(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' 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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Here we are interested in the \ question: how does the number of mates affect the proportion of resources \ that should be allocated by a hermaphrodite to male and females function in \ order to maximize individual fitness? \ \>", "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ "DEFINING THE VARIABLES...\na", StyleBox["i", FontVariations->{"CompatibilityType"->"Subscript"}], " = allocation to male function by the ith hermaphrodite in the population. \ \na = average allocation to male function in the population (this was a* in \ lecture, but I can't use the star here, because it is used exclusively by \ Mathematica as the multiplication symbol.)\nR = Total resource base for \ reproduction\nk = the number of mates\nV = The reproductive value though male \ fuction\nW", StyleBox["m", FontVariations->{"CompatibilityType"->"Subscript"}], " = Fitness through male function\nW", StyleBox["f", FontVariations->{"CompatibilityType"->"Subscript"}], " = Fitness though female function (assuming Bateman's principle).\nW", StyleBox["i", FontVariations->{"CompatibilityType"->"Subscript"}], " = Fitness of the ith individual (Wf + Wm)" }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ "\nFIRST WE DEFINE FITNESS THROUGH FEMALE FUNCTION, W", StyleBox["f.", FontVariations->{"CompatibilityType"->"Subscript"}], "\n" }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ StyleBox["(", FontFamily->"Helvetica", FontColor->RGBColor[0, 0, 1]], StyleBox[ "Put the cursor anywhere in the bracket below and press \"SHIFT-RETURN.\" \ This will store W", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1]], StyleBox["f", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" in memory)", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell["Wf = R*(1 - ai)", "Input", AspectRatioFixed->True], Cell[BoxData[ \(\((1 - ai)\)\ R\)], "Output"] }, Open ]], Cell[TextData[{ "NEXT DEFINE FITNESS THROUGH MALE FUNCTION, W", StyleBox["m. ", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[ "V is the reproductive value of male function, which will be defined \ shortly.", FontColor->RGBColor[1, 0, 0]], "\n" }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ StyleBox["(", FontFamily->"Helvetica", FontColor->RGBColor[0, 0, 1]], StyleBox[ "Put the cursor anywhere in the bracket below and press \"SHIFT-RETURN.\")", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell["Wm = V*R*ai", "Input", AspectRatioFixed->True], Cell[BoxData[ \(ai\ R\ V\)], "Output"] }, Open ]], Cell[TextData[ "\nNOW DEFINE THE VARIABLE, V, AS THE RATIO OF EGGS IN THE POPULATION TO \ SPERM IN THE POPULATION.\n"], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ StyleBox["(", FontFamily->"Helvetica", FontColor->RGBColor[0, 0, 1]], StyleBox[ "Put the cursor anywhere in the bracket below and press \"SHIFT-RETURN.\")", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell["V = (k*(1 - a))/((k - 1)*a + ai)", "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(\((1 - a)\)\ k\)\/\(ai + a\ \((\(-1\) + k)\)\)\)], "Output"] }, Open ]], Cell[TextData[ "\nTOTAL FITNESS OF THE INDIVIDUAL IS THE SUM OF FITNESS THROUGH FEMALE + \ MALE FUNCTION.\n"], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ StyleBox["(", FontFamily->"Helvetica", FontColor->RGBColor[0, 0, 1]], StyleBox[ "Put the cursor anywhere in the bracket below and press \"SHIFT-RETURN.\")", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell["Wi = Wf + Wm", "Input", AspectRatioFixed->True], Cell[BoxData[ \(\((1 - ai)\)\ R + \(\((1 - a)\)\ ai\ k\ R\)\/\(ai + a\ \((\(-1\) + \ k)\)\)\)], "Output"] }, Open ]], Cell[TextData[ "\nNOW CALCULATE THE FIRST DERIVATIVE OF INDIVIDUAL FITNESS WITH RESPECT TO \ ALLOCATION TO MALE FUNCTION.\n"], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ StyleBox["(", FontFamily->"Helvetica", FontColor->RGBColor[0, 0, 1]], StyleBox[ "Put the cursor anywhere in the bracket below and press \"SHIFT-RETURN.\")", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell["Firstder = D[Wi, ai]", "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(-R\) - \(\((1 - a)\)\ ai\ k\ R\)\/\((ai + a\ \((\(-1\) + k)\))\)\^2 + \ \(\((1 - a)\)\ k\ R\)\/\(ai + a\ \((\(-1\) + k)\)\)\)], "Output"] }, Open ]], Cell[TextData[ "\nCALCULATE THE SECOND DERIVATIVE OF INDIVIDUAL FITNESS AS A FUNCTION OF ITS \ ALLOCATION TO MALE FUNCTION. Remember, local stability requires a negative \ second derivative\n"], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ StyleBox["(", FontFamily->"Helvetica", FontColor->RGBColor[0, 0, 1]], StyleBox[ "Put the cursor anywhere in the bracket below and press \"SHIFT-RETURN.\")", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell["Secondder = D[Firstder, ai]", "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(2\ \((1 - a)\)\ ai\ k\ R\)\/\((ai + a\ \((\(-1\) + k)\))\)\^3 - \(2\ \ \((1 - a)\)\ k\ R\)\/\((ai + a\ \((\(-1\) + k)\))\)\^2\)], "Output"] }, Open ]], Cell[TextData["\nSIMPLIFY THE SECOND DERIVATIVE. "], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ StyleBox["(", FontFamily->"Helvetica", FontColor->RGBColor[0, 0, 1]], StyleBox[ "Put the cursor anywhere in the bracket below and press \"SHIFT-RETURN.\")", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell["Simplify [Secondder]", "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(2\ \((\(-1\) + a)\)\ a\ \((\(-1\) + k)\)\ k\ R\)\/\((ai + a\ \ \((\(-1\) + k)\))\)\^3\)], "Output"] }, Open ]], Cell[TextData[{ "\nNote that the second derivative is less that zero when the numerator is \ negative, which is true whenever k > 1 AND a < 1. Hence for k > 1 (i.e., \ more than one mate) and a < 1 (i.e., less than 100% allocation to male \ function in the population), we can find a locally stable ESS by setting a", StyleBox["i", FontVariations->{"CompatibilityType"->"Subscript"}], " = a and solving for the case when the first derivative equal to zero.\n\n\ SET a", StyleBox["i", FontVariations->{"CompatibilityType"->"Subscript"}], " = a. REMEMBER WHEN THE POPULATION IS AT THE ESS, INDIVIDUAL FITNESS \ SHOULD BE MAXIMIZED AT THE ESS.\n\n" }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ StyleBox["(", FontFamily->"Helvetica", FontColor->RGBColor[0, 0, 1]], StyleBox[ "Put the cursor anywhere in the bracket below and press \"SHIFT-RETURN.\")", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell["ai = a", "Input", AspectRatioFixed->True], Cell[BoxData[ \(a\)], "Output"] }, Open ]], Cell[TextData[ "\nSET THE FIRST DERIVATIVE EQUAL TO ZERO AND SOLVE FOR a. \n"], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ StyleBox["(", FontFamily->"Helvetica", FontColor->RGBColor[0, 0, 1]], StyleBox[ "Put the cursor anywhere in the bracket below and press \"SHIFT-RETURN.\")", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell["Solve [Firstder == 0, a]", "Input", AspectRatioFixed->True], Cell[BoxData[ \({{a \[Rule] \(\(-1\) + k\)\/\(\(-1\) + 2\ k\)}}\)], "Output"] }, Open ]], Cell[TextData[{ "\nHence the allocation to male function, a, at \nthe ESS is:\n\n", StyleBox[" a = (k-1)/(2k - 1)", FontColor->RGBColor[1, 0, 0]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[StyleBox["Now set the population allocation, a, at the ESS", FontFamily->"Helvetica", FontColor->RGBColor[0, 0, 1]]], "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ StyleBox["(", FontFamily->"Helvetica", FontColor->RGBColor[0, 0, 1]], StyleBox[ "Put the cursor anywhere in the bracket below and press \"SHIFT-RETURN.\")", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell["\<\ a = (k - 1)/(2*k -1)\ \>", "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(\(-1\) + k\)\/\(\(-1\) + 2\ k\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[Secondder]\)], "Input"], Cell[BoxData[ \(\(\((2 - 4\ k)\)\ R\)\/\(\((\(-1\) + k)\)\ k\)\)], "Output"] }, Open ]], Cell[TextData[StyleBox["So, the second derivative is negative for all values \ of k > 1. \nNow let the individual allocation, ai, be a variable again. So \ (ai) is no longer equal to (a). The purpose is to graph individual fitness \ as a function of ai when the population is at the ESS. We would expect that \ Wi is maximized under this condition when ai converges on a.\n\nI'm also \ going to set the Resources equal to 1 unit.\n", FontFamily->"Helvetica", FontWeight->"Bold"]], "Text", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ StyleBox["(", FontFamily->"Helvetica", FontColor->RGBColor[0, 0, 1]], StyleBox[ "Put the cursor anywhere in the bracket below and press \"SHIFT-RETURN.\")", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell["\<\ ai = . R=1\ \>", "Input", AspectRatioFixed->True], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell[TextData[StyleBox[ "\nNow, for the crux of the bisquit. Let there be two mates (k = 2).\n", FontFamily->"Helvetica", FontWeight->"Bold"]], "Text", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ StyleBox["(", FontFamily->"Helvetica", FontColor->RGBColor[0, 0, 1]], StyleBox[ "Put the cursor anywhere in the bracket below and press \"SHIFT-RETURN.\")", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell["k=2", "Input", AspectRatioFixed->True], Cell[BoxData[ \(2\)], "Output"] }, Open ]], Cell[TextData[ "The ESS can be found for two mates by simplifying the population allocation, \ a."], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0, 0, 1], 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Linear increase in fitness with increasing allocation to female \ function. (This is a consequence of our assumtion that female function is \ limited only by resources.) REMEMBER: allocation to female fitness increases \ from right to left; it's equal to 1 - ai. Bueno, Now let's plot the relationship between total fitness and the \ allocation by the ith individual to male function. 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That is when the individual allocation is at the ESS.\n\n\n\n\ Now let's let the local mating population be large, say 100 mates." }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ StyleBox["(", FontFamily->"Helvetica", FontColor->RGBColor[0, 0, 1]], StyleBox[ "Put the cursor anywhere in the bracket below and press \"SHIFT-RETURN.\")", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell["k=100", "Input", AspectRatioFixed->True], Cell[BoxData[ \(100\)], "Output"] }, Open ]], Cell[TextData[ "The new ESS for 100 mates is found by simplifying the parameter, a."], "Text",\ Evaluatable->False, AspectRatioFixed->True, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ StyleBox["(", FontFamily->"Helvetica", FontColor->RGBColor[0, 0, 1]], StyleBox[ "Put the cursor anywhere in the bracket below and press \"SHIFT-RETURN.\")", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell["Simplify [a]", "Input", AspectRatioFixed->True], Cell[BoxData[ \(99\/199\)], "Output"] }, Open ]], Cell[TextData[{ "\nHence, the ESS is at 99/199, which is about 1/2.\n\nNow, as previously, \ let's first plot the relationship between fitness through ", StyleBox["male", FontVariations->{"Underline"->True}], " function and the allocation by the ith individual to male function." }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ StyleBox["(", FontFamily->"Helvetica", FontColor->RGBColor[0, 0, 1]], StyleBox[ "Put the cursor anywhere in the bracket below and press \"SHIFT-RETURN.\")", FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell[TextData[{ "Plot [{Wm}, {ai, 0, 1}, \nAxesLabel->{\"a", StyleBox["i", FontVariations->{"CompatibilityType"->"Subscript"}], "\",\"W", StyleBox["m", FontVariations->{"CompatibilityType"->"Subscript"}], "\"}]" }], "Input", AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.952381 0.0147151 0.588604 [ [.21429 .00222 -9 -9 ] [.21429 .00222 9 0 ] [.40476 .00222 -9 -9 ] [.40476 .00222 9 0 ] [.59524 .00222 -9 -9 ] [.59524 .00222 9 0 ] [.78571 .00222 -9 -9 ] [.78571 .00222 9 0 ] [.97619 .00222 -3 -9 ] [.97619 .00222 3 0 ] [1.025 .01472 0 -6 ] [1.025 .01472 52 6 ] [.01131 .13244 -18 -4.5 ] [.01131 .13244 0 4.5 ] [.01131 .25016 -18 -4.5 ] [.01131 .25016 0 4.5 ] [.01131 .36788 -18 -4.5 ] [.01131 .36788 0 4.5 ] [.01131 .4856 -18 -4.5 ] [.01131 .4856 0 4.5 ] [.01131 .60332 -6 -4.5 ] [.01131 .60332 0 4.5 ] [.02381 .64303 -26.0625 0 ] [.02381 .64303 26.0625 12 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .21429 .01472 m .21429 .02097 L s [(0.2)] .21429 .00222 0 1 Mshowa .40476 .01472 m .40476 .02097 L s [(0.4)] .40476 .00222 0 1 Mshowa .59524 .01472 m .59524 .02097 L s [(0.6)] .59524 .00222 0 1 Mshowa .78571 .01472 m .78571 .02097 L s [(0.8)] .78571 .00222 0 1 Mshowa .97619 .01472 m .97619 .02097 L s [(1)] .97619 .00222 0 1 Mshowa .125 Mabswid .07143 .01472 m .07143 .01847 L s .11905 .01472 m .11905 .01847 L s .16667 .01472 m .16667 .01847 L s .2619 .01472 m .2619 .01847 L s .30952 .01472 m .30952 .01847 L s .35714 .01472 m .35714 .01847 L s .45238 .01472 m .45238 .01847 L s .5 .01472 m .5 .01847 L s .54762 .01472 m .54762 .01847 L s .64286 .01472 m .64286 .01847 L s .69048 .01472 m .69048 .01847 L s .7381 .01472 m .7381 .01847 L s