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Re: Mathematica Can't calculate Determinant of Matrix?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg45266] Re: Mathematica Can't calculate Determinant of Matrix?
  • From: drbob at bigfoot.com (Bobby R. Treat)
  • Date: Wed, 24 Dec 2003 17:42:21 -0500 (EST)
  • References: <bsbn7v$lsu$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Just to get an idea of what you're asking Mathematica to do, I
computed leaf-counts for matrix determinants up to 11x11, where all
the entries are simple symbols:

<< "LinearAlgebra`MatrixManip\
ulation`"
sixteen = Characters[
    "abcdefghijklmnop"]; 
many = ToExpression[
    Outer[StringJoin, 
     sixteen, sixteen]]; 
mat[n_] := SubMatrix[many, 
   {1, 1}, {n, n}]
counts = 
  (LeafCount[Det[
      mat[#1]]] & ) /@ 
   Range[11]
{1, 8, 28, 133, 781, 5401, 
  42841, 342749, 3084764, 
  30847666, 339324354}

Then I extrapolated (clumsily, perhaps) to 16x16:

N[Rest[counts]/Most[counts]]
{8., 3.5, 4.75, 
  5.87218045112782, 
  6.915492957746479, 
  7.932049620440659, 
  8.000490184636213, 
  9.00006710449921, 
  10.000008428521598, 
  11.000000907686177}

(16!/11!)*Last[counts]
N[%]
177860253392640
1.7786025339264*^14

So...

The entries in your matrix are MUCH more complicated, so the LeafCount
of your determinant is noticeably larger than 10^14.

If your machine didn't run out of memory calculating the determinant,
and if it didn't run out of memory trying to simplify it, you'd die
watching all those characters scroll by.

On the other hand, if your symbols took on numerical values, the
leafcount would be ONE.

Bobby

Arman Khalatyan <arm2arm at thsun1.jinr.ru> wrote in message news:<bsbn7v$lsu$1 at smc.vnet.net>...
> Hello.
> I have problem with Math 3.0(UNIX) after some calculations(analitical) Its give me:
> 
> Det[MyDet]
> "No more memory available.
> Mathematica kernel has shut down.
> Try quitting other applications and then retry."
> 
> MyDet is a 16x16 Matrix.
> 
> In top I can see the memory is enough.  The kernel shutted down after
> 150Mb allocation, but I have about 1Gb RAM free.
> 
> What can I do?How to solve this problem?  Please help me.  Thank You
> beforehand.
> 
> Arman.
> 
> 
> 
> 
> (***********************************************************************
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> to\ calculate\ Det*) \), "\[IndentingNewLine]", 
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> 
> Cell[CellGroupData[{
> 
> Cell[BoxData[{
>     \(<< LinearAlgebra`MatrixManipulation`\), "\n", 
>     \(CFV := {\ m\_0 \[Element] Reals, \ 
>         pvec2 \[Element] Reals, \[CapitalDelta]\ \  \[Element] 
>           Reals, \ \[Omega][_] \[Element] Reals, p\_0 \[Element] Reals, 
>         p\_1 \[Element] Reals, p\_2 \[Element] Reals, 
>         p\_3 \[Element] Reals, \[Mu] \[Element] 
>           Reals, {\[Delta]\[Mu], \[Mu]\&_, \[Mu], ga, \ gb, \ ga\__, gb\__, 
>             m\__, \ \[CurlyPhi], Absg2} \[Element] Reals, 
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>               Zero4}, \[IndentingNewLine]{Zero4, \(-g[dr]\) gamma5, Zero4, 
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>     \(\(Gop = 
>         BlockMatrix[{{Gpmo[u, ur, 1], Zero4, Zero4, Zero4}, {Zero4, 
>               Gpmo[u, ug, 1], Zero4, Zero4}, {Zero4, Zero4, Gpmo[d, dr, 1], 
>               Zero4}, {Zero4, Zero4, Zero4, 
>               Gpmo[d, dg, 1]}}];\)\[IndentingNewLine]\), "\n", 
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>               Gpmo[d, dr, \(-1\)], Zero4}, {Zero4, Zero4, Zero4, 
>               Gpmo[d, dg, \(-1\)]}}];\)\[IndentingNewLine]\n (*We\ need\ to\ \
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>     \(\(Grggr = 
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>               Gom}}];\)\[IndentingNewLine] (*\ Or\ in\ another\ "\<word\>", \ 
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>     \(\(SimpleDet1 = \(-Deltplus . Deltminus\) + 
>           Deltplus . Gop . Inverse[Deltplus] . 
>             Gom;\)\[IndentingNewLine]\[IndentingNewLine] (*Another\ \
> representation\ of\ determinant*) \n (*\(SimpleDet2 = \(-Deltminus . 
>                 Deltplus\) + 
>             Gom . Inverse[Deltplus] . Gop . 
>               Deltplus;\)*) \[IndentingNewLine]\[IndentingNewLine]\
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>         ComplexExpand[SimpleDet1] //. 
>           Rep;\)\(\[IndentingNewLine]\)\( (*\ \(SimpCompExp2 = 
>           Collect[ComplexExpand[SimpleDet2], \[Mu]\ ] //. 
>             Rep;\)*) \)\(\[IndentingNewLine]\)\)\)], "Input"],
> 
> Cell[CellGroupData[{
> 
> Cell[BoxData[
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> 
> No more memory available.
> Mathematica kernel has shut down.
> Try quitting other applications and then retry.\
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>     CellMargins->{{18, Inherited}, {6, 12}},
>     FontSize->12],
>   
>   Cell[StyleData["Section", "Printout"],
>     CellMargins->{{13, 0}, {7, 22}},
>     FontSize->14]
>   }, Closed]],
>   
>   Cell[CellGroupData[{
>   
>   Cell[StyleData["Subsection"],
>     CellDingbat->"\[FilledSmallSquare]",
>     CellMargins->{{22, Inherited}, {8, 20}},
>     CellGroupingRules->{"SectionGrouping", 40},
>     PageBreakBelow->False,
>     DefaultNewInlineCellStyle->"None",
>     InputAutoReplacements->{"TeX"->StyleBox[ 
>             RowBox[ {"T", 
>               AdjustmentBox[ 
>               "E", BoxMargins -> {{-0.074999999999999997, \
> -0.085000000000000006}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], 
>         "LaTeX"->StyleBox[ 
>             RowBox[ {"L", 
>               StyleBox[ 
>                 AdjustmentBox[ 
>                 "A", BoxMargins -> {{-0.35999999999999999, \
> -0.10000000000000001}, {0, 0}}, BoxBaselineShift -> -0.20000000000000001], 
>                 FontSize -> Smaller], "T", 
>               AdjustmentBox[ 
>               "E", BoxMargins -> {{-0.074999999999999997, \
> -0.085000000000000006}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], 
>         "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"},
>     LanguageCategory->"NaturalLanguage",
>     CounterIncrements->"Subsection",
>     CounterAssignments->{{"Subsubsection", 0}},
>     FontFamily->"Times",
>     FontSize->14,
>     FontWeight->"Bold"],
>   
>   Cell[StyleData["Subsection", "Presentation"],
>     CellMargins->{{36, 10}, {11, 32}},
>     LineSpacing->{1, 0},
>     FontSize->22],
>   
>   Cell[StyleData["Subsection", "Condensed"],
>     CellMargins->{{16, Inherited}, {6, 12}},
>     FontSize->12],
>   
>   Cell[StyleData["Subsection", "Printout"],
>     CellMargins->{{9, 0}, {7, 22}},
>     FontSize->12]
>   }, Closed]],
>   
>   Cell[CellGroupData[{
>   
>   Cell[StyleData["Subsubsection"],
>     CellDingbat->"\[FilledSmallSquare]",
>     CellMargins->{{22, Inherited}, {8, 18}},
>     CellGroupingRules->{"SectionGrouping", 50},
>     PageBreakBelow->False,
>     DefaultNewInlineCellStyle->"None",
>     InputAutoReplacements->{"TeX"->StyleBox[ 
>             RowBox[ {"T", 
>               AdjustmentBox[ 
>               "E", BoxMargins -> {{-0.074999999999999997, \
> -0.085000000000000006}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], 
>         "LaTeX"->StyleBox[ 
>             RowBox[ {"L", 
>               StyleBox[ 
>                 AdjustmentBox[ 
>                 "A", BoxMargins -> {{-0.35999999999999999, \
> -0.10000000000000001}, {0, 0}}, BoxBaselineShift -> -0.20000000000000001], 
>                 FontSize -> Smaller], "T", 
>               AdjustmentBox[ 
>               "E", BoxMargins -> {{-0.074999999999999997, \
> -0.085000000000000006}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], 
>         "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"},
>     LanguageCategory->"NaturalLanguage",
>     CounterIncrements->"Subsubsection",
>     FontFamily->"Times",
>     FontWeight->"Bold"],
>   
>   Cell[StyleData["Subsubsection", "Presentation"],
>     CellMargins->{{34, 10}, {11, 26}},
>     LineSpacing->{1, 0},
>     FontSize->18],
>   
>   Cell[StyleData["Subsubsection", "Condensed"],
>     CellMargins->{{17, Inherited}, {6, 12}},
>     FontSize->10],
>   
>   Cell[StyleData["Subsubsection", "Printout"],
>     CellMargins->{{9, 0}, {7, 14}},
>     FontSize->11]
>   }, Closed]]
>   }, Closed]],
>   
>   Cell[CellGroupData[{
>   
>   Cell["Styles for Body Text", "Section"],
>   
>   Cell[CellGroupData[{
>   
>   Cell[StyleData["Text"],
>     CellMargins->{{12, 10}, {7, 7}},
>     InputAutoReplacements->{"TeX"->StyleBox[ 
>             RowBox[ {"T", 
>               AdjustmentBox[ 
>               "E", BoxMargins -> {{-0.074999999999999997, \
> -0.085000000000000006}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], 
>         "LaTeX"->StyleBox[ 
>             RowBox[ {"L", 
>               StyleBox[ 
>                 AdjustmentBox[ 
>                 "A", BoxMargins -> {{-0.35999999999999999, \
> -0.10000000000000001}, {0, 0}}, BoxBaselineShift -> -0.20000000000000001], 
>                 FontSize -> Smaller], "T", 
>               AdjustmentBox[ 
>               "E", BoxMargins -> {{-0.074999999999999997, \
> -0.085000000000000006}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], 
>         "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"},
>     Hyphenation->True,
>     LineSpacing->{1, 3},
>     CounterIncrements->"Text"],
>   
>   Cell[StyleData["Text", "Presentation"],
>     CellMargins->{{24, 10}, {10, 10}},
>     LineSpacing->{1, 5},
>     FontSize->16],
>   
>   Cell[StyleData["Text", "Condensed"],
>     CellMargins->{{8, 10}, {6, 6}},
>     LineSpacing->{1, 1},
>     FontSize->11],
>   
>   Cell[StyleData["Text", "Printout"],
>     CellMargins->{{2, 2}, {6, 6}},
>     TextJustification->0.5,
>     FontSize->10]
>   }, Closed]],
>   
>   Cell[CellGroupData[{
>   
>   Cell[StyleData["SmallText"],
>     CellMargins->{{12, 10}, {6, 6}},
>     DefaultNewInlineCellStyle->"None",
>     Hyphenation->True,
>     LineSpacing->{1, 3},
>     LanguageCategory->"NaturalLanguage",
>     CounterIncrements->"SmallText",
>     FontFamily->"Helvetica",
>     FontSize->9],
>   
>   Cell[StyleData["SmallText", "Presentation"],
>     CellMargins->{{24, 10}, {8, 8}},
>     LineSpacing->{1, 5},
>     FontSize->12],
>   
>   Cell[StyleData["SmallText", "Condensed"],
>     CellMargins->{{8, 10}, {5, 5}},
>     LineSpacing->{1, 2},
>     FontSize->9],
>   
>   Cell[StyleData["SmallText", "Printout"],
>     CellMargins->{{2, 2}, {5, 5}},
>     TextJustification->0.5,
>     FontSize->7]
>   }, Closed]]
>   }, Closed]],
>   
>   Cell[CellGroupData[{
>   
>   Cell["Styles for Input/Output", "Section"],
>   
>   Cell["\<\
> The cells in this section define styles used for input and output \
> to the kernel.  Be careful when modifying, renaming, or removing these \
> styles, because the front end associates special meanings with these style \
> names. Some attributes for these styles are actually set in FormatType Styles \
> (in the last section of this stylesheet). \
> \>", "Text"],
>   
>   Cell[CellGroupData[{
>   
>   Cell[StyleData["Input"],
>     CellMargins->{{45, 10}, {5, 7}},
>     Evaluatable->True,
>     CellGroupingRules->"InputGrouping",
>     CellHorizontalScrolling->True,
>     PageBreakWithin->False,
>     GroupPageBreakWithin->False,
>     DefaultFormatType->DefaultInputFormatType,
>     HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"},
>     AutoItalicWords->{},
>     LanguageCategory->"Formula",
>     FormatType->InputForm,
>     ShowStringCharacters->True,
>     NumberMarks->True,
>     LinebreakAdjustments->{0.85, 2, 10, 0, 1},
>     CounterIncrements->"Input",
>     FontWeight->"Bold"],
>   
>   Cell[StyleData["Input", "Presentation"],
>     CellMargins->{{72, Inherited}, {8, 10}},
>     LineSpacing->{1, 0},
>     FontSize->16],
>   
>   Cell[StyleData["Input", "Condensed"],
>     CellMargins->{{40, 10}, {2, 3}},
>     FontSize->11],
>   
>   Cell[StyleData["Input", "Printout"],
>     CellMargins->{{39, 0}, {4, 6}},
>     LinebreakAdjustments->{0.85, 2, 10, 1, 1},
>     FontSize->9]
>   }, Closed]],
>   
>   Cell[StyleData["InputOnly"],
>     Evaluatable->True,
>     CellGroupingRules->"InputGrouping",
>     CellHorizontalScrolling->True,
>     DefaultFormatType->DefaultInputFormatType,
>     HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"},
>     AutoItalicWords->{},
>     LanguageCategory->"Formula",
>     FormatType->InputForm,
>     ShowStringCharacters->True,
>     NumberMarks->True,
>     LinebreakAdjustments->{0.85, 2, 10, 0, 1},
>     CounterIncrements->"Input",
>     StyleMenuListing->None,
>     FontWeight->"Bold"],
>   
>   Cell[CellGroupData[{
>   
>   Cell[StyleData["Output"],
>     CellMargins->{{47, 10}, {7, 5}},
>     CellEditDuplicate->True,
>     CellGroupingRules->"OutputGrouping",
>     CellHorizontalScrolling->True,
>     PageBreakWithin->False,
>     GroupPageBreakWithin->False,
>     GeneratedCell->True,
>     CellAutoOverwrite->True,
>     DefaultFormatType->DefaultOutputFormatType,
>     HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"},
>     AutoItalicWords->{},
>     LanguageCategory->"Formula",
>     FormatType->InputForm,
>     CounterIncrements->"Output"],
>   
>   Cell[StyleData["Output", "Presentation"],
>     CellMargins->{{72, Inherited}, {10, 8}},
>     LineSpacing->{1, 0},
>     FontSize->16],
>   
>   Cell[StyleData["Output", "Condensed"],
>     CellMargins->{{41, Inherited}, {3, 2}},
>     FontSize->11],
>   
>   Cell[StyleData["Output", "Printout"],
>     CellMargins->{{39, 0}, {6, 4}},
>     FontSize->9]
>   }, Closed]],
>   
>   Cell[CellGroupData[{
>   
>   Cell[StyleData["Message"],
>     CellMargins->{{45, Inherited}, {Inherited, Inherited}},
>     CellGroupingRules->"OutputGrouping",
>     PageBreakWithin->False,
>     GroupPageBreakWithin->False,
>     GeneratedCell->True,
>     CellAutoOverwrite->True,
>     ShowCellLabel->False,
>     DefaultFormatType->DefaultOutputFormatType,
>     HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"},
>     AutoItalicWords->{},
>     FormatType->InputForm,
>     CounterIncrements->"Message",
>     StyleMenuListing->None,
>     FontSize->11,
>     FontColor->RGBColor[0, 0, 1]],
>   
>   Cell[StyleData["Message", "Presentation"],
>     CellMargins->{{72, Inherited}, {Inherited, Inherited}},
>     LineSpacing->{1, 0},
>     FontSize->16],
>   
>   Cell[StyleData["Message", "Condensed"],
>     CellMargins->{{41, Inherited}, {Inherited, Inherited}},
>     FontSize->11],
>   
>   Cell[StyleData["Message", "Printout"],
>     CellMargins->{{39, Inherited}, {Inherited, Inherited}},
>     FontSize->7,
>     FontColor->GrayLevel[0]]
>   }, Closed]],
>   
>   Cell[CellGroupData[{
>   
>   Cell[StyleData["Print"],
>     CellMargins->{{45, Inherited}, {Inherited, Inherited}},
>     CellGroupingRules->"OutputGrouping",
>     CellHorizontalScrolling->True,
>     PageBreakWithin->False,
>     GroupPageBreakWithin->False,
>     GeneratedCell->True,
>     CellAutoOverwrite->True,
>     ShowCellLabel->False,
>     DefaultFormatType->DefaultOutputFormatType,
>     HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"},
>     AutoItalicWords->{},
>     FormatType->InputForm,
>     CounterIncrements->"Print",
>     StyleMenuListing->None],
>   
>   Cell[StyleData["Print", "Presentation"],
>     CellMargins->{{72, Inherited}, {Inherited, Inherited}},
>     LineSpacing->{1, 0},
>     FontSize->16],
>   
>   Cell[StyleData["Print", "Condensed"],
>     CellMargins->{{41, Inherited}, {Inherited, Inherited}},
>     FontSize->11],
>   
>   Cell[StyleData["Print", "Printout"],
>     CellMargins->{{39, Inherited}, {Inherited, Inherited}},
>     FontSize->8]
>   }, Closed]],
>   
>   Cell[CellGroupData[{
>   
>   Cell[StyleData["Graphics"],
>     CellMargins->{{4, Inherited}, {Inherited, Inherited}},
>     CellGroupingRules->"GraphicsGrouping",
>     CellHorizontalScrolling->True,
>     PageBreakWithin->False,
>     GeneratedCell->True,
>     CellAutoOverwrite->True,
>     ShowCellLabel->False,
>     DefaultFormatType->DefaultOutputFormatType,
>     LanguageCategory->None,
>     FormatType->InputForm,
>     CounterIncrements->"Graphics",
>     ImageMargins->{{43, Inherited}, {Inherited, 0}},
>     StyleMenuListing->None,
>     FontFamily->"Courier",
>     FontSize->10],
>   
>   Cell[StyleData["Graphics", "Presentation"],
>     ImageMargins->{{62, Inherited}, {Inherited, 0}}],
>   
>   Cell[StyleData["Graphics", "Condensed"],
>     ImageMargins->{{38, Inherited}, {Inherited, 0}},
>     Magnification->0.6],
>   
>   Cell[StyleData["Graphics", "Printout"],
>     ImageMargins->{{30, Inherited}, {Inherited, 0}},
>     Magnification->0.8]
>   }, Closed]],
>   
>   Cell[CellGroupData[{
>   
>   Cell[StyleData["CellLabel"],
>     StyleMenuListing->None,
>     FontFamily->"Helvetica",
>     FontSize->9,
>     FontColor->RGBColor[0, 0, 1]],
>   
>   Cell[StyleData["CellLabel", "Presentation"],
>     FontSize->12],
>   
>   Cell[StyleData["CellLabel", "Condensed"],
>     FontSize->9],
>   
>   Cell[StyleData["CellLabel", "Printout"],
>     FontFamily->"Courier",
>     FontSize->8,
>     FontSlant->"Italic",
>     FontColor->GrayLevel[0]]
>   }, Closed]]
>   }, Closed]],
>   
>   Cell[CellGroupData[{
>   
>   Cell["Inline Formatting", "Section"],
>   
>   Cell["\<\
> These styles are for modifying individual words or letters in a \
> cell exclusive of the cell tag.\
> \>", "Text"],
>   
>   Cell[StyleData["RM"],
>     StyleMenuListing->None,
>     FontWeight->"Plain",
>     FontSlant->"Plain"],
>   
>   Cell[StyleData["BF"],
>     StyleMenuListing->None,
>     FontWeight->"Bold"],
>   
>   Cell[StyleData["IT"],
>     StyleMenuListing->None,
>     FontSlant->"Italic"],
>   
>   Cell[StyleData["TR"],
>     StyleMenuListing->None,
>     FontFamily->"Times",
>     FontWeight->"Plain",
>     FontSlant->"Plain"],
>   
>   Cell[StyleData["TI"],
>     StyleMenuListing->None,
>     FontFamily->"Times",
>     FontWeight->"Plain",
>     FontSlant->"Italic"],
>   
>   Cell[StyleData["TB"],
>     StyleMenuListing->None,
>     FontFamily->"Times",
>     FontWeight->"Bold",
>     FontSlant->"Plain"],
>   
>   Cell[StyleData["TBI"],
>     StyleMenuListing->None,
>     FontFamily->"Times",
>     FontWeight->"Bold",
>     FontSlant->"Italic"],
>   
>   Cell[StyleData["MR"],
>     StyleMenuListing->None,
>     FontFamily->"Courier",
>     FontWeight->"Plain",
>     FontSlant->"Plain"],
>   
>   Cell[StyleData["MO"],
>     StyleMenuListing->None,
>     FontFamily->"Courier",
>     FontWeight->"Plain",
>     FontSlant->"Italic"],
>   
>   Cell[StyleData["MB"],
>     StyleMenuListing->None,
>     FontFamily->"Courier",
>     FontWeight->"Bold",
>     FontSlant->"Plain"],
>   
>   Cell[StyleData["MBO"],
>     StyleMenuListing->None,
>     FontFamily->"Courier",
>     FontWeight->"Bold",
>     FontSlant->"Italic"],
>   
>   Cell[StyleData["SR"],
>     StyleMenuListing->None,
>     FontFamily->"Helvetica",
>     FontWeight->"Plain",
>     FontSlant->"Plain"],
>   
>   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"],
>   
>   Cell[StyleData["SO10", "EnhancedPrintout"],
>     StyleMenuListing->None,
>     FontFamily->"Futura",
>     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"],
>     CellMargins->{{24, 10}, {10, 10}},
>     LineSpacing->{1, 5},
>     FontSize->16],
>   
>   Cell[StyleData["InlineFormula", "Condensed"],
>     CellMargins->{{8, 10}, {6, 6}},
>     LineSpacing->{1, 1},
>     FontSize->11],
>   
>   Cell[StyleData["InlineFormula", "Printout"],
>     CellMargins->{{2, 0}, {6, 6}},
>     FontSize->10]
>   }, Closed]],
>   
>   Cell[CellGroupData[{
>   
>   Cell[StyleData["DisplayFormula"],
>     CellMargins->{{42, 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", "Condensed"],
>     LineSpacing->{1, 1},
>     FontSize->11],
>   
>   Cell[StyleData["DisplayFormula", "Printout"],
>     FontSize->10]
>   }, Closed]],
>   
>   Cell[CellGroupData[{
>   
>   Cell[StyleData["Program"],
>     CellFrame->{{0, 0}, {0.5, 0.5}},
>     CellMargins->{{10, 4}, {0, 8}},
>     CellHorizontalScrolling->True,
>     Hyphenation->False,
>     LanguageCategory->"Formula",
>     ScriptLevel->1,
>     FontFamily->"Courier"],
>   
>   Cell[StyleData["Program", "Presentation"],
>     CellMargins->{{24, 10}, {10, 10}},
>     LineSpacing->{1, 5},
>     FontSize->16],
>   
>   Cell[StyleData["Program", "Condensed"],
>     CellMargins->{{8, 10}, {6, 6}},
>     LineSpacing->{1, 1},
>     FontSize->11],
>   
>   Cell[StyleData["Program", "Printout"],
>     CellMargins->{{2, 0}, {6, 6}},
>     FontSize->9]
>   }, 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, 0, 1],
>     FontVariations->{"Underline"->True},
>     ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ {
>         FrontEnd`NotebookLocate[ #2]}]&),
>     Active->True,
>     ButtonNote->ButtonData}],
>   
>   Cell[StyleData["Hyperlink", "Presentation"],
>     FontSize->16],
>   
>   Cell[StyleData["Hyperlink", "Condensed"],
>     FontSize->11],
>   
>   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, 0, 1],
>     FontVariations->{"Underline"->True},
>     ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ {
>         FrontEnd`HelpBrowserLookup[ "MainBook", #]}]&),
>     Active->True,
>     ButtonFrame->"None"}],
>   
>   Cell[StyleData["MainBookLink", "Presentation"],
>     FontSize->16],
>   
>   Cell[StyleData["MainBookLink", "Condensed"],
>     FontSize->11],
>   
>   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, 0, 1],
>     FontVariations->{"Underline"->True},
>     ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ {
>         FrontEnd`HelpBrowserLookup[ "AddOns", #]}]&),
>     Active->True,
>     ButtonFrame->"None"}],
>   
>   Cell[StyleData["AddOnsLink", "Presentation"],
>     FontSize->16],
>   
>   Cell[StyleData["AddOnsLink", "Condensed"],
>     FontSize->11],
>   
>   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, 0, 1],
>     FontVariations->{"Underline"->True},
>     ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ {
>         FrontEnd`HelpBrowserLookup[ "RefGuide", #]}]&),
>     Active->True,
>     ButtonFrame->"None"}],
>   
>   Cell[StyleData["RefGuideLink", "Presentation"],
>     FontSize->16],
>   
>   Cell[StyleData["RefGuideLink", "Condensed"],
>     FontSize->11],
>   
>   Cell[StyleData["RefGuideLink", "Printout"],
>     FontSize->10,
>     FontColor->GrayLevel[0],
>     FontVariations->{"Underline"->False}]
>   }, Closed]],
>   
>   Cell[CellGroupData[{
>   
>   Cell[StyleData["GettingStartedLink"],
>     StyleMenuListing->None,
>     ButtonStyleMenuListing->Automatic,
>     FontColor->RGBColor[0, 0, 1],
>     FontVariations->{"Underline"->True},
>     ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ {
>         FrontEnd`HelpBrowserLookup[ "GettingStarted", #]}]&),
>     Active->True,
>     ButtonFrame->"None"}],
>   
>   Cell[StyleData["GettingStartedLink", "Presentation"],
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>   
>   Cell[StyleData["GettingStartedLink", "Condensed"],
>     FontSize->11],
>   
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>     FontSize->10,
>     FontColor->GrayLevel[0],
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>   }, Closed]],
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>   Cell[CellGroupData[{
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>     ButtonStyleMenuListing->Automatic,
>     FontColor->RGBColor[0, 0, 1],
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>     ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ {
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>     ButtonFrame->"None"}],
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>     FontSize->16],
>   
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>     StyleMenuListing->None,
>     FontFamily->"Times",
>     FontSize->10]
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>   Cell[CellGroupData[{
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>     ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ {
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>   Cell[CellGroupData[{
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>   
>   Cell[CellGroupData[{
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>     StyleMenuListing->None,
>     FontSlant->"Italic",
>     FontColor->RGBColor[0.890623, 0.864698, 0.384756],
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>   Cell[StyleData["Placeholder", "Presentation"]],
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>   Cell[StyleData["Placeholder", "Condensed"]],
>   
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>   
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>     FontSlant->"Italic",
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>     TagBoxOptions->{Editable->False,
>     Selectable->False,
>     StripWrapperBoxes->False}],
>   
>   Cell[StyleData["PrimaryPlaceholder", "Presentation"]],
>   
>   Cell[StyleData["PrimaryPlaceholder", "Condensed"]],
>   
>   Cell[StyleData["PrimaryPlaceholder", "Printout"]]
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>   }, Closed]],
>   
>   Cell[CellGroupData[{
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>   Cell["FormatType Styles", "Section"],
>   
>   Cell["\<\
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> defined below, then that style is applied between the cell's style and its \
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>     CellMargins->{{6, Inherited}, {Inherited, Inherited}},
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>     ShowSpecialCharacters->False,
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>     Hyphenation->False,
>     AutoItalicWords->{},
>     StyleMenuListing->None,
>     FontFamily->"Courier",
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>     InputAutoReplacements->{},
>     AllowInlineCells->False,
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>     LineSpacing->{0.6, 1},
>     StyleMenuListing->None,
>     FontFamily->"Courier"],
>   
>   Cell[StyleData["StandardForm"],
>     InputAutoReplacements->{
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>         ">="->"\[GreaterEqual]", "!="->"\[NotEqual]", "=="->"\[Equal]", 
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>     LineSpacing->{1.25, 0},
>     StyleMenuListing->None,
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>   Cell[StyleData["TraditionalForm"],
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>     StyleMenuListing->None],
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>   Cell["\<\
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>     StyleMenuListing->None],
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>     Background->RGBColor[1, 0.749996, 0.8]]
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> 
> 
> (***********************************************************************
> Cached data follows.  If you edit this Notebook file directly, not using
> Mathematica, you must remove the line containing CacheID at the top of 
> the file.  The cache data will then be recreated when you save this file 
> from within Mathematica.
> ***********************************************************************)
> 
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> }
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> *)
> 
> 
> 
> 
> (***********************************************************************
> End of Mathematica Notebook file.
> ***********************************************************************)


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