\(\(\( (*\ 
      We\ specify\ the\ relation\ between\ the\ notation\ of\ the\ article\ and\ the\ notation\ used\ in\ this\ file, \ 
      using\ latex\ notation\ *) \)\(
\)\(
\)\( (*\ 
      Given\ the\ symmetry\ of\ all\ the\ functions\ involved, \ 
      all\ the\ computations\ are\ carried\ out\ for\ s ≥ 
        0\ *) \)\(
\)\(
\)\( (*We\ set\ the\ following\ notation*) \)\(
\)\(
\)\(
\)\( (*\ 
      All\ the\ independent\ variables\ are\ denoted\ by\ r\ *) \)\(
\)\(
\)\( (*\ 
      LegendreQ[a, c, r] = 
        Q_ {a}^{c} \((r)\)\ *) \)\(
\)\(
\)\(Q[n_, r_] := 
      LegendreQ[\((n - 1)\)/2, \((n - 3)\)/2, r]

    q[n_, r_] := 
      LegendreQ[\((n - 1)\)/2, \((n - 1)\)/2, 
        r]

 (*\ 
      LegendreP[a, c, r] = P_ {a}^{c} \((r)\)\ *) 
 (*\ 
      x\ denotes\ the\ parameter\ \tau\ *) 


    P[x_, n_, r_] := 
      LegendreP[x + \((n - 3)\)/2, \(-\((n - 3)\)\)/2, r]

    pi[x_, n_, r_] := 
      LegendreP[x + \((n - 3)\)/2, \ \(-\((n - 1)\)\)/2, 
        r]

 (*\ 
      V[x, n, r]\ denotes\ E_n\ *) 

    V[x_, n_, 
        r_] := \((P[x, n, r] + 
          P[x, n, \(-r\)])\)

 (*\ 
      Z[x, n, r]\ denotes\ Z_n\ *) 

    Z[x_, n_, 
        r_] := \((pi[x, n, r] - 
          pi[x, n, \(-r\)])\)


    S[n_] := Sign[Q[n, 0]]

 (*\ 
      The\ following\ is\ the\ formula\ for\ f_n\ *) 

    f[n_, r_] := 
      S[n] \((1 - r^2)\)^\((\(-\((n - 3)\)\)/4)\) 
        Q[n, r]

 (*The\ following\ are\ the\ formulas\ for\ respectively, \ f_n', \ f_n''*) 

    f1[n_, r_] := \(-S[n]\) \((1 - r^2)\)^\((\(-\((n - 1)\)\)/4)\) 
        q[n, r]

    f2[n_, 
        r_] := \ {\((n - 1)\) r\ f1[n, r]\  - \((n - 1)\) f[n, r]}/{\((1 - 
              r^2)\)}

 (*\ 
      The\ following\ is\ the\ formula\ for\ g_ {\(\\)\(tau\)}\ *) 

    g[x_, n_, r_] := \((1 - r^2)\)^\((\(-\((n - 3)\)\)/4)\) 
        V[x, n, r]/2

 (*\ 
      The\ following\ are\ the\ formulas\ for\ respectively, \ 
      g' _ {\(\\)\(tau\)}, \ g'' _ {\(\\)\(tau\)}\ *) 

    g1[x_, n_, r_] := \ 
      x \((x + n - 2)\) \((1 - r^2)\)^\((\(-\((n - 1)\)\)/4)\) 
        Z[x, n, r]/2

    g2[x_, n_, 
        r_] := \ {\((n - 1)\) r\ g1[x, n, r]\  - 
            x \((x + n - 2)\) g[x, n, r]}/{\((1 - 
              r^2)\)}

 (*\ 
      be[u]\ denotes\ \beta_n, \ 
      and\ u\ denotes\ \gamma_n\ *) 

    be[u_] := 
      Sqrt[\((1 - u^2)\)/\((u^2)\)]

 (*\ 
      s[n, u, x]\ denotes\ \bar {s}, \ 
      and\ ta[n, u, x]\ denotes\ \bar {r}\ *) 

    s[n_, u_, x_] := 
      u\ {\(-g[x, n, u]\)/f[n, u]}^\((1/\((1 - x)\))\)

    ta[n_, u_, x_] := 
      be[u] s[n, u, x]

 (*\ 
      A[n, u, x]\ denotes\  - 
          A \((\(\\)\(beta_n\))\) . \ 
We\ need\ to\ check\ that\ \beta_n\  + \ A\  < \ 0\ *) 

    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]}

 (*\ 
      F[n, x, r]\ denotes\ G_n\ *) 

    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

 (*\ 
      The\ following\ formulas\ are\ relevant\ to\ Step\ 2*) 

 (*\ 
      H[n, u, r, x]\ denotes\ h \((r)\)\ *) 

    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[\((r^2\  + \ 
                    ta[n, u, 
                        x]^2)\)]]

 (*\ 
      H1[n, u, r, x]\ denotes\ h' \((r)\)\ *) 

    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[\((r^2\  + \ 
                        ta[n, u, 
                            x]^2)\)]]}

 (*\ H2[n, u, r, x]\ denotes\ h'' \((r)\)\ *) 

    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)\)]]\ }

 (*\ 
      al[y, n, u, x]\ denotes\ the\ coefficient\ b_  
        7\ of\ the\ polynomial\ v*) 

    al[y_, n_, u_, 
        x_] := \(-{be[u]/\((A[n, u, x] ta[n, u, x])\) + 
              4\ y\ ta[n, u, x]^3\ }\)/\((2\ ta[n, u, 
              x])\)

 (*\ 
      ci[y, n, u, x]\ denotes\ the\ coefficient\ v_  
        7\ of\ the\ polynomial\ v*) 

    ci[y_, n_, u_, x_] := \ 
      1\  - \ y\ ta[n, u, x]^4\  - \ 
        al[y, n, u, x] 
          ta[n, u, x]^2

 (*\ 
      v[y, n, u, r, x]\ denotes\ the\ polynomial\ v . \ 
          y\ denotes\ the\ coefficient\ a_  7\ *) 

    v[y_, n_, u_, r_, x_] := \ 
      y\ r^4\  + al[y, n, u, x]\ r^2\  + \ 
        ci[y, n, u, x]

 (*b_  7\ and\ v_ 
           7\ are\ determined\ via\ the\ conditions\ \((4.3)\) - \((4.4)\)\ in\ the\ article\ *) 

 (*The\ following\ are\ the\ formulas\ for\ respectively\ v' \((r)\), \ v' \((r)\)/r, \ 
      v'' \((r)\)\ and\ v''' \((r)\)\ *) 


    v1[y_, n_, u_, r_, x_] := \ \ 4\ y\ r^3\  + \ 
        2\ al[y, n, u, x]\ r

    rv1[y_, n_, u_, r_, x_] := \ 
      4\ y\ r^2\  + 2\ al[y, n, u, x]

    v2[y_, n_, u_, r_, x_] := \ 
      12\ y\ r^2 + 2\ al[y, n, u, x]

    v3[y_] := 24\ y\ r

 (*\ 
      K4[y, n, u, r, x]\ denotes\ M\ *) 

    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]/\((1 + 
                  s[n, u, x]^2\ v1[y, n, u, r, 
                        x]^2)\)}

 (*\ 
      K5[y, n, u, r, x]\ denotes\ N\ *) 

    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] \((1 + \ s[n, u, x]^2\ v1[y, n, u, r, x]^2\  - \ 
                  s[n, u, x]^2\ v[y, n, u, r, x] 
                    v2[y, n, u, r, x])\)/\((1 + \ 
                  s[n, u, x]^2\ v1[y, n, u, r, 
                        x]^2)\)}

 (*\ 
      In\ the\ formulas\ above, \ 
      the\ derivatives\ of\ the\ function\ y\ are\ computed\ in\ temrs\ of\ the\ function\ v\ *) 

 (*\ 
      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]\ *) 

 (*\ 
      K2[y, n, u, r, x]\ denotes\ \bar {K}\ *) 

    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]

 (*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*) 


    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)\)

    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]/\((s[n, u, x] 
                A[n, u, 
                  x])\)


 (*We\ set\ the\ dimension\ equal\ to\ 7*) 

    n := 7

 (*We\ choose\ \tau\ and\ \gamma_  
        7\ so\ that\ G\ decreases\ on\ [t_  7, 
          gamma_  7]\ *) 



    x := \(-1.756\)

    u := 0.62

    t7 := 0.517331

    Plot[F[n, x, r]\  - \ F[n, x, t7], {r, 0.517331, 0.62}, 
      PlotLabel -> "F[n,x,r] - F[n,x,t7]"\ ]

 (*We\ print\ \bar {s}\ and\ \bar \(\({r}\)\(.\)\)*) 

    s[7, u, x]

    ta[7, u, 
      x]

 (*We\ check\ that\ A + \ \(\(\\)\(beta\)\)\  < 0. *) 

    A[7, u, x] + 
      be[u]

 (*Plotting\ their\ graphs\ on\ [0, \(\(\\)\(bar\)\) {s}], \ we\ check\ that\ H \((r)\) > 0, \ 
      H' \((r)\) < 0\ and\ H'' \((r)\) < 0. *) 

    Plot[H[n, u, r, x], {r, 0, 0.41000557239822627`}, \ 
      PlotLabel -> "H[n,u,r,x]"]

    Plot[H1[n, u, r, x], {r, 0, 0.41000557239822627`}, 
      PlotLabel -> "H1[n,u,r,x]"]

    Plot[H2[n, u, r, x], {r, 0, 0.41000557239822627`}, 
      PlotLabel -> "H2[n,u,r,x]"]

 (*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\ \(\(graphs\)\(.\)\)*) 

    y := \(-0.36640258708784224`\)

 (*\ 
      We\ evaluate\ b_  7\ and\ v_  7\ *) 

    al[y, n, u, x]

    ci[y, n, u, x]


    Plot[v[y, n, u, r, x], {r, 0, 0.5188566357210244`}, 
      PlotLabel -> "v[y,n,u,r,x]"]

    Plot[K2[y, n, u, r, x], {r, 0, 0.5188566357210244`}, 
      PlotLabel -> "K2[y,n,u,r,x]"]

    Plot[K4[y, n, u, r, x], {r, 0, 0.5188566357210244`}, 
      PlotLabel -> "K4[y,n,u,r,x]"]

    Plot[K5[y, n, u, r, x], {r, 0, 0.5188566357210244`}, 
      PlotLabel -> "K5[y,n,u,r,x]"]

 (*\ 
      We\ verify\ that\ \mathcal {W}\  < \ 0\ *) 

    Plot[Wt[n, u, r, x] - kt[y, n, u, r, x], {r, 0, 0.41000557239822627`}, 
      PlotLabel -> "Wt[n,u,r,x]-kt[y,n,u,r,x]"]


    \)\)\)