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Algebraic Aspects of Integrable Systems: In Memory of Irene by Mark S. Alber, Gregory G. Luther (auth.), A. S. Fokas, I. M.

By Mark S. Alber, Gregory G. Luther (auth.), A. S. Fokas, I. M. Gelfand (eds.)

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Extra resources for Algebraic Aspects of Integrable Systems: In Memory of Irene Dorfman

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Proof of Theorem 2. 7) together with the cyclic symmetry just proved implies that Tn (a, b, c; x, y, z) is invariant with respect to cyclic permutations of the three pairs of variables (a,x), (b,y), and (c,z). 8) that Tn(a, b, c; x, y, z) is an even function of b and of c. From the cyclic invariance it follows that the three variables a, b, and c play equal roles, so it is also an even function of a. 8) we see that Tn is unchanged if we interchange (b, y) and (c, z) and simultaneously replace a by -a, so we obtain also the invariance of Tn under odd permutations of (a, x), (b, y), and (c, z).

The equations L *wa = zw a, and 8ko wa = - (L kRo) ~ wa hold. Remark 2. Our definition of the mcKP differs from that in [1]'[2J and [3J where Uo = 0 and A = I.

Zagier, Modular forms and differential operators, Proc. Indian Acad. Sci. (Math. ) 104 (1994), 57-75. [Z2] D. Zagier, Some combinatorial identities occurring in the theory of modular forms, in preparation. Paula Beazley Cohen URA Geometrie-Analyse-Topologie, UFR de Mathematiques, Universite des Sciences et Technologies de Lille 59655 Villeneuve d'Ascq cedex, France Yuri Manin and Don Zagier Max-Planck Institut fur Mathematik Gottfried-Claren str. 26, 53225 Bonn, Germany Received October 1995: additions December 1995 On T -FUnctions of Zakharov-Shabat and Other Matrix Hierarchies of Integrable Equations L.

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