(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 10.3' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 3561316, 60090] NotebookOptionsPosition[ 3555714, 59908] NotebookOutlinePosition[ 3556062, 59923] CellTagsIndexPosition[ 3556019, 59920] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Gaussian Unitary Distribution (GUE) Matrices", "Title", CellChangeTimes->{{3.6620874648774157`*^9, 3.662087477495829*^9}, { 3.7251622895282545`*^9, 3.7251623112162085`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"<<", "BProbeM`", " ", RowBox[{"(*", " ", RowBox[{"load", " ", "the", " ", "package", " ", "BProbe"}], " ", "*)"}]}]], "Input", CellChangeTimes->{{3.662087507390511*^9, 3.662087525708716*^9}, { 3.66208799411233*^9, 3.662088003651579*^9}, {3.662090594100902*^9, 3.662090597016807*^9}, 3.7251626144587803`*^9}], Cell[BoxData["\<\"Loaded BProbeM. See the \ \\!\\(\\*TagBox[ButtonBox[PaneSelectorBox[{False -> \\\"\\\\\\\"documentation\ \\\\\\\"\\\", Rule[True, StyleBox[\\\"\\\\\\\"documentation\\\\\\\"\\\", \ List[\\\"HyperlinkActive\\\"]]]}, Dynamic[CurrentValue[\\\"MouseOver\\\"]], \ Rule[BaseStyle, List[\\\"Hyperlink\\\"]], Rule[FrameMargins, 0], \ Rule[ImageSize, Automatic]], Rule[BaseStyle, \\\"Hyperlink\\\"], \ Rule[ButtonData, \ List[URL[\\\"file://C:\\\\\\\\Users\\\\\\\\Timon\\\\\\\\AppData\\\\\\\\\ Roaming\\\\\\\\Mathematica\\\\\\\\Applications\\\\\\\\BProbeM\\\\\\\\\ Documentation\\\\\\\\index.html\\\"], None]], Rule[ButtonNote, \ \\\"file://C:\\\\\\\\Users\\\\\\\\Timon\\\\\\\\AppData\\\\\\\\Roaming\\\\\\\\\ Mathematica\\\\\\\\Applications\\\\\\\\BProbeM\\\\\\\\Documentation\\\\\\\\\ index.html\\\"]], Function[Annotation[Slot[1], \\\"file://C:\\\\\\\\Users\\\\\ \\\\Timon\\\\\\\\AppData\\\\\\\\Roaming\\\\\\\\Mathematica\\\\\\\\\ Applications\\\\\\\\BProbeM\\\\\\\\Documentation\\\\\\\\index.html\\\", \ \\\"Hyperlink\\\"]]]\\) for help.\"\>"], "Print", CellChangeTimes->{ 3.6620875278291693`*^9, {3.7251626153197823`*^9, 3.7251626291157827`*^9}, { 3.7251663600984383`*^9, 3.7251663729744363`*^9}}] }, Open ]], Cell["\<\ Let Mathematica generate a specified amount of random Matrices of specified \ dimension distributed according to the Gaussian Unitary Distribution:\ \>", "Text", CellChangeTimes->{{3.662087529573331*^9, 3.66208766648717*^9}, { 3.662087960433777*^9, 3.662087961109517*^9}, {3.725162322231852*^9, 3.725162343934837*^9}, {3.7251625833687778`*^9, 3.72516258667778*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", RowBox[{ "Generate", " ", "5", "x10x10", " ", "random", " ", "GUE", " ", "matrices"}], "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"RandoMatrices", "=", RowBox[{"ConstantArray", "[", RowBox[{"0", ",", RowBox[{"{", "5", "}"}]}], "]"}]}], ";"}], "\n", RowBox[{ RowBox[{"For", "[", RowBox[{ RowBox[{"i", "=", "1"}], ",", RowBox[{"i", "\[LessEqual]", "5"}], ",", RowBox[{"i", "++"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"RandoMatrices", "[", RowBox[{"[", "i", "]"}], "]"}], "=", RowBox[{"RandomVariate", "[", RowBox[{"GaussianUnitaryMatrixDistribution", "[", "10", "]"}], "]"}]}], ";"}]}], "\[IndentingNewLine]", "]"}], ";"}]}]}]], "Input", CellChangeTimes->{{3.662087642368493*^9, 3.662087685376998*^9}, { 3.662087936734487*^9, 3.6620879745921907`*^9}, {3.7251625623887844`*^9, 3.725162573098775*^9}, {3.7251633718257756`*^9, 3.7251633753357353`*^9}}], Cell[TextData[{ "Please note that these matrices are ", StyleBox["Hermitian", FontSlant->"Italic"], " and not unitary, the \[OpenCurlyQuote]unitary\[CloseCurlyQuote] in the \ name does not refer to the resulting matrices.\nSee a textbook on Random \ Matrices such as ", StyleBox["(Mehta, M.L., 2004. 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The Global \ minimum calculated by ProbeInit above is where the combined energies of all these directions is minimal but given \ what we see in these scans, we will likely primarily be interested in the structure around the origin.\ \>", "Text", CellChangeTimes->{{3.662087529573331*^9, 3.66208766648717*^9}, { 3.662087960433777*^9, 3.662087961109517*^9}, {3.725162322231852*^9, 3.725162527052035*^9}, {3.725162671907795*^9, 3.7251626892877927`*^9}, { 3.7251634912238035`*^9, 3.7251636490432053`*^9}, {3.7251636808722153`*^9, 3.7251638171212025`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Change the starting point of the scan in light of the directional scan\ \>", "Subsection", CellChangeTimes->{{3.662088291925477*^9, 3.662088335523999*^9}, { 3.725163444117753*^9, 3.7251634559037495`*^9}}], Cell["\<\ We can at this point still decide that we do not wish to use the calculated \ global minimum as the starting point, since we already know something about the matrices via the directional scans that \ suggests to us that the origin might be a better place to start the computation. We can change the scan starting \ point to the origin without recompiling the operator by using the ProbeReset command:\ \>", "Text", CellChangeTimes->{{3.662087529573331*^9, 3.66208766648717*^9}, { 3.662087960433777*^9, 3.662087961109517*^9}, {3.725162322231852*^9, 3.725162527052035*^9}, {3.725162671907795*^9, 3.7251627942468114`*^9}, 3.7251637894131947`*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"ProbeReset", "[", RowBox[{"StartingPoint", "\[Rule]", RowBox[{"{", RowBox[{"0", ",", "0", ",", "0", ",", "0", ",", "0"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.662087793113237*^9, 3.662087797224963*^9}, { 3.7251626030797806`*^9, 3.725162603259781*^9}, {3.7251627794778657`*^9, 3.7251627899768205`*^9}}], Cell[BoxData[ PanelBox[ TagBox[GridBox[{ {Cell["Energy Probe", LineColor->GrayLevel[0.25], FrontFaceColor->GrayLevel[0.25], BackFaceColor->GrayLevel[0.25], GraphicsColor->GrayLevel[0.25], FontColor->GrayLevel[0.25]], Cell["Laplace", FontWeight->Bold]}, {Cell["Starting Point 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BackFaceColor->GrayLevel[0.25], GraphicsColor->GrayLevel[0.25], FontColor->GrayLevel[0.25]], Cell["10", FontWeight->Bold]}, {Cell["Step size guess", LineColor->GrayLevel[0.25], FrontFaceColor->GrayLevel[0.25], BackFaceColor->GrayLevel[0.25], GraphicsColor->GrayLevel[0.25], FontColor->GrayLevel[0.25]], Cell["2.04188"]} }, AutoDelete->False, GridBoxAlignment->{ "Columns" -> {Right, Center}, "Rows" -> {Top, Top, Top, Top, Top, Top, Top, Top, Top}}, GridBoxDividers->{ "Columns" -> {False, {True}, False}, "Rows" -> {False, {True}, False}}, GridBoxItemSize->{"Columns" -> {Automatic, Automatic}}, GridBoxSpacings->{"Columns" -> {{3}}, "Rows" -> {{2}}}], "TextGrid"]]], "Print", CellChangeTimes->{3.725162796088812*^9, 3.725163979909274*^9, 3.7251668733761024`*^9}] }, Open ]], Cell["\<\ Note the changed starting point. Also note that in both starting points, the \ Hessian eigenvalues show no clear hierarchy. This is as one would expect, since a set of random matrices obviously will \ not contain any sensible semi-classical geometry outside of pure noise. This .nb is merely intended as showing use-cases of \ the BProbeM package.\ \>", "Text", CellChangeTimes->{{3.662087529573331*^9, 3.66208766648717*^9}, { 3.662087960433777*^9, 3.662087961109517*^9}, {3.725162322231852*^9, 3.725162527052035*^9}, {3.725162671907795*^9, 3.7251626892877927`*^9}, { 3.725162813636819*^9, 3.7251629168058357`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Start the scan with default values", "Subsection", CellChangeTimes->{{3.662088291925477*^9, 3.662088335523999*^9}}], Cell["\<\ How do we determine what a sensible step size would be? If we have no idea, \ the step size guess of BProbeM is a decent option. However, we can also look at the directional scans and observe that most of \ the interesting structure (insofar as random matrices even have any) will be occurring approximately with [-5,5] in all directions. \ Our step size should thus certainly be smaller than that. Another thing to check is the distance between approximate local minima in \ different directions, since the scanning algorithm will need to be able to jump from one to the next. Given all of this, the \ algorithm guessed step size of 2.04583 is very reasonable (it will often be like that but not always). For the sake of example we will \ still modify this default step size to a rounded down 2. It is instructive to try the same scan with different step sizes to see how \ the results change. Important note: Since there is no clear hierarchy of the Hessian EVs, we have \ to force the scan to consider a certain number of directions. In this case, since we know all directions to be just about \ equivalent, we will force a scan in all directions.\ \>", "Text", CellChangeTimes->{{3.662087529573331*^9, 3.66208766648717*^9}, { 3.662087960433777*^9, 3.662087961109517*^9}, {3.725162322231852*^9, 3.725162527052035*^9}, {3.725162671907795*^9, 3.7251626892877927`*^9}, { 3.7251629382668753`*^9, 3.7251630626878743`*^9}, {3.725164022062236*^9, 3.725164260341298*^9}, {3.725164499978032*^9, 3.725164501763034*^9}}], Cell["\<\ Depending on the comp. power of your device, this scan might take some time \ to complete. 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]], Cell[CellGroupData[{ Cell["Retrieving results ...", "Subsection", CellChangeTimes->{{3.662088386866056*^9, 3.662088413980104*^9}, { 3.662088734595255*^9, 3.6620887353166*^9}}], Cell["\<\ The obtained data here will be 5 dimensional, since we used 5 matrices and \ did not specify a subspace.\ \>", "Text", CellChangeTimes->{{3.662087529573331*^9, 3.66208766648717*^9}, { 3.662087960433777*^9, 3.662087961109517*^9}, {3.725162322231852*^9, 3.725162527052035*^9}, {3.725162671907795*^9, 3.7251626892877927`*^9}, { 3.7251630942259088`*^9, 3.725163131637151*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{"point", " ", "cloud"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"pc", " ", "=", " ", RowBox[{"ProbeGetPoints", "[", "]"}]}], ";"}]}]], "Input", CellChangeTimes->{{3.662088418775547*^9, 3.662088495015073*^9}, { 3.662089303733953*^9, 3.6620893038567944`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{"generalized", " ", "coherent", " ", "states"}], " ", "*)"}], 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When working with fuzzy \ geometry, one will often encounter spaces\nof higher dimensions than 3. To \ visualize these, we have a variety of different options. The first and almost \ always most\nuseful method is to plot ", StyleBox["projections", FontWeight->"Bold"], " of the data on 2 or more commonly 3 axes." }], "Text", CellChangeTimes->{{3.662087529573331*^9, 3.66208766648717*^9}, { 3.662087960433777*^9, 3.662087961109517*^9}, {3.725162322231852*^9, 3.725162527052035*^9}, {3.725162671907795*^9, 3.7251626892877927`*^9}, { 3.7251630942259088`*^9, 3.725163131637151*^9}, {3.7251645475480423`*^9, 3.7251646683360653`*^9}}], Cell["\<\ Here are some sample projections on various axes (for random matrices, these \ will be amorphous point clouds, i.e. noise):\ \>", "Text", CellChangeTimes->{{3.662087529573331*^9, 3.66208766648717*^9}, { 3.662087960433777*^9, 3.662087961109517*^9}, {3.725162322231852*^9, 3.725162527052035*^9}, {3.725162671907795*^9, 3.7251626892877927`*^9}, { 3.7251630942259088`*^9, 3.725163131637151*^9}, {3.725164678508071*^9, 3.7251646857160683`*^9}, {3.725167090759756*^9, 3.725167113669764*^9}}], Cell[CellGroupData[{ 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In a typical scenario, projections will result in more sensible \ visualizations, since it allows the algorithm to cross local energy maxima in this plane by moving to another plane. 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)[ Part[#, 2]]}& ), "CopiedValueFunction" -> ({ (Part[{{Identity, Identity}, {Identity, Identity}}, 1, 2][#]& )[ Part[#, 1]], (Part[{{Identity, Identity}, {Identity, Identity}}, 2, 2][#]& )[ Part[#, 2]]}& )}}, PlotRange -> {{-10., 10.}, {0, 7.251440494552831}}, PlotRangeClipping -> True, PlotRangePadding -> {{ Scaled[0.02], Scaled[0.02]}, { Scaled[0.02], Scaled[0.05]}}, Ticks -> {Automatic, Automatic}}],FormBox[ FormBox[ TemplateBox[{ TagBox[ FrameBox[ StyleBox["1", Smaller, StripOnInput -> False]], "Placeholder"], TagBox[ FrameBox[ StyleBox["2", Smaller, StripOnInput -> False]], "Placeholder"], TagBox[ FrameBox[ StyleBox["3", Smaller, StripOnInput -> False]], "Placeholder"], TagBox[ FrameBox[ StyleBox["4", Smaller, StripOnInput -> False]], "Placeholder"], TagBox[ FrameBox[ StyleBox["5", Smaller, StripOnInput -> False]], "Placeholder"]}, "PointLegend", DisplayFunction -> (FormBox[ StyleBox[ StyleBox[ PaneBox[ TagBox[ GridBox[{{ TagBox[ GridBox[{{ GraphicsBox[{{}, { Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.1], RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6]], { Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.1], RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6]], PointBox[ NCache[{ Scaled[{ Rational[1, 2], Rational[1, 2]}]}, { Scaled[{0.5, 0.5}]}]]}}}, AspectRatio -> Full, ImageSize -> {10, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #}, { GraphicsBox[{{}, { Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.1], RGBColor[0.880722, 0.611041, 0.142051], AbsoluteThickness[1.6]], { Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.1], RGBColor[0.880722, 0.611041, 0.142051], AbsoluteThickness[1.6]], PointBox[ NCache[{ Scaled[{ Rational[1, 2], Rational[1, 2]}]}, { Scaled[{0.5, 0.5}]}]]}}}, AspectRatio -> Full, ImageSize -> {10, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #2}, { GraphicsBox[{{}, { Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.1], RGBColor[0.560181, 0.691569, 0.194885], AbsoluteThickness[1.6]], { Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.1], RGBColor[0.560181, 0.691569, 0.194885], AbsoluteThickness[1.6]], PointBox[ NCache[{ Scaled[{ Rational[1, 2], Rational[1, 2]}]}, { Scaled[{0.5, 0.5}]}]]}}}, AspectRatio -> Full, ImageSize -> {10, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #3}, { GraphicsBox[{{}, { Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.1], RGBColor[0.922526, 0.385626, 0.209179], AbsoluteThickness[1.6]], { Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.1], RGBColor[0.922526, 0.385626, 0.209179], AbsoluteThickness[1.6]], PointBox[ NCache[{ Scaled[{ Rational[1, 2], Rational[1, 2]}]}, { Scaled[{0.5, 0.5}]}]]}}}, AspectRatio -> Full, ImageSize -> {10, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #4}, { GraphicsBox[{{}, { Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.1], RGBColor[0.528488, 0.470624, 0.701351], AbsoluteThickness[1.6]], { Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.1], RGBColor[0.528488, 0.470624, 0.701351], AbsoluteThickness[1.6]], PointBox[ NCache[{ Scaled[{ Rational[1, 2], Rational[1, 2]}]}, { Scaled[{0.5, 0.5}]}]]}}}, AspectRatio -> Full, ImageSize -> {10, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #5}}, GridBoxAlignment -> { "Columns" -> {Center, Left}, "Rows" -> {{Baseline}}}, AutoDelete -> False, GridBoxDividers -> { "Columns" -> {{False}}, "Rows" -> {{False}}}, GridBoxItemSize -> {"Columns" -> {{All}}, "Rows" -> {{All}}}, GridBoxSpacings -> { "Columns" -> {{0.5}}, "Rows" -> {{0.8}}}], "Grid"]}}, GridBoxAlignment -> {"Columns" -> {{Left}}, "Rows" -> {{Top}}}, AutoDelete -> False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings -> {"Columns" -> {{1}}, "Rows" -> {{0}}}], "Grid"], Alignment -> Left, AppearanceElements -> None, ImageMargins -> {{5, 5}, {5, 5}}, ImageSizeAction -> "ResizeToFit"], LineIndent -> 0, StripOnInput -> False], { FontFamily -> "Arial"}, Background -> Automatic, StripOnInput -> False], TraditionalForm]& ), InterpretationFunction :> (RowBox[{"PointLegend", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Directive", "[", RowBox[{ RowBox[{"PointSize", "[", "0.002777777777777778`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.368417, 0.506779, 0.709798], RectangleBox[{0, -1}, {2, 1}]}}, AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.24561133333333335`, 0.3378526666666667, 0.4731986666666667], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], "RGBColor[0.368417, 0.506779, 0.709798]"], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.368417, 0.506779, 0.709798]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.368417, 0.506779, 0.709798], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"PointSize", "[", "0.002777777777777778`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.880722, 0.611041, 0.142051], RectangleBox[{0, -1}, {2, 1}]}}, AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.587148, 0.40736066666666665`, 0.09470066666666668], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], "RGBColor[0.880722, 0.611041, 0.142051]"], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.880722, 0.611041, 0.142051]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.880722, 0.611041, 0.142051], Editable -> False, Selectable -> False], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"PointSize", "[", "0.002777777777777778`", "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.560181, 0.691569, 0.194885], RectangleBox[{0, -1}, {2, 1}]}}, AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.37345400000000006`, 0.461046, 0.12992333333333334`], 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Studying this behavior for random matrices can however be very useful in \ later on being able to tell apart noise from actual results contained in matrix sets, so it is recommended to play around with the GUE matrices and \ the 2-sphere in particular to get intuitions for the scanning algorithm and the BProbeM package.\ \>", "Text", CellChangeTimes->{{3.662087529573331*^9, 3.66208766648717*^9}, { 3.662087960433777*^9, 3.662087961109517*^9}, {3.725162322231852*^9, 3.725162527052035*^9}, {3.725162671907795*^9, 3.7251626892877927`*^9}, { 3.7251630942259088`*^9, 3.725163131637151*^9}, {3.725167347992838*^9, 3.725167445791832*^9}}] }, Open ]] }, Open ]] }, Open ]] }, WindowSize->{1303, 869}, WindowMargins->{{-8, Automatic}, {Automatic, -8}}, FrontEndVersion->"11.0 for Microsoft Windows (64-bit) (September 21, 2016)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) 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