The paper gives recurrence relations on the number of 2n step fully extended (i.e., full-dimensional) self-avoiding polygons in n-and (n-1)-dimensional integer lattices.

In this letter we present the set of invariant difference equations and meshes which preserve the Lie group symmetries of the equation ut = (K(u)ux)x+Q(u). All special cases of K(u) and Q(u) that extend the symmetry group admitted by the differential equation are considered. This paper completes the paper [J. Phys. A: Math. Gen. 30, Nr. 23 (1997), 8139–8155], where a few invariant models for he...

The generating functional of correlation functions for theXXZ spin chain is considered in the thermodynamic limit. We derive a system of integro-difference equations that prescribe this functional. On the basis of this system we establish the operator-valued Riemann–Hilbert problem for correlation functions of the XXZ spin chain.

Sufficient conditions are established for the oscillation of the linear two-dimensional difference system ∆xn = pn yn, ∆yn−1 = −qnxn, n ∈ N (n0) = {n0, n0 + 1, . . .}, where {pn}, {qn} are nonnegative real sequences. Our results extend the results in the literature. c © 2007 Elsevier Ltd. All rights reserved.