Authors:Vitor Balestro; Ákos G. Horváth; Horst Martini; Ralph Teixeira Pages: 201 - 236 Abstract: Abstract The concepts of angle, angle functions, and the question how to measure angles present old and well-established mathematical topics referring to the Euclidean space, and there exist also various extensions to non-Euclidean spaces of different types. In particular, it is very interesting to investigate or to combine (geometric) properties of possible concepts of angle functions and angle measures in finite-dimensional real Banach spaces (= Minkowski spaces). However, going into this direction one will observe that there is no monograph or survey reflecting the complete picture of the existing literature on such concepts in a satisfying manner. We try to close this gap. In this expository paper (containing also new results, and new proofs of known results) the reader will get a comprehensive overview of this field, including further related aspects, as well. For example, angular bisectors, their applications, and angle types which preserve certain kinds of orthogonality are discussed. The latter aspect yields, of course, an interesting link to the large variety of orthogonality types in such spaces. PubDate: 2017-04-01 DOI: 10.1007/s00010-016-0445-8 Issue No:Vol. 91, No. 2 (2017)

Authors:Jean-Luc Marichal; Bruno Teheux Pages: 265 - 277 Abstract: Abstract The so-called generalized associativity functional equation $$\begin{aligned} G(J(x,y),z) = H(x,K(y,z)) \end{aligned}$$ has been investigated under various assumptions, for instance when the unknown functions G, H, J, and K are real, continuous, and strictly monotonic in each variable. In this note we investigate the following related problem: given the functions J and K, find every function F that can be written in the form $$\begin{aligned} F(x,y,z) = G(J(x,y),z) = H(x,K(y,z)) \end{aligned}$$ for some functions G and H. We show how this problem can be solved when any of the inner functions J and K has the same range as one of its sections. PubDate: 2017-04-01 DOI: 10.1007/s00010-016-0450-y Issue No:Vol. 91, No. 2 (2017)

Authors:Henrik Stetkær Pages: 279 - 288 Abstract: Abstract Let S be a semigroup, H a 2-torsion free, abelian group and \(C^2f\) the second order Cauchy difference of a function \(f:S \rightarrow H\) . Assuming that H is uniquely 2-divisible or S is generated by its squares we prove that the solutions f of \(C^2f = 0\) are the functions of the form \(f(x) = j(x) + B(x,x)\) , where j is a solution of the symmetrized additive Cauchy equation and B is bi-additive. Under certain conditions we prove that the terms j and B are continuous, if f is. We relate the solutions f of \(C^2f = 0\) to Fréchet’s functional equation and to polynomials of degree less than or equal to 2. PubDate: 2017-04-01 DOI: 10.1007/s00010-016-0453-8 Issue No:Vol. 91, No. 2 (2017)

Authors:Imke Toborg Pages: 289 - 299 Abstract: Abstract We analyse the composite functional equation \(f(x+2f(y))=f(x)+y+f(y)\) on certain groups. In particular we give a description of solutions on abelian 3-groups and finitely generated free abelian groups. This is motivated by a work of Pál Burai, Attila Házy and Tibor Juhász, who described the solutions of the equation on uniquely 3-divisible abelian groups. PubDate: 2017-04-01 DOI: 10.1007/s00010-016-0454-7 Issue No:Vol. 91, No. 2 (2017)

Authors:Bruce Ebanks Pages: 317 - 330 Abstract: Abstract This article has two aims. First, we provide the solution to a problem posed by the author in a previous paper. Second, we consider a problem posed by Kannappan and Kurepa (Aequat Math 4: 163–175, 1970). Our results show that additive functions linked by certain types of functional equations are combinations of linear functions and derivations of various orders. We show that this is not generally the case for the problem of Kannappan and Kurepa, and we modify their problem accordingly. PubDate: 2017-04-01 DOI: 10.1007/s00010-016-0449-4 Issue No:Vol. 91, No. 2 (2017)

Authors:Matthias Henze; Romanos-Diogenes Malikiosis Pages: 331 - 352 Abstract: Abstract The goal of this paper is twofold; first, we show the equivalence between certain problems in geometry, such as view-obstructions, billiard ball motions, and the estimation of covering radii of lattice zonotopes. Second, we utilize the latter interpretation and provide upper bounds of said radii by virtue of the Flatness Theorem. Our results allow us to specify how rational dependencies in the view-direction influence the obstruction parameter. These problems are similar in nature to the famous Lonely Runner Problem for which we draw analogous conclusions. PubDate: 2017-04-01 DOI: 10.1007/s00010-016-0458-3 Issue No:Vol. 91, No. 2 (2017)

Authors:Miguel Couceiro; Michel Grabisch Pages: 353 - 371 Abstract: Abstract Integer-valued means, satisfying the decomposability condition of Kolmogoroff/Nagumo, are necessarily extremal, i.e., the mean value depends only on the minimal and maximal inputs. To overcome this severe limitation, we propose an infinite family of (weak) integer means based on the symmetric maximum and computation rules. For such means, their value depends not only on extremal inputs, but also on 2nd, 3rd, etc., extremal values as needed. In particular, we show that this family can be characterized by a weak version of decomposability. PubDate: 2017-04-01 DOI: 10.1007/s00010-016-0460-9 Issue No:Vol. 91, No. 2 (2017)

Authors:David J. Schmitz Pages: 373 - 389 Abstract: Abstract If \(f\,{:}\,G \rightarrow G\) is a bijection from a group G to itself, then the superscript \(-1\) written in proximity of f could, depending on its placement, indicate the inverse function of f or the function whose outputs are the inverses (in G) of the corresponding outputs of f. In this paper we investigate which groups G admit functions f where these two interpretations lead to the same outcome, that is, when \(f^{-1}(x) = (f(x))^{-1}\) for every \(x \in G\) . We call such functions inverse ambiguous. After deriving some preliminary results, we turn our attention to the existence of inverse ambiguous functions defined on various fields F with respect to the underlying additive structure and multiplicative structure. Our study of this notational curiosity will lead to a surprising finding: namely, in many of the standard cases, there exist either continuous inverse ambiguous functions on F with respect to addition or on \(F^\times \) with respect to multiplication, but not both. PubDate: 2017-04-01 DOI: 10.1007/s00010-016-0464-5 Issue No:Vol. 91, No. 2 (2017)

Authors:György Pál Gehér Abstract: Abstract The goal of this paper is to point out that the results obtained in the recent papers (Chen and Song in Nonlinear Anal 72:1895–1901, 2010; Chu in J Math Anal Appl 327:1041–1045, 2007; Chu et al. in Nonlinear Anal 59:1001–1011, 2004a, J. Math Anal Appl 289:666–672, 2004b) can be seriously strengthened in the sense that we can significantly relax the assumptions of the main results so that we still get the same conclusions. In order to do this first, we prove that for \(n \ge 3\) any transformation which preserves the n-norm of any n vectors is automatically plus-minus linear. This will give a re-proof of the well-known Mazur–Ulam-type result that every n-isometry is automatically affine ( \(n \ge 2\) ) which was proven in several papers, e.g. in Chu et al. (Nonlinear Anal 70:1068–1074, 2009). Second, following the work of Rassias and Šemrl (Proc Am Math Soc 118:919–925, 1993), we provide the solution of a natural Aleksandrov-type problem in n-normed spaces, namely, we show that every surjective transformation which preserves the unit n-distance in both directions ( \(n\ge 2\) ) is automatically an n-isometry. PubDate: 2017-04-07 DOI: 10.1007/s00010-017-0478-7

Authors:Dijana Mosić Abstract: Abstract We study the generalization of Jacobson’s lemma for the group inverse, Drazin inverse, generalized Drazin inverse and pseudo Drazin inverse of \(1-bd\) (or \(1-ac\) ) in a ring when \(1-ac\) (or \(1-bd\) ) has a corresponding inverse, \(acd=dbd\) and \(bdb=bac\) (or \(dba=aca\) ). Thus, we recover some recent results. PubDate: 2017-04-04 DOI: 10.1007/s00010-017-0476-9

Authors:Jacek Chudziak; Zdeněk Kočan Abstract: Abstract We determine continuous solutions of the Goła̧b–Schinzel functional equation on cylinders. PubDate: 2017-03-27 DOI: 10.1007/s00010-017-0475-x

Authors:P. A. García-Sánchez; D. Llena; A. Moscariello Abstract: Abstract This work extends the results known for the Delta sets of non-symmetric numerical semigroups with embedding dimension three to the symmetric case. Thus, we have a fast algorithm to compute the Delta set of any embedding dimension three numerical semigroup. Also, as a consequence of these results, the sets that can be realized as Delta sets of numerical semigroups of embedding dimension three are fully characterized. PubDate: 2017-03-14 DOI: 10.1007/s00010-017-0474-y

Authors:Andrzej Olbryś Abstract: Abstract In the present paper we introduce a notion of the \(\mathbb {K}\) -Riemann integral as a natural generalization of a usual Riemann integral and study its properties. The aim of this paper is to extend the classical Hermite–Hadamard inequalities to the case when the usual Riemann integral is replaced by the \(\mathbb {K}\) -Riemann integral and the convexity notion is replaced by \(\mathbb {K}\) -convexity. PubDate: 2017-03-10 DOI: 10.1007/s00010-017-0472-0

Authors:Pentti Haukkanen Abstract: Abstract It is well known that Euler’s totient function \(\phi \) satisfies the arithmetical equation \( \phi (mn)\phi ((m, n))=\phi (m)\phi (n)(m, n) \) for all positive integers m and n, where (m, n) denotes the greatest common divisor of m and n. In this paper we consider this equation in a more general setting by characterizing the arithmetical functions f with \(f(1)\ne 0\) which satisfy the arithmetical equation \( f(mn)f((m,n)) = f(m)f(n)g((m, n)) \) for all positive integers m, n with \(m,n \in A(mn)\) , where A is a regular convolution and g is an A-multiplicative function. Euler’s totient function \(\phi _A\) with respect to A is an example satisfying this equation. PubDate: 2017-03-07 DOI: 10.1007/s00010-017-0473-z

Authors:Mohamed Jleli; Donal O’Regan; Bessem Samet Abstract: Abstract In this paper, some fractional integral inequalities involving m-convex functions are established. The presented results are generalizations of the obtained inequalities in Dragomir and Toader (Babeş-Bolyai Math 38:21–28, 1993). PubDate: 2017-02-28 DOI: 10.1007/s00010-017-0470-2

Authors:El-sayed El-hady; Janusz Brzdęk; Hamed Nassar Abstract: Abstract It is a survey on functional equations of a certain type, for functions in two complex variables, which often arise in queueing models. They share a common pattern despite their apparently different forms. In particular, they invariably characterize the probability generating function of the bivariate distribution characterizing a two-queue system and their forms depend on the composition of the underlying system. Unfortunately, there is no general methodology of solving them, but rather various ad-hoc techniques depending on the nature of a particular equation; most of the techniques involve advanced complex analysis tools. Also, the known solutions to particular cases of this type of equations are in general of quite involved forms and therefore it is very difficult to apply them practically. So, it is clear that the issues connected with finding useful descriptions of solutions to these equations create a huge area of research with numerous open problems. The aim of this article is to stimulate a methodical study of this area. To this end we provide a survey of the queueing literature with such two-place functional equations. We also present several observations obtained while preparing it. We hope that in this way we will make it easier to take some steps forward on the road towards a (more or less) general solving theory for this interesting class of equations. PubDate: 2017-02-28 DOI: 10.1007/s00010-017-0471-1

Authors:Xiaowei Xu; Chen Li; Jianwei Zhu Abstract: Abstract Let n and s be integers such that \(1\le s<\frac{n}{2}\) , and let \(M_n(\mathbb {K})\) be the ring of all \(n\times n\) matrices over a field \(\mathbb {K}\) . Denote by \([\frac{n}{s}]\) the least integer m with \(m\ge \frac{n}{s}\) . In this short note, it is proved that if \(g:M_n(\mathbb {K})\rightarrow M_n(\mathbb {K})\) is a map such that \(g\left( \sum _{i=1}^{[\frac{n}{s}]}A_i\right) =\sum _{i=1}^{[\frac{n}{s}]}g(A_i)\) holds for any \([\frac{n}{s}]\) rank-s matrices \(A_1,\ldots ,A_{[\frac{n}{s}]}\in M_n(\mathbb {K})\) , then \(g(x)=f(x)+g(0)\) , \(x\in M_n(\mathbb {K})\) , for some additive map \(f:M_n(\mathbb {K})\rightarrow M_n(\mathbb {K})\) . Particularly, g is additive if \(char\mathbb {K}\not \mid \left( [\frac{n}{s}]-1\right) \) . PubDate: 2017-02-11 DOI: 10.1007/s00010-017-0469-8

Authors:Luis Barreira; Davor Dragičević; Claudia Valls Abstract: Abstract For the flow determined by a nonautonomous linear differential equation, we characterize the existence of a strong nonuniform exponential dichotomy in terms of the Fredholm property of a certain linear operator. We consider both cases of one-sided and two-sided exponential dichotomies. Moreover, we use the characterizations to establish the robustness of the notion of a strong nonuniform exponential dichotomy in a simple manner. PubDate: 2017-02-08 DOI: 10.1007/s00010-017-0468-9

Authors:Daniel Reem Abstract: Abstract This paper examines various aspects related to the Cauchy functional equation \(f(x+y)=f(x)+f(y)\) , a fundamental equation in the theory of functional equations. In particular, it considers its solvability and its stability relative to subsets of multi-dimensional Euclidean spaces and tori. Several new types of regularity conditions are introduced, such as one in which a complex exponent of the unknown function is locally measurable. An initial value approach to analyzing this equation is considered too and it yields a few by-products, such as the existence of a non-constant real function having an uncountable set of periods which are linearly independent over the rationals. The analysis is extended to related equations such as the Jensen equation, the multiplicative Cauchy equation, and the Pexider equation. The paper also includes a rather comprehensive survey of the history of the Cauchy equation. PubDate: 2017-02-07 DOI: 10.1007/s00010-016-0463-6

Authors:S. S. Linchuk; Yu. S. Linchuk Abstract: Abstract We solve generalized the generalized Rubel equation on the space of analytic functions in domains. PubDate: 2017-02-04 DOI: 10.1007/s00010-017-0467-x