(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 172729, 4640] NotebookOptionsPosition[ 163928, 4347] NotebookOutlinePosition[ 164381, 4365] CellTagsIndexPosition[ 164338, 4362] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["SP", "L"], " ", "equations", " "}], TraditionalForm]], FontSize->24]], "Title", CellChangeTimes->{{3.522644900231761*^9, 3.522644900242709*^9}, { 3.5226449513358107`*^9, 3.5226449839045687`*^9}, 3.522645020846187*^9, { 3.5226457136785097`*^9, 3.522645721370944*^9}, {3.623848546925921*^9, 3.623848550997765*^9}}], Cell[CellGroupData[{ Cell["Interior equations", "Section", CellChangeTimes->{{3.5226483838152122`*^9, 3.5226483893248262`*^9}}], Cell[TextData[StyleBox["Expansion of solution in zonal spherical harmonics \ (1D Legendre polynomials)", FontSize->16]], "Text", CellChangeTimes->{{3.522645067217414*^9, 3.522645068559268*^9}, { 3.522645113653557*^9, 3.522645154760992*^9}, {3.522648581966075*^9, 3.5226485916822042`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"\[Psi]", "[", RowBox[{"\[Mu]_", ",", "L_"}], "]"}], ":=", RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"n", "=", "0"}], "L"], RowBox[{ FractionBox[ RowBox[{ RowBox[{"2", "n"}], "+", "1"}], "2"], RowBox[{"LegendreP", "[", RowBox[{"n", ",", "\[Mu]"}], "]"}], " ", RowBox[{ SubscriptBox["\[Phi]", "n"], "[", "z", "]"}]}]}]}]], "Input", CellChangeTimes->{{3.45812214278125*^9, 3.458122149046875*^9}, { 3.458122215015625*^9, 3.458122336828125*^9}, {3.458122398546875*^9, 3.45812242484375*^9}, {3.4581233975*^9, 3.458123415328125*^9}, { 3.4581238006875*^9, 3.45812380725*^9}, 3.4581477805*^9, { 3.4582733744375*^9, 3.458273376359375*^9}, {3.511183195694076*^9, 3.511183205056978*^9}, 3.522645020846859*^9}], Cell[BoxData[ RowBox[{ RowBox[{"L", "=", "3"}], ";"}]], "Input", CellChangeTimes->{{3.511183434712954*^9, 3.511183436093316*^9}, 3.511186023414919*^9, {3.511186513983778*^9, 3.511186529977191*^9}, 3.511193503700611*^9, 3.511198703384798*^9, 3.511202493298025*^9, 3.511204091808771*^9, 3.511244027352944*^9, 3.511260122317914*^9, 3.511260181439172*^9, 3.5112602495326014`*^9, 3.511261217964011*^9, { 3.5112633297375193`*^9, 3.511263332247952*^9}, 3.511281119756188*^9, 3.511282709965703*^9, 3.511283075315803*^9, 3.511284311021352*^9, 3.511284573327456*^9, 3.511320107177246*^9, 3.5113208206250267`*^9, 3.511320999398205*^9, {3.511328357782628*^9, 3.511328357936368*^9}, { 3.511328713487506*^9, 3.511328713591861*^9}, {3.511334493233968*^9, 3.511334493286051*^9}, {3.511344521206007*^9, 3.5113445212824183`*^9}, { 3.5113445842517*^9, 3.511344605708233*^9}, {3.511344687536091*^9, 3.511344758805469*^9}, 3.511344793808539*^9, 3.511347322643062*^9, 3.5113489727381687`*^9, 3.511349024429093*^9, {3.511349460595093*^9, 3.511349461578451*^9}, 3.5113495194368973`*^9, {3.51134958530554*^9, 3.5113496396279716`*^9}, {3.5113496702757187`*^9, 3.511349670332086*^9}, 3.511351930844843*^9, {3.511352037998671*^9, 3.511352038195541*^9}, 3.514604546337442*^9, 3.514606045369644*^9, 3.514606907047196*^9, { 3.514625108106606*^9, 3.514625108159231*^9}, {3.514625160015154*^9, 3.514625160057001*^9}, {3.5146285712920628`*^9, 3.514628580168375*^9}, { 3.514634017833006*^9, 3.514634017928426*^9}, 3.51469074887234*^9, 3.514693172805154*^9, {3.514698890974867*^9, 3.514698891979401*^9}, { 3.5146989346523113`*^9, 3.5146989357181664`*^9}, {3.5147039246463547`*^9, 3.514703926276889*^9}, {3.51472610218713*^9, 3.514726103177196*^9}, { 3.5147261788897877`*^9, 3.514726179584415*^9}, 3.515952066320573*^9, 3.515952109572454*^9, 3.5168834154212027`*^9, 3.516883698717544*^9, 3.5190139468552923`*^9, 3.519099281685277*^9, 3.519105106158413*^9, 3.5226450208470993`*^9, 3.525679959909957*^9, {3.620399188395391*^9, 3.6203991898556423`*^9}, {3.620399286972267*^9, 3.620399287563558*^9}, { 3.620404080361659*^9, 3.620404081290504*^9}, {3.620407370402937*^9, 3.620407371190123*^9}, {3.620410410791429*^9, 3.6204104117806396`*^9}, 3.62041325158526*^9, 3.6204133228581467`*^9, {3.620414071659075*^9, 3.6204140721539183`*^9}, {3.620414315800043*^9, 3.620414319791226*^9}, 3.6204721047296247`*^9, 3.620474003262094*^9, {3.620490079708181*^9, 3.620490124964137*^9}, 3.62175249909418*^9, 3.621752560090125*^9, 3.621753187226018*^9, 3.6237684797312803`*^9, 3.623822432342924*^9, 3.623852352956362*^9, 3.6238524082348833`*^9}], Cell[TextData[{ Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["P", "n"], "equations"}], TraditionalForm]], FontSize->16], StyleBox[" (replace Q by \[DifferentialD]/\[DifferentialD]z)", FontSize->16] }], "Text", CellChangeTimes->{{3.522645306558002*^9, 3.5226453289294443`*^9}, { 3.5226453609678583`*^9, 3.5226453633985786`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"P", "[", "n_", "]"}], "=", RowBox[{ RowBox[{ RowBox[{ FractionBox[ RowBox[{"n", "+", "1"}], RowBox[{ RowBox[{"2", "n"}], "+", "1"}]], "Q", " ", RowBox[{ SubscriptBox["\[Phi]", RowBox[{"n", "+", "1"}]], "[", "z", "]"}]}], " ", "+", RowBox[{ FractionBox["n", RowBox[{ RowBox[{"2", "n"}], "+", "1"}]], "Q", " ", RowBox[{ SubscriptBox["\[Phi]", RowBox[{"n", "-", "1"}]], "[", "z", "]"}]}], "+", RowBox[{ RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "n"}]], "[", "z", "]"}], RowBox[{ SubscriptBox["\[Phi]", "n"], "[", "z", "]"}]}]}], "\[Equal]", RowBox[{ SubscriptBox["S", "n"], "[", "z", "]"}]}]}]], "Input", CellChangeTimes->{{3.458187139609375*^9, 3.458187280578125*^9}, { 3.458187632015625*^9, 3.458187662421875*^9}, {3.458187728984375*^9, 3.458187803234375*^9}, {3.458188008453125*^9, 3.458188075359375*^9}, { 3.458188108125*^9, 3.458188125859375*^9}, {3.458214952609375*^9, 3.458214966125*^9}, {3.45821502665625*^9, 3.4582150574375*^9}, { 3.511203090436952*^9, 3.511203132014764*^9}, {3.511203174602572*^9, 3.511203189284235*^9}, {3.5112038634838743`*^9, 3.51120387570855*^9}, 3.511357715928198*^9, {3.511357772698263*^9, 3.511357775330196*^9}, 3.522645020848256*^9}], Cell[BoxData[ RowBox[{ RowBox[{ FractionBox[ RowBox[{"n", " ", "Q", " ", RowBox[{ SubscriptBox["\[Phi]", RowBox[{ RowBox[{"-", "1"}], "+", "n"}]], "[", "z", "]"}]}], RowBox[{"1", "+", RowBox[{"2", " ", "n"}]}]], "+", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], " ", "Q", " ", RowBox[{ SubscriptBox["\[Phi]", RowBox[{"1", "+", "n"}]], "[", "z", "]"}]}], RowBox[{"1", "+", RowBox[{"2", " ", "n"}]}]], "+", RowBox[{ RowBox[{ SubscriptBox["\[Phi]", "n"], "[", "z", "]"}], " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "n"}]], "[", "z", "]"}]}]}], "\[Equal]", RowBox[{ SubscriptBox["S", "n"], "[", "z", "]"}]}]], "Output", CellChangeTimes->{ 3.522650374996583*^9, 3.522651349718388*^9, 3.5226514771418753`*^9, 3.5256794673762817`*^9, 3.525679964840897*^9, 3.5256897790391703`*^9, 3.620399199136527*^9, 3.620399289595644*^9, 3.620401061920259*^9, 3.6204028181599913`*^9, 3.620404086750493*^9, 3.620407375869812*^9, 3.62041041522281*^9, 3.6204111307729177`*^9, 3.620413253814891*^9, 3.6204133247306337`*^9, 3.620414074264928*^9, 3.620472108788485*^9, 3.6204721996221733`*^9, 3.620474006431222*^9, 3.620488276682797*^9, { 3.62049008740506*^9, 3.6204901275110407`*^9}, 3.6217525041051817`*^9, { 3.621752547650836*^9, 3.621752561548459*^9}, 3.621753264619041*^9, { 3.623768481515628*^9, 3.623768495275475*^9}, 3.623822435622315*^9, 3.623822476181375*^9, 3.6238488556517878`*^9, 3.623848908558123*^9, 3.623848958866826*^9, 3.623849013347327*^9, 3.6238505720803757`*^9, 3.623852353995593*^9, 3.623852409662582*^9}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"eqns", "=", RowBox[{ RowBox[{"Table", "[", RowBox[{ RowBox[{ RowBox[{"P", "[", "n", "]"}], "/.", RowBox[{"If", "[", RowBox[{ RowBox[{"n", "\[GreaterEqual]", "1"}], ",", " ", RowBox[{ RowBox[{ SubscriptBox["S", "n"], "[", "z", "]"}], "\[Rule]", "0"}], ",", RowBox[{ RowBox[{ SubscriptBox["S", "n"], "[", "z", "]"}], "\[Rule]", RowBox[{ SubscriptBox["S", "n"], "[", "z", "]"}]}]}], "]"}]}], ",", RowBox[{"{", RowBox[{"n", ",", "0", ",", "L"}], "}"}]}], "]"}], "/.", RowBox[{ RowBox[{ SubscriptBox["\[Phi]", RowBox[{"L", "+", "1"}]], "[", "z", "]"}], "\[Rule]", "0"}]}]}], ";"}]], "Input", CellChangeTimes->{{3.45818867740625*^9, 3.458188687046875*^9}, { 3.458188775546875*^9, 3.458188780734375*^9}, {3.45818884578125*^9, 3.458188894453125*^9}, {3.45818895790625*^9, 3.458188958328125*^9}, { 3.458227745859375*^9, 3.458227777765625*^9}, {3.4582278120625*^9, 3.45822783478125*^9}, {3.45822786678125*^9, 3.458227929453125*^9}, { 3.5111917196508913`*^9, 3.511191720074617*^9}, 3.511193072421953*^9, { 3.511193224200265*^9, 3.51119322650114*^9}, 3.522645020849934*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"MatrixForm", "[", "eqns", "]"}]], "Input", CellChangeTimes->{{3.5111930737322273`*^9, 3.5111930770133553`*^9}, 3.522645020850165*^9}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", TagBox[GridBox[{ { RowBox[{ RowBox[{ RowBox[{"Q", " ", RowBox[{ SubscriptBox["\[Phi]", "1"], "[", "z", "]"}]}], "+", RowBox[{ RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "0"}]], "[", "z", "]"}]}]}], "\[Equal]", RowBox[{ SubscriptBox["S", "0"], "[", "z", "]"}]}]}, { RowBox[{ RowBox[{ RowBox[{ FractionBox["1", "3"], " ", "Q", " ", RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}]}], "+", RowBox[{ FractionBox["2", "3"], " ", "Q", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "+", RowBox[{ RowBox[{ SubscriptBox["\[Phi]", "1"], "[", "z", "]"}], " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "1"}]], "[", "z", "]"}]}]}], "\[Equal]", "0"}]}, { RowBox[{ RowBox[{ RowBox[{ FractionBox["2", "5"], " ", "Q", " ", RowBox[{ SubscriptBox["\[Phi]", "1"], "[", "z", "]"}]}], "+", RowBox[{ FractionBox["3", "5"], " ", "Q", " ", RowBox[{ SubscriptBox["\[Phi]", "3"], "[", "z", "]"}]}], "+", RowBox[{ RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}], " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "2"}]], "[", "z", "]"}]}]}], "\[Equal]", "0"}]}, { RowBox[{ RowBox[{ RowBox[{ FractionBox["3", "7"], " ", "Q", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "+", RowBox[{ RowBox[{ SubscriptBox["\[Phi]", "3"], "[", "z", "]"}], " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "3"}]], "[", "z", "]"}]}]}], "\[Equal]", "0"}]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.5599999999999999]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], Column], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{ 3.522650375178441*^9, 3.522651349884713*^9, 3.522651477292643*^9, 3.525679467682228*^9, 3.5256799648540688`*^9, 3.52568977905268*^9, 3.620399199205152*^9, 3.620399289685996*^9, 3.6204010619870863`*^9, 3.620402818229601*^9, 3.6204040868187428`*^9, 3.620407375938766*^9, 3.620410415343788*^9, 3.620411130855154*^9, 3.620413253899328*^9, 3.62041332481537*^9, 3.620414074401692*^9, 3.6204721088967943`*^9, 3.620472199700014*^9, 3.620474006501395*^9, 3.620488276738315*^9, 3.6204900874766808`*^9, 3.620490127600861*^9, 3.621752504195196*^9, { 3.621752547718322*^9, 3.621752561615464*^9}, 3.621753264687705*^9, { 3.623768486059857*^9, 3.623768495344705*^9}, 3.623822435717731*^9, 3.623822476243401*^9, 3.62384885573315*^9, 3.6238489086144657`*^9, 3.6238489589493513`*^9, 3.623849013433112*^9, 3.623850572148856*^9, 3.623852354068809*^9, 3.623852409731594*^9}] }, Open ]], Cell[TextData[StyleBox["Expansion moments (odd/even)", FontSize->16]], "Text", CellChangeTimes->{{3.522645067217414*^9, 3.522645068559268*^9}, { 3.522645113653557*^9, 3.522645154760992*^9}, {3.5226455253291683`*^9, 3.522645537447722*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"oddn", "=", RowBox[{"Table", "[", RowBox[{ RowBox[{ RowBox[{"2", "n"}], "-", "1"}], ",", RowBox[{"{", RowBox[{"n", ",", "1", ",", RowBox[{ RowBox[{"(", RowBox[{"L", "+", "1"}], ")"}], "/", "2"}]}], "}"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"evenn", "=", RowBox[{"Table", "[", RowBox[{ RowBox[{"2", "n"}], ",", RowBox[{"{", RowBox[{"n", ",", "1", ",", RowBox[{ RowBox[{"(", RowBox[{"L", "+", "1"}], ")"}], "/", "2"}]}], "}"}]}], "]"}]}], ";"}]}], "Input", CellChangeTimes->{{3.51119177011484*^9, 3.5111918155596857`*^9}, { 3.511192684148036*^9, 3.511192690078566*^9}, {3.51119275459908*^9, 3.511192775315415*^9}, {3.511192866711084*^9, 3.5111928920744133`*^9}, 3.522645020853828*^9}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"oddmom", "=", RowBox[{ RowBox[{"Table", "[", RowBox[{ RowBox[{ SubscriptBox["\[Phi]", "n"], "[", "z", "]"}], ",", RowBox[{"{", RowBox[{"n", ",", "0", ",", "L"}], "}"}]}], "]"}], "[", RowBox[{"[", "evenn", "]"}], "]"}]}], "\[IndentingNewLine]", RowBox[{"evenmom", "=", RowBox[{ RowBox[{"Table", "[", RowBox[{ RowBox[{ SubscriptBox["\[Phi]", "n"], "[", "z", "]"}], ",", RowBox[{"{", RowBox[{"n", ",", "0", ",", "L"}], "}"}]}], "]"}], "[", RowBox[{"[", "oddn", "]"}], "]"}]}]}], "Input", CellChangeTimes->{{3.511192900976054*^9, 3.5111929646491623`*^9}, 3.522645020854443*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox["\[Phi]", "1"], "[", "z", "]"}], ",", RowBox[{ SubscriptBox["\[Phi]", "3"], "[", "z", "]"}]}], "}"}]], "Output", CellChangeTimes->{ 3.5226503752649097`*^9, 3.522651350021831*^9, 3.5226514774432*^9, 3.5256794680033073`*^9, 3.525679965007966*^9, 3.525689779326276*^9, 3.620399199285719*^9, 3.6203992897935133`*^9, 3.6204010620697603`*^9, 3.6204028183104057`*^9, 3.620404086873329*^9, 3.620407376024438*^9, 3.620410415406571*^9, 3.6204111309453506`*^9, 3.620413253981318*^9, 3.6204133248972263`*^9, 3.6204140744665194`*^9, 3.620472108978036*^9, 3.620472199786508*^9, 3.6204740065757933`*^9, 3.620488276817856*^9, 3.620490087537642*^9, 3.620490127677347*^9, 3.621752504271208*^9, { 3.621752547803879*^9, 3.621752561713209*^9}, 3.621753264769299*^9, 3.623768495425861*^9, 3.623822435789507*^9, 3.623822476312888*^9, 3.6238488558006*^9, 3.623848908678795*^9, 3.6238489590354652`*^9, 3.6238490135170307`*^9, 3.6238505722156754`*^9, 3.623852354149014*^9, 3.623852409815298*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], ",", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "}"}]], "Output", CellChangeTimes->{ 3.5226503752649097`*^9, 3.522651350021831*^9, 3.5226514774432*^9, 3.5256794680033073`*^9, 3.525679965007966*^9, 3.525689779326276*^9, 3.620399199285719*^9, 3.6203992897935133`*^9, 3.6204010620697603`*^9, 3.6204028183104057`*^9, 3.620404086873329*^9, 3.620407376024438*^9, 3.620410415406571*^9, 3.6204111309453506`*^9, 3.620413253981318*^9, 3.6204133248972263`*^9, 3.6204140744665194`*^9, 3.620472108978036*^9, 3.620472199786508*^9, 3.6204740065757933`*^9, 3.620488276817856*^9, 3.620490087537642*^9, 3.620490127677347*^9, 3.621752504271208*^9, { 3.621752547803879*^9, 3.621752561713209*^9}, 3.621753264769299*^9, 3.623768495425861*^9, 3.623822435789507*^9, 3.623822476312888*^9, 3.6238488558006*^9, 3.623848908678795*^9, 3.6238489590354652`*^9, 3.6238490135170307`*^9, 3.6238505722156754`*^9, 3.623852354149014*^9, 3.623852409817082*^9}] }, Open ]], Cell[TextData[{ StyleBox["Reduction to the \[OpenCurlyDoubleQuote]simplified ", FontSize->16], Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox["P", "L"], FontSize->16], TraditionalForm]]], " ", StyleBox["form\[CloseCurlyDoubleQuote] (second order eqns.)", FontSize->16] }], "Text", CellChangeTimes->{{3.522645067217414*^9, 3.522645068559268*^9}, { 3.522645113653557*^9, 3.522645154760992*^9}, {3.5226455253291683`*^9, 3.522645537447722*^9}, {3.522645757714418*^9, 3.522645804206421*^9}, 3.522649072559741*^9, {3.623848540766366*^9, 3.623848540766457*^9}}], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{ "express", " ", "odd", " ", "moments", " ", "using", " ", "the", " ", "even", " ", "ones"}], TraditionalForm]], FontSize->14]], "Item", CellChangeTimes->{{3.522645067217414*^9, 3.522645068559268*^9}, { 3.522645113653557*^9, 3.522645154760992*^9}, {3.5226455485449467`*^9, 3.522645578351975*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"oddWrtEven", "=", RowBox[{"Solve", "[", RowBox[{ RowBox[{"eqns", "[", RowBox[{"[", "evenn", "]"}], "]"}], ",", "oddmom"}], "]"}]}]], "Input", CellChangeTimes->{{3.4582276326875*^9, 3.45822764625*^9}, { 3.45822793853125*^9, 3.458227991609375*^9}, {3.458228163375*^9, 3.45822819828125*^9}, {3.511191734750399*^9, 3.511191767970848*^9}, 3.511191801717226*^9, {3.511192816168118*^9, 3.511192818170897*^9}, { 3.511192989617923*^9, 3.511193031050684*^9}, 3.52264502085569*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[Phi]", "1"], "[", "z", "]"}], "\[Rule]", FractionBox[ RowBox[{ RowBox[{ RowBox[{"-", "Q"}], " ", RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}]}], "-", RowBox[{"2", " ", "Q", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}]}], RowBox[{"3", " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "1"}]], "[", "z", "]"}]}]]}], ",", RowBox[{ RowBox[{ SubscriptBox["\[Phi]", "3"], "[", "z", "]"}], "\[Rule]", RowBox[{"-", FractionBox[ RowBox[{"3", " ", "Q", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], RowBox[{"7", " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "3"}]], "[", "z", "]"}]}]]}]}]}], "}"}], "}"}]], "Output", CellChangeTimes->{ 3.5226503753249083`*^9, 3.5226513501689262`*^9, 3.522651477604101*^9, 3.525679468244091*^9, 3.5256799651665707`*^9, 3.525689779338985*^9, 3.62039919934566*^9, 3.620399289833177*^9, 3.620401062130253*^9, 3.6204028183701344`*^9, 3.620404086926456*^9, 3.620407376076561*^9, 3.620410415446061*^9, 3.620411131022579*^9, 3.6204132540500383`*^9, 3.620413324983349*^9, 3.620414074519293*^9, 3.620472109058571*^9, 3.6204721998282633`*^9, 3.620474006617029*^9, 3.620488276872361*^9, 3.620490087599471*^9, 3.620490127746189*^9, 3.621752504323694*^9, { 3.621752547856861*^9, 3.621752561752541*^9}, 3.621753264811199*^9, 3.6237684954786863`*^9, 3.6238224358800097`*^9, 3.6238224763712883`*^9, 3.623848855852096*^9, 3.623848908730843*^9, 3.623848959089023*^9, 3.6238490135742483`*^9, 3.623850572269738*^9, 3.623852354202051*^9, 3.6238524098667717`*^9}] }, Open ]], Cell[TextData[StyleBox["insert into the equations", FontSize->14]], "Item", CellChangeTimes->{{3.522645067217414*^9, 3.522645068559268*^9}, { 3.522645113653557*^9, 3.522645154760992*^9}, {3.5226455253291683`*^9, 3.522645537447722*^9}, {3.522645757714418*^9, 3.522645804206421*^9}, { 3.5226487090304813`*^9, 3.522648717168818*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"SPL", "=", RowBox[{ RowBox[{ RowBox[{"eqns", "[", RowBox[{"[", "oddn", "]"}], "]"}], "/.", RowBox[{"oddWrtEven", "[", RowBox[{"[", "1", "]"}], "]"}]}], "/.", RowBox[{ RowBox[{ SubscriptBox["S", "0"], "[", "z", "]"}], "\[Rule]", RowBox[{ SubscriptBox["\[Nu]\[CapitalSigma]", "f"], RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}]}]}]}]}], ";"}]], "Input", CellChangeTimes->{{3.458228157015625*^9, 3.45822820375*^9}, { 3.45822862803125*^9, 3.45822862990625*^9}, {3.511193106327012*^9, 3.511193148589272*^9}, {3.514733415740405*^9, 3.514733452798963*^9}, 3.522645020859645*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"MatrixForm", "[", "SPL", "]"}]], "Input", CellChangeTimes->{{3.511193149864786*^9, 3.5111931532293577`*^9}, 3.522645020859894*^9}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", TagBox[GridBox[{ { RowBox[{ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "0"}]], "[", "z", "]"}]}], "+", FractionBox[ RowBox[{"Q", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "Q"}], " ", RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}]}], "-", RowBox[{"2", " ", "Q", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}]}], ")"}]}], RowBox[{"3", " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "1"}]], "[", "z", "]"}]}]]}], "\[Equal]", RowBox[{ SubscriptBox["\[Nu]\[CapitalSigma]", "f"], " ", RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}]}]}]}, { RowBox[{ RowBox[{ FractionBox[ RowBox[{"2", " ", "Q", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "Q"}], " ", RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}]}], "-", RowBox[{"2", " ", "Q", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}]}], ")"}]}], RowBox[{"15", " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "1"}]], "[", "z", "]"}]}]], "+", RowBox[{ RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}], " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "2"}]], "[", "z", "]"}]}], "-", FractionBox[ RowBox[{"9", " ", SuperscriptBox["Q", "2"], " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], RowBox[{"35", " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "3"}]], "[", "z", "]"}]}]]}], "\[Equal]", "0"}]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.5599999999999999]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], Column], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{ 3.522650375510723*^9, 3.522651350299906*^9, 3.522651477788069*^9, 3.525679468828157*^9, 3.5256799651918287`*^9, 3.525689779353943*^9, 3.620399199402336*^9, 3.6203992899113903`*^9, 3.620401062224381*^9, 3.620402818444358*^9, 3.62040408698873*^9, 3.620407376156348*^9, 3.620410415524976*^9, 3.620411131114132*^9, 3.6204132541368647`*^9, 3.6204133250494413`*^9, 3.620414074600897*^9, 3.620472109126741*^9, 3.620472199908379*^9, 3.620474006688509*^9, 3.620488276952532*^9, 3.620490087656061*^9, 3.620490127829117*^9, 3.621752504386334*^9, { 3.621752547968135*^9, 3.621752561831951*^9}, 3.621753264902121*^9, 3.62376849555783*^9, 3.623822435959474*^9, 3.623822476445939*^9, 3.623848855913312*^9, 3.623848908793263*^9, 3.623848959167267*^9, 3.623849013648638*^9, 3.623850572346718*^9, 3.623852354279943*^9, 3.623852409947501*^9}] }, Open ]], Cell[TextData[{ StyleBox["Definition of \[OpenCurlyDoubleQuote]pseudo-fluxes\ \[CloseCurlyDoubleQuote] ", FontSize->16], Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox["F", "i"], FontSize->16], TraditionalForm]]], StyleBox[" and \[OpenCurlyDoubleQuote]pseudo-diffusion-coefficients\ \[CloseCurlyDoubleQuote] ", FontSize->16], Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox["D", "i"], FontSize->16], TraditionalForm]]], StyleBox["\n", FontSize->16], StyleBox["such that \[OpenCurlyDoubleQuote]pseudo-currents\ \[CloseCurlyDoubleQuote] may be defined according to \ \[OpenCurlyDoubleQuote]pseudo-Fick\[CloseCurlyQuote]s-law\ \[CloseCurlyDoubleQuote] as ", FontSize->14], Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ StyleBox[ SubscriptBox["D", RowBox[{ RowBox[{"2", "i"}], "+", "1"}]], "InlineFormula"], FontSize->14], StyleBox[" ", FontSize->14], RowBox[{ StyleBox[ RowBox[{"\[DifferentialD]", SubscriptBox["F", "i"]}], FontSize->14], StyleBox["/", "InlineFormula", FontSize->12], RowBox[{ StyleBox["\[DifferentialD]", "InlineFormula", FontSize->12], StyleBox["z", "InlineFormula", FontSize->12], StyleBox[" ", FontSize->14]}]}]}], TraditionalForm]]], StyleBox["(here ", FontSize->14], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["D", "i"], "Q", " ", SubscriptBox["F", "i"]}], TraditionalForm]]], Cell[BoxData[")"], FontSize->14] }], "Text", CellChangeTimes->{{3.522645067217414*^9, 3.522645068559268*^9}, { 3.522645113653557*^9, 3.522645154760992*^9}, {3.5226455253291683`*^9, 3.522645537447722*^9}, {3.522645757714418*^9, 3.522645804206421*^9}, { 3.5226458615113*^9, 3.5226459447188663`*^9}, {3.522645982553526*^9, 3.522646010994257*^9}, {3.522646050970581*^9, 3.522646123120521*^9}, { 3.5226465046629467`*^9, 3.5226465396714478`*^9}, {3.6238485704102507`*^9, 3.623848572723988*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"evenPseudoFluxes", "=", RowBox[{ RowBox[{"Table", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"n", "+", "1"}], ")"}], " ", RowBox[{ SubscriptBox["\[Phi]", "n"], "[", "z", "]"}]}], " ", "+", RowBox[{ RowBox[{"(", RowBox[{"n", "+", "2"}], ")"}], " ", RowBox[{ SubscriptBox["\[Phi]", RowBox[{"n", "+", "2"}]], "[", "z", "]"}]}]}], "==", RowBox[{ SubscriptBox["F", "n"], "[", "z", "]"}]}], ",", RowBox[{"{", RowBox[{"n", ",", "0", ",", "L", ",", "2"}], "}"}]}], "]"}], "/.", RowBox[{ RowBox[{ SubscriptBox["\[Phi]", RowBox[{"L", "+", "1"}]], "[", "z", "]"}], "\[Rule]", "0"}]}]}], "\[IndentingNewLine]", RowBox[{"pseudoDiffusions", "=", RowBox[{"Table", "[", RowBox[{ RowBox[{ FractionBox["1", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"2", "n"}], "+", "1"}], ")"}], RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "n"}]], "[", "z", "]"}]}]], "\[Rule]", RowBox[{ SubscriptBox["D", "n"], "[", "z", "]"}]}], ",", RowBox[{"{", RowBox[{"n", ",", "1", ",", "L", ",", "2"}], "}"}]}], "]"}]}]}], "Input", CellChangeTimes->{{3.511198870770874*^9, 3.511198893430962*^9}, { 3.51119896743093*^9, 3.511198992928875*^9}, {3.5111991093340263`*^9, 3.511199123137375*^9}, {3.511199154122465*^9, 3.511199174917679*^9}, 3.511199248039833*^9, {3.511199339842208*^9, 3.511199377349909*^9}, { 3.511199412373128*^9, 3.511199481475945*^9}, 3.5111995140756283`*^9, { 3.5111996191394587`*^9, 3.51119964075972*^9}, {3.51119978280777*^9, 3.511199797957222*^9}, {3.51119984192183*^9, 3.511199873777314*^9}, { 3.511199946402473*^9, 3.5112000558439837`*^9}, {3.5112000961881447`*^9, 3.511200108909154*^9}, {3.511202196227047*^9, 3.511202196403117*^9}, { 3.511202485927823*^9, 3.5112024876325912`*^9}, {3.511204065899312*^9, 3.511204067014629*^9}, {3.511204183541585*^9, 3.511204199201779*^9}, { 3.5112045459606247`*^9, 3.511204547218207*^9}, {3.51124409569376*^9, 3.511244114424789*^9}, {3.511248106834757*^9, 3.511248116422564*^9}, { 3.511248561717697*^9, 3.511248562052051*^9}, {3.5112495259502153`*^9, 3.511249559213081*^9}, {3.5112564023441677`*^9, 3.511256403425394*^9}, { 3.514636836673696*^9, 3.514636896166847*^9}, 3.522645020864522*^9, { 3.620401037248993*^9, 3.620401037390915*^9}, {3.620401164717578*^9, 3.620401164836218*^9}, {3.62040233735985*^9, 3.6204023420641413`*^9}, 3.6238485877545557`*^9, {3.623848884049323*^9, 3.623848898256793*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], "+", RowBox[{"2", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}]}], "\[Equal]", RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}]}], ",", RowBox[{ RowBox[{"3", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "\[Equal]", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}]}]}], "}"}]], "Output", CellChangeTimes->{ 3.522650375525948*^9, 3.52265135046631*^9, 3.522651477921174*^9, 3.5256794691292963`*^9, 3.525679965296945*^9, 3.525689779361676*^9, 3.620399199445819*^9, 3.6203992899520893`*^9, {3.620401040843316*^9, 3.620401062260569*^9}, 3.620401167861483*^9, 3.620402818504366*^9, 3.62040408704218*^9, 3.620407376209483*^9, 3.6204104155613413`*^9, 3.620411131161474*^9, 3.620413254184952*^9, 3.620413325107617*^9, 3.620414074659274*^9, 3.62047210918401*^9, 3.6204721999459*^9, 3.620474006731588*^9, 3.620488277005869*^9, 3.620490087698781*^9, 3.620490127866515*^9, 3.621752504429459*^9, {3.621752548005726*^9, 3.621752561874942*^9}, 3.621753264942205*^9, 3.623768495610394*^9, 3.623822436016405*^9, 3.6238224765008087`*^9, 3.623848855955776*^9, 3.623848908853032*^9, 3.623848959226489*^9, 3.62384901370645*^9, 3.6238505723948*^9, 3.623852354320883*^9, 3.6238524099885807`*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ FractionBox["1", RowBox[{"3", " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "1"}]], "[", "z", "]"}]}]], "\[Rule]", RowBox[{ SubscriptBox["D", "1"], "[", "z", "]"}]}], ",", RowBox[{ FractionBox["1", RowBox[{"7", " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "3"}]], "[", "z", "]"}]}]], "\[Rule]", RowBox[{ SubscriptBox["D", "3"], "[", "z", "]"}]}]}], "}"}]], "Output", CellChangeTimes->{ 3.522650375525948*^9, 3.52265135046631*^9, 3.522651477921174*^9, 3.5256794691292963`*^9, 3.525679965296945*^9, 3.525689779361676*^9, 3.620399199445819*^9, 3.6203992899520893`*^9, {3.620401040843316*^9, 3.620401062260569*^9}, 3.620401167861483*^9, 3.620402818504366*^9, 3.62040408704218*^9, 3.620407376209483*^9, 3.6204104155613413`*^9, 3.620411131161474*^9, 3.620413254184952*^9, 3.620413325107617*^9, 3.620414074659274*^9, 3.62047210918401*^9, 3.6204721999459*^9, 3.620474006731588*^9, 3.620488277005869*^9, 3.620490087698781*^9, 3.620490127866515*^9, 3.621752504429459*^9, {3.621752548005726*^9, 3.621752561874942*^9}, 3.621753264942205*^9, 3.623768495610394*^9, 3.623822436016405*^9, 3.6238224765008087`*^9, 3.623848855955776*^9, 3.623848908853032*^9, 3.623848959226489*^9, 3.62384901370645*^9, 3.6238505723948*^9, 3.623852354320883*^9, 3.623852409990417*^9}] }, Open ]], Cell[TextData[{ StyleBox["Even expansion moments expressed using the even pseudo-fluxes\n", FontSize->16], StyleBox["the zero-th moment is the scalar flux", FontSize->14] }], "Text", CellChangeTimes->{{3.522645067217414*^9, 3.522645068559268*^9}, { 3.522645113653557*^9, 3.522645154760992*^9}, {3.5226455253291683`*^9, 3.522645537447722*^9}, {3.522645757714418*^9, 3.522645804206421*^9}, { 3.5226458615113*^9, 3.5226459447188663`*^9}, {3.522645982553526*^9, 3.522646010994257*^9}, {3.522646050970581*^9, 3.522646123120521*^9}, { 3.5226465046629467`*^9, 3.5226465396714478`*^9}, {3.522647071486895*^9, 3.5226471070132837`*^9}, {3.623848600217868*^9, 3.6238486229909487`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Table", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[Phi]", RowBox[{"2", "i"}]], "[", "z", "]"}], "/.", RowBox[{"Solve", "[", RowBox[{ RowBox[{"Eliminate", "[", RowBox[{"evenPseudoFluxes", ",", RowBox[{"Drop", "[", RowBox[{"evenmom", ",", RowBox[{"{", RowBox[{"i", "+", "1"}], "}"}]}], "]"}]}], "]"}], ",", RowBox[{ SubscriptBox["\[Phi]", RowBox[{"2", "i"}]], "[", "z", "]"}]}], "]"}]}], "//", "First"}], "//", "Expand"}], ",", RowBox[{"{", RowBox[{"i", ",", "0", ",", FractionBox[ RowBox[{"L", "-", "1"}], "2"]}], "}"}]}], "]"}], "//", "MatrixForm"}]], "Input", CellChangeTimes->{{3.515147957005156*^9, 3.515147979435075*^9}, { 3.5151480241667213`*^9, 3.5151480710490923`*^9}, {3.515148119828765*^9, 3.5151481416956253`*^9}, {3.5151481859752817`*^9, 3.515148206155452*^9}, { 3.515148241837077*^9, 3.515148560189735*^9}, 3.5226450208674107`*^9, 3.623848587788021*^9}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", TagBox[GridBox[{ { RowBox[{ RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}], "-", FractionBox[ RowBox[{"2", " ", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}]}], "3"]}]}, { FractionBox[ RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}], "3"]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.5599999999999999]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], Column], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{ 3.522650375590062*^9, 3.522651350691865*^9, 3.5226514780448*^9, 3.5256794694259243`*^9, 3.525679965496266*^9, 3.525689779509355*^9, 3.620399199492794*^9, 3.620399289999262*^9, {3.620401052794924*^9, 3.62040106230986*^9}, 3.620401170604519*^9, 3.620402818548986*^9, 3.620404087095265*^9, 3.6204073762584887`*^9, 3.6204104156178007`*^9, 3.620411131233238*^9, 3.6204132542353077`*^9, 3.620413325151917*^9, 3.620414074704316*^9, 3.6204721092392387`*^9, 3.620472199986762*^9, 3.620474006769041*^9, 3.620488277054881*^9, 3.6204900877439833`*^9, 3.6204901279169483`*^9, 3.621752504474189*^9, {3.621752548056973*^9, 3.621752561919673*^9}, 3.621753264989958*^9, 3.623768495649053*^9, 3.623822436077702*^9, 3.6238224765508337`*^9, 3.6238488560016527`*^9, 3.6238489088987293`*^9, 3.623848959272328*^9, 3.623849013755933*^9, 3.623850572458747*^9, 3.623852354361434*^9, 3.623852410033721*^9}] }, Open ]], Cell[TextData[{ StyleBox["Expressing the ", FontSize->16], Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox["SP", "N"], FontSize->16], TraditionalForm]]], StyleBox["equations in the operator form ", FontSize->16], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ StyleBox[ RowBox[{"A", " ", RowBox[{"{", SubscriptBox["F", "i"], "}"}]}], FontSize->16], StyleBox["=", FontSize->16], RowBox[{ StyleBox["{", FontSize->16], SubscriptBox[ StyleBox["b", FontSize->16], "i"], "}"}]}], StyleBox[",", FontSize->16]}], TraditionalForm]]], " ", StyleBox["where the streaming matrix ", FontSize->16], Cell[BoxData[ FormBox[ RowBox[{ StyleBox["A", FontSize->16], StyleBox["=", FontSize->16], RowBox[{ StyleBox["diag", FontSize->14], StyleBox["(", FontSize->14], RowBox[{ RowBox[{ FractionBox["\[DifferentialD]", RowBox[{"\[DifferentialD]", "z"}]], SubscriptBox["D", RowBox[{ RowBox[{"2", "i"}], "+", "1"}]]}], FractionBox["\[DifferentialD]", RowBox[{"\[DifferentialD]", "z"}]]}], ")"}]}], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.522645067217414*^9, 3.522645068559268*^9}, { 3.522645113653557*^9, 3.522645154760992*^9}, {3.5226455253291683`*^9, 3.522645537447722*^9}, {3.522645757714418*^9, 3.522645804206421*^9}, { 3.5226458615113*^9, 3.5226459447188663`*^9}, {3.522645982553526*^9, 3.522646010994257*^9}, {3.522646050970581*^9, 3.522646123120521*^9}, { 3.5226465046629467`*^9, 3.5226465396714478`*^9}, {3.522647071486895*^9, 3.5226471070132837`*^9}, {3.522647196226143*^9, 3.5226473057929897`*^9}, { 3.5226473374375267`*^9, 3.522647550210237*^9}, {3.522647591318706*^9, 3.52264759216264*^9}, {3.5226476533707314`*^9, 3.522647691867504*^9}, { 3.522648200400393*^9, 3.522648203787119*^9}, {3.62384863866026*^9, 3.62384864027973*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"lower", "=", RowBox[{"Table", "[", RowBox[{ FractionBox["n", RowBox[{ RowBox[{"2", "n"}], "+", "1"}]], ",", RowBox[{"{", RowBox[{"n", ",", "2", ",", "L", ",", "2"}], "}"}]}], "]"}]}], "\[IndentingNewLine]", RowBox[{"centr", "=", RowBox[{"Table", "[", RowBox[{ FractionBox[ RowBox[{"n", "+", "1"}], RowBox[{ RowBox[{"2", "n"}], "+", "1"}]], ",", RowBox[{"{", RowBox[{"n", ",", "0", ",", "L", ",", "2"}], "}"}]}], "]"}]}]}], "Input", CellChangeTimes->{{3.511099559635169*^9, 3.511099786572399*^9}, { 3.511252347111068*^9, 3.511252419805832*^9}, {3.511252450645809*^9, 3.51125247877311*^9}, {3.511252517000834*^9, 3.511252537606093*^9}, 3.5112525744886827`*^9, {3.511252620403894*^9, 3.51125273232136*^9}, { 3.511252844411414*^9, 3.5112529034653187`*^9}, {3.51125333405569*^9, 3.5112533688901367`*^9}, {3.5112552371421947`*^9, 3.511255241693265*^9}, { 3.511257897717341*^9, 3.5112579027375526`*^9}, {3.511319808428637*^9, 3.511319838243514*^9}, 3.522645020869926*^9}], Cell[BoxData[ RowBox[{"{", FractionBox["2", "5"], "}"}]], "Output", CellChangeTimes->{ 3.522650375617648*^9, 3.522651350890203*^9, 3.522651478327033*^9, 3.525679469450214*^9, 3.5256799656282473`*^9, 3.525689779656475*^9, 3.6203991995471478`*^9, 3.620399290049654*^9, 3.6204010623652973`*^9, 3.6204028186305933`*^9, 3.620404087143428*^9, 3.620407376308667*^9, 3.6204104156642027`*^9, 3.6204111312823687`*^9, 3.620413254290773*^9, 3.620413325201519*^9, 3.620414074756526*^9, 3.6204721092790337`*^9, 3.6204722000387506`*^9, 3.620474006815679*^9, 3.620488277105727*^9, 3.620490087796289*^9, 3.620490127968329*^9, 3.621752504524531*^9, { 3.621752548107583*^9, 3.621752561968114*^9}, 3.621753265039534*^9, 3.623768495694738*^9, 3.623822436128755*^9, 3.623822476600819*^9, 3.6238488560532303`*^9, 3.6238489089462137`*^9, 3.623848959323576*^9, 3.623849013803296*^9, 3.6238505725027447`*^9, 3.6238523544113903`*^9, 3.623852410084015*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{"1", ",", FractionBox["3", "5"]}], "}"}]], "Output", CellChangeTimes->{ 3.522650375617648*^9, 3.522651350890203*^9, 3.522651478327033*^9, 3.525679469450214*^9, 3.5256799656282473`*^9, 3.525689779656475*^9, 3.6203991995471478`*^9, 3.620399290049654*^9, 3.6204010623652973`*^9, 3.6204028186305933`*^9, 3.620404087143428*^9, 3.620407376308667*^9, 3.6204104156642027`*^9, 3.6204111312823687`*^9, 3.620413254290773*^9, 3.620413325201519*^9, 3.620414074756526*^9, 3.6204721092790337`*^9, 3.6204722000387506`*^9, 3.620474006815679*^9, 3.620488277105727*^9, 3.620490087796289*^9, 3.620490127968329*^9, 3.621752504524531*^9, { 3.621752548107583*^9, 3.621752561968114*^9}, 3.621753265039534*^9, 3.623768495694738*^9, 3.623822436128755*^9, 3.623822476600819*^9, 3.6238488560532303`*^9, 3.6238489089462137`*^9, 3.623848959323576*^9, 3.623849013803296*^9, 3.6238505725027447`*^9, 3.6238523544113903`*^9, 3.6238524100856943`*^9}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"Adata", "=", RowBox[{"If", "[", RowBox[{ RowBox[{"L", ">", "1"}], ",", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"Band", "[", RowBox[{"{", RowBox[{"2", ",", "1"}], "}"}], "]"}], "\[Rule]", "lower"}], ",", RowBox[{ RowBox[{"Band", "[", RowBox[{"{", RowBox[{"1", ",", "1"}], "}"}], "]"}], "\[Rule]", "centr"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"Band", "[", RowBox[{"{", RowBox[{"1", ",", "1"}], "}"}], "]"}], "\[Rule]", "centr"}], "}"}]}], "]"}]}], ";"}]], "Input", CellChangeTimes->{{3.511320289582745*^9, 3.5113203145331297`*^9}, 3.5226450208714237`*^9}], Cell[BoxData[ RowBox[{ RowBox[{"A", "=", RowBox[{"SparseArray", "[", RowBox[{"Adata", ",", RowBox[{ RowBox[{"Floor", "[", RowBox[{"L", "/", "2"}], "]"}], "+", "1"}]}], "]"}]}], ";"}]], "Input", CellChangeTimes->{{3.511099793187874*^9, 3.511099811008842*^9}, { 3.511099900495983*^9, 3.5110999068878717`*^9}, {3.511250206927154*^9, 3.511250211433297*^9}, {3.511252770660775*^9, 3.5112527735603247`*^9}, { 3.5112528519363728`*^9, 3.511252862988221*^9}, {3.5112552623202543`*^9, 3.511255305922925*^9}, {3.5113201916338367`*^9, 3.5113202770730057`*^9}, { 3.511320324800376*^9, 3.511320325552327*^9}, 3.522645020872095*^9}, CellID->1804033555], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"A", "//", "Normal"}], "//", "MatrixForm"}]], "Input", CellChangeTimes->{{3.511250217598606*^9, 3.51125023358689*^9}, 3.522645020872394*^9}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0"}, { FractionBox["2", "5"], FractionBox["3", "5"]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{ 3.5226503756381598`*^9, 3.522651351177065*^9, 3.522651478349552*^9, 3.525679469834527*^9, 3.52567996565206*^9, 3.5256897796942263`*^9, 3.620399199652396*^9, 3.620399290160839*^9, 3.62040106247266*^9, 3.620402818744816*^9, 3.6204040872535257`*^9, 3.620407376420032*^9, 3.620410415774437*^9, 3.620411131390358*^9, 3.6204132543975*^9, 3.620413325313533*^9, 3.6204140748688107`*^9, 3.62047210936619*^9, 3.620472200142898*^9, 3.620474006894587*^9, 3.620488277251854*^9, 3.620490087888933*^9, 3.620490128077776*^9, 3.621752504636771*^9, { 3.621752548216943*^9, 3.621752562080822*^9}, 3.621753265152206*^9, 3.6237684958060637`*^9, 3.6238224362383623`*^9, 3.623822477330809*^9, 3.6238488561643867`*^9, 3.623848909060017*^9, 3.623848959435173*^9, 3.623849013914928*^9, 3.62385057261517*^9, 3.623852354531702*^9, 3.623852410195655*^9}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"rhs", "=", RowBox[{"Table", "[", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "n"}]], "[", "z", "]"}], RowBox[{ SubscriptBox["\[Phi]", "n"], "[", "z", "]"}]}], ",", RowBox[{"{", RowBox[{"n", ",", "0", ",", "L", ",", "2"}], "}"}]}], "]"}]}], ";"}]], "Input", CellChangeTimes->{{3.511255708094071*^9, 3.5112557367042418`*^9}, { 3.511255783405711*^9, 3.511255802367193*^9}, 3.511260350277404*^9, 3.514733594698781*^9, 3.522645020876772*^9, 3.623848647144771*^9}], Cell[TextData[{ StyleBox["Eliminating the even flux expansion moments in favor of the \ unknowns: the even pseudo-fluxes ", FontSize->16], Cell[BoxData[ FormBox[ SubscriptBox["F", "i"], TraditionalForm]], FontSize->16] }], "Text", CellChangeTimes->{{3.522645067217414*^9, 3.522645068559268*^9}, { 3.522645113653557*^9, 3.522645154760992*^9}, {3.5226455253291683`*^9, 3.522645537447722*^9}, {3.522645757714418*^9, 3.522645804206421*^9}, { 3.5226458615113*^9, 3.5226459447188663`*^9}, {3.522645982553526*^9, 3.522646010994257*^9}, {3.522646050970581*^9, 3.522646123120521*^9}, { 3.5226465046629467`*^9, 3.5226465396714478`*^9}, {3.522647071486895*^9, 3.5226471070132837`*^9}, {3.522647196226143*^9, 3.5226473057929897`*^9}, { 3.5226473374375267`*^9, 3.522647550210237*^9}, {3.522647591318706*^9, 3.52264759216264*^9}, {3.5226476533707314`*^9, 3.522647691867504*^9}, { 3.522647900608212*^9, 3.522647955486134*^9}, {3.623848653333993*^9, 3.623848655110277*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"DelDDelFs", "=", RowBox[{"Table", "[", RowBox[{ SubscriptBox["DelDDelF", "n"], ",", RowBox[{"{", RowBox[{"n", ",", "0", ",", "L", ",", "2"}], "}"}]}], "]"}]}], ";"}]], "Input", CellChangeTimes->{{3.511258498689275*^9, 3.511258507210991*^9}, { 3.5112588968025084`*^9, 3.5112589006798477`*^9}, 3.5226450208771544`*^9, 3.6238486604161663`*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"eqnsForDelDDelFs1", "=", RowBox[{"Thread", "[", RowBox[{"DelDDelFs", "==", RowBox[{"LinearSolve", "[", RowBox[{"A", ",", "rhs"}], "]"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.511254998608993*^9, 3.511255057123591*^9}, { 3.511255106598044*^9, 3.5112551113618526`*^9}, {3.5112551527031803`*^9, 3.511255213844886*^9}, {3.511255327424749*^9, 3.511255341574107*^9}, 3.511255700177907*^9, {3.511255797566963*^9, 3.511255800262518*^9}, { 3.511255844586472*^9, 3.5112558510141773`*^9}, {3.511256013650725*^9, 3.511256017404396*^9}, {3.5112579338848457`*^9, 3.511257993128029*^9}, { 3.511258031270864*^9, 3.511258069094496*^9}, {3.5112581033623466`*^9, 3.5112581079310207`*^9}, {3.5112584113711042`*^9, 3.5112584363782463`*^9}, {3.511258483745788*^9, 3.511258527473763*^9}, { 3.5112589063452463`*^9, 3.511258912651373*^9}, 3.511259061853382*^9, 3.522645020877507*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox["DelDDelF", "0"], "\[Equal]", RowBox[{ RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "0"}]], "[", "z", "]"}]}]}], ",", RowBox[{ SubscriptBox["DelDDelF", "2"], "\[Equal]", RowBox[{ FractionBox["1", "3"], " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "2"}], " ", RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "0"}]], "[", "z", "]"}]}], "+", RowBox[{"5", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}], " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "2"}]], "[", "z", "]"}]}]}], ")"}]}]}]}], "}"}]], "Output", CellChangeTimes->{ 3.5226503757781687`*^9, 3.522651351569365*^9, 3.522651478508457*^9, 3.52567947043603*^9, 3.525679965760879*^9, 3.525689779864768*^9, 3.620399199756999*^9, 3.6203992902640333`*^9, 3.620401062574347*^9, 3.620402818847137*^9, 3.620404087353136*^9, 3.620407376521151*^9, 3.6204104158773108`*^9, 3.620411131492732*^9, 3.620413254498261*^9, 3.620413325415415*^9, 3.6204140749677362`*^9, 3.6204721094522457`*^9, 3.6204722002598457`*^9, 3.620474006958226*^9, 3.620488277384359*^9, 3.6204900879906797`*^9, 3.6204901281796494`*^9, 3.62175250473715*^9, { 3.621752548318368*^9, 3.621752562183442*^9}, 3.6217532652524977`*^9, 3.6237684959067163`*^9, 3.6238224364226713`*^9, 3.6238224774307137`*^9, 3.623848856264145*^9, 3.623848909160304*^9, 3.623848959534987*^9, 3.623849014017137*^9, 3.623850572715086*^9, 3.6238523546298647`*^9, 3.623852410296403*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"eqnsForDelDDelFs2", "=", RowBox[{"Map", "[", RowBox[{ RowBox[{ RowBox[{"Join", "[", RowBox[{ RowBox[{"{", "#", "}"}], ",", "evenPseudoFluxes"}], "]"}], "&"}], ",", "eqnsForDelDDelFs1"}], "]"}]}]], "Input", CellChangeTimes->{{3.511255836950974*^9, 3.511255837763493*^9}, { 3.511256043119359*^9, 3.5112560438456907`*^9}, {3.5112561501040163`*^9, 3.511256153610244*^9}, {3.511256185440885*^9, 3.511256202372899*^9}, { 3.51125629586084*^9, 3.5112563010687227`*^9}, {3.5112565258042793`*^9, 3.5112565614676657`*^9}, {3.511256686470209*^9, 3.5112567294170313`*^9}, { 3.5112569289834967`*^9, 3.5112569313321753`*^9}, {3.511257004746434*^9, 3.511257032874195*^9}, {3.511257803089201*^9, 3.5112578032807627`*^9}, { 3.5112585621606627`*^9, 3.511258591883244*^9}, {3.511258919594338*^9, 3.511258949067378*^9}, 3.511259065989482*^9, 3.5226450208810062`*^9, 3.623848587816399*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox["DelDDelF", "0"], "\[Equal]", RowBox[{ RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "0"}]], "[", "z", "]"}]}]}], ",", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], "+", RowBox[{"2", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}]}], "\[Equal]", RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}]}], ",", RowBox[{ RowBox[{"3", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "\[Equal]", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox["DelDDelF", "2"], "\[Equal]", RowBox[{ FractionBox["1", "3"], " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "2"}], " ", RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "0"}]], "[", "z", "]"}]}], "+", RowBox[{"5", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}], " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "2"}]], "[", "z", "]"}]}]}], ")"}]}]}], ",", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], "+", RowBox[{"2", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}]}], "\[Equal]", RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}]}], ",", RowBox[{ RowBox[{"3", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "\[Equal]", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}]}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{ 3.52265037584822*^9, 3.522651351832933*^9, 3.522651478632887*^9, 3.525679470702132*^9, 3.525679965775587*^9, 3.5256897798746147`*^9, 3.620399199791864*^9, 3.6203992903009653`*^9, 3.620401062612041*^9, 3.620402818907778*^9, 3.620404087395278*^9, 3.62040737656131*^9, 3.6204104159157867`*^9, 3.620411131553944*^9, 3.6204132545366917`*^9, 3.620413325452894*^9, 3.620414075007255*^9, 3.620472109505021*^9, 3.620472200304388*^9, 3.620474007000957*^9, 3.620488277446587*^9, 3.620490088032213*^9, 3.6204901282209663`*^9, 3.621752504775613*^9, { 3.6217525483574867`*^9, 3.621752562220022*^9}, 3.6217532652916403`*^9, 3.623768495948472*^9, 3.62382243648035*^9, 3.6238224774849987`*^9, 3.623848856302313*^9, 3.623848909197864*^9, 3.62384895959507*^9, 3.6238490140597553`*^9, 3.623850572754115*^9, 3.6238523546696987`*^9, 3.623852410335681*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"eqnsForDelDDelFs3", "=", RowBox[{"Map", "[", RowBox[{ RowBox[{ RowBox[{"Eliminate", "[", RowBox[{"#", ",", "evenmom"}], "]"}], "&"}], ",", "eqnsForDelDDelFs2"}], "]"}]}]], "Input", CellChangeTimes->{{3.5112568519405117`*^9, 3.511256945845723*^9}, { 3.511256976608107*^9, 3.511256977434997*^9}, {3.511257132027821*^9, 3.511257138890379*^9}, {3.511258582874502*^9, 3.511258607275258*^9}, { 3.511258952754401*^9, 3.5112589604805326`*^9}, 3.522645020889999*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"3", " ", SubscriptBox["DelDDelF", "0"]}], "+", RowBox[{"2", " ", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}], " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "0"}]], "[", "z", "]"}]}]}], "\[Equal]", RowBox[{"3", " ", RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}], " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "0"}]], "[", "z", "]"}]}]}], ",", RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", "9"}], " ", SubscriptBox["DelDDelF", "2"]}], "+", RowBox[{"4", " ", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}], " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "0"}]], "[", "z", "]"}]}], "+", RowBox[{"5", " ", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}], " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "2"}]], "[", "z", "]"}]}]}], "\[Equal]", RowBox[{"6", " ", RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}], " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "0"}]], "[", "z", "]"}]}]}]}], "}"}]], "Output", CellChangeTimes->{ 3.5226503760335913`*^9, 3.522651352088356*^9, 3.5226514788007*^9, 3.525679470966074*^9, 3.525679965864044*^9, 3.5256897798877563`*^9, 3.620399199844474*^9, 3.620399290351472*^9, 3.6204010626619053`*^9, 3.620402818976025*^9, 3.620404087451264*^9, 3.620407376611539*^9, 3.62041041597611*^9, 3.6204111316192837`*^9, 3.620413254588723*^9, 3.620413325502886*^9, 3.620414075055265*^9, 3.620472109584087*^9, 3.620472200342702*^9, 3.6204740070527163`*^9, 3.6204882775126553`*^9, 3.620490088089497*^9, 3.620490128323926*^9, 3.6217525048287163`*^9, { 3.621752548409905*^9, 3.62175256225417*^9}, 3.621753265342799*^9, 3.62376849599712*^9, 3.62382243653106*^9, 3.6238224775365353`*^9, 3.623848856354493*^9, 3.623848909247436*^9, 3.623848959643272*^9, 3.6238490141028843`*^9, 3.6238505728061666`*^9, 3.6238523547195683`*^9, 3.6238524103855143`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"coefAtDs", "=", RowBox[{"Thread", "[", RowBox[{"Coefficient", "[", RowBox[{ RowBox[{"eqnsForDelDDelFs3", "/.", RowBox[{"Equal", "\[Rule]", "Plus"}]}], ",", "DelDDelFs"}], "]"}], "]"}]}]], "Input", CellChangeTimes->{{3.5113477182397823`*^9, 3.511347730790062*^9}, { 3.511347761163702*^9, 3.5113479749499893`*^9}, {3.511348098166581*^9, 3.511348107060462*^9}, {3.511348178139757*^9, 3.511348199722951*^9}, { 3.5113491068766603`*^9, 3.5113491653331757`*^9}, {3.511349405017314*^9, 3.5113494146027718`*^9}, 3.5226450208996773`*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{"3", ",", RowBox[{"-", "9"}]}], "}"}]], "Output", CellChangeTimes->{ 3.522650376118943*^9, 3.522651352102592*^9, 3.522651478940535*^9, 3.525679471000108*^9, 3.5256799660159073`*^9, 3.5256897800405083`*^9, 3.620399199892049*^9, 3.620399290400825*^9, 3.620401062711492*^9, 3.620402819017961*^9, 3.6204040874917192`*^9, 3.6204073766596403`*^9, 3.620410416014267*^9, 3.620411131666461*^9, 3.6204132546374197`*^9, 3.620413325552126*^9, 3.620414075104591*^9, 3.620472109630933*^9, 3.620472200389389*^9, 3.6204740070995502`*^9, 3.620488277557395*^9, 3.620490088129483*^9, 3.620490128472972*^9, 3.6217525049548073`*^9, { 3.621752548458076*^9, 3.6217525623025913`*^9}, 3.621753265392659*^9, 3.6237684960449247`*^9, 3.623822436577792*^9, 3.6238224775877867`*^9, 3.623848856401641*^9, 3.6238489092991447`*^9, 3.623848959690404*^9, 3.623849014153163*^9, 3.6238505728539248`*^9, 3.623852354768683*^9, 3.6238524104362*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"eqnsForDelDDelFs4", "=", RowBox[{"Thread", "[", RowBox[{ RowBox[{"Thread", "[", RowBox[{"Expand", "[", RowBox[{ RowBox[{"eqnsForDelDDelFs3", "[", RowBox[{"[", RowBox[{"All", ",", "1"}], "]"}], "]"}], "/", "coefAtDs"}], "]"}], "]"}], "==", RowBox[{"Thread", "[", RowBox[{"Expand", "[", RowBox[{ RowBox[{"eqnsForDelDDelFs3", "[", RowBox[{"[", RowBox[{"All", ",", "2"}], "]"}], "]"}], "/", "coefAtDs"}], "]"}], "]"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.5113482452216*^9, 3.51134835164624*^9}, { 3.511348385405157*^9, 3.5113483865785923`*^9}, {3.5113484925556192`*^9, 3.511348617392295*^9}, {3.511348648290744*^9, 3.5113487088180943`*^9}, { 3.511348742466329*^9, 3.511348895941895*^9}, 3.522645020900535*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["DelDDelF", "0"], "+", RowBox[{ FractionBox["2", "3"], " ", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}], " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "0"}]], "[", "z", "]"}]}]}], "\[Equal]", RowBox[{ RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}], " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "0"}]], "[", "z", "]"}]}]}], ",", RowBox[{ RowBox[{ SubscriptBox["DelDDelF", "2"], "-", RowBox[{ FractionBox["4", "9"], " ", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}], " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "0"}]], "[", "z", "]"}]}], "-", RowBox[{ FractionBox["5", "9"], " ", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}], " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "2"}]], "[", "z", "]"}]}]}], "\[Equal]", RowBox[{ RowBox[{"-", FractionBox["2", "3"]}], " ", RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}], " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "0"}]], "[", "z", "]"}]}]}]}], "}"}]], "Output", CellChangeTimes->{ 3.5226503761934147`*^9, 3.522651352306552*^9, 3.522651479120491*^9, 3.525679471271619*^9, 3.525679966145575*^9, 3.525689780182356*^9, 3.620399199949112*^9, 3.620399290451675*^9, 3.62040106276646*^9, 3.6204028190995493`*^9, 3.620404087533564*^9, 3.620407376711768*^9, 3.620410416050674*^9, 3.6204111317168903`*^9, 3.620413254687262*^9, 3.620413325608677*^9, 3.620414075160204*^9, 3.620472109691268*^9, 3.620472200466538*^9, 3.620474007159203*^9, 3.620488277608304*^9, 3.6204900881679583`*^9, 3.620490128636281*^9, 3.621752504994163*^9, { 3.621752548508253*^9, 3.621752562354315*^9}, 3.621753265443432*^9, 3.6237684963250628`*^9, 3.6238224366320877`*^9, 3.6238224776360807`*^9, 3.623848856666305*^9, 3.6238489093472977`*^9, 3.623848959741621*^9, 3.623849014203257*^9, 3.623850572903837*^9, 3.6238523548191566`*^9, 3.6238524104851933`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"eqnsForDelDDelFs5", "=", RowBox[{"eqnsForDelDDelFs4", "/.", RowBox[{"Thread", "[", RowBox[{"DelDDelFs", "\[Rule]", RowBox[{"Table", "[", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["DelDDel", "n"], "[", "z", "]"}], RowBox[{ SubscriptBox["F", "n"], "[", "z", "]"}]}], ",", RowBox[{"{", RowBox[{"n", ",", "0", ",", "L", ",", "2"}], "}"}]}], "]"}]}], "]"}]}]}]], "Input", CellChangeTimes->{{3.511259035618959*^9, 3.51125904616938*^9}, { 3.511259090134645*^9, 3.5112591044049797`*^9}, {3.5112591435785227`*^9, 3.511259215870421*^9}, {3.511259330953957*^9, 3.511259415445286*^9}, { 3.511259448408051*^9, 3.511259575759982*^9}, {3.511259624177058*^9, 3.511259627210476*^9}, {3.5112597528430843`*^9, 3.511259771715252*^9}, { 3.5112598707649193`*^9, 3.511259899333455*^9}, {3.511348900319165*^9, 3.51134890245089*^9}, 3.52264502091551*^9, 3.623848672038365*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["DelDDel", "0"], "[", "z", "]"}], " ", RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}]}], "+", RowBox[{ FractionBox["2", "3"], " ", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}], " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "0"}]], "[", "z", "]"}]}]}], "\[Equal]", RowBox[{ RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}], " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "0"}]], "[", "z", "]"}]}]}], ",", RowBox[{ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["DelDDel", "2"], "[", "z", "]"}], " ", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}]}], "-", RowBox[{ FractionBox["4", "9"], " ", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}], " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "0"}]], "[", "z", "]"}]}], "-", RowBox[{ FractionBox["5", "9"], " ", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}], " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "2"}]], "[", "z", "]"}]}]}], "\[Equal]", RowBox[{ RowBox[{"-", FractionBox["2", "3"]}], " ", RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}], " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "0"}]], "[", "z", "]"}]}]}]}], "}"}]], "Output", CellChangeTimes->{ 3.522650376291745*^9, 3.522651352515883*^9, 3.522651479293808*^9, 3.525679471586563*^9, 3.5256799662626867`*^9, 3.525689780195779*^9, 3.6203991999950123`*^9, 3.620399290503859*^9, 3.620401062812405*^9, 3.620402819160177*^9, 3.620404087578885*^9, 3.6204073767640142`*^9, 3.620410416101367*^9, 3.6204111317934637`*^9, 3.620413254739172*^9, 3.620413325653389*^9, 3.620414075208108*^9, 3.620472109739265*^9, 3.62047220050642*^9, 3.620474007204591*^9, 3.620488277764188*^9, 3.620490088214999*^9, 3.620490128688407*^9, 3.621752505043887*^9, { 3.6217525485589857`*^9, 3.6217525624069653`*^9}, 3.6217532654926863`*^9, 3.6237684963661537`*^9, 3.623822436679665*^9, 3.623822477686157*^9, 3.6238488567038107`*^9, 3.623848909398093*^9, 3.6238489597901897`*^9, 3.623849014256775*^9, 3.623850572953878*^9, 3.6238523548690977`*^9, 3.623852410535705*^9}] }, Open ]], Cell[TextData[{ StyleBox["Rewriting the equations into a form ", FontSize->16], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"m", " ", RowBox[{"{", SubscriptBox["F", "i"], "}"}]}], "-", RowBox[{"{", SubscriptBox["b", "i"], "}"}]}], "\[Equal]", "0"}], TraditionalForm]], FontSize->16], ", ", StyleBox["where now the matrix ", FontSize->16], Cell[BoxData[ FormBox["m", TraditionalForm]], FontSize->16], StyleBox[" contains both the streaming and reaction operators ", FontSize->16], " " }], "Text", CellChangeTimes->{{3.522645067217414*^9, 3.522645068559268*^9}, { 3.522645113653557*^9, 3.522645154760992*^9}, {3.5226455253291683`*^9, 3.522645537447722*^9}, {3.522645757714418*^9, 3.522645804206421*^9}, { 3.5226458615113*^9, 3.5226459447188663`*^9}, {3.522645982553526*^9, 3.522646010994257*^9}, {3.522646050970581*^9, 3.522646123120521*^9}, { 3.5226465046629467`*^9, 3.5226465396714478`*^9}, {3.522647071486895*^9, 3.5226471070132837`*^9}, {3.522647196226143*^9, 3.5226473057929897`*^9}, { 3.5226473374375267`*^9, 3.522647550210237*^9}, {3.522647591318706*^9, 3.52264759216264*^9}, {3.5226476533707314`*^9, 3.522647691867504*^9}, { 3.522647900608212*^9, 3.522647955486134*^9}, {3.522648087495708*^9, 3.522648191535192*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"{", RowBox[{"b", ",", "m"}], "}"}], "=", RowBox[{"CoefficientArrays", "[", RowBox[{"eqnsForDelDDelFs5", ",", RowBox[{"Table", "[", RowBox[{ RowBox[{ SubscriptBox["F", "n"], "[", "z", "]"}], ",", RowBox[{"{", RowBox[{"n", ",", "0", ",", "L", ",", "2"}], "}"}]}], "]"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.511258737105991*^9, 3.5112587472782707`*^9}, { 3.511259591053022*^9, 3.511259658073683*^9}, {3.51125969281116*^9, 3.511259715330944*^9}, {3.511259797401042*^9, 3.511259814835762*^9}, { 3.5113474508464193`*^9, 3.51134745185634*^9}, 3.5113489170930433`*^9, 3.522645020929377*^9, 3.623848679175856*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ TagBox[ RowBox[{"SparseArray", "[", RowBox[{ InterpretationBox[ RowBox[{"\<\"<\"\>", "\[InvisibleSpace]", "0", "\[InvisibleSpace]", "\<\">\"\>"}], SequenceForm["<", 0, ">"], Editable->False], ",", RowBox[{"{", "2", "}"}]}], "]"}], False, Editable->False], ",", TagBox[ RowBox[{"SparseArray", "[", RowBox[{ InterpretationBox[ RowBox[{"\<\"<\"\>", "\[InvisibleSpace]", "4", "\[InvisibleSpace]", "\<\">\"\>"}], SequenceForm["<", 4, ">"], Editable->False], ",", RowBox[{"{", RowBox[{"2", ",", "2"}], "}"}]}], "]"}], False, Editable->False]}], "}"}]], "Output", CellChangeTimes->{ 3.522650376310836*^9, 3.522651352776513*^9, 3.522651479462509*^9, 3.52567947183644*^9, 3.525679966272183*^9, 3.525689780205365*^9, 3.6203992001231213`*^9, 3.620399290550848*^9, 3.62040106286065*^9, 3.6204028192058077`*^9, 3.620404087626525*^9, 3.620407376809437*^9, 3.62041041614682*^9, 3.6204111318499117`*^9, 3.6204132547750387`*^9, 3.620413325703328*^9, 3.620414075255748*^9, 3.620472109785486*^9, 3.6204722005432987`*^9, 3.6204740073047867`*^9, 3.620488277806508*^9, 3.620490088261819*^9, 3.620490128735819*^9, 3.621752505093357*^9, { 3.62175254860991*^9, 3.621752562456038*^9}, 3.621753265542605*^9, 3.623768496413705*^9, 3.6238224367323713`*^9, 3.6238224777386713`*^9, 3.623848856755032*^9, 3.623848909447378*^9, 3.623848959917918*^9, 3.6238490143660383`*^9, 3.6238505730056543`*^9, 3.623852355014535*^9, 3.623852410587016*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Normal", "[", "b", "]"}], "//", "MatrixForm"}]], "Input", CellChangeTimes->{{3.51125981842404*^9, 3.51125984237097*^9}, 3.511259954773552*^9, 3.522645020930462*^9}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", TagBox[GridBox[{ {"0"}, {"0"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.5599999999999999]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], Column], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{ 3.522650376318122*^9, 3.522651353045888*^9, 3.522651479478244*^9, 3.5256794721069098`*^9, 3.525679966281371*^9, 3.5256897802122726`*^9, 3.620399200162765*^9, 3.620399290648651*^9, 3.620401062960832*^9, 3.6204028192531347`*^9, 3.620404087677374*^9, 3.6204073769048557`*^9, 3.620410416225314*^9, 3.6204111318964663`*^9, 3.6204132548483753`*^9, 3.620413325791141*^9, 3.620414075307332*^9, 3.6204721098335342`*^9, 3.620472200640669*^9, 3.620474007455646*^9, 3.620488277862857*^9, 3.6204900883186607`*^9, 3.6204901287859373`*^9, 3.621752505144541*^9, { 3.621752548687901*^9, 3.6217525625063*^9}, 3.621753265594335*^9, 3.623768496462928*^9, 3.623822436779347*^9, 3.62382247778785*^9, 3.623848856803377*^9, 3.623848909481838*^9, 3.6238489599606533`*^9, 3.623849014425119*^9, 3.623850573132573*^9, 3.623852355056053*^9, 3.62385241076433*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Normal", "[", "m", "]"}], "//", "MatrixForm"}]], "Input", CellChangeTimes->{3.511260034546486*^9, 3.5226450209318647`*^9}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { RowBox[{ RowBox[{ SubscriptBox["DelDDel", "0"], "[", "z", "]"}], "-", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "0"}]], "[", "z", "]"}]}], RowBox[{ FractionBox["2", "3"], " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "0"}]], "[", "z", "]"}]}]}, { RowBox[{ FractionBox["2", "3"], " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "0"}]], "[", "z", "]"}]}], RowBox[{ RowBox[{ SubscriptBox["DelDDel", "2"], "[", "z", "]"}], "-", RowBox[{ FractionBox["4", "9"], " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "0"}]], "[", "z", "]"}]}], "-", RowBox[{ FractionBox["5", "9"], " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "2"}]], "[", "z", "]"}]}]}]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{ 3.5226503763310003`*^9, 3.5226513532578487`*^9, 3.522651479628311*^9, 3.52567947259875*^9, 3.525679966288731*^9, 3.525689780369074*^9, 3.620399200198471*^9, 3.620399290686143*^9, 3.620401062999426*^9, 3.620402819306303*^9, 3.6204040877566843`*^9, 3.620407376946574*^9, 3.620410416269128*^9, 3.6204111319556704`*^9, 3.620413254888647*^9, 3.6204133258932247`*^9, 3.6204140753573217`*^9, 3.620472109891489*^9, 3.620472200678439*^9, 3.620474007615087*^9, 3.6204882779081097`*^9, 3.620490088393589*^9, 3.620490128837068*^9, 3.6217525051946907`*^9, { 3.621752548725642*^9, 3.621752562582595*^9}, 3.621753265671356*^9, 3.623768496514853*^9, 3.623822436834408*^9, 3.623822477837502*^9, 3.6238488568531923`*^9, 3.623848909592659*^9, 3.623848960007265*^9, 3.623849014472711*^9, 3.623850573171237*^9, 3.623852355104351*^9, 3.6238524108049517`*^9}] }, Open ]], Cell[TextData[StyleBox["Some formal changes to improve readability", FontSize->14]], "Text", CellChangeTimes->{{3.522645067217414*^9, 3.522645068559268*^9}, { 3.522645113653557*^9, 3.522645154760992*^9}, {3.5226455253291683`*^9, 3.522645537447722*^9}, {3.522645757714418*^9, 3.522645804206421*^9}, { 3.5226458615113*^9, 3.5226459447188663`*^9}, {3.522645982553526*^9, 3.522646010994257*^9}, {3.522646050970581*^9, 3.522646123120521*^9}, { 3.5226465046629467`*^9, 3.5226465396714478`*^9}, {3.522647071486895*^9, 3.5226471070132837`*^9}, {3.522647196226143*^9, 3.5226473057929897`*^9}, { 3.5226473374375267`*^9, 3.522647550210237*^9}, {3.522647591318706*^9, 3.52264759216264*^9}, {3.5226476533707314`*^9, 3.522647691867504*^9}, { 3.522647900608212*^9, 3.522647955486134*^9}, {3.522648087495708*^9, 3.522648191535192*^9}, {3.522648250305833*^9, 3.5226482572565804`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"F", "[", "A_", "]"}], ":=", RowBox[{"TraditionalForm", "[", RowBox[{"A", "/.", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["DelDDel", "n_"], "[", "z", "]"}], "\[Rule]", SubscriptBox["DelDDel", "n"]}], ",", RowBox[{ RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "n__"}]], "[", "z", "]"}], "\[Rule]", SubscriptBox["\[CapitalSigma]", "n"]}]}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{3.5226450209431047`*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"F", "[", RowBox[{ RowBox[{"Normal", "[", "m", "]"}], "//", "MatrixForm"}], "]"}]], "Input", CellChangeTimes->{{3.5113444389977283`*^9, 3.511344441261333*^9}, 3.522645020943396*^9}], Cell[BoxData[ FormBox[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { RowBox[{ SubscriptBox["DelDDel", "0"], "-", SubscriptBox["\[CapitalSigma]", "0"]}], FractionBox[ RowBox[{"2", " ", SubscriptBox["\[CapitalSigma]", "0"]}], "3"]}, { FractionBox[ RowBox[{"2", " ", SubscriptBox["\[CapitalSigma]", "0"]}], "3"], RowBox[{ SubscriptBox["DelDDel", "2"], "-", FractionBox[ RowBox[{"4", " ", SubscriptBox["\[CapitalSigma]", "0"]}], "9"], "-", FractionBox[ RowBox[{"5", " ", SubscriptBox["\[CapitalSigma]", "2"]}], "9"]}]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]], TraditionalForm]], "Output", CellChangeTimes->{ 3.522650376603222*^9, 3.522651353676738*^9, 3.5226514799996243`*^9, 3.525679473072031*^9, 3.52567996653372*^9, 3.525689780493382*^9, 3.62039920029245*^9, 3.62039929079958*^9, 3.620401063073695*^9, 3.620402819418981*^9, 3.6204040879147778`*^9, 3.620407377133786*^9, 3.620410416417631*^9, 3.620411132103582*^9, 3.620413254966691*^9, 3.620413326108097*^9, 3.620414075794723*^9, 3.620472109969043*^9, 3.620472200740294*^9, 3.620474007774486*^9, 3.620488278028036*^9, 3.62049008852982*^9, 3.620490128987554*^9, 3.621752505277028*^9, { 3.62175254880345*^9, 3.621752562650498*^9}, 3.621753265738451*^9, 3.623768496599201*^9, 3.6238224369129677`*^9, 3.6238224779186897`*^9, 3.623848856932374*^9, 3.623848909661984*^9, 3.623848960084968*^9, 3.623849014536508*^9, 3.6238505732506733`*^9, 3.623852355181817*^9, 3.623852410883461*^9}] }, Open ]], Cell[TextData[{ StyleBox["Make sure that in the monoenergetic case, ", FontSize->16], Cell[BoxData[ FormBox["m", TraditionalForm]], FontSize->16], " ", StyleBox["is symmetric", FontSize->16] }], "Text", CellChangeTimes->{{3.522645067217414*^9, 3.522645068559268*^9}, { 3.522645113653557*^9, 3.522645154760992*^9}, {3.5226455253291683`*^9, 3.522645537447722*^9}, {3.522645757714418*^9, 3.522645804206421*^9}, { 3.5226458615113*^9, 3.5226459447188663`*^9}, {3.522645982553526*^9, 3.522646010994257*^9}, {3.522646050970581*^9, 3.522646123120521*^9}, { 3.5226465046629467`*^9, 3.5226465396714478`*^9}, {3.522647071486895*^9, 3.5226471070132837`*^9}, {3.522647196226143*^9, 3.5226473057929897`*^9}, { 3.5226473374375267`*^9, 3.522647550210237*^9}, {3.522647591318706*^9, 3.52264759216264*^9}, {3.5226476533707314`*^9, 3.522647691867504*^9}, { 3.522647900608212*^9, 3.522647955486134*^9}, {3.522648087495708*^9, 3.522648191535192*^9}, {3.5226482724320107`*^9, 3.522648295707054*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Normal", "[", RowBox[{"m", "-", RowBox[{"Transpose", "[", "m", "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.5146259430374737`*^9, 3.514625954201454*^9}, 3.522645021105043*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{ 3.5226503766141977`*^9, 3.52265135392913*^9, 3.522651480163904*^9, 3.5256794733175697`*^9, 3.525679966544669*^9, 3.525689780642908*^9, 3.6203992005953197`*^9, 3.620399290834896*^9, 3.6204010631156797`*^9, 3.620402819474002*^9, 3.620404088014263*^9, 3.620407377231276*^9, 3.620410416453281*^9, 3.620411132151855*^9, 3.6204132550040197`*^9, 3.620413326309621*^9, 3.620414075978859*^9, 3.6204721100150423`*^9, 3.620472200775382*^9, 3.620474007874482*^9, 3.620488278075494*^9, 3.620490088568429*^9, 3.620490129035945*^9, 3.621752505326892*^9, { 3.6217525488397427`*^9, 3.621752562691945*^9}, 3.621753265781658*^9, 3.623768496647071*^9, 3.623822436960395*^9, 3.623822477971301*^9, 3.623848856986457*^9, 3.623848909719954*^9, 3.6238489601437883`*^9, 3.6238490145883217`*^9, 3.623850573297291*^9, 3.623852355236288*^9, 3.623852410937458*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"m2", "=", RowBox[{ RowBox[{"-", RowBox[{"Normal", "[", "m", "]"}]}], "/.", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["DelDDel", "n_"], "[", "z", "]"}], "\[Rule]", SubscriptBox["DelDDel", "n"]}], ",", RowBox[{ RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "n__"}]], "[", "z", "]"}], "\[Rule]", SubscriptBox["\[CapitalSigma]", "n"]}]}], "}"}]}]}]], "Input", CellChangeTimes->{{3.6204109646301003`*^9, 3.620411008378586*^9}, { 3.62041109505201*^9, 3.620411112121516*^9}, 3.620411182181522*^9, { 3.6204112305772543`*^9, 3.6204112482625227`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"-", SubscriptBox["DelDDel", "0"]}], "+", SubscriptBox["\[CapitalSigma]", "0"]}], ",", RowBox[{"-", FractionBox[ RowBox[{"2", " ", SubscriptBox["\[CapitalSigma]", "0"]}], "3"]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", FractionBox[ RowBox[{"2", " ", SubscriptBox["\[CapitalSigma]", "0"]}], "3"]}], ",", RowBox[{ RowBox[{"-", SubscriptBox["DelDDel", "2"]}], "+", FractionBox[ RowBox[{"4", " ", SubscriptBox["\[CapitalSigma]", "0"]}], "9"], "+", FractionBox[ RowBox[{"5", " ", SubscriptBox["\[CapitalSigma]", "2"]}], "9"]}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{ 3.6204110982738132`*^9, 3.6204111321971693`*^9, 3.620411184685236*^9, { 3.6204112379171543`*^9, 3.620411249002533*^9}, 3.620413255055318*^9, 3.620413326509919*^9, 3.620414076262836*^9, 3.620472110066284*^9, 3.62047220081057*^9, 3.620474007976457*^9, 3.620488278148766*^9, 3.620490088602994*^9, 3.620490129091072*^9, 3.62175250536093*^9, { 3.621752548876029*^9, 3.6217525627411013`*^9}, 3.621753265827963*^9, 3.623768496697274*^9, 3.623822437011071*^9, 3.623822478008059*^9, 3.6238488570383244`*^9, 3.6238489097562437`*^9, 3.623848960191086*^9, 3.623849014638191*^9, 3.6238505733369837`*^9, 3.623852355285981*^9, 3.6238524109846067`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"m2", "//", "MatrixForm"}]], "Input", CellChangeTimes->{{3.620411100292472*^9, 3.620411106281733*^9}, { 3.62041114586408*^9, 3.620411148129545*^9}, {3.620411255503357*^9, 3.620411256101778*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { RowBox[{ RowBox[{"-", SubscriptBox["DelDDel", "0"]}], "+", SubscriptBox["\[CapitalSigma]", "0"]}], RowBox[{"-", FractionBox[ RowBox[{"2", " ", SubscriptBox["\[CapitalSigma]", "0"]}], "3"]}]}, { RowBox[{"-", FractionBox[ RowBox[{"2", " ", SubscriptBox["\[CapitalSigma]", "0"]}], "3"]}], RowBox[{ RowBox[{"-", SubscriptBox["DelDDel", "2"]}], "+", FractionBox[ RowBox[{"4", " ", SubscriptBox["\[CapitalSigma]", "0"]}], "9"], "+", FractionBox[ RowBox[{"5", " ", SubscriptBox["\[CapitalSigma]", "2"]}], "9"]}]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{{3.620411100932314*^9, 3.6204111529956923`*^9}, 3.6204111920485773`*^9, 3.620411256858202*^9, 3.620413255104453*^9, 3.620413326628395*^9, 3.62041407654825*^9, 3.6204721101188507`*^9, 3.620472200856496*^9, 3.620474008022745*^9, 3.6204882781958838`*^9, 3.620490088650653*^9, 3.620490129137988*^9, 3.621752505413332*^9, { 3.621752548927169*^9, 3.621752562789419*^9}, 3.621753265875847*^9, 3.623768496748826*^9, 3.62382243706238*^9, 3.623822478055399*^9, 3.6238488570863237`*^9, 3.623848909804894*^9, 3.623848960240676*^9, 3.623849014687714*^9, 3.623850573388936*^9, 3.623852355336995*^9, 3.623852411018517*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"MatrixForm", "[", RowBox[{"m2", "/.", RowBox[{ SubscriptBox["DelDDel", "_"], "\[Rule]", "0"}]}], "]"}]], "Input", CellChangeTimes->{{3.623918283128974*^9, 3.623918320385892*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { SubscriptBox["\[CapitalSigma]", "0"], RowBox[{"-", FractionBox[ RowBox[{"2", " ", SubscriptBox["\[CapitalSigma]", "0"]}], "3"]}]}, { RowBox[{"-", FractionBox[ RowBox[{"2", " ", SubscriptBox["\[CapitalSigma]", "0"]}], "3"]}], RowBox[{ FractionBox[ RowBox[{"4", " ", SubscriptBox["\[CapitalSigma]", "0"]}], "9"], "+", FractionBox[ RowBox[{"5", " ", SubscriptBox["\[CapitalSigma]", "2"]}], "9"]}]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{{3.623918312496723*^9, 3.62391832085128*^9}}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"Sm", "=", "0"}], ";"}]], "Input", CellChangeTimes->{{3.620412605416237*^9, 3.620412610327594*^9}, 3.620472155215521*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"For", "[", RowBox[{ RowBox[{"i", "=", "0"}], ",", RowBox[{"i", "<", "L"}], ",", RowBox[{"i", "+=", "2"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"{", RowBox[{"b", ",", "s0"}], "}"}], "=", RowBox[{"CoefficientArrays", "[", RowBox[{"m2", ",", " ", SubscriptBox["\[CapitalSigma]", "i"]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{ RowBox[{"c", "[", "i", "]"}], "=", RowBox[{"Flatten", "[", " ", RowBox[{ RowBox[{"Normal", "[", "s0", "]"}], ",", RowBox[{"{", RowBox[{"1", ",", "3"}], "}"}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"MatrixForm", "[", RowBox[{"c", "[", "i", "]"}], "]"}], "]"}], ";", "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"Eigenvalues", "[", RowBox[{"c", "[", "i", "]"}], "]"}], "]"}], ";", "\[IndentingNewLine]", RowBox[{"Sm", "=", RowBox[{"Sm", "+", RowBox[{"c", "[", "i", "]"}]}]}]}]}], "\[IndentingNewLine]", "]"}]], "Input", CellChangeTimes->{{3.620411806730609*^9, 3.620411809200247*^9}, 3.620411945961452*^9, {3.620411983561798*^9, 3.6204120100382643`*^9}, { 3.620412051452527*^9, 3.620412068117763*^9}, {3.6204121012049913`*^9, 3.6204121321245813`*^9}, {3.620412175910597*^9, 3.620412264862771*^9}, 3.620412334656172*^9, {3.6204124509493027`*^9, 3.620412506052815*^9}, { 3.620412549945539*^9, 3.62041255481819*^9}, {3.620412598216813*^9, 3.620412602097224*^9}, {3.62047216164682*^9, 3.620472162800807*^9}, { 3.62047460435806*^9, 3.620474697672391*^9}, {3.620475019302576*^9, 3.6204750702074537`*^9}, {3.620475244412112*^9, 3.620475246206048*^9}, 3.620476591860676*^9, {3.620476670353695*^9, 3.620476670515028*^9}, { 3.620476708599527*^9, 3.620476729975974*^9}, 3.620476767140719*^9, { 3.620476827463057*^9, 3.620476835007358*^9}, 3.6204789113490353`*^9, { 3.6204789787556868`*^9, 3.620479005931563*^9}, {3.620479311770315*^9, 3.620479317474914*^9}, {3.620488505681004*^9, 3.6204885067707577`*^9}, 3.623848692624874*^9}], Cell[CellGroupData[{ Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", RowBox[{"-", FractionBox["2", "3"]}]}, { RowBox[{"-", FractionBox["2", "3"]}], FractionBox["4", "9"]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Print", CellChangeTimes->{{3.6204882935171213`*^9, 3.620488348575017*^9}, { 3.620488507154011*^9, 3.620488513004909*^9}, 3.620489158692401*^9, 3.620490088758996*^9, 3.620490129250326*^9, 3.621752505508686*^9, { 3.621752549039242*^9, 3.621752562900372*^9}, 3.621753265988269*^9, 3.623768496843049*^9, 3.623822437160428*^9, 3.623822478350819*^9, 3.623848857222364*^9, 3.623848909895089*^9, 3.623848960319045*^9, 3.623849014768221*^9, 3.62385057346982*^9, 3.623852355505704*^9, 3.623852411098049*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ FractionBox["13", "9"], ",", "0"}], "}"}]], "Print", CellChangeTimes->{{3.6204882935171213`*^9, 3.620488348575017*^9}, { 3.620488507154011*^9, 3.620488513004909*^9}, 3.620489158692401*^9, 3.620490088758996*^9, 3.620490129250326*^9, 3.621752505508686*^9, { 3.621752549039242*^9, 3.621752562900372*^9}, 3.621753265988269*^9, 3.623768496843049*^9, 3.623822437160428*^9, 3.623822478350819*^9, 3.623848857222364*^9, 3.623848909895089*^9, 3.623848960319045*^9, 3.623849014768221*^9, 3.62385057346982*^9, 3.623852355505704*^9, 3.623852411099599*^9}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0"}, {"0", FractionBox["5", "9"]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Print", CellChangeTimes->{{3.6204882935171213`*^9, 3.620488348575017*^9}, { 3.620488507154011*^9, 3.620488513004909*^9}, 3.620489158692401*^9, 3.620490088758996*^9, 3.620490129250326*^9, 3.621752505508686*^9, { 3.621752549039242*^9, 3.621752562900372*^9}, 3.621753265988269*^9, 3.623768496843049*^9, 3.623822437160428*^9, 3.623822478350819*^9, 3.623848857222364*^9, 3.623848909895089*^9, 3.623848960319045*^9, 3.623849014768221*^9, 3.62385057346982*^9, 3.623852355505704*^9, 3.623852411102126*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ FractionBox["5", "9"], ",", "0"}], "}"}]], "Print", CellChangeTimes->{{3.6204882935171213`*^9, 3.620488348575017*^9}, { 3.620488507154011*^9, 3.620488513004909*^9}, 3.620489158692401*^9, 3.620490088758996*^9, 3.620490129250326*^9, 3.621752505508686*^9, { 3.621752549039242*^9, 3.621752562900372*^9}, 3.621753265988269*^9, 3.623768496843049*^9, 3.623822437160428*^9, 3.623822478350819*^9, 3.623848857222364*^9, 3.623848909895089*^9, 3.623848960319045*^9, 3.623849014768221*^9, 3.62385057346982*^9, 3.623852355505704*^9, 3.623852411104507*^9}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Eigenvalues", "[", "Sm", "]"}]], "Input", CellChangeTimes->{{3.620412714934381*^9, 3.620412723567326*^9}, 3.620472174952688*^9, {3.6240370080933237`*^9, 3.6240370082094717`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ FractionBox["5", "3"], ",", FractionBox["1", "3"]}], "}"}]], "Output", CellChangeTimes->{{3.620412717830617*^9, 3.620412724447425*^9}, 3.620413255408815*^9, 3.6204133268233347`*^9, 3.620414076796958*^9, 3.620472110303555*^9, 3.620472201060801*^9, 3.620474008207573*^9, 3.620488278525277*^9, 3.62049008908099*^9, 3.620490129792286*^9, 3.6217525065325193`*^9, {3.6217525493612556`*^9, 3.621752563372492*^9}, 3.621753266309984*^9, 3.623768497701045*^9, 3.623822437616798*^9, 3.6238224786564407`*^9, 3.62384885738326*^9, 3.6238489099499493`*^9, 3.6238489603727503`*^9, 3.623849014824608*^9, 3.623850573521153*^9, 3.623852355570384*^9, 3.623852411151883*^9, 3.6240370099505568`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"MatrixForm", "[", "Sm", "]"}]], "Input", CellChangeTimes->{{3.624037294341627*^9, 3.624037299428793*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", RowBox[{"-", FractionBox["2", "3"]}]}, { RowBox[{"-", FractionBox["2", "3"]}], "1"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{3.624037299918535*^9}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Boundary conditions", "Section", CellChangeTimes->{{3.522648424428122*^9, 3.522648427730253*^9}}], Cell[TextData[StyleBox["Legendre expansion of the outgoing partial current ", FontSize->16]], "Text", CellChangeTimes->{{3.522645067217414*^9, 3.522645068559268*^9}, { 3.522645113653557*^9, 3.522645154760992*^9}, {3.5226455253291683`*^9, 3.522645537447722*^9}, {3.522645757714418*^9, 3.522645804206421*^9}, { 3.5226458615113*^9, 3.5226459447188663`*^9}, {3.522645982553526*^9, 3.522646010994257*^9}, {3.522646050970581*^9, 3.522646123120521*^9}, { 3.5226465046629467`*^9, 3.5226465396714478`*^9}, {3.522647071486895*^9, 3.5226471070132837`*^9}, {3.522647196226143*^9, 3.5226473057929897`*^9}, { 3.5226473374375267`*^9, 3.522647550210237*^9}, {3.522647591318706*^9, 3.52264759216264*^9}, {3.5226476533707314`*^9, 3.522647691867504*^9}, { 3.522647900608212*^9, 3.522647955486134*^9}, {3.522648087495708*^9, 3.522648191535192*^9}, {3.522648536885538*^9, 3.522648558056643*^9}, { 3.522648852277603*^9, 3.5226489116663723`*^9}, 3.5226489963081408`*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ SuperscriptBox["j", "+"], "=", RowBox[{"Table", "[", RowBox[{ RowBox[{ SubsuperscriptBox["\[Integral]", "0", "1"], RowBox[{ RowBox[{"LegendreP", "[", RowBox[{"n", ",", "\[Mu]"}], "]"}], RowBox[{"\[Psi]", "[", RowBox[{"\[Mu]", ",", "L"}], "]"}], RowBox[{"\[DifferentialD]", "\[Mu]"}]}]}], ",", RowBox[{"{", RowBox[{"n", ",", "1", ",", "L", ",", "2"}], "}"}]}], "]"}]}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ FractionBox[ RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], "4"], "+", FractionBox[ RowBox[{ SubscriptBox["\[Phi]", "1"], "[", "z", "]"}], "2"], "+", FractionBox[ RowBox[{"5", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "16"]}], ",", RowBox[{ RowBox[{ RowBox[{"-", FractionBox["1", "16"]}], " ", RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}]}], "+", FractionBox[ RowBox[{"5", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "16"], "+", FractionBox[ RowBox[{ SubscriptBox["\[Phi]", "3"], "[", "z", "]"}], "2"]}]}], "}"}]], "Output", CellChangeTimes->{ 3.5226503855594254`*^9, 3.5226513611752357`*^9, 3.5226514886974993`*^9, { 3.525679453696478*^9, 3.525679483232625*^9}, 3.5256799670148487`*^9, 3.525689600953257*^9, 3.525689780822542*^9, 3.620399203564191*^9, 3.620399297562627*^9, 3.620401069874816*^9, 3.620402826217413*^9, 3.6204040975530243`*^9, 3.620407384015297*^9, 3.620410426039791*^9, 3.620411141988763*^9, 3.6204132565291367`*^9, 3.620413326993896*^9, 3.620414079954813*^9, 3.6204721109225683`*^9, 3.620472201268383*^9, 3.620474015260621*^9, 3.6204882856798687`*^9, 3.6204900987405987`*^9, 3.6204901396102753`*^9, 3.6217525072191963`*^9, {3.62175254952643*^9, 3.6217525637378387`*^9}, 3.621753266488904*^9, 3.623768500953994*^9, 3.623822437925293*^9, 3.623822479057691*^9, 3.6238488577478952`*^9, 3.623848909983658*^9, 3.6238489604281607`*^9, 3.623849014873658*^9, 3.623850573575466*^9, 3.623852358532799*^9, 3.623852411204571*^9}] }, Open ]], Cell[TextData[{ StyleBox["expressed using only the even order moments (as in the simplified \ form of the ", FontSize->14], Cell[BoxData[ FormBox[ SubscriptBox["P", "n"], TraditionalForm]], FontSize->14], StyleBox[" equations)", FontSize->14] }], "Item", CellChangeTimes->{{3.522649007804577*^9, 3.522649050041965*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ SuperscriptBox["j", "+"], "=", RowBox[{ SuperscriptBox["j", "+"], "/.", RowBox[{"oddWrtEven", "[", RowBox[{"[", "1", "]"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.458122525609375*^9, 3.45812255378125*^9}, { 3.458122593*^9, 3.45812261184375*^9}, {3.458123482421875*^9, 3.458123484734375*^9}, 3.4581238911875*^9, {3.458124040421875*^9, 3.458124091015625*^9}, {3.4581317025625*^9, 3.458131705453125*^9}, { 3.458131738421875*^9, 3.45813178303125*^9}, {3.458131819984375*^9, 3.45813182178125*^9}, {3.458147883921875*^9, 3.458147887890625*^9}, { 3.458273583375*^9, 3.4582735851875*^9}, {3.45827362234375*^9, 3.45827371415625*^9}, {3.458273783828125*^9, 3.458273793015625*^9}, { 3.511185538708654*^9, 3.5111855649085693`*^9}, {3.5111859027267027`*^9, 3.511185903332201*^9}, {3.511195158064402*^9, 3.511195161035399*^9}, 3.51126265900174*^9, 3.5226450211063757`*^9, 3.522648983074665*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ FractionBox[ RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], "4"], "+", FractionBox[ RowBox[{"5", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "16"], "+", FractionBox[ RowBox[{ RowBox[{ RowBox[{"-", "Q"}], " ", RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}]}], "-", RowBox[{"2", " ", "Q", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}]}], RowBox[{"6", " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "1"}]], "[", "z", "]"}]}]]}], ",", RowBox[{ RowBox[{ RowBox[{"-", FractionBox["1", "16"]}], " ", RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}]}], "+", FractionBox[ RowBox[{"5", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "16"], "-", FractionBox[ RowBox[{"3", " ", "Q", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], RowBox[{"14", " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "3"}]], "[", "z", "]"}]}]]}]}], "}"}]], "Output", CellChangeTimes->{ 3.522650386119969*^9, 3.522651361529142*^9, 3.522651488813242*^9, 3.525679483752784*^9, 3.525679967125881*^9, 3.525689780956871*^9, 3.62039920370156*^9, 3.620399297634508*^9, 3.6204010700119143`*^9, 3.620402826303297*^9, 3.6204040976176777`*^9, 3.6204073840936737`*^9, 3.620410426129142*^9, 3.6204111420746603`*^9, 3.620413256578052*^9, 3.620413327040975*^9, 3.6204140800159197`*^9, 3.620472110973159*^9, 3.6204722013170547`*^9, 3.620474015313404*^9, 3.620488285759712*^9, 3.620490098808395*^9, 3.62049013968051*^9, 3.621752507512413*^9, { 3.621752549564416*^9, 3.6217525637908907`*^9}, 3.62175326652861*^9, 3.62376850125949*^9, 3.623822437963545*^9, 3.623822479106112*^9, 3.6238488581394033`*^9, 3.623848910034398*^9, 3.623848960475157*^9, 3.623849014911697*^9, 3.6238505736222486`*^9, 3.623852358769454*^9, 3.623852411253686*^9}] }, Open ]], Cell[TextData[StyleBox["Extract the equations for pseudo-currents", FontSize->16]], "Text", CellChangeTimes->{{3.522645067217414*^9, 3.522645068559268*^9}, { 3.522645113653557*^9, 3.522645154760992*^9}, {3.5226455253291683`*^9, 3.522645537447722*^9}, {3.522645757714418*^9, 3.522645804206421*^9}, { 3.5226458615113*^9, 3.5226459447188663`*^9}, {3.522645982553526*^9, 3.522646010994257*^9}, {3.522646050970581*^9, 3.522646123120521*^9}, { 3.5226465046629467`*^9, 3.5226465396714478`*^9}, {3.522647071486895*^9, 3.5226471070132837`*^9}, {3.522647196226143*^9, 3.5226473057929897`*^9}, { 3.5226473374375267`*^9, 3.522647550210237*^9}, {3.522647591318706*^9, 3.52264759216264*^9}, {3.5226476533707314`*^9, 3.522647691867504*^9}, { 3.522647900608212*^9, 3.522647955486134*^9}, {3.522648087495708*^9, 3.522648191535192*^9}, {3.522648536885538*^9, 3.522648558056643*^9}, { 3.522648852277603*^9, 3.5226489116663723`*^9}, 3.5226489963081408`*^9, { 3.522649108278118*^9, 3.522649116181396*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"DDelFs", "=", RowBox[{"Table", "[", RowBox[{ SubscriptBox["DDelF", "n"], ",", RowBox[{"{", RowBox[{"n", ",", "0", ",", "L", ",", "2"}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.511262777839286*^9, 3.5112627895613003`*^9}, 3.5226450211106663`*^9, {3.522649636193574*^9, 3.522649658289812*^9}, { 3.5226497003399487`*^9, 3.5226497144436407`*^9}, {3.522651076712884*^9, 3.5226510788422737`*^9}, 3.623848718764933*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ SubscriptBox["DDelF", "0"], ",", SubscriptBox["DDelF", "2"]}], "}"}]], "Output", CellChangeTimes->{ 3.52265038653305*^9, 3.522651081204488*^9, 3.5226513617916117`*^9, 3.5226514889754477`*^9, 3.525679483759465*^9, 3.525679967133679*^9, 3.5256897809697933`*^9, 3.620399203742388*^9, 3.6203992976734657`*^9, 3.6204010700537853`*^9, 3.620402826342844*^9, 3.620404097677484*^9, 3.6204073841317596`*^9, 3.6204104261814003`*^9, 3.620411142130591*^9, 3.6204132566300583`*^9, 3.6204133270721693`*^9, 3.620414080053358*^9, 3.620472111072587*^9, 3.620472201358605*^9, 3.620474015361164*^9, 3.620488285797572*^9, 3.620490098847846*^9, 3.620490139719397*^9, 3.621752507555169*^9, {3.621752549599224*^9, 3.6217525638252993`*^9}, 3.62175326656243*^9, 3.6237685012956553`*^9, 3.6238224380124893`*^9, 3.623822479383583*^9, 3.623848858189293*^9, 3.6238489100733852`*^9, 3.623848960524444*^9, 3.6238490149543467`*^9, 3.623850573672453*^9, 3.623852358806663*^9, 3.623852411304078*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"eqnsForDDelFs1", "=", RowBox[{"Thread", "[", RowBox[{"DDelFs", "\[Equal]", RowBox[{"2", RowBox[{"Map", "[", RowBox[{ RowBox[{ RowBox[{"Delete", "[", RowBox[{"#", ",", RowBox[{"-", "1"}]}], "]"}], "&"}], ",", SuperscriptBox["j", "+"]}], "]"}]}]}], "]"}]}]], "Input", CellChangeTimes->{{3.511261750101595*^9, 3.5112617538996468`*^9}, { 3.5112618744924583`*^9, 3.5112619147458878`*^9}, {3.511261985119388*^9, 3.5112619887903*^9}, {3.5112621579118977`*^9, 3.5112622073416348`*^9}, { 3.511262237636718*^9, 3.511262242316352*^9}, 3.511262527515675*^9, { 3.511281869745502*^9, 3.511281877033084*^9}, 3.522645021111478*^9, 3.5226496661758537`*^9, 3.522651072097728*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox["DDelF", "0"], "\[Equal]", RowBox[{"2", " ", RowBox[{"(", RowBox[{ FractionBox[ RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], "4"], "+", FractionBox[ RowBox[{"5", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "16"]}], ")"}]}]}], ",", RowBox[{ SubscriptBox["DDelF", "2"], "\[Equal]", RowBox[{"2", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", FractionBox["1", "16"]}], " ", RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}]}], "+", FractionBox[ RowBox[{"5", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "16"]}], ")"}]}]}]}], "}"}]], "Output", CellChangeTimes->{ 3.522650386545113*^9, 3.522651083182682*^9, 3.522651361990919*^9, 3.522651489147017*^9, 3.525679483768229*^9, 3.5256799672310266`*^9, 3.525689781093266*^9, 3.620399203778236*^9, 3.6203992977231197`*^9, 3.620401070089878*^9, 3.6204028263906393`*^9, 3.6204040977108183`*^9, 3.620407384186211*^9, 3.6204104262339067`*^9, 3.620411142182295*^9, 3.620413256677614*^9, 3.620413327129912*^9, 3.620414080108993*^9, 3.620472111175462*^9, 3.620472201412734*^9, 3.620474015398217*^9, 3.620488285834264*^9, 3.620490098883095*^9, 3.620490139769031*^9, 3.621752507601063*^9, {3.621752549644581*^9, 3.621752563923709*^9}, 3.6217532666111927`*^9, 3.6237685013959713`*^9, 3.623822438062952*^9, 3.623822479423627*^9, 3.623848858239584*^9, 3.6238489101243553`*^9, 3.6238489605739613`*^9, 3.623849015009569*^9, 3.62385057372153*^9, 3.62385235884161*^9, 3.62385241135268*^9}] }, Open ]], Cell[TextData[StyleBox["Express the boundary equations in terms of only the \ even expansion moments and pseudo-currents", FontSize->16]], "Text", CellChangeTimes->{{3.522645067217414*^9, 3.522645068559268*^9}, { 3.522645113653557*^9, 3.522645154760992*^9}, {3.5226455253291683`*^9, 3.522645537447722*^9}, {3.522645757714418*^9, 3.522645804206421*^9}, { 3.5226458615113*^9, 3.5226459447188663`*^9}, {3.522645982553526*^9, 3.522646010994257*^9}, {3.522646050970581*^9, 3.522646123120521*^9}, { 3.5226465046629467`*^9, 3.5226465396714478`*^9}, {3.522647071486895*^9, 3.5226471070132837`*^9}, {3.522647196226143*^9, 3.5226473057929897`*^9}, { 3.5226473374375267`*^9, 3.522647550210237*^9}, {3.522647591318706*^9, 3.52264759216264*^9}, {3.5226476533707314`*^9, 3.522647691867504*^9}, { 3.522647900608212*^9, 3.522647955486134*^9}, {3.522648087495708*^9, 3.522648191535192*^9}, {3.522648536885538*^9, 3.522648558056643*^9}, { 3.522648852277603*^9, 3.5226489116663723`*^9}, 3.5226489963081408`*^9, { 3.522649134521142*^9, 3.5226491881211023`*^9}, {3.522649342096834*^9, 3.5226493813411837`*^9}, {3.62384873026683*^9, 3.623848740929593*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"eqnsForDDelFs2", "=", RowBox[{"Map", "[", RowBox[{ RowBox[{ RowBox[{"Join", "[", RowBox[{ RowBox[{"{", "#", "}"}], ",", "evenPseudoFluxes"}], "]"}], "&"}], ",", "eqnsForDDelFs1"}], "]"}]}]], "Input", CellChangeTimes->{{3.511261543942474*^9, 3.511261594947771*^9}, { 3.511261643702834*^9, 3.511261678059739*^9}, {3.511262259745359*^9, 3.511262263693705*^9}, {3.5112624296346197`*^9, 3.511262447337812*^9}, { 3.511262482023842*^9, 3.51126253026333*^9}, 3.522645021114964*^9, 3.623848587819168*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox["DDelF", "0"], "\[Equal]", RowBox[{"2", " ", RowBox[{"(", RowBox[{ FractionBox[ RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], "4"], "+", FractionBox[ RowBox[{"5", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "16"]}], ")"}]}]}], ",", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], "+", RowBox[{"2", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}]}], "\[Equal]", RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}]}], ",", RowBox[{ RowBox[{"3", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "\[Equal]", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox["DDelF", "2"], "\[Equal]", RowBox[{"2", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", FractionBox["1", "16"]}], " ", RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}]}], "+", FractionBox[ RowBox[{"5", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "16"]}], ")"}]}]}], ",", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], "+", RowBox[{"2", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}]}], "\[Equal]", RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}]}], ",", RowBox[{ RowBox[{"3", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "\[Equal]", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}]}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{ 3.5226503869580097`*^9, 3.522651085429605*^9, 3.5226513620031967`*^9, 3.522651489367037*^9, 3.525679484015777*^9, 3.525679967371092*^9, 3.5256897812592297`*^9, 3.620399203821282*^9, 3.620399297773287*^9, 3.620401070135915*^9, 3.6204028264687567`*^9, 3.620404097761405*^9, 3.62040738423214*^9, 3.62041042627915*^9, 3.6204111422288837`*^9, 3.620413256726534*^9, 3.6204133271627283`*^9, 3.620414080153626*^9, 3.620472111220641*^9, 3.620472201459063*^9, 3.620474015445898*^9, 3.62048828588208*^9, 3.6204900989312572`*^9, 3.6204901398220873`*^9, 3.6217525076529922`*^9, {3.621752549722508*^9, 3.621752563975749*^9}, 3.621753266691104*^9, 3.623768501494709*^9, 3.623822438111863*^9, 3.623822479471653*^9, 3.623848858290276*^9, 3.623848910173918*^9, 3.623848960625902*^9, 3.6238490150559397`*^9, 3.6238505737718678`*^9, 3.623852358874885*^9, 3.623852411403125*^9}] }, Open ]], Cell[TextData[StyleBox["The final Robin-type boundary conditions ", FontSize->16]], "Text", CellChangeTimes->{{3.522645067217414*^9, 3.522645068559268*^9}, { 3.522645113653557*^9, 3.522645154760992*^9}, {3.5226455253291683`*^9, 3.522645537447722*^9}, {3.522645757714418*^9, 3.522645804206421*^9}, { 3.5226458615113*^9, 3.5226459447188663`*^9}, {3.522645982553526*^9, 3.522646010994257*^9}, {3.522646050970581*^9, 3.522646123120521*^9}, { 3.5226465046629467`*^9, 3.5226465396714478`*^9}, {3.522647071486895*^9, 3.5226471070132837`*^9}, {3.522647196226143*^9, 3.5226473057929897`*^9}, { 3.5226473374375267`*^9, 3.522647550210237*^9}, {3.522647591318706*^9, 3.52264759216264*^9}, {3.5226476533707314`*^9, 3.522647691867504*^9}, { 3.522647900608212*^9, 3.522647955486134*^9}, {3.522648087495708*^9, 3.522648191535192*^9}, {3.522648536885538*^9, 3.522648558056643*^9}, { 3.522648852277603*^9, 3.5226489116663723`*^9}, 3.5226489963081408`*^9, { 3.522649134521142*^9, 3.5226491881211023`*^9}, {3.522649342096834*^9, 3.5226493813411837`*^9}, {3.522649557303268*^9, 3.522649559126287*^9}, { 3.522649916428834*^9, 3.5226499239681187`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"bc1p", "=", RowBox[{"Map", "[", RowBox[{ RowBox[{ RowBox[{"Eliminate", "[", RowBox[{"#", ",", "evenmom"}], "]"}], "&"}], ",", "eqnsForDDelFs2"}], "]"}]}]], "Input", CellChangeTimes->{{3.511262534725312*^9, 3.511262574752665*^9}, 3.522645021123007*^9, {3.522649792378985*^9, 3.52264980267071*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"4", " ", RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}]}], "-", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}]}], "\[Equal]", RowBox[{"8", " ", SubscriptBox["DDelF", "0"]}]}], ",", RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", "3"}], " ", RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}]}], "+", RowBox[{"7", " ", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}]}]}], "\[Equal]", RowBox[{"24", " ", SubscriptBox["DDelF", "2"]}]}]}], "}"}]], "Output", CellChangeTimes->{ 3.522650387335088*^9, 3.522651087931843*^9, 3.522651362234272*^9, 3.522651489483555*^9, 3.525679484261055*^9, 3.525679967473111*^9, 3.5256897812707167`*^9, 3.620399203872449*^9, 3.620399297824771*^9, 3.620401070185546*^9, 3.62040282653339*^9, 3.620404097811708*^9, 3.620407384281461*^9, 3.620410426333384*^9, 3.620411142281554*^9, 3.620413256863956*^9, 3.6204133272071466`*^9, 3.62041408020969*^9, 3.620472111268935*^9, 3.620472201497497*^9, 3.6204740154958553`*^9, 3.620488285931634*^9, 3.620490098977724*^9, 3.620490139874895*^9, 3.621752507701757*^9, {3.621752549763253*^9, 3.6217525640285273`*^9}, 3.621753266728581*^9, 3.623768501548994*^9, 3.623822438169113*^9, 3.6238224795222073`*^9, 3.623848858339361*^9, 3.623848910224531*^9, 3.623848960674618*^9, 3.623849015104925*^9, 3.623850573823523*^9, 3.623852358908721*^9, 3.623852411436054*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"bc2p", "=", RowBox[{ RowBox[{ RowBox[{"Solve", "[", RowBox[{"bc1p", ",", "DDelFs"}], "]"}], "//", "Flatten"}], "//", "Expand"}]}]], "Input", CellChangeTimes->{{3.522649796393147*^9, 3.522649819078259*^9}, { 3.522649900981268*^9, 3.522649901642701*^9}, {3.5226499985317*^9, 3.5226500018912573`*^9}, 3.5226510650896187`*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox["DDelF", "0"], "\[Rule]", RowBox[{ FractionBox[ RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}], "2"], "-", FractionBox[ RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}], "8"]}]}], ",", RowBox[{ SubscriptBox["DDelF", "2"], "\[Rule]", RowBox[{ RowBox[{ RowBox[{"-", FractionBox["1", "8"]}], " ", RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}]}], "+", FractionBox[ RowBox[{"7", " ", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}]}], "24"]}]}]}], "}"}]], "Output", CellChangeTimes->{ 3.5226503877600403`*^9, 3.522651091358552*^9, 3.5226513625768337`*^9, 3.522651489699729*^9, 3.525679484778007*^9, 3.5256799676199207`*^9, 3.525689781279889*^9, 3.620399203922814*^9, 3.620399297926647*^9, 3.6204010702352877`*^9, 3.6204028265805387`*^9, 3.620404097859486*^9, 3.620407384336239*^9, 3.620410426379486*^9, 3.620411142328916*^9, 3.620413256993555*^9, 3.6204133272619123`*^9, 3.6204140802548323`*^9, 3.620472111322081*^9, 3.6204722015419483`*^9, 3.620474015544606*^9, 3.620488285986766*^9, 3.620490099029418*^9, 3.620490139919717*^9, 3.621752507802734*^9, {3.6217525498126574`*^9, 3.621752564077448*^9}, 3.621753266780827*^9, 3.623768501599983*^9, 3.623822438215014*^9, 3.6238224795738173`*^9, 3.6238488583904057`*^9, 3.623848910274383*^9, 3.6238489607242804`*^9, 3.623849015156493*^9, 3.6238505738708563`*^9, 3.6238523589419107`*^9, 3.623852411489202*^9}] }, Open ]], Cell[TextData[StyleBox["The same as above, only for the incoming partial \ currents", FontSize->16]], "Text", CellChangeTimes->{{3.522645067217414*^9, 3.522645068559268*^9}, { 3.522645113653557*^9, 3.522645154760992*^9}, {3.5226455253291683`*^9, 3.522645537447722*^9}, {3.522645757714418*^9, 3.522645804206421*^9}, { 3.5226458615113*^9, 3.5226459447188663`*^9}, {3.522645982553526*^9, 3.522646010994257*^9}, {3.522646050970581*^9, 3.522646123120521*^9}, { 3.5226465046629467`*^9, 3.5226465396714478`*^9}, {3.522647071486895*^9, 3.5226471070132837`*^9}, {3.522647196226143*^9, 3.5226473057929897`*^9}, { 3.5226473374375267`*^9, 3.522647550210237*^9}, {3.522647591318706*^9, 3.52264759216264*^9}, {3.5226476533707314`*^9, 3.522647691867504*^9}, { 3.522647900608212*^9, 3.522647955486134*^9}, {3.522648087495708*^9, 3.522648191535192*^9}, {3.522648536885538*^9, 3.522648558056643*^9}, { 3.522648852277603*^9, 3.5226489116663723`*^9}, 3.5226489963081408`*^9, { 3.522649134521142*^9, 3.5226491881211023`*^9}, {3.522649342096834*^9, 3.5226493813411837`*^9}, {3.5226494456588*^9, 3.52264946283311*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ SuperscriptBox["j", "-"], "=", RowBox[{"Table", "[", RowBox[{ RowBox[{ RowBox[{ SubsuperscriptBox["\[Integral]", RowBox[{"-", "1"}], "0"], RowBox[{ RowBox[{"LegendreP", "[", RowBox[{"n", ",", RowBox[{"Abs", "[", "\[Mu]", "]"}]}], "]"}], RowBox[{"\[Psi]", "[", RowBox[{"\[Mu]", ",", "L"}], "]"}], RowBox[{"\[DifferentialD]", "\[Mu]"}]}]}], "//", "Distribute"}], ",", RowBox[{"{", RowBox[{"n", ",", "1", ",", "L", ",", "2"}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.525683989215228*^9, 3.5256839958432207`*^9}, { 3.525685079534404*^9, 3.5256851541284513`*^9}, {3.5256865572379427`*^9, 3.5256865911670513`*^9}, {3.5256866710567904`*^9, 3.52568671060435*^9}, 3.62384874849254*^9, 3.623848999045394*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ FractionBox[ RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], "4"], "-", FractionBox[ RowBox[{ SubscriptBox["\[Phi]", "1"], "[", "z", "]"}], "2"], "+", FractionBox[ RowBox[{"5", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "16"]}], ",", RowBox[{ RowBox[{ RowBox[{"-", FractionBox["1", "16"]}], " ", RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}]}], "+", FractionBox[ RowBox[{"5", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "16"], "-", FractionBox[ RowBox[{ SubscriptBox["\[Phi]", "3"], "[", "z", "]"}], "2"]}]}], "}"}]], "Output", CellChangeTimes->{ 3.522650392599558*^9, 3.522651101508575*^9, 3.522651367147265*^9, 3.5226514942487993`*^9, 3.525679489363946*^9, 3.5256799677696533`*^9, 3.525683997575767*^9, 3.525685083509819*^9, {3.525685118393387*^9, 3.5256851546334143`*^9}, {3.525686560400608*^9, 3.525686592068205*^9}, 3.525686674972948*^9, 3.525686712051442*^9, 3.525687509004858*^9, 3.525689582236829*^9, 3.5256897814529953`*^9, 3.620399205638116*^9, 3.6203993025455914`*^9, 3.620401074820878*^9, 3.620402831181789*^9, 3.620404104190527*^9, 3.6204073889243107`*^9, 3.620410432771134*^9, 3.620411148730009*^9, 3.620413257028343*^9, 3.6204133277022038`*^9, 3.620414081913292*^9, 3.620472111864599*^9, 3.620472201917933*^9, 3.620474020140029*^9, 3.620488290609725*^9, 3.620490105385878*^9, 3.6204901462755632`*^9, 3.621752508308639*^9, {3.6217525501982117`*^9, 3.621752564126298*^9}, 3.621753267138033*^9, 3.623768503150613*^9, 3.6238224387654943`*^9, 3.623822479995099*^9, 3.6238488588153276`*^9, 3.623848910338244*^9, 3.6238489607783623`*^9, 3.62384901553339*^9, 3.6238505739219637`*^9, 3.623852359395584*^9, 3.623852412004157*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ SuperscriptBox["j", "-"], "=", RowBox[{ SuperscriptBox["j", "-"], "/.", RowBox[{"oddWrtEven", "[", RowBox[{"[", "1", "]"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.458273780828125*^9, 3.458273804234375*^9}, { 3.5111856210726757`*^9, 3.511185679603166*^9}, {3.511185912106906*^9, 3.511185912202156*^9}, {3.511262703102982*^9, 3.511262707689557*^9}, 3.52264502112569*^9, 3.5226494150825863`*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ FractionBox[ RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], "4"], "+", FractionBox[ RowBox[{"5", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "16"], "-", FractionBox[ RowBox[{ RowBox[{ RowBox[{"-", "Q"}], " ", RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}]}], "-", RowBox[{"2", " ", "Q", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}]}], RowBox[{"6", " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "1"}]], "[", "z", "]"}]}]]}], ",", RowBox[{ RowBox[{ RowBox[{"-", FractionBox["1", "16"]}], " ", RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}]}], "+", FractionBox[ RowBox[{"5", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "16"], "+", FractionBox[ RowBox[{"3", " ", "Q", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], RowBox[{"14", " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "3"}]], "[", "z", "]"}]}]]}]}], "}"}]], "Output", CellChangeTimes->{ 3.522650392843658*^9, 3.522651109473929*^9, 3.522651367395262*^9, 3.522651494414929*^9, 3.525679489600114*^9, 3.5256799680247593`*^9, 3.525684001246387*^9, {3.525685062367817*^9, 3.525685087045763*^9}, { 3.52568511957403*^9, 3.52568515594311*^9}, {3.5256865760614023`*^9, 3.5256865934684057`*^9}, 3.525687510322588*^9, {3.525688014140573*^9, 3.525688039778775*^9}, 3.5256897815894012`*^9, 3.620399205772807*^9, 3.620399302618404*^9, 3.6204010748925133`*^9, 3.620402831252873*^9, 3.6204041042550573`*^9, 3.620407388992082*^9, 3.6204104328436127`*^9, 3.620411148809466*^9, 3.620413257060916*^9, 3.620413327742229*^9, 3.620414081972114*^9, 3.620472111944758*^9, 3.620472202051749*^9, 3.620474020191725*^9, 3.620488290709989*^9, 3.620490105444066*^9, 3.6204901463342533`*^9, 3.621752508433152*^9, {3.621752550228169*^9, 3.621752564175015*^9}, 3.62175326718167*^9, 3.623768503418116*^9, 3.623822439105544*^9, 3.6238224800397053`*^9, 3.623848859068969*^9, 3.623848910382597*^9, 3.6238489608271437`*^9, 3.623849015634239*^9, 3.623850573971506*^9, 3.623852359443701*^9, 3.6238524123709507`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"eqnsForDDelFs1", "=", RowBox[{"Thread", "[", RowBox[{"DDelFs", "\[Equal]", RowBox[{ RowBox[{"-", "2"}], RowBox[{"Map", "[", RowBox[{ RowBox[{ RowBox[{"Delete", "[", RowBox[{"#", ",", RowBox[{"-", "1"}]}], "]"}], "&"}], ",", SuperscriptBox["j", "-"]}], "]"}]}]}], "]"}]}]], "Input", CellChangeTimes->{{3.511261750101595*^9, 3.5112617538996468`*^9}, { 3.5112618744924583`*^9, 3.5112619147458878`*^9}, {3.511261985119388*^9, 3.5112619887903*^9}, {3.5112621579118977`*^9, 3.5112622073416348`*^9}, { 3.511262237636718*^9, 3.511262242316352*^9}, 3.511262527515675*^9, 3.511262803246293*^9, 3.51126283838869*^9, {3.511281897974084*^9, 3.511281906669561*^9}, 3.522645021130024*^9, 3.522649692532754*^9, 3.522651059076498*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox["DDelF", "0"], "\[Equal]", RowBox[{ RowBox[{"-", "2"}], " ", RowBox[{"(", RowBox[{ FractionBox[ RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], "4"], "+", FractionBox[ RowBox[{"5", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "16"]}], ")"}]}]}], ",", RowBox[{ SubscriptBox["DDelF", "2"], "\[Equal]", RowBox[{ RowBox[{"-", "2"}], " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", FractionBox["1", "16"]}], " ", RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}]}], "+", FractionBox[ RowBox[{"5", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "16"]}], ")"}]}]}]}], "}"}]], "Output", CellChangeTimes->{ 3.522650392863997*^9, 3.52265111564231*^9, 3.522651367641893*^9, 3.522651494531852*^9, 3.525679489610162*^9, 3.52567996814028*^9, 3.5256875163053007`*^9, 3.525688041203623*^9, 3.525689781722888*^9, 3.620399205815028*^9, 3.6203993026545258`*^9, 3.620401074931168*^9, 3.620402831291731*^9, 3.620404104293494*^9, 3.6204073890368347`*^9, 3.6204104329043417`*^9, 3.620411148870371*^9, 3.6204132570942507`*^9, 3.620413327805065*^9, 3.620414082024626*^9, 3.620472111990779*^9, 3.620472202094112*^9, 3.6204740202282267`*^9, 3.620488290747953*^9, 3.620490105483507*^9, 3.620490146394952*^9, 3.621752508524047*^9, { 3.6217525502664547`*^9, 3.62175256427544*^9}, 3.621753267215107*^9, 3.623768503454257*^9, 3.623822439154233*^9, 3.6238224800768757`*^9, 3.623848859107407*^9, 3.6238489104325323`*^9, 3.6238489609543047`*^9, 3.6238490156793213`*^9, 3.623850574017145*^9, 3.6238523594773397`*^9, 3.623852412424251*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"eqnsForDDelFs2", "=", RowBox[{"Map", "[", RowBox[{ RowBox[{ RowBox[{"Join", "[", RowBox[{ RowBox[{"{", "#", "}"}], ",", "evenPseudoFluxes"}], "]"}], "&"}], ",", "eqnsForDDelFs1"}], "]"}]}]], "Input", CellChangeTimes->{{3.511261543942474*^9, 3.511261594947771*^9}, { 3.511261643702834*^9, 3.511261678059739*^9}, {3.511262259745359*^9, 3.511262263693705*^9}, {3.5112624296346197`*^9, 3.511262447337812*^9}, { 3.511262482023842*^9, 3.51126253026333*^9}, 3.522645021134161*^9, 3.623848587820231*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox["DDelF", "0"], "\[Equal]", RowBox[{ RowBox[{"-", "2"}], " ", RowBox[{"(", RowBox[{ FractionBox[ RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], "4"], "+", FractionBox[ RowBox[{"5", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "16"]}], ")"}]}]}], ",", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], "+", RowBox[{"2", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}]}], "\[Equal]", RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}]}], ",", RowBox[{ RowBox[{"3", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "\[Equal]", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox["DDelF", "2"], "\[Equal]", RowBox[{ RowBox[{"-", "2"}], " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", FractionBox["1", "16"]}], " ", RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}]}], "+", FractionBox[ RowBox[{"5", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "16"]}], ")"}]}]}], ",", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], "+", RowBox[{"2", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}]}], "\[Equal]", RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}]}], ",", RowBox[{ RowBox[{"3", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "\[Equal]", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}]}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{ 3.5226503931308737`*^9, 3.522651119370254*^9, 3.52265136786351*^9, 3.522651494668461*^9, 3.525679489727747*^9, 3.525679968229022*^9, 3.525687517445842*^9, 3.5256880424954567`*^9, 3.52568978189769*^9, 3.620399205848884*^9, 3.620399302705501*^9, 3.620401074988234*^9, 3.620402831341552*^9, 3.620404104343707*^9, 3.620407389081751*^9, 3.620410432954125*^9, 3.620411148920348*^9, 3.62041325715514*^9, 3.620413327842942*^9, 3.6204140820598907`*^9, 3.620472112038309*^9, 3.620472202126547*^9, 3.620474020266447*^9, 3.620488290797052*^9, 3.620490105534152*^9, 3.6204901464422073`*^9, 3.621752508672797*^9, { 3.621752550300934*^9, 3.6217525643590097`*^9}, 3.6217532672639513`*^9, 3.6237685035059147`*^9, 3.623822439185403*^9, 3.623822480124247*^9, 3.623848859158646*^9, 3.623848910484733*^9, 3.623848961012907*^9, 3.623849015724379*^9, 3.623850574078562*^9, 3.6238523595098343`*^9, 3.6238524124715567`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"bc1m", "=", RowBox[{"Map", "[", RowBox[{ RowBox[{ RowBox[{"Eliminate", "[", RowBox[{"#", ",", "evenmom"}], "]"}], "&"}], ",", "eqnsForDDelFs2"}], "]"}]}]], "Input", CellChangeTimes->{{3.511262534725312*^9, 3.511262574752665*^9}, 3.522645021123007*^9, {3.522649792378985*^9, 3.52264980267071*^9}, { 3.522649842809058*^9, 3.5226498430018263`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", "8"}], " ", SubscriptBox["DDelF", "0"]}], "+", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}]}], "\[Equal]", RowBox[{"4", " ", RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}]}]}], ",", RowBox[{ RowBox[{ RowBox[{"24", " ", SubscriptBox["DDelF", "2"]}], "+", RowBox[{"7", " ", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}]}]}], "\[Equal]", RowBox[{"3", " ", RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}]}]}]}], "}"}]], "Output", CellChangeTimes->{ 3.522650393418055*^9, 3.522651125143038*^9, 3.522651368199791*^9, 3.522651494922188*^9, 3.525679489861867*^9, 3.5256799684582443`*^9, 3.525687518814999*^9, 3.5256897820392513`*^9, 3.620399205893465*^9, 3.6203993027543297`*^9, 3.620401075031333*^9, 3.620402831387466*^9, 3.6204041043929443`*^9, 3.620407389130916*^9, 3.620410433020618*^9, 3.6204111489859743`*^9, 3.620413257194296*^9, 3.620413327893268*^9, 3.620414082109576*^9, 3.620472112088932*^9, 3.620472202181834*^9, 3.6204740203127327`*^9, 3.6204882908477707`*^9, 3.6204901055826387`*^9, 3.620490146489977*^9, 3.621752508822521*^9, {3.6217525503835993`*^9, 3.621752564442443*^9}, 3.621753267315189*^9, 3.623768503555078*^9, 3.6238224392341022`*^9, 3.623822480401855*^9, 3.6238488592104883`*^9, 3.623848910534597*^9, 3.623848961058248*^9, 3.623849015773014*^9, 3.623850574218617*^9, 3.6238523595606327`*^9, 3.6238524126090307`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"bc2m", "=", RowBox[{ RowBox[{ RowBox[{"Solve", "[", RowBox[{"bc1m", ",", "DDelFs"}], "]"}], "//", "Flatten"}], "//", "Expand"}]}]], "Input", CellChangeTimes->{{3.511262534725312*^9, 3.511262574752665*^9}, { 3.5112821945741863`*^9, 3.511282201962818*^9}, 3.522645021141694*^9, 3.522649733809288*^9, {3.522649861756545*^9, 3.522649893731291*^9}, { 3.522649977844152*^9, 3.522649986095402*^9}, 3.522651128387389*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox["DDelF", "0"], "\[Rule]", RowBox[{ RowBox[{ RowBox[{"-", FractionBox["1", "2"]}], " ", RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}]}], "+", FractionBox[ RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}], "8"]}]}], ",", RowBox[{ SubscriptBox["DDelF", "2"], "\[Rule]", RowBox[{ FractionBox[ RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}], "8"], "-", FractionBox[ RowBox[{"7", " ", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}]}], "24"]}]}]}], "}"}]], "Output", CellChangeTimes->{ 3.522650393809373*^9, 3.522651129094426*^9, 3.5226513685117407`*^9, 3.5226514950568113`*^9, 3.525679489973113*^9, 3.5256799686026773`*^9, 3.5256875238811293`*^9, 3.525689782206458*^9, 3.620399205943542*^9, 3.6203993028101187`*^9, 3.6204010750858183`*^9, 3.620402831479233*^9, 3.620404104444105*^9, 3.620407389181107*^9, 3.620410433071148*^9, 3.620411149036003*^9, 3.62041325722822*^9, 3.620413327949996*^9, 3.620414082159464*^9, 3.620472112139393*^9, 3.620472202227477*^9, 3.620474020363065*^9, 3.6204882908962*^9, 3.620490105633193*^9, 3.620490146541505*^9, 3.621752508922724*^9, {3.621752550484013*^9, 3.621752564525681*^9}, 3.6217532673666058`*^9, 3.6237685036050673`*^9, 3.6238224392849207`*^9, 3.623822480444158*^9, 3.623848859257471*^9, 3.6238489106451597`*^9, 3.6238489611080523`*^9, 3.6238490158218393`*^9, 3.623850574260243*^9, 3.623852359670336*^9, 3.62385241263876*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"curs", " ", "=", " ", RowBox[{ RowBox[{"-", "DDelFs"}], "/.", "bc2m"}]}]], "Input", CellChangeTimes->{{3.623769706361328*^9, 3.62376972594489*^9}, 3.623769941000832*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ FractionBox[ RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}], "2"], "-", FractionBox[ RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}], "8"]}], ",", RowBox[{ RowBox[{ RowBox[{"-", FractionBox["1", "8"]}], " ", RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}]}], "+", FractionBox[ RowBox[{"7", " ", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}]}], "24"]}]}], "}"}]], "Output", CellChangeTimes->{3.623769728120428*^9, 3.623769941562393*^9, 3.623822439336482*^9, 3.623822480490527*^9, 3.623848859307592*^9, 3.623848910694345*^9, 3.623848961160613*^9, 3.623849015871109*^9, 3.623850574306953*^9, 3.6238523597085953`*^9, 3.6238524126888857`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"ca", "=", RowBox[{"CoefficientArrays", "[", RowBox[{"curs", ",", RowBox[{"evenmom", "/.", RowBox[{"\[Phi]", "\[Rule]", "F"}]}]}], "]"}]}]], "Input", CellChangeTimes->{{3.6237696751041117`*^9, 3.623769684080681*^9}, { 3.623769735377289*^9, 3.62376973648781*^9}, {3.623769807082581*^9, 3.623769851119483*^9}, {3.623848938941497*^9, 3.623848939366596*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ TagBox[ RowBox[{"SparseArray", "[", RowBox[{ InterpretationBox[ RowBox[{"\<\"<\"\>", "\[InvisibleSpace]", "0", "\[InvisibleSpace]", "\<\">\"\>"}], SequenceForm["<", 0, ">"], Editable->False], ",", RowBox[{"{", "2", "}"}]}], "]"}], False, Editable->False], ",", TagBox[ RowBox[{"SparseArray", "[", RowBox[{ InterpretationBox[ RowBox[{"\<\"<\"\>", "\[InvisibleSpace]", "4", "\[InvisibleSpace]", "\<\">\"\>"}], SequenceForm["<", 4, ">"], Editable->False], ",", RowBox[{"{", RowBox[{"2", ",", "2"}], "}"}]}], "]"}], False, Editable->False]}], "}"}]], "Output", CellChangeTimes->{{3.623769835851221*^9, 3.623769853176485*^9}, 3.6237699424467583`*^9, 3.623822439382833*^9, 3.6238224805428762`*^9, 3.623848859358233*^9, {3.623848910734744*^9, 3.6238489612090807`*^9}, 3.623849015927463*^9, 3.62385057435546*^9, 3.623852359760953*^9, 3.623852412723089*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Normal", "[", RowBox[{"ca", "[", RowBox[{"[", "2", "]"}], "]"}], "]"}]], "Input", CellChangeTimes->{{3.623769825649048*^9, 3.623769858878666*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ FractionBox["1", "2"], ",", RowBox[{"-", FractionBox["1", "8"]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", FractionBox["1", "8"]}], ",", FractionBox["7", "24"]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{{3.6237698273050222`*^9, 3.623769859465145*^9}, 3.623769943315426*^9, 3.623822439420066*^9, 3.623822480589869*^9, 3.623848859408636*^9, {3.623848910845036*^9, 3.6238489612599287`*^9}, 3.623849015972991*^9, 3.623850574405221*^9, 3.623852359810226*^9, 3.623852412757513*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"N", "[", RowBox[{"Eigenvalues", "[", RowBox[{"ca", "[", RowBox[{"[", "2", "]"}], "]"}], "]"}], "]"}]], "Input", CellChangeTimes->{{3.623769865935341*^9, 3.6237698782357187`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"0.5585468682480552`", ",", "0.23311979841861136`"}], "}"}]], "Output", CellChangeTimes->{{3.623769874183299*^9, 3.623769878729514*^9}, 3.623769944235021*^9, 3.623822439454218*^9, 3.623822480642846*^9, 3.623848859459935*^9, 3.623848910946731*^9, {3.6238489419238243`*^9, 3.623848961309411*^9}, 3.623849016022606*^9, 3.623850574455204*^9, 3.623852359859137*^9, 3.6238524135935907`*^9}] }, Open ]], Cell[TextData[StyleBox["Albedo boundary conditions", FontSize->16]], "Text", CellChangeTimes->{{3.522645067217414*^9, 3.522645068559268*^9}, { 3.522645113653557*^9, 3.522645154760992*^9}, {3.5226455253291683`*^9, 3.522645537447722*^9}, {3.522645757714418*^9, 3.522645804206421*^9}, { 3.5226458615113*^9, 3.5226459447188663`*^9}, {3.522645982553526*^9, 3.522646010994257*^9}, {3.522646050970581*^9, 3.522646123120521*^9}, { 3.5226465046629467`*^9, 3.5226465396714478`*^9}, {3.522647071486895*^9, 3.5226471070132837`*^9}, {3.522647196226143*^9, 3.5226473057929897`*^9}, { 3.5226473374375267`*^9, 3.522647550210237*^9}, {3.522647591318706*^9, 3.52264759216264*^9}, {3.5226476533707314`*^9, 3.522647691867504*^9}, { 3.522647900608212*^9, 3.522647955486134*^9}, {3.522648087495708*^9, 3.522648191535192*^9}, {3.522648536885538*^9, 3.522648558056643*^9}, { 3.522648852277603*^9, 3.5226489116663723`*^9}, 3.5226489963081408`*^9, { 3.522649134521142*^9, 3.5226491881211023`*^9}, {3.522649342096834*^9, 3.5226493813411837`*^9}, {3.5226494456588*^9, 3.52264946283311*^9}, { 3.623849078761973*^9, 3.623849089380601*^9}, {3.6238523449564247`*^9, 3.6238523452010727`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"alpha", "=", RowBox[{ RowBox[{"Solve", "[", RowBox[{ RowBox[{"\[Gamma]", "==", " ", FractionBox[ RowBox[{"1", "-", "\[Alpha]"}], RowBox[{"2", RowBox[{"(", RowBox[{"1", "+", "\[Alpha]"}], ")"}]}]]}], ",", "\[Alpha]"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}]], "Input", CellChangeTimes->{{3.525680442279339*^9, 3.525680488724207*^9}, { 3.525680537604802*^9, 3.525680550093995*^9}, {3.525689816293586*^9, 3.5256898340607433`*^9}, {3.62384877204681*^9, 3.62384881245055*^9}, 3.623851995273636*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{"\[Alpha]", "\[Rule]", FractionBox[ RowBox[{"1", "-", RowBox[{"2", " ", "\[Gamma]"}]}], RowBox[{"1", "+", RowBox[{"2", " ", "\[Gamma]"}]}]]}], "}"}]], "Output", CellChangeTimes->{3.6238519959132*^9, 3.623852359908347*^9, 3.623852413699957*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"aj", "=", RowBox[{"Thread", "[", RowBox[{ SuperscriptBox["j", "-"], "\[Equal]", RowBox[{"\[Alpha]", " ", SuperscriptBox["j", "+"]}]}], "]"}]}]], "Input", CellChangeTimes->{{3.525690159104034*^9, 3.525690198486574*^9}, { 3.525690340430955*^9, 3.52569034073663*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ FractionBox[ RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], "4"], "+", FractionBox[ RowBox[{"5", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "16"], "-", FractionBox[ RowBox[{ RowBox[{ RowBox[{"-", "Q"}], " ", RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}]}], "-", RowBox[{"2", " ", "Q", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}]}], RowBox[{"6", " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "1"}]], "[", "z", "]"}]}]]}], "\[Equal]", RowBox[{"\[Alpha]", " ", RowBox[{"(", RowBox[{ FractionBox[ RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], "4"], "+", FractionBox[ RowBox[{"5", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "16"], "+", FractionBox[ RowBox[{ RowBox[{ RowBox[{"-", "Q"}], " ", RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}]}], "-", RowBox[{"2", " ", "Q", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}]}], RowBox[{"6", " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "1"}]], "[", "z", "]"}]}]]}], ")"}]}]}], ",", RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", FractionBox["1", "16"]}], " ", RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}]}], "+", FractionBox[ RowBox[{"5", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "16"], "+", FractionBox[ RowBox[{"3", " ", "Q", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], RowBox[{"14", " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "3"}]], "[", "z", "]"}]}]]}], "\[Equal]", RowBox[{"\[Alpha]", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", FractionBox["1", "16"]}], " ", RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}]}], "+", FractionBox[ RowBox[{"5", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "16"], "-", FractionBox[ RowBox[{"3", " ", "Q", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], RowBox[{"14", " ", RowBox[{ SubscriptBox["\[CapitalSigma]", RowBox[{"a", ",", "3"}]], "[", "z", "]"}]}]]}], ")"}]}]}]}], "}"}]], "Output", CellChangeTimes->{ 3.525690167390123*^9, 3.525690199133114*^9, 3.525690341455398*^9, 3.525691971532394*^9, 3.620399206331531*^9, 3.620399303040556*^9, 3.62040107531669*^9, 3.620402831740964*^9, 3.6204041046855783`*^9, 3.620407389414701*^9, 3.620410433316606*^9, 3.620411149304801*^9, 3.620413257799762*^9, 3.6204133282320957`*^9, 3.6204140823986683`*^9, 3.6204721126981783`*^9, 3.620472202449562*^9, 3.620474020563628*^9, 3.620488291236128*^9, 3.620490105855954*^9, 3.620490146780336*^9, 3.621752509525693*^9, {3.6217525508806458`*^9, 3.621752564792836*^9}, 3.621753267964415*^9, 3.623768503874174*^9, 3.6238224404231243`*^9, 3.62382248109193*^9, 3.623848859538011*^9, 3.623848911030292*^9, 3.623848961445199*^9, 3.623849016101306*^9, 3.623850574535849*^9, 3.6238523599616623`*^9, 3.623852413740821*^9}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"DDelFsPos", "=", RowBox[{"1", "+", FractionBox[ RowBox[{"L", "+", "1"}], "2"]}]}], ";"}]], "Input", CellChangeTimes->{{3.623851618032578*^9, 3.623851627780673*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"DDelFsRep", "=", RowBox[{"Flatten", "[", RowBox[{"Table", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"{", RowBox[{"{", RowBox[{"i", ",", "1", ",", "DDelFsPos"}], "}"}], "}"}], "\[Rule]", FractionBox[ RowBox[{"DDelFs", "[", RowBox[{"[", "i", "]"}], "]"}], "2"]}], ",", RowBox[{ RowBox[{"{", RowBox[{"{", RowBox[{"i", ",", "2", ",", "2", ",", "DDelFsPos"}], "}"}], "}"}], "\[Rule]", RowBox[{"-", FractionBox[ RowBox[{"DDelFs", "[", RowBox[{"[", "i", "]"}], "]"}], "2"]}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"i", ",", "1", ",", FractionBox[ RowBox[{"L", "+", "1"}], "2"]}], "}"}]}], "]"}], "]"}]}]], "Input", CellChangeTimes->{{3.623851463921726*^9, 3.623851490816718*^9}, { 3.6238515465824633`*^9, 3.623851566086647*^9}, {3.623851616459722*^9, 3.623851669587076*^9}, {3.623851701930127*^9, 3.623851773767643*^9}, { 3.623851822894025*^9, 3.62385186797324*^9}, {3.623851899567865*^9, 3.6238519273202333`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"{", RowBox[{"{", RowBox[{"1", ",", "1", ",", "3"}], "}"}], "}"}], "\[Rule]", FractionBox[ SubscriptBox["DDelF", "0"], "2"]}], ",", RowBox[{ RowBox[{"{", RowBox[{"{", RowBox[{"1", ",", "2", ",", "2", ",", "3"}], "}"}], "}"}], "\[Rule]", RowBox[{"-", FractionBox[ SubscriptBox["DDelF", "0"], "2"]}]}], ",", RowBox[{ RowBox[{"{", RowBox[{"{", RowBox[{"2", ",", "1", ",", "3"}], "}"}], "}"}], "\[Rule]", FractionBox[ SubscriptBox["DDelF", "2"], "2"]}], ",", RowBox[{ RowBox[{"{", RowBox[{"{", RowBox[{"2", ",", "2", ",", "2", ",", "3"}], "}"}], "}"}], "\[Rule]", RowBox[{"-", FractionBox[ SubscriptBox["DDelF", "2"], "2"]}]}]}], "}"}]], "Output", CellChangeTimes->{{3.623851869111842*^9, 3.623851927948924*^9}, 3.623852360020945*^9, 3.623852413818068*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"aj2", "=", RowBox[{"ReplacePart", "[", RowBox[{"aj", ",", "DDelFsRep"}], "]"}]}]], "Input", CellChangeTimes->{{3.525690399742064*^9, 3.525690400920932*^9}, { 3.5256904353720913`*^9, 3.5256904515482817`*^9}, {3.525690489366888*^9, 3.525690500464184*^9}, 3.525690548702836*^9, {3.525691681913342*^9, 3.5256917588719873`*^9}, {3.5256918783065033`*^9, 3.5256919245660553`*^9}, {3.623848825803033*^9, 3.6238488301467667`*^9}, 3.623849847315168*^9, {3.6238511089979477`*^9, 3.6238511646362534`*^9}, { 3.623851313315341*^9, 3.623851329181929*^9}, 3.6238519348423862`*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ FractionBox[ SubscriptBox["DDelF", "0"], "2"], "+", FractionBox[ RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], "4"], "+", FractionBox[ RowBox[{"5", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "16"]}], "\[Equal]", RowBox[{"\[Alpha]", " ", RowBox[{"(", RowBox[{ RowBox[{"-", FractionBox[ SubscriptBox["DDelF", "0"], "2"]}], "+", FractionBox[ RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], "4"], "+", FractionBox[ RowBox[{"5", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "16"]}], ")"}]}]}], ",", RowBox[{ RowBox[{ FractionBox[ SubscriptBox["DDelF", "2"], "2"], "-", FractionBox[ RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], "16"], "+", FractionBox[ RowBox[{"5", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "16"]}], "\[Equal]", RowBox[{"\[Alpha]", " ", RowBox[{"(", RowBox[{ RowBox[{"-", FractionBox[ SubscriptBox["DDelF", "2"], "2"]}], "-", FractionBox[ RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], "16"], "+", FractionBox[ RowBox[{"5", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "16"]}], ")"}]}]}]}], "}"}]], "Output", CellChangeTimes->{ 3.525690443535924*^9, 3.525690501847478*^9, 3.525691721197926*^9, { 3.5256919153177547`*^9, 3.525691925127726*^9}, 3.525691973548847*^9, 3.620399206995955*^9, 3.620399304868864*^9, 3.6204010770601673`*^9, 3.6204028336421137`*^9, 3.620404108371337*^9, 3.6204073912047987`*^9, 3.620410436954905*^9, 3.620411152915895*^9, 3.62041325816984*^9, 3.620413328305372*^9, 3.6204140829962063`*^9, 3.6204721127921343`*^9, 3.620472202551559*^9, 3.620474022445015*^9, 3.620488293132265*^9, 3.6204901096691523`*^9, 3.620490150384562*^9, 3.621752509774961*^9, { 3.6217525509828978`*^9, 3.6217525649830217`*^9}, 3.621753268061356*^9, 3.6237685045463247`*^9, 3.6238224405220737`*^9, 3.623822481408104*^9, 3.62384886011242*^9, 3.6238489111166763`*^9, 3.623848961592258*^9, 3.623849016253792*^9, 3.623849848071069*^9, 3.623850574673955*^9, 3.623851203204341*^9, 3.623851937027989*^9, 3.623852360059815*^9, 3.6238524138561087`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"eqnsForDDelFs3", "=", RowBox[{"Map", "[", RowBox[{ RowBox[{ RowBox[{"Join", "[", RowBox[{ RowBox[{"{", "#", "}"}], ",", "evenPseudoFluxes"}], "]"}], "&"}], ",", "aj2"}], "]"}]}]], "Input", CellChangeTimes->{{3.6238513336938334`*^9, 3.6238513370616837`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ FractionBox[ SubscriptBox["DDelF", "0"], "2"], "+", FractionBox[ RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], "4"], "+", FractionBox[ RowBox[{"5", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "16"]}], "\[Equal]", RowBox[{"\[Alpha]", " ", RowBox[{"(", RowBox[{ RowBox[{"-", FractionBox[ SubscriptBox["DDelF", "0"], "2"]}], "+", FractionBox[ RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], "4"], "+", FractionBox[ RowBox[{"5", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "16"]}], ")"}]}]}], ",", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], "+", RowBox[{"2", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}]}], "\[Equal]", RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}]}], ",", RowBox[{ RowBox[{"3", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "\[Equal]", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ FractionBox[ SubscriptBox["DDelF", "2"], "2"], "-", FractionBox[ RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], "16"], "+", FractionBox[ RowBox[{"5", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "16"]}], "\[Equal]", RowBox[{"\[Alpha]", " ", RowBox[{"(", RowBox[{ RowBox[{"-", FractionBox[ SubscriptBox["DDelF", "2"], "2"]}], "-", FractionBox[ RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], "16"], "+", FractionBox[ RowBox[{"5", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "16"]}], ")"}]}]}], ",", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[Phi]", "0"], "[", "z", "]"}], "+", RowBox[{"2", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}]}], "\[Equal]", RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}]}], ",", RowBox[{ RowBox[{"3", " ", RowBox[{ SubscriptBox["\[Phi]", "2"], "[", "z", "]"}]}], "\[Equal]", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}]}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.6238513399887743`*^9, 3.623851941685835*^9, 3.6238523600934753`*^9, 3.623852413888953*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"bc1a", "=", RowBox[{"Map", "[", RowBox[{ RowBox[{ RowBox[{"Eliminate", "[", RowBox[{"#", ",", "evenmom"}], "]"}], "&"}], ",", "eqnsForDDelFs3"}], "]"}]}]], "Input", CellChangeTimes->{{3.623851396404698*^9, 3.623851402003544*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"8", " ", SubscriptBox["DDelF", "0"]}], "+", RowBox[{"8", " ", "\[Alpha]", " ", SubscriptBox["DDelF", "0"]}], "-", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}], "+", RowBox[{"\[Alpha]", " ", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}]}]}], "\[Equal]", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "4"}], "+", RowBox[{"4", " ", "\[Alpha]"}]}], ")"}], " ", RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}]}]}], ",", RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", "24"}], " ", SubscriptBox["DDelF", "2"]}], "-", RowBox[{"24", " ", "\[Alpha]", " ", SubscriptBox["DDelF", "2"]}], "-", RowBox[{"7", " ", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}]}], "+", RowBox[{"7", " ", "\[Alpha]", " ", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}]}]}], "\[Equal]", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "3"}], "+", RowBox[{"3", " ", "\[Alpha]"}]}], ")"}], " ", RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}]}]}]}], "}"}]], "Output", CellChangeTimes->{3.623851402548527*^9, 3.623851946909429*^9, 3.623852360145205*^9, 3.623852413941009*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"bc2a", "=", RowBox[{ RowBox[{"Solve", "[", RowBox[{"bc1a", ",", "DDelFs"}], "]"}], "//", "Flatten"}]}]], "Input", CellChangeTimes->{{3.6238519613750467`*^9, 3.623851969070526*^9}, 3.623852072804015*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox["DDelF", "0"], "\[Rule]", FractionBox[ RowBox[{ RowBox[{ RowBox[{"-", "4"}], " ", RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}]}], "+", RowBox[{"4", " ", "\[Alpha]", " ", RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}]}], "+", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}], "-", RowBox[{"\[Alpha]", " ", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}]}]}], RowBox[{"8", " ", RowBox[{"(", RowBox[{"1", "+", "\[Alpha]"}], ")"}]}]]}], ",", RowBox[{ SubscriptBox["DDelF", "2"], "\[Rule]", FractionBox[ RowBox[{ RowBox[{"3", " ", RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}]}], "-", RowBox[{"3", " ", "\[Alpha]", " ", RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}]}], "-", RowBox[{"7", " ", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}]}], "+", RowBox[{"7", " ", "\[Alpha]", " ", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}]}]}], RowBox[{"24", " ", RowBox[{"(", RowBox[{"1", "+", "\[Alpha]"}], ")"}]}]]}]}], "}"}]], "Output", CellChangeTimes->{{3.623851961867323*^9, 3.623851969509952*^9}, 3.62385208927075*^9, 3.6238523601935596`*^9, 3.623852413989518*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"bc3a", "=", RowBox[{ RowBox[{"bc2a", "/.", "alpha"}], "//", "FullSimplify"}]}]], "Input", CellChangeTimes->{{3.623849044592318*^9, 3.623849052893054*^9}, { 3.623852005343446*^9, 3.623852005981933*^9}, 3.623852077883823*^9, { 3.623852168807868*^9, 3.623852170224517*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox["DDelF", "0"], "\[Rule]", RowBox[{ FractionBox["1", "4"], " ", "\[Gamma]", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "4"}], " ", RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}]}], "+", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}]}], ")"}]}]}], ",", RowBox[{ SubscriptBox["DDelF", "2"], "\[Rule]", RowBox[{ FractionBox["1", "12"], " ", "\[Gamma]", " ", RowBox[{"(", RowBox[{ RowBox[{"3", " ", RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}]}], "-", RowBox[{"7", " ", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}]}]}], ")"}]}]}]}], "}"}]], "Output", CellChangeTimes->{3.62384905325537*^9, 3.623850574772242*^9, 3.623852006762484*^9, 3.623852091004784*^9, 3.62385213056376*^9, 3.623852171844431*^9, 3.623852360246614*^9, 3.623852414041911*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"cursa", " ", "=", " ", RowBox[{ RowBox[{"-", "DDelFs"}], "/.", "bc3a"}]}]], "Input", CellChangeTimes->{{3.623852079842025*^9, 3.623852103193318*^9}, 3.623852174249022*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"-", FractionBox["1", "4"]}], " ", "\[Gamma]", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "4"}], " ", RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}]}], "+", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}]}], ")"}]}], ",", RowBox[{ RowBox[{"-", FractionBox["1", "12"]}], " ", "\[Gamma]", " ", RowBox[{"(", RowBox[{ RowBox[{"3", " ", RowBox[{ SubscriptBox["F", "0"], "[", "z", "]"}]}], "-", RowBox[{"7", " ", RowBox[{ SubscriptBox["F", "2"], "[", "z", "]"}]}]}], ")"}]}]}], "}"}]], "Output", CellChangeTimes->{{3.623852092941531*^9, 3.623852132301177*^9}, 3.62385217548028*^9, 3.623852360276906*^9, 3.62385241409061*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"ca", "=", RowBox[{"CoefficientArrays", "[", RowBox[{"cursa", ",", RowBox[{"evenmom", "/.", RowBox[{"\[Phi]", "\[Rule]", "F"}]}]}], "]"}]}]], "Input", CellChangeTimes->{ 3.623852032442545*^9, {3.623852084642294*^9, 3.623852100771193*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ TagBox[ RowBox[{"SparseArray", "[", RowBox[{ InterpretationBox[ RowBox[{"\<\"<\"\>", "\[InvisibleSpace]", "0", "\[InvisibleSpace]", "\<\">\"\>"}], SequenceForm["<", 0, ">"], Editable->False], ",", RowBox[{"{", "2", "}"}]}], "]"}], False, Editable->False], ",", TagBox[ RowBox[{"SparseArray", "[", RowBox[{ InterpretationBox[ RowBox[{"\<\"<\"\>", "\[InvisibleSpace]", "4", "\[InvisibleSpace]", "\<\">\"\>"}], SequenceForm["<", 4, ">"], Editable->False], ",", RowBox[{"{", RowBox[{"2", ",", "2"}], "}"}]}], "]"}], False, Editable->False]}], "}"}]], "Output", CellChangeTimes->{{3.623852026676038*^9, 3.623852034482834*^9}, { 3.623852095531147*^9, 3.623852105460661*^9}, 3.6238521772662992`*^9, 3.6238523603118553`*^9, 3.623852414141562*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Normal", "[", RowBox[{"ca", "[", RowBox[{"[", "2", "]"}], "]"}], "]"}], "//", "MatrixForm"}]], "Input", CellChangeTimes->{{3.623852368418507*^9, 3.623852370368095*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"\[Gamma]", RowBox[{"-", FractionBox["\[Gamma]", "4"]}]}, { RowBox[{"-", FractionBox["\[Gamma]", "4"]}], FractionBox[ RowBox[{"7", " ", "\[Gamma]"}], "12"]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{ 3.62385211607728*^9, 3.623852178519928*^9, {3.623852360361672*^9, 3.62385237078874*^9}, 3.623852414190154*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Normal", "[", RowBox[{ RowBox[{"ca", "[", RowBox[{"[", "2", "]"}], "]"}], "/", "\[Gamma]"}], "]"}], "//", "MatrixForm"}]], "Input", CellChangeTimes->{{3.623852390866878*^9, 3.623852391729179*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", RowBox[{"-", FractionBox["1", "4"]}]}, { RowBox[{"-", FractionBox["1", "4"]}], FractionBox["7", "12"]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{{3.623852392436139*^9, 3.62385241424118*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Eigenvalues", "[", RowBox[{"Normal", "[", RowBox[{"ca", "[", RowBox[{"[", "2", "]"}], "]"}], "]"}], "]"}], "//", "N"}]], "Input", CellChangeTimes->{{3.6238521940401173`*^9, 3.623852218142294*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"1.1170937364961104`", " ", "\[Gamma]"}], ",", RowBox[{"0.4662395968372227`", " ", "\[Gamma]"}]}], "}"}]], "Output", CellChangeTimes->{{3.62385220260528*^9, 3.623852218576447*^9}, 3.623852360411058*^9, 3.623852414273715*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"\[Gamma]f", "[", "\[Alpha]_", "]"}], ":=", FractionBox[ RowBox[{"1", "-", "\[Alpha]"}], RowBox[{"2", RowBox[{"(", RowBox[{"1", "+", "\[Alpha]"}], ")"}]}]]}], ";"}], "\[IndentingNewLine]", RowBox[{"Plot", "[", RowBox[{ RowBox[{"\[Gamma]f", "[", "\[Alpha]", "]"}], ",", RowBox[{"{", RowBox[{"\[Alpha]", ",", "0", ",", "1"}], "}"}]}], "]"}]}], "Input", CellChangeTimes->{{3.623852235334422*^9, 3.62385229235756*^9}}], Cell[BoxData[ GraphicsBox[{{}, {}, {Hue[0.67, 0.6, 0.6], LineBox[CompressedData[" 1:eJwVxX040wkAB3CkFJfXrJbQZJTePKvWUufronLeipK6JEck6srLVe6pmDOn JHJyu0NELycPU6bCYrFTRGLeX/aT7dZs+NGRdJfd3R+f50MLOuMToqWhoeH5 n/93C1G0CUd8HbeColSrCWg7Lz9LUDfD57Tl2OwnApvyuDnlVBckTXysnJom 4N7S/SiXegAmi4+VjaoI+B2feZZBPY5v4hp15d0EBjf+5JxKjYET0eUxxyOQ b7o36So1EaUL2mvqAwmwF5bbXqFmoiskxpsrkuDQ4du8ick7KEiVMqgXB6Gn ZvvvseChFn+pXHwHQPceV9Sn8GGibE8ROvbj/Ott4E5VItrlVp3IrA9L+UYS Tc1aHNH1jKBSevF42Lm5dUcdgsL4vU+1e1AXnmt2eFQE1gWDFqpWN6ZtvjQo Of0CBTrJUTRqF7Rmiw3N6E0IXz1WZEjvBFMd+Hh/aTNKi/Skfa4daE7c/LrB rhVr68WDjefFsExXzRaNv0F6r/E4M6cdqhlC0mnfjvCbt607O9twR1ndpH9d jDM9wiUp89ugb36flMk6wJD0f87AG1w7WRGab90Flk1O1T/LWvHBqVgWFteN o8E0J9P8FsQwRHxKfQ+WvBJHF7s1gygVpbTR+jDJOrH3irQJf4d3ZK/5oR8u lMOOYamNWPuVzlxo9QD2XRor2eD3EnsKaCceLpWA3Vc7tMHwBc6bH7N6608g bqnDIY93f2DdqK9et+cQnB4Fxba/EsHEYGKh6NMQTLMkFyPz66Fdm81blfMW +XZeOhHpdYjYXN27yH0Y6l2Xrdwjn6P466D8mg/DWLZqxyZ5gBDTquWG8iwp hPZ6tq+aalCxf8+8TBcZJk821NOrBYhz5eTx5TJ4er7ntPZVgTbo1lPI+RMf 8hbkzuhXwmFrYiaVIUdWZH+DwvYJ/C1aZzXFcrBEUqddpyow3ZQ6fJb9Di6R 7NdlWeXgagfE+FgroFutMI9XPoSJpKJAS6SAyp5bPn9fGeZfYh/YFjmCRoe0 G4+3lUCjKIA3bKzE7qt9dpWWDxD1Gy/VuE6JNvO62Iyue3i2Nnmh0XcqNMvo z7m8Qhi5Ra1L0B/FditBKLU1HwPr34hjBaNoNDKz/fbHbJTdgrUsYAwnvGuf /EzexFEe08h8bgwfF0XtZGamI55zmz3wYBysaP9o1wYOShTm2foeJNZ84brS ouYcknM5gRIvEpdXTNFtg87huM8YvdSbRHVGxEHWgnNYIRCUefqRgFbwpmiv 73E17UjDtSAS6e91a92IaIQxf53UjSXxUsq1i9eKhFXiEledeyQC3hqkzeaG Yo51cXH37yRWfMwzN1aEoHdM2n6vmESuSC5wYIQg3a/86O6HJNbLTemChmCo 7XyiOQISvusuBcZNBGKgLe3WPDGJ+5cTvPWuH8TTpJlgcScJiiI8eYe1LzK3 H1tT2ENisEUrN7VqP9zvbuDvlJAouRvhlfZuH6outDSyR0hkMJVPVybtRtb6 LWneoyTGuXsp8XxnRA7nHKCRJBgvZSoLqRNWe5wihFMk0uZ2CbYkMKGt2XEn fYZEM7Gqgxyyx1DF9vDATyRSpicZ3vE2EIQXbrT/TOKGi9t4XBwFv1jqTavV JGwChJ3CkS2O/wJuDWoL "]]}}, AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948], Axes->True, AxesLabel->{None, None}, AxesOrigin->{0, 0}, Method->{}, PlotRange->{{0, 1}, {0., 0.49999997959183723`}}, PlotRangeClipping->True, PlotRangePadding->{ Scaled[0.02], Scaled[0.02]}]], "Output", CellChangeTimes->{3.623852293216588*^9, 3.623852360462247*^9, 3.6238524143252277`*^9}] }, Open ]] }, Open ]] }, Open ]] }, WindowSize->{1440, 852}, WindowMargins->{{-1407, Automatic}, {Automatic, 6}}, CellContext->Notebook, FrontEndVersion->"9.0 for Mac OS X x86 (32-bit, 64-bit Kernel) (January 25, \ 2013)", StyleDefinitions->FrontEnd`FileName[{"Report"}, "StandardReport.nb", CharacterEncoding -> "UTF-8"] ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[567, 22, 390, 8, 68, "Title"], Cell[CellGroupData[{ Cell[982, 34, 107, 1, 82, "Section"], Cell[1092, 37, 295, 5, 33, "Text"], Cell[1390, 44, 809, 20, 66, "Input"], Cell[2202, 66, 2728, 39, 37, "Input"], Cell[4933, 107, 345, 10, 35, "Text"], Cell[CellGroupData[{ Cell[5303, 121, 1390, 36, 57, "Input"], Cell[6696, 159, 1727, 41, 56, "Output"] }, Open ]], Cell[8438, 203, 1296, 32, 37, "Input"], Cell[CellGroupData[{ Cell[9759, 239, 162, 3, 37, "Input"], Cell[9924, 244, 3585, 92, 127, "Output"] }, Open ]], Cell[13524, 339, 245, 4, 33, "Text"], Cell[13772, 345, 875, 27, 55, "Input"], Cell[CellGroupData[{ Cell[14672, 376, 683, 20, 55, "Input"], Cell[15358, 398, 1096, 20, 37, "Output"], Cell[16457, 420, 1096, 20, 37, "Output"] }, Open ]], Cell[17568, 443, 586, 15, 33, "Text"], Cell[18157, 460, 357, 8, 26, "Item"], Cell[CellGroupData[{ Cell[18539, 472, 526, 10, 37, "Input"], Cell[19068, 484, 1891, 46, 61, "Output"] }, Open ]], Cell[20974, 533, 339, 5, 24, "Item"], Cell[21316, 540, 701, 19, 37, "Input"], Cell[CellGroupData[{ Cell[22042, 563, 159, 3, 37, "Input"], Cell[22204, 568, 3653, 92, 101, "Output"] }, Open ]], Cell[25872, 663, 1998, 65, 54, "Text"], Cell[CellGroupData[{ Cell[27895, 732, 2707, 61, 81, "Input"], Cell[30605, 795, 1444, 31, 37, "Output"], Cell[32052, 828, 1478, 32, 61, "Output"] }, Open ]], Cell[33545, 863, 696, 12, 52, "Text"], Cell[CellGroupData[{ Cell[34266, 879, 1139, 31, 57, "Input"], Cell[35408, 912, 1881, 43, 87, "Output"] }, Open ]], Cell[37304, 958, 1998, 64, 40, "Text"], Cell[CellGroupData[{ Cell[39327, 1026, 1094, 26, 92, "Input"], Cell[40424, 1054, 982, 16, 57, "Output"], Cell[41409, 1072, 1008, 17, 57, "Output"] }, Open ]], Cell[42432, 1092, 753, 24, 37, "Input"], Cell[43188, 1118, 688, 14, 37, "Input", CellID->1804033555], Cell[CellGroupData[{ Cell[43901, 1136, 179, 4, 37, "Input"], Cell[44083, 1142, 1530, 33, 75, "Output"] }, Open ]], Cell[45628, 1178, 600, 16, 44, "Input"], Cell[46231, 1196, 995, 19, 33, "Text"], Cell[47229, 1217, 413, 11, 37, "Input"], Cell[CellGroupData[{ Cell[47667, 1232, 941, 16, 37, "Input"], Cell[48611, 1250, 1851, 44, 57, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[50499, 1299, 959, 18, 37, "Input"], Cell[51461, 1319, 2944, 77, 57, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[54442, 1401, 519, 11, 37, "Input"], Cell[54964, 1414, 2213, 55, 44, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[57214, 1474, 594, 12, 37, "Input"], Cell[57811, 1488, 1001, 17, 37, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[58849, 1510, 852, 21, 37, "Input"], Cell[59704, 1533, 2249, 57, 57, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[61990, 1595, 984, 21, 37, "Input"], Cell[62977, 1618, 2487, 65, 57, "Output"] }, Open ]], Cell[65479, 1686, 1316, 33, 33, "Text"], Cell[CellGroupData[{ Cell[66820, 1723, 714, 17, 37, "Input"], Cell[67537, 1742, 1636, 39, 37, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[69210, 1786, 204, 4, 37, "Input"], Cell[69417, 1792, 1530, 33, 67, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[70984, 1830, 158, 3, 37, "Input"], Cell[71145, 1835, 2436, 60, 83, "Output"] }, Open ]], Cell[73596, 1898, 894, 12, 30, "Text"], Cell[74493, 1912, 556, 16, 44, "Input"], Cell[CellGroupData[{ Cell[75074, 1932, 215, 5, 37, "Input"], Cell[75292, 1939, 2096, 49, 88, "Output"] }, Open ]], Cell[77403, 1991, 1015, 20, 33, "Text"], Cell[CellGroupData[{ Cell[78443, 2015, 213, 5, 37, "Input"], Cell[78659, 2022, 1075, 20, 37, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[79771, 2047, 676, 18, 44, "Input"], Cell[80450, 2067, 1501, 39, 57, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[81988, 2111, 223, 4, 37, "Input"], Cell[82214, 2117, 1902, 48, 87, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[84153, 2170, 214, 5, 44, "Input"], Cell[84370, 2177, 1153, 34, 87, "Output"] }, Open ]], Cell[85538, 2214, 160, 4, 37, "Input"], Cell[CellGroupData[{ Cell[85723, 2222, 2140, 45, 140, "Input"], Cell[CellGroupData[{ Cell[87888, 2271, 1222, 30, 56, "Print"], Cell[89113, 2303, 612, 11, 40, "Print"], Cell[89728, 2316, 1126, 26, 46, "Print"], Cell[90857, 2344, 611, 11, 40, "Print"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[91517, 2361, 207, 3, 37, "Input"], Cell[91727, 2366, 757, 13, 57, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[92521, 2384, 130, 2, 37, "Input"], Cell[92654, 2388, 723, 22, 83, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[93426, 2416, 104, 1, 82, "Section"], Cell[93533, 2419, 987, 13, 33, "Text"], Cell[CellGroupData[{ Cell[94545, 2436, 473, 14, 56, "Input"], Cell[95021, 2452, 1718, 40, 57, "Output"] }, Open ]], Cell[96754, 2495, 334, 11, 26, "Item"], Cell[CellGroupData[{ Cell[97113, 2510, 961, 17, 39, "Input"], Cell[98077, 2529, 2137, 55, 61, "Output"] }, Open ]], Cell[100229, 2587, 1027, 14, 33, "Text"], Cell[CellGroupData[{ Cell[101281, 2605, 479, 10, 37, "Input"], Cell[101763, 2617, 1062, 18, 37, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[102862, 2640, 772, 17, 39, "Input"], Cell[103637, 2659, 1777, 44, 57, "Output"] }, Open ]], Cell[105429, 2706, 1181, 16, 33, "Text"], Cell[CellGroupData[{ Cell[106635, 2726, 569, 13, 37, "Input"], Cell[107207, 2741, 2890, 78, 57, "Output"] }, Open ]], Cell[110112, 2822, 1179, 16, 33, "Text"], Cell[CellGroupData[{ Cell[111316, 2842, 352, 9, 37, "Input"], Cell[111671, 2853, 1545, 36, 37, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[113253, 2894, 374, 9, 37, "Input"], Cell[113630, 2905, 1591, 38, 57, "Output"] }, Open ]], Cell[115236, 2946, 1142, 16, 33, "Text"], Cell[CellGroupData[{ Cell[116403, 2966, 844, 21, 56, "Input"], Cell[117250, 2989, 1928, 43, 57, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[119215, 3037, 448, 10, 37, "Input"], Cell[119666, 3049, 2404, 58, 61, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[122107, 3112, 838, 19, 37, "Input"], Cell[122948, 3133, 1873, 47, 57, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[124858, 3185, 569, 13, 37, "Input"], Cell[125430, 3200, 2980, 81, 57, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[128447, 3286, 404, 10, 37, "Input"], Cell[128854, 3298, 1583, 36, 37, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[130474, 3339, 468, 10, 37, "Input"], Cell[130945, 3351, 1605, 38, 57, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[132587, 3394, 204, 5, 37, "Input"], Cell[132794, 3401, 793, 23, 57, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[133624, 3429, 401, 8, 37, "Input"], Cell[134028, 3439, 1042, 30, 37, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[135107, 3474, 178, 4, 37, "Input"], Cell[135288, 3480, 620, 17, 57, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[135945, 3502, 215, 5, 37, "Input"], Cell[136163, 3509, 443, 8, 37, "Output"] }, Open ]], Cell[136621, 3520, 1213, 17, 33, "Text"], Cell[CellGroupData[{ Cell[137859, 3541, 600, 16, 60, "Input"], Cell[138462, 3559, 311, 9, 58, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[138810, 3573, 313, 8, 39, "Input"], Cell[139126, 3583, 3587, 100, 62, "Output"] }, Open ]], Cell[142728, 3686, 214, 6, 57, "Input"], Cell[CellGroupData[{ Cell[142967, 3696, 1157, 31, 57, "Input"], Cell[144127, 3729, 951, 30, 56, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[145115, 3764, 618, 10, 37, "Input"], Cell[145736, 3776, 2517, 67, 57, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[148290, 3848, 317, 9, 37, "Input"], Cell[148610, 3859, 2802, 88, 95, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[151449, 3952, 281, 8, 37, "Input"], Cell[151733, 3962, 1340, 42, 37, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[153110, 4009, 242, 6, 37, "Input"], Cell[153355, 4017, 1397, 42, 58, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[154789, 4064, 307, 6, 37, "Input"], Cell[155099, 4072, 989, 30, 56, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[156125, 4107, 206, 5, 37, "Input"], Cell[156334, 4114, 829, 27, 56, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[157200, 4146, 282, 7, 37, "Input"], Cell[157485, 4155, 923, 28, 37, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[158445, 4188, 213, 5, 37, "Input"], Cell[158661, 4195, 900, 26, 83, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[159598, 4226, 249, 7, 37, "Input"], Cell[159850, 4235, 772, 23, 83, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[160659, 4263, 247, 6, 37, "Input"], Cell[160909, 4271, 284, 6, 37, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[161230, 4282, 508, 15, 79, "Input"], Cell[161741, 4299, 2147, 43, 251, "Output"] }, Open ]] }, Open ]] }, Open ]] } ] *) (* End of internal cache information *)