Laguerre polynomial approach for solving linear delay difference equations

This paper presents a numerical method for the approximate solution of mth-order linear delay difference equations with variable coefficients under the mixed conditions in terms of Laguerre polynomials. The aim of this article is to present an efficient numerical procedure for solving mth-order linear delay difference equations with variable coefficients. Our method depends mainly on a Laguerre series expansion approach. This method transforms linear delay difference equations and the given conditions into matrix equation which corresponds to a system of linear algebraic equation. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments and performed on the computer algebraic system Maple.     

On the relationship between quadratic polynomial differential system and the Bernoulli equation

In this article, we have established a relationship between a quadratic polynomial differential system and a Bernoulli equation by using the method of reflective function. And applied the results to discuss the qualitative behavior of solutions of this quadratic polynomial differential system.     

A faster algorithm for 2-cyclic robotic scheduling with a fixed robot route and interval processing times

Consider an m-machine production line for processing identical parts served by a mobile robot. The problem is to find the minimum cycle time for 2-cyclic schedules, in which exactly two parts enter and two parts leave the production line during each cycle. This work treats a special case of the 2-cyclic robot scheduling problem when the robot route is given and the operation durations are to be chosen from prescribed intervals. The problem was previously proved to be polynomially solvable in O(m⁸log m) time. This paper proposes an improved algorithm with reduced complexity O(m⁴).     

A new semilocal convergence theorem for the Weierstrass method for finding zeros of a polynomial simultaneously

In this paper we study the convergence of the famous Weierstrass method for simultaneous approximation of polynomial zeros over a complete normed field. We present a new semilocal convergence theorem for the Weierstrass method under a new type of initial conditions. Our result is obtained by combining ideas of Weierstrass (1891) and Proinov (2010). A priori and a posteriori error estimates are also provided under the new initial conditions.     

Analytical properties of resource-bounded real functionals

Computable analysis is an extension of classical discrete computability by enhancing the normal Turing machine model. It investigates mathematical analysis from the computability perspective. Though it is well developed on the computability level, it is still under developed on the complexity perspective, that is, when bounding the available computational resources. Recently Kawamura and Cook developed a framework to define the computational complexity of operators arising in analysis. Our goal is to understand the effects of complexity restrictions on the analytical properties of the operator. We focus on the case of norms over C[0, 1] and introduce the notion of dependence of a norm on a point and relate it to the query complexity of the norm. We show that the dependence of almost every point is of the order of the query complexity of the norm. A norm with small complexity depends on a few points but, as compensation, highly depends on them. We briefly show how to obtain similar results for non-deterministic time complexity. We characterize the functionals that are computable using one oracle call only and discuss the uniformity of that characterization.     

Riesz-type inequalities on general sets

Sharp Riesz–Bernstein-type inequalities are proven for the derivatives of algebraic polynomials on general subsets of unit circle. The sharp Riesz–Bernstein constant involves the equilibrium density of the set in question.     

Representation of solutions of n-order Riccati equation via generalized trigonometric functions

In this work we suggest a systematic method of construction of solutions of the n-order Riccati equation with constant coefficients in a field from the set of generalized trigonometric functions. The generalized trigonometric functions satisfy the system of evolution equations generated by the companion matrix of n  -order polynomial. The set of trigonometric functions depend of (n−1)$(n-1)$ variables formally expressed by series of exponential functions. In a particular case, the second order Riccati equation with constant coefficients is isomorphic to the evolution equation generated by the companion matrix of the associated quadratic polynomial. It is shown that the n>2$n>2$ order Riccati equation with coefficients in a field is derived from a linear system of evolution equations generated by companion matrix of the associated n  -order polynomial under (n−2)$(n-2)$ constraints.     

Tauberian polynomials

In this paper we introduce and study the notion of homogeneous Tauberian polynomial, aiming at extending the concept of Tauberian operator. Such notion is characterized in terms of the polynomial topology for which we prove a Banach–Alaoglu type theorem. A number of examples show that the behavior of Tauberian polynomials differs from that of Tauberian operators.