R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.

Exactly Solved Models in Statistical Mechanics

One-dimensional Bose-gas One-dimensional Heisenberg magnet Massive Thirring model Classical r-matrix Fundamentals of inverse scattering method Algebraic Bethe ansatz Quantum field theory integral models on a lattice Theory of scalar products Form factors Mean value of operator Q Assymptotics of correlation functions Temperature correlation functions Appendices References.

Quantum Inverse Scattering Method and Correlation Functions

Preface to the second edition Preface to the first edition 1. Background 2. Orthogonal polynomials in two variables 3. General properties of orthogonal polynomials in several variables 4. Orthogonal polynomials on the unit sphere 5. Examples of orthogonal polynomials in several variables 6. Root systems and Coxeter groups 7. Spherical harmonics associated with reflection groups 8. Generalized classical orthogonal polynomials 9. Summability of orthogonal expansions 10. Orthogonal polynomials associated with symmetric groups 11. Orthogonal polynomials associated with octahedral groups and applications References Author index Symbol index Subject index.

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Orthogonal Polynomials of Several Variables

Groupes et algébres de lie

An interesting phenomenon in water channels is the appearance of waves
with length much greater than the depth of the water. In 1895 D. J. Korteweg
and G. de Vries started the mathematical theory of this phenomenon and derived
a model describing unidirectional propagation of waves of the free surface of
a shallow layer of water. This is the well-known KdV equation

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Global existence and blow-up for a shallow water equation

A general class of n‐particle difference Calogero–Moser systems with elliptic potentials is introduced. Besides the step size and two periods, the Hamiltonian depends on nine coupling constants. We prove the quantum integrability of the model for n=2 and present partial results for n≥ 3. In degenerate cases (rational, hyperbolic, or trigonometric limit), the integrability follows for arbitrary particle number from previous work connected with the multivariable q‐polynomials of Koornwinder and Macdonald. Liouville integrability of the corresponding classical systems follows as a corollary. Limit transitions lead to various well‐known models such as the nonrelativistic Calogero–Moser systems associated with classical root systems and the relativistic Calogero–Moser system.

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Integrability of difference Calogero-Moser systems

We introduce new Selberg-type multidimensional integrals built of Ruijsenaars' elliptic gamma functions. We show that the vanishing of our integrals for a specific parameter hypersurface implies closed evaluation formulas valid for the full parameter space. The resulting integration formulas contain the Macdonald-Morris constant term identities for nonreduced root systems as special limiting cases.

Elliptic Selberg integrals

We prove certain duality properties and present recurrence relations for a four-parameter family of self-dual Koornwinder-Macdonald polynomials. The recurrence relations are used to verify Macdonald's normalization conjectures for these polynomials.

Self-dual Koornwinder-Macdonald polynomials

Limits of a recently introduced n‐particle difference Calogero–Moser system with elliptic potentials are studied. We obtain hyperbolic and rational difference Calogero–Moser systems with an eight‐parameter external field and (finite) difference Toda chains with four‐parameter potentials acting on the boundary particles. Hamiltonians for a number of known integrable n‐particle systems, such as Ruijsenaars’ relativistic Calogero–Moser and Toda models and their generalizations associated with classical root systems, can be seen as special cases of the Hamiltonians considered in this paper.

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Difference Calogero-Moser systems and finite Toda chains

We present explicit generators of an algebra of commuting difference operators with trigonometric coefficients. The operators are simultaneously diagonalized by recently discovered q-polynomials (viz. Koornwinder's multivariable generalization of the Askey-Wilson polynomials). From the viewpoint of physics the algebra can be interpreted as consisting of the quantum integrals of a novel difference-type integrable sytem. This system generalizes the Calogero-Moser systems associated with non-exceptional root systems.

J. F. van Diejen

Papers

Integrability of difference Calogero-Moser systems

Elliptic Selberg integrals

Self-dual Koornwinder-Macdonald polynomials

Difference Calogero-Moser systems and finite Toda chains

Commuting difference operators with polynomial eigenfunctions