Two important algebraic proof systems are the Nullstellensatz system [1] and the polynomial calculus [2] (also called the Gröbner system). The Nullstellensatz system is a propositional proof system based on Hilbert’s Nullstellensatz, and the polynomial calculus (PC) is a proof system which allows derivations of polynomials, over some £eld. The complexity of a proof in these systems is measured ...

The one-dimensional harmonic oscillator wave functions are solutions to a SturmLiouville problem posed on the whole real line. This problem generates the Hermite polynomials. However, no other set of orthogonal polynomials can be obtained from a Sturm-Liouville problem on the whole real line. In this paper we show how to characterize an arbitrary set of polynomials orthogonal on (−∞,∞) in terms...

An effective numerical method based on Legendre polynomials is proposed for the solution of a class of variational problems with suitable boundary conditions. The Ritz spectral method is used for finding the approximate solution of the problem. By utilizing the Ritz method, the given nonlinear variational problem reduces to the problem of solving a system of algebraic equations. The advantage o...

We show that multiple orthogonal polynomials for r measures (μ1, . . . , μr) satisfy a system of linear recurrence relations only involving nearest neighbor multi-indices ~n ± ~ej, where ~ej are the standard unit vectors. The recurrence coefficients are not arbitrary but satisfy a system of partial difference equations with boundary values given by the recurrence coefficients of the orthogonal ...

‎In this paper‎, ‎we present a numerical method based on Bernstein polynomials to solve optimal control systems with constant and pantograph delays‎. ‎Constant or pantograph delays may appear in state-control or both‎. ‎We derive delay operational matrix and pantograph operational matrix for Bernstein polynomials then‎, ‎these are utilized to reduce the solution of optimal control with constant...

This note presents a deterministic integer factorization algorithm based on a system of polynomials equations. This technique exploits a new idea in the construction of irreducible polynomials with parametized roots, and recent advances in polynomial lattices reduction methods. The main result establishes a new deterministic time complexity bench mark. 1 Introduction This note presents a determ...

Generalized polynomials are functions obtained from conventional polynomials by applying the operations of taking the integer part, addition, and multiplication. We construct a system of “basic” generalized polynomials with the property that any bounded generalized polynomial is representable as a piecewise polynomial function of these basic ones. Such a representation is unique up to a functio...

In this paper, operational matrix method based upon the Bernstein polynomials is proposed to solve the variable order fractional integro-differential equations in the Caputo derivative sense. We derive the Bernstein polynomials operational matrix of fractional order integration and introduce the product operational matrix of Bernstein polynomials. A truncated the Bernstein polynomials series to...

In this paper‎, two inverse problems of determining an unknown source term in a parabolic‎ equation are considered‎. ‎First‎, ‎the unknown source term is ‎estimated in the form of a combination of Chebyshev functions‎. ‎Then‎, ‎a numerical algorithm based on Chebyshev polynomials is presented for obtaining the solution of the problem‎. ‎For solving the problem‎, ‎the operational matrices of int...