where the aν(ω), ν= 0,1, . . . ,n, are random variables defined on a fixed probability space (Ω, ,Pr) assuming real values only. During the past 40–50 years, the majority of published researches on random algebraic polynomials has concerned the estimation of Nn(R,ω). Works by Littlewood and Offord [1], Samal [2], Evans [3], and Samal and Mishra [4–6] in the main concerned cases in which the ran...

In this paper we focus on the long-term behavior of generalized polynomial chaos (gPC) and multi-element generalized polynomial chaos (ME-gPC) for partial differential equations with stochastic coefficients. First, we consider the one-dimensional advection equation with a uniform random transport velocity and derive error estimates for gPC and ME-gPC discretizations. Subsequently, we extend the...

In this paper we revisit the modular inversion hidden number problem and the inversive congruential pseudo random number generator and consider how to more efficiently attack them in terms of fewer samples or outputs. We reduce the attacking problem to finding small solutions of systems of modular polynomial equations of the form ai+bix0+cixi+x0xi = 0 (mod p), and present two strategies to cons...

We study polynomial growth of algebraic series. It is proved that if an Nalgebraic series r with arbitrarily many commuting variables has polynomial growth, then r is in fact N-rational. Also, polynomial growth is decidable. Similar results are proved forN-algebraic series with noncommuting variables having bounded supports. It is also shown that polynomial growth is not decidable for N-algebra...

let $p(lambda)$ be an $n$-square complex matrix polynomial, and $1 leq k leq n$ be a positive integer. in this paper, some algebraic and geometrical properties of the $k$-numerical range of $p(lambda)$ are investigated. in particular, the relationship between the $k$-numerical range of $p(lambda)$ and the $k$-numerical range of its companion linearization is stated. moreover, the $k$-numerical ...

We give a uniform method for the two problems of counting the connected and irreducible components of complex algebraic varieties. Our algorithms are purely algebraic, i.e., they use only the field structure of C. They work in parallel polynomial time, i.e., they can be implemented by algebraic circuits of polynomial depth. The design of our algorithms relies on the concept of algebraic differe...

. Existence of Splitting Fields for a polynomial or family of polynomials F ⊂ F[X]. Existence of Algebraic Closure. Characterizations of Algebraic Closure: A field E is algebraically closed if every nonconstant polynomial in E[X] factors as a product of linear polynomials. (Equivalently, every nonconstant polynomial in E[X] has a root in E.) A field E is an algebraic closure of a subfield F if...