Abstract: A formally correct expression at the end of Section 2 is, however, of awkward form, being not what was intended (due to my inattentiveness) and it may confuse a careful reader. PubDate: 2019-04-01 DOI: 10.1007/s00010-018-0625-9
Abstract: Abstract In this short note we show that a weak version of Bernstein’s characterization of the normal distribution implies the local integrability of a measurable solution of the Cauchy functional equation; the linearity of a solution of the Cauchy functional equation is an easy consequence of its local integrability. In its turn, this weak version of Bernstein’s theorem can be derived from Cauchy’s theorem. PubDate: 2019-04-01 DOI: 10.1007/s00010-018-0575-2
Abstract: Abstract As a generalization of a result proved independently by Kurepa and Jurkat on additive functions on \({\mathbb R}\) satisfying certain functional equations, we determine multiadditive functions on \({\mathbb R}^n\) with a positive integer n greater than 1 satisfying certain functional equations similar to those considered in the one variable case. PubDate: 2019-04-01 DOI: 10.1007/s00010-018-0627-7
Abstract: Abstract Given a probability space \( (\Omega , {\mathcal {A}}, P) \) , a complete and separable metric space X with the \( \sigma \) -algebra \( {\mathcal {B}} \) of all its Borel subsets and a \( {\mathcal {B}} \otimes {\mathcal {A}} \) -measurable \( f: X \times \Omega \rightarrow X \) we consider its iterates \( f^n\) defined on \( X \times \Omega ^{{\mathbb {N}}}\) by \(f^0(x, \omega ) = x\) and \( f^n(x, \omega ) = f\big (f^{n-1}(x, \omega ), \omega _n\big )\) for \(n \in {\mathbb {N}}\) and provide a simple criterion for the existence of a probability Borel measure \(\pi \) on X such that for every \( x \in X \) and for every Lipschitz and bounded \(\psi :X \rightarrow {\mathbb {R}}\) the sequence \(\left( \frac{1}{n}\sum _{k=0}^{n-1} \psi \big (f^k(x,\cdot )\big )\right) _{n \in {\mathbb {N}}}\) converges in probability to \(\int _X\psi (y)\pi (dy)\) . PubDate: 2019-04-01 DOI: 10.1007/s00010-018-0585-0
Abstract: Abstract In the first part of the paper we give a new integral representation for the principal branch of the Lambert W function. Then we deduce two continued fraction expansions for this branch. At the end of the paper we study the numerical behavior of the approximants of these expansions. PubDate: 2019-04-01 DOI: 10.1007/s00010-018-0559-2
Abstract: Abstract We discuss equivalence conditions for the non-existence of non-trivial meromorphic solutions to the Fermat Diophantine equation \(f^m(z)+g^n(z)=1\) with integers \(m,n\ge 2\) , from which other approaches to proving the little Picard theorem are discussed. PubDate: 2019-04-01 DOI: 10.1007/s00010-018-0614-z
Abstract: Abstract We consider variants on the classical Berz sublinearity theorem, using only DC, the Axiom of Dependent Choices, rather than AC, the Axiom of Choice, which Berz used. We consider thinned versions, in which conditions are imposed on only part of the domain of the function—results of quantifier-weakening type. There are connections with classical results on subadditivity. We close with a discussion of the extensive related literature. PubDate: 2019-04-01 DOI: 10.1007/s00010-018-0618-8
Abstract: Abstract In this paper, we prove a fixed point theorem for a system of maps on the finite product of metric spaces. Our result generalizes the result of Matkowski (Bull Acad Pol Sci Sér Sci Math Astron Phys 21:323–324, 1973), Cosentino and Vetro (Filomat 28(4):715–722, 2014) and Hardy and Rogers (Can Math Bull 16(2):201–206, 1973) and other results in the literature. Moreover, we have an application for a system of functional equations and an example to illustrate our result. PubDate: 2019-04-01 DOI: 10.1007/s00010-018-0562-7
Abstract: Abstract This is a survey on various characteristic properties of ellipsoids and convex quadrics in the family of convex hypersurfaces in \({\mathbb {R}}^n\) . The topics under consideration include planar sections and projections, planarity conditions on midsurfaces and shadow-boundaries, intersections of homothetic copies, projective centers, and invariant mappings. PubDate: 2019-04-01 DOI: 10.1007/s00010-018-0620-1
Abstract: Abstract Let G be a group, and let \(\chi \) and \(\mu \) be characters of G. We find the solutions of the functional equation \(f(xy) = f(x)\chi (y) + \mu (x)f(y)\) , \(x,y \in G\) , where \(f:G \rightarrow \mathbb {C}\) is the unknown function. This enables us to solve its Pexiderized version \(f(xy) = g(x)h_1(y) + \mu (x)h_2(y)\) , \(x,y \in G\) , in which \(f,g, h_1, h_2 :G \rightarrow \mathbb {C}\) are the unknown functions. PubDate: 2019-04-01 DOI: 10.1007/s00010-018-0584-1
Abstract: Abstract Let \( S_1 \) denote the set of all pairs (x, y) of real numbers that fulfill the condition \( x^2 - y^2 = 1 \) , and \( S_2 \) denote the set of all pairs (x, y) of real numbers that fulfill the condition \( x^2 + y^2 = 1 \,\) . In this paper we consider quadratic real functions f that satisfy the additional equation \( y^2 f(x) = x^2 f(y) \) under the condition \( (x,y) \in S_i \) \((i=1,2)\) . We prove that each of these conditions implies \( f(x) = f(1) x^2 \) for all \( x \in \mathbb {R} \) . PubDate: 2019-04-01 DOI: 10.1007/s00010-018-0591-2
Abstract: Abstract The Hermite–Hadamard inequality was extended using iterated integrals by Retkes [Acta Sci Math (Szeged) 74:95–106, 2008]. In this paper we further extend the main results of the above paper for convex and also for s-convex functions in the second sense. PubDate: 2019-03-22 DOI: 10.1007/s00010-019-00642-z
Abstract: Abstract We introduce and study the natural notion of probabilistic 1-Lipschitz maps. We use the space of all probabilistic 1-Lipschitz maps to give a new method for the construction of probabilistic metric completion (respectively of probabilistic invariant metric group completion). Our construction is of independent interest. We prove that the space of all probabilistic 1-Lipschitz maps defined on a probabilistic invariant metric group can be endowed with a monoid structure. Next, we explicit the set of all invertible elements of this monoid and characterize probabilistic invariant complete Menger groups by the space of all probabilistic 1-Lipschitz maps in the spirit of the classical Banach–Stone theorem. PubDate: 2019-03-21 DOI: 10.1007/s00010-019-00641-0
Abstract: Abstract In this paper,we introduce the weak group matrix defined by the one commutable with its weak group inverse, and consider properties and characterizations of the matrix by applying the core-EP decomposition. In particular,the set of weak group matrices is more inclusive than that of group matrices. We also derive some characterizations of p-EP matrices and i-EP matrices. PubDate: 2019-03-16 DOI: 10.1007/s00010-019-00639-8
Abstract: Abstract This paper solves a Tingley’s type problem in n-normed spaces and states that for \(n\ge 2\) , every n-isometry on the unit sphere of an n-normed space is an n-isometry on the whole space except the origin 0. Also, using analytical approach we give a short proof to show that for \(n\ge 3\) , any mapping which preserves n-norms of values one and zero is, up to pointwise multiplication by \(\pm 1\) -, a linear n-isometry. This gives a Wigner-type theorem in n-normed spaces which was proven in a recent paper. PubDate: 2019-03-05 DOI: 10.1007/s00010-019-00637-w
Abstract: Abstract Let M be a compact n-dimensional manifold with \( bB_{1} \) the set of Baire-1 self-maps of M. For \(f\in bB_{1}\) , let \(\Omega (f)=\{\omega (x,f):x\in M\}\) be the collection of \(\omega \) -limit sets generated by f, and \(\Lambda (f)=\cup _{x\in M}\omega (x,f)\) be the set of \(\omega \) -limit points of f. For a typical \(f \in bB_1\) , we show the following: for any \(x\in M\) , the \(\omega \) -limit set \(\omega (x,f)\) is contained in the set of points at which f is continuous, and \(\omega (x,f)\) is an \( \infty \) -adic adding machine; for any \(\varepsilon >0\) , there exists a natural number K such that \(f^{k}(M)\subset B_{\varepsilon }(\Lambda (f))\) whenever \(k>K\) . Moreover, \(f:\Lambda (f)\rightarrow \Lambda (f)\) is a bijection, and \(\Lambda (f)\) is closed. The Hausdorff dimension of \(\Lambda (f)\) is zero, and the collection of \(\omega \) -limit sets \(\Omega (f)\) is closed in the Hausdorff metric space. The function f is not chaotic in the sense of Li–Yorke, nor in the sense of Devaney. The function f is one-to-one, and the m-fold iterate \(f^{m}\) is an element of \(bB_{1}\) for all natural numbers m. PubDate: 2019-02-25 DOI: 10.1007/s00010-019-00640-1
Abstract: Abstract Using a simple dynamical system generated by means M and N which are considered on adjacent intervals, we show how to find their joints, that is means extending both M and N. The procedure of joining is a local version of that presented in Jarczyk (Publ. Math. Debr. 91:235–246, 2017). Among joints are those semiconjugating some functions defined by the use of the so-called marginal functions of M and N. PubDate: 2019-02-01 DOI: 10.1007/s00010-018-0596-x
Abstract: Abstract Given a set \(T\subset (0, +\infty )\) , a function \(c:T\rightarrow \mathbb R\) and a real number p we study continuous solutions \(\varphi \) of the simultaneous equations $$\begin{aligned} \varphi (tx)=\varphi (x)+c(t)x^p, \qquad t \in T. \end{aligned}$$ Here \(\varphi \) is defined on an interval \(I\subset (0, +\infty )\) , so the equations are postulated on a restricted domain: for any fixed \(t \in T\) we assume that \(x \in I\) is such that \(tx \in I\) . In the case when T is large in a sense, we determine the form of \(\varphi \) on a non-trivial subinterval of I. The research is a continuation of that of “non-restricted”, where \(I=(0,+\infty )\) , made in Jarczyk (Ann Univ Sci Budapest Sect Comp 40:353–362, 2013). PubDate: 2019-02-01 DOI: 10.1007/s00010-018-0616-x
Abstract: Abstract Let \(\mathbb {K}\) be an extension of the field \(\mathbb {Q}\) of rational numbers. We prove an algebraic characterization of the additive iterative roots of identity defined on a vector space X over the field \(\mathbb {K}\) which save a Hamel basis of X (a basis of X over \(\mathbb {Q}\) ). The method is based on a canonical decomposition theorem. PubDate: 2019-02-01 DOI: 10.1007/s00010-018-0606-z
Abstract: Abstract The purpose of this paper is to investigate the invariance of the arithmetic mean with respect to two weighted Bajraktarević means, i.e., to solve the functional equation $$\begin{aligned} \left( \frac{f}{g}\right) ^{\!\!-1}\!\!\left( \frac{tf(x)+sf(y)}{tg(x)+sg(y)}\right) +\left( \frac{h}{k}\right) ^{\!\!-1}\!\!\left( \frac{sh(x)+th(y)}{sk(x)+tk(y)}\right) =x+y \qquad (x,y\in I), \end{aligned}$$ where \(f,g,h,k:I\rightarrow \mathbb {R}\) are unknown continuous functions such that g, k are nowhere zero on I, the ratio functions f / g, h / k are strictly monotone on I, and \(t,s\in \mathbb {R}_+\) are constants different from each other. By the main result of this paper, the solutions of the above invariance equation can be expressed either in terms of hyperbolic functions or in terms of trigonometric functions and an additional weight function. For the necessity part of this result, we will assume that \(f,g,h,k:I\rightarrow \mathbb {R}\) are four times continuously differentiable. PubDate: 2019-02-01 DOI: 10.1007/s00010-018-0560-9
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