Consider the polynomial optimization problem whose objective and constraints are all described by multivariate polynomials. Under some genericity assumptions, we prove that the optimality conditions always hold on optimizers, and the coordinates of optimizers are algebraic functions of the coefficients of the input polynomials. We also give a general formula for the algebraic degree of the opti...

let $g$ be a molecular graph with vertex set $v(g)$, $d_g(u, v)$ the topological distance between vertices $u$ and $v$ in $g$. the hosoya polynomial $h(g, x)$ of $g$ is a polynomial $sumlimits_{{u, v}subseteq v(g)}x^{d_g(u, v)}$ in variable $x$. in this paper, we obtain an explicit analytical expression for the expected value of the hosoya polynomial of a random benzenoid chain with $n$ hexagon...

In this paper, a new collocation method, which is based on Boubaker polynomials, is introduced for the approximate solutions of a class of two-dimensional linear Fredholm integral equationsof the second kind. The properties of two-dimensional Boubaker functions are presented. The fundamental matrices of integration with the collocation points are utilized to reduce the solution of the integral ...

Quantitative characterization of intracellular metabolite concentrations is central to predictive models cellular functions. Current constraint-based metabolism and macromolecular expression provide genotype-phenotype predictions without accounting for using simplifying assumptions optimality principles. However, principles do not generally apply under stress conditions, rendering these unrelia...

It is well known that a system of power polynomial equations can be reduced to a single-variable polynomial equation by exploiting the so-called Newton’s identities. In this work, by further exploring Newton’s identities, we discover a binomial decomposition rule for composite elementary symmetric polynomials. Utilizing this decomposition rule, we solve three types of systems of composite power...

We describe algorithms to obtain rational parametric equations, (a polynomial equation divided by another), for degree two curves (conics) and degree two surfaces (conicoids), given the implicit equations. We further consider the rational parameterizations over t':~ fields of rationals, reals and complex numbers. In doing so, solutions are given to important subproblems of finding rational, rea...