(************** 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). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. 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[ 965516, 21522]*) (*NotebookOutlinePosition[ 966249, 21547]*) (* CellTagsIndexPosition[ 966205, 21543]*) (*WindowFrame->Normal*) Notebook[{ Cell["\<\ Homework 3I \[LongDash] Solution to Individual Exercises\ \>", \ "Title"], Cell["\<\ 3.016 Mathematical Methods for Materials Scientists and Engineers S. M. Allen October 31, 2005\ \>", "Subsubtitle"], Cell[BoxData[ \(Clear["\"]\)], "Input"], Cell[CellGroupData[{ Cell["Individual Exercise I3-1: (Kreyszig MCG 6.4, p. 77)", "Subsection"], Cell["Using the matrices provided", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(A = {{1, 3, 2}, {3, 5, 0}, {2, 0, 4}}\), "\[IndentingNewLine]", \(B = {{0, 2, 1}, {\(-2\), 0, \(-3\)}, {\(-1\), 3, 0}}\), "\[IndentingNewLine]", \(c = {{1}, {0}, {\(-2\)}}\), "\[IndentingNewLine]", \(d = {{3}, {1}, {2}}\)}], "Input"], Cell[BoxData[ \({{1, 3, 2}, {3, 5, 0}, {2, 0, 4}}\)], "Output"], Cell[BoxData[ \({{0, 2, 1}, {\(-2\), 0, \(-3\)}, {\(-1\), 3, 0}}\)], "Output"], Cell[BoxData[ \({{1}, {0}, {\(-2\)}}\)], "Output"], Cell[BoxData[ \({{3}, {1}, {2}}\)], "Output"] }, Open ]], Cell["Going to it:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(B . c\)], "Input"], Cell[BoxData[ \({{\(-2\)}, {4}, {\(-1\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(% // MatrixForm\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(-2\)}, {"4"}, {\(-1\)} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Transpose[c] . A . c\)], "Input"], Cell[BoxData[ \({{9}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(A . B . c\)], "Input"], Cell[BoxData[ \({{8}, {14}, {\(-8\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(% // MatrixForm\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"8"}, {"14"}, {\(-8\)} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Flatten[A . c] . Flatten[Transpose[B . d]]\)], "Input"], Cell[BoxData[ \(\(-48\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Transpose[d] . B . d\)], "Input"], Cell[BoxData[ \({{0}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(B . d\)], "Input"], Cell[BoxData[ \({{4}, {\(-12\)}, {0}}\)], "Output"] }, Open ]], Cell["As to why the result is zero, look at the eigensystem of B", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Eigensystem[B]\)], "Input"], Cell[BoxData[ \({{\[ImaginaryI]\ \@14, \(-\[ImaginaryI]\)\ \@14, 0}, {{3\/5 - 1\/5\ \[ImaginaryI]\ \@\(7\/2\), 1\/5 + 3\/5\ \[ImaginaryI]\ \@\(7\/2\), 1}, {3\/5 + 1\/5\ \[ImaginaryI]\ \@\(7\/2\), 1\/5 - 3\/5\ \[ImaginaryI]\ \@\(7\/2\), 1}, {\(-3\), \(-1\), 2}}}\)], "Output"] }, Open ]], Cell["\<\ Note that the third eigenvector is equal to d and its corresponding \ eigenvalue is 0. Multiplication of a matrix by its eigenvector returns a \ parallel vector scaled in length by the corresponding eigenvalue.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Individual Exercise I3-2: (Kreyszig MCG 7.10, p. 87)", "Subsection"], Cell[BoxData[ \(Clear[A, B]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[{ \(A\ = \ {{3, 4}, {4, \(-3\)}}\), "\[IndentingNewLine]", \(P\ = {{\(-4\), 2}, {3, \(-1\)}}\)}], "Input"], Cell[BoxData[ \({{3, 4}, {4, \(-3\)}}\)], "Output"], Cell[BoxData[ \({{\(-4\), 2}, {3, \(-1\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(B = Inverse[P] . A . P\)], "Input"], Cell[BoxData[ \({{\(-25\), 12}, {\(-50\), 25}}\)], "Output"] }, Open ]], Cell["\<\ Note that by definition, B is similar to A (B was obtained from A \ by a similarity transformation).\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(EigA = Eigensystem[A]\), "\[IndentingNewLine]", \(EigB = Eigensystem[B]\)}], "Input"], Cell[BoxData[ \({{\(-5\), 5}, {{\(-1\), 2}, {2, 1}}}\)], "Output"], Cell[BoxData[ RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \ error: new symbol name \\\"\\!\\(EigB\\)\\\" is similar to existing symbol \\\ \"\\!\\(EigA\\)\\\". \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"General::spell1\\\"]\\)\"\>"}]], "Message"], Cell[BoxData[ \({{\(-5\), 5}, {{3, 5}, {2, 5}}}\)], "Output"] }, Open ]], Cell["Obviously, the eigenvalues are the same.", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(Inverse[P] . EigA[\([2, 1]\)]\), "\[IndentingNewLine]", \(Inverse[P] . EigA[\([2, 2]\)]\)}], "Input"], Cell[BoxData[ \({3\/2, 5\/2}\)], "Output"], Cell[BoxData[ \({2, 5}\)], "Output"] }, Open ]], Cell["\<\ We see also that multiplication of Inverse[P] by an eigenvector of \ A yields an eigenvector of B (see Kreyszig AEM Theorem 1, p. 392).\ \>", \ "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Individual Exercise I3-3: (Kreyszig MCG 7.12, p. 87)", "Subsection"], Cell[BoxData[ \(Clear[A]\)], "Input"], Cell["Define the matrix A by", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(A\ = \ {{5, 4}, {1, 2}}\)], "Input"], Cell[BoxData[ \({{5, 4}, {1, 2}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(EiVecA = Eigenvectors[A]\)], "Input"], Cell[BoxData[ \({{4, 1}, {\(-1\), 1}}\)], "Output"] }, Open ]], Cell["\<\ Make a square matrix of eigenvectors, each eigenvector a column of \ the matrix.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(X = Transpose[EiVecA]\)], "Input"], Cell[BoxData[ \({{4, \(-1\)}, {1, 1}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(DMatrix = Inverse[X] . A . X\)], "Input"], Cell[BoxData[ \({{6, 0}, {0, 1}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Eigenvalues[A]\)], "Input"], Cell[BoxData[ \({6, 1}\)], "Output"] }, Open ]], Cell["\<\ Homework 3G \[LongDash] Solution to Group Exercises\ \>", "Title"], Cell["\<\ 3.016 Mathematical Methods for Materials Scientists and Engineers S. M. Allen November 14, 2005\ \>", "Subsubtitle"], Cell[BoxData[ \(Clear["\"]\)], "Input"], Cell["\<\ Homework 3G \[LongDash] Solution to Group Exercises\ \>", "Title"], Cell["\<\ 3.016 Mathematical Methods for Materials Scientists and Engineers S. M. Allen November 14, 2005\ \>", "Subsubtitle"], Cell[BoxData[ \(Clear["\"]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Group Exercise G3-1: Edge Dislocation", "Subsection"], Cell[CellGroupData[{ Cell["1.", "Subsubsection"], Cell["\<\ Enter the stress matrix in cartesian coordinates; use Replace to \ convert to Polar form.\ \>", "Text"], Cell[BoxData[{ \(\(\[Sigma]\_xx = \(\(G\ b\)\/\(2 \[Pi] \((1 - \[Nu])\)\)\) \(\(-y\) \ \((3\ x\^2 + y\^2)\)\)\/\((x\^2 + y\^2)\)\^2;\)\), "\[IndentingNewLine]", \(\(\[Sigma]\_xy = \(\(G\ b\)\/\(2 \[Pi] \((1 - \[Nu])\)\)\) \(x \((\ \ x\^2 - y\^2)\)\)\/\((x\^2 + y\^2)\)\^2;\)\), "\[IndentingNewLine]", \(\(\[Sigma]\_yx = \(\(G\ b\)\/\(2 \[Pi] \((1 - \[Nu])\)\)\) \(x \((\ \ x\^2 - y\^2)\)\)\/\((x\^2 + y\^2)\)\^2;\)\), "\[IndentingNewLine]", \(\(\[Sigma]\_yy = \(\(G\ b\)\/\(2 \[Pi] \((1 - \[Nu])\)\)\) \(y \((\ \ x\^2 - y\^2)\)\)\/\((x\^2 + y\^2)\)\^2;\)\)}], "Input"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(SigMatCart = {{\[Sigma]\_xx, \[Sigma]\_xy}, 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\[Sigma]\_yx /. {x \[Rule] r\ Sin[\[Theta]], y \[Rule] r\ Cos[\[Theta]]};\)\), "\[IndentingNewLine]", \(\(\[Sigma]\_\[Theta]\[Theta] = \[Sigma]\_yy /. {x \[Rule] r\ Sin[\[Theta]], y \[Rule] r\ Cos[\[Theta]]};\)\)}], "Input"], Cell[BoxData[ RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \ error: new symbol name \\\"\\!\\(\[Theta]r\\)\\\" is similar to existing \ symbol \\\"\\!\\(\[Theta]\\)\\\". \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"General::spell1\\\"]\\)\"\>"}]], "Message"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(SigMatPolar = {{\[Sigma]\_rr, \[Sigma]\_r\[Theta]}, {\[Sigma]\_\ \[Theta]r, \[Sigma]\_\[Theta]\[Theta]}};\)\), "\[IndentingNewLine]", \(MatrixForm[%]\)}], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(-\(\(b\ G\ r\ Cos[\[Theta]]\ \((r\^2\ Cos[\[Theta]]\^2 + 3\ r\^2\ Sin[\[Theta]]\^2)\)\)\/\(2\ \[Pi]\ \((1 - \ \[Nu])\)\ \((r\^2\ 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.48 g .01923 .54395 m .04077 .54006 L .05929 .53689 L .09936 .53044 L .13942 .52462 L .17949 .51942 L .21955 .51483 L .25962 .51086 L .29968 .50748 L .33974 .50467 L .37981 .50237 L .41987 .50135 L .45994 .5005 L .45994 .5 L .5 .45994 L .54006 .5 L .54006 .5005 L .58013 .50135 L .62019 .50237 L .66026 .50467 L .70032 .50748 L .74038 .51086 L .78045 .51483 L .82051 .51942 L .86058 .52462 L .90064 .53044 L .94071 .53689 L .95923 .54006 L .98077 .54395 L .98077 .98077 L .01923 .98077 L F 0 g .01923 .54395 m .04077 .54006 L .05929 .53689 L .09936 .53044 L .13942 .52462 L .17949 .51942 L .21955 .51483 L .25962 .51086 L .29968 .50748 L .33974 .50467 L .37981 .50237 L .41987 .50135 L .45994 .5005 L .45994 .5 L .5 .45994 L .54006 .5 L .54006 .5005 L .58013 .50135 L .62019 .50237 L .66026 .50467 L .70032 .50748 L .74038 .51086 L .78045 .51483 L .82051 .51942 L .86058 .52462 L .90064 .53044 L .94071 .53689 L .95923 .54006 L .98077 .54395 L s .46 g .01923 .63276 m .0409 .62019 L .05929 .61013 L .09936 .58996 L .12105 .58013 L .13942 .57215 L .17949 .55645 L .21955 .54293 L .22919 .54006 L .25962 .53131 L .29968 .52148 L .33974 .51339 L .37981 .50677 L .41987 .50381 L .45994 .50143 L .45994 .5 L .5 .45994 L .54006 .5 L .54006 .50143 L .58013 .50381 L .62019 .50677 L .66026 .51339 L .70032 .52148 L .74038 .53131 L .77081 .54006 L .78045 .54293 L .82051 .55645 L .86058 .57215 L .87895 .58013 L .90064 .58996 L .94071 .61013 L .9591 .62019 L .98077 .63276 L .98077 .98077 L .01923 .98077 L F 0 g .01923 .63276 m .0409 .62019 L .05929 .61013 L .09936 .58996 L .12105 .58013 L .13942 .57215 L .17949 .55645 L .21955 .54293 L .22919 .54006 L .25962 .53131 L .29968 .52148 L .33974 .51339 L .37981 .50677 L .41987 .50381 L .45994 .50143 L .45994 .5 L .5 .45994 L .54006 .5 L .54006 .50143 L .58013 .50381 L .62019 .50677 L .66026 .51339 L .70032 .52148 L .74038 .53131 L .77081 .54006 L .78045 .54293 L .82051 .55645 L .86058 .57215 L .87895 .58013 L .90064 .58996 L .94071 .61013 L .9591 .62019 L .98077 .63276 L s .58 g .91484 .01923 m .91881 .05929 L .91894 .09936 L .91521 .13942 L .90754 .17949 L .90064 .20634 L .89568 .21955 L .87922 .25962 L .86058 .29511 L .8576 .29968 L .82968 .33974 L .82051 .35137 L .79374 .37981 L .78045 .39275 L .74639 .41987 L .74038 .42439 L .70032 .44971 L .67798 .45994 L .66026 .4685 L .62019 .4841 L .58013 .49145 L .54006 .49681 L .5 .45994 L .45994 .49681 L .41987 .49145 L .37981 .4841 L .33974 .4685 L .32202 .45994 L .29968 .44971 L .25962 .42439 L .25361 .41987 L .21955 .39275 L .20626 .37981 L .17949 .35137 L .17032 .33974 L .1424 .29968 L .13942 .29511 L .12078 .25962 L .10432 .21955 L .09936 .20634 L .09246 .17949 L .08479 .13942 L .08106 .09936 L .08119 .05929 L .08516 .01923 L F 0 g .91484 .01923 m .91881 .05929 L .91894 .09936 L .91521 .13942 L .90754 .17949 L .90064 .20634 L .89568 .21955 L .87922 .25962 L .86058 .29511 L .8576 .29968 L .82968 .33974 L .82051 .35137 L .79374 .37981 L .78045 .39275 L .74639 .41987 L .74038 .42439 L .70032 .44971 L .67798 .45994 L .66026 .4685 L .62019 .4841 L .58013 .49145 L .54006 .49681 L .5 .45994 L .45994 .49681 L .41987 .49145 L .37981 .4841 L .33974 .4685 L .32202 .45994 L .29968 .44971 L .25962 .42439 L .25361 .41987 L .21955 .39275 L .20626 .37981 L .17949 .35137 L .17032 .33974 L .1424 .29968 L .13942 .29511 L .12078 .25962 L .10432 .21955 L .09936 .20634 L .09246 .17949 L .08479 .13942 L .08106 .09936 L .08119 .05929 L .08516 .01923 L s .42 g .90194 .98077 m .90712 .94071 L .90832 .90064 L .90559 .86058 L .90064 .82915 L .89883 .82051 L .88782 .78045 L .87221 .74038 L .86058 .71587 L .85138 .70032 L .82434 .66026 L .82051 .65515 L .78925 .62019 L .78045 .61128 L .74288 .58013 L .74038 .57819 L .70032 .5518 L .67549 .54006 L .66026 .53246 L .62019 .51638 L .58013 .50878 L .54006 .50328 L .54006 .5 L .5 .45994 L .45994 .5 L .45994 .50328 L .41987 .50878 L .37981 .51638 L .33974 .53246 L .32451 .54006 L .29968 .5518 L .25962 .57819 L .25712 .58013 L .21955 .61128 L .21075 .62019 L .17949 .65515 L .17566 .66026 L .14862 .70032 L .13942 .71587 L .12779 .74038 L .11218 .78045 L .10117 .82051 L .09936 .82915 L .09441 .86058 L .09168 .90064 L .09288 .94071 L .09806 .98077 L F 0 g .90194 .98077 m .90712 .94071 L .90832 .90064 L .90559 .86058 L .90064 .82915 L .89883 .82051 L .88782 .78045 L .87221 .74038 L .86058 .71587 L .85138 .70032 L .82434 .66026 L .82051 .65515 L .78925 .62019 L .78045 .61128 L .74288 .58013 L .74038 .57819 L .70032 .5518 L .67549 .54006 L .66026 .53246 L .62019 .51638 L .58013 .50878 L .54006 .50328 L .54006 .5 L .5 .45994 L .45994 .5 L .45994 .50328 L .41987 .50878 L .37981 .51638 L .33974 .53246 L .32451 .54006 L .29968 .5518 L .25962 .57819 L .25712 .58013 L .21955 .61128 L .21075 .62019 L .17949 .65515 L .17566 .66026 L .14862 .70032 L .13942 .71587 L .12779 .74038 L .11218 .78045 L .10117 .82051 L .09936 .82915 L .09441 .86058 L .09168 .90064 L .09288 .94071 L .09806 .98077 L s .6 g .78558 .01923 m .804 .05929 L .81633 .09936 L .82051 .11424 L .82327 .13942 L .82516 .17949 L .8221 .21955 L .82051 .22985 L .81393 .25962 L .8002 .29968 L .78045 .33951 L .78026 .33974 L .75228 .37981 L .74038 .39456 L .7134 .41987 L .70032 .43158 L .66026 .45831 L .65602 .45994 L .62019 .47868 L .58013 .48894 L .54006 .49588 L .5 .45994 L .45994 .49588 L .41987 .48894 L .37981 .47868 L .34398 .45994 L .33974 .45831 L .29968 .43158 L .2866 .41987 L .25962 .39456 L .24772 .37981 L .21974 .33974 L .21955 .33951 L .1998 .29968 L .18607 .25962 L .17949 .22985 L .1779 .21955 L .17484 .17949 L .17673 .13942 L .17949 .11424 L .18367 .09936 L .196 .05929 L .21442 .01923 L F 0 g .78558 .01923 m .804 .05929 L .81633 .09936 L .82051 .11424 L .82327 .13942 L .82516 .17949 L .8221 .21955 L .82051 .22985 L .81393 .25962 L .8002 .29968 L .78045 .33951 L .78026 .33974 L .75228 .37981 L .74038 .39456 L .7134 .41987 L .70032 .43158 L .66026 .45831 L .65602 .45994 L .62019 .47868 L .58013 .48894 L .54006 .49588 L .5 .45994 L .45994 .49588 L .41987 .48894 L .37981 .47868 L .34398 .45994 L .33974 .45831 L .29968 .43158 L .2866 .41987 L .25962 .39456 L .24772 .37981 L .21974 .33974 L .21955 .33951 L .1998 .29968 L .18607 .25962 L .17949 .22985 L .1779 .21955 L .17484 .17949 L .17673 .13942 L .17949 .11424 L .18367 .09936 L .196 .05929 L .21442 .01923 L s .4 g .7742 .98077 m .78045 .96955 L .79423 .94071 L .80782 .90064 L .81579 .86058 L .81856 .82051 L .81627 .78045 L .8088 .74038 L .79575 .70032 L .78045 .66604 L .77642 .66026 L .74913 .62019 L .74038 .60891 L .71093 .58013 L .70032 .57034 L .66026 .54247 L .65415 .54006 L .62019 .52185 L .58013 .5113 L .54006 .50421 L .54006 .5 L .5 .45994 L .45994 .5 L .45994 .50421 L .41987 .5113 L .37981 .52185 L .34585 .54006 L .33974 .54247 L .29968 .57034 L .28907 .58013 L .25962 .60891 L .25087 .62019 L .22358 .66026 L .21955 .66604 L .20425 .70032 L .1912 .74038 L .18373 .78045 L .18144 .82051 L .18421 .86058 L .19218 .90064 L .20577 .94071 L .21955 .96955 L .2258 .98077 L F 0 g .7742 .98077 m .78045 .96955 L .79423 .94071 L .80782 .90064 L .81579 .86058 L .81856 .82051 L .81627 .78045 L .8088 .74038 L .79575 .70032 L .78045 .66604 L .77642 .66026 L .74913 .62019 L .74038 .60891 L .71093 .58013 L .70032 .57034 L .66026 .54247 L .65415 .54006 L .62019 .52185 L .58013 .5113 L .54006 .50421 L .54006 .5 L .5 .45994 L .45994 .5 L .45994 .50421 L .41987 .5113 L .37981 .52185 L .34585 .54006 L .33974 .54247 L .29968 .57034 L .28907 .58013 L .25962 .60891 L .25087 .62019 L .22358 .66026 L .21955 .66604 L .20425 .70032 L .1912 .74038 L .18373 .78045 L .18144 .82051 L .18421 .86058 L .19218 .90064 L .20577 .94071 L .21955 .96955 L .2258 .98077 L s .62 g .65572 .01923 m .66026 .02265 L .69965 .05929 L .70032 .06007 L .72868 .09936 L .74038 .12146 L .748 .13942 L .75982 .17949 L .76513 .21955 L .76434 .25962 L .75736 .29968 L .74376 .33974 L .74038 .34847 L .72202 .37981 L .70032 .41092 L .68975 .41987 L .66026 .44854 L .63863 .45994 L .62019 .47264 L .58013 .48639 L .54006 .49495 L .5 .45994 L .45994 .49495 L .41987 .48639 L .37981 .47264 L .36137 .45994 L .33974 .44854 L .31025 .41987 L .29968 .41092 L .27798 .37981 L .25962 .34847 L .25624 .33974 L .24264 .29968 L .23566 .25962 L .23487 .21955 L .24018 .17949 L .252 .13942 L .25962 .12146 L .27132 .09936 L .29968 .06007 L .30035 .05929 L .33974 .02265 L .34428 .01923 L F 0 g .65572 .01923 m .66026 .02265 L .69965 .05929 L .70032 .06007 L .72868 .09936 L .74038 .12146 L .748 .13942 L .75982 .17949 L .76513 .21955 L .76434 .25962 L .75736 .29968 L .74376 .33974 L .74038 .34847 L .72202 .37981 L .70032 .41092 L .68975 .41987 L .66026 .44854 L .63863 .45994 L .62019 .47264 L .58013 .48639 L .54006 .49495 L .5 .45994 L .45994 .49495 L .41987 .48639 L .37981 .47264 L .36137 .45994 L .33974 .44854 L .31025 .41987 L .29968 .41092 L .27798 .37981 L .25962 .34847 L .25624 .33974 L .24264 .29968 L .23566 .25962 L .23487 .21955 L .24018 .17949 L .252 .13942 L .25962 .12146 L .27132 .09936 L .29968 .06007 L .30035 .05929 L .33974 .02265 L .34428 .01923 L s .38 g .6415 .98077 m .66026 .96732 L .68965 .94071 L .70032 .92865 L .72077 .90064 L .74038 .86316 L .74146 .86058 L .75428 .82051 L .7604 .78045 L .76027 .74038 L .75389 .70032 L .74086 .66026 L .74038 .65894 L .7196 .62019 L .70032 .59127 L .68782 .58013 L .66026 .55255 L .63731 .54006 L .62019 .52796 L .58013 .51384 L .54006 .50514 L .54006 .5 L .5 .45994 L .45994 .5 L .45994 .50514 L .41987 .51384 L .37981 .52796 L .36269 .54006 L .33974 .55255 L .31218 .58013 L .29968 .59127 L .2804 .62019 L .25962 .65894 L .25914 .66026 L .24611 .70032 L .23973 .74038 L .2396 .78045 L .24572 .82051 L .25854 .86058 L .25962 .86316 L .27923 .90064 L .29968 .92865 L .31035 .94071 L .33974 .96732 L .3585 .98077 L F 0 g .6415 .98077 m .66026 .96732 L .68965 .94071 L .70032 .92865 L .72077 .90064 L .74038 .86316 L .74146 .86058 L .75428 .82051 L .7604 .78045 L .76027 .74038 L .75389 .70032 L .74086 .66026 L .74038 .65894 L .7196 .62019 L .70032 .59127 L .68782 .58013 L .66026 .55255 L .63731 .54006 L .62019 .52796 L .58013 .51384 L .54006 .50514 L .54006 .5 L .5 .45994 L .45994 .5 L .45994 .50514 L .41987 .51384 L .37981 .52796 L .36269 .54006 L .33974 .55255 L .31218 .58013 L .29968 .59127 L .2804 .62019 L .25962 .65894 L .25914 .66026 L .24611 .70032 L .23973 .74038 L .2396 .78045 L .24572 .82051 L .25854 .86058 L .25962 .86316 L .27923 .90064 L .29968 .92865 L .31035 .94071 L .33974 .96732 L .3585 .98077 L s .64 g .45994 .05476 m .5 .05115 L .54006 .05476 L .56733 .05929 L .58013 .06596 L .62019 .08607 L .63918 .09936 L .66026 .11848 L .67845 .13942 L .70032 .17438 L .70281 .17949 L .7173 .21955 L .72382 .25962 L .72306 .29968 L .71492 .33974 L .70032 .37674 L .69872 .37981 L .67136 .41987 L .66026 .43466 L .6266 .45994 L .62019 .46583 L .58013 .48381 L .54006 .49402 L .5 .45994 L .45994 .49402 L .41987 .48381 L .37981 .46583 L .3734 .45994 L .33974 .43466 L .32864 .41987 L .30128 .37981 L .29968 .37674 L .28508 .33974 L .27694 .29968 L .27618 .25962 L .2827 .21955 L .29719 .17949 L .29968 .17438 L .32155 .13942 L .33974 .11848 L .36082 .09936 L .37981 .08607 L .41987 .06596 L .43267 .05929 L F 0 g .45994 .05476 m .5 .05115 L .54006 .05476 L .56733 .05929 L .58013 .06596 L .62019 .08607 L .63918 .09936 L .66026 .11848 L .67845 .13942 L .70032 .17438 L .70281 .17949 L .7173 .21955 L .72382 .25962 L .72306 .29968 L .71492 .33974 L .70032 .37674 L .69872 .37981 L .67136 .41987 L .66026 .43466 L .6266 .45994 L .62019 .46583 L .58013 .48381 L .54006 .49402 L .5 .45994 L .45994 .49402 L .41987 .48381 L .37981 .46583 L .3734 .45994 L .33974 .43466 L .32864 .41987 L .30128 .37981 L .29968 .37674 L .28508 .33974 L .27694 .29968 L .27618 .25962 L .2827 .21955 L .29719 .17949 L .29968 .17438 L .32155 .13942 L .33974 .11848 L .36082 .09936 L .37981 .08607 L .41987 .06596 L .43267 .05929 L .45994 .05476 L s .36 g .5 .45994 m .54006 .5 L .54006 .50607 L .58013 .51643 L .62019 .53484 L .62572 .54006 L .66026 .5669 L .66991 .58013 L .69675 .62019 L .70032 .62735 L .71255 .66026 L .72018 .70032 L .72038 .74038 L .71317 .78045 L .70032 .81529 L .69773 .82051 L .6719 .86058 L .66026 .87377 L .62963 .90064 L .62019 .90699 L .58013 .92745 L .54006 .93883 L .52792 .94071 L .5 .94249 L .47208 .94071 L .45994 .93883 L .41987 .92745 L .37981 .90699 L .37037 .90064 L .33974 .87377 L .3281 .86058 L .30227 .82051 L .29968 .81529 L .28683 .78045 L .27962 .74038 L .27982 .70032 L .28745 .66026 L .29968 .62735 L .30325 .62019 L .33009 .58013 L .33974 .5669 L .37428 .54006 L .37981 .53484 L .41987 .51643 L .45994 .50607 L .45994 .5 L F 0 g .5 .45994 m .54006 .5 L .54006 .50607 L .58013 .51643 L .62019 .53484 L .62572 .54006 L .66026 .5669 L .66991 .58013 L .69675 .62019 L .70032 .62735 L .71255 .66026 L .72018 .70032 L .72038 .74038 L .71317 .78045 L .70032 .81529 L .69773 .82051 L .6719 .86058 L .66026 .87377 L .62963 .90064 L .62019 .90699 L .58013 .92745 L .54006 .93883 L .52792 .94071 L .5 .94249 L .47208 .94071 L .45994 .93883 L .41987 .92745 L .37981 .90699 L .37037 .90064 L .33974 .87377 L .3281 .86058 L .30227 .82051 L .29968 .81529 L .28683 .78045 L .27962 .74038 L .27982 .70032 L .28745 .66026 L .29968 .62735 L .30325 .62019 L .33009 .58013 L .33974 .5669 L .37428 .54006 L .37981 .53484 L .41987 .51643 L .45994 .50607 L .45994 .5 L .5 .45994 L s .66 g .41987 .12868 m .45994 .11557 L .5 .1114 L .54006 .11557 L .58013 .12868 L .60136 .13942 L .62019 .15302 L .64788 .17949 L .66026 .19577 L .6742 .21955 L .68876 .25962 L .69419 .29968 L .69122 .33974 L .67933 .37981 L .66026 .41551 L .65707 .41987 L .62019 .45829 L .61766 .45994 L .58013 .48118 L .54006 .49308 L .5 .45994 L .45994 .49308 L .41987 .48118 L .38234 .45994 L .37981 .45829 L .34293 .41987 L .33974 .41551 L .32067 .37981 L .30878 .33974 L .30581 .29968 L .31124 .25962 L .3258 .21955 L .33974 .19577 L .35212 .17949 L .37981 .15302 L .39864 .13942 L F 0 g .41987 .12868 m .45994 .11557 L .5 .1114 L .54006 .11557 L .58013 .12868 L .60136 .13942 L .62019 .15302 L .64788 .17949 L .66026 .19577 L .6742 .21955 L .68876 .25962 L .69419 .29968 L .69122 .33974 L .67933 .37981 L .66026 .41551 L .65707 .41987 L .62019 .45829 L .61766 .45994 L .58013 .48118 L .54006 .49308 L .5 .45994 L .45994 .49308 L .41987 .48118 L .38234 .45994 L .37981 .45829 L .34293 .41987 L .33974 .41551 L .32067 .37981 L .30878 .33974 L .30581 .29968 L .31124 .25962 L .3258 .21955 L .33974 .19577 L .35212 .17949 L .37981 .15302 L .39864 .13942 L .41987 .12868 L s .34 g .5 .45994 m .54006 .5 L .54006 .50701 L .58013 .51906 L .61667 .54006 L .62019 .54241 L .65579 .58013 L .66026 .5864 L .67772 .62019 L .6892 .66026 L .69171 .70032 L .6857 .74038 L .67031 .78045 L .66026 .79757 L .64264 .82051 L .62019 .84154 L .59223 .86058 L .58013 .86631 L .54006 .8796 L .5 .88383 L .45994 .8796 L .41987 .86631 L .40777 .86058 L .37981 .84154 L .35736 .82051 L .33974 .79757 L .32969 .78045 L .3143 .74038 L .30829 .70032 L .3108 .66026 L .32228 .62019 L .33974 .5864 L .34421 .58013 L .37981 .54241 L .38333 .54006 L .41987 .51906 L .45994 .50701 L .45994 .5 L F 0 g .5 .45994 m .54006 .5 L .54006 .50701 L .58013 .51906 L .61667 .54006 L .62019 .54241 L .65579 .58013 L .66026 .5864 L .67772 .62019 L .6892 .66026 L .69171 .70032 L .6857 .74038 L .67031 .78045 L .66026 .79757 L .64264 .82051 L .62019 .84154 L .59223 .86058 L .58013 .86631 L .54006 .8796 L .5 .88383 L .45994 .8796 L .41987 .86631 L .40777 .86058 L .37981 .84154 L .35736 .82051 L .33974 .79757 L .32969 .78045 L .3143 .74038 L .30829 .70032 L .3108 .66026 L .32228 .62019 L .33974 .5864 L .34421 .58013 L .37981 .54241 L .38333 .54006 L .41987 .51906 L .45994 .50701 L .45994 .5 L .5 .45994 L s .68 g .41987 .17724 m .45994 .16213 L .5 .15738 L .54006 .16213 L .58013 .17724 L .58449 .17949 L .62019 .20659 L .63219 .21955 L .65681 .25962 L .66026 .26576 L .66882 .29968 L .67086 .33974 L .66342 .37981 L .66026 .3891 L .64434 .41987 L .62019 .45033 L .60844 .45994 L .58013 .47849 L .54006 .49214 L .5 .45994 L .45994 .49214 L .41987 .47849 L .39156 .45994 L .37981 .45033 L .35566 .41987 L .33974 .3891 L .33658 .37981 L .32914 .33974 L .33118 .29968 L .33974 .26576 L .34319 .25962 L .36781 .21955 L .37981 .20659 L .41551 .17949 L F 0 g .41987 .17724 m .45994 .16213 L .5 .15738 L .54006 .16213 L .58013 .17724 L .58449 .17949 L .62019 .20659 L .63219 .21955 L .65681 .25962 L .66026 .26576 L .66882 .29968 L .67086 .33974 L .66342 .37981 L .66026 .3891 L .64434 .41987 L .62019 .45033 L .60844 .45994 L .58013 .47849 L .54006 .49214 L .5 .45994 L .45994 .49214 L .41987 .47849 L .39156 .45994 L .37981 .45033 L .35566 .41987 L .33974 .3891 L .33658 .37981 L .32914 .33974 L .33118 .29968 L .33974 .26576 L .34319 .25962 L .36781 .21955 L .37981 .20659 L .41551 .17949 L .41987 .17724 L s .32 g .5 .45994 m .54006 .5 L .54006 .50795 L .58013 .52176 L .60772 .54006 L .62019 .55046 L .64328 .58013 L .66026 .61447 L .66209 .62019 L .66912 .66026 L .66661 .70032 L .66026 .72813 L .65395 .74038 L .62815 .78045 L .62019 .78893 L .58013 .81879 L .57901 .82051 L .54006 .8341 L .5 .8389 L .45994 .8341 L .42099 .82051 L .41987 .81879 L .37981 .78893 L .37185 .78045 L .34605 .74038 L .33974 .72813 L .33339 .70032 L .33088 .66026 L .33791 .62019 L .33974 .61447 L .35672 .58013 L .37981 .55046 L .39228 .54006 L .41987 .52176 L .45994 .50795 L .45994 .5 L F 0 g .5 .45994 m .54006 .5 L .54006 .50795 L .58013 .52176 L .60772 .54006 L .62019 .55046 L .64328 .58013 L .66026 .61447 L .66209 .62019 L .66912 .66026 L .66661 .70032 L .66026 .72813 L .65395 .74038 L .62815 .78045 L .62019 .78893 L .58013 .81879 L .57901 .82051 L .54006 .8341 L .5 .8389 L .45994 .8341 L .42099 .82051 L .41987 .81879 L .37981 .78893 L .37185 .78045 L .34605 .74038 L .33974 .72813 L .33339 .70032 L .33088 .66026 L .33791 .62019 L .33974 .61447 L .35672 .58013 L .37981 .55046 L .39228 .54006 L .41987 .52176 L .45994 .50795 L .45994 .5 L .5 .45994 L s .7 g .41987 .21623 m .45994 .19897 L .5 .19364 L .54006 .19897 L .58013 .21623 L .5856 .21955 L .62019 .25179 L .62602 .25962 L .64579 .29968 L .65299 .33974 L .64939 .37981 L .63383 .41987 L .62019 .4411 L .6015 .45994 L .58013 .47573 L .54006 .4912 L .5 .45994 L .45994 .4912 L .41987 .47573 L .3985 .45994 L .37981 .4411 L .36617 .41987 L .35061 .37981 L .34701 .33974 L .35421 .29968 L .37398 .25962 L .37981 .25179 L .4144 .21955 L F 0 g .41987 .21623 m .45994 .19897 L .5 .19364 L .54006 .19897 L .58013 .21623 L .5856 .21955 L .62019 .25179 L .62602 .25962 L .64579 .29968 L .65299 .33974 L .64939 .37981 L .63383 .41987 L .62019 .4411 L .6015 .45994 L .58013 .47573 L .54006 .4912 L .5 .45994 L .45994 .4912 L .41987 .47573 L .3985 .45994 L .37981 .4411 L .36617 .41987 L .35061 .37981 L .34701 .33974 L .35421 .29968 L .37398 .25962 L .37981 .25179 L .4144 .21955 L .41987 .21623 L s .3 g .5 .45994 m .54006 .5 L .54006 .50889 L .58013 .52453 L .60093 .54006 L .62019 .55985 L .63296 .58013 L .64817 .62019 L .65142 .66026 L .64373 .70032 L .62313 .74038 L .62019 .7443 L .5803 .78045 L .58013 .78055 L .54006 .798 L .5 .80338 L .45994 .798 L .41987 .78055 L .4197 .78045 L .37981 .7443 L .37687 .74038 L .35627 .70032 L .34858 .66026 L .35183 .62019 L .36704 .58013 L .37981 .55985 L .39907 .54006 L .41987 .52453 L .45994 .50889 L .45994 .5 L F 0 g .5 .45994 m .54006 .5 L .54006 .50889 L .58013 .52453 L .60093 .54006 L .62019 .55985 L .63296 .58013 L .64817 .62019 L .65142 .66026 L .64373 .70032 L .62313 .74038 L .62019 .7443 L .5803 .78045 L .58013 .78055 L .54006 .798 L .5 .80338 L .45994 .798 L .41987 .78055 L .4197 .78045 L .37981 .7443 L .37687 .74038 L .35627 .70032 L .34858 .66026 L .35183 .62019 L .36704 .58013 L .37981 .55985 L .39907 .54006 L .41987 .52453 L .45994 .50889 L .45994 .5 L .5 .45994 L s .72 g .41987 .24846 m .45994 .22886 L .5 .22292 L .54006 .22886 L .58013 .24846 L .59443 .25962 L .62019 .29245 L .62402 .29968 L .63675 .33974 L .63701 .37981 L .62516 .41987 L .62019 .42974 L .5959 .45994 L .58013 .47287 L .54006 .49025 L .5 .45994 L .45994 .49025 L .41987 .47287 L .4041 .45994 L .37981 .42974 L .37484 .41987 L .36299 .37981 L .36325 .33974 L .37598 .29968 L .37981 .29245 L .40557 .25962 L F 0 g .41987 .24846 m .45994 .22886 L .5 .22292 L .54006 .22886 L .58013 .24846 L .59443 .25962 L .62019 .29245 L .62402 .29968 L .63675 .33974 L .63701 .37981 L .62516 .41987 L .62019 .42974 L .5959 .45994 L .58013 .47287 L .54006 .49025 L .5 .45994 L .45994 .49025 L .41987 .47287 L .4041 .45994 L .37981 .42974 L .37484 .41987 L .36299 .37981 L .36325 .33974 L .37598 .29968 L .37981 .29245 L .40557 .25962 L .41987 .24846 L s .28 g .5 .45994 m .54006 .5 L .54006 .50984 L .58013 .5274 L .59543 .54006 L .62019 .57148 L .62443 .58013 L .63594 .62019 L .63532 .66026 L .62204 .70032 L .62019 .70381 L .59119 .74038 L .58013 .74883 L .54006 .76864 L .5 .77463 L .45994 .76864 L .41987 .74883 L .40881 .74038 L .37981 .70381 L .37796 .70032 L .36468 .66026 L .36406 .62019 L .37557 .58013 L .37981 .57148 L .40457 .54006 L .41987 .5274 L .45994 .50984 L .45994 .5 L F 0 g .5 .45994 m .54006 .5 L .54006 .50984 L .58013 .5274 L .59543 .54006 L .62019 .57148 L .62443 .58013 L .63594 .62019 L .63532 .66026 L .62204 .70032 L .62019 .70381 L .59119 .74038 L .58013 .74883 L .54006 .76864 L .5 .77463 L .45994 .76864 L .41987 .74883 L .40881 .74038 L .37981 .70381 L .37796 .70032 L .36468 .66026 L .36406 .62019 L .37557 .58013 L .37981 .57148 L .40457 .54006 L .41987 .5274 L .45994 .50984 L .45994 .5 L .5 .45994 L s .74 g .45994 .25365 m .5 .24717 L .54006 .25365 L .56062 .25962 L .58013 .27577 L .60301 .29968 L .62019 .3328 L .62184 .33974 L .62612 .37981 L .62019 .41291 L .6175 .41987 L .59119 .45994 L .58013 .46991 L .54006 .48929 L .5 .45994 L .45994 .48929 L .41987 .46991 L .40881 .45994 L .3825 .41987 L .37981 .41291 L .37388 .37981 L .37816 .33974 L .37981 .3328 L .39699 .29968 L .41987 .27577 L .43938 .25962 L F 0 g .45994 .25365 m .5 .24717 L .54006 .25365 L .56062 .25962 L .58013 .27577 L .60301 .29968 L .62019 .3328 L .62184 .33974 L .62612 .37981 L .62019 .41291 L .6175 .41987 L .59119 .45994 L .58013 .46991 L .54006 .48929 L .5 .45994 L .45994 .48929 L .41987 .46991 L .40881 .45994 L .3825 .41987 L .37981 .41291 L .37388 .37981 L .37816 .33974 L .37981 .3328 L .39699 .29968 L .41987 .27577 L .43938 .25962 L .45994 .25365 L s .26 g .5 .45994 m .54006 .5 L .54006 .51079 L .58013 .53038 L .59078 .54006 L .61677 .58013 L .62019 .58944 L .62517 .62019 L .6205 .66026 L .62019 .66164 L .60102 .70032 L .58013 .7219 L .55468 .74038 L .54006 .74428 L .5 .7508 L .45994 .74428 L .44532 .74038 L .41987 .7219 L .39898 .70032 L .37981 .66164 L .3795 .66026 L .37483 .62019 L .37981 .58944 L .38323 .58013 L .40922 .54006 L .41987 .53038 L .45994 .51079 L .45994 .5 L F 0 g .5 .45994 m .54006 .5 L .54006 .51079 L .58013 .53038 L .59078 .54006 L .61677 .58013 L .62019 .58944 L .62517 .62019 L .6205 .66026 L .62019 .66164 L .60102 .70032 L .58013 .7219 L .55468 .74038 L .54006 .74428 L .5 .7508 L .45994 .74428 L .44532 .74038 L .41987 .7219 L .39898 .70032 L .37981 .66164 L .3795 .66026 L .37483 .62019 L .37981 .58944 L .38323 .58013 L .40922 .54006 L .41987 .53038 L .45994 .51079 L .45994 .5 L .5 .45994 L s .76 g .41987 .29941 m .45994 .27462 L .5 .26747 L .54006 .27462 L .58013 .29941 L .58042 .29968 L .60781 .33974 L .61618 .37981 L .60995 .41987 L .58709 .45994 L .58013 .46679 L .54006 .48833 L .5 .45994 L .45994 .48833 L .41987 .46679 L .41291 .45994 L .39005 .41987 L .38382 .37981 L .39219 .33974 L .41958 .29968 L F 0 g .41987 .29941 m .45994 .27462 L .5 .26747 L .54006 .27462 L .58013 .29941 L .58042 .29968 L .60781 .33974 L .61618 .37981 L .60995 .41987 L .58709 .45994 L .58013 .46679 L .54006 .48833 L .5 .45994 L .45994 .48833 L .41987 .46679 L .41291 .45994 L .39005 .41987 L .38382 .37981 L .39219 .33974 L .41958 .29968 L .41987 .29941 L s .24 g .5 .45994 m .54006 .5 L .54006 .51176 L .58013 .5335 L .58673 .54006 L .60929 .58013 L .61528 .62019 L .60654 .66026 L .58013 .69854 L .57919 .70032 L .54006 .7236 L .5 .7308 L .45994 .7236 L .42081 .70032 L .41987 .69854 L .39346 .66026 L .38472 .62019 L .39071 .58013 L .41327 .54006 L .41987 .5335 L .45994 .51176 L .45994 .5 L F 0 g .5 .45994 m .54006 .5 L .54006 .51176 L .58013 .5335 L .58673 .54006 L .60929 .58013 L .61528 .62019 L .60654 .66026 L .58013 .69854 L .57919 .70032 L .54006 .7236 L .5 .7308 L .45994 .7236 L .42081 .70032 L .41987 .69854 L .39346 .66026 L .38472 .62019 L .39071 .58013 L .41327 .54006 L .41987 .5335 L .45994 .51176 L .45994 .5 L .5 .45994 L s .78 g .45994 .2925 m .5 .28483 L .54006 .2925 L .56009 .29968 L .58013 .32045 L .59412 .33974 L .60679 .37981 L .6032 .41987 L .58344 .45994 L .58013 .4635 L .54006 .48737 L .5 .45994 L .45994 .48737 L .41987 .4635 L .41656 .45994 L .3968 .41987 L .39321 .37981 L .40588 .33974 L .41987 .32045 L .43991 .29968 L F 0 g .45994 .2925 m .5 .28483 L .54006 .2925 L .56009 .29968 L .58013 .32045 L .59412 .33974 L .60679 .37981 L .6032 .41987 L .58344 .45994 L .58013 .4635 L .54006 .48737 L .5 .45994 L .45994 .48737 L .41987 .4635 L .41656 .45994 L .3968 .41987 L .39321 .37981 L .40588 .33974 L .41987 .32045 L .43991 .29968 L .45994 .2925 L s .22 g .5 .45994 m .54006 .5 L .54006 .51272 L .58013 .53682 L .58312 .54006 L .60261 .58013 L .60596 .62019 L .59285 .66026 L .58013 .67771 L .55671 .70032 L .54006 .70599 L .5 .7137 L .45994 .70599 L .44329 .70032 L .41987 .67771 L .40715 .66026 L .39404 .62019 L .39739 .58013 L .41688 .54006 L .41987 .53682 L .45994 .51272 L .45994 .5 L F 0 g .5 .45994 m .54006 .5 L .54006 .51272 L .58013 .53682 L .58312 .54006 L .60261 .58013 L .60596 .62019 L .59285 .66026 L .58013 .67771 L .55671 .70032 L .54006 .70599 L .5 .7137 L .45994 .70599 L .44329 .70032 L .41987 .67771 L .40715 .66026 L .39404 .62019 L .39739 .58013 L .41688 .54006 L .41987 .53682 L .45994 .51272 L .45994 .5 L .5 .45994 L s .8 g .5 .29959 m .50318 .29968 L .54006 .30804 L .58013 .33962 L .58022 .33974 L .59805 .37981 L .59717 .41987 L .58015 .45994 L .58013 .45996 L .54006 .48639 L .5 .45994 L .45994 .48639 L .41987 .45996 L .41985 .45994 L .40283 .41987 L .40195 .37981 L .41978 .33974 L .41987 .33962 L .45994 .30804 L .49682 .29968 L F 0 g .5 .29959 m .50318 .29968 L .54006 .30804 L .58013 .33962 L .58022 .33974 L .59805 .37981 L .59717 .41987 L .58015 .45994 L .58013 .45996 L .54006 .48639 L .5 .45994 L .45994 .48639 L .41987 .45996 L .41985 .45994 L .40283 .41987 L .40195 .37981 L .41978 .33974 L .41987 .33962 L .45994 .30804 L .49682 .29968 L .5 .29959 L s .2 g .5 .45994 m .54006 .5 L .54006 .5137 L .57989 .54006 L .58013 .5405 L .59665 .58013 L .59727 .62019 L .58013 .65885 L .57941 .66026 L .54006 .69061 L .5 .69912 L .45994 .69061 L .42059 .66026 L .41987 .65885 L .40273 .62019 L .40335 .58013 L .41987 .5405 L .42011 .54006 L .45994 .5137 L .45994 .5 L F 0 g .5 .45994 m .54006 .5 L .54006 .5137 L .57989 .54006 L .58013 .5405 L .59665 .58013 L .59727 .62019 L .58013 .65885 L .57941 .66026 L .54006 .69061 L .5 .69912 L .45994 .69061 L .42059 .66026 L .41987 .65885 L .40273 .62019 L .40335 .58013 L .41987 .5405 L .42011 .54006 L .45994 .5137 L .45994 .5 L .5 .45994 L s .82 g .45994 .32173 m .5 .31275 L .54006 .32173 L .57036 .33974 L .58013 .35597 L .58982 .37981 L .59179 .41987 L .58013 .45452 L .57746 .45994 L .54006 .4854 L .5 .45994 L .45994 .4854 L .42254 .45994 L .41987 .45452 L .40821 .41987 L .41018 .37981 L .41987 .35597 L .42964 .33974 L F 0 g .45994 .32173 m .5 .31275 L .54006 .32173 L .57036 .33974 L .58013 .35597 L .58982 .37981 L .59179 .41987 L .58013 .45452 L .57746 .45994 L .54006 .4854 L .5 .45994 L .45994 .4854 L .42254 .45994 L .41987 .45452 L .40821 .41987 L .41018 .37981 L .41987 .35597 L .42964 .33974 L .45994 .32173 L s .18 g .5 .45994 m .54006 .5 L .54006 .51469 L .57722 .54006 L .58013 .54604 L .59132 .58013 L .58907 .62019 L .58013 .64245 L .5692 .66026 L .54006 .6771 L .5 .68612 L .45994 .6771 L .4308 .66026 L .41987 .64245 L .41093 .62019 L .40868 .58013 L .41987 .54604 L .42278 .54006 L .45994 .51469 L .45994 .5 L F 0 g .5 .45994 m .54006 .5 L .54006 .51469 L .57722 .54006 L .58013 .54604 L .59132 .58013 L .58907 .62019 L .58013 .64245 L .5692 .66026 L .54006 .6771 L .5 .68612 L .45994 .6771 L .4308 .66026 L .41987 .64245 L .41093 .62019 L .40868 .58013 L .41987 .54604 L .42278 .54006 L .45994 .51469 L .45994 .5 L .5 .45994 L s .84 g .45994 .3337 m .5 .3242 L .54006 .3337 L .55401 .33974 L .58013 .37473 L .58192 .37981 L .58697 .41987 L .58013 .44763 L .57494 .45994 L .54006 .48441 L .5 .45994 L .45994 .48441 L .42506 .45994 L .41987 .44763 L .41303 .41987 L .41808 .37981 L .41987 .37473 L .44599 .33974 L F 0 g .45994 .3337 m .5 .3242 L .54006 .3337 L .55401 .33974 L .58013 .37473 L .58192 .37981 L .58697 .41987 L .58013 .44763 L .57494 .45994 L .54006 .48441 L .5 .45994 L .45994 .48441 L .42506 .45994 L .41987 .44763 L .41303 .41987 L .41808 .37981 L .41987 .37473 L .44599 .33974 L .45994 .3337 L s .16 g .5 .45994 m .54006 .5 L .54006 .51568 L .57471 .54006 L .58013 .55312 L .58655 .58013 L .5812 .62019 L .58013 .62328 L .55197 .66026 L .54006 .66527 L .5 .67481 L .45994 .66527 L .44803 .66026 L .41987 .62328 L .4188 .62019 L .41345 .58013 L .41987 .55312 L .42529 .54006 L .45994 .51568 L .45994 .5 L F 0 g .5 .45994 m .54006 .5 L .54006 .51568 L .57471 .54006 L .58013 .55312 L .58655 .58013 L .5812 .62019 L .58013 .62328 L .55197 .66026 L .54006 .66527 L .5 .67481 L .45994 .66527 L .44803 .66026 L .41987 .62328 L .4188 .62019 L .41345 .58013 L .41987 .55312 L .42529 .54006 L .45994 .51568 L .45994 .5 L .5 .45994 L s .86 g .5 .33418 m .52975 .33974 L .54006 .34443 L .57555 .37981 L .58013 .3933 L .58263 .41987 L .58013 .4374 L .57255 .45994 L .54006 .48341 L .5 .45994 L .45994 .48341 L .42745 .45994 L .41987 .4374 L .41737 .41987 L .41987 .3933 L .42445 .37981 L .45994 .34443 L .47025 .33974 L F 0 g .5 .33418 m .52975 .33974 L .54006 .34443 L .57555 .37981 L .58013 .3933 L .58263 .41987 L .58013 .4374 L .57255 .45994 L .54006 .48341 L .5 .45994 L .45994 .48341 L .42745 .45994 L .41987 .4374 L .41737 .41987 L .41987 .3933 L .42445 .37981 L .45994 .34443 L .47025 .33974 L .5 .33418 L s .14 g .5 .45994 m .54006 .5 L .54006 .51669 L .57234 .54006 L .58013 .56394 L .58224 .58013 L .58013 .60478 L .575 .62019 L .54006 .65461 L .5272 .66026 L .5 .66496 L .4728 .66026 L .45994 .65461 L .425 .62019 L .41987 .60478 L .41776 .58013 L .41987 .56394 L .42766 .54006 L .45994 .51669 L .45994 .5 L F 0 g .5 .45994 m .54006 .5 L .54006 .51669 L .57234 .54006 L .58013 .56394 L .58224 .58013 L .58013 .60478 L .575 .62019 L .54006 .65461 L .5272 .66026 L .5 .66496 L .4728 .66026 L .45994 .65461 L .425 .62019 L .41987 .60478 L .41776 .58013 L .41987 .56394 L .42766 .54006 L .45994 .51669 L .45994 .5 L .5 .45994 L s .88 g .45994 .35426 m .5 .34313 L .54006 .35426 L .56933 .37981 L .57852 .41987 L .57028 .45994 L .54006 .48239 L .5 .45994 L .45994 .48239 L .42972 .45994 L .42148 .41987 L .43067 .37981 L F 0 g .45994 .35426 m .5 .34313 L .54006 .35426 L .56933 .37981 L .57852 .41987 L .57028 .45994 L .54006 .48239 L .5 .45994 L .45994 .48239 L .42972 .45994 L .42148 .41987 L .43067 .37981 L .45994 .35426 L s .12 g .5 .45994 m .54006 .5 L .54006 .5177 L .57007 .54006 L .57814 .58013 L .56872 .62019 L .54006 .64489 L .5 .65606 L .45994 .64489 L .43128 .62019 L .42186 .58013 L .42993 .54006 L .45994 .5177 L .45994 .5 L F 0 g .5 .45994 m .54006 .5 L .54006 .5177 L .57007 .54006 L .57814 .58013 L .56872 .62019 L .54006 .64489 L .5 .65606 L .45994 .64489 L .43128 .62019 L .42186 .58013 L .42993 .54006 L .45994 .5177 L .45994 .5 L .5 .45994 L s .9 g .45994 .36308 m .5 .35145 L .54006 .36308 L .56223 .37981 L .57447 .41987 L .5681 .45994 L .54006 .48136 L .5 .45994 L .45994 .48136 L .4319 .45994 L .42553 .41987 L .43777 .37981 L F 0 g .45994 .36308 m .5 .35145 L .54006 .36308 L .56223 .37981 L .57447 .41987 L .5681 .45994 L .54006 .48136 L .5 .45994 L .45994 .48136 L .4319 .45994 L 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L .17949 .41551 L .20659 .37981 L .21955 .36781 L .25962 .34319 L .26576 .33974 L .29968 .33118 L s .3 g .25962 .37398 m .29968 .35421 L .33974 .34701 L .37981 .35061 L .41987 .36617 L .4411 .37981 L .45994 .3985 L .47573 .41987 L .4912 .45994 L .45994 .5 L .4912 .54006 L .47573 .58013 L .45994 .6015 L .4411 .62019 L .41987 .63383 L .37981 .64939 L .33974 .65299 L .29968 .64579 L .25962 .62602 L .25179 .62019 L .21955 .5856 L .21623 .58013 L .19897 .54006 L .19364 .5 L .19897 .45994 L .21623 .41987 L .21955 .4144 L .25179 .37981 L F 0 g .25962 .37398 m .29968 .35421 L .33974 .34701 L .37981 .35061 L .41987 .36617 L .4411 .37981 L .45994 .3985 L .47573 .41987 L .4912 .45994 L .45994 .5 L .4912 .54006 L .47573 .58013 L .45994 .6015 L .4411 .62019 L .41987 .63383 L .37981 .64939 L .33974 .65299 L .29968 .64579 L .25962 .62602 L .25179 .62019 L .21955 .5856 L .21623 .58013 L .19897 .54006 L .19364 .5 L .19897 .45994 L .21623 .41987 L .21955 .4144 L .25179 .37981 L .25962 .37398 L s .28 g .29968 .37598 m .33974 .36325 L .37981 .36299 L .41987 .37484 L .42974 .37981 L .45994 .4041 L .47287 .41987 L .49025 .45994 L .45994 .5 L .49025 .54006 L .47287 .58013 L .45994 .5959 L .42974 .62019 L .41987 .62516 L .37981 .63701 L .33974 .63675 L .29968 .62402 L .29245 .62019 L .25962 .59443 L .24846 .58013 L .22886 .54006 L .22292 .5 L .22886 .45994 L .24846 .41987 L .25962 .40557 L .29245 .37981 L F 0 g .29968 .37598 m .33974 .36325 L .37981 .36299 L .41987 .37484 L .42974 .37981 L .45994 .4041 L .47287 .41987 L .49025 .45994 L .45994 .5 L .49025 .54006 L .47287 .58013 L .45994 .5959 L .42974 .62019 L .41987 .62516 L .37981 .63701 L .33974 .63675 L .29968 .62402 L .29245 .62019 L .25962 .59443 L .24846 .58013 L .22886 .54006 L .22292 .5 L .22886 .45994 L .24846 .41987 L .25962 .40557 L .29245 .37981 L .29968 .37598 L s .26 g .33974 .37816 m .37981 .37388 L .41291 .37981 L .41987 .3825 L .45994 .40881 L .46991 .41987 L .48929 .45994 L .45994 .5 L .48929 .54006 L .46991 .58013 L .45994 .59119 L .41987 .6175 L .41291 .62019 L .37981 .62612 L .33974 .62184 L .3328 .62019 L .29968 .60301 L .27577 .58013 L .25962 .56062 L .25365 .54006 L .24717 .5 L .25365 .45994 L .25962 .43938 L .27577 .41987 L .29968 .39699 L .3328 .37981 L F 0 g .33974 .37816 m .37981 .37388 L .41291 .37981 L .41987 .3825 L .45994 .40881 L .46991 .41987 L .48929 .45994 L .45994 .5 L .48929 .54006 L .46991 .58013 L .45994 .59119 L .41987 .6175 L .41291 .62019 L .37981 .62612 L .33974 .62184 L .3328 .62019 L .29968 .60301 L .27577 .58013 L .25962 .56062 L .25365 .54006 L .24717 .5 L .25365 .45994 L .25962 .43938 L .27577 .41987 L .29968 .39699 L .3328 .37981 L .33974 .37816 L s .46 g .25232 .98077 m .25962 .97542 L .29968 .94313 L .30244 .94071 L .33974 .90434 L .34325 .90064 L .37712 .86058 L .37981 .85697 L .40559 .82051 L .41987 .79676 L .42924 .78045 L .44911 .74038 L .45994 .71325 L .46506 .70032 L .47833 .66026 L .48904 .62019 L .49394 .58013 L .49773 .54006 L .45994 .5 L .49773 .45994 L .49394 .41987 L .48904 .37981 L .47833 .33974 L .46506 .29968 L .45994 .28675 L .44911 .25962 L .42924 .21955 L .41987 .20324 L .40559 .17949 L .37981 .14303 L .37712 .13942 L .34325 .09936 L .33974 .09566 L .30244 .05929 L .29968 .05687 L .25962 .02458 L .25232 .01923 L .98077 .01923 L .98077 .98077 L F 0 g .25232 .98077 m .25962 .97542 L .29968 .94313 L .30244 .94071 L .33974 .90434 L .34325 .90064 L .37712 .86058 L .37981 .85697 L .40559 .82051 L .41987 .79676 L .42924 .78045 L .44911 .74038 L .45994 .71325 L .46506 .70032 L .47833 .66026 L .48904 .62019 L .49394 .58013 L .49773 .54006 L .45994 .5 L .49773 .45994 L .49394 .41987 L .48904 .37981 L .47833 .33974 L .46506 .29968 L .45994 .28675 L .44911 .25962 L .42924 .21955 L .41987 .20324 L .40559 .17949 L .37981 .14303 L .37712 .13942 L .34325 .09936 L .33974 .09566 L .30244 .05929 L .29968 .05687 L .25962 .02458 L .25232 .01923 L s .24 g .29968 .41958 m .33974 .39219 L .37981 .38382 L .41987 .39005 L .45994 .41291 L .46679 .41987 L .48833 .45994 L .45994 .5 L .48833 .54006 L .46679 .58013 L .45994 .58709 L .41987 .60995 L .37981 .61618 L .33974 .60781 L .29968 .58042 L .29941 .58013 L .27462 .54006 L .26747 .5 L .27462 .45994 L .29941 .41987 L F 0 g .29968 .41958 m .33974 .39219 L .37981 .38382 L .41987 .39005 L .45994 .41291 L .46679 .41987 L .48833 .45994 L .45994 .5 L .48833 .54006 L .46679 .58013 L .45994 .58709 L .41987 .60995 L .37981 .61618 L .33974 .60781 L .29968 .58042 L .29941 .58013 L .27462 .54006 L .26747 .5 L .27462 .45994 L .29941 .41987 L .29968 .41958 L s .22 g .33974 .40588 m .37981 .39321 L .41987 .3968 L .45994 .41656 L .4635 .41987 L .48737 .45994 L .45994 .5 L .48737 .54006 L .4635 .58013 L .45994 .58344 L .41987 .6032 L .37981 .60679 L .33974 .59412 L .32045 .58013 L .29968 .56009 L .2925 .54006 L .28483 .5 L .2925 .45994 L .29968 .43991 L .32045 .41987 L F 0 g .33974 .40588 m .37981 .39321 L .41987 .3968 L .45994 .41656 L .4635 .41987 L .48737 .45994 L .45994 .5 L .48737 .54006 L .4635 .58013 L .45994 .58344 L .41987 .6032 L .37981 .60679 L .33974 .59412 L .32045 .58013 L .29968 .56009 L .2925 .54006 L .28483 .5 L .2925 .45994 L .29968 .43991 L .32045 .41987 L .33974 .40588 L s .2 g .33974 .41978 m .37981 .40195 L .41987 .40283 L .45994 .41985 L .45996 .41987 L .48639 .45994 L .45994 .5 L .48639 .54006 L .45996 .58013 L .45994 .58015 L .41987 .59717 L .37981 .59805 L .33974 .58022 L .33962 .58013 L .30804 .54006 L .29968 .50318 L .29959 .5 L .29968 .49682 L .30804 .45994 L .33962 .41987 L F 0 g .33974 .41978 m .37981 .40195 L .41987 .40283 L .45994 .41985 L .45996 .41987 L .48639 .45994 L .45994 .5 L .48639 .54006 L .45996 .58013 L .45994 .58015 L .41987 .59717 L .37981 .59805 L .33974 .58022 L .33962 .58013 L .30804 .54006 L .29968 .50318 L .29959 .5 L .29968 .49682 L .30804 .45994 L .33962 .41987 L .33974 .41978 L s .18 g .37981 .41018 m .41987 .40821 L .45452 .41987 L .45994 .42254 L .4854 .45994 L .45994 .5 L .4854 .54006 L .45994 .57746 L .45452 .58013 L .41987 .59179 L .37981 .58982 L .35597 .58013 L .33974 .57036 L .32173 .54006 L .31275 .5 L .32173 .45994 L .33974 .42964 L .35597 .41987 L F 0 g .37981 .41018 m .41987 .40821 L .45452 .41987 L .45994 .42254 L .4854 .45994 L .45994 .5 L .4854 .54006 L .45994 .57746 L .45452 .58013 L .41987 .59179 L .37981 .58982 L .35597 .58013 L .33974 .57036 L .32173 .54006 L .31275 .5 L .32173 .45994 L .33974 .42964 L .35597 .41987 L .37981 .41018 L s .16 g .37981 .41808 m .41987 .41303 L .44763 .41987 L .45994 .42506 L .48441 .45994 L .45994 .5 L .48441 .54006 L .45994 .57494 L .44763 .58013 L .41987 .58697 L .37981 .58192 L .37473 .58013 L .33974 .55401 L .3337 .54006 L .3242 .5 L .3337 .45994 L .33974 .44599 L .37473 .41987 L F 0 g .37981 .41808 m .41987 .41303 L .44763 .41987 L .45994 .42506 L .48441 .45994 L .45994 .5 L .48441 .54006 L .45994 .57494 L .44763 .58013 L .41987 .58697 L .37981 .58192 L .37473 .58013 L .33974 .55401 L .3337 .54006 L .3242 .5 L .3337 .45994 L .33974 .44599 L .37473 .41987 L .37981 .41808 L s .14 g .41987 .41737 m .4374 .41987 L .45994 .42745 L .48341 .45994 L .45994 .5 L .48341 .54006 L .45994 .57255 L .4374 .58013 L .41987 .58263 L .3933 .58013 L .37981 .57555 L .34443 .54006 L .33974 .52975 L .33418 .5 L .33974 .47025 L .34443 .45994 L .37981 .42445 L .3933 .41987 L F 0 g .41987 .41737 m .4374 .41987 L .45994 .42745 L .48341 .45994 L .45994 .5 L .48341 .54006 L .45994 .57255 L .4374 .58013 L .41987 .58263 L .3933 .58013 L .37981 .57555 L .34443 .54006 L .33974 .52975 L .33418 .5 L .33974 .47025 L .34443 .45994 L .37981 .42445 L .3933 .41987 L .41987 .41737 L s .12 g .37981 .43067 m .41987 .42148 L .45994 .42972 L .48239 .45994 L .45994 .5 L .48239 .54006 L .45994 .57028 L .41987 .57852 L .37981 .56933 L .35426 .54006 L .34313 .5 L .35426 .45994 L F 0 g .37981 .43067 m .41987 .42148 L .45994 .42972 L .48239 .45994 L .45994 .5 L .48239 .54006 L .45994 .57028 L .41987 .57852 L .37981 .56933 L .35426 .54006 L .34313 .5 L .35426 .45994 L .37981 .43067 L s .1 g .37981 .43777 m .41987 .42553 L .45994 .4319 L .48136 .45994 L .45994 .5 L .48136 .54006 L .45994 .5681 L .41987 .57447 L .37981 .56223 L .36308 .54006 L .35145 .5 L .36308 .45994 L F 0 g .37981 .43777 m .41987 .42553 L .45994 .4319 L .48136 .45994 L .45994 .5 L .48136 .54006 L .45994 .5681 L .41987 .57447 L .37981 .56223 L .36308 .54006 L .35145 .5 L .36308 .45994 L .37981 .43777 L s .08 g .37981 .44635 m .41987 .42936 L .45994 .43399 L .48032 .45994 L .45994 .5 L .48032 .54006 L .45994 .56601 L .41987 .57064 L .37981 .55365 L .37102 .54006 L .35889 .5 L .37102 .45994 L F 0 g .37981 .44635 m .41987 .42936 L .45994 .43399 L .48032 .45994 L .45994 .5 L .48032 .54006 L .45994 .56601 L .41987 .57064 L .37981 .55365 L .37102 .54006 L .35889 .5 L .37102 .45994 L .37981 .44635 L s .06 g .37981 .45703 m .41987 .43301 L .45994 .436 L .47926 .45994 L .45994 .5 L .47926 .54006 L .45994 .564 L .41987 .56699 L .37981 .54297 L .37821 .54006 L .36552 .5 L .37821 .45994 L F 0 g .37981 .45703 m .41987 .43301 L .45994 .436 L .47926 .45994 L .45994 .5 L .47926 .54006 L .45994 .564 L .41987 .56699 L .37981 .54297 L .37821 .54006 L .36552 .5 L .37821 .45994 L .37981 .45703 L s .04 g .41987 .43654 m .45994 .43794 L .47818 .45994 L .45994 .5 L .47818 .54006 L .45994 .56206 L .41987 .56346 L .38493 .54006 L .37981 .53202 L .37144 .5 L .37981 .46798 L .38493 .45994 L F 0 g .41987 .43654 m .45994 .43794 L .47818 .45994 L .45994 .5 L .47818 .54006 L .45994 .56206 L .41987 .56346 L .38493 .54006 L .37981 .53202 L .37144 .5 L .37981 .46798 L .38493 .45994 L .41987 .43654 L s .48 g .37635 .98077 m .37981 .97446 L .39728 .94071 L .41592 .90064 L .41987 .89141 L .43251 .86058 L .44712 .82051 L .45972 .78045 L .45994 .77968 L .47064 .74038 L .47985 .70032 L .48744 .66026 L .49364 .62019 L .49642 .58013 L .49866 .54006 L .45994 .5 L .49866 .45994 L .49642 .41987 L .49364 .37981 L .48744 .33974 L .47985 .29968 L .47064 .25962 L .45994 .22032 L .45972 .21955 L .44712 .17949 L .43251 .13942 L .41987 .10859 L .41592 .09936 L .39728 .05929 L .37981 .02554 L .37635 .01923 L .98077 .01923 L .98077 .98077 L F 0 g .37635 .98077 m .37981 .97446 L .39728 .94071 L .41592 .90064 L .41987 .89141 L .43251 .86058 L .44712 .82051 L .45972 .78045 L .45994 .77968 L .47064 .74038 L .47985 .70032 L .48744 .66026 L .49364 .62019 L .49642 .58013 L .49866 .54006 L .45994 .5 L .49866 .45994 L .49642 .41987 L .49364 .37981 L .48744 .33974 L .47985 .29968 L .47064 .25962 L .45994 .22032 L .45972 .21955 L .44712 .17949 L .43251 .13942 L .41987 .10859 L .41592 .09936 L .39728 .05929 L .37981 .02554 L .37635 .01923 L s .02 g .41987 .43999 m .45994 .43982 L .47709 .45994 L .45994 .5 L .47709 .54006 L .45994 .56018 L .41987 .56001 L .3912 .54006 L .37981 .51847 L .37671 .5 L .37981 .48153 L .3912 .45994 L F 0 g .41987 .43999 m .45994 .43982 L .47709 .45994 L .45994 .5 L .47709 .54006 L .45994 .56018 L .41987 .56001 L .3912 .54006 L .37981 .51847 L .37671 .5 L .37981 .48153 L .3912 .45994 L .41987 .43999 L s .41987 .44343 m .45994 .44164 L .47598 .45994 L .45994 .5 L .47598 .54006 L .45994 .55836 L .41987 .55657 L .39703 .54006 L .38168 .5 L .39703 .45994 L F .41987 .44343 m .45994 .44164 L .47598 .45994 L .45994 .5 L .47598 .54006 L .45994 .55836 L .41987 .55657 L .39703 .54006 L .38168 .5 L .39703 .45994 L .41987 .44343 L s .58 g .98077 .90194 m .94071 .90712 L .90064 .90832 L .86058 .90559 L .82915 .90064 L .82051 .89883 L .78045 .88782 L .74038 .87221 L .71587 .86058 L .70032 .85138 L .66026 .82434 L .65515 .82051 L .62019 .78925 L .61128 .78045 L .58013 .74288 L .57819 .74038 L .5518 .70032 L .54006 .67549 L .53246 .66026 L .51638 .62019 L .50878 .58013 L .50328 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .50328 .45994 L .50878 .41987 L .51638 .37981 L .53246 .33974 L .54006 .32451 L .5518 .29968 L .57819 .25962 L .58013 .25712 L .61128 .21955 L .62019 .21075 L .65515 .17949 L .66026 .17566 L .70032 .14862 L .71587 .13942 L .74038 .12779 L .78045 .11218 L .82051 .10117 L .82915 .09936 L .86058 .09441 L .90064 .09168 L .94071 .09288 L .98077 .09806 L F 0 g .98077 .90194 m .94071 .90712 L .90064 .90832 L .86058 .90559 L .82915 .90064 L .82051 .89883 L .78045 .88782 L .74038 .87221 L .71587 .86058 L .70032 .85138 L .66026 .82434 L .65515 .82051 L .62019 .78925 L .61128 .78045 L .58013 .74288 L .57819 .74038 L .5518 .70032 L .54006 .67549 L .53246 .66026 L .51638 .62019 L .50878 .58013 L .50328 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .50328 .45994 L .50878 .41987 L .51638 .37981 L .53246 .33974 L .54006 .32451 L .5518 .29968 L .57819 .25962 L .58013 .25712 L .61128 .21955 L .62019 .21075 L .65515 .17949 L .66026 .17566 L .70032 .14862 L .71587 .13942 L .74038 .12779 L .78045 .11218 L .82051 .10117 L .82915 .09936 L .86058 .09441 L .90064 .09168 L .94071 .09288 L .98077 .09806 L s .6 g .98077 .7742 m .96955 .78045 L .94071 .79423 L .90064 .80782 L .86058 .81579 L .82051 .81856 L .78045 .81627 L .74038 .8088 L .70032 .79575 L .66604 .78045 L .66026 .77642 L .62019 .74913 L .60891 .74038 L .58013 .71093 L .57034 .70032 L .54247 .66026 L .54006 .65415 L .52185 .62019 L .5113 .58013 L .50421 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .50421 .45994 L .5113 .41987 L .52185 .37981 L .54006 .34585 L .54247 .33974 L .57034 .29968 L .58013 .28907 L .60891 .25962 L .62019 .25087 L .66026 .22358 L .66604 .21955 L .70032 .20425 L .74038 .1912 L .78045 .18373 L .82051 .18144 L .86058 .18421 L .90064 .19218 L .94071 .20577 L .96955 .21955 L .98077 .2258 L F 0 g .98077 .7742 m .96955 .78045 L .94071 .79423 L .90064 .80782 L .86058 .81579 L .82051 .81856 L .78045 .81627 L .74038 .8088 L .70032 .79575 L .66604 .78045 L .66026 .77642 L .62019 .74913 L .60891 .74038 L .58013 .71093 L .57034 .70032 L .54247 .66026 L .54006 .65415 L .52185 .62019 L .5113 .58013 L .50421 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .50421 .45994 L .5113 .41987 L .52185 .37981 L .54006 .34585 L .54247 .33974 L .57034 .29968 L .58013 .28907 L .60891 .25962 L .62019 .25087 L .66026 .22358 L .66604 .21955 L .70032 .20425 L .74038 .1912 L .78045 .18373 L .82051 .18144 L .86058 .18421 L .90064 .19218 L .94071 .20577 L .96955 .21955 L .98077 .2258 L s .62 g .98077 .6415 m .96732 .66026 L .94071 .68965 L .92865 .70032 L .90064 .72077 L .86316 .74038 L .86058 .74146 L .82051 .75428 L .78045 .7604 L .74038 .76027 L .70032 .75389 L .66026 .74086 L .65894 .74038 L .62019 .7196 L .59127 .70032 L .58013 .68782 L .55255 .66026 L .54006 .63731 L .52796 .62019 L .51384 .58013 L .50514 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .50514 .45994 L .51384 .41987 L .52796 .37981 L .54006 .36269 L .55255 .33974 L .58013 .31218 L .59127 .29968 L .62019 .2804 L .65894 .25962 L .66026 .25914 L .70032 .24611 L .74038 .23973 L .78045 .2396 L .82051 .24572 L .86058 .25854 L .86316 .25962 L .90064 .27923 L .92865 .29968 L .94071 .31035 L .96732 .33974 L .98077 .3585 L F 0 g .98077 .6415 m .96732 .66026 L .94071 .68965 L .92865 .70032 L .90064 .72077 L .86316 .74038 L .86058 .74146 L .82051 .75428 L .78045 .7604 L .74038 .76027 L .70032 .75389 L .66026 .74086 L .65894 .74038 L .62019 .7196 L .59127 .70032 L .58013 .68782 L .55255 .66026 L .54006 .63731 L .52796 .62019 L .51384 .58013 L .50514 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .50514 .45994 L .51384 .41987 L .52796 .37981 L .54006 .36269 L .55255 .33974 L .58013 .31218 L .59127 .29968 L .62019 .2804 L .65894 .25962 L .66026 .25914 L .70032 .24611 L .74038 .23973 L .78045 .2396 L .82051 .24572 L .86058 .25854 L .86316 .25962 L .90064 .27923 L .92865 .29968 L .94071 .31035 L .96732 .33974 L .98077 .3585 L s .52 g .54395 .98077 m .54006 .95923 L .53689 .94071 L .53044 .90064 L .52462 .86058 L .51942 .82051 L .51483 .78045 L .51086 .74038 L .50748 .70032 L .50467 .66026 L .50237 .62019 L .50135 .58013 L .5005 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .5005 .45994 L .50135 .41987 L .50237 .37981 L .50467 .33974 L .50748 .29968 L .51086 .25962 L .51483 .21955 L .51942 .17949 L .52462 .13942 L .53044 .09936 L .53689 .05929 L .54006 .04077 L .54395 .01923 L .98077 .01923 L .98077 .98077 L F 0 g .54395 .98077 m .54006 .95923 L .53689 .94071 L .53044 .90064 L .52462 .86058 L .51942 .82051 L .51483 .78045 L .51086 .74038 L .50748 .70032 L .50467 .66026 L .50237 .62019 L .50135 .58013 L .5005 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .5005 .45994 L .50135 .41987 L .50237 .37981 L .50467 .33974 L .50748 .29968 L .51086 .25962 L .51483 .21955 L .51942 .17949 L .52462 .13942 L .53044 .09936 L .53689 .05929 L .54006 .04077 L .54395 .01923 L s .5 g .4636 .98077 m .46944 .94071 L .47477 .90064 L .47959 .86058 L .4839 .82051 L .4877 .78045 L .49099 .74038 L .4938 .70032 L .49612 .66026 L .49804 .62019 L .49888 .58013 L .49958 .54006 L .45994 .5 L .49958 .45994 L .49888 .41987 L .49804 .37981 L .49612 .33974 L .4938 .29968 L .49099 .25962 L .4877 .21955 L .4839 .17949 L .47959 .13942 L .47477 .09936 L .46944 .05929 L .4636 .01923 L .98077 .01923 L .98077 .98077 L F 0 g .4636 .98077 m .46944 .94071 L .47477 .90064 L .47959 .86058 L .4839 .82051 L .4877 .78045 L .49099 .74038 L .4938 .70032 L .49612 .66026 L .49804 .62019 L .49888 .58013 L .49958 .54006 L .45994 .5 L .49958 .45994 L .49888 .41987 L .49804 .37981 L .49612 .33974 L .4938 .29968 L .49099 .25962 L .4877 .21955 L .4839 .17949 L .47959 .13942 L .47477 .09936 L .46944 .05929 L .4636 .01923 L s .54 g .63276 .98077 m .62019 .9591 L .61013 .94071 L .58996 .90064 L .58013 .87895 L .57215 .86058 L .55645 .82051 L .54293 .78045 L .54006 .77081 L .53131 .74038 L .52148 .70032 L .51339 .66026 L .50677 .62019 L .50381 .58013 L .50143 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .50143 .45994 L .50381 .41987 L .50677 .37981 L .51339 .33974 L .52148 .29968 L .53131 .25962 L .54006 .22919 L .54293 .21955 L .55645 .17949 L .57215 .13942 L .58013 .12105 L .58996 .09936 L .61013 .05929 L .62019 .0409 L .63276 .01923 L .98077 .01923 L .98077 .98077 L F 0 g .63276 .98077 m .62019 .9591 L .61013 .94071 L .58996 .90064 L .58013 .87895 L .57215 .86058 L .55645 .82051 L .54293 .78045 L .54006 .77081 L .53131 .74038 L .52148 .70032 L .51339 .66026 L .50677 .62019 L .50381 .58013 L .50143 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .50143 .45994 L .50381 .41987 L .50677 .37981 L .51339 .33974 L .52148 .29968 L .53131 .25962 L .54006 .22919 L .54293 .21955 L .55645 .17949 L .57215 .13942 L .58013 .12105 L .58996 .09936 L .61013 .05929 L .62019 .0409 L .63276 .01923 L s .56 g .76466 .98077 m .74038 .96445 L .70906 .94071 L .70032 .93328 L .66495 .90064 L .66026 .89573 L .62875 .86058 L .62019 .84961 L .5987 .82051 L .58013 .79093 L .57386 .78045 L .55299 .74038 L .54006 .70926 L .53638 .70032 L .52255 .66026 L .5114 .62019 L .50629 .58013 L .50235 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .50235 .45994 L .50629 .41987 L .5114 .37981 L .52255 .33974 L .53638 .29968 L .54006 .29074 L .55299 .25962 L .57386 .21955 L .58013 .20907 L .5987 .17949 L .62019 .15039 L .62875 .13942 L .66026 .10427 L .66495 .09936 L .70032 .06672 L .70906 .05929 L .74038 .03555 L .76466 .01923 L .98077 .01923 L .98077 .98077 L F 0 g .76466 .98077 m .74038 .96445 L .70906 .94071 L .70032 .93328 L .66495 .90064 L .66026 .89573 L .62875 .86058 L .62019 .84961 L .5987 .82051 L .58013 .79093 L .57386 .78045 L .55299 .74038 L .54006 .70926 L .53638 .70032 L .52255 .66026 L .5114 .62019 L .50629 .58013 L .50235 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .50235 .45994 L .50629 .41987 L .5114 .37981 L .52255 .33974 L .53638 .29968 L .54006 .29074 L .55299 .25962 L .57386 .21955 L .58013 .20907 L .5987 .17949 L .62019 .15039 L .62875 .13942 L .66026 .10427 L .66495 .09936 L .70032 .06672 L .70906 .05929 L .74038 .03555 L .76466 .01923 L s .64 g .66026 .28745 m .70032 .27982 L .74038 .27962 L .78045 .28683 L .81529 .29968 L .82051 .30227 L .86058 .3281 L .87377 .33974 L .90064 .37037 L .90699 .37981 L .92745 .41987 L .93883 .45994 L .94071 .47208 L .94249 .5 L .94071 .52792 L .93883 .54006 L .92745 .58013 L .90699 .62019 L .90064 .62963 L .87377 .66026 L .86058 .6719 L .82051 .69773 L .81529 .70032 L .78045 .71317 L .74038 .72038 L .70032 .72018 L .66026 .71255 L .62735 .70032 L .62019 .69675 L .58013 .66991 L .5669 .66026 L .54006 .62572 L .53484 .62019 L .51643 .58013 L .50607 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .50607 .45994 L .51643 .41987 L .53484 .37981 L .54006 .37428 L .5669 .33974 L .58013 .33009 L .62019 .30325 L .62735 .29968 L F 0 g .66026 .28745 m .70032 .27982 L .74038 .27962 L .78045 .28683 L .81529 .29968 L .82051 .30227 L .86058 .3281 L .87377 .33974 L .90064 .37037 L .90699 .37981 L .92745 .41987 L .93883 .45994 L .94071 .47208 L .94249 .5 L .94071 .52792 L .93883 .54006 L .92745 .58013 L .90699 .62019 L .90064 .62963 L .87377 .66026 L .86058 .6719 L .82051 .69773 L .81529 .70032 L .78045 .71317 L .74038 .72038 L .70032 .72018 L .66026 .71255 L .62735 .70032 L .62019 .69675 L .58013 .66991 L .5669 .66026 L .54006 .62572 L .53484 .62019 L .51643 .58013 L .50607 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .50607 .45994 L .51643 .41987 L .53484 .37981 L .54006 .37428 L .5669 .33974 L .58013 .33009 L .62019 .30325 L .62735 .29968 L .66026 .28745 L s .66 g .62019 .32228 m .66026 .3108 L .70032 .30829 L .74038 .3143 L .78045 .32969 L .79757 .33974 L .82051 .35736 L .84154 .37981 L .86058 .40777 L .86631 .41987 L .8796 .45994 L .88383 .5 L .8796 .54006 L .86631 .58013 L .86058 .59223 L .84154 .62019 L .82051 .64264 L .79757 .66026 L .78045 .67031 L .74038 .6857 L .70032 .69171 L .66026 .6892 L .62019 .67772 L .5864 .66026 L .58013 .65579 L .54241 .62019 L .54006 .61667 L .51906 .58013 L .50701 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .50701 .45994 L .51906 .41987 L .54006 .38333 L .54241 .37981 L .58013 .34421 L .5864 .33974 L F 0 g .62019 .32228 m .66026 .3108 L .70032 .30829 L .74038 .3143 L .78045 .32969 L .79757 .33974 L .82051 .35736 L .84154 .37981 L .86058 .40777 L .86631 .41987 L .8796 .45994 L .88383 .5 L .8796 .54006 L .86631 .58013 L .86058 .59223 L .84154 .62019 L .82051 .64264 L .79757 .66026 L .78045 .67031 L .74038 .6857 L .70032 .69171 L .66026 .6892 L .62019 .67772 L .5864 .66026 L .58013 .65579 L .54241 .62019 L .54006 .61667 L .51906 .58013 L .50701 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .50701 .45994 L .51906 .41987 L .54006 .38333 L .54241 .37981 L .58013 .34421 L .5864 .33974 L .62019 .32228 L s .68 g .62019 .33791 m .66026 .33088 L .70032 .33339 L .72813 .33974 L .74038 .34605 L .78045 .37185 L .78893 .37981 L .81879 .41987 L .82051 .42099 L .8341 .45994 L .8389 .5 L .8341 .54006 L .82051 .57901 L .81879 .58013 L .78893 .62019 L .78045 .62815 L .74038 .65395 L .72813 .66026 L .70032 .66661 L .66026 .66912 L .62019 .66209 L .61447 .66026 L .58013 .64328 L .55046 .62019 L .54006 .60772 L .52176 .58013 L .50795 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .50795 .45994 L .52176 .41987 L .54006 .39228 L .55046 .37981 L .58013 .35672 L .61447 .33974 L F 0 g .62019 .33791 m .66026 .33088 L .70032 .33339 L .72813 .33974 L .74038 .34605 L .78045 .37185 L .78893 .37981 L .81879 .41987 L .82051 .42099 L .8341 .45994 L .8389 .5 L .8341 .54006 L .82051 .57901 L .81879 .58013 L .78893 .62019 L .78045 .62815 L .74038 .65395 L .72813 .66026 L .70032 .66661 L .66026 .66912 L .62019 .66209 L .61447 .66026 L .58013 .64328 L .55046 .62019 L .54006 .60772 L .52176 .58013 L .50795 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .50795 .45994 L .52176 .41987 L .54006 .39228 L .55046 .37981 L .58013 .35672 L .61447 .33974 L .62019 .33791 L s .7 g .58013 .36704 m .62019 .35183 L .66026 .34858 L .70032 .35627 L .74038 .37687 L .7443 .37981 L .78045 .4197 L .78055 .41987 L .798 .45994 L .80338 .5 L .798 .54006 L .78055 .58013 L .78045 .5803 L .7443 .62019 L .74038 .62313 L .70032 .64373 L .66026 .65142 L .62019 .64817 L .58013 .63296 L .55985 .62019 L .54006 .60093 L .52453 .58013 L .50889 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .50889 .45994 L .52453 .41987 L .54006 .39907 L .55985 .37981 L F 0 g .58013 .36704 m .62019 .35183 L .66026 .34858 L .70032 .35627 L .74038 .37687 L .7443 .37981 L .78045 .4197 L .78055 .41987 L .798 .45994 L .80338 .5 L .798 .54006 L .78055 .58013 L .78045 .5803 L .7443 .62019 L .74038 .62313 L .70032 .64373 L .66026 .65142 L .62019 .64817 L .58013 .63296 L .55985 .62019 L .54006 .60093 L .52453 .58013 L .50889 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .50889 .45994 L .52453 .41987 L .54006 .39907 L .55985 .37981 L .58013 .36704 L s .72 g .58013 .37557 m .62019 .36406 L .66026 .36468 L .70032 .37796 L .70381 .37981 L .74038 .40881 L .74883 .41987 L .76864 .45994 L .77463 .5 L .76864 .54006 L .74883 .58013 L .74038 .59119 L .70381 .62019 L .70032 .62204 L .66026 .63532 L .62019 .63594 L .58013 .62443 L .57148 .62019 L .54006 .59543 L .5274 .58013 L .50984 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .50984 .45994 L .5274 .41987 L .54006 .40457 L .57148 .37981 L F 0 g .58013 .37557 m .62019 .36406 L .66026 .36468 L .70032 .37796 L .70381 .37981 L .74038 .40881 L .74883 .41987 L .76864 .45994 L .77463 .5 L .76864 .54006 L .74883 .58013 L .74038 .59119 L .70381 .62019 L .70032 .62204 L .66026 .63532 L .62019 .63594 L .58013 .62443 L .57148 .62019 L .54006 .59543 L .5274 .58013 L .50984 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .50984 .45994 L .5274 .41987 L .54006 .40457 L .57148 .37981 L .58013 .37557 L s .74 g .62019 .37483 m .66026 .3795 L .66164 .37981 L .70032 .39898 L .7219 .41987 L .74038 .44532 L .74428 .45994 L .7508 .5 L .74428 .54006 L .74038 .55468 L .7219 .58013 L .70032 .60102 L .66164 .62019 L .66026 .6205 L .62019 .62517 L .58944 .62019 L .58013 .61677 L .54006 .59078 L .53038 .58013 L .51079 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .51079 .45994 L .53038 .41987 L .54006 .40922 L .58013 .38323 L .58944 .37981 L F 0 g .62019 .37483 m .66026 .3795 L .66164 .37981 L .70032 .39898 L .7219 .41987 L .74038 .44532 L .74428 .45994 L .7508 .5 L .74428 .54006 L .74038 .55468 L .7219 .58013 L .70032 .60102 L .66164 .62019 L .66026 .6205 L .62019 .62517 L .58944 .62019 L .58013 .61677 L .54006 .59078 L .53038 .58013 L .51079 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .51079 .45994 L .53038 .41987 L .54006 .40922 L .58013 .38323 L .58944 .37981 L .62019 .37483 L s .76 g .54006 .41327 m .58013 .39071 L .62019 .38472 L .66026 .39346 L .69854 .41987 L .70032 .42081 L .7236 .45994 L .7308 .5 L .7236 .54006 L .70032 .57919 L .69854 .58013 L .66026 .60654 L .62019 .61528 L .58013 .60929 L .54006 .58673 L .5335 .58013 L .51176 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .51176 .45994 L .5335 .41987 L F 0 g .54006 .41327 m .58013 .39071 L .62019 .38472 L .66026 .39346 L .69854 .41987 L .70032 .42081 L .7236 .45994 L .7308 .5 L .7236 .54006 L .70032 .57919 L .69854 .58013 L .66026 .60654 L .62019 .61528 L .58013 .60929 L .54006 .58673 L .5335 .58013 L .51176 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .51176 .45994 L .5335 .41987 L .54006 .41327 L s .78 g .54006 .41688 m .58013 .39739 L .62019 .39404 L .66026 .40715 L .67771 .41987 L .70032 .44329 L .70599 .45994 L .7137 .5 L .70599 .54006 L .70032 .55671 L .67771 .58013 L .66026 .59285 L .62019 .60596 L .58013 .60261 L .54006 .58312 L .53682 .58013 L .51272 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .51272 .45994 L .53682 .41987 L F 0 g .54006 .41688 m .58013 .39739 L .62019 .39404 L .66026 .40715 L .67771 .41987 L .70032 .44329 L .70599 .45994 L .7137 .5 L .70599 .54006 L .70032 .55671 L .67771 .58013 L .66026 .59285 L .62019 .60596 L .58013 .60261 L .54006 .58312 L .53682 .58013 L .51272 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .51272 .45994 L .53682 .41987 L .54006 .41688 L s .8 g .58013 .40335 m .62019 .40273 L .65885 .41987 L .66026 .42059 L .69061 .45994 L .69912 .5 L .69061 .54006 L .66026 .57941 L .65885 .58013 L .62019 .59727 L .58013 .59665 L .5405 .58013 L .54006 .57989 L .5137 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .5137 .45994 L .54006 .42011 L .5405 .41987 L F 0 g .58013 .40335 m .62019 .40273 L .65885 .41987 L .66026 .42059 L .69061 .45994 L .69912 .5 L .69061 .54006 L .66026 .57941 L .65885 .58013 L .62019 .59727 L .58013 .59665 L .5405 .58013 L .54006 .57989 L .5137 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .5137 .45994 L .54006 .42011 L .5405 .41987 L .58013 .40335 L s .82 g .58013 .40868 m .62019 .41093 L .64245 .41987 L .66026 .4308 L .6771 .45994 L .68612 .5 L .6771 .54006 L .66026 .5692 L .64245 .58013 L .62019 .58907 L .58013 .59132 L .54604 .58013 L .54006 .57722 L .51469 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .51469 .45994 L .54006 .42278 L .54604 .41987 L F 0 g .58013 .40868 m .62019 .41093 L .64245 .41987 L .66026 .4308 L .6771 .45994 L .68612 .5 L .6771 .54006 L .66026 .5692 L .64245 .58013 L .62019 .58907 L .58013 .59132 L .54604 .58013 L .54006 .57722 L .51469 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .51469 .45994 L .54006 .42278 L .54604 .41987 L .58013 .40868 L s .84 g .58013 .41345 m .62019 .4188 L .62328 .41987 L .66026 .44803 L .66527 .45994 L .67481 .5 L .66527 .54006 L .66026 .55197 L .62328 .58013 L .62019 .5812 L .58013 .58655 L .55312 .58013 L .54006 .57471 L .51568 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .51568 .45994 L .54006 .42529 L .55312 .41987 L F 0 g .58013 .41345 m .62019 .4188 L .62328 .41987 L .66026 .44803 L .66527 .45994 L .67481 .5 L .66527 .54006 L .66026 .55197 L .62328 .58013 L .62019 .5812 L .58013 .58655 L .55312 .58013 L .54006 .57471 L .51568 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .51568 .45994 L .54006 .42529 L .55312 .41987 L .58013 .41345 L s .86 g .58013 .41776 m .60478 .41987 L .62019 .425 L .65461 .45994 L .66026 .4728 L .66496 .5 L .66026 .5272 L .65461 .54006 L .62019 .575 L .60478 .58013 L .58013 .58224 L .56394 .58013 L .54006 .57234 L .51669 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .51669 .45994 L .54006 .42766 L .56394 .41987 L F 0 g .58013 .41776 m .60478 .41987 L .62019 .425 L .65461 .45994 L .66026 .4728 L .66496 .5 L .66026 .5272 L .65461 .54006 L .62019 .575 L .60478 .58013 L .58013 .58224 L .56394 .58013 L .54006 .57234 L .51669 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .51669 .45994 L .54006 .42766 L .56394 .41987 L .58013 .41776 L s .88 g .54006 .42993 m .58013 .42186 L .62019 .43128 L .64489 .45994 L .65606 .5 L .64489 .54006 L .62019 .56872 L .58013 .57814 L .54006 .57007 L .5177 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .5177 .45994 L F 0 g .54006 .42993 m .58013 .42186 L .62019 .43128 L .64489 .45994 L .65606 .5 L .64489 .54006 L .62019 .56872 L .58013 .57814 L .54006 .57007 L .5177 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .5177 .45994 L .54006 .42993 L s .9 g .54006 .4321 m .58013 .4259 L .62019 .43849 L .63615 .45994 L .64783 .5 L .63615 .54006 L .62019 .56151 L .58013 .5741 L .54006 .5679 L .51873 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .51873 .45994 L F 0 g .54006 .4321 m .58013 .4259 L .62019 .43849 L .63615 .45994 L .64783 .5 L .63615 .54006 L .62019 .56151 L .58013 .5741 L .54006 .5679 L .51873 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .51873 .45994 L .54006 .4321 L s .92 g .54006 .43418 m .58013 .4297 L .62019 .44725 L .62828 .45994 L .64047 .5 L .62828 .54006 L .62019 .55275 L .58013 .5703 L .54006 .56582 L .51978 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .51978 .45994 L F 0 g .54006 .43418 m .58013 .4297 L .62019 .44725 L .62828 .45994 L .64047 .5 L .62828 .54006 L .62019 .55275 L .58013 .5703 L .54006 .56582 L .51978 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .51978 .45994 L .54006 .43418 L s .94 g .54006 .43618 m .58013 .43334 L .62019 .45815 L .62116 .45994 L .6339 .5 L .62116 .54006 L .62019 .54185 L .58013 .56666 L .54006 .56382 L .52084 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .52084 .45994 L F 0 g .54006 .43618 m .58013 .43334 L .62019 .45815 L .62116 .45994 L .6339 .5 L .62116 .54006 L .62019 .54185 L .58013 .56666 L .54006 .56382 L .52084 .54006 L .5 .54006 L .45994 .5 L .5 .45994 L .52084 .45994 L .54006 .43618 L s .96 g .54006 .43812 m .58013 .43686 L .61446 .45994 L .62019 .46905 L 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(See ", StyleBox["The Structure of Materials", FontSlant->"Italic"], " to learn more about dislocations and glide planes.)" }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Group Exercise G3-2: Entropy of Small System", "Subsection"], Cell["\<\ Credits go to the Woolf group: Jill A. Rowehl, Jin Suk Kim, \ Katherine Hartman, Talia Gershon ; and to Prof. Carter, who assisted \ them.\ \>", "Text"], Cell["\<\ Each atom can have its electron in either the n = 1 or the n = 2 \ states. The energy of an electron in one of these states depends on n \ as\ \>", "Text"], Cell[BoxData[ \(Energy[n_] := \[Epsilon]\_0\ \((1 - 1\/n\^2)\)\)], "Input"], Cell[CellGroupData[{ Cell["1 and 2.", "Subsubsection"], Cell["\<\ The distinct quantum states for the three-atom system can be \ represented\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(States = Tuples[{1, 2}, 3]\)], "Input"], Cell[BoxData[ \({{1, 1, 1}, {1, 1, 2}, {1, 2, 1}, {1, 2, 2}, {2, 1, 1}, {2, 1, 2}, {2, 2, 1}, {2, 2, 2}}\)], "Output"] }, Open ]], Cell["\<\ Make a list of energies of each of these eight states with this \ double summation\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(EnergyDist = {};\)\), "\n", \(EnergyList = Do[AppendTo[EnergyDist, Sum[1 - \((1/\((States[\([i, j]\)]^2)\))\), {j, Length[States[\([i]\)]]}]], {i, Length[States]}]\), "\n", \(EnergyDist\)}], "Input"], Cell[BoxData[ RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \ error: new symbol name \\\"\\!\\(EnergyList\\)\\\" is similar to existing \ symbol \\\"\\!\\(EnergyDist\\)\\\". \ \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\ \\\", ButtonFrame->None, ButtonData:>\\\"General::spell1\\\"]\\)\"\>"}]], \ "Message"], Cell[BoxData[ \({0, 3\/4, 3\/4, 3\/2, 3\/4, 3\/2, 3\/2, 9\/4}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(UniqueEnergies = Union[EnergyDist]\)], "Input"], Cell[BoxData[ \({0, 3\/4, 3\/2, 9\/4}\)], "Output"] }, Open ]], Cell["\<\ Here is a function that will calculate the degeneracy of energy \ states in a distribution:\ \>", "Text"], Cell[BoxData[ \(EntropyVec[Elist_] := Module[{uniques = Union[Elist], counts = {}}, For[i = 1, i \[LessEqual] Length[uniques], \(i++\), AppendTo[ counts, {Count[Elist, uniques[\([i]\)]]}]]; \[IndentingNewLine]Return[ counts]]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Degeneracy = Flatten[EntropyVec[EnergyDist]]\)], "Input"], Cell[BoxData[ \({1, 3, 3, 1}\)], "Output"] }, Open 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Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["4 (Extra Credit).", "Subsubsection"], Cell[TextData[{ "Because dU = T dS - P dV, for fixed volume ", Cell[BoxData[ \(TraditionalForm\`\((dS/dU)\)\_V\)]], " = 1/T and the slope of the plots above of S/k vs. Energy should be \ positive. Lack of accounting for high-energy states underestimates the \ entropy and at high temperatures S/k necessarily goes through a maximum and \ then falls to zero. The maximum in S/k would correspond to infinite \ temperature, and beyond the maximum would correspond to negative \ temperatures. This is unphysical." }], "Text"], Cell["Group Exercise G3-1: Hamiltonian", "Subtitle"], Cell[TextData[StyleBox["solution by WCC", FontSize->14]], "Subtitle"], Cell["Background Material", "Subsubtitle"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Here is the free electron Hamiltonian", "Section"], Cell[BoxData[ \(Ham[ phi_]\ := \(\(\ \)\(-\[HBar]\^2\)\)\/\(2\ m\)\ D[ phi, {x, 2}]\)], "Input"], Cell[CellGroupData[{ Cell[TextData[{ "Normalize x with the characteristic distance L, and define the normalized \ Hamiltonian operator as\nHamNorm = ", Cell[BoxData[ \(TraditionalForm\`H\/\(\(\(\ \)\(-\[HBar]\^2\)\)\/\(2\ m\ \ L\^2\)\)\)]], " and z = x/L" }], "Subsection"], Cell[BoxData[ \(HamNorm[phi_]\ := \ \(-D[phi, {z, 2}]\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Define the approximating basis functions; the Hamiltonian matrix \ elements, and the normalizing elements\ \>", "Subsection"], Cell[BoxData[ \(phi[i_]\ := \ z^i\ \((1 - z)\)^i\)], "Input"], Cell[BoxData[ \(H[i_, \ j_]\ := Evaluate[\ Integrate[phi[i]\ HamNorm[phi[j]], {z, 0, 1}, Assumptions -> i\ \[Element] \ Integers\ && \ j\ \[Element] \ Integers\ && \ i\ > \ 1\ && \ j\ > \ 1]]\)], "Input"], Cell[BoxData[ \(F[i_, \ j_]\ := Evaluate[\ Integrate[phi[i] phi[j], {z, 0, 1}, Assumptions -> i\ \[Element] \ Integers\ && \ j\ \[Element] \ Integers\ && \ i\ > \ 1\ && \ j\ > \ 1]]\)], "Input"], Cell["Part 1", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[H[i, j], Assumptions -> i\ \[Element] \ Integers\ && \ j\ \[Element] \ Integers\ && \ i\ > \ 1\ && \ j\ > \ 1]\)], "Input"], Cell[BoxData[ \(\(-\(\((2\^\(\(-1\) - 2\ i - 2\ j\)\ j\ \((2\^\(1 + 2\ i + 2\ j\)\ \ \((\(-1\) + j)\)\ Gamma[\(-1\) + i + j]\ Gamma[2\ \((1 + i + j)\)]\ Gamma[ 3\/2 + i + j]\ Gamma[ 2 + i + j]\ Hypergeometric2F1[\(-1\) + i + j, 3, 1 + 2\ i + 2\ j, 1] + Gamma[1 + 2\ i + 2\ j]\ \((4\ \((4\^\(i + j\) - 2\^\(1 + 2\ i + 2\ j\)\ j)\)\ Gamma[i + j]\ Gamma[ 3\/2 + i + j]\ Gamma[2 + i + j]\ Hypergeometric2F1[ i + j, 3, 2\ \((1 + i + j)\), 1] + \((\(-1\) + 2\ j)\)\ \@\[Pi]\ Gamma[ 1 + i + j]\ Gamma[ 2\ \((1 + i + j)\)]\ Hypergeometric2F1[1 + i + j, 3, 3 + 2\ i + 2\ j, 1])\))\))\)/\((Gamma[ 2\ \((1 + i + j)\)]\ Gamma[3\/2 + i + j]\ Gamma[ 1 + 2\ i + 2\ j])\)\)\)\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ We see that it is not at all obvious that H[i,j] == H[j,i] but it is plausible given the explicit values below\ \>", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(H[2, 3]\)], "Input"], Cell[BoxData[ \(1\/210\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(H[3, 2]\)], "Input"], Cell[BoxData[ \(1\/210\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["To see if this is symmetric, try and integrate by parts:", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(dv\ = \ HamNorm[phi[j]]\)], "Input"], Cell[BoxData[ \(\(-\((\(-1\) + j)\)\)\ j\ \((1 - z)\)\^j\ z\^\(\(-2\) + j\) + 2\ j\^2\ \((1 - z)\)\^\(\(-1\) + j\)\ z\^\(\(-1\) + j\) - \((\(-1\) + j)\)\ j\ \((1 - z)\)\^\(\(-2\) + j\)\ z\^j\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(v\ = \ Integrate[dv, z, Assumptions -> \ j\ \[Element] \ Integers\ && \ j\ > \ 1]\)], "Input"], Cell[BoxData[ \(\(-j\)\ \((1 - 2\ z)\)\ \((\(-\((\(-1\) + z)\)\)\ z)\)\^\(\(-1\) + j\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(u\ = \ phi[i]\)], "Input"], Cell[BoxData[ \(\((1 - z)\)\^i\ z\^i\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(du\ = \ D[phi[i], z]\)], "Input"], Cell[BoxData[ \(i\ \((1 - z)\)\^i\ z\^\(\(-1\) + i\) - i\ \((1 - z)\)\^\(\(-1\) + i\)\ z\^i\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(HNormNew = \ Simplify[\((u\ v)\) /. z -> 1, Assumptions\ -> i\ \[Element] \ Integers\ && \ j\ \[Element] \ Integers\ && \ i\ > \ 1\ && \ j\ > \ 1] - \[IndentingNewLine]Simplify[\((u\ v)\) /. z -> 0, Assumptions\ -> i\ \[Element] \ Integers\ && \ j\ \[Element] \ Integers\ && \ i\ > \ 1\ && \ j\ > \ 1] - \[IndentingNewLine]Integrate[ v\ du, {z, 0, 1}, Assumptions\ -> i\ \[Element] \ Integers\ && \ j\ \[Element] \ Integers\ && \ i\ > \ 1\ && \ j\ > \ 1]\)\(\[IndentingNewLine]\) \)\)], "Input"], Cell[BoxData[ \(i\ j\ \((Beta[1, \(-1\) + i + j, 1 + i + j] - 2\ Beta[1, i + j, 1 + i + j] + Beta[1, 1 + i + j, \(-1\) + i + j] - 2\ Beta[1, 1 + i + j, i + j])\)\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Which is obviously symmetric with respect to interchange of i and \ j. Test for consistency with above result\ \>", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[{ \(HNormNew /. {i -> 3, j -> 2}\), "\[IndentingNewLine]", \(HNormNew /. {j -> 2, \ i -> \ 3}\)}], "Input"], Cell[BoxData[ \(1\/210\)], "Output"], Cell[BoxData[ \(1\/210\)], "Output"] }, Open ]], Cell["Part 2", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Write a function to calculate E(c1,c2,... cn)", "Subsection"], Cell[BoxData[ \(Eform[ approxlevel_Integer]\ := \ \[IndentingNewLine]Sum[c[index1]\ \ c[index2]\ HNormNew /. {i -> index1, j -> index2}, {index1, approxlevel}, \ {index2, approxlevel}]\/Sum[c[index1]\ c[index2]\ F[index1, index2], {index1, \ approxlevel}, {index2, approxlevel}]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Eform[2]\)], "Input"], Cell[BoxData[ \(\(c[1]\^2\/3 + 2\/15\ c[1]\ c[2] + \(2\ c[2]\^2\)\/105\)\/\(c[1]\^2\/30 \ + 1\/70\ c[1]\ c[2] + c[2]\^2\/630\)\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Find the minimum", "Subsection"], Cell[CellGroupData[{ Cell["The hard way:", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[{ \(eqs[M_]\ := \ Table[D[Eform[M], c[k]] == \ 0, {k, M}]\), "\[IndentingNewLine]", \(vars[M_]\ := \ Table[c[k], {k, M}]\)}], "Input"], Cell[BoxData[ RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \ error: new symbol name \\\"\\!\\(vars\\)\\\" is similar to existing symbol \\\ \"\\!\\(var\\)\\\". \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\ \\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"General::spell1\\\"]\\)\"\>"}]], "Message"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(eqs[2]\)], "Input"], Cell[BoxData[ \({\(\(2\ c[1]\)\/3 + \(2\ c[2]\)\/15\)\/\(c[1]\^2\/30 + 1\/70\ c[1]\ \ c[2] + c[2]\^2\/630\) - \(\((c[1]\/15 + c[2]\/70)\)\ \((c[1]\^2\/3 + 2\/15\ \ c[1]\ c[2] + \(2\ c[2]\^2\)\/105)\)\)\/\((c[1]\^2\/30 + 1\/70\ c[1]\ c[2] + \ c[2]\^2\/630)\)\^2 \[Equal] 0, \(\(2\ c[1]\)\/15 + \(4\ c[2]\)\/105\)\/\(c[1]\^2\/30 + 1\/70\ \ c[1]\ c[2] + c[2]\^2\/630\) - \(\((c[1]\/70 + c[2]\/315)\)\ \((c[1]\^2\/3 + 2\ \/15\ c[1]\ c[2] + \(2\ c[2]\^2\)\/105)\)\)\/\((c[1]\^2\/30 + 1\/70\ c[1]\ \ c[2] + c[2]\^2\/630)\)\^2 \[Equal] 0}\)], "Output"] }, Open ]], Cell[BoxData[{ \(Clear[sols]\), "\[IndentingNewLine]", \(sols[M_]\ := \ Solve[eqs[M], vars[M]]\)}], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(sols[2]\)], "Input"], Cell[BoxData[ RowBox[{\(Solve::"svars"\), \(\(:\)\(\ \)\), "\<\"Equations may not give \ solutions for all \\\"solve\\\" variables. \\!\\(\\*ButtonBox[\\\"More\ \[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"Solve::svars\\\"]\\)\"\>"}]], "Message"], Cell[BoxData[ \({{c[1] \[Rule] 1\/21\ \((7\ c[2] - \@133\ c[2])\)}, {c[1] \[Rule] 1\/21\ \((7\ c[2] + \@133\ c[2])\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[Eform[2] /. sols[2]]\)], "Input"], Cell[BoxData[ RowBox[{\(Solve::"svars"\), \(\(:\)\(\ \)\), "\<\"Equations may not give \ solutions for all \\\"solve\\\" variables. \\!\\(\\*ButtonBox[\\\"More\ \[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"Solve::svars\\\"]\\)\"\>"}]], "Message"], Cell[BoxData[ \({\(28\ \((\(-95\) + 8\ \@133)\)\)\/\(\(-266\) + 23\ \@133\), \(28\ \ \((95 + 8\ \@133)\)\)\/\(266 + 23\ \@133\)}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["The medium way:", "Subsubsection"], Cell[BoxData[ \(solsmed[M_]\ := \ Minimize[Eform[M], vars[M]]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(solsmed[2]\)], "Input"], Cell[BoxData[ \({4\ \((14 - \@133)\), {c[1] \[Rule] \(-1\), c[2] \[Rule] 1\/4\ \((7 - \@133)\)}}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Hmm, are these two results the same?", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(FullSimplify[\ \(28\ \((95 + 8\ \@133)\)\)\/\(266 + 23\ \@133\)]\)], \ "Input"], Cell[BoxData[ \(56 - 4\ \@133\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Yes... And now the easy way", "Subsubsection"], Cell[BoxData[ \(solseasy[M_]\ := \ NMinimize[Eform[M], vars[M]]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(solseasy[2]\)], "Input"], Cell[BoxData[ \({9.869749621316819`, {c[1] \[Rule] 0.23893764353056549`, c[2] \[Rule] 0.27074995130100915`}}\)], "Output"] }, Open ]], Cell[BoxData[ \(The\ exact\ ground\ \ state\ energy\ is\ E\ = \ \(\(-\((2\ \[Pi])\)\^2\ \)\ \[HBar]\^2\)\/\(8\ m\ L\^2\)\ which\ corresponds\ to\ \ \[Pi]\^\(\(2\)\(\ \ \)\)\ in\ our\ normalized\ energy\ units, \ therefore\ we\ should\ compare\ \[Pi]\^\(\(2\)\(\ \)\)\ with\ \(28\ \((95 \ + 8\ \@133)\)\)\/\(266 + 23\ \@133\)\)], "Subsubtitle"], Cell[CellGroupData[{ Cell[BoxData[{ \(N[\(28\ \((95 + 8\ \@133)\)\)\/\(266 + 23\ \@133\)]\), "\ \[IndentingNewLine]", \(N[\[Pi]\^2]\)}], "Input"], Cell[BoxData[ \(9.869749621316817`\)], "Output"], Cell[BoxData[ \(9.869604401089358`\)], "Output"] }, Open ]], Cell[TextData[{ "Part 3\n\nThe equation between E(", Cell[BoxData[ \(TraditionalForm\`\(c\& \[Rule] \)\)]], "), H(", Cell[BoxData[ \(TraditionalForm\`\(c\& \[Rule] \)\)]], "), and can be rewritten as\nE(", Cell[BoxData[ \(TraditionalForm\`\(c\& \[Rule] \)\)]], ") F(", Cell[BoxData[ \(TraditionalForm\`\(c\& \[Rule] \)\)]], ") = H(", Cell[BoxData[ \(TraditionalForm\`\(c\& \[Rule] \)\)]], ")\n\nin our approximating scheme, this becomes a relationship between two \ quadratic forms:\nE(", Cell[BoxData[ \(TraditionalForm\`\(c\& \[Rule] \)\)]], ") ", Cell[BoxData[ \(TraditionalForm\`\(c\& \[Rule] \)\)]], "\[Bullet]", Cell[BoxData[ \(TraditionalForm\`F\+\[HorizontalLine]\)]], Cell[BoxData[ \(TraditionalForm\`\(\(\[Bullet] c\&\[RightArrow]\)\&\ \ \ \ \ \ \ \)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\(c\& \[Rule] \)\)]], "\[Bullet]", Cell[BoxData[ \(TraditionalForm\`H\+\[HorizontalLine]\)]], Cell[BoxData[ \(TraditionalForm\`\(\(\[Bullet] c\&\[RightArrow]\)\&\ \ \ \ \ \ \ \)\)]], " \n\nTake the gradient with respect to ", Cell[BoxData[ \(TraditionalForm\`c\&\[RightArrow]\)]], " and use the extremum condition \[Del]E = ", Cell[BoxData[ \(TraditionalForm\`0\&\[RightArrow]\)]], "\nE(", Cell[BoxData[ \(TraditionalForm\`\(c\& \[Rule] \)\)]], ") ", Cell[BoxData[ \(TraditionalForm\`F\+\[HorizontalLine]\)]], Cell[BoxData[ \(TraditionalForm\`\(\(\[Bullet] c\&\[RightArrow]\)\&\ \ \ \ \ \ \ \)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`H\+\[HorizontalLine]\)]], Cell[BoxData[ \(TraditionalForm\`\(\(\[Bullet] c\&\[RightArrow]\)\&\ \ \ \ \ \ \ \)\)]], " \nor as a Homogeneous equation,\n ", Cell[BoxData[ \(TraditionalForm\`\((H\+\[HorizontalLine]\)\)]], " - E(", Cell[BoxData[ \(TraditionalForm\`\(c\& \[Rule] \)\)]], ") ", Cell[BoxData[ \(TraditionalForm\`F\+\[HorizontalLine]\)]], ") ", Cell[BoxData[ \(TraditionalForm\`\(\(\[Bullet] c\&\[RightArrow]\)\&\ \ \ \ \ \ \ \)\)]], " = 0\n which has solutions if and only if det( ", Cell[BoxData[ \(TraditionalForm\`\((H\+\[HorizontalLine]\)\)]], " - E(", Cell[BoxData[ \(TraditionalForm\`\(c\& \[Rule] \)\)]], ") ", Cell[BoxData[ \(TraditionalForm\`F\+\[HorizontalLine]\)]], ") = 0" }], "Text"], Cell["Part 4", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["The larger eigenvalue is the same as the the maximum above:", \ "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\ \)\(NMaximize[Eform[2], vars[2]]\)\)\)], "Input"], Cell[BoxData[ \({102.13025037868324`, {c[1] \[Rule] \(-0.08095172988205354`\), c[2] \[Rule] 0.37506075030047326`}}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["it doesn't correlate to the other energy levels at all.", \ "Subsubsection"], Cell["Part 5", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(To\ make\ comparisons, \ let' s\ divide\ the\ difference\ between\ the\ solutions\ by\ exact\ the\ \ ground\ state\ \((normalized)\)\ energy\ \[Pi]\^2\ and\ write\ a\ function\ \ that\ returns\ this\ \(\(ratio\)\(:\)\)\)], "Subsubsection"], Cell[BoxData[ \(solsNormeasy[ M_]\ := \(\(\ 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