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Włodzimierz Fechner

Publications and source records attributed to Włodzimierz Fechner.

18 recordsLinked to original sources
arXiv2026

Second-order derivations of function spaces -- a characterization of second-order differential operators

Let $Ω\subset \mathbb{R}$ be a nonempty and open set, then for all $f, g, h\in \mathscr{C}^{2}(Ω)$ we have \begin{multline*} \diff{2}{x}(f\cdot g\cdot h) -f\diff{2}{x}(g\cdot h)-g\diff{2}{x}(f\cdot h)-h\diff{2}{x}(f\cdot g) + f\cdot g\diff{2}{x}h+f\cdot h\diff{2}{x}g+g\cdot h\diff{2}{x}f=0 \end{multline*} The aim of this paper is to consider the corresponding operator equation \[ D(f\cdot g \cdot h) - fD(g\cdot h) - gD(f\cdot h) - hD(f \cdot g) + f\cdot g D(h) + f\cdot h D(g) +g\cdot h D(f) =0 \] for operators $D\colon \mathscr{C}^{k}(Ω)\to \mathscr{C}(Ω)$, where $k$ is a given nonnegative integer and the above identity is supposed to hold for all $f, g, h \in \mathscr{C}^{k}(Ω)$. We show that besides the operators of first and second derivative, there are more solutions to this equation, and we characterize all solutions. Some special cases characterizing differential operators are also studied.

math.CA↗
arXiv2025

A direct and algebraic characterization of higher-order differential operators

This paper presents an algebraic approach to characterizing higher-order differential operators. While the foundational Leibniz rule addresses first-order derivatives, its extension to higher orders typically involves identities relating multiple distinct operators. In contrast, we introduce a novel operator equation involving only a single $n$\textsuperscript{th}-order differential operator. We demonstrate that, under certain mild conditions, this equation serves to characterize such operators. Specifically, our results show that these higher-order differential operators can be identified as particular solutions to this single-operator identity. This approach provides a framework for understanding the algebraic structure of higher-order differential operators acting on function spaces.

math.CA↗
arXiv2025

Two characterizations of quasiconvexity

We present two characterizations of quasiconvexity for radially semicontinuous mappings defined on a convex subset of a real linear space. As an application we obtain an extension of the Sion's minimax theorem, as well as a new characterization of quasiconvex risk measures.

math.OC↗
arXiv2024

On a generalized conjecture by Alzer and Matkowski

We study a recent conjecture proposed by Horst Alzer and Janusz Matkowski concerning a bilinearity property of the Cauchy exponential difference for real-to-real functions. The original conjecture was affirmatively resolved by Tomasz Małolepszy. We deal with generalizations for real or complex mappings acting on a linear space.

math.CA↗
arXiv2024

Characterizations of second-order differential operators

{Let $N, k$ be positive integers with $k\geq 2$, and $Ω\subset \mathbb{R}^{N}$ be a domain.} By the well-known properties of the Laplacian and the gradient, we have \[ Δ(f\cdot g)(x)=g(x) Δf(x)+f(x) Δg(x)+2\langle \nabla f(x), \nabla g(x)\rangle \] for all $f, g\in \mathscr{C}^{k}(Ω, \mathbb{R})$. {Due to the results of H.~König and V.~Milman, Operator relations characterizing derivatives. Birkhäuser / Springer, Cham, 2018.,} the converse is also true, i.e. this operator equation characterizes the Laplacian and the gradient under some assumptions. Thus the main aim of this paper is to provide an extension of this result and to study the corresponding equation \[ T(f\cdot g)= fT(g)+T(f)g+2B(A(f), A(g)) \qquad \left(f, g\in P\right), \] where $Q$ and $R$ are commutative rings, $P$ is a subring of $Q$ and $T\colon P\to Q$ and $A\colon P\to R$ are additive, while $B\colon R\times R\to Q$ is a symmetric and bi-additive. Related identities with one function will also be considered.

math.CA↗
arXiv2023

A characterization of differential operators in the ring of complex polynomials

The paper aims to provide a full characterization of all operators $T\colon \mathscr{P}(\mathbb{C}) \to \mathscr{P}(\mathbb{C})$ acting on the space of all complex polynomials that satisfy the Leibniz rule \[ T(f\cdot g)= T(f)\cdot g+f\cdot T(g) \] for all $f, g\in \mathscr{P}(\mathbb{C})$. We do not assume the linearity of $T$. As we will see, contrary to the well-known theorems for function spaces there are many other solutions here, not only differential operators. From our main result, we also derive two corollaries, showing that in some special cases operators that satisfy the Leibniz rule have some particular form.

math.CA↗
arXiv2023

Operator relations characterizing higher-order differential operators

Let $r$ be a positive integer, $N$ be a nonnegative integer and $Ω\subset \mathbb{R}^{r}$ be a domain. Further, for all multi-indices $α\in \mathbb{N}^{r}$, $|α|\leq N$, let us consider the partial differential operator $D^α$ defined by \[ D^α= \frac{\partial^{|α|}}{\partial x_{1}^{α_{1}}\cdots \partial x_{r}^{α_{r}}}, \] where $α= (α_{1}, \ldots, α_{r})$. Here by definition we mean $D^{0}\equiv \mathrm{id}$. An easy computation shows that if $f, g\in \mathscr{C}^{N}(Ω)$ and $α\in \mathbb{N}^{r}, |α|\leq N$, then we have \[ \tag{$\ast$} D^α(f\cdot g) = \sum_{β\leq α}\binomαβD^β(f)\cdot D^{α- β}(g). \] This paper is devoted to the study of identity $(\ast)$ in the space $\mathscr{C}(Ω)$. More precisely, if $r$ is a positive integer, $N$ is a nonnegative integer and $Ω\subset \mathbb{R}^{r}$ is a domain, then we describe those mappings $T_α \colon \mathscr{C}(Ω)\to \mathscr{C}(Ω)$, $α\in \mathbb{N}^{r}, |α|\leq N$ that satisfy identity $(\ast)$ for all possible multi-indices $α\in \mathbb{N}^{r}$, $|α|\leq N$. Our main result says that if the domain is $\mathscr{C}(Ω)$, then the mappings $T_α$ are of a rather special form. Related results in the space $\mathscr{C}^{N}(Ω)$ are also presented.

math.CA↗
arXiv2023

Delta-Sincov mappings in Banach algebras

We study solutions and approximate solutions of the multiplicative Sincov equation $$T(f, h) = T(f, g)T(g, h)$$ for mapping $T$ taking values in a commutative Banach algebra.

math.FA↗
arXiv2022

A new class of probabilities in the n-person red-and-black game

We discuss a model of a $N$-person, non-cooperative stochastic game, inspired by the discrete version of the red-and-black gambling problem introduced by Dubins and Savage in 1965. Our main theorem generalizes a result of Pontiggia from 2007 which provides conditions upon which bold strategies for all players form a Nash equilibrium. Our tool is a functional inequality introduced and discussed in the present paper. It allows us to avoid rather restrictive assumptions of super-multiplicativity and super-additivity, which appear in Pontiggia's and other authors' works. We terminate the paper with some examples which in particular show that our approach leads to a larger class of probability functions than existed in the literature so far.

math.PR↗
arXiv2020

On the $c_0$-equivalence and permutations of series

Assume that a convergent series of real numbers $\sum\limits_{n=1}^\infty a_n$ has the property that there exists a set $A\subseteq \N$ such that the series $\sum\limits_{n \in A} a_n$ is conditionally convergent. We prove that for a given arbitrary sequence $(b_n)$ of real numbers there exists a permutation $σ\colon \N \to \N$ such that $σ(n) = n$ for every $n \notin A$ and $(b_n)$ is $c_0$-equivalent to a subsequence of the sequence of partial sums of the series $\sum\limits_{n=1}^\infty a_{σ(n)}$. Moreover, we discuss a connection between our main result with the classical Riemann series theorem.

math.FA↗
arXiv2019

New inequalities for probability functions in the two-person red-and-black game

We discuss a model of a two-person, non-cooperative stochastic game, inspired by the discrete version of the red-and-black gambling problem presented by Dubins and Savage. Assume that two players hold certain amounts of money. At each stage of the game they simultaneously bid some part of their current fortune and the probability of winning or loosing depends on their bids. In many models of the red-and-black game it is assumed that the win probability is a function of the quotient of the bid of the first player and the sum of both bids. In the literature some additional properties, like concavity or super-multiplicativity are assumed in order to ensure that bold and timid strategy is the Nash equilibrium. Our aim is to provide a generalization in which the probability of winning is a two-variable function which depends on both bids. We introduce two new functional inequalities whose solutions lead to win probability functions for which a Nash equilibrium is realized by the bold-timid strategy. Since both inequalities have not easy intuitive meanings, we discuss them in a separate section of the paper and we give there some illustrating examples.

math.PR↗
arXiv2019

Convexity properties of functions defined on metric Abelian groups

The notions of quasiconvexity, Wright convexity and convexity for functions defined on a metric Abelian group are introduced. Various characterizations of such functions, the structural properties of the functions classes so obtained are established and several well-known results are extended to this new setting.

math.CA↗
arXiv2019

Convexity of sets in metric Abelian groups

In the present paper, we introduce a new concept of convexity which is generated by a family of endomorphisms of an Abelian group. In Abelian groups equipped with a translation invariant metric, we define the boundedness, the norm, the measure of injectivity and the spectral radius of endomorphisms. Beyond the investigation of their properties, our first main goal is an extension of the celebrated Rådström Cancellation Theorem. Another result generalizes the Neumann Invertibility Theorem. Next we define the convexity of sets with respect to a family of endomorphisms and we describe the set-theoretical and algebraic structure of the class of such sets. Given a subset, we also consider the family of endomorphisms that make this subset convex and we establish the basic properties of this family. Our first main result establishes conditions which imply midpoint convexity. The next main result, using our extension of the Rådström Cancellation Theorem, presents further structural properties of the family of endomorphisms that make a subset convex.

math.MG↗
arXiv2019

Sincov's inequalities on topological spaces

Assume that $X$ is a non-empty set and $T$ and $S$ are real or complex mappings defined on the product $X \times X$. Additive and multiplicative Sincov's equations are: $$T(x,z) = T(x, y ) + T(y, z)$$ and $$S(x,z) = S(x, y ) \cdot S(y, z),$$ respectively. Both equations play important roles in many areas of mathematics. In the present paper we study related inequalities. We deal with functional inequality $$ G(x,z) \leq G(x, y ) \cdot G(y, z), \quad x , y, z \in X $$ and we assume that $X$ is a topological space and $G\colon X \times X \to \mathbb{R}$ is a continuous mapping. In some our statements a considerably weaker regularity than continuity of $G$ is needed. We also study the reverse inequality: $$F(x,z) \geq F(x, y ) \cdot F(y, z), \quad x , y, z \in X $$ and the additive inequality (the triangle inequality): $$H(x,z) \leq H(x,y) + H(y,z), \quad x, y , z \in X.$$ A corollary for generalized (non-symmetric) metric is derived.

math.FA↗
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↗