Authors:George E. Andrews; Jim Lawrence Pages: 859 - 869 Abstract: 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: 2017-10-01 DOI: 10.1007/s00010-017-0493-8 Issue No:Vol. 91, No. 5 (2017)

Authors:Libo Li; Zhiwei Hao Pages: 909 - 920 Abstract: Abstract In this paper, we investigate the Hermite–Hadamard type inequality for the class of some h-convex stochastic processes, which is an extension of the Hermite–Hadamard inequality given by Barráez et al. (Math. Æterna 5:571–581, 2015). We also provide the estimates of both sides of the Hermite–Hadamard type inequality for h-convex stochastic processes, where h is any non-negative function with \(h(t)+h(1-t)\le 1\) for \(0\le t\le 1\) . PubDate: 2017-10-01 DOI: 10.1007/s00010-017-0488-5 Issue No:Vol. 91, No. 5 (2017)

Authors:J. M. Almira; E. V. Shulman Pages: 921 - 931 Abstract: Abstract We study the functional equation $$\begin{aligned} \sum _{i=1}^mf_i(b_ix+c_iy)= \sum _{k=1}^nu_k(y)v_k(x) \end{aligned}$$ with \(x,y\in \mathbb {R}^d\) and \(b_i,c_i\in {GL}(d,\mathbb {R})\) , both in the classical context of continuous complex-valued functions and in the framework of complex-valued Schwartz distributions, where these equations are properly introduced in two different ways. The solution sets are, typically, exponential polynomials and, in some particular cases, related to the so called characterization problem of the normal distribution in Probability Theory, they reduce to ordinary polynomials. PubDate: 2017-10-01 DOI: 10.1007/s00010-017-0489-4 Issue No:Vol. 91, No. 5 (2017)

Authors:György Pál Gehér Pages: 933 - 943 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: 2017-10-01 DOI: 10.1007/s00010-017-0478-7 Issue No:Vol. 91, No. 5 (2017)

Authors:Henrik Stetkær Pages: 945 - 947 Abstract: Abstract Let S be a semigroup, and \(\mathbb {F}\) a field of characteristic \(\ne 2\) . If the pair \(f,g:S \rightarrow \mathbb {F}\) is a solution of Wilson’s \(\mu \) -functional equation such that \(f \ne 0\) , then g satisfies d’Alembert’s \(\mu \) -functional equation. PubDate: 2017-10-01 DOI: 10.1007/s00010-017-0481-z Issue No:Vol. 91, No. 5 (2017)

Authors:Gennadiy Feldman Pages: 949 - 967 Abstract: Abstract Let X be a locally compact Abelian group, Y be its character group. Following A. Kagan and G. Székely we introduce a notion of Q-independence for random variables with values in X. We prove group analogues of the Cramér, Kac–Bernstein, Skitovich–Darmois and Heyde theorems for Q-independent random variables with values in X. The proofs of these theorems are reduced to solving some functional equations on the group Y. PubDate: 2017-10-01 DOI: 10.1007/s00010-017-0479-6 Issue No:Vol. 91, No. 5 (2017)

Authors:Senlin Wu; Chan He; Guang Yang Pages: 969 - 978 Abstract: 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: 2017-10-01 DOI: 10.1007/s00010-017-0494-7 Issue No:Vol. 91, No. 5 (2017)

Authors:Piotr Maćkowiak Pages: 759 - 777 Abstract: Abstract In the paper we present results on the continuity of nonlinear superposition operators acting in the space of functions of bounded variation in the sense of Jordan. It is shown that the continuity of an autonomous superposition operator is automatically guaranteed if the acting condition is met. We also give a simple proof of the fact that a nonautonomous superposition operator generated by a continuously differentiable function is uniformly continuous on bounded sets. Moreover, we present necessary and sufficient conditions for the continuity of a superposition operator (autonomous or nonautonomous) in a general setting. Thus, we give the answers to two basic open problems mentioned in the monograph (Appell et al. in Bounded variation and around, series in nonlinear analysis and application, De Gruyter, Berlin, 2014). PubDate: 2017-08-01 DOI: 10.1007/s00010-017-0491-x Issue No:Vol. 91, No. 4 (2017)

Authors:Enrique A. Sánchez Pérez Abstract: Abstract Consider a linear and continuous operator T between Banach function spaces. We prove that under certain requirements an integral inequality for T is equivalent to a factorization of T through a specific kernel operator: in other words, the operator T has what we call a Maharam-type kernel representation. In the case that the inequality provides a domination involving trigonometric functions, a special factorization through the Fourier operator is given. We apply this result to study the problem that motivates the paper: the approximation of functions in \(L^{2}[0,1]\) by means of trigonometric series whose Fourier coefficients are given by weighted trigonometric integrals. PubDate: 2017-09-19 DOI: 10.1007/s00010-017-0507-6

Authors:Josef Šlapal Abstract: Abstract We introduce and discuss a connectedness induced by n-ary relations ( \(n>1\) an integer) on their underlying sets. In particular, we focus on certain n-ary relations with the induced connectedness allowing for a definition of digital Jordan curves. For every integer \(n>1\) , we introduce one such n-ary relation on the digital plane \({\mathbb {Z}}^2\) and prove a digital analogue of the Jordan curve theorem for the induced connectedness. It follows that these n-ary relations may be used as convenient structures on the digital plane for the study of geometric properties of digital images. For \(n=2\) , such a structure coincides with the (specialization order of the) Khalimsky topology and, for \(n>2\) , it allows for a variety of Jordan curves richer than that provided by the Khalimsky topology. PubDate: 2017-09-15 DOI: 10.1007/s00010-017-0508-5

Authors:Constantin P. Niculescu; Marius Marinel Stănescu Abstract: Abstract Several applications of Abel’s partial summation formula to the convergence of series of positive vectors are presented. For example, when the norm of the ambient ordered Banach space is associated with a strong order unit, it is shown that the convergence of the series \(\sum x_{n}\) implies the convergence in density of the sequence \((nx_{n})_{n}\) to 0. This is done by extending the Koopman–von Neumann characterization of convergence in density. Also included is a new proof of the Jensen–Steffensen inequality based on Abel’s partial summation formula and a trace analogue of the Tomić–Weyl inequality of submajorization. PubDate: 2017-09-04 DOI: 10.1007/s00010-017-0504-9

Authors:Qian Zhang; Bing Xu Abstract: Abstract Given a continuous strictly monotone function \(\varphi \) defined on an open real interval I and a probability measure \(\mu \) on the Borel subsets of [0, 1], the Makó–Páles mean is defined by $$\begin{aligned} {\mathcal {M}}_{\varphi ,\mu }(x,y):=\varphi ^{-1}\left( \int ^1_0\varphi (tx+(1-t)y)\, d\mu (t)\right) ,\quad x,y\in I. \end{aligned}$$ Under some conditions on the functions \(\varphi \) and \(\psi \) defined on I, the quotient mean is given by $$\begin{aligned} Q_{\varphi ,\psi }(x,y):=\left( \frac{\varphi }{\psi }\right) ^{-1}\left( \frac{\varphi (x)}{\psi (y)}\right) , \quad x,y\in I. \end{aligned}$$ In this paper, we study some invariance of the quotient mean with respect to Makó–Páles means. PubDate: 2017-09-02 DOI: 10.1007/s00010-017-0502-y

Authors:Giedrius Alkauskas Abstract: Abstract In this second part of the work, we correct the flaw which was left in the proof of the main Theorem in the first part. This affects only a small part of the text in this first part and two consecutive papers. Yet, some additional arguments are needed to claim the validity of the classification results. With these new results, algebraic and rational flows can be much more easily and transparently classified. It also turns out that the notion of an algebraic projective flow is a very natural one. For example, we give an inductive (on dimension) method to build algebraic projective flows with rational vector fields, and ask whether these account for all such flows. Further, we expand on results concerning rational flows in dimension 2. Previously we found all such flows symmetric with respect to a linear involution \(i_{0}(x,y)=(y,x)\) . Here we find all rational flows symmetric with respect to a non-linear 1-homogeneous involution \(i(x,y)=(\frac{y^2}{x},y)\) . We also find all solenoidal rational flows. Up to linear conjugation, there appears to be exactly two non-trivial examples. PubDate: 2017-08-18 DOI: 10.1007/s00010-017-0500-0

Authors:Karol Baron Abstract: Abstract Let E be a real inner product space of dimension at least 2. We show that both the set of all orthogonally additive functions mapping E into E having orthogonally additive second iterate and its complement are dense in the space of all orthogonally additive functions from E into E with the Tychonoff topology. PubDate: 2017-08-17 DOI: 10.1007/s00010-017-0506-7

Authors:Endre Tóth; Tamás Waldhauser Abstract: Abstract Solution sets of systems of linear equations over fields are characterized as being affine subspaces. But what can we say about the “shape” of the set of all solutions of other systems of equations' We study solution sets over arbitrary algebraic structures, and we give a necessary condition for a set of n-tuples to be the set of solutions of a system of equations in n unknowns over a given algebra. In the case of Boolean equations we obtain a complete characterization, and we also characterize solution sets of systems of Boolean functional equations. PubDate: 2017-08-08 DOI: 10.1007/s00010-017-0499-2

Authors:Dirk Keppens Abstract: 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:N. H. Bingham; A. J. Ostaszewski Abstract: 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: 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: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: 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