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These can be computed directly using the ExpectedValue \ function." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(evs = Table[ExpectedValue[x^i, ChiSquareDistribution[n], x], {i, 4}]\)], "Input", CellLabel->"In[2]:="], Cell[BoxData[ \({n, n\ \((2 + n)\), n\ \((2 + n)\)\ \((4 + n)\), n\ \((2 + n)\)\ \((4 + n)\)\ \((6 + n)\)}\)], "Output", CellLabel->"Out[2]="] }, Open ]], Cell["\<\ Alternately, the moments could be could be obtained from a general \ result for expected values for positive powers of x.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(expect = ExpectedValue[x^i, ChiSquareDistribution[n], x, Assumptions \[Rule] {i > 0}]\)], "Input", CellLabel->"In[3]:="], Cell[BoxData[ \(\(2\^i\ Gamma[i + n\/2]\)\/Gamma[n\/2]\)], "Output", CellLabel->"Out[3]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(FullSimplify[expect /. i \[Rule] Range[4], Assumptions \[Rule] n > 0]\)], "Input", CellLabel->"In[4]:="], Cell[BoxData[ \({n, n\ \((2 + n)\), n\ \((2 + n)\)\ \((4 + n)\), n\ 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Transpose[sigma]; \[IndentingNewLine]mu = Table[Random[Real, {\(-5\), 5}], {5}]];\)\)], "Input", CellLabel->"In[19]:="], Cell[CellGroupData[{ Cell[BoxData[ \(sigma // MatrixForm\)], "Input", CellLabel->"In[20]:="], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"7.8409332081884395`", \(-7.0099676557425346`\), "13.295425039151258`", \(-3.176769822892843`\), \ \(-17.65560566126925`\)}, {\(-7.0099676557425346`\), "12.242415913915675`", \(-15.672576741984152`\), "3.596184878824712`", "20.19636334624787`"}, {"13.295425039151258`", \(-15.672576741984152`\), "29.207883514572586`", \(-5.131387377839062`\), \ \(-40.121123189332124`\)}, {\(-3.176769822892843`\), "3.596184878824712`", \(-5.131387377839062`\), "1.6940181192828387`", "6.610135772733947`"}, {\(-17.65560566126925`\), "20.19636334624787`", \(-40.121123189332124`\), "6.610135772733947`", "62.215248152176734`"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output", CellLabel->"Out[20]//MatrixForm="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mu\)], "Input", CellLabel->"In[21]:="], Cell[BoxData[ \({3.029566715817033`, \(-0.07706509861315691`\), \ \(-1.7009064176260722`\), \(-2.515341415883476`\), 4.247111061833191`}\)], "Output", CellLabel->"Out[21]="] }, Open ]], Cell["\<\ Here are 21 cdf values computed for this distribution. Results \ tend to be pretty fast for lower PrecisionGoal values. Sharp features in the \ integrand near the boundaries of the integral can make computation for higher \ dimensional integrals at the default PrecisionGoal much slower.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Timing[ cdfvals = Table[{x, CDF[MultinormalDistribution[mu, sigma], {8, x, 10, 0, 17}, PrecisionGoal \[Rule] 3]}, {x, \(-10\), 10, 1}]]\)], "Input", CellLabel->"In[22]:="], Cell[BoxData[ \({10.578000000000001`\ Second, {{\(-10\), 0.0003628816710511019`}, {\(-9\), 0.0013653252560795921`}, {\(-8\), 0.004307285416097439`}, {\(-7\), 0.011573792532762243`}, {\(-6\), 0.026922573214432123`}, {\(-5\), 0.05510691919485064`}, {\(-4\), 0.10077594145467546`}, {\(-3\), 0.16692954095565088`}, {\(-2\), 0.25336583974317783`}, {\(-1\), 0.35602304924607514`}, {0, 0.46725391634542296`}, {1, 0.5770740457850099`}, {2, 0.6761415046738234`}, {3, 0.7569444964205446`}, {4, 0.8157563221855053`}, {5, 0.8537734011259673`}, {6, 0.874992765825916`}, {7, 0.8851036131858028`}, {8, 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This includes clustering of numeric, Boolean, and string data." }], "Text"], Cell["Here are ten sets of bivariate normal data points.", "Text"], Cell[BoxData[ \(\(origclusters = Block[{$RandomState, muvals, sigmavals}, SeedRandom[1234546789]; \[IndentingNewLine]muvals = Table[Random[ Real, {\(-10\), 10}], {10}, {2}]; \[IndentingNewLine]sigmavals = Table[Block[{s1 = Random[], s2 = Random[], s12}, \[IndentingNewLine]s12 = Random[]*Sqrt[s1*s2]; \[IndentingNewLine]{{s1, s12}, {s12, s2}}], {10}]; \[IndentingNewLine]Join[ MapThread[ RandomArray[MultinormalDistribution[#1, #2], Random[Integer, {20, 75}]] &, {muvals, sigmavals}]]];\)\)], "Input", CellLabel->"In[29]:="], Cell["This function will differentiate lists of points by color.", "Text"], Cell[BoxData[ \(\(ClusterPlot[cl_, opts___] := Module[{colors, len = Length[cl]}, \[IndentingNewLine]colors = Map[Hue, Range[1, len]/\((len + 1)\)]; \[IndentingNewLine]Show[ Graphics[{AbsolutePointSize[5], Transpose[{colors, Map[Thread[Point /@ #] &, 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Cell[StyleData["SO"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["SB"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain"], Cell[StyleData["SBO"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Italic"], Cell[CellGroupData[{ Cell[StyleData["SO10"], StyleMenuListing->None, FontFamily->"Helvetica", FontSize->10, FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["SO10", "Printout"], StyleMenuListing->None, FontFamily->"Helvetica", FontSize->7, FontWeight->"Plain", FontSlant->"Italic"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Formulas and Programming", "Section"], Cell[CellGroupData[{ Cell[StyleData["InlineFormula"], CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", ScriptLevel->1, SingleLetterItalics->True], Cell[StyleData["InlineFormula", "Presentation"], LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["InlineFormula", "SlideShow"], LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["InlineFormula", "Printout"], CellMargins->{{2, 0}, {6, 6}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["DisplayFormula"], CellMargins->{{60, Inherited}, {Inherited, Inherited}}, CellHorizontalScrolling->True, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", ScriptLevel->0, SingleLetterItalics->True, UnderoverscriptBoxOptions->{LimitsPositioning->True}], Cell[StyleData["DisplayFormula", "Presentation"], LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["DisplayFormula", "SlideShow"], CellMargins->{{100, 50}, {Inherited, Inherited}}, LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["DisplayFormula", "Printout"], FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Program"], CellFrame->{{0, 0}, {0.5, 0.5}}, CellMargins->{{60, 4}, {0, 8}}, CellHorizontalScrolling->True, Hyphenation->False, LanguageCategory->"Formula", ScriptLevel->1, FontFamily->"Courier"], Cell[StyleData["Program", "Presentation"], CellMargins->{{24, 50}, {10, 10}}, LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["Program", "SlideShow"], CellMargins->{{100, 50}, {10, 10}}, LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["Program", "Printout"], CellMargins->{{2, 0}, {6, 6}}, FontSize->9] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Outline Styles", "Section"], Cell[CellGroupData[{ Cell[StyleData["Outline1"], CellMargins->{{60, 10}, {7, 7}}, CellGroupingRules->{"SectionGrouping", 50}, ParagraphIndent->-38, CounterIncrements->"Outline1", CounterAssignments->{{"Outline2", 0}, {"Outline3", 0}, {"Outline4", 0}}, FontSize->18, FontWeight->"Bold", CounterBoxOptions->{CounterFunction:>CapitalRomanNumeral}], Cell[StyleData["Outline1", "SlideShow"]], Cell[StyleData["Outline1", "Printout"], CounterBoxOptions->{CounterFunction:>CapitalRomanNumeral}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Outline2"], CellMargins->{{90, 10}, {7, 7}}, CellGroupingRules->{"SectionGrouping", 60}, ParagraphIndent->-27, CounterIncrements->"Outline2", CounterAssignments->{{"Outline3", 0}, {"Outline4", 0}}, FontSize->15, FontWeight->"Bold", CounterBoxOptions->{CounterFunction:>(Part[ CharacterRange[ "A", "Z"], #]&)}], Cell[StyleData["Outline2", "SlideShow"]], Cell[StyleData["Outline2", "Printout"], CounterBoxOptions->{CounterFunction:>(Part[ CharacterRange[ "A", "Z"], #]&)}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Outline3"], CellMargins->{{120, 10}, {7, 7}}, CellGroupingRules->{"SectionGrouping", 70}, ParagraphIndent->-21, CounterIncrements->"Outline3", CounterAssignments->{{"Outline4", 0}}, FontSize->12, CounterBoxOptions->{CounterFunction:>Identity}], Cell[StyleData["Outline3", "SlideShow"]], Cell[StyleData["Outline3", "Printout"], CounterBoxOptions->{CounterFunction:>Identity}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Outline4"], CellMargins->{{150, 10}, {7, 7}}, CellGroupingRules->{"SectionGrouping", 80}, ParagraphIndent->-18, CounterIncrements->"Outline4", FontSize->10, CounterBoxOptions->{CounterFunction:>(Part[ CharacterRange[ "a", "z"], #]&)}], Cell[StyleData["Outline4", "SlideShow"]], Cell[StyleData["Outline4", "Printout"]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Hyperlink Styles", "Section"], Cell["\<\ The cells below define styles useful for making hypertext \ ButtonBoxes. The \"Hyperlink\" style is for links within the same Notebook, \ or between Notebooks.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Hyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonFrame->"None", ButtonNote->ButtonData}], Cell[StyleData["Hyperlink", "Presentation"], FontSize->16], Cell[StyleData["Hyperlink", "SlideShow"]], Cell[StyleData["Hyperlink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell["\<\ The following styles are for linking automatically to the on-line \ help system.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["MainBookLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "MainBook", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["MainBookLink", "Presentation"], FontSize->16], Cell[StyleData["MainBookLink", "SlideShow"]], Cell[StyleData["MainBookLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["AddOnsLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "AddOns", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["AddOnsLink", "Presentation"], FontSize->16], Cell[StyleData["AddOnsLink", "SlideShow"]], Cell[StyleData["AddOnsLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["RefGuideLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "RefGuide", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["RefGuideLink", "Presentation"], FontSize->16], Cell[StyleData["RefGuideLink", "SlideShow"]], Cell[StyleData["RefGuideLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["RefGuideLinkText"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "RefGuide", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["RefGuideLinkText", "Presentation"], FontSize->16], Cell[StyleData["RefGuideLinkText", "SlideShow"]], Cell[StyleData["RefGuideLinkText", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["GettingStartedLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "GettingStarted", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["GettingStartedLink", "Presentation"], FontSize->16], Cell[StyleData["GettingStartedLink", "SlideShow"]], Cell[StyleData["GettingStartedLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["DemosLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "Demos", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["DemosLink", "SlideShow"]], Cell[StyleData["DemosLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["TourLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "Tour", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["TourLink", "SlideShow"]], Cell[StyleData["TourLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["OtherInformationLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "OtherInformation", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["OtherInformationLink", "Presentation"], FontSize->16], Cell[StyleData["OtherInformationLink", "SlideShow"]], Cell[StyleData["OtherInformationLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["MasterIndexLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "MasterIndex", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["MasterIndexLink", "SlideShow"]], Cell[StyleData["MasterIndexLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Headers and Footers", "Section"], Cell[StyleData["Header"], CellMargins->{{0, 0}, {4, 1}}, DefaultNewInlineCellStyle->"None", LanguageCategory->"NaturalLanguage", StyleMenuListing->None, FontSize->10, FontSlant->"Italic"], Cell[StyleData["Footer"], CellMargins->{{0, 0}, {0, 4}}, DefaultNewInlineCellStyle->"None", LanguageCategory->"NaturalLanguage", StyleMenuListing->None, FontSize->9, FontSlant->"Italic"], Cell[StyleData["PageNumber"], CellMargins->{{0, 0}, {4, 1}}, StyleMenuListing->None, FontFamily->"Times", FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell["Palette Styles", "Section"], Cell["\<\ The cells below define styles that define standard \ ButtonFunctions, for use in palette buttons.\ \>", "Text"], Cell[StyleData["Paste"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, Placeholder]}]&)}], Cell[StyleData["Evaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["EvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionMove[ FrontEnd`InputNotebook[ ], All, Cell, 1], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}] }, Closed]], Cell[CellGroupData[{ Cell["Placeholder Styles", "Section"], Cell["\<\ The cells below define styles useful for making placeholder \ objects in palette templates.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Placeholder"], Placeholder->True, StyleMenuListing->None, FontSlant->"Italic", FontColor->RGBColor[0.890623, 0.864698, 0.384756], TagBoxOptions->{Editable->False, Selectable->False, StripWrapperBoxes->False}], Cell[StyleData["Placeholder", "Presentation"]], Cell[StyleData["Placeholder", "SlideShow"]], Cell[StyleData["Placeholder", "Printout"]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["PrimaryPlaceholder"], StyleMenuListing->None, DrawHighlighted->True, FontSlant->"Italic", Background->RGBColor[0.912505, 0.891798, 0.507774], TagBoxOptions->{Editable->False, Selectable->False, StripWrapperBoxes->False}], Cell[StyleData["PrimaryPlaceholder", "Presentation"]], Cell[StyleData["PrimaryPlaceholder", "SlideShow"]], Cell[StyleData["PrimaryPlaceholder", "Printout"]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["FormatType Styles", "Section"], Cell["\<\ The cells below define styles that are mixed in with the styles \ of most cells. If a cell's FormatType matches the name of one of the styles \ defined below, then that style is applied between the cell's style and its \ own options. This is particularly true of Input and Output.\ \>", "Text"], Cell[StyleData["CellExpression"], PageWidth->Infinity, CellMargins->{{6, Inherited}, {Inherited, Inherited}}, ShowCellLabel->False, ShowSpecialCharacters->False, AllowInlineCells->False, Hyphenation->False, AutoItalicWords->{}, StyleMenuListing->None, FontFamily->"Courier", FontSize->12, Background->GrayLevel[1]], Cell[StyleData["InputForm"], InputAutoReplacements->{}, AllowInlineCells->False, Hyphenation->False, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["OutputForm"], PageWidth->Infinity, TextAlignment->Left, LineSpacing->{0.6, 1}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["StandardForm"], InputAutoReplacements->{ "->"->"\[Rule]", ":>"->"\[RuleDelayed]", "<="->"\[LessEqual]", ">="->"\[GreaterEqual]", "!="->"\[NotEqual]", "=="->"\[Equal]", Inherited}, "TwoByteSyntaxCharacterAutoReplacement"->True, LineSpacing->{1.25, 0}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["TraditionalForm"], InputAutoReplacements->{ "->"->"\[Rule]", ":>"->"\[RuleDelayed]", "<="->"\[LessEqual]", ">="->"\[GreaterEqual]", "!="->"\[NotEqual]", "=="->"\[Equal]", Inherited}, "TwoByteSyntaxCharacterAutoReplacement"->True, LineSpacing->{1.25, 0}, SingleLetterItalics->True, TraditionalFunctionNotation->True, DelimiterMatching->None, StyleMenuListing->None], Cell["\<\ The style defined below is mixed in to any cell that is in an \ inline cell within another.\ \>", "Text"], Cell[StyleData["InlineCell"], LanguageCategory->"Formula", ScriptLevel->1, StyleMenuListing->None], Cell[StyleData["InlineCellEditing"], StyleMenuListing->None, Background->RGBColor[0.964706, 0.929412, 0.839216]] }, Closed]], Cell[CellGroupData[{ Cell["Automatic Styles", "Section"], Cell["\<\ The cells below define styles that are used to affect the display \ of certain types of objects in typeset expressions. For example, \ \"UnmatchedBracket\" style defines how unmatched bracket, curly bracket, and \ parenthesis characters are displayed (typically by coloring them to make them \ stand out).\ \>", "Text"], Cell[StyleData["UnmatchedBracket"], StyleMenuListing->None, FontColor->RGBColor[0.760006, 0.330007, 0.8]], Cell[StyleData["Completions"], StyleMenuListing->None, FontFamily->"Courier"] }, Closed]], Cell[CellGroupData[{ Cell["Styles from HelpBrowser", "Section"], Cell[CellGroupData[{ Cell[StyleData["MathCaption"], CellFrame->{{0, 0}, {0, 0.5}}, CellMargins->{{66, 12}, {2, 24}}, PageBreakBelow->False, CellFrameMargins->{{8, 8}, {8, 2}}, CellFrameColor->GrayLevel[0.700008], CellFrameLabelMargins->4, LineSpacing->{1, 1}, ParagraphSpacing->{0, 8}, StyleMenuListing->None, FontColor->GrayLevel[0.2]], Cell[StyleData["MathCaption", "Presentation"], FontSize->18], Cell[StyleData["MathCaption", "SlideShow"], CellMargins->{{100, 50}, {Inherited, Inherited}}], Cell[StyleData["MathCaption", "Printout"], CellMargins->{{39, 0}, {0, 14}}, Hyphenation->True, FontSize->9, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["ObjectName"], ShowCellBracket->True, CellMargins->{{66, 4}, {8, 8}}, Evaluatable->True, CellGroupingRules->"InputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, CellLabelAutoDelete->False, CellLabelMargins->{{14, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, ShowSpecialCharacters->Automatic, "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Mathematica", FormatType->StandardForm, ShowStringCharacters->True, NumberMarks->True, StyleMenuListing->None, FontWeight->"Bold"], Cell[StyleData["ObjectName", "Presentation"], FontSize->18], Cell[StyleData["ObjectName", "SlideShow"], CellMargins->{{100, 50}, {Inherited, Inherited}}], Cell[StyleData["ObjectName", "Printout"], ShowCellBracket->False, CellMargins->{{39, 0}, {6, 6}}, FontSize->9] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Usage"], ShowCellBracket->True, CellMargins->{{66, 4}, {8, 8}}, Evaluatable->True, CellGroupingRules->"InputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, CellLabelAutoDelete->False, CellLabelMargins->{{14, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, ShowSpecialCharacters->Automatic, "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Mathematica", FormatType->StandardForm, ShowStringCharacters->True, NumberMarks->True, StyleMenuListing->None, FontWeight->"Bold"], Cell[StyleData["Usage", "Presentation"], FontSize->18], Cell[StyleData["Usage", "SlideShow"], CellMargins->{{100, 50}, {Inherited, Inherited}}], Cell[StyleData["Usage", "Printout"], ShowCellBracket->False, CellMargins->{{39, 0}, {6, 6}}, FontSize->9] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Notes"], ShowCellBracket->True, CellMargins->{{66, 4}, {8, 8}}, Evaluatable->True, CellGroupingRules->"InputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, CellLabelAutoDelete->False, CellLabelMargins->{{14, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, ShowSpecialCharacters->Automatic, "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Mathematica", FormatType->StandardForm, ShowStringCharacters->True, NumberMarks->True, StyleMenuListing->None, FontWeight->"Bold"], Cell[StyleData["Notes", "Presentation"], FontSize->18], Cell[StyleData["Notes", "SlideShow"], CellMargins->{{100, 50}, {Inherited, Inherited}}], Cell[StyleData["Notes", "Printout"], ShowCellBracket->False, CellMargins->{{39, 0}, {6, 6}}, FontSize->9] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["InlineOutput"], ShowCellBracket->True, CellMargins->{{66, 4}, {8, 8}}, Evaluatable->True, CellGroupingRules->"InputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, CellLabelAutoDelete->False, CellLabelMargins->{{14, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, ShowSpecialCharacters->Automatic, "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Mathematica", FormatType->StandardForm, ShowStringCharacters->True, NumberMarks->True, StyleMenuListing->None, FontWeight->"Bold"], Cell[StyleData["InlineOutput", "Presentation"], FontSize->18], Cell[StyleData["InlineOutput", "SlideShow"], CellMargins->{{100, 50}, {Inherited, Inherited}}], Cell[StyleData["InlineOutput", "Printout"], ShowCellBracket->False, CellMargins->{{39, 0}, {6, 6}}, FontSize->9] }, Closed]], Cell[CellGroupData[{ Cell["Emphasis Boxes and Pictures", "Subsection"], Cell[CellGroupData[{ Cell[StyleData["Box"], CellFrame->0.5, CellMargins->{{27, 12}, {0, 8}}, CellHorizontalScrolling->True, CellFrameColor->RGBColor[0.74902, 0.694118, 0.552941], StyleMenuListing->None, Background->RGBColor[0.964706, 0.929412, 0.839216], FrameBoxOptions->{BoxFrame->0.5, BoxMargins->True}, GridBoxOptions->{ColumnSpacings->1}], Cell[StyleData["Box", "Presentation"], FontSize->18], Cell[StyleData["Box", "SlideShow"], CellMargins->{{100, 50}, {Inherited, Inherited}}], Cell[StyleData["Box", "Printout"], CellMargins->{{2, 0}, {0, 8}}, FontSize->10, Background->GrayLevel[0.900008]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["DoubleBox"], CellFrame->0.5, CellMargins->{{27, 12}, {0, 8}}, CellHorizontalScrolling->True, CellFrameColor->RGBColor[0.74902, 0.694118, 0.552941], StyleMenuListing->None, Background->RGBColor[0.964706, 0.929412, 0.839216], FrameBoxOptions->{BoxFrame->0.5, BoxMargins->True}, GridBoxOptions->{ColumnSpacings->2, RowAlignments->Top}], Cell[StyleData["DoubleBox", "Presentation"], FontSize->18], Cell[StyleData["DoubleBox", "SlideShow"], CellMargins->{{100, 50}, {Inherited, Inherited}}], Cell[StyleData["DoubleBox", "Printout"], CellMargins->{{2, 0}, {0, 8}}, FontSize->10, Background->GrayLevel[0.900008]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["1ColumnBox"], CellFrame->0.5, CellMargins->{{27, 12}, {0, 8}}, CellHorizontalScrolling->True, CellFrameColor->RGBColor[0.74902, 0.694118, 0.552941], LineIndent->0, StyleMenuListing->None, Background->RGBColor[0.964706, 0.929412, 0.839216], FrameBoxOptions->{BoxFrame->0.5, BoxMargins->True}, GridBoxOptions->{ColumnSpacings->1}], Cell[StyleData["1ColumnBox", "Presentation"], FontSize->18], Cell[StyleData["1ColumnBox", "SlideShow"], CellMargins->{{100, 50}, {Inherited, Inherited}}], Cell[StyleData["1ColumnBox", "Printout"], CellMargins->{{2, 0}, {0, 8}}, FontSize->10, Background->GrayLevel[0.900008]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["2ColumnBox"], CellFrame->0.5, CellMargins->{{27, 12}, {0, 8}}, CellHorizontalScrolling->True, CellFrameColor->RGBColor[0.74902, 0.694118, 0.552941], SingleLetterItalics->False, LineIndent->0, StyleMenuListing->None, Background->RGBColor[0.964706, 0.929412, 0.839216], FrameBoxOptions->{BoxFrame->0.5, BoxMargins->True}, GridBoxOptions->{ColumnWidths->{0.31, 0.67}}], Cell[StyleData["2ColumnBox", "Presentation"], FontSize->18], Cell[StyleData["2ColumnBox", "SlideShow"], CellMargins->{{100, 50}, {Inherited, Inherited}}], Cell[StyleData["2ColumnBox", "Printout"], CellMargins->{{2, 0}, {0, 8}}, FontSize->9, Background->GrayLevel[0.900008]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["2ColumnEvenBox"], CellFrame->0.5, CellMargins->{{27, 12}, {0, 8}}, CellHorizontalScrolling->True, CellFrameColor->RGBColor[0.74902, 0.694118, 0.552941], LineIndent->0, StyleMenuListing->None, Background->RGBColor[0.964706, 0.929412, 0.839216], FrameBoxOptions->{BoxFrame->0.5, BoxMargins->True}, GridBoxOptions->{ColumnWidths->0.46}], Cell[StyleData["2ColumnEvenBox", "Presentation"], FontSize->18], Cell[StyleData["2ColumnEvenBox", "SlideShow"], 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