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Abstract: We produce a series of results extending information-theoretical inequalities (discussed by Dembo–Cover–Thomas in 1988–1991) to a weighted version of entropy. Most of the resulting inequalities involve the Gaussian weighted entropy; they imply a number of new relations for determinants of positive-definite matrices. Unlike the Shannon entropy where the contribution of an outcome depends only upon its probability, the weighted (or context-dependent) entropy takes into account a ‘value’ of an outcome determined by a given weight function \({\varphi }\) . An example of a new result is a weighted version of the strong Hadamard inequality (SHI) between the determinants of a positive-definite \(d\times d\) matrix and its square blocks (sub-matrices) of different sizes. When \({\varphi }\equiv 1\) , the weighted inequality becomes a ‘standard’ SHI; in general, the weighted version requires some assumptions upon \({\varphi }\) . The SHI and its weighted version generalize a widely known ‘usual’ Hadamard inequality \({\mathrm{det}}\,{{\mathbf {C}}}\le \prod \nolimits _{j=1}^dC_{jj}\) . PubDate: 2022-01-22
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Abstract: We examine the family of all (at most) three-point symmetric quadratures on \([-1,1]\) which are exact on polynomials of order 3 to find all possible inequalities between them in the class of 3-convex functions. Next we optimise them by using convex combinations of the quadratures considered. We find the optimal quadrature and use it to construct the adaptive method of approximate integration. An effective method to estimate the error of this method is also given. It needs a considerably fewer number of subdivisions of the interval of integration than the classical adaptive methods as well as the method developed by the second-named author in his recent paper (Wa̧sowicz in Aequ Math 94(5):887–898, 2020). PubDate: 2022-01-17
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Abstract: A global forcing set for maximal matchings of a graph \(G=(V(G), E(G))\) is a set \(S \subseteq E(G)\) such that \(M_1\cap S \ne M_2 \cap S\) for each pair of maximal matchings \(M_1\) and \(M_2\) of G. The smallest such set is called a minimum global forcing set, its size being the global forcing number for maximal matchings \(\phi _{gm}(G)\) of G. In this paper, we establish lower and upper bounds on the forcing number for maximal matchings of the corona product of graphs. We also introduce an integer linear programming model for computing the forcing number for maximal matchings of graphs. PubDate: 2022-01-15
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Abstract: Let X be a Banach space. Fix a torsion-free commutative and cancellative semigroup S whose torsion-free rank is the same as the density of \(X^{**}\) . We then show that X is complemented in \(X^{**}\) if and only if there exists an invariant mean \(M:\ell _\infty (S,X)\rightarrow X\) . This improves upon previous results due to Bustos Domecq (J Math Anal Appl 275(2):512–520, 2002), Kania (J Math Anal Appl 445:797–802, 2017), Goucher and Kania (Studia Math 260:91–101, 2021). PubDate: 2022-01-11
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Abstract: For a finite field \({\mathbb {F}}\) of characteristic 2, an integer \(n\ge 3\) , and for a function \(f : {\mathbb {F}}\rightarrow {\mathbb {F}}\) , if there is a function \(h: {\mathbb {F}} \rightarrow {\mathbb {F}}\) such that the divided difference \(f[x_1, \dots , x_n]\) on any n distinct elements of \({\mathbb {F}}\) satisfies \(f[x_1,\ldots ,x_n] = h(x_1 +\cdots + x_n)\) , then f is a polynomial of degree at most n over \({\mathbb {F}}\) . This makes earlier work of Davies and Rousseau complete. PubDate: 2022-01-09
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Abstract: We axiomatically characterize the \(\chi ^{2}\) dissimilarity measure. To this end, we solve a new generalization of a functional equation discussed in Aczel (Lectures on functional equations and their applications. Academic Press, 1966). PubDate: 2022-01-09
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Abstract: The section method provides a technique for solving certain functional equations with the help of their characteristic equations. We prove here that a large class of such equations are functionally stable in the hypothesis of the stability of the corresponding characteristic equations. We use these results to solve two equations that have composite functions as solutions and to give classes of control functions that ensure their generalized stability. PubDate: 2022-01-04
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Abstract: Let S be a semigroup. We describe the solutions \(f,g:S \rightarrow \mathbb {C}\) of the functional equation $$\begin{aligned} f(xy) = f(x)g(y) + g(x)f(y) - g(x)g(y), \ x,y \in S, \end{aligned}$$ in terms of multiplicative functions on S and solutions of the particular case $$\begin{aligned} \varphi (xy) = \varphi (x)\chi (y) + \varphi (y)\chi (x), \ x,y \in S, \end{aligned}$$ of the sine addition law in which \(\chi :S \rightarrow \mathbb {C}\) is a non-zero multiplicative function and \(\varphi :S \rightarrow \mathbb {C}\) . PubDate: 2022-01-04
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Abstract: Generalized Berwald manifolds are Finsler manifolds admitting linear connections such that the parallel transports preserve the Finslerian length of tangent vectors (compatibility condition). It is known (Vincze in J AMAPN 21:199–204, 2005) that such a linear connection must be metrical with respect to the averaged Riemannian metric given by integration of the Riemann-Finsler metric on the indicatrix hypersurfaces. Therefore the linear connection (preserving the Finslerian length of tangent vectors) is uniquely determined by its torsion. If the torsion is zero then we have a classical Berwald manifold. Otherwise, the torsion is some strange data we need to express in terms of the intrinsic quantities of the Finsler manifold. The paper presents the idea of the extremal compatible linear connection of a generalized Berwald manifold by minimizing the pointwise length of its torsion tensor. It is uniquely determined because the number of the Lagrange multipliers is equal to the number of the equations for the compatibility of the linear connection with the Finslerian metric. Using the reference element method, the extremal compatible linear connection can be expressed in terms of the canonical data as well. It is an intrinsic algorithm to check the existence of compatible linear connections on a Finsler manifold because it is equivalent to the existence of the extremal compatible linear connection. PubDate: 2021-12-29
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Abstract: Despite the fact that Eri Jabotinsky (1910–1969) published only few (i.e. fourteen) mathematical papers, some of them had a remarkable influence in iteration theory. But also his life was remakable. Eri was the son of the famous Zionist Revisionist leader Vladimir Ze’ev Jabotinsky. Eri Jabotinsky was active in the Zionist movement and later as parlamentarian in the Knesset. Here we give an outline of his live and a complete list of his publications. PubDate: 2021-12-01 DOI: 10.1007/s00010-021-00779-w
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Abstract: We study k-valued solutions of Wilson’s \(\mu \) -functional equation on semigroups and monoids where k is an algebraically closed field of characteristic \(\ne 2\) . As applications we solve the functional equation on some finite groups and find the continuous, complex valued solutions on compact groups. PubDate: 2021-12-01 DOI: 10.1007/s00010-021-00784-z
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Abstract: The purpose of this paper is to prove that if on a commutative hypergroup an exponential monomial has the property that the linear subspace of all sine functions in its variety is one dimensional, then this exponential monomial is a linear combination of generalized moment functions. PubDate: 2021-12-01 DOI: 10.1007/s00010-021-00827-5
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Abstract: Based on iteration of random-valued functions we study the problem of solvability in the class of continuous and Hölder continuous functions \(\varphi \) of the equations $$\begin{aligned} \varphi (x)=F(x)-\int _{\Omega }\varphi \big (f(x,\omega )\big )P(d\omega ),\\ \varphi (x)=F(x)+\int _{\Omega }\varphi \big (f(x,\omega )\big )P(d\omega ), \end{aligned}$$ where P is a probability measure on a \(\sigma \) -algebra of subsets of \(\Omega \) . PubDate: 2021-12-01 DOI: 10.1007/s00010-021-00794-x
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Abstract: On the occasion of his 80th birthday, Ludwig Reich, editor-in-chief of Aequationes Mathematicae from 1996 to 2008, gave an interview conducted via Zoom. This paper presents his answers to several questions regarding his education and academic life. PubDate: 2021-12-01 DOI: 10.1007/s00010-021-00853-3
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Abstract: Let G be an Abelian group, and let \({{\mathbb {C}}}^G\) denote the set of complex valued functions defined on G. A map \(D: {{\mathbb {C}}}^G \rightarrow {{\mathbb {C}}}^G\) is a difference operator, if there are complex numbers \(a_i\) and elements \(b_i \in G\) \((i=1,\ldots , n)\) such that \((Df)(x)=\sum _{i=1}^n a_i f(x+b_i)\) for every \(f\in {{\mathbb {C}}}^G \) and \(x\in G\) . By a system of difference equations we mean a set of equations \(\{ D_i f=g_i : i\in I\}\) , where I is an arbitrary set of indices, \(D_i\) is a difference operator and \(g_i \in {{\mathbb {C}}}^G\) is a given function for every \(i\in I\) , and f is the unknown function. The solvability cardinal \(\mathrm{sc} \,({{\mathcal {F}}})\) of a class of functions \({{\mathcal {F}}} \subset {{\mathbb {C}}}^G\) is the smallest cardinal number \(\kappa \) with the following property: whenever S is a system of difference equations on G such that each subsystem of S of cardinality \(<\kappa \) has a solution in \({{\mathcal {F}}}\) , then S itself has a solution in \({{\mathcal {F}}}\) . The behaviour of \(\mathrm{sc} \,({{\mathcal {F}}})\) is rather erratic, even for classes of functions defined on \({{\mathbb {R}}}\) . For example, \(\mathrm{sc} \,({{\mathbb {C}}}[x])=3\) , but \(\mathrm{sc} \,({\mathcal {TP}}) =\omega _1\) , where \({\mathcal {TP}}\) is the set of trigonometric polynomials; \(\mathrm{sc} \,({{\mathbb {C}}}^{{\mathbb {R}}})=\omega \) , but \(\mathrm{sc} \,({\mathcal {DF}}) =(2^\omega )^+\) , where \({\mathcal {DF}}\) is the set of functions having the Darboux property. Our aim is to determine or to estimate the solvability cardinal of the class of polynomials defined on \({{{\mathbb {R}}}}^n\) , on normed linear spaces and, in general, on topological Abelian groups. Let PubDate: 2021-12-01 DOI: 10.1007/s00010-021-00818-6
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Abstract: Let \(I\subset {\mathbb {R}}\) be an interval that is closed under addition, and \( k\in {\mathbb {N}}\) , \(k\ge 2\,\) . For a function \(f:I\rightarrow \left( 0,\infty \right) \) such that \(F\left( x\right) :=\frac{f\left( kx\right) }{ kf\left( x\right) }\) is invertible in I, the k-variable function \( M_{f}:I^{k}\rightarrow I,\) $$\begin{aligned} M_{f}\left( x_{1},\ldots ,x_{k}\right) :=F^{-1}\left( \frac{f\left( x_{1}+\cdots +x_{k}\right) }{f\left( x_{1}\right) +\cdots +f\left( x_{k}\right) } \right) , \end{aligned}$$ is a premean in I, and it is referred to as a quasi Cauchy quotient of the additive type of generator f. Three classes of means of this type generated by the exponential, logarithmic, and power functions, are examined. The suitable quasi Cauchy quotients of the exponential types (for continuous additive, logarithmic, and power functions) are considered. When I is closed under multiplication, the quasi Cauchy quotient means of logarithmic and multiplicative type are studied. The equalities of premeans within each of these classes are discussed and some open problems are proposed. PubDate: 2021-12-01 DOI: 10.1007/s00010-021-00856-0
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Abstract: The d-dimensional analogue \({\mathcal J}^{(d)}(\mathbf{k})\) of the Jennings group ( \(d=1\) ) of substitutions of mappings of formal power series with coefficients in an arbitrary field \(\mathbf{k}\) is defined and studied. More precisely, we find the commutants in the group \({\mathcal J}^{(d)}(\mathbf{k})\) for \(d\ge 2\) . The case \(d=1\) was done before. PubDate: 2021-12-01 DOI: 10.1007/s00010-021-00783-0
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Abstract: Assume that \(\Omega \subset \mathbb {R}^k\) is an open set, V is a real separable Banach space and \(f_1,\ldots ,f_N :\Omega \rightarrow \Omega \) , \(g_1,\ldots , g_N:\Omega \rightarrow \mathbb {R}\) , \(h_0:\Omega \rightarrow V\) are given functions. We are interested in the existence and uniqueness of solutions \(\varphi :\Omega \rightarrow V\) of the linear equation \(\varphi =\sum _{k=1}^{N}g_k\cdot (\varphi \circ f_k)+h_0\) in generalized Orlicz spaces. PubDate: 2021-12-01 DOI: 10.1007/s00010-021-00851-5
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Abstract: It was proved in Forti and Schwaiger (C R Math Acad Sci Soc R Can 11(6):215–220, 1989), Schwaiger (Aequ Math 35:120–121, 1988) and with different methods in Schwaiger (Developments in functional equations and related topics. Selected papers based on the presentations at the 16th international conference on functional equations and inequalities, ICFEI, Bȩdlewo, Poland, May 17–23, 2015, Springer, Cham, pp 275–295, 2017) that under the assumption that every function defined on suitable abelian semigroups with values in a normed space such that the norm of its Cauchy difference is bounded by a constant (function) is close to some additive function, i.e., the norm of the difference between the given function and that additive function is also bounded by a constant, the normed space must necessarily be complete. By Schwaiger (Ann Math Sil 34:151–163, 2020) this is also true in the non-archimedean case. Here we discuss the situation when the bound is a suitable non-constant function. PubDate: 2021-12-01 DOI: 10.1007/s00010-021-00804-y
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Abstract: We prove that every K–subadditive set–valued map weakly K–upper bounded on a “large” set (e.g. not null–finite, not Haar–null or not a Haar–meager set), as well as any K–superadditive set–valued map K–lower bounded on a “large” set, is locally K–lower bounded and locally weakly K–upper bounded at every point of its domain. PubDate: 2021-10-07 DOI: 10.1007/s00010-021-00850-6
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