Equations In Several Variables Research Articles

The resolution of three exponential Diophantine equations in several variables

We find all solutions of three exponential Diophantine equations, arising from certain quadratic, cubic and quartic identities. The first identity comes from a painting of the famous Russian painter Nikolay Bogdanov-Belsky, highlighted by Ja. I. Perelman. The equations have five, four and six terms, respectively, so they cannot be handled by classical tools based upon Baker's method. To solve the equations we use our method developed earlier, which is based upon Skolem's conjecture, local considerations and a computational approach.

On Hurwitz stability for families of polynomials

AbstractThe robustness of a linear system in the view of parametric variations requires a stability analysis of a family of polynomials. If the parameters vary in a compact set , then obtaining necessary and sufficient conditions to determine stability of the family is one of the most important tasks in the field of robust control. Three interesting classes of families arise when is a diamond, a box or a ball of dimension . These families will be denoted by , , and , respectively. In this article, a study is presented to contribute to the understanding of Hurwitz stability of families of polynomials . As a result of this study and the use of classical results found in the literature, it is shown the existence of an extremal polynomial whose stability determines the stability of the entire family . In this case comes from minimizing determinants and in some cases coincides with a Kharitonov's polynomial. Thus another extremal property of Kharitonov's polynomials has been found. To illustrate our approach, it is applied to families such as , , and with . The study is also used to obtain the maximum robustness of the parameters of a polynomial. To exemplify the proposed results, first, a family is taken from the literature to compare and corroborate the effectiveness and the advantage of our perspective. Followed by two examples where the maximum robustness of the parameters of polynomials of degree 3 and 4 are obtained. Lastly, a family is proposed whose extreme polynomial is not necessarily a Kharitonov's polynomial. Finally, a family is used to exemplify that if the boundary of is given by a polynomial equation in several variables, the number of candidates to be an extremal polynomial is finite.

Stability of a General Functional Equation in m-Banach Spaces

In this note, the Ulam stability of a general functional equation in several variables is investigated. It is shown that this equation is Ulam stable in m-Banach spaces. Since a particular case of the considered equation is, among others, a functional equation introduced by Ji et al. and Zhao et al. for a characterization of the so-called multi-quadratic mapping, a result on its stability is also presented. Moreover, some other applications are provided.

On perturbations of two general equations in several variables

In this paper, we deal with perturbations of two general functional equations in several variables. Namely, we prove the generalized, in the spirit of Bourgin, Ulam stability of these equations in Banach spaces. In order to do this, we use the fixed point method. Moreover, as corollaries from our main results, we get several outcomes on approximate solutions of a few important classic equations. They include, among others, the functional equations which characterize multi-additive and multi-quadratic mappings. In consequence, the perturbation of homomorphisms of Banach spaces and quadratic mappings is also treated.

Linear Diophantine equations in several variables

Let R be a ring and let (a1,…,an)∈Rn be a unimodular vector, where n≥2 and each ai is in the center of R. Consider the linear equation a1X1+⋯+anXn=0, with solution set S. Then S=S1+⋯+Sn, where each Si is naturally derived from (a1,…,an), and we give a presentation of S in terms of generators taken from the Si and appropriate relations. Moreover, under suitable assumptions, we elucidate the structure of each quotient module S/Si. Furthermore, assuming that R is a principal ideal domain, we provide a simple way to construct a basis of S and, as an application, we determine the structure of the quotient module S/Ui, where each Ui is a specific module containing Si.

Ulam stability of functional equations in 2-Banach spaces via the fixed point method

Using the fixed point method, we prove the Ulam stability of two general functional equations in several variables in 2-Banach spaces. As corollaries from our main results, some outcomes on the stability of a few known equations being special cases of the considered ones will be presented. In particular, we extend several recent results on the Ulam stability of functional equations in 2-Banach spaces.

On hyperstability of generalized linear equations in several variables in quasi-normed spaces

In this paper, we state and prove the hyperstability of generalized linear equations in several variables in quasi-normed spaces. As applications, we deduce some known results and some particular cases of generalized linear equations in several variables.

An efficient line search trust-region for systems of nonlinear equations

An improved derivative-free trust-region method to solve systems of nonlinear equations in several variables is presented, combined with the Wolfe conditions to update the trust-region radius. We believe that producing step-sizes by the Wolfe conditions can control the trust-region radius. The new algorithm for which strong global convergence properties are proved is robust and efficient enough to solve systems of nonlinear equations.

HYPERSTABILITY OF GENERALISED LINEAR FUNCTIONAL EQUATIONS IN SEVERAL VARIABLES

Zhang [‘On hyperstability of generalised linear functional equations in several variables’, Bull. Aust. Math. Soc.92 (2015), 259–267] proved a hyperstability result for generalised linear functional equations in several variables by using Brzdęk’s fixed point theorem. We complete and extend Zhang’s result. We illustrate our results for general linear equations in two variables and Fréchet equations.

Maximal Independent Sets and Maximal Matchings in Series-Parallel and Related Graph Classes

The goal of this paper is to obtain quantitative results on the number and on the size of maximal independent sets and maximal matchings in several block-stable graph classes that satisfy a proper sub-criticality condition. In particular we cover trees, cacti graphs and series-parallel graphs. The proof methods are based on a generating function approach and a proper singularity analysis of solutions of implicit systems of functional equations in several variables. As a byproduct, this method extends previous results of Meir and Moon for trees [Meir, Moon: On maximal independent sets of nodes in trees, Journal of Graph Theory 1988].

Hyers–Ulam–Rassias Stability of Set Valued Additive and Cubic Functional Equations in Several Variables

In this paper, we establish Hyers–Ulam–Rassias stability results belonging to two different set valued functional equations in several variables, namely additive and cubic. The results are obtained in the contexts of Banach spaces. The work is in the domain of set valued analysis.

Skeletons and moduli of Stokes torsors

We prove an analogue for Stokes torsors of Deligne's skeleton conjecture and deduce from it the representability of the functor of relative Stokes torsors by an affine scheme of finite type over C. This provides, in characteristic 0, a local analogue of the existence of a coarse moduli for skeletons with bounded ramification, due to Deligne. As an application, we use the geometry of this moduli to derive quite strong finiteness results for integrable systems of differential equations in several variables which did not have any analogue in one variable.

Approximating roots of nonlinear systems by α-dense curves

We extend an algorithm due to Khamisov (Math. Notes 98(3/4), 484–491, 2015) to approximate, if any exists, a root of a single variable function. For this goal, using the so called α-dense curves, we transform a system of equations of several variables into a single variable equation. The feasibility and limitations of the proposed method are discussed.

Functional equations with division and regular operations

Functional equations are equations in which the unknown (or unknowns) are functions. We consider equations of generalized associativity, mediality (bisymmetry, entropy), paramediality, transitivity as well as the generalized Kolmogoroff equation. Their usefulness was proved in applications both in mathematics and in other disciplines, particularly in economics and social sciences (see J. Aczél, On mean values, Bull. Amer. Math. Soc. 54 (1948) 392–400; J. Aczél, Remarques algebriques sur la solution donner par M. Frechet a l’equation de Kolmogoroff, Pupl. Math. Debrecen 4 (1955) 33–42; J. Aczél, A Short Course on Functional Equations Based Upon Recent Applications to the Social and Behavioral Sciences, Theory and decision library, Series B: Mathematical and statistical methods (D. Reidel Publishing Company, Dordrecht, Boston, Lancaster, Tokyo, 1987); J. Aczél, Lectures on Functional Equations and Their Applications (Supplemented by the author, ed. H. Oser) (Dover Publications, Mineola, New York, 2006); J. Aczél, V. D. Belousov and M. Hosszu, Generalized associativity and bisymmetry on quasigroups, Acta Math. Acad. Sci. Hungar. 11 (1960) 127–136; J. Aczél and J. Dhombres, Functional Equations in Several Variables (Cambridge University Press, New York, 1991); J. Aczél and T. L. Saaty, Procedures for synthesizing ratio judgements, J. Math. Psych. 27(1) (1983) 93–102). We use unifying approach to solve these equations for division and regular operations generalizing the classical quasigroup case.

On a fixed point theorem in 2-Banach spaces and some of its applications

The aim of this article is to prove a fixed point theorem in 2-Banach spaces and show its applications to the Ulam stability of functional equations. The obtained stability results concern both some single variable equations and the most important functional equation in several variables, namely, the Cauchy equation. Moreover, a few corollaries corresponding to some known hyperstability outcomes are presented.

Non-archimedean stability and non-stability of quadratic reciprocal functional equation in several variables

Non-archimedean stability and non-stability of quadratic reciprocal functional equation in several variables

Non-Archimedean stability of a generalized reciprocal-quadratic functional equation in several variables by direct and fixed point methods

This study is aimed to determine various stabilities of a generalized reciprocal-quadratic functional equation of the form r (?pj=1 ?juj) = ?pj=1 r (uj) [?pj=1 ?j ?pk=1,k?j ?r(uk)] 2 connected with Ulam, Hyers, T. M. Rassias, J. M. Rassias and Gavruta in non-Archimedean fields, where ?j?0; j = 1,2,..., p are arbitrary real numbers and 0 < ?1+?2+...+?p = ?pj=1 ?j=??1 in non-Archimedean fields by direct and fixed point methods.

Non-Archimedean stability of a generalized reciprocal-quadratic functional equation in several variables by direct and fixed point methods

This study is aimed to determine various stabilities of a generalized reciprocal-quadratic functional equation of the form r (Σpj=1 βjuj) = Πpj=1 r (uj) [Σpj=1 βj Πpk=1,k≠j √r(uk)] 2 connected with Ulam, Hyers, T. M. Rassias, J. M. Rassias and Gavruta in non-Archimedean fields, where βj≠0; j = 1,2,..., p are arbitrary real numbers and 0 < β1+β2+...+βp = Σpj=1 βj=β≠1 in non-Archimedean fields by direct and fixed point methods.

A Fixed Point Approach to the Stability of a Mean Value Type Functional Equation

We prove the generalized Hyers–Ulam stability of a mean value type functional equation f ( x ) − g ( y ) = ( x − y ) h ( x + y ) by applying a method originated from fixed point theory.

Stability problem for the composite type functional equations

Composite type functional equations in several variables play a significant role in various branches of mathematics and they have several interesting applications. Therefore their stability properties are of interest. The aim of this paper is to present a survey of some results and methods concerning stability of several of such equations; especially those somehow connected to the well known Gołąb–Schinzel equation.