Authors:Ákos G. Horváth Pages: 401 - 418 Abstract: In this paper we give lower and upper bounds for the volume growth of a regular hyperbolic simplex, namely for the ratio of the n-dimensional volume of a regular simplex and the \((n-1)\) -dimensional volume of its facets. In addition to the methods of U. Haagerup and M. Munkholm we use a third volume form based on the hyperbolic orthogonal coordinates of a body. In the case of the ideal, regular simplex our upper bound gives the best known upper bound. On the other hand, also in the ideal case our general lower bound, improved the best known one for \(n=3\) . PubDate: 2017-06-01 DOI: 10.1007/s00010-016-0465-4 Issue No:Vol. 91, No. 3 (2017)

Authors:Andrzej Olbryś Pages: 429 - 444 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: 2017-06-01 DOI: 10.1007/s00010-017-0472-0 Issue No:Vol. 91, No. 3 (2017)

Authors:El-sayed El-hady; Janusz Brzdęk; Hamed Nassar Pages: 445 - 477 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: 2017-06-01 DOI: 10.1007/s00010-017-0471-1 Issue No:Vol. 91, No. 3 (2017)

Authors:Mohamed Jleli; Donal O’Regan; Bessem Samet Pages: 479 - 490 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: 2017-06-01 DOI: 10.1007/s00010-017-0470-2 Issue No:Vol. 91, No. 3 (2017)

Authors:Teerapong Suksumran; Keng Wiboonton Pages: 491 - 503 Abstract: Möbius addition is defined on the complex open unit disk by $$\begin{aligned} a\oplus _M b = \dfrac{a+b}{1+\bar{a}b} \end{aligned}$$ and Möbius’s exponential equation takes the form \(L(a\oplus _M b) = L(a)L(b)\) , where L is a complex-valued function defined on the complex unit disk. In the present article, we indicate how Möbius’s exponential equation is connected to Cauchy’s exponential equation. Möbius’s exponential equation arises when one determines the irreducible linear representations of the unit disk equipped with Möbius addition, considered as a nonassociative group-like structure. This suggests studying Schur’s lemma in a more general setting. PubDate: 2017-06-01 DOI: 10.1007/s00010-016-0452-9 Issue No:Vol. 91, No. 3 (2017)

Authors:Tibor Kiss; Zsolt Páles Pages: 505 - 525 Abstract: It is well-known that if a real valued function acting on a convex set satisfies the n-variable Jensen inequality, for some natural number \(n\ge 2\) , then, for all \(k\in \{1,\dots , n\}\) , it fulfills the k-variable Jensen inequality as well. In other words, the arithmetic mean and the Jensen inequality (as a convexity property) are both reducible. Motivated by this phenomenon, we investigate this property concerning more general means and convexity notions. We introduce a wide class of means which generalize the well-known means for arbitrary linear spaces and enjoy a so-called reducibility property. Finally, we give a sufficient condition for the reducibility of the (M, N)-convexity property of functions and also for Hölder–Minkowski type inequalities. PubDate: 2017-06-01 DOI: 10.1007/s00010-016-0459-2 Issue No:Vol. 91, No. 3 (2017)

Authors:Pentti Haukkanen Pages: 527 - 536 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: 2017-06-01 DOI: 10.1007/s00010-017-0473-z Issue No:Vol. 91, No. 3 (2017)

Authors:S. S. Linchuk; Yu. S. Linchuk Pages: 537 - 545 Abstract: We solve generalized the generalized Rubel equation on the space of analytic functions in domains. PubDate: 2017-06-01 DOI: 10.1007/s00010-017-0467-x Issue No:Vol. 91, No. 3 (2017)

Authors:Daniel Eremita Pages: 563 - 578 Abstract: Let \(T_n(R)\) be the upper triangular matrix ring over a unital ring R. Suppose that \(B:T_n(R)\times T_n(R) \rightarrow T_n(R)\) is a biadditive map such that \(B(X,X)X = XB(X,X)\) for all \(X \in T_n(R)\) . We consider the problem of describing the form of the map \(X\mapsto B(X,X)\) . PubDate: 2017-06-01 DOI: 10.1007/s00010-016-0462-7 Issue No:Vol. 91, No. 3 (2017)

Authors:Dirk Keppens Abstract: In Keppens (Innov. Incidence Geom. 15: 119–139, 2017) we gave a state of the art concerning “projective planes” over finite rings. The current paper gives a complementary overview for “affine planes” over rings (including the important subclass of desarguesian affine Klingenberg and Hjelmslev planes). No essentially new material is presented here but we give a summary of known results with special attention to the finite case, filling a gap in the literature. PubDate: 2017-07-06 DOI: 10.1007/s00010-017-0497-4

Authors:Hamid Reza Moradi; Mohsen Erfanian Omidvar; Muhammad Adil Khan; Kazimierz Nikodem Abstract: In this paper we use basic properties of strongly convex functions to obtain new inequalities including Jensen type and Jensen–Mercer type inequalities. Applications for special means are pointed out as well. We also give a Jensen’s operator inequality for strongly convex functions. As a corollary, we improve the Hölder-McCarthy inequality under suitable conditions. More precisely we show that if \(Sp\left( A \right) \subset \left( 1,\infty \right) \) , then $$\begin{aligned} {{\left\langle Ax,x \right\rangle }^{r}}\le \left\langle {{A}^{r}}x,x \right\rangle -\frac{{{r}^{2}}-r}{2}\left( \left\langle {{A}^{2}}x,x \right\rangle -{{\left\langle Ax,x \right\rangle }^{2}} \right) ,\quad r\ge 2 \end{aligned}$$ and if \(Sp\left( A \right) \subset \left( 0,1 \right) \) , then $$\begin{aligned} \left\langle {{A}^{r}}x,x \right\rangle \le {{\left\langle Ax,x \right\rangle }^{r}}+\frac{r-{{r}^{2}}}{2}\left( {{\left\langle Ax,x \right\rangle }^{2}}-\left\langle {{A}^{2}}x,x \right\rangle \right) ,\quad 0<r<1 \end{aligned}$$ for each positive operator A and \(x\in \mathcal {H}\) with \(\left\ x \right\ =1\) . PubDate: 2017-07-04 DOI: 10.1007/s00010-017-0496-5

Authors:N. H. Bingham; A. J. Ostaszewski Abstract: Category-measure duality concerns applications of Baire-category methods that have measure-theoretic analogues. The set-theoretic axiom needed in connection with the Baire category theorem is the Axiom of Dependent Choice, DC rather than the Axiom of Choice, AC. Berz used the Hahn–Banach theorem over \({\mathbb {Q}}\) to prove that the graph of a measurable sublinear function that is \({\mathbb {Q}}_{+}\) -homogeneous consists of two half-lines through the origin. We give a category form of the Berz theorem. Our proof is simpler than that of the classical measure-theoretic Berz theorem, our result contains Berz’s theorem rather than simply being an analogue of it, and we use only DC rather than AC. Furthermore, the category form easily generalizes: the graph of a Baire sublinear function defined on a Banach space is a cone. The results are seen to be of automatic-continuity type. We use Christensen Haar null sets to extend the category approach beyond the locally compact setting where Haar measure exists. We extend Berz’s result from Euclidean to Banach spaces, and beyond. Passing from sublinearity to convexity, we extend the Bernstein–Doetsch theorem and related continuity results, allowing our conditions to be ‘local’—holding off some exceptional set. PubDate: 2017-07-01 DOI: 10.1007/s00010-017-0486-7

Authors:Sokol Bush Kaliaj Abstract: In this paper, we utilize some fixed point theorems of contractive type to present a few existence and uniqueness theorems for a functional equation arising in dynamic programming of continuous multistage decision processes. PubDate: 2017-06-06 DOI: 10.1007/s00010-017-0495-6

Authors:Senlin Wu; Chan He; Guang Yang Abstract: It is proved that a normed space, whose dimension is at least three, admitting a nonzero linear operator reversing Birkhoff orthogonality is an inner product space, which releases the smoothness condition in one of J. Chmieliński’s results. Further characterizations of inner product spaces are obtained by studying properties of linear operators related to Birkhoff orthogonality and isosceles orthogonality. PubDate: 2017-05-31 DOI: 10.1007/s00010-017-0494-7

Authors:Paweł Siedlecki Abstract: A generalized solution operator is a mapping abstractly describing a computational problem and its approximate solutions. It assigns a set of \(\varepsilon \) -approximations of a solution to the problem instance f and accuracy of approximation \(\varepsilon \) . In this paper we study generalized solution operators for which the accuracy of approximation is described by elements of a complete lattice equipped with a compatible monoid structure, namely, a quantale. We provide examples of computational problems for which the accuracy of approximation of a solution is measured by such objects. We show that the sets of \(\varepsilon \) -approximations are, roughly, closed balls with radii \(\varepsilon \) with respect to a certain family of quantale-valued generalized metrics induced by a generalized solution operator. PubDate: 2017-05-29 DOI: 10.1007/s00010-017-0485-8

Authors:M. Rajesh Kannan Abstract: In this article we introduce the notion of P-proper splitting for square matrices. For an inconsistent linear system of equations \(Ax =b\) , we associate an iterative method based on a P-proper splitting of A, which if convergent, converges to the best least squares solution of this system. We extend a result of Stein, using which we prove that if A is positive semidefinite, then the said iterative method converges. Also, we generalize Sylvester’s law of inertia and as an application of this generalization we establish some properties of P-proper splittings. Finally, we prove a comparison theorem for iterative methods associated with P-proper splittings of a positive semidefinite matrix. PubDate: 2017-05-27 DOI: 10.1007/s00010-017-0492-9

Authors:George E. Andrews; Jim Lawrence Abstract: This paper delves into the number of partitions of positive integers n into powers of 2 in which exactly m powers of 2 are used an odd number of times. The study of these numbers is motivated by their connections with the f-vectors of the binary partition polytopes. PubDate: 2017-05-27 DOI: 10.1007/s00010-017-0493-8

Authors:Dijana Mosić 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: 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: 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