the topological index of a graph g is a numeric quantity related to g which is invariant underautomorphisms of g. the vertex pi polynomial is defined as piv (g)  euv nu (e)  nv (e).then omega polynomial (g,x) for counting qoc strips in g is defined as (g,x) =cm(g,c)xc with m(g,c) being the number of strips of length c. in this paper, a new infiniteclass of fullerenes is constructed. the ...

We present a general semidefinite relaxation scheme for general n-variate quartic polynomial optimization under homogeneous quadratic constraints. Unlike the existing sum-of-squares (SOS) approach which relaxes the quartic optimization problems to a sequence of (typically large) linear semidefinite programs (SDP), our relaxation scheme leads to a (possibly nonconvex) quadratic optimization prob...

The Lasserre hierarchy of semidefinite programming approximations to convex polynomial optimization problems is known to converge finitely under some assumptions. [J.B. Lasserre. Convexity in semialgebraic geometry and polynomial optimization. SIAM J. Optim. 19, 1995–2014, 2009.] We give a new proof of the finite convergence property, that does not require the assumption that the Hessian of the...

Direction-of-arrival (DOA) estimation is widely applied in acoustic source localization. A multi-frequency model suitable for characterizing the broadband structure signals. In this paper, continuous (gridless) DOA problem with multiple frequencies considered. This formulated as an atomic norm minimization (ANM) problem. The ANM equivalent to a semi-definite program (SDP) which can be solved by...

We consider Murphy’s polynomial selection algorithm for the general number field sieve. One of the steps in this algorithm consists of finding a minimum of an integral. However, the size of the polynomial coefficients causes the classical steepest descent algorithm to be ineffective. This article brings an idea how to improve the steepest descent algorithm so that it converges better and faster...

Positive definite homogeneous multivariate forms play an important role in polynomial problems and medical imaging, the definiteness of can be tested using structured tensors. In this paper, we state equivalence between positive corresponding tensors, explain connection tensors with H-tensors. Then, based on notion diagonally dominant some criteria for H-tensors are presented. Meanwhile, these ...

The geometric mean and the function (det(·))1/m (on the m-by-m positive definite matrices) are examples of “hyperbolic means”: functions of the form p1/m , where p is a hyperbolic polynomial of degree m. (A homogeneous polynomial p is “hyperbolic” with respect to a vector d if the polynomial t → p(x+ td) has only real roots for every vector x.) Any hyperbolic mean is positively homogeneous and ...

We determine new sufficient conditions in terms of the coefficients for a polynomial f ∈ R[X] of degree 2d (d ≥ 1) in n ≥ 1 variables to be a sum of squares of polynomials, thereby strengthening results of Fidalgo and Kovacec [2] and of Lasserre [6]. Exploiting these results, we determine, for any polynomial f ∈ R[X] of degree 2d whose highest degree term is an interior point in the cone of sum...

We consider the problem of finding exact sums of squares (SOS) decompositions for certain classes of non-negative multivariate polynomials, relying on semidefinite programming (SDP) solvers. We start by providing a hybrid numeric-symbolic algorithm computing exact rational SOS decompositions for polynomials lying in the interior of the SOS cone. It computes an approximate SOS decomposition for ...

The following is a special case of the problem to be considered in this paper : Given a function cb(x) of the real variable x, continuous on a finite interval (a, b) ; to determine the polynomial p(x) of given degree n, which gives the closest approximation to the given function cp on the interval (a, b). This problem becomes definite only when the meaning of the phrase " closest approximation ...