(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 63160, 1473] NotebookOptionsPosition[ 60040, 1363] NotebookOutlinePosition[ 60399, 1379] CellTagsIndexPosition[ 60356, 1376] WindowFrame->Normal ContainsDynamic->True *) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["BTM 6.2.12", "Section", CellChangeTimes->{{3.4526959014300003`*^9, 3.452695923876*^9}}], Cell["asks for all the solutions to the equation", "Text", CellChangeTimes->{{3.4526959459040003`*^9, 3.452695960954*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"eqn", " ", "=", " ", RowBox[{ RowBox[{ RowBox[{"Exp", "[", "z", "]"}], " ", "+", " ", "2"}], " ", "\[Equal]", "0"}]}]], "Input", CellChangeTimes->{{3.4526959620550003`*^9, 3.452695967649*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"2", "+", SuperscriptBox["\[ExponentialE]", "z"]}], "\[Equal]", "0"}]], "Output", CellChangeTimes->{3.452695993907*^9}] }, Open ]], Cell["\<\ If we just ask we get one solution, good old Log[-2], and warning that there \ may be other solutions:\ \>", "Text", CellChangeTimes->{{3.452695970762*^9, 3.4526960072*^9}, { 3.4526961491809998`*^9, 3.452696156277*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"soln", " ", "=", RowBox[{"Solve", "[", RowBox[{"eqn", ",", "z"}], "]"}]}]], "Input", CellChangeTimes->{{3.452695976498*^9, 3.452695979842*^9}, {3.452696018152*^9, 3.45269601944*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Solve", "::", "\<\"ifun\"\>"}], RowBox[{ ":", " "}], "\<\"Inverse functions are being used by \\!\\(Solve\\), so \ some solutions may not be found; use Reduce for complete solution \ information. \\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", \ ButtonStyle->\\\"Link\\\", ButtonFrame->None, \ ButtonData:>\\\"paclet:ref/message/Solve/ifun\\\", ButtonNote -> \ \\\"Solve::ifun\\\"]\\)\"\>"}]], "Message", "MSG", CellChangeTimes->{{3.4526959803459997`*^9, 3.4526960198199997`*^9}, 3.452696158959*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"z", "\[Rule]", RowBox[{ RowBox[{"\[ImaginaryI]", " ", "\[Pi]"}], "+", RowBox[{"Log", "[", "2", "]"}]}]}], "}"}], "}"}]], "Output", CellChangeTimes->{{3.452695980347*^9, 3.452696019823*^9}, 3.4526961589630003`*^9}] }, Open ]], Cell["And indeed this z does give", "Text", CellChangeTimes->{{3.452696197832*^9, 3.4526962065360003`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Exp", "[", "z", "]"}], "/.", RowBox[{"soln", "[", RowBox[{"[", "1", "]"}], "]"}]}]], "Input", CellChangeTimes->{{3.452696207892*^9, 3.452696212913*^9}}], Cell[BoxData[ RowBox[{"-", "2"}]], "Output", CellChangeTimes->{3.452696213506*^9}] }, Open ]], Cell["\<\ But then, so will z's with any odd integer multiple of I\[Pi]:\ \>", "Text", CellChangeTimes->{{3.452696221109*^9, 3.4526962380559998`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"z", "[", "n_", "]"}], " ", "=", " ", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"2", "n"}], "+", "1"}], ")"}], " ", "I", " ", "\[Pi]"}], " ", "+", " ", RowBox[{"Log", "[", "2", "]"}]}]}]], "Input", CellChangeTimes->{{3.452696239318*^9, 3.452696258133*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"\[ImaginaryI]", " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{"2", " ", "n"}]}], ")"}], " ", "\[Pi]"}], "+", RowBox[{"Log", "[", "2", "]"}]}]], "Output", CellChangeTimes->{3.452696259915*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Exp", "[", RowBox[{"z", "[", "n", "]"}], "]"}]], "Input", CellChangeTimes->{{3.4526962926689997`*^9, 3.452696296165*^9}}], Cell[BoxData[ RowBox[{"2", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{"2", " ", "n"}]}], ")"}], " ", "\[Pi]"}]]}]], "Output", CellChangeTimes->{3.452696296641*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{ RowBox[{"Exp", "[", RowBox[{"z", "[", "n", "]"}], "]"}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{"Element", "[", RowBox[{"n", ",", "Integers"}], "]"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.4526962617869997`*^9, 3.4526962838710003`*^9}}], Cell[BoxData[ RowBox[{"-", "2"}]], "Output", CellChangeTimes->{3.452696284297*^9}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Morin 6.27", "Section", CellChangeTimes->{{3.452693936131*^9, 3.45269394037*^9}}], Cell["\<\ A coffee cup of mass M is connected to a mass m by a string which runs \ through a frictionless pulley. The mass m is initially held with the string horizontal and of length r0. We seek the motion. In particular we seek the local first local minimum of \ r[t].\ \>", "Text", CellChangeTimes->{{3.452693994526*^9, 3.4526940340620003`*^9}, { 3.452694079729*^9, 3.452694162584*^9}, {3.452695261473*^9, 3.452695291674*^9}}], Cell["First a drawing function.", "Text", CellChangeTimes->{{3.4526941651280003`*^9, 3.4526941782279997`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"draw", "[", RowBox[{"r_", ",", "\[Theta]_"}], "]"}], " ", ":=", " ", RowBox[{"Show", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"ListPlot", "[", RowBox[{ RowBox[{"{", RowBox[{"r", RowBox[{"{", RowBox[{ RowBox[{"-", RowBox[{"Cos", "[", "\[Theta]", "]"}]}], ",", RowBox[{"-", RowBox[{"Sin", "[", "\[Theta]", "]"}]}]}], "}"}]}], "}"}], ",", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"PointSize", "[", ".02", "]"}], ",", "Red"}], "}"}]}], ",", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "1.1"}], ",", "1.1"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", ".2"}], "}"}]}], "}"}]}], ",", RowBox[{"AspectRatio", "\[Rule]", "1"}], ",", RowBox[{"Axes", "\[Rule]", "False"}]}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"ListPlot", "[", RowBox[{ RowBox[{"{", RowBox[{"{", RowBox[{"0", ",", RowBox[{"r", "-", "2"}]}], "}"}], "}"}], ",", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"{", RowBox[{"Blue", ",", RowBox[{"PointSize", "[", ".04", "]"}]}], "}"}]}]}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"ListPlot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"r", RowBox[{"{", RowBox[{ RowBox[{"-", RowBox[{"Cos", "[", "\[Theta]", "]"}]}], ",", RowBox[{"-", RowBox[{"Sin", "[", "\[Theta]", "]"}]}]}], "}"}]}], ",", RowBox[{"{", RowBox[{"0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", RowBox[{"r", "-", "2"}]}], "}"}]}], "}"}], ",", RowBox[{"Joined", "\[Rule]", "True"}]}], "]"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.452694184433*^9, 3.452694192955*^9}, {3.452694459238*^9, 3.452694495446*^9}, {3.452694526895*^9, 3.452694594703*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"draw", "[", RowBox[{"1", ",", RowBox[{"\[Pi]", "/", "10"}]}], "]"}]], "Input", CellChangeTimes->{{3.452694461509*^9, 3.4526944641210003`*^9}, { 3.4526945144779997`*^9, 3.452694519005*^9}, {3.452694609013*^9, 3.452694611083*^9}}], Cell[BoxData[ GraphicsBox[{ {RGBColor[1, 0, 0], PointSize[0.02], PointBox[{{-0.9510565162951535, -0.30901699437494745`}}]}, {RGBColor[0, 0, 1], PointSize[0.04], PointBox[{{0., -1.}}]}, {Hue[0.67, 0.6, 0.6], LineBox[{{-0.9510565162951535, -0.30901699437494745`}, {0., 0.}, { 0., -1.}}]}}, AspectRatio->1, PlotRange->{{-1.1, 1.1}, {-2, 0.2}}, PlotRangeClipping->True]], "Output", CellChangeTimes->{3.452696432375*^9}] }, Open ]], Cell["\<\ Next get the equations of motion. In terms of the variables r and \[Theta], \ the Lagrangian (kinetic energy - potential energy) is:\ \>", "Text", CellChangeTimes->{{3.452694628597*^9, 3.4526946799519997`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"L", " ", "=", " ", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"1", "/", "2"}], ")"}], " ", "m", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{ RowBox[{"r", "'"}], "[", "t", "]"}], "^", "2"}], " ", "+", " ", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"r", "[", "t", "]"}], " ", RowBox[{ RowBox[{"\[Theta]", "'"}], "[", "t", "]"}]}], ")"}], "^", "2"}]}], ")"}]}], " ", "+", " ", RowBox[{ RowBox[{"(", RowBox[{"1", "/", "2"}], ")"}], " ", "M", " ", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"r", "'"}], "[", "t", "]"}], ")"}], "^", "2"}]}], " ", "+", " ", RowBox[{"m", " ", "g", " ", RowBox[{"r", "[", "t", "]"}], " ", RowBox[{"Sin", "[", RowBox[{"\[Theta]", "[", "t", "]"}], "]"}]}], " ", "-", " ", RowBox[{"M", " ", "g", " ", RowBox[{"r", "[", "t", "]"}]}]}]}]], "Input", CellChangeTimes->{{3.452694682441*^9, 3.452694697467*^9}, {3.452694731224*^9, 3.4526947899709997`*^9}, {3.4526948671*^9, 3.452694867466*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"-", "g"}], " ", "M", " ", RowBox[{"r", "[", "t", "]"}]}], "+", RowBox[{"g", " ", "m", " ", RowBox[{"r", "[", "t", "]"}], " ", RowBox[{"Sin", "[", RowBox[{"\[Theta]", "[", "t", "]"}], "]"}]}], "+", RowBox[{ FractionBox["1", "2"], " ", "M", " ", SuperscriptBox[ RowBox[{ SuperscriptBox["r", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "2"]}], "+", RowBox[{ FractionBox["1", "2"], " ", "m", " ", RowBox[{"(", RowBox[{ SuperscriptBox[ RowBox[{ SuperscriptBox["r", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "2"], "+", RowBox[{ SuperscriptBox[ RowBox[{"r", "[", "t", "]"}], "2"], " ", SuperscriptBox[ RowBox[{ SuperscriptBox["\[Theta]", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "2"]}]}], ")"}]}]}]], "Output", CellChangeTimes->{3.452694868283*^9}] }, Open ]], Cell["The two Euler-Lagrange equations are then:", "Text", CellChangeTimes->{{3.452694896266*^9, 3.45269491017*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"eqnr", " ", "=", " ", RowBox[{ RowBox[{"D", "[", RowBox[{"L", ",", RowBox[{"\[Theta]", "[", "t", "]"}]}], "]"}], "\[Equal]", " ", RowBox[{"D", "[", RowBox[{ RowBox[{"D", "[", RowBox[{"L", ",", RowBox[{ RowBox[{"\[Theta]", "'"}], "[", "t", "]"}]}], "]"}], ",", "t"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.452694911834*^9, 3.452694948568*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"g", " ", "m", " ", RowBox[{"Cos", "[", RowBox[{"\[Theta]", "[", "t", "]"}], "]"}], " ", RowBox[{"r", "[", "t", "]"}]}], "\[Equal]", RowBox[{ RowBox[{"2", " ", "m", " ", RowBox[{"r", "[", "t", "]"}], " ", RowBox[{ SuperscriptBox["r", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], " ", RowBox[{ SuperscriptBox["\[Theta]", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}], "+", RowBox[{"m", " ", SuperscriptBox[ RowBox[{"r", "[", "t", "]"}], "2"], " ", RowBox[{ SuperscriptBox["\[Theta]", "\[Prime]\[Prime]", MultilineFunction->None], "[", "t", "]"}]}]}]}]], "Output", CellChangeTimes->{{3.4526949387320004`*^9, 3.452694949006*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"eqn\[Theta]", " ", "=", " ", RowBox[{ RowBox[{"D", "[", RowBox[{"L", ",", RowBox[{"r", "[", "t", "]"}]}], "]"}], "\[Equal]", " ", RowBox[{"D", "[", RowBox[{ RowBox[{"D", "[", RowBox[{"L", ",", RowBox[{ RowBox[{"r", "'"}], "[", "t", "]"}]}], "]"}], ",", "t"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.45269495317*^9, 3.452694975799*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", "g"}], " ", "M"}], "+", RowBox[{"g", " ", "m", " ", RowBox[{"Sin", "[", RowBox[{"\[Theta]", "[", "t", "]"}], "]"}]}], "+", RowBox[{"m", " ", RowBox[{"r", "[", "t", "]"}], " ", SuperscriptBox[ RowBox[{ SuperscriptBox["\[Theta]", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "2"]}]}], "\[Equal]", RowBox[{ RowBox[{"m", " ", RowBox[{ SuperscriptBox["r", "\[Prime]\[Prime]", MultilineFunction->None], "[", "t", "]"}]}], "+", RowBox[{"M", " ", RowBox[{ SuperscriptBox["r", "\[Prime]\[Prime]", MultilineFunction->None], "[", "t", "]"}]}]}]}]], "Output", CellChangeTimes->{3.4526949765179996`*^9}] }, Open ]], Cell["Pick some parameters, choosing m/M = 1/10.", "Text", CellChangeTimes->{{3.4526949994440002`*^9, 3.4526950254449997`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"p", " ", "=", " ", RowBox[{"{", RowBox[{ RowBox[{"r0", "\[Rule]", "1"}], ",", RowBox[{"g", "\[Rule]", "1"}], ",", RowBox[{"M", "\[Rule]", "1"}], ",", RowBox[{"m", "\[Rule]", RowBox[{"1", "/", "10"}]}]}], "}"}]}]], "Input", CellChangeTimes->{{3.452695026582*^9, 3.452695052877*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"r0", "\[Rule]", "1"}], ",", RowBox[{"g", "\[Rule]", "1"}], ",", RowBox[{"M", "\[Rule]", "1"}], ",", RowBox[{"m", "\[Rule]", FractionBox["1", "10"]}]}], "}"}]], "Output", CellChangeTimes->{3.452695053678*^9}] }, Open ]], Cell["And solve numerically:", "Text", CellChangeTimes->{{3.4526950562200003`*^9, 3.4526950610039997`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"s", "=", RowBox[{"NDSolve", "[", RowBox[{ RowBox[{ RowBox[{"{", RowBox[{"eqnr", ",", "eqn\[Theta]", ",", RowBox[{ RowBox[{ RowBox[{"r", "'"}], "[", "0", "]"}], "\[Equal]", "0"}], ",", RowBox[{ RowBox[{ RowBox[{"\[Theta]", "'"}], "[", "0", "]"}], "\[Equal]", "0"}], ",", RowBox[{ RowBox[{"\[Theta]", "[", "0", "]"}], "\[Equal]", "0"}], ",", RowBox[{ RowBox[{"r", "[", "0", "]"}], "\[Equal]", "r0"}]}], "}"}], "/.", "p"}], ",", RowBox[{"{", RowBox[{ RowBox[{"r", "[", "t", "]"}], ",", RowBox[{"\[Theta]", "[", "t", "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "100"}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.452695062605*^9, 3.452695111986*^9}, {3.452695750586*^9, 3.4526957510369997`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"r", "[", "t", "]"}], "\[Rule]", RowBox[{ TagBox[ RowBox[{"InterpolatingFunction", "[", RowBox[{ RowBox[{"{", RowBox[{"{", RowBox[{"0.`", ",", "100.`"}], "}"}], "}"}], ",", "\<\"<>\"\>"}], "]"}], False, Editable->False], "[", "t", "]"}]}], ",", RowBox[{ RowBox[{"\[Theta]", "[", "t", "]"}], "\[Rule]", RowBox[{ TagBox[ RowBox[{"InterpolatingFunction", "[", RowBox[{ RowBox[{"{", RowBox[{"{", RowBox[{"0.`", ",", "100.`"}], "}"}], "}"}], ",", "\<\"<>\"\>"}], "]"}], False, Editable->False], "[", "t", "]"}]}]}], "}"}], "}"}]], "Output", CellChangeTimes->{{3.452695109026*^9, 3.4526951124639997`*^9}, 3.452695751733*^9}] }, Open ]], Cell["Save a copy as \"myr\" etc.", "Text", CellChangeTimes->{{3.452695302271*^9, 3.452695307421*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"myr", "[", "t_", "]"}], " ", "=", " ", RowBox[{ RowBox[{"r", "[", "t", "]"}], "/.", RowBox[{"s", "[", RowBox[{"[", "1", "]"}], "]"}]}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"my\[Theta]", "[", "t_", "]"}], "=", " ", RowBox[{ RowBox[{"\[Theta]", "[", "t", "]"}], "/.", RowBox[{"s", "[", RowBox[{"[", "1", "]"}], "]"}]}]}]}], "Input", CellChangeTimes->{{3.452695114152*^9, 3.4526951511870003`*^9}}], Cell[BoxData[ RowBox[{ TagBox[ RowBox[{"InterpolatingFunction", "[", RowBox[{ RowBox[{"{", RowBox[{"{", RowBox[{"0.`", ",", "100.`"}], "}"}], "}"}], ",", "\<\"<>\"\>"}], "]"}], False, Editable->False], "[", "t", "]"}]], "Output", CellChangeTimes->{3.452695204403*^9, 3.452695754024*^9}], Cell[BoxData[ RowBox[{ TagBox[ RowBox[{"InterpolatingFunction", "[", RowBox[{ RowBox[{"{", RowBox[{"{", RowBox[{"0.`", ",", "100.`"}], "}"}], "}"}], ",", "\<\"<>\"\>"}], "]"}], False, Editable->False], "[", "t", "]"}]], "Output", CellChangeTimes->{3.452695204403*^9, 3.4526957540290003`*^9}] }, Open ]], Cell["Which we can plot and seek the first minimum:", "Text", CellChangeTimes->{{3.4526953100150003`*^9, 3.452695323591*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{"myr", "[", "t", "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "10"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.452695215896*^9, 3.4526952209110003`*^9}, { 3.452695330337*^9, 3.452695346915*^9}, {3.4526955376210003`*^9, 3.452695537993*^9}, {3.452695761885*^9, 3.452695770516*^9}}], Cell[BoxData[ GraphicsBox[{{}, {}, {Hue[0.67, 0.6, 0.6], LineBox[CompressedData[" 1:eJwV13c81d8bAHB773XNi6sQKkpSqudpWNFSipKEMpKkpFB8kSRKsiqhUDIK UZHKzJaVXWSPe91r7/w+P/94vV/38zmfc87zPGco2l4xu8DCxMTkxsrE9P// L+7YcmjZP9hjtvb/PzqUNSuYnlJwgq1p0qzzhNd+fAtQULgFu+kest2EBSI+ ZvG0+8P9yEyD94RfFIXJsyvch9hiM383whp0+4drDo/g0EeNn/KEfwvE/1tp i4AC24Mbv/6jw7rX3DpLy1FAZf8WZ0TYZbfH5Xn5p1C/5YVcySodPrT0Js/s j4OMVqasDYSXLpl2TzokQNkW9hO+K3RIZO6+P6//Eph+VXGXLtPhQN3j9XNt r2Dk2dXGmSU6jMQaFs84JUNfLiVdjHCo/arV9HIKeLzliKEs0qF52TGSIf8W Pv2Te801T4cbFWRNenYaOOjmVfXO0kH6SUsNbX8GdHAbMr2ZoYOtGrKOO7yH jijDN6uTdGBZ9vo545MF9ZLbJB8y6NDSay43o58NZdXSL/jodLiZwZs/1ZYD KtOM/uZxOhyMGOKYSvwA6/i9X8uO0UH2ZvGJSadcuJKteef4CB2K9nsy6Mt5 kLfnpuuDATpwdvWp0OTzwZ2UYbWumw7tRV89qCP50HQms+l3Bx3SXseWjmcX gDb/kM3dNjocdj98bmx/Idw2L6+Ib6JDNHdB9LDDd7D086pdqqBDrudB92uS RdA9ekZ4fTkd4sa2S037FMFVAxv33SV0cG0QcZjSL4YAL54HuoV0EImrZJls KwHOQlCxeE+HZYGPb912lYLCLq+z0+l0GPgv6SgjsRQMm1JzfVLp8NHhTjzd qQyWBh9VWr6kw+mt2jsnlstBcFLi2e4IOrysTrg6Ll8F6fZaYRludJiJzVPr Ca0Cxd0Keysv0cHAoaa/abEKlG/lsvy8SAcq67x5QXM1PGDx/PDCig66uw/v vH+vFpo9dtPAkA4N71dYVOg/QbvMtj9WkugvcFvzb2qAn0YHuS+K0EFPOXLn PfsGiPslaUbmo8PDmbfTXo0NkDUVZnZ0bQK0H/+6YJveCIc/b9ArGJgA32qN g1rnmsHudh/lTfoEpGZ/Uk6Paob59RG7VJInoCl2H+v62mbIE02JjIibgHUO FoWSO1pgYXcJp3rYBFSzBm5iEv0FOdd5tthdngCJ3V0iP3+0gsynUrepDRPw 7v39LpdNncB++x77iyc0GBMZCz14vhOUx6W7tENooHLj4B7VyE4wfn+uq8CP Bom7eF/2L3ZC+smvFx5fpkFEVejF0+VdUC+fPexoQIMb/Y8mDa1+Q55VxGjp DBUSeGsChPV7oU0xZWT3ASrs2e1t5WfdCzb3sx0SdanQ7aq+je7ZC7EH/K9M alBBsunBUG1aL5zdnaNyTpwKj2NNjYOF/oKi6Y1Ep8FxCFhfL7DW9Re+5uvy 7PEfBwdofEp174c/AcdXmd+PAdOruYKuB/1wnfY+zf/lGMSyyXZXJ/fDWelU keknY1BZdZH8trUftngpKifdHAPVEyuvLuwcAJpG+AjuHYMRR5WMP8yD8Dbt oZfnz1Fwjrj9rSF8CCwPo9Nizwi4Dm7oz303Cp/FJNs6m4ag17qDq7hqFAzF mqJYi4bgeHvwprqBUbBTGbGXyRwC3ZrhW4PSY6Cn//qy5L0hYM1OEZK4NwZk G69K451DEOujuMfz3DhEOcrZx7wYhGJRqac7hGjwJCnH/LTFAPAoBqr06DJg 5d72mrtJf+GhUI3aS5tpKLz9e9PyrXbI3y6Xd9h1Gj4NsXNtN2mHAesrsOI9 DXveVmsEybaDXqboiVMx01BNYQu1+94GI8ZWt/nrp8FMj9aNHG2wP4D285be DDz6yJzz6+kvcE2D0yrGMzDYwdk64/ILnjU+Hmg5OQPJUHIQ8Rcw5HUWN7vP gE34q2674RaIL7xDGUqdgX33My0kdIm8mxX0MJOYBTdk05YebAIlOdu1NaVZ 8B24P8VU2ASHD+Tez9SahRzFiEXdJ02QEmERz3VoFvTmnFVS9zWB2eaXFd8C ZuGTeajPr5RGuHNy+pjL41nICDEPifVthLe39bulEmYhd8MpUrllIzDVjjKu F8xCfqgNu6NgI2Q6bpFWn5yF7KkkZR2/Bmh/FJjc/m8WpvhtMlLONgDbp9ZN QXxz0PsVWZ/rNcBpdu/9f1XmIPT8zwHTqZ/A9arUJcZ6DhaDSa1Wc/WwtUp8 /oDLHLw4+xSvF9SDNcPhv6lbc/C1OU+kzbeeWBf5og9FzYHl3NljiQL1YNt1 4jtr7Rywj3Zv3bi7Dnie/718tWMOtn9rqXfmr4Oc066yPUNz0OcZkavaUwus nfduFTDPwxuqi6Q+sY5kPBVTVRWcB5nOwoMfztbCCcuXrVGy8/B9p59K8bZa SG4v2Hp1+zx4OSvpdY7VgGmsYd+fA/OQ51hgPFNRAzOnWsJNzebBPeRjas3r GogjnYeCc/PEvhNo4XCvBvTbaDSVy/Og1ii7vt6pBiaiveKivObhtifvS47D NRB9ktOENXgehCsm5UnaNbBHInLRLWoepr1lstlla2Dol0Lqn1fzEPOhQbqd vQYeRmWeNM2aBw/hly1PJqtBx3wne8HXeai568myp6ca/ohVfFCpmQcppF7p qq+Gey3HbaPa58EmaSrEragaNCN7hViH5uFsadk55txqaD9++bvb9DyklydE Pn5bDX6iS5f/MC0AU9mVC+teVoNqc5CsqcACbBt2Syx8Vg0NEaI1+TILxL74 6ZVVdDXcNEu8pbJhAU692GzOEVkNCiIbVaN0FsDp+ym1L0+qoaoxv5XlwAJE ZwnK+kRVg/tjg7tuxxbg82nBNOOn1SB9rHnrH+sFuMD3ol0poRpKhGz6TFwW IIZptybPm2pwbqCG599agL4R56/LWdUgEn4LVO4tgLNdndB8YTUUHOGYiIxc gHcxX3CpuhpsBZ/EsbxagJF9LGMsXdXA+1PexO39AgRdY2gJ0Koh92HG4u/C BeDxO1olyVwDVod3pJpUL8ASy54JMqkGMurM2FUGF+DRT4cPfMY1cCKs50Pk 1AJ0PqnPHbOvgRVTF1sWpkU4GftcINu/Bg7V3v3+W3oRvkRcse0prYG5ByKu JqqLwBfeV7dzuAbiTRJk87ctQmD1Ud8LfLUwUf35VuTRReCt5DXhs6qF6BB9 VRbrRUil3wy7FVQLew42tV65tAidmtpXY3Nq4VHV+FaToEXAqL2kGsE60Kwk TzAXLsJsW4mAzO86aL+XHnelahG2hL0R2SRWD36Guia/WxfhXYGI4XvTemj4 cSz18+QitI6NC3UU18PV8kDbKypLYCJzK7P5y08gDxhwzG1dAoOX4sWMhZ9Q w8Kd5oNLsH1LC2uwTAOsx7DJ+5ZLYKwv/V32TAN0FET5JT9YAqFjwpJfexpg b9br+A76EuQXuw6wrzSC8PPKzv2fl8H3U6hjrUsLZLvxHZc0XoXHNz/7vi7o gNpnorcyT67CW84ohv9oBwyVSSfss1+FrXcLnp6T7ARpqQ3jLr6r0CLa6SR+ oxMCi/UDivNW4dHCpvOntLrgpIjvB2fKP9AT8m5yed0NSx8mRb8u/4Prc4/j 8u/2wEvvS2Oap5hwu8i9t/J3+kCziHvh3lkm/DLIQ0l83AdFbKnsPXZMyNj3 Zq9ISh/0hg0ohLkxoYJvl0xNTR+QE8+eGr3PhJm5Ed2hkv3wrOxI2ctCJow2 Sh7e874fIvi140UozCiZ+8utr2EAKGZN6U6qzLhpwXrXtb4ByI52yy/axIwc +yIeTE8PQIN8ZourHjM2b3ANKJEYBIEtyrw1J5gx0543JPbMIISclLwZcI8Z t9ACKTJ9g+CfsHJshsqMvxLImtv6huCo5uer6tPMqJIj7nlicgjIxdce2y4y o9ePZ4W2TMNQ0DfW0MDOgnH7OlVMycMwtb7jSCaZBZVNSsuOWQ6DbUbeoYtH WfBEmEY0pXYY9ua7Grd/YEF2rRbxD8kjIHBQzUmwgAUD9tjFv84ege7OwWCD Ihbc9zPtevC3Ebi5YlWZW8uC3y3XL0q0j0AWmBg+HmTBLwUWVGueUVD4oaJ/ kMSKX91f/Yu+NApMzb1Y6MWKuN4r+KHCGASOfnvp9R8rLln891ZDYwy4mV6w 7AhmxQyp6Ct528dAZKNlWV4UK7a23+gOOjwG6+81Gr7PYsVrK6mtqt5jcFCv 5PCrIVZcNkg5d7FhDOqPJb63obGikVuSbmbnGBx3vCMkP8OKOVZK/n8GxsAq amfTc2Y2pMgsnGZaHANXeo55lCwbaqZqtaDCOLHPJ1kFm7Hh4QeGgTnO4yBZ 8N9XA0s2tM8Llkq4Ng5xDefI7DZs2Lv1yYyHzzi8XpXp/e8yG7Ie93DvCxuH z6ci7byD2fCJ6jrTj+/HoZv3rrPrNzZMNRsLMKSPgw3FrkajnA2NfhVdaJsb h0HdvRrjNWzI43nY+di/cZi4sEpz6GDDzMn0ZlZ+KjAXeVw9P8OGT3N8J89v oILy9Ys3T6ixo4eaaSGzNRU27uw5nqDJjiFRX+5W2VFBm8li85gOO65e+ahx x4kK+0KNh3z3s2N5/jz/u+tUsE7WOJFuxY7uDgKLKiFUuOCcsnnOlh0/RqwP OfmICi6aZN69TuyY23l94mokFbwKBUtaPdiR1t2ofiWeCtEtU5tZHrGjeXXT qYBsKrx4don3UBQ78n/rtZH8SIVkm4GhmOfseOxe8tZnBVTIof56sTGVHX2Z f2jZl1Khni2f17KYHXWyrK4Kt1CBfZvv8Ptpdvz13FSgYZIKfMvzJUuL7Cge HaepNUsFkWK3eH0mDiQnlPD6LlBB4ZCteRcfByLnh5Hef1TYdUG/lEOZA/m/ xlC+8tBgv/q3+GMaHHi9eZERwE+Dg5M6XnFbOHAwMcdtO3GOs7itqrUFOLA8 6YmfvTgNrkfxJpy14MDMvd5VrWQaeJ8J9Eq15kDvXePcrIo08FdcNZ+258CS ku8zMko0CM+c4Lt/lQMpxw/8EFWhQcw1h5FmTw7033k0fUqVBvE7ekvJdzjQ +ezN7d/UaJBR3uiVG8KBMcZje2U20eDDg4Mn18I5sPtAwreMzTQoOFaqdTCG AzV+GfSoadGgmLSLP+oFB0pryyRFbaFB5e/ckZ4kDrxwyYKbtpUGP5M2lqml caC26S5hrW00aHV6neCRxYGVjhwF53VoMDAbc5L3Kwd2GJ8YD9alwau9W6/9 LuXA0L8p+5/soMG5sPpH76s58PjYs19BO2kg2+GU8V8jBz50MU6006NB5zr2 quPtHMj1uerVhl00iHVLHFzfw4HhZ3W62wmfLNRjWRjkwPuqGSZXiSuWGFcb uZrKgX6SBxlThBuPu+vFTXNgp6VW1dk9NHiYwG/husSBh/Tu/sombDqeeh2Z OVFW+ZIYnTDP9gOPRbg48XYsz11xIMbr35M5IMCJXAphyusJB9V7VX8U58RN J8gLcoT3S0sMB8ty4quG8YU14n2mi9msZ5Q4sXHzetUawt+yTRU2qnHiMfaV QF/CPqvDu9Y0OdGsPUlAjvAO4wDLxu2cKEg2K0si+jsXSb6RtIcTUXtPiijh 3N78CA99TtQ58zLLhRivu4b5e0NTThzI/Dj4jpifzTcZNVLHifdlvuzvJOaP WvpgZNySE6/Y0KomiPlNE1Rh/2bDiXPsDz2pxPw7nilRDHcgxmPZY9K8nQbr 35zdY+vKicUO6w2SiHj1TS2c1vbgxK0rKResiHgm7on05PDhxJSY1Lf/iHhb h2yObPfnxPbLN4RDiHyQba3OSrvPiftsTsf90yTy7zLz2OEYTkx2iJVM2kiD E/lxHIrxnPjYM5yrWZ0GIuy6StPJnMgW1ilJ3UDEI87VKiaHE09ty4npWE/E Y4T7llM+J2qdH2N7R+Q3t3ZKlF4RJ/5XWPXoEpH/gTVd9T11nNg607/8UpYG +0ie49ktnLh891eXtDQN1mxFuAK7OLH2dVerD4kGXktGe1VHOfGwUJLMijAN 3DZ8/ODGxoXf57rqh9lpsMnjWMM+Xi5U2LH+xlsWGowXUaliIlxoCq+2nlij wkULJeV8eS6Mp/dO2BP1fvZeeCyzHhe2L58+eW2UCtLN6nnNe7lQUFIp4fYg FdrJFY0pRlxYvxyx6PaXCic+rvIcPMmFGh/1/op0UOHgoPPtiKtcWFL4zmV/ BRUMbitlHLjJhcWpTaYZJcT6KN7dOXeHC0/m7Rdc+0qFnfqHdM+EcqFJofwO 21wqqKVsnlZ6w4V/L3mJyidSgffCjGNeNxc+by33dr9GBa7VjBiHfi78d2TG 6NZlKrBFXfghNcaFKXrqk04OVFgpa1W6M8+Fmz4yN3KdoQJ1Xf4fAxFuNDmS UbJrLxVqB26faDfkRpZXD0Rvc1Oh0kcnIOQwN74MMbJRZKFCmRg9e5c5N35X S//v/dI4FB6wEXxpy40lwQN4f3wcMpL3VTvd5kbL14Ovt9SOQ5g9EZscbgxO ulhsdX8cDg+Eq5PJPLi9c/UX7/QY1KoPtFit48HOzM0Pno6Ogcm17Xeeq/Fg bvDihGjvGBix/GmQ3M6D/Z22xfW1Y7BPQf2G6FEeXP+R+YFVyhhssyov4vLn wQlptzvtZmMg27JoPj3Ig6p8mrZHE0bhmcyhNS0qD45GWfpcezIKUnaJqW5T POhds29PwL1RkJgyWKb948GZ+vDPjldGQUgoMnGExIt0jV//mewZBVbTTeO/ jXnxtHHzbjfiPDFWautXmcmLvfltitv+DYP843bnh7m8mCEwnPhxYhhOWB82 P/GFFxPkSk8p9gzDt4Udar2VvBg0IOGS830YIjYKt8z386Lj6McNpb7DoBfz XUVFmg9zPffVxy8PQaiz7M+7QXwooVF0czdxvireHpFvEsaHO1fvRw3+HIQ5 Nq5k4Ug+nD7gPHDj6yDYJEzffPGSD/eH1JDMYwdBu6WaklfIhy7pmQ1rpoPw e/etGwNTfGgEnx0FcgZgk3Cr3P5z/Ojm0LtR8FI/DIUN+Udd4Ef5pKGbhqf6 4QXP/PDwJX48IsrSd2l/P/CxSeaE3uTHkB8CArdk+mFsztKg9TE/9gulVCdX 90FK929XpzJ+vKRy//ve9X0gmzr4/dEGAfRf+kQPr+kFHpg93z0tgFLidP+1 z11wc68q18clAbTgLCyyCOiCof1n3j1iFsS2baz7Ag51QYlR8dI+QUEMcPUW MP3bCV5mYU/eqgniTHRSTQZHJ4xfWF9+47wgHmBN/pZm1A61oeaqwj8FUeHr Ae7Yty2w81Fw/dgvQfwZLRkq5doCqY+/XC/rFsQiqD7qsqUFAqMViz3HBHGO bV3i3YJmIDfMadBZhPDrtOjpe5VN0NpRUm5OFsJg1sgLFV0NYEg7PU8xF0LX B6Uht75VQbxYQLOvhRDmNV1DzT1VMKOX/r77jBC2K87+C/1aCYkhyw7RtkK4 8J9Viu6XClhSiWvndhPCtO4ECcuccsiw/Z3PCBHCSzE6HmWPioElhD360EPi e7fIr/5wF4NF9kb3tMdCmJ3w2D4hoAjYme6o2ccKoVak1NoG6jewfkF+3pYi hK+osSffZBaAUPs5n29FQmhkKt/+QjAHLq7ds5ApE0IfuwN9LQtZUKicpX2z QggD2iOOVZe8BycPJppWPfG9U0Jhu8+nQ4nIy7MpXUIYapVbfyb8FUjurNrB 0iOEUftmHgxYJoLr+Unxc31CqGxw6wanQRzIZO2tlxwVQueJZfOHl5+Ae5tT mgdVCJsPk3abiT6Eyn+Pg5roQnhYXjzpjPtdICsX2G6eFkLF7lrG1kQvuH6o b0/onBC2Hnf66n/UFaqv88iMLgrhsy+5ji2y10Ehbsu8/irRn9o7yvph/8GN 0tPNr5iE0cfvg907zhCoG/N/v8YqjOKLZEVx28ewTiT9gRWnMNZztU2J10aB 145mh3weYeSIOOSQufMZKAcrKVwTFkZu7jmyZ9hLaC+xZz6tIYz3SzddZnqa DYy9ndK3Nwujinj4vEp8DnAWH9FO3CKMoY4/cutffQCd7zsdhnSFERl3nCPT 8iDyi1Cdu74w5p/tNBz+kA9HcgtjH1gL47bvLSfiW4vgR4qY5tdwwgXZsVlx lZAXXGatMCuMntyNFlcymyH1YI3kvQVhPN4WwrUw1gzP+JqaaMvC2NT95aqF agv4hfcYfGERwYQruTcSX7WAaezSppNCItg44O55KOoX9L3RXHugLoLXS5V+ yF5rA4GKuIT58yKYWNPYFSfUBRfZPXrr60VwPCHG4sdoL0jLV7UtN4ogH/UG uYX5L9Tryv1U/SWChjyTIQ1Sf0HHpfyrf5cILs3Kn4ky/gsczeLPdUZEsCnH kXE49S+kJH40j2cWRbcbQln+tn0woLdQfVlbFP9pTSdPVfWDnbt3Ht8zUdxi YfZV+cIQ6Gz5k7fhhSgaUDr4/W8OAc8UfjRIFEUN8q7R2gdDkHWV45Pfa1Hk VNt4QCtnCFbcHn+eySHeN6Emaf8bgugrb75014jip1n/X8pPhqHKpbk4Y1UU jc0kHb9njUCchk5JFZMYruTfbuMvGQE3amzJEKsY4hcpuknzCEi4WJcq8Ihh z8tP0jGzI2B7abQsUkIMzd5RbzvtGIUVp7UKn81ieFLVwWjt6yhsdlCvNz0v hgnHOxJnC8bgj5P8lUx7MbT0ahy6WzMGoS6iQgKOYnj3LFmctXsMRq8uH/vp Koaynws2l62MQZJPza9jPmJYsKjjd373OJAiLv0+GSOG1xjRb7m/jMNaYTrV pk4Mw55UC0ulUuHd94Sw4gYxHI+72WFE3E+sSp5sorSIoenWezkXyqiQX+Ht 1t8phnIpnPEXe6lwrclk5sKIGN5h8/G9S5xnhofHl51ZxbFFYTVS1584f4to 8HrsEMfUOR3n28YTsOfy5dHnu8Rxy72j24NPTMC7incVJSCOliiQ6X9uAh76 aAUKGYgj16GJaAOPCTAd0llNNxPH0LfxATyJE1Cdv5fed0kcNyWPFQZMTUC5 zanmo/HieLLONHn/Azpof4nN9nwpjvfzdD7pR9IhWbzzUXyyOAaX7o7Z/oIO gdVWptQ0cWwcNCkde0eHfdq25fc+iWN9sxT1YSMdijgvf/rWII6d0q9Lh7kZ 8OWd/3MNVgl8X+1MkjNgAIepuXQMhwQy8134OmTMgGOjKk+ZeSRwM8vGudeH GDCsVBfdKiSBxmu7lHjNGSD6lBThR5bAAzL8/9naMeBSQEZwyw4JtKY1LR66 w4CPCr6csJto7+fohv/+YwDzt2NBb1EC988aL6YFMiBmYS7gjqEE2kUL3OoL YUDp5b2+quYSOGjXIvI3mgEyFq0ePm4SmL07+rxcJgMuzqbODF2TQO3WyMGC 9wzIjvC+dsxTAkc2JYgfyWGAYZ3CVeU7EvgoUvj2mU8MuLbvkkvDAwlU1pXw YytmwLee3eM7H0lgZa+U96lSBnDfFnJOiZDAuYgU1ZflDEj4lOfg9VQC15Jj 4yWrGVCjwWS37o0EHnzSyhfbxACJ6qa/D9Mk8OFT5g+JLQw475Bis5gpgU2G +0USWxkw//KgdX0u8bzDqTX/TgYokaIsb5ZIoEsgNa3jLwNccx3a+8olMChm RjO2nwH5x3aeOlQlgWTv3BuHBhlwJLTnBKVBAo3asw/EjTDAm1ntaO1vCdzi LsBSM8GAoQO87cp/JVBUkGdQl8GAo8HUc/8NSKBg7e/+F5MMWCf4/orOuAR+ q9uje2iGAQ/NwufDJySw9NSFRzGzDFiIuuo7PimBf0Kj2TvmGFArq/0ocYFo L/Kw2d5FBujYiJOWlyXQbNhNzXGJAYlJc/HmaxKIIy3rgpYZ4KGW/46Hg4Ql /HEBb1YZQNbdU39dlISs3nVrvsyTEOwtf/KnBAmz2l4y2bBMwtQ3pj8bpEk4 67lVbTvrJFTol9L+KJAwIL9xqZptErbcT/bYsY6E2Zl+SffYJyGu9u7qExUS pjzQuqnHMQlux434jTeRMLCtNzWYcxI6ozdEJWkRHgngoHBNwoFOHrl/2iRs 3+D+JIfwOzlqsoUuCXPe3Dm6k3sSpM7XaXzQI6Fxvs+efMKBye9y+YGEfMxo vZlnEiaGH+1y3EfCZpe4rDjCFupXy0r0SUjSuajDzDsJJa5mpnLGJFTsu8qw IqyRs7XF05Rob19kz3vC0bNiVk1HSHjN4xXbEmGmHXP9GsdJuDp81UaPbxIu +bRduneShNvM2qevEW79/nn6ryUJH35O/55EGFmfee86S8KxSyVl1YTTDLxZ Y2xIOMw9zTpKWCzE6sGkHQk1bovd+Uf4Tt1uUVMHEn5YXd3Kyz8JI0Lyz187 E/0VvqcsSNjsBJMSsysJ/4TcO8lD+GvM37QzV0lYEdpWtkK8r9pVsuXjdRKK 3zviPkQ4gpxcIHSThF26lecqCK+cv7vvkjcJjbKkHyYQvphysbr8DglnDir8 cyXcMGJopuBPQvWPOVnbCO/U2NDpdZeEO+jpr6eJ8Sdf4bH9FUzCj/uHelMJ C34YH90cSszvsV025oS95mqvhjwi4Wuu++sXifke2PFucSCChB6pb7QiCX8u cuN+9pSEbXpVillEvChsZo9n4kgYaRTIqUU41HCr1JFEEu68+t/2t0S8z9fP qrK9IearOe+WL5EftcJtWdZpJOK++ibsN5E/OuafdfMzSdggf4y+hTBPt5eR ay4JmU/sTygl8k/H8y3/o08kXKeZ5MZH+LxIe9P7AuL3fZq5h4h8/Wy07exk EQll/d5E5hP5PdBvRxEpI6Ek9+qWASZifL4Rw1sqSDjx30kDTsIX8+hXr9cR 8SqbqNf5x4CIo+TtUQ0kdJ58pr+fqJev46Yrec0kbGldr220wgAxSlrQfAcJ bRzyvPWIeit+aP/ca4iIV3BHZQVRr9QNT2yej5Kw9UmMSCxRz5LlxesLqSTs tZvrsJkm1pNlctbqFAmrLvgMdBHrgaxjR5nfGgnTEllXPccZxHn/8ESQpCT6 nsnpMephQHK3z4c3MpK45s965uVvBvz0TL9ZSZbE3fczz011MUD5HRcrz3pJ bJNn47rTzoBm6VJSmJYkjl9/MqrXyIBNM9v3PjkoifGM6RgzYj09/egiR+4h STzXfZtF+zsD7qpF1bQclcTLv2xZBL4yoNtmylzilCQGcRiXp39mQEhdhvNT O0nclT3k8pBYvwdfK0Ym+EiiwudbdMZzYryBfxdZ/SRxjyxtQOkpA6ZsX55z DJDEyMfCmoeJ/YBJXlFdK4To/xfnmuBwBkjHKJSUxEjixE/7LdHEfnL0njxj MEcSjc6YJYlcYsCpCz3mBz9K4hJTkMsXBwZY70/48i5fEpm2P31+xp7Yr9bI 9zyLJPH7CZ/N/mcZEORJJnPXE89bK5s4HmFAoYOcqcaoJG70c9F7qkXsT/q/ s8OpkugZa9jzeSMDqpVekGbpkvjWqnDx5wYGtPXI9n+dk8Rvdh+jBxSJ/p+S 9TrCJoXFfiMP3wgzQNVIJtVdXgpJni9a1HroEKkqxZZvLoVb7f3ZtA7Swcpa +U2/hRSq/adZvoR0WBe59aCAlRRaFv3sy9tOh9y1Q+F2tlLo3X1pF9N6OrS0 +ssKXJFCn0WN+MF/EyB6l6ptFyyFTNk8VTPvJyDi7/cL/F+k0CpS32ML5wSc JtVx6X6TQtrZlGj1FRpQDnWm2xZLYcma6gmJSRrkfJ6Z/FQhhbwLM+lFnTRo erThjm2LFE79usPtlEkD4T1Poj/RpHD5iuRpnqM0CH92seK8gjRuWtmSFx9C Bc8b/td8lKSxp3a/gdAdKlibxcvHKEtjkY694fWrVNDgafWs1ZDGqI+/Xola UKHqlr7q9h3SeKlU5d/7dVRgsVx3n89MGg/fTzp0nji/eZD6Dn4KkEYONc2t n/6MwenIs/V8w9KoshSuXLY6ArIfeGQujUnjw+eSFx5QR6Cn8ZNDFU0abfrT b+3vGgF7QRHmoBlpHK4J0gn6PAKuIT+2rjHL4MbvTj4n3UfA32/z00lZGbSd /WOs3D8Mb12Y7VvNZFBgzk75df4QLBxIWUz4JoMaW58euaU3AP2D+26W3JPF HkN7xwMif2BfQv0rtQeyeDydbSmo7zckWpyui3goizENfn8/5vwG65qrFLso WSzfad3RY/YbOrMSa1mTiOefCs5YR3RDk/c/Bf1vsvjT3CT9rABx/xYuqKqY kcXP8w6D5qttoFijP7N5QRbHbI1y2WrbwC+wkRy7LIu9RSrV75+1wZ75kWuO LHK46fu31tXtbVDQTSJzC8khk5r7kXD3Vsh54+F+UF0O5VxmJGxGWuDVbi2Z uvNyKPnZbdixpxFKDvMELNvLYWLRZsfZnEboO9c/tsFRDvdUXpN9HdQISv5R BUGucmg4PuTts6kRkn8sWqK3HK4FH7+4y68BUo6UxH6IlMPanLr+NKt6+GHz nKkvRg5133JFJPPUw9DV645Cz+VQdfRDH+bXgXKksu7ll3J4yC8j0U+yDl63 h7Qpv5PDttt5Fly9NZB6/rjEswo5tAyRnc96XgWV7hp3KqvlMEecT1ndogpG AtiH5urk0P2w699x8SpQff0p73iLHFaJinneiKoEo0/hcv5tcuij6j545lQl OFY63c3qlMONS+NM36Ur4e2YjDn/XznUmw38qvu6AqqXZwr1BuRwZ/kEXHat gDG++nXOw3K4Yb0o5YxuBaht9pv5QZNDkdH+1sTGH3AQLa1mGXK48AabSC9/ gPOxLWVKM3LoEfPcPsj9B6RdG3jitySHXvwS4cEyP6A68Ovyu1U5nBTtndk5 XQ5jUdF2v5nIGPao0o5SVw48b67U8LKRcTCJ+dmht+Wg/tlo605OMrH+7qRV 3ysH0yrF5448ZGRytNBJdSwHl84llhh+MppnFX2ZMimH0PFm53IhMlY0xsh8 0iqHjJWMpmlRMkp9frGNWbocavmDdlJIZORyWyfRzVYOVPK5V0elyfiP7WXv saky4NPU5fGVI2NK/N3CS31loLFX2D1TgYyGy+PFWr/KwNRsrKNLiYx7Oa+N plWXgYtd6V4eFTIe0/Jd6S4pg9DrcW911ciYpcFyv/ZrGWTe9RB22EjG5w8K Wn2+lAF111n/u5pkXOe1vmmIsMbMgemkrWSUl5gtWPhWBpfSNexLdMgYd6hn LrusDNJsxX717iDjSN5KtlBdGYxJreiv7SJjn73ZX+H2MtjQ2P9RDsmY2yxq nD1YBo7BNSq79pNR9nV94OhsGaTCh9jTBmT82H9MJJerHEbmnnHfMiajDe/f s9zkclB55+8VY0rGV3MLDb3byuHiBefxvCNkjDl/q0/zaDm8ljWzajEj4/Ds lArdpRyGmnfUTZmT8ULKOkeh0HJY/0Bxj7AlGe0bz7wLziyH5EWG/OFzxPzr FxmazJfDQFZ7uIstGUW7hwq85H+AkmMR84MLZOSdan9eY/wDXrU+6q+4RMaQ d5YPY5J/QMKHTW/23iTjvtD8xW1XK+CPs4SkjTcZf6ftcGxMrQAy5V/wnTtk TDD6HiHRVwFx4XXOXwLJuMKpEGVrWQnPLrts2vaYjFU/wn9esqyCznUnEo5H klHsbX/zt7gqkOrWE3KPIaOSwdD9g3+rIPYg79S7F2R8JlbG33G5GqJU0vJU 04j5Ziz89ntSA+F/h3fJlpJx00aXhCjOerhnbmfCOkvGc78CTxhmNMKrI91W FxaI741ByXJXIxQam7tWLJPxSKXOv0beJpjabfj4AYs8Pjl4pbHjUhNYKau3 iQrJY8/7XorExmbYMj9lu05dHnPsmOXu5rTAn1h/L/3z8hjEf7m091sbbOt+ +fZenTyuXO5Qn8z4DUKbHBfeN8hj8MXaU11tv4Hqu8mwvVkef29aDXrH8geS lb4MqHbKI+PmemVBiz8gdqlFvnpIHnVvqOnmMvfAkg+rCQeTAgaeKtLSVu2F Dl6dYiMpBXQbHdi9//JfiN3wPLPORAHt7px+vLe+H2amDVICDingbQWvwL6/ /XDk21TcjiMK2J2tH+A+2w8cxw+GppgpYFgPOfyc7AB4+Cw637ZUQL8B5YuX nQbA7Ocp1Y0OCsix93zACaZB4PMQSwr1V0DiclWopzgEjlD0bF+gAvKMN53/ tGUISrldIhbuKmBPaepnxQND4BVf+p/9fQU8m+Fxs+TiEIxWuNvsCldAbPZS 004fgh/SjXLUFwo4tbpttE1zGHyLwmJN8hWwWjJqTWHzCIz9LopcLFDA7OGt ZPU9I3ByeTr8TaECilx1iFh3aAQ0dE7fZy1SwOWWRN9B5xFoT1f2+vJDAXUr DEMTUkZgS8z3M+otCvhL3OreLGkUhi5PknnpCrh776fsrYxROPZgnUw+QwEp dyq01VdHoTD1FMlhSgEjp9pBhGcMnvR/FSydVUDXIsu5d0pjQCzkTN4rCnhj Q5aC0skxeHZAqW+MWxEjrgacXfg0BodkzFOq1yni4FTtZocr4xA7sKnojLIi pqdrNcbeGoeBTK4uqooi7jYnny8MGAcfLBQSVFfEcRNH846Ycci4oORzXEsR U34cqXf+Pg48WZNm3bsV0Z1NaI3GS4WKAw+ZJk4qYtudv3V3n1FBRMBRxtdC EYcNjb8sviLOJ217dYROK+LeH61959KpMOc0e2nLWUX0DbswxlJIBeVHVm03 7BRx9k+ZYnI3FQI71N6tXVHE0QwHZ3MZGux1rTgjfJ/oz38x1+8+pMFS+6hB WYgi6uunvWuNokHufr4tnqGKqEZekZR8QQNlqWNcvx8p4u1j9AeX0mnAV9aZ mxqtiOueHzloXkGDdqkJfkxWRCW/ufXbifNZRKDQ4lSKIq70rV8NZZkAE/qW gZQ3imjXbaTXzDUB38o8C3jTFTF0bvOSpvgEcf9idmjLVsS1Uu5+rY0T4FYu VuT6XRG3/DG5aHp6AtQ0t6crFitiUvfXg2nnJqD/mWV0SwnR/i67ugX7CTjp Fu+y84ciNq3IfnG4MgG7ZFSlOOoUkToi+TQsYAK4ruq5v+hURBYYsF9JnYCS rrNnj3YrYl+YbM2FzAnwNvAzYv2jiMpdRRxF2RMwIVNOdvqriN26/HyGBRPQ 8uNwjfaIIkqG/XyXUD0BCbK262pnFTF3o/XxOyMTsHOr+ibbeeJ9p0/rCseJ 541nti8sKGJWuRuNOkF83zPIZN2KIm4q4nRTmSXG15jm7sNCwe7VrCAmZjrw jFzzEWGjIKfq8rlWVjok/9sVlMpOwZquKIMEDjq0q/982sJFwW9Drwx4+OiA QdNFGoIUbGNXt9aXoENnXGF1iRAFT7UKTnyQpMP1D3dbLEQoOH5sNlFchg6p vaSRQHEKDv0K9MqRp4Ow3i7B3zIU1PAqNTRTpUPGMXapa3IU5FEp3XlajQ4G jvUUbnkKbtVUvWiuQQevKBudbRQKhox9cVuvSYc+euDZMFUKfhKjx2sQ53sf jsMOSmoU4v7PlVCvSwcJOdLVfHUKauZFUu120uHgwbeBg5so+MH5upfjbjoM 2Lg/9Nak4Nnti9/b9tDhjqderPAWCt7WNj+7k7g/5CTXpe/eRsHM9oG3Hfvo YPolOq9Zh4KB3rutJQ7QYajx3HcnXWL8ep+uG+jTQXptsilKj4JJn/Wb/AyJ +4b4l2713RT8xexOfmBEh8MagUPFeyiodGuiNdiYDgGWEku0vRR8NjwdbW1C B7JbD2vgfgr+Ox+UpGNKh89BqfzS+hQc4fBRZjlEB7MXV0lZBhTcxmNOKSZM /bBT0cCI+L3s0dNrh+kQVM2q3m1MQces1IdSR+ig8LdW292Egvnnpdk+EC6Y j9rDdYiCxYFnV+AoHU4InDOKP0zBe20jt4oIT6xTNdM+SkHn9Vq+2sfoEKw3 eab6GNHevrcCcYQpZgUXbI5T0IKjRG2ecKFjwJW5ExRUMxr7pW9Gh5O+prdC T1JwfcdDkWDCjCjxAIoFBZX9pPu/Eg7J+BP62ZKCPTOzxsOE15W+iT58hoJi XfeM2Y7ToV5jIbnbioKk+K3vxQjfjDb64GxNwUS5oo1ShClMT4sXzlEwKNS5 QphwjdPoz6DzFPSQd73zj2jvevOOP2J2FFThZjPvIUzeHUJ9ZU/BdzfsDuUS rnzduaR5kYIfHxQ73SbsLqTO/d2BiI/1sYydhGW9vEmHnChYO7ufRCXGV95f s77LmYKNeyszHhN2PSSr7eRCxFuG94o6YclPLvvmL1OwFbfb5BPzV6Lw9ejd KxQUPezhr0fYJYT/nOhVCl78Ot6WTcz/97PvvDdfpyD2Ca33JuLlWPHv/lcP CvZ/ZlasJ+IponUk1sSTguc03luIE77Ixsh18CLqZzo9wI/IDyFXLJ31puDA Z3XvZOK+m98W3hhwm4JnZio/FhL5xJeuNZHgR8ErlUks1UT+fRTzX9noT8Gf iWsS3w3oYHOniacwgBi/Y55nKpGvH45dU2kPouDjomPKFvvpcHoh10YojKi/ J9/c1Il6YLNlvxL/kILz8pcUSnfR4V2N+W2NcArqLt/SOKJHB+aEuadGTyho rBx1/gBRb6n6us1+T4n6fyJ+I0mLyNf3wX8FnlMwu3QH/NlMh2XJDnpcHDFf lI3X+TcR9UC9xZefQLRnxbXBmKj32Ygv+pMpFHTYkDO1jUKH+BXeE75vKLhI 8z8spkAHo4tWtvxvKWjJPrFuWI4OcTtW72zIoKCdb4WRsxQd9vXu+Xw+h4JZ QbbxQ4J0CN9Yotb0jYILfCc485cmIFX3DLd/EQX16vc4vpyfgO/7Z4a1Sih4 /57gaZ8ZYr21VE55XE7B1/8+LXAT66VJUIj8sVoi3oyXZwJ7J4Cjx0ysoYOC PgnCnvMlEyA3Nj7l20XBQ39OtXN/nwDt2cDGzb8pyHaZNCbwZQLseD89fNRL QV+wVp7MmYCi7TLcR4YpuMHkvbbEK2K9D+9frZuhYMKF25oDtycg4rlP9+05 ChqR+i9Tbk1A2mvxLxsXiPg8tvc8fn0C2gsNb4YtU9Bws+RsmPME6IymT5my KCFJiPHD/uQEMPZeH64RVMITbtViz9Un4OI0W2OVmhK+oY/ya9fSwDrtQL2V hhKu8Bb9t+sHDU6eD6yhb1TC7Svn4nYU0cDgJ+sPcS0lbCyKqRLOJfbbdJYv 57cr4e7bgbI7n9Ng0JYpZXG/EposHLe55EAD+6blmxvOKmEBu89H83kqWN3X u1ForYQirjzhM3QqnEDva0dslJCTsks1eIQKB94tXfawU8KAxwPRYR1UWBey aFvspIQvfpndFv5Chf6986aWN5TQ+O71kVVvKthmTyncf6SEmktLb11mx6GS 23ti92MldNG21xweH4eNtqyFUxFKGJf959GJvnFYEBE7ZRWthMw6/N2cP8fh 0fVtYZtfKKHbI/jU8GYcCrffXPyVpoSrAqOD4yfHQfzbaiPlhxIyPez+zyR9 DCqruQIKV5RQ7r34i3qNUXDZ/ze8ivDtgByaC2Ghwvz4VsIK707q8RK2yHQu YBAWE8irM1QfheFHtZPrVonxa1bXF20YBQ7zcJswwlc+n9qXozwK+3tIe6z/ KWFZIz89QoE4H55imFwivNP4aa8m4ZCGSsubhH+f1PhZLz8KTcW3PCIId+xy SuIhbJvclfGDsI/LHlV/uVHwc0qQ2bymhIYUiyQ36VFY1+e5YRfhYcbZCwKE K08f3W5M+MSNS+sypIj+mzIftyNcm/ni6bDkKCRssguJIVzKdc7Gmjiv7nuj F5tM2LriFc+KBNE/ebHX2YS3HZrIfkp4k3B5cQ1h2se4+RZxon/3X/xsJzwZ vxblTvgGy43fg4R9TjprCRGW9j48PkX4VU93VabYKHybVl5cI+yx69Q5E8L/ A4rxMaQ= "]]}}, AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948], Axes->True, AxesOrigin->{0, 0}, PlotRange->{{0, 10}, {0., 1.0656882767662574`}}, PlotRangeClipping->True, PlotRangePadding->{ Scaled[0.02], Scaled[0.02]}]], "Output", CellChangeTimes->{ 3.452695221293*^9, {3.452695331917*^9, 3.452695347226*^9}, 3.45269553834*^9, {3.452695756133*^9, 3.452695771025*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"FindMinimum", "[", RowBox[{ RowBox[{"myr", "[", "t", "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "1.5"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.452695413954*^9, 3.45269544351*^9}, {3.452695550396*^9, 3.452695555999*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"0.2076294285652612`", ",", RowBox[{"{", RowBox[{"t", "\[Rule]", "1.512157356808945`"}], "}"}]}], "}"}]], "Output",\ CellChangeTimes->{ 3.452695443921*^9, {3.4526955521809998`*^9, 3.452695556439*^9}}] }, Open ]], Cell["\<\ ... which agrees with the books assertion that the min is \"about .208\". The full motion is kind of crazy:\ \>", "Text", CellChangeTimes->{{3.45269545724*^9, 3.45269548149*^9}, {3.45269560217*^9, 3.452695609115*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"ParametricPlot", "[", RowBox[{ RowBox[{ RowBox[{"myr", "[", "t", "]"}], "*", RowBox[{"{", RowBox[{ RowBox[{"-", RowBox[{"Cos", "[", RowBox[{"my\[Theta]", "[", "t", "]"}], "]"}]}], ",", RowBox[{"-", RowBox[{"Sin", "[", RowBox[{"my\[Theta]", "[", "t", "]"}], "]"}]}]}], "}"}]}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "4"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.452695158586*^9, 3.452695200032*^9}, {3.452695377267*^9, 3.4526953776280003`*^9}}], Cell[BoxData[ GraphicsBox[{{}, {}, {Hue[0.67, 0.6, 0.6], LineBox[CompressedData[" 1:eJwl2Hc81e/7OPAiIakoRUYykpB3Eam4KCsrGmQlpKJhZmaPkIgKWZGMZGeP bnsc4RzzDOMcMs6hCIko39fn9/PPeTzPuF+v13Vf93Xdt6M2jlftmLZt2xbC vG3b/17//98i+n8vHfM1K5Jvn//bWkBKh9fZdZM+fnlNbVf+g/nJim1XnHHn l6O2T3fSMefczmDO1KB/uUHIPP4Bs0JG8LKjGjvac3T508t/C0jV5+tXaVVJ 9NVa+drFP9h4mb+Lr8FlpEPUj2aiL6CMlZX0PxfsUX+kY+q75AU0MlHGk2rx HM2z/7JIEV5ANT1Ryf5Cacg1yLmd3f0HCl0cHBWSzEOFphphBwK/I/HLQt6K jWWIf8z35C6reeTYWJ2QaVePjhzYnzVDYqCEfgFLAeEWFHObbJvEQkfUjWGV p/w4FJSaZWPzexplip603sGFR/eD74Z/+D6NeDKK+R4K4dHacKby+OQ0uvje aW1QCo+Y4xzUVPDTqIZPd6tIE49C2ci873On0f3rp3Y/fYpHJlm6upsm00hx u2+OwyweMT6MNPYXTyEB+bULRo0EpGW2sNJy+RuqqanbZe/Wj7Yp3gtsU/mG vHZOIe7AfmTtfVCzQe4bqox8c6f2RT8yuuhokyL4DfX3Z11iz+lH1GhHg9XF SfTk71jzG1I/ciKxi0jHT6JbpWYtd1QGEM5pzNqTPIFITH+3P9w5iFhxttLu PROoPCUYP8g9iBTuPfJ+0DSBwqsPPb9wZBDZhoZGXMibQO+G5723nR1E94gW PR5eE2i6P/TcbftBdIFDsF7/4AQ6blZj/bhrEFUTUhf0LtPQsW8FzApRQ+hl kHOx8wUa+nyXZ42cMITknoXEvJClIaezN9y9M4fQrTzu2DweGjJVCKnIrRlC J7uV/nnSqKgoWrMNzQ4hc35+axsPKhr6cSSi+dIw4t2we/j07Dj61hRzw2Vl GA0nW3S5C4yj5TN3DJW2hlEA7cCi5bZxFCvZ8+E3OxFlzRhbLLWPIZG04inT I0RkMj3rc9p4DJ0fNowouUxEN157t7M4jiIl78JXcSlEZLTrY+HHWArKdGAv UjxHQqlLccQXThRUup23wEudhGIOF9eYXqGg1fEj8fkGJNQ+nOGSu5uCcBEh d7/ZkFAmPUVdOIyMjK+kvomPJKETKkx3HruRUB+T/w3iEAmROYd5OoxIyM+a rs5FJaEjv5mq2WRJ6G/Tm3FFOgllRJzPVqET0fIP3Bn9DRIK8t5H97EgItmg WyJxQmQkPhnMKqcyjMzu3RA3siajxNHvo6t8w+iYR6LxZXsy+svnzJLxawgx Ez85SzmTkZh7/7Hc/CH0LlfHOj+AjOhCh8gBfEOIISJHf5tGRsHKLbfcVwZR mhFRlpxFRifOHl836h1Ec48rtZkKyIip/KFxbQg2700FJLZaMrq4/uag8Y8B 1BXSWDwzSEatLC15Ph0DSFmO0HprhIyivI5IPXs/gBKkwh/VTpCRbE8N7Yrx AOK6laottkBGPa28Azl1/eh5BA3/aScFFcn/cNt804/i7pqJamFxHEiW/6Hg 2I/Ozc2mtnJR0J4H8fcsj/ajg1bvOhwEKEihY8HHKaQPjVRorOv9R0E3rf4d eWvVhzLtdGI+ylPQhPss86dzfegA7/EuxlkKEnrT/TpykYCSHW/yH1ajoDVe X62bXQTkOpuTwapBQRfb3DT2ZhOQx+WOEqI2Bb0b3CGnZE5AGoFzemKGmB9G SOWfISBZbXGNtGsUFP7idCDbPgJaZk/J+GNMQcZiAWa+LXgkeeivmaklBWmu PWO8ScMjs+Tzwua3KUjO2n8xwROPdudbHlaxpaDDLw8+vSmNR3HvDntk36eg 8sFBrkM78eguB7uy1AMKulHz0scP9aJBiYB/MY8oqMrOnfu9Sy/y6It9MOBI QT63/m4kivei6O2f5Nacsfil9unZEnvQQkEcy4YrBdUlVR7e/rwHHQi7FzPy hIK0q8Kfuij3INtraX4pHhRUdv9iZOVCN2IV3x5wzouCnO7utMRndKMEv0Ko 8KYg8s6je+uvdaPGUF77vU8pyDB3T50vSzcSlt2Ro+FLQfsvBAZwV35FDxzC M2/6UdDJeTtvn/tf0XkDTjZtfwpaHmV0VPF9RdGXdH33B1CQO1doFA7XhYxO 6Xyqwfzm1up0sU8X2si5raQcSEFuMdc2Hkh3oUpOuZ4kzGVvd/3YGMEhjq7S vf2YfVdnv91+gUMunoKZU5hP8omwJivjEBPnpnQfZp2dxr553zuR8x4Ro0TM V3YlOsekdqK8pvxSRcxLmU/26uh3IrIt00oRdn2bC1vB5M0OdG8HoWIbZpUy wl/V/A60nLI7RRK7/x0lggW+5h2obOP7pRPY8xX0Fja/3NWBsiSW9Jmw52fZ WHLzrm5Hee2kmyU+2Px/27567n47uhqVy6SIxS/43QffvoPtiJTyYTbOk4KG bSzV1VrbkAurcU6LOwVdcHAMeebahozPhJG73Cjo09a0V9bRNux7IJjrQkHq uRk2yb2tiMT/ld0UqzuL7sZ+9r6t6EddhdIYlg+aoWl79km1orT1Dg1FLF+M 4/4axxBb0O97HG22WD6NdzBSZkJb0BZXq5GtHRbfU5fUxKjNaJV38RvpFgVd /lUoz/6iGa1cFDtkaE5BSTfqr3QoNSO+69fzEkwoqOEzt8pQXBN6rYpU32J1 blZ/WvXcpUbEE2UjZKZMQZ19U3unZhpQhEB6NlWRgvaVsn28G9WADpmVaUqf pqDoLc171I8I6fhGh+w7RkFidzJuLP38gk40N/3JOUJBvH2WC+VKX5B0TZ4S Ex8FPUrtfG7cUYcSLForNndRUJtt2cCZvXVoA31vSWamoPWJLze6btSiqPny 8xsbZBTzIntCcKIaCRwO+L02R0arb+bEQn9XoFOXuZ+OdZDRIc6qX0oXKpDG sao8R0RG7MKNqUUB5YjA3XilrpyMQq+mOkyylSH17b5OoRlYPb091p14sARl 26X/bXlCRmlXZhb8TYtR4PnA4SMOZKRVlZZjf7YIbcvW/vXfLTIa2zjwn9DF fHTyhbyRqwYZObKa/3QzykbD/q9z7faRkeebX9HhAx9Q1urtNismMho2V+gz NslEwn/tanatkJDuubuJ8hbpSJ1F75YT1g8EBcaTZtpeIxfrixt3E0hoSUxR obwzFlU2WygMhJFQVyfz0dDEF4jgsJ49/4SEaKKT7s8zAtHljWxdgWskNJDB 7Cxx3h5e953ytGEnIaFXwh0VbzxhCK+rxLNKRELErNQ6zSAI+qt1/9EEEf36 m/8uSjkatOc2RTlqiEj8RV6FPncyfLPc9XLmDhF1aFbtEXT6CGHC7E3+ucMo 7WOUl+G3PBgZle/KjRlGifl7TnLczIcvkzJqju7DSHdFqX1VuQjMDc3LCBeH 0d4D8hq3A0uhOiMqjTQ0hNz3l0aKKX0GurHtEhnr795cFR2Ri5+BVnJG0y1t CNlmaQycuFUOhx7+1Xp4Zwjx7D23JSdfBXmxfY+t5gYROYXNMXi2CpwKKt33 fB1E28sr5/RTqiHo2LYbhvmDaN2SMavCVAuf2tMFIx8MIgkcU+/vtnoo2FG4 u/XbAJLMvEy6kdAI4necI0iN/egwq3fd1lwjtFn9q3BK7kfo4xFtRdUmCIB7 f2Kx/dabK/Vv1Waa4O+1rW3O4v0omkNz2kq2BfqHhvLeBPah3XoW0gcCWmCn NxuHi0kfGmUhn1bCtwB+UtlxULoPpUUob2t73Ao1i3MH/xsiIEcZBYb3hzao ++03LSNKQNntKVMai21w91Gx5vlfWN0/ISxufb4dxNe8Gd1teHQwCM/v0dsO X/B6CUH38Yjw97rphx8dQDfRrDPU60VNfiUGT+Q7ocXJlL1+tgfRHKstXbw6 oX/h07BcSA8q/4NLHt2GgxMClNMXq7qRyqkRmYUdXWBuevjDjwNf0W3yj9a7 0AUXtt/X3SzuQp/+i6hj9uqCh7u54KxeF/qmeh6XQu8CLdGDh1QCcOhhnote H/oKO9xyG5LGOpCWzQTr5tJXWBb6HC3n1oGuFqmLHxfvhlthT1OZ2DvQboGW K0/CukGmmHNe+7921H2hbtP6Qg/8lWply3jSijznDmQK3usBz3zX3QdZWlGz ynvripc9wHiFHva8akHdvtnfH1B7gH6cJYarqBkFx5tw+u/sBRftExXp55ux uh7AMJfqhfhwCZan7U3IvS4RrJx7gXt7JcvR0UYk5phE8YnrhZ5jp38u3G1E hUU8fgalvTBTdduLZ7EB8Tx0/LH9Ry+Qo+o5nLY1oH+SVKv5HXiw3j3gYLwX oYX4mt7j3HiYsIv3iBP4goT6jGojT+CBV9aX6C5Xh84eSPrv0xk8HDhZZiKj XItUfjamc6viITjlxWyrRg3603ol/MtlPBycfXdRW78aUb0v3cm/igcVK6Hb ddeqkMW2h6UkMzw0xZD9TphWovh9DT8v2eDBw1TtaLJlBVI8X/N+9j4eFtWv 7T1kXY5koUqr9TEeMj8mmOTYliHObku7YVc8EDJ2Gt6w+4z+cW8ECXjiQYJ1 Iv7E3VIU3fbrRJwPHqQ9svefuluC7A6emTvjhwffkoXfLnbFqJ839gZLAB42 3NMCfrUXIsJHjW3/MLvwKDhkN+ej6Yr7YYKBeBA0vze050seOrlqG26NfR6M D4rQqMxFKztc8tux8Zadeyeti7JR9prQO/2n2PM5j0/4ZH9A65++7v+J3Y88 YU4kP+U9ejaq1FrkhgdvAXEqR1w6MiTsdn/miIcYh4IFe85URHtv/fOJPR5Y Isf6gn8lIlcOCw4vLB7aeqOFySOvkVVMXUY0Fi+xX4776E0v0cAOr5QyIzwc jzqcfNkoAt28c/PHrBYexPn2c3ILBKLGotg3ksp4KK4VSVHvdkYHxR+luJ3C g5fVep3XSQUkfXSOr0MMDx+0rmX/4ngMm6WPToscwuI9KnTf7qE/HGtb4mv7 2QspgX6rThwvYZXNoySruxdab7qKH/Z+DdFiIjwncnphbKrVtHM6Eczfvsni utYLtYwgH/uRdODgMQt8JN4LlgnKzTV7MuGAc81zq1898E7vsKIJZEHGesD4 +oseuH/6lFxx0kdweTT3JsW4B3wi/GnsrZ9gciLTFCfQA8u8R/fQ5wug9IBi 2peMbhAI6Q2XP1wC4jsyXwfe7gYXcpwqXrIUHuCuhXcIdMNIjNr1VsXP0GH7 n8fYi69A9opZ0jcoB+5eC+X8i19hyly6/aNxBdSupCZvrHSBXm5V7IJFJdiY zqsLX+0C9ZndU6021XCi0ODv6iYOEhsjeQRsaqA98KTM1Swc1KSvxs3cqoUj ny+2pvzoBPb6VwV3DeuhbNdK+MuYTghtvNc5qv4FghZe+nOd7ITxV3uUjBQQ jLRZHOqw7YBALq8J4rMGQP5cvd6lbWDC4znrqdcEsq/pGtqqbaC21pW/VNUE bbNGkh1drWBzzSr7kWgzbIgK6v6htED0rvoTx5eaQaBlFmpvt4BT0iUlbeMW +Bn+ZZ/eZDOYXNx6fqKqBZKEn61u0ZrAyea/Z7/dWsG2zzv5zC3MjBDx5t5W UBUWUrlNbASPreOb3MfbQOb5HXbU1gBGV2+6MBHaQBEVV7CfbwCVQ4L/go+2 Q7vi96F5MwSQ9TZT37EdrmxY0vB89UDg3sHhz9QBnE6nQ/et1IKBLKcKTqsD KPJMF/7hakBzMMhdLLIDfj8X21aZVg1rN0kcAZ0dYPWZv+2mYxW4bjak9rJ0 ggXzTW8m5Up4vm2Ldyd0Av188r9+1gr456hdd/BJJ4x2pVa6vPkMvV0a0m+G O0EU53or5GYpvPzqcH6OGQfDBlIlmnwl4LhPafu8NA5C3y8b1BUVwa3rgWlR V3Hw6USqNPV4ATTmTb1vcsMBv23jdY20PKg7/YIr5BUOBj+vNCvty4UZ6UG7 9kIcKMcIcHL6Z0H21ZGE4DYcTHBy6e2bew8se9aFi8k4sBL7btIVkA7nFdMX NBk4OHz7crKKZAr4/njiWrGKA88DIivv9RNgj9k1pvUtHPSGOukaOsYBf3f6 bfadXfDHOFAkT/A53JHkb5lg64LTvKorQaaBQA4JHArA3JyxdvC8/mM4x3ey ahzrW1XHliot0ozQE+8Yps2/OJgTL5xQ/+WBOjJDcH3LOJAVTXvC3hCKHKlJ 762nsc/faTEIQTHoprl0avYgDhJ2dHtMpb9GstKWBumNOKjXJtOovm+RzO25 bv08HOi0jjM1mqShT++HqgpjcPDhRIzfr/gMNPp49nSzM7YODp4uo21loquz B2afG+IgkwmdM7mbjRaLSzl3YfFO+cnDzobLRSNT4bawAwc8q2hn9olP6J5l qfxJUidUiLgJTT4rQDU2+jdJeZ2gP5pg5EwrQhoai6fUsb79LDXXLpC/BP1b Xi9xUe8EhkNiw1OHz8iKYCMhQ+iAQ3kGWjbRZehULK2+Oa4DWNXf3igqKkel Rx5dkDPqgG/VoeGz9ErkpHTobFVLO/SuUmsimKpRArNx6YhXOxj1XcvcPFSD uGtydq9JtYMw5ye2trN1qIq5XU7xWRuwHHrAGLhUjxpObFN4/F8bpHcr3Dqm +wVFNrj5Nw21Qkn0PyKzRAPqW7Yf+3G4FQStmpXyPjQgzT2N4mPYetPVius+ KdyI/hwa4Oa73gK3Jk6Iv9rXhA5/OGiVFdQM9+n2wtdDm9AxL423XAeb4WiC fmv57yYkH5R2fCu7CbqUEihHB5tRxCFcdGJjIwiNubTuU21BqufZAm10G4E6 qW1on9uCpo2yf9P7GmDcm1dG3LkVNV2LlZqsQEDYqH/nRGhF47hIZfHqL3Br aXP918k2tPmx6qtdZT20BRTsC6G1IUHvaP2HubUwaESc9VRoR8NSG96eyTUg 2Foy6R/ejtTiGiP3Pq+GTeXYs+9FsfNeMiHF8XYlUOr6JgsediCzBw9lgjQr 4Aq7s3F+aQfy5qv6WCpZDjmRWvsTfnWgHe8E44/tKoMH2rPH75/pRFktt0uY Z0tBy3bj1GGXTmQou8xwbC6B7iOPbuR+6kQ3b1w2jU0phtYriuraPDg0/z3o MV9sPhzjZKycuoxD7A4pf1Z25EFNy8Ipcy8cMhmtKRh1y4EHdr2WjVk45GCu tSRP+wDaPtcTbXpwKIpb85+5znsYryPXKS/j0G8rR6d71u9ApeDq7asHulB9 8cBZJckkKH3G3pt+qgvlFmlofp9/Ddtn5Had0O1C8d+T0kPMYuDukHzrwu0u 9CtnSntuMASu9+fYrLh0oS+V+sMcyW7wYbfsz3OBXYiZafg3IVkI7f7Jerz9 eRcqIdky1cY+QXe27G1T47pQqfaVPduuh6L6nc3pTW+6EH01T+PsaAyyUezy voDZ/ZoUc9ypN4jGbyfOFduFkuLy/5NzSULh6Kq9VkQX2tshutRf9A71lze9 m/btQksjR6um3N4jv4DKgtXHXYgi1zlXvTcLSUpd3fK36EKC7hwXeHJz0Glv oeDnml3o6c3ziznKeehB0cUsEZku1O3JtCmKz0cvuqN0ruzrQqr+1ruVrYrQ zwCHLv5FHLqpN3f5+q9iNCLW8yPuKw69Cutl7FEqRZkezFMIi/fwH7a1PO/P aHwt7GKhDw4pdLcvsdSUodO7N/7YG+DQSEC20b/VcrTfktliSxCHdgnwh7ie qkTXnXjYfBidiMBT9kvDvgoJ7hO7burdieZ6fdUi8DXoy0/30iHlTnTpmP8q 2qpFhWo6PY7/OlDScafMcal6VPNW00HGowM9/RD51ckbIba/N4Ify3ag19Oa 4jcNsHN49Fzgjql25OUgZ1Mz2oBeyc2cddBuRycsR37x/W5EHr58nSarbUjn 7jNrpsAm5ET0mxrMaEOTKjdcmdmbkVRZaUDDz1bEIa+kILW/BTEW3Sw23FvQ GbZgm5N721DyxtsqXux9y9Mmqb/C29Bf9W3b1QqakfGhk/Zvt7UjnfvPHy5Q mtD5ki+h1vR2tPRRcu2VUxMqL/E1fWrWgdqsvpwLYG5Cn2sONdzu7EBDbhuU B2KNSE23dNUjvRN9b4LV8NIGFJkrkfmBDYcehAYpSUIDUrYpjI9+jEPfljVa zHK/IAt8u2GcXBfS/56bmT1ag+z5Hl078/4r+vXaTPoLvRqJVTrdTVr9itiI Jfm9y1VIj9nvH067G+3sKqeoM1WiEfWTOrHT3ShUybqLtLMCyXoZLh4/3YPk qPz/dbGXo137KBsxPj2o+PGZ9qMcn9GK0/H5PtZeNEpnOV/OXorCzd/VlOr0 onRNjg/bWUtQPvl2vf3zXvR2rXHH+c0C1Jn9slOOGY8KHH9M32vMQmE8Hyy9 nfFo75+s2o7CTHRe75KTYyQeSRqo/ZpOykCUtFNlFu/xqOt1uf7keBKqOjeZ JdGLR0peRccMSuOR4mC+Pfs3PPL86JBqEhyHFnqWwud+4xH391t8xeohWD5F FVULEND8oQe7rjR7IJYbJZfyZAioU7LzgteCORpYvqT7QZmATvzhCf+ZYAtX FGZP5OkRkL3fiduXXnnDj6pJoy9mBJR53CQiMj8UCL3GSpP3CEgirZR8aU80 hGo9teV3IyCtx+eGTyq+gvNvFszs/QnIwPZN7Jm7CfBMc72kN4KA4nYmnt7+ Ohnaz49uN3pFQCZZFtdvNbyDn49lGAvJBPSzbf3Zi18ZYMYjvq8ok4DwZ6Wf Mu/6AAXybEKxeQTkF5sS6SWYDTuZIpoSiwmoerDtB1E2F+r55Io7ygkoq4Lp LYdaHuyKHkkRrSEgxRc5UpuG+aCJBNTy6wko+sLjtTdWhRAW3GZi10BA5Eox fxuFYjBjvpNq1ERAFgevLp+7UAJPH9b0ujYT0PHrZ7frqpWC2aerBT2YG9jj MyM0PsOiJuuvO5jdPs6J/9AuA95HCXZK2O/TmJZFnHTLQUYM6g2w8VXyuINZ 9StA8PX16hzs+pPODO98/UoQ6Zvh0cLuL1HiSKSJfhWckrR5IVuBnfcfkr22 61XD4aeyRNsSApKv0DLPvFwDru8c8qc/ERAj90CYgmYt6D16UV+TRUBjsovx ZWp1MHlAvG4iDXveLrbHhy7UA29C5A3rBAIqioiUMT3zBc7cfienFENAFaHi Z91kEAxxvNh8GEZAHH/HE05xN4CpUOjVracEtN8qwvA6ayMs/5Pi07lPQOn2 021/AhrB69jjKlZLAvLKMs3a/qcRdpGeqp00IqBVCSat44wm4DxcdLZBkYB4 itk8dU2b4RvHqd1CUgRkxnR5/2h7M/ivPNrcJkRA3rSsfZqZLWCr1S9zi4mA 5v4znTq8rxWs5v2sR5bxSFwuis30aSu0STXFz2D5zCf7dGL3tTb4qZZVnd+K R1LntBIC6tvArsXniUM5Hp2j4d+YSrSDyXel3roPeOSeyyJwdr0dWsorW44F 4dH7hvsXX73pBI7uPUEXz+LRlvcunMBGJ8T1l/ISxfAoyLTaYedtHNCKCtP2 ceERXb3Yck2iCyRGT4RN4HsRPafFyaD4KwQd/16tcgxbv/Lh/zLe98LPy69T RD93o97ndWwH/vUCbTX+hmxQN3qdP6rrIo2H68dCy0oNu5GN5FFOozA8lEl1 ekvNf0VL7SROXQUCvHf98ek/wa/osStVxPRZH3BLMvW02eOQYMXxD3zFfbDj 6epi6Ckcsn/j7xdF7IOwrKWBb2udaOb7M8vI4/0gc8k6PzKsE7n64nYttPSD bn+bcWlqB+J+kdiWuTAA9y/8aZOvb0NLR9zayw4OguzDzds13phT5wuclAdh PKWWa0WxDXGGhOyZixiE2hhZ+6iSVmRWkhz7W2gIysNUS55ntKCW+C35MaVh 4Lp0xu6pexOyjnsmSjQfBr+02p/y/zUh541DsyG+w6Alu02xgd6IvgpIMK2j YdAZlnx4w6IRhWtGzJmqEYErXfJmNlan287JqrgokOA6T5DJ1W11qNzxl+Pw NRJwBBlcCHKsRbd2SJX/diLBUc/EA6FjNSg4RjX4QR4Jvg2kynHWVaOStWqz k4fJcEHg7sd7TypRkUso4+V3Mrw6t/VPbLIUebDUh0jfGwGti/gFG8dMxLE0 sdfTZwSI4UkHP1m+R5wrG4TXMSOQTh/4zKaXgcptPOwuVY5AW+XsT96raYh+ kPmV345RKEll4iN3vEHe03ufVMePQtO4guEVLldECj9dq5Y7CrY4kmz6CTsE +d/epVePwoJjk+r6kCoq8GiKnKSMQoprm4lR3iPIv9xaGCY0Bs/sv72+bBgO REdqXFjKGPzhrnVkrKbAzdDYhtRPYxDi/y780Z53YFQzYP2yZgwIEuMypXHp YByQLsxOHIPp53x3+3LfQ4rWA3NDrnHI2dhrkELIhik7OSsX33FIeqHCCDtQ BM4OR2amhagQEr7H4khBJURnCyteEaPCT9bwg+pSVWDHcXLgoyQVjA8e8Tv/ sQr6WI9aKchTYXN09F7Gh2rg0fZ+WaRNhXGZu/6X3tbCl/u9rmcdqTCvKydR 5oXgJtejCIIrFfY/8phmrCAg5jH53PKkwg5xu/Q1xQYY+0Y1uB5IBbXpQ4cq Shug5OGDs6VxVGiOD/bnyGkE4hNLJ53PVHjBuge4nzcDr5R2s1QlFfZFxWeI YHWiw1XIkKmWCr+r6kf4mFug89j0/ZgmKoy27Cmu826B3vS/bVIEKvRGd+f0 3W+F0AKxk37zVIjIJrOdVW2H5RMtRnULVFBKJwWXeLfDt/At2uISFeKePTbk Km+H8OWYGOV1KoiEF3YES3bAZimJ/R4LDfjDagrluDrh7N9dZuwCNHD9r5St joKD8/eS9y0K0aDt8ZtMB74uiDdfQV+P0oBIl2j9Y9wFjTPZN+9L0ODQ2bB4 ROgCJtHONydO0wD4NMworV/h1NuE7jkNGrz4YPPKPLcHxtoUOM5r0+DlBYsg kW89oN/x4mGgDg0ko7dt7zrSC1F3vcaXDWhw6/fJqob4XiDcfGDtbEIDI3ve z+/U8LDHNkVu9C4N2Ec74jSlCKC1vOmefZ8GE1fJVoraBBDOCqTdc6ABEvjC w2dHgLd/f9/rf0SDX7nyWkVpBIjovDFs4EYD+u77+yW4+uAoPfgn+QkNqDqc Xu9k+gCq/HOsPGjw3KxlnE2nD9Kp/ULXvGkw5ttcXxjUBwK7yLWT/jTY29wq oLXcBz9YQUEhkAZP/S7Pae3th5po1erAIBqwZu4eVpLqB3NmPc4doTQYf3aP 56dNP3SE5l6vjKDBrN+ntwWEfiiPnPTpj6SBz8quL4rf+8FA4yIX/TkNBKuY j3xmG4DHAaKV26JpUC2vlOMLAyDGV68yGUuDcL8tDZv8AdDZ9bKzLo4GXbKr h3NbB2D5Tsr3l69o4AjEU7SxASgk2R6TfEODorIMORmuQSjiyDM9nIjF847l vPSTQZBaxBV0Yn4aFn+GM3oQxBV0Jl3f0oDg0TNKzR6EdUthnuokbP6ijprZ EAdBZiw8mDUVm9/0qAc954aAxunRF42ZstLPo3NtCEJF/DW50mjQsG6gXP1g CP4o+H5hfYeNN6S83yVlCD7YJfX5YNaxsu+sLRuCqf3C0nOYUwcVd69/HYIV Xqbe2nQaRF000jL4OwTboliOC2bQYMD5kdZdnmEQ2dkw5YV5Mq9n0llmGIhz fwvE32PxCz2y/Y7lMMgYFbS7YZZ5uRKh82QYrDcf3EGY90qRvoi9GIZklS51 rUws/vylF8rqhuG51dGEEMyLhskv7AeG4fHRpYh6zNvjfJO454fB2c5NbAlz II/JvSJmIhYv/CORDzR4YyGxBvxEsHou8sgAc/DBBcOW00RIlnCXdMc8yFz8 5IIOEeibuI9vMW9QHjrkWhNh+0fuX1WYzRUk5di8sN//d4p7APOrMnq/+Usi rPzbZJ3D7KhYZpiZQwRRVhj9izmlLaZg7AsRnk6WJu7OokGVY8gPziEinF3b qXYIs/2x5EOnvhNB8vXIiCBm7axRae0dJDDk6nARxjxzxfj0NX4SsP8yZD+C uYNnn/TV0yTo37uUzYf5lPFBIY3LJKhK4bu6D/Ngi9tu6dskELgtycWMueeK 4sZODxI4t7kzfmL3M8Di8GPwBQk0mXS/jWBe/cvLSPxAglXJyR0tmBfBcOlK LQl23nC6moP5VAE/xwYB63PVh/rDMK8wws4mzWJ9rm0pygazin6qn8wWCZgv 7Q07h9mryZ5WxkOG7PuOLXswSzSs3vlPmoytZ3aDcWw+GLIGnOkXyVDghDue j5nvltcYsykZ5r6/NHuCOXYpYMTckQyiVz3Jm9j8e49qOM4mk8HW73tEOWbr RPHfa5/JIEWwZ3PAnPkrqHCjiwwyQadj2rD8Enos3zC8QQaNXZVZjzH3bLSI leyngK/55xxuzDyDut1PpShgkTAZYYDla9bvnH8MMwpobR/xmMby2dGtKPaF CwVkfAcfemM+5czlKxJJgZ2stWHx2PqoDFB0Fq2mANfVCbn0FCxej6Ytgg6O QAAM7+fDrF6Uo1olMwLatwxko5NpMHJKIYuqPgIO6rtOP8LWo71C9w521xEQ Fhw9sJVAA/3G3x+rekZgx/U7/maYd16+JBw8NQIa7aXGpfHY9e7ogermCCzN WNqaYPXA+Yrd8zjJUZCvtk30wOrHqmHbVELwKHRHL+6uxOpL5RlR6bW3o7BX I0xg+SWWf6LxF3WLR6FVhUfUOgar90cVljtHRiHN1OIeTxTm10xzkmfGINZV 230yhAbv+FPue06OQUVp4dGFYBokcj1ZYl0fAzjrVPcbq4/fzX8lRu4Zh1n/ iv6tABowz2cb2CuNw++bB7nmn9Lg3Hu58ajocYhy55o5i9XrQOVEg7P8VFDf rDrM74o9/zlcgr04FSrJSxl/nLH+xWHS81qWCg+D0mtLHLH8Yv3DQbpEhTdR HS+2PcDy6ZHZDr6HVHgPi6Ji1tj8LRNrPbE++5LVz2nUCqs33ZGDl1upcGxS 58rrWzTo//Hl74FeKpwLuNb3x5wGCuoo8O0EFSgZgdc+GdPA9o/KS3N2GtQL kUxSdbF8DBgMar2B5bG3r+sprJ9lVF9zPYuNmyUgZ9qM9bsDWZTYLKxP7bX5 HkPF+mGzfvHtxz5YH+LmYtmuSoM0+1zvGiyP+Pc39IzI/W++HNRVGDQoHChr O3CYBr8tvf5t85qA4szTf0J4aXBWvXzrWOAESAgbvVo+iK1vooKodsQEPOFZ +de1HxsvO2nI5+0EdF4rWLXnpAHTTDTPx+oJaNH7KmexDbtf32hxgfUJmLYS TpaYogKd8xPlkdsk7Ljw6c79SSo0SIw3/PGZBN9zA4QcGhWMVt4uBQVPggHT eqrIGBVyj75TDY2bhBodJMM2jO2vhiM/GhVPwrFK5g+vO6jwheLUsGsO87cx qS95VNjyUjTeafkN9E4oWhbkUuHX0tWcN3e+Qe6Lof+Ss6lgmVWQeuThN3Bx ON/m8h4bf67bX9TnG9yTkxrhSqLClcfH1eYSv0EwUyBwRVAhQfy/BZ/+b9Du EZ0rfJcKpXNS9p/Up6BUn91zxpYK23+bsMXpTsFo3qZQgTW2HxpP8nC9OgWH AzTX5S2x/RCPfdex21Nw6+5nNsXrVLhT+dfhis8UhDmVDrJcxK6X25nysXgK run64oYFqOAcbicSeWgapudfO344TIXE2dUPNoLToB8nfdaJlwp3jxg1K4hO A/t+iiLLASyeTzaGCCengfvS+G5xDir8lVBy7deYhiXJqK9eS+OwxnVag8V1 GjioIgU5H8dhzrvymiRuGg692hbMmTUOfP+Mt9x7p0HUoozsnD4Oy3z7bRoG pmHfuuFBhYRxuMvPa6g5Pg2nTjbO54eMQ++P/G9CK9OQz3Im6IbVOOQlmqYz C85A5GLDudT941BaegulP5gBp5+vPH9g605jb1eHgdMMBLO/c1fZNQ7OtkFd a24zcMSgqIq8bRx8P4UiVb8ZyCS88trxYwzED+a4pb+cAUGlj3uFOsbgPPfH nHdlMzC99RNcfMag2PNIecb6DHD75Kyluo/BusMvKfzfGWAu30zscB6Dbf5J xX+2z4JNLsebw/fH4NH9+qlLu2ahYO/epJLrY/Dd9uy+nMOzwJvxrjtfZgwW TcYZxedmQS/2wKXxsVHQ6eBsHvKYBTE/jd++pFGQ5ZgxI/rMQqfNUjj/wChk 8jDZDvrPwpXbBZevdY5CBqWprfHZLPinddiUfx6FCI/iBdsE7Pqml7fUw0ch qErGqqd8Fjxyo76//W8UeubDrqkuzIJyn2L8QalRCBnVhpKlWdjd2qsaJz4K SxZaxYKrs/CWb+ld2OFRUPZ1FqNvzgLVif7eFjsvvfycdldnFx1uBsWnLQyP QJTER5KCGB1iPfjVjP1GwOW5aWLtdTqcoJt+4PEagWsSuX8emtDh2GUdjQGs jlvc+x522IwOVUmil67Yj8DrZ+JHHKzoUK5bf/HC9RHw/VFjRrGnw+nZfz+W JEegJLf89k4/OiTuca4bHqDAxr+x7MBMzBUG0gG9FCgIbBOqzqLDbJv9ggSO Aod01Xrmc+iweWfI+QmigMFK1KB2Ph1K5gc92fIoIFpd9LK/jA6qkxGH9vpT AIVkc8m30oFvtrLs/XEKlMQX/xSdpAPH48DtTSIUiI8NCGebokNM+IW3VAEK EMmltbPTdLiuq53Pz0UB60O868kMOoiSu10j1slQFdE00v2TDkEeH4XUcWTQ uSdbYLVFB65H7g9utpChOaRP9d82Bry+QuZ8+IUMIY/3eSYyMUDsysqN2FIy /BYV7EcsDPCIFWMZSML2ARdztcgcDHg0kuJw7gEZWmxqI1UOMbBz7Ki5uh0Z krz6nifwMmCI/jpf34oM4uNmivN8DDBjaX186xoZ8Kb3Y8IFGJDlnxLmeR47 P1/dMnhxlAGrZiwhvmfIoF4oXdQswoBCyZdTgbJk+Pt2f/0vUQYccbs+HC5K Br/ET5N6xxigFecYGsOB7SPU4F77CQbcr2j5HMNCBpdEd1WKFAPWNmt1YrB9 TWHUWxxDmgE8oS1lUcskcJ+UX948iY1fHRQeQCGBU3Vx+vfTDBgn9mbrfSLB i0mR5m3nGGA52LP6MYsEEtPP1JswO/op4Xamk+C0swt7wHkGnGf/u/nlNQk4 d+61WrzAgJcNfamSfiQw3k5S+wDY+CvtB4I9SfC8f2xGS5UBw62u3CMuJIhv J/TPYNa/HpUZdY8EdY8K/QQuMoDbcUVi1pAEenKBx03VGWC3ODCurEsCG4X/ do5j7mdLPvpKAxuPW1zaWoMBb13vqV84R4J6C7arZprY83vIH3spTwKScuRV AuYkd7dXkydJEDHmmaKuxYC9ekNcEaIkaO5u4BTWZoBPXK0oRZAE91s/nQnH PFOe1CTNSwL53mO585iX93Xm9uwmQebm90d5lxmw/TlxUYiVBBYsuEZmHQYI 37dpf7ydBI2MG9ammI8+sz7KuUoE/qKkkHXM4ouvsswXiSBcTmNT12VABkdt 9UcGEa5Mr5AiMXN7yd/6/Y0IBl8Pr3VjFvCwfKU+ToS/1W9tOfWwz7/FmcaS iPChq1LgMuYOlVMlo/1EODBTIxqIeYTIe/BJBxFCW8IEpjHTTvOJNTQRIUeD vGe/PgOMihdbdtUTwfTKccMLmEW/r69dryRCbl051RrzkHhMQ1oJEebl2huC MU8p7hKY/USE5WMZixmYce9bOE5lE4G3/5lTHeaD4YdeeqUTIVINqfZjfpdv /rExiQg6Ws+tpzEHeNJN2N8QwTdUtv8X5uNrvCmGMUR4Yr/wZrsBAyJG1NwT IojwQ3pPHjvmWp/a6dFgIpwf6tm7B/MIYeS7qB8Wn7Nu7f/zvsDN5/aeRNi/ rErgwBzf59RY6EKEu64eMiyYqRUx0csPiXBv2XTkD3a9022Fy4r3iBDrfWJs HrOCmuhPH+zcMz16Qo6COZbr4jNkTgSuhhRKK+YDOKsKJmMi5MU2DudjLs/B +2kYEkEhlizxEvPN2FnqM+wcFdZ1/Ksj5hHejZFOdSJU8n9v0cVs72PuygFE eCh1n1sMc+n3K972SkSoF/BPWMfmw/6hiHC7HBFsz6jEd2E+7ytsJ3aSCJNk 2p4kzGU3Ey4HHidCo0b2djvM1+b6u0dFiBAU2P5YGnP+oMiSkiD2/YeBxotY fuyoOi7wk4sIiz8K4h9hrnX1ENHbTYQEu4SFY5j37mHDZe8kgo0ea8sIln/D /Pz/zDaG4aDOXaoy5sp+GSb0bRh4ciO/D2L53ddfevjQ+DDouXFLumLeHY8b ekwahpHdK7TdmE9bDp4T7BmGbaKfvp7B1s/8ceWaJ5XDoHtUJsoSW3/4B/Ab VzIMF6s7qsaw9ekttZ98JH8YTg+G3bHAXK91Oa4jfRjukXXP6V9iwLfNhAGe yGEwHOWd2qGGra+DeK37IcMw7UPf54DVA/7drLY1fsMQsJJU34XVj1eKwXGW rsPwJvFpUogKA2y2i4y+sxgGWZrC/a9Y/eE3aEhYMB6GmykcGTz/s773porR MPy0fWJhjtWrM/90Wkc0hkFHddx85CyWn32vzA/IDkMUZSqo7Ay23uU1H1tL YvdTqTxEk2dATKuwTKHoMETuGizgwGw+cqVYk3cYRL1eM65j9ZHyO7bfdfsw 2E3tSa/H6qc+1LV86R+Chapzx2exegxeq6WR3UNgbPhWuF2cAcQaqUvG7UPQ ohTL/F6MAV6edTrzNUOw+ko/Uher7/Ra0r/974cg0fFnxiNBBogkHFG+6jQE +EyOUz+4GZCnn/WY12EI3uZ9XHzOxQChHUTNMdsh6KB/TRPfxwCnl3fn75sM gbJ7QrseJwOmNXce8FYZguB7NCMHVqxeSurtj9o9BD/mhnwn/mD7A9adBL2d Q1C1tfsirNPBo6hfiXPbEOjxhq4l/KbDoHYN54vlQdh+S0xZeQXrjzw8beHk QVivcJMy+Y71X0vJz065gzD+zrOnY4wOF0yGL+5QH4QgScGKD1/oEC7frXRM ZRCqT7lPfaijQ7V543ets4NwIWo+LqOGDh/z772IlB6E+JzR+y8rsP2H1Vrx bp5BmP8byKJeSIcylrXSHdMDEHdp9RhzKh28lPhnW54NwCWDApkJDzq8vipB HgscwMbd7eLyhA7+LOvRaz4DkFlSprTlgu1vPJLVpJwG4PMN/2t7HtPhq0FC 74ubA6AceLmc6Q4dyBJfPmtJDsDC86Zr6lfoQOl4jmJw/XBszJ1VWpQO3sLW dqL/+39+rO7NvcLY9R/ZdFbU94NlgVX8DwE68K8Wfh0p7oeU2X/k9IN0+MmF bxVP6AfVwCPPB9ix/Yzfu/xPtv1grZlf6Int577vuREfvtkHGlwPzpVXzsL8 yfqklzJ9MFLDWcN2YRYy6+zc5CT6wNNw9s+G4ixweajyDQn3gfIn6s05uVlY vCvcevhAHyhM1T38IjULnFnRHzP+EGDfTcG1M/yzwCI51pLVToATjzbqHLD9 Lx/P5wvh1gTQWWJbnSudAeOe98/qX+OhkX4m5vrhGUiy+eXTFI0Hpczc59k8 M2AqGnO5PRwPW6ElMr/2zcBL9zUn/FM88I7QbcJZse8XwjL1Dh4UWApfh2H7 +99VGuQ/8nhIEheQmuiehvnDldnieb2wT/V090G/aVgo4wguluyBA36JTD4D U/D7cou45zcc2P8rofxUwM5PuLAL7P3tYPXtZmvu0wm4eeLq34XuFui6ePpc djwV9HcL7Twc0gj/UYUW8G9HwLyj78nqeB0c4pzQvT4zDMZduBQ4WQFHCxbG tm72wfnMXQ9cZCqgesrI0QCz+BsFpSzpCrhttvA3xaQPWtquDu6SqgCN3xSe c8Z9kGGasntIogI8//j853atD+zZ3TweHq0Aka2hfbP6ffCIJqfy9kAFSLT+ 9u25iI0Xvsr+dX8FaEs2Dgpg5lPLGvzHXQG9za1SD9T6wJbe9vAOVwU8Sg7o Y1XtA2EO/iTZPRVA7q3bpabcB6GmO3+2slZApNiVY58V+0CENbJ2bWcFfCZ1 WW/HzN28HiqF+UZTcPIVhT5wv5N6OHZHBTjJZLHPy/fBgsavixbbK6DRrhAv droP6ixJu2O2VcBAyY5/Lqf6IDXw41DjVjn8kY2XbPyvD17NczpI/CuH940O PpayfcAv917O7G85tA0UZHw62Qd+ISJ/ozbLQYZPum0dy8PX1Og2tFEOxdFT s1qYj2vNxCz9KQeJM2T2eOk++D/ZXIC3 "]]}}, Axes->True, AxesOrigin->{0, 0}, PlotRange->{{-1., 0.5514070741487931}, {-0.42980328492365694`, 0.822028531607211}}, PlotRangeClipping->True, PlotRangePadding->{ Scaled[0.02], Scaled[0.02]}]], "Output", CellChangeTimes->{{3.452695191243*^9, 3.4526952061809998`*^9}, 3.4526953779379997`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Animate", "[", RowBox[{ RowBox[{"draw", "[", RowBox[{ RowBox[{"myr", "[", "t", "]"}], ",", RowBox[{"my\[Theta]", "[", "t", "]"}]}], "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "100"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.4526956107390003`*^9, 3.452695651148*^9}, { 3.45269578115*^9, 3.45269578196*^9}}], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`t$$ = 8.233621221100439, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[$CellContext`t$$], 0, 100}}, Typeset`size$$ = {360., {178., 182.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`t$6358$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`t$$ = 0}, "ControllerVariables" :> { Hold[$CellContext`t$$, $CellContext`t$6358$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> $CellContext`draw[ $CellContext`myr[$CellContext`t$$], $CellContext`my\[Theta][$CellContext`t$$]], "Specifications" :> {{$CellContext`t$$, 0, 100, AppearanceElements -> { "ProgressSlider", "PlayPauseButton", "FasterSlowerButtons", "DirectionButton"}}}, "Options" :> { ControlType -> Animator, AppearanceElements -> None, SynchronousUpdating -> True, ShrinkingDelay -> 10.}, "DefaultOptions" :> {}], ImageSizeCache->{408., {216., 221.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{{3.452695630928*^9, 3.45269565156*^9}, { 3.4526956893459997`*^9, 3.452695690818*^9}, {3.4526957822720003`*^9, 3.452695787651*^9}, 3.45269586024*^9}] }, Open ]], Cell["To make an animated GIF, we create a list of frames:", "Text", CellChangeTimes->{{3.45269659245*^9, 3.4526966102200003`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"frames", " ", "=", " ", RowBox[{"Table", "[", RowBox[{ RowBox[{"draw", "[", RowBox[{ RowBox[{"myr", "[", "t", "]"}], ",", RowBox[{"my\[Theta]", "[", "t", "]"}]}], "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "20", ",", ".1"}], "}"}]}], "]"}]}], ";"}]], "Input", CellChangeTimes->{{3.452696611782*^9, 3.452696665755*^9}}], Cell["And export to a file, whcih we then can post on the webpage:", "Text", CellChangeTimes->{{3.4526971417720003`*^9, 3.452697155105*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Export", "[", RowBox[{"\"\\"", ",", "frames"}], "]"}]], "Input", CellChangeTimes->{{3.4526966696470003`*^9, 3.4526967019560003`*^9}}], Cell[BoxData["\<\"coffeecup.gif\"\>"], "Output", CellChangeTimes->{3.452696762862*^9}] }, Open ]] }, Open ]] }, WindowSize->{678, 357}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, ShowSelection->True, FrontEndVersion->"6.0 for Microsoft Windows (32-bit) (May 21, 2008)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[590, 23, 94, 1, 71, "Section"], Cell[687, 26, 123, 1, 29, "Text"], Cell[CellGroupData[{ Cell[835, 31, 231, 6, 31, "Input"], Cell[1069, 39, 158, 4, 30, "Output"] }, Open ]], Cell[1242, 46, 232, 5, 29, "Text"], Cell[CellGroupData[{ Cell[1499, 55, 217, 5, 31, "Input"], Cell[1719, 62, 541, 11, 39, "Message"], Cell[2263, 75, 291, 8, 30, "Output"] }, Open ]], Cell[2569, 86, 108, 1, 29, "Text"], Cell[CellGroupData[{ Cell[2702, 91, 195, 5, 31, "Input"], Cell[2900, 98, 84, 2, 30, "Output"] }, Open ]], Cell[2999, 103, 151, 3, 29, "Text"], Cell[CellGroupData[{ Cell[3175, 110, 327, 10, 31, "Input"], Cell[3505, 122, 243, 7, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[3785, 134, 149, 3, 31, "Input"], Cell[3937, 139, 255, 7, 32, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[4229, 151, 324, 8, 31, "Input"], Cell[4556, 161, 84, 2, 30, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[4689, 169, 88, 1, 71, "Section"], Cell[4780, 172, 436, 9, 65, "Text"], Cell[5219, 183, 111, 1, 29, "Text"], Cell[5333, 186, 2138, 61, 112, "Input"], Cell[CellGroupData[{ Cell[7496, 251, 265, 6, 31, "Input"], Cell[7764, 259, 449, 11, 375, "Output"] }, Open ]], Cell[8228, 273, 222, 4, 29, "Text"], Cell[CellGroupData[{ Cell[8475, 281, 1104, 33, 52, "Input"], Cell[9582, 316, 968, 31, 45, "Output"] }, Open ]], Cell[10565, 350, 117, 1, 29, "Text"], Cell[CellGroupData[{ Cell[10707, 355, 430, 13, 31, "Input"], Cell[11140, 370, 767, 21, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[11944, 396, 422, 13, 31, "Input"], Cell[12369, 411, 754, 23, 30, "Output"] }, Open ]], Cell[13138, 437, 128, 1, 29, "Text"], Cell[CellGroupData[{ Cell[13291, 442, 338, 9, 31, "Input"], Cell[13632, 453, 277, 8, 45, "Output"] }, Open ]], Cell[13924, 464, 108, 1, 29, "Text"], Cell[CellGroupData[{ Cell[14057, 469, 889, 25, 52, "Input"], Cell[14949, 496, 882, 29, 50, "Output"] }, Open ]], Cell[15846, 528, 103, 1, 29, "Text"], Cell[CellGroupData[{ Cell[15974, 533, 472, 13, 52, "Input"], Cell[16449, 548, 329, 11, 30, "Output"], Cell[16781, 561, 334, 11, 30, "Output"] }, Open ]], Cell[17130, 575, 126, 1, 29, "Text"], Cell[CellGroupData[{ Cell[17281, 580, 367, 8, 31, "Input"], Cell[17651, 590, 18617, 312, 243, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[36305, 907, 271, 7, 31, "Input"], Cell[36579, 916, 252, 7, 30, "Output"] }, Open ]], Cell[36846, 926, 231, 5, 47, "Text"], Cell[CellGroupData[{ Cell[37102, 935, 564, 16, 31, "Input"], Cell[37669, 953, 18934, 317, 304, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[56640, 1275, 380, 10, 31, "Input"], Cell[57023, 1287, 1987, 41, 454, "Output"] }, Open ]], Cell[59025, 1331, 132, 1, 29, "Text"], Cell[59160, 1334, 416, 12, 31, "Input"], Cell[59579, 1348, 141, 1, 29, "Text"], Cell[CellGroupData[{ Cell[59745, 1353, 177, 3, 31, "Input"], Cell[59925, 1358, 87, 1, 30, "Output"] }, Open ]] }, Open ]] } ] *) (* End of internal cache information *)