(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 68377, 2457]*) (*NotebookOutlinePosition[ 69327, 2487]*) (* CellTagsIndexPosition[ 69283, 2483]*) (*WindowFrame->Normal*) Notebook[{ Cell["Rep.nb", "Title"], Cell["\<\ A notebook for calculations of central functions of \ SL(2,C)-character varieties using representative functions.\ \>", "Subtitle"], Cell["W.Goldman, 16 December 2003", "Subsubtitle"], Cell[CellGroupData[{ Cell["Preamble:", "Section"], Cell[CellGroupData[{ Cell["Turn off the annoying warning messages:", "Text"], Cell[BoxData[{ \(Off[General::"\"]\), "\n", \(Off[General::"\"]\)}], "Input"], Cell["A sample matrix:", "Text"], Cell[BoxData[ \(\(Mm = {{a, b}, {c, d}};\)\)], "Input"] }, Open ]] }, Open ]], Cell["Here are the heads for monomials in the symmetric algebra:", "Text"], Cell[BoxData[{ \(\(Protect[pp, PP, qq, QQ];\)\), "\n", \(HHeadQ[x_]\ := \ MemberQ[SymHeads = {pp, qq, QQ, PP}, x]\)}], "Input"], Cell[CellGroupData[{ Cell["The Fricke slice ", "Section"], Cell[CellGroupData[{ Cell["\<\ Here are the two elements of SL(2,C) which realize the character \ (x,y,z). We work in the quadratic extension C[x, y, z1, z2] of C[x,y,z] \ with embedding given by \t1 = z1 + z2 \tz = z1 z2\ \>", "Subsection"], Cell[BoxData[ \(Xx\ = \ {{x, \(-1\)}, {1, 0}}; \ Yy\ = \ {{0, z1}, {\(-1\) z2, y}};\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Map[MatrixForm, GeneratorList = {Xx, \ Yy, \ Zz\ = \ Xx . Yy}]\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"x", \(-1\)}, {"1", "0"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]], ",", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "z1"}, {\(-z2\), "y"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]], ",", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"z2", \(\(-y\) + x\ z1\)}, {"0", "z1"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]}], "}"}]], "Output"] }, Open ]], Cell["These three matrices have the desired traces:", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(Map[Tr, GeneratorList] /. {z1\ z2\ \[Rule] 1, \ z2\ \[Rule] \ z - z1}\)], "Input"], Cell[BoxData[ \({x, y, z}\)], "Output"] }, Open ]] }, Open ]] }, Open ]], Cell["\<\ The function zPoly (defined below) implements this substitution to \ put all polynomials in terms of the variables x, y, and z.\ \>", \ "Subsubsection"], Cell[CellGroupData[{ Cell["Bilinear and linear mappings", "Section"], Cell["\<\ One basic operation is tensor product (\[CircleTimes]). \ Bilinearity is its defining property:\ \>", "Subsection"], Cell[CellGroupData[{ Cell["Make it bilinear:", "Subsubsection"], Cell[BoxData[{ \(TensorProduct[X_\ + \ Y_, Z_\ ]\ := \ TensorProduct[X, Z]\ + \ TensorProduct[Y, Z]\), "\n", \(TensorProduct[X_, Y_\ + \ Z_]\ := \ TensorProduct[X, Y]\ + \ TensorProduct[X, Z]\), "\n", \(TensorProduct[s_Integer\ \ X_, Y_]\ := \ s\ TensorProduct[X, Y]\), "\n", \(TensorProduct[s_Rational\ \ X_, Y_]\ := \ s\ TensorProduct[X, Y]\), "\[IndentingNewLine]", \(TensorProduct[X_, s_Integer\ Y_]\ := \ s\ TensorProduct[X, Y]\), "\[IndentingNewLine]", \(TensorProduct[X_, s_Rational\ Y_]\ := \ s\ TensorProduct[X, Y]\)}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Use this function to display tensor products using \ \[CircleTimes]:\ \>", "Subsubsection"], Cell[BoxData[ \(DisplayT[x_] := x /. \((TensorProduct \[Rule] CircleTimes)\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Example:", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(TensorProduct[4\ x\ + \ 5\ y, \ 2\ z\ - 3\ w]\)], "Input"], Cell[BoxData[ \(\(-12\)\ TensorProduct[x, w] + 8\ TensorProduct[x, z] - 15\ TensorProduct[y, w] + 10\ TensorProduct[y, z]\)], "Output"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(TensorProduct[4\ x\ + \ 5\ y, \ 2\ z\ - 3\ w] // DisplayT\)], "Input"], Cell[BoxData[ \(\(-12\)\ x\[CircleTimes]w + 8\ x\[CircleTimes]z - 15\ y\[CircleTimes]w + 10\ y\[CircleTimes]z\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["The", FontWeight->"Plain"], " ", StyleBox["bilinear map", FontWeight->"Plain"], " ScalarProduct ", StyleBox["implements duality", FontWeight->"Plain"], ":" }], "Subsection"], Cell[CellGroupData[{ Cell[BoxData[{ \(ScalarProduct[X_\ + \ Y_, Z_\ ]\ := \ ScalarProduct[X, Z]\ + \ ScalarProduct[Y, Z]\), "\n", \(ScalarProduct[X_, Y_\ + \ Z_]\ := \ ScalarProduct[X, Y]\ + \ ScalarProduct[X, Z]\), "\n", \(ScalarProduct[s_Rational\ \ X_, Y_]\ := \ s\ ScalarProduct[X, Y]\), "\n", \(ScalarProduct[X_, s_Rational\ Y_]\ := \ s\ ScalarProduct[X, Y]\), "\n", \(ScalarProduct[s_Integer\ \ X_, Y_]\ := \ s\ ScalarProduct[X, Y]\), "\n", \(ScalarProduct[X_, s_Integer\ Y_]\ := \ s\ ScalarProduct[X, Y]\)}], "Input"], Cell[BoxData[ \(ScalarProduct[TensorProduct[a1_, b1_], TensorProduct[a2_, b2_]] := ScalarProduct[a1, a2] ScalarProduct[b1, b2]\)], "Input"], Cell[BoxData[{ \(ScalarProduct[s_\ \ f_[a___], Y_]\ := \ s\ ScalarProduct[f[a], Y] /; HHeadQ[f]\), "\[IndentingNewLine]", \(ScalarProduct[X_, s_\ f_[a___]]\ := \ s\ ScalarProduct[X, f[a]] /; HHeadQ[f]\)}], "Input"], Cell[TextData[{ "ScalarProduct implements the pairing between the symmetric power of a \ vector space and the symmetric power\nof its dual. The ", Cell[BoxData[ \(TraditionalForm\`\(\(i\^th\)\(\ \)\)\)]], "basic monomial ", Cell[BoxData[ \(TraditionalForm\`\[Phi]\_i\)]], " in the ", Cell[BoxData[ \(TraditionalForm\`m\^th\)]], " symmetric power ", Cell[BoxData[ \(TraditionalForm\`V\_m\)]], " is the symmetrization of the ", Cell[BoxData[ \(TraditionalForm\`\(\(i\^th\)\(\ \)\)\)]], "basic monomial in the ", Cell[BoxData[ \(TraditionalForm\`m\^th\)]], " tensor power ", Cell[BoxData[ \(TraditionalForm\`\(\[CircleTimes]\^m V\)\)]], ". In ", StyleBox["Mathematica", FontSlant->"Italic"], " we denote this element by pp[i,m]." }], "Subsubsection"], Cell[TextData[{ "Similarly, the dual (", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[CircleTimes]\^m V\)\()\)\)\^*\)\ canonically\ identifies\ with\ \(\ \[CircleTimes]\^m\)\((\(V\^*\))\)\)]], ", we obtain a corresponding basis \[CapitalPhi]", Cell[BoxData[ \(TraditionalForm\`\(\(\_\(\(0\)\(,\)\(\ \)\(...\)\(,\)\(\ \)\)\)\(\ \[CapitalPhi]\_\(\(m\)\(\ \)\)\)\)\)]], "of ", Cell[BoxData[ \(TraditionalForm\`\(\[CircleTimes]\^m\((\(V\^*\))\)\)\)]], "\nobtained by symmetrizing basic monomials in ", Cell[BoxData[ \(TraditionalForm\`\(\[CircleTimes]\^m\((\(V\^*\))\)\)\)]], " (denoted in ", StyleBox["Mathematica", FontSlant->"Italic"], " by PP[0,m], ..., PP[m,m]). However these bases are not dual, due to the \ symmetrizations: The scalar product of ", Cell[BoxData[ \(TraditionalForm\`\[Phi]\_i\)]], "with ", Cell[BoxData[ \(TraditionalForm\`\[CapitalPhi]\_\(\(j\)\(\ \)\)\)]], "equals ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[ RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ {"m"}, {"i"} }], "\[NegativeThinSpace]", ")"}], "TraditionalForm"], \(\[Delta]\_ij\)}], TraditionalForm]]], "." }], "Subsubsection"], Cell[BoxData[{ \(ScalarProduct[pp[i_, m_], PP[j_, n_]] := If[i \[Equal] j && m \[Equal] n, 1/Binomial[m, i], 0]\), "\[IndentingNewLine]", \(ScalarProduct[qq[i_, m_], QQ[j_, n_]] := If[i \[Equal] j && m \[Equal] n, 1/Binomial[m, i], 0]\)}], "Input"], Cell[CellGroupData[{ Cell["GL(2) induces linear maps on the symmetric powers:", "Subsection"], Cell[BoxData[{ \(InducedMap[Mat_, X_\ + \ Y_\ ]\ := \ InducedMap[Mat, X\ ] + \ InducedMap[Mat, \ Y]\), "\n", \(InducedMap[Mat_, s_Integer\ X_]\ := \ s\ InducedMap[Mat, X]\), "\n", \(InducedMap[Mat_, s_Rational\ X_]\ := \ s\ InducedMap[Mat, X]\)}], "Input"] }, Open ]], Cell["Internal tensor multiplication", "Subsection"], Cell[CellGroupData[{ Cell["\<\ The next function implement internal multiplication of symmetric \ tensors:\ \>", "Subsubsection"], Cell[BoxData[ \(TensorTimes[TensorProduct[a_[i1_, m1_], b_[j1_, n1_]], TensorProduct[a_[i2_, m2_], b_[j2_, n2_]]] := \[IndentingNewLine]TensorProduct[ a[i1 + i2, m1 + m2], b[j1 + j2, n1 + n2]] /; HHeadQ[a] && HHeadQ[b]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(TensorTimes[TensorProduct[pp[0, 0], QQ[1, 1]], TensorProduct[pp[3, 3], QQ[4, 4]]] // DisplayT\)], "Input"], Cell[BoxData[ \(pp[3, 3]\[CircleTimes]QQ[5, 5]\)], "Output"] }, Open ]], Cell["Make it bilinear:", "Subsubsection"], Cell[BoxData[{ \(TensorTimes[X_ + Y_, Z_] := TensorTimes[X, Z] + TensorTimes[Y, Z]\), "\[IndentingNewLine]", \(TensorTimes[X_, Y_ + Z_]\ := TensorTimes[X, Y] + TensorTimes[X, Z]\), "\[IndentingNewLine]", \(TensorTimes[s_\ X_, \ Y_]\ := \ s\ TensorTimes[X, Y]\), "\[IndentingNewLine]", \(TensorTimes[X_, s_\ \ Y_]\ := \ s\ TensorTimes[X, Y]\)}], "Input"] }, Open ]], Cell["Later on, we will want the following function to be linear:", \ "Subsection"], Cell[BoxData[{ \(EvalMatCoeff[X_ + Y_, {pMatrix_, qMatrix_}] := EvalMatCoeff[X, {pMatrix, qMatrix}] + EvalMatCoeff[Y, {pMatrix, qMatrix}]\), "\[IndentingNewLine]", \(EvalMatCoeff[s_Integer\ \ X_, {pMatrix_, qMatrix_}] := \ s\ EvalMatCoeff[X, {pMatrix, qMatrix}]\), "\[IndentingNewLine]", \(EvalMatCoeff[s_Rational\ X_, {pMatrix_, qMatrix_}] := s\ EvalMatCoeff[X, {pMatrix, qMatrix}]\)}], "Input"], Cell["Symmetric powers", "Section"], Cell[TextData[{ "The function SymmetricPower returns the action of an element of GL(2) on \ the ", Cell[BoxData[ \(TraditionalForm\`m\^th\)]], " symmetric power ", Cell[BoxData[ \(TraditionalForm\`V\_m\)]], ". First we compute the induced map on the basis pp[i,m] of ", Cell[BoxData[ \(TraditionalForm\`\(\(V\_m\)\(,\)\(\ \)\)\)]], "and then we use the basis PP[i,m] of ", Cell[BoxData[ \(TraditionalForm\`\((\(V\^*\))\)\_m\)]], " to compute matrix coefficients." }], "Subsection"], Cell[CellGroupData[{ Cell["\<\ The next function provides a list corresponding to the basis of the \ symmetric monomials:\ \>", "Subsubsection"], Cell[BoxData[ \(LList[f_, m_] := Map[f[#, m] &, Range[0, m]]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(LList[pp, 3]\)], "Input"], Cell[BoxData[ \({pp[0, 3], pp[1, 3], pp[2, 3], pp[3, 3]}\)], "Output"] }, Open ]] }, Open ]], Cell[TextData[{ "The function supplied by ", StyleBox["Mathematica", FontSlant->"Italic"], " for obtaining the coefficients of a polynomial\nis inadequate if the \ polynomial has missing terms:" }], "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \({CoefficientList[0\ x^4\ + \ 2\ x^3\ + \ 3\ x\ \ , x], CoefficientList[0\ x^4\ + \ 2\ x^3\ + \ 3\ x\ + \ 4, x], CoefficientList[x^4\ + \ 2\ x^3\ + \ 3\ x\ + \ 4, x]}\)], "Input"], Cell[BoxData[ \({{0, 3, 0, 2}, {4, 3, 0, 2}, {4, 3, 0, 2, 1}}\)], "Output"] }, Open ]], Cell[BoxData[ \(NewCoefficientList[f_, xxx_, n_] := \ Map[Coefficient[xxx\ f, xxx^#] &, Range[1, n + 1]]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(NewCoefficientList[0\ x^4\ + \ 2\ \ x^3\ + \ 3 x, x, 4]\)], "Input"], Cell[BoxData[ \({0, 3, 0, 2, 0}\)], "Output"] }, Open ]], Cell["\<\ The next function computes the map induced on a symmetric power by \ a 2x2 matrix Mat:\ \>", "Subsubsection"], Cell[BoxData[ \(InducedMap[Mat_, f_[i_, m_]] := Inner[Times, NewCoefficientList[ Expand[\((\((Mat . {1, xx})\)[\([1]\)])\)^\((m - i)\) \((\((Mat . {1, xx})\)[\([2]\)])\)^i], xx, m], Map[f[#, m] &, Range[0, m]]]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(InducedMap[Mm, pp[0, 2]]\)], "Input"], Cell[BoxData[ \(a\^2\ pp[0, 2] + 2\ a\ b\ pp[1, 2] + b\^2\ pp[2, 2]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(InducedMap[Mm, pp[1, 1]]\)], "Input"], Cell[BoxData[ \(c\ pp[0, 1] + d\ pp[1, 1]\)], "Output"] }, Open ]], Cell[TextData[{ "The next function gives the symmetric power in terms of the bases of ", Cell[BoxData[ \(TraditionalForm\`\(\(V\_m\)\(\ \)\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\((\(V\^*\))\)\_m\)]], "." }], "Subsubsection"], Cell[BoxData[ \(SymmetricPower[Mat_, m_] := Outer[ScalarProduct, Map[InducedMap[Mat, #] &, LList[pp, m]], LList[PP, m]]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Map[MatrixForm[SymmetricPower[Mm, #]] &, Range[0, 3]]\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]], ",", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"a", "b"}, {"c", "d"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]], ",", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(a\^2\), \(a\ b\), \(b\^2\)}, {\(a\ c\), \(1\/2\ \((b\ c + a\ d)\)\), \(b\ d\)}, {\(c\^2\), \(c\ d\), \(d\^2\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]], ",", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(a\^3\), \(a\^2\ b\), \(a\ b\^2\), \(b\^3\)}, {\(a\^2\ c\), \(1\/3\ \((2\ a\ b\ c + a\^2\ d)\)\), \(1\/3\ \((b\^2\ c + 2\ a\ b\ d)\)\), \(b\^2\ d\)}, {\(a\ c\^2\), \(1\/3\ \((b\ c\^2 + 2\ a\ c\ d)\)\), \(1\/3\ \((2\ b\ c\ d + a\ d\^2)\)\), \(b\ d\^2\)}, {\(c\^3\), \(c\^2\ d\), \(c\ d\^2\), \(d\^3\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]}], "}"}]], "Output"] }, Open ]], Cell[TextData[{ "Recall that the ", Cell[BoxData[ \(TraditionalForm\`\((i, j)\)\^th\)]], " entry ", Cell[BoxData[ \(TraditionalForm\`M\_\(\(i\)\(\ \)\(j\)\(\ \)\)\ = \ M[\([i, j]\)]\)]], " ", Cell[BoxData[ \(TraditionalForm\`\((\(i\^\(\(th\)\(\ \)\)\) row, \ j\^\(\(th\)\(\ \)\)\ column)\)\)]], " of a matrix M equals the ", Cell[BoxData[ \(TraditionalForm\`i\^\(\(th\)\(\ \)\)\)]], "basic covector evaluated on the image of the ", Cell[BoxData[ \(TraditionalForm\`j\^\(\(th\)\(\ \)\)\)]], " basis vector under M." }], "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(\((P2 = {0, 1})\) . MatrixForm[Mm] . MatrixForm[p1]\ == MatrixForm[P2 . Mm . \((p1 = {1, 0})\)]\)], "Input"], Cell[BoxData[ RowBox[{ RowBox[{\({0, 1}\), ".", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"a", "b"}, {"c", "d"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]], ".", TagBox["p1", Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]}], "\[Equal]", TagBox["c", Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Mm[\([2, 1]\)]\)], "Input"], Cell[BoxData[ \(c\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["The Clebsch-Gordan Summand", "Section"], Cell[CellGroupData[{ Cell[TextData[{ "Suppose integers m, n, and r satisfy 0 \[LessEqual] m, n, r. Then the \ representation ", Cell[BoxData[ \(TraditionalForm\`\(\(V\_r\)\(\ \)\)\)]], "includes into ", Cell[BoxData[ \(TraditionalForm\`V\_m\)]], Cell[BoxData[ \(TraditionalForm\`\[CircleTimes]\)]], " ", Cell[BoxData[ \(TraditionalForm\`V\_n\)]], " if and only if |m-n| \[LessEqual] r \[LessEqual] m+n and m+n-r even. \ In that case the inclusion is\nunique, and the Clebsch-Gordan summand ", Cell[BoxData[ \(TraditionalForm\`V\_r\%\(m, n\)\)], "Input"], " is the image of the inclusion\nof the representation ", Cell[BoxData[ \(TraditionalForm\`V\_r\)]], " into ", Cell[BoxData[ \(TraditionalForm\`V\_m\)]], Cell[BoxData[ \(TraditionalForm\`\[CircleTimes]\)]], " ", Cell[BoxData[ \(TraditionalForm\`V\_n\)]], ". " }], "Subsection"], Cell["The following tests whether a summand exists.", "Subsubsection"], Cell[BoxData[ \(ClebschGordanQ[{m_, n_, r_}] := IntegerQ[m] && IntegerQ[n] && IntegerQ[r] && m \[GreaterEqual] 0 && n \[GreaterEqual] 0 && r \[GreaterEqual] 0 && \ IntegerQ[\((m + n - r)\)/2] && \ \((r \[LessEqual] \ m + n)\)\ && \ Abs[m - n] \[LessEqual] r\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(ClebschGordanQ[{2, 2, 2}]\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]] }, Open ]] }, Open ]], Cell["\<\ The following is the syntax for including summands in the tensor \ product\ \>", "Subsubsection"], Cell[BoxData[{ \(IncludeSummand[{m_, n_, r_}, k_] := IncludeSummand[{m, n, r}, k, {pp, qq}] /; ClebschGordanQ[{m, n, r}] && \ 0 \[LessEqual] k && k \[LessEqual] r\), "\n", \(IncludeSummand[{m_, n_, r_}] := Map[IncludeSummand[{m, n, r}, #] &, Range[0, r]]\)}], "Input"], Cell[TextData[{ "Denote the basis of ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FormBox[\(\(V\_m\)\(\ \)\(by\)\(\ \)\), "TraditionalForm"], \(\[CurlyPhi]\_0\)}], " ", ",", " ", "...", ",", " ", \(\[CurlyPhi]\_m\), " ", ",", " ", \(\(the\)\(\ \)\(basis\)\(\ \)\(of\)\(\ \)\(V\_n\)\(\ \)\(by\)\ \(\ \)\)}], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`\[Psi]\_\(\(0\)\(,\)\(\ \)\) ... , \ \[Psi]\_n\ \ \ and\ the\ basis\ of\)]], " ", Cell[BoxData[ \(TraditionalForm\`\(\(V\_r\)\(\ \)\(by\)\(\ \)\)\)]], "\n", Cell[BoxData[ \(TraditionalForm\`\[Tau]\_\(\(0\)\(,\)\(\ \)\)\ ... , \ \(\(\[Tau]\_r\ \)\(.\)\(\ \ \)\)\)]], " If r = m + n, the inclusion ", Cell[BoxData[ \(TraditionalForm\`V\_r\)]], " into ", Cell[BoxData[ \(TraditionalForm\`V\_m\)]], Cell[BoxData[ \(TraditionalForm\`\[CircleTimes]\)]], " ", Cell[BoxData[ \(TraditionalForm\`V\_n\)]], " is given by\n\n ", Cell[BoxData[ FormBox[ RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ {\(m + n\)}, {"k"} }], "\[NegativeThinSpace]", ")"}], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`\[Tau]\_\(\(k\)\(\ \ \ \ \ \ \ \)\)\)]], "\[RightTeeArrow]", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ UnderscriptBox[ StyleBox["\[Sum]", FontSize->48], \(\(\ \)\(0 \[LessEqual] \ i \[LessEqual] m\[IndentingNewLine] \ 0 \[LessEqual] j \[LessEqual] n\[IndentingNewLine] \ i + j = k\)\)], " "}]}], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ {"m"}, {"i"} }], "\[NegativeThinSpace]", ")"}]}], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\` \[CurlyPhi]\_i\)]], " \[CircleTimes] ", Cell[BoxData[ FormBox[ RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ {"n"}, {"j"} }], "\[NegativeThinSpace]", ")"}], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`\[Psi]\_\(\(j\)\(\ \)\)\)]] }], "Subsection"], Cell[BoxData[ \(IncludeSummand[{m_, n_, r_}, k_, {a_, b_}] := Apply[Plus, Map[\[IndentingNewLine]Binomial[m, #] Binomial[n, k - #]/Binomial[m + n, k]\ TensorProduct[a[#, m], b[k - #, n]]\[IndentingNewLine] &, Range[0, m + n]]] /; ClebschGordanQ[{m, n, r}] && \((r == m + n)\) && HHeadQ[a] && HHeadQ[b]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(IncludeSummand[{1, 1, 2}] // DisplayT\) // MatrixForm\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(pp[0, 1]\[CircleTimes]qq[0, 1]\)}, {\(1\/2\ pp[0, 1]\[CircleTimes]qq[1, 1] + 1\/2\ pp[1, 1]\[CircleTimes]qq[0, 1]\)}, {\(pp[1, 1]\[CircleTimes]qq[1, 1]\)} }], "\[NoBreak]", ")"}], MatrixForm[ { CircleTimes[ pp[ 0, 1], qq[ 0, 1]], Plus[ Times[ Rational[ 1, 2], CircleTimes[ pp[ 0, 1], qq[ 1, 1]]], Times[ Rational[ 1, 2], CircleTimes[ pp[ 1, 1], qq[ 0, 1]]]], CircleTimes[ pp[ 1, 1], qq[ 1, 1]]}]]], "Output"] }, Open ]], Cell[TextData[{ "In general the inclusion ", Cell[BoxData[ \(TraditionalForm\`V\_r\)]], " into ", Cell[BoxData[ \(TraditionalForm\`V\_m\)]], Cell[BoxData[ \(TraditionalForm\`\[CircleTimes]\)]], " ", Cell[BoxData[ \(TraditionalForm\`V\_n\)]], " is given by\n ", Cell[BoxData[ FormBox[ RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ {"r"}, {"k"} }], "\[NegativeThinSpace]", ")"}], TraditionalForm]]], Cell[BoxData[ FormBox[ StyleBox[\(\[Tau]\_\(\(k\)\(\ \)\)\), FontSize->24], TraditionalForm]]], "\[RightTeeArrow] ", StyleBox["(", FontSize->48, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ RowBox[{" ", UnderscriptBox[ StyleBox["\[Sum]", FontSize->48], \(\(\ \)\(0 \[LessEqual] \ i \[LessEqual] m - s\[IndentingNewLine] \ 0 \[LessEqual] j \[LessEqual] n - s\[IndentingNewLine] \ i + j = k\)\)]}], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ {\(m - s\)}, {"i"} }], "\[NegativeThinSpace]", ")"}]}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{" ", StyleBox[\(\[CurlyPhi]\_i\), FontSize->24]}], TraditionalForm]]], " \[CircleTimes] ", Cell[BoxData[ FormBox[ RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ {\(n - s\)}, {"j"} }], "\[NegativeThinSpace]", ")"}], TraditionalForm]]], Cell[BoxData[ FormBox[ StyleBox[\(\[Psi]\_\(\(j\)\(\ \)\)\), FontSize->24], TraditionalForm]]], StyleBox[")", FontSize->48, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox[" ", FontSize->16], Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{ FormBox[ RowBox[{" ", StyleBox[\(\[CurlyPhi]\_0\), FontSize->24]}], "TraditionalForm"], "\[CircleTimes]", " ", FormBox[ StyleBox[\(\[Psi]\_\(\(1\)\(\ \)\)\), FontSize->24], "TraditionalForm"]}], "-", " ", RowBox[{ FormBox[ RowBox[{" ", StyleBox[\(\[CurlyPhi]\_1\), FontSize->24]}], "TraditionalForm"], "\[CircleTimes]", " ", FormBox[ StyleBox[\(\[Psi]\_\(\(o\)\(\ \)\)\), FontSize->24], "TraditionalForm"]}]}], ")"}], "s"], TraditionalForm]], FontSize->16], "\n where 0 \[LessEqual] k \[LessEqual] r, s = (m + n - r) /2 and \ multiplication is internal mutiplication of tensors.", StyleBox[" ", FontSize->48] }], "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(SymDetTensor[{a_, b_}] := TensorProduct[a[0, 1], b[1, 1]] - TensorProduct[a[1, 1], b[0, 1]] /; HHeadQ[a] && HHeadQ[b]\)], "Input"], Cell[BoxData[ \(TensorTimesSymDetTensor[t_, n_, {a_, b_}]\ := \ Nest[TensorTimes[SymDetTensor[{a, b}], #] &, t, n] /; HHeadQ[a] && HHeadQ[b]\)], "Input"], Cell[BoxData[ \(IncludeSummand[{m_, n_, r_}, k_, {a_, b_}] := Module[{s\ = \ \((m + n\ - r)\)/ 2}, \ \[IndentingNewLine]TensorTimesSymDetTensor[ IncludeSummand[{m - s, n - s, r}, k, {a, b}\ ], s, {a, b}]\ ] /; HHeadQ[a] && HHeadQ[b]\)], "Input"] }, Open ]], Cell["Including the Dual Summand", "Section"], Cell[BoxData[ \(DualSummandsInnerProduct[{m_, n_, r_}, k_] := ScalarProduct[IncludeSummand[{m, n, r}, k], IncludeSummand[{m, n, r}, k, {PP, QQ}]]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Map[DualSummandsInnerProduct[{2, 1, 3}, #] &, Range[0, 3]]\)], "Input"], Cell[BoxData[ \({1, 1\/3, 1\/3, 1}\)], "Output"] }, Open ]], Cell[BoxData[ \(IncludeDualSummand[{m_, n_, r_}] := Map[IncludeDualSummand[{m, n, r}, #] &, Range[0, r]]\)], "Input"], Cell[BoxData[ \(IncludeDualSummand[{m_, n_, r_}, k_] := Expand[IncludeSummand[{m, n, r}, k, {PP, QQ}]/ DualSummandsInnerProduct[{m, n, r}, k]]\)], "Input"], Cell[BoxData[ \(IncludeDualSummand[{m_, n_, r_}] := Map[IncludeDualSummand[{m, n, r}, #] &, Range[0, r]]\)], "Input"], Cell["Computing Central Functions", "Section"], Cell[BoxData[ \(EvalMatCoeff[ TensorProduct[TensorProduct[pp[i1_, m_], qq[j1_, n_]], TensorProduct[PP[i2_, m_], QQ[j2_, n_]]\ \[IndentingNewLine]]] := \(SymmetricPower[Xx, m]\)[\([i2 + 1, i1 + 1]\)]\ \(SymmetricPower[Yy, n]\)[\([j2 + 1, j1 + 1]\)]\)], "Input"], Cell[BoxData[ \(EvalMatCoeff[ TensorProduct[TensorProduct[pp[i1_, m_], qq[j1_, n_]], TensorProduct[PP[i2_, m_], QQ[j2_, n_]]\ \[IndentingNewLine]], {pMatrix_, qMatrix_}] := pMatrix[\([i2 + 1, i1 + 1]\)]\ qMatrix[\([j2 + 1, j1 + 1]\)]\)], "Input"], Cell[BoxData[ \(pqMatrix[{m_, n_, r_}] := {SymmetricPower[Xx, m], SymmetricPower[Yy, n]}\)], "Input"], Cell["\<\ The sample matrices X, Y are defined over the quadratic extension \ of the polynomial ring C[x,y,z] generated by z1, z2 such that z1 z2 = 1 and z1 + z2 = z. The \ routine zPoly expresses a polynomial in x, y, z1, z2 which is symmetric in z1 \ and z2 as a polynomial in x, y, z.\ \>", "Subsection"], Cell["The next routine gives the term without any z's.", "Subsubsection"], Cell[BoxData[ \(ConstantTerm[f_] := Coefficient[Coefficient[f, z1, 0], z2, 0]\)], "Input"], Cell[TextData[{ "The ", Cell[BoxData[ \(TraditionalForm\`\(\(n\^th\)\(\ \)\)\)]], "Chebychev polynomial Cheb[n,z] expresses ", Cell[BoxData[ \(TraditionalForm\`\((z1)\)\^n\ + \ \((z2)\)\^n\ as\ a\ polynomial\ in\ \ z\ = \ z1\ + \ z\)]], "2 where z1 z2 = 1." }], "Subsubsection"], Cell[BoxData[{ \(\(Cheb[0, z_] = 2;\)\), "\[IndentingNewLine]", \(\(Cheb[1, z_]\ = \ z;\)\), "\[IndentingNewLine]", \(Cheb[n_, z_]\ := \ Expand[z\ Cheb[n - 1, z]\ - \ Cheb[n - 2, z]]\)}], "Input"], Cell[BoxData[ \(zPoly[f_] := \ Module[{f1 = f /. {\ z2\ \[Rule] \ 1/z1}}, Expand[ConstantTerm[f1] + Apply[Plus, Map[Cheb[#, z]\ Coefficient[f1, z1, #] &, Range[Exponent[f, z1]]]]]]\)], "Input"], Cell[CellGroupData[{ Cell["Put it all together:", "Subsection"], Cell[BoxData[ \(Casimir[{m_, n_, r_}] := Inner[TensorProduct, IncludeSummand[{m, n, r}], IncludeDualSummand[{m, n, r}]]\)], "Input"], Cell[BoxData[ \(CentralFunction[{m_, n_, r_}] := zPoly[EvalMatCoeff[Casimir[{m, n, r}], pqMatrix[{m, n, r}]]]\)], "Input"], Cell[BoxData[ \(Central[m_, n_, r_] := CentralFunction[{m, n, r}]\)], "Input"] }, Open ]], Cell["Some Calculations:", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \({c000, c101, c011} = {Central[0, 0, 0], Central[1, 0, 1], Central[0, 1, 1]}\)], "Input"], Cell[BoxData[ \({1, x, y}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(({c202, c112, c110, c022} = {Central[2, 0, 2], Central[1, 1, 2], Central[1, 1, 0], Central[0, 2, 2]})\) // MatrixForm\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(-1\) + x\^2\)}, {\(\(x\ y\)\/2 + z\/2\)}, {\(\(x\ y\)\/2 - z\/2\)}, {\(\(-1\) + y\^2\)} }], "\[NoBreak]", ")"}], MatrixForm[ { Plus[ -1, Power[ x, 2]], Plus[ Times[ Rational[ 1, 2], x, y], Times[ Rational[ 1, 2], z]], Plus[ Times[ Rational[ 1, 2], x, y], Times[ Rational[ -1, 2], z]], Plus[ -1, Power[ y, 2]]}]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(({c303, c213, c211, c123, c121, c033} = {Central[3, 0, 3], Central[2, 1, 3], Central[2, 1, 1], Central[1, 2, 3], Central[1, 2, 1], Central[0, 3, 3]})\) // MatrixForm\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(-2\)\ x + x\^3\)}, {\(\(-\(\(2\ y\)\/3\)\) + \(x\^2\ y\)\/3 + \(2\ x\ z\)\/3\)}, {\(\(-\(y\/3\)\) + \(2\ x\^2\ y\)\/3 - \(2\ x\ z\)\/3\)}, {\(\(-\(\(2\ x\)\/3\)\) + \(x\ y\^2\)\/3 + \(2\ y\ z\)\/3\)}, {\(\(-\(x\/3\)\) + \(2\ x\ y\^2\)\/3 - \(2\ y\ z\)\/3\)}, {\(\(-2\)\ y + y\^3\)} }], "\[NoBreak]", ")"}], MatrixForm[ { Plus[ Times[ -2, x], Power[ x, 3]], Plus[ Times[ Rational[ -2, 3], y], Times[ Rational[ 1, 3], Power[ x, 2], y], Times[ Rational[ 2, 3], x, z]], Plus[ Times[ Rational[ -1, 3], y], Times[ Rational[ 2, 3], Power[ x, 2], y], Times[ Rational[ -2, 3], x, z]], Plus[ Times[ Rational[ -2, 3], x], Times[ Rational[ 1, 3], x, Power[ y, 2]], Times[ Rational[ 2, 3], y, z]], Plus[ Times[ Rational[ -1, 3], x], Times[ Rational[ 2, 3], x, Power[ y, 2]], Times[ Rational[ -2, 3], y, z]], Plus[ Times[ -2, y], Power[ y, 3]]}]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[{c404, c044} = {Central[4, 0, 4], Central[0, 4, 4]}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(1 - 3\ x\^2 + x\^4\)}, {\(1 - 3\ y\^2 + y\^4\)} }], "\[NoBreak]", ")"}], MatrixForm[ { Plus[ 1, Times[ -3, Power[ x, 2]], Power[ x, 4]], Plus[ 1, Times[ -3, Power[ y, 2]], Power[ y, 4]]}]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Central[3, 1, 2]\)], "Input"], Cell[BoxData[ \(\(-x\)\ y + \(3\ x\^3\ y\)\/4 + z\/2 - \(3\ x\^2\ z\)\/4\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[{c314, c312} = {Central[3, 1, 4], Central[3, 1, 2]}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(-x\)\ y + \(x\^3\ y\)\/4 - z\/2 + \(3\ x\^2\ z\)\/4\)}, {\(\(-x\)\ y + \(3\ x\^3\ y\)\/4 + z\/2 - \(3\ x\^2\ z\)\/4\)} }], "\[NoBreak]", ")"}], MatrixForm[ { Plus[ Times[ -1, x, y], Times[ Rational[ 1, 4], Power[ x, 3], y], Times[ Rational[ -1, 2], z], Times[ Rational[ 3, 4], Power[ x, 2], z]], Plus[ Times[ -1, x, y], Times[ Rational[ 3, 4], Power[ x, 3], y], Times[ Rational[ 1, 2], z], Times[ Rational[ -3, 4], Power[ x, 2], z]]}]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[{c224, c222, c220} = {Central[2, 2, 4], Central[2, 2, 2], Central[2, 2, 0]}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(1\/3 - x\^2\/2 - y\^2\/2 + \(x\^2\ y\^2\)\/6 + \(2\ x\ y\ z\)\/3 + z\^2\/6\)}, {\(1 - x\^2\/2 - y\^2\/2 + \(x\^2\ y\^2\)\/2 - z\^2\/2\)}, {\(\(-\(1\/3\)\) + \(x\^2\ y\^2\)\/3 - \(2\ x\ y\ z\)\/3 + z\^2\/3\)} }], "\[NoBreak]", ")"}], MatrixForm[ { Plus[ Rational[ 1, 3], Times[ Rational[ -1, 2], Power[ x, 2]], Times[ Rational[ -1, 2], Power[ y, 2]], Times[ Rational[ 1, 6], Power[ x, 2], Power[ y, 2]], Times[ Rational[ 2, 3], x, y, z], Times[ Rational[ 1, 6], Power[ z, 2]]], Plus[ 1, Times[ Rational[ -1, 2], Power[ x, 2]], Times[ Rational[ -1, 2], Power[ y, 2]], Times[ Rational[ 1, 2], Power[ x, 2], Power[ y, 2]], Times[ Rational[ -1, 2], Power[ z, 2]]], Plus[ Rational[ -1, 3], Times[ Rational[ 1, 3], Power[ x, 2], Power[ y, 2]], Times[ Rational[ -2, 3], x, y, z], Times[ Rational[ 1, 3], Power[ z, 2]]]}]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[{c134, c132} = {Central[1, 3, 4], Central[1, 3, 2]}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(-x\)\ y + \(x\ y\^3\)\/4 - z\/2 + \(3\ y\^2\ z\)\/4\)}, {\(\(-x\)\ y + \(3\ x\ y\^3\)\/4 + z\/2 - \(3\ y\^2\ z\)\/4\)} }], "\[NoBreak]", ")"}], MatrixForm[ { Plus[ Times[ -1, x, y], Times[ Rational[ 1, 4], x, Power[ y, 3]], Times[ Rational[ -1, 2], z], Times[ Rational[ 3, 4], Power[ y, 2], z]], Plus[ Times[ -1, x, y], Times[ Rational[ 3, 4], x, Power[ y, 3]], Times[ Rational[ 1, 2], z], Times[ Rational[ -3, 4], Power[ y, 2], z]]}]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Central[0, 5, 5]\)], "Input"], Cell[BoxData[ \(3\ y - 4\ y\^3 + y\^5\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \({\((Map[Central[1, 4, #] &, {5, 3}] // MatrixForm)\), \((Map[Central[2, 3, #] &, {5, 3, 1}] // MatrixForm)\)}\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(3\ x\)\/5 - \(6\ x\ y\^2\)\/5 + \(x\ y\^4\)\/5 - \(6\ y\ \ z\)\/5 + \(4\ y\^3\ z\)\/5\)}, {\(\(2\ x\)\/5 - \(9\ x\ y\^2\)\/5 + \(4\ x\ y\^4\)\/5 + \(6\ \ y\ z\)\/5 - \(4\ y\^3\ z\)\/5\)} }], "\[NoBreak]", ")"}], MatrixForm[ { Plus[ Times[ Rational[ 3, 5], x], Times[ Rational[ -6, 5], x, Power[ y, 2]], Times[ Rational[ 1, 5], x, Power[ y, 4]], Times[ Rational[ -6, 5], y, z], Times[ Rational[ 4, 5], Power[ y, 3], z]], Plus[ Times[ Rational[ 2, 5], x], Times[ Rational[ -9, 5], x, Power[ y, 2]], Times[ Rational[ 4, 5], x, Power[ y, 4]], Times[ Rational[ 6, 5], y, z], Times[ Rational[ -4, 5], Power[ y, 3], z]]}]], ",", InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(3\ y\)\/5 - \(3\ x\^2\ y\)\/5 - \(2\ y\^3\)\/5 + \(x\^2\ \ y\^3\)\/10 - \(3\ x\ z\)\/5 + 3\/5\ x\ y\^2\ z + \(3\ y\ z\^2\)\/10\)}, {\(\(26\ y\)\/15 - \(16\ x\^2\ y\)\/15 - \(3\ y\^3\)\/5 + \(2\ \ x\^2\ y\^3\)\/5 + \(4\ x\ z\)\/15 + 2\/5\ x\ y\^2\ z - \(4\ y\ z\^2\)\/5\)}, {\(\(-\(y\/3\)\) - \(x\^2\ y\)\/3 + \(x\^2\ y\^3\)\/2 + \(x\ \ z\)\/3 - x\ y\^2\ z + \(y\ z\^2\)\/2\)} }], "\[NoBreak]", ")"}], MatrixForm[ { Plus[ Times[ Rational[ 3, 5], y], Times[ Rational[ -3, 5], Power[ x, 2], y], Times[ Rational[ -2, 5], Power[ y, 3]], Times[ Rational[ 1, 10], Power[ x, 2], Power[ y, 3]], Times[ Rational[ -3, 5], x, z], Times[ Rational[ 3, 5], x, Power[ y, 2], z], Times[ Rational[ 3, 10], y, Power[ z, 2]]], Plus[ Times[ Rational[ 26, 15], y], Times[ Rational[ -16, 15], Power[ x, 2], y], Times[ Rational[ -3, 5], Power[ y, 3]], Times[ Rational[ 2, 5], Power[ x, 2], Power[ y, 3]], Times[ Rational[ 4, 15], x, z], Times[ Rational[ 2, 5], x, Power[ y, 2], z], Times[ Rational[ -4, 5], y, Power[ z, 2]]], Plus[ Times[ Rational[ -1, 3], y], Times[ Rational[ -1, 3], Power[ x, 2], y], Times[ Rational[ 1, 2], Power[ x, 2], Power[ y, 3]], Times[ Rational[ 1, 3], x, z], Times[ -1, x, Power[ y, 2], z], Times[ Rational[ 1, 2], y, Power[ z, 2]]]}]]}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Map[Central[2, 3, #] &, {5, 3, 1}] // MatrixForm\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(3\ y\)\/5 - \(3\ x\^2\ y\)\/5 - \(2\ y\^3\)\/5 + \(x\^2\ \ y\^3\)\/10 - \(3\ x\ z\)\/5 + 3\/5\ x\ y\^2\ z + \(3\ y\ z\^2\)\/10\)}, {\(\(26\ y\)\/15 - \(16\ x\^2\ y\)\/15 - \(3\ y\^3\)\/5 + \(2\ \ x\^2\ y\^3\)\/5 + \(4\ x\ z\)\/15 + 2\/5\ x\ y\^2\ z - \(4\ y\ z\^2\)\/5\)}, {\(\(-\(y\/3\)\) - \(x\^2\ y\)\/3 + \(x\^2\ y\^3\)\/2 + \(x\ \ z\)\/3 - x\ y\^2\ z + \(y\ z\^2\)\/2\)} }], "\[NoBreak]", ")"}], MatrixForm[ { Plus[ Times[ Rational[ 3, 5], y], Times[ Rational[ -3, 5], Power[ x, 2], y], Times[ Rational[ -2, 5], Power[ y, 3]], Times[ Rational[ 1, 10], Power[ x, 2], Power[ y, 3]], Times[ Rational[ -3, 5], x, z], Times[ Rational[ 3, 5], x, Power[ y, 2], z], Times[ Rational[ 3, 10], y, Power[ z, 2]]], Plus[ Times[ Rational[ 26, 15], y], Times[ Rational[ -16, 15], Power[ x, 2], y], Times[ Rational[ -3, 5], Power[ y, 3]], Times[ Rational[ 2, 5], Power[ x, 2], Power[ y, 3]], Times[ Rational[ 4, 15], x, z], Times[ Rational[ 2, 5], x, Power[ y, 2], z], Times[ Rational[ -4, 5], y, Power[ z, 2]]], Plus[ Times[ Rational[ -1, 3], y], Times[ Rational[ -1, 3], Power[ x, 2], y], Times[ Rational[ 1, 2], Power[ x, 2], Power[ y, 3]], Times[ Rational[ 1, 3], x, z], Times[ -1, x, Power[ y, 2], z], Times[ Rational[ 1, 2], y, Power[ z, 2]]]}]]], "Output"] }, Open ]], Cell["\<\ Here are some trace polynomials, written as linear combinations of \ central functions:\ \>", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(Map[Expand, {c101, c011, c112 - c110}]\)], "Input"], Cell[BoxData[ \({x, y, z}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Map[ Expand, {2\ c110, c213 - \ 1/2\ c211\ - \ 1/2\ c011, c123\ - \ 1/2\ c121\ - \ 1/2\ c101}]\)], "Input"], Cell[BoxData[ \({x\ y - z, \(-y\) + x\ z, \(-x\) + y\ z}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[\(-c222\)\ + \ 3/2\ \ c220\ + \ 1/2\ c202\ + \ 1/2\ c022\ + \ 1/2]\)], "Input"], Cell[BoxData[ \(\(-2\) + x\^2 + y\^2 - x\ y\ z + z\^2\)], "Output"] }, Open ]] }, Open ]], Cell["New Stuff", "Section"], Cell[CellGroupData[{ Cell[BoxData[ \(c112\ + \ c110\)], "Input"], Cell[BoxData[ \(x\ y\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(c112\)], "Input"], Cell[BoxData[ \(\(x\ y\)\/2 + z\/2\)], "Output"] }, Open ]], Cell[BoxData[ \(ToC[x] = Cent[{1, 0, 1}]; ToC[y] = Cent[{0, 1, 1}]; ToC[x^2] = Cent[{2, 0, 2}] + Cent[{0, 0, 0}]; ToC[1]\ = \ Cent[{0, 0, 0}]; ToC[z] = Cent[{1, 1, 2}] - Cent[{1, 1, 0}]; ToC[y^2]\ = \ Cent[{0, 2, 0}]\ + \ Cent[{0, 0, 0}]; \ ToC[x\ y] = Cent[{1, 1, 2}] + Cent[{1, 1, 0}];\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(c213\ + \ c211\ + \ c011\)], "Input"], Cell[BoxData[ \(x\^2\ y\)], "Output"] }, Open ]], Cell[BoxData[ \(\(ToC[x\ z]\ = \ Cent[{2, 1, 3}]\ - 1/2\ Cent[{2, 1, 1}]\ + \ 1/2\ Cent[{0, 1, 1}];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(ToC[y\ z]\ = \ Cent[{1, 2, 3}]\ - \ 1/2\ Cent[{1, 2, 1}]\ + \ 1/2\ Cent[{1, 0, 1}]\)], "Input"], Cell[BoxData[ \(1\/2\ Cent[{1, 0, 1}] - 1\/2\ Cent[{1, 2, 1}] + Cent[{1, 2, 3}]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \({c110, c220, c330 = Central[3, 3, 0], c440 = Central[4, 4, 0]}\)], "Input"], Cell[BoxData[ \({\(x\ y\)\/2 - z\/2, \(-\(1\/3\)\) + \(x\^2\ y\^2\)\/3 - \(2\ x\ y\ z\)\/3 + z\^2\/3, \(-\(\(x\ y\)\/2\)\) + \(x\^3\ y\^3\)\/4 + z\/2 - 3\/4\ x\^2\ y\^2\ z + 3\/4\ x\ y\ z\^2 - z\^3\/4, 1\/5 - \(3\ x\^2\ y\^2\)\/5 + \(x\^4\ y\^4\)\/5 + \(6\ x\ y\ z\)\/5 - 4\/5\ x\^3\ y\^3\ z - \(3\ z\^2\)\/5 + 6\/5\ x\^2\ y\^2\ z\^2 - 4\/5\ x\ y\ z\^3 + z\^4\/5}\)], "Output"] }, Open ]], Cell[BoxData[ \(RestrictToDiagonal[f_] := Expand[f /. {y \[Rule] x, z \[Rule] x^2 - 2}]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(RestrictToDiagonal[c213]\)], "Input"], Cell[BoxData[ \(\(-2\)\ x + x\^3\)], "Output"] }, Open ]], Cell[BoxData[ \(RestrictToX[f_] := Expand[f /. {y \[Rule] 2, z \[Rule] x}]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Map[RestrictToX, {c110, c112, c213, c211, c123, c121}]\)], "Input"], Cell[BoxData[ \({x\/2, \(3\ x\)\/2, \(-\(4\/3\)\) + \(4\ x\^2\)\/3, \(-\(2\/3\)\) + \(2\ \ x\^2\)\/3, 2\ x, x}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Map[ RestrictToDiagonal, {c110, c112, c213, c211, c123, c121}]\)], "Input"], Cell[BoxData[ \({1, \(-1\) + x\^2, \(-2\)\ x + x\^3, x, \(-2\)\ x + x\^3, x}\)], "Output"] }, Open ]], Cell[BoxData[ \(XRes[m_, n_, r_] := Expand[\ RestrictToX[ Central[m, n, r]]\ - \((r + 1)\)/\((m + 1)\)\ Central[m, 0, m]]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(XRes[5, 5, 6]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \({3\ c123, 3\ c121} // Expand\)], "Input"], Cell[BoxData[ \({\(-2\)\ x + x\ y\^2 + 2\ y\ z, \(-x\) + 2\ x\ y\^2 - 2\ y\ z}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[c011\ c112\ - \ 3/4\ c121]\)], "Input"], Cell[BoxData[ \(x\/4 + y\ z\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[c011\ c112]\)], "Input"], Cell[BoxData[ \(\(x\ y\^2\)\/2 + \(y\ z\)\/2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(c123\)], "Input"], Cell[BoxData[ \(\(-\(\(2\ x\)\/3\)\) + \(x\ y\^2\)\/3 + \(2\ y\ z\)\/3\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[c011\ c112 - 3/2\ c123]\)], "Input"], Cell[BoxData[ \(x - \(y\ z\)\/2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ToC[y\ z]\)], "Input"], Cell[BoxData[ \(1\/2\ Cent[{1, 0, 1}] - 1\/2\ Cent[{1, 2, 1}] + Cent[{1, 2, 3}]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(c101\ - \ c121\ + \ 2\ c123 // Expand\)], "Input"], Cell[BoxData[ \(2\ y\ z\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[ c011\ c112 - 3/2\ c123\ - c101\ + \ 1/4\ c101\ - \ 1/4\ c121\ + \ 1/2\ c123]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[ c011\ c112 - \ c123\ - 3/4\ c101\ \ - \ 1/4\ c121\ ]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[{c011\ c110\ , \ c121}\ ]\)], "Input"], Cell[BoxData[ \({\(x\ y\^2\)\/2 - \(y\ z\)\/2, \(-\(x\/3\)\) + \(2\ x\ y\^2\)\/3 - \(2\ \ y\ z\)\/3}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[{c011\ c110\ \ - 3/4\ c121}\ ]\)], "Input"], Cell[BoxData[ \({x\/4}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[{c011\ c220\ , c231}] // MatrixForm\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(-\(y\/3\)\) + \(x\^2\ y\^3\)\/3 - 2\/3\ x\ y\^2\ z + \(y\ z\^2\)\/3\)}, {"c231"} }], "\[NoBreak]", ")"}], MatrixForm[ { Plus[ Times[ Rational[ -1, 3], y], Times[ Rational[ 1, 3], Power[ x, 2], Power[ y, 3]], Times[ Rational[ -2, 3], x, Power[ y, 2], z], Times[ Rational[ 1, 3], y, Power[ z, 2]]], c231}]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[{c011\ c220\ - 2/3\ c231 - 1/3 c211}]\)], "Input"], Cell[BoxData[ \({\(-\(\(2\ c231\)\/3\)\) - \(2\ y\)\/9 - \(2\ x\^2\ y\)\/9 + \(x\^2\ \ y\^3\)\/3 + \(2\ x\ z\)\/9 - 2\/3\ x\ y\^2\ z + \(y\ z\^2\)\/3}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(c321 = Central[3, 2, 1]\)], "Input"], Cell[BoxData[ \(\(-\(x\/3\)\) - \(x\ y\^2\)\/3 + \(x\^3\ y\^2\)\/2 + \(y\ z\)\/3 - x\^2\ y\ z + \(x\ z\^2\)\/2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(c330\)], "Input"], Cell[BoxData[ \(\(-\(\(x\ y\)\/2\)\) + \(x\^3\ y\^3\)\/4 + z\/2 - 3\/4\ x\^2\ y\^2\ z + 3\/4\ x\ y\ z\^2 - z\^3\/4\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[{y\ c330, c341 = Central[3, 4, 1]}] // MatrixForm\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(-\(\(x\ y\^2\)\/2\)\) + \(x\^3\ y\^4\)\/4 + \(y\ z\)\/2 - 3\/4\ x\^2\ y\^3\ z + 3\/4\ x\ y\^2\ z\^2 - \(y\ z\^3\)\/4\)}, {\(x\/5 - \(3\ x\ y\^2\)\/5 - \(3\ x\^3\ y\^2\)\/10 + \(2\ x\^3\ \ y\^4\)\/5 + \(3\ y\ z\)\/5 + 3\/5\ x\^2\ y\ z - 6\/5\ x\^2\ y\^3\ z - \(3\ x\ z\^2\)\/10 + 6\/5\ x\ y\^2\ z\^2 - \(2\ y\ z\^3\)\/5\)} }], "\[NoBreak]", ")"}], MatrixForm[ { Plus[ Times[ Rational[ -1, 2], x, Power[ y, 2]], Times[ Rational[ 1, 4], Power[ x, 3], Power[ y, 4]], Times[ Rational[ 1, 2], y, z], Times[ Rational[ -3, 4], Power[ x, 2], Power[ y, 3], z], Times[ Rational[ 3, 4], x, Power[ y, 2], Power[ z, 2]], Times[ Rational[ -1, 4], y, Power[ z, 3]]], Plus[ Times[ Rational[ 1, 5], x], Times[ Rational[ -3, 5], x, Power[ y, 2]], Times[ Rational[ -3, 10], Power[ x, 3], Power[ y, 2]], Times[ Rational[ 2, 5], Power[ x, 3], Power[ y, 4]], Times[ Rational[ 3, 5], y, z], Times[ Rational[ 3, 5], Power[ x, 2], y, z], Times[ Rational[ -6, 5], Power[ x, 2], Power[ y, 3], z], Times[ Rational[ -3, 10], x, Power[ z, 2]], Times[ Rational[ 6, 5], x, Power[ y, 2], Power[ z, 2]], Times[ Rational[ -2, 5], y, Power[ z, 3]]]}]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[{y\ c330\ - 5/8\ c341, c321}] // MatrixForm\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(-\(x\/8\)\) - \(x\ y\^2\)\/8 + \(3\ x\^3\ y\^2\)\/16 + \(y\ \ z\)\/8 - 3\/8\ x\^2\ y\ z + \(3\ x\ z\^2\)\/16\)}, {\(\(-\(x\/3\)\) - \(x\ y\^2\)\/3 + \(x\^3\ y\^2\)\/2 + \(y\ \ z\)\/3 - x\^2\ y\ z + \(x\ z\^2\)\/2\)} }], "\[NoBreak]", ")"}], MatrixForm[ { Plus[ Times[ Rational[ -1, 8], x], Times[ Rational[ -1, 8], x, Power[ y, 2]], Times[ Rational[ 3, 16], Power[ x, 3], Power[ y, 2]], Times[ Rational[ 1, 8], y, z], Times[ Rational[ -3, 8], Power[ x, 2], y, z], Times[ Rational[ 3, 16], x, Power[ z, 2]]], Plus[ Times[ Rational[ -1, 3], x], Times[ Rational[ -1, 3], x, Power[ y, 2]], Times[ Rational[ 1, 2], Power[ x, 3], Power[ y, 2]], Times[ Rational[ 1, 3], y, z], Times[ -1, Power[ x, 2], y, z], Times[ Rational[ 1, 2], x, Power[ z, 2]]]}]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[y\ c330\ - 5/8\ c341\ - 3/8 c321]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[Expand[{y\ c211, c222, c220, c202}]]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(-\(y\^2\/3\)\) + \(2\ x\^2\ y\^2\)\/3 - \(2\ x\ y\ \ z\)\/3\)}, {\(1 - x\^2\/2 - y\^2\/2 + \(x\^2\ y\^2\)\/2 - z\^2\/2\)}, {\(\(-\(1\/3\)\) + \(x\^2\ y\^2\)\/3 - \(2\ x\ y\ z\)\/3 + z\^2\/3\)}, {\(\(-1\) + x\^2\)} }], "\[NoBreak]", ")"}], MatrixForm[ { Plus[ Times[ Rational[ -1, 3], Power[ y, 2]], Times[ Rational[ 2, 3], Power[ x, 2], Power[ y, 2]], Times[ Rational[ -2, 3], x, y, z]], Plus[ 1, Times[ Rational[ -1, 2], Power[ x, 2]], Times[ Rational[ -1, 2], Power[ y, 2]], Times[ Rational[ 1, 2], Power[ x, 2], Power[ y, 2]], Times[ Rational[ -1, 2], Power[ z, 2]]], Plus[ Rational[ -1, 3], Times[ Rational[ 1, 3], Power[ x, 2], Power[ y, 2]], Times[ Rational[ -2, 3], x, y, z], Times[ Rational[ 1, 3], Power[ z, 2]]], Plus[ -1, Power[ x, 2]]}]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[y\ c211\ - \ c220\ - \ 2/3 c222\ - \ 1/3\ c202]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[Expand[{y\ c121, c132}]]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(-\(\(x\ y\)\/3\)\) + \(2\ x\ y\^3\)\/3 - \(2\ y\^2\ \ z\)\/3\)}, {\(\(-x\)\ y + \(3\ x\ y\^3\)\/4 + z\/2 - \(3\ y\^2\ z\)\/4\)} }], "\[NoBreak]", ")"}], MatrixForm[ { Plus[ Times[ Rational[ -1, 3], x, y], Times[ Rational[ 2, 3], x, Power[ y, 3]], Times[ Rational[ -2, 3], Power[ y, 2], z]], Plus[ Times[ -1, x, y], Times[ Rational[ 3, 4], x, Power[ y, 3]], Times[ Rational[ 1, 2], z], Times[ Rational[ -3, 4], Power[ y, 2], z]]}]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[Expand[{y\ c121 - 8/9\ c132, c112, c110}]]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(5\ x\ y\)\/9 - \(4\ z\)\/9\)}, {\(\(x\ y\)\/2 + z\/2\)}, {\(\(x\ y\)\/2 - z\/2\)} }], "\[NoBreak]", ")"}], MatrixForm[ { Plus[ Times[ Rational[ 5, 9], x, y], Times[ Rational[ -4, 9], z]], Plus[ Times[ Rational[ 1, 2], x, y], Times[ Rational[ 1, 2], z]], Plus[ Times[ Rational[ 1, 2], x, y], Times[ Rational[ -1, 2], z]]}]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Central[2, 5, 7]\)], "Input"], Cell[BoxData[ \(\(-\(\(4\ y\)\/7\)\) + \(6\ x\^2\ y\)\/7 + \(20\ y\^3\)\/21 - \(4\ x\^2\ \ y\^3\)\/7 - \(2\ y\^5\)\/7 + \(x\^2\ y\^5\)\/21 + \(4\ x\ z\)\/7 - 12\/7\ x\ y\^2\ z + 10\/21\ x\ y\^4\ z - \(4\ y\ z\^2\)\/7 + \(10\ y\^3\ z\^2\)\/21\)], \ "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[y\ c202\ - \ c213\ - \ c211]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[ y\ c303\ - \ \((c314 = \ Central[3, 1, 4])\)\ - \ \((c312 = Central[3, 1, 2])\)]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[y\ c121\ - 8/9\ c132\ - 1/9\ c112\ - \ c110]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[ Expand[{y\ c213, c222, c220, c202, \ 2\ c222\ + \ 3\ c220, \ c224, c112}]]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(-\(\(2\ y\^2\)\/3\)\) + \(x\^2\ y\^2\)\/3 + \(2\ x\ y\ \ z\)\/3\)}, {\(1 - x\^2\/2 - y\^2\/2 + \(x\^2\ y\^2\)\/2 - z\^2\/2\)}, {\(\(-\(1\/3\)\) + \(x\^2\ y\^2\)\/3 - \(2\ x\ y\ z\)\/3 + z\^2\/3\)}, {\(\(-1\) + x\^2\)}, {\(1 - x\^2 - y\^2 + 2\ x\^2\ y\^2 - 2\ x\ y\ z\)}, {\(1\/3 - x\^2\/2 - y\^2\/2 + \(x\^2\ y\^2\)\/6 + \(2\ x\ y\ z\)\/3 + z\^2\/6\)}, {\(\(x\ y\)\/2 + z\/2\)} }], "\[NoBreak]", ")"}], MatrixForm[ { Plus[ Times[ Rational[ -2, 3], Power[ y, 2]], Times[ Rational[ 1, 3], Power[ x, 2], Power[ y, 2]], Times[ Rational[ 2, 3], x, y, z]], Plus[ 1, Times[ Rational[ -1, 2], Power[ x, 2]], Times[ Rational[ -1, 2], Power[ y, 2]], Times[ Rational[ 1, 2], Power[ x, 2], Power[ y, 2]], Times[ Rational[ -1, 2], Power[ z, 2]]], Plus[ Rational[ -1, 3], Times[ Rational[ 1, 3], Power[ x, 2], Power[ y, 2]], Times[ Rational[ -2, 3], x, y, z], Times[ Rational[ 1, 3], Power[ z, 2]]], Plus[ -1, Power[ x, 2]], Plus[ 1, Times[ -1, Power[ x, 2]], Times[ -1, Power[ y, 2]], Times[ 2, Power[ x, 2], Power[ y, 2]], Times[ -2, x, y, z]], Plus[ Rational[ 1, 3], Times[ Rational[ -1, 2], Power[ x, 2]], Times[ Rational[ -1, 2], Power[ y, 2]], Times[ Rational[ 1, 6], Power[ x, 2], Power[ y, 2]], Times[ Rational[ 2, 3], x, y, z], Times[ Rational[ 1, 6], Power[ z, 2]]], Plus[ Times[ Rational[ 1, 2], x, y], Times[ Rational[ 1, 2], z]]}]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[Expand[{3\ y\ c213\ + c220}]]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(-\(1\/3\)\) - 2\ y\^2 + \(4\ x\^2\ y\^2\)\/3 + \(4\ x\ y\ z\)\/3 + z\^2\/3\)} }], "\[NoBreak]", ")"}], MatrixForm[ { Plus[ Rational[ -1, 3], Times[ -2, Power[ y, 2]], Times[ Rational[ 4, 3], Power[ x, 2], Power[ y, 2]], Times[ Rational[ 4, 3], x, y, z], Times[ Rational[ 1, 3], Power[ z, 2]]]}]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[Expand[{y\ c213, \ c224, \ c222, \ c220}]]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(-\(\(2\ y\^2\)\/3\)\) + \(x\^2\ y\^2\)\/3 + \(2\ x\ y\ \ z\)\/3\)}, {\(1\/3 - x\^2\/2 - y\^2\/2 + \(x\^2\ y\^2\)\/6 + \(2\ x\ y\ z\)\/3 + z\^2\/6\)}, {\(1 - x\^2\/2 - y\^2\/2 + \(x\^2\ y\^2\)\/2 - z\^2\/2\)}, {\(\(-\(1\/3\)\) + \(x\^2\ y\^2\)\/3 - \(2\ x\ y\ z\)\/3 + z\^2\/3\)} }], "\[NoBreak]", ")"}], MatrixForm[ { Plus[ Times[ Rational[ -2, 3], Power[ y, 2]], Times[ Rational[ 1, 3], Power[ x, 2], Power[ y, 2]], Times[ Rational[ 2, 3], x, y, z]], Plus[ Rational[ 1, 3], Times[ Rational[ -1, 2], Power[ x, 2]], Times[ Rational[ -1, 2], Power[ y, 2]], Times[ Rational[ 1, 6], Power[ x, 2], Power[ y, 2]], Times[ Rational[ 2, 3], x, y, z], Times[ Rational[ 1, 6], Power[ z, 2]]], Plus[ 1, Times[ Rational[ -1, 2], Power[ x, 2]], Times[ Rational[ -1, 2], Power[ y, 2]], Times[ Rational[ 1, 2], Power[ x, 2], Power[ y, 2]], Times[ Rational[ -1, 2], Power[ z, 2]]], Plus[ Rational[ -1, 3], Times[ Rational[ 1, 3], Power[ x, 2], Power[ y, 2]], Times[ Rational[ -2, 3], x, y, z], Times[ Rational[ 1, 3], Power[ z, 2]]]}]]], "Output", FontSize->18] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Collect[ Expand[\ y\ c213\ - \ a\ c224\ - \ b\ c222\ - \ c\ c220], {x, y, z}]\)], "Input"], Cell[BoxData[ \(\(-\(a\/3\)\) - b + c\/3 + \((\(-\(2\/3\)\) + a\/2 + b\/2)\)\ y\^2 + x\^2\ \((a\/2 + b\/2 + \((1\/3 - a\/6 - b\/2 - c\/3)\)\ y\^2)\) + \((2\/3 - \(2\ a\)\/3 + \(2\ c\)\/3)\)\ \ x\ y\ z + \((\(-\(a\/6\)\) + b\/2 - c\/3)\)\ z\^2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[{\((1\/3 - a\/6 - b\/2 - c\/3)\) \[Equal] 0, \((2\/3 - \(2\ a\)\/3 + \(2\ c\)\/3)\) \[Equal] 0, \((\(-\(a\/6\)\) + b\/2 - c\/3)\) \[Equal] 0}, {a, b, c}]\)], "Input"], Cell[BoxData[ \({{a \[Rule] 1, b \[Rule] 1\/3, c \[Rule] 0}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[\ y\ c213\ - \ \ c224\ - \ 1/3\ c222\ \ - \ 2/3\ c202]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[ Expand[{y\ c123, \ c134\ = \ Central[1, 3, 4], c132, c112, c110}]]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(-\(\(2\ x\ y\)\/3\)\) + \(x\ y\^3\)\/3 + \(2\ y\^2\ \ z\)\/3\)}, {\(\(-x\)\ y + \(x\ y\^3\)\/4 - z\/2 + \(3\ y\^2\ z\)\/4\)}, {\(\(-x\)\ y + \(3\ x\ y\^3\)\/4 + z\/2 - \(3\ y\^2\ z\)\/4\)}, {\(\(x\ y\)\/2 + z\/2\)}, {\(\(x\ y\)\/2 - z\/2\)} }], "\[NoBreak]", ")"}], MatrixForm[ { Plus[ Times[ Rational[ -2, 3], x, y], Times[ Rational[ 1, 3], x, Power[ y, 3]], Times[ Rational[ 2, 3], Power[ y, 2], z]], Plus[ Times[ -1, x, y], Times[ Rational[ 1, 4], x, Power[ y, 3]], Times[ Rational[ -1, 2], z], Times[ Rational[ 3, 4], Power[ y, 2], z]], Plus[ Times[ -1, x, y], Times[ Rational[ 3, 4], x, Power[ y, 3]], Times[ Rational[ 1, 2], z], Times[ Rational[ -3, 4], Power[ y, 2], z]], Plus[ Times[ Rational[ 1, 2], x, y], Times[ Rational[ 1, 2], z]], Plus[ Times[ Rational[ 1, 2], x, y], Times[ Rational[ -1, 2], z]]}]]], "Output", FontSize->36] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Collect[y\ c123\ \ \ - \ a\ c134\ - \ b\ c132, {x, y, z}]\)], "Input"], Cell[BoxData[ \(x\ \((\((\(-\(2\/3\)\) + a + b)\)\ y + \((1\/3 - a\/4 - \(3\ b\)\/4)\)\ y\^3)\) + \((a\/2 - b\/2)\)\ z + \((2\/3 - \(3\ a\)\/4 + \(3\ b\)\/4)\)\ y\^2\ z\)], \ "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[{\((1\/3 - a\/4 - \(3\ b\)\/4)\) \[Equal] 0, \((2\/3 - \(3\ a\)\/4 + \(3\ b\)\/4)\) \[Equal] 0}, {a, b, }]\)], "Input"], Cell[BoxData[ RowBox[{\(Solve::"svars"\), \(\(:\)\(\ \)\), "\<\"Equations may not give \ solutions for all \\\"solve\\\" variables. \\!\\(\\*ButtonBox[\\\"More\ \[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"Solve::svars\\\"]\\)\"\>"}]], "Message"], Cell[BoxData[ \({{a \[Rule] 1, b \[Rule] 1\/9}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[y\ c123\ - \ c134\ - \ 1/9\ c132 - \ 8/9\ c112]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[BoxData[ \(Test[m_, n_] := \((0 \[Equal] \ Expand[Central[0, n, n] Central[m, 0, m]\ ] - Expand[Apply[Plus, Map[Central[m, n, #] &, Range[Abs[m - n], m + n, 2]]]])\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \({Test[7, 5], Test[6, 6]}\)], "Input"], Cell[BoxData[ \({True, True}\)], "Output"] }, Open ]], Cell["Recursion limit 256 exceeded for larger values", "Subsection"] }, FrontEndVersion->"5.0 for X", ScreenRectangle->{{0, 1024}, {0, 768}}, CellGrouping->Manual, WindowSize->{958, 692}, WindowMargins->{{23, Automatic}, {Automatic, 4}}, PrintingPageRange->{Automatic, Automatic}, PrintingOptions->{"PaperSize"->{612, 792}, "PaperOrientation"->"Portrait", "PostScriptOutputFile":>FrontEnd`FileName[{$RootDirectory, "home", "wmg", \ "Mathematica"}, "Rep.nb.ps", CharacterEncoding -> "iso8859-1"], "Magnification"->1}, Magnification->1.25 ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. *******************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[1754, 51, 23, 0, 122, "Title"], Cell[1780, 53, 141, 3, 109, "Subtitle"], Cell[1924, 58, 50, 0, 40, "Subsubtitle"], Cell[CellGroupData[{ Cell[1999, 62, 28, 0, 95, "Section"], Cell[CellGroupData[{ Cell[2052, 66, 55, 0, 39, "Text"], Cell[2110, 68, 103, 2, 56, "Input"], Cell[2216, 72, 32, 0, 39, "Text"], Cell[2251, 74, 59, 1, 35, "Input"] }, Open ]] }, Open ]], Cell[2337, 79, 74, 0, 39, "Text"], Cell[2414, 81, 137, 2, 56, "Input"], Cell[CellGroupData[{ Cell[2576, 87, 36, 0, 95, "Section"], Cell[CellGroupData[{ Cell[2637, 91, 227, 7, 140, "Subsection"], Cell[2867, 100, 107, 2, 35, "Input"], Cell[CellGroupData[{ Cell[2999, 106, 103, 2, 35, "Input"], Cell[3105, 110, 796, 23, 49, "Output"] }, Open ]], Cell[3916, 136, 67, 0, 48, "Subsection"], Cell[CellGroupData[{ Cell[4008, 140, 113, 2, 35, "Input"], Cell[4124, 144, 43, 1, 35, "Output"] }, Open ]] }, Open ]] }, Open ]], Cell[4206, 150, 163, 4, 37, "Subsubsection"], Cell[CellGroupData[{ Cell[4394, 158, 47, 0, 95, "Section"], Cell[4444, 160, 126, 3, 48, "Subsection"], Cell[CellGroupData[{ Cell[4595, 167, 42, 0, 37, "Subsubsection"], Cell[4640, 169, 618, 12, 140, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[5295, 186, 101, 3, 37, "Subsubsection"], Cell[5399, 191, 93, 1, 35, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[5529, 197, 33, 0, 37, "Subsubsection"], Cell[CellGroupData[{ Cell[5587, 201, 80, 1, 35, "Input"], Cell[5670, 204, 146, 2, 35, "Output"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[5877, 213, 92, 1, 35, "Input"], Cell[5972, 216, 134, 2, 35, "Output"] }, Open ]], Cell[6121, 221, 223, 10, 48, "Subsection"], Cell[CellGroupData[{ Cell[6369, 235, 584, 12, 140, "Input"], Cell[6956, 249, 150, 2, 56, "Input"], Cell[7109, 253, 239, 4, 56, "Input"], Cell[7351, 259, 827, 27, 81, "Subsubsection"], Cell[8181, 288, 1301, 37, 95, "Subsubsection"], Cell[9485, 327, 280, 5, 56, "Input"], Cell[CellGroupData[{ Cell[9790, 336, 72, 0, 48, "Subsection"], Cell[9865, 338, 288, 5, 77, "Input"] }, Open ]], Cell[10168, 346, 52, 0, 48, "Subsection"], Cell[CellGroupData[{ Cell[10245, 350, 109, 3, 37, "Subsubsection"], Cell[10357, 355, 274, 5, 77, "Input"], Cell[CellGroupData[{ Cell[10656, 364, 135, 2, 56, "Input"], Cell[10794, 368, 64, 1, 35, "Output"] }, Open ]], Cell[10873, 372, 42, 0, 37, "Subsubsection"], Cell[10918, 374, 396, 7, 98, "Input"] }, Open ]], Cell[11329, 384, 83, 1, 48, "Subsection"], Cell[11415, 387, 440, 7, 98, "Input"], Cell[11858, 396, 35, 0, 95, "Section"], Cell[11896, 398, 523, 15, 94, "Subsection"], Cell[CellGroupData[{ Cell[12444, 417, 123, 3, 37, "Subsubsection"], Cell[12570, 422, 77, 1, 35, "Input"], Cell[CellGroupData[{ Cell[12672, 427, 45, 1, 35, "Input"], Cell[12720, 430, 74, 1, 35, "Output"] }, Open ]] }, Open ]], Cell[12821, 435, 225, 6, 59, "Subsubsection"], Cell[CellGroupData[{ Cell[13071, 445, 220, 3, 56, "Input"], Cell[13294, 450, 79, 1, 35, "Output"] }, Open ]], Cell[13388, 454, 128, 2, 35, "Input"], Cell[CellGroupData[{ Cell[13541, 460, 91, 1, 35, "Input"], Cell[13635, 463, 49, 1, 35, "Output"] }, Open ]], Cell[13699, 467, 119, 3, 37, "Subsubsection"], Cell[13821, 472, 283, 6, 77, "Input"], Cell[CellGroupData[{ Cell[14129, 482, 57, 1, 35, "Input"], Cell[14189, 485, 85, 1, 36, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[14311, 491, 57, 1, 35, "Input"], Cell[14371, 494, 59, 1, 35, "Output"] }, Open ]], Cell[14445, 498, 257, 8, 37, "Subsubsection"], Cell[14705, 508, 152, 3, 56, "Input"], Cell[CellGroupData[{ Cell[14882, 515, 86, 1, 35, "Input"], Cell[14971, 518, 1494, 36, 115, "Output"] }, Open ]], Cell[16480, 557, 608, 19, 59, "Subsubsection"], Cell[CellGroupData[{ Cell[17113, 580, 136, 2, 35, "Input"], Cell[17252, 584, 502, 15, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[17791, 604, 47, 1, 35, "Input"], Cell[17841, 607, 35, 1, 35, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[17913, 613, 45, 0, 95, "Section"], Cell[CellGroupData[{ Cell[17983, 617, 905, 29, 117, "Subsection"], Cell[18891, 648, 70, 0, 37, "Subsubsection"], Cell[18964, 650, 309, 5, 77, "Input"], Cell[CellGroupData[{ Cell[19298, 659, 58, 1, 35, "Input"], Cell[19359, 662, 38, 1, 35, "Output"] }, Open ]] }, Open ]] }, Open ]], Cell[19436, 668, 107, 3, 37, "Subsubsection"], Cell[19546, 673, 311, 6, 77, "Input"], Cell[19860, 681, 2346, 71, 189, "Subsection"], Cell[22209, 754, 392, 7, 77, "Input"], Cell[CellGroupData[{ Cell[22626, 765, 88, 1, 35, "Input"], Cell[22717, 768, 766, 25, 96, "Output"] }, Open ]], Cell[23498, 796, 3083, 97, 240, "Subsection"], Cell[CellGroupData[{ Cell[26606, 897, 166, 3, 56, "Input"], Cell[26775, 902, 174, 3, 56, "Input"], Cell[26952, 907, 295, 5, 77, "Input"] }, Open ]], Cell[27262, 915, 45, 0, 95, "Section"], Cell[27310, 917, 178, 3, 56, "Input"], Cell[CellGroupData[{ Cell[27513, 924, 91, 1, 35, "Input"], Cell[27607, 927, 52, 1, 50, "Output"] }, Open ]], Cell[27674, 931, 128, 2, 35, "Input"], Cell[27805, 935, 176, 3, 56, "Input"], Cell[27984, 940, 128, 2, 35, "Input"], Cell[28115, 944, 46, 0, 95, "Section"], Cell[28164, 946, 322, 6, 77, "Input"], Cell[28489, 954, 289, 5, 77, "Input"], Cell[28781, 961, 114, 2, 35, "Input"], Cell[28898, 965, 311, 6, 94, "Subsection"], Cell[29212, 973, 73, 0, 37, "Subsubsection"], Cell[29288, 975, 101, 2, 35, "Input"], Cell[29392, 979, 303, 9, 37, "Subsubsection"], Cell[29698, 990, 220, 4, 77, "Input"], Cell[29921, 996, 259, 6, 77, "Input"], Cell[CellGroupData[{ Cell[30205, 1006, 42, 0, 48, "Subsection"], Cell[30250, 1008, 152, 3, 56, "Input"], Cell[30405, 1013, 133, 2, 35, "Input"], Cell[30541, 1017, 82, 1, 35, "Input"] }, Open ]], Cell[30638, 1021, 40, 0, 48, "Subsection"], Cell[CellGroupData[{ Cell[30703, 1025, 117, 2, 35, "Input"], Cell[30823, 1029, 43, 1, 35, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[30903, 1035, 165, 2, 56, "Input"], Cell[31071, 1039, 627, 22, 128, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[31735, 1066, 226, 3, 77, "Input"], Cell[31964, 1071, 1456, 48, 206, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[33457, 1124, 107, 2, 35, "Input"], Cell[33567, 1128, 412, 14, 74, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[34016, 1147, 49, 1, 35, "Input"], Cell[34068, 1150, 90, 1, 54, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[34195, 1156, 107, 2, 35, "Input"], Cell[34305, 1160, 809, 26, 94, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[35151, 1191, 131, 2, 35, "Input"], Cell[35285, 1195, 1501, 51, 128, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[36823, 1251, 107, 2, 35, "Input"], Cell[36933, 1255, 809, 26, 94, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[37779, 1286, 49, 1, 35, "Input"], Cell[37831, 1289, 55, 1, 36, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[37923, 1295, 164, 3, 56, "Input"], Cell[38090, 1300, 3639, 108, 111, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[41766, 1413, 81, 1, 35, "Input"], Cell[41850, 1416, 2105, 69, 128, "Output"] }, Open ]], Cell[43970, 1488, 118, 3, 48, "Subsection"], Cell[CellGroupData[{ Cell[44113, 1495, 71, 1, 35, "Input"], Cell[44187, 1498, 43, 1, 35, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[44267, 1504, 145, 3, 35, "Input"], Cell[44415, 1509, 74, 1, 35, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[44526, 1515, 119, 2, 35, "Input"], Cell[44648, 1519, 71, 1, 36, "Output"] }, Open ]] }, Open ]], Cell[44746, 1524, 28, 0, 95, "Section"], Cell[CellGroupData[{ Cell[44799, 1528, 48, 1, 35, "Input"], Cell[44850, 1531, 38, 1, 35, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[44925, 1537, 37, 1, 35, "Input"], Cell[44965, 1540, 52, 1, 48, "Output"] }, Open ]], Cell[45032, 1544, 327, 5, 98, "Input"], Cell[CellGroupData[{ Cell[45384, 1553, 59, 1, 35, "Input"], Cell[45446, 1556, 41, 1, 36, "Output"] }, Open ]], Cell[45502, 1560, 141, 3, 35, "Input"], Cell[CellGroupData[{ Cell[45668, 1567, 134, 3, 35, "Input"], Cell[45805, 1572, 104, 2, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[45946, 1579, 102, 2, 35, "Input"], Cell[46051, 1583, 433, 7, 97, "Output"] }, Open ]], Cell[46499, 1593, 111, 2, 35, "Input"], Cell[CellGroupData[{ Cell[46635, 1599, 57, 1, 35, "Input"], Cell[46695, 1602, 50, 1, 36, "Output"] }, Open ]], Cell[46760, 1606, 91, 1, 35, "Input"], Cell[CellGroupData[{ Cell[46876, 1611, 87, 1, 35, "Input"], Cell[46966, 1614, 128, 2, 54, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[47131, 1621, 101, 2, 35, "Input"], Cell[47235, 1625, 101, 2, 36, "Output"] }, Open ]], Cell[47351, 1630, 173, 4, 35, "Input"], Cell[CellGroupData[{ Cell[47549, 1638, 46, 1, 35, "Input"], Cell[47598, 1641, 35, 1, 35, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[47670, 1647, 61, 1, 35, "Input"], Cell[47734, 1650, 105, 2, 36, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[47876, 1657, 67, 1, 35, "Input"], Cell[47946, 1660, 45, 1, 48, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[48028, 1666, 51, 1, 35, "Input"], Cell[48082, 1669, 62, 1, 54, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[48181, 1675, 37, 1, 35, "Input"], Cell[48221, 1678, 88, 1, 54, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[48346, 1684, 63, 1, 35, "Input"], Cell[48412, 1687, 49, 1, 48, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[48498, 1693, 42, 1, 35, "Input"], Cell[48543, 1696, 104, 2, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[48684, 1703, 72, 1, 35, "Input"], Cell[48759, 1706, 41, 1, 35, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[48837, 1712, 136, 3, 35, "Input"], Cell[48976, 1717, 35, 1, 35, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[49048, 1723, 101, 2, 35, "Input"], Cell[49152, 1727, 35, 1, 35, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[49224, 1733, 65, 1, 35, "Input"], Cell[49292, 1736, 118, 2, 54, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[49447, 1743, 71, 1, 35, "Input"], Cell[49521, 1746, 40, 1, 48, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[49598, 1752, 75, 1, 35, "Input"], Cell[49676, 1755, 616, 20, 80, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[50329, 1780, 79, 1, 35, "Input"], Cell[50411, 1783, 168, 2, 54, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[50616, 1790, 56, 1, 35, "Input"], Cell[50675, 1793, 135, 2, 54, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[50847, 1800, 37, 1, 35, "Input"], Cell[50887, 1803, 139, 2, 54, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[51063, 1810, 89, 1, 35, "Input"], Cell[51155, 1813, 1982, 65, 94, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[53174, 1883, 84, 1, 35, "Input"], Cell[53261, 1886, 1307, 43, 94, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[54605, 1934, 76, 1, 35, "Input"], Cell[54684, 1937, 35, 1, 35, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[54756, 1943, 80, 1, 35, "Input"], Cell[54839, 1946, 1369, 47, 150, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[56245, 1998, 91, 1, 35, "Input"], Cell[56339, 2001, 35, 1, 35, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[56411, 2007, 68, 1, 35, "Input"], Cell[56482, 2010, 793, 26, 94, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[57312, 2041, 86, 1, 35, "Input"], Cell[57401, 2044, 643, 22, 112, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[58081, 2071, 49, 1, 35, "Input"], Cell[58133, 2074, 275, 5, 54, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[58445, 2084, 70, 1, 35, "Input"], Cell[58518, 2087, 35, 1, 35, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[58590, 2093, 142, 3, 35, "Input"], Cell[58735, 2098, 35, 1, 35, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[58807, 2104, 87, 1, 35, "Input"], Cell[58897, 2107, 35, 1, 35, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[58969, 2113, 137, 3, 35, "Input"], Cell[59109, 2118, 2370, 82, 234, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[61516, 2205, 74, 1, 35, "Input"], Cell[61593, 2208, 610, 20, 62, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[62240, 2233, 86, 1, 35, "Input"], Cell[62329, 2236, 1856, 64, 186, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[64222, 2305, 128, 3, 35, "Input"], Cell[64353, 2310, 299, 5, 89, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[64689, 2320, 223, 4, 57, "Input"], Cell[64915, 2326, 78, 1, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[65030, 2332, 104, 2, 35, "Input"], Cell[65137, 2336, 35, 1, 35, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[65209, 2342, 129, 3, 35, "Input"], Cell[65341, 2347, 1460, 50, 389, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[66838, 2402, 92, 1, 35, "Input"], Cell[66933, 2405, 220, 4, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[67190, 2414, 165, 3, 57, "Input"], Cell[67358, 2419, 298, 4, 28, "Message"], Cell[67659, 2425, 65, 1, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[67761, 2431, 89, 1, 35, "Input"], Cell[67853, 2434, 35, 1, 35, "Output"] }, Open ]], Cell[67903, 2438, 256, 6, 77, "Input"], Cell[CellGroupData[{ Cell[68184, 2448, 57, 1, 35, "Input"], Cell[68244, 2451, 46, 1, 35, "Output"] }, Open ]], Cell[68305, 2455, 68, 0, 48, "Subsection"] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)