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копироватьфайл краткоепредставлениеошибки лев макс местноевремя месяц мин минута монопольныйрежим найти найтинедопустимыесимволыxml найтиокнопонавигационнойссылке найтипомеченныенаудаление найтипоссылкам найтифайлы началогода началодня началоквартала началомесяца началоминуты началонедели началочаса начатьзапросразрешенияпользователя начатьзапускприложения начатькопированиефайла начатьперемещениефайла начатьподключениевнешнейкомпоненты начатьподключениерасширенияработыскриптографией начатьподключениерасширенияработысфайлами начатьпоискфайлов начатьполучениекаталогавременныхфайлов начатьполучениекаталогадокументов начатьполучениерабочегокаталогаданныхпользователя начатьполучениефайлов начатьпомещениефайла начатьпомещениефайлов начатьсозданиедвоичныхданныхизфайла начатьсозданиекаталога начатьтранзакцию начатьудалениефайлов начатьустановкувнешнейкомпоненты начатьустановкурасширенияработыскриптографией начатьустановкурасширенияработысфайлами неделягода необходимостьзавершениясоединения номерсеансаинформационнойбазы номерсоединенияинформационнойбазы нрег нстр обновитьинтерфейс обновитьнумерациюобъектов обновитьповторноиспользуемыезначения обработкапрерыванияпользователя объединитьфайлы окр описаниеошибки оповестить оповеститьобизменении отключитьобработчикзапросанастроекклиенталицензирования отключитьобработчикожидания отключитьобработчикоповещения открытьзначение открытьиндекссправки открытьсодержаниесправки открытьсправку открытьформу открытьформумодально отменитьтранзакцию очиститьжурналрегистрации очиститьнастройкипользователя очиститьсообщения параметрыдоступа перейтипонавигационнойссылке переместитьфайл подключитьвнешнююкомпоненту подключитьобработчикзапросанастроекклиенталицензирования подключитьобработчикожидания подключитьобработчикоповещения подключитьрасширениеработыскриптографией подключитьрасширениеработысфайлами подробноепредставлениеошибки показатьвводдаты показатьвводзначения показатьвводстроки показатьвводчисла показатьвопрос показатьзначение показатьинформациюобошибке показатьнакарте показатьоповещениепользователя показатьпредупреждение полноеимяпользователя получитьcomобъект получитьxmlтип получитьадреспоместоположению получитьблокировкусеансов получитьвремязавершенияспящегосеанса получитьвремязасыпанияпассивногосеанса получитьвремяожиданияблокировкиданных получитьданныевыбора получитьдополнительныйпараметрклиенталицензирования получитьдопустимыекодылокализации получитьдопустимыечасовыепояса получитьзаголовокклиентскогоприложения получитьзаголовоксистемы получитьзначенияотборажурналарегистрации получитьидентификаторконфигурации получитьизвременногохранилища получитьимявременногофайла получитьимяклиенталицензирования получитьинформациюэкрановклиента получитьиспользованиежурналарегистрации получитьиспользованиесобытияжурналарегистрации получитькраткийзаголовокприложения получитьмакетоформления получитьмаскувсефайлы получитьмаскувсефайлыклиента получитьмаскувсефайлысервера получитьместоположениепоадресу получитьминимальнуюдлинупаролейпользователей получитьнавигационнуюссылку получитьнавигационнуюссылкуинформационнойбазы получитьобновлениеконфигурациибазыданных получитьобновлениепредопределенныхданныхинформационнойбазы получитьобщиймакет получитьобщуюформу получитьокна получитьоперативнуюотметкувремени получитьотключениебезопасногорежима получитьпараметрыфункциональныхопцийинтерфейса получитьполноеимяпредопределенногозначения получитьпредставлениянавигационныхссылок получитьпроверкусложностипаролейпользователей получитьразделительпути получитьразделительпутиклиента получитьразделительпутисервера получитьсеансыинформационнойбазы получитьскоростьклиентскогосоединения получитьсоединенияинформационнойбазы получитьсообщенияпользователю получитьсоответствиеобъектаиформы получитьсоставстандартногоинтерфейсаodata получитьструктурухранениябазыданных получитьтекущийсеансинформационнойбазы получитьфайл получитьфайлы получитьформу получитьфункциональнуюопцию получитьфункциональнуюопциюинтерфейса 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стрразделить стрсоединить стрсравнить стрчисловхождений стрчислострок стршаблон текущаядата текущаядатасеанса текущаяуниверсальнаядата текущаяуниверсальнаядатавмиллисекундах текущийвариантинтерфейсаклиентскогоприложения текущийвариантосновногошрифтаклиентскогоприложения текущийкодлокализации текущийрежимзапуска текущийязык текущийязыксистемы тип типзнч транзакцияактивна трег удалитьданныеинформационнойбазы удалитьизвременногохранилища удалитьобъекты удалитьфайлы универсальноевремя установитьбезопасныйрежим установитьбезопасныйрежимразделенияданных установитьблокировкусеансов установитьвнешнююкомпоненту установитьвремязавершенияспящегосеанса установитьвремязасыпанияпассивногосеанса установитьвремяожиданияблокировкиданных установитьзаголовокклиентскогоприложения установитьзаголовоксистемы установитьиспользованиежурналарегистрации установитьиспользованиесобытияжурналарегистрации установитькраткийзаголовокприложения установитьминимальнуюдлинупаролейпользователей установитьмонопольныйрежим установитьнастройкиклиенталицензирования установитьобновлениепредопределенныхданныхинформационнойбазы установитьотключениебезопасногорежима установитьпараметрыфункциональныхопцийинтерфейса установитьпривилегированныйрежим установитьпроверкусложностипаролейпользователей установитьрасширениеработыскриптографией установитьрасширениеработысфайлами установитьсоединениесвнешнимисточникомданных установитьсоответствиеобъектаиформы установитьсоставстандартногоинтерфейсаodata установитьчасовойпоясинформационнойбазы установитьчасовойпояссеанса формат цел час часовойпояс часовойпояссеанса число числопрописью этоадресвременногохранилища wsссылки библиотекакартинок библиотекамакетовоформлениякомпоновкиданных библиотекастилей бизнеспроцессы внешниеисточникиданных внешниеобработки внешниеотчеты встроенныепокупки главныйинтерфейс главныйстиль документы доставляемыеуведомления журналыдокументов задачи информацияобинтернетсоединении использованиерабочейдаты историяработыпользователя константы критерииотбора метаданные обработки отображениерекламы отправкадоставляемыхуведомлений отчеты панельзадачос параметрзапуска параметрысеанса перечисления планывидоврасчета планывидовхарактеристик планыобмена планысчетов полнотекстовыйпоиск пользователиинформационнойбазы последовательности проверкавстроенныхпокупок рабочаядата расширенияконфигурации регистрыбухгалтерии регистрынакопления регистрырасчета регистрысведений регламентныезадания сериализаторxdto справочники средствагеопозиционирования средствакриптографии средствамультимедиа средстваотображениярекламы средствапочты средствателефонии фабрикаxdto файловыепотоки фоновыезадания хранилищанастроек хранилищевариантовотчетов хранилищенастроекданныхформ хранилищеобщихнастроек хранилищепользовательскихнастроекдинамическихсписков хранилищепользовательскихнастроекотчетов хранилищесистемныхнастроек ",class:"webцвета windowsцвета windowsшрифты библиотекакартинок рамкистиля символы цветастиля шрифтыстиля 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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 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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 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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 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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 keyboard_virtual_height",literal:"self other all noone global local undefined pointer_invalid pointer_null path_action_stop path_action_restart path_action_continue path_action_reverse true false pi GM_build_date GM_version GM_runtime_version timezone_local timezone_utc gamespeed_fps gamespeed_microseconds ev_create ev_destroy ev_step ev_alarm ev_keyboard ev_mouse ev_collision ev_other ev_draw ev_draw_begin ev_draw_end ev_draw_pre ev_draw_post ev_keypress ev_keyrelease ev_trigger ev_left_button ev_right_button ev_middle_button ev_no_button ev_left_press ev_right_press ev_middle_press ev_left_release ev_right_release ev_middle_release ev_mouse_enter ev_mouse_leave ev_mouse_wheel_up ev_mouse_wheel_down ev_global_left_button ev_global_right_button ev_global_middle_button ev_global_left_press ev_global_right_press ev_global_middle_press ev_global_left_release ev_global_right_release ev_global_middle_release ev_joystick1_left ev_joystick1_right ev_joystick1_up ev_joystick1_down ev_joystick1_button1 ev_joystick1_button2 ev_joystick1_button3 ev_joystick1_button4 ev_joystick1_button5 ev_joystick1_button6 ev_joystick1_button7 ev_joystick1_button8 ev_joystick2_left ev_joystick2_right ev_joystick2_up ev_joystick2_down ev_joystick2_button1 ev_joystick2_button2 ev_joystick2_button3 ev_joystick2_button4 ev_joystick2_button5 ev_joystick2_button6 ev_joystick2_button7 ev_joystick2_button8 ev_outside ev_boundary ev_game_start ev_game_end ev_room_start ev_room_end ev_no_more_lives ev_animation_end ev_end_of_path ev_no_more_health ev_close_button ev_user0 ev_user1 ev_user2 ev_user3 ev_user4 ev_user5 ev_user6 ev_user7 ev_user8 ev_user9 ev_user10 ev_user11 ev_user12 ev_user13 ev_user14 ev_user15 ev_step_normal ev_step_begin 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this._contexts[e]:this._contexts=Object(ue.a)(Object(ue.a)({},this._contexts),((r={})[e]=t,r)),this._notifyScopeListeners(),this},e.prototype.setSpan=function(e){return this._span=e,this._notifyScopeListeners(),this},e.prototype.getSpan=function(){return this._span},e.prototype.getTransaction=function(){var e,t,r,n,o=this.getSpan();return(null===(e=o)||void 0===e?void 0:e.transaction)?null===(t=o)||void 0===t?void 0:t.transaction:(null===(n=null===(r=o)||void 0===r?void 0:r.spanRecorder)||void 0===n?void 0:n.spans[0])?o.spanRecorder.spans[0]:void 0},e.prototype.setSession=function(e){return e?this._session=e:delete this._session,this._notifyScopeListeners(),this},e.prototype.getSession=function(){return this._session},e.prototype.update=function(t){if(!t)return this;if("function"==typeof t){var r=t(this);return r instanceof e?r:this}return t instanceof e?(this._tags=Object(ue.a)(Object(ue.a)({},this._tags),t._tags),this._extra=Object(ue.a)(Object(ue.a)({},this._extra),t._extra),this._contexts=Object(ue.a)(Object(ue.a)({},this._contexts),t._contexts),t._user&&Object.keys(t._user).length&&(this._user=t._user),t._level&&(this._level=t._level),t._fingerprint&&(this._fingerprint=t._fingerprint)):Object(me.h)(t)&&(t=t,this._tags=Object(ue.a)(Object(ue.a)({},this._tags),t.tags),this._extra=Object(ue.a)(Object(ue.a)({},this._extra),t.extra),this._contexts=Object(ue.a)(Object(ue.a)({},this._contexts),t.contexts),t.user&&(this._user=t.user),t.level&&(this._level=t.level),t.fingerprint&&(this._fingerprint=t.fingerprint)),this},e.prototype.clear=function(){return this._breadcrumbs=[],this._tags={},this._extra={},this._user={},this._contexts={},this._level=void 0,this._transactionName=void 0,this._fingerprint=void 0,this._span=void 0,this._session=void 0,this._notifyScopeListeners(),this},e.prototype.addBreadcrumb=function(e,t){var r=Object(ue.a)({timestamp:Object(ge.a)()},e);return this._breadcrumbs=void 0!==t&&t>=0?Object(ue.e)(this._breadcrumbs,[r]).slice(-t):Object(ue.e)(this._breadcrumbs,[r]),this._notifyScopeListeners(),this},e.prototype.clearBreadcrumbs=function(){return this._breadcrumbs=[],this._notifyScopeListeners(),this},e.prototype.applyToEvent=function(e,t){var r;if(this._extra&&Object.keys(this._extra).length&&(e.extra=Object(ue.a)(Object(ue.a)({},this._extra),e.extra)),this._tags&&Object.keys(this._tags).length&&(e.tags=Object(ue.a)(Object(ue.a)({},this._tags),e.tags)),this._user&&Object.keys(this._user).length&&(e.user=Object(ue.a)(Object(ue.a)({},this._user),e.user)),this._contexts&&Object.keys(this._contexts).length&&(e.contexts=Object(ue.a)(Object(ue.a)({},this._contexts),e.contexts)),this._level&&(e.level=this._level),this._transactionName&&(e.transaction=this._transactionName),this._span){e.contexts=Object(ue.a)({trace:this._span.getTraceContext()},e.contexts);var o=null===(r=this._span.transaction)||void 0===r?void 0:r.name;o&&(e.tags=Object(ue.a)({transaction:o},e.tags))}return this._applyFingerprint(e),e.breadcrumbs=Object(ue.e)(e.breadcrumbs||[],this._breadcrumbs),e.breadcrumbs=e.breadcrumbs.length>0?e.breadcrumbs:void 0,this._notifyEventProcessors(Object(ue.e)(n(),this._eventProcessors),e,t)},e.prototype._notifyEventProcessors=function(e,t,r,n){var o=this;return void 0===n&&(n=0),new fe((function(a,i){var s=e[n];if(null===t||"function"!=typeof s)a(t);else{var l=s(Object(ue.a)({},t),r);Object(me.m)(l)?l.then((function(t){return o._notifyEventProcessors(e,t,r,n+1).then(a)})).then(null,i):o._notifyEventProcessors(e,l,r,n+1).then(a).then(null,i)}}))},e.prototype._notifyScopeListeners=function(){var e=this;this._notifyingListeners||(this._notifyingListeners=!0,this._scopeListeners.forEach((function(t){t(e)})),this._notifyingListeners=!1)},e.prototype._applyFingerprint=function(e){e.fingerprint=e.fingerprint?Array.isArray(e.fingerprint)?e.fingerprint:[e.fingerprint]:[],this._fingerprint&&(e.fingerprint=e.fingerprint.concat(this._fingerprint)),e.fingerprint&&!e.fingerprint.length&&delete e.fingerprint},e}(),he=Object(Ee.e)(),Se="Sentry Logger ",Te=function(){function e(){this._enabled=!1}return e.prototype.disable=function(){this._enabled=!1},e.prototype.enable=function(){this._enabled=!0},e.prototype.log=function(){for(var e=[],t=0;t1&&(_=u.slice(0,-1).join("/"),d=u.pop()),d){var p=d.match(/^\d+/);p&&(d=p[0])}this._fromComponents({host:s,pass:i,path:_,projectId:d,port:c,protocol:n,user:o})},e.prototype._fromComponents=function(e){this.protocol=e.protocol,this.user=e.user,this.pass=e.pass||"",this.host=e.host,this.port=e.port||"",this.path=e.path||"",this.projectId=e.projectId},e.prototype._validate=function(){var e=this;if(["protocol","user","host","projectId"].forEach((function(t){if(!e[t])throw new Me("Invalid Dsn: "+t+" missing")})),!this.projectId.match(/^\d+$/))throw new Me("Invalid Dsn: Invalid projectId "+this.projectId);if("http"!==this.protocol&&"https"!==this.protocol)throw new Me("Invalid Dsn: Invalid protocol "+this.protocol);if(this.port&&isNaN(parseInt(this.port,10)))throw new Me("Invalid Dsn: Invalid port "+this.port)},e}(),ke=r("dbjm"),je=[],Ue=function(){function e(e,t){this._integrations={},this._processing=0,this._backend=new e(t),this._options=t,t.dsn&&(this._dsn=new Pe(t.dsn))}return e.prototype.captureException=function(e,t,r){var n=this,o=t&&t.event_id;return this._process(this._getBackend().eventFromException(e,t).then((function(e){return n._captureEvent(e,t,r)})).then((function(e){o=e}))),o},e.prototype.captureMessage=function(e,t,r,n){var o=this,a=r&&r.event_id,i=Object(me.i)(e)?this._getBackend().eventFromMessage(String(e),t,r):this._getBackend().eventFromException(e,r);return this._process(i.then((function(e){return o._captureEvent(e,r,n)})).then((function(e){a=e}))),a},e.prototype.captureEvent=function(e,t,r){var n=t&&t.event_id;return this._process(this._captureEvent(e,t,r).then((function(e){n=e}))),n},e.prototype.captureSession=function(e){e.release?this._sendSession(e):Ce.warn("Discarded session because of missing release")},e.prototype.getDsn=function(){return this._dsn},e.prototype.getOptions=function(){return this._options},e.prototype.flush=function(e){var t=this;return this._isClientProcessing(e).then((function(r){return t._getBackend().getTransport().close(e).then((function(e){return r&&e}))}))},e.prototype.close=function(e){var t=this;return this.flush(e).then((function(e){return t.getOptions().enabled=!1,e}))},e.prototype.setupIntegrations=function(){this._isEnabled()&&(this._integrations=R(this._options))},e.prototype.getIntegration=function(e){try{return this._integrations[e.id]||null}catch(t){return Ce.warn("Cannot retrieve integration "+e.id+" from the current Client"),null}},e.prototype._updateSessionFromEvent=function(e,t){var r,n,o,a=!1,i=!1,s=t.exception&&t.exception.values;if(s){i=!0;try{for(var l=Object(ue.f)(s),c=l.next();!c.done;c=l.next()){var _=c.value.mechanism;if(_&&!1===_.handled){a=!0;break}}}catch(e){r={error:e}}finally{try{c&&!c.done&&(n=l.return)&&n.call(l)}finally{if(r)throw r.error}}}var d=t.user;if(!e.userAgent){var u=t.request?t.request.headers:{};for(var p in u)if("user-agent"===p.toLowerCase()){o=u[p];break}}e.update(Object(ue.a)(Object(ue.a)({},a&&{status:ve.Crashed}),{user:d,userAgent:o,errors:e.errors+Number(i||a)}))},e.prototype._sendSession=function(e){this._getBackend().sendSession(e)},e.prototype._isClientProcessing=function(e){var t=this;return new fe((function(r){var n=0,o=setInterval((function(){0==t._processing?(clearInterval(o),r(!0)):(n+=1,e&&n>=e&&(clearInterval(o),r(!1)))}),1)}))},e.prototype._getBackend=function(){return this._backend},e.prototype._isEnabled=function(){return!1!==this.getOptions().enabled&&void 0!==this._dsn},e.prototype._prepareEvent=function(e,t,r){var n=this,o=this.getOptions().normalizeDepth,a=void 0===o?3:o,i=Object(ue.a)(Object(ue.a)({},e),{event_id:e.event_id||(r&&r.event_id?r.event_id:Object(Ee.i)()),timestamp:e.timestamp||Object(ge.a)()});this._applyClientOptions(i),this._applyIntegrationsMetadata(i);var s=t;r&&r.captureContext&&(s=be.clone(s).update(r.captureContext));var l=fe.resolve(i);return s&&(l=s.applyToEvent(i,r)),l.then((function(e){return"number"==typeof a&&a>0?n._normalizeEvent(e,a):e}))},e.prototype._normalizeEvent=function(e,t){if(!e)return null;var r=Object(ue.a)(Object(ue.a)(Object(ue.a)(Object(ue.a)(Object(ue.a)({},e),e.breadcrumbs&&{breadcrumbs:e.breadcrumbs.map((function(e){return Object(ue.a)(Object(ue.a)({},e),e.data&&{data:Object(Ie.d)(e.data,t)})}))}),e.user&&{user:Object(Ie.d)(e.user,t)}),e.contexts&&{contexts:Object(Ie.d)(e.contexts,t)}),e.extra&&{extra:Object(Ie.d)(e.extra,t)});return e.contexts&&e.contexts.trace&&(r.contexts.trace=e.contexts.trace),r},e.prototype._applyClientOptions=function(e){var t=this.getOptions(),r=t.release,n=t.dist,o=t.maxValueLength,a=void 0===o?250:o;"environment"in e||(e.environment="environment"in t?t.environment:"production"),void 0===e.release&&void 0!==r&&(e.release=r),void 0===e.dist&&void 0!==n&&(e.dist=n),e.message&&(e.message=Object(ke.d)(e.message,a));var i=e.exception&&e.exception.values&&e.exception.values[0];i&&i.value&&(i.value=Object(ke.d)(i.value,a));var s=e.request;s&&s.url&&(s.url=Object(ke.d)(s.url,a))},e.prototype._applyIntegrationsMetadata=function(e){var t=e.sdk,r=Object.keys(this._integrations);t&&r.length>0&&(t.integrations=r)},e.prototype._sendEvent=function(e){this._getBackend().sendEvent(e)},e.prototype._captureEvent=function(e,t,r){return this._processEvent(e,t,r).then((function(e){return e.event_id}),(function(e){Ce.error(e)}))},e.prototype._processEvent=function(e,t,r){var n=this,o=this.getOptions(),a=o.beforeSend,i=o.sampleRate;if(!this._isEnabled())return fe.reject(new Me("SDK not enabled, will not send event."));var s="transaction"===e.type;return!s&&"number"==typeof i&&Math.random()>i?fe.reject(new Me("Discarding event because it's not included in the random sample (sampling rate = "+i+")")):this._prepareEvent(e,r,t).then((function(e){if(null===e)throw new Me("An event processor returned null, will not send 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"+e)}))},e.prototype.sendSession=function(e){this._transport.sendSession?this._transport.sendSession(e).then(null,(function(e){Ce.error("Error while sending session: "+e)})):Ce.warn("Dropping session because custom transport doesn't implement sendSession")},e.prototype.getTransport=function(){return this._transport},e.prototype._setupTransport=function(){return new Fe},e}(),Be=/^\s*at (?:(.*?) ?\()?((?:file|https?|blob|chrome-extension|address|native|eval|webpack||[-a-z]+:|.*bundle|\/).*?)(?::(\d+))?(?::(\d+))?\)?\s*$/i,Ye=/^\s*(.*?)(?:\((.*?)\))?(?:^|@)?((?:file|https?|blob|chrome|webpack|resource|moz-extension|capacitor).*?:\/.*?|\[native code\]|[^@]*(?:bundle|\d+\.js)|\/[\w\-. /=]+)(?::(\d+))?(?::(\d+))?\s*$/i,Ve=/^\s*at (?:((?:\[object object\])?.+) )?\(?((?:file|ms-appx|https?|webpack|blob):.*?):(\d+)(?::(\d+))?\)?\s*$/i,He=/(\S+) line (\d+)(?: > eval line \d+)* > eval/i,ze=/\((\S*)(?::(\d+))(?::(\d+))\)/,qe=/Minified React error #\d+;/i,We=function(){function e(e){this.dsn=e,this._dsnObject=new Pe(e)}return e.prototype.getDsn=function(){return this._dsnObject},e.prototype.getBaseApiEndpoint=function(){var e=this._dsnObject;return(e.protocol?e.protocol+":":"")+"//"+e.host+(e.port?":"+e.port:"")+(e.path?"/"+e.path:"")+"/api/"},e.prototype.getStoreEndpoint=function(){return this._getIngestEndpoint("store")},e.prototype.getStoreEndpointWithUrlEncodedAuth=function(){return this.getStoreEndpoint()+"?"+this._encodedAuth()},e.prototype.getEnvelopeEndpointWithUrlEncodedAuth=function(){return this._getEnvelopeEndpoint()+"?"+this._encodedAuth()},e.prototype.getStoreEndpointPath=function(){var e=this._dsnObject;return(e.path?"/"+e.path:"")+"/api/"+e.projectId+"/store/"},e.prototype.getRequestHeaders=function(e,t){var r=this._dsnObject,n=["Sentry sentry_version=7"];return n.push("sentry_client="+e+"/"+t),n.push("sentry_key="+r.user),r.pass&&n.push("sentry_secret="+r.pass),{"Content-Type":"application/json","X-Sentry-Auth":n.join(", 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t,r,n,o,a=Date.now(),i=e["x-sentry-rate-limits"],s=e["retry-after"];if(i){try{for(var l=Object(ue.f)(i.trim().split(",")),c=l.next();!c.done;c=l.next()){var _=c.value.split(":",2),d=parseInt(_[0],10),u=1e3*(isNaN(d)?60:d);try{for(var p=(n=void 0,Object(ue.f)(_[1].split(";"))),m=p.next();!m.done;m=p.next()){this._rateLimits[m.value||"all"]=new Date(a+u)}}catch(e){n={error:e}}finally{try{m&&!m.done&&(o=p.return)&&o.call(p)}finally{if(n)throw n.error}}}}catch(e){t={error:e}}finally{try{c&&!c.done&&(r=l.return)&&r.call(l)}finally{if(t)throw t.error}}return!0}return!!s&&(this._rateLimits.all=new Date(a+Object(Ee.g)(a,s)),!0)},e}(),$e=Object(Ee.e)(),Xe=function(e){function t(){return null!==e&&e.apply(this,arguments)||this}return Object(ue.b)(t,e),t.prototype.sendEvent=function(e){return this._sendRequest(G(e,this._api),e)},t.prototype.sendSession=function(e){return this._sendRequest(F(e,this._api),e)},t.prototype._sendRequest=function(e,t){var r=this;if(this._isRateLimited(e.type))return 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this._isRateLimited(e.type)?Promise.reject({event:t,type:e.type,reason:"Transport locked till "+this._disabledUntil(e.type)+" due to too many requests.",status:429}):this._buffer.add(new fe((function(t,n){var o=new XMLHttpRequest;for(var a in o.onreadystatechange=function(){if(4===o.readyState){var a={"x-sentry-rate-limits":o.getResponseHeader("X-Sentry-Rate-Limits"),"retry-after":o.getResponseHeader("Retry-After")};r._handleResponse({requestType:e.type,response:o,headers:a,resolve:t,reject:n})}},o.open("POST",e.url),r.options.headers)r.options.headers.hasOwnProperty(a)&&o.setRequestHeader(a,r.options.headers[a]);o.send(e.body)})))},t}(Ke),Je=function(e){function t(){return null!==e&&e.apply(this,arguments)||this}return Object(ue.b)(t,e),t.prototype.eventFromException=function(e,t){return P(this._options,e,t)},t.prototype.eventFromMessage=function(e,t,r){return void 0===t&&(t=_e.Info),k(this._options,e,t,r)},t.prototype._setupTransport=function(){if(!this._options.dsn)return 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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 // IRPF90 special built_ins\n 'IRP_ALIGN irp_here'\n };\n return {\n case_insensitive: true,\n keywords: F_KEYWORDS,\n illegal: /\\/\\*/,\n contains: [\n hljs.inherit(hljs.APOS_STRING_MODE, {className: 'string', relevance: 0}),\n hljs.inherit(hljs.QUOTE_STRING_MODE, {className: 'string', relevance: 0}),\n {\n className: 'function',\n beginKeywords: 'subroutine function program',\n illegal: '[${=\\\\n]',\n contains: [hljs.UNDERSCORE_TITLE_MODE, PARAMS]\n },\n hljs.COMMENT('!', '$', {relevance: 0}),\n hljs.COMMENT('begin_doc', 'end_doc', {relevance: 10}),\n {\n className: 'number',\n begin: '(?=\\\\b|\\\\+|\\\\-|\\\\.)(?=\\\\.\\\\d|\\\\d)(?:\\\\d+)?(?:\\\\.?\\\\d*)(?:[de][+-]?\\\\d+)?\\\\b\\\\.?',\n relevance: 0\n }\n ]\n };\n};","module.exports = function(hljs) {\n var XML_IDENT_RE = '[A-Za-z0-9\\\\._:-]+';\n var TAG_INTERNALS = {\n endsWithParent: true,\n illegal: /`]+/}\n ]\n }\n ]\n }\n ]\n };\n return {\n aliases: ['html', 'xhtml', 'rss', 'atom', 'xjb', 'xsd', 'xsl', 'plist', 'wsf'],\n case_insensitive: true,\n contains: [\n {\n className: 'meta',\n begin: '',\n relevance: 10,\n contains: [{begin: '\\\\[', end: '\\\\]'}]\n },\n hljs.COMMENT(\n '',\n {\n relevance: 10\n }\n ),\n {\n begin: '<\\\\!\\\\[CDATA\\\\[', end: '\\\\]\\\\]>',\n relevance: 10\n },\n {\n className: 'meta',\n begin: /<\\?xml/, end: /\\?>/, relevance: 10\n },\n {\n begin: /<\\?(php)?/, end: /\\?>/,\n subLanguage: 'php',\n contains: [\n // We don't want the php closing tag ?> to close the PHP block when\n // inside any of the following blocks:\n {begin: '/\\\\*', end: '\\\\*/', skip: true},\n {begin: 'b\"', end: '\"', skip: true},\n {begin: 'b\\'', end: '\\'', skip: true},\n hljs.inherit(hljs.APOS_STRING_MODE, {illegal: null, className: null, contains: null, skip: true}),\n hljs.inherit(hljs.QUOTE_STRING_MODE, {illegal: null, className: null, contains: null, skip: true})\n ]\n },\n {\n className: 'tag',\n /*\n The lookahead pattern (?=...) ensures that 'begin' only matches\n '|$)', end: '>',\n keywords: {name: 'style'},\n contains: [TAG_INTERNALS],\n starts: {\n end: '', returnEnd: true,\n subLanguage: ['css', 'xml']\n }\n },\n {\n className: 'tag',\n // See the comment in the
\ No newline at end of file
diff --git a/docs/manifest.json b/docs/manifest.json
new file mode 100644
index 0000000..2ebcb6e
--- /dev/null
+++ b/docs/manifest.json
@@ -0,0 +1,15 @@
+{
+ "name": "Unisteffo Old",
+ "short_name": "Unisteffo",
+ "start_url": "/",
+ "display": "browser",
+ "orientation": "portrait",
+ "background_color": "#0d193b",
+ "theme_color": "#1c2b4f",
+ "icons": [
+ {
+ "src": "/favicon.ico",
+ "type": "image/x-icon"
+ }
+ ]
+}
diff --git a/docs/opengraph.png b/docs/opengraph.png
new file mode 100644
index 0000000..19f01ce
Binary files /dev/null and b/docs/opengraph.png differ
diff --git a/docs/polyfills.1efa5.esm.js b/docs/polyfills.1efa5.esm.js
new file mode 100644
index 0000000..405d939
--- /dev/null
+++ b/docs/polyfills.1efa5.esm.js
@@ -0,0 +1,2 @@
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i < len; i++) {\n handle(self, self._deferreds[i]);\n }\n self._deferreds = null;\n}\n\n/**\n * @constructor\n */\nfunction Handler(onFulfilled, onRejected, promise) {\n this.onFulfilled = typeof onFulfilled === 'function' ? onFulfilled : null;\n this.onRejected = typeof onRejected === 'function' ? onRejected : null;\n this.promise = promise;\n}\n\n/**\n * Take a potentially misbehaving resolver function and make sure\n * onFulfilled and onRejected are only called once.\n *\n * Makes no guarantees about asynchrony.\n */\nfunction doResolve(fn, self) {\n var done = false;\n try {\n fn(\n function(value) {\n if (done) return;\n done = true;\n resolve(self, value);\n },\n function(reason) {\n if (done) return;\n done = true;\n reject(self, reason);\n }\n );\n } catch (ex) {\n if (done) return;\n done = true;\n reject(self, ex);\n }\n}\n\nPromise.prototype['catch'] = function(onRejected) {\n return this.then(null, onRejected);\n};\n\nPromise.prototype.then = function(onFulfilled, onRejected) {\n // @ts-ignore\n var prom = new this.constructor(noop);\n\n handle(this, new Handler(onFulfilled, onRejected, prom));\n return prom;\n};\n\nPromise.prototype['finally'] = promiseFinally;\n\nPromise.all = function(arr) {\n return new Promise(function(resolve, reject) {\n if (!isArray(arr)) {\n return reject(new TypeError('Promise.all accepts an array'));\n }\n\n var args = Array.prototype.slice.call(arr);\n if (args.length === 0) return resolve([]);\n var remaining = args.length;\n\n function res(i, val) {\n try {\n if (val && (typeof val === 'object' || typeof val === 'function')) {\n var then = val.then;\n if (typeof then === 'function') {\n then.call(\n val,\n function(val) {\n res(i, val);\n },\n reject\n );\n return;\n }\n }\n args[i] = val;\n if (--remaining === 0) {\n resolve(args);\n }\n } catch (ex) {\n reject(ex);\n }\n }\n\n for (var i = 0; i < args.length; i++) {\n res(i, args[i]);\n }\n });\n};\n\nPromise.allSettled = allSettled;\n\nPromise.resolve = function(value) {\n if (value && typeof value === 'object' && value.constructor === Promise) {\n return value;\n }\n\n return new Promise(function(resolve) {\n resolve(value);\n });\n};\n\nPromise.reject = function(value) {\n return new Promise(function(resolve, reject) {\n reject(value);\n });\n};\n\nPromise.race = function(arr) {\n return new Promise(function(resolve, reject) {\n if (!isArray(arr)) {\n return reject(new TypeError('Promise.race accepts an array'));\n }\n\n for (var i = 0, len = arr.length; i < len; i++) {\n Promise.resolve(arr[i]).then(resolve, reject);\n }\n });\n};\n\n// Use polyfill for setImmediate for performance gains\nPromise._immediateFn =\n // @ts-ignore\n (typeof setImmediate === 'function' &&\n function(fn) {\n // @ts-ignore\n setImmediate(fn);\n }) ||\n function(fn) {\n setTimeoutFunc(fn, 0);\n };\n\nPromise._unhandledRejectionFn = function _unhandledRejectionFn(err) {\n if (typeof console !== 'undefined' && console) {\n console.warn('Possible Unhandled Promise Rejection:', err); // eslint-disable-line no-console\n }\n};\n\nexport default Promise;\n","/**\n * @this {Promise}\n */\nfunction finallyConstructor(callback) {\n var constructor = this.constructor;\n return this.then(\n function(value) {\n // @ts-ignore\n return constructor.resolve(callback()).then(function() {\n return value;\n });\n },\n function(reason) {\n // @ts-ignore\n return constructor.resolve(callback()).then(function() {\n // @ts-ignore\n return constructor.reject(reason);\n });\n }\n );\n}\n\nexport default finallyConstructor;\n","function allSettled(arr) {\n var P = this;\n return new P(function(resolve, reject) {\n if (!(arr && typeof arr.length !== 'undefined')) {\n return reject(\n new TypeError(\n typeof arr +\n ' ' +\n arr +\n ' is not iterable(cannot read property Symbol(Symbol.iterator))'\n )\n );\n }\n var args = Array.prototype.slice.call(arr);\n if (args.length === 0) return resolve([]);\n var remaining = args.length;\n\n function res(i, val) {\n if (val && (typeof val === 'object' || typeof val === 'function')) {\n var then = val.then;\n if (typeof then === 'function') {\n then.call(\n val,\n function(val) {\n res(i, val);\n },\n function(e) {\n args[i] = { status: 'rejected', reason: e };\n if (--remaining === 0) {\n resolve(args);\n }\n }\n );\n return;\n }\n }\n args[i] = { status: 'fulfilled', value: val };\n if (--remaining === 0) {\n resolve(args);\n }\n }\n\n for (var i = 0; i < args.length; i++) {\n res(i, args[i]);\n }\n });\n}\n\nexport default allSettled;\n","export default function(url, options) {\n\toptions = options || {};\n\treturn new Promise( (resolve, reject) => {\n\t\tconst request = new XMLHttpRequest();\n\t\tconst keys = [];\n\t\tconst all = [];\n\t\tconst headers = {};\n\n\t\tconst response = () => ({\n\t\t\tok: (request.status/100|0) == 2,\t\t// 200-299\n\t\t\tstatusText: request.statusText,\n\t\t\tstatus: request.status,\n\t\t\turl: request.responseURL,\n\t\t\ttext: () => Promise.resolve(request.responseText),\n\t\t\tjson: () => Promise.resolve(request.responseText).then(JSON.parse),\n\t\t\tblob: () => Promise.resolve(new Blob([request.response])),\n\t\t\tclone: response,\n\t\t\theaders: {\n\t\t\t\tkeys: () => keys,\n\t\t\t\tentries: () => all,\n\t\t\t\tget: n => headers[n.toLowerCase()],\n\t\t\t\thas: n => n.toLowerCase() in headers\n\t\t\t}\n\t\t});\n\n\t\trequest.open(options.method || 'get', url, true);\n\n\t\trequest.onload = () => {\n\t\t\trequest.getAllResponseHeaders().replace(/^(.*?):[^\\S\\n]*([\\s\\S]*?)$/gm, (m, key, value) => {\n\t\t\t\tkeys.push(key = key.toLowerCase());\n\t\t\t\tall.push([key, value]);\n\t\t\t\theaders[key] = headers[key] ? `${headers[key]},${value}` : value;\n\t\t\t});\n\t\t\tresolve(response());\n\t\t};\n\n\t\trequest.onerror = reject;\n\n\t\trequest.withCredentials = options.credentials=='include';\n\n\t\tfor (const i in options.headers) {\n\t\t\trequest.setRequestHeader(i, options.headers[i]);\n\t\t}\n\n\t\trequest.send(options.body || null);\n\t});\n}\n","if (!global.Promise) global.Promise = require('promise-polyfill').default;\nif (!global.fetch) global.fetch = require('isomorphic-unfetch');\n","var g;\n\n// This works in non-strict mode\ng = (function() {\n\treturn this;\n})();\n\ntry {\n\t// This works if eval is allowed (see CSP)\n\tg = g || new Function(\"return this\")();\n} catch (e) {\n\t// This works if the window reference is available\n\tif (typeof window === \"object\") g = window;\n}\n\n// g can still be undefined, but nothing to do about it...\n// We return undefined, instead of nothing here, so it's\n// easier to handle this case. if(!global) { ...}\n\nmodule.exports = g;\n","// This will be replaced\n"],"sourceRoot":""}
\ No newline at end of file
diff --git a/docs/polyfills.60541.js b/docs/polyfills.60541.js
new file mode 100644
index 0000000..d1201ec
--- /dev/null
+++ b/docs/polyfills.60541.js
@@ -0,0 +1,2 @@
+!function(e){function t(r){if(n[r])return n[r].exports;var o=n[r]={i:r,l:!1,exports:{}};return e[r].call(o.exports,o,o.exports,t),o.l=!0,o.exports}var n={};t.m=e,t.c=n,t.d=function(e,n,r){t.o(e,n)||Object.defineProperty(e,n,{enumerable:!0,get:r})},t.r=function(e){"undefined"!=typeof Symbol&&Symbol.toStringTag&&Object.defineProperty(e,Symbol.toStringTag,{value:"Module"}),Object.defineProperty(e,"__esModule",{value:!0})},t.t=function(e,n){if(1&n&&(e=t(e)),8&n)return e;if(4&n&&"object"==typeof e&&e&&e.__esModule)return e;var r=Object.create(null);if(t.r(r),Object.defineProperty(r,"default",{enumerable:!0,value:e}),2&n&&"string"!=typeof e)for(var o in e)t.d(r,o,function(t){return e[t]}.bind(null,o));return r},t.n=function(e){var n=e&&e.__esModule?function(){return e.default}:function(){return e};return t.d(n,"a",n),n},t.o=function(e,t){return Object.prototype.hasOwnProperty.call(e,t)},t.p="/",t(t.s=1)}({"+TSP":function(e,t,n){e.exports=self.fetch||(self.fetch=n("Cf1v").default||n("Cf1v"))},1:function(e,t,n){n("xhzY"),e.exports=n("Ve17")},"5+HP":function(e,t,n){"use strict";function r(e){return Boolean(e&&void 0!==e.length)}function o(){}function i(e){if(!(this instanceof i))throw new TypeError("Promises must be constructed via new");if("function"!=typeof e)throw new TypeError("not a function");this._state=0,this._handled=!1,this._value=void 0,this._deferreds=[],l(e,this)}function u(e,t){for(;3===e._state;)e=e._value;0!==e._state?(e._handled=!0,i._immediateFn((function(){var n=1===e._state?t.onFulfilled:t.onRejected;if(null!==n){var r;try{r=n(e._value)}catch(e){return void c(t.promise,e)}f(t.promise,r)}else(1===e._state?f:c)(t.promise,e._value)}))):e._deferreds.push(t)}function f(e,t){try{if(t===e)throw new TypeError("A promise cannot be resolved with itself.");if(t&&("object"==typeof t||"function"==typeof t)){var n=t.then;if(t instanceof i)return e._state=3,e._value=t,void s(e);if("function"==typeof n)return void l((r=n,o=t,function(){r.apply(o,arguments)}),e)}e._state=1,e._value=t,s(e)}catch(t){c(e,t)}var r,o}function c(e,t){e._state=2,e._value=t,s(e)}function s(e){2===e._state&&0===e._deferreds.length&&i._immediateFn((function(){e._handled||i._unhandledRejectionFn(e._value)}));for(var t=0,n=e._deferreds.length;t} */\n this._deferreds = [];\n\n doResolve(fn, this);\n}\n\nfunction handle(self, deferred) {\n while (self._state === 3) {\n self = self._value;\n }\n if (self._state === 0) {\n self._deferreds.push(deferred);\n return;\n }\n self._handled = true;\n Promise._immediateFn(function() {\n var cb = self._state === 1 ? deferred.onFulfilled : deferred.onRejected;\n if (cb === null) {\n (self._state === 1 ? resolve : reject)(deferred.promise, self._value);\n return;\n }\n var ret;\n try {\n ret = cb(self._value);\n } catch (e) {\n reject(deferred.promise, e);\n return;\n }\n resolve(deferred.promise, ret);\n });\n}\n\nfunction resolve(self, newValue) {\n try {\n // Promise Resolution Procedure: https://github.com/promises-aplus/promises-spec#the-promise-resolution-procedure\n if (newValue === self)\n throw new TypeError('A promise cannot be resolved with itself.');\n if (\n newValue &&\n (typeof newValue === 'object' || typeof newValue === 'function')\n ) {\n var then = newValue.then;\n if (newValue instanceof Promise) {\n self._state = 3;\n self._value = newValue;\n finale(self);\n return;\n } else if (typeof then === 'function') {\n doResolve(bind(then, newValue), self);\n return;\n }\n }\n self._state = 1;\n self._value = newValue;\n finale(self);\n } catch (e) {\n reject(self, e);\n }\n}\n\nfunction reject(self, newValue) {\n self._state = 2;\n self._value = newValue;\n finale(self);\n}\n\nfunction finale(self) {\n if (self._state === 2 && self._deferreds.length === 0) {\n Promise._immediateFn(function() {\n if (!self._handled) {\n Promise._unhandledRejectionFn(self._value);\n }\n });\n }\n\n for (var i = 0, len = self._deferreds.length; i < len; i++) {\n handle(self, self._deferreds[i]);\n }\n self._deferreds = null;\n}\n\n/**\n * @constructor\n */\nfunction Handler(onFulfilled, onRejected, promise) {\n this.onFulfilled = typeof onFulfilled === 'function' ? onFulfilled : null;\n this.onRejected = typeof onRejected === 'function' ? onRejected : null;\n this.promise = promise;\n}\n\n/**\n * Take a potentially misbehaving resolver function and make sure\n * onFulfilled and onRejected are only called once.\n *\n * Makes no guarantees about asynchrony.\n */\nfunction doResolve(fn, self) {\n var done = false;\n try {\n fn(\n function(value) {\n if (done) return;\n done = true;\n resolve(self, value);\n },\n function(reason) {\n if (done) return;\n done = true;\n reject(self, reason);\n }\n );\n } catch (ex) {\n if (done) return;\n done = true;\n reject(self, ex);\n }\n}\n\nPromise.prototype['catch'] = function(onRejected) {\n return this.then(null, onRejected);\n};\n\nPromise.prototype.then = function(onFulfilled, onRejected) {\n // @ts-ignore\n var prom = new this.constructor(noop);\n\n handle(this, new Handler(onFulfilled, onRejected, prom));\n return prom;\n};\n\nPromise.prototype['finally'] = promiseFinally;\n\nPromise.all = function(arr) {\n return new Promise(function(resolve, reject) {\n if (!isArray(arr)) {\n return reject(new TypeError('Promise.all accepts an array'));\n }\n\n var args = Array.prototype.slice.call(arr);\n if (args.length === 0) return resolve([]);\n var remaining = args.length;\n\n function res(i, val) {\n try {\n if (val && (typeof val === 'object' || typeof val === 'function')) {\n var then = val.then;\n if (typeof then === 'function') {\n then.call(\n val,\n function(val) {\n res(i, val);\n },\n reject\n );\n return;\n }\n }\n args[i] = val;\n if (--remaining === 0) {\n resolve(args);\n }\n } catch (ex) {\n reject(ex);\n }\n }\n\n for (var i = 0; i < args.length; i++) {\n res(i, args[i]);\n }\n });\n};\n\nPromise.allSettled = allSettled;\n\nPromise.resolve = function(value) {\n if (value && typeof value === 'object' && value.constructor === Promise) {\n return value;\n }\n\n return new Promise(function(resolve) {\n resolve(value);\n });\n};\n\nPromise.reject = function(value) {\n return new Promise(function(resolve, reject) {\n reject(value);\n });\n};\n\nPromise.race = function(arr) {\n return new Promise(function(resolve, reject) {\n if (!isArray(arr)) {\n return reject(new TypeError('Promise.race accepts an array'));\n }\n\n for (var i = 0, len = arr.length; i < len; i++) {\n Promise.resolve(arr[i]).then(resolve, reject);\n }\n });\n};\n\n// Use polyfill for setImmediate for performance gains\nPromise._immediateFn =\n // @ts-ignore\n (typeof setImmediate === 'function' &&\n function(fn) {\n // @ts-ignore\n setImmediate(fn);\n }) ||\n function(fn) {\n setTimeoutFunc(fn, 0);\n };\n\nPromise._unhandledRejectionFn = function _unhandledRejectionFn(err) {\n if (typeof console !== 'undefined' && console) {\n console.warn('Possible Unhandled Promise Rejection:', err); // eslint-disable-line no-console\n }\n};\n\nexport default Promise;\n","/**\n * @this {Promise}\n */\nfunction finallyConstructor(callback) {\n var constructor = this.constructor;\n return this.then(\n function(value) {\n // @ts-ignore\n return constructor.resolve(callback()).then(function() {\n return value;\n });\n },\n function(reason) {\n // @ts-ignore\n return constructor.resolve(callback()).then(function() {\n // @ts-ignore\n return constructor.reject(reason);\n });\n }\n );\n}\n\nexport default finallyConstructor;\n","function allSettled(arr) {\n var P = this;\n return new P(function(resolve, reject) {\n if (!(arr && typeof arr.length !== 'undefined')) {\n return reject(\n new TypeError(\n typeof arr +\n ' ' +\n arr +\n ' is not iterable(cannot read property Symbol(Symbol.iterator))'\n )\n );\n }\n var args = Array.prototype.slice.call(arr);\n if (args.length === 0) return resolve([]);\n var remaining = args.length;\n\n function res(i, val) {\n if (val && (typeof val === 'object' || typeof val === 'function')) {\n var then = val.then;\n if (typeof then === 'function') {\n then.call(\n val,\n function(val) {\n res(i, val);\n },\n function(e) {\n args[i] = { status: 'rejected', reason: e };\n if (--remaining === 0) {\n resolve(args);\n }\n }\n );\n return;\n }\n }\n args[i] = { status: 'fulfilled', value: val };\n if (--remaining === 0) {\n resolve(args);\n }\n }\n\n for (var i = 0; i < args.length; i++) {\n res(i, args[i]);\n }\n });\n}\n\nexport default allSettled;\n","export default function(url, options) {\n\toptions = options || {};\n\treturn new Promise( (resolve, reject) => {\n\t\tconst request = new XMLHttpRequest();\n\t\tconst keys = [];\n\t\tconst all = [];\n\t\tconst headers = {};\n\n\t\tconst response = () => ({\n\t\t\tok: (request.status/100|0) == 2,\t\t// 200-299\n\t\t\tstatusText: request.statusText,\n\t\t\tstatus: request.status,\n\t\t\turl: request.responseURL,\n\t\t\ttext: () => Promise.resolve(request.responseText),\n\t\t\tjson: () => Promise.resolve(request.responseText).then(JSON.parse),\n\t\t\tblob: () => Promise.resolve(new Blob([request.response])),\n\t\t\tclone: response,\n\t\t\theaders: {\n\t\t\t\tkeys: () => keys,\n\t\t\t\tentries: () => all,\n\t\t\t\tget: n => headers[n.toLowerCase()],\n\t\t\t\thas: n => n.toLowerCase() in headers\n\t\t\t}\n\t\t});\n\n\t\trequest.open(options.method || 'get', url, true);\n\n\t\trequest.onload = () => {\n\t\t\trequest.getAllResponseHeaders().replace(/^(.*?):[^\\S\\n]*([\\s\\S]*?)$/gm, (m, key, value) => {\n\t\t\t\tkeys.push(key = key.toLowerCase());\n\t\t\t\tall.push([key, value]);\n\t\t\t\theaders[key] = headers[key] ? `${headers[key]},${value}` : value;\n\t\t\t});\n\t\t\tresolve(response());\n\t\t};\n\n\t\trequest.onerror = reject;\n\n\t\trequest.withCredentials = options.credentials=='include';\n\n\t\tfor (const i in options.headers) {\n\t\t\trequest.setRequestHeader(i, options.headers[i]);\n\t\t}\n\n\t\trequest.send(options.body || null);\n\t});\n}\n","if (!global.Promise) global.Promise = require('promise-polyfill').default;\nif (!global.fetch) global.fetch = require('isomorphic-unfetch');\n","var g;\n\n// This works in non-strict mode\ng = (function() {\n\treturn this;\n})();\n\ntry {\n\t// This works if eval is allowed (see CSP)\n\tg = g || new Function(\"return this\")();\n} catch (e) {\n\t// This works if the window reference is available\n\tif (typeof window === \"object\") g = window;\n}\n\n// g can still be undefined, but nothing to do about it...\n// We return undefined, instead of nothing here, so it's\n// easier to handle this case. if(!global) { ...}\n\nmodule.exports = g;\n","// This will be replaced\n"],"sourceRoot":""}
\ No newline at end of file
diff --git a/docs/preact_prerender_data.json b/docs/preact_prerender_data.json
new file mode 100644
index 0000000..e4aee46
--- /dev/null
+++ b/docs/preact_prerender_data.json
@@ -0,0 +1 @@
+{"url":"/"}
\ No newline at end of file
diff --git a/docs/push-manifest.json b/docs/push-manifest.json
new file mode 100644
index 0000000..eb8258e
--- /dev/null
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+(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.r,null,n(o.p,null,t(a||(a=r`
+ # Algoritmi e Strutture Dati
+
+ Docente: [**Manuela Montangero**](mailto:manuela.montangero@unimore.it)
+
+ Crediti: **9 CFU** (72 ore di lezione)
+
+ Orario di ricevimento: **Giovedì dalle 14:30 alle 16:30**
+ _Mandare una mail prima, altrimenti potrebbe andarsene_
+
+
+ #### Regole particolari per le email
+
+ - Oggetto "ASD"
+ - Mail **firmata** con **nome e cognome**
+ - Mail spedita dall'**account Unimore**
+
+ ### Materiale
+
+ Libri:
+ - **Introduzione agli Algoritmi e Strutture Dati** di _T.H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein_
+
+ **[Dolly (FIM)](https://dolly.fim.unimore.it/2018/course/view.php?id=26)**
+
+ ### Tutorato
+
+ Tutor: [**Gianluca d'Addese**](mailto:tutoratoalgoritmi@gmail.com)
+
+ Cosa: **Esercizi sugli argomenti visti a lezione** e **preparatori all'esame**
+
+ Quando: **Mercoledì dalle 09:00 alle 11:00**...?
+
+ ### Esame
+
+ Per iscriversi all'esame, bisogna aver passato:
+
+ - Analisi matematica (propedeutica)
+ - Programmazione I
+
+ Formato:
+
+ - Prima prova scritta
+ - Risolvere problemi con algoritmi proposti a lezione
+ - E' un test sulla preparazione
+ - Dura 1h30m
+ - Non si può usare nessun tipo di materiale
+ - Seconda prova scritta
+ - Proponi soluzioni per nuovi problemi non studiati a lezione
+ - Domande teoriche sugli argomenti studiati
+ - Dura 2h
+ - E' il giorno dopo la prima prova
+ - Si può utilizzare qualsiasi materiale, ma non deve permettere di comunicare
+ - Orale facoltativo
+ - Solo per chi supera entrambi gli scritti
+ - Domande su quello che abbiamo visto a lezione ("perchè gli algoritmi funzionano?")
+ - Informare via email entro 3 giorni dalla pubblicazione degli esiti della seconda prova
+ - L'orale potrebbe migliorare o peggiorare il voto (anche "molto")
+
+ **Attenzione:** Vengono verbalizzati anche i voti insufficienti; ricordarsi di rifiutarli!
+
+ Ci sono i seguenti appelli:
+
+ - 3 appelli tra Giugno e Luglio
+ - 1 appello a Settembre
+ - 2 appelli a Gennaio e Febbraio
+
+ Vale il salto di appello se:
+
+ - Uno studente regolarmente iscritto **non si presenta** all'appello e non ha avvisato via email almeno il giorno prima.
+ - Uno studente ha riportato una **grave insufficienza** in uno dei due scritti, e l'appello successivo è nella stessa sessione
+
+
+ ### Note
+
+ Il corso è ben collegato con quello di Programmazione 2.
+
+ Dormire non fa bene!
+ `)))),n(o.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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 %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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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 \n ")),n(o.r,null,n(o.p,null,"\n # Caso particolare del Master Theorem\n \n ## Ipotesi\n \n Se: \n \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 \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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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.r,null,n(o.p,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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\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 %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 \n `}\n \n \n {`\n # Caso particolare del Master Theorem\n \n ## Ipotesi\n \n Se: \n \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 \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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+(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.r,null,n(a.p,null,r(o()))),n(a.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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 %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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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 \n ")),n(a.r,null,n(a.p,null,"\n # Caso particolare del Master Theorem\n \n ## Ipotesi\n \n Se: \n \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 \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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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.r,null,n(a.p,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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\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 %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 \n `}\n \n \n {`\n # Caso particolare del Master Theorem\n \n ## Ipotesi\n \n Se: \n \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 \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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",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.s,{title:"Sistemi lineari"},i(n.r,{title:"Cosa sono?"},i("p",null,"Sistemi dinamici i cui cambiamenti sono descritti da ",i("b",null,"funzioni lineari"),".")),i(n.r,{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.i,null,$(r||(r=K`\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.r,{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.s,{title:"Agenti"},i(n.r,{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.r,{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.r,{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.s,{title:"Benefici degli agenti"},i(n.r,{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.s,{title:"Architetture di agente"},i(n.r,{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.r,{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.r,{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.s,{title:"Sistemi multi-agente"},i(n.r,{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.r,{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.r,{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.s,{title:"Evoluzione"},i(n.r,{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.r,{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.r,{title:"Cromosoma"},i("p",null,i("b",null,"Sequenza di valori")," che definisce uno o più tratti di un individuo.")),i(n.r,{title:"Popolazione"},i("p",null,"Un ",i("b",null,"insieme di individui")," aventi tutti gli stessi cromosomi.")),i(n.r,{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.r,{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.r,{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.r,{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.s,{title:"Sistema a classificatori"},i(n.r,{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.s,null,i(n.r,{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.r,{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.r,{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.r,{title:"Effettori"},i("p",null,i("b",null,"Motori")," che rispondono ai ",i("i",null,"messaggi")," effettuando una qualche azione nell'ambiente."))),i(n.s,null,i(n.r,{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.r,{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.r,{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.s,null,i(n.r,{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.r,{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.s,{title:"Reti neurali"},i(n.r,{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.s,{title:"Modello booleano"},i(n.r,{title:"Cos'è?"},i("p",null,"Un modello semplificato di rete neurale in cui vengono considerati ",i("b",null,"solo tempi discreti")," (",i(a.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.r,{title:"Neuroni"},i("p",null,"I neuroni ",i("b",null,"si attivano")," in un dato ",i(a.a,null)," se la ",i("b",null,"somma dei loro impulsi")," nel ",i(a.a,null)," precedente è ",i("b",null,"maggiore o uguale a 1"),".")),i(n.r,{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.s,{title:"Funzioni logiche nel modello booleano"},i(n.r,{title:"NOT"},i("p",null,"Un neurone con una sinapsi entrante con intensità ",i(n.i,null,$(s||(s=K`-1`))),".")),i(n.r,{title:"OR"},i("p",null,"Un neurone con due o più sinapsi entranti con intensità ",i(n.i,null,$(c||(c=K`1`))),".")),i(n.r,{title:"AND"},i("p",null,"Un neurone con due o più sinapsi entranti con intensità ",i(n.i,null,$(d||(d=K`\frac{1}{numero\ sinapsi}`))),"."))),i(n.s,{title:"Modello di Hopfield"},i(n.r,{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.t,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,$(p||(p=K`t`)))),i("td",null,i(a.a,null,"Tick")," attuale")),i("tr",null,i("td",null),i("td",null,i(n.i,null,$(m||(m=K`n`))),", ",i(n.i,null,$(g||(g=K`m`)))),i("td",null,"Identificatore di un neurone specifico")),i("tr",null,i("td",null,i(n.a,null,$(b||(b=K`\mathbf{W}`)))),i("td",null,i(n.a,null,$(f||(f=K`w_{nm}`)))),i("td",null,"Intensità della sinapsi diretta da ",i(n.i,null,$(v||(v=K`n`)))," verso ",i(n.i,null,$(h||(h=K`m`))))),i("tr",null,i("td",null,i(n.a,null,$(z||(z=K`\mathbf{\Theta}`)))),i("td",null,i(n.a,null,$(_||(_=K`\theta_n`)))),i("td",null,"Soglia di attivazione di un neurone")),i("tr",null,i("td",null,i(n.a,null,$(w||(w=K`\mathbf{X}(t)`)))),i("td",null,i(n.a,null,$(S||(S=K`x_n(t)`)))),i("td",null,"Emissione di un neurone")),i("tr",null,i("td",null,i(n.a,null,$(q||(q=K`\mathbf{I}(t)`)))),i("td",null,i(n.a,null,$(k||(k=K`i_n(t)`)))),i("td",null,"Somma degli ingressi di un neurone")),i("tr",null,i("td",null),i("td",null,i(n.a,null,$(C||(C=K`E`)))),i("td",null,"Energia del sistema")),i("tr",null,i("td",null,i(n.a,null,$(I||(I=K`\mathbf{A}`)))),i("td",null,i(n.a,null,$(A||(A=K`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,$(N||(N=K`Q(\mathbf{A}, \mathbf{B})`)))),i("td",null,"Sovrapposizione tra due pattern"))))),i(n.s,null,i(n.r,{title:"Emissione"},i("p",null,"In ogni ",i(a.a,null),", i neuroni:"),i("ul",null,i("li",null,"Emettono ",i(n.i,null,$(E||(E=K`0`)))," se gli input nel ",i(a.a,null)," precedente ",i("b",null,"erano inferiori")," alla soglia di attivazione"),i("li",null,"Emettono ",i(n.i,null,$(P||(P=K`1`)))," se gli input nel ",i(a.a,null)," precedente ",i("b",null,"superavano")," la soglia di attivazione"),i("li",null,"Non cambiano stato se gli input nel ",i(a.a,null)," precedente ",i("b",null,"erano uguali")," alla soglia di attivazione"))),i(n.r,{title:"Sinapsi"},i("p",null,i("b",null,"Tutti")," i neuroni del modello sono intercollegati tra loro da sinapsi."),i("p",null,"I neuroni non possono essere collegati a loro stessi."),i("p",null,"Questo porta il ",i("b",null,"costo computazionale")," del modello ad essere ",i(n.i,null,$(U||(U=K`O(n^2)`))),".")),i(n.r,{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.q,null,$(x||(x=K`
+ E = - \frac{1}{2} \sum_{n, m} ( w_{nm} \cdot x_n \cdot x_m ) + \sum_n ( \theta_n \cdot x_n )
+ `))))),i(n.s,null,i(n.r,{title:"Apprendimento hebbiano"},i("p",null,"Un metodo per realizzare l'apprendimento nel modello di Hopfield."),i("p",null,"Si incrementa l'intensità delle sinapsi che connettono neuroni nello stesso stato, e invece si decrementa l'intensità di quelle che connettono neuroni in stati opposti."),i("p",null,"Considerando i neuroni spenti e quelli accesi come ",i(n.i,null,$(L||(L=K`0`)))," e ",i(n.i,null,$(M||(M=K`1`)))," rispettivamente, si ha che per ogni pattern:"),i(n.q,null,$(T||(T=K`
+ \Delta w_{ik} = (2 \cdot A_i - 1)(2 \cdot A_k - 1)
+ `))),i(t.a,null,"Così facendo, si insegna sia il pattern normale sia il suo complementare!")),i(n.r,{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(n.r,{title:"Overlap di due pattern"},i("p",null,"Il numero di neuroni attivati in entrambi i pattern."),i(n.q,null,$(y||(y=K`
+ Q(A, B) = \sum_{i = 1}^n A_i B_i
+ `)))),i(n.r,{title:"Interferenza"},i("p",null,"Più pattern vengono imparati da un modello, più è facile che essi interferiscano tra loro."),i("p",null,"In caso di pattern completamente scorrelati tra loro, il limite di pattern imparabili è circa:"),i(n.q,null,$(D||(D=K`0.14 \cdot N`)))),i(n.r,{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.s,{title:"Modello a percettroni"},i(n.r,{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.t,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,$(H||(H=K`N`)))),i("td",null,"Numero totale di neuroni nel sistema")),i("tr",null,i("td",null,i(n.a,null,$(B||(B=K`n`)))),i("td",null,"Numero di un neurone specifico")),i("tr",null,i("td",null,i(n.a,null,$(R||(R=K`w_{nm}`)))),i("td",null,"Intensità della sinapsi diretta da ",i(n.i,null,$(G||(G=K`n`)))," verso ",i(n.i,null,$(V||(V=K`m`))))),i("tr",null,i("td",null,i(n.a,null,$(F||(F=K`x_n`)))),i("td",null,"Emissione del neurone ",i(n.i,null,$(O||(O=K`n`))))),i("tr",null,i("td",null,i(n.a,null,$(W||(W=K`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,$(J||(J=K`\sum_1^N ( w_n \cdot x_n )`)))),i("td",null,"Somma degli input di un neurone")),i("tr",null,i("td",null,i(n.a,null,$(Q||(Q=K`b`)))),i("td",null,"Bias di un neurone"))))),i(n.s,null,i(n.r,{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.i,null,$(X||(X=K`H`))),":"),i(n.q,null,$(Y||(Y=K`
+ x_n = H \left( \sum_1^N ( w_n \cdot x_n + b) \right)
+ `))),i("p",null,i(n.i,null,$(Z||(Z=K`b`)))," è una costante configurabile, detta ",i("i",null,"bias"),", che rappresenta il valore di partenza della somma degli input.")),i(n.r,{title:"Percettrone booleano"},i("p",null,"Un percettrone la cui funzione di emissione è:"),i(n.q,null,$(j||(j=K`
+ \begin{cases}
+ 1 \qquad se\ v > 0\\
+ 0 \qquad se\ v = 0\\
+ -1 \qquad se\ v < 0
+ \end{cases}
+ `)))),i(n.r,{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.s,{title:"Rete feed-forward"},i(n.r,{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)},T2GU:function(i,l,e){"use strict";(function(i){var n=e("5aVd"),t=e.n(n);l.a=function(l){return i("ul",{class:t.a.menulist},l.children)}}).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"}}}]);
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\n
Apprendimento ed evoluzione in sistemi artificiali
\n \n \n
\n
Prof. Marco Villani
\n
\n \n \n
\n
Scegli 2 tra 6 domande possibili
\n
90 minuti
\n
\n \n \n \n \n
\n NetLogo è un software di modellazione sistemi multiagente, da noi usato per le lezioni di\n laboratorio.\n
\n Il suo codice sorgente è disponibile su GitHub, e ha una pagina di documentazione.\n
\n \n
Appunti di NetLogo
\n \n \n \n \n \n
\n Sistemi naturali o artificiali che si basano su leggi reversibili e deterministiche.\n
\n
\n In natura, alcuni leggi possono sembrare irreversibili a livello macroscopico, ma sono in realtà\n reversibili a livello microscopico.\n
\n \n
\n Urne di Ehrenfest:\n due urne con N palline; estraggo una pallina da una urna casuale ad ogni passo e la sposto\n 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\n attrattore.\n
\n \n L'evoluzione biologica crea nuove specie partendo da quelle precedenti di maggiore successo fino\n 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\n attrattore?\n \n \n \n \n \n
\n Sistemi dinamici i cui cambiamenti sono descritti da funzioni lineari.\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\n cambiamenti di quest'ultimo\n
\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\n 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\n dell'ambiente\n \n Razionalità limitata: gli agenti non conoscono completamente l'ambiente, e\n 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\n mondo (belief), i propri desideri (desire) e le\n sue intenzioni correnti (intention)\n
\n
Layered: utilizza diversi strati di capacità cognitive per giungere a una\n decisione\n
\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\n percezioni interne\n
\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\n in fretta zone più dense di informazione\n
\n
Rappresentazione: i sistemi multi-agente modellano più accuratamente il mondo reale\n
\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\n agenti semplici.\n
\n \n \n \n \n
\n Meccanismi simili a quelli evolutivi umani che permettono ai tratti degli agenti\n di convergere verso un valore.\n
\n \n \n
\n Inizialmente definita come numero di discendenti fertili, solitamente indica quanto è\n 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\n tagliati nello stesso punto scelto a caso, e per costruire quelli del figlio viene presa una\n 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\n 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\n valore è irrilevante.\n
\n
\n Gli algoritmi genetici permettono di trovare gli schemi con la fitness più alta in\n assoluto in un tempo relativamente breve: il sistema generalmente favorisce gli\n 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\n sistema.\n
\n
\n Si può impedire con vari metodi: con la mutazione, introducendo requisiti di\n 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\n categorie.\n
\n
\n Sono formati da classificatori, liste\n 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\n 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:\n sono generati da detettori e classificatori,\n e consumati da classificatori ed effettori.\n
\n \n \n
\n Sensori che percepiscono lo stato dell'ambiente esterno e lo riportano sotto forma\n 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\n 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\n 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\n che trasforma quel messaggio in un output valido.\n
\n \n \n \n \n
\n Agenti che possono collegarsi tra loro tramite sinapsi (dirette)\n e ricevere ed emettereimpulsi lungo di esse.\n
\n
\n Gli impulsi ricevuti vengono temporaneamente memorizzati dal neurone attraverso valori\n che decadono nel tempo.\n
\n
\n Se la somma dei valori di tutti gli impulsi ricevuti è maggiore di una certa soglia,\n 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\n discreti (ticks), e non è presente la memorizzazione degli impulsi nel tempo.\n
\n \n È importante perchè dimostra che le reti neurali possono elaborare qualsiasi cosa, ma\n 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\n 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\n 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\n intensità {r`\\frac{1}{numero\\ sinapsi}`}.\n
\n \n \n \n \n
\n Un'estensione del modello booleano per permettere l'apprendimento automatico delle\n configurazioni giuste di neuroni.\n
\n
\n È stato sviluppato da John\n 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
\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\n inferiori alla soglia di attivazione\n
\n
Emettono {r`1`} se gli input nel precedente superavano la\n soglia di attivazione\n
\n
Non cambiano stato se gli input nel precedente erano uguali alla soglia di\n attivazione\n
\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\n tutte le connessioni.\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\n decrementa l'intensità di quelle che connettono neuroni in stati opposti.\n
\n
\n Considerando i neuroni spenti e quelli accesi\n come {r`0`} e {r`1`} rispettivamente, si ha che per ogni\n 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\n decrescente, e che quindi che tenderà a un punto fisso!\n
\n \n \n
\n Il numero di neuroni attivati in entrambi i pattern.\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:\n si insegna più volte il pattern originale applicandoci una minima quantità di interferenza\n casuale.\n
\n \n \n \n \n
\n Un modello di rete neurale che supporta l'apprendimento e che presenta più strati di\n 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
\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\n 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\n sbagliato, si alterano le intensità delle sinapsi in direzione della sua classificazione\n corretta.\n
\n
\n Nel caso che vi siano più strati di neuroni, allora sarà necessario ricorrere alla backpropagation, che stima l'errore\n 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\n pozzo.\n
\n \n \n
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\n
Apprendimento ed evoluzione in sistemi artificiali
\n \n \n
\n
Prof. Marco Villani
\n
\n \n \n
\n
Scegli 2 tra 6 domande possibili
\n
90 minuti
\n
\n \n \n \n \n
\n NetLogo è un software di modellazione sistemi multiagente, da noi usato per le lezioni di\n laboratorio.\n
\n Il suo codice sorgente è disponibile su GitHub, e ha una pagina di documentazione.\n
\n \n
Appunti di NetLogo
\n \n \n \n \n \n
\n Sistemi naturali o artificiali che si basano su leggi reversibili e deterministiche.\n
\n
\n In natura, alcuni leggi possono sembrare irreversibili a livello macroscopico, ma sono in realtà\n reversibili a livello microscopico.\n
\n \n
\n Urne di Ehrenfest:\n due urne con N palline; estraggo una pallina da una urna casuale ad ogni passo e la sposto\n 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\n attrattore.\n
\n \n L'evoluzione biologica crea nuove specie partendo da quelle precedenti di maggiore successo fino\n 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\n attrattore?\n \n \n \n \n \n
\n Sistemi dinamici i cui cambiamenti sono descritti da funzioni lineari.\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\n cambiamenti di quest'ultimo\n
\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\n 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\n dell'ambiente\n \n Razionalità limitata: gli agenti non conoscono completamente l'ambiente, e\n 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\n mondo (belief), i propri desideri (desire) e le\n sue intenzioni correnti (intention)\n
\n
Layered: utilizza diversi strati di capacità cognitive per giungere a una\n decisione\n
\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\n percezioni interne\n
\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\n in fretta zone più dense di informazione\n
\n
Rappresentazione: i sistemi multi-agente modellano più accuratamente il mondo reale\n
\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\n agenti semplici.\n
\n \n \n \n \n
\n Meccanismi simili a quelli evolutivi umani che permettono ai tratti degli agenti\n di convergere verso un valore.\n
\n \n \n
\n Inizialmente definita come numero di discendenti fertili, solitamente indica quanto è\n 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\n tagliati nello stesso punto scelto a caso, e per costruire quelli del figlio viene presa una\n 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\n 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\n valore è irrilevante.\n
\n
\n Gli algoritmi genetici permettono di trovare gli schemi con la fitness più alta in\n assoluto in un tempo relativamente breve: il sistema generalmente favorisce gli\n 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\n sistema.\n
\n
\n Si può impedire con vari metodi: con la mutazione, introducendo requisiti di\n 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\n categorie.\n
\n
\n Sono formati da classificatori, liste\n 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\n 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:\n sono generati da detettori e classificatori,\n e consumati da classificatori ed effettori.\n
\n \n \n
\n Sensori che percepiscono lo stato dell'ambiente esterno e lo riportano sotto forma\n 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\n 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\n 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\n che trasforma quel messaggio in un output valido.\n
\n \n \n \n \n
\n Agenti che possono collegarsi tra loro tramite sinapsi (dirette)\n e ricevere ed emettereimpulsi lungo di esse.\n
\n
\n Gli impulsi ricevuti vengono temporaneamente memorizzati dal neurone attraverso valori\n che decadono nel tempo.\n
\n
\n Se la somma dei valori di tutti gli impulsi ricevuti è maggiore di una certa soglia,\n 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\n discreti (ticks), e non è presente la memorizzazione degli impulsi nel tempo.\n
\n \n È importante perchè dimostra che le reti neurali possono elaborare qualsiasi cosa, ma\n 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\n 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\n 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\n intensità {r`\\frac{1}{numero\\ sinapsi}`}.\n
\n \n \n \n \n
\n Un'estensione del modello booleano per permettere l'apprendimento automatico delle\n configurazioni giuste di neuroni.\n
\n
\n È stato sviluppato da John\n 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
\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\n inferiori alla soglia di attivazione\n
\n
Emettono {r`1`} se gli input nel precedente superavano la\n soglia di attivazione\n
\n
Non cambiano stato se gli input nel precedente erano uguali alla soglia di\n attivazione\n
\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\n tutte le connessioni.\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\n decrementa l'intensità di quelle che connettono neuroni in stati opposti.\n
\n
\n Considerando i neuroni spenti e quelli accesi\n come {r`0`} e {r`1`} rispettivamente, si ha che per ogni\n 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\n decrescente, e che quindi che tenderà a un punto fisso!\n
\n \n \n
\n Il numero di neuroni attivati in entrambi i pattern.\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:\n si insegna più volte il pattern originale applicandoci una minima quantità di interferenza\n casuale.\n
\n \n \n \n \n
\n Un modello di rete neurale che supporta l'apprendimento e che presenta più strati di\n 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
\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\n 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\n sbagliato, si alterano le intensità delle sinapsi in direzione della sua classificazione\n corretta.\n
\n
\n Nel caso che vi siano più strati di neuroni, allora sarà necessario ricorrere alla backpropagation, che stima l'errore\n 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\n pozzo.\n
\n \n \n
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x_{(k)} \right|}{left| x_{(k+1)} \right|} leq \tau"],["\\frac{\\left| x_{(k+1)} - x_{(k)} \\right|}{\\left| x_{(k+1)} \\right|} \\leq \\tau"]);return z=function(){return n},n}function q(){var n=un(["\tau"],["\\tau"]);return q=function(){return n},n}function k(){var n=un(["x_{(k+1)} = g( x_{(k)} )"]);return k=function(){return n},n}function A(){var n=un(["g(x) = x - phi(x) cdot f(x)"],["g(x) = x - \\phi(x) \\cdot f(x)"]);return A=function(){return n},n}function L(){var n=un(["f"]);return L=function(){return n},n}function y(){var n=un(["phi(x)"],["\\phi(x)"]);return y=function(){return n},n}function w(){var n=un(["f"]);return w=function(){return n},n}function S(){var n=un(["g(x_{(star)}) = x_{(star)}"],["g(x_{(\\star)}) = x_{(\\star)}"]);return S=function(){return n},n}function G(){var n=un(["x_{(k+1)} = g( x_{(k)} )"]);return G=function(){return n},n}function O(){var n=un(["x = x - phi(x) cdot f(x)"],["x = x - \\phi(x) \\cdot f(x)"]);return O=function(){return n},n}function C(){var n=un(["phi"],["\\phi"]);return C=function(){return n},n}function R(){var n=un(["[a_{(n+1)}, b_{(n+1)}]"]);return R=function(){return n},n}function I(){var n=un(["[c_{(n)}, b_{(n)}]"]);return I=function(){return n},n}function M(){var n=un(["[a_{(n)}, c_{(n)}]"]);return M=function(){return n},n}function F(){var n=un(["c_{(n)}"]);return F=function(){return n},n}function Y(){var n=un(["c_{(n)} = b_{(n)} - \frac{f(b_{(n)})}{\frac{f(b_{(n)}) - f(a_{(n)})}{b_{(n)} - a_{(n)}}}"],["c_{(n)} = b_{(n)} - \\frac{f(b_{(n)})}{\\frac{f(b_{(n)}) - f(a_{(n)})}{b_{(n)} - a_{(n)}}}"]);return Y=function(){return n},n}function Q(){var n=un(["a_{(n)}, b_{(n)}"]);return Q=function(){return n},n}function P(){var n=un(["[a_{(n+1)}, b_{(n+1)}]"]);return P=function(){return n},n}function U(){var n=un(["[c_{(n)}, b_{(n)}]"]);return U=function(){return n},n}function E(){var n=un(["[a_{(n)}, c_{(n)}]"]);return E=function(){return n},n}function T(){var n=un(["c_{(n)}"]);return T=function(){return n},n}function D(){var n=un(["c_{(n)} = a_{(n)} + \frac{b_{(n)} - a_{(n)}}{2}"],["c_{(n)} = a_{(n)} + \\frac{b_{(n)} - a_{(n)}}{2}"]);return D=function(){return n},n}function N(){var n=un(["[a_{(n)}, b_{(n)}]"]);return N=function(){return n},n}function j(){var n=un(["[a, b]"]);return j=function(){return n},n}function H(){var n=un(["i geq log_2 left( \frac{b - a}{\tau} \right)"],["i \\geq \\log_2 \\left( \\frac{b - a}{\\tau} \\right)"]);return H=function(){return n},n}function V(){var n=un(["C = \frac{1}{2}, p = 1"],["C = \\frac{1}{2}, p = 1"]);return V=function(){return n},n}function J(){var n=un(["b_{(i)} - a_{(i)} = \frac{b - a}{2^{i - 1}}"],["b_{(i)} - a_{(i)} = \\frac{b - a}{2^{i - 1}}"]);return J=function(){return n},n}function X(){var n=un(["C = 0"]);return X=function(){return n},n}function B(){var n=un(["p = 2"]);return B=function(){return n},n}function Z(){var n=un(["0 < C < 1"]);return Z=function(){return n},n}function K(){var n=un(["p = 2"]);return K=function(){return n},n}function W(){var n=un(["C = 0"]);return W=function(){return n},n}function $(){var n=un(["p = 1"]);return $=function(){return n},n}function nn(){var n=un(["0 < C < 1"]);return nn=function(){return n},n}function ln(){var n=un(["p = 1"]);return ln=function(){return n},n}function en(){var n=un(["lim_{i \to +infty} \frac{ left| x_{(i+1)} - x_{(star)} \right| }{ left| x_{(k)} - x_{(star)} \right|^{color{Orange} p}}"],["\\lim_{i \\to +\\infty} \\frac{ \\left| x_{(i+1)} - x_{(\\star)} \\right| }{ \\left| x_{(k)} - x_{(\\star)} \\right|^{\\color{Orange} p}}"]);return en=function(){return n},n}function rn(){var n=un(["{color{Orange} p}"],["{\\color{Orange} p}"]);return rn=function(){return n},n}function tn(){var n=un(["f'(x_{(star)}) simeq 0 implies mal condizionato"],["f'(x_{(\\star)}) \\simeq 0 \\implies mal\\ condizionato"]);return tn=function(){return n},n}function on(){var n=un(["x_{(star)}"],["x_{(\\star)}"]);return on=function(){return n},n}function an(){var n=un(["f(a) cdot f(b) leq 0"],["f(a) \\cdot f(b) \\leq 0"]);return an=function(){return n},n}function un(n,i){return i||(i=n.slice(0)),n.raw=i,n}var cn=l("hosL"),sn=l("mbOI"),dn=l("ke5e"),fn=String.raw;i.a=function(){return n(cn.Fragment,null,n(sn.s,{title:"Problema: Ricerca degli zeri di funzione"},n(sn.r,{title:"Descrizione"},n("p",null,"Si vogliono trovare i punti (",n("i",null,"zeri"),") in cui una funzione ",n("b",null,"continua")," ",n(sn.i,null,"f : [a, b] \\to R")," vale ",n(sn.i,null,"0"),"."),n("p",null,"Per il ",n("b",null,"teorema del valore medio"),", se ",n(sn.i,null,fn(an())),", allora esiste sicuramente un punto in cui la funzione vale 0."),n("p",null,"Denominiamo il punto in cui la funzione vale ",n(sn.i,null,"0")," come ",n(sn.i,null,fn(on())),".")),n(sn.r,{title:"Condizionamento"},n("p",null,"Più la ",n("b",null,"derivata prima")," della funzione ",n("b",null,"si avvicina allo 0"),", ",n("b",null,"peggio")," il problema sarà condizionato."),n(sn.q,null,fn(tn())))),n(sn.s,null,n(sn.r,{title:"Ordine di convergenza"},n("p",null,"Indice ",n(sn.i,null,fn(rn()))," di quanto in fretta una successione converge alla soluzione."),n(sn.q,null,fn(en())),n("ul",null,n("li",null,n("u",null,"Convergenza lineare"),": ",n(sn.i,null,fn(ln()))," e ",n(sn.i,null,fn(nn()))),n("li",null,n("u",null,"Convergenza superlineare"),": ",n(sn.i,null,fn($()))," e ",n(sn.i,null,fn(W()))),n("li",null,n("u",null,"Convergenza quadratica"),": ",n(sn.i,null,fn(K()))," e ",n(sn.i,null,fn(Z()))),n("li",null,n("u",null,"Convergenza superquadratica"),": ",n(sn.i,null,fn(B()))," e ",n(sn.i,null,fn(X()))),n("li",null,"...")))),n(sn.s,{title:"Metodi dicotomici"},n(sn.r,{title:"Cosa sono?"},n("p",null,"Sono ",n("b",null,"metodi iterativi")," in grado di ridurre sempre di più l'intervallo in cui è definita la funzione, facendolo convergere allo zero desiderato."),n("p",null,"Alcuni di essi sono il ",n("i",null,"metodo dicotomico")," e il ",n("i",null,"metodo regula falsi"),"."),n("p",null,"Richiedono ",n("b",null,"una valutazione di funzione non-lineare")," ad ogni iterazione."),n("p",null,"Ad ogni iterazione, l'intervallo viene sempre ",n("i",null,"almeno")," ",n("b",null,"dimezzato"),"; si ha, pertanto, che:"),n(sn.q,null,fn(J())),n("p",null,"Hanno quindi ",n("b",null,"convergenza lineare")," (",n(sn.i,null,fn(V())),")."),n("p",null,"Il loro ",n("i",null,"criterio di arresto")," è un ",n("b",null,"numero di iterazioni prefissato")," che dipende dalla ",n("b",null,"tolleranza")," sull'errore:"),n(sn.q,null,fn(H())),n(dn.a,null,"Dividi l'intervallo ",n(sn.i,null,fn(j()))," in tante parti grandi quanto la tolleranza. L'algoritmo di bisezione ne escluderà metà ad ogni iterazione; la tolleranza sarà raggiunta quando rimarrà una parte sola!"))),n(sn.s,null,n(sn.r,{title:"Metodo di bisezione"},n("ol",null,n("li",null,"Finchè non sono state compiute il numero di iterazioni prefissate:",n("ol",null,n("li",null,"Calcoliamo il ",n("b",null,"punto medio")," dell'intervallo ",n(sn.i,null,fn(N())),":",n(sn.q,null,fn(D()))),n("li",null,"Dividiamo l'intervallo in due parti, separate da ",n(sn.i,null,fn(T())),":",n("ul",null,n("li",null,n(sn.i,null,fn(E()))," è la ",n("b",null,"metà")," sinistra"),n("li",null,n(sn.i,null,fn(U()))," è la ",n("b",null,"metà")," destra"))),n("li",null,"Teniamo l'intervallo in cui i valori della funzione ai due estremi sono discordi, e rinominiamolo in ",n(sn.i,null,fn(P())),"."))))),n(sn.r,{title:"Metodo regula falsi"},n("ol",null,n("li",null,"Finchè non sono state compiute il numero di iterazioni prefissate:",n("ol",null,n("li",null,"Calcoliamo l'",n("b",null,"intersezione")," tra la ",n("b",null,"retta che congiunge i due estremi"),n(sn.i,null,fn(Q()))," e l'",n("b",null,"asse X"),":",n(sn.q,null,fn(Y()))),n("li",null,"Dividiamo l'intervallo in due parti, separate da ",n(sn.i,null,fn(F())),":",n("ul",null,n("li",null,n(sn.i,null,fn(M()))," è la parte sinistra"),n("li",null,n(sn.i,null,fn(I()))," è la parte destra"))),n("li",null,"Teniamo l'intervallo in cui i valori della funzione ai due estremi sono discordi, e rinominiamolo in ",n(sn.i,null,fn(R())),".")))))),n(sn.s,{title:"Metodo delle approssimazioni successive"},n(sn.r,{title:"Metodi delle approssimazioni successive"},n("p",null,"Sono ",n("b",null,"metodi iterativi")," che funzionano in modo molto simile ai metodi iterativi per i sistemi lineari, utilizzando una funzione ",n(sn.i,null,fn(C())),' come "metodo".'),n(sn.q,null,fn(O())),n("p",null,"Che diventa:"),n(sn.q,null,fn(G())),n("p",null,"Sfruttano i ",n("b",null,"punti fissi")," ",n(sn.i,null,fn(S()))," della funzione ",n(sn.i,null,fn(w()))," per convergere:",n("br",null),"se ",n(sn.i,null,fn(y()))," non ha zeri, allora i punti fissi ",n("b",null,"coincideranno")," con gli ",n("b",null,"zeri")," della funzione ",n(sn.i,null,fn(L())),"."),n(sn.q,null,fn(A())),n("p",null,"Si può raggiungere iterativamente ad un punto fisso attraverso la formula:"),n(sn.q,null,fn(k())),n("p",null,"Non si conosce in anticipo il numero di iterazioni necessarie per soddisfare la tolleranza ",n(sn.i,null,fn(q())),"; ad ogni iterazione, si controlla se la tolleranza è soddisfatta:"),n("ul",null,n("li",null,"Nella differenza tra due iterate: ",n(sn.i,null,fn(z()))),n("li",null,"Nel ",n("i",null,"residuo")," del problema: ",n(sn.i,null,fn(h()))))),n(sn.r,{title:"Teorema della mappa contrattiva"},n("p",null,"Se:"),n("ul",null,n("li",null,"Tutti i valori restituiti dalla funzione ",n(sn.i,null,fn(g()))," rientrano nel suo stesso dominio:",n(sn.q,null,fn(b()))),n("li",null,n("p",null,"La funzione ",n(sn.i,null,fn(x()))," è una contrazione, ovvero restringe l'intervallo ",n(sn.i,null,fn(v())),":"),n(sn.q,null,fn(_())),n("p",null,"(dove ",n(sn.i,null,fn(m())),")"))),n("p",null,"Allora:"),n("ul",null,n("li",null,n("p",null,"Il punto fisso esiste ed è unico:"),n(sn.q,null,fn(p()))),n("li",null,"Il metodo delle approssimazioni successive converge per qualsiasi punto di partenza."),n("li",null,n("p",null,"Vale la seguente disequazione di ",n("i",null,"maggiorazione dell'errore"),":"),n(sn.q,null,fn(f())))),n("p",null,"Più è piccolo ",n(sn.i,null,"L"),", più il metodo convergerà in fretta."),n(dn.a,null,n(sn.i,null,"L")," è molto simile al raggio spettrale ",n(sn.i,null,fn(d()))," dei metodi iterativi per i sistemi lineari!"))),n(sn.s,null,n(sn.r,{title:"Metodo di Newton"},n("p",null,"Sfrutta la ",n("b",null,"continuità")," delle funzioni per ottenere una convergenza di ordine più alto."),n(sn.q,null,fn(s())),n(sn.q,null,fn(c())),n(dn.a,null,"Geometricamente, corrisponde a prolungare una retta nel punto ",n(sn.i,null,fn(u()))," con pendenza ",n(sn.i,null,fn(a())),", e prendendo come nuovo punto la sua intersezione con l'asse X e la sua corrispettiva immagine nella funzione."),n("p",null,"Ha costo computazionale di ",n("b",null,"2 valutazioni di funzione")," più ",n("b",null,"2 valutazioni di derivata"),"."),n("p",null,"Ha ",n("b",null,"convergenza quadratica"),".")),n(sn.r,{title:"Metodo delle secanti"},n("p",null,"È come il metodo di Newton, ma usa il ",n("b",null,"rapporto incrementale"),", in modo da poter essere applicato a funzioni non continue."),n(sn.q,null,fn(o())),n(sn.q,null,fn(t())),n(dn.a,null,"Geometricamente, corrisponde a costruire una retta che attraversa i punti ",n(sn.i,null,fn(r()))," e ",n(sn.i,null,fn(e())),", e prendendo come nuovo punto la sua intersezione con l'asse X e la sua corrispettiva immagine nella funzione."),n("p",null,"Ha costo computazionale di ",n("b",null,"3 valutazioni di funzione"),"."),n("p",null,"Ha ",n("b",null,"convergenza superlineare"),"."))),n(sn.s,null,n(sn.r,{title:"Approssimare sistemi non-lineari"},n("p",null,"È possibile usare questi metodi per ",n("b",null,"approssimare le soluzioni di sistemi non-lineari"),"."))))}}).call(this,l("hosL").h)},LHkx:function(n,i,l){"use strict";(function(n){function e(){var n=I(["h"]);return e=function(){return n},n}function r(){var n=I(["int_a^b ( s''(x) )^2 dx leq int_a^b ( f''(x) )^2 dx"],["\\int_a^b ( s''(x) )^2 dx \\leq \\int_a^b ( f''(x) )^2 dx"]);return r=function(){return n},n}function t(){var n=I(["[x_{m-1}, x_{m+1}]"]);return t=function(){return n},n}function o(){var n=I(["[x_0, x_2]"]);return o=function(){return n},n}function a(){var n=I(["s''(x) = s''(m+1)"]);return a=function(){return n},n}function u(){var n=I(["s'(x) = s'(m+1)"]);return u=function(){return n},n}function c(){var n=I(["s(x) = s(m+1)"]);return c=function(){return n},n}function s(){var n=I(["s''(x_0) = s''(x_{m+1}) = 0"]);return s=function(){return n},n}function d(){var n=I(["\n c =\n \begin{pmatrix}\n b_0 - h_1 \beta_0\\\\\n b_1\\\\\n \vdots\\\\\n b_{m-2}\\\\\n b_{m-1} - h_{m-1} \beta_{m+1}\n end{pmatrix}\n "],["\n c =\n \\begin{pmatrix}\n b_0 - h_1 \\beta_0\\\\\\\\\n b_1\\\\\\\\\n \\vdots\\\\\\\\\n b_{m-2}\\\\\\\\\n b_{m-1} - h_{m-1} \\beta_{m+1}\n \\end{pmatrix}\n "]);return d=function(){return n},n}function f(){var n=I(["\n z =\n \begin{pmatrix}\n \beta_1\\\\\n \beta_2\\\\\n \vdots\\\\\n \beta_{m-1}\\\\\n \beta_{m}\n end{pmatrix}\n "],["\n z =\n \\begin{pmatrix}\n \\beta_1\\\\\\\\\n \\beta_2\\\\\\\\\n \\vdots\\\\\\\\\n \\beta_{m-1}\\\\\\\\\n \\beta_{m}\n \\end{pmatrix}\n "]);return f=function(){return n},n}function p(){var n=I(["\n T = \n \begin{pmatrix}\n 2 (h_0 + 2 h_1) & h_0 & {color{Gray} 0} & {color{Gray} 0} & {color{Gray} 0} \\\\\n h_2 & 2 (h_1 + h_2) & h_1 & {color{Gray} 0} & {color{Gray} 0} \\\\\n {color{Gray} 0} & ddots & ddots & ddots & {color{Gray} 0} \\\\\n {color{Gray} 0} & {color{Gray} 0} & h_{m-1} & 2 (h_{m-2} + h_{m-1}) & h_{m-2} \\\\\n {color{Gray} 0} & {color{Gray} 0} & {color{Gray} 0} & h_m & 2 (h_{m-1} + h_m)\n end{pmatrix}\n "],["\n T = \n \\begin{pmatrix}\n 2 (h_0 + 2 h_1) & h_0 & {\\color{Gray} 0} & {\\color{Gray} 0} & {\\color{Gray} 0} \\\\\\\\\n h_2 & 2 (h_1 + h_2) & h_1 & {\\color{Gray} 0} & {\\color{Gray} 0} \\\\\\\\\n {\\color{Gray} 0} & \\ddots & \\ddots & \\ddots & {\\color{Gray} 0} \\\\\\\\\n {\\color{Gray} 0} & {\\color{Gray} 0} & h_{m-1} & 2 (h_{m-2} + h_{m-1}) & h_{m-2} \\\\\\\\\n {\\color{Gray} 0} & {\\color{Gray} 0} & {\\color{Gray} 0} & h_m & 2 (h_{m-1} + h_m)\n \\end{pmatrix}\n "]);return p=function(){return n},n}function m(){var n=I(["\n b_i = h_{i+1} \beta_i + 2 ( h_i + h_{i+1} ) + h_{i} \beta_i+2\n "],["\n b_i = h_{i+1} \\beta_i + 2 ( h_i + h_{i+1} ) + h_{i} \\beta_i+2\n "]);return m=function(){return n},n}function _(){var n=I(["T z = c"]);return _=function(){return n},n}function v(){var n=I(["\beta_{m+1}"],["\\beta_{m+1}"]);return v=function(){return n},n}function x(){var 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migliore ottenibile, considerando la precisione di macchina."},"*")," di un sistema lineare."),n("p",null,"Tipicamente prevedono l'applicazione ripetuta di un ",n("b",null,"metodo"),", in base al quale cambia la ",n("b",null,"velocità di convergenza")," alla soluzione."),n("p",null,"Generalmente hanno una complessità temporale ",n(Fn.i,null,Un(yn())),"."))),n(Fn.s,{title:"Metodi diretti"},n(Fn.r,{title:"Divisione"},n("p",null,"Se la matrice dei coefficienti del sistema è ",n("b",null,"diagonale"),", allora è possibile trovare la soluzione ",n("i",null,"dividendo")," ogni termine noto per l'unico coefficiente diverso da zero presente nella sua riga:"),n(Fn.q,null,Un(Ln()))),n(Fn.r,{title:"Sostituzione"},n("p",null,"Se la matrice dei coefficienti del sistema è ",n("b",null,"triangolare")," inferiore o superiore, allora è possibile trovare la soluzione effettuando una ",n("i",null,"sostituzione")," all'avanti oppure all'indietro:"),n(Fn.q,null,Un(An())),n(Fn.q,null,Un(kn())))),n(Fn.s,null,n(Fn.r,{title:n("span",null,"Fattorizzazione ",n(Fn.i,null,Un(qn())))},n("p",null,"Se tutti i valori sulla diagonale di ",n(Fn.i,null,Un(zn()))," sono ",n("b",null,"diversi da 0 ",n("small",null,"(eccetto l'ultimo)"))," allora è possibile ",n("i",null,"fattorizzarla")," in due matrici: una ",n(Fn.i,null,Un(hn()))," ",n("b",null,"triangolare inferiore"),", e una ",n(Fn.i,null,Un(gn()))," ",n("b",null,"triangolare superiore"),"."),n(Fn.q,null,Un(bn())),n(Qn.a,null,"Abbiamo fatto questo metodo in Algebra Lineare, chiamandolo ",n("b",null,"metodo di Gauss"),"."),n("p",null,"La matrice ",n(Fn.i,null,Un(xn()))," è così composta:"),n(Fn.q,null,Un(vn())),n(Qn.a,null,"Sono i moltiplicatori usati per rendere annullare il triangolo inferiore!"),n("p",null,"La matrice ",n(Fn.i,null,Un(_n()))," è così composta:"),n(Fn.q,null,Un(mn())),n(Qn.a,null,"È la parte triangolare superiore di ",n(Fn.i,null,Un(pn())),"!"),n("p",null,"Il sistema può essere poi risolto applicando due volte il metodo di sostituzione (all'avanti e all'indietro):"),n(Fn.q,null,Un(fn())),n("p",null,"Questo metodo ha costo computazionale:"),n(Fn.q,null,Un(dn()))),n(Fn.r,{title:n("span",null,"Fattorizzazione ",n(Fn.i,null,Un(sn()))," con pivoting parziale")},n("p",null,"È possibile applicare la fattorizzazione ",n(Fn.i,null,Un(cn()))," a ",n("b",null,"qualsiasi matrice non-singolare")," permettendo lo scambio (",n("i",null,"pivoting"),") delle righe, potenzialmente ",n("b",null,"aumentando la stabilità")," dell'algoritmo."),n(Qn.a,null,"Abbiamo fatto questo metodo in Algebra Lineare, chiamandolo ",n("b",null,"metodo di Gauss-Jordan"),"!"),n("p",null,"Alla formula precedente si aggiunge una ",n("b",null,"matrice di permutazione")," che indica quali righe sono state scambiate:"),n(Fn.q,null,Un(un())),n("p",null,"Per massimizzare la stabilità, si cerca di ",n("b",null,"usare come perno l'elemento più grande")," della colonna."),n("p",null,"Questo metodo ha costo computazionale:"),n(Fn.q,null,Un(an()))),n(Fn.r,{title:n("span",null,"Fattorizzazione ",n(Fn.i,null,Un(on()))," con pivoting totale")},n("p",null,"È possibile anche permettere il ",n("i",null,"pivoting")," ",n("b",null,"sulle colonne")," per ",n("b",null,"aumentare ulteriormente la stabilità")," dell'algoritmo, a costo di maggiore costo computazionale:"),n(Fn.q,null,Un(tn())),n("p",null,"Per massimizzare la stabilità, si cerca di ",n("b",null,"ordinare in modo decrescente la diagonale"),", assicurandoci che il primo perno sia più grande del secondo e così via."),n("p",null,"Questo metodo ha costo computazionale:"),n(Fn.q,null,Un(rn())))),n(Fn.s,null,n(Fn.r,{title:n("span",null,"Fattorizzazione ",n(Fn.i,null,Un(en()))," a banda")},n("p",null,"Se la matrice ",n(Fn.i,null,Un(ln()))," è ",n("b",null,"a banda"),", è possibile risparmiare spazio durante la fattorizzazione, in quanto sia ",n(Fn.i,null,Un(nn()))," sia ",n(Fn.i,null,Un($()))," saranno a banda!")),n(Fn.r,{title:n("span",null,"Fattorizzazione ",n(Fn.i,null,Un(W()))," sparsa")},n("p",null,"Se la matrice ",n(Fn.i,null,Un(K()))," è ",n("b",null,"sparsa"),", non è detto che ",n(Fn.i,null,Un(Z()))," e ",n(Fn.i,null,Un(B()))," siano sparse a loro volta."),n("p",null,"Per evitare il ",n("u",null,"fill-in"),", è necessario ",n("b",null,"riordinare")," la matrice ",n(Fn.i,null,Un(X()))," in modo che sia il più possibile simile a una matrice a banda."))),n(Fn.s,null,n(Fn.r,{title:n("span",null,"Fattorizzazione ",n(Fn.i,null,Un(J())))},n("p",null,"È possibile ",n("b",null,"ridurre la complessità computazionale")," della fattorizzazione ",n(Fn.i,null,Un(V()))," se la matrice dei coefficienti è ",n("b",null,"simmetrica"),":"),n(Fn.q,null,Un(H())),n("p",null,"In questo caso, si calcola solo la matrice L, utilizzando il ",n("b",null,"metodo di pavimentazione"),"."),n(Fn.q,null,Un(j())),n(Qn.a,null,n("p",null,"La prima colonna della matrice sarà:"),n(Fn.q,null,Un(N())),n("p",null,"La seconda colonna della matrice sarà:"),n(Fn.q,null,Un(D()))),n("p",null,"Questo metodo ha costo computazionale:"),n(Fn.q,null,Un(T()))),n(Fn.r,{title:n("span",null,"Fattorizzazione ",n(Fn.i,null,Un(E())))},n("p",null,"È possibile dare ",n("b",null,"stabilità forte")," alla fattorizzazione ",n(Fn.i,null,Un(U()))," se la matrice dei coefficienti è ",n("b",null,"simmetrica definita positiva"),":"),n(Fn.q,null,Un(P())),n("p",null,"Il ",n("b",null,"metodo di pavimentazione")," diventa:"),n(Fn.q,null,Un(Q())),n("p",null,"Questo metodo ha costo computazionale:"),n(Fn.q,null,Un(Y())))),n(Fn.s,null,n(Fn.r,{title:"Trasformazione di Householder"},n("p",null,"Matrice ricavata dalla seguente formula, dove ",n(Fn.i,null,Un(F()))," è la colonna di un'altra matrice:"),n(Fn.q,null,Un(M())),n("p",null,"Se moltiplicata per per la matrice da cui proviene ",n(Fn.i,null,Un(I())),", sostituirà la colonna ",n(Fn.i,null,Un(R()))," con la colonna:"),n(Fn.q,null,Un(C())),n("p",null,"Si calcola con una complessità computazionale nell'ordine di ",n(Fn.i,null,Un(O())),".")),n(Fn.r,{title:n("span",null,"Fattorizzazione ",n(Fn.i,null,Un(G())))},n("p",null,"Metodo che fornisce una ",n("b",null,"maggiore stabilità")," a costo di una ",n("b",null,"maggiore complessità computazionale"),"."),n("p",null,"La matrice ",n(Fn.i,null,Un(S()))," viene ",n("i",null,"fattorizzata")," in due matrici, una ",n("b",null,"ortogonale")," ",n(Fn.i,null,Un(w()))," e una ",n("b",null,"triangolare superiore"),n(Fn.i,null,Un(y())),":"),n(Fn.q,null,Un(L())),n("p",null,"Le matrici si ottengono dal ",n("b",null,"prodotto delle trasformazioni di Householder")," (che concatenate formano ",n(Fn.i,null,Un(A())),") sulla matrice ",n(Fn.i,null,Un(k()))," necessarie a trasformarla in una matrice triangolare superiore (",n(Fn.i,null,Un(q())),")."),n(Qn.a,null,"C'è un bell'esempietto ",n(Yn.a,{href:"https://web.archive.org/web/20200828003151/https://rpubs.com/aaronsc32/qr-decomposition-householder"},"qui"),"."),n("p",null,"Una volta fattorizzata, il sistema si può risolvere con:"),n(Fn.q,null,Un(z())),n("p",null,"Questo metodo ha costo computazionale:"),n(Fn.q,null,Un(h())))),n(Fn.s,{title:"Metodi iterativi"},n(Fn.r,{title:"Forma generale"},n("p",null,"Se si pone che:"),n(Fn.q,null,Un(g())),n("p",null,"Allora la formula generale di un sistema lineare può anche essere scritta in questo modo:"),n(Fn.q,null,Un(b())),n("p",null,"È particolarmente utile perchè ci permette di definire un ",n("b",null,"algoritmo ricorsivo")," che trovi ",n(Fn.i,null,Un(x())),":"),n(Fn.q,null,Un(v())),n("p",null,n(Fn.i,null,Un(_()))," è il ",n("b",null,"metodo"),", e in base ad esso cambiano stabilità e velocità di convergenza."),n("p",null,"Ponendo ",n(Fn.i,null,Un(m())),", la formula può essere scritta anche in questo modo:"),n(Fn.q,null,Un(p())),n("p",null,"Possiamo ottenere alcuni metodi separando ",n(Fn.i,null,"A")," in tre matrici:"),n("ul",null,n("li",null,"La parte diagonale ",n(Fn.i,null,Un(f()))),n("li",null,"L'opposto del triangolo inferiore ",n(Fn.i,null,Un(d()))),n("li",null,"L'opposto del triangolo superiore ",n(Fn.i,null,Un(s())))),n(Fn.q,null,Un(c()))),n(Fn.r,{title:"Convergenza di un metodo"},n("p",null,"Un metodo è convergente se e solo se:"),n(Fn.q,null,Un(u())),n("p",null,"(dove ",n(Fn.i,null,Un(a()))," è il ",n("b",null,"raggio spettrale"),", il massimo autovalore della matrice)"),n("p",null,"Perchè un metodo sia convergente, è sufficiente che:"),n(Fn.q,null,Un(o())))),n(Fn.s,null,n(Fn.r,{title:"Metodo di Jacobi"},n("p",null,"Il metodo di Jacobi si ottiene ponendo:"),n(Fn.q,null,Un(t())),n("p",null,n("u",null,"Spostamenti simultanei"),": Permette di ottenere ogni componente di ",n(Fn.i,null,Un(r()))," indipendentemente dagli altri: è ",n("b",null,"parallelizzabile"),"."),n("p",null,"Se la matrice è ",n("b",null,"diagonale dominante"),", allora il metodo di Jacobi ",n("b",null,"converge")," sicuramente.")),n(Fn.r,{title:"Metodo di Gauss-Seidel"},n("p",null,"Il metodo di Gauss-Seidel si ottiene ponendo:"),n(Fn.q,null,Un(e())),n("p",null,"Ha una velocità di convergenza ",n("b",null,"maggiore o uguale")," rispetto al metodo di Jacobi."),n("p",null,n("u",null,"Spostamenti successivi"),": Non è parallelizzabile, perchè ogni componente ",n("b",null,"dipende da quelle calcolate in precedenza"),"."),n("p",null,"Se la matrice è ",n("b",null,"diagonale dominante"),", allora il metodo di Gauss-Seidel ",n("b",null,"converge")," sicuramente."))))}}).call(this,l("hosL").h)},qXt2:function(n,i,l){"use strict";l.r(i),function(n){var e=l("sl5E"),r=l("lijF"),t=l("FEtp"),o=l("31Ft"),a=l("LHkx"),u=l("uqDQ");i.default=function(){return n("div",null,n("h1",null,"Calcolo Numerico"),n(e.a,null),n(r.a,null),n(t.a,null),n(o.a,null),n(a.a,null),n(u.a,null))}}.call(this,l("hosL").h)},sl5E:function(n,i,l){"use strict";(function(n){function e(){var n=A(["x = fl left( \frac{2}{4} \right)"],["x = fl \\left( \\frac{2}{4} \\right)"]);return e=function(){return n},n}function r(){var n=A(["x = fl ( 2 cdot t )"],["x = fl ( 2 \\cdot t )"]);return r=function(){return n},n}function t(){var n=A(["t = fl left( \frac{1}{4} \right)"],["t = fl \\left( \\frac{1}{4} \\right)"]);return t=function(){return n},n}function o(){var n=A(["2x^star = 4"],["2x^\\star = 4"]);return o=function(){return n},n}function a(){var n=A(["y = \frac{1}{x}"],["y = \\frac{1}{x}"]);return a=function(){return n},n}function u(){var n=A(["epsilon_{1}"],["\\epsilon_{1}"]);return u=function(){return n},n}function c(){var n=A(["epsilon_{num_passo}"],["\\epsilon_{num\\_passo}"]);return c=function(){return n},n}function s(){var n=A(["epsilon_{x}"],["\\epsilon_{x}"]);return s=function(){return n},n}function d(){var n=A(["epsilon_{nome_var}"],["\\epsilon_{nome\\_var}"]);return d=function(){return n},n}function f(){var n=A(["mathbb{F}"],["\\mathbb{F}"]);return f=function(){return 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z=function(){return n},n}function q(){var n=A(["alpha^star"],["\\alpha^\\star"]);return q=function(){return n},n}function k(){var n=A(["alpha"],["\\alpha"]);return k=function(){return n},n}function A(n,i){return i||(i=n.slice(0)),n.raw=i,n}var L=l("mbOI"),y=l("YNhk"),w=l("T2GU"),S=l("ke5e"),G=l("hosL"),O=String.raw;i.a=function(){return n(G.Fragment,null,n(L.s,{title:"Esame"},n(L.r,{title:"Contatti"},n("ul",null,n("li",null,n(y.a,{href:"mailto:silvia.bonettini@unimore.it"},"Prof.ssa Silvia Bonettini")))),n(L.r,{title:"Orale"},n("p",null,"E' composto da:"),n("ul",null,n("li",null,"2 domande sugli argomenti teorici"),n("li",null,"1 domanda di implementazione algoritmo in MATLAB")))),n(L.s,{title:"Extra"},n(L.r,{title:"Ripasso di Algebra Lineare"},n("p",null,"Prima di iniziare a studiare Calcolo Numerico, potrebbe essere una buona idea ripassare un pochino Algebra Lineare:"),n(w.a,null,n("li",null,n("a",{href:"/calcolonumerico/ripassodialgebralineare"},"Ripasso di Algebra Lineare")," ",n("small",null,"(per studenti sperduti di Calcolo Numerico)")))),n(L.r,{title:"Anki"},n("p",null,"Ho provato a fare un deck ",n(y.a,{href:"https://apps.ankiweb.net/"},"Anki")," con tutte le formule di Calcolo numerico."),n("p",null,"Sappiatemi dire com'è! ",n("small",null,"Io non l'ho usato moltissimo perchè ho studiato scrivendo questa pagina di Appuntiweb...")),n(w.a,null,n("li",null,n(y.a,{href:"https://ryg.s3.fr-par.scw.cloud/builds/anki-decks/formule-di-calcolo-numerico.apkg"},"Scarica il deck Anki"))))),n(L.s,{title:"Algoritmi"},n(L.r,{title:"Algoritmi numerici"},n("p",null,"Particolari algoritmi che hanno:"),n("ul",null,n("li",null,"numeri reali in input e output"),n("li",null,"successioni delle quattro operazioni aritmetiche fondamentali come passi")))),n(L.s,{title:"Errore di rappresentazione"},n(L.r,{title:"Cos'è?"},n("p",null,"Con i numeri floating point può capitare che un certo numero ",n(L.i,null,O(k()))," non sia rappresentato correttamente."),n("p",null,"In tal caso, il numero si indica con ",n(L.i,null,O(q())),"."))),n(L.s,null,n(L.r,{title:"Errore assoluto"},n("p",null,"È la differenza tra il numero desiderato e il numero rappresentato:"),n(L.q,null,O(z()))),n(L.r,{title:"Errore relativo"},n("p",null,"Indica quanto il numero rappresentato differisce dal numero desiderato:"),n(L.q,null,O(h())))),n(L.s,null,n(L.r,{title:"Troncamento"},n("p",null,"Metodo con cui gestire gli ",n("b",null,"underflow floating point"),": le cifre meno significative vengono ",n("b",null,"rimosse"),"."),n(S.a,null,n("pre",null,"1.00 → 1.0",n("br",null),"1.01 → 1.0",n("br",null),"1.10 → 1.1",n("br",null),"1.11 → 1.1"))),n(L.r,{title:"Arrotondamento"},n("p",null,"Metodo con cui gestire gli ",n("b",null,"underflow floating point"),": se la cifra più significativa di quelle che devono essere rimosse è 1, allora ",n("b",null,"aumenta di 1")," anche quella meno signficativa che viene tenuta."),n(S.a,null,n("pre",null,"1.00 → 1.0",n("br",null),"1.01 → 1.0",n("br",null),"1.10 → 1.1",n("br",null),"1.11 → 10.")))),n(L.s,null,n(L.r,{title:"Precisione di macchina"},n("p",null,"Un numero reale rappresentato in ",n("b",null,"virgola mobile")," ha un ",n("b",null,"errore relativo")," minore o uguale alla ",n("i",null,"precisione di macchina"),":"),n("p",null,n(L.i,null,O(g()))),n("ul",null,n("li",null,n(L.i,null,"\\beta")," è uguale alla base utilizzata (solitamente 2)."),n("li",null,n(L.i,null,"t")," è uguale al numero di cifre della mantissa."),n("li",null,n(L.i,null,"k")," è uguale a ",n(L.i,null,"1")," se il numero viene rappresentato per troncamento oppure a ",n(L.i,null,O(b()))," se viene rappresentato per arrotondamento."))),n(L.r,{title:"La funzione fl"},n("p",null,"Associa un valore reale al suo ",n("b",null,"corrispondente valore floating point"),", utilizzando uno dei due metodi di gestione dell'undeflow."),n(L.q,null,O(x())),n(S.a,null,"Indica che un valore è soggetto alla precisione di macchina.",n(L.q,null,O(v()))))),n(L.s,null,n(L.r,{title:"Un nuovo insieme"},n("p",null,"L'insieme ",n(L.i,null,O(_()))," è il sottoinsieme dei numeri reali rappresentabili in floating point dalla macchina che stiamo usando."),n("p",null,"Operazioni tra elementi di ",n(L.i,null,O(m()))," producono risultati in ",n(L.i,null,O(p())),", che però decaderanno nuovamente a elementi di ",n(L.i,null,O(f())),", perdendo informazioni."),n("p",null,"Il teorema della precisione di macchina si applica quindi anche ai risultati delle operazioni.")),n(L.r,{title:"Caratteristiche delle operazioni di macchina"},n("ul",null,n("li",null,"Hanno ",n("b",null,"più elementi neutri"),"."),n("li",null,"Un numero ha ",n("b",null,"più opposti"),"."),n("li",null,n("b",null,"Non")," sono associative."),n("li",null,n("b",null,"Non")," sono distributive."),n("li",null,n("b",null,"Non")," vale la legge di annullamento del prodotto.")))),n(L.s,{title:"Errori nelle operazioni di macchina"},n(L.r,{title:"Errore inerente"},n("p",null,"Errore derivato da underflow sui ",n("b",null,"dati"),"."),n("p",null,"Si indica con ",n(L.i,null,O(d())),"."),n(S.a,null,"L'errore sulla variabile ",n(L.i,null,"x")," si indica con ",n(L.i,null,O(s())),".")),n(L.r,{title:"Errore algoritmico"},n("p",null,"Errore derivato da underflow durante l'",n("b",null,"esecuzione dell'algoritmo"),"."),n("p",null,"Si indica con ",n(L.i,null,O(c())),"."),n(S.a,null,"L'errore al primo passo dell'algoritmo si indica con ",n(L.i,null,O(u())),"."))),n(L.s,null,n(L.r,{title:"Condizionamento"},n("p",null,"Sensibilità di un problema all'",n("b",null,"errore inerente"),"."),n(S.a,null,n(L.i,null,O(a()))," è mal condizionato intorno allo 0 e ben condizionato lontano dallo 0.")),n(L.r,{title:"Stabilità"},n("p",null,"Sensibilità di un problema all'",n("b",null,"errore algoritmico"),"."),n(S.a,null,n("p",null,"Cerchiamo un algoritmo che risolva ",n(L.i,null,O(o())),"."),n("p",null,"Calcolare prima ",n(L.i,null,O(t()))," e poi ",n(L.i,null,O(r()))," porta a una perdita di precisione."),n("p",null,"Calcolare direttamente ",n(L.i,null,O(e()))," non ha alcuna perdita di precisione e rende l'algoritmo ",n("b",null,"più stabile")," del precedente.")))),n(L.s,null,n(L.r,{title:"Indice di condizionamento"},n("p",null,"È il coefficiente di proporzionalità 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n=on(["A = Q cdot \begin{pmatrix} R\\ 0 end{pmatrix}"],["A = Q \\cdot \\begin{pmatrix} R\\\\ 0 \\end{pmatrix}"]);return m=function(){return n},n}function _(){var n=on(["A"]);return _=function(){return n},n}function v(){var n=on(["{color{Red} Q } = | A cdot alpha - y |^2"],["{\\color{Red} Q } = \\| A \\cdot \\alpha - y \\|^2"]);return v=function(){return n},n}function x(){var n=on(["{color{Yellow} q(x) } = A cdot alpha"],["{\\color{Yellow} q(x) } = A \\cdot \\alpha"]);return x=function(){return n},n}function b(){var n=on(["\n alpha =\n \begin{pmatrix}\n a_0\\\\\n a_1\\\\\n \vdots\\\\\n a_{n-1}\n end{pmatrix}\n "],["\n \\alpha =\n \\begin{pmatrix}\n a_0\\\\\\\\\n a_1\\\\\\\\\n \\vdots\\\\\\\\\n a_{n-1}\n \\end{pmatrix}\n "]);return b=function(){return n},n}function g(){var n=on(["alpha"],["\\alpha"]);return g=function(){return n},n}function h(){var n=on(["\n A =\n \begin{pmatrix}\n phi_0(x_1) & phi_1(x_1) & phi_2(x_1) & dots & phi_{n_1}(x_1) \\\\\n phi_0(x_2) & phi_1(x_2) & phi_2(x_2) & 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costruire una ",n("b",null,"funzione di regressione")," che, dati molti più dati del grado della funzione, minimizzi il quadrato della distanza tra i punti sperimentali e i punti della funzione di regressione."),n("p",null,"Denominiamo:"),n("ul",null,n("li",null,n(un.i,null,sn(tn())),": la ",n("b",null,'funzione "effettiva"')," del fenomeno"),n("li",null,n(un.i,null,sn(rn())),": la ",n("b",null,"funzione di regressione")," che costruiamo per approssimarlo"),n("li",null,n(un.i,null,sn(en())),": la ",n("b",null,'funzione "errore di regressione"')," da minimizzare"),n("li",null,n(un.i,null,sn(ln())),": i ",n("b",null,"punti sperimentali"))),n("p",null,"L'obiettivo è minimizzare l'",n("b",null,"errore di approssimazione")," ",n(un.i,null,sn(nn())),", ovvero:"),n(un.q,null,sn($())))),n(un.s,null,n(un.r,{title:"Regressione lineare"},n("p",null,"Trova la ",n("b",null,"retta")," ",n(un.i,null,sn(W()))," che meglio approssima tutti gli ",n(un.i,null,sn(K()))," dati sperimentali."),n("p",null,"Essendo una retta, avrà ",n("b",null,"due parametri"),": il termine noto ",n(un.i,null,sn(Z())),", e la pendenza ",n(un.i,null,"a_1"),"."),n(un.q,null,sn(B())),n("p",null,"L'errore da minimizzare per ricavare i parametri sarà:"),n(un.q,null,sn(X()))),n(un.r,{title:"Regressione lineare matriciale"},n("p",null,"Possiamo costruire una ",n("b",null,"matrice di regressione")," ",n(un.i,null,sn(J()))," contenente tutti i ",n("b",null,"punti sperimentali"),":"),n(un.q,null,sn(V())),n("p",null,"Inoltre, se costruiamo il ",n("b",null,"vettore dei parametri")," ",n(un.i,null,sn(H())),":"),n(un.q,null,sn(j())),n("p",null,"Avremo che:"),n(un.q,null,sn(N())),n("p",null,"Inoltre, potremo calcolare l'errore attraverso la norma:"),n(un.q,null,sn(D())))),n(un.s,null,n(un.r,{title:"Regressione polinomiale"},n("p",null,"Trova il ",n("b",null,"polinomio")," ",n(un.i,null,sn(T()))," di grado ",n(un.i,null,sn(E()))," che meglio approssima tutti gli ",n(un.i,null,sn(U()))," dati sperimentali."),n("p",null,"Essendo un polinomio di grado ",n(un.i,null,sn(P())),", avrà ",n(un.i,null,sn(Q()))," parametri."),n(un.q,null,sn(Y())),n(cn.a,null,n("p",null,"La regressione lineare è un caso particolare di regressione generale in cui i parametri sono 2!")),n("p",null,"L'errore da minimizzare per ricavare i parametri sarà:"),n(un.q,null,sn(F()))),n(un.r,{title:"Regressione polinomiale matriciale"},n("p",null,"Possiamo costruire una ",n("b",null,"matrice di regressione")," ",n(un.i,null,sn(M()))," contenente tutti i ",n("b",null,"punti sperimentali")," a tutti i gradi del polinomio:"),n(un.q,null,sn(I())),n("p",null,"Inoltre, se costruiamo il ",n("b",null,"vettore dei parametri")," ",n(un.i,null,sn(R())),":"),n(un.q,null,sn(C())),n("p",null,"Avremo che:"),n(un.q,null,sn(O())),n("p",null,"Inoltre, potremo calcolare l'errore attraverso la norma:"),n(un.q,null,sn(G())),n(cn.a,null,"Normalmente, i dati sono molti di più, ma se il numero di parametri ",n(un.i,null,sn(S()))," fosse uguale al numero di dati ",n(un.i,null,sn(w())),", allora si otterrebbe il ",n("b",null,"polinomio di interpolazione"),"!"))),n(un.s,null,n(un.r,{title:"Regressione generale"},n("p",null,"Trova i ",n("b",null,"coefficienti della combinazione lineare"),n(un.i,null,sn(y()))," che meglio approssima tutti gli ",n(un.i,null,sn(L()))," dati sperimentali."),n(un.q,null,sn(A())),n(cn.a,null,n("p",null,"La regressione polinomiale è un caso particolare di regressione generale in cui:"),n(un.q,null,sn(k()))),n("p",null,"L'errore da minimizzare per ricavare i parametri sarà:"),n(un.q,null,sn(q()))),n(un.r,{title:"Regressione polinomiale generale"},n("p",null,"Possiamo costruire una ",n("b",null,"matrice di regressione")," ",n(un.i,null,sn(z()))," contenente tutti i ",n("b",null,"punti sperimentali")," a tutti i gradi del polinomio:"),n(un.q,null,sn(h())),n("p",null,"Inoltre, se costruiamo il ",n("b",null,"vettore dei parametri")," ",n(un.i,null,sn(g())),":"),n(un.q,null,sn(b())),n("p",null,"Avremo che:"),n(un.q,null,sn(x())),n("p",null,"Inoltre, potremo calcolare l'errore attraverso la norma:"),n(un.q,null,sn(v())))),n(un.s,{title:"Trovare i parametri"},n(un.r,{title:"Caso non degenere"},n("p",null,"Caso che prevede che le colonne di ",n(un.i,null,sn(_()))," siano ",n("b",null,"linearmente indipendenti"),"."),n("p",null,"La soluzione ",n("b",null,"esiste")," sempre, ed è ",n("b",null,"unica"),"."),n("p",null,"Per trovarla:"),n("ul",null,n("li",null,"Fattorizziamo ",n(un.i,null,sn(m())),"."),n("li",null,"Calcoliamo ",n(un.i,null,sn(p())),"."),n("li",null,"Teniamo solo i primi ",n(un.i,null,"n")," valori di ",n(un.i,null,sn(f()))," e mettiamoli in ",n(un.i,null,sn(d())),"."),n("li",null,"Calcoliamo ",n(un.i,null,sn(s())),"."))),n(un.r,{title:"Caso generale"},n("p",null,"Caso che non preclude alcuna composizione di ",n(un.i,null,sn(c())),"."),n("p",null,"Ci sono ",n("b",null,"infinite")," soluzioni, con ",n(un.i,null,"n-k")," gradi di libertà."),n("p",null,"Si cerca sempre di trovare la ",n("i",null,"soluzione di norma minima"),", che, se ",n(un.i,null,sn(u())),", allora è ",n("b",null,"unica"),"."),n("p",null,"Per trovarla:"),n("ul",null,n("li",null,"Fattorizziamo ",n(un.i,null,sn(a()))," con la ",n("i",null,"fattorizzazione SVD")),n("li",null,"Calcoliamo ",n(un.i,null,sn(o()))),n("li",null,n("p",null,"Calcoliamo:"),n(un.q,null,sn(t()))),n("li",null,"Calcoliamo ",n(un.i,null,sn(r())))),n(cn.a,null,"Gli zeri nella ",n(un.i,null,sn(e()))," sono i gradi di libertà, sono zero in modo che essi diano la norma minima."))))}}).call(this,l("hosL").h)}}]);
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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 {Fragment} from \"preact\";\nimport {ILatex, Panel, PLatex, Section, Todo} from \"bluelib\";\nimport Example from \"../../components/Example\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n Si vuole trovare una funzione in grado di approssimarne un'altra, di cui si conoscono\n però solo alcuni punti.\n
\n \n È utile in un sacco di casi! Ad esempio, quando si vuole scalare un'immagine.\n \n
\n I punti sono detti nodi{r`(x_i, y_i)`}, mentre la funzione costruita su\n di essi è detta interpolante{r`g`}:\n
\n {r`g(x_i) = y_i`}\n
\n Dato un insieme di punti, esistono infinite funzioni interpolanti.\n
\n \n \n
\n Il teorema fondamentale dell'algebra dice che esiste una sola\n interpolante polinomiale che interpola un dato insieme di punti.\n
\n
\n Con n+1 punti, l'interpolante sarà al massimo di grado n, e\n viene detta {r`p_n`}.\n
\n {r`\n b =\n \\begin{pmatrix}\n y_0\\\\\\\\\n y_1\\\\\\\\\n y_2\\\\\\\\\n \\vdots\\\\\\\\\n y_n\n \\end{pmatrix}\n `}\n \n Per trovare il polinomio di interpolazione è sufficiente risolvere il problema!\n \n
\n È efficace perchè una volta calcolati i coefficienti essi valgono per tutti i punti, ma\n ha come svantaggio che la matrice di Vandermonde è spesso malcondizionata.\n
\n \n \n
\n È possibile scrivere il polinomio di interpolazione raccogliendo\n le {r`y`}:\n
\n I polinomi {r`L_k`} sono detti polinomi di Lagrange, e hanno le seguenti\n proprietà:\n
\n
\n
\n Valgono 1 in corrispondenza del nodo con lo stesso\n indice, 0 in corrispondenza dei nodi con indice diverso\n e {r`0 < n < 1`} in tutti gli altri casi.\n\n {r`\n \\begin{cases}\n L_k(x_k) = 1 \\qquad (nel\\ nodo)\\\\\n L_k(x_j) = 0 \\qquad (altri\\ nodi)\n \\end{cases}\n `}
\n È l'errore compiuto durante l'interpolazione.\n
\n
\n Se la funzione f è interpolata da p_n, allora esso varrà:\n
\n {r`R_n(x) = f(x) - p_n(x)`}\n
\n In particolare, è interessante la sua norma a\n infinito, {r`\\| f - p_n \\|_\\infty`}, che corrisponde alla distanza massima tra\n le due funzioni.\n
\n Hanno quindi convergenza lineare ({r`C = \\frac{1}{2}, p = 1`}).\n
\n
\n Il loro criterio di arresto è un numero di iterazioni prefissato che dipende\n dalla tolleranza sull'errore:\n
\n {r`i \\geq \\log_2 \\left( \\frac{b - a}{\\tau} \\right)`}\n \n Dividi l'intervallo {r`[a, b]`} in tante parti grandi quanto la tolleranza.\n L'algoritmo di bisezione ne escluderà metà ad ogni iterazione; la tolleranza sarà raggiunta\n quando rimarrà una parte sola!\n \n \n \n \n \n \n
Finchè non sono state compiute il numero di iterazioni prefissate:\n \n
\n Calcoliamo il punto\n medio dell'intervallo {r`[a_{(n)}, b_{(n)}]`}:\n {r`c_{(n)} = a_{(n)} + \\frac{b_{(n)} - a_{(n)}}{2}`}\n
\n
\n Dividiamo l'intervallo in due parti, separate da {r`c_{(n)}`}:\n
\n
{r`[a_{(n)}, c_{(n)}]`} è la metà sinistra
\n
{r`[c_{(n)}, b_{(n)}]`} è la metà destra
\n
\n
\n
\n Teniamo l'intervallo in cui i valori della funzione ai due estremi sono discordi, e\n rinominiamolo in {r`[a_{(n+1)}, b_{(n+1)}]`}.\n
\n \n
\n \n \n \n \n
Finchè non sono state compiute il numero di iterazioni prefissate:\n \n
\n Calcoliamo l'intersezione tra la retta che congiunge i due estremi\n {r`a_{(n)}, b_{(n)}`} e l'asse X:\n {r`c_{(n)} = b_{(n)} - \\frac{f(b_{(n)})}{\\frac{f(b_{(n)}) - f(a_{(n)})}{b_{(n)} - a_{(n)}}}`}\n
\n
\n Dividiamo l'intervallo in due parti, separate da {r`c_{(n)}`}:\n
\n
{r`[a_{(n)}, c_{(n)}]`} è la parte sinistra
\n
{r`[c_{(n)}, b_{(n)}]`} è la parte destra
\n
\n
\n
\n Teniamo l'intervallo in cui i valori della funzione ai due estremi sono discordi, e\n rinominiamolo in {r`[a_{(n+1)}, b_{(n+1)}]`}.\n
\n \n
\n \n \n \n \n \n
\n Sono metodi iterativi che funzionano in modo molto simile ai metodi iterativi per i\n sistemi lineari, utilizzando una funzione {r`\\phi`} come \"metodo\".\n
\n {r`x = x - \\phi(x) \\cdot f(x)`}\n
\n Che diventa:\n
\n {r`x_{(k+1)} = g( x_{(k)} )`}\n
\n Sfruttano i punti fissi{r`g(x_{(\\star)}) = x_{(\\star)}`} della\n funzione {r`f`} per convergere: \n se {r`\\phi(x)`} non ha zeri, allora i punti fissi coincideranno con\n gli zeri della funzione {r`f`}.\n
\n {r`g(x) = x - \\phi(x) \\cdot f(x)`}\n
\n Si può raggiungere iterativamente ad un punto fisso attraverso la formula:\n
\n {r`x_{(k+1)} = g( x_{(k)} )`}\n
\n Non si conosce in anticipo il numero di iterazioni necessarie per soddisfare la\n tolleranza {r`\\tau`}; ad ogni iterazione, si controlla se la tolleranza è\n soddisfatta:\n
\n
\n
Nella differenza tra due\n iterate: {r`\\frac{\\left| x_{(k+1)} - x_{(k)} \\right|}{\\left| x_{(k+1)} \\right|} \\leq \\tau`}\n
\n
Nel residuo del problema: {r`\\left| f(x_{(k)}) \\right| \\leq \\tau`}\n
\n
\n \n \n
\n Se:\n
\n
\n
\n Tutti i valori restituiti dalla funzione {r`g`} rientrano nel suo stesso\n dominio:\n {r`g : [a, b] \\to [a, b]`}\n
\n
\n
\n La funzione {r`g`} è una contrazione, ovvero restringe\n l'intervallo {r`[a, b]`}:\n
\n {r`\\forall (x, y) \\in [a, b], | g(x) - g(y) | \\leq L \\cdot | x - y |`}\n
\n (dove {r`0 < L < 1`})\n
\n
\n
\n
\n Allora:\n
\n
\n
\n
\n Il punto fisso esiste ed è unico:\n
\n {r`\\exists! x_{(\\star)}`}\n
\n
\n Il metodo delle approssimazioni successive converge per qualsiasi punto di partenza.\n
\n
\n
\n Vale la seguente disequazione di maggiorazione dell'errore:\n
\n Più è piccolo L, più il metodo convergerà in fretta.\n
\n \n L è molto simile al raggio spettrale {r`\\rho(M)`} dei metodi\n iterativi per i sistemi lineari!\n \n \n \n \n \n
\n Sfrutta la continuità delle funzioni per ottenere una convergenza di ordine più alto.\n
\n {r`\\phi (x) = \\frac{1}{f' (x)}`}\n {r`x_{(k+1)} = x_{(k)} - \\frac{ f(x_{(k)}) }{ f'(x_{(k)}) }`}\n \n Geometricamente, corrisponde a prolungare una retta nel\n punto {r`(x_{(k)}, f(x_{(k)}))`} con pendenza {r`f'(x_{(k)})`},\n e prendendo come nuovo punto la sua intersezione con l'asse X e la sua corrispettiva immagine\n nella funzione.\n \n
\n Ha costo computazionale di 2 valutazioni di funzione più 2 valutazioni di derivata.\n
\n
\n Ha convergenza quadratica.\n
\n \n \n
\n È come il metodo di Newton, ma usa il rapporto incrementale, in modo da poter essere\n applicato a funzioni non continue.\n
\n {r`\\phi (x) = \\frac{ 1 }{ \\frac{ f(x_{(k)}) - f(x_{(k-1)}) }{ x_{(k)} - x_{(k-1)} } }`}\n {r`x_{(k+1)} = x_{(k)} - \\frac{ f(x_{(k)}) }{ \\frac{ f(x_{(k)}) - f(x_{(k-1)}) }{ x_{(k)} - x_{(k-1)} } }`}\n \n Geometricamente, corrisponde a costruire una retta che attraversa i\n punti {r`(x_{(k)}, f(x_{(k)}))`} e {r`(x_{(k-1)}, f(x_{(k-1)}))`},\n e prendendo come nuovo punto la sua intersezione con l'asse X e la sua corrispettiva immagine\n nella funzione.\n \n
\n Ha costo computazionale di 3 valutazioni di funzione.\n
\n
\n Ha convergenza superlineare.\n
\n \n \n \n \n
\n È possibile usare questi metodi per approssimare le soluzioni di sistemi non-lineari.\n
\n \n \n \n )\n}\n","import {Fragment} from \"preact\";\nimport {ILatex, Panel, PLatex, Section} from \"bluelib\";\nimport Example from \"../../components/Example\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n Invece che costruire una singola funzione che interpola tutti i punti, per ogni intervallo\n tra due punti (sottointervallo) si costruisce una funzione apposta.\n
\n \n \n \n \n
\n Interpolanti che:\n
\n
\n
sono polinomiali di grado massimo {r`n`}
\n
sono continue fino al grado {r`n - 1`}
\n
connettono {r`m + 2`} punti, e\n hanno {r`m`} sottointervalli\n
\n
hanno funzioni definite appositamente per ogni sottointervallo
\n
\n \n
\n Significa che agli estremi dell'intervallo, i valori di tutte le derivate fino al\n grado {r`n - 1`} devono essere uguali:\n
\n {r`\\forall \\ k \\leq n-1, \\forall \\ i \\in \\{intervalli\\}, \\quad s_i^{(k)} (x_{i+1}) = s_i^{(k)} (x_{i+1})`}\n \n
\n Hanno {r`n + m + 1`} gradi di libertà.\n
\n \n Esistono infinite spline di grado {r`n \\geq 2`}!\n \n \n \n \n \n \n Sono anche dette interpolanti lineari a tratti.\n \n
\n Per ogni sottointervallo, costruiamo una funzione lineare passante per i due estremi:\n
\n {r`s_i(x) = y_i + \\frac{ y_{i + 1} - y_i }{ x_{i + 1} - x_i } \\cdot (x - x_i)`}\n \n È una linea spezzata!\n \n
\n Il loro errore è:\n
\n {r`\\| R \\|_\\infty = \\| f - s \\|_\\infty \\leq \\frac{1}{8} \\cdot \\max_{y \\in [a, b]} \\left| f''(y) \\right| \\cdot \\left( \\max_{i \\in \\{intervalli\\}} (x_{i+1} - x_{i}) \\right)^2`}\n
\n Ha come vantaggi complessità computazionale molto più bassa e l'assenza del\n fenomeno di Runge, ma allo stesso tempo si perde la derivabilità della funzione.\n
\n
\n Non hanno gradi di libertà.\n
\n \n \n
\n Spline con {r`n = 3`}, che soddisfano le seguenti uguaglianze:\n
\n {r`s_i(x) = \\alpha_i + \\beta_i \\ ( x - x_i ) + \\gamma_i \\ ( x - x_i )^2 + \\delta_i \\ ( x - x_i )^3`}\n \n Spesso si indica con {r`h`} la distanza orizzontale tra due punti di un\n sottointervallo.\n \n \n \n \n \n
Negli\n intervalli {r`[x_0, x_2]`} e {r`[x_{m-1}, x_{m+1}]`} si\n presenta obbligatoriamente un polinomio di grado 3.\n
\n
\n
\n È unica.\n
\n \n \n \n \n
\n Tra tutte le funzioni che interpolano dei punti, le tre classi di funzioni sopraelencate sono\n quelle che interpolano la funzione più \"dolcemente\".\n
\n )\n}","import style from \"./Example.less\";\n\nexport default function (props) {\n return (\n
\n {props.children}\n
\n );\n}\n","import {ILatex, Panel, PLatex, Section, Todo} from \"bluelib\";\nimport Link from \"../../components/Link\";\nimport Example from \"../../components/Example\";\nimport {Fragment} from \"preact\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n Dato un sistema di equazioni lineari, si vuole trovare la sua soluzione.\n
\n
\n In forma matriciale, avrà una matrice dei coefficienti{r`A`}, un vettore\n dei termini noti{r`b`} e un vettore delle incognite\n {r`x`}.\n
\n
\n L'equazione matriciale del sistema è:\n
\n {r`A \\cdot x = b`}\n \n \n
\n Il condizionamento della risoluzione di sistemi lineari è:\n
\n {r`\\frac{{\\color{yellow} \\|A\\| \\cdot \\|A^{-1}\\|} \\cdot \\| \\Delta b \\|}{\\| b \\|}`}\n
\n In particolare, è segnato in giallo nella formula il numero di condizionamento:\n
\n Metodi che trovano la soluzione esatta* di un sistema lineare.\n
\n
\n Tipicamente prevedono la fattorizzazione della matrice dei coefficienti in due\n sottomatrici più facili da risolvere.\n
\n
\n Generalmente hanno una complessità temporale {r`O(n^3)`}.\n
\n \n \n
\n Metodi che trovano una soluzione imperfetta* di\n un sistema lineare.\n
\n
\n Tipicamente prevedono l'applicazione ripetuta di un metodo, in base al quale cambia\n la velocità di convergenza alla soluzione.\n
\n
\n Generalmente hanno una complessità temporale {r`O(n^2)`}.\n
\n \n \n \n \n
\n Se la matrice dei coefficienti del sistema è diagonale, allora è possibile trovare la\n soluzione dividendo ogni termine noto per l'unico coefficiente diverso da zero presente\n nella sua riga:\n
\n {r`x_i = \\frac{b_i}{A_{ii}}`}\n \n \n
\n Se la matrice dei coefficienti del sistema è triangolare inferiore o superiore, allora è\n possibile trovare la soluzione effettuando una sostituzione all'avanti oppure\n all'indietro:\n
\n Se tutti i valori sulla diagonale di {r`A`} sono diversi da 0 (eccetto\n l'ultimo) allora è possibile fattorizzarla in due matrici:\n una {r`L`}triangolare inferiore, e una {r`U`}triangolare\n superiore.\n
\n {r`A = L \\cdot U`}\n \n Abbiamo fatto questo metodo in Algebra Lineare, chiamandolo metodo di Gauss.\n \n
\n La matrice {r`L`} è così composta:\n
\n {r`\n \\begin{cases}\n L_{ii} = 1 \\qquad \\qquad (diagonale)\\\\\n L_{ik} = -\\frac{A_{ik}}{A_{kk}} \\qquad (tri.\\ infer.)\n \\end{cases}\n `}\n \n Sono i moltiplicatori usati per rendere annullare il triangolo inferiore!\n \n
\n La matrice {r`U`} è così composta:\n
\n {r`\n \\begin{cases}\n U_{ik} = A_{ik} \\quad se\\ i \\leq k \\quad (tri.\\ super.)\\\\\n U_{ik} = 0 \\qquad se\\ i > k \\quad (tri.\\ infer.)\n \\end{cases}\n `}\n \n È la parte triangolare superiore di {r`A`}!\n \n
\n Il sistema può essere poi risolto applicando due volte il metodo di sostituzione (all'avanti e\n all'indietro):\n
\n {r`\n \\begin{cases}\n L \\cdot y = b\\\\\n U \\cdot x = y\n \\end{cases}\n `}\n
\n È possibile applicare la fattorizzazione {r`LU`} a qualsiasi matrice\n non-singolare permettendo lo scambio (pivoting) delle righe, potenzialmente aumentando\n la stabilità dell'algoritmo.\n
\n \n Abbiamo fatto questo metodo in Algebra Lineare, chiamandolo metodo di Gauss-Jordan!\n \n
\n Alla formula precedente si aggiunge una matrice di permutazione che indica quali righe\n sono state scambiate:\n
\n {r`P \\cdot A = L \\cdot U`}\n
\n Per massimizzare la stabilità, si cerca di usare come perno l'elemento più grande della\n colonna.\n
\n È possibile anche permettere il pivotingsulle colonne per aumentare\n ulteriormente la stabilità dell'algoritmo, a costo di maggiore costo computazionale:\n
\n {r`P \\cdot A \\cdot Q = L \\cdot U`}\n
\n Per massimizzare la stabilità, si cerca di ordinare in modo decrescente la diagonale,\n assicurandoci che il primo perno sia più grande del secondo e così via.\n
\n Si calcola con una complessità computazionale nell'ordine di {r`O(n)`}.\n
\n \n Fattorizzazione {r`QR`}}>\n
\n Metodo che fornisce una maggiore stabilità a costo di una maggiore complessità\n computazionale.\n
\n
\n La matrice {r`A`} viene fattorizzata in due matrici,\n una ortogonale{r`Q`} e una triangolare superiore\n {r`R`}:\n
\n {r`A = Q \\cdot R`}\n
\n Le matrici si ottengono dal prodotto delle trasformazioni di Householder (che concatenate\n formano {r`Q`}) sulla matrice {r`A`} necessarie a trasformarla\n in una matrice triangolare superiore ({r`R`}).\n
\n \n C'è un bell'esempietto qui.\n \n
\n Una volta fattorizzata, il sistema si può risolvere con:\n
\n {r`\n \\begin{cases}\n y = Q^T \\cdot b\\\\\n R \\cdot x = y\n \\end{cases}\n `}\n
\n {r`\n \\begin{cases}\n G = I - M^{-1} \\cdot A\\\\\n c = M^{-1} \\cdot b\n \\end{cases}\n `}\n
\n Allora la formula generale di un sistema lineare può anche essere scritta in questo modo:\n
\n {r`x = G \\cdot x + c`}\n
\n È particolarmente utile perchè ci permette di definire un algoritmo ricorsivo che\n trovi {r`x`}:\n
\n {r`x_{(i+1)} = G \\cdot x_{(i)} + c`}\n
\n {r`G`} è il metodo, e in base ad esso cambiano stabilità e velocità di\n convergenza.\n
\n
\n Ponendo {r`A = M - N`}, la formula può essere scritta anche in questo modo:\n
\n {r`M \\cdot x_{(i+1)} = N \\cdot x_{(i)} + b`}\n
\n Possiamo ottenere alcuni metodi separando A in tre matrici:\n
\n
\n
La parte diagonale {r`D`}
\n
L'opposto del triangolo inferiore {r`E`}
\n
L'opposto del triangolo superiore {r`F`}
\n
\n {r`A = D - E - F`}\n \n \n
\n Un metodo è convergente se e solo se:\n
\n {r`\\rho (M) < 1`}\n
\n (dove {r`\\rho`} è il raggio spettrale, il massimo autovalore della\n matrice)\n
\n
\n Perchè un metodo sia convergente, è sufficiente che:\n
\n {r`\\| M \\| < 1`}\n \n \n \n \n
\n Il metodo di Jacobi si ottiene ponendo:\n
\n {r`\n \\begin{cases}\n M = D\\\\\n N = E + F\n \\end{cases}\n `}\n
\n Spostamenti simultanei: Permette di ottenere ogni componente\n di {r`x`} indipendentemente dagli altri: è parallelizzabile.\n
\n
\n Se la matrice è diagonale dominante, allora il metodo di\n Jacobi converge sicuramente.\n
\n \n \n
\n Il metodo di Gauss-Seidel si ottiene ponendo:\n
\n {r`\n \\begin{cases}\n M = D - E\\\\\n N = F\n \\end{cases}\n `}\n
\n Ha una velocità di convergenza maggiore o uguale rispetto al metodo di Jacobi.\n
\n
\n Spostamenti successivi: Non è parallelizzabile, perchè ogni componente dipende da\n quelle calcolate in precedenza.\n
\n
\n Se la matrice è diagonale dominante, allora il metodo di\n Gauss-Seidel converge sicuramente.\n
\n \n \n \n )\n}\n","import Intro from \"./00_Intro\";\nimport SistemiLineari from \"./01_SistemiLineari\";\nimport ZeriDiFunzione from \"./02_ZeriDiFunzione\";\nimport Interpolazione from \"./03_Interpolazione\";\nimport InterpolazioneATratti from \"./04_InterpolazioneATratti\";\nimport ApprossimazioneDatiSperimentali from \"./05_ApprossimazioneDatiSperimentali\";\n\n\nexport default function () {\n return (\n
\n
Calcolo Numerico
\n \n \n \n \n \n \n
\n )\n}\n","import {ILatex, Panel, PLatex, Section} from \"bluelib\";\nimport Link from \"../../components/Link\";\nimport MenuList from \"../../components/MenuList\";\nimport Example from \"../../components/Example\";\nimport {Fragment} from \"preact\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n
Prof.ssa Silvia Bonettini
\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 \n
\n Prima di iniziare a studiare Calcolo Numerico, potrebbe essere una buona idea ripassare un\n pochino Algebra Lineare:\n
\n Metodo con cui gestire gli underflow floating point: se la cifra più significativa di\n quelle che devono essere rimosse è 1, allora aumenta di 1 anche quella meno signficativa\n che viene tenuta.\n
\n Un numero reale rappresentato in virgola mobile ha un errore relativo minore o\n uguale alla precisione\n di macchina:\n
\n
\n {r`\\epsilon_x \\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\n troncamento oppure a {r`\\frac{1}{2}`} se viene rappresentato per\n arrotondamento.\n
\n
\n \n \n
\n Associa un valore reale al suo corrispondente valore floating point, utilizzando uno dei\n 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\n floating point dalla macchina che stiamo usando.\n
\n
\n Operazioni tra elementi di {r`\\mathbb{F}`} producono risultati\n in {r`\\mathbb{R}`}, che però decaderanno nuovamente a elementi\n 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 elementi neutri.
\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_{nome\\_var}`}.\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_{num\\_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\n 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^\\star = 4`}.\n
\n
\n Calcolare prima {r`t = fl \\left( \\frac{1}{4} \\right)`} e\n 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\n 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","import {Fragment} from \"preact\";\nimport {ILatex, Panel, PLatex, Section} from \"bluelib\";\nimport Example from \"../../components/Example\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n Interpolare dati sperimentali non fornisce quasi mai un modello del fenomeno.\n
\n
\n Vogliamo costruire una funzione di regressione che, dati molti più dati del grado della\n funzione, minimizzi il quadrato della distanza tra i punti sperimentali e i punti della funzione\n di regressione.\n
\n
\n Denominiamo:\n
\n
\n
{r`{\\color{Orange} f}`}: la funzione \"effettiva\" del fenomeno
\n
{r`{\\color{Yellow} q}`}: la funzione di regressione che costruiamo\n per approssimarlo\n
\n
{r`{\\color{Red} Q }`}: la funzione \"errore di regressione\" da\n minimizzare\n
\n
{r`(\\ x_i, f(x_i)\\ )`}: i punti sperimentali
\n
\n
\n L'obiettivo è minimizzare l'errore di approssimazione{r`Q`}, ovvero:\n
\n {r`{\\color{Yellow} q(x) } = A \\cdot \\alpha`}\n
\n Inoltre, potremo calcolare l'errore attraverso la norma:\n
\n {r`{\\color{Red} Q } = \\| A \\cdot \\alpha - y \\|^2`}\n \n Normalmente, i dati sono molti di più, ma se il numero di\n parametri {r`n`} fosse uguale al numero di dati {r`m`}, allora\n si otterrebbe il polinomio di interpolazione!\n \n \n \n \n \n
\n Trova i coefficienti della combinazione lineare\n {r`{\\color{Yellow} q}`} che meglio approssima tutti\n gli {r`m`} dati sperimentali.\n
\n \n Gli zeri nella {r`\\gamma`} sono i gradi di libertà, sono zero in modo che essi\n diano la norma minima.\n \n \n \n \n )\n}\n"],"sourceRoot":""}
\ No newline at end of file
diff --git a/docs/route-CalcoloNumerico.chunk.feb65.esm.js b/docs/route-CalcoloNumerico.chunk.feb65.esm.js
new file mode 100644
index 0000000..dabc1f0
--- /dev/null
+++ b/docs/route-CalcoloNumerico.chunk.feb65.esm.js
@@ -0,0 +1,207 @@
+(window.webpackJsonp=window.webpackJsonp||[]).push([[5],{"2w3n":function(l){l.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"}},"31Ft":function(l,i,n){"use strict";(function(l){var e=n("hosL"),a=n("mbOI"),o=n("ke5e");let t,r,u,s,c,d,p,m,_,b,f,g,z,h,x,v,q,k,L,A,S,y,C,w,I,O,G,R=l=>l;const F=String.raw;i.a=function(){return l(e.Fragment,null,l(a.s,{title:"Problema: Interpolazione"},l(a.r,{title:"Descrizione"},l("p",null,"Si vuole trovare una funzione in grado di ",l("b",null,"approssimarne")," un'altra, di cui si conoscono però solo alcuni punti."),l(o.a,null,"È utile in un sacco di casi! Ad esempio, quando si vuole scalare un'immagine."),l("p",null,"I punti sono detti ",l("b",null,"nodi")," ",l(a.i,null,F(t||(t=R`(x_i, y_i)`))),", mentre la funzione costruita su di essi è detta ",l("b",null,"interpolante")," ",l(a.i,null,F(r||(r=R`g`))),":"),l(a.q,null,F(u||(u=R`g(x_i) = y_i`))),l("p",null,"Dato un insieme di punti, esistono ",l("b",null,"infinite")," funzioni interpolanti.")),l(a.r,{title:"Interpolazione polinomiale"},l("p",null,"Il ",l("u",null,"teorema fondamentale dell'algebra")," dice che ",l("b",null,"esiste una sola interpolante ",l("i",null,"polinomiale"))," che interpola un dato insieme di punti."),l("p",null,"Con ",l(a.i,null,"n+1")," punti, l'interpolante sarà al massimo di grado ",l(a.i,null,"n"),", e viene detta ",l(a.i,null,F(s||(s=R`p_n`))),"."),l("p",null,"La sua ",l("b",null,"forma canonica")," sarà:"),l(a.q,null,F(c||(c=R`p_n(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \dots + a_n x^n`))))),l(a.s,{title:"Metodi di interpolazione"},l(a.r,{title:"Metodo dei coefficienti indeterminati"},l("p",null,"È possibile scrivere la forma canonica come ",l("b",null,"matrice"),":"),l(a.q,null,F(d||(d=R`A \cdot x = b`))),l("p",null,"Costruiamo la ",l("b",null,"matrice di Vandermonde"),":"),l(a.q,null,F(p||(p=R`
+ A =
+ \begin{pmatrix}
+ 1 & x_0 & x_0^2 & \dots & x_0^n\\\\
+ 1 & x_1 & x_1^2 & \dots & x_1^n\\\\
+ 1 & x_2 & x_2^2 & \dots & x_2^n\\\\
+ \vdots & \vdots & \vdots & \ddots & \vdots\\\\
+ 1 & x_n & x_n^2 & \dots & x_n^n
+ \end{pmatrix}
+ `))),l("p",null,"Costruiamo il ",l("b",null,"vettore delle incognite"),":"),l(a.q,null,F(m||(m=R`
+ x =
+ \begin{pmatrix}
+ a_0\\\\
+ a_1\\\\
+ a_2\\\\
+ \vdots\\\\
+ a_n
+ \end{pmatrix}
+ `))),l("p",null,"Costruiamo il ",l("b",null,"vettore dei termini noti"),":"),l(a.q,null,F(_||(_=R`
+ b =
+ \begin{pmatrix}
+ y_0\\\\
+ y_1\\\\
+ y_2\\\\
+ \vdots\\\\
+ y_n
+ \end{pmatrix}
+ `))),l(o.a,null,"Per trovare il polinomio di interpolazione è sufficiente risolvere il problema!"),l("p",null,"È efficace perchè una volta calcolati i coefficienti essi ",l("b",null,"valgono per tutti i punti"),", ma ha come svantaggio che la matrice di Vandermonde è ",l("b",null,"spesso malcondizionata."))),l(a.r,{title:"Metodo di Lagrange"},l("p",null,"È possibile scrivere il polinomio di interpolazione ",l("b",null,"raccogliendo le ",l(a.i,null,F(b||(b=R`y`)))),":"),l(a.q,null,F(f||(f=R`p_n (x) = y_0 L_0 + y_1 L_1 + y_2 L_2 + \dots + y_n L_n`))),l("p",null,"I polinomi ",l(a.i,null,F(g||(g=R`L_k`)))," sono detti ",l("b",null,"polinomi di Lagrange"),", e hanno le seguenti proprietà:"),l("ul",null,l("li",null,"Valgono ",l(a.i,null,"1")," in corrispondenza del nodo con lo stesso indice, ",l(a.i,null,"0")," in corrispondenza dei nodi con indice diverso e ",l(a.i,null,F(z||(z=R`0 < n < 1`)))," in tutti gli altri casi.",l(a.q,null,F(h||(h=R`
+ \begin{cases}
+ L_k(x_k) = 1 \qquad (nel\ nodo)\\
+ L_k(x_j) = 0 \qquad (altri\ nodi)
+ \end{cases}
+ `)))),l("li",null,"Si compongono con questo prodotto:",l(a.q,null,F(x||(x=R`L_k = \frac{(x - x_0) \cdot \dots \cdot (x - x_{k-1}) \cdot (x - x_{k+1}) \cdot \dots \cdot (x_k - x_n)}{(x_k - x_0) \cdot \dots \cdot (x_k - x_{k-1}) \cdot (x_k - x_{k+1}) \cdot \dots \cdot (x_k - x_n)}`))))),l(o.a,null,"Non c'è il termine con ",l(a.i,null,F(v||(v=R`x_k`))),"!"),l("p",null,"Tutti insieme formano la ",l("b",null,"base di Lagrange"),"."),l(o.a,null,"Si chiama base perchè sono ",l("b",null,"linearmente indipendenti"),"!"),l("p",null,"Questo metodo permette di calcolare il valore del polinomio di interpolazione ",l("b",null,"in un singolo punto"),":"),l(o.a,null,l("p",null,"Si può risparmiare tempo di calcolo calcolando una singola volta il numeratore con ",l("i",null,"tutti")," i termini:"),l(a.q,null,F(q||(q=R`\omega_n = (x - x_0) \cdot (x - x_1) \cdot \dots \cdot (x - x_n)`))),l("p",null,"E poi dividendo per il termine che andrebbe escluso:"),l(a.q,null,F(k||(k=R`L_k(x) = \frac{ \omega_n }{ (x - x_k) \cdot \prod_{i=0, i \neq k} (x_k - x_i) }`)))),l("p",null,"Ha costo computazionale ",l(a.i,null,F(L||(L=R`O(n^2)`))),"."))),l(a.s,{title:"Resto di interpolazione"},l(a.r,{title:"Definizione"},l("p",null,"È l'",l("b",null,"errore compiuto durante l'interpolazione"),"."),l("p",null,"Se la funzione ",l(a.i,null,"f")," è interpolata da ",l(a.i,null,"p_n"),", allora esso varrà:"),l(a.q,null,F(A||(A=R`R_n(x) = f(x) - p_n(x)`))),l("p",null,"In particolare, è interessante la sua norma a infinito, ",l(a.i,null,F(S||(S=R`\| f - p_n \|_\infty`))),", che corrisponde alla distanza massima tra le due funzioni."),l("p",null,"Un teorema dice che esso è uguale a:"),l(a.q,null,F(y||(y=R`R_n(x) = \frac{ \omega_n(x) }{ (n + 1)! } \cdot f^{(n+1)}(\xi)`)))),l(a.r,{title:"Stima"},l("p",null,"L'errore nell'interpolazione dipende principalmente da due fattori:"),l("ul",null,l("li",null,"Come sono ",l("b",null,"distribuiti sull'asse X")," i punti da interpolare"),l("li",null,"Il grado del polinomio di interpolazione")))),l(a.s,null,l(a.r,{title:"Fenomeno di Runge"},l("p",null,"Fenomeno che si verifica cercando di interpolare la ",l("i",null,"funzione di Runge")," (",l(a.i,null,F(C||(C=R`\frac{1}{1 + 25x^2}`))),")."),l("p",null,"Scegliendo ",l("b",null,"nodi equispaziati"),", l'errore di interpolazione sarà ",l("b",null,"enorme")," vicino ai due estremi dell'intervallo."),l(o.a,null,"Addirittura, più nodi verranno scelti, più esso sarà alto!"),l("p",null,"Si evita scegliendo i nodi in una maniera diversa.")),l(a.r,{title:"Nodi di Chebychev"},l("p",null,"Nodi ottenuti partizionando una ",l("b",null,"semicirconferenza"),", e proiettando le partizioni sul diametro."),l("p",null,"La formula usata per ottenere ",l(a.i,null,F(w||(w=R`n`)))," punti è:"),l(a.q,null,F(I||(I=R`x_i = \cos \left( \frac{ (2 \cdot i + 1) \cdot \pi }{ 2 \cdot (n+1) } \right)`))),l("p",null,l("u",null,"Proprietà di min-max"),": sono la ",l("b",null,"scelta ottimale")," dei punti di interpolazione."),l(a.q,null,F(O||(O=R`\omega_n(\star) = \max_{x \in [a, b]} \left| \omega_n(x) \right|`))),l("p",null,"In particolare, si ha che:"),l(a.q,null,F(G||(G=R`\omega_n(\star) = 2 \left( \frac{b-a}{4} \right)^{n+1}`))))))}}).call(this,n("hosL").h)},"5aVd":function(l){l.exports={menulist:"menulist__2Cmnq"}},FEtp:function(l,i,n){"use strict";(function(l){var e=n("hosL"),a=n("mbOI"),o=n("ke5e");let t,r,u,s,c,d,p,m,_,b,f,g,z,h,x,v,q,k,L,A,S,y,C,w,I,O,G,R,F,P,M,E,U,Q,N,T,D,Y,H,j,V,J,X,B,Z,K,W,$,ll,il,nl,el,al,ol,tl,rl,ul,sl,cl=l=>l;const dl=String.raw;i.a=function(){return l(e.Fragment,null,l(a.s,{title:"Problema: Ricerca degli zeri di funzione"},l(a.r,{title:"Descrizione"},l("p",null,"Si vogliono trovare i punti (",l("i",null,"zeri"),") in cui una funzione ",l("b",null,"continua")," ",l(a.i,null,"f : [a, b] \\to R")," vale ",l(a.i,null,"0"),"."),l("p",null,"Per il ",l("b",null,"teorema del valore medio"),", se ",l(a.i,null,dl(t||(t=cl`f(a) \cdot f(b) \leq 0`))),", allora esiste sicuramente un punto in cui la funzione vale 0."),l("p",null,"Denominiamo il punto in cui la funzione vale ",l(a.i,null,"0")," come ",l(a.i,null,dl(r||(r=cl`x_{(\star)}`))),".")),l(a.r,{title:"Condizionamento"},l("p",null,"Più la ",l("b",null,"derivata prima")," della funzione ",l("b",null,"si avvicina allo 0"),", ",l("b",null,"peggio")," il problema sarà condizionato."),l(a.q,null,dl(u||(u=cl`f'(x_{(\star)}) \simeq 0 \implies mal\ condizionato`))))),l(a.s,null,l(a.r,{title:"Ordine di convergenza"},l("p",null,"Indice ",l(a.i,null,dl(s||(s=cl`{\color{Orange} p}`)))," di quanto in fretta una successione converge alla soluzione."),l(a.q,null,dl(c||(c=cl`\lim_{i \to +\infty} \frac{ \left| x_{(i+1)} - x_{(\star)} \right| }{ \left| x_{(k)} - x_{(\star)} \right|^{\color{Orange} p}}`))),l("ul",null,l("li",null,l("u",null,"Convergenza lineare"),": ",l(a.i,null,dl(d||(d=cl`p = 1`)))," e ",l(a.i,null,dl(p||(p=cl`0 < C < 1`)))),l("li",null,l("u",null,"Convergenza superlineare"),": ",l(a.i,null,dl(m||(m=cl`p = 1`)))," e ",l(a.i,null,dl(_||(_=cl`C = 0`)))),l("li",null,l("u",null,"Convergenza quadratica"),": ",l(a.i,null,dl(b||(b=cl`p = 2`)))," e ",l(a.i,null,dl(f||(f=cl`0 < C < 1`)))),l("li",null,l("u",null,"Convergenza superquadratica"),": ",l(a.i,null,dl(g||(g=cl`p = 2`)))," e ",l(a.i,null,dl(z||(z=cl`C = 0`)))),l("li",null,"...")))),l(a.s,{title:"Metodi dicotomici"},l(a.r,{title:"Cosa sono?"},l("p",null,"Sono ",l("b",null,"metodi iterativi")," in grado di ridurre sempre di più l'intervallo in cui è definita la funzione, facendolo convergere allo zero desiderato."),l("p",null,"Alcuni di essi sono il ",l("i",null,"metodo dicotomico")," e il ",l("i",null,"metodo regula falsi"),"."),l("p",null,"Richiedono ",l("b",null,"una valutazione di funzione non-lineare")," ad ogni iterazione."),l("p",null,"Ad ogni iterazione, l'intervallo viene sempre ",l("i",null,"almeno")," ",l("b",null,"dimezzato"),"; si ha, pertanto, che:"),l(a.q,null,dl(h||(h=cl`b_{(i)} - a_{(i)} = \frac{b - a}{2^{i - 1}}`))),l("p",null,"Hanno quindi ",l("b",null,"convergenza lineare")," (",l(a.i,null,dl(x||(x=cl`C = \frac{1}{2}, p = 1`))),")."),l("p",null,"Il loro ",l("i",null,"criterio di arresto")," è un ",l("b",null,"numero di iterazioni prefissato")," che dipende dalla ",l("b",null,"tolleranza")," sull'errore:"),l(a.q,null,dl(v||(v=cl`i \geq \log_2 \left( \frac{b - a}{\tau} \right)`))),l(o.a,null,"Dividi l'intervallo ",l(a.i,null,dl(q||(q=cl`[a, b]`)))," in tante parti grandi quanto la tolleranza. L'algoritmo di bisezione ne escluderà metà ad ogni iterazione; la tolleranza sarà raggiunta quando rimarrà una parte sola!"))),l(a.s,null,l(a.r,{title:"Metodo di bisezione"},l("ol",null,l("li",null,"Finchè non sono state compiute il numero di iterazioni prefissate:",l("ol",null,l("li",null,"Calcoliamo il ",l("b",null,"punto medio")," dell'intervallo ",l(a.i,null,dl(k||(k=cl`[a_{(n)}, b_{(n)}]`))),":",l(a.q,null,dl(L||(L=cl`c_{(n)} = a_{(n)} + \frac{b_{(n)} - a_{(n)}}{2}`)))),l("li",null,"Dividiamo l'intervallo in due parti, separate da ",l(a.i,null,dl(A||(A=cl`c_{(n)}`))),":",l("ul",null,l("li",null,l(a.i,null,dl(S||(S=cl`[a_{(n)}, c_{(n)}]`)))," è la ",l("b",null,"metà")," sinistra"),l("li",null,l(a.i,null,dl(y||(y=cl`[c_{(n)}, b_{(n)}]`)))," è la ",l("b",null,"metà")," destra"))),l("li",null,"Teniamo l'intervallo in cui i valori della funzione ai due estremi sono discordi, e rinominiamolo in ",l(a.i,null,dl(C||(C=cl`[a_{(n+1)}, b_{(n+1)}]`))),"."))))),l(a.r,{title:"Metodo regula falsi"},l("ol",null,l("li",null,"Finchè non sono state compiute il numero di iterazioni prefissate:",l("ol",null,l("li",null,"Calcoliamo l'",l("b",null,"intersezione")," tra la ",l("b",null,"retta che congiunge i due estremi"),l(a.i,null,dl(w||(w=cl`a_{(n)}, b_{(n)}`)))," e l'",l("b",null,"asse X"),":",l(a.q,null,dl(I||(I=cl`c_{(n)} = b_{(n)} - \frac{f(b_{(n)})}{\frac{f(b_{(n)}) - f(a_{(n)})}{b_{(n)} - a_{(n)}}}`)))),l("li",null,"Dividiamo l'intervallo in due parti, separate da ",l(a.i,null,dl(O||(O=cl`c_{(n)}`))),":",l("ul",null,l("li",null,l(a.i,null,dl(G||(G=cl`[a_{(n)}, c_{(n)}]`)))," è la parte sinistra"),l("li",null,l(a.i,null,dl(R||(R=cl`[c_{(n)}, b_{(n)}]`)))," è la parte destra"))),l("li",null,"Teniamo l'intervallo in cui i valori della funzione ai due estremi sono discordi, e rinominiamolo in ",l(a.i,null,dl(F||(F=cl`[a_{(n+1)}, b_{(n+1)}]`))),".")))))),l(a.s,{title:"Metodo delle approssimazioni successive"},l(a.r,{title:"Metodi delle approssimazioni successive"},l("p",null,"Sono ",l("b",null,"metodi iterativi")," che funzionano in modo molto simile ai metodi iterativi per i sistemi lineari, utilizzando una funzione ",l(a.i,null,dl(P||(P=cl`\phi`))),' come "metodo".'),l(a.q,null,dl(M||(M=cl`x = x - \phi(x) \cdot f(x)`))),l("p",null,"Che diventa:"),l(a.q,null,dl(E||(E=cl`x_{(k+1)} = g( x_{(k)} )`))),l("p",null,"Sfruttano i ",l("b",null,"punti fissi")," ",l(a.i,null,dl(U||(U=cl`g(x_{(\star)}) = x_{(\star)}`)))," della funzione ",l(a.i,null,dl(Q||(Q=cl`f`)))," per convergere:",l("br",null),"se ",l(a.i,null,dl(N||(N=cl`\phi(x)`)))," non ha zeri, allora i punti fissi ",l("b",null,"coincideranno")," con gli ",l("b",null,"zeri")," della funzione ",l(a.i,null,dl(T||(T=cl`f`))),"."),l(a.q,null,dl(D||(D=cl`g(x) = x - \phi(x) \cdot f(x)`))),l("p",null,"Si può raggiungere iterativamente ad un punto fisso attraverso la formula:"),l(a.q,null,dl(Y||(Y=cl`x_{(k+1)} = g( x_{(k)} )`))),l("p",null,"Non si conosce in anticipo il numero di iterazioni necessarie per soddisfare la tolleranza ",l(a.i,null,dl(H||(H=cl`\tau`))),"; ad ogni iterazione, si controlla se la tolleranza è soddisfatta:"),l("ul",null,l("li",null,"Nella differenza tra due iterate: ",l(a.i,null,dl(j||(j=cl`\frac{\left| x_{(k+1)} - x_{(k)} \right|}{\left| x_{(k+1)} \right|} \leq \tau`)))),l("li",null,"Nel ",l("i",null,"residuo")," del problema: ",l(a.i,null,dl(V||(V=cl`\left| f(x_{(k)}) \right| \leq \tau`)))))),l(a.r,{title:"Teorema della mappa contrattiva"},l("p",null,"Se:"),l("ul",null,l("li",null,"Tutti i valori restituiti dalla funzione ",l(a.i,null,dl(J||(J=cl`g`)))," rientrano nel suo stesso dominio:",l(a.q,null,dl(X||(X=cl`g : [a, b] \to [a, b]`)))),l("li",null,l("p",null,"La funzione ",l(a.i,null,dl(B||(B=cl`g`)))," è una contrazione, ovvero restringe l'intervallo ",l(a.i,null,dl(Z||(Z=cl`[a, b]`))),":"),l(a.q,null,dl(K||(K=cl`\forall (x, y) \in [a, b], | g(x) - g(y) | \leq L \cdot | x - y |`))),l("p",null,"(dove ",l(a.i,null,dl(W||(W=cl`0 < L < 1`))),")"))),l("p",null,"Allora:"),l("ul",null,l("li",null,l("p",null,"Il punto fisso esiste ed è unico:"),l(a.q,null,dl($||($=cl`\exists! x_{(\star)}`)))),l("li",null,"Il metodo delle approssimazioni successive converge per qualsiasi punto di partenza."),l("li",null,l("p",null,"Vale la seguente disequazione di ",l("i",null,"maggiorazione dell'errore"),":"),l(a.q,null,dl(ll||(ll=cl`\left| x_{(k)} - x_{(\star)} \right| \leq \frac{ L^k }{ 1 - L } \cdot \left| x_{(1)} - x_{(0)} \right|`))))),l("p",null,"Più è piccolo ",l(a.i,null,"L"),", più il metodo convergerà in fretta."),l(o.a,null,l(a.i,null,"L")," è molto simile al raggio spettrale ",l(a.i,null,dl(il||(il=cl`\rho(M)`)))," dei metodi iterativi per i sistemi lineari!"))),l(a.s,null,l(a.r,{title:"Metodo di Newton"},l("p",null,"Sfrutta la ",l("b",null,"continuità")," delle funzioni per ottenere una convergenza di ordine più alto."),l(a.q,null,dl(nl||(nl=cl`\phi (x) = \frac{1}{f' (x)}`))),l(a.q,null,dl(el||(el=cl`x_{(k+1)} = x_{(k)} - \frac{ f(x_{(k)}) }{ f'(x_{(k)}) }`))),l(o.a,null,"Geometricamente, corrisponde a prolungare una retta nel punto ",l(a.i,null,dl(al||(al=cl`(x_{(k)}, f(x_{(k)}))`)))," con pendenza ",l(a.i,null,dl(ol||(ol=cl`f'(x_{(k)})`))),", e prendendo come nuovo punto la sua intersezione con l'asse X e la sua corrispettiva immagine nella funzione."),l("p",null,"Ha costo computazionale di ",l("b",null,"2 valutazioni di funzione")," più ",l("b",null,"2 valutazioni di derivata"),"."),l("p",null,"Ha ",l("b",null,"convergenza quadratica"),".")),l(a.r,{title:"Metodo delle secanti"},l("p",null,"È come il metodo di Newton, ma usa il ",l("b",null,"rapporto incrementale"),", in modo da poter essere applicato a funzioni non continue."),l(a.q,null,dl(tl||(tl=cl`\phi (x) = \frac{ 1 }{ \frac{ f(x_{(k)}) - f(x_{(k-1)}) }{ x_{(k)} - x_{(k-1)} } }`))),l(a.q,null,dl(rl||(rl=cl`x_{(k+1)} = x_{(k)} - \frac{ f(x_{(k)}) }{ \frac{ f(x_{(k)}) - f(x_{(k-1)}) }{ x_{(k)} - x_{(k-1)} } }`))),l(o.a,null,"Geometricamente, corrisponde a costruire una retta che attraversa i punti ",l(a.i,null,dl(ul||(ul=cl`(x_{(k)}, f(x_{(k)}))`)))," e ",l(a.i,null,dl(sl||(sl=cl`(x_{(k-1)}, f(x_{(k-1)}))`))),", e prendendo come nuovo punto la sua intersezione con l'asse X e la sua corrispettiva immagine nella funzione."),l("p",null,"Ha costo computazionale di ",l("b",null,"3 valutazioni di funzione"),"."),l("p",null,"Ha ",l("b",null,"convergenza superlineare"),"."))),l(a.s,null,l(a.r,{title:"Approssimare sistemi non-lineari"},l("p",null,"È possibile usare questi metodi per ",l("b",null,"approssimare le soluzioni di sistemi non-lineari"),"."))))}}).call(this,n("hosL").h)},LHkx:function(l,i,n){"use strict";(function(l){var e=n("hosL"),a=n("mbOI"),o=n("ke5e");let t,r,u,s,c,d,p,m,_,b,f,g,z,h,x,v,q,k,L,A,S,y,C,w,I,O,G,R,F,P,M=l=>l;const E=String.raw;i.a=function(){return l(e.Fragment,null,l(a.s,{title:"Problema: Interpolazione a tratti"},l(a.r,{title:"Come funziona?"},l("p",null,"Invece che costruire una singola funzione che interpola tutti i punti, per ",l("b",null,"ogni intervallo tra due punti")," (",l("i",null,"sottointervallo"),") si costruisce ",l("b",null,"una funzione apposta"),"."))),l(a.s,{title:"Splines"},l(a.r,{title:"Cosa sono?"},l("p",null,"Interpolanti che:"),l("ul",null,l("li",null,"sono ",l("b",null,"polinomiali")," di grado massimo ",l(a.i,null,E(t||(t=M`n`)))),l("li",null,"sono ",l("b",null,"continue")," fino al grado ",l(a.i,null,E(r||(r=M`n - 1`)))),l("li",null,"connettono ",l(a.i,null,E(u||(u=M`m + 2`)))," punti, e hanno ",l(a.i,null,E(s||(s=M`m`)))," sottointervalli"),l("li",null,"hanno funzioni ",l("b",null,"definite appositamente")," per ogni sottointervallo")),l(o.a,null,l("p",null,"Significa che agli estremi dell'intervallo, i valori di tutte le derivate fino al grado ",l(a.i,null,E(c||(c=M`n - 1`)))," devono essere uguali:"),l(a.q,null,E(d||(d=M`\forall \ k \leq n-1, \forall \ i \in \{intervalli\}, \quad s_i^{(k)} (x_{i+1}) = s_i^{(k)} (x_{i+1})`)))),l("p",null,"Hanno ",l(a.i,null,E(p||(p=M`n + m + 1`)))," gradi di libertà."),l(o.a,null,"Esistono infinite spline di grado ",l(a.i,null,E(m||(m=M`n \geq 2`))),"!"))),l(a.s,null,l(a.r,{title:"Spline lineari"},l(o.a,null,"Sono anche dette ",l("b",null,"interpolanti lineari a tratti"),"."),l("p",null,"Per ogni sottointervallo, costruiamo una ",l("b",null,"funzione lineare")," passante per i due estremi:"),l(a.q,null,E(_||(_=M`s_i(x) = y_i + \frac{ y_{i + 1} - y_i }{ x_{i + 1} - x_i } \cdot (x - x_i)`))),l(o.a,null,"È una linea spezzata!"),l("p",null,"Il loro errore è:"),l(a.q,null,E(b||(b=M`\| R \|_\infty = \| f - s \|_\infty \leq \frac{1}{8} \cdot \max_{y \in [a, b]} \left| f''(y) \right| \cdot \left( \max_{i \in \{intervalli\}} (x_{i+1} - x_{i}) \right)^2`))),l("p",null,"Ha come vantaggi complessità computazionale ",l("b",null,"molto più bassa")," e l'",l("b",null,"assenza")," del fenomeno di Runge, ma allo stesso tempo si ",l("b",null,"perde la derivabilità della funzione.")),l("p",null,l("b",null,"Non")," hanno gradi di libertà.")),l(a.r,{title:"Spline cubiche"},l("p",null,"Spline con ",l(a.i,null,E(f||(f=M`n = 3`))),", che soddisfano le seguenti uguaglianze:"),l(a.q,null,E(g||(g=M`
+ \forall \ i \in \{0,\ \dots\ ,\ m - 1\},\
+ \begin{cases}
+ s_i (x_{i+1}) = s_{i+1} (x_{i+1})\\\\
+ s'_i (x_{i+1}) = s'_{i+1} (x_{i+1})\\\\
+ s''_i (x_{i+1}) = s''_{i+1} (x_{i+1})
+ \end{cases}
+ `))),l(a.q,null,E(z||(z=M`
+ \forall \ i \in \{0,\ \dots\ ,\ m + 1\},\
+ \begin{cases}
+ s_i(x_i) = y_i
+ \end{cases}
+ `))),l("p",null,"Esse hanno la seguente equazione:"),l(a.q,null,E(h||(h=M`s_i(x) = \alpha_i + \beta_i \ ( x - x_i ) + \gamma_i \ ( x - x_i )^2 + \delta_i \ ( x - x_i )^3`))),l(o.a,null,"Spesso si indica con ",l(a.i,null,E(x||(x=M`h`)))," la distanza orizzontale tra due punti di un sottointervallo."))),l(a.s,null,l(a.r,{title:"Spline cubica vincolata"},l("p",null,"Classe di spline cubiche in cui:"),l("ul",null,l("li",null,l(a.i,null,E(v||(v=M`\beta_0`)))," e ",l(a.i,null,E(q||(q=M`\beta_{m+1}`)))," sono prefissati")),l("p",null,"È ",l("b",null,"unica"),"."),l("p",null,"Forma il seguente sistema di equazioni:"),l(a.q,null,E(k||(k=M`T z = c`))),l(a.q,null,E(L||(L=M`
+ b_i = h_{i+1} \beta_i + 2 ( h_i + h_{i+1} ) + h_{i} \beta_i+2
+ `))),l(a.q,null,E(A||(A=M`
+ T =
+ \begin{pmatrix}
+ 2 (h_0 + 2 h_1) & h_0 & {\color{Gray} 0} & {\color{Gray} 0} & {\color{Gray} 0} \\\\
+ h_2 & 2 (h_1 + h_2) & h_1 & {\color{Gray} 0} & {\color{Gray} 0} \\\\
+ {\color{Gray} 0} & \ddots & \ddots & \ddots & {\color{Gray} 0} \\\\
+ {\color{Gray} 0} & {\color{Gray} 0} & h_{m-1} & 2 (h_{m-2} + h_{m-1}) & h_{m-2} \\\\
+ {\color{Gray} 0} & {\color{Gray} 0} & {\color{Gray} 0} & h_m & 2 (h_{m-1} + h_m)
+ \end{pmatrix}
+ `))),l(a.q,null,E(S||(S=M`
+ z =
+ \begin{pmatrix}
+ \beta_1\\\\
+ \beta_2\\\\
+ \vdots\\\\
+ \beta_{m-1}\\\\
+ \beta_{m}
+ \end{pmatrix}
+ `))),l(a.q,null,E(y||(y=M`
+ c =
+ \begin{pmatrix}
+ b_0 - h_1 \beta_0\\\\
+ b_1\\\\
+ \vdots\\\\
+ b_{m-2}\\\\
+ b_{m-1} - h_{m-1} \beta_{m+1}
+ \end{pmatrix}
+ `))))),l(a.s,null,l(a.r,{title:"Spline cubica naturale"},l("p",null,"Classe di spline cubiche in cui:"),l("ul",null,l("li",null,l(a.i,null,E(C||(C=M`s''(x_0) = s''(x_{m+1}) = 0`))))),l("p",null,"È ",l("b",null,"unica"),".")),l(a.r,{title:"Spline cubica periodica"},l("p",null,"Classe di spline cubiche in cui:"),l("ul",null,l("li",null,l(a.i,null,E(w||(w=M`s(x) = s(m+1)`)))),l("li",null,l(a.i,null,E(I||(I=M`s'(x) = s'(m+1)`)))),l("li",null,l(a.i,null,E(O||(O=M`s''(x) = s''(m+1)`))))),l("p",null,"È ",l("b",null,"unica"),".")),l(a.r,{title:"Spline cubica not-a-knot"},l("p",null,"Classe di spline cubiche in cui:"),l("ul",null,l("li",null,"Negli intervalli ",l(a.i,null,E(G||(G=M`[x_0, x_2]`)))," e ",l(a.i,null,E(R||(R=M`[x_{m-1}, x_{m+1}]`)))," si presenta ",l("b",null,"obbligatoriamente")," un polinomio di ",l("b",null,"grado 3"),".")),l("p",null,"È ",l("b",null,"unica"),"."))),l(a.s,null,l(a.r,{title:"Proprietà di minima curvatura"},l("p",null,'Tra tutte le funzioni che interpolano dei punti, le tre classi di funzioni sopraelencate sono quelle che interpolano la funzione più "dolcemente".'),l("p",null,"Per loro è valida la seguente proprietà:"),l(a.q,null,E(F||(F=M`\int_a^b ( s''(x) )^2 dx \leq \int_a^b ( f''(x) )^2 dx`)))),l(a.r,{title:"Errore di interpolazione"},l("p",null,"Più diminuisce la lunghezza ",l(a.i,null,E(P||(P=M`h`)))," degli intervalli, più aumenta l'accuratezza."),l("p",null,l("b",null,"Non")," si verifica il fenomeno di Runge."),l("p",null,"Si ha un'interpolazione anche della ",l("b",null,"derivata prima"),"."))))}}).call(this,n("hosL").h)},T2GU:function(l,i,n){"use strict";(function(l){var e=n("5aVd"),a=n.n(e);i.a=function(i){return l("ul",{class:a.a.menulist},i.children)}}).call(this,n("hosL").h)},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)},lijF:function(l,i,n){"use strict";(function(l){var e=n("mbOI"),a=n("YNhk"),o=n("ke5e"),t=n("hosL");let r,u,s,c,d,p,m,_,b,f,g,z,h,x,v,q,k,L,A,S,y,C,w,I,O,G,R,F,P,M,E,U,Q,N,T,D,Y,H,j,V,J,X,B,Z,K,W,$,ll,il,nl,el,al,ol,tl,rl,ul,sl,cl,dl,pl,ml,_l,bl,fl,gl,zl,hl,xl,vl,ql,kl,Ll,Al,Sl,yl,Cl,wl,Il,Ol,Gl,Rl,Fl,Pl,Ml,El=l=>l;const Ul=String.raw;i.a=function(){return l(t.Fragment,null,l(e.s,{title:"Problema: Risoluzione di sistemi lineari"},l(e.r,{title:"Descrizione"},l("p",null,"Dato un sistema di equazioni lineari, si vuole trovare la sua soluzione."),l("p",null,"In forma matriciale, avrà una ",l("b",null,"matrice dei coefficienti")," ",l(e.i,null,Ul(r||(r=El`A`))),", un ",l("b",null,"vettore dei termini noti")," ",l(e.i,null,Ul(u||(u=El`b`)))," e un ",l("b",null,"vettore delle incognite"),l(e.i,null,Ul(s||(s=El`x`))),"."),l("p",null,"L'equazione matriciale del sistema è:"),l(e.q,null,Ul(c||(c=El`A \cdot x = b`)))),l(e.r,{title:"Condizionamento"},l("p",null,"Il condizionamento della risoluzione di sistemi lineari è:"),l(e.q,null,Ul(d||(d=El`\frac{{\color{yellow} \|A\| \cdot \|A^{-1}\|} \cdot \| \Delta b \|}{\| b \|}`))),l("p",null,"In particolare, è segnato in giallo nella formula il ",l("b",null,"numero di condizionamento"),":"),l(e.q,null,Ul(p||(p=El`k(A) = \| A \| \cdot \| A^{-1} \|`))))),l(e.s,null,l(e.r,{title:"Metodi diretti"},l("p",null,"Metodi che trovano la soluzione esatta",l("abbr",{title:"Per quanto possibile nell'algebra di macchina."},"*")," di un sistema lineare."),l("p",null,"Tipicamente prevedono la ",l("b",null,"fattorizzazione")," della matrice dei coefficienti in due sottomatrici più facili da risolvere."),l("p",null,"Generalmente hanno una complessità temporale ",l(e.i,null,Ul(m||(m=El`O(n^3)`))),".")),l(e.r,{title:"Metodi iterativi"},l("p",null,"Metodi che trovano una soluzione imperfetta",l("abbr",{title:"Che però può essere la migliore ottenibile, considerando la precisione di macchina."},"*")," di un sistema lineare."),l("p",null,"Tipicamente prevedono l'applicazione ripetuta di un ",l("b",null,"metodo"),", in base al quale cambia la ",l("b",null,"velocità di convergenza")," alla soluzione."),l("p",null,"Generalmente hanno una complessità temporale ",l(e.i,null,Ul(_||(_=El`O(n^2)`))),"."))),l(e.s,{title:"Metodi diretti"},l(e.r,{title:"Divisione"},l("p",null,"Se la matrice dei coefficienti del sistema è ",l("b",null,"diagonale"),", allora è possibile trovare la soluzione ",l("i",null,"dividendo")," ogni termine noto per l'unico coefficiente diverso da zero presente nella sua riga:"),l(e.q,null,Ul(b||(b=El`x_i = \frac{b_i}{A_{ii}}`)))),l(e.r,{title:"Sostituzione"},l("p",null,"Se la matrice dei coefficienti del sistema è ",l("b",null,"triangolare")," inferiore o superiore, allora è possibile trovare la soluzione effettuando una ",l("i",null,"sostituzione")," all'avanti oppure all'indietro:"),l(e.q,null,Ul(f||(f=El`x_i = \frac{b_i - \sum_{k = 1}^{i - 1} (x_k \cdot A_{ik})}{A_{ii}}`))),l(e.q,null,Ul(g||(g=El`x_i = \frac{b_i - \sum_{k = i - 1}^{n} (x_k \cdot A_{ik})}{A_{ii}}`))))),l(e.s,null,l(e.r,{title:l("span",null,"Fattorizzazione ",l(e.i,null,Ul(z||(z=El`LU`))))},l("p",null,"Se tutti i valori sulla diagonale di ",l(e.i,null,Ul(h||(h=El`A`)))," sono ",l("b",null,"diversi da 0 ",l("small",null,"(eccetto l'ultimo)"))," allora è possibile ",l("i",null,"fattorizzarla")," in due matrici: una ",l(e.i,null,Ul(x||(x=El`L`)))," ",l("b",null,"triangolare inferiore"),", e una ",l(e.i,null,Ul(v||(v=El`U`)))," ",l("b",null,"triangolare superiore"),"."),l(e.q,null,Ul(q||(q=El`A = L \cdot U`))),l(o.a,null,"Abbiamo fatto questo metodo in Algebra Lineare, chiamandolo ",l("b",null,"metodo di Gauss"),"."),l("p",null,"La matrice ",l(e.i,null,Ul(k||(k=El`L`)))," è così composta:"),l(e.q,null,Ul(L||(L=El`
+ \begin{cases}
+ L_{ii} = 1 \qquad \qquad (diagonale)\\
+ L_{ik} = -\frac{A_{ik}}{A_{kk}} \qquad (tri.\ infer.)
+ \end{cases}
+ `))),l(o.a,null,"Sono i moltiplicatori usati per rendere annullare il triangolo inferiore!"),l("p",null,"La matrice ",l(e.i,null,Ul(A||(A=El`U`)))," è così composta:"),l(e.q,null,Ul(S||(S=El`
+ \begin{cases}
+ U_{ik} = A_{ik} \quad se\ i \leq k \quad (tri.\ super.)\\
+ U_{ik} = 0 \qquad se\ i > k \quad (tri.\ infer.)
+ \end{cases}
+ `))),l(o.a,null,"È la parte triangolare superiore di ",l(e.i,null,Ul(y||(y=El`A`))),"!"),l("p",null,"Il sistema può essere poi risolto applicando due volte il metodo di sostituzione (all'avanti e all'indietro):"),l(e.q,null,Ul(C||(C=El`
+ \begin{cases}
+ L \cdot y = b\\
+ U \cdot x = y
+ \end{cases}
+ `))),l("p",null,"Questo metodo ha costo computazionale:"),l(e.q,null,Ul(w||(w=El`{\color{Yellow} O\left(\frac{n^3}{3}\right)} + 2 \cdot O\left(\frac{n^2}{2}\right)`)))),l(e.r,{title:l("span",null,"Fattorizzazione ",l(e.i,null,Ul(I||(I=El`LU`)))," con pivoting parziale")},l("p",null,"È possibile applicare la fattorizzazione ",l(e.i,null,Ul(O||(O=El`LU`)))," a ",l("b",null,"qualsiasi matrice non-singolare")," permettendo lo scambio (",l("i",null,"pivoting"),") delle righe, potenzialmente ",l("b",null,"aumentando la stabilità")," dell'algoritmo."),l(o.a,null,"Abbiamo fatto questo metodo in Algebra Lineare, chiamandolo ",l("b",null,"metodo di Gauss-Jordan"),"!"),l("p",null,"Alla formula precedente si aggiunge una ",l("b",null,"matrice di permutazione")," che indica quali righe sono state scambiate:"),l(e.q,null,Ul(G||(G=El`P \cdot A = L \cdot U`))),l("p",null,"Per massimizzare la stabilità, si cerca di ",l("b",null,"usare come perno l'elemento più grande")," della colonna."),l("p",null,"Questo metodo ha costo computazionale:"),l(e.q,null,Ul(R||(R=El`{\color{Yellow} O\left(\frac{n^2}{2}\right)} + O\left(\frac{n^3}{3}\right) + 2 \cdot O\left(\frac{n^2}{2}\right)`)))),l(e.r,{title:l("span",null,"Fattorizzazione ",l(e.i,null,Ul(F||(F=El`LU`)))," con pivoting totale")},l("p",null,"È possibile anche permettere il ",l("i",null,"pivoting")," ",l("b",null,"sulle colonne")," per ",l("b",null,"aumentare ulteriormente la stabilità")," dell'algoritmo, a costo di maggiore costo computazionale:"),l(e.q,null,Ul(P||(P=El`P \cdot A \cdot Q = L \cdot U`))),l("p",null,"Per massimizzare la stabilità, si cerca di ",l("b",null,"ordinare in modo decrescente la diagonale"),", assicurandoci che il primo perno sia più grande del secondo e così via."),l("p",null,"Questo metodo ha costo computazionale:"),l(e.q,null,Ul(M||(M=El`{\color{Yellow} O\left(\frac{n^3}{3}\right)} + O\left(\frac{n^3}{3}\right) + 2 \cdot O\left(\frac{n^2}{2}\right)`))))),l(e.s,null,l(e.r,{title:l("span",null,"Fattorizzazione ",l(e.i,null,Ul(E||(E=El`LU`)))," a banda")},l("p",null,"Se la matrice ",l(e.i,null,Ul(U||(U=El`A`)))," è ",l("b",null,"a banda"),", è possibile risparmiare spazio durante la fattorizzazione, in quanto sia ",l(e.i,null,Ul(Q||(Q=El`L`)))," sia ",l(e.i,null,Ul(N||(N=El`U`)))," saranno a banda!")),l(e.r,{title:l("span",null,"Fattorizzazione ",l(e.i,null,Ul(T||(T=El`LU`)))," sparsa")},l("p",null,"Se la matrice ",l(e.i,null,Ul(D||(D=El`A`)))," è ",l("b",null,"sparsa"),", non è detto che ",l(e.i,null,Ul(Y||(Y=El`L`)))," e ",l(e.i,null,Ul(H||(H=El`U`)))," siano sparse a loro volta."),l("p",null,"Per evitare il ",l("u",null,"fill-in"),", è necessario ",l("b",null,"riordinare")," la matrice ",l(e.i,null,Ul(j||(j=El`A`)))," in modo che sia il più possibile simile a una matrice a banda."))),l(e.s,null,l(e.r,{title:l("span",null,"Fattorizzazione ",l(e.i,null,Ul(V||(V=El`LDL^{-1}`))))},l("p",null,"È possibile ",l("b",null,"ridurre la complessità computazionale")," della fattorizzazione ",l(e.i,null,Ul(J||(J=El`LU`)))," se la matrice dei coefficienti è ",l("b",null,"simmetrica"),":"),l(e.q,null,Ul(X||(X=El`A = L \cdot D \cdot L^{-1}`))),l("p",null,"In questo caso, si calcola solo la matrice L, utilizzando il ",l("b",null,"metodo di pavimentazione"),"."),l(e.q,null,Ul(B||(B=El`
+ \begin{cases}
+ d_{ii} = A_{ii} - \sum_{k=1}^{i-1} ( d_{kk} \cdot (l_{jk})^2 )\\
+ \\
+ l_{ij} = \frac{A_{ij} - \sum_{k=1}^{j-1} l_{ik} \cdot d_{kk} \cdot l_{jk}}{d_{ii}}
+ \end{cases}
+ `))),l(o.a,null,l("p",null,"La prima colonna della matrice sarà:"),l(e.q,null,Ul(Z||(Z=El`
+ \begin{cases}
+ d_{11} = A_{11}\\
+ \\
+ l_{i1} = \frac{A_{i1}}{d_{11}}
+ \end{cases}
+ `))),l("p",null,"La seconda colonna della matrice sarà:"),l(e.q,null,Ul(K||(K=El`
+ \begin{cases}
+ d_{22} = A_{22} - d_{11} \cdot (l_{21})^2\\
+ \\
+ l_{i2} = \frac{A_{i2} - l_{i1} \cdot d_{11} \cdot l_{21}}{d_{ii}}
+ \end{cases}
+ `)))),l("p",null,"Questo metodo ha costo computazionale:"),l(e.q,null,Ul(W||(W=El`{\color{Yellow} O\left(\frac{n^3}{6}\right)} + O\left(\frac{n^3}{3}\right) + 2 \cdot O\left(\frac{n^2}{2}\right)`)))),l(e.r,{title:l("span",null,"Fattorizzazione ",l(e.i,null,Ul($||($=El`\mathcal{L} \mathcal{L}^{-1}`))))},l("p",null,"È possibile dare ",l("b",null,"stabilità forte")," alla fattorizzazione ",l(e.i,null,Ul(ll||(ll=El`LDL^{-1}`)))," se la matrice dei coefficienti è ",l("b",null,"simmetrica definita positiva"),":"),l(e.q,null,Ul(il||(il=El`A = \mathcal{L} \cdot \mathcal{L}^{-1}`))),l("p",null,"Il ",l("b",null,"metodo di pavimentazione")," diventa:"),l(e.q,null,Ul(nl||(nl=El`
+ \begin{cases}
+ l_{ii} = \sqrt{A_{ii} - \sum_{k=1}^{i-1} (l_{ik})^2 }\\
+ \\
+ l_{ij} = \frac{A_{ij} - \sum_{k=1}^{j-1} l_{ik} \cdot l_{jk}}{l_{ii}}
+ \end{cases}
+ `))),l("p",null,"Questo metodo ha costo computazionale:"),l(e.q,null,Ul(el||(el=El`O\left(\frac{n^3}{3}\right) + O\left(\frac{n^3}{3}\right) + 2 \cdot O\left(\frac{n^2}{2}\right)`))))),l(e.s,null,l(e.r,{title:"Trasformazione di Householder"},l("p",null,"Matrice ricavata dalla seguente formula, dove ",l(e.i,null,Ul(al||(al=El`v`)))," è la colonna di un'altra matrice:"),l(e.q,null,Ul(ol||(ol=El`U(v) = \mathbf{I} - \frac{2 \cdot v \cdot v^T}{\| v \|_{(2)}^2}`))),l("p",null,"Se moltiplicata per per la matrice da cui proviene ",l(e.i,null,Ul(tl||(tl=El`v`))),", sostituirà la colonna ",l(e.i,null,Ul(rl||(rl=El`v`)))," con la colonna:"),l(e.q,null,Ul(ul||(ul=El`
+ \begin{pmatrix}
+ - \| v \|\\\\
+ 0\\\\
+ 0\\\\
+ \vdots\\\\
+ 0
+ \end{pmatrix}
+ `))),l("p",null,"Si calcola con una complessità computazionale nell'ordine di ",l(e.i,null,Ul(sl||(sl=El`O(n)`))),".")),l(e.r,{title:l("span",null,"Fattorizzazione ",l(e.i,null,Ul(cl||(cl=El`QR`))))},l("p",null,"Metodo che fornisce una ",l("b",null,"maggiore stabilità")," a costo di una ",l("b",null,"maggiore complessità computazionale"),"."),l("p",null,"La matrice ",l(e.i,null,Ul(dl||(dl=El`A`)))," viene ",l("i",null,"fattorizzata")," in due matrici, una ",l("b",null,"ortogonale")," ",l(e.i,null,Ul(pl||(pl=El`Q`)))," e una ",l("b",null,"triangolare superiore"),l(e.i,null,Ul(ml||(ml=El`R`))),":"),l(e.q,null,Ul(_l||(_l=El`A = Q \cdot R`))),l("p",null,"Le matrici si ottengono dal ",l("b",null,"prodotto delle trasformazioni di Householder")," (che concatenate formano ",l(e.i,null,Ul(bl||(bl=El`Q`))),") sulla matrice ",l(e.i,null,Ul(fl||(fl=El`A`)))," necessarie a trasformarla in una matrice triangolare superiore (",l(e.i,null,Ul(gl||(gl=El`R`))),")."),l(o.a,null,"C'è un bell'esempietto ",l(a.a,{href:"https://web.archive.org/web/20200828003151/https://rpubs.com/aaronsc32/qr-decomposition-householder"},"qui"),"."),l("p",null,"Una volta fattorizzata, il sistema si può risolvere con:"),l(e.q,null,Ul(zl||(zl=El`
+ \begin{cases}
+ y = Q^T \cdot b\\
+ R \cdot x = y
+ \end{cases}
+ `))),l("p",null,"Questo metodo ha costo computazionale:"),l(e.q,null,Ul(hl||(hl=El`{\color{Yellow} O\left(\frac{2 \cdot n^3}{3}\right)} + 2 \cdot O\left(\frac{n^2}{2}\right)`))))),l(e.s,{title:"Metodi iterativi"},l(e.r,{title:"Forma generale"},l("p",null,"Se si pone che:"),l(e.q,null,Ul(xl||(xl=El`
+ \begin{cases}
+ G = I - M^{-1} \cdot A\\
+ c = M^{-1} \cdot b
+ \end{cases}
+ `))),l("p",null,"Allora la formula generale di un sistema lineare può anche essere scritta in questo modo:"),l(e.q,null,Ul(vl||(vl=El`x = G \cdot x + c`))),l("p",null,"È particolarmente utile perchè ci permette di definire un ",l("b",null,"algoritmo ricorsivo")," che trovi ",l(e.i,null,Ul(ql||(ql=El`x`))),":"),l(e.q,null,Ul(kl||(kl=El`x_{(i+1)} = G \cdot x_{(i)} + c`))),l("p",null,l(e.i,null,Ul(Ll||(Ll=El`G`)))," è il ",l("b",null,"metodo"),", e in base ad esso cambiano stabilità e velocità di convergenza."),l("p",null,"Ponendo ",l(e.i,null,Ul(Al||(Al=El`A = M - N`))),", la formula può essere scritta anche in questo modo:"),l(e.q,null,Ul(Sl||(Sl=El`M \cdot x_{(i+1)} = N \cdot x_{(i)} + b`))),l("p",null,"Possiamo ottenere alcuni metodi separando ",l(e.i,null,"A")," in tre matrici:"),l("ul",null,l("li",null,"La parte diagonale ",l(e.i,null,Ul(yl||(yl=El`D`)))),l("li",null,"L'opposto del triangolo inferiore ",l(e.i,null,Ul(Cl||(Cl=El`E`)))),l("li",null,"L'opposto del triangolo superiore ",l(e.i,null,Ul(wl||(wl=El`F`))))),l(e.q,null,Ul(Il||(Il=El`A = D - E - F`)))),l(e.r,{title:"Convergenza di un metodo"},l("p",null,"Un metodo è convergente se e solo se:"),l(e.q,null,Ul(Ol||(Ol=El`\rho (M) < 1`))),l("p",null,"(dove ",l(e.i,null,Ul(Gl||(Gl=El`\rho`)))," è il ",l("b",null,"raggio spettrale"),", il massimo autovalore della matrice)"),l("p",null,"Perchè un metodo sia convergente, è sufficiente che:"),l(e.q,null,Ul(Rl||(Rl=El`\| M \| < 1`))))),l(e.s,null,l(e.r,{title:"Metodo di Jacobi"},l("p",null,"Il metodo di Jacobi si ottiene ponendo:"),l(e.q,null,Ul(Fl||(Fl=El`
+ \begin{cases}
+ M = D\\
+ N = E + F
+ \end{cases}
+ `))),l("p",null,l("u",null,"Spostamenti simultanei"),": Permette di ottenere ogni componente di ",l(e.i,null,Ul(Pl||(Pl=El`x`)))," indipendentemente dagli altri: è ",l("b",null,"parallelizzabile"),"."),l("p",null,"Se la matrice è ",l("b",null,"diagonale dominante"),", allora il metodo di Jacobi ",l("b",null,"converge")," sicuramente.")),l(e.r,{title:"Metodo di Gauss-Seidel"},l("p",null,"Il metodo di Gauss-Seidel si ottiene ponendo:"),l(e.q,null,Ul(Ml||(Ml=El`
+ \begin{cases}
+ M = D - E\\
+ N = F
+ \end{cases}
+ `))),l("p",null,"Ha una velocità di convergenza ",l("b",null,"maggiore o uguale")," rispetto al metodo di Jacobi."),l("p",null,l("u",null,"Spostamenti successivi"),": Non è parallelizzabile, perchè ogni componente ",l("b",null,"dipende da quelle calcolate in precedenza"),"."),l("p",null,"Se la matrice è ",l("b",null,"diagonale dominante"),", allora il metodo di Gauss-Seidel ",l("b",null,"converge")," sicuramente."))))}}).call(this,n("hosL").h)},qXt2:function(l,i,n){"use strict";n.r(i),function(l){var e=n("sl5E"),a=n("lijF"),o=n("FEtp"),t=n("31Ft"),r=n("LHkx"),u=n("uqDQ");i.default=function(){return l("div",null,l("h1",null,"Calcolo Numerico"),l(e.a,null),l(a.a,null),l(o.a,null),l(t.a,null),l(r.a,null),l(u.a,null))}}.call(this,n("hosL").h)},sl5E:function(l,i,n){"use strict";(function(l){var e=n("mbOI"),a=n("YNhk"),o=n("T2GU"),t=n("ke5e"),r=n("hosL");let u,s,c,d,p,m,_,b,f,g,z,h,x,v,q,k,L,A,S,y,C,w=l=>l;const I=String.raw;i.a=function(){return l(r.Fragment,null,l(e.s,{title:"Esame"},l(e.r,{title:"Contatti"},l("ul",null,l("li",null,l(a.a,{href:"mailto:silvia.bonettini@unimore.it"},"Prof.ssa Silvia Bonettini")))),l(e.r,{title:"Orale"},l("p",null,"E' composto da:"),l("ul",null,l("li",null,"2 domande sugli argomenti teorici"),l("li",null,"1 domanda di implementazione algoritmo in MATLAB")))),l(e.s,{title:"Extra"},l(e.r,{title:"Ripasso di Algebra Lineare"},l("p",null,"Prima di iniziare a studiare Calcolo Numerico, potrebbe essere una buona idea ripassare un pochino Algebra Lineare:"),l(o.a,null,l("li",null,l("a",{href:"/calcolonumerico/ripassodialgebralineare"},"Ripasso di Algebra Lineare")," ",l("small",null,"(per studenti sperduti di Calcolo Numerico)")))),l(e.r,{title:"Anki"},l("p",null,"Ho provato a fare un deck ",l(a.a,{href:"https://apps.ankiweb.net/"},"Anki")," con tutte le formule di Calcolo numerico."),l("p",null,"Sappiatemi dire com'è! ",l("small",null,"Io non l'ho usato moltissimo perchè ho studiato scrivendo questa pagina di Appuntiweb...")),l(o.a,null,l("li",null,l(a.a,{href:"https://ryg.s3.fr-par.scw.cloud/builds/anki-decks/formule-di-calcolo-numerico.apkg"},"Scarica il deck Anki"))))),l(e.s,{title:"Algoritmi"},l(e.r,{title:"Algoritmi numerici"},l("p",null,"Particolari algoritmi che hanno:"),l("ul",null,l("li",null,"numeri reali in input e output"),l("li",null,"successioni delle quattro operazioni aritmetiche fondamentali come passi")))),l(e.s,{title:"Errore di rappresentazione"},l(e.r,{title:"Cos'è?"},l("p",null,"Con i numeri floating point può capitare che un certo numero ",l(e.i,null,I(u||(u=w`\alpha`)))," non sia rappresentato correttamente."),l("p",null,"In tal caso, il numero si indica con ",l(e.i,null,I(s||(s=w`\alpha^\star`))),"."))),l(e.s,null,l(e.r,{title:"Errore assoluto"},l("p",null,"È la differenza tra il numero desiderato e il numero rappresentato:"),l(e.q,null,I(c||(c=w`E_a = \left | \alpha - \alpha^\star \right |`)))),l(e.r,{title:"Errore relativo"},l("p",null,"Indica quanto il numero rappresentato differisce dal numero desiderato:"),l(e.q,null,I(d||(d=w`\forall \alpha \neq 0, E_r = \frac{E_a}{\left | \alpha \right |}`))))),l(e.s,null,l(e.r,{title:"Troncamento"},l("p",null,"Metodo con cui gestire gli ",l("b",null,"underflow floating point"),": le cifre meno significative vengono ",l("b",null,"rimosse"),"."),l(t.a,null,l("pre",null,"1.00 → 1.0",l("br",null),"1.01 → 1.0",l("br",null),"1.10 → 1.1",l("br",null),"1.11 → 1.1"))),l(e.r,{title:"Arrotondamento"},l("p",null,"Metodo con cui gestire gli ",l("b",null,"underflow floating point"),": se la cifra più significativa di quelle che devono essere rimosse è 1, allora ",l("b",null,"aumenta di 1")," anche quella meno signficativa che viene tenuta."),l(t.a,null,l("pre",null,"1.00 → 1.0",l("br",null),"1.01 → 1.0",l("br",null),"1.10 → 1.1",l("br",null),"1.11 → 10.")))),l(e.s,null,l(e.r,{title:"Precisione di macchina"},l("p",null,"Un numero reale rappresentato in ",l("b",null,"virgola mobile")," ha un ",l("b",null,"errore relativo")," minore o uguale alla ",l("i",null,"precisione di macchina"),":"),l("p",null,l(e.i,null,I(p||(p=w`\epsilon_x \leq k \cdot \beta^{1-t}`)))),l("ul",null,l("li",null,l(e.i,null,"\\beta")," è uguale alla base utilizzata (solitamente 2)."),l("li",null,l(e.i,null,"t")," è uguale al numero di cifre della mantissa."),l("li",null,l(e.i,null,"k")," è uguale a ",l(e.i,null,"1")," se il numero viene rappresentato per troncamento oppure a ",l(e.i,null,I(m||(m=w`\frac{1}{2}`)))," se viene rappresentato per arrotondamento."))),l(e.r,{title:"La funzione fl"},l("p",null,"Associa un valore reale al suo ",l("b",null,"corrispondente valore floating point"),", utilizzando uno dei due metodi di gestione dell'undeflow."),l(e.q,null,I(_||(_=w`fl(x) = (x)(1 + \epsilon_x)`))),l(t.a,null,"Indica che un valore è soggetto alla precisione di macchina.",l(e.q,null,I(b||(b=w`fl(1.11) = 1.1`)))))),l(e.s,null,l(e.r,{title:"Un nuovo insieme"},l("p",null,"L'insieme ",l(e.i,null,I(f||(f=w`\mathbb{F}`)))," è il sottoinsieme dei numeri reali rappresentabili in floating point dalla macchina che stiamo usando."),l("p",null,"Operazioni tra elementi di ",l(e.i,null,I(g||(g=w`\mathbb{F}`)))," producono risultati in ",l(e.i,null,I(z||(z=w`\mathbb{R}`))),", che però decaderanno nuovamente a elementi di ",l(e.i,null,I(h||(h=w`\mathbb{F}`))),", perdendo informazioni."),l("p",null,"Il teorema della precisione di macchina si applica quindi anche ai risultati delle operazioni.")),l(e.r,{title:"Caratteristiche delle operazioni di macchina"},l("ul",null,l("li",null,"Hanno ",l("b",null,"più elementi neutri"),"."),l("li",null,"Un numero ha ",l("b",null,"più opposti"),"."),l("li",null,l("b",null,"Non")," sono associative."),l("li",null,l("b",null,"Non")," sono distributive."),l("li",null,l("b",null,"Non")," vale la legge di annullamento del prodotto.")))),l(e.s,{title:"Errori nelle operazioni di macchina"},l(e.r,{title:"Errore inerente"},l("p",null,"Errore derivato da underflow sui ",l("b",null,"dati"),"."),l("p",null,"Si indica con ",l(e.i,null,I(x||(x=w`\epsilon_{nome\_var}`))),"."),l(t.a,null,"L'errore sulla variabile ",l(e.i,null,"x")," si indica con ",l(e.i,null,I(v||(v=w`\epsilon_{x}`))),".")),l(e.r,{title:"Errore algoritmico"},l("p",null,"Errore derivato da underflow durante l'",l("b",null,"esecuzione dell'algoritmo"),"."),l("p",null,"Si indica con ",l(e.i,null,I(q||(q=w`\epsilon_{num\_passo}`))),"."),l(t.a,null,"L'errore al primo passo dell'algoritmo si indica con ",l(e.i,null,I(k||(k=w`\epsilon_{1}`))),"."))),l(e.s,null,l(e.r,{title:"Condizionamento"},l("p",null,"Sensibilità di un problema all'",l("b",null,"errore inerente"),"."),l(t.a,null,l(e.i,null,I(L||(L=w`y = \frac{1}{x}`)))," è mal condizionato intorno allo 0 e ben condizionato lontano dallo 0.")),l(e.r,{title:"Stabilità"},l("p",null,"Sensibilità di un problema all'",l("b",null,"errore algoritmico"),"."),l(t.a,null,l("p",null,"Cerchiamo un algoritmo che risolva ",l(e.i,null,I(A||(A=w`2x^\star = 4`))),"."),l("p",null,"Calcolare prima ",l(e.i,null,I(S||(S=w`t = fl \left( \frac{1}{4} \right)`)))," e poi ",l(e.i,null,I(y||(y=w`x = fl ( 2 \cdot t )`)))," porta a una perdita di precisione."),l("p",null,"Calcolare direttamente ",l(e.i,null,I(C||(C=w`x = fl \left( \frac{2}{4} \right)`)))," non ha alcuna perdita di precisione e rende l'algoritmo ",l("b",null,"più stabile")," del precedente.")))),l(e.s,null,l(e.r,{title:"Indice di condizionamento"},l("p",null,"È il coefficiente di proporzionalità tra i dati e l'",l("b",null,"errore inerente"),"."),l("p",null,"Essendo sempre maggiore di uno, si può dire che sia un coefficiente di amplificazione."),l("p",null,"Minore è l'indice di condizionamento, meglio condizionato è un problema.")),l(e.r,{title:"Indice algoritmico"},l("p",null,"È il coefficiente di proporzionalità tra i dati e l'",l("b",null,"errore algoritmico"),"."),l("p",null,"Essendo sempre maggiore di uno, si può dire che sia un coefficiente di amplificazione."))))}}).call(this,n("hosL").h)},uqDQ:function(l,i,n){"use strict";(function(l){var e=n("hosL"),a=n("mbOI"),o=n("ke5e");let t,r,u,s,c,d,p,m,_,b,f,g,z,h,x,v,q,k,L,A,S,y,C,w,I,O,G,R,F,P,M,E,U,Q,N,T,D,Y,H,j,V,J,X,B,Z,K,W,$,ll,il,nl,el,al,ol,tl,rl,ul=l=>l;const sl=String.raw;i.a=function(){return l(e.Fragment,null,l(a.s,{title:"Problema: Approssimazione di dati sperimentali"},l(a.r,{title:"Perchè?"},l("p",null,"Interpolare dati sperimentali non fornisce quasi mai un modello del fenomeno."),l("p",null,"Vogliamo costruire una ",l("b",null,"funzione di regressione")," che, dati molti più dati del grado della funzione, minimizzi il quadrato della distanza tra i punti sperimentali e i punti della funzione di regressione."),l("p",null,"Denominiamo:"),l("ul",null,l("li",null,l(a.i,null,sl(t||(t=ul`{\color{Orange} f}`))),": la ",l("b",null,'funzione "effettiva"')," del fenomeno"),l("li",null,l(a.i,null,sl(r||(r=ul`{\color{Yellow} q}`))),": la ",l("b",null,"funzione di regressione")," che costruiamo per approssimarlo"),l("li",null,l(a.i,null,sl(u||(u=ul`{\color{Red} Q }`))),": la ",l("b",null,'funzione "errore di regressione"')," da minimizzare"),l("li",null,l(a.i,null,sl(s||(s=ul`(\ x_i, f(x_i)\ )`))),": i ",l("b",null,"punti sperimentali"))),l("p",null,"L'obiettivo è minimizzare l'",l("b",null,"errore di approssimazione")," ",l(a.i,null,sl(c||(c=ul`Q`))),", ovvero:"),l(a.q,null,sl(d||(d=ul`\min {\color{Red} Q } = \sum_{i = 1}^m (\ {\color{Yellow} q(x_i)} - {\color{Orange} f(x_i)}\ )^2 `))))),l(a.s,null,l(a.r,{title:"Regressione lineare"},l("p",null,"Trova la ",l("b",null,"retta")," ",l(a.i,null,sl(p||(p=ul`{\color{Yellow} q}`)))," che meglio approssima tutti gli ",l(a.i,null,sl(m||(m=ul`m`)))," dati sperimentali."),l("p",null,"Essendo una retta, avrà ",l("b",null,"due parametri"),": il termine noto ",l(a.i,null,sl(_||(_=ul`a_0`))),", e la pendenza ",l(a.i,null,"a_1"),"."),l(a.q,null,sl(b||(b=ul`{\color{Yellow} q(x) } = a_0 + a_1 \cdot {\color{Green} x}`))),l("p",null,"L'errore da minimizzare per ricavare i parametri sarà:"),l(a.q,null,sl(f||(f=ul`
+ \min {\color{Red} Q } = \sum_{i = 1}^m ( {\color{Yellow} a_0 + a_1 \cdot x_i} - {\color{Orange} f(x_i)} )^2
+ `)))),l(a.r,{title:"Regressione lineare matriciale"},l("p",null,"Possiamo costruire una ",l("b",null,"matrice di regressione")," ",l(a.i,null,sl(g||(g=ul`A`)))," contenente tutti i ",l("b",null,"punti sperimentali"),":"),l(a.q,null,sl(z||(z=ul`
+ A =
+ \begin{pmatrix}
+ 1 & x_1\\\\
+ 1 & x_2\\\\
+ \vdots & \vdots\\\\
+ 1 & x_m
+ \end{pmatrix}
+ `))),l("p",null,"Inoltre, se costruiamo il ",l("b",null,"vettore dei parametri")," ",l(a.i,null,sl(h||(h=ul`\alpha`))),":"),l(a.q,null,sl(x||(x=ul`
+ \alpha =
+ \begin{pmatrix}
+ a_0\\\\
+ a_1
+ \end{pmatrix}
+ `))),l("p",null,"Avremo che:"),l(a.q,null,sl(v||(v=ul`{\color{Yellow} q(x) } = A \cdot \alpha`))),l("p",null,"Inoltre, potremo calcolare l'errore attraverso la norma:"),l(a.q,null,sl(q||(q=ul`{\color{Red} Q } = \| A \cdot \alpha - y \|^2`))))),l(a.s,null,l(a.r,{title:"Regressione polinomiale"},l("p",null,"Trova il ",l("b",null,"polinomio")," ",l(a.i,null,sl(k||(k=ul`{\color{Yellow} q}`)))," di grado ",l(a.i,null,sl(L||(L=ul`n-1`)))," che meglio approssima tutti gli ",l(a.i,null,sl(A||(A=ul`m`)))," dati sperimentali."),l("p",null,"Essendo un polinomio di grado ",l(a.i,null,sl(S||(S=ul`n-1`))),", avrà ",l(a.i,null,sl(y||(y=ul`n`)))," parametri."),l(a.q,null,sl(C||(C=ul`{\color{Yellow} q(x) } = a_0 + a_1 \cdot {\color{Green} x} + a_2 \cdot {\color{Green} x^2} +\ \dots \ + a_{n-1} \cdot {\color{Green} x^{n-1}`))),l(o.a,null,l("p",null,"La regressione lineare è un caso particolare di regressione generale in cui i parametri sono 2!")),l("p",null,"L'errore da minimizzare per ricavare i parametri sarà:"),l(a.q,null,sl(w||(w=ul`
+ \min {\color{Red} Q} = \sum_{i = 1}^m ( {\color{Yellow} a_0 + a_1 \cdot x_i + a_2 \cdot x_i^2 +\ \dots \ + a_{n-1} \cdot x_i^{n-1}} - {\color{Orange} y_i} )^2
+ `)))),l(a.r,{title:"Regressione polinomiale matriciale"},l("p",null,"Possiamo costruire una ",l("b",null,"matrice di regressione")," ",l(a.i,null,sl(I||(I=ul`A`)))," contenente tutti i ",l("b",null,"punti sperimentali")," a tutti i gradi del polinomio:"),l(a.q,null,sl(O||(O=ul`
+ A =
+ \begin{pmatrix}
+ 1 & x_1 & x_1^2 & \dots & x_1^{n-1} \\\\
+ 1 & x_2 & x_2^2 & \dots & x_2^{n-1} \\\\
+ \vdots & \vdots & \vdots & \ddots & \vdots \\\\
+ 1 & x_m & x_m^2 & \dots & x_m^{n-1}
+ \end{pmatrix}
+ `))),l("p",null,"Inoltre, se costruiamo il ",l("b",null,"vettore dei parametri")," ",l(a.i,null,sl(G||(G=ul`\alpha`))),":"),l(a.q,null,sl(R||(R=ul`
+ \alpha =
+ \begin{pmatrix}
+ a_0\\\\
+ a_1\\\\
+ \vdots\\\\
+ a_{n-1}
+ \end{pmatrix}
+ `))),l("p",null,"Avremo che:"),l(a.q,null,sl(F||(F=ul`{\color{Yellow} q(x) } = A \cdot \alpha`))),l("p",null,"Inoltre, potremo calcolare l'errore attraverso la norma:"),l(a.q,null,sl(P||(P=ul`{\color{Red} Q } = \| A \cdot \alpha - y \|^2`))),l(o.a,null,"Normalmente, i dati sono molti di più, ma se il numero di parametri ",l(a.i,null,sl(M||(M=ul`n`)))," fosse uguale al numero di dati ",l(a.i,null,sl(E||(E=ul`m`))),", allora si otterrebbe il ",l("b",null,"polinomio di interpolazione"),"!"))),l(a.s,null,l(a.r,{title:"Regressione generale"},l("p",null,"Trova i ",l("b",null,"coefficienti della combinazione lineare"),l(a.i,null,sl(U||(U=ul`{\color{Yellow} q}`)))," che meglio approssima tutti gli ",l(a.i,null,sl(Q||(Q=ul`m`)))," dati sperimentali."),l(a.q,null,sl(N||(N=ul`{\color{Yellow} q(x) } = a_0 \cdot {\color{Green} \phi_0 (x)} + a_1 \cdot {\color{Green} \phi_1 (x)} + \dots + a_2 \cdot {\color{Green} \phi_2 (x)} +\ \dots\ + a_{n-1} \cdot {\color{Green} \phi_{n-1} (x)}`))),l(o.a,null,l("p",null,"La regressione polinomiale è un caso particolare di regressione generale in cui:"),l(a.q,null,sl(T||(T=ul`{\color{Green} \phi_{n} (x)} = x^n`)))),l("p",null,"L'errore da minimizzare per ricavare i parametri sarà:"),l(a.q,null,sl(D||(D=ul`
+ \min {\color{Red} Q } = \sum_{i = 1}^m ( {\color{Yellow} a_0 \cdot \phi_0 (x) + a_1 \cdot \phi_1 (x) + \dots + a_2 \cdot \phi_2 (x) +\ \dots\ + a_{n-1} \cdot \phi_{n-1} (x)} - {\color{Orange} f(x_i)} )^2
+ `)))),l(a.r,{title:"Regressione polinomiale generale"},l("p",null,"Possiamo costruire una ",l("b",null,"matrice di regressione")," ",l(a.i,null,sl(Y||(Y=ul`A`)))," contenente tutti i ",l("b",null,"punti sperimentali")," a tutti i gradi del polinomio:"),l(a.q,null,sl(H||(H=ul`
+ A =
+ \begin{pmatrix}
+ \phi_0(x_1) & \phi_1(x_1) & \phi_2(x_1) & \dots & \phi_{n_1}(x_1) \\\\
+ \phi_0(x_2) & \phi_1(x_2) & \phi_2(x_2) & \dots & \phi_{n-1}(x_2) \\\\
+ \vdots & \vdots & \vdots & \ddots & \vdots \\\\
+ \phi_0(x_m) & \phi_1(x_m) & \phi_2(x_m) & \dots & \phi_{n-1}(x_m)
+ \end{pmatrix}
+ `))),l("p",null,"Inoltre, se costruiamo il ",l("b",null,"vettore dei parametri")," ",l(a.i,null,sl(j||(j=ul`\alpha`))),":"),l(a.q,null,sl(V||(V=ul`
+ \alpha =
+ \begin{pmatrix}
+ a_0\\\\
+ a_1\\\\
+ \vdots\\\\
+ a_{n-1}
+ \end{pmatrix}
+ `))),l("p",null,"Avremo che:"),l(a.q,null,sl(J||(J=ul`{\color{Yellow} q(x) } = A \cdot \alpha`))),l("p",null,"Inoltre, potremo calcolare l'errore attraverso la norma:"),l(a.q,null,sl(X||(X=ul`{\color{Red} Q } = \| A \cdot \alpha - y \|^2`))))),l(a.s,{title:"Trovare i parametri"},l(a.r,{title:"Caso non degenere"},l("p",null,"Caso che prevede che le colonne di ",l(a.i,null,sl(B||(B=ul`A`)))," siano ",l("b",null,"linearmente indipendenti"),"."),l("p",null,"La soluzione ",l("b",null,"esiste")," sempre, ed è ",l("b",null,"unica"),"."),l("p",null,"Per trovarla:"),l("ul",null,l("li",null,"Fattorizziamo ",l(a.i,null,sl(Z||(Z=ul`A = Q \cdot \begin{pmatrix} R\\ 0 \end{pmatrix}`))),"."),l("li",null,"Calcoliamo ",l(a.i,null,sl(K||(K=ul`w = Q^T \cdot y`))),"."),l("li",null,"Teniamo solo i primi ",l(a.i,null,"n")," valori di ",l(a.i,null,sl(W||(W=ul`w`)))," e mettiamoli in ",l(a.i,null,sl($||($=ul`w_1`))),"."),l("li",null,"Calcoliamo ",l(a.i,null,sl(ll||(ll=ul`R \cdot \alpha = w_1`))),"."))),l(a.r,{title:"Caso generale"},l("p",null,"Caso che non preclude alcuna composizione di ",l(a.i,null,sl(il||(il=ul`A`))),"."),l("p",null,"Ci sono ",l("b",null,"infinite")," soluzioni, con ",l(a.i,null,"n-k")," gradi di libertà."),l("p",null,"Si cerca sempre di trovare la ",l("i",null,"soluzione di norma minima"),", che, se ",l(a.i,null,sl(nl||(nl=ul`k \leq n \leq m`))),", allora è ",l("b",null,"unica"),"."),l("p",null,"Per trovarla:"),l("ul",null,l("li",null,"Fattorizziamo ",l(a.i,null,sl(el||(el=ul`A = U \cdot \Sigma \cdot V^T`)))," con la ",l("i",null,"fattorizzazione SVD")),l("li",null,"Calcoliamo ",l(a.i,null,sl(al||(al=ul`\zeta = U^T \cdot y`)))),l("li",null,l("p",null,"Calcoliamo:"),l(a.q,null,sl(ol||(ol=ul`
+ \gamma =
+ \begin{pmatrix}
+ \frac{z_1}{\sigma_1}\\\\
+ \frac{z_2}{\sigma_2}\\\\
+ \vdots\\\\
+ \frac{z_{k-1}}{\sigma_{k-1}}\\\\
+ \frac{z_k}{\sigma_k}\\\\
+ 0\\\\
+ 0\\\\
+ \vdots\\\\
+ 0
+ \end{pmatrix}
+ `)))),l("li",null,"Calcoliamo ",l(a.i,null,sl(tl||(tl=ul`\alpha = V \cdot \gamma`))))),l(o.a,null,"Gli zeri nella ",l(a.i,null,sl(rl||(rl=ul`\gamma`)))," sono i gradi di libertà, sono zero in modo che essi diano la norma minima."))))}}).call(this,n("hosL").h)}}]);
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new file mode 100644
index 0000000..215a270
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+++ b/docs/route-CalcoloNumerico.chunk.feb65.esm.js.map
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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 {Fragment} from \"preact\";\nimport {ILatex, Panel, PLatex, Section, Todo} from \"bluelib\";\nimport Example from \"../../components/Example\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n Si vuole trovare una funzione in grado di approssimarne un'altra, di cui si conoscono\n però solo alcuni punti.\n
\n \n È utile in un sacco di casi! Ad esempio, quando si vuole scalare un'immagine.\n \n
\n I punti sono detti nodi{r`(x_i, y_i)`}, mentre la funzione costruita su\n di essi è detta interpolante{r`g`}:\n
\n {r`g(x_i) = y_i`}\n
\n Dato un insieme di punti, esistono infinite funzioni interpolanti.\n
\n \n \n
\n Il teorema fondamentale dell'algebra dice che esiste una sola\n interpolante polinomiale che interpola un dato insieme di punti.\n
\n
\n Con n+1 punti, l'interpolante sarà al massimo di grado n, e\n viene detta {r`p_n`}.\n
\n {r`\n b =\n \\begin{pmatrix}\n y_0\\\\\\\\\n y_1\\\\\\\\\n y_2\\\\\\\\\n \\vdots\\\\\\\\\n y_n\n \\end{pmatrix}\n `}\n \n Per trovare il polinomio di interpolazione è sufficiente risolvere il problema!\n \n
\n È efficace perchè una volta calcolati i coefficienti essi valgono per tutti i punti, ma\n ha come svantaggio che la matrice di Vandermonde è spesso malcondizionata.\n
\n \n \n
\n È possibile scrivere il polinomio di interpolazione raccogliendo\n le {r`y`}:\n
\n I polinomi {r`L_k`} sono detti polinomi di Lagrange, e hanno le seguenti\n proprietà:\n
\n
\n
\n Valgono 1 in corrispondenza del nodo con lo stesso\n indice, 0 in corrispondenza dei nodi con indice diverso\n e {r`0 < n < 1`} in tutti gli altri casi.\n\n {r`\n \\begin{cases}\n L_k(x_k) = 1 \\qquad (nel\\ nodo)\\\\\n L_k(x_j) = 0 \\qquad (altri\\ nodi)\n \\end{cases}\n `}
\n È l'errore compiuto durante l'interpolazione.\n
\n
\n Se la funzione f è interpolata da p_n, allora esso varrà:\n
\n {r`R_n(x) = f(x) - p_n(x)`}\n
\n In particolare, è interessante la sua norma a\n infinito, {r`\\| f - p_n \\|_\\infty`}, che corrisponde alla distanza massima tra\n le due funzioni.\n
\n Hanno quindi convergenza lineare ({r`C = \\frac{1}{2}, p = 1`}).\n
\n
\n Il loro criterio di arresto è un numero di iterazioni prefissato che dipende\n dalla tolleranza sull'errore:\n
\n {r`i \\geq \\log_2 \\left( \\frac{b - a}{\\tau} \\right)`}\n \n Dividi l'intervallo {r`[a, b]`} in tante parti grandi quanto la tolleranza.\n L'algoritmo di bisezione ne escluderà metà ad ogni iterazione; la tolleranza sarà raggiunta\n quando rimarrà una parte sola!\n \n \n \n \n \n \n
Finchè non sono state compiute il numero di iterazioni prefissate:\n \n
\n Calcoliamo il punto\n medio dell'intervallo {r`[a_{(n)}, b_{(n)}]`}:\n {r`c_{(n)} = a_{(n)} + \\frac{b_{(n)} - a_{(n)}}{2}`}\n
\n
\n Dividiamo l'intervallo in due parti, separate da {r`c_{(n)}`}:\n
\n
{r`[a_{(n)}, c_{(n)}]`} è la metà sinistra
\n
{r`[c_{(n)}, b_{(n)}]`} è la metà destra
\n
\n
\n
\n Teniamo l'intervallo in cui i valori della funzione ai due estremi sono discordi, e\n rinominiamolo in {r`[a_{(n+1)}, b_{(n+1)}]`}.\n
\n \n
\n \n \n \n \n
Finchè non sono state compiute il numero di iterazioni prefissate:\n \n
\n Calcoliamo l'intersezione tra la retta che congiunge i due estremi\n {r`a_{(n)}, b_{(n)}`} e l'asse X:\n {r`c_{(n)} = b_{(n)} - \\frac{f(b_{(n)})}{\\frac{f(b_{(n)}) - f(a_{(n)})}{b_{(n)} - a_{(n)}}}`}\n
\n
\n Dividiamo l'intervallo in due parti, separate da {r`c_{(n)}`}:\n
\n
{r`[a_{(n)}, c_{(n)}]`} è la parte sinistra
\n
{r`[c_{(n)}, b_{(n)}]`} è la parte destra
\n
\n
\n
\n Teniamo l'intervallo in cui i valori della funzione ai due estremi sono discordi, e\n rinominiamolo in {r`[a_{(n+1)}, b_{(n+1)}]`}.\n
\n \n
\n \n \n \n \n \n
\n Sono metodi iterativi che funzionano in modo molto simile ai metodi iterativi per i\n sistemi lineari, utilizzando una funzione {r`\\phi`} come \"metodo\".\n
\n {r`x = x - \\phi(x) \\cdot f(x)`}\n
\n Che diventa:\n
\n {r`x_{(k+1)} = g( x_{(k)} )`}\n
\n Sfruttano i punti fissi{r`g(x_{(\\star)}) = x_{(\\star)}`} della\n funzione {r`f`} per convergere: \n se {r`\\phi(x)`} non ha zeri, allora i punti fissi coincideranno con\n gli zeri della funzione {r`f`}.\n
\n {r`g(x) = x - \\phi(x) \\cdot f(x)`}\n
\n Si può raggiungere iterativamente ad un punto fisso attraverso la formula:\n
\n {r`x_{(k+1)} = g( x_{(k)} )`}\n
\n Non si conosce in anticipo il numero di iterazioni necessarie per soddisfare la\n tolleranza {r`\\tau`}; ad ogni iterazione, si controlla se la tolleranza è\n soddisfatta:\n
\n
\n
Nella differenza tra due\n iterate: {r`\\frac{\\left| x_{(k+1)} - x_{(k)} \\right|}{\\left| x_{(k+1)} \\right|} \\leq \\tau`}\n
\n
Nel residuo del problema: {r`\\left| f(x_{(k)}) \\right| \\leq \\tau`}\n
\n
\n \n \n
\n Se:\n
\n
\n
\n Tutti i valori restituiti dalla funzione {r`g`} rientrano nel suo stesso\n dominio:\n {r`g : [a, b] \\to [a, b]`}\n
\n
\n
\n La funzione {r`g`} è una contrazione, ovvero restringe\n l'intervallo {r`[a, b]`}:\n
\n {r`\\forall (x, y) \\in [a, b], | g(x) - g(y) | \\leq L \\cdot | x - y |`}\n
\n (dove {r`0 < L < 1`})\n
\n
\n
\n
\n Allora:\n
\n
\n
\n
\n Il punto fisso esiste ed è unico:\n
\n {r`\\exists! x_{(\\star)}`}\n
\n
\n Il metodo delle approssimazioni successive converge per qualsiasi punto di partenza.\n
\n
\n
\n Vale la seguente disequazione di maggiorazione dell'errore:\n
\n Più è piccolo L, più il metodo convergerà in fretta.\n
\n \n L è molto simile al raggio spettrale {r`\\rho(M)`} dei metodi\n iterativi per i sistemi lineari!\n \n \n \n \n \n
\n Sfrutta la continuità delle funzioni per ottenere una convergenza di ordine più alto.\n
\n {r`\\phi (x) = \\frac{1}{f' (x)}`}\n {r`x_{(k+1)} = x_{(k)} - \\frac{ f(x_{(k)}) }{ f'(x_{(k)}) }`}\n \n Geometricamente, corrisponde a prolungare una retta nel\n punto {r`(x_{(k)}, f(x_{(k)}))`} con pendenza {r`f'(x_{(k)})`},\n e prendendo come nuovo punto la sua intersezione con l'asse X e la sua corrispettiva immagine\n nella funzione.\n \n
\n Ha costo computazionale di 2 valutazioni di funzione più 2 valutazioni di derivata.\n
\n
\n Ha convergenza quadratica.\n
\n \n \n
\n È come il metodo di Newton, ma usa il rapporto incrementale, in modo da poter essere\n applicato a funzioni non continue.\n
\n {r`\\phi (x) = \\frac{ 1 }{ \\frac{ f(x_{(k)}) - f(x_{(k-1)}) }{ x_{(k)} - x_{(k-1)} } }`}\n {r`x_{(k+1)} = x_{(k)} - \\frac{ f(x_{(k)}) }{ \\frac{ f(x_{(k)}) - f(x_{(k-1)}) }{ x_{(k)} - x_{(k-1)} } }`}\n \n Geometricamente, corrisponde a costruire una retta che attraversa i\n punti {r`(x_{(k)}, f(x_{(k)}))`} e {r`(x_{(k-1)}, f(x_{(k-1)}))`},\n e prendendo come nuovo punto la sua intersezione con l'asse X e la sua corrispettiva immagine\n nella funzione.\n \n
\n Ha costo computazionale di 3 valutazioni di funzione.\n
\n
\n Ha convergenza superlineare.\n
\n \n \n \n \n
\n È possibile usare questi metodi per approssimare le soluzioni di sistemi non-lineari.\n
\n \n \n \n )\n}\n","import {Fragment} from \"preact\";\nimport {ILatex, Panel, PLatex, Section} from \"bluelib\";\nimport Example from \"../../components/Example\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n Invece che costruire una singola funzione che interpola tutti i punti, per ogni intervallo\n tra due punti (sottointervallo) si costruisce una funzione apposta.\n
\n \n \n \n \n
\n Interpolanti che:\n
\n
\n
sono polinomiali di grado massimo {r`n`}
\n
sono continue fino al grado {r`n - 1`}
\n
connettono {r`m + 2`} punti, e\n hanno {r`m`} sottointervalli\n
\n
hanno funzioni definite appositamente per ogni sottointervallo
\n
\n \n
\n Significa che agli estremi dell'intervallo, i valori di tutte le derivate fino al\n grado {r`n - 1`} devono essere uguali:\n
\n {r`\\forall \\ k \\leq n-1, \\forall \\ i \\in \\{intervalli\\}, \\quad s_i^{(k)} (x_{i+1}) = s_i^{(k)} (x_{i+1})`}\n \n
\n Hanno {r`n + m + 1`} gradi di libertà.\n
\n \n Esistono infinite spline di grado {r`n \\geq 2`}!\n \n \n \n \n \n \n Sono anche dette interpolanti lineari a tratti.\n \n
\n Per ogni sottointervallo, costruiamo una funzione lineare passante per i due estremi:\n
\n {r`s_i(x) = y_i + \\frac{ y_{i + 1} - y_i }{ x_{i + 1} - x_i } \\cdot (x - x_i)`}\n \n È una linea spezzata!\n \n
\n Il loro errore è:\n
\n {r`\\| R \\|_\\infty = \\| f - s \\|_\\infty \\leq \\frac{1}{8} \\cdot \\max_{y \\in [a, b]} \\left| f''(y) \\right| \\cdot \\left( \\max_{i \\in \\{intervalli\\}} (x_{i+1} - x_{i}) \\right)^2`}\n
\n Ha come vantaggi complessità computazionale molto più bassa e l'assenza del\n fenomeno di Runge, ma allo stesso tempo si perde la derivabilità della funzione.\n
\n
\n Non hanno gradi di libertà.\n
\n \n \n
\n Spline con {r`n = 3`}, che soddisfano le seguenti uguaglianze:\n
\n {r`s_i(x) = \\alpha_i + \\beta_i \\ ( x - x_i ) + \\gamma_i \\ ( x - x_i )^2 + \\delta_i \\ ( x - x_i )^3`}\n \n Spesso si indica con {r`h`} la distanza orizzontale tra due punti di un\n sottointervallo.\n \n \n \n \n \n
Negli\n intervalli {r`[x_0, x_2]`} e {r`[x_{m-1}, x_{m+1}]`} si\n presenta obbligatoriamente un polinomio di grado 3.\n
\n
\n
\n È unica.\n
\n \n \n \n \n
\n Tra tutte le funzioni che interpolano dei punti, le tre classi di funzioni sopraelencate sono\n quelle che interpolano la funzione più \"dolcemente\".\n
\n )\n}","import style from \"./Example.less\";\n\nexport default function (props) {\n return (\n
\n {props.children}\n
\n );\n}\n","import {ILatex, Panel, PLatex, Section, Todo} from \"bluelib\";\nimport Link from \"../../components/Link\";\nimport Example from \"../../components/Example\";\nimport {Fragment} from \"preact\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n Dato un sistema di equazioni lineari, si vuole trovare la sua soluzione.\n
\n
\n In forma matriciale, avrà una matrice dei coefficienti{r`A`}, un vettore\n dei termini noti{r`b`} e un vettore delle incognite\n {r`x`}.\n
\n
\n L'equazione matriciale del sistema è:\n
\n {r`A \\cdot x = b`}\n \n \n
\n Il condizionamento della risoluzione di sistemi lineari è:\n
\n {r`\\frac{{\\color{yellow} \\|A\\| \\cdot \\|A^{-1}\\|} \\cdot \\| \\Delta b \\|}{\\| b \\|}`}\n
\n In particolare, è segnato in giallo nella formula il numero di condizionamento:\n
\n Metodi che trovano la soluzione esatta* di un sistema lineare.\n
\n
\n Tipicamente prevedono la fattorizzazione della matrice dei coefficienti in due\n sottomatrici più facili da risolvere.\n
\n
\n Generalmente hanno una complessità temporale {r`O(n^3)`}.\n
\n \n \n
\n Metodi che trovano una soluzione imperfetta* di\n un sistema lineare.\n
\n
\n Tipicamente prevedono l'applicazione ripetuta di un metodo, in base al quale cambia\n la velocità di convergenza alla soluzione.\n
\n
\n Generalmente hanno una complessità temporale {r`O(n^2)`}.\n
\n \n \n \n \n
\n Se la matrice dei coefficienti del sistema è diagonale, allora è possibile trovare la\n soluzione dividendo ogni termine noto per l'unico coefficiente diverso da zero presente\n nella sua riga:\n
\n {r`x_i = \\frac{b_i}{A_{ii}}`}\n \n \n
\n Se la matrice dei coefficienti del sistema è triangolare inferiore o superiore, allora è\n possibile trovare la soluzione effettuando una sostituzione all'avanti oppure\n all'indietro:\n
\n Se tutti i valori sulla diagonale di {r`A`} sono diversi da 0 (eccetto\n l'ultimo) allora è possibile fattorizzarla in due matrici:\n una {r`L`}triangolare inferiore, e una {r`U`}triangolare\n superiore.\n
\n {r`A = L \\cdot U`}\n \n Abbiamo fatto questo metodo in Algebra Lineare, chiamandolo metodo di Gauss.\n \n
\n La matrice {r`L`} è così composta:\n
\n {r`\n \\begin{cases}\n L_{ii} = 1 \\qquad \\qquad (diagonale)\\\\\n L_{ik} = -\\frac{A_{ik}}{A_{kk}} \\qquad (tri.\\ infer.)\n \\end{cases}\n `}\n \n Sono i moltiplicatori usati per rendere annullare il triangolo inferiore!\n \n
\n La matrice {r`U`} è così composta:\n
\n {r`\n \\begin{cases}\n U_{ik} = A_{ik} \\quad se\\ i \\leq k \\quad (tri.\\ super.)\\\\\n U_{ik} = 0 \\qquad se\\ i > k \\quad (tri.\\ infer.)\n \\end{cases}\n `}\n \n È la parte triangolare superiore di {r`A`}!\n \n
\n Il sistema può essere poi risolto applicando due volte il metodo di sostituzione (all'avanti e\n all'indietro):\n
\n {r`\n \\begin{cases}\n L \\cdot y = b\\\\\n U \\cdot x = y\n \\end{cases}\n `}\n
\n È possibile applicare la fattorizzazione {r`LU`} a qualsiasi matrice\n non-singolare permettendo lo scambio (pivoting) delle righe, potenzialmente aumentando\n la stabilità dell'algoritmo.\n
\n \n Abbiamo fatto questo metodo in Algebra Lineare, chiamandolo metodo di Gauss-Jordan!\n \n
\n Alla formula precedente si aggiunge una matrice di permutazione che indica quali righe\n sono state scambiate:\n
\n {r`P \\cdot A = L \\cdot U`}\n
\n Per massimizzare la stabilità, si cerca di usare come perno l'elemento più grande della\n colonna.\n
\n È possibile anche permettere il pivotingsulle colonne per aumentare\n ulteriormente la stabilità dell'algoritmo, a costo di maggiore costo computazionale:\n
\n {r`P \\cdot A \\cdot Q = L \\cdot U`}\n
\n Per massimizzare la stabilità, si cerca di ordinare in modo decrescente la diagonale,\n assicurandoci che il primo perno sia più grande del secondo e così via.\n
\n Si calcola con una complessità computazionale nell'ordine di {r`O(n)`}.\n
\n \n Fattorizzazione {r`QR`}}>\n
\n Metodo che fornisce una maggiore stabilità a costo di una maggiore complessità\n computazionale.\n
\n
\n La matrice {r`A`} viene fattorizzata in due matrici,\n una ortogonale{r`Q`} e una triangolare superiore\n {r`R`}:\n
\n {r`A = Q \\cdot R`}\n
\n Le matrici si ottengono dal prodotto delle trasformazioni di Householder (che concatenate\n formano {r`Q`}) sulla matrice {r`A`} necessarie a trasformarla\n in una matrice triangolare superiore ({r`R`}).\n
\n \n C'è un bell'esempietto qui.\n \n
\n Una volta fattorizzata, il sistema si può risolvere con:\n
\n {r`\n \\begin{cases}\n y = Q^T \\cdot b\\\\\n R \\cdot x = y\n \\end{cases}\n `}\n
\n {r`\n \\begin{cases}\n G = I - M^{-1} \\cdot A\\\\\n c = M^{-1} \\cdot b\n \\end{cases}\n `}\n
\n Allora la formula generale di un sistema lineare può anche essere scritta in questo modo:\n
\n {r`x = G \\cdot x + c`}\n
\n È particolarmente utile perchè ci permette di definire un algoritmo ricorsivo che\n trovi {r`x`}:\n
\n {r`x_{(i+1)} = G \\cdot x_{(i)} + c`}\n
\n {r`G`} è il metodo, e in base ad esso cambiano stabilità e velocità di\n convergenza.\n
\n
\n Ponendo {r`A = M - N`}, la formula può essere scritta anche in questo modo:\n
\n {r`M \\cdot x_{(i+1)} = N \\cdot x_{(i)} + b`}\n
\n Possiamo ottenere alcuni metodi separando A in tre matrici:\n
\n
\n
La parte diagonale {r`D`}
\n
L'opposto del triangolo inferiore {r`E`}
\n
L'opposto del triangolo superiore {r`F`}
\n
\n {r`A = D - E - F`}\n \n \n
\n Un metodo è convergente se e solo se:\n
\n {r`\\rho (M) < 1`}\n
\n (dove {r`\\rho`} è il raggio spettrale, il massimo autovalore della\n matrice)\n
\n
\n Perchè un metodo sia convergente, è sufficiente che:\n
\n {r`\\| M \\| < 1`}\n \n \n \n \n
\n Il metodo di Jacobi si ottiene ponendo:\n
\n {r`\n \\begin{cases}\n M = D\\\\\n N = E + F\n \\end{cases}\n `}\n
\n Spostamenti simultanei: Permette di ottenere ogni componente\n di {r`x`} indipendentemente dagli altri: è parallelizzabile.\n
\n
\n Se la matrice è diagonale dominante, allora il metodo di\n Jacobi converge sicuramente.\n
\n \n \n
\n Il metodo di Gauss-Seidel si ottiene ponendo:\n
\n {r`\n \\begin{cases}\n M = D - E\\\\\n N = F\n \\end{cases}\n `}\n
\n Ha una velocità di convergenza maggiore o uguale rispetto al metodo di Jacobi.\n
\n
\n Spostamenti successivi: Non è parallelizzabile, perchè ogni componente dipende da\n quelle calcolate in precedenza.\n
\n
\n Se la matrice è diagonale dominante, allora il metodo di\n Gauss-Seidel converge sicuramente.\n
\n \n \n \n )\n}\n","import Intro from \"./00_Intro\";\nimport SistemiLineari from \"./01_SistemiLineari\";\nimport ZeriDiFunzione from \"./02_ZeriDiFunzione\";\nimport Interpolazione from \"./03_Interpolazione\";\nimport InterpolazioneATratti from \"./04_InterpolazioneATratti\";\nimport ApprossimazioneDatiSperimentali from \"./05_ApprossimazioneDatiSperimentali\";\n\n\nexport default function () {\n return (\n
\n
Calcolo Numerico
\n \n \n \n \n \n \n
\n )\n}\n","import {ILatex, Panel, PLatex, Section} from \"bluelib\";\nimport Link from \"../../components/Link\";\nimport MenuList from \"../../components/MenuList\";\nimport Example from \"../../components/Example\";\nimport {Fragment} from \"preact\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n
Prof.ssa Silvia Bonettini
\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 \n
\n Prima di iniziare a studiare Calcolo Numerico, potrebbe essere una buona idea ripassare un\n pochino Algebra Lineare:\n
\n Metodo con cui gestire gli underflow floating point: se la cifra più significativa di\n quelle che devono essere rimosse è 1, allora aumenta di 1 anche quella meno signficativa\n che viene tenuta.\n
\n Un numero reale rappresentato in virgola mobile ha un errore relativo minore o\n uguale alla precisione\n di macchina:\n
\n
\n {r`\\epsilon_x \\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\n troncamento oppure a {r`\\frac{1}{2}`} se viene rappresentato per\n arrotondamento.\n
\n
\n \n \n
\n Associa un valore reale al suo corrispondente valore floating point, utilizzando uno dei\n 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\n floating point dalla macchina che stiamo usando.\n
\n
\n Operazioni tra elementi di {r`\\mathbb{F}`} producono risultati\n in {r`\\mathbb{R}`}, che però decaderanno nuovamente a elementi\n 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 elementi neutri.
\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_{nome\\_var}`}.\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_{num\\_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\n 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^\\star = 4`}.\n
\n
\n Calcolare prima {r`t = fl \\left( \\frac{1}{4} \\right)`} e\n 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\n 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","import {Fragment} from \"preact\";\nimport {ILatex, Panel, PLatex, Section} from \"bluelib\";\nimport Example from \"../../components/Example\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n Interpolare dati sperimentali non fornisce quasi mai un modello del fenomeno.\n
\n
\n Vogliamo costruire una funzione di regressione che, dati molti più dati del grado della\n funzione, minimizzi il quadrato della distanza tra i punti sperimentali e i punti della funzione\n di regressione.\n
\n
\n Denominiamo:\n
\n
\n
{r`{\\color{Orange} f}`}: la funzione \"effettiva\" del fenomeno
\n
{r`{\\color{Yellow} q}`}: la funzione di regressione che costruiamo\n per approssimarlo\n
\n
{r`{\\color{Red} Q }`}: la funzione \"errore di regressione\" da\n minimizzare\n
\n
{r`(\\ x_i, f(x_i)\\ )`}: i punti sperimentali
\n
\n
\n L'obiettivo è minimizzare l'errore di approssimazione{r`Q`}, ovvero:\n
\n {r`{\\color{Yellow} q(x) } = A \\cdot \\alpha`}\n
\n Inoltre, potremo calcolare l'errore attraverso la norma:\n
\n {r`{\\color{Red} Q } = \\| A \\cdot \\alpha - y \\|^2`}\n \n Normalmente, i dati sono molti di più, ma se il numero di\n parametri {r`n`} fosse uguale al numero di dati {r`m`}, allora\n si otterrebbe il polinomio di interpolazione!\n \n \n \n \n \n
\n Trova i coefficienti della combinazione lineare\n {r`{\\color{Yellow} q}`} che meglio approssima tutti\n gli {r`m`} dati sperimentali.\n
\n \n Gli zeri nella {r`\\gamma`} sono i gradi di libertà, sono zero in modo che essi\n diano la norma minima.\n \n \n \n \n )\n}\n"],"sourceRoot":""}
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trigonometria:"),e("p",null,e(wn.k,null,Vn(In()))),e("p",null,e(wn.k,null,Vn(Cn()))),e("p",null,e(wn.k,null,Vn(Pn())))),e(wn.r,{title:"Somma"},e("p",null,"Scomponi in componenti, poi sommali:"),e("p",null,e(wn.k,null,Vn(An()))),e("p",null,"Produce il vettore risultante dall'applicazione della regola del parallelogramma.")),e(wn.r,{title:"Differenza"},e("p",null,"Alla fine è sempre una somma:"),e("p",null,e(wn.k,null,Vn(En()))),e("p",null,"Produce il vettore che parte da ",e(wn.k,null,"w")," e arriva a ",e(wn.k,null,"v"),".")),e(wn.r,{title:"Prodotto scalare"},e("p",null,"Si chiama scalare perchè il risultato è uno scalare, non un vettore."),e("p",null,e(wn.k,null,Vn(Fn()))),e("p",null,"Produce il modulo della proiezione di ",e(wn.k,null,Vn(Ln()))," su ",e(wn.k,null,Vn(qn())),".")),e(wn.r,{title:"Prodotto vettoriale"},e("p",null,"Si chiama vettoriale perchè il risultato è un altro vettore."),e("ul",null,e("li",null,e(wn.k,null,Vn(Dn()))),e("li",null,e(wn.k,null,Vn(bn()))),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.s,{title:"Leggi di Newton"},e(wn.r,{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.k,null,Vn(kn())))),e(wn.r,{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.k,null,Vn(zn())))),e(wn.r,{title:"3ᵃ: Azione e reazione"},e("p",null,"Due corpi esercitano forze uguali e opposte uno sull'altro."),e("p",null,e(wn.k,null,Vn(hn()))))),e(wn.s,{title:"Forza di gravità"},e(wn.r,{title:"Tra due corpi"},e("p",null,"Due corpi puntiformi si attirano uno verso l'altro con forza:"),e("p",null,e(wn.k,null,Vn(_n()))),e("p",null,e(wn.k,null,"G")," è la ",e("i",null,"costante di gravitazione universale")," e vale:"),e("p",null,e(wn.k,null,Vn(gn())))),e(wn.r,{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.k,null,Vn(mn()))),e("p",null,e(wn.k,null,"g")," è la ",e("i",null,"costante di gravità")," della Terra, e vale:"),e("p",null,e(wn.k,null,Vn(vn())))),e(wn.r,{title:"Su pianeti diversi"},e("p",null,"Per pianeti diversi dalla Terra vale la stessa regola:"),e("p",null,e(wn.k,null,Vn(fn()))),e("p",null,"L'unica differenza è che cambia la ",e("i",null,"costante di gravità"),":"),e("p",null,e(wn.k,null,Vn(dn()))),e("p",null,e(wn.k,null,Vn(pn()))))),e(wn.s,{title:"Forze di contatto"},e(wn.r,{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.r,{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.k,null,Vn(sn())))),e(wn.r,{title:"Attrito dinamico"},e("p",null,"Rallenta i corpi che si stanno muovendo finchè essi non si fermano:"),e("p",null,e(wn.k,null,Vn(cn())))),e(wn.r,{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.r,{title:"Elastica"},e("p",null,"Una molla cerca sempre di tornare alla sua posizione indeformata con forza:"),e("p",null,e(wn.k,null,Vn(un()))),e("p",null,"(E' negativa perchè la forza è opposta a quella applicata per deformarla.)"))),e(wn.s,{title:"Cinematica"},e(wn.r,{title:"Spostamento"},e("p",null,"È un vettore che indica la posizione di un corpo rispetto a un'origine."),e("p",null,e(wn.k,null,Vn(on())))),e(wn.r,{title:"Velocità"},e("p",null,"È un vettore che misura la variazione di posizione nel tempo."),e("p",null,e(wn.k,null,Vn(rn()))),e("p",null,"Se si considera un intervallo di tempo infinitesimale si dice ",e("i",null,"velocità istantanea"),":"),e("p",null,e(wn.k,null,Vn(an())))),e(wn.r,{title:"Accelerazione"},e("p",null,"È un vettore che misura la variazione di velocità nel tempo."),e("p",null,e(wn.k,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(wn.k,null,Vn(tn())))),e(wn.r,{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.k,null,Vn(nn()))),e("p",null,"Se la forza risultante è nulla, la quantità di moto non cambia."),e("p",null,e(wn.k,null,Vn(en()))))),e(wn.s,{title:"Moto rettilineo uniforme"},e(wn.r,{title:"Spostamento"},e("p",null,"La ",e("i",null,"legge oraria")," è:"),e("p",null,e(wn.k,null,Vn($e())))),e(wn.r,{title:"Velocità"},e("p",null,"È costante:"),e("p",null,e(wn.k,null,Vn(Ye())))),e(wn.r,{title:"Accelerazione"},e("p",null,"La velocità non varia:"),e("p",null,e(wn.k,null,Vn(Ke())))),e(wn.r,{title:"Forze"},e("p",null,"Si applica la prima legge di Newton:"),e("p",null,e(wn.k,null,"f(t) = 0")))),e(wn.s,{title:"Moto rettilineo uniformemente accelerato"},e(wn.r,{title:"Spostamento"},e("p",null,"La ",e("i",null,"legge oraria")," è:"),e("p",null,e(wn.k,null,Vn(Xe())))),e(wn.r,{title:"Velocità"},e("p",null,"È una retta:"),e("p",null,e(wn.k,null,Vn(Ze())))),e(wn.r,{title:"Accelerazione"},e("p",null,"È costante:"),e("p",null,e(wn.k,null,Vn(He())))),e(wn.r,{title:"Forze"},e("p",null,"Si applica la prima legge di Newton:"),e("p",null,e(wn.k,null,"f(t) = m a")))),e(wn.s,{title:"Moto armonico semplice"},e(wn.r,{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.r,{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.k,null,Vn(je())))),e(wn.r,{title:"Spostamento"},e("p",null,"E' una sinusoide:"),e("p",null,e(wn.k,null,Vn(Je())))),e(wn.r,{title:"Velocità"},e("p",null,"E' la sinusoide dello spostamento, sfasata di ",e(wn.k,null,Vn(Qe())),":"),e("p",null,e(wn.k,null,Vn(We())))),e(wn.r,{title:"Accelerazione"},e("p",null,"E' la sinusoide della velocità, sfasata di ",e(wn.k,null,Vn(Ge())),":"),e("p",null,e(wn.k,null,Vn(Oe())))),e(wn.r,{title:"Forze"},e("p",null,"Si applica la prima legge di Newton:"),e("p",null,e(wn.k,null,"f(t) = m a")))),e(wn.s,{title:"Moti composti"},e(wn.r,{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.r,{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.k,null,Vn(Me())),", sull'asse Y."))),e(wn.s,{title:"Moto circolare uniforme"},e(wn.r,null,e("h3",null,"Velocità angolare"),e("p",null,"Quanto cambia la fase nel tempo."),e("p",null,e(wn.k,null,Vn(Ue())))),e(wn.r,{title:"Fase"},e("p",null,"E' l'angolo percorso dal corpo rispetto alla posizione iniziale."),e("p",null,"Si indica con ",e(wn.k,null,Vn(ye())),", e generalmente si usa in radianti.")),e(wn.r,{title:"Velocità"},e("p",null,"Si applicano le formule per la circonferenza:"),e("p",null,e(wn.k,null,Vn(xe())))),e(wn.r,{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.k,null,Vn(Te())))),e(wn.r,{title:"Forza centripeta"},e("p",null,"È verso il centro e si calcola con:"),e("p",null,e(wn.k,null,Vn(Re()))))),e(wn.s,{title:"Lavoro ed energia"},e(wn.r,{title:"Lavoro"},e("p",null,"E' compiuto da una forza che sposta un corpo."),e("p",null,e(wn.k,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(wn.r,{title:"Energia cinetica"},e("p",null,"Un corpo ha energia cinetica in ogni momento uguale a:"),e("p",null,e(wn.k,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(wn.k,null,Vn(Be())))),e(wn.r,{title:"Energia potenziale gravitazionale"},e("p",null,"Un corpo ha energia potenziale in ogni momento pari a:"),e("p",null,e(wn.k,null,Vn(we()))),e("p",null,"(Con ",e(wn.k,null,"h")," uguale a un altezza scelta come punto di riferimento.)")),e(wn.r,{title:"Energia potenziale elastica"},e("p",null,"Una molla ha sempre energia potenziale elastica pari a:"),e("p",null,e(wn.k,null,Vn(Se())))),e(wn.r,{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.k,null,Vn(Ie())))),e(wn.r,{title:"Potenza"},e("p",null,"È la velocità di trasferimento di energia:"),e("p",null,e(wn.k,null,Vn(Ce()))))),e(wn.s,{title:"Elettrostatica"},e(wn.r,{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(wn.k,null,Vn(Pe())),"."),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(wn.r,{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.s,{title:"Polarizzazione"},e(wn.r,{title:"Polarizzazione"},e("p",null,"E' possibile polarizzare un corpo per accumulare la carica di un segno in una certa zona."))),e(wn.s,null,e(wn.r,{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(wn.s,null,e(wn.r,{title:"Polarizzazione per strofinio"},e("p",null,"Strofinando tra loro due corpi isolanti, essi si ",e("i",null,"polarizzeranno per strofinio"),".")),e(wn.r,{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.r,{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(wn.s,{title:"Forza elettrica"},e(wn.r,{title:"Legge di Coulomb"},e("p",null,"Due corpi carichi si attraggono tra loro con forza:"),e("p",null,e(wn.k,null,Vn(Ae()))),e("p",null,e(wn.k,null,Vn(Ee()))," è la ",e("i",null,"costante di Coulomb"),", e vale ",e(wn.k,null,Vn(Fe())),".")),e(wn.r,{title:"Permeabilità dello spazio vuoto"},e("p",null,"La costante ",e(wn.k,null,Vn(Le()))," è in realtà dipendente da un altra costante, ",e(wn.k,null,Vn(qe())),", la ",e("i",null,"permeabilità del vuoto"),"."),e("p",null,e(wn.k,null,Vn(De()))),e("p",null,e(wn.k,null,Vn(be())))),e(wn.r,{title:"Campo elettrico"},e("p",null,"Misura che forza viene applicata in ogni punto su una carica unitaria:"),e("p",null,e(wn.k,null,Vn(ke())))),e(wn.r,{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(wn.k,null,Vn(ze()))),e("p",null,"Se il campo elettrico è uniforme, se ne può calcolare facilmente il valore:"),e("p",null,e(wn.k,null,Vn(he()))),e("p",null,e(wn.u,null,"Circa. E' una specie di integrale..."))),e(wn.r,{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.k,null,Vn(_e()))),e("p",null,"Ovvero, i campi elettrostatici sono generati dalle cariche elettriche."))),e(wn.s,{title:"Energia elettrica"},e(wn.r,{title:"Energia potenziale elettrica"},e("p",null,"Un corpo carico vicino ad altre cariche possiede un'",e("i",null,"energia potenziale elettrica"),e(wn.k,null,Vn(ge())),"."))),e(wn.s,{title:"Circuiti elettrici"},e(wn.r,{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.k,null,Vn(me()))),e("p",null,"La sua unità di misura è il Volt (",e(wn.k,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(wn.r,{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.k,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(wn.k,null,Vn(de())),").")),e(wn.r,{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.r,{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.r,{title:"Potenza elettrica"},e("p",null,"Possiamo calcolare la potenza di un circuito:"),e("p",null,e(wn.k,null,Vn(pe()))))),e(wn.s,{title:"Elementi di un circuito"},e(wn.r,{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.k,null,Vn(se()))),e("p",null,"Dove ",e(wn.k,null,Vn(ce()))," è una costante detta ",e("i",null,"resistenza")," con unità di misura Ohm (",e(wn.k,null,Vn(ue())),")."),e("p",null,"La resistenza di un conduttore vale:"),e("p",null,e(wn.k,null,Vn(oe()))),e("p",null,e(wn.k,null,Vn(re()))," è la ",e("i",null,"resistività")," del materiale, e varia in base alla temperatura:"),e("p",null,e(wn.k,null,Vn(ie())))),e(wn.r,{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(wn.k,null,Vn(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.k,null,Vn(le()))),e("p",null,"Dove ",e(wn.k,null,Vn(te()))," è la ",e("i",null,"costante dielettrica relativa")," del materiale inserito, ",e(wn.k,null,Vn(ne()))," l'area di una armatura e ",e(wn.k,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(wn.k,null,Vn($())),")")),e(wn.r,{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.r,{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.s,{title:"Principi di Kirchhoff"},e(wn.r,{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.r,{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.s,{title:"Serie e Parallelo"},e(wn.r,{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.r,{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.s,{title:"Resistenze equivalenti"},e(wn.r,{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.k,null,Vn(Y())))),e(wn.r,{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.k,null,Vn(K()))))),e(wn.s,{title:"Condensatori equivalenti"},e(wn.r,{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.k,null,Vn(X())))),e(wn.r,{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.k,null,Vn(Z()))))),e(wn.s,{title:"Magnetismo"},e(wn.r,{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.k,null,Vn(H()))," (",e(wn.k,null,Vn(j())),")")),e(wn.r,{title:"Campo magnetico"},e("p",null,"Come un campo elettrico, ma per i magneti."),e("p",null,"Il suo simbolo è ",e(wn.k,null,Vn(J())),", e la sua unità di misura è il Tesla (",e(wn.k,null,"T"),").")),e(wn.r,{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.k,null,Vn(Q()))),e("p",null,e(wn.k,null,Vn(W()))),e("p",null,"La sua unità di misura è il Weber (",e(wn.k,null,Vn(G())),").")),e(wn.r,{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.r,{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.k,null,Vn(O()))))),e(wn.s,{title:"Forze magnetiche"},e(wn.r,{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.k,null,Vn(M()))),e("p",null,"Dove ",e(wn.k,null,Vn(U()))," è l'intensità del campo magnetico e ",e(wn.k,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(wn.r,{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.k,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(wn.k,null,Vn(T()))," è la corrente elettrica, ",e(wn.k,null,Vn(R()))," è un vettore che punta nella direzione di scorrimento della corrente e ha come modulo la lunghezza del conduttore."))),e(wn.s,{title:"Campi magnetici"},e(wn.r,{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.r,{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.k,null,Vn(V())))),e(wn.r,{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.k,null,"B")," prodotto da un filo in cui passa una corrente continua ",e(wn.k,null,"I")," alla distanza ",e(wn.k,null,"s")," è:"),e("p",null,e(wn.k,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(wn.s,{title:"Induzione elettromagnetica"},e(wn.r,{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.k,null,Vn(B()))),e("p",null,"Dove ",e(wn.k,null,"v")," è la velocità del conduttore, ",e(wn.k,null,"B")," è l'intensità del campo magnetico ed ",e(wn.k,null,"L")," è la lunghezza del conduttore.")),e(wn.r,{title:"Flusso magnetico in una spira"},e("p",null,"In un campo magnetico ",e(wn.k,null,Vn(w()))," uniforme e perpendicolare al piano di una spira di area ",e(wn.k,null,Vn(S())),", il flusso magnetico si può determinare con la ",e("i",null,"Legge di Faraday-Neumann-Lenz"),":"),e("p",null,e(wn.k,null,Vn(I()))))),e(wn.s,null,e(wn.r,{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.k,null,Vn(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.r,{title:"Faraday in un solenoide"},e("p",null,"In un solenoide, la forza elettromotrice indotta è uguale a:"),e("p",null,e(wn.k,null,Vn(P()))),e("p",null,"Dove ",e(wn.k,null,Vn(A()))," è il numero delle spire del solenoide.")),e(wn.r,{title:"Legge di Ampère-Maxwell"},e("p",null,"Correnti o campi elettrici variabili creano un campo magnetico."))),e(wn.s,{title:"Elettromagnetismo"},e(wn.r,{title:"Onde elettromagnetiche"},e("p",null,"Nel vuoto, il campo elettrico ",e(wn.k,null,Vn(E()))," e il campo magnetico ",e(wn.k,null,Vn(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.k,null,Vn(L()))),e("p",null,"Dove ",e(wn.k,null,Vn(q()))," è la velocità delle onde (luce) nel vuoto, e a sua volta è uguale a:"),e("p",null,e(wn.k,null,Vn(D())))),e(wn.r,{title:"Formula delle onde"},e("p",null,e(wn.k,null,Vn(b()))),e("p",null,"Dove ",e(wn.k,null,Vn(k()))," è l'ampiezza massima che può avere l'onda, ",e(wn.k,null,Vn(z()))," è il vettore d'onda, ",e(wn.k,null,Vn(h()))," la frequenza angolare e ",e(wn.k,null,Vn(_()))," la fase."))),e(wn.s,{title:"Spettroscopia"},e(wn.r,{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.k,null,Vn(g()))),e("p",null,"Con ",e(wn.k,null,Vn(m())),", detta costante di Rydberg, e ",e(wn.k,null,Vn(v()))," un numero intero.")),e(wn.r,{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.k,null,Vn(f())),", la costante di Planck, ovvero il valore minimo possibile per la carica (talvolta espressa come ",e(wn.k,null,Vn(d())),"."))),e(wn.s,null,e(wn.r,{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.k,null,Vn(p()))),e("p",null,"Ancora, il raggio delle orbite è uguale a:"),e("p",null,e(wn.k,null,Vn(s()))),e("p",null,"Con ",e(wn.k,null,Vn(c())),"."),e("p",null,"Infine, in ogni stato, l'energia è pari a:"),e("p",null,e(wn.k,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(wn.s,null,e(wn.r,{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.s,{title:"Semiconduttori"},e(wn.r,{title:"Semiconduttori"},e("p",null,e(wn.u,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.k,null,Vn(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.k,null,Vn(r())),", allora il corpo è un semiconduttore.")),e(wn.r,{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(wn.r,{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(wn.r,{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(wn.s,{title:e("span",null,"Ottica ",e("small",null,"(non l'abbiamo fatta)"))},e(wn.r,{title:"Assorbimento e riflessione"},e("p",null,"I corpi possono assorbire o riflettere le onde elettromagnetiche che li colpiscono.")),e(wn.r,{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.k,null,Vn(i()))," variabile in base alla temperatura."),e("p",null,e(wn.k,null,Vn(a()))," è costante.")),e(wn.r,{title:"Teoria di Planck per il corpo nero"},e("p",null,"L'energia assorbita e emessa dai corpi neri è quantizzata.")),e(wn.r,{title:"Fotone"},e("p",null,"Un onda magnetica con un quanto di energia è detta ",e("i",null,"fotone"),":"),e("p",null,e(wn.k,null,Vn(l())))),e(wn.r,{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 e("span",{class:a.a.plus},n.children)}}).call(this,t("hosL").h)}}]);
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{Latex, LatexDefaultDisplay, LatexDefaultInline, LatexDisplay, Panel, Section, Todo} 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 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\n 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\n 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 Cariche opposte si attraggono;\n cariche uguali si respingono.\n
\n \n \n
\n Più ioni ha un corpo, meglio la carica\n si muove attraverso di esso.\n
\n
\n I corpi in cui la carica si muove bene sono conduttori, mentre quelli in cui si\n 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\n zona.\n
\n \n \n \n \n
\n Se un corpo conduttore è in contatto con la Terra, le cariche su di esso\n saranno equilibrate e il corpo diventerà elettricamente neutro (con stesso numero\n 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\n contatto.\n
\n \n \n
\n Se un corpo conduttore ha cariche \"esterne\" di un certo segno vicino, esso\n avrà tutte le cariche del segno opposto in equilibrio vicino alle cariche\n esterne, e tutte le cariche dello stesso segno più lontano possibile da\n esse.\n
\n
\n Mettendo a terra il conduttore, nuove cariche del segno opposto saranno\n 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\n 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 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\n {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\n 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\n 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\n 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\n 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 Dove {r`\\kappa`} è la costante dielettrica relativa del materiale\n inserito, {r`A`} l'area di una armatura e {r`s`} la\n distanza tra le due armature.\n
\n
\n Se il campo elettrico creatosi tra le due armature supera la rigidità\n dielettrica del condensatore, la carica immagazzinata viene persa e ha luogo\n 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\n 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\n 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\n 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\n 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\n equivalente dalla resistenza di:\n
\n Nei circuiti in parallelo, tutte i condensatori possono essere sostituite con uno\n 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\n facilmente.\n
\n Dove {r`\\vec{B}`} è l'intensità del campo magnetico\n 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\n 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\n vettore che punta nella direzione di scorrimento della corrente e ha come modulo la\n lunghezza del conduttore.\n
\n \n \n \n \n
\n Una spira in cui passa corrente produce un campo magnetico perpendicolare al piano\n 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 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\n attraversati da correnti\n opposte si respingono.\n
\n \n \n \n \n
\n Un conduttore perpendicolare ad un campo magnetico può ottenere una differenza di\n potenziale se messo in movimento in un direzione perpendicolare alla direzione del\n conduttore e del campo.\n
\n
\n La differenza di potenziale si crea a causa della forza magnetica, che fa spostare tutti\n 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\n 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\n spira di area {r`A`}, il flusso magnetico si può determinare con la Legge\n 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\n nel tempo del flusso magnetico nello stesso percorso.\n
\n Il meno è dovuto alla Legge di\n 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\n magnetico {r`B`} sono perpendicolari tra loro e la direzione di\n 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\n uguale a:\n
\n Dove {r`A_{max}`} è l'ampiezza massima che può avere\n l'onda, {r`\\frac{2 \\pi}{\\lambda} = \\left | \\vec{k} \\right |`} è il\n vettore d'onda, {r`\\omega`} la frequenza angolare\n 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 Con {r`R = 1.097 \\cdot 10^7 \\frac{1}{m}`}, detta costante di Rydberg,\n 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\n 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\n minimo possibile per la carica (talvolta espressa\n 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 Due elettroni non possono occupare lo stesso stato.\n
\n
\n Questo modello funziona solo per atomi con numero atomico basso. Atomi con molti\n 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\n 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\n gap dell'ordine di grandezza maggiore di {r`10^1 eV`}, allora il corpo è\n un isolante.\n
\n
\n Se invece la banda di emissione si sovrappone a un altra, allora il corpo è un\n conduttore.\n
\n
\n Se il gap è invece dell'ordine di grandezza di {r`1 eV`}, allora il corpo\n è un semiconduttore.\n
\n \n \n
\n Legami in cui mancano elettroni.\n
\n
\n Elettroni di altri legami possono spostarsi per colmare\n le lacune, creandone altre, e spostandole in direzione opposta a quella\n della corrente.\n
\n \n \n
\n Se si inserisce in un cristallo semiconduttore si inserisce un atomo con numero atomico\n diverso, si otterrà:\n
\n
\n
Con numero atomico maggiore, un semiconduttore di tipo N con elettroni\n in eccesso liberi di scorrere.\n
\n
Con numero atomico minore, un semiconduttore di tipo P con lacune\n in eccesso libere di catturare elettroni da altri legami.\n
\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\n 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\n rifletterne nessuna.\n
\n
\n Le onde assorbite vengono poi riemesse sotto forma di un onda\n 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\n 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\n 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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\vec{v}_y \right | = \left | \vec{v} \right | \cos \alpha`))))),l(i.r,{title:"Somma"},l("p",null,"Scomponi in componenti, poi sommali:"),l("p",null,l(i.k,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(i.r,{title:"Differenza"},l("p",null,"Alla fine è sempre una somma:"),l("p",null,l(i.k,null,Ne(s||(s=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(i.k,null,"w")," e arriva a ",l(i.k,null,"v"),".")),l(i.r,{title:"Prodotto scalare"},l("p",null,"Si chiama scalare perchè il risultato è uno scalare, non un vettore."),l("p",null,l(i.k,null,Ne(p||(p=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(i.k,null,Ne(d||(d=Be`\vec{a}`)))," su ",l(i.k,null,Ne(m||(m=Be`\vec{b}`))),".")),l(i.r,{title:"Prodotto vettoriale"},l("p",null,"Si chiama vettoriale perchè il risultato è un altro vettore."),l("ul",null,l("li",null,l(i.k,null,Ne(v||(v=Be`\vec{c} = \vec{a} \times \vec{b}`)))),l("li",null,l(i.k,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(i.s,{title:"Leggi di Newton"},l(i.r,{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.k,null,Ne(f||(f=Be`\Sigma \vec{F} = 0 \Longleftrightarrow \Delta v = 0`))))),l(i.r,{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.k,null,Ne(z||(z=Be`\Sigma \vec{F} = m \vec{a}`))))),l(i.r,{title:"3ᵃ: Azione e reazione"},l("p",null,"Due corpi esercitano forze uguali e opposte uno sull'altro."),l("p",null,l(i.k,null,Ne(h||(h=Be`\vec{F}_{21} = -\vec{F}_{12}`)))))),l(i.s,{title:"Forza di gravità"},l(i.r,{title:"Tra due corpi"},l("p",null,"Due corpi puntiformi si attirano uno verso l'altro con forza:"),l("p",null,l(i.k,null,Ne(k||(k=Be`\left | \vec{F} \right | = G \frac{m_1 m_2}{s^2}`)))),l("p",null,l(i.k,null,"G")," è la ",l("i",null,"costante di gravitazione universale")," e vale:"),l("p",null,l(i.k,null,Ne(_||(_=Be`G = 6.67 \cdot 10^{-11} \frac{N m^2}{{kg}^2}`))))),l(i.r,{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.k,null,Ne(b||(b=Be`\left | \vec{F} \right | = g m`)))),l("p",null,l(i.k,null,"g")," è la ",l("i",null,"costante di gravità")," della Terra, e vale:"),l("p",null,l(i.k,null,Ne(q||(q=Be`g = 9.81 \frac{m}{s^2}`))))),l(i.r,{title:"Su pianeti diversi"},l("p",null,"Per pianeti diversi dalla Terra vale la stessa regola:"),l("p",null,l(i.k,null,Ne(L||(L=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(i.k,null,Ne(D||(D=Be`g_{luna} = 1.62 \frac{m}{s^2}`)))),l("p",null,l(i.k,null,Ne(F||(F=Be`g_{marte} = 3.71 \frac{m}{s^2}`)))))),l(i.s,{title:"Forze di contatto"},l(i.r,{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.r,{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.k,null,Ne(S||(S=Be`\left | \vec{F} \right | \leq \mu_{s} \left | \vec{F}_{normale} \right |`))))),l(i.r,{title:"Attrito dinamico"},l("p",null,"Rallenta i corpi che si stanno muovendo finchè essi non si fermano:"),l("p",null,l(i.k,null,Ne(E||(E=Be`\left | \vec{F} \right | \leq \mu_{d} \left | \vec{F}_{normale} \right |`))))),l(i.r,{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.r,{title:"Elastica"},l("p",null,"Una molla cerca sempre di tornare alla sua posizione indeformata con forza:"),l("p",null,l(i.k,null,Ne(P||(P=Be`F = -k x`)))),l("p",null,"(E' negativa perchè la forza è opposta a quella applicata per deformarla.)"))),l(i.s,{title:"Cinematica"},l(i.r,{title:"Spostamento"},l("p",null,"È un vettore che indica la posizione di un corpo rispetto a un'origine."),l("p",null,l(i.k,null,Ne(C||(C=Be`\Delta \vec{s} = \vec{s}(fine) - \vec{s}(inizio)`))))),l(i.r,{title:"Velocità"},l("p",null,"È un vettore che misura la variazione di posizione nel tempo."),l("p",null,l(i.k,null,Ne(I||(I=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(i.k,null,Ne(A||(A=Be`\vec{v} = \lim_{\Delta t \to 0} \frac{\Delta \vec{s}}{\Delta t} = \frac{d \vec{s}}{dt}`))))),l(i.r,{title:"Accelerazione"},l("p",null,"È un vettore che misura la variazione di velocità nel tempo."),l("p",null,l(i.k,null,Ne(w||(w=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(i.k,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(i.r,{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.k,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(i.k,null,Ne(V||(V=Be`\Sigma \vec{F} = 0 \Longleftrightarrow \Delta \vec{p} = 0`)))))),l(i.s,{title:"Moto rettilineo uniforme"},l(i.r,{title:"Spostamento"},l("p",null,"La ",l("i",null,"legge oraria")," è:"),l("p",null,l(i.k,null,Ne(R||(R=Be`s(t) = v \cdot \Delta t + s(0)`))))),l(i.r,{title:"Velocità"},l("p",null,"È costante:"),l("p",null,l(i.k,null,Ne(T||(T=Be`v(t) = k`))))),l(i.r,{title:"Accelerazione"},l("p",null,"La velocità non varia:"),l("p",null,l(i.k,null,Ne(y||(y=Be`a(t) = 0`))))),l(i.r,{title:"Forze"},l("p",null,"Si applica la prima legge di Newton:"),l("p",null,l(i.k,null,"f(t) = 0")))),l(i.s,{title:"Moto rettilineo uniformemente accelerato"},l(i.r,{title:"Spostamento"},l("p",null,"La ",l("i",null,"legge oraria")," è:"),l("p",null,l(i.k,null,Ne(M||(M=Be`s(t) = \frac{1}{2} a \cdot (\Delta t)^2 + v(0) \cdot (\Delta t) + s(0)`))))),l(i.r,{title:"Velocità"},l("p",null,"È una retta:"),l("p",null,l(i.k,null,Ne(U||(U=Be`v(t) = a \Delta t + v(0)`))))),l(i.r,{title:"Accelerazione"},l("p",null,"È costante:"),l("p",null,l(i.k,null,Ne(x||(x=Be`a(t) = k`))))),l(i.r,{title:"Forze"},l("p",null,"Si applica la prima legge di Newton:"),l("p",null,l(i.k,null,"f(t) = m a")))),l(i.s,{title:"Moto armonico semplice"},l(i.r,{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.r,{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.k,null,Ne(O||(O=Be`\omega = \frac{2 \pi}{T}`))))),l(i.r,{title:"Spostamento"},l("p",null,"E' una sinusoide:"),l("p",null,l(i.k,null,Ne(G||(G=Be`s(t) = A \sin (\omega \cdot t + \phi)`))))),l(i.r,{title:"Velocità"},l("p",null,"E' la sinusoide dello spostamento, sfasata di ",l(i.k,null,Ne(Q||(Q=Be`\frac{\pi}{2}`))),":"),l("p",null,l(i.k,null,Ne(J||(J=Be`v(t) = A \sin (\omega \cdot t + \phi + \frac{\pi}{2})`))))),l(i.r,{title:"Accelerazione"},l("p",null,"E' la sinusoide della velocità, sfasata di ",l(i.k,null,Ne(W||(W=Be`\pi`))),":"),l("p",null,l(i.k,null,Ne(j||(j=Be`a(t) = A \sin (\omega \cdot t + \phi + \pi)`))))),l(i.r,{title:"Forze"},l("p",null,"Si applica la prima legge di Newton:"),l("p",null,l(i.k,null,"f(t) = m a")))),l(i.s,{title:"Moti composti"},l(i.r,{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.r,{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.k,null,Ne(H||(H=Be`\frac{\pi}{2}`))),", sull'asse Y."))),l(i.s,{title:"Moto circolare uniforme"},l(i.r,null,l("h3",null,"Velocità angolare"),l("p",null,"Quanto cambia la fase nel tempo."),l("p",null,l(i.k,null,Ne(Z||(Z=Be`\omega = \frac{2 \pi}{T}`))))),l(i.r,{title:"Fase"},l("p",null,"E' l'angolo percorso dal corpo rispetto alla posizione iniziale."),l("p",null,"Si indica con ",l(i.k,null,Ne(X||(X=Be`\phi`))),", e generalmente si usa in radianti.")),l(i.r,{title:"Velocità"},l("p",null,"Si applicano le formule per la circonferenza:"),l("p",null,l(i.k,null,Ne(K||(K=Be`v = \frac{\Delta s}{t} = \frac{2 \pi \cdot r}{T} = \omega r`))))),l(i.r,{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.k,null,Ne(Y||(Y=Be`a = \frac{v^2}{r} = r \cdot \omega^2 = v \cdot \omega`))))),l(i.r,{title:"Forza centripeta"},l("p",null,"È verso il centro e si calcola con:"),l("p",null,l(i.k,null,Ne($||($=Be`F = m \cdot a`)))))),l(i.s,{title:"Lavoro ed energia"},l(i.r,{title:"Lavoro"},l("p",null,"E' compiuto da una forza che sposta un corpo."),l("p",null,l(i.k,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(i.r,{title:"Energia cinetica"},l("p",null,"Un corpo ha energia cinetica in ogni momento uguale a:"),l("p",null,l(i.k,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(i.k,null,Ne(al||(al=Be`\Delta E_c = W`))))),l(i.r,{title:"Energia potenziale gravitazionale"},l("p",null,"Un corpo ha energia potenziale in ogni momento pari a:"),l("p",null,l(i.k,null,Ne(il||(il=Be`E_{p_g} = m \cdot g \cdot h`)))),l("p",null,"(Con ",l(i.k,null,"h")," uguale a un altezza scelta come punto di riferimento.)")),l(i.r,{title:"Energia potenziale elastica"},l("p",null,"Una molla ha sempre energia potenziale elastica pari a:"),l("p",null,l(i.k,null,Ne(nl||(nl=Be`E_{p_e} = \frac{1}{2} k x^2`))))),l(i.r,{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.k,null,Ne(tl||(tl=Be`E = E_k + E_p`))))),l(i.r,{title:"Potenza"},l("p",null,"È la velocità di trasferimento di energia:"),l("p",null,l(i.k,null,Ne(ol||(ol=Be`P = \frac{\Delta E}{\Delta t}`)))))),l(i.s,{title:"Elettrostatica"},l(i.r,{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.k,null,Ne(rl||(rl=Be`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.r,{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.s,{title:"Polarizzazione"},l(i.r,{title:"Polarizzazione"},l("p",null,"E' possibile polarizzare un corpo per accumulare la carica di un segno in una certa zona."))),l(i.s,null,l(i.r,{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.s,null,l(i.r,{title:"Polarizzazione per strofinio"},l("p",null,"Strofinando tra loro due corpi isolanti, essi si ",l("i",null,"polarizzeranno per strofinio"),".")),l(i.r,{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.r,{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.s,{title:"Forza elettrica"},l(i.r,{title:"Legge di Coulomb"},l("p",null,"Due corpi carichi si attraggono tra loro con forza:"),l("p",null,l(i.k,null,Ne(ul||(ul=Be`\left | \vec{F}_{elettrica} \right | = \frac{-k \cdot q_1 \cdot q_2}{s^2}`)))),l("p",null,l(i.k,null,Ne(cl||(cl=Be`k`)))," è la ",l("i",null,"costante di Coulomb"),", e vale ",l(i.k,null,Ne(sl||(sl=Be`k = 8.99 \cdot 10^9 \frac{N \cdot m^2}{C^2}`))),".")),l(i.r,{title:"Permeabilità dello spazio vuoto"},l("p",null,"La costante ",l(i.k,null,Ne(pl||(pl=Be`k`)))," è in realtà dipendente da un altra costante, ",l(i.k,null,Ne(dl||(dl=Be`\epsilon_0`))),", la ",l("i",null,"permeabilità del vuoto"),"."),l("p",null,l(i.k,null,Ne(ml||(ml=Be`k = \frac{1}{4 \pi \cdot \epsilon_0}`)))),l("p",null,l(i.k,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(i.r,{title:"Campo elettrico"},l("p",null,"Misura che forza viene applicata in ogni punto su una carica unitaria:"),l("p",null,l(i.k,null,Ne(gl||(gl=Be`\vec{E} = \frac{\vec{F}_{elettrica}}{q} = \frac{-k \cdot q}{s^2}`))))),l(i.r,{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.k,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(i.k,null,Ne(zl||(zl=Be`\Phi_E = \vec{E} \cdot \vec{A} = E_\perp \cdot A \cdot \cos(\alpha)`)))),l("p",null,l(i.u,null,"Circa. E' una specie di integrale..."))),l(i.r,{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.k,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(i.s,{title:"Energia elettrica"},l(i.r,{title:"Energia potenziale elettrica"},l("p",null,"Un corpo carico vicino ad altre cariche possiede un'",l("i",null,"energia potenziale elettrica"),l(i.k,null,Ne(kl||(kl=Be`U_e`))),"."))),l(i.s,{title:"Circuiti elettrici"},l(i.r,{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.k,null,Ne(_l||(_l=Be`V = \frac{U_e}{q}`)))),l("p",null,"La sua unità di misura è il Volt (",l(i.k,null,Ne(bl||(bl=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(i.r,{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.k,null,Ne(ql||(ql=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(i.k,null,Ne(Ll||(Ll=Be`A`))),").")),l(i.r,{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.r,{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.r,{title:"Potenza elettrica"},l("p",null,"Possiamo calcolare la potenza di un circuito:"),l("p",null,l(i.k,null,Ne(Dl||(Dl=Be`P = \frac{\Delta U_e}{\Delta t} = I \cdot \Delta V = I^2 \cdot R = \frac{(\Delta V)^2}{R}`)))))),l(i.s,{title:"Elementi di un circuito"},l(i.r,{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.k,null,Ne(Fl||(Fl=Be`V = R \cdot I`)))),l("p",null,"Dove ",l(i.k,null,Ne(Sl||(Sl=Be`R`)))," è una costante detta ",l("i",null,"resistenza")," con unità di misura Ohm (",l(i.k,null,Ne(El||(El=Be`\Omega`))),")."),l("p",null,"La resistenza di un conduttore vale:"),l("p",null,l(i.k,null,Ne(Pl||(Pl=Be`R = \rho \frac{L_{unghezza}}{A_{rea}}`)))),l("p",null,l(i.k,null,Ne(Cl||(Cl=Be`\rho`)))," è la ",l("i",null,"resistività")," del materiale, e varia in base alla temperatura:"),l("p",null,l(i.k,null,Ne(Il||(Il=Be`\rho = \rho_0 (1 + \alpha(T - T_0))`))))),l(i.r,{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.k,null,Ne(Al||(Al=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(i.k,null,Ne(wl||(wl=Be`C_{nuova} = \kappa \cdot \frac{\epsilon_0 \cdot A}{s}`)))),l("p",null,"Dove ",l(i.k,null,Ne(Bl||(Bl=Be`\kappa`)))," è la ",l("i",null,"costante dielettrica relativa")," del materiale inserito, ",l(i.k,null,Ne(Nl||(Nl=Be`A`)))," l'area di una armatura e ",l(i.k,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(i.k,null,Ne(Rl||(Rl=Be`Fa`))),")")),l(i.r,{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.r,{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.s,{title:"Principi di Kirchhoff"},l(i.r,{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.r,{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.s,{title:"Serie e Parallelo"},l(i.r,{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.r,{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.s,{title:"Resistenze equivalenti"},l(i.r,{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.k,null,Ne(Tl||(Tl=Be`R_{serie} = \sum_{i=1}^{n} R_i`))))),l(i.r,{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.k,null,Ne(yl||(yl=Be`R_{parallelo} = \frac{1}{\sum_{i=1}^{n} \frac{1}{R_i}}`)))))),l(i.s,{title:"Condensatori equivalenti"},l(i.r,{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.k,null,Ne(Ml||(Ml=Be`C_{serie} = \frac{1}{\sum_{i=1}^{n} \frac{1}{C_i}}`))))),l(i.r,{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.k,null,Ne(Ul||(Ul=Be`C_{parallelo} = \sum_{i=1}^{n} C_n`)))))),l(i.s,{title:"Magnetismo"},l(i.r,{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.k,null,Ne(xl||(xl=Be`\mu_0 = 4 \pi \cdot 10^{-7} \frac{H}{m}`)))," (",l(i.k,null,Ne(Ol||(Ol=Be`\frac{N}{A^2}`))),")")),l(i.r,{title:"Campo magnetico"},l("p",null,"Come un campo elettrico, ma per i magneti."),l("p",null,"Il suo simbolo è ",l(i.k,null,Ne(Gl||(Gl=Be`B`))),", e la sua unità di misura è il Tesla (",l(i.k,null,"T"),").")),l(i.r,{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.k,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(i.k,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(i.k,null,Ne(Wl||(Wl=Be`Wb = T \cdot m^2`))),").")),l(i.r,{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.r,{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.k,null,Ne(jl||(jl=Be`\Phi_B = \mu_0 \cdot I`)))))),l(i.s,{title:"Forze magnetiche"},l(i.r,{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.k,null,Ne(Hl||(Hl=Be`\vec{F}_{B} = q \cdot (\vec{v} \times \vec{B})`)))),l("p",null,"Dove ",l(i.k,null,Ne(Zl||(Zl=Be`\vec{B}`)))," è l'intensità del campo magnetico e ",l(i.k,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(i.r,{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.k,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(i.k,null,Ne(Yl||(Yl=Be`I`)))," è la corrente elettrica, ",l(i.k,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(i.s,{title:"Campi magnetici"},l(i.r,{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.r,{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.k,null,Ne(le||(le=Be`\left | \vec{B} \right | = \mu_0 \cdot I \cdot \frac{A_{vvolgimenti}}{L_{unghezzafilo}}`))))),l(i.r,{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.k,null,"B")," prodotto da un filo in cui passa una corrente continua ",l(i.k,null,"I")," alla distanza ",l(i.k,null,"s")," è:"),l("p",null,l(i.k,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(t.a,null,"enti"),l(n.a,null,"opp"),l(t.a,null,"oste")," si respingono."))),l(i.s,{title:"Induzione elettromagnetica"},l(i.r,{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.k,null,Ne(ae||(ae=Be`\Delta V_{indotta} = v \cdot B \cdot L`)))),l("p",null,"Dove ",l(i.k,null,"v")," è la velocità del conduttore, ",l(i.k,null,"B")," è l'intensità del campo magnetico ed ",l(i.k,null,"L")," è la lunghezza del conduttore.")),l(i.r,{title:"Flusso magnetico in una spira"},l("p",null,"In un campo magnetico ",l(i.k,null,Ne(ie||(ie=Be`B`)))," uniforme e perpendicolare al piano di una spira di area ",l(i.k,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(i.k,null,Ne(te||(te=Be`\Phi_B = \vec{B} \cdot \vec{A} = B \cdot A \cdot \cos(\alpha)`)))))),l(i.s,null,l(i.r,{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.k,null,Ne(oe||(oe=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(i.r,{title:"Faraday in un solenoide"},l("p",null,"In un solenoide, la forza elettromotrice indotta è uguale a:"),l("p",null,l(i.k,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(i.k,null,Ne(ue||(ue=Be`N`)))," è il numero delle spire del solenoide.")),l(i.r,{title:"Legge di Ampère-Maxwell"},l("p",null,"Correnti o campi elettrici variabili creano un campo magnetico."))),l(i.s,{title:"Elettromagnetismo"},l(i.r,{title:"Onde elettromagnetiche"},l("p",null,"Nel vuoto, il campo elettrico ",l(i.k,null,Ne(ce||(ce=Be`E`)))," e il campo magnetico ",l(i.k,null,Ne(se||(se=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(i.k,null,Ne(pe||(pe=Be`E = c \cdot B`)))),l("p",null,"Dove ",l(i.k,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(i.k,null,Ne(me||(me=Be`c = \frac{1}{\sqrt{\epsilon_0 \cdot \mu_0}} = 3.00 \cdot 10^8 \frac{m}{s}`))))),l(i.r,{title:"Formula delle onde"},l("p",null,l(i.k,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(i.k,null,Ne(ge||(ge=Be`A_{max}`)))," è l'ampiezza massima che può avere l'onda, ",l(i.k,null,Ne(fe||(fe=Be`\frac{2 \pi}{\lambda} = \left | \vec{k} \right |`)))," è il vettore d'onda, ",l(i.k,null,Ne(ze||(ze=Be`\omega`)))," la frequenza angolare e ",l(i.k,null,Ne(he||(he=Be`\phi`)))," la fase."))),l(i.s,{title:"Spettroscopia"},l(i.r,{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.k,null,Ne(ke||(ke=Be`\frac{1}{\lambda} = R \left ( \frac{1}{4} - \frac{1}{n^2} \right )`)))),l("p",null,"Con ",l(i.k,null,Ne(_e||(_e=Be`R = 1.097 \cdot 10^7 \frac{1}{m}`))),", detta costante di Rydberg, e ",l(i.k,null,Ne(be||(be=Be`n`)))," un numero intero.")),l(i.r,{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.k,null,Ne(qe||(qe=Be`h`))),", la costante di Planck, ovvero il valore minimo possibile per la carica (talvolta espressa come ",l(i.k,null,Ne(Le||(Le=Be`\hbar = \left ( \frac{h}{2 \pi} \right )`))),"."))),l(i.s,null,l(i.r,{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.k,null,Ne(De||(De=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(i.k,null,Ne(Fe||(Fe=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(i.k,null,Ne(Se||(Se=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(i.k,null,Ne(Ee||(Ee=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(i.s,null,l(i.r,{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.s,{title:"Semiconduttori"},l(i.r,{title:"Semiconduttori"},l("p",null,l(i.u,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.k,null,Ne(Pe||(Pe=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(i.k,null,Ne(Ce||(Ce=Be`1 eV`))),", allora il corpo è un semiconduttore.")),l(i.r,{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.r,{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.r,{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.s,{title:l("span",null,"Ottica ",l("small",null,"(non l'abbiamo fatta)"))},l(i.r,{title:"Assorbimento e riflessione"},l("p",null,"I corpi possono assorbire o riflettere le onde elettromagnetiche che li colpiscono.")),l(i.r,{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.k,null,Ne(Ie||(Ie=Be`\lambda`)))," variabile in base alla temperatura."),l("p",null,l(i.k,null,Ne(Ae||(Ae=Be`\lambda_{max} \cdot T`)))," è costante.")),l(i.r,{title:"Teoria di Planck per il corpo nero"},l("p",null,"L'energia assorbita e emessa dai corpi neri è quantizzata.")),l(i.r,{title:"Fotone"},l("p",null,"Un onda magnetica con un quanto di energia è detta ",l("i",null,"fotone"),":"),l("p",null,l(i.k,null,Ne(we||(we=Be`E_{fotone} = h \cdot f`))))),l(i.r,{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 l("span",{class:n.a.minus},e.children)}}).call(this,a("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_"}},o8Jj:function(l,e,a){"use strict";(function(l){var i=a("jZax"),n=a.n(i);e.a=function(e){return l("span",{class:n.a.plus},e.children)}}).call(this,a("hosL").h)}}]);
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{Latex, LatexDefaultDisplay, LatexDefaultInline, LatexDisplay, Panel, Section, Todo} 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 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\n 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\n 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 Cariche opposte si attraggono;\n cariche uguali si respingono.\n
\n \n \n
\n Più ioni ha un corpo, meglio la carica\n si muove attraverso di esso.\n
\n
\n I corpi in cui la carica si muove bene sono conduttori, mentre quelli in cui si\n 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\n zona.\n
\n \n \n \n \n
\n Se un corpo conduttore è in contatto con la Terra, le cariche su di esso\n saranno equilibrate e il corpo diventerà elettricamente neutro (con stesso numero\n 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\n contatto.\n
\n \n \n
\n Se un corpo conduttore ha cariche \"esterne\" di un certo segno vicino, esso\n avrà tutte le cariche del segno opposto in equilibrio vicino alle cariche\n esterne, e tutte le cariche dello stesso segno più lontano possibile da\n esse.\n
\n
\n Mettendo a terra il conduttore, nuove cariche del segno opposto saranno\n 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\n 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 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\n {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\n 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\n 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\n 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\n 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 Dove {r`\\kappa`} è la costante dielettrica relativa del materiale\n inserito, {r`A`} l'area di una armatura e {r`s`} la\n distanza tra le due armature.\n
\n
\n Se il campo elettrico creatosi tra le due armature supera la rigidità\n dielettrica del condensatore, la carica immagazzinata viene persa e ha luogo\n 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\n 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\n 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\n 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\n 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\n equivalente dalla resistenza di:\n
\n Nei circuiti in parallelo, tutte i condensatori possono essere sostituite con uno\n 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\n facilmente.\n
\n Dove {r`\\vec{B}`} è l'intensità del campo magnetico\n 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\n 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\n vettore che punta nella direzione di scorrimento della corrente e ha come modulo la\n lunghezza del conduttore.\n
\n \n \n \n \n
\n Una spira in cui passa corrente produce un campo magnetico perpendicolare al piano\n 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 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\n attraversati da correnti\n opposte si respingono.\n
\n \n \n \n \n
\n Un conduttore perpendicolare ad un campo magnetico può ottenere una differenza di\n potenziale se messo in movimento in un direzione perpendicolare alla direzione del\n conduttore e del campo.\n
\n
\n La differenza di potenziale si crea a causa della forza magnetica, che fa spostare tutti\n 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\n 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\n spira di area {r`A`}, il flusso magnetico si può determinare con la Legge\n 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\n nel tempo del flusso magnetico nello stesso percorso.\n
\n Il meno è dovuto alla Legge di\n 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\n magnetico {r`B`} sono perpendicolari tra loro e la direzione di\n 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\n uguale a:\n
\n Dove {r`A_{max}`} è l'ampiezza massima che può avere\n l'onda, {r`\\frac{2 \\pi}{\\lambda} = \\left | \\vec{k} \\right |`} è il\n vettore d'onda, {r`\\omega`} la frequenza angolare\n 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 Con {r`R = 1.097 \\cdot 10^7 \\frac{1}{m}`}, detta costante di Rydberg,\n 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\n 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\n minimo possibile per la carica (talvolta espressa\n 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 Due elettroni non possono occupare lo stesso stato.\n
\n
\n Questo modello funziona solo per atomi con numero atomico basso. Atomi con molti\n 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\n 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\n gap dell'ordine di grandezza maggiore di {r`10^1 eV`}, allora il corpo è\n un isolante.\n
\n
\n Se invece la banda di emissione si sovrappone a un altra, allora il corpo è un\n conduttore.\n
\n
\n Se il gap è invece dell'ordine di grandezza di {r`1 eV`}, allora il corpo\n è un semiconduttore.\n
\n \n \n
\n Legami in cui mancano elettroni.\n
\n
\n Elettroni di altri legami possono spostarsi per colmare\n le lacune, creandone altre, e spostandole in direzione opposta a quella\n della corrente.\n
\n \n \n
\n Se si inserisce in un cristallo semiconduttore si inserisce un atomo con numero atomico\n diverso, si otterrà:\n
\n
\n
Con numero atomico maggiore, un semiconduttore di tipo N con elettroni\n in eccesso liberi di scorrere.\n
\n
Con numero atomico minore, un semiconduttore di tipo P con lacune\n in eccesso libere di catturare elettroni da altri legami.\n
\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\n 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\n rifletterne nessuna.\n
\n
\n Le onde assorbite vengono poi riemesse sotto forma di un onda\n 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\n 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\n 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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{Fragment} from \"preact\";\nimport {Section, Panel, ILatex, BLatex, PLatex, Image} from \"bluelib\";\nimport Example from \"../../components/Example\";\nimport Link from \"../../components/Link\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n Le summary sono brevi spezzoni di testo che permettono all'utente di capire quali documenti siano rilevanti per lui.\n
\n \n \n \n \n \n
\n Summary fisse che dipendono solo dal documento.\n
\n
\n Possono essere generate dalle prime parole del documento, dalle frasi più importanti o da metadati del documento stesso (OpenGraph).\n
\n \n Opinione personale: le summary generate da OpenGraph solitamente fanno schifo :(\n \n \n \n
\n Summary che dipendono sia dal documento, sia dalla query effettuata.\n
\n
\n Generalmente, estraggono le frasi vicino alle occorrenze dei termini della query.\n
\n \n \n \n )\n}\n","import {Fragment} from \"preact\";\nimport {Section, Panel, ILatex, BLatex, PLatex} from \"bluelib\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n Un supplemento al thesaurus specifico a un certo testo: aggiunge ad ogni token {r`c`} la probabilità{r`P(c)`} che un termine selezionato a caso del testo appartenga al sottoalbero formato dal token e i suoi discendenti.\n
\n
\n La radice del thesaurus ha sempre probabilità {r`1`}.\n
\n \n \n
\n Una misura probabilistica di quanto un certo token {r`c`} sia specifico in un certo contesto:\n
\n È anche detta sorpresa o informazione di Shannon.\n
\n \n \n \n )\n}\n","import {Fragment} from \"preact\";\nimport {Section, Panel, ILatex, BLatex, PLatex, Todo} from \"bluelib\";\nimport Example from \"../../components/Example\";\nimport Link from \"../../components/Link\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n Un albero di token organizzati in una gerarchia di significato.\n
\n \n Il token \"mela\" avrà come genitore \"frutto\" e come figli \"Renetta\", \"Fuji\" e \"Granny Smith\".\n \n \n WordNet è un thesaurus generale per la lingua inglese, mentre Nuovo soggettario è un thesaurus generale per la lingua italiana.\n \n \n \n \n )\n}\n","import {Fragment} from \"preact\";\nimport {Section, Panel, ILatex, BLatex, PLatex, Code, BaseLink} from \"bluelib\";\nimport Example from \"../../components/Example\";\nimport Link from \"../../components/Link\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n Una struttura dati che permette di velocizzare le ricerche su testi grandi e semi-statici.\n
\n \n
\n Si crea un vocabolario dall'insieme dei token, e ad ogni token del vocabolario si associa una lista ordinata di tutte le sue occorrenze (posting list).\n
\n
\n L'efficacia ed efficienza dell'indice dipendono dalle strutture di indicizzazione utilizzate per serializzarlo.\n
\n \n \n \n \n
\n Insieme di tutti i token ricercabili.\n
\n
\n Può essere implementato con diverse strutture dati:\n
\n
\n
\n Array ordinato, utilizzando la bisezione per gli accessi;\n {r`\n ciao → R1, R15, R123\n steffo → R1, R14\n ciano → R1231\n `}\n
\n
\n Prefix tree (trie), utilizzando le proprietà degli alberi per gli accessi;\n {r`\n - [root]\n - c\n - i\n - a\n - o\n - R1\n - R15\n - R123\n - no\n - R1231\n - steffo\n - R1\n - R14\n `}\n
\n
\n Suffix tree, come il prefix tree ma partendo dall'ultima lettera\n
\n
\n B+ tree, utilizzando le proprietà degli alberi B+ per gli accessi\n \n Visualizzazione di un B+ tree\n \n \n È la stessa struttura utilizzata dai database relazionali.\n \n
\n
\n Dizionario, utilizzando gli hash per gli accessi.\n
\n
\n
\n Per permettere pattern-based query, è possibile utilizzare più vocabolari per un singolo documento.\n
\n \n \n
\n Insieme di tutte le occorrenze di un determinato token.\n
\n
\n Viene implementata attraverso una lista ordinata (sorted list).\n
\n
\n Essendo ordinata, è efficiente {r`O(n + m)`} nelle operazioni di unione e intersezione tra più posting list: si mantiene su ognuna un cursore che avanza quando l'occorrenza a cui punta è stata superata.\n
\n
\n Non è però efficiente nelle operazioni di negazione.\n
\n
\n È possibile aggiungere skip pointers alle posting list, in modo da rendere più efficiente l'intersezione.\n
\n \n \n \n \n
\n Per effettuare una ricerca utilizzando l'indice, sono necessari i seguenti passi:\n
\n \n
Si cercano individualmente i termini della query nel vocabolario
\n
Si accede alle posting list dei vari termini
\n
Si effettuano operazioni logiche sui dati delle posting list
\n
Si restituisce il sottoinsieme di token risultante
\n \n \n \n \n )\n}\n","// 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 Intro from \"./00_Intro\";\nimport InformationRetrieval from \"./01_InformationRetrival\";\nimport Similarity from \"./02_Similarity\";\nimport Queries from \"./03_Queries\";\nimport DocumentProcessing from \"./04_DocumentProcessing\";\nimport Thesaurus from \"./05_Thesaurus\";\nimport InformationContent from \"./06_InformationContent\";\nimport AnalisiLessicale from \"./07_AnalisiLessicale\";\nimport Indexing from \"./08_Indexing\";\nimport Modeling from \"./09_Modeling\";\nimport Evaluation from \"./10_Evaluation\";\nimport Presentation from \"./11_Presentation\";\n\n\nexport default function () {\n return (\n
\n
Gestione dell'informazione
\n \n \n \n \n \n \n \n \n \n \n \n \n
\n )\n}\n","import {Fragment} from \"preact\";\nimport {Section, Panel, ILatex, BLatex, PLatex} from \"bluelib\";\nimport ToBeContinued from \"../../components/ToBeContinued\";\nimport Example from \"../../components/Example\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n Una procedura che ottimizza un documento per la ricerca.\n
\n \n \n \n \n
\n Passo che trasforma il documento in un insieme di parole (token).\n
\n
\n Deve decidere come considerare i simboli tipografici, il case delle lettere e le cifre.\n
\n
\n Inoltre, è possibile che disambigui tra i vari significati delle parole (word sense disambiguation).\n
\n \n \n
\n Passo che elimina dalla lista i token inutili ai fini delle ricerche.\n
\n
\n Esse solitamente sono gli articoli e le preposizioni.\n
\n
\n Con l'avanzare degli anni, le stopwords hanno iniziato a diminuire sempre di più.\n
\n \n I motori di ricerca moderni non usano nessuna stopword.\n \n \n \n
\n Passo che riduce alla radice tutti i token della lista.\n
\n
\n Converte i plurali in singolari, passati ai presenti, e così via.\n
\n \n \n
\n Passo che rimuove i caratteri speciali contenuti nei token.\n
\n
\n Duplica le parole separate da trattino nella versione con spazio e senza spazio, rimuove gli accenti non significativi, etc.\n
\n \n \n
\n Passo che seleziona i termini più importanti della lista.\n
\n
\n Generalmente sfrutta dei parser o dei tagger.\n
\n \n \n \n \n
\n Passo che classifica il documento in una o più categorie del thesaurus.\n
\n \n \n \n \n )\n}\n","import {Fragment} from \"preact\";\nimport {Section, Panel, ILatex, BLatex, PLatex} from \"bluelib\";\nimport ToBeContinued from \"../../components/ToBeContinued\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n synonymy\n
\n
\n Due parole sono sinonimi quando hanno lo stesso identico significato.\n
\n \n \n
\n similarity\n
\n
\n Due parole sono simili rappresentano concetti con caratteristiche in comune.\n
\n \n \n
\n relatedness\n
\n
\n Due parole sono correlate quando rappresentano concetti vengono usati insieme spesso.\n
\n \n \n \n \n
\n Un modo per misurare la somiglianza tra due parole confrontando solo la loro posizione nel thesaurus.\n
\n \n \n \n \n
\n Calcola la similarità basandosi sull'inverso della distanza tra le due parole {r`c_1`} e {r`c_2`} nell'albero:\n
\n Questa sezione è incompleta. Torna più avanti!\n
\n {props.children}\n \n )\n}\n","import {Fragment} from \"preact\";\nimport {Section, Panel, ILatex, BLatex, PLatex} from \"bluelib\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n Procedura che disambigua tra più significati dello stesso token.\n
\n
\n Si può realizzare confrontando la somiglianza di ogni significato con tutti gli altri nomi presenti nella frase.\n
\n \n \n \n )\n}\n","import {Panel, Section} from \"bluelib\";\nimport {Fragment} from \"preact\";\nimport Example from \"../../components/Example\";\nimport Link from \"../../components/Link\";\n\nconst r = String.raw;\n\n\nexport default function (props) {\n return (\n \n \n \n
\n Processo di raccolta e recupero approssimato di informazioni spesso non strutturate.\n
\n
\n Riceve in input una query, elabora documenti e restituisce una risposta più rilevante possibile.\n
\n
\n Un'applicazione che effettua IR si chiama motore di ricerca.\n
\n \n \n
\n Raggruppamenti di informazioni che possono avere varie forme.\n
\n \n Potrebbe essere un documento un file, oppure un gruppo di informazioni contenute in quel file (email, slides)...\n \n \n \n \n )\n}\n","import style from \"./Example.less\";\n\nexport default function (props) {\n return (\n
\n {props.children}\n
\n );\n}\n","import {Fragment} from \"preact\";\nimport {Section, Panel, ILatex, BLatex, PLatex, BoxColors} from \"bluelib\";\nimport Example from \"../../components/Example\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n Modelli matematici che ordinano per rilevanza (ranking) i risultati di una query.\n
\n
\n Formalmente, sono composti da:\n
\n
\n
I documenti {r`D`}
\n
La query {r`Q`}
\n
Una funzione di ranking{r`rank(q_i, d_j) \\to \\mathbb{R}`} che quantifica la similarità
\n
\n \n \n \n \n \n I motori di ricerca web usano generalmente questo modello.\n \n
\n Modello IR che rappresenta ogni documento come un insieme ordinato di parole chiave (index terms).\n
\n
\n Associa un peso a ogni token, e ordina l'insieme in base a quel peso.\n
\n
\n Generalmente, i pesi sono indipendenti uno dall'altro, e questo permette di calcolarli parallelamente.\n
\n
\n Bag-of-words: L'ordine delle parole nel documento viene solitamente ignorato dal ranking.\n
\n \n \n \n grep è a tutti gli effetti un modello booleano.\n \n
\n Modello IR classico che dà un peso booleano a ogni documento: {r`1`} se soddisfa tutti i requisiti della query, {r`0`} se non li soddisfa.\n
\n
\n Ha una precisione perfetta, ma richiede query ben preparate e specifiche.\n
\n
\n È anche facile da implementare utilizzando una posting list.\n
\n \n \n \n Apache Lucene si basa su questo modello.\n \n
\n Modello IR in cui ogni caratteristica di un documento è rappresentata con un valore floating point e l'intero documento è quindi rappresentato da una n-pla di tutte le sue caratteristiche.\n
\n
\n Similarità cosinusoidale: il ranking si basa sull'angolo tra il \"vettore query\" e il \"vettore documento\":\n
\n Spesso si associa un peso a ogni dimensione in modo da variare l'importanza di una data caratteristica; comunemente, per i testi, si usano come pesi le frequenza dei token normalizzate (df) o logaritmiche (idf).\n
\n
\n Come il modello classico, ignora l'ordine delle parole.\n
\n \n \n
\n Modello IR che effettua il ranking in base alla probabilità di comparsa dei termini della query nei documenti.\n
\n
\n Si basa sul calcolare un peso {r`c_i`} per ogni termine della query. Quest'ultimo diventa più grande se il termine compare in documenti rilevanti, e più piccolo se compare in documenti irrilevanti. Se il termine compare in ugual modo in entrambi, allora esso varrà {r`0`}.\n
\n
\n Ci sarebbe una dimostrazione complessa che ho omesso per brevità.\n
\n \n \n
\n Estensione del modello booleano che ammette la possibilità che certe condizioni siano parzialmente soddisfatte.\n
\n
\n Invece che usare solo {r`0`} e {r`1`}, permette ai documenti di avere valori intermedi tra quei due.\n
\n
\n Le operazioni booleane vengono quindi modificate:\n
\n {r`\n \\begin{cases}\n a \\land b = \\min(a, b)\\\\\n a \\lor b = \\max(a, b)\\\\\n \\neg\\ a = 1 - a\n \\end{cases} \n `}\n \n \n
\n Modello IR simile ai modelli classici che però tiene conto anche della frequenza dei termini nei documenti e della lunghezza dei documenti.\n
\n
\n Permette anche di tenere in considerazione la frequenza dei termini nella query, nel caso essa sia molto lunga.\n
\n
\n Ci sono formule che ho omesso per brevità.\n
\n \n \n \n )\n}\n","import {Fragment} from \"preact\";\nimport {Section, Panel, ILatex, BLatex, PLatex} from \"bluelib\";\nimport Example from \"../../components/Example\";\nimport Link from \"../../components/Link\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n La richiesta di informazioni effettuata da un utente in linguaggio naturale.\n
\n \n \n
\n La traduzione dell'UIN in un linguaggio specifico al motore di ricerca.\n
\n
\n La complessità di questo linguaggio dipende dall'utilizzatore del motore di ricerca: il suo livello di esperienza, il suo background di conoscenze e il livello di approfondimento desiderato.\n
\n \n \n \n \n \n Wikipedia e la ricerca di Telegram usano un linguaggio di query keyword-based.\n \n
\n Definisce parole chiave da cercare all'interno dei documenti.\n
\n
\n In esse, è possibile cercare molteplici parole chiave, concatenarle per formare una frase oppure cercare parole a una data prossimità l'una dall'altra.\n
\n
\n Spesso processa il linguaggio naturale per trasformarne le parole specifiche in parole chiave più generiche.\n
\n \n Cercando su Telegram un plurale inglese, verranno restituiti anche i messaggi che riportano la stessa parola ma al singolare.\n \n \n \n \n Le regex e i glob sono due dei pattern-based query languages più comuni.\n \n
\n Definisce parti di testo da cercare all'interno dei documenti.\n
\n
\n Permettono dunque la ricerca di prefissi, suffissi, sottostringhe e intervalli.\n
\n \n \n \n \n \n Google, DuckDuckGo, Ecosia usano tutti un linguaggio di query keyword-based booleano.\n \n
\n Estendono gli altri tipi di query permettendo l'inserimento di operatori booleani (AND, OR e NOT) tra le parole chiave.\n
\n \n Per esempi sulla sintassi per effettuare ricerche booleane su Google, visita Google Advanced Search.\n \n \n \n \n Google Books permette di sfruttare la struttura fissa delle informazioni sui libri (titolo, autore, editore...) per ottenere risultati più precisi.,\n \n
\n Permettono di effettuare i precedenti tipi di query su specifiche parti della struttura dei documenti.\n
\n \n \n \n MeSH cataloga i paper di medicina caricati su PubMed organizzandoli in una gerarchia di categorie.\n \n
\n Limitano le keyword a un vocabolario predefinito utilizzato anche dai documenti.\n
\n \n \n \n )\n}\n","import {Fragment} from \"preact\";\nimport {Section, Panel, ILatex, BLatex, PLatex, BaseLink, Code} from \"bluelib\";\nimport Link from \"../../components/Link\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n
Prof. Federica Mandreoli
\n
\n \n \n
\n
Pagina Dolly Unimore
\n
\n \n \n
\n Puoi scaricare le videolezioni di Gestione dell'informazione con bbbdl:\n
\n \n bbbdl sync -r https://raw.githubusercontent.com/Steffo99/unimoreinfo-wiki/main/gestionedellinformazione.json\n \n \n \n \n )\n}\n","import {Fragment} from \"preact\";\nimport {Section, Panel, ILatex, BLatex, PLatex, BoxColors} from \"bluelib\";\nimport Example from \"../../components/Example\";\nimport ToBeContinued from \"../../components/ToBeContinued\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n La frazione dei documenti rilevanti totali restituita:\n
\n {r`\n \\frac{R \\cap A}{R}\n `}\n \n \n
\n La frazione dei documenti restituiti che sono rilevanti:\n
\n {r`\n \\frac{R \\cap A}{A}\n `}\n \n \n \n \n
\n Si effettuano test e si valutano precisione e richiamo ottenuti.\n
\n \n
\n Il modello IR ideale ha valori di richiamo e precisione di 1, ma nella realtà questi valori tendono ad essere inversamente proporzionali.\n
\n
\n Un fattore significativo nel richiamo e nella precisione è la forma del documento.\n
\n \n Libri interi, capitoli, pagine o frasi; possono essere tutti documenti, cambia solo la loro granularità.\n \n \n \n
\n Si fissano un certo numero di livelli di richiamo desiderati e si misura la precisione che ha il modello in quei punti, creando una curva di richiamo standard.\n
\n
\n È possibile ottenere una curva di richiamo interpolata prendendo per ogni punto il valore di massima precisione tra esso e i punti precedenti.\n
\n \n Interpolata mi sembra un termine orribile... Non sarebbe molto meglio monotonica?\n \n \n \n
\n È possibile misurare la curva di richiamo media effettuando la media su più query dei vari valori, e la curva di richiamo a valore singolo effettuando una query sola.\n
\n
\n Sono entrambe importanti per effettuare una valutazione, perchè la curva media potrebbe mascherare problemi di specifiche query nei modelli.\n
\n \n \n
\n La media tra i valori di più curve di richiamo.\n
\n
\n È utile per le valutazioni in cui l'utente sta cercando più documenti rilevanti alla volta.\n
\n \n \n
\n Valori di recall e precision riguardo i primi {r`R`} documenti restituiti.\n
\n \n La prima pagina di Google (10 documenti) considererà i valori di 10-recall e 10-precision.\n \n \n \n \n \n
\n È anche detta media armonica.\n
\n
\n Un singolo valore che tiene conto sia della precisione {r`p`} sia del richiamo {r`r`}:\n
\n {r`\n F = \\frac{2}{\\frac{1}{p} + \\frac{1}{r}}\n `}\n \n \n
\n Una versione più configurabile della Misura F che permette di dare priorità{r`b`} a precisione ({r`b > 1`}) o a richiamo ({r`b < 1`}) in base alle proprie esigenze:\n
\n Misura per valutare i motori di ricerca online che tiene conto dell'ordine in cui vengono restituiti i documenti.\n
\n
\n Si basa su valutazioni di rilevanza e su pesi posizionali dei risultati.\n
\n \n Valuta quanto è stata utile questa ricerca:\n \n ☆☆☆☆☆\n \n \n \n La valutazione del primo risultato viene moltiplicata per {r`\\frac{1}{\\log_2(1)}`}, la seconda per {r`\\frac{1}{\\log_2(2)}`}, la terza per {r`\\frac{1}{\\log_2(3)}`} e così via.\n \n \n \n
\n Miglioramento alla DCG che supporta quantità di risultati differenti.\n
\n \n Non dice la formula, e l'esempio non esemplifica un bel niente...\n \n \n \n \n )\n}\n"],"sourceRoot":""}
\ No newline at end of file
diff --git a/docs/route-GestioneDellInformazione.chunk.a1161.css b/docs/route-GestioneDellInformazione.chunk.a1161.css
new file mode 100644
index 0000000..3c4dea0
--- /dev/null
+++ b/docs/route-GestioneDellInformazione.chunk.a1161.css
@@ -0,0 +1 @@
+.red__2y1B_{color:#ff7d7d}.orange__dD2kx{color:#ffbb7d}.yellow__OEpwl{color:#ffff7d}.lime__CVe41{color:#7dff7d}.cyan__26ZAg{color:#7dffff}.blue__LO7Xm{color:#7d7dff}.magenta__1Akee{color:#ff7dff}.example__2PzAa{color:#d3a1ff;padding:4px;border-radius:4px;margin:4px 0}.example__2PzAa,.example__2PzAa table{background-color:rgba(211,161,255,.05)}.example__2PzAa table{border-spacing:0;border:2px solid rgba(211,161,255,.1);border-collapse:collapse}.example__2PzAa table tbody td,.example__2PzAa table tbody th,.example__2PzAa table thead td,.example__2PzAa table thead th{padding:4px;border:1px solid rgba(211,161,255,.1)}.example__2PzAa table thead{background-color:rgba(211,161,255,.1);color:#fff}
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diff --git a/docs/route-GestioneDellInformazione.chunk.af84a.esm.js b/docs/route-GestioneDellInformazione.chunk.af84a.esm.js
new file mode 100644
index 0000000..a037945
--- /dev/null
+++ b/docs/route-GestioneDellInformazione.chunk.af84a.esm.js
@@ -0,0 +1,44 @@
+(window.webpackJsonp=window.webpackJsonp||[]).push([[7],{"0tML":function(e,l,i){"use strict";(function(e){var n=i("hosL"),a=i("mbOI"),t=i("ke5e"),o=i("YNhk");l.a=function(){return e(n.Fragment,null,e(a.s,{title:"Summaries"},e(a.r,{title:"Cosa sono?"},e("p",null,"Le summary sono ",e("b",null,"brevi spezzoni di testo")," che permettono all'utente di capire quali documenti siano rilevanti per lui."),e(t.a,null,e(a.j,{src:"https://i.imgur.com/8aMrIDY.png"}))),e(a.r,{title:"Static summaries"},e("p",null,"Summary fisse che dipendono ",e("b",null,"solo dal documento"),"."),e("p",null,"Possono essere generate dalle ",e("b",null,"prime parole del documento"),", dalle ",e("b",null,"frasi più importanti")," o da ",e("b",null,"metadati")," del documento stesso (",e(o.a,{href:"https://ogp.me/"},"OpenGraph"),")."),e(t.a,null,"Opinione personale: le summary generate da OpenGraph solitamente fanno schifo :(")),e(a.r,{title:"Dynamic summaries"},e("p",null,"Summary che dipendono ",e("b",null,"sia dal documento"),", ",e("b",null,"sia dalla query effettuata"),"."),e("p",null,"Generalmente, estraggono le frasi vicino alle occorrenze dei termini della query."))))}}).call(this,i("hosL").h)},"0vRf":function(e,l,i){"use strict";(function(e){var n=i("hosL"),a=i("mbOI");let t,o,r,u,s,c=e=>e;const d=String.raw;l.a=function(){return e(n.Fragment,null,e(a.s,{title:"Un approccio probabilistico"},e(a.r,{title:"Probabilità dei token"},e("p",null,"Un supplemento al thesaurus specifico a un certo testo: aggiunge ad ogni token ",e(a.i,null,d(t||(t=c`c`)))," la ",e("b",null,"probabilità")," ",e(a.i,null,d(o||(o=c`P(c)`)))," che un ",e("b",null,"termine selezionato a caso del testo appartenga al sottoalbero")," formato dal token e i suoi discendenti."),e("p",null,"La radice del thesaurus ha sempre probabilità ",e(a.i,null,d(r||(r=c`1`))),".")),e(a.r,{title:"Information content"},e("p",null,"Una misura ",e("b",null,"probabilistica")," di quanto un certo token ",e(a.i,null,d(u||(u=c`c`)))," sia ",e("b",null,"specifico")," in un certo contesto:"),e(a.q,null,d(s||(s=c`
+ IC(c) = - \log \left( P(c) \right)
+ `))),e("p",null,"È anche detta ",e("i",null,"sorpresa")," o ",e("i",null,"informazione di Shannon"),"."))))}}).call(this,i("hosL").h)},"1Gqa":function(e,l,i){"use strict";(function(e){var n=i("hosL"),a=i("mbOI"),t=i("ke5e"),o=i("YNhk");l.a=function(){return e(n.Fragment,null,e(a.s,{title:"Thesaurus"},e(a.r,{title:"Cos'è?"},e("p",null,"Un ",e("b",null,"albero di token")," organizzati in una gerarchia di significato."),e(t.a,null,'Il token "mela" avrà come genitore "frutto" e come figli ',e(o.a,{href:"https://it.wikipedia.org/wiki/Renetta"},'"Renetta"'),", ",e(o.a,{href:"https://it.wikipedia.org/wiki/Fuji_(mela)"},'"Fuji"')," e ",e(o.a,{href:"https://it.wikipedia.org/wiki/Granny_Smith"},'"Granny Smith"'),"."),e(t.a,null,e(o.a,{href:"https://wordnet.princeton.edu/"},"WordNet")," è un thesaurus generale per la lingua inglese, mentre ",e(o.a,{href:"https://thes.bncf.firenze.sbn.it/thes-dati.htm"},"Nuovo soggettario")," è un thesaurus generale per la lingua italiana."))))}}).call(this,i("hosL").h)},"29je":function(e,l,i){"use strict";(function(e){var n=i("hosL"),a=i("mbOI"),t=i("ke5e"),o=i("YNhk");let r,u,s,c=e=>e;const d=String.raw;l.a=function(){return e(n.Fragment,null,e(a.s,{title:"Indicizzazione"},e(a.r,{title:"Inverted index"},e("p",null,"Una ",e("b",null,"struttura dati")," che permette di velocizzare le ricerche su testi ",e("b",null,"grandi")," e ",e("b",null,"semi-statici"),"."),e("aside",null,e("u",null,"Testo semi-statico"),": un testo che viene aggiornato raramente."),e("p",null,"Si crea un ",e("i",null,"vocabolario")," dall'",e("b",null,"insieme dei token"),", e ad ogni ",e("b",null,"token")," del vocabolario si associa una ",e("b",null,"lista ordinata di tutte le sue occorrenze")," (",e("i",null,"posting list"),")."),e("p",null,"L'efficacia ed efficienza dell'indice dipendono dalle ",e("i",null,"strutture di indicizzazione")," utilizzate per serializzarlo."))),e(a.s,null,e(a.r,{title:"Vocabolario"},e("p",null,e("b",null,"Insieme")," di ",e("b",null,"tutti i token")," ricercabili."),e("p",null,"Può essere implementato con diverse strutture dati:"),e("ul",null,e("li",null,e("b",null,"Array ordinato"),", utilizzando la ",e("b",null,"bisezione")," per gli accessi;",e(a.f,null,d(r||(r=c`
+ ciao → R1, R15, R123
+ steffo → R1, R14
+ ciano → R1231
+ `)))),e("li",null,e("b",null,"Prefix tree")," (",e("i",null,"trie"),"), utilizzando le ",e("b",null,"proprietà degli alberi")," per gli accessi;",e(a.f,null,d(u||(u=c`
+ - [root]
+ - c
+ - i
+ - a
+ - o
+ - R1
+ - R15
+ - R123
+ - no
+ - R1231
+ - steffo
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+ - R14
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kDACwC,+CADxC,KAGA,sF","file":"route-GestioneDellInformazione.chunk.af84a.esm.js","sourcesContent":["import {Fragment} from \"preact\";\nimport {Section, Panel, ILatex, BLatex, PLatex, Image} from \"bluelib\";\nimport Example from \"../../components/Example\";\nimport Link from \"../../components/Link\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n Le summary sono brevi spezzoni di testo che permettono all'utente di capire quali documenti siano rilevanti per lui.\n
\n \n \n \n \n \n
\n Summary fisse che dipendono solo dal documento.\n
\n
\n Possono essere generate dalle prime parole del documento, dalle frasi più importanti o da metadati del documento stesso (OpenGraph).\n
\n \n Opinione personale: le summary generate da OpenGraph solitamente fanno schifo :(\n \n \n \n
\n Summary che dipendono sia dal documento, sia dalla query effettuata.\n
\n
\n Generalmente, estraggono le frasi vicino alle occorrenze dei termini della query.\n
\n \n \n \n )\n}\n","import {Fragment} from \"preact\";\nimport {Section, Panel, ILatex, BLatex, PLatex} from \"bluelib\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n Un supplemento al thesaurus specifico a un certo testo: aggiunge ad ogni token {r`c`} la probabilità{r`P(c)`} che un termine selezionato a caso del testo appartenga al sottoalbero formato dal token e i suoi discendenti.\n
\n
\n La radice del thesaurus ha sempre probabilità {r`1`}.\n
\n \n \n
\n Una misura probabilistica di quanto un certo token {r`c`} sia specifico in un certo contesto:\n
\n È anche detta sorpresa o informazione di Shannon.\n
\n \n \n \n )\n}\n","import {Fragment} from \"preact\";\nimport {Section, Panel, ILatex, BLatex, PLatex, Todo} from \"bluelib\";\nimport Example from \"../../components/Example\";\nimport Link from \"../../components/Link\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n Un albero di token organizzati in una gerarchia di significato.\n
\n \n Il token \"mela\" avrà come genitore \"frutto\" e come figli \"Renetta\", \"Fuji\" e \"Granny Smith\".\n \n \n WordNet è un thesaurus generale per la lingua inglese, mentre Nuovo soggettario è un thesaurus generale per la lingua italiana.\n \n \n \n \n )\n}\n","import {Fragment} from \"preact\";\nimport {Section, Panel, ILatex, BLatex, PLatex, Code, BaseLink} from \"bluelib\";\nimport Example from \"../../components/Example\";\nimport Link from \"../../components/Link\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n Una struttura dati che permette di velocizzare le ricerche su testi grandi e semi-statici.\n
\n \n
\n Si crea un vocabolario dall'insieme dei token, e ad ogni token del vocabolario si associa una lista ordinata di tutte le sue occorrenze (posting list).\n
\n
\n L'efficacia ed efficienza dell'indice dipendono dalle strutture di indicizzazione utilizzate per serializzarlo.\n
\n \n \n \n \n
\n Insieme di tutti i token ricercabili.\n
\n
\n Può essere implementato con diverse strutture dati:\n
\n
\n
\n Array ordinato, utilizzando la bisezione per gli accessi;\n {r`\n ciao → R1, R15, R123\n steffo → R1, R14\n ciano → R1231\n `}\n
\n
\n Prefix tree (trie), utilizzando le proprietà degli alberi per gli accessi;\n {r`\n - [root]\n - c\n - i\n - a\n - o\n - R1\n - R15\n - R123\n - no\n - R1231\n - steffo\n - R1\n - R14\n `}\n
\n
\n Suffix tree, come il prefix tree ma partendo dall'ultima lettera\n
\n
\n B+ tree, utilizzando le proprietà degli alberi B+ per gli accessi\n \n Visualizzazione di un B+ tree\n \n \n È la stessa struttura utilizzata dai database relazionali.\n \n
\n
\n Dizionario, utilizzando gli hash per gli accessi.\n
\n
\n
\n Per permettere pattern-based query, è possibile utilizzare più vocabolari per un singolo documento.\n
\n \n \n
\n Insieme di tutte le occorrenze di un determinato token.\n
\n
\n Viene implementata attraverso una lista ordinata (sorted list).\n
\n
\n Essendo ordinata, è efficiente {r`O(n + m)`} nelle operazioni di unione e intersezione tra più posting list: si mantiene su ognuna un cursore che avanza quando l'occorrenza a cui punta è stata superata.\n
\n
\n Non è però efficiente nelle operazioni di negazione.\n
\n
\n È possibile aggiungere skip pointers alle posting list, in modo da rendere più efficiente l'intersezione.\n
\n \n \n \n \n
\n Per effettuare una ricerca utilizzando l'indice, sono necessari i seguenti passi:\n
\n \n
Si cercano individualmente i termini della query nel vocabolario
\n
Si accede alle posting list dei vari termini
\n
Si effettuano operazioni logiche sui dati delle posting list
\n
Si restituisce il sottoinsieme di token risultante
\n \n \n \n \n )\n}\n","// 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 Intro from \"./00_Intro\";\nimport InformationRetrieval from \"./01_InformationRetrival\";\nimport Similarity from \"./02_Similarity\";\nimport Queries from \"./03_Queries\";\nimport DocumentProcessing from \"./04_DocumentProcessing\";\nimport Thesaurus from \"./05_Thesaurus\";\nimport InformationContent from \"./06_InformationContent\";\nimport AnalisiLessicale from \"./07_AnalisiLessicale\";\nimport Indexing from \"./08_Indexing\";\nimport Modeling from \"./09_Modeling\";\nimport Evaluation from \"./10_Evaluation\";\nimport Presentation from \"./11_Presentation\";\n\n\nexport default function () {\n return (\n
\n
Gestione dell'informazione
\n \n \n \n \n \n \n \n \n \n \n \n \n
\n )\n}\n","import {Fragment} from \"preact\";\nimport {Section, Panel, ILatex, BLatex, PLatex} from \"bluelib\";\nimport ToBeContinued from \"../../components/ToBeContinued\";\nimport Example from \"../../components/Example\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n Una procedura che ottimizza un documento per la ricerca.\n
\n \n \n \n \n
\n Passo che trasforma il documento in un insieme di parole (token).\n
\n
\n Deve decidere come considerare i simboli tipografici, il case delle lettere e le cifre.\n
\n
\n Inoltre, è possibile che disambigui tra i vari significati delle parole (word sense disambiguation).\n
\n \n \n
\n Passo che elimina dalla lista i token inutili ai fini delle ricerche.\n
\n
\n Esse solitamente sono gli articoli e le preposizioni.\n
\n
\n Con l'avanzare degli anni, le stopwords hanno iniziato a diminuire sempre di più.\n
\n \n I motori di ricerca moderni non usano nessuna stopword.\n \n \n \n
\n Passo che riduce alla radice tutti i token della lista.\n
\n
\n Converte i plurali in singolari, passati ai presenti, e così via.\n
\n \n \n
\n Passo che rimuove i caratteri speciali contenuti nei token.\n
\n
\n Duplica le parole separate da trattino nella versione con spazio e senza spazio, rimuove gli accenti non significativi, etc.\n
\n \n \n
\n Passo che seleziona i termini più importanti della lista.\n
\n
\n Generalmente sfrutta dei parser o dei tagger.\n
\n \n \n \n \n
\n Passo che classifica il documento in una o più categorie del thesaurus.\n
\n \n \n \n \n )\n}\n","import {Fragment} from \"preact\";\nimport {Section, Panel, ILatex, BLatex, PLatex} from \"bluelib\";\nimport ToBeContinued from \"../../components/ToBeContinued\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n synonymy\n
\n
\n Due parole sono sinonimi quando hanno lo stesso identico significato.\n
\n \n \n
\n similarity\n
\n
\n Due parole sono simili rappresentano concetti con caratteristiche in comune.\n
\n \n \n
\n relatedness\n
\n
\n Due parole sono correlate quando rappresentano concetti vengono usati insieme spesso.\n
\n \n \n \n \n
\n Un modo per misurare la somiglianza tra due parole confrontando solo la loro posizione nel thesaurus.\n
\n \n \n \n \n
\n Calcola la similarità basandosi sull'inverso della distanza tra le due parole {r`c_1`} e {r`c_2`} nell'albero:\n
\n Questa sezione è incompleta. Torna più avanti!\n
\n {props.children}\n \n )\n}\n","import {Fragment} from \"preact\";\nimport {Section, Panel, ILatex, BLatex, PLatex} from \"bluelib\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n Procedura che disambigua tra più significati dello stesso token.\n
\n
\n Si può realizzare confrontando la somiglianza di ogni significato con tutti gli altri nomi presenti nella frase.\n
\n \n \n \n )\n}\n","import {Panel, Section} from \"bluelib\";\nimport {Fragment} from \"preact\";\nimport Example from \"../../components/Example\";\nimport Link from \"../../components/Link\";\n\nconst r = String.raw;\n\n\nexport default function (props) {\n return (\n \n \n \n
\n Processo di raccolta e recupero approssimato di informazioni spesso non strutturate.\n
\n
\n Riceve in input una query, elabora documenti e restituisce una risposta più rilevante possibile.\n
\n
\n Un'applicazione che effettua IR si chiama motore di ricerca.\n
\n \n \n
\n Raggruppamenti di informazioni che possono avere varie forme.\n
\n \n Potrebbe essere un documento un file, oppure un gruppo di informazioni contenute in quel file (email, slides)...\n \n \n \n \n )\n}\n","import style from \"./Example.less\";\n\nexport default function (props) {\n return (\n
\n {props.children}\n
\n );\n}\n","import {Fragment} from \"preact\";\nimport {Section, Panel, ILatex, BLatex, PLatex, BoxColors} from \"bluelib\";\nimport Example from \"../../components/Example\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n Modelli matematici che ordinano per rilevanza (ranking) i risultati di una query.\n
\n
\n Formalmente, sono composti da:\n
\n
\n
I documenti {r`D`}
\n
La query {r`Q`}
\n
Una funzione di ranking{r`rank(q_i, d_j) \\to \\mathbb{R}`} che quantifica la similarità
\n
\n \n \n \n \n \n I motori di ricerca web usano generalmente questo modello.\n \n
\n Modello IR che rappresenta ogni documento come un insieme ordinato di parole chiave (index terms).\n
\n
\n Associa un peso a ogni token, e ordina l'insieme in base a quel peso.\n
\n
\n Generalmente, i pesi sono indipendenti uno dall'altro, e questo permette di calcolarli parallelamente.\n
\n
\n Bag-of-words: L'ordine delle parole nel documento viene solitamente ignorato dal ranking.\n
\n \n \n \n grep è a tutti gli effetti un modello booleano.\n \n
\n Modello IR classico che dà un peso booleano a ogni documento: {r`1`} se soddisfa tutti i requisiti della query, {r`0`} se non li soddisfa.\n
\n
\n Ha una precisione perfetta, ma richiede query ben preparate e specifiche.\n
\n
\n È anche facile da implementare utilizzando una posting list.\n
\n \n \n \n Apache Lucene si basa su questo modello.\n \n
\n Modello IR in cui ogni caratteristica di un documento è rappresentata con un valore floating point e l'intero documento è quindi rappresentato da una n-pla di tutte le sue caratteristiche.\n
\n
\n Similarità cosinusoidale: il ranking si basa sull'angolo tra il \"vettore query\" e il \"vettore documento\":\n
\n Spesso si associa un peso a ogni dimensione in modo da variare l'importanza di una data caratteristica; comunemente, per i testi, si usano come pesi le frequenza dei token normalizzate (df) o logaritmiche (idf).\n
\n
\n Come il modello classico, ignora l'ordine delle parole.\n
\n \n \n
\n Modello IR che effettua il ranking in base alla probabilità di comparsa dei termini della query nei documenti.\n
\n
\n Si basa sul calcolare un peso {r`c_i`} per ogni termine della query. Quest'ultimo diventa più grande se il termine compare in documenti rilevanti, e più piccolo se compare in documenti irrilevanti. Se il termine compare in ugual modo in entrambi, allora esso varrà {r`0`}.\n
\n
\n Ci sarebbe una dimostrazione complessa che ho omesso per brevità.\n
\n \n \n
\n Estensione del modello booleano che ammette la possibilità che certe condizioni siano parzialmente soddisfatte.\n
\n
\n Invece che usare solo {r`0`} e {r`1`}, permette ai documenti di avere valori intermedi tra quei due.\n
\n
\n Le operazioni booleane vengono quindi modificate:\n
\n {r`\n \\begin{cases}\n a \\land b = \\min(a, b)\\\\\n a \\lor b = \\max(a, b)\\\\\n \\neg\\ a = 1 - a\n \\end{cases} \n `}\n \n \n
\n Modello IR simile ai modelli classici che però tiene conto anche della frequenza dei termini nei documenti e della lunghezza dei documenti.\n
\n
\n Permette anche di tenere in considerazione la frequenza dei termini nella query, nel caso essa sia molto lunga.\n
\n
\n Ci sono formule che ho omesso per brevità.\n
\n \n \n \n )\n}\n","import {Fragment} from \"preact\";\nimport {Section, Panel, ILatex, BLatex, PLatex} from \"bluelib\";\nimport Example from \"../../components/Example\";\nimport Link from \"../../components/Link\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n La richiesta di informazioni effettuata da un utente in linguaggio naturale.\n
\n \n \n
\n La traduzione dell'UIN in un linguaggio specifico al motore di ricerca.\n
\n
\n La complessità di questo linguaggio dipende dall'utilizzatore del motore di ricerca: il suo livello di esperienza, il suo background di conoscenze e il livello di approfondimento desiderato.\n
\n \n \n \n \n \n Wikipedia e la ricerca di Telegram usano un linguaggio di query keyword-based.\n \n
\n Definisce parole chiave da cercare all'interno dei documenti.\n
\n
\n In esse, è possibile cercare molteplici parole chiave, concatenarle per formare una frase oppure cercare parole a una data prossimità l'una dall'altra.\n
\n
\n Spesso processa il linguaggio naturale per trasformarne le parole specifiche in parole chiave più generiche.\n
\n \n Cercando su Telegram un plurale inglese, verranno restituiti anche i messaggi che riportano la stessa parola ma al singolare.\n \n \n \n \n Le regex e i glob sono due dei pattern-based query languages più comuni.\n \n
\n Definisce parti di testo da cercare all'interno dei documenti.\n
\n
\n Permettono dunque la ricerca di prefissi, suffissi, sottostringhe e intervalli.\n
\n \n \n \n \n \n Google, DuckDuckGo, Ecosia usano tutti un linguaggio di query keyword-based booleano.\n \n
\n Estendono gli altri tipi di query permettendo l'inserimento di operatori booleani (AND, OR e NOT) tra le parole chiave.\n
\n \n Per esempi sulla sintassi per effettuare ricerche booleane su Google, visita Google Advanced Search.\n \n \n \n \n Google Books permette di sfruttare la struttura fissa delle informazioni sui libri (titolo, autore, editore...) per ottenere risultati più precisi.,\n \n
\n Permettono di effettuare i precedenti tipi di query su specifiche parti della struttura dei documenti.\n
\n \n \n \n MeSH cataloga i paper di medicina caricati su PubMed organizzandoli in una gerarchia di categorie.\n \n
\n Limitano le keyword a un vocabolario predefinito utilizzato anche dai documenti.\n
\n \n \n \n )\n}\n","import {Fragment} from \"preact\";\nimport {Section, Panel, ILatex, BLatex, PLatex, BaseLink, Code} from \"bluelib\";\nimport Link from \"../../components/Link\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n
Prof. Federica Mandreoli
\n
\n \n \n
\n
Pagina Dolly Unimore
\n
\n \n \n
\n Puoi scaricare le videolezioni di Gestione dell'informazione con bbbdl:\n
\n \n bbbdl sync -r https://raw.githubusercontent.com/Steffo99/unimoreinfo-wiki/main/gestionedellinformazione.json\n \n \n \n \n )\n}\n","import {Fragment} from \"preact\";\nimport {Section, Panel, ILatex, BLatex, PLatex, BoxColors} from \"bluelib\";\nimport Example from \"../../components/Example\";\nimport ToBeContinued from \"../../components/ToBeContinued\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n La frazione dei documenti rilevanti totali restituita:\n
\n {r`\n \\frac{R \\cap A}{R}\n `}\n \n \n
\n La frazione dei documenti restituiti che sono rilevanti:\n
\n {r`\n \\frac{R \\cap A}{A}\n `}\n \n \n \n \n
\n Si effettuano test e si valutano precisione e richiamo ottenuti.\n
\n \n
\n Il modello IR ideale ha valori di richiamo e precisione di 1, ma nella realtà questi valori tendono ad essere inversamente proporzionali.\n
\n
\n Un fattore significativo nel richiamo e nella precisione è la forma del documento.\n
\n \n Libri interi, capitoli, pagine o frasi; possono essere tutti documenti, cambia solo la loro granularità.\n \n \n \n
\n Si fissano un certo numero di livelli di richiamo desiderati e si misura la precisione che ha il modello in quei punti, creando una curva di richiamo standard.\n
\n
\n È possibile ottenere una curva di richiamo interpolata prendendo per ogni punto il valore di massima precisione tra esso e i punti precedenti.\n
\n \n Interpolata mi sembra un termine orribile... Non sarebbe molto meglio monotonica?\n \n \n \n
\n È possibile misurare la curva di richiamo media effettuando la media su più query dei vari valori, e la curva di richiamo a valore singolo effettuando una query sola.\n
\n
\n Sono entrambe importanti per effettuare una valutazione, perchè la curva media potrebbe mascherare problemi di specifiche query nei modelli.\n
\n \n \n
\n La media tra i valori di più curve di richiamo.\n
\n
\n È utile per le valutazioni in cui l'utente sta cercando più documenti rilevanti alla volta.\n
\n \n \n
\n Valori di recall e precision riguardo i primi {r`R`} documenti restituiti.\n
\n \n La prima pagina di Google (10 documenti) considererà i valori di 10-recall e 10-precision.\n \n \n \n \n \n
\n È anche detta media armonica.\n
\n
\n Un singolo valore che tiene conto sia della precisione {r`p`} sia del richiamo {r`r`}:\n
\n {r`\n F = \\frac{2}{\\frac{1}{p} + \\frac{1}{r}}\n `}\n \n \n
\n Una versione più configurabile della Misura F che permette di dare priorità{r`b`} a precisione ({r`b > 1`}) o a richiamo ({r`b < 1`}) in base alle proprie esigenze:\n
\n Misura per valutare i motori di ricerca online che tiene conto dell'ordine in cui vengono restituiti i documenti.\n
\n
\n Si basa su valutazioni di rilevanza e su pesi posizionali dei risultati.\n
\n \n Valuta quanto è stata utile questa ricerca:\n \n ☆☆☆☆☆\n \n \n \n La valutazione del primo risultato viene moltiplicata per {r`\\frac{1}{\\log_2(1)}`}, la seconda per {r`\\frac{1}{\\log_2(2)}`}, la terza per {r`\\frac{1}{\\log_2(3)}`} e così via.\n \n \n \n
\n Miglioramento alla DCG che supporta quantità di risultati differenti.\n
\n \n Non dice la formula, e l'esempio non esemplifica un bel niente...\n \n \n \n \n )\n}\n"],"sourceRoot":""}
\ No newline at end of file
diff --git a/docs/route-Home.chunk.1743e.css b/docs/route-Home.chunk.1743e.css
new file mode 100644
index 0000000..a3c6f0d
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+.menulist__2Cmnq{font-size:large}.menulist__2Cmnq small{font-size:small}
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diff --git a/docs/route-Home.chunk.9c5d6.esm.js b/docs/route-Home.chunk.9c5d6.esm.js
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+++ b/docs/route-Home.chunk.9c5d6.esm.js
@@ -0,0 +1,2 @@
+(window.webpackJsonp=window.webpackJsonp||[]).push([[8],{"5aVd":function(i){i.exports={menulist:"menulist__2Cmnq"}},T2GU:function(i,e,t){"use strict";(function(i){var l=t("5aVd"),r=t.n(l);e.a=function(e){return i("ul",{class:r.a.menulist},e.children)}}).call(this,t("hosL").h)},bJge:function(i,e,t){"use strict";t.r(e),function(i){var l=t("mbOI"),r=t("YNhk"),n=t("T2GU");e.default=function(){return i("div",null,i(l.s,{title:"Benvenuto ad Unisteffo!"},i(l.r,{title:"Benvenuto!"},i("p",null,"Ciao! Benvenuto ad ",i("b",null,"Unisteffo"),", il sito dove carico tutto il materiale che uso per studiare!"),i("p",null,"Se trovi un errore negli appunti, ",i("a",{href:"https://github.com/Steffo99/appuntiweb-alpha/issues/new"},"segnalamelo su GitHub")," :)"),i("p",null,"Se trovi un bug del sito... Pazienza, ormai l'ho ",i("a",{href:"https://github.com/Steffo99/appuntiweb"},"riscritto da 0"),"!"))),i(l.s,{title:"Primo anno"},i(l.r,{title:"Primo semestre"},i(n.a,null,i("li",null,i(r.a,{href:"https://github.com/Steffo99/appunti-universitari/tree/master/2018_Analisi1"},"Analisi matematica")),i("li",null,i(r.a,{href:"https://github.com/Steffo99/appunti-universitari/tree/master/2018_AlgebraLineare"},"Algebra lineare")))),i(l.r,{title:"Secondo semestre"},i(n.a,null,i("li",null,i(r.a,{href:"https://github.com/Steffo99/appunti-universitari/tree/master/2018_ArchitetturaDeiCalcolatori"},"Architettura dei calcolatori")),i("li",null,i(l.b,{href:"/algoritmiestrutturedati"},"Algoritmi e strutture dati"))))),i(l.s,{title:"Secondo anno"},i(l.r,{title:"Terzo semestre"},i(n.a,null,i("li",null,i(l.b,{href:"/fisica"},"Fisica")),i("li",null,i(r.a,{href:"https://github.com/Steffo99/cleaver"},"Programmazione a oggetti")),i("li",null,i(r.a,{href:"https://github.com/Steffo99/appunti-universitari/tree/master/2019_SistemiOperativi/Arzigogoli"},"Sistemi operativi")),i("li",null,i(l.b,{href:"/statistica"},"Statistica ed elementi di probabilità")))),i(l.r,{title:"Quarto semestre"},i(n.a,null,i("li",null,i(l.b,{href:"/apprendimento"},"Apprendim. ed evol. in sist. artificiali")),i("li",null,i(l.b,{href:"/basididati"},"Basi di dati")),i("li",null,i(l.b,{href:"/calcolonumerico"},"Calcolo numerico")),i("li",null,i(l.b,{href:"/ottimizzazionelineare"},"Ottimizzazione lineare intera"))))),i(l.s,{title:"Terzo anno"},i(l.r,{title:"Quinto semestre"},i(n.a,null,i("li",null,i(l.b,{href:"/linguaggidinamici"},"Linguaggi dinamici")),i("li",null,i(l.b,{href:"/protocollidirete"},"Protocolli e architett. di rete")),i("li",null,i(l.b,{href:"/gestionedellinformazione"},"Gestione dell'informazione")))),i(l.r,{title:"Ultimo semestre"},i("p",null,i("i",null,"Non c'è ancora nulla qui...")))),i(l.s,{title:"Altro"},i(l.r,{title:"Altri appunti"},i(n.a,null,i("li",null,i(l.b,{href:"/mingwinstall"},"Come installare MinGW")," ",i("small",null,"(per studenti sperduti di Programmazione 2)")),i("li",null,i(l.b,{href:"/vldigeometria"},"Videolezioni di Geometria")," ",i("small",null,"(per studenti sperduti di Algebra Lineare)")))),i(l.r,{title:"Collegamenti utili"},i(n.a,null,i("li",null,i(r.a,{href:"https://erre2.fermitech.info/dashboard"},"Erre2")),i("li",null,i("small",null,"Tag GitHub: "),i(r.a,{href:"https://github.com/topics/unimore-informatica"},"unimore-informatica")),i("li",null,i("small",null,"Network Telegram: "),i(r.a,{href:"https://thor.steffo.eu"},"Unimore Informatica")),i("li",null,i("small",null,"Solo studenti Unimore: "),i(r.a,{href:"https://drive.google.com/drive/folders/1gqY-QIe4UeOSHpcho0R-Nvh2IRAlTFmf"},"Archivio Lezioni"))))))}}.call(this,t("hosL").h)}}]);
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\n Questa sezione è incompleta. Torna più avanti!\n
\n {props.children}\n \n )\n}\n","import {Fragment} from \"preact\";\nimport {Section, Panel, ILatex, BLatex, PLatex} from \"bluelib\";\nimport Link from \"../../components/Link\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n
Prof. Mauro Leoncini
\n
\n \n \n \n )\n}\n","import {Fragment} from \"preact\";\nimport {Section, Panel, ILatex, BLatex, PLatex, CodeDefaultLanguage, Code, BoxColors} from \"bluelib\";\nimport Link from \"../../components/Link\";\nimport ToBeContinued from \"../../components/ToBeContinued\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n\n \n \n
\n Su questo sito non sono presenti i concetti base di Python e della programmazione.\n
\n
\n Se ne avete bisogno, potete leggervi A Byte of Python, un ottimo libro gratuito e veloce a riguardo :)\n
\n \n \n \n\n \n )\n}\n","import Intro from \"./00_Intro\";\nimport Python from \"./01_Python\";\n\n\nexport default function () {\n return (\n
\n
Linguaggi dinamici
\n \n \n
\n )\n}\n"],"sourceRoot":""}
\ No newline at end of file
diff --git a/docs/route-MingwInstall.chunk.fba30.esm.js b/docs/route-MingwInstall.chunk.fba30.esm.js
new file mode 100644
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\n
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\n
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Una volta installato, dobbiamo aggiungere g++ ai programmi eseguibili da Prompt dei\n Comandi: premete il\n tasto Windows, e scrivete PATH. Windows dovrebbe trovarvi automaticamente\n quell'opzione.
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\n
Nel nostro caso, dovrebbero servirci mingw32-base-bin (per il C e alcune librerie C++)\n e\n mingw32-gcc-g++-bin (per il C++). Cliccate, quindi, sui due quadratini corrispondenti,\n e premete\n Mark for Installation. Dovrebbe comparire una freccia gialla sul quadratino.
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\n \n
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"]);return k=function(){return n},n}function T(){var n=D(["\n a geq b Leftrightarrow a - s = b\n "],["\n a \\geq b \\Leftrightarrow a - s = b\n "]);return T=function(){return n},n}function S(){var n=D(["\n a leq b Leftrightarrow a + s = b\n "],["\n a \\leq b \\Leftrightarrow a + s = b\n "]);return S=function(){return n},n}function y(){var n=D(["s"]);return y=function(){return n},n}function I(){var n=D(["\n a = b Leftrightarrow\n \begin{cases}\n a leq b\\\n a geq b\n end{cases}\n "],["\n a = b \\Leftrightarrow\n \\begin{cases}\n a \\leq b\\\\\n a \\geq b\n \\end{cases}\n "]);return I=function(){return n},n}function L(){var n=D(["min left{ mathbf{c}^T mathbf{x} : mathbf{A} mathbf{x} = b,quad x_j geq 0,quad j = 1 dots n \right}"],["min \\left\\{ \\mathbf{c}^T \\mathbf{x} : \\mathbf{A} \\mathbf{x} = b,\\quad x_j \\geq 0,\\quad j = 1 \\dots n \\right\\}"]);return L=function(){return n},n}function A(){var n=D(["min left{ mathbf{c}^T mathbf{x} : mathbf{A} mathbf{x} geq b,quad x_j geq 0,quad j = 1 dots n \right}"],["min \\left\\{ \\mathbf{c}^T \\mathbf{x} : \\mathbf{A} \\mathbf{x} \\geq b,\\quad x_j \\geq 0,\\quad j = 1 \\dots n \\right\\}"]);return A=function(){return n},n}function P(){var n=D(["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}"],["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\\}"]);return P=function(){return n},n}function w(){var n=D(["mathbf{c}^T mathbf{x}"],["\\mathbf{c}^T \\mathbf{x}"]);return w=function(){return n},n}function C(){var n=D(["mathbf{b}"],["\\mathbf{b}"]);return C=function(){return n},n}function V(){var n=D(["mathbf{A}"],["\\mathbf{A}"]);return V=function(){return n},n}function j(){var n=D(["\nabla z = (2, 3, 4)"],["\\nabla z = (2, 3, 4)"]);return j=function(){return n},n}function F(){var n=D(["mathbf{I}"],["\\mathbf{I}"]);return F=function(){return n},n}function U(){var n=D(["\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"],["\\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"]);return U=function(){return n},n}function R(){var n=D(["z(mathbf{x}) = c_1 cdot x_1 + c_2 cdot x_2 + dots + c_n cdot x_n"],["z(\\mathbf{x}) = c_1 \\cdot x_1 + c_2 \\cdot x_2 + \\dots + c_n \\cdot x_n"]);return R=function(){return n},n}function G(){var n=D(["mathbf{x}"],["\\mathbf{x}"]);return G=function(){return n},n}function O(){var n=D(["mathbf{c}"],["\\mathbf{c}"]);return O=function(){return n},n}function D(n,l){return l||(l=n.slice(0)),n.raw=l,n}var M=i("hosL"),E=i("mbOI"),B=i("fWbf"),N=i("RPi6"),J=i("ke5e"),Z=i("RtuP"),K=i("Th1s"),X=i("dIxp"),H=i("o8Jj"),W=i("ZHMS"),Y=i("mGtH"),Q=String.raw;l.a=function(){return n(M.Fragment,null,n(E.s,{title:"Problemi di ottimizzazione lineare"},n(E.r,{title:"Cosa sono?"},n("p",null,"Problemi che cercano di ",n(B.a,null,"minimizzare"),"/",n(N.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(E.r,{title:"Funzione obiettivo"},n("p",null,"La funzione da ",n(B.a,null,"minimizzare"),"/",n(N.a,null,"massimizzare"),"."),n("p",null,"Il vettore dei suoi coefficienti è detto ",n(E.k,null,Q(O())),", mentre quello delle sue incognite ",n(E.k,null,Q(G())),"."),n("p",null,"In genere, la funzione obiettivo è scritta in forma di ",n("b",null,"combinazione lineare")," tra le ",n("b",null,"incognite")," e i ",n("b",null,"coefficienti"),":"),n("p",null,n(E.k,null,Q(R())))),n(E.r,{title:"Gradiente"},n("p",null,n("b",null,"Funzione")," della funzione obiettivo che restituisce la direzione del suo aumento più veloce."),n("p",null,n(E.k,null,Q(U()))),n(J.a,null,"La matrice ",n(E.k,null,Q(F()))," è la matrice identità."),n(J.a,null,"Se la funzione obiettivo è ",n(E.k,null,"z = 2w + 3x + 4y"),", il suo gradiente è ",n(E.k,null,Q(j())),".")),n(E.r,{title:"Vincoli"},n("p",null,"Equazioni e disequazioni a cui devono sottostare le incognite perchè esse formino una soluzione valida."),n("p",null,"I loro coefficienti sono contenuti nella matrice ",n(E.k,null,Q(V())),", mentre i loro termini noti nel vettore ",n(E.k,null,Q(C())),".")),n(E.r,{title:"Poliedro"},n("p",null,"L'",n("b",null,"insieme")," che racchiunde tutte le ",n("b",null,"soluzioni ammissibili")," di un problema."),n("p",null,"Può essere ",n("i",null,n(Z.a,null)),", ",n("i",null,n(K.a,null))," oppure ",n("i",null,n(X.a,null)),"."),n(J.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(E.r,{title:"Valore ottimo"},n("p",null,"La ",n("b",null,"soluzione")," di un problema, ricavabile dal prodotto ",n(E.k,null,Q(w())),"."),n("p",null,"In particolare, il valore ottimo è un ",n("b",null,"vertice")," del poliedro, detto ",n("i",null,"vertice ottimo"),"."))),n(E.s,{title:"Forme di un problema di ottimizzazione"},n(E.r,{title:"Forma generale"},n("p",null,"Un problema con:"),n("ul",null,n("li",null,n(H.a,null,"Equazioni e disequazioni")),n("li",null,n(H.a,null,"Variabili non vincolate"))),n(E.q,null,Q(P()))),n(E.r,{title:"Forma canonica"},n("p",null,"Un problema con:"),n("ul",null,n("li",null,n(H.a,null,"Solo disequazioni")),n("li",null,n(W.a,null,"Vincoli di non-negatività sulle incognite"))),n(E.q,null,Q(A()))),n(E.r,{title:"Forma standard"},n("p",null,"Un problema con:"),n("ul",null,n("li",null,n(W.a,null,"Solo equazioni")),n("li",null,n(W.a,null,"Vincoli di non-negatività sulle incognite"))),n(E.q,null,Q(L())))),n(E.s,{title:"Conversioni tra le forme"},n(E.r,{title:"Standard e generale"},n("p",null,"Applica questa conversione a ogni equazione nel sistema:"),n("p",null,n(E.k,{inline:!1},Q(I()))),n(J.a,null,"Serve solo nella teoria per dimostrare che le forme sono equivalenti.")),n(E.r,{title:"Canonica e standard"},n("p",null,"Aggiungi una ",n("i",null,"variabile slack")," ",n(E.k,null,Q(y()))," ",n("b",null,"non-vincolata")," a ogni disequazione nel sistema:"),n("p",null,n(E.k,{inline:!1},Q(S()))),n("p",null,n(E.k,{inline:!1},Q(T())))),n(E.r,{title:"Generale e canonica"},n("p",null,"Sdoppia ogni variabile non-vincolata in due variabili con vincolo di non-negatività:"),n("p",null,n(E.k,{inline:!1},Q(k()))))),n(E.s,{title:"Tableau"},n(E.r,{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(E.r,{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(J.a,null,"Suona familiare? Sì, lo abbiamo fatto anche in Algebra Lineare.")),n(E.r,{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(E.r,{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(E.s,null,n(E.r,{color:Y.a,title:"Un esempio"},n("p",null,"Il sistema:"),n(E.q,null,Q(q())),n("p",null,"Diventa il tableau:"),n("table",{className:"right"},n("thead",null,n("tr",null,n("th",null,n(E.k,null,"x_1")),n("th",null,n(E.k,null,"x_2")),n("th",null,n(E.k,null,"s_1")),n("th",null,n(E.k,null,"s_2")),n("th",null,n(E.k,null,"s_3")),n("th",null,n("abbr",{title:"Termine noto"},"TN")))),n("tbody",null,n("tr",null,n("td",null,n(E.k,null,"1")),n("td",null,n(E.k,null,"0")),n("td",{style:"background-color: rgba(0, 255, 255, 0.1);"},n(E.k,null,"1")),n("td",{style:"background-color: rgba(0, 255, 255, 0.1);"},n(E.k,null,"0")),n("td",{style:"background-color: rgba(0, 255, 255, 0.1);"},n(E.k,null,"0")),n("td",{style:"background-color: rgba(255, 0, 0, 0.1);"},n(E.k,null,"3"))),n("tr",null,n("td",null,n(E.k,null,"0")),n("td",null,n(E.k,null,"1")),n("td",{style:"background-color: rgba(0, 255, 255, 0.1);"},n(E.k,null,"0")),n("td",{style:"background-color: rgba(0, 255, 255, 0.1);"},n(E.k,null,"1")),n("td",{style:"background-color: rgba(0, 255, 255, 0.1);"},n(E.k,null,"0")),n("td",{style:"background-color: rgba(255, 0, 0, 0.1);"},n(E.k,null,"3"))),n("tr",null,n("td",null,n(E.k,null,"2")),n("td",null,n(E.k,null,"2")),n("td",{style:"background-color: rgba(0, 255, 255, 0.1);"},n(E.k,null,"0")),n("td",{style:"background-color: rgba(0, 255, 255, 0.1);"},n(E.k,null,"0")),n("td",{style:"background-color: rgba(0, 255, 255, 0.1);"},n(E.k,null,"1")),n("td",{style:"background-color: rgba(255, 0, 0, 0.1);"},n(E.k,null,"7"))),n("tr",{style:"background-color: rgba(0, 255, 0, 0.1);"},n("td",null,n(E.k,null,"2000")),n("td",null,n(E.k,null,"1000")),n("td",null,n(E.k,null,"0")),n("td",null,n(E.k,null,"0")),n("td",null,n(E.k,null,"0")),n("td",{style:"background-color: rgba(255, 255, 0, 0.2);"},n(E.k,null,"0"))))),n("p",null,"Con i seguenti elementi:"),n("ul",null,n("li",null,n("u",{style:"color: #7dff7d;"},"Funzione obiettivo")),n("li",null,n("u",{style:"color: #ffff7d;"},"Valore attuale")),n("li",null,n("u",{style:"color: #ff7d7d;"},"Termini noti")),n("li",null,n("u",{style:"color: #7dffff;"},"Variabili slack"))))),n(E.s,{title:"Simplex primale"},n(E.r,{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(J.a,null,"Ricordi ",n(E.b,{href:"/calcolonumerico"},"Gauss"),"? Il Simplex è la stessa cosa, in cui però si cerca di ",n(B.a,null,"minimizzare"),"/",n(N.a,null,"massimizzare")," il termine noto della funzione obiettivo."),n(J.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(K.a,null),".")),n(E.r,{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(B.a,null,"positivi"),"/",n(N.a,null,"negativi")," nella funzione obiettivo:",n("ol",null,n("li",null,n("p",null,n("b",null,"Scegli")," la prima variabile fuori base con coefficiente ",n(B.a,null,"positivo"),"/",n(N.a,null,"negativo")," nella funzione obiettivo: essa è la ",n("i",null,"variabile entrante"),"."),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("p",null,n("b",null,"Scegli")," la variabile in base con il minor rapporto positivo:"),n(E.q,null,Q(x())),n("aside",null,"Se non sei riuscito a trovare nessuna variabile con un rapporto positivo, significa che il poliedro è ",n(X.a,null),".")),n("li",null,n("p",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(Z.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(E.r,{title:"Soluzioni di base degenerata"},n("p",null,"Una soluzione con almeno una variabile di valore ",n(E.k,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(E.s,{title:"Metodo delle due fasi"},n(E.r,{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(E.k,null,Q(z())),".")),n(E.r,{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(K.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(E.s,{title:"Rilassamento"},n(E.r,{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(E.r,{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(E.k,null,Q(_())),"."),n(J.a,null,n("p",null,"Il sistema:"),n(E.k,{inline:!1},Q(g())),n("p",null,"diventa:"),n(E.k,{inline:!1},Q(v()))))),n(E.s,{title:"Dualità"},n(E.r,{title:"Duale"},n("p",null,"Il sistema che ",n("b",null,n(B.a,null,"massimizza"),"/",n(N.a,null,"minimizza")," i moltiplicatori di rilassamento")," di un problema detto ",n("i",null,"primale"),".")),n(E.r,{title:"In termini matriciali"},n("p",null,"Possiamo ",n("b",null,"trasporre")," il tableau e sostituire le variabili ",n(E.k,null,Q(h()))," con variabili ",n(E.k,null,Q(p()))," per ottenere il sistema duale!"),n("p",null,"I maggiori e minori dei vincoli diventeranno maggiori e minori delle variabili e viceversa.")),n(E.r,{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(K.a,null)),", allora il suo duale potrà essere ",n(K.a,null)," oppure ",n(X.a,null),"."),n("li",null,"Se un problema è ",n("b",null,n(X.a,null)),", allora il suo duale sarà certamente ",n(K.a,null),"."))),n(E.r,{title:"Variabili e vincoli"},n("p",null,"Variabili e vincoli del duale corrispondono rispettivamente a vincoli e variabili del primale."),n("p",null,"In 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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\"};","// extracted by mini-css-extract-plugin\nmodule.exports = {\"menulist\":\"menulist__2Cmnq\"};","import Intro from \"./00_Intro\";\nimport OttimizzazioneLineare from \"./01_OttimizzazioneLineare\";\nimport OttimizzazioneLineareIntera from \"./02_OttimizzazioneLineareIntera\";\nimport TeoriaDeiGrafi from \"./03_TeoriaDeiGrafi\";\n\n\nexport default function () {\n return (\n
\n
Ottimizzazione Lineare Intera
\n \n \n \n \n
\n )\n}\n","import {Fragment} from \"preact\";\nimport {Section, Panel, BLatex, TablePanel} from \"bluelib\";\nimport Link from \"../../components/Link\";\nimport MenuList from \"../../components/MenuList\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n
Prof. Stefano Novellani
\n
\n \n \n
\n Esame digitale in 3 fasi:\n
\n
\n
25 domande di teoria stile vero/falso
\n
2 domande a risposta aperta
\n
2 modelli matematici da rappresentare tramite GLPK/MathProg
\n
\n \n \n \n \n
\n Ti serve una mano anche in GLPK? Che in realtà si chiama GMPL, ma vabbè...\n
\n
\n Ho fatto una pagina apposta:\n
\n \n
GLPK/MathProg/GMPL
\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","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 \"./MenuList.less\";\n\nexport default function(props) {\n return (\n
\n {props.children}\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 {Fragment} from \"preact\";\nimport {Section, Panel, ILatex, BLatex, PLatex, Todo, BaseLink, Image} from \"bluelib\";\nimport Example from \"../../components/Example\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n Terminologia dei grafi TODO: migliorare}>\n \n
\n Insieme di nodi{r`N`} e archi{r`E`} che li\n 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\n 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\n 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
Aggiungi l'arco di costo minimo all'albero.
\n
Finchè mancano ancora archi:\n \n
Trova tutti gli archi che aggiungerebbero un nuovo nodo all'albero.
\n
Seleziona l'arco di costo minore.
\n \n
\n \n \n \n
\n Trova l'ordine topologico di un albero.\n
\n \n
Ripeti finchè ci sono nodi nel grafo:\n \n
Assegna un numero sequenziale a un nodo senza archi entranti.
\n
Elimina il nodo a cui hai assegnato il numero.
\n
Elimina tutti gli archi incidenti sul nodo che hai eliminato.
\n \n
\n \n \n \n
\n Trova i percorsi di costo minimo in un albero.\n
\n \n
Trova l'ordine topologico dell'albero.
\n
Invece che provare ogni singola combinazione di nodi, prova solo i nodi che hanno un numero\n topologico maggiore di quello del nodo attuale.\n
\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\n 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","import {Fragment} from \"preact\";\nimport {Section, Panel, ILatex, BLatex, PLatex, Latex, BaseLink, Todo} from \"bluelib\";\nimport Min from \"../../components/OttimizzazioneLineare/Min\";\nimport Max from \"../../components/OttimizzazioneLineare/Max\";\nimport Example from \"../../components/Example\";\nimport Finite from \"../../components/OttimizzazioneLineare/Finite\";\nimport Empty from \"../../components/OttimizzazioneLineare/Empty\";\nimport Unbounded from \"../../components/OttimizzazioneLineare/Unbounded\";\nimport Plus from \"../../components/Fisica/Plus\";\nimport Minus from \"../../components/Fisica/Minus\";\nimport ExampleBoxColor from \"../../components/ExampleBoxColor\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n Problemi che cercano di minimizzare/massimizzare il valore di una funzione\n 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\n incognite {r`\\mathbf{x}`}.\n
\n
\n In genere, la funzione obiettivo è scritta in forma di combinazione lineare tra le incognite e i coefficienti:\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\n 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\n valida.\n
\n
\n I loro coefficienti sono contenuti nella matrice {r`\\mathbf{A}`}, mentre i loro\n 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\n più lati, ovvero un poliedro.\n \n \n \n
\n La soluzione di un problema, ricavabile dal\n 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 Un modo per rappresentare sistemi in forma standard, anche noto come matrice equivalente\n 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\n valessero 0.\n
\n
\n Procedendo nella risoluzione (descritta in seguito) del tableau, questo valore aumenterà, fino a\n raggiungere il valore ottimo quando la risoluzione sarà completata.\n
\n Un algoritmo per trovare efficientemente il valore ottimo e le coordinate di un vertice\n ottimo in problemi di ottimizzazione lineare.\n
\n \n Ricordi Gauss? Il Simplex è la stessa cosa,\n in cui però si cerca di minimizzare/massimizzare il termine noto della funzione obiettivo.\n \n \n Questa è la soluzione passo per passo del\n problema 3 del file Ex_LP_testo.\n \n
\n Perchè sia possibile effettuare il Simplex è necessario che l'origine sia nel poliedro:\n pertanto, non è possibile che un problema risolto con il Simplex sia .\n
\n \n \n \n
Trasforma il sistema in forma standard.
\n
Trova tante variabili linearmente indipendenti quante siano le righe: esse saranno\n la base iniziale.\n
\n
Finchè ci sono variabili con coefficienti positivi/negativi nella\n funzione obiettivo:\n \n
\n
\n Scegli la prima variabile fuori base con\n coefficiente positivo/negativo nella funzione obiettivo: essa\n è la variabile entrante.\n
\n \n
\n
\n
\n Scegli la variabile in base con il minor rapporto\n positivo:\n
\n {r`\\frac{b_i}{A_ik}`}\n \n
\n
\n
\n Pivot: trasforma tutte le funzioni del sistema in modo che abbiano 0\n nella colonna della variabile entrante, tranne nella riga della variabile uscente,\n in cui avrà 1.\n
\n
\n \n
\n
Il poliedro è : i termini noti dei vincoli sono le coordinate del suo\n vertice ottimo, mentre il termine noto della funzione obiettivo è il valore ottimo.\n
\n \n \n \n
\n Una soluzione con almeno una variabile di valore 0, dovuta a uno o più vincoli\n ridondanti.\n
\n
\n Senza Regola di Bland e in presenza di vincoli ridondanti si rischia di trovarsi a fare\n pivot infiniti.\n
\n \n \n \n \n
\n Un estensione del Simplex per permettere la risoluzione di problemi la cui origine non\n è una soluzione ammissibile.\n
\n
\n Prevede l'introduzione di un problema ausiliario, le cui incognite sono\n dette artificiali.\n
\n
\n Il vettore delle incognite artificiali è solitamente chiamato {r`\\mathbf{y}`}.\n
\n \n \n \n
Crea un nuovo tableau, aggiungendo variabili artificiali in modo da avere una base\n ammissibile.\n
\n
Sostituisci la vecchia funzione obiettivo con una nuova che minimizzi la somma di\n tutte le variabili artificiali.\n
\n
Fase 1: Risolvi il nuovo problema con il Simplex primale.
\n
Se il Simplex termina quando ci sono ancora variabili artificiali nella base, allora\n il poliedro è .\n
\n
Una volta che le variabili artificiali sono fuori base, elimina le loro colonne e la\n nuova funzione obiettivo.
\n
Riporta il tableau in forma base compiendo operazioni per azzerare i\n coefficienti delle variabili di base nella funzione obiettivo.\n
\n
Fase 2: Risolvi il tableau con il Simplex primale.
\n \n \n \n \n \n
\n Una versione semplificata di un problema nella quale si ignora la violazione di uno o più\n 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\n obiettivo.\n
\n
\n Il vettore dei coefficienti di rilassamento solitamente è indicato\n con {r`\\mathbf{u}`}.\n
\n Il sistema che massimizza/minimizza i moltiplicatori di\n rilassamento di un problema detto primale.\n
\n \n \n
\n Possiamo trasporre il tableau e sostituire le variabili {r`x_n`} con\n 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\n essere oppure .\n
\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
Min
\n
Max
\n
\n \n \n
\n
Vincolo \\leq
\n
Variabile \\leq
\n
\n
\n
Vincolo \\geq
\n
Variabile \\geq
\n
\n
\n
Vincolo =
\n
Variabile libera
\n
\n
\n
Variabile \\leq
\n
Vincolo \\geq
\n
\n
\n
Variabile \\geq
\n
Vincolo \\leq
\n
\n
\n
Variabile libera
\n
Vincolo =
\n
\n \n
\n \n \n \n \n
\n Una disuguaglianza lineare {r`c_0 \\leq \\mathbf{c}^T \\mathbf{x}`} è verificata da\n tutti i punti di un poliedro non- se e solo se esiste un\n vettore {r`u \\in \\mathfrak{R}^m`} tale che:\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\n tableau) è sempre minore o uguale/maggiore o uguale alla soluzione del\n 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\n primale. TODO: credo?\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\n vincolo {r`b_i`} o il coefficiente della funzione\n obiettivo {r`c_i`} prima che la base degeneri.\n
\n \n \n \n )\n}\n","import style from \"./Minus.less\";\n\nexport default function (props) {\n return {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\"};","// 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\";\n\nexport default function (props) {\n return (\n
\n {props.children}\n
\n );\n}\n","import style from \"./ExampleBoxColor.less\";\n\nexport default style.example;\n","import style from \"./Plus.less\";\n\nexport default function (props) {\n return {props.children};\n}\n","import {Fragment} from \"preact\";\nimport {Section, Panel, ILatex, BLatex, PLatex} from \"bluelib\";\nimport Min from \"../../components/OttimizzazioneLineare/Min\";\nimport Max from \"../../components/OttimizzazioneLineare/Max\";\nimport Example from \"../../components/Example\";\nimport Empty from \"../../components/OttimizzazioneLineare/Empty\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n Particolari problemi di ottimizzazione lineare in cui le variabili sono vincolate ad essere\n 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\n continua.\n
\n \n \n \n \n
\n Un modo per passare dalla soluzione del rilassamento alla soluzione intera di un problema\n di ILP.\n
\n
\n Consiste nel calcolare la soluzione di ogni singolo punto incluso nel poliedro, e selezionare\n la minore/maggiore.\n
\n
\n Trova sicuramente la soluzione giusta, ma il costo computazionale è\n esponenziale O(n^k)!\n
\n \n \n
\n Un altro modo per passare dalla soluzione del rilassamento alla soluzione intera di un\n problema di ILP.\n
\n
\n Consiste nell'arrotondare tutte le variabili al loro valore intero più vicino, e\n calcolarne il valore ottimo.\n
\n
\n Funziona bene per valori grandi, ma più essi si avvicinano allo 0 più l'errore diventa\n grande.\n
\n \n \n
\n Un altro modo ancora per passare dalla soluzione del rilassamento alla soluzione intera\n di un problema di ILP.\n
\n
\n Consiste nel tagliare il poliedro con nuovi vincoli (piani secanti) che riducono le\n possibili soluzioni continue ma non quelle intere.\n
\n
\n Per selezionare i vincoli, si usano i tagli di Gomory:\n
\n Per ogni valore noto frazionario si viene quindi a creare una nuova variabile in base e\n 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
{r`x_1`}
\n
{r`x_2`}
\n
{r`s_1`}
\n
{r`s_2`}
\n
TN
\n
\n \n \n
\n
{r`1`}
\n
{r`1`}
\n
{r`0`}
\n
{r`0`}
\n
{r`0`}
\n
\n
\n
{r`1`}
\n
{r`0`}
\n
{r`1`}
\n
{r`0`}
\n
{r`3`}
\n
\n
\n
{r`\\frac{3}{2}`}
\n
{r`\\frac{1}{2}`}
\n
{r`0`}
\n
{r`1`}
\n
{r`\\frac{6}{5}`}
\n
\n \n
\n \n
\n Diventa:\n
\n \n
\n
{r`x_1`}
\n
{r`x_2`}
\n
{r`s_1`}
\n
{r`s_2`}
\n
{r`s_3`}\n
\n
TN
\n
\n \n \n
\n
{r`1`}
\n
{r`1`}
\n
{r`0`}
\n
{r`0`}
\n
{r`0`}
\n
{r`0`}
\n
\n
\n
{r`1`}
\n
{r`0`}
\n
{r`1`}
\n
{r`0`}
\n
{r`0`}
\n
{r`3`}
\n
\n
\n
{r`\\frac{3}{2}`}
\n
{r`\\frac{1}{2}`}
\n
{r`0`}
\n
{r`1`}
\n
{r`0`}
\n
{r`\\frac{6}{5}`}
\n
\n
\n
{r`-\\frac{1}{2}`}
\n
{r`-\\frac{1}{2}`}
\n
{r`0`}
\n
{r`0`}
\n
{r`1`}
\n
{r`-\\frac{1}{5}`}
\n
\n \n
\n \n \n \n \n
\n È possibile usare la tecnica divide et impera per rendere più efficiente l'enumerazione\n totale.\n
\n
\n Si divide il problema principale (trovare il valore ottimo di un problema di ILP) in più\n sottoproblemi (trovare il valore ottimo di un problema di ILP con una variabile impostata a un\n 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\n 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\n (ma non ottimale)\n
\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\n 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\n iterazioni si applica il divide et impera.\n
\n \n \n \n )\n}\n","// extracted by mini-css-extract-plugin\nmodule.exports = {\"red\":\"red__3r1Jz\",\"orange\":\"orange__2q18p\",\"yellow\":\"yellow__3cyw6\",\"lime\":\"lime__2GJfI\",\"cyan\":\"cyan__k_5Y_\",\"blue\":\"blue__ejrIP\",\"magenta\":\"magenta__1Zdi7\",\"example\":\"example__22j7L\"};"],"sourceRoot":""}
\ No newline at end of file
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new file mode 100644
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+++ b/docs/route-OttimizzazioneLineare.chunk.dee78.esm.js
@@ -0,0 +1,43 @@
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grafi"},l(a.r,{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(a.r,{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(a.r,{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 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\quad x_j \geq 0,\quad j = 1 \dots n \right\}`)))),l(a.r,{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(m.a,null,"Vincoli di non-negatività sulle incognite"))),l(a.q,null,Q(S||(S=Y`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(a.r,{title:"Forma standard"},l("p",null,"Un problema con:"),l("ul",null,l("li",null,l(m.a,null,"Solo equazioni")),l("li",null,l(m.a,null,"Vincoli di non-negatività sulle incognite"))),l(a.q,null,Q(y||(y=Y`min \left\{ \mathbf{c}^T \mathbf{x} : \mathbf{A} \mathbf{x} = b,\quad x_j \geq 0,\quad j = 1 \dots n \right\}`))))),l(a.s,{title:"Conversioni tra le forme"},l(a.r,{title:"Standard e generale"},l("p",null,"Applica questa conversione a ogni equazione nel sistema:"),l("p",null,l(a.k,{inline:!1},Q(P||(P=Y`
+ a = b \Leftrightarrow
+ \begin{cases}
+ a \leq b\\
+ a \geq b
+ \end{cases}
+ `)))),l(u.a,null,"Serve solo nella teoria per dimostrare che le forme sono equivalenti.")),l(a.r,{title:"Canonica e standard"},l("p",null,"Aggiungi una ",l("i",null,"variabile slack")," ",l(a.k,null,Q(I||(I=Y`s`)))," ",l("b",null,"non-vincolata")," a ogni disequazione nel sistema:"),l("p",null,l(a.k,{inline:!1},Q(L||(L=Y`
+ a \leq b \Leftrightarrow a + s = b
+ `)))),l("p",null,l(a.k,{inline:!1},Q(T||(T=Y`
+ a \geq b \Leftrightarrow a - s = b
+ `))))),l(a.r,{title:"Generale e canonica"},l("p",null,"Sdoppia ogni variabile non-vincolata in due variabili con vincolo di non-negatività:"),l("p",null,l(a.k,{inline:!1},Q(A||(A=Y`
+ \begin{cases}
+ a = a^+ - a^-\\
+ a^+ \geq 0\\
+ a^- \geq 0
+ \end{cases}
+ `)))))),l(a.s,{title:"Tableau"},l(a.r,{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(a.r,{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(u.a,null,"Suona familiare? Sì, lo abbiamo fatto anche in Algebra Lineare.")),l(a.r,{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(a.r,{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(a.s,null,l(a.r,{color:b.a,title:"Un esempio"},l("p",null,"Il sistema:"),l(a.q,null,Q(V||(V=Y`
+ \begin{cases}
+ 1x_1\ {\color{Cyan} \leq}\ {\color{Red} 3}\\
+ 1x_2\ {\color{Cyan} \leq}\ {\color{Red} 3}\\
+ 2x_1 + 2x_2\ {\color{Cyan} \leq}\ {\color{Red} 7}\\
+ {\color{Yellow} \min}\ {\color{Green} 2000x_1 + 1000x_2}
+ \end{cases}
+ `))),l("p",null,"Diventa il tableau:"),l("table",{className:"right"},l("thead",null,l("tr",null,l("th",null,l(a.k,null,"x_1")),l("th",null,l(a.k,null,"x_2")),l("th",null,l(a.k,null,"s_1")),l("th",null,l(a.k,null,"s_2")),l("th",null,l(a.k,null,"s_3")),l("th",null,l("abbr",{title:"Termine noto"},"TN")))),l("tbody",null,l("tr",null,l("td",null,l(a.k,null,"1")),l("td",null,l(a.k,null,"0")),l("td",{style:"background-color: rgba(0, 255, 255, 0.1);"},l(a.k,null,"1")),l("td",{style:"background-color: rgba(0, 255, 255, 0.1);"},l(a.k,null,"0")),l("td",{style:"background-color: rgba(0, 255, 255, 0.1);"},l(a.k,null,"0")),l("td",{style:"background-color: rgba(255, 0, 0, 0.1);"},l(a.k,null,"3"))),l("tr",null,l("td",null,l(a.k,null,"0")),l("td",null,l(a.k,null,"1")),l("td",{style:"background-color: rgba(0, 255, 255, 0.1);"},l(a.k,null,"0")),l("td",{style:"background-color: rgba(0, 255, 255, 0.1);"},l(a.k,null,"1")),l("td",{style:"background-color: rgba(0, 255, 255, 0.1);"},l(a.k,null,"0")),l("td",{style:"background-color: rgba(255, 0, 0, 0.1);"},l(a.k,null,"3"))),l("tr",null,l("td",null,l(a.k,null,"2")),l("td",null,l(a.k,null,"2")),l("td",{style:"background-color: rgba(0, 255, 255, 0.1);"},l(a.k,null,"0")),l("td",{style:"background-color: rgba(0, 255, 255, 0.1);"},l(a.k,null,"0")),l("td",{style:"background-color: rgba(0, 255, 255, 0.1);"},l(a.k,null,"1")),l("td",{style:"background-color: rgba(255, 0, 0, 0.1);"},l(a.k,null,"7"))),l("tr",{style:"background-color: rgba(0, 255, 0, 0.1);"},l("td",null,l(a.k,null,"2000")),l("td",null,l(a.k,null,"1000")),l("td",null,l(a.k,null,"0")),l("td",null,l(a.k,null,"0")),l("td",null,l(a.k,null,"0")),l("td",{style:"background-color: rgba(255, 255, 0, 0.2);"},l(a.k,null,"0"))))),l("p",null,"Con i seguenti elementi:"),l("ul",null,l("li",null,l("u",{style:"color: #7dff7d;"},"Funzione obiettivo")),l("li",null,l("u",{style:"color: #ffff7d;"},"Valore attuale")),l("li",null,l("u",{style:"color: #ff7d7d;"},"Termini noti")),l("li",null,l("u",{style:"color: #7dffff;"},"Variabili slack"))))),l(a.s,{title:"Simplex primale"},l(a.r,{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(u.a,null,"Ricordi ",l(a.b,{href:"/calcolonumerico"},"Gauss"),"? Il Simplex è la stessa cosa, in cui però si cerca di ",l(t.a,null,"minimizzare"),"/",l(o.a,null,"massimizzare")," il termine noto della funzione obiettivo."),l(u.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(s.a,null),".")),l(a.r,{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(t.a,null,"positivi"),"/",l(o.a,null,"negativi")," nella funzione obiettivo:",l("ol",null,l("li",null,l("p",null,l("b",null,"Scegli")," la prima variabile fuori base con coefficiente ",l(t.a,null,"positivo"),"/",l(o.a,null,"negativo")," nella funzione obiettivo: essa è la ",l("i",null,"variabile entrante"),"."),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("p",null,l("b",null,"Scegli")," la variabile in base con il minor rapporto positivo:"),l(a.q,null,Q(C||(C=Y`\frac{b_i}{A_ik}`))),l("aside",null,"Se non sei riuscito a trovare nessuna variabile con un rapporto positivo, significa che il poliedro è ",l(c.a,null),".")),l("li",null,l("p",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(r.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(a.r,{title:"Soluzioni di base degenerata"},l("p",null,"Una soluzione con almeno una variabile di valore ",l(a.k,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(a.s,{title:"Metodo delle due fasi"},l(a.r,{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(a.k,null,Q(w||(w=Y`\mathbf{y}`))),".")),l(a.r,{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(s.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(a.s,{title:"Rilassamento"},l(a.r,{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(a.r,{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(a.k,null,Q(F||(F=Y`\mathbf{u}`))),"."),l(u.a,null,l("p",null,"Il sistema:"),l(a.k,{inline:!1},Q(U||(U=Y`
+ \begin{cases}
+ z = 3 x_1 + 5 x_2\\
+ 2 x_1 + 3 x_2 \geq 12\\
+ - x_1 + 3 x_2 \geq 3\\
+ x_1 \geq 0\\
+ x_2 \geq 0
+ \end{cases}
+ `))),l("p",null,"diventa:"),l(a.k,{inline:!1},Q(j||(j=Y`
+ \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(a.s,{title:"Dualità"},l(a.r,{title:"Duale"},l("p",null,"Il sistema che ",l("b",null,l(t.a,null,"massimizza"),"/",l(o.a,null,"minimizza")," i moltiplicatori di rilassamento")," di un problema detto ",l("i",null,"primale"),".")),l(a.r,{title:"In termini matriciali"},l("p",null,"Possiamo ",l("b",null,"trasporre")," il tableau e sostituire le variabili ",l(a.k,null,Q(G||(G=Y`x_n`)))," con variabili ",l(a.k,null,Q(O||(O=Y`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(a.r,{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(s.a,null)),", allora il suo duale potrà essere ",l(s.a,null)," oppure ",l(c.a,null),"."),l("li",null,"Se un problema è ",l("b",null,l(c.a,null)),", allora il suo duale sarà certamente ",l(s.a,null),"."))),l(a.r,{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(t.a,null,"Min")),l("th",null,l(o.a,null,"Max")))),l("tbody",null,l("tr",null,l("td",null,"Vincolo ",l(a.i,null,"\\leq")),l("td",null,"Variabile ",l(a.i,null,"\\leq"))),l("tr",null,l("td",null,"Vincolo ",l(a.i,null,"\\geq")),l("td",null,"Variabile ",l(a.i,null,"\\geq"))),l("tr",null,l("td",null,"Vincolo ",l(a.i,null,"=")),l("td",null,"Variabile ",l("b",null,"libera"))),l("tr",null,l("td",null,"Variabile ",l(a.i,null,"\\leq")),l("td",null,"Vincolo ",l(a.i,null,"\\geq"))),l("tr",null,l("td",null,"Variabile ",l(a.i,null,"\\geq")),l("td",null,"Vincolo ",l(a.i,null,"\\leq"))),l("tr",null,l("td",null,"Variabile ",l("b",null,"libera")),l("td",null,"Vincolo ",l(a.i,null,"="))))))),l(a.s,{title:"Un po' di teoria"},l(a.r,{title:"Lemma di Farkas"},l("p",null,"Una disuguaglianza lineare ",l(a.k,null,Q(R||(R=Y`c_0 \leq \mathbf{c}^T \mathbf{x}`)))," è verificata da tutti i punti di un poliedro non-",l(s.a,null)," se e solo se esiste un vettore ",l(a.k,null,Q(D||(D=Y`u \in \mathfrak{R}^m`)))," tale che:"),l(a.q,null,Q(M||(M=Y`\mathbf{c}^T \geq \mathbf{u}^T \mathbf{A}`))),l(a.q,null,Q(E||(E=Y`c_0 \leq \mathbf{u}^T \mathbf{b}`)))),l(a.r,{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(a.k,null,Q(N||(N=Y`\mathbf{c}^T \mathbf{x} = \mathbf{u}^T \mathbf{b}`)))),l("p",null,l(a.u,null,"TODO: Anche qui c'è una lunga dimostrazione..."))),l(a.r,{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(t.a,null,"minore o uguale"),"/",l(o.a,null,"maggiore o uguale")," alla soluzione del corrispettivo primale."),l("p",null,l(a.u,null,"TODO: Dimostrazione cortina, ma sembra complicata."))),l(a.r,{title:"Condizioni di ottimalità"},l("p",null,"Il teorema che ci permette di passare dalla soluzione del duale alla soluzione del primale. ",l(a.u,null,"TODO: credo?")),l("p",null,"Si deriva combinando le seguenti condizioni:"),l("ul",null,l("li",null,"Ammissibilità del primale: ",l(a.k,null,Q(J||(J=Y`\mathbf{A} \mathbf{X} \geq \mathbf{b}, \quad \mathbf{x} \geq 0`)))),l("li",null,"Ammissibilità del duale: ",l(a.k,null,Q(B||(B=Y`\mathbf{u}^T \mathbf{A} \leq \mathbf{c}^T, \quad \mathbf{u} \geq 0`)))),l("li",null,"Teorema della dualità forte: ",l(a.k,null,Q(K||(K=Y`\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(a.q,null,Q(Z||(Z=Y`\left( \mathbf{c}^T - \mathbf{u}^T \mathbf{A} \right) \mathbf{x} = 0`))),l(a.q,null,Q(H||(H=Y`\mathbf{u}^T \left( \mathbf{A} \mathbf{x} - \mathbf{b} \right) = 0`))))),l(a.s,{title:"Simplex duale"},l(a.r,{title:"Cos'è?"},l("p",null,"Un'estensione al Simplex primale che opera sul problema duale.")),l(a.r,{title:"Come funziona?"},l("p",null,"Funziona esattamente come il Simplex primale, ma opera sul duale."))),l(a.s,{title:"Analisi di sensibilità"},l(a.r,{title:"Cos'è?"},l("p",null,"Un procedimento che misura di ",l("b",null,"quanto può variare")," il termine noto di un vincolo ",l(a.k,null,Q(W||(W=Y`b_i`)))," o il coefficiente della funzione obiettivo ",l(a.k,null,Q(X||(X=Y`c_i`)))," prima che la base degeneri."))))}}).call(this,i("hosL").h)},ZHMS:function(l,n,i){"use strict";(function(l){var e=i("gJgL"),a=i.n(e);n.a=function(n){return l("span",{class:a.a.minus},n.children)}}).call(this,i("hosL").h)},dIxp:function(l,n,i){"use strict";(function(l){var e=i("4fSx"),a=i.n(e);n.a=function(n){return l("abbr",{class:a.a.unbounded,title:"I punti del poliedro sono infiniti."},n.children?n.children:"illimitato")}}).call(this,i("hosL").h)},fWbf:function(l,n,i){"use strict";(function(l){var e=i("4fSx"),a=i.n(e);n.a=function(n){return l("abbr",{class:a.a.min,title:"In problemi in cui il primale è di minimizzazione."},n.children?n.children:"min")}}).call(this,i("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,n,i){"use strict";(function(l){var e=i("2w3n"),a=i.n(e);n.a=function(n){return l("div",{class:a.a.example},n.children)}}).call(this,i("hosL").h)},mGtH:function(l,n,i){"use strict";var e=i("uzfk");n.a=i.n(e).a.example},o8Jj:function(l,n,i){"use strict";(function(l){var e=i("jZax"),a=i.n(e);n.a=function(n){return l("span",{class:a.a.plus},n.children)}}).call(this,i("hosL").h)},sEPs:function(l,n,i){"use strict";(function(l){var e=i("hosL"),a=i("mbOI"),t=i("fWbf"),o=i("RPi6"),u=i("ke5e"),r=i("Th1s");let s,c,d,m,b,p,f,h,g,v,_,z,x,k,q,S,y,P,I,L,T,A,V,C,w,F,U,j,G,O,R,D,M,E,N,J,B,K,Z,H,W,X,Y,Q,$,ll,nl,il,el,al,tl=l=>l;const ol=String.raw;n.a=function(){return l(e.Fragment,null,l(a.s,{title:"Ottimizzazione lineare intera"},l(a.r,{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(a.q,null,ol(s||(s=tl`
+ \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(a.r,{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(a.s,{title:"Dal rilassamento alla soluzione"},l(a.r,{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(t.a,null,"minore"),"/",l(o.a,null,"maggiore"),"."),l("p",null,"Trova ",l("b",null,"sicuramente")," la soluzione giusta, ma il costo computazionale è esponenziale ",l(a.i,null,"O(n^k)"),"!")),l(a.r,{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(a.r,{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(a.q,null,ol(c||(c=tl`
+ \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(u.a,null,l("p",null,"Il tableau:",l("table",null,l("thead",null,l("tr",null,l("th",null,l(a.a,null,ol(d||(d=tl`x_1`)))),l("th",null,l(a.a,null,ol(m||(m=tl`x_2`)))),l("th",null,l(a.a,null,ol(b||(b=tl`s_1`)))),l("th",null,l(a.a,null,ol(p||(p=tl`s_2`)))),l("th",null,l("abbr",{title:"Termine noto"},"TN")))),l("tbody",null,l("tr",null,l("td",null,l(a.a,null,ol(f||(f=tl`1`)))),l("td",null,l(a.a,null,ol(h||(h=tl`1`)))),l("td",null,l(a.a,null,ol(g||(g=tl`0`)))),l("td",null,l(a.a,null,ol(v||(v=tl`0`)))),l("td",null,l(a.a,null,ol(_||(_=tl`0`))))),l("tr",null,l("td",null,l(a.a,null,ol(z||(z=tl`1`)))),l("td",null,l(a.a,null,ol(x||(x=tl`0`)))),l("td",null,l(a.a,null,ol(k||(k=tl`1`)))),l("td",null,l(a.a,null,ol(q||(q=tl`0`)))),l("td",null,l(a.a,null,ol(S||(S=tl`3`))))),l("tr",null,l("td",null,l(a.a,null,ol(y||(y=tl`\frac{3}{2}`)))),l("td",null,l(a.a,null,ol(P||(P=tl`\frac{1}{2}`)))),l("td",null,l(a.a,null,ol(I||(I=tl`0`)))),l("td",null,l(a.a,null,ol(L||(L=tl`1`)))),l("td",null,l(a.a,null,ol(T||(T=tl`\frac{6}{5}`)))))))),l("p",null,"Diventa:",l("table",null,l("thead",null,l("tr",null,l("th",null,l(a.a,null,ol(A||(A=tl`x_1`)))),l("th",null,l(a.a,null,ol(V||(V=tl`x_2`)))),l("th",null,l(a.a,null,ol(C||(C=tl`s_1`)))),l("th",null,l(a.a,null,ol(w||(w=tl`s_2`)))),l("th",{style:"background-color: rgba(255, 255, 0, 0.1);"},l(a.a,null,ol(F||(F=tl`s_3`)))),l("th",null,l("abbr",{title:"Termine noto"},"TN")))),l("tbody",null,l("tr",null,l("td",null,l(a.a,null,ol(U||(U=tl`1`)))),l("td",null,l(a.a,null,ol(j||(j=tl`1`)))),l("td",null,l(a.a,null,ol(G||(G=tl`0`)))),l("td",null,l(a.a,null,ol(O||(O=tl`0`)))),l("td",{style:"background-color: rgba(255, 255, 0, 0.1);"},l(a.a,null,ol(R||(R=tl`0`)))),l("td",null,l(a.a,null,ol(D||(D=tl`0`))))),l("tr",null,l("td",null,l(a.a,null,ol(M||(M=tl`1`)))),l("td",null,l(a.a,null,ol(E||(E=tl`0`)))),l("td",null,l(a.a,null,ol(N||(N=tl`1`)))),l("td",null,l(a.a,null,ol(J||(J=tl`0`)))),l("td",{style:"background-color: rgba(255, 255, 0, 0.1);"},l(a.a,null,ol(B||(B=tl`0`)))),l("td",null,l(a.a,null,ol(K||(K=tl`3`))))),l("tr",null,l("td",null,l(a.a,null,ol(Z||(Z=tl`\frac{3}{2}`)))),l("td",null,l(a.a,null,ol(H||(H=tl`\frac{1}{2}`)))),l("td",null,l(a.a,null,ol(W||(W=tl`0`)))),l("td",null,l(a.a,null,ol(X||(X=tl`1`)))),l("td",{style:"background-color: rgba(255, 255, 0, 0.1);"},l(a.a,null,ol(Y||(Y=tl`0`)))),l("td",null,l(a.a,null,ol(Q||(Q=tl`\frac{6}{5}`))))),l("tr",{style:"background-color: rgba(255, 255, 0, 0.1);"},l("td",null,l(a.a,null,ol($||($=tl`-\frac{1}{2}`)))),l("td",null,l(a.a,null,ol(ll||(ll=tl`-\frac{1}{2}`)))),l("td",null,l(a.a,null,ol(nl||(nl=tl`0`)))),l("td",null,l(a.a,null,ol(il||(il=tl`0`)))),l("td",{style:"background-color: rgba(255, 255, 0, 0.2);"},l(a.a,null,ol(el||(el=tl`1`)))),l("td",null,l(a.a,null,ol(al||(al=tl`-\frac{1}{5}`)))))))))),l(a.r,{title:"Divide et impera"},l("p",null,"È possibile usare la tecnica ",l("i",null,"divide et impera")," per rendere più efficiente 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(r.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(a.r,{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"),"."))))}}).call(this,i("hosL").h)},uzfk:function(l){l.exports={red:"red__3r1Jz",orange:"orange__2q18p",yellow:"yellow__3cyw6",lime:"lime__2GJfI",cyan:"cyan__k_5Y_",blue:"blue__ejrIP",magenta:"magenta__1Zdi7",example:"example__22j7L"}}}]);
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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\"};","// extracted by mini-css-extract-plugin\nmodule.exports = {\"menulist\":\"menulist__2Cmnq\"};","import Intro from \"./00_Intro\";\nimport OttimizzazioneLineare from \"./01_OttimizzazioneLineare\";\nimport OttimizzazioneLineareIntera from \"./02_OttimizzazioneLineareIntera\";\nimport TeoriaDeiGrafi from \"./03_TeoriaDeiGrafi\";\n\n\nexport default function () {\n return (\n
\n
Ottimizzazione Lineare Intera
\n \n \n \n \n
\n )\n}\n","import {Fragment} from \"preact\";\nimport {Section, Panel, BLatex, TablePanel} from \"bluelib\";\nimport Link from \"../../components/Link\";\nimport MenuList from \"../../components/MenuList\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n
Prof. Stefano Novellani
\n
\n \n \n
\n Esame digitale in 3 fasi:\n
\n
\n
25 domande di teoria stile vero/falso
\n
2 domande a risposta aperta
\n
2 modelli matematici da rappresentare tramite GLPK/MathProg
\n
\n \n \n \n \n
\n Ti serve una mano anche in GLPK? Che in realtà si chiama GMPL, ma vabbè...\n
\n
\n Ho fatto una pagina apposta:\n
\n \n
GLPK/MathProg/GMPL
\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","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 \"./MenuList.less\";\n\nexport default function(props) {\n return (\n
\n {props.children}\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 {Fragment} from \"preact\";\nimport {Section, Panel, ILatex, BLatex, PLatex, Todo, BaseLink, Image} from \"bluelib\";\nimport Example from \"../../components/Example\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n Terminologia dei grafi TODO: migliorare}>\n \n
\n Insieme di nodi{r`N`} e archi{r`E`} che li\n 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\n 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\n 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
Aggiungi l'arco di costo minimo all'albero.
\n
Finchè mancano ancora archi:\n \n
Trova tutti gli archi che aggiungerebbero un nuovo nodo all'albero.
\n
Seleziona l'arco di costo minore.
\n \n
\n \n \n \n
\n Trova l'ordine topologico di un albero.\n
\n \n
Ripeti finchè ci sono nodi nel grafo:\n \n
Assegna un numero sequenziale a un nodo senza archi entranti.
\n
Elimina il nodo a cui hai assegnato il numero.
\n
Elimina tutti gli archi incidenti sul nodo che hai eliminato.
\n \n
\n \n \n \n
\n Trova i percorsi di costo minimo in un albero.\n
\n \n
Trova l'ordine topologico dell'albero.
\n
Invece che provare ogni singola combinazione di nodi, prova solo i nodi che hanno un numero\n topologico maggiore di quello del nodo attuale.\n
\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\n 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","import {Fragment} from \"preact\";\nimport {Section, Panel, ILatex, BLatex, PLatex, Latex, BaseLink, Todo} from \"bluelib\";\nimport Min from \"../../components/OttimizzazioneLineare/Min\";\nimport Max from \"../../components/OttimizzazioneLineare/Max\";\nimport Example from \"../../components/Example\";\nimport Finite from \"../../components/OttimizzazioneLineare/Finite\";\nimport Empty from \"../../components/OttimizzazioneLineare/Empty\";\nimport Unbounded from \"../../components/OttimizzazioneLineare/Unbounded\";\nimport Plus from \"../../components/Fisica/Plus\";\nimport Minus from \"../../components/Fisica/Minus\";\nimport ExampleBoxColor from \"../../components/ExampleBoxColor\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n Problemi che cercano di minimizzare/massimizzare il valore di una funzione\n 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\n incognite {r`\\mathbf{x}`}.\n
\n
\n In genere, la funzione obiettivo è scritta in forma di combinazione lineare tra le incognite e i coefficienti:\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\n 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\n valida.\n
\n
\n I loro coefficienti sono contenuti nella matrice {r`\\mathbf{A}`}, mentre i loro\n 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\n più lati, ovvero un poliedro.\n \n \n \n
\n La soluzione di un problema, ricavabile dal\n 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 Un modo per rappresentare sistemi in forma standard, anche noto come matrice equivalente\n 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\n valessero 0.\n
\n
\n Procedendo nella risoluzione (descritta in seguito) del tableau, questo valore aumenterà, fino a\n raggiungere il valore ottimo quando la risoluzione sarà completata.\n
\n Un algoritmo per trovare efficientemente il valore ottimo e le coordinate di un vertice\n ottimo in problemi di ottimizzazione lineare.\n
\n \n Ricordi Gauss? Il Simplex è la stessa cosa,\n in cui però si cerca di minimizzare/massimizzare il termine noto della funzione obiettivo.\n \n \n Questa è la soluzione passo per passo del\n problema 3 del file Ex_LP_testo.\n \n
\n Perchè sia possibile effettuare il Simplex è necessario che l'origine sia nel poliedro:\n pertanto, non è possibile che un problema risolto con il Simplex sia .\n
\n \n \n \n
Trasforma il sistema in forma standard.
\n
Trova tante variabili linearmente indipendenti quante siano le righe: esse saranno\n la base iniziale.\n
\n
Finchè ci sono variabili con coefficienti positivi/negativi nella\n funzione obiettivo:\n \n
\n
\n Scegli la prima variabile fuori base con\n coefficiente positivo/negativo nella funzione obiettivo: essa\n è la variabile entrante.\n
\n \n
\n
\n
\n Scegli la variabile in base con il minor rapporto\n positivo:\n
\n {r`\\frac{b_i}{A_ik}`}\n \n
\n
\n
\n Pivot: trasforma tutte le funzioni del sistema in modo che abbiano 0\n nella colonna della variabile entrante, tranne nella riga della variabile uscente,\n in cui avrà 1.\n
\n
\n \n
\n
Il poliedro è : i termini noti dei vincoli sono le coordinate del suo\n vertice ottimo, mentre il termine noto della funzione obiettivo è il valore ottimo.\n
\n \n \n \n
\n Una soluzione con almeno una variabile di valore 0, dovuta a uno o più vincoli\n ridondanti.\n
\n
\n Senza Regola di Bland e in presenza di vincoli ridondanti si rischia di trovarsi a fare\n pivot infiniti.\n
\n \n \n \n \n
\n Un estensione del Simplex per permettere la risoluzione di problemi la cui origine non\n è una soluzione ammissibile.\n
\n
\n Prevede l'introduzione di un problema ausiliario, le cui incognite sono\n dette artificiali.\n
\n
\n Il vettore delle incognite artificiali è solitamente chiamato {r`\\mathbf{y}`}.\n
\n \n \n \n
Crea un nuovo tableau, aggiungendo variabili artificiali in modo da avere una base\n ammissibile.\n
\n
Sostituisci la vecchia funzione obiettivo con una nuova che minimizzi la somma di\n tutte le variabili artificiali.\n
\n
Fase 1: Risolvi il nuovo problema con il Simplex primale.
\n
Se il Simplex termina quando ci sono ancora variabili artificiali nella base, allora\n il poliedro è .\n
\n
Una volta che le variabili artificiali sono fuori base, elimina le loro colonne e la\n nuova funzione obiettivo.
\n
Riporta il tableau in forma base compiendo operazioni per azzerare i\n coefficienti delle variabili di base nella funzione obiettivo.\n
\n
Fase 2: Risolvi il tableau con il Simplex primale.
\n \n \n \n \n \n
\n Una versione semplificata di un problema nella quale si ignora la violazione di uno o più\n 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\n obiettivo.\n
\n
\n Il vettore dei coefficienti di rilassamento solitamente è indicato\n con {r`\\mathbf{u}`}.\n
\n Il sistema che massimizza/minimizza i moltiplicatori di\n rilassamento di un problema detto primale.\n
\n \n \n
\n Possiamo trasporre il tableau e sostituire le variabili {r`x_n`} con\n 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\n essere oppure .\n
\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
Min
\n
Max
\n
\n \n \n
\n
Vincolo \\leq
\n
Variabile \\leq
\n
\n
\n
Vincolo \\geq
\n
Variabile \\geq
\n
\n
\n
Vincolo =
\n
Variabile libera
\n
\n
\n
Variabile \\leq
\n
Vincolo \\geq
\n
\n
\n
Variabile \\geq
\n
Vincolo \\leq
\n
\n
\n
Variabile libera
\n
Vincolo =
\n
\n \n
\n \n \n \n \n
\n Una disuguaglianza lineare {r`c_0 \\leq \\mathbf{c}^T \\mathbf{x}`} è verificata da\n tutti i punti di un poliedro non- se e solo se esiste un\n vettore {r`u \\in \\mathfrak{R}^m`} tale che:\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\n tableau) è sempre minore o uguale/maggiore o uguale alla soluzione del\n 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\n primale. TODO: credo?\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\n vincolo {r`b_i`} o il coefficiente della funzione\n obiettivo {r`c_i`} prima che la base degeneri.\n
\n \n \n \n )\n}\n","import style from \"./Minus.less\";\n\nexport default function (props) {\n return {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\"};","// 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\";\n\nexport default function (props) {\n return (\n
\n {props.children}\n
\n );\n}\n","import style from \"./ExampleBoxColor.less\";\n\nexport default style.example;\n","import style from \"./Plus.less\";\n\nexport default function (props) {\n return {props.children};\n}\n","import {Fragment} from \"preact\";\nimport {Section, Panel, ILatex, BLatex, PLatex} from \"bluelib\";\nimport Min from \"../../components/OttimizzazioneLineare/Min\";\nimport Max from \"../../components/OttimizzazioneLineare/Max\";\nimport Example from \"../../components/Example\";\nimport Empty from \"../../components/OttimizzazioneLineare/Empty\";\n\nconst r = String.raw;\n\n\nexport default function () {\n return (\n \n \n \n
\n Particolari problemi di ottimizzazione lineare in cui le variabili sono vincolate ad essere\n 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\n continua.\n
\n \n \n \n \n
\n Un modo per passare dalla soluzione del rilassamento alla soluzione intera di un problema\n di ILP.\n
\n
\n Consiste nel calcolare la soluzione di ogni singolo punto incluso nel poliedro, e selezionare\n la minore/maggiore.\n
\n
\n Trova sicuramente la soluzione giusta, ma il costo computazionale è\n esponenziale O(n^k)!\n
\n \n \n
\n Un altro modo per passare dalla soluzione del rilassamento alla soluzione intera di un\n problema di ILP.\n
\n
\n Consiste nell'arrotondare tutte le variabili al loro valore intero più vicino, e\n calcolarne il valore ottimo.\n
\n
\n Funziona bene per valori grandi, ma più essi si avvicinano allo 0 più l'errore diventa\n grande.\n
\n \n \n
\n Un altro modo ancora per passare dalla soluzione del rilassamento alla soluzione intera\n di un problema di ILP.\n
\n
\n Consiste nel tagliare il poliedro con nuovi vincoli (piani secanti) che riducono le\n possibili soluzioni continue ma non quelle intere.\n
\n
\n Per selezionare i vincoli, si usano i tagli di Gomory:\n
\n Per ogni valore noto frazionario si viene quindi a creare una nuova variabile in base e\n 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
{r`x_1`}
\n
{r`x_2`}
\n
{r`s_1`}
\n
{r`s_2`}
\n
TN
\n
\n \n \n
\n
{r`1`}
\n
{r`1`}
\n
{r`0`}
\n
{r`0`}
\n
{r`0`}
\n
\n
\n
{r`1`}
\n
{r`0`}
\n
{r`1`}
\n
{r`0`}
\n
{r`3`}
\n
\n
\n
{r`\\frac{3}{2}`}
\n
{r`\\frac{1}{2}`}
\n
{r`0`}
\n
{r`1`}
\n
{r`\\frac{6}{5}`}
\n
\n \n
\n \n
\n Diventa:\n
\n \n
\n
{r`x_1`}
\n
{r`x_2`}
\n
{r`s_1`}
\n
{r`s_2`}
\n
{r`s_3`}\n
\n
TN
\n
\n \n \n
\n
{r`1`}
\n
{r`1`}
\n
{r`0`}
\n
{r`0`}
\n
{r`0`}
\n
{r`0`}
\n
\n
\n
{r`1`}
\n
{r`0`}
\n
{r`1`}
\n
{r`0`}
\n
{r`0`}
\n
{r`3`}
\n
\n
\n
{r`\\frac{3}{2}`}
\n
{r`\\frac{1}{2}`}
\n
{r`0`}
\n
{r`1`}
\n
{r`0`}
\n
{r`\\frac{6}{5}`}
\n
\n
\n
{r`-\\frac{1}{2}`}
\n
{r`-\\frac{1}{2}`}
\n
{r`0`}
\n
{r`0`}
\n
{r`1`}
\n
{r`-\\frac{1}{5}`}
\n
\n \n
\n \n \n \n \n
\n È possibile usare la tecnica divide et impera per rendere più efficiente l'enumerazione\n totale.\n
\n
\n Si divide il problema principale (trovare il valore ottimo di un problema di ILP) in più\n sottoproblemi (trovare il valore ottimo di un problema di ILP con una variabile impostata a un\n 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\n 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\n (ma non ottimale)\n
\n
best-first: permette di raggiungere più velocemente alla soluzione corretta
\n
\n \n \n \n
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\n Matrice con un asse di simmetria lungo la diagonale; gli elementi nel triangolo superiore sono\n però l'opposto di quelli del triangolo inferiore.\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 Funzione che associa un valore reale positivo a ogni funzione.\n
\n \n \n
\n Valore massimo che assume la funzione nel suo dominio.\n
\n {r`\\| f \\|_\\infty = max | f(x) |`}\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 L'errore, ovvero la massima distanza tra due funzioni, si ottiene con:\n
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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 Xn=function(){return n},n}function gn(){var n=We(["Y = a X + b Longleftrightarrow | \rho_{X, Y} | = 1"],["Y = a X + b \\Longleftrightarrow | \\rho_{X, Y} | = 1"]);return gn=function(){return n},n}function hn(){var n=We(["-1 leq \rho_{X, Y} leq 1"],["-1 \\leq \\rho_{X, Y} \\leq 1"]);return hn=function(){return n},n}function qn(){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 qn=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 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 \\cdot \\beta \\cdot Cov(X, Y)"]);return Fn=function(){return n},n}function yn(){var n=We(["Cov(X, X) = Var(X)"]);return yn=function(){return n},n}function Nn(){var n=We(["Cov(X, Y) = Cov(Y, X)"]);return Nn=function(){return n},n}function Vn(){var n=We(["Cov(X, alpha) = 0"],["Cov(X, \\alpha) = 0"]);return Vn=function(){return n},n}function Sn(){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 Sn=function(){return n},n}function Yn(){var n=We(["E(X + Y) = E(X) + E(Y)"]);return Yn=function(){return n},n}function wn(){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 wn=function(){return n},n}function Cn(){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 Cn=function(){return n},n}function In(){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 In=function(){return n},n}function Dn(){var n=We(["p_{X, Y} (x, y) = P(X = x, Y = y)"]);return Dn=function(){return n},n}function Mn(){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 Mn=function(){return n},n}function Un(){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 Un=function(){return n},n}function Bn(){var n=We(["X, Y"]);return Bn=function(){return n},n}function An(){var n=We(["\boldsymbol{X}"],["\\boldsymbol{X}"]);return An=function(){return n},n}function Tn(){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 Tn=function(){return n},n}function Gn(){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 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 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 sl(){var n=We(["Var(X) = \frac{(b - a)^2}{12}"],["Var(X) = \\frac{(b - a)^2}{12}"]);return sl=function(){return n},n}function pl(){var n=We(["E(X) = \frac{a + b}{2}"],["E(X) = \\frac{a + b}{2}"]);return pl=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 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}"]);return ml=function(){return n},n}function fl(){var n=We(["\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 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 "]);return fl=function(){return n},n}function vl(){var n=We(["P(X in (c, d)) = \frac{d - c}{b - a}"],["P(X \\in (c, d)) = \\frac{d - c}{b - a}"]);return vl=function(){return n},n}function bl(){var n=We(["Uni(a, b)"]);return bl=function(){return n},n}function _l(){var n=We(["[a, b]"]);return _l=function(){return n},n}function kl(){var n=We(["Var(X) = \frac{alpha}{lambda^2}"],["Var(X) = \\frac{\\alpha}{\\lambda^2}"]);return kl=function(){return n},n}function Xl(){var n=We(["E(X) = \frac{alpha}{lambda}"],["E(X) = \\frac{\\alpha}{\\lambda}"]);return Xl=function(){return n},n}function gl(){var n=We(["m_X (t) : ( t < lambda ) in mathbb{R} = left( \frac{lambda}{lambda - t} \right) ^alpha"],["m_X (t) : ( t < \\lambda ) \\in \\mathbb{R} = \\left( \\frac{\\lambda}{\\lambda - t} \\right) ^\\alpha"]);return gl=function(){return n},n}function hl(){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 hl=function(){return n},n}function ql(){var n=We(["Gamma(n, lambda)"],["\\Gamma(n, \\lambda)"]);return ql=function(){return n},n}function zl(){var n=We(["lambda"],["\\lambda"]);return zl=function(){return n},n}function El(){var n=We(["P([X > s + t | X > s]) = P([X > t])"]);return El=function(){return n},n}function Pl(){var n=We(["Var(X) = \frac{1}{lambda^2}"],["Var(X) = \\frac{1}{\\lambda^2}"]);return Pl=function(){return n},n}function xl(){var n=We(["E(X) = \frac{1}{lambda}"],["E(X) = \\frac{1}{\\lambda}"]);return xl=function(){return n},n}function Ll(){var n=We(["m_X (t) : { t | t < lambda } in mathbb{R} = \frac{lambda}{lambda - t}"],["m_X (t) : \\{ t | t < \\lambda \\} \\in \\mathbb{R} = \\frac{\\lambda}{\\lambda - t}"]);return Ll=function(){return n},n}function Fl(){var n=We(["\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 F_X (t) = \\begin{cases}\n 0 \\qquad \\qquad t < 0\\\\\n 1 - e^{-\\lambda \\cdot t} \\quad t \\geq 0\n \\end{cases}"]);return Fl=function(){return n},n}function yl(){var n=We(["\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 f_X (x) = \\begin{cases}\n 0 \\qquad \\qquad x < 0\\\\\n \\lambda \\cdot e^{-\\lambda \\cdot x} \\quad x > 0\n \\end{cases}"]);return yl=function(){return n},n}function Nl(){var n=We(["Esp(lambda)"],["Esp(\\lambda)"]);return Nl=function(){return n},n}function Vl(){var n=We(["lambda"],["\\lambda"]);return Vl=function(){return n},n}function Sl(){var n=We(["Poi(t cdot lambda)"],["Poi(t \\cdot \\lambda)"]);return Sl=function(){return n},n}function Yl(){var n=We(["mu = t cdot lambda"],["\\mu = t \\cdot \\lambda"]);return Yl=function(){return n},n}function wl(){var n=We(["lambda"],["\\lambda"]);return wl=function(){return n},n}function Cl(){var n=We(["lambda"],["\\lambda"]);return Cl=function(){return n},n}function Il(){var n=We(["E(X^2) = mu^2 + mu"],["E(X^2) = \\mu^2 + \\mu"]);return Il=function(){return n},n}function Dl(){var n=We(["Var(X) = mu"],["Var(X) = \\mu"]);return Dl=function(){return n},n}function Ml(){var n=We(["E(X) = mu"],["E(X) = \\mu"]);return Ml=function(){return n},n}function Ul(){var n=We(["m_X (t) = e^{mu cdot (e^t - 1)}"],["m_X (t) = e^{\\mu \\cdot (e^t - 1)}"]);return Ul=function(){return n},n}function Bl(){var n=We(["f_X (k) : mathbb{N} = \frac{e^{-mu} cdot mu^k}{k!}"],["f_X (k) : \\mathbb{N} = \\frac{e^{-\\mu} \\cdot \\mu^k}{k!}"]);return Bl=function(){return n},n}function Al(){var n=We(["Poi(mu)"],["Poi(\\mu)"]);return Al=function(){return n},n}function Tl(){var n=We(["E(X) = n cdot p \to mu \neq 0"],["E(X) = n \\cdot p \\to \\mu \\neq 0"]);return Tl=function(){return n},n}function Gl(){var n=We(["p \to 0"],["p \\to 0"]);return Gl=function(){return n},n}function Ol(){var n=We(["n \to +infty"],["n \\to +\\infty"]);return Ol=function(){return n},n}function Rl(){var n=We(["X sim Bin(n, p)"],["X \\sim Bin(n, p)"]);return Rl=function(){return n},n}function Kl(){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 Kl=function(){return n},n}function jl(){var n=We(["E(X) = n cdot \frac{K}{N}"],["E(X) = n \\cdot \\frac{K}{N}"]);return jl=function(){return n},n}function Zl(){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 Zl=function(){return n},n}function Ql(){var n=We(["Var(X) = \frac{n cdot q}{p^2}"],["Var(X) = \\frac{n \\cdot q}{p^2}"]);return Ql=function(){return n},n}function Wl(){var n=We(["E(X) = \frac{n cdot q}{p}"],["E(X) = \\frac{n \\cdot q}{p}"]);return Wl=function(){return n},n}function Jl(){var n=We(["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"],["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"]);return Jl=function(){return n},n}function Hl(){var n=We(["f_X (k) : mathbb{N} = \binom{k + n - 1}{n - 1} cdot p^n cdot q^k "],["f_X (k) : \\mathbb{N} = \\binom{k + n - 1}{n - 1} \\cdot p^n \\cdot q^k "]);return Hl=function(){return n},n}function $l(){var n=We(["overline{Bin}(n, p)"],["\\overline{Bin}(n, p)"]);return $l=function(){return n},n}function ni(){var n=We(["P([X = i + j | X > i ]) = P([X = j])"]);return ni=function(){return n},n}function li(){var n=We(["Var(X) = \frac{q}{p^2}"],["Var(X) = \\frac{q}{p^2}"]);return li=function(){return n},n}function ii(){var n=We(["E(X) = \frac{q}{p}"],["E(X) = \\frac{q}{p}"]);return ii=function(){return n},n}function ei(){var n=We(["m_X (t) : left{ t < ln left( \frac{1}{q} \right) \right} = \frac{p}{1 - q cdot e^t}"],["m_X (t) : \\left\\{ t < ln \\left( \\frac{1}{q} \\right) \\right\\} = \\frac{p}{1 - q \\cdot 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",n("b",null,"vuoto"),"."),n("p",null,n(Je.k,null,la(Ue())))),n(Je.r,{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.k,null,la(Me()))))),n(Je.s,null,n(Je.r,{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.k,null,la(De()))),n("p",null,"Qualsiasi sottoinsieme appartenente a ",n(Je.k,null,la(Ie()))," è considerato un evento.")),n(Je.r,{title:n("span",null,n(Je.k,null,la(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.k,null,la(we())),"-algebra"),":"),n("ol",null,n("li",null,"Lo spazio campionario è un evento: ",n(Je.k,null,la(Ye()))),n("li",null,"Se un sottoinsieme è un evento, allora anche il suo complementare lo è: ",n(Je.k,null,la(Se()))),n("li",null,"Se due sottoinsiemi sono eventi, allora lo sono anche la loro unione e intersezione: ",n(Je.k,null,la(Ve())))),n("p",null,"Un esempio: ",n(Je.k,null,la(Ne()))))),n(Je.s,null,n(Je.r,{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.k,null,la(ye()))," è composta dagli eventi ",n(Je.k,null,la(Fe())),", ",n(Je.k,null,la(Le())),", ",n(Je.k,null,la(xe())),", fino a ",n(Je.k,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.s,{title:"Assiomi della probabilità"},n(Je.r,{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.k,null,la(Ee())))),n(Je.r,{title:"Secondo assioma della probabilità"},n("p",null,"La probabilità dello spazio campionario è sempre 1."),n("p",null,n(Je.k,null,la(ze())))),n(Je.r,{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.k,null,la(qe()))))),n(Je.s,{title:"Conseguenze degli assiomi"},n(Je.r,{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.k,null,la(he())))),n(Je.r,{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.k,null,la(ge())))),n(Je.r,{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.k,null,la(Xe()))),n(He.a,null,"Sommando le probabilità dei due eventi, l'intersezione viene contata due volte, e va quindi rimossa!"))),n(Je.s,{title:"Spazi equiprobabili"},n(Je.r,{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.k,null,la(ke())))),n(Je.r,{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.s,{title:"Calcolo combinatorio"},n(Je.r,{title:"Disposizioni"},n("p",null,"Estraggo un numero, da un sacchetto con ",n(Je.k,null,"n")," numeri, mi segno che numero ho estratto e lo ",n("b",null,"tengo fuori dal sacchetto"),". Ripeto per ",n(Je.k,null,"k")," volte."),n("p",null,n("b",null,"Tengo conto")," dell'ordine in cui ho estratto i numeri."),n("p",null,n(Je.k,null,la(_e())))),n(Je.r,{title:"Disposizioni con ripetizione"},n("p",null,"Estraggo un numero, da un sacchetto con ",n(Je.k,null,"n")," numeri, mi segno che numero ho estratto e lo ",n("b",null,"rimetto nel sacchetto"),". Ripeto per ",n(Je.k,null,"k")," volte."),n("p",null,n("b",null,"Tengo conto")," dell'ordine in cui ho estratto i numeri."),n("p",null,n(Je.k,null,la(be())))),n(Je.r,{title:"Combinazioni"},n("p",null,"Estraggo un numero, da un sacchetto con ",n(Je.k,null,"n")," numeri, mi segno che numero ho estratto e lo ",n("b",null,"tengo fuori dal sacchetto"),". Ripeto per ",n(Je.k,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.k,null,la(ve())))),n(Je.r,{title:"Combinazioni con ripetizione"},n("p",null,"Estraggo un numero, da un sacchetto con ",n(Je.k,null,"n")," numeri, mi segno che numero ho estratto e lo ",n("b",null,"rimetto nel sacchetto"),". Ripeto per ",n(Je.k,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.k,null,la(fe())))),n(Je.r,{title:"Permutazioni"},n("p",null,"Estraggo ",n(Je.k,null,"n")," numeri e guardo in quanti ordini diversi li posso mettere."),n("p",null,n(Je.k,null,la(me()))))),n(Je.s,{title:"Probabilità condizionata"},n(Je.r,{title:"Eventi condizionati"},n("blockquote",null,'"E dato F"'),n("p",null,"La probabilità che si verifichi ",n(Je.k,null,"E")," sapendo che ",n("b",null,"si è già verificato"),n(Je.k,null,"F"),"."),n("p",null,n(Je.k,null,la(de()))),n(He.a,null,"Ricorda vagamente le pipe di ",n("code",null,"bash"),", però al contrario...")),n(Je.r,{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.k,null,la(pe()))))),n(Je.s,null,n(Je.r,{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.k,null,la(se()))))),n(Je.s,{title:"Le alternative"},n(Je.r,{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.k,null,la(ce())))),n(Je.r,{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.k,null,la(ue())))),n(Je.r,{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.k,null,la(oe()))),n(He.a,null,"In pratica, invertiamo gli eventi."))),n(Je.s,{title:"Eventi indipendenti"},n(Je.r,{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.k,null,la(re())))),n(Je.r,{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.k,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.r,{title:"Famiglia di eventi indipendenti"},n("p",null,"Un insieme di ",n(Je.k,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.s,{title:"Variabili aleatorie"},n(Je.r,{title:"Variabile aleatoria"},n("p",null,"Una funzione che fa corrispondere un numero reale a ogni possibile esito dello spazio campionario. ",n(Je.k,null,la(ae())),".")),n(Je.r,{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.k,null,la(ee())),", che contengono tutti gli esiti a cui la variabile aleatoria associa un valore minore o uguale a ",n(Je.k,null,"t"),"."),n("p",null,"Per definizione, tutte le variabili aleatorie devono rispettare questa condizione:"),n("p",null,n(Je.k,null,la(ie()))),n(He.a,null,"All'aumentare di t, l'insieme conterrà sempre più elementi.")),n(Je.r,{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.k,null,"x_0")," appartiene al supporto di ",n(Je.k,null,"X"),", si usa la notazione ",n(Je.k,null,"X \\mapsto x_0"),"."))),n(Je.s,{title:"Densità"},n(Je.r,{title:"Funzione probabilità"},n("p",null,"La ",n("i",null,"funzione probabilità")," ",n(Je.k,null,la(le()))," di una variabile aleatoria ",n("b",null,"discreta")," ",n(Je.k,null,"X")," è la funzione che associa ad ogni esito la sua probabilità:"),n("p",null,n(Je.k,null,la(ne())))),n(Je.r,{title:"Funzione densità"},n("p",null,"La ",n("i",null,"funzione densità")," ",n(Je.k,null,la($i()))," di una variabile aleatoria ",n("b",null,"continua")," ",n(Je.k,null,"X")," è l'equivalente continuo della funzione probabilità:"),n("p",null,n(Je.k,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.s,{title:"Funzione di ripartizione"},n(Je.r,{title:"Definizione"},n("p",null,"Ogni variabile aleatoria ha una ",n("i",null,"funzione di ripartizione"),n(Je.k,null,la(Ji()))," associata, che rappresenta la probabilità che la variabile aleatoria assuma un valore minore o uguale a ",n(Je.k,null,"t"),":"),n("p",null,"Si può dire che essa rappresenti la probabilità dell'evento ",n(Je.k,null,la(Wi())),":"),n("p",null,n(Je.k,null,la(Qi())))),n(Je.r,{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.k,null,"-\\infty")," e ",n("b",null,"1")," a ",n(Je.k,null,"+\\infty"),"."),n("br",null),n("li",null,"È ",n("b",null,"continua da destra"),": ",n(Je.k,null,la(Zi()))))),n(Je.r,{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.k,null,la(ji()))))),n(Je.s,{title:"Trasformazioni di variabili aleatorie"},n(Je.r,{title:"Nel discreto"},n("p",null,"Nel discreto basta abbinare un nuovo valore a ogni valore della variabile originale.")),n(Je.r,{title:"Nel continuo (invertibile)"},n("p",null,"Nel continuo applichiamo la formula dell'integrazione per sostituzione:"),n("p",null,n(Je.k,null,la(Ki())))),n(Je.r,{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.s,{title:"Informazioni delle variabili aleatorie"},n(Je.r,{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.k,null,la(Ri()))),n("p",null,"Nel discreto, si può calcolare con:"),n("p",null,n(Je.k,null,la(Oi()))),n("p",null,"Nel continuo, si può calcolare con:"),n("p",null,n(Je.k,null,la(Gi()))))),n(Je.s,null,n(Je.r,{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.r,{title:"Quantili"},n("p",null,"Il ",n("i",null,"quantile")," ",n(Je.k,null,la(Ti()))," di ordine ",n(Je.k,null,la(Ai()))," della variabile aleatoria ",n(Je.k,null,"X")," è il più piccolo numero tale che:"),n("p",null,n(Je.k,null,la(Bi()))),n("p",null),n("p",null,"Il quantile di ordine 0.5 ",n(Je.k,null,la(Ui()))," è detto ",n("i",null,"mediana"),"."),n("p",null,"I quantili di ordine 0.25 ",n(Je.k,null,la(Mi()))," e 0.75 ",n(Je.k,null,la(Di()))," sono detti ",n("i",null,"quartili"),"."),n("p",null,"I quantili di ordine ",n(Je.k,null,la(Ii()))," sono detti ",n("i",null,n(Je.k,null,"n"),"-esima percentile"),".")),n(Je.r,{title:"Varianza"},n("p",null,"È un valore che indica quanto la variabile aleatoria si discosta generalmente dalla media:"),n("p",null,n(Je.k,null,la(Ci()))))),n(Je.s,{title:"Disuguaglianze notevoli"},n(Je.r,{title:"Disuguaglianza di Markov"},n("p",null,"Data una variabile aleatoria non-negativa:"),n("p",null,n(Je.k,null,la(wi()))),n("p",null,"Divide in due parti (",n(Je.k,null,la(Yi()))," e ",n(Je.k,null,la(Si())),") la funzione X, la cui media risulterà uguale a:"),n("p",null,n(Je.k,null,la(Vi())))),n(Je.r,{title:"Disuguaglianza di Čebyšëv"},n("blockquote",null,'"disuguaglianza di cebicev"'),n("p",null,"Se la variabile aleatoria ",n(Je.k,null,"X")," ha media e varianza, allora la probabilità che essa abbia un valore a più di ",n(Je.k,null,la(Ni()))," di distanza dal valore medio è minore o uguale a ",n(Je.k,null,la(yi())),"."),n("p",null,n(Je.k,null,la(Fi()))),n("p",null,"E anche:"),n("p",null,n(Je.k,null,la(Li()))),n(He.a,null,"Serve per semplificare i calcoli quando la funzione di ripartizione è difficile da calcolare!"))),n(Je.s,{title:"Un momento...!"},n(Je.r,{title:"Momento"},n("p",null,"Il ",n("i",null,"momento")," ",n(Je.k,null,"k"),"-esimo di una variabile aleatoria è:"),n("p",null,n(Je.k,null,la(xi()))),n(He.a,null,"La media di una variabile aleatoria è anche il suo primo momento.")),n(Je.r,{title:"Funzione generatrice dei momenti"},n("p",null,"La ",n("i",null,"funzione generatrice dei momenti")," è:"),n("p",null,n(Je.k,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.r,{title:"Funzione caratteristica"},n("p",null,"La ",n("i",null,"funzione caratteristica")," è:"),n("p",null,n(Je.k,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.s,{title:"Prove e schemi"},n(Je.r,{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.k,null,la(zi())))),n(Je.r,{title:"Prova di Bernoulli"},n("p",null,"Una prova con solo due possibili esiti: ",n($e.a,null,"successo")," e ",n(na.a,null,"insuccesso"),".")),n(Je.r,{title:"Schema di Bernoulli"},n("p",null,"Una sequenza di prove di Bernoulli per le quali le probabilità di successo e fallimento rimangono invariate."))),n(Je.s,{title:"Bernoulliana"},n(Je.r,{title:"Distribuzione bernoulliana"},n("p",null,"Una variabile aleatoria che rappresenta una prova di Bernoulli:"),n("ul",null,n("li",null,"vale ",n($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.k,null,la(qi())))),n(Je.r,{title:"Densità della bernoulliana"},n("p",null,"La distribuzione bernoulliana ha come densità:"),n("p",null,n(Je.k,null,la(hi()))))),n(Je.s,{title:"Binomiale"},n(Je.r,{title:"Distribuzione binomiale"},n("p",null,"Una variabile aleatoria che conta il numero di successi di ",n(Je.k,null,"n")," prove di uno schema di Bernoulli."),n("p",null,"Il suo simbolo è ",n(Je.k,null,la(gi())),".")),n(Je.r,{title:"Densità della binomiale"},n("p",null,"La binomiale ha come densità:"),n("p",null,n(Je.k,null,la(Xi())))),n(Je.r,{title:"Momenti della binomiale"},n("p",null,"La ",n("b",null,"funzione generatrice dei momenti")," della binomiale è:"),n("p",null,n(Je.k,null,la(ki()))),n("p",null,"La ",n("b",null,"media")," di una binomiale è:"),n("p",null,n(Je.k,null,la(_i()))),n("p",null,"La ",n("b",null,"varianza")," di una binomiale è:"),n("p",null,n(Je.k,null,la(bi()))))),n(Je.s,{title:"Geometrica"},n(Je.r,{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.k,null,"Geo(p)"),".")),n(Je.r,{title:"Densità della geometrica"},n("p",null,"La geometrica ha come densità:"),n("p",null,n(Je.k,null,la(vi())))),n(Je.r,{title:"Momenti della geometrica"},n("p",null,"La ",n("b",null,"funzione generatrice dei momenti")," della geometrica è:"),n("p",null,n(Je.k,null,la(fi()))),n("p",null,"La ",n("b",null,"media")," della geometrica è:"),n("p",null,n(Je.k,null,la(mi()))),n("p",null,"La ",n("b",null,"varianza")," della geometrica è:"),n("p",null,n(Je.k,null,la(di())))),n(Je.r,{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.k,null,la(pi()))),n(He.a,null,"Ovvero, riscalando opportunamente l'asse Y posso prendere come 0 qualsiasi punto dell'asse X."))),n(Je.s,{title:"Binomiale negativa"},n(Je.r,{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.k,null,"n"),"-esimo successo."),n("p",null,"Il suo simbolo è ",n(Je.k,null,la(si())),".")),n(Je.r,{title:"Densità della binomiale negativa"},n("p",null,"La binomiale negativa ha come densità:"),n("p",null,n(Je.k,null,la(ci())))),n(Je.r,{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.k,null,la(ui()))),n("p",null,"La ",n("b",null,"media")," della binomiale negativa è:"),n("p",null,n(Je.k,null,la(oi()))),n("p",null,"La ",n("b",null,"varianza")," della binomiale negativa è:"),n("p",null,n(Je.k,null,la(ri())))))),n(Je.s,{title:"Geometrica traslata"},n(Je.r,{title:"Distribuzione geometrica traslata"},n("p",null,"Una variabile aleatoria che conta il numero ",n(Je.k,null,"k")," di insuccessi consecutivi in uno schema di Bernoulli:"),n("p",null,"Il suo simbolo rimane ",n(Je.k,null,la(ti())),".")),n(Je.r,{title:"Densità della geometrica tralsata"},n("p",null,"La geometrica traslata ha come densità:"),n("p",null,n(Je.k,null,la(ai())))),n(Je.r,{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.k,null,la(ei()))),n("p",null,"La ",n("b",null,"media")," della geometrica traslata è:"),n("p",null,n(Je.k,null,la(ii()))),n("p",null,"La ",n("b",null,"varianza")," della geometrica è:"),n("p",null,n(Je.k,null,la(li())))),n(Je.r,{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.k,null,la(ni()))),n(He.a,null,"Ovvero, riscalando opportunamente l'asse Y posso prendere come 0 qualsiasi punto dell'asse X."))),n(Je.s,{title:"Binomiale negativa traslata"},n(Je.r,{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.k,null,"n"),"-esimo successo."),n("p",null,"Il suo simbolo rimane ",n(Je.k,null,la($l())),".")),n(Je.r,{title:"Densità della binomiale negativa traslata"},n("p",null,"La binomiale negativa traslata ha come densità:"),n("p",null,n(Je.k,null,la(Hl())))),n(Je.r,{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.k,null,la(Jl()))),n("p",null,"La ",n("b",null,"media")," della binomiale negativa traslata è:"),n("p",null,n(Je.k,null,la(Wl()))),n("p",null,"La ",n("b",null,"varianza")," della binomiale negativa traslata è:"),n("p",null,n(Je.k,null,la(Ql())))))),n(Je.s,{title:"Ipergeometrica"},n(Je.r,{title:"Distribuzione ipergeometrica"},n("p",null,"Una variabile aleatoria che, sapendo il numero di successi ",n(Je.k,null,"K")," e di insuccessi ",n(Je.k,null,"N-K"),", conta quanti successi si otterrebbero se se ne estraessero ",n(Je.k,null,"n")," in blocco."),n("p",null,"Il suo simbolo è ",n(Je.k,null,"Ipe(N, K, n)"),".")),n(Je.r,{title:"Densità della ipergeometrica"},n("p",null,"La ipergeometrica ha come densità:"),n("p",null,n(Je.k,null,la(Zl())))),n(Je.r,{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.k,null,la(jl()))),n("p",null,"La ",n("b",null,"varianza")," della ipergeometrica è:"),n("p",null,n(Je.k,null,la(Kl())))))),n(Je.s,{title:"Poissoniana"},n(Je.r,{title:"Distribuzione poissoniana"},n("p",null,"Una variabile aleatoria che soddisfa tutte le seguenti caratteristiche:"),n("ul",null,n("li",null,"Binomiale: ",n(Je.k,null,la(Rl()))),n("li",null,"Il numero di prove tende a infinito: ",n(Je.k,null,la(Ol()))),n("li",null,"La probabilità di successo tende a 0: ",n(Je.k,null,la(Gl()))),n("li",null,"La media è finita: ",n(Je.k,null,la(Tl())))),n("p",null,"Il suo simbolo è ",n(Je.k,null,la(Al())))),n(Je.r,{title:"Densità della poissoniana"},n("p",null,"La poissoniana ha come densità:"),n("p",null,n(Je.k,null,la(Bl())))),n(Je.r,{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.k,null,la(Ul()))),n("p",null,"La ",n("b",null,"media")," della poissoniana è:"),n("p",null,n(Je.k,null,la(Ml()))),n("p",null,"La ",n("b",null,"varianza")," della poissoniana è:"),n("p",null,n(Je.k,null,la(Dl()))),n("p",null,"Gli altri momenti della poissoniana sono:"),n("ol",{start:2},n("li",null,n(Je.k,null,la(Il()))))))),n(Je.s,{title:"Un altro schema"},n(Je.r,{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.k,null,la(Cl()))," costante."),n("li",null,"avvengono indipendentemente gli uni dagli altri."))),n(Je.r,{title:"Processo di Poisson"},n("p",null,"Una variabile aleatoria ",n(Je.k,null,"N_t")," che conta il numero di arrivi di uno schema di Poisson di intensità ",n(Je.k,null,la(wl()))," in un intervallo di tempo di durata ",n(Je.k,null,"t"),"."),n("p",null,"E' una distribuzione poissoniana con ",n(Je.k,null,la(Yl())),": ",n(Je.k,null,la(Sl()))),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.s,{title:"Esponenziale"},n(Je.r,{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.k,null,la(Vl())),"."),n("p",null,"Il suo simbolo è ",n(Je.k,null,la(Nl())),".")),n(Je.r,{title:"Densità dell'esponenziale"},n("p",null,"L'esponenziale ha come ",n("b",null,"densità"),":"),n("p",null,n(Je.k,null,la(yl()))),n("p",null,"L'esponenziale ha come ",n("b",null,"funzione di ripartizione"),":"),n("p",null,n(Je.k,null,la(Fl())))),n(Je.r,{title:"Momenti dell'esponenziale"},n("p",null,"La ",n("b",null,"funzione generatrice dei momenti")," dell'esponenziale è:"),n("p",null,n(Je.k,null,la(Ll()))),n("p",null,"La ",n("b",null,"media")," dell'esponenziale è:"),n("p",null,n(Je.k,null,la(xl()))),n("p",null,"La ",n("b",null,"varianza")," dell'esponenziale è:"),n("p",null,n(Je.k,null,la(Pl())))),n(Je.r,{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.k,null,la(El()))),n(He.a,null,"Ovvero, riscalando opportunamente l'asse Y posso prendere come 0 qualsiasi punto dell'asse X."))),n(Je.s,{title:"Legge gamma"},n(Je.r,{title:"Distribuzione gamma"},n("p",null,"Una variabile aleatoria che conta il tempo diwidehattesa prima dell'",n(Je.k,null,"n"),"-esimo arrivo di un processo di Poisson di intensità ",n(Je.k,null,la(zl())),"."),n("p",null,"Il suo simbolo è ",n(Je.k,null,la(ql())),".")),n(Je.r,{title:"Densità della legge gamma"},n("p",null,"La legge gamma ha come densità:"),n("p",null,n(Je.k,null,la(hl())))),n(Je.r,{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.k,null,la(gl()))),n("p",null,"La ",n("b",null,"media")," della legge gamma è:"),n("p",null,n(Je.k,null,la(Xl()))),n("p",null,"La ",n("b",null,"varianza")," della legge gamma è:"),n("p",null,n(Je.k,null,la(kl())))))),n(Je.s,{title:"Uniforme"},n(Je.r,{title:"Distribuzione uniforme"},n("p",null,"Una variabile aleatoria che può assumere qualsiasi valore in un intervallo ",n(Je.k,null,la(_l()))," in modo equiprobabile."),n("p",null,"Il suo simbolo è ",n(Je.k,null,la(bl()))),n("p",null,"Su di essa vale la seguente proprietà:"),n("p",null,n(Je.k,null,la(vl())))),n(Je.r,{title:"Densità della distribuzione uniforme"},n("p",null,"La distribuzione uniforme ha come ",n("b",null,"densità"),":"),n("p",null,n(Je.k,null,la(fl()))),n("p",null,"La distribuzione uniforme ha come ",n("b",null,"funzione di ripartizione"),":"),n("p",null,n(Je.k,null,la(ml())))),n(Je.r,{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.k,null,la(dl()))),n("p",null,"La ",n("b",null,"media")," della distribuzione uniforme è:"),n("p",null,n(Je.k,null,la(pl()))),n("p",null,"La ",n("b",null,"varianza")," della distribuzione uniforme è:"),n("p",null,n(Je.k,null,la(sl())))))),n(Je.s,{title:"Normale o Gaussiana"},n(Je.r,{title:"Distribuzione normale"},n("p",null,"Una variabile aleatoria con una specifica distribuzione."),n("p",null,"Il suo simbolo è ",n(Je.k,null,la(cl())),"."),n(He.a,null,n(Je.k,null,"\\mu")," e ",n(Je.k,null,"\\sigma^2")," sono rispettivamente la media e la varianza della distribuzione!")),n(Je.r,{title:"Densità della distribuzione normale"},n("p",null,"La distribuzione normale ha come densità:"),n("p",null,n(Je.k,null,la(ul())))),n(Je.r,{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.k,null,la(ol()))),n("p",null,"La ",n("b",null,"media")," della distribuzione normale è:"),n("p",null,n(Je.k,null,la(rl()))),n("p",null,"La ",n("b",null,"varianza")," della distribuzione normale è:"),n("p",null,n(Je.k,null,la(tl())))))),n(Je.s,null,n(Je.r,{title:"Trasformazione della normale"},n("p",null,"Qualsiasi normale può essere trasformata in qualsiasi altra normale:"),n("p",null,n(Je.k,null,la(al())))),n(Je.r,{title:"Normale standard"},n("p",null,"La distribuzione normale standard ",n(Je.k,null,"Z")," è:"),n("p",null,n(Je.k,null,"Z \\sim Nor(0, 1)")),n("p",null,"La sua funzione di ripartizione è detta ",n(Je.k,null,la(el()))," e vale:"),n("p",null,n(Je.k,null,la(il())))),n(Je.r,{title:"Quantili normali"},n("p",null,"Da un quantile ",n(Je.k,null,la(ll()))," della normale standard è possibile risalire allo stesso quantile di qualsiasi altra normale:"),n("p",null,n(Je.k,null,la(nl()))))),n(Je.s,null,n(Je.r,{title:"Gamma e normale"},n("p",null,"La distribuzione normale ha una particolare relazione con la distribuzione Gamma:"),n("p",null,n(Je.k,null,la($n())))),n(Je.r,{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.k,null,la(Hn()))),n("p",null,"Più chi-quadro possono essere sommate per aumentare i loro gradi di libertà:"),n("p",null,n(Je.k,null,la(Jn())))),n(Je.r,{title:"T di Student"},n("p",null,"Un'altra funzione particolare è la funzione T di Student:"),n("p",null,n(Je.k,null,la(Wn()))))),n(Je.s,{title:"Approssimazioni notevoli"},n(Je.r,{title:"Ipergeometrica e binomiale"},n("p",null,"La binomiale è come una ipergeometrica ma con ripetizioni, quindi per valori molto grandi di ",n(Je.k,null,"N")," rispetto a ",n(Je.k,null,"n"),", si può dire che:"),n("p",null,n(Je.k,null,la(Qn())))),n(Je.r,{title:"Binomiale e poissoniana"},n("p",null,"La binomiale non è altro che una poissoniana a tempo discreto, quindi, se ",n(Je.k,null,"n")," è grande e ",n(Je.k,null,"n \\cdot p")," è nell'ordine di grandezza delle unità, allora:"),n("p",null,n(Je.k,null,la(Zn())))),n(Je.r,{title:"Binomiale e normale"},n("p",null,"Per il Teorema di De Moivre-Laplace, se una binomiale ha una ",n(Je.k,null,"n")," grande e ",n(Je.k,null,"p")," non vicina a 0 o 1, si può approssimare con:"),n("p",null,n(Je.k,null,la(jn())))),n(Je.r,{title:"Correzione di Yates"},n("p",null,"Passando da una variabile discreta ",n(Je.k,null,"X")," a una continua ",n(Je.k,null,"Y"),", per ogni valore discreto ",n(Je.k,null,"k"),' la probabilità viene "spalmata" su tutto l\'intervallo ',n(Je.k,null,la(Kn())),":"),n("ul",null,n("li",null,n(Je.k,null,la(Rn()))),n("li",null,n(Je.k,null,la(On()))),n("li",null,n(Je.k,null,la(Gn()))),n("li",null,n(Je.k,null,la(Tn())))))),n(Je.s,{title:"Vettori aleatori"},n(Je.r,{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.k,null,la(An()))," oppure ",n(Je.k,null,la(Bn())),".")),n(Je.r,{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.k,null,la(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.k,null,la(Mn())))),n(Je.r,{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.k,null,la(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.k,null,la(In()))))),n(Je.s,{title:"Più variabili aleatorie"},n(Je.r,{title:"Indipendenza delle variabili aleatorie"},n("p",null,"Più variabili aleatorie sono indipendenti se, per qualsiasi scelta di intervalli ",n(Je.k,null,"A_i"),":"),n("p",null,n(Je.k,null,la(Cn())))),n(Je.r,{title:"Media dei vettori aleatori"},n("p",null,"E' possibile calcolare la media di qualsiasi funzione ",n(Je.k,null,"g(X, Y)")," avente elementi del vettore come variabili:"),n("p",null,n(Je.k,null,la(wn()))),n(He.a,null,"Solitamente si calcola la media di ",n(Je.k,null,"x \\cdot y"),"."),n("p",null,"Le medie di più variabili aleatorie si possono sommare:"),n("p",null,n(Je.k,null,la(Yn()))))),n(Je.s,null,n(Je.r,{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.k,null,la(Sn()))),n("p",null,"Ha diverse proprietà:"),n("ul",null,n("li",null,"Il suo ",n("b",null,"valore nullo")," è 0: ",n(Je.k,null,la(Vn()))),n("li",null,"E' ",n("b",null,"commutativa"),": ",n(Je.k,null,la(Nn()))),n("li",null,"E' ",n("b",null,"semplificabile"),": ",n(Je.k,null,la(yn()))),n("li",null,"E' ",n("b",null,"lineare"),": ",n(Je.k,null,la(Fn()))),n("li",null,"E' ",n("b",null,"distributiva"),": ",n(Je.k,null,la(Ln()))))),n(Je.r,{title:"Variabili incorrelate"},n("p",null,"Due variabili sono ",n("i",null,"variabili incorrelate")," se:"),n("p",null,n(Je.k,null,la(xn()))),n("p",null,"Variabili indipendenti sono sempre incorrelate.")),n(Je.r,{title:"Matrice di covarianza"},n("p",null,"Una matrice ",n(Je.k,null,la(Pn()))," che contiene la covarianza tra tutte le variabili di un vettore aleatorio ",n(Je.k,null,la(En())),":"),n("p",null,n(Je.k,null,la(zn()))),n("p",null,"E' sempre simmetrica e semidefinita positiva (tutti gli autovalori sono ",n(Je.k,null,"\\geq 0"),".")),n(Je.r,{title:"Coefficiente di correlazione"},n("p",null,"Un valore che misura come due variabili aleatorie sono correlate:"),n("p",null,n(Je.k,null,la(qn()))),n("p",null,"E' sempre compreso tra -1 e 1:"),n("p",null,n(Je.k,null,la(hn()))),n("p",null,"Vale esattamente -1 o 1 solo se esiste un legame lineare tra le due variaibli:"),n("p",null,n(Je.k,null,la(gn())))),n(Je.r,{title:"Varianza di variabili aleatorie sommate"},n("p",null,"La varianza di due variabili aleatorie sommate è:"),n("p",null,n(Je.k,null,la(Xn()))),n(He.a,null,"Si dimostra applicando le proprietà della covarianza!"),n("p",null,"Se più variabili aleatorie ",n(Je.k,null,"X_i")," sono ",n("b",null,"indipendenti")," (",n(Je.k,null,la(kn())),"), allora:"),n("p",null,n(Je.k,null,la(_n()))))),n(Je.s,{title:"Campioni"},n(Je.r,{title:"Campione casuale"},n("p",null,"Una ",n("b",null,"n-pla")," di variabili aleatorie con la stessa distribuzione della variabile aleatoria ",n(Je.k,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.r,{title:"Momento campionario"},n("p",null,"Il valore dato dalla media aritmetica degli ",n(Je.k,null,"n")," elementi del campione elevati alla potenza ",n(Je.k,null,"k"),":"),n("p",null,n(Je.k,null,la(bn()))),n("p",null,"Il momento campionario di primo ordine è la ",n("i",null,"media campionaria"),n(Je.k,null,la(vn())),".")),n(Je.r,{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.k,null,la(fn()))," di X:"),n("p",null,n(Je.k,null,la(mn()))),n("p",null,"Altrimenti:"),n("p",null,n(Je.k,null,la(dn()))))),n(Je.s,{title:"Media-ception"},n(Je.r,{title:"Media campionaria"},n("p",null,"Se calcoliamo la media della media campionaria, risulterà vero che:"),n("p",null,n(Je.k,null,la(pn()))),n(He.a,null,"Quindi, è possibile usare i campioni per trovare la media di una variabile aleatoria!")),n(Je.r,{title:"Varianza campionaria"},n("p",null,"Se calcoliamo la varianza della media campionaria, risulterà vero che:"),n("p",null,n(Je.k,null,la(sn()))),n(He.a,null,"Quindi, possiamo stimare l'errore della media calcolata tramite campioni!")),n(Je.r,{title:"Correzione campionaria"},n("p",null,"Se calcoliamo la media della varianza campionaria, risulterà vero che:"),n("p",null,n(Je.k,null,la(cn()))),n(He.a,null,"Quindi, possiamo stimare l'errore della media calcolata tramite campioni!"))),n(Je.s,{title:"Campionamento di una distribuzione normale"},n(Je.r,{title:"Campionamento di una distribuzione normale"},n("p",null,"Se la popolazione ",n(Je.k,null,"X")," ha una distribuzione normale (",n(Je.k,null,la(un())),")...")),n(Je.r,{title:"Distribuzione della media campionaria"},n("p",null,"...allora sappiamo anche la distribuzione della media campionaria!"),n("p",null,n(Je.k,null,la(on())))),n(Je.r,{title:"Distribuzione della varianza campionaria"},n("p",null,"...e anche della varianza campionaria!"),n("p",null,n(Je.k,null,la(rn()))),n("p",null,n(Je.k,null,la(tn())))),n(Je.r,{title:"Indipendenza"},n("p",null,"...e che media campionaria e varianza campionaria sono indipendenti tra loro!"))),n(Je.s,{title:"Quando i campioni hanno dimensioni infinite"},n(Je.r,{title:"Convergenza in distribuzione"},n("p",null,"Se la successione di variabili aleatorie ",n(Je.k,null,"X_n")," all'infinito ha la ",n("b",null,"stessa funzione di ripartizione")," della popolazione ",n(Je.k,null,"X"),", allora essa ",n("i",null,"converge in distribuzione"),"."),n("p",null,n(Je.k,null,"\\lim_{n \\to +\\infty} F_{X_n} (x) = F_X (x) \\implies X_n \\xrightarrow{d} X"))),n(Je.r,{title:"Convergenza in probabilità"},n("p",null,"Se la successione di variabili aleatorie ",n(Je.k,null,"X_n")," all'infinito ha la ",n("b",null,"stessa probabilità")," della popolazione ",n(Je.k,null,"X"),", allora essa ",n("i",null,"converge in probabilità"),"."),n("p",null,n(Je.k,null,"\\forall \\epsilon > 0, \\lim_{n \\to +\\infty} P( | X_n - X | < \\epsilon) = 1 \\implies X_n \\xrightarrow{p} X"))),n(Je.r,{title:"Convergenza quasi certa"},n("p",null,"Se la successione di variabili aleatorie ",n(Je.k,null,"X_n")," all'infinito ha la ",n("b",null,"stessa probabilità a ")," della popolazione ",n(Je.k,null,"X"),", allora essa ",n("i",null,"converge quasi certamente"),"."),n("p",null,n(Je.k,null,"\\forall \\epsilon > 0, P left( \\lim_{n \\to +\\infty} | X_n - X | < \\epsilon) \right) = 1 \\implies X_n \\xrightarrow{qc} X"))),n(Je.r,{title:"Convergenza in media quadratica"},n("p",null,"Se la successione di variabili aleatorie ",n(Je.k,null,"X_n")," all'infinito ha la ",n("b",null,"media del quadrato della distanza")," tra la successione e la popolazione ",n(Je.k,null,"X")," ",n("b",null,"uguale a 0"),", allora essa ",n("i",null,"converge in media quadratica"),"."),n("p",null,n(Je.k,null,"\\lim_{n \\to +\\infty} E( | X_n - X |^2 = 0 \\implies X_n \\xrightarrow{mq} X"))),n(Je.r,{title:"Gerarchia delle convergenze"},n("p",null,n(Je.k,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.k,null,"X_n \\xrightarrow{p} x \\Longleftrightarrow X_n \\xrightarrow{d} x")))),n(Je.s,{title:"I grandi numeri"},n(Je.r,{title:"Legge debole dei grandi numeri"},n("p",null,"La successione delle medie campionarie ",n(Je.k,null,la(an()))," ",n("b",null,"converge in probabilità")," alla media della popolazione ",n(Je.k,null,la(en())),", se essa esiste."),n("p",null,n(Je.k,null,"\\overline{X}_n \\xrightarrow{p} X")),n("p",null,"Ovvero:"),n("p",null,n(Je.k,null,la(ln()))),n("p",null,n(Je.k,null,la(nn())))),n(Je.r,{title:"Legge forte dei grandi numeri"},n("p",null,"La successione delle medie campionarie ",n(Je.k,null,la($()))," ",n("b",null,"converge quasi certamente")," alla media della popolazione ",n(Je.k,null,la(H())),", se essa esiste."),n("p",null,n(Je.k,null,"\\overline{X}_n \\xrightarrow{qc} X")),n("p",null,"Ovvero:"),n("p",null,n(Je.k,null,la(J()))),n(He.a,null,"Dimostra che l'interpretazione frequentista della probabilità è valida!"))),n(Je.s,{title:"Al limite"},n(Je.r,{title:"Teorema centrale del limite"},n("p",null,"La successione delle medie campionarie ",n(Je.k,null,la(W()))," ",n("b",null,"converge in distribuzione")," a ",n(Je.k,null,la(Q())),"."),n("p",null,n(Je.k,null,la(Z()))),n("p",null,"Ovvero:"),n("p",null,n(Je.k,null,la(j()))))),n(Je.s,{title:"Altre approsimazioni"},n(Je.r,{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.k,null,la(K())))),n(Je.r,{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.k,null,la(R())))),n(Je.r,{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.k,null,la(O())))),n(Je.r,{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.k,null,la(G())))),n(Je.r,{title:"In generale"},n("p",null,"Se ",n(Je.k,null,"n")," è grande, allora:"),n("p",null,n(Je.k,null,la(T()))))),n(Je.s,{title:"Actually statistica"},n(Je.r,{title:"Parametri sconosciuti"},n("p",null,"Per indicare parametri sconosciuti di una legge si usa ",n(Je.k,null,"\\theta"),".")),n(Je.r,{title:"Statistica"},n("p",null,"Una variabile aleatoria funzione di un campione:"),n("p",null,n(Je.k,null,la(A()))),n(He.a,null,"Ad esempio, sono statistiche media e varianza campionaria, così come il campione stesso ",n(Je.k,null,la(B())),"."))),n(Je.s,{title:"Stimatori"},n(Je.r,{title:"Stimatore"},n("p",null,"Una statistica ",n(Je.k,null,"T_n")," ottenuta da ",n(Je.k,null,"n")," osservazioni, che stimi i parametri di una legge e sia indipendente da essi.")),n(Je.r,{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.k,null,la(U())))),n(Je.r,{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.k,null,la(M())))),n(Je.r,{title:"Consistente in media quadratica"},n("p",null,"Uno stimatore è ",n("i",null,"consistente in media quadratica")," se:"),n("p",null,n(Je.k,null,la(D())))),n(Je.r,{title:"Consistente in probabilità"},n("p",null,"Uno stimatore è ",n("i",null,"consistente in probabilità")," se:"),n("p",null,n(Je.k,null,la(I())))),n(Je.r,{title:"Asintoticamente normale"},n("p",null,"Uno stimatore è ",n("i",null,"asintoticamente normale")," se:"),n("p",null,n(Je.k,null,la(C()))))),n(Je.s,{title:"Metodo dei momenti"},n(Je.r,{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.k,null,"X"),"."),n("p",null,"Lo stimatore di ",n(Je.k,null,la(w()))," così ottenuto sarà indicato aggiungendo un cappellino e una ",n(Je.k,null,"M")," a ",n(Je.k,null,"\\theta"),": ",n(Je.k,null,la(Y()))),n("p",null,"Visto che:"),n("ul",null,n("li",null,n(Je.k,null,la(S()))),n("li",null,n(Je.k,null,la(V())))),n("p",null,"Allora:"),n("p",null,n(Je.k,null,la(N()))),n("p",null,"Se ",n(Je.k,null,la(y()))," non è esprimibile in termini di ",n(Je.k,null,la(F())),", si possono usare i momenti successivi ",n(Je.k,null,la(L())),", ",n(Je.k,null,la(x())),", ",n(Je.k,null,la(P())),"..."))),n(Je.s,{title:"Metodo della massima verosomiglianza"},n(Je.r,{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.k,null,"X"),"."),n("p",null,"Lo stimatore di ",n(Je.k,null,la(E()))," così ottenuto sarà indicato aggiungendo un cappellino e una ",n(Je.k,null,"L")," a ",n(Je.k,null,"\\theta"),": ",n(Je.k,null,la(z()))),n("p",null,"Consiste nel trovare il massimo assoluto ",n(Je.k,null,la(q()))," della la funzione di verosomiglianza ",n(Je.k,null,la(h())),":"),n("p",null,n(Je.k,null,la(g()))),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.r,{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.k,null,la(X())))))),n(Je.s,{title:"Nuove stime notevoli"},n(Je.r,{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.k,null,la(k())))),n(Je.r,{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.k,null,la(_())))),n(Je.r,{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.k,null,la(b())))),n(Je.r,{title:"Stima di una normale"},n("p",null,"Per il metodo della massima verosomiglianza:"),n("ul",null,n("li",null,n(Je.k,null,la(v()))),n("br",null),n("li",null,n(Je.k,null,la(f())))))),n(Je.s,{title:"Intervalli di confidenza"},n(Je.r,{title:"Confidenza"},n("blockquote",null,'"intervallo di confidenza al 95%"'),n("p",null,"L'intervallo di valori di ",n(Je.k,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.k,null,la(m()))," è l'intervallo ",n(Je.k,null,"]a, b[")," tale che:"),n("p",null,n(Je.k,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.s,{title:"Confidenza nella media di una normale"},n(Je.r,{title:"Varianza nota"},n("p",null,"Se conosciamo la varianza di una normale, allora possiamo ricavare velocemente gli intervalli di confidenza all'",n(Je.k,null,"\\alpha"),"% con queste formule:"),n("ul",null,n("li",null,"Intervalli bilateri: ",n(Je.k,null,la(p()))),n("li",null,"Intervallo unilatero da sinistra: ",n(Je.k,null,la(s()))),n("li",null,"Intervallo unilatero da destra: ",n(Je.k,null,la(c()))))),n(Je.r,{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.k,null,"\\alpha"),"% con queste formule:"),n("ul",null,n("li",null,"Intervalli bilateri: ",n(Je.k,null,la(u()))),n("li",null,"Intervallo unilatero da sinistra: ",n(Je.k,null,la(o()))),n("li",null,"Intervallo unilatero da destra: ",n(Je.k,null,la(r())))),n("p",null,n(Je.k,null,la(t()))," è un quantile della distribuzione di Student di parametro ",n(Je.k,null,"v"),"."))),n(Je.s,{title:"Confidenza per la proporzione di una bernoulliana"},n(Je.r,{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.k,null,la(a()))))),n(Je.s,{title:"Confidenza per la media di qualsiasi popolazione"},n(Je.r,{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.k,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)}}]);
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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 return {props.children};\n}\n","import {Latex, LatexDefaultDisplay, LatexDefaultInline, LatexDisplay, Panel, Section} 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 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\n detta famiglia degli eventi.\n
\n
\n {r`\\mathcal{F}`}\n
\n
\n Qualsiasi sottoinsieme appartenente a {r`\\mathcal{F}`} è considerato un\n 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\n detta {r`\\sigma`}-algebra:\n
\n \n
\n Lo spazio campionario è un evento: {r`\\Omega \\in \\mathcal{F}`}\n
\n
\n Se un sottoinsieme è un evento, allora anche il suo complementare lo\n è: {r`E \\in \\mathcal{F} \\implies \\bar{E} \\in \\mathcal{F}`}\n
\n
\n Se due sottoinsiemi sono eventi, allora lo sono anche la loro unione e\n intersezione: {r`(E, F) \\in \\mathcal{F} \\implies (E \\cup F, E \\cap F) \\in \\mathcal{F}`}\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\n eventi {r`E_1`}, {r`E_2`}, {r`E_3`}, fino\n a {r`E_n`}.\n
\n \n Se lo spazio campionario fosse una torta, una sua partizione sarebbe l'insieme delle\n 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\n probabilità.\n
\n \n Sommando le probabilità dei due eventi, l'intersezione viene contata due volte, e va\n 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\n 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\n 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\n 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 Tramite la formula di Bayes possiamo risalire alla probabilità di un evento\n condizionato a un altro partendo dalla probabilità di quest'ultimo condizionato al\n primo:\n
\n Eventi indipendenti a due a due non sono per forza indipendenti a tre a tre, e\n viceversa.\n
\n \n \n
\n Un insieme di n eventi è una famiglia di eventi indipendenti se,\n 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\n 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\n campionario. {r`X(\\omega) : \\Omega \\to \\mathbb{R}`}.\n
\n \n Insieme di ripartizione}>\n
\n Ad ogni variabile aleatoria sono associati gli\n eventi {r`A_t = \\{ \\omega | X(\\omega) \\leq t \\}`}, che contengono tutti\n gli esiti a cui la variabile aleatoria associa un valore minore o uguale\n 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,\n 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\n aleatoria discretaX è la funzione che associa ad ogni esito la\n 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\n aleatoria continuaX è l'equivalente continuo della funzione\n probabilità:\n
\n A differenza della funzione probabilità, è possibile che la funzione densità non\n 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\n {r`F_X : \\mathbb{R} \\to [0, 1]`} associata, che rappresenta la\n probabilità che la variabile aleatoria assuma un valore minore o uguale\n a t:\n
\n
\n Si può dire che essa rappresenti la probabilità dell'evento {r`A_t`}:\n
\n Trasformare variabili aleatorie è molto utile nell'informatica per creare distribuzioni\n partendo da una funzione random() che\n 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\n finito ha anche una media (o valore medio o atteso):\n
\n Se la variabile aleatoria X ha media e varianza, allora la probabilità\n che essa abbia un valore a più di {r`\\epsilon`} di distanza dal valore\n medio è minore o uguale a {r`\\frac{Var(X)}{\\epsilon^2}`}.\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\n insuccessi N-K, conta quanti successi si otterrebbero se se ne\n estraessero n in blocco.\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
{r`E(X^2) = \\mu^2 + \\mu`}
\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\n di Poisson di intensità {r`\\lambda`} in un intervallo di tempo di\n durata t.\n
\n
\n E' una distribuzione poissoniana\n 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\n 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\n processo di Poisson di intensità {r`\\lambda`}.\n
\n La binomiale non è altro che una poissoniana a tempo discreto, quindi,\n se n è grande e n \\cdot p è nell'ordine di grandezza delle\n 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\n 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\n ogni valore discreto k la probabilità viene \"spalmata\" su tutto\n 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\n 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\n numero di parametri.\n
\n
\n Se il numero di parametri coincide con la dimensione del vettore aleatorio, allora la\n funzione sarà una funzione di ripartizione congiunta:\n
\n Una n-pla di variabili aleatorie con la stessa distribuzione della variabile\n aleatoria X (\"popolazione\") ma indipendenti tra loro.\n
\n \n Le variabili aleatorie sono come un lazy-load in programmazione; quando ci sarà bisogno\n 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\n elevati alla potenza k:\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\n funzione di ripartizione della popolazione X, allora essa converge\n in distribuzione.\n
\n Se la successione di variabili aleatorie X_n all'infinito ha la stessa\n probabilità a della popolazione X, allora essa converge quasi\n certamente.\n
\n Se la successione di variabili aleatorie X_n all'infinito ha la media\n del quadrato della distanza tra la successione e la popolazione Xuguale\n a 0, allora essa converge in media quadratica.\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\n 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\n 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\n 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\n valore atteso coincide con quello dei parametri che stima:\n
\n Se non conosciamo la varianza di una normale, allora possiamo ricavare velocemente gli\n intervalli di confidenza all'\\alpha% con queste formule:\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\";\n\nexport default function (props) {\n return (\n
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u,r,s,p,d,m,c,b,v,k,g,f,_,h,z,X,q,E,P,L,x,y,F,S,I,C,N,V,w,D,Y,U,M,B,A,T,G,O,R,j,K,Q,Z,W,J,H,$,ll,il,nl,el,al,ol,tl,ul,rl,sl,pl,dl,ml,cl,bl,vl,kl,gl,fl,_l,hl,zl,Xl,ql,El,Pl,Ll,xl,yl,Fl,Sl,Il,Cl,Nl,Vl,wl,Dl,Yl,Ul,Ml,Bl,Al,Tl,Gl,Ol,Rl,jl,Kl,Ql,Zl,Wl,Jl,Hl,$l,li,ii,ni,ei,ai,oi,ti,ui,ri,si,pi,di,mi,ci,bi,vi,ki,gi,fi,_i,hi,zi,Xi,qi,Ei,Pi,Li,xi,yi,Fi,Si,Ii,Ci,Ni,Vi,wi,Di,Yi,Ui,Mi,Bi,Ai,Ti,Gi,Oi,Ri,ji,Ki,Qi,Zi,Wi,Ji,Hi,$i,ln,nn,en,an,on,tn,un,rn,sn,pn,dn,mn,cn,bn,vn,kn,gn,fn,_n,hn,zn,Xn,qn,En,Pn,Ln,xn,yn,Fn,Sn,In,Cn,Nn,Vn,wn,Dn,Yn,Un,Mn,Bn,An,Tn,Gn,On,Rn,jn,Kn,Qn,Zn,Wn,Jn,Hn,$n,le,ie,ne,ee,ae,oe,te,ue,re,se,pe,de,me,ce,be,ve,ke,ge,fe,_e,he,ze,Xe,qe,Ee,Pe,Le,xe,ye,Fe,Se,Ie,Ce,Ne,Ve,we,De,Ye,Ue,Me,Be,Ae,Te,Ge,Oe,Re,je,Ke,Qe,Ze,We,Je,He,$e,la=l=>l;const ia=String.raw;i.default=function(){return l(e.m.Provider,{value:!1},l(e.l.Provider,{value:e.n.INLINE},l("div",null,l("h1",null,"Statistica ed Elementi di Probabilità"),l(e.s,{title:"Tipi di probabilità"},l(e.r,{title:"Classica"},l("p",null,l(e.k,null,ia(u||(u=la`P(E) = \frac{casi\ favorevoli}{casi\ possibili}`))))),l(e.r,{title:"Frequentista"},l("p",null,l(e.k,null,ia(r||(r=la`P(E) = \frac{successi}{prove\ totali}`))))),l(e.r,{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.s,{title:"Linguaggio matematico"},l(e.r,{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.k,null,ia(s||(s=la`\Omega = \left \{ 1, 2, 3, 4, 5, 6 \right \}`))))),l(e.r,{title:"Esito"},l("blockquote",null,'"omeghino"'),l("p",null,"Un ",l("b",null,"elemento")," dello spazio campionario."),l("p",null,l(e.k,null,ia(p||(p=la`\omega = 1`))))),l(e.r,{title:"Evento"},l("blockquote",null,'"e"'),l("p",null,"Un ",l("b",null,"sottoinsieme")," dello spazio campionario."),l("p",null,l(e.k,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.r,{title:"Not"},l("blockquote",null,'"not e"'),l("p",null,"Il ",l("b",null,"complementare")," di un sottoinsieme."),l("p",null,l(e.k,null,ia(m||(m=la`\bar{E} = \left \{ 3, 4, 5, 6 \right \}`))))),l(e.r,{title:"And"},l("blockquote",null,'"e intersecato effe"'),l("p",null,"L'",l("b",null,"intersezione")," di più sottoinsiemi."),l("p",null,l(e.k,null,ia(c||(c=la`E \cap F = \left \{ 1 \right \}`))))),l(e.r,{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.k,null,ia(b||(b=la`E \cup F = \left \{ 1, 2, 3, 4 \right \}`))))),l(e.r,{title:"Differenza"},l("blockquote",null,'"e meno effe"'),l("p",null,l(e.k,null,ia(v||(v=la`E \setminus F = E \cap \bar{F}`))))),l(e.r,{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.k,null,ia(k||(k=la`E \subseteq F`)))),l("p",null,"Se si verifica ",l(e.k,null,"E"),", allora si verifica anche ",l(e.k,null,"F"),".")),l(e.r,{title:"Evento impossibile"},l("blockquote",null,'"e è impossibile"'),l("p",null,"Un sottoinsieme ",l("b",null,"vuoto"),"."),l("p",null,l(e.k,null,ia(g||(g=la`E = \emptyset`))))),l(e.r,{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.k,null,ia(f||(f=la`E \cap F = \emptyset`)))))),l(e.s,null,l(e.r,{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.k,null,ia(_||(_=la`\mathcal{F}`)))),l("p",null,"Qualsiasi sottoinsieme appartenente a ",l(e.k,null,ia(h||(h=la`\mathcal{F}`)))," è considerato un evento.")),l(e.r,{title:l("span",null,l(e.k,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.k,null,ia(X||(X=la`\sigma`))),"-algebra"),":"),l("ol",null,l("li",null,"Lo spazio campionario è un evento: ",l(e.k,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.k,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.k,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.k,null,ia(L||(L=la`E \in \mathcal{F} \implies \mathcal{F} = \{ \emptyset, E, \bar{E}, \Omega \}`)))))),l(e.s,null,l(e.r,{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.k,null,ia(x||(x=la`E_i`)))," è composta dagli eventi ",l(e.k,null,ia(y||(y=la`E_1`))),", ",l(e.k,null,ia(F||(F=la`E_2`))),", ",l(e.k,null,ia(S||(S=la`E_3`))),", fino a ",l(e.k,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.s,{title:"Assiomi della probabilità"},l(e.r,{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.k,null,ia(C||(C=la`\forall E \in \mathcal{F}, 0 \leq P(E) \leq 1`))))),l(e.r,{title:"Secondo assioma della probabilità"},l("p",null,"La probabilità dello spazio campionario è sempre 1."),l("p",null,l(e.k,null,ia(N||(N=la`P(\Omega) = 1`))))),l(e.r,{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.k,null,ia(V||(V=la`P \left ( \bigcup_i E_i \right ) = \sum_i P ( E_i )`)))))),l(e.s,{title:"Conseguenze degli assiomi"},l(e.r,{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.k,null,ia(w||(w=la`P(\bar{E}) = 1 - P({E})`))))),l(e.r,{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.k,null,ia(D||(D=la`F \subseteq E \implies P(F) \leq P(E)`))))),l(e.r,{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.k,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.s,{title:"Spazi equiprobabili"},l(e.r,{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.k,null,ia(U||(U=la`P(E) = \frac{len(E)}{len(\Omega)}`))))),l(e.r,{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.s,{title:"Calcolo combinatorio"},l(e.r,{title:"Disposizioni"},l("p",null,"Estraggo un numero, da un sacchetto con ",l(e.k,null,"n")," numeri, mi segno che numero ho estratto e lo ",l("b",null,"tengo fuori dal sacchetto"),". Ripeto per ",l(e.k,null,"k")," volte."),l("p",null,l("b",null,"Tengo conto")," dell'ordine in cui ho estratto i numeri."),l("p",null,l(e.k,null,ia(M||(M=la`\boldsymbol{D}_{n, k} = \frac{n!}{(n - k)!}`))))),l(e.r,{title:"Disposizioni con ripetizione"},l("p",null,"Estraggo un numero, da un sacchetto con ",l(e.k,null,"n")," numeri, mi segno che numero ho estratto e lo ",l("b",null,"rimetto nel sacchetto"),". Ripeto per ",l(e.k,null,"k")," volte."),l("p",null,l("b",null,"Tengo conto")," dell'ordine in cui ho estratto i numeri."),l("p",null,l(e.k,null,ia(B||(B=la`\boldsymbol{D}^{r}_{n, k} = n^k`))))),l(e.r,{title:"Combinazioni"},l("p",null,"Estraggo un numero, da un sacchetto con ",l(e.k,null,"n")," numeri, mi segno che numero ho estratto e lo ",l("b",null,"tengo fuori dal sacchetto"),". Ripeto per ",l(e.k,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.k,null,ia(A||(A=la`\boldsymbol{C}_{n, k} = \binom{n}{k} = \frac{n!}{(k)! \cdot (n - k)!}`))))),l(e.r,{title:"Combinazioni con ripetizione"},l("p",null,"Estraggo un numero, da un sacchetto con ",l(e.k,null,"n")," numeri, mi segno che numero ho estratto e lo ",l("b",null,"rimetto nel sacchetto"),". Ripeto per ",l(e.k,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.k,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.r,{title:"Permutazioni"},l("p",null,"Estraggo ",l(e.k,null,"n")," numeri e guardo in quanti ordini diversi li posso mettere."),l("p",null,l(e.k,null,ia(G||(G=la`\boldsymbol{P}_n = n!`)))))),l(e.s,{title:"Probabilità condizionata"},l(e.r,{title:"Eventi condizionati"},l("blockquote",null,'"E dato F"'),l("p",null,"La probabilità che si verifichi ",l(e.k,null,"E")," sapendo che ",l("b",null,"si è già verificato"),l(e.k,null,"F"),"."),l("p",null,l(e.k,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.r,{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.k,null,ia(R||(R=la`E \cap F = \emptyset \Longleftrightarrow P(E|F) = P(F|E) = 0`)))))),l(e.s,null,l(e.r,{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.k,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.s,{title:"Le alternative"},l(e.r,{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.k,null,ia(K||(K=la`P(F) = \sum_{i} P(F|E_i) \cdot P(E_i)`))))),l(e.r,{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.k,null,ia(Q||(Q=la`P(F|G) = \sum_i P(F|E_i \cap G) \cdot P(E_i | G)`))))),l(e.r,{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.k,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.s,{title:"Eventi indipendenti"},l(e.r,{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.k,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.r,{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.k,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.r,{title:"Famiglia di eventi indipendenti"},l("p",null,"Un insieme di ",l(e.k,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.s,{title:"Variabili aleatorie"},l(e.r,{title:"Variabile aleatoria"},l("p",null,"Una funzione che fa corrispondere un numero reale a ogni possibile esito dello spazio campionario. ",l(e.k,null,ia(H||(H=la`X(\omega) : \Omega \to \mathbb{R}`))),".")),l(e.r,{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.k,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.k,null,"t"),"."),l("p",null,"Per definizione, tutte le variabili aleatorie devono rispettare questa condizione:"),l("p",null,l(e.k,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.r,{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.k,null,"x_0")," appartiene al supporto di ",l(e.k,null,"X"),", si usa la notazione ",l(e.k,null,"X \\mapsto x_0"),"."))),l(e.s,{title:"Densità"},l(e.r,{title:"Funzione probabilità"},l("p",null,"La ",l("i",null,"funzione probabilità")," ",l(e.k,null,ia(il||(il=la`p_X : X \to [0, 1]`)))," di una variabile aleatoria ",l("b",null,"discreta")," ",l(e.k,null,"X")," è la funzione che associa ad ogni esito la sua probabilità:"),l("p",null,l(e.k,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.r,{title:"Funzione densità"},l("p",null,"La ",l("i",null,"funzione densità")," ",l(e.k,null,ia(el||(el=la`f_X : X \to [0, 1]`)))," di una variabile aleatoria ",l("b",null,"continua")," ",l(e.k,null,"X")," è l'equivalente continuo della funzione probabilità:"),l("p",null,l(e.k,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.s,{title:"Funzione di ripartizione"},l(e.r,{title:"Definizione"},l("p",null,"Ogni variabile aleatoria ha una ",l("i",null,"funzione di ripartizione"),l(e.k,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.k,null,"t"),":"),l("p",null,"Si può dire che essa rappresenti la probabilità dell'evento ",l(e.k,null,ia(tl||(tl=la`A_t`))),":"),l("p",null,l(e.k,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.r,{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.k,null,"-\\infty")," e ",l("b",null,"1")," a ",l(e.k,null,"+\\infty"),"."),l("br",null),l("li",null,"È ",l("b",null,"continua da destra"),": ",l(e.k,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.r,{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.k,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.s,{title:"Trasformazioni di variabili aleatorie"},l(e.r,{title:"Nel discreto"},l("p",null,"Nel discreto basta abbinare un nuovo valore a ogni valore della variabile originale.")),l(e.r,{title:"Nel continuo (invertibile)"},l("p",null,"Nel continuo applichiamo la formula dell'integrazione per sostituzione:"),l("p",null,l(e.k,null,ia(pl||(pl=la`f_Y (y) = \int_{g(a)}^{g(b)} f_X ( g^{-1} (x) ) g^{-2} (x)`))))),l(e.r,{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.s,{title:"Informazioni delle variabili aleatorie"},l(e.r,{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.k,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.k,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.k,null,ia(cl||(cl=la`E(X) = \int_{-\infty}^{+\infty} f_X (x) \cdot x \cdot dx`)))))),l(e.s,null,l(e.r,{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.r,{title:"Quantili"},l("p",null,"Il ",l("i",null,"quantile")," ",l(e.k,null,ia(bl||(bl=la`x_{\alpha}`)))," di ordine ",l(e.k,null,ia(vl||(vl=la`0 \leq \alpha \leq 1`)))," della variabile aleatoria ",l(e.k,null,"X")," è il più piccolo numero tale che:"),l("p",null,l(e.k,null,ia(kl||(kl=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.k,null,ia(gl||(gl=la`x_{0.5}`)))," è detto ",l("i",null,"mediana"),"."),l("p",null,"I quantili di ordine 0.25 ",l(e.k,null,ia(fl||(fl=la`x_{0.25}`)))," e 0.75 ",l(e.k,null,ia(_l||(_l=la`x_{0.75}`)))," sono detti ",l("i",null,"quartili"),"."),l("p",null,"I quantili di ordine ",l(e.k,null,ia(hl||(hl=la`\frac{n}{100}`)))," sono detti ",l("i",null,l(e.k,null,"n"),"-esima percentile"),".")),l(e.r,{title:"Varianza"},l("p",null,"È un valore che indica quanto la variabile aleatoria si discosta generalmente dalla media:"),l("p",null,l(e.k,null,ia(zl||(zl=la`Var(X) = E( (X - E(X) )^2 ) = E ( X^2 ) - (E(X))^2`)))))),l(e.s,{title:"Disuguaglianze notevoli"},l(e.r,{title:"Disuguaglianza di Markov"},l("p",null,"Data una variabile aleatoria non-negativa:"),l("p",null,l(e.k,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.k,null,ia(ql||(ql=la`P(X < k)`)))," e ",l(e.k,null,ia(El||(El=la`P(X \geq k)`))),") la funzione X, la cui media risulterà uguale a:"),l("p",null,l(e.k,null,ia(Pl||(Pl=la`E(X) = \overline{k} \cdot P(X < k) + k \cdot P(X \geq k)`))))),l(e.r,{title:"Disuguaglianza di Čebyšëv"},l("blockquote",null,'"disuguaglianza di cebicev"'),l("p",null,"Se la variabile aleatoria ",l(e.k,null,"X")," ha media e varianza, allora la probabilità che essa abbia un valore a più di ",l(e.k,null,ia(Ll||(Ll=la`\epsilon`)))," di distanza dal valore medio è minore o uguale a ",l(e.k,null,ia(xl||(xl=la`\frac{Var(X)}{\epsilon^2}`))),"."),l("p",null,l(e.k,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.k,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.s,{title:"Un momento...!"},l(e.r,{title:"Momento"},l("p",null,"Il ",l("i",null,"momento")," ",l(e.k,null,"k"),"-esimo di una variabile aleatoria è:"),l("p",null,l(e.k,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.r,{title:"Funzione generatrice dei momenti"},l("p",null,"La ",l("i",null,"funzione generatrice dei momenti")," è:"),l("p",null,l(e.k,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.r,{title:"Funzione caratteristica"},l("p",null,"La ",l("i",null,"funzione caratteristica")," è:"),l("p",null,l(e.k,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.s,{title:"Prove e schemi"},l(e.r,{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.k,null,ia(Nl||(Nl=la`X \sim Distribuzione()`))))),l(e.r,{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.r,{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.s,{title:"Bernoulliana"},l(e.r,{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.k,null,ia(Vl||(Vl=la`Ber(p)`))))),l(e.r,{title:"Densità della bernoulliana"},l("p",null,"La distribuzione bernoulliana ha come densità:"),l("p",null,l(e.k,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.s,{title:"Binomiale"},l(e.r,{title:"Distribuzione binomiale"},l("p",null,"Una variabile aleatoria che conta il numero di successi di ",l(e.k,null,"n")," prove di uno schema di Bernoulli."),l("p",null,"Il suo simbolo è ",l(e.k,null,ia(Dl||(Dl=la`Bin(n, p)`))),".")),l(e.r,{title:"Densità della binomiale"},l("p",null,"La binomiale ha come densità:"),l("p",null,l(e.k,null,ia(Yl||(Yl=la`f_X (k) : \{0..n\} = \binom{n}{k} \cdot p^k \cdot q^{n - k}`))))),l(e.r,{title:"Momenti della binomiale"},l("p",null,"La ",l("b",null,"funzione generatrice dei momenti")," della binomiale è:"),l("p",null,l(e.k,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.k,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.k,null,ia(Bl||(Bl=la`Var(X) = n \cdot p \cdot q`)))))),l(e.s,{title:"Geometrica"},l(e.r,{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.k,null,"Geo(p)"),".")),l(e.r,{title:"Densità della geometrica"},l("p",null,"La geometrica ha come densità:"),l("p",null,l(e.k,null,ia(Al||(Al=la`f_X (k) : \mathbb{N} = q^{k - 1} p`))))),l(e.r,{title:"Momenti della geometrica"},l("p",null,"La ",l("b",null,"funzione generatrice dei momenti")," della geometrica è:"),l("p",null,l(e.k,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.k,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.k,null,ia(Ol||(Ol=la`Var(X) = \frac{q}{p^2}`))))),l(e.r,{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.k,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.s,{title:"Binomiale negativa"},l(e.r,{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.k,null,"n"),"-esimo successo."),l("p",null,"Il suo simbolo è ",l(e.k,null,ia(jl||(jl=la`\overline{Bin}(n, p)`))),".")),l(e.r,{title:"Densità della binomiale negativa"},l("p",null,"La binomiale negativa ha come densità:"),l("p",null,l(e.k,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.r,{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.k,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.k,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.k,null,ia(Wl||(Wl=la`Var(X) = \frac{n \cdot q}{p^2}`))))))),l(e.s,{title:"Geometrica traslata"},l(e.r,{title:"Distribuzione geometrica traslata"},l("p",null,"Una variabile aleatoria che conta il numero ",l(e.k,null,"k")," di insuccessi consecutivi in uno schema di Bernoulli:"),l("p",null,"Il suo simbolo rimane ",l(e.k,null,ia(Jl||(Jl=la`Geo(p)`))),".")),l(e.r,{title:"Densità della geometrica tralsata"},l("p",null,"La geometrica traslata ha come densità:"),l("p",null,l(e.k,null,ia(Hl||(Hl=la`f_X (k) : \mathbb{N} = p \cdot q^k `))))),l(e.r,{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.k,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.k,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.k,null,ia(ii||(ii=la`Var(X) = \frac{q}{p^2}`))))),l(e.r,{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.k,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.s,{title:"Binomiale negativa traslata"},l(e.r,{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.k,null,"n"),"-esimo successo."),l("p",null,"Il suo simbolo rimane ",l(e.k,null,ia(ei||(ei=la`\overline{Bin}(n, p)`))),".")),l(e.r,{title:"Densità della binomiale negativa traslata"},l("p",null,"La binomiale negativa traslata ha come densità:"),l("p",null,l(e.k,null,ia(ai||(ai=la`f_X (k) : \mathbb{N} = \binom{k + n - 1}{n - 1} \cdot p^n \cdot q^k `))))),l(e.r,{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.k,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.k,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.k,null,ia(ui||(ui=la`Var(X) = \frac{n \cdot q}{p^2}`))))))),l(e.s,{title:"Ipergeometrica"},l(e.r,{title:"Distribuzione ipergeometrica"},l("p",null,"Una variabile aleatoria che, sapendo il numero di successi ",l(e.k,null,"K")," e di insuccessi ",l(e.k,null,"N-K"),", conta quanti successi si otterrebbero se se ne estraessero ",l(e.k,null,"n")," in blocco."),l("p",null,"Il suo simbolo è ",l(e.k,null,"Ipe(N, K, n)"),".")),l(e.r,{title:"Densità della ipergeometrica"},l("p",null,"La ipergeometrica ha come densità:"),l("p",null,l(e.k,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.r,{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.k,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.k,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.s,{title:"Poissoniana"},l(e.r,{title:"Distribuzione poissoniana"},l("p",null,"Una variabile aleatoria che soddisfa tutte le seguenti caratteristiche:"),l("ul",null,l("li",null,"Binomiale: ",l(e.k,null,ia(di||(di=la`X \sim Bin(n, p)`)))),l("li",null,"Il numero di prove tende a infinito: ",l(e.k,null,ia(mi||(mi=la`n \to +\infty`)))),l("li",null,"La probabilità di successo tende a 0: ",l(e.k,null,ia(ci||(ci=la`p \to 0`)))),l("li",null,"La media è finita: ",l(e.k,null,ia(bi||(bi=la`E(X) = n \cdot p \to \mu \neq 0`))))),l("p",null,"Il suo simbolo è ",l(e.k,null,ia(vi||(vi=la`Poi(\mu)`))))),l(e.r,{title:"Densità della poissoniana"},l("p",null,"La poissoniana ha come densità:"),l("p",null,l(e.k,null,ia(ki||(ki=la`f_X (k) : \mathbb{N} = \frac{e^{-\mu} \cdot \mu^k}{k!}`))))),l(e.r,{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.k,null,ia(gi||(gi=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.k,null,ia(fi||(fi=la`E(X) = \mu`)))),l("p",null,"La ",l("b",null,"varianza")," della poissoniana è:"),l("p",null,l(e.k,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.k,null,ia(hi||(hi=la`E(X^2) = \mu^2 + \mu`)))))))),l(e.s,{title:"Un altro schema"},l(e.r,{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.k,null,ia(zi||(zi=la`\lambda`)))," costante."),l("li",null,"avvengono indipendentemente gli uni dagli altri."))),l(e.r,{title:"Processo di Poisson"},l("p",null,"Una variabile aleatoria ",l(e.k,null,"N_t")," che conta il numero di arrivi di uno schema di Poisson di intensità ",l(e.k,null,ia(Xi||(Xi=la`\lambda`)))," in un intervallo di tempo di durata ",l(e.k,null,"t"),"."),l("p",null,"E' una distribuzione poissoniana con ",l(e.k,null,ia(qi||(qi=la`\mu = t \cdot \lambda`))),": ",l(e.k,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.s,{title:"Esponenziale"},l(e.r,{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.k,null,ia(Pi||(Pi=la`\lambda`))),"."),l("p",null,"Il suo simbolo è ",l(e.k,null,ia(Li||(Li=la`Esp(\lambda)`))),".")),l(e.r,{title:"Densità dell'esponenziale"},l("p",null,"L'esponenziale ha come ",l("b",null,"densità"),":"),l("p",null,l(e.k,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.k,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.r,{title:"Momenti dell'esponenziale"},l("p",null,"La ",l("b",null,"funzione generatrice dei momenti")," dell'esponenziale è:"),l("p",null,l(e.k,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.k,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.k,null,ia(Ii||(Ii=la`Var(X) = \frac{1}{\lambda^2}`))))),l(e.r,{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.k,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.s,{title:"Legge gamma"},l(e.r,{title:"Distribuzione gamma"},l("p",null,"Una variabile aleatoria che conta il tempo diwidehattesa prima dell'",l(e.k,null,"n"),"-esimo arrivo di un processo di Poisson di intensità ",l(e.k,null,ia(Ni||(Ni=la`\lambda`))),"."),l("p",null,"Il suo simbolo è ",l(e.k,null,ia(Vi||(Vi=la`\Gamma(n, \lambda)`))),".")),l(e.r,{title:"Densità della legge gamma"},l("p",null,"La legge gamma ha come densità:"),l("p",null,l(e.k,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.r,{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.k,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.k,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.k,null,ia(Ui||(Ui=la`Var(X) = \frac{\alpha}{\lambda^2}`))))))),l(e.s,{title:"Uniforme"},l(e.r,{title:"Distribuzione uniforme"},l("p",null,"Una variabile aleatoria che può assumere qualsiasi valore in un intervallo ",l(e.k,null,ia(Mi||(Mi=la`[a, b]`)))," in modo equiprobabile."),l("p",null,"Il suo simbolo è ",l(e.k,null,ia(Bi||(Bi=la`Uni(a, b)`)))),l("p",null,"Su di essa vale la seguente proprietà:"),l("p",null,l(e.k,null,ia(Ai||(Ai=la`P(X \in (c, d)) = \frac{d - c}{b - a}`))))),l(e.r,{title:"Densità della distribuzione uniforme"},l("p",null,"La distribuzione uniforme ha come ",l("b",null,"densità"),":"),l("p",null,l(e.k,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.k,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.r,{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.k,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.k,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.k,null,ia(ji||(ji=la`Var(X) = \frac{(b - a)^2}{12}`))))))),l(e.s,{title:"Normale o Gaussiana"},l(e.r,{title:"Distribuzione normale"},l("p",null,"Una variabile aleatoria con una specifica distribuzione."),l("p",null,"Il suo simbolo è ",l(e.k,null,ia(Ki||(Ki=la`Nor(\mu, \sigma^2)`))),"."),l(a.a,null,l(e.k,null,"\\mu")," e ",l(e.k,null,"\\sigma^2")," sono rispettivamente la media e la varianza della distribuzione!")),l(e.r,{title:"Densità della distribuzione normale"},l("p",null,"La distribuzione normale ha come densità:"),l("p",null,l(e.k,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.r,{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.k,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.k,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.k,null,ia(Ji||(Ji=la`Var(X) = \sigma^2`))))))),l(e.s,null,l(e.r,{title:"Trasformazione della normale"},l("p",null,"Qualsiasi normale può essere trasformata in qualsiasi altra normale:"),l("p",null,l(e.k,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.r,{title:"Normale standard"},l("p",null,"La distribuzione normale standard ",l(e.k,null,"Z")," è:"),l("p",null,l(e.k,null,"Z \\sim Nor(0, 1)")),l("p",null,"La sua funzione di ripartizione è detta ",l(e.k,null,ia($i||($i=la`\phi(z)`)))," e vale:"),l("p",null,l(e.k,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.r,{title:"Quantili normali"},l("p",null,"Da un quantile ",l(e.k,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.k,null,ia(en||(en=la`x_\alpha = \mu + z_\alpha \cdot \sqrt{\sigma^2}`)))))),l(e.s,null,l(e.r,{title:"Gamma e normale"},l("p",null,"La distribuzione normale ha una particolare relazione con la distribuzione Gamma:"),l("p",null,l(e.k,null,ia(an||(an=la`Z^2 \sim \chi^2 (v = 1)`))))),l(e.r,{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.k,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.k,null,ia(tn||(tn=la`\chi^2 (n) + \chi^2 (m) = \chi^2 (n + m)`))))),l(e.r,{title:"T di Student"},l("p",null,"Un'altra funzione particolare è la funzione T di Student:"),l("p",null,l(e.k,null,ia(un||(un=la`T(v) = \frac{Nor(0, 1)}{\sqrt{\frac{\chi^2(v)}{v}}}`)))))),l(e.s,{title:"Approssimazioni notevoli"},l(e.r,{title:"Ipergeometrica e binomiale"},l("p",null,"La binomiale è come una ipergeometrica ma con ripetizioni, quindi per valori molto grandi di ",l(e.k,null,"N")," rispetto a ",l(e.k,null,"n"),", si può dire che:"),l("p",null,l(e.k,null,ia(rn||(rn=la`Ipe(N, K, n) \approx Bin(n, \frac{K}{N})`))))),l(e.r,{title:"Binomiale e poissoniana"},l("p",null,"La binomiale non è altro che una poissoniana a tempo discreto, quindi, se ",l(e.k,null,"n")," è grande e ",l(e.k,null,"n \\cdot p")," è nell'ordine di grandezza delle unità, allora:"),l("p",null,l(e.k,null,ia(sn||(sn=la`Bin(n, p) \approx Poi(n \cdot p)`))))),l(e.r,{title:"Binomiale e normale"},l("p",null,"Per il Teorema di De Moivre-Laplace, se una binomiale ha una ",l(e.k,null,"n")," grande e ",l(e.k,null,"p")," non vicina a 0 o 1, si può approssimare con:"),l("p",null,l(e.k,null,ia(pn||(pn=la`Bin(n, p) \approx Nor(n \cdot p, n \cdot p \cdot q)`))))),l(e.r,{title:"Correzione di Yates"},l("p",null,"Passando da una variabile discreta ",l(e.k,null,"X")," a una continua ",l(e.k,null,"Y"),", per ogni valore discreto ",l(e.k,null,"k"),' la probabilità viene "spalmata" su tutto l\'intervallo ',l(e.k,null,ia(dn||(dn=la`(k - \frac{1}{2}, k + \frac{1}{2})`))),":"),l("ul",null,l("li",null,l(e.k,null,ia(mn||(mn=la`P(X < k) \simeq P(Y \leq k - \frac{1}{2})`)))),l("li",null,l(e.k,null,ia(cn||(cn=la`P(X \leq k) \simeq P(Y \leq k + \frac{1}{2})`)))),l("li",null,l(e.k,null,ia(bn||(bn=la`P(X \geq k) \simeq P(Y \geq k - \frac{1}{2})`)))),l("li",null,l(e.k,null,ia(vn||(vn=la`P(X > k) \simeq P(Y \geq k + \frac{1}{2})`))))))),l(e.s,{title:"Vettori aleatori"},l(e.r,{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.k,null,ia(kn||(kn=la`\boldsymbol{X}`)))," oppure ",l(e.k,null,ia(gn||(gn=la`X, Y`))),".")),l(e.r,{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.k,null,ia(fn||(fn=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.k,null,ia(_n||(_n=la`F_X (x) = P(X \leq x) = \lim_{y \to +\infty} F_{X, Y} (x, y)`))))),l(e.r,{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.k,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.k,null,ia(zn||(zn=la`p_X (x) = \sum_j p_{X, Y} (x_i, y_j)`)))))),l(e.s,{title:"Più variabili aleatorie"},l(e.r,{title:"Indipendenza delle variabili aleatorie"},l("p",null,"Più variabili aleatorie sono indipendenti se, per qualsiasi scelta di intervalli ",l(e.k,null,"A_i"),":"),l("p",null,l(e.k,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.r,{title:"Media dei vettori aleatori"},l("p",null,"E' possibile calcolare la media di qualsiasi funzione ",l(e.k,null,"g(X, Y)")," avente elementi del vettore come variabili:"),l("p",null,l(e.k,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.k,null,"x \\cdot y"),"."),l("p",null,"Le medie di più variabili aleatorie si possono sommare:"),l("p",null,l(e.k,null,ia(En||(En=la`E(X + Y) = E(X) + E(Y)`)))))),l(e.s,null,l(e.r,{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.k,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.k,null,ia(Ln||(Ln=la`Cov(X, \alpha) = 0`)))),l("li",null,"E' ",l("b",null,"commutativa"),": ",l(e.k,null,ia(xn||(xn=la`Cov(X, Y) = Cov(Y, X)`)))),l("li",null,"E' ",l("b",null,"semplificabile"),": ",l(e.k,null,ia(yn||(yn=la`Cov(X, X) = Var(X)`)))),l("li",null,"E' ",l("b",null,"lineare"),": ",l(e.k,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.k,null,ia(Sn||(Sn=la`Cov(X + Y, V + W) = Cov(X, Y) + Cov(X, W) + Cov(Y, V) + Cov(Y, W)`)))))),l(e.r,{title:"Variabili incorrelate"},l("p",null,"Due variabili sono ",l("i",null,"variabili incorrelate")," se:"),l("p",null,l(e.k,null,ia(In||(In=la`Cov(X, Y) = 0`)))),l("p",null,"Variabili indipendenti sono sempre incorrelate.")),l(e.r,{title:"Matrice di covarianza"},l("p",null,"Una matrice ",l(e.k,null,ia(Cn||(Cn=la`\boldsymbol{C_X}`)))," che contiene la covarianza tra tutte le variabili di un vettore aleatorio ",l(e.k,null,ia(Nn||(Nn=la`\boldsymbol{X}`))),":"),l("p",null,l(e.k,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.k,null,"\\geq 0"),".")),l(e.r,{title:"Coefficiente di correlazione"},l("p",null,"Un valore che misura come due variabili aleatorie sono correlate:"),l("p",null,l(e.k,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.k,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.k,null,ia(Yn||(Yn=la`Y = a X + b \Longleftrightarrow | \rho_{X, Y} | = 1`))))),l(e.r,{title:"Varianza di variabili aleatorie sommate"},l("p",null,"La varianza di due variabili aleatorie sommate è:"),l("p",null,l(e.k,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.k,null,"X_i")," sono ",l("b",null,"indipendenti")," (",l(e.k,null,ia(Mn||(Mn=la`Cov(X, Y) = 0`))),"), allora:"),l("p",null,l(e.k,null,ia(Bn||(Bn=la`Var \left( \sum_i X_i \right) = \sum_i Var(X_i)`)))))),l(e.s,{title:"Campioni"},l(e.r,{title:"Campione casuale"},l("p",null,"Una ",l("b",null,"n-pla")," di variabili aleatorie con la stessa distribuzione della variabile aleatoria ",l(e.k,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.r,{title:"Momento campionario"},l("p",null,"Il valore dato dalla media aritmetica degli ",l(e.k,null,"n")," elementi del campione elevati alla potenza ",l(e.k,null,"k"),":"),l("p",null,l(e.k,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.k,null,ia(Tn||(Tn=la`\overline{X}_n`))),".")),l(e.r,{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.k,null,ia(Gn||(Gn=la`m = E(X)`)))," di X:"),l("p",null,l(e.k,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.k,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.s,{title:"Media-ception"},l(e.r,{title:"Media campionaria"},l("p",null,"Se calcoliamo la media della media campionaria, risulterà vero che:"),l("p",null,l(e.k,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.r,{title:"Varianza campionaria"},l("p",null,"Se calcoliamo la varianza della media campionaria, risulterà vero che:"),l("p",null,l(e.k,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.r,{title:"Correzione campionaria"},l("p",null,"Se calcoliamo la media della varianza campionaria, risulterà vero che:"),l("p",null,l(e.k,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.s,{title:"Campionamento di una distribuzione normale"},l(e.r,{title:"Campionamento di una distribuzione normale"},l("p",null,"Se la popolazione ",l(e.k,null,"X")," ha una distribuzione normale (",l(e.k,null,ia(Zn||(Zn=la`X \sim Nor(\mu, \sigma^2)`))),")...")),l(e.r,{title:"Distribuzione della media campionaria"},l("p",null,"...allora sappiamo anche la distribuzione della media campionaria!"),l("p",null,l(e.k,null,ia(Wn||(Wn=la`\overline{X}_n \sim Nor \left( \mu, \frac{\sigma^2}{n} \right)`))))),l(e.r,{title:"Distribuzione della varianza campionaria"},l("p",null,"...e anche della varianza campionaria!"),l("p",null,l(e.k,null,ia(Jn||(Jn=la`S_0^2 \sim \frac{\sigma^2}{n} \cdot \chi^2 (n)`)))),l("p",null,l(e.k,null,ia(Hn||(Hn=la`S_n^2 \sim \frac{\sigma^2}{n - 1} \cdot \chi^2 (n-1)`))))),l(e.r,{title:"Indipendenza"},l("p",null,"...e che media campionaria e varianza campionaria sono indipendenti tra loro!"))),l(e.s,{title:"Quando i campioni hanno dimensioni infinite"},l(e.r,{title:"Convergenza in distribuzione"},l("p",null,"Se la successione di variabili aleatorie ",l(e.k,null,"X_n")," all'infinito ha la ",l("b",null,"stessa funzione di ripartizione")," della popolazione ",l(e.k,null,"X"),", allora essa ",l("i",null,"converge in distribuzione"),"."),l("p",null,l(e.k,null,"\\lim_{n \\to +\\infty} F_{X_n} (x) = F_X (x) \\implies X_n \\xrightarrow{d} X"))),l(e.r,{title:"Convergenza in probabilità"},l("p",null,"Se la successione di variabili aleatorie ",l(e.k,null,"X_n")," all'infinito ha la ",l("b",null,"stessa probabilità")," della popolazione ",l(e.k,null,"X"),", allora essa ",l("i",null,"converge in probabilità"),"."),l("p",null,l(e.k,null,"\\forall \\epsilon > 0, \\lim_{n \\to +\\infty} P( | X_n - X | < \\epsilon) = 1 \\implies X_n \\xrightarrow{p} X"))),l(e.r,{title:"Convergenza quasi certa"},l("p",null,"Se la successione di variabili aleatorie ",l(e.k,null,"X_n")," all'infinito ha la ",l("b",null,"stessa probabilità a ")," della popolazione ",l(e.k,null,"X"),", allora essa ",l("i",null,"converge quasi certamente"),"."),l("p",null,l(e.k,null,"\\forall \\epsilon > 0, P left( \\lim_{n \\to +\\infty} | X_n - X | < \\epsilon) \right) = 1 \\implies X_n \\xrightarrow{qc} X"))),l(e.r,{title:"Convergenza in media quadratica"},l("p",null,"Se la successione di variabili aleatorie ",l(e.k,null,"X_n")," all'infinito ha la ",l("b",null,"media del quadrato della distanza")," tra la successione e la popolazione ",l(e.k,null,"X")," ",l("b",null,"uguale a 0"),", allora essa ",l("i",null,"converge in media quadratica"),"."),l("p",null,l(e.k,null,"\\lim_{n \\to +\\infty} E( | X_n - X |^2 = 0 \\implies X_n \\xrightarrow{mq} X"))),l(e.r,{title:"Gerarchia delle convergenze"},l("p",null,l(e.k,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.k,null,"X_n \\xrightarrow{p} x \\Longleftrightarrow X_n \\xrightarrow{d} x")))),l(e.s,{title:"I grandi numeri"},l(e.r,{title:"Legge debole dei grandi numeri"},l("p",null,"La successione delle medie campionarie ",l(e.k,null,ia($n||($n=la`\overline{X}_n`)))," ",l("b",null,"converge in probabilità")," alla media della popolazione ",l(e.k,null,ia(le||(le=la`E(X)`))),", se essa esiste."),l("p",null,l(e.k,null,"\\overline{X}_n \\xrightarrow{p} X")),l("p",null,"Ovvero:"),l("p",null,l(e.k,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.k,null,ia(ne||(ne=la`P( | \overline{X}_n - E(X) | < \epsilon) \to 1`))))),l(e.r,{title:"Legge forte dei grandi numeri"},l("p",null,"La successione delle medie campionarie ",l(e.k,null,ia(ee||(ee=la`\overline{X}_n`)))," ",l("b",null,"converge quasi certamente")," alla media della popolazione ",l(e.k,null,ia(ae||(ae=la`E(X)`))),", se essa esiste."),l("p",null,l(e.k,null,"\\overline{X}_n \\xrightarrow{qc} X")),l("p",null,"Ovvero:"),l("p",null,l(e.k,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.s,{title:"Al limite"},l(e.r,{title:"Teorema centrale del limite"},l("p",null,"La successione delle medie campionarie ",l(e.k,null,ia(te||(te=la`\overline{X}_n`)))," ",l("b",null,"converge in distribuzione")," a ",l(e.k,null,ia(ue||(ue=la`Nor(0, 1) = \Phi()`))),"."),l("p",null,l(e.k,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.k,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.s,{title:"Altre approsimazioni"},l(e.r,{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.k,null,ia(pe||(pe=la`Bin(n, p) \approx Nor(n \cdot p, n \cdot p \cdot q)`))))),l(e.r,{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.k,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.r,{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.k,null,ia(me||(me=la`Poi(\lambda) \approx Nor(\lambda, \lambda)`))))),l(e.r,{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.k,null,ia(ce||(ce=la`\Gamma (\alpha, \lambda) \approx Nor \left( \frac{\alpha}{\lambda}, \frac{\alpha}{\lambda^2} \right)`))))),l(e.r,{title:"In generale"},l("p",null,"Se ",l(e.k,null,"n")," è grande, allora:"),l("p",null,l(e.k,null,ia(be||(be=la`Y = \sum_{i=1}^{n} X_i`)))))),l(e.s,{title:"Actually statistica"},l(e.r,{title:"Parametri sconosciuti"},l("p",null,"Per indicare parametri sconosciuti di una legge si usa ",l(e.k,null,"\\theta"),".")),l(e.r,{title:"Statistica"},l("p",null,"Una variabile aleatoria funzione di un campione:"),l("p",null,l(e.k,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.k,null,ia(ke||(ke=la`T(\boldsymbol{X}) = \boldsymbol{X}`))),"."))),l(e.s,{title:"Stimatori"},l(e.r,{title:"Stimatore"},l("p",null,"Una statistica ",l(e.k,null,"T_n")," ottenuta da ",l(e.k,null,"n")," osservazioni, che stimi i parametri di una legge e sia indipendente da essi.")),l(e.r,{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.k,null,ia(ge||(ge=la`E(T_n) = \theta`))))),l(e.r,{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.k,null,ia(fe||(fe=la`\lim_{n \to +\infty} E(T_n) = \theta`))))),l(e.r,{title:"Consistente in media quadratica"},l("p",null,"Uno stimatore è ",l("i",null,"consistente in media quadratica")," se:"),l("p",null,l(e.k,null,ia(_e||(_e=la`\lim_{n \to +\infty} E((T_n - \theta)^2) = 0`))))),l(e.r,{title:"Consistente in probabilità"},l("p",null,"Uno stimatore è ",l("i",null,"consistente in probabilità")," se:"),l("p",null,l(e.k,null,ia(he||(he=la`\forall \epsilon > 0, \lim_{n \to +\infty} P( |T_n - \theta| < \epsilon) = 1`))))),l(e.r,{title:"Asintoticamente normale"},l("p",null,"Uno stimatore è ",l("i",null,"asintoticamente normale")," se:"),l("p",null,l(e.k,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.s,{title:"Metodo dei momenti"},l(e.r,{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.k,null,"X"),"."),l("p",null,"Lo stimatore di ",l(e.k,null,ia(Xe||(Xe=la`\theta`)))," così ottenuto sarà indicato aggiungendo un cappellino e una ",l(e.k,null,"M")," a ",l(e.k,null,"\\theta"),": ",l(e.k,null,ia(qe||(qe=la`\widehat{\theta}_M`)))),l("p",null,"Visto che:"),l("ul",null,l("li",null,l(e.k,null,ia(Ee||(Ee=la`\theta = g(E(X))`)))),l("li",null,l(e.k,null,ia(Pe||(Pe=la`\widehat{E(X)} = \overline{X}_n`))))),l("p",null,"Allora:"),l("p",null,l(e.k,null,ia(Le||(Le=la`\widehat{\theta}_M = g( \overline{X}_n )`)))),l("p",null,"Se ",l(e.k,null,ia(xe||(xe=la`\theta`)))," non è esprimibile in termini di ",l(e.k,null,ia(ye||(ye=la`E(X)`))),", si possono usare i momenti successivi ",l(e.k,null,ia(Fe||(Fe=la`M_n^2`))),", ",l(e.k,null,ia(Se||(Se=la`M_n^3`))),", ",l(e.k,null,ia(Ie||(Ie=la`M_n^3`))),"..."))),l(e.s,{title:"Metodo della massima verosomiglianza"},l(e.r,{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.k,null,"X"),"."),l("p",null,"Lo stimatore di ",l(e.k,null,ia(Ce||(Ce=la`\theta`)))," così ottenuto sarà indicato aggiungendo un cappellino e una ",l(e.k,null,"L")," a ",l(e.k,null,"\\theta"),": ",l(e.k,null,ia(Ne||(Ne=la`\widehat{\theta}_L`)))),l("p",null,"Consiste nel trovare il massimo assoluto ",l(e.k,null,ia(Ve||(Ve=la`\widehat{\theta}_L`)))," della la funzione di verosomiglianza ",l(e.k,null,ia(we||(we=la`L`))),":"),l("p",null,l(e.k,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.r,{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.k,null,ia(Ye||(Ye=la`\widehat{g(\theta)}_L = g(\widehat{\theta}_L)`))))))),l(e.s,{title:"Nuove stime notevoli"},l(e.r,{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.k,null,ia(Ue||(Ue=la`\widehat{p}_M = \widehat{p}_L = \overline{X}_n`))))),l(e.r,{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.k,null,ia(Me||(Me=la`\widehat{\mu}_M = \widehat{\mu}_L = \overline{X}_n`))))),l(e.r,{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.k,null,ia(Be||(Be=la`\widehat{\lambda}_M = \widehat{\lambda}_L = \frac{1}{\overline{X}_n}`))))),l(e.r,{title:"Stima di una normale"},l("p",null,"Per il metodo della massima verosomiglianza:"),l("ul",null,l("li",null,l(e.k,null,ia(Ae||(Ae=la`\widehat{\mu}_L = \overline{X}_n`)))),l("br",null),l("li",null,l(e.k,null,ia(Te||(Te=la`\widehat{\sigma^2}_L = \frac{\sum (X_i - \overline{X}_n)^2 }{n}`))))))),l(e.s,{title:"Intervalli di confidenza"},l(e.r,{title:"Confidenza"},l("blockquote",null,'"intervallo di confidenza al 95%"'),l("p",null,"L'intervallo di valori di ",l(e.k,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.k,null,ia(Ge||(Ge=la`\widehat{W}`)))," è l'intervallo ",l(e.k,null,"]a, b[")," tale che:"),l("p",null,l(e.k,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.s,{title:"Confidenza nella media di una normale"},l(e.r,{title:"Varianza nota"},l("p",null,"Se conosciamo la varianza di una normale, allora possiamo ricavare velocemente gli intervalli di confidenza all'",l(e.k,null,"\\alpha"),"% con queste formule:"),l("ul",null,l("li",null,"Intervalli bilateri: ",l(e.k,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.k,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.k,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.r,{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.k,null,"\\alpha"),"% con queste formule:"),l("ul",null,l("li",null,"Intervalli bilateri: ",l(e.k,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.k,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.k,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.k,null,ia(Je||(Je=la`t_{\alpha, v}`)))," è un quantile della distribuzione di Student di parametro ",l(e.k,null,"v"),"."))),l(e.s,{title:"Confidenza per la proporzione di una bernoulliana"},l(e.r,{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.k,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.s,{title:"Confidenza per la media di qualsiasi popolazione"},l(e.r,{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.k,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)}}]);
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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 return {props.children};\n}\n","import {Latex, LatexDefaultDisplay, LatexDefaultInline, LatexDisplay, Panel, Section} 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 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\n detta famiglia degli eventi.\n
\n
\n {r`\\mathcal{F}`}\n
\n
\n Qualsiasi sottoinsieme appartenente a {r`\\mathcal{F}`} è considerato un\n 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\n detta {r`\\sigma`}-algebra:\n
\n \n
\n Lo spazio campionario è un evento: {r`\\Omega \\in \\mathcal{F}`}\n
\n
\n Se un sottoinsieme è un evento, allora anche il suo complementare lo\n è: {r`E \\in \\mathcal{F} \\implies \\bar{E} \\in \\mathcal{F}`}\n
\n
\n Se due sottoinsiemi sono eventi, allora lo sono anche la loro unione e\n intersezione: {r`(E, F) \\in \\mathcal{F} \\implies (E \\cup F, E \\cap F) \\in \\mathcal{F}`}\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\n eventi {r`E_1`}, {r`E_2`}, {r`E_3`}, fino\n a {r`E_n`}.\n
\n \n Se lo spazio campionario fosse una torta, una sua partizione sarebbe l'insieme delle\n 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\n probabilità.\n
\n \n Sommando le probabilità dei due eventi, l'intersezione viene contata due volte, e va\n 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\n 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\n 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\n 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 Tramite la formula di Bayes possiamo risalire alla probabilità di un evento\n condizionato a un altro partendo dalla probabilità di quest'ultimo condizionato al\n primo:\n
\n Eventi indipendenti a due a due non sono per forza indipendenti a tre a tre, e\n viceversa.\n
\n \n \n
\n Un insieme di n eventi è una famiglia di eventi indipendenti se,\n 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\n 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\n campionario. {r`X(\\omega) : \\Omega \\to \\mathbb{R}`}.\n
\n \n Insieme di ripartizione}>\n
\n Ad ogni variabile aleatoria sono associati gli\n eventi {r`A_t = \\{ \\omega | X(\\omega) \\leq t \\}`}, che contengono tutti\n gli esiti a cui la variabile aleatoria associa un valore minore o uguale\n 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,\n 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\n aleatoria discretaX è la funzione che associa ad ogni esito la\n 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\n aleatoria continuaX è l'equivalente continuo della funzione\n probabilità:\n
\n A differenza della funzione probabilità, è possibile che la funzione densità non\n 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\n {r`F_X : \\mathbb{R} \\to [0, 1]`} associata, che rappresenta la\n probabilità che la variabile aleatoria assuma un valore minore o uguale\n a t:\n
\n
\n Si può dire che essa rappresenti la probabilità dell'evento {r`A_t`}:\n
\n Trasformare variabili aleatorie è molto utile nell'informatica per creare distribuzioni\n partendo da una funzione random() che\n 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\n finito ha anche una media (o valore medio o atteso):\n
\n Se la variabile aleatoria X ha media e varianza, allora la probabilità\n che essa abbia un valore a più di {r`\\epsilon`} di distanza dal valore\n medio è minore o uguale a {r`\\frac{Var(X)}{\\epsilon^2}`}.\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\n insuccessi N-K, conta quanti successi si otterrebbero se se ne\n estraessero n in blocco.\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
{r`E(X^2) = \\mu^2 + \\mu`}
\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\n di Poisson di intensità {r`\\lambda`} in un intervallo di tempo di\n durata t.\n
\n
\n E' una distribuzione poissoniana\n 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\n 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\n processo di Poisson di intensità {r`\\lambda`}.\n
\n La binomiale non è altro che una poissoniana a tempo discreto, quindi,\n se n è grande e n \\cdot p è nell'ordine di grandezza delle\n 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\n 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\n ogni valore discreto k la probabilità viene \"spalmata\" su tutto\n 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\n 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\n numero di parametri.\n
\n
\n Se il numero di parametri coincide con la dimensione del vettore aleatorio, allora la\n funzione sarà una funzione di ripartizione congiunta:\n
\n Una n-pla di variabili aleatorie con la stessa distribuzione della variabile\n aleatoria X (\"popolazione\") ma indipendenti tra loro.\n
\n \n Le variabili aleatorie sono come un lazy-load in programmazione; quando ci sarà bisogno\n 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\n elevati alla potenza k:\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\n funzione di ripartizione della popolazione X, allora essa converge\n in distribuzione.\n
\n Se la successione di variabili aleatorie X_n all'infinito ha la stessa\n probabilità a della popolazione X, allora essa converge quasi\n certamente.\n
\n Se la successione di variabili aleatorie X_n all'infinito ha la media\n del quadrato della distanza tra la successione e la popolazione Xuguale\n a 0, allora essa converge in media quadratica.\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\n 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\n 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\n 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\n valore atteso coincide con quello dei parametri che stima:\n
\n Se non conosciamo la varianza di una normale, allora possiamo ricavare velocemente gli\n intervalli di confidenza all'\\alpha% con queste formule:\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\";\n\nexport default function (props) {\n return (\n
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diff --git a/docs/route-VlDiGeometria.chunk.06e35.js b/docs/route-VlDiGeometria.chunk.06e35.js
new file mode 100644
index 0000000..718fc67
--- /dev/null
+++ b/docs/route-VlDiGeometria.chunk.06e35.js
@@ -0,0 +1,2 @@
+(window.webpackJsonp=window.webpackJsonp||[]).push([[16],{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.r,null,t(a.p,null,n(o()))))}}.call(this,e("hosL").h)}}]);
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+{"version":3,"sources":["webpack:///./routes/VlDiGeometria.js"],"names":["r","String","raw"],"mappings":"g1GAEMA,EAAIC,OAAOC,IAEF,qBACX,OACI,aACI,yCACA,EAAC,IAAD,KACI,EAAC,IAAD,KAAWF,EAAX,U","file":"route-VlDiGeometria.chunk.06e35.js","sourcesContent":["import {Markdown, Panel} from \"bluelib\";\n\nconst r = String.raw;\n\nexport default function (props) {\n return (\n
\n
Videolezioni di Geometria
\n \n {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 \n
\n );\n}\n"],"sourceRoot":""}
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diff --git a/docs/route-VlDiGeometria.chunk.ba9ee.esm.js b/docs/route-VlDiGeometria.chunk.ba9ee.esm.js
new file mode 100644
index 0000000..3d6ee39
--- /dev/null
+++ b/docs/route-VlDiGeometria.chunk.ba9ee.esm.js
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+(window.webpackJsonp=window.webpackJsonp||[]).push([[16],{UvCR:function(t,i,e){"use strict";e.r(i),function(t){var o=e("mbOI");let a,w=t=>t;const c=String.raw;i.default=function(){return t("div",null,t("h1",null,"Videolezioni di Geometria"),t(o.r,null,t(o.p,null,c(a||(a=w`
+ Tutte 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).
+
+ Per comodità, ho estratto l'url sorgente del video dall'embed presente nella rispettiva pagina.
+
+ 1. [Definizione di Spazio Vettoriale](https://www.youtube.com/watch?v=7eHEzf4403c) (1:17:29)
+ 2. [Sottospazi vettoriali I](https://www.youtube.com/watch?v=FPqrULk5HBU) (37:15)
+ 3. [Sottospazi vettoriali II](https://www.youtube.com/watch?v=ubDWUw9hk0k) (43:26)
+ 4. [Sottospazi vettoriali III](https://www.youtube.com/watch?v=381n4NPb6Oc) (40:29)
+ 5. [Lineare dipendenza e indipendenza](https://www.youtube.com/watch?v=9YVQ5olYrh0) (56:12)
+ 6. [Basi di uno spazio vettoriale I](https://www.youtube.com/watch?v=mEF_lcTzEoE) (25:52)
+ 7. [Basi di uno spazio vettoriale II](https://www.youtube.com/watch?v=k1r9JfXY53k) (48:24)
+ 8. [Teorema di Grassmann](https://www.youtube.com/watch?v=3sqB-MMyCWM) (32:36)
+ 9. [Basi e Matrici](https://www.youtube.com/watch?v=Rd6AB_jE7YI) (27:06)
+ 10. [Definizione di Applicazioni Lineari](https://www.youtube.com/watch?v=rmd7ffZeVYk) (16:23)
+ 11. [Proprietà delle Applicazioni Lineari](https://www.youtube.com/watch?v=MH7ztQGkqmw) (31:58)
+ 12. [Definizione di determinante](https://www.youtube.com/watch?v=EwubcLwBdzk) (36:43)
+ 13. [Proprietà e metodo di triangolazione](https://www.youtube.com/watch?v=SFusGarV6HI) (22:36)
+ 14. [Teorema di Laplace](https://www.youtube.com/watch?v=BqZDWnKl2nQ) (29:03)
+ 15. [4 applicazioni del Teorema di Laplace](https://www.youtube.com/watch?v=2tr3y725GY0) (47:53)
+ 16. [Spazi vettoriali euclidei reali - Parte 1](https://www.youtube.com/watch?v=W7Z1hm-jwMM) (28:46)
+ 17. [Spazi vettoriali euclidei reali - Parte 2](https://www.youtube.com/watch?v=zjmKE9TMGm8) (27:17)
+ 18. [Autovalori e autovettori](https://www.youtube.com/watch?v=XlrlcnvcTtQ) (33:00)
+ 19. [Polinomio caratteristico](https://www.youtube.com/watch?v=61icRbgWTdI) (31:31)
+ 20. [Teorema diagonalizzabilità](https://www.youtube.com/watch?v=wm5V6en9OFo) (18:49)
+ 21. [Spazi affini](https://player.vimeo.com/video/291457587) (20:46)
+ 22. [Sottospazi affini](https://player.vimeo.com/video/291458991) (21:32)
+ 23. [Parallelismo e Riferimenti Affini](https://player.vimeo.com/video/291510181) (16:57)
+ 24. [Rappresentazione di Sottospazi Affini](https://player.vimeo.com/video/291510296) (31:17)
+ 25. [Spazi Euclidei](https://player.vimeo.com/video/291510612) (35:57)
+ 26. [Teoria dei ranghi](https://player.vimeo.com/video/291510964) (9:44)
+ 27. [Teoria dei ranghi 2](https://player.vimeo.com/video/291510862) (14:44)
+
+ Nell'anno accademico 2018/2019 non sono stati trattati gli argomenti nei video 21, 22 e 23.
+ `)))))}}.call(this,e("hosL").h)}}]);
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Videolezioni di Geometria
\n \n {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 \n
\n );\n}\n"],"sourceRoot":""}
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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
',showdown:"S"},s.Converter=function(e){"use strict";function t(e,t){if(t=t||null,s.helper.isString(e)){if(t=e=s.helper.stdExtName(e),s.extensions[e])return console.warn("DEPRECATION WARNING: "+e+" is an old extension that uses a deprecated loading method.Please inform the developer that the extension should be updated!"),void function(e,t){"function"==typeof e&&(e=e(new s.Converter));s.helper.isArray(e)||(e=[e]);var r=a(e,t);if(!r.valid)throw Error(r.error);for(var n=0;n![\"\"/](\"https://i.imgur.com/mDZSqjV.png\")
\n ![\"\"/](\"https://i.imgur.com/aPTwrxz.png\")
\n ![\"\"/](\"https://i.imgur.com/Lq9IepY.png\")
\n ![\"\"/](\"https://i.imgur.com/dy3b5Ub.png\")
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\n ![Questa]({\"https://i.imgur.com/1r405Mb.jpg\"}){kind=link}
![Esempi\n su Wikipedia]({\"https://it.wikipedia.org/wiki/Norma_(matematica)#/media/File:Vector_norms.svg\"}){kind=link}