(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 12.1' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 109340, 2266] NotebookOptionsPosition[ 105698, 2195] NotebookOutlinePosition[ 106096, 2211] CellTagsIndexPosition[ 106053, 2208] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Quantum Signal Processing", "Title", CellChangeTimes->{{3.8547894062431803`*^9, 3.854789411276634*^9}},ExpressionUUID->"93f1078a-c17d-4647-8ecc-\ 253a2973747c"], Cell["\<\ QuIC Seminar, March 1st, 2022 By Jacob Watkins\ \>", "Subtitle", CellChangeTimes->{{3.8547894155496407`*^9, 3.85478943762259*^9}},ExpressionUUID->"4868ce54-430c-4a3c-99ec-\ d2b5daa59e7c"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Creating QSP Polynomials ", "Section"], Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["P", "TI"], ",", StyleBox["Q", "TI"]}], TraditionalForm], "errors" -> {}, "input" -> "P,Q", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "eca4dc07-761f-462e-a89d-c92f08cf1ad7"], StyleBox[" from angles ", "Section"], Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ OverscriptBox["\[Phi]", "\[RightVector]"], TraditionalForm], "errors" -> {}, "input" -> "\\vec{\\phi}", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "4d64bb61-a913-4f63-bfd8-a6b85a021794"], StyleBox[" ", "Section"] }], "Section", CellChangeTimes->{{3.802024486342704*^9, 3.80202449393602*^9}, { 3.8547894698355217`*^9, 3.854789517513175*^9}, {3.8547895606341867`*^9, 3.854789572257175*^9}, {3.854789646403391*^9, 3.8547896914125757`*^9}, { 3.8547897305984364`*^9, 3.854789743257827*^9}, {3.854789794492003*^9, 3.854789901778776*^9}, {3.854795319145369*^9, 3.85479531959656*^9}, { 3.854795423856771*^9, 3.8547954249187307`*^9}, {3.8547957469184113`*^9, 3.8547957580285788`*^9}},ExpressionUUID->"a28626c4-cf14-472c-a780-\ 9af7fed69c80"], Cell[CellGroupData[{ Cell[TextData[{ "Define signal-rotation operator ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["W", "TI"], "(", StyleBox["a", "TI"], ")"}], TraditionalForm], "errors" -> {}, "input" -> "W(a)", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "9be3ce74-e48a-4eae-9e46-8ba826982936"], " and signal-processing rotation ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["S", "TI"], "(", "\[Phi]", ")"}], TraditionalForm], "errors" -> {}, "input" -> "S(\\phi)", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "4baba047-a76e-446b-b433-5b2818357ed0"] }], "Subsubsection", CellChangeTimes->{{3.8547957141158047`*^9, 3.854795729434824*^9}},ExpressionUUID->"792f85c3-5d7d-4453-ba0a-\ 30b355101975"], Cell[BoxData[{ StyleBox[ RowBox[{ RowBox[{"W", "[", "a_", "]"}], ":=", RowBox[{"(", GridBox[{ {"a", RowBox[{"I", " ", SqrtBox[ RowBox[{"1", "-", SuperscriptBox["a", "2"]}]]}]}, { RowBox[{"I", SqrtBox[ RowBox[{"1", "-", SuperscriptBox["a", "2"]}]]}], "a"} }], ")"}]}], "Code", FontSize->12, FontColor->GrayLevel[0]], "\[LineSeparator]", StyleBox[ RowBox[{ RowBox[{"S", "[", "\[Phi]_", "]"}], ":=", RowBox[{"(", GridBox[{ { SuperscriptBox["\[ExponentialE]", RowBox[{"I", " ", "\[Phi]"}]], "0"}, {"0", SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"-", "I"}], " ", "\[Phi]"}]]} }], StyleBox[")", "Code", FontSize->12, FontColor->GrayLevel[0]]}]}], "Code", FontSize->12, FontColor->GrayLevel[0]]}], "Input", CellLabel->"In[45]:=",ExpressionUUID->"f694f58b-9327-4951-8079-6c38c7f65c05"], Cell[TextData[{ StyleBox["Choose signal angles ", "Subsubsection"], StyleBox[Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ OverscriptBox["\[Phi]", "\[RightVector]"], TraditionalForm], "errors" -> {}, "input" -> "\\vec{\\phi}", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]], "Subsubsection", ExpressionUUID->"eced5f5b-da17-424b-86ee-25418fec7de0"], "Subsubsection"] }], "Text", CellChangeTimes->{{3.854795498243672*^9, 3.854795510565425*^9}},ExpressionUUID->"b99515c8-5824-4aa7-a91f-\ c486424de483"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[Phi]", "=", RowBox[{"\[Pi]", "*", RowBox[{"{", RowBox[{"0", ",", "0", ",", "0", ",", "0", ",", "0"}], "}"}]}]}]], "Input",\ CellChangeTimes->{{3.802024497633624*^9, 3.8020245418604517`*^9}, { 3.802084095361841*^9, 3.802084108653825*^9}, {3.8020841409745407`*^9, 3.802084161670167*^9}, 3.802084212501238*^9, {3.802084244652742*^9, 3.802084278020651*^9}, {3.802084423094756*^9, 3.8020846298402977`*^9}, { 3.8020849581323833`*^9, 3.802084994306072*^9}, {3.802085050767692*^9, 3.802085163990142*^9}, {3.802085202584613*^9, 3.802085251665312*^9}, 3.802085307377961*^9, {3.8020853486829643`*^9, 3.802085352330017*^9}, { 3.802085412982503*^9, 3.802085413507552*^9}, {3.8020862949066896`*^9, 3.802086378522773*^9}, {3.802086419758582*^9, 3.802086420467738*^9}, { 3.8020864521444483`*^9, 3.802086455710369*^9}, {3.8020864973077593`*^9, 3.802086542077324*^9}, {3.802086599556776*^9, 3.802086805617375*^9}, 3.80208783340549*^9, {3.8020879041098413`*^9, 3.802088142007761*^9}, { 3.802088178651374*^9, 3.80208822518908*^9}, {3.802088894055563*^9, 3.8020888982080603`*^9}, {3.8020889340426083`*^9, 3.802088951494369*^9}, { 3.802089002146434*^9, 3.80208900393335*^9}, {3.80208915422486*^9, 3.802089176573462*^9}, {3.8020895319809847`*^9, 3.802089546420405*^9}, { 3.802089626200128*^9, 3.802089990969708*^9}, {3.802090049382504*^9, 3.802090174649485*^9}, {3.802090217734236*^9, 3.802090233521098*^9}, { 3.8020902703924093`*^9, 3.802090351054804*^9}, 3.802090825928948*^9, 3.802090919018063*^9, {3.802091037601871*^9, 3.802091136133688*^9}, { 3.8020912205292892`*^9, 3.8020912572495728`*^9}, {3.802091496735574*^9, 3.802091497836735*^9}, {3.8020916303004417`*^9, 3.8020916542671137`*^9}, { 3.802091708229781*^9, 3.802091710306443*^9}, {3.802091754425097*^9, 3.802091788472343*^9}, {3.8020919239265757`*^9, 3.802091956125485*^9}, { 3.80209201247558*^9, 3.80209208120962*^9}, 3.8020921637950983`*^9, { 3.8020922289969063`*^9, 3.802092264054653*^9}, 3.802092316719646*^9, { 3.8024577879505*^9, 3.802457866582061*^9}, {3.854789904145109*^9, 3.854789949024129*^9}, {3.854790059546473*^9, 3.8547900764804077`*^9}, { 3.854790483284277*^9, 3.85479051286814*^9}, {3.854790755498229*^9, 3.85479076683178*^9}, {3.854792495132019*^9, 3.8547925031797037`*^9}, { 3.854792743694083*^9, 3.8547927587807493`*^9}, {3.854792845654024*^9, 3.854792873144697*^9}, {3.854795516971635*^9, 3.854795525524777*^9}, { 3.854795800553784*^9, 3.854795841521721*^9}, {3.8547958734325323`*^9, 3.85479588815247*^9}, {3.8547994234846373`*^9, 3.854799424765991*^9}, { 3.8547994803245573`*^9, 3.854799482629807*^9}, {3.854799799850148*^9, 3.8547998009259243`*^9}, {3.8548118695409107`*^9, 3.854811889621092*^9}, { 3.854812021347793*^9, 3.854812023772965*^9}, {3.854812084826228*^9, 3.854812086045744*^9}, {3.85481251620217*^9, 3.854812525085142*^9}, { 3.855140049830862*^9, 3.855140077639813*^9}, {3.8551401893351583`*^9, 3.8551402011465693`*^9}, {3.855140389813345*^9, 3.855140402583091*^9}, { 3.855140470126319*^9, 3.8551404716194887`*^9}}, CellLabel->"In[23]:=",ExpressionUUID->"51015464-2f55-489a-88da-cb1da7045c6d"], Cell[BoxData[ RowBox[{"{", RowBox[{"0", ",", "0", ",", "0", ",", "0", ",", "0"}], "}"}]], "Output", CellChangeTimes->{3.855140201445368*^9, 3.855140285214162*^9, 3.855140403191189*^9, 3.855140474212027*^9}, CellLabel->"Out[23]=",ExpressionUUID->"66d28cf7-c24f-4705-a371-f3084bc3c417"] }, Open ]], Cell[BoxData[""], "Input", CellChangeTimes->{{3.8551401724416027`*^9, 3.855140174616989*^9}}, CellLabel->"In[2]:=",ExpressionUUID->"b2063c4a-0841-459d-9a75-615f5ad20eb3"], Cell["NMR BB1 Protocol", "Text", CellChangeTimes->{{3.802024497633624*^9, 3.8020245418604517`*^9}, { 3.802084095361841*^9, 3.802084108653825*^9}, {3.8020841409745407`*^9, 3.802084161670167*^9}, 3.802084212501238*^9, {3.802084244652742*^9, 3.802084278020651*^9}, {3.802084423094756*^9, 3.8020846298402977`*^9}, { 3.8020849581323833`*^9, 3.802084994306072*^9}, {3.802085050767692*^9, 3.802085163990142*^9}, {3.802085202584613*^9, 3.802085251665312*^9}, 3.802085307377961*^9, {3.8020853486829643`*^9, 3.802085352330017*^9}, { 3.802085412982503*^9, 3.802085413507552*^9}, {3.8020862949066896`*^9, 3.802086378522773*^9}, {3.802086419758582*^9, 3.802086420467738*^9}, { 3.8020864521444483`*^9, 3.802086455710369*^9}, {3.8020864973077593`*^9, 3.802086542077324*^9}, {3.802086599556776*^9, 3.802086805617375*^9}, 3.80208783340549*^9, {3.8020879041098413`*^9, 3.802088142007761*^9}, { 3.802088178651374*^9, 3.80208822518908*^9}, {3.802088894055563*^9, 3.8020888982080603`*^9}, {3.8020889340426083`*^9, 3.802088951494369*^9}, { 3.802089002146434*^9, 3.80208900393335*^9}, {3.80208915422486*^9, 3.802089176573462*^9}, {3.8020895319809847`*^9, 3.802089546420405*^9}, { 3.802089626200128*^9, 3.802089990969708*^9}, {3.802090049382504*^9, 3.802090174649485*^9}, {3.802090217734236*^9, 3.802090233521098*^9}, { 3.8020902703924093`*^9, 3.802090351054804*^9}, 3.802090825928948*^9, 3.802090919018063*^9, {3.802091037601871*^9, 3.802091136133688*^9}, { 3.8020912205292892`*^9, 3.8020912572495728`*^9}, {3.802091496735574*^9, 3.802091497836735*^9}, {3.8020916303004417`*^9, 3.8020916542671137`*^9}, { 3.802091708229781*^9, 3.802091710306443*^9}, {3.802091754425097*^9, 3.802091788472343*^9}, {3.8020919239265757`*^9, 3.802091956125485*^9}, { 3.80209201247558*^9, 3.80209208120962*^9}, 3.8020921637950983`*^9, { 3.8020922289969063`*^9, 3.802092264054653*^9}, 3.802092316719646*^9, { 3.8024577879505*^9, 3.802457866582061*^9}, {3.854789904145109*^9, 3.854789949024129*^9}, {3.854790059546473*^9, 3.8547900764804077`*^9}, { 3.854790483284277*^9, 3.85479051286814*^9}, {3.854790755498229*^9, 3.85479076683178*^9}, {3.854792495132019*^9, 3.8547925031797037`*^9}, { 3.854792743694083*^9, 3.8547927587807493`*^9}, {3.854792845654024*^9, 3.854792873144697*^9}, {3.854795516971635*^9, 3.854795525524777*^9}, { 3.854795800553784*^9, 3.854795837382207*^9}},ExpressionUUID->"5c22b2e4-0eec-4ba2-b557-\ 68f9bbd38e18"], Cell[BoxData[ RowBox[{"(*", RowBox[{ RowBox[{"\[Eta]", "=", RowBox[{".5", " ", RowBox[{"ArcCos", "[", RowBox[{"-", ".25"}], "]"}]}]}], ";", "\[IndentingNewLine]", RowBox[{"\[Phi]", "=", RowBox[{"{", RowBox[{ RowBox[{"\[Pi]", "/", "2"}], ",", RowBox[{"\[Minus]", "\[Eta]"}], ",", RowBox[{"2", "\[Eta]"}], ",", "0", ",", RowBox[{ RowBox[{"\[Minus]", "2"}], "\[Eta]"}], ",", "\[Eta]"}], "}"}]}], ";"}], "*)"}]], "Input", CellChangeTimes->{{3.802024497633624*^9, 3.8020245418604517`*^9}, { 3.802084095361841*^9, 3.802084108653825*^9}, {3.8020841409745407`*^9, 3.802084161670167*^9}, 3.802084212501238*^9, {3.802084244652742*^9, 3.802084278020651*^9}, {3.802084423094756*^9, 3.8020846298402977`*^9}, { 3.8020849581323833`*^9, 3.802084994306072*^9}, {3.802085050767692*^9, 3.802085163990142*^9}, {3.802085202584613*^9, 3.802085251665312*^9}, 3.802085307377961*^9, {3.8020853486829643`*^9, 3.802085352330017*^9}, { 3.802085412982503*^9, 3.802085413507552*^9}, {3.8020862949066896`*^9, 3.802086378522773*^9}, {3.802086419758582*^9, 3.802086420467738*^9}, { 3.8020864521444483`*^9, 3.802086455710369*^9}, {3.8020864973077593`*^9, 3.802086542077324*^9}, {3.802086599556776*^9, 3.802086805617375*^9}, 3.80208783340549*^9, {3.8020879041098413`*^9, 3.802088142007761*^9}, { 3.802088178651374*^9, 3.80208822518908*^9}, {3.802088894055563*^9, 3.8020888982080603`*^9}, {3.8020889340426083`*^9, 3.802088951494369*^9}, { 3.802089002146434*^9, 3.80208900393335*^9}, {3.80208915422486*^9, 3.802089176573462*^9}, {3.8020895319809847`*^9, 3.802089546420405*^9}, { 3.802089626200128*^9, 3.802089990969708*^9}, {3.802090049382504*^9, 3.802090174649485*^9}, {3.802090217734236*^9, 3.802090233521098*^9}, { 3.8020902703924093`*^9, 3.802090351054804*^9}, 3.802090825928948*^9, 3.802090919018063*^9, {3.802091037601871*^9, 3.802091136133688*^9}, { 3.8020912205292892`*^9, 3.8020912572495728`*^9}, {3.802091496735574*^9, 3.802091497836735*^9}, {3.8020916303004417`*^9, 3.8020916542671137`*^9}, { 3.802091708229781*^9, 3.802091710306443*^9}, {3.802091754425097*^9, 3.802091788472343*^9}, {3.8020919239265757`*^9, 3.802091956125485*^9}, { 3.80209201247558*^9, 3.80209208120962*^9}, 3.8020921637950983`*^9, { 3.8020922289969063`*^9, 3.802092264054653*^9}, 3.802092316719646*^9, { 3.8024577879505*^9, 3.802457866582061*^9}, {3.854789904145109*^9, 3.854789949024129*^9}, {3.854790059546473*^9, 3.8547900764804077`*^9}, { 3.854790483284277*^9, 3.85479051286814*^9}, {3.854790755498229*^9, 3.85479076683178*^9}, {3.854792495132019*^9, 3.8547925031797037`*^9}, { 3.854792743694083*^9, 3.8547927587807493`*^9}, {3.854792845654024*^9, 3.854792873144697*^9}, {3.854795516971635*^9, 3.854795525524777*^9}, { 3.854795800553784*^9, 3.85479582227068*^9}}, CellLabel->"In[3]:=",ExpressionUUID->"7940378a-4705-4128-9764-d1d637aa5616"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Construct QSP operator ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ SubscriptBox[ StyleBox["U", "TI"], OverscriptBox["\[Phi]", "\[RightVector]"]], TraditionalForm], "errors" -> {}, "input" -> "U_{\\vec{\\phi}}", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "31208da7-2b39-4934-bc3d-33acec276e81"] }], "Subsubsection", CellChangeTimes->{{3.854795678816122*^9, 3.854795692037191*^9}, { 3.854795876688034*^9, 3.85479587771037*^9}},ExpressionUUID->"5e27256d-67e4-45b6-8e0e-\ 91fcd6c5f221"], Cell[BoxData[ RowBox[{ RowBox[{"U", "[", "a_", "]"}], " ", ":=", " ", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{"result", " ", "=", " ", RowBox[{"S", "[", RowBox[{"\[Phi]", "[", RowBox[{"[", "1", "]"}], "]"}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"For", "[", RowBox[{ RowBox[{"k", "=", "2"}], ",", RowBox[{"k", "\[LessEqual]", " ", RowBox[{"Length", "[", "\[Phi]", "]"}]}], ",", RowBox[{"k", "++"}], ",", "\[IndentingNewLine]", RowBox[{"result", "=", RowBox[{ RowBox[{"S", "[", RowBox[{"\[Phi]", "[", RowBox[{"[", "k", "]"}], "]"}], "]"}], ".", RowBox[{"W", "[", "a", "]"}], ".", "result"}]}]}], "\[IndentingNewLine]", "]"}], ";", "\[IndentingNewLine]", RowBox[{"Expand", "[", RowBox[{"Simplify", "[", "result", "]"}], "]"}]}]}], "]"}]}]], "Input", CellChangeTimes->{{3.802024497633624*^9, 3.8020245418604517`*^9}, { 3.802084095361841*^9, 3.802084108653825*^9}, {3.8020841409745407`*^9, 3.802084161670167*^9}, 3.802084212501238*^9, {3.802084244652742*^9, 3.802084278020651*^9}, {3.802084423094756*^9, 3.8020846298402977`*^9}, { 3.8020849581323833`*^9, 3.802084994306072*^9}, {3.802085050767692*^9, 3.802085163990142*^9}, {3.802085202584613*^9, 3.802085251665312*^9}, 3.802085307377961*^9, {3.8020853486829643`*^9, 3.802085352330017*^9}, { 3.802085412982503*^9, 3.802085413507552*^9}, {3.8020862949066896`*^9, 3.802086378522773*^9}, {3.802086419758582*^9, 3.802086420467738*^9}, { 3.8020864521444483`*^9, 3.802086455710369*^9}, {3.8020864973077593`*^9, 3.802086542077324*^9}, {3.802086599556776*^9, 3.802086805617375*^9}, 3.80208783340549*^9, {3.8020879041098413`*^9, 3.802088142007761*^9}, { 3.802088178651374*^9, 3.80208822518908*^9}, {3.802088894055563*^9, 3.8020888982080603`*^9}, {3.8020889340426083`*^9, 3.802088951494369*^9}, { 3.802089002146434*^9, 3.80208900393335*^9}, {3.80208915422486*^9, 3.802089176573462*^9}, {3.8020895319809847`*^9, 3.802089546420405*^9}, { 3.802089626200128*^9, 3.802089990969708*^9}, {3.802090049382504*^9, 3.802090174649485*^9}, {3.802090217734236*^9, 3.802090233521098*^9}, { 3.8020902703924093`*^9, 3.802090351054804*^9}, 3.802090825928948*^9, 3.802090919018063*^9, {3.802091037601871*^9, 3.802091136133688*^9}, { 3.8020912205292892`*^9, 3.8020912572495728`*^9}, {3.802091496735574*^9, 3.802091497836735*^9}, {3.8020916303004417`*^9, 3.8020916542671137`*^9}, { 3.802091708229781*^9, 3.802091710306443*^9}, {3.802091754425097*^9, 3.802091788472343*^9}, {3.8020919239265757`*^9, 3.802091956125485*^9}, { 3.80209201247558*^9, 3.80209208120962*^9}, 3.8020921637950983`*^9, { 3.8020922289969063`*^9, 3.802092264054653*^9}, 3.802092316719646*^9, { 3.8024577879505*^9, 3.802457866582061*^9}, {3.854789904145109*^9, 3.854789949024129*^9}, {3.854790059546473*^9, 3.854790474549716*^9}, { 3.854790623744232*^9, 3.8547906902616158`*^9}, {3.854790737208364*^9, 3.8547907403510733`*^9}, {3.854790786060149*^9, 3.854790788642679*^9}, { 3.854791025155115*^9, 3.854791070629614*^9}, {3.854791119440732*^9, 3.8547911382443438`*^9}, {3.854791444701264*^9, 3.854791607614956*^9}, { 3.854791825686335*^9, 3.854791827731265*^9}, {3.8547918802170467`*^9, 3.854791887827241*^9}, {3.8547922457242737`*^9, 3.8547922616243973`*^9}, { 3.854792365324345*^9, 3.8547923836317997`*^9}, {3.8547955319516993`*^9, 3.854795534321845*^9}}, CellLabel->"In[24]:=",ExpressionUUID->"b05155ee-f50c-4152-a715-d999a2e18624"] }, Open ]], Cell[CellGroupData[{ Cell["Extract polynomials", "Subsubsection", CellChangeTimes->{{3.854795569512101*^9, 3.8547955866975594`*^9}},ExpressionUUID->"7a19e667-2266-4f80-b2d8-\ 2945f0bc8a3e"], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"P", "[", "a_", "]"}], ":=", RowBox[{ RowBox[{"U", "[", "a", "]"}], "[", RowBox[{"[", RowBox[{"1", ",", "1"}], "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Q", "[", "a_", "]"}], ":=", RowBox[{"Expand", "[", RowBox[{"Simplify", "[", RowBox[{ RowBox[{"-", "I"}], " ", RowBox[{ RowBox[{ RowBox[{"U", "[", "a", "]"}], "[", RowBox[{"[", RowBox[{"1", ",", "2"}], "]"}], "]"}], " ", "/", SqrtBox[ RowBox[{"1", "-", SuperscriptBox["a", "2"]}]]}]}], "]"}], "]"}]}], ";"}], "\[IndentingNewLine]"}], "Input", CellChangeTimes->{{3.802024497633624*^9, 3.8020245418604517`*^9}, { 3.802084095361841*^9, 3.802084108653825*^9}, {3.8020841409745407`*^9, 3.802084161670167*^9}, 3.802084212501238*^9, {3.802084244652742*^9, 3.802084278020651*^9}, {3.802084423094756*^9, 3.8020846298402977`*^9}, { 3.8020849581323833`*^9, 3.802084994306072*^9}, {3.802085050767692*^9, 3.802085163990142*^9}, {3.802085202584613*^9, 3.802085251665312*^9}, 3.802085307377961*^9, {3.8020853486829643`*^9, 3.802085352330017*^9}, { 3.802085412982503*^9, 3.802085413507552*^9}, {3.8020862949066896`*^9, 3.802086378522773*^9}, {3.802086419758582*^9, 3.802086420467738*^9}, { 3.8020864521444483`*^9, 3.802086455710369*^9}, {3.8020864973077593`*^9, 3.802086542077324*^9}, {3.802086599556776*^9, 3.802086805617375*^9}, 3.80208783340549*^9, {3.8020879041098413`*^9, 3.802088142007761*^9}, { 3.802088178651374*^9, 3.80208822518908*^9}, {3.802088894055563*^9, 3.8020888982080603`*^9}, {3.8020889340426083`*^9, 3.802088951494369*^9}, { 3.802089002146434*^9, 3.80208900393335*^9}, {3.80208915422486*^9, 3.802089176573462*^9}, {3.8020895319809847`*^9, 3.802089546420405*^9}, { 3.802089626200128*^9, 3.802089990969708*^9}, {3.802090049382504*^9, 3.802090174649485*^9}, {3.802090217734236*^9, 3.802090233521098*^9}, { 3.8020902703924093`*^9, 3.802090351054804*^9}, 3.802090825928948*^9, 3.802090919018063*^9, {3.802091037601871*^9, 3.802091136133688*^9}, { 3.8020912205292892`*^9, 3.8020912572495728`*^9}, {3.802091496735574*^9, 3.802091497836735*^9}, {3.8020916303004417`*^9, 3.8020916542671137`*^9}, { 3.802091708229781*^9, 3.802091710306443*^9}, {3.802091754425097*^9, 3.802091788472343*^9}, {3.8020919239265757`*^9, 3.802091956125485*^9}, { 3.80209201247558*^9, 3.80209208120962*^9}, 3.8020921637950983`*^9, { 3.8020922289969063`*^9, 3.802092264054653*^9}, 3.802092316719646*^9, { 3.8024577879505*^9, 3.802457866582061*^9}, {3.854789904145109*^9, 3.854789949024129*^9}, {3.854790059546473*^9, 3.854790474549716*^9}, { 3.854790623744232*^9, 3.8547906902616158`*^9}, {3.854790737208364*^9, 3.8547907403510733`*^9}, {3.854790786060149*^9, 3.854790788642679*^9}, { 3.854791025155115*^9, 3.854791070629614*^9}, {3.854791119440732*^9, 3.8547911382443438`*^9}, {3.854791444701264*^9, 3.854791607614956*^9}, { 3.854791825686335*^9, 3.854791827731265*^9}, {3.8547918802170467`*^9, 3.854791887827241*^9}, {3.8547922457242737`*^9, 3.8547922616243973`*^9}, { 3.854792365324345*^9, 3.854792393540859*^9}, {3.854795588891655*^9, 3.854795589411965*^9}}, CellLabel->"In[25]:=",ExpressionUUID->"73cdef54-8362-4f2f-b3a3-94c8e3f8d667"] }, Open ]], Cell[CellGroupData[{ Cell["Plot the results", "Subsubsection", CellChangeTimes->{{3.8547996859835997`*^9, 3.854799687837446*^9}},ExpressionUUID->"bbd5bf82-cfc0-42bc-939b-\ ba7a983da4f9"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"(*", RowBox[{"Plot", "[", RowBox[{ RowBox[{"{", RowBox[{ SuperscriptBox[ RowBox[{"Abs", "[", RowBox[{"P", "[", "a", "]"}], "]"}], "2"], ",", SuperscriptBox[ RowBox[{"Abs", "[", RowBox[{"Q", "[", "a", "]"}], "]"}], "2"]}], "}"}], ",", RowBox[{"{", RowBox[{"a", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", " ", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{"0", ",", "3"}], "}"}]}], ",", RowBox[{"PlotLegends", "\[Rule]", "\"\\""}]}], "]"}], "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Re", "[", RowBox[{"P", "[", "a", "]"}], "]"}], ",", RowBox[{"Im", "[", RowBox[{"P", "[", "a", "]"}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"a", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", " ", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", "2"}], "}"}]}], ",", RowBox[{"PlotLegends", "\[Rule]", "\"\\""}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Plot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Re", "[", RowBox[{"Q", "[", "a", "]"}], "]"}], ",", RowBox[{"Im", "[", RowBox[{"Q", "[", "a", "]"}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"a", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", " ", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", "2"}], "}"}]}], ",", RowBox[{"PlotLegends", "\[Rule]", "\"\\""}]}], "]"}]}]}]], "Input", CellChangeTimes->{{3.854790811666819*^9, 3.854790813774249*^9}, { 3.854791464801591*^9, 3.854791465646448*^9}, {3.854791502065489*^9, 3.854791535929309*^9}, {3.854791585500169*^9, 3.854791601405529*^9}, { 3.854791633767523*^9, 3.8547917214276257`*^9}, {3.854792397120655*^9, 3.8547924373187523`*^9}, {3.854793051865789*^9, 3.85479321258151*^9}, { 3.85479562117111*^9, 3.854795631307437*^9}, {3.854795774018279*^9, 3.854795786505724*^9}, {3.854795862678804*^9, 3.854795863505371*^9}, { 3.854799387665194*^9, 3.854799401081017*^9}, {3.854799433943449*^9, 3.854799435665477*^9}, {3.854799470190534*^9, 3.85479947470617*^9}, { 3.854799669267357*^9, 3.854799724094225*^9}}, CellLabel->"In[27]:=",ExpressionUUID->"fcf3e09e-0746-44cf-a89d-de454fee2400"], Cell[BoxData[ TemplateBox[{ GraphicsBox[{{{{}, {}, TagBox[{ Directive[ Opacity[1.], RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6]], LineBox[CompressedData[" 1:eJw1mnk01d/3/6Ui8jaVRkqUkFJIUdpISJFUKolImoQkKRmSRKNUUlSmypSh yJDsDCFUZpmne1/3utNJZUji+/qs9fv9c+96rNdZ5+7huc/e567XMmcPm2PC QkJCLvTH/77XwDzO1BTBjcNH4yenCHj8UQ77M04wRmK/VH4mgay361R+DRNk 6fpey3IkoKVq6UxxCR5fpLvK4aMANsiEtNe2EnyTKl6eEsIHw4Gh6seZBAcP vvbiL+CCzfUvadqOBB/OSS+/4kHBhW8hZ44XC5AtaTHr2vMOuMX1af2dL8BN 10//lpPtgATRk0bBbwVYPvvEntXX2qEWLOWeJgvQRdXwisCtDZSy5D40RArw 1Oo6oxzjVvga8VJiy3EBekSeSFwk2ggqNlVpc6UFOPFmbWfAwCfY7F4olyAu wIz0hJDDg+Ww+0Z6oOZMer+bwpn6P8rgcmnEHotxPtZRF0sFQqXQoHXwbyCD j74j0QJn+yLwn8Ox4ObzMdkppn60OxWammZzSpz4mLWGmRD64zVeccktTrPn 43fxW9+f3svCNb8d7j/cz8eJhELpfWpvMGzOm80nLen9d4We6NyTiwY2B+5K 6/FxVtHWy84P3+PLb0k6jtJ8fO56ROg8swx9qw0CJ4p5aPS9buPx4G+4wo61 hyrgoa7HAUbFojpsGIxQrcvhYVjQ6NDIuTrUEGPUJ6by0Puks9SaFfXYZ3Zj +Y4oHvJMZNT+BTegxaeWz4/P8DCtz8Rjm0YzKnz0mLtBnodO23Jds3XbsVZ2 vbf+fB5alDAtM4Lb8dKx8UYDWR7OW7tvwe6v7dgifu2+ySwefuSpjIe5dGDE vieyNr+5aKFzybXiVidO55ZLn6nl4vtW/bMxtd04KLfov8TLXEyuxZcu1/ow +kSP28sLXJxbyY5gv+hD0/dJtSleXOROxP00qujDhCOat7JOcHGjxQ238yL9 aJ9mMrt4LxeTqqef6Avpx2+GHmJtGjQHfe7pvjiAuafLZ0p1cfBy+yKVcRMm blR0X/eylYPFgvjfCc5MLGyaf9iggYMP9uc+lw9i4keD0zmnKzl4yGIWCS5k YrWU7NGqbA6uyRlVW7Kawt63jh+vhHJQ45/lhyQxFopPjF/6rcnBbcLjOc/i 2Hg7K+nlTXUOXmvJ50bms1H6mFWD0goOdsalyxyqY6Pc13i13Ys4OE94463j U2xUjDP//noGBxNdWH3nDg3iepOo9cfbBnEy2F/TYzYHj9xa96MteBAnjpZb RO/g4qWcyNAv/oOYjxYyE/ZcfNj5S77EdxDbPVF6szsXqzXebU92H0TBfXPR 3RFc1P6in+hjN4gL/zZz6xq4KCJpYjtHaxAXTBpa6e3moaLuC57I6kFU+lUQ e/IIDzc5iFwdXzmItQWF+y548NAzoyqzT2EQNQZ09qy6zcM2K0uxLLFBZNxz uDNawcP0u7ZFO/vY+DEw/uY6HT5W5OXZGHaysaxlgaORER/7ehYMareykSk3 UK1pxad10iG3+AsbzdsPv0s+zsfAOkePQTpO5uuj2BKP+Wgjc1IpNIKNXuX9 4pnDfHTTq86/eJONvhx/5WwhAV53WrXrTCgbj6ScmIycLcCibL7f3sts1KVU to0oCnCFjVez0gk2xg6YLjawEOBYpF8YAhu9+TEzO6MEyJIv4Zjq0/vpzhxJ iBNgy0sRy686bFzquNJlW6oA3xbek+lSY2PczmvzFD4I0L3/1ZPxOWwUuip+ eFGfAO3d+BNXJNlo7F1ajoMC3DGi5SgmxsaG4rCjRj8FqCperLxgkoUm9tcT 2oQJDqxrSl/PZqHW0/lpR5QINrxfKPWhn4WzJtdZc9QIftzmeNaki4Xqa48w rNcRfHaQs35PAwtLpj9anwsED1wRQs8iFiqZxf/ttCNoNtt02dg7FprWZz+2 diKo+/Dm1cBsFo4l3OyKP05wTso88zsvWRg7PzK76xzBr3Ua9WkRLMxbck3k ww2CRXZe2to3Wbgp4LmVYgTBNEbew8JrLGx7zB5yekgwbMzY7vMlFi6c2pcU 9JzgheCwIuvzLLz74kHv4SSCrhJfl3z3YGFRiW7DwhSCWxUPDlAuLHyyN05z zRuCWqnPtrk7srB/5YGq0HcEFXUYr4YPsnCR+l7P4kKCk2YebjN2sVA/B81q Sgnme1wbXqbHwjXp30IP1xN89ad6f4o27c/VjRLZTQSjrkoXrl3Dwpa/qXsY dN849ygmAJRZWF2WdoXTRdB5WV9vhQILv3ukP/jQS9A6TWWr1QIWUudiSj0H CK7BbNHD/7Ewyo9UXmATlN8+epIhysLx9tiaGg7B2Y2ba08Ls7Bz+2djIT7B cfvgNb8mKBRpublVjhBkU5URl0YpNLUcGRAfItjq+d+vaT8pPGggp9/3k2DF uM2+cB6Fk7/IoejfBBMluxdG91GYn9ZhlzJKMPIQiOp1UuglWHro3xjB4OS4 320tFH7NWhOrTfdNr+Fp/ZfqKYx7Uqm18y/tj/HRb4trKVxy7oW22QRBm7vl RUUVFLYrBrxd/o+gceeK1MMlFH5MEa5k0Kyldv3R5HsKVbR7gq9PElTyYYc8 f0fhm2nv/0rQfVq2bLuXYTaFvhc3mp6nWVg6zbEvjcJuhWGfjzT/tJewDH5J Yfq98AQBzf0pZ/SV4ymUeBTe9b8+3zDydWV5DIUhUiGbf9NcunWt3LEoCkf+ CjfV0Pwm4p6wyD0Kky/fzA+lOaHrJ3l5k0L9+qeTy2mOVN/bZRZKYeqB3MKX tH3BF3Kr2UEUHrpnPk2CZq/yefnhfhTOWPiZ2kv74yzj+0Ldh8KK9MTAK//z 36EtssaTwkVqhr336PgYp+kHuZ2msCl8mkIoHT+tsZgz/7lSuJazzcLhD8Fl 2/7ZZRyhsDzE9fIiOt7SkQ7muw5RePltXmXhCEGhHlz/Yx8dHxJmCPSc8mPV MuV71hTObVj+99Uvgr2+wdJaOyh8K1s/e5jOd92ngX8N2yicCPsaqvKDrlfZ bdxzhnR+Vx0M2iwgGJcuWpG7nsLn7nx6MiIY8efEW9u1FOo+tl3TziIYZFod N6pO4cqZErxgJsEjvbf89BRpf6WYK67+T6+rBSfaF1G4V8uoup3Ws+GlXbZ+ crT98XZ35ncQXDpXdt0HcQoZVUIVhs0EJZ3OLXGYSSFH6LnT6ga6nl43zZ6a YmLlNYUzU18Jdps9ogyH6b72LVl5bxV9nvjJx5b3MDEw18WgtIDgnSr/8GPt TLz8+X5oZy7BALkeH5FmJnr8N9XdmU3QITN+t3k1E1+d6pCIoOtbYUBFtDaH iSu2NIaWPiIYa7HOqzGciUf2Tuic9CC4fLZFdV0IEydmV3/ZeYo+T2qclb4G MnHumDdz0TGChTsfNFSdp58rVtb50OdXm9WoVrETE8UbrWw0TQjO2/PhV7Ie E4M7HGqz5tD2zmnZ8VKHiVTlyoKt/xFc0SRITNRkoplDjmKpCEGdfYp7n61g YiI74tmVPwK02X81574ME1+aN4r91y3Au4e2nw8YZOC9OVt0jyYKcJ688xc/ BgPTS/MW5sYI8FnnpRUXexiotFXd89d9es47nN58rpmBO8uvvV0fIsBaRynd kyUMvLNH3rjhqABnuTSP7HnMwEVbWvp9lwow2M3JV207AzmBcveoMD5233GI LdzKQPN1n0/ODuCjfvahjzu2MNAz9+6Vxef4+HN43yx3bQZGq58t/EfPfU6B FtFvFBhofW/VsVpNPhpGaudv+jmA0Qo89cR6Hk7mzRyzjBlAm37hbwUiPLRr F5bveTiAJ/hqoqf/cPHdxBR4RgzgBdGwZjEeF92Nxq9HXhtA074LAdL1XOyq Fsxv9RjABff5wiZPuFjU1brhiMkAHs9Ma9yjxkW/6akXvPj9uEXayPuhPgfd HzuIHGH3Y+wYOC3X4KCz5pyHlgP9GHnkpN4LBQ5aHLr8RrWtH9emCzMvC3Fw wVsrfnd5P47/poqufRrEHOdfzjti+9FYym9KzJKeO0o2Wy3fQa/XcQ4xtGGj U9C35S3JfRi3z8HXZyOFDz8wo2NP9WKbBXdtTFIfFvaoSqQrdaNEr7PDuYPf UVna0Hjvsw482jjQXLGmBt+oh53qH/iOdc8i9XoTYrDnU7airm4Lcv4GtYqG lAFT46EWvmvEQ3oXjONFm0AmIEV3zbR6/G637oPWiW7Q0CUaL37V4iergz55 kQPguiYwsKCrFgPaXfsDkwbguYpUw5fKWhxacbjH4N0AyM7XvDASU4t+SQae T9oGYHTUvcTMpBb9a8M1mpcyoLRAsI/zoAZN9wfP25HKgAMGggBN3WrsC5g+ JZbHhKsm/LoC30rUNOcMeWSwoNSct0vnaCVuUYu6qfyBBUKW3K8ZlpXoYyMX WVPDgsv7BmsTlSpx0TqTujE2C3xcmVW3ayvw3DQf/0olNpwI6/ropFiB49HN s9c9YMPO2i9ZYlXluPlYWqCM+yDcqKvVDHlTjmGq00rM/QahqqkmYzK2HMum bf56LmwQtnV9Tv91thx7V93Se5EwCFsEn5K7FpejZn1rnXfLIKyVLo7L9ijD xnCTidJNHJDbmxFxcH4pOngGLJyY4MAblZuF26aXYrezl0mnKBes/xxnrCMl +OBi0b5sWS7ceLZso3hlCS5xn7lEX5ULk+wHPYU+JSipvXjwkw0XKH+/NfIt H7HNVPp+2QsuuO+6F/1BDVFgr21mtJUHyb8teA+nIbKT78nk7+RB/+MZhu5t xTg0QByUbXlgy7jAWhJejMIR+hdaT/AALjroBrE/oBbzg1XubR5IJ65qMn5V hCHVB1dta+bBm5FPUlXLC3E8RZafcIgPd1ITSrf8KcCCtbGUtQsfTjsEns/9 UoA1sj5/f7jxYXnFxo4EnwJUy6ZOiQbwIeph2kv/qnxMjz0jcfs5Hy7p3DPQ dsvD83uuVI318MGWfeZHimEeFjY8nFPJ4oN2rEWiolweLh/233Gd8IE3fYaY ZPE7ZDR1+XRN8sGh0aeJLfUO/7y85dkrLwBjL/vTz97mYETJrJdbbAWwREVv iVxYDqZKnLiiflgAf9vk6m/a52A9y0hxlosAcoy+6V6cmYPKg6XPos8KQEXG WHjv/rfoujTFXvKWAMSzVj4W+5uNbncOtVsUC+BFB/P2YEgGXvSs7xqbT6D9 8rznT7UzcMPGI9tL5QlILjHL2t3/Gi+M2ckELiPgcyS5oQBe46azDVkN6gRM mafm3xhPQw8HW5H+zQTYAhKn7pmCy/25nc8dCMhHKL7pXpKCqxwfnnrmTMB6 3e6yyC/J6HjpBES4Esg/94Y5rp6Mp+1LCyzdCYSPeavXMF9iwdVXdZr+BNSF x9+ePpSEwtcU7t6NJuCQqP5pqXgS5rQIzSqNIRBpcqilMT8Rfxx/8471jMB4 aNHYpnmJ+ET/TbbECwI1swMNJOrj8RSxn/Ehi4DQ60wr+u6B0eu/yYS/JaBj 1evovSYen6m2WZq/IxAbYRTceTMOn005W8W9J/Bdfdadg7ef4YzbQ3bvPhEw PHMyq4TzFO2mkisWVxFIzqxuUDN/ilb+7+28qwlc1Lk9f1w4FtcHKvyZ/Epg 0RbZ+Ce+jzHw6sDfi60EgoPOlQm3RONK5rFHvm0EuKVNzFPa0fgyQcj2VAeB IrNH6psEUciRUNVf2UPg8G75nI6jD1B3b94dEYoA79ew3vny+6gk/Hl3FIvA 5ahvKLniPuruo94uHKTt7wiuMaLuYXJ8yJ6/XAIaAfa7O0zvoebfNREH+QTe K+q2er+KwE2FG06lCQjk1DR82lV/B4W/d/9WGyLQJB1aHxx5G2XPlSXZ/iTw a59eV+6eW+jCi/zh84uAVu/z34tbwnFj/Eu7yGECu1fsEdr1KAyNhy19b48Q 8DwlIhF84DomHXi59vIogczfbsrs9hA0XH3AU+cPga96ipqLY6/ixvvrRv/R LAho1Lc6HIzDQ+/nvx8n8F95qOmVJVcww/xj/em/BA7IeiedwECU05q9QnqC wKPY9vdfjfzxEMNNJpnmFhWjRp3yS2ixriBK+x+BudmvOE9MfXFhaGXOG5r3 bJIUnvb5PMZzvU6vmKT18sl74fEd51BF5EnhTZrrdnWs/fLFE6/JLU1g0rwk 52qEa5wbyg+zlmlPEYipskatF8exYnG3njfNC7oUBJMpzjgeIxhMpnnVQLNe 1OFD+FhZXKueZhUJz0drju/GpRPLZfk0BzWpNi0uM8Bg8Q03JmjWkOo6NXP1 cjD6tukJPS9Dy1Dwc+kxc3Bz1TIeofnHrpU/Pbxs4fp1heBemv3um1af1jkC Sdem2xXTPBlwMea55jFw3SX4fPd/+59Kd2tUPwW97tz6vTRPt+0xEFXxgEIx Ca//aC5TDL/+VNoLlva5pBbQ/oVEKXluuOMNJbsl/Q7SbCpRdKB+9gVo2LOc waPjJRq8z+h0+EWQ86tgnqO5alSgNlP0MtSMzA8S0PG+cSZM9nlIAOTeg7f2 NL8civCXvxIEqm+OBxbT+Tp+XN313WQwXFE5P2BP59Nc6YLdVYMQeGZd4veY zrdqV5mV9eVrkJ/e8frzGAGOzeENnD/XQd7Ws3cGrZfq/1JX5W0Mh6Z97n1S tJ7SqkaWhly4Aft5Rv6StN7ctkTMWjJ8C84PRYdStD4t/3ROcLTvQJKg6DfS +l2TozaU53UXOp6PTt76QWDsQR9PdSwC9nMWDk6j9V4g7tdcev4+3M8vWLKf TSButVxjptIDyHpyMDCNrq9Q68y62G8PoK71vdUQg8DeqIEaH7UoKFVX37y7 j87Psp2l6l3RsPzSxaPOdP22mlA4/8ZjmHOlaYcxXd/Fx4M+zNjwBPLs9ZLn NBO4+TonvzsiBoxB+eONOlofegqZkSbPwAMSgqsrCEjY56UHDj2DnMA9y3TL 6XoL2J3q9uw53Gt7qH+/hMCR/lIb5ctxUO7m2busiPZPddTyX3M8bKBE5t6j z7PPbx1NssOTQGXhmam8KAKbP2mUrexNAiXPh8Wb7hPIavlj9Ez3BcyolRfL vksg6s99uDnwAi79UN/mGUbABar0jxm8gue7bl82ukhgqnrtuoVDKdD3RXbO 2gMEvDr/Zd41S4Wuj+IrvuwhQPGr14g8SwV23Zrvh3YRqJU+pvHLIg02nVVQ tTIl8MT28covSekgSIp5EqpNYH3/tCVXDmRC6pt8ffPZtP2F+4fWpmTCagdC 2YnQ8Yh8Xd77JxO8zV6pOE0jMDs+fX2cZRasbo92NR8VwIT1gc4pQRbIuW6f E9wvgK6sTDVc9wbOeQcHJ74TwHNPh3KDvBwY07B7kLpPALH2YhkPe3KguPdg j9YuAUSb5zzii+aC9q4ZwxnmAohQFD/99EAurDq7OSx4kwCC6nJl/v3JBav7 W0MfKQrAae1/jkWb8+Bv7anXl9h8UP7x/o9+aQEEtfxt0vHgw+UvW0N+8wvg o6P09zuufGhJrZHMWFgIl3W+tLQf5sONYx3Ky84WwsTTd43mO/nwo33cUlTx PXhbv1b2V+VD8Se9hEb/Iph/+OKP7d08sIvJs3DTQ8h1DPt5E3iQ4wvNK44h iCoJta5azwNJ20rHngiEs2Vflxer86BEutWbHpNBqzpkdqEcD+Zu9dsVu+Mj +FXM9JTicsEv/8nF4PGP8CBedkL7Phe2J7Z9sTpQCkcWPTgR1MaBH9cyXD+d KQUN0X93X3/hQPSJq1ObrpaCyRln+a8lHGCvXr1OLaMUvvb3eA+mcCAsP+jB 9BllcFO5/0n4RQ5UfVl5KD+zDEzi2i6fm8cBs7Hz7GWin0CibOtLCTN6frSU nT6cWwnRq+J49iEsuOMkNvTncyV0i9lqXvdhwffzQj2TXZWg+2R/yKsTLDj1 TFA4S6QKAjRKfap30utJtZe8bRUEe8ePPZ5Lr793tX/rSBWsvv53/9EECk61 /C6J1K0GHUHuFXzHhLuOrUFr82pBMkfSJ6VsACqtFP6zlq2H8p9im+6pdQN/ mVxAw5NGWMSSPvPHpwnOe69yujuvBR6MJJU/ulMOrZ3SNq5V3yHC5bRv7ZoH eOTiZpbozg7QfKY+q8ulCstWbbhRNrMb3NLTJxf2tqKKWBE7ybwX5KVqL7OX 9qFC0eltM8P6oCJ++599FfQ9duH4uPvtPvhj9c/YrI2JEj5hWd8j+yDOqe2j Bo++12q+WJz+tA98fhXNb5KhsD2xe8jmbR+8LR49HmlPYdTN3c/iuvvgixVH EwmFkoc2jG5a3w++ysu2HpNgo0j+p/QX+v1wjrtW5oMCG//N3essZdgPz7a5 TIlospH31fNLv0U/mMi4jHnuZmO1cWpi+JF+cB//GRTzkI3X1RWsW2/2w96T y/9tWjyIQuPCKV79/bDE9pyi1mIOHhlo+xXI6gef4amfSWoc/FibteU2rx+O dxZ+E9vIwaDnDk2vRvrhzbW77il7OThlUjjVKT4Ak6mp9EIOTt49a2umPQDv vJuzc/5ycGJF73T5kAEQPSSqd/ILF+2l8qzUwgeAJzSyf1o7fd8cu/1Y984A KN5zywqnuHi5ZpPm7ugBMBZ0Npya4uJfz0d2oekD8M36GJxZy8Px91bZP5oG oOqMS96xuzwc3f3B4dMKBrw7q3+4yJCPwW2ZXVtWMWBB5G3ZpTv5KOmUYJ+/ lgHLqi0lzuzno4rndbu0TQxI+HY+sPoMH/fe2W17bzcDfkb1bA1/zMesaqbl YX8GxClkac7k83GzzffalmAG6BzeOGvOKB8r26p3WIcxQPXoTp74NAF2szO3 b73PgJe6K8Py5wpQQvTSNrUUBti78RmzNgnw5FZJg+EmBlQfT2h0uyrA39VC Re7tDLBaUb9Q9pYAg2x+6bN7GFBPXFfHPRBgtNP3jR0c+vdMQ918XgiwIjBB p0SICSleifPFKgSoVKSrcVuDCUVn7vjmChPM2KqWJqLFBJHQZ0rdYgT1ahar B21gQnn4a1uBNEHrdiHVc8ZMmDdXd/anJQQDxqqVDx5gAjuv+uR2PYIdOo6L V4QwYcGacPHYkwRd/6yXuhbOhCg9t66FHgR/fJCYzrzDBLtfEr0B3gRnmBVy kh4zYUx9gcpkAME1B+UKl2cyQZi58e63SIL5CrzXITlM4LVKFoY8ImjcXxrP KGCCdJXJBcVYgvtPe4YnlTOhqrv84OwXBIP8aw4sb2fC9YrsB+m5BMWNE3aG 9DDhXfE21QcFBB+KXDRkMJgwXTIx0+kDwdS7KqpJhAlKPjIWGeUEdfb+Wzxj mAleG4zXr64iWLygScplnPYvzq/8Xg3Bxvgro8ozKdD4JRM4u4Ggg+sB7lVx ClqPHQ9WbCbIVtfsGZCiwPXBu7ZF3wlOvO2sSFxEQY/7P23sIhjq+7ZwuiIF M0IrJE/2EpQxuJFxdAUFdzQ1+KP9BFUqNkQpr6VAyVTfpZxFMOuG5I2r6yn4 OnWraRqHoP4upv+APgVzmXWjyjyC5XOKzm41pCCr/2uBuoDgru+RxxK3URA+ bYfIvB8E22NPHpy+g4IM1xmfmUMEXZwMLY9aU3BkTy7/8S+Cvhy+jvIhCoz3 fZ2dOUJQOLNc9eoRCsovRnZKjRG8fS5GfuAYBcEmIsV7/xBcsNFLeutpCiKv 1sT4jxNMmDCfkehJgf6yQKcbfwlqlCwdE/ahgKFcMH5pguC7ayNcZz8KtDIk 9uz6R9DQ4ktPaRAFnElRW9FJgjWSSY1KoRT89dQhCTTbNl6qDL5JQYjfPHGl KYK9j3a/74+goCl7dWQIzafsVTONoyg4UPbbq5rm34pTCQkxFKBf36NfNAcw m6OE4ymoKHXh/e//Y7HU9BvOLynwjOUcIjQ/cL8aUJpGQZ9qe30JzUu07byU silwSXqod4Hm5NG1rsHvKPA9filAimatIlG7/vcUlFK9N2/Q9hUFdVsal1Cw 6K28JZv2x2xbrlFCBQV+Fy+/VqW5QezWeuFaCiSnW0TvoP23/+qs5lxPgfr6 6jFrOj5UpJ5CaQsFe2UMCjfS8fPcLy2j1EnB+tyuSmE6vs7Jmk9W9FGwsS7+ S8UowX1/rJTVKAoKjxa2etL5MbdwT9fgUiAuZtgsNEzrIeb2+rU/KHjywiyX vk/hal56sfYwBT9P9bs20PlWNKg12zBOwZsWhbY5tB5m9ojbbZlB98m5T96b 0foZ01QfMBJjwXKHATF9Wl+coO1u2yRZUGnQ3STDJlinFBawcwEL7H/VxZ4d IPjEdWainQYL3C115v+g9X0rb7mGwzoWOJhlnJFoIRg4yyTXSZcFI+Pb8/9r pPWWGlx5wpAFLaYHmMW1BDUFU1yfvSyQV1AobismqARLz186yAIrv9nNi98T nBuxZdLfgQW6H57ZGucRHF/nL32N7utiKzP3bs6k9X5+XOf+ZRYkNYUe6H9K 66tiYXHUFRYk+3/q3faYYMp8PbMnoSz4qA/bbz8geKfgwsH4CBZA5onzlTcI Hvz32z8ziQWeB4/tYZ0nuNNqrujbFNp+1xMjxz0JbnmuHfEugwULincJak4R XG7klfAhnwUSd9uiDBwJCkJIRU0tC9avcO/9ZEqwr1nS+ls9C/L7jAwdDQk2 qaxpa2hhweSKdyc66POwoMqN29bLAgvxBV/8NAiGSHCk2L9Z8GPbghE7+jxd +IBxYIYCGxiHY3ddbRDgxqtL53krsWFCLMv69mcB2nrZNQ6sZIOE/4GUcx8F GGldZ1mmxYY3y86LNL+m+8N/RcbB5myQTpH/yQoV4OS1+xpC3mxI9d0Toa8l QPnzXwc9LtLPzV7vn1IR4CYXsVc9AWz4XR046/liAfoaBy7DcDYY1+ytdp8h wKHJU/MCnrNBZWSWuWUTHxkXjIUmqmnuVWQOufFR+Lj/h9N1bAh5FXvKzpGP irb5lzqa2WBSa+sau5uPh3XWDBf2skHLe8bR9PV8bPmxkHNxhA3eN570/fnL w88nhxpHlw2CRHfEMd0rPMw4FPfql+8g8CQHxSPOcFGnVXlcJGAQTCMlia49 F9/bvNq56OogpHyUTSmw4GLV9owhw9uD4Gx429pmJRcHNhRtuhM/CGvHd2z/ 2s3BBXO/f1OtHgSr+A25PaYcDK6VGju8mAN/j/aYdI+zcZb5fYuzihxIZi3B NgYb75bNexqyggNJySFRGV/ZGFuoYJymyYEx61cSjHg25iWvujW2lQOKQRkM P1N6HgoxU7zvxgHbgxqplmEs3G8QaF71gQNFu6f/5PKZ2DlMSZWWcqDZIcHI u5GJzhlWre8rOWCZ3yDUnM9Et6VLXDPrOXBlzeOgpVeZGDztw9UoJgfcFtcf b5FjYsan8WKX/7jwrqKsrVCbgTOsfNZPO8yFl9O7c5ut+/GGSPfEuBMXvKaN GJmu60cp3Fb+25ULSU/31d6S6cdFa+X2sD25oNpRfyumvg81ZXM8v4VwQcrL MPLDrj60axlKe5rOhUg/g13CJr2Y7XBGSX+CC9FSNRkzF3Sic+4PhbhpPJjv M+vql8oOnCvhvVBElAeihoW3NC50oE/BJekGGR68W9V8NrOpHfXnhk2eXMmD XjOt9tIbbVj6OaH9sQ0P7FPXfvjZ34JNOt8j/yTzYPjk8B0x43q8dvPQHccM HtSRUwZ7GHWo298d/uktDzq8Vj9eHVqH0XeZQfeKeTDh8kts5b1veJjz012t iQeWy8zKN0fUIvX8vx0Hp3iwf9Dz9HSXChwT3zq9YB8fkqxnGygXv0FZgeSD Ujs+HNGJkRlTz8ZV9e3Lax35ULWzoOOoRiY6PPIy7TnJB193xb+J35KxTDkh fGYAvb7s7reP3Y/wzuZp0jYv+WDdN7SvpT0WVrijPHeED6FpTo5eJ0pgy+6b r3//5UPvRkvpoztL4YDO/i2TQgJoupBzUGNdGdwYFzjKzBaA0rtzChb/yuFH 6JLEDUsF0CviZVj0vAqKnvurhZgJYP8D3hBDpA5s6jfpKkQLoDx1/HFwVSuI b5pdUvRUAE/ZbcRo/3coSWrfYZ8ogOG/sjZ3qe+w1veiU0yGAO6+HtBvFmkH yaV5txZ+EoDgVUXMXKtO+OymNSD3UwA+dnGttX69YDhLLUJyJ4HkzbVzDe3p Oevs2KKM3QTGzr/4XJzOhMyOyheW+wnY5y+6HfaPCUsyXd/fciYgW+pR+u85 BRP7kihx+h7v1q16yYvNgrzEpQaiLwmozd1YG3+dA+7/kcqXaQTuwP3y3k4O qFwotjHNJlCTdfO7mxYXHlgcPnGtiIDqR6dWt25at0NP7k9vJGCnav5VSp8P 6odOLUn8TmCmnLjR+3t86CvXSzHuJjBvt/z3nEE6D9Hfi68MEvC9NT4a+UQA s4STty8jBJ7G6JVd+CUAPH2h6eNvAl0LMzek0X76NJs6HhknkKKT6LOatvP/ vS8G//99sf8DhYJMZg== "]]}, Annotation[#, "Charting`Private`Tag$10467#1"]& ], TagBox[{ Directive[ Opacity[1.], RGBColor[0.880722, 0.611041, 0.142051], AbsoluteThickness[1.6]], LineBox[CompressedData[" 1:eJxF1H0sFHAYB/DjMjp5XTTlyOu8zVCdRcYUc3LGjdHEOS01L6lbujOb0LWu prGwHam1nZNLu0slO47juJ0ak87LcTruuMudzpEUZlNten7P9t13n7++/z0e eSXkq+YYDIb0N/86ONrZsL9vEmMOrmTHi7Wzi/zmXajv5hZymB8pT7eKHO7A nBudQY5Z2vjUJEAmPxjrOEVBpo8zi6/1r4EnJ60Ng1QjGD9QcjTc9Tu4q3DY wu6rAZxbE7o+W60Hbz8pZ4mjV8AcW5ULW60DtySG0uQPteDqIirDn7gMLse+ otOMGjC1ctx7ul0NbuzTslsKFsE9C35HXnuqwF72MbFpz5XgtwGsAs2SArwg 7TxJIEyDtUGNYeIPcrBDBY8QbDYBDiKYgribo+B7F4yfhQwZ2CmNX3fpmATt /ZLajXj3gLlK7WM9kw/OTnV9r7zSABbiyqckpfXR/31GY+ZWlSkAe6337kRI hGAiZ3YsOVMCjiM5Yre6ZOBaykxlSPcoWJaMt0lxnAAbPZwqvjTLwaW3A6m1 ztPgmXl7cv6IApxbdu6bZZISPBQY/mjIQgX2PSxaaU1YBONFhXEWLDUYs2vO o2k04N+pfTlSn2Ww8jTlhA9TC76ZYe/gOa8DuzQsZx7Cr4D5WS9ebjL04Iyo uwkjfQZwZ06xZ8TeKngbdx4rTDeCyRORBDx7DYyLtB4UPUMebJ27eJmDHMIo oz7lI9u6d9e4SJE/FoUtOf1AjrHyr7NNMqH9W9vH+anIAqWMS8pAdhPk99bk Ie+lt+pwZcjdHPcoyzbkGzYmWVsHsi+9nxzfidyQmH39vgiZttFcj5UjB2QV uHEUyOrhs7xYFXIKW9FfpUe2Mm8nepiQxYX0yYGfyHem4im5u8gH/wv8B+0M Suw= "]]}, Annotation[#, "Charting`Private`Tag$10467#2"]& ]}}, {}}, { DisplayFunction -> Identity, Ticks -> {Automatic, Automatic}, AxesOrigin -> {0, 0}, FrameTicks -> {{Automatic, Automatic}, {Automatic, Automatic}}, GridLines -> {None, None}, DisplayFunction -> Identity, PlotRangePadding -> {{ Scaled[0.02], Scaled[0.02]}, {0, 0}}, PlotRangeClipping -> True, ImagePadding -> All, DisplayFunction -> Identity, AspectRatio -> NCache[GoldenRatio^(-1), 0.6180339887498948], Axes -> {True, True}, AxesLabel -> {None, None}, AxesOrigin -> {0, 0}, DisplayFunction :> Identity, Frame -> {{False, False}, {False, False}}, FrameLabel -> {{None, None}, {None, None}}, FrameTicks -> {{Automatic, Automatic}, {Automatic, Automatic}}, GridLines -> {None, None}, GridLinesStyle -> Directive[ GrayLevel[0.5, 0.4]], Method -> { "DefaultBoundaryStyle" -> Automatic, "DefaultGraphicsInteraction" -> { "Version" -> 1.2, "TrackMousePosition" -> {True, False}, "Effects" -> { "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> {"ratio" -> 2}, "Droplines" -> { "freeformCursorMode" -> True, "placement" -> {"x" -> "All", "y" -> "None"}}}}, "DefaultMeshStyle" -> AbsolutePointSize[6], "ScalingFunctions" -> None, "CoordinatesToolOptions" -> {"DisplayFunction" -> ({ (Identity[#]& )[ Part[#, 1]], (Identity[#]& )[ Part[#, 2]]}& ), "CopiedValueFunction" -> ({ (Identity[#]& )[ Part[#, 1]], (Identity[#]& )[ Part[#, 2]]}& )}}, PlotRange -> {{-0.9999999591836735, 0.9999999591836735}, {-2., 2.}}, PlotRangeClipping -> True, PlotRangePadding -> {{ Scaled[0.02], Scaled[0.02]}, {Automatic, Automatic}}, Ticks -> {Automatic, Automatic}}], FormBox[ FormBox[ TemplateBox[{ RowBox[{"Re", "(", RowBox[{"P", "(", TagBox["a", HoldForm], ")"}], ")"}], RowBox[{"Im", "(", RowBox[{"P", "(", TagBox["a", HoldForm], ")"}], ")"}]}, "LineLegend", DisplayFunction -> (FormBox[ StyleBox[ StyleBox[ PaneBox[ TagBox[ GridBox[{{ TagBox[ GridBox[{{ GraphicsBox[{{ Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6]], { LineBox[{{0, 10}, {20, 10}}]}}, { Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, ImageSize -> {20, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #}, { GraphicsBox[{{ Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.880722, 0.611041, 0.142051], AbsoluteThickness[1.6]], { LineBox[{{0, 10}, {20, 10}}]}}, { Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.880722, 0.611041, 0.142051], AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, ImageSize -> {20, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #2}}, GridBoxAlignment -> { "Columns" -> {Center, Left}, "Rows" -> {{Baseline}}}, AutoDelete -> False, GridBoxDividers -> { "Columns" -> {{False}}, "Rows" -> {{False}}}, GridBoxItemSize -> {"Columns" -> {{All}}, "Rows" -> {{All}}}, GridBoxSpacings -> { "Columns" -> {{0.5}}, "Rows" -> {{0.8}}}], "Grid"]}}, GridBoxAlignment -> {"Columns" -> {{Left}}, "Rows" -> {{Top}}}, AutoDelete -> False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings -> {"Columns" -> {{1}}, "Rows" -> {{0}}}], "Grid"], Alignment -> Left, AppearanceElements -> None, ImageMargins -> {{5, 5}, {5, 5}}, ImageSizeAction -> "ResizeToFit"], LineIndent -> 0, StripOnInput -> False], { FontFamily -> "Arial"}, Background -> Automatic, StripOnInput -> False], TraditionalForm]& ), InterpretationFunction :> (RowBox[{"LineLegend", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", TemplateBox[<| "color" -> RGBColor[0.368417, 0.506779, 0.709798]|>, "RGBColorSwatchTemplate"], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", TemplateBox[<| "color" -> RGBColor[0.880722, 0.611041, 0.142051]|>, "RGBColorSwatchTemplate"], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{ TagBox[#, HoldForm], ",", TagBox[#2, HoldForm]}], "}"}], ",", RowBox[{"LegendMarkers", "\[Rule]", "None"}], ",", RowBox[{"LabelStyle", "\[Rule]", RowBox[{"{", "}"}]}], ",", RowBox[{"LegendLayout", "\[Rule]", "\"Column\""}]}], "]"}]& ), Editable -> True], TraditionalForm], TraditionalForm]}, "Legended", DisplayFunction->(GridBox[{{ TagBox[ ItemBox[ PaneBox[ TagBox[#, "SkipImageSizeLevel"], Alignment -> {Center, Baseline}, BaselinePosition -> Baseline], DefaultBaseStyle -> "Labeled"], "SkipImageSizeLevel"], ItemBox[#2, DefaultBaseStyle -> "LabeledLabel"]}}, GridBoxAlignment -> {"Columns" -> {{Center}}, "Rows" -> {{Center}}}, AutoDelete -> False, GridBoxItemSize -> Automatic, BaselinePosition -> {1, 1}]& ), Editable->True, InterpretationFunction->(RowBox[{"Legended", "[", RowBox[{#, ",", RowBox[{"Placed", "[", RowBox[{#2, ",", "After"}], "]"}]}], "]"}]& )]], "Output", CellChangeTimes->{{3.85479971844681*^9, 3.854799724619273*^9}, 3.854799807577301*^9, 3.854811848706294*^9, 3.854811965354575*^9, 3.854812029293939*^9, 3.8548121000683193`*^9, 3.8551401406559143`*^9, 3.8551402426035357`*^9, {3.855140285553207*^9, 3.8551402905760927`*^9}, 3.855140341435275*^9, 3.855140409112891*^9, 3.855140479510516*^9}, CellLabel->"Out[27]=",ExpressionUUID->"7d6456f3-b6c4-4d9e-acf5-3fd42f1d99c5"], Cell[BoxData[ TemplateBox[{ GraphicsBox[{{{{}, {}, TagBox[{ Directive[ Opacity[1.], RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6]], LineBox[CompressedData[" 1:eJwt2Xk0ld/3OHCF0kRzhN4lU2ggUsguEREhswyZNagUiUIqCQmVpFmmECVl SG0apDIP1x24k3l+HoqQfJ/P+v3uP3e91j3rnufss/c+Z61ng9spS8+5AgIC GXMEBP73bS+i/Dnt4CAK/L9PuXT5qZWaUoOoLPPQvyHmH749/kVYrL0fXTZm EPzZ3+gaq0owIvrQ87TPjYbsAfyTGBKF0Iv3OxeVy9S0YY9URf9+rV5UyH+8 W8etDWkZ80xr1Xux2T+w+P4EC9+UJixr39SL/LZon/UbWejHz0yZWtGLVnOO CjleYmCHanOuRm8PSr6Of39el4a19SoNOfE9+LCi/ci3rjp8LsqWSOZ1Y8H5 cq/Mq68w0RHm72rrRssVQb1B+/IwIuvpLwatG/3HnPISwnLQTc+9TrK6Gzl2 C4/dnElHmcDeq0/edSP/QV/z3g1JmNo+OpIR0407m4cMHZl34Gnu/Mq3Gt2o /kjYtGemBB4aq/o33ejCp/ue3G/YWg+yi4x/1F/twh4fN37SuXrI+ekmUxvW hc/ydkVuKqmH0oN3GqsCutAu4JBo0d4GYJhNqH082oXRXfuOnTNvhNWHP4xl 7erCKoFXqREezXDL8UBAaF8nBj5IfAredIg4cTRo04FO9FM+qVjP4AA7zvlh 6b5OnHAQ0r62iAtarx3LTXQ7ka/003aeEhdGf1uL+G3vxGc/rRMKvbhwNMw4 uUC6E7k72mgVbC7sSdxerD3agYldSZDxnQf/ioT/mD7owA1blyj8jO4AB+Zc Kc7dDpTRTH/0O70D3v2dhdPxHcgw/zolVNEBfnunride60DrlPzIxvEOaP8x vKb1VAeW57wo2nS0E8raWzVd9TvwxKTU8rptXRAimH3ef4iPgT8PGCmVdYPf fed5rr18fEJbE3a8sRvctq64a9rBx5S1Y713e7vB2PFigSKDjy7eRn4ZK3tA /I3ZEPsLHzVr8vvfHe+BQrcxN5OHfPwSHyb0YFUvDFfomMma8PHnloTYFLs+ 4NuSbcv28zF3XdG/JSf6oGUo/fjsHj5mrn3ufTysD8okxG4wd/BR30J/R2d6 H8Sc4X2Ol+GjkP/QmVqyDxQ3RO6ameThHaPesuhr/XA0vE6WlsVDzcvDZ2oe DoAN44lu+3Me9hE3Lmi+HABjtdN2nY95OBidI3bzwwCody6NHb3DwwPEzX9T 7QMgYmQ5JnqZh4fcFKP40oOQJ9ZSYWjPQ9HRYOuu+4Mw/ZjpXLKAhyy1ui38 iCEgJrKDyoV4WOxlo/o9fgg6zUMSv81ycaf99+CHj4egdq5UZctvLt5VCZuc XzoEqV5Om0d5XHzmd8VwbGQIjLZwp5VKuRjq93bberthuPuhK/nhMS621U3c nJYYAeeglFBNTy5yr3nfPCU/AorbD3k0unBx6dGoLTVqI1CaVbxVxJqLPtFm Zw1MRoB9O6byLHDRt1ywRSlkBBSOqf4yXsnFqsAfvnW0ESBkuxldolyc6Yye 4PJGoISTguELuMi7Vv6wbXAE7mWLrD45w8FY1VC5A3MI4Ibn7dNv5KBvg9fc i6sJ8N889Xg0hIM7gw3zGboECD13P6Z4hoM+F04PFO4jIEm8RsPZi4PZ4jv8 rhkR8F7wafV3cw4qlhyymWdJjWcaTD2V42Dlkqo1UR7U+MgE60N1bJx135Qc d52AUo7i4lwZNkJxgkXFVwIu6sx5mSLOxpIpg/VLvxOw+z7D9IYoGzlzRGat qwkot4y+5TXVjstY2X8qGwn49nVg+YbGdhTdKLNNl0NAc06eRNLldnxnH216 dZyaT+R66dXz7ciO4lwxmCTA1tPF8ezJdpQ7LT41M00Ac93SR+b27TjaoVRg NYcEXsKZ9YtU2zHsR1Ta/UUkDJ9XVwjntuHHDtbWq+tIeNW8uMqP1oZlO1R/ lq4n4Yxql49TdRuaauyh9cqQ8Kv/brZWcRvOLQg3VFYgYdppYvPvW21YsUKv /8BWEspK62o7rrWhzWZJWx1VEkLXZJ1qDGlD7cFkE/ntJMxtsHud702Np51v 5e4gQUS/VP0YtGFD1YVaYjcJP54mtthptGF14JbqJ0BC7MyxQEPlNqz97O+7 fy8JYkWSxbJr2rC06ElcsD4Jq5UuaXOGWbjvzN7WUGMS6JHWbTWdLDxxeZPc gAkJ9zs2XypjstBaKVbpkCkJUo/YH+9XslDr8zvvf4dI2Lh0j57VYxZ6h9jO TlqRELeNF3zsNgslvmcs3G1Dwh/ziDfhUSyk8X4EBtqSUJ3wRfalPwsnc6rD q+1J2FHg6fTZm4WKk4byfQ4kPG2cl8Q4wsJRoybtv44knFtxYP48QxYyFKwr hZ1JYG/vBykdFn6y7NP7S9nIKiZITZWF9UVVun0uJEjfrelzlmQh4Z32OeMo CVFv/WQClrLwu5mu1Xk3EkZbxBxjhFlol+l8VtedhMrVltVFI0zc2HXyxCsP ErZpjgnVdjJR3mvK1NGThBTbO7s7GUz0izeq+UtZKEgjcKqWiRxVheE7XiSc SqblLf3CRHtxz8KN3iQwi8/3yJcwsWJZw8YXlPUZ4ut35zHx38CuPXI+JORN ltgdfs7E4XN+S5Ipi691TPBNZmJQ8O5bc3xJiND6+z3sJhPDfwR8P0p50OHR 3KQIJvaqdWMxZZsQXe3c80xMuuwUOO8YCRUPOGc/nWDisavZhDFllbLwXPpR JhoveKgWSTmpbUPXsA0TJdgCe4opC8x8khY+yMSy+kwpHuVj0h42knuZuDzf pVrgOAnNu4Vvqe5gopDhhOUayrrOGd8MlZkY4LW7UJZyVqihgPN6JhqwBH4p Ul7xpHfnuVVMdDy7dPX/fg/FG2eiFzIx+a/WutWUezlK2U9nGcjXP7R0lprP UqCa/+4XAyXXShFsymXrT0rW9DFQwfh0xVvK8ntFrTrYDBxMXRt1hXL80fzY ySYGGo32GRpRnrps/lXsOwM3znspJETZM5WckfvIwKO5ap/eUfGp/5S4Q+cN NV/BgauulLU6tp+yzGJgXE+52f/imybYkunziIGG4t5y96n4i8oGckMTGTi0 fNFiRcpB+msk7l5noG96yLw8av/4HsUWORcZmPLourgy5aL0qU+tXgwsHM9N XEjtv0zlg+khRwa6j7nPPUnlR2y3jrqQBQOtb5LJlVQ+uSqEpW/TZuDD+p8H XKh8+2G4nr1/GwPfrnp3/DGVj+o+Faud5BiYtrXgczOVrwteCEbdEGNgtISk nIIrCQVKUcf4HXT0NTYTdz9Cwn8NriXKDDrCqJPmUao+Ys/vEgmopWP1Dstp G6p+PL/0p88voWPLd4vwTXZUfjmb8lTi6GgecbPj2GESrgrJbwu8QsfLwalx 8pYkkNn/QjGIjoLn/aYZ5tTzTeRLWbrTkfXqjqKiGQmXEpbbnd9Jx9OJUxpu RiT0aw5klG+m4+81WYfb9lP5yP78e8FGOp5awDlkZkDVj3LA7YdL6LjYw3JU Qo+K35fW2gp+K75026pvrk3V65+HBotvtuIQ86uejgoJbx8H3LGOaMVlas6T GkpUPA3MOh6fb0XvL7JVCopUf0yYDVd1a8UrtellYxup+lF2K7PRbEWxDluv zZIkrHRR2P6MR8Npt7AOJRESLgsLRPTTaPgq7FBNpjDVn3Po9duracjIGBWR FiTh258bfpXvaGhpJHmVmCEgOHHwxUAMDeP1aMPaYwRwvr5ev2MHDYt8x8u8 26jz5vCk/k4VGspHc1WfMAgQ5+3x1ZKh4bDaY6V6GgHmf+te64rS8N49/Tfr GqjzR21Yz7C7Bb8JLjvrR51XT58oedoltaCv5D5tVi4Bbheevwgeb8a3XTy3 f4EE/Js3UHNxsBkvqRZPHz9LQModtdFQfjPK0boCmk4R0JhfoXWlthlV0iYk 4n0I2NfN+RmT0Yw3XlT5l9sTIHtYavihTTPqLHn/e1aLgC6Vu2r4rgk1X548 Lj06Akqh6cO9OU3o5yvVWt47An51b7OXP2tCvlpAiQ1nBP6cocl4xTTh2JrJ WfvqEVhYtGbFEtcm1LVa44npI7Btb8qYw4Im7OGuXr3YZgRCrJ8Ujjs2om/G if283GFYFvpix5Y5DXi84muwlMYQRL5TrFo6UY/rsqs2+G0agqnhTPuxwXq0 NNvy6K30EPBdMi4W0+vxmbT7Jvl5Q1Cw9/knvVf1eEVeZ28cbRAshR+Z2brU o4DNeFlYwCAoSGyoEQipQ9VGts3y3AFQ2TGikj5WjbZnpDMPzPTBFf2h+pKg bxh1SOxPamcXrLLKi7df8wl3nYp6Z+7Mg4Lxr2JVsqXoZrlu5Zb7DEhndd3s u5qHoY1LznqY1oKThVQhy/0OfpWwXVTsUgglC0NaPgXchous0OxN/m9Qgz9n 3WW7fPjB5OXZS9XhRuL9pNanEhi9HnS3UIeBB54zaszsPoHbQGSx6z0eGpgu F/z99hs45b60cV/VjbdcWsO3FVVD5Pachbub+jByUAe6q6uhSC852oXfh6EX Umce8KvhZZ+9RBDZh363TwbPF62BC6WuBjFi/Wj2TfAc26MGEqPShHeZ9KPo 1m3esctrgck2SM4p78e4f1GmvSfr4F/hxK3i5wMYGTO86PGVOqDf0/WMez2A oeJWPw7fr4NFI9ft7XEA/dT+M8QvdSDxTuFyLXMAFd44rP+5pB6qvn2eUl82 iPN7CIlUj3rIn5MvMhgyiN/MpJeYL2+A3KUOErVGQ7hyC2+1v1QDBHNyVb2s htB1Sfr6O/INoFX+5N+YyxBO/VRRp2s1wMdtgo+JwCFUOaDj6OreAObSYkYn 04Ywfp/ji9OFDdD77OWY38wQOuy8b5Bg0wiZe+//9ckcxow1TofeuDbCw3F1 Rs7rYRwbX2/fcqwR9kceyex4P4yxb7NOSIQ1wre8wVnV+mFEteLbzzIbYe05 lwmjP8Mou7mV9/pPI2w2nZPtsH8EhzasCm1MaYL6DT/O8BkjuDPk2ZmKtCYo z/+lH8MfwYjmzZ6v8ppA0upNmeLACK6+vv9g3KcmyB8bi9j7dwR1B8+vNe5v gr+LmJ2/RAm89Y75rmJXMxisZ0tFbSVQ1eTpyCt6M2RpKuzcf4LAi2kqHU/4 zdCdUejAOUVg5UwxLW6wGXpPZZ31P0ugw6uGDycEWkDw/v5LkcEERqwSjFVU bAGXxeOPPaMIbOJ4Kj493wLhj3qSZ1IJDDinfPTWahrsSt5vGt5E4OqFI2U2 G2jw2kadeYtGYPGTAvF1KjRIUf7Jvscg8O+PXfW5e2kwX9twTwKH+v8NRvDj BA2kuR9nFAcIjKv1kBb+TAP1TyyVPgESt3koXqiuoUH5gPjFUEESGyYHmm/T adBYaCKyeB6JK+XOxsoM0wANu6rEFpGYEnJ5GiRaIdN1y9WSlSRmKD5mBJ9q hafthi8T5Uk0+nhUQy+4FewFH63IViSx77BcwoJrrZBuoX6xRIlElbBco+SU VtCNvmhQvoXEgpbSordfW+GAY4nFyR0kWh0PXXGpvhXC33q66+8k8beA3il9 Vis0XpyyX6FF4k6V7/JNRCuIBmzrSd5N4seI1ruEJB3qpldI6umT6Cr+gCyS p8PXvANDrQYkzslzNg1TpYP7CtHrnoYk6jO6hEQN6VC/ImmujzGJP7b+Oqvs TwdbXm/BAnMST3wtqhu9SAebZUwHJwsSlziGKJdep8MVTYGXWZYkmkfO7TB6 RIfsKN298tYktrYttfSqooPolu+fyuxJNI7xpJ9rpMP48v9s3zuQWLar1PlK Gx1ijz6KL3Ak8WmS+/GnBB1UeQo90U4kLjcoJvOm6DDlofvvuDOJV8cWB30Q YgB3IiHPwIVEH/N3V5niDAgghOr4riQy/y1c1CfDgLasLbuo+y0efOmSMKHC gPpV2dvc3EhUXbjg8cq9DKgryH7/2Z3EtGIn2Y0mDLijoXXA14PE1d4F2arW DDi8fJvdfE8Spz47Fpn5MmBHmPuwihe1fv9Xu53OMmCZW86pQsrs9cJfjl9i gJ2+n5e6N7X+Onvj4OsM8Dn0ui6X8qdLefVRCQxIyb2QJe1DorqKoO29Bwww uVY9FEk5g2nbnp7OAPrc/Ie9lMVv5LoX5lPjzeQL9HxJjNGc0/+phAGjCZrK dyn/67I+3fCZep7xHiEu5dN3ssc5NQyoTNPR2XiMRL7e7MXhVgZMVWrVOlO2 Ig8LzvAYoBrZU5RIufJJ1o1Fgwz4usho/CPlnWYzYmvHGbDq8tHIDsrZfy2S FAWYECS2y5O676JUToaU5kImmHGbbq2iHGc/nWqwkgmElKbgRsoCIuabrNYx oWzYu4y676L/u7R8N0UmLA47XixPudNjUuOMGhN0Jvb/kqJss8KsLEyHCZnh sycWUa6qSNWL28+EXP2U/0ap+bVOT1Q9NGdCxgmpBQ2UX647eCjHgZpvQ4x8 FuX1NU9bSjyY0HFz4Nx5yokhvx2r/JjAYeiRupSFlIz5tCAm7NmalPyPik8g /bFPVwQTjhT1e1P3X+yNHBsei2XCvQcGR7woO2gYBcy9xwQD4Vx/Ucq6ieTl /3Ko9Yu+mtxP7cerPftFtrxlQqGAnUcrtZ8yIylxOsgEmvT6LhfK8w7qP7Bv YoJWqJy0PZUPQVPJG3zamaC+8mTtdyp/+rOGMgN7mHA1uO+GGuU64XuFt6eZ kOh0d8UglX96hQNaqcIsmL+5jqVDudBtT8UrMRbs87dMvUblazL21dRsZEGg oPG6WSq/XS/o9Mw/yAJm5c3QBKo+lDyqWoZtWCDj7+1+8wiJY2ZWX1qOsoA1 uulzGFVPkbInnj0/z4IEOa1OQ6r+cuseOMJzFuTJ6+h5WlHxKVUwls9jgbTV sjsrD5MI6W92LilhAa/4tF4pVd+NwT9XsWpZcITxOL3XjMRJuem6wCkWfDp4 UrLBiMrvpdc/Ogm3waVFB3rUqP4RO738pf7SNgh0Nyejqf6yvkEperl8G8yL L3glrUfi/osO+nkWbXCo4IJ/DdWvEhtLiruy2iC+viTfkup/Rz4YZFa/aQNn zuek1VR/lMtquPvmYxuomeiE1cuRWHypz/9ycxvQ+40M5DdQ9acosVlaoB0q 10dq26+h4hEW9MzKth2UpTRDGXNI/OUrFK/t1g4vih4UTv8j8INVfKjMyXaY PPs6f8VfAs2VshyJiHYYmy3vlB0nMLCldVVsXjvMaMtxf/YR+FlZM/qzMBsu TGZ9V6wj0Hmr1XNSjA2/LPI9eD8J/KN2puy/tWwwuSzJia8iUEUrZzhkCxvy Pao8mysIvGP032F1WzaYQZ/70BsCvTznS6VnsWHMZMTPI4lAAV9ZjaYCNpxc ffLmjUQCU07sNZvzgQ22xW326XEE1p4NCXdqYMP9CHWiPJJAzYiRzlVTbGCr apddDyRQ5Enry0gTDoQKmkpbWBGYmvqrstCaA6oH5mZxDhGok7GMy3fhQLeX Rr6HCYGnX5osh3McCFIqlrfUI5DxHgMnHnLApCnNvZQ6r7PpmeAzzIG3bz65 fhAhUH5BWW+aERe8uM+iR7xGcPTblmOpplx4n/63hO44gh8jnw08seRCgK4P L898BG2Erg+nHOHC7kHZOTJaI3ht1uJX/GkuiBlJ8GQWj2Dn757Zi8lcWHRh 9cLo3GFM5a9YY93LBemIjF1f2UPo9ywy2XKIC5If6zfoNQyhluukhPkoF9LY zw1zPg9hYztbyuQvF5LqJtT1soZwDuOFzN6lPOCGaY9dPk3dt+pgy+adPNBI dOjaNzOI0mXHDYSjeDAA2oys+YO4UmJqyu8mD0SyD3w0GB/AxYFRr+iJPEgT Ia//6BzAv1vTJXMf8YD8ZZoTUzGAzOds0vINDxS+rnEbuDCASTEWj5+yeRAi nS66tpu6bzpqTmhr8KExOMBeILcP5xV/zU3X4oP59O5jMff6cGallZvYHj6s mSk5MxPRh4O1p2v4xnxQpokZJtn34Q+97Oc3XPlwWWKhdopwH15XkjZvjeHD o4+jlZ12vSgwNfeFP58PEYfcPcwHu9G1gzEW1sMHpN103EPrxvLqV7o3B/ng 8Gf+W4nybgx/4tycOc6HF8ofA6Nud+Osfuls28IOWLvrT4z1rm78d+uMjeH2 Djgn0/gmK7wL/8pxBaWudsDoj6RskX8dOGHxwfmrXCessNhWfqSAixGM/HZd 5U5oihWbXneHi6JHU48Ub+uEoihx7pcALsqfvu6Qo90Jm/o7brzfyUWrOAub BItOWNe3cGFmMAdf/egydbrUCal2uz6YRbSj7z7R3b+bO4GW+i3ewouBLHUX SbmrXdAWrnL526V69JrUELt2owvKDTaOz1OvR+LDYsGuuC5YfD5xs3hzHQoZ lvan3afG+zbsrc2uxS32q0pl87tAKQQvf7CsxvBLP+1kmV1A7pzqiov9hvKV mkkbt3WD4XhBn5vZezxtu3SZTFs3iCqKv07VzAOJO512QtK94Dpz1nK+MhPy HJ9mjgX1wYIdt4+laQ+A7e4wo6oP/ZBoucRdVvM3vHY+KaP1dwD8f41HdKfN wMnxsgITk0H4/++T9vwfnx9pig== "]]}, Annotation[#, "Charting`Private`Tag$11248#1"]& ], TagBox[{ Directive[ Opacity[1.], RGBColor[0.880722, 0.611041, 0.142051], AbsoluteThickness[1.6]], LineBox[CompressedData[" 1:eJxF1H0sFHAYB/DjMjp5XTTlyOu8zVCdRcYUc3LGjdHEOS01L6lbujOb0LWu prGwHam1nZNLu0slO47juJ0ak87LcTruuMudzpEUZlNten7P9t13n7++/z0e eSXkq+YYDIb0N/86ONrZsL9vEmMOrmTHi7Wzi/zmXajv5hZymB8pT7eKHO7A nBudQY5Z2vjUJEAmPxjrOEVBpo8zi6/1r4EnJ60Ng1QjGD9QcjTc9Tu4q3DY wu6rAZxbE7o+W60Hbz8pZ4mjV8AcW5ULW60DtySG0uQPteDqIirDn7gMLse+ otOMGjC1ctx7ul0NbuzTslsKFsE9C35HXnuqwF72MbFpz5XgtwGsAs2SArwg 7TxJIEyDtUGNYeIPcrBDBY8QbDYBDiKYgribo+B7F4yfhQwZ2CmNX3fpmATt /ZLajXj3gLlK7WM9kw/OTnV9r7zSABbiyqckpfXR/31GY+ZWlSkAe6337kRI hGAiZ3YsOVMCjiM5Yre6ZOBaykxlSPcoWJaMt0lxnAAbPZwqvjTLwaW3A6m1 ztPgmXl7cv6IApxbdu6bZZISPBQY/mjIQgX2PSxaaU1YBONFhXEWLDUYs2vO o2k04N+pfTlSn2Ww8jTlhA9TC76ZYe/gOa8DuzQsZx7Cr4D5WS9ebjL04Iyo uwkjfQZwZ06xZ8TeKngbdx4rTDeCyRORBDx7DYyLtB4UPUMebJ27eJmDHMIo oz7lI9u6d9e4SJE/FoUtOf1AjrHyr7NNMqH9W9vH+anIAqWMS8pAdhPk99bk Ie+lt+pwZcjdHPcoyzbkGzYmWVsHsi+9nxzfidyQmH39vgiZttFcj5UjB2QV uHEUyOrhs7xYFXIKW9FfpUe2Mm8nepiQxYX0yYGfyHem4im5u8gH/wv8B+0M Suw= "]]}, Annotation[#, "Charting`Private`Tag$11248#2"]& ]}}, {}}, { DisplayFunction -> Identity, Ticks -> {Automatic, Automatic}, AxesOrigin -> {0, 0}, FrameTicks -> {{Automatic, Automatic}, {Automatic, Automatic}}, GridLines -> {None, None}, DisplayFunction -> Identity, PlotRangePadding -> {{ Scaled[0.02], Scaled[0.02]}, {0, 0}}, PlotRangeClipping -> True, ImagePadding -> All, DisplayFunction -> Identity, AspectRatio -> NCache[GoldenRatio^(-1), 0.6180339887498948], Axes -> {True, True}, AxesLabel -> {None, None}, AxesOrigin -> {0, 0}, DisplayFunction :> Identity, Frame -> {{False, False}, {False, False}}, FrameLabel -> {{None, None}, {None, None}}, FrameTicks -> {{Automatic, Automatic}, {Automatic, Automatic}}, GridLines -> {None, None}, GridLinesStyle -> Directive[ GrayLevel[0.5, 0.4]], Method -> { "DefaultBoundaryStyle" -> Automatic, "DefaultGraphicsInteraction" -> { "Version" -> 1.2, "TrackMousePosition" -> {True, False}, "Effects" -> { "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> {"ratio" -> 2}, "Droplines" -> { "freeformCursorMode" -> True, "placement" -> {"x" -> "All", "y" -> "None"}}}}, "DefaultMeshStyle" -> AbsolutePointSize[6], "ScalingFunctions" -> None, "CoordinatesToolOptions" -> {"DisplayFunction" -> ({ (Identity[#]& )[ Part[#, 1]], (Identity[#]& )[ Part[#, 2]]}& ), "CopiedValueFunction" -> ({ (Identity[#]& )[ Part[#, 1]], (Identity[#]& )[ Part[#, 2]]}& )}}, PlotRange -> {{-0.9999999591836735, 0.9999999591836735}, {-2., 2.}}, PlotRangeClipping -> True, PlotRangePadding -> {{ Scaled[0.02], Scaled[0.02]}, {Automatic, Automatic}}, Ticks -> {Automatic, Automatic}}], FormBox[ FormBox[ TemplateBox[{ RowBox[{"Re", "(", RowBox[{"Q", "(", TagBox["a", HoldForm], ")"}], ")"}], RowBox[{"Im", "(", RowBox[{"Q", "(", TagBox["a", HoldForm], ")"}], ")"}]}, "LineLegend", DisplayFunction -> (FormBox[ StyleBox[ StyleBox[ PaneBox[ TagBox[ GridBox[{{ TagBox[ GridBox[{{ GraphicsBox[{{ Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6]], { LineBox[{{0, 10}, {20, 10}}]}}, { Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, ImageSize -> {20, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #}, { GraphicsBox[{{ Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.880722, 0.611041, 0.142051], AbsoluteThickness[1.6]], { LineBox[{{0, 10}, {20, 10}}]}}, { Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.880722, 0.611041, 0.142051], AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, ImageSize -> {20, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #2}}, GridBoxAlignment -> { "Columns" -> {Center, Left}, "Rows" -> {{Baseline}}}, AutoDelete -> False, GridBoxDividers -> { "Columns" -> {{False}}, "Rows" -> {{False}}}, GridBoxItemSize -> {"Columns" -> {{All}}, "Rows" -> {{All}}}, GridBoxSpacings -> { "Columns" -> {{0.5}}, "Rows" -> {{0.8}}}], "Grid"]}}, GridBoxAlignment -> {"Columns" -> {{Left}}, "Rows" -> {{Top}}}, AutoDelete -> False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings -> {"Columns" -> {{1}}, "Rows" -> {{0}}}], "Grid"], Alignment -> Left, AppearanceElements -> None, ImageMargins -> {{5, 5}, {5, 5}}, ImageSizeAction -> "ResizeToFit"], LineIndent -> 0, StripOnInput -> False], { FontFamily -> "Arial"}, Background -> Automatic, StripOnInput -> False], TraditionalForm]& ), InterpretationFunction :> (RowBox[{"LineLegend", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", TemplateBox[<| "color" -> RGBColor[0.368417, 0.506779, 0.709798]|>, "RGBColorSwatchTemplate"], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", TemplateBox[<| "color" -> RGBColor[0.880722, 0.611041, 0.142051]|>, "RGBColorSwatchTemplate"], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{ TagBox[#, HoldForm], ",", TagBox[#2, HoldForm]}], "}"}], ",", RowBox[{"LegendMarkers", "\[Rule]", "None"}], ",", RowBox[{"LabelStyle", "\[Rule]", RowBox[{"{", "}"}]}], ",", RowBox[{"LegendLayout", "\[Rule]", "\"Column\""}]}], "]"}]& ), Editable -> True], TraditionalForm], TraditionalForm]}, "Legended", DisplayFunction->(GridBox[{{ TagBox[ ItemBox[ PaneBox[ TagBox[#, "SkipImageSizeLevel"], Alignment -> {Center, Baseline}, BaselinePosition -> Baseline], DefaultBaseStyle -> "Labeled"], "SkipImageSizeLevel"], ItemBox[#2, DefaultBaseStyle -> "LabeledLabel"]}}, GridBoxAlignment -> {"Columns" -> {{Center}}, "Rows" -> {{Center}}}, AutoDelete -> False, GridBoxItemSize -> Automatic, BaselinePosition -> {1, 1}]& ), Editable->True, InterpretationFunction->(RowBox[{"Legended", "[", RowBox[{#, ",", RowBox[{"Placed", "[", RowBox[{#2, ",", "After"}], "]"}]}], "]"}]& )]], "Output", CellChangeTimes->{{3.85479971844681*^9, 3.854799724619273*^9}, 3.854799807577301*^9, 3.854811848706294*^9, 3.854811965354575*^9, 3.854812029293939*^9, 3.8548121000683193`*^9, 3.8551401406559143`*^9, 3.8551402426035357`*^9, {3.855140285553207*^9, 3.8551402905760927`*^9}, 3.855140341435275*^9, 3.855140409112891*^9, 3.85514047956218*^9}, CellLabel->"Out[28]=",ExpressionUUID->"acb4d93c-0f41-4dcb-ae7e-ccc3e67c7dfc"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Check \[OpenCurlyDoubleQuote]normalization\[CloseCurlyDoubleQuote] \ condition ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{"|", StyleBox["P", "TI"], SuperscriptBox["|", "2"], "+", RowBox[{"(", RowBox[{"1", "-", SuperscriptBox[ StyleBox["a", "TI"], "2"]}], ")"}], "|", StyleBox["Q", "TI"], SuperscriptBox["|", "2"], "\[LongEqual]", "1"}], TraditionalForm], "errors" -> {}, "input" -> "|P|^2 + (1-a^2) |Q|^2 = 1", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "8df0ab74-4311-4c2a-9568-6ece8b27c0f5"] }], "Subsubsection", CellChangeTimes->{{3.854799738839718*^9, 3.8547997703109407`*^9}},ExpressionUUID->"a9c578e8-d664-43c2-b184-\ 01b1619ccb23"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"Conjugate", "[", RowBox[{"P", "[", "a", "]"}], "]"}], RowBox[{"P", "[", "a", "]"}]}], "+", RowBox[{ RowBox[{"(", RowBox[{"1", "-", SuperscriptBox["a", "2"]}], ")"}], RowBox[{"Conjugate", "[", RowBox[{"Q", "[", "a", "]"}], "]"}], " ", RowBox[{"Q", "[", "a", "]"}]}]}], ",", RowBox[{"a", "\[Element]", TemplateBox[{}, "Reals"]}]}], "]"}]], "Input", CellChangeTimes->{{3.802024497633624*^9, 3.8020245418604517`*^9}, { 3.802084095361841*^9, 3.802084108653825*^9}, {3.8020841409745407`*^9, 3.802084161670167*^9}, 3.802084212501238*^9, {3.802084244652742*^9, 3.802084278020651*^9}, {3.802084423094756*^9, 3.8020846298402977`*^9}, { 3.8020849581323833`*^9, 3.802084994306072*^9}, {3.802085050767692*^9, 3.802085163990142*^9}, {3.802085202584613*^9, 3.802085251665312*^9}, 3.802085307377961*^9, {3.8020853486829643`*^9, 3.802085352330017*^9}, { 3.802085412982503*^9, 3.802085413507552*^9}, {3.8020862949066896`*^9, 3.802086378522773*^9}, {3.802086419758582*^9, 3.802086420467738*^9}, { 3.8020864521444483`*^9, 3.802086455710369*^9}, {3.8020864973077593`*^9, 3.802086542077324*^9}, {3.802086599556776*^9, 3.802086805617375*^9}, 3.80208783340549*^9, {3.8020879041098413`*^9, 3.802088142007761*^9}, { 3.802088178651374*^9, 3.80208822518908*^9}, {3.802088894055563*^9, 3.8020888982080603`*^9}, {3.8020889340426083`*^9, 3.802088951494369*^9}, { 3.802089002146434*^9, 3.80208900393335*^9}, {3.80208915422486*^9, 3.802089176573462*^9}, {3.8020895319809847`*^9, 3.802089546420405*^9}, { 3.802089626200128*^9, 3.802089990969708*^9}, {3.802090049382504*^9, 3.802090174649485*^9}, {3.802090217734236*^9, 3.802090233521098*^9}, { 3.8020902703924093`*^9, 3.802090351054804*^9}, 3.802090825928948*^9, 3.802090919018063*^9, {3.802091037601871*^9, 3.802091136133688*^9}, { 3.8020912205292892`*^9, 3.8020912572495728`*^9}, {3.802091496735574*^9, 3.802091497836735*^9}, {3.8020916303004417`*^9, 3.8020916542671137`*^9}, { 3.802091708229781*^9, 3.802091710306443*^9}, {3.802091754425097*^9, 3.802091788472343*^9}, {3.8020919239265757`*^9, 3.802091956125485*^9}, { 3.80209201247558*^9, 3.80209208120962*^9}, 3.8020921637950983`*^9, { 3.8020922289969063`*^9, 3.802092264054653*^9}, 3.802092316719646*^9, { 3.8024577879505*^9, 3.802457866582061*^9}, {3.854789904145109*^9, 3.854789949024129*^9}, {3.854790059546473*^9, 3.854790428067522*^9}, { 3.854791736268067*^9, 3.854791737832754*^9}, {3.854791771374771*^9, 3.854791776278699*^9}, {3.854792294683609*^9, 3.8547923550670233`*^9}, { 3.854792444977457*^9, 3.8547924893606977`*^9}, {3.854792577733321*^9, 3.854792622204557*^9}, {3.8547997953586483`*^9, 3.854799795852812*^9}}, CellLabel->"In[22]:=",ExpressionUUID->"6ce02805-2b03-400c-83c8-13a50a6c772f"], Cell[BoxData[ RowBox[{ RowBox[{"(", RowBox[{"0.9999999999999993`", "\[VeryThinSpace]", "+", RowBox[{"0.`", " ", "\[ImaginaryI]"}]}], ")"}], "-", RowBox[{ RowBox[{"(", RowBox[{"8.881784197001252`*^-16", "+", RowBox[{"2.220446049250313`*^-16", " ", "\[ImaginaryI]"}]}], ")"}], " ", SuperscriptBox["a", "4"]}], "-", RowBox[{ RowBox[{"(", RowBox[{"1.7763568394002505`*^-15", "+", RowBox[{"2.220446049250313`*^-16", " ", "\[ImaginaryI]"}]}], ")"}], " ", SuperscriptBox["a", "6"]}], "+", RowBox[{ RowBox[{"(", RowBox[{"1.3322676295501878`*^-15", "+", RowBox[{"0.`", " ", "\[ImaginaryI]"}]}], ")"}], " ", SuperscriptBox["a", "8"]}]}]], "Output", CellChangeTimes->{{3.85479249004858*^9, 3.854792517484809*^9}, { 3.854792582648171*^9, 3.85479259354699*^9}, {3.854792639973487*^9, 3.854792664677833*^9}, 3.854795913721328*^9, 3.854799379305612*^9, { 3.854799792710635*^9, 3.854799810826932*^9}, 3.854811848774515*^9, 3.854812000101851*^9, 3.855140140759308*^9, 3.8551402856915407`*^9, 3.855140437513804*^9}, CellLabel->"Out[22]=",ExpressionUUID->"d76ed83d-6b49-4217-8679-5ebd22283d8c"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Application to Hamiltonian simulation", "Section", CellChangeTimes->{{3.854793268276285*^9, 3.854793293042036*^9}},ExpressionUUID->"3e09731d-05d0-481d-96c5-\ 2007266765ea"], Cell[TextData[{ "\[Bullet] For quantum dynamics, we need to construct a polynomial \ approximation to ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ SuperscriptBox[ StyleBox["e", "TI"], RowBox[{"-", StyleBox["i", "TI"], "\[Lambda]", StyleBox["t", "TI"]}]], TraditionalForm], "errors" -> {}, "input" -> "e^{-i \\lambda t}", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "07c0da96-d5ad-422d-83e7-ca95c27334e7"], ". \n\[Bullet] We also need to assume ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{"\[Lambda]", "\[LessEqual]", "1"}], TraditionalForm], "errors" -> {}, "input" -> "\\lambda \\leq 1", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "5426c015-b773-4a72-a332-4597fdd834c3"], ", which can be done with appropriate rescaling. \n\nJacobi Auger expansion \ (similar to plane wave expansion in scattering theory): " }], "Text", CellChangeTimes->{{3.854793305199485*^9, 3.854793370417179*^9}, { 3.854793672802127*^9, 3.854793795887865*^9}, {3.854793837513523*^9, 3.854793920294013*^9}, {3.854794907351964*^9, 3.854794915244308*^9}, { 3.854795920270471*^9, 3.854795935218124*^9}},ExpressionUUID->"bcba399b-d003-4a64-a4fa-\ fbc8f45baf41"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox[Cell[BoxData[ FormBox[ RowBox[{ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ SuperscriptBox[ StyleBox["e", "TI"], RowBox[{"-", StyleBox["i", "TI"], "\[Lambda]", StyleBox["t", "TI"]}]], "\[LongEqual]"}], TraditionalForm], "errors" -> {}, "input" -> "e^{-i\\lambda t} =", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TemplateBox[<|"boxes" -> FormBox[ RowBox[{ SubscriptBox[ StyleBox["J", "TI"], "0"], RowBox[{"(", StyleBox["t", "TI"], ")"}], "+", "2", UnderoverscriptBox["\[Sum]", RowBox[{ StyleBox["k", "TI"], "\[LongEqual]", "1"}], "\[Infinity]", LimitsPositioning -> True]}], TraditionalForm], "errors" -> {}, "input" -> "J_0(t) + 2 \\sum_{k=1}^\\infty", "state" -> "Boxes"|>, "TeXAssistantTemplate"]}], TraditionalForm]], "Text", FontSize->24, FontColor->GrayLevel[0],ExpressionUUID-> "527c4392-cc8f-41e5-98a2-cdbf4faab0d7"], "Text"], StyleBox[Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ SuperscriptBox[ StyleBox["i", "TI"], StyleBox["k", "TI"]], SubscriptBox[ StyleBox["J", "TI"], StyleBox["k", "TI"]], RowBox[{"(", StyleBox["t", "TI"], ")"}]}], TraditionalForm], "errors" -> {}, "input" -> "i^k J_k(t)", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]], "Text", FontSize->24, FontColor->GrayLevel[0],ExpressionUUID-> "2f3224d1-8b97-496a-939c-b7edd3272398"], "Text"], StyleBox[Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ SubscriptBox[ StyleBox["T", "TI"], StyleBox["k", "TI"]], "(", "\[Lambda]", ")"}], TraditionalForm], "errors" -> {}, "input" -> "T_k(\\lambda)", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]], "Text", FontSize->24, FontColor->RGBColor[0, 0, 1],ExpressionUUID-> "79ae9eaa-520e-44bb-8b2e-d975d14f79c8"], "Text"], StyleBox["\n\nWhere ", FontColor->GrayLevel[0]], Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ SubscriptBox[ StyleBox["J", "TI"], StyleBox["k", "TI"]], TraditionalForm], "errors" -> {}, "input" -> "J_k", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]], FontColor->GrayLevel[0],ExpressionUUID-> "04cf42fe-253c-4985-b79b-65f56a3a85f5"], StyleBox[" are Bessel and ", FontColor->GrayLevel[0]], Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ SubscriptBox[ StyleBox["T", "TI"], StyleBox["k", "TI"]], TraditionalForm], "errors" -> {}, "input" -> "T_k", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]], FontColor->GrayLevel[0],ExpressionUUID-> "4194fff8-470f-4473-a353-4b501b0891d4"], StyleBox[" are Chebyshev. ", FontColor->GrayLevel[0]], StyleBox["Truncate for polynomial approximation", FontColor->RGBColor[1, 0, 0]] }], "Subsection", CellChangeTimes->{{3.8020244674793577`*^9, 3.802024469257287*^9}, { 3.8547927356939983`*^9, 3.854792736134763*^9}, {3.854793922747849*^9, 3.8547939267333593`*^9}, {3.854793965608692*^9, 3.8547940391079807`*^9}, { 3.854794094955572*^9, 3.854794235922949*^9}, {3.8547951185570393`*^9, 3.854795198659296*^9}, {3.854795257772464*^9, 3.8547953005087442`*^9}, { 3.8547960150826273`*^9, 3.854796022107719*^9}}, TextAlignment->Center,ExpressionUUID->"39e28940-1687-4773-b5af-97ef90e0a9cd"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"P", "[", RowBox[{"\[Lambda]_", ",", "t_", ",", " ", "trunc_"}], "]"}], ":=", RowBox[{ RowBox[{"BesselJ", "[", RowBox[{"0", ",", "t"}], "]"}], "+", " ", RowBox[{"2", " ", RowBox[{"Sum", "[", RowBox[{ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "I"}], ")"}], "k"], " ", RowBox[{"BesselJ", "[", RowBox[{"k", ",", "t"}], "]"}], " ", RowBox[{"ChebyshevT", "[", RowBox[{"k", ",", "\[Lambda]"}], "]"}]}], ",", RowBox[{"{", RowBox[{"k", ",", "1", ",", "trunc"}], "}"}]}], "]"}]}]}]}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{"With", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"\[Lambda]", " ", "=", " ", ".5"}], ",", " ", RowBox[{"tmax", " ", "=", " ", "10"}], ",", RowBox[{"trunc", "=", "10"}]}], "}"}], ",", RowBox[{"Plot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Re", "[", SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"-", "I"}], " ", "\[Lambda]", " ", "t"}]], "]"}], ",", " ", RowBox[{"Re", "[", RowBox[{"P", "[", RowBox[{"\[Lambda]", ",", " ", "t", ",", " ", "trunc"}], "]"}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "tmax"}], "}"}], ",", " ", RowBox[{"PlotLegends", "\[Rule]", RowBox[{"{", RowBox[{"\"\\"", ",", " ", "\"\\""}], "}"}]}]}], "]"}]}], "]"}]}], "Input", CellChangeTimes->{ 3.8547953128779783`*^9, {3.8547960249546013`*^9, 3.854796326986027*^9}, { 3.854796403754579*^9, 3.854796573135689*^9}, {3.8548123894006233`*^9, 3.8548124263748417`*^9}, {3.855140606362084*^9, 3.8551406465520906`*^9}}, CellLabel->"In[43]:=",ExpressionUUID->"780da9ab-38bc-447b-89c8-22a0155ca44a"], Cell[BoxData[ TemplateBox[{ GraphicsBox[{{{{}, {}, TagBox[{ Directive[ Opacity[1.], RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6]], LineBox[CompressedData[" 1:eJwV13k4VO0bB3BbG5J1zBm7CSVp0UKL51a8lZSlSPJWRpYQRdKisoZkiRYi W/ImpZSltDy2kn0N2bIk28wce9b8zm/+metzzcyZ53qe733f5yix3Mzs+Hh4 eEx5eXj+//74BmvppjOhunWL/3+RqKRB0eiY4ll0qO958ALlxa+f/RUVryCd 2SblKcoiUbmvBVv80EVWYPUg5ccFYQpLFEOQi8C70BbKGuSZ8EWHCKQQcsGq iHKHSMLf+eYoJGeds+M/yqvTVmybnbuPAjOCNIIou+z2PPdHIRYtc+RsYFF+ 29iVOrE3Hv3UZ+trU551NmofdUhELXm3nVdQTuJtD/ljkIwOzNQ8+f6XRPpV d1WmmlOQpEAJJ57yQMy+womzqei/+24HTlG+c2bBenzuKfqzvj1XlnLDnOO9 EYV0dAAplwYvkOhSqfxGMus5+h4g7rSdMiO6sYKz9wXa6DQq3z1PIpY68A87 vEJTy7M+q1Dmm7taM+H9GnFTMtML5kjU2GUuN2GQhcoeN6ZZUL78Quj9WPMb 5Fnf1uwxSyLDqN9Lx5LeonCeLqG5GRLJXi48Ono2G0ldNjO7Trlgr9cIOZeD PtyqpXlMk2hZW48aR+E9Wu5z04c5RaKWgk+e7IH3iLEUGfpPkuh5WkzxcFY+ Qgw5tc4JEh12P3xqaO9H9OlaHCNwnEQPVuQ/6HfA6OCd2OGHIyTK9jJ096AX INc4MK0lSRQ/tJ0Y9y5AIWL95fyUXWvFHcYMCtHBFVmzJzgkEo//xjfaXIRe XW8zrhsk0ZxIbvr5XcWorIJxvGeARL98n5iMJBUjkz4nL24/iXIdbiSQZ0uQ zL3G8T99JLLS2rKDO/cFHedljNR0kyi5PPHCsEIZmjtmKHOvhUQTMTnqP++U obC8zbE7m0n0j0NFb/1MGRpazDDr+E4iNv8f8/yGchQob3JlVQOJtHcf3hES VIkKmiLjNatIVPtqnk+NrEErece9LAuo9aIVJ1dq1qKFEbL6yWcS7VS9tyPo TC3apDJmN/iRROET6eNX66jPOfvMbd+TaMvd73asjDoUYKG7R/wNiW6Waxhu OtWAVJcNJ1mnkOhZVp5qxv0GdG9GXNEoiUT1MXv4VSobkJsq/rU1gcqvg+VH uk4jqty/BqZjSVTOH6DJI/EdbfaUNNS+SyLa7jbxmq9NiDl1XiDtBokyX4W0 uWi2ovgQF8ZOcxINiQ/dMbRpRXxuwWeemJFI7ZKh7pp7rSjc2PLjUhMq37uE kntnWtFy9Q8PPhuSKKrsjr3VlzakKXGQbxxRee2NGN1n3YHKN4tJua0lUaJQ hb+YQRfKOT1vnjfNRbq7r1n7nOxC1eUGm/0muajddd1W0qsLJa6bVt8/xkX0 +tDflc+7kNupFI8vw1x0N8boQLBoNzr51Pd8YCcX+atUiyy2daMr3nHhMsVc 5IDqYtnuvag/TWxbajAX8aRM5beF9iIn677O3wFcFCMg216e2ota/Vr6Vvty 0bcye/n0pl6k22snGn2Fi9YcnU+x2/ELBYbpDWx34qIBR7UXnbx9qNjPVTXV kIucoq5/ro38jTTNe2WFlnORa9/a3uzMQfS3qXzndk8O6jr5Y3lh2SBiXEeH 2G4cdKQlWLPq1yAyaSyQjXfiIO2K/it9jCFUp587M3yKg/iznorSgoaQy2Wd r0aGHBTjraTrdWoYfbeS85eQ56BCCSJWR5SDdtophXwsZCNBpQC1n9ojSHzj prd5M8Po2667G24YjCDFnIime+PD6JZlwnY5sxGkw1uZdI4zjPgi3+074TyC Sr++WS/ePYxm/7Idmh+PIFrhRn21b8NouN38WQ3fKBrr6t7td38YVcaorS2o HEU7Ytst+NYPo3DRCvXk0+Poodal7jPGQ4jVdhTzV04h9nIgnm4fQBe+BLDc 1GaR73ak3PCrD2WdFz5CP7CATuoFOez360XJ15yHNh7jgcMyT2zO5XYhv8R5 0wk2L0SrWbqLdLQhnoYu+HiVH+BFwrH9Kk1I9aL95aPqS+B7hfvgFqca9Gvy oYXQp6XADGvapC1VjAz7nK5HXVgOew87KloLpaHDvyLXycsLQpq17bs9N1/h oWKWz7eXQtDgyf/cdX8J1hRrktt7aiVY5huxfItqsSCatGkfF4FHigarpkua 8D6O1R9lc1GoGZB4YLesA7cUneG10hADpk5kI8+5bpwTXHJScVIM3thpRUlN 92L7JZ5d1dXioKrxx9XC7ze2db+WI/xIAhxEVJK60wbwBod11UY2kqAes+SR fcAQ7jyr4PbyjCRYcQ2ZE6FD+I6LhKiIoyTI/tIf9Y4ewoMX5kxrXCVBjTVt 7pMyhJ94V3w39ZaE4f5p2cMFQ1g6yrnD4qEkXI7kuyIwN4QXP2awT1dJwrUq hSMNLsO4RlxDyFNHCsBGN+etHhvrnjs3GLdLCoby0vbF72PjzNLM0iIkBa06 5pd8DrFxuPemANF/pGB+eUfqzuNsbPR720KGmRRoiqxZf/08G5e/1yN7nKWg TddFSSOBjb+cPtZgkiAFM7rl7lpTbLzlQ0yWV7IU/Cp7/T1sjo1TpVojElKl gKtLvuvl4eCAcmsj9nMpoIWvXvAV4uA9W1hfgvKkIO+yi12wIgcXLDuX97lW CjJn4hY/HODgD5l+cRr8NLj/qbD2XQwHLzUyZzxcSgPujhSVgcccbDqoFssr SINjvrKS4k84uJ9Z9aBJlAZdhjL55i85WCJWOspHngZ+m5ii8YUc7Oz/IrhR hwZ39z6lrR7kYBnLJk/v8zTYKVJzRGYLF9tPPpv47UGD+iUM2XhtLs6KuuZh 6kUDnqYyG+ndXLyvSvGC6g0aeO2XZC0YcLHHHmeX2lAa5Be/UQg6xsUVGjy2 q/+jwfMRi47IK1x8jVfdpLKDBueMLFdcfs/Fv/WFWlS7adD0fHzX1k9cbBLM PuX7iwbG616t4BRw8epVr9y2DdNA/8qQs9E3Lq6U3RKRNE2Dz9UsvfImLpbX 1q2+KCENrj8XUgrHuLjI1cxI7oA0FOiJL49TJbHGG61GLyNpcGDQNJ+vJfGD SUnremNp0O1Rb32rQWJn72bnIAtp6IiwK8rdTGLJ29aho7bSEB1wtuSqLont n9qXf7khDTeaVGXnjpJYsP3qftdsadgoK/T+x3USb/NKXxmRJw2VZ2d4RX1I bCPeUv8qXxrmLotc1fMj8bv9W/8dLZAGmq7F1P1b1PVyyAsXq6Sh2FSNtiSC xIXhZ+Ku/paGRJuojLkEErPXRp+OG5SGmyNGBb+SSEz/UqjykS0N2mZrF7+l kNh1Tv71wpg02KkrzvqlkVjW8UeJzyL1+w6Zjo8vSXxZ7zD3Fp0ON/ztGfs/ kDi13fvtfzJ0uILXvO79SOIar4zL3+TpECz7xfnqZxKrZi7nF1Shg+Pue0di C0ncwCiWDttEhxOFnM74UhL/zRltf7mFDoR3R7xYGYnVTRVTqrfTwSaj/IZv OYl9gq5riOrS4cz6Ty/NqkisObFdL9qQDqZfBd/k15PYKsJ+afYhOljNDIQs byRxoPr9ikYTOqzxkQ4w/U7i9tNj5rRjdGgjHnc2NpP4dtULp1hbOszLz6gm t5O4L03pXqI3HZ50tLYb91H7E9A9w+9Dh0bmFjW73yQeYyWfcvSnw8GJf3Iu 9pOYR0Fp3abbdGCONMX5DZKY8VCxqOghHaR23g47ziGxkmeXmlocHc75R40A l8RrjiSFhSbQIT5p6L4ySeKtqxSPH31KBxXGYFrjCIlNghRG+t7QYaXCSMf4 OImP2f00N8ylQ+kgGL+cIPHJvYkfMt/TYX3ve16bSSpPi/JBXgV0cNFMXZs3 RWL3jk52WzEdsueaE47/oc7nQ4IZlNIhnPvu6BTlW17y8iuq6fAwVtxPfobE Yead/ufq6HBdzGImnfI9rYTBukY6uGq2v94wS+JkUi7nURsdLm817Vo7R+Jn VR2MxU46aKyWsEygnJnx2Me2hw6qhZ9WrZwn8UcHOSONQTp4uN7Q/UG52KAj K5JNh+JvbjnbFkhcznwsPUnS4V87HpdwyrW8/14/Pk6HFTcVWV2Um3/K9n6a osOh44UPNf6SuPNT+37lWTq4wXdhD8q/4uIzby3Q4VKIceFbysNXrCWHeQjQ iFqdw6U8dkz2qrEAAasLDwwwF0k8s7X959tlBOQE5lgdocwjGW9AFyJgieUp 4euUl46dyPAWIWDxHVpIoryyVkasW4wAm80m6zFlycy2SwZSBLyxCI1vpsy4 E9eeTifAsJBtOERZyenEHhFZAkxoF7T/UF6zX+aZuwIBH/lodn8pb1BtW9ms TID5rtZa6nkIbxOI89ipSq3H99O1Ocq7eqx+JK4lQP9Rgd0o5b0FDCSwnoDd Fn3h3ZQNE1pTHTcS8CJc808FZRPvR4JVWgQ4qSQlvKZ8zMrq/KbtBCgtQYER lE9qM5ru76DWpySW4UjZjta6c3Y3AeqX5ER3UXaZiE0+qUdAw3KXbOr5B7vX H19WrE+AYv+SmDpq/668Js6p7ScgV20yP5qyT/iP+tCDBPAO6cuZUL7lEqs9 cpgALQtevJTyvTWEwHtzAn5MTxawqPOzPqn6X68ltX86MhYilFff0zIUsSaA zx8Gsqk8ZC8eirRlEdC/pIR/nMqP99YTWuF2BLQkQthtyvrOjk3vHAnYIjAr IUe5sclPVsSN2t9IR0KLymO8cCTWdicg8PSemFwqr2f2PGbZehIgZJQiuZXy xMu8Z++uEaDn+M9S5jSJJQLZW2yDCTh7OKSmiKqX1vyZ5rBQAhIkpnfJU04Z WXrtXTgBT/WePrtI1ddma6XClfcJ4OlXuSZO1Z+ZlsWhd8kEdD3/sUqcqk/G WduRnlQC0u8YuxtQ9duTcD565TMC6i25DR5UfbsLhv5gZRLwoEn9fiGbxFHd 2G7lBwIijjlKKFH9wUq6arn2ZwLkFwfcNQZIrHyoNYNVSMC13a/qN1P95M27 idG8UgJ25aVHb6T6T33E2husRgKMBWoURrpI/OjLNqWwZgLoRmRgw08Ss+b2 luS1EtA89YST1Unl3/6k4MpuAsbf+xSdovqZmG70gzwOQd0Mbgi6SfW7Fo8k nZ4RAgq0pRZ3NJE4Kf1lu/AEdf4ioldHqX65UeqbMmuWgLJSyavGVD81Yc9n Ci9jgPeOTc9qKkkc+ci+1EaRQfXzeU5XAYm9Lvl5eDMZsOav69ujmMqfWYLC Q1Xq+77bsoo/UfNQsMmrUoMBe/7LsYjOJ3HZFYM123UYUH14l8nMWxLzHV8d ImzGALuxJdu6npJ4cAtsVTNnwM7GlIRlqVR/ELXu1rNkgF/K673q1DxK/Bat 43WSAVHaSN+Gml+7tPmHup0YsKRzk9qDByT2lO4xzPNnwGlniTwZav5Zjy9M 1d1iwNkxHsE//lS+aogn7BAGjBhpJVT5klg8yHROKZIBHjezQ12o+Zo5VZBx J54BgcPuARcuknjge6IwK4cB3S+CdGJsqPO792+1cD8DcuHNrhPa1Dx8Kyjj PET9n0BAsu1WEv+sy3Mo4zCgvGz+uj0138+sEue9NUGtr58v6QQ1/11vf9Va 5JUBt7WeD/kVSeznsyF2VFYGhHm/1hguJXG6C++ZJjMZyF361yu/motdQjNf b7GQAYMFt2HxCi7WfH5iIeq4DOz5bTNoV8rFb/pzHhiflgHHGpugOczFH1lO Zd/OyUBAz6GTfa+5uNayQfNDsAyY0rtFsu9y8bT+05nEzzLgYVhht8eYi70/ JPqXFMrA/qN+o9sPcjHP5kcrB0tk4G6XQ+2afVy8TCFCaXOFDKS/gNvzulxM m7l8oKSZul5q16KzJhdrvTSKHRiRAf75MT9nYS52lZrQ3sSUBdUfVeGjJRzc 27fnclGQLEj5Vv2jt4a630ysTlEPlYUT746rJitzcJKlVVVUuCzs7Nv6aU6W g09WXFC2vS8LQil/JJPFOLj1dVIl/xNZOFYVHPJilo3rr/1VNPgsC50t+hdK K9i4SCy/rHRCFjbwEgWSTmycsnuTTJWNHLgfSo4xejiMiw4L+s+dkYNGFTuT lshh3HOqd2itoxzMi+u+OHV7GDP97uffcpWDfz7uO2pzfRinfp05DtfkgKf0 jt9G22H81Lgo5u09OTBrz4LF9cP4mc0R2qNSObDgNL20o+7/XwZ6ijmsl4fC 5D+N0S2DOMjc9iD/pDxEjP8OV+3tx1vbk9ODqhTgCFvFudy3D8esjXtZdVAR tvQumbgo1otvFoTFHHyvCBceEDevbu/Ch2TMn5avVoKeCvME3SutWM+19IRY iBIIdIiBbVojTpRlra6cVIKhDQqpajsq8eri/x4cPqEMkgkigSJeGEeuL1Kv /6wMOmdhLDz9EbYfF6grU2dCUu1Ru45D2YiVNaYYEsGEg7yHbo+kfEUufkfm De8yYXpH3c9/Wr8izyPZzcLRTHgTD5UxYqUoeNIzPPIBE8SGNNAGn1KUqTMz 9/AxE07n+rBU/v2GZgoXm9IymPD0zjqj9bRyFFkvHFZSyoTR/gIDm4Aq9OjJ OcdbZUxoetnZKvWhCj25WL13fwUTVuyTeFsyWoVyaZGzFdVMeL4ulE/sVDVq t5JwbPjOhM513lMsnRqk1kvs7ellwt6c9TcuXKpFG7Ovyqf2MeGOwbWzcXdr 0Y7Athm7fiZYxq5TLnpRi4zUHr8eHGLCgjH+vaS3Frk7K8qPjjLB3FpJ3d64 Dnnv8p15M86EXAWS4+lUhwJX9jRenGTC63ADc7/AOhTz6kno9DQTXoVlzER+ qEMpPgIO+bNMsDc20ohuqkMZpnZ7vOeZoKAcw7k7Woeylb/K6f5lwrzBce0w 4Xr0eVx1ZnGRCdvTrJYFqtWj/wHOa8b5 "]]}, Annotation[#, "Charting`Private`Tag$15014#1"]& ], TagBox[{ Directive[ Opacity[1.], RGBColor[0.880722, 0.611041, 0.142051], AbsoluteThickness[1.6]], LineBox[CompressedData[" 1:eJwV13k8VN0bAHDipWxvtjF3ZqwjSvtKm/NU/JK0KSVJZc+WlKhUskuWFFLI lpJSikg4yL5lp6hIkjEzd2wVkt997z/38/2c8zn3nnue8zznqlufMbWbJyQk tEtYSOi/e+JVa7HVtmH6LXP/XSSqaFMzOaJ2Gu0dfBIyS3muqsRfTe0i2jjd qfGTskz06xcS3X7I0zqwaZhyYmm46j9qochFtCCsm/Iy0jZiziESqYSetSin /Ekm6e+frmikbJm36RFlzYwFG6ZnYlBgVvCyYMouWz1df6nGo/mOvJXWlF+1 96VP7EhAfQZcAz3K084mvaMOD1B3/g3nBZSThXtDfxmmoF1T79M6/pLIoPHW op9dqUhetIKXQPnH3Z1lE6fTUUbMmV0nKN+0nbUcn3mIfi3vfc2i3DbjeEeg mol2IY3qkFkSXahWWUXmPEHtAXJOupQZt9vreTueolVOoyr9f0hkrQMiIw7P 0c/5OSWLKM+bufR+wucF4qdmZ5bOkKi9z0x5wjAH1SS2Zxym7P1U8s1Y10vk 2drTdW6aRMbR38XGkl+hSKE+yZkpErG8yw6Nns5Fit6mplcol+7wEpAzeago qJl27jeJxHu+avNU36AFvtd82T9J1F1a7Mn98QYxxZCx/ySJnmTcfTeSU4gQ Q1n78wSJ9nrsPcHZUYSKL99nBI6TKHZBYeyQA0a7b8aPxAlIlOtl7HGOXopc 78OBZpJECRxdYtynFIXKDtWJUHZrlnMYMyxDxgtypo/xSCSXUDNvtKscZV/p 2dcyTKIZmdeZ7lveoZp6xtGvP0j07XrafkHyO7R/0MmLP0Si1w5Xk8jTFYhx p3381yCJLNau28SfqUQWwgzB+34SpdQ9ODuiWoumjxgz73STaOJuns6Xm7Uo PH9N/OYuEv3PoX6gdaoWDc9lmX7qIBFX5JdZYVsdClTZf/HfNhLpbd27KTS4 AZV2RiWsaCRR8/M/87TJ90hKeNzLvJR6X7TASnpFM5oVkE1pJSTarHVnU7Bt M1q9aMxuuIhEEROZ45daqHbeTjObNyRad6vDzjqrBfkf1t8u95JE1+qWGa8+ 0Ya0xEeSLVNJ9DgnXysrpg3FTMmpmSSTqPXudpFFDW3ojBb+tj6Jil8H8yL6 xnbUYLQYfseTqE4kYIWQfAda46lgrHeLRLStPXLvqzqR5k930YyrJMp+Htrj suIjSgh1YWw2IxFHjnPT+NRHJHomxDbNlETaF4z1F9/5iCL2mReJ7afie4tk ysDURySu8za2xJhE0bU37S0qe9By+d3zxhEVrwORozstP6HaNbKKZ5aQ6IFk vb+sYR/KPfnHLP83H+lvvWzpa9WHmuoM1/hN8lGv29L1pFcfSlz6W8dojI/o rWHfG570IZcTqecqR/jo1l2TXSEL+5HFw+vugZ/5yH9Rk8xcTz/y8LkfwXzH Rw6oJZ7rMYCaM2Q3pIfwkVDqz8KesAF0wHLw8/cAProryuqtSx9ApX7dg5rX +aim1l4ls3MAqQ3YLbx9kY8WH/qTarfpGzoVvu2HrhMf/XDUfvpZeBCF+Llp pRvzkVP0lZLmqO8o/dAAS3I+H7kNLhnIzR5GXu11m3U9eajP6sP8stphdOYS 2sM9w0MHu0NWNH4bRgEtpawEJx7Sqx+6OMjgoOJtr6dGTvCQSM7DhbRgDnpx fmOViTEP3fVR1/c6MYLszJT95VV4qEyeiN+4kIeYFuqhRWVcJKEeoP1FT4DC xVa/yp8aQTVbbq28aihA8jciO++Mj6Ag8yRdZVMBim2vT3bljaB5UQU7jzkL UGPsy+Vy/SNo+i/XoStRgIpDVxlo14ygkV6zx+/njaIXcf1b/WJGUMNd7SWl DaNoB7P38LzlIyhiYb1OyslxJNd9vt92HwdZ9xzCIg0/0Z64DcRD3R/obGWA 9RntafTopYxG27dBlOMudZC+axbJazTaG/kNoJTLzpxVR4Sg5tQvK9fXfcjv wZ8DE1xheLr6pLXMpx4k1NYHRZdEYIaQWm60qBNpnbf3PqTzD+wPuCS+zuk9 +jYZd1iyWAw+LTyutUHxHTIedLoSfXY+eJ8fJXZJZqC936KWqqhIQHJs3Xvn a88x5521b80zSdjDuVrtZ1SBV8h2Ku84IQ1Pr2ociClvxhJo8lTvuAzs6d7h yKrsxDt5Fr80zBbCYIwWfiz+CXeX2wpbLJMFv980zS2u/TgvpMJKbVIWXkY+ f+z1ewDb/+PZ19QkB/dOqcjO+H3HNh6X86TuyYPzBV/T749+4JUOS5tMTimA z8Ntpr8COfjzadUzz2wVQFnTrSA4nINvusgvlHFUgLPDUw7yMRw8fHbmwHs3 BTgZOJLISufgNJ/6jgM+CmCIw+FnGQcrRTt/OhynAFeH7PUr/3DwXFEW92Sj Aux1f6Vb6zaC38stk/TcqAgTWmE8TQMu1nd1Hb6/RRH0iiW9ZY25OLs6u7oc KcJioYj0qX1cHOGzOmDh/xQhMVqqr+gYF5t83zCbZaoIP0ZNamTOcXHdm23k V2dFkHaSuP8lmYsrTx5p25+kCHYG6eYrp7h43du7OV4pirBILfJixF8uTlf8 GJmUrgh5c9fVh0V4OKDO0oT7RBF0iAMit2R4ePs668rgfEVY2nHh9xM2D5eK u+aXNCvCq2tHp8X28PDbbL/7y0RosOPupIxFAg+LmZgx4sRooOb25npkCg8f GNaOF5agQe+4XhzO4OEhdmNs50IadIh1Zfybw8Py8UrRvio0iJpXjM5U8rCz /9OQ9o008HP+J2Ypj4eZ5p2ePu408M75wYnW42P7yccT38/R4NpxeWnmVj7O ib587oAXDXZPmeUnbuPjnY1qZ7Wu0uDykEJKtDEfn9vu7NIcRoPrhqa5myz5 uH6ZkI3mIxoE6Uqt87/Kx5eFdfY3fKKB56LrqYISPv5uINmt1U+DbWU63q/L +Xh/CPfE9W80GIufGbtQxcea/z4/s2GEBk9cXnQMNfJxA2tdZPJvGihUc/VD e/lYRU+/6by8EnCsVqrV/ebjcjdTE+VdSlBe/GkvLCfxspdr271MlEA2523u 0VUkjp1UsGzdpwQ8Zny261oSO/t0OQcfVoJZ97OWgRtJrHDDMmzURgk2SWbd PmdIYvuH9nWVV5XgSPt6qd2WJJbovWTklqsEbenrNw8EkniDV6Z0ZL4SRN9S btEMIfEpue7W54VUe4ZW+qkbJC4wWn98tFQJYpJ8hRojqPHyyLPnG5Xgp0TJ Dac4EpdF2N6/9F0JHvsbZpU+IjF3ye2T94ep/s6p43mZJKZXli0q4ipBg5i0 Z0YWid1mVF7MjilBt7SGxsXnJGY5fqjwnVOC1w9oqoOvSey9bS8/iE6Hx9os z64KEqf3+rx6xKQDszZ5nW8Vid97ZXnXqNDhQvHwEs0aEmtlzxeRWEQHWRG5 vOP1JG5jvFMKX02HFZs5RnEtJF4xobvttjEdAv5Wbdf8RGKLSHux3D10GO18 b+z7mcSBOjH17fvp0NWiEtT1hcS9J8fMaEfo4OEvGev1lcQ3Gp86xdvQYVHH S6WrQyQezFC/88CHDv/TcQ9NF1DzDeifEvGlAyftiX3dKInHrFNOOPrTwdXr UwJnjMRCqupLV9+ggyarSp4xSWJGnFp5eRwdMoWUlqyfIrG6Z5+29n06NC7z rV8+TeLFB5PDw5LoYO6wpkpthsTr/1U7eughHYy9V1ZM/iHx/mBVweBLOtgu 8hEsFxLgI3ZfzIxf0+HeOvHU25Stdjx4m/2GDqFP/a79pOw8pxLsVUqHrCOb S14JC7DHp8/cnnd00F/poSozT4C93yaZQjUdogpuPrWjHOSlorKgiQ59RvbG UiICHG722d+1hQ6HHNnHjlG+szZpuKWdDuVOpXGPKKeQynn3eujg6JRwR1dU gB83fmLMfaaD3uSnQ5cpZ2cl+tp8pcOJq6LbiigXOSibLBumw1CNZOL6fwT4 neGnnCguHcQixqXcKNexE5UmSer55k2P0yg3Cx+/cnScDmYLU9w7KXd9YQ0U /6TDHp2ztmJiAvy5uNdIY5oOwVkGAWspf7ufkB00S4e72hrNxymPHWFd2idK wEodOjeT8tT63i+vxAmQSIOiespCCgmGdEkCzs1EF3Aoi40dy/KRIcAwmt4v Ji7A0s1M2X5ZAizMetarUVbI7rlgqEhAfNng6w2UGTfv92bSCdgysN3WmLK6 07HtMiwCos3+bj1GebER87GHKgGN2zUNTlNeqdUj3aVBQLdt2YXzlDeI3j+3 WYuAC6IDHT6Ut3y1+PBgCQHVU7Gn/CjvKGUg0eXU+E9HiEDKxkkf0x1XERAk J/gbQHm/zz2JxrUEXHv5auF//Y9YWLiv1iUgIeV/+/4bz0qP0RmziYCDAW8K z1G2o33cPL2VgLYI5YOOlF0m4lOstlHzDwigW1D2aD0q/s6AAH6+kOQuyhdf EK7aRgTEST7UWU/ZN+JDa9huAsRHA7xUKAe5xOsJ9hKw/XTuqCjlcOOjSYdM CfgmahT/g/qedxYTom/MCPg+cMitlrKlldajAXMC/JzU6/5bD807a41lLKn3 j/d6EUyZWws8vRMEuI2dVrehnDu3J8rGmoBpG0nlLZR91h9bG2FHwOddZ9Lk KBs4O3YWOBJQ4JKZ+52Kj/ZOP5bMGQL0n416BVNOkIrCeh7UeFeW6ByibLs9 0drGk4B35A0HFcoTz/IfF1wmYOt7jYAsKj7lA7nrbEIIWDSktfMxFd8fC6e6 wsMIKN+4xOwU5VSB2OWCCAJWuCwklSivsVQvk44hoFV7vOoKtX9M1x7eU5BC wOr+4QXLqP3GOG0j+JpOwCxTcLGF2o9fk9xvSz8moOZtzoXzlD0kwj5YZxOQ Lp233H6OxNH92E76LRWvTiIc6n8OWyg1ztcrIWBUAiXsovKBxp6PWdZlBDAU O047UvniZcHEaH41Ae5OcuZxv0ncGrnkqnU7ATHCC0MKJ0h8r3KDengXAQ29 h1n54yS2ntlRkf+RAIOjck3ZVH4as7eSkO6n1j9YNCySymey+rdj83kEJBe2 etC4JO4+l7zxq4AA88d50yMcEidnPuuVmiCgar5VatEwiVcp1mhYT1PxUapr uo/Kj/u5f7KlxBmQzK7fsonKn1H37KtPqTHAZL5qkFYXib0u+J3zYTOgIoyW kddBYivTJNU4LQZ87Y9iQjtV7yQ6vRqWMUDug8P8nVT+rr1ouFh3IwM+1qBx FpXf5x3VDJUyZUDqnbXHh4tJPLwO1mubMWCW5Tu3vIjEzQst+7eZM0D/X7lZ 10ISP6i5vdHLigHSJ6Ml+6j6skVPhNPvxIBVEmLXb1L1x1Ppq3G+PwOiLpsv Fk8mseX47M+WIAbc/LDguXASiQ3eE2ncUAbQw2/G/bpPYrngAzPqUQwIOlMR 2UHVv+yfpVk3Exhg4HRiiUUkiX90PJCyzmOAQ9AX+6Ar1PrdOd4kNcSAmhvE KbfDVL17JcF05jCge/XgrsSDJP7Sku9Qy2OAzfXp9qr9JLb9V044aIIBrmuK 2xfspurljaq1c8JMkNOaDd+HSOznuzJ+lMWEsZHKwI9aJM50EbbtNGXCRKjK 6/FRPnYJy36x7jATdO0X7/Pm8/GKJ8dmo48yQSLGJvUXh49fDuXF7jvJhKNC ry59H+DjImun2hpXJlwt7lgQ2MHHzeZtK96GMOH2gadWVfl8/Nvg4dSDEiac 3tTw94Q3H/u8feBfUcaEuh+hcQHn+VhozT3p4QomGNCKc9Ld+VhcNVJ9TT0T Ko/cc+1y5GPalPeuii4mFJycposf5eO1z0zifwiY8Ic6d0ZS5zM3xQm91WwW HC68uHvPBA8PDG73Lg9mwfzVTiMZVtR58kFTqk4YC5YsyEjfeZSHk80tGqMj WFARAkcGDvKwVf1ZDZsYFhif9n8muYuHP75IbhBJY8GF7Yxtamt5uPXyXzXD EhZYHvysYiHGw+WyhbXVEyyIZjtjIpOLU7euZjaeUgaTNI6NQt8IfhboKeuw XAWSbZZOSOhycLCZzW6RSRVIXWYUyTv2A6/vTckMblSFms7zt6qUv+O7S+4/ a9ytBucuhavf7RjA10rD7+5+owZa32PwwVX9eA/T7GGdpjpYLtY/WTjUi7e5 VR+TDVWHtgurinFSF37AstZsmFSHQ8ZHT8lVtmDNd49i9x7TAIXS3CLP0Boc tbxcp7VEA75tyyUWpbzF9uOiLbU6bFBT8j+bueM2ts4ZUwuNZENzwQr/jsDn yMXv4B/jW2wQ+lAd28p6gTwP5nZJ3WaDi9mTnY8uvEAhk54RUbFs8Jg/0EfT yUHZG6dm4hLZkLwu110q6iWaKpvrzMhiQ4H0gOxW8zwU1SoVXlFN9T/+9K3i h0J0L83VMaiWDazI7XMuq96itPNNO4zq2aCvWGWcHfwWvaZFTdc3scH0oFcy f30R6rWQd2zrYIMg/FuXza1ipD1A7Pg6wIbFVYtviGmUolW5l1TSB9mwz/Gk 3qxJKdoU2DNlN8QG99a4jT+8SpGJduKLYQ4bDnOWLbjfWIo8nNVURkfZUM/O eOLhXYZ8tlyfejnOhqbqKyUSaWUoUPpr+/lJNvh4CBpjG8vQ3edpYb9/s0Gy Z3nuDY1ylOor6lA4zYbzNTs8x0zKUdYBu+0+f9hQLRImtterHOVqVCnr/2VD C7nqeGJKOSoZ15qam2PDhmE7t776cvR/NUBMyA== "]]}, Annotation[#, "Charting`Private`Tag$15014#2"]& ]}}, {}}, { DisplayFunction -> Identity, Ticks -> {Automatic, Automatic}, AxesOrigin -> {0, 0}, FrameTicks -> {{Automatic, Automatic}, {Automatic, Automatic}}, GridLines -> {None, None}, DisplayFunction -> Identity, PlotRangePadding -> {{ Scaled[0.02], Scaled[0.02]}, { Scaled[0.05], Scaled[0.05]}}, PlotRangeClipping -> True, ImagePadding -> All, DisplayFunction -> Identity, AspectRatio -> NCache[GoldenRatio^(-1), 0.6180339887498948], Axes -> {True, True}, AxesLabel -> {None, None}, AxesOrigin -> {0, 0}, DisplayFunction :> Identity, Frame -> {{False, False}, {False, False}}, FrameLabel -> {{None, None}, {None, None}}, FrameTicks -> {{Automatic, Automatic}, {Automatic, Automatic}}, GridLines -> {None, None}, GridLinesStyle -> Directive[ GrayLevel[0.5, 0.4]], Method -> { "DefaultBoundaryStyle" -> Automatic, "DefaultGraphicsInteraction" -> { "Version" -> 1.2, "TrackMousePosition" -> {True, False}, "Effects" -> { "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> {"ratio" -> 2}, "Droplines" -> { "freeformCursorMode" -> True, "placement" -> {"x" -> "All", "y" -> "None"}}}}, "DefaultMeshStyle" -> AbsolutePointSize[6], "ScalingFunctions" -> None, "CoordinatesToolOptions" -> {"DisplayFunction" -> ({ (Identity[#]& )[ Part[#, 1]], (Identity[#]& )[ Part[#, 2]]}& ), "CopiedValueFunction" -> ({ (Identity[#]& )[ Part[#, 1]], (Identity[#]& )[ Part[#, 2]]}& )}}, PlotRange -> {{0, 10}, {-1.00183591962083, 0.9999999999999948}}, PlotRangeClipping -> True, PlotRangePadding -> {{ Scaled[0.02], Scaled[0.02]}, { Scaled[0.02], Scaled[0.02]}}, Ticks -> {Automatic, Automatic}}], FormBox[ FormBox[ TemplateBox[{"\"cos(\[Lambda]t)\"", "\"Re[P(t)]\""}, "LineLegend", DisplayFunction -> (FormBox[ StyleBox[ StyleBox[ PaneBox[ TagBox[ GridBox[{{ TagBox[ GridBox[{{ GraphicsBox[{{ Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6]], { LineBox[{{0, 10}, {20, 10}}]}}, { Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, ImageSize -> {20, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #}, { GraphicsBox[{{ Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.880722, 0.611041, 0.142051], AbsoluteThickness[1.6]], { LineBox[{{0, 10}, {20, 10}}]}}, { Directive[ EdgeForm[ Directive[ Opacity[0.3], GrayLevel[0]]], PointSize[0.5], Opacity[1.], RGBColor[0.880722, 0.611041, 0.142051], AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, ImageSize -> {20, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #2}}, GridBoxAlignment -> { "Columns" -> {Center, Left}, "Rows" -> {{Baseline}}}, AutoDelete -> False, GridBoxDividers -> { "Columns" -> {{False}}, "Rows" -> {{False}}}, GridBoxItemSize -> {"Columns" -> {{All}}, "Rows" -> {{All}}}, GridBoxSpacings -> { "Columns" -> {{0.5}}, "Rows" -> {{0.8}}}], "Grid"]}}, GridBoxAlignment -> {"Columns" -> {{Left}}, "Rows" -> {{Top}}}, AutoDelete -> False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings -> {"Columns" -> {{1}}, "Rows" -> {{0}}}], "Grid"], Alignment -> Left, AppearanceElements -> None, ImageMargins -> {{5, 5}, {5, 5}}, ImageSizeAction -> "ResizeToFit"], LineIndent -> 0, StripOnInput -> False], { FontFamily -> "Arial"}, Background -> Automatic, StripOnInput -> False], TraditionalForm]& ), InterpretationFunction :> (RowBox[{"LineLegend", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", TemplateBox[<| "color" -> RGBColor[0.368417, 0.506779, 0.709798]|>, "RGBColorSwatchTemplate"], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1.`", "]"}], ",", TemplateBox[<| "color" -> RGBColor[0.880722, 0.611041, 0.142051]|>, "RGBColorSwatchTemplate"], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{#, ",", #2}], "}"}], ",", RowBox[{"LegendMarkers", "\[Rule]", "None"}], ",", RowBox[{"LabelStyle", "\[Rule]", RowBox[{"{", "}"}]}], ",", RowBox[{"LegendLayout", "\[Rule]", "\"Column\""}]}], "]"}]& ), Editable -> True], TraditionalForm], TraditionalForm]}, "Legended", DisplayFunction->(GridBox[{{ TagBox[ ItemBox[ PaneBox[ TagBox[#, "SkipImageSizeLevel"], Alignment -> {Center, Baseline}, BaselinePosition -> Baseline], DefaultBaseStyle -> "Labeled"], "SkipImageSizeLevel"], ItemBox[#2, DefaultBaseStyle -> "LabeledLabel"]}}, GridBoxAlignment -> {"Columns" -> {{Center}}, "Rows" -> {{Center}}}, AutoDelete -> False, GridBoxItemSize -> Automatic, BaselinePosition -> {1, 1}]& ), Editable->True, InterpretationFunction->(RowBox[{"Legended", "[", RowBox[{#, ",", RowBox[{"Placed", "[", RowBox[{#2, ",", "After"}], "]"}]}], "]"}]& )]], "Output", CellChangeTimes->{{3.854796214764694*^9, 3.8547962298758793`*^9}, { 3.8547962677450333`*^9, 3.8547962983547497`*^9}, {3.8547964040164557`*^9, 3.854796487142726*^9}, {3.8547965341813107`*^9, 3.854796574243944*^9}, 3.8548118488294153`*^9, {3.854812360295925*^9, 3.8548124269364433`*^9}, 3.8551401411019983`*^9, {3.8551405886333227`*^9, 3.855140646898128*^9}}, CellLabel->"Out[44]=",ExpressionUUID->"dc47d687-eeab-4ada-b718-60a215f45c93"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"s", "[", "x_", "]"}], "=", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"-", "\[ImaginaryI]"}], " ", SuperscriptBox["\[ExponentialE]", FractionBox[ RowBox[{"5", " ", "\[ImaginaryI]", " ", "\[Pi]"}], "6"]], " ", "x"}], ",", RowBox[{"-", "1"}]}], "}"}]}], ";"}], " ", RowBox[{"(*", RowBox[{"{", RowBox[{ RowBox[{"p", "[", "x", "]"}], ",", RowBox[{"q", "[", "x", "]"}]}], "}"}], "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"InvT", "[", RowBox[{"x_", ",", "phi_"}], "]"}], ":=", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"-", "\[ImaginaryI]"}], " ", "phi"}]], " ", "x"}], RowBox[{ RowBox[{"-", SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", "phi"}]]}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SuperscriptBox["x", "2"]}], ")"}]}]}, { RowBox[{"-", SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"-", "\[ImaginaryI]"}], " ", "phi"}]]}], RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", "phi"}]], " ", "x"}]} }, GridBoxAlignment->{"Columns" -> {{Center}}, "Rows" -> {{Baseline}}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"phases", "=", RowBox[{"{", "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"While", "[", RowBox[{ RowBox[{ RowBox[{"Exponent", "[", RowBox[{ RowBox[{ RowBox[{"s", "[", "x", "]"}], "[", RowBox[{"[", "1", "]"}], "]"}], ",", "x"}], "]"}], "\[GreaterEqual]", " ", "1"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"phase", "=", RowBox[{"Simplify", "[", RowBox[{ FractionBox["1", RowBox[{"2", " ", "\[ImaginaryI]"}]], RowBox[{"Log", "[", FractionBox[ RowBox[{"Coefficient", "[", RowBox[{ RowBox[{ RowBox[{"s", "[", "x", "]"}], "[", RowBox[{"[", "1", "]"}], "]"}], ",", "x", ",", RowBox[{"Exponent", "[", RowBox[{ RowBox[{ RowBox[{"s", "[", "x", "]"}], "[", RowBox[{"[", "1", "]"}], "]"}], ",", "x"}], "]"}]}], "]"}], RowBox[{"Coefficient", "[", RowBox[{ RowBox[{ RowBox[{"s", "[", "x", "]"}], "[", RowBox[{"[", "2", "]"}], "]"}], ",", "x", ",", RowBox[{"Exponent", "[", RowBox[{ RowBox[{ RowBox[{"s", "[", "x", "]"}], "[", RowBox[{"[", "2", "]"}], "]"}], ",", "x"}], "]"}]}], "]"}]], "]"}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"PrependTo", "[", RowBox[{"phases", ",", "phase"}], "]"}], ";", "\[IndentingNewLine]", RowBox[{ RowBox[{"s", "[", "x_", "]"}], "=", RowBox[{"Simplify", "[", RowBox[{ RowBox[{ RowBox[{"InvT", "[", RowBox[{"x", ",", "phase"}], "]"}], ".", RowBox[{"s", "[", "x", "]"}]}], ",", RowBox[{"{", RowBox[{"0", "<", "x", "<", "1"}], "}"}]}], "]"}]}], ";"}]}], "\[IndentingNewLine]", "]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"phase", " ", "=", " ", RowBox[{"Simplify", "[", RowBox[{ RowBox[{"-", "\[ImaginaryI]"}], " ", RowBox[{"Log", "[", RowBox[{ RowBox[{"s", "[", "x", "]"}], "[", RowBox[{"[", "1", "]"}], "]"}], "]"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"PrependTo", "[", RowBox[{"phases", ",", "phase"}], "]"}], ";"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"s", "[", "x", "]"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.802020767970352*^9, 3.802020896231595*^9}, { 3.802021096834921*^9, 3.802021125987208*^9}, {3.8020211648156443`*^9, 3.8020212631311827`*^9}, 3.802021776321725*^9, {3.8020221289389477`*^9, 3.8020221972072678`*^9}, {3.802022235314743*^9, 3.8020223116869*^9}, { 3.802022343358479*^9, 3.802022384573765*^9}, {3.80202255912642*^9, 3.802022574104383*^9}, {3.802022622535842*^9, 3.8020226809701767`*^9}, { 3.80202273837965*^9, 3.8020227675051928`*^9}, {3.802022899704442*^9, 3.802022925930895*^9}, {3.80202297261862*^9, 3.802023169435871*^9}, { 3.8020233042724457`*^9, 3.802023316048911*^9}, {3.8020233486714287`*^9, 3.802023402312436*^9}, {3.802023444262382*^9, 3.802023568078896*^9}, { 3.802024143485313*^9, 3.802024203694549*^9}, {3.802024272040656*^9, 3.8020242777201233`*^9}, {3.802088686608314*^9, 3.8020888710701036`*^9}, { 3.8020889265858307`*^9, 3.802088926679277*^9}, {3.8020890538073387`*^9, 3.8020890712289553`*^9}, {3.8020891185921164`*^9, 3.802089148514141*^9}, { 3.8020907623535423`*^9, 3.802090768578217*^9}, {3.802091268991807*^9, 3.802091315306542*^9}, {3.80209146845155*^9, 3.8020916235662518`*^9}, 3.802091673217185*^9, 3.802091722387987*^9, 3.802091813550157*^9, { 3.85412777564189*^9, 3.854127775995962*^9}, {3.854796340515751*^9, 3.8547963406710443`*^9}}, CellLabel->"In[10]:=",ExpressionUUID->"c3ba0f0a-6dfb-4b58-b45d-3741914e1ecc"], Cell[BoxData[ RowBox[{"{", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], RowBox[{"2", "/", "3"}]], ",", "0"}], "}"}]], "Print", CellChangeTimes->{ 3.802091623967437*^9, 3.802091674940975*^9, 3.802091723975075*^9, 3.802091814348111*^9, 3.8020922799649076`*^9, {3.854127776468775*^9, 3.85412777963096*^9}, 3.8548118488480177`*^9, 3.8551401411198673`*^9}, CellLabel-> "During evaluation of \ In[10]:=",ExpressionUUID->"20169a18-67dd-465c-82f2-6c40e5a70224"] }, Open ]], Cell[BoxData[ RowBox[{"ClearAll", "[", "t", "]"}]], "Input", CellChangeTimes->{{3.854796412464613*^9, 3.854796414572763*^9}}, CellLabel->"In[17]:=",ExpressionUUID->"13ea9c80-b164-49a0-9700-ef4a7df24f79"] }, Open ]] }, Open ]] }, Open ]] }, WindowSize->{Full, Full}, WindowMargins->{{12, Automatic}, {Automatic, 24}}, FrontEndVersion->"13.0 for Mac OS X ARM (64-bit) (December 2, 2021)", StyleDefinitions->"Default.nb", ExpressionUUID->"c7c2dec5-45c8-4e9b-81ba-3f5325655145" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[580, 22, 170, 3, 98, "Title",ExpressionUUID->"93f1078a-c17d-4647-8ecc-253a2973747c"], Cell[753, 27, 201, 6, 85, "Subtitle",ExpressionUUID->"4868ce54-430c-4a3c-99ec-d2b5daa59e7c"], Cell[CellGroupData[{ Cell[979, 37, 1312, 28, 69, "Section",ExpressionUUID->"a28626c4-cf14-472c-a780-9af7fed69c80"], Cell[CellGroupData[{ Cell[2316, 69, 900, 23, 47, "Subsubsection",ExpressionUUID->"792f85c3-5d7d-4453-ba0a-30b355101975"], Cell[3219, 94, 980, 35, 127, "Input",ExpressionUUID->"f694f58b-9327-4951-8079-6c38c7f65c05"], Cell[4202, 131, 558, 12, 43, "Text",ExpressionUUID->"b99515c8-5824-4aa7-a91f-c486424de483"], Cell[CellGroupData[{ Cell[4785, 147, 3247, 47, 30, "Input",ExpressionUUID->"51015464-2f55-489a-88da-cb1da7045c6d"], Cell[8035, 196, 293, 5, 34, "Output",ExpressionUUID->"66d28cf7-c24f-4705-a371-f3084bc3c417"] }, Open ]], Cell[8343, 204, 172, 2, 30, "Input",ExpressionUUID->"b2063c4a-0841-459d-9a75-615f5ad20eb3"], Cell[8518, 208, 2501, 35, 35, "Text",ExpressionUUID->"5c22b2e4-0eec-4ba2-b557-68f9bbd38e18"], Cell[11022, 245, 2980, 49, 52, "Input",ExpressionUUID->"7940378a-4705-4128-9764-d1d637aa5616"] }, Open ]], Cell[CellGroupData[{ Cell[14039, 299, 603, 15, 51, "Subsubsection",ExpressionUUID->"5e27256d-67e4-45b6-8e0e-91fcd6c5f221"], Cell[14645, 316, 3655, 63, 115, "Input",ExpressionUUID->"b05155ee-f50c-4152-a715-d999a2e18624"] }, Open ]], Cell[CellGroupData[{ Cell[18337, 384, 172, 3, 45, "Subsubsection",ExpressionUUID->"7a19e667-2266-4f80-b2d8-2945f0bc8a3e"], Cell[18512, 389, 3380, 60, 82, "Input",ExpressionUUID->"73cdef54-8362-4f2f-b3a3-94c8e3f8d667"] }, Open ]], Cell[CellGroupData[{ Cell[21929, 454, 169, 3, 45, "Subsubsection",ExpressionUUID->"bbd5bf82-cfc0-42bc-939b-ba7a983da4f9"], Cell[CellGroupData[{ Cell[22123, 461, 2565, 66, 75, "Input",ExpressionUUID->"fcf3e09e-0746-44cf-a89d-de454fee2400"], Cell[24691, 529, 21946, 415, 253, "Output",ExpressionUUID->"7d6456f3-b6c4-4d9e-acf5-3fd42f1d99c5"], Cell[46640, 946, 18049, 351, 253, "Output",ExpressionUUID->"acb4d93c-0f41-4dcb-ae7e-ccc3e67c7dfc"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[64738, 1303, 845, 22, 49, "Subsubsection",ExpressionUUID->"a9c578e8-d664-43c2-b184-01b1619ccb23"], Cell[CellGroupData[{ Cell[65608, 1329, 2947, 50, 33, "Input",ExpressionUUID->"6ce02805-2b03-400c-83c8-13a50a6c772f"], Cell[68558, 1381, 1178, 28, 37, "Output",ExpressionUUID->"d76ed83d-6b49-4217-8679-5ebd22283d8c"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[69797, 1416, 182, 3, 67, "Section",ExpressionUUID->"3e09731d-05d0-481d-96c5-2007266765ea"], Cell[69982, 1421, 1347, 30, 108, "Text",ExpressionUUID->"bcba399b-d003-4a64-a4fa-fbc8f45baf41"], Cell[CellGroupData[{ Cell[71354, 1455, 3603, 92, 148, "Subsection",ExpressionUUID->"39e28940-1687-4773-b5af-97ef90e0a9cd"], Cell[CellGroupData[{ Cell[74982, 1551, 1926, 52, 110, "Input",ExpressionUUID->"780da9ab-38bc-447b-89c8-22a0155ca44a"], Cell[76911, 1605, 22074, 414, 247, "Output",ExpressionUUID->"dc47d687-eeab-4ada-b718-60a215f45c93"] }, Open ]], Cell[CellGroupData[{ Cell[99022, 2024, 5903, 145, 366, "Input",ExpressionUUID->"c3ba0f0a-6dfb-4b58-b45d-3741914e1ecc"], Cell[104928, 2171, 509, 13, 26, "Print",ExpressionUUID->"20169a18-67dd-465c-82f2-6c40e5a70224"] }, Open ]], Cell[105452, 2187, 206, 3, 30, "Input",ExpressionUUID->"13ea9c80-b164-49a0-9700-ef4a7df24f79"] }, Open ]] }, Open ]] }, Open ]] } ] *)