Authors:J. A. Martínez-Cadena; R. G. Wilson Pages: 259 - 270 Abstract: Abstract A topological space X is densely countably compact if it possesses a dense subspace D with the property that every infinite subset of D has an accumulation point in X. We study topologies which are maximal with respect to this property; in particular we show that a T 1 densely countably compact space is maximal densely countably compact if and only if it is a scattered Fréchet SC-space of scattering order 2. PubDate: 2017-04-01 DOI: 10.1007/s10474-016-0684-0 Issue No:Vol. 151, No. 2 (2017)

Authors:Z. Garbouj; H. Skhiri Pages: 328 - 360 Abstract: Abstract For a closed linear relation in a Hilbert space the notions of minimum modulus, essential g-ascent, essential ascent and essential descent are introduced and studied. We prove that some results of E. Chafai and M. Mnif [3] related to the stability of the essential descent and descent of a linear relation T everywhere defined such that \({T(0)\subseteq \mathsf{ker}(T)}\) by a finite rank operator F commuting with T, remain valid when F is an everywhere defined linear relation and without the assumption that \({T(0)\subseteq \mathsf{ker}(T)}\) . We studied also the stability of the essential g-ascent and the essential ascent under a finite rank relation. Motivated by the recent work of T. Álvarez and A. Sandovici [1], we extend to a closed linear relation, the well known notion of minimum modulus of a linear operator (H. A. Gindler and A. E. Taylor [7]). Also, we introduce and study the new notion of minimum g-modulus for a linear relation. PubDate: 2017-04-01 DOI: 10.1007/s10474-016-0683-1 Issue No:Vol. 151, No. 2 (2017)

Authors:M. Aggarwal; S. Kundu Pages: 391 - 408 Abstract: Abstract Some classes of metric spaces satisfying properties stronger than completeness but weaker than compactness have been studied by many authors over the years. One such significant family consists of those metric spaces on which every real-valued continuous function is uniformly continuous, which are widely known as Atsuji spaces or UC spaces. Recently in 2014, two new kinds of complete metric spaces are introduced, namely Bourbaki-complete and cofinally Bourbaki-complete metric spaces, whose idea has come from some new classes of sequences acting as generalizations of Cauchy sequences. Our major goal is to give several new equivalent conditions for metric spaces whose completions are one of the aforesaid spaces, especially in terms of some functions, sequences and geometric functionals. PubDate: 2017-04-01 DOI: 10.1007/s10474-016-0682-2 Issue No:Vol. 151, No. 2 (2017)

Authors:H. Halas Pages: 462 - 481 Abstract: Abstract The line-inversion and pedal transformation are defined in the quasi-hyperbolic plane and certain properties of these transformations are shown with regard to analogous transformations in the Euclidean [1, 3, 10, 12, 20], hyperbolic [4, 15, 18] isotropic [17, 19] and pseudo-Euclidean plane [5, 6, 7, 14]. As it is natural to observe class curves in the quasi-hyperbolic plane, i.e. line envelopes, the construction of a tangent point on any line of the class curve obtained by the line-inversion and pedal transformation is shown. PubDate: 2017-04-01 DOI: 10.1007/s10474-016-0686-y Issue No:Vol. 151, No. 2 (2017)

Authors:A. Gut; U. Stadtmüller Pages: 510 - 530 Abstract: Abstract Various methods of summation for divergent series have been extended to analogs for sums of i.i.d. random variables. The present paper deals with a special class of matrix weighted sums of i.i.d. random variables where the weights \({a_{n,k}}\) are defined as the weights from Cesàro summability, i.e., \({a_{n,k}=\binom{n-k+\alpha-1}{n-k}/\binom{n+\alpha}{n}}\) , where \({\alpha > 0}\) . A strong law of large numbers (SLLN) has been shown to hold in this setting iff \({E { X }^{1/\alpha}<\infty}\) , but a law of the iterated logarithm (LIL) has been shown for the case \({\alpha \geqq 1}\) only. We will study the case \({0 < \alpha < 1}\) in more detail, giving an LIL for \({1/2 < \alpha < 1}\) and some additional strong limit theorems under appropriate moment conditions for \({1/2 \leqq \alpha < 1}\) . PubDate: 2017-04-01 DOI: 10.1007/s10474-016-0685-z Issue No:Vol. 151, No. 2 (2017)

Authors:L. Aiemsomboon; W. Sintunavarat Pages: 35 - 46 Abstract: Abstract Using a fixed point result and an approach to stability of functional equations presented in [8], we investigate a new type of stability for the radical quadratic functional equation of the form $$ f(\sqrt{x^2+y^2}) = f(x) + f(y), $$ where f is a self-mapping on the set of real numbers. We generalize, extend, and complement some earlier classical results concerning the Hyers–Ulam stability for that functional equations. PubDate: 2017-02-01 DOI: 10.1007/s10474-016-0666-2 Issue No:Vol. 151, No. 1 (2017)

Authors:T. Natkaniec; P. Szuca Pages: 69 - 81 Abstract: Abstract For any Borel ideal \({\mathcal{I}}\) we describe the discrete \({\mathcal{I}}\) -Baire system generated by the family of quasi-continuous real-valued functions. We characterize Borel ideals \({\mathcal{I}}\) for which ideal and ordinary discrete Baire systems coincide. PubDate: 2017-02-01 DOI: 10.1007/s10474-016-0673-3 Issue No:Vol. 151, No. 1 (2017)

Authors:N. Abughazalah Abstract: Abstract Every semigroup which is a finite disjoint union of copies of the free monogenic semigroup (natural numbers under addition) has soluble word problem and soluble membership problem. Efficient algorithms are given for both problems. PubDate: 2017-03-02 DOI: 10.1007/s10474-017-0687-5

Authors:D. E. Otera; F. G. Russo; C. Tanasi Abstract: Abstract We study the notion of wgsc inverse-representation of finitely presented groups and use the “ \({(\Phi,\Psi)}\) -technique” of Poénaru, in order to prove that the universal cover of a closed 3-manifold admitting a wgsc inverse-representation with an extra finiteness condition is simply connected at infinity. Furthermore, we investigate some new relations between wgsc inverse-representations and the qsf property for groups. PubDate: 2017-02-21 DOI: 10.1007/s10474-017-0698-2

Authors:H. Mishou; H. Nagoshi Abstract: Abstract We establish a joint universality theorem for pairs of functions in the Selberg class under certain conditions. This theorem generalizes and unifies several previous results, which were shown individually. We also give further examples of pairs of jointly universal L-functions, and actually extend the known universality theorem for the symmetric power L-function \({L(s, \mathrm{sym}^m f)}\) associated to a holomorphic Hecke eigen cusp form f for \({\mathrm{SL}_{2} (\mathbb{Z})}\) with \({1 \le m \le 4}\) . PubDate: 2017-02-21 DOI: 10.1007/s10474-017-0696-4

Authors:K. Aloui; Ch. Mauduit; M. Mkaouar Abstract: Résumé L’objet de cet article est de généraliser plusieurs résultats concernant la répartition dans les progressions arithmétiques de la fonction somme des chiffres au cas des nombres palindromes ellipséphiques. PubDate: 2017-02-21 DOI: 10.1007/s10474-017-0688-4

Authors:A. Artigue; D. Carrasco-Olivera; I. Monteverde 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: 2017-02-20 DOI: 10.1007/s10474-017-0689-3

Authors:A. Domokos; J. M. Ingram; M. M. Marsh 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: 2017-02-20 DOI: 10.1007/s10474-017-0691-9

Authors:E. de Amo; M. Díaz Carrillo; J. Fernández-Sánchez Abstract: Abstract Among the members of the celebrated family of functions introduced by Salem in the mid 20th century, there is a particular and very interesting one that we use to relate the dyadic system of numbers representation with the modified Engel system. Various properties are studied for this function, including derivatives and fractal dimensions. PubDate: 2017-02-20 DOI: 10.1007/s10474-017-0690-x

Authors:L. Montejano; E. Roldán-Pensado Abstract: Abstract We develop a concrete way to construct bodies of constant width in dimension three. They are constructed from special embeddings of self-dual graphs. PubDate: 2017-02-20 DOI: 10.1007/s10474-017-0697-3

Authors:P. Frankl; V. Rödl; A. Ruciński Abstract: Abstract Erdős [1] conjectured that for all \({k \geq 2}\) , \({s \geq 1}\) and \({n \geq {k(s+1)}}\) , an n-vertex k-uniform hypergraph \({\mathcal{F}}\) with \({\nu(\mathcal{F})=s}\) cannot have more than \({\max\{\binom{sk+k-1}k,\binom nk-\binom{n-s}k\}}\) edges. It took almost fifty years to prove it for triple systems. In [5] we proved the conjecture for all s and all \({n \geq 4(s+1)}\) . Then Łuczak and Mieczkowska [6] proved the conjecture for sufficiently large s and all n. Soon after, Frankl proved it for all s. Here we present a simpler version of that proof which yields Erdős’ conjecture for \({s \geq 33}\) . Our motivation is to lay down foundations for a possible proof in the much harder case k = 4, at least for large s. PubDate: 2017-02-20 DOI: 10.1007/s10474-017-0692-8

Authors:J. Makó Abstract: Abstract The generalized convexity of the Takagi function was proved by Z. Boros [7]. We give an another proof of this result, which is more transparent. PubDate: 2017-02-20 DOI: 10.1007/s10474-017-0695-5

Authors:N. K. Govil; P. Kumar Abstract: 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: 2017-02-20 DOI: 10.1007/s10474-017-0693-7

Authors:A. Hernandez; L. Kadison; S. A. Lopes Abstract: 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: 2017-02-20 DOI: 10.1007/s10474-017-0694-6

Authors:J. Cilleruelo Abstract: Abstract We study the probability that the edges of a random cycle of k vertices in the lattice \({\{1,\ldots,n\}^s}\) do not contain more lattice points than the k vertices of the cycle. Then we introduce the chromatic zeta function of a graph to generalize this problem to other configurations induced by a given graph \({\mathcal H}\) . PubDate: 2016-12-19 DOI: 10.1007/s10474-016-0678-y