(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 4.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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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[ 16103, 468]*) (*NotebookOutlinePosition[ 53138, 1706]*) (* CellTagsIndexPosition[ 53094, 1702]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Shooting", "Section", FontSize->24], Cell["Set up the right-hand side for the widget-manufacturing ODE:", "Text", FontSize->19], Cell[TextData[{ "y ' = v\nv ' = ", Cell[BoxData[ \(TraditionalForm\`125\)]], "- ", Cell[BoxData[ \(TraditionalForm\`\(3\ \@y\)\/40\)]] }], "Text", TextAlignment->Center, FontSize->24], Cell[BoxData[ \(Clear[x, t]; f[t_, x_]\ := \ {x[\([2]\)], 125 - \(3\ \@x[\([1]\)]\)\/40}\)], "Input"], Cell["\<\ Define a function finalprod[v0_] which, given an initial value v0 for v, runs \ Runge-Kutta with y(0)=0, v(0)=v0, and then outputs the value of y at t=200.\ \>", "Text", FontSize->19], Cell[BoxData[ RowBox[{ RowBox[{\(finalprod[v0_]\), " ", ":=", "\[IndentingNewLine]", RowBox[{"Block", "[", RowBox[{\({h, ti, tf, nsteps, x, t, k1, k2, k3, k4}\), ",", "\[IndentingNewLine]", RowBox[{\(h = 0.2\), ";", \(ti = 0\), ";", \(tf = 200\), ";", \(nsteps = \((tf - ti)\)/h\), ";", "\[IndentingNewLine]", StyleBox[\(x = {0, v0}\), FontSize->24], StyleBox[";", FontSize->24], StyleBox[\(t = 0\), FontSize->24], StyleBox[";", FontSize->24], "\[IndentingNewLine]", \(Do[\[IndentingNewLine]k1 = h*f[t, x]; \[IndentingNewLine]k2 = h*f[t + h/2, x + k1/2]; \[IndentingNewLine]k3 = h*f[t + h/2, x + k2/2]; \[IndentingNewLine]k4 = h*f[t + h, x + k3]; \[IndentingNewLine]x = x + \((k1 + 2*k2 + 2*k3 + k4)\)/6; \[IndentingNewLine]t = t + h, \[IndentingNewLine]{j, 1, nsteps}]\), ";", "\[IndentingNewLine]", \(x[\([1]\)]\)}]}], "\[IndentingNewLine]", "]"}]}], "\[IndentingNewLine]"}]], "Input"], Cell["\<\ Our goal is to find the value of v0 for which finalprod[v0] = 2000000, since \ the problem says we must produce 2000000 widgets after 200 days. To start, \ we can experiment with some values for v0 by hand:\ \>", "Text", FontSize->19], Cell[BoxData[ \(finalprod[10000]\)], "Input"], Cell["\<\ Then we can set up a secant-method iteration to solve phi[v0] = \ finalprod[v0]-2000000=0.\ \>", "Text", FontSize->19], Cell[BoxData[ \(phi[v0_]\ := \ finalprod[v0] - 2000000\)], "Input"], Cell[BoxData[{ \(\(ztwoago\ = \ 1000;\)\), "\[IndentingNewLine]", \(\(Print[ztwoago];\)\), "\[IndentingNewLine]", \(\(zprev\ = \ 2000;\)\), "\[IndentingNewLine]", \(\(Print[zprev];\)\), "\[IndentingNewLine]", \(While[ Abs[zprev - ztwoago]\ > \ 1, \[IndentingNewLine]z\ = \ zprev\ - \ \(phi[zprev] \((zprev - ztwoago)\)\)\/\(phi[zprev] - \ phi[ztwoago]\); \[IndentingNewLine]Print[z]; \[IndentingNewLine]ztwoago = zprev; \[IndentingNewLine]zprev = z]\), "\[IndentingNewLine]", \(Print["\", \((zprev + ztwoago)\)/2]\)}], "Input"], Cell["\<\ Now that we know what initial value of v we should use, we can re-run \ Runge-Kutta with that value and output the entire function y(t):\ \>", "Text", FontSize->19], Cell[BoxData[{ \(ti = 0; tf = 200; h = 0.2; nsteps = \((tf - ti)\)/h;\), "\[IndentingNewLine]", \(xlist = Table[{0, 0}, {nsteps + 1}]; tlist = Table[{}, {nsteps + 1}];\), "\[IndentingNewLine]", \(xlist[\([1]\)] = {0, 1818.87}; tlist[\([1]\)] = 0;\), "\[IndentingNewLine]", \(Do[\[IndentingNewLine]x = xlist[\([j]\)]; t = tlist[\([j]\)]; \[IndentingNewLine]k1 = h*f[t, x]; \[IndentingNewLine]k2 = h*f[t + h/2, x + k1/2]; \[IndentingNewLine]k3 = h*f[t + h/2, x + k2/2]; \[IndentingNewLine]k4 = h*f[t + h, x + k3]; \[IndentingNewLine]xlist[\([j + 1]\)] = x + \((k1 + 2*k2 + 2*k3 + k4)\)/ 6; \[IndentingNewLine]tlist[\([j + 1]\)] = t + h, \[IndentingNewLine]{j, 1, nsteps}]\), "\[IndentingNewLine]", \(\(ylist\ = \ \((Transpose[ xlist])\)[\([1]\)];\)\), "\[IndentingNewLine]", \(\(plshoot = ListPlot[Transpose[{tlist, ylist}], PlotJoined \[Rule] True];\)\)}], "Input"], Cell["", "Subsubtitle"] }, Closed]], Cell[CellGroupData[{ Cell["Finite Differences", "Section", FontSize->24], Cell[TextData[{ "We can solve the same problem by finite differences. First we subdivide \ the time interval [0,200] into some number of subintervals, let's say N=20. \ So, the intermediate times will be ", Cell[BoxData[ \(TraditionalForm\`t\_0\)]], "=0, ", Cell[BoxData[ \(TraditionalForm\`t\_1\)]], "=10, ", Cell[BoxData[ \(TraditionalForm\`t\_2\)]], "=20, ..., ", Cell[BoxData[ \(TraditionalForm\`t\_19\)]], "= 190, ", Cell[BoxData[ \(TraditionalForm\`t\_20\)]], "= 200. Our fundamental unknowns will be ", Cell[BoxData[ \(TraditionalForm\`y\_1\)]], "= y(", Cell[BoxData[ \(TraditionalForm\`t\_1\)]], "), ... ", Cell[BoxData[ \(TraditionalForm\`y\_19\)]], "= y(", Cell[BoxData[ \(TraditionalForm\`t\_19\)]], ").\nAs above, the fundamental differential equation is:" }], "Text", FontSize->19], Cell[BoxData[ RowBox[{\(y''\), "=", RowBox[{"125", "-", FormBox[\(\(3\ \@y\)\/40\), "TraditionalForm"]}]}]], "Text", TextAlignment->Center, FontSize->19], Cell["When discretized by finite differences, we find:", "Text", FontSize->19], Cell[BoxData[ \(\(g\_i\) \((y\_1, y\_2, \ ... \ , \ y\_\(N - 1\))\)\ = \ \(\(y\_\(i + 1\) - 2 y\_i + y\_\(i - \ 1\)\)\/h\^2 - \ \((125\ - \ \(3 \@ y\_i\)\/40)\) = 0\)\)], "Text", TextAlignment->Center, FontSize->17], Cell[TextData[{ "Let's define these N-1 functions into a single vector function g. (Note \ that for i=1, ", Cell[BoxData[ \(TraditionalForm\`y\_\(i - 1\)\)]], "= ", Cell[BoxData[ \(TraditionalForm\`y\_0\)]], " is known to be 0, and for i=19, ", Cell[BoxData[ \(TraditionalForm\`y\_\(i + 1\)\)]], "= ", Cell[BoxData[ \(TraditionalForm\`y\_20\)]], " is known to be 2000000:" }], "Text", FontSize->19], Cell[BoxData[ \(g[y_]\ := \ \[IndentingNewLine]Block[{vec, h, i, prev, next}, \[IndentingNewLine]vec\ = \ Table[0, {19}]; \[IndentingNewLine]h = 10; \[IndentingNewLine]Do[\ \[IndentingNewLine]If[i \[Equal] 1, prev = 0, prev = y[\([i - 1]\)]]; \[IndentingNewLine]If[ i \[Equal] 19, next = 2000000, next = y[\([i + 1]\)]]; \[IndentingNewLine]vec[\([i]\)]\ = \ \(next - \ 2*y[\([i]\)] + prev\)\/h\^2\ - \ \((125\ - \ \(3 \@ \ Abs[y[\([i]\)]]\)\/40)\), \[IndentingNewLine]{i, 1, 19}\[IndentingNewLine]]; \ \[IndentingNewLine]vec\[IndentingNewLine]]\)], "Input"], Cell["Now we define the Jacobian of g:", "Text", FontSize->19], Cell[BoxData[ \(<< LinearAlgebra`MatrixManipulation`\)], "Input"], Cell[BoxData[ \(Jg[y_]\ := \ \[IndentingNewLine]Block[{mat, h, i, prev, next}, \[IndentingNewLine]mat\ = \ ZeroMatrix[19]; \[IndentingNewLine]h = 10; \[IndentingNewLine]mat[\([1, 2]\)] = 1\/h\^2; mat[\([1, 1]\)] = \(-\(2\/h\^2\)\) + \ 3\/\(80 \@ Abs[y[\([1]\)]]\); \[IndentingNewLine]Do[\ \ \[IndentingNewLine]mat[\([i, i - 1]\)]\ = \ 1\/h\^2\ ; \[IndentingNewLine]mat[\([i, i]\)] = \(-\(2\/h\^2\)\) + \ 3\/\(80 \@ Abs[y[\([i]\)]]\); \[IndentingNewLine]mat[\([i, i + 1]\)]\ = \ 1\/h\^2\ , \[IndentingNewLine]{i, 2, 18}\[IndentingNewLine]]; \[IndentingNewLine]mat[\([19, 18]\)] = 1\/h\^2; mat[\([19, 19]\)] = \(-\(2\/h\^2\)\) + \ 3\/\(80 \@ Abs[y[\([19]\)]]\); \[IndentingNewLine]mat\ \[IndentingNewLine]]\)], "Input"], Cell[TextData[{ "Now run Newton's Method. As an initial guess, assume that y(t) is a \ linear function connecting y(0)=0 to y(200)=2000000, i.e., y(t) = 10000 t, so \ that ", Cell[BoxData[ \(TraditionalForm\`y\_i\)]], " = y(ih) = y(10i) = 100000 i:" }], "Text", FontSize->19], Cell[BoxData[{ \(\(y\ = \ Table[100000. *i, {i, 1, 19}];\)\), "\[IndentingNewLine]", \(\(gy\ = \ g[y];\)\), "\[IndentingNewLine]", \(\(j = 0;\)\), "\[IndentingNewLine]", \(\(While[ Max[Abs[gy]]\ > \ 10\ && \ j < 10, \[IndentingNewLine]y\ = \ y\ - \ LinearSolve[Jg[y], gy]; \[IndentingNewLine]gy\ = \ g[y]; \[IndentingNewLine]j = j + 1;\[IndentingNewLine]];\)\), "\[IndentingNewLine]", \(\(If[j < 10, Print["\", \ y], Print["\"]];\)\)}], "Input"], Cell[BoxData[{ \(\(tlist = Table[10*i, {i, 0, 20}];\)\), "\[IndentingNewLine]", \(\(ylist\ = \ Prepend[y, 0];\)\), "\[IndentingNewLine]", \(\(ylist\ = \ Append[ylist, 2000000];\)\), "\[IndentingNewLine]", \(plfd = ListPlot[Transpose[{tlist, ylist}], PlotJoined \[Rule] True]\)}], "Input"], Cell[BoxData[ \(Show[plshoot, plfd]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Finite Differences (Linear Case)", "Section", FontSize->24], Cell[TextData[{ "Now let's solve a linear ODE by finite differences. We will do a \ variation of the widget problem (see below). As before, first we subdivide \ the time interval [0,200] into some number of subintervals, let's say N=20. \ So, the intermediate times will be ", Cell[BoxData[ \(TraditionalForm\`t\_0\)]], "=0, ", Cell[BoxData[ \(TraditionalForm\`t\_1\)]], "=10, ", Cell[BoxData[ \(TraditionalForm\`t\_2\)]], "=20, ..., ", Cell[BoxData[ \(TraditionalForm\`t\_19\)]], "= 190, ", Cell[BoxData[ \(TraditionalForm\`t\_20\)]], "= 200. Our fundamental unknowns will be ", Cell[BoxData[ \(TraditionalForm\`y\_1\)]], "= y(", Cell[BoxData[ \(TraditionalForm\`t\_1\)]], "), ... ", Cell[BoxData[ \(TraditionalForm\`y\_19\)]], "= y(", Cell[BoxData[ \(TraditionalForm\`t\_19\)]], ").\nNow take the fundamental differential equation to be a variation of \ the widget differential equation:" }], "Text", FontSize->19], Cell[BoxData[ RowBox[{\(y''\), "=", RowBox[{"125", "-", FormBox[\(y\/8000\), "TraditionalForm"]}]}]], "Text", TextAlignment->Center, FontSize->19], Cell["When discretized by finite differences, we find:", "Text", FontSize->19], Cell[BoxData[ \(\(g\_i\) \((y\_1, y\_2, \ ... \ , \ y\_\(N - 1\))\)\ = \ \(\(y\_\(i + 1\) - 2 y\_i + y\_\(i - \ 1\)\)\/h\^2 - \ \((125\ - \ y\_i\/8000)\) = 0\)\)], "Text", TextAlignment->Center, FontSize->17], Cell[TextData[{ "Let's define these N-1 functions into a single vector function g. (Note \ that for i=1, ", Cell[BoxData[ \(TraditionalForm\`y\_\(i - 1\)\)]], "= ", Cell[BoxData[ \(TraditionalForm\`y\_0\)]], " is known to be 0, and for i=19, ", Cell[BoxData[ \(TraditionalForm\`y\_\(i + 1\)\)]], "= ", Cell[BoxData[ \(TraditionalForm\`y\_20\)]], " is known to be 2000000:" }], "Text", FontSize->19], Cell[BoxData[ \(g[y_]\ := \ \[IndentingNewLine]Block[{vec, h, i, prev, next}, \[IndentingNewLine]vec\ = \ Table[0, {19}]; \[IndentingNewLine]h = 10; \[IndentingNewLine]Do[\ \[IndentingNewLine]If[i \[Equal] 1, prev = 0, prev = y[\([i - 1]\)]]; \[IndentingNewLine]If[ i \[Equal] 19, next = 2000000, next = y[\([i + 1]\)]]; \[IndentingNewLine]vec[\([i]\)]\ = \ \(next - \ 2*y[\([i]\)] + prev\)\/h\^2\ - \ \((125\ - \ y[\([i]\)]\/8000)\), \[IndentingNewLine]{i, 1, 19}\[IndentingNewLine]]; \ \[IndentingNewLine]vec\[IndentingNewLine]]\)], "Input"], Cell["Now we define the Jacobian of g:", "Text", FontSize->19], Cell[BoxData[ \(<< LinearAlgebra`MatrixManipulation`\)], "Input"], Cell[BoxData[ \(Jg[y_]\ := \ \[IndentingNewLine]Block[{mat, h, i, prev, next}, \[IndentingNewLine]mat\ = \ ZeroMatrix[19]; \[IndentingNewLine]h = 10; \[IndentingNewLine]mat[\([1, 2]\)] = 1\/h\^2; mat[\([1, 1]\)] = \(-\(2\/h\^2\)\) + \ 1\/8000; \[IndentingNewLine]Do[\ \[IndentingNewLine]mat[\([i, i - 1]\)]\ = \ 1\/h\^2\ ; \[IndentingNewLine]mat[\([i, i]\)] = \(-\(2\/h\^2\)\) + \ 1\/8000; \[IndentingNewLine]mat[\([i, i + 1]\)]\ = \ 1\/h\^2\ , \[IndentingNewLine]{i, 2, 18}\[IndentingNewLine]]; \[IndentingNewLine]mat[\([19, 18]\)] = 1\/h\^2; mat[\([19, 19]\)] = \(-\(2\/h\^2\)\) + \ 1\/8000; \[IndentingNewLine]mat\[IndentingNewLine]]\)], "Input"], Cell[TextData[{ "Now run Newton's Method. 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FontFamily->"Times", FontSize->13, FontWeight->"Bold"], Cell[StyleData["Subsubsection", "Presentation"], CellMargins->{{29, 10}, {8, 26}}, LineSpacing->{1, 0}, FontSize->18], Cell[StyleData["Subsubsection", "Condensed"], CellMargins->{{22, Inherited}, {2, 12}}, FontSize->10], Cell[StyleData["Subsubsection", "Printout"], CellMargins->{{21, 0}, {4, 20}}, FontSize->11] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Body Text", "Section"], Cell[CellGroupData[{ Cell[StyleData["Text"], CellMargins->{{12, 10}, {5, 5}}, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, -0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"}, Hyphenation->True, LineSpacing->{1, 3}, ParagraphSpacing->{0, 12}, CounterIncrements->"Text", FontFamily->"Times"], Cell[StyleData["Text", "Presentation"], CellMargins->{{13, 10}, {8, 8}}, LineSpacing->{1, 5}, ParagraphSpacing->{0, 12}], Cell[StyleData["Text", "Condensed"], CellMargins->{{8, 10}, {4, 4}}, LineSpacing->{1, 1}, ParagraphSpacing->{0, 4}], Cell[StyleData["Text", "Printout"], CellMargins->{{9, 0}, {4, 4}}, ParagraphSpacing->{0, 6}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SmallText"], CellMargins->{{12, 10}, {5, 5}}, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, -0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"}, Hyphenation->True, LineSpacing->{1, 3}, ParagraphSpacing->{0, 6}, CounterIncrements->"SmallText", FontFamily->"Helvetica", FontSize->9], Cell[StyleData["SmallText", "Presentation"], CellMargins->{{13, 10}, {8, 8}}, LineSpacing->{1, 5}, FontSize->12], Cell[StyleData["SmallText", "Condensed"], CellMargins->{{8, 10}, {2, 2}}, LineSpacing->{1, 2}, FontSize->9], Cell[StyleData["SmallText", "Printout"], CellMargins->{{9, 0}, {4, 4}}, 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.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Input"], CellFrame->{{3, 0}, {0, 0}}, CellMargins->{{52, 10}, {8, 8}}, Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, CellLabelMargins->{{5, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, LinebreakAdjustments->{0.85, 2, 10, 0, 1}, CounterIncrements->"Input", FontWeight->"Bold", Background->RGBColor[1, 0.700008, 0.4]], Cell[StyleData["Input", "Presentation"], CellMargins->{{62, Inherited}, {10, 10}}, LineSpacing->{1, 0}], Cell[StyleData["Input", "Condensed"], CellMargins->{{40, 10}, {4, 4}}], Cell[StyleData["Input", "Printout"], CellMargins->{{44, 0}, {6, 6}}, LinebreakAdjustments->{0.85, 2, 10, 1, 1}, Background->GrayLevel[0.8]] }, Closed]], Cell[StyleData["InlineInput"], Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, CounterIncrements->"Input", FontWeight->"Bold"], Cell[CellGroupData[{ Cell[StyleData["Output"], CellFrame->{{3, 0}, {0, 0}}, CellMargins->{{52, 10}, {8, 8}}, CellEditDuplicate->True, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, CellLabelMargins->{{3, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", FormatType->InputForm, CounterIncrements->"Output", Background->RGBColor[0, 1, 1]], Cell[StyleData["Output", "Presentation"], CellMargins->{{62, Inherited}, {12, 5}}, LineSpacing->{1, 0}], Cell[StyleData["Output", "Condensed"], CellMargins->{{40, Inherited}, {4, 1}}], Cell[StyleData["Output", "Printout"], CellMargins->{{44, 0}, {6, 2}}, Background->GrayLevel[0.900008]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Message"], CellMargins->{{62, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, FormatType->InputForm, CounterIncrements->"Message", StyleMenuListing->None, FontColor->RGBColor[1, 0, 0]], Cell[StyleData["Message", "Presentation"], CellMargins->{{74, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Message", "Condensed"], CellMargins->{{50, Inherited}, {Inherited, Inherited}}], Cell[StyleData["Message", "Printout"], CellMargins->{{54, Inherited}, {Inherited, Inherited}}, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Print"], CellMargins->{{62, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, FormatType->InputForm, CounterIncrements->"Print", StyleMenuListing->None, Background->RGBColor[0, 1, 0]], Cell[StyleData["Print", "Presentation"], CellMargins->{{74, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Print", "Condensed"], CellMargins->{{50, Inherited}, {Inherited, Inherited}}], Cell[StyleData["Print", "Printout"], CellMargins->{{54, Inherited}, {Inherited, Inherited}}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Graphics"], CellMargins->{{62, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"GraphicsGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, CounterIncrements->"Graphics", StyleMenuListing->None], Cell[StyleData["Graphics", "Presentation"], CellMargins->{{74, Inherited}, {Inherited, Inherited}}], Cell[StyleData["Graphics", "Condensed"], CellMargins->{{52, Inherited}, {Inherited, Inherited}}, ImageSize->{175, 175}], Cell[StyleData["Graphics", "Printout"], CellMargins->{{54, Inherited}, {Inherited, Inherited}}, ImageSize->{250, 250}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["CellLabel"], StyleMenuListing->None, FontFamily->"Helvetica", FontSize->11, FontWeight->"Bold", FontColor->RGBColor[0.571389, 0.19675, 0.570504]], Cell[StyleData["CellLabel", "Presentation"], FontSize->12], Cell[StyleData["CellLabel", "Condensed"], FontSize->8], Cell[StyleData["CellLabel", "Printout"], FontSize->8, FontColor->GrayLevel[0]] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Unique Styles", "Section"], Cell[CellGroupData[{ Cell[StyleData["Author"], ShowCellBracket->False, CellMargins->{{10, 4}, {2, 10}}, LineSpacing->{1, 5}, FontSize->16, FontSlant->"Italic"], Cell[StyleData["Author", "Presentation"], CellMargins->{{12, 10}, {2, 12}}, LineSpacing->{1, 5}, ParagraphSpacing->{0, 12}, FontSize->20], Cell[StyleData["Author", "Condensed"], CellMargins->{{8, 10}, {1, 4}}, LineSpacing->{1, 1}, ParagraphSpacing->{0, 4}, FontSize->12], Cell[StyleData["Author", "Printout"], CellMargins->{{9, 0}, {4, 12}}, ParagraphSpacing->{0, 6}, FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Copyright"], ShowCellBracket->False, CellMargins->{{10, 10}, {40, 2}}, FontFamily->"Helvetica", FontSize->9], Cell[StyleData["Copyright", "Presentation"], CellMargins->{{12, 10}, {50, 2}}, LineSpacing->{1, 5}, FontSize->12], Cell[StyleData["Copyright", "Condensed"], CellMargins->{{8, 10}, {12, 1}}, LineSpacing->{1, 2}, FontSize->9], Cell[StyleData["Copyright", "Printout"], CellMargins->{{9, 0}, {72, 4}}, FontSize->7] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Example"], CellMargins->{{12, 10}, {5, 12}}, LineSpacing->{1, 3}, ParagraphSpacing->{0, 12}, CounterIncrements->"Example", FontFamily->"Times", FontWeight->"Bold"], Cell[StyleData["Example", "Presentation"], CellMargins->{{18, 10}, {8, 20}}, LineSpacing->{1, 5}, ParagraphSpacing->{0, 12}], Cell[StyleData["Example", "Condensed"], CellMargins->{{8, 10}, {4, 8}}, LineSpacing->{1, 1}, ParagraphSpacing->{0, 4}], Cell[StyleData["Example", "Printout"], CellMargins->{{9, 0}, {4, 10}}, ParagraphSpacing->{0, 6}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Definition"], CellFrame->{{3, 0}, {0, 0}}, CellMargins->{{52, 10}, {8, 8}}, PageBreakWithin->False, GroupPageBreakWithin->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, Hyphenation->True, ShowStringCharacters->True, CounterIncrements->"Definition", FontFamily->"Helvetica", FontWeight->"Bold", FontColor->GrayLevel[1], Background->RGBColor[0.2, 0.700008, 0.700008]], Cell[StyleData["Definition", "Presentation"], CellMargins->{{62, Inherited}, {5, 12}}, LineSpacing->{1, 0}], Cell[StyleData["Definition", "Condensed"], CellMargins->{{40, 10}, {1, 4}}], Cell[StyleData["Definition", "Printout"], CellMargins->{{44, 0}, {2, 6}}, Background->GrayLevel[0.6]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Theorem"], CellFrame->{{3, 0}, {0, 0}}, CellMargins->{{52, 10}, {8, 8}}, PageBreakWithin->False, GroupPageBreakWithin->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, Hyphenation->True, ShowStringCharacters->True, CounterIncrements->"Theorem", FontFamily->"Helvetica", FontWeight->"Bold", FontColor->GrayLevel[1], Background->RGBColor[0.571389, 0.19675, 0.570504]], Cell[StyleData["Theorem", "Presentation"], CellMargins->{{62, Inherited}, {5, 12}}, LineSpacing->{1, 0}], Cell[StyleData["Theorem", "Condensed"], CellMargins->{{40, 10}, {1, 4}}], Cell[StyleData["Theorem", "Printout"], CellMargins->{{44, 0}, {2, 6}}, Background->GrayLevel[0.4]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["ExerciseMain"], CellFrame->{{6, 0}, {0, 1}}, CellMargins->{{12, Inherited}, {4, 20}}, CellGroupingRules->{"SectionGrouping", 30}, PageBreakBelow->False, CellFrameColor->RGBColor[0.571389, 0.19675, 0.570504], LineSpacing->{1, 7}, CounterIncrements->"Subsection", CounterAssignments->{{"Subsubsection", 0}}, FontFamily->"Helvetica", FontSize->16, FontWeight->"Bold", FontColor->RGBColor[0.571389, 0.19675, 0.570504]], Cell[StyleData["ExerciseMain", "Presentation"], CellMargins->{{18, 10}, {8, 32}}, LineSpacing->{1, 2}, FontSize->24, FontTracking->"Condensed"], Cell[StyleData["ExerciseMain", "Condensed"], CellMargins->{{8, Inherited}, {2, 12}}, FontSize->12], Cell[StyleData["ExerciseMain", "Printout"], CellMargins->{{9, 0}, {2, 50}}, CellFrameColor->GrayLevel[0.500008], FontSize->14, FontTracking->"Plain", FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Exercise"], CellDingbat->"\[FilledDownTriangle]", CellMargins->{{23, Inherited}, {4, 18}}, CellGroupingRules->{"SectionGrouping", 50}, PageBreakBelow->False, Hyphenation->True, LineSpacing->{1, 7}, CounterIncrements->"Subsubsection", FontFamily->"Times", FontSize->13, FontWeight->"Bold", FontColor->RGBColor[0.571389, 0.19675, 0.570504]], Cell[StyleData["Exercise", "Presentation"], CellMargins->{{33, 10}, {8, 26}}, LineSpacing->{1, 0}, FontSize->18], Cell[StyleData["Exercise", "Condensed"], CellMargins->{{17, Inherited}, {2, 12}}, FontSize->10], Cell[StyleData["Exercise", "Printout"], CellFrame->{{0, 0}, {0.5, 0}}, CellDingbat->None, CellMargins->{{9, 0}, {6, 20}}, FontSize->11, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["ExerciseText"], CellMargins->{{62, 10}, {5, 5}}, Hyphenation->True, LineSpacing->{1, 3}, ParagraphSpacing->{0, 8}, CounterIncrements->"ExcersiceText", FontFamily->"Times"], Cell[StyleData["ExerciseText", "Presentation"], CellMargins->{{74, 10}, {8, 8}}, LineSpacing->{1, 5}, ParagraphSpacing->{0, 12}], Cell[StyleData["ExerciseText", "Condensed"], CellMargins->{{52, 10}, {2, 2}}, LineSpacing->{1, 1}, ParagraphSpacing->{0, 4}], Cell[StyleData["ExerciseText", "Printout"], CellMargins->{{54, 0}, {4, 4}}, ParagraphSpacing->{0, 6}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Formulas and Programming", "Section"], Cell[CellGroupData[{ Cell[StyleData["DisplayFormula"], CellMargins->{{62, 10}, {2, 10}}, CellHorizontalScrolling->True, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", ScriptLevel->0, SingleLetterItalics->True, UnderoverscriptBoxOptions->{LimitsPositioning->True}], Cell[StyleData["DisplayFormula", "Presentation"], CellMargins->{{74, 10}, {2, 10}}, FontSize->10], Cell[StyleData["DisplayFormula", "Condensed"], CellMargins->{{52, 10}, {2, 10}}, FontSize->10], Cell[StyleData["DisplayFormula", "Printout"], CellMargins->{{54, 10}, {2, 10}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["ChemicalFormula"], CellMargins->{{62, 10}, {2, 10}}, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", AutoSpacing->False, ScriptLevel->1, ScriptBaselineShifts->{0.6, Automatic}, SingleLetterItalics->False, ZeroWidthTimes->True], Cell[StyleData["ChemicalFormula", "Presentation"], CellMargins->{{74, 10}, {2, 10}}, FontSize->10], Cell[StyleData["ChemicalFormula", "Condensed"], CellMargins->{{52, 10}, {2, 10}}, FontSize->10], Cell[StyleData["ChemicalFormula", "Printout"], CellMargins->{{54, 10}, {2, 10}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Program"], CellMargins->{{12, 10}, {Inherited, 6}}, Hyphenation->False, LanguageCategory->"Formula", FontFamily->"Courier"], Cell[StyleData["Program", "Presentation"], CellMargins->{{13, 30}, {Inherited, 4}}, FontSize->9.5], Cell[StyleData["Program", "Condensed"], CellMargins->{{8, 10}, {Inherited, 4}}, FontSize->9.5], Cell[StyleData["Program", "Printout"], CellMargins->{{9, 0}, {Inherited, 4}}, FontSize->9.5] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Automatic Numbering", "Section"], Cell["\<\ The following styles are useful for numbered equations, figures, etc. They \ automatically give the cell a FrameLabel containing a reference to a \ particular counter, and also increment that counter.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["NumberedEquation"], CellMargins->{{62, 10}, {Inherited, Inherited}}, CellFrameLabels->{{None, Cell[ TextData[ {"(", CounterBox[ "NumberedEquation"], ")"}]]}, {None, None}}, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, CounterIncrements->"NumberedEquation", FormatTypeAutoConvert->False], Cell[StyleData["NumberedEquation", "Presentation"], CellMargins->{{74, 10}, {Inherited, Inherited}}], Cell[StyleData["NumberedEquation", "Condensed"], CellMargins->{{52, 10}, {Inherited, Inherited}}], Cell[StyleData["NumberedEquation", "Printout"], CellMargins->{{54, 0}, {Inherited, Inherited}}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["NumberedFigure"], CellMargins->{{62, 145}, {Inherited, Inherited}}, CellFrameLabels->{{None, None}, {Cell[ TextData[ {"Figure ", CounterBox[ "NumberedFigure"]}]], None}}, CounterIncrements->"NumberedFigure", ImageMargins->{{43, Inherited}, {Inherited, 0}}, FormatTypeAutoConvert->False], Cell[StyleData["NumberedFigure", "Presentation"]], Cell[StyleData["NumberedFigure", "Condensed"]], Cell[StyleData["NumberedFigure", "Printout"]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["NumberedTable"], CellMargins->{{62, 145}, {Inherited, Inherited}}, CellFrameLabels->{{None, None}, {Cell[ TextData[ {"Table ", CounterBox[ "NumberedTable"]}]], None}}, TextAlignment->Center, CounterIncrements->"NumberedTable", FormatTypeAutoConvert->False], Cell[StyleData["NumberedTable", "Presentation"]], Cell[StyleData["NumberedTable", "Condensed"]], Cell[StyleData["NumberedTable", "Printout"]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Headers and Footers", "Section"], Cell[StyleData["Header"], CellMargins->{{0, 0}, {4, 1}}, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->9, FontSlant->"Italic"], Cell[StyleData["Footer"], CellMargins->{{0, 0}, {0, 4}}, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->6], Cell[StyleData["PageNumber"], CellMargins->{{0, 0}, {4, 1}}, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->9, FontWeight->"Bold"] }, 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->GrayLevel[1], Background->RGBColor[1, 0.4, 0], ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonFrame->"None", ButtonNote->ButtonData}], Cell[StyleData["Hyperlink", "Presentation"]], Cell[StyleData["Hyperlink", "Condensed"]], Cell[StyleData["Hyperlink", "Printout"], FontColor->GrayLevel[0], Background->GrayLevel[1]] }, 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->GrayLevel[1], Background->RGBColor[1, 0.4, 0], 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], Background->GrayLevel[1]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["AddOnsLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->GrayLevel[1], Background->RGBColor[1, 0.4, 0], 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], Background->GrayLevel[1]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["RefGuideLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->GrayLevel[1], Background->RGBColor[1, 0.4, 0], 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], Background->GrayLevel[1]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["GettingStartedLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->GrayLevel[1], Background->RGBColor[1, 0.4, 0], 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], Background->GrayLevel[1]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["OtherInformationLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->GrayLevel[1], Background->RGBColor[1, 0.4, 0], 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], Background->GrayLevel[1]] }, Closed]] }, 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[ ], #, 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, 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