Volume 12, Issue Supplement 2, December 2005

Foreword

Pages: 1 - 14

It is shown that discrete analogs of zc
and log(z), defined via particular "integrable"
circle patterns, have the same asymptotic behavior as their smooth counterparts.
These discrete maps are described in terms of special solutions of discrete Painlevé-II
equations, asymptotics of these solutions...

Pages: 15 - 27

By associating polynomials and power series expansions with sln(C) modules we dscribe the theory of Padé approximants in terms of tensor products of representations
and interpret their recurrence relations algebraically. The treatment links with the
theory of Hirota derivatives and discrete integrable...

Pages: 28 - 35

First, we recall the algebro-geometric method of construction of finite field valued solutions of the discrete KP equation, and next we perform a reduction of dKP to the discrete 1D Toda equation.

Pages: 36 - 45

The theory of dynamic inclusions on a time scale is introduced, hence accommodating
the special cases of differential inclusions and difference inclusions. Fixed point theory
for set-valued upper semicontinuous maps, Green's functions, and upper and lower
solutions are used to establish existence...

Pages: 46 - 62

The Somos 4 sequences are a family of sequences satisfying a fourth order bilinear
recurrence relation. In recent work, one of us has proved that the general term in
such sequences can be expressed in terms of the Weierstrass sigma function for an
associated elliptic curve. Here we derive the analogous...

Pages: 63 - 76

We describe a few families of orthogonal matrix polynomials of size N × N satisfying
first order differential equations. This problem differs from the recent efforts reported
for instance in [7] (Orthogonal matrix polynomials satisfying second order differential
equations, Internat. Math. Research...

Pages: 77 - 94

We show that the m-dimensional EulerManakov top on so
(m) can be
represented as a Poisson reduction of an integrable Hamiltonian system on a
symplectic extended Stiefel variety ¯V(k, m), and present its Lax representation
with a rational parameter.
We also describe an integrable two-valued symplectic...

Pages: 95 - 136

We prove bispectral duality for the generalized CalogeroMoserSutherland systems
related to configurations An,2(m), Cn(l, m). The trigonometric axiomatics of the
BakerAkhiezer function is modified, the dual difference operators of rational Madonald type and the BakerAkhiezer functions related to...

Pages: 137 - 146

It is known that many equations of interest in Mathematical Physics display solutions
which are only asymptotically invariant under transformations (e.g. scaling and/or
translations) which are not symmetries of the considered equation. In this note we
extend the approach to asymptotic symmetries for...

Pages: 147 - 152

Pages: 155 - 168

A family of real Hamiltonian forms (RHF) for the special class of affine 1 + dimensional Toda field theories is constructed. Thus the method, proposed in [1] for
systems with finite number of degrees of freedom is generalized to infinite-dimensional
Hamiltonian systems. We show that each of these RHF...

Pages: 169 - 179

It is shown that the 2-discrete dimensional Lotka-Volterra lattice, the two dmensional Toda lattice equation and the recent 2-discrete dimensional Toda lattice
equation of Santini et al can be obtained from a 2-discrete 2-continuous dimensional
Lotka-Volterra lattice.

Pages: 180 - 196

In a recent paper we introduced a new 2 + 1-dimensional non-isospectral extension
of the Volterra lattice hierarchy, along with its corresponding hierarchy of underlying
linear problems. Here we consider reductions of this lattice hierarchy to hierarchies of
discrete equations, which we obtain once...

Pages: 197 - 205

The problem of the classification of integrable truncations of the Toda chain is dicussed. A new example of the cutting off constraint is found.

Pages: 206 - 222

In this paper one considers the problem of finding solutions to a number of Todtype hierarchies. All of them are associated with a commutative subalgebra of the
k×k-matrices. The first one is formulated in terms of upper triangular Z×Z-matrices,
the second one in terms of lower triangular ones and...

Pages: 223 - 230

For two-dimensional lattice equations the standard definition of integrability is that it
should be possible to extend the map consistently to three dimensions, i.e., that it is
"consistent around a cube" (CAC). Recently Adler, Bobenko and Suris conducted a
search based on this principle, together...

Pages: 231 - 238

We present q-discretizations of a second order differential equation in two independent
variables that not only go to the differential counterpart as q goes to 1 but admit
Moutard-Darboux transformations as well.

Pages: 240 - 252

We consider a longrange homogeneous chain where the local variables are the geerators of the direct sum of N e(3) interacting Lagrange tops. We call this classical
integrable model rational "Lagrange chain" showing how one can obtain it starting
from su(2) rational Gaudin models. Moreover we construct...

Pages: 253 - 294

For positive parameters a+ and a- the commuting difference operators exp(ia±d/dz)
+ exp(2z/a), acting on meromorphic functions f(z), z = x + iy, are formally
self-adjoint on the Hilbert space H = L2
(R, dx). Volkov showed that they admit
joint eigenfunctions. We prove that the joint eigenfunctions...

Pages: 295 - 299

In this paper an extension of the q-deformed Volterra equation associated with linear
rescaling to the general non-linear rescaling is obtained.

Pages: 300 - 315

In this paper, we prove some effects concerning a Fuzzy Difference Equation of a
rational form.

Pages: 316 - 332

In this paper, we first present the Casorati and grammian determinant solutions to a
special two-dimensional lattice by Blaszak and Szum. Then, by using the pfaffianiztion procedure of Hirota and Ohta, a new integrable coupled system is generated from
the special lattice. Moreover, gram-type pfaffian...

Pages: 333 - 356

We consider the generalized eigenvalue problem A = B for two operators A, B.
Self-similar closure of this problem under a simplest Darboux transformation gives rise
to two possible types of regular algebras of dimension 2 with generators A, B. Realiztion of the operators A, B by tri-diagonal operators...