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Abstract: Abstract In this article we study the difference between orthogonality induced by norm derivatives (known as \(\rho \) -orthogonality) and Birkhoff-James orthogonality in a normed linear space \(\mathbb {X}\) by introducing a new geometric constant, denoted by \(\Gamma (\mathbb {X}).\) We explore the relation between various geometric properties of the space and the constant \(\Gamma (\mathbb {X}).\) We also investigate the left symmetric and right symmetric elements of a normed linear space with respect to \(\rho \) -orthogonality and obtain a characterization of the same. We characterize inner product spaces among normed linear spaces using the symmetricity of \(\rho \) -orthogonality. Finally, we provide a complete description of both left symmetric and right symmetric elements with respect to \(\rho \) -orthogonality for some particular Banach spaces. PubDate: 2025-01-28 DOI: 10.1007/s00010-025-01154-9
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Abstract: Abstract In the first part of the paper we prove a necessary and sufficient condition for the existence of the composition of formal power series in the case when the outer series is a series of one variable while the inner one is a series of multiple variables. The aim of the second part is to remove ambiguities connected with the Right Distributive Law for formal power series of one variable as well as to provide analogues of that law in the multivariable case. PubDate: 2025-01-23 DOI: 10.1007/s00010-024-01152-3
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Abstract: Abstract In this article, we introduce several new moduli of convexity connected with \(\rho _{\pm }\) -orthogonalities and semi-orthogonality, which are closely related to the modulus of convexity \(\delta _{X}(\varepsilon )\) . In particular, these new parameters are computed for X being some specific spaces. Moreover, we give some applications of these two coefficients \(\delta _{\perp }(\rho _{\pm }, X)\) and investigate the relation between these two parameters and the approximate symmetry of \(\rho _{-}\) -orthogonality. In the meantime, we give characterization of the Radon plane with an affine-hexagonal unit sphere in terms of these new moduli of convexity. Moreover, we also consider the moduli of smoothness related to \(\rho _{\pm }\) -orthgonalities and semi-orthogonality. In the end, we discuss some applications of these new moduli of smoothness and study some relationships between these new parameters and uniform non-squareness, uniform convexity. PubDate: 2025-01-20 DOI: 10.1007/s00010-025-01153-w
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Abstract: Abstract The main objective of this study is to introduce unique representations and characterizations for the several classes of weighted generalized inverses of matrices. Proposed representations of the matrix-weighted core inverse will help us to discuss some results associated with the reverse order law for these inverses. Furthermore, this paper introduces an extension of the concepts of generalized bilateral inverse and their respective dual for complex rectangular matrices. Characteristics that lead to self-duality in weighted bilateral inverses are also examined. In addition, a W-weighted index-MP, W-weighted MP-index, and W-weighted MP-index-MP matrices for rectangular complex matrices are introduced. PubDate: 2025-01-20 DOI: 10.1007/s00010-024-01151-4
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Abstract: Abstract This paper explores the connections between vector-valued Banach limits and weak compactness in Banach spaces. We show that a Banach space, \({X}\) , is reflexive if it admits a Banach limit on bounded \({X}\) -valued sequences such that, for any input sequence, the corresponding limit vector lies in the closed linear span of that sequence. This conclusion is based on proving that the existence of a vector-valued Banach limit with the aforementioned linear span property implies the weak compactness of the closed unit ball of the underlying Banach space. Furthermore, we extend the above result by establishing a characterisation of the relative weak compactness of bounded sets in Banach spaces. The characterisation states that a bounded set is relatively weakly compact if, for every sequence in the set, there exists a vector-valued Banach limit on the smallest shift-invariant linear space containing the sequence and all vector-valued constant sequences, such that, for any input sequence, the corresponding limit vector lies in the closed linear span of that sequence. PubDate: 2025-01-19 DOI: 10.1007/s00010-024-01140-7
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Abstract: Abstract We discuss some combinatorics associated with 1-away permutations, where an element can be displaced from its correct position by at most one location. Specifically, we look at a sorting algorithm for such permutations and analyze its number of comparisons, \(C_n\) . We find that the mean is a certain combination of two-fold convolutions of Fibonacci numbers and the variance is a certain combination of three-fold convolutions of Fibonacci numbers, with corresponding asymptotics (as \(n\rightarrow \infty \) ): $${\mathbb {E}}[C_n] \sim \frac{5 + \sqrt{5}}{10}\, n, \qquad {\mathbb {V}\textrm{ar}}[C_n]\sim \frac{\sqrt{5}}{25} \, n.$$ The proofs contain finer asymptotics down to exponentially small error terms. The relatively small variance admits a weak law and a central limit theorem via a super moment generating function. In view of the special nature of the data, such a specialized algorithm outperforms general comparison-based sorting algorithms. PubDate: 2025-01-19 DOI: 10.1007/s00010-024-01146-1
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Abstract: Abstract Probabilistic divergence measures the statistical distance between two probability distributions. Traditionally, they are used in probability theory and information theory. Nowadays, many machine learning algorithms rely on such divergences to learn models and distributions of parameters, enabling them to perform a wide range of automated tasks. This small article proposes a new family of symmetric probabilistic divergences generated using a novel functional generator. The generator uses monotonically increasing and decreasing functions to create a variety of probabilistic divergences. While it is possible to generate a variety of probabilistic divergences based on the suitable choices of functions, here the focus on six new probabilistic divergences. PubDate: 2025-01-15 DOI: 10.1007/s00010-024-01148-z
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Abstract: Abstract In this paper, we apply a direct method instead of a limit approach, for proving the Levin–Cochran–Lee inequalities. First, we state and prove Levin–Cochran–Lee type inequalities on a homogeneous group \(\mathbb {G}\) with parameters \(0<p\le q<\infty \) . Furthermore, for the case \(p=q\) , we prove the sharp inequalities with power weights and derive some other new inequalities. PubDate: 2025-01-13 DOI: 10.1007/s00010-024-01143-4
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Abstract: Abstract We introduce the concept of d-Young distance function with respect to the 5-uplet \((p,q,\tau ,\kappa ,\xi )\) , where d is a metric on a certain set \(\Lambda \) , \(1<p,q<\infty \) with \(\frac{1}{p}+\frac{1}{q}=1\) , \(\tau : \Lambda \times \Lambda \rightarrow [0,\infty )\) , \(\kappa >0\) , and \(\xi : [0,\infty )\rightarrow [0,\infty )\) satisfies the condition \(\inf _{t>0} \frac{\xi (t)}{t^\kappa }>0\) . We establish some properties of the introduced distance function. Next, we study the existence and uniqueness of fixed points for some classes of mappings \(F: \Lambda \rightarrow \Lambda \) satisfying contractions involving the d-Young distance function. In particular, for a special choice of the 5-uplet \((p,q,\tau ,\kappa ,\xi )\) , we recover the Banach fixed point theorem. We also provide an example, where our approach can be used, but the Banach fixed point theorem is inapplicable. PubDate: 2025-01-13 DOI: 10.1007/s00010-024-01144-3
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Abstract: Abstract After reviewing various notions of symmetry in graph theory, which are typically defined by the connections between vertices, we demonstrate that traditional concepts of symmetry, such as vertex transitivity, can be too restrictive for certain applications. For instance, in some areas of graph analysis, symmetry based on metric properties (such as average distances between vertices) may be more appropriate, particularly in social network analysis or economic fraud detection. This paper focuses on developing metric-based symmetry concepts by introducing mathematical analysis tools, all related to the central idea of the distance distribution function, to group vertices according to their distance-related properties within the graph. In particular, we prove several results that show, under certain compactness properties for the set of distribution functions of all the vertices in an infinite graph, that it is always possible to group these vertices into a finite number of classes with the desired accuracy based on distances. PubDate: 2025-01-10 DOI: 10.1007/s00010-024-01145-2
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Abstract: Abstract Certain generalization of Euler numbers was defined in 1935 by Lehmer using cubic roots of unity, as a natural generalization of Bernoulli and Euler numbers. In this paper, Lehmer’s generalized Euler numbers are studied to give certain congruence properties together with recurrence and explicit formulas of the numbers. We also show a new polynomial sequence and its properties. Some identities including Euler and central factorial numbers are obtained. PubDate: 2025-01-10 DOI: 10.1007/s00010-024-01150-5
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Abstract: Abstract In this paper we present some coincidence results between lower semicontinuous type maps, one of which has decomposable values. Our argument relies on fixed point theory combined with continuous (or upper semicontinuous) selection theory. PubDate: 2025-01-10 DOI: 10.1007/s00010-024-01149-y
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Abstract: Abstract We consider a probability model in which the hull of a sample of i.i.d. uniform random points from a convex disc K is formed by the intersection of all translates of another suitable fixed convex disc L that contain the sample. Such an object is called a random L-polygon in K. We assume that both K and L have \(C^2_+\) smooth boundaries, and we prove upper bounds on the variance of the number of vertices and missed area of random L-polygons assuming different curvature conditions. We also transfer some of our result to a circumscribed variant of this model. PubDate: 2025-01-10 DOI: 10.1007/s00010-024-01147-0
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Abstract: Abstract In 1953 von Neumann proved that every \(n\times n\) doubly substochastic matrix A can be increased to a doubly stochastic matrix, i.e., there is an \(n\times n\) doubly stochastic matrix D for which \(A\le D.\) In this paper, we will discuss this result for a class of \(I\times I\) doubly substochastic matrices. In fact, by a constructive method, we find an equivalent condition for the existence of a doubly stochastic matrix D which satisfies \(A\le D,\) for all \(A\in {\mathcal {A}},\) where \({\mathcal { A}}\) is assumed to be a class of (finite or infinite) doubly substochastic matrices. Such a matrix D is called a cover of \(\mathcal {A}.\) The uniqueness of the cover will also be discussed. Then we obtain an application of this concept to a system of (infinite) linear equations and inequalities. PubDate: 2025-01-06 DOI: 10.1007/s00010-024-01141-6
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Abstract: Abstract We prove that whenever \(M_1,\dots ,M_n:I^k \rightarrow I\) , ( \(n,k \in \mathbb {N}\) ) are symmetric, continuous means on the interval I and \(S_1,\dots ,S_m:I^k \rightarrow I\) ( \(m < n\) ) satisfy a sort of embeddability assumptions then for every continuous function \(\mu :I^n \rightarrow \mathbb {R}\) which is strictly monotone in each coordinate, the functional equation $$ \mu (S_1(v),\dots ,S_m(v),\underbrace{F(v),\dots ,F(v)}_{(n-m)\text { times}})=\mu (M_1(v),\dots ,M_n(v)) $$ has the unique solution \(F=F_\mu :I^k \rightarrow I\) which is a mean. We deliver some sufficient conditions so that \(F_\mu \) is well-defined (in particular uniquely determined) and study its properties. The aim of this research is to provide a broad overview of the family of Beta-type means introduced in (Himmel and Matkowski, 2018). PubDate: 2025-01-03 DOI: 10.1007/s00010-024-01139-0
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Abstract: Abstract Given a semigroup S equipped with an involutive automorphic \(\sigma :S \rightarrow S\) , we determine the complex-valued solutions of the following generalization of the Kannappan-sine addition law $$f(x\sigma (y)z_0)=f(x)g(y)+f(y)g(x),\; x,y \in S. $$ As an application we obtain the solutions of the following functional equation $$f(x\sigma (y)z_0)=f(x)f(z_1y)+f(z_1x)f(y),\; x,y \in S, $$ where \(z_0, z_1\) are two fixed elements in S such that \(z_0\ne z_1\) . The continuous solutions on topological semigroups are given. We illustrate the main result with two examples. PubDate: 2024-12-26 DOI: 10.1007/s00010-024-01138-1
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Abstract: Abstract This is the second of three sequels to (Ostaszewski in Aequat Math 90:427–448, 2016)—the third of the resulting quartet—concerning the real-valued continuous solutions of the multivariate Goldie functional equation (GFE) below of Levi–Civita type. Following on from the preceding paper (Bingham and Ostaszewski in Homomorphisms from Functional Equations: II. The Goldie Equation, arXiv:1910.05816), in which these solutions are described explicitly, here we characterize (GFE) as expressing homomorphy (in all but some exceptional “improper” cases) between multivariate Popa groups, defined and characterized earlier in the sequence. The group operation involves a form of affine addition (with local scalar acceleration) similar to the circle operation of ring theory. We show the affine action in \((GFE)\ \) may be replaced by a general continuous acceleration yielding a functional equation (GGE) which it emerges has the same solution structure as (GFE). The final member of the sequence (Bingham and Ostaszewski, The Gołąb–Schinzel and Goldie functional equations in Banach algebras, arXiv:2105.07794) considers the richer framework of a Banach algebra which allows vectorial acceleration, giving the closest possible similarity to the circle operation. PubDate: 2024-12-09 DOI: 10.1007/s00010-024-01133-6
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Abstract: Abstract In the paper the notion of weakly K-subadditive set-valued maps is introduced in such a way that F is weakly K-superadditive if and only if \(-F\) is weakly K-subadditive. This new definition is a natural generalization of K-subadditive set-valued maps from Jabłońska and Nikodem (Aequ Math 95:1221–1231, 2021), for which opposite set-valued maps need not be K-subadditive. Among others, we prove that every weakly K-subadditive set-valued map which is K–upper bounded on a “large” set has to be locally weakly K-upper bounded and weakly K-lower bounded at every point of the domain. This theorem completes an analogous result for K-subadditive set-valued maps which are weakly K-upper bounded on “large” sets from Jabłońska and Nikodem (Aequ Math 95:1221–1231, 2021). PubDate: 2024-12-01 DOI: 10.1007/s00010-024-01083-z
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Abstract: Abstract We show that every surjective mapping f between the unit spheres of two real \(\mathcal {L}^\infty (\Gamma )\) -type spaces satisfies $$\begin{aligned} \min \{\Vert f(x)+f(y)\Vert ,\Vert f(x)-f(y)\Vert \}=\min \{\Vert x+y\Vert ,\Vert x-y\Vert \}\quad (x,y\in S_X) \end{aligned}$$ if and only if f is phase-equivalent to an isometry, i.e., there is a phase-function \(\varepsilon \) from the unit sphere of the \(\mathcal {L}^\infty (\Gamma )\) -type space onto \(\{-1,1\}\) such that \(\varepsilon \cdot f\) is a surjective isometry between the unit spheres of two real \(\mathcal {L}^\infty (\Gamma )\) -type spaces, and furthermore, this isometry can be extended to a linear isometry on the whole space \(\mathcal {L}^\infty (\Gamma )\) . We also give an example to show that these are not true if “min” is replaced by “max”. PubDate: 2024-12-01 DOI: 10.1007/s00010-024-01119-4
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