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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 21612, 577]*) (*NotebookOutlinePosition[ 51647, 1635]*) (* CellTagsIndexPosition[ 51603, 1631]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[StyleBox[ "The length of a quaternion in curved spacetime: a close relative of the \ affine parameter of general relativity"]], "Subtitle", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox["doug "]], "Subsubtitle", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ "The affine parameter of general relativity\nLength in flat spacetime\nLength \ in curved spacetime\nImplications"]], "Text", FontFamily->"Times New Roman"], Cell[CellGroupData[{ Cell[TextData[StyleBox[ " The affine parameter of general relativity"]], "Subsection", CellMargins->{{0, Inherited}, {Inherited, Inherited}}], Cell["\<\ The affine parameter is defined in Misner, Thorne and Wheeler as a \ multiple of the proper time plus a displacement.\ \>", "Text"], Cell[BoxData[ \(\(\[Lambda]\ = \ a\ \[Tau]\ + \ b; \)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(\[Lambda]\ = \ a\ \[Tau]\ + \ b\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["\<\ The affine parameter is used to determine length in curved \ spacetime. In this notebook, the length of a quaternion in curved spacetime \ will be analyzed. Under certain approximations, this length will depend on \ the square of the affine parameter, but the two measures are slightly \ different.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[" Length in flat spacetime"]], "Subsection", CellMargins->{{0, Inherited}, {Inherited, Inherited}}], Cell["\<\ Calculating the square of the interval between two events in flat \ spacetime was straightforward: take the difference between two quaternions \ and square it.\ \>", "Text"], Cell[BoxData[{ \(q\_1\ = \ q[t\_1, x\_1, y\_1, z\_1]; \n q\_2\ = \ q[t\_2, x\_2, y\_2, z\_2]; \n \((L\_flat = Simplify[\n\t \((\((q\_2 - \ q\_1)\).\((q\_2 - \ q\_1)\))\) /. \ toDelta])\).{ 1, 0, 0, 0}\), RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox[ RowBox[{"(", InterpretationBox[\(\[CapitalDelta] t\), Prefix[ f[ t], \[CapitalDelta]], Editable->False], ")"}], "2"], "-", SuperscriptBox[ RowBox[{"(", InterpretationBox[\(\[CapitalDelta] x\), Prefix[ f[ x], \[CapitalDelta]], Editable->False], ")"}], "2"], "-", SuperscriptBox[ RowBox[{"(", InterpretationBox[\(\[CapitalDelta] y\), Prefix[ f[ y], \[CapitalDelta]], Editable->False], ")"}], "2"], "-", SuperscriptBox[ RowBox[{"(", InterpretationBox[\(\[CapitalDelta] z\), Prefix[ f[ z], \[CapitalDelta]], Editable->False], ")"}], "2"]}], ",", "\n", "\t", RowBox[{"2", " ", RowBox[{"(", InterpretationBox[\(\[CapitalDelta] t\), Prefix[ f[ t], \[CapitalDelta]], Editable->False], ")"}], " ", RowBox[{"(", InterpretationBox[\(\[CapitalDelta] x\), Prefix[ f[ x], \[CapitalDelta]], Editable->False], ")"}]}], ",", RowBox[{"2", " ", RowBox[{"(", InterpretationBox[\(\[CapitalDelta] t\), Prefix[ f[ t], \[CapitalDelta]], Editable->False], ")"}], " ", RowBox[{"(", InterpretationBox[\(\[CapitalDelta] y\), Prefix[ f[ y], \[CapitalDelta]], Editable->False], ")"}]}], ",", RowBox[{"2", " ", RowBox[{"(", InterpretationBox[\(\[CapitalDelta] t\), Prefix[ f[ t], \[CapitalDelta]], Editable->False], ")"}], " ", RowBox[{"(", InterpretationBox[\(\[CapitalDelta] z\), Prefix[ f[ z], \[CapitalDelta]], Editable->False], ")"}]}]}], "}"}]}], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(L\_flat = \(\((q\ - \ q\^\[Prime])\)\^2 = \ \((dt\^2\ - \ d X\&\[RightVector]\^2, \ 2\ dt\ d X\&\[RightVector]) \)\)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[TextData[StyleBox[ "The first term is the square of the interval. Spacetime is flat in the \ sense that the first term is exactly like the Minkowski metric in spacetime. \ There are quaternions which preserve the interval, and those quaternions were \ used to solve problems in special relativity. "]], "Text"], Cell[TextData[StyleBox[ "Although not important in this context, it is significant that the value of \ the vector portion depends upon the observer. This gives a way to \ distinguish between various frequencies of light for example."]], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[" Length in curved spacetime"]], "Subsection", CellMargins->{{0, Inherited}, {Inherited, Inherited}}], Cell[TextData[StyleBox[ "Consider if the origin is located at two different locations in spacetime. \ Characterize each origin as a quaternion, calling the o and o'. In flat \ spacetime, the two origins would be identical. Calculate the interval as \ done above, but account for the change in the origin."]], "Text"], Cell[BoxData[{ \(\((L\_curved = Simplify[\n\t \((\((\((\((q\_2 + \ origin\_2)\) - \ \((q\_1 + \ origin\_1)\))\). \n\t\t\t\(( \((q\_2 + \ origin\_2)\) - \ \((q\_1 + \ origin\_1)\)) \))\)\n\t\t\t /. \ toDelta)\) /. \ toDelta])\).{1, 0, 0, 0}\), RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ InterpretationBox[\(\[CapitalDelta] ot\), Prefix[ f[ ot], \[CapitalDelta]], Editable->False], "+", InterpretationBox[\(\[CapitalDelta] t\), Prefix[ f[ t], \[CapitalDelta]], Editable->False]}], ")"}], "2"], "-", SuperscriptBox[ RowBox[{"(", RowBox[{ InterpretationBox[\(\[CapitalDelta] ox\), Prefix[ f[ ox], \[CapitalDelta]], Editable->False], "+", InterpretationBox[\(\[CapitalDelta] x\), Prefix[ f[ x], \[CapitalDelta]], Editable->False]}], ")"}], "2"], "-", SuperscriptBox[ RowBox[{"(", RowBox[{ InterpretationBox[\(\[CapitalDelta] oy\), Prefix[ f[ oy], \[CapitalDelta]], Editable->False], "+", InterpretationBox[\(\[CapitalDelta] y\), Prefix[ f[ y], \[CapitalDelta]], Editable->False]}], ")"}], "2"], "-", SuperscriptBox[ RowBox[{"(", RowBox[{ InterpretationBox[\(\[CapitalDelta] oz\), Prefix[ f[ oz], \[CapitalDelta]], Editable->False], "+", InterpretationBox[\(\[CapitalDelta] z\), Prefix[ f[ z], \[CapitalDelta]], Editable->False]}], ")"}], "2"]}], ",", "\n", "\t", RowBox[{"2", " ", RowBox[{"(", RowBox[{ InterpretationBox[\(\[CapitalDelta] ot\), Prefix[ f[ ot], \[CapitalDelta]], Editable->False], "+", InterpretationBox[\(\[CapitalDelta] t\), Prefix[ f[ t], \[CapitalDelta]], Editable->False]}], ")"}], " ", RowBox[{"(", RowBox[{ InterpretationBox[\(\[CapitalDelta] ox\), Prefix[ f[ ox], \[CapitalDelta]], Editable->False], "+", InterpretationBox[\(\[CapitalDelta] x\), Prefix[ f[ x], \[CapitalDelta]], Editable->False]}], ")"}]}], ",", "\n", "\t", RowBox[{"2", " ", RowBox[{"(", RowBox[{ InterpretationBox[\(\[CapitalDelta] ot\), Prefix[ f[ ot], \[CapitalDelta]], Editable->False], "+", InterpretationBox[\(\[CapitalDelta] t\), Prefix[ f[ t], \[CapitalDelta]], Editable->False]}], ")"}], " ", RowBox[{"(", RowBox[{ InterpretationBox[\(\[CapitalDelta] oy\), Prefix[ f[ oy], \[CapitalDelta]], Editable->False], "+", InterpretationBox[\(\[CapitalDelta] y\), Prefix[ f[ y], \[CapitalDelta]], Editable->False]}], ")"}]}], ",", "\n", "\t", RowBox[{"2", " ", RowBox[{"(", RowBox[{ InterpretationBox[\(\[CapitalDelta] ot\), Prefix[ f[ ot], \[CapitalDelta]], Editable->False], "+", InterpretationBox[\(\[CapitalDelta] t\), Prefix[ f[ t], \[CapitalDelta]], Editable->False]}], ")"}], " ", RowBox[{"(", RowBox[{ InterpretationBox[\(\[CapitalDelta] oz\), Prefix[ f[ oz], \[CapitalDelta]], Editable->False], "+", InterpretationBox[\(\[CapitalDelta] z\), Prefix[ f[ z], \[CapitalDelta]], Editable->False]}], ")"}]}]}], "}"}]}], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(L\_curved = \(\((\((q + o)\)\ - \ \((q\^\[Prime] + o\^\[Prime])\))\)\^2\n = \((d \((t + t\_o)\)\^2\ - \ d \((X\&\[RightVector] + X\&\[RightVector]\_o)\)\^2, \ 2\ d \((t + t\_o)\)\ d \((X\&\[RightVector] + X\&\[RightVector]\_o)\))\)\)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[TextData[StyleBox[ "Examine the first term more closely by expanding it."]], "Text"], Cell[BoxData[{\(Expand[L\_curved[[1, 1]]]\), RowBox[{ SuperscriptBox[ RowBox[{"(", InterpretationBox[\(\[CapitalDelta] ot\), Prefix[ f[ ot], \[CapitalDelta]], Editable->False], ")"}], "2"], "-", SuperscriptBox[ RowBox[{"(", InterpretationBox[\(\[CapitalDelta] ox\), Prefix[ f[ ox], \[CapitalDelta]], Editable->False], ")"}], "2"], "-", SuperscriptBox[ RowBox[{"(", InterpretationBox[\(\[CapitalDelta] oy\), Prefix[ f[ oy], \[CapitalDelta]], Editable->False], ")"}], "2"], "-", SuperscriptBox[ RowBox[{"(", InterpretationBox[\(\[CapitalDelta] oz\), Prefix[ f[ oz], \[CapitalDelta]], Editable->False], ")"}], "2"], "+", "\n", RowBox[{"2", " ", RowBox[{"(", InterpretationBox[\(\[CapitalDelta] ot\), Prefix[ f[ ot], \[CapitalDelta]], Editable->False], ")"}], " ", RowBox[{"(", InterpretationBox[\(\[CapitalDelta] t\), Prefix[ f[ t], \[CapitalDelta]], Editable->False], ")"}]}], "-", RowBox[{"2", " ", RowBox[{"(", InterpretationBox[\(\[CapitalDelta] ox\), Prefix[ f[ ox], \[CapitalDelta]], Editable->False], ")"}], " ", RowBox[{"(", InterpretationBox[\(\[CapitalDelta] x\), Prefix[ f[ x], \[CapitalDelta]], Editable->False], ")"}]}], "-", RowBox[{"2", " ", RowBox[{"(", InterpretationBox[\(\[CapitalDelta] oy\), Prefix[ f[ oy], \[CapitalDelta]], Editable->False], ")"}], " ", RowBox[{"(", InterpretationBox[\(\[CapitalDelta] y\), Prefix[ f[ y], \[CapitalDelta]], Editable->False], ")"}]}], "-", RowBox[{"2", " ", RowBox[{"(", InterpretationBox[\(\[CapitalDelta] oz\), Prefix[ f[ oz], \[CapitalDelta]], Editable->False], ")"}], " ", RowBox[{"(", InterpretationBox[\(\[CapitalDelta] z\), Prefix[ f[ z], \[CapitalDelta]], Editable->False], ")"}]}], "+", "\n", SuperscriptBox[ RowBox[{"(", InterpretationBox[\(\[CapitalDelta] t\), Prefix[ f[ t], \[CapitalDelta]], Editable->False], ")"}], "2"], "-", SuperscriptBox[ RowBox[{"(", InterpretationBox[\(\[CapitalDelta] x\), Prefix[ f[ x], \[CapitalDelta]], Editable->False], ")"}], "2"], "-", SuperscriptBox[ RowBox[{"(", InterpretationBox[\(\[CapitalDelta] y\), Prefix[ f[ y], \[CapitalDelta]], Editable->False], ")"}], "2"], "-", SuperscriptBox[ RowBox[{"(", InterpretationBox[\(\[CapitalDelta] z\), Prefix[ f[ z], \[CapitalDelta]], Editable->False], ")"}], "2"]}]}], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(\((dt\^2 - d X\&\[RightVector]\^2)\)\ + \ \((dt\_o\^2 - d X\&\[RightVector]\_o\^2)\) + \ 2\ dt\ dt\_o\ - 2\ d X\&\[RightVector]\ \ d X\&\[RightVector]\_o\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[TextData[StyleBox[ "The length in curved spacetime is the square of the interval (invariant \ under a boost) between the two origins, plus the square of the interval \ between the two events, plus a cross term, which will not be invariant under \ a boost. The length is symmetric under exchange of the event with the origin \ translation."]], "Text"], Cell["Lcurved looks similar to the square of the affine parameter:", "Text"], Cell[BoxData[{ \(Expand[\[Lambda]\^2]\), \(b\^2 + 2\ a\ b\ \[Tau] + a\^2\ \[Tau]\^2\)}], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(\[Lambda]\^2 = b\^2 + 2\ a\ b\ \[Tau] + a\^2\ \[Tau]\^2\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[TextData[StyleBox[ "In this case, b^2 is the origin interval squared and a = 1. There is a \ difference in the cross terms. However, in the small curvature limit, delta \ to >> delta Xo, so tau ~ delta to. Under this approximation, the square of \ the affine parameter and Lcurved are the same."]], "Text"], Cell[TextData[StyleBox[ "For a strong gravitational field, Lcurved will be different than the square \ of the affine parameter. The difference will be solely in the nature of the \ cross term. In general relativity, b and tau are invariant under a boost. \ For Lcurved, the cross term should be covariant. Whether this has any \ effects that can be measured needs to be explored."]], "Text"], Cell[TextData[StyleBox[ "There exist quaternions which preserve Lcurved because quaternions are a \ field (I haven't found them yet because the math is getting tough at this \ point!) It is my hope that those quaternions will help solve problems in \ general relativity, as was the case in special relativity."]], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[" Implications"]], "Subsection", CellMargins->{{0, Inherited}, {Inherited, Inherited}}], Cell[TextData[StyleBox[ "A connection to the curved geometry of general relativity was sketched. It \ should be possible to solve problems with this \"curved\" measure. As \ always, all the objects employed were quaternions. Therefore any of the \ previously outline techniques should be applicable. In particular, it will \ be fun in the future to think about things like"]], "Text"], Cell[BoxData[{ \(\((L\_curved = Simplify[\n\t \((\((Transpose[\n\t\t\t\t\t\t \((\((q\_2 + \ origin\_2)\) - \ \((q\_1 + \ origin\_1)\))\)].\n\t\t\t \((\((q\_2 + \ origin\_2)\) - \ \((q\_1 + \ origin\_1)\))\))\)\n\t\t\t /. \ toDelta) \) /. \ toDelta])\).{1, 0, 0, 0}\), RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ InterpretationBox[\(\[CapitalDelta] ot\), Prefix[ f[ ot], \[CapitalDelta]], Editable->False], "+", InterpretationBox[\(\[CapitalDelta] t\), Prefix[ f[ t], \[CapitalDelta]], Editable->False]}], ")"}], "2"], "+", SuperscriptBox[ RowBox[{"(", RowBox[{ InterpretationBox[\(\[CapitalDelta] ox\), Prefix[ f[ ox], \[CapitalDelta]], Editable->False], "+", InterpretationBox[\(\[CapitalDelta] x\), Prefix[ f[ x], \[CapitalDelta]], Editable->False]}], ")"}], "2"], "+", SuperscriptBox[ RowBox[{"(", RowBox[{ InterpretationBox[\(\[CapitalDelta] oy\), Prefix[ f[ oy], \[CapitalDelta]], Editable->False], "+", InterpretationBox[\(\[CapitalDelta] y\), Prefix[ f[ y], \[CapitalDelta]], Editable->False]}], ")"}], "2"], "+", SuperscriptBox[ RowBox[{"(", RowBox[{ InterpretationBox[\(\[CapitalDelta] oz\), Prefix[ f[ oz], \[CapitalDelta]], Editable->False], "+", InterpretationBox[\(\[CapitalDelta] z\), Prefix[ f[ z], \[CapitalDelta]], Editable->False]}], ")"}], "2"]}], ",", "0", ",", "0", ",", "0"}], "}"}]}], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(\(\((\((q + o)\)\ - \ \((q\^\[Prime] + o\^\[Prime])\))\)\^*\) \((\((q + o)\)\ - \ \((q\^\[Prime] + o\^\[Prime])\))\)\n = \(\((d \((t + t\_o)\)\^2\ + \ d \((X\&\[RightVector] + X\&\[RightVector]\_o)\)\^2, \ 2\ d \((t + t\_o)\)\ d \((X\&\[RightVector] + X\&\[RightVector]\_o)\))\)\n = \((\((dt\^2 - d X\&\[RightVector]\^2)\)\ + \ \((dt\_o\^2 - d X\&\[RightVector]\_o\^2)\) + \ 2\ dt\ dt\_o\ + 2\ d X\&\[RightVector]\ \ d X\&\[RightVector]\_o, \ ...)\)\)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[TextData[StyleBox[ "which could open the door to a quantum approach to curvature."]], "Text"] }, Open ]] }, Open ]] }, FrontEndVersion->"Microsoft Windows 3.0", ScreenRectangle->{{0, 640}, {0, 451}}, AutoGeneratedPackage->None, WindowToolbars->{"RulerBar", "EditBar"}, CellGrouping->Automatic, WindowSize->{563, 223}, WindowMargins->{{-5, Automatic}, {Automatic, 5}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, PageHeaders->{{Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"], Inherited, Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"]}, {Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"], Inherited, Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"]}}, PageHeaderLines->{False, Inherited}, PrintingOptions->{"PrintingMargins"->{{72, 57.5625}, {57.5625, 72}}, "PrintCellBrackets"->False, "PrintRegistrationMarks"->False, "PrintMultipleHorizontalPages"->False, "FirstPageHeader"->False}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, 128}}, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, CharacterEncoding->"MacintoshAutomaticEncoding", StyleDefinitions -> Notebook[{ Cell[CellGroupData[{ Cell[TextData[StyleBox["Style Definitions"]], "Subtitle"], Cell[TextData[StyleBox[ "Modify the definitions below to change the default appearance of all cells \ in a given style. 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Cell[TextData[StyleBox[ "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."]], "Text", FontFamily->"Courier", FontSize->14], Cell[CellGroupData[{ Cell[StyleData["Input"], PageWidth->Infinity, CellMargins->{{42, 10}, {5, 7}}, Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, LineSpacing->{1, 0}, AutoItalicWords->{}, FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, CounterIncrements->"Input", AspectRatioFixed->True, FontFamily->"Courier", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[StyleData["Input", "Presentation"], CellMargins->{{72, Inherited}, {8, 10}}, LineSpacing->{1, 0}, FontFamily->"Courier", FontSize->14, FontWeight->"Plain"], Cell[StyleData["Input", "Condensed"], CellMargins->{{40, 10}, {2, 3}}, FontFamily->"Courier", FontSize->14, FontWeight->"Plain"], Cell[StyleData["Input", "Printout"], CellMargins->{{39, 0}, {4, 6}}, FontFamily->"Courier", FontSize->14, FontWeight->"Plain"] }, Closed]], Cell[StyleData["InputOnly"], Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, DefaultFormatType->DefaultInputFormatType, AutoItalicWords->{}, FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, CounterIncrements->"Input", StyleMenuListing->None, FontFamily->"Courier", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell[StyleData["Output"], PageWidth->Infinity, CellMargins->{{42, 10}, {7, 5}}, CellEditDuplicate->True, Evaluatable->False, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, LineSpacing->{1, 0}, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Output", AspectRatioFixed->True, FontFamily->"Courier", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[StyleData["Output", "Presentation"], CellMargins->{{72, Inherited}, {10, 8}}, LineSpacing->{1, 0}, FontFamily->"Courier", FontSize->14], Cell[StyleData["Output", "Condensed"], CellMargins->{{41, Inherited}, {3, 2}}, FontFamily->"Courier", FontSize->14], Cell[StyleData["Output", "Printout"], CellMargins->{{39, 0}, {6, 4}}, FontFamily->"Courier", FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Message"], PageWidth->Infinity, CellMargins->{{42, Inherited}, {Inherited, Inherited}}, Evaluatable->False, CellGroupingRules->"OutputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, LineSpacing->{1, 0}, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Message", AspectRatioFixed->True, StyleMenuListing->None, FontFamily->"Courier", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0.500008, 0, 0], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[StyleData["Message", "Presentation"], CellMargins->{{72, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}, FontFamily->"Courier", FontSize->14], Cell[StyleData["Message", "Condensed"], CellMargins->{{41, Inherited}, {Inherited, Inherited}}, FontFamily->"Courier", FontSize->14], Cell[StyleData["Message", "Printout"], CellMargins->{{39, Inherited}, {Inherited, Inherited}}, FontFamily->"Courier", FontSize->14, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Print"], PageWidth->Infinity, CellMargins->{{42, Inherited}, {Inherited, Inherited}}, Evaluatable->False, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, LineSpacing->{1, 0}, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Print", AspectRatioFixed->True, StyleMenuListing->None, FontFamily->"Courier", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[StyleData["Print", "Presentation"], CellMargins->{{72, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}, FontFamily->"Courier", FontSize->14], Cell[StyleData["Print", "Condensed"], CellMargins->{{41, Inherited}, {Inherited, Inherited}}, FontFamily->"Courier", FontSize->14], Cell[StyleData["Print", "Printout"], CellMargins->{{39, Inherited}, {Inherited, Inherited}}, FontFamily->"Courier", FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Graphics"], PageWidth->Infinity, CellMargins->{{7, Inherited}, {Inherited, Inherited}}, Evaluatable->False, CellGroupingRules->"GraphicsGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, CounterIncrements->"Graphics", AspectRatioFixed->True, ImageSize->{387, 393}, ImageMargins->{{34, Inherited}, {Inherited, 0}}, StyleMenuListing->None, FontFamily->"Courier", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[StyleData["Graphics", "Presentation"], ImageMargins->{{62, Inherited}, {Inherited, 0}}, FontFamily->"Courier", FontSize->14], Cell[StyleData["Graphics", "Condensed"], ImageSize->{175, 175}, ImageMargins->{{38, Inherited}, {Inherited, 0}}, FontFamily->"Courier", FontSize->14], Cell[StyleData["Graphics", "Printout"], ImageSize->{250, 250}, ImageMargins->{{30, Inherited}, {Inherited, 0}}, FontFamily->"Courier", FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["CellLabel"], StyleMenuListing->None, FontFamily->"Courier", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[StyleData["CellLabel", "Presentation"], FontFamily->"Courier", FontSize->14], Cell[StyleData["CellLabel", "Condensed"], FontFamily->"Courier", FontSize->14], Cell[StyleData["CellLabel", "Printout"], FontFamily->"Courier", FontSize->14, FontColor->GrayLevel[0]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Formulas and Programming"]], "Section"], Cell[CellGroupData[{ Cell[StyleData["InlineFormula"], CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, ScriptLevel->1, SingleLetterItalics->True], Cell[StyleData["InlineFormula", "Presentation"], CellMargins->{{24, 10}, {10, 10}}, LineSpacing->{1, 5}], Cell[StyleData["InlineFormula", "Condensed"], CellMargins->{{8, 10}, {6, 6}}, LineSpacing->{1, 1}], Cell[StyleData["InlineFormula", "Printout"], CellMargins->{{2, 0}, {6, 6}}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["DisplayFormula"], CellMargins->{{42, Inherited}, {Inherited, Inherited}}, CellHorizontalScrolling->True, DefaultFormatType->DefaultInputFormatType, ScriptLevel->0, SingleLetterItalics->True, UnderoverscriptBoxOptions->{LimitsPositioning->True}], Cell[StyleData["DisplayFormula", "Presentation"], LineSpacing->{1, 5}], Cell[StyleData["DisplayFormula", "Condensed"], LineSpacing->{1, 1}], Cell[StyleData["DisplayFormula", "Printout"]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Styles for Headers and Footers"]], "Section"], Cell[StyleData["Header"], CellMargins->{{7, 0}, {4, 1}}, Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True, StyleMenuListing->None, FontFamily->"Times", FontSize->12, FontSlant->"Italic"], Cell[StyleData["Footer"], CellMargins->{{7, 0}, {0, 4}}, Evaluatable->False, PageBreakWithin->Automatic, TextAlignment->Center, AspectRatioFixed->True, StyleMenuListing->None, FontFamily->"Times", FontSize->12, FontSlant->"Italic"], Cell[StyleData["PageNumber"], CellMargins->{{0, 0}, {4, 1}}, StyleMenuListing->None, FontFamily->"Times", FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Palette Styles"]], "Section"], Cell[TextData[StyleBox[ "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[ ], #, After]}]&)}], Cell[StyleData["Evaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], 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[TextData[StyleBox["Hyperlink Styles"]], "Section"], Cell[TextData[StyleBox[ "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"]], Cell[StyleData["Hyperlink", "Condensed"]], Cell[StyleData["Hyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[TextData[StyleBox[ "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"]], Cell[StyleData["MainBookLink", "Condensed"]], Cell[StyleData["MainBookLink", "Printout"], 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"]], Cell[StyleData["AddOnsLink", "Condensed"]], Cell[StyleData["AddOnLink", "Printout"], 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[ "RefGuideLink", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["RefGuideLink", "Presentation"]], Cell[StyleData["RefGuideLink", "Condensed"]], Cell[StyleData["RefGuideLink", "Printout"], 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"]], Cell[StyleData["GettingStartedLink", "Condensed"]], Cell[StyleData["GettingStartedLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["OtherInformationLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "OtherInformation", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["OtherInformationLink", "Presentation"]], Cell[StyleData["OtherInformationLink", "Condensed"]], Cell[StyleData["OtherInformationLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Placeholder Styles"]], "Section"], Cell[TextData[StyleBox[ "The cells below define styles useful for making placeholder objects in \ palette templates."]], "Text"], Cell[CellGroupData[{ Cell[StyleData["Placeholder"], Editable->False, Selectable->False, StyleBoxAutoDelete->True, Placeholder->True, StyleMenuListing->None], Cell[StyleData["Placeholder", "Presentation"]], Cell[StyleData["Placeholder", "Condensed"]], Cell[StyleData["Placeholder", "Printout"]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SelectionPlaceholder"], Editable->False, Selectable->False, StyleBoxAutoDelete->True, Placeholder->Primary, StyleMenuListing->None, DrawHighlighted->True], Cell[StyleData["SelectionPlaceholder", "Presentation"]], Cell[StyleData["SelectionPlaceholder", "Condensed"]], Cell[StyleData["SelectionPlaceholder", "Printout"]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["FormatType Styles"]], "Section"], Cell[TextData[StyleBox[ "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."]], "Text"], Cell[StyleData["CellExpression"], PageWidth->Infinity, CellMargins->{{6, Inherited}, {Inherited, Inherited}}, ShowCellLabel->False, ShowSpecialCharacters->False, AllowInlineCells->False, AutoItalicWords->{}, StyleMenuListing->None, FontFamily->"Courier", Background->GrayLevel[1]], Cell[StyleData["InputForm"], AllowInlineCells->False, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["OutputForm"], PageWidth->Infinity, TextAlignment->Left, LineSpacing->{1, -5}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["StandardForm"], LineSpacing->{1.25, 0}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["TraditionalForm"], LineSpacing->{1.25, 0}, SingleLetterItalics->True, TraditionalFunctionNotation->True, DelimiterMatching->None, StyleMenuListing->None], Cell[TextData[StyleBox[ "The style defined below is mixed in to any cell that is in an inline cell \ within another."]], "Text"], Cell[StyleData["InlineCell"], TextAlignment->Left, ScriptLevel->1, StyleMenuListing->None], Cell[StyleData["InlineCellEditing"], StyleMenuListing->None, Background->RGBColor[1, 0.749996, 0.8]] }, Closed]] }, Open ]] }] ] (*********************************************************************** 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. 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