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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[ 225327, 5864]*) (*NotebookOutlinePosition[ 225973, 5886]*) (* CellTagsIndexPosition[ 225929, 5882]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ We\ specify\ the\ relation\ between\ the\ notation\ of\ the\ article\ \ and\ the\ notation\ used\ in\ this\ file, \ using\ latex\ notation\ *) \)\(\[IndentingNewLine]\)\(\ \[IndentingNewLine]\)\( (*\ Given\ the\ symmetry\ of\ all\ the\ functions\ involved, \ all\ the\ computations\ are\ carried\ out\ for\ s \[GreaterEqual] 0\ *) \)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\( (*We\ set\ \ the\ following\ notation*) \)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\ \[IndentingNewLine]\)\( (*\ All\ the\ independent\ variables\ are\ denoted\ by\ r\ *) \)\(\ \[IndentingNewLine]\)\(\[IndentingNewLine]\)\( (*\ LegendreQ[a, c, r] = Q_ {a}^{c} \((r)\)\ *) \)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\ \)\(Q[n_, r_] := LegendreQ[\((n - 1)\)/2, \((n - 3)\)/2, r]\[IndentingNewLine] q[n_, r_] := LegendreQ[\((n - 1)\)/2, \((n - 1)\)/2, r]\[IndentingNewLine]\[IndentingNewLine] (*\ LegendreP[a, c, r] = P_ {a}^{c} \((r)\)\ *) \[IndentingNewLine] (*\ x\ denotes\ the\ parameter\ \\tau\ *) \[IndentingNewLine]\ \[IndentingNewLine] P[x_, n_, r_] := LegendreP[x + \((n - 3)\)/2, \(-\((n - 3)\)\)/2, r]\[IndentingNewLine] pi[x_, n_, r_] := LegendreP[x + \((n - 3)\)/2, \ \(-\((n - 1)\)\)/2, r]\[IndentingNewLine]\[IndentingNewLine] (*\ V[x, n, r]\ denotes\ E_n\ *) \[IndentingNewLine] V[x_, n_, r_] := \((P[x, n, r] + P[x, n, \(-r\)])\)\[IndentingNewLine]\[IndentingNewLine] (*\ Z[x, n, r]\ denotes\ Z_n\ *) \[IndentingNewLine] Z[x_, n_, r_] := \((pi[x, n, r] - pi[x, n, \(-r\)])\)\[IndentingNewLine]\[IndentingNewLine] S[n_] := Sign[Q[n, 0]]\[IndentingNewLine]\[IndentingNewLine] (*\ The\ following\ is\ the\ formula\ for\ f_n\ *) \[IndentingNewLine] f[n_, r_] := S[n] \((1 - r^2)\)^\((\(-\((n - 3)\)\)/4)\) Q[n, r]\[IndentingNewLine]\[IndentingNewLine] (*The\ following\ are\ \ the\ formulas\ for\ respectively, \ f_n', \ f_n''*) \[IndentingNewLine] f1[n_, r_] := \(-S[n]\) \((1 - r^2)\)^\((\(-\((n - 1)\)\)/4)\) q[n, r]\[IndentingNewLine] f2[n_, r_] := \ {\((n - 1)\) r\ f1[n, r]\ - \((n - 1)\) f[n, r]}/{\((1 - r^2)\)}\[IndentingNewLine]\[IndentingNewLine] (*\ The\ following\ is\ the\ formula\ for\ g_ {\(\\\)\(tau\)}\ \ *) \[IndentingNewLine] g[x_, n_, r_] := \((1 - r^2)\)^\((\(-\((n - 3)\)\)/4)\) V[x, n, r]/2\[IndentingNewLine]\[IndentingNewLine] (*\ The\ following\ are\ the\ formulas\ for\ respectively, \ g' _ {\(\\\)\(tau\)}, \ g'' _ {\(\\\)\(tau\)}\ *) \[IndentingNewLine] g1[x_, n_, r_] := \ x \((x + n - 2)\) \((1 - r^2)\)^\((\(-\((n - 1)\)\)/4)\) Z[x, n, r]/2\[IndentingNewLine] g2[x_, n_, r_] := \ {\((n - 1)\) r\ g1[x, n, r]\ - x \((x + n - 2)\) g[x, n, r]}/{\((1 - r^2)\)}\[IndentingNewLine]\[IndentingNewLine] (*\ be[u]\ denotes\ \\beta_n, \ and\ u\ denotes\ \\gamma_n\ *) \[IndentingNewLine] be[u_] := Sqrt[\((1 - u^2)\)/\((u^2)\)]\[IndentingNewLine]\[IndentingNewLine] (*\ s[n, u, x]\ denotes\ \\bar {s}, \ and\ ta[n, u, x]\ denotes\ \\bar {r}\ *) \[IndentingNewLine] s[n_, u_, x_] := u\ {\(-g[x, n, u]\)/f[n, u]}^\((1/\((1 - x)\))\)\[IndentingNewLine] ta[n_, u_, x_] := be[u] s[n, u, x]\[IndentingNewLine]\[IndentingNewLine] (*\ A[n, u, x]\ denotes\ - A \((\(\\\)\(beta_n\))\) . \ \[IndentingNewLine]We\ need\ to\ check\ \ that\ \\beta_n\ + \ A\ < \ 0\ *) \[IndentingNewLine] A[n_, u_, x_] := {u\ f[n, u]\ + \ be[u]^2\ u^2\ f1[n, u] + \ x\ s[n, u, x]^\((x - 1)\) u^\((2 - x)\) g[x, n, u] + \ be[u]^2\ s[n, u, x]^\((x - 1)\) u^\((3 - x)\) g1[x, n, u]}/{u\ be[u] f[n, u]\ - \ be[u] u^2\ f1[n, u] + x\ be[u] s[n, u, x]^\((x - 1)\) u^\((2 - x)\)\ g[x, n, u] - be[u] s[n, u, x]^\((x - 1)\) u^\((3 - x)\)\ g1[x, n, u]}\[IndentingNewLine]\[IndentingNewLine] (*\ F[n, x, r]\ denotes\ G_n\ *) \[IndentingNewLine] F[n_, x_, r_] := \((1 - x)\)^2\ f[n, r]^2\ + \ \((1 - r^2)\) \((f1[n, r] - g1[x, n, r] f[n, r]/ g[x, n, r])\)^2\[IndentingNewLine]\[IndentingNewLine] (*\ The\ following\ formulas\ are\ relevant\ to\ Step\ 2\ *) \[IndentingNewLine]\[IndentingNewLine] (*\ H[n, u, r, x]\ denotes\ h \((r)\)\ *) \[IndentingNewLine] H[n_, u_, r_, x_] := Sqrt[\((r^2\ + \ ta[n, u, x]^2)\)] f[n, r/Sqrt[\((r^2\ + \ ta[n, u, x]^2)\)]]\ + \((r^2\ + \ ta[n, u, x]^2)\)^\((x/2)\) g[x, n, r/ Sqrt[\((r^2\ + \ ta[n, u, x]^2)\)]]\[IndentingNewLine]\[IndentingNewLine] (*\ H1[n, u, r, x]\ denotes\ h' \((r)\)\ *) \[IndentingNewLine] H1[n_, u_, r_, x_] := {r\ f[n, r/Sqrt[\((r^2\ + \ ta[n, u, x]^2)\)]]/ Sqrt[\((r^2\ + \ ta[n, u, x]^2)\)] + \ ta[n, u, x]^2\ /\((r^2\ + \ ta[n, u, x]^2)\)\ f1[n, r/Sqrt[\((r^2\ + \ ta[n, u, x]^2)\)]]} + \ {r\ x\ \ g[x, n, r/Sqrt[\((r^2\ + \ ta[n, u, x]^2)\)]] \((r^2\ + \ ta[n, u, x]^2)\)^\((\((x - 2)\)/2)\) + \ ta[n, u, x]^2\ \((r^2\ + \ ta[n, u, x]^2)\)^\((\((x - 3)\)/2)\)\ g1[ x, n, r/ Sqrt[\((r^2\ + \ ta[n, u, x]^2)\)]]}\[IndentingNewLine]\[IndentingNewLine] \ (*\ H2[n, u, r, x]\ denotes\ h'' \((r)\)\ *) \[IndentingNewLine] H2[n_, u_, r_, x_] := {ta[n, u, x]^2\ /\((\((r^2\ + \ ta[n, u, x]^2)\)^\((3/2)\)\ )\) f[n, r/Sqrt[\((r^2\ + \ ta[n, u, x]^2)\)]]\ - \ r\ ta[n, u, x]^2\ /\((\((r^2\ + \ ta[n, u, x]^2)\)^\((2)\)\ )\) f1[n, r/Sqrt[\((r^2\ + \ ta[n, u, x]^2)\)]]\ + \ ta[n, u, x]^4\ /\((\((r^2\ + \ ta[n, u, x]^2)\)^\((5/2)\)\ )\) f2[n, r/Sqrt[\((r^2\ + \ ta[n, u, x]^2)\)]]\ } + {x\ \((ta[n, u, x]^2\ + \((x - 1)\) r^2)\) \((\((r^2\ + \ ta[n, u, x]^2)\)^\((\((x - 4)\)/ 2)\)\ )\) g[x, n, r/Sqrt[\((r^2\ + \ ta[n, u, x]^2)\)]]\ + \ \((2 x - 3)\)\ r\ ta[n, u, x]^2\ \((\((r^2\ + \ ta[n, u, x]^2)\)^\((\((x - 5)\)/ 2)\)\ )\) g1[x, n, r/Sqrt[\((r^2\ + \ ta[n, u, x]^2)\)]]\ + \ ta[n, u, x]^4\ \((\((r^2\ + \ ta[n, u, x]^2)\)^\((\((x - 6)\)/ 2)\)\ )\) g2[x, n, r/Sqrt[\((r^2\ + \ ta[n, u, x]^2)\)]]\ \ }\[IndentingNewLine]\[IndentingNewLine] (*\ al[y, n, u, x]\ denotes\ the\ coefficient\ b_ 7\ of\ the\ polynomial\ v*) \[IndentingNewLine] al[y_, n_, u_, x_] := \(-{be[u]/\((A[n, u, x] ta[n, u, x])\) + 4\ y\ ta[n, u, x]^3\ }\)/\((2\ ta[n, u, x])\)\[IndentingNewLine]\[IndentingNewLine] (*\ ci[y, n, u, x]\ denotes\ the\ coefficient\ v_ 7\ of\ the\ polynomial\ v*) \[IndentingNewLine] ci[y_, n_, u_, x_] := \ 1\ - \ y\ ta[n, u, x]^4\ - \ al[y, n, u, x] ta[n, u, x]^2\[IndentingNewLine]\[IndentingNewLine] (*\ v[y, n, u, r, x]\ denotes\ the\ polynomial\ v . \ y\ denotes\ the\ coefficient\ a_ 7\ *) \[IndentingNewLine] v[y_, n_, u_, r_, x_] := \ y\ r^4\ + al[y, n, u, x]\ r^2\ + \ ci[y, n, u, x]\[IndentingNewLine]\[IndentingNewLine] (*b_ 7\ and\ v_ 7\ are\ determined\ via\ the\ conditions\ \((4.3)\) - \((4.4)\)\ \ in\ the\ article\ *) \[IndentingNewLine]\[IndentingNewLine] (*The\ following\ \ are\ the\ formulas\ for\ respectively\ v' \((r)\), \ v' \((r)\)/r, \ v'' \((r)\)\ and\ v''' \((r)\)\ *) \[IndentingNewLine]\ \[IndentingNewLine] v1[y_, n_, u_, r_, x_] := \ \ 4\ y\ r^3\ + \ 2\ al[y, n, u, x]\ r\[IndentingNewLine] rv1[y_, n_, u_, r_, x_] := \ 4\ y\ r^2\ + 2\ al[y, n, u, x]\[IndentingNewLine] v2[y_, n_, u_, r_, x_] := \ 12\ y\ r^2 + 2\ al[y, n, u, x]\[IndentingNewLine] v3[y_] := 24\ y\ r\[IndentingNewLine]\[IndentingNewLine] (*\ K4[y, n, u, r, x]\ denotes\ M\ *) \[IndentingNewLine] K4[y_, n_, u_, r_, x_] := \ {v2[y, n, u, r, x] + \((n - 2)\) rv1[y, n, u, r, x] - 2\ s[n, u, x]^2\ v1[y, n, u, r, x]^2\ v2[y, n, u, r, x]/\((1 + s[n, u, x]^2\ v1[y, n, u, r, x]^2)\)}\[IndentingNewLine]\[IndentingNewLine] (*\ K5[y, n, u, r, x]\ denotes\ N\ *) \[IndentingNewLine] K5[y_, n_, u_, r_, x_] := \ {{v2[y, n, u, r, x] \((1 - 2 s[n, u, x]^2\ v1[y, n, u, r, x]^2)\) \((1 + \ s[n, u, x]^2\ v1[y, n, u, r, x]^2\ - \ s[n, u, x]^2\ v[y, n, u, r, x] v2[y, n, u, r, x]\ )\)\ + \ s[n, u, x]^2\ v1[y, n, u, r, x] \((1 + \ s[n, u, x]^2\ v1[y, n, u, r, x]^2)\) \((v1[y, n, u, r, x] v2[y, n, u, r, x] - v[y, n, u, r, x] v3[y])\)\ }/\((\((1 + \ s[n, u, x]^2\ v1[y, n, u, r, x]^2)\)^2)\)\ + \ \((n - 2)\) rv1[y, n, u, r, x] \((1 + \ s[n, u, x]^2\ v1[y, n, u, r, x]^2\ - \ s[n, u, x]^2\ v[y, n, u, r, x] v2[y, n, u, r, x])\)/\((1 + \ s[n, u, x]^2\ v1[y, n, u, r, x]^2)\)}\[IndentingNewLine]\[IndentingNewLine] (*\ In\ the\ formulas\ above, \ the\ derivatives\ of\ the\ function\ y\ are\ computed\ in\ temrs\ of\ \ the\ function\ v\ *) \[IndentingNewLine]\[IndentingNewLine] (*\ We\ will\ check\ that\ on\ [0, \ \(\(\\\)\(bar\)\) {r}], M \((r)\)\ and\ N \((r)\)\ assume\ their\ absolute\ maximum\ at\ 0, \ that\ is\ \\bar {M} = K4[y, n, u, 0, x], \ \(\(\\\)\(bar\)\) {N} = K5[y, n, u, 0, x]\ *) \[IndentingNewLine]\[IndentingNewLine] (*\ K2[y, n, u, r, x]\ denotes\ \\bar {K}\ *) \[IndentingNewLine] K2[y_, n_, u_, r_, x_] := \ H2[n, u, r, x] - r\ H1[n, u, r, x] K4[y, n, u, 0, x] + H[n, u, r, x] K5[y, n, u, 0, x]\[IndentingNewLine]\[IndentingNewLine] (*The\ following\ are\ \ the\ formulas\ for\ respectively\ \\partial_r {w_ 1 \((s, r)\)}\ and\ \\partial_r {w_ 2 \((s, r)\)}, \ evaluated\ at\ r = \(\(\\\)\(bar\)\) {r} . \ The\ independent\ variable\ s\ is\ denoted\ with\ r, \ as\ usually*) \[IndentingNewLine]\[IndentingNewLine] Wt[n, u_, r_, x_] := ta[n, u, x] f[n, r/Sqrt[\((r^2\ + \ ta[n, u, x]^2)\)]]/ Sqrt[\((r^2\ + \ ta[n, u, x]^2)\)]\ - \ r\ ta[n, u, x] f1[n, r/Sqrt[\((r^2\ + \ ta[n, u, x]^2)\)]]\ /\((r^2\ + \ ta[n, u, x]^2)\)\ + \ x\ ta[n, u, x] g[x, n, r/ Sqrt[\((r^2\ + \ ta[n, u, x]^2)\)]] \((r^2\ + \ ta[n, u, x]^2)\)^\((\((x - 2)\)/2)\) - \ r\ ta[n, u, x] g1[x, n, r/Sqrt[\((r^2\ + \ ta[n, u, x]^2)\)]]\ \((r^2\ + \ ta[n, u, x]^2)\)^\ \((\((x - 3)\)/2)\)\[IndentingNewLine] kt[y_, n_, u_, r_, x_] := \ \(-\ H[n, u, r, x]\)/\((s[n, u, x] A[n, u, x])\) + \ s[n, u, x] A[n, u, x] v2[y, n, u, ta[n, u, x], x]/\((1 + A[n, u, x]^2)\) H[n, u, r, x]\ + \ r\ H1[n, u, r, x]/\((s[n, u, x] A[n, u, x])\)\[IndentingNewLine]\[IndentingNewLine]\ \[IndentingNewLine] (*We\ set\ the\ dimension\ equal\ to\ 7\ *) \[IndentingNewLine] n := 7\[IndentingNewLine]\[IndentingNewLine] (*We\ choose\ \\tau\ and\ \ \\gamma_ 7\ so\ that\ G\ decreases\ on\ [t_ 7, gamma_ 7]\ *) \[IndentingNewLine]\[IndentingNewLine]\ \[IndentingNewLine] x := \(-1.756\)\[IndentingNewLine] u := 0.62\[IndentingNewLine] t7 := 0.517331\[IndentingNewLine] Plot[F[n, x, r]\ - \ F[n, x, t7], {r, 0.517331, 0.62}, PlotLabel -> "\"\ ]\[IndentingNewLine]\ \[IndentingNewLine] (*We\ print\ \\bar {s}\ and\ \\bar \(\({r}\)\(.\)\)\ *) \[IndentingNewLine] s[7, u, x]\[IndentingNewLine] ta[7, u, x]\[IndentingNewLine]\[IndentingNewLine] (*We\ check\ that\ A + \ \ \(\(\\\)\(beta\)\)\ < 0. *) \[IndentingNewLine] A[7, u, x] + be[u]\[IndentingNewLine]\[IndentingNewLine] (*Plotting\ their\ graphs\ \ on\ [0, \(\(\\\)\(bar\)\) {s}], \ we\ check\ that\ H \((r)\) > 0, \ H' \((r)\) < 0\ and\ H'' \((r)\) < 0. *) \[IndentingNewLine] Plot[H[n, u, r, x], {r, 0, 0.41000557239822627`}, \ PlotLabel \[Rule] "\"]\[IndentingNewLine] Plot[H1[n, u, r, x], {r, 0, 0.41000557239822627`}, PlotLabel \[Rule] "\"]\[IndentingNewLine] Plot[H2[n, u, r, x], {r, 0, 0.41000557239822627`}, PlotLabel \[Rule] "\"]\[IndentingNewLine]\ \[IndentingNewLine] (*We\ pick\ y = a_ 7\ so\ that\ \ v, \ K2 = \(\(\\\)\(bar\)\) {K}, \ K4 = \(M\ and\ K5 = N\ have\ all\ of\ the\ required\ properties\), \ which\ we\ verify\ by\ plotting\ their\ \(\(graphs\)\(.\)\)\ *) \[IndentingNewLine] y := \(-0.36640258708784224`\)\[IndentingNewLine]\[IndentingNewLine] (*\ We\ evaluate\ b_ 7\ and\ v_ 7\ *) \[IndentingNewLine] al[y, n, u, x]\[IndentingNewLine] ci[y, n, u, x]\[IndentingNewLine]\[IndentingNewLine] Plot[v[y, n, u, r, x], {r, 0, 0.5188566357210244`}, PlotLabel \[Rule] "\"]\[IndentingNewLine] Plot[K2[y, n, u, r, x], {r, 0, 0.5188566357210244`}, PlotLabel \[Rule] "\"]\[IndentingNewLine] Plot[K4[y, n, u, r, x], {r, 0, 0.5188566357210244`}, PlotLabel \[Rule] "\"]\[IndentingNewLine] Plot[K5[y, n, u, r, x], {r, 0, 0.5188566357210244`}, PlotLabel \[Rule] "\"]\[IndentingNewLine]\ \[IndentingNewLine] (*\ We\ verify\ that\ \\mathcal {W}\ < \ 0\ *) \[IndentingNewLine] Plot[Wt[n, u, r, x] - kt[y, n, u, r, x], {r, 0, 0.41000557239822627`}, PlotLabel \[Rule] "\"]\[IndentingNewLine]\ \[IndentingNewLine] \)\)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations -4.77507 9.27623 0.603312 0.0515192 [ [.04857 .59081 -12 -9 ] [.04857 .59081 12 0 ] [.23409 .59081 -12 -9 ] [.23409 .59081 12 0 ] [.41962 .59081 -12 -9 ] [.41962 .59081 12 0 ] [.60514 .59081 -12 -9 ] [.60514 .59081 12 0 ] [.97619 .59081 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