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Gergo Kiss

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Papers by Gergo Kiss

On functional equations characterizing derivations: methods and examples

arXiv (Cornell University), Aug 25, 2017

Functional equations satisfied by additive functions have a special interest not only in the theo... more 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. This method allows us to refine the process of computing the solutions of univariate functional equations of the form n k=1 x p k f k (x q k) = 0, where p k and q k (k = 1,. .. , n) are given nonnegative integers and the unknown functions f 1 ,. .. , f n : R → 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 it is spanned by differential operators. The results are uniformly based on the investigation of the multivariate version of the functional equations.

Characterizations of biselective operations

arXiv (Cornell University), Jun 6, 2018

Let X be a nonempty set and let i, j ∈ {1, 2, 3, 4}. We say that a binary operation F ∶ X 2 → X i... more Let X be a nonempty set and let i, j ∈ {1, 2, 3, 4}. We say that a binary operation F ∶ X 2 → 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 ∈ 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.

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

arXiv (Cornell University), Apr 16, 2017

As a continuation of our previous work [18] the aim of the recent paper is to investigate the sol... more As a continuation of our previous work [18] 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 [3]. 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 C and differential operators. We apply the general theory to some inhomogeneous problems motivated by quadrature rules of approximate integration [6], see also [7] and [8].

Isometric rigidity of Wasserstein spaces: the graph metric case

arXiv (Cornell University), Jan 4, 2022

The aim of this paper is to prove that the p-Wasserstein space W p (X) is isometrically rigid for... more The aim of this paper is to prove that the p-Wasserstein space W p (X) is isometrically rigid for all p ≥ 1 whenever X is a countable graph metric space. As a consequence, we obtain that for every countable group H and any p ≥ 1 there exists a p-Wasserstein space whose isometry group is isomorphic to H.

$Research paper thumbnail of Fuglede's conjecture holds on $\mathbb{Z}_p^2 \times \mathbb{Z}_q$$

Fuglede's conjecture holds on $\mathbb{Z}_p^2 \times \mathbb{Z}_q$

arXiv (Cornell University), Dec 15, 2019

The study of Fuglede's conjecture on the direct product of elementary abelian groups was initiate... more The study of Fuglede's conjecture on the direct product of elementary abelian groups was initiated by Iosevich et al. For the product of two elementary abelian groups the conjecture holds. For Z 3 p the problem is still open if p ≥ 11. In connection we prove that Fuglede's conjecture holds on Z 2 p × Z q by developing a method based on ideas from discrete geometry.

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

arXiv (Cornell University), Sep 1, 2016

We investigate monotone idempotent n-ary semigroups and provide a generalization of the Czogala-D... 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.

$Research paper thumbnail of Pointwise regularity of parameterized affine zipper fractal curves$

Pointwise regularity of parameterized affine zipper fractal curves

arXiv (Cornell University), Aug 16, 2016

We study the pointwise regularity of zipper fractal curves generated by affine mappings. Under th... 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.

Visual characterization of associative quasitrivial nondecreasing operations on finite chains

arXiv (Cornell University), Sep 21, 2017

In this paper we provide visual characterization of associative quasitrivial nondecreasing operat... more In this paper we provide visual characterization of associative quasitrivial nondecreasing operations on finite chains. We also provide a characterization of bisymmetric quasitrivial nondecreasing binary operations on finite chains. Finally, we estimate the number of operations belonging to the previous classes.

Translation invariant linear spaces of polynomials

arXiv (Cornell University), Aug 17, 2021

A set of polynomials M is called a submodule of C[x 1 ,. .. , x n ] if M is a translation invaria... more A set of polynomials M is called a submodule of C[x 1 ,. .. , x n ] if M is a translation invariant linear subspace of C[x 1 ,. .. , x n ]. We present a description of the submodules of C[x, y] in terms of a special type of submodules. We say that the submodule M of C[x, y] is an Lmodule of order s if, whenever F (x, y) = N n=0 f n (x) • y n ∈ M is such that f 0 =. .. = f s−1 = 0, then F = 0. We show that the proper submodules of C[x, y] are the sums M d + M , where M d = {F ∈ C[x, y] : deg x F < d}, and M is an L-module. We give a construction of L-modules parametrized by sequences of complex numbers. A submodule M ⊂ C[x 1 ,. .. , x n ] is decomposable if it is the sum of finitely many proper submodules of M. Otherwise M is indecomposable. It is easy to see that every submodule of C[x 1 ,. .. , x n ] is the sum of finitely many indecomposable submodules. In C[x, y] every indecomposable submodule is either an L-module or equals M d for some d. In the other direction we show that M d is indecomposable for every d, and so is every L-module of order 1. Finally, we prove that there exists a submodule of C[x, y] (in fact, an L-module of order 1) which is not relatively closed in C[x, y]. This answers a problem posed by L. Székelyhidi in 2011.

On a class of linear functional equations without range condition

arXiv (Cornell University), Mar 19, 2019

The main purpose of this work is to provide the general solutions of a class of linear functional... more The main purpose of this work is to provide the general solutions of a class of linear functional equations. Let n ≥ 2 be an arbitrarily fixed integer, let further X and Y be linear spaces over the field K and let α i , β i ∈ K, i = 1,. .. , n be arbitrarily fixed constants. We will describe all those functions f, f i, j : X ×Y → K,

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

arXiv (Cornell University), Apr 16, 2017

The aim of the paper is to investigate the solutions of special inhomogeneous linear functional e... more 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 [2], see also G. Kiss and A. Varga [1]. We are going to adopt the main theoretical tools to solve some inhomogeneous problems due to T. Szostok [5], see also [6] and [7]. They are motivated by quadrature rules of approximate integration.

Associative idempotent nondecreasing functions are reducible

arXiv (Cornell University), Jul 13, 2017

An n-ary associative function is called reducible if it can be written as a composition of a bina... more 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.

Separately polynomial functions

arXiv (Cornell University), Jan 8, 2021

It is known that if f : R 2 → R is a polynomial in each variable, then f is a polynomial. We pres... more It is known that if f : R 2 → R is a polynomial in each variable, then f is a polynomial. We present generalizations of this fact, when R 2 is replaced by G × H, where G and H are topological Abelian groups. We show, e.g., that the conclusion holds (with generalized polynomials in place of polynomials) if G is a connected Baire space and H has a dense subgroup of finite rank or, for continuous functions, if G and H are connected Baire spaces. The condition of continuity can be omitted if G and H are locally compact or one of them is metrizable. We present several examples showing that the results are not far from being optimal.

$Research paper thumbnail of Spectral sets and tiles in $\mathbb{Z}_p^2 \times \mathbb{Z}_q^2$$

Spectral sets and tiles in $\mathbb{Z}_p^2 \times \mathbb{Z}_q^2$

arXiv (Cornell University), May 21, 2021

In this paper we study Fuglede's conjecture on the direct product of two cyclic groups. After col... more

Functional equation that characterize higher order derivations

A characterisation of associative idempotent nondecreasing functions with neutral elements

Fuglede's conjecture holds for cyclic groups of order $pqrs$

Repository of the Academy's Library (Library of the Hungarian Academy of Sciences), Nov 18, 2020

The tile-spectral direction of the discrete Fuglede-conjecture is well-known for cyclic groups of... more The tile-spectral direction of the discrete Fuglede-conjecture is well-known for cyclic groups of square-free order, initiated by Laba and Meyerowitz, but the spectral-tile direction is far from being well-understood. The product of at most three primes as the order of the cyclic group was studied intensely in the last couple of years. In this paper we study the case when the order of the cyclic group is the product of four different primes and prove that Fuglede's conjecture holds in this case.

Characterization of field homomorphisms through Pexiderized functional equations

arXiv (Cornell University), Oct 29, 2018

The aim of this paper is to prove characterization theorems for field homomorphisms. More precise... more The aim of this paper is to prove characterization theorems for field homomorphisms. More precisely, the main result investigates the following problem. Let n ∈ N be arbitrary, K a field and f 1 ,. .. , f n : K → C additive functions. Suppose further that equation n i=1 f q i i x p i = 0 (x ∈ K) is also satisfied. Then the functions f 1 ,. .. , f n are linear combinations of field homomorphisms from K to C.

Sincov's and other functional equations and negative interest rates

arXiv (Cornell University), Oct 5, 2022

Investigating the future value (Ã × Ø) of a capital Ã invested at date Ë at date Ø the "natural" ... more Investigating the future value (Ã × Ø) of a capital Ã invested at date Ë at date Ø the "natural" condition (Ã × Ø) ≥ Ã has lost its naturality because of the strange fact of negative interest rates. This leads to the task of describing the possible solutions of the multiplicative Sincov equation (× Ù) = (× Ø) (Ø Ù) for × ≤ Ø ≤ Ù where (× Ø) = 0 may happen. In this paper we solve this task and discuss connections to the theory of investments.

$Research paper thumbnail of Tiling and weak tiling in $$(\mathbb {Z}_p)^d$$$

Tiling and weak tiling in $$(\mathbb {Z}_p)^d$$

Sampling Theory, Signal Processing, and Data Analysis, Dec 3, 2023

We discuss the relation of tiling, weak tiling and spectral sets in finite abelian groups. In par... more We discuss the relation of tiling, weak tiling and spectral sets in finite abelian groups. In particular, in elementary p-groups (Z p) d , we introduce an averaging procedure that leads to a natural object of study: a 4-tuple of functions which can be regarded as a common generalization of tiles and spectral sets. We characterize such 4-tuples for d = 1, 2, and prove some partial results for d = 3.

On functional equations characterizing derivations: methods and examples

arXiv (Cornell University), Aug 25, 2017

Functional equations satisfied by additive functions have a special interest not only in the theo... more 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. This method allows us to refine the process of computing the solutions of univariate functional equations of the form n k=1 x p k f k (x q k) = 0, where p k and q k (k = 1,. .. , n) are given nonnegative integers and the unknown functions f 1 ,. .. , f n : R → 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 it is spanned by differential operators. The results are uniformly based on the investigation of the multivariate version of the functional equations.

Characterizations of biselective operations

arXiv (Cornell University), Jun 6, 2018

Let X be a nonempty set and let i, j ∈ {1, 2, 3, 4}. We say that a binary operation F ∶ X 2 → X i... more Let X be a nonempty set and let i, j ∈ {1, 2, 3, 4}. We say that a binary operation F ∶ X 2 → 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 ∈ 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.

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

arXiv (Cornell University), Apr 16, 2017

As a continuation of our previous work [18] the aim of the recent paper is to investigate the sol... more As a continuation of our previous work [18] 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 [3]. 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 C and differential operators. We apply the general theory to some inhomogeneous problems motivated by quadrature rules of approximate integration [6], see also [7] and [8].

Isometric rigidity of Wasserstein spaces: the graph metric case

arXiv (Cornell University), Jan 4, 2022

The aim of this paper is to prove that the p-Wasserstein space W p (X) is isometrically rigid for... more The aim of this paper is to prove that the p-Wasserstein space W p (X) is isometrically rigid for all p ≥ 1 whenever X is a countable graph metric space. As a consequence, we obtain that for every countable group H and any p ≥ 1 there exists a p-Wasserstein space whose isometry group is isomorphic to H.

$Research paper thumbnail of Fuglede's conjecture holds on $\mathbb{Z}_p^2 \times \mathbb{Z}_q$$

Fuglede's conjecture holds on $\mathbb{Z}_p^2 \times \mathbb{Z}_q$

arXiv (Cornell University), Dec 15, 2019

The study of Fuglede's conjecture on the direct product of elementary abelian groups was initiate... more The study of Fuglede's conjecture on the direct product of elementary abelian groups was initiated by Iosevich et al. For the product of two elementary abelian groups the conjecture holds. For Z 3 p the problem is still open if p ≥ 11. In connection we prove that Fuglede's conjecture holds on Z 2 p × Z q by developing a method based on ideas from discrete geometry.

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

arXiv (Cornell University), Sep 1, 2016

We investigate monotone idempotent n-ary semigroups and provide a generalization of the Czogala-D... 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.

$Research paper thumbnail of Pointwise regularity of parameterized affine zipper fractal curves$

Pointwise regularity of parameterized affine zipper fractal curves

arXiv (Cornell University), Aug 16, 2016

We study the pointwise regularity of zipper fractal curves generated by affine mappings. Under th... 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.

Visual characterization of associative quasitrivial nondecreasing operations on finite chains

arXiv (Cornell University), Sep 21, 2017

In this paper we provide visual characterization of associative quasitrivial nondecreasing operat... more In this paper we provide visual characterization of associative quasitrivial nondecreasing operations on finite chains. We also provide a characterization of bisymmetric quasitrivial nondecreasing binary operations on finite chains. Finally, we estimate the number of operations belonging to the previous classes.

Translation invariant linear spaces of polynomials

arXiv (Cornell University), Aug 17, 2021

A set of polynomials M is called a submodule of C[x 1 ,. .. , x n ] if M is a translation invaria... more A set of polynomials M is called a submodule of C[x 1 ,. .. , x n ] if M is a translation invariant linear subspace of C[x 1 ,. .. , x n ]. We present a description of the submodules of C[x, y] in terms of a special type of submodules. We say that the submodule M of C[x, y] is an Lmodule of order s if, whenever F (x, y) = N n=0 f n (x) • y n ∈ M is such that f 0 =. .. = f s−1 = 0, then F = 0. We show that the proper submodules of C[x, y] are the sums M d + M , where M d = {F ∈ C[x, y] : deg x F < d}, and M is an L-module. We give a construction of L-modules parametrized by sequences of complex numbers. A submodule M ⊂ C[x 1 ,. .. , x n ] is decomposable if it is the sum of finitely many proper submodules of M. Otherwise M is indecomposable. It is easy to see that every submodule of C[x 1 ,. .. , x n ] is the sum of finitely many indecomposable submodules. In C[x, y] every indecomposable submodule is either an L-module or equals M d for some d. In the other direction we show that M d is indecomposable for every d, and so is every L-module of order 1. Finally, we prove that there exists a submodule of C[x, y] (in fact, an L-module of order 1) which is not relatively closed in C[x, y]. This answers a problem posed by L. Székelyhidi in 2011.

On a class of linear functional equations without range condition

arXiv (Cornell University), Mar 19, 2019

The main purpose of this work is to provide the general solutions of a class of linear functional... more The main purpose of this work is to provide the general solutions of a class of linear functional equations. Let n ≥ 2 be an arbitrarily fixed integer, let further X and Y be linear spaces over the field K and let α i , β i ∈ K, i = 1,. .. , n be arbitrarily fixed constants. We will describe all those functions f, f i, j : X ×Y → K,

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

arXiv (Cornell University), Apr 16, 2017

The aim of the paper is to investigate the solutions of special inhomogeneous linear functional e... more 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 [2], see also G. Kiss and A. Varga [1]. We are going to adopt the main theoretical tools to solve some inhomogeneous problems due to T. Szostok [5], see also [6] and [7]. They are motivated by quadrature rules of approximate integration.

Associative idempotent nondecreasing functions are reducible

arXiv (Cornell University), Jul 13, 2017

An n-ary associative function is called reducible if it can be written as a composition of a bina... more 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.

Separately polynomial functions

arXiv (Cornell University), Jan 8, 2021

It is known that if f : R 2 → R is a polynomial in each variable, then f is a polynomial. We pres... more It is known that if f : R 2 → R is a polynomial in each variable, then f is a polynomial. We present generalizations of this fact, when R 2 is replaced by G × H, where G and H are topological Abelian groups. We show, e.g., that the conclusion holds (with generalized polynomials in place of polynomials) if G is a connected Baire space and H has a dense subgroup of finite rank or, for continuous functions, if G and H are connected Baire spaces. The condition of continuity can be omitted if G and H are locally compact or one of them is metrizable. We present several examples showing that the results are not far from being optimal.

$Research paper thumbnail of Spectral sets and tiles in $\mathbb{Z}_p^2 \times \mathbb{Z}_q^2$$

Spectral sets and tiles in $\mathbb{Z}_p^2 \times \mathbb{Z}_q^2$

arXiv (Cornell University), May 21, 2021

In this paper we study Fuglede's conjecture on the direct product of two cyclic groups. After col... more

Functional equation that characterize higher order derivations

A characterisation of associative idempotent nondecreasing functions with neutral elements

Fuglede's conjecture holds for cyclic groups of order $pqrs$

Repository of the Academy's Library (Library of the Hungarian Academy of Sciences), Nov 18, 2020

The tile-spectral direction of the discrete Fuglede-conjecture is well-known for cyclic groups of... more The tile-spectral direction of the discrete Fuglede-conjecture is well-known for cyclic groups of square-free order, initiated by Laba and Meyerowitz, but the spectral-tile direction is far from being well-understood. The product of at most three primes as the order of the cyclic group was studied intensely in the last couple of years. In this paper we study the case when the order of the cyclic group is the product of four different primes and prove that Fuglede's conjecture holds in this case.

Characterization of field homomorphisms through Pexiderized functional equations

arXiv (Cornell University), Oct 29, 2018

The aim of this paper is to prove characterization theorems for field homomorphisms. More precise... more The aim of this paper is to prove characterization theorems for field homomorphisms. More precisely, the main result investigates the following problem. Let n ∈ N be arbitrary, K a field and f 1 ,. .. , f n : K → C additive functions. Suppose further that equation n i=1 f q i i x p i = 0 (x ∈ K) is also satisfied. Then the functions f 1 ,. .. , f n are linear combinations of field homomorphisms from K to C.

Sincov's and other functional equations and negative interest rates

arXiv (Cornell University), Oct 5, 2022

Investigating the future value (Ã × Ø) of a capital Ã invested at date Ë at date Ø the "natural" ... more Investigating the future value (Ã × Ø) of a capital Ã invested at date Ë at date Ø the "natural" condition (Ã × Ø) ≥ Ã has lost its naturality because of the strange fact of negative interest rates. This leads to the task of describing the possible solutions of the multiplicative Sincov equation (× Ù) = (× Ø) (Ø Ù) for × ≤ Ø ≤ Ù where (× Ø) = 0 may happen. In this paper we solve this task and discuss connections to the theory of investments.

$Research paper thumbnail of Tiling and weak tiling in $$(\mathbb {Z}_p)^d$$$

Tiling and weak tiling in $$(\mathbb {Z}_p)^d$$

Sampling Theory, Signal Processing, and Data Analysis, Dec 3, 2023

We discuss the relation of tiling, weak tiling and spectral sets in finite abelian groups. In par... more We discuss the relation of tiling, weak tiling and spectral sets in finite abelian groups. In particular, in elementary p-groups (Z p) d , we introduce an averaging procedure that leads to a natural object of study: a 4-tuple of functions which can be regarded as a common generalization of tiles and spectral sets. We characterize such 4-tuples for d = 1, 2, and prove some partial results for d = 3.