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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[ 21989, 426]*) (*NotebookOutlinePosition[ 22633, 448]*) (* CellTagsIndexPosition[ 22589, 444]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[{ \(Clear[k, x, y, Bnd, Cff, Poly, Polys, PrmeCalc, ConstCalc]\[IndentingNewLine]\[IndentingNewLine] (*\ Bnd : \ Input\ a\ list\ of\ variables\ {\ a_ 1\ , \ ... \ , \ a_ {k}\ }, \ and\ an\ integer\ tot . \ Gives\ a\ list\ of\ ranges\ for\ summing\ the\ variables\ in\ the\ \ list\ subject\ to\ each\ variable\ being\ \[GreaterEqual] 1\ and\ the\ sum\ of\ the\ variables\ being\ < tot\ *) \ \), "\[IndentingNewLine]", \(\(Bnd[lst_, tot_] := Module[{i1, i2, i3, i4, i5}, Table[{lst[\([i1]\)], 1, tot - 1 - Sum[lst[\([i2]\)], {i2, 1, i1 - 1}] - \((Length[lst] - i1)\)}, {i1, 1, Length[lst]}]\[IndentingNewLine] (*\ Since\ each\ variable\ is\ \[GreaterEqual] 1, \ the\ lower\ bound\ is\ 1. \ If\ we\ alread\ know\ the\ values\ of\ \ a_ 1\ , \ ... \ , \ a_ {m - 1}\ then\ since\ sum_ {i = 1}^{k}\ a_i\ \[LessEqual] \ tot - 1\ we\ have\[IndentingNewLine]a_m\ \[LessEqual] \ sum_ {i = 1}^k\ a_i\ - \ sum_ {i = 1}^{m - 1}\ a_i\ - \ sum_ {i = m + 1}^k\ a_i\ \[IndentingNewLine] \[LessEqual] \ tot\ - \ 1\ - \ sum_ {i = 1}^k\ a_i\ - \ sum_ {i = m + 1}^k\ 1\[IndentingNewLine] \[LessEqual] \ tot\ - \ 1\ - \ sum_ {i = 1}^ka_i\ - \ \((k - m)\), \ which\ is\ our\ upper\ bound*) \[IndentingNewLine]];\)\ \[IndentingNewLine]\[IndentingNewLine] (*\ Cff : \ Input\ a\ list\ of\ variables\ {\ a_ 1\ , \ ... \ , \ a_ {k}\ }, \ and\ an\ integer\ tot . \ Gives\ the\ summand\ \(tot!\)\ *\ prod_ {i = 1}^{k + 1}\ \(\((2 a_i)\)!\)/\(a_i!\)\ with\ a_ {k + 1} = tot\ - \ sum_ {i = 1}^{k}\ a_i\ *) \), "\[IndentingNewLine]", \(\(Cff[lst_, tot_] := Module[{i1, i2, i3}, Product[Factorial[2*lst[\([i1]\)]]/Factorial[lst[\([i1]\)]], {i1, 1, Length[lst]}]*Factorial[tot]* Factorial[2*tot - 2*Sum[lst[\([i2]\)], {i2, 1, Length[lst]}]]/ Factorial[ tot - Sum[ lst[\([i3]\)], {i3, 1, Length[lst]}]]];\)\[IndentingNewLine]\[IndentingNewLine]\ (*\ Poly : \ Input\ an\ integer\ n < 40\ and\ a\ variable\ k . \ Gives\ the\ n^{th}\ polynomial\ G_ {n, 2} \((k)\)\ in\ the\ variable\ k\ *) \), "\[IndentingNewLine]", \ \(\(Poly[n_, k_] := Module[{A = {a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12, a13, a14, a15, a16, a17, a18, a19, a20, a21, a22, a23, a24, a25, a26, a27, a28, a29, a30, a31, a32, a33, a34, a35, a36, a37, a38, a39, a40}, X = k*Factorial[2*n], i, Lst}, \[IndentingNewLine]If[n \[Equal] 0, X = 1, For[i = 1, i \[LessEqual] n - 1, \(i++\), Lst = Take[A, i]; X = X + Binomial[k, i + 1]* Apply[Sum, Join[{Cff[Lst, n]}, Bnd[Lst, n]]];]; \[IndentingNewLine] (*\ Use\ our\ formula\ to\ write\ the\ polynomial\ as\ a\ sum\ of\ \ binomial\ coefficients\ in\ k\ multiplied\ by\ some\ constants, \ which\ we\ calculate\ by\ summing\ Cff\ over\ the\ range\ Bnd\ \ *) \[IndentingNewLine]Expand[ X]]];\)\[IndentingNewLine]\[IndentingNewLine] (*\ It\ is\ convenient\ to\ precompute\ the\ polynomials\ Poly[n, k]\ we\ will\ use\ to\ speed\ up\ calculation\ *) \), "\ \[IndentingNewLine]", \(\(Polys[k_] = Table[Poly[n, k], {n, 0, 11}];\)\[IndentingNewLine]\[IndentingNewLine] (*\ ConstCalc : \ Input\ a\ polynomial\ expression\ Expr\ in\ x\ and\ y, \ and\ an\ integer\ k . \ x\ represents\ 1 - P_ 1 = \(1 - \(\(\\\)\(sum_\)\) {i = 1}^ k\ t_i\ and\ y\ represents\ P_ 2 = \(\(\\\)\(sum_\)\) {i = 1}^k\ t_i^2\[IndentingNewLine] Output\ is\ the\ result\ of\ integrating\ Expr^2\ with\ respect\ to\ \ t_ 1\), \(\(\\\)\(dots\)\)\ t_k\ over\ the\ region\ \ \[IndentingNewLine]\\mathcal {R} _k\ = \ \(\(\\\)\({\ \((t_ 1, \(\(\\\)\(dots\)\), t_k)\)\ \\in\ [0, 1]^\(k\ : \ \(\(\\\)\(sum_\)\) {i = 1}^ k\ t_i\ < \(\(1\)\(\\\)\)\)} . \ Assumes\)\)\ that\ the\ relevent\ Polynomials\ G_ {c, 2}\ have\ been\ precomputed\ in\ the\ list\ Polys\ *) \), "\ \[IndentingNewLine]", \(ConstCalc[Expr_, k_] := Module[{Sum = 0, IntExpr = Expand[Expr^2], xdeg, ydeg, b, c, tmp, Coeff}, \[IndentingNewLine] (*we\ are\ integrating\ IntExpr, \ which\ is\ Expr\ squared*) \[IndentingNewLine]xdeg = Exponent[IntExpr, x]; \[IndentingNewLine] (*we\ let\ xdeg\ be\ the\ maximum\ degree\ \ of\ IntExpr\ with\ respect\ to\ x*) \[IndentingNewLine] (*we\ now\ split\ \ our\ expression\ IntExpr\ into\ monomial\ terms\ x^b* y^c*) \[IndentingNewLine]For[b = 0, b \[LessEqual] xdeg, \(b++\), tmp = Coefficient[IntExpr, x, b]; ydeg = Exponent[tmp, y]; For[c = 0, c \[LessEqual] ydeg, \(c++\), Coeff = Coefficient[tmp, y, c]; \[IndentingNewLine] (*Coeff\ is\ the\ coefficient\ of\ x^ b*y^c\ in\ IntExpr*) Sum = Sum + Coeff* Factorial[b]/ Factorial[k + b + 2*c]*\(Polys[ k]\)[\([c + 1]\)];];]; \[IndentingNewLine] (*we\ use\ our\ \ formula\ from\ Lemma\ 7.1\ to\ count\ the\ contribution\ from\ this\ term\ *) \[IndentingNewLine]Sum]\[IndentingNewLine]\[IndentingNewLine] (*PrimeCalc \ : \ Input\ a\ polynomial\ expression\ Expr\ in\ x\ and\ y\ and\ an\ integer\ \ k . \ \[IndentingNewLine]x\ represents\ 1 - P_ 1 = \(1 - \(\(\\\)\(sum_\)\) {i = 1}^ k\ t_i\ and\ y\ represents\ P_ 2 = \(\(\\\)\(sum_\)\) {i = 1}^k\ t_i^2\ \[IndentingNewLine] Output\ is\ a\ polynomial\ expression\ which\ is\ the\ result\ of\ \ integrating\ Expr\ with\ resepect\ to\ t1\ between\ 0\ and\ 1 - \ \(\(\\\)\(sum_\)\) {i = 2}^k\ t_i . \ The\ polynomial\ expression\ outputted\ is\ given\ in\ terms\ \ of\ x\ and\ y\), \ where\ x\ represents\ 1 - P_ 1'\ and\ y\ represents\ P_ 2', \ where\ P_j' = \(\(\\\)\(sum_\)\) {i = 2}^\(kt_i^j\) . \ Calls\ constcalc, \ and\ so\ assumes\ the\ relevent\ polynomials\ have\ been\ precomputed\ \[IndentingNewLine]*) \), "\[IndentingNewLine]", \(\(PrmeCalc[Expr_, k_] := Module[{NewExpr = 0, FirstExpr = Expand[Expr], n1, n2, b, c, cp, tmp, Coeff}, \[IndentingNewLine]n1 = Exponent[Expr, x]; \[IndentingNewLine] (*\ We\ let\ n1\ be\ the\ maximum\ degree\ of\ the\ polynomial\ with\ \ respect\ to\ x\ *) \[IndentingNewLine] (*\ We\ now\ wish\ to\ split\ our\ polynomial\ into\ terms\ of\ the\ \ form\ x^b\ y^c\ *) \[IndentingNewLine]For[b = 0, b \[LessEqual] n1, \(b++\), tmp = Coefficient[FirstExpr, x, b]; \[IndentingNewLine] (*\ We\ extract\ the\ part\ of\ the\ polynomial\ which\ has\ degree\ \ exactly\ b\ with\ respect\ to\ x\ *) \[IndentingNewLine]n2 = Exponent[tmp, y]; \[IndentingNewLine] (*\ We\ let\ n2\ be\ the\ maximum\ degree\ of\ this\ part\ of\ the\ \ polynomial\ with\ respect\ to\ y\ *) \[IndentingNewLine]For[c = 0, c \[LessEqual] n2, \(c++\), \[IndentingNewLine]Coeff = Coefficient[tmp, y, c]; \[IndentingNewLine] (*\ Coeff\ is\ the\ coefficient\ of\ x^b\ y^ c\ in\ Expr\ *) \[IndentingNewLine]NewExpr = NewExpr + Coeff*Sum[ x^\((b + 2 c - 2 cp + 1)\)*y^\((cp)\)*Binomial[c, cp]* Factorial[b]* Factorial[2*c - 2*cp]/ Factorial[b + 2*c - 2*cp + 1], {cp, 0, c}]; \[IndentingNewLine] (*\ We\ use\ our\ formula\ from\ equation\ 7.8\ to\ produce\ the\ \ polynomial\ which\ we\ get\ after\ integrating\ with\ respect\ to\ t1\ \ *) \[IndentingNewLine] (*\ NewExpr\ is\ the\ sum\ of\ all\ of\ these\ polynomials, \ and\ so\ the\ result\ of\ integrating\ Expr\ with\ respect\ \ to\ t1 . \ Here\ x\ now\ refers\ to\ 1 - P_ 1'\ and\ y\ to\ P_ 2'\ *) \[IndentingNewLine]];\[IndentingNewLine]]; \ \[IndentingNewLine]ConstCalc[NewExpr, k - 1]\[IndentingNewLine] (*\ We\ now\ intgrate\ our\ polynomial\ squared\ over\ $\\mathcal {R} \ _ {k - 1} $\ using\ our\ function\ defined\ above\ *) \[IndentingNewLine]];\)\ \[IndentingNewLine]\[IndentingNewLine] (*\ p[n] : \ Input\ an\ integer\ n, \ output\ a\ polynomial\ which\ is\ a\ linear\ combination\ of\ all\ \ monomials\ x^b\ *\ y^c\ with\ b + 2 c \[LessEqual] 2 n + 1. \ The\ coefficients\ are\ given\ by\ A[\([1]\)], ... \ *) \ \), "\[IndentingNewLine]", \(\(yExponents[n_] := Module[{S = {}, i, tmp}, For[i = 0, i \[LessEqual] n, \(i++\), tmp = Append[Reverse[Range[i]], 0]; \[IndentingNewLine]S = Join[S, tmp, tmp]; ]; \[IndentingNewLine]S];\)\), "\[IndentingNewLine]", \(\(xExponents[n_] := Module[{S = {}, i, tmp}, For[i = 1, i \[LessEqual] n + 1, \(i++\), tmp = 2*Range[i]; \[IndentingNewLine]S = Join[S, tmp - 2, tmp - 1]; ]; \[IndentingNewLine]S];\)\), \ "\[IndentingNewLine]", \(\(p[n_] := Module[{X = xExponents[n], Y = yExponents[n], i}, Sum[A[\([\)\(i\)\(]\)]*x^X[\([\)\(i\)\(]\)]* y^Y[\([\)\(i\)\(]\)], {i, 1, Length[X]}]];\)\[IndentingNewLine]\[IndentingNewLine] (*\ We\ now\ perform\ the\ actual\ computation\ for\ k = 105\ to\ calculate\ a\ lower\ bound\ for\ M_ {105}\ *) \), "\ \[IndentingNewLine]", \(\(k = 105;\)\), "\[IndentingNewLine]", \(poly = p[5]\), "\[IndentingNewLine]", \(\(vars = DeleteCases[DeleteCases[Variables[poly], x], y];\)\[IndentingNewLine] (*\ take\ our\ polynomial\ to\ be\ all\ monomial\ combination\ \((1 - P_ 1)\)^b\ *\ P_ 2^c\ with\ b + 2 c \[LessEqual] 11. \ Let\ vars\ be\ a\ vector\ of\ the\ coefficients\ \((A[\([1]\)]\ \ , \ ... \ , \ A[\([42]\)])\)\ in\ some\ order\ *) \), "\[IndentingNewLine]", \(\(M1 = \(CoefficientArrays[ConstCalc[poly, k], vars, Symmetric \[Rule] True]\)[\([3]\)];\)\), "\[IndentingNewLine]", \(\(M2 = \(CoefficientArrays[PrmeCalc[poly, k], vars, Symmetric \[Rule] True]\)[\([3]\)];\)\[IndentingNewLine] (*\ Calculate\ the\ positive\ definite\ real\ symmetric\ matrices\ M1\ and\ \ M2\ corresponding\ to\ the\ I_k\ and\ J_k\ integrals\ *) \), "\ \[IndentingNewLine]", \(\(M3 = Inverse[M1] . 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