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NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 3636520, 130936]*) (*NotebookOutlinePosition[ 3637258, 130962]*) (* CellTagsIndexPosition[ 3637214, 130958]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Exploring the Theory of Geometric Bifurcation"]], "Title", PageWidth->PaperWidth, TextAlignment->Center], Cell["\<\ Jay H. Wolkowisky Department of Mathematics University of Colorado Boulder, Colorado 80309, USA wolkowis@euclid.colorado.edu\ \>", "Subtitle", PageWidth->PaperWidth, TextAlignment->Center], Cell[CellGroupData[{ Cell["Abstract", "SectionFirst", PageWidth->PaperWidth, TextAlignment->Center], Cell[TextData[{ "This paper will deal with the Theory of Geometric Bifurcation which the \ author developed in 1986. The theory developed in those papers is very \ general and abstract. So, until a symbolic software program such as ", StyleBox["Mathematica", FontSlant->"Italic"], " came along, it was very difficult to examine concrete examples which \ would illustrate and explain the theory. In this paper we will look at \ examples of bifurcating branches of solutions of nonlinear: algebraic \ equations, ordinary differential equations, and partial differential \ equations." }], "Text", PageWidth->PaperWidth] }, Open ]], Cell[CellGroupData[{ Cell["1. Introduction", "Section", PageWidth->PaperWidth], Cell[TextData[{ "This paper will investigate nonlinear eigenvalue problems of the form", StyleBox[" \nLw - \[Lambda] w + F(\[Lambda],w) = 0", FontWeight->"Bold"], ", where ", StyleBox["L", FontWeight->"Bold"], " is a linear operator in a Hilbert space ", StyleBox["H", FontWeight->"Bold"], ", with ", StyleBox["\[Lambda]", FontWeight->"Bold"], " being a scalar parameter, and", StyleBox[" F(\[Lambda],w) = o(||w||)", FontWeight->"Bold"], ". Such equations arise in bifurcation (branching) problems where a \ solution becomes non-unique and bifurcates into more than one solution. For \ example, in the buckling of rods, plates, and shells. Also in the field of \ fluid dynamics, such as the Bernard and Taylor problems.\n\nIn this paper we \ will look at examples of the following types:\n(a) ", StyleBox["L = A", FontWeight->"Bold"], ", ", StyleBox["A", FontWeight->"Bold"], " is a n x n matrix,\n(b)", StyleBox[" L =", FontWeight->"Bold"], Cell[BoxData[ FormBox[ StyleBox[ FractionBox[ RowBox[{" ", FormBox[\(\[DoubleStruckD]\^2\), "TraditionalForm"]}], \(\[DoubleStruckD]x\^2\)], FontSize->15], TraditionalForm]], FontSize->14, FontWeight->"Bold"], ",\n(c) ", StyleBox["L = ", FontWeight->"Bold"], Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["\[EmptyDownTriangle]", FontSize->14], "2"], TraditionalForm]], FontWeight->"Bold"], ", the Laplace operator.\n\nWe will investigate the existence(or \ non-existence) of global branches of solutions of the above equations. We \ also will show how to construct these branches. The method used to accomplish \ this is based on the\"Theory of Geometric Bifurcation\" developed by the \ author [", StyleBox["1", FontWeight->"Bold"], ",", StyleBox["2", FontWeight->"Bold"], "]. The above equation has ", StyleBox["w \[Congruent] 0", FontWeight->"Bold"], " as a solution for all values of ", StyleBox["\[Lambda].", FontWeight->"Bold"], " The \"linearized\" equation ", StyleBox["Lw - \[Lambda] w = 0", FontWeight->"Bold"], ", besides having this ", StyleBox["trivial solution ", FontWeight->"Bold"], "also has non-trivial solutions at the eigenvalues of ", StyleBox["L", FontWeight->"Bold"], ", ", StyleBox["\[Lambda] = ", FontWeight->"Bold"], Cell[BoxData[ FormBox[ SubscriptBox["\[Lambda]", StyleBox["i", FontSize->14]], TraditionalForm]], FontWeight->"Bold"], ". The main question we will be interested in is whether the nonlinear \ equation ", StyleBox[" Lw - \[Lambda] w + F(\[Lambda],w) = 0 ", FontWeight->"Bold"], "has non-trivial solutions bifurcating from the trivial solution at the \ eigenvalues ", StyleBox["\[Lambda] = ", FontWeight->"Bold"], Cell[BoxData[ FormBox[ SubscriptBox["\[Lambda]", StyleBox["i", FontSize->14]], TraditionalForm]], FontWeight->"Bold"], ". (It is easy to show that if the nonlinear equation has non-trivial \ solutions they must bifurcate from the trivial solution at the eigenvalues of \ the linearized equation). We will be restricting our attention to the case \ that the operator ", StyleBox["L", FontWeight->"Bold"], " is self-adjoint (symmetric), which is the case for most physical models. \ This will allow the results proved in [", StyleBox["1", FontWeight->"Bold"], ",", StyleBox["2", FontWeight->"Bold"], "] to be stated more simply. We will use ", StyleBox["dim ", FontWeight->"Bold"], "and", StyleBox[" null ", FontWeight->"Bold"], "for dimensiom and null space respectively. When ", StyleBox["dim null", FontWeight->"Bold"], "(L - ", Cell[BoxData[ FormBox[ SubscriptBox["\[Lambda]", StyleBox["i", FontSize->14]], TraditionalForm]]], "I) = 1 (", StyleBox["multiplicity of", FontWeight->"Bold"], " ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox["\[Lambda]", StyleBox["i", FontSize->14]], FontWeight->"Bold"], TraditionalForm]]], " ", StyleBox["= 1", FontWeight->"Bold"], "), then bifurcation from the eigenvalue ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox["\[Lambda]", StyleBox["i", FontSize->14]], FontWeight->"Bold"], TraditionalForm]]], " always occurs. This case is very simple to handle and not very \ interesting. We will only be looking at the case when the ", StyleBox["multiplicity of ", FontWeight->"Bold"], Cell[BoxData[ FormBox[ SubscriptBox["\[Lambda]", StyleBox["i", FontSize->14]], TraditionalForm]], FontWeight->"Bold"], StyleBox[" is \[GreaterEqual] 2", FontWeight->"Bold"], ". If we let\n ", StyleBox[" w = \[Epsilon] (y + \[Eta])", FontWeight->"Bold"], ", (1)\nwhere \n \ ", StyleBox["y", FontWeight->"Bold"], " \[Element] ", StyleBox["N", FontWeight->"Bold"], "= ", StyleBox["null", FontWeight->"Bold"], "(L - ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_0\)]], "I), \n ", StyleBox[" ||y|| = 1", FontWeight->"Bold"], ",\n and \n \ ", StyleBox[" \[Eta]", FontWeight->"Bold"], " \[Element] ", Cell[BoxData[ FormBox[ StyleBox[ SuperscriptBox[ StyleBox["N", FontSlant->"Plain"], "\[UpTee]"], FontWeight->"Bold"], TraditionalForm]]], " (the orthogonal compliment of", StyleBox[" N, H =", FontWeight->"Bold"], " ", StyleBox["N ", FontWeight->"Bold"], "\[CirclePlus] ", Cell[BoxData[ FormBox[ StyleBox[ SuperscriptBox[ StyleBox["N", FontSlant->"Plain"], "\[UpTee]"], FontWeight->"Bold"], TraditionalForm]]], ").\n \nWhere ", StyleBox["\[Epsilon]", FontWeight->"Bold"], " is a small parameter, and ", Cell[BoxData[ FormBox[ StyleBox[\(\[Lambda]\_0\), FontWeight->"Bold"], TraditionalForm]]], " is an eigenvalue of ", StyleBox["L", FontWeight->"Bold"], " with \n ", StyleBox["dim", FontWeight->"Bold"], " ", StyleBox["N ", FontWeight->"Bold"], " ", StyleBox["\[GreaterEqual] 2.", FontWeight->"Bold"], "\nWe can now state the above mentioned results. But first we define ", StyleBox["T", FontWeight->"Bold"], "[y], which is the projection operator of ", StyleBox["H", FontWeight->"Bold"], " onto the tangent plane to the unit sphere in ", StyleBox["N", FontWeight->"Bold"], " at y, i.e.,\n", StyleBox["T", FontWeight->"Bold"], "[y]", StyleBox[" :", FontWeight->"Bold"], " ", StyleBox["H ", FontWeight->"Bold"], "\[RightArrow] ", StyleBox["\[Tau]", FontSize->18], "[y] = {x | \[LeftAngleBracket]x , y\[RightAngleBracket] = 0, x and y \ \[Element] ", StyleBox["N", FontWeight->"Bold"], ", ", StyleBox["|", FontWeight->"Bold"], "|y", StyleBox["||", FontWeight->"Bold"], " = 1}, \nwith \[LeftAngleBracket] , \[RightAngleBracket] being the inner \ product on ", StyleBox["H", FontWeight->"Bold"], ". \n", StyleBox["Theorem 1 ", FontWeight->"Bold", FontVariations->{"Underline"->True}], StyleBox["\n", FontWeight->"Bold"], " ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_0\)]], " ", StyleBox["is a bifurcation point of", FontSlant->"Italic"], " \n Lw - \[Lambda] w + \ F(\[Lambda],w) = 0 (2)\n", StyleBox["if and only if for each", FontSlant->"Italic"], " \[Epsilon] \[Succeeds] 0 ", StyleBox["sufficiently small there exists a", FontSlant->"Italic"], " y \[Element] ", StyleBox["N", FontWeight->"Bold"], " ", StyleBox["with", FontSlant->"Italic"], " ", StyleBox["||", FontWeight->"Bold"], "y", StyleBox["||", FontWeight->"Bold"], " = 1", StyleBox[" such that it satisfies", FontSlant->"Italic"], "\n ", StyleBox["T", FontWeight->"Bold"], "[y]F( ", Cell[BoxData[ \(TraditionalForm\`\(\[Lambda]\&\[Wedge]\)\)]], "( y, \[Epsilon] ), \[Epsilon] (y + ", Cell[BoxData[ \(TraditionalForm\`\(\[Eta]\&\[Wedge]\)\)]], "( y, \[Epsilon] ))) = 0. (3) \n", StyleBox["Where", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\(\[Lambda]\&\[Wedge]\)\)]], "( y, \[Epsilon] ) ", StyleBox["and", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\(\[Eta]\&\[Wedge]\)\)]], "( y, \[Epsilon] ) ", StyleBox["are solutions of the equations", FontSlant->"Italic"], "\n L\[Eta] - ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_0\)]], " \[Eta] - (\[Lambda] - ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_0\)]], ") (y + \[Eta]) + ", Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\^\(-1\)\)]], " F(\[Lambda] , \[Epsilon] (y + \[Eta])) = 0 (4)\n \ \[Lambda] - ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_0\)]], " - \[LeftAngleBracket] ", Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\^\(-1\)\)]], " F(\[Lambda] , \[Epsilon] (y + \[Eta])) , y\[RightAngleBracket] = 0 , \ (5)\n", StyleBox["for a fixed", FontSlant->"Italic"], " y ", StyleBox["and", FontSlant->"Italic"], " \[Epsilon].\n\nLoosely speaking, this theorem says that solutions of (2) \ will bifurcate from the zero solution at an eigenvalue ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_0\)]], " of L if and only if the tangential vector field ", StyleBox["\nT", FontWeight->"Bold"], "[y]F( ", Cell[BoxData[ \(TraditionalForm\`\(\[Lambda]\&\[Wedge]\)\)]], "( y, \[Epsilon] ), \[Epsilon] (y + ", Cell[BoxData[ \(TraditionalForm\`\(\[Eta]\&\[Wedge]\)\)]], "( y, \[Epsilon] )))", StyleBox[" ", FontWeight->"Bold"], "has a zero for each \[Epsilon] sufficiently small .\n\nIt can be shown \ that ", Cell[BoxData[ \(TraditionalForm\`\(\[Lambda]\&\[Wedge]\)\)]], "(y,0) = ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_0\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\(\[Eta]\&\[Wedge]\)\)]], "(y,0) = 0. In addition, if we assume that\n \ F(\[Lambda], \[Epsilon] w) = ", Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\^\[Alpha]\)]], " F(\[Lambda],w), with \[Alpha]\[Succeeds]1, \nthen the leading term in \ approximating Equation (3) is\n ", StyleBox["T", FontWeight->"Bold"], "[y]F( ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_0\)]], ",y ) = 0 . 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A ", StyleBox["transverse zero ", FontWeight->"Bold"], "of a vector field is a zero of the vector field such that any small \ perturbation of the vector field will not eliminate the zero. For example, \ in one dimension ", Cell[BoxData[ \(TraditionalForm\`\(x\^3\ \)\)]], "has a transverse zero at x = 0, while ", Cell[BoxData[ \(TraditionalForm\`x\^2\)]], " does not.\n\nIt is shown in [", StyleBox["1", FontWeight->"Bold"], "] that if the multiplicity of an eigenvalue ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_0\)]], " is", StyleBox[" ", FontVariations->{"Underline"->True}], StyleBox["odd", FontWeight->"Bold", FontVariations->{"Underline"->True}], ", then bifurcation from ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_0\)]], " will occur. In [", StyleBox["2", FontWeight->"Bold"], "] it is shown that if F(\[Lambda],w) is an ", StyleBox["even", FontWeight->"Bold", FontVariations->{"Underline"->True}], " function of the variable w, then bifurcation from ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_0\)]], " will also occur. An even function being one that satisfies the condition \ F(\[Lambda],-w) = F(\[Lambda],w).", StyleBox[ " It is for these reasons that, in this paper, we will be looking at \ examples where the", FontWeight->"Bold"], " ", StyleBox["multiplicity of ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_0\)], FontWeight->"Bold"], StyleBox[" is ", FontWeight->"Bold"], StyleBox["even", FontWeight->"Bold", FontVariations->{"Underline"->True}], StyleBox[" and F(\[Lambda],w) is an ", FontWeight->"Bold"], StyleBox["odd", FontWeight->"Bold", FontVariations->{"Underline"->True}], StyleBox[" function of w", FontWeight->"Bold"], ". ", StyleBox["In this case bifurcation from ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_0\)], FontWeight->"Bold"], StyleBox[" may or may not occur, as the following examples will show.", FontWeight->"Bold"] }], "Text", PageWidth->PaperWidth], Cell[TextData[{ StyleBox["2. L = A, an n \[Times]n matrix and H = ", "Section"], Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["R", FontSize->16], "n"], TraditionalForm]], FontSize->18, FontWeight->"Bold"], " ." }], "Text", PageWidth->PaperWidth, FontSize->13], Cell[TextData[{ "We first consider the following example for", StyleBox[" n=4.", FontWeight->"Bold"], "\n" }], "Text", PageWidth->PaperWidth, FontSize->13], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"A", "=", TagBox[ RowBox[{"(", GridBox[{ {"1", "0", "0", "0"}, {"0", "1", "0", "0"}, {"0", "0", "2", "0"}, {"0", "0", "0", "2"} }, ColumnAlignments->{Decimal}], ")"}], (MatrixForm[ #]&)]}], ";"}], TraditionalForm]], "Input", CellLabel->"In[48]:=", PageWidth->PaperWidth, FontSize->13], Cell[BoxData[ FormBox[ RowBox[{"w", "=", RowBox[{"(", GridBox[{ {"x"}, {"y"}, {"u"}, {"v"} }], ")"}]}], TraditionalForm]], "Input", CellLabel->"In[5]:=", PageWidth->PaperWidth], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ StyleBox["F", FontSlant->"Plain"], "[", \(\[Lambda], w\), "]"}], "=", RowBox[{"(", GridBox[{ {\(x\^3 + y\^3 + u\ y\^2 + v\ y\^2\)}, {\(x\^3 + v\ x\^2 + y\^2\ x + u\ y\ x\)}, {\(x\^3 + u\ y\ x + y\^3 + v\ y\^2\)}, {\(x\^3 + u\ y\ x + y\^3 + v\ y\^2 + u\ v\ y\)} }], ")"}]}], ";"}], TraditionalForm]], "Input", CellLabel->"In[19]:=", PageWidth->PaperWidth], Cell["So that equation (2) becomes", "Text", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell[BoxData[ \(A . w - \[Lambda]\ w + F[\[Lambda], w] == 0\)], "Input", CellLabel->"In[20]:=", PageWidth->PaperWidth], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", GridBox[{ {\(x\^3 + x + y\^3 + u\ y\^2 + v\ y\^2 - x\ \[Lambda]\)}, {\(x\^3 + v\ x\^2 + y\^2\ x + u\ y\ x + y - y\ \[Lambda]\)}, {\(x\^3 + u\ y\ x + y\^3 + v\ y\^2 + 2\ u - u\ \[Lambda]\)}, { \(x\^3 + u\ y\ x + y\^3 + v\ y\^2 + 2\ v + u\ v\ y - v\ \[Lambda]\)} }], ")"}], "==", "0"}], TraditionalForm]], "Output", CellLabel->"Out[20]=", PageWidth->PaperWidth] }, Open ]], Cell[TextData[{ "The eigenvalues of ", StyleBox["A", FontWeight->"Bold"], " are ", StyleBox["{1, 1, 2, 2}", FontWeight->"Bold"], ". We will be investigating the bifurcation from the double eigenvalue" }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \[Lambda]\_0 = \ 1\)\)], "Input", PageWidth->PaperWidth], Cell[TextData[{ "Therefore ", StyleBox["dim", FontWeight->"Bold"], " ", StyleBox["N", FontWeight->"Bold"], " = ", StyleBox["dim", FontWeight->"Bold"], Cell[BoxData[ FormBox[ RowBox[{" ", FormBox[ StyleBox[ SuperscriptBox[ StyleBox["N", FontSlant->"Plain"], "\[UpTee]"], FontWeight->"Bold"], "TraditionalForm"]}], TraditionalForm]]], " ", StyleBox["= 2", FontWeight->"Bold"], "." }], "Text", PageWidth->PaperWidth], Cell[BoxData[ FormBox[ RowBox[{\(In\ this\ case\), ",", " ", RowBox[{"bases", " ", "for", " ", StyleBox["N", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], " ", "and", " ", FormBox[ StyleBox[ 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\[Epsilon] - \[Epsilon]\ \[Lambda]\ \(cos(\[Alpha])\)\)}, { \(\(\(cos\^3\)(\[Alpha])\)\ \[Epsilon]\^3 + \[Eta]2\ \(\(cos\^2\)(\[Alpha])\)\ \[Epsilon]\^3 + \(cos(\[Alpha])\)\ \(\(sin\^2\)(\[Alpha])\)\ \[Epsilon]\^3 + \[Eta]1\ \(cos(\[Alpha])\)\ \(sin(\[Alpha])\)\ \[Epsilon]\^3 + \(sin(\[Alpha])\)\ \[Epsilon] - \[Epsilon]\ \[Lambda]\ \(sin(\[Alpha])\)\)}, { \(\(\(cos\^3\)(\[Alpha])\)\ \[Epsilon]\^3 + \(\(sin\^3\)(\[Alpha])\)\ \[Epsilon]\^3 + \[Eta]2\ \(\(sin\^2\)(\[Alpha])\)\ \[Epsilon]\^3 + \[Eta]1\ \(cos(\[Alpha])\)\ \(sin(\[Alpha])\)\ \[Epsilon]\^3 + 2\ \[Eta]1\ \[Epsilon] - \[Epsilon]\ \[Eta]1\ \[Lambda]\)}, { \(\(\(cos\^3\)(\[Alpha])\)\ \[Epsilon]\^3 + \(\(sin\^3\)(\[Alpha])\)\ \[Epsilon]\^3 + \[Eta]2\ \(\(sin\^2\)(\[Alpha])\)\ \[Epsilon]\^3 + \[Eta]1\ \[Eta]2\ \(sin(\[Alpha])\)\ \[Epsilon]\^3 + \[Eta]1\ \(cos(\[Alpha])\)\ \(sin(\[Alpha])\)\ \[Epsilon]\^3 + 2\ \[Eta]2\ \[Epsilon] - \[Epsilon]\ \[Eta]2\ \[Lambda]\)} }], ")"}], "==", RowBox[{"(", GridBox[{ {"0"}, {"0"}, {"0"}, {"0"} }, ColumnAlignments->{Decimal}], ")"}]}], TraditionalForm]], "Output",\ CellLabel->"Out[34]="] }, Closed]], Cell[TextData[{ "Since ", Cell[BoxData[ RowBox[{"(", GridBox[{ {\(Sin[\[Alpha]]\)}, {\(-Cos[\[Alpha]]\)}, {"0"}, {"0"} }], ")"}]], FontWeight->"Bold"], "is orthogonal to ", Cell[BoxData[ StyleBox[ RowBox[{"(", GridBox[{ {\(Cos[\[Alpha]]\)}, {\(Sin[\[Alpha]]\)}, {"0"}, {"0"} }], ")"}], FontWeight->"Bold"]]], " then Equation (3) becomes " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"Dot", "[", RowBox[{ RowBox[{"Flatten", "[", RowBox[{"(", GridBox[{ {\(Sin[\[Alpha]]\)}, {\(-Cos[\[Alpha]]\)}, {"0"}, {"0"} }], ")"}], "]"}], ",", RowBox[{"Flatten", "[", RowBox[{ RowBox[{"(", GridBox[{ {\(x\^3 + y\^3 + u\ y\^2 + v\ y\^2\)}, {\(x\^3 + v\ x\^2 + y\^2\ x + u\ y\ x\)}, {\(x\^3 + u\ y\ x + y\^3 + v\ y\^2\)}, {\(x\^3 + u\ y\ x + y\^3 + v\ y\^2 + u\ v\ y\)} }], ")"}], "/.", \({x -> \[Epsilon]\ Cos[\[Alpha]], y -> \[Epsilon]\ Sin[\[Alpha]], u -> \[Epsilon]\ 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Editable->False]], "Output", CellLabel->"Out[21]="] }, Closed]], Cell[TextData[{ "This shows, using the above Theorem and the remark about ", StyleBox["transverse zeros", FontWeight->"Bold"], " , that bifurcation will occur for ", StyleBox["\[Lambda]=1", FontWeight->"Bold"], " at each of the four zeros of ", StyleBox["t[\[Alpha]]", FontWeight->"Bold"], ". We can calculate these zeros by" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Alpha]1 = \[Alpha] /. FindRoot[t[\[Alpha]], {\[Alpha], .8}]\)], "Input", CellLabel->"In[21]:="], Cell[BoxData[ \(0.785398165043810614`\)], "Output", CellLabel->"Out[21]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Alpha]2 = \[Alpha] /. FindRoot[t[\[Alpha]], {\[Alpha], 2.2}]\)], "Input", CellLabel->"In[22]:="], Cell[BoxData[ \(2.16956303205093758`\)], "Output", CellLabel->"Out[22]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Alpha]3 = \[Alpha] /. FindRoot[t[\[Alpha]], {\[Alpha], 3.8}]\)], "Input", CellLabel->"In[41]:="], Cell[BoxData[ \(3.92699081701547356`\)], "Output", CellLabel->"Out[41]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Alpha]4 = \[Alpha] /. FindRoot[t[\[Alpha]], {\[Alpha], 5.3}]\)], "Input", CellLabel->"In[42]:="], Cell[BoxData[ \(5.31115568565196927`\)], "Output", CellLabel->"Out[42]="] }, Closed]], Cell[TextData[{ "We note that ", StyleBox["\[Alpha]1 ", FontWeight->"Bold"], "-", StyleBox[" \[Alpha]3 = \[Pi]", FontWeight->"Bold"], " and ", StyleBox["\[Alpha]2 - \[Alpha]4 = \[Pi]", FontWeight->"Bold"], ", which means that the solution branches bifurcate in pairs in opposite \ directions. This is expected since the ", StyleBox["nonlinearity is cubic", FontWeight->"Bold"], ", so that if ", StyleBox["(x,y,u,v)", FontFamily->"Terminal", FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"], "is a solution then ", StyleBox["(-x,-y,-u,-v)", FontFamily->"Terminal", FontWeight->"Bold"], " is also a solution." }], "Text"], Cell["\<\ Now that we know bifurcation will occur, we can construct the bifurcating \ branches.\ \>", "Text"], Cell[CellGroupData[{ Cell["2.1 Constructing the Solution Branches", "Subsection"], Cell[TextData[{ "We first find ", StyleBox["\[Eta]1", FontWeight->"Bold"], " and ", StyleBox["\[Eta]2", FontWeight->"Bold"], " for small ", StyleBox["\[Epsilon]", FontWeight->"Bold"], "." }], "Text"], Cell[TextData[{ "Equation (4) can be written, using, ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ StyleBox[\(\[Lambda]\_0\), FontWeight->"Bold"], " ", StyleBox["=", FontWeight->"Bold"], StyleBox["1", FontWeight->"Bold"]}], ","}], TraditionalForm]]], " as" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"A", ".", RowBox[{"(", GridBox[{ {"0"}, {"0"}, {"\[Eta]1"}, {"\[Eta]2"} }], ")"}]}], " ", "-", " ", RowBox[{"(", GridBox[{ {"0"}, {"0"}, {"\[Eta]1"}, {"\[Eta]2"} }], ")"}], " ", "-", " ", RowBox[{\((\[Lambda]\ - \ 1)\), " ", RowBox[{"(", GridBox[{ {\(Cos[\[Alpha]]\)}, {\(Sin[\[Alpha]]\)}, {"\[Eta]1"}, {"\[Eta]2"} }], ")"}]}], " ", "+", RowBox[{\(1\/\[Epsilon]\), " ", RowBox[{"(", GridBox[{ {\(x\^3 + y\^3 + u\ y\^2 + v\ y\^2\)}, {\(x\^3 + v\ x\^2 + y\^2\ x + u\ y\ x\)}, {\(x\^3 + u\ y\ x + y\^3 + v\ y\^2\)}, {\(x\^3 + u\ y\ x + y\^3 + v\ y\^2 + u\ v\ y\)} }], ")"}]}]}], "/.", \({x -> \[Epsilon]\ Cos[\[Alpha]], y -> \[Epsilon]\ Sin[\[Alpha]], u -> \[Epsilon]\ \[Eta]1, v -> \[Epsilon]\ \[Eta]2}\)}], ")"}], "==", RowBox[{"(", GridBox[{ {"0"}, {"0"}, {"0"}, {"0"} }], ")"}]}], " ", \( (*8*) \)}]], "Input", CellLabel->"In[49]:="], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", GridBox[{ { \(\(\(\(cos\^3\)(\[Alpha])\)\ \[Epsilon]\^3 + \(\(sin\^3\)(\[Alpha])\)\ \[Epsilon]\^3 + \[Eta]1\ \(\(sin\^2\)(\[Alpha])\)\ \[Epsilon]\^3 + \[Eta]2\ \(\(sin\^2\)(\[Alpha])\)\ \[Epsilon]\^3\)\/\[Epsilon] - \((\[Lambda] - 1)\)\ \(cos(\[Alpha])\)\)}, { \(\(\(\(cos\^3\)(\[Alpha])\)\ \[Epsilon]\^3 + \[Eta]2\ \(\(cos\^2\)(\[Alpha])\)\ \[Epsilon]\^3 + \(cos(\[Alpha])\)\ \(\(sin\^2\)(\[Alpha])\)\ \[Epsilon]\^3 + \[Eta]1\ \(cos(\[Alpha])\)\ \(sin(\[Alpha])\)\ \[Epsilon]\^3\)\/\[Epsilon] - \((\[Lambda] - 1)\)\ \(sin(\[Alpha])\)\)}, { \(\[Eta]1 - \[Eta]1\ \((\[Lambda] - 1)\) + \(\(\(cos\^3\)(\[Alpha])\)\ \[Epsilon]\^3 + \(\(sin\^3\)(\[Alpha])\)\ \[Epsilon]\^3 + \[Eta]2\ \(\(sin\^2\)(\[Alpha])\)\ \[Epsilon]\^3 + \[Eta]1\ \(cos(\[Alpha])\)\ \(sin(\[Alpha])\)\ \[Epsilon]\^3\)\/\[Epsilon]\)}, { \(\[Eta]2 - \[Eta]2\ \((\[Lambda] - 1)\) + \(\(\(cos\^3\)(\[Alpha])\)\ \[Epsilon]\^3 + \(\(sin\^3\)(\[Alpha])\)\ \[Epsilon]\^3 + \[Eta]2\ \(\(sin\^2\)(\[Alpha])\)\ \[Epsilon]\^3 + \[Eta]1\ \[Eta]2\ \(sin(\[Alpha])\)\ \[Epsilon]\^3 + \[Eta]1\ \(cos(\[Alpha])\)\ \(sin(\[Alpha])\)\ \[Epsilon]\^3\)\/\[Epsilon]\)} }], ")"}], "==", RowBox[{"(", GridBox[{ {"0"}, {"0"}, {"0"}, {"0"} }, ColumnAlignments->{Decimal}], ")"}]}], TraditionalForm]], "Output",\ CellLabel->"Out[49]="] }, Closed]], Cell["The first two elements in the above vector equation are ", "Text"], Cell[BoxData[ \(Cos[\[Alpha]]\^3\ \[Epsilon]\^2 + Sin[\[Alpha]]\^3\ \[Epsilon]\^2 + \[Eta]1\ Sin[\[Alpha]]\^2\ \[Epsilon]\^2 + \[Eta]2\ Sin[\[Alpha]]\^2\ \[Epsilon]\^2 - \((\[Lambda] - 1)\)\ Cos[\[Alpha]] == 0\)], "Input"], Cell["and", "Text"], Cell[BoxData[ \(Cos[\[Alpha]]\^3\ \[Epsilon]\^2 + \[Eta]2\ Cos[\[Alpha]]\^2\ \[Epsilon]\^2 + Cos[\[Alpha]]\ Sin[\[Alpha]]\^2\ \[Epsilon]\^2 + \[Eta]1\ Cos[\[Alpha]]\ Sin[\[Alpha]]\ \[Epsilon]\^2 - \((\[Lambda] - 1)\)\ sin\ \[Alpha] == 0\)], "Input"], Cell[TextData[{ "Solving each of these for ", StyleBox["(\[Lambda] - 1)", FontWeight->"Bold"], ", we get" }], "Text"], Cell[BoxData[ \(\((\[Lambda] - 1)\)\ == \(Cos[\[Alpha]]\^3\ \[Epsilon]\^2 + Sin[\[Alpha]]\^3\ \[Epsilon]\^2 + \[Eta]1\ Sin[\[Alpha]]\^2\ \[Epsilon]\^2 + \[Eta]2\ Sin[\[Alpha]]\^2\ \[Epsilon]\^2\)\/Cos[\[Alpha]]\)], "Input", CellLabel->"In[4]:="], Cell["and", "Text"], Cell[BoxData[ \(\((\[Lambda] - 1)\)\ == \(Cos[\[Alpha]]\^3\ \[Epsilon]\^2 + \[Eta]2\ Cos[\[Alpha]]\^2\ \[Epsilon]\^2 + Cos[\[Alpha]]\ Sin[\[Alpha]]\^2\ \[Epsilon]\^2 + \[Eta]1\ Cos[\[Alpha]]\ Sin[\[Alpha]]\ \[Epsilon]\^2\)\/Sin[ \[Alpha]]\)], "Input"], Cell["Equating these we get", "Text"], Cell[BoxData[ \(\(Cos[\[Alpha]]\^3\ \[Epsilon]\^2 + Sin[\[Alpha]]\^3\ \[Epsilon]\^2 + \[Eta]1\ Sin[\[Alpha]]\^2\ \[Epsilon]\^2 + \[Eta]2\ Sin[\[Alpha]]\^2\ \[Epsilon]\^2\)\/Cos[\[Alpha]] == \(1\/Sin[\[Alpha]]\(( Cos[\[Alpha]]\^3\ \[Epsilon]\^2 + \[Eta]2\ Cos[\[Alpha]]\^2\ \[Epsilon]\^2 + Cos[\[Alpha]]\ Sin[\[Alpha]]\^2\ \[Epsilon]\^2 + \[Eta]1\ Cos[\[Alpha]]\ Sin[\[Alpha]]\ \[Epsilon]\^2)\)\)\)], "Input"], Cell[TextData[{ "We note that ", StyleBox["this equation is", FontWeight->"Bold"], " ", StyleBox["the same as", FontWeight->"Bold"], " Equation ", StyleBox["(7)", FontWeight->"Bold"], ", which means that it's leading term for small ", StyleBox["\[Epsilon],", FontWeight->"Bold"], " (", StyleBox["\[Eta]1 = 0 , \[Eta]2 = 0", FontWeight->"Bold"], "), is" }], "Text"], Cell[BoxData[ \(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ t[\[Alpha]]\)\)], "Input"], Cell[TextData[{ "and therefore is satisfied for ", StyleBox["\[Alpha] = \[Alpha]i ", FontFamily->"Terminal", FontWeight->"Bold"], ", ", StyleBox["i = 1,2,3,4", FontWeight->"Bold"], ". " }], "Text"], Cell[TextData[{ "So we may use", StyleBox[" either", FontWeight->"Bold"], " one of the above expressions for ", StyleBox["(\[Lambda] - 1) ", FontWeight->"Bold"], ". This makes sense since none of the ", StyleBox["\[Alpha]i", FontWeight->"Bold"], " 's are equal to ", StyleBox["\[Pi] ", FontWeight->"Bold"], "or", StyleBox[" ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\[Pi]\/2\)], FontWeight->"Bold"], "." }], "Text"], Cell[TextData[{ "We next substitute the 1st expression for ", StyleBox["(\[Lambda] - 1) ", FontWeight->"Bold"], "into the 3rd and 4th elements of \nEquation (8)", StyleBox[".", FontWeight->"Bold"] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\((Cos[\[Alpha]]\^3\ \[Epsilon]\^2 + Sin[\[Alpha]]\^3\ \[Epsilon]\^2 + \[Eta]2\ Sin[\[Alpha]]\^2\ \[Epsilon]\^2 + \[Eta]1\ Cos[\[Alpha]]\ Sin[\[Alpha]]\ \[Epsilon]\^2 + \[Eta]1 - \[Eta]1\ \((\[Lambda] - 1)\) /. \n\t \((\[Lambda]\ - \ 1)\) -> \(Cos[\[Alpha]]\^3\ \[Epsilon]\^2 + Sin[\[Alpha]]\^3\ \[Epsilon]\^2 + \[Eta]1\ Sin[\[Alpha]]\^2\ \[Epsilon]\^2 + \[Eta]2\ Sin[\[Alpha]]\^2\ \[Epsilon]\^2\)\/Cos[\[Alpha]]) \) == 0\)], "Input", CellLabel->"In[153]:="], Cell[BoxData[ \(\[Eta]1 + \[Epsilon]\^2\ Cos[\[Alpha]]\^3 + \[Epsilon]\^2\ \[Eta]1\ Cos[\[Alpha]]\ Sin[\[Alpha]] + \[Epsilon]\^2\ \[Eta]2\ Sin[\[Alpha]]\^2 + \[Epsilon]\^2\ Sin[\[Alpha]]\^3 - \[Eta]1\ Sec[\[Alpha]]\ \((\[Epsilon]\^2\ Cos[\[Alpha]]\^3 + \[Epsilon]\^2\ \[Eta]1\ Sin[\[Alpha]]\^2 + \[Epsilon]\^2\ \[Eta]2\ Sin[\[Alpha]]\^2 + \[Epsilon]\^2\ Sin[\[Alpha]]\^3)\) == 0\)], "Output", CellLabel->"Out[153]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(\((Cos[\[Alpha]]\^3\ \[Epsilon]\^2 + Sin[\[Alpha]]\^3\ \[Epsilon]\^2 + \[Eta]2\ Sin[\[Alpha]]\^2\ \[Epsilon]\^2 + \[Eta]1\ \[Eta]2\ Sin[\[Alpha]]\ \[Epsilon]\^2 + \[Eta]1\ Cos[\[Alpha]]\ Sin[\[Alpha]]\ \[Epsilon]\^2 + \[Eta]2 - \[Eta]2\ \((\[Lambda] - 1)\) /. \t\n\t\t \((\[Lambda]\ - \ 1)\) -> \(Cos[\[Alpha]]\^3\ \[Epsilon]\^2 + Sin[\[Alpha]]\^3\ \[Epsilon]\^2 + \[Eta]1\ Sin[\[Alpha]]\^2\ \[Epsilon]\^2 + \[Eta]2\ Sin[\[Alpha]]\^2\ \[Epsilon]\^2\)\/Cos[\[Alpha]]) \) == 0\)], "Input", CellLabel->"In[154]:="], Cell[BoxData[ \(\[Eta]2 + \[Epsilon]\^2\ Cos[\[Alpha]]\^3 + \[Epsilon]\^2\ \[Eta]1\ \[Eta]2\ Sin[\[Alpha]] + \[Epsilon]\^2\ \[Eta]1\ Cos[\[Alpha]]\ Sin[\[Alpha]] + \[Epsilon]\^2\ \[Eta]2\ Sin[\[Alpha]]\^2 + \[Epsilon]\^2\ Sin[\[Alpha]]\^3 - \[Eta]2\ Sec[\[Alpha]]\ \((\[Epsilon]\^2\ Cos[\[Alpha]]\^3 + \[Epsilon]\^2\ \[Eta]1\ Sin[\[Alpha]]\^2 + \[Epsilon]\^2\ \[Eta]2\ Sin[\[Alpha]]\^2 + \[Epsilon]\^2\ Sin[\[Alpha]]\^3)\) == 0\)], "Output", CellLabel->"Out[154]="] }, Closed]], Cell[TextData[{ "It is not hard to see from these equations that ", StyleBox["\[Eta]1\[RightArrow]0", FontWeight->"Bold"], " and ", StyleBox["\[Eta]2 \[RightArrow]0", FontWeight->"Bold"], " as ", StyleBox["\[Epsilon]\[RightArrow]0", FontWeight->"Bold"], ". Therefore the leading terms in their series will be \n", StyleBox["\[Eta]1", FontFamily->"Terminal", FontWeight->"Bold"], StyleBox[" \[TildeTilde] - ", FontWeight->"Bold"], Cell[BoxData[ \(\[Epsilon]\^2\ Cos[\[Alpha]]\^3\)], FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"], Cell[BoxData[ \(\(-\ \[Epsilon]\^2\)\ Sin[\[Alpha]]\^3\)], FontWeight->"Bold"], " and ", StyleBox["\[Eta]2", FontFamily->"Terminal", FontWeight->"Bold"], " \[TildeTilde] ", StyleBox[" - ", FontWeight->"Bold"], Cell[BoxData[ \(\[Epsilon]\^2\ Cos[\[Alpha]]\^3\)], FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"], Cell[BoxData[ \(\(-\ \[Epsilon]\^2\)\ Sin[\[Alpha]]\^3\)], FontWeight->"Bold"], " . Therefore we define the following." }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{\(\[Eta]1[\[Alpha]_, \[Epsilon]_]\), "=", RowBox[{ RowBox[{ RowBox[{ StyleBox["-", FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"], \(\[Epsilon]\^2\)}], " ", \(Cos[\[Alpha]]\^3\)}], StyleBox[" ", FontWeight->"Bold"], "-", " ", \(\[Epsilon]\^2\ Sin[\[Alpha]]\^3\)}]}], ";"}]], "Input", CellLabel->"In[17]:="], Cell[BoxData[ RowBox[{ RowBox[{\(\[Eta]2[\[Alpha]_, \[Epsilon]_]\), "=", RowBox[{ RowBox[{ RowBox[{ StyleBox["-", FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"], \(\[Epsilon]\^2\)}], " ", \(Cos[\[Alpha]]\^3\)}], StyleBox[" ", FontWeight->"Bold"], "-", " ", \(\[Epsilon]\^2\ Sin[\[Alpha]]\^3\)}]}], ";"}]], "Input", CellLabel->"In[18]:="], Cell[TextData[{ "From this we see that the leading term for ", StyleBox["(\[Lambda] - 1) \[TildeTilde] ", FontWeight->"Bold"], Cell[BoxData[ FormBox[ StyleBox[ \(\(Cos[\[Alpha]]\^3\ \[Epsilon]\^2\ + \ Sin[\[Alpha]]\^3\ \[Epsilon]\^2\)\/Cos[\[Alpha]]\), FontFamily->"Terminal"], TraditionalForm]]], " , where we have used the first of the two expressions for ", StyleBox["(\[Lambda] - 1)", FontWeight->"Bold"], ". Therefore we define" }], "Text"], Cell[BoxData[ \(\(\[Lambda][\[Alpha]_, \[Epsilon]_] = 1\ + \ \ \(Cos[\[Alpha]]\^3\ \[Epsilon]\^2\ + \ Sin[\[Alpha]]\^3\ \[Epsilon]\^2\)\/Cos[\[Alpha]]; \)\)], "Input", CellLabel->"In[19]:="], Cell[TextData[{ "These three functions, ", StyleBox["\[Eta]1[\[Alpha]_,\[Epsilon]_]", FontFamily->"Terminal", FontWeight->"Bold"], StyleBox[", ", FontFamily->"Terminal"], StyleBox["\[Eta]2[\[Alpha]_,\[Epsilon]_]", FontFamily->"Terminal", FontWeight->"Bold"], ", and ", StyleBox["\[Lambda][\[Alpha]_,\[Epsilon]_]", FontFamily->"Terminal", FontWeight->"Bold"], ", will be used to start the iteration scheme for a continuation process \ which will construct the solution branches bifurcating from ", StyleBox["\[Lambda]", FontWeight->"Bold"], " = 1. This will be done as follows." }], "Text"], Cell[BoxData[ \(eqs[x_, y_, u_, v_, \[Lambda]_] := \n \t{x\^3 + x + y\^3 + u\ y\^2 + v\ y\^2 - x\ \[Lambda] == 0, x\^3 + v\ x\^2 + y\^2\ x + u\ y\ x + y - y\ \[Lambda] == 0, x\^3 + u\ y\ x + y\^3 + v\ y\^2 + 2\ u - u\ \[Lambda] == 0, x\^3 + u\ y\ x + y\^3 + v\ y\^2 + 2\ v + u\ v\ y - v\ \[Lambda] == 0} \)], "Input", CellLabel->"In[4]:="], Cell[BoxData[ \(lhseqs[x_, y_, u_, v_, \[Lambda]_] := \n \t{x\^3 + x + y\^3 + u\ y\^2 + v\ y\^2 - x\ \[Lambda], x\^3 + v\ x\^2 + y\^2\ x + u\ y\ x + y - y\ \[Lambda], x\^3 + u\ y\ x + y\^3 + v\ y\^2 + 2\ u - u\ \[Lambda], x\^3 + u\ y\ x + y\^3 + v\ y\^2 + 2\ v + u\ v\ y - v\ \[Lambda]}\)], "Input", CellLabel->"In[5]:="], Cell[TextData[{ "We calculate the determinant of the Jacobian matrix of ", StyleBox["lhseqs", FontFamily->"Terminal", FontWeight->"Bold"], " so that we can see if it gets near zero. This would mean the map is \ becoming singular, and we may be near a bifurcation point." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(jacobian[x_, y_, u_, v_, \[Lambda]_] = {{ \[PartialD]\_x \( lhseqs[x, y, u, v, \[Lambda]]\)\[LeftDoubleBracket]1 \[RightDoubleBracket], \[PartialD]\_y \( lhseqs[x, y, u, v, \[Lambda]]\)\[LeftDoubleBracket]1 \[RightDoubleBracket], \[PartialD]\_u \( lhseqs[x, y, u, v, \[Lambda]]\)\[LeftDoubleBracket]1 \[RightDoubleBracket], \[PartialD]\_v \( lhseqs[x, y, u, v, \[Lambda]]\)\[LeftDoubleBracket]1 \[RightDoubleBracket]}, { \[PartialD]\_x \( lhseqs[x, y, u, v, \[Lambda]]\)\[LeftDoubleBracket]2 \[RightDoubleBracket], \[PartialD]\_y \( lhseqs[x, y, u, v, \[Lambda]]\)\[LeftDoubleBracket]2 \[RightDoubleBracket], \[PartialD]\_u \( lhseqs[x, y, u, v, \[Lambda]]\)\[LeftDoubleBracket]2 \[RightDoubleBracket], \[PartialD]\_v \( lhseqs[x, y, u, v, \[Lambda]]\)\[LeftDoubleBracket]2 \[RightDoubleBracket]}, { \[PartialD]\_x \( lhseqs[x, y, u, v, \[Lambda]]\)\[LeftDoubleBracket]3 \[RightDoubleBracket], \[PartialD]\_y \( lhseqs[x, y, u, v, \[Lambda]]\)\[LeftDoubleBracket]3 \[RightDoubleBracket], \[PartialD]\_u \( lhseqs[x, y, u, v, \[Lambda]]\)\[LeftDoubleBracket]3 \[RightDoubleBracket], \[PartialD]\_v \( lhseqs[x, y, u, v, \[Lambda]]\)\[LeftDoubleBracket]3 \[RightDoubleBracket]}, { \[PartialD]\_x \( lhseqs[x, y, u, v, \[Lambda]]\)\[LeftDoubleBracket]4 \[RightDoubleBracket], \[PartialD]\_y \( lhseqs[x, y, u, v, \[Lambda]]\)\[LeftDoubleBracket]4 \[RightDoubleBracket], \[PartialD]\_u \( lhseqs[x, y, u, v, \[Lambda]]\)\[LeftDoubleBracket]4 \[RightDoubleBracket], \[PartialD]\_v \( lhseqs[x, y, u, v, \[Lambda]]\)\[LeftDoubleBracket]4 \[RightDoubleBracket]}}\)], "Input", CellLabel->"In[6]:="], Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ {\(3\ x\^2 - \[Lambda] + 1\), \(3\ y\^2 + 2\ u\ y + 2\ v\ y\), \(y\^2\), \(y\^2\)}, {\(3\ x\^2 + 2\ v\ x + y\^2 + u\ y\), \(u\ x + 2\ y\ x - \[Lambda] + 1\), \(x\ y\), \(x\^2\)}, {\(3\ x\^2 + u\ y\), \(3\ y\^2 + 2\ v\ y + u\ x\), \(x\ y - \[Lambda] + 2\), \(y\^2\)}, {\(3\ x\^2 + u\ y\), \(3\ y\^2 + 2\ v\ y + u\ v + u\ x\), \(v\ y + x\ y\), \(y\^2 + u\ y - \[Lambda] + 2\)} }], ")"}], TraditionalForm]], "Output", CellLabel->"Out[6]="] }, Open ]], Cell[BoxData[ \(\(jdet[{x_, y_, u_, v_}, \[Lambda]_] = Det[jacobian[x, y, u, v, \[Lambda]]]; \)\)], "Input", CellLabel->"In[7]:="], Cell[BoxData[ \(findroot[{x0_, y0_, u0_, v0_}, \[Lambda]_] := {x, y, u, v} /. FindRoot[ Evaluate[eqs[x, y, u, v, \[Lambda]]], {x, x0}, {y, y0}, {u, u0}, { v, v0}, MaxIterations -> 20]\)], "Input", CellLabel->"In[8]:="], Cell[TextData[{ "In order to follow a branch of solutions as we increment ", StyleBox["\[Lambda]", FontFamily->"Terminal", FontWeight->"Bold"], ", we define the following function ", StyleBox["bifbranch", FontFamily->"Terminal", FontWeight->"Bold"], ". In the arguments of this function, ", StyleBox["x0, y0, u0, v0", FontFamily->"Terminal", FontWeight->"Bold"], StyleBox[", and ", FontWeight->"Bold"], StyleBox["\[Lambda]00", FontFamily->"Terminal", FontWeight->"Bold"], StyleBox[" ", FontFamily->"Terminal"], "represent the starting values of ", StyleBox["x, y, u, v,", FontFamily->"Terminal", FontWeight->"Bold"], StyleBox[" and ", FontWeight->"Bold"], StyleBox["\[Lambda]", FontFamily->"Terminal", FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"], "respectively, with ", StyleBox["d\[Lambda]", FontFamily->"Terminal", FontWeight->"Bold"], " being the size of the increment of ", StyleBox["\[Lambda]", FontFamily->"Terminal", FontWeight->"Bold"], " and ", StyleBox["n", FontWeight->"Bold"], " the number of increments to perform . " }], "Text"], Cell[BoxData[ \(\(\nbifbranch[x0_, y0_, u0_, v0_, \[Lambda]00_, \ d\[Lambda]_, n_]\ := \ \n\ \ Module[{sol, jd}, \ \n\ \ sol[0] := {x0, y0, u0, v0}; \ \n\ \ \ sol[1] := findroot[sol[0], \[Lambda]00 + \ d\[Lambda]]; \ \n\ \ \ sol[i_] := \((sol[i] = findroot[2*sol[i - 1] - sol[i - 2], \[Lambda]00 + i*\ d\[Lambda]])\) /; i >= 2; \n\ \ \ jd[i_]\ := \ \(jd[i]\ = \ jdet[sol[i], \[Lambda]00 + i*\ d\[Lambda]]\); \ \n\t\t Table[{\[Lambda]00 + j*\ d\[Lambda], sol[j], jd[j]}, {j, 1, n}]\n \t\ \ \ \ \ \ \ \ \ \ \ ]\)\)], "Input", CellLabel->"In[9]:="], Cell[BoxData[ \(branch[\[Alpha]_, \[Epsilon]_, d\[Lambda]_, n_] := \n\t bifbranch[\[Epsilon]\ Cos[\[Alpha]], \[Epsilon]\ Sin[\[Alpha]], \[Epsilon]\ \[Eta]1[\[Alpha], \[Epsilon]], \[Eta]2[\[Alpha], \[Epsilon]], \[Lambda][\[Alpha], \[Epsilon]], d\[Lambda], n]\)], "Input", CellLabel->"In[84]:="], Cell[TextData[{ "We first find the branch for ", StyleBox["\[Alpha] = \[Alpha]1", FontFamily->"Terminal", FontWeight->"Bold"], StyleBox[".", FontFamily->"Terminal"] }], "Text"], Cell[BoxData[ \(\(bbranch = branch[\[Alpha]1, .02, .0005, 650]; \)\)], "Input", CellLabel->"In[1]:="], Cell[TextData[{ "The following functions will enable us to plot each of the variables ", StyleBox["x,y,u,v", FontFamily->"Terminal", FontWeight->"Bold"], " versus ", StyleBox["\[Lambda]", FontFamily->"Terminal", FontWeight->"Bold"], "." }], "Text"], Cell[BoxData[{ \(\nplotsolx[\[Alpha]_, c_] := \ \ \ \n\ \ \ \ \ \ \ \ ListPlot[Table[{bbranch[\([i, 1]\)], bbranch[\([i, 2, 1]\)]}, {i, Length[bbranch]}\n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ], \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ Axes -> Automatic, AxesLabel -> {\[Lambda], x}, PlotJoined -> True, \n \t\t\ \ \ \ \ \ \ \ PlotLabel -> StringForm["\<\[Alpha]=``\>", \[Alpha]], \ PlotRange -> All\ , PlotStyle -> Hue[c]]\n\), \(plotsoly[\[Alpha]_, c_] := \ \ \ \n\ \ \ \ \ \ \ \ ListPlot[Table[{bbranch[\([i, 1]\)], bbranch[\([i, 2, 2]\)]}, {i, Length[bbranch]}\n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ], \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ Axes -> Automatic, AxesLabel -> {\[Lambda], y}, PlotJoined -> True, \n \t\t\ \ \ \ \ \ \ \ PlotLabel -> StringForm["\<\[Alpha]=``\>", \[Alpha]], \ PlotRange -> All\ , PlotStyle -> Hue[c]]\n\), \(plotsolu[\[Alpha]_, c_] := \ \ \ \n\ \ \ \ \ \ \ \ ListPlot[Table[{bbranch[\([i, 1]\)], bbranch[\([i, 2, 3]\)]}, {i, Length[bbranch]}\n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ], \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ Axes -> Automatic, AxesLabel -> {\[Lambda], u}, PlotJoined -> True, \n \t\t\ \ \ \ \ \ \ \ PlotLabel -> StringForm["\<\[Alpha]=``\>", \[Alpha]], \ PlotRange -> All\ , PlotStyle -> Hue[c]]\n\), \(plotsolv[\[Alpha]_, c_] := \ \ \ \n\ \ \ \ \ \ \ \ ListPlot[Table[{bbranch[\([i, 1]\)], bbranch[\([i, 2, 4]\)]}, {i, Length[bbranch]}\n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ], \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ Axes -> Automatic, 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Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output", CellLabel->"Out[92]="] }, Closed]], Cell[TextData[{ "We see from these graphs that the derivatives with respect to ", StyleBox["\[Lambda]", FontFamily->"Terminal", FontWeight->"Bold"], " are getting large, which means that the map is becoming singular. In \ order to get around this difficulty we change variables. Instead of having ", StyleBox["\[Lambda]", FontFamily->"Terminal", FontWeight->"Bold"], " as the independent variable we change to a new independent variable, say \ ", StyleBox["x", FontFamily->"Terminal", FontWeight->"Bold"], ". Now we can think of the unknowns being ", StyleBox["(\[Lambda],y,u,v)", FontFamily->"Terminal", FontWeight->"Bold"], " instead of ", StyleBox["(x,y,u,v", FontFamily->"Terminal", FontWeight->"Bold"], StyleBox[")", FontFamily->"Terminal"], ". We accomplish this with the following." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(jacobianx[x_, y_, u_, v_, \[Lambda]_] = {{ \[PartialD]\_\[Lambda]\( lhseqs[x, y, u, v, \[Lambda]] \)\[LeftDoubleBracket]1\[RightDoubleBracket], \[PartialD]\_y \( lhseqs[x, y, u, v, \[Lambda]]\)\[LeftDoubleBracket]1 \[RightDoubleBracket], \[PartialD]\_u \( lhseqs[x, y, u, v, \[Lambda]]\)\[LeftDoubleBracket]1 \[RightDoubleBracket], \[PartialD]\_v \( lhseqs[x, y, u, v, \[Lambda]]\)\[LeftDoubleBracket]1 \[RightDoubleBracket]}, { \[PartialD]\_\[Lambda]\( lhseqs[x, y, u, v, \[Lambda]] \)\[LeftDoubleBracket]2\[RightDoubleBracket], \[PartialD]\_y \( lhseqs[x, y, u, v, \[Lambda]]\)\[LeftDoubleBracket]2 \[RightDoubleBracket], \[PartialD]\_u \( lhseqs[x, y, u, v, \[Lambda]]\)\[LeftDoubleBracket]2 \[RightDoubleBracket], \[PartialD]\_v \( lhseqs[x, y, u, v, \[Lambda]]\)\[LeftDoubleBracket]2 \[RightDoubleBracket]}, { \[PartialD]\_\[Lambda]\( lhseqs[x, y, u, v, \[Lambda]] \)\[LeftDoubleBracket]3\[RightDoubleBracket], \[PartialD]\_y \( lhseqs[x, y, u, v, \[Lambda]]\)\[LeftDoubleBracket]3 \[RightDoubleBracket], \[PartialD]\_u \( lhseqs[x, y, u, v, \[Lambda]]\)\[LeftDoubleBracket]3 \[RightDoubleBracket], \[PartialD]\_v \( lhseqs[x, y, u, v, \[Lambda]]\)\[LeftDoubleBracket]3 \[RightDoubleBracket]}, { \[PartialD]\_\[Lambda]\( lhseqs[x, y, u, v, \[Lambda]] \)\[LeftDoubleBracket]4\[RightDoubleBracket], \[PartialD]\_y \( lhseqs[x, y, u, v, \[Lambda]]\)\[LeftDoubleBracket]4 \[RightDoubleBracket], \[PartialD]\_u \( lhseqs[x, y, u, v, \[Lambda]]\)\[LeftDoubleBracket]4 \[RightDoubleBracket], \[PartialD]\_v \( lhseqs[x, y, u, v, \[Lambda]]\)\[LeftDoubleBracket]4 \[RightDoubleBracket]}}\)], "Input", CellLabel->"In[10]:="], Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ {\(-x\), \(3\ y\^2 + 2\ u\ y + 2\ v\ y\), \(y\^2\), \(y\^2\)}, {\(-y\), \(u\ x + 2\ y\ x - \[Lambda] + 1\), \(x\ y\), \(x\^2\)}, {\(-u\), \(3\ y\^2 + 2\ v\ y + u\ x\), \(x\ y - \[Lambda] + 2\), \(y\^2\)}, {\(-v\), \(3\ y\^2 + 2\ v\ y + u\ v + u\ x\), \(v\ y + x\ y\), \(y\^2 + u\ y - \[Lambda] + 2\)} }], ")"}], TraditionalForm]], "Output", CellLabel->"Out[10]="] }, Open ]], Cell[BoxData[ \(\(jdetx[{\[Lambda]_, y_, u_, v_}, x_] = Det[jacobianx[x, y, u, v, \[Lambda]]]; \)\)], "Input", CellLabel->"In[93]:="], Cell[BoxData[ \(findrootx[{\[Lambda]00_, y0_, u0_, v0_}, x_] := {\[Lambda], y, u, v} /. FindRoot[ Evaluate[eqs[x, y, u, v, \[Lambda]]], {\[Lambda], \[Lambda]00}, {y, y0}, {u, u0}, {v, v0}, MaxIterations -> 20]\)], "Input", CellLabel->"In[94]:="], Cell[BoxData[ \(\(\nbifbranchx[\[Lambda]00_, y0_, u0_, v0_, x0_, \ dx_, n_]\ := \ \n \ \ Module[{sol, jd}, \ \n\ \ sol[0] := {\[Lambda]00, y0, u0, v0}; \ \n \ \ \ sol[1] := findrootx[sol[0], x0 + \ dx]; \ \n\ \ \ sol[i_] := \((sol[i] = findrootx[2*sol[i - 1] - sol[i - 2], x0 + i*\ dx])\) /; i >= 2; \n\ \ \ jd[i_]\ := \ \(jd[i]\ = \ jdetx[sol[i], x0 + i*\ dx]\); \ \n\t\t Table[{x0 + j*\ dx, sol[j], jd[j]}, {j, 1, n}]\n \t\ \ \ \ \ \ \ \ \ \ \ ]\)\)], "Input", CellLabel->"In[95]:="], Cell["\<\ We start this branch with the ending values in the previous calculation.\ \>", "Text"], Cell[BoxData[ \(\(bbranch = \n bifbranchx[1.3, 0.533956558414711235`, \(-0.228838525034876294`\), \(-0.277231132406344116\), 0.533956558414711235`, .001, 3000]; \)\)], "Input", CellLabel->"In[96]:="], Cell["We define a new set of plot functions for this case.", "Text"], Cell[BoxData[{ \(\nplotsolxx[\[Alpha]_, c_] := \ \ \ \n\ \ \ \ \ \ \ \ ListPlot[Table[{bbranch[\([i, 2, 1]\)], bbranch[\([i, 1]\)]}, {i, Length[bbranch]}\n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ], \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ Axes -> Automatic, AxesLabel -> {\[Lambda], x}, PlotJoined -> True, \n \t\t\ \ \ \ \ \ \ \ PlotLabel -> StringForm["\<\[Alpha]=``\>", \[Alpha]], \ PlotRange -> All\ , PlotStyle -> Hue[c]]\n\), \(plotsolxy[\[Alpha]_, c_] := \ \ \ \n\ \ \ \ \ \ \ \ ListPlot[Table[{bbranch[\([i, 2, 1]\)], bbranch[\([i, 2, 2]\)]}, {i, Length[bbranch]}\n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ], \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ Axes -> Automatic, AxesLabel -> {\[Lambda], y}, PlotJoined -> True, \n \t\t\ \ \ \ \ \ \ \ PlotLabel -> StringForm["\<\[Alpha]=``\>", \[Alpha]], \ PlotRange -> All\ , PlotStyle -> Hue[c]]\n\), \(plotsolxu[\[Alpha]_, c_] := \ \ \ \n\ \ \ \ \ \ \ \ ListPlot[Table[{bbranch[\([i, 2, 1]\)], bbranch[\([i, 2, 3]\)]}, {i, Length[bbranch]}\n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ], \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ Axes -> Automatic, AxesLabel -> {\[Lambda], u}, PlotJoined -> True, \n \t\t\ \ \ \ \ \ \ \ PlotLabel -> StringForm["\<\[Alpha]=``\>", \[Alpha]], \ PlotRange -> All\ , PlotStyle -> Hue[c]]\n\), \(plotsolxv[\[Alpha]_, c_] := \ \ \ \n\ \ \ \ \ \ \ \ ListPlot[Table[{bbranch[\([i, 2, 1]\)], bbranch[\([i, 2, 4]\)]}, {i, Length[bbranch]}\n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ], \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ Axes -> Automatic, AxesLabel -> {\[Lambda], v}, PlotJoined -> True, \n \t\t\ \ \ \ \ \ \ \ PlotLabel -> StringForm["\<\[Alpha]=``\>", \[Alpha]], \ PlotRange -> All\ , PlotStyle -> Hue[c]]\)}], "Input", CellLabel->"In[29]:="], Cell[CellGroupData[{ Cell[BoxData[ \(plotsolxx[\[Alpha]1, 0]\)], "Input", CellLabel->"In[97]:="], Cell[GraphicsData["PostScript", 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.87615 .10519 L .87555 .10539 L .87495 .10559 L .87435 .10578 L .87375 .10598 L .87314 .10618 L .87254 .10637 L .87194 .10657 L .87133 .10676 L .87073 .10696 L .87012 .10716 L .86952 .10735 L .86891 .10755 L .8683 .10775 L .86769 .10794 L .86709 .10814 L .86648 .10833 L .86587 .10853 L .86526 .10873 L .86465 .10892 L .86403 .10912 L .86342 .10932 L .86281 .10951 L .8622 .10971 L .86158 .1099 L .86097 .1101 L .86035 .1103 L .85974 .11049 L .85912 .11069 L .85851 .11089 L .85789 .11108 L .85727 .11128 L .85666 .11147 L .85604 .11167 L .85542 .11187 L .8548 .11206 L .85418 .11226 L .85356 .11246 L .85294 .11265 L Mistroke .85232 .11285 L .8517 .11304 L .85108 .11324 L .85046 .11344 L .84983 .11363 L .84921 .11383 L .84859 .11403 L .84796 .11422 L .84734 .11442 L .84672 .11461 L .84609 .11481 L .84547 .11501 L .84484 .1152 L .84421 .1154 L .84359 .1156 L .84296 .11579 L .84233 .11599 L .84171 .11618 L .84108 .11638 L .84045 .11658 L .83982 .11677 L .83919 .11697 L .83857 .11717 L .83794 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.79786 .12973 L .79722 .12992 L .79658 .13012 L .79594 .13032 L .7953 .13051 L .79465 .13071 L .79401 .1309 L .79337 .1311 L .79273 .1313 L .79209 .13149 L .79145 .13169 L .79081 .13189 L .79017 .13208 L .78953 .13228 L Mistroke .78889 .13248 L .78825 .13267 L .78761 .13287 L .78697 .13306 L .78633 .13326 L .78569 .13346 L .78505 .13365 L .78441 .13385 L .78377 .13405 L .78313 .13424 L .78249 .13444 L .78184 .13463 L .7812 .13483 L .78056 .13503 L .77992 .13522 L .77928 .13542 L .77864 .13562 L .778 .13581 L .77736 .13601 L .77672 .1362 L .77608 .1364 L .77544 .1366 L .7748 .13679 L .77416 .13699 L .77352 .13719 L .77288 .13738 L .77224 .13758 L .7716 .13777 L .77096 .13797 L .77032 .13817 L .76969 .13836 L .76905 .13856 L .76841 .13876 L .76777 .13895 L .76713 .13915 L .76649 .13934 L .76585 .13954 L .76521 .13974 L .76457 .13993 L .76393 .14013 L .7633 .14033 L .76266 .14052 L .76202 .14072 L .76138 .14091 L .76074 .14111 L .76011 .14131 L .75947 .1415 L .75883 .1417 L .75819 .1419 L .75755 .14209 L Mistroke .75692 .14229 L .75628 .14248 L .75564 .14268 L .75501 .14288 L .75437 .14307 L .75373 .14327 L .7531 .14347 L .75246 .14366 L .75182 .14386 L .75119 .14405 L .75055 .14425 L .74991 .14445 L .74928 .14464 L .74864 .14484 L .74801 .14504 L .74737 .14523 L .74674 .14543 L .7461 .14562 L .74547 .14582 L .74483 .14602 L .7442 .14621 L .74357 .14641 L .74293 .14661 L .7423 .1468 L .74166 .147 L .74103 .1472 L .7404 .14739 L .73976 .14759 L .73913 .14778 L .7385 .14798 L .73786 .14818 L .73723 .14837 L .7366 .14857 L .73597 .14877 L .73534 .14896 L .7347 .14916 L .73407 .14935 L .73344 .14955 L .73281 .14975 L .73218 .14994 L .73155 .15014 L .73092 .15034 L .73029 .15053 L .72966 .15073 L .72903 .15092 L .7284 .15112 L .72777 .15132 L .72714 .15151 L .72651 .15171 L .72588 .15191 L Mistroke .72525 .1521 L .72462 .1523 L .724 .15249 L .72337 .15269 L .72274 .15289 L .72211 .15308 L .72149 .15328 L .72086 .15348 L .72023 .15367 L .7196 .15387 L .71898 .15406 L .71835 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.16663 L .67883 .16682 L .67822 .16702 L .67761 .16721 L .677 .16741 L .6764 .16761 L .67579 .1678 L .67518 .168 L .67458 .1682 L .67397 .16839 L .67336 .16859 L .67276 .16878 L .67215 .16898 L .67155 .16918 L .67094 .16937 L .67034 .16957 L .66974 .16977 L .66913 .16996 L .66853 .17016 L .66793 .17035 L .66732 .17055 L .66672 .17075 L .66612 .17094 L .66552 .17114 L .66492 .17134 L .66432 .17153 L Mistroke .66372 .17173 L .66312 .17192 L .66252 .17212 L .66192 .17232 L .66132 .17251 L .66072 .17271 L .66012 .17291 L .65952 .1731 L .65892 .1733 L .65832 .17349 L .65773 .17369 L .65713 .17389 L .65653 .17408 L .65594 .17428 L .65534 .17448 L .65475 .17467 L .65415 .17487 L .65356 .17506 L .65296 .17526 L .65237 .17546 L .65177 .17565 L .65118 .17585 L .65059 .17605 L .64999 .17624 L .6494 .17644 L .64881 .17664 L .64822 .17683 L .64763 .17703 L .64703 .17722 L .64644 .17742 L .64585 .17762 L .64526 .17781 L .64467 .17801 L .64408 .17821 L .6435 .1784 L .64291 .1786 L .64232 .17879 L .64173 .17899 L .64114 .17919 L .64056 .17938 L .63997 .17958 L .63938 .17978 L .6388 .17997 L .63821 .18017 L .63762 .18036 L .63704 .18056 L .63645 .18076 L .63587 .18095 L .63529 .18115 L .6347 .18135 L Mistroke .63412 .18154 L .63354 .18174 L .63295 .18193 L .63237 .18213 L .63179 .18233 L .63121 .18252 L .63063 .18272 L .63004 .18292 L .62946 .18311 L .62888 .18331 L .6283 .1835 L .62772 .1837 L .62715 .1839 L .62657 .18409 L .62599 .18429 L .62541 .18449 L .62483 .18468 L .62426 .18488 L .62368 .18507 L .6231 .18527 L .62253 .18547 L .62195 .18566 L .62137 .18586 L .6208 .18606 L .62023 .18625 L .61965 .18645 L .61908 .18664 L .6185 .18684 L .61793 .18704 L .61736 .18723 L .61678 .18743 L .61621 .18763 L .61564 .18782 L .61507 .18802 L .6145 .18821 L .61393 .18841 L .61336 .18861 L .61279 .1888 L .61222 .189 L .61165 .1892 L .61108 .18939 L .61051 .18959 L .60994 .18978 L .60938 .18998 L .60881 .19018 L .60824 .19037 L .60768 .19057 L .60711 .19077 L .60654 .19096 L .60598 .19116 L Mistroke .60541 .19136 L .60485 .19155 L .60428 .19175 L .60372 .19194 L .60316 .19214 L .60259 .19234 L .60203 .19253 L .60147 .19273 L .60091 .19293 L .60034 .19312 L .59978 .19332 L .59922 .19351 L .59866 .19371 L .5981 .19391 L .59754 .1941 L .59698 .1943 L .59642 .1945 L .59586 .19469 L .59531 .19489 L .59475 .19508 L .59419 .19528 L .59363 .19548 L .59308 .19567 L .59252 .19587 L .59196 .19607 L .59141 .19626 L .59085 .19646 L .5903 .19665 L .58974 .19685 L .58919 .19705 L .58864 .19724 L .58808 .19744 L .58753 .19764 L .58698 .19783 L .58643 .19803 L .58587 .19822 L .58532 .19842 L .58477 .19862 L .58422 .19881 L .58367 .19901 L .58312 .19921 L .58257 .1994 L .58202 .1996 L .58147 .19979 L .58093 .19999 L .58038 .20019 L .57983 .20038 L .57928 .20058 L .57874 .20078 L .57819 .20097 L Mistroke .57764 .20117 L .5771 .20136 L .57655 .20156 L .57601 .20176 L .57546 .20195 L .57492 .20215 L .57437 .20235 L .57383 .20254 L .57329 .20274 L .57275 .20293 L .5722 .20313 L .57166 .20333 L .57112 .20352 L .57058 .20372 L .57004 .20392 L .5695 .20411 L .56896 .20431 L .56842 .2045 L .56788 .2047 L .56734 .2049 L .5668 .20509 L .56627 .20529 L .56573 .20549 L .56519 .20568 L .56465 .20588 L .56412 .20608 L .56358 .20627 L .56305 .20647 L .56251 .20666 L .56198 .20686 L .56144 .20706 L .56091 .20725 L .56037 .20745 L .55984 .20765 L .55931 .20784 L .55877 .20804 L .55824 .20823 L .55771 .20843 L .55718 .20863 L .55665 .20882 L .55612 .20902 L .55559 .20922 L .55506 .20941 L .55453 .20961 L .554 .2098 L .55347 .21 L .55294 .2102 L .55241 .21039 L .55189 .21059 L .55136 .21079 L Mistroke .55083 .21098 L .55031 .21118 L .54978 .21137 L .54925 .21157 L .54873 .21177 L .5482 .21196 L .54768 .21216 L .54716 .21236 L .54663 .21255 L .54611 .21275 L .54559 .21294 L .54506 .21314 L .54454 .21334 L .54402 .21353 L .5435 .21373 L .54298 .21393 L .54246 .21412 L .54194 .21432 L .54142 .21451 L .5409 .21471 L .54038 .21491 L .53986 .2151 L .53934 .2153 L .53882 .2155 L .53831 .21569 L .53779 .21589 L .53727 .21608 L .53676 .21628 L .53624 .21648 L .53573 .21667 L .53521 .21687 L .5347 .21707 L .53418 .21726 L .53367 .21746 L .53315 .21765 L .53264 .21785 L .53213 .21805 L .53161 .21824 L .5311 .21844 L .53059 .21864 L .53008 .21883 L .52957 .21903 L .52906 .21922 L .52855 .21942 L .52804 .21962 L .52753 .21981 L .52702 .22001 L .52651 .22021 L .526 .2204 L .5255 .2206 L Mistroke .52499 .2208 L .52448 .22099 L .52397 .22119 L .52347 .22138 L .52296 .22158 L .52246 .22178 L .52195 .22197 L .52145 .22217 L .52094 .22237 L .52044 .22256 L .51993 .22276 L .51943 .22295 L .51893 .22315 L .51843 .22335 L .51792 .22354 L .51742 .22374 L .51692 .22394 L .51642 .22413 L .51592 .22433 L .51542 .22452 L .51492 .22472 L .51442 .22492 L .51392 .22511 L .51342 .22531 L .51293 .22551 L .51243 .2257 L .51193 .2259 L .51143 .22609 L .51094 .22629 L .51044 .22649 L .50994 .22668 L .50945 .22688 L .50895 .22708 L .50846 .22727 L .50796 .22747 L .50747 .22766 L .50698 .22786 L .50648 .22806 L .50599 .22825 L .5055 .22845 L .50501 .22865 L .50451 .22884 L .50402 .22904 L .50353 .22923 L .50304 .22943 L .50255 .22963 L .50206 .22982 L .50157 .23002 L .50108 .23022 L .50059 .23041 L Mistroke .50011 .23061 L .49962 .2308 L .49913 .231 L .49864 .2312 L .49816 .23139 L .49767 .23159 L .49718 .23179 L .4967 .23198 L .49621 .23218 L .49573 .23237 L .49524 .23257 L .49476 .23277 L .49428 .23296 L .49379 .23316 L .49331 .23336 L .49283 .23355 L .49234 .23375 L .49186 .23394 L .49138 .23414 L .4909 .23434 L .49042 .23453 L .48994 .23473 L .48946 .23493 L .48898 .23512 L .4885 .23532 L .48802 .23552 L .48754 .23571 L .48706 .23591 L .48659 .2361 L .48611 .2363 L .48563 .2365 L .48516 .23669 L .48468 .23689 L .4842 .23709 L .48373 .23728 L .48325 .23748 L .48278 .23767 L .4823 .23787 L .48183 .23807 L .48136 .23826 L .48088 .23846 L .48041 .23866 L .47994 .23885 L .47947 .23905 L .47899 .23924 L .47852 .23944 L .47805 .23964 L .47758 .23983 L .47711 .24003 L .47664 .24023 L Mistroke .47617 .24042 L .4757 .24062 L .47523 .24081 L .47477 .24101 L .4743 .24121 L .47383 .2414 L .47336 .2416 L .4729 .2418 L .47243 .24199 L .47196 .24219 L .4715 .24238 L .47103 .24258 L .47057 .24278 L .4701 .24297 L .46964 .24317 L .46917 .24337 L .46871 .24356 L .46825 .24376 L .46778 .24395 L .46732 .24415 L .46686 .24435 L .4664 .24454 L .46594 .24474 L .46548 .24494 L .46502 .24513 L .46456 .24533 L .4641 .24552 L .46364 .24572 L .46318 .24592 L .46272 .24611 L .46226 .24631 L .4618 .24651 L .46134 .2467 L .46089 .2469 L .46043 .24709 L .45997 .24729 L .45952 .24749 L .45906 .24768 L .45861 .24788 L .45815 .24808 L .4577 .24827 L .45724 .24847 L .45679 .24866 L .45633 .24886 L .45588 .24906 L .45543 .24925 L .45497 .24945 L .45452 .24965 L .45407 .24984 L .45362 .25004 L Mistroke .45317 .25024 L .45272 .25043 L .45227 .25063 L .45182 .25082 L .45137 .25102 L .45092 .25122 L .45047 .25141 L .45002 .25161 L .44957 .25181 L .44912 .252 L .44868 .2522 L .44823 .25239 L .44778 .25259 L .44734 .25279 L .44689 .25298 L .44644 .25318 L .446 .25338 L .44555 .25357 L .44511 .25377 L .44467 .25396 L .44422 .25416 L .44378 .25436 L .44333 .25455 L .44289 .25475 L .44245 .25495 L .44201 .25514 L .44157 .25534 L .44112 .25553 L .44068 .25573 L .44024 .25593 L .4398 .25612 L .43936 .25632 L .43892 .25652 L .43848 .25671 L .43804 .25691 L .43761 .2571 L .43717 .2573 L .43673 .2575 L .43629 .25769 L .43586 .25789 L .43542 .25809 L .43498 .25828 L .43455 .25848 L .43411 .25867 L .43367 .25887 L .43324 .25907 L .43281 .25926 L .43237 .25946 L .43194 .25966 L .4315 .25985 L Mistroke .43107 .26005 L .43064 .26024 L .4302 .26044 L .42977 .26064 L .42934 .26083 L .42891 .26103 L .42848 .26123 L .42805 .26142 L .42762 .26162 L .42719 .26181 L .42676 .26201 L .42633 .26221 L .4259 .2624 L .42547 .2626 L .42504 .2628 L .42461 .26299 L .42419 .26319 L .42376 .26338 L .42333 .26358 L .4229 .26378 L .42248 .26397 L .42205 .26417 L .42163 .26437 L .4212 .26456 L .42078 .26476 L .42035 .26496 L .41993 .26515 L .4195 .26535 L .41908 .26554 L .41866 .26574 L .41824 .26594 L .41781 .26613 L .41739 .26633 L .41697 .26653 L .41655 .26672 L .41613 .26692 L .41571 .26711 L .41529 .26731 L .41487 .26751 L .41445 .2677 L .41403 .2679 L .41361 .2681 L .41319 .26829 L .41277 .26849 L .41235 .26868 L .41193 .26888 L .41152 .26908 L .4111 .26927 L .41068 .26947 L .41027 .26967 L Mistroke .40985 .26986 L .40944 .27006 L .40902 .27025 L .40861 .27045 L .40819 .27065 L .40778 .27084 L .40736 .27104 L .40695 .27124 L .40654 .27143 L .40612 .27163 L .40571 .27182 L .4053 .27202 L .40489 .27222 L .40447 .27241 L .40406 .27261 L .40365 .27281 L .40324 .273 L .40283 .2732 L .40242 .27339 L .40201 .27359 L .4016 .27379 L .40119 .27398 L .40079 .27418 L .40038 .27438 L .39997 .27457 L .39956 .27477 L .39916 .27496 L .39875 .27516 L .39834 .27536 L .39794 .27555 L .39753 .27575 L .39712 .27595 L .39672 .27614 L .39631 .27634 L .39591 .27653 L .39551 .27673 L .3951 .27693 L .3947 .27712 L .3943 .27732 L .39389 .27752 L .39349 .27771 L .39309 .27791 L .39269 .2781 L .39228 .2783 L .39188 .2785 L .39148 .27869 L .39108 .27889 L .39068 .27909 L .39028 .27928 L .38988 .27948 L Mistroke .38948 .27968 L .38908 .27987 L .38869 .28007 L .38829 .28026 L .38789 .28046 L .38749 .28066 L .38709 .28085 L .3867 .28105 L .3863 .28125 L .3859 .28144 L .38551 .28164 L .38511 .28183 L .38472 .28203 L .38432 .28223 L .38393 .28242 L .38353 .28262 L .38314 .28282 L .38275 .28301 L .38235 .28321 L .38196 .2834 L .38157 .2836 L .38117 .2838 L .38078 .28399 L .38039 .28419 L .38 .28439 L .37961 .28458 L .37922 .28478 L .37883 .28497 L .37844 .28517 L .37805 .28537 L .37766 .28556 L .37727 .28576 L .37688 .28596 L .37649 .28615 L .3761 .28635 L .37571 .28654 L .37533 .28674 L .37494 .28694 L .37455 .28713 L .37417 .28733 L .37378 .28753 L .37339 .28772 L .37301 .28792 L .37262 .28811 L .37224 .28831 L .37185 .28851 L .37147 .2887 L .37108 .2889 L .3707 .2891 L .37032 .28929 L Mistroke .36993 .28949 L .36955 .28968 L .36917 .28988 L .36879 .29008 L .36841 .29027 L .36802 .29047 L .36764 .29067 L .36726 .29086 L .36688 .29106 L .3665 .29125 L .36612 .29145 L .36574 .29165 L .36536 .29184 L .36498 .29204 L .3646 .29224 L .36423 .29243 L .36385 .29263 L .36347 .29282 L .36309 .29302 L .36271 .29322 L .36234 .29341 L .36196 .29361 L .36159 .29381 L .36121 .294 L .36083 .2942 L .36046 .2944 L .36008 .29459 L .35971 .29479 L .35933 .29498 L .35896 .29518 L .35859 .29538 L .35821 .29557 L .35784 .29577 L .35747 .29597 L .35709 .29616 L .35672 .29636 L .35635 .29655 L .35598 .29675 L .35561 .29695 L .35524 .29714 L .35487 .29734 L .3545 .29754 L .35413 .29773 L .35376 .29793 L .35339 .29812 L .35302 .29832 L .35265 .29852 L .35228 .29871 L .35191 .29891 L .35154 .29911 L Mistroke .35118 .2993 L .35081 .2995 L .35044 .29969 L .35007 .29989 L .34971 .30009 L .34934 .30028 L .34898 .30048 L .34861 .30068 L .34825 .30087 L .34788 .30107 L .34752 .30126 L .34715 .30146 L .34679 .30166 L .34642 .30185 L .34606 .30205 L .3457 .30225 L .34533 .30244 L .34497 .30264 L .34461 .30283 L .34425 .30303 L .34389 .30323 L .34353 .30342 L .34316 .30362 L .3428 .30382 L .34244 .30401 L .34208 .30421 L .34172 .3044 L .34136 .3046 L .341 .3048 L .34065 .30499 L .34029 .30519 L .33993 .30539 L .33957 .30558 L .33921 .30578 L .33886 .30597 L .3385 .30617 L .33814 .30637 L .33778 .30656 L .33743 .30676 L .33707 .30696 L .33672 .30715 L .33636 .30735 L .33601 .30754 L .33565 .30774 L .3353 .30794 L .33494 .30813 L .33459 .30833 L .33424 .30853 L .33388 .30872 L .33353 .30892 L Mistroke .33318 .30912 L .33282 .30931 L .33247 .30951 L .33212 .3097 L .33177 .3099 L .33142 .3101 L .33107 .31029 L .33071 .31049 L .33036 .31069 L .33001 .31088 L .32966 .31108 L .32931 .31127 L .32897 .31147 L .32862 .31167 L .32827 .31186 L .32792 .31206 L .32757 .31226 L .32722 .31245 L .32688 .31265 L .32653 .31284 L .32618 .31304 L .32584 .31324 L .32549 .31343 L .32514 .31363 L .3248 .31383 L .32445 .31402 L .32411 .31422 L .32376 .31441 L .32342 .31461 L .32307 .31481 L .32273 .315 L .32238 .3152 L .32204 .3154 L .3217 .31559 L .32135 .31579 L .32101 .31598 L .32067 .31618 L .32033 .31638 L .31998 .31657 L .31964 .31677 L .3193 .31697 L .31896 .31716 L .31862 .31736 L .31828 .31755 L .31794 .31775 L .3176 .31795 L .31726 .31814 L .31692 .31834 L .31658 .31854 L .31624 .31873 L Mistroke .3159 .31893 L .31557 .31912 L .31523 .31932 L .31489 .31952 L .31455 .31971 L .31422 .31991 L .31388 .32011 L .31354 .3203 L .31321 .3205 L .31287 .32069 L .31253 .32089 L .3122 .32109 L .31186 .32128 L .31153 .32148 L .31119 .32168 L .31086 .32187 L .31053 .32207 L .31019 .32226 L .30986 .32246 L .30953 .32266 L .30919 .32285 L .30886 .32305 L .30853 .32325 L .3082 .32344 L .30786 .32364 L .30753 .32384 L .3072 .32403 L .30687 .32423 L .30654 .32442 L .30621 .32462 L .30588 .32482 L .30555 .32501 L .30522 .32521 L .30489 .32541 L .30456 .3256 L .30423 .3258 L .3039 .32599 L .30357 .32619 L .30325 .32639 L .30292 .32658 L .30259 .32678 L .30226 .32698 L .30194 .32717 L .30161 .32737 L .30128 .32756 L .30096 .32776 L .30063 .32796 L .30031 .32815 L .29998 .32835 L .29966 .32855 L Mistroke .29933 .32874 L .29901 .32894 L .29868 .32913 L .29836 .32933 L .29803 .32953 L .29771 .32972 L .29739 .32992 L .29706 .33012 L .29674 .33031 L .29642 .33051 L .2961 .3307 L .29578 .3309 L .29545 .3311 L .29513 .33129 L .29481 .33149 L .29449 .33169 L .29417 .33188 L .29385 .33208 L .29353 .33227 L .29321 .33247 L .29289 .33267 L .29257 .33286 L .29225 .33306 L .29193 .33326 L .29161 .33345 L .2913 .33365 L .29098 .33384 L .29066 .33404 L .29034 .33424 L .29003 .33443 L .28971 .33463 L .28939 .33483 L .28908 .33502 L .28876 .33522 L .28844 .33541 L .28813 .33561 L .28781 .33581 L .2875 .336 L .28718 .3362 L .28687 .3364 L .28655 .33659 L .28624 .33679 L .28593 .33698 L .28561 .33718 L .2853 .33738 L .28499 .33757 L .28467 .33777 L .28436 .33797 L .28405 .33816 L .28374 .33836 L Mistroke .28343 .33856 L .28311 .33875 L .2828 .33895 L .28249 .33914 L .28218 .33934 L .28187 .33954 L .28156 .33973 L .28125 .33993 L .28094 .34013 L .28063 .34032 L .28032 .34052 L .28001 .34071 L .2797 .34091 L .2794 .34111 L .27909 .3413 L .27878 .3415 L .27847 .3417 L .27816 .34189 L .27786 .34209 L .27755 .34228 L .27724 .34248 L .27694 .34268 L .27663 .34287 L .27633 .34307 L .27602 .34327 L .27571 .34346 L .27541 .34366 L .2751 .34385 L .2748 .34405 L .27449 .34425 L .27419 .34444 L .27389 .34464 L .27358 .34484 L .27328 .34503 L .27298 .34523 L .27267 .34542 L .27237 .34562 L .27207 .34582 L .27177 .34601 L .27146 .34621 L .27116 .34641 L .27086 .3466 L .27056 .3468 L .27026 .34699 L .26996 .34719 L .26966 .34739 L .26936 .34758 L .26906 .34778 L .26876 .34798 L .26846 .34817 L Mistroke .26816 .34837 L .26786 .34856 L .26756 .34876 L .26726 .34896 L .26696 .34915 L .26667 .34935 L .26637 .34955 L .26607 .34974 L .26577 .34994 L .26548 .35013 L .26518 .35033 L .26488 .35053 L .26459 .35072 L .26429 .35092 L .264 .35112 L .2637 .35131 L .2634 .35151 L .26311 .3517 L .26281 .3519 L .26252 .3521 L .26223 .35229 L .26193 .35249 L .26164 .35269 L .26134 .35288 L .26105 .35308 L .26076 .35328 L .26046 .35347 L .26017 .35367 L .25988 .35386 L .25959 .35406 L .2593 .35426 L .259 .35445 L .25871 .35465 L .25842 .35485 L .25813 .35504 L .25784 .35524 L .25755 .35543 L .25726 .35563 L .25697 .35583 L .25668 .35602 L .25639 .35622 L .2561 .35642 L .25581 .35661 L .25552 .35681 L .25523 .357 L .25494 .3572 L .25466 .3574 L .25437 .35759 L .25408 .35779 L .25379 .35799 L Mistroke .25351 .35818 L .25322 .35838 L .25293 .35857 L .25264 .35877 L .25236 .35897 L .25207 .35916 L .25179 .35936 L .2515 .35956 L .25122 .35975 L .25093 .35995 L .25065 .36014 L .25036 .36034 L .25008 .36054 L .24979 .36073 L .24951 .36093 L .24922 .36113 L .24894 .36132 L .24866 .36152 L .24837 .36171 L .24809 .36191 L .24781 .36211 L .24753 .3623 L .24724 .3625 L .24696 .3627 L .24668 .36289 L .2464 .36309 L .24612 .36328 L .24584 .36348 L .24556 .36368 L .24528 .36387 L .245 .36407 L .24471 .36427 L .24444 .36446 L .24416 .36466 L .24388 .36485 L .2436 .36505 L .24332 .36525 L .24304 .36544 L .24276 .36564 L .24248 .36584 L .2422 .36603 L .24193 .36623 L .24165 .36643 L .24137 .36662 L .24109 .36682 L .24082 .36701 L .24054 .36721 L .24026 .36741 L .23999 .3676 L .23971 .3678 L Mistroke .23944 .368 L .23916 .36819 L .23888 .36839 L .23861 .36858 L .23833 .36878 L .23806 .36898 L .23779 .36917 L .23751 .36937 L .23724 .36957 L .23696 .36976 L .23669 .36996 L .23642 .37015 L .23614 .37035 L .23587 .37055 L .2356 .37074 L .23532 .37094 L .23505 .37114 L .23478 .37133 L .23451 .37153 L .23424 .37172 L .23397 .37192 L .23369 .37212 L .23342 .37231 L .23315 .37251 L .23288 .37271 L .23261 .3729 L .23234 .3731 L .23207 .37329 L .2318 .37349 L .23153 .37369 L .23126 .37388 L .23099 .37408 L .23073 .37428 L .23046 .37447 L .23019 .37467 L .22992 .37486 L .22965 .37506 L .22939 .37526 L .22912 .37545 L .22885 .37565 L .22858 .37585 L .22832 .37604 L .22805 .37624 L .22778 .37643 L .22752 .37663 L .22725 .37683 L .22699 .37702 L .22672 .37722 L .22645 .37742 L .22619 .37761 L Mistroke .22592 .37781 L .22566 .378 L .2254 .3782 L .22513 .3784 L .22487 .37859 L .2246 .37879 L .22434 .37899 L .22408 .37918 L .22381 .37938 L .22355 .37957 L .22329 .37977 L .22302 .37997 L .22276 .38016 L .2225 .38036 L .22224 .38056 L .22198 .38075 L .22172 .38095 L .22145 .38115 L .22119 .38134 L .22093 .38154 L .22067 .38173 L .22041 .38193 L .22015 .38213 L .21989 .38232 L .21963 .38252 L .21937 .38272 L .21911 .38291 L .21885 .38311 L .21859 .3833 L .21833 .3835 L .21808 .3837 L .21782 .38389 L .21756 .38409 L .2173 .38429 L .21704 .38448 L .21679 .38468 L .21653 .38487 L .21627 .38507 L .21601 .38527 L .21576 .38546 L .2155 .38566 L .21525 .38586 L .21499 .38605 L .21473 .38625 L .21448 .38644 L .21422 .38664 L .21397 .38684 L .21371 .38703 L .21346 .38723 L .2132 .38743 L Mistroke .21295 .38762 L .21269 .38782 L .21244 .38801 L .21219 .38821 L .21193 .38841 L .21168 .3886 L .21143 .3888 L .21117 .389 L .21092 .38919 L .21067 .38939 L .21041 .38958 L .21016 .38978 L .20991 .38998 L .20966 .39017 L .20941 .39037 L .20916 .39057 L .2089 .39076 L .20865 .39096 L .2084 .39115 L .20815 .39135 L .2079 .39155 L .20765 .39174 L .2074 .39194 L .20715 .39214 L .2069 .39233 L .20665 .39253 L .2064 .39272 L .20615 .39292 L .2059 .39312 L .20566 .39331 L .20541 .39351 L .20516 .39371 L .20491 .3939 L .20466 .3941 L .20442 .39429 L .20417 .39449 L .20392 .39469 L .20367 .39488 L .20343 .39508 L .20318 .39528 L .20293 .39547 L .20269 .39567 L .20244 .39587 L .2022 .39606 L .20195 .39626 L .2017 .39645 L .20146 .39665 L .20121 .39685 L .20097 .39704 L .20072 .39724 L Mistroke .20048 .39744 L .20024 .39763 L .19999 .39783 L .19975 .39802 L .1995 .39822 L .19926 .39842 L .19902 .39861 L .19877 .39881 L .19853 .39901 L .19829 .3992 L .19805 .3994 L .1978 .39959 L .19756 .39979 L .19732 .39999 L .19708 .40018 L .19684 .40038 L .19659 .40058 L .19635 .40077 L .19611 .40097 L .19587 .40116 L .19563 .40136 L .19539 .40156 L .19515 .40175 L .19491 .40195 L .19467 .40215 L .19443 .40234 L .19419 .40254 L .19395 .40273 L .19371 .40293 L .19347 .40313 L .19324 .40332 L .193 .40352 L .19276 .40372 L .19252 .40391 L .19228 .40411 L .19204 .4043 L .19181 .4045 L .19157 .4047 L .19133 .40489 L .1911 .40509 L .19086 .40529 L .19062 .40548 L .19039 .40568 L .19015 .40587 L .18991 .40607 L .18968 .40627 L .18944 .40646 L .18921 .40666 L .18897 .40686 L .18874 .40705 L Mistroke .1885 .40725 L .18827 .40744 L .18803 .40764 L .1878 .40784 L .18756 .40803 L .18733 .40823 L .18709 .40843 L .18686 .40862 L .18663 .40882 L .18639 .40901 L .18616 .40921 L .18593 .40941 L .18569 .4096 L .18546 .4098 L .18523 .41 L .185 .41019 L .18477 .41039 L .18453 .41059 L .1843 .41078 L .18407 .41098 L .18384 .41117 L .18361 .41137 L .18338 .41157 L .18315 .41176 L .18292 .41196 L .18269 .41216 L .18246 .41235 L .18223 .41255 L .182 .41274 L .18177 .41294 L .18154 .41314 L .18131 .41333 L .18108 .41353 L .18085 .41373 L .18062 .41392 L .18039 .41412 L .18016 .41431 L .17993 .41451 L .17971 .41471 L .17948 .4149 L .17925 .4151 L .17902 .4153 L .1788 .41549 L .17857 .41569 L .17834 .41588 L .17812 .41608 L .17789 .41628 L .17766 .41647 L .17744 .41667 L .17721 .41687 L Mistroke .17698 .41706 L .17676 .41726 L .17653 .41745 L .17631 .41765 L .17608 .41785 L .17586 .41804 L .17563 .41824 L .17541 .41844 L .17518 .41863 L .17496 .41883 L .17474 .41902 L .17451 .41922 L .17429 .41942 L .17406 .41961 L .17384 .41981 L .17362 .42001 L .17339 .4202 L .17317 .4204 L .17295 .42059 L .17273 .42079 L .1725 .42099 L .17228 .42118 L .17206 .42138 L .17184 .42158 L .17162 .42177 L .17139 .42197 L .17117 .42216 L .17095 .42236 L .17073 .42256 L .17051 .42275 L .17029 .42295 L .17007 .42315 L .16985 .42334 L .16963 .42354 L .16941 .42373 L .16919 .42393 L .16897 .42413 L .16875 .42432 L .16853 .42452 L .16831 .42472 L .16809 .42491 L .16787 .42511 L .16766 .42531 L .16744 .4255 L .16722 .4257 L .167 .42589 L .16678 .42609 L .16656 .42629 L .16635 .42648 L .16613 .42668 L Mistroke .16591 .42688 L .1657 .42707 L .16548 .42727 L .16526 .42746 L .16505 .42766 L .16483 .42786 L .16461 .42805 L .1644 .42825 L .16418 .42845 L .16397 .42864 L .16375 .42884 L .16353 .42903 L .16332 .42923 L .1631 .42943 L .16289 .42962 L .16267 .42982 L .16246 .43002 L .16225 .43021 L .16203 .43041 L .16182 .4306 L .1616 .4308 L .16139 .431 L .16118 .43119 L .16096 .43139 L .16075 .43159 L .16054 .43178 L .16032 .43198 L .16011 .43217 L .1599 .43237 L .15969 .43257 L .15947 .43276 L .15926 .43296 L .15905 .43316 L .15884 .43335 L .15863 .43355 L .15842 .43374 L .1582 .43394 L .15799 .43414 L .15778 .43433 L .15757 .43453 L .15736 .43473 L .15715 .43492 L .15694 .43512 L .15673 .43531 L .15652 .43551 L .15631 .43571 L .1561 .4359 L .15589 .4361 L .15568 .4363 L .15547 .43649 L Mistroke .15527 .43669 L .15506 .43688 L .15485 .43708 L .15464 .43728 L .15443 .43747 L .15422 .43767 L .15401 .43787 L .15381 .43806 L .1536 .43826 L .15339 .43845 L .15318 .43865 L .15298 .43885 L .15277 .43904 L .15256 .43924 L .15236 .43944 L .15215 .43963 L .15194 .43983 L .15174 .44003 L .15153 .44022 L .15133 .44042 L .15112 .44061 L .15091 .44081 L .15071 .44101 L .1505 .4412 L .1503 .4414 L .15009 .4416 L .14989 .44179 L .14968 .44199 L .14948 .44218 L .14928 .44238 L .14907 .44258 L .14887 .44277 L .14866 .44297 L .14846 .44317 L .14826 .44336 L .14805 .44356 L .14785 .44375 L .14765 .44395 L .14744 .44415 L .14724 .44434 L .14704 .44454 L .14684 .44474 L .14663 .44493 L .14643 .44513 L .14623 .44532 L .14603 .44552 L .14583 .44572 L .14563 .44591 L .14542 .44611 L .14522 .44631 L Mistroke .14502 .4465 L .14482 .4467 L .14462 .44689 L .14442 .44709 L .14422 .44729 L .14402 .44748 L .14382 .44768 L .14362 .44788 L .14342 .44807 L .14322 .44827 L .14302 .44846 L .14282 .44866 L .14262 .44886 L .14242 .44905 L .14222 .44925 L .14203 .44945 L .14183 .44964 L .14163 .44984 L .14143 .45003 L .14123 .45023 L .14103 .45043 L .14084 .45062 L .14064 .45082 L .14044 .45102 L .14024 .45121 L .14005 .45141 L .13985 .4516 L .13965 .4518 L .13946 .452 L .13926 .45219 L .13906 .45239 L .13887 .45259 L .13867 .45278 L .13847 .45298 L .13828 .45317 L .13808 .45337 L .13789 .45357 L .13769 .45376 L .1375 .45396 L .1373 .45416 L .13711 .45435 L .13691 .45455 L .13672 .45475 L .13652 .45494 L .13633 .45514 L .13613 .45533 L .13594 .45553 L .13575 .45573 L .13555 .45592 L .13536 .45612 L Mistroke .13517 .45632 L .13497 .45651 L .13478 .45671 L .13459 .4569 L .13439 .4571 L .1342 .4573 L .13401 .45749 L .13382 .45769 L .13362 .45789 L .13343 .45808 L .13324 .45828 L .13305 .45847 L .13286 .45867 L .13266 .45887 L .13247 .45906 L .13228 .45926 L .13209 .45946 L .1319 .45965 L .13171 .45985 L .13152 .46004 L .13133 .46024 L .13114 .46044 L .13095 .46063 L .13076 .46083 L .13057 .46103 L .13038 .46122 L 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.71086 .15662 L .71023 .15681 L .70961 .15701 L .70899 .1572 L .70837 .1574 L .70775 .1576 L .70712 .15779 L .7065 .15799 L .70588 .15819 L .70526 .15838 L .70464 .15858 L .70402 .15877 L .7034 .15897 L .70278 .15917 L .70216 .15936 L .70154 .15956 L .70092 .15976 L .7003 .15995 L .69968 .16015 L .69907 .16034 L .69845 .16054 L .69783 .16074 L .69721 .16093 L .6966 .16113 L .69598 .16133 L .69536 .16152 L .69475 .16172 L Mistroke .69413 .16192 L .69352 .16211 L .6929 .16231 L .69228 .1625 L .69167 .1627 L .69106 .1629 L .69044 .16309 L .68983 .16329 L .68921 .16349 L .6886 .16368 L .68799 .16388 L .68737 .16407 L .68676 .16427 L .68615 .16447 L .68554 .16466 L .68493 .16486 L .68431 .16506 L .6837 .16525 L .68309 .16545 L .68248 .16564 L .68187 .16584 L .68126 .16604 L .68065 .16623 L .68004 .16643 L .67944 .16663 L .67883 .16682 L .67822 .16702 L .67761 .16721 L .677 .16741 L .6764 .16761 L .67579 .1678 L .67518 .168 L .67458 .1682 L .67397 .16839 L .67336 .16859 L .67276 .16878 L .67215 .16898 L .67155 .16918 L .67094 .16937 L .67034 .16957 L .66974 .16977 L .66913 .16996 L .66853 .17016 L .66793 .17035 L .66732 .17055 L .66672 .17075 L .66612 .17094 L .66552 .17114 L .66492 .17134 L .66432 .17153 L Mistroke .66372 .17173 L .66312 .17192 L .66252 .17212 L .66192 .17232 L .66132 .17251 L .66072 .17271 L .66012 .17291 L .65952 .1731 L .65892 .1733 L .65832 .17349 L .65773 .17369 L .65713 .17389 L .65653 .17408 L .65594 .17428 L .65534 .17448 L .65475 .17467 L .65415 .17487 L .65356 .17506 L .65296 .17526 L .65237 .17546 L .65177 .17565 L .65118 .17585 L .65059 .17605 L .64999 .17624 L .6494 .17644 L .64881 .17664 L .64822 .17683 L .64763 .17703 L .64703 .17722 L .64644 .17742 L .64585 .17762 L .64526 .17781 L .64467 .17801 L .64408 .17821 L .6435 .1784 L .64291 .1786 L .64232 .17879 L .64173 .17899 L .64114 .17919 L .64056 .17938 L .63997 .17958 L .63938 .17978 L .6388 .17997 L .63821 .18017 L .63762 .18036 L .63704 .18056 L .63645 .18076 L .63587 .18095 L .63529 .18115 L .6347 .18135 L Mistroke .63412 .18154 L .63354 .18174 L .63295 .18193 L .63237 .18213 L .63179 .18233 L .63121 .18252 L .63063 .18272 L .63004 .18292 L .62946 .18311 L .62888 .18331 L .6283 .1835 L .62772 .1837 L .62715 .1839 L .62657 .18409 L .62599 .18429 L .62541 .18449 L .62483 .18468 L .62426 .18488 L .62368 .18507 L .6231 .18527 L .62253 .18547 L .62195 .18566 L .62137 .18586 L .6208 .18606 L .62023 .18625 L .61965 .18645 L .61908 .18664 L .6185 .18684 L .61793 .18704 L .61736 .18723 L .61678 .18743 L .61621 .18763 L .61564 .18782 L .61507 .18802 L .6145 .18821 L .61393 .18841 L .61336 .18861 L .61279 .1888 L .61222 .189 L .61165 .1892 L .61108 .18939 L .61051 .18959 L .60994 .18978 L .60938 .18998 L .60881 .19018 L .60824 .19037 L .60768 .19057 L .60711 .19077 L .60654 .19096 L .60598 .19116 L Mistroke .60541 .19136 L .60485 .19155 L .60428 .19175 L .60372 .19194 L .60316 .19214 L .60259 .19234 L .60203 .19253 L .60147 .19273 L .60091 .19293 L .60034 .19312 L .59978 .19332 L .59922 .19351 L .59866 .19371 L .5981 .19391 L .59754 .1941 L .59698 .1943 L .59642 .1945 L .59586 .19469 L .59531 .19489 L .59475 .19508 L .59419 .19528 L .59363 .19548 L .59308 .19567 L .59252 .19587 L .59196 .19607 L .59141 .19626 L .59085 .19646 L .5903 .19665 L .58974 .19685 L .58919 .19705 L .58864 .19724 L .58808 .19744 L .58753 .19764 L .58698 .19783 L .58643 .19803 L .58587 .19822 L .58532 .19842 L .58477 .19862 L .58422 .19881 L .58367 .19901 L .58312 .19921 L .58257 .1994 L .58202 .1996 L .58147 .19979 L .58093 .19999 L .58038 .20019 L .57983 .20038 L .57928 .20058 L .57874 .20078 L .57819 .20097 L Mistroke .57764 .20117 L .5771 .20136 L .57655 .20156 L .57601 .20176 L .57546 .20195 L .57492 .20215 L .57437 .20235 L .57383 .20254 L .57329 .20274 L .57275 .20293 L .5722 .20313 L .57166 .20333 L .57112 .20352 L .57058 .20372 L .57004 .20392 L .5695 .20411 L .56896 .20431 L .56842 .2045 L .56788 .2047 L .56734 .2049 L .5668 .20509 L .56627 .20529 L .56573 .20549 L .56519 .20568 L .56465 .20588 L .56412 .20608 L .56358 .20627 L .56305 .20647 L .56251 .20666 L .56198 .20686 L .56144 .20706 L .56091 .20725 L .56037 .20745 L .55984 .20765 L .55931 .20784 L .55877 .20804 L .55824 .20823 L .55771 .20843 L .55718 .20863 L .55665 .20882 L .55612 .20902 L .55559 .20922 L .55506 .20941 L .55453 .20961 L .554 .2098 L .55347 .21 L .55294 .2102 L .55241 .21039 L .55189 .21059 L .55136 .21079 L Mistroke .55083 .21098 L .55031 .21118 L .54978 .21137 L .54925 .21157 L .54873 .21177 L .5482 .21196 L .54768 .21216 L .54716 .21236 L .54663 .21255 L .54611 .21275 L .54559 .21294 L .54506 .21314 L .54454 .21334 L .54402 .21353 L .5435 .21373 L .54298 .21393 L .54246 .21412 L .54194 .21432 L .54142 .21451 L .5409 .21471 L .54038 .21491 L .53986 .2151 L .53934 .2153 L .53882 .2155 L .53831 .21569 L .53779 .21589 L .53727 .21608 L .53676 .21628 L .53624 .21648 L .53573 .21667 L .53521 .21687 L .5347 .21707 L .53418 .21726 L .53367 .21746 L .53315 .21765 L .53264 .21785 L .53213 .21805 L .53161 .21824 L .5311 .21844 L .53059 .21864 L .53008 .21883 L .52957 .21903 L .52906 .21922 L .52855 .21942 L .52804 .21962 L .52753 .21981 L .52702 .22001 L .52651 .22021 L .526 .2204 L .5255 .2206 L Mistroke .52499 .2208 L .52448 .22099 L .52397 .22119 L .52347 .22138 L .52296 .22158 L .52246 .22178 L .52195 .22197 L .52145 .22217 L .52094 .22237 L .52044 .22256 L .51993 .22276 L .51943 .22295 L .51893 .22315 L .51843 .22335 L .51792 .22354 L .51742 .22374 L .51692 .22394 L .51642 .22413 L .51592 .22433 L .51542 .22452 L .51492 .22472 L .51442 .22492 L .51392 .22511 L .51342 .22531 L .51293 .22551 L .51243 .2257 L .51193 .2259 L .51143 .22609 L .51094 .22629 L .51044 .22649 L .50994 .22668 L .50945 .22688 L .50895 .22708 L .50846 .22727 L .50796 .22747 L .50747 .22766 L .50698 .22786 L .50648 .22806 L .50599 .22825 L .5055 .22845 L .50501 .22865 L .50451 .22884 L .50402 .22904 L .50353 .22923 L .50304 .22943 L .50255 .22963 L .50206 .22982 L .50157 .23002 L .50108 .23022 L .50059 .23041 L Mistroke .50011 .23061 L .49962 .2308 L .49913 .231 L .49864 .2312 L .49816 .23139 L .49767 .23159 L .49718 .23179 L .4967 .23198 L .49621 .23218 L .49573 .23237 L .49524 .23257 L .49476 .23277 L .49428 .23296 L .49379 .23316 L .49331 .23336 L .49283 .23355 L .49234 .23375 L .49186 .23394 L .49138 .23414 L .4909 .23434 L .49042 .23453 L .48994 .23473 L .48946 .23493 L .48898 .23512 L .4885 .23532 L .48802 .23552 L .48754 .23571 L .48706 .23591 L .48659 .2361 L .48611 .2363 L .48563 .2365 L .48516 .23669 L .48468 .23689 L .4842 .23709 L .48373 .23728 L .48325 .23748 L .48278 .23767 L .4823 .23787 L .48183 .23807 L .48136 .23826 L .48088 .23846 L .48041 .23866 L .47994 .23885 L .47947 .23905 L .47899 .23924 L .47852 .23944 L .47805 .23964 L .47758 .23983 L .47711 .24003 L .47664 .24023 L Mistroke .47617 .24042 L .4757 .24062 L .47523 .24081 L .47477 .24101 L .4743 .24121 L .47383 .2414 L .47336 .2416 L .4729 .2418 L .47243 .24199 L .47196 .24219 L .4715 .24238 L .47103 .24258 L .47057 .24278 L .4701 .24297 L .46964 .24317 L .46917 .24337 L .46871 .24356 L .46825 .24376 L .46778 .24395 L .46732 .24415 L .46686 .24435 L .4664 .24454 L .46594 .24474 L .46548 .24494 L .46502 .24513 L .46456 .24533 L .4641 .24552 L .46364 .24572 L .46318 .24592 L .46272 .24611 L .46226 .24631 L .4618 .24651 L .46134 .2467 L .46089 .2469 L .46043 .24709 L .45997 .24729 L .45952 .24749 L .45906 .24768 L .45861 .24788 L .45815 .24808 L .4577 .24827 L .45724 .24847 L .45679 .24866 L .45633 .24886 L .45588 .24906 L .45543 .24925 L .45497 .24945 L .45452 .24965 L .45407 .24984 L .45362 .25004 L Mistroke .45317 .25024 L .45272 .25043 L .45227 .25063 L .45182 .25082 L .45137 .25102 L .45092 .25122 L .45047 .25141 L .45002 .25161 L .44957 .25181 L .44912 .252 L .44868 .2522 L .44823 .25239 L .44778 .25259 L .44734 .25279 L .44689 .25298 L .44644 .25318 L .446 .25338 L .44555 .25357 L .44511 .25377 L .44467 .25396 L .44422 .25416 L .44378 .25436 L .44333 .25455 L .44289 .25475 L .44245 .25495 L .44201 .25514 L .44157 .25534 L .44112 .25553 L .44068 .25573 L .44024 .25593 L .4398 .25612 L .43936 .25632 L .43892 .25652 L .43848 .25671 L .43804 .25691 L .43761 .2571 L .43717 .2573 L .43673 .2575 L .43629 .25769 L .43586 .25789 L .43542 .25809 L .43498 .25828 L .43455 .25848 L .43411 .25867 L .43367 .25887 L .43324 .25907 L .43281 .25926 L .43237 .25946 L .43194 .25966 L .4315 .25985 L Mistroke .43107 .26005 L .43064 .26024 L .4302 .26044 L .42977 .26064 L .42934 .26083 L .42891 .26103 L .42848 .26123 L .42805 .26142 L .42762 .26162 L .42719 .26181 L .42676 .26201 L .42633 .26221 L .4259 .2624 L .42547 .2626 L .42504 .2628 L .42461 .26299 L .42419 .26319 L .42376 .26338 L .42333 .26358 L .4229 .26378 L .42248 .26397 L .42205 .26417 L .42163 .26437 L .4212 .26456 L .42078 .26476 L .42035 .26496 L .41993 .26515 L .4195 .26535 L .41908 .26554 L .41866 .26574 L .41824 .26594 L .41781 .26613 L .41739 .26633 L .41697 .26653 L .41655 .26672 L .41613 .26692 L .41571 .26711 L .41529 .26731 L .41487 .26751 L .41445 .2677 L .41403 .2679 L .41361 .2681 L .41319 .26829 L .41277 .26849 L .41235 .26868 L .41193 .26888 L .41152 .26908 L .4111 .26927 L .41068 .26947 L .41027 .26967 L Mistroke .40985 .26986 L .40944 .27006 L .40902 .27025 L .40861 .27045 L .40819 .27065 L .40778 .27084 L .40736 .27104 L .40695 .27124 L .40654 .27143 L .40612 .27163 L .40571 .27182 L .4053 .27202 L .40489 .27222 L .40447 .27241 L .40406 .27261 L .40365 .27281 L .40324 .273 L .40283 .2732 L .40242 .27339 L .40201 .27359 L .4016 .27379 L .40119 .27398 L .40079 .27418 L .40038 .27438 L .39997 .27457 L .39956 .27477 L .39916 .27496 L .39875 .27516 L .39834 .27536 L .39794 .27555 L .39753 .27575 L .39712 .27595 L .39672 .27614 L .39631 .27634 L .39591 .27653 L .39551 .27673 L .3951 .27693 L .3947 .27712 L .3943 .27732 L .39389 .27752 L .39349 .27771 L .39309 .27791 L .39269 .2781 L .39228 .2783 L .39188 .2785 L .39148 .27869 L .39108 .27889 L .39068 .27909 L .39028 .27928 L .38988 .27948 L Mistroke .38948 .27968 L .38908 .27987 L .38869 .28007 L .38829 .28026 L .38789 .28046 L .38749 .28066 L .38709 .28085 L .3867 .28105 L .3863 .28125 L .3859 .28144 L .38551 .28164 L .38511 .28183 L .38472 .28203 L .38432 .28223 L .38393 .28242 L .38353 .28262 L .38314 .28282 L .38275 .28301 L .38235 .28321 L .38196 .2834 L .38157 .2836 L .38117 .2838 L .38078 .28399 L .38039 .28419 L .38 .28439 L .37961 .28458 L .37922 .28478 L .37883 .28497 L .37844 .28517 L .37805 .28537 L .37766 .28556 L .37727 .28576 L .37688 .28596 L .37649 .28615 L .3761 .28635 L .37571 .28654 L .37533 .28674 L .37494 .28694 L .37455 .28713 L .37417 .28733 L .37378 .28753 L .37339 .28772 L .37301 .28792 L .37262 .28811 L .37224 .28831 L .37185 .28851 L .37147 .2887 L .37108 .2889 L .3707 .2891 L .37032 .28929 L Mistroke .36993 .28949 L .36955 .28968 L .36917 .28988 L .36879 .29008 L .36841 .29027 L .36802 .29047 L .36764 .29067 L .36726 .29086 L .36688 .29106 L .3665 .29125 L .36612 .29145 L .36574 .29165 L .36536 .29184 L .36498 .29204 L .3646 .29224 L .36423 .29243 L .36385 .29263 L .36347 .29282 L .36309 .29302 L .36271 .29322 L .36234 .29341 L .36196 .29361 L .36159 .29381 L .36121 .294 L .36083 .2942 L .36046 .2944 L .36008 .29459 L .35971 .29479 L .35933 .29498 L .35896 .29518 L .35859 .29538 L .35821 .29557 L .35784 .29577 L .35747 .29597 L .35709 .29616 L .35672 .29636 L .35635 .29655 L .35598 .29675 L .35561 .29695 L .35524 .29714 L .35487 .29734 L .3545 .29754 L .35413 .29773 L .35376 .29793 L .35339 .29812 L .35302 .29832 L .35265 .29852 L .35228 .29871 L .35191 .29891 L .35154 .29911 L Mistroke .35118 .2993 L .35081 .2995 L .35044 .29969 L .35007 .29989 L .34971 .30009 L .34934 .30028 L .34898 .30048 L .34861 .30068 L .34825 .30087 L .34788 .30107 L .34752 .30126 L .34715 .30146 L .34679 .30166 L .34642 .30185 L .34606 .30205 L .3457 .30225 L .34533 .30244 L .34497 .30264 L .34461 .30283 L .34425 .30303 L .34389 .30323 L .34353 .30342 L .34316 .30362 L .3428 .30382 L .34244 .30401 L .34208 .30421 L .34172 .3044 L .34136 .3046 L .341 .3048 L .34065 .30499 L .34029 .30519 L .33993 .30539 L .33957 .30558 L .33921 .30578 L .33886 .30597 L .3385 .30617 L .33814 .30637 L .33778 .30656 L .33743 .30676 L .33707 .30696 L .33672 .30715 L .33636 .30735 L .33601 .30754 L .33565 .30774 L .3353 .30794 L .33494 .30813 L .33459 .30833 L .33424 .30853 L .33388 .30872 L .33353 .30892 L Mistroke .33318 .30912 L .33282 .30931 L .33247 .30951 L .33212 .3097 L .33177 .3099 L .33142 .3101 L .33107 .31029 L .33071 .31049 L .33036 .31069 L .33001 .31088 L .32966 .31108 L .32931 .31127 L .32897 .31147 L .32862 .31167 L .32827 .31186 L .32792 .31206 L .32757 .31226 L .32722 .31245 L .32688 .31265 L .32653 .31284 L .32618 .31304 L .32584 .31324 L .32549 .31343 L .32514 .31363 L .3248 .31383 L .32445 .31402 L .32411 .31422 L .32376 .31441 L .32342 .31461 L .32307 .31481 L .32273 .315 L .32238 .3152 L .32204 .3154 L .3217 .31559 L .32135 .31579 L .32101 .31598 L .32067 .31618 L .32033 .31638 L .31998 .31657 L .31964 .31677 L .3193 .31697 L .31896 .31716 L .31862 .31736 L .31828 .31755 L .31794 .31775 L .3176 .31795 L .31726 .31814 L .31692 .31834 L .31658 .31854 L .31624 .31873 L Mistroke .3159 .31893 L .31557 .31912 L .31523 .31932 L .31489 .31952 L .31455 .31971 L .31422 .31991 L .31388 .32011 L .31354 .3203 L .31321 .3205 L .31287 .32069 L .31253 .32089 L .3122 .32109 L .31186 .32128 L .31153 .32148 L .31119 .32168 L .31086 .32187 L .31053 .32207 L .31019 .32226 L .30986 .32246 L .30953 .32266 L .30919 .32285 L .30886 .32305 L .30853 .32325 L .3082 .32344 L .30786 .32364 L .30753 .32384 L .3072 .32403 L .30687 .32423 L .30654 .32442 L .30621 .32462 L .30588 .32482 L .30555 .32501 L .30522 .32521 L .30489 .32541 L .30456 .3256 L .30423 .3258 L .3039 .32599 L .30357 .32619 L .30325 .32639 L .30292 .32658 L .30259 .32678 L .30226 .32698 L .30194 .32717 L .30161 .32737 L .30128 .32756 L .30096 .32776 L .30063 .32796 L .30031 .32815 L .29998 .32835 L .29966 .32855 L Mistroke .29933 .32874 L .29901 .32894 L .29868 .32913 L .29836 .32933 L .29803 .32953 L .29771 .32972 L .29739 .32992 L .29706 .33012 L .29674 .33031 L .29642 .33051 L .2961 .3307 L .29578 .3309 L .29545 .3311 L .29513 .33129 L .29481 .33149 L .29449 .33169 L .29417 .33188 L .29385 .33208 L .29353 .33227 L .29321 .33247 L .29289 .33267 L .29257 .33286 L .29225 .33306 L .29193 .33326 L .29161 .33345 L .2913 .33365 L .29098 .33384 L .29066 .33404 L .29034 .33424 L .29003 .33443 L .28971 .33463 L .28939 .33483 L .28908 .33502 L .28876 .33522 L .28844 .33541 L .28813 .33561 L .28781 .33581 L .2875 .336 L .28718 .3362 L .28687 .3364 L .28655 .33659 L .28624 .33679 L .28593 .33698 L .28561 .33718 L .2853 .33738 L .28499 .33757 L .28467 .33777 L .28436 .33797 L .28405 .33816 L .28374 .33836 L Mistroke .28343 .33856 L .28311 .33875 L .2828 .33895 L .28249 .33914 L .28218 .33934 L .28187 .33954 L .28156 .33973 L .28125 .33993 L .28094 .34013 L .28063 .34032 L .28032 .34052 L .28001 .34071 L .2797 .34091 L .2794 .34111 L .27909 .3413 L .27878 .3415 L .27847 .3417 L .27816 .34189 L .27786 .34209 L .27755 .34228 L .27724 .34248 L .27694 .34268 L .27663 .34287 L .27633 .34307 L .27602 .34327 L .27571 .34346 L .27541 .34366 L .2751 .34385 L .2748 .34405 L .27449 .34425 L .27419 .34444 L .27389 .34464 L .27358 .34484 L .27328 .34503 L .27298 .34523 L .27267 .34542 L .27237 .34562 L .27207 .34582 L .27177 .34601 L .27146 .34621 L .27116 .34641 L .27086 .3466 L .27056 .3468 L .27026 .34699 L .26996 .34719 L .26966 .34739 L .26936 .34758 L .26906 .34778 L .26876 .34798 L .26846 .34817 L Mistroke .26816 .34837 L .26786 .34856 L .26756 .34876 L .26726 .34896 L .26696 .34915 L .26667 .34935 L .26637 .34955 L .26607 .34974 L .26577 .34994 L .26548 .35013 L .26518 .35033 L .26488 .35053 L .26459 .35072 L .26429 .35092 L .264 .35112 L .2637 .35131 L .2634 .35151 L .26311 .3517 L .26281 .3519 L .26252 .3521 L .26223 .35229 L .26193 .35249 L .26164 .35269 L .26134 .35288 L .26105 .35308 L .26076 .35328 L .26046 .35347 L .26017 .35367 L .25988 .35386 L .25959 .35406 L .2593 .35426 L .259 .35445 L .25871 .35465 L .25842 .35485 L .25813 .35504 L .25784 .35524 L .25755 .35543 L .25726 .35563 L .25697 .35583 L .25668 .35602 L .25639 .35622 L .2561 .35642 L .25581 .35661 L .25552 .35681 L .25523 .357 L .25494 .3572 L .25466 .3574 L .25437 .35759 L .25408 .35779 L .25379 .35799 L Mistroke .25351 .35818 L .25322 .35838 L .25293 .35857 L .25264 .35877 L .25236 .35897 L .25207 .35916 L .25179 .35936 L .2515 .35956 L .25122 .35975 L .25093 .35995 L .25065 .36014 L .25036 .36034 L .25008 .36054 L .24979 .36073 L .24951 .36093 L .24922 .36113 L .24894 .36132 L .24866 .36152 L .24837 .36171 L .24809 .36191 L .24781 .36211 L .24753 .3623 L .24724 .3625 L .24696 .3627 L .24668 .36289 L .2464 .36309 L .24612 .36328 L .24584 .36348 L .24556 .36368 L .24528 .36387 L .245 .36407 L .24471 .36427 L .24444 .36446 L .24416 .36466 L .24388 .36485 L .2436 .36505 L .24332 .36525 L .24304 .36544 L .24276 .36564 L .24248 .36584 L .2422 .36603 L .24193 .36623 L .24165 .36643 L .24137 .36662 L .24109 .36682 L .24082 .36701 L .24054 .36721 L .24026 .36741 L .23999 .3676 L .23971 .3678 L Mistroke .23944 .368 L .23916 .36819 L .23888 .36839 L .23861 .36858 L .23833 .36878 L .23806 .36898 L .23779 .36917 L .23751 .36937 L .23724 .36957 L .23696 .36976 L .23669 .36996 L .23642 .37015 L .23614 .37035 L .23587 .37055 L .2356 .37074 L .23532 .37094 L .23505 .37114 L .23478 .37133 L .23451 .37153 L .23424 .37172 L .23397 .37192 L .23369 .37212 L .23342 .37231 L .23315 .37251 L .23288 .37271 L .23261 .3729 L .23234 .3731 L .23207 .37329 L .2318 .37349 L .23153 .37369 L .23126 .37388 L .23099 .37408 L .23073 .37428 L .23046 .37447 L .23019 .37467 L .22992 .37486 L .22965 .37506 L .22939 .37526 L .22912 .37545 L .22885 .37565 L .22858 .37585 L .22832 .37604 L .22805 .37624 L .22778 .37643 L .22752 .37663 L .22725 .37683 L .22699 .37702 L .22672 .37722 L .22645 .37742 L .22619 .37761 L Mistroke .22592 .37781 L .22566 .378 L .2254 .3782 L .22513 .3784 L .22487 .37859 L .2246 .37879 L .22434 .37899 L .22408 .37918 L .22381 .37938 L .22355 .37957 L .22329 .37977 L .22302 .37997 L .22276 .38016 L .2225 .38036 L .22224 .38056 L .22198 .38075 L .22172 .38095 L .22145 .38115 L .22119 .38134 L .22093 .38154 L .22067 .38173 L .22041 .38193 L .22015 .38213 L .21989 .38232 L .21963 .38252 L .21937 .38272 L .21911 .38291 L .21885 .38311 L .21859 .3833 L .21833 .3835 L .21808 .3837 L .21782 .38389 L .21756 .38409 L .2173 .38429 L .21704 .38448 L .21679 .38468 L .21653 .38487 L .21627 .38507 L .21601 .38527 L .21576 .38546 L .2155 .38566 L .21525 .38586 L .21499 .38605 L .21473 .38625 L .21448 .38644 L .21422 .38664 L .21397 .38684 L .21371 .38703 L .21346 .38723 L .2132 .38743 L Mistroke .21295 .38762 L .21269 .38782 L .21244 .38801 L .21219 .38821 L .21193 .38841 L .21168 .3886 L .21143 .3888 L .21117 .389 L .21092 .38919 L .21067 .38939 L .21041 .38958 L .21016 .38978 L .20991 .38998 L .20966 .39017 L .20941 .39037 L .20916 .39057 L .2089 .39076 L .20865 .39096 L .2084 .39115 L .20815 .39135 L .2079 .39155 L .20765 .39174 L .2074 .39194 L .20715 .39214 L .2069 .39233 L .20665 .39253 L .2064 .39272 L .20615 .39292 L .2059 .39312 L .20566 .39331 L .20541 .39351 L .20516 .39371 L .20491 .3939 L .20466 .3941 L .20442 .39429 L .20417 .39449 L .20392 .39469 L .20367 .39488 L .20343 .39508 L .20318 .39528 L .20293 .39547 L .20269 .39567 L .20244 .39587 L .2022 .39606 L .20195 .39626 L .2017 .39645 L .20146 .39665 L .20121 .39685 L .20097 .39704 L .20072 .39724 L Mistroke .20048 .39744 L .20024 .39763 L .19999 .39783 L .19975 .39802 L .1995 .39822 L .19926 .39842 L .19902 .39861 L .19877 .39881 L .19853 .39901 L .19829 .3992 L .19805 .3994 L .1978 .39959 L .19756 .39979 L .19732 .39999 L .19708 .40018 L .19684 .40038 L .19659 .40058 L .19635 .40077 L .19611 .40097 L .19587 .40116 L .19563 .40136 L .19539 .40156 L .19515 .40175 L .19491 .40195 L .19467 .40215 L .19443 .40234 L .19419 .40254 L .19395 .40273 L .19371 .40293 L .19347 .40313 L .19324 .40332 L .193 .40352 L .19276 .40372 L .19252 .40391 L .19228 .40411 L .19204 .4043 L .19181 .4045 L .19157 .4047 L .19133 .40489 L .1911 .40509 L .19086 .40529 L .19062 .40548 L .19039 .40568 L .19015 .40587 L .18991 .40607 L .18968 .40627 L .18944 .40646 L .18921 .40666 L .18897 .40686 L .18874 .40705 L Mistroke .1885 .40725 L .18827 .40744 L .18803 .40764 L .1878 .40784 L .18756 .40803 L .18733 .40823 L .18709 .40843 L .18686 .40862 L .18663 .40882 L .18639 .40901 L .18616 .40921 L .18593 .40941 L .18569 .4096 L .18546 .4098 L .18523 .41 L .185 .41019 L .18477 .41039 L .18453 .41059 L .1843 .41078 L .18407 .41098 L .18384 .41117 L .18361 .41137 L .18338 .41157 L .18315 .41176 L .18292 .41196 L .18269 .41216 L .18246 .41235 L .18223 .41255 L .182 .41274 L .18177 .41294 L .18154 .41314 L .18131 .41333 L .18108 .41353 L .18085 .41373 L .18062 .41392 L .18039 .41412 L .18016 .41431 L .17993 .41451 L .17971 .41471 L .17948 .4149 L .17925 .4151 L .17902 .4153 L .1788 .41549 L .17857 .41569 L .17834 .41588 L .17812 .41608 L .17789 .41628 L .17766 .41647 L .17744 .41667 L .17721 .41687 L Mistroke .17698 .41706 L .17676 .41726 L .17653 .41745 L .17631 .41765 L .17608 .41785 L .17586 .41804 L .17563 .41824 L .17541 .41844 L .17518 .41863 L .17496 .41883 L .17474 .41902 L .17451 .41922 L .17429 .41942 L .17406 .41961 L .17384 .41981 L .17362 .42001 L .17339 .4202 L .17317 .4204 L .17295 .42059 L .17273 .42079 L .1725 .42099 L .17228 .42118 L .17206 .42138 L .17184 .42158 L .17162 .42177 L .17139 .42197 L .17117 .42216 L .17095 .42236 L .17073 .42256 L .17051 .42275 L .17029 .42295 L .17007 .42315 L .16985 .42334 L .16963 .42354 L .16941 .42373 L .16919 .42393 L .16897 .42413 L .16875 .42432 L .16853 .42452 L .16831 .42472 L .16809 .42491 L .16787 .42511 L .16766 .42531 L .16744 .4255 L .16722 .4257 L .167 .42589 L .16678 .42609 L .16656 .42629 L .16635 .42648 L .16613 .42668 L Mistroke .16591 .42688 L .1657 .42707 L .16548 .42727 L .16526 .42746 L .16505 .42766 L .16483 .42786 L .16461 .42805 L .1644 .42825 L .16418 .42845 L .16397 .42864 L .16375 .42884 L .16353 .42903 L .16332 .42923 L .1631 .42943 L .16289 .42962 L .16267 .42982 L .16246 .43002 L .16225 .43021 L .16203 .43041 L .16182 .4306 L .1616 .4308 L .16139 .431 L .16118 .43119 L .16096 .43139 L .16075 .43159 L .16054 .43178 L .16032 .43198 L .16011 .43217 L .1599 .43237 L .15969 .43257 L .15947 .43276 L .15926 .43296 L .15905 .43316 L .15884 .43335 L .15863 .43355 L .15842 .43374 L .1582 .43394 L .15799 .43414 L .15778 .43433 L .15757 .43453 L .15736 .43473 L .15715 .43492 L .15694 .43512 L .15673 .43531 L .15652 .43551 L .15631 .43571 L .1561 .4359 L .15589 .4361 L .15568 .4363 L .15547 .43649 L Mistroke .15527 .43669 L .15506 .43688 L .15485 .43708 L .15464 .43728 L .15443 .43747 L .15422 .43767 L .15401 .43787 L .15381 .43806 L .1536 .43826 L .15339 .43845 L .15318 .43865 L .15298 .43885 L .15277 .43904 L .15256 .43924 L .15236 .43944 L .15215 .43963 L .15194 .43983 L .15174 .44003 L .15153 .44022 L .15133 .44042 L .15112 .44061 L .15091 .44081 L .15071 .44101 L .1505 .4412 L .1503 .4414 L .15009 .4416 L .14989 .44179 L .14968 .44199 L .14948 .44218 L .14928 .44238 L .14907 .44258 L .14887 .44277 L .14866 .44297 L .14846 .44317 L .14826 .44336 L .14805 .44356 L .14785 .44375 L .14765 .44395 L .14744 .44415 L .14724 .44434 L .14704 .44454 L .14684 .44474 L .14663 .44493 L .14643 .44513 L .14623 .44532 L .14603 .44552 L .14583 .44572 L .14563 .44591 L .14542 .44611 L .14522 .44631 L Mistroke .14502 .4465 L .14482 .4467 L .14462 .44689 L .14442 .44709 L .14422 .44729 L .14402 .44748 L .14382 .44768 L .14362 .44788 L .14342 .44807 L .14322 .44827 L .14302 .44846 L .14282 .44866 L .14262 .44886 L .14242 .44905 L .14222 .44925 L .14203 .44945 L .14183 .44964 L .14163 .44984 L .14143 .45003 L .14123 .45023 L .14103 .45043 L .14084 .45062 L .14064 .45082 L .14044 .45102 L .14024 .45121 L .14005 .45141 L .13985 .4516 L .13965 .4518 L .13946 .452 L .13926 .45219 L .13906 .45239 L .13887 .45259 L .13867 .45278 L .13847 .45298 L .13828 .45317 L .13808 .45337 L .13789 .45357 L .13769 .45376 L .1375 .45396 L .1373 .45416 L .13711 .45435 L .13691 .45455 L .13672 .45475 L .13652 .45494 L .13633 .45514 L .13613 .45533 L .13594 .45553 L .13575 .45573 L .13555 .45592 L .13536 .45612 L Mistroke .13517 .45632 L .13497 .45651 L .13478 .45671 L .13459 .4569 L .13439 .4571 L .1342 .4573 L .13401 .45749 L .13382 .45769 L .13362 .45789 L .13343 .45808 L .13324 .45828 L .13305 .45847 L .13286 .45867 L .13266 .45887 L .13247 .45906 L .13228 .45926 L .13209 .45946 L .1319 .45965 L .13171 .45985 L .13152 .46004 L .13133 .46024 L .13114 .46044 L .13095 .46063 L .13076 .46083 L .13057 .46103 L .13038 .46122 L .13019 .46142 L .13 .46161 L .12981 .46181 L .12962 .46201 L .12943 .4622 L .12924 .4624 L .12905 .4626 L .12886 .46279 L .12868 .46299 L .12849 .46318 L .1283 .46338 L .12811 .46358 L .12792 .46377 L .12774 .46397 L .12755 .46417 L .12736 .46436 L .12717 .46456 L .12699 .46475 L .1268 .46495 L .12661 .46515 L .12642 .46534 L .12624 .46554 L .12605 .46574 L .12587 .46593 L Mistroke .12568 .46613 L .12549 .46632 L .12531 .46652 L .12512 .46672 L .12494 .46691 L .12475 .46711 L .12456 .46731 L .12438 .4675 L .12419 .4677 L .12401 .46789 L .12382 .46809 L .12364 .46829 L .12346 .46848 L .12327 .46868 L .12309 .46888 L .1229 .46907 L .12272 .46927 L .12253 .46947 L .12235 .46966 L .12217 .46986 L .12198 .47005 L .1218 .47025 L .12162 .47045 L .12143 .47064 L .12125 .47084 L .12107 .47104 L .12089 .47123 L .1207 .47143 L .12052 .47162 L .12034 .47182 L .12016 .47202 L .11998 .47221 L .11979 .47241 L .11961 .47261 L .11943 .4728 L .11925 .473 L .11907 .47319 L .11889 .47339 L .11871 .47359 L .11853 .47378 L .11834 .47398 L .11816 .47418 L .11798 .47437 L .1178 .47457 L .11762 .47476 L .11744 .47496 L .11726 .47516 L .11708 .47535 L .1169 .47555 L .11672 .47575 L Mistroke .11655 .47594 L .11637 .47614 L .11619 .47633 L .11601 .47653 L .11583 .47673 L .11565 .47692 L .11547 .47712 L .11529 .47732 L .11512 .47751 L .11494 .47771 L .11476 .4779 L .11458 .4781 L .1144 .4783 L .11423 .47849 L .11405 .47869 L .11387 .47889 L .11369 .47908 L .11352 .47928 L .11334 .47947 L .11316 .47967 L .11299 .47987 L .11281 .48006 L .11263 .48026 L .11246 .48046 L .11228 .48065 L .11211 .48085 L .11193 .48104 L .11175 .48124 L .11158 .48144 L .1114 .48163 L .11123 .48183 L .11105 .48203 L .11088 .48222 L .1107 .48242 L .11053 .48261 L .11035 .48281 L .11018 .48301 L .11 .4832 L .10983 .4834 L .10966 .4836 L .10948 .48379 L .10931 .48399 L .10913 .48419 L .10896 .48438 L .10879 .48458 L .10861 .48477 L .10844 .48497 L .10827 .48517 L .10809 .48536 L .10792 .48556 L Mistroke .10775 .48576 L .10758 .48595 L .1074 .48615 L .10723 .48634 L .10706 .48654 L .10689 .48674 L .10671 .48693 L .10654 .48713 L .10637 .48733 L .1062 .48752 L .10603 .48772 L .10586 .48791 L .10569 .48811 L .10551 .48831 L .10534 .4885 L .10517 .4887 L .105 .4889 L .10483 .48909 L .10466 .48929 L .10449 .48948 L .10432 .48968 L .10415 .48988 L .10398 .49007 L .10381 .49027 L .10364 .49047 L .10347 .49066 L .1033 .49086 L .10313 .49105 L .10296 .49125 L .10279 .49145 L .10263 .49164 L .10246 .49184 L .10229 .49204 L .10212 .49223 L .10195 .49243 L .10178 .49262 L .10161 .49282 L .10145 .49302 L .10128 .49321 L .10111 .49341 L .10094 .49361 L .10078 .4938 L .10061 .494 L .10044 .49419 L .10027 .49439 L .10011 .49459 L .09994 .49478 L .09977 .49498 L .09961 .49518 L .09944 .49537 L Mistroke .09927 .49557 L .09911 .49576 L .09894 .49596 L .09877 .49616 L .09861 .49635 L .09844 .49655 L .09828 .49675 L .09811 .49694 L .09795 .49714 L .09778 .49733 L .09762 .49753 L .09745 .49773 L .09729 .49792 L .09712 .49812 L .09696 .49832 L .09679 .49851 L .09663 .49871 L .09646 .49891 L .0963 .4991 L .09613 .4993 L .09597 .49949 L .09581 .49969 L .09564 .49989 L .09548 .50008 L .09532 .50028 L .09515 .50048 L .09499 .50067 L .09483 .50087 L .09466 .50106 L .0945 .50126 L .09434 .50146 L .09417 .50165 L .09401 .50185 L .09385 .50205 L .09369 .50224 L .09353 .50244 L .09336 .50263 L .0932 .50283 L .09304 .50303 L .09288 .50322 L .09272 .50342 L .09255 .50362 L .09239 .50381 L .09223 .50401 L .09207 .5042 L .09191 .5044 L .09175 .5046 L .09159 .50479 L .09143 .50499 L .09127 .50519 L Mistroke .09111 .50538 L .09095 .50558 L .09079 .50577 L .09063 .50597 L .09047 .50617 L .09031 .50636 L .09015 .50656 L .08999 .50676 L .08983 .50695 L .08967 .50715 L .08951 .50734 L .08935 .50754 L .08919 .50774 L .08903 .50793 L .08887 .50813 L .08871 .50833 L .08856 .50852 L .0884 .50872 L .08824 .50891 L .08808 .50911 L .08792 .50931 L .08776 .5095 L .08761 .5097 L .08745 .5099 L .08729 .51009 L .08713 .51029 L .08698 .51048 L .08682 .51068 L .08666 .51088 L .08651 .51107 L .08635 .51127 L .08619 .51147 L .08604 .51166 L .08588 .51186 L .08572 .51205 L .08557 .51225 L .08541 .51245 L .08525 .51264 L .0851 .51284 L .08494 .51304 L .08479 .51323 L .08463 .51343 L .08447 .51363 L .08432 .51382 L .08416 .51402 L .08401 .51421 L .08385 .51441 L .0837 .51461 L .08354 .5148 L .08339 .515 L Mistroke .08323 .5152 L .08308 .51539 L .08293 .51559 L .08277 .51578 L .08262 .51598 L .08246 .51618 L .08231 .51637 L .08215 .51657 L .082 .51677 L .08185 .51696 L .08169 .51716 L .08154 .51735 L .08139 .51755 L .08123 .51775 L .08108 .51794 L .08093 .51814 L .08077 .51834 L .08062 .51853 L .08047 .51873 L .08032 .51892 L .08016 .51912 L .08001 .51932 L .07986 .51951 L .07971 .51971 L .07956 .51991 L .0794 .5201 L .07925 .5203 L .0791 .52049 L .07895 .52069 L .0788 .52089 L .07865 .52108 L .0785 .52128 L .07834 .52148 L .07819 .52167 L .07804 .52187 L .07789 .52206 L .07774 .52226 L .07759 .52246 L .07744 .52265 L .07729 .52285 L .07714 .52305 L .07699 .52324 L .07684 .52344 L .07669 .52363 L .07654 .52383 L .07639 .52403 L .07624 .52422 L .07609 .52442 L .07594 .52462 L .07579 .52481 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.58864 .06662 L .58808 .06686 L .58753 .0671 L .58698 .06735 L .58643 .06759 L .58587 .06783 L .58532 .06808 L .58477 .06832 L .58422 .06857 L .58367 .06881 L .58312 .06906 L .58257 .0693 L .58202 .06955 L .58147 .0698 L .58093 .07004 L .58038 .07029 L .57983 .07054 L .57928 .07078 L .57874 .07103 L .57819 .07128 L Mistroke .57764 .07153 L .5771 .07177 L .57655 .07202 L .57601 .07227 L .57546 .07252 L .57492 .07277 L .57437 .07302 L .57383 .07327 L .57329 .07352 L .57275 .07377 L .5722 .07402 L .57166 .07427 L .57112 .07452 L .57058 .07477 L .57004 .07502 L .5695 .07527 L .56896 .07552 L .56842 .07578 L .56788 .07603 L .56734 .07628 L .5668 .07653 L .56627 .07678 L .56573 .07704 L .56519 .07729 L .56465 .07754 L .56412 .0778 L .56358 .07805 L .56305 .07831 L .56251 .07856 L .56198 .07881 L .56144 .07907 L .56091 .07932 L .56037 .07958 L .55984 .07983 L .55931 .08009 L .55877 .08035 L .55824 .0806 L .55771 .08086 L .55718 .08111 L .55665 .08137 L .55612 .08163 L .55559 .08189 L .55506 .08214 L .55453 .0824 L .554 .08266 L .55347 .08291 L .55294 .08317 L .55241 .08343 L .55189 .08369 L .55136 .08395 L Mistroke .55083 .08421 L .55031 .08447 L .54978 .08472 L .54925 .08498 L .54873 .08524 L .5482 .0855 L .54768 .08576 L .54716 .08602 L .54663 .08628 L .54611 .08654 L .54559 .0868 L .54506 .08706 L .54454 .08732 L .54402 .08759 L .5435 .08785 L .54298 .08811 L .54246 .08837 L .54194 .08863 L .54142 .08889 L .5409 .08916 L .54038 .08942 L .53986 .08968 L .53934 .08994 L .53882 .09021 L .53831 .09047 L .53779 .09073 L .53727 .09099 L .53676 .09126 L .53624 .09152 L .53573 .09179 L .53521 .09205 L .5347 .09231 L .53418 .09258 L .53367 .09284 L .53315 .09311 L .53264 .09337 L .53213 .09364 L .53161 .0939 L .5311 .09417 L .53059 .09443 L .53008 .0947 L .52957 .09496 L .52906 .09523 L .52855 .09549 L .52804 .09576 L .52753 .09602 L .52702 .09629 L .52651 .09656 L .526 .09682 L .5255 .09709 L Mistroke .52499 .09736 L .52448 .09762 L .52397 .09789 L .52347 .09816 L .52296 .09842 L .52246 .09869 L .52195 .09896 L .52145 .09923 L .52094 .0995 L .52044 .09976 L .51993 .10003 L .51943 .1003 L .51893 .10057 L .51843 .10084 L .51792 .1011 L .51742 .10137 L .51692 .10164 L .51642 .10191 L .51592 .10218 L .51542 .10245 L .51492 .10272 L .51442 .10299 L .51392 .10326 L .51342 .10353 L .51293 .1038 L .51243 .10407 L .51193 .10434 L .51143 .10461 L .51094 .10488 L .51044 .10515 L .50994 .10542 L .50945 .10569 L .50895 .10596 L .50846 .10623 L .50796 .1065 L .50747 .10677 L .50698 .10704 L .50648 .10731 L .50599 .10758 L .5055 .10786 L .50501 .10813 L .50451 .1084 L .50402 .10867 L .50353 .10894 L .50304 .10921 L .50255 .10949 L .50206 .10976 L .50157 .11003 L .50108 .1103 L .50059 .11057 L Mistroke .50011 .11085 L .49962 .11112 L .49913 .11139 L .49864 .11167 L .49816 .11194 L .49767 .11221 L .49718 .11248 L .4967 .11276 L .49621 .11303 L .49573 .1133 L .49524 .11358 L .49476 .11385 L .49428 .11412 L .49379 .1144 L .49331 .11467 L .49283 .11494 L .49234 .11522 L .49186 .11549 L .49138 .11577 L .4909 .11604 L .49042 .11631 L .48994 .11659 L .48946 .11686 L .48898 .11714 L .4885 .11741 L .48802 .11769 L .48754 .11796 L .48706 .11824 L .48659 .11851 L .48611 .11879 L .48563 .11906 L .48516 .11934 L .48468 .11961 L .4842 .11989 L .48373 .12016 L .48325 .12044 L .48278 .12071 L .4823 .12099 L .48183 .12126 L .48136 .12154 L .48088 .12181 L .48041 .12209 L .47994 .12237 L .47947 .12264 L .47899 .12292 L .47852 .12319 L .47805 .12347 L .47758 .12374 L .47711 .12402 L .47664 .1243 L Mistroke .47617 .12457 L .4757 .12485 L .47523 .12513 L .47477 .1254 L .4743 .12568 L .47383 .12596 L .47336 .12623 L .4729 .12651 L .47243 .12679 L .47196 .12706 L .4715 .12734 L .47103 .12762 L .47057 .12789 L .4701 .12817 L .46964 .12845 L .46917 .12872 L .46871 .129 L .46825 .12928 L .46778 .12955 L .46732 .12983 L .46686 .13011 L .4664 .13039 L .46594 .13066 L .46548 .13094 L .46502 .13122 L .46456 .1315 L .4641 .13177 L .46364 .13205 L .46318 .13233 L .46272 .13261 L .46226 .13288 L .4618 .13316 L .46134 .13344 L .46089 .13372 L .46043 .134 L .45997 .13427 L .45952 .13455 L .45906 .13483 L .45861 .13511 L .45815 .13539 L .4577 .13566 L .45724 .13594 L .45679 .13622 L .45633 .1365 L .45588 .13678 L .45543 .13706 L .45497 .13733 L .45452 .13761 L .45407 .13789 L .45362 .13817 L Mistroke .45317 .13845 L .45272 .13873 L .45227 .139 L .45182 .13928 L .45137 .13956 L .45092 .13984 L .45047 .14012 L .45002 .1404 L .44957 .14068 L .44912 .14096 L .44868 .14123 L .44823 .14151 L .44778 .14179 L .44734 .14207 L .44689 .14235 L .44644 .14263 L .446 .14291 L .44555 .14319 L .44511 .14346 L .44467 .14374 L .44422 .14402 L .44378 .1443 L .44333 .14458 L .44289 .14486 L .44245 .14514 L .44201 .14542 L .44157 .1457 L .44112 .14598 L .44068 .14626 L .44024 .14653 L .4398 .14681 L .43936 .14709 L .43892 .14737 L .43848 .14765 L .43804 .14793 L .43761 .14821 L .43717 .14849 L .43673 .14877 L .43629 .14905 L .43586 .14933 L .43542 .14961 L .43498 .14989 L .43455 .15016 L .43411 .15044 L .43367 .15072 L .43324 .151 L .43281 .15128 L .43237 .15156 L .43194 .15184 L .4315 .15212 L Mistroke .43107 .1524 L .43064 .15268 L .4302 .15296 L .42977 .15324 L .42934 .15352 L .42891 .1538 L .42848 .15408 L .42805 .15436 L .42762 .15463 L .42719 .15491 L .42676 .15519 L .42633 .15547 L .4259 .15575 L .42547 .15603 L .42504 .15631 L .42461 .15659 L .42419 .15687 L .42376 .15715 L .42333 .15743 L .4229 .15771 L .42248 .15799 L .42205 .15827 L .42163 .15855 L .4212 .15883 L .42078 .15911 L .42035 .15939 L .41993 .15966 L .4195 .15994 L .41908 .16022 L .41866 .1605 L .41824 .16078 L .41781 .16106 L .41739 .16134 L .41697 .16162 L .41655 .1619 L .41613 .16218 L .41571 .16246 L .41529 .16274 L .41487 .16302 L .41445 .1633 L .41403 .16358 L .41361 .16386 L .41319 .16414 L .41277 .16441 L .41235 .16469 L .41193 .16497 L .41152 .16525 L .4111 .16553 L .41068 .16581 L .41027 .16609 L Mistroke .40985 .16637 L .40944 .16665 L .40902 .16693 L .40861 .16721 L .40819 .16749 L .40778 .16777 L .40736 .16804 L .40695 .16832 L .40654 .1686 L .40612 .16888 L .40571 .16916 L .4053 .16944 L .40489 .16972 L .40447 .17 L .40406 .17028 L .40365 .17056 L .40324 .17084 L .40283 .17111 L .40242 .17139 L .40201 .17167 L .4016 .17195 L .40119 .17223 L .40079 .17251 L .40038 .17279 L .39997 .17307 L .39956 .17335 L .39916 .17362 L .39875 .1739 L .39834 .17418 L .39794 .17446 L .39753 .17474 L .39712 .17502 L .39672 .1753 L .39631 .17558 L .39591 .17585 L .39551 .17613 L .3951 .17641 L .3947 .17669 L .3943 .17697 L .39389 .17725 L .39349 .17753 L .39309 .1778 L .39269 .17808 L .39228 .17836 L .39188 .17864 L .39148 .17892 L .39108 .1792 L .39068 .17947 L .39028 .17975 L .38988 .18003 L Mistroke .38948 .18031 L .38908 .18059 L .38869 .18087 L .38829 .18114 L .38789 .18142 L .38749 .1817 L .38709 .18198 L .3867 .18226 L .3863 .18253 L .3859 .18281 L .38551 .18309 L .38511 .18337 L .38472 .18365 L .38432 .18392 L .38393 .1842 L .38353 .18448 L .38314 .18476 L .38275 .18503 L .38235 .18531 L .38196 .18559 L .38157 .18587 L .38117 .18615 L .38078 .18642 L .38039 .1867 L .38 .18698 L .37961 .18726 L .37922 .18753 L .37883 .18781 L .37844 .18809 L .37805 .18836 L .37766 .18864 L .37727 .18892 L .37688 .1892 L .37649 .18947 L .3761 .18975 L .37571 .19003 L .37533 .19031 L .37494 .19058 L .37455 .19086 L .37417 .19114 L .37378 .19141 L .37339 .19169 L .37301 .19197 L .37262 .19224 L .37224 .19252 L .37185 .1928 L .37147 .19307 L .37108 .19335 L .3707 .19363 L .37032 .1939 L Mistroke .36993 .19418 L .36955 .19446 L .36917 .19473 L .36879 .19501 L .36841 .19529 L .36802 .19556 L .36764 .19584 L .36726 .19611 L .36688 .19639 L .3665 .19667 L .36612 .19694 L .36574 .19722 L .36536 .19749 L .36498 .19777 L .3646 .19805 L .36423 .19832 L .36385 .1986 L .36347 .19887 L .36309 .19915 L .36271 .19942 L .36234 .1997 L .36196 .19998 L .36159 .20025 L .36121 .20053 L .36083 .2008 L .36046 .20108 L .36008 .20135 L .35971 .20163 L .35933 .2019 L .35896 .20218 L .35859 .20245 L .35821 .20273 L .35784 .203 L .35747 .20328 L .35709 .20355 L .35672 .20383 L .35635 .2041 L .35598 .20438 L .35561 .20465 L .35524 .20493 L .35487 .2052 L .3545 .20548 L .35413 .20575 L .35376 .20603 L .35339 .2063 L .35302 .20658 L .35265 .20685 L .35228 .20712 L .35191 .2074 L .35154 .20767 L Mistroke .35118 .20795 L .35081 .20822 L .35044 .2085 L .35007 .20877 L .34971 .20904 L .34934 .20932 L .34898 .20959 L .34861 .20986 L .34825 .21014 L .34788 .21041 L .34752 .21069 L .34715 .21096 L .34679 .21123 L .34642 .21151 L .34606 .21178 L .3457 .21205 L .34533 .21233 L .34497 .2126 L .34461 .21287 L .34425 .21315 L .34389 .21342 L .34353 .21369 L .34316 .21397 L .3428 .21424 L .34244 .21451 L .34208 .21478 L .34172 .21506 L .34136 .21533 L .341 .2156 L .34065 .21587 L .34029 .21615 L .33993 .21642 L .33957 .21669 L .33921 .21696 L .33886 .21724 L .3385 .21751 L .33814 .21778 L .33778 .21805 L .33743 .21833 L .33707 .2186 L .33672 .21887 L .33636 .21914 L .33601 .21941 L .33565 .21968 L .3353 .21996 L .33494 .22023 L .33459 .2205 L .33424 .22077 L .33388 .22104 L .33353 .22131 L Mistroke .33318 .22159 L .33282 .22186 L .33247 .22213 L .33212 .2224 L .33177 .22267 L .33142 .22294 L .33107 .22321 L .33071 .22348 L .33036 .22375 L .33001 .22402 L .32966 .2243 L .32931 .22457 L .32897 .22484 L .32862 .22511 L .32827 .22538 L .32792 .22565 L .32757 .22592 L .32722 .22619 L .32688 .22646 L .32653 .22673 L .32618 .227 L .32584 .22727 L .32549 .22754 L .32514 .22781 L .3248 .22808 L .32445 .22835 L .32411 .22862 L .32376 .22889 L .32342 .22916 L .32307 .22943 L .32273 .2297 L .32238 .22997 L .32204 .23024 L .3217 .23051 L .32135 .23077 L .32101 .23104 L .32067 .23131 L .32033 .23158 L .31998 .23185 L .31964 .23212 L .3193 .23239 L .31896 .23266 L .31862 .23293 L .31828 .23319 L .31794 .23346 L .3176 .23373 L .31726 .234 L .31692 .23427 L .31658 .23454 L .31624 .2348 L Mistroke .3159 .23507 L .31557 .23534 L .31523 .23561 L .31489 .23588 L .31455 .23614 L .31422 .23641 L .31388 .23668 L .31354 .23695 L .31321 .23722 L .31287 .23748 L .31253 .23775 L .3122 .23802 L .31186 .23829 L .31153 .23855 L .31119 .23882 L .31086 .23909 L .31053 .23935 L .31019 .23962 L .30986 .23989 L .30953 .24015 L .30919 .24042 L .30886 .24069 L .30853 .24095 L .3082 .24122 L .30786 .24149 L .30753 .24175 L .3072 .24202 L .30687 .24229 L .30654 .24255 L .30621 .24282 L .30588 .24309 L .30555 .24335 L .30522 .24362 L .30489 .24388 L .30456 .24415 L .30423 .24441 L .3039 .24468 L .30357 .24495 L .30325 .24521 L .30292 .24548 L .30259 .24574 L .30226 .24601 L .30194 .24627 L .30161 .24654 L .30128 .2468 L .30096 .24707 L .30063 .24733 L .30031 .2476 L .29998 .24786 L .29966 .24813 L Mistroke .29933 .24839 L .29901 .24866 L .29868 .24892 L .29836 .24919 L .29803 .24945 L .29771 .24971 L .29739 .24998 L .29706 .25024 L .29674 .25051 L .29642 .25077 L .2961 .25103 L .29578 .2513 L .29545 .25156 L .29513 .25183 L .29481 .25209 L .29449 .25235 L .29417 .25262 L .29385 .25288 L .29353 .25314 L .29321 .25341 L .29289 .25367 L .29257 .25393 L .29225 .25419 L .29193 .25446 L .29161 .25472 L .2913 .25498 L .29098 .25525 L .29066 .25551 L .29034 .25577 L .29003 .25603 L .28971 .2563 L .28939 .25656 L .28908 .25682 L .28876 .25708 L .28844 .25734 L .28813 .25761 L .28781 .25787 L .2875 .25813 L .28718 .25839 L .28687 .25865 L .28655 .25892 L .28624 .25918 L .28593 .25944 L .28561 .2597 L .2853 .25996 L .28499 .26022 L .28467 .26048 L .28436 .26075 L .28405 .26101 L .28374 .26127 L Mistroke .28343 .26153 L .28311 .26179 L .2828 .26205 L .28249 .26231 L .28218 .26257 L .28187 .26283 L .28156 .26309 L .28125 .26335 L .28094 .26361 L .28063 .26387 L .28032 .26413 L .28001 .26439 L .2797 .26465 L .2794 .26491 L .27909 .26517 L .27878 .26543 L .27847 .26569 L .27816 .26595 L .27786 .26621 L .27755 .26647 L .27724 .26673 L .27694 .26699 L .27663 .26725 L .27633 .26751 L .27602 .26777 L .27571 .26802 L .27541 .26828 L .2751 .26854 L .2748 .2688 L .27449 .26906 L .27419 .26932 L .27389 .26958 L .27358 .26983 L .27328 .27009 L .27298 .27035 L .27267 .27061 L .27237 .27087 L .27207 .27112 L .27177 .27138 L .27146 .27164 L .27116 .2719 L .27086 .27216 L .27056 .27241 L .27026 .27267 L .26996 .27293 L .26966 .27319 L .26936 .27344 L .26906 .2737 L .26876 .27396 L .26846 .27421 L Mistroke .26816 .27447 L .26786 .27473 L .26756 .27498 L .26726 .27524 L .26696 .2755 L .26667 .27575 L .26637 .27601 L .26607 .27627 L .26577 .27652 L .26548 .27678 L .26518 .27703 L .26488 .27729 L .26459 .27755 L .26429 .2778 L .264 .27806 L .2637 .27831 L .2634 .27857 L .26311 .27883 L .26281 .27908 L .26252 .27934 L .26223 .27959 L .26193 .27985 L .26164 .2801 L .26134 .28036 L .26105 .28061 L .26076 .28087 L .26046 .28112 L .26017 .28138 L .25988 .28163 L .25959 .28188 L .2593 .28214 L .259 .28239 L .25871 .28265 L .25842 .2829 L .25813 .28316 L .25784 .28341 L .25755 .28366 L .25726 .28392 L .25697 .28417 L .25668 .28443 L .25639 .28468 L .2561 .28493 L .25581 .28519 L .25552 .28544 L .25523 .28569 L .25494 .28595 L .25466 .2862 L .25437 .28645 L .25408 .28671 L .25379 .28696 L Mistroke .25351 .28721 L .25322 .28746 L .25293 .28772 L .25264 .28797 L .25236 .28822 L .25207 .28847 L .25179 .28873 L .2515 .28898 L .25122 .28923 L .25093 .28948 L .25065 .28973 L .25036 .28999 L .25008 .29024 L .24979 .29049 L .24951 .29074 L .24922 .29099 L .24894 .29124 L .24866 .2915 L .24837 .29175 L .24809 .292 L .24781 .29225 L .24753 .2925 L .24724 .29275 L .24696 .293 L .24668 .29325 L .2464 .2935 L .24612 .29375 L .24584 .294 L .24556 .29425 L .24528 .2945 L .245 .29476 L .24471 .29501 L .24444 .29526 L .24416 .29551 L .24388 .29576 L .2436 .29601 L .24332 .29625 L .24304 .2965 L .24276 .29675 L .24248 .297 L .2422 .29725 L .24193 .2975 L .24165 .29775 L .24137 .298 L .24109 .29825 L .24082 .2985 L .24054 .29875 L .24026 .299 L .23999 .29925 L .23971 .29949 L Mistroke .23944 .29974 L .23916 .29999 L .23888 .30024 L .23861 .30049 L .23833 .30074 L .23806 .30098 L .23779 .30123 L .23751 .30148 L .23724 .30173 L .23696 .30198 L .23669 .30222 L .23642 .30247 L .23614 .30272 L .23587 .30297 L .2356 .30321 L .23532 .30346 L .23505 .30371 L .23478 .30395 L .23451 .3042 L .23424 .30445 L .23397 .3047 L .23369 .30494 L .23342 .30519 L .23315 .30544 L .23288 .30568 L .23261 .30593 L .23234 .30617 L .23207 .30642 L .2318 .30667 L .23153 .30691 L .23126 .30716 L .23099 .30741 L .23073 .30765 L .23046 .3079 L .23019 .30814 L .22992 .30839 L .22965 .30863 L .22939 .30888 L .22912 .30912 L .22885 .30937 L .22858 .30961 L .22832 .30986 L .22805 .3101 L .22778 .31035 L .22752 .31059 L .22725 .31084 L .22699 .31108 L .22672 .31133 L .22645 .31157 L .22619 .31182 L Mistroke .22592 .31206 L .22566 .3123 L .2254 .31255 L .22513 .31279 L .22487 .31304 L .2246 .31328 L .22434 .31352 L .22408 .31377 L .22381 .31401 L .22355 .31426 L .22329 .3145 L .22302 .31474 L .22276 .31499 L .2225 .31523 L .22224 .31547 L .22198 .31571 L .22172 .31596 L .22145 .3162 L .22119 .31644 L .22093 .31669 L .22067 .31693 L .22041 .31717 L .22015 .31741 L .21989 .31765 L .21963 .3179 L .21937 .31814 L .21911 .31838 L .21885 .31862 L .21859 .31886 L .21833 .31911 L .21808 .31935 L .21782 .31959 L .21756 .31983 L .2173 .32007 L .21704 .32031 L .21679 .32055 L .21653 .3208 L .21627 .32104 L .21601 .32128 L .21576 .32152 L .2155 .32176 L .21525 .322 L .21499 .32224 L .21473 .32248 L .21448 .32272 L .21422 .32296 L .21397 .3232 L .21371 .32344 L .21346 .32368 L .2132 .32392 L Mistroke .21295 .32416 L .21269 .3244 L .21244 .32464 L .21219 .32488 L .21193 .32512 L .21168 .32536 L .21143 .3256 L .21117 .32584 L .21092 .32608 L .21067 .32632 L .21041 .32656 L .21016 .3268 L .20991 .32703 L .20966 .32727 L .20941 .32751 L .20916 .32775 L .2089 .32799 L .20865 .32823 L .2084 .32847 L .20815 .3287 L .2079 .32894 L .20765 .32918 L .2074 .32942 L .20715 .32966 L .2069 .32989 L .20665 .33013 L .2064 .33037 L .20615 .33061 L .2059 .33084 L .20566 .33108 L .20541 .33132 L .20516 .33156 L .20491 .33179 L .20466 .33203 L .20442 .33227 L .20417 .3325 L .20392 .33274 L .20367 .33298 L .20343 .33321 L .20318 .33345 L .20293 .33369 L .20269 .33392 L .20244 .33416 L .2022 .3344 L .20195 .33463 L .2017 .33487 L .20146 .3351 L .20121 .33534 L .20097 .33557 L .20072 .33581 L Mistroke .20048 .33605 L .20024 .33628 L .19999 .33652 L .19975 .33675 L .1995 .33699 L .19926 .33722 L .19902 .33746 L .19877 .33769 L .19853 .33793 L .19829 .33816 L .19805 .3384 L .1978 .33863 L .19756 .33887 L .19732 .3391 L .19708 .33933 L .19684 .33957 L .19659 .3398 L .19635 .34004 L .19611 .34027 L .19587 .3405 L .19563 .34074 L .19539 .34097 L .19515 .34121 L .19491 .34144 L .19467 .34167 L .19443 .34191 L .19419 .34214 L .19395 .34237 L .19371 .3426 L .19347 .34284 L .19324 .34307 L .193 .3433 L .19276 .34354 L .19252 .34377 L .19228 .344 L .19204 .34423 L .19181 .34447 L .19157 .3447 L .19133 .34493 L .1911 .34516 L .19086 .34539 L .19062 .34563 L .19039 .34586 L .19015 .34609 L .18991 .34632 L .18968 .34655 L .18944 .34678 L .18921 .34702 L .18897 .34725 L .18874 .34748 L Mistroke .1885 .34771 L .18827 .34794 L .18803 .34817 L .1878 .3484 L .18756 .34863 L .18733 .34886 L .18709 .34909 L .18686 .34933 L .18663 .34956 L .18639 .34979 L .18616 .35002 L .18593 .35025 L .18569 .35048 L .18546 .35071 L .18523 .35094 L .185 .35117 L .18477 .3514 L .18453 .35163 L .1843 .35186 L .18407 .35208 L .18384 .35231 L .18361 .35254 L .18338 .35277 L .18315 .353 L .18292 .35323 L .18269 .35346 L .18246 .35369 L .18223 .35392 L .182 .35415 L .18177 .35437 L .18154 .3546 L .18131 .35483 L .18108 .35506 L .18085 .35529 L .18062 .35552 L .18039 .35574 L .18016 .35597 L .17993 .3562 L .17971 .35643 L .17948 .35666 L .17925 .35688 L .17902 .35711 L .1788 .35734 L .17857 .35757 L .17834 .35779 L .17812 .35802 L .17789 .35825 L .17766 .35847 L .17744 .3587 L .17721 .35893 L Mistroke .17698 .35915 L .17676 .35938 L .17653 .35961 L .17631 .35983 L .17608 .36006 L .17586 .36029 L .17563 .36051 L .17541 .36074 L .17518 .36097 L .17496 .36119 L .17474 .36142 L .17451 .36164 L .17429 .36187 L .17406 .36209 L .17384 .36232 L .17362 .36255 L .17339 .36277 L .17317 .363 L .17295 .36322 L .17273 .36345 L .1725 .36367 L .17228 .3639 L .17206 .36412 L .17184 .36435 L .17162 .36457 L .17139 .3648 L .17117 .36502 L .17095 .36524 L .17073 .36547 L .17051 .36569 L .17029 .36592 L .17007 .36614 L .16985 .36636 L .16963 .36659 L .16941 .36681 L .16919 .36704 L .16897 .36726 L .16875 .36748 L .16853 .36771 L .16831 .36793 L .16809 .36815 L .16787 .36838 L .16766 .3686 L .16744 .36882 L .16722 .36905 L .167 .36927 L .16678 .36949 L .16656 .36971 L .16635 .36994 L .16613 .37016 L Mistroke .16591 .37038 L .1657 .3706 L .16548 .37083 L .16526 .37105 L .16505 .37127 L .16483 .37149 L .16461 .37171 L .1644 .37194 L .16418 .37216 L .16397 .37238 L .16375 .3726 L .16353 .37282 L .16332 .37304 L .1631 .37326 L .16289 .37349 L .16267 .37371 L .16246 .37393 L .16225 .37415 L .16203 .37437 L .16182 .37459 L .1616 .37481 L .16139 .37503 L .16118 .37525 L .16096 .37547 L .16075 .37569 L .16054 .37591 L .16032 .37613 L .16011 .37635 L .1599 .37657 L .15969 .37679 L .15947 .37701 L .15926 .37723 L .15905 .37745 L .15884 .37767 L .15863 .37789 L .15842 .37811 L .1582 .37833 L .15799 .37855 L .15778 .37877 L .15757 .37899 L .15736 .37921 L .15715 .37942 L .15694 .37964 L .15673 .37986 L .15652 .38008 L .15631 .3803 L .1561 .38052 L .15589 .38074 L .15568 .38095 L .15547 .38117 L Mistroke .15527 .38139 L .15506 .38161 L .15485 .38183 L .15464 .38204 L .15443 .38226 L .15422 .38248 L .15401 .3827 L .15381 .38291 L .1536 .38313 L .15339 .38335 L .15318 .38357 L .15298 .38378 L .15277 .384 L .15256 .38422 L .15236 .38443 L .15215 .38465 L .15194 .38487 L .15174 .38508 L .15153 .3853 L .15133 .38552 L .15112 .38573 L .15091 .38595 L .15071 .38617 L .1505 .38638 L .1503 .3866 L .15009 .38681 L .14989 .38703 L .14968 .38725 L .14948 .38746 L .14928 .38768 L .14907 .38789 L .14887 .38811 L .14866 .38832 L .14846 .38854 L .14826 .38875 L .14805 .38897 L .14785 .38918 L .14765 .3894 L .14744 .38961 L .14724 .38983 L .14704 .39004 L .14684 .39026 L .14663 .39047 L .14643 .39069 L .14623 .3909 L .14603 .39112 L .14583 .39133 L .14563 .39154 L .14542 .39176 L .14522 .39197 L Mistroke .14502 .39219 L .14482 .3924 L .14462 .39261 L .14442 .39283 L .14422 .39304 L .14402 .39325 L .14382 .39347 L .14362 .39368 L .14342 .39389 L .14322 .39411 L .14302 .39432 L .14282 .39453 L .14262 .39474 L .14242 .39496 L .14222 .39517 L .14203 .39538 L .14183 .39559 L .14163 .39581 L .14143 .39602 L .14123 .39623 L .14103 .39644 L .14084 .39666 L .14064 .39687 L .14044 .39708 L .14024 .39729 L .14005 .3975 L .13985 .39771 L .13965 .39793 L .13946 .39814 L .13926 .39835 L .13906 .39856 L .13887 .39877 L .13867 .39898 L .13847 .39919 L .13828 .3994 L .13808 .39961 L .13789 .39983 L .13769 .40004 L .1375 .40025 L .1373 .40046 L .13711 .40067 L .13691 .40088 L .13672 .40109 L .13652 .4013 L .13633 .40151 L .13613 .40172 L .13594 .40193 L .13575 .40214 L .13555 .40235 L .13536 .40256 L Mistroke .13517 .40277 L .13497 .40298 L .13478 .40319 L .13459 .4034 L .13439 .4036 L .1342 .40381 L .13401 .40402 L .13382 .40423 L .13362 .40444 L .13343 .40465 L .13324 .40486 L .13305 .40507 L .13286 .40528 L .13266 .40548 L .13247 .40569 L .13228 .4059 L .13209 .40611 L .1319 .40632 L .13171 .40652 L .13152 .40673 L .13133 .40694 L .13114 .40715 L .13095 .40736 L .13076 .40756 L .13057 .40777 L .13038 .40798 L .13019 .40819 L .13 .40839 L .12981 .4086 L .12962 .40881 L .12943 .40902 L .12924 .40922 L .12905 .40943 L .12886 .40964 L .12868 .40984 L .12849 .41005 L .1283 .41026 L .12811 .41046 L .12792 .41067 L .12774 .41087 L .12755 .41108 L .12736 .41129 L .12717 .41149 L .12699 .4117 L .1268 .41191 L .12661 .41211 L .12642 .41232 L .12624 .41252 L .12605 .41273 L .12587 .41293 L Mistroke .12568 .41314 L .12549 .41334 L .12531 .41355 L .12512 .41375 L .12494 .41396 L .12475 .41416 L .12456 .41437 L .12438 .41457 L .12419 .41478 L .12401 .41498 L .12382 .41519 L .12364 .41539 L .12346 .4156 L .12327 .4158 L .12309 .41601 L .1229 .41621 L .12272 .41641 L .12253 .41662 L .12235 .41682 L .12217 .41703 L .12198 .41723 L .1218 .41743 L .12162 .41764 L .12143 .41784 L .12125 .41804 L .12107 .41825 L .12089 .41845 L .1207 .41865 L .12052 .41886 L .12034 .41906 L .12016 .41926 L .11998 .41947 L .11979 .41967 L .11961 .41987 L .11943 .42007 L .11925 .42028 L .11907 .42048 L .11889 .42068 L .11871 .42088 L .11853 .42109 L .11834 .42129 L .11816 .42149 L .11798 .42169 L .1178 .42189 L .11762 .42209 L .11744 .4223 L .11726 .4225 L .11708 .4227 L .1169 .4229 L .11672 .4231 L Mistroke .11655 .4233 L .11637 .42351 L .11619 .42371 L .11601 .42391 L .11583 .42411 L .11565 .42431 L .11547 .42451 L .11529 .42471 L .11512 .42491 L .11494 .42511 L .11476 .42531 L .11458 .42551 L .1144 .42571 L .11423 .42591 L .11405 .42611 L .11387 .42631 L .11369 .42651 L .11352 .42671 L .11334 .42691 L .11316 .42711 L .11299 .42731 L .11281 .42751 L .11263 .42771 L .11246 .42791 L .11228 .42811 L .11211 .42831 L .11193 .42851 L .11175 .42871 L .11158 .42891 L .1114 .42911 L .11123 .42931 L .11105 .4295 L .11088 .4297 L .1107 .4299 L .11053 .4301 L .11035 .4303 L .11018 .4305 L .11 .43069 L .10983 .43089 L .10966 .43109 L .10948 .43129 L .10931 .43149 L .10913 .43169 L .10896 .43188 L .10879 .43208 L .10861 .43228 L .10844 .43248 L .10827 .43267 L .10809 .43287 L .10792 .43307 L Mistroke .10775 .43327 L .10758 .43346 L .1074 .43366 L .10723 .43386 L .10706 .43405 L .10689 .43425 L .10671 .43445 L .10654 .43464 L .10637 .43484 L .1062 .43504 L .10603 .43523 L .10586 .43543 L .10569 .43563 L .10551 .43582 L .10534 .43602 L .10517 .43621 L .105 .43641 L .10483 .43661 L .10466 .4368 L .10449 .437 L .10432 .43719 L .10415 .43739 L .10398 .43758 L .10381 .43778 L .10364 .43798 L .10347 .43817 L .1033 .43837 L .10313 .43856 L .10296 .43876 L .10279 .43895 L .10263 .43915 L .10246 .43934 L .10229 .43954 L .10212 .43973 L .10195 .43992 L .10178 .44012 L .10161 .44031 L .10145 .44051 L .10128 .4407 L .10111 .4409 L .10094 .44109 L .10078 .44128 L .10061 .44148 L .10044 .44167 L .10027 .44187 L .10011 .44206 L .09994 .44225 L .09977 .44245 L .09961 .44264 L .09944 .44283 L Mistroke .09927 .44303 L .09911 .44322 L .09894 .44341 L .09877 .44361 L .09861 .4438 L .09844 .44399 L .09828 .44419 L .09811 .44438 L .09795 .44457 L .09778 .44476 L .09762 .44496 L .09745 .44515 L .09729 .44534 L .09712 .44553 L .09696 .44572 L .09679 .44592 L .09663 .44611 L .09646 .4463 L .0963 .44649 L .09613 .44668 L .09597 .44688 L .09581 .44707 L .09564 .44726 L .09548 .44745 L .09532 .44764 L .09515 .44783 L .09499 .44803 L .09483 .44822 L .09466 .44841 L .0945 .4486 L .09434 .44879 L .09417 .44898 L .09401 .44917 L .09385 .44936 L .09369 .44955 L .09353 .44974 L .09336 .44993 L .0932 .45012 L .09304 .45031 L .09288 .4505 L .09272 .45069 L .09255 .45088 L .09239 .45107 L .09223 .45126 L .09207 .45145 L .09191 .45164 L .09175 .4518