Related papers: Hermite-Pad\'{e} approximation, multiple orthogona…
We formulate a well-posedness and approximation theory for a class of generalised saddle point problems with a specific form of constraints. In this way we develop an approach to a class of fourth order elliptic partial differential…
We discuss geometrical aspects of Toda Fields generalizing the links between Liouville gravity and uniformization of Riemann surfaces of genus greater than one. The framework is the interplay between the hermitian and the holomorphic…
In this paper we study sequences of vector orthogonal polynomials. The vector orthogonality presented here provides a reinterpretation of what is known in the literature as matrix orthogonality. These systems of orthogonal polynomials…
This paper is concerned with space-time homogenization problems for damped wave equations with spatially periodic oscillating elliptic coefficients and temporally (arithmetic) quasi-periodic oscillating viscosity coefficients. Main results…
Burchnall's method to invert the Feldheim-Watson linearization formula for the Hermite polynomials is extended to all polynomial families in the Askey-scheme and its $q$-analogue. The resulting expansion formulas are made explicit for…
Some properties and relations satisfied by the polynomial solutions of a bispectral problem are studied. Given a finite order differential operator, under certain restrictions, its polynomial eigenfunctions are explicitly obtained, as well…
The complex Plateau problem is analogous, in a Hermitian complex manifold, to the classical Plateau problem in 3 dimensional real space: it is a geometrical problem of extension of a closed real manifold into a complex analytic subvariety,…
Matrix functions are a central topic of linear algebra, and problems requiring their numerical approximation appear increasingly often in scientific computing. We review various limited-memory methods for the approximation of the action of…
Orthogonal polynomials of two real variables can often be represented in complex variables. We explore the connection between the two types of representations and study the structural relations of complex orthogonal polynomials. The complex…
The correspondence between a high-order non symmetric difference operator with complex coefficients and the evolution of an operator defined by a Lax pair is established. The solution of the discrete dynamical system is studied, giving…
We present a variational theory of integrable differential-difference equations (semi-discrete integrable systems). This is a natural extension of the ideas known by the names "Lagrangian multiforms" and "Pluri-Lagrangian systems", which…
Exact rational solutions of the generalized Hunter-Saxton equation are obtained using Pad\'e approximant approach for the traveling-wave and self-similarity reduction. A larger class of algebraic solutions are also obtained by extending a…
We discuss a fast approximate solution to the associated classical -- classical orthogonal polynomial connection problem. We first show that associated classical orthogonal polynomials are solutions to a fourth-order quadratic eigenvalue…
A new direct method of obtaining reductions of the Toda equation is described. We find a canonical and complete class of all possible reductions under certain assumptions. The resulting equations are ordinary differential-difference…
We consider a two dimensional extension of the so-called linearizable mappings. In particular, we start from the Heideman-Hogan recurrence, which is known as one of the linearizable Somos-like recurrences, and introduce one of its two…
It is shown that the problem of calculating form factors in ADE affine Toda field theories can be reduced to the nonperturbative recursive calculation of polynomials symmetric in each sort of variables. We determine these recursion…
The new method for obtaining a variety of extensions of Hermite polynomials is given. As a first example a family of orthogonal polynomial systems which includes the generalized Hermite polynomials is considered. Apparently, either these…
We show that the formalism of hybrid polynomials, interpolating between Hermite and Laguerre polynomials, is very useful in the study of Motzkin numbers and central trinomial coefficients. These sequences are identified as special values of…
By applying the Hamiltonian reduction method to the cotangent bundle over loop groups we recover the well-known classical trigonometric $r$-matrices of the periodic Toda lattice.
This book is mainly an exposition of the author's works and his joint works with his former students on explicit representations of finite-dimensional simple Lie algebras, related partial differential equations, linear orthogonal algebraic…