Authors:P. Ali; J. P. Mazorodze; S. Mukwembi; T. Vetrík Pages: 11 - 24 Abstract: Abstract To bound the size (the number of edges) of a graph in terms of other parameters of a graph forms an important family of problems in the extremal graph theory. We present a number of upper bounds on the size of general graphs and triangle-free graphs. We bound the size of any graph and of any triangle-free graph in terms of its order (number of vertices), diameter and edge-connectivity. We also give an upper bound on the size of triangle-free graphs of given order, diameter and minimum degree. All bounds presented in this paper are asymptotically sharp. PubDate: 2017-06-01 DOI: 10.1007/s10474-017-0699-1 Issue No:Vol. 152, No. 1 (2017)

Authors:A. Biró Pages: 58 - 71 Abstract: Abstract We consider a certain definite integral involving the product of two classical hypergeometric functions having complicated arguments. We show the surprising fact that this integral does not depend on the parameters of the hypergeometric functions. PubDate: 2017-06-01 DOI: 10.1007/s10474-017-0700-z Issue No:Vol. 152, No. 1 (2017)

Authors:A. Domokos; J. M. Ingram; M. M. Marsh Pages: 114 - 129 Abstract: Abstract Let X be a real Hilbert space. We give necessary and sufficient algebraic conditions for a mapping \({F\colon X \to X}\) with a closed image set to be the metric projection mapping onto a closed convex set. We provide examples that illustrate the necessity of each of the conditions. Our characterizations generalize several results related to projections onto closed convex sets. PubDate: 2017-06-01 DOI: 10.1007/s10474-017-0691-9 Issue No:Vol. 152, No. 1 (2017)

Authors:A. Artigue; D. Carrasco-Olivera; I. Monteverde Pages: 140 - 149 Abstract: Abstract We study the polynomial entropy of homeomorphisms on compact metric spaces. We construct a homeomorphism on a compact metric space with vanishing polynomial entropy that it is not equicontinuous. Also we give examples with arbitrarily small polynomial entropy. Finally, we show that expansive homeomorphisms and positively expansive maps of compact metric spaces with infinitely many points have polynomial entropy greater than or equal to 1. PubDate: 2017-06-01 DOI: 10.1007/s10474-017-0689-3 Issue No:Vol. 152, No. 1 (2017)

Authors:P. Klinga; A. Nowik Pages: 150 - 160 Abstract: Abstract We continue our work on the ideal version of the Lévy–Steinitz theorem on conditionally convergent series of vectors. In particular, we prove that for each series \({\sum_{n\in\omega}v_n}\) , \({(v_n)_{n\in\omega} \subset\mathbb{R}^2}\) , such that its sum range is \({\mathbb{R}^2}\) and its set of Lévy vectors is of power at least 3, it is possible to find \({A\in\mathcal{I}}\) such that the sum range of \({\sum_{n\in A}v_n}\) is still \({\mathbb{R}^2}\) , for some proper ideal \({\mathcal{I}\subset\mathcal{P}(\omega)}\) . We also work on the summability of certain known ideals as well as introduce the cardinal number \({\kappa_{M}}\) as the minimal number of summable ideals required to cover an ideal, and prove some basic properties of it. PubDate: 2017-06-01 DOI: 10.1007/s10474-017-0704-8 Issue No:Vol. 152, No. 1 (2017)

Authors:P. Komjáth Pages: 161 - 165 Abstract: Abstract We give complete proofs of the results of Galvin and Nagy on a problem of Erdős and Hajnal concerning certain combinatorial games of transfinite length. PubDate: 2017-06-01 DOI: 10.1007/s10474-017-0705-7 Issue No:Vol. 152, No. 1 (2017)

Authors:A. Yang Pages: 186 - 200 Abstract: Abstract This paper is devoted to studying the boundedness of sublinear operators on vector-valued weak Orlicz martingale spaces. These results closely depend on the geometrical properties of the Banach space in which the martingales take values. Also the results obtained here extend the corresponding known results from scalar-valued setting to vector-valued setting. PubDate: 2017-06-01 DOI: 10.1007/s10474-017-0710-x Issue No:Vol. 152, No. 1 (2017)

Authors:V. Totik; Y. Zhou Pages: 227 - 242 Abstract: Abstract The best asymptotic constant for k-th order Markov inequality on a general compact set is determined. PubDate: 2017-06-01 DOI: 10.1007/s10474-017-0709-3 Issue No:Vol. 152, No. 1 (2017)

Authors:A. Liu Pages: 243 - 256 Abstract: Abstract We investigate a new spectrum property ( \({W_E}\) ), which extends the generalized Weyl theorem. Using the property of consistence in Fredholm and index, we establish for a bounded linear operator T defined on a Hilbert space sufficient and necessary conditions for which the property \({(W_E)}\) holds. We also explore conditions on Hilbert operators T and S so that property \({(W_E)}\) holds for \({T\oplus S}\) . Moreover, we study the permanence of property \({(W_E)}\) under perturbations by power finite rank operators commuting with T and discuss the relation between property ( \({W_E}\) ) and hypercyclic operators. PubDate: 2017-06-01 DOI: 10.1007/s10474-017-0707-5 Issue No:Vol. 152, No. 1 (2017)

Authors:V. V. Andrievskii Abstract: Abstract The estimates of the uniform norm of the Chebyshev polynomials associated with a compact set K in the complex plane are established. These estimates are exact (up to a constant factor) in the case where K consists of a finite number of quasiconformal curves or arcs. The case where K is a uniformly perfect subset of the real line is also studied. PubDate: 2017-07-19 DOI: 10.1007/s10474-017-0720-8

Authors:R. Thornton Abstract: Abstract We study the relationship between the sizes of sets B, S in \({\mathbb{R}^n}\) where B contains the k-skeleton of an axes-parallel cube around each point in S, generalizing the results of Keleti, Nagy, and Shmerkin [6] about such sets in the plane. We find sharp estimates for the possible packing and box-counting dimensions for B and S. These estimates follow from related cardinality bounds for sets containing the discrete skeleta of cubes around a finite set of a given size. The Katona–Kruskal Theorem from hypergraph theory plays an important role. We also find partial results for the Haussdorff dimension and settle an analogous question for the dual polytope of the cube, the orthoplex. PubDate: 2017-07-19 DOI: 10.1007/s10474-017-0729-z

Authors:M. Kiss Abstract: Abstract We consider an inverse eigenvalue problem for Dirac operators on finite intervals. We show that if for a \({\mu\in\mathbb{C}}\) the system \({\{\exp{2i\lambda_nx}}\) , \({\exp{2i\mu x}\}}\) is closed in \({L^p[-\pi,\pi]}\) , then there is at most one \({L^p}\) -potential with the eigenvalues \({\lambda_n}\) . The result corresponds to the case of Schrödinger operators. PubDate: 2017-06-21 DOI: 10.1007/s10474-017-0733-3

Authors:B. H. Sadathoseyni; S. M. Tabatabaie Abstract: Abstract We initiate the study of coorbit spaces on locally compact hypergroups and give some necessary and sufficient conditions for the extended coorbit spaces to be complete. PubDate: 2017-06-20 DOI: 10.1007/s10474-017-0736-0

Authors:P. Kórus Abstract: Abstract We present an improved version of Young’s inequality as well as an operator inequality version of it. Our result is compared to the latest refinements. PubDate: 2017-06-20 DOI: 10.1007/s10474-017-0735-1

Authors:H. F. de Lima; A. M. S. Oliveira; M. S. Santos; M. A. L. Velásquez Abstract: Abstract We establish the notions of f-stability and strong f-stability concerning closed spacelike hypersurfaces immersed with constant f-mean curvature in a conformally stationary spacetime endowed with a conformal timelike vector field V and a weight function f. When V is closed, with the aid of the f-Laplacian of a suitable support function, we characterize f-stable closed spacelike hypersurfaces through the analysis of the first eigenvalue of the Jacobi operator associated to the corresponding variational problem. Furthermore, we obtain sufficient conditions which assure that a strongly f-stable closed spacelike hypersurface must be either f-maximal or isometric to a leaf orthogonal to V. PubDate: 2017-06-20 DOI: 10.1007/s10474-017-0731-5

Authors:D. Głazowska; P. Leonetti; J. Matkowski; S. Tringali Abstract: Abstract Let \({(X, \mathscr{L}, \lambda)}\) and \({(Y, \mathscr{M}, \mu)}\) be finite measure spaces for which there exist \({A \in \mathscr{L}}\) and \({B \in \mathscr{M}}\) with \({0 < \lambda(A) < \lambda(X)}\) and \({0 < \mu(B) < \mu(Y)}\) , and let \({I\subseteq \mathbf{R}}\) be a non-empty interval. We prove that, if f and g are continuous bijections \({I \to \mathbf{R}^+}\) , then the equation $$f^{-1}\Big(\int_X f\Big(g^{-1}\Big(\int_Y g \circ h \,d\mu\Big)\Big)d \lambda\Big) = g^{-1}\Big(\int_Y g\Big(f^{-1}\Big(\int_X f \circ h \,d\lambda\Big)\Big)d \mu\Big)$$ is satisfied by every \({\mathscr{L} \otimes \mathscr{M}}\) -measurable simple function \({h\colon X \times Y \to I}\) if and only if f = cg for some \({c \in \mathbf{R}^+}\) (it is easy to see that the equation is well posed). An analogous, but essentially different result, with f and g replaced by continuous injections \({I \to \mathbf R}\) and \({\lambda(X)=\mu(Y)=1}\) , was recently obtained in [7]. PubDate: 2017-06-20 DOI: 10.1007/s10474-017-0734-2

Authors:C. Baiocchi; V. Komornik; P. Loreti Abstract: Abstract Cantor’s ternary function is generalized to arbitrary base-change functions in non-integer bases. They turn out to have radically different continuity, differentiability and monotonicity properties, depending on the particular bases involved in their definition. PubDate: 2017-06-20 DOI: 10.1007/s10474-017-0732-4

Authors:V. Gutev Abstract: Abstract Using the framework of weak selections, Nagao and Shakhmatov introduced topological king spaces, and extended the classical “King Chicken Theorem” by showing that each compact space with a continuous weak selection is a king space. They also obtained that several king spaces are compact, and raised the question whether every locally compact (or locally pseudocompact) king space must be compact. In the present paper, we settle this question in the affirmative. PubDate: 2017-04-18 DOI: 10.1007/s10474-017-0713-7

Authors:P. Mafuta; S. Mukwembi; S. Munyira; T. Vetrík Abstract: Abstract We prove a new sufficient condition for a connected graph to be Hamiltonian in terms of the leaf number and the minimum degree. Our results give solutions to conjectures on the Hamiltonicity and traceability of graphs. We considerably generalize known results in the area by showing that if G is a connected graph having minimum degree \({\delta}\) and leaf number L such that \({\delta \ge \frac{L}{2}+1}\) , then G is Hamiltonian and thus traceable. PubDate: 2017-04-18 DOI: 10.1007/s10474-017-0716-4

Authors:N. K. Agbeko; W. Fechner; E. Rak Abstract: Abstract The main purpose of this paper is to study certain lattice-valued maps through associated functional equations and inequalities. We deal with morphisms between an algebraic structure and an ordered structure. Next, we solve a separation problem for the inequalities studied. Moreover, we discuss the Hyers-Ulam stability of our main equation. Our research is motivated by the notion of optimal average, which was introduced by the first author in 1994. PubDate: 2017-04-18 DOI: 10.1007/s10474-017-0719-1