Alex au pays des chiffres by Alex Bellos

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By Alex Bellos

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For example, J23,24 = adj N23,24 = adj = a22 a32 A22 A24 a24 a34 A32 . A34 The Jacobi identity on the minors of adj A is given by the following theorem: Theorem. qr , 1 ≤ r ≤ n − 1. 7, it is seen that if A = 0, r > 1, then J = 0. The right-hand side of the above identity is also zero. Hence, in this particular case, the theorem is valid but trivial. When r = 1, the theorem degenerates into the definition of Ap1 q1 and is again trivial. It therefore remains to prove the theorem when A = 0, r > 1.

16) Hence, (n+1) (n+1) (n) An P = An+1 A(n) rr An,n+1;r,n+1 − Anr Ar,n+1;r,n+1 . 6 The Jacobi Identity and Variants (n+1) (n) But Ai,n+1;j,n+1 = Aij . Hence, An P = 0. The result follows. 45 ✷ Three particular cases of (B) are required for the proof of the next theorem. Put (i, p, q) = (r, r, n), (n − 1, r, n), (n, r, n) in turn: (n) (n) Arr (n+1) An+1,r (n+1) Arn (n+1) − An Ar,n+1;rn = 0, An+1,n An−1,r (n) An−1,n (n+1) An+1,r (n) (n+1) An+1,n (n) (B1 ) (n+1) − An An−1,n+1;rn = 0, (B2 ) (n) Anr (n+1) An+1,r (n+1) Ann (n+1) − An An,n+1;rn = 0.

Intermediate Determinant Theory Hence, removing the factor An from each row, |cij |n = Ann δij xi + Hij An n which yields the stated result. 4 on the K dV equation. 1. qr Ap1 q1 Ap2 q1 · · · Apr q1 Ap1 q2 Ap2 q2 · · · Apr q2 = ......................... Ap1 qr Ap2 qr · · · Apr qr . 1) r J is a minor of adj A. For example, J23,24 = adj N23,24 = adj = a22 a32 A22 A24 a24 a34 A32 . A34 The Jacobi identity on the minors of adj A is given by the following theorem: Theorem. qr , 1 ≤ r ≤ n − 1. 7, it is seen that if A = 0, r > 1, then J = 0.

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