,null,ia(_n||(_n=la`F_X (x) = P(X \leq x) = \lim_{y \to +\infty} F_{X, Y} (x, y)`))))),l(e.p,{title:"Densità discreta"},l("p",null,"I vettori aleatori ",l("b",null,"discreti")," hanno più densità che si differenziano in base al numero di parametri."),l("p",null,"Se il numero di parametri coincide con la dimensione del vettore aleatorio, allora la funzione sarà una ",l("i",null,"densità congiunta"),":"),l("p",null,l(e.j,null,ia(hn||(hn=la`p_{X, Y} (x, y) = P(X = x, Y = y)`)))),l("p",null,"Se il numero di parametri è minore della dimensione del vettore aleatorio, allora la funzione sarà una ",l("i",null,"densità marginale"),":"),l("p",null,l(e.j,null,ia(zn||(zn=la`p_X (x) = \sum_j p_{X, Y} (x_i, y_j)`)))))),l(e.q,{title:"Più variabili aleatorie"},l(e.p,{title:"Indipendenza delle variabili aleatorie"},l("p",null,"Più variabili aleatorie sono indipendenti se, per qualsiasi scelta di intervalli ",l(e.j,null,"A_i"),":"),l("p",null,l(e.j,null,ia(Xn||(Xn=la`P(X_1 \in A_1, \dots, X_n \in A_n) = P(X_1 \in A_1) \times \dots \times P(X_n \in A_n)`))))),l(e.p,{title:"Media dei vettori aleatori"},l("p",null,"E' possibile calcolare la media di qualsiasi funzione ",l(e.j,null,"g(X, Y)")," avente elementi del vettore come variabili:"),l("p",null,l(e.j,null,ia(qn||(qn=la`E(g(X, Y)) = \sum_{i, j} g(x_i, y_i) \cdot p_{X, Y} (x_i, y_i)`)))),l(a.a,null,"Solitamente si calcola la media di ",l(e.j,null,"x \\cdot y"),"."),l("p",null,"Le medie di più variabili aleatorie si possono sommare:"),l("p",null,l(e.j,null,ia(En||(En=la`E(X + Y) = E(X) + E(Y)`)))))),l(e.q,null,l(e.p,{title:"Covarianza"},l("p",null,"Un ",l("b",null,"operatore")," che misura la correlazione di due variabili aleatorie."),l("p",null,"Si calcola con il valore atteso dei prodotti delle distanze dalla media:"),l("p",null,l(e.j,null,ia(Pn||(Pn=la`Cov(X, Y) = E((X - E(X) \cdot (Y - E(Y)) = E(XY) - E(X) \cdot E(Y)`)))),l("p",null,"Ha diverse proprietà:"),l("ul",null,l("li",null,"Il suo ",l("b",null,"valore nullo")," è 0: ",l(e.j,null,ia(Ln||(Ln=la`Cov(X, \alpha) = 0`)))),l("li",null,"E' ",l("b",null,"commutativa"),": ",l(e.j,null,ia(xn||(xn=la`Cov(X, Y) = Cov(Y, X)`)))),l("li",null,"E' ",l("b",null,"semplificabile"),": ",l(e.j,null,ia(kn||(kn=la`Cov(X, X) = Var(X)`)))),l("li",null,"E' ",l("b",null,"lineare"),": ",l(e.j,null,ia(yn||(yn=la`Cov(\alpha X, \beta Y) = \alpha \cdot \beta \cdot Cov(X, Y)`)))),l("li",null,"E' ",l("b",null,"distributiva"),": ",l(e.j,null,ia(Fn||(Fn=la`Cov(X + Y, V + W) = Cov(X, Y) + Cov(X, W) + Cov(Y, V) + Cov(Y, W)`)))))),l(e.p,{title:"Variabili incorrelate"},l("p",null,"Due variabili sono ",l("i",null,"variabili incorrelate")," se:"),l("p",null,l(e.j,null,ia(Sn||(Sn=la`Cov(X, Y) = 0`)))),l("p",null,"Variabili indipendenti sono sempre incorrelate.")),l(e.p,{title:"Matrice di covarianza"},l("p",null,"Una matrice ",l(e.j,null,ia(In||(In=la`\boldsymbol{C_X}`)))," che contiene la covarianza tra tutte le variabili di un vettore aleatorio ",l(e.j,null,ia(Cn||(Cn=la`\boldsymbol{X}`))),":"),l("p",null,l(e.j,null,ia(Nn||(Nn=la` + \boldsymbol{C_X} = + \begin{bmatrix} + Var(X_1) & Cov(X_1, X_2) & Cov(X_1, X_3)\\ + Cov(X_2, X_1) & Var(X_2) & Cov(X_2, X_3)\\ + Cov(X_3, X_1) & Cov(X_3, X_2) & Var(X_3) + \end{bmatrix} + `)))),l("p",null,"E' sempre simmetrica e semidefinita positiva (tutti gli autovalori sono ",l(e.j,null,"\\geq 0"),".")),l(e.p,{title:"Coefficiente di correlazione"},l("p",null,"Un valore che misura come due variabili aleatorie sono correlate:"),l("p",null,l(e.j,null,ia(Vn||(Vn=la`\rho_{X, Y} = \frac{Cov(X, Y)}{\sqrt{Var(X)} \cdot \sqrt{Var(Y)}}`)))),l("p",null,"E' sempre compreso tra -1 e 1:"),l("p",null,l(e.j,null,ia(wn||(wn=la`-1 \leq \rho_{X, Y} \leq 1`)))),l("p",null,"Vale esattamente -1 o 1 solo se esiste un legame lineare tra le due variaibli:"),l("p",null,l(e.j,null,ia(Dn||(Dn=la`Y = a X + b \Longleftrightarrow | \rho_{X, Y} | = 1`))))),l(e.p,{title:"Varianza di variabili aleatorie sommate"},l("p",null,"La varianza di due variabili aleatorie sommate è:"),l("p",null,l(e.j,null,ia(Yn||(Yn=la`Var(X + Y) = Var(X) + Var(Y) + 2 \cdot Cov(X, Y)`)))),l(a.a,null,"Si dimostra applicando le proprietà della covarianza!"),l("p",null,"Se più variabili aleatorie ",l(e.j,null,"X_i")," sono ",l("b",null,"indipendenti")," (",l(e.j,null,ia(Un||(Un=la`Cov(X, Y) = 0`))),"), allora:"),l("p",null,l(e.j,null,ia(Mn||(Mn=la`Var \left( \sum_i X_i \right) = \sum_i Var(X_i)`)))))),l(e.q,{title:"Campioni"},l(e.p,{title:"Campione casuale"},l("p",null,"Una ",l("b",null,"n-pla")," di variabili aleatorie con la stessa distribuzione della variabile aleatoria ",l(e.j,null,"X"),' ("popolazione") ma ',l("b",null,"indipendenti")," tra loro."),l(a.a,null,"Le variabili aleatorie sono come un lazy-load in programmazione; quando ci sarà bisogno del loro valore numerico, esse si ",l("b",null,"realizzeranno")," nel loro valore.")),l(e.p,{title:"Momento campionario"},l("p",null,"Il valore dato dalla media aritmetica degli ",l(e.j,null,"n")," elementi del campione elevati alla potenza ",l(e.j,null,"k"),":"),l("p",null,l(e.j,null,ia(Bn||(Bn=la`M^{(k)}_n = \frac{1}{n} \cdot \sum_{i = 1}^n X_i^k `)))),l("p",null,"Il momento campionario di primo ordine è la ",l("i",null,"media campionaria")," ",l(e.j,null,ia(An||(An=la`\overline{X}_n`))),".")),l(e.p,{title:"Varianza campionaria"},l("p",null,"La media aritmetica dello scarto quadratico medio degli elementi del campione."),l("p",null,"Se è noto il valore medio ",l(e.j,null,ia(Tn||(Tn=la`m = E(X)`)))," di X:"),l("p",null,l(e.j,null,ia(Gn||(Gn=la`S_0^2 = \frac{1}{n} \cdot \sum_{i = 1}^n (X_i - m)^2 = M_n^(2) - 2 \cdot m \cdot \overline{X}_n + m^2`)))),l("p",null,"Altrimenti:"),l("p",null,l(e.j,null,ia(On||(On=la`S_n^2 = \frac{1}{n - 1} \cdot \sum_{i = 1}^n (X_i - \overline{X}_n)^2 = \frac{1}{n - 1} \cdot ( n \cdot M_2^{(2)} - n \cdot \overline{X}_n^2)`)))))),l(e.q,{title:"Media-ception"},l(e.p,{title:"Media campionaria"},l("p",null,"Se calcoliamo la media della media campionaria, risulterà vero che:"),l("p",null,l(e.j,null,ia(Rn||(Rn=la`E(\overline{X}_n) = E(X)`)))),l(a.a,null,"Quindi, è possibile usare i campioni per trovare la media di una variabile aleatoria!")),l(e.p,{title:"Varianza campionaria"},l("p",null,"Se calcoliamo la varianza della media campionaria, risulterà vero che:"),l("p",null,l(e.j,null,ia(Kn||(Kn=la`Var(\overline{X}_n) = \frac{Var(X)}{n}`)))),l(a.a,null,"Quindi, possiamo stimare l'errore della media calcolata tramite campioni!")),l(e.p,{title:"Correzione campionaria"},l("p",null,"Se calcoliamo la media della varianza campionaria, risulterà vero che:"),l("p",null,l(e.j,null,ia(Qn||(Qn=la`E(S_0^2) = E(S_n^2) = Var(X)`)))),l(a.a,null,"Quindi, possiamo stimare l'errore della media calcolata tramite campioni!"))),l(e.q,{title:"Campionamento di una distribuzione normale"},l(e.p,{title:"Campionamento di una distribuzione normale"},l("p",null,"Se la popolazione ",l(e.j,null,"X")," ha una distribuzione normale (",l(e.j,null,ia(Zn||(Zn=la`X \sim Nor(\mu, \sigma^2)`))),")...")),l(e.p,{title:"Distribuzione della media campionaria"},l("p",null,"...allora sappiamo anche la distribuzione della media campionaria!"),l("p",null,l(e.j,null,ia(Wn||(Wn=la`\overline{X}_n \sim Nor \left( \mu, \frac{\sigma^2}{n} \right)`))))),l(e.p,{title:"Distribuzione della varianza campionaria"},l("p",null,"...e anche della varianza campionaria!"),l("p",null,l(e.j,null,ia(Jn||(Jn=la`S_0^2 \sim \frac{\sigma^2}{n} \cdot \chi^2 (n)`)))),l("p",null,l(e.j,null,ia(Hn||(Hn=la`S_n^2 \sim \frac{\sigma^2}{n - 1} \cdot \chi^2 (n-1)`))))),l(e.p,{title:"Indipendenza"},l("p",null,"...e che media campionaria e varianza campionaria sono indipendenti tra loro!"))),l(e.q,{title:"Quando i campioni hanno dimensioni infinite"},l(e.p,{title:"Convergenza in distribuzione"},l("p",null,"Se la successione di variabili aleatorie ",l(e.j,null,"X_n")," all'infinito ha la ",l("b",null,"stessa funzione di ripartizione")," della popolazione ",l(e.j,null,"X"),", allora essa ",l("i",null,"converge in distribuzione"),"."),l("p",null,l(e.j,null,"\\lim_{n \\to +\\infty} F_{X_n} (x) = F_X (x) \\implies X_n \\xrightarrow{d} X"))),l(e.p,{title:"Convergenza in probabilità"},l("p",null,"Se la successione di variabili aleatorie ",l(e.j,null,"X_n")," all'infinito ha la ",l("b",null,"stessa probabilità")," della popolazione ",l(e.j,null,"X"),", allora essa ",l("i",null,"converge in probabilità"),"."),l("p",null,l(e.j,null,"\\forall \\epsilon > 0, \\lim_{n \\to +\\infty} P( | X_n - X | < \\epsilon) = 1 \\implies X_n \\xrightarrow{p} X"))),l(e.p,{title:"Convergenza quasi certa"},l("p",null,"Se la successione di variabili aleatorie ",l(e.j,null,"X_n")," all'infinito ha la ",l("b",null,"stessa probabilità a ")," della popolazione ",l(e.j,null,"X"),", allora essa ",l("i",null,"converge quasi certamente"),"."),l("p",null,l(e.j,null,"\\forall \\epsilon > 0, P left( \\lim_{n \\to +\\infty} | X_n - X | < \\epsilon) \right) = 1 \\implies X_n \\xrightarrow{qc} X"))),l(e.p,{title:"Convergenza in media quadratica"},l("p",null,"Se la successione di variabili aleatorie ",l(e.j,null,"X_n")," all'infinito ha la ",l("b",null,"media del quadrato della distanza")," tra la successione e la popolazione ",l(e.j,null,"X")," ",l("b",null,"uguale a 0"),", allora essa ",l("i",null,"converge in media quadratica"),"."),l("p",null,l(e.j,null,"\\lim_{n \\to +\\infty} E( | X_n - X |^2 = 0 \\implies X_n \\xrightarrow{mq} X"))),l(e.p,{title:"Gerarchia delle convergenze"},l("p",null,l(e.j,null,"\n \\begin{matrix}\n X_n \\xrightarrow{mq} X\\\\\n X_n \\xrightarrow{qc} X\n \\end{matrix} \\implies X_n \\xrightarrow{p} X \\implies X_n \\xrightarrow{d} X")),l("p",null,"In più:"),l("p",null,l(e.j,null,"X_n \\xrightarrow{p} x \\Longleftrightarrow X_n \\xrightarrow{d} x")))),l(e.q,{title:"I grandi numeri"},l(e.p,{title:"Legge debole dei grandi numeri"},l("p",null,"La successione delle medie campionarie ",l(e.j,null,ia($n||($n=la`\overline{X}_n`)))," ",l("b",null,"converge in probabilità")," alla media della popolazione ",l(e.j,null,ia(le||(le=la`E(X)`))),", se essa esiste."),l("p",null,l(e.j,null,"\\overline{X}_n \\xrightarrow{p} X")),l("p",null,"Ovvero:"),l("p",null,l(e.j,null,ia(ie||(ie=la`\forall \epsilon > 0, \lim_{n \to +\infty} P( | \overline{X}_n - E(X) | < \epsilon) = 1`)))),l("p",null,l(e.j,null,ia(ne||(ne=la`P( | \overline{X}_n - E(X) | < \epsilon) \to 1`))))),l(e.p,{title:"Legge forte dei grandi numeri"},l("p",null,"La successione delle medie campionarie ",l(e.j,null,ia(ee||(ee=la`\overline{X}_n`)))," ",l("b",null,"converge quasi certamente")," alla media della popolazione ",l(e.j,null,ia(ae||(ae=la`E(X)`))),", se essa esiste."),l("p",null,l(e.j,null,"\\overline{X}_n \\xrightarrow{qc} X")),l("p",null,"Ovvero:"),l("p",null,l(e.j,null,ia(oe||(oe=la`\forall \epsilon > 0, P \left( \lim_{n \to +\infty} | \overline{X}_n - E(X) | < \epsilon \right) = 1`)))),l(a.a,null,"Dimostra che l'interpretazione frequentista della probabilità è valida!"))),l(e.q,{title:"Al limite"},l(e.p,{title:"Teorema centrale del limite"},l("p",null,"La successione delle medie campionarie ",l(e.j,null,ia(te||(te=la`\overline{X}_n`)))," ",l("b",null,"converge in distribuzione")," a ",l(e.j,null,ia(ue||(ue=la`Nor(0, 1) = \Phi()`))),"."),l("p",null,l(e.j,null,ia(re||(re=la`\overline{X}_n \approx Nor \left(E(X), \frac{Var(X)}{n} \right)`)))),l("p",null,"Ovvero:"),l("p",null,l(e.j,null,ia(pe||(pe=la`\forall x \in \mathbb{R}, \lim_{n \to +\infty} P \left( \frac{\overline{X}_n - E(X)}{\sqrt{\frac{Var(X)}{n}}} \leq x \right) = \Phi(x)`)))))),l(e.q,{title:"Altre approsimazioni"},l(e.p,{title:"Binomiale e normale"},l("p",null,"E' una somma di ",l("b",null,"bernoulliane"),", e quindi si approssima a una normale:"),l("p",null,l(e.j,null,ia(se||(se=la`Bin(n, p) \approx Nor(n \cdot p, n \cdot p \cdot q)`))))),l(e.p,{title:"Binomiale negativa e normale"},l("p",null,"E' una somma di ",l("b",null,"geometriche"),", e quindi si approssima a una normale:"),l("p",null,l(e.j,null,ia(de||(de=la`\overline{Bin} (n, p) \approx Nor \left( \frac{n}{p}, \frac{n \cdot (1 - p)}{p^2} \right)`))))),l(e.p,{title:"Poissoniana e normale"},l("p",null,"E' una somma di altre ",l("b",null,"poissoniane"),", e quindi si approssima a una normale:"),l("p",null,l(e.j,null,ia(me||(me=la`Poi(\lambda) \approx Nor(\lambda, \lambda)`))))),l(e.p,{title:"Gamma e normale"},l("p",null,"E' una somma di ",l("b",null,"esponenziali"),", e quindi si approssima a una normale:"),l("p",null,l(e.j,null,ia(ce||(ce=la`\Gamma (\alpha, \lambda) \approx Nor \left( \frac{\alpha}{\lambda}, \frac{\alpha}{\lambda^2} \right)`))))),l(e.p,{title:"In generale"},l("p",null,"Se ",l(e.j,null,"n")," è grande, allora:"),l("p",null,l(e.j,null,ia(be||(be=la`Y = \sum_{i=1}^{n} X_i`)))))),l(e.q,{title:"Actually statistica"},l(e.p,{title:"Parametri sconosciuti"},l("p",null,"Per indicare parametri sconosciuti di una legge si usa ",l(e.j,null,"\\theta"),".")),l(e.p,{title:"Statistica"},l("p",null,"Una variabile aleatoria funzione di un campione:"),l("p",null,l(e.j,null,ia(ve||(ve=la`T(\boldsymbol{X})`)))),l(a.a,null,"Ad esempio, sono statistiche media e varianza campionaria, così come il campione stesso ",l(e.j,null,ia(ge||(ge=la`T(\boldsymbol{X}) = \boldsymbol{X}`))),"."))),l(e.q,{title:"Stimatori"},l(e.p,{title:"Stimatore"},l("p",null,"Una statistica ",l(e.j,null,"T_n")," ottenuta da ",l(e.j,null,"n")," osservazioni, che stimi i parametri di una legge e sia indipendente da essi.")),l(e.p,{title:"Corretto"},l("p",null,"Uno stimatore è ",l("i",null,"corretto")," se il suo valore atteso coincide con quello dei parametri che stima:"),l("p",null,l(e.j,null,ia(fe||(fe=la`E(T_n) = \theta`))))),l(e.p,{title:"Asintoticamente corretto"},l("p",null,"Uno stimatore è ",l("i",null,"asintoticamente corretto")," se, per infinite osservazioni, il suo valore atteso coincide con quello dei parametri che stima:"),l("p",null,l(e.j,null,ia(je||(je=la`\lim_{n \to +\infty} E(T_n) = \theta`))))),l(e.p,{title:"Consistente in media quadratica"},l("p",null,"Uno stimatore è ",l("i",null,"consistente in media quadratica")," se:"),l("p",null,l(e.j,null,ia(_e||(_e=la`\lim_{n \to +\infty} E((T_n - \theta)^2) = 0`))))),l(e.p,{title:"Consistente in probabilità"},l("p",null,"Uno stimatore è ",l("i",null,"consistente in probabilità")," se:"),l("p",null,l(e.j,null,ia(he||(he=la`\forall \epsilon > 0, \lim_{n \to +\infty} P( |T_n - \theta| < \epsilon) = 1`))))),l(e.p,{title:"Asintoticamente normale"},l("p",null,"Uno stimatore è ",l("i",null,"asintoticamente normale")," se:"),l("p",null,l(e.j,null,ia(ze||(ze=la`\lim_{n \to +\infty} \frac{T_n - E(T_n)}{\sqrt{Var(T_n)}} \sim Nor(0, 1)`)))))),l(e.q,{title:"Metodo dei momenti"},l(e.p,{title:"Metodo dei momenti"},l("p",null,"Si può usare il ",l("i",null,"metodo dei momenti")," per ottenere uno stimatore di una popolazione ",l(e.j,null,"X"),"."),l("p",null,"Lo stimatore di ",l(e.j,null,ia(Xe||(Xe=la`\theta`)))," così ottenuto sarà indicato aggiungendo un cappellino e una ",l(e.j,null,"M")," a ",l(e.j,null,"\\theta"),": ",l(e.j,null,ia(qe||(qe=la`\widehat{\theta}_M`)))),l("p",null,"Visto che:"),l("ul",null,l("li",null,l(e.j,null,ia(Ee||(Ee=la`\theta = g(E(X))`)))),l("li",null,l(e.j,null,ia(Pe||(Pe=la`\widehat{E(X)} = \overline{X}_n`))))),l("p",null,"Allora:"),l("p",null,l(e.j,null,ia(Le||(Le=la`\widehat{\theta}_M = g( \overline{X}_n )`)))),l("p",null,"Se ",l(e.j,null,ia(xe||(xe=la`\theta`)))," non è esprimibile in termini di ",l(e.j,null,ia(ke||(ke=la`E(X)`))),", si possono usare i momenti successivi ",l(e.j,null,ia(ye||(ye=la`M_n^2`))),", ",l(e.j,null,ia(Fe||(Fe=la`M_n^3`))),", ",l(e.j,null,ia(Se||(Se=la`M_n^3`))),"..."))),l(e.q,{title:"Metodo della massima verosomiglianza"},l(e.p,{title:"Metodo della massima verosomiglianza"},l("p",null,"Si può usare il ",l("i",null,"metodo della massima verosomiglianza")," per ottenere uno stimatore di una popolazione ",l(e.j,null,"X"),"."),l("p",null,"Lo stimatore di ",l(e.j,null,ia(Ie||(Ie=la`\theta`)))," così ottenuto sarà indicato aggiungendo un cappellino e una ",l(e.j,null,"L")," a ",l(e.j,null,"\\theta"),": ",l(e.j,null,ia(Ce||(Ce=la`\widehat{\theta}_L`)))),l("p",null,"Consiste nel trovare il massimo assoluto ",l(e.j,null,ia(Ne||(Ne=la`\widehat{\theta}_L`)))," della la funzione di verosomiglianza ",l(e.j,null,ia(Ve||(Ve=la`L`))),":"),l("p",null,l(e.j,null,ia(we||(we=la`L(x_1, ..., x_n; \theta) = \prod_{i=1}^n f_X(x_i; \theta)`)))),l("p",null,"Gli stimatori di massima verosomiglianza sono ",l("b",null,"asintoticamente corretti"),", ",l("b",null,"consistenti in probabilità")," e ",l("b",null,"asintoticamente normali"),".")),l(e.p,{title:"Proprietà degli stimatori di massima verosomiglianza"},l("p",null,"Gli stimatori di massima verosomiglianza godono delle seguenti proprietà:"),l("ul",null,l("li",null,"Sono ",l("b",null,"asintoticamente corretti"),"."),l("li",null,"Sono ",l("b",null,"consistenti in probabilità"),"."),l("li",null,"Sono ",l("b",null,"asintoticamente normali"),"."),l("li",null,"Sono ",l("b",null,"invarianti"),": ",l(e.j,null,ia(De||(De=la`\widehat{g(\theta)}_L = g(\widehat{\theta}_L)`))))))),l(e.q,{title:"Nuove stime notevoli"},l(e.p,{title:"Stima di una bernoulliana"},l("p",null,"Per il metodo dei momenti oppure per il metodo della massima verosomiglianza:"),l("p",null,l(e.j,null,ia(Ye||(Ye=la`\widehat{p}_M = \widehat{p}_L = \overline{X}_n`))))),l(e.p,{title:"Stima di una poissoniana"},l("p",null,"Per il metodo dei momenti oppure per il metodo della massima verosomiglianza:"),l("p",null,l(e.j,null,ia(Ue||(Ue=la`\widehat{\mu}_M = \widehat{\mu}_L = \overline{X}_n`))))),l(e.p,{title:"Stima di una esponenziale"},l("p",null,"Per il metodo dei momenti oppure per il metodo della massima verosomiglianza:"),l("p",null,l(e.j,null,ia(Me||(Me=la`\widehat{\lambda}_M = \widehat{\lambda}_L = \frac{1}{\overline{X}_n}`))))),l(e.p,{title:"Stima di una normale"},l("p",null,"Per il metodo della massima verosomiglianza:"),l("ul",null,l("li",null,l(e.j,null,ia(Be||(Be=la`\widehat{\mu}_L = \overline{X}_n`)))),l("br",null),l("li",null,l(e.j,null,ia(Ae||(Ae=la`\widehat{\sigma^2}_L = \frac{\sum (X_i - \overline{X}_n)^2 }{n}`))))))),l(e.q,{title:"Intervalli di confidenza"},l(e.p,{title:"Confidenza"},l("blockquote",null,'"intervallo di confidenza al 95%"'),l("p",null,"L'intervallo di valori di ",l(e.j,null,"\\theta"),' all\'interno del quale siamo "più o meno sicuri" si trovi il valore effettivo:'),l("p",null,"L'intervallo di confidenza a N della stima ",l(e.j,null,ia(Te||(Te=la`\widehat{W}`)))," è l'intervallo ",l(e.j,null,"]a, b[")," tale che:"),l("p",null,l(e.j,null,ia(Ge||(Ge=la`P( a < W < b ) = N`)))),l("p",null,"Può anche essere ",l("b",null,"unilatero")," nel caso limiti la stima in una sola direzione, positiva o negativa."))),l(e.q,{title:"Confidenza nella media di una normale"},l(e.p,{title:"Varianza nota"},l("p",null,"Se conosciamo la varianza di una normale, allora possiamo ricavare velocemente gli intervalli di confidenza all'",l(e.j,null,"\\alpha"),"% con queste formule:"),l("ul",null,l("li",null,"Intervalli bilateri: ",l(e.j,null,ia(Oe||(Oe=la`\mu \in \left[ \overline{x}_n - z_{1 - \frac{\alpha}{2}} \cdot \sqrt{\frac{\sigma^2}{n}}, \overline{x}_n + z_{1 - \frac{\alpha}{2}} \cdot \sqrt{\frac{\sigma^2}{n}} \right]`)))),l("li",null,"Intervallo unilatero da sinistra: ",l(e.j,null,ia(Re||(Re=la`\mu \in \left( -\infty, \overline{x}_n + z_{1 - \frac{\alpha}{2}} \cdot \sqrt{\frac{\sigma^2}{n}} \right]`)))),l("li",null,"Intervallo unilatero da destra: ",l(e.j,null,ia(Ke||(Ke=la`\mu \in \left[ \overline{x}_n - z_{1 - \frac{\alpha}{2}} \cdot \sqrt{\frac{\sigma^2}{n}}, +\infty \right)`)))))),l(e.p,{title:"Varianza incognita"},l("p",null,"Se non conosciamo la varianza di una normale, allora possiamo ricavare velocemente gli intervalli di confidenza all'",l(e.j,null,"\\alpha"),"% con queste formule:"),l("ul",null,l("li",null,"Intervalli bilateri: ",l(e.j,null,ia(Qe||(Qe=la`\mu \in \left[ \overline{x}_n - t_{1 - \frac{\alpha}{2}; n-1} \cdot \sqrt{\frac{s_n^2}{n}}, \overline{x}_n + t_{1 - \frac{\alpha}{2}; n-1} \cdot \sqrt{\frac{s_n^2}{n}} \right]`)))),l("li",null,"Intervallo unilatero da sinistra: ",l(e.j,null,ia(Ze||(Ze=la`\mu \in \left( -\infty, \overline{x}_n + t_{1 - \frac{\alpha}{2}; n-1} \cdot \sqrt{\frac{s_n^2}{n}} \right]`)))),l("li",null,"Intervallo unilatero da destra: ",l(e.j,null,ia(We||(We=la`\mu \in \left[ \overline{x}_n - t_{1 - \frac{\alpha}{2}; n-1} \cdot \sqrt{\frac{s_n^2}{n}}, +\infty \right)`))))),l("p",null,l(e.j,null,ia(Je||(Je=la`t_{\alpha, v}`)))," è un quantile della distribuzione di Student di parametro ",l(e.j,null,"v"),"."))),l(e.q,{title:"Confidenza per la proporzione di una bernoulliana"},l(e.p,{title:"Terzo metodo corretto"},l("p",null,"L'intervallo di confidenza per la proprorzione di una bernoulliana qualsiasi si ottiene da questa formula:"),l("p",null,l(e.j,null,ia(He||(He=la`p \in \left[ \overline{p} - z_{1 - \frac{\alpha}{2}} \cdot \sqrt{\frac{\overline{p} \cdot (1 - \overline{p})}{n+4}}, \overline{p} + z_{1 - \frac{\alpha}{2}} \cdot \sqrt{\frac{\overline{p} \cdot (1 - \overline{p})}{n+4}} \right]`)))))),l(e.q,{title:"Confidenza per la media di qualsiasi popolazione"},l(e.p,{title:"Approssimando con la normale"},l("p",null,"L'intervallo di confidenza per la media di una qualsiasi popolazione si ottiene da questa formula:"),l("p",null,l(e.j,null,ia($e||($e=la`m \in \left[ \overline{x}_n - z_{1 - \frac{\alpha}{2}} \cdot 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\\overline{x}_n + t_{1 - \\frac{\\alpha}{2}; n-1} \\cdot \\sqrt{\\frac{s_n^2}{n}} \\right]"]);return u=function(){return n},n}function c(){var n=We(["mu in left[ overline{x}_n - z_{1 - \frac{alpha}{2}} cdot sqrt{\frac{sigma^2}{n}}, +infty \right)"],["\\mu \\in \\left[ \\overline{x}_n - z_{1 - \\frac{\\alpha}{2}} \\cdot \\sqrt{\\frac{\\sigma^2}{n}}, +\\infty \\right)"]);return c=function(){return n},n}function p(){var n=We(["mu in left( -infty, overline{x}_n + z_{1 - \frac{alpha}{2}} cdot sqrt{\frac{sigma^2}{n}} \right]"],["\\mu \\in \\left( -\\infty, \\overline{x}_n + z_{1 - \\frac{\\alpha}{2}} \\cdot \\sqrt{\\frac{\\sigma^2}{n}} \\right]"]);return p=function(){return n},n}function s(){var n=We(["mu in left[ overline{x}_n - z_{1 - \frac{alpha}{2}} cdot sqrt{\frac{sigma^2}{n}}, overline{x}_n + z_{1 - \frac{alpha}{2}} cdot sqrt{\frac{sigma^2}{n}} \right]"],["\\mu \\in \\left[ \\overline{x}_n - z_{1 - \\frac{\\alpha}{2}} \\cdot \\sqrt{\\frac{\\sigma^2}{n}}, \\overline{x}_n + z_{1 - \\frac{\\alpha}{2}} \\cdot \\sqrt{\\frac{\\sigma^2}{n}} \\right]"]);return s=function(){return n},n}function d(){var n=We(["P( a < W < b ) = N"]);return d=function(){return n},n}function m(){var n=We(["widehat{W}"],["\\widehat{W}"]);return m=function(){return n},n}function f(){var n=We(["widehat{sigma^2}_L = \frac{sum (X_i - overline{X}_n)^2 }{n}"],["\\widehat{\\sigma^2}_L = \\frac{\\sum (X_i - \\overline{X}_n)^2 }{n}"]);return f=function(){return n},n}function v(){var n=We(["widehat{mu}_L = overline{X}_n"],["\\widehat{\\mu}_L = \\overline{X}_n"]);return v=function(){return n},n}function b(){var n=We(["widehat{lambda}_M = widehat{lambda}_L = \frac{1}{overline{X}_n}"],["\\widehat{\\lambda}_M = \\widehat{\\lambda}_L = \\frac{1}{\\overline{X}_n}"]);return b=function(){return n},n}function _(){var n=We(["widehat{mu}_M = widehat{mu}_L = overline{X}_n"],["\\widehat{\\mu}_M = \\widehat{\\mu}_L = \\overline{X}_n"]);return _=function(){return n},n}function X(){var n=We(["widehat{p}_M = widehat{p}_L = overline{X}_n"],["\\widehat{p}_M = \\widehat{p}_L = \\overline{X}_n"]);return X=function(){return n},n}function g(){var n=We(["widehat{g(\theta)}_L = g(widehat{\theta}_L)"],["\\widehat{g(\\theta)}_L = g(\\widehat{\\theta}_L)"]);return g=function(){return n},n}function h(){var n=We(["L(x_1, ..., x_n; \theta) = prod_{i=1}^n f_X(x_i; \theta)"],["L(x_1, ..., x_n; \\theta) = \\prod_{i=1}^n f_X(x_i; \\theta)"]);return h=function(){return n},n}function q(){var n=We(["L"]);return q=function(){return n},n}function j(){var n=We(["widehat{\theta}_L"],["\\widehat{\\theta}_L"]);return j=function(){return n},n}function z(){var n=We(["widehat{\theta}_L"],["\\widehat{\\theta}_L"]);return z=function(){return n},n}function E(){var n=We(["\theta"],["\\theta"]);return E=function(){return n},n}function P(){var n=We(["M_n^3"]);return P=function(){return n},n}function x(){var n=We(["M_n^3"]);return x=function(){return n},n}function k(){var n=We(["M_n^2"]);return k=function(){return n},n}function L(){var n=We(["E(X)"]);return L=function(){return n},n}function F(){var n=We(["\theta"],["\\theta"]);return F=function(){return n},n}function y(){var n=We(["widehat{\theta}_M = g( overline{X}_n )"],["\\widehat{\\theta}_M = g( \\overline{X}_n )"]);return y=function(){return n},n}function N(){var n=We(["widehat{E(X)} = overline{X}_n"],["\\widehat{E(X)} = \\overline{X}_n"]);return N=function(){return n},n}function V(){var n=We(["\theta = g(E(X))"],["\\theta = g(E(X))"]);return V=function(){return n},n}function S(){var n=We(["widehat{\theta}_M"],["\\widehat{\\theta}_M"]);return S=function(){return n},n}function Y(){var n=We(["\theta"],["\\theta"]);return Y=function(){return n},n}function w(){var n=We(["lim_{n \to +infty} \frac{T_n - E(T_n)}{sqrt{Var(T_n)}} sim Nor(0, 1)"],["\\lim_{n \\to +\\infty} \\frac{T_n - E(T_n)}{\\sqrt{Var(T_n)}} \\sim Nor(0, 1)"]);return w=function(){return n},n}function C(){var n=We(["\forall epsilon > 0, lim_{n \to +infty} P( |T_n - \theta| < epsilon) = 1"],["\\forall \\epsilon > 0, \\lim_{n \\to +\\infty} P( |T_n - \\theta| < \\epsilon) = 1"]);return C=function(){return n},n}function I(){var n=We(["lim_{n \to +infty} E((T_n - \theta)^2) = 0"],["\\lim_{n \\to +\\infty} E((T_n - \\theta)^2) = 0"]);return I=function(){return n},n}function D(){var n=We(["lim_{n \to +infty} E(T_n) = \theta"],["\\lim_{n \\to +\\infty} E(T_n) = \\theta"]);return D=function(){return n},n}function M(){var n=We(["E(T_n) = \theta"],["E(T_n) = \\theta"]);return M=function(){return n},n}function U(){var n=We(["T(\boldsymbol{X}) = \boldsymbol{X}"],["T(\\boldsymbol{X}) = \\boldsymbol{X}"]);return U=function(){return n},n}function B(){var n=We(["T(\boldsymbol{X})"],["T(\\boldsymbol{X})"]);return B=function(){return n},n}function A(){var n=We(["Y = sum_{i=1}^{n} X_i"],["Y = \\sum_{i=1}^{n} X_i"]);return A=function(){return n},n}function T(){var n=We(["Gamma (alpha, lambda) approx Nor left( \frac{alpha}{lambda}, \frac{alpha}{lambda^2} \right)"],["\\Gamma (\\alpha, \\lambda) \\approx Nor \\left( \\frac{\\alpha}{\\lambda}, \\frac{\\alpha}{\\lambda^2} \\right)"]);return T=function(){return n},n}function G(){var n=We(["Poi(lambda) approx Nor(lambda, lambda)"],["Poi(\\lambda) \\approx Nor(\\lambda, \\lambda)"]);return G=function(){return n},n}function O(){var n=We(["overline{Bin} (n, p) approx Nor left( \frac{n}{p}, \frac{n cdot (1 - p)}{p^2} \right)"],["\\overline{Bin} (n, p) \\approx Nor \\left( \\frac{n}{p}, \\frac{n \\cdot (1 - p)}{p^2} \\right)"]);return O=function(){return n},n}function R(){var n=We(["Bin(n, p) approx Nor(n cdot p, n cdot p cdot q)"],["Bin(n, p) \\approx Nor(n \\cdot p, n \\cdot p \\cdot q)"]);return R=function(){return n},n}function K(){var n=We(["\forall x in mathbb{R}, lim_{n \to +infty} P left( \frac{overline{X}_n - E(X)}{sqrt{\frac{Var(X)}{n}}} leq x \right) = Phi(x)"],["\\forall x \\in \\mathbb{R}, \\lim_{n \\to +\\infty} P \\left( \\frac{\\overline{X}_n - E(X)}{\\sqrt{\\frac{Var(X)}{n}}} \\leq x \\right) = \\Phi(x)"]);return K=function(){return n},n}function Z(){var n=We(["overline{X}_n approx Nor left(E(X), \frac{Var(X)}{n} \right)"],["\\overline{X}_n \\approx Nor \\left(E(X), \\frac{Var(X)}{n} \\right)"]);return Z=function(){return n},n}function Q(){var n=We(["Nor(0, 1) = Phi()"],["Nor(0, 1) = \\Phi()"]);return Q=function(){return n},n}function W(){var n=We(["overline{X}_n"],["\\overline{X}_n"]);return W=function(){return n},n}function J(){var n=We(["\forall epsilon > 0, P left( lim_{n \to +infty} | overline{X}_n - E(X) | < epsilon \right) = 1"],["\\forall \\epsilon > 0, P \\left( \\lim_{n \\to +\\infty} | \\overline{X}_n - E(X) | < \\epsilon \\right) = 1"]);return J=function(){return n},n}function H(){var n=We(["E(X)"]);return H=function(){return n},n}function $(){var n=We(["overline{X}_n"],["\\overline{X}_n"]);return $=function(){return n},n}function nn(){var n=We(["P( | overline{X}_n - E(X) | < epsilon) \to 1"],["P( | \\overline{X}_n - E(X) | < \\epsilon) \\to 1"]);return nn=function(){return n},n}function ln(){var n=We(["\forall epsilon > 0, lim_{n \to +infty} P( | overline{X}_n - E(X) | < epsilon) = 1"],["\\forall \\epsilon > 0, \\lim_{n \\to +\\infty} P( | \\overline{X}_n - E(X) | < \\epsilon) = 1"]);return ln=function(){return n},n}function en(){var n=We(["E(X)"]);return en=function(){return n},n}function an(){var n=We(["overline{X}_n"],["\\overline{X}_n"]);return an=function(){return n},n}function tn(){var n=We(["S_n^2 sim \frac{sigma^2}{n - 1} cdot chi^2 (n-1)"],["S_n^2 \\sim \\frac{\\sigma^2}{n - 1} \\cdot \\chi^2 (n-1)"]);return tn=function(){return n},n}function rn(){var n=We(["S_0^2 sim \frac{sigma^2}{n} cdot chi^2 (n)"],["S_0^2 \\sim \\frac{\\sigma^2}{n} \\cdot \\chi^2 (n)"]);return rn=function(){return n},n}function on(){var n=We(["overline{X}_n sim Nor left( mu, \frac{sigma^2}{n} \right)"],["\\overline{X}_n \\sim Nor \\left( \\mu, \\frac{\\sigma^2}{n} \\right)"]);return on=function(){return n},n}function un(){var n=We(["X sim Nor(mu, sigma^2)"],["X \\sim Nor(\\mu, \\sigma^2)"]);return un=function(){return n},n}function cn(){var n=We(["E(S_0^2) = E(S_n^2) = Var(X)"]);return cn=function(){return n},n}function pn(){var n=We(["Var(overline{X}_n) = \frac{Var(X)}{n}"],["Var(\\overline{X}_n) = \\frac{Var(X)}{n}"]);return pn=function(){return n},n}function sn(){var n=We(["E(overline{X}_n) = E(X)"],["E(\\overline{X}_n) = E(X)"]);return sn=function(){return n},n}function dn(){var n=We(["S_n^2 = \frac{1}{n - 1} cdot sum_{i = 1}^n (X_i - overline{X}_n)^2 = \frac{1}{n - 1} cdot ( n cdot M_2^{(2)} - n cdot overline{X}_n^2)"],["S_n^2 = \\frac{1}{n - 1} \\cdot \\sum_{i = 1}^n (X_i - \\overline{X}_n)^2 = \\frac{1}{n - 1} \\cdot ( n \\cdot M_2^{(2)} - n \\cdot \\overline{X}_n^2)"]);return dn=function(){return n},n}function mn(){var n=We(["S_0^2 = \frac{1}{n} cdot sum_{i = 1}^n (X_i - m)^2 = M_n^(2) - 2 cdot m cdot overline{X}_n + m^2"],["S_0^2 = \\frac{1}{n} \\cdot \\sum_{i = 1}^n (X_i - m)^2 = M_n^(2) - 2 \\cdot m \\cdot \\overline{X}_n + m^2"]);return mn=function(){return n},n}function fn(){var n=We(["m = E(X)"]);return fn=function(){return n},n}function vn(){var n=We(["overline{X}_n"],["\\overline{X}_n"]);return vn=function(){return n},n}function bn(){var n=We(["M^{(k)}_n = \frac{1}{n} cdot sum_{i = 1}^n X_i^k "],["M^{(k)}_n = \\frac{1}{n} \\cdot \\sum_{i = 1}^n X_i^k "]);return bn=function(){return n},n}function _n(){var n=We(["Var left( sum_i X_i \right) = sum_i Var(X_i)"],["Var \\left( \\sum_i X_i \\right) = \\sum_i Var(X_i)"]);return _n=function(){return n},n}function Xn(){var n=We(["Cov(X, Y) = 0"]);return Xn=function(){return n},n}function gn(){var n=We(["Var(X + Y) = Var(X) + Var(Y) + 2 cdot Cov(X, Y)"],["Var(X + Y) = Var(X) + Var(Y) + 2 \\cdot Cov(X, Y)"]);return gn=function(){return n},n}function hn(){var n=We(["Y = a X + b Longleftrightarrow | \rho_{X, Y} | = 1"],["Y = a X + b \\Longleftrightarrow | \\rho_{X, Y} | = 1"]);return hn=function(){return n},n}function qn(){var n=We(["-1 leq \rho_{X, Y} leq 1"],["-1 \\leq \\rho_{X, Y} \\leq 1"]);return qn=function(){return n},n}function jn(){var n=We(["\rho_{X, Y} = \frac{Cov(X, Y)}{sqrt{Var(X)} cdot sqrt{Var(Y)}}"],["\\rho_{X, Y} = \\frac{Cov(X, Y)}{\\sqrt{Var(X)} \\cdot \\sqrt{Var(Y)}}"]);return jn=function(){return n},n}function zn(){var n=We(["\n \boldsymbol{C_X} = \n \begin{bmatrix}\n Var(X_1) & Cov(X_1, X_2) & Cov(X_1, X_3)\\\n Cov(X_2, X_1) & Var(X_2) & Cov(X_2, X_3)\\\n Cov(X_3, X_1) & Cov(X_3, X_2) & Var(X_3)\n end{bmatrix}\n "],["\n \\boldsymbol{C_X} = \n \\begin{bmatrix}\n Var(X_1) & Cov(X_1, X_2) & Cov(X_1, X_3)\\\\\n Cov(X_2, X_1) & Var(X_2) & Cov(X_2, X_3)\\\\\n Cov(X_3, X_1) & Cov(X_3, X_2) & Var(X_3)\n \\end{bmatrix}\n "]);return zn=function(){return n},n}function En(){var n=We(["\boldsymbol{X}"],["\\boldsymbol{X}"]);return En=function(){return n},n}function Pn(){var n=We(["\boldsymbol{C_X}"],["\\boldsymbol{C_X}"]);return Pn=function(){return n},n}function xn(){var n=We(["Cov(X, Y) = 0"]);return xn=function(){return n},n}function kn(){var n=We(["Cov(X + Y, V + W) = Cov(X, Y) + Cov(X, W) + Cov(Y, V) + Cov(Y, W)"]);return kn=function(){return n},n}function Ln(){var n=We(["Cov(alpha X, \beta Y) = alpha cdot \beta cdot Cov(X, Y)"],["Cov(\\alpha X, \\beta Y) = \\alpha \\cdot \\beta \\cdot Cov(X, Y)"]);return Ln=function(){return n},n}function Fn(){var n=We(["Cov(X, X) = Var(X)"]);return Fn=function(){return n},n}function yn(){var n=We(["Cov(X, Y) = Cov(Y, X)"]);return yn=function(){return n},n}function Nn(){var n=We(["Cov(X, alpha) = 0"],["Cov(X, \\alpha) = 0"]);return Nn=function(){return n},n}function Vn(){var n=We(["Cov(X, Y) = E((X - E(X) cdot (Y - E(Y)) = E(XY) - E(X) cdot E(Y)"],["Cov(X, Y) = E((X - E(X) \\cdot (Y - E(Y)) = E(XY) - E(X) \\cdot E(Y)"]);return Vn=function(){return n},n}function Sn(){var n=We(["E(X + Y) = E(X) + E(Y)"]);return Sn=function(){return n},n}function Yn(){var n=We(["E(g(X, Y)) = sum_{i, j} g(x_i, y_i) cdot p_{X, Y} (x_i, y_i)"],["E(g(X, Y)) = \\sum_{i, j} g(x_i, y_i) \\cdot p_{X, Y} (x_i, y_i)"]);return Yn=function(){return n},n}function wn(){var n=We(["P(X_1 in A_1, dots, X_n in A_n) = P(X_1 in A_1) \times dots \times P(X_n in A_n)"],["P(X_1 \\in A_1, \\dots, X_n \\in A_n) = P(X_1 \\in A_1) \\times \\dots \\times P(X_n \\in A_n)"]);return wn=function(){return n},n}function Cn(){var n=We(["p_X (x) = sum_j p_{X, Y} (x_i, y_j)"],["p_X (x) = \\sum_j p_{X, Y} (x_i, y_j)"]);return Cn=function(){return n},n}function In(){var n=We(["p_{X, Y} (x, y) = P(X = x, Y = y)"]);return In=function(){return n},n}function Dn(){var n=We(["F_X (x) = P(X leq x) = lim_{y \to +infty} F_{X, Y} (x, y)"],["F_X (x) = P(X \\leq x) = \\lim_{y \\to +\\infty} F_{X, Y} (x, y)"]);return Dn=function(){return n},n}function Mn(){var n=We(["F_{X, Y} (x, y) = P(X leq x, Y leq y)"],["F_{X, Y} (x, y) = P(X \\leq x, Y \\leq y)"]);return Mn=function(){return n},n}function Un(){var n=We(["X, Y"]);return Un=function(){return n},n}function Bn(){var n=We(["\boldsymbol{X}"],["\\boldsymbol{X}"]);return Bn=function(){return n},n}function An(){var n=We(["P(X > k) simeq P(Y geq k + \frac{1}{2})"],["P(X > k) \\simeq P(Y \\geq k + \\frac{1}{2})"]);return An=function(){return n},n}function Tn(){var n=We(["P(X geq k) simeq P(Y geq k - \frac{1}{2})"],["P(X \\geq k) \\simeq P(Y \\geq k - \\frac{1}{2})"]);return Tn=function(){return n},n}function Gn(){var n=We(["P(X leq k) simeq P(Y leq k + \frac{1}{2})"],["P(X \\leq k) \\simeq P(Y \\leq k + \\frac{1}{2})"]);return Gn=function(){return n},n}function On(){var n=We(["P(X < k) simeq P(Y leq k - \frac{1}{2})"],["P(X < k) \\simeq P(Y \\leq k - \\frac{1}{2})"]);return On=function(){return n},n}function Rn(){var n=We(["(k - \frac{1}{2}, k + \frac{1}{2})"],["(k - \\frac{1}{2}, k + \\frac{1}{2})"]);return Rn=function(){return n},n}function Kn(){var n=We(["Bin(n, p) approx Nor(n cdot p, n cdot p cdot q)"],["Bin(n, p) \\approx Nor(n \\cdot p, n \\cdot p \\cdot q)"]);return Kn=function(){return n},n}function Zn(){var n=We(["Bin(n, p) approx Poi(n cdot p)"],["Bin(n, p) \\approx Poi(n \\cdot p)"]);return Zn=function(){return n},n}function Qn(){var n=We(["Ipe(N, K, n) approx Bin(n, \frac{K}{N})"],["Ipe(N, K, n) \\approx Bin(n, \\frac{K}{N})"]);return Qn=function(){return n},n}function Wn(){var n=We(["T(v) = \frac{Nor(0, 1)}{sqrt{\frac{chi^2(v)}{v}}}"],["T(v) = \\frac{Nor(0, 1)}{\\sqrt{\\frac{\\chi^2(v)}{v}}}"]);return Wn=function(){return n},n}function Jn(){var n=We(["chi^2 (n) + chi^2 (m) = chi^2 (n + m)"],["\\chi^2 (n) + \\chi^2 (m) = \\chi^2 (n + m)"]);return Jn=function(){return n},n}function Hn(){var n=We(["Gamma left( \frac{1}{2}, \frac{1}{2} \right) = chi^2 (v = 1)"],["\\Gamma \\left( \\frac{1}{2}, \\frac{1}{2} \\right) = \\chi^2 (v = 1)"]);return Hn=function(){return n},n}function $n(){var n=We(["Z^2 sim chi^2 (v = 1)"],["Z^2 \\sim \\chi^2 (v = 1)"]);return $n=function(){return n},n}function nl(){var n=We(["x_alpha = mu + z_alpha cdot sqrt{sigma^2}"],["x_\\alpha = \\mu + z_\\alpha \\cdot \\sqrt{\\sigma^2}"]);return nl=function(){return n},n}function ll(){var n=We(["z_alpha"],["z_\\alpha"]);return ll=function(){return n},n}function il(){var n=We(["F_Z(z) = phi(z) = \frac{1}{sqrt{2 pi}} int_{-infty}^{z} e^{-\frac{x^2}{2}} dx"],["F_Z(z) = \\phi(z) = \\frac{1}{\\sqrt{2 \\pi}} \\int_{-\\infty}^{z} e^{-\\frac{x^2}{2}} dx"]);return il=function(){return n},n}function el(){var n=We(["phi(z)"],["\\phi(z)"]);return el=function(){return n},n}function al(){var n=We(["X sim Nor(m, v^2) implies alpha X + \beta sim Nor(alpha m + \beta, (alpha v)^2)"],["X \\sim Nor(m, v^2) \\implies \\alpha X + \\beta \\sim Nor(\\alpha m + \\beta, (\\alpha v)^2)"]);return al=function(){return n},n}function tl(){var n=We(["Var(X) = sigma^2"],["Var(X) = \\sigma^2"]);return tl=function(){return n},n}function rl(){var n=We(["E(X) = mu"],["E(X) = \\mu"]);return rl=function(){return n},n}function ol(){var n=We(["m_X (t) = e^{mu cdot t + \frac{sigma^2 cdot t^2}{2}}"],["m_X (t) = e^{\\mu \\cdot t + \\frac{\\sigma^2 \\cdot t^2}{2}}"]);return ol=function(){return n},n}function ul(){var n=We(["f_X (x) = \frac{e^{-\frac{(x - mu)^2}{2 sigma^2}}}{sqrt{2 pi cdot sigma^2}}"],["f_X (x) = \\frac{e^{-\\frac{(x - \\mu)^2}{2 \\sigma^2}}}{\\sqrt{2 \\pi \\cdot \\sigma^2}}"]);return ul=function(){return n},n}function cl(){var n=We(["Nor(mu, sigma^2)"],["Nor(\\mu, \\sigma^2)"]);return cl=function(){return n},n}function pl(){var n=We(["Var(X) = \frac{(b - a)^2}{12}"],["Var(X) = \\frac{(b - a)^2}{12}"]);return pl=function(){return n},n}function sl(){var n=We(["E(X) = \frac{a + b}{2}"],["E(X) = \\frac{a + b}{2}"]);return sl=function(){return n},n}function dl(){var n=We(["m_X (t) = \frac{e^{b cdot t} - e^{a cdot t}}{(b - a) cdot t}"],["m_X (t) = \\frac{e^{b \\cdot t} - e^{a \\cdot t}}{(b - a) \\cdot t}"]);return dl=function(){return n},n}function ml(){var n=We(["\n f_X (x) = \begin{cases}\n 0 qquad quad x < a \n \frac{1}{b - a} qquad a leq x leq b\\\n 1 qquad quad x > b\n end{cases}"],["\n 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q}{p^2}"],["Var(X) = \\frac{n \\cdot q}{p^2}"]);return ri=function(){return n},n}function oi(){var n=We(["E(X) = \frac{n}{p}"],["E(X) = \\frac{n}{p}"]);return oi=function(){return n},n}function ui(){var n=We(["m_X (t) : { t < ln(\frac{1}{q}) } = left( \frac{p cdot e^t}{1 - q cdot e^t} \right) ^n"],["m_X (t) : \\{ t < ln(\\frac{1}{q}) \\} = \\left( \\frac{p \\cdot e^t}{1 - q \\cdot e^t} \\right) ^n"]);return ui=function(){return n},n}function ci(){var n=We(["f_X (k) : { n .. +infty } in mathbb{N} = \binom{k - 1}{n - 1} cdot p^n cdot q^{k - n} "],["f_X (k) : \\{ n .. +\\infty \\} \\in \\mathbb{N} = \\binom{k - 1}{n - 1} \\cdot p^n \\cdot q^{k - n} "]);return ci=function(){return n},n}function pi(){var n=We(["overline{Bin}(n, p)"],["\\overline{Bin}(n, p)"]);return pi=function(){return n},n}function si(){var n=We(["P([X = i + j | X > i ]) = P([X = j])"]);return si=function(){return n},n}function di(){var n=We(["Var(X) = \frac{q}{p^2}"],["Var(X) = \\frac{q}{p^2}"]);return di=function(){return n},n}function mi(){var n=We(["E(X) = \frac{1}{p}"],["E(X) = \\frac{1}{p}"]);return mi=function(){return n},n}function fi(){var n=We(["m_X (t) = \frac{p cdot e^t}{1 - q cdot e^t}"],["m_X (t) = \\frac{p \\cdot e^t}{1 - q \\cdot e^t}"]);return fi=function(){return n},n}function vi(){var n=We(["f_X (k) : mathbb{N} = q^{k - 1} p"],["f_X (k) : \\mathbb{N} = q^{k - 1} p"]);return vi=function(){return n},n}function bi(){var n=We(["Var(X) = n cdot p cdot q"],["Var(X) = n \\cdot p \\cdot q"]);return bi=function(){return n},n}function _i(){var n=We(["E(X) = n cdot p"],["E(X) = n \\cdot p"]);return _i=function(){return n},n}function Xi(){var n=We(["m_X (t) = (q + p cdot e^t) ^ n"],["m_X (t) = (q + p \\cdot e^t) ^ n"]);return Xi=function(){return n},n}function gi(){var n=We(["f_X (k) : {0..n} = \binom{n}{k} cdot p^k cdot q^{n - k}"],["f_X (k) : \\{0..n\\} = \\binom{n}{k} \\cdot p^k \\cdot q^{n - k}"]);return gi=function(){return n},n}function hi(){var n=We(["Bin(n, p)"]);return hi=function(){return n},n}function qi(){var n=We(["\n f_X (k) : {0, 1} = \begin{cases}\n p quad se k = 1\\\n q quad se k = 0\\\n 0 quad altrimenti\n end{cases} = p^x cdot q^{1 - k}"],["\n f_X (k) : \\{0, 1\\} = \\begin{cases}\n p \\quad se\\ k = 1\\\\\n q \\quad se\\ k = 0\\\\\n 0 \\quad altrimenti\n \\end{cases} = p^x \\cdot q^{1 - k}"]);return qi=function(){return n},n}function ji(){var n=We(["Ber(p)"]);return ji=function(){return n},n}function zi(){var n=We(["X sim Distribuzione()"],["X \\sim Distribuzione()"]);return zi=function(){return n},n}function Ei(){var n=We(["H_X (t) = E ( e^{i cdot t cdot X} )"],["H_X (t) = E ( e^{i \\cdot t \\cdot X} )"]);return Ei=function(){return n},n}function Pi(){var n=We(["m_X (t) = E( e^{t cdot X} )"],["m_X (t) = E( e^{t \\cdot X} )"]);return Pi=function(){return n},n}function xi(){var n=We(["\n mu_k = E ( X^k ) = \begin{cases}\n sum_i x_i^k p_X (x_i) qquad nel discreto\\\n \\\n int_{-infty}^{+infty} x^k f_X (x) dx qquad nel continuo\n end{cases}"],["\n \\mu_k = E ( X^k ) = \\begin{cases}\n \\sum_i x_i^k p_X (x_i) \\qquad nel\\ discreto\\\\\n \\\\\n \\int_{-\\infty}^{+\\infty} x^k f_X (x) dx \\qquad nel\\ continuo\n \\end{cases}"]);return xi=function(){return n},n}function ki(){var n=We(["\forall epsilon > 0, P([ left| X - E(X) \right| < epsilon]) geq 1 - \frac{Var(X)}{epsilon^2}"],["\\forall \\epsilon > 0, P([ \\left| X - E(X) \\right| < \\epsilon]) \\geq 1 - \\frac{Var(X)}{\\epsilon^2}"]);return ki=function(){return n},n}function Li(){var n=We(["\forall epsilon > 0, P([ left| X - E(X) \right| geq epsilon]) leq \frac{Var(X)}{epsilon^2}"],["\\forall \\epsilon > 0, P([ \\left| X - E(X) \\right| \\geq \\epsilon]) \\leq \\frac{Var(X)}{\\epsilon^2}"]);return Li=function(){return n},n}function Fi(){var n=We(["\frac{Var(X)}{epsilon^2}"],["\\frac{Var(X)}{\\epsilon^2}"]);return Fi=function(){return n},n}function yi(){var n=We(["epsilon"],["\\epsilon"]);return yi=function(){return n},n}function Ni(){var n=We(["E(X) = overline{k} cdot P(X < k) + k cdot P(X geq k)"],["E(X) = \\overline{k} \\cdot P(X < k) + k \\cdot P(X \\geq k)"]);return Ni=function(){return n},n}function Vi(){var n=We(["P(X geq k)"],["P(X \\geq k)"]);return Vi=function(){return n},n}function Si(){var n=We(["P(X < k)"]);return Si=function(){return n},n}function Yi(){var n=We(["\forall k > 0, P([X geq k]) leq \frac{E(X)}{k}"],["\\forall k > 0, P([X \\geq k]) \\leq \\frac{E(X)}{k}"]);return Yi=function(){return n},n}function wi(){var n=We(["Var(X) = E( (X - E(X) )^2 ) = E ( X^2 ) - (E(X))^2"]);return wi=function(){return n},n}function Ci(){var n=We(["\frac{n}{100}"],["\\frac{n}{100}"]);return Ci=function(){return n},n}function Ii(){var n=We(["x_{0.75}"]);return Ii=function(){return n},n}function Di(){var n=We(["x_{0.25}"]);return Di=function(){return n},n}function Mi(){var n=We(["x_{0.5}"]);return Mi=function(){return n},n}function Ui(){var n=We(["P([X < x_{alpha}]) leq alpha leq P([X leq x_{alpha}])"],["P([X < x_{\\alpha}]) \\leq \\alpha \\leq P([X \\leq x_{\\alpha}])"]);return Ui=function(){return n},n}function Bi(){var n=We(["0 leq alpha leq 1"],["0 \\leq \\alpha \\leq 1"]);return Bi=function(){return n},n}function Ai(){var n=We(["x_{alpha}"],["x_{\\alpha}"]);return Ai=function(){return n},n}function Ti(){var n=We(["E(X) = int_{-infty}^{+infty} f_X (x) cdot x cdot dx"],["E(X) = \\int_{-\\infty}^{+\\infty} f_X (x) \\cdot x \\cdot dx"]);return Ti=function(){return n},n}function Gi(){var n=We(["E(X) = sum_i P(X = x_i) cdot x_i"],["E(X) = \\sum_i P(X = x_i) \\cdot x_i"]);return Gi=function(){return n},n}function Oi(){var n=We(["E(X) = int_0^{+infty} (1 - F_X (t)) dt - int_{-infty}^{0} F_X (t) dt"],["E(X) = \\int_0^{+infty} (1 - F_X (t)) dt - \\int_{-\\infty}^{0} F_X (t) dt"]);return Oi=function(){return n},n}function Ri(){var n=We(["f_Y (y) = int_{g(a)}^{g(b)} f_X ( g^{-1} (x) ) g^{-2} (x)"],["f_Y (y) = \\int_{g(a)}^{g(b)} f_X ( g^{-1} (x) ) g^{-2} (x)"]);return Ri=function(){return n},n}function Ki(){var n=We(["P([X = x_0]) = lim_{t \to x^+_0} F_X (t) - lim_{t \to x^-_0} F_X (t)"],["P([X = x_0]) = \\lim_{t \\to x^+_0} F_X (t) - \\lim_{t \\to x^-_0} F_X (t)"]);return Ki=function(){return n},n}function Zi(){var n=We(["\forall x_0 in mathbb{R}, F_X (x_0) = lim_{t \to x^+_0} F_X (t)"],["\\forall x_0 \\in \\mathbb{R}, F_X (x_0) = \\lim_{t \\to x^+_0} F_X (t)"]);return Zi=function(){return n},n}function Qi(){var n=We(["\n F_X (t) = P(A_t) = \begin{cases}\n sum_{i = 0}^{t} p_X (x_i) quad nel discreto\\\n \\\n int_{-infty}^t f_X (x) dx quad nel continuo\n end{cases}\n "],["\n F_X (t) = P(A_t) = \\begin{cases}\n \\sum_{i = 0}^{t} p_X (x_i) \\quad nel\\ discreto\\\\\n \\\\\n \\int_{-\\infty}^t f_X (x) dx \\quad nel\\ continuo\n \\end{cases}\n "]);return Qi=function(){return n},n}function Wi(){var n=We(["A_t"]);return Wi=function(){return n},n}function Ji(){var n=We(["F_X : mathbb{R} \to [0, 1]"],["F_X : \\mathbb{R} \\to [0, 1]"]);return Ji=function(){return n},n}function Hi(){var n=We(["P([a < X leq b]) = int_a^b f_X (x) dx"],["P([a < X \\leq b]) = \\int_a^b f_X (x) dx"]);return Hi=function(){return n},n}function $i(){var n=We(["f_X : X \to [0, 1]"],["f_X : X \\to [0, 1]"]);return $i=function(){return n},n}function ne(){var n=We(["\n p_X (x) = \begin{cases}\n P([X = x]) quad se X mapsto x \\\n 0 qquad qquad quad se X \notmapsto x\n end{cases} \n "],["\n p_X (x) = \\begin{cases}\n P([X = x]) \\quad se\\ X \\mapsto x \\\\\n 0 \\qquad \\qquad \\quad se\\ X \\not\\mapsto x\n \\end{cases} \n "]);return ne=function(){return n},n}function le(){var n=We(["p_X : X \to [0, 1]"],["p_X : X \\to [0, 1]"]);return le=function(){return n},n}function ie(){var n=We(["\forall t in mathbb{R}, A_t in mathcal{F}"],["\\forall t \\in \\mathbb{R}, A_t \\in \\mathcal{F}"]);return ie=function(){return n},n}function ee(){var n=We(["A_t = { omega | X(omega) leq t }"],["A_t = \\{ \\omega | X(\\omega) \\leq t \\}"]);return ee=function(){return n},n}function ae(){var n=We(["X(omega) : Omega \to mathbb{R}"],["X(\\omega) : \\Omega \\to \\mathbb{R}"]);return ae=function(){return n},n}function te(){var n=We(["P(E cap F cap G) = P(E) cdot P(F) cdot P(G)"],["P(E \\cap F \\cap G) = P(E) \\cdot P(F) \\cdot P(G)"]);return te=function(){return n},n}function re(){var n=We(["P(E cap F) = P(E) cdot P(F) Longleftrightarrow P(E|F) = P(E) Longleftrightarrow P(F|E) = P(F)"],["P(E \\cap F) = P(E) \\cdot P(F) \\Longleftrightarrow P(E|F) = P(E) \\Longleftrightarrow P(F|E) = P(F)"]);return re=function(){return n},n}function oe(){var n=We(["P(E_h | F) = \frac{P(F | E_h) cdot P(E_h)}{P(F)}"],["P(E_h | F) = \\frac{P(F | E_h) \\cdot P(E_h)}{P(F)}"]);return oe=function(){return n},n}function ue(){var n=We(["P(F|G) = sum_i P(F|E_i cap G) cdot P(E_i | G)"],["P(F|G) = \\sum_i P(F|E_i \\cap G) \\cdot P(E_i | G)"]);return ue=function(){return n},n}function ce(){var n=We(["P(F) = sum_{i} P(F|E_i) cdot P(E_i)"],["P(F) = \\sum_{i} P(F|E_i) \\cdot P(E_i)"]);return ce=function(){return n},n}function pe(){var n=We(["P(E_1 cap \times cap E_n) = P(E_1) \times P(E_2 | E_1) \times dots \times P(E_n | E_1 cap E_2 cap dots cap E_{n-1})"],["P(E_1 \\cap \\times \\cap E_n) = P(E_1) \\times P(E_2 | E_1) \\times \\dots \\times P(E_n | E_1 \\cap E_2 \\cap \\dots \\cap E_{n-1})"]);return pe=function(){return n},n}function se(){var n=We(["E cap F = emptyset Longleftrightarrow P(E|F) = P(F|E) = 0"],["E \\cap F = \\emptyset \\Longleftrightarrow P(E|F) = P(F|E) = 0"]);return se=function(){return n},n}function de(){var n=We(["P(E|F) = \frac{P(E cap F)}{P(F)}"],["P(E|F) = \\frac{P(E \\cap F)}{P(F)}"]);return de=function(){return n},n}function me(){var n=We(["\boldsymbol{P}_n = n!"],["\\boldsymbol{P}_n = n!"]);return me=function(){return n},n}function fe(){var n=We(["\boldsymbol{C}^{r}_{n, k} = \binom{n + k - 1}{k} = \frac{(n + k - 1)!}{(k)! cdot (n - 1)!}"],["\\boldsymbol{C}^{r}_{n, k} = \\binom{n + k - 1}{k} = \\frac{(n + k - 1)!}{(k)! \\cdot (n - 1)!}"]);return fe=function(){return n},n}function ve(){var n=We(["\boldsymbol{C}_{n, k} = \binom{n}{k} = \frac{n!}{(k)! cdot (n - k)!}"],["\\boldsymbol{C}_{n, k} = \\binom{n}{k} = \\frac{n!}{(k)! \\cdot (n - k)!}"]);return ve=function(){return n},n}function be(){var n=We(["\boldsymbol{D}^{r}_{n, k} = n^k"],["\\boldsymbol{D}^{r}_{n, k} = n^k"]);return be=function(){return n},n}function _e(){var n=We(["\boldsymbol{D}_{n, k} = \frac{n!}{(n - k)!}"],["\\boldsymbol{D}_{n, k} = \\frac{n!}{(n - k)!}"]);return _e=function(){return n},n}function Xe(){var n=We(["P(E) = \frac{len(E)}{len(Omega)}"],["P(E) = \\frac{len(E)}{len(\\Omega)}"]);return Xe=function(){return n},n}function ge(){var n=We(["P(E cup F) = P(E) + P(F) - P(E cap F)"],["P(E \\cup F) = P(E) + P(F) - P(E \\cap F)"]);return ge=function(){return n},n}function he(){var n=We(["F subseteq E implies P(F) leq P(E)"],["F \\subseteq E \\implies P(F) \\leq P(E)"]);return he=function(){return n},n}function qe(){var n=We(["P(\bar{E}) = 1 - P({E})"],["P(\\bar{E}) = 1 - P({E})"]);return qe=function(){return n},n}function je(){var n=We(["P left ( \bigcup_i E_i \right ) = sum_i P ( E_i )"],["P \\left ( \\bigcup_i E_i \\right ) = \\sum_i P ( E_i )"]);return je=function(){return n},n}function ze(){var n=We(["P(Omega) = 1"],["P(\\Omega) = 1"]);return ze=function(){return n},n}function Ee(){var n=We(["\forall E in mathcal{F}, 0 leq P(E) leq 1"],["\\forall E \\in \\mathcal{F}, 0 \\leq P(E) \\leq 1"]);return Ee=function(){return n},n}function Pe(){var n=We(["E_n"]);return Pe=function(){return n},n}function xe(){var n=We(["E_3"]);return xe=function(){return n},n}function ke(){var n=We(["E_2"]);return ke=function(){return n},n}function Le(){var n=We(["E_1"]);return Le=function(){return n},n}function Fe(){var n=We(["E_i"]);return Fe=function(){return n},n}function ye(){var n=We(["E in mathcal{F} implies mathcal{F} = { emptyset, E, \bar{E}, Omega }"],["E \\in \\mathcal{F} \\implies \\mathcal{F} = \\{ \\emptyset, E, \\bar{E}, \\Omega \\}"]);return ye=function(){return n},n}function Ne(){var n=We(["(E, F) in mathcal{F} implies (E cup F, E cap F) in mathcal{F}"],["(E, F) \\in \\mathcal{F} \\implies (E \\cup F, E \\cap F) \\in \\mathcal{F}"]);return Ne=function(){return n},n}function Ve(){var n=We(["E in mathcal{F} implies \bar{E} in mathcal{F}"],["E \\in \\mathcal{F} \\implies \\bar{E} \\in \\mathcal{F}"]);return Ve=function(){return n},n}function Se(){var n=We(["Omega in mathcal{F}"],["\\Omega \\in \\mathcal{F}"]);return Se=function(){return n},n}function Ye(){var n=We(["sigma"],["\\sigma"]);return Ye=function(){return n},n}function we(){var n=We(["sigma"],["\\sigma"]);return we=function(){return n},n}function Ce(){var n=We(["mathcal{F}"],["\\mathcal{F}"]);return Ce=function(){return n},n}function Ie(){var n=We(["mathcal{F}"],["\\mathcal{F}"]);return Ie=function(){return n},n}function De(){var n=We(["E cap F = emptyset"],["E \\cap F = \\emptyset"]);return De=function(){return n},n}function Me(){var n=We(["E = emptyset"],["E = \\emptyset"]);return Me=function(){return n},n}function Ue(){var n=We(["E subseteq F"],["E \\subseteq F"]);return Ue=function(){return n},n}function Be(){var n=We(["E setminus F = E cap \bar{F}"],["E \\setminus F = E \\cap \\bar{F}"]);return Be=function(){return n},n}function Ae(){var n=We(["E cup F = left { 1, 2, 3, 4 \right }"],["E \\cup F = \\left \\{ 1, 2, 3, 4 \\right \\}"]);return Ae=function(){return n},n}function Te(){var n=We(["E cap F = left { 1 \right }"],["E \\cap F = \\left \\{ 1 \\right \\}"]);return Te=function(){return n},n}function Ge(){var n=We(["\bar{E} = left { 3, 4, 5, 6 \right }"],["\\bar{E} = \\left \\{ 3, 4, 5, 6 \\right \\}"]);return Ge=function(){return n},n}function Oe(){var n=We(["E = left { 1, 2 \right }"],["E = \\left \\{ 1, 2 \\right \\}"]);return Oe=function(){return n},n}function Re(){var n=We(["omega = 1"],["\\omega = 1"]);return Re=function(){return n},n}function Ke(){var n=We(["Omega = left { 1, 2, 3, 4, 5, 6 \right }"],["\\Omega = \\left \\{ 1, 2, 3, 4, 5, 6 \\right \\}"]);return Ke=function(){return n},n}function Ze(){var n=We(["P(E) = \frac{successi}{prove totali}"],["P(E) = \\frac{successi}{prove\\ totali}"]);return Ze=function(){return n},n}function Qe(){var n=We(["P(E) = \frac{casi favorevoli}{casi possibili}"],["P(E) = \\frac{casi\\ favorevoli}{casi\\ possibili}"]);return Qe=function(){return n},n}function We(n,l){return l||(l=n.slice(0)),n.raw=l,n}var Je=i("mbOI"),He=i("ke5e"),$e=i("o8Jj"),na=i("ZHMS"),la=String.raw;l.default=function(){return n(Je.l.Provider,{value:!1},n(Je.k.Provider,{value:Je.m.INLINE},n("div",null,n("h1",null,"Statistica ed Elementi di Probabilità"),n(Je.q,{title:"Tipi di probabilità"},n(Je.p,{title:"Classica"},n("p",null,n(Je.j,null,la(Qe())))),n(Je.p,{title:"Frequentista"},n("p",null,n(Je.j,null,la(Ze())))),n(Je.p,{title:"Soggettiva"},n("p",null,"Il prezzo che un individuo coerente riterrebbe equo per ricevere ",n("b",null,"1")," nel caso l'evento si verificasse e ",n("b",null,"0")," nel caso l'evento non si verificasse."))),n(Je.q,{title:"Linguaggio matematico"},n(Je.p,{title:"Spazio campionario"},n("blockquote",null,'"omegone"'),n("p",null,"L'",n("b",null,"insieme")," di tutti gli esiti possibili di un esperimento."),n("p",null,n(Je.j,null,la(Ke())))),n(Je.p,{title:"Esito"},n("blockquote",null,'"omeghino"'),n("p",null,"Un ",n("b",null,"elemento")," dello spazio campionario."),n("p",null,n(Je.j,null,la(Re())))),n(Je.p,{title:"Evento"},n("blockquote",null,'"e"'),n("p",null,"Un ",n("b",null,"sottoinsieme")," dello spazio campionario."),n("p",null,n(Je.j,null,la(Oe()))),n("p",null,"Lo spazio campionario stesso è un ",n("b",null,"evento certo"),".")),n(Je.p,{title:"Not"},n("blockquote",null,'"not e"'),n("p",null,"Il ",n("b",null,"complementare")," di un sottoinsieme."),n("p",null,n(Je.j,null,la(Ge())))),n(Je.p,{title:"And"},n("blockquote",null,'"e intersecato effe"'),n("p",null,"L'",n("b",null,"intersezione")," di più sottoinsiemi."),n("p",null,n(Je.j,null,la(Te())))),n(Je.p,{title:"Or"},n("blockquote",null,'"e unito a effe"'),n("p",null,"L'",n("b",null,"unione")," di più sottoinsiemi."),n("p",null,n(Je.j,null,la(Ae())))),n(Je.p,{title:"Differenza"},n("blockquote",null,'"e meno effe"'),n("p",null,n(Je.j,null,la(Be())))),n(Je.p,{title:"Implicazione"},n("blockquote",null,'"e contenuto in effe"'),n("p",null,"L'",n("b",null,"inclusione")," del primo insieme in un altro."),n("p",null,n(Je.j,null,la(Ue()))),n("p",null,"Se si verifica ",n(Je.j,null,"E"),", allora si verifica anche ",n(Je.j,null,"F"),".")),n(Je.p,{title:"Evento impossibile"},n("blockquote",null,'"e è impossibile"'),n("p",null,"Un sottoinsieme ",n("b",null,"vuoto"),"."),n("p",null,n(Je.j,null,la(Me())))),n(Je.p,{title:"Mutua esclusione"},n("blockquote",null,'"e ed effe si escludono mutualmente"'),n("p",null,"La ",n("b",null,"disgiunzione")," di due insiemi."),n("p",null,n(Je.j,null,la(De()))))),n(Je.q,null,n(Je.p,{title:"Famiglia degli eventi"},n("blockquote",null,'"famiglia effe"'),n("p",null,"I sottoinsiemi dello spazio campionario formano una ",n("b",null,"famiglia")," di sottoinsiemi detta ",n("i",null,"famiglia degli eventi"),"."),n("p",null,n(Je.j,null,la(Ie()))),n("p",null,"Qualsiasi sottoinsieme appartenente a ",n(Je.j,null,la(Ce()))," è considerato un evento.")),n(Je.p,{title:n("span",null,n(Je.j,null,la(we())),"-algebra")},n("blockquote",null,'"sigma algebra"'),n("p",null,"Se la famiglia degli eventi soddisfa questi tre requisiti, allora viene detta ",n("i",null,n(Je.j,null,la(Ye())),"-algebra"),":"),n("ol",null,n("li",null,"Lo spazio campionario è un evento: ",n(Je.j,null,la(Se()))),n("li",null,"Se un sottoinsieme è un evento, allora anche il suo complementare lo è: ",n(Je.j,null,la(Ve()))),n("li",null,"Se due sottoinsiemi sono eventi, allora lo sono anche la loro unione e intersezione: ",n(Je.j,null,la(Ne())))),n("p",null,"Un esempio: ",n(Je.j,null,la(ye()))))),n(Je.q,null,n(Je.p,{title:"Partizione"},n("blockquote",null,'"la partizione e composta da e uno, e due, e tre..."'),n("p",null,"Un insieme di esiti e eventi:"),n("ul",null,n("li",null,n("b",null,"Finito"),"."),n("li",null,"In cui tutti gli eventi hanno ",n("b",null,"probabilità diversa da 0"),"."),n("li",null,"In cui tutti gli eventi sono ",n("b",null,"mutualmente esclusivi"),"."),n("li",null,"In cui l'unione di tutti i suoi elementi ",n("b",null,"copre lo spazio campionario"),".")),n("p",null,"La partizione ",n(Je.j,null,la(Fe()))," è composta dagli eventi ",n(Je.j,null,la(Le())),", ",n(Je.j,null,la(ke())),", ",n(Je.j,null,la(xe())),", fino a ",n(Je.j,null,la(Pe())),"."),n(He.a,null,"Se lo spazio campionario fosse una torta, una sua partizione sarebbe l'insieme delle fette di uno dei modi in cui si potrebbe tagliare."))),n(Je.q,{title:"Assiomi della probabilità"},n(Je.p,{title:"Primo assioma della probabilità"},n("p",null,"La probabilità di un evento è un numero tra 0 e 1."),n("p",null,n(Je.j,null,la(Ee())))),n(Je.p,{title:"Secondo assioma della probabilità"},n("p",null,"La probabilità dello spazio campionario è sempre 1."),n("p",null,n(Je.j,null,la(ze())))),n(Je.p,{title:"Terzo assioma della probabilità"},n("p",null,"La probabilità dell'unione di eventi indipendenti è uguale alla somma delle loro probabilità."),n("p",null,n(Je.j,null,la(je()))))),n(Je.q,{title:"Conseguenze degli assiomi"},n(Je.p,{title:"Probabilità di un evento negato"},n("p",null,"La probabilità di un evento negato è uguale a 1 meno la probabilità dell'evento non negato."),n("p",null,n(Je.j,null,la(qe())))),n(Je.p,{title:"Probabilità di un evento incluso"},n("p",null,"La probabilità di un evento incluso in un altro è sempre minore o uguale alla probabilità dell'evento in cui è incluso."),n("p",null,n(Je.j,null,la(he())))),n(Je.p,{title:"Unione"},n("p",null,"La probabilità di un evento unito a un altro è uguale alla somma delle probabilità dei due eventi meno la probabilità della loro intersezione."),n("p",null,n(Je.j,null,la(ge()))),n(He.a,null,"Sommando le probabilità dei due eventi, l'intersezione viene contata due volte, e va quindi rimossa!"))),n(Je.q,{title:"Spazi equiprobabili"},n(Je.p,{title:"Cosa sono?"},n("p",null,"Spazi campionari in cui ci sono un numero finito di esiti e ogni esito ha la stessa probabilità di verificarsi."),n("p",null,n(Je.j,null,la(Xe())))),n(Je.p,{title:"Spazi equiprobabili geometrici"},n("p",null,"Gli spazi campionari possono avere un numero infinito di esiti: sono ",n("i",null,"equiprobabili geometrici")," se nessun esito è privilegiato rispetto agli altri."))),n(Je.q,{title:"Calcolo combinatorio"},n(Je.p,{title:"Disposizioni"},n("p",null,"Estraggo un numero, da un sacchetto con ",n(Je.j,null,"n")," numeri, mi segno che numero ho estratto e lo ",n("b",null,"tengo fuori dal sacchetto"),". Ripeto per ",n(Je.j,null,"k")," volte."),n("p",null,n("b",null,"Tengo conto")," dell'ordine in cui ho estratto i numeri."),n("p",null,n(Je.j,null,la(_e())))),n(Je.p,{title:"Disposizioni con ripetizione"},n("p",null,"Estraggo un numero, da un sacchetto con ",n(Je.j,null,"n")," numeri, mi segno che numero ho estratto e lo ",n("b",null,"rimetto nel sacchetto"),". Ripeto per ",n(Je.j,null,"k")," volte."),n("p",null,n("b",null,"Tengo conto")," dell'ordine in cui ho estratto i numeri."),n("p",null,n(Je.j,null,la(be())))),n(Je.p,{title:"Combinazioni"},n("p",null,"Estraggo un numero, da un sacchetto con ",n(Je.j,null,"n")," numeri, mi segno che numero ho estratto e lo ",n("b",null,"tengo fuori dal sacchetto"),". Ripeto per ",n(Je.j,null,"k")," volte."),n("p",null,n("b",null,"Non mi interessa")," l'ordine in cui ho estratto i numeri."),n("p",null,n(Je.j,null,la(ve())))),n(Je.p,{title:"Combinazioni con ripetizione"},n("p",null,"Estraggo un numero, da un sacchetto con ",n(Je.j,null,"n")," numeri, mi segno che numero ho estratto e lo ",n("b",null,"rimetto nel sacchetto"),". Ripeto per ",n(Je.j,null,"k")," volte."),n("p",null,n("b",null,"Non mi interessa")," l'ordine in cui ho estratto i numeri."),n("p",null,n(Je.j,null,la(fe())))),n(Je.p,{title:"Permutazioni"},n("p",null,"Estraggo ",n(Je.j,null,"n")," numeri e guardo in quanti ordini diversi li posso mettere."),n("p",null,n(Je.j,null,la(me()))))),n(Je.q,{title:"Probabilità condizionata"},n(Je.p,{title:"Eventi condizionati"},n("blockquote",null,'"E dato F"'),n("p",null,"La probabilità che si verifichi ",n(Je.j,null,"E")," sapendo che ",n("b",null,"si è già verificato")," ",n(Je.j,null,"F"),"."),n("p",null,n(Je.j,null,la(de()))),n(He.a,null,"Ricorda vagamente le pipe di ",n("code",null,"bash"),", però al contrario...")),n(Je.p,{title:"Eventi mutualmente esclusivi"},n("p",null,"Se due eventi sono mutualmente esclusivi, entrambe le loro probabilità condizionate saranno uguali a 0."),n("p",null,n(Je.j,null,la(se()))))),n(Je.q,null,n(Je.p,{title:"Regola della catena"},n("p",null,"Si può sfruttare la formula inversa della probabilità condizionata per calcolare catene di intersezioni:"),n("p",null,n(Je.j,null,la(pe()))))),n(Je.q,{title:"Le alternative"},n(Je.p,{title:"Legge delle alternative"},n("p",null,"La probabilità che si verifichi un evento è pari alla somma delle probabilità dell'evento stesso dati tutti gli eventi di una partizione."),n("p",null,n(Je.j,null,la(ce())))),n(Je.p,{title:"Legge condizionata delle alternative"},n("p",null,"La legge delle alternative funziona anche quando ad essere partizionato è un ",n("b",null,"evento"),":"),n("p",null,n(Je.j,null,la(ue())))),n(Je.p,{title:"Formula di Bayes"},n("p",null,"Tramite la ",n("i",null,"formula di Bayes")," possiamo risalire alla probabilità di un evento condizionato a un altro partendo dalla probabilità di quest'ultimo condizionato al primo:"),n("p",null,n(Je.j,null,la(oe()))),n(He.a,null,"In pratica, invertiamo gli eventi."))),n(Je.q,{title:"Eventi indipendenti"},n(Je.p,{title:"Due eventi indipendenti"},n("blockquote",null,'"eventi indipendenti a due a due"'),n("p",null,"Se due eventi sono indipendenti, sapere che uno dei due si è verificato non influisce sulle probabilità che si sia verificato l'altro."),n("p",null,n(Je.j,null,la(re())))),n(Je.p,{title:"Più eventi indipendenti"},n("blockquote",null,'"eventi indipendenti a tre a tre, a quattro a quattro, a cinque a cinque..."'),n("p",null,"Si può verificare l'indipendenza di più eventi alla volta:"),n("p",null,n(Je.j,null,la(te()))),n("p",null,"Eventi indipendenti a due a due non sono per forza indipendenti a tre a tre, e viceversa.")),n(Je.p,{title:"Famiglia di eventi indipendenti"},n("p",null,"Un insieme di ",n(Je.j,null,"n")," eventi è una ",n("i",null,"famiglia di eventi indipendenti")," se, preso un qualsiasi numero di eventi da essa, essi risulteranno indipendenti."),n(He.a,null,"Tutti gli eventi provenienti da essa saranno indipendenti sia a due a due, sia a tre a tre, sia a quattro a quattro, e così via!"))),n(Je.q,{title:"Variabili aleatorie"},n(Je.p,{title:"Variabile aleatoria"},n("p",null,"Una funzione che fa corrispondere un numero reale a ogni possibile esito dello spazio campionario. ",n(Je.j,null,la(ae())),".")),n(Je.p,{title:n("abbr",{title:"Nome artigianale dato da Steffo."},"Insieme di ripartizione")},n("p",null,"Ad ogni variabile aleatoria sono associati gli eventi ",n(Je.j,null,la(ee())),", che contengono tutti gli esiti a cui la variabile aleatoria associa un valore minore o uguale a ",n(Je.j,null,"t"),"."),n("p",null,"Per definizione, tutte le variabili aleatorie devono rispettare questa condizione:"),n("p",null,n(Je.j,null,la(ie()))),n(He.a,null,"All'aumentare di t, l'insieme conterrà sempre più elementi.")),n(Je.p,{title:"Supporto"},n("blockquote",null,'"supporto di X"'),n("p",null,"Il ",n("b",null,"codominio")," della variabile aleatoria è il suo ",n("i",null,"supporto"),"."),n("p",null,"Per indicare che un valore ",n(Je.j,null,"x_0")," appartiene al supporto di ",n(Je.j,null,"X"),", si usa la notazione ",n(Je.j,null,"X \\mapsto x_0"),"."))),n(Je.q,{title:"Densità"},n(Je.p,{title:"Funzione probabilità"},n("p",null,"La ",n("i",null,"funzione probabilità")," ",n(Je.j,null,la(le()))," di una variabile aleatoria ",n("b",null,"discreta")," ",n(Je.j,null,"X")," è la funzione che associa ad ogni esito la sua probabilità:"),n("p",null,n(Je.j,null,la(ne())))),n(Je.p,{title:"Funzione densità"},n("p",null,"La ",n("i",null,"funzione densità")," ",n(Je.j,null,la($i()))," di una variabile aleatoria ",n("b",null,"continua")," ",n(Je.j,null,"X")," è l'equivalente continuo della funzione probabilità:"),n("p",null,n(Je.j,null,la(Hi()))),n("p",null,"A differenza della funzione probabilità, è possibile che la funzione densità ",n("b",null,"non esista")," per una certa variabile aleatoria."),n(He.a,null,"Rappresenta \"quanta\" probabilità c'è in un'unità di x!"))),n(Je.q,{title:"Funzione di ripartizione"},n(Je.p,{title:"Definizione"},n("p",null,"Ogni variabile aleatoria ha una ",n("i",null,"funzione di ripartizione")," ",n(Je.j,null,la(Ji()))," associata, che rappresenta la probabilità che la variabile aleatoria assuma un valore minore o uguale a ",n(Je.j,null,"t"),":"),n("p",null,"Si può dire che essa rappresenti la probabilità dell'evento ",n(Je.j,null,la(Wi())),":"),n("p",null,n(Je.j,null,la(Qi())))),n(Je.p,{title:"Proprietà della funzione"},n("ul",null,n("li",null,"È sempre ",n("b",null,"monotona crescente")," (non strettamente)."),n("br",null),n("li",null,"Vale ",n("b",null,"0")," a ",n(Je.j,null,"-\\infty")," e ",n("b",null,"1")," a ",n(Je.j,null,"+\\infty"),"."),n("br",null),n("li",null,"È ",n("b",null,"continua da destra"),": ",n(Je.j,null,la(Zi()))))),n(Je.p,{title:"Probabilità di un valore"},n("p",null,"Possiamo usare la funzione di ripartizione per calcolare la probabilità di un certo valore reale:"),n("p",null,n(Je.j,null,la(Ki()))))),n(Je.q,{title:"Trasformazioni di variabili aleatorie"},n(Je.p,{title:"Nel discreto"},n("p",null,"Nel discreto basta abbinare un nuovo valore a ogni valore della variabile originale.")),n(Je.p,{title:"Nel continuo (invertibile)"},n("p",null,"Nel continuo applichiamo la formula dell'integrazione per sostituzione:"),n("p",null,n(Je.j,null,la(Ri())))),n(Je.p,{title:"Nel... digitale"},n("p",null,"Trasformare variabili aleatorie è molto utile nell'informatica per creare distribuzioni partendo da una funzione ",n("a",{href:"https://docs.python.org/3/library/random.html#random.random"},n("code",null,"random()"))," che restituisce numeri da 0 a 1 con una distribuzione lineare."))),n(Je.q,{title:"Informazioni delle variabili aleatorie"},n(Je.p,{title:"Media"},n("p",null,"Ogni variabile aleatoria che ha una ",n("b",null,"funzione di ripartizione")," e un ",n("b",null,"supporto finito")," ha anche una ",n("i",null,"media")," (o ",n("i",null,"valore medio")," o ",n("i",null,"atteso"),"):"),n("p",null,n(Je.j,null,la(Oi()))),n("p",null,"Nel discreto, si può calcolare con:"),n("p",null,n(Je.j,null,la(Gi()))),n("p",null,"Nel continuo, si può calcolare con:"),n("p",null,n(Je.j,null,la(Ti()))))),n(Je.q,null,n(Je.p,{title:"Moda"},n("p",null,"Valore per cui la ",n("b",null,"funzione probabilità")," o ",n("b",null,"funzione densità")," è ",n("b",null,"massima"),".")),n(Je.p,{title:"Quantili"},n("p",null,"Il ",n("i",null,"quantile")," ",n(Je.j,null,la(Ai()))," di ordine ",n(Je.j,null,la(Bi()))," della variabile aleatoria ",n(Je.j,null,"X")," è il più piccolo numero tale che:"),n("p",null,n(Je.j,null,la(Ui()))),n("p",null),n("p",null,"Il quantile di ordine 0.5 ",n(Je.j,null,la(Mi()))," è detto ",n("i",null,"mediana"),"."),n("p",null,"I quantili di ordine 0.25 ",n(Je.j,null,la(Di()))," e 0.75 ",n(Je.j,null,la(Ii()))," sono detti ",n("i",null,"quartili"),"."),n("p",null,"I quantili di ordine ",n(Je.j,null,la(Ci()))," sono detti ",n("i",null,n(Je.j,null,"n"),"-esima percentile"),".")),n(Je.p,{title:"Varianza"},n("p",null,"È un valore che indica quanto la variabile aleatoria si discosta generalmente dalla media:"),n("p",null,n(Je.j,null,la(wi()))))),n(Je.q,{title:"Disuguaglianze notevoli"},n(Je.p,{title:"Disuguaglianza di Markov"},n("p",null,"Data una variabile aleatoria non-negativa:"),n("p",null,n(Je.j,null,la(Yi()))),n("p",null,"Divide in due parti (",n(Je.j,null,la(Si()))," e ",n(Je.j,null,la(Vi())),") la funzione X, la cui media risulterà uguale a:"),n("p",null,n(Je.j,null,la(Ni())))),n(Je.p,{title:"Disuguaglianza di Čebyšëv"},n("blockquote",null,'"disuguaglianza di cebicev"'),n("p",null,"Se la variabile aleatoria ",n(Je.j,null,"X")," ha media e varianza, allora la probabilità che essa abbia un valore a più di ",n(Je.j,null,la(yi()))," di distanza dal valore medio è minore o uguale a ",n(Je.j,null,la(Fi())),"."),n("p",null,n(Je.j,null,la(Li()))),n("p",null,"E anche:"),n("p",null,n(Je.j,null,la(ki()))),n(He.a,null,"Serve per semplificare i calcoli quando la funzione di ripartizione è difficile da calcolare!"))),n(Je.q,{title:"Un momento...!"},n(Je.p,{title:"Momento"},n("p",null,"Il ",n("i",null,"momento")," ",n(Je.j,null,"k"),"-esimo di una variabile aleatoria è:"),n("p",null,n(Je.j,null,la(xi()))),n(He.a,null,"La media di una variabile aleatoria è anche il suo primo momento.")),n(Je.p,{title:"Funzione generatrice dei momenti"},n("p",null,"La ",n("i",null,"funzione generatrice dei momenti")," è:"),n("p",null,n(Je.j,null,la(Pi()))),n("p",null,"Se due variabile aleatorie hanno la stessa funzione generatrice dei momenti, allora esse hanno la ",n("b",null,"stessa distribuzione"),"."),n("p",null,"E' la ",n("b",null,"trasformata di Laplace")," della variabile aleatoria di X.")),n(Je.p,{title:"Funzione caratteristica"},n("p",null,"La ",n("i",null,"funzione caratteristica")," è:"),n("p",null,n(Je.j,null,la(Ei()))),n("p",null,"Se due variabile aleatorie hanno la stessa funzione caratteristica, allora esse hanno la ",n("b",null,"stessa distribuzione"),"."),n("p",null,"E' la ",n("b",null,"trasformata di Fourier")," della variabile aleatoria di X."))),n(Je.q,{title:"Prove e schemi"},n(Je.p,{title:"Variabile con distribuzione"},n("p",null,"Per dire che una variabile ha una certa distribuzione, si usa la notazione:"),n("p",null,n(Je.j,null,la(zi())))),n(Je.p,{title:"Prova di Bernoulli"},n("p",null,"Una prova con solo due possibili esiti: ",n($e.a,null,"successo")," e ",n(na.a,null,"insuccesso"),".")),n(Je.p,{title:"Schema di Bernoulli"},n("p",null,"Una sequenza di prove di Bernoulli per le quali le probabilità di successo e fallimento rimangono invariate."))),n(Je.q,{title:"Bernoulliana"},n(Je.p,{title:"Distribuzione bernoulliana"},n("p",null,"Una variabile aleatoria che rappresenta una prova di Bernoulli:"),n("ul",null,n("li",null,"vale ",n($e.a,null,"1")," in caso di ",n($e.a,null,"successo"),"."),n("li",null,"vale ",n(na.a,null,"0")," in caso di ",n(na.a,null,"insuccesso"),".")),n("p",null,"Il suo simbolo è ",n(Je.j,null,la(ji())))),n(Je.p,{title:"Densità della bernoulliana"},n("p",null,"La distribuzione bernoulliana ha come densità:"),n("p",null,n(Je.j,null,la(qi()))))),n(Je.q,{title:"Binomiale"},n(Je.p,{title:"Distribuzione binomiale"},n("p",null,"Una variabile aleatoria che conta il numero di successi di ",n(Je.j,null,"n")," prove di uno schema di Bernoulli."),n("p",null,"Il suo simbolo è ",n(Je.j,null,la(hi())),".")),n(Je.p,{title:"Densità della binomiale"},n("p",null,"La binomiale ha come densità:"),n("p",null,n(Je.j,null,la(gi())))),n(Je.p,{title:"Momenti della binomiale"},n("p",null,"La ",n("b",null,"funzione generatrice dei momenti")," della binomiale è:"),n("p",null,n(Je.j,null,la(Xi()))),n("p",null,"La ",n("b",null,"media")," di una binomiale è:"),n("p",null,n(Je.j,null,la(_i()))),n("p",null,"La ",n("b",null,"varianza")," di una binomiale è:"),n("p",null,n(Je.j,null,la(bi()))))),n(Je.q,{title:"Geometrica"},n(Je.p,{title:"Distribuzione geometrica"},n("p",null,"Una variabile aleatoria che conta il numero di prove in uno schema di Bernoulli fino alla comparsa del primo successo."),n("p",null,"Il suo simbolo è ",n(Je.j,null,"Geo(p)"),".")),n(Je.p,{title:"Densità della geometrica"},n("p",null,"La geometrica ha come densità:"),n("p",null,n(Je.j,null,la(vi())))),n(Je.p,{title:"Momenti della geometrica"},n("p",null,"La ",n("b",null,"funzione generatrice dei momenti")," della geometrica è:"),n("p",null,n(Je.j,null,la(fi()))),n("p",null,"La ",n("b",null,"media")," della geometrica è:"),n("p",null,n(Je.j,null,la(mi()))),n("p",null,"La ",n("b",null,"varianza")," della geometrica è:"),n("p",null,n(Je.j,null,la(di())))),n(Je.p,{title:"Assenza di memoria della geometrica"},n("p",null,"La geometrica non tiene conto degli eventi avvenuti in passato: ha la proprietà dell'assenza di memoria:"),n("p",null,n(Je.j,null,la(si()))),n(He.a,null,"Ovvero, riscalando opportunamente l'asse Y posso prendere come 0 qualsiasi punto dell'asse X."))),n(Je.q,{title:"Binomiale negativa"},n(Je.p,{title:"Distribuzione binomiale negativa"},n("p",null,"Una variabile aleatoria che conta il numero di prove in uno schema di Bernoulli necessarie perchè si verifichi l'",n(Je.j,null,"n"),"-esimo successo."),n("p",null,"Il suo simbolo è ",n(Je.j,null,la(pi())),".")),n(Je.p,{title:"Densità della binomiale negativa"},n("p",null,"La binomiale negativa ha come densità:"),n("p",null,n(Je.j,null,la(ci())))),n(Je.p,{title:"Momenti della binomiale negativa"},n("p",null,n("p",null,"La ",n("b",null,"funzione generatrice dei momenti")," della binomiale negativa è:"),n("p",null,n(Je.j,null,la(ui()))),n("p",null,"La ",n("b",null,"media")," della binomiale negativa è:"),n("p",null,n(Je.j,null,la(oi()))),n("p",null,"La ",n("b",null,"varianza")," della binomiale negativa è:"),n("p",null,n(Je.j,null,la(ri())))))),n(Je.q,{title:"Geometrica traslata"},n(Je.p,{title:"Distribuzione geometrica traslata"},n("p",null,"Una variabile aleatoria che conta il numero ",n(Je.j,null,"k")," di insuccessi consecutivi in uno schema di Bernoulli:"),n("p",null,"Il suo simbolo rimane ",n(Je.j,null,la(ti())),".")),n(Je.p,{title:"Densità della geometrica tralsata"},n("p",null,"La geometrica traslata ha come densità:"),n("p",null,n(Je.j,null,la(ai())))),n(Je.p,{title:"Momenti della geometrica traslata"},n("p",null,"La ",n("b",null,"funzione generatrice dei momenti")," della geometrica traslata è:"),n("p",null,n(Je.j,null,la(ei()))),n("p",null,"La ",n("b",null,"media")," della geometrica traslata è:"),n("p",null,n(Je.j,null,la(ii()))),n("p",null,"La ",n("b",null,"varianza")," della geometrica è:"),n("p",null,n(Je.j,null,la(li())))),n(Je.p,{title:"Assenza di memoria della geometrica traslata"},n("p",null,"La geometrica traslata non tiene conto degli eventi avvenuti in passato: ha la proprietà dell'assenza di memoria:"),n("p",null,n(Je.j,null,la(ni()))),n(He.a,null,"Ovvero, riscalando opportunamente l'asse Y posso prendere come 0 qualsiasi punto dell'asse X."))),n(Je.q,{title:"Binomiale negativa traslata"},n(Je.p,{title:"Distribuzione binomiale negativa traslata"},n("p",null,"Una variabile aleatoria che conta il numero di insuccessi in uno schema di Bernoulli prima che si verifichi l'",n(Je.j,null,"n"),"-esimo successo."),n("p",null,"Il suo simbolo rimane ",n(Je.j,null,la($l())),".")),n(Je.p,{title:"Densità della binomiale negativa traslata"},n("p",null,"La binomiale negativa traslata ha come densità:"),n("p",null,n(Je.j,null,la(Hl())))),n(Je.p,{title:"Momenti della binomiale negativa traslata"},n("p",null,n("p",null,"La ",n("b",null,"funzione generatrice dei momenti")," della binomiale negativa traslata è:"),n("p",null,n(Je.j,null,la(Jl()))),n("p",null,"La ",n("b",null,"media")," della binomiale negativa traslata è:"),n("p",null,n(Je.j,null,la(Wl()))),n("p",null,"La ",n("b",null,"varianza")," della binomiale negativa traslata è:"),n("p",null,n(Je.j,null,la(Ql())))))),n(Je.q,{title:"Ipergeometrica"},n(Je.p,{title:"Distribuzione ipergeometrica"},n("p",null,"Una variabile aleatoria che, sapendo il numero di successi ",n(Je.j,null,"K")," e di insuccessi ",n(Je.j,null,"N-K"),", conta quanti successi si otterrebbero se se ne estraessero ",n(Je.j,null,"n")," in blocco."),n("p",null,"Il suo simbolo è ",n(Je.j,null,"Ipe(N, K, n)"),".")),n(Je.p,{title:"Densità della ipergeometrica"},n("p",null,"La ipergeometrica ha come densità:"),n("p",null,n(Je.j,null,la(Zl())))),n(Je.p,{title:"Momenti della ipergeometrica"},n("p",null,n("p",null,"La ",n("b",null,"funzione generatrice dei momenti")," della ipergeometrica è trascurabile."),n("p",null,"La ",n("b",null,"media")," della ipergeometrica è:"),n("p",null,n(Je.j,null,la(Kl()))),n("p",null,"La ",n("b",null,"varianza")," della ipergeometrica è:"),n("p",null,n(Je.j,null,la(Rl())))))),n(Je.q,{title:"Poissoniana"},n(Je.p,{title:"Distribuzione poissoniana"},n("p",null,"Una variabile aleatoria che soddisfa tutte le seguenti caratteristiche:"),n("ul",null,n("li",null,"Binomiale: ",n(Je.j,null,la(Ol()))),n("li",null,"Il numero di prove tende a infinito: ",n(Je.j,null,la(Gl()))),n("li",null,"La probabilità di successo tende a 0: ",n(Je.j,null,la(Tl()))),n("li",null,"La media è finita: ",n(Je.j,null,la(Al())))),n("p",null,"Il suo simbolo è ",n(Je.j,null,la(Bl())))),n(Je.p,{title:"Densità della poissoniana"},n("p",null,"La poissoniana ha come densità:"),n("p",null,n(Je.j,null,la(Ul())))),n(Je.p,{title:"Momenti della poissoniana"},n("p",null,n("p",null,"La ",n("b",null,"funzione generatrice dei momenti")," della poissoniana è:"),n("p",null,n(Je.j,null,la(Ml()))),n("p",null,"La ",n("b",null,"media")," della poissoniana è:"),n("p",null,n(Je.j,null,la(Dl()))),n("p",null,"La ",n("b",null,"varianza")," della poissoniana è:"),n("p",null,n(Je.j,null,la(Il()))),n("p",null,"Gli altri momenti della poissoniana sono:"),n("ol",{start:2},n("li",null,n(Je.j,null,la(Cl()))))))),n(Je.q,{title:"Un altro schema"},n(Je.p,{title:"Schema di Poisson"},n("p",null,"Una successione di ",n("b",null,"arrivi")," avvenuti in un certo arco temporale che:"),n("ul",null,n("li",null,"non sono sovrapposti."),n("li",null,"hanno intensità ",n(Je.j,null,la(wl()))," costante."),n("li",null,"avvengono indipendentemente gli uni dagli altri."))),n(Je.p,{title:"Processo di Poisson"},n("p",null,"Una variabile aleatoria ",n(Je.j,null,"N_t")," che conta il numero di arrivi di uno schema di Poisson di intensità ",n(Je.j,null,la(Yl()))," in un intervallo di tempo di durata ",n(Je.j,null,"t"),"."),n("p",null,"E' una distribuzione poissoniana con ",n(Je.j,null,la(Sl())),": ",n(Je.j,null,la(Vl()))),n(He.a,null,"E' paragonabile a una bernoulliana: ogni successo corrisponde a un arrivo, mentre il tempo è il numero di prove effettuate (ma nel continuo)."))),n(Je.q,{title:"Esponenziale"},n(Je.p,{title:"Distribuzione esponenziale"},n("p",null,"Una variabile aleatoria che conta il tempo diwidehattesa prima del primo arrivo di un processo di Poisson di intensità ",n(Je.j,null,la(Nl())),"."),n("p",null,"Il suo simbolo è ",n(Je.j,null,la(yl())),".")),n(Je.p,{title:"Densità dell'esponenziale"},n("p",null,"L'esponenziale ha come ",n("b",null,"densità"),":"),n("p",null,n(Je.j,null,la(Fl()))),n("p",null,"L'esponenziale ha come ",n("b",null,"funzione di ripartizione"),":"),n("p",null,n(Je.j,null,la(Ll())))),n(Je.p,{title:"Momenti dell'esponenziale"},n("p",null,"La ",n("b",null,"funzione generatrice dei momenti")," dell'esponenziale è:"),n("p",null,n(Je.j,null,la(kl()))),n("p",null,"La ",n("b",null,"media")," dell'esponenziale è:"),n("p",null,n(Je.j,null,la(xl()))),n("p",null,"La ",n("b",null,"varianza")," dell'esponenziale è:"),n("p",null,n(Je.j,null,la(Pl())))),n(Je.p,{title:"Assenza di memoria della esponenziale"},n("p",null,"L'esponenziale non tiene conto degli eventi avvenuti in passato: ha la proprietà dell'assenza di memoria:"),n("p",null,n(Je.j,null,la(El()))),n(He.a,null,"Ovvero, riscalando opportunamente l'asse Y posso prendere come 0 qualsiasi punto dell'asse X."))),n(Je.q,{title:"Legge gamma"},n(Je.p,{title:"Distribuzione gamma"},n("p",null,"Una variabile aleatoria che conta il tempo diwidehattesa prima dell'",n(Je.j,null,"n"),"-esimo arrivo di un processo di Poisson di intensità ",n(Je.j,null,la(zl())),"."),n("p",null,"Il suo simbolo è ",n(Je.j,null,la(jl())),".")),n(Je.p,{title:"Densità della legge gamma"},n("p",null,"La legge gamma ha come densità:"),n("p",null,n(Je.j,null,la(ql())))),n(Je.p,{title:"Momenti della legge gamma"},n("p",null,n("p",null,"La ",n("b",null,"funzione generatrice dei momenti")," della legge gamma è:"),n("p",null,n(Je.j,null,la(hl()))),n("p",null,"La ",n("b",null,"media")," della legge gamma è:"),n("p",null,n(Je.j,null,la(gl()))),n("p",null,"La ",n("b",null,"varianza")," della legge gamma è:"),n("p",null,n(Je.j,null,la(Xl())))))),n(Je.q,{title:"Uniforme"},n(Je.p,{title:"Distribuzione uniforme"},n("p",null,"Una variabile aleatoria che può assumere qualsiasi valore in un intervallo ",n(Je.j,null,la(_l()))," in modo equiprobabile."),n("p",null,"Il suo simbolo è ",n(Je.j,null,la(bl()))),n("p",null,"Su di essa vale la seguente proprietà:"),n("p",null,n(Je.j,null,la(vl())))),n(Je.p,{title:"Densità della distribuzione uniforme"},n("p",null,"La distribuzione uniforme ha come ",n("b",null,"densità"),":"),n("p",null,n(Je.j,null,la(fl()))),n("p",null,"La distribuzione uniforme ha come ",n("b",null,"funzione di ripartizione"),":"),n("p",null,n(Je.j,null,la(ml())))),n(Je.p,{title:"Momenti della distribuzione uniforme"},n("p",null,n("p",null,"La ",n("b",null,"funzione generatrice dei momenti")," della distribuzione uniforme è:"),n("p",null,n(Je.j,null,la(dl()))),n("p",null,"La ",n("b",null,"media")," della distribuzione uniforme è:"),n("p",null,n(Je.j,null,la(sl()))),n("p",null,"La ",n("b",null,"varianza")," della distribuzione uniforme è:"),n("p",null,n(Je.j,null,la(pl())))))),n(Je.q,{title:"Normale o Gaussiana"},n(Je.p,{title:"Distribuzione normale"},n("p",null,"Una variabile aleatoria con una specifica distribuzione."),n("p",null,"Il suo simbolo è ",n(Je.j,null,la(cl())),"."),n(He.a,null,n(Je.j,null,"\\mu")," e ",n(Je.j,null,"\\sigma^2")," sono rispettivamente la media e la varianza della distribuzione!")),n(Je.p,{title:"Densità della distribuzione normale"},n("p",null,"La distribuzione normale ha come densità:"),n("p",null,n(Je.j,null,la(ul())))),n(Je.p,{title:"Momenti della distribuzione normale"},n("p",null,n("p",null,"La ",n("b",null,"funzione generatrice dei momenti")," della distribuzione normale è:"),n("p",null,n(Je.j,null,la(ol()))),n("p",null,"La ",n("b",null,"media")," della distribuzione normale è:"),n("p",null,n(Je.j,null,la(rl()))),n("p",null,"La ",n("b",null,"varianza")," della distribuzione normale è:"),n("p",null,n(Je.j,null,la(tl())))))),n(Je.q,null,n(Je.p,{title:"Trasformazione della normale"},n("p",null,"Qualsiasi normale può essere trasformata in qualsiasi altra normale:"),n("p",null,n(Je.j,null,la(al())))),n(Je.p,{title:"Normale standard"},n("p",null,"La distribuzione normale standard ",n(Je.j,null,"Z")," è:"),n("p",null,n(Je.j,null,"Z \\sim Nor(0, 1)")),n("p",null,"La sua funzione di ripartizione è detta ",n(Je.j,null,la(el()))," e vale:"),n("p",null,n(Je.j,null,la(il())))),n(Je.p,{title:"Quantili normali"},n("p",null,"Da un quantile ",n(Je.j,null,la(ll()))," della normale standard è possibile risalire allo stesso quantile di qualsiasi altra normale:"),n("p",null,n(Je.j,null,la(nl()))))),n(Je.q,null,n(Je.p,{title:"Gamma e normale"},n("p",null,"La distribuzione normale ha una particolare relazione con la distribuzione Gamma:"),n("p",null,n(Je.j,null,la($n())))),n(Je.p,{title:"La funzione Chi"},n("blockquote",null,'"chi-quadro a un grado di libertà"'),n("p",null,"Esiste una distribuzione Gamma particolare:"),n("p",null,n(Je.j,null,la(Hn()))),n("p",null,"Più chi-quadro possono essere sommate per aumentare i loro gradi di libertà:"),n("p",null,n(Je.j,null,la(Jn())))),n(Je.p,{title:"T di Student"},n("p",null,"Un'altra funzione particolare è la funzione T di Student:"),n("p",null,n(Je.j,null,la(Wn()))))),n(Je.q,{title:"Approssimazioni notevoli"},n(Je.p,{title:"Ipergeometrica e binomiale"},n("p",null,"La binomiale è come una ipergeometrica ma con ripetizioni, quindi per valori molto grandi di ",n(Je.j,null,"N")," rispetto a ",n(Je.j,null,"n"),", si può dire che:"),n("p",null,n(Je.j,null,la(Qn())))),n(Je.p,{title:"Binomiale e poissoniana"},n("p",null,"La binomiale non è altro che una poissoniana a tempo discreto, quindi, se ",n(Je.j,null,"n")," è grande e ",n(Je.j,null,"n \\cdot p")," è nell'ordine di grandezza delle unità, allora:"),n("p",null,n(Je.j,null,la(Zn())))),n(Je.p,{title:"Binomiale e normale"},n("p",null,"Per il Teorema di De Moivre-Laplace, se una binomiale ha una ",n(Je.j,null,"n")," grande e ",n(Je.j,null,"p")," non vicina a 0 o 1, si può approssimare con:"),n("p",null,n(Je.j,null,la(Kn())))),n(Je.p,{title:"Correzione di Yates"},n("p",null,"Passando da una variabile discreta ",n(Je.j,null,"X")," a una continua ",n(Je.j,null,"Y"),", per ogni valore discreto ",n(Je.j,null,"k"),' la probabilità viene "spalmata" su tutto l\'intervallo ',n(Je.j,null,la(Rn())),":"),n("ul",null,n("li",null,n(Je.j,null,la(On()))),n("li",null,n(Je.j,null,la(Gn()))),n("li",null,n(Je.j,null,la(Tn()))),n("li",null,n(Je.j,null,la(An())))))),n(Je.q,{title:"Vettori aleatori"},n(Je.p,{title:"Vettore aleatorio"},n("p",null,"Un vettore ",n("b",null,"composto da variabili aleatorie"),"."),n("p",null,"Il suo simbolo generalmente è ",n(Je.j,null,la(Bn()))," oppure ",n(Je.j,null,la(Un())),".")),n(Je.p,{title:"Funzioni di ripartizione"},n("p",null,"I vettori aleatori hanno più funzioni di ripartizione che si differenziano in base al numero di parametri."),n("p",null,"Se il numero di parametri coincide con la dimensione del vettore aleatorio, allora la funzione sarà una ",n("i",null,"funzione di ripartizione congiunta"),":"),n("p",null,n(Je.j,null,la(Mn()))),n("p",null,"Se il numero di parametri è minore della dimensione del vettore aleatorio, allora la funzione sarà una ",n("i",null,"funzione di ripartizione marginale"),":"),n("p",null,n(Je.j,null,la(Dn())))),n(Je.p,{title:"Densità discreta"},n("p",null,"I vettori aleatori ",n("b",null,"discreti")," hanno più densità che si differenziano in base al numero di parametri."),n("p",null,"Se il numero di parametri coincide con la dimensione del vettore aleatorio, allora la funzione sarà una ",n("i",null,"densità congiunta"),":"),n("p",null,n(Je.j,null,la(In()))),n("p",null,"Se il numero di parametri è minore della dimensione del vettore aleatorio, allora la funzione sarà una ",n("i",null,"densità marginale"),":"),n("p",null,n(Je.j,null,la(Cn()))))),n(Je.q,{title:"Più variabili aleatorie"},n(Je.p,{title:"Indipendenza delle variabili aleatorie"},n("p",null,"Più variabili aleatorie sono indipendenti se, per qualsiasi scelta di intervalli ",n(Je.j,null,"A_i"),":"),n("p",null,n(Je.j,null,la(wn())))),n(Je.p,{title:"Media dei vettori aleatori"},n("p",null,"E' possibile calcolare la media di qualsiasi funzione ",n(Je.j,null,"g(X, Y)")," avente elementi del vettore come variabili:"),n("p",null,n(Je.j,null,la(Yn()))),n(He.a,null,"Solitamente si calcola la media di ",n(Je.j,null,"x \\cdot y"),"."),n("p",null,"Le medie di più variabili aleatorie si possono sommare:"),n("p",null,n(Je.j,null,la(Sn()))))),n(Je.q,null,n(Je.p,{title:"Covarianza"},n("p",null,"Un ",n("b",null,"operatore")," che misura la correlazione di due variabili aleatorie."),n("p",null,"Si calcola con il valore atteso dei prodotti delle distanze dalla media:"),n("p",null,n(Je.j,null,la(Vn()))),n("p",null,"Ha diverse proprietà:"),n("ul",null,n("li",null,"Il suo ",n("b",null,"valore nullo")," è 0: ",n(Je.j,null,la(Nn()))),n("li",null,"E' ",n("b",null,"commutativa"),": ",n(Je.j,null,la(yn()))),n("li",null,"E' ",n("b",null,"semplificabile"),": ",n(Je.j,null,la(Fn()))),n("li",null,"E' ",n("b",null,"lineare"),": ",n(Je.j,null,la(Ln()))),n("li",null,"E' ",n("b",null,"distributiva"),": ",n(Je.j,null,la(kn()))))),n(Je.p,{title:"Variabili incorrelate"},n("p",null,"Due variabili sono ",n("i",null,"variabili incorrelate")," se:"),n("p",null,n(Je.j,null,la(xn()))),n("p",null,"Variabili indipendenti sono sempre incorrelate.")),n(Je.p,{title:"Matrice di covarianza"},n("p",null,"Una matrice ",n(Je.j,null,la(Pn()))," che contiene la covarianza tra tutte le variabili di un vettore aleatorio ",n(Je.j,null,la(En())),":"),n("p",null,n(Je.j,null,la(zn()))),n("p",null,"E' sempre simmetrica e semidefinita positiva (tutti gli autovalori sono ",n(Je.j,null,"\\geq 0"),".")),n(Je.p,{title:"Coefficiente di correlazione"},n("p",null,"Un valore che misura come due variabili aleatorie sono correlate:"),n("p",null,n(Je.j,null,la(jn()))),n("p",null,"E' sempre compreso tra -1 e 1:"),n("p",null,n(Je.j,null,la(qn()))),n("p",null,"Vale esattamente -1 o 1 solo se esiste un legame lineare tra le due variaibli:"),n("p",null,n(Je.j,null,la(hn())))),n(Je.p,{title:"Varianza di variabili aleatorie sommate"},n("p",null,"La varianza di due variabili aleatorie sommate è:"),n("p",null,n(Je.j,null,la(gn()))),n(He.a,null,"Si dimostra applicando le proprietà della covarianza!"),n("p",null,"Se più variabili aleatorie ",n(Je.j,null,"X_i")," sono ",n("b",null,"indipendenti")," (",n(Je.j,null,la(Xn())),"), allora:"),n("p",null,n(Je.j,null,la(_n()))))),n(Je.q,{title:"Campioni"},n(Je.p,{title:"Campione casuale"},n("p",null,"Una ",n("b",null,"n-pla")," di variabili aleatorie con la stessa distribuzione della variabile aleatoria ",n(Je.j,null,"X"),' ("popolazione") ma ',n("b",null,"indipendenti")," tra loro."),n(He.a,null,"Le variabili aleatorie sono come un lazy-load in programmazione; quando ci sarà bisogno del loro valore numerico, esse si ",n("b",null,"realizzeranno")," nel loro valore.")),n(Je.p,{title:"Momento campionario"},n("p",null,"Il valore dato dalla media aritmetica degli ",n(Je.j,null,"n")," elementi del campione elevati alla potenza ",n(Je.j,null,"k"),":"),n("p",null,n(Je.j,null,la(bn()))),n("p",null,"Il momento campionario di primo ordine è la ",n("i",null,"media campionaria")," ",n(Je.j,null,la(vn())),".")),n(Je.p,{title:"Varianza campionaria"},n("p",null,"La media aritmetica dello scarto quadratico medio degli elementi del campione."),n("p",null,"Se è noto il valore medio ",n(Je.j,null,la(fn()))," di X:"),n("p",null,n(Je.j,null,la(mn()))),n("p",null,"Altrimenti:"),n("p",null,n(Je.j,null,la(dn()))))),n(Je.q,{title:"Media-ception"},n(Je.p,{title:"Media campionaria"},n("p",null,"Se calcoliamo la media della media campionaria, risulterà vero che:"),n("p",null,n(Je.j,null,la(sn()))),n(He.a,null,"Quindi, è possibile usare i campioni per trovare la media di una variabile aleatoria!")),n(Je.p,{title:"Varianza campionaria"},n("p",null,"Se calcoliamo la varianza della media campionaria, risulterà vero che:"),n("p",null,n(Je.j,null,la(pn()))),n(He.a,null,"Quindi, possiamo stimare l'errore della media calcolata tramite campioni!")),n(Je.p,{title:"Correzione campionaria"},n("p",null,"Se calcoliamo la media della varianza campionaria, risulterà vero che:"),n("p",null,n(Je.j,null,la(cn()))),n(He.a,null,"Quindi, possiamo stimare l'errore della media calcolata tramite campioni!"))),n(Je.q,{title:"Campionamento di una distribuzione normale"},n(Je.p,{title:"Campionamento di una distribuzione normale"},n("p",null,"Se la popolazione ",n(Je.j,null,"X")," ha una distribuzione normale (",n(Je.j,null,la(un())),")...")),n(Je.p,{title:"Distribuzione della media campionaria"},n("p",null,"...allora sappiamo anche la distribuzione della media campionaria!"),n("p",null,n(Je.j,null,la(on())))),n(Je.p,{title:"Distribuzione della varianza campionaria"},n("p",null,"...e anche della varianza campionaria!"),n("p",null,n(Je.j,null,la(rn()))),n("p",null,n(Je.j,null,la(tn())))),n(Je.p,{title:"Indipendenza"},n("p",null,"...e che media campionaria e varianza campionaria sono indipendenti tra loro!"))),n(Je.q,{title:"Quando i campioni hanno dimensioni infinite"},n(Je.p,{title:"Convergenza in distribuzione"},n("p",null,"Se la successione di variabili aleatorie ",n(Je.j,null,"X_n")," all'infinito ha la ",n("b",null,"stessa funzione di ripartizione")," della popolazione ",n(Je.j,null,"X"),", allora essa ",n("i",null,"converge in distribuzione"),"."),n("p",null,n(Je.j,null,"\\lim_{n \\to +\\infty} F_{X_n} (x) = F_X (x) \\implies X_n \\xrightarrow{d} X"))),n(Je.p,{title:"Convergenza in probabilità"},n("p",null,"Se la successione di variabili aleatorie ",n(Je.j,null,"X_n")," all'infinito ha la ",n("b",null,"stessa probabilità")," della popolazione ",n(Je.j,null,"X"),", allora essa ",n("i",null,"converge in probabilità"),"."),n("p",null,n(Je.j,null,"\\forall \\epsilon > 0, \\lim_{n \\to +\\infty} P( | X_n - X | < \\epsilon) = 1 \\implies X_n \\xrightarrow{p} X"))),n(Je.p,{title:"Convergenza quasi certa"},n("p",null,"Se la successione di variabili aleatorie ",n(Je.j,null,"X_n")," all'infinito ha la ",n("b",null,"stessa probabilità a ")," della popolazione ",n(Je.j,null,"X"),", allora essa ",n("i",null,"converge quasi certamente"),"."),n("p",null,n(Je.j,null,"\\forall \\epsilon > 0, P left( \\lim_{n \\to +\\infty} | X_n - X | < \\epsilon) \right) = 1 \\implies X_n \\xrightarrow{qc} X"))),n(Je.p,{title:"Convergenza in media quadratica"},n("p",null,"Se la successione di variabili aleatorie ",n(Je.j,null,"X_n")," all'infinito ha la ",n("b",null,"media del quadrato della distanza")," tra la successione e la popolazione ",n(Je.j,null,"X")," ",n("b",null,"uguale a 0"),", allora essa ",n("i",null,"converge in media quadratica"),"."),n("p",null,n(Je.j,null,"\\lim_{n \\to +\\infty} E( | X_n - X |^2 = 0 \\implies X_n \\xrightarrow{mq} X"))),n(Je.p,{title:"Gerarchia delle convergenze"},n("p",null,n(Je.j,null,"\n \\begin{matrix}\n X_n \\xrightarrow{mq} X\\\\\n X_n \\xrightarrow{qc} X\n \\end{matrix} \\implies X_n \\xrightarrow{p} X \\implies X_n \\xrightarrow{d} X")),n("p",null,"In più:"),n("p",null,n(Je.j,null,"X_n \\xrightarrow{p} x \\Longleftrightarrow X_n \\xrightarrow{d} x")))),n(Je.q,{title:"I grandi numeri"},n(Je.p,{title:"Legge debole dei grandi numeri"},n("p",null,"La successione delle medie campionarie ",n(Je.j,null,la(an()))," ",n("b",null,"converge in probabilità")," alla media della popolazione ",n(Je.j,null,la(en())),", se essa esiste."),n("p",null,n(Je.j,null,"\\overline{X}_n \\xrightarrow{p} X")),n("p",null,"Ovvero:"),n("p",null,n(Je.j,null,la(ln()))),n("p",null,n(Je.j,null,la(nn())))),n(Je.p,{title:"Legge forte dei grandi numeri"},n("p",null,"La successione delle medie campionarie ",n(Je.j,null,la($()))," ",n("b",null,"converge quasi certamente")," alla media della popolazione ",n(Je.j,null,la(H())),", se essa esiste."),n("p",null,n(Je.j,null,"\\overline{X}_n \\xrightarrow{qc} X")),n("p",null,"Ovvero:"),n("p",null,n(Je.j,null,la(J()))),n(He.a,null,"Dimostra che l'interpretazione frequentista della probabilità è valida!"))),n(Je.q,{title:"Al limite"},n(Je.p,{title:"Teorema centrale del limite"},n("p",null,"La successione delle medie campionarie ",n(Je.j,null,la(W()))," ",n("b",null,"converge in distribuzione")," a ",n(Je.j,null,la(Q())),"."),n("p",null,n(Je.j,null,la(Z()))),n("p",null,"Ovvero:"),n("p",null,n(Je.j,null,la(K()))))),n(Je.q,{title:"Altre approsimazioni"},n(Je.p,{title:"Binomiale e normale"},n("p",null,"E' una somma di ",n("b",null,"bernoulliane"),", e quindi si approssima a una normale:"),n("p",null,n(Je.j,null,la(R())))),n(Je.p,{title:"Binomiale negativa e normale"},n("p",null,"E' una somma di ",n("b",null,"geometriche"),", e quindi si approssima a una normale:"),n("p",null,n(Je.j,null,la(O())))),n(Je.p,{title:"Poissoniana e normale"},n("p",null,"E' una somma di altre ",n("b",null,"poissoniane"),", e quindi si approssima a una normale:"),n("p",null,n(Je.j,null,la(G())))),n(Je.p,{title:"Gamma e normale"},n("p",null,"E' una somma di ",n("b",null,"esponenziali"),", e quindi si approssima a una normale:"),n("p",null,n(Je.j,null,la(T())))),n(Je.p,{title:"In generale"},n("p",null,"Se ",n(Je.j,null,"n")," è grande, allora:"),n("p",null,n(Je.j,null,la(A()))))),n(Je.q,{title:"Actually statistica"},n(Je.p,{title:"Parametri sconosciuti"},n("p",null,"Per indicare parametri sconosciuti di una legge si usa ",n(Je.j,null,"\\theta"),".")),n(Je.p,{title:"Statistica"},n("p",null,"Una variabile aleatoria funzione di un campione:"),n("p",null,n(Je.j,null,la(B()))),n(He.a,null,"Ad esempio, sono statistiche media e varianza campionaria, così come il campione stesso ",n(Je.j,null,la(U())),"."))),n(Je.q,{title:"Stimatori"},n(Je.p,{title:"Stimatore"},n("p",null,"Una statistica ",n(Je.j,null,"T_n")," ottenuta da ",n(Je.j,null,"n")," osservazioni, che stimi i parametri di una legge e sia indipendente da essi.")),n(Je.p,{title:"Corretto"},n("p",null,"Uno stimatore è ",n("i",null,"corretto")," se il suo valore atteso coincide con quello dei parametri che stima:"),n("p",null,n(Je.j,null,la(M())))),n(Je.p,{title:"Asintoticamente corretto"},n("p",null,"Uno stimatore è ",n("i",null,"asintoticamente corretto")," se, per infinite osservazioni, il suo valore atteso coincide con quello dei parametri che stima:"),n("p",null,n(Je.j,null,la(D())))),n(Je.p,{title:"Consistente in media quadratica"},n("p",null,"Uno stimatore è ",n("i",null,"consistente in media quadratica")," se:"),n("p",null,n(Je.j,null,la(I())))),n(Je.p,{title:"Consistente in probabilità"},n("p",null,"Uno stimatore è ",n("i",null,"consistente in probabilità")," se:"),n("p",null,n(Je.j,null,la(C())))),n(Je.p,{title:"Asintoticamente normale"},n("p",null,"Uno stimatore è ",n("i",null,"asintoticamente normale")," se:"),n("p",null,n(Je.j,null,la(w()))))),n(Je.q,{title:"Metodo dei momenti"},n(Je.p,{title:"Metodo dei momenti"},n("p",null,"Si può usare il ",n("i",null,"metodo dei momenti")," per ottenere uno stimatore di una popolazione ",n(Je.j,null,"X"),"."),n("p",null,"Lo stimatore di ",n(Je.j,null,la(Y()))," così ottenuto sarà indicato aggiungendo un cappellino e una ",n(Je.j,null,"M")," a ",n(Je.j,null,"\\theta"),": ",n(Je.j,null,la(S()))),n("p",null,"Visto che:"),n("ul",null,n("li",null,n(Je.j,null,la(V()))),n("li",null,n(Je.j,null,la(N())))),n("p",null,"Allora:"),n("p",null,n(Je.j,null,la(y()))),n("p",null,"Se ",n(Je.j,null,la(F()))," non è esprimibile in termini di ",n(Je.j,null,la(L())),", si possono usare i momenti successivi ",n(Je.j,null,la(k())),", ",n(Je.j,null,la(x())),", ",n(Je.j,null,la(P())),"..."))),n(Je.q,{title:"Metodo della massima verosomiglianza"},n(Je.p,{title:"Metodo della massima verosomiglianza"},n("p",null,"Si può usare il ",n("i",null,"metodo della massima verosomiglianza")," per ottenere uno stimatore di una popolazione ",n(Je.j,null,"X"),"."),n("p",null,"Lo stimatore di ",n(Je.j,null,la(E()))," così ottenuto sarà indicato aggiungendo un cappellino e una ",n(Je.j,null,"L")," a ",n(Je.j,null,"\\theta"),": ",n(Je.j,null,la(z()))),n("p",null,"Consiste nel trovare il massimo assoluto ",n(Je.j,null,la(j()))," della la funzione di verosomiglianza ",n(Je.j,null,la(q())),":"),n("p",null,n(Je.j,null,la(h()))),n("p",null,"Gli stimatori di massima verosomiglianza sono ",n("b",null,"asintoticamente corretti"),", ",n("b",null,"consistenti in probabilità")," e ",n("b",null,"asintoticamente normali"),".")),n(Je.p,{title:"Proprietà degli stimatori di massima verosomiglianza"},n("p",null,"Gli stimatori di massima verosomiglianza godono delle seguenti proprietà:"),n("ul",null,n("li",null,"Sono ",n("b",null,"asintoticamente corretti"),"."),n("li",null,"Sono ",n("b",null,"consistenti in probabilità"),"."),n("li",null,"Sono ",n("b",null,"asintoticamente normali"),"."),n("li",null,"Sono ",n("b",null,"invarianti"),": ",n(Je.j,null,la(g())))))),n(Je.q,{title:"Nuove stime notevoli"},n(Je.p,{title:"Stima di una bernoulliana"},n("p",null,"Per il metodo dei momenti oppure per il metodo della massima verosomiglianza:"),n("p",null,n(Je.j,null,la(X())))),n(Je.p,{title:"Stima di una poissoniana"},n("p",null,"Per il metodo dei momenti oppure per il metodo della massima verosomiglianza:"),n("p",null,n(Je.j,null,la(_())))),n(Je.p,{title:"Stima di una esponenziale"},n("p",null,"Per il metodo dei momenti oppure per il metodo della massima verosomiglianza:"),n("p",null,n(Je.j,null,la(b())))),n(Je.p,{title:"Stima di una normale"},n("p",null,"Per il metodo della massima verosomiglianza:"),n("ul",null,n("li",null,n(Je.j,null,la(v()))),n("br",null),n("li",null,n(Je.j,null,la(f())))))),n(Je.q,{title:"Intervalli di confidenza"},n(Je.p,{title:"Confidenza"},n("blockquote",null,'"intervallo di confidenza al 95%"'),n("p",null,"L'intervallo di valori di ",n(Je.j,null,"\\theta"),' all\'interno del quale siamo "più o meno sicuri" si trovi il valore effettivo:'),n("p",null,"L'intervallo di confidenza a N della stima ",n(Je.j,null,la(m()))," è l'intervallo ",n(Je.j,null,"]a, b[")," tale che:"),n("p",null,n(Je.j,null,la(d()))),n("p",null,"Può anche essere ",n("b",null,"unilatero")," nel caso limiti la stima in una sola direzione, positiva o negativa."))),n(Je.q,{title:"Confidenza nella media di una normale"},n(Je.p,{title:"Varianza nota"},n("p",null,"Se conosciamo la varianza di una normale, allora possiamo ricavare velocemente gli intervalli di confidenza all'",n(Je.j,null,"\\alpha"),"% con queste formule:"),n("ul",null,n("li",null,"Intervalli bilateri: ",n(Je.j,null,la(s()))),n("li",null,"Intervallo unilatero da sinistra: ",n(Je.j,null,la(p()))),n("li",null,"Intervallo unilatero da destra: ",n(Je.j,null,la(c()))))),n(Je.p,{title:"Varianza incognita"},n("p",null,"Se non conosciamo la varianza di una normale, allora possiamo ricavare velocemente gli intervalli di confidenza all'",n(Je.j,null,"\\alpha"),"% con queste formule:"),n("ul",null,n("li",null,"Intervalli bilateri: ",n(Je.j,null,la(u()))),n("li",null,"Intervallo unilatero da sinistra: ",n(Je.j,null,la(o()))),n("li",null,"Intervallo unilatero da destra: ",n(Je.j,null,la(r())))),n("p",null,n(Je.j,null,la(t()))," è un quantile della distribuzione di Student di parametro ",n(Je.j,null,"v"),"."))),n(Je.q,{title:"Confidenza per la proporzione di una bernoulliana"},n(Je.p,{title:"Terzo metodo corretto"},n("p",null,"L'intervallo di confidenza per la proprorzione di una bernoulliana qualsiasi si ottiene da questa formula:"),n("p",null,n(Je.j,null,la(a()))))),n(Je.q,{title:"Confidenza per la media di qualsiasi popolazione"},n(Je.p,{title:"Approssimando con la normale"},n("p",null,"L'intervallo di confidenza per la media di una qualsiasi popolazione si ottiene da questa formula:"),n("p",null,n(Je.j,null,la(e()))))))))}}.call(this,i("hosL").h)},gJgL:function(n){n.exports={red:"red__3phX0",orange:"orange__1OSkI",yellow:"yellow__2pQOr",lime:"lime__3vRUg",cyan:"cyan__1cBtG",blue:"blue__1G1wb",magenta:"magenta__3Wagv",minus:"minus__2qLyv"}},jZax:function(n){n.exports={red:"red__ipSoq",orange:"orange__1nmkj",yellow:"yellow__353ge",lime:"lime__3FW1E",cyan:"cyan__2xhu7",blue:"blue__3MvDe",magenta:"magenta__i03ie",plus:"plus__1eV4_"}},ke5e:function(n,l,i){"use strict";(function(n){var e=i("2w3n"),a=i.n(e);l.a=function(l){return n("div",{class:a.a.example},l.children)}}).call(this,i("hosL").h)},o8Jj:function(n,l,i){"use strict";(function(n){var e=i("jZax"),a=i.n(e);l.a=function(l){return n("span",{class:a.a.plus},l.children)}}).call(this,i("hosL").h)}}]); +//# sourceMappingURL=route-Statistica.chunk.6a8b4.js.map \ No newline at end of file diff --git a/docs/route-Statistica.chunk.a6da0.js.map b/docs/route-Statistica.chunk.6a8b4.js.map similarity index 99% rename from docs/route-Statistica.chunk.a6da0.js.map rename to docs/route-Statistica.chunk.6a8b4.js.map index 510cb90..a337636 100644 --- a/docs/route-Statistica.chunk.a6da0.js.map +++ b/docs/route-Statistica.chunk.6a8b4.js.map @@ -1 +1 @@ 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extracted by mini-css-extract-plugin\nmodule.exports = {\"red\":\"red__2y1B_\",\"orange\":\"orange__dD2kx\",\"yellow\":\"yellow__OEpwl\",\"lime\":\"lime__CVe41\",\"cyan\":\"cyan__26ZAg\",\"blue\":\"blue__LO7Xm\",\"magenta\":\"magenta__1Akee\",\"example\":\"example__2PzAa\"};","import style from \"./Minus.less\";\n\nexport default function(props) {\n\treturn {props.children};\n}\n","import {Latex, LatexDisplay, Panel, Section, LatexDefaultInline, LatexDefaultDisplay} from \"bluelib\";\nimport Example from \"../components/Example\";\n\nimport Plus from \"../components/Fisica/Plus\";\nimport Minus from \"../components/Fisica/Minus\";\n\nconst r = String.raw;\n\n\nexport default function(props) {\n return (\n \n \n
\n

Statistica ed Elementi di Probabilità

\n
\n \n

\n {r`P(E) = \\frac{casi\\ favorevoli}{casi\\ possibili}`}\n

\n \n \n

\n {r`P(E) = \\frac{successi}{prove\\ totali}`}\n

\n \n \n

\n Il prezzo che un individuo coerente riterrebbe equo per ricevere 1 nel caso l'evento si verificasse e 0 nel caso l'evento non si verificasse.\n

\n \n
\n
\n \n
\n \"omegone\"\n
\n

\n L'insieme di tutti gli esiti possibili di un esperimento.\n

\n

\n {r`\\Omega = \\left \\{ 1, 2, 3, 4, 5, 6 \\right \\}`}\n

\n \n \n
\n \"omeghino\"\n
\n

\n Un elemento dello spazio campionario.\n

\n

\n {r`\\omega = 1`}\n

\n \n \n
\n \"e\"\n
\n

\n Un sottoinsieme dello spazio campionario.\n

\n

\n {r`E = \\left \\{ 1, 2 \\right \\}`}\n

\n

\n Lo spazio campionario stesso è un evento certo.\n

\n \n \n
\n \"not e\"\n
\n

\n Il complementare di un sottoinsieme.\n

\n

\n {r`\\bar{E} = \\left \\{ 3, 4, 5, 6 \\right \\}`}\n

\n \n \n
\n \"e intersecato effe\"\n
\n

\n L'intersezione di più sottoinsiemi.\n

\n

\n {r`E \\cap F = \\left \\{ 1 \\right \\}`}\n

\n \n \n
\n \"e unito a effe\"\n
\n

\n L'unione di più sottoinsiemi.\n

\n

\n {r`E \\cup F = \\left \\{ 1, 2, 3, 4 \\right \\}`}\n

\n \n \n
\n \"e meno effe\"\n
\n

\n {r`E \\setminus F = E \\cap \\bar{F}`}\n

\n \n \n
\n \"e contenuto in effe\"\n
\n

\n L'inclusione del primo insieme in un altro.\n

\n

\n {r`E \\subseteq F`}\n

\n

\n Se si verifica E, allora si verifica anche F.\n

\n \n \n
\n \"e è impossibile\"\n
\n

\n Un sottoinsieme vuoto.\n

\n

\n {r`E = \\emptyset`}\n

\n \n \n
\n \"e ed effe si escludono mutualmente\"\n
\n

\n La disgiunzione di due insiemi.\n

\n

\n {r`E \\cap F = \\emptyset`}\n

\n \n
\n
\n \n
\n \"famiglia effe\"\n
\n

\n I sottoinsiemi dello spazio campionario formano una famiglia di sottoinsiemi detta famiglia degli eventi.\n

\n

\n {r`\\mathcal{F}`}\n

\n

\n Qualsiasi sottoinsieme appartenente a {r`\\mathcal{F}`} è considerato un evento.\n

\n \n {r`\\sigma`}-algebra}>\n
\n \"sigma algebra\"\n
\n

\n Se la famiglia degli eventi soddisfa questi tre requisiti, allora viene detta {r`\\sigma`}-algebra:\n

\n
    \n
  1. \n Lo spazio campionario è un evento: {r`\\Omega \\in \\mathcal{F}`}\n
    2. \n
  2. \n Se un sottoinsieme è un evento, allora anche il suo complementare lo è: {r`E \\in \\mathcal{F} \\implies \\bar{E} \\in \\mathcal{F}`}\n
    3. \n
  3. \n Se due sottoinsiemi sono eventi, allora lo sono anche la loro unione e intersezione: {r`(E, F) \\in \\mathcal{F} \\implies (E \\cup F, E \\cap F) \\in \\mathcal{F}`}\n
    4. \n
\n

\n Un esempio: {r`E \\in \\mathcal{F} \\implies \\mathcal{F} = \\{ \\emptyset, E, \\bar{E}, \\Omega \\}`}\n

\n \n
\n
\n \n
\n \"la partizione e composta da e uno, e due, e tre...\"\n
\n

\n Un insieme di esiti e eventi:\n

\n
    \n
  • Finito.
    • \n
  • In cui tutti gli eventi hanno probabilità diversa da 0.
    • \n
  • In cui tutti gli eventi sono mutualmente esclusivi.
    • \n
  • In cui l'unione di tutti i suoi elementi copre lo spazio campionario.
    • \n
\n

\n La partizione {r`E_i`} è composta dagli eventi {r`E_1`}, {r`E_2`}, {r`E_3`}, fino a {r`E_n`}.\n

\n \n Se lo spazio campionario fosse una torta, una sua partizione sarebbe l'insieme delle fette di uno dei modi in cui si potrebbe tagliare.\n \n \n
\n
\n \n

\n La probabilità di un evento è un numero tra 0 e 1.\n

\n

\n {r`\\forall E \\in \\mathcal{F}, 0 \\leq P(E) \\leq 1`}\n

\n \n \n

\n La probabilità dello spazio campionario è sempre 1.\n

\n

\n {r`P(\\Omega) = 1`}\n

\n \n \n

\n La probabilità dell'unione di eventi indipendenti è uguale alla somma delle loro probabilità.\n

\n

\n {r`P \\left ( \\bigcup_i E_i \\right ) = \\sum_i P ( E_i )`}\n

\n \n
\n
\n \n

\n La probabilità di un evento negato è uguale a 1 meno la probabilità dell'evento non negato.\n

\n

\n {r`P(\\bar{E}) = 1 - P({E})`}\n

\n \n \n

\n La probabilità di un evento incluso in un altro è sempre minore o uguale alla probabilità dell'evento in cui è incluso.\n

\n

\n {r`F \\subseteq E \\implies P(F) \\leq P(E)`}\n

\n \n \n

\n La probabilità di un evento unito a un altro è uguale alla somma delle probabilità dei due eventi meno la probabilità della loro intersezione.\n

\n

\n {r`P(E \\cup F) = P(E) + P(F) - P(E \\cap F)`}\n

\n \n Sommando le probabilità dei due eventi, l'intersezione viene contata due volte, e va quindi rimossa!\n \n \n
\n
\n \n

\n Spazi campionari in cui ci sono un numero finito di esiti e ogni esito ha la stessa probabilità di verificarsi.\n

\n

\n {r`P(E) = \\frac{len(E)}{len(\\Omega)}`}\n

\n \n \n

\n Gli spazi campionari possono avere un numero infinito di esiti: sono equiprobabili geometrici se nessun esito è privilegiato rispetto agli altri.\n

\n \n
\n
\n \n

\n Estraggo un numero, da un sacchetto con n numeri, mi segno che numero ho estratto e lo tengo fuori dal sacchetto. Ripeto per k volte.\n

\n

\n Tengo conto dell'ordine in cui ho estratto i numeri.\n

\n

\n {r`\\boldsymbol{D}_{n, k} = \\frac{n!}{(n - k)!}`}\n

\n \n \n

\n Estraggo un numero, da un sacchetto con n numeri, mi segno che numero ho estratto e lo rimetto nel sacchetto. Ripeto per k volte.\n

\n

\n Tengo conto dell'ordine in cui ho estratto i numeri.\n

\n

\n {r`\\boldsymbol{D}^{r}_{n, k} = n^k`}\n

\n \n \n

\n Estraggo un numero, da un sacchetto con n numeri, mi segno che numero ho estratto e lo tengo fuori dal sacchetto. Ripeto per k volte.\n

\n

\n Non mi interessa l'ordine in cui ho estratto i numeri.\n

\n

\n {r`\\boldsymbol{C}_{n, k} = \\binom{n}{k} = \\frac{n!}{(k)! \\cdot (n - k)!}`}\n

\n \n \n

\n Estraggo un numero, da un sacchetto con n numeri, mi segno che numero ho estratto e lo rimetto nel sacchetto. Ripeto per k volte.\n

\n

\n Non mi interessa l'ordine in cui ho estratto i numeri.\n

\n

\n {r`\\boldsymbol{C}^{r}_{n, k} = \\binom{n + k - 1}{k} = \\frac{(n + k - 1)!}{(k)! \\cdot (n - 1)!}`}\n

\n \n \n

\n Estraggo n numeri e guardo in quanti ordini diversi li posso mettere.\n

\n

\n {r`\\boldsymbol{P}_n = n!`}\n

\n \n
\n
\n \n
\n \"E dato F\"\n
\n

\n La probabilità che si verifichi E sapendo che si è già verificato F.\n

\n

\n {r`P(E|F) = \\frac{P(E \\cap F)}{P(F)}`}\n

\n \n Ricorda vagamente le pipe di bash, però al contrario...\n \n \n \n

\n Se due eventi sono mutualmente esclusivi, entrambe le loro probabilità condizionate saranno uguali a 0.\n

\n

\n {r`E \\cap F = \\emptyset \\Longleftrightarrow P(E|F) = P(F|E) = 0`}\n

\n \n
\n
\n \n

\n Si può sfruttare la formula inversa della probabilità condizionata per calcolare catene di intersezioni:\n

\n

\n {r`P(E_1 \\cap \\times \\cap E_n) = P(E_1) \\times P(E_2 | E_1) \\times \\dots \\times P(E_n | E_1 \\cap E_2 \\cap \\dots \\cap E_{n-1})`}\n

\n \n
\n
\n \n

\n La probabilità che si verifichi un evento è pari alla somma delle probabilità dell'evento stesso dati tutti gli eventi di una partizione.\n

\n

\n {r`P(F) = \\sum_{i} P(F|E_i) \\cdot P(E_i)`}\n

\n \n \n

\n La legge delle alternative funziona anche quando ad essere partizionato è un evento:\n

\n

\n {r`P(F|G) = \\sum_i P(F|E_i \\cap G) \\cdot P(E_i | G)`}\n

\n \n \n

\n Tramite la formula di Bayes possiamo risalire alla probabilità di un evento condizionato a un altro partendo dalla probabilità di quest'ultimo condizionato al primo:\n

\n

\n {r`P(E_h | F) = \\frac{P(F | E_h) \\cdot P(E_h)}{P(F)}`}\n

\n \n In pratica, invertiamo gli eventi.\n \n \n
\n
\n \n
\n \"eventi indipendenti a due a due\"\n
\n

\n Se due eventi sono indipendenti, sapere che uno dei due si è verificato non influisce sulle probabilità che si sia verificato l'altro.\n

\n

\n {r`P(E \\cap F) = P(E) \\cdot P(F) \\Longleftrightarrow P(E|F) = P(E) \\Longleftrightarrow P(F|E) = P(F)`}\n

\n \n \n
\n \"eventi indipendenti a tre a tre, a quattro a quattro, a cinque a cinque...\"\n
\n

\n Si può verificare l'indipendenza di più eventi alla volta:\n

\n

\n {r`P(E \\cap F \\cap G) = P(E) \\cdot P(F) \\cdot P(G)`}\n

\n

\n Eventi indipendenti a due a due non sono per forza indipendenti a tre a tre, e viceversa.\n

\n \n \n

\n Un insieme di n eventi è una famiglia di eventi indipendenti se, preso un qualsiasi numero di eventi da essa, essi risulteranno indipendenti.\n

\n \n Tutti gli eventi provenienti da essa saranno indipendenti sia a due a due, sia a tre a tre, sia a quattro a quattro, e così via!\n \n \n
\n
\n \n

\n Una funzione che fa corrispondere un numero reale a ogni possibile esito dello spazio campionario. {r`X(\\omega) : \\Omega \\to \\mathbb{R}`}.\n

\n \n Insieme di ripartizione}>\n

\n Ad ogni variabile aleatoria sono associati gli eventi {r`A_t = \\{ \\omega | X(\\omega) \\leq t \\}`}, che contengono tutti gli esiti a cui la variabile aleatoria associa un valore minore o uguale a t.\n

\n

\n Per definizione, tutte le variabili aleatorie devono rispettare questa condizione:\n

\n

\n {r`\\forall t \\in \\mathbb{R}, A_t \\in \\mathcal{F}`}\n

\n \n All'aumentare di t, l'insieme conterrà sempre più elementi.\n \n \n \n
\n \"supporto di X\"\n
\n

\n Il codominio della variabile aleatoria è il suo supporto.\n

\n

\n Per indicare che un valore x_0 appartiene al supporto di X, si usa la notazione X \\mapsto x_0.\n

\n \n
\n
\n \n

\n La funzione probabilità {r`p_X : X \\to [0, 1]`} di una variabile aleatoria discreta X è la funzione che associa ad ogni esito la sua probabilità:\n

\n

\n {r`\n p_X (x) = \\begin{cases}\n P([X = x]) \\quad se\\ X \\mapsto x \\\\\n 0 \\qquad \\qquad \\quad se\\ X \\not\\mapsto x\n \\end{cases} \n `}\n

\n \n \n

\n La funzione densità {r`f_X : X \\to [0, 1]`} di una variabile aleatoria continua X è l'equivalente continuo della funzione probabilità:\n

\n

\n {r`P([a < X \\leq b]) = \\int_a^b f_X (x) dx`}\n

\n

\n A differenza della funzione probabilità, è possibile che la funzione densità non esista per una certa variabile aleatoria.\n

\n \n Rappresenta \"quanta\" probabilità c'è in un'unità di x!\n \n \n
\n
\n \n

\n Ogni variabile aleatoria ha una funzione di ripartizione {r`F_X : \\mathbb{R} \\to [0, 1]`} associata, che rappresenta la probabilità che la variabile aleatoria assuma un valore minore o uguale a t:\n

\n

\n Si può dire che essa rappresenti la probabilità dell'evento {r`A_t`}:\n

\n

\n {r`\n F_X (t) = P(A_t) = \\begin{cases}\n \\sum_{i = 0}^{t} p_X (x_i) \\quad nel\\ discreto\\\\\n \\\\\n \\int_{-\\infty}^t f_X (x) dx \\quad nel\\ continuo\n \\end{cases}\n `}\n

\n \n \n
    \n
  • È sempre monotona crescente (non strettamente).

    • \n
  • Vale 0 a -\\infty e 1 a +\\infty.

    • \n
  • È continua da destra: {r`\\forall x_0 \\in \\mathbb{R}, F_X (x_0) = \\lim_{t \\to x^+_0} F_X (t)`}
    • \n
\n \n \n

\n Possiamo usare la funzione di ripartizione per calcolare la probabilità di un certo valore reale:\n

\n

\n {r`P([X = x_0]) = \\lim_{t \\to x^+_0} F_X (t) - \\lim_{t \\to x^-_0} F_X (t)`}\n

\n \n
\n
\n \n

\n Nel discreto basta abbinare un nuovo valore a ogni valore della variabile originale.\n

\n \n \n

\n Nel continuo applichiamo la formula dell'integrazione per sostituzione:\n

\n

\n {r`f_Y (y) = \\int_{g(a)}^{g(b)} f_X ( g^{-1} (x) ) g^{-2} (x)`}\n

\n \n \n

\n Trasformare variabili aleatorie è molto utile nell'informatica per creare distribuzioni partendo da una funzione random() che restituisce numeri da 0 a 1 con una distribuzione lineare.\n

\n \n
\n
\n \n

\n Ogni variabile aleatoria che ha una funzione di ripartizione e un supporto finito ha anche una media (o valore medio o atteso):\n

\n

\n {r`E(X) = \\int_0^{+infty} (1 - F_X (t)) dt - \\int_{-\\infty}^{0} F_X (t) dt`}\n

\n

\n Nel discreto, si può calcolare con:\n

\n

\n {r`E(X) = \\sum_i P(X = x_i) \\cdot x_i`}\n

\n

\n Nel continuo, si può calcolare con:\n

\n

\n {r`E(X) = \\int_{-\\infty}^{+\\infty} f_X (x) \\cdot x \\cdot dx`}\n

\n \n
\n
\n \n

\n Valore per cui la funzione probabilità o funzione densità è massima.\n

\n \n \n

\n Il quantile {r`x_{\\alpha}`} di ordine {r`0 \\leq \\alpha \\leq 1`} della variabile aleatoria X è il più piccolo numero tale che:\n

\n

\n {r`P([X < x_{\\alpha}]) \\leq \\alpha \\leq P([X \\leq x_{\\alpha}])`}\n

\n

\n\n

\n

\n Il quantile di ordine 0.5 {r`x_{0.5}`} è detto mediana.\n

\n

\n I quantili di ordine 0.25 {r`x_{0.25}`} e 0.75 {r`x_{0.75}`} sono detti quartili.\n

\n

\n I quantili di ordine {r`\\frac{n}{100}`} sono detti n-esima percentile.\n

\n \n \n

\n È un valore che indica quanto la variabile aleatoria si discosta generalmente dalla media:\n

\n

\n {r`Var(X) = E( (X - E(X) )^2 ) = E ( X^2 ) - (E(X))^2`}\n

\n \n
\n
\n \n

\n Data una variabile aleatoria non-negativa:\n

\n

\n {r`\\forall k > 0, P([X \\geq k]) \\leq \\frac{E(X)}{k}`}\n

\n

\n Divide in due parti ({r`P(X < k)`} e {r`P(X \\geq k)`}) la funzione X, la cui media risulterà uguale a:\n

\n

\n {r`E(X) = \\overline{k} \\cdot P(X < k) + k \\cdot P(X \\geq k)`}\n

\n \n \n
\n \"disuguaglianza di cebicev\"\n
\n

\n Se la variabile aleatoria X ha media e varianza, allora la probabilità che essa abbia un valore a più di {r`\\epsilon`} di distanza dal valore medio è minore o uguale a {r`\\frac{Var(X)}{\\epsilon^2}`}.\n

\n

\n {r`\\forall \\epsilon > 0, P([ \\left| X - E(X) \\right| \\geq \\epsilon]) \\leq \\frac{Var(X)}{\\epsilon^2}`}\n

\n

\n E anche:\n

\n

\n {r`\\forall \\epsilon > 0, P([ \\left| X - E(X) \\right| < \\epsilon]) \\geq 1 - \\frac{Var(X)}{\\epsilon^2}`}\n

\n \n Serve per semplificare i calcoli quando la funzione di ripartizione è difficile da calcolare!\n \n \n
\n
\n \n

\n Il momento k-esimo di una variabile aleatoria è:\n

\n

\n {r`\n \\mu_k = E ( X^k ) = \\begin{cases}\n \\sum_i x_i^k p_X (x_i) \\qquad nel\\ discreto\\\\\n \\\\\n \\int_{-\\infty}^{+\\infty} x^k f_X (x) dx \\qquad nel\\ continuo\n \\end{cases}`\n }\n

\n \n La media di una variabile aleatoria è anche il suo primo momento.\n \n \n \n

\n La funzione generatrice dei momenti è:\n

\n

\n {r`m_X (t) = E( e^{t \\cdot X} )`}\n

\n

\n Se due variabile aleatorie hanno la stessa funzione generatrice dei momenti, allora esse hanno la stessa distribuzione.\n

\n

\n E' la trasformata di Laplace della variabile aleatoria di X.\n

\n \n \n

\n La funzione caratteristica è:\n

\n

\n {r`H_X (t) = E ( e^{i \\cdot t \\cdot X} )`}\n

\n

\n Se due variabile aleatorie hanno la stessa funzione caratteristica, allora esse hanno la stessa distribuzione.\n

\n

\n E' la trasformata di Fourier della variabile aleatoria di X.\n

\n \n
\n
\n \n

\n Per dire che una variabile ha una certa distribuzione, si usa la notazione:\n

\n

\n {r`X \\sim Distribuzione()`}\n

\n \n \n

\n Una prova con solo due possibili esiti: successo e insuccesso.\n

\n \n \n

\n Una sequenza di prove di Bernoulli per le quali le probabilità di successo e fallimento rimangono invariate.\n

\n \n
\n
\n \n

\n Una variabile aleatoria che rappresenta una prova di Bernoulli:\n

\n
    \n
  • vale 1 in caso di successo.
    • \n
  • vale 0 in caso di insuccesso.
    • \n
\n

\n Il suo simbolo è {r`Ber(p)`}\n

\n \n \n

\n La distribuzione bernoulliana ha come densità:\n

\n

\n {r`\n f_X (k) : \\{0, 1\\} = \\begin{cases}\n p \\quad se\\ k = 1\\\\\n q \\quad se\\ k = 0\\\\\n 0 \\quad altrimenti\n \\end{cases} = p^x \\cdot q^{1 - k}`\n }\n

\n \n
\n
\n \n

\n Una variabile aleatoria che conta il numero di successi di n prove di uno schema di Bernoulli.\n

\n

\n Il suo simbolo è {r`Bin(n, p)`}.\n

\n \n \n

\n La binomiale ha come densità:\n

\n

\n {r`f_X (k) : \\{0..n\\} = \\binom{n}{k} \\cdot p^k \\cdot q^{n - k}`}\n

\n \n \n

\n La funzione generatrice dei momenti della binomiale è:\n

\n

\n {r`m_X (t) = (q + p \\cdot e^t) ^ n`}\n

\n

\n La media di una binomiale è:\n

\n

\n {r`E(X) = n \\cdot p`}\n

\n

\n La varianza di una binomiale è:\n

\n

\n {r`Var(X) = n \\cdot p \\cdot q`}\n

\n \n
\n
\n \n

\n Una variabile aleatoria che conta il numero di prove in uno schema di Bernoulli fino alla comparsa del primo successo.\n

\n

\n Il suo simbolo è Geo(p).\n

\n \n \n

\n La geometrica ha come densità:\n

\n

\n {r`f_X (k) : \\mathbb{N} = q^{k - 1} p`}\n

\n \n \n

\n La funzione generatrice dei momenti della geometrica è:\n

\n

\n {r`m_X (t) = \\frac{p \\cdot e^t}{1 - q \\cdot e^t}`}\n

\n

\n La media della geometrica è:\n

\n

\n {r`E(X) = \\frac{1}{p}`}\n

\n

\n La varianza della geometrica è:\n

\n

\n {r`Var(X) = \\frac{q}{p^2}`}\n

\n \n \n

\n La geometrica non tiene conto degli eventi avvenuti in passato: ha la proprietà dell'assenza di memoria:\n

\n

\n {r`P([X = i + j | X > i ]) = P([X = j])`}\n

\n \n Ovvero, riscalando opportunamente l'asse Y posso prendere come 0 qualsiasi punto dell'asse X.\n \n \n
\n
\n \n

\n Una variabile aleatoria che conta il numero di prove in uno schema di Bernoulli necessarie perchè si verifichi l'n-esimo successo.\n

\n

\n Il suo simbolo è {r`\\overline{Bin}(n, p)`}.\n

\n \n \n

\n La binomiale negativa ha come densità:\n

\n

\n {r`f_X (k) : \\{ n .. +\\infty \\} \\in \\mathbb{N} = \\binom{k - 1}{n - 1} \\cdot p^n \\cdot q^{k - n} `}\n

\n \n \n

\n

\n La funzione generatrice dei momenti della binomiale negativa è:\n

\n

\n {r`m_X (t) : \\{ t < ln(\\frac{1}{q}) \\} = \\left( \\frac{p \\cdot e^t}{1 - q \\cdot e^t} \\right) ^n`}\n

\n

\n La media della binomiale negativa è:\n

\n

\n {r`E(X) = \\frac{n}{p}`}\n

\n

\n La varianza della binomiale negativa è:\n

\n

\n {r`Var(X) = \\frac{n \\cdot q}{p^2}`}\n

\n

\n \n
\n
\n \n

\n Una variabile aleatoria che conta il numero k di insuccessi consecutivi in uno schema di Bernoulli:\n

\n

\n Il suo simbolo rimane {r`Geo(p)`}.\n

\n \n \n

\n La geometrica traslata ha come densità:\n

\n

\n {r`f_X (k) : \\mathbb{N} = p \\cdot q^k `}\n

\n \n \n

\n La funzione generatrice dei momenti della geometrica traslata è:\n

\n

\n {r`m_X (t) : \\left\\{ t < ln \\left( \\frac{1}{q} \\right) \\right\\} = \\frac{p}{1 - q \\cdot e^t}`}\n

\n

\n La media della geometrica traslata è:\n

\n

\n {r`E(X) = \\frac{q}{p}`}\n

\n

\n La varianza della geometrica è:\n

\n

\n {r`Var(X) = \\frac{q}{p^2}`}\n

\n \n \n

\n La geometrica traslata non tiene conto degli eventi avvenuti in passato: ha la proprietà dell'assenza di memoria:\n

\n

\n {r`P([X = i + j | X > i ]) = P([X = j])`}\n

\n \n Ovvero, riscalando opportunamente l'asse Y posso prendere come 0 qualsiasi punto dell'asse X.\n \n \n
\n
\n \n

\n Una variabile aleatoria che conta il numero di insuccessi in uno schema di Bernoulli prima che si verifichi l'n-esimo successo.\n

\n

\n Il suo simbolo rimane {r`\\overline{Bin}(n, p)`}.\n

\n \n \n

\n La binomiale negativa traslata ha come densità:\n

\n

\n {r`f_X (k) : \\mathbb{N} = \\binom{k + n - 1}{n - 1} \\cdot p^n \\cdot q^k `}\n

\n \n \n

\n

\n La funzione generatrice dei momenti della binomiale negativa traslata è:\n

\n

\n {r`m_X (t) : \\left\\{ t < ln \\left( \\frac{1}{q} \\right) \\right\\} = \\left( \\frac{p \\cdot e^t}{1 - q \\cdot e^t} \\right) ^n`}\n

\n

\n La media della binomiale negativa traslata è:\n

\n

\n {r`E(X) = \\frac{n \\cdot q}{p}`}\n

\n

\n La varianza della binomiale negativa traslata è:\n

\n

\n {r`Var(X) = \\frac{n \\cdot q}{p^2}`}\n

\n

\n \n
\n
\n \n

\n Una variabile aleatoria che, sapendo il numero di successi K e di insuccessi N-K, conta quanti successi si otterrebbero se se ne estraessero n in blocco.\n

\n

\n Il suo simbolo è Ipe(N, K, n).\n

\n \n \n

\n La ipergeometrica ha come densità:\n

\n

\n {r`f_X (k) : \\{0..n\\} \\in \\mathbb{N} = \\frac{\\binom{K}{k} \\cdot \\binom{N - K}{n - k}}{\\binom{N}{n}}`}\n

\n \n \n

\n

\n La funzione generatrice dei momenti della ipergeometrica è trascurabile.\n

\n

\n La media della ipergeometrica è:\n

\n

\n {r`E(X) = n \\cdot \\frac{K}{N}`}\n

\n

\n La varianza della ipergeometrica è:\n

\n

\n {r`Var(X) = n \\cdot \\frac{K}{N} \\cdot \\frac{N - K}{N} \\cdot \\frac{N - n}{N - 1}`}\n

\n

\n \n
\n
\n \n

\n Una variabile aleatoria che soddisfa tutte le seguenti caratteristiche:\n

\n
    \n
  • Binomiale: {r`X \\sim Bin(n, p)`}
    • \n
  • Il numero di prove tende a infinito: {r`n \\to +\\infty`}
    • \n
  • La probabilità di successo tende a 0: {r`p \\to 0`}
    • \n
  • La media è finita: {r`E(X) = n \\cdot p \\to \\mu \\neq 0`}
    • \n
\n

\n Il suo simbolo è {r`Poi(\\mu)`}\n

\n \n \n

\n La poissoniana ha come densità:\n

\n

\n {r`f_X (k) : \\mathbb{N} = \\frac{e^{-\\mu} \\cdot \\mu^k}{k!}`}\n

\n \n \n

\n

\n La funzione generatrice dei momenti della poissoniana è:\n

\n

\n {r`m_X (t) = e^{\\mu \\cdot (e^t - 1)}`}\n

\n

\n La media della poissoniana è:\n

\n

\n {r`E(X) = \\mu`}\n

\n

\n La varianza della poissoniana è:\n

\n

\n {r`Var(X) = \\mu`}\n

\n

\n Gli altri momenti della poissoniana sono:\n

\n
    \n
  1. {r`E(X^2) = \\mu^2 + \\mu`}
    2. \n
\n

\n \n
\n
\n \n

\n Una successione di arrivi avvenuti in un certo arco temporale che:\n

\n
    \n
  • non sono sovrapposti.
    • \n
  • hanno intensità {r`\\lambda`} costante.
    • \n
  • avvengono indipendentemente gli uni dagli altri.
    • \n
\n \n \n

\n Una variabile aleatoria N_t che conta il numero di arrivi di uno schema di Poisson di intensità {r`\\lambda`} in un intervallo di tempo di durata t.\n

\n

\n E' una distribuzione poissoniana con {r`\\mu = t \\cdot \\lambda`}: {r`Poi(t \\cdot \\lambda)`}\n

\n \n E' paragonabile a una bernoulliana: ogni successo corrisponde a un arrivo, mentre il tempo è il numero di prove effettuate (ma nel continuo).\n \n \n
\n
\n \n

\n Una variabile aleatoria che conta il tempo diwidehattesa prima del primo arrivo di un processo di Poisson di intensità {r`\\lambda`}.\n

\n

\n Il suo simbolo è {r`Esp(\\lambda)`}.\n

\n \n \n

\n L'esponenziale ha come densità:\n

\n

\n {r`\n f_X (x) = \\begin{cases}\n 0 \\qquad \\qquad x < 0\\\\\n \\lambda \\cdot e^{-\\lambda \\cdot x} \\quad x > 0\n \\end{cases}`\n }\n

\n

\n L'esponenziale ha come funzione di ripartizione:\n

\n

\n {r`\n F_X (t) = \\begin{cases}\n 0 \\qquad \\qquad t < 0\\\\\n 1 - e^{-\\lambda \\cdot t} \\quad t \\geq 0\n \\end{cases}`\n }\n

\n \n \n

\n La funzione generatrice dei momenti dell'esponenziale è:\n

\n

\n {r`m_X (t) : \\{ t | t < \\lambda \\} \\in \\mathbb{R} = \\frac{\\lambda}{\\lambda - t}`}\n

\n

\n La media dell'esponenziale è:\n

\n

\n {r`E(X) = \\frac{1}{\\lambda}`}\n

\n

\n La varianza dell'esponenziale è:\n

\n

\n {r`Var(X) = \\frac{1}{\\lambda^2}`}\n

\n \n \n

\n L'esponenziale non tiene conto degli eventi avvenuti in passato: ha la proprietà dell'assenza di memoria:\n

\n

\n {r`P([X > s + t | X > s]) = P([X > t])`}\n

\n \n Ovvero, riscalando opportunamente l'asse Y posso prendere come 0 qualsiasi punto dell'asse X.\n \n \n
\n
\n \n

\n Una variabile aleatoria che conta il tempo diwidehattesa prima dell'n-esimo arrivo di un processo di Poisson di intensità {r`\\lambda`}.\n

\n

\n Il suo simbolo è {r`\\Gamma(n, \\lambda)`}.\n

\n \n \n

\n La legge gamma ha come densità:\n

\n

\n {r`\n f_X (x) = \\begin{cases}\n 0 \\qquad \\qquad \\qquad \\qquad \\qquad x < 0\\\\\n \\frac{1}{(n-1)!} \\cdot \\lambda^n \\cdot x^{n-1} \\cdot e^{-\\lambda \\cdot x} \\quad k > 0\n \\end{cases}`\n }\n

\n \n \n

\n

\n La funzione generatrice dei momenti della legge gamma è:\n

\n

\n {r`m_X (t) : ( t < \\lambda ) \\in \\mathbb{R} = \\left( \\frac{\\lambda}{\\lambda - t} \\right) ^\\alpha`}\n

\n

\n La media della legge gamma è:\n

\n

\n {r`E(X) = \\frac{\\alpha}{\\lambda}`}\n

\n

\n La varianza della legge gamma è:\n

\n

\n {r`Var(X) = \\frac{\\alpha}{\\lambda^2}`}\n

\n

\n \n
\n
\n \n

\n Una variabile aleatoria che può assumere qualsiasi valore in un intervallo {r`[a, b]`} in modo equiprobabile.\n

\n

\n Il suo simbolo è {r`Uni(a, b)`}\n

\n

\n Su di essa vale la seguente proprietà:\n

\n

\n {r`P(X \\in (c, d)) = \\frac{d - c}{b - a}`}\n

\n \n \n

\n La distribuzione uniforme ha come densità:\n

\n

\n {r`\n f_X (x) = \\begin{cases}\n \\frac{1}{b - a} \\qquad a \\leq x \\leq b\\\\\n 0 \\qquad \\quad altrimenti \n \\end{cases}\n `}\n

\n

\n La distribuzione uniforme ha come funzione di ripartizione:\n

\n

\n {r`\n f_X (x) = \\begin{cases}\n 0 \\qquad \\quad x < a \n \\frac{1}{b - a} \\qquad a \\leq x \\leq b\\\\\n 1 \\qquad \\quad x > b\n \\end{cases}`\n }\n

\n \n \n

\n

\n La funzione generatrice dei momenti della distribuzione uniforme è:\n

\n

\n {r`m_X (t) = \\frac{e^{b \\cdot t} - e^{a \\cdot t}}{(b - a) \\cdot t}`}\n

\n

\n La media della distribuzione uniforme è:\n

\n

\n {r`E(X) = \\frac{a + b}{2}`}\n

\n

\n La varianza della distribuzione uniforme è:\n

\n

\n {r`Var(X) = \\frac{(b - a)^2}{12}`}\n

\n

\n \n
\n
\n \n

\n Una variabile aleatoria con una specifica distribuzione.\n

\n

\n Il suo simbolo è {r`Nor(\\mu, \\sigma^2)`}.\n

\n \n \\mu e \\sigma^2 sono rispettivamente la media e la varianza della distribuzione!\n \n \n \n

\n La distribuzione normale ha come densità:\n

\n

\n {r`f_X (x) = \\frac{e^{-\\frac{(x - \\mu)^2}{2 \\sigma^2}}}{\\sqrt{2 \\pi \\cdot \\sigma^2}}`}\n

\n \n \n

\n

\n La funzione generatrice dei momenti della distribuzione normale è:\n

\n

\n {r`m_X (t) = e^{\\mu \\cdot t + \\frac{\\sigma^2 \\cdot t^2}{2}}`}\n

\n

\n La media della distribuzione normale è:\n

\n

\n {r`E(X) = \\mu`}\n

\n

\n La varianza della distribuzione normale è:\n

\n

\n {r`Var(X) = \\sigma^2`}\n

\n

\n \n
\n
\n \n

\n Qualsiasi normale può essere trasformata in qualsiasi altra normale:\n

\n

\n {r`X \\sim Nor(m, v^2) \\implies \\alpha X + \\beta \\sim Nor(\\alpha m + \\beta, (\\alpha v)^2)`}\n

\n \n \n

\n La distribuzione normale standard Z è:\n

\n

\n Z \\sim Nor(0, 1)\n

\n

\n La sua funzione di ripartizione è detta {r`\\phi(z)`} e vale:\n

\n

\n {r`F_Z(z) = \\phi(z) = \\frac{1}{\\sqrt{2 \\pi}} \\int_{-\\infty}^{z} e^{-\\frac{x^2}{2}} dx`}\n

\n \n \n

\n Da un quantile {r`z_\\alpha`} della normale standard è possibile risalire allo stesso quantile di qualsiasi altra normale:\n

\n

\n {r`x_\\alpha = \\mu + z_\\alpha \\cdot \\sqrt{\\sigma^2}`}\n

\n \n
\n
\n \n

\n La distribuzione normale ha una particolare relazione con la distribuzione Gamma:\n

\n

\n {r`Z^2 \\sim \\chi^2 (v = 1)`}\n

\n \n \n
\n \"chi-quadro a un grado di libertà\"\n
\n

\n Esiste una distribuzione Gamma particolare:\n

\n

\n {r`\\Gamma \\left( \\frac{1}{2}, \\frac{1}{2} \\right) = \\chi^2 (v = 1)`}\n

\n

\n Più chi-quadro possono essere sommate per aumentare i loro gradi di libertà:\n

\n

\n {r`\\chi^2 (n) + \\chi^2 (m) = \\chi^2 (n + m)`}\n

\n \n \n

\n Un'altra funzione particolare è la funzione T di Student:\n

\n

\n {r`T(v) = \\frac{Nor(0, 1)}{\\sqrt{\\frac{\\chi^2(v)}{v}}}`}\n

\n \n
\n
\n \n

\n La binomiale è come una ipergeometrica ma con ripetizioni, quindi per valori molto grandi di N rispetto a n, si può dire che:\n

\n

\n {r`Ipe(N, K, n) \\approx Bin(n, \\frac{K}{N})`}\n

\n \n \n

\n La binomiale non è altro che una poissoniana a tempo discreto, quindi, se n è grande e n \\cdot p è nell'ordine di grandezza delle unità, allora:\n

\n

\n {r`Bin(n, p) \\approx Poi(n \\cdot p)`}\n

\n \n \n

\n Per il Teorema di De Moivre-Laplace, se una binomiale ha una n grande e p non vicina a 0 o 1, si può approssimare con:\n

\n

\n {r`Bin(n, p) \\approx Nor(n \\cdot p, n \\cdot p \\cdot q)`}\n

\n \n \n

\n Passando da una variabile discreta X a una continua Y, per ogni valore discreto k la probabilità viene \"spalmata\" su tutto l'intervallo {r`(k - \\frac{1}{2}, k + \\frac{1}{2})`}:\n

\n
    \n
  • {r`P(X < k) \\simeq P(Y \\leq k - \\frac{1}{2})`}
    • \n
  • {r`P(X \\leq k) \\simeq P(Y \\leq k + \\frac{1}{2})`}
    • \n
  • {r`P(X \\geq k) \\simeq P(Y \\geq k - \\frac{1}{2})`}
    • \n
  • {r`P(X > k) \\simeq P(Y \\geq k + \\frac{1}{2})`}
    • \n
\n \n
\n
\n \n

\n Un vettore composto da variabili aleatorie.\n

\n

\n Il suo simbolo generalmente è {r`\\boldsymbol{X}`} oppure {r`X, Y`}.\n

\n \n \n

\n I vettori aleatori hanno più funzioni di ripartizione che si differenziano in base al numero di parametri.\n

\n

\n Se il numero di parametri coincide con la dimensione del vettore aleatorio, allora la funzione sarà una funzione di ripartizione congiunta:\n

\n

\n {r`F_{X, Y} (x, y) = P(X \\leq x, Y \\leq y)`}\n

\n

\n Se il numero di parametri è minore della dimensione del vettore aleatorio, allora la funzione sarà una funzione di ripartizione marginale:\n

\n

\n {r`F_X (x) = P(X \\leq x) = \\lim_{y \\to +\\infty} F_{X, Y} (x, y)`}\n

\n \n \n

\n I vettori aleatori discreti hanno più densità che si differenziano in base al numero di parametri.\n

\n

\n Se il numero di parametri coincide con la dimensione del vettore aleatorio, allora la funzione sarà una densità congiunta:\n

\n

\n {r`p_{X, Y} (x, y) = P(X = x, Y = y)`}\n

\n

\n Se il numero di parametri è minore della dimensione del vettore aleatorio, allora la funzione sarà una densità marginale:\n

\n

\n {r`p_X (x) = \\sum_j p_{X, Y} (x_i, y_j)`}\n

\n \n
\n
\n \n

\n Più variabili aleatorie sono indipendenti se, per qualsiasi scelta di intervalli A_i:\n

\n

\n {r`P(X_1 \\in A_1, \\dots, X_n \\in A_n) = P(X_1 \\in A_1) \\times \\dots \\times P(X_n \\in A_n)`}\n

\n \n \n

\n E' possibile calcolare la media di qualsiasi funzione g(X, Y) avente elementi del vettore come variabili:\n

\n

\n {r`E(g(X, Y)) = \\sum_{i, j} g(x_i, y_i) \\cdot p_{X, Y} (x_i, y_i)`}\n

\n \n Solitamente si calcola la media di x \\cdot y.\n \n

\n Le medie di più variabili aleatorie si possono sommare:\n

\n

\n {r`E(X + Y) = E(X) + E(Y)`}\n

\n \n
\n
\n \n

\n Un operatore che misura la correlazione di due variabili aleatorie.\n

\n

\n Si calcola con il valore atteso dei prodotti delle distanze dalla media:\n

\n

\n {r`Cov(X, Y) = E((X - E(X) \\cdot (Y - E(Y)) = E(XY) - E(X) \\cdot E(Y)`}\n

\n

\n Ha diverse proprietà:\n

\n
    \n
  • Il suo valore nullo è 0: {r`Cov(X, \\alpha) = 0`}
    • \n
  • E' commutativa: {r`Cov(X, Y) = Cov(Y, X)`}
    • \n
  • E' semplificabile: {r`Cov(X, X) = Var(X)`}
    • \n
  • E' lineare: {r`Cov(\\alpha X, \\beta Y) = \\alpha \\cdot \\beta \\cdot Cov(X, Y)`}
    • \n
  • E' distributiva: {r`Cov(X + Y, V + W) = Cov(X, Y) + Cov(X, W) + Cov(Y, V) + Cov(Y, W)`}
    • \n
\n \n \n

\n Due variabili sono variabili incorrelate se:\n

\n

\n {r`Cov(X, Y) = 0`}\n

\n

\n Variabili indipendenti sono sempre incorrelate.\n

\n \n \n

\n Una matrice {r`\\boldsymbol{C_X}`} che contiene la covarianza tra tutte le variabili di un vettore aleatorio {r`\\boldsymbol{X}`}:\n

\n

\n {r`\n \\boldsymbol{C_X} = \n \\begin{bmatrix}\n Var(X_1) & Cov(X_1, X_2) & Cov(X_1, X_3)\\\\\n Cov(X_2, X_1) & Var(X_2) & Cov(X_2, X_3)\\\\\n Cov(X_3, X_1) & Cov(X_3, X_2) & Var(X_3)\n \\end{bmatrix}\n `}\n

\n

\n E' sempre simmetrica e semidefinita positiva (tutti gli autovalori sono \\geq 0.\n

\n \n \n

\n Un valore che misura come due variabili aleatorie sono correlate:\n

\n

\n {r`\\rho_{X, Y} = \\frac{Cov(X, Y)}{\\sqrt{Var(X)} \\cdot \\sqrt{Var(Y)}}`}\n

\n

\n E' sempre compreso tra -1 e 1:\n

\n

\n {r`-1 \\leq \\rho_{X, Y} \\leq 1`}\n

\n

\n Vale esattamente -1 o 1 solo se esiste un legame lineare tra le due variaibli:\n

\n

\n {r`Y = a X + b \\Longleftrightarrow | \\rho_{X, Y} | = 1`}\n

\n \n \n

\n La varianza di due variabili aleatorie sommate è:\n

\n

\n {r`Var(X + Y) = Var(X) + Var(Y) + 2 \\cdot Cov(X, Y)`}\n

\n \n Si dimostra applicando le proprietà della covarianza!\n \n

\n Se più variabili aleatorie X_i sono indipendenti ({r`Cov(X, Y) = 0`}), allora:\n

\n

\n {r`Var \\left( \\sum_i X_i \\right) = \\sum_i Var(X_i)`}\n

\n \n
\n
\n \n

\n Una n-pla di variabili aleatorie con la stessa distribuzione della variabile aleatoria X (\"popolazione\") ma indipendenti tra loro.\n

\n \n Le variabili aleatorie sono come un lazy-load in programmazione; quando ci sarà bisogno del loro valore numerico, esse si realizzeranno nel loro valore.\n \n \n \n

\n Il valore dato dalla media aritmetica degli n elementi del campione elevati alla potenza k:\n

\n

\n {r`M^{(k)}_n = \\frac{1}{n} \\cdot \\sum_{i = 1}^n X_i^k `}\n

\n

\n Il momento campionario di primo ordine è la media campionaria {r`\\overline{X}_n`}.\n

\n \n \n

\n La media aritmetica dello scarto quadratico medio degli elementi del campione.\n

\n

\n Se è noto il valore medio {r`m = E(X)`} di X:\n

\n

\n {r`S_0^2 = \\frac{1}{n} \\cdot \\sum_{i = 1}^n (X_i - m)^2 = M_n^(2) - 2 \\cdot m \\cdot \\overline{X}_n + m^2`}\n

\n

\n Altrimenti:\n

\n

\n {r`S_n^2 = \\frac{1}{n - 1} \\cdot \\sum_{i = 1}^n (X_i - \\overline{X}_n)^2 = \\frac{1}{n - 1} \\cdot ( n \\cdot M_2^{(2)} - n \\cdot \\overline{X}_n^2)`}\n

\n \n
\n
\n \n

\n Se calcoliamo la media della media campionaria, risulterà vero che:\n

\n

\n {r`E(\\overline{X}_n) = E(X)`}\n

\n \n Quindi, è possibile usare i campioni per trovare la media di una variabile aleatoria!\n \n \n \n

\n Se calcoliamo la varianza della media campionaria, risulterà vero che:\n

\n

\n {r`Var(\\overline{X}_n) = \\frac{Var(X)}{n}`}\n

\n \n Quindi, possiamo stimare l'errore della media calcolata tramite campioni!\n \n \n \n

\n Se calcoliamo la media della varianza campionaria, risulterà vero che:\n

\n

\n {r`E(S_0^2) = E(S_n^2) = Var(X)`}\n

\n \n Quindi, possiamo stimare l'errore della media calcolata tramite campioni!\n \n \n
\n
\n \n

\n Se la popolazione X ha una distribuzione normale ({r`X \\sim Nor(\\mu, \\sigma^2)`})...\n

\n \n \n

\n ...allora sappiamo anche la distribuzione della media campionaria!\n

\n

\n {r`\\overline{X}_n \\sim Nor \\left( \\mu, \\frac{\\sigma^2}{n} \\right)`}\n

\n \n \n

\n ...e anche della varianza campionaria!\n

\n

\n {r`S_0^2 \\sim \\frac{\\sigma^2}{n} \\cdot \\chi^2 (n)`}\n

\n

\n {r`S_n^2 \\sim \\frac{\\sigma^2}{n - 1} \\cdot \\chi^2 (n-1)`}\n

\n \n \n

\n ...e che media campionaria e varianza campionaria sono indipendenti tra loro!\n

\n \n
\n
\n \n

\n Se la successione di variabili aleatorie X_n all'infinito ha la stessa funzione di ripartizione della popolazione X, allora essa converge in distribuzione.\n

\n

\n {`\\\\lim_{n \\\\to +\\\\infty} F_{X_n} (x) = F_X (x) \\\\implies X_n \\\\xrightarrow{d} X`}\n

\n \n \n

\n Se la successione di variabili aleatorie X_n all'infinito ha la stessa probabilità della popolazione X, allora essa converge in probabilità.\n

\n

\n {`\\\\forall \\\\epsilon > 0, \\\\lim_{n \\\\to +\\\\infty} P( | X_n - X | < \\\\epsilon) = 1 \\\\implies X_n \\\\xrightarrow{p} X`}\n

\n \n \n

\n Se la successione di variabili aleatorie X_n all'infinito ha la stessa probabilità a della popolazione X, allora essa converge quasi certamente.\n

\n

\n {`\\\\forall \\\\epsilon > 0, P \\left( \\\\lim_{n \\\\to +\\\\infty} | X_n - X | < \\\\epsilon) \\right) = 1 \\\\implies X_n \\\\xrightarrow{qc} X`}\n

\n \n \n

\n Se la successione di variabili aleatorie X_n all'infinito ha la media del quadrato della distanza tra la successione e la popolazione X uguale a 0, allora essa converge in media quadratica.\n

\n

\n {`\\\\lim_{n \\\\to +\\\\infty} E( | X_n - X |^2 = 0 \\\\implies X_n \\\\xrightarrow{mq} X`}\n

\n \n \n

\n {`\n \\\\begin{matrix}\n X_n \\\\xrightarrow{mq} X\\\\\\\\\n X_n \\\\xrightarrow{qc} X\n \\\\end{matrix} \\\\implies X_n \\\\xrightarrow{p} X \\\\implies X_n \\\\xrightarrow{d} X`\n }\n

\n

\n In più:\n

\n

\n {`X_n \\\\xrightarrow{p} x \\\\Longleftrightarrow X_n \\\\xrightarrow{d} x`}\n

\n \n
\n
\n \n

\n La successione delle medie campionarie {r`\\overline{X}_n`} converge in probabilità alla media della popolazione {r`E(X)`}, se essa esiste.\n

\n

\n {`\\\\overline{X}_n \\\\xrightarrow{p} X`}\n

\n

\n Ovvero:\n

\n

\n {r`\\forall \\epsilon > 0, \\lim_{n \\to +\\infty} P( | \\overline{X}_n - E(X) | < \\epsilon) = 1`}\n

\n

\n {r`P( | \\overline{X}_n - E(X) | < \\epsilon) \\to 1`}\n

\n \n \n

\n La successione delle medie campionarie {r`\\overline{X}_n`} converge quasi certamente alla media della popolazione {r`E(X)`}, se essa esiste.\n

\n

\n {`\\\\overline{X}_n \\\\xrightarrow{qc} X`}\n

\n

\n Ovvero:\n

\n

\n {r`\\forall \\epsilon > 0, P \\left( \\lim_{n \\to +\\infty} | \\overline{X}_n - E(X) | < \\epsilon \\right) = 1`}\n

\n \n Dimostra che l'interpretazione frequentista della probabilità è valida!\n \n \n
\n
\n \n

\n La successione delle medie campionarie {r`\\overline{X}_n`} converge in distribuzione a {r`Nor(0, 1) = \\Phi()`}.\n

\n

\n {r`\\overline{X}_n \\approx Nor \\left(E(X), \\frac{Var(X)}{n} \\right)`}\n

\n

\n Ovvero:\n

\n

\n {r`\\forall x \\in \\mathbb{R}, \\lim_{n \\to +\\infty} P \\left( \\frac{\\overline{X}_n - E(X)}{\\sqrt{\\frac{Var(X)}{n}}} \\leq x \\right) = \\Phi(x)`}\n

\n \n
\n
\n \n

\n E' una somma di bernoulliane, e quindi si approssima a una normale:\n

\n

\n {r`Bin(n, p) \\approx Nor(n \\cdot p, n \\cdot p \\cdot q)`}\n

\n \n \n

\n E' una somma di geometriche, e quindi si approssima a una normale:\n

\n

\n {r`\\overline{Bin} (n, p) \\approx Nor \\left( \\frac{n}{p}, \\frac{n \\cdot (1 - p)}{p^2} \\right)`}\n

\n \n \n

\n E' una somma di altre poissoniane, e quindi si approssima a una normale:\n

\n

\n {r`Poi(\\lambda) \\approx Nor(\\lambda, \\lambda)`}\n

\n \n \n

\n E' una somma di esponenziali, e quindi si approssima a una normale:\n

\n

\n {r`\\Gamma (\\alpha, \\lambda) \\approx Nor \\left( \\frac{\\alpha}{\\lambda}, \\frac{\\alpha}{\\lambda^2} \\right)`}\n

\n \n \n

\n Se n è grande, allora:\n

\n

\n {r`Y = \\sum_{i=1}^{n} X_i`}\n

\n \n
\n
\n \n

\n Per indicare parametri sconosciuti di una legge si usa \\theta.\n

\n \n \n

\n Una variabile aleatoria funzione di un campione:\n

\n

\n {r`T(\\boldsymbol{X})`}\n

\n \n Ad esempio, sono statistiche media e varianza campionaria, così come il campione stesso {r`T(\\boldsymbol{X}) = \\boldsymbol{X}`}.\n \n \n
\n
\n \n

\n Una statistica T_n ottenuta da n osservazioni, che stimi i parametri di una legge e sia indipendente da essi.\n

\n \n \n

\n Uno stimatore è corretto se il suo valore atteso coincide con quello dei parametri che stima:\n

\n

\n {r`E(T_n) = \\theta`}\n

\n \n \n

\n Uno stimatore è asintoticamente corretto se, per infinite osservazioni, il suo valore atteso coincide con quello dei parametri che stima:\n

\n

\n {r`\\lim_{n \\to +\\infty} E(T_n) = \\theta`}\n

\n \n \n

\n Uno stimatore è consistente in media quadratica se:\n

\n

\n {r`\\lim_{n \\to +\\infty} E((T_n - \\theta)^2) = 0`}\n

\n \n \n

\n Uno stimatore è consistente in probabilità se:\n

\n

\n {r`\\forall \\epsilon > 0, \\lim_{n \\to +\\infty} P( |T_n - \\theta| < \\epsilon) = 1`}\n

\n \n \n

\n Uno stimatore è asintoticamente normale se:\n

\n

\n {r`\\lim_{n \\to +\\infty} \\frac{T_n - E(T_n)}{\\sqrt{Var(T_n)}} \\sim Nor(0, 1)`}\n

\n \n
\n
\n \n

\n Si può usare il metodo dei momenti per ottenere uno stimatore di una popolazione X.\n

\n

\n Lo stimatore di {r`\\theta`} così ottenuto sarà indicato aggiungendo un cappellino e una M a \\theta: {r`\\widehat{\\theta}_M`}\n

\n

\n Visto che:\n

\n
    \n
  • {r`\\theta = g(E(X))`}
    • \n
  • {r`\\widehat{E(X)} = \\overline{X}_n`}
    • \n
\n

\n Allora:\n

\n

\n {r`\\widehat{\\theta}_M = g( \\overline{X}_n )`}\n

\n

\n Se {r`\\theta`} non è esprimibile in termini di {r`E(X)`}, si possono usare i momenti successivi {r`M_n^2`}, {r`M_n^3`}, {r`M_n^3`}...\n

\n \n
\n
\n \n

\n Si può usare il metodo della massima verosomiglianza per ottenere uno stimatore di una popolazione X.\n

\n

\n Lo stimatore di {r`\\theta`} così ottenuto sarà indicato aggiungendo un cappellino e una L a \\theta: {r`\\widehat{\\theta}_L`}\n

\n

\n Consiste nel trovare il massimo assoluto {r`\\widehat{\\theta}_L`} della la funzione di verosomiglianza {r`L`}:\n

\n

\n {r`L(x_1, ..., x_n; \\theta) = \\prod_{i=1}^n f_X(x_i; \\theta)`}\n

\n

\n Gli stimatori di massima verosomiglianza sono asintoticamente corretti, consistenti in probabilità e asintoticamente normali.\n

\n \n \n

\n Gli stimatori di massima verosomiglianza godono delle seguenti proprietà:\n

\n
    \n
  • Sono asintoticamente corretti.
    • \n
  • Sono consistenti in probabilità.
    • \n
  • Sono asintoticamente normali.
    • \n
  • Sono invarianti: {r`\\widehat{g(\\theta)}_L = g(\\widehat{\\theta}_L)`}
    • \n
\n \n
\n
\n \n

\n Per il metodo dei momenti oppure per il metodo della massima verosomiglianza:\n

\n

\n {r`\\widehat{p}_M = \\widehat{p}_L = \\overline{X}_n`}\n

\n \n \n

\n Per il metodo dei momenti oppure per il metodo della massima verosomiglianza:\n

\n

\n {r`\\widehat{\\mu}_M = \\widehat{\\mu}_L = \\overline{X}_n`}\n

\n \n \n

\n Per il metodo dei momenti oppure per il metodo della massima verosomiglianza:\n

\n

\n {r`\\widehat{\\lambda}_M = \\widehat{\\lambda}_L = \\frac{1}{\\overline{X}_n}`}\n

\n \n \n

\n Per il metodo della massima verosomiglianza:\n

\n
    \n
  • {r`\\widehat{\\mu}_L = \\overline{X}_n`}

    • \n
  • {r`\\widehat{\\sigma^2}_L = \\frac{\\sum (X_i - \\overline{X}_n)^2 }{n}`}
    • \n
\n \n
\n
\n \n
\n \"intervallo di confidenza al 95%\"\n
\n

\n L'intervallo di valori di \\theta all'interno del quale siamo \"più o meno sicuri\" si trovi il valore effettivo:\n

\n

\n L'intervallo di confidenza a N della stima {r`\\widehat{W}`} è l'intervallo ]a, b[ tale che:\n

\n

\n {r`P( a < W < b ) = N`}\n

\n

\n Può anche essere unilatero nel caso limiti la stima in una sola direzione, positiva o negativa.\n

\n \n
\n
\n \n

\n Se conosciamo la varianza di una normale, allora possiamo ricavare velocemente gli intervalli di confidenza all'\\alpha% con queste formule:\n

\n
    \n
  • Intervalli bilateri: {r`\\mu \\in \\left[ \\overline{x}_n - z_{1 - \\frac{\\alpha}{2}} \\cdot \\sqrt{\\frac{\\sigma^2}{n}}, \\overline{x}_n + z_{1 - \\frac{\\alpha}{2}} \\cdot \\sqrt{\\frac{\\sigma^2}{n}} \\right]`}
    • \n
  • Intervallo unilatero da sinistra: {r`\\mu \\in \\left( -\\infty, \\overline{x}_n + z_{1 - \\frac{\\alpha}{2}} \\cdot \\sqrt{\\frac{\\sigma^2}{n}} \\right]`}
    • \n
  • Intervallo unilatero da destra: {r`\\mu \\in \\left[ \\overline{x}_n - z_{1 - \\frac{\\alpha}{2}} \\cdot \\sqrt{\\frac{\\sigma^2}{n}}, +\\infty \\right)`}
    • \n
\n \n \n

\n Se non conosciamo la varianza di una normale, allora possiamo ricavare velocemente gli intervalli di confidenza all'\\alpha% con queste formule:\n

\n
    \n
  • Intervalli bilateri: {r`\\mu \\in \\left[ \\overline{x}_n - t_{1 - \\frac{\\alpha}{2}; n-1} \\cdot \\sqrt{\\frac{s_n^2}{n}}, \\overline{x}_n + t_{1 - \\frac{\\alpha}{2}; n-1} \\cdot \\sqrt{\\frac{s_n^2}{n}} \\right]`}
    • \n
  • Intervallo unilatero da sinistra: {r`\\mu \\in \\left( -\\infty, \\overline{x}_n + t_{1 - \\frac{\\alpha}{2}; n-1} \\cdot \\sqrt{\\frac{s_n^2}{n}} \\right]`}
    • \n
  • Intervallo unilatero da destra: {r`\\mu \\in \\left[ \\overline{x}_n - t_{1 - \\frac{\\alpha}{2}; n-1} \\cdot \\sqrt{\\frac{s_n^2}{n}}, +\\infty \\right)`}
    • \n
\n

\n {r`t_{\\alpha, v}`} è un quantile della distribuzione di Student di parametro v.\n

\n \n
\n
\n \n

\n L'intervallo di confidenza per la proprorzione di una bernoulliana qualsiasi si ottiene da questa formula:\n

\n

\n {r`p \\in \\left[ \\overline{p} - z_{1 - \\frac{\\alpha}{2}} \\cdot \\sqrt{\\frac{\\overline{p} \\cdot (1 - \\overline{p})}{n+4}}, \\overline{p} + z_{1 - \\frac{\\alpha}{2}} \\cdot \\sqrt{\\frac{\\overline{p} \\cdot (1 - \\overline{p})}{n+4}} \\right]`}\n

\n \n
\n
\n \n

\n L'intervallo di confidenza per la media di una qualsiasi popolazione si ottiene da questa formula:\n

\n

\n {r`m \\in \\left[ \\overline{x}_n - z_{1 - \\frac{\\alpha}{2}} \\cdot \\sqrt{\\frac{s^2_n}{n}}, \\overline{x}_n + z_{1 - \\frac{\\alpha}{2}} \\cdot \\sqrt{\\frac{s^2_n}{n}} \\right]`}\n

\n \n
\n
\n \n \n )\n}\n","// extracted by mini-css-extract-plugin\nmodule.exports = {\"red\":\"red__3phX0\",\"orange\":\"orange__1OSkI\",\"yellow\":\"yellow__2pQOr\",\"lime\":\"lime__3vRUg\",\"cyan\":\"cyan__1cBtG\",\"blue\":\"blue__1G1wb\",\"magenta\":\"magenta__3Wagv\",\"minus\":\"minus__2qLyv\"};","// extracted by mini-css-extract-plugin\nmodule.exports = {\"red\":\"red__ipSoq\",\"orange\":\"orange__1nmkj\",\"yellow\":\"yellow__353ge\",\"lime\":\"lime__3FW1E\",\"cyan\":\"cyan__2xhu7\",\"blue\":\"blue__3MvDe\",\"magenta\":\"magenta__i03ie\",\"plus\":\"plus__1eV4_\"};","import style from \"./Example.less\";\r\n\r\nexport default function(props) {\r\n return (\r\n
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extracted by mini-css-extract-plugin\nmodule.exports = {\"red\":\"red__2y1B_\",\"orange\":\"orange__dD2kx\",\"yellow\":\"yellow__OEpwl\",\"lime\":\"lime__CVe41\",\"cyan\":\"cyan__26ZAg\",\"blue\":\"blue__LO7Xm\",\"magenta\":\"magenta__1Akee\",\"example\":\"example__2PzAa\"};","import style from \"./Minus.less\";\n\nexport default function(props) {\n\treturn {props.children};\n}\n","import {Latex, LatexDisplay, Panel, Section, LatexDefaultInline, LatexDefaultDisplay} from \"bluelib\";\nimport Example from \"../components/Example\";\n\nimport Plus from \"../components/Fisica/Plus\";\nimport Minus from \"../components/Fisica/Minus\";\n\nconst r = String.raw;\n\n\nexport default function(props) {\n return (\n \n \n
\n

Statistica ed Elementi di Probabilità

\n
\n \n

\n {r`P(E) = \\frac{casi\\ favorevoli}{casi\\ possibili}`}\n

\n \n \n

\n {r`P(E) = \\frac{successi}{prove\\ totali}`}\n

\n \n \n

\n Il prezzo che un individuo coerente riterrebbe equo per ricevere 1 nel caso l'evento si verificasse e 0 nel caso l'evento non si verificasse.\n

\n \n
\n
\n \n
\n \"omegone\"\n
\n

\n L'insieme di tutti gli esiti possibili di un esperimento.\n

\n

\n {r`\\Omega = \\left \\{ 1, 2, 3, 4, 5, 6 \\right \\}`}\n

\n \n \n
\n \"omeghino\"\n
\n

\n Un elemento dello spazio campionario.\n

\n

\n {r`\\omega = 1`}\n

\n \n \n
\n \"e\"\n
\n

\n Un sottoinsieme dello spazio campionario.\n

\n

\n {r`E = \\left \\{ 1, 2 \\right \\}`}\n

\n

\n Lo spazio campionario stesso è un evento certo.\n

\n \n \n
\n \"not e\"\n
\n

\n Il complementare di un sottoinsieme.\n

\n

\n {r`\\bar{E} = \\left \\{ 3, 4, 5, 6 \\right \\}`}\n

\n \n \n
\n \"e intersecato effe\"\n
\n

\n L'intersezione di più sottoinsiemi.\n

\n

\n {r`E \\cap F = \\left \\{ 1 \\right \\}`}\n

\n \n \n
\n \"e unito a effe\"\n
\n

\n L'unione di più sottoinsiemi.\n

\n

\n {r`E \\cup F = \\left \\{ 1, 2, 3, 4 \\right \\}`}\n

\n \n \n
\n \"e meno effe\"\n
\n

\n {r`E \\setminus F = E \\cap \\bar{F}`}\n

\n \n \n
\n \"e contenuto in effe\"\n
\n

\n L'inclusione del primo insieme in un altro.\n

\n

\n {r`E \\subseteq F`}\n

\n

\n Se si verifica E, allora si verifica anche F.\n

\n \n \n
\n \"e è impossibile\"\n
\n

\n Un sottoinsieme vuoto.\n

\n

\n {r`E = \\emptyset`}\n

\n \n \n
\n \"e ed effe si escludono mutualmente\"\n
\n

\n La disgiunzione di due insiemi.\n

\n

\n {r`E \\cap F = \\emptyset`}\n

\n \n
\n
\n \n
\n \"famiglia effe\"\n
\n

\n I sottoinsiemi dello spazio campionario formano una famiglia di sottoinsiemi detta famiglia degli eventi.\n

\n

\n {r`\\mathcal{F}`}\n

\n

\n Qualsiasi sottoinsieme appartenente a {r`\\mathcal{F}`} è considerato un evento.\n

\n \n {r`\\sigma`}-algebra}>\n
\n \"sigma algebra\"\n
\n

\n Se la famiglia degli eventi soddisfa questi tre requisiti, allora viene detta {r`\\sigma`}-algebra:\n

\n
    \n
  1. \n Lo spazio campionario è un evento: {r`\\Omega \\in \\mathcal{F}`}\n
    2. \n
  2. \n Se un sottoinsieme è un evento, allora anche il suo complementare lo è: {r`E \\in \\mathcal{F} \\implies \\bar{E} \\in \\mathcal{F}`}\n
    3. \n
  3. \n Se due sottoinsiemi sono eventi, allora lo sono anche la loro unione e intersezione: {r`(E, F) \\in \\mathcal{F} \\implies (E \\cup F, E \\cap F) \\in \\mathcal{F}`}\n
    4. \n
\n

\n Un esempio: {r`E \\in \\mathcal{F} \\implies \\mathcal{F} = \\{ \\emptyset, E, \\bar{E}, \\Omega \\}`}\n

\n \n
\n
\n \n
\n \"la partizione e composta da e uno, e due, e tre...\"\n
\n

\n Un insieme di esiti e eventi:\n

\n
    \n
  • Finito.
    • \n
  • In cui tutti gli eventi hanno probabilità diversa da 0.
    • \n
  • In cui tutti gli eventi sono mutualmente esclusivi.
    • \n
  • In cui l'unione di tutti i suoi elementi copre lo spazio campionario.
    • \n
\n

\n La partizione {r`E_i`} è composta dagli eventi {r`E_1`}, {r`E_2`}, {r`E_3`}, fino a {r`E_n`}.\n

\n \n Se lo spazio campionario fosse una torta, una sua partizione sarebbe l'insieme delle fette di uno dei modi in cui si potrebbe tagliare.\n \n \n
\n
\n \n

\n La probabilità di un evento è un numero tra 0 e 1.\n

\n

\n {r`\\forall E \\in \\mathcal{F}, 0 \\leq P(E) \\leq 1`}\n

\n \n \n

\n La probabilità dello spazio campionario è sempre 1.\n

\n

\n {r`P(\\Omega) = 1`}\n

\n \n \n

\n La probabilità dell'unione di eventi indipendenti è uguale alla somma delle loro probabilità.\n

\n

\n {r`P \\left ( \\bigcup_i E_i \\right ) = \\sum_i P ( E_i )`}\n

\n \n
\n
\n \n

\n La probabilità di un evento negato è uguale a 1 meno la probabilità dell'evento non negato.\n

\n

\n {r`P(\\bar{E}) = 1 - P({E})`}\n

\n \n \n

\n La probabilità di un evento incluso in un altro è sempre minore o uguale alla probabilità dell'evento in cui è incluso.\n

\n

\n {r`F \\subseteq E \\implies P(F) \\leq P(E)`}\n

\n \n \n

\n La probabilità di un evento unito a un altro è uguale alla somma delle probabilità dei due eventi meno la probabilità della loro intersezione.\n

\n

\n {r`P(E \\cup F) = P(E) + P(F) - P(E \\cap F)`}\n

\n \n Sommando le probabilità dei due eventi, l'intersezione viene contata due volte, e va quindi rimossa!\n \n \n
\n
\n \n

\n Spazi campionari in cui ci sono un numero finito di esiti e ogni esito ha la stessa probabilità di verificarsi.\n

\n

\n {r`P(E) = \\frac{len(E)}{len(\\Omega)}`}\n

\n \n \n

\n Gli spazi campionari possono avere un numero infinito di esiti: sono equiprobabili geometrici se nessun esito è privilegiato rispetto agli altri.\n

\n \n
\n
\n \n

\n Estraggo un numero, da un sacchetto con n numeri, mi segno che numero ho estratto e lo tengo fuori dal sacchetto. Ripeto per k volte.\n

\n

\n Tengo conto dell'ordine in cui ho estratto i numeri.\n

\n

\n {r`\\boldsymbol{D}_{n, k} = \\frac{n!}{(n - k)!}`}\n

\n \n \n

\n Estraggo un numero, da un sacchetto con n numeri, mi segno che numero ho estratto e lo rimetto nel sacchetto. Ripeto per k volte.\n

\n

\n Tengo conto dell'ordine in cui ho estratto i numeri.\n

\n

\n {r`\\boldsymbol{D}^{r}_{n, k} = n^k`}\n

\n \n \n

\n Estraggo un numero, da un sacchetto con n numeri, mi segno che numero ho estratto e lo tengo fuori dal sacchetto. Ripeto per k volte.\n

\n

\n Non mi interessa l'ordine in cui ho estratto i numeri.\n

\n

\n {r`\\boldsymbol{C}_{n, k} = \\binom{n}{k} = \\frac{n!}{(k)! \\cdot (n - k)!}`}\n

\n \n \n

\n Estraggo un numero, da un sacchetto con n numeri, mi segno che numero ho estratto e lo rimetto nel sacchetto. Ripeto per k volte.\n

\n

\n Non mi interessa l'ordine in cui ho estratto i numeri.\n

\n

\n {r`\\boldsymbol{C}^{r}_{n, k} = \\binom{n + k - 1}{k} = \\frac{(n + k - 1)!}{(k)! \\cdot (n - 1)!}`}\n

\n \n \n

\n Estraggo n numeri e guardo in quanti ordini diversi li posso mettere.\n

\n

\n {r`\\boldsymbol{P}_n = n!`}\n

\n \n
\n
\n \n
\n \"E dato F\"\n
\n

\n La probabilità che si verifichi E sapendo che si è già verificato F.\n

\n

\n {r`P(E|F) = \\frac{P(E \\cap F)}{P(F)}`}\n

\n \n Ricorda vagamente le pipe di bash, però al contrario...\n \n \n \n

\n Se due eventi sono mutualmente esclusivi, entrambe le loro probabilità condizionate saranno uguali a 0.\n

\n

\n {r`E \\cap F = \\emptyset \\Longleftrightarrow P(E|F) = P(F|E) = 0`}\n

\n \n
\n
\n \n

\n Si può sfruttare la formula inversa della probabilità condizionata per calcolare catene di intersezioni:\n

\n

\n {r`P(E_1 \\cap \\times \\cap E_n) = P(E_1) \\times P(E_2 | E_1) \\times \\dots \\times P(E_n | E_1 \\cap E_2 \\cap \\dots \\cap E_{n-1})`}\n

\n \n
\n
\n \n

\n La probabilità che si verifichi un evento è pari alla somma delle probabilità dell'evento stesso dati tutti gli eventi di una partizione.\n

\n

\n {r`P(F) = \\sum_{i} P(F|E_i) \\cdot P(E_i)`}\n

\n \n \n

\n La legge delle alternative funziona anche quando ad essere partizionato è un evento:\n

\n

\n {r`P(F|G) = \\sum_i P(F|E_i \\cap G) \\cdot P(E_i | G)`}\n

\n \n \n

\n Tramite la formula di Bayes possiamo risalire alla probabilità di un evento condizionato a un altro partendo dalla probabilità di quest'ultimo condizionato al primo:\n

\n

\n {r`P(E_h | F) = \\frac{P(F | E_h) \\cdot P(E_h)}{P(F)}`}\n

\n \n In pratica, invertiamo gli eventi.\n \n \n
\n
\n \n
\n \"eventi indipendenti a due a due\"\n
\n

\n Se due eventi sono indipendenti, sapere che uno dei due si è verificato non influisce sulle probabilità che si sia verificato l'altro.\n

\n

\n {r`P(E \\cap F) = P(E) \\cdot P(F) \\Longleftrightarrow P(E|F) = P(E) \\Longleftrightarrow P(F|E) = P(F)`}\n

\n \n \n
\n \"eventi indipendenti a tre a tre, a quattro a quattro, a cinque a cinque...\"\n
\n

\n Si può verificare l'indipendenza di più eventi alla volta:\n

\n

\n {r`P(E \\cap F \\cap G) = P(E) \\cdot P(F) \\cdot P(G)`}\n

\n

\n Eventi indipendenti a due a due non sono per forza indipendenti a tre a tre, e viceversa.\n

\n \n \n

\n Un insieme di n eventi è una famiglia di eventi indipendenti se, preso un qualsiasi numero di eventi da essa, essi risulteranno indipendenti.\n

\n \n Tutti gli eventi provenienti da essa saranno indipendenti sia a due a due, sia a tre a tre, sia a quattro a quattro, e così via!\n \n \n
\n
\n \n

\n Una funzione che fa corrispondere un numero reale a ogni possibile esito dello spazio campionario. {r`X(\\omega) : \\Omega \\to \\mathbb{R}`}.\n

\n \n Insieme di ripartizione}>\n

\n Ad ogni variabile aleatoria sono associati gli eventi {r`A_t = \\{ \\omega | X(\\omega) \\leq t \\}`}, che contengono tutti gli esiti a cui la variabile aleatoria associa un valore minore o uguale a t.\n

\n

\n Per definizione, tutte le variabili aleatorie devono rispettare questa condizione:\n

\n

\n {r`\\forall t \\in \\mathbb{R}, A_t \\in \\mathcal{F}`}\n

\n \n All'aumentare di t, l'insieme conterrà sempre più elementi.\n \n \n \n
\n \"supporto di X\"\n
\n

\n Il codominio della variabile aleatoria è il suo supporto.\n

\n

\n Per indicare che un valore x_0 appartiene al supporto di X, si usa la notazione X \\mapsto x_0.\n

\n \n
\n
\n \n

\n La funzione probabilità {r`p_X : X \\to [0, 1]`} di una variabile aleatoria discreta X è la funzione che associa ad ogni esito la sua probabilità:\n

\n

\n {r`\n p_X (x) = \\begin{cases}\n P([X = x]) \\quad se\\ X \\mapsto x \\\\\n 0 \\qquad \\qquad \\quad se\\ X \\not\\mapsto x\n \\end{cases} \n `}\n

\n \n \n

\n La funzione densità {r`f_X : X \\to [0, 1]`} di una variabile aleatoria continua X è l'equivalente continuo della funzione probabilità:\n

\n

\n {r`P([a < X \\leq b]) = \\int_a^b f_X (x) dx`}\n

\n

\n A differenza della funzione probabilità, è possibile che la funzione densità non esista per una certa variabile aleatoria.\n

\n \n Rappresenta \"quanta\" probabilità c'è in un'unità di x!\n \n \n
\n
\n \n

\n Ogni variabile aleatoria ha una funzione di ripartizione {r`F_X : \\mathbb{R} \\to [0, 1]`} associata, che rappresenta la probabilità che la variabile aleatoria assuma un valore minore o uguale a t:\n

\n

\n Si può dire che essa rappresenti la probabilità dell'evento {r`A_t`}:\n

\n

\n {r`\n F_X (t) = P(A_t) = \\begin{cases}\n \\sum_{i = 0}^{t} p_X (x_i) \\quad nel\\ discreto\\\\\n \\\\\n \\int_{-\\infty}^t f_X (x) dx \\quad nel\\ continuo\n \\end{cases}\n `}\n

\n \n \n
    \n
  • È sempre monotona crescente (non strettamente).

    • \n
  • Vale 0 a -\\infty e 1 a +\\infty.

    • \n
  • È continua da destra: {r`\\forall x_0 \\in \\mathbb{R}, F_X (x_0) = \\lim_{t \\to x^+_0} F_X (t)`}
    • \n
\n \n \n

\n Possiamo usare la funzione di ripartizione per calcolare la probabilità di un certo valore reale:\n

\n

\n {r`P([X = x_0]) = \\lim_{t \\to x^+_0} F_X (t) - \\lim_{t \\to x^-_0} F_X (t)`}\n

\n \n
\n
\n \n

\n Nel discreto basta abbinare un nuovo valore a ogni valore della variabile originale.\n

\n \n \n

\n Nel continuo applichiamo la formula dell'integrazione per sostituzione:\n

\n

\n {r`f_Y (y) = \\int_{g(a)}^{g(b)} f_X ( g^{-1} (x) ) g^{-2} (x)`}\n

\n \n \n

\n Trasformare variabili aleatorie è molto utile nell'informatica per creare distribuzioni partendo da una funzione random() che restituisce numeri da 0 a 1 con una distribuzione lineare.\n

\n \n
\n
\n \n

\n Ogni variabile aleatoria che ha una funzione di ripartizione e un supporto finito ha anche una media (o valore medio o atteso):\n

\n

\n {r`E(X) = \\int_0^{+infty} (1 - F_X (t)) dt - \\int_{-\\infty}^{0} F_X (t) dt`}\n

\n

\n Nel discreto, si può calcolare con:\n

\n

\n {r`E(X) = \\sum_i P(X = x_i) \\cdot x_i`}\n

\n

\n Nel continuo, si può calcolare con:\n

\n

\n {r`E(X) = \\int_{-\\infty}^{+\\infty} f_X (x) \\cdot x \\cdot dx`}\n

\n \n
\n
\n \n

\n Valore per cui la funzione probabilità o funzione densità è massima.\n

\n \n \n

\n Il quantile {r`x_{\\alpha}`} di ordine {r`0 \\leq \\alpha \\leq 1`} della variabile aleatoria X è il più piccolo numero tale che:\n

\n

\n {r`P([X < x_{\\alpha}]) \\leq \\alpha \\leq P([X \\leq x_{\\alpha}])`}\n

\n

\n\n

\n

\n Il quantile di ordine 0.5 {r`x_{0.5}`} è detto mediana.\n

\n

\n I quantili di ordine 0.25 {r`x_{0.25}`} e 0.75 {r`x_{0.75}`} sono detti quartili.\n

\n

\n I quantili di ordine {r`\\frac{n}{100}`} sono detti n-esima percentile.\n

\n \n \n

\n È un valore che indica quanto la variabile aleatoria si discosta generalmente dalla media:\n

\n

\n {r`Var(X) = E( (X - E(X) )^2 ) = E ( X^2 ) - (E(X))^2`}\n

\n \n
\n
\n \n

\n Data una variabile aleatoria non-negativa:\n

\n

\n {r`\\forall k > 0, P([X \\geq k]) \\leq \\frac{E(X)}{k}`}\n

\n

\n Divide in due parti ({r`P(X < k)`} e {r`P(X \\geq k)`}) la funzione X, la cui media risulterà uguale a:\n

\n

\n {r`E(X) = \\overline{k} \\cdot P(X < k) + k \\cdot P(X \\geq k)`}\n

\n \n \n
\n \"disuguaglianza di cebicev\"\n
\n

\n Se la variabile aleatoria X ha media e varianza, allora la probabilità che essa abbia un valore a più di {r`\\epsilon`} di distanza dal valore medio è minore o uguale a {r`\\frac{Var(X)}{\\epsilon^2}`}.\n

\n

\n {r`\\forall \\epsilon > 0, P([ \\left| X - E(X) \\right| \\geq \\epsilon]) \\leq \\frac{Var(X)}{\\epsilon^2}`}\n

\n

\n E anche:\n

\n

\n {r`\\forall \\epsilon > 0, P([ \\left| X - E(X) \\right| < \\epsilon]) \\geq 1 - \\frac{Var(X)}{\\epsilon^2}`}\n

\n \n Serve per semplificare i calcoli quando la funzione di ripartizione è difficile da calcolare!\n \n \n
\n
\n \n

\n Il momento k-esimo di una variabile aleatoria è:\n

\n

\n {r`\n \\mu_k = E ( X^k ) = \\begin{cases}\n \\sum_i x_i^k p_X (x_i) \\qquad nel\\ discreto\\\\\n \\\\\n \\int_{-\\infty}^{+\\infty} x^k f_X (x) dx \\qquad nel\\ continuo\n \\end{cases}`\n }\n

\n \n La media di una variabile aleatoria è anche il suo primo momento.\n \n \n \n

\n La funzione generatrice dei momenti è:\n

\n

\n {r`m_X (t) = E( e^{t \\cdot X} )`}\n

\n

\n Se due variabile aleatorie hanno la stessa funzione generatrice dei momenti, allora esse hanno la stessa distribuzione.\n

\n

\n E' la trasformata di Laplace della variabile aleatoria di X.\n

\n \n \n

\n La funzione caratteristica è:\n

\n

\n {r`H_X (t) = E ( e^{i \\cdot t \\cdot X} )`}\n

\n

\n Se due variabile aleatorie hanno la stessa funzione caratteristica, allora esse hanno la stessa distribuzione.\n

\n

\n E' la trasformata di Fourier della variabile aleatoria di X.\n

\n \n
\n
\n \n

\n Per dire che una variabile ha una certa distribuzione, si usa la notazione:\n

\n

\n {r`X \\sim Distribuzione()`}\n

\n \n \n

\n Una prova con solo due possibili esiti: successo e insuccesso.\n

\n \n \n

\n Una sequenza di prove di Bernoulli per le quali le probabilità di successo e fallimento rimangono invariate.\n

\n \n
\n
\n \n

\n Una variabile aleatoria che rappresenta una prova di Bernoulli:\n

\n
    \n
  • vale 1 in caso di successo.
    • \n
  • vale 0 in caso di insuccesso.
    • \n
\n

\n Il suo simbolo è {r`Ber(p)`}\n

\n \n \n

\n La distribuzione bernoulliana ha come densità:\n

\n

\n {r`\n f_X (k) : \\{0, 1\\} = \\begin{cases}\n p \\quad se\\ k = 1\\\\\n q \\quad se\\ k = 0\\\\\n 0 \\quad altrimenti\n \\end{cases} = p^x \\cdot q^{1 - k}`\n }\n

\n \n
\n
\n \n

\n Una variabile aleatoria che conta il numero di successi di n prove di uno schema di Bernoulli.\n

\n

\n Il suo simbolo è {r`Bin(n, p)`}.\n

\n \n \n

\n La binomiale ha come densità:\n

\n

\n {r`f_X (k) : \\{0..n\\} = \\binom{n}{k} \\cdot p^k \\cdot q^{n - k}`}\n

\n \n \n

\n La funzione generatrice dei momenti della binomiale è:\n

\n

\n {r`m_X (t) = (q + p \\cdot e^t) ^ n`}\n

\n

\n La media di una binomiale è:\n

\n

\n {r`E(X) = n \\cdot p`}\n

\n

\n La varianza di una binomiale è:\n

\n

\n {r`Var(X) = n \\cdot p \\cdot q`}\n

\n \n
\n
\n \n

\n Una variabile aleatoria che conta il numero di prove in uno schema di Bernoulli fino alla comparsa del primo successo.\n

\n

\n Il suo simbolo è Geo(p).\n

\n \n \n

\n La geometrica ha come densità:\n

\n

\n {r`f_X (k) : \\mathbb{N} = q^{k - 1} p`}\n

\n \n \n

\n La funzione generatrice dei momenti della geometrica è:\n

\n

\n {r`m_X (t) = \\frac{p \\cdot e^t}{1 - q \\cdot e^t}`}\n

\n

\n La media della geometrica è:\n

\n

\n {r`E(X) = \\frac{1}{p}`}\n

\n

\n La varianza della geometrica è:\n

\n

\n {r`Var(X) = \\frac{q}{p^2}`}\n

\n \n \n

\n La geometrica non tiene conto degli eventi avvenuti in passato: ha la proprietà dell'assenza di memoria:\n

\n

\n {r`P([X = i + j | X > i ]) = P([X = j])`}\n

\n \n Ovvero, riscalando opportunamente l'asse Y posso prendere come 0 qualsiasi punto dell'asse X.\n \n \n
\n
\n \n

\n Una variabile aleatoria che conta il numero di prove in uno schema di Bernoulli necessarie perchè si verifichi l'n-esimo successo.\n

\n

\n Il suo simbolo è {r`\\overline{Bin}(n, p)`}.\n

\n \n \n

\n La binomiale negativa ha come densità:\n

\n

\n {r`f_X (k) : \\{ n .. +\\infty \\} \\in \\mathbb{N} = \\binom{k - 1}{n - 1} \\cdot p^n \\cdot q^{k - n} `}\n

\n \n \n

\n

\n La funzione generatrice dei momenti della binomiale negativa è:\n

\n

\n {r`m_X (t) : \\{ t < ln(\\frac{1}{q}) \\} = \\left( \\frac{p \\cdot e^t}{1 - q \\cdot e^t} \\right) ^n`}\n

\n

\n La media della binomiale negativa è:\n

\n

\n {r`E(X) = \\frac{n}{p}`}\n

\n

\n La varianza della binomiale negativa è:\n

\n

\n {r`Var(X) = \\frac{n \\cdot q}{p^2}`}\n

\n

\n \n
\n
\n \n

\n Una variabile aleatoria che conta il numero k di insuccessi consecutivi in uno schema di Bernoulli:\n

\n

\n Il suo simbolo rimane {r`Geo(p)`}.\n

\n \n \n

\n La geometrica traslata ha come densità:\n

\n

\n {r`f_X (k) : \\mathbb{N} = p \\cdot q^k `}\n

\n \n \n

\n La funzione generatrice dei momenti della geometrica traslata è:\n

\n

\n {r`m_X (t) : \\left\\{ t < ln \\left( \\frac{1}{q} \\right) \\right\\} = \\frac{p}{1 - q \\cdot e^t}`}\n

\n

\n La media della geometrica traslata è:\n

\n

\n {r`E(X) = \\frac{q}{p}`}\n

\n

\n La varianza della geometrica è:\n

\n

\n {r`Var(X) = \\frac{q}{p^2}`}\n

\n \n \n

\n La geometrica traslata non tiene conto degli eventi avvenuti in passato: ha la proprietà dell'assenza di memoria:\n

\n

\n {r`P([X = i + j | X > i ]) = P([X = j])`}\n

\n \n Ovvero, riscalando opportunamente l'asse Y posso prendere come 0 qualsiasi punto dell'asse X.\n \n \n
\n
\n \n

\n Una variabile aleatoria che conta il numero di insuccessi in uno schema di Bernoulli prima che si verifichi l'n-esimo successo.\n

\n

\n Il suo simbolo rimane {r`\\overline{Bin}(n, p)`}.\n

\n \n \n

\n La binomiale negativa traslata ha come densità:\n

\n

\n {r`f_X (k) : \\mathbb{N} = \\binom{k + n - 1}{n - 1} \\cdot p^n \\cdot q^k `}\n

\n \n \n

\n

\n La funzione generatrice dei momenti della binomiale negativa traslata è:\n

\n

\n {r`m_X (t) : \\left\\{ t < ln \\left( \\frac{1}{q} \\right) \\right\\} = \\left( \\frac{p \\cdot e^t}{1 - q \\cdot e^t} \\right) ^n`}\n

\n

\n La media della binomiale negativa traslata è:\n

\n

\n {r`E(X) = \\frac{n \\cdot q}{p}`}\n

\n

\n La varianza della binomiale negativa traslata è:\n

\n

\n {r`Var(X) = \\frac{n \\cdot q}{p^2}`}\n

\n

\n \n
\n
\n \n

\n Una variabile aleatoria che, sapendo il numero di successi K e di insuccessi N-K, conta quanti successi si otterrebbero se se ne estraessero n in blocco.\n

\n

\n Il suo simbolo è Ipe(N, K, n).\n

\n \n \n

\n La ipergeometrica ha come densità:\n

\n

\n {r`f_X (k) : \\{0..n\\} \\in \\mathbb{N} = \\frac{\\binom{K}{k} \\cdot \\binom{N - K}{n - k}}{\\binom{N}{n}}`}\n

\n \n \n

\n

\n La funzione generatrice dei momenti della ipergeometrica è trascurabile.\n

\n

\n La media della ipergeometrica è:\n

\n

\n {r`E(X) = n \\cdot \\frac{K}{N}`}\n

\n

\n La varianza della ipergeometrica è:\n

\n

\n {r`Var(X) = n \\cdot \\frac{K}{N} \\cdot \\frac{N - K}{N} \\cdot \\frac{N - n}{N - 1}`}\n

\n

\n \n
\n
\n \n

\n Una variabile aleatoria che soddisfa tutte le seguenti caratteristiche:\n

\n
    \n
  • Binomiale: {r`X \\sim Bin(n, p)`}
    • \n
  • Il numero di prove tende a infinito: {r`n \\to +\\infty`}
    • \n
  • La probabilità di successo tende a 0: {r`p \\to 0`}
    • \n
  • La media è finita: {r`E(X) = n \\cdot p \\to \\mu \\neq 0`}
    • \n
\n

\n Il suo simbolo è {r`Poi(\\mu)`}\n

\n \n \n

\n La poissoniana ha come densità:\n

\n

\n {r`f_X (k) : \\mathbb{N} = \\frac{e^{-\\mu} \\cdot \\mu^k}{k!}`}\n

\n \n \n

\n

\n La funzione generatrice dei momenti della poissoniana è:\n

\n

\n {r`m_X (t) = e^{\\mu \\cdot (e^t - 1)}`}\n

\n

\n La media della poissoniana è:\n

\n

\n {r`E(X) = \\mu`}\n

\n

\n La varianza della poissoniana è:\n

\n

\n {r`Var(X) = \\mu`}\n

\n

\n Gli altri momenti della poissoniana sono:\n

\n
    \n
  1. {r`E(X^2) = \\mu^2 + \\mu`}
    2. \n
\n

\n \n
\n
\n \n

\n Una successione di arrivi avvenuti in un certo arco temporale che:\n

\n
    \n
  • non sono sovrapposti.
    • \n
  • hanno intensità {r`\\lambda`} costante.
    • \n
  • avvengono indipendentemente gli uni dagli altri.
    • \n
\n \n \n

\n Una variabile aleatoria N_t che conta il numero di arrivi di uno schema di Poisson di intensità {r`\\lambda`} in un intervallo di tempo di durata t.\n

\n

\n E' una distribuzione poissoniana con {r`\\mu = t \\cdot \\lambda`}: {r`Poi(t \\cdot \\lambda)`}\n

\n \n E' paragonabile a una bernoulliana: ogni successo corrisponde a un arrivo, mentre il tempo è il numero di prove effettuate (ma nel continuo).\n \n \n
\n
\n \n

\n Una variabile aleatoria che conta il tempo diwidehattesa prima del primo arrivo di un processo di Poisson di intensità {r`\\lambda`}.\n

\n

\n Il suo simbolo è {r`Esp(\\lambda)`}.\n

\n \n \n

\n L'esponenziale ha come densità:\n

\n

\n {r`\n f_X (x) = \\begin{cases}\n 0 \\qquad \\qquad x < 0\\\\\n \\lambda \\cdot e^{-\\lambda \\cdot x} \\quad x > 0\n \\end{cases}`\n }\n

\n

\n L'esponenziale ha come funzione di ripartizione:\n

\n

\n {r`\n F_X (t) = \\begin{cases}\n 0 \\qquad \\qquad t < 0\\\\\n 1 - e^{-\\lambda \\cdot t} \\quad t \\geq 0\n \\end{cases}`\n }\n

\n \n \n

\n La funzione generatrice dei momenti dell'esponenziale è:\n

\n

\n {r`m_X (t) : \\{ t | t < \\lambda \\} \\in \\mathbb{R} = \\frac{\\lambda}{\\lambda - t}`}\n

\n

\n La media dell'esponenziale è:\n

\n

\n {r`E(X) = \\frac{1}{\\lambda}`}\n

\n

\n La varianza dell'esponenziale è:\n

\n

\n {r`Var(X) = \\frac{1}{\\lambda^2}`}\n

\n \n \n

\n L'esponenziale non tiene conto degli eventi avvenuti in passato: ha la proprietà dell'assenza di memoria:\n

\n

\n {r`P([X > s + t | X > s]) = P([X > t])`}\n

\n \n Ovvero, riscalando opportunamente l'asse Y posso prendere come 0 qualsiasi punto dell'asse X.\n \n \n
\n
\n \n

\n Una variabile aleatoria che conta il tempo diwidehattesa prima dell'n-esimo arrivo di un processo di Poisson di intensità {r`\\lambda`}.\n

\n

\n Il suo simbolo è {r`\\Gamma(n, \\lambda)`}.\n

\n \n \n

\n La legge gamma ha come densità:\n

\n

\n {r`\n f_X (x) = \\begin{cases}\n 0 \\qquad \\qquad \\qquad \\qquad \\qquad x < 0\\\\\n \\frac{1}{(n-1)!} \\cdot \\lambda^n \\cdot x^{n-1} \\cdot e^{-\\lambda \\cdot x} \\quad k > 0\n \\end{cases}`\n }\n

\n \n \n

\n

\n La funzione generatrice dei momenti della legge gamma è:\n

\n

\n {r`m_X (t) : ( t < \\lambda ) \\in \\mathbb{R} = \\left( \\frac{\\lambda}{\\lambda - t} \\right) ^\\alpha`}\n

\n

\n La media della legge gamma è:\n

\n

\n {r`E(X) = \\frac{\\alpha}{\\lambda}`}\n

\n

\n La varianza della legge gamma è:\n

\n

\n {r`Var(X) = \\frac{\\alpha}{\\lambda^2}`}\n

\n

\n \n
\n
\n \n

\n Una variabile aleatoria che può assumere qualsiasi valore in un intervallo {r`[a, b]`} in modo equiprobabile.\n

\n

\n Il suo simbolo è {r`Uni(a, b)`}\n

\n

\n Su di essa vale la seguente proprietà:\n

\n

\n {r`P(X \\in (c, d)) = \\frac{d - c}{b - a}`}\n

\n \n \n

\n La distribuzione uniforme ha come densità:\n

\n

\n {r`\n f_X (x) = \\begin{cases}\n \\frac{1}{b - a} \\qquad a \\leq x \\leq b\\\\\n 0 \\qquad \\quad altrimenti \n \\end{cases}\n `}\n

\n

\n La distribuzione uniforme ha come funzione di ripartizione:\n

\n

\n {r`\n f_X (x) = \\begin{cases}\n 0 \\qquad \\quad x < a \n \\frac{1}{b - a} \\qquad a \\leq x \\leq b\\\\\n 1 \\qquad \\quad x > b\n \\end{cases}`\n }\n

\n \n \n

\n

\n La funzione generatrice dei momenti della distribuzione uniforme è:\n

\n

\n {r`m_X (t) = \\frac{e^{b \\cdot t} - e^{a \\cdot t}}{(b - a) \\cdot t}`}\n

\n

\n La media della distribuzione uniforme è:\n

\n

\n {r`E(X) = \\frac{a + b}{2}`}\n

\n

\n La varianza della distribuzione uniforme è:\n

\n

\n {r`Var(X) = \\frac{(b - a)^2}{12}`}\n

\n

\n \n
\n
\n \n

\n Una variabile aleatoria con una specifica distribuzione.\n

\n

\n Il suo simbolo è {r`Nor(\\mu, \\sigma^2)`}.\n

\n \n \\mu e \\sigma^2 sono rispettivamente la media e la varianza della distribuzione!\n \n \n \n

\n La distribuzione normale ha come densità:\n

\n

\n {r`f_X (x) = \\frac{e^{-\\frac{(x - \\mu)^2}{2 \\sigma^2}}}{\\sqrt{2 \\pi \\cdot \\sigma^2}}`}\n

\n \n \n

\n

\n La funzione generatrice dei momenti della distribuzione normale è:\n

\n

\n {r`m_X (t) = e^{\\mu \\cdot t + \\frac{\\sigma^2 \\cdot t^2}{2}}`}\n

\n

\n La media della distribuzione normale è:\n

\n

\n {r`E(X) = \\mu`}\n

\n

\n La varianza della distribuzione normale è:\n

\n

\n {r`Var(X) = \\sigma^2`}\n

\n

\n \n
\n
\n \n

\n Qualsiasi normale può essere trasformata in qualsiasi altra normale:\n

\n

\n {r`X \\sim Nor(m, v^2) \\implies \\alpha X + \\beta \\sim Nor(\\alpha m + \\beta, (\\alpha v)^2)`}\n

\n \n \n

\n La distribuzione normale standard Z è:\n

\n

\n Z \\sim Nor(0, 1)\n

\n

\n La sua funzione di ripartizione è detta {r`\\phi(z)`} e vale:\n

\n

\n {r`F_Z(z) = \\phi(z) = \\frac{1}{\\sqrt{2 \\pi}} \\int_{-\\infty}^{z} e^{-\\frac{x^2}{2}} dx`}\n

\n \n \n

\n Da un quantile {r`z_\\alpha`} della normale standard è possibile risalire allo stesso quantile di qualsiasi altra normale:\n

\n

\n {r`x_\\alpha = \\mu + z_\\alpha \\cdot \\sqrt{\\sigma^2}`}\n

\n \n
\n
\n \n

\n La distribuzione normale ha una particolare relazione con la distribuzione Gamma:\n

\n

\n {r`Z^2 \\sim \\chi^2 (v = 1)`}\n

\n \n \n
\n \"chi-quadro a un grado di libertà\"\n
\n

\n Esiste una distribuzione Gamma particolare:\n

\n

\n {r`\\Gamma \\left( \\frac{1}{2}, \\frac{1}{2} \\right) = \\chi^2 (v = 1)`}\n

\n

\n Più chi-quadro possono essere sommate per aumentare i loro gradi di libertà:\n

\n

\n {r`\\chi^2 (n) + \\chi^2 (m) = \\chi^2 (n + m)`}\n

\n \n \n

\n Un'altra funzione particolare è la funzione T di Student:\n

\n

\n {r`T(v) = \\frac{Nor(0, 1)}{\\sqrt{\\frac{\\chi^2(v)}{v}}}`}\n

\n \n
\n
\n \n

\n La binomiale è come una ipergeometrica ma con ripetizioni, quindi per valori molto grandi di N rispetto a n, si può dire che:\n

\n

\n {r`Ipe(N, K, n) \\approx Bin(n, \\frac{K}{N})`}\n

\n \n \n

\n La binomiale non è altro che una poissoniana a tempo discreto, quindi, se n è grande e n \\cdot p è nell'ordine di grandezza delle unità, allora:\n

\n

\n {r`Bin(n, p) \\approx Poi(n \\cdot p)`}\n

\n \n \n

\n Per il Teorema di De Moivre-Laplace, se una binomiale ha una n grande e p non vicina a 0 o 1, si può approssimare con:\n

\n

\n {r`Bin(n, p) \\approx Nor(n \\cdot p, n \\cdot p \\cdot q)`}\n

\n \n \n

\n Passando da una variabile discreta X a una continua Y, per ogni valore discreto k la probabilità viene \"spalmata\" su tutto l'intervallo {r`(k - \\frac{1}{2}, k + \\frac{1}{2})`}:\n

\n
    \n
  • {r`P(X < k) \\simeq P(Y \\leq k - \\frac{1}{2})`}
    • \n
  • {r`P(X \\leq k) \\simeq P(Y \\leq k + \\frac{1}{2})`}
    • \n
  • {r`P(X \\geq k) \\simeq P(Y \\geq k - \\frac{1}{2})`}
    • \n
  • {r`P(X > k) \\simeq P(Y \\geq k + \\frac{1}{2})`}
    • \n
\n \n
\n
\n \n

\n Un vettore composto da variabili aleatorie.\n

\n

\n Il suo simbolo generalmente è {r`\\boldsymbol{X}`} oppure {r`X, Y`}.\n

\n \n \n

\n I vettori aleatori hanno più funzioni di ripartizione che si differenziano in base al numero di parametri.\n

\n

\n Se il numero di parametri coincide con la dimensione del vettore aleatorio, allora la funzione sarà una funzione di ripartizione congiunta:\n

\n

\n {r`F_{X, Y} (x, y) = P(X \\leq x, Y \\leq y)`}\n

\n

\n Se il numero di parametri è minore della dimensione del vettore aleatorio, allora la funzione sarà una funzione di ripartizione marginale:\n

\n

\n {r`F_X (x) = P(X \\leq x) = \\lim_{y \\to +\\infty} F_{X, Y} (x, y)`}\n

\n \n \n

\n I vettori aleatori discreti hanno più densità che si differenziano in base al numero di parametri.\n

\n

\n Se il numero di parametri coincide con la dimensione del vettore aleatorio, allora la funzione sarà una densità congiunta:\n

\n

\n {r`p_{X, Y} (x, y) = P(X = x, Y = y)`}\n

\n

\n Se il numero di parametri è minore della dimensione del vettore aleatorio, allora la funzione sarà una densità marginale:\n

\n

\n {r`p_X (x) = \\sum_j p_{X, Y} (x_i, y_j)`}\n

\n \n
\n
\n \n

\n Più variabili aleatorie sono indipendenti se, per qualsiasi scelta di intervalli A_i:\n

\n

\n {r`P(X_1 \\in A_1, \\dots, X_n \\in A_n) = P(X_1 \\in A_1) \\times \\dots \\times P(X_n \\in A_n)`}\n

\n \n \n

\n E' possibile calcolare la media di qualsiasi funzione g(X, Y) avente elementi del vettore come variabili:\n

\n

\n {r`E(g(X, Y)) = \\sum_{i, j} g(x_i, y_i) \\cdot p_{X, Y} (x_i, y_i)`}\n

\n \n Solitamente si calcola la media di x \\cdot y.\n \n

\n Le medie di più variabili aleatorie si possono sommare:\n

\n

\n {r`E(X + Y) = E(X) + E(Y)`}\n

\n \n
\n
\n \n

\n Un operatore che misura la correlazione di due variabili aleatorie.\n

\n

\n Si calcola con il valore atteso dei prodotti delle distanze dalla media:\n

\n

\n {r`Cov(X, Y) = E((X - E(X) \\cdot (Y - E(Y)) = E(XY) - E(X) \\cdot E(Y)`}\n

\n

\n Ha diverse proprietà:\n

\n
    \n
  • Il suo valore nullo è 0: {r`Cov(X, \\alpha) = 0`}
    • \n
  • E' commutativa: {r`Cov(X, Y) = Cov(Y, X)`}
    • \n
  • E' semplificabile: {r`Cov(X, X) = Var(X)`}
    • \n
  • E' lineare: {r`Cov(\\alpha X, \\beta Y) = \\alpha \\cdot \\beta \\cdot Cov(X, Y)`}
    • \n
  • E' distributiva: {r`Cov(X + Y, V + W) = Cov(X, Y) + Cov(X, W) + Cov(Y, V) + Cov(Y, W)`}
    • \n
\n \n \n

\n Due variabili sono variabili incorrelate se:\n

\n

\n {r`Cov(X, Y) = 0`}\n

\n

\n Variabili indipendenti sono sempre incorrelate.\n

\n \n \n

\n Una matrice {r`\\boldsymbol{C_X}`} che contiene la covarianza tra tutte le variabili di un vettore aleatorio {r`\\boldsymbol{X}`}:\n

\n

\n {r`\n \\boldsymbol{C_X} = \n \\begin{bmatrix}\n Var(X_1) & Cov(X_1, X_2) & Cov(X_1, X_3)\\\\\n Cov(X_2, X_1) & Var(X_2) & Cov(X_2, X_3)\\\\\n Cov(X_3, X_1) & Cov(X_3, X_2) & Var(X_3)\n \\end{bmatrix}\n `}\n

\n

\n E' sempre simmetrica e semidefinita positiva (tutti gli autovalori sono \\geq 0.\n

\n \n \n

\n Un valore che misura come due variabili aleatorie sono correlate:\n

\n

\n {r`\\rho_{X, Y} = \\frac{Cov(X, Y)}{\\sqrt{Var(X)} \\cdot \\sqrt{Var(Y)}}`}\n

\n

\n E' sempre compreso tra -1 e 1:\n

\n

\n {r`-1 \\leq \\rho_{X, Y} \\leq 1`}\n

\n

\n Vale esattamente -1 o 1 solo se esiste un legame lineare tra le due variaibli:\n

\n

\n {r`Y = a X + b \\Longleftrightarrow | \\rho_{X, Y} | = 1`}\n

\n \n \n

\n La varianza di due variabili aleatorie sommate è:\n

\n

\n {r`Var(X + Y) = Var(X) + Var(Y) + 2 \\cdot Cov(X, Y)`}\n

\n \n Si dimostra applicando le proprietà della covarianza!\n \n

\n Se più variabili aleatorie X_i sono indipendenti ({r`Cov(X, Y) = 0`}), allora:\n

\n

\n {r`Var \\left( \\sum_i X_i \\right) = \\sum_i Var(X_i)`}\n

\n \n
\n
\n \n

\n Una n-pla di variabili aleatorie con la stessa distribuzione della variabile aleatoria X (\"popolazione\") ma indipendenti tra loro.\n

\n \n Le variabili aleatorie sono come un lazy-load in programmazione; quando ci sarà bisogno del loro valore numerico, esse si realizzeranno nel loro valore.\n \n \n \n

\n Il valore dato dalla media aritmetica degli n elementi del campione elevati alla potenza k:\n

\n

\n {r`M^{(k)}_n = \\frac{1}{n} \\cdot \\sum_{i = 1}^n X_i^k `}\n

\n

\n Il momento campionario di primo ordine è la media campionaria {r`\\overline{X}_n`}.\n

\n \n \n

\n La media aritmetica dello scarto quadratico medio degli elementi del campione.\n

\n

\n Se è noto il valore medio {r`m = E(X)`} di X:\n

\n

\n {r`S_0^2 = \\frac{1}{n} \\cdot \\sum_{i = 1}^n (X_i - m)^2 = M_n^(2) - 2 \\cdot m \\cdot \\overline{X}_n + m^2`}\n

\n

\n Altrimenti:\n

\n

\n {r`S_n^2 = \\frac{1}{n - 1} \\cdot \\sum_{i = 1}^n (X_i - \\overline{X}_n)^2 = \\frac{1}{n - 1} \\cdot ( n \\cdot M_2^{(2)} - n \\cdot \\overline{X}_n^2)`}\n

\n \n
\n
\n \n

\n Se calcoliamo la media della media campionaria, risulterà vero che:\n

\n

\n {r`E(\\overline{X}_n) = E(X)`}\n

\n \n Quindi, è possibile usare i campioni per trovare la media di una variabile aleatoria!\n \n \n \n

\n Se calcoliamo la varianza della media campionaria, risulterà vero che:\n

\n

\n {r`Var(\\overline{X}_n) = \\frac{Var(X)}{n}`}\n

\n \n Quindi, possiamo stimare l'errore della media calcolata tramite campioni!\n \n \n \n

\n Se calcoliamo la media della varianza campionaria, risulterà vero che:\n

\n

\n {r`E(S_0^2) = E(S_n^2) = Var(X)`}\n

\n \n Quindi, possiamo stimare l'errore della media calcolata tramite campioni!\n \n \n
\n
\n \n

\n Se la popolazione X ha una distribuzione normale ({r`X \\sim Nor(\\mu, \\sigma^2)`})...\n

\n \n \n

\n ...allora sappiamo anche la distribuzione della media campionaria!\n

\n

\n {r`\\overline{X}_n \\sim Nor \\left( \\mu, \\frac{\\sigma^2}{n} \\right)`}\n

\n \n \n

\n ...e anche della varianza campionaria!\n

\n

\n {r`S_0^2 \\sim \\frac{\\sigma^2}{n} \\cdot \\chi^2 (n)`}\n

\n

\n {r`S_n^2 \\sim \\frac{\\sigma^2}{n - 1} \\cdot \\chi^2 (n-1)`}\n

\n \n \n

\n ...e che media campionaria e varianza campionaria sono indipendenti tra loro!\n

\n \n
\n
\n \n

\n Se la successione di variabili aleatorie X_n all'infinito ha la stessa funzione di ripartizione della popolazione X, allora essa converge in distribuzione.\n

\n

\n {`\\\\lim_{n \\\\to +\\\\infty} F_{X_n} (x) = F_X (x) \\\\implies X_n \\\\xrightarrow{d} X`}\n

\n \n \n

\n Se la successione di variabili aleatorie X_n all'infinito ha la stessa probabilità della popolazione X, allora essa converge in probabilità.\n

\n

\n {`\\\\forall \\\\epsilon > 0, \\\\lim_{n \\\\to +\\\\infty} P( | X_n - X | < \\\\epsilon) = 1 \\\\implies X_n \\\\xrightarrow{p} X`}\n

\n \n \n

\n Se la successione di variabili aleatorie X_n all'infinito ha la stessa probabilità a della popolazione X, allora essa converge quasi certamente.\n

\n

\n {`\\\\forall \\\\epsilon > 0, P \\left( \\\\lim_{n \\\\to +\\\\infty} | X_n - X | < \\\\epsilon) \\right) = 1 \\\\implies X_n \\\\xrightarrow{qc} X`}\n

\n \n \n

\n Se la successione di variabili aleatorie X_n all'infinito ha la media del quadrato della distanza tra la successione e la popolazione X uguale a 0, allora essa converge in media quadratica.\n

\n

\n {`\\\\lim_{n \\\\to +\\\\infty} E( | X_n - X |^2 = 0 \\\\implies X_n \\\\xrightarrow{mq} X`}\n

\n \n \n

\n {`\n \\\\begin{matrix}\n X_n \\\\xrightarrow{mq} X\\\\\\\\\n X_n \\\\xrightarrow{qc} X\n \\\\end{matrix} \\\\implies X_n \\\\xrightarrow{p} X \\\\implies X_n \\\\xrightarrow{d} X`\n }\n

\n

\n In più:\n

\n

\n {`X_n \\\\xrightarrow{p} x \\\\Longleftrightarrow X_n \\\\xrightarrow{d} x`}\n

\n \n
\n
\n \n

\n La successione delle medie campionarie {r`\\overline{X}_n`} converge in probabilità alla media della popolazione {r`E(X)`}, se essa esiste.\n

\n

\n {`\\\\overline{X}_n \\\\xrightarrow{p} X`}\n

\n

\n Ovvero:\n

\n

\n {r`\\forall \\epsilon > 0, \\lim_{n \\to +\\infty} P( | \\overline{X}_n - E(X) | < \\epsilon) = 1`}\n

\n

\n {r`P( | \\overline{X}_n - E(X) | < \\epsilon) \\to 1`}\n

\n \n \n

\n La successione delle medie campionarie {r`\\overline{X}_n`} converge quasi certamente alla media della popolazione {r`E(X)`}, se essa esiste.\n

\n

\n {`\\\\overline{X}_n \\\\xrightarrow{qc} X`}\n

\n

\n Ovvero:\n

\n

\n {r`\\forall \\epsilon > 0, P \\left( \\lim_{n \\to +\\infty} | \\overline{X}_n - E(X) | < \\epsilon \\right) = 1`}\n

\n \n Dimostra che l'interpretazione frequentista della probabilità è valida!\n \n \n
\n
\n \n

\n La successione delle medie campionarie {r`\\overline{X}_n`} converge in distribuzione a {r`Nor(0, 1) = \\Phi()`}.\n

\n

\n {r`\\overline{X}_n \\approx Nor \\left(E(X), \\frac{Var(X)}{n} \\right)`}\n

\n

\n Ovvero:\n

\n

\n {r`\\forall x \\in \\mathbb{R}, \\lim_{n \\to +\\infty} P \\left( \\frac{\\overline{X}_n - E(X)}{\\sqrt{\\frac{Var(X)}{n}}} \\leq x \\right) = \\Phi(x)`}\n

\n \n
\n
\n \n

\n E' una somma di bernoulliane, e quindi si approssima a una normale:\n

\n

\n {r`Bin(n, p) \\approx Nor(n \\cdot p, n \\cdot p \\cdot q)`}\n

\n \n \n

\n E' una somma di geometriche, e quindi si approssima a una normale:\n

\n

\n {r`\\overline{Bin} (n, p) \\approx Nor \\left( \\frac{n}{p}, \\frac{n \\cdot (1 - p)}{p^2} \\right)`}\n

\n \n \n

\n E' una somma di altre poissoniane, e quindi si approssima a una normale:\n

\n

\n {r`Poi(\\lambda) \\approx Nor(\\lambda, \\lambda)`}\n

\n \n \n

\n E' una somma di esponenziali, e quindi si approssima a una normale:\n

\n

\n {r`\\Gamma (\\alpha, \\lambda) \\approx Nor \\left( \\frac{\\alpha}{\\lambda}, \\frac{\\alpha}{\\lambda^2} \\right)`}\n

\n \n \n

\n Se n è grande, allora:\n

\n

\n {r`Y = \\sum_{i=1}^{n} X_i`}\n

\n \n
\n
\n \n

\n Per indicare parametri sconosciuti di una legge si usa \\theta.\n

\n \n \n

\n Una variabile aleatoria funzione di un campione:\n

\n

\n {r`T(\\boldsymbol{X})`}\n

\n \n Ad esempio, sono statistiche media e varianza campionaria, così come il campione stesso {r`T(\\boldsymbol{X}) = \\boldsymbol{X}`}.\n \n \n
\n
\n \n

\n Una statistica T_n ottenuta da n osservazioni, che stimi i parametri di una legge e sia indipendente da essi.\n

\n \n \n

\n Uno stimatore è corretto se il suo valore atteso coincide con quello dei parametri che stima:\n

\n

\n {r`E(T_n) = \\theta`}\n

\n \n \n

\n Uno stimatore è asintoticamente corretto se, per infinite osservazioni, il suo valore atteso coincide con quello dei parametri che stima:\n

\n

\n {r`\\lim_{n \\to +\\infty} E(T_n) = \\theta`}\n

\n \n \n

\n Uno stimatore è consistente in media quadratica se:\n

\n

\n {r`\\lim_{n \\to +\\infty} E((T_n - \\theta)^2) = 0`}\n

\n \n \n

\n Uno stimatore è consistente in probabilità se:\n

\n

\n {r`\\forall \\epsilon > 0, \\lim_{n \\to +\\infty} P( |T_n - \\theta| < \\epsilon) = 1`}\n

\n \n \n

\n Uno stimatore è asintoticamente normale se:\n

\n

\n {r`\\lim_{n \\to +\\infty} \\frac{T_n - E(T_n)}{\\sqrt{Var(T_n)}} \\sim Nor(0, 1)`}\n

\n \n
\n
\n \n

\n Si può usare il metodo dei momenti per ottenere uno stimatore di una popolazione X.\n

\n

\n Lo stimatore di {r`\\theta`} così ottenuto sarà indicato aggiungendo un cappellino e una M a \\theta: {r`\\widehat{\\theta}_M`}\n

\n

\n Visto che:\n

\n
    \n
  • {r`\\theta = g(E(X))`}
    • \n
  • {r`\\widehat{E(X)} = \\overline{X}_n`}
    • \n
\n

\n Allora:\n

\n

\n {r`\\widehat{\\theta}_M = g( \\overline{X}_n )`}\n

\n

\n Se {r`\\theta`} non è esprimibile in termini di {r`E(X)`}, si possono usare i momenti successivi {r`M_n^2`}, {r`M_n^3`}, {r`M_n^3`}...\n

\n \n
\n
\n \n

\n Si può usare il metodo della massima verosomiglianza per ottenere uno stimatore di una popolazione X.\n

\n

\n Lo stimatore di {r`\\theta`} così ottenuto sarà indicato aggiungendo un cappellino e una L a \\theta: {r`\\widehat{\\theta}_L`}\n

\n

\n Consiste nel trovare il massimo assoluto {r`\\widehat{\\theta}_L`} della la funzione di verosomiglianza {r`L`}:\n

\n

\n {r`L(x_1, ..., x_n; \\theta) = \\prod_{i=1}^n f_X(x_i; \\theta)`}\n

\n

\n Gli stimatori di massima verosomiglianza sono asintoticamente corretti, consistenti in probabilità e asintoticamente normali.\n

\n \n \n

\n Gli stimatori di massima verosomiglianza godono delle seguenti proprietà:\n

\n
    \n
  • Sono asintoticamente corretti.
    • \n
  • Sono consistenti in probabilità.
    • \n
  • Sono asintoticamente normali.
    • \n
  • Sono invarianti: {r`\\widehat{g(\\theta)}_L = g(\\widehat{\\theta}_L)`}
    • \n
\n \n
\n
\n \n

\n Per il metodo dei momenti oppure per il metodo della massima verosomiglianza:\n

\n

\n {r`\\widehat{p}_M = \\widehat{p}_L = \\overline{X}_n`}\n

\n \n \n

\n Per il metodo dei momenti oppure per il metodo della massima verosomiglianza:\n

\n

\n {r`\\widehat{\\mu}_M = \\widehat{\\mu}_L = \\overline{X}_n`}\n

\n \n \n

\n Per il metodo dei momenti oppure per il metodo della massima verosomiglianza:\n

\n

\n {r`\\widehat{\\lambda}_M = \\widehat{\\lambda}_L = \\frac{1}{\\overline{X}_n}`}\n

\n \n \n

\n Per il metodo della massima verosomiglianza:\n

\n
    \n
  • {r`\\widehat{\\mu}_L = \\overline{X}_n`}

    • \n
  • {r`\\widehat{\\sigma^2}_L = \\frac{\\sum (X_i - \\overline{X}_n)^2 }{n}`}
    • \n
\n \n
\n
\n \n
\n \"intervallo di confidenza al 95%\"\n
\n

\n L'intervallo di valori di \\theta all'interno del quale siamo \"più o meno sicuri\" si trovi il valore effettivo:\n

\n

\n L'intervallo di confidenza a N della stima {r`\\widehat{W}`} è l'intervallo ]a, b[ tale che:\n

\n

\n {r`P( a < W < b ) = N`}\n

\n

\n Può anche essere unilatero nel caso limiti la stima in una sola direzione, positiva o negativa.\n

\n \n
\n
\n \n

\n Se conosciamo la varianza di una normale, allora possiamo ricavare velocemente gli intervalli di confidenza all'\\alpha% con queste formule:\n

\n
    \n
  • Intervalli bilateri: {r`\\mu \\in \\left[ \\overline{x}_n - z_{1 - \\frac{\\alpha}{2}} \\cdot \\sqrt{\\frac{\\sigma^2}{n}}, \\overline{x}_n + z_{1 - \\frac{\\alpha}{2}} \\cdot \\sqrt{\\frac{\\sigma^2}{n}} \\right]`}
    • \n
  • Intervallo unilatero da sinistra: {r`\\mu \\in \\left( -\\infty, \\overline{x}_n + z_{1 - \\frac{\\alpha}{2}} \\cdot \\sqrt{\\frac{\\sigma^2}{n}} \\right]`}
    • \n
  • Intervallo unilatero da destra: {r`\\mu \\in \\left[ \\overline{x}_n - z_{1 - \\frac{\\alpha}{2}} \\cdot \\sqrt{\\frac{\\sigma^2}{n}}, +\\infty \\right)`}
    • \n
\n \n \n

\n Se non conosciamo la varianza di una normale, allora possiamo ricavare velocemente gli intervalli di confidenza all'\\alpha% con queste formule:\n

\n
    \n
  • Intervalli bilateri: {r`\\mu \\in \\left[ \\overline{x}_n - t_{1 - \\frac{\\alpha}{2}; n-1} \\cdot \\sqrt{\\frac{s_n^2}{n}}, \\overline{x}_n + t_{1 - \\frac{\\alpha}{2}; n-1} \\cdot \\sqrt{\\frac{s_n^2}{n}} \\right]`}
    • \n
  • Intervallo unilatero da sinistra: {r`\\mu \\in \\left( -\\infty, \\overline{x}_n + t_{1 - \\frac{\\alpha}{2}; n-1} \\cdot \\sqrt{\\frac{s_n^2}{n}} \\right]`}
    • \n
  • Intervallo unilatero da destra: {r`\\mu \\in \\left[ \\overline{x}_n - t_{1 - \\frac{\\alpha}{2}; n-1} \\cdot \\sqrt{\\frac{s_n^2}{n}}, +\\infty \\right)`}
    • \n
\n

\n {r`t_{\\alpha, v}`} è un quantile della distribuzione di Student di parametro v.\n

\n \n
\n
\n \n

\n L'intervallo di confidenza per la proprorzione di una bernoulliana qualsiasi si ottiene da questa formula:\n

\n

\n {r`p \\in \\left[ \\overline{p} - z_{1 - \\frac{\\alpha}{2}} \\cdot \\sqrt{\\frac{\\overline{p} \\cdot (1 - \\overline{p})}{n+4}}, \\overline{p} + z_{1 - \\frac{\\alpha}{2}} \\cdot \\sqrt{\\frac{\\overline{p} \\cdot (1 - \\overline{p})}{n+4}} \\right]`}\n

\n \n
\n
\n \n

\n L'intervallo di confidenza per la media di una qualsiasi popolazione si ottiene da questa formula:\n

\n

\n {r`m \\in \\left[ \\overline{x}_n - z_{1 - \\frac{\\alpha}{2}} \\cdot \\sqrt{\\frac{s^2_n}{n}}, \\overline{x}_n + z_{1 - \\frac{\\alpha}{2}} \\cdot \\sqrt{\\frac{s^2_n}{n}} \\right]`}\n

\n \n
\n
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\\overline{x}_n - z_{1 - \\frac{\\alpha}{2}} \\cdot \\sqrt{\\frac{\\sigma^2}{n}}, +\\infty \\right)"]);return c=function(){return n},n}function s(){var n=We(["mu in left( -infty, overline{x}_n + z_{1 - \frac{alpha}{2}} cdot sqrt{\frac{sigma^2}{n}} \right]"],["\\mu \\in \\left( -\\infty, \\overline{x}_n + z_{1 - \\frac{\\alpha}{2}} \\cdot \\sqrt{\\frac{\\sigma^2}{n}} \\right]"]);return s=function(){return n},n}function p(){var n=We(["mu in left[ overline{x}_n - z_{1 - \frac{alpha}{2}} cdot sqrt{\frac{sigma^2}{n}}, overline{x}_n + z_{1 - \frac{alpha}{2}} cdot sqrt{\frac{sigma^2}{n}} \right]"],["\\mu \\in \\left[ \\overline{x}_n - z_{1 - \\frac{\\alpha}{2}} \\cdot \\sqrt{\\frac{\\sigma^2}{n}}, \\overline{x}_n + z_{1 - \\frac{\\alpha}{2}} \\cdot \\sqrt{\\frac{\\sigma^2}{n}} \\right]"]);return p=function(){return n},n}function d(){var n=We(["P( a < W < b ) = N"]);return d=function(){return n},n}function m(){var n=We(["widehat{W}"],["\\widehat{W}"]);return m=function(){return n},n}function 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zn=function(){return n},n}function En(){var n=We(["\n \boldsymbol{C_X} = \n \begin{bmatrix}\n Var(X_1) & Cov(X_1, X_2) & Cov(X_1, X_3)\\\n Cov(X_2, X_1) & Var(X_2) & Cov(X_2, X_3)\\\n Cov(X_3, X_1) & Cov(X_3, X_2) & Var(X_3)\n end{bmatrix}\n "],["\n \\boldsymbol{C_X} = \n \\begin{bmatrix}\n Var(X_1) & Cov(X_1, X_2) & Cov(X_1, X_3)\\\\\n Cov(X_2, X_1) & Var(X_2) & Cov(X_2, X_3)\\\\\n Cov(X_3, X_1) & Cov(X_3, X_2) & Var(X_3)\n \\end{bmatrix}\n "]);return En=function(){return n},n}function Pn(){var n=We(["\boldsymbol{X}"],["\\boldsymbol{X}"]);return Pn=function(){return n},n}function xn(){var n=We(["\boldsymbol{C_X}"],["\\boldsymbol{C_X}"]);return xn=function(){return n},n}function kn(){var n=We(["Cov(X, Y) = 0"]);return kn=function(){return n},n}function Ln(){var n=We(["Cov(X + Y, V + W) = Cov(X, Y) + Cov(X, W) + Cov(Y, V) + Cov(Y, W)"]);return Ln=function(){return n},n}function Fn(){var n=We(["Cov(alpha X, \beta Y) = alpha cdot \beta cdot Cov(X, Y)"],["Cov(\\alpha X, \\beta Y) = \\alpha 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\frac{1}{2})"],["P(X \\geq k) \\simeq P(Y \\geq k - \\frac{1}{2})"]);return Gn=function(){return n},n}function On(){var n=We(["P(X leq k) simeq P(Y leq k + \frac{1}{2})"],["P(X \\leq k) \\simeq P(Y \\leq k + \\frac{1}{2})"]);return On=function(){return n},n}function Rn(){var n=We(["P(X < k) simeq P(Y leq k - \frac{1}{2})"],["P(X < k) \\simeq P(Y \\leq k - \\frac{1}{2})"]);return Rn=function(){return n},n}function Kn(){var n=We(["(k - \frac{1}{2}, k + \frac{1}{2})"],["(k - \\frac{1}{2}, k + \\frac{1}{2})"]);return Kn=function(){return n},n}function jn(){var n=We(["Bin(n, p) approx Nor(n cdot p, n cdot p cdot q)"],["Bin(n, p) \\approx Nor(n \\cdot p, n \\cdot p \\cdot q)"]);return jn=function(){return n},n}function Zn(){var n=We(["Bin(n, p) approx Poi(n cdot p)"],["Bin(n, p) \\approx Poi(n \\cdot p)"]);return Zn=function(){return n},n}function Qn(){var n=We(["Ipe(N, K, n) approx Bin(n, \frac{K}{N})"],["Ipe(N, K, n) \\approx Bin(n, \\frac{K}{N})"]);return Qn=function(){return 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qi(){var n=We(["\n f_X (x) = \begin{cases}\n 0 qquad qquad qquad qquad qquad x < 0\\\n \frac{1}{(n-1)!} cdot lambda^n cdot x^{n-1} cdot e^{-lambda cdot x} quad k > 0\n end{cases}"],["\n f_X (x) = \\begin{cases}\n 0 \\qquad \\qquad \\qquad \\qquad \\qquad x < 0\\\\\n \\frac{1}{(n-1)!} \\cdot \\lambda^n \\cdot x^{n-1} \\cdot e^{-\\lambda \\cdot x} \\quad k > 0\n \\end{cases}"]);return qi=function(){return n},n}function zi(){var n=We(["Gamma(n, lambda)"],["\\Gamma(n, \\lambda)"]);return zi=function(){return n},n}function Ei(){var n=We(["lambda"],["\\lambda"]);return Ei=function(){return n},n}function Pi(){var n=We(["P([X > s + t | X > s]) = P([X > t])"]);return Pi=function(){return n},n}function xi(){var n=We(["Var(X) = \frac{1}{lambda^2}"],["Var(X) = \\frac{1}{\\lambda^2}"]);return xi=function(){return n},n}function ki(){var n=We(["E(X) = \frac{1}{lambda}"],["E(X) = \\frac{1}{\\lambda}"]);return ki=function(){return n},n}function Li(){var n=We(["m_X (t) : { t | t < lambda } in mathbb{R} 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n=We(["E(X) = n cdot p \to mu \neq 0"],["E(X) = n \\cdot p \\to \\mu \\neq 0"]);return Ti=function(){return n},n}function Gi(){var n=We(["p \to 0"],["p \\to 0"]);return Gi=function(){return n},n}function Oi(){var n=We(["n \to +infty"],["n \\to +\\infty"]);return Oi=function(){return n},n}function Ri(){var n=We(["X sim Bin(n, p)"],["X \\sim Bin(n, p)"]);return Ri=function(){return n},n}function Ki(){var n=We(["Var(X) = n cdot \frac{K}{N} cdot \frac{N - K}{N} cdot \frac{N - n}{N - 1}"],["Var(X) = n \\cdot \\frac{K}{N} \\cdot \\frac{N - K}{N} \\cdot \\frac{N - n}{N - 1}"]);return Ki=function(){return n},n}function ji(){var n=We(["E(X) = n cdot \frac{K}{N}"],["E(X) = n \\cdot \\frac{K}{N}"]);return ji=function(){return n},n}function Zi(){var n=We(["f_X (k) : {0..n} in mathbb{N} = \frac{\binom{K}{k} cdot \binom{N - K}{n - k}}{\binom{N}{n}}"],["f_X (k) : \\{0..n\\} \\in \\mathbb{N} = \\frac{\\binom{K}{k} \\cdot \\binom{N - K}{n - k}}{\\binom{N}{n}}"]);return Zi=function(){return 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q^{k - 1} p"],["f_X (k) : \\mathbb{N} = q^{k - 1} p"]);return vl=function(){return n},n}function bl(){var n=We(["Var(X) = n cdot p cdot q"],["Var(X) = n \\cdot p \\cdot q"]);return bl=function(){return n},n}function _l(){var n=We(["E(X) = n cdot p"],["E(X) = n \\cdot p"]);return _l=function(){return n},n}function Xl(){var n=We(["m_X (t) = (q + p cdot e^t) ^ n"],["m_X (t) = (q + p \\cdot e^t) ^ n"]);return Xl=function(){return n},n}function gl(){var n=We(["f_X (k) : {0..n} = \binom{n}{k} cdot p^k cdot q^{n - k}"],["f_X (k) : \\{0..n\\} = \\binom{n}{k} \\cdot p^k \\cdot q^{n - k}"]);return gl=function(){return n},n}function hl(){var n=We(["Bin(n, p)"]);return hl=function(){return n},n}function ql(){var n=We(["\n f_X (k) : {0, 1} = \begin{cases}\n p quad se k = 1\\\n q quad se k = 0\\\n 0 quad altrimenti\n end{cases} = p^x cdot q^{1 - k}"],["\n f_X (k) : \\{0, 1\\} = \\begin{cases}\n p \\quad se\\ k = 1\\\\\n q \\quad se\\ k = 0\\\\\n 0 \\quad altrimenti\n \\end{cases} = p^x \\cdot q^{1 - k}"]);return ql=function(){return n},n}function zl(){var n=We(["Ber(p)"]);return zl=function(){return n},n}function El(){var n=We(["X sim Distribuzione()"],["X \\sim Distribuzione()"]);return El=function(){return n},n}function Pl(){var n=We(["H_X (t) = E ( e^{i cdot t cdot X} )"],["H_X (t) = E ( e^{i \\cdot t \\cdot X} )"]);return Pl=function(){return n},n}function xl(){var n=We(["m_X (t) = E( e^{t cdot X} )"],["m_X (t) = E( e^{t \\cdot X} )"]);return xl=function(){return n},n}function kl(){var n=We(["\n mu_k = E ( X^k ) = \begin{cases}\n sum_i x_i^k p_X (x_i) qquad nel discreto\\\n \\\n int_{-infty}^{+infty} x^k f_X (x) dx qquad nel continuo\n end{cases}"],["\n \\mu_k = E ( X^k ) = \\begin{cases}\n \\sum_i x_i^k p_X (x_i) \\qquad nel\\ discreto\\\\\n \\\\\n \\int_{-\\infty}^{+\\infty} x^k f_X (x) dx \\qquad nel\\ continuo\n \\end{cases}"]);return kl=function(){return n},n}function Ll(){var n=We(["\forall epsilon > 0, P([ left| X - E(X) \right| < epsilon]) geq 1 - \frac{Var(X)}{epsilon^2}"],["\\forall \\epsilon > 0, P([ \\left| X - E(X) \\right| < \\epsilon]) \\geq 1 - \\frac{Var(X)}{\\epsilon^2}"]);return Ll=function(){return n},n}function Fl(){var n=We(["\forall epsilon > 0, P([ left| X - E(X) \right| geq epsilon]) leq \frac{Var(X)}{epsilon^2}"],["\\forall \\epsilon > 0, P([ \\left| X - E(X) \\right| \\geq \\epsilon]) \\leq \\frac{Var(X)}{\\epsilon^2}"]);return Fl=function(){return n},n}function yl(){var n=We(["\frac{Var(X)}{epsilon^2}"],["\\frac{Var(X)}{\\epsilon^2}"]);return yl=function(){return n},n}function Nl(){var n=We(["epsilon"],["\\epsilon"]);return Nl=function(){return n},n}function Vl(){var n=We(["E(X) = overline{k} cdot P(X < k) + k cdot P(X geq k)"],["E(X) = \\overline{k} \\cdot P(X < k) + k \\cdot P(X \\geq k)"]);return Vl=function(){return n},n}function Sl(){var n=We(["P(X geq k)"],["P(X \\geq k)"]);return Sl=function(){return n},n}function Yl(){var n=We(["P(X < k)"]);return Yl=function(){return n},n}function wl(){var n=We(["\forall k > 0, P([X geq k]) leq \frac{E(X)}{k}"],["\\forall k > 0, P([X \\geq k]) \\leq \\frac{E(X)}{k}"]);return wl=function(){return n},n}function Cl(){var n=We(["Var(X) = E( (X - E(X) )^2 ) = E ( X^2 ) - (E(X))^2"]);return Cl=function(){return n},n}function Il(){var n=We(["\frac{n}{100}"],["\\frac{n}{100}"]);return Il=function(){return n},n}function Dl(){var n=We(["x_{0.75}"]);return Dl=function(){return n},n}function Ml(){var n=We(["x_{0.25}"]);return Ml=function(){return n},n}function Ul(){var n=We(["x_{0.5}"]);return Ul=function(){return n},n}function Bl(){var n=We(["P([X < x_{alpha}]) leq alpha leq P([X leq x_{alpha}])"],["P([X < x_{\\alpha}]) \\leq \\alpha \\leq P([X \\leq x_{\\alpha}])"]);return Bl=function(){return n},n}function Al(){var n=We(["0 leq alpha leq 1"],["0 \\leq \\alpha \\leq 1"]);return Al=function(){return n},n}function Tl(){var n=We(["x_{alpha}"],["x_{\\alpha}"]);return Tl=function(){return n},n}function Gl(){var n=We(["E(X) = int_{-infty}^{+infty} f_X (x) cdot x cdot dx"],["E(X) = \\int_{-\\infty}^{+\\infty} f_X (x) \\cdot x \\cdot dx"]);return Gl=function(){return n},n}function Ol(){var n=We(["E(X) = sum_i P(X = x_i) cdot x_i"],["E(X) = \\sum_i P(X = x_i) \\cdot x_i"]);return Ol=function(){return n},n}function Rl(){var n=We(["E(X) = int_0^{+infty} (1 - F_X (t)) dt - int_{-infty}^{0} F_X (t) dt"],["E(X) = \\int_0^{+infty} (1 - F_X (t)) dt - \\int_{-\\infty}^{0} F_X (t) dt"]);return Rl=function(){return n},n}function Kl(){var n=We(["f_Y (y) = int_{g(a)}^{g(b)} f_X ( g^{-1} (x) ) g^{-2} (x)"],["f_Y (y) = \\int_{g(a)}^{g(b)} f_X ( g^{-1} (x) ) g^{-2} (x)"]);return Kl=function(){return n},n}function jl(){var n=We(["P([X = x_0]) = lim_{t \to x^+_0} F_X (t) - lim_{t \to x^-_0} F_X (t)"],["P([X = x_0]) = \\lim_{t \\to x^+_0} F_X (t) - \\lim_{t \\to x^-_0} F_X (t)"]);return jl=function(){return n},n}function Zl(){var n=We(["\forall x_0 in mathbb{R}, F_X (x_0) = lim_{t \to x^+_0} F_X (t)"],["\\forall x_0 \\in \\mathbb{R}, F_X (x_0) = 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\\right \\}"]);return Te=function(){return n},n}function Ge(){var n=We(["E cap F = left { 1 \right }"],["E \\cap F = \\left \\{ 1 \\right \\}"]);return Ge=function(){return n},n}function Oe(){var n=We(["\bar{E} = left { 3, 4, 5, 6 \right }"],["\\bar{E} = \\left \\{ 3, 4, 5, 6 \\right \\}"]);return Oe=function(){return n},n}function Re(){var n=We(["E = left { 1, 2 \right }"],["E = \\left \\{ 1, 2 \\right \\}"]);return Re=function(){return n},n}function Ke(){var n=We(["omega = 1"],["\\omega = 1"]);return Ke=function(){return n},n}function je(){var n=We(["Omega = left { 1, 2, 3, 4, 5, 6 \right }"],["\\Omega = \\left \\{ 1, 2, 3, 4, 5, 6 \\right \\}"]);return je=function(){return n},n}function Ze(){var n=We(["P(E) = \frac{successi}{prove totali}"],["P(E) = \\frac{successi}{prove\\ totali}"]);return Ze=function(){return n},n}function Qe(){var n=We(["P(E) = \frac{casi favorevoli}{casi possibili}"],["P(E) = \\frac{casi\\ favorevoli}{casi\\ possibili}"]);return Qe=function(){return n},n}function We(n,i){return i||(i=n.slice(0)),n.raw=i,n}var Je=l("mbOI"),He=l("ke5e"),$e=l("o8Jj"),na=l("ZHMS"),ia=String.raw;i.default=function(){return n(Je.k.Provider,{value:!1},n(Je.j.Provider,{value:Je.l.INLINE},n("div",null,n("h1",null,"Statistica ed Elementi di Probabilità"),n(Je.p,{title:"Tipi di probabilità"},n(Je.o,{title:"Classica"},n("p",null,n(Je.i,null,ia(Qe())))),n(Je.o,{title:"Frequentista"},n("p",null,n(Je.i,null,ia(Ze())))),n(Je.o,{title:"Soggettiva"},n("p",null,"Il prezzo che un individuo coerente riterrebbe equo per ricevere ",n("b",null,"1")," nel caso l'evento si verificasse e ",n("b",null,"0")," nel caso l'evento non si verificasse."))),n(Je.p,{title:"Linguaggio matematico"},n(Je.o,{title:"Spazio campionario"},n("blockquote",null,'"omegone"'),n("p",null,"L'",n("b",null,"insieme")," di tutti gli esiti possibili di un esperimento."),n("p",null,n(Je.i,null,ia(je())))),n(Je.o,{title:"Esito"},n("blockquote",null,'"omeghino"'),n("p",null,"Un ",n("b",null,"elemento")," dello spazio campionario."),n("p",null,n(Je.i,null,ia(Ke())))),n(Je.o,{title:"Evento"},n("blockquote",null,'"e"'),n("p",null,"Un ",n("b",null,"sottoinsieme")," dello spazio campionario."),n("p",null,n(Je.i,null,ia(Re()))),n("p",null,"Lo spazio campionario stesso è un ",n("b",null,"evento certo"),".")),n(Je.o,{title:"Not"},n("blockquote",null,'"not e"'),n("p",null,"Il ",n("b",null,"complementare")," di un sottoinsieme."),n("p",null,n(Je.i,null,ia(Oe())))),n(Je.o,{title:"And"},n("blockquote",null,'"e intersecato effe"'),n("p",null,"L'",n("b",null,"intersezione")," di più sottoinsiemi."),n("p",null,n(Je.i,null,ia(Ge())))),n(Je.o,{title:"Or"},n("blockquote",null,'"e unito a effe"'),n("p",null,"L'",n("b",null,"unione")," di più sottoinsiemi."),n("p",null,n(Je.i,null,ia(Te())))),n(Je.o,{title:"Differenza"},n("blockquote",null,'"e meno effe"'),n("p",null,n(Je.i,null,ia(Ae())))),n(Je.o,{title:"Implicazione"},n("blockquote",null,'"e contenuto in effe"'),n("p",null,"L'",n("b",null,"inclusione")," del primo insieme in un altro."),n("p",null,n(Je.i,null,ia(Be()))),n("p",null,"Se si verifica ",n(Je.i,null,"E"),", allora si verifica anche ",n(Je.i,null,"F"),".")),n(Je.o,{title:"Evento impossibile"},n("blockquote",null,'"e è impossibile"'),n("p",null,"Un sottoinsieme ",n("b",null,"vuoto"),"."),n("p",null,n(Je.i,null,ia(Ue())))),n(Je.o,{title:"Mutua esclusione"},n("blockquote",null,'"e ed effe si escludono mutualmente"'),n("p",null,"La ",n("b",null,"disgiunzione")," di due insiemi."),n("p",null,n(Je.i,null,ia(Me()))))),n(Je.p,null,n(Je.o,{title:"Famiglia degli eventi"},n("blockquote",null,'"famiglia effe"'),n("p",null,"I sottoinsiemi dello spazio campionario formano una ",n("b",null,"famiglia")," di sottoinsiemi detta ",n("i",null,"famiglia degli eventi"),"."),n("p",null,n(Je.i,null,ia(De()))),n("p",null,"Qualsiasi sottoinsieme appartenente a ",n(Je.i,null,ia(Ie()))," è considerato un evento.")),n(Je.o,{title:n("span",null,n(Je.i,null,ia(Ce())),"-algebra")},n("blockquote",null,'"sigma algebra"'),n("p",null,"Se la famiglia degli eventi soddisfa questi tre requisiti, allora viene detta ",n("i",null,n(Je.i,null,ia(we())),"-algebra"),":"),n("ol",null,n("li",null,"Lo spazio campionario è un evento: ",n(Je.i,null,ia(Ye()))),n("li",null,"Se un sottoinsieme è un evento, allora anche il suo complementare lo è: ",n(Je.i,null,ia(Se()))),n("li",null,"Se due sottoinsiemi sono eventi, allora lo sono anche la loro unione e intersezione: ",n(Je.i,null,ia(Ve())))),n("p",null,"Un esempio: ",n(Je.i,null,ia(Ne()))))),n(Je.p,null,n(Je.o,{title:"Partizione"},n("blockquote",null,'"la partizione e composta da e uno, e due, e tre..."'),n("p",null,"Un insieme di esiti e eventi:"),n("ul",null,n("li",null,n("b",null,"Finito"),"."),n("li",null,"In cui tutti gli eventi hanno ",n("b",null,"probabilità diversa da 0"),"."),n("li",null,"In cui tutti gli eventi sono ",n("b",null,"mutualmente 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probabilità."),n("p",null,n(Je.i,null,ia(ze()))))),n(Je.p,{title:"Conseguenze degli assiomi"},n(Je.o,{title:"Probabilità di un evento negato"},n("p",null,"La probabilità di un evento negato è uguale a 1 meno la probabilità dell'evento non negato."),n("p",null,n(Je.i,null,ia(qe())))),n(Je.o,{title:"Probabilità di un evento incluso"},n("p",null,"La probabilità di un evento incluso in un altro è sempre minore o uguale alla probabilità dell'evento in cui è incluso."),n("p",null,n(Je.i,null,ia(he())))),n(Je.o,{title:"Unione"},n("p",null,"La probabilità di un evento unito a un altro è uguale alla somma delle probabilità dei due eventi meno la probabilità della loro intersezione."),n("p",null,n(Je.i,null,ia(ge()))),n(He.a,null,"Sommando le probabilità dei due eventi, l'intersezione viene contata due volte, e va quindi rimossa!"))),n(Je.p,{title:"Spazi equiprobabili"},n(Je.o,{title:"Cosa sono?"},n("p",null,"Spazi campionari in cui ci sono un numero finito di esiti e ogni esito ha la stessa probabilità di verificarsi."),n("p",null,n(Je.i,null,ia(Xe())))),n(Je.o,{title:"Spazi equiprobabili geometrici"},n("p",null,"Gli spazi campionari possono avere un numero infinito di esiti: sono ",n("i",null,"equiprobabili geometrici")," se nessun esito è privilegiato rispetto agli altri."))),n(Je.p,{title:"Calcolo combinatorio"},n(Je.o,{title:"Disposizioni"},n("p",null,"Estraggo un numero, da un sacchetto con ",n(Je.i,null,"n")," numeri, mi segno che numero ho estratto e lo ",n("b",null,"tengo fuori dal sacchetto"),". Ripeto per ",n(Je.i,null,"k")," volte."),n("p",null,n("b",null,"Tengo conto")," dell'ordine in cui ho estratto i numeri."),n("p",null,n(Je.i,null,ia(_e())))),n(Je.o,{title:"Disposizioni con ripetizione"},n("p",null,"Estraggo un numero, da un sacchetto con ",n(Je.i,null,"n")," numeri, mi segno che numero ho estratto e lo ",n("b",null,"rimetto nel sacchetto"),". Ripeto per ",n(Je.i,null,"k")," volte."),n("p",null,n("b",null,"Tengo conto")," dell'ordine in cui ho estratto i numeri."),n("p",null,n(Je.i,null,ia(be())))),n(Je.o,{title:"Combinazioni"},n("p",null,"Estraggo un numero, da un sacchetto con ",n(Je.i,null,"n")," numeri, mi segno che numero ho estratto e lo ",n("b",null,"tengo fuori dal sacchetto"),". Ripeto per ",n(Je.i,null,"k")," volte."),n("p",null,n("b",null,"Non mi interessa")," l'ordine in cui ho estratto i numeri."),n("p",null,n(Je.i,null,ia(ve())))),n(Je.o,{title:"Combinazioni con ripetizione"},n("p",null,"Estraggo un numero, da un sacchetto con ",n(Je.i,null,"n")," numeri, mi segno che numero ho estratto e lo ",n("b",null,"rimetto nel sacchetto"),". Ripeto per ",n(Je.i,null,"k")," volte."),n("p",null,n("b",null,"Non mi interessa")," l'ordine in cui ho estratto i numeri."),n("p",null,n(Je.i,null,ia(fe())))),n(Je.o,{title:"Permutazioni"},n("p",null,"Estraggo ",n(Je.i,null,"n")," numeri e guardo in quanti ordini diversi li posso mettere."),n("p",null,n(Je.i,null,ia(me()))))),n(Je.p,{title:"Probabilità condizionata"},n(Je.o,{title:"Eventi condizionati"},n("blockquote",null,'"E dato F"'),n("p",null,"La probabilità che si verifichi ",n(Je.i,null,"E")," sapendo che ",n("b",null,"si è già verificato")," ",n(Je.i,null,"F"),"."),n("p",null,n(Je.i,null,ia(de()))),n(He.a,null,"Ricorda vagamente le pipe di ",n("code",null,"bash"),", però al contrario...")),n(Je.o,{title:"Eventi mutualmente esclusivi"},n("p",null,"Se due eventi sono mutualmente esclusivi, entrambe le loro probabilità condizionate saranno uguali a 0."),n("p",null,n(Je.i,null,ia(pe()))))),n(Je.p,null,n(Je.o,{title:"Regola della catena"},n("p",null,"Si può sfruttare la formula inversa della probabilità condizionata per calcolare catene di intersezioni:"),n("p",null,n(Je.i,null,ia(se()))))),n(Je.p,{title:"Le alternative"},n(Je.o,{title:"Legge delle alternative"},n("p",null,"La probabilità che si verifichi un evento è pari alla somma delle probabilità dell'evento stesso dati tutti gli eventi di una partizione."),n("p",null,n(Je.i,null,ia(ce())))),n(Je.o,{title:"Legge condizionata delle alternative"},n("p",null,"La legge delle alternative funziona anche quando ad essere partizionato è un ",n("b",null,"evento"),":"),n("p",null,n(Je.i,null,ia(ue())))),n(Je.o,{title:"Formula di Bayes"},n("p",null,"Tramite la ",n("i",null,"formula di Bayes")," possiamo risalire alla probabilità di un evento condizionato a un altro partendo dalla probabilità di quest'ultimo condizionato al primo:"),n("p",null,n(Je.i,null,ia(oe()))),n(He.a,null,"In pratica, invertiamo gli eventi."))),n(Je.p,{title:"Eventi indipendenti"},n(Je.o,{title:"Due eventi indipendenti"},n("blockquote",null,'"eventi indipendenti a due a due"'),n("p",null,"Se due eventi sono indipendenti, sapere che uno dei due si è verificato non influisce sulle probabilità che si sia verificato l'altro."),n("p",null,n(Je.i,null,ia(re())))),n(Je.o,{title:"Più eventi indipendenti"},n("blockquote",null,'"eventi indipendenti a tre a tre, a quattro a quattro, a cinque a cinque..."'),n("p",null,"Si può verificare l'indipendenza di più eventi alla volta:"),n("p",null,n(Je.i,null,ia(te()))),n("p",null,"Eventi indipendenti a due a due non sono per forza indipendenti a tre a tre, e viceversa.")),n(Je.o,{title:"Famiglia di eventi indipendenti"},n("p",null,"Un insieme di ",n(Je.i,null,"n")," eventi è una ",n("i",null,"famiglia di eventi indipendenti")," se, preso un qualsiasi numero di eventi da essa, essi risulteranno indipendenti."),n(He.a,null,"Tutti gli eventi provenienti da essa saranno indipendenti sia a due a due, sia a tre a tre, sia a quattro a quattro, e così via!"))),n(Je.p,{title:"Variabili aleatorie"},n(Je.o,{title:"Variabile aleatoria"},n("p",null,"Una funzione che fa corrispondere un numero reale a ogni possibile esito dello spazio campionario. ",n(Je.i,null,ia(ae())),".")),n(Je.o,{title:n("abbr",{title:"Nome artigianale dato da Steffo."},"Insieme di ripartizione")},n("p",null,"Ad ogni variabile aleatoria sono associati gli eventi ",n(Je.i,null,ia(ee())),", che contengono tutti gli esiti a cui la variabile aleatoria associa un valore minore o uguale a ",n(Je.i,null,"t"),"."),n("p",null,"Per definizione, tutte le variabili aleatorie devono rispettare questa condizione:"),n("p",null,n(Je.i,null,ia(le()))),n(He.a,null,"All'aumentare di t, l'insieme conterrà sempre più elementi.")),n(Je.o,{title:"Supporto"},n("blockquote",null,'"supporto di X"'),n("p",null,"Il ",n("b",null,"codominio")," della variabile aleatoria è il suo ",n("i",null,"supporto"),"."),n("p",null,"Per indicare che un valore ",n(Je.i,null,"x_0")," appartiene al supporto di ",n(Je.i,null,"X"),", si usa la notazione ",n(Je.i,null,"X \\mapsto x_0"),"."))),n(Je.p,{title:"Densità"},n(Je.o,{title:"Funzione probabilità"},n("p",null,"La ",n("i",null,"funzione probabilità")," ",n(Je.i,null,ia(ie()))," di una variabile aleatoria ",n("b",null,"discreta")," ",n(Je.i,null,"X")," è la funzione che associa ad ogni esito la sua probabilità:"),n("p",null,n(Je.i,null,ia(ne())))),n(Je.o,{title:"Funzione densità"},n("p",null,"La ",n("i",null,"funzione densità")," ",n(Je.i,null,ia($l()))," di una variabile aleatoria ",n("b",null,"continua")," ",n(Je.i,null,"X")," è l'equivalente continuo della funzione probabilità:"),n("p",null,n(Je.i,null,ia(Hl()))),n("p",null,"A differenza della funzione probabilità, è possibile che la funzione densità ",n("b",null,"non esista")," per una certa variabile aleatoria."),n(He.a,null,"Rappresenta \"quanta\" probabilità c'è in un'unità di x!"))),n(Je.p,{title:"Funzione di ripartizione"},n(Je.o,{title:"Definizione"},n("p",null,"Ogni variabile aleatoria ha una ",n("i",null,"funzione di ripartizione")," ",n(Je.i,null,ia(Jl()))," associata, che rappresenta la probabilità che la variabile aleatoria assuma un valore minore o uguale a ",n(Je.i,null,"t"),":"),n("p",null,"Si può dire che essa rappresenti la probabilità dell'evento ",n(Je.i,null,ia(Wl())),":"),n("p",null,n(Je.i,null,ia(Ql())))),n(Je.o,{title:"Proprietà della funzione"},n("ul",null,n("li",null,"È sempre ",n("b",null,"monotona crescente")," (non strettamente)."),n("br",null),n("li",null,"Vale ",n("b",null,"0")," a ",n(Je.i,null,"-\\infty")," e ",n("b",null,"1")," a ",n(Je.i,null,"+\\infty"),"."),n("br",null),n("li",null,"È ",n("b",null,"continua da destra"),": ",n(Je.i,null,ia(Zl()))))),n(Je.o,{title:"Probabilità di un valore"},n("p",null,"Possiamo usare la funzione di ripartizione per calcolare la probabilità di un certo valore reale:"),n("p",null,n(Je.i,null,ia(jl()))))),n(Je.p,{title:"Trasformazioni di variabili aleatorie"},n(Je.o,{title:"Nel discreto"},n("p",null,"Nel discreto basta abbinare un nuovo valore a ogni valore della variabile originale.")),n(Je.o,{title:"Nel continuo (invertibile)"},n("p",null,"Nel continuo applichiamo la formula dell'integrazione per sostituzione:"),n("p",null,n(Je.i,null,ia(Kl())))),n(Je.o,{title:"Nel... digitale"},n("p",null,"Trasformare variabili aleatorie è molto utile nell'informatica per creare distribuzioni partendo da una funzione ",n("a",{href:"https://docs.python.org/3/library/random.html#random.random"},n("code",null,"random()"))," che restituisce numeri da 0 a 1 con una distribuzione lineare."))),n(Je.p,{title:"Informazioni delle variabili aleatorie"},n(Je.o,{title:"Media"},n("p",null,"Ogni variabile aleatoria che ha una ",n("b",null,"funzione di ripartizione")," e un ",n("b",null,"supporto finito")," ha anche una ",n("i",null,"media")," (o ",n("i",null,"valore medio")," o ",n("i",null,"atteso"),"):"),n("p",null,n(Je.i,null,ia(Rl()))),n("p",null,"Nel discreto, si può calcolare con:"),n("p",null,n(Je.i,null,ia(Ol()))),n("p",null,"Nel continuo, si può calcolare con:"),n("p",null,n(Je.i,null,ia(Gl()))))),n(Je.p,null,n(Je.o,{title:"Moda"},n("p",null,"Valore per cui la ",n("b",null,"funzione probabilità")," o ",n("b",null,"funzione densità")," è ",n("b",null,"massima"),".")),n(Je.o,{title:"Quantili"},n("p",null,"Il ",n("i",null,"quantile")," ",n(Je.i,null,ia(Tl()))," di ordine ",n(Je.i,null,ia(Al()))," della variabile aleatoria ",n(Je.i,null,"X")," è il più piccolo numero tale che:"),n("p",null,n(Je.i,null,ia(Bl()))),n("p",null),n("p",null,"Il quantile di ordine 0.5 ",n(Je.i,null,ia(Ul()))," è detto ",n("i",null,"mediana"),"."),n("p",null,"I quantili di ordine 0.25 ",n(Je.i,null,ia(Ml()))," e 0.75 ",n(Je.i,null,ia(Dl()))," sono detti ",n("i",null,"quartili"),"."),n("p",null,"I quantili di ordine ",n(Je.i,null,ia(Il()))," sono detti ",n("i",null,n(Je.i,null,"n"),"-esima percentile"),".")),n(Je.o,{title:"Varianza"},n("p",null,"È un valore che indica quanto la variabile aleatoria si discosta generalmente dalla media:"),n("p",null,n(Je.i,null,ia(Cl()))))),n(Je.p,{title:"Disuguaglianze notevoli"},n(Je.o,{title:"Disuguaglianza di Markov"},n("p",null,"Data una variabile aleatoria non-negativa:"),n("p",null,n(Je.i,null,ia(wl()))),n("p",null,"Divide in due parti (",n(Je.i,null,ia(Yl()))," e ",n(Je.i,null,ia(Sl())),") la funzione X, la cui media risulterà uguale a:"),n("p",null,n(Je.i,null,ia(Vl())))),n(Je.o,{title:"Disuguaglianza di Čebyšëv"},n("blockquote",null,'"disuguaglianza di cebicev"'),n("p",null,"Se la variabile aleatoria ",n(Je.i,null,"X")," ha media e varianza, allora la probabilità che essa abbia un valore a più di ",n(Je.i,null,ia(Nl()))," di distanza dal valore medio è minore o uguale a ",n(Je.i,null,ia(yl())),"."),n("p",null,n(Je.i,null,ia(Fl()))),n("p",null,"E anche:"),n("p",null,n(Je.i,null,ia(Ll()))),n(He.a,null,"Serve per semplificare i calcoli quando la funzione di ripartizione è difficile da calcolare!"))),n(Je.p,{title:"Un momento...!"},n(Je.o,{title:"Momento"},n("p",null,"Il ",n("i",null,"momento")," ",n(Je.i,null,"k"),"-esimo di una variabile aleatoria è:"),n("p",null,n(Je.i,null,ia(kl()))),n(He.a,null,"La media di una variabile aleatoria è anche il suo primo momento.")),n(Je.o,{title:"Funzione generatrice dei momenti"},n("p",null,"La ",n("i",null,"funzione generatrice dei momenti")," è:"),n("p",null,n(Je.i,null,ia(xl()))),n("p",null,"Se due variabile aleatorie hanno la stessa funzione generatrice dei momenti, allora esse hanno la ",n("b",null,"stessa distribuzione"),"."),n("p",null,"E' la ",n("b",null,"trasformata di Laplace")," della variabile aleatoria di X.")),n(Je.o,{title:"Funzione caratteristica"},n("p",null,"La ",n("i",null,"funzione caratteristica")," è:"),n("p",null,n(Je.i,null,ia(Pl()))),n("p",null,"Se due variabile aleatorie hanno la stessa funzione caratteristica, allora esse hanno la ",n("b",null,"stessa distribuzione"),"."),n("p",null,"E' la ",n("b",null,"trasformata di Fourier")," della variabile aleatoria di X."))),n(Je.p,{title:"Prove e schemi"},n(Je.o,{title:"Variabile con distribuzione"},n("p",null,"Per dire che una variabile ha una certa distribuzione, si usa la notazione:"),n("p",null,n(Je.i,null,ia(El())))),n(Je.o,{title:"Prova di Bernoulli"},n("p",null,"Una prova con solo due possibili esiti: ",n($e.a,null,"successo")," e ",n(na.a,null,"insuccesso"),".")),n(Je.o,{title:"Schema di Bernoulli"},n("p",null,"Una sequenza di prove di Bernoulli per le quali le probabilità di successo e fallimento rimangono invariate."))),n(Je.p,{title:"Bernoulliana"},n(Je.o,{title:"Distribuzione bernoulliana"},n("p",null,"Una variabile aleatoria che rappresenta una prova di Bernoulli:"),n("ul",null,n("li",null,"vale ",n($e.a,null,"1")," in caso di ",n($e.a,null,"successo"),"."),n("li",null,"vale ",n(na.a,null,"0")," in caso di ",n(na.a,null,"insuccesso"),".")),n("p",null,"Il suo simbolo è ",n(Je.i,null,ia(zl())))),n(Je.o,{title:"Densità della bernoulliana"},n("p",null,"La distribuzione bernoulliana ha come densità:"),n("p",null,n(Je.i,null,ia(ql()))))),n(Je.p,{title:"Binomiale"},n(Je.o,{title:"Distribuzione binomiale"},n("p",null,"Una variabile aleatoria che conta il numero di successi di ",n(Je.i,null,"n")," prove di uno schema di Bernoulli."),n("p",null,"Il suo simbolo è ",n(Je.i,null,ia(hl())),".")),n(Je.o,{title:"Densità della binomiale"},n("p",null,"La binomiale ha come densità:"),n("p",null,n(Je.i,null,ia(gl())))),n(Je.o,{title:"Momenti della binomiale"},n("p",null,"La ",n("b",null,"funzione generatrice dei momenti")," della binomiale è:"),n("p",null,n(Je.i,null,ia(Xl()))),n("p",null,"La ",n("b",null,"media")," di una binomiale è:"),n("p",null,n(Je.i,null,ia(_l()))),n("p",null,"La ",n("b",null,"varianza")," di una binomiale è:"),n("p",null,n(Je.i,null,ia(bl()))))),n(Je.p,{title:"Geometrica"},n(Je.o,{title:"Distribuzione geometrica"},n("p",null,"Una variabile aleatoria che conta il numero di prove in uno schema di Bernoulli fino alla comparsa del primo successo."),n("p",null,"Il suo simbolo è ",n(Je.i,null,"Geo(p)"),".")),n(Je.o,{title:"Densità della geometrica"},n("p",null,"La geometrica ha come densità:"),n("p",null,n(Je.i,null,ia(vl())))),n(Je.o,{title:"Momenti della geometrica"},n("p",null,"La ",n("b",null,"funzione generatrice dei momenti")," della geometrica è:"),n("p",null,n(Je.i,null,ia(fl()))),n("p",null,"La ",n("b",null,"media")," della geometrica è:"),n("p",null,n(Je.i,null,ia(ml()))),n("p",null,"La ",n("b",null,"varianza")," della geometrica è:"),n("p",null,n(Je.i,null,ia(dl())))),n(Je.o,{title:"Assenza di memoria della geometrica"},n("p",null,"La geometrica non tiene conto degli eventi avvenuti in passato: ha la proprietà dell'assenza di memoria:"),n("p",null,n(Je.i,null,ia(pl()))),n(He.a,null,"Ovvero, riscalando opportunamente l'asse Y posso prendere come 0 qualsiasi punto dell'asse X."))),n(Je.p,{title:"Binomiale negativa"},n(Je.o,{title:"Distribuzione binomiale negativa"},n("p",null,"Una variabile aleatoria che conta il numero di prove in uno schema di Bernoulli necessarie perchè si verifichi l'",n(Je.i,null,"n"),"-esimo successo."),n("p",null,"Il suo simbolo è ",n(Je.i,null,ia(sl())),".")),n(Je.o,{title:"Densità della binomiale negativa"},n("p",null,"La binomiale negativa ha come densità:"),n("p",null,n(Je.i,null,ia(cl())))),n(Je.o,{title:"Momenti della binomiale negativa"},n("p",null,n("p",null,"La ",n("b",null,"funzione generatrice dei momenti")," della binomiale negativa è:"),n("p",null,n(Je.i,null,ia(ul()))),n("p",null,"La ",n("b",null,"media")," della binomiale negativa è:"),n("p",null,n(Je.i,null,ia(ol()))),n("p",null,"La ",n("b",null,"varianza")," della binomiale negativa è:"),n("p",null,n(Je.i,null,ia(rl())))))),n(Je.p,{title:"Geometrica traslata"},n(Je.o,{title:"Distribuzione geometrica traslata"},n("p",null,"Una variabile aleatoria che conta il numero ",n(Je.i,null,"k")," di insuccessi consecutivi in uno schema di Bernoulli:"),n("p",null,"Il suo simbolo rimane ",n(Je.i,null,ia(tl())),".")),n(Je.o,{title:"Densità della geometrica tralsata"},n("p",null,"La geometrica traslata ha come densità:"),n("p",null,n(Je.i,null,ia(al())))),n(Je.o,{title:"Momenti della geometrica traslata"},n("p",null,"La ",n("b",null,"funzione generatrice dei momenti")," della geometrica traslata è:"),n("p",null,n(Je.i,null,ia(el()))),n("p",null,"La ",n("b",null,"media")," della geometrica traslata è:"),n("p",null,n(Je.i,null,ia(ll()))),n("p",null,"La ",n("b",null,"varianza")," della geometrica è:"),n("p",null,n(Je.i,null,ia(il())))),n(Je.o,{title:"Assenza di memoria della geometrica traslata"},n("p",null,"La geometrica traslata non tiene conto degli eventi avvenuti in passato: ha la proprietà dell'assenza di memoria:"),n("p",null,n(Je.i,null,ia(nl()))),n(He.a,null,"Ovvero, riscalando opportunamente l'asse Y posso prendere come 0 qualsiasi punto dell'asse X."))),n(Je.p,{title:"Binomiale negativa traslata"},n(Je.o,{title:"Distribuzione binomiale negativa traslata"},n("p",null,"Una variabile aleatoria che conta il numero di insuccessi in uno schema di Bernoulli prima che si verifichi l'",n(Je.i,null,"n"),"-esimo successo."),n("p",null,"Il suo simbolo rimane ",n(Je.i,null,ia($i())),".")),n(Je.o,{title:"Densità della binomiale negativa traslata"},n("p",null,"La binomiale negativa traslata ha come densità:"),n("p",null,n(Je.i,null,ia(Hi())))),n(Je.o,{title:"Momenti della binomiale negativa traslata"},n("p",null,n("p",null,"La ",n("b",null,"funzione generatrice dei momenti")," della binomiale negativa traslata è:"),n("p",null,n(Je.i,null,ia(Ji()))),n("p",null,"La ",n("b",null,"media")," della binomiale negativa traslata è:"),n("p",null,n(Je.i,null,ia(Wi()))),n("p",null,"La ",n("b",null,"varianza")," della binomiale negativa traslata è:"),n("p",null,n(Je.i,null,ia(Qi())))))),n(Je.p,{title:"Ipergeometrica"},n(Je.o,{title:"Distribuzione ipergeometrica"},n("p",null,"Una variabile aleatoria che, sapendo il numero di successi ",n(Je.i,null,"K")," e di insuccessi ",n(Je.i,null,"N-K"),", conta quanti successi si otterrebbero se se ne estraessero ",n(Je.i,null,"n")," in blocco."),n("p",null,"Il suo simbolo è ",n(Je.i,null,"Ipe(N, K, n)"),".")),n(Je.o,{title:"Densità della ipergeometrica"},n("p",null,"La ipergeometrica ha come densità:"),n("p",null,n(Je.i,null,ia(Zi())))),n(Je.o,{title:"Momenti della ipergeometrica"},n("p",null,n("p",null,"La ",n("b",null,"funzione generatrice dei momenti")," della ipergeometrica è trascurabile."),n("p",null,"La ",n("b",null,"media")," della ipergeometrica è:"),n("p",null,n(Je.i,null,ia(ji()))),n("p",null,"La ",n("b",null,"varianza")," della ipergeometrica è:"),n("p",null,n(Je.i,null,ia(Ki())))))),n(Je.p,{title:"Poissoniana"},n(Je.o,{title:"Distribuzione poissoniana"},n("p",null,"Una variabile aleatoria che soddisfa tutte le seguenti caratteristiche:"),n("ul",null,n("li",null,"Binomiale: ",n(Je.i,null,ia(Ri()))),n("li",null,"Il numero di prove tende a infinito: ",n(Je.i,null,ia(Oi()))),n("li",null,"La probabilità di successo tende a 0: ",n(Je.i,null,ia(Gi()))),n("li",null,"La media è finita: ",n(Je.i,null,ia(Ti())))),n("p",null,"Il suo simbolo è ",n(Je.i,null,ia(Ai())))),n(Je.o,{title:"Densità della poissoniana"},n("p",null,"La poissoniana ha come densità:"),n("p",null,n(Je.i,null,ia(Bi())))),n(Je.o,{title:"Momenti della poissoniana"},n("p",null,n("p",null,"La ",n("b",null,"funzione generatrice dei momenti")," della poissoniana è:"),n("p",null,n(Je.i,null,ia(Ui()))),n("p",null,"La ",n("b",null,"media")," della poissoniana è:"),n("p",null,n(Je.i,null,ia(Mi()))),n("p",null,"La ",n("b",null,"varianza")," della poissoniana è:"),n("p",null,n(Je.i,null,ia(Di()))),n("p",null,"Gli altri momenti della poissoniana sono:"),n("ol",{start:2},n("li",null,n(Je.i,null,ia(Ii()))))))),n(Je.p,{title:"Un altro schema"},n(Je.o,{title:"Schema di Poisson"},n("p",null,"Una successione di ",n("b",null,"arrivi")," avvenuti in un certo arco temporale che:"),n("ul",null,n("li",null,"non sono sovrapposti."),n("li",null,"hanno intensità ",n(Je.i,null,ia(Ci()))," costante."),n("li",null,"avvengono indipendentemente gli uni dagli altri."))),n(Je.o,{title:"Processo di Poisson"},n("p",null,"Una variabile aleatoria ",n(Je.i,null,"N_t")," che conta il numero di arrivi di uno schema di Poisson di intensità ",n(Je.i,null,ia(wi()))," in un intervallo di tempo di durata ",n(Je.i,null,"t"),"."),n("p",null,"E' una distribuzione poissoniana con ",n(Je.i,null,ia(Yi())),": ",n(Je.i,null,ia(Si()))),n(He.a,null,"E' paragonabile a una bernoulliana: ogni successo corrisponde a un arrivo, mentre il tempo è il numero di prove effettuate (ma nel continuo)."))),n(Je.p,{title:"Esponenziale"},n(Je.o,{title:"Distribuzione esponenziale"},n("p",null,"Una variabile aleatoria che conta il tempo diwidehattesa prima del primo arrivo di un processo di Poisson di intensità ",n(Je.i,null,ia(Vi())),"."),n("p",null,"Il suo simbolo è ",n(Je.i,null,ia(Ni())),".")),n(Je.o,{title:"Densità dell'esponenziale"},n("p",null,"L'esponenziale ha come ",n("b",null,"densità"),":"),n("p",null,n(Je.i,null,ia(yi()))),n("p",null,"L'esponenziale ha come ",n("b",null,"funzione di ripartizione"),":"),n("p",null,n(Je.i,null,ia(Fi())))),n(Je.o,{title:"Momenti dell'esponenziale"},n("p",null,"La ",n("b",null,"funzione generatrice dei momenti")," dell'esponenziale è:"),n("p",null,n(Je.i,null,ia(Li()))),n("p",null,"La ",n("b",null,"media")," dell'esponenziale è:"),n("p",null,n(Je.i,null,ia(ki()))),n("p",null,"La ",n("b",null,"varianza")," dell'esponenziale è:"),n("p",null,n(Je.i,null,ia(xi())))),n(Je.o,{title:"Assenza di memoria della esponenziale"},n("p",null,"L'esponenziale non tiene conto degli eventi avvenuti in passato: ha la proprietà dell'assenza di memoria:"),n("p",null,n(Je.i,null,ia(Pi()))),n(He.a,null,"Ovvero, riscalando opportunamente l'asse Y posso prendere come 0 qualsiasi punto dell'asse X."))),n(Je.p,{title:"Legge gamma"},n(Je.o,{title:"Distribuzione gamma"},n("p",null,"Una variabile aleatoria che conta il tempo diwidehattesa prima dell'",n(Je.i,null,"n"),"-esimo arrivo di un processo di Poisson di intensità ",n(Je.i,null,ia(Ei())),"."),n("p",null,"Il suo simbolo è ",n(Je.i,null,ia(zi())),".")),n(Je.o,{title:"Densità della legge gamma"},n("p",null,"La legge gamma ha come densità:"),n("p",null,n(Je.i,null,ia(qi())))),n(Je.o,{title:"Momenti della legge gamma"},n("p",null,n("p",null,"La ",n("b",null,"funzione generatrice dei momenti")," della legge gamma è:"),n("p",null,n(Je.i,null,ia(hi()))),n("p",null,"La ",n("b",null,"media")," della legge gamma è:"),n("p",null,n(Je.i,null,ia(gi()))),n("p",null,"La ",n("b",null,"varianza")," della legge gamma è:"),n("p",null,n(Je.i,null,ia(Xi())))))),n(Je.p,{title:"Uniforme"},n(Je.o,{title:"Distribuzione uniforme"},n("p",null,"Una variabile aleatoria che può assumere qualsiasi valore in un intervallo ",n(Je.i,null,ia(_i()))," in modo equiprobabile."),n("p",null,"Il suo simbolo è ",n(Je.i,null,ia(bi()))),n("p",null,"Su di essa vale la seguente proprietà:"),n("p",null,n(Je.i,null,ia(vi())))),n(Je.o,{title:"Densità della distribuzione uniforme"},n("p",null,"La distribuzione uniforme ha come ",n("b",null,"densità"),":"),n("p",null,n(Je.i,null,ia(fi()))),n("p",null,"La distribuzione uniforme ha come ",n("b",null,"funzione di ripartizione"),":"),n("p",null,n(Je.i,null,ia(mi())))),n(Je.o,{title:"Momenti della distribuzione uniforme"},n("p",null,n("p",null,"La ",n("b",null,"funzione generatrice dei momenti")," della distribuzione uniforme è:"),n("p",null,n(Je.i,null,ia(di()))),n("p",null,"La ",n("b",null,"media")," della distribuzione uniforme è:"),n("p",null,n(Je.i,null,ia(pi()))),n("p",null,"La ",n("b",null,"varianza")," della distribuzione uniforme è:"),n("p",null,n(Je.i,null,ia(si())))))),n(Je.p,{title:"Normale o Gaussiana"},n(Je.o,{title:"Distribuzione normale"},n("p",null,"Una variabile aleatoria con una specifica distribuzione."),n("p",null,"Il suo simbolo è ",n(Je.i,null,ia(ci())),"."),n(He.a,null,n(Je.i,null,"\\mu")," e ",n(Je.i,null,"\\sigma^2")," sono rispettivamente la media e la varianza della distribuzione!")),n(Je.o,{title:"Densità della distribuzione normale"},n("p",null,"La distribuzione normale ha come densità:"),n("p",null,n(Je.i,null,ia(ui())))),n(Je.o,{title:"Momenti della distribuzione normale"},n("p",null,n("p",null,"La ",n("b",null,"funzione generatrice dei momenti")," della distribuzione normale è:"),n("p",null,n(Je.i,null,ia(oi()))),n("p",null,"La ",n("b",null,"media")," della distribuzione normale è:"),n("p",null,n(Je.i,null,ia(ri()))),n("p",null,"La ",n("b",null,"varianza")," della distribuzione normale è:"),n("p",null,n(Je.i,null,ia(ti())))))),n(Je.p,null,n(Je.o,{title:"Trasformazione della normale"},n("p",null,"Qualsiasi normale può essere trasformata in qualsiasi altra normale:"),n("p",null,n(Je.i,null,ia(ai())))),n(Je.o,{title:"Normale standard"},n("p",null,"La distribuzione normale standard ",n(Je.i,null,"Z")," è:"),n("p",null,n(Je.i,null,"Z \\sim Nor(0, 1)")),n("p",null,"La sua funzione di ripartizione è detta ",n(Je.i,null,ia(ei()))," e vale:"),n("p",null,n(Je.i,null,ia(li())))),n(Je.o,{title:"Quantili normali"},n("p",null,"Da un quantile ",n(Je.i,null,ia(ii()))," della normale standard è possibile risalire allo stesso quantile di qualsiasi altra normale:"),n("p",null,n(Je.i,null,ia(ni()))))),n(Je.p,null,n(Je.o,{title:"Gamma e normale"},n("p",null,"La distribuzione normale ha una particolare relazione con la distribuzione Gamma:"),n("p",null,n(Je.i,null,ia($n())))),n(Je.o,{title:"La funzione Chi"},n("blockquote",null,'"chi-quadro a un grado di libertà"'),n("p",null,"Esiste una distribuzione Gamma particolare:"),n("p",null,n(Je.i,null,ia(Hn()))),n("p",null,"Più chi-quadro possono essere sommate per aumentare i loro gradi di libertà:"),n("p",null,n(Je.i,null,ia(Jn())))),n(Je.o,{title:"T di Student"},n("p",null,"Un'altra funzione particolare è la funzione T di Student:"),n("p",null,n(Je.i,null,ia(Wn()))))),n(Je.p,{title:"Approssimazioni notevoli"},n(Je.o,{title:"Ipergeometrica e binomiale"},n("p",null,"La binomiale è come una ipergeometrica ma con ripetizioni, quindi per valori molto grandi di ",n(Je.i,null,"N")," rispetto a ",n(Je.i,null,"n"),", si può dire che:"),n("p",null,n(Je.i,null,ia(Qn())))),n(Je.o,{title:"Binomiale e poissoniana"},n("p",null,"La binomiale non è altro che una poissoniana a tempo discreto, quindi, se ",n(Je.i,null,"n")," è grande e ",n(Je.i,null,"n \\cdot p")," è nell'ordine di grandezza delle unità, allora:"),n("p",null,n(Je.i,null,ia(Zn())))),n(Je.o,{title:"Binomiale e normale"},n("p",null,"Per il Teorema di De Moivre-Laplace, se una binomiale ha una ",n(Je.i,null,"n")," grande e ",n(Je.i,null,"p")," non vicina a 0 o 1, si può approssimare con:"),n("p",null,n(Je.i,null,ia(jn())))),n(Je.o,{title:"Correzione di Yates"},n("p",null,"Passando da una variabile discreta ",n(Je.i,null,"X")," a una continua ",n(Je.i,null,"Y"),", per ogni valore discreto ",n(Je.i,null,"k"),' la probabilità viene "spalmata" su tutto l\'intervallo ',n(Je.i,null,ia(Kn())),":"),n("ul",null,n("li",null,n(Je.i,null,ia(Rn()))),n("li",null,n(Je.i,null,ia(On()))),n("li",null,n(Je.i,null,ia(Gn()))),n("li",null,n(Je.i,null,ia(Tn())))))),n(Je.p,{title:"Vettori aleatori"},n(Je.o,{title:"Vettore aleatorio"},n("p",null,"Un vettore ",n("b",null,"composto da variabili aleatorie"),"."),n("p",null,"Il suo simbolo generalmente è ",n(Je.i,null,ia(An()))," oppure ",n(Je.i,null,ia(Bn())),".")),n(Je.o,{title:"Funzioni di ripartizione"},n("p",null,"I vettori aleatori hanno più funzioni di ripartizione che si differenziano in base al numero di parametri."),n("p",null,"Se il numero di parametri coincide con la dimensione del vettore aleatorio, allora la funzione sarà una ",n("i",null,"funzione di ripartizione congiunta"),":"),n("p",null,n(Je.i,null,ia(Un()))),n("p",null,"Se il numero di parametri è minore della dimensione del vettore aleatorio, allora la funzione sarà una ",n("i",null,"funzione di ripartizione marginale"),":"),n("p",null,n(Je.i,null,ia(Mn())))),n(Je.o,{title:"Densità discreta"},n("p",null,"I vettori aleatori ",n("b",null,"discreti")," hanno più densità che si differenziano in base al numero di parametri."),n("p",null,"Se il numero di parametri coincide con la dimensione del vettore aleatorio, allora la funzione sarà una ",n("i",null,"densità congiunta"),":"),n("p",null,n(Je.i,null,ia(Dn()))),n("p",null,"Se il numero di parametri è minore della dimensione del vettore aleatorio, allora la funzione sarà una ",n("i",null,"densità marginale"),":"),n("p",null,n(Je.i,null,ia(In()))))),n(Je.p,{title:"Più variabili aleatorie"},n(Je.o,{title:"Indipendenza delle variabili aleatorie"},n("p",null,"Più variabili aleatorie sono indipendenti se, per qualsiasi scelta di intervalli ",n(Je.i,null,"A_i"),":"),n("p",null,n(Je.i,null,ia(Cn())))),n(Je.o,{title:"Media dei vettori aleatori"},n("p",null,"E' possibile calcolare la media di qualsiasi funzione ",n(Je.i,null,"g(X, Y)")," avente elementi del vettore come variabili:"),n("p",null,n(Je.i,null,ia(wn()))),n(He.a,null,"Solitamente si calcola la media di ",n(Je.i,null,"x \\cdot y"),"."),n("p",null,"Le medie di più variabili aleatorie si possono sommare:"),n("p",null,n(Je.i,null,ia(Yn()))))),n(Je.p,null,n(Je.o,{title:"Covarianza"},n("p",null,"Un ",n("b",null,"operatore")," che misura la correlazione di due variabili aleatorie."),n("p",null,"Si calcola con il valore atteso dei prodotti delle distanze dalla media:"),n("p",null,n(Je.i,null,ia(Sn()))),n("p",null,"Ha diverse proprietà:"),n("ul",null,n("li",null,"Il suo ",n("b",null,"valore nullo")," è 0: ",n(Je.i,null,ia(Vn()))),n("li",null,"E' ",n("b",null,"commutativa"),": ",n(Je.i,null,ia(Nn()))),n("li",null,"E' ",n("b",null,"semplificabile"),": ",n(Je.i,null,ia(yn()))),n("li",null,"E' ",n("b",null,"lineare"),": ",n(Je.i,null,ia(Fn()))),n("li",null,"E' ",n("b",null,"distributiva"),": ",n(Je.i,null,ia(Ln()))))),n(Je.o,{title:"Variabili incorrelate"},n("p",null,"Due variabili sono ",n("i",null,"variabili incorrelate")," se:"),n("p",null,n(Je.i,null,ia(kn()))),n("p",null,"Variabili indipendenti sono sempre incorrelate.")),n(Je.o,{title:"Matrice di covarianza"},n("p",null,"Una matrice ",n(Je.i,null,ia(xn()))," che contiene la covarianza tra tutte le variabili di un vettore aleatorio ",n(Je.i,null,ia(Pn())),":"),n("p",null,n(Je.i,null,ia(En()))),n("p",null,"E' sempre simmetrica e semidefinita positiva (tutti gli autovalori sono ",n(Je.i,null,"\\geq 0"),".")),n(Je.o,{title:"Coefficiente di correlazione"},n("p",null,"Un valore che misura come due variabili aleatorie sono correlate:"),n("p",null,n(Je.i,null,ia(zn()))),n("p",null,"E' sempre compreso tra -1 e 1:"),n("p",null,n(Je.i,null,ia(qn()))),n("p",null,"Vale esattamente -1 o 1 solo se esiste un legame lineare tra le due variaibli:"),n("p",null,n(Je.i,null,ia(hn())))),n(Je.o,{title:"Varianza di variabili aleatorie sommate"},n("p",null,"La varianza di due variabili aleatorie sommate è:"),n("p",null,n(Je.i,null,ia(gn()))),n(He.a,null,"Si dimostra applicando le proprietà della covarianza!"),n("p",null,"Se più variabili aleatorie ",n(Je.i,null,"X_i")," sono ",n("b",null,"indipendenti")," (",n(Je.i,null,ia(Xn())),"), allora:"),n("p",null,n(Je.i,null,ia(_n()))))),n(Je.p,{title:"Campioni"},n(Je.o,{title:"Campione casuale"},n("p",null,"Una ",n("b",null,"n-pla")," di variabili aleatorie con la stessa distribuzione della variabile aleatoria ",n(Je.i,null,"X"),' ("popolazione") ma ',n("b",null,"indipendenti")," tra loro."),n(He.a,null,"Le variabili aleatorie sono come un lazy-load in programmazione; quando ci sarà bisogno del loro valore numerico, esse si ",n("b",null,"realizzeranno")," nel loro valore.")),n(Je.o,{title:"Momento campionario"},n("p",null,"Il valore dato dalla media aritmetica degli ",n(Je.i,null,"n")," elementi del campione elevati alla potenza ",n(Je.i,null,"k"),":"),n("p",null,n(Je.i,null,ia(bn()))),n("p",null,"Il momento campionario di primo ordine è la ",n("i",null,"media campionaria")," ",n(Je.i,null,ia(vn())),".")),n(Je.o,{title:"Varianza campionaria"},n("p",null,"La media aritmetica dello scarto quadratico medio degli elementi del campione."),n("p",null,"Se è noto il valore medio ",n(Je.i,null,ia(fn()))," di X:"),n("p",null,n(Je.i,null,ia(mn()))),n("p",null,"Altrimenti:"),n("p",null,n(Je.i,null,ia(dn()))))),n(Je.p,{title:"Media-ception"},n(Je.o,{title:"Media campionaria"},n("p",null,"Se calcoliamo la media della media campionaria, risulterà vero che:"),n("p",null,n(Je.i,null,ia(pn()))),n(He.a,null,"Quindi, è possibile usare i campioni per trovare la media di una variabile aleatoria!")),n(Je.o,{title:"Varianza campionaria"},n("p",null,"Se calcoliamo la varianza della media campionaria, risulterà vero che:"),n("p",null,n(Je.i,null,ia(sn()))),n(He.a,null,"Quindi, possiamo stimare l'errore della media calcolata tramite campioni!")),n(Je.o,{title:"Correzione campionaria"},n("p",null,"Se calcoliamo la media della varianza campionaria, risulterà vero che:"),n("p",null,n(Je.i,null,ia(cn()))),n(He.a,null,"Quindi, possiamo stimare l'errore della media calcolata tramite campioni!"))),n(Je.p,{title:"Campionamento di una distribuzione normale"},n(Je.o,{title:"Campionamento di una distribuzione normale"},n("p",null,"Se la popolazione ",n(Je.i,null,"X")," ha una distribuzione normale (",n(Je.i,null,ia(un())),")...")),n(Je.o,{title:"Distribuzione della media campionaria"},n("p",null,"...allora sappiamo anche la distribuzione della media campionaria!"),n("p",null,n(Je.i,null,ia(on())))),n(Je.o,{title:"Distribuzione della varianza campionaria"},n("p",null,"...e anche della varianza campionaria!"),n("p",null,n(Je.i,null,ia(rn()))),n("p",null,n(Je.i,null,ia(tn())))),n(Je.o,{title:"Indipendenza"},n("p",null,"...e che media campionaria e varianza campionaria sono indipendenti tra loro!"))),n(Je.p,{title:"Quando i campioni hanno dimensioni infinite"},n(Je.o,{title:"Convergenza in distribuzione"},n("p",null,"Se la successione di variabili aleatorie ",n(Je.i,null,"X_n")," all'infinito ha la ",n("b",null,"stessa funzione di ripartizione")," della popolazione ",n(Je.i,null,"X"),", allora essa ",n("i",null,"converge in distribuzione"),"."),n("p",null,n(Je.i,null,"\\lim_{n \\to +\\infty} F_{X_n} (x) = F_X (x) \\implies X_n \\xrightarrow{d} X"))),n(Je.o,{title:"Convergenza in probabilità"},n("p",null,"Se la successione di variabili aleatorie ",n(Je.i,null,"X_n")," all'infinito ha la ",n("b",null,"stessa probabilità")," della popolazione ",n(Je.i,null,"X"),", allora essa ",n("i",null,"converge in probabilità"),"."),n("p",null,n(Je.i,null,"\\forall \\epsilon > 0, \\lim_{n \\to +\\infty} P( | X_n - X | < \\epsilon) = 1 \\implies X_n \\xrightarrow{p} X"))),n(Je.o,{title:"Convergenza quasi certa"},n("p",null,"Se la successione di variabili aleatorie ",n(Je.i,null,"X_n")," all'infinito ha la ",n("b",null,"stessa probabilità a ")," della popolazione ",n(Je.i,null,"X"),", allora essa ",n("i",null,"converge quasi certamente"),"."),n("p",null,n(Je.i,null,"\\forall \\epsilon > 0, P left( \\lim_{n \\to +\\infty} | X_n - X | < \\epsilon) \right) = 1 \\implies X_n \\xrightarrow{qc} X"))),n(Je.o,{title:"Convergenza in media quadratica"},n("p",null,"Se la successione di variabili aleatorie ",n(Je.i,null,"X_n")," all'infinito ha la ",n("b",null,"media del quadrato della distanza")," tra la successione e la popolazione ",n(Je.i,null,"X")," ",n("b",null,"uguale a 0"),", allora essa ",n("i",null,"converge in media quadratica"),"."),n("p",null,n(Je.i,null,"\\lim_{n \\to +\\infty} E( | X_n - X |^2 = 0 \\implies X_n \\xrightarrow{mq} X"))),n(Je.o,{title:"Gerarchia delle convergenze"},n("p",null,n(Je.i,null,"\n \\begin{matrix}\n X_n \\xrightarrow{mq} X\\\\\n X_n \\xrightarrow{qc} X\n \\end{matrix} \\implies X_n \\xrightarrow{p} X \\implies X_n \\xrightarrow{d} X")),n("p",null,"In più:"),n("p",null,n(Je.i,null,"X_n \\xrightarrow{p} x \\Longleftrightarrow X_n \\xrightarrow{d} x")))),n(Je.p,{title:"I grandi numeri"},n(Je.o,{title:"Legge debole dei grandi numeri"},n("p",null,"La successione delle medie campionarie ",n(Je.i,null,ia(an()))," ",n("b",null,"converge in probabilità")," alla media della popolazione ",n(Je.i,null,ia(en())),", se essa esiste."),n("p",null,n(Je.i,null,"\\overline{X}_n \\xrightarrow{p} X")),n("p",null,"Ovvero:"),n("p",null,n(Je.i,null,ia(ln()))),n("p",null,n(Je.i,null,ia(nn())))),n(Je.o,{title:"Legge forte dei grandi numeri"},n("p",null,"La successione delle medie campionarie ",n(Je.i,null,ia($()))," ",n("b",null,"converge quasi certamente")," alla media della popolazione ",n(Je.i,null,ia(H())),", se essa esiste."),n("p",null,n(Je.i,null,"\\overline{X}_n \\xrightarrow{qc} X")),n("p",null,"Ovvero:"),n("p",null,n(Je.i,null,ia(J()))),n(He.a,null,"Dimostra che l'interpretazione frequentista della probabilità è valida!"))),n(Je.p,{title:"Al limite"},n(Je.o,{title:"Teorema centrale del limite"},n("p",null,"La successione delle medie campionarie ",n(Je.i,null,ia(W()))," ",n("b",null,"converge in distribuzione")," a ",n(Je.i,null,ia(Q())),"."),n("p",null,n(Je.i,null,ia(Z()))),n("p",null,"Ovvero:"),n("p",null,n(Je.i,null,ia(j()))))),n(Je.p,{title:"Altre approsimazioni"},n(Je.o,{title:"Binomiale e normale"},n("p",null,"E' una somma di ",n("b",null,"bernoulliane"),", e quindi si approssima a una normale:"),n("p",null,n(Je.i,null,ia(K())))),n(Je.o,{title:"Binomiale negativa e normale"},n("p",null,"E' una somma di ",n("b",null,"geometriche"),", e quindi si approssima a una normale:"),n("p",null,n(Je.i,null,ia(R())))),n(Je.o,{title:"Poissoniana e normale"},n("p",null,"E' una somma di altre ",n("b",null,"poissoniane"),", e quindi si approssima a una normale:"),n("p",null,n(Je.i,null,ia(O())))),n(Je.o,{title:"Gamma e normale"},n("p",null,"E' una somma di ",n("b",null,"esponenziali"),", e quindi si approssima a una normale:"),n("p",null,n(Je.i,null,ia(G())))),n(Je.o,{title:"In generale"},n("p",null,"Se ",n(Je.i,null,"n")," è grande, allora:"),n("p",null,n(Je.i,null,ia(T()))))),n(Je.p,{title:"Actually statistica"},n(Je.o,{title:"Parametri sconosciuti"},n("p",null,"Per indicare parametri sconosciuti di una legge si usa ",n(Je.i,null,"\\theta"),".")),n(Je.o,{title:"Statistica"},n("p",null,"Una variabile aleatoria funzione di un campione:"),n("p",null,n(Je.i,null,ia(A()))),n(He.a,null,"Ad esempio, sono statistiche media e varianza campionaria, così come il campione stesso ",n(Je.i,null,ia(B())),"."))),n(Je.p,{title:"Stimatori"},n(Je.o,{title:"Stimatore"},n("p",null,"Una statistica ",n(Je.i,null,"T_n")," ottenuta da ",n(Je.i,null,"n")," osservazioni, che stimi i parametri di una legge e sia indipendente da essi.")),n(Je.o,{title:"Corretto"},n("p",null,"Uno stimatore è ",n("i",null,"corretto")," se il suo valore atteso coincide con quello dei parametri che stima:"),n("p",null,n(Je.i,null,ia(U())))),n(Je.o,{title:"Asintoticamente corretto"},n("p",null,"Uno stimatore è ",n("i",null,"asintoticamente corretto")," se, per infinite osservazioni, il suo valore atteso coincide con quello dei parametri che stima:"),n("p",null,n(Je.i,null,ia(M())))),n(Je.o,{title:"Consistente in media quadratica"},n("p",null,"Uno stimatore è ",n("i",null,"consistente in media quadratica")," se:"),n("p",null,n(Je.i,null,ia(D())))),n(Je.o,{title:"Consistente in probabilità"},n("p",null,"Uno stimatore è ",n("i",null,"consistente in probabilità")," se:"),n("p",null,n(Je.i,null,ia(I())))),n(Je.o,{title:"Asintoticamente normale"},n("p",null,"Uno stimatore è ",n("i",null,"asintoticamente normale")," se:"),n("p",null,n(Je.i,null,ia(C()))))),n(Je.p,{title:"Metodo dei momenti"},n(Je.o,{title:"Metodo dei momenti"},n("p",null,"Si può usare il ",n("i",null,"metodo dei momenti")," per ottenere uno stimatore di una popolazione ",n(Je.i,null,"X"),"."),n("p",null,"Lo stimatore di ",n(Je.i,null,ia(w()))," così ottenuto sarà indicato aggiungendo un cappellino e una ",n(Je.i,null,"M")," a ",n(Je.i,null,"\\theta"),": ",n(Je.i,null,ia(Y()))),n("p",null,"Visto che:"),n("ul",null,n("li",null,n(Je.i,null,ia(S()))),n("li",null,n(Je.i,null,ia(V())))),n("p",null,"Allora:"),n("p",null,n(Je.i,null,ia(N()))),n("p",null,"Se ",n(Je.i,null,ia(y()))," non è esprimibile in termini di ",n(Je.i,null,ia(F())),", si possono usare i momenti successivi ",n(Je.i,null,ia(L())),", ",n(Je.i,null,ia(k())),", ",n(Je.i,null,ia(x())),"..."))),n(Je.p,{title:"Metodo della massima verosomiglianza"},n(Je.o,{title:"Metodo della massima verosomiglianza"},n("p",null,"Si può usare il ",n("i",null,"metodo della massima verosomiglianza")," per ottenere uno stimatore di una popolazione ",n(Je.i,null,"X"),"."),n("p",null,"Lo stimatore di ",n(Je.i,null,ia(P()))," così ottenuto sarà indicato aggiungendo un cappellino e una ",n(Je.i,null,"L")," a ",n(Je.i,null,"\\theta"),": ",n(Je.i,null,ia(E()))),n("p",null,"Consiste nel trovare il massimo assoluto ",n(Je.i,null,ia(z()))," della la funzione di verosomiglianza 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probabilità"},l(e.o,{title:"Classica"},l("p",null,l(e.i,null,ia(u||(u=la`P(E) = \frac{casi\ favorevoli}{casi\ possibili}`))))),l(e.o,{title:"Frequentista"},l("p",null,l(e.i,null,ia(r||(r=la`P(E) = \frac{successi}{prove\ totali}`))))),l(e.o,{title:"Soggettiva"},l("p",null,"Il prezzo che un individuo coerente riterrebbe equo per ricevere ",l("b",null,"1")," nel caso l'evento si verificasse e ",l("b",null,"0")," nel caso l'evento non si verificasse."))),l(e.p,{title:"Linguaggio matematico"},l(e.o,{title:"Spazio campionario"},l("blockquote",null,'"omegone"'),l("p",null,"L'",l("b",null,"insieme")," di tutti gli esiti possibili di un esperimento."),l("p",null,l(e.i,null,ia(s||(s=la`\Omega = \left \{ 1, 2, 3, 4, 5, 6 \right \}`))))),l(e.o,{title:"Esito"},l("blockquote",null,'"omeghino"'),l("p",null,"Un ",l("b",null,"elemento")," dello spazio campionario."),l("p",null,l(e.i,null,ia(p||(p=la`\omega = 1`))))),l(e.o,{title:"Evento"},l("blockquote",null,'"e"'),l("p",null,"Un 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contenuto in effe"'),l("p",null,"L'",l("b",null,"inclusione")," del primo insieme in un altro."),l("p",null,l(e.i,null,ia(g||(g=la`E \subseteq F`)))),l("p",null,"Se si verifica ",l(e.i,null,"E"),", allora si verifica anche ",l(e.i,null,"F"),".")),l(e.o,{title:"Evento impossibile"},l("blockquote",null,'"e è impossibile"'),l("p",null,"Un sottoinsieme ",l("b",null,"vuoto"),"."),l("p",null,l(e.i,null,ia(f||(f=la`E = \emptyset`))))),l(e.o,{title:"Mutua esclusione"},l("blockquote",null,'"e ed effe si escludono mutualmente"'),l("p",null,"La ",l("b",null,"disgiunzione")," di due insiemi."),l("p",null,l(e.i,null,ia(_||(_=la`E \cap F = \emptyset`)))))),l(e.p,null,l(e.o,{title:"Famiglia degli eventi"},l("blockquote",null,'"famiglia effe"'),l("p",null,"I sottoinsiemi dello spazio campionario formano una ",l("b",null,"famiglia")," di sottoinsiemi detta ",l("i",null,"famiglia degli eventi"),"."),l("p",null,l(e.i,null,ia(h||(h=la`\mathcal{F}`)))),l("p",null,"Qualsiasi sottoinsieme appartenente a ",l(e.i,null,ia(z||(z=la`\mathcal{F}`)))," è considerato un evento.")),l(e.o,{title:l("span",null,l(e.i,null,ia(X||(X=la`\sigma`))),"-algebra")},l("blockquote",null,'"sigma algebra"'),l("p",null,"Se la famiglia degli eventi soddisfa questi tre requisiti, allora viene detta ",l("i",null,l(e.i,null,ia(q||(q=la`\sigma`))),"-algebra"),":"),l("ol",null,l("li",null,"Lo spazio campionario è un evento: ",l(e.i,null,ia(E||(E=la`\Omega \in \mathcal{F}`)))),l("li",null,"Se un sottoinsieme è un evento, allora anche il suo complementare lo è: ",l(e.i,null,ia(P||(P=la`E \in \mathcal{F} \implies \bar{E} \in \mathcal{F}`)))),l("li",null,"Se due sottoinsiemi sono eventi, allora lo sono anche la loro unione e intersezione: ",l(e.i,null,ia(L||(L=la`(E, F) \in \mathcal{F} \implies (E \cup F, E \cap F) \in \mathcal{F}`))))),l("p",null,"Un esempio: ",l(e.i,null,ia(x||(x=la`E \in \mathcal{F} \implies \mathcal{F} = \{ \emptyset, E, \bar{E}, \Omega \}`)))))),l(e.p,null,l(e.o,{title:"Partizione"},l("blockquote",null,'"la partizione e composta da e uno, e due, e tre..."'),l("p",null,"Un insieme di esiti e eventi:"),l("ul",null,l("li",null,l("b",null,"Finito"),"."),l("li",null,"In cui tutti gli eventi hanno ",l("b",null,"probabilità diversa da 0"),"."),l("li",null,"In cui tutti gli eventi sono ",l("b",null,"mutualmente esclusivi"),"."),l("li",null,"In cui l'unione di tutti i suoi elementi ",l("b",null,"copre lo spazio campionario"),".")),l("p",null,"La partizione ",l(e.i,null,ia(k||(k=la`E_i`)))," è composta dagli eventi ",l(e.i,null,ia(y||(y=la`E_1`))),", ",l(e.i,null,ia(F||(F=la`E_2`))),", ",l(e.i,null,ia(S||(S=la`E_3`))),", fino a ",l(e.i,null,ia(I||(I=la`E_n`))),"."),l(a.a,null,"Se lo spazio campionario fosse una torta, una sua partizione sarebbe l'insieme delle fette di uno dei modi in cui si potrebbe tagliare."))),l(e.p,{title:"Assiomi della probabilità"},l(e.o,{title:"Primo assioma della probabilità"},l("p",null,"La 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incluso."),l("p",null,l(e.i,null,ia(D||(D=la`F \subseteq E \implies P(F) \leq P(E)`))))),l(e.o,{title:"Unione"},l("p",null,"La probabilità di un evento unito a un altro è uguale alla somma delle probabilità dei due eventi meno la probabilità della loro intersezione."),l("p",null,l(e.i,null,ia(Y||(Y=la`P(E \cup F) = P(E) + P(F) - P(E \cap F)`)))),l(a.a,null,"Sommando le probabilità dei due eventi, l'intersezione viene contata due volte, e va quindi rimossa!"))),l(e.p,{title:"Spazi equiprobabili"},l(e.o,{title:"Cosa sono?"},l("p",null,"Spazi campionari in cui ci sono un numero finito di esiti e ogni esito ha la stessa probabilità di verificarsi."),l("p",null,l(e.i,null,ia(U||(U=la`P(E) = \frac{len(E)}{len(\Omega)}`))))),l(e.o,{title:"Spazi equiprobabili geometrici"},l("p",null,"Gli spazi campionari possono avere un numero infinito di esiti: sono ",l("i",null,"equiprobabili geometrici")," se nessun esito è privilegiato rispetto agli altri."))),l(e.p,{title:"Calcolo 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Ripeto per ",l(e.i,null,"k")," volte."),l("p",null,l("b",null,"Tengo conto")," dell'ordine in cui ho estratto i numeri."),l("p",null,l(e.i,null,ia(M||(M=la`\boldsymbol{D}_{n, k} = \frac{n!}{(n - k)!}`))))),l(e.o,{title:"Disposizioni con ripetizione"},l("p",null,"Estraggo un numero, da un sacchetto con ",l(e.i,null,"n")," numeri, mi segno che numero ho estratto e lo ",l("b",null,"rimetto nel sacchetto"),". Ripeto per ",l(e.i,null,"k")," volte."),l("p",null,l("b",null,"Tengo conto")," dell'ordine in cui ho estratto i numeri."),l("p",null,l(e.i,null,ia(B||(B=la`\boldsymbol{D}^{r}_{n, k} = n^k`))))),l(e.o,{title:"Combinazioni"},l("p",null,"Estraggo un numero, da un sacchetto con ",l(e.i,null,"n")," numeri, mi segno che numero ho estratto e lo ",l("b",null,"tengo fuori dal sacchetto"),". Ripeto per ",l(e.i,null,"k")," volte."),l("p",null,l("b",null,"Non mi interessa")," l'ordine in cui ho estratto i numeri."),l("p",null,l(e.i,null,ia(A||(A=la`\boldsymbol{C}_{n, k} = \binom{n}{k} = \frac{n!}{(k)! \cdot (n - k)!}`))))),l(e.o,{title:"Combinazioni con ripetizione"},l("p",null,"Estraggo un numero, da un sacchetto con ",l(e.i,null,"n")," numeri, mi segno che numero ho estratto e lo ",l("b",null,"rimetto nel sacchetto"),". Ripeto per ",l(e.i,null,"k")," volte."),l("p",null,l("b",null,"Non mi interessa")," l'ordine in cui ho estratto i numeri."),l("p",null,l(e.i,null,ia(T||(T=la`\boldsymbol{C}^{r}_{n, k} = \binom{n + k - 1}{k} = \frac{(n + k - 1)!}{(k)! \cdot (n - 1)!}`))))),l(e.o,{title:"Permutazioni"},l("p",null,"Estraggo ",l(e.i,null,"n")," numeri e guardo in quanti ordini diversi li posso mettere."),l("p",null,l(e.i,null,ia(G||(G=la`\boldsymbol{P}_n = n!`)))))),l(e.p,{title:"Probabilità condizionata"},l(e.o,{title:"Eventi condizionati"},l("blockquote",null,'"E dato F"'),l("p",null,"La probabilità che si verifichi ",l(e.i,null,"E")," sapendo che ",l("b",null,"si è già verificato")," ",l(e.i,null,"F"),"."),l("p",null,l(e.i,null,ia(O||(O=la`P(E|F) = \frac{P(E \cap F)}{P(F)}`)))),l(a.a,null,"Ricorda vagamente le pipe di ",l("code",null,"bash"),", però al contrario...")),l(e.o,{title:"Eventi mutualmente esclusivi"},l("p",null,"Se due eventi sono mutualmente esclusivi, entrambe le loro probabilità condizionate saranno uguali a 0."),l("p",null,l(e.i,null,ia(R||(R=la`E \cap F = \emptyset \Longleftrightarrow P(E|F) = P(F|E) = 0`)))))),l(e.p,null,l(e.o,{title:"Regola della catena"},l("p",null,"Si può sfruttare la formula inversa della probabilità condizionata per calcolare catene di intersezioni:"),l("p",null,l(e.i,null,ia(j||(j=la`P(E_1 \cap \times \cap E_n) = P(E_1) \times P(E_2 | E_1) \times \dots \times P(E_n | E_1 \cap E_2 \cap \dots \cap E_{n-1})`)))))),l(e.p,{title:"Le alternative"},l(e.o,{title:"Legge delle alternative"},l("p",null,"La probabilità che si verifichi un evento è pari alla somma delle probabilità dell'evento stesso dati tutti gli eventi di una partizione."),l("p",null,l(e.i,null,ia(K||(K=la`P(F) = \sum_{i} P(F|E_i) \cdot P(E_i)`))))),l(e.o,{title:"Legge condizionata delle alternative"},l("p",null,"La legge delle alternative funziona anche quando ad essere partizionato è un ",l("b",null,"evento"),":"),l("p",null,l(e.i,null,ia(Q||(Q=la`P(F|G) = \sum_i P(F|E_i \cap G) \cdot P(E_i | G)`))))),l(e.o,{title:"Formula di Bayes"},l("p",null,"Tramite la ",l("i",null,"formula di Bayes")," possiamo risalire alla probabilità di un evento condizionato a un altro partendo dalla probabilità di quest'ultimo condizionato al primo:"),l("p",null,l(e.i,null,ia(Z||(Z=la`P(E_h | F) = \frac{P(F | E_h) \cdot P(E_h)}{P(F)}`)))),l(a.a,null,"In pratica, invertiamo gli eventi."))),l(e.p,{title:"Eventi indipendenti"},l(e.o,{title:"Due eventi indipendenti"},l("blockquote",null,'"eventi indipendenti a due a due"'),l("p",null,"Se due eventi sono indipendenti, sapere che uno dei due si è verificato non influisce sulle probabilità che si sia verificato l'altro."),l("p",null,l(e.i,null,ia(W||(W=la`P(E \cap F) = P(E) \cdot P(F) \Longleftrightarrow P(E|F) = P(E) \Longleftrightarrow P(F|E) = P(F)`))))),l(e.o,{title:"Più eventi indipendenti"},l("blockquote",null,'"eventi indipendenti a tre a tre, a quattro a quattro, a cinque a cinque..."'),l("p",null,"Si può verificare l'indipendenza di più eventi alla volta:"),l("p",null,l(e.i,null,ia(J||(J=la`P(E \cap F \cap G) = P(E) \cdot P(F) \cdot P(G)`)))),l("p",null,"Eventi indipendenti a due a due non sono per forza indipendenti a tre a tre, e viceversa.")),l(e.o,{title:"Famiglia di eventi indipendenti"},l("p",null,"Un insieme di ",l(e.i,null,"n")," eventi è una ",l("i",null,"famiglia di eventi indipendenti")," se, preso un qualsiasi numero di eventi da essa, essi risulteranno indipendenti."),l(a.a,null,"Tutti gli eventi provenienti da essa saranno indipendenti sia a due a due, sia a tre a tre, sia a quattro a quattro, e così via!"))),l(e.p,{title:"Variabili aleatorie"},l(e.o,{title:"Variabile aleatoria"},l("p",null,"Una funzione che fa corrispondere un numero reale a ogni possibile esito dello spazio campionario. ",l(e.i,null,ia(H||(H=la`X(\omega) : \Omega \to \mathbb{R}`))),".")),l(e.o,{title:l("abbr",{title:"Nome artigianale dato da Steffo."},"Insieme di ripartizione")},l("p",null,"Ad ogni variabile aleatoria sono associati gli eventi ",l(e.i,null,ia($||($=la`A_t = \{ \omega | X(\omega) \leq t \}`))),", che contengono tutti gli esiti a cui la variabile aleatoria associa un valore minore o uguale a ",l(e.i,null,"t"),"."),l("p",null,"Per definizione, tutte le variabili aleatorie devono rispettare questa condizione:"),l("p",null,l(e.i,null,ia(ll||(ll=la`\forall t \in \mathbb{R}, A_t \in \mathcal{F}`)))),l(a.a,null,"All'aumentare di t, l'insieme conterrà sempre più elementi.")),l(e.o,{title:"Supporto"},l("blockquote",null,'"supporto di X"'),l("p",null,"Il ",l("b",null,"codominio")," della variabile aleatoria è il suo ",l("i",null,"supporto"),"."),l("p",null,"Per indicare che un valore ",l(e.i,null,"x_0")," appartiene al supporto di ",l(e.i,null,"X"),", si usa la notazione ",l(e.i,null,"X \\mapsto x_0"),"."))),l(e.p,{title:"Densità"},l(e.o,{title:"Funzione probabilità"},l("p",null,"La ",l("i",null,"funzione probabilità")," ",l(e.i,null,ia(il||(il=la`p_X : X \to [0, 1]`)))," di una variabile aleatoria ",l("b",null,"discreta")," ",l(e.i,null,"X")," è la funzione che associa ad ogni esito la sua probabilità:"),l("p",null,l(e.i,null,ia(nl||(nl=la` - p_X (x) = \begin{cases} - P([X = x]) \quad se\ X \mapsto x \\ - 0 \qquad \qquad \quad se\ X \not\mapsto x - \end{cases} - `))))),l(e.o,{title:"Funzione densità"},l("p",null,"La ",l("i",null,"funzione densità")," ",l(e.i,null,ia(el||(el=la`f_X : X \to [0, 1]`)))," di una variabile aleatoria ",l("b",null,"continua")," ",l(e.i,null,"X")," è l'equivalente continuo della funzione probabilità:"),l("p",null,l(e.i,null,ia(al||(al=la`P([a < X \leq b]) = \int_a^b f_X (x) dx`)))),l("p",null,"A differenza della funzione probabilità, è possibile che la funzione densità ",l("b",null,"non esista")," per una certa variabile aleatoria."),l(a.a,null,"Rappresenta \"quanta\" probabilità c'è in un'unità di x!"))),l(e.p,{title:"Funzione di ripartizione"},l(e.o,{title:"Definizione"},l("p",null,"Ogni variabile aleatoria ha una ",l("i",null,"funzione di ripartizione")," ",l(e.i,null,ia(ol||(ol=la`F_X : \mathbb{R} \to [0, 1]`)))," associata, che rappresenta la probabilità che la variabile aleatoria assuma un valore minore o uguale a ",l(e.i,null,"t"),":"),l("p",null,"Si può dire che essa rappresenti la probabilità dell'evento ",l(e.i,null,ia(tl||(tl=la`A_t`))),":"),l("p",null,l(e.i,null,ia(ul||(ul=la` - F_X (t) = P(A_t) = \begin{cases} - \sum_{i = 0}^{t} p_X (x_i) \quad nel\ discreto\\ - \\ - \int_{-\infty}^t f_X (x) dx \quad nel\ continuo - \end{cases} - `))))),l(e.o,{title:"Proprietà della funzione"},l("ul",null,l("li",null,"È sempre ",l("b",null,"monotona crescente")," (non strettamente)."),l("br",null),l("li",null,"Vale ",l("b",null,"0")," a ",l(e.i,null,"-\\infty")," e ",l("b",null,"1")," a ",l(e.i,null,"+\\infty"),"."),l("br",null),l("li",null,"È ",l("b",null,"continua da destra"),": ",l(e.i,null,ia(rl||(rl=la`\forall x_0 \in \mathbb{R}, F_X (x_0) = \lim_{t \to x^+_0} F_X (t)`)))))),l(e.o,{title:"Probabilità di un valore"},l("p",null,"Possiamo usare la funzione di ripartizione per calcolare la probabilità di un certo valore reale:"),l("p",null,l(e.i,null,ia(sl||(sl=la`P([X = x_0]) = \lim_{t \to x^+_0} F_X (t) - \lim_{t \to x^-_0} F_X (t)`)))))),l(e.p,{title:"Trasformazioni di variabili aleatorie"},l(e.o,{title:"Nel discreto"},l("p",null,"Nel discreto basta abbinare un nuovo valore a ogni valore della variabile originale.")),l(e.o,{title:"Nel continuo (invertibile)"},l("p",null,"Nel continuo applichiamo la formula dell'integrazione per sostituzione:"),l("p",null,l(e.i,null,ia(pl||(pl=la`f_Y (y) = \int_{g(a)}^{g(b)} f_X ( g^{-1} (x) ) g^{-2} (x)`))))),l(e.o,{title:"Nel... digitale"},l("p",null,"Trasformare variabili aleatorie è molto utile nell'informatica per creare distribuzioni partendo da una funzione ",l("a",{href:"https://docs.python.org/3/library/random.html#random.random"},l("code",null,"random()"))," che restituisce numeri da 0 a 1 con una distribuzione lineare."))),l(e.p,{title:"Informazioni delle variabili aleatorie"},l(e.o,{title:"Media"},l("p",null,"Ogni variabile aleatoria che ha una ",l("b",null,"funzione di ripartizione")," e un ",l("b",null,"supporto finito")," ha anche una ",l("i",null,"media")," (o ",l("i",null,"valore medio")," o ",l("i",null,"atteso"),"):"),l("p",null,l(e.i,null,ia(dl||(dl=la`E(X) = \int_0^{+infty} (1 - F_X (t)) dt - \int_{-\infty}^{0} F_X (t) dt`)))),l("p",null,"Nel discreto, si può calcolare con:"),l("p",null,l(e.i,null,ia(ml||(ml=la`E(X) = \sum_i P(X = x_i) \cdot x_i`)))),l("p",null,"Nel continuo, si può calcolare con:"),l("p",null,l(e.i,null,ia(cl||(cl=la`E(X) = \int_{-\infty}^{+\infty} f_X (x) \cdot x \cdot dx`)))))),l(e.p,null,l(e.o,{title:"Moda"},l("p",null,"Valore per cui la ",l("b",null,"funzione probabilità")," o ",l("b",null,"funzione densità")," è ",l("b",null,"massima"),".")),l(e.o,{title:"Quantili"},l("p",null,"Il ",l("i",null,"quantile")," ",l(e.i,null,ia(bl||(bl=la`x_{\alpha}`)))," di ordine ",l(e.i,null,ia(vl||(vl=la`0 \leq \alpha \leq 1`)))," della variabile aleatoria ",l(e.i,null,"X")," è il più piccolo numero tale che:"),l("p",null,l(e.i,null,ia(gl||(gl=la`P([X < x_{\alpha}]) \leq \alpha \leq P([X \leq x_{\alpha}])`)))),l("p",null),l("p",null,"Il quantile di ordine 0.5 ",l(e.i,null,ia(fl||(fl=la`x_{0.5}`)))," è detto ",l("i",null,"mediana"),"."),l("p",null,"I quantili di ordine 0.25 ",l(e.i,null,ia(_l||(_l=la`x_{0.25}`)))," e 0.75 ",l(e.i,null,ia(hl||(hl=la`x_{0.75}`)))," sono detti ",l("i",null,"quartili"),"."),l("p",null,"I quantili di ordine ",l(e.i,null,ia(zl||(zl=la`\frac{n}{100}`)))," sono detti ",l("i",null,l(e.i,null,"n"),"-esima percentile"),".")),l(e.o,{title:"Varianza"},l("p",null,"È un valore che indica quanto la variabile aleatoria si discosta generalmente dalla media:"),l("p",null,l(e.i,null,ia(Xl||(Xl=la`Var(X) = E( (X - E(X) )^2 ) = E ( X^2 ) - (E(X))^2`)))))),l(e.p,{title:"Disuguaglianze notevoli"},l(e.o,{title:"Disuguaglianza di Markov"},l("p",null,"Data una variabile aleatoria non-negativa:"),l("p",null,l(e.i,null,ia(ql||(ql=la`\forall k > 0, P([X \geq k]) \leq \frac{E(X)}{k}`)))),l("p",null,"Divide in due parti (",l(e.i,null,ia(El||(El=la`P(X < k)`)))," e ",l(e.i,null,ia(Pl||(Pl=la`P(X \geq k)`))),") la funzione X, la cui media risulterà uguale a:"),l("p",null,l(e.i,null,ia(Ll||(Ll=la`E(X) = \overline{k} \cdot P(X < k) + k \cdot P(X \geq k)`))))),l(e.o,{title:"Disuguaglianza di Čebyšëv"},l("blockquote",null,'"disuguaglianza di cebicev"'),l("p",null,"Se la variabile aleatoria ",l(e.i,null,"X")," ha media e varianza, allora la probabilità che essa abbia un valore a più di ",l(e.i,null,ia(xl||(xl=la`\epsilon`)))," di distanza dal valore medio è minore o uguale a ",l(e.i,null,ia(kl||(kl=la`\frac{Var(X)}{\epsilon^2}`))),"."),l("p",null,l(e.i,null,ia(yl||(yl=la`\forall \epsilon > 0, P([ \left| X - E(X) \right| \geq \epsilon]) \leq \frac{Var(X)}{\epsilon^2}`)))),l("p",null,"E anche:"),l("p",null,l(e.i,null,ia(Fl||(Fl=la`\forall \epsilon > 0, P([ \left| X - E(X) \right| < \epsilon]) \geq 1 - \frac{Var(X)}{\epsilon^2}`)))),l(a.a,null,"Serve per semplificare i calcoli quando la funzione di ripartizione è difficile da calcolare!"))),l(e.p,{title:"Un momento...!"},l(e.o,{title:"Momento"},l("p",null,"Il ",l("i",null,"momento")," ",l(e.i,null,"k"),"-esimo di una variabile aleatoria è:"),l("p",null,l(e.i,null,ia(Sl||(Sl=la` - \mu_k = E ( X^k ) = \begin{cases} - \sum_i x_i^k p_X (x_i) \qquad nel\ discreto\\ - \\ - \int_{-\infty}^{+\infty} x^k f_X (x) dx \qquad nel\ continuo - \end{cases}`)))),l(a.a,null,"La media di una variabile aleatoria è anche il suo primo momento.")),l(e.o,{title:"Funzione generatrice dei momenti"},l("p",null,"La ",l("i",null,"funzione generatrice dei momenti")," è:"),l("p",null,l(e.i,null,ia(Il||(Il=la`m_X (t) = E( e^{t \cdot X} )`)))),l("p",null,"Se due variabile aleatorie hanno la stessa funzione generatrice dei momenti, allora esse hanno la ",l("b",null,"stessa distribuzione"),"."),l("p",null,"E' la ",l("b",null,"trasformata di Laplace")," della variabile aleatoria di X.")),l(e.o,{title:"Funzione caratteristica"},l("p",null,"La ",l("i",null,"funzione caratteristica")," è:"),l("p",null,l(e.i,null,ia(Cl||(Cl=la`H_X (t) = E ( e^{i \cdot t \cdot X} )`)))),l("p",null,"Se due variabile aleatorie hanno la stessa funzione caratteristica, allora esse hanno la ",l("b",null,"stessa distribuzione"),"."),l("p",null,"E' la ",l("b",null,"trasformata di Fourier")," della variabile aleatoria di X."))),l(e.p,{title:"Prove e schemi"},l(e.o,{title:"Variabile con distribuzione"},l("p",null,"Per dire che una variabile ha una certa distribuzione, si usa la notazione:"),l("p",null,l(e.i,null,ia(Nl||(Nl=la`X \sim Distribuzione()`))))),l(e.o,{title:"Prova di Bernoulli"},l("p",null,"Una prova con solo due possibili esiti: ",l(o.a,null,"successo")," e ",l(t.a,null,"insuccesso"),".")),l(e.o,{title:"Schema di Bernoulli"},l("p",null,"Una sequenza di prove di Bernoulli per le quali le probabilità di successo e fallimento rimangono invariate."))),l(e.p,{title:"Bernoulliana"},l(e.o,{title:"Distribuzione bernoulliana"},l("p",null,"Una variabile aleatoria che rappresenta una prova di Bernoulli:"),l("ul",null,l("li",null,"vale ",l(o.a,null,"1")," in caso di ",l(o.a,null,"successo"),"."),l("li",null,"vale ",l(t.a,null,"0")," in caso di ",l(t.a,null,"insuccesso"),".")),l("p",null,"Il suo simbolo è ",l(e.i,null,ia(Vl||(Vl=la`Ber(p)`))))),l(e.o,{title:"Densità della bernoulliana"},l("p",null,"La distribuzione bernoulliana ha come densità:"),l("p",null,l(e.i,null,ia(wl||(wl=la` - f_X (k) : \{0, 1\} = \begin{cases} - p \quad se\ k = 1\\ - q \quad se\ k = 0\\ - 0 \quad altrimenti - \end{cases} = p^x \cdot q^{1 - k}`)))))),l(e.p,{title:"Binomiale"},l(e.o,{title:"Distribuzione binomiale"},l("p",null,"Una variabile aleatoria che conta il numero di successi di ",l(e.i,null,"n")," prove di uno schema di Bernoulli."),l("p",null,"Il suo simbolo è ",l(e.i,null,ia(Dl||(Dl=la`Bin(n, p)`))),".")),l(e.o,{title:"Densità della binomiale"},l("p",null,"La binomiale ha come densità:"),l("p",null,l(e.i,null,ia(Yl||(Yl=la`f_X (k) : \{0..n\} = \binom{n}{k} \cdot p^k \cdot q^{n - k}`))))),l(e.o,{title:"Momenti della binomiale"},l("p",null,"La ",l("b",null,"funzione generatrice dei momenti")," della binomiale è:"),l("p",null,l(e.i,null,ia(Ul||(Ul=la`m_X (t) = (q + p \cdot e^t) ^ n`)))),l("p",null,"La ",l("b",null,"media")," di una binomiale è:"),l("p",null,l(e.i,null,ia(Ml||(Ml=la`E(X) = n \cdot p`)))),l("p",null,"La ",l("b",null,"varianza")," di una binomiale è:"),l("p",null,l(e.i,null,ia(Bl||(Bl=la`Var(X) = n \cdot p \cdot q`)))))),l(e.p,{title:"Geometrica"},l(e.o,{title:"Distribuzione geometrica"},l("p",null,"Una variabile aleatoria che conta il numero di prove in uno schema di Bernoulli fino alla comparsa del primo successo."),l("p",null,"Il suo simbolo è ",l(e.i,null,"Geo(p)"),".")),l(e.o,{title:"Densità della geometrica"},l("p",null,"La geometrica ha come densità:"),l("p",null,l(e.i,null,ia(Al||(Al=la`f_X (k) : \mathbb{N} = q^{k - 1} p`))))),l(e.o,{title:"Momenti della geometrica"},l("p",null,"La ",l("b",null,"funzione generatrice dei momenti")," della geometrica è:"),l("p",null,l(e.i,null,ia(Tl||(Tl=la`m_X (t) = \frac{p \cdot e^t}{1 - q \cdot e^t}`)))),l("p",null,"La ",l("b",null,"media")," della geometrica è:"),l("p",null,l(e.i,null,ia(Gl||(Gl=la`E(X) = \frac{1}{p}`)))),l("p",null,"La ",l("b",null,"varianza")," della geometrica è:"),l("p",null,l(e.i,null,ia(Ol||(Ol=la`Var(X) = \frac{q}{p^2}`))))),l(e.o,{title:"Assenza di memoria della geometrica"},l("p",null,"La geometrica non tiene conto degli eventi avvenuti in passato: ha la proprietà dell'assenza di memoria:"),l("p",null,l(e.i,null,ia(Rl||(Rl=la`P([X = i + j | X > i ]) = P([X = j])`)))),l(a.a,null,"Ovvero, riscalando opportunamente l'asse Y posso prendere come 0 qualsiasi punto dell'asse X."))),l(e.p,{title:"Binomiale negativa"},l(e.o,{title:"Distribuzione binomiale negativa"},l("p",null,"Una variabile aleatoria che conta il numero di prove in uno schema di Bernoulli necessarie perchè si verifichi l'",l(e.i,null,"n"),"-esimo successo."),l("p",null,"Il suo simbolo è ",l(e.i,null,ia(jl||(jl=la`\overline{Bin}(n, p)`))),".")),l(e.o,{title:"Densità della binomiale negativa"},l("p",null,"La binomiale negativa ha come densità:"),l("p",null,l(e.i,null,ia(Kl||(Kl=la`f_X (k) : \{ n .. +\infty \} \in \mathbb{N} = \binom{k - 1}{n - 1} \cdot p^n \cdot q^{k - n} `))))),l(e.o,{title:"Momenti della binomiale negativa"},l("p",null,l("p",null,"La ",l("b",null,"funzione generatrice dei momenti")," della binomiale negativa è:"),l("p",null,l(e.i,null,ia(Ql||(Ql=la`m_X (t) : \{ t < ln(\frac{1}{q}) \} = \left( \frac{p \cdot e^t}{1 - q \cdot e^t} \right) ^n`)))),l("p",null,"La ",l("b",null,"media")," della binomiale negativa è:"),l("p",null,l(e.i,null,ia(Zl||(Zl=la`E(X) = \frac{n}{p}`)))),l("p",null,"La ",l("b",null,"varianza")," della binomiale negativa è:"),l("p",null,l(e.i,null,ia(Wl||(Wl=la`Var(X) = \frac{n \cdot q}{p^2}`))))))),l(e.p,{title:"Geometrica traslata"},l(e.o,{title:"Distribuzione geometrica traslata"},l("p",null,"Una variabile aleatoria che conta il numero ",l(e.i,null,"k")," di insuccessi consecutivi in uno schema di Bernoulli:"),l("p",null,"Il suo simbolo rimane ",l(e.i,null,ia(Jl||(Jl=la`Geo(p)`))),".")),l(e.o,{title:"Densità della geometrica tralsata"},l("p",null,"La geometrica traslata ha come densità:"),l("p",null,l(e.i,null,ia(Hl||(Hl=la`f_X (k) : \mathbb{N} = p \cdot q^k `))))),l(e.o,{title:"Momenti della geometrica traslata"},l("p",null,"La ",l("b",null,"funzione generatrice dei momenti")," della geometrica traslata è:"),l("p",null,l(e.i,null,ia($l||($l=la`m_X (t) : \left\{ t < ln \left( \frac{1}{q} \right) \right\} = \frac{p}{1 - q \cdot e^t}`)))),l("p",null,"La ",l("b",null,"media")," della geometrica traslata è:"),l("p",null,l(e.i,null,ia(li||(li=la`E(X) = \frac{q}{p}`)))),l("p",null,"La ",l("b",null,"varianza")," della geometrica è:"),l("p",null,l(e.i,null,ia(ii||(ii=la`Var(X) = \frac{q}{p^2}`))))),l(e.o,{title:"Assenza di memoria della geometrica traslata"},l("p",null,"La geometrica traslata non tiene conto degli eventi avvenuti in passato: ha la proprietà dell'assenza di memoria:"),l("p",null,l(e.i,null,ia(ni||(ni=la`P([X = i + j | X > i ]) = P([X = j])`)))),l(a.a,null,"Ovvero, riscalando opportunamente l'asse Y posso prendere come 0 qualsiasi punto dell'asse X."))),l(e.p,{title:"Binomiale negativa traslata"},l(e.o,{title:"Distribuzione binomiale negativa traslata"},l("p",null,"Una variabile aleatoria che conta il numero di insuccessi in uno schema di Bernoulli prima che si verifichi l'",l(e.i,null,"n"),"-esimo successo."),l("p",null,"Il suo simbolo rimane ",l(e.i,null,ia(ei||(ei=la`\overline{Bin}(n, p)`))),".")),l(e.o,{title:"Densità della binomiale negativa traslata"},l("p",null,"La binomiale negativa traslata ha come densità:"),l("p",null,l(e.i,null,ia(ai||(ai=la`f_X (k) : \mathbb{N} = \binom{k + n - 1}{n - 1} \cdot p^n \cdot q^k `))))),l(e.o,{title:"Momenti della binomiale negativa traslata"},l("p",null,l("p",null,"La ",l("b",null,"funzione generatrice dei momenti")," della binomiale negativa traslata è:"),l("p",null,l(e.i,null,ia(oi||(oi=la`m_X (t) : \left\{ t < ln \left( \frac{1}{q} \right) \right\} = \left( \frac{p \cdot e^t}{1 - q \cdot e^t} \right) ^n`)))),l("p",null,"La ",l("b",null,"media")," della binomiale negativa traslata è:"),l("p",null,l(e.i,null,ia(ti||(ti=la`E(X) = \frac{n \cdot q}{p}`)))),l("p",null,"La ",l("b",null,"varianza")," della binomiale negativa traslata è:"),l("p",null,l(e.i,null,ia(ui||(ui=la`Var(X) = \frac{n \cdot q}{p^2}`))))))),l(e.p,{title:"Ipergeometrica"},l(e.o,{title:"Distribuzione ipergeometrica"},l("p",null,"Una variabile aleatoria che, sapendo il numero di successi ",l(e.i,null,"K")," e di insuccessi ",l(e.i,null,"N-K"),", conta quanti successi si otterrebbero se se ne estraessero ",l(e.i,null,"n")," in blocco."),l("p",null,"Il suo simbolo è ",l(e.i,null,"Ipe(N, K, n)"),".")),l(e.o,{title:"Densità della ipergeometrica"},l("p",null,"La ipergeometrica ha come densità:"),l("p",null,l(e.i,null,ia(ri||(ri=la`f_X (k) : \{0..n\} \in \mathbb{N} = \frac{\binom{K}{k} \cdot \binom{N - K}{n - k}}{\binom{N}{n}}`))))),l(e.o,{title:"Momenti della ipergeometrica"},l("p",null,l("p",null,"La ",l("b",null,"funzione generatrice dei momenti")," della ipergeometrica è trascurabile."),l("p",null,"La ",l("b",null,"media")," della ipergeometrica è:"),l("p",null,l(e.i,null,ia(si||(si=la`E(X) = n \cdot \frac{K}{N}`)))),l("p",null,"La ",l("b",null,"varianza")," della ipergeometrica è:"),l("p",null,l(e.i,null,ia(pi||(pi=la`Var(X) = n \cdot \frac{K}{N} \cdot \frac{N - K}{N} \cdot \frac{N - n}{N - 1}`))))))),l(e.p,{title:"Poissoniana"},l(e.o,{title:"Distribuzione poissoniana"},l("p",null,"Una variabile aleatoria che soddisfa tutte le seguenti caratteristiche:"),l("ul",null,l("li",null,"Binomiale: ",l(e.i,null,ia(di||(di=la`X \sim Bin(n, p)`)))),l("li",null,"Il numero di prove tende a infinito: ",l(e.i,null,ia(mi||(mi=la`n \to +\infty`)))),l("li",null,"La probabilità di successo tende a 0: ",l(e.i,null,ia(ci||(ci=la`p \to 0`)))),l("li",null,"La media è finita: ",l(e.i,null,ia(bi||(bi=la`E(X) = n \cdot p \to \mu \neq 0`))))),l("p",null,"Il suo simbolo è ",l(e.i,null,ia(vi||(vi=la`Poi(\mu)`))))),l(e.o,{title:"Densità della poissoniana"},l("p",null,"La poissoniana ha come densità:"),l("p",null,l(e.i,null,ia(gi||(gi=la`f_X (k) : \mathbb{N} = \frac{e^{-\mu} \cdot \mu^k}{k!}`))))),l(e.o,{title:"Momenti della poissoniana"},l("p",null,l("p",null,"La ",l("b",null,"funzione generatrice dei momenti")," della poissoniana è:"),l("p",null,l(e.i,null,ia(fi||(fi=la`m_X (t) = e^{\mu \cdot (e^t - 1)}`)))),l("p",null,"La ",l("b",null,"media")," della poissoniana è:"),l("p",null,l(e.i,null,ia(_i||(_i=la`E(X) = \mu`)))),l("p",null,"La ",l("b",null,"varianza")," della poissoniana è:"),l("p",null,l(e.i,null,ia(hi||(hi=la`Var(X) = \mu`)))),l("p",null,"Gli altri momenti della poissoniana sono:"),l("ol",{start:2},l("li",null,l(e.i,null,ia(zi||(zi=la`E(X^2) = \mu^2 + \mu`)))))))),l(e.p,{title:"Un altro schema"},l(e.o,{title:"Schema di Poisson"},l("p",null,"Una successione di ",l("b",null,"arrivi")," avvenuti in un certo arco temporale che:"),l("ul",null,l("li",null,"non sono sovrapposti."),l("li",null,"hanno intensità ",l(e.i,null,ia(Xi||(Xi=la`\lambda`)))," costante."),l("li",null,"avvengono indipendentemente gli uni dagli altri."))),l(e.o,{title:"Processo di Poisson"},l("p",null,"Una variabile aleatoria ",l(e.i,null,"N_t")," che conta il numero di arrivi di uno schema di Poisson di intensità ",l(e.i,null,ia(qi||(qi=la`\lambda`)))," in un intervallo di tempo di durata ",l(e.i,null,"t"),"."),l("p",null,"E' una distribuzione poissoniana con ",l(e.i,null,ia(Ei||(Ei=la`\mu = t \cdot \lambda`))),": ",l(e.i,null,ia(Pi||(Pi=la`Poi(t \cdot \lambda)`)))),l(a.a,null,"E' paragonabile a una bernoulliana: ogni successo corrisponde a un arrivo, mentre il tempo è il numero di prove effettuate (ma nel continuo)."))),l(e.p,{title:"Esponenziale"},l(e.o,{title:"Distribuzione esponenziale"},l("p",null,"Una variabile aleatoria che conta il tempo diwidehattesa prima del primo arrivo di un processo di Poisson di intensità ",l(e.i,null,ia(Li||(Li=la`\lambda`))),"."),l("p",null,"Il suo simbolo è ",l(e.i,null,ia(xi||(xi=la`Esp(\lambda)`))),".")),l(e.o,{title:"Densità dell'esponenziale"},l("p",null,"L'esponenziale ha come ",l("b",null,"densità"),":"),l("p",null,l(e.i,null,ia(ki||(ki=la` - f_X (x) = \begin{cases} - 0 \qquad \qquad x < 0\\ - \lambda \cdot e^{-\lambda \cdot x} \quad x > 0 - \end{cases}`)))),l("p",null,"L'esponenziale ha come ",l("b",null,"funzione di ripartizione"),":"),l("p",null,l(e.i,null,ia(yi||(yi=la` - F_X (t) = \begin{cases} - 0 \qquad \qquad t < 0\\ - 1 - e^{-\lambda \cdot t} \quad t \geq 0 - \end{cases}`))))),l(e.o,{title:"Momenti dell'esponenziale"},l("p",null,"La ",l("b",null,"funzione generatrice dei momenti")," dell'esponenziale è:"),l("p",null,l(e.i,null,ia(Fi||(Fi=la`m_X (t) : \{ t | t < \lambda \} \in \mathbb{R} = \frac{\lambda}{\lambda - t}`)))),l("p",null,"La ",l("b",null,"media")," dell'esponenziale è:"),l("p",null,l(e.i,null,ia(Si||(Si=la`E(X) = \frac{1}{\lambda}`)))),l("p",null,"La ",l("b",null,"varianza")," dell'esponenziale è:"),l("p",null,l(e.i,null,ia(Ii||(Ii=la`Var(X) = \frac{1}{\lambda^2}`))))),l(e.o,{title:"Assenza di memoria della esponenziale"},l("p",null,"L'esponenziale non tiene conto degli eventi avvenuti in passato: ha la proprietà dell'assenza di memoria:"),l("p",null,l(e.i,null,ia(Ci||(Ci=la`P([X > s + t | X > s]) = P([X > t])`)))),l(a.a,null,"Ovvero, riscalando opportunamente l'asse Y posso prendere come 0 qualsiasi punto dell'asse X."))),l(e.p,{title:"Legge gamma"},l(e.o,{title:"Distribuzione gamma"},l("p",null,"Una variabile aleatoria che conta il tempo diwidehattesa prima dell'",l(e.i,null,"n"),"-esimo arrivo di un processo di Poisson di intensità ",l(e.i,null,ia(Ni||(Ni=la`\lambda`))),"."),l("p",null,"Il suo simbolo è ",l(e.i,null,ia(Vi||(Vi=la`\Gamma(n, \lambda)`))),".")),l(e.o,{title:"Densità della legge gamma"},l("p",null,"La legge gamma ha come densità:"),l("p",null,l(e.i,null,ia(wi||(wi=la` - f_X (x) = \begin{cases} - 0 \qquad \qquad \qquad \qquad \qquad x < 0\\ - \frac{1}{(n-1)!} \cdot \lambda^n \cdot x^{n-1} \cdot e^{-\lambda \cdot x} \quad k > 0 - \end{cases}`))))),l(e.o,{title:"Momenti della legge gamma"},l("p",null,l("p",null,"La ",l("b",null,"funzione generatrice dei momenti")," della legge gamma è:"),l("p",null,l(e.i,null,ia(Di||(Di=la`m_X (t) : ( t < \lambda ) \in \mathbb{R} = \left( \frac{\lambda}{\lambda - t} \right) ^\alpha`)))),l("p",null,"La ",l("b",null,"media")," della legge gamma è:"),l("p",null,l(e.i,null,ia(Yi||(Yi=la`E(X) = \frac{\alpha}{\lambda}`)))),l("p",null,"La ",l("b",null,"varianza")," della legge gamma è:"),l("p",null,l(e.i,null,ia(Ui||(Ui=la`Var(X) = \frac{\alpha}{\lambda^2}`))))))),l(e.p,{title:"Uniforme"},l(e.o,{title:"Distribuzione uniforme"},l("p",null,"Una variabile aleatoria che può assumere qualsiasi valore in un intervallo ",l(e.i,null,ia(Mi||(Mi=la`[a, b]`)))," in modo equiprobabile."),l("p",null,"Il suo simbolo è ",l(e.i,null,ia(Bi||(Bi=la`Uni(a, b)`)))),l("p",null,"Su di essa vale la seguente proprietà:"),l("p",null,l(e.i,null,ia(Ai||(Ai=la`P(X \in (c, d)) = \frac{d - c}{b - a}`))))),l(e.o,{title:"Densità della distribuzione uniforme"},l("p",null,"La distribuzione uniforme ha come ",l("b",null,"densità"),":"),l("p",null,l(e.i,null,ia(Ti||(Ti=la` - f_X (x) = \begin{cases} - \frac{1}{b - a} \qquad a \leq x \leq b\\ - 0 \qquad \quad altrimenti - \end{cases} - `)))),l("p",null,"La distribuzione uniforme ha come ",l("b",null,"funzione di ripartizione"),":"),l("p",null,l(e.i,null,ia(Gi||(Gi=la` - f_X (x) = \begin{cases} - 0 \qquad \quad x < a - \frac{1}{b - a} \qquad a \leq x \leq b\\ - 1 \qquad \quad x > b - \end{cases}`))))),l(e.o,{title:"Momenti della distribuzione uniforme"},l("p",null,l("p",null,"La ",l("b",null,"funzione generatrice dei momenti")," della distribuzione uniforme è:"),l("p",null,l(e.i,null,ia(Oi||(Oi=la`m_X (t) = \frac{e^{b \cdot t} - e^{a \cdot t}}{(b - a) \cdot t}`)))),l("p",null,"La ",l("b",null,"media")," della distribuzione uniforme è:"),l("p",null,l(e.i,null,ia(Ri||(Ri=la`E(X) = \frac{a + b}{2}`)))),l("p",null,"La ",l("b",null,"varianza")," della distribuzione uniforme è:"),l("p",null,l(e.i,null,ia(ji||(ji=la`Var(X) = \frac{(b - a)^2}{12}`))))))),l(e.p,{title:"Normale o Gaussiana"},l(e.o,{title:"Distribuzione normale"},l("p",null,"Una variabile aleatoria con una specifica distribuzione."),l("p",null,"Il suo simbolo è ",l(e.i,null,ia(Ki||(Ki=la`Nor(\mu, \sigma^2)`))),"."),l(a.a,null,l(e.i,null,"\\mu")," e ",l(e.i,null,"\\sigma^2")," sono rispettivamente la media e la varianza della distribuzione!")),l(e.o,{title:"Densità della distribuzione normale"},l("p",null,"La distribuzione normale ha come densità:"),l("p",null,l(e.i,null,ia(Qi||(Qi=la`f_X (x) = \frac{e^{-\frac{(x - \mu)^2}{2 \sigma^2}}}{\sqrt{2 \pi \cdot \sigma^2}}`))))),l(e.o,{title:"Momenti della distribuzione normale"},l("p",null,l("p",null,"La ",l("b",null,"funzione generatrice dei momenti")," della distribuzione normale è:"),l("p",null,l(e.i,null,ia(Zi||(Zi=la`m_X (t) = e^{\mu \cdot t + \frac{\sigma^2 \cdot t^2}{2}}`)))),l("p",null,"La ",l("b",null,"media")," della distribuzione normale è:"),l("p",null,l(e.i,null,ia(Wi||(Wi=la`E(X) = \mu`)))),l("p",null,"La ",l("b",null,"varianza")," della distribuzione normale è:"),l("p",null,l(e.i,null,ia(Ji||(Ji=la`Var(X) = \sigma^2`))))))),l(e.p,null,l(e.o,{title:"Trasformazione della normale"},l("p",null,"Qualsiasi normale può essere trasformata in qualsiasi altra normale:"),l("p",null,l(e.i,null,ia(Hi||(Hi=la`X \sim Nor(m, v^2) \implies \alpha X + \beta \sim Nor(\alpha m + \beta, (\alpha v)^2)`))))),l(e.o,{title:"Normale standard"},l("p",null,"La distribuzione normale standard ",l(e.i,null,"Z")," è:"),l("p",null,l(e.i,null,"Z \\sim Nor(0, 1)")),l("p",null,"La sua funzione di ripartizione è detta ",l(e.i,null,ia($i||($i=la`\phi(z)`)))," e vale:"),l("p",null,l(e.i,null,ia(ln||(ln=la`F_Z(z) = \phi(z) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{z} e^{-\frac{x^2}{2}} dx`))))),l(e.o,{title:"Quantili normali"},l("p",null,"Da un quantile ",l(e.i,null,ia(nn||(nn=la`z_\alpha`)))," della normale standard è possibile risalire allo stesso quantile di qualsiasi altra normale:"),l("p",null,l(e.i,null,ia(en||(en=la`x_\alpha = \mu + z_\alpha \cdot \sqrt{\sigma^2}`)))))),l(e.p,null,l(e.o,{title:"Gamma e normale"},l("p",null,"La distribuzione normale ha una particolare relazione con la distribuzione Gamma:"),l("p",null,l(e.i,null,ia(an||(an=la`Z^2 \sim \chi^2 (v = 1)`))))),l(e.o,{title:"La funzione Chi"},l("blockquote",null,'"chi-quadro a un grado di libertà"'),l("p",null,"Esiste una distribuzione Gamma particolare:"),l("p",null,l(e.i,null,ia(on||(on=la`\Gamma \left( \frac{1}{2}, \frac{1}{2} \right) = \chi^2 (v = 1)`)))),l("p",null,"Più chi-quadro possono essere sommate per aumentare i loro gradi di libertà:"),l("p",null,l(e.i,null,ia(tn||(tn=la`\chi^2 (n) + \chi^2 (m) = \chi^2 (n + m)`))))),l(e.o,{title:"T di Student"},l("p",null,"Un'altra funzione particolare è la funzione T di Student:"),l("p",null,l(e.i,null,ia(un||(un=la`T(v) = \frac{Nor(0, 1)}{\sqrt{\frac{\chi^2(v)}{v}}}`)))))),l(e.p,{title:"Approssimazioni notevoli"},l(e.o,{title:"Ipergeometrica e binomiale"},l("p",null,"La binomiale è come una ipergeometrica ma con ripetizioni, quindi per valori molto grandi di ",l(e.i,null,"N")," rispetto a ",l(e.i,null,"n"),", si può dire che:"),l("p",null,l(e.i,null,ia(rn||(rn=la`Ipe(N, K, n) \approx Bin(n, \frac{K}{N})`))))),l(e.o,{title:"Binomiale e poissoniana"},l("p",null,"La binomiale non è altro che una poissoniana a tempo discreto, quindi, se ",l(e.i,null,"n")," è grande e ",l(e.i,null,"n \\cdot p")," è nell'ordine di grandezza delle unità, allora:"),l("p",null,l(e.i,null,ia(sn||(sn=la`Bin(n, p) \approx Poi(n \cdot p)`))))),l(e.o,{title:"Binomiale e normale"},l("p",null,"Per il Teorema di De Moivre-Laplace, se una binomiale ha una ",l(e.i,null,"n")," grande e ",l(e.i,null,"p")," non vicina a 0 o 1, si può approssimare con:"),l("p",null,l(e.i,null,ia(pn||(pn=la`Bin(n, p) \approx Nor(n \cdot p, n \cdot p \cdot q)`))))),l(e.o,{title:"Correzione di Yates"},l("p",null,"Passando da una variabile discreta ",l(e.i,null,"X")," a una continua ",l(e.i,null,"Y"),", per ogni valore discreto ",l(e.i,null,"k"),' la probabilità viene "spalmata" su tutto l\'intervallo ',l(e.i,null,ia(dn||(dn=la`(k - \frac{1}{2}, k + \frac{1}{2})`))),":"),l("ul",null,l("li",null,l(e.i,null,ia(mn||(mn=la`P(X < k) \simeq P(Y \leq k - \frac{1}{2})`)))),l("li",null,l(e.i,null,ia(cn||(cn=la`P(X \leq k) \simeq P(Y \leq k + \frac{1}{2})`)))),l("li",null,l(e.i,null,ia(bn||(bn=la`P(X \geq k) \simeq P(Y \geq k - \frac{1}{2})`)))),l("li",null,l(e.i,null,ia(vn||(vn=la`P(X > k) \simeq P(Y \geq k + \frac{1}{2})`))))))),l(e.p,{title:"Vettori aleatori"},l(e.o,{title:"Vettore aleatorio"},l("p",null,"Un vettore ",l("b",null,"composto da variabili aleatorie"),"."),l("p",null,"Il suo simbolo generalmente è ",l(e.i,null,ia(gn||(gn=la`\boldsymbol{X}`)))," oppure ",l(e.i,null,ia(fn||(fn=la`X, Y`))),".")),l(e.o,{title:"Funzioni di ripartizione"},l("p",null,"I vettori aleatori hanno più funzioni di ripartizione che si differenziano in base al numero di parametri."),l("p",null,"Se il numero di parametri coincide con la dimensione del vettore aleatorio, allora la funzione sarà una ",l("i",null,"funzione di ripartizione congiunta"),":"),l("p",null,l(e.i,null,ia(_n||(_n=la`F_{X, Y} (x, y) = P(X \leq x, Y \leq y)`)))),l("p",null,"Se il numero di parametri è minore della dimensione del vettore aleatorio, allora la funzione sarà una ",l("i",null,"funzione di ripartizione marginale"),":"),l("p",null,l(e.i,null,ia(hn||(hn=la`F_X (x) = P(X \leq x) = \lim_{y \to +\infty} F_{X, Y} (x, y)`))))),l(e.o,{title:"Densità discreta"},l("p",null,"I vettori aleatori ",l("b",null,"discreti")," hanno più densità che si differenziano in base al numero di parametri."),l("p",null,"Se il numero di parametri coincide con la dimensione del vettore aleatorio, allora la funzione sarà una ",l("i",null,"densità congiunta"),":"),l("p",null,l(e.i,null,ia(zn||(zn=la`p_{X, Y} (x, y) = P(X = x, Y = y)`)))),l("p",null,"Se il numero di parametri è minore della dimensione del vettore aleatorio, allora la funzione sarà una ",l("i",null,"densità marginale"),":"),l("p",null,l(e.i,null,ia(Xn||(Xn=la`p_X (x) = \sum_j p_{X, Y} (x_i, y_j)`)))))),l(e.p,{title:"Più variabili aleatorie"},l(e.o,{title:"Indipendenza delle variabili aleatorie"},l("p",null,"Più variabili aleatorie sono indipendenti se, per qualsiasi scelta di intervalli ",l(e.i,null,"A_i"),":"),l("p",null,l(e.i,null,ia(qn||(qn=la`P(X_1 \in A_1, \dots, X_n \in A_n) = P(X_1 \in A_1) \times \dots \times P(X_n \in A_n)`))))),l(e.o,{title:"Media dei vettori aleatori"},l("p",null,"E' possibile calcolare la media di qualsiasi funzione ",l(e.i,null,"g(X, Y)")," avente elementi del vettore come variabili:"),l("p",null,l(e.i,null,ia(En||(En=la`E(g(X, Y)) = \sum_{i, j} g(x_i, y_i) \cdot p_{X, Y} (x_i, y_i)`)))),l(a.a,null,"Solitamente si calcola la media di ",l(e.i,null,"x \\cdot y"),"."),l("p",null,"Le medie di più variabili aleatorie si possono sommare:"),l("p",null,l(e.i,null,ia(Pn||(Pn=la`E(X + Y) = E(X) + E(Y)`)))))),l(e.p,null,l(e.o,{title:"Covarianza"},l("p",null,"Un ",l("b",null,"operatore")," che misura la correlazione di due variabili aleatorie."),l("p",null,"Si calcola con il valore atteso dei prodotti delle distanze dalla media:"),l("p",null,l(e.i,null,ia(Ln||(Ln=la`Cov(X, Y) = E((X - E(X) \cdot (Y - E(Y)) = E(XY) - E(X) \cdot E(Y)`)))),l("p",null,"Ha diverse proprietà:"),l("ul",null,l("li",null,"Il suo ",l("b",null,"valore nullo")," è 0: ",l(e.i,null,ia(xn||(xn=la`Cov(X, \alpha) = 0`)))),l("li",null,"E' ",l("b",null,"commutativa"),": ",l(e.i,null,ia(kn||(kn=la`Cov(X, Y) = Cov(Y, X)`)))),l("li",null,"E' ",l("b",null,"semplificabile"),": ",l(e.i,null,ia(yn||(yn=la`Cov(X, X) = Var(X)`)))),l("li",null,"E' ",l("b",null,"lineare"),": ",l(e.i,null,ia(Fn||(Fn=la`Cov(\alpha X, \beta Y) = \alpha \cdot \beta \cdot Cov(X, Y)`)))),l("li",null,"E' ",l("b",null,"distributiva"),": ",l(e.i,null,ia(Sn||(Sn=la`Cov(X + Y, V + W) = Cov(X, Y) + Cov(X, W) + Cov(Y, V) + Cov(Y, W)`)))))),l(e.o,{title:"Variabili incorrelate"},l("p",null,"Due variabili sono ",l("i",null,"variabili incorrelate")," se:"),l("p",null,l(e.i,null,ia(In||(In=la`Cov(X, Y) = 0`)))),l("p",null,"Variabili indipendenti sono sempre incorrelate.")),l(e.o,{title:"Matrice di covarianza"},l("p",null,"Una matrice ",l(e.i,null,ia(Cn||(Cn=la`\boldsymbol{C_X}`)))," che contiene la covarianza tra tutte le variabili di un vettore aleatorio ",l(e.i,null,ia(Nn||(Nn=la`\boldsymbol{X}`))),":"),l("p",null,l(e.i,null,ia(Vn||(Vn=la` - \boldsymbol{C_X} = - \begin{bmatrix} - Var(X_1) & Cov(X_1, X_2) & Cov(X_1, X_3)\\ - Cov(X_2, X_1) & Var(X_2) & Cov(X_2, X_3)\\ - Cov(X_3, X_1) & Cov(X_3, X_2) & Var(X_3) - \end{bmatrix} - `)))),l("p",null,"E' sempre simmetrica e semidefinita positiva (tutti gli autovalori sono ",l(e.i,null,"\\geq 0"),".")),l(e.o,{title:"Coefficiente di correlazione"},l("p",null,"Un valore che misura come due variabili aleatorie sono correlate:"),l("p",null,l(e.i,null,ia(wn||(wn=la`\rho_{X, Y} = \frac{Cov(X, Y)}{\sqrt{Var(X)} \cdot \sqrt{Var(Y)}}`)))),l("p",null,"E' sempre compreso tra -1 e 1:"),l("p",null,l(e.i,null,ia(Dn||(Dn=la`-1 \leq \rho_{X, Y} \leq 1`)))),l("p",null,"Vale esattamente -1 o 1 solo se esiste un legame lineare tra le due variaibli:"),l("p",null,l(e.i,null,ia(Yn||(Yn=la`Y = a X + b \Longleftrightarrow | \rho_{X, Y} | = 1`))))),l(e.o,{title:"Varianza di variabili aleatorie sommate"},l("p",null,"La varianza di due variabili aleatorie sommate è:"),l("p",null,l(e.i,null,ia(Un||(Un=la`Var(X + Y) = Var(X) + Var(Y) + 2 \cdot Cov(X, Y)`)))),l(a.a,null,"Si dimostra applicando le proprietà della covarianza!"),l("p",null,"Se più variabili aleatorie ",l(e.i,null,"X_i")," sono ",l("b",null,"indipendenti")," (",l(e.i,null,ia(Mn||(Mn=la`Cov(X, Y) = 0`))),"), allora:"),l("p",null,l(e.i,null,ia(Bn||(Bn=la`Var \left( \sum_i X_i \right) = \sum_i Var(X_i)`)))))),l(e.p,{title:"Campioni"},l(e.o,{title:"Campione casuale"},l("p",null,"Una ",l("b",null,"n-pla")," di variabili aleatorie con la stessa distribuzione della variabile aleatoria ",l(e.i,null,"X"),' ("popolazione") ma ',l("b",null,"indipendenti")," tra loro."),l(a.a,null,"Le variabili aleatorie sono come un lazy-load in programmazione; quando ci sarà bisogno del loro valore numerico, esse si ",l("b",null,"realizzeranno")," nel loro valore.")),l(e.o,{title:"Momento campionario"},l("p",null,"Il valore dato dalla media aritmetica degli ",l(e.i,null,"n")," elementi del campione elevati alla potenza ",l(e.i,null,"k"),":"),l("p",null,l(e.i,null,ia(An||(An=la`M^{(k)}_n = \frac{1}{n} \cdot \sum_{i = 1}^n X_i^k `)))),l("p",null,"Il momento campionario di primo ordine è la ",l("i",null,"media campionaria")," ",l(e.i,null,ia(Tn||(Tn=la`\overline{X}_n`))),".")),l(e.o,{title:"Varianza campionaria"},l("p",null,"La media aritmetica dello scarto quadratico medio degli elementi del campione."),l("p",null,"Se è noto il valore medio ",l(e.i,null,ia(Gn||(Gn=la`m = E(X)`)))," di X:"),l("p",null,l(e.i,null,ia(On||(On=la`S_0^2 = \frac{1}{n} \cdot \sum_{i = 1}^n (X_i - m)^2 = M_n^(2) - 2 \cdot m \cdot \overline{X}_n + m^2`)))),l("p",null,"Altrimenti:"),l("p",null,l(e.i,null,ia(Rn||(Rn=la`S_n^2 = \frac{1}{n - 1} \cdot \sum_{i = 1}^n (X_i - \overline{X}_n)^2 = \frac{1}{n - 1} \cdot ( n \cdot M_2^{(2)} - n \cdot \overline{X}_n^2)`)))))),l(e.p,{title:"Media-ception"},l(e.o,{title:"Media campionaria"},l("p",null,"Se calcoliamo la media della media campionaria, risulterà vero che:"),l("p",null,l(e.i,null,ia(jn||(jn=la`E(\overline{X}_n) = E(X)`)))),l(a.a,null,"Quindi, è possibile usare i campioni per trovare la media di una variabile aleatoria!")),l(e.o,{title:"Varianza campionaria"},l("p",null,"Se calcoliamo la varianza della media campionaria, risulterà vero che:"),l("p",null,l(e.i,null,ia(Kn||(Kn=la`Var(\overline{X}_n) = \frac{Var(X)}{n}`)))),l(a.a,null,"Quindi, possiamo stimare l'errore della media calcolata tramite campioni!")),l(e.o,{title:"Correzione campionaria"},l("p",null,"Se calcoliamo la media della varianza campionaria, risulterà vero che:"),l("p",null,l(e.i,null,ia(Qn||(Qn=la`E(S_0^2) = E(S_n^2) = Var(X)`)))),l(a.a,null,"Quindi, possiamo stimare l'errore della media calcolata tramite campioni!"))),l(e.p,{title:"Campionamento di una distribuzione normale"},l(e.o,{title:"Campionamento di una distribuzione normale"},l("p",null,"Se la popolazione ",l(e.i,null,"X")," ha una distribuzione normale (",l(e.i,null,ia(Zn||(Zn=la`X \sim Nor(\mu, \sigma^2)`))),")...")),l(e.o,{title:"Distribuzione della media campionaria"},l("p",null,"...allora sappiamo anche la distribuzione della media campionaria!"),l("p",null,l(e.i,null,ia(Wn||(Wn=la`\overline{X}_n \sim Nor \left( \mu, \frac{\sigma^2}{n} \right)`))))),l(e.o,{title:"Distribuzione della varianza campionaria"},l("p",null,"...e anche della varianza campionaria!"),l("p",null,l(e.i,null,ia(Jn||(Jn=la`S_0^2 \sim \frac{\sigma^2}{n} \cdot \chi^2 (n)`)))),l("p",null,l(e.i,null,ia(Hn||(Hn=la`S_n^2 \sim \frac{\sigma^2}{n - 1} \cdot \chi^2 (n-1)`))))),l(e.o,{title:"Indipendenza"},l("p",null,"...e che media campionaria e varianza campionaria sono indipendenti tra loro!"))),l(e.p,{title:"Quando i campioni hanno dimensioni infinite"},l(e.o,{title:"Convergenza in distribuzione"},l("p",null,"Se la successione di variabili aleatorie ",l(e.i,null,"X_n")," all'infinito ha la ",l("b",null,"stessa funzione di ripartizione")," della popolazione ",l(e.i,null,"X"),", allora essa ",l("i",null,"converge in distribuzione"),"."),l("p",null,l(e.i,null,"\\lim_{n \\to +\\infty} F_{X_n} (x) = F_X (x) \\implies X_n \\xrightarrow{d} X"))),l(e.o,{title:"Convergenza in probabilità"},l("p",null,"Se la successione di variabili aleatorie ",l(e.i,null,"X_n")," all'infinito ha la ",l("b",null,"stessa probabilità")," della popolazione ",l(e.i,null,"X"),", allora essa ",l("i",null,"converge in probabilità"),"."),l("p",null,l(e.i,null,"\\forall \\epsilon > 0, \\lim_{n \\to +\\infty} P( | X_n - X | < \\epsilon) = 1 \\implies X_n \\xrightarrow{p} X"))),l(e.o,{title:"Convergenza quasi certa"},l("p",null,"Se la successione di variabili aleatorie ",l(e.i,null,"X_n")," all'infinito ha la ",l("b",null,"stessa probabilità a ")," della popolazione ",l(e.i,null,"X"),", allora essa ",l("i",null,"converge quasi certamente"),"."),l("p",null,l(e.i,null,"\\forall \\epsilon > 0, P left( \\lim_{n \\to +\\infty} | X_n - X | < \\epsilon) \right) = 1 \\implies X_n \\xrightarrow{qc} X"))),l(e.o,{title:"Convergenza in media quadratica"},l("p",null,"Se la successione di variabili aleatorie ",l(e.i,null,"X_n")," all'infinito ha la ",l("b",null,"media del quadrato della distanza")," tra la successione e la popolazione ",l(e.i,null,"X")," ",l("b",null,"uguale a 0"),", allora essa ",l("i",null,"converge in media quadratica"),"."),l("p",null,l(e.i,null,"\\lim_{n \\to +\\infty} E( | X_n - X |^2 = 0 \\implies X_n \\xrightarrow{mq} X"))),l(e.o,{title:"Gerarchia delle convergenze"},l("p",null,l(e.i,null,"\n \\begin{matrix}\n X_n \\xrightarrow{mq} X\\\\\n X_n \\xrightarrow{qc} X\n \\end{matrix} \\implies X_n \\xrightarrow{p} X \\implies X_n \\xrightarrow{d} X")),l("p",null,"In più:"),l("p",null,l(e.i,null,"X_n \\xrightarrow{p} x \\Longleftrightarrow X_n \\xrightarrow{d} x")))),l(e.p,{title:"I grandi numeri"},l(e.o,{title:"Legge debole dei grandi numeri"},l("p",null,"La successione delle medie campionarie ",l(e.i,null,ia($n||($n=la`\overline{X}_n`)))," ",l("b",null,"converge in probabilità")," alla media della popolazione ",l(e.i,null,ia(le||(le=la`E(X)`))),", se essa esiste."),l("p",null,l(e.i,null,"\\overline{X}_n \\xrightarrow{p} X")),l("p",null,"Ovvero:"),l("p",null,l(e.i,null,ia(ie||(ie=la`\forall \epsilon > 0, \lim_{n \to +\infty} P( | \overline{X}_n - E(X) | < \epsilon) = 1`)))),l("p",null,l(e.i,null,ia(ne||(ne=la`P( | \overline{X}_n - E(X) | < \epsilon) \to 1`))))),l(e.o,{title:"Legge forte dei grandi numeri"},l("p",null,"La successione delle medie campionarie ",l(e.i,null,ia(ee||(ee=la`\overline{X}_n`)))," ",l("b",null,"converge quasi certamente")," alla media della popolazione ",l(e.i,null,ia(ae||(ae=la`E(X)`))),", se essa esiste."),l("p",null,l(e.i,null,"\\overline{X}_n \\xrightarrow{qc} X")),l("p",null,"Ovvero:"),l("p",null,l(e.i,null,ia(oe||(oe=la`\forall \epsilon > 0, P \left( \lim_{n \to +\infty} | \overline{X}_n - E(X) | < \epsilon \right) = 1`)))),l(a.a,null,"Dimostra che l'interpretazione frequentista della probabilità è valida!"))),l(e.p,{title:"Al limite"},l(e.o,{title:"Teorema centrale del limite"},l("p",null,"La successione delle medie campionarie ",l(e.i,null,ia(te||(te=la`\overline{X}_n`)))," ",l("b",null,"converge in distribuzione")," a ",l(e.i,null,ia(ue||(ue=la`Nor(0, 1) = \Phi()`))),"."),l("p",null,l(e.i,null,ia(re||(re=la`\overline{X}_n \approx Nor \left(E(X), \frac{Var(X)}{n} \right)`)))),l("p",null,"Ovvero:"),l("p",null,l(e.i,null,ia(se||(se=la`\forall x \in \mathbb{R}, \lim_{n \to +\infty} P \left( \frac{\overline{X}_n - E(X)}{\sqrt{\frac{Var(X)}{n}}} \leq x \right) = \Phi(x)`)))))),l(e.p,{title:"Altre approsimazioni"},l(e.o,{title:"Binomiale e normale"},l("p",null,"E' una somma di ",l("b",null,"bernoulliane"),", e quindi si approssima a una normale:"),l("p",null,l(e.i,null,ia(pe||(pe=la`Bin(n, p) \approx Nor(n \cdot p, n \cdot p \cdot q)`))))),l(e.o,{title:"Binomiale negativa e normale"},l("p",null,"E' una somma di ",l("b",null,"geometriche"),", e quindi si approssima a una normale:"),l("p",null,l(e.i,null,ia(de||(de=la`\overline{Bin} (n, p) \approx Nor \left( \frac{n}{p}, \frac{n \cdot (1 - p)}{p^2} \right)`))))),l(e.o,{title:"Poissoniana e normale"},l("p",null,"E' una somma di altre ",l("b",null,"poissoniane"),", e quindi si approssima a una normale:"),l("p",null,l(e.i,null,ia(me||(me=la`Poi(\lambda) \approx Nor(\lambda, \lambda)`))))),l(e.o,{title:"Gamma e normale"},l("p",null,"E' una somma di ",l("b",null,"esponenziali"),", e quindi si approssima a una normale:"),l("p",null,l(e.i,null,ia(ce||(ce=la`\Gamma (\alpha, \lambda) \approx Nor \left( \frac{\alpha}{\lambda}, \frac{\alpha}{\lambda^2} \right)`))))),l(e.o,{title:"In generale"},l("p",null,"Se ",l(e.i,null,"n")," è grande, allora:"),l("p",null,l(e.i,null,ia(be||(be=la`Y = \sum_{i=1}^{n} X_i`)))))),l(e.p,{title:"Actually statistica"},l(e.o,{title:"Parametri sconosciuti"},l("p",null,"Per indicare parametri sconosciuti di una legge si usa ",l(e.i,null,"\\theta"),".")),l(e.o,{title:"Statistica"},l("p",null,"Una variabile aleatoria funzione di un campione:"),l("p",null,l(e.i,null,ia(ve||(ve=la`T(\boldsymbol{X})`)))),l(a.a,null,"Ad esempio, sono statistiche media e varianza campionaria, così come il campione stesso ",l(e.i,null,ia(ge||(ge=la`T(\boldsymbol{X}) = \boldsymbol{X}`))),"."))),l(e.p,{title:"Stimatori"},l(e.o,{title:"Stimatore"},l("p",null,"Una statistica ",l(e.i,null,"T_n")," ottenuta da ",l(e.i,null,"n")," osservazioni, che stimi i parametri di una legge e sia indipendente da essi.")),l(e.o,{title:"Corretto"},l("p",null,"Uno stimatore è ",l("i",null,"corretto")," se il suo valore atteso coincide con quello dei parametri che stima:"),l("p",null,l(e.i,null,ia(fe||(fe=la`E(T_n) = \theta`))))),l(e.o,{title:"Asintoticamente corretto"},l("p",null,"Uno stimatore è ",l("i",null,"asintoticamente corretto")," se, per infinite osservazioni, il suo valore atteso coincide con quello dei parametri che stima:"),l("p",null,l(e.i,null,ia(_e||(_e=la`\lim_{n \to +\infty} E(T_n) = \theta`))))),l(e.o,{title:"Consistente in media quadratica"},l("p",null,"Uno stimatore è ",l("i",null,"consistente in media quadratica")," se:"),l("p",null,l(e.i,null,ia(he||(he=la`\lim_{n \to +\infty} E((T_n - \theta)^2) = 0`))))),l(e.o,{title:"Consistente in probabilità"},l("p",null,"Uno stimatore è ",l("i",null,"consistente in probabilità")," se:"),l("p",null,l(e.i,null,ia(ze||(ze=la`\forall \epsilon > 0, \lim_{n \to +\infty} P( |T_n - \theta| < \epsilon) = 1`))))),l(e.o,{title:"Asintoticamente normale"},l("p",null,"Uno stimatore è ",l("i",null,"asintoticamente normale")," se:"),l("p",null,l(e.i,null,ia(Xe||(Xe=la`\lim_{n \to +\infty} \frac{T_n - E(T_n)}{\sqrt{Var(T_n)}} \sim Nor(0, 1)`)))))),l(e.p,{title:"Metodo dei momenti"},l(e.o,{title:"Metodo dei momenti"},l("p",null,"Si può usare il ",l("i",null,"metodo dei momenti")," per ottenere uno stimatore di una popolazione ",l(e.i,null,"X"),"."),l("p",null,"Lo stimatore di ",l(e.i,null,ia(qe||(qe=la`\theta`)))," così ottenuto sarà indicato aggiungendo un cappellino e una ",l(e.i,null,"M")," a ",l(e.i,null,"\\theta"),": ",l(e.i,null,ia(Ee||(Ee=la`\widehat{\theta}_M`)))),l("p",null,"Visto che:"),l("ul",null,l("li",null,l(e.i,null,ia(Pe||(Pe=la`\theta = g(E(X))`)))),l("li",null,l(e.i,null,ia(Le||(Le=la`\widehat{E(X)} = \overline{X}_n`))))),l("p",null,"Allora:"),l("p",null,l(e.i,null,ia(xe||(xe=la`\widehat{\theta}_M = g( \overline{X}_n )`)))),l("p",null,"Se ",l(e.i,null,ia(ke||(ke=la`\theta`)))," non è esprimibile in termini di ",l(e.i,null,ia(ye||(ye=la`E(X)`))),", si possono usare i momenti successivi ",l(e.i,null,ia(Fe||(Fe=la`M_n^2`))),", ",l(e.i,null,ia(Se||(Se=la`M_n^3`))),", ",l(e.i,null,ia(Ie||(Ie=la`M_n^3`))),"..."))),l(e.p,{title:"Metodo della massima verosomiglianza"},l(e.o,{title:"Metodo della massima verosomiglianza"},l("p",null,"Si può usare il ",l("i",null,"metodo della massima verosomiglianza")," per ottenere uno stimatore di una popolazione ",l(e.i,null,"X"),"."),l("p",null,"Lo stimatore di ",l(e.i,null,ia(Ce||(Ce=la`\theta`)))," così ottenuto sarà indicato aggiungendo un cappellino e una ",l(e.i,null,"L")," a ",l(e.i,null,"\\theta"),": ",l(e.i,null,ia(Ne||(Ne=la`\widehat{\theta}_L`)))),l("p",null,"Consiste nel trovare il massimo assoluto ",l(e.i,null,ia(Ve||(Ve=la`\widehat{\theta}_L`)))," della la funzione di verosomiglianza ",l(e.i,null,ia(we||(we=la`L`))),":"),l("p",null,l(e.i,null,ia(De||(De=la`L(x_1, ..., x_n; \theta) = \prod_{i=1}^n f_X(x_i; \theta)`)))),l("p",null,"Gli stimatori di massima verosomiglianza sono ",l("b",null,"asintoticamente corretti"),", ",l("b",null,"consistenti in probabilità")," e ",l("b",null,"asintoticamente normali"),".")),l(e.o,{title:"Proprietà degli stimatori di massima verosomiglianza"},l("p",null,"Gli stimatori di massima verosomiglianza godono delle seguenti proprietà:"),l("ul",null,l("li",null,"Sono ",l("b",null,"asintoticamente corretti"),"."),l("li",null,"Sono ",l("b",null,"consistenti in probabilità"),"."),l("li",null,"Sono ",l("b",null,"asintoticamente normali"),"."),l("li",null,"Sono ",l("b",null,"invarianti"),": ",l(e.i,null,ia(Ye||(Ye=la`\widehat{g(\theta)}_L = g(\widehat{\theta}_L)`))))))),l(e.p,{title:"Nuove stime notevoli"},l(e.o,{title:"Stima di una bernoulliana"},l("p",null,"Per il metodo dei momenti oppure per il metodo della massima verosomiglianza:"),l("p",null,l(e.i,null,ia(Ue||(Ue=la`\widehat{p}_M = \widehat{p}_L = \overline{X}_n`))))),l(e.o,{title:"Stima di una poissoniana"},l("p",null,"Per il metodo dei momenti oppure per il metodo della massima verosomiglianza:"),l("p",null,l(e.i,null,ia(Me||(Me=la`\widehat{\mu}_M = \widehat{\mu}_L = \overline{X}_n`))))),l(e.o,{title:"Stima di una esponenziale"},l("p",null,"Per il metodo dei momenti oppure per il metodo della massima verosomiglianza:"),l("p",null,l(e.i,null,ia(Be||(Be=la`\widehat{\lambda}_M = \widehat{\lambda}_L = \frac{1}{\overline{X}_n}`))))),l(e.o,{title:"Stima di una normale"},l("p",null,"Per il metodo della massima verosomiglianza:"),l("ul",null,l("li",null,l(e.i,null,ia(Ae||(Ae=la`\widehat{\mu}_L = \overline{X}_n`)))),l("br",null),l("li",null,l(e.i,null,ia(Te||(Te=la`\widehat{\sigma^2}_L = \frac{\sum (X_i - \overline{X}_n)^2 }{n}`))))))),l(e.p,{title:"Intervalli di confidenza"},l(e.o,{title:"Confidenza"},l("blockquote",null,'"intervallo di confidenza al 95%"'),l("p",null,"L'intervallo di valori di ",l(e.i,null,"\\theta"),' all\'interno del quale siamo "più o meno sicuri" si trovi il valore effettivo:'),l("p",null,"L'intervallo di confidenza a N della stima ",l(e.i,null,ia(Ge||(Ge=la`\widehat{W}`)))," è l'intervallo ",l(e.i,null,"]a, b[")," tale che:"),l("p",null,l(e.i,null,ia(Oe||(Oe=la`P( a < W < b ) = N`)))),l("p",null,"Può anche essere ",l("b",null,"unilatero")," nel caso limiti la stima in una sola direzione, positiva o negativa."))),l(e.p,{title:"Confidenza nella media di una normale"},l(e.o,{title:"Varianza nota"},l("p",null,"Se conosciamo la varianza di una normale, allora possiamo ricavare velocemente gli intervalli di confidenza all'",l(e.i,null,"\\alpha"),"% con queste formule:"),l("ul",null,l("li",null,"Intervalli bilateri: ",l(e.i,null,ia(Re||(Re=la`\mu \in \left[ \overline{x}_n - z_{1 - \frac{\alpha}{2}} \cdot \sqrt{\frac{\sigma^2}{n}}, \overline{x}_n + z_{1 - \frac{\alpha}{2}} \cdot \sqrt{\frac{\sigma^2}{n}} \right]`)))),l("li",null,"Intervallo unilatero da sinistra: ",l(e.i,null,ia(je||(je=la`\mu \in \left( -\infty, \overline{x}_n + z_{1 - \frac{\alpha}{2}} \cdot \sqrt{\frac{\sigma^2}{n}} \right]`)))),l("li",null,"Intervallo unilatero da destra: ",l(e.i,null,ia(Ke||(Ke=la`\mu \in \left[ \overline{x}_n - z_{1 - \frac{\alpha}{2}} \cdot \sqrt{\frac{\sigma^2}{n}}, +\infty \right)`)))))),l(e.o,{title:"Varianza incognita"},l("p",null,"Se non conosciamo la varianza di una normale, allora possiamo ricavare velocemente gli intervalli di confidenza all'",l(e.i,null,"\\alpha"),"% con queste formule:"),l("ul",null,l("li",null,"Intervalli bilateri: ",l(e.i,null,ia(Qe||(Qe=la`\mu \in \left[ \overline{x}_n - t_{1 - \frac{\alpha}{2}; n-1} \cdot \sqrt{\frac{s_n^2}{n}}, \overline{x}_n + t_{1 - \frac{\alpha}{2}; n-1} \cdot \sqrt{\frac{s_n^2}{n}} \right]`)))),l("li",null,"Intervallo unilatero da sinistra: ",l(e.i,null,ia(Ze||(Ze=la`\mu \in \left( -\infty, \overline{x}_n + t_{1 - \frac{\alpha}{2}; n-1} \cdot \sqrt{\frac{s_n^2}{n}} \right]`)))),l("li",null,"Intervallo unilatero da destra: ",l(e.i,null,ia(We||(We=la`\mu \in \left[ \overline{x}_n - t_{1 - \frac{\alpha}{2}; n-1} \cdot \sqrt{\frac{s_n^2}{n}}, +\infty \right)`))))),l("p",null,l(e.i,null,ia(Je||(Je=la`t_{\alpha, v}`)))," è un quantile della distribuzione di Student di parametro ",l(e.i,null,"v"),"."))),l(e.p,{title:"Confidenza per la proporzione di una bernoulliana"},l(e.o,{title:"Terzo metodo corretto"},l("p",null,"L'intervallo di confidenza per la proprorzione di una bernoulliana qualsiasi si ottiene da questa formula:"),l("p",null,l(e.i,null,ia(He||(He=la`p \in \left[ \overline{p} - z_{1 - \frac{\alpha}{2}} \cdot \sqrt{\frac{\overline{p} \cdot (1 - \overline{p})}{n+4}}, \overline{p} + z_{1 - \frac{\alpha}{2}} \cdot \sqrt{\frac{\overline{p} \cdot (1 - \overline{p})}{n+4}} \right]`)))))),l(e.p,{title:"Confidenza per la media di qualsiasi popolazione"},l(e.o,{title:"Approssimando con la normale"},l("p",null,"L'intervallo di confidenza per la media di una qualsiasi popolazione si ottiene da questa formula:"),l("p",null,l(e.i,null,ia($e||($e=la`m \in \left[ \overline{x}_n - z_{1 - \frac{\alpha}{2}} \cdot \sqrt{\frac{s^2_n}{n}}, \overline{x}_n + z_{1 - \frac{\alpha}{2}} \cdot \sqrt{\frac{s^2_n}{n}} \right]`)))))))))}}.call(this,n("hosL").h)},gJgL:function(l){l.exports={red:"red__3phX0",orange:"orange__1OSkI",yellow:"yellow__2pQOr",lime:"lime__3vRUg",cyan:"cyan__1cBtG",blue:"blue__1G1wb",magenta:"magenta__3Wagv",minus:"minus__2qLyv"}},jZax:function(l){l.exports={red:"red__ipSoq",orange:"orange__1nmkj",yellow:"yellow__353ge",lime:"lime__3FW1E",cyan:"cyan__2xhu7",blue:"blue__3MvDe",magenta:"magenta__i03ie",plus:"plus__1eV4_"}},ke5e:function(l,i,n){"use strict";(function(l){var e=n("2w3n"),a=n.n(e);i.a=function(i){return l("div",{class:a.a.example},i.children)}}).call(this,n("hosL").h)},o8Jj:function(l,i,n){"use strict";(function(l){var e=n("jZax"),a=n.n(e);i.a=function(i){return l("span",{class:a.a.plus},i.children)}}).call(this,n("hosL").h)}}]); -//# sourceMappingURL=route-Statistica.chunk.ae230.esm.js.map \ No newline at end of file diff --git a/docs/route-Statistica.chunk.ae230.esm.js.map b/docs/route-Statistica.chunk.ae230.esm.js.map deleted file mode 100644 index daa84c3..0000000 --- a/docs/route-Statistica.chunk.ae230.esm.js.map +++ /dev/null @@ -1 +0,0 @@ -{"version":3,"sources":[],"names":[],"mappings":"","file":"route-Statistica.chunk.ae230.esm.js","sourceRoot":""} \ No newline at end of file diff --git a/docs/route-VlDiGeometria.chunk.b1428.js b/docs/route-VlDiGeometria.chunk.615ff.js similarity index 93% rename from docs/route-VlDiGeometria.chunk.b1428.js rename to docs/route-VlDiGeometria.chunk.615ff.js index 2400aab..4df21dd 100644 --- a/docs/route-VlDiGeometria.chunk.b1428.js +++ b/docs/route-VlDiGeometria.chunk.615ff.js @@ -1,2 +1,2 @@ -(window.webpackJsonp=window.webpackJsonp||[]).push([[12],{UvCR:function(t,i,e){"use strict";e.r(i),function(t){function o(){var t=function(t,i){i||(i=t.slice(0));return t.raw=i,t}(["\n\t\t\t\t\tTutte le videolezioni sono state pubblicate sotto licenza [CC BY-NC-SA 4.0](https://creativecommons.org/licenses/by-nc-sa/4.0/) dalla Prof.ssa Beatrice Ruini nell'anno accademico 2018/2019 sul [portale Dolly 2018](https://dolly.fim.unimore.it/2018/course/view.php?id=14#section-0) (Moodle).\n\t\t\t\t\t\n\t\t\t\t\tPer comodità, ho estratto l'url sorgente del video dall'embed presente nella rispettiva pagina.\n\t\t\t\t\t\n\t\t\t\t\t1. [Definizione di Spazio Vettoriale](https://www.youtube.com/watch?v=7eHEzf4403c) (1:17:29)\n\t\t\t\t\t2. [Sottospazi vettoriali I](https://www.youtube.com/watch?v=FPqrULk5HBU) (37:15)\n\t\t\t\t\t3. [Sottospazi vettoriali II](https://www.youtube.com/watch?v=ubDWUw9hk0k) (43:26)\n\t\t\t\t\t4. [Sottospazi vettoriali III](https://www.youtube.com/watch?v=381n4NPb6Oc) (40:29)\n\t\t\t\t\t5. [Lineare dipendenza e indipendenza](https://www.youtube.com/watch?v=9YVQ5olYrh0) (56:12)\n\t\t\t\t\t6. [Basi di uno spazio vettoriale I](https://www.youtube.com/watch?v=mEF_lcTzEoE) (25:52)\n\t\t\t\t\t7. [Basi di uno spazio vettoriale II](https://www.youtube.com/watch?v=k1r9JfXY53k) (48:24)\n\t\t\t\t\t8. [Teorema di Grassmann](https://www.youtube.com/watch?v=3sqB-MMyCWM) (32:36)\n\t\t\t\t\t9. [Basi e Matrici](https://www.youtube.com/watch?v=Rd6AB_jE7YI) (27:06)\n\t\t\t\t\t10. [Definizione di Applicazioni Lineari](https://www.youtube.com/watch?v=rmd7ffZeVYk) (16:23)\n\t\t\t\t\t11. [Proprietà delle Applicazioni Lineari](https://www.youtube.com/watch?v=MH7ztQGkqmw) (31:58)\n\t\t\t\t\t12. [Definizione di determinante](https://www.youtube.com/watch?v=EwubcLwBdzk) (36:43)\n\t\t\t\t\t13. [Proprietà e metodo di triangolazione](https://www.youtube.com/watch?v=SFusGarV6HI) (22:36)\n\t\t\t\t\t14. [Teorema di Laplace](https://www.youtube.com/watch?v=BqZDWnKl2nQ) (29:03)\n\t\t\t\t\t15. [4 applicazioni del Teorema di Laplace](https://www.youtube.com/watch?v=2tr3y725GY0) (47:53)\n\t\t\t\t\t16. [Spazi vettoriali euclidei reali - Parte 1](https://www.youtube.com/watch?v=W7Z1hm-jwMM) (28:46)\n\t\t\t\t\t17. [Spazi vettoriali euclidei reali - Parte 2](https://www.youtube.com/watch?v=zjmKE9TMGm8) (27:17)\n\t\t\t\t\t18. [Autovalori e autovettori](https://www.youtube.com/watch?v=XlrlcnvcTtQ) (33:00)\n\t\t\t\t\t19. [Polinomio caratteristico](https://www.youtube.com/watch?v=61icRbgWTdI) (31:31)\n\t\t\t\t\t20. [Teorema diagonalizzabilità](https://www.youtube.com/watch?v=wm5V6en9OFo) (18:49)\n\t\t\t\t\t21. [Spazi affini](https://player.vimeo.com/video/291457587) (20:46)\n\t\t\t\t\t22. [Sottospazi affini](https://player.vimeo.com/video/291458991) (21:32)\n\t\t\t\t\t23. [Parallelismo e Riferimenti Affini](https://player.vimeo.com/video/291510181) (16:57)\n\t\t\t\t\t24. [Rappresentazione di Sottospazi Affini](https://player.vimeo.com/video/291510296) (31:17)\n\t\t\t\t\t25. [Spazi Euclidei](https://player.vimeo.com/video/291510612) (35:57)\n\t\t\t\t\t26. [Teoria dei ranghi](https://player.vimeo.com/video/291510964) (9:44)\n\t\t\t\t\t27. [Teoria dei ranghi 2](https://player.vimeo.com/video/291510862) (14:44)\n\t\t\t\t\t\n\t\t\t\t\tNell'anno accademico 2018/2019 non sono stati trattati gli argomenti nei video 21, 22 e 23.\n\t\t\t\t"]);return o=function(){return t},t}var a=e("mbOI"),n=String.raw;i.default=function(){return t("div",null,t("h1",null,"Videolezioni di Geometria"),t(a.o,null,t(a.m,null,n(o()))))}}.call(this,e("hosL").h)}}]); -//# sourceMappingURL=route-VlDiGeometria.chunk.b1428.js.map \ No newline at end of file +(window.webpackJsonp=window.webpackJsonp||[]).push([[13],{UvCR:function(t,i,e){"use strict";e.r(i),function(t){function o(){var t=function(t,i){i||(i=t.slice(0));return t.raw=i,t}(["\n\t\t\t\t\tTutte le videolezioni sono state pubblicate sotto licenza [CC BY-NC-SA 4.0](https://creativecommons.org/licenses/by-nc-sa/4.0/) dalla Prof.ssa Beatrice Ruini nell'anno accademico 2018/2019 sul [portale Dolly 2018](https://dolly.fim.unimore.it/2018/course/view.php?id=14#section-0) (Moodle).\n\t\t\t\t\t\n\t\t\t\t\tPer comodità, ho estratto l'url sorgente del video dall'embed presente nella rispettiva pagina.\n\t\t\t\t\t\n\t\t\t\t\t1. [Definizione di Spazio Vettoriale](https://www.youtube.com/watch?v=7eHEzf4403c) (1:17:29)\n\t\t\t\t\t2. [Sottospazi vettoriali I](https://www.youtube.com/watch?v=FPqrULk5HBU) (37:15)\n\t\t\t\t\t3. [Sottospazi vettoriali II](https://www.youtube.com/watch?v=ubDWUw9hk0k) (43:26)\n\t\t\t\t\t4. [Sottospazi vettoriali III](https://www.youtube.com/watch?v=381n4NPb6Oc) (40:29)\n\t\t\t\t\t5. [Lineare dipendenza e indipendenza](https://www.youtube.com/watch?v=9YVQ5olYrh0) (56:12)\n\t\t\t\t\t6. [Basi di uno spazio vettoriale I](https://www.youtube.com/watch?v=mEF_lcTzEoE) (25:52)\n\t\t\t\t\t7. [Basi di uno spazio vettoriale II](https://www.youtube.com/watch?v=k1r9JfXY53k) (48:24)\n\t\t\t\t\t8. [Teorema di Grassmann](https://www.youtube.com/watch?v=3sqB-MMyCWM) (32:36)\n\t\t\t\t\t9. [Basi e Matrici](https://www.youtube.com/watch?v=Rd6AB_jE7YI) (27:06)\n\t\t\t\t\t10. [Definizione di Applicazioni Lineari](https://www.youtube.com/watch?v=rmd7ffZeVYk) (16:23)\n\t\t\t\t\t11. [Proprietà delle Applicazioni Lineari](https://www.youtube.com/watch?v=MH7ztQGkqmw) (31:58)\n\t\t\t\t\t12. [Definizione di determinante](https://www.youtube.com/watch?v=EwubcLwBdzk) (36:43)\n\t\t\t\t\t13. [Proprietà e metodo di triangolazione](https://www.youtube.com/watch?v=SFusGarV6HI) (22:36)\n\t\t\t\t\t14. [Teorema di Laplace](https://www.youtube.com/watch?v=BqZDWnKl2nQ) (29:03)\n\t\t\t\t\t15. [4 applicazioni del Teorema di Laplace](https://www.youtube.com/watch?v=2tr3y725GY0) (47:53)\n\t\t\t\t\t16. [Spazi vettoriali euclidei reali - Parte 1](https://www.youtube.com/watch?v=W7Z1hm-jwMM) (28:46)\n\t\t\t\t\t17. [Spazi vettoriali euclidei reali - Parte 2](https://www.youtube.com/watch?v=zjmKE9TMGm8) (27:17)\n\t\t\t\t\t18. [Autovalori e autovettori](https://www.youtube.com/watch?v=XlrlcnvcTtQ) (33:00)\n\t\t\t\t\t19. [Polinomio caratteristico](https://www.youtube.com/watch?v=61icRbgWTdI) (31:31)\n\t\t\t\t\t20. [Teorema diagonalizzabilità](https://www.youtube.com/watch?v=wm5V6en9OFo) (18:49)\n\t\t\t\t\t21. [Spazi affini](https://player.vimeo.com/video/291457587) (20:46)\n\t\t\t\t\t22. [Sottospazi affini](https://player.vimeo.com/video/291458991) (21:32)\n\t\t\t\t\t23. [Parallelismo e Riferimenti Affini](https://player.vimeo.com/video/291510181) (16:57)\n\t\t\t\t\t24. [Rappresentazione di Sottospazi Affini](https://player.vimeo.com/video/291510296) (31:17)\n\t\t\t\t\t25. [Spazi Euclidei](https://player.vimeo.com/video/291510612) (35:57)\n\t\t\t\t\t26. [Teoria dei ranghi](https://player.vimeo.com/video/291510964) (9:44)\n\t\t\t\t\t27. [Teoria dei ranghi 2](https://player.vimeo.com/video/291510862) (14:44)\n\t\t\t\t\t\n\t\t\t\t\tNell'anno accademico 2018/2019 non sono stati trattati gli argomenti nei video 21, 22 e 23.\n\t\t\t\t"]);return o=function(){return t},t}var a=e("mbOI"),n=String.raw;i.default=function(){return t("div",null,t("h1",null,"Videolezioni di Geometria"),t(a.p,null,t(a.n,null,n(o()))))}}.call(this,e("hosL").h)}}]); +//# sourceMappingURL=route-VlDiGeometria.chunk.615ff.js.map \ No newline at end of file diff --git a/docs/route-VlDiGeometria.chunk.b1428.js.map b/docs/route-VlDiGeometria.chunk.615ff.js.map similarity index 98% rename from docs/route-VlDiGeometria.chunk.b1428.js.map rename to docs/route-VlDiGeometria.chunk.615ff.js.map index b519c7e..9754104 100644 --- a/docs/route-VlDiGeometria.chunk.b1428.js.map +++ b/docs/route-VlDiGeometria.chunk.615ff.js.map @@ -1 +1 @@ -{"version":3,"sources":["webpack:///./routes/VlDiGeometria.js"],"names":["r","String","raw"],"mappings":"g1GAEMA,EAAIC,OAAOC,IAEF,qBACd,OACC,aACC,yCACA,EAAC,IAAD,KACC,EAAC,IAAD,KAAWF,EAAX,U","file":"route-VlDiGeometria.chunk.b1428.js","sourcesContent":["import {Markdown, Panel} from \"bluelib\";\n\nconst r = String.raw;\n\nexport default function(props) {\n\treturn (\n\t\t
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Videolezioni di Geometria

\n\t\t\t\n\t\t\t\t{r`\n\t\t\t\t\tTutte le videolezioni sono state pubblicate sotto licenza [CC BY-NC-SA 4.0](https://creativecommons.org/licenses/by-nc-sa/4.0/) dalla Prof.ssa Beatrice Ruini nell'anno accademico 2018/2019 sul [portale Dolly 2018](https://dolly.fim.unimore.it/2018/course/view.php?id=14#section-0) (Moodle).\n\t\t\t\t\t\n\t\t\t\t\tPer comodità, ho estratto l'url sorgente del video dall'embed presente nella rispettiva pagina.\n\t\t\t\t\t\n\t\t\t\t\t1. [Definizione di Spazio Vettoriale](https://www.youtube.com/watch?v=7eHEzf4403c) (1:17:29)\n\t\t\t\t\t2. [Sottospazi vettoriali I](https://www.youtube.com/watch?v=FPqrULk5HBU) (37:15)\n\t\t\t\t\t3. [Sottospazi vettoriali II](https://www.youtube.com/watch?v=ubDWUw9hk0k) (43:26)\n\t\t\t\t\t4. [Sottospazi vettoriali III](https://www.youtube.com/watch?v=381n4NPb6Oc) (40:29)\n\t\t\t\t\t5. [Lineare dipendenza e indipendenza](https://www.youtube.com/watch?v=9YVQ5olYrh0) (56:12)\n\t\t\t\t\t6. [Basi di uno spazio vettoriale I](https://www.youtube.com/watch?v=mEF_lcTzEoE) (25:52)\n\t\t\t\t\t7. [Basi di uno spazio vettoriale II](https://www.youtube.com/watch?v=k1r9JfXY53k) (48:24)\n\t\t\t\t\t8. [Teorema di Grassmann](https://www.youtube.com/watch?v=3sqB-MMyCWM) (32:36)\n\t\t\t\t\t9. [Basi e Matrici](https://www.youtube.com/watch?v=Rd6AB_jE7YI) (27:06)\n\t\t\t\t\t10. [Definizione di Applicazioni Lineari](https://www.youtube.com/watch?v=rmd7ffZeVYk) (16:23)\n\t\t\t\t\t11. [Proprietà delle Applicazioni Lineari](https://www.youtube.com/watch?v=MH7ztQGkqmw) (31:58)\n\t\t\t\t\t12. [Definizione di determinante](https://www.youtube.com/watch?v=EwubcLwBdzk) (36:43)\n\t\t\t\t\t13. [Proprietà e metodo di triangolazione](https://www.youtube.com/watch?v=SFusGarV6HI) (22:36)\n\t\t\t\t\t14. [Teorema di Laplace](https://www.youtube.com/watch?v=BqZDWnKl2nQ) (29:03)\n\t\t\t\t\t15. [4 applicazioni del Teorema di Laplace](https://www.youtube.com/watch?v=2tr3y725GY0) (47:53)\n\t\t\t\t\t16. [Spazi vettoriali euclidei reali - Parte 1](https://www.youtube.com/watch?v=W7Z1hm-jwMM) (28:46)\n\t\t\t\t\t17. [Spazi vettoriali euclidei reali - Parte 2](https://www.youtube.com/watch?v=zjmKE9TMGm8) (27:17)\n\t\t\t\t\t18. [Autovalori e autovettori](https://www.youtube.com/watch?v=XlrlcnvcTtQ) (33:00)\n\t\t\t\t\t19. [Polinomio caratteristico](https://www.youtube.com/watch?v=61icRbgWTdI) (31:31)\n\t\t\t\t\t20. [Teorema diagonalizzabilità](https://www.youtube.com/watch?v=wm5V6en9OFo) (18:49)\n\t\t\t\t\t21. [Spazi affini](https://player.vimeo.com/video/291457587) (20:46)\n\t\t\t\t\t22. [Sottospazi affini](https://player.vimeo.com/video/291458991) (21:32)\n\t\t\t\t\t23. [Parallelismo e Riferimenti Affini](https://player.vimeo.com/video/291510181) (16:57)\n\t\t\t\t\t24. [Rappresentazione di Sottospazi Affini](https://player.vimeo.com/video/291510296) (31:17)\n\t\t\t\t\t25. [Spazi Euclidei](https://player.vimeo.com/video/291510612) (35:57)\n\t\t\t\t\t26. [Teoria dei ranghi](https://player.vimeo.com/video/291510964) (9:44)\n\t\t\t\t\t27. [Teoria dei ranghi 2](https://player.vimeo.com/video/291510862) (14:44)\n\t\t\t\t\t\n\t\t\t\t\tNell'anno accademico 2018/2019 non sono stati trattati gli argomenti nei video 21, 22 e 23.\n\t\t\t\t`}\n\t\t\t\n\t\t
\n\t);\n}\n"],"sourceRoot":""} \ No newline at end of file +{"version":3,"sources":["webpack:///./routes/VlDiGeometria.js"],"names":["r","String","raw"],"mappings":"g1GAEMA,EAAIC,OAAOC,IAEF,qBACd,OACC,aACC,yCACA,EAAC,IAAD,KACC,EAAC,IAAD,KAAWF,EAAX,U","file":"route-VlDiGeometria.chunk.615ff.js","sourcesContent":["import {Markdown, Panel} from \"bluelib\";\n\nconst r = String.raw;\n\nexport default function(props) {\n\treturn (\n\t\t
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Videolezioni di Geometria

\n\t\t\t\n\t\t\t\t{r`\n\t\t\t\t\tTutte le videolezioni sono state pubblicate sotto licenza [CC BY-NC-SA 4.0](https://creativecommons.org/licenses/by-nc-sa/4.0/) dalla Prof.ssa Beatrice Ruini nell'anno accademico 2018/2019 sul [portale Dolly 2018](https://dolly.fim.unimore.it/2018/course/view.php?id=14#section-0) (Moodle).\n\t\t\t\t\t\n\t\t\t\t\tPer comodità, ho estratto l'url sorgente del video dall'embed presente nella rispettiva pagina.\n\t\t\t\t\t\n\t\t\t\t\t1. [Definizione di Spazio Vettoriale](https://www.youtube.com/watch?v=7eHEzf4403c) (1:17:29)\n\t\t\t\t\t2. [Sottospazi vettoriali I](https://www.youtube.com/watch?v=FPqrULk5HBU) (37:15)\n\t\t\t\t\t3. [Sottospazi vettoriali II](https://www.youtube.com/watch?v=ubDWUw9hk0k) (43:26)\n\t\t\t\t\t4. [Sottospazi vettoriali III](https://www.youtube.com/watch?v=381n4NPb6Oc) (40:29)\n\t\t\t\t\t5. [Lineare dipendenza e indipendenza](https://www.youtube.com/watch?v=9YVQ5olYrh0) (56:12)\n\t\t\t\t\t6. [Basi di uno spazio vettoriale I](https://www.youtube.com/watch?v=mEF_lcTzEoE) (25:52)\n\t\t\t\t\t7. [Basi di uno spazio vettoriale II](https://www.youtube.com/watch?v=k1r9JfXY53k) (48:24)\n\t\t\t\t\t8. [Teorema di Grassmann](https://www.youtube.com/watch?v=3sqB-MMyCWM) (32:36)\n\t\t\t\t\t9. [Basi e Matrici](https://www.youtube.com/watch?v=Rd6AB_jE7YI) (27:06)\n\t\t\t\t\t10. [Definizione di Applicazioni Lineari](https://www.youtube.com/watch?v=rmd7ffZeVYk) (16:23)\n\t\t\t\t\t11. [Proprietà delle Applicazioni Lineari](https://www.youtube.com/watch?v=MH7ztQGkqmw) (31:58)\n\t\t\t\t\t12. [Definizione di determinante](https://www.youtube.com/watch?v=EwubcLwBdzk) (36:43)\n\t\t\t\t\t13. [Proprietà e metodo di triangolazione](https://www.youtube.com/watch?v=SFusGarV6HI) (22:36)\n\t\t\t\t\t14. [Teorema di Laplace](https://www.youtube.com/watch?v=BqZDWnKl2nQ) (29:03)\n\t\t\t\t\t15. [4 applicazioni del Teorema di Laplace](https://www.youtube.com/watch?v=2tr3y725GY0) (47:53)\n\t\t\t\t\t16. [Spazi vettoriali euclidei reali - Parte 1](https://www.youtube.com/watch?v=W7Z1hm-jwMM) (28:46)\n\t\t\t\t\t17. [Spazi vettoriali euclidei reali - Parte 2](https://www.youtube.com/watch?v=zjmKE9TMGm8) (27:17)\n\t\t\t\t\t18. [Autovalori e autovettori](https://www.youtube.com/watch?v=XlrlcnvcTtQ) (33:00)\n\t\t\t\t\t19. [Polinomio caratteristico](https://www.youtube.com/watch?v=61icRbgWTdI) (31:31)\n\t\t\t\t\t20. [Teorema diagonalizzabilità](https://www.youtube.com/watch?v=wm5V6en9OFo) (18:49)\n\t\t\t\t\t21. [Spazi affini](https://player.vimeo.com/video/291457587) (20:46)\n\t\t\t\t\t22. [Sottospazi affini](https://player.vimeo.com/video/291458991) (21:32)\n\t\t\t\t\t23. [Parallelismo e Riferimenti Affini](https://player.vimeo.com/video/291510181) (16:57)\n\t\t\t\t\t24. [Rappresentazione di Sottospazi Affini](https://player.vimeo.com/video/291510296) (31:17)\n\t\t\t\t\t25. [Spazi Euclidei](https://player.vimeo.com/video/291510612) (35:57)\n\t\t\t\t\t26. [Teoria dei ranghi](https://player.vimeo.com/video/291510964) (9:44)\n\t\t\t\t\t27. [Teoria dei ranghi 2](https://player.vimeo.com/video/291510862) (14:44)\n\t\t\t\t\t\n\t\t\t\t\tNell'anno accademico 2018/2019 non sono stati trattati gli argomenti nei video 21, 22 e 23.\n\t\t\t\t`}\n\t\t\t\n\t\t
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r(e[0])}catch(e){return n(e)}}),n)}.bind(this))},o.getAll=function(e,t,r){return new Promise(function(n,o){return Promise.resolve(this.getAllMatching(e,{query:t,count:r})).then(n,o)}.bind(this))},o.getAllKeys=function(e,t,r){return new Promise(function(n,o){return Promise.resolve(this.getAllMatching(e,{query:t,count:r,includeKeys:!0})).then((function(e){try{return n(e.map((function(e){return e.key})))}catch(e){return o(e)}}),o)}.bind(this))},o.getAllMatching=function(e,t){return new Promise(function(r,n){var o,i,u,c,s,a,l,h,f;return i=(o=void 0===t?{}:t).index,c=void 0===(u=o.query)?null:u,a=void 0===(s=o.direction)?"next":s,l=o.count,f=void 0!==(h=o.includeKeys)&&h,Promise.resolve(this.transaction([e],"readonly",(function(t,r){var n=t.objectStore(e),o=i?n.index(i):n,u=[],s=o.openCursor(c,a);s.onsuccess=function(){var e=s.result;e?(u.push(f?e:e.value),l&&u.length>=l?r(u):e.continue()):r(u)}}))).then(r,n)}.bind(this))},o.transaction=function(e,t,r){return new Promise(function(n,o){var i;return i=this,Promise.resolve(this.open()).then((function(){try{return Promise.resolve(new Promise((function(n,o){var u=i._db.transaction(e,t);u.onabort=function(){return o(u.error)},u.oncomplete=function(){return n()},r(u,(function(e){return n(e)}))}))).then(n,o)}catch(e){return o(e)}}),o)}.bind(this))},o._call=function(e,t,r){var n=arguments;return new Promise(function(o,i){var u,c,s;for(u=n.length,c=new Array(u>3?u-3:0),s=3;s1?u-1:0),s=1;s} [options.ignoreURLParametersMatching=[/^utm_/]] An\n * array of regex's to remove search params when looking for a cache match.\n * @param {boolean} [options.cleanURLs=true] The `cleanURLs` option will\n * check the cache for the URL with a `.html` added to the end of the end.\n * @param {module:workbox-precaching~urlManipulation} [options.urlManipulation]\n * This is a function that should take a URL and return an array of\n * alternative URLs that should be checked for precache matches.\n *\n * @memberof module:workbox-precaching\n */\nfunction addRoute(options) {\n if (!listenerAdded) {\n addFetchListener(options);\n listenerAdded = true;\n }\n}\nexport { addRoute };\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { cacheNames } from 'workbox-core/_private/cacheNames.js';\nimport { getFriendlyURL } from 'workbox-core/_private/getFriendlyURL.js';\nimport { logger } from 'workbox-core/_private/logger.js';\nimport { getCacheKeyForURL } from './getCacheKeyForURL.js';\nimport '../_version.js';\n/**\n * Adds a `fetch` listener to the service worker that will\n * respond to\n * [network requests]{@link https://developer.mozilla.org/en-US/docs/Web/API/Service_Worker_API/Using_Service_Workers#Custom_responses_to_requests}\n * with precached assets.\n *\n * Requests for assets that aren't precached, the `FetchEvent` will not be\n * responded to, allowing the event to fall through to other `fetch` event\n * listeners.\n *\n * NOTE: when called more than once this method will replace the previously set\n * configuration options. Calling it more than once is not recommended outside\n * of tests.\n *\n * @private\n * @param {Object} [options]\n * @param {string} [options.directoryIndex=index.html] The `directoryIndex` will\n * check cache entries for a URLs ending with '/' to see if there is a hit when\n * appending the `directoryIndex` value.\n * @param {Array} [options.ignoreURLParametersMatching=[/^utm_/]] An\n * array of regex's to remove search params when looking for a cache match.\n * @param {boolean} [options.cleanURLs=true] The `cleanURLs` option will\n * check the cache for the URL with a `.html` added to the end of the end.\n * @param {workbox.precaching~urlManipulation} [options.urlManipulation]\n * This is a function that should take a URL and return an array of\n * alternative URLs that should be checked for precache matches.\n */\nexport const addFetchListener = ({ ignoreURLParametersMatching = [/^utm_/], directoryIndex = 'index.html', cleanURLs = true, urlManipulation, } = {}) => {\n const cacheName = cacheNames.getPrecacheName();\n // See https://github.com/Microsoft/TypeScript/issues/28357#issuecomment-436484705\n self.addEventListener('fetch', ((event) => {\n const precachedURL = getCacheKeyForURL(event.request.url, {\n cleanURLs,\n directoryIndex,\n ignoreURLParametersMatching,\n urlManipulation,\n });\n if (!precachedURL) {\n if (process.env.NODE_ENV !== 'production') {\n logger.debug(`Precaching did not find a match for ` +\n getFriendlyURL(event.request.url));\n }\n return;\n }\n let responsePromise = self.caches.open(cacheName).then((cache) => {\n return cache.match(precachedURL);\n }).then((cachedResponse) => {\n if (cachedResponse) {\n return cachedResponse;\n }\n // Fall back to the network if we don't have a cached response\n // (perhaps due to manual cache cleanup).\n if (process.env.NODE_ENV !== 'production') {\n logger.warn(`The precached response for ` +\n `${getFriendlyURL(precachedURL)} in ${cacheName} was not found. ` +\n `Falling back to the network instead.`);\n }\n return fetch(precachedURL);\n });\n if (process.env.NODE_ENV !== 'production') {\n responsePromise = responsePromise.then((response) => {\n // Workbox is going to handle the route.\n // print the routing details to the console.\n logger.groupCollapsed(`Precaching is responding to: ` +\n getFriendlyURL(event.request.url));\n logger.log(`Serving the precached url: ${precachedURL}`);\n logger.groupCollapsed(`View request details here.`);\n logger.log(event.request);\n logger.groupEnd();\n logger.groupCollapsed(`View response details here.`);\n logger.log(response);\n logger.groupEnd();\n logger.groupEnd();\n return response;\n });\n }\n event.respondWith(responsePromise);\n }));\n};\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { getOrCreatePrecacheController } from './getOrCreatePrecacheController.js';\nimport { generateURLVariations } from './generateURLVariations.js';\nimport '../_version.js';\n/**\n * This function will take the request URL and manipulate it based on the\n * configuration options.\n *\n * @param {string} url\n * @param {Object} options\n * @return {string} Returns the URL in the cache that matches the request,\n * if possible.\n *\n * @private\n */\nexport const getCacheKeyForURL = (url, options) => {\n const precacheController = getOrCreatePrecacheController();\n const urlsToCacheKeys = precacheController.getURLsToCacheKeys();\n for (const possibleURL of generateURLVariations(url, options)) {\n const possibleCacheKey = urlsToCacheKeys.get(possibleURL);\n if (possibleCacheKey) {\n return possibleCacheKey;\n }\n }\n};\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { removeIgnoredSearchParams } from './removeIgnoredSearchParams.js';\nimport '../_version.js';\n/**\n * Generator function that yields possible variations on the original URL to\n * check, one at a time.\n *\n * @param {string} url\n * @param {Object} options\n *\n * @private\n * @memberof module:workbox-precaching\n */\nexport function* generateURLVariations(url, { ignoreURLParametersMatching, directoryIndex, cleanURLs, urlManipulation, } = {}) {\n const urlObject = new URL(url, location.href);\n urlObject.hash = '';\n yield urlObject.href;\n const urlWithoutIgnoredParams = removeIgnoredSearchParams(urlObject, ignoreURLParametersMatching);\n yield urlWithoutIgnoredParams.href;\n if (directoryIndex && urlWithoutIgnoredParams.pathname.endsWith('/')) {\n const directoryURL = new URL(urlWithoutIgnoredParams.href);\n directoryURL.pathname += directoryIndex;\n yield directoryURL.href;\n }\n if (cleanURLs) {\n const cleanURL = new URL(urlWithoutIgnoredParams.href);\n cleanURL.pathname += '.html';\n yield cleanURL.href;\n }\n if (urlManipulation) {\n const additionalURLs = urlManipulation({ url: urlObject });\n for (const urlToAttempt of additionalURLs) {\n yield urlToAttempt.href;\n }\n }\n}\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\n/**\n * Removes any URL search parameters that should be ignored.\n *\n * @param {URL} urlObject The original URL.\n * @param {Array} ignoreURLParametersMatching RegExps to test against\n * each search parameter name. Matches mean that the search parameter should be\n * ignored.\n * @return {URL} The URL with any ignored search parameters removed.\n *\n * @private\n * @memberof module:workbox-precaching\n */\nexport function removeIgnoredSearchParams(urlObject, ignoreURLParametersMatching = []) {\n // Convert the iterable into an array at the start of the loop to make sure\n // deletion doesn't mess up iteration.\n for (const paramName of [...urlObject.searchParams.keys()]) {\n if (ignoreURLParametersMatching.some((regExp) => regExp.test(paramName))) {\n urlObject.searchParams.delete(paramName);\n }\n }\n return urlObject;\n}\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { addRoute } from './addRoute.js';\nimport { precache } from './precache.js';\nimport './_version.js';\n/**\n * This method will add entries to the precache list and add a route to\n * respond to fetch events.\n *\n * This is a convenience method that will call\n * [precache()]{@link module:workbox-precaching.precache} and\n * [addRoute()]{@link module:workbox-precaching.addRoute} in a single call.\n *\n * @param {Array} entries Array of entries to precache.\n * @param {Object} [options] See\n * [addRoute() options]{@link module:workbox-precaching.addRoute}.\n *\n * @memberof module:workbox-precaching\n */\nfunction precacheAndRoute(entries, options) {\n precache(entries);\n addRoute(options);\n}\nexport { precacheAndRoute };\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { logger } from 'workbox-core/_private/logger.js';\nimport { getOrCreatePrecacheController } from './utils/getOrCreatePrecacheController.js';\nimport { precachePlugins } from './utils/precachePlugins.js';\nimport './_version.js';\nconst installListener = (event) => {\n const precacheController = getOrCreatePrecacheController();\n const plugins = precachePlugins.get();\n event.waitUntil(precacheController.install({ event, plugins })\n .catch((error) => {\n if (process.env.NODE_ENV !== 'production') {\n logger.error(`Service worker installation failed. It will ` +\n `be retried automatically during the next navigation.`);\n }\n // Re-throw the error to ensure installation fails.\n throw error;\n }));\n};\nconst activateListener = (event) => {\n const precacheController = getOrCreatePrecacheController();\n event.waitUntil(precacheController.activate());\n};\n/**\n * Adds items to the precache list, removing any duplicates and\n * stores the files in the\n * [\"precache cache\"]{@link module:workbox-core.cacheNames} when the service\n * worker installs.\n *\n * This method can be called multiple times.\n *\n * Please note: This method **will not** serve any of the cached files for you.\n * It only precaches files. To respond to a network request you call\n * [addRoute()]{@link module:workbox-precaching.addRoute}.\n *\n * If you have a single array of files to precache, you can just call\n * [precacheAndRoute()]{@link module:workbox-precaching.precacheAndRoute}.\n *\n * @param {Array} [entries=[]] Array of entries to precache.\n *\n * @memberof module:workbox-precaching\n */\nfunction precache(entries) {\n const precacheController = getOrCreatePrecacheController();\n precacheController.addToCacheList(entries);\n if (entries.length > 0) {\n // NOTE: these listeners will only be added once (even if the `precache()`\n // method is called multiple times) because event listeners are implemented\n // as a set, where each listener must be unique.\n // See https://github.com/Microsoft/TypeScript/issues/28357#issuecomment-436484705\n self.addEventListener('install', installListener);\n self.addEventListener('activate', activateListener);\n }\n}\nexport { precache };\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { Router } from '../Router.js';\nimport '../_version.js';\nlet defaultRouter;\n/**\n * Creates a new, singleton Router instance if one does not exist. If one\n * does already exist, that instance is returned.\n *\n * @private\n * @return {Router}\n */\nexport const getOrCreateDefaultRouter = () => {\n if (!defaultRouter) {\n defaultRouter = new Router();\n // The helpers that use the default Router assume these listeners exist.\n defaultRouter.addFetchListener();\n defaultRouter.addCacheListener();\n }\n return defaultRouter;\n};\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { PrecacheController } from '../PrecacheController.js';\nimport '../_version.js';\nlet precacheController;\n/**\n * @return {PrecacheController}\n * @private\n */\nexport const getOrCreatePrecacheController = () => {\n if (!precacheController) {\n precacheController = new PrecacheController();\n }\n return precacheController;\n};\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { messages } from './messages.js';\nimport '../../_version.js';\nconst fallback = (code, ...args) => {\n let msg = code;\n if (args.length > 0) {\n msg += ` :: ${JSON.stringify(args)}`;\n }\n return msg;\n};\nconst generatorFunction = (code, details = {}) => {\n const message = messages[code];\n if (!message) {\n throw new Error(`Unable to find message for code '${code}'.`);\n }\n return message(details);\n};\nexport const messageGenerator = (process.env.NODE_ENV === 'production') ?\n fallback : generatorFunction;\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { messageGenerator } from '../models/messages/messageGenerator.js';\nimport '../_version.js';\n/**\n * Workbox errors should be thrown with this class.\n * This allows use to ensure the type easily in tests,\n * helps developers identify errors from workbox\n * easily and allows use to optimise error\n * messages correctly.\n *\n * @private\n */\nclass WorkboxError extends Error {\n /**\n *\n * @param {string} errorCode The error code that\n * identifies this particular error.\n * @param {Object=} details Any relevant arguments\n * that will help developers identify issues should\n * be added as a key on the context object.\n */\n constructor(errorCode, details) {\n const message = messageGenerator(errorCode, details);\n super(message);\n this.name = errorCode;\n this.details = details;\n }\n}\nexport { WorkboxError };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { assert } from 'workbox-core/_private/assert.js';\nimport '../_version.js';\n/**\n * @param {function()|Object} handler Either a function, or an object with a\n * 'handle' method.\n * @return {Object} An object with a handle method.\n *\n * @private\n */\nexport const normalizeHandler = (handler) => {\n if (handler && typeof handler === 'object') {\n if (process.env.NODE_ENV !== 'production') {\n assert.hasMethod(handler, 'handle', {\n moduleName: 'workbox-routing',\n className: 'Route',\n funcName: 'constructor',\n paramName: 'handler',\n });\n }\n return handler;\n }\n else {\n if (process.env.NODE_ENV !== 'production') {\n assert.isType(handler, 'function', {\n moduleName: 'workbox-routing',\n className: 'Route',\n funcName: 'constructor',\n paramName: 'handler',\n });\n }\n return { handle: handler };\n }\n};\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { assert } from 'workbox-core/_private/assert.js';\nimport { defaultMethod, validMethods } from './utils/constants.js';\nimport { normalizeHandler } from './utils/normalizeHandler.js';\nimport './_version.js';\n/**\n * A `Route` consists of a pair of callback functions, \"match\" and \"handler\".\n * The \"match\" callback determine if a route should be used to \"handle\" a\n * request by returning a non-falsy value if it can. The \"handler\" callback\n * is called when there is a match and should return a Promise that resolves\n * to a `Response`.\n *\n * @memberof module:workbox-routing\n */\nclass Route {\n /**\n * Constructor for Route class.\n *\n * @param {module:workbox-routing~matchCallback} match\n * A callback function that determines whether the route matches a given\n * `fetch` event by returning a non-falsy value.\n * @param {module:workbox-routing~handlerCallback} handler A callback\n * function that returns a Promise resolving to a Response.\n * @param {string} [method='GET'] The HTTP method to match the Route\n * against.\n */\n constructor(match, handler, method = defaultMethod) {\n if (process.env.NODE_ENV !== 'production') {\n assert.isType(match, 'function', {\n moduleName: 'workbox-routing',\n className: 'Route',\n funcName: 'constructor',\n paramName: 'match',\n });\n if (method) {\n assert.isOneOf(method, validMethods, { paramName: 'method' });\n }\n }\n // These values are referenced directly by Router so cannot be\n // altered by minificaton.\n this.handler = normalizeHandler(handler);\n this.match = match;\n this.method = method;\n }\n}\nexport { Route };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\n/**\n * The default HTTP method, 'GET', used when there's no specific method\n * configured for a route.\n *\n * @type {string}\n *\n * @private\n */\nexport const defaultMethod = 'GET';\n/**\n * The list of valid HTTP methods associated with requests that could be routed.\n *\n * @type {Array}\n *\n * @private\n */\nexport const validMethods = [\n 'DELETE',\n 'GET',\n 'HEAD',\n 'PATCH',\n 'POST',\n 'PUT',\n];\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { assert } from 'workbox-core/_private/assert.js';\nimport { logger } from 'workbox-core/_private/logger.js';\nimport { Route } from './Route.js';\nimport './_version.js';\n/**\n * RegExpRoute makes it easy to create a regular expression based\n * [Route]{@link module:workbox-routing.Route}.\n *\n * For same-origin requests the RegExp only needs to match part of the URL. For\n * requests against third-party servers, you must define a RegExp that matches\n * the start of the URL.\n *\n * [See the module docs for info.]{@link https://developers.google.com/web/tools/workbox/modules/workbox-routing}\n *\n * @memberof module:workbox-routing\n * @extends module:workbox-routing.Route\n */\nclass RegExpRoute extends Route {\n /**\n * If the regular expression contains\n * [capture groups]{@link https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/RegExp#grouping-back-references},\n * the captured values will be passed to the\n * [handler's]{@link module:workbox-routing~handlerCallback} `params`\n * argument.\n *\n * @param {RegExp} regExp The regular expression to match against URLs.\n * @param {module:workbox-routing~handlerCallback} handler A callback\n * function that returns a Promise resulting in a Response.\n * @param {string} [method='GET'] The HTTP method to match the Route\n * against.\n */\n constructor(regExp, handler, method) {\n if (process.env.NODE_ENV !== 'production') {\n assert.isInstance(regExp, RegExp, {\n moduleName: 'workbox-routing',\n className: 'RegExpRoute',\n funcName: 'constructor',\n paramName: 'pattern',\n });\n }\n const match = ({ url }) => {\n const result = regExp.exec(url.href);\n // Return immediately if there's no match.\n if (!result) {\n return;\n }\n // Require that the match start at the first character in the URL string\n // if it's a cross-origin request.\n // See https://github.com/GoogleChrome/workbox/issues/281 for the context\n // behind this behavior.\n if ((url.origin !== location.origin) && (result.index !== 0)) {\n if (process.env.NODE_ENV !== 'production') {\n logger.debug(`The regular expression '${regExp}' only partially matched ` +\n `against the cross-origin URL '${url}'. RegExpRoute's will only ` +\n `handle cross-origin requests if they match the entire URL.`);\n }\n return;\n }\n // If the route matches, but there aren't any capture groups defined, then\n // this will return [], which is truthy and therefore sufficient to\n // indicate a match.\n // If there are capture groups, then it will return their values.\n return result.slice(1);\n };\n super(match, handler, method);\n }\n}\nexport { RegExpRoute };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\nconst getFriendlyURL = (url) => {\n const urlObj = new URL(String(url), location.href);\n // See https://github.com/GoogleChrome/workbox/issues/2323\n // We want to include everything, except for the origin if it's same-origin.\n return urlObj.href.replace(new RegExp(`^${location.origin}`), '');\n};\nexport { getFriendlyURL };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { assert } from 'workbox-core/_private/assert.js';\nimport { logger } from 'workbox-core/_private/logger.js';\nimport { WorkboxError } from 'workbox-core/_private/WorkboxError.js';\nimport { getFriendlyURL } from 'workbox-core/_private/getFriendlyURL.js';\nimport { normalizeHandler } from './utils/normalizeHandler.js';\nimport './_version.js';\n/**\n * The Router can be used to process a FetchEvent through one or more\n * [Routes]{@link module:workbox-routing.Route} responding with a Request if\n * a matching route exists.\n *\n * If no route matches a given a request, the Router will use a \"default\"\n * handler if one is defined.\n *\n * Should the matching Route throw an error, the Router will use a \"catch\"\n * handler if one is defined to gracefully deal with issues and respond with a\n * Request.\n *\n * If a request matches multiple routes, the **earliest** registered route will\n * be used to respond to the request.\n *\n * @memberof module:workbox-routing\n */\nclass Router {\n /**\n * Initializes a new Router.\n */\n constructor() {\n this._routes = new Map();\n }\n /**\n * @return {Map>} routes A `Map` of HTTP\n * method name ('GET', etc.) to an array of all the corresponding `Route`\n * instances that are registered.\n */\n get routes() {\n return this._routes;\n }\n /**\n * Adds a fetch event listener to respond to events when a route matches\n * the event's request.\n */\n addFetchListener() {\n // See https://github.com/Microsoft/TypeScript/issues/28357#issuecomment-436484705\n self.addEventListener('fetch', ((event) => {\n const { request } = event;\n const responsePromise = this.handleRequest({ request, event });\n if (responsePromise) {\n event.respondWith(responsePromise);\n }\n }));\n }\n /**\n * Adds a message event listener for URLs to cache from the window.\n * This is useful to cache resources loaded on the page prior to when the\n * service worker started controlling it.\n *\n * The format of the message data sent from the window should be as follows.\n * Where the `urlsToCache` array may consist of URL strings or an array of\n * URL string + `requestInit` object (the same as you'd pass to `fetch()`).\n *\n * ```\n * {\n * type: 'CACHE_URLS',\n * payload: {\n * urlsToCache: [\n * './script1.js',\n * './script2.js',\n * ['./script3.js', {mode: 'no-cors'}],\n * ],\n * },\n * }\n * ```\n */\n addCacheListener() {\n // See https://github.com/Microsoft/TypeScript/issues/28357#issuecomment-436484705\n self.addEventListener('message', ((event) => {\n if (event.data && event.data.type === 'CACHE_URLS') {\n const { payload } = event.data;\n if (process.env.NODE_ENV !== 'production') {\n logger.debug(`Caching URLs from the window`, payload.urlsToCache);\n }\n const requestPromises = Promise.all(payload.urlsToCache.map((entry) => {\n if (typeof entry === 'string') {\n entry = [entry];\n }\n const request = new Request(...entry);\n return this.handleRequest({ request });\n // TODO(philipwalton): TypeScript errors without this typecast for\n // some reason (probably a bug). The real type here should work but\n // doesn't: `Array | undefined>`.\n })); // TypeScript\n event.waitUntil(requestPromises);\n // If a MessageChannel was used, reply to the message on success.\n if (event.ports && event.ports[0]) {\n requestPromises.then(() => event.ports[0].postMessage(true));\n }\n }\n }));\n }\n /**\n * Apply the routing rules to a FetchEvent object to get a Response from an\n * appropriate Route's handler.\n *\n * @param {Object} options\n * @param {Request} options.request The request to handle (this is usually\n * from a fetch event, but it does not have to be).\n * @param {FetchEvent} [options.event] The event that triggered the request,\n * if applicable.\n * @return {Promise|undefined} A promise is returned if a\n * registered route can handle the request. If there is no matching\n * route and there's no `defaultHandler`, `undefined` is returned.\n */\n handleRequest({ request, event }) {\n if (process.env.NODE_ENV !== 'production') {\n assert.isInstance(request, Request, {\n moduleName: 'workbox-routing',\n className: 'Router',\n funcName: 'handleRequest',\n paramName: 'options.request',\n });\n }\n const url = new URL(request.url, location.href);\n if (!url.protocol.startsWith('http')) {\n if (process.env.NODE_ENV !== 'production') {\n logger.debug(`Workbox Router only supports URLs that start with 'http'.`);\n }\n return;\n }\n const { params, route } = this.findMatchingRoute({ url, request, event });\n let handler = route && route.handler;\n const debugMessages = [];\n if (process.env.NODE_ENV !== 'production') {\n if (handler) {\n debugMessages.push([\n `Found a route to handle this request:`, route,\n ]);\n if (params) {\n debugMessages.push([\n `Passing the following params to the route's handler:`, params,\n ]);\n }\n }\n }\n // If we don't have a handler because there was no matching route, then\n // fall back to defaultHandler if that's defined.\n if (!handler && this._defaultHandler) {\n if (process.env.NODE_ENV !== 'production') {\n debugMessages.push(`Failed to find a matching route. Falling ` +\n `back to the default handler.`);\n }\n handler = this._defaultHandler;\n }\n if (!handler) {\n if (process.env.NODE_ENV !== 'production') {\n // No handler so Workbox will do nothing. If logs is set of debug\n // i.e. verbose, we should print out this information.\n logger.debug(`No route found for: ${getFriendlyURL(url)}`);\n }\n return;\n }\n if (process.env.NODE_ENV !== 'production') {\n // We have a handler, meaning Workbox is going to handle the route.\n // print the routing details to the console.\n logger.groupCollapsed(`Router is responding to: ${getFriendlyURL(url)}`);\n debugMessages.forEach((msg) => {\n if (Array.isArray(msg)) {\n logger.log(...msg);\n }\n else {\n logger.log(msg);\n }\n });\n logger.groupEnd();\n }\n // Wrap in try and catch in case the handle method throws a synchronous\n // error. It should still callback to the catch handler.\n let responsePromise;\n try {\n responsePromise = handler.handle({ url, request, event, params });\n }\n catch (err) {\n responsePromise = Promise.reject(err);\n }\n if (responsePromise instanceof Promise && this._catchHandler) {\n responsePromise = responsePromise.catch((err) => {\n if (process.env.NODE_ENV !== 'production') {\n // Still include URL here as it will be async from the console group\n // and may not make sense without the URL\n logger.groupCollapsed(`Error thrown when responding to: ` +\n ` ${getFriendlyURL(url)}. Falling back to Catch Handler.`);\n logger.error(`Error thrown by:`, route);\n logger.error(err);\n logger.groupEnd();\n }\n return this._catchHandler.handle({ url, request, event });\n });\n }\n return responsePromise;\n }\n /**\n * Checks a request and URL (and optionally an event) against the list of\n * registered routes, and if there's a match, returns the corresponding\n * route along with any params generated by the match.\n *\n * @param {Object} options\n * @param {URL} options.url\n * @param {Request} options.request The request to match.\n * @param {Event} [options.event] The corresponding event (unless N/A).\n * @return {Object} An object with `route` and `params` properties.\n * They are populated if a matching route was found or `undefined`\n * otherwise.\n */\n findMatchingRoute({ url, request, event }) {\n if (process.env.NODE_ENV !== 'production') {\n assert.isInstance(url, URL, {\n moduleName: 'workbox-routing',\n className: 'Router',\n funcName: 'findMatchingRoute',\n paramName: 'options.url',\n });\n assert.isInstance(request, Request, {\n moduleName: 'workbox-routing',\n className: 'Router',\n funcName: 'findMatchingRoute',\n paramName: 'options.request',\n });\n }\n const routes = this._routes.get(request.method) || [];\n for (const route of routes) {\n let params;\n const matchResult = route.match({ url, request, event });\n if (matchResult) {\n // See https://github.com/GoogleChrome/workbox/issues/2079\n params = matchResult;\n if (Array.isArray(matchResult) && matchResult.length === 0) {\n // Instead of passing an empty array in as params, use undefined.\n params = undefined;\n }\n else if ((matchResult.constructor === Object &&\n Object.keys(matchResult).length === 0)) {\n // Instead of passing an empty object in as params, use undefined.\n params = undefined;\n }\n else if (typeof matchResult === 'boolean') {\n // For the boolean value true (rather than just something truth-y),\n // don't set params.\n // See https://github.com/GoogleChrome/workbox/pull/2134#issuecomment-513924353\n params = undefined;\n }\n // Return early if have a match.\n return { route, params };\n }\n }\n // If no match was found above, return and empty object.\n return {};\n }\n /**\n * Define a default `handler` that's called when no routes explicitly\n * match the incoming request.\n *\n * Without a default handler, unmatched requests will go against the\n * network as if there were no service worker present.\n *\n * @param {module:workbox-routing~handlerCallback} handler A callback\n * function that returns a Promise resulting in a Response.\n */\n setDefaultHandler(handler) {\n this._defaultHandler = normalizeHandler(handler);\n }\n /**\n * If a Route throws an error while handling a request, this `handler`\n * will be called and given a chance to provide a response.\n *\n * @param {module:workbox-routing~handlerCallback} handler A callback\n * function that returns a Promise resulting in a Response.\n */\n setCatchHandler(handler) {\n this._catchHandler = normalizeHandler(handler);\n }\n /**\n * Registers a route with the router.\n *\n * @param {module:workbox-routing.Route} route The route to register.\n */\n registerRoute(route) {\n if (process.env.NODE_ENV !== 'production') {\n assert.isType(route, 'object', {\n moduleName: 'workbox-routing',\n className: 'Router',\n funcName: 'registerRoute',\n paramName: 'route',\n });\n assert.hasMethod(route, 'match', {\n moduleName: 'workbox-routing',\n className: 'Router',\n funcName: 'registerRoute',\n paramName: 'route',\n });\n assert.isType(route.handler, 'object', {\n moduleName: 'workbox-routing',\n className: 'Router',\n funcName: 'registerRoute',\n paramName: 'route',\n });\n assert.hasMethod(route.handler, 'handle', {\n moduleName: 'workbox-routing',\n className: 'Router',\n funcName: 'registerRoute',\n paramName: 'route.handler',\n });\n assert.isType(route.method, 'string', {\n moduleName: 'workbox-routing',\n className: 'Router',\n funcName: 'registerRoute',\n paramName: 'route.method',\n });\n }\n if (!this._routes.has(route.method)) {\n this._routes.set(route.method, []);\n }\n // Give precedence to all of the earlier routes by adding this additional\n // route to the end of the array.\n this._routes.get(route.method).push(route);\n }\n /**\n * Unregisters a route with the router.\n *\n * @param {module:workbox-routing.Route} route The route to unregister.\n */\n unregisterRoute(route) {\n if (!this._routes.has(route.method)) {\n throw new WorkboxError('unregister-route-but-not-found-with-method', {\n method: route.method,\n });\n }\n const routeIndex = this._routes.get(route.method).indexOf(route);\n if (routeIndex > -1) {\n this._routes.get(route.method).splice(routeIndex, 1);\n }\n else {\n throw new WorkboxError('unregister-route-route-not-registered');\n }\n }\n}\nexport { Router };\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\nconst plugins = [];\nexport const precachePlugins = {\n /*\n * @return {Array}\n * @private\n */\n get() {\n return plugins;\n },\n /*\n * @param {Array} newPlugins\n * @private\n */\n add(newPlugins) {\n plugins.push(...newPlugins);\n },\n};\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\nconst _cacheNameDetails = {\n googleAnalytics: 'googleAnalytics',\n precache: 'precache-v2',\n prefix: 'workbox',\n runtime: 'runtime',\n suffix: typeof registration !== 'undefined' ? registration.scope : '',\n};\nconst _createCacheName = (cacheName) => {\n return [_cacheNameDetails.prefix, cacheName, _cacheNameDetails.suffix]\n .filter((value) => value && value.length > 0)\n .join('-');\n};\nconst eachCacheNameDetail = (fn) => {\n for (const key of Object.keys(_cacheNameDetails)) {\n fn(key);\n }\n};\nexport const cacheNames = {\n updateDetails: (details) => {\n eachCacheNameDetail((key) => {\n if (typeof details[key] === 'string') {\n _cacheNameDetails[key] = details[key];\n }\n });\n },\n getGoogleAnalyticsName: (userCacheName) => {\n return userCacheName || _createCacheName(_cacheNameDetails.googleAnalytics);\n },\n getPrecacheName: (userCacheName) => {\n return userCacheName || _createCacheName(_cacheNameDetails.precache);\n },\n getPrefix: () => {\n return _cacheNameDetails.prefix;\n },\n getRuntimeName: (userCacheName) => {\n return userCacheName || _createCacheName(_cacheNameDetails.runtime);\n },\n getSuffix: () => {\n return _cacheNameDetails.suffix;\n },\n};\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\n// Callbacks to be executed whenever there's a quota error.\nconst quotaErrorCallbacks = new Set();\nexport { quotaErrorCallbacks };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\nexport const pluginUtils = {\n filter: (plugins, callbackName) => {\n return plugins.filter((plugin) => callbackName in plugin);\n },\n};\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { assert } from './assert.js';\nimport { executeQuotaErrorCallbacks } from './executeQuotaErrorCallbacks.js';\nimport { getFriendlyURL } from './getFriendlyURL.js';\nimport { logger } from './logger.js';\nimport { pluginUtils } from '../utils/pluginUtils.js';\nimport { WorkboxError } from './WorkboxError.js';\nimport '../_version.js';\n/**\n * Checks the list of plugins for the cacheKeyWillBeUsed callback, and\n * executes any of those callbacks found in sequence. The final `Request` object\n * returned by the last plugin is treated as the cache key for cache reads\n * and/or writes.\n *\n * @param {Object} options\n * @param {Request} options.request\n * @param {string} options.mode\n * @param {Array} [options.plugins=[]]\n * @return {Promise}\n *\n * @private\n * @memberof module:workbox-core\n */\nconst _getEffectiveRequest = async ({ request, mode, plugins = [], }) => {\n const cacheKeyWillBeUsedPlugins = pluginUtils.filter(plugins, \"cacheKeyWillBeUsed\" /* CACHE_KEY_WILL_BE_USED */);\n let effectiveRequest = request;\n for (const plugin of cacheKeyWillBeUsedPlugins) {\n effectiveRequest = await plugin[\"cacheKeyWillBeUsed\" /* CACHE_KEY_WILL_BE_USED */].call(plugin, { mode, request: effectiveRequest });\n if (typeof effectiveRequest === 'string') {\n effectiveRequest = new Request(effectiveRequest);\n }\n if (process.env.NODE_ENV !== 'production') {\n assert.isInstance(effectiveRequest, Request, {\n moduleName: 'Plugin',\n funcName: \"cacheKeyWillBeUsed\" /* CACHE_KEY_WILL_BE_USED */,\n isReturnValueProblem: true,\n });\n }\n }\n return effectiveRequest;\n};\n/**\n * This method will call cacheWillUpdate on the available plugins (or use\n * status === 200) to determine if the Response is safe and valid to cache.\n *\n * @param {Object} options\n * @param {Request} options.request\n * @param {Response} options.response\n * @param {Event} [options.event]\n * @param {Array} [options.plugins=[]]\n * @return {Promise}\n *\n * @private\n * @memberof module:workbox-core\n */\nconst _isResponseSafeToCache = async ({ request, response, event, plugins = [], }) => {\n let responseToCache = response;\n let pluginsUsed = false;\n for (const plugin of plugins) {\n if (\"cacheWillUpdate\" /* CACHE_WILL_UPDATE */ in plugin) {\n pluginsUsed = true;\n const pluginMethod = plugin[\"cacheWillUpdate\" /* CACHE_WILL_UPDATE */];\n responseToCache = await pluginMethod.call(plugin, {\n request,\n response: responseToCache,\n event,\n });\n if (process.env.NODE_ENV !== 'production') {\n if (responseToCache) {\n assert.isInstance(responseToCache, Response, {\n moduleName: 'Plugin',\n funcName: \"cacheWillUpdate\" /* CACHE_WILL_UPDATE */,\n isReturnValueProblem: true,\n });\n }\n }\n if (!responseToCache) {\n break;\n }\n }\n }\n if (!pluginsUsed) {\n if (process.env.NODE_ENV !== 'production') {\n if (responseToCache) {\n if (responseToCache.status !== 200) {\n if (responseToCache.status === 0) {\n logger.warn(`The response for '${request.url}' is an opaque ` +\n `response. The caching strategy that you're using will not ` +\n `cache opaque responses by default.`);\n }\n else {\n logger.debug(`The response for '${request.url}' returned ` +\n `a status code of '${response.status}' and won't be cached as a ` +\n `result.`);\n }\n }\n }\n }\n responseToCache = responseToCache && responseToCache.status === 200 ?\n responseToCache : undefined;\n }\n return responseToCache ? responseToCache : null;\n};\n/**\n * This is a wrapper around cache.match().\n *\n * @param {Object} options\n * @param {string} options.cacheName Name of the cache to match against.\n * @param {Request} options.request The Request that will be used to look up\n * cache entries.\n * @param {Event} [options.event] The event that prompted the action.\n * @param {Object} [options.matchOptions] Options passed to cache.match().\n * @param {Array} [options.plugins=[]] Array of plugins.\n * @return {Response} A cached response if available.\n *\n * @private\n * @memberof module:workbox-core\n */\nconst matchWrapper = async ({ cacheName, request, event, matchOptions, plugins = [], }) => {\n const cache = await self.caches.open(cacheName);\n const effectiveRequest = await _getEffectiveRequest({\n plugins, request, mode: 'read'\n });\n let cachedResponse = await cache.match(effectiveRequest, matchOptions);\n if (process.env.NODE_ENV !== 'production') {\n if (cachedResponse) {\n logger.debug(`Found a cached response in '${cacheName}'.`);\n }\n else {\n logger.debug(`No cached response found in '${cacheName}'.`);\n }\n }\n for (const plugin of plugins) {\n if (\"cachedResponseWillBeUsed\" /* CACHED_RESPONSE_WILL_BE_USED */ in plugin) {\n const pluginMethod = plugin[\"cachedResponseWillBeUsed\" /* CACHED_RESPONSE_WILL_BE_USED */];\n cachedResponse = await pluginMethod.call(plugin, {\n cacheName,\n event,\n matchOptions,\n cachedResponse,\n request: effectiveRequest,\n });\n if (process.env.NODE_ENV !== 'production') {\n if (cachedResponse) {\n assert.isInstance(cachedResponse, Response, {\n moduleName: 'Plugin',\n funcName: \"cachedResponseWillBeUsed\" /* CACHED_RESPONSE_WILL_BE_USED */,\n isReturnValueProblem: true,\n });\n }\n }\n }\n }\n return cachedResponse;\n};\n/**\n * Wrapper around cache.put().\n *\n * Will call `cacheDidUpdate` on plugins if the cache was updated, using\n * `matchOptions` when determining what the old entry is.\n *\n * @param {Object} options\n * @param {string} options.cacheName\n * @param {Request} options.request\n * @param {Response} options.response\n * @param {Event} [options.event]\n * @param {Array} [options.plugins=[]]\n * @param {Object} [options.matchOptions]\n *\n * @private\n * @memberof module:workbox-core\n */\nconst putWrapper = async ({ cacheName, request, response, event, plugins = [], matchOptions, }) => {\n if (process.env.NODE_ENV !== 'production') {\n if (request.method && request.method !== 'GET') {\n throw new WorkboxError('attempt-to-cache-non-get-request', {\n url: getFriendlyURL(request.url),\n method: request.method,\n });\n }\n }\n const effectiveRequest = await _getEffectiveRequest({\n plugins, request, mode: 'write'\n });\n if (!response) {\n if (process.env.NODE_ENV !== 'production') {\n logger.error(`Cannot cache non-existent response for ` +\n `'${getFriendlyURL(effectiveRequest.url)}'.`);\n }\n throw new WorkboxError('cache-put-with-no-response', {\n url: getFriendlyURL(effectiveRequest.url),\n });\n }\n const responseToCache = await _isResponseSafeToCache({\n event,\n plugins,\n response,\n request: effectiveRequest,\n });\n if (!responseToCache) {\n if (process.env.NODE_ENV !== 'production') {\n logger.debug(`Response '${getFriendlyURL(effectiveRequest.url)}' will ` +\n `not be cached.`, responseToCache);\n }\n return;\n }\n const cache = await self.caches.open(cacheName);\n const updatePlugins = pluginUtils.filter(plugins, \"cacheDidUpdate\" /* CACHE_DID_UPDATE */);\n const oldResponse = updatePlugins.length > 0 ?\n await matchWrapper({ cacheName, matchOptions, request: effectiveRequest }) :\n null;\n if (process.env.NODE_ENV !== 'production') {\n logger.debug(`Updating the '${cacheName}' cache with a new Response for ` +\n `${getFriendlyURL(effectiveRequest.url)}.`);\n }\n try {\n await cache.put(effectiveRequest, responseToCache);\n }\n catch (error) {\n // See https://developer.mozilla.org/en-US/docs/Web/API/DOMException#exception-QuotaExceededError\n if (error.name === 'QuotaExceededError') {\n await executeQuotaErrorCallbacks();\n }\n throw error;\n }\n for (const plugin of updatePlugins) {\n await plugin[\"cacheDidUpdate\" /* CACHE_DID_UPDATE */].call(plugin, {\n cacheName,\n event,\n oldResponse,\n newResponse: responseToCache,\n request: effectiveRequest,\n });\n }\n};\nexport const cacheWrapper = {\n put: putWrapper,\n match: matchWrapper,\n};\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { logger } from '../_private/logger.js';\nimport { quotaErrorCallbacks } from '../models/quotaErrorCallbacks.js';\nimport '../_version.js';\n/**\n * Runs all of the callback functions, one at a time sequentially, in the order\n * in which they were registered.\n *\n * @memberof module:workbox-core\n * @private\n */\nasync function executeQuotaErrorCallbacks() {\n if (process.env.NODE_ENV !== 'production') {\n logger.log(`About to run ${quotaErrorCallbacks.size} ` +\n `callbacks to clean up caches.`);\n }\n for (const callback of quotaErrorCallbacks) {\n await callback();\n if (process.env.NODE_ENV !== 'production') {\n logger.log(callback, 'is complete.');\n }\n }\n if (process.env.NODE_ENV !== 'production') {\n logger.log('Finished running callbacks.');\n }\n}\nexport { executeQuotaErrorCallbacks };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { WorkboxError } from './WorkboxError.js';\nimport { logger } from './logger.js';\nimport { assert } from './assert.js';\nimport { getFriendlyURL } from '../_private/getFriendlyURL.js';\nimport { pluginUtils } from '../utils/pluginUtils.js';\nimport '../_version.js';\n/**\n * Wrapper around the fetch API.\n *\n * Will call requestWillFetch on available plugins.\n *\n * @param {Object} options\n * @param {Request|string} options.request\n * @param {Object} [options.fetchOptions]\n * @param {ExtendableEvent} [options.event]\n * @param {Array} [options.plugins=[]]\n * @return {Promise}\n *\n * @private\n * @memberof module:workbox-core\n */\nconst wrappedFetch = async ({ request, fetchOptions, event, plugins = [], }) => {\n if (typeof request === 'string') {\n request = new Request(request);\n }\n // We *should* be able to call `await event.preloadResponse` even if it's\n // undefined, but for some reason, doing so leads to errors in our Node unit\n // tests. To work around that, explicitly check preloadResponse's value first.\n if (event instanceof FetchEvent && event.preloadResponse) {\n const possiblePreloadResponse = await event.preloadResponse;\n if (possiblePreloadResponse) {\n if (process.env.NODE_ENV !== 'production') {\n logger.log(`Using a preloaded navigation response for ` +\n `'${getFriendlyURL(request.url)}'`);\n }\n return possiblePreloadResponse;\n }\n }\n if (process.env.NODE_ENV !== 'production') {\n assert.isInstance(request, Request, {\n paramName: 'request',\n expectedClass: Request,\n moduleName: 'workbox-core',\n className: 'fetchWrapper',\n funcName: 'wrappedFetch',\n });\n }\n const failedFetchPlugins = pluginUtils.filter(plugins, \"fetchDidFail\" /* FETCH_DID_FAIL */);\n // If there is a fetchDidFail plugin, we need to save a clone of the\n // original request before it's either modified by a requestWillFetch\n // plugin or before the original request's body is consumed via fetch().\n const originalRequest = failedFetchPlugins.length > 0 ?\n request.clone() : null;\n try {\n for (const plugin of plugins) {\n if (\"requestWillFetch\" /* REQUEST_WILL_FETCH */ in plugin) {\n const pluginMethod = plugin[\"requestWillFetch\" /* REQUEST_WILL_FETCH */];\n const requestClone = request.clone();\n request = await pluginMethod.call(plugin, {\n request: requestClone,\n event,\n });\n if (process.env.NODE_ENV !== 'production') {\n if (request) {\n assert.isInstance(request, Request, {\n moduleName: 'Plugin',\n funcName: \"cachedResponseWillBeUsed\" /* CACHED_RESPONSE_WILL_BE_USED */,\n isReturnValueProblem: true,\n });\n }\n }\n }\n }\n }\n catch (err) {\n throw new WorkboxError('plugin-error-request-will-fetch', {\n thrownError: err,\n });\n }\n // The request can be altered by plugins with `requestWillFetch` making\n // the original request (Most likely from a `fetch` event) to be different\n // to the Request we make. Pass both to `fetchDidFail` to aid debugging.\n const pluginFilteredRequest = request.clone();\n try {\n let fetchResponse;\n // See https://github.com/GoogleChrome/workbox/issues/1796\n if (request.mode === 'navigate') {\n fetchResponse = await fetch(request);\n }\n else {\n fetchResponse = await fetch(request, fetchOptions);\n }\n if (process.env.NODE_ENV !== 'production') {\n logger.debug(`Network request for ` +\n `'${getFriendlyURL(request.url)}' returned a response with ` +\n `status '${fetchResponse.status}'.`);\n }\n for (const plugin of plugins) {\n if (\"fetchDidSucceed\" /* FETCH_DID_SUCCEED */ in plugin) {\n fetchResponse = await plugin[\"fetchDidSucceed\" /* FETCH_DID_SUCCEED */]\n .call(plugin, {\n event,\n request: pluginFilteredRequest,\n response: fetchResponse,\n });\n if (process.env.NODE_ENV !== 'production') {\n if (fetchResponse) {\n assert.isInstance(fetchResponse, Response, {\n moduleName: 'Plugin',\n funcName: \"fetchDidSucceed\" /* FETCH_DID_SUCCEED */,\n isReturnValueProblem: true,\n });\n }\n }\n }\n }\n return fetchResponse;\n }\n catch (error) {\n if (process.env.NODE_ENV !== 'production') {\n logger.error(`Network request for ` +\n `'${getFriendlyURL(request.url)}' threw an error.`, error);\n }\n for (const plugin of failedFetchPlugins) {\n await plugin[\"fetchDidFail\" /* FETCH_DID_FAIL */].call(plugin, {\n error,\n event,\n originalRequest: originalRequest.clone(),\n request: pluginFilteredRequest.clone(),\n });\n }\n throw error;\n }\n};\nconst fetchWrapper = {\n fetch: wrappedFetch,\n};\nexport { fetchWrapper };\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { assert } from 'workbox-core/_private/assert.js';\nimport { cacheNames } from 'workbox-core/_private/cacheNames.js';\nimport { cacheWrapper } from 'workbox-core/_private/cacheWrapper.js';\nimport { fetchWrapper } from 'workbox-core/_private/fetchWrapper.js';\nimport { logger } from 'workbox-core/_private/logger.js';\nimport { WorkboxError } from 'workbox-core/_private/WorkboxError.js';\nimport { copyResponse } from 'workbox-core/copyResponse.js';\nimport { createCacheKey } from './utils/createCacheKey.js';\nimport { printCleanupDetails } from './utils/printCleanupDetails.js';\nimport { printInstallDetails } from './utils/printInstallDetails.js';\nimport './_version.js';\n/**\n * Performs efficient precaching of assets.\n *\n * @memberof module:workbox-precaching\n */\nclass PrecacheController {\n /**\n * Create a new PrecacheController.\n *\n * @param {string} [cacheName] An optional name for the cache, to override\n * the default precache name.\n */\n constructor(cacheName) {\n this._cacheName = cacheNames.getPrecacheName(cacheName);\n this._urlsToCacheKeys = new Map();\n this._urlsToCacheModes = new Map();\n this._cacheKeysToIntegrities = new Map();\n }\n /**\n * This method will add items to the precache list, removing duplicates\n * and ensuring the information is valid.\n *\n * @param {\n * Array\n * } entries Array of entries to precache.\n */\n addToCacheList(entries) {\n if (process.env.NODE_ENV !== 'production') {\n assert.isArray(entries, {\n moduleName: 'workbox-precaching',\n className: 'PrecacheController',\n funcName: 'addToCacheList',\n paramName: 'entries',\n });\n }\n const urlsToWarnAbout = [];\n for (const entry of entries) {\n // See https://github.com/GoogleChrome/workbox/issues/2259\n if (typeof entry === 'string') {\n urlsToWarnAbout.push(entry);\n }\n else if (entry && entry.revision === undefined) {\n urlsToWarnAbout.push(entry.url);\n }\n const { cacheKey, url } = createCacheKey(entry);\n const cacheMode = (typeof entry !== 'string' && entry.revision) ?\n 'reload' : 'default';\n if (this._urlsToCacheKeys.has(url) &&\n this._urlsToCacheKeys.get(url) !== cacheKey) {\n throw new WorkboxError('add-to-cache-list-conflicting-entries', {\n firstEntry: this._urlsToCacheKeys.get(url),\n secondEntry: cacheKey,\n });\n }\n if (typeof entry !== 'string' && entry.integrity) {\n if (this._cacheKeysToIntegrities.has(cacheKey) &&\n this._cacheKeysToIntegrities.get(cacheKey) !== entry.integrity) {\n throw new WorkboxError('add-to-cache-list-conflicting-integrities', {\n url,\n });\n }\n this._cacheKeysToIntegrities.set(cacheKey, entry.integrity);\n }\n this._urlsToCacheKeys.set(url, cacheKey);\n this._urlsToCacheModes.set(url, cacheMode);\n if (urlsToWarnAbout.length > 0) {\n const warningMessage = `Workbox is precaching URLs without revision ` +\n `info: ${urlsToWarnAbout.join(', ')}\\nThis is generally NOT safe. ` +\n `Learn more at https://bit.ly/wb-precache`;\n if (process.env.NODE_ENV === 'production') {\n // Use console directly to display this warning without bloating\n // bundle sizes by pulling in all of the logger codebase in prod.\n console.warn(warningMessage);\n }\n else {\n logger.warn(warningMessage);\n }\n }\n }\n }\n /**\n * Precaches new and updated assets. Call this method from the service worker\n * install event.\n *\n * @param {Object} options\n * @param {Event} [options.event] The install event (if needed).\n * @param {Array} [options.plugins] Plugins to be used for fetching\n * and caching during install.\n * @return {Promise}\n */\n async install({ event, plugins } = {}) {\n if (process.env.NODE_ENV !== 'production') {\n if (plugins) {\n assert.isArray(plugins, {\n moduleName: 'workbox-precaching',\n className: 'PrecacheController',\n funcName: 'install',\n paramName: 'plugins',\n });\n }\n }\n const toBePrecached = [];\n const alreadyPrecached = [];\n const cache = await self.caches.open(this._cacheName);\n const alreadyCachedRequests = await cache.keys();\n const existingCacheKeys = new Set(alreadyCachedRequests.map((request) => request.url));\n for (const [url, cacheKey] of this._urlsToCacheKeys) {\n if (existingCacheKeys.has(cacheKey)) {\n alreadyPrecached.push(url);\n }\n else {\n toBePrecached.push({ cacheKey, url });\n }\n }\n const precacheRequests = toBePrecached.map(({ cacheKey, url }) => {\n const integrity = this._cacheKeysToIntegrities.get(cacheKey);\n const cacheMode = this._urlsToCacheModes.get(url);\n return this._addURLToCache({\n cacheKey,\n cacheMode,\n event,\n integrity,\n plugins,\n url,\n });\n });\n await Promise.all(precacheRequests);\n const updatedURLs = toBePrecached.map((item) => item.url);\n if (process.env.NODE_ENV !== 'production') {\n printInstallDetails(updatedURLs, alreadyPrecached);\n }\n return {\n updatedURLs,\n notUpdatedURLs: alreadyPrecached,\n };\n }\n /**\n * Deletes assets that are no longer present in the current precache manifest.\n * Call this method from the service worker activate event.\n *\n * @return {Promise}\n */\n async activate() {\n const cache = await self.caches.open(this._cacheName);\n const currentlyCachedRequests = await cache.keys();\n const expectedCacheKeys = new Set(this._urlsToCacheKeys.values());\n const deletedURLs = [];\n for (const request of currentlyCachedRequests) {\n if (!expectedCacheKeys.has(request.url)) {\n await cache.delete(request);\n deletedURLs.push(request.url);\n }\n }\n if (process.env.NODE_ENV !== 'production') {\n printCleanupDetails(deletedURLs);\n }\n return { deletedURLs };\n }\n /**\n * Requests the entry and saves it to the cache if the response is valid.\n * By default, any response with a status code of less than 400 (including\n * opaque responses) is considered valid.\n *\n * If you need to use custom criteria to determine what's valid and what\n * isn't, then pass in an item in `options.plugins` that implements the\n * `cacheWillUpdate()` lifecycle event.\n *\n * @private\n * @param {Object} options\n * @param {string} options.cacheKey The string to use a cache key.\n * @param {string} options.url The URL to fetch and cache.\n * @param {string} [options.cacheMode] The cache mode for the network request.\n * @param {Event} [options.event] The install event (if passed).\n * @param {Array} [options.plugins] An array of plugins to apply to\n * fetch and caching.\n * @param {string} [options.integrity] The value to use for the `integrity`\n * field when making the request.\n */\n async _addURLToCache({ cacheKey, url, cacheMode, event, plugins, integrity }) {\n const request = new Request(url, {\n integrity,\n cache: cacheMode,\n credentials: 'same-origin',\n });\n let response = await fetchWrapper.fetch({\n event,\n plugins,\n request,\n });\n // Allow developers to override the default logic about what is and isn't\n // valid by passing in a plugin implementing cacheWillUpdate(), e.g.\n // a `CacheableResponsePlugin` instance.\n let cacheWillUpdatePlugin;\n for (const plugin of (plugins || [])) {\n if ('cacheWillUpdate' in plugin) {\n cacheWillUpdatePlugin = plugin;\n }\n }\n const isValidResponse = cacheWillUpdatePlugin ?\n // Use a callback if provided. It returns a truthy value if valid.\n // NOTE: invoke the method on the plugin instance so the `this` context\n // is correct.\n await cacheWillUpdatePlugin.cacheWillUpdate({ event, request, response }) :\n // Otherwise, default to considering any response status under 400 valid.\n // This includes, by default, considering opaque responses valid.\n response.status < 400;\n // Consider this a failure, leading to the `install` handler failing, if\n // we get back an invalid response.\n if (!isValidResponse) {\n throw new WorkboxError('bad-precaching-response', {\n url,\n status: response.status,\n });\n }\n // Redirected responses cannot be used to satisfy a navigation request, so\n // any redirected response must be \"copied\" rather than cloned, so the new\n // response doesn't contain the `redirected` flag. See:\n // https://bugs.chromium.org/p/chromium/issues/detail?id=669363&desc=2#c1\n if (response.redirected) {\n response = await copyResponse(response);\n }\n await cacheWrapper.put({\n event,\n plugins,\n response,\n // `request` already uses `url`. We may be able to reuse it.\n request: cacheKey === url ? request : new Request(cacheKey),\n cacheName: this._cacheName,\n matchOptions: {\n ignoreSearch: true,\n },\n });\n }\n /**\n * Returns a mapping of a precached URL to the corresponding cache key, taking\n * into account the revision information for the URL.\n *\n * @return {Map} A URL to cache key mapping.\n */\n getURLsToCacheKeys() {\n return this._urlsToCacheKeys;\n }\n /**\n * Returns a list of all the URLs that have been precached by the current\n * service worker.\n *\n * @return {Array} The precached URLs.\n */\n getCachedURLs() {\n return [...this._urlsToCacheKeys.keys()];\n }\n /**\n * Returns the cache key used for storing a given URL. If that URL is\n * unversioned, like `/index.html', then the cache key will be the original\n * URL with a search parameter appended to it.\n *\n * @param {string} url A URL whose cache key you want to look up.\n * @return {string} The versioned URL that corresponds to a cache key\n * for the original URL, or undefined if that URL isn't precached.\n */\n getCacheKeyForURL(url) {\n const urlObject = new URL(url, location.href);\n return this._urlsToCacheKeys.get(urlObject.href);\n }\n /**\n * This acts as a drop-in replacement for [`cache.match()`](https://developer.mozilla.org/en-US/docs/Web/API/Cache/match)\n * with the following differences:\n *\n * - It knows what the name of the precache is, and only checks in that cache.\n * - It allows you to pass in an \"original\" URL without versioning parameters,\n * and it will automatically look up the correct cache key for the currently\n * active revision of that URL.\n *\n * E.g., `matchPrecache('index.html')` will find the correct precached\n * response for the currently active service worker, even if the actual cache\n * key is `'/index.html?__WB_REVISION__=1234abcd'`.\n *\n * @param {string|Request} request The key (without revisioning parameters)\n * to look up in the precache.\n * @return {Promise}\n */\n async matchPrecache(request) {\n const url = request instanceof Request ? request.url : request;\n const cacheKey = this.getCacheKeyForURL(url);\n if (cacheKey) {\n const cache = await self.caches.open(this._cacheName);\n return cache.match(cacheKey);\n }\n return undefined;\n }\n /**\n * Returns a function that can be used within a\n * {@link module:workbox-routing.Route} that will find a response for the\n * incoming request against the precache.\n *\n * If for an unexpected reason there is a cache miss for the request,\n * this will fall back to retrieving the `Response` via `fetch()` when\n * `fallbackToNetwork` is `true`.\n *\n * @param {boolean} [fallbackToNetwork=true] Whether to attempt to get the\n * response from the network if there's a precache miss.\n * @return {module:workbox-routing~handlerCallback}\n */\n createHandler(fallbackToNetwork = true) {\n return async ({ request }) => {\n try {\n const response = await this.matchPrecache(request);\n if (response) {\n return response;\n }\n // This shouldn't normally happen, but there are edge cases:\n // https://github.com/GoogleChrome/workbox/issues/1441\n throw new WorkboxError('missing-precache-entry', {\n cacheName: this._cacheName,\n url: request instanceof Request ? request.url : request,\n });\n }\n catch (error) {\n if (fallbackToNetwork) {\n if (process.env.NODE_ENV !== 'production') {\n logger.debug(`Unable to respond with precached response. ` +\n `Falling back to network.`, error);\n }\n return fetch(request);\n }\n throw error;\n }\n };\n }\n /**\n * Returns a function that looks up `url` in the precache (taking into\n * account revision information), and returns the corresponding `Response`.\n *\n * If for an unexpected reason there is a cache miss when looking up `url`,\n * this will fall back to retrieving the `Response` via `fetch()` when\n * `fallbackToNetwork` is `true`.\n *\n * @param {string} url The precached URL which will be used to lookup the\n * `Response`.\n * @param {boolean} [fallbackToNetwork=true] Whether to attempt to get the\n * response from the network if there's a precache miss.\n * @return {module:workbox-routing~handlerCallback}\n */\n createHandlerBoundToURL(url, fallbackToNetwork = true) {\n const cacheKey = this.getCacheKeyForURL(url);\n if (!cacheKey) {\n throw new WorkboxError('non-precached-url', { url });\n }\n const handler = this.createHandler(fallbackToNetwork);\n const request = new Request(url);\n return () => handler({ request });\n }\n}\nexport { PrecacheController };\n","export const isNav = event => event.request.mode === 'navigate';\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\nexport const cacheOkAndOpaquePlugin = {\n /**\n * Returns a valid response (to allow caching) if the status is 200 (OK) or\n * 0 (opaque).\n *\n * @param {Object} options\n * @param {Response} options.response\n * @return {Response|null}\n *\n * @private\n */\n cacheWillUpdate: async ({ response }) => {\n if (response.status === 200 || response.status === 0) {\n return response;\n }\n return null;\n },\n};\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { assert } from 'workbox-core/_private/assert.js';\nimport { cacheNames } from 'workbox-core/_private/cacheNames.js';\nimport { cacheWrapper } from 'workbox-core/_private/cacheWrapper.js';\nimport { fetchWrapper } from 'workbox-core/_private/fetchWrapper.js';\nimport { getFriendlyURL } from 'workbox-core/_private/getFriendlyURL.js';\nimport { logger } from 'workbox-core/_private/logger.js';\nimport { WorkboxError } from 'workbox-core/_private/WorkboxError.js';\nimport { messages } from './utils/messages.js';\nimport { cacheOkAndOpaquePlugin } from './plugins/cacheOkAndOpaquePlugin.js';\nimport './_version.js';\n/**\n * An implementation of a\n * [network first]{@link https://developers.google.com/web/fundamentals/instant-and-offline/offline-cookbook/#network-falling-back-to-cache}\n * request strategy.\n *\n * By default, this strategy will cache responses with a 200 status code as\n * well as [opaque responses]{@link https://developers.google.com/web/tools/workbox/guides/handle-third-party-requests}.\n * Opaque responses are are cross-origin requests where the response doesn't\n * support [CORS]{@link https://enable-cors.org/}.\n *\n * If the network request fails, and there is no cache match, this will throw\n * a `WorkboxError` exception.\n *\n * @memberof module:workbox-strategies\n */\nclass NetworkFirst {\n /**\n * @param {Object} options\n * @param {string} options.cacheName Cache name to store and retrieve\n * requests. Defaults to cache names provided by\n * [workbox-core]{@link module:workbox-core.cacheNames}.\n * @param {Array} options.plugins [Plugins]{@link https://developers.google.com/web/tools/workbox/guides/using-plugins}\n * to use in conjunction with this caching strategy.\n * @param {Object} options.fetchOptions Values passed along to the\n * [`init`](https://developer.mozilla.org/en-US/docs/Web/API/WindowOrWorkerGlobalScope/fetch#Parameters)\n * of all fetch() requests made by this strategy.\n * @param {Object} options.matchOptions [`CacheQueryOptions`](https://w3c.github.io/ServiceWorker/#dictdef-cachequeryoptions)\n * @param {number} options.networkTimeoutSeconds If set, any network requests\n * that fail to respond within the timeout will fallback to the cache.\n *\n * This option can be used to combat\n * \"[lie-fi]{@link https://developers.google.com/web/fundamentals/performance/poor-connectivity/#lie-fi}\"\n * scenarios.\n */\n constructor(options = {}) {\n this._cacheName = cacheNames.getRuntimeName(options.cacheName);\n if (options.plugins) {\n const isUsingCacheWillUpdate = options.plugins.some((plugin) => !!plugin.cacheWillUpdate);\n this._plugins = isUsingCacheWillUpdate ?\n options.plugins : [cacheOkAndOpaquePlugin, ...options.plugins];\n }\n else {\n // No plugins passed in, use the default plugin.\n this._plugins = [cacheOkAndOpaquePlugin];\n }\n this._networkTimeoutSeconds = options.networkTimeoutSeconds || 0;\n if (process.env.NODE_ENV !== 'production') {\n if (this._networkTimeoutSeconds) {\n assert.isType(this._networkTimeoutSeconds, 'number', {\n moduleName: 'workbox-strategies',\n className: 'NetworkFirst',\n funcName: 'constructor',\n paramName: 'networkTimeoutSeconds',\n });\n }\n }\n this._fetchOptions = options.fetchOptions;\n this._matchOptions = options.matchOptions;\n }\n /**\n * This method will perform a request strategy and follows an API that\n * will work with the\n * [Workbox Router]{@link module:workbox-routing.Router}.\n *\n * @param {Object} options\n * @param {Request|string} options.request A request to run this strategy for.\n * @param {Event} [options.event] The event that triggered the request.\n * @return {Promise}\n */\n async handle({ event, request }) {\n const logs = [];\n if (typeof request === 'string') {\n request = new Request(request);\n }\n if (process.env.NODE_ENV !== 'production') {\n assert.isInstance(request, Request, {\n moduleName: 'workbox-strategies',\n className: 'NetworkFirst',\n funcName: 'handle',\n paramName: 'makeRequest',\n });\n }\n const promises = [];\n let timeoutId;\n if (this._networkTimeoutSeconds) {\n const { id, promise } = this._getTimeoutPromise({ request, event, logs });\n timeoutId = id;\n promises.push(promise);\n }\n const networkPromise = this._getNetworkPromise({ timeoutId, request, event, logs });\n promises.push(networkPromise);\n // Promise.race() will resolve as soon as the first promise resolves.\n let response = await Promise.race(promises);\n // If Promise.race() resolved with null, it might be due to a network\n // timeout + a cache miss. If that were to happen, we'd rather wait until\n // the networkPromise resolves instead of returning null.\n // Note that it's fine to await an already-resolved promise, so we don't\n // have to check to see if it's still \"in flight\".\n if (!response) {\n response = await networkPromise;\n }\n if (process.env.NODE_ENV !== 'production') {\n logger.groupCollapsed(messages.strategyStart('NetworkFirst', request));\n for (const log of logs) {\n logger.log(log);\n }\n messages.printFinalResponse(response);\n logger.groupEnd();\n }\n if (!response) {\n throw new WorkboxError('no-response', { url: request.url });\n }\n return response;\n }\n /**\n * @param {Object} options\n * @param {Request} options.request\n * @param {Array} options.logs A reference to the logs array\n * @param {Event} [options.event]\n * @return {Promise}\n *\n * @private\n */\n _getTimeoutPromise({ request, logs, event }) {\n let timeoutId;\n const timeoutPromise = new Promise((resolve) => {\n const onNetworkTimeout = async () => {\n if (process.env.NODE_ENV !== 'production') {\n logs.push(`Timing out the network response at ` +\n `${this._networkTimeoutSeconds} seconds.`);\n }\n resolve(await this._respondFromCache({ request, event }));\n };\n timeoutId = setTimeout(onNetworkTimeout, this._networkTimeoutSeconds * 1000);\n });\n return {\n promise: timeoutPromise,\n id: timeoutId,\n };\n }\n /**\n * @param {Object} options\n * @param {number|undefined} options.timeoutId\n * @param {Request} options.request\n * @param {Array} options.logs A reference to the logs Array.\n * @param {Event} [options.event]\n * @return {Promise}\n *\n * @private\n */\n async _getNetworkPromise({ timeoutId, request, logs, event }) {\n let error;\n let response;\n try {\n response = await fetchWrapper.fetch({\n request,\n event,\n fetchOptions: this._fetchOptions,\n plugins: this._plugins,\n });\n }\n catch (err) {\n error = err;\n }\n if (timeoutId) {\n clearTimeout(timeoutId);\n }\n if (process.env.NODE_ENV !== 'production') {\n if (response) {\n logs.push(`Got response from network.`);\n }\n else {\n logs.push(`Unable to get a response from the network. Will respond ` +\n `with a cached response.`);\n }\n }\n if (error || !response) {\n response = await this._respondFromCache({ request, event });\n if (process.env.NODE_ENV !== 'production') {\n if (response) {\n logs.push(`Found a cached response in the '${this._cacheName}'` +\n ` cache.`);\n }\n else {\n logs.push(`No response found in the '${this._cacheName}' cache.`);\n }\n }\n }\n else {\n // Keep the service worker alive while we put the request in the cache\n const responseClone = response.clone();\n const cachePut = cacheWrapper.put({\n cacheName: this._cacheName,\n request,\n response: responseClone,\n event,\n plugins: this._plugins,\n });\n if (event) {\n try {\n // The event has been responded to so we can keep the SW alive to\n // respond to the request\n event.waitUntil(cachePut);\n }\n catch (err) {\n if (process.env.NODE_ENV !== 'production') {\n logger.warn(`Unable to ensure service worker stays alive when ` +\n `updating cache for '${getFriendlyURL(request.url)}'.`);\n }\n }\n }\n }\n return response;\n }\n /**\n * Used if the network timeouts or fails to make the request.\n *\n * @param {Object} options\n * @param {Request} request The request to match in the cache\n * @param {Event} [options.event]\n * @return {Promise}\n *\n * @private\n */\n _respondFromCache({ event, request }) {\n return cacheWrapper.match({\n cacheName: this._cacheName,\n request,\n event,\n matchOptions: this._matchOptions,\n plugins: this._plugins,\n });\n }\n}\nexport { NetworkFirst };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\n/**\n * A class that wraps common IndexedDB functionality in a promise-based API.\n * It exposes all the underlying power and functionality of IndexedDB, but\n * wraps the most commonly used features in a way that's much simpler to use.\n *\n * @private\n */\nexport class DBWrapper {\n /**\n * @param {string} name\n * @param {number} version\n * @param {Object=} [callback]\n * @param {!Function} [callbacks.onupgradeneeded]\n * @param {!Function} [callbacks.onversionchange] Defaults to\n * DBWrapper.prototype._onversionchange when not specified.\n * @private\n */\n constructor(name, version, { onupgradeneeded, onversionchange, } = {}) {\n this._db = null;\n this._name = name;\n this._version = version;\n this._onupgradeneeded = onupgradeneeded;\n this._onversionchange = onversionchange || (() => this.close());\n }\n /**\n * Returns the IDBDatabase instance (not normally needed).\n * @return {IDBDatabase|undefined}\n *\n * @private\n */\n get db() {\n return this._db;\n }\n /**\n * Opens a connected to an IDBDatabase, invokes any onupgradedneeded\n * callback, and added an onversionchange callback to the database.\n *\n * @return {IDBDatabase}\n * @private\n */\n async open() {\n if (this._db)\n return;\n this._db = await new Promise((resolve, reject) => {\n // This flag is flipped to true if the timeout callback runs prior\n // to the request failing or succeeding. Note: we use a timeout instead\n // of an onblocked handler since there are cases where onblocked will\n // never never run. A timeout better handles all possible scenarios:\n // https://github.com/w3c/IndexedDB/issues/223\n let openRequestTimedOut = false;\n setTimeout(() => {\n openRequestTimedOut = true;\n reject(new Error('The open request was blocked and timed out'));\n }, this.OPEN_TIMEOUT);\n const openRequest = indexedDB.open(this._name, this._version);\n openRequest.onerror = () => reject(openRequest.error);\n openRequest.onupgradeneeded = (evt) => {\n if (openRequestTimedOut) {\n openRequest.transaction.abort();\n openRequest.result.close();\n }\n else if (typeof this._onupgradeneeded === 'function') {\n this._onupgradeneeded(evt);\n }\n };\n openRequest.onsuccess = () => {\n const db = openRequest.result;\n if (openRequestTimedOut) {\n db.close();\n }\n else {\n db.onversionchange = this._onversionchange.bind(this);\n resolve(db);\n }\n };\n });\n return this;\n }\n /**\n * Polyfills the native `getKey()` method. Note, this is overridden at\n * runtime if the browser supports the native method.\n *\n * @param {string} storeName\n * @param {*} query\n * @return {Array}\n * @private\n */\n async getKey(storeName, query) {\n return (await this.getAllKeys(storeName, query, 1))[0];\n }\n /**\n * Polyfills the native `getAll()` method. Note, this is overridden at\n * runtime if the browser supports the native method.\n *\n * @param {string} storeName\n * @param {*} query\n * @param {number} count\n * @return {Array}\n * @private\n */\n async getAll(storeName, query, count) {\n return await this.getAllMatching(storeName, { query, count });\n }\n /**\n * Polyfills the native `getAllKeys()` method. Note, this is overridden at\n * runtime if the browser supports the native method.\n *\n * @param {string} storeName\n * @param {*} query\n * @param {number} count\n * @return {Array}\n * @private\n */\n async getAllKeys(storeName, query, count) {\n const entries = await this.getAllMatching(storeName, { query, count, includeKeys: true });\n return entries.map((entry) => entry.key);\n }\n /**\n * Supports flexible lookup in an object store by specifying an index,\n * query, direction, and count. This method returns an array of objects\n * with the signature .\n *\n * @param {string} storeName\n * @param {Object} [opts]\n * @param {string} [opts.index] The index to use (if specified).\n * @param {*} [opts.query]\n * @param {IDBCursorDirection} [opts.direction]\n * @param {number} [opts.count] The max number of results to return.\n * @param {boolean} [opts.includeKeys] When true, the structure of the\n * returned objects is changed from an array of values to an array of\n * objects in the form {key, primaryKey, value}.\n * @return {Array}\n * @private\n */\n async getAllMatching(storeName, { index, query = null, // IE/Edge errors if query === `undefined`.\n direction = 'next', count, includeKeys = false, } = {}) {\n return await this.transaction([storeName], 'readonly', (txn, done) => {\n const store = txn.objectStore(storeName);\n const target = index ? store.index(index) : store;\n const results = [];\n const request = target.openCursor(query, direction);\n request.onsuccess = () => {\n const cursor = request.result;\n if (cursor) {\n results.push(includeKeys ? cursor : cursor.value);\n if (count && results.length >= count) {\n done(results);\n }\n else {\n cursor.continue();\n }\n }\n else {\n done(results);\n }\n };\n });\n }\n /**\n * Accepts a list of stores, a transaction type, and a callback and\n * performs a transaction. A promise is returned that resolves to whatever\n * value the callback chooses. The callback holds all the transaction logic\n * and is invoked with two arguments:\n * 1. The IDBTransaction object\n * 2. A `done` function, that's used to resolve the promise when\n * when the transaction is done, if passed a value, the promise is\n * resolved to that value.\n *\n * @param {Array} storeNames An array of object store names\n * involved in the transaction.\n * @param {string} type Can be `readonly` or `readwrite`.\n * @param {!Function} callback\n * @return {*} The result of the transaction ran by the callback.\n * @private\n */\n async transaction(storeNames, type, callback) {\n await this.open();\n return await new Promise((resolve, reject) => {\n const txn = this._db.transaction(storeNames, type);\n txn.onabort = () => reject(txn.error);\n txn.oncomplete = () => resolve();\n callback(txn, (value) => resolve(value));\n });\n }\n /**\n * Delegates async to a native IDBObjectStore method.\n *\n * @param {string} method The method name.\n * @param {string} storeName The object store name.\n * @param {string} type Can be `readonly` or `readwrite`.\n * @param {...*} args The list of args to pass to the native method.\n * @return {*} The result of the transaction.\n * @private\n */\n async _call(method, storeName, type, ...args) {\n const callback = (txn, done) => {\n const objStore = txn.objectStore(storeName);\n // TODO(philipwalton): Fix this underlying TS2684 error.\n // @ts-ignore\n const request = objStore[method].apply(objStore, args);\n request.onsuccess = () => done(request.result);\n };\n return await this.transaction([storeName], type, callback);\n }\n /**\n * Closes the connection opened by `DBWrapper.open()`. Generally this method\n * doesn't need to be called since:\n * 1. It's usually better to keep a connection open since opening\n * a new connection is somewhat slow.\n * 2. Connections are automatically closed when the reference is\n * garbage collected.\n * The primary use case for needing to close a connection is when another\n * reference (typically in another tab) needs to upgrade it and would be\n * blocked by the current, open connection.\n *\n * @private\n */\n close() {\n if (this._db) {\n this._db.close();\n this._db = null;\n }\n }\n}\n// Exposed on the prototype to let users modify the default timeout on a\n// per-instance or global basis.\nDBWrapper.prototype.OPEN_TIMEOUT = 2000;\n// Wrap native IDBObjectStore methods according to their mode.\nconst methodsToWrap = {\n readonly: ['get', 'count', 'getKey', 'getAll', 'getAllKeys'],\n readwrite: ['add', 'put', 'clear', 'delete'],\n};\nfor (const [mode, methods] of Object.entries(methodsToWrap)) {\n for (const method of methods) {\n if (method in IDBObjectStore.prototype) {\n // Don't use arrow functions here since we're outside of the class.\n DBWrapper.prototype[method] =\n async function (storeName, ...args) {\n return await this._call(method, storeName, mode, ...args);\n };\n }\n }\n}\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { getOrCreateDefaultRouter } from './utils/getOrCreateDefaultRouter.js';\nimport './_version.js';\n/**\n * If a Route throws an error while handling a request, this `handler`\n * will be called and given a chance to provide a response.\n *\n * @param {module:workbox-routing~handlerCallback} handler A callback\n * function that returns a Promise resulting in a Response.\n *\n * @memberof module:workbox-routing\n */\nfunction setCatchHandler(handler) {\n const defaultRouter = getOrCreateDefaultRouter();\n defaultRouter.setCatchHandler(handler);\n}\nexport { setCatchHandler };\n","import { registerRoute, setCatchHandler } from 'workbox-routing';\nimport { precacheAndRoute, getCacheKeyForURL } from 'workbox-precaching';\nimport { isNav } from './utils';\nimport { NETWORK_HANDLER, PRECACHING_OPTIONS } from './contants';\n\nexport function getFiles() {\n\treturn self.__WB_MANIFEST;\n}\n\nexport function setupPrecaching(precacheFiles, precachingOptions) {\n\tprecacheAndRoute(precacheFiles, precachingOptions || PRECACHING_OPTIONS);\n}\n\nexport function setupRouting() {\n\t/**\n\t * Adding this before `precacheAndRoute` lets us handle all\n\t * the navigation requests even if they are in precache.\n\t */\n\tregisterRoute(({ event }) => isNav(event), NETWORK_HANDLER);\n\n\tsetCatchHandler(({ event }) => {\n\t\tif (isNav(event)) {\n\t\t\treturn caches.match(getCacheKeyForURL('/index.html'));\n\t\t}\n\t\treturn Response.error();\n\t});\n}\n\nexport { PRECACHING_OPTIONS };\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { cacheNames as _cacheNames } from './_private/cacheNames.js';\nimport './_version.js';\n/**\n * Get the current cache names and prefix/suffix used by Workbox.\n *\n * `cacheNames.precache` is used for precached assets,\n * `cacheNames.googleAnalytics` is used by `workbox-google-analytics` to\n * store `analytics.js`, and `cacheNames.runtime` is used for everything else.\n *\n * `cacheNames.prefix` can be used to retrieve just the current prefix value.\n * `cacheNames.suffix` can be used to retrieve just the current suffix value.\n *\n * @return {Object} An object with `precache`, `runtime`, `prefix`, and\n * `googleAnalytics` properties.\n *\n * @memberof module:workbox-core\n */\nconst cacheNames = {\n get googleAnalytics() {\n return _cacheNames.getGoogleAnalyticsName();\n },\n get precache() {\n return _cacheNames.getPrecacheName();\n },\n get prefix() {\n return _cacheNames.getPrefix();\n },\n get runtime() {\n return _cacheNames.getRuntimeName();\n },\n get suffix() {\n return _cacheNames.getSuffix();\n },\n};\nexport { cacheNames };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { assert } from 'workbox-core/_private/assert.js';\nimport { WorkboxError } from 'workbox-core/_private/WorkboxError.js';\nimport { getFriendlyURL } from 'workbox-core/_private/getFriendlyURL.js';\nimport { logger } from 'workbox-core/_private/logger.js';\nimport './_version.js';\n/**\n * This class allows you to set up rules determining what\n * status codes and/or headers need to be present in order for a\n * [`Response`](https://developer.mozilla.org/en-US/docs/Web/API/Response)\n * to be considered cacheable.\n *\n * @memberof module:workbox-cacheable-response\n */\nclass CacheableResponse {\n /**\n * To construct a new CacheableResponse instance you must provide at least\n * one of the `config` properties.\n *\n * If both `statuses` and `headers` are specified, then both conditions must\n * be met for the `Response` to be considered cacheable.\n *\n * @param {Object} config\n * @param {Array} [config.statuses] One or more status codes that a\n * `Response` can have and be considered cacheable.\n * @param {Object} [config.headers] A mapping of header names\n * and expected values that a `Response` can have and be considered cacheable.\n * If multiple headers are provided, only one needs to be present.\n */\n constructor(config = {}) {\n if (process.env.NODE_ENV !== 'production') {\n if (!(config.statuses || config.headers)) {\n throw new WorkboxError('statuses-or-headers-required', {\n moduleName: 'workbox-cacheable-response',\n className: 'CacheableResponse',\n funcName: 'constructor',\n });\n }\n if (config.statuses) {\n assert.isArray(config.statuses, {\n moduleName: 'workbox-cacheable-response',\n className: 'CacheableResponse',\n funcName: 'constructor',\n paramName: 'config.statuses',\n });\n }\n if (config.headers) {\n assert.isType(config.headers, 'object', {\n moduleName: 'workbox-cacheable-response',\n className: 'CacheableResponse',\n funcName: 'constructor',\n paramName: 'config.headers',\n });\n }\n }\n this._statuses = config.statuses;\n this._headers = config.headers;\n }\n /**\n * Checks a response to see whether it's cacheable or not, based on this\n * object's configuration.\n *\n * @param {Response} response The response whose cacheability is being\n * checked.\n * @return {boolean} `true` if the `Response` is cacheable, and `false`\n * otherwise.\n */\n isResponseCacheable(response) {\n if (process.env.NODE_ENV !== 'production') {\n assert.isInstance(response, Response, {\n moduleName: 'workbox-cacheable-response',\n className: 'CacheableResponse',\n funcName: 'isResponseCacheable',\n paramName: 'response',\n });\n }\n let cacheable = true;\n if (this._statuses) {\n cacheable = this._statuses.includes(response.status);\n }\n if (this._headers && cacheable) {\n cacheable = Object.keys(this._headers).some((headerName) => {\n return response.headers.get(headerName) === this._headers[headerName];\n });\n }\n if (process.env.NODE_ENV !== 'production') {\n if (!cacheable) {\n logger.groupCollapsed(`The request for ` +\n `'${getFriendlyURL(response.url)}' returned a response that does ` +\n `not meet the criteria for being cached.`);\n logger.groupCollapsed(`View cacheability criteria here.`);\n logger.log(`Cacheable statuses: ` +\n JSON.stringify(this._statuses));\n logger.log(`Cacheable headers: ` +\n JSON.stringify(this._headers, null, 2));\n logger.groupEnd();\n const logFriendlyHeaders = {};\n response.headers.forEach((value, key) => {\n logFriendlyHeaders[key] = value;\n });\n logger.groupCollapsed(`View response status and headers here.`);\n logger.log(`Response status: ` + response.status);\n logger.log(`Response headers: ` +\n JSON.stringify(logFriendlyHeaders, null, 2));\n logger.groupEnd();\n logger.groupCollapsed(`View full response details here.`);\n logger.log(response.headers);\n logger.log(response);\n logger.groupEnd();\n logger.groupEnd();\n }\n }\n return cacheable;\n }\n}\nexport { CacheableResponse };\n","import { NetworkFirst } from 'workbox-strategies';\nimport { cacheNames } from 'workbox-core';\nimport { CacheableResponsePlugin } from 'workbox-cacheable-response';\n\nexport const PRECACHING_OPTIONS = {};\n\nexport const NETWORK_HANDLER = new NetworkFirst({\n\t// this cache is plunged with every new service worker deploy so we dont need to care about purging the cache.\n\tcacheName: cacheNames.precache,\n\tnetworkTimeoutSeconds: 5, // if u dont start getting headers within 5 sec fallback to cache.\n\tplugins: [\n\t\tnew CacheableResponsePlugin({\n\t\t\tstatuses: [200], // only cache valid responses, not opaque responses e.g. wifi portal.\n\t\t}),\n\t],\n});\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { logger } from 'workbox-core/_private/logger.js';\nimport { WorkboxError } from 'workbox-core/_private/WorkboxError.js';\nimport { Route } from './Route.js';\nimport { RegExpRoute } from './RegExpRoute.js';\nimport { getOrCreateDefaultRouter } from './utils/getOrCreateDefaultRouter.js';\nimport './_version.js';\n/**\n * Easily register a RegExp, string, or function with a caching\n * strategy to a singleton Router instance.\n *\n * This method will generate a Route for you if needed and\n * call [registerRoute()]{@link module:workbox-routing.Router#registerRoute}.\n *\n * @param {RegExp|string|module:workbox-routing.Route~matchCallback|module:workbox-routing.Route} capture\n * If the capture param is a `Route`, all other arguments will be ignored.\n * @param {module:workbox-routing~handlerCallback} [handler] A callback\n * function that returns a Promise resulting in a Response. This parameter\n * is required if `capture` is not a `Route` object.\n * @param {string} [method='GET'] The HTTP method to match the Route\n * against.\n * @return {module:workbox-routing.Route} The generated `Route`(Useful for\n * unregistering).\n *\n * @memberof module:workbox-routing\n */\nfunction registerRoute(capture, handler, method) {\n let route;\n if (typeof capture === 'string') {\n const captureUrl = new URL(capture, location.href);\n if (process.env.NODE_ENV !== 'production') {\n if (!(capture.startsWith('/') || capture.startsWith('http'))) {\n throw new WorkboxError('invalid-string', {\n moduleName: 'workbox-routing',\n funcName: 'registerRoute',\n paramName: 'capture',\n });\n }\n // We want to check if Express-style wildcards are in the pathname only.\n // TODO: Remove this log message in v4.\n const valueToCheck = capture.startsWith('http') ?\n captureUrl.pathname : capture;\n // See https://github.com/pillarjs/path-to-regexp#parameters\n const wildcards = '[*:?+]';\n if ((new RegExp(`${wildcards}`)).exec(valueToCheck)) {\n logger.debug(`The '$capture' parameter contains an Express-style wildcard ` +\n `character (${wildcards}). Strings are now always interpreted as ` +\n `exact matches; use a RegExp for partial or wildcard matches.`);\n }\n }\n const matchCallback = ({ url }) => {\n if (process.env.NODE_ENV !== 'production') {\n if ((url.pathname === captureUrl.pathname) &&\n (url.origin !== captureUrl.origin)) {\n logger.debug(`${capture} only partially matches the cross-origin URL ` +\n `${url}. This route will only handle cross-origin requests ` +\n `if they match the entire URL.`);\n }\n }\n return url.href === captureUrl.href;\n };\n // If `capture` is a string then `handler` and `method` must be present.\n route = new Route(matchCallback, handler, method);\n }\n else if (capture instanceof RegExp) {\n // If `capture` is a `RegExp` then `handler` and `method` must be present.\n route = new RegExpRoute(capture, handler, method);\n }\n else if (typeof capture === 'function') {\n // If `capture` is a function then `handler` and `method` must be present.\n route = new Route(capture, handler, method);\n }\n else if (capture instanceof Route) {\n route = capture;\n }\n else {\n throw new WorkboxError('unsupported-route-type', {\n moduleName: 'workbox-routing',\n funcName: 'registerRoute',\n paramName: 'capture',\n });\n }\n const defaultRouter = getOrCreateDefaultRouter();\n defaultRouter.registerRoute(route);\n return route;\n}\nexport { registerRoute };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { CacheableResponse } from './CacheableResponse.js';\nimport './_version.js';\n/**\n * A class implementing the `cacheWillUpdate` lifecycle callback. This makes it\n * easier to add in cacheability checks to requests made via Workbox's built-in\n * strategies.\n *\n * @memberof module:workbox-cacheable-response\n */\nclass CacheableResponsePlugin {\n /**\n * To construct a new CacheableResponsePlugin instance you must provide at\n * least one of the `config` properties.\n *\n * If both `statuses` and `headers` are specified, then both conditions must\n * be met for the `Response` to be considered cacheable.\n *\n * @param {Object} config\n * @param {Array} [config.statuses] One or more status codes that a\n * `Response` can have and be considered cacheable.\n * @param {Object} [config.headers] A mapping of header names\n * and expected values that a `Response` can have and be considered cacheable.\n * If multiple headers are provided, only one needs to be present.\n */\n constructor(config) {\n /**\n * @param {Object} options\n * @param {Response} options.response\n * @return {Response|null}\n * @private\n */\n this.cacheWillUpdate = async ({ response }) => {\n if (this._cacheableResponse.isResponseCacheable(response)) {\n return response;\n }\n return null;\n };\n this._cacheableResponse = new CacheableResponse(config);\n }\n}\nexport { CacheableResponsePlugin };\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { getOrCreatePrecacheController } from './utils/getOrCreatePrecacheController.js';\nimport './_version.js';\n/**\n * Takes in a URL, and returns the corresponding URL that could be used to\n * lookup the entry in the precache.\n *\n * If a relative URL is provided, the location of the service worker file will\n * be used as the base.\n *\n * For precached entries without revision information, the cache key will be the\n * same as the original URL.\n *\n * For precached entries with revision information, the cache key will be the\n * original URL with the 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\"p5Qz\");\n","\"use strict\";\n// @ts-ignore\ntry {\n self['workbox:cacheable-response:5.1.3'] && _();\n}\ncatch (e) { }\n","\"use strict\";\n// @ts-ignore\ntry {\n self['workbox:precaching:5.1.3'] && _();\n}\ncatch (e) { }\n","\"use strict\";\n// @ts-ignore\ntry {\n self['workbox:routing:5.1.3'] && _();\n}\ncatch (e) { }\n","\"use strict\";\n// @ts-ignore\ntry {\n self['workbox:strategies:5.1.3'] && _();\n}\ncatch (e) { }\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { canConstructResponseFromBodyStream } from './_private/canConstructResponseFromBodyStream.js';\nimport './_version.js';\n/**\n * Allows developers to copy a response and modify its `headers`, `status`,\n * or `statusText` values (the values settable via a\n * [`ResponseInit`]{@link https://developer.mozilla.org/en-US/docs/Web/API/Response/Response#Syntax}\n * object in the 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That\n * function will be invoked with a single object with the response properties\n * `{headers, status, statusText}`. The return value of this function will\n * be used as the `ResponseInit` for the new `Response`. To change the values\n * either modify the passed parameter(s) and return it, or return a totally\n * new object.\n *\n * @param {Response} response\n * @param {Function} modifier\n * @memberof module:workbox-core\n */\nasync function copyResponse(response, modifier) {\n const clonedResponse = response.clone();\n // Create a fresh `ResponseInit` object by cloning the headers.\n const responseInit = {\n headers: new Headers(clonedResponse.headers),\n status: clonedResponse.status,\n statusText: clonedResponse.statusText,\n };\n // Apply any user modifications.\n const modifiedResponseInit = modifier ? modifier(responseInit) : responseInit;\n // Create the new response from the body stream and `ResponseInit`\n // modifications. 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Return as-is.\n if (typeof entry === 'string') {\n const urlObject = new URL(entry, location.href);\n return {\n cacheKey: urlObject.href,\n url: urlObject.href,\n };\n }\n const { revision, url } = entry;\n if (!url) {\n throw new WorkboxError('add-to-cache-list-unexpected-type', { entry });\n }\n // If there's just a URL and no revision, then it's also assumed to be a\n // versioned URL.\n if (!revision) {\n const urlObject = new URL(url, location.href);\n return {\n cacheKey: urlObject.href,\n url: urlObject.href,\n };\n }\n // Otherwise, construct a properly versioned URL using the custom Workbox\n // search parameter along with the revision info.\n const cacheKeyURL = new URL(url, location.href);\n const originalURL = new URL(url, location.href);\n cacheKeyURL.searchParams.set(REVISION_SEARCH_PARAM, revision);\n return {\n cacheKey: cacheKeyURL.href,\n url: originalURL.href,\n };\n}\n","/*\n Copyright 2019 Google LLC\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { addFetchListener } from './utils/addFetchListener.js';\nimport './_version.js';\nlet listenerAdded = false;\n/**\n * Add a `fetch` listener to the service worker that will\n * respond to\n * [network requests]{@link https://developer.mozilla.org/en-US/docs/Web/API/Service_Worker_API/Using_Service_Workers#Custom_responses_to_requests}\n * with precached assets.\n *\n * Requests for assets that aren't precached, the `FetchEvent` will not be\n * responded to, allowing the event to fall through to other `fetch` event\n * listeners.\n *\n * @param {Object} [options]\n * @param {string} [options.directoryIndex=index.html] The `directoryIndex` will\n * check cache entries for a URLs ending with '/' to see if there is a hit when\n * appending the `directoryIndex` value.\n * @param {Array} [options.ignoreURLParametersMatching=[/^utm_/]] An\n * array of regex's to remove search params when looking for a cache match.\n * @param {boolean} [options.cleanURLs=true] The `cleanURLs` option will\n * check the cache for the URL with a `.html` added to the end of the end.\n * @param {module:workbox-precaching~urlManipulation} [options.urlManipulation]\n * This is a function that should take a URL and return an array of\n * alternative URLs that should be checked for precache matches.\n *\n * @memberof module:workbox-precaching\n */\nfunction addRoute(options) {\n if (!listenerAdded) {\n addFetchListener(options);\n listenerAdded = true;\n }\n}\nexport { addRoute };\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { cacheNames } from 'workbox-core/_private/cacheNames.js';\nimport { getFriendlyURL } from 'workbox-core/_private/getFriendlyURL.js';\nimport { logger } from 'workbox-core/_private/logger.js';\nimport { getCacheKeyForURL } from './getCacheKeyForURL.js';\nimport '../_version.js';\n/**\n * Adds a `fetch` listener to the service worker that will\n * respond to\n * [network requests]{@link https://developer.mozilla.org/en-US/docs/Web/API/Service_Worker_API/Using_Service_Workers#Custom_responses_to_requests}\n * with precached assets.\n *\n * Requests for assets that aren't precached, the `FetchEvent` will not be\n * responded to, allowing the event to fall through to other `fetch` event\n * listeners.\n *\n * NOTE: when called more than once this method will replace the previously set\n * configuration options. Calling it more than once is not recommended outside\n * of tests.\n *\n * @private\n * @param {Object} [options]\n * @param {string} [options.directoryIndex=index.html] The `directoryIndex` will\n * check cache entries for a URLs ending with '/' to see if there is a hit when\n * appending the `directoryIndex` value.\n * @param {Array} [options.ignoreURLParametersMatching=[/^utm_/]] An\n * array of regex's to remove search params when looking for a cache match.\n * @param {boolean} [options.cleanURLs=true] The `cleanURLs` option will\n * check the cache for the URL with a `.html` added to the end of the end.\n * @param {workbox.precaching~urlManipulation} [options.urlManipulation]\n * This is a function that should take a URL and return an array of\n * alternative URLs that should be checked for precache matches.\n */\nexport const addFetchListener = ({ ignoreURLParametersMatching = [/^utm_/], directoryIndex = 'index.html', cleanURLs = true, urlManipulation, } = {}) => {\n const cacheName = cacheNames.getPrecacheName();\n // See https://github.com/Microsoft/TypeScript/issues/28357#issuecomment-436484705\n self.addEventListener('fetch', ((event) => {\n const precachedURL = getCacheKeyForURL(event.request.url, {\n cleanURLs,\n directoryIndex,\n ignoreURLParametersMatching,\n urlManipulation,\n });\n if (!precachedURL) {\n if (process.env.NODE_ENV !== 'production') {\n logger.debug(`Precaching did not find a match for ` +\n getFriendlyURL(event.request.url));\n }\n return;\n }\n let responsePromise = self.caches.open(cacheName).then((cache) => {\n return cache.match(precachedURL);\n }).then((cachedResponse) => {\n if (cachedResponse) {\n return cachedResponse;\n }\n // Fall back to the network if we don't have a cached response\n // (perhaps due to manual cache cleanup).\n if (process.env.NODE_ENV !== 'production') {\n logger.warn(`The precached response for ` +\n `${getFriendlyURL(precachedURL)} in ${cacheName} was not found. ` +\n `Falling back to the network instead.`);\n }\n return fetch(precachedURL);\n });\n if (process.env.NODE_ENV !== 'production') {\n responsePromise = responsePromise.then((response) => {\n // Workbox is going to handle the route.\n // print the routing details to the console.\n logger.groupCollapsed(`Precaching is responding to: ` +\n getFriendlyURL(event.request.url));\n logger.log(`Serving the precached url: ${precachedURL}`);\n logger.groupCollapsed(`View request details here.`);\n logger.log(event.request);\n logger.groupEnd();\n logger.groupCollapsed(`View response details here.`);\n logger.log(response);\n logger.groupEnd();\n logger.groupEnd();\n return response;\n });\n }\n event.respondWith(responsePromise);\n }));\n};\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { getOrCreatePrecacheController } from './getOrCreatePrecacheController.js';\nimport { generateURLVariations } from './generateURLVariations.js';\nimport '../_version.js';\n/**\n * This function will take the request URL and manipulate it based on the\n * configuration options.\n *\n * @param {string} url\n * @param {Object} options\n * @return {string} Returns the URL in the cache that matches the request,\n * if possible.\n *\n * @private\n */\nexport const getCacheKeyForURL = (url, options) => {\n const precacheController = getOrCreatePrecacheController();\n const urlsToCacheKeys = precacheController.getURLsToCacheKeys();\n for (const possibleURL of generateURLVariations(url, options)) {\n const possibleCacheKey = urlsToCacheKeys.get(possibleURL);\n if (possibleCacheKey) {\n return possibleCacheKey;\n }\n }\n};\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { removeIgnoredSearchParams } from './removeIgnoredSearchParams.js';\nimport '../_version.js';\n/**\n * Generator function that yields possible variations on the original URL to\n * check, one at a time.\n *\n * @param {string} url\n * @param {Object} options\n *\n * @private\n * @memberof module:workbox-precaching\n */\nexport function* generateURLVariations(url, { ignoreURLParametersMatching, directoryIndex, cleanURLs, urlManipulation, } = {}) {\n const urlObject = new URL(url, location.href);\n urlObject.hash = '';\n yield urlObject.href;\n const urlWithoutIgnoredParams = removeIgnoredSearchParams(urlObject, ignoreURLParametersMatching);\n yield urlWithoutIgnoredParams.href;\n if (directoryIndex && urlWithoutIgnoredParams.pathname.endsWith('/')) {\n const directoryURL = new URL(urlWithoutIgnoredParams.href);\n directoryURL.pathname += directoryIndex;\n yield directoryURL.href;\n }\n if (cleanURLs) {\n const cleanURL = new URL(urlWithoutIgnoredParams.href);\n cleanURL.pathname += '.html';\n yield cleanURL.href;\n }\n if (urlManipulation) {\n const additionalURLs = urlManipulation({ url: urlObject });\n for (const urlToAttempt of additionalURLs) {\n yield urlToAttempt.href;\n }\n }\n}\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\n/**\n * Removes any URL search parameters that should be ignored.\n *\n * @param {URL} urlObject The original URL.\n * @param {Array} ignoreURLParametersMatching RegExps to test against\n * each search parameter name. Matches mean that the search parameter should be\n * ignored.\n * @return {URL} The URL with any ignored search parameters removed.\n *\n * @private\n * @memberof module:workbox-precaching\n */\nexport function removeIgnoredSearchParams(urlObject, ignoreURLParametersMatching = []) {\n // Convert the iterable into an array at the start of the loop to make sure\n // deletion doesn't mess up iteration.\n for (const paramName of [...urlObject.searchParams.keys()]) {\n if (ignoreURLParametersMatching.some((regExp) => regExp.test(paramName))) {\n urlObject.searchParams.delete(paramName);\n }\n }\n return urlObject;\n}\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { addRoute } from './addRoute.js';\nimport { precache } from './precache.js';\nimport './_version.js';\n/**\n * This method will add entries to the precache list and add a route to\n * respond to fetch events.\n *\n * This is a convenience method that will call\n * [precache()]{@link module:workbox-precaching.precache} and\n * [addRoute()]{@link module:workbox-precaching.addRoute} in a single call.\n *\n * @param {Array} entries Array of entries to precache.\n * @param {Object} [options] See\n * [addRoute() options]{@link module:workbox-precaching.addRoute}.\n *\n * @memberof module:workbox-precaching\n */\nfunction precacheAndRoute(entries, options) {\n precache(entries);\n addRoute(options);\n}\nexport { precacheAndRoute };\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { logger } from 'workbox-core/_private/logger.js';\nimport { getOrCreatePrecacheController } from './utils/getOrCreatePrecacheController.js';\nimport { precachePlugins } from './utils/precachePlugins.js';\nimport './_version.js';\nconst installListener = (event) => {\n const precacheController = getOrCreatePrecacheController();\n const plugins = precachePlugins.get();\n event.waitUntil(precacheController.install({ event, plugins })\n .catch((error) => {\n if (process.env.NODE_ENV !== 'production') {\n logger.error(`Service worker installation failed. It will ` +\n `be retried automatically during the next navigation.`);\n }\n // Re-throw the error to ensure installation fails.\n throw error;\n }));\n};\nconst activateListener = (event) => {\n const precacheController = getOrCreatePrecacheController();\n event.waitUntil(precacheController.activate());\n};\n/**\n * Adds items to the precache list, removing any duplicates and\n * stores the files in the\n * [\"precache cache\"]{@link module:workbox-core.cacheNames} when the service\n * worker installs.\n *\n * This method can be called multiple times.\n *\n * Please note: This method **will not** serve any of the cached files for you.\n * It only precaches files. To respond to a network request you call\n * [addRoute()]{@link module:workbox-precaching.addRoute}.\n *\n * If you have a single array of files to precache, you can just call\n * [precacheAndRoute()]{@link module:workbox-precaching.precacheAndRoute}.\n *\n * @param {Array} [entries=[]] Array of entries to precache.\n *\n * @memberof module:workbox-precaching\n */\nfunction precache(entries) {\n const precacheController = getOrCreatePrecacheController();\n precacheController.addToCacheList(entries);\n if (entries.length > 0) {\n // NOTE: these listeners will only be added once (even if the `precache()`\n // method is called multiple times) because event listeners are implemented\n // as a set, where each listener must be unique.\n // See https://github.com/Microsoft/TypeScript/issues/28357#issuecomment-436484705\n self.addEventListener('install', installListener);\n self.addEventListener('activate', activateListener);\n }\n}\nexport { precache };\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { Router } from '../Router.js';\nimport '../_version.js';\nlet defaultRouter;\n/**\n * Creates a new, singleton Router instance if one does not exist. If one\n * does already exist, that instance is returned.\n *\n * @private\n * @return {Router}\n */\nexport const getOrCreateDefaultRouter = () => {\n if (!defaultRouter) {\n defaultRouter = new Router();\n // The helpers that use the default Router assume these listeners exist.\n defaultRouter.addFetchListener();\n defaultRouter.addCacheListener();\n }\n return defaultRouter;\n};\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { PrecacheController } from '../PrecacheController.js';\nimport '../_version.js';\nlet precacheController;\n/**\n * @return {PrecacheController}\n * @private\n */\nexport const getOrCreatePrecacheController = () => {\n if (!precacheController) {\n precacheController = new PrecacheController();\n }\n return precacheController;\n};\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { messages } from './messages.js';\nimport '../../_version.js';\nconst fallback = (code, ...args) => {\n let msg = code;\n if (args.length > 0) {\n msg += ` :: ${JSON.stringify(args)}`;\n }\n return msg;\n};\nconst generatorFunction = (code, details = {}) => {\n const message = messages[code];\n if (!message) {\n throw new Error(`Unable to find message for code '${code}'.`);\n }\n return message(details);\n};\nexport const messageGenerator = (process.env.NODE_ENV === 'production') ?\n fallback : generatorFunction;\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { messageGenerator } from '../models/messages/messageGenerator.js';\nimport '../_version.js';\n/**\n * Workbox errors should be thrown with this class.\n * This allows use to ensure the type easily in tests,\n * helps developers identify errors from workbox\n * easily and allows use to optimise error\n * messages correctly.\n *\n * @private\n */\nclass WorkboxError extends Error {\n /**\n *\n * @param {string} errorCode The error code that\n * identifies this particular error.\n * @param {Object=} details Any relevant arguments\n * that will help developers identify issues should\n * be added as a key on the context object.\n */\n constructor(errorCode, details) {\n const message = messageGenerator(errorCode, details);\n super(message);\n this.name = errorCode;\n this.details = details;\n }\n}\nexport { WorkboxError };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { assert } from 'workbox-core/_private/assert.js';\nimport '../_version.js';\n/**\n * @param {function()|Object} handler Either a function, or an object with a\n * 'handle' method.\n * @return {Object} An object with a handle method.\n *\n * @private\n */\nexport const normalizeHandler = (handler) => {\n if (handler && typeof handler === 'object') {\n if (process.env.NODE_ENV !== 'production') {\n assert.hasMethod(handler, 'handle', {\n moduleName: 'workbox-routing',\n className: 'Route',\n funcName: 'constructor',\n paramName: 'handler',\n });\n }\n return handler;\n }\n else {\n if (process.env.NODE_ENV !== 'production') {\n assert.isType(handler, 'function', {\n moduleName: 'workbox-routing',\n className: 'Route',\n funcName: 'constructor',\n paramName: 'handler',\n });\n }\n return { handle: handler };\n }\n};\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { assert } from 'workbox-core/_private/assert.js';\nimport { defaultMethod, validMethods } from './utils/constants.js';\nimport { normalizeHandler } from './utils/normalizeHandler.js';\nimport './_version.js';\n/**\n * A `Route` consists of a pair of callback functions, \"match\" and \"handler\".\n * The \"match\" callback determine if a route should be used to \"handle\" a\n * request by returning a non-falsy value if it can. The \"handler\" callback\n * is called when there is a match and should return a Promise that resolves\n * to a `Response`.\n *\n * @memberof module:workbox-routing\n */\nclass Route {\n /**\n * Constructor for Route class.\n *\n * @param {module:workbox-routing~matchCallback} match\n * A callback function that determines whether the route matches a given\n * `fetch` event by returning a non-falsy value.\n * @param {module:workbox-routing~handlerCallback} handler A callback\n * function that returns a Promise resolving to a Response.\n * @param {string} [method='GET'] The HTTP method to match the Route\n * against.\n */\n constructor(match, handler, method = defaultMethod) {\n if (process.env.NODE_ENV !== 'production') {\n assert.isType(match, 'function', {\n moduleName: 'workbox-routing',\n className: 'Route',\n funcName: 'constructor',\n paramName: 'match',\n });\n if (method) {\n assert.isOneOf(method, validMethods, { paramName: 'method' });\n }\n }\n // These values are referenced directly by Router so cannot be\n // altered by minificaton.\n this.handler = normalizeHandler(handler);\n this.match = match;\n this.method = method;\n }\n}\nexport { Route };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\n/**\n * The default HTTP method, 'GET', used when there's no specific method\n * configured for a route.\n *\n * @type {string}\n *\n * @private\n */\nexport const defaultMethod = 'GET';\n/**\n * The list of valid HTTP methods associated with requests that could be routed.\n *\n * @type {Array}\n *\n * @private\n */\nexport const validMethods = [\n 'DELETE',\n 'GET',\n 'HEAD',\n 'PATCH',\n 'POST',\n 'PUT',\n];\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { assert } from 'workbox-core/_private/assert.js';\nimport { logger } from 'workbox-core/_private/logger.js';\nimport { Route } from './Route.js';\nimport './_version.js';\n/**\n * RegExpRoute makes it easy to create a regular expression based\n * [Route]{@link module:workbox-routing.Route}.\n *\n * For same-origin requests the RegExp only needs to match part of the URL. For\n * requests against third-party servers, you must define a RegExp that matches\n * the start of the URL.\n *\n * [See the module docs for info.]{@link https://developers.google.com/web/tools/workbox/modules/workbox-routing}\n *\n * @memberof module:workbox-routing\n * @extends module:workbox-routing.Route\n */\nclass RegExpRoute extends Route {\n /**\n * If the regular expression contains\n * [capture groups]{@link https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/RegExp#grouping-back-references},\n * the captured values will be passed to the\n * [handler's]{@link module:workbox-routing~handlerCallback} `params`\n * argument.\n *\n * @param {RegExp} regExp The regular expression to match against URLs.\n * @param {module:workbox-routing~handlerCallback} handler A callback\n * function that returns a Promise resulting in a Response.\n * @param {string} [method='GET'] The HTTP method to match the Route\n * against.\n */\n constructor(regExp, handler, method) {\n if (process.env.NODE_ENV !== 'production') {\n assert.isInstance(regExp, RegExp, {\n moduleName: 'workbox-routing',\n className: 'RegExpRoute',\n funcName: 'constructor',\n paramName: 'pattern',\n });\n }\n const match = ({ url }) => {\n const result = regExp.exec(url.href);\n // Return immediately if there's no match.\n if (!result) {\n return;\n }\n // Require that the match start at the first character in the URL string\n // if it's a cross-origin request.\n // See https://github.com/GoogleChrome/workbox/issues/281 for the context\n // behind this behavior.\n if ((url.origin !== location.origin) && (result.index !== 0)) {\n if (process.env.NODE_ENV !== 'production') {\n logger.debug(`The regular expression '${regExp}' only partially matched ` +\n `against the cross-origin URL '${url}'. RegExpRoute's will only ` +\n `handle cross-origin requests if they match the entire URL.`);\n }\n return;\n }\n // If the route matches, but there aren't any capture groups defined, then\n // this will return [], which is truthy and therefore sufficient to\n // indicate a match.\n // If there are capture groups, then it will return their values.\n return result.slice(1);\n };\n super(match, handler, method);\n }\n}\nexport { RegExpRoute };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\nconst getFriendlyURL = (url) => {\n const urlObj = new URL(String(url), location.href);\n // See https://github.com/GoogleChrome/workbox/issues/2323\n // We want to include everything, except for the origin if it's same-origin.\n return urlObj.href.replace(new RegExp(`^${location.origin}`), '');\n};\nexport { getFriendlyURL };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { assert } from 'workbox-core/_private/assert.js';\nimport { logger } from 'workbox-core/_private/logger.js';\nimport { WorkboxError } from 'workbox-core/_private/WorkboxError.js';\nimport { getFriendlyURL } from 'workbox-core/_private/getFriendlyURL.js';\nimport { normalizeHandler } from './utils/normalizeHandler.js';\nimport './_version.js';\n/**\n * The Router can be used to process a FetchEvent through one or more\n * [Routes]{@link module:workbox-routing.Route} responding with a Request if\n * a matching route exists.\n *\n * If no route matches a given a request, the Router will use a \"default\"\n * handler if one is defined.\n *\n * Should the matching Route throw an error, the Router will use a \"catch\"\n * handler if one is defined to gracefully deal with issues and respond with a\n * Request.\n *\n * If a request matches multiple routes, the **earliest** registered route will\n * be used to respond to the request.\n *\n * @memberof module:workbox-routing\n */\nclass Router {\n /**\n * Initializes a new Router.\n */\n constructor() {\n this._routes = new Map();\n }\n /**\n * @return {Map>} routes A `Map` of HTTP\n * method name ('GET', etc.) to an array of all the corresponding `Route`\n * instances that are registered.\n */\n get routes() {\n return this._routes;\n }\n /**\n * Adds a fetch event listener to respond to events when a route matches\n * the event's request.\n */\n addFetchListener() {\n // See https://github.com/Microsoft/TypeScript/issues/28357#issuecomment-436484705\n self.addEventListener('fetch', ((event) => {\n const { request } = event;\n const responsePromise = this.handleRequest({ request, event });\n if (responsePromise) {\n event.respondWith(responsePromise);\n }\n }));\n }\n /**\n * Adds a message event listener for URLs to cache from the window.\n * This is useful to cache resources loaded on the page prior to when the\n * service worker started controlling it.\n *\n * The format of the message data sent from the window should be as follows.\n * Where the `urlsToCache` array may consist of URL strings or an array of\n * URL string + `requestInit` object (the same as you'd pass to `fetch()`).\n *\n * ```\n * {\n * type: 'CACHE_URLS',\n * payload: {\n * urlsToCache: [\n * './script1.js',\n * './script2.js',\n * ['./script3.js', {mode: 'no-cors'}],\n * ],\n * },\n * }\n * ```\n */\n addCacheListener() {\n // See https://github.com/Microsoft/TypeScript/issues/28357#issuecomment-436484705\n self.addEventListener('message', ((event) => {\n if (event.data && event.data.type === 'CACHE_URLS') {\n const { payload } = event.data;\n if (process.env.NODE_ENV !== 'production') {\n logger.debug(`Caching URLs from the window`, payload.urlsToCache);\n }\n const requestPromises = Promise.all(payload.urlsToCache.map((entry) => {\n if (typeof entry === 'string') {\n entry = [entry];\n }\n const request = new Request(...entry);\n return this.handleRequest({ request });\n // TODO(philipwalton): TypeScript errors without this typecast for\n // some reason (probably a bug). The real type here should work but\n // doesn't: `Array | undefined>`.\n })); // TypeScript\n event.waitUntil(requestPromises);\n // If a MessageChannel was used, reply to the message on success.\n if (event.ports && event.ports[0]) {\n requestPromises.then(() => event.ports[0].postMessage(true));\n }\n }\n }));\n }\n /**\n * Apply the routing rules to a FetchEvent object to get a Response from an\n * appropriate Route's handler.\n *\n * @param {Object} options\n * @param {Request} options.request The request to handle (this is usually\n * from a fetch event, but it does not have to be).\n * @param {FetchEvent} [options.event] The event that triggered the request,\n * if applicable.\n * @return {Promise|undefined} A promise is returned if a\n * registered route can handle the request. If there is no matching\n * route and there's no `defaultHandler`, `undefined` is returned.\n */\n handleRequest({ request, event }) {\n if (process.env.NODE_ENV !== 'production') {\n assert.isInstance(request, Request, {\n moduleName: 'workbox-routing',\n className: 'Router',\n funcName: 'handleRequest',\n paramName: 'options.request',\n });\n }\n const url = new URL(request.url, location.href);\n if (!url.protocol.startsWith('http')) {\n if (process.env.NODE_ENV !== 'production') {\n logger.debug(`Workbox Router only supports URLs that start with 'http'.`);\n }\n return;\n }\n const { params, route } = this.findMatchingRoute({ url, request, event });\n let handler = route && route.handler;\n const debugMessages = [];\n if (process.env.NODE_ENV !== 'production') {\n if (handler) {\n debugMessages.push([\n `Found a route to handle this request:`, route,\n ]);\n if (params) {\n debugMessages.push([\n `Passing the following params to the route's handler:`, params,\n ]);\n }\n }\n }\n // If we don't have a handler because there was no matching route, then\n // fall back to defaultHandler if that's defined.\n if (!handler && this._defaultHandler) {\n if (process.env.NODE_ENV !== 'production') {\n debugMessages.push(`Failed to find a matching route. Falling ` +\n `back to the default handler.`);\n }\n handler = this._defaultHandler;\n }\n if (!handler) {\n if (process.env.NODE_ENV !== 'production') {\n // No handler so Workbox will do nothing. If logs is set of debug\n // i.e. verbose, we should print out this information.\n logger.debug(`No route found for: ${getFriendlyURL(url)}`);\n }\n return;\n }\n if (process.env.NODE_ENV !== 'production') {\n // We have a handler, meaning Workbox is going to handle the route.\n // print the routing details to the console.\n logger.groupCollapsed(`Router is responding to: ${getFriendlyURL(url)}`);\n debugMessages.forEach((msg) => {\n if (Array.isArray(msg)) {\n logger.log(...msg);\n }\n else {\n logger.log(msg);\n }\n });\n logger.groupEnd();\n }\n // Wrap in try and catch in case the handle method throws a synchronous\n // error. It should still callback to the catch handler.\n let responsePromise;\n try {\n responsePromise = handler.handle({ url, request, event, params });\n }\n catch (err) {\n responsePromise = Promise.reject(err);\n }\n if (responsePromise instanceof Promise && this._catchHandler) {\n responsePromise = responsePromise.catch((err) => {\n if (process.env.NODE_ENV !== 'production') {\n // Still include URL here as it will be async from the console group\n // and may not make sense without the URL\n logger.groupCollapsed(`Error thrown when responding to: ` +\n ` ${getFriendlyURL(url)}. Falling back to Catch Handler.`);\n logger.error(`Error thrown by:`, route);\n logger.error(err);\n logger.groupEnd();\n }\n return this._catchHandler.handle({ url, request, event });\n });\n }\n return responsePromise;\n }\n /**\n * Checks a request and URL (and optionally an event) against the list of\n * registered routes, and if there's a match, returns the corresponding\n * route along with any params generated by the match.\n *\n * @param {Object} options\n * @param {URL} options.url\n * @param {Request} options.request The request to match.\n * @param {Event} [options.event] The corresponding event (unless N/A).\n * @return {Object} An object with `route` and `params` properties.\n * They are populated if a matching route was found or `undefined`\n * otherwise.\n */\n findMatchingRoute({ url, request, event }) {\n if (process.env.NODE_ENV !== 'production') {\n assert.isInstance(url, URL, {\n moduleName: 'workbox-routing',\n className: 'Router',\n funcName: 'findMatchingRoute',\n paramName: 'options.url',\n });\n assert.isInstance(request, Request, {\n moduleName: 'workbox-routing',\n className: 'Router',\n funcName: 'findMatchingRoute',\n paramName: 'options.request',\n });\n }\n const routes = this._routes.get(request.method) || [];\n for (const route of routes) {\n let params;\n const matchResult = route.match({ url, request, event });\n if (matchResult) {\n // See https://github.com/GoogleChrome/workbox/issues/2079\n params = matchResult;\n if (Array.isArray(matchResult) && matchResult.length === 0) {\n // Instead of passing an empty array in as params, use undefined.\n params = undefined;\n }\n else if ((matchResult.constructor === Object &&\n Object.keys(matchResult).length === 0)) {\n // Instead of passing an empty object in as params, use undefined.\n params = undefined;\n }\n else if (typeof matchResult === 'boolean') {\n // For the boolean value true (rather than just something truth-y),\n // don't set params.\n // See https://github.com/GoogleChrome/workbox/pull/2134#issuecomment-513924353\n params = undefined;\n }\n // Return early if have a match.\n return { route, params };\n }\n }\n // If no match was found above, return and empty object.\n return {};\n }\n /**\n * Define a default `handler` that's called when no routes explicitly\n * match the incoming request.\n *\n * Without a default handler, unmatched requests will go against the\n * network as if there were no service worker present.\n *\n * @param {module:workbox-routing~handlerCallback} handler A callback\n * function that returns a Promise resulting in a Response.\n */\n setDefaultHandler(handler) {\n this._defaultHandler = normalizeHandler(handler);\n }\n /**\n * If a Route throws an error while handling a request, this `handler`\n * will be called and given a chance to provide a response.\n *\n * @param {module:workbox-routing~handlerCallback} handler A callback\n * function that returns a Promise resulting in a Response.\n */\n setCatchHandler(handler) {\n this._catchHandler = normalizeHandler(handler);\n }\n /**\n * Registers a route with the router.\n *\n * @param {module:workbox-routing.Route} route The route to register.\n */\n registerRoute(route) {\n if (process.env.NODE_ENV !== 'production') {\n assert.isType(route, 'object', {\n moduleName: 'workbox-routing',\n className: 'Router',\n funcName: 'registerRoute',\n paramName: 'route',\n });\n assert.hasMethod(route, 'match', {\n moduleName: 'workbox-routing',\n className: 'Router',\n funcName: 'registerRoute',\n paramName: 'route',\n });\n assert.isType(route.handler, 'object', {\n moduleName: 'workbox-routing',\n className: 'Router',\n funcName: 'registerRoute',\n paramName: 'route',\n });\n assert.hasMethod(route.handler, 'handle', {\n moduleName: 'workbox-routing',\n className: 'Router',\n funcName: 'registerRoute',\n paramName: 'route.handler',\n });\n assert.isType(route.method, 'string', {\n moduleName: 'workbox-routing',\n className: 'Router',\n funcName: 'registerRoute',\n paramName: 'route.method',\n });\n }\n if (!this._routes.has(route.method)) {\n this._routes.set(route.method, []);\n }\n // Give precedence to all of the earlier routes by adding this additional\n // route to the end of the array.\n this._routes.get(route.method).push(route);\n }\n /**\n * Unregisters a route with the router.\n *\n * @param {module:workbox-routing.Route} route The route to unregister.\n */\n unregisterRoute(route) {\n if (!this._routes.has(route.method)) {\n throw new WorkboxError('unregister-route-but-not-found-with-method', {\n method: route.method,\n });\n }\n const routeIndex = this._routes.get(route.method).indexOf(route);\n if (routeIndex > -1) {\n this._routes.get(route.method).splice(routeIndex, 1);\n }\n else {\n throw new WorkboxError('unregister-route-route-not-registered');\n }\n }\n}\nexport { Router };\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\nconst plugins = [];\nexport const precachePlugins = {\n /*\n * @return {Array}\n * @private\n */\n get() {\n return plugins;\n },\n /*\n * @param {Array} newPlugins\n * @private\n */\n add(newPlugins) {\n plugins.push(...newPlugins);\n },\n};\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\nconst _cacheNameDetails = {\n googleAnalytics: 'googleAnalytics',\n precache: 'precache-v2',\n prefix: 'workbox',\n runtime: 'runtime',\n suffix: typeof registration !== 'undefined' ? registration.scope : '',\n};\nconst _createCacheName = (cacheName) => {\n return [_cacheNameDetails.prefix, cacheName, _cacheNameDetails.suffix]\n .filter((value) => value && value.length > 0)\n .join('-');\n};\nconst eachCacheNameDetail = (fn) => {\n for (const key of Object.keys(_cacheNameDetails)) {\n fn(key);\n }\n};\nexport const cacheNames = {\n updateDetails: (details) => {\n eachCacheNameDetail((key) => {\n if (typeof details[key] === 'string') {\n _cacheNameDetails[key] = details[key];\n }\n });\n },\n getGoogleAnalyticsName: (userCacheName) => {\n return userCacheName || _createCacheName(_cacheNameDetails.googleAnalytics);\n },\n getPrecacheName: (userCacheName) => {\n return userCacheName || _createCacheName(_cacheNameDetails.precache);\n },\n getPrefix: () => {\n return _cacheNameDetails.prefix;\n },\n getRuntimeName: (userCacheName) => {\n return userCacheName || _createCacheName(_cacheNameDetails.runtime);\n },\n getSuffix: () => {\n return _cacheNameDetails.suffix;\n },\n};\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\n// Callbacks to be executed whenever there's a quota error.\nconst quotaErrorCallbacks = new Set();\nexport { quotaErrorCallbacks };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\nexport const pluginUtils = {\n filter: (plugins, callbackName) => {\n return plugins.filter((plugin) => callbackName in plugin);\n },\n};\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { assert } from './assert.js';\nimport { executeQuotaErrorCallbacks } from './executeQuotaErrorCallbacks.js';\nimport { getFriendlyURL } from './getFriendlyURL.js';\nimport { logger } from './logger.js';\nimport { pluginUtils } from '../utils/pluginUtils.js';\nimport { WorkboxError } from './WorkboxError.js';\nimport '../_version.js';\n/**\n * Checks the list of plugins for the cacheKeyWillBeUsed callback, and\n * executes any of those callbacks found in sequence. The final `Request` object\n * returned by the last plugin is treated as the cache key for cache reads\n * and/or writes.\n *\n * @param {Object} options\n * @param {Request} options.request\n * @param {string} options.mode\n * @param {Array} [options.plugins=[]]\n * @return {Promise}\n *\n * @private\n * @memberof module:workbox-core\n */\nconst _getEffectiveRequest = async ({ request, mode, plugins = [], }) => {\n const cacheKeyWillBeUsedPlugins = pluginUtils.filter(plugins, \"cacheKeyWillBeUsed\" /* CACHE_KEY_WILL_BE_USED */);\n let effectiveRequest = request;\n for (const plugin of cacheKeyWillBeUsedPlugins) {\n effectiveRequest = await plugin[\"cacheKeyWillBeUsed\" /* CACHE_KEY_WILL_BE_USED */].call(plugin, { mode, request: effectiveRequest });\n if (typeof effectiveRequest === 'string') {\n effectiveRequest = new Request(effectiveRequest);\n }\n if (process.env.NODE_ENV !== 'production') {\n assert.isInstance(effectiveRequest, Request, {\n moduleName: 'Plugin',\n funcName: \"cacheKeyWillBeUsed\" /* CACHE_KEY_WILL_BE_USED */,\n isReturnValueProblem: true,\n });\n }\n }\n return effectiveRequest;\n};\n/**\n * This method will call cacheWillUpdate on the available plugins (or use\n * status === 200) to determine if the Response is safe and valid to cache.\n *\n * @param {Object} options\n * @param {Request} options.request\n * @param {Response} options.response\n * @param {Event} [options.event]\n * @param {Array} [options.plugins=[]]\n * @return {Promise}\n *\n * @private\n * @memberof module:workbox-core\n */\nconst _isResponseSafeToCache = async ({ request, response, event, plugins = [], }) => {\n let responseToCache = response;\n let pluginsUsed = false;\n for (const plugin of plugins) {\n if (\"cacheWillUpdate\" /* CACHE_WILL_UPDATE */ in plugin) {\n pluginsUsed = true;\n const pluginMethod = plugin[\"cacheWillUpdate\" /* CACHE_WILL_UPDATE */];\n responseToCache = await pluginMethod.call(plugin, {\n request,\n response: responseToCache,\n event,\n });\n if (process.env.NODE_ENV !== 'production') {\n if (responseToCache) {\n assert.isInstance(responseToCache, Response, {\n moduleName: 'Plugin',\n funcName: \"cacheWillUpdate\" /* CACHE_WILL_UPDATE */,\n isReturnValueProblem: true,\n });\n }\n }\n if (!responseToCache) {\n break;\n }\n }\n }\n if (!pluginsUsed) {\n if (process.env.NODE_ENV !== 'production') {\n if (responseToCache) {\n if (responseToCache.status !== 200) {\n if (responseToCache.status === 0) {\n logger.warn(`The response for '${request.url}' is an opaque ` +\n `response. The caching strategy that you're using will not ` +\n `cache opaque responses by default.`);\n }\n else {\n logger.debug(`The response for '${request.url}' returned ` +\n `a status code of '${response.status}' and won't be cached as a ` +\n `result.`);\n }\n }\n }\n }\n responseToCache = responseToCache && responseToCache.status === 200 ?\n responseToCache : undefined;\n }\n return responseToCache ? responseToCache : null;\n};\n/**\n * This is a wrapper around cache.match().\n *\n * @param {Object} options\n * @param {string} options.cacheName Name of the cache to match against.\n * @param {Request} options.request The Request that will be used to look up\n * cache entries.\n * @param {Event} [options.event] The event that prompted the action.\n * @param {Object} [options.matchOptions] Options passed to cache.match().\n * @param {Array} [options.plugins=[]] Array of plugins.\n * @return {Response} A cached response if available.\n *\n * @private\n * @memberof module:workbox-core\n */\nconst matchWrapper = async ({ cacheName, request, event, matchOptions, plugins = [], }) => {\n const cache = await self.caches.open(cacheName);\n const effectiveRequest = await _getEffectiveRequest({\n plugins, request, mode: 'read'\n });\n let cachedResponse = await cache.match(effectiveRequest, matchOptions);\n if (process.env.NODE_ENV !== 'production') {\n if (cachedResponse) {\n logger.debug(`Found a cached response in '${cacheName}'.`);\n }\n else {\n logger.debug(`No cached response found in '${cacheName}'.`);\n }\n }\n for (const plugin of plugins) {\n if (\"cachedResponseWillBeUsed\" /* CACHED_RESPONSE_WILL_BE_USED */ in plugin) {\n const pluginMethod = plugin[\"cachedResponseWillBeUsed\" /* CACHED_RESPONSE_WILL_BE_USED */];\n cachedResponse = await pluginMethod.call(plugin, {\n cacheName,\n event,\n matchOptions,\n cachedResponse,\n request: effectiveRequest,\n });\n if (process.env.NODE_ENV !== 'production') {\n if (cachedResponse) {\n assert.isInstance(cachedResponse, Response, {\n moduleName: 'Plugin',\n funcName: \"cachedResponseWillBeUsed\" /* CACHED_RESPONSE_WILL_BE_USED */,\n isReturnValueProblem: true,\n });\n }\n }\n }\n }\n return cachedResponse;\n};\n/**\n * Wrapper around cache.put().\n *\n * Will call `cacheDidUpdate` on plugins if the cache was updated, using\n * `matchOptions` when determining what the old entry is.\n *\n * @param {Object} options\n * @param {string} options.cacheName\n * @param {Request} options.request\n * @param {Response} options.response\n * @param {Event} [options.event]\n * @param {Array} [options.plugins=[]]\n * @param {Object} [options.matchOptions]\n *\n * @private\n * @memberof module:workbox-core\n */\nconst putWrapper = async ({ cacheName, request, response, event, plugins = [], matchOptions, }) => {\n if (process.env.NODE_ENV !== 'production') {\n if (request.method && request.method !== 'GET') {\n throw new WorkboxError('attempt-to-cache-non-get-request', {\n url: getFriendlyURL(request.url),\n method: request.method,\n });\n }\n }\n const effectiveRequest = await _getEffectiveRequest({\n plugins, request, mode: 'write'\n });\n if (!response) {\n if (process.env.NODE_ENV !== 'production') {\n logger.error(`Cannot cache non-existent response for ` +\n `'${getFriendlyURL(effectiveRequest.url)}'.`);\n }\n throw new WorkboxError('cache-put-with-no-response', {\n url: getFriendlyURL(effectiveRequest.url),\n });\n }\n const responseToCache = await _isResponseSafeToCache({\n event,\n plugins,\n response,\n request: effectiveRequest,\n });\n if (!responseToCache) {\n if (process.env.NODE_ENV !== 'production') {\n logger.debug(`Response '${getFriendlyURL(effectiveRequest.url)}' will ` +\n `not be cached.`, responseToCache);\n }\n return;\n }\n const cache = await self.caches.open(cacheName);\n const updatePlugins = pluginUtils.filter(plugins, \"cacheDidUpdate\" /* CACHE_DID_UPDATE */);\n const oldResponse = updatePlugins.length > 0 ?\n await matchWrapper({ cacheName, matchOptions, request: effectiveRequest }) :\n null;\n if (process.env.NODE_ENV !== 'production') {\n logger.debug(`Updating the '${cacheName}' cache with a new Response for ` +\n `${getFriendlyURL(effectiveRequest.url)}.`);\n }\n try {\n await cache.put(effectiveRequest, responseToCache);\n }\n catch (error) {\n // See https://developer.mozilla.org/en-US/docs/Web/API/DOMException#exception-QuotaExceededError\n if (error.name === 'QuotaExceededError') {\n await executeQuotaErrorCallbacks();\n }\n throw error;\n }\n for (const plugin of updatePlugins) {\n await plugin[\"cacheDidUpdate\" /* CACHE_DID_UPDATE */].call(plugin, {\n cacheName,\n event,\n oldResponse,\n newResponse: responseToCache,\n request: effectiveRequest,\n });\n }\n};\nexport const cacheWrapper = {\n put: putWrapper,\n match: matchWrapper,\n};\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { logger } from '../_private/logger.js';\nimport { quotaErrorCallbacks } from '../models/quotaErrorCallbacks.js';\nimport '../_version.js';\n/**\n * Runs all of the callback functions, one at a time sequentially, in the order\n * in which they were registered.\n *\n * @memberof module:workbox-core\n * @private\n */\nasync function executeQuotaErrorCallbacks() {\n if (process.env.NODE_ENV !== 'production') {\n logger.log(`About to run ${quotaErrorCallbacks.size} ` +\n `callbacks to clean up caches.`);\n }\n for (const callback of quotaErrorCallbacks) {\n await callback();\n if (process.env.NODE_ENV !== 'production') {\n logger.log(callback, 'is complete.');\n }\n }\n if (process.env.NODE_ENV !== 'production') {\n logger.log('Finished running callbacks.');\n }\n}\nexport { executeQuotaErrorCallbacks };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { WorkboxError } from './WorkboxError.js';\nimport { logger } from './logger.js';\nimport { assert } from './assert.js';\nimport { getFriendlyURL } from '../_private/getFriendlyURL.js';\nimport { pluginUtils } from '../utils/pluginUtils.js';\nimport '../_version.js';\n/**\n * Wrapper around the fetch API.\n *\n * Will call requestWillFetch on available plugins.\n *\n * @param {Object} options\n * @param {Request|string} options.request\n * @param {Object} [options.fetchOptions]\n * @param {ExtendableEvent} [options.event]\n * @param {Array} [options.plugins=[]]\n * @return {Promise}\n *\n * @private\n * @memberof module:workbox-core\n */\nconst wrappedFetch = async ({ request, fetchOptions, event, plugins = [], }) => {\n if (typeof request === 'string') {\n request = new Request(request);\n }\n // We *should* be able to call `await event.preloadResponse` even if it's\n // undefined, but for some reason, doing so leads to errors in our Node unit\n // tests. To work around that, explicitly check preloadResponse's value first.\n if (event instanceof FetchEvent && event.preloadResponse) {\n const possiblePreloadResponse = await event.preloadResponse;\n if (possiblePreloadResponse) {\n if (process.env.NODE_ENV !== 'production') {\n logger.log(`Using a preloaded navigation response for ` +\n `'${getFriendlyURL(request.url)}'`);\n }\n return possiblePreloadResponse;\n }\n }\n if (process.env.NODE_ENV !== 'production') {\n assert.isInstance(request, Request, {\n paramName: 'request',\n expectedClass: Request,\n moduleName: 'workbox-core',\n className: 'fetchWrapper',\n funcName: 'wrappedFetch',\n });\n }\n const failedFetchPlugins = pluginUtils.filter(plugins, \"fetchDidFail\" /* FETCH_DID_FAIL */);\n // If there is a fetchDidFail plugin, we need to save a clone of the\n // original request before it's either modified by a requestWillFetch\n // plugin or before the original request's body is consumed via fetch().\n const originalRequest = failedFetchPlugins.length > 0 ?\n request.clone() : null;\n try {\n for (const plugin of plugins) {\n if (\"requestWillFetch\" /* REQUEST_WILL_FETCH */ in plugin) {\n const pluginMethod = plugin[\"requestWillFetch\" /* REQUEST_WILL_FETCH */];\n const requestClone = request.clone();\n request = await pluginMethod.call(plugin, {\n request: requestClone,\n event,\n });\n if (process.env.NODE_ENV !== 'production') {\n if (request) {\n assert.isInstance(request, Request, {\n moduleName: 'Plugin',\n funcName: \"cachedResponseWillBeUsed\" /* CACHED_RESPONSE_WILL_BE_USED */,\n isReturnValueProblem: true,\n });\n }\n }\n }\n }\n }\n catch (err) {\n throw new WorkboxError('plugin-error-request-will-fetch', {\n thrownError: err,\n });\n }\n // The request can be altered by plugins with `requestWillFetch` making\n // the original request (Most likely from a `fetch` event) to be different\n // to the Request we make. Pass both to `fetchDidFail` to aid debugging.\n const pluginFilteredRequest = request.clone();\n try {\n let fetchResponse;\n // See https://github.com/GoogleChrome/workbox/issues/1796\n if (request.mode === 'navigate') {\n fetchResponse = await fetch(request);\n }\n else {\n fetchResponse = await fetch(request, fetchOptions);\n }\n if (process.env.NODE_ENV !== 'production') {\n logger.debug(`Network request for ` +\n `'${getFriendlyURL(request.url)}' returned a response with ` +\n `status '${fetchResponse.status}'.`);\n }\n for (const plugin of plugins) {\n if (\"fetchDidSucceed\" /* FETCH_DID_SUCCEED */ in plugin) {\n fetchResponse = await plugin[\"fetchDidSucceed\" /* FETCH_DID_SUCCEED */]\n .call(plugin, {\n event,\n request: pluginFilteredRequest,\n response: fetchResponse,\n });\n if (process.env.NODE_ENV !== 'production') {\n if (fetchResponse) {\n assert.isInstance(fetchResponse, Response, {\n moduleName: 'Plugin',\n funcName: \"fetchDidSucceed\" /* FETCH_DID_SUCCEED */,\n isReturnValueProblem: true,\n });\n }\n }\n }\n }\n return fetchResponse;\n }\n catch (error) {\n if (process.env.NODE_ENV !== 'production') {\n logger.error(`Network request for ` +\n `'${getFriendlyURL(request.url)}' threw an error.`, error);\n }\n for (const plugin of failedFetchPlugins) {\n await plugin[\"fetchDidFail\" /* FETCH_DID_FAIL */].call(plugin, {\n error,\n event,\n originalRequest: originalRequest.clone(),\n request: pluginFilteredRequest.clone(),\n });\n }\n throw error;\n }\n};\nconst fetchWrapper = {\n fetch: wrappedFetch,\n};\nexport { fetchWrapper };\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { assert } from 'workbox-core/_private/assert.js';\nimport { cacheNames } from 'workbox-core/_private/cacheNames.js';\nimport { cacheWrapper } from 'workbox-core/_private/cacheWrapper.js';\nimport { fetchWrapper } from 'workbox-core/_private/fetchWrapper.js';\nimport { logger } from 'workbox-core/_private/logger.js';\nimport { WorkboxError } from 'workbox-core/_private/WorkboxError.js';\nimport { copyResponse } from 'workbox-core/copyResponse.js';\nimport { createCacheKey } from './utils/createCacheKey.js';\nimport { printCleanupDetails } from './utils/printCleanupDetails.js';\nimport { printInstallDetails } from './utils/printInstallDetails.js';\nimport './_version.js';\n/**\n * Performs efficient precaching of assets.\n *\n * @memberof module:workbox-precaching\n */\nclass PrecacheController {\n /**\n * Create a new PrecacheController.\n *\n * @param {string} [cacheName] An optional name for the cache, to override\n * the default precache name.\n */\n constructor(cacheName) {\n this._cacheName = cacheNames.getPrecacheName(cacheName);\n this._urlsToCacheKeys = new Map();\n this._urlsToCacheModes = new Map();\n this._cacheKeysToIntegrities = new Map();\n }\n /**\n * This method will add items to the precache list, removing duplicates\n * and ensuring the information is valid.\n *\n * @param {\n * Array\n * } entries Array of entries to precache.\n */\n addToCacheList(entries) {\n if (process.env.NODE_ENV !== 'production') {\n assert.isArray(entries, {\n moduleName: 'workbox-precaching',\n className: 'PrecacheController',\n funcName: 'addToCacheList',\n paramName: 'entries',\n });\n }\n const urlsToWarnAbout = [];\n for (const entry of entries) {\n // See https://github.com/GoogleChrome/workbox/issues/2259\n if (typeof entry === 'string') {\n urlsToWarnAbout.push(entry);\n }\n else if (entry && entry.revision === undefined) {\n urlsToWarnAbout.push(entry.url);\n }\n const { cacheKey, url } = createCacheKey(entry);\n const cacheMode = (typeof entry !== 'string' && entry.revision) ?\n 'reload' : 'default';\n if (this._urlsToCacheKeys.has(url) &&\n this._urlsToCacheKeys.get(url) !== cacheKey) {\n throw new WorkboxError('add-to-cache-list-conflicting-entries', {\n firstEntry: this._urlsToCacheKeys.get(url),\n secondEntry: cacheKey,\n });\n }\n if (typeof entry !== 'string' && entry.integrity) {\n if (this._cacheKeysToIntegrities.has(cacheKey) &&\n this._cacheKeysToIntegrities.get(cacheKey) !== entry.integrity) {\n throw new WorkboxError('add-to-cache-list-conflicting-integrities', {\n url,\n });\n }\n this._cacheKeysToIntegrities.set(cacheKey, entry.integrity);\n }\n this._urlsToCacheKeys.set(url, cacheKey);\n this._urlsToCacheModes.set(url, cacheMode);\n if (urlsToWarnAbout.length > 0) {\n const warningMessage = `Workbox is precaching URLs without revision ` +\n `info: ${urlsToWarnAbout.join(', ')}\\nThis is generally NOT safe. ` +\n `Learn more at https://bit.ly/wb-precache`;\n if (process.env.NODE_ENV === 'production') {\n // Use console directly to display this warning without bloating\n // bundle sizes by pulling in all of the logger codebase in prod.\n console.warn(warningMessage);\n }\n else {\n logger.warn(warningMessage);\n }\n }\n }\n }\n /**\n * Precaches new and updated assets. Call this method from the service worker\n * install event.\n *\n * @param {Object} options\n * @param {Event} [options.event] The install event (if needed).\n * @param {Array} [options.plugins] Plugins to be used for fetching\n * and caching during install.\n * @return {Promise}\n */\n async install({ event, plugins } = {}) {\n if (process.env.NODE_ENV !== 'production') {\n if (plugins) {\n assert.isArray(plugins, {\n moduleName: 'workbox-precaching',\n className: 'PrecacheController',\n funcName: 'install',\n paramName: 'plugins',\n });\n }\n }\n const toBePrecached = [];\n const alreadyPrecached = [];\n const cache = await self.caches.open(this._cacheName);\n const alreadyCachedRequests = await cache.keys();\n const existingCacheKeys = new Set(alreadyCachedRequests.map((request) => request.url));\n for (const [url, cacheKey] of this._urlsToCacheKeys) {\n if (existingCacheKeys.has(cacheKey)) {\n alreadyPrecached.push(url);\n }\n else {\n toBePrecached.push({ cacheKey, url });\n }\n }\n const precacheRequests = toBePrecached.map(({ cacheKey, url }) => {\n const integrity = this._cacheKeysToIntegrities.get(cacheKey);\n const cacheMode = this._urlsToCacheModes.get(url);\n return this._addURLToCache({\n cacheKey,\n cacheMode,\n event,\n integrity,\n plugins,\n url,\n });\n });\n await Promise.all(precacheRequests);\n const updatedURLs = toBePrecached.map((item) => item.url);\n if (process.env.NODE_ENV !== 'production') {\n printInstallDetails(updatedURLs, alreadyPrecached);\n }\n return {\n updatedURLs,\n notUpdatedURLs: alreadyPrecached,\n };\n }\n /**\n * Deletes assets that are no longer present in the current precache manifest.\n * Call this method from the service worker activate event.\n *\n * @return {Promise}\n */\n async activate() {\n const cache = await self.caches.open(this._cacheName);\n const currentlyCachedRequests = await cache.keys();\n const expectedCacheKeys = new Set(this._urlsToCacheKeys.values());\n const deletedURLs = [];\n for (const request of currentlyCachedRequests) {\n if (!expectedCacheKeys.has(request.url)) {\n await cache.delete(request);\n deletedURLs.push(request.url);\n }\n }\n if (process.env.NODE_ENV !== 'production') {\n printCleanupDetails(deletedURLs);\n }\n return { deletedURLs };\n }\n /**\n * Requests the entry and saves it to the cache if the response is valid.\n * By default, any response with a status code of less than 400 (including\n * opaque responses) is considered valid.\n *\n * If you need to use custom criteria to determine what's valid and what\n * isn't, then pass in an item in `options.plugins` that implements the\n * `cacheWillUpdate()` lifecycle event.\n *\n * @private\n * @param {Object} options\n * @param {string} options.cacheKey The string to use a cache key.\n * @param {string} options.url The URL to fetch and cache.\n * @param {string} [options.cacheMode] The cache mode for the network request.\n * @param {Event} [options.event] The install event (if passed).\n * @param {Array} [options.plugins] An array of plugins to apply to\n * fetch and caching.\n * @param {string} [options.integrity] The value to use for the `integrity`\n * field when making the request.\n */\n async _addURLToCache({ cacheKey, url, cacheMode, event, plugins, integrity }) {\n const request = new Request(url, {\n integrity,\n cache: cacheMode,\n credentials: 'same-origin',\n });\n let response = await fetchWrapper.fetch({\n event,\n plugins,\n request,\n });\n // Allow developers to override the default logic about what is and isn't\n // valid by passing in a plugin implementing cacheWillUpdate(), e.g.\n // a `CacheableResponsePlugin` instance.\n let cacheWillUpdatePlugin;\n for (const plugin of (plugins || [])) {\n if ('cacheWillUpdate' in plugin) {\n cacheWillUpdatePlugin = plugin;\n }\n }\n const isValidResponse = cacheWillUpdatePlugin ?\n // Use a callback if provided. It returns a truthy value if valid.\n // NOTE: invoke the method on the plugin instance so the `this` context\n // is correct.\n await cacheWillUpdatePlugin.cacheWillUpdate({ event, request, response }) :\n // Otherwise, default to considering any response status under 400 valid.\n // This includes, by default, considering opaque responses valid.\n response.status < 400;\n // Consider this a failure, leading to the `install` handler failing, if\n // we get back an invalid response.\n if (!isValidResponse) {\n throw new WorkboxError('bad-precaching-response', {\n url,\n status: response.status,\n });\n }\n // Redirected responses cannot be used to satisfy a navigation request, so\n // any redirected response must be \"copied\" rather than cloned, so the new\n // response doesn't contain the `redirected` flag. See:\n // https://bugs.chromium.org/p/chromium/issues/detail?id=669363&desc=2#c1\n if (response.redirected) {\n response = await copyResponse(response);\n }\n await cacheWrapper.put({\n event,\n plugins,\n response,\n // `request` already uses `url`. We may be able to reuse it.\n request: cacheKey === url ? request : new Request(cacheKey),\n cacheName: this._cacheName,\n matchOptions: {\n ignoreSearch: true,\n },\n });\n }\n /**\n * Returns a mapping of a precached URL to the corresponding cache key, taking\n * into account the revision information for the URL.\n *\n * @return {Map} A URL to cache key mapping.\n */\n getURLsToCacheKeys() {\n return this._urlsToCacheKeys;\n }\n /**\n * Returns a list of all the URLs that have been precached by the current\n * service worker.\n *\n * @return {Array} The precached URLs.\n */\n getCachedURLs() {\n return [...this._urlsToCacheKeys.keys()];\n }\n /**\n * Returns the cache key used for storing a given URL. If that URL is\n * unversioned, like `/index.html', then the cache key will be the original\n * URL with a search parameter appended to it.\n *\n * @param {string} url A URL whose cache key you want to look up.\n * @return {string} The versioned URL that corresponds to a cache key\n * for the original URL, or undefined if that URL isn't precached.\n */\n getCacheKeyForURL(url) {\n const urlObject = new URL(url, location.href);\n return this._urlsToCacheKeys.get(urlObject.href);\n }\n /**\n * This acts as a drop-in replacement for [`cache.match()`](https://developer.mozilla.org/en-US/docs/Web/API/Cache/match)\n * with the following differences:\n *\n * - It knows what the name of the precache is, and only checks in that cache.\n * - It allows you to pass in an \"original\" URL without versioning parameters,\n * and it will automatically look up the correct cache key for the currently\n * active revision of that URL.\n *\n * E.g., `matchPrecache('index.html')` will find the correct precached\n * response for the currently active service worker, even if the actual cache\n * key is `'/index.html?__WB_REVISION__=1234abcd'`.\n *\n * @param {string|Request} request The key (without revisioning parameters)\n * to look up in the precache.\n * @return {Promise}\n */\n async matchPrecache(request) {\n const url = request instanceof Request ? request.url : request;\n const cacheKey = this.getCacheKeyForURL(url);\n if (cacheKey) {\n const cache = await self.caches.open(this._cacheName);\n return cache.match(cacheKey);\n }\n return undefined;\n }\n /**\n * Returns a function that can be used within a\n * {@link module:workbox-routing.Route} that will find a response for the\n * incoming request against the precache.\n *\n * If for an unexpected reason there is a cache miss for the request,\n * this will fall back to retrieving the `Response` via `fetch()` when\n * `fallbackToNetwork` is `true`.\n *\n * @param {boolean} [fallbackToNetwork=true] Whether to attempt to get the\n * response from the network if there's a precache miss.\n * @return {module:workbox-routing~handlerCallback}\n */\n createHandler(fallbackToNetwork = true) {\n return async ({ request }) => {\n try {\n const response = await this.matchPrecache(request);\n if (response) {\n return response;\n }\n // This shouldn't normally happen, but there are edge cases:\n // https://github.com/GoogleChrome/workbox/issues/1441\n throw new WorkboxError('missing-precache-entry', {\n cacheName: this._cacheName,\n url: request instanceof Request ? request.url : request,\n });\n }\n catch (error) {\n if (fallbackToNetwork) {\n if (process.env.NODE_ENV !== 'production') {\n logger.debug(`Unable to respond with precached response. ` +\n `Falling back to network.`, error);\n }\n return fetch(request);\n }\n throw error;\n }\n };\n }\n /**\n * Returns a function that looks up `url` in the precache (taking into\n * account revision information), and returns the corresponding `Response`.\n *\n * If for an unexpected reason there is a cache miss when looking up `url`,\n * this will fall back to retrieving the `Response` via `fetch()` when\n * `fallbackToNetwork` is `true`.\n *\n * @param {string} url The precached URL which will be used to lookup the\n * `Response`.\n * @param {boolean} [fallbackToNetwork=true] Whether to attempt to get the\n * response from the network if there's a precache miss.\n * @return {module:workbox-routing~handlerCallback}\n */\n createHandlerBoundToURL(url, fallbackToNetwork = true) {\n const cacheKey = this.getCacheKeyForURL(url);\n if (!cacheKey) {\n throw new WorkboxError('non-precached-url', { url });\n }\n const handler = this.createHandler(fallbackToNetwork);\n const request = new Request(url);\n return () => handler({ request });\n }\n}\nexport { PrecacheController };\n","export const isNav = event => event.request.mode === 'navigate';\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\nexport const cacheOkAndOpaquePlugin = {\n /**\n * Returns a valid response (to allow caching) if the status is 200 (OK) or\n * 0 (opaque).\n *\n * @param {Object} options\n * @param {Response} options.response\n * @return {Response|null}\n *\n * @private\n */\n cacheWillUpdate: async ({ response }) => {\n if (response.status === 200 || response.status === 0) {\n return response;\n }\n return null;\n },\n};\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { assert } from 'workbox-core/_private/assert.js';\nimport { cacheNames } from 'workbox-core/_private/cacheNames.js';\nimport { cacheWrapper } from 'workbox-core/_private/cacheWrapper.js';\nimport { fetchWrapper } from 'workbox-core/_private/fetchWrapper.js';\nimport { getFriendlyURL } from 'workbox-core/_private/getFriendlyURL.js';\nimport { logger } from 'workbox-core/_private/logger.js';\nimport { WorkboxError } from 'workbox-core/_private/WorkboxError.js';\nimport { messages } from './utils/messages.js';\nimport { cacheOkAndOpaquePlugin } from './plugins/cacheOkAndOpaquePlugin.js';\nimport './_version.js';\n/**\n * An implementation of a\n * [network first]{@link https://developers.google.com/web/fundamentals/instant-and-offline/offline-cookbook/#network-falling-back-to-cache}\n * request strategy.\n *\n * By default, this strategy will cache responses with a 200 status code as\n * well as [opaque responses]{@link https://developers.google.com/web/tools/workbox/guides/handle-third-party-requests}.\n * Opaque responses are are cross-origin requests where the response doesn't\n * support [CORS]{@link https://enable-cors.org/}.\n *\n * If the network request fails, and there is no cache match, this will throw\n * a `WorkboxError` exception.\n *\n * @memberof module:workbox-strategies\n */\nclass NetworkFirst {\n /**\n * @param {Object} options\n * @param {string} options.cacheName Cache name to store and retrieve\n * requests. Defaults to cache names provided by\n * [workbox-core]{@link module:workbox-core.cacheNames}.\n * @param {Array} options.plugins [Plugins]{@link https://developers.google.com/web/tools/workbox/guides/using-plugins}\n * to use in conjunction with this caching strategy.\n * @param {Object} options.fetchOptions Values passed along to the\n * [`init`](https://developer.mozilla.org/en-US/docs/Web/API/WindowOrWorkerGlobalScope/fetch#Parameters)\n * of all fetch() requests made by this strategy.\n * @param {Object} options.matchOptions [`CacheQueryOptions`](https://w3c.github.io/ServiceWorker/#dictdef-cachequeryoptions)\n * @param {number} options.networkTimeoutSeconds If set, any network requests\n * that fail to respond within the timeout will fallback to the cache.\n *\n * This option can be used to combat\n * \"[lie-fi]{@link https://developers.google.com/web/fundamentals/performance/poor-connectivity/#lie-fi}\"\n * scenarios.\n */\n constructor(options = {}) {\n this._cacheName = cacheNames.getRuntimeName(options.cacheName);\n if (options.plugins) {\n const isUsingCacheWillUpdate = options.plugins.some((plugin) => !!plugin.cacheWillUpdate);\n this._plugins = isUsingCacheWillUpdate ?\n options.plugins : [cacheOkAndOpaquePlugin, ...options.plugins];\n }\n else {\n // No plugins passed in, use the default plugin.\n this._plugins = [cacheOkAndOpaquePlugin];\n }\n this._networkTimeoutSeconds = options.networkTimeoutSeconds || 0;\n if (process.env.NODE_ENV !== 'production') {\n if (this._networkTimeoutSeconds) {\n assert.isType(this._networkTimeoutSeconds, 'number', {\n moduleName: 'workbox-strategies',\n className: 'NetworkFirst',\n funcName: 'constructor',\n paramName: 'networkTimeoutSeconds',\n });\n }\n }\n this._fetchOptions = options.fetchOptions;\n this._matchOptions = options.matchOptions;\n }\n /**\n * This method will perform a request strategy and follows an API that\n * will work with the\n * [Workbox Router]{@link module:workbox-routing.Router}.\n *\n * @param {Object} options\n * @param {Request|string} options.request A request to run this strategy for.\n * @param {Event} [options.event] The event that triggered the request.\n * @return {Promise}\n */\n async handle({ event, request }) {\n const logs = [];\n if (typeof request === 'string') {\n request = new Request(request);\n }\n if (process.env.NODE_ENV !== 'production') {\n assert.isInstance(request, Request, {\n moduleName: 'workbox-strategies',\n className: 'NetworkFirst',\n funcName: 'handle',\n paramName: 'makeRequest',\n });\n }\n const promises = [];\n let timeoutId;\n if (this._networkTimeoutSeconds) {\n const { id, promise } = this._getTimeoutPromise({ request, event, logs });\n timeoutId = id;\n promises.push(promise);\n }\n const networkPromise = this._getNetworkPromise({ timeoutId, request, event, logs });\n promises.push(networkPromise);\n // Promise.race() will resolve as soon as the first promise resolves.\n let response = await Promise.race(promises);\n // If Promise.race() resolved with null, it might be due to a network\n // timeout + a cache miss. If that were to happen, we'd rather wait until\n // the networkPromise resolves instead of returning null.\n // Note that it's fine to await an already-resolved promise, so we don't\n // have to check to see if it's still \"in flight\".\n if (!response) {\n response = await networkPromise;\n }\n if (process.env.NODE_ENV !== 'production') {\n logger.groupCollapsed(messages.strategyStart('NetworkFirst', request));\n for (const log of logs) {\n logger.log(log);\n }\n messages.printFinalResponse(response);\n logger.groupEnd();\n }\n if (!response) {\n throw new WorkboxError('no-response', { url: request.url });\n }\n return response;\n }\n /**\n * @param {Object} options\n * @param {Request} options.request\n * @param {Array} options.logs A reference to the logs array\n * @param {Event} [options.event]\n * @return {Promise}\n *\n * @private\n */\n _getTimeoutPromise({ request, logs, event }) {\n let timeoutId;\n const timeoutPromise = new Promise((resolve) => {\n const onNetworkTimeout = async () => {\n if (process.env.NODE_ENV !== 'production') {\n logs.push(`Timing out the network response at ` +\n `${this._networkTimeoutSeconds} seconds.`);\n }\n resolve(await this._respondFromCache({ request, event }));\n };\n timeoutId = setTimeout(onNetworkTimeout, this._networkTimeoutSeconds * 1000);\n });\n return {\n promise: timeoutPromise,\n id: timeoutId,\n };\n }\n /**\n * @param {Object} options\n * @param {number|undefined} options.timeoutId\n * @param {Request} options.request\n * @param {Array} options.logs A reference to the logs Array.\n * @param {Event} [options.event]\n * @return {Promise}\n *\n * @private\n */\n async _getNetworkPromise({ timeoutId, request, logs, event }) {\n let error;\n let response;\n try {\n response = await fetchWrapper.fetch({\n request,\n event,\n fetchOptions: this._fetchOptions,\n plugins: this._plugins,\n });\n }\n catch (err) {\n error = err;\n }\n if (timeoutId) {\n clearTimeout(timeoutId);\n }\n if (process.env.NODE_ENV !== 'production') {\n if (response) {\n logs.push(`Got response from network.`);\n }\n else {\n logs.push(`Unable to get a response from the network. Will respond ` +\n `with a cached response.`);\n }\n }\n if (error || !response) {\n response = await this._respondFromCache({ request, event });\n if (process.env.NODE_ENV !== 'production') {\n if (response) {\n logs.push(`Found a cached response in the '${this._cacheName}'` +\n ` cache.`);\n }\n else {\n logs.push(`No response found in the '${this._cacheName}' cache.`);\n }\n }\n }\n else {\n // Keep the service worker alive while we put the request in the cache\n const responseClone = response.clone();\n const cachePut = cacheWrapper.put({\n cacheName: this._cacheName,\n request,\n response: responseClone,\n event,\n plugins: this._plugins,\n });\n if (event) {\n try {\n // The event has been responded to so we can keep the SW alive to\n // respond to the request\n event.waitUntil(cachePut);\n }\n catch (err) {\n if (process.env.NODE_ENV !== 'production') {\n logger.warn(`Unable to ensure service worker stays alive when ` +\n `updating cache for '${getFriendlyURL(request.url)}'.`);\n }\n }\n }\n }\n return response;\n }\n /**\n * Used if the network timeouts or fails to make the request.\n *\n * @param {Object} options\n * @param {Request} request The request to match in the cache\n * @param {Event} [options.event]\n * @return {Promise}\n *\n * @private\n */\n _respondFromCache({ event, request }) {\n return cacheWrapper.match({\n cacheName: this._cacheName,\n request,\n event,\n matchOptions: this._matchOptions,\n plugins: this._plugins,\n });\n }\n}\nexport { NetworkFirst };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\n/**\n * A class that wraps common IndexedDB functionality in a promise-based API.\n * It exposes all the underlying power and functionality of IndexedDB, but\n * wraps the most commonly used features in a way that's much simpler to use.\n *\n * @private\n */\nexport class DBWrapper {\n /**\n * @param {string} name\n * @param {number} version\n * @param {Object=} [callback]\n * @param {!Function} [callbacks.onupgradeneeded]\n * @param {!Function} [callbacks.onversionchange] Defaults to\n * DBWrapper.prototype._onversionchange when not specified.\n * @private\n */\n constructor(name, version, { onupgradeneeded, onversionchange, } = {}) {\n this._db = null;\n this._name = name;\n this._version = version;\n this._onupgradeneeded = onupgradeneeded;\n this._onversionchange = onversionchange || (() => this.close());\n }\n /**\n * Returns the IDBDatabase instance (not normally needed).\n * @return {IDBDatabase|undefined}\n *\n * @private\n */\n get db() {\n return this._db;\n }\n /**\n * Opens a connected to an IDBDatabase, invokes any onupgradedneeded\n * callback, and added an onversionchange callback to the database.\n *\n * @return {IDBDatabase}\n * @private\n */\n async open() {\n if (this._db)\n return;\n this._db = await new Promise((resolve, reject) => {\n // This flag is flipped to true if the timeout callback runs prior\n // to the request failing or succeeding. Note: we use a timeout instead\n // of an onblocked handler since there are cases where onblocked will\n // never never run. A timeout better handles all possible scenarios:\n // https://github.com/w3c/IndexedDB/issues/223\n let openRequestTimedOut = false;\n setTimeout(() => {\n openRequestTimedOut = true;\n reject(new Error('The open request was blocked and timed out'));\n }, this.OPEN_TIMEOUT);\n const openRequest = indexedDB.open(this._name, this._version);\n openRequest.onerror = () => reject(openRequest.error);\n openRequest.onupgradeneeded = (evt) => {\n if (openRequestTimedOut) {\n openRequest.transaction.abort();\n openRequest.result.close();\n }\n else if (typeof this._onupgradeneeded === 'function') {\n this._onupgradeneeded(evt);\n }\n };\n openRequest.onsuccess = () => {\n const db = openRequest.result;\n if (openRequestTimedOut) {\n db.close();\n }\n else {\n db.onversionchange = this._onversionchange.bind(this);\n resolve(db);\n }\n };\n });\n return this;\n }\n /**\n * Polyfills the native `getKey()` method. Note, this is overridden at\n * runtime if the browser supports the native method.\n *\n * @param {string} storeName\n * @param {*} query\n * @return {Array}\n * @private\n */\n async getKey(storeName, query) {\n return (await this.getAllKeys(storeName, query, 1))[0];\n }\n /**\n * Polyfills the native `getAll()` method. Note, this is overridden at\n * runtime if the browser supports the native method.\n *\n * @param {string} storeName\n * @param {*} query\n * @param {number} count\n * @return {Array}\n * @private\n */\n async getAll(storeName, query, count) {\n return await this.getAllMatching(storeName, { query, count });\n }\n /**\n * Polyfills the native `getAllKeys()` method. Note, this is overridden at\n * runtime if the browser supports the native method.\n *\n * @param {string} storeName\n * @param {*} query\n * @param {number} count\n * @return {Array}\n * @private\n */\n async getAllKeys(storeName, query, count) {\n const entries = await this.getAllMatching(storeName, { query, count, includeKeys: true });\n return entries.map((entry) => entry.key);\n }\n /**\n * Supports flexible lookup in an object store by specifying an index,\n * query, direction, and count. This method returns an array of objects\n * with the signature .\n *\n * @param {string} storeName\n * @param {Object} [opts]\n * @param {string} [opts.index] The index to use (if specified).\n * @param {*} [opts.query]\n * @param {IDBCursorDirection} [opts.direction]\n * @param {number} [opts.count] The max number of results to return.\n * @param {boolean} [opts.includeKeys] When true, the structure of the\n * returned objects is changed from an array of values to an array of\n * objects in the form {key, primaryKey, value}.\n * @return {Array}\n * @private\n */\n async getAllMatching(storeName, { index, query = null, // IE/Edge errors if query === `undefined`.\n direction = 'next', count, includeKeys = false, } = {}) {\n return await this.transaction([storeName], 'readonly', (txn, done) => {\n const store = txn.objectStore(storeName);\n const target = index ? store.index(index) : store;\n const results = [];\n const request = target.openCursor(query, direction);\n request.onsuccess = () => {\n const cursor = request.result;\n if (cursor) {\n results.push(includeKeys ? cursor : cursor.value);\n if (count && results.length >= count) {\n done(results);\n }\n else {\n cursor.continue();\n }\n }\n else {\n done(results);\n }\n };\n });\n }\n /**\n * Accepts a list of stores, a transaction type, and a callback and\n * performs a transaction. A promise is returned that resolves to whatever\n * value the callback chooses. The callback holds all the transaction logic\n * and is invoked with two arguments:\n * 1. The IDBTransaction object\n * 2. A `done` function, that's used to resolve the promise when\n * when the transaction is done, if passed a value, the promise is\n * resolved to that value.\n *\n * @param {Array} storeNames An array of object store names\n * involved in the transaction.\n * @param {string} type Can be `readonly` or `readwrite`.\n * @param {!Function} callback\n * @return {*} The result of the transaction ran by the callback.\n * @private\n */\n async transaction(storeNames, type, callback) {\n await this.open();\n return await new Promise((resolve, reject) => {\n const txn = this._db.transaction(storeNames, type);\n txn.onabort = () => reject(txn.error);\n txn.oncomplete = () => resolve();\n callback(txn, (value) => resolve(value));\n });\n }\n /**\n * Delegates async to a native IDBObjectStore method.\n *\n * @param {string} method The method name.\n * @param {string} storeName The object store name.\n * @param {string} type Can be `readonly` or `readwrite`.\n * @param {...*} args The list of args to pass to the native method.\n * @return {*} The result of the transaction.\n * @private\n */\n async _call(method, storeName, type, ...args) {\n const callback = (txn, done) => {\n const objStore = txn.objectStore(storeName);\n // TODO(philipwalton): Fix this underlying TS2684 error.\n // @ts-ignore\n const request = objStore[method].apply(objStore, args);\n request.onsuccess = () => done(request.result);\n };\n return await this.transaction([storeName], type, callback);\n }\n /**\n * Closes the connection opened by `DBWrapper.open()`. Generally this method\n * doesn't need to be called since:\n * 1. It's usually better to keep a connection open since opening\n * a new connection is somewhat slow.\n * 2. Connections are automatically closed when the reference is\n * garbage collected.\n * The primary use case for needing to close a connection is when another\n * reference (typically in another tab) needs to upgrade it and would be\n * blocked by the current, open connection.\n *\n * @private\n */\n close() {\n if (this._db) {\n this._db.close();\n this._db = null;\n }\n }\n}\n// Exposed on the prototype to let users modify the default timeout on a\n// per-instance or global basis.\nDBWrapper.prototype.OPEN_TIMEOUT = 2000;\n// Wrap native IDBObjectStore methods according to their mode.\nconst methodsToWrap = {\n readonly: ['get', 'count', 'getKey', 'getAll', 'getAllKeys'],\n readwrite: ['add', 'put', 'clear', 'delete'],\n};\nfor (const [mode, methods] of Object.entries(methodsToWrap)) {\n for (const method of methods) {\n if (method in IDBObjectStore.prototype) {\n // Don't use arrow functions here since we're outside of the class.\n DBWrapper.prototype[method] =\n async function (storeName, ...args) {\n return await this._call(method, storeName, mode, ...args);\n };\n }\n }\n}\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { getOrCreateDefaultRouter } from './utils/getOrCreateDefaultRouter.js';\nimport './_version.js';\n/**\n * If a Route throws an error while handling a request, this `handler`\n * will be called and given a chance to provide a response.\n *\n * @param {module:workbox-routing~handlerCallback} handler A callback\n * function that returns a Promise resulting in a Response.\n *\n * @memberof module:workbox-routing\n */\nfunction setCatchHandler(handler) {\n const defaultRouter = getOrCreateDefaultRouter();\n defaultRouter.setCatchHandler(handler);\n}\nexport { setCatchHandler };\n","import { registerRoute, setCatchHandler } from 'workbox-routing';\nimport { precacheAndRoute, getCacheKeyForURL } from 'workbox-precaching';\nimport { isNav } from './utils';\nimport { NETWORK_HANDLER, PRECACHING_OPTIONS } from './contants';\n\nexport function getFiles() {\n\treturn self.__WB_MANIFEST;\n}\n\nexport function setupPrecaching(precacheFiles, precachingOptions) {\n\tprecacheAndRoute(precacheFiles, precachingOptions || PRECACHING_OPTIONS);\n}\n\nexport function setupRouting() {\n\t/**\n\t * Adding this before `precacheAndRoute` lets us handle all\n\t * the navigation requests even if they are in precache.\n\t */\n\tregisterRoute(({ event }) => isNav(event), NETWORK_HANDLER);\n\n\tsetCatchHandler(({ event }) => {\n\t\tif (isNav(event)) {\n\t\t\treturn caches.match(getCacheKeyForURL('/index.html'));\n\t\t}\n\t\treturn Response.error();\n\t});\n}\n\nexport { PRECACHING_OPTIONS };\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { cacheNames as _cacheNames } from './_private/cacheNames.js';\nimport './_version.js';\n/**\n * Get the current cache names and prefix/suffix used by Workbox.\n *\n * `cacheNames.precache` is used for precached assets,\n * `cacheNames.googleAnalytics` is used by `workbox-google-analytics` to\n * store `analytics.js`, and `cacheNames.runtime` is used for everything else.\n *\n * `cacheNames.prefix` can be used to retrieve just the current prefix value.\n * `cacheNames.suffix` can be used to retrieve just the current suffix value.\n *\n * @return {Object} An object with `precache`, `runtime`, `prefix`, and\n * `googleAnalytics` properties.\n *\n * @memberof module:workbox-core\n */\nconst cacheNames = {\n get googleAnalytics() {\n return _cacheNames.getGoogleAnalyticsName();\n },\n get precache() {\n return _cacheNames.getPrecacheName();\n },\n get prefix() {\n return _cacheNames.getPrefix();\n },\n get runtime() {\n return _cacheNames.getRuntimeName();\n },\n get suffix() {\n return _cacheNames.getSuffix();\n },\n};\nexport { cacheNames };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { assert } from 'workbox-core/_private/assert.js';\nimport { WorkboxError } from 'workbox-core/_private/WorkboxError.js';\nimport { getFriendlyURL } from 'workbox-core/_private/getFriendlyURL.js';\nimport { logger } from 'workbox-core/_private/logger.js';\nimport './_version.js';\n/**\n * This class allows you to set up rules determining what\n * status codes and/or headers need to be present in order for a\n * [`Response`](https://developer.mozilla.org/en-US/docs/Web/API/Response)\n * to be considered cacheable.\n *\n * @memberof module:workbox-cacheable-response\n */\nclass CacheableResponse {\n /**\n * To construct a new CacheableResponse instance you must provide at least\n * one of the `config` properties.\n *\n * If both `statuses` and `headers` are specified, then both conditions must\n * be met for the `Response` to be considered cacheable.\n *\n * @param {Object} config\n * @param {Array} [config.statuses] One or more status codes that a\n * `Response` can have and be considered cacheable.\n * @param {Object} [config.headers] A mapping of header names\n * and expected values that a `Response` can have and be considered cacheable.\n * If multiple headers are provided, only one needs to be present.\n */\n constructor(config = {}) {\n if (process.env.NODE_ENV !== 'production') {\n if (!(config.statuses || config.headers)) {\n throw new WorkboxError('statuses-or-headers-required', {\n moduleName: 'workbox-cacheable-response',\n className: 'CacheableResponse',\n funcName: 'constructor',\n });\n }\n if (config.statuses) {\n assert.isArray(config.statuses, {\n moduleName: 'workbox-cacheable-response',\n className: 'CacheableResponse',\n funcName: 'constructor',\n paramName: 'config.statuses',\n });\n }\n if (config.headers) {\n assert.isType(config.headers, 'object', {\n moduleName: 'workbox-cacheable-response',\n className: 'CacheableResponse',\n funcName: 'constructor',\n paramName: 'config.headers',\n });\n }\n }\n this._statuses = config.statuses;\n this._headers = config.headers;\n }\n /**\n * Checks a response to see whether it's cacheable or not, based on this\n * object's configuration.\n *\n * @param {Response} response The response whose cacheability is being\n * checked.\n * @return {boolean} `true` if the `Response` is cacheable, and `false`\n * otherwise.\n */\n isResponseCacheable(response) {\n if (process.env.NODE_ENV !== 'production') {\n assert.isInstance(response, Response, {\n moduleName: 'workbox-cacheable-response',\n className: 'CacheableResponse',\n funcName: 'isResponseCacheable',\n paramName: 'response',\n });\n }\n let cacheable = true;\n if (this._statuses) {\n cacheable = this._statuses.includes(response.status);\n }\n if (this._headers && cacheable) {\n cacheable = Object.keys(this._headers).some((headerName) => {\n return response.headers.get(headerName) === this._headers[headerName];\n });\n }\n if (process.env.NODE_ENV !== 'production') {\n if (!cacheable) {\n logger.groupCollapsed(`The request for ` +\n `'${getFriendlyURL(response.url)}' returned a response that does ` +\n `not meet the criteria for being cached.`);\n logger.groupCollapsed(`View cacheability criteria here.`);\n logger.log(`Cacheable statuses: ` +\n JSON.stringify(this._statuses));\n logger.log(`Cacheable headers: ` +\n JSON.stringify(this._headers, null, 2));\n logger.groupEnd();\n const logFriendlyHeaders = {};\n response.headers.forEach((value, key) => {\n logFriendlyHeaders[key] = value;\n });\n logger.groupCollapsed(`View response status and headers here.`);\n logger.log(`Response status: ` + response.status);\n logger.log(`Response headers: ` +\n JSON.stringify(logFriendlyHeaders, null, 2));\n logger.groupEnd();\n logger.groupCollapsed(`View full response details here.`);\n logger.log(response.headers);\n logger.log(response);\n logger.groupEnd();\n logger.groupEnd();\n }\n }\n return cacheable;\n }\n}\nexport { CacheableResponse };\n","import { NetworkFirst } from 'workbox-strategies';\nimport { cacheNames } from 'workbox-core';\nimport { CacheableResponsePlugin } from 'workbox-cacheable-response';\n\nexport const PRECACHING_OPTIONS = {};\n\nexport const NETWORK_HANDLER = new NetworkFirst({\n\t// this cache is plunged with every new service worker deploy so we dont need to care about purging the cache.\n\tcacheName: cacheNames.precache,\n\tnetworkTimeoutSeconds: 5, // if u dont start getting headers within 5 sec fallback to cache.\n\tplugins: [\n\t\tnew CacheableResponsePlugin({\n\t\t\tstatuses: [200], // only cache valid responses, not opaque responses e.g. wifi portal.\n\t\t}),\n\t],\n});\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { logger } from 'workbox-core/_private/logger.js';\nimport { WorkboxError } from 'workbox-core/_private/WorkboxError.js';\nimport { Route } from './Route.js';\nimport { RegExpRoute } from './RegExpRoute.js';\nimport { getOrCreateDefaultRouter } from './utils/getOrCreateDefaultRouter.js';\nimport './_version.js';\n/**\n * Easily register a RegExp, string, or function with a caching\n * strategy to a singleton Router instance.\n *\n * This method will generate a Route for you if needed and\n * call [registerRoute()]{@link module:workbox-routing.Router#registerRoute}.\n *\n * @param {RegExp|string|module:workbox-routing.Route~matchCallback|module:workbox-routing.Route} capture\n * If the capture param is a `Route`, all other arguments will be ignored.\n * @param {module:workbox-routing~handlerCallback} [handler] A callback\n * function that returns a Promise resulting in a Response. This parameter\n * is required if `capture` is not a `Route` object.\n * @param {string} [method='GET'] The HTTP method to match the Route\n * against.\n * @return {module:workbox-routing.Route} The generated `Route`(Useful for\n * unregistering).\n *\n * @memberof module:workbox-routing\n */\nfunction registerRoute(capture, handler, method) {\n let route;\n if (typeof capture === 'string') {\n const captureUrl = new URL(capture, location.href);\n if (process.env.NODE_ENV !== 'production') {\n if (!(capture.startsWith('/') || capture.startsWith('http'))) {\n throw new WorkboxError('invalid-string', {\n moduleName: 'workbox-routing',\n funcName: 'registerRoute',\n paramName: 'capture',\n });\n }\n // We want to check if Express-style wildcards are in the pathname only.\n // TODO: Remove this log message in v4.\n const valueToCheck = capture.startsWith('http') ?\n captureUrl.pathname : capture;\n // See https://github.com/pillarjs/path-to-regexp#parameters\n const wildcards = '[*:?+]';\n if ((new RegExp(`${wildcards}`)).exec(valueToCheck)) {\n logger.debug(`The '$capture' parameter contains an Express-style wildcard ` +\n `character (${wildcards}). Strings are now always interpreted as ` +\n `exact matches; use a RegExp for partial or wildcard matches.`);\n }\n }\n const matchCallback = ({ url }) => {\n if (process.env.NODE_ENV !== 'production') {\n if ((url.pathname === captureUrl.pathname) &&\n (url.origin !== captureUrl.origin)) {\n logger.debug(`${capture} only partially matches the cross-origin URL ` +\n `${url}. This route will only handle cross-origin requests ` +\n `if they match the entire URL.`);\n }\n }\n return url.href === captureUrl.href;\n };\n // If `capture` is a string then `handler` and `method` must be present.\n route = new Route(matchCallback, handler, method);\n }\n else if (capture instanceof RegExp) {\n // If `capture` is a `RegExp` then `handler` and `method` must be present.\n route = new RegExpRoute(capture, handler, method);\n }\n else if (typeof capture === 'function') {\n // If `capture` is a function then `handler` and `method` must be present.\n route = new Route(capture, handler, method);\n }\n else if (capture instanceof Route) {\n route = capture;\n }\n else {\n throw new WorkboxError('unsupported-route-type', {\n moduleName: 'workbox-routing',\n funcName: 'registerRoute',\n paramName: 'capture',\n });\n }\n const defaultRouter = getOrCreateDefaultRouter();\n defaultRouter.registerRoute(route);\n return route;\n}\nexport { registerRoute };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { CacheableResponse } from './CacheableResponse.js';\nimport './_version.js';\n/**\n * A class implementing the `cacheWillUpdate` lifecycle callback. This makes it\n * easier to add in cacheability checks to requests made via Workbox's built-in\n * strategies.\n *\n * @memberof module:workbox-cacheable-response\n */\nclass CacheableResponsePlugin {\n /**\n * To construct a new CacheableResponsePlugin instance you must provide at\n * least one of the `config` properties.\n *\n * If both `statuses` and `headers` are specified, then both conditions must\n * be met for the `Response` to be considered cacheable.\n *\n * @param {Object} config\n * @param {Array} [config.statuses] One or more status codes that a\n * `Response` can have and be considered cacheable.\n * @param {Object} [config.headers] A mapping of header names\n * and expected values that a `Response` can have and be considered cacheable.\n * If multiple headers are provided, only one needs to be present.\n */\n constructor(config) {\n /**\n * @param {Object} options\n * @param {Response} options.response\n * @return {Response|null}\n * @private\n */\n this.cacheWillUpdate = async ({ response }) => {\n if (this._cacheableResponse.isResponseCacheable(response)) {\n return response;\n }\n return null;\n };\n this._cacheableResponse = new CacheableResponse(config);\n }\n}\nexport { CacheableResponsePlugin };\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { getOrCreatePrecacheController } from './utils/getOrCreatePrecacheController.js';\nimport './_version.js';\n/**\n * Takes in a URL, and returns the corresponding URL that could be used to\n * lookup the entry in the precache.\n *\n * If a relative URL is provided, the location of the service worker file will\n * be used as the base.\n *\n * For precached entries without revision information, the cache key will be the\n * same as the original URL.\n *\n * For precached entries with revision information, the cache key will be the\n * original URL with the addition of a query parameter used for keeping track of\n * the revision info.\n *\n * @param {string} url The URL whose cache key to look up.\n * @return {string} The cache key that corresponds to that URL.\n *\n * @memberof module:workbox-precaching\n */\nfunction getCacheKeyForURL(url) {\n const precacheController = getOrCreatePrecacheController();\n return precacheController.getCacheKeyForURL(url);\n}\nexport { getCacheKeyForURL };\n","\"use strict\";\n// @ts-ignore\ntry {\n self['workbox:core:5.1.3'] && _();\n}\ncatch (e) { }\n"]} \ No newline at end of file diff --git a/docs/sw.js b/docs/sw.js index 973cf0b..6ece800 100644 --- a/docs/sw.js +++ b/docs/sw.js @@ -1,2 +1,2 @@ -!function(e){function t(n){if(r[n])return r[n].exports;var o=r[n]={i:n,l:!1,exports:{}};return e[n].call(o.exports,o,o.exports,t),o.l=!0,o.exports}var r={};t.m=e,t.c=r,t.d=function(e,r,n){t.o(e,r)||Object.defineProperty(e,r,{enumerable:!0,get:n})},t.r=function(e){"undefined"!=typeof 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Promise(function(e,t){var r,n,o,i,u,c,s;return Promise.resolve(self.caches.open(this._cacheName)).then(function(a){try{return r=a,Promise.resolve(r.keys()).then(function(a){try{{var l;function h(){if((c=u()).done)return[1];{if(!o.has((s=c.value).url))return Promise.resolve(r.delete(s)).then(function(){try{return i.push(s.url),e.call(this)}catch(e){return t(e)}}.bind(this),t);function e(){return h}return e.call(this)}}return n=a,o=new Set(this._urlsToCacheKeys.values()),i=[],u=P(n),(l=function(e){for(;e;){if(e.then)return void e.then(l,t);try{if(e.pop){if(e.length)return e.pop()?f.call(this):e;e=h}else e=e.call(this)}catch(e){return t(e)}}}.bind(this))(h);function f(){return e({deletedURLs:i})}}}catch(e){return t(e)}}.bind(this),t)}catch(e){return t(e)}}.bind(this),t)}.bind(this))},t._addURLToCache=function(e){return new Promise(function(t,r){var n,o,i,u,c,s,a,l,h,f;return n=e.cacheKey,o=e.url,i=e.event,u=e.plugins,c=new Request(o,{integrity:e.integrity,cache:e.cacheMode,credentials:"same-origin"}),Promise.resolve(ie({event:i,plugins:u,request:c})).then(function(e){try{for(s=e,l=P(u||[]);!(h=l()).done;)"cacheWillUpdate"in(f=h.value)&&(a=f);return Promise.resolve(new Promise((function(e,t){return a?Promise.resolve(a.cacheWillUpdate({event:i,request:c,response:s})).then(e,t):e(s.status<400)}))).then(function(e){try{if(!e)return r(new E("bad-precaching-response",{url:o,status:s.status}));if(s.redirected)return Promise.resolve(w(s)).then(function(e){try{return s=e,a.call(this)}catch(e){return r(e)}}.bind(this),r);function a(){return Promise.resolve(ne({event:i,plugins:u,response:s,request:n===o?c:new Request(n),cacheName:this._cacheName,matchOptions:{ignoreSearch:!0}})).then((function(){try{return t()}catch(e){return r(e)}}),r)}return a.call(this)}catch(e){return r(e)}}.bind(this),r)}catch(e){return r(e)}}.bind(this),r)}.bind(this))},t.getURLsToCacheKeys=function(){return this._urlsToCacheKeys},t.getCachedURLs=function(){return[].concat(this._urlsToCacheKeys.keys())},t.getCacheKeyForURL=function(e){var t=new URL(e,location.href);return this._urlsToCacheKeys.get(t.href)},t.matchPrecache=function(e){return new Promise(function(t,r){var n,o;return n=e instanceof Request?e.url:e,(o=this.getCacheKeyForURL(n))?Promise.resolve(self.caches.open(this._cacheName)).then((function(e){try{return t(e.match(o))}catch(e){return r(e)}}),r):t(void 0)}.bind(this))},t.createHandler=function(e){var t=this;return void 0===e&&(e=!0),function(r){return new Promise((function(n,o){var i,u;i=r.request;var c=function(t){try{if(e)return n(fetch(i));throw t}catch(e){return o(e)}};try{return Promise.resolve(t.matchPrecache(i)).then((function(e){try{if(u=e)return n(u);throw new E("missing-precache-entry",{cacheName:t._cacheName,url:i instanceof Request?i.url:i})}catch(e){return c(e)}}),c)}catch(e){c(e)}}))}},t.createHandlerBoundToURL=function(e,t){if(void 0===t&&(t=!0),!this.getCacheKeyForURL(e))throw new E("non-precached-url",{url:e});var r=this.createHandler(t),n=new Request(e);return function(){return r({request:n})}},e}(),ce=function(){return K||(K=new ue),K},se=!1,ae=function(e){var t=ce(),r=$();e.waitUntil(t.install({event:e,plugins:r}).catch((function(e){throw e})))},le=function(e){var t=ce();e.waitUntil(t.activate())},he=function(e){return"navigate"===e.request.mode},fe=(r("myed"),{cacheWillUpdate:function(e){return new Promise((function(t){var r=e.response;return t(200===r.status||0===r.status?r:null)}))}}),de=function(){function e(e){if(void 0===e&&(e={}),this._cacheName=V(e.cacheName),e.plugins){var t=e.plugins.some((function(e){return!!e.cacheWillUpdate}));this._plugins=t?e.plugins:[fe].concat(e.plugins)}else this._plugins=[fe];this._networkTimeoutSeconds=e.networkTimeoutSeconds||0,this._fetchOptions=e.fetchOptions,this._matchOptions=e.matchOptions}var t=e.prototype;return t.handle=function(e){return new Promise(function(t,r){var n,o,i,u,c,s,a,l;return n=e.event,i=[],"string"==typeof(o=e.request)&&(o=new Request(o)),u=[],this._networkTimeoutSeconds&&(c=(s=this._getTimeoutPromise({request:o,event:n,logs:i})).id,u.push(s.promise)),a=this._getNetworkPromise({timeoutId:c,request:o,event:n,logs:i}),u.push(a),Promise.resolve(Promise.race(u)).then(function(e){try{if(!(l=e))return Promise.resolve(a).then(function(e){try{return l=e,n.call(this)}catch(e){return r(e)}}.bind(this),r);function n(){return l?t(l):r(new E("no-response",{url:o.url}))}return n.call(this)}catch(e){return r(e)}}.bind(this),r)}.bind(this))},t._getTimeoutPromise=function(e){var t,r=this,n=e.request,o=e.event;return{promise:new Promise((function(e){t=setTimeout((function(){return new Promise((function(t,i){return Promise.resolve(r._respondFromCache({request:n,event:o})).then((function(r){try{return e(r),t()}catch(e){return i(e)}}),i)}))}),1e3*r._networkTimeoutSeconds)})),id:t}},t._getNetworkPromise=function(e){return new Promise(function(t,r){var n,o,i,u,c,s,a;n=e.timeoutId,o=e.request,i=e.event;var l=function(){try{if(n&&clearTimeout(n),u||!c)return Promise.resolve(this._respondFromCache({request:o,event:i})).then(function(t){try{return c=t,e.call(this)}catch(e){return r(e)}}.bind(this),r);if(s=c.clone(),a=ne({cacheName:this._cacheName,request:o,response:s,event:i,plugins:this._plugins}),i)try{i.waitUntil(a)}catch(e){0}return e.call(this);function e(){return t(c)}}catch(e){return r(e)}}.bind(this),h=function(e){try{return u=e,l()}catch(e){return r(e)}};try{return Promise.resolve(ie({request:o,event:i,fetchOptions:this._fetchOptions,plugins:this._plugins})).then((function(e){try{return c=e,l()}catch(e){return h(e)}}),h)}catch(e){h(e)}}.bind(this))},t._respondFromCache=function(e){return oe({cacheName:this._cacheName,request:e.request,event:e.event,matchOptions:this._matchOptions,plugins:this._plugins})},e}(),pe=function(){function e(e,t,r){var n=this,o=void 0===r?{}:r,i=o.onupgradeneeded,u=o.onversionchange;this._db=null,this._name=e,this._version=t,this._onupgradeneeded=i,this._onversionchange=u||function(){return n.close()}}var t,r,n,o=e.prototype;return o.open=function(){return new Promise(function(e,t){var r;return r=this,this._db?e():Promise.resolve(new Promise((function(e,t){var n=!1;setTimeout((function(){n=!0,t(new Error("The open request was blocked and timed out"))}),r.OPEN_TIMEOUT);var o=indexedDB.open(r._name,r._version);o.onerror=function(){return t(o.error)},o.onupgradeneeded=function(e){n?(o.transaction.abort(),o.result.close()):"function"==typeof r._onupgradeneeded&&r._onupgradeneeded(e)},o.onsuccess=function(){var t=o.result;n?t.close():(t.onversionchange=r._onversionchange.bind(r),e(t))}}))).then(function(r){try{return this._db=r,e(this)}catch(e){return t(e)}}.bind(this),t)}.bind(this))},o.getKey=function(e,t){return new Promise(function(r,n){return Promise.resolve(this.getAllKeys(e,t,1)).then((function(e){try{return r(e[0])}catch(e){return n(e)}}),n)}.bind(this))},o.getAll=function(e,t,r){return new Promise(function(n,o){return Promise.resolve(this.getAllMatching(e,{query:t,count:r})).then(n,o)}.bind(this))},o.getAllKeys=function(e,t,r){return new Promise(function(n,o){return Promise.resolve(this.getAllMatching(e,{query:t,count:r,includeKeys:!0})).then((function(e){try{return n(e.map((function(e){return e.key})))}catch(e){return o(e)}}),o)}.bind(this))},o.getAllMatching=function(e,t){return new Promise(function(r,n){var o,i,u,c,s,a,l,h,f;return i=(o=void 0===t?{}:t).index,c=void 0===(u=o.query)?null:u,a=void 0===(s=o.direction)?"next":s,l=o.count,f=void 0!==(h=o.includeKeys)&&h,Promise.resolve(this.transaction([e],"readonly",(function(t,r){var n=t.objectStore(e),o=i?n.index(i):n,u=[],s=o.openCursor(c,a);s.onsuccess=function(){var e=s.result;e?(u.push(f?e:e.value),l&&u.length>=l?r(u):e.continue()):r(u)}}))).then(r,n)}.bind(this))},o.transaction=function(e,t,r){return new Promise(function(n,o){var i;return i=this,Promise.resolve(this.open()).then((function(){try{return Promise.resolve(new Promise((function(n,o){var u=i._db.transaction(e,t);u.onabort=function(){return o(u.error)},u.oncomplete=function(){return n()},r(u,(function(e){return n(e)}))}))).then(n,o)}catch(e){return o(e)}}),o)}.bind(this))},o._call=function(e,t,r){var n=arguments;return new Promise(function(o,i){var u,c,s;for(u=n.length,c=new Array(u>3?u-3:0),s=3;s1?u-1:0),s=1;s} [options.ignoreURLParametersMatching=[/^utm_/]] An\n * array of regex's to remove search params when looking for a cache match.\n * @param {boolean} [options.cleanURLs=true] The `cleanURLs` option will\n * check the cache for the URL with a `.html` added to the end of the end.\n * @param {module:workbox-precaching~urlManipulation} [options.urlManipulation]\n * This is a function that should take a URL and return an array of\n * alternative URLs that should be checked for precache matches.\n *\n * @memberof module:workbox-precaching\n */\nfunction addRoute(options) {\n if (!listenerAdded) {\n addFetchListener(options);\n listenerAdded = true;\n }\n}\nexport { addRoute };\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { cacheNames } from 'workbox-core/_private/cacheNames.js';\nimport { getFriendlyURL } from 'workbox-core/_private/getFriendlyURL.js';\nimport { logger } from 'workbox-core/_private/logger.js';\nimport { getCacheKeyForURL } from './getCacheKeyForURL.js';\nimport '../_version.js';\n/**\n * Adds a `fetch` listener to the service worker that will\n * respond to\n * [network requests]{@link https://developer.mozilla.org/en-US/docs/Web/API/Service_Worker_API/Using_Service_Workers#Custom_responses_to_requests}\n * with precached assets.\n *\n * Requests for assets that aren't precached, the `FetchEvent` will not be\n * responded to, allowing the event to fall through to other `fetch` event\n * listeners.\n *\n * NOTE: when called more than once this method will replace the previously set\n * configuration options. Calling it more than once is not recommended outside\n * of tests.\n *\n * @private\n * @param {Object} [options]\n * @param {string} [options.directoryIndex=index.html] The `directoryIndex` will\n * check cache entries for a URLs ending with '/' to see if there is a hit when\n * appending the `directoryIndex` value.\n * @param {Array} [options.ignoreURLParametersMatching=[/^utm_/]] An\n * array of regex's to remove search params when looking for a cache match.\n * @param {boolean} [options.cleanURLs=true] The `cleanURLs` option will\n * check the cache for the URL with a `.html` added to the end of the end.\n * @param {workbox.precaching~urlManipulation} [options.urlManipulation]\n * This is a function that should take a URL and return an array of\n * alternative URLs that should be checked for precache matches.\n */\nexport const addFetchListener = ({ ignoreURLParametersMatching = [/^utm_/], directoryIndex = 'index.html', cleanURLs = true, urlManipulation, } = {}) => {\n const cacheName = cacheNames.getPrecacheName();\n // See https://github.com/Microsoft/TypeScript/issues/28357#issuecomment-436484705\n self.addEventListener('fetch', ((event) => {\n const precachedURL = getCacheKeyForURL(event.request.url, {\n cleanURLs,\n directoryIndex,\n ignoreURLParametersMatching,\n urlManipulation,\n });\n if (!precachedURL) {\n if (process.env.NODE_ENV !== 'production') {\n logger.debug(`Precaching did not find a match for ` +\n getFriendlyURL(event.request.url));\n }\n return;\n }\n let responsePromise = self.caches.open(cacheName).then((cache) => {\n return cache.match(precachedURL);\n }).then((cachedResponse) => {\n if (cachedResponse) {\n return cachedResponse;\n }\n // Fall back to the network if we don't have a cached response\n // (perhaps due to manual cache cleanup).\n if (process.env.NODE_ENV !== 'production') {\n logger.warn(`The precached response for ` +\n `${getFriendlyURL(precachedURL)} in ${cacheName} was not found. ` +\n `Falling back to the network instead.`);\n }\n return fetch(precachedURL);\n });\n if (process.env.NODE_ENV !== 'production') {\n responsePromise = responsePromise.then((response) => {\n // Workbox is going to handle the route.\n // print the routing details to the console.\n logger.groupCollapsed(`Precaching is responding to: ` +\n getFriendlyURL(event.request.url));\n logger.log(`Serving the precached url: ${precachedURL}`);\n logger.groupCollapsed(`View request details here.`);\n logger.log(event.request);\n logger.groupEnd();\n logger.groupCollapsed(`View response details here.`);\n logger.log(response);\n logger.groupEnd();\n logger.groupEnd();\n return response;\n });\n }\n event.respondWith(responsePromise);\n }));\n};\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { getOrCreatePrecacheController } from './getOrCreatePrecacheController.js';\nimport { generateURLVariations } from './generateURLVariations.js';\nimport '../_version.js';\n/**\n * This function will take the request URL and manipulate it based on the\n * configuration options.\n *\n * @param {string} url\n * @param {Object} options\n * @return {string} Returns the URL in the cache that matches the request,\n * if possible.\n *\n * @private\n */\nexport const getCacheKeyForURL = (url, options) => {\n const precacheController = getOrCreatePrecacheController();\n const urlsToCacheKeys = precacheController.getURLsToCacheKeys();\n for (const possibleURL of generateURLVariations(url, options)) {\n const possibleCacheKey = urlsToCacheKeys.get(possibleURL);\n if (possibleCacheKey) {\n return possibleCacheKey;\n }\n }\n};\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { removeIgnoredSearchParams } from './removeIgnoredSearchParams.js';\nimport '../_version.js';\n/**\n * Generator function that yields possible variations on the original URL to\n * check, one at a time.\n *\n * @param {string} url\n * @param {Object} options\n *\n * @private\n * @memberof module:workbox-precaching\n */\nexport function* generateURLVariations(url, { ignoreURLParametersMatching, directoryIndex, cleanURLs, urlManipulation, } = {}) {\n const urlObject = new URL(url, location.href);\n urlObject.hash = '';\n yield urlObject.href;\n const urlWithoutIgnoredParams = removeIgnoredSearchParams(urlObject, ignoreURLParametersMatching);\n yield urlWithoutIgnoredParams.href;\n if (directoryIndex && urlWithoutIgnoredParams.pathname.endsWith('/')) {\n const directoryURL = new URL(urlWithoutIgnoredParams.href);\n directoryURL.pathname += directoryIndex;\n yield directoryURL.href;\n }\n if (cleanURLs) {\n const cleanURL = new URL(urlWithoutIgnoredParams.href);\n cleanURL.pathname += '.html';\n yield cleanURL.href;\n }\n if (urlManipulation) {\n const additionalURLs = urlManipulation({ url: urlObject });\n for (const urlToAttempt of additionalURLs) {\n yield urlToAttempt.href;\n }\n }\n}\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\n/**\n * Removes any URL search parameters that should be ignored.\n *\n * @param {URL} urlObject The original URL.\n * @param {Array} ignoreURLParametersMatching RegExps to test against\n * each search parameter name. Matches mean that the search parameter should be\n * ignored.\n * @return {URL} The URL with any ignored search parameters removed.\n *\n * @private\n * @memberof module:workbox-precaching\n */\nexport function removeIgnoredSearchParams(urlObject, ignoreURLParametersMatching = []) {\n // Convert the iterable into an array at the start of the loop to make sure\n // deletion doesn't mess up iteration.\n for (const paramName of [...urlObject.searchParams.keys()]) {\n if (ignoreURLParametersMatching.some((regExp) => regExp.test(paramName))) {\n urlObject.searchParams.delete(paramName);\n }\n }\n return urlObject;\n}\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { addRoute } from './addRoute.js';\nimport { precache } from './precache.js';\nimport './_version.js';\n/**\n * This method will add entries to the precache list and add a route to\n * respond to fetch events.\n *\n * This is a convenience method that will call\n * [precache()]{@link module:workbox-precaching.precache} and\n * [addRoute()]{@link module:workbox-precaching.addRoute} in a single call.\n *\n * @param {Array} entries Array of entries to precache.\n * @param {Object} [options] See\n * [addRoute() options]{@link module:workbox-precaching.addRoute}.\n *\n * @memberof module:workbox-precaching\n */\nfunction precacheAndRoute(entries, options) {\n precache(entries);\n addRoute(options);\n}\nexport { precacheAndRoute };\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { logger } from 'workbox-core/_private/logger.js';\nimport { getOrCreatePrecacheController } from './utils/getOrCreatePrecacheController.js';\nimport { precachePlugins } from './utils/precachePlugins.js';\nimport './_version.js';\nconst installListener = (event) => {\n const precacheController = getOrCreatePrecacheController();\n const plugins = precachePlugins.get();\n event.waitUntil(precacheController.install({ event, plugins })\n .catch((error) => {\n if (process.env.NODE_ENV !== 'production') {\n logger.error(`Service worker installation failed. It will ` +\n `be retried automatically during the next navigation.`);\n }\n // Re-throw the error to ensure installation fails.\n throw error;\n }));\n};\nconst activateListener = (event) => {\n const precacheController = getOrCreatePrecacheController();\n event.waitUntil(precacheController.activate());\n};\n/**\n * Adds items to the precache list, removing any duplicates and\n * stores the files in the\n * [\"precache cache\"]{@link module:workbox-core.cacheNames} when the service\n * worker installs.\n *\n * This method can be called multiple times.\n *\n * Please note: This method **will not** serve any of the cached files for you.\n * It only precaches files. To respond to a network request you call\n * [addRoute()]{@link module:workbox-precaching.addRoute}.\n *\n * If you have a single array of files to precache, you can just call\n * [precacheAndRoute()]{@link module:workbox-precaching.precacheAndRoute}.\n *\n * @param {Array} [entries=[]] Array of entries to precache.\n *\n * @memberof module:workbox-precaching\n */\nfunction precache(entries) {\n const precacheController = getOrCreatePrecacheController();\n precacheController.addToCacheList(entries);\n if (entries.length > 0) {\n // NOTE: these listeners will only be added once (even if the `precache()`\n // method is called multiple times) because event listeners are implemented\n // as a set, where each listener must be unique.\n // See https://github.com/Microsoft/TypeScript/issues/28357#issuecomment-436484705\n self.addEventListener('install', installListener);\n self.addEventListener('activate', activateListener);\n }\n}\nexport { precache };\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { Router } from '../Router.js';\nimport '../_version.js';\nlet defaultRouter;\n/**\n * Creates a new, singleton Router instance if one does not exist. If one\n * does already exist, that instance is returned.\n *\n * @private\n * @return {Router}\n */\nexport const getOrCreateDefaultRouter = () => {\n if (!defaultRouter) {\n defaultRouter = new Router();\n // The helpers that use the default Router assume these listeners exist.\n defaultRouter.addFetchListener();\n defaultRouter.addCacheListener();\n }\n return defaultRouter;\n};\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { PrecacheController } from '../PrecacheController.js';\nimport '../_version.js';\nlet precacheController;\n/**\n * @return {PrecacheController}\n * @private\n */\nexport const getOrCreatePrecacheController = () => {\n if (!precacheController) {\n precacheController = new PrecacheController();\n }\n return precacheController;\n};\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { messages } from './messages.js';\nimport '../../_version.js';\nconst fallback = (code, ...args) => {\n let msg = code;\n if (args.length > 0) {\n msg += ` :: ${JSON.stringify(args)}`;\n }\n return msg;\n};\nconst generatorFunction = (code, details = {}) => {\n const message = messages[code];\n if (!message) {\n throw new Error(`Unable to find message for code '${code}'.`);\n }\n return message(details);\n};\nexport const messageGenerator = (process.env.NODE_ENV === 'production') ?\n fallback : generatorFunction;\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { messageGenerator } from '../models/messages/messageGenerator.js';\nimport '../_version.js';\n/**\n * Workbox errors should be thrown with this class.\n * This allows use to ensure the type easily in tests,\n * helps developers identify errors from workbox\n * easily and allows use to optimise error\n * messages correctly.\n *\n * @private\n */\nclass WorkboxError extends Error {\n /**\n *\n * @param {string} errorCode The error code that\n * identifies this particular error.\n * @param {Object=} details Any relevant arguments\n * that will help developers identify issues should\n * be added as a key on the context object.\n */\n constructor(errorCode, details) {\n const message = messageGenerator(errorCode, details);\n super(message);\n this.name = errorCode;\n this.details = details;\n }\n}\nexport { WorkboxError };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { assert } from 'workbox-core/_private/assert.js';\nimport '../_version.js';\n/**\n * @param {function()|Object} handler Either a function, or an object with a\n * 'handle' method.\n * @return {Object} An object with a handle method.\n *\n * @private\n */\nexport const normalizeHandler = (handler) => {\n if (handler && typeof handler === 'object') {\n if (process.env.NODE_ENV !== 'production') {\n assert.hasMethod(handler, 'handle', {\n moduleName: 'workbox-routing',\n className: 'Route',\n funcName: 'constructor',\n paramName: 'handler',\n });\n }\n return handler;\n }\n else {\n if (process.env.NODE_ENV !== 'production') {\n assert.isType(handler, 'function', {\n moduleName: 'workbox-routing',\n className: 'Route',\n funcName: 'constructor',\n paramName: 'handler',\n });\n }\n return { handle: handler };\n }\n};\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { assert } from 'workbox-core/_private/assert.js';\nimport { defaultMethod, validMethods } from './utils/constants.js';\nimport { normalizeHandler } from './utils/normalizeHandler.js';\nimport './_version.js';\n/**\n * A `Route` consists of a pair of callback functions, \"match\" and \"handler\".\n * The \"match\" callback determine if a route should be used to \"handle\" a\n * request by returning a non-falsy value if it can. The \"handler\" callback\n * is called when there is a match and should return a Promise that resolves\n * to a `Response`.\n *\n * @memberof module:workbox-routing\n */\nclass Route {\n /**\n * Constructor for Route class.\n *\n * @param {module:workbox-routing~matchCallback} match\n * A callback function that determines whether the route matches a given\n * `fetch` event by returning a non-falsy value.\n * @param {module:workbox-routing~handlerCallback} handler A callback\n * function that returns a Promise resolving to a Response.\n * @param {string} [method='GET'] The HTTP method to match the Route\n * against.\n */\n constructor(match, handler, method = defaultMethod) {\n if (process.env.NODE_ENV !== 'production') {\n assert.isType(match, 'function', {\n moduleName: 'workbox-routing',\n className: 'Route',\n funcName: 'constructor',\n paramName: 'match',\n });\n if (method) {\n assert.isOneOf(method, validMethods, { paramName: 'method' });\n }\n }\n // These values are referenced directly by Router so cannot be\n // altered by minificaton.\n this.handler = normalizeHandler(handler);\n this.match = match;\n this.method = method;\n }\n}\nexport { Route };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\n/**\n * The default HTTP method, 'GET', used when there's no specific method\n * configured for a route.\n *\n * @type {string}\n *\n * @private\n */\nexport const defaultMethod = 'GET';\n/**\n * The list of valid HTTP methods associated with requests that could be routed.\n *\n * @type {Array}\n *\n * @private\n */\nexport const validMethods = [\n 'DELETE',\n 'GET',\n 'HEAD',\n 'PATCH',\n 'POST',\n 'PUT',\n];\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { assert } from 'workbox-core/_private/assert.js';\nimport { logger } from 'workbox-core/_private/logger.js';\nimport { Route } from './Route.js';\nimport './_version.js';\n/**\n * RegExpRoute makes it easy to create a regular expression based\n * [Route]{@link module:workbox-routing.Route}.\n *\n * For same-origin requests the RegExp only needs to match part of the URL. For\n * requests against third-party servers, you must define a RegExp that matches\n * the start of the URL.\n *\n * [See the module docs for info.]{@link https://developers.google.com/web/tools/workbox/modules/workbox-routing}\n *\n * @memberof module:workbox-routing\n * @extends module:workbox-routing.Route\n */\nclass RegExpRoute extends Route {\n /**\n * If the regular expression contains\n * [capture groups]{@link https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/RegExp#grouping-back-references},\n * the captured values will be passed to the\n * [handler's]{@link module:workbox-routing~handlerCallback} `params`\n * argument.\n *\n * @param {RegExp} regExp The regular expression to match against URLs.\n * @param {module:workbox-routing~handlerCallback} handler A callback\n * function that returns a Promise resulting in a Response.\n * @param {string} [method='GET'] The HTTP method to match the Route\n * against.\n */\n constructor(regExp, handler, method) {\n if (process.env.NODE_ENV !== 'production') {\n assert.isInstance(regExp, RegExp, {\n moduleName: 'workbox-routing',\n className: 'RegExpRoute',\n funcName: 'constructor',\n paramName: 'pattern',\n });\n }\n const match = ({ url }) => {\n const result = regExp.exec(url.href);\n // Return immediately if there's no match.\n if (!result) {\n return;\n }\n // Require that the match start at the first character in the URL string\n // if it's a cross-origin request.\n // See https://github.com/GoogleChrome/workbox/issues/281 for the context\n // behind this behavior.\n if ((url.origin !== location.origin) && (result.index !== 0)) {\n if (process.env.NODE_ENV !== 'production') {\n logger.debug(`The regular expression '${regExp}' only partially matched ` +\n `against the cross-origin URL '${url}'. RegExpRoute's will only ` +\n `handle cross-origin requests if they match the entire URL.`);\n }\n return;\n }\n // If the route matches, but there aren't any capture groups defined, then\n // this will return [], which is truthy and therefore sufficient to\n // indicate a match.\n // If there are capture groups, then it will return their values.\n return result.slice(1);\n };\n super(match, handler, method);\n }\n}\nexport { RegExpRoute };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\nconst getFriendlyURL = (url) => {\n const urlObj = new URL(String(url), location.href);\n // See https://github.com/GoogleChrome/workbox/issues/2323\n // We want to include everything, except for the origin if it's same-origin.\n return urlObj.href.replace(new RegExp(`^${location.origin}`), '');\n};\nexport { getFriendlyURL };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { assert } from 'workbox-core/_private/assert.js';\nimport { logger } from 'workbox-core/_private/logger.js';\nimport { WorkboxError } from 'workbox-core/_private/WorkboxError.js';\nimport { getFriendlyURL } from 'workbox-core/_private/getFriendlyURL.js';\nimport { normalizeHandler } from './utils/normalizeHandler.js';\nimport './_version.js';\n/**\n * The Router can be used to process a FetchEvent through one or more\n * [Routes]{@link module:workbox-routing.Route} responding with a Request if\n * a matching route exists.\n *\n * If no route matches a given a request, the Router will use a \"default\"\n * handler if one is defined.\n *\n * Should the matching Route throw an error, the Router will use a \"catch\"\n * handler if one is defined to gracefully deal with issues and respond with a\n * Request.\n *\n * If a request matches multiple routes, the **earliest** registered route will\n * be used to respond to the request.\n *\n * @memberof module:workbox-routing\n */\nclass Router {\n /**\n * Initializes a new Router.\n */\n constructor() {\n this._routes = new Map();\n }\n /**\n * @return {Map>} routes A `Map` of HTTP\n * method name ('GET', etc.) to an array of all the corresponding `Route`\n * instances that are registered.\n */\n get routes() {\n return this._routes;\n }\n /**\n * Adds a fetch event listener to respond to events when a route matches\n * the event's request.\n */\n addFetchListener() {\n // See https://github.com/Microsoft/TypeScript/issues/28357#issuecomment-436484705\n self.addEventListener('fetch', ((event) => {\n const { request } = event;\n const responsePromise = this.handleRequest({ request, event });\n if (responsePromise) {\n event.respondWith(responsePromise);\n }\n }));\n }\n /**\n * Adds a message event listener for URLs to cache from the window.\n * This is useful to cache resources loaded on the page prior to when the\n * service worker started controlling it.\n *\n * The format of the message data sent from the window should be as follows.\n * Where the `urlsToCache` array may consist of URL strings or an array of\n * URL string + `requestInit` object (the same as you'd pass to `fetch()`).\n *\n * ```\n * {\n * type: 'CACHE_URLS',\n * payload: {\n * urlsToCache: [\n * './script1.js',\n * './script2.js',\n * ['./script3.js', {mode: 'no-cors'}],\n * ],\n * },\n * }\n * ```\n */\n addCacheListener() {\n // See https://github.com/Microsoft/TypeScript/issues/28357#issuecomment-436484705\n self.addEventListener('message', ((event) => {\n if (event.data && event.data.type === 'CACHE_URLS') {\n const { payload } = event.data;\n if (process.env.NODE_ENV !== 'production') {\n logger.debug(`Caching URLs from the window`, payload.urlsToCache);\n }\n const requestPromises = Promise.all(payload.urlsToCache.map((entry) => {\n if (typeof entry === 'string') {\n entry = [entry];\n }\n const request = new Request(...entry);\n return this.handleRequest({ request });\n // TODO(philipwalton): TypeScript errors without this typecast for\n // some reason (probably a bug). The real type here should work but\n // doesn't: `Array | undefined>`.\n })); // TypeScript\n event.waitUntil(requestPromises);\n // If a MessageChannel was used, reply to the message on success.\n if (event.ports && event.ports[0]) {\n requestPromises.then(() => event.ports[0].postMessage(true));\n }\n }\n }));\n }\n /**\n * Apply the routing rules to a FetchEvent object to get a Response from an\n * appropriate Route's handler.\n *\n * @param {Object} options\n * @param {Request} options.request The request to handle (this is usually\n * from a fetch event, but it does not have to be).\n * @param {FetchEvent} [options.event] The event that triggered the request,\n * if applicable.\n * @return {Promise|undefined} A promise is returned if a\n * registered route can handle the request. If there is no matching\n * route and there's no `defaultHandler`, `undefined` is returned.\n */\n handleRequest({ request, event }) {\n if (process.env.NODE_ENV !== 'production') {\n assert.isInstance(request, Request, {\n moduleName: 'workbox-routing',\n className: 'Router',\n funcName: 'handleRequest',\n paramName: 'options.request',\n });\n }\n const url = new URL(request.url, location.href);\n if (!url.protocol.startsWith('http')) {\n if (process.env.NODE_ENV !== 'production') {\n logger.debug(`Workbox Router only supports URLs that start with 'http'.`);\n }\n return;\n }\n const { params, route } = this.findMatchingRoute({ url, request, event });\n let handler = route && route.handler;\n const debugMessages = [];\n if (process.env.NODE_ENV !== 'production') {\n if (handler) {\n debugMessages.push([\n `Found a route to handle this request:`, route,\n ]);\n if (params) {\n debugMessages.push([\n `Passing the following params to the route's handler:`, params,\n ]);\n }\n }\n }\n // If we don't have a handler because there was no matching route, then\n // fall back to defaultHandler if that's defined.\n if (!handler && this._defaultHandler) {\n if (process.env.NODE_ENV !== 'production') {\n debugMessages.push(`Failed to find a matching route. Falling ` +\n `back to the default handler.`);\n }\n handler = this._defaultHandler;\n }\n if (!handler) {\n if (process.env.NODE_ENV !== 'production') {\n // No handler so Workbox will do nothing. If logs is set of debug\n // i.e. verbose, we should print out this information.\n logger.debug(`No route found for: ${getFriendlyURL(url)}`);\n }\n return;\n }\n if (process.env.NODE_ENV !== 'production') {\n // We have a handler, meaning Workbox is going to handle the route.\n // print the routing details to the console.\n logger.groupCollapsed(`Router is responding to: ${getFriendlyURL(url)}`);\n debugMessages.forEach((msg) => {\n if (Array.isArray(msg)) {\n logger.log(...msg);\n }\n else {\n logger.log(msg);\n }\n });\n logger.groupEnd();\n }\n // Wrap in try and catch in case the handle method throws a synchronous\n // error. It should still callback to the catch handler.\n let responsePromise;\n try {\n responsePromise = handler.handle({ url, request, event, params });\n }\n catch (err) {\n responsePromise = Promise.reject(err);\n }\n if (responsePromise instanceof Promise && this._catchHandler) {\n responsePromise = responsePromise.catch((err) => {\n if (process.env.NODE_ENV !== 'production') {\n // Still include URL here as it will be async from the console group\n // and may not make sense without the URL\n logger.groupCollapsed(`Error thrown when responding to: ` +\n ` ${getFriendlyURL(url)}. Falling back to Catch Handler.`);\n logger.error(`Error thrown by:`, route);\n logger.error(err);\n logger.groupEnd();\n }\n return this._catchHandler.handle({ url, request, event });\n });\n }\n return responsePromise;\n }\n /**\n * Checks a request and URL (and optionally an event) against the list of\n * registered routes, and if there's a match, returns the corresponding\n * route along with any params generated by the match.\n *\n * @param {Object} options\n * @param {URL} options.url\n * @param {Request} options.request The request to match.\n * @param {Event} [options.event] The corresponding event (unless N/A).\n * @return {Object} An object with `route` and `params` properties.\n * They are populated if a matching route was found or `undefined`\n * otherwise.\n */\n findMatchingRoute({ url, request, event }) {\n if (process.env.NODE_ENV !== 'production') {\n assert.isInstance(url, URL, {\n moduleName: 'workbox-routing',\n className: 'Router',\n funcName: 'findMatchingRoute',\n paramName: 'options.url',\n });\n assert.isInstance(request, Request, {\n moduleName: 'workbox-routing',\n className: 'Router',\n funcName: 'findMatchingRoute',\n paramName: 'options.request',\n });\n }\n const routes = this._routes.get(request.method) || [];\n for (const route of routes) {\n let params;\n const matchResult = route.match({ url, request, event });\n if (matchResult) {\n // See https://github.com/GoogleChrome/workbox/issues/2079\n params = matchResult;\n if (Array.isArray(matchResult) && matchResult.length === 0) {\n // Instead of passing an empty array in as params, use undefined.\n params = undefined;\n }\n else if ((matchResult.constructor === Object &&\n Object.keys(matchResult).length === 0)) {\n // Instead of passing an empty object in as params, use undefined.\n params = undefined;\n }\n else if (typeof matchResult === 'boolean') {\n // For the boolean value true (rather than just something truth-y),\n // don't set params.\n // See https://github.com/GoogleChrome/workbox/pull/2134#issuecomment-513924353\n params = undefined;\n }\n // Return early if have a match.\n return { route, params };\n }\n }\n // If no match was found above, return and empty object.\n return {};\n }\n /**\n * Define a default `handler` that's called when no routes explicitly\n * match the incoming request.\n *\n * Without a default handler, unmatched requests will go against the\n * network as if there were no service worker present.\n *\n * @param {module:workbox-routing~handlerCallback} handler A callback\n * function that returns a Promise resulting in a Response.\n */\n setDefaultHandler(handler) {\n this._defaultHandler = normalizeHandler(handler);\n }\n /**\n * If a Route throws an error while handling a request, this `handler`\n * will be called and given a chance to provide a response.\n *\n * @param {module:workbox-routing~handlerCallback} handler A callback\n * function that returns a Promise resulting in a Response.\n */\n setCatchHandler(handler) {\n this._catchHandler = normalizeHandler(handler);\n }\n /**\n * Registers a route with the router.\n *\n * @param {module:workbox-routing.Route} route The route to register.\n */\n registerRoute(route) {\n if (process.env.NODE_ENV !== 'production') {\n assert.isType(route, 'object', {\n moduleName: 'workbox-routing',\n className: 'Router',\n funcName: 'registerRoute',\n paramName: 'route',\n });\n assert.hasMethod(route, 'match', {\n moduleName: 'workbox-routing',\n className: 'Router',\n funcName: 'registerRoute',\n paramName: 'route',\n });\n assert.isType(route.handler, 'object', {\n moduleName: 'workbox-routing',\n className: 'Router',\n funcName: 'registerRoute',\n paramName: 'route',\n });\n assert.hasMethod(route.handler, 'handle', {\n moduleName: 'workbox-routing',\n className: 'Router',\n funcName: 'registerRoute',\n paramName: 'route.handler',\n });\n assert.isType(route.method, 'string', {\n moduleName: 'workbox-routing',\n className: 'Router',\n funcName: 'registerRoute',\n paramName: 'route.method',\n });\n }\n if (!this._routes.has(route.method)) {\n this._routes.set(route.method, []);\n }\n // Give precedence to all of the earlier routes by adding this additional\n // route to the end of the array.\n this._routes.get(route.method).push(route);\n }\n /**\n * Unregisters a route with the router.\n *\n * @param {module:workbox-routing.Route} route The route to unregister.\n */\n unregisterRoute(route) {\n if (!this._routes.has(route.method)) {\n throw new WorkboxError('unregister-route-but-not-found-with-method', {\n method: route.method,\n });\n }\n const routeIndex = this._routes.get(route.method).indexOf(route);\n if (routeIndex > -1) {\n this._routes.get(route.method).splice(routeIndex, 1);\n }\n else {\n throw new WorkboxError('unregister-route-route-not-registered');\n }\n }\n}\nexport { Router };\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\nconst plugins = [];\nexport const precachePlugins = {\n /*\n * @return {Array}\n * @private\n */\n get() {\n return plugins;\n },\n /*\n * @param {Array} newPlugins\n * @private\n */\n add(newPlugins) {\n plugins.push(...newPlugins);\n },\n};\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\nconst _cacheNameDetails = {\n googleAnalytics: 'googleAnalytics',\n precache: 'precache-v2',\n prefix: 'workbox',\n runtime: 'runtime',\n suffix: typeof registration !== 'undefined' ? registration.scope : '',\n};\nconst _createCacheName = (cacheName) => {\n return [_cacheNameDetails.prefix, cacheName, _cacheNameDetails.suffix]\n .filter((value) => value && value.length > 0)\n .join('-');\n};\nconst eachCacheNameDetail = (fn) => {\n for (const key of Object.keys(_cacheNameDetails)) {\n fn(key);\n }\n};\nexport const cacheNames = {\n updateDetails: (details) => {\n eachCacheNameDetail((key) => {\n if (typeof details[key] === 'string') {\n _cacheNameDetails[key] = details[key];\n }\n });\n },\n getGoogleAnalyticsName: (userCacheName) => {\n return userCacheName || _createCacheName(_cacheNameDetails.googleAnalytics);\n },\n getPrecacheName: (userCacheName) => {\n return userCacheName || _createCacheName(_cacheNameDetails.precache);\n },\n getPrefix: () => {\n return _cacheNameDetails.prefix;\n },\n getRuntimeName: (userCacheName) => {\n return userCacheName || _createCacheName(_cacheNameDetails.runtime);\n },\n getSuffix: () => {\n return _cacheNameDetails.suffix;\n },\n};\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\n// Callbacks to be executed whenever there's a quota error.\nconst quotaErrorCallbacks = new Set();\nexport { quotaErrorCallbacks };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\nexport const pluginUtils = {\n filter: (plugins, callbackName) => {\n return plugins.filter((plugin) => callbackName in plugin);\n },\n};\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { assert } from './assert.js';\nimport { executeQuotaErrorCallbacks } from './executeQuotaErrorCallbacks.js';\nimport { getFriendlyURL } from './getFriendlyURL.js';\nimport { logger } from './logger.js';\nimport { pluginUtils } from '../utils/pluginUtils.js';\nimport { WorkboxError } from './WorkboxError.js';\nimport '../_version.js';\n/**\n * Checks the list of plugins for the cacheKeyWillBeUsed callback, and\n * executes any of those callbacks found in sequence. The final `Request` object\n * returned by the last plugin is treated as the cache key for cache reads\n * and/or writes.\n *\n * @param {Object} options\n * @param {Request} options.request\n * @param {string} options.mode\n * @param {Array} [options.plugins=[]]\n * @return {Promise}\n *\n * @private\n * @memberof module:workbox-core\n */\nconst _getEffectiveRequest = async ({ request, mode, plugins = [], }) => {\n const cacheKeyWillBeUsedPlugins = pluginUtils.filter(plugins, \"cacheKeyWillBeUsed\" /* CACHE_KEY_WILL_BE_USED */);\n let effectiveRequest = request;\n for (const plugin of cacheKeyWillBeUsedPlugins) {\n effectiveRequest = await plugin[\"cacheKeyWillBeUsed\" /* CACHE_KEY_WILL_BE_USED */].call(plugin, { mode, request: effectiveRequest });\n if (typeof effectiveRequest === 'string') {\n effectiveRequest = new Request(effectiveRequest);\n }\n if (process.env.NODE_ENV !== 'production') {\n assert.isInstance(effectiveRequest, Request, {\n moduleName: 'Plugin',\n funcName: \"cacheKeyWillBeUsed\" /* CACHE_KEY_WILL_BE_USED */,\n isReturnValueProblem: true,\n });\n }\n }\n return effectiveRequest;\n};\n/**\n * This method will call cacheWillUpdate on the available plugins (or use\n * status === 200) to determine if the Response is safe and valid to cache.\n *\n * @param {Object} options\n * @param {Request} options.request\n * @param {Response} options.response\n * @param {Event} [options.event]\n * @param {Array} [options.plugins=[]]\n * @return {Promise}\n *\n * @private\n * @memberof module:workbox-core\n */\nconst _isResponseSafeToCache = async ({ request, response, event, plugins = [], }) => {\n let responseToCache = response;\n let pluginsUsed = false;\n for (const plugin of plugins) {\n if (\"cacheWillUpdate\" /* CACHE_WILL_UPDATE */ in plugin) {\n pluginsUsed = true;\n const pluginMethod = plugin[\"cacheWillUpdate\" /* CACHE_WILL_UPDATE */];\n responseToCache = await pluginMethod.call(plugin, {\n request,\n response: responseToCache,\n event,\n });\n if (process.env.NODE_ENV !== 'production') {\n if (responseToCache) {\n assert.isInstance(responseToCache, Response, {\n moduleName: 'Plugin',\n funcName: \"cacheWillUpdate\" /* CACHE_WILL_UPDATE */,\n isReturnValueProblem: true,\n });\n }\n }\n if (!responseToCache) {\n break;\n }\n }\n }\n if (!pluginsUsed) {\n if (process.env.NODE_ENV !== 'production') {\n if (responseToCache) {\n if (responseToCache.status !== 200) {\n if (responseToCache.status === 0) {\n logger.warn(`The response for '${request.url}' is an opaque ` +\n `response. The caching strategy that you're using will not ` +\n `cache opaque responses by default.`);\n }\n else {\n logger.debug(`The response for '${request.url}' returned ` +\n `a status code of '${response.status}' and won't be cached as a ` +\n `result.`);\n }\n }\n }\n }\n responseToCache = responseToCache && responseToCache.status === 200 ?\n responseToCache : undefined;\n }\n return responseToCache ? responseToCache : null;\n};\n/**\n * This is a wrapper around cache.match().\n *\n * @param {Object} options\n * @param {string} options.cacheName Name of the cache to match against.\n * @param {Request} options.request The Request that will be used to look up\n * cache entries.\n * @param {Event} [options.event] The event that prompted the action.\n * @param {Object} [options.matchOptions] Options passed to cache.match().\n * @param {Array} [options.plugins=[]] Array of plugins.\n * @return {Response} A cached response if available.\n *\n * @private\n * @memberof module:workbox-core\n */\nconst matchWrapper = async ({ cacheName, request, event, matchOptions, plugins = [], }) => {\n const cache = await self.caches.open(cacheName);\n const effectiveRequest = await _getEffectiveRequest({\n plugins, request, mode: 'read'\n });\n let cachedResponse = await cache.match(effectiveRequest, matchOptions);\n if (process.env.NODE_ENV !== 'production') {\n if (cachedResponse) {\n logger.debug(`Found a cached response in '${cacheName}'.`);\n }\n else {\n logger.debug(`No cached response found in '${cacheName}'.`);\n }\n }\n for (const plugin of plugins) {\n if (\"cachedResponseWillBeUsed\" /* CACHED_RESPONSE_WILL_BE_USED */ in plugin) {\n const pluginMethod = plugin[\"cachedResponseWillBeUsed\" /* CACHED_RESPONSE_WILL_BE_USED */];\n cachedResponse = await pluginMethod.call(plugin, {\n cacheName,\n event,\n matchOptions,\n cachedResponse,\n request: effectiveRequest,\n });\n if (process.env.NODE_ENV !== 'production') {\n if (cachedResponse) {\n assert.isInstance(cachedResponse, Response, {\n moduleName: 'Plugin',\n funcName: \"cachedResponseWillBeUsed\" /* CACHED_RESPONSE_WILL_BE_USED */,\n isReturnValueProblem: true,\n });\n }\n }\n }\n }\n return cachedResponse;\n};\n/**\n * Wrapper around cache.put().\n *\n * Will call `cacheDidUpdate` on plugins if the cache was updated, using\n * `matchOptions` when determining what the old entry is.\n *\n * @param {Object} options\n * @param {string} options.cacheName\n * @param {Request} options.request\n * @param {Response} options.response\n * @param {Event} [options.event]\n * @param {Array} [options.plugins=[]]\n * @param {Object} [options.matchOptions]\n *\n * @private\n * @memberof module:workbox-core\n */\nconst putWrapper = async ({ cacheName, request, response, event, plugins = [], matchOptions, }) => {\n if (process.env.NODE_ENV !== 'production') {\n if (request.method && request.method !== 'GET') {\n throw new WorkboxError('attempt-to-cache-non-get-request', {\n url: getFriendlyURL(request.url),\n method: request.method,\n });\n }\n }\n const effectiveRequest = await _getEffectiveRequest({\n plugins, request, mode: 'write'\n });\n if (!response) {\n if (process.env.NODE_ENV !== 'production') {\n logger.error(`Cannot cache non-existent response for ` +\n `'${getFriendlyURL(effectiveRequest.url)}'.`);\n }\n throw new WorkboxError('cache-put-with-no-response', {\n url: getFriendlyURL(effectiveRequest.url),\n });\n }\n const responseToCache = await _isResponseSafeToCache({\n event,\n plugins,\n response,\n request: effectiveRequest,\n });\n if (!responseToCache) {\n if (process.env.NODE_ENV !== 'production') {\n logger.debug(`Response '${getFriendlyURL(effectiveRequest.url)}' will ` +\n `not be cached.`, responseToCache);\n }\n return;\n }\n const cache = await self.caches.open(cacheName);\n const updatePlugins = pluginUtils.filter(plugins, \"cacheDidUpdate\" /* CACHE_DID_UPDATE */);\n const oldResponse = updatePlugins.length > 0 ?\n await matchWrapper({ cacheName, matchOptions, request: effectiveRequest }) :\n null;\n if (process.env.NODE_ENV !== 'production') {\n logger.debug(`Updating the '${cacheName}' cache with a new Response for ` +\n `${getFriendlyURL(effectiveRequest.url)}.`);\n }\n try {\n await cache.put(effectiveRequest, responseToCache);\n }\n catch (error) {\n // See https://developer.mozilla.org/en-US/docs/Web/API/DOMException#exception-QuotaExceededError\n if (error.name === 'QuotaExceededError') {\n await executeQuotaErrorCallbacks();\n }\n throw error;\n }\n for (const plugin of updatePlugins) {\n await plugin[\"cacheDidUpdate\" /* CACHE_DID_UPDATE */].call(plugin, {\n cacheName,\n event,\n oldResponse,\n newResponse: responseToCache,\n request: effectiveRequest,\n });\n }\n};\nexport const cacheWrapper = {\n put: putWrapper,\n match: matchWrapper,\n};\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { logger } from '../_private/logger.js';\nimport { quotaErrorCallbacks } from '../models/quotaErrorCallbacks.js';\nimport '../_version.js';\n/**\n * Runs all of the callback functions, one at a time sequentially, in the order\n * in which they were registered.\n *\n * @memberof module:workbox-core\n * @private\n */\nasync function executeQuotaErrorCallbacks() {\n if (process.env.NODE_ENV !== 'production') {\n logger.log(`About to run ${quotaErrorCallbacks.size} ` +\n `callbacks to clean up caches.`);\n }\n for (const callback of quotaErrorCallbacks) {\n await callback();\n if (process.env.NODE_ENV !== 'production') {\n logger.log(callback, 'is complete.');\n }\n }\n if (process.env.NODE_ENV !== 'production') {\n logger.log('Finished running callbacks.');\n }\n}\nexport { executeQuotaErrorCallbacks };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { WorkboxError } from './WorkboxError.js';\nimport { logger } from './logger.js';\nimport { assert } from './assert.js';\nimport { getFriendlyURL } from '../_private/getFriendlyURL.js';\nimport { pluginUtils } from '../utils/pluginUtils.js';\nimport '../_version.js';\n/**\n * Wrapper around the fetch API.\n *\n * Will call requestWillFetch on available plugins.\n *\n * @param {Object} options\n * @param {Request|string} options.request\n * @param {Object} [options.fetchOptions]\n * @param {ExtendableEvent} [options.event]\n * @param {Array} [options.plugins=[]]\n * @return {Promise}\n *\n * @private\n * @memberof module:workbox-core\n */\nconst wrappedFetch = async ({ request, fetchOptions, event, plugins = [], }) => {\n if (typeof request === 'string') {\n request = new Request(request);\n }\n // We *should* be able to call `await event.preloadResponse` even if it's\n // undefined, but for some reason, doing so leads to errors in our Node unit\n // tests. To work around that, explicitly check preloadResponse's value first.\n if (event instanceof FetchEvent && event.preloadResponse) {\n const possiblePreloadResponse = await event.preloadResponse;\n if (possiblePreloadResponse) {\n if (process.env.NODE_ENV !== 'production') {\n logger.log(`Using a preloaded navigation response for ` +\n `'${getFriendlyURL(request.url)}'`);\n }\n return possiblePreloadResponse;\n }\n }\n if (process.env.NODE_ENV !== 'production') {\n assert.isInstance(request, Request, {\n paramName: 'request',\n expectedClass: Request,\n moduleName: 'workbox-core',\n className: 'fetchWrapper',\n funcName: 'wrappedFetch',\n });\n }\n const failedFetchPlugins = pluginUtils.filter(plugins, \"fetchDidFail\" /* FETCH_DID_FAIL */);\n // If there is a fetchDidFail plugin, we need to save a clone of the\n // original request before it's either modified by a requestWillFetch\n // plugin or before the original request's body is consumed via fetch().\n const originalRequest = failedFetchPlugins.length > 0 ?\n request.clone() : null;\n try {\n for (const plugin of plugins) {\n if (\"requestWillFetch\" /* REQUEST_WILL_FETCH */ in plugin) {\n const pluginMethod = plugin[\"requestWillFetch\" /* REQUEST_WILL_FETCH */];\n const requestClone = request.clone();\n request = await pluginMethod.call(plugin, {\n request: requestClone,\n event,\n });\n if (process.env.NODE_ENV !== 'production') {\n if (request) {\n assert.isInstance(request, Request, {\n moduleName: 'Plugin',\n funcName: \"cachedResponseWillBeUsed\" /* CACHED_RESPONSE_WILL_BE_USED */,\n isReturnValueProblem: true,\n });\n }\n }\n }\n }\n }\n catch (err) {\n throw new WorkboxError('plugin-error-request-will-fetch', {\n thrownError: err,\n });\n }\n // The request can be altered by plugins with `requestWillFetch` making\n // the original request (Most likely from a `fetch` event) to be different\n // to the Request we make. Pass both to `fetchDidFail` to aid debugging.\n const pluginFilteredRequest = request.clone();\n try {\n let fetchResponse;\n // See https://github.com/GoogleChrome/workbox/issues/1796\n if (request.mode === 'navigate') {\n fetchResponse = await fetch(request);\n }\n else {\n fetchResponse = await fetch(request, fetchOptions);\n }\n if (process.env.NODE_ENV !== 'production') {\n logger.debug(`Network request for ` +\n `'${getFriendlyURL(request.url)}' returned a response with ` +\n `status '${fetchResponse.status}'.`);\n }\n for (const plugin of plugins) {\n if (\"fetchDidSucceed\" /* FETCH_DID_SUCCEED */ in plugin) {\n fetchResponse = await plugin[\"fetchDidSucceed\" /* FETCH_DID_SUCCEED */]\n .call(plugin, {\n event,\n request: pluginFilteredRequest,\n response: fetchResponse,\n });\n if (process.env.NODE_ENV !== 'production') {\n if (fetchResponse) {\n assert.isInstance(fetchResponse, Response, {\n moduleName: 'Plugin',\n funcName: \"fetchDidSucceed\" /* FETCH_DID_SUCCEED */,\n isReturnValueProblem: true,\n });\n }\n }\n }\n }\n return fetchResponse;\n }\n catch (error) {\n if (process.env.NODE_ENV !== 'production') {\n logger.error(`Network request for ` +\n `'${getFriendlyURL(request.url)}' threw an error.`, error);\n }\n for (const plugin of failedFetchPlugins) {\n await plugin[\"fetchDidFail\" /* FETCH_DID_FAIL */].call(plugin, {\n error,\n event,\n originalRequest: originalRequest.clone(),\n request: pluginFilteredRequest.clone(),\n });\n }\n throw error;\n }\n};\nconst fetchWrapper = {\n fetch: wrappedFetch,\n};\nexport { fetchWrapper };\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { assert } from 'workbox-core/_private/assert.js';\nimport { cacheNames } from 'workbox-core/_private/cacheNames.js';\nimport { cacheWrapper } from 'workbox-core/_private/cacheWrapper.js';\nimport { fetchWrapper } from 'workbox-core/_private/fetchWrapper.js';\nimport { logger } from 'workbox-core/_private/logger.js';\nimport { WorkboxError } from 'workbox-core/_private/WorkboxError.js';\nimport { copyResponse } from 'workbox-core/copyResponse.js';\nimport { createCacheKey } from './utils/createCacheKey.js';\nimport { printCleanupDetails } from './utils/printCleanupDetails.js';\nimport { printInstallDetails } from './utils/printInstallDetails.js';\nimport './_version.js';\n/**\n * Performs efficient precaching of assets.\n *\n * @memberof module:workbox-precaching\n */\nclass PrecacheController {\n /**\n * Create a new PrecacheController.\n *\n * @param {string} [cacheName] An optional name for the cache, to override\n * the default precache name.\n */\n constructor(cacheName) {\n this._cacheName = cacheNames.getPrecacheName(cacheName);\n this._urlsToCacheKeys = new Map();\n this._urlsToCacheModes = new Map();\n this._cacheKeysToIntegrities = new Map();\n }\n /**\n * This method will add items to the precache list, removing duplicates\n * and ensuring the information is valid.\n *\n * @param {\n * Array\n * } entries Array of entries to precache.\n */\n addToCacheList(entries) {\n if (process.env.NODE_ENV !== 'production') {\n assert.isArray(entries, {\n moduleName: 'workbox-precaching',\n className: 'PrecacheController',\n funcName: 'addToCacheList',\n paramName: 'entries',\n });\n }\n const urlsToWarnAbout = [];\n for (const entry of entries) {\n // See https://github.com/GoogleChrome/workbox/issues/2259\n if (typeof entry === 'string') {\n urlsToWarnAbout.push(entry);\n }\n else if (entry && entry.revision === undefined) {\n urlsToWarnAbout.push(entry.url);\n }\n const { cacheKey, url } = createCacheKey(entry);\n const cacheMode = (typeof entry !== 'string' && entry.revision) ?\n 'reload' : 'default';\n if (this._urlsToCacheKeys.has(url) &&\n this._urlsToCacheKeys.get(url) !== cacheKey) {\n throw new WorkboxError('add-to-cache-list-conflicting-entries', {\n firstEntry: this._urlsToCacheKeys.get(url),\n secondEntry: cacheKey,\n });\n }\n if (typeof entry !== 'string' && entry.integrity) {\n if (this._cacheKeysToIntegrities.has(cacheKey) &&\n this._cacheKeysToIntegrities.get(cacheKey) !== entry.integrity) {\n throw new WorkboxError('add-to-cache-list-conflicting-integrities', {\n url,\n });\n }\n this._cacheKeysToIntegrities.set(cacheKey, entry.integrity);\n }\n this._urlsToCacheKeys.set(url, cacheKey);\n this._urlsToCacheModes.set(url, cacheMode);\n if (urlsToWarnAbout.length > 0) {\n const warningMessage = `Workbox is precaching URLs without revision ` +\n `info: ${urlsToWarnAbout.join(', ')}\\nThis is generally NOT safe. ` +\n `Learn more at https://bit.ly/wb-precache`;\n if (process.env.NODE_ENV === 'production') {\n // Use console directly to display this warning without bloating\n // bundle sizes by pulling in all of the logger codebase in prod.\n console.warn(warningMessage);\n }\n else {\n logger.warn(warningMessage);\n }\n }\n }\n }\n /**\n * Precaches new and updated assets. Call this method from the service worker\n * install event.\n *\n * @param {Object} options\n * @param {Event} [options.event] The install event (if needed).\n * @param {Array} [options.plugins] Plugins to be used for fetching\n * and caching during install.\n * @return {Promise}\n */\n async install({ event, plugins } = {}) {\n if (process.env.NODE_ENV !== 'production') {\n if (plugins) {\n assert.isArray(plugins, {\n moduleName: 'workbox-precaching',\n className: 'PrecacheController',\n funcName: 'install',\n paramName: 'plugins',\n });\n }\n }\n const toBePrecached = [];\n const alreadyPrecached = [];\n const cache = await self.caches.open(this._cacheName);\n const alreadyCachedRequests = await cache.keys();\n const existingCacheKeys = new Set(alreadyCachedRequests.map((request) => request.url));\n for (const [url, cacheKey] of this._urlsToCacheKeys) {\n if (existingCacheKeys.has(cacheKey)) {\n alreadyPrecached.push(url);\n }\n else {\n toBePrecached.push({ cacheKey, url });\n }\n }\n const precacheRequests = toBePrecached.map(({ cacheKey, url }) => {\n const integrity = this._cacheKeysToIntegrities.get(cacheKey);\n const cacheMode = this._urlsToCacheModes.get(url);\n return this._addURLToCache({\n cacheKey,\n cacheMode,\n event,\n integrity,\n plugins,\n url,\n });\n });\n await Promise.all(precacheRequests);\n const updatedURLs = toBePrecached.map((item) => item.url);\n if (process.env.NODE_ENV !== 'production') {\n printInstallDetails(updatedURLs, alreadyPrecached);\n }\n return {\n updatedURLs,\n notUpdatedURLs: alreadyPrecached,\n };\n }\n /**\n * Deletes assets that are no longer present in the current precache manifest.\n * Call this method from the service worker activate event.\n *\n * @return {Promise}\n */\n async activate() {\n const cache = await self.caches.open(this._cacheName);\n const currentlyCachedRequests = await cache.keys();\n const expectedCacheKeys = new Set(this._urlsToCacheKeys.values());\n const deletedURLs = [];\n for (const request of currentlyCachedRequests) {\n if (!expectedCacheKeys.has(request.url)) {\n await cache.delete(request);\n deletedURLs.push(request.url);\n }\n }\n if (process.env.NODE_ENV !== 'production') {\n printCleanupDetails(deletedURLs);\n }\n return { deletedURLs };\n }\n /**\n * Requests the entry and saves it to the cache if the response is valid.\n * By default, any response with a status code of less than 400 (including\n * opaque responses) is considered valid.\n *\n * If you need to use custom criteria to determine what's valid and what\n * isn't, then pass in an item in `options.plugins` that implements the\n * `cacheWillUpdate()` lifecycle event.\n *\n * @private\n * @param {Object} options\n * @param {string} options.cacheKey The string to use a cache key.\n * @param {string} options.url The URL to fetch and cache.\n * @param {string} [options.cacheMode] The cache mode for the network request.\n * @param {Event} [options.event] The install event (if passed).\n * @param {Array} [options.plugins] An array of plugins to apply to\n * fetch and caching.\n * @param {string} [options.integrity] The value to use for the `integrity`\n * field when making the request.\n */\n async _addURLToCache({ cacheKey, url, cacheMode, event, plugins, integrity }) {\n const request = new Request(url, {\n integrity,\n cache: cacheMode,\n credentials: 'same-origin',\n });\n let response = await fetchWrapper.fetch({\n event,\n plugins,\n request,\n });\n // Allow developers to override the default logic about what is and isn't\n // valid by passing in a plugin implementing cacheWillUpdate(), e.g.\n // a `CacheableResponsePlugin` instance.\n let cacheWillUpdatePlugin;\n for (const plugin of (plugins || [])) {\n if ('cacheWillUpdate' in plugin) {\n cacheWillUpdatePlugin = plugin;\n }\n }\n const isValidResponse = cacheWillUpdatePlugin ?\n // Use a callback if provided. It returns a truthy value if valid.\n // NOTE: invoke the method on the plugin instance so the `this` context\n // is correct.\n await cacheWillUpdatePlugin.cacheWillUpdate({ event, request, response }) :\n // Otherwise, default to considering any response status under 400 valid.\n // This includes, by default, considering opaque responses valid.\n response.status < 400;\n // Consider this a failure, leading to the `install` handler failing, if\n // we get back an invalid response.\n if (!isValidResponse) {\n throw new WorkboxError('bad-precaching-response', {\n url,\n status: response.status,\n });\n }\n // Redirected responses cannot be used to satisfy a navigation request, so\n // any redirected response must be \"copied\" rather than cloned, so the new\n // response doesn't contain the `redirected` flag. See:\n // https://bugs.chromium.org/p/chromium/issues/detail?id=669363&desc=2#c1\n if (response.redirected) {\n response = await copyResponse(response);\n }\n await cacheWrapper.put({\n event,\n plugins,\n response,\n // `request` already uses `url`. We may be able to reuse it.\n request: cacheKey === url ? request : new Request(cacheKey),\n cacheName: this._cacheName,\n matchOptions: {\n ignoreSearch: true,\n },\n });\n }\n /**\n * Returns a mapping of a precached URL to the corresponding cache key, taking\n * into account the revision information for the URL.\n *\n * @return {Map} A URL to cache key mapping.\n */\n getURLsToCacheKeys() {\n return this._urlsToCacheKeys;\n }\n /**\n * Returns a list of all the URLs that have been precached by the current\n * service worker.\n *\n * @return {Array} The precached URLs.\n */\n getCachedURLs() {\n return [...this._urlsToCacheKeys.keys()];\n }\n /**\n * Returns the cache key used for storing a given URL. If that URL is\n * unversioned, like `/index.html', then the cache key will be the original\n * URL with a search parameter appended to it.\n *\n * @param {string} url A URL whose cache key you want to look up.\n * @return {string} The versioned URL that corresponds to a cache key\n * for the original URL, or undefined if that URL isn't precached.\n */\n getCacheKeyForURL(url) {\n const urlObject = new URL(url, location.href);\n return this._urlsToCacheKeys.get(urlObject.href);\n }\n /**\n * This acts as a drop-in replacement for [`cache.match()`](https://developer.mozilla.org/en-US/docs/Web/API/Cache/match)\n * with the following differences:\n *\n * - It knows what the name of the precache is, and only checks in that cache.\n * - It allows you to pass in an \"original\" URL without versioning parameters,\n * and it will automatically look up the correct cache key for the currently\n * active revision of that URL.\n *\n * E.g., `matchPrecache('index.html')` will find the correct precached\n * response for the currently active service worker, even if the actual cache\n * key is `'/index.html?__WB_REVISION__=1234abcd'`.\n *\n * @param {string|Request} request The key (without revisioning parameters)\n * to look up in the precache.\n * @return {Promise}\n */\n async matchPrecache(request) {\n const url = request instanceof Request ? request.url : request;\n const cacheKey = this.getCacheKeyForURL(url);\n if (cacheKey) {\n const cache = await self.caches.open(this._cacheName);\n return cache.match(cacheKey);\n }\n return undefined;\n }\n /**\n * Returns a function that can be used within a\n * {@link module:workbox-routing.Route} that will find a response for the\n * incoming request against the precache.\n *\n * If for an unexpected reason there is a cache miss for the request,\n * this will fall back to retrieving the `Response` via `fetch()` when\n * `fallbackToNetwork` is `true`.\n *\n * @param {boolean} [fallbackToNetwork=true] Whether to attempt to get the\n * response from the network if there's a precache miss.\n * @return {module:workbox-routing~handlerCallback}\n */\n createHandler(fallbackToNetwork = true) {\n return async ({ request }) => {\n try {\n const response = await this.matchPrecache(request);\n if (response) {\n return response;\n }\n // This shouldn't normally happen, but there are edge cases:\n // https://github.com/GoogleChrome/workbox/issues/1441\n throw new WorkboxError('missing-precache-entry', {\n cacheName: this._cacheName,\n url: request instanceof Request ? request.url : request,\n });\n }\n catch (error) {\n if (fallbackToNetwork) {\n if (process.env.NODE_ENV !== 'production') {\n logger.debug(`Unable to respond with precached response. ` +\n `Falling back to network.`, error);\n }\n return fetch(request);\n }\n throw error;\n }\n };\n }\n /**\n * Returns a function that looks up `url` in the precache (taking into\n * account revision information), and returns the corresponding `Response`.\n *\n * If for an unexpected reason there is a cache miss when looking up `url`,\n * this will fall back to retrieving the `Response` via `fetch()` when\n * `fallbackToNetwork` is `true`.\n *\n * @param {string} url The precached URL which will be used to lookup the\n * `Response`.\n * @param {boolean} [fallbackToNetwork=true] Whether to attempt to get the\n * response from the network if there's a precache miss.\n * @return {module:workbox-routing~handlerCallback}\n */\n createHandlerBoundToURL(url, fallbackToNetwork = true) {\n const cacheKey = this.getCacheKeyForURL(url);\n if (!cacheKey) {\n throw new WorkboxError('non-precached-url', { url });\n }\n const handler = this.createHandler(fallbackToNetwork);\n const request = new Request(url);\n return () => handler({ request });\n }\n}\nexport { PrecacheController };\n","export const isNav = event => event.request.mode === 'navigate';\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\nexport const cacheOkAndOpaquePlugin = {\n /**\n * Returns a valid response (to allow caching) if the status is 200 (OK) or\n * 0 (opaque).\n *\n * @param {Object} options\n * @param {Response} options.response\n * @return {Response|null}\n *\n * @private\n */\n cacheWillUpdate: async ({ response }) => {\n if (response.status === 200 || response.status === 0) {\n return response;\n }\n return null;\n },\n};\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { assert } from 'workbox-core/_private/assert.js';\nimport { cacheNames } from 'workbox-core/_private/cacheNames.js';\nimport { cacheWrapper } from 'workbox-core/_private/cacheWrapper.js';\nimport { fetchWrapper } from 'workbox-core/_private/fetchWrapper.js';\nimport { getFriendlyURL } from 'workbox-core/_private/getFriendlyURL.js';\nimport { logger } from 'workbox-core/_private/logger.js';\nimport { WorkboxError } from 'workbox-core/_private/WorkboxError.js';\nimport { messages } from './utils/messages.js';\nimport { cacheOkAndOpaquePlugin } from './plugins/cacheOkAndOpaquePlugin.js';\nimport './_version.js';\n/**\n * An implementation of a\n * [network first]{@link https://developers.google.com/web/fundamentals/instant-and-offline/offline-cookbook/#network-falling-back-to-cache}\n * request strategy.\n *\n * By default, this strategy will cache responses with a 200 status code as\n * well as [opaque responses]{@link https://developers.google.com/web/tools/workbox/guides/handle-third-party-requests}.\n * Opaque responses are are cross-origin requests where the response doesn't\n * support [CORS]{@link https://enable-cors.org/}.\n *\n * If the network request fails, and there is no cache match, this will throw\n * a `WorkboxError` exception.\n *\n * @memberof module:workbox-strategies\n */\nclass NetworkFirst {\n /**\n * @param {Object} options\n * @param {string} options.cacheName Cache name to store and retrieve\n * requests. Defaults to cache names provided by\n * [workbox-core]{@link module:workbox-core.cacheNames}.\n * @param {Array} options.plugins [Plugins]{@link https://developers.google.com/web/tools/workbox/guides/using-plugins}\n * to use in conjunction with this caching strategy.\n * @param {Object} options.fetchOptions Values passed along to the\n * [`init`](https://developer.mozilla.org/en-US/docs/Web/API/WindowOrWorkerGlobalScope/fetch#Parameters)\n * of all fetch() requests made by this strategy.\n * @param {Object} options.matchOptions [`CacheQueryOptions`](https://w3c.github.io/ServiceWorker/#dictdef-cachequeryoptions)\n * @param {number} options.networkTimeoutSeconds If set, any network requests\n * that fail to respond within the timeout will fallback to the cache.\n *\n * This option can be used to combat\n * \"[lie-fi]{@link https://developers.google.com/web/fundamentals/performance/poor-connectivity/#lie-fi}\"\n * scenarios.\n */\n constructor(options = {}) {\n this._cacheName = cacheNames.getRuntimeName(options.cacheName);\n if (options.plugins) {\n const isUsingCacheWillUpdate = options.plugins.some((plugin) => !!plugin.cacheWillUpdate);\n this._plugins = isUsingCacheWillUpdate ?\n options.plugins : [cacheOkAndOpaquePlugin, ...options.plugins];\n }\n else {\n // No plugins passed in, use the default plugin.\n this._plugins = [cacheOkAndOpaquePlugin];\n }\n this._networkTimeoutSeconds = options.networkTimeoutSeconds || 0;\n if (process.env.NODE_ENV !== 'production') {\n if (this._networkTimeoutSeconds) {\n assert.isType(this._networkTimeoutSeconds, 'number', {\n moduleName: 'workbox-strategies',\n className: 'NetworkFirst',\n funcName: 'constructor',\n paramName: 'networkTimeoutSeconds',\n });\n }\n }\n this._fetchOptions = options.fetchOptions;\n this._matchOptions = options.matchOptions;\n }\n /**\n * This method will perform a request strategy and follows an API that\n * will work with the\n * [Workbox Router]{@link module:workbox-routing.Router}.\n *\n * @param {Object} options\n * @param {Request|string} options.request A request to run this strategy for.\n * @param {Event} [options.event] The event that triggered the request.\n * @return {Promise}\n */\n async handle({ event, request }) {\n const logs = [];\n if (typeof request === 'string') {\n request = new Request(request);\n }\n if (process.env.NODE_ENV !== 'production') {\n assert.isInstance(request, Request, {\n moduleName: 'workbox-strategies',\n className: 'NetworkFirst',\n funcName: 'handle',\n paramName: 'makeRequest',\n });\n }\n const promises = [];\n let timeoutId;\n if (this._networkTimeoutSeconds) {\n const { id, promise } = this._getTimeoutPromise({ request, event, logs });\n timeoutId = id;\n promises.push(promise);\n }\n const networkPromise = this._getNetworkPromise({ timeoutId, request, event, logs });\n promises.push(networkPromise);\n // Promise.race() will resolve as soon as the first promise resolves.\n let response = await Promise.race(promises);\n // If Promise.race() resolved with null, it might be due to a network\n // timeout + a cache miss. If that were to happen, we'd rather wait until\n // the networkPromise resolves instead of returning null.\n // Note that it's fine to await an already-resolved promise, so we don't\n // have to check to see if it's still \"in flight\".\n if (!response) {\n response = await networkPromise;\n }\n if (process.env.NODE_ENV !== 'production') {\n logger.groupCollapsed(messages.strategyStart('NetworkFirst', request));\n for (const log of logs) {\n logger.log(log);\n }\n messages.printFinalResponse(response);\n logger.groupEnd();\n }\n if (!response) {\n throw new WorkboxError('no-response', { url: request.url });\n }\n return response;\n }\n /**\n * @param {Object} options\n * @param {Request} options.request\n * @param {Array} options.logs A reference to the logs array\n * @param {Event} [options.event]\n * @return {Promise}\n *\n * @private\n */\n _getTimeoutPromise({ request, logs, event }) {\n let timeoutId;\n const timeoutPromise = new Promise((resolve) => {\n const onNetworkTimeout = async () => {\n if (process.env.NODE_ENV !== 'production') {\n logs.push(`Timing out the network response at ` +\n `${this._networkTimeoutSeconds} seconds.`);\n }\n resolve(await this._respondFromCache({ request, event }));\n };\n timeoutId = setTimeout(onNetworkTimeout, this._networkTimeoutSeconds * 1000);\n });\n return {\n promise: timeoutPromise,\n id: timeoutId,\n };\n }\n /**\n * @param {Object} options\n * @param {number|undefined} options.timeoutId\n * @param {Request} options.request\n * @param {Array} options.logs A reference to the logs Array.\n * @param {Event} [options.event]\n * @return {Promise}\n *\n * @private\n */\n async _getNetworkPromise({ timeoutId, request, logs, event }) {\n let error;\n let response;\n try {\n response = await fetchWrapper.fetch({\n request,\n event,\n fetchOptions: this._fetchOptions,\n plugins: this._plugins,\n });\n }\n catch (err) {\n error = err;\n }\n if (timeoutId) {\n clearTimeout(timeoutId);\n }\n if (process.env.NODE_ENV !== 'production') {\n if (response) {\n logs.push(`Got response from network.`);\n }\n else {\n logs.push(`Unable to get a response from the network. Will respond ` +\n `with a cached response.`);\n }\n }\n if (error || !response) {\n response = await this._respondFromCache({ request, event });\n if (process.env.NODE_ENV !== 'production') {\n if (response) {\n logs.push(`Found a cached response in the '${this._cacheName}'` +\n ` cache.`);\n }\n else {\n logs.push(`No response found in the '${this._cacheName}' cache.`);\n }\n }\n }\n else {\n // Keep the service worker alive while we put the request in the cache\n const responseClone = response.clone();\n const cachePut = cacheWrapper.put({\n cacheName: this._cacheName,\n request,\n response: responseClone,\n event,\n plugins: this._plugins,\n });\n if (event) {\n try {\n // The event has been responded to so we can keep the SW alive to\n // respond to the request\n event.waitUntil(cachePut);\n }\n catch (err) {\n if (process.env.NODE_ENV !== 'production') {\n logger.warn(`Unable to ensure service worker stays alive when ` +\n `updating cache for '${getFriendlyURL(request.url)}'.`);\n }\n }\n }\n }\n return response;\n }\n /**\n * Used if the network timeouts or fails to make the request.\n *\n * @param {Object} options\n * @param {Request} request The request to match in the cache\n * @param {Event} [options.event]\n * @return {Promise}\n *\n * @private\n */\n _respondFromCache({ event, request }) {\n return cacheWrapper.match({\n cacheName: this._cacheName,\n request,\n event,\n matchOptions: this._matchOptions,\n plugins: this._plugins,\n });\n }\n}\nexport { NetworkFirst };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\n/**\n * A class that wraps common IndexedDB functionality in a promise-based API.\n * It exposes all the underlying power and functionality of IndexedDB, but\n * wraps the most commonly used features in a way that's much simpler to use.\n *\n * @private\n */\nexport class DBWrapper {\n /**\n * @param {string} name\n * @param {number} version\n * @param {Object=} [callback]\n * @param {!Function} [callbacks.onupgradeneeded]\n * @param {!Function} [callbacks.onversionchange] Defaults to\n * DBWrapper.prototype._onversionchange when not specified.\n * @private\n */\n constructor(name, version, { onupgradeneeded, onversionchange, } = {}) {\n this._db = null;\n this._name = name;\n this._version = version;\n this._onupgradeneeded = onupgradeneeded;\n this._onversionchange = onversionchange || (() => this.close());\n }\n /**\n * Returns the IDBDatabase instance (not normally needed).\n * @return {IDBDatabase|undefined}\n *\n * @private\n */\n get db() {\n return this._db;\n }\n /**\n * Opens a connected to an IDBDatabase, invokes any onupgradedneeded\n * callback, and added an onversionchange callback to the database.\n *\n * @return {IDBDatabase}\n * @private\n */\n async open() {\n if (this._db)\n return;\n this._db = await new Promise((resolve, reject) => {\n // This flag is flipped to true if the timeout callback runs prior\n // to the request failing or succeeding. Note: we use a timeout instead\n // of an onblocked handler since there are cases where onblocked will\n // never never run. A timeout better handles all possible scenarios:\n // https://github.com/w3c/IndexedDB/issues/223\n let openRequestTimedOut = false;\n setTimeout(() => {\n openRequestTimedOut = true;\n reject(new Error('The open request was blocked and timed out'));\n }, this.OPEN_TIMEOUT);\n const openRequest = indexedDB.open(this._name, this._version);\n openRequest.onerror = () => reject(openRequest.error);\n openRequest.onupgradeneeded = (evt) => {\n if (openRequestTimedOut) {\n openRequest.transaction.abort();\n openRequest.result.close();\n }\n else if (typeof this._onupgradeneeded === 'function') {\n this._onupgradeneeded(evt);\n }\n };\n openRequest.onsuccess = () => {\n const db = openRequest.result;\n if (openRequestTimedOut) {\n db.close();\n }\n else {\n db.onversionchange = this._onversionchange.bind(this);\n resolve(db);\n }\n };\n });\n return this;\n }\n /**\n * Polyfills the native `getKey()` method. Note, this is overridden at\n * runtime if the browser supports the native method.\n *\n * @param {string} storeName\n * @param {*} query\n * @return {Array}\n * @private\n */\n async getKey(storeName, query) {\n return (await this.getAllKeys(storeName, query, 1))[0];\n }\n /**\n * Polyfills the native `getAll()` method. Note, this is overridden at\n * runtime if the browser supports the native method.\n *\n * @param {string} storeName\n * @param {*} query\n * @param {number} count\n * @return {Array}\n * @private\n */\n async getAll(storeName, query, count) {\n return await this.getAllMatching(storeName, { query, count });\n }\n /**\n * Polyfills the native `getAllKeys()` method. Note, this is overridden at\n * runtime if the browser supports the native method.\n *\n * @param {string} storeName\n * @param {*} query\n * @param {number} count\n * @return {Array}\n * @private\n */\n async getAllKeys(storeName, query, count) {\n const entries = await this.getAllMatching(storeName, { query, count, includeKeys: true });\n return entries.map((entry) => entry.key);\n }\n /**\n * Supports flexible lookup in an object store by specifying an index,\n * query, direction, and count. This method returns an array of objects\n * with the signature .\n *\n * @param {string} storeName\n * @param {Object} [opts]\n * @param {string} [opts.index] The index to use (if specified).\n * @param {*} [opts.query]\n * @param {IDBCursorDirection} [opts.direction]\n * @param {number} [opts.count] The max number of results to return.\n * @param {boolean} [opts.includeKeys] When true, the structure of the\n * returned objects is changed from an array of values to an array of\n * objects in the form {key, primaryKey, value}.\n * @return {Array}\n * @private\n */\n async getAllMatching(storeName, { index, query = null, // IE/Edge errors if query === `undefined`.\n direction = 'next', count, includeKeys = false, } = {}) {\n return await this.transaction([storeName], 'readonly', (txn, done) => {\n const store = txn.objectStore(storeName);\n const target = index ? store.index(index) : store;\n const results = [];\n const request = target.openCursor(query, direction);\n request.onsuccess = () => {\n const cursor = request.result;\n if (cursor) {\n results.push(includeKeys ? cursor : cursor.value);\n if (count && results.length >= count) {\n done(results);\n }\n else {\n cursor.continue();\n }\n }\n else {\n done(results);\n }\n };\n });\n }\n /**\n * Accepts a list of stores, a transaction type, and a callback and\n * performs a transaction. A promise is returned that resolves to whatever\n * value the callback chooses. The callback holds all the transaction logic\n * and is invoked with two arguments:\n * 1. The IDBTransaction object\n * 2. A `done` function, that's used to resolve the promise when\n * when the transaction is done, if passed a value, the promise is\n * resolved to that value.\n *\n * @param {Array} storeNames An array of object store names\n * involved in the transaction.\n * @param {string} type Can be `readonly` or `readwrite`.\n * @param {!Function} callback\n * @return {*} The result of the transaction ran by the callback.\n * @private\n */\n async transaction(storeNames, type, callback) {\n await this.open();\n return await new Promise((resolve, reject) => {\n const txn = this._db.transaction(storeNames, type);\n txn.onabort = () => reject(txn.error);\n txn.oncomplete = () => resolve();\n callback(txn, (value) => resolve(value));\n });\n }\n /**\n * Delegates async to a native IDBObjectStore method.\n *\n * @param {string} method The method name.\n * @param {string} storeName The object store name.\n * @param {string} type Can be `readonly` or `readwrite`.\n * @param {...*} args The list of args to pass to the native method.\n * @return {*} The result of the transaction.\n * @private\n */\n async _call(method, storeName, type, ...args) {\n const callback = (txn, done) => {\n const objStore = txn.objectStore(storeName);\n // TODO(philipwalton): Fix this underlying TS2684 error.\n // @ts-ignore\n const request = objStore[method].apply(objStore, args);\n request.onsuccess = () => done(request.result);\n };\n return await this.transaction([storeName], type, callback);\n }\n /**\n * Closes the connection opened by `DBWrapper.open()`. Generally this method\n * doesn't need to be called since:\n * 1. It's usually better to keep a connection open since opening\n * a new connection is somewhat slow.\n * 2. Connections are automatically closed when the reference is\n * garbage collected.\n * The primary use case for needing to close a connection is when another\n * reference (typically in another tab) needs to upgrade it and would be\n * blocked by the current, open connection.\n *\n * @private\n */\n close() {\n if (this._db) {\n this._db.close();\n this._db = null;\n }\n }\n}\n// Exposed on the prototype to let users modify the default timeout on a\n// per-instance or global basis.\nDBWrapper.prototype.OPEN_TIMEOUT = 2000;\n// Wrap native IDBObjectStore methods according to their mode.\nconst methodsToWrap = {\n readonly: ['get', 'count', 'getKey', 'getAll', 'getAllKeys'],\n readwrite: ['add', 'put', 'clear', 'delete'],\n};\nfor (const [mode, methods] of Object.entries(methodsToWrap)) {\n for (const method of methods) {\n if (method in IDBObjectStore.prototype) {\n // Don't use arrow functions here since we're outside of the class.\n DBWrapper.prototype[method] =\n async function (storeName, ...args) {\n return await this._call(method, storeName, mode, ...args);\n };\n }\n }\n}\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { getOrCreateDefaultRouter } from './utils/getOrCreateDefaultRouter.js';\nimport './_version.js';\n/**\n * If a Route throws an error while handling a request, this `handler`\n * will be called and given a chance to provide a response.\n *\n * @param {module:workbox-routing~handlerCallback} handler A callback\n * function that returns a Promise resulting in a Response.\n *\n * @memberof module:workbox-routing\n */\nfunction setCatchHandler(handler) {\n const defaultRouter = getOrCreateDefaultRouter();\n defaultRouter.setCatchHandler(handler);\n}\nexport { setCatchHandler };\n","import { registerRoute, setCatchHandler } from 'workbox-routing';\nimport { precacheAndRoute, getCacheKeyForURL } from 'workbox-precaching';\nimport { isNav } from './utils';\nimport { NETWORK_HANDLER, PRECACHING_OPTIONS } from './contants';\n\nexport function getFiles() {\n\treturn self.__WB_MANIFEST;\n}\n\nexport function setupPrecaching(precacheFiles, precachingOptions) {\n\tprecacheAndRoute(precacheFiles, precachingOptions || PRECACHING_OPTIONS);\n}\n\nexport function setupRouting() {\n\t/**\n\t * Adding this before `precacheAndRoute` lets us handle all\n\t * the navigation requests even if they are in precache.\n\t */\n\tregisterRoute(({ event }) => isNav(event), NETWORK_HANDLER);\n\n\tsetCatchHandler(({ event }) => {\n\t\tif (isNav(event)) {\n\t\t\treturn caches.match(getCacheKeyForURL('/index.html'));\n\t\t}\n\t\treturn Response.error();\n\t});\n}\n\nexport { PRECACHING_OPTIONS };\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { cacheNames as _cacheNames } from './_private/cacheNames.js';\nimport './_version.js';\n/**\n * Get the current cache names and prefix/suffix used by Workbox.\n *\n * `cacheNames.precache` is used for precached assets,\n * `cacheNames.googleAnalytics` is used by `workbox-google-analytics` to\n * store `analytics.js`, and `cacheNames.runtime` is used for everything else.\n *\n * `cacheNames.prefix` can be used to retrieve just the current prefix value.\n * `cacheNames.suffix` can be used to retrieve just the current suffix value.\n *\n * @return {Object} An object with `precache`, `runtime`, `prefix`, and\n * `googleAnalytics` properties.\n *\n * @memberof module:workbox-core\n */\nconst cacheNames = {\n get googleAnalytics() {\n return _cacheNames.getGoogleAnalyticsName();\n },\n get precache() {\n return _cacheNames.getPrecacheName();\n },\n get prefix() {\n return _cacheNames.getPrefix();\n },\n get runtime() {\n return _cacheNames.getRuntimeName();\n },\n get suffix() {\n return _cacheNames.getSuffix();\n },\n};\nexport { cacheNames };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { assert } from 'workbox-core/_private/assert.js';\nimport { WorkboxError } from 'workbox-core/_private/WorkboxError.js';\nimport { getFriendlyURL } from 'workbox-core/_private/getFriendlyURL.js';\nimport { logger } from 'workbox-core/_private/logger.js';\nimport './_version.js';\n/**\n * This class allows you to set up rules determining what\n * status codes and/or headers need to be present in order for a\n * [`Response`](https://developer.mozilla.org/en-US/docs/Web/API/Response)\n * to be considered cacheable.\n *\n * @memberof module:workbox-cacheable-response\n */\nclass CacheableResponse {\n /**\n * To construct a new CacheableResponse instance you must provide at least\n * one of the `config` properties.\n *\n * If both `statuses` and `headers` are specified, then both conditions must\n * be met for the `Response` to be considered cacheable.\n *\n * @param {Object} config\n * @param {Array} [config.statuses] One or more status codes that a\n * `Response` can have and be considered cacheable.\n * @param {Object} [config.headers] A mapping of header names\n * and expected values that a `Response` can have and be considered cacheable.\n * If multiple headers are provided, only one needs to be present.\n */\n constructor(config = {}) {\n if (process.env.NODE_ENV !== 'production') {\n if (!(config.statuses || config.headers)) {\n throw new WorkboxError('statuses-or-headers-required', {\n moduleName: 'workbox-cacheable-response',\n className: 'CacheableResponse',\n funcName: 'constructor',\n });\n }\n if (config.statuses) {\n assert.isArray(config.statuses, {\n moduleName: 'workbox-cacheable-response',\n className: 'CacheableResponse',\n funcName: 'constructor',\n paramName: 'config.statuses',\n });\n }\n if (config.headers) {\n assert.isType(config.headers, 'object', {\n moduleName: 'workbox-cacheable-response',\n className: 'CacheableResponse',\n funcName: 'constructor',\n paramName: 'config.headers',\n });\n }\n }\n this._statuses = config.statuses;\n this._headers = config.headers;\n }\n /**\n * Checks a response to see whether it's cacheable or not, based on this\n * object's configuration.\n *\n * @param {Response} response The response whose cacheability is being\n * checked.\n * @return {boolean} `true` if the `Response` is cacheable, and `false`\n * otherwise.\n */\n isResponseCacheable(response) {\n if (process.env.NODE_ENV !== 'production') {\n assert.isInstance(response, Response, {\n moduleName: 'workbox-cacheable-response',\n className: 'CacheableResponse',\n funcName: 'isResponseCacheable',\n paramName: 'response',\n });\n }\n let cacheable = true;\n if (this._statuses) {\n cacheable = this._statuses.includes(response.status);\n }\n if (this._headers && cacheable) {\n cacheable = Object.keys(this._headers).some((headerName) => {\n return response.headers.get(headerName) === this._headers[headerName];\n });\n }\n if (process.env.NODE_ENV !== 'production') {\n if (!cacheable) {\n logger.groupCollapsed(`The request for ` +\n `'${getFriendlyURL(response.url)}' returned a response that does ` +\n `not meet the criteria for being cached.`);\n logger.groupCollapsed(`View cacheability criteria here.`);\n logger.log(`Cacheable statuses: ` +\n JSON.stringify(this._statuses));\n logger.log(`Cacheable headers: ` +\n JSON.stringify(this._headers, null, 2));\n logger.groupEnd();\n const logFriendlyHeaders = {};\n response.headers.forEach((value, key) => {\n logFriendlyHeaders[key] = value;\n });\n logger.groupCollapsed(`View response status and headers here.`);\n logger.log(`Response status: ` + response.status);\n logger.log(`Response headers: ` +\n JSON.stringify(logFriendlyHeaders, null, 2));\n logger.groupEnd();\n logger.groupCollapsed(`View full response details here.`);\n logger.log(response.headers);\n logger.log(response);\n logger.groupEnd();\n logger.groupEnd();\n }\n }\n return cacheable;\n }\n}\nexport { CacheableResponse };\n","import { NetworkFirst } from 'workbox-strategies';\nimport { cacheNames } from 'workbox-core';\nimport { CacheableResponsePlugin } from 'workbox-cacheable-response';\n\nexport const PRECACHING_OPTIONS = {};\n\nexport const NETWORK_HANDLER = new NetworkFirst({\n\t// this cache is plunged with every new service worker deploy so we dont need to care about purging the cache.\n\tcacheName: cacheNames.precache,\n\tnetworkTimeoutSeconds: 5, // if u dont start getting headers within 5 sec fallback to cache.\n\tplugins: [\n\t\tnew CacheableResponsePlugin({\n\t\t\tstatuses: [200], // only cache valid responses, not opaque responses e.g. wifi portal.\n\t\t}),\n\t],\n});\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { logger } from 'workbox-core/_private/logger.js';\nimport { WorkboxError } from 'workbox-core/_private/WorkboxError.js';\nimport { Route } from './Route.js';\nimport { RegExpRoute } from './RegExpRoute.js';\nimport { getOrCreateDefaultRouter } from './utils/getOrCreateDefaultRouter.js';\nimport './_version.js';\n/**\n * Easily register a RegExp, string, or function with a caching\n * strategy to a singleton Router instance.\n *\n * This method will generate a Route for you if needed and\n * call [registerRoute()]{@link module:workbox-routing.Router#registerRoute}.\n *\n * @param {RegExp|string|module:workbox-routing.Route~matchCallback|module:workbox-routing.Route} capture\n * If the capture param is a `Route`, all other arguments will be ignored.\n * @param {module:workbox-routing~handlerCallback} [handler] A callback\n * function that returns a Promise resulting in a Response. This parameter\n * is required if `capture` is not a `Route` object.\n * @param {string} [method='GET'] The HTTP method to match the Route\n * against.\n * @return {module:workbox-routing.Route} The generated `Route`(Useful for\n * unregistering).\n *\n * @memberof module:workbox-routing\n */\nfunction registerRoute(capture, handler, method) {\n let route;\n if (typeof capture === 'string') {\n const captureUrl = new URL(capture, location.href);\n if (process.env.NODE_ENV !== 'production') {\n if (!(capture.startsWith('/') || capture.startsWith('http'))) {\n throw new WorkboxError('invalid-string', {\n moduleName: 'workbox-routing',\n funcName: 'registerRoute',\n paramName: 'capture',\n });\n }\n // We want to check if Express-style wildcards are in the pathname only.\n // TODO: Remove this log message in v4.\n const valueToCheck = capture.startsWith('http') ?\n captureUrl.pathname : capture;\n // See https://github.com/pillarjs/path-to-regexp#parameters\n const wildcards = '[*:?+]';\n if ((new RegExp(`${wildcards}`)).exec(valueToCheck)) {\n logger.debug(`The '$capture' parameter contains an Express-style wildcard ` +\n `character (${wildcards}). Strings are now always interpreted as ` +\n `exact matches; use a RegExp for partial or wildcard matches.`);\n }\n }\n const matchCallback = ({ url }) => {\n if (process.env.NODE_ENV !== 'production') {\n if ((url.pathname === captureUrl.pathname) &&\n (url.origin !== captureUrl.origin)) {\n logger.debug(`${capture} only partially matches the cross-origin URL ` +\n `${url}. This route will only handle cross-origin requests ` +\n `if they match the entire URL.`);\n }\n }\n return url.href === captureUrl.href;\n };\n // If `capture` is a string then `handler` and `method` must be present.\n route = new Route(matchCallback, handler, method);\n }\n else if (capture instanceof RegExp) {\n // If `capture` is a `RegExp` then `handler` and `method` must be present.\n route = new RegExpRoute(capture, handler, method);\n }\n else if (typeof capture === 'function') {\n // If `capture` is a function then `handler` and `method` must be present.\n route = new Route(capture, handler, method);\n }\n else if (capture instanceof Route) {\n route = capture;\n }\n else {\n throw new WorkboxError('unsupported-route-type', {\n moduleName: 'workbox-routing',\n funcName: 'registerRoute',\n paramName: 'capture',\n });\n }\n const defaultRouter = getOrCreateDefaultRouter();\n defaultRouter.registerRoute(route);\n return route;\n}\nexport { registerRoute };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { CacheableResponse } from './CacheableResponse.js';\nimport './_version.js';\n/**\n * A class implementing the `cacheWillUpdate` lifecycle callback. This makes it\n * easier to add in cacheability checks to requests made via Workbox's built-in\n * strategies.\n *\n * @memberof module:workbox-cacheable-response\n */\nclass CacheableResponsePlugin {\n /**\n * To construct a new CacheableResponsePlugin instance you must provide at\n * least one of the `config` properties.\n *\n * If both `statuses` and `headers` are specified, then both conditions must\n * be met for the `Response` to be considered cacheable.\n *\n * @param {Object} config\n * @param {Array} [config.statuses] One or more status codes that a\n * `Response` can have and be considered cacheable.\n * @param {Object} [config.headers] A mapping of header names\n * and expected values that a `Response` can have and be considered cacheable.\n * If multiple headers are provided, only one needs to be present.\n */\n constructor(config) {\n /**\n * @param {Object} options\n * @param {Response} options.response\n * @return {Response|null}\n * @private\n */\n this.cacheWillUpdate = async ({ response }) => {\n if (this._cacheableResponse.isResponseCacheable(response)) {\n return response;\n }\n return null;\n };\n this._cacheableResponse = new CacheableResponse(config);\n }\n}\nexport { CacheableResponsePlugin };\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { getOrCreatePrecacheController } from './utils/getOrCreatePrecacheController.js';\nimport './_version.js';\n/**\n * Takes in a URL, and returns the corresponding URL that could be used to\n * lookup the entry in the precache.\n *\n * If a relative URL is provided, the location of the service worker file will\n * be used as the base.\n *\n * For precached entries without revision information, the cache key will be the\n * same as the original URL.\n *\n * For precached entries with revision information, the cache key will be the\n * original URL with the 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\"p5Qz\");\n","\"use strict\";\n// @ts-ignore\ntry {\n self['workbox:cacheable-response:5.1.3'] && _();\n}\ncatch (e) { }\n","\"use strict\";\n// @ts-ignore\ntry {\n self['workbox:precaching:5.1.3'] && _();\n}\ncatch (e) { }\n","\"use strict\";\n// @ts-ignore\ntry {\n self['workbox:routing:5.1.3'] && _();\n}\ncatch (e) { }\n","\"use strict\";\n// @ts-ignore\ntry {\n self['workbox:strategies:5.1.3'] && _();\n}\ncatch (e) { }\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { canConstructResponseFromBodyStream } from './_private/canConstructResponseFromBodyStream.js';\nimport './_version.js';\n/**\n * Allows developers to copy a response and modify its `headers`, `status`,\n * or `statusText` values (the values settable via a\n * [`ResponseInit`]{@link https://developer.mozilla.org/en-US/docs/Web/API/Response/Response#Syntax}\n * object in the constructor).\n * To modify these values, pass a function as the second argument. That\n * function will be invoked with a single object with the response properties\n * `{headers, status, statusText}`. The return value of this function will\n * be used as the `ResponseInit` for the new `Response`. To change the values\n * either modify the passed parameter(s) and return it, or return a totally\n * new object.\n *\n * @param {Response} response\n * @param {Function} modifier\n * @memberof module:workbox-core\n */\nasync function copyResponse(response, modifier) {\n const clonedResponse = response.clone();\n // Create a fresh `ResponseInit` object by cloning the headers.\n const responseInit = {\n headers: new Headers(clonedResponse.headers),\n status: clonedResponse.status,\n statusText: clonedResponse.statusText,\n };\n // Apply any user modifications.\n const modifiedResponseInit = modifier ? modifier(responseInit) : responseInit;\n // Create the new response from the body stream and `ResponseInit`\n // modifications. 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Return as-is.\n if (typeof entry === 'string') {\n const urlObject = new URL(entry, location.href);\n return {\n cacheKey: urlObject.href,\n url: urlObject.href,\n };\n }\n const { revision, url } = entry;\n if (!url) {\n throw new WorkboxError('add-to-cache-list-unexpected-type', { entry });\n }\n // If there's just a URL and no revision, then it's also assumed to be a\n // versioned URL.\n if (!revision) {\n const urlObject = new URL(url, location.href);\n return {\n cacheKey: urlObject.href,\n url: urlObject.href,\n };\n }\n // Otherwise, construct a properly versioned URL using the custom Workbox\n // search parameter along with the revision info.\n const cacheKeyURL = new URL(url, location.href);\n const originalURL = new URL(url, location.href);\n cacheKeyURL.searchParams.set(REVISION_SEARCH_PARAM, revision);\n return {\n cacheKey: cacheKeyURL.href,\n url: originalURL.href,\n };\n}\n","/*\n Copyright 2019 Google LLC\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { addFetchListener } from './utils/addFetchListener.js';\nimport './_version.js';\nlet listenerAdded = false;\n/**\n * Add a `fetch` listener to the service worker that will\n * respond to\n * [network requests]{@link https://developer.mozilla.org/en-US/docs/Web/API/Service_Worker_API/Using_Service_Workers#Custom_responses_to_requests}\n * with precached assets.\n *\n * Requests for assets that aren't precached, the `FetchEvent` will not be\n * responded to, allowing the event to fall through to other `fetch` event\n * listeners.\n *\n * @param {Object} [options]\n * @param {string} [options.directoryIndex=index.html] The `directoryIndex` will\n * check cache entries for a URLs ending with '/' to see if there is a hit when\n * appending the `directoryIndex` value.\n * @param {Array} [options.ignoreURLParametersMatching=[/^utm_/]] An\n * array of regex's to remove search params when looking for a cache match.\n * @param {boolean} [options.cleanURLs=true] The `cleanURLs` option will\n * check the cache for the URL with a `.html` added to the end of the end.\n * @param {module:workbox-precaching~urlManipulation} [options.urlManipulation]\n * This is a function that should take a URL and return an array of\n * alternative URLs that should be checked for precache matches.\n *\n * @memberof module:workbox-precaching\n */\nfunction addRoute(options) {\n if (!listenerAdded) {\n addFetchListener(options);\n listenerAdded = true;\n }\n}\nexport { addRoute };\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { cacheNames } from 'workbox-core/_private/cacheNames.js';\nimport { getFriendlyURL } from 'workbox-core/_private/getFriendlyURL.js';\nimport { logger } from 'workbox-core/_private/logger.js';\nimport { getCacheKeyForURL } from './getCacheKeyForURL.js';\nimport '../_version.js';\n/**\n * Adds a `fetch` listener to the service worker that will\n * respond to\n * [network requests]{@link https://developer.mozilla.org/en-US/docs/Web/API/Service_Worker_API/Using_Service_Workers#Custom_responses_to_requests}\n * with precached assets.\n *\n * Requests for assets that aren't precached, the `FetchEvent` will not be\n * responded to, allowing the event to fall through to other `fetch` event\n * listeners.\n *\n * NOTE: when called more than once this method will replace the previously set\n * configuration options. Calling it more than once is not recommended outside\n * of tests.\n *\n * @private\n * @param {Object} [options]\n * @param {string} [options.directoryIndex=index.html] The `directoryIndex` will\n * check cache entries for a URLs ending with '/' to see if there is a hit when\n * appending the `directoryIndex` value.\n * @param {Array} [options.ignoreURLParametersMatching=[/^utm_/]] An\n * array of regex's to remove search params when looking for a cache match.\n * @param {boolean} [options.cleanURLs=true] The `cleanURLs` option will\n * check the cache for the URL with a `.html` added to the end of the end.\n * @param {workbox.precaching~urlManipulation} [options.urlManipulation]\n * This is a function that should take a URL and return an array of\n * alternative URLs that should be checked for precache matches.\n */\nexport const addFetchListener = ({ ignoreURLParametersMatching = [/^utm_/], directoryIndex = 'index.html', cleanURLs = true, urlManipulation, } = {}) => {\n const cacheName = cacheNames.getPrecacheName();\n // See https://github.com/Microsoft/TypeScript/issues/28357#issuecomment-436484705\n self.addEventListener('fetch', ((event) => {\n const precachedURL = getCacheKeyForURL(event.request.url, {\n cleanURLs,\n directoryIndex,\n ignoreURLParametersMatching,\n urlManipulation,\n });\n if (!precachedURL) {\n if (process.env.NODE_ENV !== 'production') {\n logger.debug(`Precaching did not find a match for ` +\n getFriendlyURL(event.request.url));\n }\n return;\n }\n let responsePromise = self.caches.open(cacheName).then((cache) => {\n return cache.match(precachedURL);\n }).then((cachedResponse) => {\n if (cachedResponse) {\n return cachedResponse;\n }\n // Fall back to the network if we don't have a cached response\n // (perhaps due to manual cache cleanup).\n if (process.env.NODE_ENV !== 'production') {\n logger.warn(`The precached response for ` +\n `${getFriendlyURL(precachedURL)} in ${cacheName} was not found. ` +\n `Falling back to the network instead.`);\n }\n return fetch(precachedURL);\n });\n if (process.env.NODE_ENV !== 'production') {\n responsePromise = responsePromise.then((response) => {\n // Workbox is going to handle the route.\n // print the routing details to the console.\n logger.groupCollapsed(`Precaching is responding to: ` +\n getFriendlyURL(event.request.url));\n logger.log(`Serving the precached url: ${precachedURL}`);\n logger.groupCollapsed(`View request details here.`);\n logger.log(event.request);\n logger.groupEnd();\n logger.groupCollapsed(`View response details here.`);\n logger.log(response);\n logger.groupEnd();\n logger.groupEnd();\n return response;\n });\n }\n event.respondWith(responsePromise);\n }));\n};\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { getOrCreatePrecacheController } from './getOrCreatePrecacheController.js';\nimport { generateURLVariations } from './generateURLVariations.js';\nimport '../_version.js';\n/**\n * This function will take the request URL and manipulate it based on the\n * configuration options.\n *\n * @param {string} url\n * @param {Object} options\n * @return {string} Returns the URL in the cache that matches the request,\n * if possible.\n *\n * @private\n */\nexport const getCacheKeyForURL = (url, options) => {\n const precacheController = getOrCreatePrecacheController();\n const urlsToCacheKeys = precacheController.getURLsToCacheKeys();\n for (const possibleURL of generateURLVariations(url, options)) {\n const possibleCacheKey = urlsToCacheKeys.get(possibleURL);\n if (possibleCacheKey) {\n return possibleCacheKey;\n }\n }\n};\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { removeIgnoredSearchParams } from './removeIgnoredSearchParams.js';\nimport '../_version.js';\n/**\n * Generator function that yields possible variations on the original URL to\n * check, one at a time.\n *\n * @param {string} url\n * @param {Object} options\n *\n * @private\n * @memberof module:workbox-precaching\n */\nexport function* generateURLVariations(url, { ignoreURLParametersMatching, directoryIndex, cleanURLs, urlManipulation, } = {}) {\n const urlObject = new URL(url, location.href);\n urlObject.hash = '';\n yield urlObject.href;\n const urlWithoutIgnoredParams = removeIgnoredSearchParams(urlObject, ignoreURLParametersMatching);\n yield urlWithoutIgnoredParams.href;\n if (directoryIndex && urlWithoutIgnoredParams.pathname.endsWith('/')) {\n const directoryURL = new URL(urlWithoutIgnoredParams.href);\n directoryURL.pathname += directoryIndex;\n yield directoryURL.href;\n }\n if (cleanURLs) {\n const cleanURL = new URL(urlWithoutIgnoredParams.href);\n cleanURL.pathname += '.html';\n yield cleanURL.href;\n }\n if (urlManipulation) {\n const additionalURLs = urlManipulation({ url: urlObject });\n for (const urlToAttempt of additionalURLs) {\n yield urlToAttempt.href;\n }\n }\n}\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\n/**\n * Removes any URL search parameters that should be ignored.\n *\n * @param {URL} urlObject The original URL.\n * @param {Array} ignoreURLParametersMatching RegExps to test against\n * each search parameter name. Matches mean that the search parameter should be\n * ignored.\n * @return {URL} The URL with any ignored search parameters removed.\n *\n * @private\n * @memberof module:workbox-precaching\n */\nexport function removeIgnoredSearchParams(urlObject, ignoreURLParametersMatching = []) {\n // Convert the iterable into an array at the start of the loop to make sure\n // deletion doesn't mess up iteration.\n for (const paramName of [...urlObject.searchParams.keys()]) {\n if (ignoreURLParametersMatching.some((regExp) => regExp.test(paramName))) {\n urlObject.searchParams.delete(paramName);\n }\n }\n return urlObject;\n}\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { addRoute } from './addRoute.js';\nimport { precache } from './precache.js';\nimport './_version.js';\n/**\n * This method will add entries to the precache list and add a route to\n * respond to fetch events.\n *\n * This is a convenience method that will call\n * [precache()]{@link module:workbox-precaching.precache} and\n * [addRoute()]{@link module:workbox-precaching.addRoute} in a single call.\n *\n * @param {Array} entries Array of entries to precache.\n * @param {Object} [options] See\n * [addRoute() options]{@link module:workbox-precaching.addRoute}.\n *\n * @memberof module:workbox-precaching\n */\nfunction precacheAndRoute(entries, options) {\n precache(entries);\n addRoute(options);\n}\nexport { precacheAndRoute };\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { logger } from 'workbox-core/_private/logger.js';\nimport { getOrCreatePrecacheController } from './utils/getOrCreatePrecacheController.js';\nimport { precachePlugins } from './utils/precachePlugins.js';\nimport './_version.js';\nconst installListener = (event) => {\n const precacheController = getOrCreatePrecacheController();\n const plugins = precachePlugins.get();\n event.waitUntil(precacheController.install({ event, plugins })\n .catch((error) => {\n if (process.env.NODE_ENV !== 'production') {\n logger.error(`Service worker installation failed. It will ` +\n `be retried automatically during the next navigation.`);\n }\n // Re-throw the error to ensure installation fails.\n throw error;\n }));\n};\nconst activateListener = (event) => {\n const precacheController = getOrCreatePrecacheController();\n event.waitUntil(precacheController.activate());\n};\n/**\n * Adds items to the precache list, removing any duplicates and\n * stores the files in the\n * [\"precache cache\"]{@link module:workbox-core.cacheNames} when the service\n * worker installs.\n *\n * This method can be called multiple times.\n *\n * Please note: This method **will not** serve any of the cached files for you.\n * It only precaches files. To respond to a network request you call\n * [addRoute()]{@link module:workbox-precaching.addRoute}.\n *\n * If you have a single array of files to precache, you can just call\n * [precacheAndRoute()]{@link module:workbox-precaching.precacheAndRoute}.\n *\n * @param {Array} [entries=[]] Array of entries to precache.\n *\n * @memberof module:workbox-precaching\n */\nfunction precache(entries) {\n const precacheController = getOrCreatePrecacheController();\n precacheController.addToCacheList(entries);\n if (entries.length > 0) {\n // NOTE: these listeners will only be added once (even if the `precache()`\n // method is called multiple times) because event listeners are implemented\n // as a set, where each listener must be unique.\n // See https://github.com/Microsoft/TypeScript/issues/28357#issuecomment-436484705\n self.addEventListener('install', installListener);\n self.addEventListener('activate', activateListener);\n }\n}\nexport { precache };\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { Router } from '../Router.js';\nimport '../_version.js';\nlet defaultRouter;\n/**\n * Creates a new, singleton Router instance if one does not exist. If one\n * does already exist, that instance is returned.\n *\n * @private\n * @return {Router}\n */\nexport const getOrCreateDefaultRouter = () => {\n if (!defaultRouter) {\n defaultRouter = new Router();\n // The helpers that use the default Router assume these listeners exist.\n defaultRouter.addFetchListener();\n defaultRouter.addCacheListener();\n }\n return defaultRouter;\n};\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { PrecacheController } from '../PrecacheController.js';\nimport '../_version.js';\nlet precacheController;\n/**\n * @return {PrecacheController}\n * @private\n */\nexport const getOrCreatePrecacheController = () => {\n if (!precacheController) {\n precacheController = new PrecacheController();\n }\n return precacheController;\n};\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { messages } from './messages.js';\nimport '../../_version.js';\nconst fallback = (code, ...args) => {\n let msg = code;\n if (args.length > 0) {\n msg += ` :: ${JSON.stringify(args)}`;\n }\n return msg;\n};\nconst generatorFunction = (code, details = {}) => {\n const message = messages[code];\n if (!message) {\n throw new Error(`Unable to find message for code '${code}'.`);\n }\n return message(details);\n};\nexport const messageGenerator = (process.env.NODE_ENV === 'production') ?\n fallback : generatorFunction;\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { messageGenerator } from '../models/messages/messageGenerator.js';\nimport '../_version.js';\n/**\n * Workbox errors should be thrown with this class.\n * This allows use to ensure the type easily in tests,\n * helps developers identify errors from workbox\n * easily and allows use to optimise error\n * messages correctly.\n *\n * @private\n */\nclass WorkboxError extends Error {\n /**\n *\n * @param {string} errorCode The error code that\n * identifies this particular error.\n * @param {Object=} details Any relevant arguments\n * that will help developers identify issues should\n * be added as a key on the context object.\n */\n constructor(errorCode, details) {\n const message = messageGenerator(errorCode, details);\n super(message);\n this.name = errorCode;\n this.details = details;\n }\n}\nexport { WorkboxError };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { assert } from 'workbox-core/_private/assert.js';\nimport '../_version.js';\n/**\n * @param {function()|Object} handler Either a function, or an object with a\n * 'handle' method.\n * @return {Object} An object with a handle method.\n *\n * @private\n */\nexport const normalizeHandler = (handler) => {\n if (handler && typeof handler === 'object') {\n if (process.env.NODE_ENV !== 'production') {\n assert.hasMethod(handler, 'handle', {\n moduleName: 'workbox-routing',\n className: 'Route',\n funcName: 'constructor',\n paramName: 'handler',\n });\n }\n return handler;\n }\n else {\n if (process.env.NODE_ENV !== 'production') {\n assert.isType(handler, 'function', {\n moduleName: 'workbox-routing',\n className: 'Route',\n funcName: 'constructor',\n paramName: 'handler',\n });\n }\n return { handle: handler };\n }\n};\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { assert } from 'workbox-core/_private/assert.js';\nimport { defaultMethod, validMethods } from './utils/constants.js';\nimport { normalizeHandler } from './utils/normalizeHandler.js';\nimport './_version.js';\n/**\n * A `Route` consists of a pair of callback functions, \"match\" and \"handler\".\n * The \"match\" callback determine if a route should be used to \"handle\" a\n * request by returning a non-falsy value if it can. The \"handler\" callback\n * is called when there is a match and should return a Promise that resolves\n * to a `Response`.\n *\n * @memberof module:workbox-routing\n */\nclass Route {\n /**\n * Constructor for Route class.\n *\n * @param {module:workbox-routing~matchCallback} match\n * A callback function that determines whether the route matches a given\n * `fetch` event by returning a non-falsy value.\n * @param {module:workbox-routing~handlerCallback} handler A callback\n * function that returns a Promise resolving to a Response.\n * @param {string} [method='GET'] The HTTP method to match the Route\n * against.\n */\n constructor(match, handler, method = defaultMethod) {\n if (process.env.NODE_ENV !== 'production') {\n assert.isType(match, 'function', {\n moduleName: 'workbox-routing',\n className: 'Route',\n funcName: 'constructor',\n paramName: 'match',\n });\n if (method) {\n assert.isOneOf(method, validMethods, { paramName: 'method' });\n }\n }\n // These values are referenced directly by Router so cannot be\n // altered by minificaton.\n this.handler = normalizeHandler(handler);\n this.match = match;\n this.method = method;\n }\n}\nexport { Route };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\n/**\n * The default HTTP method, 'GET', used when there's no specific method\n * configured for a route.\n *\n * @type {string}\n *\n * @private\n */\nexport const defaultMethod = 'GET';\n/**\n * The list of valid HTTP methods associated with requests that could be routed.\n *\n * @type {Array}\n *\n * @private\n */\nexport const validMethods = [\n 'DELETE',\n 'GET',\n 'HEAD',\n 'PATCH',\n 'POST',\n 'PUT',\n];\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { assert } from 'workbox-core/_private/assert.js';\nimport { logger } from 'workbox-core/_private/logger.js';\nimport { Route } from './Route.js';\nimport './_version.js';\n/**\n * RegExpRoute makes it easy to create a regular expression based\n * [Route]{@link module:workbox-routing.Route}.\n *\n * For same-origin requests the RegExp only needs to match part of the URL. For\n * requests against third-party servers, you must define a RegExp that matches\n * the start of the URL.\n *\n * [See the module docs for info.]{@link https://developers.google.com/web/tools/workbox/modules/workbox-routing}\n *\n * @memberof module:workbox-routing\n * @extends module:workbox-routing.Route\n */\nclass RegExpRoute extends Route {\n /**\n * If the regular expression contains\n * [capture groups]{@link https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/RegExp#grouping-back-references},\n * the captured values will be passed to the\n * [handler's]{@link module:workbox-routing~handlerCallback} `params`\n * argument.\n *\n * @param {RegExp} regExp The regular expression to match against URLs.\n * @param {module:workbox-routing~handlerCallback} handler A callback\n * function that returns a Promise resulting in a Response.\n * @param {string} [method='GET'] The HTTP method to match the Route\n * against.\n */\n constructor(regExp, handler, method) {\n if (process.env.NODE_ENV !== 'production') {\n assert.isInstance(regExp, RegExp, {\n moduleName: 'workbox-routing',\n className: 'RegExpRoute',\n funcName: 'constructor',\n paramName: 'pattern',\n });\n }\n const match = ({ url }) => {\n const result = regExp.exec(url.href);\n // Return immediately if there's no match.\n if (!result) {\n return;\n }\n // Require that the match start at the first character in the URL string\n // if it's a cross-origin request.\n // See https://github.com/GoogleChrome/workbox/issues/281 for the context\n // behind this behavior.\n if ((url.origin !== location.origin) && (result.index !== 0)) {\n if (process.env.NODE_ENV !== 'production') {\n logger.debug(`The regular expression '${regExp}' only partially matched ` +\n `against the cross-origin URL '${url}'. RegExpRoute's will only ` +\n `handle cross-origin requests if they match the entire URL.`);\n }\n return;\n }\n // If the route matches, but there aren't any capture groups defined, then\n // this will return [], which is truthy and therefore sufficient to\n // indicate a match.\n // If there are capture groups, then it will return their values.\n return result.slice(1);\n };\n super(match, handler, method);\n }\n}\nexport { RegExpRoute };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\nconst getFriendlyURL = (url) => {\n const urlObj = new URL(String(url), location.href);\n // See https://github.com/GoogleChrome/workbox/issues/2323\n // We want to include everything, except for the origin if it's same-origin.\n return urlObj.href.replace(new RegExp(`^${location.origin}`), '');\n};\nexport { getFriendlyURL };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { assert } from 'workbox-core/_private/assert.js';\nimport { logger } from 'workbox-core/_private/logger.js';\nimport { WorkboxError } from 'workbox-core/_private/WorkboxError.js';\nimport { getFriendlyURL } from 'workbox-core/_private/getFriendlyURL.js';\nimport { normalizeHandler } from './utils/normalizeHandler.js';\nimport './_version.js';\n/**\n * The Router can be used to process a FetchEvent through one or more\n * [Routes]{@link module:workbox-routing.Route} responding with a Request if\n * a matching route exists.\n *\n * If no route matches a given a request, the Router will use a \"default\"\n * handler if one is defined.\n *\n * Should the matching Route throw an error, the Router will use a \"catch\"\n * handler if one is defined to gracefully deal with issues and respond with a\n * Request.\n *\n * If a request matches multiple routes, the **earliest** registered route will\n * be used to respond to the request.\n *\n * @memberof module:workbox-routing\n */\nclass Router {\n /**\n * Initializes a new Router.\n */\n constructor() {\n this._routes = new Map();\n }\n /**\n * @return {Map>} routes A `Map` of HTTP\n * method name ('GET', etc.) to an array of all the corresponding `Route`\n * instances that are registered.\n */\n get routes() {\n return this._routes;\n }\n /**\n * Adds a fetch event listener to respond to events when a route matches\n * the event's request.\n */\n addFetchListener() {\n // See https://github.com/Microsoft/TypeScript/issues/28357#issuecomment-436484705\n self.addEventListener('fetch', ((event) => {\n const { request } = event;\n const responsePromise = this.handleRequest({ request, event });\n if (responsePromise) {\n event.respondWith(responsePromise);\n }\n }));\n }\n /**\n * Adds a message event listener for URLs to cache from the window.\n * This is useful to cache resources loaded on the page prior to when the\n * service worker started controlling it.\n *\n * The format of the message data sent from the window should be as follows.\n * Where the `urlsToCache` array may consist of URL strings or an array of\n * URL string + `requestInit` object (the same as you'd pass to `fetch()`).\n *\n * ```\n * {\n * type: 'CACHE_URLS',\n * payload: {\n * urlsToCache: [\n * './script1.js',\n * './script2.js',\n * ['./script3.js', {mode: 'no-cors'}],\n * ],\n * },\n * }\n * ```\n */\n addCacheListener() {\n // See https://github.com/Microsoft/TypeScript/issues/28357#issuecomment-436484705\n self.addEventListener('message', ((event) => {\n if (event.data && event.data.type === 'CACHE_URLS') {\n const { payload } = event.data;\n if (process.env.NODE_ENV !== 'production') {\n logger.debug(`Caching URLs from the window`, payload.urlsToCache);\n }\n const requestPromises = Promise.all(payload.urlsToCache.map((entry) => {\n if (typeof entry === 'string') {\n entry = [entry];\n }\n const request = new Request(...entry);\n return this.handleRequest({ request });\n // TODO(philipwalton): TypeScript errors without this typecast for\n // some reason (probably a bug). The real type here should work but\n // doesn't: `Array | undefined>`.\n })); // TypeScript\n event.waitUntil(requestPromises);\n // If a MessageChannel was used, reply to the message on success.\n if (event.ports && event.ports[0]) {\n requestPromises.then(() => event.ports[0].postMessage(true));\n }\n }\n }));\n }\n /**\n * Apply the routing rules to a FetchEvent object to get a Response from an\n * appropriate Route's handler.\n *\n * @param {Object} options\n * @param {Request} options.request The request to handle (this is usually\n * from a fetch event, but it does not have to be).\n * @param {FetchEvent} [options.event] The event that triggered the request,\n * if applicable.\n * @return {Promise|undefined} A promise is returned if a\n * registered route can handle the request. If there is no matching\n * route and there's no `defaultHandler`, `undefined` is returned.\n */\n handleRequest({ request, event }) {\n if (process.env.NODE_ENV !== 'production') {\n assert.isInstance(request, Request, {\n moduleName: 'workbox-routing',\n className: 'Router',\n funcName: 'handleRequest',\n paramName: 'options.request',\n });\n }\n const url = new URL(request.url, location.href);\n if (!url.protocol.startsWith('http')) {\n if (process.env.NODE_ENV !== 'production') {\n logger.debug(`Workbox Router only supports URLs that start with 'http'.`);\n }\n return;\n }\n const { params, route } = this.findMatchingRoute({ url, request, event });\n let handler = route && route.handler;\n const debugMessages = [];\n if (process.env.NODE_ENV !== 'production') {\n if (handler) {\n debugMessages.push([\n `Found a route to handle this request:`, route,\n ]);\n if (params) {\n debugMessages.push([\n `Passing the following params to the route's handler:`, params,\n ]);\n }\n }\n }\n // If we don't have a handler because there was no matching route, then\n // fall back to defaultHandler if that's defined.\n if (!handler && this._defaultHandler) {\n if (process.env.NODE_ENV !== 'production') {\n debugMessages.push(`Failed to find a matching route. Falling ` +\n `back to the default handler.`);\n }\n handler = this._defaultHandler;\n }\n if (!handler) {\n if (process.env.NODE_ENV !== 'production') {\n // No handler so Workbox will do nothing. If logs is set of debug\n // i.e. verbose, we should print out this information.\n logger.debug(`No route found for: ${getFriendlyURL(url)}`);\n }\n return;\n }\n if (process.env.NODE_ENV !== 'production') {\n // We have a handler, meaning Workbox is going to handle the route.\n // print the routing details to the console.\n logger.groupCollapsed(`Router is responding to: ${getFriendlyURL(url)}`);\n debugMessages.forEach((msg) => {\n if (Array.isArray(msg)) {\n logger.log(...msg);\n }\n else {\n logger.log(msg);\n }\n });\n logger.groupEnd();\n }\n // Wrap in try and catch in case the handle method throws a synchronous\n // error. It should still callback to the catch handler.\n let responsePromise;\n try {\n responsePromise = handler.handle({ url, request, event, params });\n }\n catch (err) {\n responsePromise = Promise.reject(err);\n }\n if (responsePromise instanceof Promise && this._catchHandler) {\n responsePromise = responsePromise.catch((err) => {\n if (process.env.NODE_ENV !== 'production') {\n // Still include URL here as it will be async from the console group\n // and may not make sense without the URL\n logger.groupCollapsed(`Error thrown when responding to: ` +\n ` ${getFriendlyURL(url)}. Falling back to Catch Handler.`);\n logger.error(`Error thrown by:`, route);\n logger.error(err);\n logger.groupEnd();\n }\n return this._catchHandler.handle({ url, request, event });\n });\n }\n return responsePromise;\n }\n /**\n * Checks a request and URL (and optionally an event) against the list of\n * registered routes, and if there's a match, returns the corresponding\n * route along with any params generated by the match.\n *\n * @param {Object} options\n * @param {URL} options.url\n * @param {Request} options.request The request to match.\n * @param {Event} [options.event] The corresponding event (unless N/A).\n * @return {Object} An object with `route` and `params` properties.\n * They are populated if a matching route was found or `undefined`\n * otherwise.\n */\n findMatchingRoute({ url, request, event }) {\n if (process.env.NODE_ENV !== 'production') {\n assert.isInstance(url, URL, {\n moduleName: 'workbox-routing',\n className: 'Router',\n funcName: 'findMatchingRoute',\n paramName: 'options.url',\n });\n assert.isInstance(request, Request, {\n moduleName: 'workbox-routing',\n className: 'Router',\n funcName: 'findMatchingRoute',\n paramName: 'options.request',\n });\n }\n const routes = this._routes.get(request.method) || [];\n for (const route of routes) {\n let params;\n const matchResult = route.match({ url, request, event });\n if (matchResult) {\n // See https://github.com/GoogleChrome/workbox/issues/2079\n params = matchResult;\n if (Array.isArray(matchResult) && matchResult.length === 0) {\n // Instead of passing an empty array in as params, use undefined.\n params = undefined;\n }\n else if ((matchResult.constructor === Object &&\n Object.keys(matchResult).length === 0)) {\n // Instead of passing an empty object in as params, use undefined.\n params = undefined;\n }\n else if (typeof matchResult === 'boolean') {\n // For the boolean value true (rather than just something truth-y),\n // don't set params.\n // See https://github.com/GoogleChrome/workbox/pull/2134#issuecomment-513924353\n params = undefined;\n }\n // Return early if have a match.\n return { route, params };\n }\n }\n // If no match was found above, return and empty object.\n return {};\n }\n /**\n * Define a default `handler` that's called when no routes explicitly\n * match the incoming request.\n *\n * Without a default handler, unmatched requests will go against the\n * network as if there were no service worker present.\n *\n * @param {module:workbox-routing~handlerCallback} handler A callback\n * function that returns a Promise resulting in a Response.\n */\n setDefaultHandler(handler) {\n this._defaultHandler = normalizeHandler(handler);\n }\n /**\n * If a Route throws an error while handling a request, this `handler`\n * will be called and given a chance to provide a response.\n *\n * @param {module:workbox-routing~handlerCallback} handler A callback\n * function that returns a Promise resulting in a Response.\n */\n setCatchHandler(handler) {\n this._catchHandler = normalizeHandler(handler);\n }\n /**\n * Registers a route with the router.\n *\n * @param {module:workbox-routing.Route} route The route to register.\n */\n registerRoute(route) {\n if (process.env.NODE_ENV !== 'production') {\n assert.isType(route, 'object', {\n moduleName: 'workbox-routing',\n className: 'Router',\n funcName: 'registerRoute',\n paramName: 'route',\n });\n assert.hasMethod(route, 'match', {\n moduleName: 'workbox-routing',\n className: 'Router',\n funcName: 'registerRoute',\n paramName: 'route',\n });\n assert.isType(route.handler, 'object', {\n moduleName: 'workbox-routing',\n className: 'Router',\n funcName: 'registerRoute',\n paramName: 'route',\n });\n assert.hasMethod(route.handler, 'handle', {\n moduleName: 'workbox-routing',\n className: 'Router',\n funcName: 'registerRoute',\n paramName: 'route.handler',\n });\n assert.isType(route.method, 'string', {\n moduleName: 'workbox-routing',\n className: 'Router',\n funcName: 'registerRoute',\n paramName: 'route.method',\n });\n }\n if (!this._routes.has(route.method)) {\n this._routes.set(route.method, []);\n }\n // Give precedence to all of the earlier routes by adding this additional\n // route to the end of the array.\n this._routes.get(route.method).push(route);\n }\n /**\n * Unregisters a route with the router.\n *\n * @param {module:workbox-routing.Route} route The route to unregister.\n */\n unregisterRoute(route) {\n if (!this._routes.has(route.method)) {\n throw new WorkboxError('unregister-route-but-not-found-with-method', {\n method: route.method,\n });\n }\n const routeIndex = this._routes.get(route.method).indexOf(route);\n if (routeIndex > -1) {\n this._routes.get(route.method).splice(routeIndex, 1);\n }\n else {\n throw new WorkboxError('unregister-route-route-not-registered');\n }\n }\n}\nexport { Router };\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\nconst plugins = [];\nexport const precachePlugins = {\n /*\n * @return {Array}\n * @private\n */\n get() {\n return plugins;\n },\n /*\n * @param {Array} newPlugins\n * @private\n */\n add(newPlugins) {\n plugins.push(...newPlugins);\n },\n};\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\nconst _cacheNameDetails = {\n googleAnalytics: 'googleAnalytics',\n precache: 'precache-v2',\n prefix: 'workbox',\n runtime: 'runtime',\n suffix: typeof registration !== 'undefined' ? registration.scope : '',\n};\nconst _createCacheName = (cacheName) => {\n return [_cacheNameDetails.prefix, cacheName, _cacheNameDetails.suffix]\n .filter((value) => value && value.length > 0)\n .join('-');\n};\nconst eachCacheNameDetail = (fn) => {\n for (const key of Object.keys(_cacheNameDetails)) {\n fn(key);\n }\n};\nexport const cacheNames = {\n updateDetails: (details) => {\n eachCacheNameDetail((key) => {\n if (typeof details[key] === 'string') {\n _cacheNameDetails[key] = details[key];\n }\n });\n },\n getGoogleAnalyticsName: (userCacheName) => {\n return userCacheName || _createCacheName(_cacheNameDetails.googleAnalytics);\n },\n getPrecacheName: (userCacheName) => {\n return userCacheName || _createCacheName(_cacheNameDetails.precache);\n },\n getPrefix: () => {\n return _cacheNameDetails.prefix;\n },\n getRuntimeName: (userCacheName) => {\n return userCacheName || _createCacheName(_cacheNameDetails.runtime);\n },\n getSuffix: () => {\n return _cacheNameDetails.suffix;\n },\n};\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\n// Callbacks to be executed whenever there's a quota error.\nconst quotaErrorCallbacks = new Set();\nexport { quotaErrorCallbacks };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\nexport const pluginUtils = {\n filter: (plugins, callbackName) => {\n return plugins.filter((plugin) => callbackName in plugin);\n },\n};\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { assert } from './assert.js';\nimport { executeQuotaErrorCallbacks } from './executeQuotaErrorCallbacks.js';\nimport { getFriendlyURL } from './getFriendlyURL.js';\nimport { logger } from './logger.js';\nimport { pluginUtils } from '../utils/pluginUtils.js';\nimport { WorkboxError } from './WorkboxError.js';\nimport '../_version.js';\n/**\n * Checks the list of plugins for the cacheKeyWillBeUsed callback, and\n * executes any of those callbacks found in sequence. The final `Request` object\n * returned by the last plugin is treated as the cache key for cache reads\n * and/or writes.\n *\n * @param {Object} options\n * @param {Request} options.request\n * @param {string} options.mode\n * @param {Array} [options.plugins=[]]\n * @return {Promise}\n *\n * @private\n * @memberof module:workbox-core\n */\nconst _getEffectiveRequest = async ({ request, mode, plugins = [], }) => {\n const cacheKeyWillBeUsedPlugins = pluginUtils.filter(plugins, \"cacheKeyWillBeUsed\" /* CACHE_KEY_WILL_BE_USED */);\n let effectiveRequest = request;\n for (const plugin of cacheKeyWillBeUsedPlugins) {\n effectiveRequest = await plugin[\"cacheKeyWillBeUsed\" /* CACHE_KEY_WILL_BE_USED */].call(plugin, { mode, request: effectiveRequest });\n if (typeof effectiveRequest === 'string') {\n effectiveRequest = new Request(effectiveRequest);\n }\n if (process.env.NODE_ENV !== 'production') {\n assert.isInstance(effectiveRequest, Request, {\n moduleName: 'Plugin',\n funcName: \"cacheKeyWillBeUsed\" /* CACHE_KEY_WILL_BE_USED */,\n isReturnValueProblem: true,\n });\n }\n }\n return effectiveRequest;\n};\n/**\n * This method will call cacheWillUpdate on the available plugins (or use\n * status === 200) to determine if the Response is safe and valid to cache.\n *\n * @param {Object} options\n * @param {Request} options.request\n * @param {Response} options.response\n * @param {Event} [options.event]\n * @param {Array} [options.plugins=[]]\n * @return {Promise}\n *\n * @private\n * @memberof module:workbox-core\n */\nconst _isResponseSafeToCache = async ({ request, response, event, plugins = [], }) => {\n let responseToCache = response;\n let pluginsUsed = false;\n for (const plugin of plugins) {\n if (\"cacheWillUpdate\" /* CACHE_WILL_UPDATE */ in plugin) {\n pluginsUsed = true;\n const pluginMethod = plugin[\"cacheWillUpdate\" /* CACHE_WILL_UPDATE */];\n responseToCache = await pluginMethod.call(plugin, {\n request,\n response: responseToCache,\n event,\n });\n if (process.env.NODE_ENV !== 'production') {\n if (responseToCache) {\n assert.isInstance(responseToCache, Response, {\n moduleName: 'Plugin',\n funcName: \"cacheWillUpdate\" /* CACHE_WILL_UPDATE */,\n isReturnValueProblem: true,\n });\n }\n }\n if (!responseToCache) {\n break;\n }\n }\n }\n if (!pluginsUsed) {\n if (process.env.NODE_ENV !== 'production') {\n if (responseToCache) {\n if (responseToCache.status !== 200) {\n if (responseToCache.status === 0) {\n logger.warn(`The response for '${request.url}' is an opaque ` +\n `response. The caching strategy that you're using will not ` +\n `cache opaque responses by default.`);\n }\n else {\n logger.debug(`The response for '${request.url}' returned ` +\n `a status code of '${response.status}' and won't be cached as a ` +\n `result.`);\n }\n }\n }\n }\n responseToCache = responseToCache && responseToCache.status === 200 ?\n responseToCache : undefined;\n }\n return responseToCache ? responseToCache : null;\n};\n/**\n * This is a wrapper around cache.match().\n *\n * @param {Object} options\n * @param {string} options.cacheName Name of the cache to match against.\n * @param {Request} options.request The Request that will be used to look up\n * cache entries.\n * @param {Event} [options.event] The event that prompted the action.\n * @param {Object} [options.matchOptions] Options passed to cache.match().\n * @param {Array} [options.plugins=[]] Array of plugins.\n * @return {Response} A cached response if available.\n *\n * @private\n * @memberof module:workbox-core\n */\nconst matchWrapper = async ({ cacheName, request, event, matchOptions, plugins = [], }) => {\n const cache = await self.caches.open(cacheName);\n const effectiveRequest = await _getEffectiveRequest({\n plugins, request, mode: 'read'\n });\n let cachedResponse = await cache.match(effectiveRequest, matchOptions);\n if (process.env.NODE_ENV !== 'production') {\n if (cachedResponse) {\n logger.debug(`Found a cached response in '${cacheName}'.`);\n }\n else {\n logger.debug(`No cached response found in '${cacheName}'.`);\n }\n }\n for (const plugin of plugins) {\n if (\"cachedResponseWillBeUsed\" /* CACHED_RESPONSE_WILL_BE_USED */ in plugin) {\n const pluginMethod = plugin[\"cachedResponseWillBeUsed\" /* CACHED_RESPONSE_WILL_BE_USED */];\n cachedResponse = await pluginMethod.call(plugin, {\n cacheName,\n event,\n matchOptions,\n cachedResponse,\n request: effectiveRequest,\n });\n if (process.env.NODE_ENV !== 'production') {\n if (cachedResponse) {\n assert.isInstance(cachedResponse, Response, {\n moduleName: 'Plugin',\n funcName: \"cachedResponseWillBeUsed\" /* CACHED_RESPONSE_WILL_BE_USED */,\n isReturnValueProblem: true,\n });\n }\n }\n }\n }\n return cachedResponse;\n};\n/**\n * Wrapper around cache.put().\n *\n * Will call `cacheDidUpdate` on plugins if the cache was updated, using\n * `matchOptions` when determining what the old entry is.\n *\n * @param {Object} options\n * @param {string} options.cacheName\n * @param {Request} options.request\n * @param {Response} options.response\n * @param {Event} [options.event]\n * @param {Array} [options.plugins=[]]\n * @param {Object} [options.matchOptions]\n *\n * @private\n * @memberof module:workbox-core\n */\nconst putWrapper = async ({ cacheName, request, response, event, plugins = [], matchOptions, }) => {\n if (process.env.NODE_ENV !== 'production') {\n if (request.method && request.method !== 'GET') {\n throw new WorkboxError('attempt-to-cache-non-get-request', {\n url: getFriendlyURL(request.url),\n method: request.method,\n });\n }\n }\n const effectiveRequest = await _getEffectiveRequest({\n plugins, request, mode: 'write'\n });\n if (!response) {\n if (process.env.NODE_ENV !== 'production') {\n logger.error(`Cannot cache non-existent response for ` +\n `'${getFriendlyURL(effectiveRequest.url)}'.`);\n }\n throw new WorkboxError('cache-put-with-no-response', {\n url: getFriendlyURL(effectiveRequest.url),\n });\n }\n const responseToCache = await _isResponseSafeToCache({\n event,\n plugins,\n response,\n request: effectiveRequest,\n });\n if (!responseToCache) {\n if (process.env.NODE_ENV !== 'production') {\n logger.debug(`Response '${getFriendlyURL(effectiveRequest.url)}' will ` +\n `not be cached.`, responseToCache);\n }\n return;\n }\n const cache = await self.caches.open(cacheName);\n const updatePlugins = pluginUtils.filter(plugins, \"cacheDidUpdate\" /* CACHE_DID_UPDATE */);\n const oldResponse = updatePlugins.length > 0 ?\n await matchWrapper({ cacheName, matchOptions, request: effectiveRequest }) :\n null;\n if (process.env.NODE_ENV !== 'production') {\n logger.debug(`Updating the '${cacheName}' cache with a new Response for ` +\n `${getFriendlyURL(effectiveRequest.url)}.`);\n }\n try {\n await cache.put(effectiveRequest, responseToCache);\n }\n catch (error) {\n // See https://developer.mozilla.org/en-US/docs/Web/API/DOMException#exception-QuotaExceededError\n if (error.name === 'QuotaExceededError') {\n await executeQuotaErrorCallbacks();\n }\n throw error;\n }\n for (const plugin of updatePlugins) {\n await plugin[\"cacheDidUpdate\" /* CACHE_DID_UPDATE */].call(plugin, {\n cacheName,\n event,\n oldResponse,\n newResponse: responseToCache,\n request: effectiveRequest,\n });\n }\n};\nexport const cacheWrapper = {\n put: putWrapper,\n match: matchWrapper,\n};\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { logger } from '../_private/logger.js';\nimport { quotaErrorCallbacks } from '../models/quotaErrorCallbacks.js';\nimport '../_version.js';\n/**\n * Runs all of the callback functions, one at a time sequentially, in the order\n * in which they were registered.\n *\n * @memberof module:workbox-core\n * @private\n */\nasync function executeQuotaErrorCallbacks() {\n if (process.env.NODE_ENV !== 'production') {\n logger.log(`About to run ${quotaErrorCallbacks.size} ` +\n `callbacks to clean up caches.`);\n }\n for (const callback of quotaErrorCallbacks) {\n await callback();\n if (process.env.NODE_ENV !== 'production') {\n logger.log(callback, 'is complete.');\n }\n }\n if (process.env.NODE_ENV !== 'production') {\n logger.log('Finished running callbacks.');\n }\n}\nexport { executeQuotaErrorCallbacks };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { WorkboxError } from './WorkboxError.js';\nimport { logger } from './logger.js';\nimport { assert } from './assert.js';\nimport { getFriendlyURL } from '../_private/getFriendlyURL.js';\nimport { pluginUtils } from '../utils/pluginUtils.js';\nimport '../_version.js';\n/**\n * Wrapper around the fetch API.\n *\n * Will call requestWillFetch on available plugins.\n *\n * @param {Object} options\n * @param {Request|string} options.request\n * @param {Object} [options.fetchOptions]\n * @param {ExtendableEvent} [options.event]\n * @param {Array} [options.plugins=[]]\n * @return {Promise}\n *\n * @private\n * @memberof module:workbox-core\n */\nconst wrappedFetch = async ({ request, fetchOptions, event, plugins = [], }) => {\n if (typeof request === 'string') {\n request = new Request(request);\n }\n // We *should* be able to call `await event.preloadResponse` even if it's\n // undefined, but for some reason, doing so leads to errors in our Node unit\n // tests. To work around that, explicitly check preloadResponse's value first.\n if (event instanceof FetchEvent && event.preloadResponse) {\n const possiblePreloadResponse = await event.preloadResponse;\n if (possiblePreloadResponse) {\n if (process.env.NODE_ENV !== 'production') {\n logger.log(`Using a preloaded navigation response for ` +\n `'${getFriendlyURL(request.url)}'`);\n }\n return possiblePreloadResponse;\n }\n }\n if (process.env.NODE_ENV !== 'production') {\n assert.isInstance(request, Request, {\n paramName: 'request',\n expectedClass: Request,\n moduleName: 'workbox-core',\n className: 'fetchWrapper',\n funcName: 'wrappedFetch',\n });\n }\n const failedFetchPlugins = pluginUtils.filter(plugins, \"fetchDidFail\" /* FETCH_DID_FAIL */);\n // If there is a fetchDidFail plugin, we need to save a clone of the\n // original request before it's either modified by a requestWillFetch\n // plugin or before the original request's body is consumed via fetch().\n const originalRequest = failedFetchPlugins.length > 0 ?\n request.clone() : null;\n try {\n for (const plugin of plugins) {\n if (\"requestWillFetch\" /* REQUEST_WILL_FETCH */ in plugin) {\n const pluginMethod = plugin[\"requestWillFetch\" /* REQUEST_WILL_FETCH */];\n const requestClone = request.clone();\n request = await pluginMethod.call(plugin, {\n request: requestClone,\n event,\n });\n if (process.env.NODE_ENV !== 'production') {\n if (request) {\n assert.isInstance(request, Request, {\n moduleName: 'Plugin',\n funcName: \"cachedResponseWillBeUsed\" /* CACHED_RESPONSE_WILL_BE_USED */,\n isReturnValueProblem: true,\n });\n }\n }\n }\n }\n }\n catch (err) {\n throw new WorkboxError('plugin-error-request-will-fetch', {\n thrownError: err,\n });\n }\n // The request can be altered by plugins with `requestWillFetch` making\n // the original request (Most likely from a `fetch` event) to be different\n // to the Request we make. Pass both to `fetchDidFail` to aid debugging.\n const pluginFilteredRequest = request.clone();\n try {\n let fetchResponse;\n // See https://github.com/GoogleChrome/workbox/issues/1796\n if (request.mode === 'navigate') {\n fetchResponse = await fetch(request);\n }\n else {\n fetchResponse = await fetch(request, fetchOptions);\n }\n if (process.env.NODE_ENV !== 'production') {\n logger.debug(`Network request for ` +\n `'${getFriendlyURL(request.url)}' returned a response with ` +\n `status '${fetchResponse.status}'.`);\n }\n for (const plugin of plugins) {\n if (\"fetchDidSucceed\" /* FETCH_DID_SUCCEED */ in plugin) {\n fetchResponse = await plugin[\"fetchDidSucceed\" /* FETCH_DID_SUCCEED */]\n .call(plugin, {\n event,\n request: pluginFilteredRequest,\n response: fetchResponse,\n });\n if (process.env.NODE_ENV !== 'production') {\n if (fetchResponse) {\n assert.isInstance(fetchResponse, Response, {\n moduleName: 'Plugin',\n funcName: \"fetchDidSucceed\" /* FETCH_DID_SUCCEED */,\n isReturnValueProblem: true,\n });\n }\n }\n }\n }\n return fetchResponse;\n }\n catch (error) {\n if (process.env.NODE_ENV !== 'production') {\n logger.error(`Network request for ` +\n `'${getFriendlyURL(request.url)}' threw an error.`, error);\n }\n for (const plugin of failedFetchPlugins) {\n await plugin[\"fetchDidFail\" /* FETCH_DID_FAIL */].call(plugin, {\n error,\n event,\n originalRequest: originalRequest.clone(),\n request: pluginFilteredRequest.clone(),\n });\n }\n throw error;\n }\n};\nconst fetchWrapper = {\n fetch: wrappedFetch,\n};\nexport { fetchWrapper };\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { assert } from 'workbox-core/_private/assert.js';\nimport { cacheNames } from 'workbox-core/_private/cacheNames.js';\nimport { cacheWrapper } from 'workbox-core/_private/cacheWrapper.js';\nimport { fetchWrapper } from 'workbox-core/_private/fetchWrapper.js';\nimport { logger } from 'workbox-core/_private/logger.js';\nimport { WorkboxError } from 'workbox-core/_private/WorkboxError.js';\nimport { copyResponse } from 'workbox-core/copyResponse.js';\nimport { createCacheKey } from './utils/createCacheKey.js';\nimport { printCleanupDetails } from './utils/printCleanupDetails.js';\nimport { printInstallDetails } from './utils/printInstallDetails.js';\nimport './_version.js';\n/**\n * Performs efficient precaching of assets.\n *\n * @memberof module:workbox-precaching\n */\nclass PrecacheController {\n /**\n * Create a new PrecacheController.\n *\n * @param {string} [cacheName] An optional name for the cache, to override\n * the default precache name.\n */\n constructor(cacheName) {\n this._cacheName = cacheNames.getPrecacheName(cacheName);\n this._urlsToCacheKeys = new Map();\n this._urlsToCacheModes = new Map();\n this._cacheKeysToIntegrities = new Map();\n }\n /**\n * This method will add items to the precache list, removing duplicates\n * and ensuring the information is valid.\n *\n * @param {\n * Array\n * } entries Array of entries to precache.\n */\n addToCacheList(entries) {\n if (process.env.NODE_ENV !== 'production') {\n assert.isArray(entries, {\n moduleName: 'workbox-precaching',\n className: 'PrecacheController',\n funcName: 'addToCacheList',\n paramName: 'entries',\n });\n }\n const urlsToWarnAbout = [];\n for (const entry of entries) {\n // See https://github.com/GoogleChrome/workbox/issues/2259\n if (typeof entry === 'string') {\n urlsToWarnAbout.push(entry);\n }\n else if (entry && entry.revision === undefined) {\n urlsToWarnAbout.push(entry.url);\n }\n const { cacheKey, url } = createCacheKey(entry);\n const cacheMode = (typeof entry !== 'string' && entry.revision) ?\n 'reload' : 'default';\n if (this._urlsToCacheKeys.has(url) &&\n this._urlsToCacheKeys.get(url) !== cacheKey) {\n throw new WorkboxError('add-to-cache-list-conflicting-entries', {\n firstEntry: this._urlsToCacheKeys.get(url),\n secondEntry: cacheKey,\n });\n }\n if (typeof entry !== 'string' && entry.integrity) {\n if (this._cacheKeysToIntegrities.has(cacheKey) &&\n this._cacheKeysToIntegrities.get(cacheKey) !== entry.integrity) {\n throw new WorkboxError('add-to-cache-list-conflicting-integrities', {\n url,\n });\n }\n this._cacheKeysToIntegrities.set(cacheKey, entry.integrity);\n }\n this._urlsToCacheKeys.set(url, cacheKey);\n this._urlsToCacheModes.set(url, cacheMode);\n if (urlsToWarnAbout.length > 0) {\n const warningMessage = `Workbox is precaching URLs without revision ` +\n `info: ${urlsToWarnAbout.join(', ')}\\nThis is generally NOT safe. ` +\n `Learn more at https://bit.ly/wb-precache`;\n if (process.env.NODE_ENV === 'production') {\n // Use console directly to display this warning without bloating\n // bundle sizes by pulling in all of the logger codebase in prod.\n console.warn(warningMessage);\n }\n else {\n logger.warn(warningMessage);\n }\n }\n }\n }\n /**\n * Precaches new and updated assets. Call this method from the service worker\n * install event.\n *\n * @param {Object} options\n * @param {Event} [options.event] The install event (if needed).\n * @param {Array} [options.plugins] Plugins to be used for fetching\n * and caching during install.\n * @return {Promise}\n */\n async install({ event, plugins } = {}) {\n if (process.env.NODE_ENV !== 'production') {\n if (plugins) {\n assert.isArray(plugins, {\n moduleName: 'workbox-precaching',\n className: 'PrecacheController',\n funcName: 'install',\n paramName: 'plugins',\n });\n }\n }\n const toBePrecached = [];\n const alreadyPrecached = [];\n const cache = await self.caches.open(this._cacheName);\n const alreadyCachedRequests = await cache.keys();\n const existingCacheKeys = new Set(alreadyCachedRequests.map((request) => request.url));\n for (const [url, cacheKey] of this._urlsToCacheKeys) {\n if (existingCacheKeys.has(cacheKey)) {\n alreadyPrecached.push(url);\n }\n else {\n toBePrecached.push({ cacheKey, url });\n }\n }\n const precacheRequests = toBePrecached.map(({ cacheKey, url }) => {\n const integrity = this._cacheKeysToIntegrities.get(cacheKey);\n const cacheMode = this._urlsToCacheModes.get(url);\n return this._addURLToCache({\n cacheKey,\n cacheMode,\n event,\n integrity,\n plugins,\n url,\n });\n });\n await Promise.all(precacheRequests);\n const updatedURLs = toBePrecached.map((item) => item.url);\n if (process.env.NODE_ENV !== 'production') {\n printInstallDetails(updatedURLs, alreadyPrecached);\n }\n return {\n updatedURLs,\n notUpdatedURLs: alreadyPrecached,\n };\n }\n /**\n * Deletes assets that are no longer present in the current precache manifest.\n * Call this method from the service worker activate event.\n *\n * @return {Promise}\n */\n async activate() {\n const cache = await self.caches.open(this._cacheName);\n const currentlyCachedRequests = await cache.keys();\n const expectedCacheKeys = new Set(this._urlsToCacheKeys.values());\n const deletedURLs = [];\n for (const request of currentlyCachedRequests) {\n if (!expectedCacheKeys.has(request.url)) {\n await cache.delete(request);\n deletedURLs.push(request.url);\n }\n }\n if (process.env.NODE_ENV !== 'production') {\n printCleanupDetails(deletedURLs);\n }\n return { deletedURLs };\n }\n /**\n * Requests the entry and saves it to the cache if the response is valid.\n * By default, any response with a status code of less than 400 (including\n * opaque responses) is considered valid.\n *\n * If you need to use custom criteria to determine what's valid and what\n * isn't, then pass in an item in `options.plugins` that implements the\n * `cacheWillUpdate()` lifecycle event.\n *\n * @private\n * @param {Object} options\n * @param {string} options.cacheKey The string to use a cache key.\n * @param {string} options.url The URL to fetch and cache.\n * @param {string} [options.cacheMode] The cache mode for the network request.\n * @param {Event} [options.event] The install event (if passed).\n * @param {Array} [options.plugins] An array of plugins to apply to\n * fetch and caching.\n * @param {string} [options.integrity] The value to use for the `integrity`\n * field when making the request.\n */\n async _addURLToCache({ cacheKey, url, cacheMode, event, plugins, integrity }) {\n const request = new Request(url, {\n integrity,\n cache: cacheMode,\n credentials: 'same-origin',\n });\n let response = await fetchWrapper.fetch({\n event,\n plugins,\n request,\n });\n // Allow developers to override the default logic about what is and isn't\n // valid by passing in a plugin implementing cacheWillUpdate(), e.g.\n // a `CacheableResponsePlugin` instance.\n let cacheWillUpdatePlugin;\n for (const plugin of (plugins || [])) {\n if ('cacheWillUpdate' in plugin) {\n cacheWillUpdatePlugin = plugin;\n }\n }\n const isValidResponse = cacheWillUpdatePlugin ?\n // Use a callback if provided. It returns a truthy value if valid.\n // NOTE: invoke the method on the plugin instance so the `this` context\n // is correct.\n await cacheWillUpdatePlugin.cacheWillUpdate({ event, request, response }) :\n // Otherwise, default to considering any response status under 400 valid.\n // This includes, by default, considering opaque responses valid.\n response.status < 400;\n // Consider this a failure, leading to the `install` handler failing, if\n // we get back an invalid response.\n if (!isValidResponse) {\n throw new WorkboxError('bad-precaching-response', {\n url,\n status: response.status,\n });\n }\n // Redirected responses cannot be used to satisfy a navigation request, so\n // any redirected response must be \"copied\" rather than cloned, so the new\n // response doesn't contain the `redirected` flag. See:\n // https://bugs.chromium.org/p/chromium/issues/detail?id=669363&desc=2#c1\n if (response.redirected) {\n response = await copyResponse(response);\n }\n await cacheWrapper.put({\n event,\n plugins,\n response,\n // `request` already uses `url`. We may be able to reuse it.\n request: cacheKey === url ? request : new Request(cacheKey),\n cacheName: this._cacheName,\n matchOptions: {\n ignoreSearch: true,\n },\n });\n }\n /**\n * Returns a mapping of a precached URL to the corresponding cache key, taking\n * into account the revision information for the URL.\n *\n * @return {Map} A URL to cache key mapping.\n */\n getURLsToCacheKeys() {\n return this._urlsToCacheKeys;\n }\n /**\n * Returns a list of all the URLs that have been precached by the current\n * service worker.\n *\n * @return {Array} The precached URLs.\n */\n getCachedURLs() {\n return [...this._urlsToCacheKeys.keys()];\n }\n /**\n * Returns the cache key used for storing a given URL. If that URL is\n * unversioned, like `/index.html', then the cache key will be the original\n * URL with a search parameter appended to it.\n *\n * @param {string} url A URL whose cache key you want to look up.\n * @return {string} The versioned URL that corresponds to a cache key\n * for the original URL, or undefined if that URL isn't precached.\n */\n getCacheKeyForURL(url) {\n const urlObject = new URL(url, location.href);\n return this._urlsToCacheKeys.get(urlObject.href);\n }\n /**\n * This acts as a drop-in replacement for [`cache.match()`](https://developer.mozilla.org/en-US/docs/Web/API/Cache/match)\n * with the following differences:\n *\n * - It knows what the name of the precache is, and only checks in that cache.\n * - It allows you to pass in an \"original\" URL without versioning parameters,\n * and it will automatically look up the correct cache key for the currently\n * active revision of that URL.\n *\n * E.g., `matchPrecache('index.html')` will find the correct precached\n * response for the currently active service worker, even if the actual cache\n * key is `'/index.html?__WB_REVISION__=1234abcd'`.\n *\n * @param {string|Request} request The key (without revisioning parameters)\n * to look up in the precache.\n * @return {Promise}\n */\n async matchPrecache(request) {\n const url = request instanceof Request ? request.url : request;\n const cacheKey = this.getCacheKeyForURL(url);\n if (cacheKey) {\n const cache = await self.caches.open(this._cacheName);\n return cache.match(cacheKey);\n }\n return undefined;\n }\n /**\n * Returns a function that can be used within a\n * {@link module:workbox-routing.Route} that will find a response for the\n * incoming request against the precache.\n *\n * If for an unexpected reason there is a cache miss for the request,\n * this will fall back to retrieving the `Response` via `fetch()` when\n * `fallbackToNetwork` is `true`.\n *\n * @param {boolean} [fallbackToNetwork=true] Whether to attempt to get the\n * response from the network if there's a precache miss.\n * @return {module:workbox-routing~handlerCallback}\n */\n createHandler(fallbackToNetwork = true) {\n return async ({ request }) => {\n try {\n const response = await this.matchPrecache(request);\n if (response) {\n return response;\n }\n // This shouldn't normally happen, but there are edge cases:\n // https://github.com/GoogleChrome/workbox/issues/1441\n throw new WorkboxError('missing-precache-entry', {\n cacheName: this._cacheName,\n url: request instanceof Request ? request.url : request,\n });\n }\n catch (error) {\n if (fallbackToNetwork) {\n if (process.env.NODE_ENV !== 'production') {\n logger.debug(`Unable to respond with precached response. ` +\n `Falling back to network.`, error);\n }\n return fetch(request);\n }\n throw error;\n }\n };\n }\n /**\n * Returns a function that looks up `url` in the precache (taking into\n * account revision information), and returns the corresponding `Response`.\n *\n * If for an unexpected reason there is a cache miss when looking up `url`,\n * this will fall back to retrieving the `Response` via `fetch()` when\n * `fallbackToNetwork` is `true`.\n *\n * @param {string} url The precached URL which will be used to lookup the\n * `Response`.\n * @param {boolean} [fallbackToNetwork=true] Whether to attempt to get the\n * response from the network if there's a precache miss.\n * @return {module:workbox-routing~handlerCallback}\n */\n createHandlerBoundToURL(url, fallbackToNetwork = true) {\n const cacheKey = this.getCacheKeyForURL(url);\n if (!cacheKey) {\n throw new WorkboxError('non-precached-url', { url });\n }\n const handler = this.createHandler(fallbackToNetwork);\n const request = new Request(url);\n return () => handler({ request });\n }\n}\nexport { PrecacheController };\n","export const isNav = event => event.request.mode === 'navigate';\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\nexport const cacheOkAndOpaquePlugin = {\n /**\n * Returns a valid response (to allow caching) if the status is 200 (OK) or\n * 0 (opaque).\n *\n * @param {Object} options\n * @param {Response} options.response\n * @return {Response|null}\n *\n * @private\n */\n cacheWillUpdate: async ({ response }) => {\n if (response.status === 200 || response.status === 0) {\n return response;\n }\n return null;\n },\n};\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { assert } from 'workbox-core/_private/assert.js';\nimport { cacheNames } from 'workbox-core/_private/cacheNames.js';\nimport { cacheWrapper } from 'workbox-core/_private/cacheWrapper.js';\nimport { fetchWrapper } from 'workbox-core/_private/fetchWrapper.js';\nimport { getFriendlyURL } from 'workbox-core/_private/getFriendlyURL.js';\nimport { logger } from 'workbox-core/_private/logger.js';\nimport { WorkboxError } from 'workbox-core/_private/WorkboxError.js';\nimport { messages } from './utils/messages.js';\nimport { cacheOkAndOpaquePlugin } from './plugins/cacheOkAndOpaquePlugin.js';\nimport './_version.js';\n/**\n * An implementation of a\n * [network first]{@link https://developers.google.com/web/fundamentals/instant-and-offline/offline-cookbook/#network-falling-back-to-cache}\n * request strategy.\n *\n * By default, this strategy will cache responses with a 200 status code as\n * well as [opaque responses]{@link https://developers.google.com/web/tools/workbox/guides/handle-third-party-requests}.\n * Opaque responses are are cross-origin requests where the response doesn't\n * support [CORS]{@link https://enable-cors.org/}.\n *\n * If the network request fails, and there is no cache match, this will throw\n * a `WorkboxError` exception.\n *\n * @memberof module:workbox-strategies\n */\nclass NetworkFirst {\n /**\n * @param {Object} options\n * @param {string} options.cacheName Cache name to store and retrieve\n * requests. Defaults to cache names provided by\n * [workbox-core]{@link module:workbox-core.cacheNames}.\n * @param {Array} options.plugins [Plugins]{@link https://developers.google.com/web/tools/workbox/guides/using-plugins}\n * to use in conjunction with this caching strategy.\n * @param {Object} options.fetchOptions Values passed along to the\n * [`init`](https://developer.mozilla.org/en-US/docs/Web/API/WindowOrWorkerGlobalScope/fetch#Parameters)\n * of all fetch() requests made by this strategy.\n * @param {Object} options.matchOptions [`CacheQueryOptions`](https://w3c.github.io/ServiceWorker/#dictdef-cachequeryoptions)\n * @param {number} options.networkTimeoutSeconds If set, any network requests\n * that fail to respond within the timeout will fallback to the cache.\n *\n * This option can be used to combat\n * \"[lie-fi]{@link https://developers.google.com/web/fundamentals/performance/poor-connectivity/#lie-fi}\"\n * scenarios.\n */\n constructor(options = {}) {\n this._cacheName = cacheNames.getRuntimeName(options.cacheName);\n if (options.plugins) {\n const isUsingCacheWillUpdate = options.plugins.some((plugin) => !!plugin.cacheWillUpdate);\n this._plugins = isUsingCacheWillUpdate ?\n options.plugins : [cacheOkAndOpaquePlugin, ...options.plugins];\n }\n else {\n // No plugins passed in, use the default plugin.\n this._plugins = [cacheOkAndOpaquePlugin];\n }\n this._networkTimeoutSeconds = options.networkTimeoutSeconds || 0;\n if (process.env.NODE_ENV !== 'production') {\n if (this._networkTimeoutSeconds) {\n assert.isType(this._networkTimeoutSeconds, 'number', {\n moduleName: 'workbox-strategies',\n className: 'NetworkFirst',\n funcName: 'constructor',\n paramName: 'networkTimeoutSeconds',\n });\n }\n }\n this._fetchOptions = options.fetchOptions;\n this._matchOptions = options.matchOptions;\n }\n /**\n * This method will perform a request strategy and follows an API that\n * will work with the\n * [Workbox Router]{@link module:workbox-routing.Router}.\n *\n * @param {Object} options\n * @param {Request|string} options.request A request to run this strategy for.\n * @param {Event} [options.event] The event that triggered the request.\n * @return {Promise}\n */\n async handle({ event, request }) {\n const logs = [];\n if (typeof request === 'string') {\n request = new Request(request);\n }\n if (process.env.NODE_ENV !== 'production') {\n assert.isInstance(request, Request, {\n moduleName: 'workbox-strategies',\n className: 'NetworkFirst',\n funcName: 'handle',\n paramName: 'makeRequest',\n });\n }\n const promises = [];\n let timeoutId;\n if (this._networkTimeoutSeconds) {\n const { id, promise } = this._getTimeoutPromise({ request, event, logs });\n timeoutId = id;\n promises.push(promise);\n }\n const networkPromise = this._getNetworkPromise({ timeoutId, request, event, logs });\n promises.push(networkPromise);\n // Promise.race() will resolve as soon as the first promise resolves.\n let response = await Promise.race(promises);\n // If Promise.race() resolved with null, it might be due to a network\n // timeout + a cache miss. If that were to happen, we'd rather wait until\n // the networkPromise resolves instead of returning null.\n // Note that it's fine to await an already-resolved promise, so we don't\n // have to check to see if it's still \"in flight\".\n if (!response) {\n response = await networkPromise;\n }\n if (process.env.NODE_ENV !== 'production') {\n logger.groupCollapsed(messages.strategyStart('NetworkFirst', request));\n for (const log of logs) {\n logger.log(log);\n }\n messages.printFinalResponse(response);\n logger.groupEnd();\n }\n if (!response) {\n throw new WorkboxError('no-response', { url: request.url });\n }\n return response;\n }\n /**\n * @param {Object} options\n * @param {Request} options.request\n * @param {Array} options.logs A reference to the logs array\n * @param {Event} [options.event]\n * @return {Promise}\n *\n * @private\n */\n _getTimeoutPromise({ request, logs, event }) {\n let timeoutId;\n const timeoutPromise = new Promise((resolve) => {\n const onNetworkTimeout = async () => {\n if (process.env.NODE_ENV !== 'production') {\n logs.push(`Timing out the network response at ` +\n `${this._networkTimeoutSeconds} seconds.`);\n }\n resolve(await this._respondFromCache({ request, event }));\n };\n timeoutId = setTimeout(onNetworkTimeout, this._networkTimeoutSeconds * 1000);\n });\n return {\n promise: timeoutPromise,\n id: timeoutId,\n };\n }\n /**\n * @param {Object} options\n * @param {number|undefined} options.timeoutId\n * @param {Request} options.request\n * @param {Array} options.logs A reference to the logs Array.\n * @param {Event} [options.event]\n * @return {Promise}\n *\n * @private\n */\n async _getNetworkPromise({ timeoutId, request, logs, event }) {\n let error;\n let response;\n try {\n response = await fetchWrapper.fetch({\n request,\n event,\n fetchOptions: this._fetchOptions,\n plugins: this._plugins,\n });\n }\n catch (err) {\n error = err;\n }\n if (timeoutId) {\n clearTimeout(timeoutId);\n }\n if (process.env.NODE_ENV !== 'production') {\n if (response) {\n logs.push(`Got response from network.`);\n }\n else {\n logs.push(`Unable to get a response from the network. Will respond ` +\n `with a cached response.`);\n }\n }\n if (error || !response) {\n response = await this._respondFromCache({ request, event });\n if (process.env.NODE_ENV !== 'production') {\n if (response) {\n logs.push(`Found a cached response in the '${this._cacheName}'` +\n ` cache.`);\n }\n else {\n logs.push(`No response found in the '${this._cacheName}' cache.`);\n }\n }\n }\n else {\n // Keep the service worker alive while we put the request in the cache\n const responseClone = response.clone();\n const cachePut = cacheWrapper.put({\n cacheName: this._cacheName,\n request,\n response: responseClone,\n event,\n plugins: this._plugins,\n });\n if (event) {\n try {\n // The event has been responded to so we can keep the SW alive to\n // respond to the request\n event.waitUntil(cachePut);\n }\n catch (err) {\n if (process.env.NODE_ENV !== 'production') {\n logger.warn(`Unable to ensure service worker stays alive when ` +\n `updating cache for '${getFriendlyURL(request.url)}'.`);\n }\n }\n }\n }\n return response;\n }\n /**\n * Used if the network timeouts or fails to make the request.\n *\n * @param {Object} options\n * @param {Request} request The request to match in the cache\n * @param {Event} [options.event]\n * @return {Promise}\n *\n * @private\n */\n _respondFromCache({ event, request }) {\n return cacheWrapper.match({\n cacheName: this._cacheName,\n request,\n event,\n matchOptions: this._matchOptions,\n plugins: this._plugins,\n });\n }\n}\nexport { NetworkFirst };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport '../_version.js';\n/**\n * A class that wraps common IndexedDB functionality in a promise-based API.\n * It exposes all the underlying power and functionality of IndexedDB, but\n * wraps the most commonly used features in a way that's much simpler to use.\n *\n * @private\n */\nexport class DBWrapper {\n /**\n * @param {string} name\n * @param {number} version\n * @param {Object=} [callback]\n * @param {!Function} [callbacks.onupgradeneeded]\n * @param {!Function} [callbacks.onversionchange] Defaults to\n * DBWrapper.prototype._onversionchange when not specified.\n * @private\n */\n constructor(name, version, { onupgradeneeded, onversionchange, } = {}) {\n this._db = null;\n this._name = name;\n this._version = version;\n this._onupgradeneeded = onupgradeneeded;\n this._onversionchange = onversionchange || (() => this.close());\n }\n /**\n * Returns the IDBDatabase instance (not normally needed).\n * @return {IDBDatabase|undefined}\n *\n * @private\n */\n get db() {\n return this._db;\n }\n /**\n * Opens a connected to an IDBDatabase, invokes any onupgradedneeded\n * callback, and added an onversionchange callback to the database.\n *\n * @return {IDBDatabase}\n * @private\n */\n async open() {\n if (this._db)\n return;\n this._db = await new Promise((resolve, reject) => {\n // This flag is flipped to true if the timeout callback runs prior\n // to the request failing or succeeding. Note: we use a timeout instead\n // of an onblocked handler since there are cases where onblocked will\n // never never run. A timeout better handles all possible scenarios:\n // https://github.com/w3c/IndexedDB/issues/223\n let openRequestTimedOut = false;\n setTimeout(() => {\n openRequestTimedOut = true;\n reject(new Error('The open request was blocked and timed out'));\n }, this.OPEN_TIMEOUT);\n const openRequest = indexedDB.open(this._name, this._version);\n openRequest.onerror = () => reject(openRequest.error);\n openRequest.onupgradeneeded = (evt) => {\n if (openRequestTimedOut) {\n openRequest.transaction.abort();\n openRequest.result.close();\n }\n else if (typeof this._onupgradeneeded === 'function') {\n this._onupgradeneeded(evt);\n }\n };\n openRequest.onsuccess = () => {\n const db = openRequest.result;\n if (openRequestTimedOut) {\n db.close();\n }\n else {\n db.onversionchange = this._onversionchange.bind(this);\n resolve(db);\n }\n };\n });\n return this;\n }\n /**\n * Polyfills the native `getKey()` method. Note, this is overridden at\n * runtime if the browser supports the native method.\n *\n * @param {string} storeName\n * @param {*} query\n * @return {Array}\n * @private\n */\n async getKey(storeName, query) {\n return (await this.getAllKeys(storeName, query, 1))[0];\n }\n /**\n * Polyfills the native `getAll()` method. Note, this is overridden at\n * runtime if the browser supports the native method.\n *\n * @param {string} storeName\n * @param {*} query\n * @param {number} count\n * @return {Array}\n * @private\n */\n async getAll(storeName, query, count) {\n return await this.getAllMatching(storeName, { query, count });\n }\n /**\n * Polyfills the native `getAllKeys()` method. Note, this is overridden at\n * runtime if the browser supports the native method.\n *\n * @param {string} storeName\n * @param {*} query\n * @param {number} count\n * @return {Array}\n * @private\n */\n async getAllKeys(storeName, query, count) {\n const entries = await this.getAllMatching(storeName, { query, count, includeKeys: true });\n return entries.map((entry) => entry.key);\n }\n /**\n * Supports flexible lookup in an object store by specifying an index,\n * query, direction, and count. This method returns an array of objects\n * with the signature .\n *\n * @param {string} storeName\n * @param {Object} [opts]\n * @param {string} [opts.index] The index to use (if specified).\n * @param {*} [opts.query]\n * @param {IDBCursorDirection} [opts.direction]\n * @param {number} [opts.count] The max number of results to return.\n * @param {boolean} [opts.includeKeys] When true, the structure of the\n * returned objects is changed from an array of values to an array of\n * objects in the form {key, primaryKey, value}.\n * @return {Array}\n * @private\n */\n async getAllMatching(storeName, { index, query = null, // IE/Edge errors if query === `undefined`.\n direction = 'next', count, includeKeys = false, } = {}) {\n return await this.transaction([storeName], 'readonly', (txn, done) => {\n const store = txn.objectStore(storeName);\n const target = index ? store.index(index) : store;\n const results = [];\n const request = target.openCursor(query, direction);\n request.onsuccess = () => {\n const cursor = request.result;\n if (cursor) {\n results.push(includeKeys ? cursor : cursor.value);\n if (count && results.length >= count) {\n done(results);\n }\n else {\n cursor.continue();\n }\n }\n else {\n done(results);\n }\n };\n });\n }\n /**\n * Accepts a list of stores, a transaction type, and a callback and\n * performs a transaction. A promise is returned that resolves to whatever\n * value the callback chooses. The callback holds all the transaction logic\n * and is invoked with two arguments:\n * 1. The IDBTransaction object\n * 2. A `done` function, that's used to resolve the promise when\n * when the transaction is done, if passed a value, the promise is\n * resolved to that value.\n *\n * @param {Array} storeNames An array of object store names\n * involved in the transaction.\n * @param {string} type Can be `readonly` or `readwrite`.\n * @param {!Function} callback\n * @return {*} The result of the transaction ran by the callback.\n * @private\n */\n async transaction(storeNames, type, callback) {\n await this.open();\n return await new Promise((resolve, reject) => {\n const txn = this._db.transaction(storeNames, type);\n txn.onabort = () => reject(txn.error);\n txn.oncomplete = () => resolve();\n callback(txn, (value) => resolve(value));\n });\n }\n /**\n * Delegates async to a native IDBObjectStore method.\n *\n * @param {string} method The method name.\n * @param {string} storeName The object store name.\n * @param {string} type Can be `readonly` or `readwrite`.\n * @param {...*} args The list of args to pass to the native method.\n * @return {*} The result of the transaction.\n * @private\n */\n async _call(method, storeName, type, ...args) {\n const callback = (txn, done) => {\n const objStore = txn.objectStore(storeName);\n // TODO(philipwalton): Fix this underlying TS2684 error.\n // @ts-ignore\n const request = objStore[method].apply(objStore, args);\n request.onsuccess = () => done(request.result);\n };\n return await this.transaction([storeName], type, callback);\n }\n /**\n * Closes the connection opened by `DBWrapper.open()`. Generally this method\n * doesn't need to be called since:\n * 1. It's usually better to keep a connection open since opening\n * a new connection is somewhat slow.\n * 2. Connections are automatically closed when the reference is\n * garbage collected.\n * The primary use case for needing to close a connection is when another\n * reference (typically in another tab) needs to upgrade it and would be\n * blocked by the current, open connection.\n *\n * @private\n */\n close() {\n if (this._db) {\n this._db.close();\n this._db = null;\n }\n }\n}\n// Exposed on the prototype to let users modify the default timeout on a\n// per-instance or global basis.\nDBWrapper.prototype.OPEN_TIMEOUT = 2000;\n// Wrap native IDBObjectStore methods according to their mode.\nconst methodsToWrap = {\n readonly: ['get', 'count', 'getKey', 'getAll', 'getAllKeys'],\n readwrite: ['add', 'put', 'clear', 'delete'],\n};\nfor (const [mode, methods] of Object.entries(methodsToWrap)) {\n for (const method of methods) {\n if (method in IDBObjectStore.prototype) {\n // Don't use arrow functions here since we're outside of the class.\n DBWrapper.prototype[method] =\n async function (storeName, ...args) {\n return await this._call(method, storeName, mode, ...args);\n };\n }\n }\n}\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { getOrCreateDefaultRouter } from './utils/getOrCreateDefaultRouter.js';\nimport './_version.js';\n/**\n * If a Route throws an error while handling a request, this `handler`\n * will be called and given a chance to provide a response.\n *\n * @param {module:workbox-routing~handlerCallback} handler A callback\n * function that returns a Promise resulting in a Response.\n *\n * @memberof module:workbox-routing\n */\nfunction setCatchHandler(handler) {\n const defaultRouter = getOrCreateDefaultRouter();\n defaultRouter.setCatchHandler(handler);\n}\nexport { setCatchHandler };\n","import { registerRoute, setCatchHandler } from 'workbox-routing';\nimport { precacheAndRoute, getCacheKeyForURL } from 'workbox-precaching';\nimport { isNav } from './utils';\nimport { NETWORK_HANDLER, PRECACHING_OPTIONS } from './contants';\n\nexport function getFiles() {\n\treturn self.__WB_MANIFEST;\n}\n\nexport function setupPrecaching(precacheFiles, precachingOptions) {\n\tprecacheAndRoute(precacheFiles, precachingOptions || PRECACHING_OPTIONS);\n}\n\nexport function setupRouting() {\n\t/**\n\t * Adding this before `precacheAndRoute` lets us handle all\n\t * the navigation requests even if they are in precache.\n\t */\n\tregisterRoute(({ event }) => isNav(event), NETWORK_HANDLER);\n\n\tsetCatchHandler(({ event }) => {\n\t\tif (isNav(event)) {\n\t\t\treturn caches.match(getCacheKeyForURL('/index.html'));\n\t\t}\n\t\treturn Response.error();\n\t});\n}\n\nexport { PRECACHING_OPTIONS };\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { cacheNames as _cacheNames } from './_private/cacheNames.js';\nimport './_version.js';\n/**\n * Get the current cache names and prefix/suffix used by Workbox.\n *\n * `cacheNames.precache` is used for precached assets,\n * `cacheNames.googleAnalytics` is used by `workbox-google-analytics` to\n * store `analytics.js`, and `cacheNames.runtime` is used for everything else.\n *\n * `cacheNames.prefix` can be used to retrieve just the current prefix value.\n * `cacheNames.suffix` can be used to retrieve just the current suffix value.\n *\n * @return {Object} An object with `precache`, `runtime`, `prefix`, and\n * `googleAnalytics` properties.\n *\n * @memberof module:workbox-core\n */\nconst cacheNames = {\n get googleAnalytics() {\n return _cacheNames.getGoogleAnalyticsName();\n },\n get precache() {\n return _cacheNames.getPrecacheName();\n },\n get prefix() {\n return _cacheNames.getPrefix();\n },\n get runtime() {\n return _cacheNames.getRuntimeName();\n },\n get suffix() {\n return _cacheNames.getSuffix();\n },\n};\nexport { cacheNames };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { assert } from 'workbox-core/_private/assert.js';\nimport { WorkboxError } from 'workbox-core/_private/WorkboxError.js';\nimport { getFriendlyURL } from 'workbox-core/_private/getFriendlyURL.js';\nimport { logger } from 'workbox-core/_private/logger.js';\nimport './_version.js';\n/**\n * This class allows you to set up rules determining what\n * status codes and/or headers need to be present in order for a\n * [`Response`](https://developer.mozilla.org/en-US/docs/Web/API/Response)\n * to be considered cacheable.\n *\n * @memberof module:workbox-cacheable-response\n */\nclass CacheableResponse {\n /**\n * To construct a new CacheableResponse instance you must provide at least\n * one of the `config` properties.\n *\n * If both `statuses` and `headers` are specified, then both conditions must\n * be met for the `Response` to be considered cacheable.\n *\n * @param {Object} config\n * @param {Array} [config.statuses] One or more status codes that a\n * `Response` can have and be considered cacheable.\n * @param {Object} [config.headers] A mapping of header names\n * and expected values that a `Response` can have and be considered cacheable.\n * If multiple headers are provided, only one needs to be present.\n */\n constructor(config = {}) {\n if (process.env.NODE_ENV !== 'production') {\n if (!(config.statuses || config.headers)) {\n throw new WorkboxError('statuses-or-headers-required', {\n moduleName: 'workbox-cacheable-response',\n className: 'CacheableResponse',\n funcName: 'constructor',\n });\n }\n if (config.statuses) {\n assert.isArray(config.statuses, {\n moduleName: 'workbox-cacheable-response',\n className: 'CacheableResponse',\n funcName: 'constructor',\n paramName: 'config.statuses',\n });\n }\n if (config.headers) {\n assert.isType(config.headers, 'object', {\n moduleName: 'workbox-cacheable-response',\n className: 'CacheableResponse',\n funcName: 'constructor',\n paramName: 'config.headers',\n });\n }\n }\n this._statuses = config.statuses;\n this._headers = config.headers;\n }\n /**\n * Checks a response to see whether it's cacheable or not, based on this\n * object's configuration.\n *\n * @param {Response} response The response whose cacheability is being\n * checked.\n * @return {boolean} `true` if the `Response` is cacheable, and `false`\n * otherwise.\n */\n isResponseCacheable(response) {\n if (process.env.NODE_ENV !== 'production') {\n assert.isInstance(response, Response, {\n moduleName: 'workbox-cacheable-response',\n className: 'CacheableResponse',\n funcName: 'isResponseCacheable',\n paramName: 'response',\n });\n }\n let cacheable = true;\n if (this._statuses) {\n cacheable = this._statuses.includes(response.status);\n }\n if (this._headers && cacheable) {\n cacheable = Object.keys(this._headers).some((headerName) => {\n return response.headers.get(headerName) === this._headers[headerName];\n });\n }\n if (process.env.NODE_ENV !== 'production') {\n if (!cacheable) {\n logger.groupCollapsed(`The request for ` +\n `'${getFriendlyURL(response.url)}' returned a response that does ` +\n `not meet the criteria for being cached.`);\n logger.groupCollapsed(`View cacheability criteria here.`);\n logger.log(`Cacheable statuses: ` +\n JSON.stringify(this._statuses));\n logger.log(`Cacheable headers: ` +\n JSON.stringify(this._headers, null, 2));\n logger.groupEnd();\n const logFriendlyHeaders = {};\n response.headers.forEach((value, key) => {\n logFriendlyHeaders[key] = value;\n });\n logger.groupCollapsed(`View response status and headers here.`);\n logger.log(`Response status: ` + response.status);\n logger.log(`Response headers: ` +\n JSON.stringify(logFriendlyHeaders, null, 2));\n logger.groupEnd();\n logger.groupCollapsed(`View full response details here.`);\n logger.log(response.headers);\n logger.log(response);\n logger.groupEnd();\n logger.groupEnd();\n }\n }\n return cacheable;\n }\n}\nexport { CacheableResponse };\n","import { NetworkFirst } from 'workbox-strategies';\nimport { cacheNames } from 'workbox-core';\nimport { CacheableResponsePlugin } from 'workbox-cacheable-response';\n\nexport const PRECACHING_OPTIONS = {};\n\nexport const NETWORK_HANDLER = new NetworkFirst({\n\t// this cache is plunged with every new service worker deploy so we dont need to care about purging the cache.\n\tcacheName: cacheNames.precache,\n\tnetworkTimeoutSeconds: 5, // if u dont start getting headers within 5 sec fallback to cache.\n\tplugins: [\n\t\tnew CacheableResponsePlugin({\n\t\t\tstatuses: [200], // only cache valid responses, not opaque responses e.g. wifi portal.\n\t\t}),\n\t],\n});\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { logger } from 'workbox-core/_private/logger.js';\nimport { WorkboxError } from 'workbox-core/_private/WorkboxError.js';\nimport { Route } from './Route.js';\nimport { RegExpRoute } from './RegExpRoute.js';\nimport { getOrCreateDefaultRouter } from './utils/getOrCreateDefaultRouter.js';\nimport './_version.js';\n/**\n * Easily register a RegExp, string, or function with a caching\n * strategy to a singleton Router instance.\n *\n * This method will generate a Route for you if needed and\n * call [registerRoute()]{@link module:workbox-routing.Router#registerRoute}.\n *\n * @param {RegExp|string|module:workbox-routing.Route~matchCallback|module:workbox-routing.Route} capture\n * If the capture param is a `Route`, all other arguments will be ignored.\n * @param {module:workbox-routing~handlerCallback} [handler] A callback\n * function that returns a Promise resulting in a Response. This parameter\n * is required if `capture` is not a `Route` object.\n * @param {string} [method='GET'] The HTTP method to match the Route\n * against.\n * @return {module:workbox-routing.Route} The generated `Route`(Useful for\n * unregistering).\n *\n * @memberof module:workbox-routing\n */\nfunction registerRoute(capture, handler, method) {\n let route;\n if (typeof capture === 'string') {\n const captureUrl = new URL(capture, location.href);\n if (process.env.NODE_ENV !== 'production') {\n if (!(capture.startsWith('/') || capture.startsWith('http'))) {\n throw new WorkboxError('invalid-string', {\n moduleName: 'workbox-routing',\n funcName: 'registerRoute',\n paramName: 'capture',\n });\n }\n // We want to check if Express-style wildcards are in the pathname only.\n // TODO: Remove this log message in v4.\n const valueToCheck = capture.startsWith('http') ?\n captureUrl.pathname : capture;\n // See https://github.com/pillarjs/path-to-regexp#parameters\n const wildcards = '[*:?+]';\n if ((new RegExp(`${wildcards}`)).exec(valueToCheck)) {\n logger.debug(`The '$capture' parameter contains an Express-style wildcard ` +\n `character (${wildcards}). Strings are now always interpreted as ` +\n `exact matches; use a RegExp for partial or wildcard matches.`);\n }\n }\n const matchCallback = ({ url }) => {\n if (process.env.NODE_ENV !== 'production') {\n if ((url.pathname === captureUrl.pathname) &&\n (url.origin !== captureUrl.origin)) {\n logger.debug(`${capture} only partially matches the cross-origin URL ` +\n `${url}. This route will only handle cross-origin requests ` +\n `if they match the entire URL.`);\n }\n }\n return url.href === captureUrl.href;\n };\n // If `capture` is a string then `handler` and `method` must be present.\n route = new Route(matchCallback, handler, method);\n }\n else if (capture instanceof RegExp) {\n // If `capture` is a `RegExp` then `handler` and `method` must be present.\n route = new RegExpRoute(capture, handler, method);\n }\n else if (typeof capture === 'function') {\n // If `capture` is a function then `handler` and `method` must be present.\n route = new Route(capture, handler, method);\n }\n else if (capture instanceof Route) {\n route = capture;\n }\n else {\n throw new WorkboxError('unsupported-route-type', {\n moduleName: 'workbox-routing',\n funcName: 'registerRoute',\n paramName: 'capture',\n });\n }\n const defaultRouter = getOrCreateDefaultRouter();\n defaultRouter.registerRoute(route);\n return route;\n}\nexport { registerRoute };\n","/*\n Copyright 2018 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { CacheableResponse } from './CacheableResponse.js';\nimport './_version.js';\n/**\n * A class implementing the `cacheWillUpdate` lifecycle callback. This makes it\n * easier to add in cacheability checks to requests made via Workbox's built-in\n * strategies.\n *\n * @memberof module:workbox-cacheable-response\n */\nclass CacheableResponsePlugin {\n /**\n * To construct a new CacheableResponsePlugin instance you must provide at\n * least one of the `config` properties.\n *\n * If both `statuses` and `headers` are specified, then both conditions must\n * be met for the `Response` to be considered cacheable.\n *\n * @param {Object} config\n * @param {Array} [config.statuses] One or more status codes that a\n * `Response` can have and be considered cacheable.\n * @param {Object} [config.headers] A mapping of header names\n * and expected values that a `Response` can have and be considered cacheable.\n * If multiple headers are provided, only one needs to be present.\n */\n constructor(config) {\n /**\n * @param {Object} options\n * @param {Response} options.response\n * @return {Response|null}\n * @private\n */\n this.cacheWillUpdate = async ({ response }) => {\n if (this._cacheableResponse.isResponseCacheable(response)) {\n return response;\n }\n return null;\n };\n this._cacheableResponse = new CacheableResponse(config);\n }\n}\nexport { CacheableResponsePlugin };\n","/*\n Copyright 2019 Google LLC\n\n Use of this source code is governed by an MIT-style\n license that can be found in the LICENSE file or at\n https://opensource.org/licenses/MIT.\n*/\nimport { getOrCreatePrecacheController } from './utils/getOrCreatePrecacheController.js';\nimport './_version.js';\n/**\n * Takes in a URL, and returns the corresponding URL that could be used to\n * lookup the entry in the precache.\n *\n * If a relative URL is provided, the location of the service worker file will\n * be used as the base.\n *\n * For precached entries without revision information, the cache key will be the\n * same as the original URL.\n *\n * For precached entries with revision information, the cache key will be the\n * original URL with the addition of a query parameter used for keeping track of\n * the revision info.\n *\n * @param {string} url The URL whose cache key to look up.\n * @return {string} The cache key that corresponds to that URL.\n *\n * @memberof module:workbox-precaching\n */\nfunction getCacheKeyForURL(url) {\n const precacheController = getOrCreatePrecacheController();\n return precacheController.getCacheKeyForURL(url);\n}\nexport { getCacheKeyForURL };\n","\"use strict\";\n// @ts-ignore\ntry {\n self['workbox:core:5.1.3'] && _();\n}\ncatch (e) { }\n"]} \ No newline at end of file diff --git a/package-lock.json b/package-lock.json index 1ea8cf1..e26928d 100644 --- a/package-lock.json +++ b/package-lock.json @@ -1459,12 +1459,12 @@ "dev": true }, "@types/react": { - "version": "16.9.41", - "resolved": "https://registry.npmjs.org/@types/react/-/react-16.9.41.tgz", - "integrity": "sha512-6cFei7F7L4wwuM+IND/Q2cV1koQUvJ8iSV+Gwn0c3kvABZ691g7sp3hfEQHOUBJtccl1gPi+EyNjMIl9nGA0ug==", + "version": "16.9.46", + "resolved": "https://registry.npmjs.org/@types/react/-/react-16.9.46.tgz", + "integrity": "sha512-dbHzO3aAq1lB3jRQuNpuZ/mnu+CdD3H0WVaaBQA8LTT3S33xhVBUj232T8M3tAhSWJs/D/UqORYUlJNl/8VQZg==", "requires": { "@types/prop-types": "*", - "csstype": "^2.2.0" + "csstype": "^3.0.2" } }, "@types/resolve": { @@ -2267,9 +2267,9 @@ "dev": true }, "bluelib": { - "version": "0.14.2", - "resolved": "https://registry.npmjs.org/bluelib/-/bluelib-0.14.2.tgz", - "integrity": "sha512-3EQ0k800HD+L8LUFMShop6b2ZNpg+fky5+DQ76vZ8ODmJ5ucAxOC+tlBgmDDBR+oAX2d6qZXeGD2Olx2QTwmBA==", + "version": "0.15.1", + "resolved": "https://registry.npmjs.org/bluelib/-/bluelib-0.15.1.tgz", + "integrity": "sha512-ZedqpBtoAYsXfOpJOeZfXraYMH0k6HxRFplaQWNm7G5PXZssmRWO/nP7X+rWSlI5LBHDdmjycNA6VbtCNd6DNg==", "requires": { "@fortawesome/fontawesome-svg-core": "^1.2.29", "@fortawesome/free-brands-svg-icons": "^5.13.1", @@ -4070,9 +4070,9 @@ } }, "csstype": { - "version": "2.6.11", - "resolved": "https://registry.npmjs.org/csstype/-/csstype-2.6.11.tgz", - "integrity": "sha512-l8YyEC9NBkSm783PFTvh0FmJy7s5pFKrDp49ZL7zBGX3fWkO+N4EEyan1qqp8cwPLDcD0OSdyY6hAMoxp34JFw==" + "version": "3.0.2", + "resolved": "https://registry.npmjs.org/csstype/-/csstype-3.0.2.tgz", + "integrity": "sha512-ofovWglpqoqbfLNOTBNZLSbMuGrblAf1efvvArGKOZMBrIoJeu5UsAipQolkijtyQx5MtAzT/J9IHj/CEY1mJw==" }, "cyclist": { "version": "1.0.1", @@ -4317,9 +4317,9 @@ "dev": true }, "dequal": { - "version": "1.0.0", - "resolved": "https://registry.npmjs.org/dequal/-/dequal-1.0.0.tgz", - "integrity": "sha512-/Nd1EQbQbI9UbSHrMiKZjFLrXSnU328iQdZKPQf78XQI6C+gutkFUeoHpG5J08Ioa6HeRbRNFpSIclh1xyG0mw==" + "version": "1.0.1", + "resolved": "https://registry.npmjs.org/dequal/-/dequal-1.0.1.tgz", + "integrity": "sha512-Fx8jxibzkJX2aJgyfSdLhr9tlRoTnHKrRJuu2XHlAgKioN2j19/Bcbe0d4mFXYZ3+wpE2KVobUVTfDutcD17xQ==" }, "des.js": { "version": "1.0.1", @@ -9077,9 +9077,9 @@ "integrity": "sha1-v77VbV6ad2ZF9LH/eqGjrE+jw4U=" }, "prismjs": { - "version": "1.20.0", - "resolved": "https://registry.npmjs.org/prismjs/-/prismjs-1.20.0.tgz", - "integrity": "sha512-AEDjSrVNkynnw6A+B1DsFkd6AVdTnp+/WoUixFRULlCLZVRZlVQMVWio/16jv7G1FscUxQxOQhWwApgbnxr6kQ==", + "version": "1.21.0", + "resolved": "https://registry.npmjs.org/prismjs/-/prismjs-1.21.0.tgz", + "integrity": "sha512-uGdSIu1nk3kej2iZsLyDoJ7e9bnPzIgY0naW/HdknGj61zScaprVEVGHrPoXqI+M9sP0NDnTK2jpkvmldpuqDw==", "requires": { "clipboard": "^2.0.0" } diff --git a/package.json b/package.json index 0bc8952..6d2d92d 100644 --- a/package.json +++ b/package.json @@ -1,7 +1,7 @@ { "private": true, "name": "appuntiweb", - "version": "0.5.8", + "version": "0.6.0", "license": "AGPL-3.0-or-later", "scripts": { "start": "preact watch --template src/template.html", @@ -21,7 +21,7 @@ "@fortawesome/react-fontawesome": "^0.1.11", "@sentry/browser": "^5.19.0", "@sentry/integrations": "^5.19.0", - "bluelib": "^0.14.2", + "bluelib": "^0.15.1", "chart.js": "^2.9.3", "css-loader": "^3.6.0", "file-loader": "^5.1.0", diff --git a/size-plugin.json b/size-plugin.json index edfde83..7244c07 100644 --- a/size-plugin.json +++ b/size-plugin.json @@ -1 +1 @@ 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diff --git a/src/index.js b/src/index.js index c1636cb..c03e822 100644 --- a/src/index.js +++ b/src/index.js @@ -1,4 +1,6 @@ // Import debugging tools +import RipassoDiAlgebraLineare from "./routes/RipassoDiAlgebraLineare"; + let Sentry = null; if(process.env.NODE_ENV === "development") { console.debug("Initializing Preact Debugger...") @@ -24,7 +26,7 @@ else if(process.env.NODE_ENV === "production") { // noinspection ES6UnusedImports import "bluelib/dist/index.css"; -import {Box, BoxColors, Panel, theme} from "bluelib"; +import {Bluelib, Box, BoxColors, Panel} from "bluelib"; import './meta/manifest.json'; import './meta/CNAME'; import './meta/.nojekyll'; @@ -58,7 +60,7 @@ export default function(props) { return ( -
+

Appuntiweb di Steffo

@@ -72,6 +74,7 @@ export default function(props) { + @@ -79,7 +82,7 @@ export default function(props) {
-
+ ); diff --git a/src/routes/CalcoloNumerico.js b/src/routes/CalcoloNumerico.js index 80fa767..55fddf7 100644 --- a/src/routes/CalcoloNumerico.js +++ b/src/routes/CalcoloNumerico.js @@ -1,4 +1,4 @@ -import {Section, Latex, Panel, Todo, Timer, LatexDefaultInline} from "bluelib"; +import {Section, ILatex, PLatex, BLatex, Panel, Todo, Timer, LatexDefaultInline} from "bluelib"; import Example from "../components/Example"; const r = String.raw; @@ -6,317 +6,299 @@ const r = String.raw; export default function (props) { return ( - -
-

Calcolo Numerico

-
- - - - +
+

Calcolo Numerico

+
+ + + + +

+ Se sei uno studente dell'Unimore, puoi accedere all'archivio del corso su Google Drive.. +

+ + +

+ Prima di studiare Calcolo Numerico, guardati i prerequisiti di Algebra Lineare! +

+ +
+
+ +

+ E' composto da: +

+
    +
  • 2 domande sugli argomenti teorici
    • +
  • 1 domanda di implementazione algoritmo in MATLAB
    • +
+ + +
    +
    2. +
+ +
+
+ +

+ Algoritmi che hanno: +

+
    +
  • numeri reali in input e output
    • +
  • successioni delle quattro operazioni aritmetiche fondamentali come passi
    • +
+ +
+
+ +

+ Con i numeri floating point può capitare che un certo numero {r`\alpha`} non sia rappresentato correttamente. +

+

+ In tal caso, il numero si indica con {r`\alpha^*`}. +

+ +
+
+ +

+ È la differenza tra il numero desiderato e il numero rappresentato: +

+ {r`E_a = \left | \alpha - \alpha^* \right |`} + + +

+ Indica quanto il numero rappresentato differisce dal numero desiderato: +

+ {r`\forall \alpha \neq 0, E_r = \frac{E_a}{\left | \alpha \right |}`} + +
+
+ +

+ Metodo con cui gestire gli underflow floating point: le cifre meno significative vengono rimosse. +

+ + 
+                            1.00  →  1.0

+                            1.01  →  1.0

+                            1.10  →  1.1

+                            1.11  →  1.1
+
+ + + +

+ Metodo con cui gestire gli underflow floating point: se la cifra più significativa di quelle che devono essere rimosse è 1, allora aumenta di 1 anche quella meno signficativa che viene tenuta. +

+ + 
+                            1.00  →  1.0

+                            1.01  →  1.0

+                            1.10  →  1.1

+                            1.11  → 10.
+
+ + +
+
+ +

+ L'errore relativo di un numero reale rappresentato in virgola mobile è minore o uguale alla precisione di macchina: +

+

+ {r`E_r \leq k \cdot \beta^{1-t}`} +

+
    +
  • + \beta è uguale alla base utilizzata (solitamente 2). +
    • +
  • + t è uguale al numero di cifre della mantissa. +
    • +
  • + k è uguale a 1 se il numero viene rappresentato per troncamento oppure a {r`\frac{1}{2}`} se viene rappresentato per arrotondamento. +
    • +
+ + +

+ Associa un valore reale al suo corrispondente valore floating point, utilizzando uno dei due metodi di gestione dell'undeflow. +

+ {r`fl(x) = (x)(1 + \epsilon_x)`} + + Indica che un valore è soggetto alla precisione di macchina. + {r`fl(1.11) = 1.1`} + + +
+
+ +

+ L'insieme {r`\mathbb{F}`} è il sottoinsieme dei numeri reali rappresentabili in floating point dalla macchina che stiamo usando. +

+

+ Operazioni tra elementi di {r`\mathbb{F}`} producono risultati in {r`\mathbb{R}`}, che però decaderanno nuovamente a elementi di {r`\mathbb{F}`}, perdendo informazioni. +

+

+ Il teorema della precisione di macchina si applica quindi anche ai risultati delle operazioni. +

+ + +
    +
  • Hanno più di un elemento neutro.
    • +
  • Un numero ha più opposti.
    • +
  • Non sono associative.
    • +
  • Non sono distributive.
    • +
  • Non vale la legge di annullamento del prodotto.
    • +
+ +
+
+ +

+ Errore derivato da underflow sui dati. +

+

+ Si indica con {r`\epsilon_{variabile}`}. +

+ + L'errore sulla variabile x si indica con {r`\epsilon_{x}`}. + + + +

+ Errore derivato da underflow durante l'esecuzione dell'algoritmo. +

+

+ Si indica con {r`\epsilon_{passo}`}. +

+ + L'errore al primo passo dell'algoritmo si indica con {r`\epsilon_{1}`}. + + +
+
+ +

+ Sensibilità di un problema all'errore inerente. +

+ + {r`y = \frac{1}{x}`} è mal condizionato intorno allo 0 e ben condizionato lontano dallo 0. + + + +

+ Sensibilità di un problema all'errore algoritmico. +

+

- Se sei uno studente dell'Unimore, puoi accedere all'archivio del corso su Google Drive.. -

- -
-
- -

- E' composto da: -

-
    -
  • 2 domande sugli argomenti teorici
    • -
  • 1 domanda di implementazione algoritmo in MATLAB
    • -
- - -
    -
    2. -
    3. -
    4. -
    5. -
    6. -
    7. -
- -
-
- -

- Algoritmi che hanno: -

-
    -
  • numeri reali in input e output
    • -
  • successioni delle quattro operazioni aritmetiche fondamentali come passi
    • -
- -
-
- -

- Con i numeri floating point può capitare che un certo numero {r`\alpha`} non sia rappresentato correttamente. + Cerchiamo un algoritmo che risolva {r`2x^* = 4`}.

- In tal caso, il numero si indica con {r`\alpha^*`}. -

- - -

- È la differenza tra il numero desiderato e il numero rappresentato: + Calcolare prima {r`t = fl \left( \frac{1}{4} \right)`} e poi {r`x = fl ( 2 \cdot t )`} porta a una perdita di precisione.

- {r`E_a = \left | \alpha - \alpha^* \right |`} + Calcolare direttamente {r`x = fl \left( \frac{2}{4} \right)`} non ha alcuna perdita di precisione e rende l'algoritmo più stabile del precedente.

- - Vale sempre 0, tranne in caso di underflow o overflow. - - - -

- Indica quanto il numero rappresentato differisce dal numero desiderato: -

-

- {r`\forall \alpha \neq 0, E_r = \frac{E_a}{\left | \alpha \right |}`} -

- -
-
- -

- L'errore relativo di un numero reale rappresentato in virgola mobile è minore o uguale alla precisione di macchina: -

-

- {r`E_r \leq k \cdot \beta^{1-t}`} -

-
    -
  • - \beta è uguale alla base utilizzata (solitamente 2). -
    • -
  • - t è uguale al numero di cifre della mantissa. -
    • -
  • - k è uguale a 1 se il numero viene rappresentato per troncamento oppure a {r`\frac{1}{2}`} se viene rappresentato per arrotondamento. -
    • -
- -
-
- -

- L'insieme F è il sottoinsieme dei numeri reali rappresentabili in floating point dalla macchina che stiamo usando. -

-

- Operazioni tra elementi di F producono risultati in {r`\mathbb{R}`}, che però decaderanno nuovamente a elementi di F, perdendo informazioni. -

-

- Il teorema della precisione di macchina si applica quindi anche ai risultati delle operazioni. -

- - -
    -
  • Hanno più di un elemento neutro.
    • -
  • Un numero ha più opposti.
    • -
  • Non sono associative.
    • -
  • Non sono distributive.
    • -
  • Non vale la legge di annullamento del prodotto.
    • -
- - -

- Indica che un valore è soggetto alla precisione di macchina: -

-

- {r`fl(x) = (x)(1 + \epsilon_x)`} -

- -
-
- -

- Errore dipendente solo dai dati. -

-

- TODO: qual è la formula? -

-

- È l'errore che si presenterebbe se {r`\epsilon_1 = \epsilon_2 = \dots = 0`}. -

- - -

- Errore dipendente solo dalle operazioni effettuate. -

-

- TODO: qual è la formula? -

-

- È l'errore che si presenterebbe se {r`\epsilon_x = \epsilon_y = \dots = 0`}. -

- -
-
- -

- Un problema poco sensibile all'errore inerente. -

- - {r`y = \frac{1}{x}`} è mal condizionato intorno allo 0 e ben condizionato lontano dallo 0. - - - -

- Un algoritmo poco sensibile all'errore algoritmico. -

- -

- Cerchiamo un algoritmo che risolva {r`2x = 4`}. -

-

- Calcolare prima {r`t = fl \left( \frac{1}{4} \right)`} e poi {r`x^* = fl ( 2 \cdot t )`} porta a una perdita di precisione. -

-

- Calcolare direttamente {r`x^* = fl \left( \frac{2}{4} \right)`} non ha alcuna perdita di precisione e rende l'algoritmo più stabile del precedente. -

- - -
-
- -

- È il coefficiente di proporzionalità tra i dati e l'errore inerente. -

-

- Essendo sempre maggiore di uno, si può dire che sia un coefficiente di amplificazione. -

-

- Minore è l'indice di condizionamento, meglio condizionato è un problema. -

- - -

- È il coefficiente di proporzionalità tra i dati e l'errore algoritmico. -

-

- Essendo sempre maggiore di uno, si può dire che sia un coefficiente di amplificazione. -

- -
-
- -

- Matrice con determinante 0. -

-

- Le matrici singolari non sono invertibili. -

- -
-
- -

- Funzione che associa un valore positivo a ogni vettore diverso da 0, e 0 al vettore zero. -

- - Esempi su Wikipedia - - - -

- Massimo dei valori assoluti di tutti gli elementi del vettore. -

-

- {r`\Vert x \Vert_\infty = max_{i = 1..n} | x_i |`} -

- - -

- Somma dei valori assoluti di tutti gli elementi del vettore. -

-

- {r`\Vert x \Vert_1 = \sum_{i = 1}^n | x_i |`} -

- - -

- Radice quadrata della somma dei quadrati di tutti gli elementi del vettore. -

-

- {r`\Vert x \Vert_2 = \sqrt{\sum_{i = 1}^n x_i^2}`} -

- -
-
- -

- Funzione che associa un valore positivo a ogni matrice diversa da 0, e 0 alla matrice zero. -

-

- Si ricavano dalle norme vettoriali: -

-

- {r`\Vert A \Vert = sup_{x \in \mathbb{R}, x \neq 0} \frac{\Vert A \cdot x \Vert}{\Vert x \Vert}`} -

- - sup è l'estremo superiore di un insieme. E' molto simile al massimo: ricordi le prime lezioni di Analisi? - - - -

- Massimo delle somme dei valori assoluti di tutti gli elementi di ogni riga di una matrice. -

-

- {r`\Vert A \Vert_\infty = max_{i = 1..n} \sum_{j = 1}^n | a_{ij} |`} -

- - -

- Massimo delle somme dei valori assoluti di tutti gli elementi di ogni colonna di una matrice. -

-

- {r`\Vert A \Vert_1 = max_{j = 1..n} \sum_{i = 1}^n | a_{ij} |`} -

- - -

- Radice quadrata del rango del prodotto tra una matrice e la sua trasposta. -

-

- {r`\Vert A \Vert_2 = \sqrt{\rho ( A^T \times A ) }`} -

- -
-
- -

- Le norme sono usate per calcolare l'errore relativo tra due vettori o matrici: -

-

- {r`\frac{\Vert x - y \Vert}{\Vert x \Vert}`} -

- -
-
- -

- La fattorizzazione è il processo che permette di risolvere sistemi di equazioni lineari rappresentati in forma di matrice. -

-

- Esistono molteplici algoritmi in grado di realizzarla: mentre tutti portano alla stessa soluzione, possono avere velocità e indici algoritmici diversi. -

-

- Il sistema lineare da risolvere viene diviso in due parti: la matrice dei coefficienti e il vettore termine noto. -

- - -

+ + +

+
+ +

+ È il coefficiente di proporzionalità tra i dati e l'errore inerente. +

+

+ Essendo sempre maggiore di uno, si può dire che sia un coefficiente di amplificazione. +

+

+ Minore è l'indice di condizionamento, meglio condizionato è un problema. +

+ + +

+ È il coefficiente di proporzionalità tra i dati e l'errore algoritmico. +

+

+ Essendo sempre maggiore di uno, si può dire che sia un coefficiente di amplificazione. +

+ +
-

- -
-
- +
+ +

+ Funzione che associa un valore positivo a ogni matrice diversa da 0, e 0 alla matrice zero. +

+

+ Si ricavano dalle norme vettoriali: +

+

+ {r`\Vert A \Vert = sup_{x \in \mathbb{R}, x \neq 0} \frac{\Vert A \cdot x \Vert}{\Vert x \Vert}`} +

+ + sup è l'estremo superiore di un insieme. E' molto simile al massimo: ricordi le prime lezioni di Analisi? + + + +

+ Massimo delle somme dei valori assoluti di tutti gli elementi di ogni riga di una matrice. +

+

+ {r`\Vert A \Vert_\infty = max_{i = 1..n} \sum_{j = 1}^n | a_{ij} |`} +

+ + +

+ Massimo delle somme dei valori assoluti di tutti gli elementi di ogni colonna di una matrice. +

+

+ {r`\Vert A \Vert_1 = max_{j = 1..n} \sum_{i = 1}^n | a_{ij} |`} +

+ + +

+ Radice quadrata del rango del prodotto tra una matrice e la sua trasposta. +

+

+ {r`\Vert A \Vert_2 = \sqrt{\rho ( A^T \times A ) }`} +

+ +
+
+ + + {r`\frac{{\color{yellow} \|A\| \cdot \|A^{-1}\|} \cdot \| \Delta b \|}{\| b \|}`} +

+ In particolare, le numero di condizionamento: +

+ + {r`k(A) = `} + + +
+
+ +

+ La fattorizzazione è il processo che permette di risolvere sistemi di equazioni lineari rappresentati in forma di matrice. +

+

+ Esistono molteplici algoritmi in grado di realizzarla: mentre tutti portano alla stessa soluzione, possono avere velocità e indici algoritmici diversi. +

+

+ Il sistema lineare da risolvere viene diviso in due parti: la matrice dei coefficienti e il vettore termine noto. +

+ + +

+ +

+ +
+ ) } diff --git a/src/routes/RipassoDiAlgebraLineare.js b/src/routes/RipassoDiAlgebraLineare.js new file mode 100644 index 0000000..6375ba1 --- /dev/null +++ b/src/routes/RipassoDiAlgebraLineare.js @@ -0,0 +1,213 @@ +import {ILatex, Panel, PLatex, Section} from "bluelib"; +import Example from "../components/Example"; + +const r = String.raw; + + +export default function(params) { + return ( +
+

Ripasso di Algebra Lineare per Calcolo Numerico

+
+ +

+ Elemento neutro della moltiplicazione matriciale. +

+ + {r` + \begin{pmatrix} + {\color{Yellow} 1} & {\color{Yellow} 0} & {\color{Yellow} 0}\\ + {\color{Yellow} 0} & {\color{Yellow} 1} & {\color{Yellow} 0}\\ + {\color{Yellow} 0} & {\color{Yellow} 0} & {\color{Yellow} 1} + \end{pmatrix} + `} + + + +

+ Matrice con elementi diversi da 0 solo sulla diagonale. +

+ + {r` + \begin{pmatrix} + {\color{Yellow} 3} & {\color{Gray} 0} & {\color{Gray} 0}\\ + {\color{Gray} 0} & {\color{Yellow} 4} & {\color{Gray} 0}\\ + {\color{Gray} 0} & {\color{Gray} 0} & {\color{Yellow} 5} + \end{pmatrix} + `} + + + +

+ Matrice con elementi diversi da 0 sopra la diagonale. +

+ + {r` + \begin{pmatrix} + {\color{Yellow} 3} & {\color{Gray} 0} & {\color{Gray} 0}\\ + {\color{Orange} 4} & {\color{Yellow} 4} & {\color{Gray} 0}\\ + {\color{Orange} 5} & {\color{Orange} 5} & {\color{Yellow} 5} + \end{pmatrix} + `} + + + +

+ Matrice con elementi diversi da 0 sotto la diagonale. +

+ + {r` + \begin{pmatrix} + {\color{Yellow} 3} & {\color{Orange} 3} & {\color{Orange} 3}\\ + {\color{Gray} 0} & {\color{Yellow} 4} & {\color{Orange} 4}\\ + {\color{Gray} 0} & {\color{Gray} 0} & {\color{Yellow} 5} + \end{pmatrix} + `} + + + +

+ Matrice con determinante diverso da 0. +

+ {r`det(A) \neq 0`} +

+ Sono anche dette matrici linearmente indipendenti o matrici invertibili. +

+ + {r` + \begin{pmatrix} + {\color{Yellow} 1} & {\color{Yellow} 1} & {\color{Yellow} 2}\\ + {\color{Orange} 2} & {\color{Orange} 1} & {\color{Orange} 1}\\ + {\color{Red} 1} & {\color{Red} 2} & {\color{Red} 1} + \end{pmatrix} + `} + + + +

+ Matrice con un asse di simmetria lungo la diagonale. +

+ {r`A = A^T`} + + {r` + \begin{pmatrix} + 1 & {\color{Yellow} 2} & {\color{Orange} 4}\\ + {\color{Yellow} 2} & 3 & {\color{Red} 5}\\ + {\color{Orange} 4} & {\color{Red} 5} & 6 + \end{pmatrix} + `} + + + +

+ Matrice con un asse di simmetria lungo la diagonale; gli elementi nel triangolo superiore sono però l'opposto di quelli del triangolo inferiore. +

+

+ Ha sempre degli 0 lungo la diagonale. +

+ {r`A = -A^T`} + + {r` + \begin{pmatrix} + {\color{Gray} 0} & {\color{Yellow} -2} & {\color{Orange} -4}\\ + {\color{Yellow} 2} & {\color{Gray} 0} & {\color{Red} -5}\\ + {\color{Orange} 4} & {\color{Red} 5} & {\color{Gray} 0} + \end{pmatrix} + `} + + + +

+ Matrice in cui i valori della diagonale sono maggiori della somma di tutti gli altri nella riga/colonna. +

+ + {r` + \begin{pmatrix} + {\color{Orange} 9} & 1 & 2\\ + 1 & {\color{Orange} 8} & 1\\ + 1 & 2 & {\color{Orange} 7} + \end{pmatrix} + `} + + + +

+ Matrice che se moltiplicata per la sua trasposta dà come risultato la matrice identità. +

+ {r`A^T \cdot A = I`} + + {r` + \begin{pmatrix} + \frac{1}{3} & \frac{2}{3} & -\frac{2}{3}\\ + \frac{2}{3} & \frac{1}{3} & \frac{2}{3}\\ + \frac{2}{3} & -\frac{2}{3} & -\frac{1}{3}\\ + \end{pmatrix} + `} + + + +

+ Matrice tale che: +

+ {r`A^{-1} \cdot A = I`} + + +

+ Matrice con pochissimi valori diversi da 0. +

+ + {r` + \begin{pmatrix} + {\color{Gray} 0} & 1 & {\color{Gray} 0}\\ + 1 & 1 & {\color{Gray} 0}\\ + {\color{Gray} 0} & {\color{Gray} 0} & 1 + \end{pmatrix} + `} + + +
+
+ +

+ Funzione che associa un valore positivo a ogni vettore diverso da 0, e 0 al vettore zero. +

+ + Esempi su Wikipedia + + + +

+ Massimo dei valori assoluti di tutti gli elementi del vettore. +

+

+ {r`\Vert x \Vert_\infty = max_{i = 1..n} | x_i |`} +

+ + +

+ Somma dei valori assoluti di tutti gli elementi del vettore. +

+

+ {r`\Vert x \Vert_1 = \sum_{i = 1}^n | x_i |`} +

+ + +

+ Radice quadrata della somma dei quadrati di tutti gli elementi del vettore. +

+

+ {r`\Vert x \Vert_2 = \sqrt{\sum_{i = 1}^n x_i^2}`} +

+ + +

+ Le norme sono usate per calcolare l'errore relativo tra due vettori o matrici: +

+

+ {r`\frac{\Vert x - y \Vert}{\Vert x \Vert}`} +

+ +
+
+ ) +} \ No newline at end of file diff --git a/src/styles/constants.less b/src/styles/constants.less index 4476463..8d19b83 100644 --- a/src/styles/constants.less +++ b/src/styles/constants.less @@ -25,11 +25,6 @@ @disabledfg: #808080; @disabledbg: #1f1f1f; -@sans: "Arial", sans-serif; -@text: "Calibri", sans-serif; -@title: "Verdana", sans-serif; -@mono: "Consolas", "SFMono-Regular", "Liberation Mono", "Menlo", monospace; - @example: #d3a1ff;