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

We list the so-called Askey-scheme of hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence relation and generating functions of all classes of orthogonal polynomials in this scheme. In chapeter 2 we give all limit relation between different classes of orthogonal polynomials listed in the Askey-scheme. 
In chapter 3 we list the q-analogues of the polynomials in the Askey-scheme. We give their definition, orthogonality relation, three term recurrence relation and generating functions. In chapter 4 we give the limit relations between those basic hypergeometric orthogonal polynomials. Finally in chapter 5 we point out how the `classical` hypergeometric orthogonal polynomials of the Askey-scheme can be obtained from their q-analogues.

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The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue

Combinatorial Group Theory: COMBINATORIAL GROUP THEORY

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

Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or regularity) and expanding the breadth of its universality class. Over the past 25 years a new universality class has emerged to describe a host of important physical and probabilistic models (including one-dimensional interface growth processes, interacting particle systems and polymers in random environments) which display characteristic, though unusual, scalings and new statistics. This class is called the Kardar–Parisi–Zhang (KPZ) universality class and underlying it is, again, a continuum object — a non-linear stochastic partial differential equation — known as the KPZ equation. The purpose of this survey is to explain the context for, as well as the content of a number of mathematical breakthroughs which have culminated in the derivation of the exact formula for the distribution function of the KPZ equation started with narrow wedge initial data. In particular we emphasize three topics: (1) The approximation of the KPZ equation through the weakly asymmetric simple exclusion process; (2) The derivation of the exact one-point distribution of the solution to the KPZ equation with narrow wedge initial data; (3) Connections with directed polymers in random media. As the purpose of this article is to survey and review, we make precise statements but provide only heuristic arguments with indications of the technical complexities necessary to make such arguments mathematically rigorous.

The kardar-parisi-zhang equation and universality class

Macdonald's Symmetric Polynomials as Zonal Spherical Functions on Some Quantum Homogeneous Spaces

A new class of representations of affine Weyl groups on rational functions are constructed, in order to formulate discrete dynamical systems associated with affine root systems. As an application, some examples of difference and differential systems of Painleve type are discussed.

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Affine Weyl Groups, Discrete Dynamical Systems and Painlevé Equations

A series of systems of nonlinear equations with affine Weyl group symmetry of type $A^{(1)}_l$ is studied. This series gives a generalization of Painleve equations $P_{IV}$ and $P_{V}$ to higher orders.

Higher order Painlevé equations of type $A^{(1)}_l$

The fourth Painlev\'e equation $P_{IV}$ is known to have symmetry of the affine Weyl group of type $A^{(1)}_2$ with respect to the Backlund transformations. We introduce a new representation of $P_{IV}$, called the symmetric form , by taking the three fundamental invariant divisors as the dependent variables. A complete description of the symmetry of $P_{IV}$ is given in terms of this representation. Through the symmetric form, it turns out that $P_{IV}$ is obtained as a similarity reduction of the 3-reduced modified KP hierarchy. It is proved in particular that the special polynomials for rational solutions $P_{IV}$, called Okamoto polynomials , are expressible in terms of the 3-reduced Schur functions.

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Masatoshi Noumi

Papers

Macdonald's Symmetric Polynomials as Zonal Spherical Functions on Some Quantum Homogeneous Spaces

Affine Weyl Groups, Discrete Dynamical Systems and Painlevé Equations

Higher order Painlevé equations of type $A^{(1)}_l$

Symmetries in the fourth Painlevé equation and Okamoto polynomials

Painlevé Equations through Symmetry