In Shamir’s scheme that the security based on the numbers of the field of a prime number P which the coefficients' polynomial reduced to modulo P (takes a value from some field p Z , where P is a large prime number). Thus, the adversary must know only the free coefficient of the polynomial in order to break the scheme. Our scheme which based on representation integer using the so-called g a d i...

X.S. Lin’s original definition of twisted Alexander knot polynomial is generalized for arbitrary finitely presented groups. J. Cha’s fibering obstruction theorem is generalized. The group of a nontrivial virtual knot shown by L. Kauffman to have trivial Jones polynomial is seen also to have a faithful representation that yields a trivial twisted Alexander polynomial.

The main purpose of this work is to construct the forward and backward solution space of a nonregular discrete time AR-representation i.e. A(σ)ξ(k) = 0, in a closed interval [0, N ] where A(σ) is a polynomial matrix and σ is the forward shift operator. The construction of the behavior is based on the structural invariants of the polynomial matrix which describe the AR-representation, i.e. the f...

We prove that the colored HOMFLY polynomial of a link, colored by symmetric or exterior powers of the fundamental representation, is q-holonomic with respect to the color parameters. As a result, we obtain the existence of an (a, q) super-polynomial of all knots in 3-space. Our result has implications on the quantization of the SL(2,C) character variety of knots using ideal triangulations or th...

It is well known that the principal specialization of the Schur function, sλ(1, q, . . . , q ), is a unimodal polynomial. A brief representation theoretic proof consists in identifying sλ(1, q, . . . , q ) as the character of a homogeneous polynomial representation of GL(2,C) evaluated at diag(1, q), see [3, p. 67]. Recently O’Hara gave a combinatorial proof [4] of the unimodality of the Gaussi...

The (asymptotic) dimension growth functions of groups were introduced by Gromov in 1999. In this paper, we show connections between dimension growth and expansion properties of graphs, Ramsey theory and the Kolmogorov-Ostrand dimension of groups and prove that all solvable subgroups of the R.Thompson group F have polynomial dimension growth. We introduce controlled dimension growth function and...

We prove that a two-sided sub-Gaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay.

This paper deals with the recoverable robust spanning tree problem under interval uncertainty representations. A strongly polynomial time, combinatorial algorithm for the recoverable spanning tree problem is first constructed. This problem generalizes the incremental spanning tree problem, previously discussed in literature. The algorithm built is then applied to solve the recoverable robust sp...

In this paper we give a new and simple algorithm to put any multivariate polynomial into a normal determinant form in which each entry has the form i i i b x a  , and in each column the same variable appears. We also apply the algorithm to obtain a triangular determinant representation, a reduced determinant representation, and a uniform determinant representation of any multivariable polynomi...