Authors:P. Ali; J. P. Mazorodze; S. Mukwembi; T. Vetrík Pages: 11 - 24 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:V. Komornik; M. Pedicini Pages: 25 - 57 Abstract: Glendinning and Sidorov discovered an important feature of the Komornik–Loreti constant \({q' \approx 1.78723}\) in non-integer base expansions on two-letter alphabets: in bases \({1 < q < q'}\) only countably numbers have unique expansions, while for \({q \geq q'}\) there is a continuum of such numbers. We investigate the analogous question for ternary alphabets. PubDate: 2017-06-01 DOI: 10.1007/s10474-017-0706-6 Issue No:Vol. 152, No. 1 (2017)

Authors:A. Biró Pages: 58 - 71 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:K. Doi; N. Shimaru; K. Takashima Pages: 109 - 113 Abstract: We give an upper bound for the discrepancy of irrational rotations \({\{n\alpha\}}\) in terms of the continued fraction expansion of \({\alpha}\) and the related Ostrowski expansion. Our result improves earlier bounds in the literature and substantially simplifies their proofs. PubDate: 2017-06-01 DOI: 10.1007/s10474-017-0702-x Issue No:Vol. 152, No. 1 (2017)

Authors:A. Domokos; J. M. Ingram; M. M. Marsh Pages: 114 - 129 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:N. K. Govil; P. Kumar Pages: 130 - 139 Abstract: We prove that if P(z) is a polynomial of degree n with zeros z m , that satisfy \({ z_m \geq K_m \geq 1}\) , \({1 \leq m \leq n}\) , then for any p > 0, and for every complex number α, with \({ \alpha \geq 1}\) , we have $$\bigg\{\int_0^{2\pi} D_{\alpha}\{P(e^{i \theta})\} ^{p}\,d\theta\bigg\}^{{1}/{p}}\leq n( \alpha +t_0)G_{p}\bigg\{\int_0^{2\pi} P(e^{i\theta}) ^{p}\,d\theta\bigg\}^{{1}/{p}},$$ where \({G_{p}=\big\{\frac{2\pi}{\int_0^{2\pi} t_0+e^{i\theta} ^{p}\,d\theta}\big\}^{{1}/{p}}}\) , and \({t_0=\big\{1+\frac{n}{\sum_{m=1}^n\frac{1}{K_m-1}}\big\}}\) if \({K_{m}>1}\) \({(1\leq m \leq n)}\) , and t 0 = 1 if K m = 1 for some m, \({1\leq m\leq n}\) . Our results generalize and sharpen several of the known results. PubDate: 2017-06-01 DOI: 10.1007/s10474-017-0693-7 Issue No:Vol. 152, No. 1 (2017)

Authors:A. Artigue; D. Carrasco-Olivera; I. Monteverde Pages: 140 - 149 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: 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: 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. Hernandez; L. Kadison; S. A. Lopes Pages: 166 - 185 Abstract: The Green ring of the half quantum group \({H=U_n(q)}\) is computed in [9]. The tensor product formulas between indecomposables may be used for a generalized subgroup depth computation in the setting of quantum groups—to compute the depth of the Hopf subalgebra H in its Drinfeld double D(H). In this paper the Hopf subalgebra quotient module Q (a generalization of the permutation module of cosets for a group extension) is computed and, as H-modules, Q and its second tensor power are decomposed into a direct sum of indecomposables. We note that the least power n, referred to as depth, for which \({Q^{\otimes (n)}}\) has the same indecomposable constituents as \({Q^{\otimes (n+1)}}\) is \({n = 2}\) , since \({ Q^{\otimes (2)}}\) contains all H-module indecomposables, which determines the minimum even depth \({d_{\mathrm{ev}}(H,D(H)) = 6}\) . PubDate: 2017-06-01 DOI: 10.1007/s10474-017-0694-6 Issue No:Vol. 152, No. 1 (2017)

Authors:A. Yang Pages: 186 - 200 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:G. Gát; U. Goginava Pages: 201 - 216 Abstract: We study approximation by rectangular partial sums of double Fourier series on unbounded Vilenkin groups in the spaces C and L 1. From these results we obtain criterions of the uniform convergence and L-convergence of double Vilenkin–Fourier series. We also prove that these results are sharp. PubDate: 2017-06-01 DOI: 10.1007/s10474-017-0703-9 Issue No:Vol. 152, No. 1 (2017)

Authors:V. Totik; Y. Zhou Pages: 227 - 242 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: 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:L. Pereira Pages: 257 - 268 Abstract: We prove that morasses are not only compatible with supercompact cardinals, but, moreover, simplified morasses can be elements of the supercompact ultrafilter. We also give an application of this fact to the construction of continuous tree-like scales. PubDate: 2017-06-01 DOI: 10.1007/s10474-017-0708-4 Issue No:Vol. 152, No. 1 (2017)

Authors:I. Ghenciu Abstract: A bounded subset K of X is defined to be a weak reciprocal Dunford–Pettis set (or wRDP set) if T(K) is relatively weakly compact for each completely continuous operator \({T \colon X \to c_0}\) . A Banach space X has the R * property if every wRDP subset of X is relatively weakly compact. In this paper we study weak reciprocal Dunford–Pettis sets and Banach spaces with property R *. PubDate: 2017-05-19 DOI: 10.1007/s10474-017-0723-5

Authors:V. Gutev 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: 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: 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

Authors:Z. Buczolich Abstract: We show that there are functions f in the Hölder class \({C^{\alpha}[0,1]}\) , \({1 < \alpha < 2}\) such that \({f _{A}}\) is neither convex nor concave for any \({A\subset [0,1]}\) with \({\overline{{\rm dim}}_\textsc{M}\,A > \alpha-1}\) . Our earlier result shows that for the typical/generic \({f\in C_1^{\alpha}[0,1]}\) , \({0\leq \alpha < 2}\) there is always a set \({A\subset [0,1]}\) such that \({f _A}\) is convex and \({\overline{{\rm dim}}_\textsc{M}\,A=1}\) . The analogous statement for monotone restrictions is the following: there are functions \({f}\) in the Hölder class \({C^{\alpha}[0,1]}\) , \({1/2 \leq \alpha < 1}\) such that \({f _{A}}\) is not monotone on \({A\subset [0,1]}\) with \({\overline{{\rm dim}}_\textsc{M}\,A > \alpha}\) . This statement is not true for the range of parameters \({\alpha < 1/2}\) and the main theorem of this paper for the parameter range \({1< \alpha < 3/2}\) cannot be obtained by integration of the result about monotone restrictions. PubDate: 2017-04-18 DOI: 10.1007/s10474-017-0712-8