The motivation of this paper comes from statistical physics as well as from combinatorics and topology. The general setup in statistical mechanics can be outlined as follows. Let G be a graph and let C be a finite set of “states” or “colors”. We think of G as a crystal in which either the edges or the vertices are regarded as “sites” which can have states from C. In the first case we speak abou...

Introduction Recently I.Macdonald deened a family of systems of orthogonal symmetric poly-nomials depending on two parameters q; k which interpolate between Schur's symmetric functions and certain spherical functions on SL(n) over the real and p-adic elds M]. These polynomials are labeled by dominant integral weights of SL(n), and (as was shown by I.Macdonald) are uniquely deened by two conditi...

It is shown that a (curved) projective structure on a smooth manifold determines on the Poisson algebra of smooth functions on the cotangent bundle, fiberwise-polynomial of bounded degree, a one-parameter family of graded star products. For a particular value of the parameter (corresponding to half-densities) the star product is symmetric, and specializes in the projectively flat case to the on...

The quadratic shortest path problem (QSPP) is the problem of finding a path with prespecified start vertex s and end vertex t in a digraph such that the sum of weights of arcs and the sum of interaction costs over all pairs of arcs on the path is minimized. We first consider a variant of the QSPP known as the adjacent QSPP. It was recently proven that the adjacent QSPP on cyclic digraphs cannot...

1. Weighted enumeration For a vector x = (x1, . . . , xk) of k distinct real numbers, define Π(x) to be the standardization of the vector x, that is, the unique permutation σ = (σ1, . . . , σk) in Sk such that for all indices 1 ≤ i < j ≤ k the inequality xi < xj is equivalent to σi < σj. Let S be a set of permutations in the symmetric group Sm+1. We say that a permutation π in Sn avoids the set...

commutative properties of four-dimensional generalized symmetric pseudo-riemannian manifolds were considered. specially, in this paper, we studied skew-tsankov and jacobi-tsankov conditions in 4-dimensional pseudo-riemannian generalized symmetric manifolds.

We solve a class of isoperimetric problems on ℝ + 2 with respect to monomial weights. Let α and β be real numbers such that 0 ≤ &lt; 1, . show that, among all smooth sets Ω in fixed weighted measure ∬ y d x , the perimeter ∫ ∂ s achieves its minimum for set which is symmetric w.r.t. -axis, explicitly given. Our results also imply an estimate Cheeger constant bound eigenvalues some nonlinear pro...

a graph is textit{symmetric}, if its automorphism group is transitive on the set of its arcs. in this paper, we  classifyall the connected cubic symmetric  graphs of order $36p$  and $36p^{2}$, for each prime $p$, of which the proof depends on the classification of finite simple groups.

we give a survey of recent applications of group rings to combinatorics and tocryptography, including their use in the di erential cryptanalysis of block ciphers.