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Eszter Gselmann

Publications and source records attributed to Eszter Gselmann.

At least 37 records · Page 2Linked to original sources
arXiv2020

The global symmetry group of first order differential equations and the global rectification theorem

Symmetry analysis can provide a suitable change of variables, i.e., in geometric terms, a suitable diffeomorphism that simplifies the given direction field, which can help significantly in solving or studying differential equations. Roughly speaking this is the so-called rectification theorem. The local version of this result is a well-known theorem in the field of ordinary differential equations. In this note we prove a global counterpart when the equation fulfils the Lipschitz condition. Then we use this result to determine the global symmetry group of such an ordinary differential equation. It turns out that, assuming the Lipschitz condition, the full symmetry group is a smooth wreath product of two diffeomorphism groups, and does not depend on the form of the equation, at all.

math.CA↗
arXiv2020

Moment functions on groups

The main purpose of this work is to prove characterization theorems for generalized moment functions on groups. According one of the main results these are exponential polynomials that can be described with the aid of complete (exponential) Bell polynomials. These characterizations will be immediate consequences of our main result about the characterization of generalized moment functions of higher rank.

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arXiv2020

Remarks on the notion of homo-derivations

The purpose of this paper is to study the (different) notions of homo-derivations. These are additive mappings $f$ of a ring $R$ that also fulfill the identity \[ f(xy)=f(x)y+xf(y)+f(x)f(y) \qquad \left(x, y\in R\right), \] or (in case of the other notion) the system of equations \[ f(xy)=f(x)f(y)\] \[f(xy)=f(x)y+xf(y) \qquad \left(x, y\in R\right).\] Our primary aim is to investigate the above equations without additivity as well as the following Pexiderized equation \[ f(xy)=h(x)h(y)+xk(y)+k(x)y. \] The obtained results show that under rather mild assumptions homo-derivations can be fully characterized, even without the additivity assumption.

math.RA↗
arXiv2019

Characterizations of derivations

The main purpose of this work is to characterize derivations through functional equations. This work consists of five chapters. In the first one, we summarize the most important notions and results from the theory of functional equations. In Chapter 2 we collect all the definitions and results regarding derivations that are essential while studying this area. In Chapter 3 we intend to show that derivations can be characterized by one single functional equation. More exactly, we study here the following problem. Let $Q$ be a commutative ring and let $P$ be a subring of $Q$. Let $λ, μ\in Q\setminus\left\{0\right\}$ be arbitrary, $f\colon P\rightarrow Q$ be a function and consider the equation \[ λ\left[f(x+y)-f(x)-f(y)\right]+ μ\left[f(xy)-xf(y)-yf(x)\right]=0 \quad \left(x, y\in P\right). \] In this chapter it will be proved that under some assumptions on the rings $P$ and $Q$, derivations can be characterized via the above equation. Chapter 4 is devoted to the additive solvability of a system of functional equations. Moreover, the linear dependence and independence of the additive solutions $d_{0},d_{1},\dots,d_{n} \colon\mathbb{R}\to\mathbb{R}$ of the above system of equations is characterized. Finally, the closing chapter deals with the following problem. Assume that $ξ\colon \mathbb{R}\to \mathbb{R}$ is a given differentiable function and for the additive function $f\colon \mathbb{R}\to \mathbb{R}$, the mapping \[ φ(x)=f\left(ξ(x)\right)-ξ'(x)f(x) \] fulfills some regularity condition on its domain. Is it true that in such a case $f$ is a sum of a derivation and a linear function?

math.FA↗
arXiv2019

On a class of linear functional equations without range condition

The main purpose of this work is to provide the general solutions of a class of linear functional equations. Let $n\geq 2$ be an arbitrarily fixed integer, let further $X$ and $Y$ be linear spaces over the field $\mathbb{K}$ and let $α_{i}, β_{i}\in \mathbb{K}$, $i=1, \ldots, n$ be arbitrarily fixed constants. We will describe all those functions $f, f_{i, j}\colon X\times Y\to \mathbb{K}$, $i, j=1, \ldots, n$ that fulfill functional equation \[ f\left(\sum_{i=1}^n α_i x_i, \sum_{i=1}^n β_i y_i\right)= \sum_{i, j=1}^{n}f_{i, j}(x_i, y_j) \qquad \left(x_i \in X, y_i \in Y, i=1, \ldots, n\right). \] Additionally, necessary and sufficient conditions will also be given that guarantee the solutions to be non-trivial.

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arXiv2018

Characterization of field homomorphisms through Pexiderized functional equations

The aim of this paper is to prove characterization theorems for field homomorphisms. More precisely, the main result investigates the following problem. Let $n\in \mathbb{N}$ be arbitrary, $\mathbb{K}$ a field and $f_{1}, \ldots, f_{n}\colon \mathbb{K}\to \mathbb{C}$ additive functions. Suppose further that equation \[ \sum_{i=1}^{n}f^{q_{i}}_{i}\left(x^{p_{i}}\right)=0 \qquad \left(x\in \mathbb{K}\right) \] is also satisfied. Then the functions $f_{1}, \ldots, f_{n}$ are linear combinations of field homomorphisms from $\mathbb{K}$ to $\mathbb{C}$.

math.AC↗
arXiv2018

On functional equations characterizing derivations: methods and examples

Functional equations satisfied by additive functions have a special interest not only in the theory of functional equations, but also in the theory of (commutative) algebra because the fundamental notions such as derivations and automorphisms are additive functions satisfying some further functional equations as well. It is an important question that how these morphisms can be characterized among additive mappings in general. The paper contains some multivariate characterizations of higher order derivations. The univariate characterizations are given as consequences by the diagonalization of the multivariate formulas. The method allows us to refine the process of computing the solutions of univariate functional equations of the form \[ \sum_{k=1}^{n}x^{p_{k}}f_{k}(x^{q_{k}})=0, \] where $p_k$ and $q_k$ ($k=1, \ldots, n$) are given nonnegative integers and the unknown functions $f_{1}, \ldots, f_{n}\colon R\to R$ are supposed to be additive on the ring $R$. It is illustrated by some explicit examples too. As another application of the multivariate setting we use spectral analysis and spectral synthesis in the space of the additive solutions to prove that such a functional equation characterizes derivations of higher order. The results are uniformly based on the investigation of the multivariate version of the functional equations.

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arXiv2016

Stability of the entropy equation

In this paper we prove that the so--called entropy equation, i.e., \[ H\left(x, y, z\right)=H\left(x+y, 0, z\right)+H\left(x, y, 0\right) \] is stable in the sense of Hyers and Ulam on the positive cone of $\mathbb{R}^{3}$, assuming that the function $H$ is approximatively symmetric in each variable and approximatively homogeneous of degree $α$, where $α$ is an arbitrarily fixed real number.

math.CA↗
arXiv2016

Stability in the class of first order delay differential equations

The main aim of this paper is the investigation of the stability problem for ordinary delay differential equations. More precisely, we would like to study the following problem. Assume that for a continuous function a given delay differential equation is fulfilled only approximately. Is it true that in this case this function is close to an exact solution of this delay differential equation?

math.CA↗
arXiv2016

Approximate derivations of order $n$

The aim of this paper is to prove characterization theorems for higher order derivations. Among others we prove that the system defining higher order derivations is stable. Further characterization theorems in the spirit of N.~G.~de Bruijn will also be presented.

math.CA↗
arXiv2014

Additive solvability and linear independence of the solutions of a system of functional equations

The aim of this paper is twofold. On one hand, the additive solvability of the system of functional equations \[d_{k}(xy)=\sum_{i=0}^{k}Γ(i,k-i) d_{i}(x)d_{k-i}(y) \qquad (x,y\in \R,\,k\in\{0,\ldots,n\}) \] is studied, where $Δ_n:=\big\{(i,j)\in\Z\times\Z\mid 0\leq i,j\mbox{and}i+j\leq n\big\}$ and $Γ\colonΔ_n\to\R$ is a symmetric function such that $Γ(i,j)=1$ whenever $i\cdot j=0$. On the other hand, the linear dependence and independence of the additive solutions $d_{0},d_{1},\dots,d_{n}\colon \R\to\R$ of the above system of equations is characterized. As a consequence of the main result, for any nonzero real derivation $d\colon\R\to\R$, the iterates $d^0,d^1,\dots,d^n$ of $d$ are shown to be linearly independent, and the graph of the mapping $x\mapsto (x,d^1(x),\dots,d^n(x))$ to be dense in $\R^{n+1}$.

math.AC↗
arXiv2013

Characterization of derivations through their actions on certain elementary functions

The main aim of this note is to provide characterization theorems concerning real derivations. Among others the following implication will be verified: Assume that $ξ\colon \mathbb{R}\to \mathbb{R}$ is a given differentiable function and for the additive function $d\colon \mathbb{R}\to \mathbb{R}$, the mapping \[ x\longmapsto d(ξ(x))-ξ'(x)d(x) \] is regular (e. g. measurable, continuous, locally bounded). Then $d$ is a sum of a derivation and a linear function.

math.CA↗
arXiv2013

Notes on the characterization of derivations

Although the characterization of ring derivations has an extensive literature, up to now, all of the characterizations have had the following form: additivity and another property imply that the function in question is a derivation. The aim of this note is to point out that derivations can be described via a single equation.

math.CA↗
arXiv2013

Entropy functions and functional equations

The purpose of this note is to give the general solution of two functional equations connected to the Shannon entropy and also to the Tsallis entropy. As a result of this, we present the regular solution of these equations, as well. Furthermore, we point out that the regularity assumptions used in previous works can substantially be weakened.

math.CA↗
arXiv2013

General and alien solutions of a functional equation and of a functional inequality

The purpose of the present paper is to solve (under some assumption on the domain) the equation $$ g(x+y)-g(x)-g(y)=xf(y)+yf(x). $$ After determining the general solutions, we will investigate the so--called alien solutions. %More precisely, we will examine the cases when the above equation implies that %$g(x+y)=g(x)+g(y)$ and $xf(y)+yf(x)=0$. %Concerning this, necessary and sufficient conditions will be provided. Finally, we will discuss the real solutions of the following related functional inequality: $$ g(x+y)-g(x)-g(y)\geq xf(y)+yf(x). $$

math.CA↗
arXiv2013

On the modified entropy equation

The object of this paper is to solve the so--called modified entropy equation \[ f =f + μ f<\mathbf{0}, \frac{y}{y+z}, \frac{z}{y+z}>, \] on the positive cone of $\R^{k}$, where $μ$ is a given multiplicative function on this cone. After that the regular solutions of this equation are determined. Furthermore we investigate its connection between the entropy equation and other equations, as well.

math.CA↗
arXiv2013

Recent results on the stability of the parametric fundamental equation of information

The purpose of this paper is to summarize the recent results on the stability of the parametric fundamental equation of information. Furthermore, by the help of a modification of a method we used in \cite{GM08} we shall give a unified proof for the Hyers--Ulam stability of the equation in question, assuming that the parameter does not equal to 1. As a corollary of the main result, a system of equations, that defines the recursive and semi--symmetric information measures is also discussed.

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