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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 998711, 43720]*) (*NotebookOutlinePosition[ 999837, 43755]*) (* CellTagsIndexPosition[ 999793, 43751]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData["Linear Systems"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[ "Transforming a Sysytem into an N-th order Equation"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Solve the initial value problem\n", StyleBox[ " x1'[t] == 5 x1[t] + 3 x2[t],\n x2'[t] == -3 x1[t] - 2 x2[t],\n \ x1[0] == 0, x2[0] == 1.\n", FontFamily->"Courier", FontWeight->"Bold"], "by transforming it into a second order equation as in ", StyleBox["a)", FontSlant->"Italic"], " and plot the solution for ", StyleBox["t>=0", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Solution"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Let ", StyleBox["y[t] == x1[t]", FontFamily->"Courier", FontWeight->"Bold"], ", ", StyleBox["y'[t] == x1'[t]", FontFamily->"Courier", FontWeight->"Bold"], ". 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Open ]], Cell[TextData["In one step:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Clear[x1,x2,t]", "Input", AspectRatioFixed->True], Cell[CellGroupData[{ Cell["\<\ DSolve[{x1'[t] == 5 x1[t] + 3 x2[t], x2'[t] == -3 x1[t] - 2 x2[t], x1[0] == 0, x2[0] == 1}, {x1[t],x2[t]}, t]//Simplify \ \>", "Input", AspectRatioFixed->True], Cell[BoxData[ \({{x1[t] \[Rule] \(3\ E\^\(\(-\(1\/2\)\)\ \((\(-3\) + \@13)\)\ t\)\ \(( \(-1\) + E\^\(\@13\ t\))\)\)\/\@13, x2[t] \[Rule] 1\/26\ \((13 + 7\ \@13)\)\ E\^\(\(-\(1\/2\)\)\ \((\(-3\) + \@13)\)\ t\) + \((1\/2 - 7\/\(2\ \@13\))\)\ E\^\(1\/2\ \((3 + \@13)\)\ t\)}}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Solving Systems of Equations"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ "Solve the system of linear equations\n", StyleBox[ " 3x1 + 2x2 + x3 == 4,\n 2x1 + x2 + x4 == 4,\n \ x1 + x3 + 2x4 == 4,\n x2 + 2x3 + 3x4 == 4,\n", FontFamily->"Courier", FontWeight->"Bold"], "by using ", StyleBox["Solve[]", FontFamily->"Courier", FontWeight->"Bold"], ", ", StyleBox["LinearSolve[]", FontFamily->"Courier", FontWeight->"Bold"], ", and by using the inverse of the matrix of coefficients.", StyleBox["\n ", FontFamily->"Courier", FontWeight->"Bold"] }], "Text", CellMargins->{{18, 126}, {Inherited, Inherited}}, Evaluatable->False, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Solution"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["\<\ Solve[{3x1 + 2x2 + x3 == 4, 2x1 + x2 + x4 == 4, x1 + x3 + 2x4 == 4, x2 + 2x3 + 3x4 == 4}] \ \>", "Input", AspectRatioFixed->True], Cell[BoxData[ \({{x1 \[Rule] 4\/3, x2 \[Rule] 0, x3 \[Rule] 0, x4 \[Rule] 4\/3}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[TextData["?LinearSolve"], "Input", AspectRatioFixed->True], Cell[TextData[ "LinearSolve[m, b] gives the vector x which solves the matrix equation\n \ m.x==b."], "Print", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell["\<\ LinearSolve[{{3,2,1,0}, {2,1,0,1}, {1,0,1,2}, {0,1,2,3}},{4,3,2,1}]\ \>", "Input", AspectRatioFixed->True], Cell[BoxData[ \({4\/3, 0, 0, 1\/3}\)], "Output"] }, Open ]], Cell[TextData["The matrix of coefficients is"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ A = {{3,2,1,0}, {2,1,0,1}, {1,0,1,2}, {0,1,2,3}}; \ \>", "Input", AspectRatioFixed->True], Cell[TextData[{ "The inverse of ", StyleBox["A", FontFamily->"Courier", FontWeight->"Bold"], " is:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["\<\ B = Inverse[A]; B//MatrixForm\ \>", "Input", AspectRatioFixed->True], Cell[BoxData[ TagBox[ RowBox[{"(", GridBox[{ {\(1\/6\), "0", \(1\/2\), \(-\(1\/3\)\)}, {"0", \(1\/2\), \(-1\), \(1\/2\)}, {\(1\/2\), \(-1\), \(1\/2\), "0"}, {\(-\(1\/3\)\), \(1\/2\), "0", \(1\/6\)} }], ")"}], (MatrixForm[ #]&)]], "Output"] }, Open ]], Cell[TextData[{ "Applying this inverse to the vector on the RHS of the equation gives the \ same solution as ", StyleBox["LinearSolve[]", FontFamily->"Courier", FontWeight->"Bold"], ":" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["B.{4,3,2,1}", "Input", AspectRatioFixed->True], Cell[BoxData[ \({4\/3, 0, 0, 1\/3}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Finding Eigenvalues and Eigenvectors"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ "Find the eigenvalues and eigenvectors of the matrix\n", StyleBox[ " A = {{4, 3, -1},\n {2, 5, -1},\n {4, 6, 0}}\n", FontFamily->"Courier", FontWeight->"Bold"], "Compare your results to the output of ", StyleBox["Eigensystem[]", FontFamily->"Courier", FontWeight->"Bold"], ". Then find a matrix ", StyleBox["T", FontFamily->"Courier", FontWeight->"Bold"], " such that ", StyleBox["Inverse[T].A.T == d", FontFamily->"Courier", FontWeight->"Bold"], " where ", StyleBox["d", FontFamily->"Courier", FontWeight->"Bold"], " is a diagonal matrix." }], "Text", CellMargins->{{18, 126}, {Inherited, Inherited}}, Evaluatable->False, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell["Solution", "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell["\<\ A = {{4, 3, -1}, {2, 5, -1}, {4, 6, 0}};\ \>", "Input", AspectRatioFixed->True], Cell[TextData["First compute the eigenvalues by finding the roots of"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Det[A - r*IdentityMatrix[3]] == 0", "Input", AspectRatioFixed->True], Cell[BoxData[ \(20 - 24\ r + 9\ r\^2 - r\^3 == 0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Solve[Det[A - r*IdentityMatrix[3]] == 0]", "Input", AspectRatioFixed->True], Cell[BoxData[ \({{r \[Rule] 2}, {r \[Rule] 2}, {r \[Rule] 5}}\)], "Output"] }, Open ]], Cell[TextData[ "Next find the eigenvectors corresponding to each of the eigenvalues by \ solving the equations."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["\<\ eqns = (A-r*IdentityMatrix[3]).{v1,v2,v3} == {0,0,0} //Simplify \ \ \>", "Input", AspectRatioFixed->True], Cell[BoxData[ \({4\ v1 - r\ v1 + 3\ v2 - v3, 2\ v1 + 5\ v2 - r\ v2 - v3, 4\ v1 + 6\ v2 - r\ v3} == {0, 0, 0}\)], "Output"] }, Open ]], Cell[TextData[{ "For the eigenvalue ", StyleBox["2", FontFamily->"Courier", FontWeight->"Bold"], ":" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Solve[eqns /. {r->2}]", "Input", AspectRatioFixed->True], Cell[BoxData[ \({{v1 \[Rule] \(-\(\(3\ v2\)\/2\)\) + v3\/2}}\)], "Output"] }, Open ]], Cell[TextData[{ "The values of ", StyleBox["v2", FontFamily->"Courier", FontWeight->"Bold"], " and ", StyleBox["v3", FontFamily->"Courier", FontWeight->"Bold"], " can be chosen freely, say constants ", StyleBox["2a", FontFamily->"Courier", FontWeight->"Bold"], " and ", StyleBox["2b", FontFamily->"Courier", FontWeight->"Bold"], ", respectively. The solution vector is then\n", StyleBox[" {-3a + b, 2a, 2b} == a{-3,2,0} + b{1,0,2}\n", FontFamily->"Courier", FontWeight->"Bold"], "Two eigenvectors corresponding to the repeated eigenvalue ", StyleBox["2", FontFamily->"Courier", FontWeight->"Bold"], " are thus" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["X1 = {-3,2,0}; X2 = {1,0,2};", "Input", AspectRatioFixed->True], Cell[TextData[{ "For the eigenvalue ", StyleBox["5", FontFamily->"Courier", FontWeight->"Bold"], ":" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Solve[eqns /. {r->5}]", "Input", AspectRatioFixed->True], Cell[BoxData[ \({{v1 \[Rule] v3\/2, v2 \[Rule] v3\/2}}\)], "Output"] }, Open ]], Cell[TextData[{ "The value of ", StyleBox["v3", FontFamily->"Courier", FontWeight->"Bold"], " can be chosen freely, say ", StyleBox["2c", FontFamily->"Courier", FontWeight->"Bold"], ". The solution vector is then\n", StyleBox[" {c,c,2c} == c{1,1,2}\n", FontFamily->"Courier", FontWeight->"Bold"], "An eigenvector corresponding to the eigenvalue ", StyleBox["5", FontFamily->"Courier", FontWeight->"Bold"], " is thus" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["X3 = {1,1,2};", "Input", AspectRatioFixed->True], Cell[TextData[ "Let's check that the vectors we have found are in fact eigenvectors:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["A.X1 == 2 X1", "Input", AspectRatioFixed->True], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["A.X2 == 2 X2", "Input", AspectRatioFixed->True], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["A.X3 == 5 X3", "Input", AspectRatioFixed->True], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[TextData[ "Finding eigenvalues and eigenvectors can be done in one step:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["?Eigensystem", "Input", AspectRatioFixed->True], Cell[BoxData[ \("Eigensystem[m] gives a list {values, vectors} of the eigenvalues and \ eigenvectors of the square matrix m."\)], "Print"] }, Open ]], Cell[CellGroupData[{ Cell["Eigensystem[A]", "Input", AspectRatioFixed->True], Cell[BoxData[ \({{2, 2, 5}, {{1, 0, 2}, {\(-3\), 2, 0}, {1, 1, 2}}}\)], "Output"] }, Open ]], Cell[TextData[{ "By chance ", StyleBox["Eigensystem[]", FontFamily->"Courier", FontWeight->"Bold"], " gives the same two eigenvectors for the repeated eigenvalue ", StyleBox["2", FontFamily->"Courier", FontWeight->"Bold"], ". There are infintely many choices possible for these two vectors by \ varying the constants ", StyleBox["a", FontFamily->"Courier", FontWeight->"Bold"], " and ", StyleBox["b", FontFamily->"Courier", FontWeight->"Bold"], " above. For example:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["X11 = 2{-3,2,0} + {1,0,2}", "Input", AspectRatioFixed->True], Cell[BoxData[ \({\(-5\), 4, 2}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["X22 = {-3,2,0} + 2{1,0,2}", "Input", AspectRatioFixed->True], Cell[BoxData[ \({\(-1\), 2, 4}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["A.X11 == 2 X11", "Input", AspectRatioFixed->True], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["A.X22 == 2 X22", "Input", AspectRatioFixed->True], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[TextData[{ "To get a diagonal matrix ", StyleBox["d == Inverse[T].A.T", FontFamily->"Courier", FontWeight->"Bold"], ", just let ", StyleBox["T", FontFamily->"Courier", FontWeight->"Bold"], " be the matrix whose columns are the eigenvectors ", StyleBox["X1", FontFamily->"Courier", FontWeight->"Bold"], ", ", StyleBox["X2", FontFamily->"Courier", FontWeight->"Bold"], ", and ", StyleBox["X3", FontFamily->"Courier", FontWeight->"Bold"], ", see p. 362." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["\<\ T = Transpose[{X1,X2,X3}]; T//MatrixForm\ \>", "Input", AspectRatioFixed->True], Cell[BoxData[ TagBox[ RowBox[{"(", GridBox[{ {\(-3\), "1", "1"}, {"2", "0", "1"}, {"0", "2", "2"} }], ")"}], (MatrixForm[ #]&)]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ d = Inverse[T].A.T; d//MatrixForm\ \>", "Input", AspectRatioFixed->True], Cell[BoxData[ TagBox[ RowBox[{"(", GridBox[{ {"2", "0", "0"}, {"0", "2", "0"}, {"0", "0", "5"} }], ")"}], (MatrixForm[ #]&)]], "Output"] }, Open ]], Cell[TextData[ "The diagonal entries are the corresponding eigenvectors."], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Wronskians and Solutions of Systems of Differential Equations\ \>", "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ "Calculate the Wronskian of the vector functions\n", StyleBox[ " X1 = {t^2-t, t, t-1};\n X2 = { 0, t^3, t };\n X3 = { t, t^2, \ 0 };", FontFamily->"Courier", FontWeight->"Bold"], "\nFor what values of ", StyleBox["t", FontFamily->"Courier", FontWeight->"Bold"], " are the vectors ", StyleBox["x1", FontFamily->"Courier", FontWeight->"Bold"], ", ", StyleBox["x2", FontFamily->"Courier", FontWeight->"Bold"], ", ", StyleBox["x3", FontFamily->"Courier", FontWeight->"Bold"], " linearly independent? On what intervals are the vector functions ", StyleBox["x1", FontFamily->"Courier", FontWeight->"Bold"], ", ", StyleBox["x2", FontFamily->"Courier", FontWeight->"Bold"], ", ", StyleBox["x3", FontFamily->"Courier", FontWeight->"Bold"], " linearly independent? Find a homogeneous system of first order equations \ satisfied by ", StyleBox["x1", FontFamily->"Courier", FontWeight->"Bold"], ", ", StyleBox["x2", FontFamily->"Courier", FontWeight->"Bold"], ", ", StyleBox["x3", FontFamily->"Courier", FontWeight->"Bold"], " and check where the coefficients are continuous." }], "Text", CellMargins->{{18, 126}, {Inherited, Inherited}}, Evaluatable->False, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Solution"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[ "First we caclulate the Wronskian of the vector functions."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Clear[X1,X2,X3]", "Input", AspectRatioFixed->True], Cell["\<\ X1 = {t^2-t, t, t-1}; X2 = { 0, t^3, t }; X3 = { t, t^2, 0 };\ \>", "Input", AspectRatioFixed->True], Cell[CellGroupData[{ Cell["w = Det[{X1,X2,X3}]", "Input", AspectRatioFixed->True], Cell[BoxData[ \(t\^3 + 2\ t\^4 - 2\ t\^5\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Solve[w == 0]", "Input", AspectRatioFixed->True], Cell[BoxData[ \({{t \[Rule] 0}, {t \[Rule] 0}, {t \[Rule] 0}, { t \[Rule] 1\/2\ \((1 - \@3)\)}, {t \[Rule] 1\/2\ \((1 + \@3)\)}}\)], "Output"] }, Open ]], Cell[TextData[{ "As ", StyleBox["vectors", FontSlant->"Italic"], ", ", StyleBox["X1", FontFamily->"Courier", FontWeight->"Bold"], ", ", StyleBox["X2", FontFamily->"Courier", FontWeight->"Bold"], ", ", StyleBox["X3", FontFamily->"Courier", FontWeight->"Bold"], " are linearly independent as long as ", StyleBox["t", FontFamily->"Courier", FontWeight->"Bold"], " is not ", StyleBox["0", FontFamily->"Courier", FontWeight->"Bold"], ", or\n", StyleBox["(2 \[PlusMinus] 2*3^(1/2))/4", FontFamily->"Courier", FontWeight->"Bold"], ". As vector functions they are linearly independent on any interval. The \ only linear comination that equals ", StyleBox["{0,0,0}", "Input"], " for ", StyleBox["all", FontSlant->"Italic"], " ", StyleBox["t", "Input"], " is when the coeficients are all zero:\n" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["\<\ eqns = { c1 X1 + c2 X2 + c3 X3 == {0,0,0} /. t->0, c1 X1 + c2 X2 + c3 X3 == {0,0,0} /. t->1, c1 X1 + c2 X2 + c3 X3 == {0,0,0} /. t->2}\ \>", "Input"], Cell[BoxData[ \({{0, 0, \(-c1\)} == {0, 0, 0}, {c3, c1 + c2 + c3, c2} == {0, 0, 0}, { 2\ c1 + 2\ c3, 2\ c1 + 8\ c2 + 4\ c3, c1 + 2\ c2} == {0, 0, 0}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Solve[eqns]", "Input"], Cell[BoxData[ \({{c1 \[Rule] 0, c2 \[Rule] 0, c3 \[Rule] 0}}\)], "Output"] }, Open ]], Cell[TextData[{ "To find a homogeneous system that is satisfied by ", StyleBox["X1", FontFamily->"Courier", FontWeight->"Bold"], ", ", StyleBox["X2", FontFamily->"Courier", FontWeight->"Bold"], ", ", StyleBox["X3", FontFamily->"Courier", FontWeight->"Bold"], ", we need only solve the following equations for the coefficients:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ P = {{p11,p12,p13}, {p21,p22,p23}, {p31,p32,p33}};\ \>", "Input"], Cell[CellGroupData[{ Cell["\<\ eqns = { D[X1,t] == P.X1, D[X2,t] == P.X2, D[X3,t] == P.X3}\ \>", "Input", AspectRatioFixed->True], Cell[BoxData[ \({{\(-1\) + 2\ t, 1, 1} == { p13\ \((\(-1\) + t)\) + p12\ t + p11\ \((\(-t\) + t\^2)\), p23\ \((\(-1\) + t)\) + p22\ t + p21\ \((\(-t\) + t\^2)\), p33\ \((\(-1\) + t)\) + p32\ t + p31\ \((\(-t\) + t\^2)\)}, {0, 3\ t\^2, 1} == {p13\ t + p12\ t\^3, p23\ t + p22\ t\^3, p33\ t + p32\ t\^3}, {1, 2\ t, 0} == {p11\ t + p12\ t\^2, p21\ t + p22\ t\^2, p31\ t + p32\ t\^2}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Solve[eqns]//Simplify", "Input", AspectRatioFixed->True], Cell[BoxData[ \({{p13 \[Rule] t\^2\/\(\(-1\) - 2\ t + 2\ t\^2\), p23 \[Rule] 1\/2\ \((t + \(3\ t\)\/\(1 + 2\ t - 2\ t\^2\))\), p33 \[Rule] \(1 + t - 2\ t\^2\)\/\(t + 2\ t\^2 - 2\ t\^3\), p21 \[Rule] 1\/2\ \((\(-1\) + 3\/\(1 + 2\ t - 2\ t\^2\))\), p31 \[Rule] 1\/\(\(-t\) - 2\ t\^2 + 2\ t\^3\), p22 \[Rule] \(1 + 5\ t - 5\ t\^2\)\/\(t + 2\ t\^2 - 2\ t\^3\), p11 \[Rule] \(1 + 2\ t - 3\ t\^2\)\/\(t + 2\ t\^2 - 2\ t\^3\), p12 \[Rule] 1\/\(1 + 2\ t - 2\ t\^2\), p32 \[Rule] 1\/\(t\^2 + 2\ t\^3 - 2\ t\^4\)}}\)], "Output"] }, Open ]], Cell[TextData[{ "We see that the coefficients are not all continuous at ", StyleBox["t == 0", FontFamily->"Courier", FontWeight->"Bold"], " and", StyleBox[" (1 \[PlusMinus] 3^(1/2))/2", FontFamily->"Courier", FontWeight->"Bold"], ". Note that if they were continuous at these points, then Theorem 7.4.3 \ would imply that the Wronskian would not vanish for these values of ", StyleBox["t", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[ "Solving Homogeneous Systems with Constant Coefficients and Plotting \ Trajectories"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Saddle (eigenvalues of different sign)"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Find the general solution and plot several trajectories of ", StyleBox["X'[t] = A.X[t]", FontFamily->"Courier", FontWeight->"Bold"], " where" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ A = {{ 7 , 5}, {-10,-8}};\ \>", "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Solution"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Find the eigenvalues and eigenvectors of ", StyleBox["A", FontFamily->"Courier", FontWeight->"Bold"], ":" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Eigensystem[A]", "Input", AspectRatioFixed->True], Cell[BoxData[ \({{\(-3\), 2}, {{\(-1\), 2}, {\(-1\), 1}}}\)], "Output"] }, Open ]], Cell[TextData["This gives the two fundamental solutions"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ X1[t_] = {-1,1}E^(2t); X2[t_] = {-1,2}E^(-3t);\ \>", "Input", AspectRatioFixed->True], Cell[TextData["The general solution is thus"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["X[t_] = c1 X1[t] + c2 X2[t]", "Input", AspectRatioFixed->True], Cell[BoxData[ \({\(-c2\)\ E\^\(\(-3\)\ t\) - c1\ E\^\(2\ t\), 2\ c2\ E\^\(\(-3\)\ t\) + c1\ E\^\(2\ t\)}\)], "Output"] }, Open ]], Cell[TextData[{ "Let us plot the fundamental solutions, whose trajectories are lines, in \ different colors. 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Solutions that start on the blue line (multiples of ", StyleBox["X2[t]", FontFamily->"Courier", FontWeight->"Bold"], ") are the only exceptions and they approach the origin." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Note that the system can also be solved using ", StyleBox["DSolve[]", FontFamily->"Courier", FontWeight->"Bold"], ", although the constants come out arranged differently than above." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Clear[x1,x2,x3]", "Input"], Cell[CellGroupData[{ Cell["\<\ DSolve[{x1'[t] == 7x1[t] + 5x2[t], x2'[t] ==-10x1[t] - 8x2[t]}, {x1[t],x2[t]}, t]\ \>", "Input", AspectRatioFixed->True], Cell[BoxData[ \({{x1[t] \[Rule] \(-E\^\(\(-3\)\ t\)\)\ C[1] - E\^\(2\ t\)\ C[2], x2[t] \[Rule] 2\ E\^\(\(-3\)\ t\)\ C[1] + E\^\(2\ t\)\ C[2]}}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Stable Node (eigenvalues both negative)"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Find the general solution and plot several trajectories of ", StyleBox["X'[t] = A.X[t]", FontFamily->"Courier", FontWeight->"Bold"], " where" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ A = {{1,-2}, {3,-4}};\ \>", "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Solution"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["We follow the same steps as in the first example:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Eigensystem[A]", "Input", AspectRatioFixed->True], Cell[BoxData[ \({{\(-2\), \(-1\)}, {{2, 3}, {1, 1}}}\)], "Output"] }, Open ]], Cell[TextData["This gives the two fundamental solutions"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ X1[t_] = {1,1}E^(-t); X2[t_] = {2,3}E^(-2t);\ \>", "Input", AspectRatioFixed->True], Cell[TextData["The general solution is thus"], "Text", Evaluatable->False, AspectRatioFixed->True], 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.43771 .47267 L .43764 .4719 L .43758 .47108 L .43754 .4703 L .43752 .46988 L .43751 .4697 L .43751 .4695 L .4375 .46915 L .4375 .46882 L .4375 .46851 L .4375 .46823 L .43751 .46793 L .43752 .46761 L .43754 .46728 L .43755 .46698 L .43759 .46646 L .43765 .46594 L .43777 .46506 L .43793 .46419 L .43816 .46333 L .43843 .46251 L .43873 .46182 L .43938 .46073 L .43979 .46022 L .4402 .45981 L .44066 .45943 L .44119 .4591 L .44169 .45885 L .44217 .45867 L .44266 .45853 L .44314 .45843 L .44342 .45839 L .44357 .45838 L .44372 .45836 L .44385 .45835 L .444 .45834 L .44414 .45834 L .44426 .45834 L Mistroke .44441 .45833 L .44455 .45833 L .44463 .45834 L .44471 .45834 L .44479 .45834 L .44488 .45834 L .44504 .45835 L .44519 .45836 L .44554 .45839 L .44583 .45843 L .4461 .45847 L .44672 .45857 L .44785 .45884 L .44914 .45924 L .45051 .45975 L .45551 .46221 L .46071 .46548 L .46543 .46887 L .47005 .47247 L .47405 .47578 L .47749 .47874 L .47759 .47882 L Mfstroke .75014 1 m .65731 .84398 L .58741 .72325 L .53809 .63493 L .50763 .57756 L .48772 .53718 L .47675 .51218 L .47125 .49695 L .46977 .49156 L .46926 .48913 L .46896 .48723 L .46886 .4864 L .46882 .48597 L .46879 .48559 L .46877 .48527 L .46877 .48512 L .46876 .48495 L .46875 .48477 L .46875 .4846 L .46875 .48444 L .46875 .48429 L .46875 .48414 L .46876 .48398 L .46876 .48383 L .46877 .4837 L .46878 .48339 L .4688 .48325 L .46881 .48309 L .46884 .48281 L .46889 .48252 L .46898 .48203 L .4691 .48158 L .46924 .48121 L .4694 .48084 L .4696 .4805 L .46997 .48003 L .4702 .47982 L .47044 .47965 L .47067 .47951 L .47095 .47938 L .47121 .4793 L .47134 .47926 L .47147 .47924 L .47159 .47921 L .47173 .47919 L .47179 .47919 L .47186 .47918 L .47192 .47918 L .47197 .47917 L .47204 .47917 L Mistroke .47211 .47917 L .47217 .47917 L .47224 .47917 L .47231 .47917 L .47238 .47917 L .47245 .47917 L .47253 .47918 L .47267 .47919 L .47282 .4792 L .4731 .47924 L .4734 .47929 L .47373 .47937 L .4749 .47974 L .47618 .48028 L .47755 .48099 L .48006 .48254 L Mfstroke .16667 0 m .22831 .09246 L .30631 .20947 L .36015 .29023 L .40108 .35162 L .42914 .39371 L .44859 .42288 L .46346 .44519 L .47369 .46054 L .48145 .47217 L .48675 .48013 L .49042 .48562 L .49321 .48981 L .49504 .49256 L s .16667 0 m .24021 .11031 L .31333 .22 L .36415 .29623 L .40316 .35473 L .43008 .39511 L .45054 .42581 L .46457 .44685 L .47429 .46144 L .48173 .47259 L .48685 .48027 L .49072 .48609 L .49338 .49006 L .49521 .49281 L .4966 .49491 L .49752 .49628 L s .5 .5 m .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L s .83333 1 m .75979 .88969 L .68667 .78 L .63585 .70377 L .59684 .64527 L .56992 .60489 L .54946 .57419 L .53543 .55315 L .52571 .53856 L .51827 .52741 L .51315 .51973 L .50928 .51391 L .50662 .50994 L .50479 .50719 L .5034 .50509 L .50248 .50372 L s .83333 1 m .77169 .90754 L .69369 .79053 L .63985 .70977 L .59892 .64838 L .57086 .60629 L .55141 .57712 L .53654 .55481 L .52631 .53946 L .51855 .52783 L .51325 .51987 L .50958 .51438 L .50679 .51019 L .50496 .50744 L s .24986 0 m .34269 .15602 L .41259 .27675 L .46191 .36507 L .49237 .42244 L .51228 .46282 L .52325 .48782 L .52875 .50305 L .53023 .50844 L .53074 .51087 L .53104 .51277 L .53114 .5136 L .53118 .51403 L .53121 .51441 L .53123 .51473 L .53123 .51488 L .53124 .51505 L .53125 .51523 L .53125 .5154 L .53125 .51556 L .53125 .51571 L .53125 .51586 L .53124 .51602 L .53124 .51617 L .53123 .5163 L .53122 .51661 L .5312 .51675 L .53119 .51691 L .53116 .51719 L .53111 .51748 L .53102 .51797 L .5309 .51842 L .53076 .51879 L .5306 .51916 L .5304 .5195 L .53003 .51997 L .5298 .52018 L .52956 .52035 L .52933 .52049 L .52905 .52062 L .52879 .5207 L .52866 .52074 L .52853 .52076 L .52841 .52079 L .52827 .52081 L .52821 .52081 L .52814 .52082 L .52808 .52082 L .52803 .52083 L .52796 .52083 L Mistroke .52789 .52083 L .52783 .52083 L .52776 .52083 L .52769 .52083 L .52762 .52083 L .52755 .52083 L .52747 .52082 L .52733 .52081 L .52718 .5208 L .5269 .52076 L .5266 .52071 L .52627 .52063 L .5251 .52026 L .52382 .51972 L .52245 .51901 L .51994 .51746 L Mfstroke .29403 0 m .3499 .09707 L .43638 .25328 L .49506 .36516 L .52936 .43602 L .54844 .48052 L .55486 .49794 L .55875 .51033 L .5601 .51539 L .56118 .52021 L .56189 .52415 L .56213 .52586 L .56229 .52733 L .56236 .5281 L .56242 .52892 L .56246 .5297 L .56248 .53012 L .56249 .5303 L .56249 .5305 L .5625 .53085 L .5625 .53118 L .5625 .53149 L .5625 .53177 L .56249 .53207 L .56248 .53239 L .56246 .53272 L .56245 .53302 L .56241 .53354 L .56235 .53406 L .56223 .53494 L .56207 .53581 L .56184 .53667 L .56157 .53749 L .56127 .53818 L .56062 .53927 L .56021 .53978 L .5598 .54019 L .55934 .54057 L .55881 .5409 L .55831 .54115 L .55783 .54133 L .55734 .54147 L .55686 .54157 L .55658 .54161 L .55643 .54162 L .55628 .54164 L .55615 .54165 L .556 .54166 L .55586 .54166 L .55574 .54166 L Mistroke .55559 .54167 L .55545 .54167 L .55537 .54166 L .55529 .54166 L .55521 .54166 L .55512 .54166 L .55496 .54165 L .55481 .54164 L .55446 .54161 L .55417 .54157 L .5539 .54153 L .55328 .54143 L .55215 .54116 L .55086 .54076 L .54949 .54025 L .54449 .53779 L .53929 .53452 L .53457 .53113 L .52995 .52753 L .52595 .52422 L .52251 .52126 L .52241 .52118 L Mfstroke 1 1 m .92046 .92046 L .85554 .85554 L .80257 .80257 L .75485 .75485 L .71603 .71603 L .68429 .68429 L .6556 .6556 L .63222 .63222 L .6112 .6112 L .59412 .59412 L .58017 .58017 L .56759 .56759 L .55734 .55734 L .54815 .54815 L .5407 .5407 L .53461 .53461 L .52914 .52914 L .52489 .52489 L s .90569 1 m .90269 .99603 L .82013 .88806 L .75244 .80087 L .70214 .7371 L .66066 .68538 L .62955 .64726 L .60587 .61873 L .58586 .59499 L .5705 .57707 L .55743 .56207 L .54732 .55063 L .5394 .5418 L .53253 .53423 L .52737 .52861 L s .88874 1 m .84929 .94613 L .77207 .84199 L .71011 .75957 L .66498 .70041 L .63158 .65729 L .60413 .6224 L .58365 .5968 L .56671 .57598 L .55395 .56057 L .5442 .54899 L .53593 .53932 L .52985 .53233 L s .36912 0 m .37275 .00656 L .49011 .23032 L .55871 .37205 L .59688 .46103 L .60972 .49588 L .61751 .52067 L .62019 .53078 L .62237 .54042 L .62378 .5483 L .62425 .55172 L .62458 .55466 L .62472 .55621 L .62484 .55784 L .62493 .55941 L .62496 .56024 L .62497 .5606 L .62498 .561 L .62499 .5617 L .625 .56235 L .625 .56297 L .62499 .56354 L .62498 .56413 L .62496 .56478 L .62493 .56543 L .62489 .56604 L .62481 .56707 L .62471 .56812 L .62446 .56988 L .62413 .57162 L .62369 .57335 L .62313 .57498 L .62253 .57636 L .62124 .57854 L .62042 .57956 L .61961 .58037 L .61867 .58113 L .61761 .58181 L .61661 .5823 L .61566 .58266 L .61468 .58294 L .61371 .58313 L .61316 .58321 L .61286 .58325 L .61257 .58327 L .61229 .58329 L .612 .58331 L .61173 .58332 L .61148 .58333 L .61118 .58333 L Mistroke .61089 .58333 L .61073 .58333 L .61058 .58333 L .61041 .58332 L .61025 .58331 L .60992 .5833 L .60962 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.5061 L .50442 .50617 L .50442 .50625 L .50442 .50632 L .50442 .50638 L .50441 .50645 L .50441 .50653 L .50439 .50662 L .50438 .50671 L .50434 .5069 L .50428 .50707 L .50421 .50726 L .50404 .50758 L .50378 .50795 L .50345 .50829 L .503 .50866 L .50251 .50897 L .50186 .5093 L .50099 .50966 L .49984 .51001 L .49851 .51033 L .49706 .51059 L .49532 .51082 L .49433 .51092 L .49314 .51101 L .49204 .51108 L .49139 .51112 L .49076 .51114 L .49014 .51116 L .48945 .51118 L Mistroke .4888 .5112 L .48817 .51121 L .48734 .51121 L .48686 .51122 L .4864 .51122 L .48554 .51121 L .48459 .5112 L .4837 .51118 L .48285 .51116 L .48079 .51109 L .47838 .51099 L .47601 .51086 L .46998 .51044 L .45637 .50912 L .43977 .50707 L .38176 .49791 L .12319 .44507 L Mfstroke .12319 .44507 m 0 .41594 L s .50026 .50022 m .50043 .50036 L .50072 .50061 L .50118 .50101 L .50188 .50162 L .50308 .50269 L .50488 .50435 L .50777 .5072 L .51164 .51156 L .51366 .51428 L .5148 .51606 L .51581 .51789 L .51667 .51986 L .51707 .52103 L .51737 .52215 L .51749 .52278 L .51758 .52337 L .51762 .52372 L .51765 .52403 L .51766 .52421 L .51768 .5244 L .51768 .52458 L .51769 .52475 L .51769 .52506 L .51768 .52535 L .51766 .52567 L .51765 .52585 L .51763 .52602 L .51759 .52635 L .51754 .52672 L .5174 .52739 L .51723 .52802 L .51699 .5287 L .5164 .52994 L .5154 .53145 L .51415 .53284 L .51255 .53422 L .5103 .53573 L .5073 .53728 L .50381 .53867 L .49966 .53997 L .49417 .54129 L .48789 .54241 L .48101 .5433 L .47376 .54397 L .46916 .54428 L .4645 .54451 L .46184 .54462 L .45921 .5447 L .45676 .54476 L Mistroke .454 .54481 L .45089 .54485 L .4491 .54486 L .44744 .54486 L .44412 .54486 L .44087 .54483 L .43746 .54478 L .43368 .5447 L .42647 .5445 L .41722 .54413 L .40812 .54367 L .38585 .54221 L .36127 .54014 L .308 .53448 L .23242 .5246 L Mfstroke .23242 .5246 m 0 .48207 L s .50039 .50033 m .50064 .50054 L .50108 .50092 L .50178 .50151 L .50285 .50244 L .50469 .50405 L .50751 .50657 L .51221 .51095 L .51899 .51778 L .52773 .52797 L .53267 .53509 L .53507 .53929 L .53629 .54178 L .53735 .54426 L .5382 .54664 L .53859 .54794 L .53896 .54935 L .53925 .55067 L .53947 .5519 L .53955 .5525 L .53963 .55315 L .53969 .55373 L .53974 .55435 L .53977 .55499 L .53979 .55559 L .5398 .5563 L .53978 .55696 L .53975 .55765 L .53969 .5584 L .53961 .55912 L .53951 .55979 L .53929 .56099 L .53897 .56232 L .53853 .56373 L .53802 .56506 L .53656 .56796 L .53432 .57118 L .53112 .57458 L .52723 .57775 L .51751 .58337 L .51021 .58643 L .50218 .5891 L .49196 .59179 L .47895 .59441 L .46581 .5964 L .44888 .59827 L .4381 .59914 L .42736 .5998 L .42105 .60011 L .41384 .6004 L Mistroke .40671 .60062 L .3998 .60077 L .39253 .60088 L .38541 .60093 L .37877 .60094 L .37133 .60091 L .36286 .60081 L .35847 .60074 L .35354 .60065 L .33584 .60016 L .31642 .59942 L .29357 .59828 L .24689 .59524 L .13851 .58545 L Mfstroke .13851 .58545 m 0 .56883 L s .4996 .49967 m .49936 .49946 L .4989 .49908 L .49819 .49848 L .49706 .49754 L .49506 .49588 L .49183 .49324 L .48593 .48852 L .47592 .48079 L .45888 .46825 L .42515 .44517 L .36129 .40552 L .23678 .33659 L s .23678 .33659 m 0 .22158 L s .49974 .49978 m .49957 .49964 L .49927 .49938 L .49879 .49898 L .49803 .49835 L .49668 .49725 L .49447 .49547 L .49037 .49228 L .48327 .487 L .4708 .47833 L .4451 .46203 L .39405 .43321 L .28954 .3812 L .03507 .27015 L s .03507 .27015 m 0 .25656 L s .49987 .49989 m .49978 .49982 L .49963 .49969 L .49939 .49949 L .499 .49917 L .49829 .49861 L .4971 .4977 L .49481 .49603 L .49062 .49321 L .48272 .48841 L .46505 .4789 L .4268 .46091 L .34231 .42582 L .12273 .34426 L s .12273 .34426 m 0 .30253 L s .5 .5 m .5 .5 L .5 .5 L .49999 .5 L .49996 .49999 L .4999 .49997 L .49974 .49993 L .49926 .49979 L .49797 .49943 L .49464 .49849 L .485 .49577 L .45956 .4886 L .39507 .47043 L .21038 .41838 L s .21038 .41838 m 0 .35908 L s .50013 .50011 m .50021 .50018 L .50036 .50031 L .50059 .5005 L .50093 .50081 L .50151 .50134 L .50237 .50215 L .5037 .50355 L .50452 .50453 L .50531 .50564 L .50604 .50694 L .50634 .50768 L .50647 .50813 L .50652 .50836 L .50656 .50857 L .5066 .50879 L .50661 .50891 L .50662 .50902 L .50662 .50908 L .50663 .50914 L .50663 .50921 L .50663 .50928 L .50663 .5094 L .50663 .50946 L .50663 .50953 L .50662 .50965 L .50661 .50977 L .5066 .50987 L .50658 .50999 L .50653 .51025 L .50646 .51049 L .5063 .51093 L .50604 .51141 L .50567 .51193 L .5052 .51242 L .5046 .51292 L .50376 .51346 L .50266 .51401 L .50139 .51452 L .49998 .51496 L .4984 .51536 L .49624 .51578 L .49391 .51612 L .49114 .51641 L .48937 .51654 L .48756 .51665 L .48652 .5167 L .48549 .51674 L .48453 .51677 L .48345 .51679 L Mistroke .48281 .5168 L .48222 .51681 L .48153 .51682 L .48086 .51682 L .47962 .51682 L .47891 .51682 L .47824 .51682 L .47709 .5168 L .4758 .51678 L .47455 .51675 L .47338 .51672 L .47034 .51661 L .46725 .51647 L .46044 .51606 L .45191 .5154 L .43173 .51336 L .40208 .50957 L .30762 .49422 L Mfstroke .30762 .49422 m 0 .43028 L s .50026 .50022 m .50043 .50036 L .50072 .50061 L .50118 .50101 L .5019 .50163 L .50313 .5027 L .50501 .50438 L .50814 .5073 L .51266 .51185 L .51849 .51865 L .52178 .5234 L .52338 .5262 L .5242 .52785 L .5249 .52951 L .52547 .53109 L .52573 .53196 L .52597 .5329 L .52617 .53378 L .52631 .5346 L .52637 .535 L .52642 .53544 L .52646 .53582 L .52649 .53623 L .52652 .53666 L .52653 .53706 L .52653 .53753 L .52652 .53797 L .5265 .53843 L .52646 .53894 L .5264 .53942 L .52634 .53986 L .52619 .54066 L .52598 .54155 L .52569 .54248 L .52535 .54337 L .52437 .54531 L .52288 .54745 L .52075 .54972 L .51815 .55183 L .51167 .55558 L .5068 .55762 L .50146 .5594 L .49464 .56119 L .48597 .56294 L .47721 .56427 L .46592 .56551 L .45873 .56609 L .45157 .56653 L .44737 .56674 L .44256 .56693 L Mistroke .43781 .56708 L .4332 .56718 L .42835 .56725 L .4236 .56729 L .41918 .5673 L .41422 .56727 L .40857 .56721 L .40564 .56716 L .40236 .5671 L .39056 .56678 L .37762 .56628 L .36238 .56552 L .33126 .5635 L .25901 .55697 L .14568 .54337 L .00462 .52295 L Mfstroke .00462 .52295 m 0 .52207 L s .50039 .50033 m .50064 .50054 L .50109 .50092 L .50178 .50152 L .50287 .50244 L .50474 .50406 L .50764 .50661 L .51258 .51106 L .52001 .51806 L .53041 .52873 L .53695 .53617 L .54485 .54638 L .54851 .55184 L .55228 .55831 L .55545 .56498 L .55678 .56843 L .55783 .57169 L .55877 .57547 L .55916 .57761 L .55944 .57966 L .55955 .58077 L .5596 .58137 L .55963 .58194 L .55968 .58296 L .55969 .5835 L .55969 .58408 L .55969 .58471 L .55967 .58539 L .55965 .58604 L .55961 .58664 L .55951 .58779 L .55937 .58902 L .55901 .59112 L .55847 .5934 L .5578 .59558 L .55702 .59761 L .55459 .60237 L .55091 .60746 L .5463 .61222 L .54094 .61655 L .53313 .62153 L .52389 .62615 L .50203 .63401 L .48601 .63812 L .46843 .64161 L .44731 .64478 L .43412 .64633 L .4205 .64765 L .40682 .64873 L .3907 .64972 L Mistroke .38318 .65009 L .37488 .65044 L .36735 .65071 L .35914 .65095 L .35387 .65107 L .34797 .65119 L .33726 .65134 L .33102 .65139 L .32419 .65141 L .31758 .65141 L .31143 .65139 L .29912 .65128 L .29195 .65118 L .28518 .65107 L .27237 .65078 L .25824 .65039 L .2308 .64939 L .20318 .6481 L .13274 .64377 L .05798 .63782 L Mfstroke .05798 .63782 m 0 .63226 L s .49961 .49967 m .49936 .49946 L .49891 .49908 L .4982 .49848 L .49708 .49754 L .49511 .4959 L .49196 .49328 L .4863 .48863 L .47694 .48108 L .46156 .46901 L .43265 .44728 L .38151 .41121 L .28925 .35138 L .09222 .23684 L s .09222 .23684 m 0 .18962 L s .49974 .49978 m .49957 .49964 L .49927 .49938 L .49879 .49898 L .49805 .49836 L .49673 .49726 L .4946 .49551 L .49074 .49238 L .48429 .48729 L .47348 .47909 L .4526 .46415 L .41427 .43891 L .34201 .39599 L .17988 .31096 L s .17988 .31096 m 0 .2283 L s .49987 .49989 m .49979 .49982 L .49963 .49969 L .49939 .49949 L .49901 .49918 L .49834 .49862 L .49723 .49773 L .49519 .49614 L .49164 .4935 L .4854 .48917 L .47255 .48102 L .44702 .46661 L .39477 .4406 L .26754 .38507 L s .26754 .38507 m 0 .28146 L s .5 .5 m .5 .5 L .5 .5 L .49999 .5 L .49998 .5 L .49995 .49999 L .49987 .49996 L .49963 .4999 L .49898 .49971 L .49732 .49925 L .4925 .49789 L .47978 .4943 L .44753 .48521 L .35519 .45919 L .1153 .39158 L s .1153 .39158 m 0 .35908 L s .50013 .50011 m .50021 .50018 L .50036 .50031 L .50059 .5005 L .50095 .50081 L .50156 .50135 L .5025 .50219 L .50407 .50365 L .50633 .50593 L .50924 .50932 L .51089 .5117 L .51169 .5131 L .5121 .51393 L .51245 .51475 L .51273 .51555 L .51286 .51598 L .51299 .51645 L .51308 .51689 L .51316 .5173 L .51318 .5175 L .51321 .51772 L .51323 .51791 L .51325 .51812 L .51326 .51833 L .51326 .51853 L .51327 .51877 L .51326 .51899 L .51325 .51922 L .51323 .51947 L .5132 .51971 L .51317 .51993 L .5131 .52033 L .51299 .52077 L .51284 .52124 L .51267 .52169 L .51219 .52265 L .51144 .52373 L .51037 .52486 L .50908 .52592 L .50584 .52779 L .5034 .52881 L .50073 .5297 L .49732 .5306 L .49298 .53147 L .4886 .53213 L .48296 .53276 L .47937 .53305 L .47579 .53327 L .47368 .53337 L .47128 .53347 L Mistroke .4689 .53354 L .4666 .53359 L .46418 .53363 L .4618 .53364 L .45959 .53365 L .45711 .53364 L .45429 .5336 L .45282 .53358 L .45118 .53355 L .44528 .53339 L .43881 .53314 L .43119 .53276 L .41563 .53175 L .3795 .52848 L .32284 .52169 L .25231 .51148 L Mfstroke .25231 .51148 m 0 .46339 L s .50026 .50022 m .50043 .50036 L .50073 .50061 L .50119 .50101 L .50192 .50163 L .50318 .50271 L .50514 .50442 L .50851 .50741 L .51368 .51214 L .52116 .5194 L .5324 .53162 L .53903 .54005 L .54529 .54969 L .54811 .5551 L .55042 .56067 L .55146 .56392 L .55192 .56569 L .55234 .56761 L .55266 .56942 L .55287 .5711 L .55294 .57191 L .55301 .5728 L .55304 .57365 L .55306 .57444 L .55306 .57539 L .55303 .5763 L .55298 .57712 L .5529 .57802 L .55279 .579 L .55263 .58006 L .55224 .58197 L .55177 .58375 L .55121 .58544 L .54946 .58934 L .54672 .59366 L .54331 .59767 L .53838 .6021 L .53164 .60675 L .52387 .61092 L .51523 .61462 L .50309 .61874 L .48918 .62241 L .47382 .62554 L .45727 .62815 L .4361 .63062 L .42409 .63168 L .40976 .63269 L .39466 .63348 L .38556 .63385 L .37675 .63412 L Mistroke .36792 .63433 L .35961 .63447 L .34963 .63457 L .33986 .63459 L .33059 .63456 L .32043 .63446 L .31146 .63432 L .30129 .63412 L .27727 .63342 L .25316 .63246 L .19167 .62903 L .12652 .62416 L Mfstroke .12652 .62416 m 0 .6114 L s .50039 .50033 m .50064 .50054 L .50109 .50092 L .50179 .50152 L .50289 .50245 L .50479 .50408 L .50778 .50665 L .51296 .51116 L .52102 .51835 L .53309 .52948 L .55235 .54849 L .57805 .57739 L .59278 .5972 L .60582 .61905 L .61201 .63271 L .61442 .6394 L .61658 .6468 L .61755 .65104 L .61828 .65496 L .61857 .65687 L .61884 .65897 L .61905 .66095 L .6192 .66281 L .61932 .665 L .61936 .66617 L .61938 .66743 L .61939 .66876 L .61937 .67002 L .61933 .67116 L .61927 .67241 L .61911 .67468 L .619 .67583 L .61885 .67711 L .61853 .67939 L .61816 .68153 L .61723 .68572 L .61587 .69036 L .61416 .69493 L .61189 .69986 L .60643 .70893 L .59977 .71727 L .5896 .72708 L .57515 .7377 L .55764 .74763 L .53759 .75659 L .51422 .76492 L .48393 .77343 L .44968 .78088 L .41288 .78705 L .36218 .7933 L Mistroke .3294 .79627 L .29609 .79861 L .27847 .7996 L .26118 .80042 L .24486 .80107 L .22698 .80166 L .20663 .80217 L .18718 .80251 L .1654 .80275 L .15265 .80282 L .14049 .80283 L .11711 .80275 L .10496 .80264 L .09127 .80247 L .06635 .80205 L .04209 .8015 L Mfstroke .04209 .8015 m 0 .80017 L s .49961 .49967 m .49936 .49946 L .49891 .49908 L .4982 .49848 L .4971 .49755 L .49516 .49591 L .49209 .49332 L .48667 .48873 L .47796 .48136 L .46424 .46976 L .44015 .44939 L .40173 .41691 L .34171 .36617 L .23703 .27766 L .07138 .1376 L s .07138 .1376 m 0 .07725 L s .49974 .49978 m .49957 .49964 L .49927 .49938 L .4988 .49899 L .49806 .49836 L .49678 .49727 L .49473 .49554 L .49112 .49249 L .48531 .48758 L .47616 .47984 L .4601 .46626 L .43449 .44461 L .39447 .41078 L .32469 .35177 L .21425 .2584 L .01964 .09385 L s .01964 .09385 m 0 .07725 L s .49987 .49989 m .49979 .49982 L .49964 .49969 L .4994 .49949 L .49903 .49918 L .49839 .49864 L .49736 .49777 L .49556 .49624 L .49265 .49379 L .48808 .48992 L .48005 .48313 L .46724 .4723 L .44724 .45539 L .41234 .42589 L .35713 .3792 L .25982 .29693 L .10388 .16508 L s .10388 .16508 m 0 .07725 L s .5 .5 m .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L s .50013 .50011 m .50021 .50018 L .50036 .50031 L .5006 .50051 L .50097 .50082 L .50161 .50136 L .50264 .50223 L .50444 .50376 L .50735 .50621 L .51192 .51008 L .51995 .51687 L .53276 .5277 L .55276 .54461 L .58766 .57411 L .64287 .6208 L .74018 .70307 L .89612 .83492 L s .89612 .83492 m 1 .92275 L s .50026 .50022 m .50043 .50036 L .50073 .50062 L .5012 .50101 L .50194 .50164 L .50322 .50273 L .50527 .50446 L .50888 .50751 L .51469 .51242 L .52384 .52016 L .5399 .53374 L .56551 .55539 L .60553 .58922 L .67531 .64823 L .78575 .7416 L .98036 .90615 L s .98036 .90615 m 1 .92275 L s .50039 .50033 m .50064 .50054 L .50109 .50092 L .5018 .50152 L .5029 .50245 L .50484 .50409 L .50791 .50668 L .51333 .51127 L .52204 .51864 L .53576 .53024 L .55985 .55061 L .59827 .58309 L .65829 .63383 L .76297 .72234 L .92862 .8624 L s .92862 .8624 m 1 .92275 L s .49961 .49967 m .49936 .49946 L .49891 .49908 L .49821 .49848 L .49711 .49755 L .49521 .49592 L .49222 .49335 L .48704 .48884 L .47898 .48165 L .46691 .47052 L .44765 .45151 L .42195 .42261 L .40722 .4028 L .39418 .38095 L .38799 .36729 L .38558 .3606 L .38342 .3532 L .38245 .34896 L .38172 .34504 L .38143 .34313 L .38116 .34103 L .38095 .33905 L .3808 .33719 L .38068 .335 L .38064 .33383 L .38062 .33257 L .38061 .33124 L .38063 .32998 L .38067 .32884 L .38073 .32759 L .38089 .32532 L .381 .32417 L .38115 .32289 L .38147 .32061 L .38184 .31847 L .38277 .31428 L .38413 .30964 L .38584 .30507 L .38811 .30014 L .39357 .29107 L .40023 .28273 L .4104 .27292 L .42485 .2623 L .44236 .25237 L .46241 .24341 L .48578 .23508 L .51607 .22657 L .55032 .21912 L .58712 .21295 L .63782 .2067 L Mistroke .6706 .20373 L .70391 .20139 L .72153 .2004 L .73882 .19958 L .75514 .19893 L .77302 .19834 L .79337 .19783 L .81282 .19749 L .8346 .19725 L .84735 .19718 L .85951 .19717 L .88289 .19725 L .89504 .19736 L .90873 .19753 L .93365 .19795 L .95791 .1985 L Mfstroke .95791 .1985 m 1 .19983 L s .49974 .49978 m .49957 .49964 L .49927 .49939 L .49881 .49899 L .49808 .49837 L .49682 .49729 L .49486 .49558 L .49149 .49259 L .48632 .48786 L .47884 .4806 L .4676 .46838 L .46097 .45995 L .45471 .45031 L .45189 .4449 L .44958 .43933 L .44854 .43608 L .44808 .43431 L .44766 .43239 L .44734 .43058 L .44713 .4289 L .44706 .42809 L .44699 .4272 L .44696 .42635 L .44694 .42556 L .44694 .42461 L .44697 .4237 L .44702 .42288 L .4471 .42198 L .44721 .421 L .44737 .41994 L .44776 .41803 L .44823 .41625 L .44879 .41456 L .45054 .41066 L .45328 .40634 L .45669 .40233 L .46162 .3979 L .46836 .39325 L .47613 .38908 L .48477 .38538 L .49691 .38126 L .51082 .37759 L .52618 .37446 L .54273 .37185 L .5639 .36938 L .57591 .36832 L .59024 .36731 L .60534 .36652 L .61444 .36615 L .62325 .36588 L Mistroke .63208 .36567 L .64039 .36553 L .65037 .36543 L .66014 .36541 L .66941 .36544 L .67957 .36554 L .68854 .36568 L .69871 .36588 L .72273 .36658 L .74684 .36754 L .80833 .37097 L .87348 .37584 L Mfstroke .87348 .37584 m 1 .3886 L s .49987 .49989 m .49979 .49982 L .49964 .49969 L .49941 .4995 L .49905 .49919 L .49844 .49865 L .4975 .49781 L .49593 .49635 L .49367 .49407 L .49076 .49068 L .48911 .4883 L .48831 .4869 L .4879 .48607 L .48755 .48525 L .48727 .48445 L .48714 .48402 L .48701 .48355 L .48692 .48311 L .48684 .4827 L .48682 .4825 L .48679 .48228 L .48677 .48209 L .48675 .48188 L .48674 .48167 L .48674 .48147 L .48673 .48123 L .48674 .48101 L .48675 .48078 L .48677 .48053 L .4868 .48029 L .48683 .48007 L .4869 .47967 L .48701 .47923 L .48716 .47876 L .48733 .47831 L .48781 .47735 L .48856 .47627 L .48963 .47514 L .49092 .47408 L .49416 .47221 L .4966 .47119 L .49927 .4703 L .50268 .4694 L .50702 .46853 L .5114 .46787 L .51704 .46724 L .52063 .46695 L .52421 .46673 L .52632 .46663 L .52872 .46653 L Mistroke .5311 .46646 L .5334 .46641 L .53582 .46637 L .5382 .46636 L .54041 .46635 L .54289 .46636 L .54571 .4664 L .54718 .46642 L .54882 .46645 L .55472 .46661 L .56119 .46686 L .56881 .46724 L .58437 .46825 L .6205 .47152 L .67716 .47831 L .74769 .48852 L Mfstroke .74769 .48852 m 1 .53661 L s .5 .5 m .5 .5 L .5 .5 L .50001 .5 L .50002 .5 L .50005 .50001 L .50013 .50004 L .50037 .5001 L .50102 .50029 L .50268 .50075 L .5075 .50211 L .52022 .5057 L .55247 .51479 L .64481 .54081 L .8847 .60842 L s .8847 .60842 m 1 .64092 L s .50013 .50011 m .50021 .50018 L .50037 .50031 L .50061 .50051 L .50099 .50082 L .50166 .50138 L .50277 .50227 L .50481 .50386 L .50836 .5065 L .5146 .51083 L .52745 .51898 L .55298 .53339 L .60523 .5594 L .73246 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.41396 L .44039 .41336 L .44049 .41221 L .44063 .41098 L .44099 .40888 L .44153 .4066 L .4422 .40442 L .44298 .40239 L .44541 .39763 L .44909 .39254 L .4537 .38778 L .45906 .38345 L .46687 .37847 L .47611 .37385 L .49797 .36599 L .51399 .36188 L .53157 .35839 L .55269 .35522 L .56588 .35367 L .5795 .35235 L .59318 .35127 L .6093 .35028 L Mistroke .61682 .34991 L .62512 .34956 L .63265 .34929 L .64086 .34905 L .64613 .34893 L .65203 .34881 L .66274 .34866 L .66898 .34861 L .67581 .34859 L .68242 .34859 L .68857 .34861 L .70088 .34872 L .70805 .34882 L .71482 .34893 L .72763 .34922 L .74176 .34961 L .7692 .35061 L .79682 .3519 L .86726 .35623 L .94202 .36218 L Mfstroke .94202 .36218 m 1 .36774 L s .49974 .49978 m .49957 .49964 L .49928 .49939 L .49882 .49899 L .4981 .49837 L .49687 .4973 L .49499 .49562 L .49186 .4927 L .48734 .48815 L .48151 .48135 L .47822 .4766 L .47662 .4738 L .4758 .47215 L .4751 .47049 L .47453 .46891 L .47427 .46804 L .47403 .4671 L .47383 .46622 L .47369 .4654 L .47363 .465 L .47358 .46456 L .47354 .46418 L .47351 .46377 L .47348 .46334 L .47347 .46294 L .47347 .46247 L .47348 .46203 L .4735 .46157 L .47354 .46106 L .4736 .46058 L .47366 .46014 L .47381 .45934 L .47402 .45845 L .47431 .45752 L .47465 .45663 L .47563 .45469 L .47712 .45255 L .47925 .45028 L .48185 .44817 L .48833 .44442 L .4932 .44238 L .49854 .4406 L .50536 .43881 L .51403 .43706 L .52279 .43573 L .53408 .43449 L .54127 .43391 L .54843 .43347 L .55263 .43326 L .55744 .43307 L Mistroke .56219 .43292 L .5668 .43282 L .57165 .43275 L .5764 .43271 L .58082 .4327 L .58578 .43273 L .59143 .43279 L .59436 .43284 L .59764 .4329 L .60944 .43322 L .62238 .43372 L .63762 .43448 L .66874 .4365 L .74099 .44303 L .85432 .45663 L .99538 .47705 L Mfstroke .99538 .47705 m 1 .47793 L s .49987 .49989 m .49979 .49982 L .49964 .49969 L .49941 .4995 L .49907 .49919 L .49849 .49866 L .49763 .49785 L .4963 .49645 L .49548 .49547 L .49469 .49436 L .49396 .49306 L .49366 .49232 L .49353 .49187 L .49348 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.50021 L .50203 .50057 L .50536 .50151 L .515 .50423 L .54044 .5114 L .60493 .52957 L .78962 .58162 L s .78962 .58162 m 1 .64092 L s .50013 .50011 m .50022 .50018 L .50037 .50031 L .50061 .50051 L .501 .50083 L .50171 .50139 L .5029 .5023 L .50519 .50397 L .50938 .50679 L .51728 .51159 L .53495 .5211 L .5732 .53909 L .65769 .57418 L .87727 .65574 L s .87727 .65574 m 1 .69747 L s .50026 .50022 m .50043 .50036 L .50073 .50062 L .50121 .50102 L .50197 .50165 L .50332 .50275 L .50553 .50453 L .50963 .50772 L .51673 .513 L .5292 .52167 L .5549 .53797 L .60595 .56679 L .71046 .6188 L .96493 .72985 L s .96493 .72985 m 1 .74344 L s .5004 .50033 m .50064 .50054 L .5011 .50092 L .50181 .50152 L .50294 .50246 L .50494 .50412 L .50817 .50676 L .51407 .51148 L .52408 .51921 L .54112 .53175 L .57485 .55483 L .63871 .59448 L .76322 .66341 L s .76322 .66341 m 1 .77842 L s .49961 .49967 m .49936 .49946 L .49892 .49908 L .49822 .49849 L .49715 .49756 L .49531 .49595 L .49249 .49343 L .48779 .48905 L 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.11862 m .24353 .33317 L s s s s s s s s s s P P p p .004 w .50004 .50011 m .50006 .50018 L .5001 .5003 L .50016 .5005 L .50027 .50082 L .50044 .50135 L .50071 .5022 L .50113 .50359 L .50176 .50582 L .50215 .50737 L .5026 .50929 L .50307 .51166 L .50329 .51303 L .50349 .51452 L .50357 .51532 L .50365 .51615 L .50368 .51658 L .5037 .51701 L .50372 .51746 L .50373 .51768 L .50374 .51791 L .50374 .51814 L .50375 .51837 L .50375 .5186 L .50375 .51883 L .50375 .51907 L .50374 .51931 L .50374 .51955 L .50373 .51979 L .50371 .52028 L .50368 .52077 L .50363 .52127 L .50358 .52178 L .50343 .5228 L .50322 .52384 L .50295 .52489 L .50258 .52595 L .50213 .527 L .50156 .52803 L .50085 .52904 L .5 .53 L .49897 .53091 L .49774 .53173 L .49703 .53211 L .49627 .53246 L .49544 .53277 L .49453 .53305 L .49355 .53329 L .49302 .53339 L .49248 .53348 L Mistroke .49191 .53356 L .49132 .53363 L .49071 .53368 L .49007 .53372 L .48941 .53374 L .48871 .53375 L .488 .53374 L .48725 .53371 L .48647 .53367 L .48567 .53361 L .48396 .53342 L .48211 .53314 L .48012 .53277 L .47567 .53172 L .46453 .52804 L .44966 .52163 L .42994 .51149 L .26594 .40039 L Mfstroke 0 .18338 m .26594 .40039 L s s s s s s s s s s P P p p .004 w .49996 .49989 m .49994 .49982 L .4999 .4997 L .49983 .4995 L .49972 .49917 L .49954 .49863 L .49923 .49773 L .4987 .49624 L .49779 .49373 L .49616 .48946 L .49313 .48209 L .48722 .46903 L .475 .445 L .44809 .39862 L .38534 .30379 L .23192 .09747 L s .15231 0 m .23192 .09747 L s s s s s s s s s s P P p p .004 w .49998 .49993 m .49996 .49988 L .49993 .4998 L .49989 .49967 L .49981 .49945 L .49969 .49908 L .49948 .49848 L .49911 .49747 L .49846 .49576 L .49727 .49281 L .49497 .48761 L .49026 .47813 L .48 .46 L .45633 .42336 L .39893 .34456 L .25433 .16469 L s .11178 0 m .25433 .16469 L s s s s s s s s s s P P p p .004 w .49999 .49996 m .49998 .49994 L .49997 .4999 L .49994 .49983 L .49991 .49972 L .49984 .49954 L .49973 .49923 L .49952 .4987 L .49914 .49779 L .49839 .49616 L .49681 .49313 L .49329 .48722 L .485 .475 L .46457 .44809 L .41252 .38534 L .27674 .23192 L s .06216 0 m .27674 .23192 L s s s s s s s s s s P P p p .004 w .5 .5 m .5 .5 L .5 .5 L .5 .5 L .5 .5 L .49999 .49999 L .49998 .49998 L .49993 .49993 L .49982 .49982 L .4995 .4995 L .49865 .49865 L .49632 .49632 L .49 .49 L .47282 .47282 L .42611 .42611 L .29914 .29914 L s 0 0 m .29914 .29914 L s s s s s s s s s s P P p p .004 w .50001 .50004 m .50002 .50006 L .50003 .5001 L .50005 .50017 L .50009 .50027 L .50014 .50044 L .50022 .50072 L .50034 .50116 L .50042 .50147 L .50049 .50185 L .50053 .50207 L .50057 .5023 L .50058 .50243 L .5006 .50257 L .50061 .5027 L .50061 .50278 L .50062 .50285 L .50062 .50289 L .50062 .50292 L .50062 .50296 L .50062 .503 L .50062 .50304 L .50062 .50308 L .50062 .50311 L .50062 .50315 L .50062 .50319 L .50062 .50323 L .50062 .50327 L .50062 .50331 L .50062 .50339 L .50061 .50348 L .5006 .50356 L .50059 .50364 L .50057 .50382 L .50053 .50399 L .50049 .50416 L .50042 .50434 L .50035 .50452 L .50025 .50469 L .50013 .50485 L .49999 .50501 L .49981 .50516 L .4996 .5053 L .49948 .50536 L .49935 .50542 L .49921 .50547 L .49906 .50552 L .49889 .50555 L .4988 .50557 L .49871 .50559 L Mistroke .49862 .5056 L .49852 .50561 L .49841 .50562 L .49831 .50562 L .49819 .50562 L .49808 .50562 L .49795 .50562 L .49783 .50562 L .4977 .50561 L .49756 .5056 L .49727 .50556 L .49696 .50551 L .49662 .50545 L .49587 .50527 L .495 .505 L .49283 .50416 L .48993 .50277 L .48106 .49755 L .4397 .46688 L .32155 .36637 L Mfstroke 0 .0731 m .32155 .36637 L s s s s s s s s s s P P p p .004 w .50002 .50007 m .50004 .50012 L .50007 .5002 L .50011 .50033 L .50018 .50055 L .50029 .5009 L .50047 .50147 L .50075 .5024 L .50117 .50388 L .50144 .50491 L .50173 .5062 L .50204 .50777 L .50219 .50869 L .50233 .50968 L .50238 .51021 L .50243 .51077 L .50245 .51105 L .50247 .51134 L .50248 .51164 L .50249 .51179 L .50249 .51194 L .5025 .51209 L .5025 .51225 L .5025 .5124 L .5025 .51256 L .5025 .51271 L .5025 .51287 L .50249 .51303 L .50249 .51319 L .50247 .51352 L .50245 .51385 L .50242 .51418 L .50239 .51452 L .50229 .5152 L .50215 .5159 L .50196 .5166 L .50172 .5173 L .50142 .518 L .50104 .51869 L .50057 .51936 L .5 .52 L .49931 .5206 L .49849 .52115 L .49802 .52141 L .49751 .52164 L .49696 .52185 L .49635 .52203 L .4957 .52219 L .49535 .52226 L .49499 .52232 L Mistroke .49461 .52238 L .49421 .52242 L .49381 .52245 L .49338 .52248 L .49294 .5225 L .49248 .5225 L .492 .52249 L .4915 .52248 L .49098 .52245 L .49044 .5224 L .4893 .52228 L .48807 .5221 L .48675 .52185 L .48378 .52114 L .47635 .51869 L .46644 .51442 L .45329 .50766 L .34396 .4336 L .02791 .17569 L Mfstroke 0 .15079 m .02791 .17569 L s s s s s s s s s P P p p .004 w .50004 .50011 m .50006 .50018 L .5001 .5003 L .50017 .5005 L .50027 .50082 L .50044 .50135 L .50072 .50222 L .50116 .50363 L .50185 .50591 L .50285 .50954 L .50348 .51207 L .50416 .5152 L .50452 .51702 L .50485 .51903 L .50516 .52122 L .5053 .52239 L .50542 .52362 L .50547 .52424 L .50552 .52489 L .50555 .52554 L .50559 .5262 L .5056 .52654 L .50561 .52688 L .50562 .52722 L .50562 .52757 L .50562 .52792 L .50562 .52827 L .50562 .52862 L .50562 .52898 L .50561 .52934 L .5056 .5297 L .50556 .53043 L .50551 .53117 L .50545 .53192 L .50537 .53268 L .50527 .53345 L .505 .535 L .50464 .53657 L .50416 .53815 L .50354 .53973 L .50277 .54129 L .50182 .54283 L .50065 .5443 L .49924 .54571 L .49755 .54701 L .49552 .54818 L .49437 .5487 L .49312 .54917 L .49176 .54959 L .49028 .54994 L Mistroke .48949 .5501 L .48868 .55023 L .48782 .55035 L .48694 .55045 L .48602 .55053 L .48506 .55058 L .48406 .55062 L .48302 .55062 L .48194 .55061 L .48082 .55057 L .47965 .5505 L .47844 .5504 L .47587 .55012 L .4731 .5497 L .46688 .54843 L .45967 .54648 L .45133 .54373 L .43053 .53524 L .36637 .50082 L .06485 .28653 L Mfstroke 0 .23244 m .06485 .28653 L s s s s s s s s s P P p p .004 w .49996 .49989 m .49994 .49982 L .4999 .4997 L .49983 .4995 L .49972 .49917 L .49954 .49864 L .49924 .49775 L .49874 .49627 L .49788 .49382 L .4964 .48971 L .49381 .48277 L .48906 .47087 L .48 .45 L .46168 .41222 L .42228 .34073 L .33235 .1979 L s .19134 0 m .33235 .1979 L s s s s s s s s s s P P p p .004 w .49998 .49993 m .49996 .49988 L .49993 .4998 L .49989 .49967 L .49982 .49945 L .49969 .49909 L .49949 .49849 L .49915 .4975 L .49856 .49585 L .49752 .49306 L .49564 .48829 L .4921 .47996 L .485 .465 L .46992 .43695 L .43587 .38151 L .35476 .26512 L .15312 .00534 L s .14862 0 m .15312 .00534 L s s s s s s s s s P P p p .004 w .49999 .49996 m .49998 .49994 L .49997 .4999 L .49994 .49983 L .49991 .49972 L .49984 .49954 L .49974 .49924 L .49956 .49874 L .49923 .49788 L .49864 .4964 L .49748 .49381 L .49513 .48906 L .49 .48 L .47816 .46168 L .44946 .42228 L .37716 .33235 L .19006 .11617 L s .08418 0 m .19006 .11617 L s s s s s s s s s P P p p .004 w .5 .5 m .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .49999 .49999 L .49997 .49997 L .49991 .49991 L .49975 .49975 L .49932 .49932 L .49816 .49816 L .495 .495 L .48641 .48641 L .46305 .46305 L .39957 .39957 L .22701 .22701 L s 0 0 m .22701 .22701 L s s s s s s s s s P P p p .004 w .50001 .50004 m .50002 .50006 L .50003 .5001 L .50005 .50017 L .50009 .50027 L .50015 .50045 L .50024 .50073 L .50038 .5012 L .50059 .50194 L .50072 .50246 L .50087 .5031 L .50102 .50389 L .5011 .50434 L .50116 .50484 L .50119 .50511 L .50122 .50538 L .50123 .50553 L .50123 .50567 L .50124 .50582 L .50124 .50589 L .50125 .50597 L .50125 .50605 L .50125 .50612 L .50125 .5062 L .50125 .50628 L .50125 .50636 L .50125 .50644 L .50125 .50652 L .50124 .5066 L .50124 .50676 L .50123 .50692 L .50121 .50709 L .50119 .50726 L .50114 .5076 L .50107 .50795 L .50098 .5083 L .50086 .50865 L .50071 .509 L .50052 .50934 L .50028 .50968 L .5 .51 L .49966 .5103 L .49925 .51058 L .49901 .5107 L .49876 .51082 L .49848 .51092 L .49818 .51102 L .49785 .5111 L .49767 .51113 L .49749 .51116 L Mistroke .4973 .51119 L .49711 .51121 L .4969 .51123 L .49669 .51124 L .49647 .51125 L .49624 .51125 L .496 .51125 L .49575 .51124 L .49549 .51122 L .49522 .5112 L .49465 .51114 L .49404 .51105 L .49337 .51092 L .49189 .51057 L .48818 .50935 L .48322 .50721 L .47665 .50383 L .42198 .4668 L .26395 .33785 L Mfstroke 0 .10232 m .26395 .33785 L s s s s s s s s s P P p p .004 w .50002 .50007 m .50004 .50012 L .50007 .5002 L .50011 .50033 L .50018 .50055 L .5003 .5009 L .50049 .50148 L .50079 .50243 L .50126 .50397 L .50198 .50644 L .503 .51036 L .50361 .51306 L .50423 .51636 L .50451 .51826 L .50464 .51927 L .50476 .52033 L .50485 .52143 L .5049 .522 L .50493 .52258 L .50496 .52317 L .50497 .52347 L .50498 .52377 L .50499 .52407 L .505 .52438 L .505 .52469 L .505 .525 L .505 .52531 L .50499 .52563 L .50499 .52595 L .50498 .52627 L .50497 .52659 L .50495 .52692 L .50491 .52757 L .50486 .52824 L .50479 .52891 L .5046 .53028 L .50433 .53166 L .50396 .53306 L .50349 .53447 L .5029 .53587 L .50215 .53725 L .50123 .5386 L .50011 .53989 L .49876 .5411 L .49714 .54221 L .49622 .54272 L .49522 .54319 L .49412 .54362 L .49293 .544 L .49164 .54433 L Mistroke .49095 .54448 L .49024 .5446 L .48949 .54471 L .48872 .54481 L .48791 .54489 L .48707 .54494 L .4862 .54498 L .48529 .545 L .48435 .54499 L .48336 .54497 L .48234 .54492 L .48128 .54484 L .47903 .54461 L .47661 .54427 L .47399 .5438 L .46813 .54244 L .46134 .54044 L .44439 .53402 L .39197 .50706 L .3009 .44868 L Mfstroke 0 .21628 m .3009 .44868 L s s s s s s s s s P P p p .004 w .50004 .50011 m .50006 .50018 L .5001 .5003 L .50017 .5005 L .50027 .50082 L .50045 .50135 L .50073 .50223 L .5012 .50366 L .50194 .506 L .5031 .50979 L .50484 .51588 L .50726 .52545 L .50865 .53201 L .50934 .53582 L .51 .54 L .5103 .54224 L .51058 .54457 L .51082 .54701 L .51092 .54826 L .51102 .54954 L .5111 .55084 L .51116 .55217 L .51119 .55284 L .51121 .55352 L .51123 .5542 L .51124 .55489 L .51125 .55558 L .51125 .55629 L .51125 .55699 L .51124 .5577 L .51122 .55842 L .5112 .55914 L .51117 .55987 L .51114 .5606 L .51105 .56208 L .51092 .56358 L .51077 .56509 L .51057 .56662 L .51006 .56972 L .50935 .57286 L .50841 .57602 L .50721 .57918 L .5057 .58231 L .50383 .58538 L .50154 .58835 L .49876 .59117 L .49543 .59379 L .49144 .59615 L .48918 .59721 L .48671 .59817 L Mistroke .48403 .59903 L .48111 .59977 L .47956 .60009 L .47795 .60037 L .47627 .60062 L .47453 .60083 L .47271 .601 L .47081 .60113 L .46885 .60121 L .4668 .60125 L .46467 .60124 L .46245 .60117 L .46015 .60106 L .45776 .60088 L .45269 .60036 L .44722 .59958 L .43497 .59715 L .42074 .59338 L .38521 .58083 L .33785 .55952 L Mfstroke 0 .34399 m .33785 .55952 L s s s s s s s s s P P p p .004 w .49996 .49989 m .49994 .49982 L .4999 .4997 L .49983 .4995 L .49973 .49918 L .49955 .49864 L .49925 .49776 L .49877 .49631 L .49797 .49391 L .49665 .48996 L .49448 .48345 L .4909 .47271 L .485 .455 L .47527 .42581 L .45923 .37768 L .43277 .29832 L .38916 .16749 L s .33333 0 m .38916 .16749 L s s s s s s s s s P P p p .004 w .49998 .49993 m .49996 .49988 L .49993 .4998 L .49989 .49967 L .49982 .49945 L .4997 .49909 L .4995 .49851 L .49918 .49754 L .49865 .49594 L .49777 .49331 L .49632 .48896 L .49393 .4818 L .49 .47 L .48351 .45054 L .47282 .41845 L .45518 .36555 L .42611 .27833 L .37818 .13453 L s .33333 0 m .37818 .13453 L s s s s s s s s P P p p .004 w .49999 .49996 m .49998 .49994 L .49997 .4999 L .49994 .49983 L .49991 .49973 L .49985 .49955 L .49975 .49925 L .49959 .49877 L .49932 .49797 L .49888 .49665 L .49816 .49448 L .49697 .4909 L .495 .485 L .49176 .47527 L .48641 .45923 L .47759 .43277 L .46305 .38916 L .43909 .31726 L .39957 .19872 L .33442 .00327 L s .33333 0 m .33442 .00327 L s s s s s s P P p p .004 w .5 .5 m .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L s P P p p .004 w .50001 .50004 m .50002 .50006 L .50003 .5001 L .50006 .50017 L .50009 .50027 L .50015 .50045 L .50025 .50075 L .50041 .50123 L .50068 .50203 L .50112 .50335 L .50184 .50552 L .50303 .5091 L .505 .515 L .50824 .52473 L .51359 .54077 L .52241 .56723 L .53695 .61084 L .56091 .68274 L .60043 .80128 L .66558 .99673 L s .66667 1 m .66558 .99673 L s s s s s s P P p p .004 w .50002 .50007 m .50004 .50012 L .50007 .5002 L .50011 .50033 L .50018 .50055 L .5003 .50091 L .5005 .50149 L .50082 .50246 L .50135 .50406 L .50223 .50669 L .50368 .51104 L .50607 .5182 L .51 .53 L .51649 .54946 L .52718 .58155 L .54482 .63445 L .57389 .72167 L .62182 .86547 L s .66667 1 m .62182 .86547 L s s s s s s s s P P p p .004 w .50004 .50011 m .50006 .50018 L .5001 .5003 L .50017 .5005 L .50027 .50082 L .50045 .50136 L .50075 .50224 L .50123 .50369 L .50203 .50609 L .50335 .51004 L .50552 .51655 L .5091 .52729 L .515 .545 L .52473 .57419 L .54077 .62232 L .56723 .70168 L .61084 .83251 L s .66667 1 m .61084 .83251 L s s s s s s s s s P P p p .004 w .49996 .49989 m .49994 .49982 L .4999 .4997 L .49983 .4995 L .49973 .49918 L .49955 .49865 L .49927 .49777 L .4988 .49634 L .49806 .494 L .4969 .49021 L .49516 .48412 L .49274 .47455 L .49135 .46799 L .49066 .46418 L .49 .46 L .4897 .45776 L .48942 .45543 L .48918 .45299 L .48908 .45174 L .48898 .45046 L .4889 .44916 L .48884 .44783 L .48881 .44716 L .48879 .44648 L .48877 .4458 L .48876 .44511 L .48875 .44442 L .48875 .44371 L .48875 .44301 L .48876 .4423 L .48878 .44158 L .4888 .44086 L .48883 .44013 L .48886 .4394 L .48895 .43792 L .48908 .43642 L .48923 .43491 L .48943 .43338 L .48994 .43028 L .49065 .42714 L .49159 .42398 L .49279 .42082 L .4943 .41769 L .49617 .41462 L .49846 .41165 L .50124 .40883 L .50457 .40621 L .50856 .40385 L .51082 .40279 L .51329 .40183 L Mistroke .51597 .40097 L .51889 .40023 L .52044 .39991 L .52205 .39963 L .52373 .39938 L .52547 .39917 L .52729 .399 L .52919 .39887 L .53115 .39879 L .5332 .39875 L .53533 .39876 L .53755 .39883 L .53985 .39894 L .54224 .39912 L .54731 .39964 L .55278 .40042 L .56503 .40285 L .57926 .40662 L .61479 .41917 L .66215 .44048 L Mfstroke 1 .65601 m .66215 .44048 L s s s s s s s s s P P p p .004 w .49998 .49993 m .49996 .49988 L .49993 .4998 L .49989 .49967 L .49982 .49945 L .4997 .4991 L .49951 .49852 L .49921 .49757 L .49874 .49603 L .49802 .49356 L .497 .48964 L .49639 .48694 L .49577 .48364 L .49549 .48174 L .49536 .48073 L .49524 .47967 L .49515 .47857 L .4951 .478 L .49507 .47742 L .49504 .47683 L .49503 .47653 L .49502 .47623 L .49501 .47593 L .495 .47562 L .495 .47531 L .495 .475 L .495 .47469 L .49501 .47437 L .49501 .47405 L .49502 .47373 L .49503 .47341 L .49505 .47308 L .49509 .47243 L .49514 .47176 L .49521 .47109 L .4954 .46972 L .49567 .46834 L .49604 .46694 L .49651 .46553 L .4971 .46413 L .49785 .46275 L .49877 .4614 L .49989 .46011 L .50124 .4589 L .50286 .45779 L .50378 .45728 L .50478 .45681 L .50588 .45638 L .50707 .456 L .50836 .45567 L Mistroke .50905 .45552 L .50976 .4554 L .51051 .45529 L .51128 .45519 L .51209 .45511 L .51293 .45506 L .5138 .45502 L .51471 .455 L .51565 .45501 L .51664 .45503 L .51766 .45508 L .51872 .45516 L .52097 .45539 L .52339 .45573 L .52601 .4562 L .53187 .45756 L .53866 .45956 L .55561 .46598 L .60803 .49294 L .6991 .55132 L Mfstroke 1 .78372 m .6991 .55132 L s s s s s s s s s P P p p .004 w .49999 .49996 m .49998 .49994 L .49997 .4999 L .49995 .49983 L .49991 .49973 L .49985 .49955 L .49976 .49927 L .49962 .4988 L .49941 .49806 L .49928 .49754 L .49913 .4969 L .49898 .49611 L .4989 .49566 L .49884 .49516 L .49881 .49489 L .49878 .49462 L .49877 .49447 L .49877 .49433 L .49876 .49418 L .49876 .49411 L .49875 .49403 L .49875 .49395 L .49875 .49388 L .49875 .4938 L .49875 .49372 L .49875 .49364 L .49875 .49356 L .49875 .49348 L .49876 .4934 L .49876 .49324 L .49877 .49308 L .49879 .49291 L .49881 .49274 L .49886 .4924 L .49893 .49205 L .49902 .4917 L .49914 .49135 L .49929 .491 L .49948 .49066 L .49972 .49032 L .5 .49 L .50034 .4897 L .50075 .48942 L .50099 .4893 L .50124 .48918 L .50152 .48908 L .50182 .48898 L .50215 .4889 L .50233 .48887 L .50251 .48884 L Mistroke .5027 .48881 L .50289 .48879 L .5031 .48877 L .50331 .48876 L .50353 .48875 L .50376 .48875 L .504 .48875 L .50425 .48876 L .50451 .48878 L .50478 .4888 L .50535 .48886 L .50596 .48895 L .50663 .48908 L .50811 .48943 L .51182 .49065 L .51678 .49279 L .52335 .49617 L .57802 .5332 L .73605 .66215 L Mfstroke 1 .89768 m .73605 .66215 L s s s s s s s s s P P p p .004 w .5 .5 m .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .50001 .50001 L .50003 .50003 L .50009 .50009 L .50025 .50025 L .50068 .50068 L .50184 .50184 L .505 .505 L .51359 .51359 L .53695 .53695 L .60043 .60043 L .77299 .77299 L s 1 1 m .77299 .77299 L s s s s s s s s s P P p p .004 w .50001 .50004 m .50002 .50006 L .50003 .5001 L .50006 .50017 L .50009 .50028 L .50016 .50046 L .50026 .50076 L .50044 .50126 L .50077 .50212 L .50136 .5036 L .50252 .50619 L .50487 .51094 L .51 .52 L .52184 .53832 L .55054 .57772 L .62284 .66765 L .80994 .88383 L s .91582 1 m .80994 .88383 L s s s s s s s s s P P p p .004 w .50002 .50007 m .50004 .50012 L .50007 .5002 L .50011 .50033 L .50018 .50055 L .50031 .50091 L .50051 .50151 L .50085 .5025 L .50144 .50415 L .50248 .50694 L .50436 .51171 L .5079 .52004 L .515 .535 L .53008 .56305 L .56413 .61849 L .64524 .73488 L .84688 .99466 L s .85138 1 m .84688 .99466 L s s s s s s s s s P P p p .004 w .50004 .50011 m .50006 .50018 L .5001 .5003 L .50017 .5005 L .50028 .50083 L .50046 .50136 L .50076 .50225 L .50126 .50373 L .50212 .50618 L .5036 .51029 L .50619 .51723 L .51094 .52913 L .52 .55 L .53832 .58778 L .57772 .65927 L .66765 .8021 L s .80866 1 m .66765 .8021 L s s s s s s s s s s P P p p .004 w .49996 .49989 m .49994 .49982 L .4999 .4997 L .49983 .4995 L .49973 .49918 L .49956 .49865 L .49928 .49778 L .49884 .49637 L .49815 .49409 L .49715 .49046 L .49652 .48793 L .49584 .4848 L .49548 .48298 L .49515 .48097 L .49484 .47878 L .4947 .47761 L .49458 .47638 L .49453 .47576 L .49448 .47511 L .49445 .47446 L .49441 .4738 L .4944 .47346 L .49439 .47312 L .49438 .47278 L .49438 .47243 L .49438 .47208 L .49438 .47173 L .49438 .47138 L .49438 .47102 L .49439 .47066 L .4944 .4703 L .49444 .46957 L .49449 .46883 L .49455 .46808 L .49463 .46732 L .49473 .46655 L .495 .465 L .49536 .46343 L .49584 .46185 L .49646 .46027 L .49723 .45871 L .49818 .45717 L .49935 .4557 L .50076 .45429 L .50245 .45299 L .50448 .45182 L .50563 .4513 L .50688 .45083 L .50824 .45041 L .50972 .45006 L Mistroke .51051 .4499 L .51132 .44977 L .51218 .44965 L .51306 .44955 L .51398 .44947 L .51494 .44942 L .51594 .44938 L .51698 .44938 L .51806 .44939 L .51918 .44943 L .52035 .4495 L .52156 .4496 L .52413 .44988 L .5269 .4503 L .53312 .45157 L .54033 .45352 L .54867 .45627 L .56947 .46476 L .63363 .49918 L .93515 .71347 L Mfstroke 1 .76756 m .93515 .71347 L s s s s s s s s s P P p p .004 w .49998 .49993 m .49996 .49988 L .49993 .4998 L .49989 .49967 L .49982 .49945 L .49971 .4991 L .49953 .49853 L .49925 .4976 L .49883 .49612 L .49856 .49509 L .49827 .4938 L .49796 .49223 L .49781 .49131 L .49767 .49032 L .49762 .48979 L .49757 .48923 L .49755 .48895 L .49753 .48866 L .49752 .48836 L .49751 .48821 L .49751 .48806 L .4975 .48791 L .4975 .48775 L .4975 .4876 L .4975 .48744 L .4975 .48729 L .4975 .48713 L .49751 .48697 L .49751 .48681 L .49753 .48648 L .49755 .48615 L .49758 .48582 L .49761 .48548 L .49771 .4848 L .49785 .4841 L .49804 .4834 L .49828 .4827 L .49858 .482 L .49896 .48131 L .49943 .48064 L .5 .48 L .50069 .4794 L .50151 .47885 L .50198 .47859 L .50249 .47836 L .50304 .47815 L .50365 .47797 L .5043 .47781 L .50465 .47774 L .50501 .47768 L Mistroke .50539 .47762 L .50579 .47758 L .50619 .47755 L .50662 .47752 L .50706 .4775 L .50752 .4775 L .508 .47751 L .5085 .47752 L .50902 .47755 L .50956 .4776 L .5107 .47772 L .51193 .4779 L .51325 .47815 L .51622 .47886 L .52365 .48131 L .53356 .48558 L .54671 .49234 L .65604 .5664 L .97209 .82431 L Mfstroke 1 .84921 m .97209 .82431 L s s s s s s s s s P P p p .004 w .49999 .49996 m .49998 .49994 L .49997 .4999 L .49995 .49983 L .49991 .49973 L .49986 .49956 L .49978 .49928 L .49966 .49884 L .49958 .49853 L .49951 .49815 L .49947 .49793 L .49943 .4977 L .49942 .49757 L .4994 .49743 L .49939 .4973 L .49939 .49722 L .49938 .49715 L .49938 .49711 L .49938 .49708 L .49938 .49704 L .49938 .497 L .49938 .49696 L .49938 .49692 L .49938 .49689 L .49938 .49685 L .49938 .49681 L .49938 .49677 L .49938 .49673 L .49938 .49669 L .49938 .49661 L .49939 .49652 L .4994 .49644 L .49941 .49636 L .49943 .49618 L .49947 .49601 L .49951 .49584 L .49958 .49566 L .49965 .49548 L .49975 .49531 L .49987 .49515 L .50001 .49499 L .50019 .49484 L .5004 .4947 L .50052 .49464 L .50065 .49458 L .50079 .49453 L .50094 .49448 L .50111 .49445 L .5012 .49443 L .50129 .49441 L Mistroke .50138 .4944 L .50148 .49439 L .50159 .49438 L .50169 .49438 L .50181 .49438 L .50192 .49438 L .50205 .49438 L .50217 .49438 L .5023 .49439 L .50244 .4944 L .50273 .49444 L .50304 .49449 L .50338 .49455 L .50413 .49473 L .505 .495 L .50717 .49584 L .51007 .49723 L .51894 .50245 L .5603 .53312 L .67845 .63363 L Mfstroke 1 .9269 m .67845 .63363 L s s s s s s s s s s P P p p .004 w .5 .5 m .5 .5 L .5 .5 L .5 .5 L .5 .5 L .50001 .50001 L .50002 .50002 L .50007 .50007 L .50018 .50018 L .5005 .5005 L .50135 .50135 L .50368 .50368 L .51 .51 L .52718 .52718 L .57389 .57389 L .70086 .70086 L s 1 1 m .70086 .70086 L s s s s s s s s s s P P p p .004 w .50001 .50004 m .50002 .50006 L .50003 .5001 L .50006 .50017 L .50009 .50028 L .50016 .50046 L .50027 .50077 L .50048 .5013 L .50086 .50221 L .50161 .50384 L .50319 .50687 L .50671 .51278 L .515 .525 L .53543 .55191 L .58748 .61466 L .72326 .76808 L s .93784 1 m .72326 .76808 L s s s s s s s s s s P P p p .004 w .50002 .50007 m .50004 .50012 L .50007 .5002 L .50011 .50033 L .50019 .50055 L .50031 .50092 L .50052 .50152 L .50089 .50253 L .50154 .50424 L .50273 .50719 L .50503 .51239 L .50974 .52187 L .52 .54 L .54367 .57664 L .60107 .65544 L .74567 .83531 L s .88822 1 m .74567 .83531 L s s s s s s s s s s P P p p .004 w .50004 .50011 m .50006 .50018 L .5001 .5003 L .50017 .5005 L .50028 .50083 L .50046 .50137 L .50077 .50227 L .5013 .50376 L .50221 .50627 L .50384 .51054 L .50687 .51791 L .51278 .53097 L .525 .555 L .55191 .60138 L .61466 .69621 L .76808 .90253 L s .84769 1 m .76808 .90253 L s s s s s s s s s s P P p p .004 w .49996 .49989 m .49994 .49982 L .4999 .4997 L .49984 .4995 L .49973 .49918 L .49956 .49865 L .49929 .4978 L .49887 .49641 L .49824 .49418 L .49785 .49263 L .4974 .49071 L .49693 .48834 L .49671 .48697 L .49651 .48548 L .49643 .48468 L .49635 .48385 L .49632 .48342 L .4963 .48299 L .49628 .48254 L .49627 .48232 L .49626 .48209 L .49626 .48186 L .49625 .48163 L .49625 .4814 L .49625 .48117 L .49625 .48093 L .49626 .48069 L .49626 .48045 L .49627 .48021 L .49629 .47972 L .49632 .47923 L .49637 .47873 L .49642 .47822 L .49657 .4772 L .49678 .47616 L .49705 .47511 L .49742 .47405 L .49787 .473 L .49844 .47197 L .49915 .47096 L .5 .47 L .50103 .46909 L .50226 .46827 L .50297 .46789 L .50373 .46754 L .50456 .46723 L .50547 .46695 L .50645 .46671 L .50698 .46661 L .50752 .46652 L Mistroke .50809 .46644 L .50868 .46637 L .50929 .46632 L .50993 .46628 L .51059 .46626 L .51129 .46625 L .512 .46626 L .51275 .46629 L .51353 .46633 L .51433 .46639 L .51604 .46658 L .51789 .46686 L .51988 .46723 L .52433 .46828 L .53547 .47196 L .55034 .47837 L .57006 .48851 L .73406 .59961 L Mfstroke 1 .81662 m .73406 .59961 L s s s s s s s s s s P P p p .004 w .49998 .49993 m .49996 .49988 L .49993 .4998 L .49989 .49967 L .49982 .49946 L .49971 .49911 L .49954 .49854 L .49928 .49764 L .49892 .49621 L .49872 .49524 L .49861 .49467 L .49852 .49405 L .49847 .49372 L .49843 .49337 L .4984 .49301 L .49838 .49283 L .49837 .49264 L .49835 .49244 L .49835 .49225 L .49834 .49215 L .49834 .49205 L .49834 .49194 L .49833 .49184 L .49833 .49174 L .49833 .49163 L .49833 .49153 L .49834 .49142 L .49834 .49132 L .49834 .49121 L .49835 .4911 L .49835 .49099 L .49837 .49077 L .49838 .49055 L .49841 .49033 L .49844 .4901 L .49852 .48964 L .49862 .48918 L .49876 .48871 L .49894 .48824 L .49917 .48778 L .49945 .48732 L .4998 .48688 L .50021 .48647 L .50071 .48608 L .50131 .48573 L .50165 .48558 L .50202 .48544 L .50242 .48531 L .50286 .48521 L .50309 .48516 L Mistroke .50333 .48512 L .50358 .48508 L .50384 .48506 L .50412 .48503 L .5044 .48501 L .50469 .485 L .505 .485 L .50532 .485 L .50565 .48502 L .50599 .48503 L .50635 .48506 L .50711 .48514 L .50793 .48527 L .50976 .48564 L .51189 .48621 L .51721 .48811 L .52429 .49131 L .58365 .52929 L .75647 .66683 L Mfstroke 1 .88138 m .75647 .66683 L s s s s s s s s s s P P p p .004 w .49999 .49996 m .49998 .49994 L .49997 .4999 L .49995 .49984 L .49991 .49973 L .49986 .49956 L .49979 .49929 L .49974 .4991 L .49969 .49887 L .49966 .49873 L .49964 .49859 L .49962 .49842 L .49961 .49834 L .4996 .49824 L .49959 .4982 L .49959 .49815 L .49959 .4981 L .49959 .49808 L .49959 .49805 L .49959 .49803 L .49958 .498 L .49958 .49798 L .49958 .49795 L .49958 .49793 L .49958 .4979 L .49958 .49787 L .49958 .49785 L .49958 .49782 L .49959 .49779 L .49959 .49774 L .49959 .49768 L .4996 .49763 L .4996 .49757 L .49961 .49751 L .49963 .4974 L .49966 .49728 L .49969 .49717 L .49974 .49705 L .4998 .49693 L .49987 .49682 L .49996 .49671 L .50006 .49661 L .50019 .49651 L .50034 .49643 L .50043 .49639 L .50052 .49635 L .50062 .49632 L .50073 .4963 L .50079 .49629 L .50086 .49628 L Mistroke .50092 .49627 L .50099 .49626 L .50105 .49626 L .50113 .49625 L .5012 .49625 L .50128 .49625 L .50136 .49625 L .50144 .49626 L .50153 .49626 L .50162 .49627 L .50181 .49629 L .50202 .49632 L .50249 .49642 L .50302 .49657 L .50365 .49678 L .5052 .49742 L .51 .5 L .53253 .51604 L .59724 .57006 L .77887 .73406 L Mfstroke 1 .94241 m .77887 .73406 L s s s s s s s s s s P P p p .004 w .5 .5 m .5 .5 L .5 .5 L .5 .5 L .50001 .50001 L .50001 .50001 L .50004 .50004 L .5001 .5001 L .50027 .50027 L .50075 .50075 L .50203 .50203 L .50552 .50552 L .515 .515 L .54077 .54077 L .61084 .61084 L .80128 .80128 L s 1 1 m .80128 .80128 L s s s s s s s s s s P P p p .004 w .50001 .50004 m .50002 .50006 L .50003 .5001 L .50006 .50017 L .5001 .50028 L .50016 .50047 L .50029 .50078 L .50051 .50133 L .50095 .5023 L .50186 .50409 L .50387 .50755 L .50855 .51462 L .52 .53 L .54902 .56551 L .62443 .65161 L .82369 .86851 L s .94837 1 m .82369 .86851 L s s s s s s s s s s P P p p .004 w .50002 .50007 m .50004 .50012 L .50007 .5002 L .50011 .50034 L .50019 .50055 L .50032 .50092 L .50054 .50153 L .50092 .50256 L .50163 .50433 L .50298 .50744 L .50571 .51307 L .51158 .52371 L .525 .545 L .55726 .59024 L .63802 .69238 L .8461 .93573 L s .90419 1 m .8461 .93573 L s s s s s s s s s s P P p p .004 w .50004 .50011 m .50006 .50018 L .5001 .5003 L .50017 .5005 L .50028 .50083 L .50047 .50137 L .50078 .50228 L .50133 .50379 L .5023 .50636 L .50409 .51079 L .50755 .51858 L .51462 .53281 L .53 .56 L .56551 .61497 L .65161 .73316 L s .86613 1 m .65161 .73316 L s s s s s s s s s s s P P P p p 1 0 0 r p .004 w .5 .5 m .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .50001 .50001 L .50003 .50003 L .50009 .50009 L .50025 .50025 L .50068 .50068 L .50184 .50184 L .505 .505 L .51359 .51359 L .53695 .53695 L .60043 .60043 L .77299 .77299 L s 1 1 m .77299 .77299 L s s s s s s s s s P P p 1 0 0 r p .004 w .5 .5 m .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .49999 .49999 L .49997 .49997 L .49991 .49991 L .49975 .49975 L .49932 .49932 L .49816 .49816 L .495 .495 L .48641 .48641 L .46305 .46305 L .39957 .39957 L .22701 .22701 L s 0 0 m .22701 .22701 L s s s s s s s s s P P P p p 0 0 1 r p .004 w .5 .5 m .5 .5 L .5 .5 L .5 .5 L .5 .50001 L .5 .50001 L .50001 .50004 L .50003 .5001 L .50009 .50027 L .50025 .50075 L .50068 .50203 L .50184 .50552 L .505 .515 L .51359 .54077 L .53695 .61084 L .60043 .80128 L s .66667 1 m .60043 .80128 L s s s s s s s s s s P P p 0 0 1 r p .004 w .5 .5 m .5 .5 L .5 .5 L .5 .5 L .5 .49999 L .5 .49999 L .49999 .49996 L .49997 .4999 L .49991 .49973 L .49975 .49925 L .49932 .49797 L .49816 .49448 L .495 .485 L .48641 .45923 L .46305 .38916 L .39957 .19872 L s .33333 0 m .39957 .19872 L s s s s s s s s s s P P P P % End of Graphics MathPictureEnd \ \>"], "Graphics", Evaluatable->False, AspectRatioFixed->True, ImageSize->{250, 250}, ImageMargins->{{34, Inherited}, {Inherited, Inherited}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHggYjN[Vi^OShn03ooeGooj[ooonIgYjN[Vi^OShn03o oclo?ginO[fm_Oclo5D00:X00?l00?l0VOl0I_l0CXf=SLcCXf=SLcII03=VoalO7eiN GYfMWMcLg02Z020003o`0000D000?o00000P000ol0001i0000 3P001Ol0003o00D00_l20003o`0000P0000004o`0000?o1@000ol0 001i000040001Ol0003o00H000Co003o2P000b?o000800;o1002o`8000Wo0000o`3o0?l01@000ol0 000B0003om8000l000Co003o1`000ol0001i00004@001Ol0003o00H000Co003o2P000b?o00070003 o`3o00@00_l20006o`000?l00_l60003o`00018000?B00002`04o`801?l40003o`0007T0000B0004 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.52832 L .43902 .52525 L .43845 .52394 L .43793 .52251 L .43756 .52126 L .43724 .51994 L .4371 .51921 L .437 .51856 L .43692 .5179 L .43687 .51729 L Mistroke .43683 .51672 L .43682 .51613 L .43681 .51562 L .43683 .51507 L .43686 .51449 L .43691 .51386 L .43698 .51327 L .43705 .51274 L .43728 .51155 L .43755 .51047 L .43839 .50809 L .43947 .50587 L .44089 .50363 L .44417 .4998 L .44851 .49618 L .45317 .49328 L .45779 .49105 L .46164 .48956 L .46585 .48825 L .47049 .48714 L .47479 .48638 L Mfstroke 0 .12812 m .02926 .12643 L .04562 .12569 L .06334 .12504 L .07954 .1246 L .09447 .12431 L .10863 .12414 L .1236 .12407 L .13828 .12413 L .14656 .12421 L .15425 .12432 L .16233 .12447 L .16966 .12463 L .18602 .12511 L .19983 .12562 L .2147 .1263 L .24136 .12783 L .27189 .13008 L .30353 .13302 L .35825 .13958 L .40641 .14705 L .44935 .15515 L .53882 .17711 L .61128 .20111 L .67623 .22915 L .72262 .25468 L .764 .28361 L .79778 .31487 L .81981 .34314 L .82868 .35858 L .83244 .36662 L .83576 .37503 L .83811 .38231 L .84013 .39022 L .84084 .39366 L .84147 .39732 L .84195 .40071 L .8423 .40388 L .84245 .40565 L .84259 .40756 L .84274 .41087 L .84279 .41274 L .8428 .4147 L .84273 .41819 L .84256 .4214 L .84232 .42437 L .84196 .4276 L .84147 .43102 L .83997 .43863 L .83807 .44566 L Mistroke .83329 .45838 L .82738 .46995 L .82083 .48012 L .80264 .5013 L .78236 .51851 L .76099 .53255 L .73561 .54564 L .70681 .55707 L .67891 .56543 L .6641 .56894 L .64839 .57201 L .63346 .57434 L .62012 .57595 L .61376 .57656 L .60695 .57711 L .60056 .57751 L .59481 .5778 L .5917 .57791 L .58878 .578 L .58572 .57807 L .584 .5781 L .58241 .57812 L .57979 .57814 L .57692 .57815 L .57394 .57812 L .57116 .57808 L .56809 .57802 L .56489 .57792 L .55925 .57768 L .55328 .57734 L .5469 .57685 L .53539 .57568 L .52371 .57407 L .50433 .57033 L .48706 .56565 L .47098 .55978 L .45834 .55369 L .44881 .54776 L .44126 .54166 L .43591 .53593 L .43356 .53267 L .43184 .52972 L .43106 .52811 L .43035 .52637 L .42979 .52474 L .42939 .52326 L .42908 .52175 L .42896 .52097 L .42886 .52015 L .4288 .51944 L Mistroke .42876 .51878 L .42874 .51807 L .42874 .51767 L .42874 .51731 L .42877 .51664 L .42881 .51603 L .42887 .51537 L .42896 .51467 L .42916 .51351 L .42945 .51225 L .42987 .51085 L .43032 .50959 L .4316 .50686 L .433 .50457 L .43473 .50228 L .43847 .49844 L .44316 .49485 L .44807 .49195 L .45283 .48973 L .45848 .48765 L .46379 .48615 L .4696 .48495 L .47249 .48452 L .47562 .48415 L .47772 .48397 L .47874 .48391 L .47968 .48386 L .4805 .48382 L .4814 .48379 L .48187 .48378 L .4823 .48377 L .48276 .48376 L .48325 .48376 L .48374 .48375 L .4842 .48376 L .4847 .48376 L .48523 .48377 L .48578 .48378 L .48629 .48379 L .48725 .48383 L .48814 .48388 L .48894 .48392 L .49073 .48407 L Mfstroke .35256 0 m .47066 .02304 L .59016 .05553 L .68801 .09094 L .76315 .12559 L .83138 .16537 L .88472 .20531 L .92929 .24982 L .94632 .27208 L .96085 .29581 L .97109 .31786 L .97485 .3283 L .97765 .33768 L .97997 .34747 L .98158 .3566 L .98223 .36156 L .98269 .36611 L .98304 .37107 L .98315 .37355 L .98322 .37627 L .98324 .3808 L .98314 .3849 L .98289 .3896 L .98252 .39403 L .98206 .39807 L .98154 .4018 L .98002 .41012 L .97765 .41976 L .97486 .42864 L .9683 .4447 L .95937 .46122 L .94775 .47812 L .92353 .50483 L .89192 .53052 L .85462 .5533 L .81711 .57083 L .77677 .58528 L .738 .59569 L .70334 .6025 L .68404 .60534 L .66647 .60736 L .64916 .60881 L .63959 .60939 L .63509 .60961 L .63094 .60978 L .62678 .60992 L .62302 .61002 L .61888 .6101 L .61449 .61015 L .61069 .61017 L Mistroke .6066 .61016 L .60277 .61012 L .59927 .61006 L .59523 .60996 L .59094 .60983 L .5833 .6095 L .57438 .60897 L .56642 .60837 L .54968 .60667 L .53459 .6046 L .50843 .59968 L .48377 .59318 L .46532 .5868 L .44777 .57908 L .43257 .57051 L .42134 .56237 L .41189 .55332 L .40822 .54878 L .40498 .54385 L .40266 .53927 L .40176 .53705 L .40107 .53502 L .40057 .53323 L .40014 .53128 L .39982 .52934 L .3997 .52829 L .39962 .52732 L .39956 .52628 L .39955 .5257 L .39954 .52516 L .39956 .52418 L .39958 .52363 L .3996 .52314 L .39968 .52222 L .39979 .52123 L .40006 .51945 L .40051 .51741 L .40113 .51529 L .40258 .51164 L .40436 .5083 L .40628 .50541 L .41152 .49941 L .41734 .49451 L .42351 .49046 L .43075 .48672 L .43884 .48347 L .44593 .48126 L .45378 .47941 L .45776 .47869 L .4619 .47808 L Mistroke .46538 .47768 L .46912 .47737 L .4712 .47725 L .47307 .47717 L .47409 .47714 L .47467 .47712 L .47519 .47711 L .47616 .4771 L .47664 .4771 L .47716 .4771 L .4781 .4771 L .47896 .47711 L .47942 .47712 L .47993 .47713 L .48085 .47716 L .48259 .47723 L .48417 .47732 L .48721 .47756 L .49036 .47791 L .4932 .47832 L .49569 .47875 L .50101 .47995 L .50592 .48147 L .50666 .48175 L Mfstroke .98549 0 m 1 .00971 L s 1 .59208 m .99082 .59678 L .92778 .62135 L .8668 .63903 L .8108 .65081 L .78376 .6551 L .75521 .65868 L .74153 .66005 L .72872 .66114 L .71742 .66194 L .70509 .66264 L .69169 .6632 L .68443 .66341 L .67768 .66355 L .67142 .66364 L .66574 .66367 L .65923 .66366 L .65309 .6636 L .64654 .66349 L .64291 .66341 L .63963 .66332 L .62736 .66285 L .62058 .66251 L .61332 .66208 L .60038 .66114 L .57781 .65894 L .55763 .65634 L .5348 .65262 L .51513 .64867 L .47846 .63922 L .44449 .62738 L .41845 .61549 L .39841 .60385 L .38077 .59066 L .37288 .58327 L .36678 .5765 L .3583 .56422 L .35518 .55789 L .35392 .55471 L .35282 .55134 L .35205 .54839 L .3515 .54575 L .35129 .54448 L .35111 .54311 L .35097 .54184 L .35087 .54066 L .35079 .53929 L .35075 .53779 L .35076 .53635 L .35078 .53554 L Mistroke .35082 .53481 L .35091 .53351 L .35104 .5321 L .35121 .53079 L .3514 .52958 L .352 .52663 L .35272 .52397 L .35487 .51818 L .35734 .5133 L .36049 .5084 L .36713 .50053 L .37586 .49285 L .38629 .48587 L .39663 .48049 L .4092 .47541 L .42144 .47169 L .43311 .46906 L .43895 .46806 L .4453 .4672 L .45062 .46665 L .45357 .46642 L .45624 .46625 L .45857 .46614 L .45989 .46608 L .46108 .46605 L .46234 .46601 L .46367 .46599 L .4649 .46598 L .46605 .46598 L .46728 .46598 L .4686 .466 L .46983 .46602 L .47094 .46605 L .47224 .46609 L .47364 .46615 L .47614 .46629 L .47871 .46647 L .48138 .4667 L .48601 .46722 L .49368 .46845 L .50133 .47017 L .50799 .47219 L .51332 .47426 L .5183 .47671 L .5226 .47943 L Mfstroke 0 .28529 m .04417 .27684 L .09163 .26958 L .13573 .26446 L .1566 .26257 L .17537 .26117 L .19494 .26001 L .2132 .25921 L .22313 .25888 L .23222 .25865 L .24213 .25848 L .2471 .25843 L .25254 .25839 L .26159 .25838 L .2698 .25843 L .27921 .25855 L .28806 .25874 L .29615 .25897 L .30359 .25923 L .32024 .25999 L .33953 .26118 L .35728 .26257 L .3894 .26585 L .42245 .27031 L .45624 .27612 L .50965 .28823 L .56103 .30404 L .60659 .32269 L .64166 .34145 L .67056 .36162 L .69139 .381 L .705 .39833 L .71068 .40798 L .71471 .41677 L .71762 .42542 L .71878 .4302 L .71922 .43245 L .71956 .43453 L .71984 .43661 L .72004 .43849 L .7202 .44056 L .7203 .44275 L .72033 .44466 L .72031 .4467 L .72023 .44861 L .72012 .45036 L .71992 .45238 L .71965 .45453 L .719 .45835 L .71795 .46281 L Mistroke .71674 .46679 L .71334 .47516 L .70921 .48271 L .69936 .49578 L .68636 .50812 L .6736 .51734 L .65817 .52612 L .64102 .53372 L .62473 .53933 L .60663 .54406 L .59755 .54589 L .58771 .54751 L .57854 .54867 L .5741 .54912 L .57004 .54946 L .56646 .54971 L .56256 .54993 L .55869 .55009 L .55659 .55015 L .55464 .55019 L .55256 .55022 L .5514 .55023 L .55032 .55023 L .54836 .55022 L .54727 .55021 L .54629 .5502 L .54444 .55016 L .54245 .55011 L .53891 .54997 L .53481 .54974 L .53058 .54943 L .52327 .54871 L .5166 .54782 L .51081 .54686 L .49882 .54424 L .48902 .54133 L .48092 .53824 L .47344 .53462 L .46693 .53058 L .46252 .52703 L .45882 .52311 L .45737 .52112 L .45616 .51905 L .45537 .51731 L .45474 .51544 L .4545 .5144 L .45434 .51347 L .45428 .51295 L .45425 .51267 L .45423 .5124 L Mistroke .4542 .51192 L .4542 .51168 L .4542 .51142 L .45421 .51095 L .45423 .51052 L .45424 .51029 L .45427 .51003 L .45432 .50958 L .45447 .50871 L .45464 .50792 L .45513 .50639 L .45582 .50482 L .45663 .5034 L .4575 .50216 L .45991 .49949 L .46295 .49704 L .46349 .49667 L Mfstroke 0 .37863 m .03487 .3656 L .09801 .34747 L .13364 .33945 L .16718 .33325 L .19783 .32867 L .23064 .32491 L .24491 .32363 L .2599 .32252 L .27256 .32177 L .28638 .32115 L .29408 .3209 L .29826 .32078 L .30213 .3207 L .30895 .32058 L .31634 .32052 L .32048 .32051 L .32437 .32052 L .33159 .32058 L .33561 .32064 L .3393 .32071 L .34757 .32092 L .35417 .32115 L .36029 .32141 L .37402 .32214 L .38713 .32306 L .40091 .32426 L .42689 .32719 L .44982 .33056 L .49751 .34023 L .53447 .35073 L .56844 .36346 L .59678 .37729 L .62135 .39305 L .63903 .4083 L .65081 .4223 L .6551 .42906 L .65868 .4362 L .66005 .43962 L .66114 .44282 L .66194 .44565 L .66264 .44873 L .6632 .45208 L .66341 .45389 L .66355 .45558 L .66364 .45714 L .66367 .45856 L .66366 .46019 L .6636 .46173 L .66349 .46336 L Mistroke .66341 .46427 L .66332 .46509 L .66285 .46816 L .66251 .46986 L .66208 .47167 L .66114 .4749 L .65894 .48055 L .65634 .48559 L .65262 .4913 L .64867 .49622 L .63922 .50539 L .62738 .51388 L .61549 .52039 L .60385 .5254 L .59066 .52981 L .58327 .53178 L .5765 .53331 L .56422 .53542 L .55789 .53621 L .55471 .53652 L .55134 .5368 L .54839 .53699 L .54575 .53712 L .54448 .53718 L .54311 .53722 L .54184 .53726 L .54066 .53728 L .53929 .5373 L .53779 .53731 L .53635 .53731 L .53554 .5373 L .53481 .5373 L .53351 .53727 L .5321 .53724 L .53079 .5372 L .52958 .53715 L .52663 .537 L .52397 .53682 L .51818 .53628 L .5133 .53566 L .5084 .53488 L .50053 .53322 L .49285 .53104 L .48587 .52843 L .48049 .52584 L .47541 .5227 L .47169 .51964 L .46906 .51672 L .46806 .51526 L .4672 .51367 L Mistroke .46665 .51234 L .46642 .51161 L .46625 .51094 L .46614 .51036 L .46608 .51003 L .46605 .50973 L .46601 .50942 L .46599 .50908 L .46598 .50878 L .46598 .50849 L .46598 .50818 L .466 .50785 L .46602 .50754 L .46605 .50726 L .46609 .50694 L .46615 .50659 L .46629 .50596 L .46647 .50532 L .4667 .50466 L .46722 .5035 L .46845 .50158 L .47017 .49967 L .47219 .498 L .47426 .49667 L .47671 .49543 L .47943 .49435 L Mfstroke 0 .36444 m .04463 .34946 L .11285 .33259 L .15111 .32544 L .18601 .32027 L .21608 .31682 L .23313 .31528 L .24856 .31414 L .26463 .31322 L .27281 .31285 L .28167 .31252 L .28977 .3123 L .29723 .31215 L .30431 .31207 L .3118 .31204 L .31914 .31206 L .32328 .3121 L .32712 .31216 L .33116 .31223 L .33483 .31232 L .34301 .31255 L .34992 .31281 L .35735 .31315 L .37068 .31391 L .38594 .31504 L .40176 .31651 L .42913 .31979 L .45321 .32352 L .47468 .32758 L .51941 .33855 L .55564 .35055 L .58812 .36457 L .61131 .37734 L .632 .3918 L .64889 .40744 L .65991 .42157 L .66434 .42929 L .66622 .43331 L .66788 .43752 L .66906 .44116 L .67007 .44511 L .67042 .44683 L .67073 .44866 L .67097 .45036 L .67115 .45194 L .67123 .45283 L .67129 .45378 L .67137 .45544 L .67139 .45637 L .6714 .45735 L Mistroke .67136 .4591 L .67128 .4607 L .67116 .46218 L .67098 .4638 L .67074 .46551 L .66999 .46931 L .66903 .47283 L .66664 .47919 L .66369 .48497 L .66041 .49006 L .65132 .50065 L .64118 .50925 L .6305 .51627 L .6178 .52282 L .60341 .52853 L .58945 .53271 L .58205 .53447 L .57419 .536 L .56673 .53717 L .56006 .53797 L .55688 .53828 L .55348 .53855 L .55028 .53876 L .5474 .5389 L .54585 .53896 L .54439 .539 L .54286 .53904 L .542 .53905 L .54121 .53906 L .5399 .53907 L .53846 .53907 L .53697 .53906 L .53558 .53904 L .53405 .53901 L .53245 .53896 L .52962 .53884 L .52664 .53867 L .52345 .53843 L .5177 .53784 L .51185 .53704 L .50216 .53517 L .49353 .53283 L .48549 .52989 L .47917 .52684 L .47441 .52388 L .47063 .52083 L .46796 .51797 L .46678 .51634 L .46592 .51486 L .46553 .51406 L Mistroke .46517 .51319 L .4649 .51237 L .4647 .51163 L .46454 .51088 L .46448 .51048 L .46443 .51007 L .4644 .50972 L .46438 .50939 L .46437 .50903 L .46437 .50884 L .46437 .50865 L .46439 .50832 L .46441 .50802 L .46444 .50768 L .46448 .50733 L .46458 .50675 L .46473 .50613 L .46493 .50542 L .46516 .50479 L .4658 .50343 L .4665 .50228 L .46736 .50114 L .46924 .49922 L .47158 .49743 L .47403 .49598 L .47641 .49486 L .47924 .49382 L .48189 .49308 L .4848 .49248 L .48625 .49226 L .48781 .49208 L .48886 .49199 L .48937 .49195 L .48984 .49193 L .49025 .49191 L .4907 .49189 L .49093 .49189 L .49115 .49188 L .49138 .49188 L .49162 .49188 L .49187 .49188 L .4921 .49188 L .49235 .49188 L .49261 .49188 L .49289 .49189 L .49315 .4919 L .49363 .49192 L .49407 .49194 L .49447 .49196 L .49536 .49203 L Mistroke Mfstroke 0 .10917 m .01058 .10862 L .02759 .10788 L .04331 .10733 L .06048 .10686 L .06951 .10667 L .07912 .10651 L .08918 .1064 L .09863 .10634 L .10708 .10633 L .11621 .10635 L .13155 .10649 L .1458 .10673 L .16131 .1071 L .17782 .10764 L .21097 .10917 L .24067 .11104 L .29612 .11589 L .34659 .12189 L .44251 .13796 L .49182 .14892 L .53376 .1599 L .60402 .18237 L .66839 .20866 L .72579 .23866 L .76881 .26744 L .80506 .299 L .8211 .31682 L .8335 .33359 L .84265 .34891 L .85019 .36532 L .85274 .37245 L .85496 .37995 L .85646 .38628 L .8577 .39319 L .85821 .39704 L .85843 .39913 L .85861 .40106 L .85883 .40447 L .85896 .40817 L .85898 .41024 L .85896 .41218 L .85884 .41579 L .85873 .41781 L .85858 .41965 L .85816 .42378 L .8577 .42708 L .85719 .43014 L .85571 .43701 L .85388 .44357 L Mistroke .85149 .45046 L .84563 .46344 L .83889 .47491 L .81954 .49876 L .79854 .51724 L .77308 .53422 L .74541 .54839 L .71389 .56067 L .6834 .56951 L .6554 .5754 L .64188 .57755 L .6276 .57934 L .62077 .58003 L .61436 .58057 L .60871 .58097 L .60254 .58132 L .59584 .5816 L .59221 .58171 L .58884 .58178 L .58571 .58182 L .58287 .58184 L .57961 .58183 L .57654 .5818 L .57327 .58175 L .57146 .5817 L .56982 .58166 L .56368 .58143 L .56029 .58126 L .55666 .58104 L .55019 .58057 L .5389 .57947 L .52881 .57817 L .5174 .57631 L .50756 .57434 L .48923 .56961 L .47224 .56369 L .45923 .55775 L .44921 .55192 L .44038 .54533 L .43644 .54163 L .43339 .53825 L .42915 .53211 L .42759 .52894 L .42696 .52735 L .42641 .52567 L .42602 .5242 L .42575 .52287 L .42565 .52224 L .42555 .52156 L .42548 .52092 L Mistroke .42543 .52033 L .4254 .51964 L .42538 .5189 L .42538 .51817 L .42539 .51777 L .42541 .5174 L .42545 .51675 L .42552 .51605 L .42561 .51539 L .4257 .51479 L .426 .51332 L .42636 .51199 L .42743 .50909 L .42867 .50665 L .43025 .5042 L .43357 .50027 L .43793 .49643 L .44314 .49294 L .44831 .49025 L .4546 .48771 L .46072 .48584 L .46655 .48453 L .46947 .48403 L .47265 .4836 L .47531 .48333 L .47678 .48321 L .47812 .48313 L .47928 .48307 L .47994 .48304 L .48054 .48302 L .48117 .48301 L .48184 .483 L .48245 .48299 L .48302 .48299 L .48364 .48299 L .4843 .483 L .48491 .48301 L .48547 .48302 L .48612 .48305 L .48682 .48308 L .48807 .48314 L .48935 .48323 L .49069 .48335 L .49301 .48361 L .49684 .48422 L .50067 .48509 L .504 .4861 L .50666 .48713 L .50915 .48836 L .5113 .48971 L Mistroke Mfstroke .89121 0 m .95883 .03947 L s .95883 .03947 m 1 .07101 L s 1 .56686 m .98371 .5764 L .93817 .59742 L .88694 .61604 L .83589 .63024 L .77992 .6415 L .75215 .64555 L .73703 .64733 L .72283 .64875 L .70979 .64982 L .69575 .65074 L .68962 .65106 L .68315 .65135 L .67716 .65157 L .67162 .65173 L .66551 .65186 L .6623 .65191 L .65887 .65195 L .65529 .65197 L .65192 .65198 L .64565 .65195 L .63983 .65188 L .63449 .65178 L .62868 .65162 L .62258 .6514 L .61015 .6508 L .59874 .65006 L .57363 .64774 L .55287 .64508 L .53477 .64216 L .49728 .63407 L .4663 .62489 L .44131 .61529 L .41806 .6039 L .398 .59111 L .38314 .57851 L .37698 .57186 L .37167 .56487 L .36798 .55881 L .36491 .55226 L .36388 .54943 L .36299 .54642 L .36233 .54361 L .36189 .54109 L .36171 .53976 L .36159 .53853 L .36149 .53718 L .36143 .53574 L .36141 .53432 L .36143 .53361 L Mistroke .36145 .53284 L .36153 .53148 L .36164 .53022 L .36179 .52896 L .36197 .52778 L .36238 .52559 L .36295 .52327 L .3637 .52081 L .36576 .51574 L .36857 .51063 L .37503 .50218 L .38259 .49506 L .39193 .48839 L .40162 .48303 L .41099 .47892 L .42191 .47515 L .43217 .47246 L .44336 .47034 L .44951 .46951 L .45213 .46923 L .45498 .46897 L .45748 .46878 L .46013 .46862 L .46158 .46855 L .46289 .4685 L .4642 .46846 L .46543 .46843 L .46656 .46842 L .46774 .46841 L .46875 .46841 L .46985 .46841 L .47103 .46843 L .47228 .46845 L .47345 .46849 L .47452 .46853 L .47689 .46864 L .47906 .46878 L .48382 .46919 L .48825 .46973 L .49274 .47044 L .5004 .47209 L .50763 .47425 L .51344 .47658 L .5179 .4789 L .52087 .48082 L .52349 .48292 L .52573 .48525 L .52723 .4874 L Mfstroke 0 .39294 m .00936 .38869 L .06722 .368 L .12975 .35111 L .18629 .34009 L .21717 .33566 L .23323 .33378 L .25006 .33212 L .26462 .33094 L .28045 .32993 L .28732 .32958 L .29465 .32927 L .30143 .32903 L .30776 .32885 L .31131 .32877 L .31512 .32871 L .32174 .32863 L .32548 .32861 L .3294 .3286 L .33638 .32864 L .3428 .32872 L .34874 .32884 L .35519 .32902 L .36204 .32926 L .37725 .33001 L .39131 .33097 L .41677 .33336 L .43989 .33631 L .46025 .33959 L .50261 .34868 L .53702 .35882 L .5651 .3695 L .59128 .3822 L .61413 .39659 L .63086 .41055 L .63787 .41795 L .64402 .42581 L .64867 .43327 L .65189 .43994 L .65312 .44312 L .65421 .44652 L .65503 .44972 L .65559 .4526 L .65583 .45415 L .65601 .45561 L .65615 .45714 L .65621 .458 L .65625 .45879 L .65629 .4601 L .65629 .46154 L Mistroke .65625 .46303 L .65617 .46442 L .65603 .46595 L .65584 .46755 L .65536 .47038 L .65467 .47336 L .65371 .47655 L .65137 .4823 L .64815 .48815 L .64067 .49784 L .63131 .50647 L .61956 .51451 L .60737 .52083 L .59552 .52559 L .58333 .52937 L .57186 .53204 L .56534 .53322 L .55945 .53408 L .55622 .53447 L .55274 .53483 L .54948 .5351 L .54651 .5353 L .54351 .53546 L .54194 .53552 L .5403 .53557 L .53888 .5356 L .53757 .53562 L .53614 .53563 L .53534 .53563 L .53462 .53563 L .53329 .53561 L .53207 .53559 L .53073 .53556 L .52934 .53552 L .52702 .53542 L .5245 .53527 L .52169 .53507 L .51918 .53484 L .51371 .5342 L .50913 .5335 L .50457 .53264 L .49688 .53076 L .4897 .52842 L .4839 .52597 L .47945 .52359 L .47529 .52076 L .47231 .51811 L .46991 .5152 L .46903 .51375 L .46831 .51219 L Mistroke .46795 .51114 L .46781 .51063 L .46771 .51016 L .46764 .50975 L .46758 .5093 L .46755 .50907 L .46754 .50885 L .46752 .50862 L .46751 .50838 L .46751 .50813 L .46751 .5079 L .46752 .50765 L .46754 .50739 L .46756 .50711 L .46759 .50685 L .46766 .50637 L .46775 .50593 L .46785 .50553 L .46813 .50464 L Mfstroke 1 .91039 m .98965 .91082 L .9726 .91138 L .95685 .91178 L .94032 .91207 L .92385 .91222 L .91455 .91226 L .9059 .91225 L .89035 .91214 L .87626 .91194 L .86013 .91159 L .84497 .91114 L .81563 .90995 L .78853 .90845 L .72851 .90374 L .67251 .89751 L .61497 .88914 L .51902 .87015 L .4403 .849 L .36357 .82219 L .30232 .79483 L .24628 .76297 L .19892 .72769 L .16518 .69357 L .15087 .67444 L .14053 .65699 L .13365 .64196 L .13056 .63343 L .12828 .62572 L .12643 .61769 L .12569 .6136 L .12504 .60917 L .1246 .60511 L .12431 .60138 L .12414 .59784 L .12407 .5941 L .12413 .59043 L .12421 .58836 L .12432 .58644 L .12447 .58442 L .12463 .58258 L .12511 .57849 L .12562 .57504 L .1263 .57132 L .12783 .56466 L .13008 .55703 L .13302 .54912 L .13958 .53544 L .14705 .5234 L .15515 .51266 L Mistroke .17711 .49029 L .20111 .47218 L .22915 .45594 L .25468 .44435 L .28361 .434 L .31487 .42555 L .34314 .42005 L .35858 .41783 L .36662 .41689 L .37503 .41606 L .38231 .41547 L .39022 .41497 L .39366 .41479 L .39732 .41463 L .40071 .41451 L .40388 .41443 L .40565 .41439 L .40756 .41435 L .41087 .41431 L .41274 .4143 L .4147 .4143 L .41819 .41432 L .4214 .41436 L .42437 .41442 L .4276 .41451 L .43102 .41463 L .43863 .41501 L .44566 .41548 L .45838 .41668 L .46995 .41816 L .48012 .41979 L .5013 .42434 L .51851 .42941 L .53255 .43475 L .54564 .4411 L .55707 .4483 L .56543 .45527 L .56894 .45897 L .57201 .4629 L .57434 .46663 L .57595 .46997 L .57656 .47156 L .57711 .47326 L .57751 .47486 L .5778 .4763 L .57791 .47707 L .578 .4778 L .57807 .47857 L .5781 .479 L .57812 .4794 L Mistroke .57814 .48005 L .57815 .48077 L .57812 .48152 L .57808 .48221 L .57802 .48298 L .57792 .48378 L .57768 .48519 L .57734 .48668 L .57685 .48828 L .57568 .49115 L .57407 .49407 L .57033 .49892 L .56565 .50323 L .55978 .50725 L .55369 .51042 L .54776 .5128 L .54166 .51468 L .53593 .51602 L .53267 .51661 L .52972 .51704 L .52811 .51723 L .52637 .51741 L .52474 .51755 L .52326 .51765 L .52175 .51773 L .52097 .51776 L .52015 .51778 L .51944 .5178 L .51878 .51781 L .51807 .51781 L .51767 .51782 L .51731 .51781 L .51664 .51781 L .51603 .5178 L .51537 .51778 L .51467 .51776 L .51351 .51771 L .51225 .51764 L .51085 .51753 L .50959 .51742 L .50686 .5171 L .50457 .51675 L .50228 .51632 L .49844 .51538 L .49485 .51421 L .49195 .51298 L .48973 .51179 L .48765 .51038 L .48615 .50905 L .48495 .5076 L Mistroke .48452 .50688 L .48415 .5061 L .48397 .50557 L .48391 .50532 L .48386 .50508 L .48382 .50487 L .48379 .50465 L .48378 .50453 L .48377 .50442 L .48376 .50431 L .48376 .50419 L .48375 .50406 L .48376 .50395 L .48376 .50383 L .48377 .50369 L .48378 .50356 L .48379 .50343 L .48383 .50319 L .48388 .50297 L .48392 .50276 L .48407 .50232 L Mfstroke .5 .5 m .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L s 0 .08961 m .01035 .08918 L .0274 .08862 L .04315 .08822 L .05968 .08793 L .07615 .08778 L .08545 .08774 L .0941 .08775 L .10965 .08786 L .12374 .08806 L .13987 .08841 L .15503 .08886 L .18437 .09005 L .21147 .09155 L .27149 .09626 L .32749 .10249 L .38503 .11086 L .48098 .12985 L .5597 .151 L .63643 .17781 L .69768 .20517 L .75372 .23703 L .80108 .27231 L .83482 .30643 L .84913 .32556 L .85947 .34301 L .86635 .35804 L .86944 .36657 L .87172 .37428 L .87357 .38231 L .87431 .3864 L .87496 .39083 L .8754 .39489 L .87569 .39862 L .87586 .40216 L .87593 .4059 L .87587 .40957 L .87579 .41164 L .87568 .41356 L .87553 .41558 L .87537 .41742 L .87489 .42151 L .87438 .42496 L .8737 .42868 L .87217 .43534 L .86992 .44297 L .86698 .45088 L .86042 .46456 L .85295 .4766 L .84485 .48734 L Mistroke .82289 .50971 L .79889 .52782 L .77085 .54406 L .74532 .55565 L .71639 .566 L .68513 .57445 L .65686 .57995 L .64142 .58217 L .63338 .58311 L .62497 .58394 L .61769 .58453 L .60978 .58503 L .60634 .58521 L .60268 .58537 L .59929 .58549 L .59612 .58557 L .59435 .58561 L .59244 .58565 L .58913 .58569 L .58726 .5857 L .5853 .5857 L .58181 .58568 L .5786 .58564 L .57563 .58558 L .5724 .58549 L .56898 .58537 L .56137 .58499 L .55434 .58452 L .54162 .58332 L .53005 .58184 L .51988 .58021 L .4987 .57566 L .48149 .57059 L .46745 .56525 L .45436 .5589 L .44293 .5517 L .43457 .54473 L .43106 .54103 L .42799 .5371 L .42566 .53337 L .42405 .53003 L .42344 .52844 L .42289 .52674 L .42249 .52514 L .4222 .5237 L .42209 .52293 L .422 .5222 L .42193 .52143 L .4219 .521 L .42188 .5206 L Mistroke .42186 .51995 L .42185 .51923 L .42188 .51848 L .42192 .51779 L .42198 .51702 L .42208 .51622 L .42232 .51481 L .42266 .51332 L .42315 .51172 L .42432 .50885 L .42593 .50593 L .42967 .50108 L .43435 .49677 L .44022 .49275 L .44631 .48958 L .45224 .4872 L .45834 .48532 L .46407 .48398 L .46733 .48339 L .47028 .48296 L .47189 .48277 L .47363 .48259 L .47526 .48245 L .47674 .48235 L .47825 .48227 L .47903 .48224 L .47985 .48222 L .48056 .4822 L .48122 .48219 L .48193 .48219 L .48233 .48218 L .48269 .48219 L .48336 .48219 L .48397 .4822 L .48463 .48222 L .48533 .48224 L .48649 .48229 L .48775 .48236 L .48915 .48247 L .49041 .48258 L .49314 .4829 L .49543 .48325 L .49772 .48368 L .50156 .48462 L .50515 .48579 L .50805 .48702 L .51027 .48821 L .51235 .48962 L .51385 .49095 L .51505 .4924 L Mistroke .51548 .49312 L .51585 .4939 L .51603 .49443 L .51609 .49468 L .51614 .49492 L .51618 .49513 L .51621 .49535 L .51622 .49547 L .51623 .49558 L .51624 .49569 L .51624 .49581 L .51625 .49594 L .51624 .49605 L .51624 .49617 L .51623 .49631 L .51622 .49644 L .51621 .49657 L .51617 .49681 L .51612 .49703 L .51608 .49724 L .51593 .49768 L Mfstroke 1 .60706 m .99064 .61131 L .93278 .632 L .87025 .64889 L .81371 .65991 L .78283 .66434 L .76677 .66622 L .74994 .66788 L .73538 .66906 L .71955 .67007 L .71268 .67042 L .70535 .67073 L .69857 .67097 L .69224 .67115 L .68869 .67123 L .68488 .67129 L .67826 .67137 L .67452 .67139 L .6706 .6714 L .66362 .67136 L .6572 .67128 L .65126 .67116 L .64481 .67098 L .63796 .67074 L .62275 .66999 L .60869 .66903 L .58323 .66664 L .56011 .66369 L .53975 .66041 L .49739 .65132 L .46298 .64118 L .4349 .6305 L .40872 .6178 L .38587 .60341 L .36914 .58945 L .36213 .58205 L .35598 .57419 L .35133 .56673 L .34811 .56006 L .34688 .55688 L .34579 .55348 L .34497 .55028 L .34441 .5474 L .34417 .54585 L .34399 .54439 L .34385 .54286 L .34379 .542 L .34375 .54121 L .34371 .5399 L .34371 .53846 L Mistroke .34375 .53697 L .34383 .53558 L .34397 .53405 L .34416 .53245 L .34464 .52962 L .34533 .52664 L .34629 .52345 L .34863 .5177 L .35185 .51185 L .35933 .50216 L .36869 .49353 L .38044 .48549 L .39263 .47917 L .40448 .47441 L .41667 .47063 L .42814 .46796 L .43466 .46678 L .44055 .46592 L .44378 .46553 L .44726 .46517 L .45052 .4649 L .45349 .4647 L .45649 .46454 L .45806 .46448 L .4597 .46443 L .46112 .4644 L .46243 .46438 L .46386 .46437 L .46466 .46437 L .46538 .46437 L .46671 .46439 L .46793 .46441 L .46927 .46444 L .47066 .46448 L .47298 .46458 L .4755 .46473 L .47831 .46493 L .48082 .46516 L .48629 .4658 L .49087 .4665 L .49543 .46736 L .50312 .46924 L .5103 .47158 L .5161 .47403 L .52055 .47641 L .52471 .47924 L .52769 .48189 L .53009 .4848 L .53097 .48625 L .53169 .48781 L Mistroke .53205 .48886 L .53219 .48937 L .53229 .48984 L .53236 .49025 L .53242 .4907 L .53245 .49093 L .53246 .49115 L .53248 .49138 L .53249 .49162 L .53249 .49187 L .53249 .4921 L .53248 .49235 L .53246 .49261 L .53244 .49289 L .53241 .49315 L .53234 .49363 L .53225 .49407 L .53215 .49447 L .53187 .49536 L Mfstroke .10879 1 m .04117 .96053 L s .04117 .96053 m 0 .92899 L s 0 .43314 m .01629 .4236 L .06183 .40258 L .11306 .38396 L .16411 .36976 L .22008 .3585 L .24785 .35445 L .26297 .35267 L .27717 .35125 L .29021 .35018 L .30425 .34926 L .31038 .34894 L .31685 .34865 L .32284 .34843 L .32838 .34827 L .33449 .34814 L .3377 .34809 L .34113 .34805 L .34471 .34803 L .34808 .34802 L .35435 .34805 L .36017 .34812 L .36551 .34822 L .37132 .34838 L .37742 .3486 L .38985 .3492 L .40126 .34994 L .42637 .35226 L .44713 .35492 L .46523 .35784 L .50272 .36593 L .5337 .37511 L .55869 .38471 L .58194 .3961 L .602 .40889 L .61686 .42149 L .62302 .42814 L .62833 .43513 L .63202 .44119 L .63509 .44774 L .63612 .45057 L .63701 .45358 L .63767 .45639 L .63811 .45891 L .63829 .46024 L .63841 .46147 L .63851 .46282 L .63857 .46426 L .63859 .46568 L .63857 .46639 L Mistroke .63855 .46716 L .63847 .46852 L .63836 .46978 L .63821 .47104 L .63803 .47222 L .63762 .47441 L .63705 .47673 L .6363 .47919 L .63424 .48426 L .63143 .48937 L .62497 .49782 L .61741 .50494 L .60807 .51161 L .59838 .51697 L .58901 .52108 L .57809 .52485 L .56783 .52754 L .55664 .52966 L .55049 .53049 L .54787 .53077 L .54502 .53103 L .54252 .53122 L .53987 .53138 L .53842 .53145 L .53711 .5315 L .5358 .53154 L .53457 .53157 L .53344 .53158 L .53226 .53159 L .53125 .53159 L .53015 .53159 L .52897 .53157 L .52772 .53155 L .52655 .53151 L .52548 .53147 L .52311 .53136 L .52094 .53122 L .51618 .53081 L .51175 .53027 L .50726 .52956 L .4996 .52791 L .49237 .52575 L .48656 .52342 L .4821 .5211 L .47913 .51918 L .47651 .51708 L .47427 .51475 L .47277 .5126 L Mfstroke 1 .89083 m .98942 .89138 L .97241 .89212 L .95669 .89267 L .93952 .89314 L .93049 .89333 L .92088 .89349 L .91082 .8936 L .90137 .89366 L .89292 .89367 L .88379 .89365 L .86845 .89351 L .8542 .89327 L .83869 .8929 L .82218 .89236 L .78903 .89083 L .75933 .88896 L .70388 .88411 L .65341 .87811 L .55749 .86204 L .50818 .85108 L .46624 .8401 L .39598 .81763 L .33161 .79134 L .27421 .76134 L .23119 .73256 L .19494 .701 L .1789 .68318 L .1665 .66641 L .15735 .65109 L .14981 .63468 L .14726 .62755 L .14504 .62005 L .14354 .61372 L .1423 .60681 L .14179 .60296 L .14157 .60087 L .14139 .59894 L .14117 .59553 L .14104 .59183 L .14102 .58976 L .14104 .58782 L .14116 .58421 L .14127 .58219 L .14142 .58035 L .14184 .57622 L .1423 .57292 L .14281 .56986 L .14429 .56299 L .14612 .55643 L Mistroke .14851 .54954 L .15437 .53656 L .16111 .52509 L .18046 .50124 L .20146 .48276 L .22692 .46578 L .25459 .45161 L .28611 .43933 L .3166 .43049 L .3446 .4246 L .35812 .42245 L .3724 .42066 L .37923 .41997 L .38564 .41943 L .39129 .41903 L .39746 .41868 L .40416 .4184 L .40779 .41829 L .41116 .41822 L .41429 .41818 L .41713 .41816 L .42039 .41817 L .42346 .4182 L .42673 .41825 L .42854 .4183 L .43018 .41834 L .43632 .41857 L .43971 .41874 L .44334 .41896 L .44981 .41943 L .4611 .42053 L .47119 .42183 L .4826 .42369 L .49244 .42566 L .51077 .43039 L .52776 .43631 L .54077 .44225 L .55079 .44808 L .55962 .45467 L .56356 .45837 L .56661 .46175 L .57085 .46789 L .57241 .47106 L .57304 .47265 L .57359 .47433 L .57398 .4758 L .57425 .47713 L .57435 .47776 L .57445 .47844 L .57452 .47908 L Mistroke .57457 .47967 L .5746 .48036 L .57462 .4811 L .57462 .48183 L .57461 .48223 L .57459 .4826 L .57455 .48325 L .57448 .48395 L .57439 .48461 L .5743 .48521 L .574 .48668 L .57364 .48801 L .57257 .49091 L .57133 .49335 L .56975 .4958 L .56643 .49973 L .56207 .50357 L .55686 .50706 L .55169 .50975 L .5454 .51229 L .53928 .51416 L .53345 .51547 L .53053 .51597 L .52735 .5164 L .52469 .51667 L .52322 .51679 L .52188 .51687 L .52072 .51693 L .52006 .51696 L .51946 .51698 L .51883 .51699 L .51816 .517 L .51755 .51701 L .51698 .51701 L .51636 .51701 L .5157 .517 L .51509 .51699 L .51453 .51698 L .51388 .51695 L .51318 .51692 L .51193 .51686 L .51065 .51677 L .50931 .51665 L .50699 .51639 L .50316 .51578 L .49933 .51491 L .496 .5139 L .49334 .51287 L .49085 .51164 L .4887 .51029 L Mistroke Mfstroke 1 .63556 m .95537 .65054 L .88715 .66741 L .84889 .67456 L .81399 .67973 L .78392 .68318 L .76687 .68472 L .75144 .68586 L .73537 .68678 L .72719 .68715 L .71833 .68748 L .71023 .6877 L .70277 .68785 L .69569 .68793 L .6882 .68796 L .68086 .68794 L .67672 .6879 L .67288 .68784 L .66884 .68777 L .66517 .68768 L .65699 .68745 L .65008 .68719 L .64265 .68685 L .62932 .68609 L .61406 .68496 L .59824 .68349 L .57087 .68021 L .54679 .67648 L .52532 .67242 L .48059 .66145 L .44436 .64945 L .41188 .63543 L .38869 .62266 L .368 .6082 L .35111 .59256 L .34009 .57843 L .33566 .57071 L .33378 .56669 L .33212 .56248 L .33094 .55884 L .32993 .55489 L .32958 .55317 L .32927 .55134 L .32903 .54964 L .32885 .54806 L .32877 .54717 L .32871 .54622 L .32863 .54456 L .32861 .54363 L .3286 .54265 L Mistroke .32864 .5409 L .32872 .5393 L .32884 .53782 L .32902 .5362 L .32926 .53449 L .33001 .53069 L .33097 .52717 L .33336 .52081 L .33631 .51503 L .33959 .50994 L .34868 .49935 L .35882 .49075 L .3695 .48373 L .3822 .47718 L .39659 .47147 L .41055 .46729 L .41795 .46553 L .42581 .464 L .43327 .46283 L .43994 .46203 L .44312 .46172 L .44652 .46145 L .44972 .46124 L .4526 .4611 L .45415 .46104 L .45561 .461 L .45714 .46096 L .458 .46095 L .45879 .46094 L .4601 .46093 L .46154 .46093 L .46303 .46094 L .46442 .46096 L .46595 .46099 L .46755 .46104 L .47038 .46116 L .47336 .46133 L .47655 .46157 L .4823 .46216 L .48815 .46296 L .49784 .46483 L .50647 .46717 L .51451 .47011 L .52083 .47316 L .52559 .47612 L .52937 .47917 L .53204 .48203 L .53322 .48366 L .53408 .48514 L .53447 .48594 L Mistroke .53483 .48681 L .5351 .48763 L .5353 .48837 L .53546 .48912 L .53552 .48952 L .53557 .48993 L .5356 .49028 L .53562 .49061 L .53563 .49097 L .53563 .49116 L .53563 .49135 L .53561 .49168 L .53559 .49198 L .53556 .49232 L .53552 .49267 L .53542 .49325 L .53527 .49387 L .53507 .49458 L .53484 .49521 L .5342 .49657 L .5335 .49772 L .53264 .49886 L .53076 .50078 L .52842 .50257 L .52597 .50402 L .52359 .50514 L .52076 .50618 L .51811 .50692 L .5152 .50752 L .51375 .50774 L .51219 .50792 L .51114 .50801 L .51063 .50805 L .51016 .50807 L .50975 .50809 L .5093 .50811 L .50907 .50811 L .50885 .50812 L .50862 .50812 L .50838 .50812 L .50813 .50812 L .5079 .50812 L .50765 .50812 L .50739 .50812 L .50711 .50811 L .50685 .5081 L .50637 .50808 L .50593 .50806 L .50553 .50804 L .50464 .50797 L Mistroke Mfstroke 1 .62137 m .96513 .6344 L .90199 .65253 L .86636 .66055 L .83282 .66675 L .80217 .67133 L .76936 .67509 L .75509 .67637 L .7401 .67748 L .72744 .67823 L .71362 .67885 L .70592 .6791 L .70174 .67922 L .69787 .6793 L .69105 .67942 L .68366 .67948 L .67952 .67949 L .67563 .67948 L .66841 .67942 L .66439 .67936 L .6607 .67929 L .65243 .67908 L .64583 .67885 L .63971 .67859 L .62598 .67786 L .61287 .67694 L .59909 .67574 L .57311 .67281 L .55018 .66944 L .50249 .65977 L .46553 .64927 L .43156 .63654 L .40322 .62271 L .37865 .60695 L .36097 .5917 L .34919 .5777 L .3449 .57094 L .34132 .5638 L .33995 .56038 L .33886 .55718 L .33806 .55435 L .33736 .55127 L .3368 .54792 L .33659 .54611 L .33645 .54442 L .33636 .54286 L .33633 .54144 L .33634 .53981 L .3364 .53827 L .33651 .53664 L Mistroke .33659 .53573 L .33668 .53491 L .33715 .53184 L .33749 .53014 L .33792 .52833 L .33886 .5251 L .34106 .51945 L .34366 .51441 L .34738 .5087 L .35133 .50378 L .36078 .49461 L .37262 .48612 L .38451 .47961 L .39615 .4746 L .40934 .47019 L .41673 .46822 L .4235 .46669 L .43578 .46458 L .44211 .46379 L .44529 .46348 L .44866 .4632 L .45161 .46301 L .45425 .46288 L .45552 .46282 L .45689 .46278 L .45816 .46274 L .45934 .46272 L .46071 .4627 L .46221 .46269 L .46365 .46269 L .46446 .4627 L .46519 .4627 L .46649 .46273 L .4679 .46276 L .46921 .4628 L .47042 .46285 L .47337 .463 L .47603 .46318 L .48182 .46372 L .4867 .46434 L .4916 .46512 L .49947 .46678 L .50715 .46896 L .51413 .47157 L .51951 .47416 L .52459 .4773 L .52831 .48036 L .53094 .48328 L .53194 .48474 L .5328 .48633 L Mistroke .53335 .48766 L .53358 .48839 L .53375 .48906 L .53386 .48964 L .53392 .48997 L .53395 .49027 L .53399 .49058 L .53401 .49092 L .53402 .49122 L .53402 .49151 L .53402 .49182 L .534 .49215 L .53398 .49246 L .53395 .49274 L .53391 .49306 L .53385 .49341 L .53371 .49404 L .53353 .49468 L .5333 .49534 L .53278 .4965 L .53155 .49842 L .52983 .50033 L .52781 .502 L .52574 .50333 L .52329 .50457 L .52057 .50565 L Mfstroke 1 .71471 m .95583 .72316 L .90837 .73042 L .86427 .73554 L .8434 .73743 L .82463 .73883 L .80506 .73999 L .7868 .74079 L .77687 .74112 L .76778 .74135 L .75787 .74152 L .7529 .74157 L .74746 .74161 L .73841 .74162 L .7302 .74157 L .72079 .74145 L .71194 .74126 L .70385 .74103 L .69641 .74077 L .67976 .74001 L .66047 .73882 L .64272 .73743 L .6106 .73415 L .57755 .72969 L .54376 .72388 L .49035 .71177 L .43897 .69596 L .39341 .67731 L .35834 .65855 L .32944 .63838 L .30861 .619 L .295 .60167 L .28932 .59202 L .28529 .58323 L .28238 .57458 L .28122 .5698 L .28078 .56755 L .28044 .56547 L .28016 .56339 L .27996 .56151 L .2798 .55944 L .2797 .55725 L .27967 .55534 L .27969 .5533 L .27977 .55139 L .27988 .54964 L .28008 .54762 L .28035 .54547 L .281 .54165 L .28205 .53719 L Mistroke .28326 .53321 L .28666 .52484 L .29079 .51729 L .30064 .50422 L .31364 .49188 L .3264 .48266 L .34183 .47388 L .35898 .46628 L .37527 .46067 L .39337 .45594 L .40245 .45411 L .41229 .45249 L .42146 .45133 L .4259 .45088 L .42996 .45054 L .43354 .45029 L .43744 .45007 L .44131 .44991 L .44341 .44985 L .44536 .44981 L .44744 .44978 L .4486 .44977 L .44968 .44977 L .45164 .44978 L .45273 .44979 L .45371 .4498 L .45556 .44984 L .45755 .44989 L .46109 .45003 L .46519 .45026 L .46942 .45057 L .47673 .45129 L .4834 .45218 L .48919 .45314 L .50118 .45576 L .51098 .45867 L .51908 .46176 L .52656 .46538 L .53307 .46942 L .53748 .47297 L .54118 .47689 L .54263 .47888 L .54384 .48095 L .54463 .48269 L .54526 .48456 L .5455 .4856 L .54566 .48653 L .54572 .48705 L .54575 .48733 L .54577 .4876 L Mistroke .5458 .48808 L .5458 .48832 L .5458 .48858 L .54579 .48905 L .54577 .48948 L .54576 .48971 L .54573 .48997 L .54568 .49042 L .54553 .49129 L .54536 .49208 L .54487 .49361 L .54418 .49518 L .54337 .4966 L .5425 .49784 L .54009 .50051 L .53705 .50296 L .53651 .50333 L Mfstroke .01451 1 m 0 .99029 L s 0 .40792 m .00918 .40322 L .07222 .37865 L .1332 .36097 L .1892 .34919 L .21624 .3449 L .24479 .34132 L .25847 .33995 L .27128 .33886 L .28258 .33806 L .29491 .33736 L .30831 .3368 L .31557 .33659 L .32232 .33645 L .32858 .33636 L .33426 .33633 L .34077 .33634 L .34691 .3364 L .35346 .33651 L .35709 .33659 L .36037 .33668 L .37264 .33715 L .37942 .33749 L .38668 .33792 L .39962 .33886 L .42219 .34106 L .44237 .34366 L .4652 .34738 L .48487 .35133 L .52154 .36078 L .55551 .37262 L .58155 .38451 L .60159 .39615 L .61923 .40934 L .62712 .41673 L .63322 .4235 L .6417 .43578 L .64482 .44211 L .64608 .44529 L .64718 .44866 L .64795 .45161 L .6485 .45425 L .64871 .45552 L .64889 .45689 L .64903 .45816 L .64913 .45934 L .64921 .46071 L .64925 .46221 L .64924 .46365 L .64922 .46446 L Mistroke .64918 .46519 L .64909 .46649 L .64896 .4679 L .64879 .46921 L .6486 .47042 L .648 .47337 L .64728 .47603 L .64513 .48182 L .64266 .4867 L .63951 .4916 L .63287 .49947 L .62414 .50715 L .61371 .51413 L .60337 .51951 L .5908 .52459 L .57856 .52831 L .56689 .53094 L .56105 .53194 L .5547 .5328 L .54938 .53335 L .54643 .53358 L .54376 .53375 L .54143 .53386 L .54011 .53392 L .53892 .53395 L .53766 .53399 L .53633 .53401 L .5351 .53402 L .53395 .53402 L .53272 .53402 L .5314 .534 L .53017 .53398 L .52906 .53395 L .52776 .53391 L .52636 .53385 L .52386 .53371 L .52129 .53353 L .51862 .5333 L .51399 .53278 L .50632 .53155 L .49867 .52983 L .49201 .52781 L .48668 .52574 L .4817 .52329 L .4774 .52057 L Mfstroke .64744 1 m .52934 .97696 L .40984 .94447 L .31199 .90906 L .23685 .87441 L .16862 .83463 L .11528 .79469 L .07071 .75018 L .05368 .72792 L .03915 .70419 L .02891 .68214 L .02515 .6717 L .02235 .66232 L .02003 .65253 L .01842 .6434 L .01777 .63844 L .01731 .63389 L .01696 .62893 L .01685 .62645 L .01678 .62373 L .01676 .6192 L .01686 .6151 L .01711 .6104 L .01748 .60597 L .01794 .60193 L .01846 .5982 L .01998 .58988 L .02235 .58024 L .02514 .57136 L .0317 .5553 L .04063 .53878 L .05225 .52188 L .07647 .49517 L .10808 .46948 L .14538 .4467 L .18289 .42917 L .22323 .41472 L .262 .40431 L .29666 .3975 L .31596 .39466 L .33353 .39264 L .35084 .39119 L .36041 .39061 L .36491 .39039 L .36906 .39022 L .37322 .39008 L .37698 .38998 L .38112 .3899 L .38551 .38985 L .38931 .38983 L Mistroke .3934 .38984 L .39723 .38988 L .40073 .38994 L .40477 .39004 L .40906 .39017 L .4167 .3905 L .42562 .39103 L .43358 .39163 L .45032 .39333 L .46541 .3954 L .49157 .40032 L .51623 .40682 L .53468 .4132 L .55223 .42092 L .56743 .42949 L .57866 .43763 L .58811 .44668 L .59178 .45122 L .59502 .45615 L .59734 .46073 L .59824 .46295 L .59893 .46498 L .59943 .46677 L .59986 .46872 L .60018 .47066 L .6003 .47171 L .60038 .47268 L .60044 .47372 L .60045 .4743 L .60046 .47484 L .60044 .47582 L .60042 .47637 L .6004 .47686 L .60032 .47778 L .60021 .47877 L .59994 .48055 L .59949 .48259 L .59887 .48471 L .59742 .48836 L .59564 .4917 L .59372 .49459 L .58848 .50059 L .58266 .50549 L .57649 .50954 L .56925 .51328 L .56116 .51653 L .55407 .51874 L .54622 .52059 L .54224 .52131 L .5381 .52192 L Mistroke .53462 .52232 L .53088 .52263 L .5288 .52275 L .52693 .52283 L .52591 .52286 L .52533 .52288 L .52481 .52289 L .52384 .5229 L .52336 .5229 L .52284 .5229 L .5219 .5229 L .52104 .52289 L .52058 .52288 L .52007 .52287 L .51915 .52284 L .51741 .52277 L .51583 .52268 L .51279 .52244 L .50964 .52209 L .5068 .52168 L .50431 .52125 L .49899 .52005 L .49408 .51853 L .49334 .51825 L Mfstroke 1 .87188 m .97074 .87357 L .95438 .87431 L .93666 .87496 L .92046 .8754 L .90553 .87569 L .89137 .87586 L .8764 .87593 L .86172 .87587 L .85344 .87579 L .84575 .87568 L .83767 .87553 L .83034 .87537 L .81398 .87489 L .80017 .87438 L .7853 .8737 L .75864 .87217 L .72811 .86992 L .69647 .86698 L .64175 .86042 L .59359 .85295 L .55065 .84485 L .46118 .82289 L .38872 .79889 L .32377 .77085 L .27738 .74532 L .236 .71639 L .20222 .68513 L .18019 .65686 L .17132 .64142 L .16756 .63338 L .16424 .62497 L .16189 .61769 L .15987 .60978 L .15916 .60634 L .15853 .60268 L .15805 .59929 L .1577 .59612 L .15755 .59435 L .15741 .59244 L .15726 .58913 L .15721 .58726 L .1572 .5853 L .15727 .58181 L .15744 .5786 L .15768 .57563 L .15804 .5724 L .15853 .56898 L .16003 .56137 L .16193 .55434 L Mistroke .16671 .54162 L .17262 .53005 L .17917 .51988 L .19736 .4987 L .21764 .48149 L .23901 .46745 L .26439 .45436 L .29319 .44293 L .32109 .43457 L .3359 .43106 L .35161 .42799 L .36654 .42566 L .37988 .42405 L .38624 .42344 L .39305 .42289 L .39944 .42249 L .40519 .4222 L .4083 .42209 L .41122 .422 L .41428 .42193 L .416 .4219 L .41759 .42188 L .42021 .42186 L .42308 .42185 L .42606 .42188 L .42884 .42192 L .43191 .42198 L .43511 .42208 L .44075 .42232 L .44672 .42266 L .4531 .42315 L .46461 .42432 L .47629 .42593 L .49567 .42967 L .51294 .43435 L .52902 .44022 L .54166 .44631 L .55119 .45224 L .55874 .45834 L .56409 .46407 L .56644 .46733 L .56816 .47028 L .56894 .47189 L .56965 .47363 L .57021 .47526 L .57061 .47674 L .57092 .47825 L .57104 .47903 L .57114 .47985 L .5712 .48056 L Mistroke .57124 .48122 L .57126 .48193 L .57126 .48233 L .57126 .48269 L .57123 .48336 L .57119 .48397 L .57113 .48463 L .57104 .48533 L .57084 .48649 L .57055 .48775 L .57013 .48915 L .56968 .49041 L .5684 .49314 L .567 .49543 L .56527 .49772 L .56153 .50156 L .55684 .50515 L .55193 .50805 L .54717 .51027 L .54152 .51235 L .53621 .51385 L .5304 .51505 L .52751 .51548 L .52438 .51585 L .52228 .51603 L .52126 .51609 L .52032 .51614 L .5195 .51618 L .5186 .51621 L .51813 .51622 L .5177 .51623 L .51724 .51624 L .51675 .51624 L .51626 .51625 L .5158 .51624 L .5153 .51624 L .51477 .51623 L .51422 .51622 L .51371 .51621 L .51275 .51617 L .51186 .51612 L .51106 .51608 L .50927 .51593 L Mfstroke 1 .82504 m .98898 .82622 L .96189 .82847 L .93722 .83015 L .90899 .83164 L .89429 .83223 L .87889 .83272 L .86558 .83302 L .85324 .83321 L .83981 .83332 L .83232 .83333 L .82553 .83332 L .81324 .83321 L .8018 .83303 L .78934 .83275 L .77604 .83234 L .74933 .83117 L .72532 .82974 L .68261 .82624 L .63944 .82141 L .59444 .81488 L .55469 .80774 L .47229 .78822 L .40291 .76562 L .34719 .74185 L .30515 .71908 L .26792 .69347 L .23951 .66794 L .21699 .63996 L .2089 .62608 L .20533 .61852 L .2025 .61141 L .20036 .60489 L .19852 .59787 L .19788 .59481 L .1973 .59157 L .19687 .58858 L .19654 .58581 L .19628 .58276 L .19618 .58115 L .1961 .57944 L .19606 .57765 L .19604 .57596 L .19609 .57282 L .19624 .56991 L .19645 .56725 L .19676 .56434 L .1972 .56129 L .19839 .55507 L .19988 .54937 L Mistroke .20451 .53682 L .20984 .52644 L .21569 .51739 L .23186 .49864 L .25022 .48315 L .26941 .47065 L .2922 .45903 L .31778 .449 L .34299 .44157 L .35629 .43849 L .37027 .43583 L .38238 .43399 L .39548 .43246 L .40115 .43194 L .40716 .43149 L .41277 .43116 L .41783 .43094 L .42048 .43086 L .42295 .43079 L .42564 .43074 L .42853 .43071 L .43136 .43071 L .43278 .43071 L .43433 .43073 L .43705 .43077 L .43956 .43082 L .44208 .4309 L .44445 .43098 L .44882 .43119 L .45345 .43147 L .45838 .43185 L .46853 .43288 L .47875 .43428 L .49565 .43752 L .50989 .44129 L .52323 .44596 L .53395 .45081 L .54217 .45549 L .54969 .46095 L .55508 .46608 L .55932 .47168 L .56098 .47475 L .56155 .47606 L .56207 .47749 L .56244 .47874 L .56276 .48006 L .5629 .48079 L .563 .48144 L .56308 .4821 L .56313 .48271 L Mistroke .56317 .48328 L .56318 .48387 L .56319 .48438 L .56317 .48493 L .56314 .48551 L .56309 .48614 L .56302 .48673 L .56295 .48726 L .56272 .48845 L .56245 .48953 L .56161 .49191 L .56053 .49413 L .55911 .49637 L .55583 .5002 L .55149 .50382 L .54683 .50672 L .54221 .50895 L .53836 .51044 L .53415 .51175 L .52951 .51286 L .52521 .51362 L Mfstroke 1 .8535 m .9802 .85496 L .95487 .85646 L .92724 .8577 L .91184 .85821 L .90348 .85843 L .89575 .85861 L .88211 .85883 L .86731 .85896 L .85904 .85898 L .85127 .85896 L .83682 .85884 L .82878 .85873 L .8214 .85858 L .80486 .85816 L .79166 .8577 L .77942 .85719 L .75196 .85571 L .72574 .85388 L .69817 .85149 L .64622 .84563 L .60036 .83889 L .50497 .81954 L .43106 .79854 L .36312 .77308 L .30645 .74541 L .25731 .71389 L .22194 .6834 L .19838 .6554 L .1898 .64188 L .18264 .6276 L .1799 .62077 L .17772 .61436 L .17612 .60871 L .17472 .60254 L .1736 .59584 L .17318 .59221 L .17289 .58884 L .17273 .58571 L .17266 .58287 L .17268 .57961 L .17279 .57654 L .17301 .57327 L .17319 .57146 L .17337 .56982 L .17429 .56368 L .17497 .56029 L .17583 .55666 L .17771 .55019 L .18211 .5389 L Mistroke .18732 .52881 L .19477 .5174 L .20266 .50756 L .22157 .48923 L .24525 .47224 L .26902 .45923 L .29231 .44921 L .31869 .44038 L .33346 .43644 L .34701 .43339 L .37156 .42915 L .38422 .42759 L .39059 .42696 L .39732 .42641 L .40321 .42602 L .40851 .42575 L .41104 .42565 L .41377 .42555 L .41632 .42548 L .41869 .42543 L .42143 .4254 L .42442 .42538 L .42731 .42538 L .42892 .42539 L .43039 .42541 L .43299 .42545 L .43579 .42552 L .43843 .42561 L .44085 .4257 L .44674 .426 L .45205 .42636 L .46365 .42743 L .4734 .42867 L .48319 .43025 L .49894 .43357 L .5143 .43793 L .52825 .44314 L .53901 .44831 L .54917 .4546 L .55663 .46072 L .56187 .46655 L .56387 .46947 L .5656 .47265 L .56669 .47531 L .56716 .47678 L .5675 .47812 L .56773 .47928 L .56783 .47994 L .56791 .48054 L .56797 .48117 L Mistroke .56802 .48184 L .56804 .48245 L .56805 .48302 L .56804 .48364 L .56801 .4843 L .56796 .48491 L .5679 .48547 L .56781 .48612 L .5677 .48682 L .56743 .48807 L .56706 .48935 L .5666 .49069 L .56555 .49301 L .56311 .49684 L .55965 .50067 L .55561 .504 L .55147 .50666 L .54657 .50915 L 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.49573 .49188 L .49581 .49188 L .4959 .49188 L .49605 .49188 L .49622 .49188 L .49639 .49189 L .49654 .49189 L .49671 .4919 L .49689 .49192 L .49724 .49195 L .49807 .49206 L .49893 .49222 L .49977 .4924 L .50185 .49299 L .50438 .49389 L .50962 .49632 L .51535 .49971 L .52169 .50436 L .52746 .50966 L .53205 .51506 L .53412 .51819 L .53568 .52119 L .53633 .52283 L .5368 .52433 L .53699 .52511 L .53715 .52596 L .53721 .52638 L .53723 .5266 L .53726 .52683 L .53728 .52704 L .53729 .52726 L .53731 .52764 L .53731 .52803 L .53731 .52839 L .5373 .52857 L .53728 .52878 L .53727 .52899 L .53725 .52918 L .5372 .52954 L .53715 .52986 L .53699 .53058 L .53681 .5312 L .53655 .53186 L Mistroke .53596 .533 L .5355 .53367 L .53503 .53424 L .53375 .53544 L .53229 .53643 L .53066 .53723 L .52887 .53788 L .52774 .53819 L .52663 .53844 L .52538 .53866 L .52417 .53883 L .52347 .5389 L .52269 .53896 L .52194 .53901 L .52122 .53904 L .52059 .53906 L .51991 .53907 L .51924 .53907 L .51863 .53907 L .51793 .53905 L .51715 .53902 L .51631 .53897 L .51552 .53892 L .51383 .53878 L .51223 .5386 L .50845 .53806 L .50442 .5373 L .49986 .53625 L .49094 .53369 L .46743 .52446 L .44171 .51105 L .41566 .49426 L .38873 .47319 L .36113 .44629 L .34792 .43036 L .33742 .41508 L .32937 .40026 L .32617 .39278 L .32341 .38458 L .32242 .38079 L .32168 .37729 L .32109 .37367 L .32086 .37177 L .32075 .37072 L .32067 .36976 L .32057 .36804 L .32052 .3662 L .32052 .36458 L .32057 .36288 L .32066 .36138 L Mistroke .32078 .35998 L .32096 .35845 L .32121 .35683 L .32196 .35327 L .3225 .35129 L .3231 .34948 L .32451 .34597 L .32606 .34289 L .32981 .33729 L .33495 .33179 L .34098 .32706 L .3486 .32267 L .35617 .31944 L .36562 .31651 L .37148 .31516 L .37718 .31413 L .38047 .31364 L .38416 .31318 L .38745 .31285 L .39119 .31254 L .39472 .31232 L .39812 .31217 L .4018 .31208 L .40392 .31205 L .40588 .31204 L .40924 .31206 L .41299 .31214 L .417 .31228 L .42082 .31247 L .42499 .31274 L .42951 .31309 L .43786 .31392 L .45797 .31669 L .47671 .32015 L .49876 .32516 L .58958 .35411 L .7029 .40463 L .84035 .48352 L .97779 .58151 L Mfstroke .97779 .58151 m 1 .6013 L s .50464 .49435 m .50929 .49654 L .51474 .49977 L .52101 .50441 L .52661 .50962 L .53097 .51489 L .53293 .51798 L .53432 .52081 L .53485 .52226 L .53509 .52305 L .53528 .52378 L .53542 .52445 L .53551 .52506 L .53556 .5254 L .53559 .52573 L .53561 .52607 L .53563 .52644 L .53563 .52663 L .53563 .52683 L .53563 .52703 L .53562 .5272 L .5356 .52754 L .53557 .52788 L .53552 .52826 L .53546 .5286 L .53528 .52937 L .53507 .53001 L .53483 .53059 L .53453 .53118 L .53414 .53182 L .53328 .53288 L .53214 .53391 L .53086 .53477 L .52947 .53547 L .52858 .53583 L .52757 .53617 L .52559 .53668 L .52437 .5369 L .52318 .53706 L .52246 .53714 L .52179 .5372 L .52141 .53723 L .52099 .53725 L .52059 .53727 L .52021 .53729 L .51952 .5373 L .51887 .53731 L .51815 .53731 L .51736 .53729 L .51664 .53727 L Mistroke .51587 .53723 L .51444 .53714 L .51253 .53696 L .51065 .53673 L .50693 .53615 L .49847 .53425 L .48808 .53109 L .46597 .522 L .43912 .50745 L .38697 .46837 L .36228 .44286 L .35047 .42759 L .34134 .4131 L .33427 .39829 L .33159 .39054 L .33055 .38667 L .32964 .38244 L .32934 .38065 L .32908 .37879 L .32889 .37718 L .32875 .37548 L .32869 .37452 L .32865 .37366 L .3286 .37197 L .3286 .37103 L .32862 .37002 L .32865 .36907 L .3287 .3682 L .32882 .36666 L .32898 .36523 L .3292 .36368 L .3295 .36203 L .33029 .35872 L .33125 .35577 L .33384 .35009 L .3355 .34735 L .33763 .34444 L .34214 .3396 L .34728 .33546 L .35448 .33114 L .36221 .32774 L .37201 .3247 L .37745 .32346 L .38383 .32233 L .38977 .32156 L .39326 .32122 L .39659 .32096 L .39957 .32078 L .40293 .32063 L .40465 .32058 L Mistroke .40653 .32054 L .40999 .32051 L .4134 .32053 L .41721 .32061 L .42089 .32075 L .42438 .32092 L .43115 .32139 L .43865 .32209 L .44789 .32318 L .45832 .3247 L .47843 .3284 L .52006 .33871 L .56466 .35301 L .61869 .37401 L .74184 .43448 L .86796 .51278 L Mfstroke .86796 .51278 m 1 .61422 L s .51492 .49667 m .52204 .50118 L .5297 .50714 L .53703 .51426 L .54072 .51872 L .54359 .52293 L .54471 .52488 L .54576 .52701 L .54653 .52885 L .54719 .53086 L .54746 .53187 L .54768 .53295 L .54778 .53354 L .54785 .53408 L .54788 .53434 L .54791 .53462 L .54794 .53513 L .54795 .53539 L .54796 .53563 L .54796 .53609 L .54795 .53657 L .54794 .53684 L .54792 .53709 L .54788 .5375 L .54783 .53795 L .54775 .53842 L .54766 .53885 L .54743 .53976 L .54714 .54059 L .54675 .54147 L .54626 .54237 L .54521 .5438 L .54375 .54526 L .54217 .54642 L .54016 .54753 L .53795 .54842 L .53566 .5491 L .53422 .54942 L .53283 .54967 L .53114 .5499 L .53033 .54999 L .52945 .55007 L .52868 .55012 L .52781 .55017 L .527 .5502 L .52622 .55022 L .52529 .55023 L .52442 .55023 L .5234 .55021 L .5224 .55017 L Mistroke .52129 .55012 L .52026 .55006 L .51788 .54986 L .51558 .54961 L .51333 .54931 L .50799 .54841 L .50176 .54707 L .4901 .54386 L .47584 .5389 L .44281 .52396 L .40827 .50391 L .36936 .47586 L .33101 .44112 L .31495 .42348 L .2997 .40393 L .28784 .38554 L .28254 .37569 L .27771 .36509 L .27416 .35544 L .27289 .35124 L .27171 .34669 L .27085 .34269 L .27013 .33845 L .26982 .33612 L .2696 .33402 L .26943 .33192 L .26933 .32996 L .26927 .32784 L .26928 .32558 L .26935 .32347 L .26947 .32156 L .26969 .31933 L .27 .31694 L .27039 .3147 L .27082 .31265 L .27189 .30865 L .27341 .30433 L .27538 .29996 L .27761 .29599 L .28256 .2892 L .28821 .28342 L .29566 .27764 L .30499 .27223 L .31605 .26755 L .32755 .26405 L .33451 .26245 L .34256 .26099 L .35064 .2599 L .35438 .25951 L .35851 .25915 L Mistroke .36244 .25887 L .36612 .25867 L .36989 .25852 L .37355 .25843 L .37789 .25838 L .38199 .25839 L .38659 .25846 L .39161 .25862 L .39661 .25885 L .40214 .25918 L .41249 .26001 L .42297 .26111 L .43295 .26239 L .45597 .26608 L .4824 .27146 L .5347 .28511 L .67442 .33668 L .84389 .42212 L Mfstroke .84389 .42212 m 1 .52166 L s .47943 .49536 m .4745 .49071 L .47007 .48526 L .46825 .48228 L .46698 .47948 L .46669 .47866 L .46646 .47789 L .46629 .4772 L .46615 .47647 L .46609 .47609 L .46604 .47568 L .46602 .47551 L .46601 .47531 L .46599 .47497 L .46598 .47476 L .46598 .47454 L .46598 .47433 L .46598 .47414 L .46599 .47395 L .466 .47378 L .46602 .47358 L .46603 .47339 L .46607 .47309 L .46612 .47277 L .46624 .47221 L .46644 .47155 L .46666 .47096 L .46695 .47037 L .46732 .46975 L .46815 .46868 L .46911 .46777 L .47035 .4669 L .47159 .46625 L .47316 .46562 L .47505 .46509 L .47609 .46487 L .47729 .46468 L .47839 .46455 L .47903 .46449 L .47962 .46445 L .48019 .46442 L .48082 .46439 L .48143 .46438 L .48201 .46437 L .48268 .46437 L .48343 .46439 L .48415 .46441 L .48482 .46444 L .48562 .46449 L .48652 .46456 L Mistroke .4882 .46472 L .49208 .46526 L .49596 .46599 L .50483 .46823 L .51422 .47132 L .53426 .47984 L .55888 .49334 L .58309 .50972 L .60986 .53197 L .63265 .55593 L .65031 .58087 L .65427 .5882 L .65745 .59514 L .65996 .60181 L .66176 .60801 L .66244 .61101 L .663 .61421 L .66324 .61592 L .66343 .61772 L .66356 .61924 L .66364 .6209 L .66366 .6218 L .66367 .62264 L .66365 .62422 L .66357 .6259 L .6635 .62684 L .66342 .6277 L .66325 .62913 L .66301 .63069 L .66236 .63378 L .66153 .63664 L .66058 .63919 L .65936 .64185 L .65778 .64467 L .65433 .64943 L .65026 .65361 L .64476 .65786 L .63865 .66141 L .63086 .66475 L .62597 .66637 L .62115 .66767 L .61644 .6687 L .61112 .66963 L .60791 .67008 L .60491 .67043 L .60198 .67072 L .59879 .67097 L .59717 .67107 L .59536 .67117 L .59212 .6713 L Mistroke .59026 .67135 L .58823 .67138 L .58629 .6714 L .58448 .6714 L .58095 .67135 L .57891 .6713 L .57699 .67123 L .57342 .67107 L .56953 .67084 L .56155 .67019 L .55275 .66923 L .536 .66677 L .51659 .66302 L .49363 .65752 L .40121 .62632 L .29486 .57655 L .16755 .50059 L .04156 .40736 L Mfstroke .04156 .40736 m 0 .36877 L s .48971 .49768 m .48725 .49536 L .48503 .49263 L .48412 .49114 L .48349 .48974 L .48335 .48933 L .48323 .48895 L .48315 .4886 L .48307 .48823 L .48305 .48805 L .48302 .48784 L .48301 .48775 L .483 .48766 L .48299 .48748 L .48299 .48738 L .48299 .48727 L .48299 .48717 L .48299 .48707 L .48299 .48697 L .483 .48689 L .48301 .48679 L .48302 .4867 L .48304 .48655 L .48306 .48639 L .48312 .4861 L .48322 .48577 L .48333 .48548 L .48347 .48518 L .48366 .48487 L .48407 .48434 L .48456 .48388 L .48518 .48345 L .4858 .48312 L .48658 .48281 L .48752 .48255 L .48804 .48244 L .48865 .48234 L .48919 .48228 L .48951 .48225 L .48981 .48222 L .4901 .48221 L .49041 .4822 L .49072 .48219 L .491 .48218 L .49134 .48219 L .49171 .48219 L .49207 .48221 L .49241 .48222 L .49281 .48225 L .49326 .48228 L Mistroke .4941 .48236 L .49604 .48263 L .49798 .48299 L .50241 .48411 L .50711 .48566 L .51713 .48992 L .52944 .49667 L .54154 .50486 L .55493 .51599 L .56633 .52796 L .57515 .54043 L .57713 .5441 L .57873 .54757 L .57998 .55091 L .58088 .554 L .58122 .5555 L .5815 .5571 L .58162 .55796 L .58172 .55886 L .58178 .55962 L .58182 .56045 L .58183 .5609 L .58184 .56132 L .58183 .56211 L .58179 .56295 L .58175 .56342 L .58171 .56385 L .58163 .56457 L .58151 .56534 L .58118 .56689 L .58076 .56832 L .58029 .56959 L .57968 .57093 L .57889 .57234 L .57716 .57471 L .57513 .5768 L .57238 .57893 L .56933 .5807 L .56543 .58238 L .56298 .58318 L .56057 .58383 L .55822 .58435 L .55556 .58481 L .55396 .58504 L .55245 .58522 L .55099 .58536 L .54939 .58548 L .54858 .58554 L .54768 .58558 L .54606 .58565 L Mistroke .54513 .58567 L .54411 .58569 L .54314 .5857 L .54224 .5857 L .54047 .58567 L .53945 .58565 L .5385 .58562 L .53671 .58553 L .53477 .58542 L .53078 .58509 L .52637 .58461 L .518 .58339 L .50829 .58151 L .49681 .57876 L .4506 .56316 L .39743 .53828 L .33377 .50029 L .27078 .45368 L .2051 .39268 L .17755 .36136 L .15177 .32672 L .13185 .29352 L .11805 .26299 L .11299 .24788 L .11094 .24026 L .10914 .23218 L .10836 .22795 L .10776 .22413 L .10726 .22026 L .10689 .21671 L .10657 .2126 L .10646 .21044 L .10637 .20817 L .10633 .20437 L .10635 .20223 L .10641 .20023 L .10659 .19677 L .10691 .19303 L .10732 .18953 L .1078 .18632 L .10898 .18033 L .11063 .17407 L .11311 .16686 L .11598 .16027 L .1228 .14848 L .13195 .13704 L .14451 .12558 L .16056 .11494 L .17832 .1064 L .19751 .0997 L Mistroke .21716 .09479 L .22894 .0926 L .24249 .09067 L .24896 .08993 L .25599 .08927 L .26271 .08876 L .26909 .08838 L .27641 .08805 L .28461 .08783 L .292 .08775 L .30007 .08778 L .30818 .08794 L .31599 .0882 L .32314 .08852 L .33102 .08898 L .34979 .09045 L .3705 .09265 L .40988 .09829 L .44911 .10558 L .48801 .11426 L .58446 .14103 L .8241 .23307 L Mfstroke .8241 .23307 m 1 .32249 L s .5 .5 m .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L .5 .5 L s .51029 .50232 m .51275 .50464 L .51497 .50737 L .51588 .50886 L .51651 .51026 L .51665 .51067 L .51677 .51105 L .51685 .5114 L .51693 .51177 L .51695 .51195 L .51698 .51216 L .51699 .51225 L .517 .51234 L .51701 .51252 L .51701 .51262 L .51701 .51273 L .51701 .51283 L .51701 .51293 L .51701 .51303 L .517 .51311 L .51699 .51321 L .51698 .5133 L .51696 .51345 L .51694 .51361 L .51688 .5139 L .51678 .51423 L .51667 .51452 L .51653 .51482 L .51634 .51513 L .51593 .51566 L .51544 .51612 L .51482 .51655 L .5142 .51688 L .51342 .51719 L .51248 .51745 L .51196 .51756 L .51135 .51766 L .51081 .51772 L .51049 .51775 L .51019 .51778 L .5099 .51779 L .50959 .5178 L .50928 .51781 L .509 .51782 L .50866 .51781 L .50829 .51781 L .50793 .51779 L .50759 .51778 L .50719 .51775 L .50674 .51772 L Mistroke .5059 .51764 L .50396 .51737 L .50202 .51701 L .49759 .51589 L .49289 .51434 L .48287 .51008 L .47056 .50333 L .45846 .49514 L .44507 .48401 L .43367 .47204 L .42485 .45957 L .42287 .4559 L .42127 .45243 L .42002 .44909 L .41912 .446 L .41878 .4445 L .4185 .4429 L .41838 .44204 L .41828 .44114 L .41822 .44038 L .41818 .43955 L .41817 .4391 L .41816 .43868 L .41817 .43789 L .41821 .43705 L .41825 .43658 L .41829 .43615 L .41837 .43543 L .41849 .43466 L .41882 .43311 L .41924 .43168 L .41971 .43041 L .42032 .42907 L .42111 .42766 L .42284 .42529 L .42487 .4232 L .42762 .42107 L .43067 .4193 L .43457 .41762 L .43702 .41682 L .43943 .41617 L .44178 .41565 L .44444 .41519 L .44604 .41496 L .44755 .41478 L .44901 .41464 L .45061 .41452 L .45142 .41446 L .45232 .41442 L .45394 .41435 L Mistroke .45487 .41433 L .45589 .41431 L .45686 .4143 L .45776 .4143 L .45953 .41433 L .46055 .41435 L .4615 .41438 L .46329 .41447 L .46523 .41458 L .46922 .41491 L .47363 .41539 L .482 .41661 L .49171 .41849 L .50319 .42124 L .5494 .43684 L .60257 .46172 L .66623 .49971 L .72922 .54632 L .7949 .60732 L .82245 .63864 L .84823 .67328 L .86815 .70648 L .88195 .73701 L .88701 .75212 L .88906 .75974 L .89086 .76782 L .89164 .77205 L .89224 .77587 L .89274 .77974 L .89311 .78329 L .89343 .7874 L .89354 .78956 L .89363 .79183 L .89367 .79563 L .89365 .79777 L .89359 .79977 L .89341 .80323 L .89309 .80697 L .89268 .81047 L .8922 .81368 L .89102 .81967 L .88937 .82593 L .88689 .83314 L .88402 .83973 L .8772 .85152 L .86805 .86296 L .85549 .87442 L .83944 .88506 L .82168 .8936 L .80249 .9003 L Mistroke .78284 .90521 L .77106 .9074 L .75751 .90933 L .75104 .91007 L .74401 .91073 L .73729 .91124 L .73091 .91162 L .72359 .91195 L .71539 .91217 L .708 .91225 L .69993 .91222 L .69182 .91206 L .68401 .9118 L .67686 .91148 L .66898 .91102 L .65021 .90955 L .6295 .90735 L .59012 .90171 L .55089 .89442 L .51199 .88574 L .41554 .85897 L .1759 .76693 L Mfstroke .1759 .76693 m 0 .67751 L s .52057 .50464 m .5255 .50929 L .52993 .51474 L .53175 .51772 L .53302 .52052 L .53331 .52134 L .53354 .52211 L .53371 .5228 L .53385 .52353 L .53391 .52391 L .53396 .52432 L .53398 .52449 L .53399 .52469 L .53401 .52503 L .53402 .52524 L .53402 .52546 L .53402 .52567 L .53402 .52586 L .53401 .52605 L .534 .52622 L .53398 .52642 L .53397 .52661 L .53393 .52691 L .53388 .52723 L .53376 .52779 L .53356 .52845 L .53334 .52904 L .53305 .52963 L .53268 .53025 L .53185 .53132 L .53089 .53223 L .52965 .5331 L .52841 .53375 L .52684 .53438 L .52495 .53491 L .52391 .53513 L .52271 .53532 L .52161 .53545 L .52097 .53551 L .52038 .53555 L .51981 .53558 L .51918 .53561 L .51857 .53562 L .51799 .53563 L .51732 .53563 L .51657 .53561 L .51585 .53559 L .51518 .53556 L .51438 .53551 L .51348 .53544 L Mistroke .5118 .53528 L .50792 .53474 L .50404 .53401 L .49517 .53177 L .48578 .52868 L .46574 .52016 L .44112 .50666 L .41691 .49028 L .39014 .46803 L .36735 .44407 L .34969 .41913 L .34573 .4118 L .34255 .40486 L .34004 .39819 L .33824 .39199 L .33756 .38899 L .337 .38579 L .33676 .38408 L .33657 .38228 L .33644 .38076 L .33636 .3791 L .33634 .3782 L .33633 .37736 L .33635 .37578 L .33643 .3741 L .3365 .37316 L .33658 .3723 L .33675 .37087 L .33699 .36931 L .33764 .36622 L .33847 .36336 L .33942 .36081 L .34064 .35815 L .34222 .35533 L .34567 .35057 L .34974 .34639 L .35524 .34214 L .36135 .33859 L .36914 .33525 L .37403 .33363 L .37885 .33233 L .38356 .3313 L .38888 .33037 L .39209 .32992 L .39509 .32957 L .39802 .32928 L .40121 .32903 L .40283 .32893 L .40464 .32883 L .40788 .3287 L Mistroke .40974 .32865 L .41177 .32862 L .41371 .3286 L .41552 .3286 L .41905 .32865 L .42109 .3287 L .42301 .32877 L .42658 .32893 L .43047 .32916 L .43845 .32981 L .44725 .33077 L .464 .33323 L .48341 .33698 L .50637 .34248 L .59879 .37368 L .70514 .42345 L .83245 .49941 L .95844 .59264 L Mfstroke .95844 .59264 m 1 .63123 L s .48508 .50333 m .47796 .49882 L .4703 .49286 L .46297 .48574 L .45928 .48128 L .45641 .47707 L .45529 .47512 L .45424 .47299 L .45347 .47115 L .45281 .46914 L .45254 .46813 L .45232 .46705 L .45222 .46646 L .45215 .46592 L .45212 .46566 L .45209 .46538 L .45206 .46487 L .45205 .46461 L .45204 .46437 L .45204 .46391 L .45205 .46343 L .45206 .46316 L .45208 .46291 L .45212 .4625 L .45217 .46205 L .45225 .46158 L .45234 .46115 L .45257 .46024 L .45286 .45941 L .45325 .45853 L .45374 .45763 L .45479 .4562 L .45625 .45474 L .45783 .45358 L .45984 .45247 L .46205 .45158 L .46434 .4509 L .46578 .45058 L .46717 .45033 L .46886 .4501 L .46967 .45001 L .47055 .44993 L .47132 .44988 L .47219 .44983 L .473 .4498 L .47378 .44978 L .47471 .44977 L .47558 .44977 L .4766 .44979 L .4776 .44983 L Mistroke .47871 .44988 L .47974 .44994 L .48212 .45014 L .48442 .45039 L .48667 .45069 L .49201 .45159 L .49824 .45293 L .5099 .45614 L .52416 .4611 L .55719 .47604 L .59173 .49609 L .63064 .52414 L .66899 .55888 L .68505 .57652 L .7003 .59607 L .71216 .61446 L .71746 .62431 L .72229 .63491 L .72584 .64456 L .72711 .64876 L .72829 .65331 L .72915 .65731 L .72987 .66155 L .73018 .66388 L .7304 .66598 L .73057 .66808 L .73067 .67004 L .73073 .67216 L .73072 .67442 L .73065 .67653 L .73053 .67844 L .73031 .68067 L .73 .68306 L .72961 .6853 L .72918 .68735 L .72811 .69135 L .72659 .69567 L .72462 .70004 L .72239 .70401 L .71744 .7108 L .71179 .71658 L .70434 .72236 L .69501 .72777 L .68395 .73245 L .67245 .73595 L .66549 .73755 L .65744 .73901 L .64936 .7401 L .64562 .74049 L .64149 .74085 L Mistroke .63756 .74113 L .63388 .74133 L .63011 .74148 L .62645 .74157 L .62211 .74162 L .61801 .74161 L .61341 .74154 L .60839 .74138 L .60339 .74115 L .59786 .74082 L .58751 .73999 L .57703 .73889 L .56705 .73761 L .54403 .73392 L .5176 .72854 L .4653 .71489 L .32558 .66332 L .15611 .57788 L Mfstroke .15611 .57788 m 0 .47834 L s .49536 .50565 m .49071 .50346 L .48526 .50023 L .47899 .49559 L .47339 .49038 L .46903 .48511 L .46707 .48202 L .46568 .47919 L .46515 .47774 L .46491 .47695 L .46472 .47622 L .46458 .47555 L .46449 .47494 L .46444 .4746 L .46441 .47427 L .46439 .47393 L .46437 .47356 L .46437 .47337 L .46437 .47317 L .46437 .47297 L .46438 .4728 L .4644 .47246 L .46443 .47212 L .46448 .47174 L .46454 .4714 L .46472 .47063 L .46493 .46999 L .46517 .46941 L .46547 .46882 L .46586 .46818 L .46672 .46712 L .46786 .46609 L .46914 .46523 L .47053 .46453 L .47142 .46417 L .47243 .46383 L .47441 .46332 L .47563 .4631 L .47682 .46294 L .47754 .46286 L .47821 .4628 L .47859 .46277 L .47901 .46275 L .47941 .46273 L .47979 .46271 L .48048 .4627 L .48113 .46269 L .48185 .46269 L .48264 .46271 L .48336 .46273 L Mistroke .48413 .46277 L .48556 .46286 L .48747 .46304 L .48935 .46327 L .49307 .46385 L .50153 .46575 L .51192 .46891 L .53403 .478 L .56088 .49255 L .61303 .53163 L .63772 .55714 L .64953 .57241 L .65866 .5869 L .66573 .60171 L .66841 .60946 L .66945 .61333 L .67036 .61756 L .67066 .61935 L .67092 .62121 L .67111 .62282 L .67125 .62452 L .67131 .62548 L .67135 .62634 L .6714 .62803 L .6714 .62897 L .67138 .62998 L .67135 .63093 L .6713 .6318 L .67118 .63334 L .67102 .63477 L .6708 .63632 L .6705 .63797 L .66971 .64128 L .66875 .64423 L .66616 .64991 L .6645 .65265 L .66237 .65556 L .65786 .6604 L .65272 .66454 L .64552 .66886 L .63779 .67226 L .62799 .6753 L .62255 .67654 L .61617 .67767 L .61023 .67844 L .60674 .67878 L .60341 .67904 L .60043 .67922 L .59707 .67937 L .59535 .67942 L Mistroke .59347 .67946 L .59001 .67949 L .5866 .67947 L .58279 .67939 L .57911 .67925 L .57562 .67908 L .56885 .67861 L .56135 .67791 L .55211 .67682 L .54168 .6753 L .52157 .6716 L .47994 .66129 L .43534 .64699 L .38131 .62599 L .25816 .56552 L .13204 .48722 L Mfstroke .13204 .48722 m 0 .38578 L s .50565 .50797 m .50543 .50801 L .50519 .50805 L .50494 .50808 L .50471 .5081 L .50456 .50811 L .50442 .50812 L .50427 .50812 L .50419 .50812 L .5041 .50812 L .50395 .50812 L .50378 .50812 L .50361 .50811 L .50346 .50811 L .50329 .5081 L .50311 .50808 L .50276 .50805 L .50193 .50794 L .50107 .50778 L .50023 .5076 L .49815 .50701 L .49562 .50611 L .49038 .50368 L .48465 .50029 L .47831 .49564 L .47254 .49034 L .46795 .48494 L .46588 .48181 L .46432 .47881 L .46367 .47717 L .4632 .47567 L .46301 .47489 L .46285 .47404 L .46279 .47362 L .46277 .4734 L .46274 .47317 L .46272 .47296 L .46271 .47274 L .46269 .47236 L .46269 .47197 L .46269 .47161 L .4627 .47143 L .46272 .47122 L .46273 .47101 L .46275 .47082 L .4628 .47046 L .46285 .47014 L .46301 .46942 L .46319 .4688 L .46345 .46814 L Mistroke .46404 .467 L .4645 .46633 L .46497 .46576 L .46625 .46456 L .46771 .46357 L .46934 .46277 L .47113 .46212 L .47226 .46181 L .47337 .46156 L .47462 .46134 L .47583 .46117 L .47653 .4611 L .47731 .46104 L .47806 .46099 L .47878 .46096 L .47941 .46094 L .48009 .46093 L .48076 .46093 L .48137 .46093 L .48207 .46095 L .48285 .46098 L .48369 .46103 L .48448 .46108 L .48617 .46122 L .48777 .4614 L .49155 .46194 L .49558 .4627 L .50014 .46375 L .50906 .46631 L .53257 .47554 L .55829 .48895 L .58434 .50574 L .61127 .52681 L .63887 .55371 L .65208 .56964 L .66258 .58492 L .67063 .59974 L .67383 .60722 L .67659 .61542 L .67758 .61921 L .67832 .62271 L .67891 .62633 L .67914 .62823 L .67925 .62928 L .67933 .63024 L .67943 .63196 L .67948 .6338 L .67948 .63542 L .67943 .63712 L .67934 .63862 L Mistroke .67922 .64002 L .67904 .64155 L .67879 .64317 L .67804 .64673 L .6775 .64871 L .6769 .65052 L .67549 .65403 L .67394 .65711 L .67019 .66271 L .66505 .66821 L .65902 .67294 L .6514 .67733 L .64383 .68056 L .63438 .68349 L .62852 .68484 L .62282 .68587 L .61953 .68636 L .61584 .68682 L .61255 .68715 L .60881 .68746 L .60528 .68768 L .60188 .68783 L .5982 .68792 L .59608 .68795 L .59412 .68796 L .59076 .68794 L .58701 .68786 L .583 .68772 L .57918 .68753 L .57501 .68726 L .57049 .68691 L .56214 .68608 L .54203 .68331 L .52329 .67985 L .50124 .67484 L .41042 .64589 L .2971 .59537 L .15965 .51648 L .02221 .41849 L Mfstroke .02221 .41849 m 0 .3987 L s .51593 .51029 m .51602 .51059 L .5161 .51091 L .51616 .51119 L .5162 .51149 L .51622 .51167 L .51623 .51184 L .51624 .51199 L .51625 .51216 L .51624 .51231 L .51624 .51244 L .51623 .51259 L .51621 .51275 L .51619 .51291 L .51616 .51306 L .51608 .51339 L .51599 .51368 L .51587 .51397 L .51556 .51451 L .5152 .51497 L .51465 .51547 L .51403 .51588 L .51322 .51626 L .51276 .51643 L .51222 .51659 L .51171 .51671 L .5112 .51681 L .5109 .51686 L .51057 .5169 L .51029 .51693 L .50996 .51696 L .50965 .51698 L .50935 .51699 L .50903 .51701 L .50885 .51701 L .50867 .51701 L .50838 .51701 L .50805 .51701 L .5077 .517 L .50736 .51698 L .50699 .51696 L .50658 .51693 L .50584 .51686 L .505 .51677 L .50404 .51663 L .50038 .51589 L .49638 .51477 L .49222 .51333 L .48156 .50859 L .47057 .50236 L Mistroke .45761 .49329 L .44576 .48304 L .43542 .47177 L .43056 .46513 L .42686 .4589 L .42528 .45564 L .42386 .45208 L .42279 .44854 L .42237 .44663 L .42222 .44578 L .42209 .44486 L .42199 .444 L .42192 .44321 L .42188 .44246 L .42185 .44167 L .42186 .44097 L .42188 .44032 L .42192 .43961 L .42199 .43885 L .42209 .43809 L .42223 .43728 L .4226 .43566 L .42307 .43421 L .42361 .4329 L .4251 .43025 L .42616 .42884 L .42727 .4276 L .42969 .42548 L .43277 .42348 L .43601 .42192 L .43946 .42068 L .44373 .41959 L .44635 .4191 L .44888 .41874 L .45141 .41848 L .4529 .41836 L .45434 .41828 L .45589 .41822 L .45735 .41818 L .45899 .41816 L .45983 .41817 L .46076 .41818 L .46236 .41821 L .46385 .41827 L .46561 .41836 L .4673 .41847 L .4704 .41872 L .47389 .41909 L .48233 .42029 L .492 .42215 L Mistroke .51124 .42707 L .55565 .44321 L .60422 .4668 L .71978 .54496 L .7813 .60215 L .80685 .63109 L .83098 .66318 L .851 .69614 L .86466 .72613 L .86949 .7405 L .87152 .74799 L .87326 .75584 L .87444 .76266 L .87493 .76625 L .87535 .77006 L .87564 .77356 L .87581 .77672 L .87591 .77988 L .87592 .78285 L .87585 .78621 L .87566 .7898 L .87537 .79316 L .87501 .79621 L .87451 .79957 L .87384 .8032 L .87207 .8104 L .86975 .81733 L .86666 .82452 L .85939 .83684 L .8498 .84833 L .83857 .85823 L .82661 .86627 L .81158 .87398 L .79299 .88099 L .77323 .88627 L .76085 .88869 L .74886 .89051 L .74195 .89134 L .7342 .89211 L .72666 .8927 L .71948 .89312 L .71547 .8933 L .71174 .89344 L .70759 .89355 L .70315 .89363 L .69553 .89367 L .69113 .89364 L .68698 .89358 L .67907 .89337 L .67172 .89307 L Mistroke .66305 .8926 L .65467 .89202 L .63543 .8903 L .61688 .88815 L .57222 .8812 L .53258 .87322 L .4871 .86225 L .39756 .83581 L .1608 .74135 L Mfstroke .1608 .74135 m 0 .65581 L s .52622 .5126 m .52769 .51488 L .52896 .51739 L .52945 .51865 L .52966 .51934 L .52984 .51999 L .52997 .52059 L .53002 .5209 L .53007 .52123 L .53011 .52151 L .53013 .52181 L .53015 .52208 L .53016 .52234 L .53017 .52249 L .53017 .52266 L .53016 .52295 L .53016 .52313 L .53014 .5233 L .53013 .52345 L .53012 .52361 L .53008 .52391 L .53002 .52423 L .5299 .52478 L .52973 .5253 L .52955 .52578 L .52899 .52681 L .5286 .52735 L .52818 .52785 L .52723 .52871 L .52598 .52953 L .52517 .52994 L .52434 .53029 L .52285 .53077 L .52192 .531 L .52102 .53118 L .52018 .53131 L .51922 .53143 L .51872 .53147 L .51816 .53152 L .51763 .53155 L .51713 .53157 L .51652 .53159 L .5162 .53159 L .51585 .53159 L .51522 .53159 L .51461 .53158 L .51424 .53156 L .51388 .53155 L .5132 .53151 L .51245 .53146 L Mistroke .5116 .53139 L .50851 .53103 L .50541 .53052 L .49781 .52876 L .48973 .52626 L .47235 .51917 L .45083 .50774 L .42949 .4937 L .40564 .47445 L .38504 .45354 L .3687 .43159 L .36184 .41895 L .35952 .41348 L .35746 .40748 L .35668 .40467 L .35599 .40167 L .35568 .40003 L .35544 .39853 L .35523 .39702 L .35508 .39562 L .35499 .39442 L .35491 .39314 L .35488 .39173 L .35488 .39046 L .35492 .38914 L .35501 .38772 L .35513 .3864 L .35529 .38519 L .35578 .38248 L .35641 .38 L .35731 .37735 L .35851 .37459 L .36005 .3718 L .36177 .36925 L .36572 .36477 L .3701 .3611 L .37589 .35749 L .38304 .35425 L .3867 .35298 L .39082 .35178 L .39461 .35086 L .399 .35 L .40321 .34934 L .40732 .34884 L .40992 .3486 L .41243 .3484 L .41474 .34826 L .4173 .34814 L .42016 .34806 L .42326 .34802 L Mistroke .42597 .34803 L .42903 .34808 L .43227 .34819 L .43536 .34834 L .43881 .34855 L .44258 .34884 L .4498 .34954 L .45661 .35038 L .47123 .35267 L .48595 .35559 L .52206 .36491 L .56604 .37963 L .65907 .42038 L .77066 .48378 L .98257 .64959 L Mfstroke .98257 .64959 m 1 .66862 L s .49073 .5113 m .48143 .50693 L .47053 .50046 L .45799 .49119 L .44679 .48076 L .43805 .47021 L .43413 .46404 L .43137 .45838 L .43029 .45548 L .42982 .45391 L .42944 .45244 L .42917 .4511 L .42897 .44988 L .42889 .4492 L .42883 .44855 L .42878 .44786 L .42875 .44713 L .42874 .44674 L .42874 .44633 L .42875 .44595 L .42876 .44559 L .4288 .44493 L .42887 .44423 L .42896 .44349 L .42908 .4428 L .42944 .44127 L .42986 .43997 L .43034 .43883 L .43094 .43763 L .43173 .43636 L .43344 .43423 L .43572 .43217 L .43827 .43046 L .44107 .42905 L .44284 .42833 L .44486 .42765 L .44883 .42665 L .45125 .4262 L .45365 .42587 L .45508 .42572 L .45642 .4256 L .45718 .42555 L .45802 .4255 L .45881 .42546 L .45957 .42543 L .46096 .42539 L .46226 .42538 L .46369 .42538 L .46529 .42541 L .46671 .42546 L Mistroke .46826 .42553 L .47112 .42572 L .47494 .42608 L .4787 .42653 L .48614 .42771 L .50306 .43149 L .52385 .43783 L .56805 .45601 L .62176 .48511 L .72607 .56325 L .77544 .61428 L .79906 .64482 L .81733 .67381 L .83146 .70343 L .83681 .71892 L .83891 .72666 L .84071 .73512 L .84132 .73869 L .84185 .74242 L .84221 .74563 L .8425 .74903 L .84263 .75095 L .84271 .75269 L .84279 .75605 L .8428 .75793 L .84276 .75995 L .84269 .76185 L .8426 .7636 L .84235 .76668 L .84204 .76954 L .84159 .77265 L .841 .77595 L .83941 .78257 L .83751 .78846 L .83232 .79982 L .82899 .8053 L .82474 .81113 L .81572 .82079 L .80543 .82908 L .79105 .83772 L .77557 .84452 L .75599 .85061 L .74509 .85309 L .73235 .85533 L .72047 .85687 L .71347 .85756 L .70683 .85808 L .70087 .85844 L .69414 .85873 L .69069 .85884 L Mistroke .68694 .85892 L .68002 .85898 L .6732 .85894 L .66558 .85877 L .65822 .8585 L .65125 .85815 L .63769 .85722 L .6227 .85583 L .60422 .85364 L .58335 .85059 L .54314 .84321 L .45988 .82258 L .37069 .79399 L .26263 .75197 L .01632 .63105 L Mfstroke .01632 .63105 m 0 .62091 L s .50101 .51362 m .49793 .51281 L .49418 .51157 L .48549 .50783 L .47459 .50169 L .46377 .49406 L .45398 .48555 L .44448 .47503 L .44021 .46893 L .43861 .46621 L .43708 .4632 L .43595 .46057 L .43499 .45775 L .43457 .45621 L .43427 .45481 L .43415 .45415 L .43404 .45342 L .43396 .45279 L .4339 .45212 L .43388 .45178 L .43386 .45141 L .43383 .45077 L .43383 .45041 L .43383 .45003 L .43384 .44967 L .43386 .44934 L .4339 .44871 L .43393 .44835 L .43397 .44803 L .43406 .4474 L .43416 .44681 L .4344 .44576 L .43475 .44462 L .43516 .44353 L .43571 .4424 L .43719 .4401 L .43812 .43902 L .43927 .43789 L .44167 .43605 L .44435 .43455 L .4476 .43321 L .44965 .43257 L .45168 .43206 L .45374 .43164 L .45577 .43132 L .45801 .43105 L .45924 .43094 L .4606 .43085 L .46185 .43078 L .46304 .43074 L Mistroke .46438 .43071 L .46514 .43071 L .46583 .43071 L .46714 .43072 L .46857 .43076 L .46997 .43082 L .47127 .43089 L .47378 .43108 L .47664 .43135 L .48273 .43214 L .49029 .43347 L .49797 .43516 L .51372 .43951 L .55516 .45544 L .60127 .47919 L .64763 .50881 L .69968 .54958 L .74557 .59448 L .78263 .64171 L .79884 .66921 L .80986 .69444 L .81383 .70712 L .81554 .71414 L .81674 .72045 L .81757 .72629 L .8179 .72956 L .81812 .73255 L .81824 .73515 L .8183 .73799 L .81828 .74094 L .81818 .74367 L .81797 .74683 L .8178 .74861 L .81761 .75024 L .81719 .75316 L .81662 .75635 L .81533 .76178 L .81355 .76742 L .81113 .77335 L .80793 .77955 L .80052 .79014 L .79145 .79942 L .78117 .80724 L .76778 .81483 L .75093 .82166 L .74142 .82457 L .73033 .82729 L .71909 .82942 L .70805 .83099 L .69755 .83208 L Mistroke .69135 .83255 L .68539 .83289 L .67895 .83315 L .67292 .83329 L .66953 .83332 L .66578 .83333 L .65883 .83326 L .65531 .83318 L .65155 .83307 L .64471 .83279 L .63718 .83237 L .62871 .83176 L .61301 .83031 L .59776 .8285 L .56768 .82393 L .53318 .81724 L .4464 .79474 L .35727 .76479 L .16294 .6805 L Mfstroke .16294 .6805 m 0 .58921 L s .5113 .51593 m .51086 .51602 L .51037 .5161 L .50989 .51616 L .50942 .5162 L .50913 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This gives one fundamental solution" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["X1[t_] = {1,1}E^(3*t)", "Input", AspectRatioFixed->True], Cell[BoxData[ \({E\^\(3\ t\), E\^\(3\ t\)}\)], "Output"] }, Open ]], Cell[TextData[ "To find another fundamental solution we need to find a generalized \ eigenevector. To do this we solve:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Solve[(A - 3*IdentityMatrix[2]).{w1,w2} == {1,1}]", "Input", AspectRatioFixed->True], Cell[BoxData[ \({{w1 \[Rule] \(-1\) + w2}}\)], "Output"] }, Open ]], Cell[TextData[{ "So ", StyleBox["{-1,0}", FontFamily->"Courier", FontWeight->"Bold"], " is one generalized eigenvector giving us a second fundamental solution:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["X2[t_] = {1,1} t E^(3*t) + {-1, 0} E^(3*t)", "Input", AspectRatioFixed->True], Cell[BoxData[ \({\(-E\^\(3\ t\)\) + E\^\(3\ t\)\ t, E\^\(3\ t\)\ t}\)], "Output"] }, Open ]], Cell[TextData["The general solution is:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["X[t_] = c1 X1[t] + c2 X2[t]", "Input", AspectRatioFixed->True], Cell[BoxData[ \({c1\ E\^\(3\ t\) + c2\ \((\(-E\^\(3\ t\)\) + E\^\(3\ t\)\ t)\), c1\ E\^\(3\ t\) + c2\ E\^\(3\ t\)\ t}\)], "Output"] }, 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.31086 L .54385 .30925 L .54749 .30807 L .55162 .30646 L .55964 .30244 L .56441 .29941 L .56979 .29532 L .57458 .29098 L .57994 .28516 L .58997 .27052 L .59521 .25994 L .59975 .24798 L .60179 .2413 L .60383 .23332 L Mistroke .60545 .22563 L .60696 .21671 L .60819 .20707 L .60905 .19759 L .60937 .19274 L .60951 .19012 L .60964 .1873 L .60973 .18477 L .60981 .18195 L .60986 .17933 L .6099 .17685 L .60991 .17364 L .60991 .172 L .6099 .17016 L .60986 .16687 L .6098 .16372 L .60969 .15993 L .60956 .15639 L .60915 .14814 L .60868 .14111 L .60803 .13312 L .60624 .11597 L .60084 .07927 L .59299 .03959 L Mfstroke .59299 .03959 m .58221 0 L s .5005 .30917 m .50074 .30923 L .50091 .30926 L .50114 .3093 L .5014 .30933 L .50169 .30936 L .50186 .30938 L .50204 .30939 L .50209 .30939 L .50215 .30939 L .50219 .30939 L .50225 .30939 L .50228 .30939 L .5023 .30939 L .50233 .30939 L .50236 .30939 L .50238 .30939 L .50239 .30939 L .50241 .30939 L .50243 .30939 L .50244 .30939 L .50246 .30939 L .50248 .30939 L .50249 .30939 L .50252 .30939 L .50255 .30939 L .50258 .30939 L .5026 .30939 L .50267 .30939 L .50273 .30939 L .50284 .30939 L .50297 .30938 L .50311 .30937 L .50325 .30936 L .50358 .30932 L .50391 .30927 L .50428 .3092 L .50502 .309 L .50543 .30886 L .5059 .30867 L .50681 .30819 L .50735 .30783 L .50795 .30735 L .50849 .30683 L .50909 .30615 L .5102 .30442 L .51078 .30317 L .51127 .30177 L .51148 .30098 L .5117 .30004 L Mistroke .51187 .29914 L .51202 .29809 L .51208 .29752 L .51214 .29696 L .51218 .29639 L .51221 .29584 L .51224 .29531 L .51225 .29503 L .51225 .29472 L .51226 .2944 L .51226 .29405 L .51226 .29372 L .51226 .29341 L .51226 .29306 L .51225 .29272 L .51224 .29235 L .51222 .29193 L .51218 .29106 L .51212 .29023 L .51198 .28862 L .51177 .28679 L .51108 .28243 L .50988 .27679 L .5061 .26369 L .49883 .24436 L .46755 .18104 L .39763 .06702 L Mfstroke .39763 .06702 m .3503 0 L s .49975 .30886 m .4996 .30877 L .49931 .30859 L .49887 .30832 L .49818 .30789 L .49696 .30714 L .49503 .30595 L .49163 .30384 L .48615 .30046 L .47753 .29513 L .4624 .28578 L .43827 .27086 L .40056 .24756 L .3348 .20692 L .23074 .14261 L .04736 .02927 L s .04736 .02927 m 0 0 L s .49901 .30856 m .49845 .30831 L .49749 .30789 L .49605 .30728 L .49392 .30639 L .49038 .30495 L .4851 .30288 L .47633 .29956 L .46318 .29482 L .44388 .28822 L .41254 .2782 L .36661 .26473 L .30094 .24745 L .19725 .224 L .05038 .19755 L s .05038 .19755 m 0 .19055 L s .49826 .30825 m .49731 .30785 L .49566 .30719 L .49324 .30623 L .48966 .30488 L .48381 .30276 L .47516 .2998 L .46104 .29529 L .44021 .28918 L .41023 .28131 L .36268 .27063 L .33091 .26455 L .29495 .2586 L .25103 .25257 L .20132 .24733 L .1717 .24497 L .155 .24389 L .13634 .24289 L .11829 .24214 L .10124 .24161 L .09276 .24141 L .08823 .24133 L .0834 .24125 L .07913 .2412 L .0744 .24115 L .07005 .24112 L .06596 .2411 L .06075 .24109 L .0581 .24109 L .05516 .24109 L .04992 .24112 L .04496 .24116 L .04209 .24119 L .03907 .24123 L .03361 .24131 L .02109 .24157 L .01001 .24188 L s .01001 .24188 m 0 .24223 L s .49752 .30794 m .49616 .30739 L .49384 .30648 L .49042 .30519 L .4854 .30338 L .47723 .30058 L .46522 .29673 L .44575 .29101 L .41724 .28354 L .37658 .2744 L .34935 .26913 L .31282 .26305 L .27012 .25726 L .2233 .25246 L .19511 .25033 L .16591 .24871 L .14888 .24803 L .14047 .24778 L .13238 .24757 L .1277 .24748 L .12257 .24739 L .11826 .24733 L .11347 .24728 L .11076 .24725 L .10822 .24724 L .10552 .24722 L .1026 .24722 L .09987 .24721 L .09727 .24722 L .09236 .24723 L .08796 .24726 L .08317 .24731 L .07861 .24737 L .07438 .24744 L .06442 .24765 L .05372 .24795 L .03292 .24876 L .00919 .25005 L s .00919 .25005 m 0 .2507 L s .50223 .30994 m .50343 .31039 L .50547 .31113 L .50845 .31215 L .51278 .31353 L .51973 .31558 L .52981 .31823 L .54588 .32185 L .55676 .32393 L .56892 .32594 L .58404 .32799 L .60095 .32975 L .61024 .33048 L .6211 .33112 L .6277 .3314 L .63108 .33151 L .63428 .3316 L .63723 .33166 L .63876 .33169 L .64044 .33171 L .64141 .33172 L .6423 .33173 L .64408 .33174 L .64571 .33175 L .64745 .33175 L .64916 .33175 L .65013 .33174 L .65104 .33174 L .65271 .33172 L .6545 .3317 L .65765 .33165 L .66119 .33157 L .66503 .33145 L .66926 .3313 L .67716 .33091 L .68493 .33042 L .69342 .32975 L .70944 .3281 L .72814 .32555 L .74961 .32176 L .79234 .31132 L .81848 .3029 L .84869 .2911 L .90828 .26045 L .97226 .21437 L s .97226 .21437 m 1 .18869 L s .50149 .30963 m .50229 .30993 L .50365 .31042 L .50563 .3111 L .50852 .31203 L .51315 .31339 L .51987 .31516 L .53059 .31757 L .53784 .31896 L .54594 .3203 L .55603 .32167 L .5673 .32284 L .57349 .32332 L .58073 .32375 L .58513 .32394 L .58738 .32401 L .58952 .32407 L .59149 .32411 L .59251 .32413 L .59363 .32415 L .59427 .32415 L .59487 .32416 L .59605 .32417 L .59714 .32417 L .5983 .32417 L .59944 .32417 L .60009 .32417 L .60069 .32417 L .60181 .32416 L .603 .32414 L .6051 .32411 L .60746 .32405 L .61002 .32398 L .61284 .32387 L .6181 .32361 L .62328 .32328 L .62895 .32284 L .63963 .32174 L .65209 .32004 L .6664 .31751 L .69489 .31055 L .71232 .30494 L .73246 .29707 L .77219 .27664 L .81484 .24592 L .8373 .22512 L .8624 .19707 L .9063 .13093 L .92968 .08209 L .94937 .0283 L Mistroke Mfstroke .94937 .0283 m .95769 0 L s .50074 .30932 m .50114 .30948 L .50182 .30972 L .50282 .31006 L .50426 .31052 L .50658 .3112 L .50994 .31209 L .51529 .31329 L .51892 .31399 L .52297 .31466 L .52801 .31534 L .53365 .31593 L .53675 .31617 L .54037 .31638 L .54257 .31648 L .54369 .31652 L .54476 .31654 L .54574 .31657 L .54625 .31657 L .54681 .31658 L .54714 .31659 L .54743 .31659 L .54803 .31659 L .54857 .3166 L .54915 .3166 L .54972 .31659 L .55004 .31659 L .55035 .31659 L .5509 .31659 L .5515 .31658 L .55255 .31656 L .55373 .31653 L .55501 .3165 L .55642 .31644 L .55905 .31632 L .56164 .31615 L .56447 .31593 L .56981 .31538 L .57605 .31453 L .5832 .31326 L .59745 .30978 L .60616 .30698 L .61623 .30304 L .63609 .29283 L .65742 .27747 L .66865 .26707 L .6812 .25304 L .70315 .21997 L .71484 .19555 L .72468 .16866 L Mistroke .729 .15399 L .73323 .13696 L .73653 .12098 L .73967 .10229 L .74134 .09001 L .7427 .07818 L .7438 .06659 L .74479 .0533 L .74515 .04733 L .74548 .0408 L .74573 .03504 L .74595 .02867 L .74605 .02486 L .74613 .0213 L .74625 .01418 L .7463 .00757 L .74629 .00042 L Mfstroke .74629 .00042 m .74629 0 L s .5 .30902 m .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L s .49926 .30871 m .49886 .30856 L .49818 .30831 L .49718 .30797 L .49574 .30751 L .49342 .30683 L .49006 .30595 L .48471 .30474 L .48108 .30405 L .47703 .30338 L .47199 .30269 L .46635 .30211 L .46325 .30186 L .45963 .30165 L .45743 .30156 L .45631 .30152 L .45524 .30149 L .45426 .30147 L .45375 .30146 L .45319 .30145 L .45286 .30145 L .45257 .30145 L .45197 .30144 L .45143 .30144 L .45085 .30144 L .45028 .30144 L .44996 .30144 L .44965 .30144 L .4491 .30145 L .4485 .30146 L .44745 .30147 L .44627 .3015 L .44499 .30154 L .44358 .30159 L .44095 .30172 L .43836 .30188 L .43553 .30211 L .43019 .30265 L .42395 .30351 L .4168 .30477 L .40255 .30825 L .39384 .31105 L .38377 .31499 L .36391 .3252 L .34258 .34057 L .33135 .35097 L .3188 .36499 L .29685 .39806 L .28516 .42248 L .27532 .44938 L Mistroke .271 .46405 L .26677 .48108 L .26347 .49706 L .26033 .51575 L .25866 .52802 L .2573 .53985 L .2562 .55144 L .25521 .56473 L .25485 .57071 L .25452 .57723 L .25427 .58299 L .25405 .58937 L .25395 .59317 L .25387 .59673 L .25375 .60385 L .2537 .61046 L .25371 .61761 L Mfstroke .25371 .61761 m .25371 .61803 L s .49851 .3084 m .49771 .3081 L .49635 .30761 L .49437 .30693 L .49148 .30601 L .48685 .30464 L .48013 .30287 L .46941 .30046 L .46216 .29908 L .45406 .29774 L .44397 .29637 L .4327 .2952 L .42651 .29471 L .41927 .29428 L .41487 .2941 L .41262 .29402 L .41048 .29396 L .40851 .29392 L .40749 .2939 L .40637 .29389 L .40573 .29388 L .40513 .29387 L .40395 .29387 L .40286 .29386 L .4017 .29386 L .40056 .29386 L .39991 .29387 L .39931 .29387 L .39819 .29388 L .397 .29389 L .3949 .29393 L .39254 .29398 L .38998 .29406 L .38716 .29416 L .3819 .29442 L .37672 .29475 L .37105 .2952 L .36037 .29629 L .34791 .29799 L .3336 .30052 L .30511 .30748 L .28768 .31309 L .26754 .32096 L .22781 .34139 L .18516 .37212 L .1627 .39291 L .1376 .42096 L .0937 .48711 L .07032 .53595 L .05063 .58974 L Mistroke Mfstroke .05063 .58974 m .04231 .61803 L s .49777 .3081 m .49657 .30764 L .49453 .30691 L .49155 .30589 L .48722 .3045 L .48027 .30246 L .47019 .2998 L .45412 .29618 L .44324 .2941 L .43108 .2921 L .41596 .29004 L .39905 .28828 L .38976 .28756 L .3789 .28692 L .3723 .28663 L .36892 .28652 L .36572 .28644 L .36277 .28637 L .36124 .28635 L .35956 .28632 L .35859 .28631 L .3577 .2863 L .35592 .28629 L .35429 .28628 L .35255 .28628 L .35084 .28628 L .34987 .28629 L .34896 .28629 L .34729 .28631 L .3455 .28633 L .34235 .28638 L .33881 .28646 L .33497 .28658 L .33074 .28674 L .32284 .28712 L .31507 .28762 L .30658 .28829 L .29056 .28993 L .27186 .29248 L .25039 .29627 L .20766 .30672 L .18152 .31513 L .15131 .32693 L .09172 .35758 L .02774 .40367 L s .02774 .40367 m 0 .42935 L s .50248 .31009 m .50384 .31064 L .50616 .31155 L .50958 .31284 L .5146 .31466 L .52277 .31746 L .53478 .3213 L .55425 .32703 L .58276 .33449 L .62342 .34364 L .65065 .3489 L .68718 .35499 L .72988 .36078 L .7767 .36557 L .80489 .3677 L .83409 .36933 L .85112 .37 L .85953 .37026 L .86762 .37046 L .8723 .37056 L .87743 .37065 L .88174 .37071 L .88653 .37076 L .88924 .37078 L .89178 .3708 L .89448 .37081 L .8974 .37082 L .90013 .37082 L .90273 .37082 L .90764 .3708 L .91204 .37077 L .91683 .37072 L .92139 .37066 L .92562 .37059 L .93558 .37039 L .94628 .37008 L .96708 .36927 L .99081 .36798 L s .99081 .36798 m 1 .36734 L s .50174 .30978 m .50269 .31018 L .50434 .31085 L .50676 .3118 L .51034 .31315 L .51619 .31527 L .52484 .31823 L .53896 .32275 L .55979 .32885 L .58977 .33672 L .63732 .34741 L .66909 .35349 L .70505 .35944 L .74897 .36546 L .79868 .3707 L .8283 .37306 L .845 .37414 L .86366 .37514 L .88171 .3759 L .89876 .37643 L .90724 .37662 L .91177 .3767 L .9166 .37678 L .92087 .37684 L .9256 .37689 L .92995 .37692 L .93404 .37694 L .93925 .37695 L .9419 .37695 L .94484 .37694 L .95008 .37691 L .95504 .37687 L .95791 .37684 L .96093 .37681 L .96639 .37672 L .97891 .37647 L .98999 .37616 L s .98999 .37616 m 1 .37581 L s .50099 .30948 m .50155 .30972 L .50251 .31014 L .50395 .31076 L .50608 .31165 L .50962 .31308 L .5149 .31516 L .52367 .31847 L .53682 .32321 L .55612 .32981 L .58746 .33983 L .63339 .3533 L .69906 .37059 L .80275 .39403 L .94962 .42048 L s .94962 .42048 m 1 .42749 L s .50025 .30917 m .5004 .30927 L .50069 .30944 L .50113 .30971 L .50182 .31014 L .50304 .31089 L .50497 .31209 L .50837 .31419 L .51385 .31757 L .52247 .3229 L .5376 .33225 L .56173 .34717 L .59944 .37047 L .6652 .41111 L .76926 .47543 L .95264 .58876 L s .95264 .58876 m 1 .61803 L s .4995 .30886 m .49926 .30881 L .49909 .30878 L .49886 .30874 L .4986 .3087 L .49831 .30867 L .49814 .30866 L .49796 .30865 L .49791 .30865 L .49785 .30864 L .49781 .30864 L .49775 .30864 L .49772 .30864 L .4977 .30864 L .49767 .30864 L .49764 .30864 L .49762 .30864 L .49761 .30864 L .49759 .30864 L .49757 .30864 L .49756 .30864 L .49754 .30864 L .49752 .30864 L .49751 .30864 L .49748 .30864 L .49745 .30864 L .49742 .30864 L .4974 .30864 L .49733 .30864 L .49727 .30864 L .49716 .30865 L .49703 .30866 L .49689 .30866 L .49675 .30868 L .49642 .30871 L .49609 .30876 L .49572 .30883 L .49498 .30903 L .49457 .30917 L .4941 .30936 L .49319 .30984 L .49265 .3102 L .49205 .31069 L .49151 .3112 L .49091 .31189 L .4898 .31362 L .48922 .31486 L .48873 .31627 L .48852 .31705 L .4883 .31799 L Mistroke .48813 .3189 L .48798 .31994 L .48792 .32052 L .48786 .32108 L .48782 .32165 L .48779 .32219 L .48776 .32273 L .48775 .323 L .48775 .32331 L .48774 .32363 L .48774 .32398 L .48774 .32431 L .48774 .32462 L .48774 .32498 L .48775 .32531 L .48776 .32569 L .48778 .3261 L .48782 .32697 L .48788 .3278 L .48802 .32942 L .48823 .33125 L .48892 .3356 L .49012 .34124 L .4939 .35435 L .50117 .37368 L .53245 .437 L .60237 .55101 L Mfstroke .60237 .55101 m .6497 .61803 L s .49876 .30856 m .49811 .30835 L .49704 .30803 L .4955 .30763 L .49331 .30714 L .49168 .30682 L .48988 .30652 L .48761 .30621 L .48509 .30594 L .48368 .30583 L .48204 .30573 L .48114 .30569 L .48025 .30566 L .47973 .30565 L .47924 .30564 L .47897 .30564 L .47882 .30564 L .47866 .30564 L .47851 .30564 L .47837 .30564 L .47823 .30563 L .4781 .30563 L .47786 .30564 L .4776 .30564 L .47735 .30564 L .47712 .30564 L .47685 .30564 L .47655 .30565 L .47599 .30566 L .47541 .30568 L .47477 .30571 L .47358 .30576 L .47245 .30584 L .47136 .30592 L .46873 .30619 L .46585 .30659 L .46259 .30718 L .45615 .30878 L .45251 .30997 L .44838 .31157 L .44036 .31559 L .43559 .31862 L .43021 .32271 L .42542 .32706 L .42006 .33287 L .41003 .34752 L .40479 .3581 L .40025 .37005 L .39821 .37673 L .39617 .38471 L Mistroke .39455 .3924 L .39304 .40132 L .39181 .41096 L .39095 .42044 L .39063 .42529 L .39049 .42791 L .39036 .43074 L .39027 .43327 L .39019 .43609 L .39014 .4387 L .3901 .44119 L .39009 .44439 L .39009 .44604 L .3901 .44787 L .39014 .45117 L .3902 .45432 L .39031 .4581 L .39044 .46164 L .39085 .46989 L .39132 .47692 L .39197 .48491 L .39376 .50207 L .39916 .53876 L .40701 .57845 L Mfstroke .40701 .57845 m .41779 .61803 L s .49802 .30825 m .49697 .30789 L .49521 .30733 L .49268 .30659 L .48905 .30563 L .48331 .30433 L .47516 .30287 L .46925 .30206 L .46612 .3017 L .46249 .30136 L .45885 .30108 L .45679 .30096 L .45447 .30084 L .45224 .30076 L .45115 .30072 L .45012 .3007 L .44907 .30068 L .44851 .30067 L .4479 .30066 L .44732 .30066 L 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.31052 L .50658 .3112 L .50994 .31209 L .51529 .31329 L .51892 .31399 L .52297 .31466 L .52801 .31534 L .53365 .31593 L .53675 .31617 L .54037 .31638 L .54257 .31648 L .54369 .31652 L .54476 .31654 L .54574 .31657 L .54625 .31657 L .54681 .31658 L .54714 .31659 L .54743 .31659 L .54803 .31659 L .54857 .3166 L .54915 .3166 L .54972 .31659 L .55004 .31659 L .55035 .31659 L .5509 .31659 L .5515 .31658 L .55255 .31656 L .55373 .31653 L .55501 .3165 L .55642 .31644 L .55905 .31632 L .56164 .31615 L .56447 .31593 L .56981 .31538 L .57605 .31453 L .5832 .31326 L .59745 .30978 L .60616 .30698 L .61623 .30304 L .63609 .29283 L .65742 .27747 L .66865 .26707 L .6812 .25304 L .70315 .21997 L .71484 .19555 L .72468 .16866 L Mistroke .729 .15399 L .73323 .13696 L .73653 .12098 L .73967 .10229 L .74134 .09001 L .7427 .07818 L .7438 .06659 L .74479 .0533 L .74515 .04733 L .74548 .0408 L .74573 .03504 L .74595 .02867 L .74605 .02486 L .74613 .0213 L .74625 .01418 L .7463 .00757 L .74629 .00042 L Mfstroke .74629 .00042 m .74629 0 L s .5 .30902 m .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L s .49926 .30871 m .49886 .30856 L .49818 .30831 L .49718 .30797 L .49574 .30751 L .49342 .30683 L .49006 .30595 L .48471 .30474 L .48108 .30405 L .47703 .30338 L .47199 .30269 L .46635 .30211 L .46325 .30186 L .45963 .30165 L .45743 .30156 L .45631 .30152 L .45524 .30149 L .45426 .30147 L .45375 .30146 L .45319 .30145 L .45286 .30145 L .45257 .30145 L .45197 .30144 L .45143 .30144 L .45085 .30144 L .45028 .30144 L .44996 .30144 L .44965 .30144 L .4491 .30145 L .4485 .30146 L .44745 .30147 L .44627 .3015 L .44499 .30154 L .44358 .30159 L .44095 .30172 L .43836 .30188 L .43553 .30211 L .43019 .30265 L .42395 .30351 L .4168 .30477 L .40255 .30825 L .39384 .31105 L .38377 .31499 L .36391 .3252 L .34258 .34057 L .33135 .35097 L .3188 .36499 L .29685 .39806 L .28516 .42248 L .27532 .44938 L Mistroke .271 .46405 L .26677 .48108 L .26347 .49706 L .26033 .51575 L .25866 .52802 L .2573 .53985 L .2562 .55144 L .25521 .56473 L .25485 .57071 L .25452 .57723 L .25427 .58299 L .25405 .58937 L .25395 .59317 L .25387 .59673 L .25375 .60385 L .2537 .61046 L .25371 .61761 L Mfstroke .25371 .61761 m .25371 .61803 L s .49851 .3084 m .49771 .3081 L .49635 .30761 L .49437 .30693 L .49148 .30601 L .48685 .30464 L .48013 .30287 L .46941 .30046 L .46216 .29908 L .45406 .29774 L .44397 .29637 L .4327 .2952 L .42651 .29471 L .41927 .29428 L .41487 .2941 L .41262 .29402 L .41048 .29396 L .40851 .29392 L .40749 .2939 L .40637 .29389 L .40573 .29388 L .40513 .29387 L .40395 .29387 L .40286 .29386 L .4017 .29386 L .40056 .29386 L .39991 .29387 L .39931 .29387 L .39819 .29388 L .397 .29389 L .3949 .29393 L .39254 .29398 L .38998 .29406 L .38716 .29416 L .3819 .29442 L .37672 .29475 L .37105 .2952 L .36037 .29629 L .34791 .29799 L .3336 .30052 L .30511 .30748 L .28768 .31309 L .26754 .32096 L .22781 .34139 L .18516 .37212 L .1627 .39291 L .1376 .42096 L .0937 .48711 L .07032 .53595 L .05063 .58974 L Mistroke Mfstroke .05063 .58974 m .04231 .61803 L s .49777 .3081 m .49657 .30764 L .49453 .30691 L .49155 .30589 L .48722 .3045 L .48027 .30246 L .47019 .2998 L .45412 .29618 L .44324 .2941 L .43108 .2921 L .41596 .29004 L .39905 .28828 L .38976 .28756 L .3789 .28692 L .3723 .28663 L .36892 .28652 L .36572 .28644 L .36277 .28637 L .36124 .28635 L .35956 .28632 L .35859 .28631 L .3577 .2863 L .35592 .28629 L .35429 .28628 L .35255 .28628 L .35084 .28628 L .34987 .28629 L .34896 .28629 L .34729 .28631 L .3455 .28633 L .34235 .28638 L .33881 .28646 L .33497 .28658 L .33074 .28674 L .32284 .28712 L .31507 .28762 L .30658 .28829 L .29056 .28993 L .27186 .29248 L .25039 .29627 L .20766 .30672 L .18152 .31513 L .15131 .32693 L .09172 .35758 L .02774 .40367 L s .02774 .40367 m 0 .42935 L s .50248 .31009 m .50384 .31064 L .50616 .31155 L .50958 .31284 L .5146 .31466 L .52277 .31746 L .53478 .3213 L .55425 .32703 L .58276 .33449 L .62342 .34364 L .65065 .3489 L .68718 .35499 L .72988 .36078 L .7767 .36557 L .80489 .3677 L .83409 .36933 L .85112 .37 L .85953 .37026 L .86762 .37046 L .8723 .37056 L .87743 .37065 L .88174 .37071 L .88653 .37076 L .88924 .37078 L .89178 .3708 L .89448 .37081 L .8974 .37082 L .90013 .37082 L .90273 .37082 L .90764 .3708 L .91204 .37077 L .91683 .37072 L .92139 .37066 L .92562 .37059 L .93558 .37039 L .94628 .37008 L .96708 .36927 L .99081 .36798 L s .99081 .36798 m 1 .36734 L s .50174 .30978 m .50269 .31018 L .50434 .31085 L .50676 .3118 L .51034 .31315 L .51619 .31527 L .52484 .31823 L .53896 .32275 L .55979 .32885 L .58977 .33672 L .63732 .34741 L .66909 .35349 L .70505 .35944 L .74897 .36546 L .79868 .3707 L .8283 .37306 L .845 .37414 L .86366 .37514 L .88171 .3759 L .89876 .37643 L .90724 .37662 L .91177 .3767 L .9166 .37678 L .92087 .37684 L .9256 .37689 L .92995 .37692 L .93404 .37694 L .93925 .37695 L .9419 .37695 L .94484 .37694 L .95008 .37691 L .95504 .37687 L .95791 .37684 L .96093 .37681 L .96639 .37672 L .97891 .37647 L .98999 .37616 L s .98999 .37616 m 1 .37581 L s .50099 .30948 m .50155 .30972 L .50251 .31014 L .50395 .31076 L .50608 .31165 L .50962 .31308 L .5149 .31516 L .52367 .31847 L .53682 .32321 L .55612 .32981 L .58746 .33983 L .63339 .3533 L .69906 .37059 L .80275 .39403 L .94962 .42048 L s .94962 .42048 m 1 .42749 L s .50025 .30917 m .5004 .30927 L .50069 .30944 L .50113 .30971 L .50182 .31014 L .50304 .31089 L .50497 .31209 L .50837 .31419 L .51385 .31757 L .52247 .3229 L .5376 .33225 L .56173 .34717 L .59944 .37047 L .6652 .41111 L .76926 .47543 L .95264 .58876 L s .95264 .58876 m 1 .61803 L s .4995 .30886 m .49926 .30881 L .49909 .30878 L .49886 .30874 L .4986 .3087 L .49831 .30867 L .49814 .30866 L .49796 .30865 L .49791 .30865 L .49785 .30864 L .49781 .30864 L .49775 .30864 L .49772 .30864 L .4977 .30864 L .49767 .30864 L .49764 .30864 L .49762 .30864 L .49761 .30864 L .49759 .30864 L .49757 .30864 L .49756 .30864 L .49754 .30864 L .49752 .30864 L .49751 .30864 L .49748 .30864 L .49745 .30864 L .49742 .30864 L .4974 .30864 L .49733 .30864 L .49727 .30864 L .49716 .30865 L .49703 .30866 L .49689 .30866 L .49675 .30868 L .49642 .30871 L .49609 .30876 L .49572 .30883 L .49498 .30903 L .49457 .30917 L .4941 .30936 L .49319 .30984 L .49265 .3102 L .49205 .31069 L .49151 .3112 L .49091 .31189 L .4898 .31362 L .48922 .31486 L .48873 .31627 L .48852 .31705 L .4883 .31799 L Mistroke .48813 .3189 L .48798 .31994 L .48792 .32052 L .48786 .32108 L .48782 .32165 L .48779 .32219 L .48776 .32273 L .48775 .323 L .48775 .32331 L .48774 .32363 L .48774 .32398 L .48774 .32431 L .48774 .32462 L .48774 .32498 L .48775 .32531 L .48776 .32569 L .48778 .3261 L .48782 .32697 L .48788 .3278 L .48802 .32942 L .48823 .33125 L .48892 .3356 L .49012 .34124 L .4939 .35435 L .50117 .37368 L .53245 .437 L .60237 .55101 L Mfstroke .60237 .55101 m .6497 .61803 L s .49876 .30856 m .49811 .30835 L .49704 .30803 L .4955 .30763 L .49331 .30714 L .49168 .30682 L .48988 .30652 L .48761 .30621 L .48509 .30594 L .48368 .30583 L .48204 .30573 L .48114 .30569 L .48025 .30566 L .47973 .30565 L .47924 .30564 L .47897 .30564 L .47882 .30564 L .47866 .30564 L .47851 .30564 L .47837 .30564 L .47823 .30563 L .4781 .30563 L .47786 .30564 L .4776 .30564 L .47735 .30564 L .47712 .30564 L .47685 .30564 L .47655 .30565 L .47599 .30566 L .47541 .30568 L .47477 .30571 L .47358 .30576 L .47245 .30584 L .47136 .30592 L .46873 .30619 L .46585 .30659 L .46259 .30718 L .45615 .30878 L .45251 .30997 L .44838 .31157 L .44036 .31559 L .43559 .31862 L .43021 .32271 L .42542 .32706 L .42006 .33287 L .41003 .34752 L .40479 .3581 L .40025 .37005 L .39821 .37673 L .39617 .38471 L Mistroke .39455 .3924 L .39304 .40132 L .39181 .41096 L .39095 .42044 L .39063 .42529 L .39049 .42791 L .39036 .43074 L .39027 .43327 L .39019 .43609 L .39014 .4387 L .3901 .44119 L .39009 .44439 L .39009 .44604 L .3901 .44787 L .39014 .45117 L .3902 .45432 L .39031 .4581 L .39044 .46164 L .39085 .46989 L .39132 .47692 L .39197 .48491 L .39376 .50207 L .39916 .53876 L .40701 .57845 L Mfstroke .40701 .57845 m .41779 .61803 L s .49802 .30825 m .49697 .30789 L .49521 .30733 L .49268 .30659 L .48905 .30563 L .48331 .30433 L .47516 .30287 L .46925 .30206 L .46612 .3017 L .46249 .30136 L .45885 .30108 L .45679 .30096 L .45447 .30084 L .45224 .30076 L .45115 .30072 L .45012 .3007 L .44907 .30068 L .44851 .30067 L .4479 .30066 L .44732 .30066 L .44676 .30065 L .44626 .30065 L .44571 .30065 L .44507 .30065 L .44448 .30066 L .4438 .30066 L .44314 .30067 L .44195 .30069 L .44064 .30072 L .43917 .30077 L .43756 .30083 L .4346 .30098 L .43141 .30118 L .4283 .30144 L .42076 .30227 L .4135 .30335 L .40528 .30493 L .38906 .3092 L .37991 .31233 L .36955 .31656 L .34945 .32709 L .33751 .33502 L .32409 .34571 L .31218 .35704 L .29891 .37218 L .2742 .41026 L .26146 .43773 L .25054 .46874 L .24569 .48607 L .24089 .50676 L Mistroke .23715 .52669 L .23376 .54979 L .23229 .56239 L .23109 .57476 L .2301 .58731 L .22936 .5993 L .22881 .61115 L .2286 .61727 L Mfstroke .2286 .61727 m .22858 .61803 L s .50273 .31024 m .50424 .31089 L .50685 .31198 L .51071 .31354 L .51642 .31579 L .52581 .31933 L .53974 .32437 L .56263 .3322 L .59661 .34305 L .64588 .35752 L .72477 .37822 L .83844 .40373 L .99774 .43227 L s .99774 .43227 m 1 .43259 L s .50198 .30994 m .5031 .31043 L .50502 .31127 L .50789 .3125 L .51216 .31428 L .51923 .31715 L .52981 .3213 L .54733 .32792 L .57364 .33741 L .61223 .35061 L .67492 .37065 L .76678 .39759 L .89812 .43216 L s .89812 .43216 m 1 .45519 L s .50124 .30963 m .50195 .30997 L .5032 .31057 L .50508 .31146 L .50791 .31278 L .51265 .31496 L .51987 .31823 L .53204 .32364 L .55067 .33177 L .57858 .3437 L .62506 .36307 L .69512 .39146 L .7985 .43204 L .96795 .49613 L s .96795 .49613 m 1 .50772 L s .5005 .30932 m .50081 .30952 L .50137 .30986 L .50226 .31041 L .50365 .31127 L .50608 .31277 L .50993 .31516 L .51674 .31937 L .52769 .32613 L .54493 .33679 L .5752 .35549 L .62346 .38532 L .69887 .43193 L .8304 .51321 L s .8304 .51321 m 1 .61803 L s .49975 .30902 m .49971 .30903 L .49966 .30906 L .49961 .3091 L .49958 .30913 L .49955 .30916 L .49952 .3092 L .49949 .30925 L .49947 .3093 L .49944 .30937 L .49942 .30944 L .49941 .30949 L .4994 .30954 L .4994 .30959 L .49939 .30964 L .49939 .30967 L .49939 .30968 L .49939 .3097 L .49939 .30972 L .49939 .30973 L .49939 .30975 L .49939 .30976 L .49939 .30978 L .49939 .3098 L .49939 .30982 L .49939 .30983 L .49939 .30985 L .49939 .30987 L .49939 .30991 L .4994 .30996 L .4994 .31001 L .49941 .3101 L .49944 .31029 L .49949 .31053 L .49965 .31113 L .49998 .31203 L .50138 .31496 L .50454 .32021 L .51085 .32928 L .52431 .34665 L .54957 .37663 L .60121 .43393 L .69554 .53308 L s .69554 .53308 m .77996 .61803 L s .49901 .30871 m .49852 .3086 L .49818 .30853 L .49772 .30846 L .4972 .30839 L .49662 .30833 L .49628 .3083 L .49592 .30828 L .49582 .30828 L .49571 .30827 L .49561 .30827 L .49551 .30827 L .49545 .30827 L .49539 .30826 L .49533 .30826 L .49528 .30826 L .49525 .30826 L .49522 .30826 L .49518 .30826 L .49515 .30826 L .49511 .30826 L .49508 .30826 L .49505 .30826 L .49502 .30826 L .49496 .30826 L .4949 .30826 L .49485 .30826 L .49479 .30827 L .49466 .30827 L .49454 .30827 L .49431 .30828 L .49406 .30829 L .49378 .30831 L .4935 .30834 L .49283 .30841 L .49217 .30851 L .49143 .30865 L .48996 .30904 L .48914 .30933 L .4882 .30971 L .48639 .31067 L .48531 .31139 L .48409 .31236 L .48302 .31339 L .48182 .31476 L .47959 .31821 L .47845 .32071 L .47746 .32352 L .47703 .32509 L .4766 .32697 L Mistroke .47627 .32878 L .47596 .33087 L .47583 .33201 L .47573 .33314 L .47564 .33427 L .47557 .33536 L .47553 .33644 L .47551 .33699 L .47549 .33761 L .47548 .33825 L .47548 .33894 L .47548 .33961 L .47548 .34023 L .47549 .34093 L .4755 .34161 L .47553 .34236 L .47556 .34318 L .47565 .34493 L .47576 .34659 L .47604 .34982 L .47647 .35347 L .47784 .36219 L .48025 .37347 L .48781 .39967 L .50234 .43834 L .5649 .56498 L Mfstroke .5649 .56498 m .59744 .61803 L s .49826 .3084 m .49737 .30814 L .4959 .30775 L .49381 .30728 L .49087 .30676 L .4889 .30648 L .48635 .30621 L .48491 .3061 L .48411 .30605 L .48321 .30601 L .48238 .30598 L .4819 .30596 L .48144 .30595 L .48122 .30595 L .481 .30595 L .48081 .30594 L .4806 .30594 L .48036 .30594 L .48023 .30594 L .48011 .30594 L .47988 .30594 L .47966 .30594 L .47943 .30594 L .47919 .30595 L .47896 .30595 L .47875 .30595 L .47822 .30596 L .47771 .30598 L .47666 .30602 L .47547 .30609 L .4744 .30616 L .47322 .30627 L .47098 .30652 L .46845 .30689 L .46552 .30744 L .45967 .30893 L .45633 .31005 L .45244 .3116 L .44394 .31608 L .43535 .32246 L .43033 .32728 L .42572 .33264 L .41695 .34624 L .41228 .35629 L .40833 .36744 L .40665 .37338 L .40499 .38032 L .4037 .38683 L .40247 .39451 L .40182 .39948 L Mistroke .4013 .40431 L .40085 .40956 L .40051 .41452 L .40038 .41694 L .40026 .41962 L .40017 .42198 L .4001 .42461 L .40004 .42748 L .40002 .42907 L .40001 .43057 L .4 .43331 L .40002 .43629 L .40006 .43926 L .4001 .44097 L .40014 .44259 L .40025 .44592 L .40039 .44958 L .40076 .45643 L .40121 .46296 L .4027 .47902 L .40502 .49753 L .41103 .5331 L .42119 .5789 L Mfstroke .42119 .5789 m .43275 .61803 L s .50297 .3104 m .50464 .31114 L .50753 .3124 L .51184 .31424 L .51825 .31691 L .52885 .32121 L .54471 .32744 L .571 .33737 L .61046 .35161 L .66835 .37141 L .76237 .40146 L .90017 .44188 L s .90017 .44188 m 1 .46815 L s .50223 .31009 m .5035 .31068 L .50571 .3117 L .50902 .3132 L .51399 .31541 L .52227 .31903 L .53477 .32437 L .5557 .3331 L .58748 .34597 L .6347 .3645 L .71251 .39388 L .82851 .43574 L .99756 .49361 L s .99756 .49361 m 1 .4944 L s .50149 .30978 m .50235 .31022 L .50388 .31099 L .5062 .31215 L .50973 .3139 L .51569 .31684 L .52484 .3213 L .54041 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AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Solution"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Eigensystem[A]", "Input", AspectRatioFixed->True], Cell[BoxData[ \({{\(-3\), \(-3\), \(-3\)}, {{1, 1, 1}, {0, 0, 0}, {0, 0, 0}}}\)], "Output"] }, Open ]], Cell[TextData[{ "The eigenvalue ", StyleBox["-3", FontFamily->"Courier", FontWeight->"Bold"], " is repeated three times. Only one non-trivial eigenvector, ", StyleBox["{1,1,1}", FontFamily->"Courier", FontWeight->"Bold"], ", is found which gives a fundamental solution:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["X1[t_] = {1,1,1}E^(-3t);", "Input", AspectRatioFixed->True], Cell[TextData[ "We now try to find two generalized eigenvectors. First solve:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Solve[(A-(-3)IdentityMatrix[3]).{w1,w2,w3}=={1,1,1}]", "Input", AspectRatioFixed->True], Cell[BoxData[ \({{w1 \[Rule] \(-1\) + w3, w2 \[Rule] w3}}\)], "Output"] }, Open ]], Cell[TextData[{ "Choose ", StyleBox["w3 == 0", FontFamily->"Courier", FontWeight->"Bold"], " to get the generalized eigenvector ", StyleBox["{-1,0,0}", FontFamily->"Courier", FontWeight->"Bold"], " and a corresponding fundamental solution:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["X2[t_] = {1,1,1}t*E^(-3t) + {-1,0,0}E^(-3t);", "Input", AspectRatioFixed->True], Cell[TextData["To find another, solve:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Solve[(A-(-3)IdentityMatrix[3]).{u1,u2,u3}=={-1,0,0}]", "Input", AspectRatioFixed->True], Cell[BoxData[ \({{u1 \[Rule] 1\/3 + u3, u2 \[Rule] \(-\(1\/3\)\) + u3}}\)], "Output"] }, Open ]], Cell[TextData[{ "Choose ", StyleBox["u3 == 0", FontFamily->"Courier", FontWeight->"Bold"], " to get the second generalized eigenvector ", StyleBox["{1/3,-1/3,0}", FontFamily->"Courier", FontWeight->"Bold"], ". A third fundamental solution is then given by" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ X3[t_] = {1,1,1}(t^2/2)*E^(-3t) + {-1,0,0}t*E^(-3t) + {1/3,-1/3,0}E^(-3t);\ \>", "Input", AspectRatioFixed->True], Cell[TextData["And the general solution is:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["X[t_] = c1 X1[t] + c2 X2[t] + c3 X3[t]//Simplify", "Input", AspectRatioFixed->True], Cell[BoxData[ \({1\/6\ E\^\(\(-3\)\ t\)\ \(( 6\ c1 + 6\ c2\ \((\(-1\) + t)\) + c3\ \((2 - 6\ t + 3\ t\^2)\))\), 1\/6\ E\^\(\(-3\)\ t\)\ \(( 6\ c1 + 6\ c2\ t + c3\ \((\(-2\) + 3\ t\^2)\))\), 1\/2\ E\^\(\(-3\)\ t\)\ \((2\ c1 + t\ \((2\ c2 + c3\ t)\))\)}\)], "Output"] }, Open ]], Cell[TextData["Let's check:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["X'[t]//Simplify", "Input", AspectRatioFixed->True], Cell[BoxData[ \({\(-\(1\/2\)\)\ E\^\(\(-3\)\ t\)\ \(( 6\ c1 + 2\ c2\ \((\(-4\) + 3\ t)\) + c3\ \((4 - 8\ t + 3\ t\^2)\))\), \(-\(1\/2\)\)\ E\^\(\(-3\)\ t\)\ \(( 6\ c1 + c2\ \((\(-2\) + 6\ t)\) + c3\ \((\(-2\) - 2\ t + 3\ t\^2)\)) \), \(-\(1\/2\)\)\ E\^\(\(-3\)\ t\)\ \(( 6\ c1 + c3\ t\ \((\(-2\) + 3\ t)\) + c2\ \((\(-2\) + 6\ t)\))\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["A.X[t]//Simplify", "Input", AspectRatioFixed->True], Cell[BoxData[ \({\(-\(1\/2\)\)\ E\^\(\(-3\)\ t\)\ \(( 6\ c1 + 2\ c2\ \((\(-4\) + 3\ t)\) + c3\ \((4 - 8\ t + 3\ t\^2)\))\), \(-\(1\/2\)\)\ E\^\(\(-3\)\ t\)\ \(( 6\ c1 + c2\ \((\(-2\) + 6\ t)\) + c3\ \((\(-2\) - 2\ t + 3\ t\^2)\)) \), \(-\(1\/2\)\)\ E\^\(\(-3\)\ t\)\ \(( 6\ c1 + c3\ t\ \((\(-2\) + 3\ t)\) + c2\ \((\(-2\) + 6\ t)\))\)}\)], "Output"] }, Open ]], Cell["\<\ solns = Flatten[Table[X[t],{c1,-1,1},{c2,-1,1},{c3,-1,1}],2];\ \>", "Input", AspectRatioFixed->True], Cell[CellGroupData[{ Cell["p = ParametricPlot3D[Evaluate[solns], {t,-2,2}]", "Input", AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .85869 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StyleBox["Phi[t]", FontFamily->"Courier", FontWeight->"Bold"], " such that ", StyleBox["Phi[0]", FontFamily->"Courier", FontWeight->"Bold"], " is the identity matrix for the system\n", StyleBox["X'[t] == A.X[t]", FontFamily->"Courier", FontWeight->"Bold"], "where" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ A = {{-4, -3, -1}, {-1, 0, 5}, { 1, 3, -2}};\ \>", "Input", AspectRatioFixed->True], Cell[TextData[{ "Use ", StyleBox["Phi[t]", FontFamily->"Courier", FontWeight->"Bold"], " to find a solution satisfying the given initial condition. ", StyleBox["X[0]=={1,0,1}", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Solution"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Eigensystem[A]", "Input", AspectRatioFixed->True], Cell[BoxData[ \({{\(-6\), \(-3\), 3}, {{\(-1\), \(-1\), 1}, {2, \(-1\), 1}, {\(-1\), 2, 1}}}\)], "Output"] }, Open ]], Cell["\<\ X1[t_] = {-1, -1, 1}E^(-6t); X2[t_] = { 2, -1, 1}E^(-3t); X3[t_] = {-1, 2, 1}E^( 3t);\ \>", "Input", AspectRatioFixed->True], Cell[TextData[{ "A fundamental matrix has columns given by ", StyleBox["X1[t]", FontFamily->"Courier", FontWeight->"Bold"], ", ", StyleBox["X2[t]", FontFamily->"Courier", FontWeight->"Bold"], ", and ", StyleBox["X3[t]", FontFamily->"Courier", FontWeight->"Bold"], ". We use ", StyleBox["Transpose[]", FontFamily->"Courier", FontWeight->"Bold"], " to covert these vectors from rows into columns." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["\<\ Xi[t_] = Transpose[{X1[t], X2[t], X3[t]}]; Xi[t]//MatrixForm\ \>", "Input", AspectRatioFixed->True], Cell[BoxData[ TagBox[ RowBox[{"(", GridBox[{ {\(-E\^\(\(-6\)\ t\)\), \(2\ E\^\(\(-3\)\ t\)\), \(-E\^\(3\ t\)\)}, {\(-E\^\(\(-6\)\ t\)\), \(-E\^\(\(-3\)\ t\)\), \(2\ E\^\(3\ t\)\)}, {\(E\^\(\(-6\)\ t\)\), \(E\^\(\(-3\)\ t\)\), \(E\^\(3\ t\)\)} }], ")"}], (MatrixForm[ #]&)]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Xi[0]//MatrixForm", "Input", AspectRatioFixed->True], Cell[BoxData[ TagBox[ RowBox[{"(", GridBox[{ {\(-1\), "2", \(-1\)}, {\(-1\), \(-1\), "2"}, {"1", "1", "1"} }], ")"}], (MatrixForm[ #]&)]], "Output"] }, Open ]], Cell[TextData[{ "To get a fundamental matrix ", StyleBox["Phi[t]", FontFamily->"Courier", FontWeight->"Bold"], " such that ", StyleBox["Phi[0]", FontFamily->"Courier", FontWeight->"Bold"], " is the identity matrix, we multiply ", StyleBox["Xi[t]", FontFamily->"Courier", FontWeight->"Bold"], " on the right by the inverse of ", StyleBox["Xi[0]", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["\<\ Phi[t_] = Xi[t].Inverse[Xi[0]]; Phi[t]//MatrixForm\ \>", "Input", AspectRatioFixed->True], Cell[BoxData[ TagBox[ RowBox[{"(", GridBox[{ {\(E\^\(\(-6\)\ t\)\/3 + \(2\ E\^\(\(-3\)\ t\)\)\/3\), \(E\^\(\(-6\)\ t\)\/3 - E\^\(3\ t\)\/3\), \(\(-\(1\/3\)\)\ E\^\(\(-6\)\ t\) + \(2\ E\^\(\(-3\)\ t\)\)\/3 - E\^\(3\ t\)\/3\)}, {\(E\^\(\(-6\)\ t\)\/3 - E\^\(\(-3\)\ t\)\/3\), \(E\^\(\(-6\)\ t\)\/3 + \(2\ E\^\(3\ t\)\)\/3\), \(\(-\(1\/3\)\)\ E\^\(\(-6\)\ t\) - E\^\(\(-3\)\ t\)\/3 + \(2\ E\^\(3\ t\)\)\/3\)}, {\(\(-\(1\/3\)\)\ E\^\(\(-6\)\ t\) + E\^\(\(-3\)\ t\)\/3\), \(\(-\(1\/3\)\)\ E\^\(\(-6\)\ t\) + E\^\(3\ t\)\/3\), \(E\^\(\(-6\)\ t\)\/3 + E\^\(\(-3\)\ t\)\/3 + E\^\(3\ t\)\/3\)} }], ")"}], (MatrixForm[ #]&)]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Phi[0]//MatrixForm", "Input", AspectRatioFixed->True], Cell[BoxData[ TagBox[ RowBox[{"(", GridBox[{ {"1", "0", "0"}, {"0", "1", "0"}, {"0", "0", "1"} }], ")"}], (MatrixForm[ #]&)]], "Output"] }, Open ]], Cell[TextData[{ "Let's check that ", StyleBox["Phi[t]", FontFamily->"Courier", FontWeight->"Bold"], " satisfies the equation:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Phi'[t]//Simplify//MatrixForm", "Input", AspectRatioFixed->True], Cell[BoxData[ TagBox[ RowBox[{"(", GridBox[{ {\(\(-2\)\ E\^\(\(-6\)\ t\)\ \((1 + E\^\(3\ t\))\)\), \(\(-E\^\(\(-6\)\ t\)\)\ \((2 + E\^\(9\ t\))\)\), \(\(-E\^\(\(-6\)\ t\)\)\ \((\(-2\) + 2\ E\^\(3\ t\) + E\^\(9\ t\))\)\)}, {\(E\^\(\(-6\)\ t\)\ \((\(-2\) + E\^\(3\ t\))\)\), \(2\ E\^\(\(-6\)\ t\)\ \((\(-1\) + E\^\(9\ t\))\)\), \(E\^\(\(-6\)\ t\)\ \((2 + E\^\(3\ t\) + 2\ E\^\(9\ t\))\)\)}, {\(\(-E\^\(\(-6\)\ t\)\)\ \((\(-2\) + E\^\(3\ t\))\)\), \(2\ E\^\(\(-6\)\ t\) + E\^\(3\ t\)\), \(E\^\(\(-6\)\ t\)\ \((\(-2\) - E\^\(3\ t\) + E\^\(9\ t\))\)\)} }], ")"}], (MatrixForm[ #]&)]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["A.Phi[t]//Simplify//MatrixForm", "Input", AspectRatioFixed->True], Cell[BoxData[ TagBox[ RowBox[{"(", GridBox[{ {\(\(-2\)\ E\^\(\(-6\)\ t\)\ \((1 + E\^\(3\ t\))\)\), \(\(-E\^\(\(-6\)\ t\)\)\ \((2 + E\^\(9\ t\))\)\), \(\(-E\^\(\(-6\)\ t\)\)\ \((\(-2\) + 2\ E\^\(3\ t\) + E\^\(9\ t\))\)\)}, {\(E\^\(\(-6\)\ t\)\ \((\(-2\) + E\^\(3\ t\))\)\), \(2\ E\^\(\(-6\)\ t\)\ \((\(-1\) + E\^\(9\ t\))\)\), \(E\^\(\(-6\)\ t\)\ \((2 + E\^\(3\ t\) + 2\ E\^\(9\ t\))\)\)}, {\(\(-E\^\(\(-6\)\ t\)\)\ \((\(-2\) + E\^\(3\ t\))\)\), \(2\ E\^\(\(-6\)\ t\) + E\^\(3\ t\)\), \(E\^\(\(-6\)\ t\)\ \((\(-2\) - E\^\(3\ t\) + E\^\(9\ t\))\)\)} }], ")"}], (MatrixForm[ #]&)]], "Output"] }, Open ]], Cell[TextData[ "The solution satisfying the given initial condition is simply:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["X[t_] = Phi[t].{1,0,1}//Simplify", "Input", AspectRatioFixed->True], Cell[BoxData[ \({\(-\(1\/3\)\)\ E\^\(\(-3\)\ t\)\ \((\(-4\) + E\^\(6\ t\))\), 2\/3\ E\^\(\(-3\)\ t\)\ \((\(-1\) + E\^\(6\ t\))\), 1\/3\ E\^\(\(-3\)\ t\)\ \((2 + E\^\(6\ t\))\)}\)], "Output"] }, Open ]], Cell[TextData[{ "Compare with ", StyleBox["DSolve[]", FontFamily->"Courier", FontWeight->"Bold"], ":" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["\<\ DSolve[ {x1'[t] == -4x1[t]-3x2[t]- x3[t], x2'[t] == - x1[t] +5x3[t], x3'[t] == x1[t]+3x2[t]-2x3[t], x1[0]==1,x2[0]==0,x3[0]==1}, {x1[t],x2[t],x3[t]},t]//Simplify\ \>", "Input", AspectRatioFixed->True], Cell[BoxData[ \({{x1[t] \[Rule] \(-\(1\/3\)\)\ E\^\(\(-3\)\ t\)\ \((\(-4\) + E\^\(6\ t\))\), x2[t] \[Rule] 2\/3\ E\^\(\(-3\)\ t\)\ \((\(-1\) + E\^\(6\ t\))\), x3[t] \[Rule] 1\/3\ E\^\(\(-3\)\ t\)\ \((2 + E\^\(6\ t\))\)}}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Variation of Parameters"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ "Use the method of variation of parameters to find the general solution of \ the non-homogeneous system\n", StyleBox["X'[t] == A.X[t] + G[t]", FontFamily->"Courier", FontWeight->"Bold"], ",", StyleBox[" X[0] == {1,0,1}", FontFamily->"Courier", FontWeight->"Bold"], ",\nwhere" }], "Text", CellMargins->{{18, 126}, {Inherited, Inherited}}, Evaluatable->False, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True] }, Open ]], Cell["\<\ A = {{-4, -3, -1}, {-1, 0, 5}, { 1, 3, -2}}; G[t_] = {1,t,t^2}; \ \>", "Input", AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Solution"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ "A solution to the associated homogeneous problem, ", StyleBox["X'[t] == A.X[t]", FontFamily->"Courier", FontWeight->"Bold"], ", was found in in the previous problem:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["\<\ X1[t_] = {-1, -1, 1}E^(-6t); X2[t_] = { 2, -1, 1}E^(-3t); X3[t_] = {-1, 2, 1}E^( 3t); Xi[t_] = Transpose[{X1[t],X2[t],X3[t]}]; IXi[t_] = Inverse[Xi[t]]//Simplify; IXi[t]//MatrixForm\ \>", "Input", AspectRatioFixed->True], Cell[BoxData[ TagBox[ RowBox[{"(", GridBox[{ {\(-\(E\^\(6\ t\)\/3\)\), \(-\(E\^\(6\ t\)\/3\)\), \(E\^\(6\ t\)\/3\)}, {\(E\^\(3\ t\)\/3\), "0", \(E\^\(3\ t\)\/3\)}, {"0", \(E\^\(\(-3\)\ t\)\/3\), \(E\^\(\(-3\)\ t\)\/3\)} }], ")"}], (MatrixForm[ #]&)]], "Output"] }, Open ]], Cell[TextData[{ "A particular solution to the non-homogeneous problem is given by ", StyleBox["Y[t_] = Xi[t].U[t]", FontFamily->"Courier", FontWeight->"Bold"], " where ", StyleBox["U[t]", FontFamily->"Courier", FontWeight->"Bold"], " satisfies ", StyleBox["U'[t] == IXi[t].G[t]", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["IXi[t].G[t]", "Input", AspectRatioFixed->True], Cell[BoxData[ \({\(-\(E\^\(6\ t\)\/3\)\) - 1\/3\ E\^\(6\ t\)\ t + 1\/3\ E\^\(6\ t\)\ t\^2, E\^\(3\ t\)\/3 + 1\/3\ E\^\(3\ t\)\ t\^2, 1\/3\ E\^\(\(-3\)\ t\)\ t + 1\/3\ E\^\(\(-3\)\ t\)\ t\^2}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["U[t_] = Integrate[IXi[t].G[t], t]//Simplify", "Input", AspectRatioFixed->True], Cell[BoxData[ \({1\/162\ E\^\(6\ t\)\ \((\(-7\) - 12\ t + 9\ t\^2)\), 1\/81\ E\^\(3\ t\)\ \((11 - 6\ t + 9\ t\^2)\), \(-\(1\/81\)\)\ E\^\(\(-3\)\ t\)\ \((5 + 15\ t + 9\ t\^2)\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Y[t_] = Xi[t].U[t]//Simplify", "Input", AspectRatioFixed->True], Cell[BoxData[ \({1\/162\ \((61 + 18\ t + 45\ t\^2)\), 1\/162\ \((\(-35\) - 36\ t - 63\ t\^2)\), 1\/162\ \((5 - 54\ t + 9\ t\^2)\)}\)], "Output"] }, Open ]], Cell[TextData["The general solution to the given system is then"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["X[t_] = c1 X1[t] + c2 X2[t] + c3 X3[t] + Y[t]", "Input", AspectRatioFixed->True], Cell[BoxData[ \({\(-c1\)\ E\^\(\(-6\)\ t\) + 2\ c2\ E\^\(\(-3\)\ t\) - c3\ E\^\(3\ t\) + 1\/162\ \((61 + 18\ t + 45\ t\^2)\), \(-c1\)\ E\^\(\(-6\)\ t\) - c2\ E\^\(\(-3\)\ t\) + 2\ c3\ E\^\(3\ t\) + 1\/162\ \((\(-35\) - 36\ t - 63\ t\^2)\), c1\ E\^\(\(-6\)\ t\) + c2\ E\^\(\(-3\)\ t\) + c3\ E\^\(3\ t\) + 1\/162\ \((5 - 54\ t + 9\ t\^2)\)}\)], "Output"] }, Open ]], Cell[TextData["Let's check:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["X'[t]//Simplify", "Input", AspectRatioFixed->True], Cell[BoxData[ \({1\/9 + 6\ c1\ E\^\(\(-6\)\ t\) - 6\ c2\ E\^\(\(-3\)\ t\) - 3\ c3\ E\^\(3\ t\) + \(5\ t\)\/9, \(-\(2\/9\)\) + 6\ c1\ E\^\(\(-6\)\ t\) + 3\ c2\ E\^\(\(-3\)\ t\) + 6\ c3\ E\^\(3\ t\) - \(7\ t\)\/9, \(-6\)\ c1\ E\^\(\(-6\)\ t\) - 3\ c2\ E\^\(\(-3\)\ t\) + 1\/9\ \((\(-3\) + 27\ c3\ E\^\(3\ t\) + t)\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["A.X[t] + G[t]//Simplify", "Input", AspectRatioFixed->True], Cell[BoxData[ \({1\/9 + 6\ c1\ E\^\(\(-6\)\ t\) - 6\ c2\ E\^\(\(-3\)\ t\) - 3\ c3\ E\^\(3\ t\) + \(5\ t\)\/9, \(-\(2\/9\)\) + 6\ c1\ E\^\(\(-6\)\ t\) + 3\ c2\ E\^\(\(-3\)\ t\) + 6\ c3\ E\^\(3\ t\) - \(7\ t\)\/9, \(-6\)\ c1\ E\^\(\(-6\)\ t\) - 3\ c2\ E\^\(\(-3\)\ t\) + 1\/9\ \((\(-3\) + 27\ c3\ E\^\(3\ t\) + t)\)}\)], "Output"] }, Open ]], Cell[TextData[{ "We now solve for the constants ", StyleBox["c1", FontFamily->"Courier", FontWeight->"Bold"], ", ", StyleBox["c2", FontFamily->"Courier", FontWeight->"Bold"], ", and ", StyleBox["c3", FontFamily->"Courier", FontWeight->"Bold"], " that satisfy the given initial condition." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["cs = Solve[X[0] == {1,0,1}]", "Input", AspectRatioFixed->True], Cell[BoxData[ \({{c1 \[Rule] 7\/162, c2 \[Rule] 43\/81, c3 \[Rule] 32\/81}}\)], "Output"] }, Open ]], Cell[TextData["The solution to the initial value problem is thus"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["X[t_] = X[t] /. 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