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Gergely Kiss

Publications and source records attributed to Gergely Kiss.

50 records · Page 3Linked to original sources
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

Associative idempotent nondecreasing functions are reducible

An $n$-ary associative function is called reducible if it can be written as a composition of a binary associative function. We summarize known results when the function is defined on a chain and is nondecreasing. Our main result shows that associative idempotent and nondecreasing functions are uniquely reducible.

math.RA↗
arXiv2018

Characterizations of biselective operations

Let $X$ be a nonempty set and let $i,j \in \{1,2,3,4\}$. We say that a binary operation $F:X^2\to X$ is $(i,j)$-selective if $$ F(F(x_1,x_2),F(x_3,x_4))~=~F(x_i,x_j), $$ for all $x_1,x_2,x_3,x_4\in X$. In this paper we provide characterizations of the class of $(i,j)$-selective operations. We also investigate some subclasses by adding algebraic properties such as associativity or bisymmetry.

math.RA↗
arXiv2018

A generalization of the concept of distance based on the simplex inequality

We introduce and discuss the concept of \emph{$n$-distance}, a generalization to $n$ elements of the classical notion of distance obtained by replacing the triangle inequality with the so-called simplex inequality $$ d(x_1, \ldots, x_n)~\leq~K\, \sum_{i=1}^n d(x_1, \ldots, x_n)_i^z{\,}, \qquad x_1, \ldots, x_n, z \in X, $$ where $K=1$. Here $d(x_1,\ldots,x_n)_i^z$ is obtained from the function $d(x_1,\ldots,x_n)$ by setting its $i$th variable to $z$. We provide several examples of $n$-distances, and for each of them we investigate the infimum of the set of real numbers $K\in\left]0,1\right]$ for which the inequality above holds. We also introduce a generalization of the concept of $n$-distance obtained by replacing in the simplex inequality the sum function with an arbitrary symmetric function.

math.MG↗
arXiv2018

Derivations and differential operators on rings and fields

Let $R$ be an integral domain of characteristic zero. We prove that a function $D\colon R\to R$ is a derivation of order $n$ if and only if $D$ belongs to the closure of the set of differential operators of degree $n$ in the product topology of $R^R$, where the image space is endowed with the discrete topology. In other words, $f$ is a derivation of order $n$ if and only if, for every finite set $F\subset R$, there is a differential operator $D$ of degree $n$ such that $f=D$ on $F$. We also prove that if $d_1, \dots, d_n$ are nonzero derivations on $R$, then $d_1 \circ \ldots \circ d_n$ is a derivation of exact order $n$.

math.RA↗
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.

math.CA↗
arXiv2017

A characterization of $n$-associative, monotone, idempotent functions on an interval that have neutral elements

We investigate monotone idempotent $n$-ary semigroups. One of the main result of this article is the generalisation of Czogala-Drewniak Theorem, which describes the idempotent monotone associative functions having neutral element. Furthermore we present the full characterisation of idempotent, monotone, $n$-associative functions on an interval which have neutral elements. %{\bl defined}. Our description provides that monotone, idempotent, $n$-ary semigroups are quasitrivial.

math.GR↗
arXiv2017

Pointwise regularity of parameterized affine zipper fractal curves

We study the pointwise regularity of zipper fractal curves generated by affine mappings. Under the assumption of dominated splitting of index-1, we calculate the Hausdorff dimension of the level sets of the pointwise Hölder exponent for a subinterval of the spectrum. We give an equivalent characterization for the existence of regular pointwise Hölder exponent for Lebesgue almost every point. In this case, we extend the multifractal analysis to the full spectrum. In particular, our results apply for de Rham's curve.

math.DS↗
arXiv2017

On spectral analysis in varieties containing the solutions of inhomogeneous linear functional equations

The aim of the paper is to investigate the solutions of special inhomogeneous linear functional equations by using spectral analysis in a translation invariant closed linear subspace of additive/multiadditive functions containing the restrictions of the solutions to finitely generated fields. The application of spectral analysis in some related varieties is a new and important trend in the theory of functional equations; especially they have successful applications in case of homogeneous linear functional equations. The foundation of the theory can be found in M. Laczkovich and G. Kiss \cite{KL}, see also G. Kiss and A. Varga \cite{KV}. We are going to adopt the main theoretical tools to solve some inhomogeneous problems due to T. Szostok \cite{KKSZ08}, see also \cite{KKSZ} and \cite{KKSZW}. They are motivated by quadrature rules of approximate integration.

math.CV↗
arXiv2017

On spectral synthesis in varieties containing the solutions of inhomogeneous linear functional equations

As a continuation of our previous work \cite{KV2} the aim of the recent paper is to investigate the solutions of special inhomogeneous linear functional equations by using spectral synthesis in translation invariant closed linear subspaces of additive/multiadditive functions containing the restrictions of the solutions to finitely generated fields. The idea is based on the fundamental work of M. Laczkovich and G. Kiss \cite{KL}. Using spectral analysis in some related varieties we can prove the existence of special solutions (automorphisms) of the functional equation but the spectral synthesis allows us to describe the entire space of solutions on a large class of finitely generated fields. It is spanned by the so-called exponential monomials which can be given in terms of automorphisms of $\cc$ and differential operators. We apply the general theory to some inhomogeneous problems motivated by quadrature rules of approximate integration \cite{KKSZ08}, see also \cite{KKSZ} and \cite{KKSZW}.

math.CV↗
arXiv2016

The discrete Pompeiu problem on the plane

We say that a finite subset $E$ of the Euclidean plane $\mathbb{R}^2$ has the discrete Pompeiu property with respect to isometries (similarities), if, whenever $f:\mathbb{R}^2\to \mathbb{C}$ is such that the sum of the values of $f$ on any congruent (similar) copy of $E$ is zero, then $f$ is identically zero. We show that every parallelogram and every quadrangle with rational coordinates has the discrete Pompeiu property w.r.t. isometries. We also present a family of quadrangles depending on a continuous parameter having the same property. We investigate the weighted version of the discrete Pompeiu property as well, and show that every finite linear set with commensurable distances has the weighted discrete Pompeiu property w.r.t. isometries, and every finite set has the weighted discrete Pompeiu property w.r.t. similarities.

math.FA↗
arXiv2014

Decomposition of balls in $\mathbb{R}^d$

We investigate the decomposition problem of balls into finitely many congruent pieces in dimension $d=2k$. In addition, we prove that the $d$ dimensional unit ball $B_d$ can be divided into finitely many congruent pieces if $d=4$ or $d\ge 6$. We show that the minimal number of required pieces is less than $20d$ if $d \ge 10$.

math.MG↗
arXiv2012

How large dimension guarantees a given angle?

We study the following two problems: (1) Given $n\ge 2$ and $\al$, how large Hausdorff dimension can a compact set $A\su\Rn$ have if $A$ does not contain three points that form an angle $\al$? (2) Given $\al$ and $\de$, how large Hausdorff dimension can a %compact subset $A$ of a Euclidean space have if $A$ does not contain three points that form an angle in the $\de$-neighborhood of $\al$? An interesting phenomenon is that different angles show different behaviour in the above problems. Apart from the clearly special extreme angles 0 and $180^\circ$, the angles $60^\circ,90^\circ$ and $120^\circ$ also play special role in problem (2): the maximal dimension is smaller for these special angles than for the other angles. In problem (1) the angle $90^\circ$ seems to behave differently from other angles.

math.CA↗