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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
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Abstract: This article is devoted to the study of certain nonautonomous and nonlinear difference equations of higher order. Our main objective is to formulate sufficient conditions under which the class of difference equations we consider exhibits Hyers–Ulam stability. Our methods rely on the relationship between Hyers–Ulam stability and hyperbolicity for nonautonomous systems. PubDate: 2021-10-01
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Abstract: Let X be a normed linear space. We prove that if the norm on X is almost transitive and if there exists a unit vector u satisfying that, for each point y in the unit ball of X that is isosceles orthogonal to u, there always exists \(\alpha \in (0,1)\) so that u is isosceles orthogonal to \(\alpha y\) , then X is an inner product space. PubDate: 2021-10-01
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Abstract: Kleinewillinghöfer classified Laguerre planes with respect to central automorphisms. Polster and Steinke investigated 2-dimensional Laguerre planes and their so-called Kleinewillinghöfer types, that is, the Kleinewillinghöfer types with respect to the full automorphism group. For one of these types the existence question remained open. We provide examples of such planes of type I.A.2, which are obtained by interchanging some circles between certain 2-dimensional Laguerre planes of type IV.A.1. This completes the classification of Kleinewillinghöfer types of 2-dimensional Laguerre planes. PubDate: 2021-10-01
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Abstract: Let \(\mathbb {N}\) be the set of nonnegative integer numbers. A plane monoid is a submonoid of \((\mathbb {N}^2,+)\) . Let M be a plane monoid and \(p,q\in \mathbb {N}\) . We will say that an integer number n is M(p, q)-bounded if there is \((a,b)\in M\) such that \(a+p\le n \le b-q\) . We will denote by \({\mathrm A}(M(p,q))=\{n\in \mathbb {N}\mid n \text { is } M(p,q)\text {-bounded}\}.\) An \(\mathcal {A}(p,q)\) -semigroup is a numerical semigroup S such that \(S= {\mathrm A}(M(p,q))\cup \{0\}\) for some plane monoid M. In this work we will study these kinds of numerical semigroups. PubDate: 2021-10-01
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Abstract: We take a categorical approach to describe ternary derivations and ternary automorphisms of triangular algebras. New classes of automorphisms and derivations of triangular algebras are also introduced and studied. PubDate: 2021-10-01
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Abstract: In this paper the concept of symmetrized convex stochastic processes is introduced. Some characterizations involving Hermite–Hadamard type inequalities and a stability result for symmetrized convex stochastic processes are proved. PubDate: 2021-10-01
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Abstract: We investigate the orthogonality preserving property for pairs of operators on inner product \(C^*\) -modules. Employing the fact that the \(C^*\) -valued inner product structure of a Hilbert \(C^*\) -module is determined essentially by the module structure and by the orthogonality structure, pairs of linear and local orthogonality-preserving operators are investigated, not a priori bounded. We obtain that if \({\mathscr {A}}\) is a \(C^{*}\) -algebra and \(T, S:{\mathscr {E}}\rightarrow {\mathscr {F}}\) are two bounded \({{\mathscr {A}}}\) -linear operators between full Hilbert \({\mathscr {A}}\) -modules, then \(\langle x, y\rangle = 0\) implies \(\langle T(x), S(y)\rangle = 0\) for all \(x, y\in {\mathscr {E}}\) if and only if there exists an element \(\gamma \) of the center \(Z(M({{\mathscr {A}}}))\) of the multiplier algebra \(M({{\mathscr {A}}})\) of \({{\mathscr {A}}}\) such that \(\langle T(x), S(y)\rangle = \gamma \langle x, y\rangle \) for all \(x, y\in {\mathscr {E}}\) . Varying the conditions on the operators T and S we obtain further affirmative results for local operators and for pairs of a bounded and an unbounded \({{\mathscr {A}}}\) -linear operator with bounded inverse. PubDate: 2021-10-01
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Abstract: This paper studies the conditional distributivity for T-uninorms over uninorms in the most general setting, transforming it into the (conditional) distributivity equation involving two uninorms. PubDate: 2021-10-01
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Abstract: It is well known that Heron’s equality provides an explicit formula for the area of a triangle, as a symmetric function of the lengths of its edges. It has been extended by Brahmagupta to quadrilaterals inscribed in a circle (cyclic quadrilaterals). A natural problem is trying to further generalize the result to cyclic polygons with a larger number of edges. Surprisingly, this has proved to be far from simple, and no explicit solutions exist for cyclic polygons having \(n>4\) edges. In this paper we investigate such a problem by following a new and elementary approach, based on the idea that the simple geometry underlying Heron’s and Brahmagupta’s equalities hides the real players of the game. In details, we propose to focus on the dissection of the edges determined by the incircles of a suitable triangulation of the cyclic polygon, showing that this approach leads to an explicit formula for the area as a symmetric function of the lengths of these segments. We also show that such a symmetry can be rediscovered in Heron’s and Brahmagupta’s results, which consequently represent special cases of the provided general equality. PubDate: 2021-10-01
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Abstract: In the present paper we are concerned with the problem of characterization of maps which can be expressed as an affine difference i.e. a map of the form $$\begin{aligned} tf(x)+(1-t)f(y)-f(tx+(1-t)y), \end{aligned}$$ where \(t\in (0,1)\) is a given number. We give a general solution of the functional equation associated with this problem. PubDate: 2021-10-01
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Abstract: The cosine function is a classical tool for measuring angles in inner product spaces, and it has various extensions to normed linear spaces. In this paper, we investigate a cosine function for the convex angle formed by two nonzero elements of a complex normed linear space, in connection with recent results on the Birkhoff-James approximate orthogonality sets. PubDate: 2021-10-01
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Abstract: The present article is devoted to generalized Salem functions, the generalized shift operator, and certain related problems. A description of further investigations of the author of this article is given. These investigations (in terms of various representations of real numbers) include generalized Salem functions and generalizations of the Gauss–Kuzmin problem. PubDate: 2021-10-01
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Abstract: Let \(k\ge 2\) be an integer and \(S=\{1\}\cup S'\) be a Goldbach-type set such that \(S'\subseteq 2{\mathbb {N}}\) and \(\{2,4,6,8,12,18\}\in S'\) . If a multiplicative function f satisfies $$\begin{aligned} f(x_1+x_2+\cdots +x_k)=f(x_1)+f(x_2)+\cdots +f(x_k) \end{aligned}$$ for arbitrary \(x_1,\ldots ,x_k\in S\) , then f is the identity function \(f(n)=n\) for all \(n\in {\mathbb {N}}\) . In particular, the set of practical numbers is a k-additive uniqueness set. PubDate: 2021-09-27
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Abstract: In this chapter we present a brief survey of theory of measures of noncompactness and discuss some fixed point theorems of Darbo’s type. We describe some applications in the solvability of infinite systems of differential equations in classical sequence spaces. PubDate: 2021-09-27
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Abstract: Steinitz’s theorem states that a graph G is the edge-graph of a 3-dimensional convex polyhedron if and only if, G is simple, plane and 3-connected. We prove an analogue of this theorem for ball polyhedra, that is, for intersections of finitely many unit balls in \(\mathbb {R}^3\) . PubDate: 2021-09-24
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Abstract: In this work, we give some characterizations or representations of set-valued solutions defined on a commutative monoid \((M,+)\) with values in a Hausdorff topological vector space of the following two-variable functional equation with involutions: $$\begin{aligned} F(x+y,z+w)+F(x+\sigma (y),z+\tau (w)) =\alpha F(x,z)+\beta F(y,w), \end{aligned}$$ where \(\alpha ,\;\beta \) are fixed nonnegative real numbers and \(\sigma ,\tau : M\rightarrow M\) are involutions (i.e., \(\sigma (x+y)=\sigma (x)+\sigma (y)\) and \(\sigma \circ \sigma (x)=x\) for all \(x,y\in M\) ). PubDate: 2021-09-13
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Abstract: It is known that if a bivariate mean K is symmetric, continuous and strictly increasing in each variable, then for every mean M there is a unique mean \(N\,\) such that K is invariant with respect to the mean-type mapping \(\left( M,N\right) ,\) which means that \(K\circ \left( M,N\right) =K\) and N is called a K-complementary mean for M (Matkowski in Aequ Math 57(1):87–107, 1999). This paper extends this result for a large class of nonsymmetric means. As an application, the limits of the sequences of iterates of the related mean-type mappings are determined, which allows us to find all continuous solutions of some functional equations. PubDate: 2021-09-09
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Abstract: In this work, Levinson type inequalities involving two types of data points are proved using Green functions and the Hermite interpolating polynomial for the class of n-convex functions. In seek of applications to information theory some estimates for new functionals are obtained, based on \(\mathbf {f}\) -divergence. Moreover, some inequalities involving Shannon entropies are deduced as well. PubDate: 2021-08-25
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Abstract: The issue of distributivity of aggregation operations is crucial for many different areas such as utility theory and integration theory. Of special interest are aggregation operations with annihilator. This paper is focused on the problem of distributivity for some so called associative, commutative aggregation operations with annihilator a, known as associative a-CAOA, and uninorms. The full characterization of distributive pairs for T-uninorms, S-uninorms and bi-uninorms is given. PubDate: 2021-08-10
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