let $g$ be a simple graph of order $n$‎. ‎we consider the‎ ‎independence polynomial and the domination polynomial of a graph‎ ‎$g$‎. ‎the value of a graph polynomial at a specific point can give‎ ‎sometimes a very surprising information about the structure of the‎ ‎graph‎. ‎in this paper we investigate independence and domination‎ ‎polynomial at $-1$ and $1$‎.

has positive coefficients when m = 6 and A and B are any two 3-by-3 complex Hermitian positive definite matrices. This case is the first that is not covered by prior, general results. This problem arises from a conjecture raised by Bessis, Moussa and Villani in connection with a long-standing problem in theoretical physics. The full conjecture, as shown recently by Lieb and Seiringer, is equiva...

Sometimes the assumption on the form (1.1) of the solution to the scattered data interpolation Problem 1.2.1 is extended by adding certain polynomials to the expansion,

We introduce a method for constructing a polynomial preconditioner using a nonlinear least squares (NLLS) algorithm. We show that this polynomial-based NLLS-optimized (PBNO) preconditioner significantly improves the performance of 2-D continuous Galerkin (CG) and discontinuous Galerkin (DG) fluid dynamical research models when run in an implicit-explicit time integration mode. When employed in ...

In Multi-Input Multi-Output (MIMO) systems, maximum-likelihood (ML) decoding is equivalent to finding the closest lattice point in an N -dimensional complex space. In general, this algorithm is known to be NP hard. In this paper, we propose a quasi-maximum likelihood algorithm based on Semi-Definite Programming (SDP). We introduce several SDP relaxation models for MIMO systems, with increasing ...

Let Q(z, w) = − ∏n k=1[(z − ak)(w̄ − āk) − R]. The main result of the paper states that in the case when the nodes aj are situated at the vertices of a regular n-gon inscribed in the unit circle, the matrix Q(ai, aj) is positive definite if and only if R < ρn, where z = 2ρ 2 n − 1 is the smallest 6= −1 zero of the Jacobi polynomial Pn−2ν,−1 ν (z), ν = [n/2].

We show how to bound the accuracy of a family of semi-definite programming relaxations for the problem of polynomial optimization on the hypersphere. Our method is inspired by a set of results from quantum information known as quantum de Finetti theorems. In particular, we prove a de Finetti theorem for a special class of real symmetric matrices to establish the existence of approximate represe...

This paper is concerned with the study of diagonal Diophantine inequalities fractional degree θ, where θ > 2 real and non-integral. For fixed non-zero numbers λ i not all same sign, we write F ( x ) = 1 + ⋯ s . a positive number τ, give an asymptotic formula for integer solutions inequality | < τ inside box side length P. Moreover, investigate problem representing large by definite generalised ...

We compute the Newton step for the characteristic polynomial and for the even and odd characteristic polynomials of a symmetric positive definite Toeplitz matrix as the reciprocal of the trace of an appropriate matrix. We show that, after the Yule–Walker equations are solved, this trace can be computed in O(n) additional arithmetic operations, which is in contrast to existing methods, which rel...