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A generalization of the concept of distance based on the simplex inequality Kiss, Gergely ; Marichal, Jean-Luc ; Teheux, Bruno in Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry (in press) 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 ... [more ▼] 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. [less ▲] Detailed reference viewed: 54 (14 UL)The discrete Pompeiu problem on the plane Kiss, Gergely ; ; in Monatshefte für Mathematik (in press) We say that a finite subset $E$ of the Euclidean plane $\R^2$ has the discrete Pompeiu property with respect to isometries (similarities), if, whenever $f:\R^2\to \C$ is such that the sum of the values of ... [more ▼] We say that a finite subset $E$ of the Euclidean plane $\R^2$ has the discrete Pompeiu property with respect to isometries (similarities), if, whenever $f:\R^2\to \C$ 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 with respect to 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 with respect to isometries, and every finite set has the weighted discrete Pompeiu property with respect to similarities. [less ▲] Detailed reference viewed: 19 (2 UL)Derivations and differential operators on rings and fields Kiss, Gergely in Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae. Sectio Computatorica (2018) Let R be an integral domain of characteristic zero. We prove that a function D : R → 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 ... [more ▼] Let R be an integral domain of characteristic zero. We prove that a function D : R → 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 ⊂ R, there is a differential operator D of degree n such that f = D on F. We also prove that if d1, . . . , dn are nonzero derivations on R, then d1 ◦ . . . ◦ dn is a derivation of exact order n. [less ▲] Detailed reference viewed: 25 (1 UL)Pointwise regularity of parameterized affine zipper fractal curves fractal curves ; Kiss, Gergely ; in Nonlinearity (2018), 31(5), 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 ... [more ▼] 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, we apply our results for de Rham’s curve. [less ▲] Detailed reference viewed: 21 (2 UL)The discrete Pompeiu problem on the plane Kiss, Gergely Presentation (2017, November 27) The discrete Pompeiu problem is stemmed from an integral-geometric question on the plane. The problem is whether we can reconstruct a function if we know the average values of the function on every ... [more ▼] The discrete Pompeiu problem is stemmed from an integral-geometric question on the plane. The problem is whether we can reconstruct a function if we know the average values of the function on every congruent copy of a given pattern. After introducing the theory of spectral analysis on discrete Abelian groups, I show some results for the discrete version of the problem. One of the arguments is connected to a coloring problem of the plane. One of them is a geometric construction and some others based on some geometric and combinatoric properties of the plane. I also mention some unsolved questions of the topic. My talk is based on a joint work with M. Laczkovich and Cs. Vincze. [less ▲] Detailed reference viewed: 36 (3 UL)Visual characterization of associative quasitrivial nondecreasing functions on finite chains Kiss, Gergely E-print/Working paper (2017) Detailed reference viewed: 17 (3 UL)On functional equations characterizing derivations: methods and examples ; Kiss, Gergely ; E-print/Working paper (2017) Detailed reference viewed: 11 (3 UL)Associative idempotent nondecreasing functions are reducible Kiss, Gergely ; E-print/Working paper (2017) An n-variable associative function is called reducible if it can be written as a composition of a binary associative function. In this paper we summarize the known results when the function is defined on ... [more ▼] An n-variable associative function is called reducible if it can be written as a composition of a binary associative function. In this paper we summarize the known results when the function is defined on a chain and nondecreasing. The main result of this paper shows that associative idempotent and nondecreasing functions are uniquely reducible. [less ▲] Detailed reference viewed: 5 (0 UL)Recent results on conservative and symmetric n-ary semigroups Kiss, Gergely ; Devillet, Jimmy ; Marichal, Jean-Luc Scientific Conference (2017, June 16) See attached file Detailed reference viewed: 48 (13 UL)Generalization of Czoga\l a-Drewniak Theorem for $n$-ary semigroups Kiss, Gergely ; in Torra, Vicenç; Mesiar, Radko; De Baets, Bernard (Eds.) Aggregation Functions in Theory and in Practice (2017) We investigate n-ary semigroups as a natural generalization of binary semigroups. We refer it as a pair (X,F_n), where X is a set and an n-associative function F_n : X^n -> X is defined on X. We show that ... [more ▼] We investigate n-ary semigroups as a natural generalization of binary semigroups. We refer it as a pair (X,F_n), where X is a set and an n-associative function F_n : X^n -> X is defined on X. We show that if F_n is idempotent, n-associative function which is monotone in each of its variables, defined on an interval I and has a neutral element, then F_n is combination of the minimum and maximum operation. Moreover we can characterize the n-ary semigroups (I,F_n) where F_n has the previous properties. [less ▲] Detailed reference viewed: 24 (3 UL)Characterizations of quasitrivial symmetric nondecreasing associative operations Devillet, Jimmy ; Kiss, Gergely ; Marichal, Jean-Luc E-print/Working paper (2017) In this paper we are interested in the class of n-ary operations on an arbitrary chain that are quasitrivial, symmetric, nondecreasing, and associative. We first provide a description of these operations ... [more ▼] In this paper we are interested in the class of n-ary operations on an arbitrary chain that are quasitrivial, symmetric, nondecreasing, and associative. We first provide a description of these operations. We then prove that associativity can be replaced with bisymmetry in the definition of this class. Finally we investigate the special situation where the chain is finite. [less ▲] Detailed reference viewed: 18 (9 UL)On spectral analysis in varieties containing the solutions of inhomogeneous linear functional equations Kiss, Gergely ; in Aequationes Mathematicae (2017) The aim of the paper is to investigate the solutions of special inhomogeneous linear functional equations using spectral analysis in a translation invariant closed linear subspace of additive ... [more ▼] The aim of the paper is to investigate the solutions of special inhomogeneous linear functional equations 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 the case of homogeneous linear functional equations. The foundations of the theory can be found in Kiss and Varga (Aequat Math 88(1):151–162, 2014) and Kiss and Laczkovich (Aequat Math 89(2):301–328, 2015). We are going to adopt the main theoretical tools to solve some inhomogeneous problems due to Koclȩga-Kulpa and Szostok (Ann Math Sylesianae 22:27–40, 2008), see also Koclȩga-Kulpa and Szostok (Georgian Math J 16:725–736, 2009; Acta Math Hung 130(4):340–348, 2011). They are motivated by quadrature rules of approximate integration. [less ▲] Detailed reference viewed: 29 (0 UL)A characterization of n-associative, monotone, idempotent functions on an interval that have neutral elements Kiss, Gergely ; in Semigroup Forum (2017) We investigate monotone idempotent n-ary semigroups and provide a generalization of the Czogala–Drewniak Theorem, which describes the idempotent monotone associative functions having a neutral element. We ... [more ▼] We investigate monotone idempotent n-ary semigroups and provide a generalization of the Czogala–Drewniak Theorem, which describes the idempotent monotone associative functions having a neutral element. We also present a complete characterization of idempotent monotone n-associative functions on an interval that have neutral elements. [less ▲] Detailed reference viewed: 80 (38 UL)On spectral synthesis in varieties containing the solutions of inhomogeneous linear functional equations Kiss, Gergely ; in Aequationes Mathematicae (2017) As a continuation of our previous work [2] the aim of the recent paper is to investigate the solutions of special inhomogeneous linear functional equations by using spectral synthesis in translation ... [more ▼] As a continuation of our previous work [2] 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 [5]. Using spectral analysis in some related varieties we can prove the existence of special solutions (automorphisms) of the functional equation but 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 [8], see also [7, 9]. [less ▲] Detailed reference viewed: 21 (3 UL)Discrete Pompeiu problem on the plain Kiss, Gergely Presentation (2016, December) Detailed reference viewed: 26 (3 UL)A characterisation of associative idempotent nondecreasing functions with neutral elements Kiss, Gergely ; ; Marichal, Jean-Luc et al Scientific Conference (2016, June) Detailed reference viewed: 61 (14 UL)International Symposium on Aggregation and Structures (ISAS 2016) - Book of abstracts Kiss, Gergely ; Marichal, Jean-Luc ; Teheux, Bruno Book published by NA (2016) Detailed reference viewed: 226 (11 UL)An extension of the concept of distance as functions of several variables Kiss, Gergely ; Marichal, Jean-Luc ; Teheux, Bruno in De Baets, Bernard; Mesiar, Radko; Saminger-Platz, Susanne (Eds.) et al 36th Linz Seminar on Fuzzy Set Theory (LINZ 2016) - Functional Equations and Inequalities (2016, February) Extensions of the concept of distance to more than two elements have been recently proposed in the literature to measure to which extent the elements of a set are spread out. Such extensions may be ... [more ▼] Extensions of the concept of distance to more than two elements have been recently proposed in the literature to measure to which extent the elements of a set are spread out. Such extensions may be particularly useful to define dispersion measures for instance in statistics or data analysis. In this note we provide and discuss an extension of the concept of distance, called n-distance, as functions of n variables. The key feature of this extension is a natural generalization of the triangle inequality. We also provide some examples of n-distances that involve geometric and graph theoretic constructions. [less ▲] Detailed reference viewed: 74 (12 UL)Decomposition of balls in R^d Kiss, Gergely ; in Mathematika (2016), 62(2), 378-405 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 ... [more ▼] 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 ≥ 6. We show that the minimal number of required pieces is less than 20d if d ≥ 10. [less ▲] Detailed reference viewed: 39 (3 UL)Algebraic methods for the solution of linear functional equations Kiss, Gergely ; ; in Acta Mathematica Hungarica (2015), 146(1), 128141 Detailed reference viewed: 29 (6 UL) |
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