For any given polynomial f over the finite field Fq with degree at most q−1, we associate it with a q×q matrix A(f) = (aik) consisting of coefficients of its powers (f(x)) = ∑q−1 i=0 aikx i modulo x−x for k = 0, 1, . . . , q−1. This matrix has some interesting properties such as A(g◦f) = A(f)A(g) where (g◦f)(x) = g(f(x)) is the composition of the polynomial g with the polynomial f . In particul...

It is shown that the finite dimensional irreducible representaions of the quantum matrix algebra Mq,p(2) ( the coordinate ring of GLq,p(2)) exist only when both q and p are roots of unity. In this case th e space of states has either the topology of a torus or a cylinder which may be thought of as generalizations of cyclic representations.

For a finite group G, let Z(G) be the centre of G. Then non-commuting graph on denoted by ΓG, has G\Z(G) as its vertex set with two distinct vertices vp and vq joined an edge whenever vpvq ≠ vqvp. The degree sum matrix is square whose (p,q)-th entry dvp + dvq p different from q, otherwise, it zero, where dvi vi. This study presents general formula for energy, EDS (ΓG), dihedral groups order 2n,...

We demonstrate that the transfer matrix of the inhomogeneous N -state chiral Potts model with two vertical superintegrable rapidities serves as the Q-operator of XXZ chain model for a cyclic representation of Uq(sl2) with Nth root-of-unity q and representation-parameter. The symmetry problem of XXZ chain with a general cyclic Uq(sl2)-representation is mapped onto the problem of studying Q-opera...

Power residue symbols and their reciprocity laws have applications not only in number theory, but also in other fields like cryptography. A crucial ingredient in certain public key cryptosystems is a fast algorithm for computing power residue symbols. Such algorithms have only been devised for the Jacobi symbol as well as for cubic and quintic power residue symbols, but for no higher powers. In...

The present paper investigates the embedding problem for time-homogeneous Markov chains. A discretetime Markov chain with time unit 1 is embeddable in case there exists a compatible Markov chain regarding time unit 1 m (with m ∈ N,m ≥ 2). An embeddable Markov chain has a transition matrix for which there exists an m-th root that is a probability matrix. The present paper examines the embedding ...

By a quasi-permutation matrix we mean a square matrix over the complex field C with non-negative integral trace. Thus every permutation matrix over C is a quasipermutation matrix. For a given finite group G, let p(G) denote the minimal degree of a faithful permutation representation of G (or of a faithful representation of G by permutation matrices), let q(G) denote the minimal degree of a fai...