Authors:M.-J. Deng; J. Guo Abstract: Abstract Let \({(m^2 - n^2, 2mn, m^2 + n^2)}\) be a primitive Pythagorean triple such that m, n are positive integers with \({ \gcd (m,n)=1}\) , \({m > n}\) , \({m\not\equiv n\pmod{2}}\) . In 1956, Jeśmanowicz conjectured that the only positive integer solution to the exponential Diophantine equation \({(m^2-n^2)^x + (2mn)^y = (m^2+n^2)^z}\) is x = y = z = 2. Let \({(m,n)\equiv(u,v)\pmod{d}}\) denote \({m\equiv u\pmod{d}}\) and \({n\equiv v\pmod{d}}\) . Using the theory of quartic residue character and elementary method, we first prove Jeśmanowicz’ conjecture in the following cases. (i) \({(m,n)\equiv(1,2)\pmod{4}}\) . (ii) \({(m,n)\equiv(3,2)}\) , \({(7,6)\pmod{8}}\) or \({(m,n)\equiv(3,6)}\) , (7,2), (11,14), (15,10) \({(\mod{16})}\) . (iii) \({(m,n)\equiv(3,14)}\) , (7,10), (11,6), \({(15,2)\pmod{16}}\) and \({y > 1}\) . Then, by using the above results, two lemmas that based on Laurent’s deep result and computer assistance, for \({n\equiv2\pmod{4}}\) with \({n < 600}\) , we prove the conjecture without any assumption on m. PubDate: 2017-09-15 DOI: 10.1007/s10474-017-0751-1
Authors:Cs. Nagy Abstract: Abstract We consider branched coverings which are simple in the sense that any point of the target has at most one singular preimage. The cobordism classes of k-fold simple branched coverings between n-manifolds form an abelian group \({{\rm Cob}^1(n, k)}\) . Moreover, \({{\rm Cob}^1(*, k) = \bigoplus_{n = 0}^{\infty}{\rm Cob}^1(n, k)}\) is a module over \({\Omega^{SO}_{*}}\) . We construct a universal k-fold simple branched covering, and use it to compute this module rationally. As a corollary, we determine the rank of the groups \({{\rm Cob}^1(n, k)}\) . In the case n = 2 we compute the group \({{\rm Cob}^1(2, k)}\) , give a complete set of invariants and construct generators. PubDate: 2017-09-15 DOI: 10.1007/s10474-017-0755-x
Authors:T. Komatsu; P. Yuan Abstract: Abstract For positive integers N and M, the general hypergeometric Cauchy polynomials c M,N,n (z) (M, N ≥ 1; n ≥ 0) are defined by $$\frac{1}{(1+t)^z} \frac{1}{{}_2F_1(M,N;N+1;-t)}=\sum_{n=0}^\infty c_{M,N,n}(z)\, \frac{t^n}{n!}\,, $$ where \({{}_2 F_1(a,b;c;z)}\) is the Gauss hypergeometric function. When M = N = 1, c n = c 1,1,n are the classical Cauchy numbers. In 1875, Glaisher gave several interesting determinant expressions of numbers, including Bernoulli, Cauchy and Euler numbers. In the aspect of determinant expressions, hypergeometric Cauchy numbers are the natural extension of the classical Cauchy numbers, though many kinds of generalizations of the Cauchy numbers have been considered by many authors. In this paper, we show some interesting expressions of generalized hypergeometric Cauchy numbers. We also give a convolution identity for generalized hypergeometric Cauchy polynomials. PubDate: 2017-09-15 DOI: 10.1007/s10474-017-0744-0
Authors:F. Weisz Abstract: Abstract Under some conditions on \({\theta}\) , we characterize the set of convergence of the Marcinkiewicz- \({\theta\mbox{-}}\) means of a function \({f \in L_1(\mathbb{T}^d)}\) . More exactly, the \({\theta\mbox{-}}\) means converge to f at each modified strong Lebesgue point. The same holds for a weaker version of Lebesgue points, for the so called modified Lebesgue points of \({f \in L_p(\mathbb{T}^d)}\) , whenever \({1 < p < \infty}\) . The \({\theta\mbox{-}}\) summability includes the Fejér, Abel, Cesàro and some other summations. As an application we give simple proofs for the classical one-dimensional strong summability results of Hardy and Littlewood, Marcinkiewicz, Zygmund and Gabisoniya and generalize them for strong \({\theta\mbox{-}}\) summability. PubDate: 2017-09-14 DOI: 10.1007/s10474-017-0737-z
Authors:Z. Masáková; M. Tinková Abstract: Abstract We study finiteness property in numeration systems with cubic Pisot unit base. A base β > 1 is said to satisfy property (F), if the set Fin (β) of numbers with finite β-expansions forms a ring. We show that in every real cubic field which is not totally real, there exists a cubic Pisot unit satisfying (F). On the other hand, there exist totally real cubic fields without such a unit. In such fields, however, one finds a cubic Pisot unit β > 1 satisfying property (−F), i.e., the set Fin (−β) of finite (−β)-expansions forms a ring. PubDate: 2017-09-14 DOI: 10.1007/s10474-017-0757-8
Authors:P. Komjáth Abstract: Abstract Let \({\mu \geq \omega}\) be regular, assume the Generalized Continuum Hypothesis and the principle \({\square_\lambda}\) holds for every singular \({\lambda}\) with \({{\rm cf}(\lambda) \leq \mu}\) . Let X be a graph with chromatic number greater than \({\mu^+}\) . Then X contains a \({\mu}\) -connected subgraph Y of X whose chromatic number is greater than \({\mu^+}\) . PubDate: 2017-09-14 DOI: 10.1007/s10474-017-0752-0
Authors:Z. D. Mitrović; S. Radenović Abstract: Abstract We give a proof for the common fixed point theorem of Jungck in rectangular b-metric spaces. As a corollary, we obtain well known common fixed point theorems in b-metric spaces. PubDate: 2017-09-14 DOI: 10.1007/s10474-017-0750-2
Authors:V. V. Tkachuk Abstract: Abstract Given a sequence \({\mathcal{U} =\{U_n: n \in \omega\}}\) of non-empty open subsets of a space X, a set \({\{x_n : n \in \omega\}}\) is a selection of \({\mathcal{U}}\) if \({x_n \in U_n}\) for every \({n \in \omega}\) . We show that a space X is uncountable if and only if every sequence of non-empty open subsets of C p (X) has a closed discrete selection. The same statement is not true for \({C_p(X,[0,1])}\) so we study when the above selection property (which we call discrete selectivity) holds in \({C_p(X,[0,1])}\) . We prove, among other things, that \({C_p(X, [0,1])}\) is discretely selective if X is an uncountable Lindelöf \({\Sigma}\) -space. We also give a characterization, in terms of the topology of X, of discrete selectivity of \({C_p(X,[0,1])}\) if X is an \({\omega}\) -monolithic space of countable tightness. PubDate: 2017-09-14 DOI: 10.1007/s10474-017-0756-9
Authors:Z. Hao; L. Li Abstract: Abstract We introduce grand Hardy spaces defined on a probability space. Analogous to the classical theory, we prove Doob’s maximal inequality and obtain atomic characterization of grand Hardy martingale spaces. Finally, we investigate the John–Nirenberg theorem in the frame of grand Hardy spaces. PubDate: 2017-09-14 DOI: 10.1007/s10474-017-0741-3
Authors:X.-W. Ma Abstract: Abstract We study the distribution of the sequence of vectors \({(p^{{\alpha}_1},\ldots,p^{{\alpha}_k})}\) modulo one. Here p runs over prime numbers, k ≥ 2 is a fixed integer, α1, . . . , α k are fixed real numbers lying in the interval (s, s + 1) with s being a sufficiently large integer. PubDate: 2017-09-14 DOI: 10.1007/s10474-017-0746-y
Abstract: Abstract Let \({(X, m, \leq)}\) be a partially ordered metric space, that is, a metric space (X, m) equipped with a partial order \({\leq}\) on X. We say that a T 0-quasi-metric d on X is m-splitting provided that \({d\vee d^{-1}=m}\) . Furthermore d is said to be \({(X, m, \leq)}\) -producing provided that d is m-splitting and the specialization partial preorder of d is equal to \({\leq}\) . It is known and easy to see that if \({(X, m, \leq)}\) is a partially ordered metric space that is produced by a T 0-quasi-metric and \({\leq}\) is a total order, then there exists exactly one producing T 0-quasi-metric on X. We first will give an example that shows that a partially ordered metric space can be uniquely produced by a T 0-quasi-metric although \({\leq}\) is not total. Then we present solutions to the following two problems: Let \({(X, m, \leq)}\) be a partially ordered metric space and A a subset of X. (1) Suppose that d is a T 0-quasi-metric on A which is \({m\vert ({A \times A})}\) -splitting. When can d be extended to an m-splitting T 0-quasi-metric \({\widetilde{d}}\) on X' (2) Suppose that d is a T 0-quasi-metric on A which is \({(A, m\vert (A\times A)}\) , \({{\leq}\vert ({A\times A}))}\) -producing. When can d be extended to a T 0-uasi-metric \({\widetilde{d}}\) on X that produces \({(X, m, \leq)'}\) PubDate: 2017-09-08 DOI: 10.1007/s10474-017-0753-z
Abstract: Abstract We use suitably Page’s theorem to get effective results for interesting problems, by avoiding the ineffective Siegel’s theorem. PubDate: 2017-09-07 DOI: 10.1007/s10474-017-0745-z
Abstract: Abstract Over any field \({\mathbb{K}}\) , there is a bijection between regular spreads of the projective space \({{\rm PG}(3,\mathbb{K})}\) and 0-secant lines of the Klein quadric in \({{\rm PG}(5,\mathbb{K})}\) . Under this bijection, regular parallelisms of \({{\rm PG}(3,\mathbb{K})}\) correspond to hyperflock determining line sets (hfd line sets) with respect to the Klein quadric. An hfd line set is defined to be pencilled if it is composed of pencils of lines. We present a construction of pencilled hfd line sets, which is then shown to determine all such sets. Based on these results, we describe the corresponding regular parallelisms. These are also termed as being pencilled. Any Clifford parallelism is regular and pencilled. From this, we derive necessary and sufficient algebraic conditions for the existence of pencilled hfd line sets. PubDate: 2017-09-07 DOI: 10.1007/s10474-017-0742-2
Abstract: Abstract The classical Lebedev index transform [3] involving squares and products of the Legendre functions is generalized on the associated Legendre functions of an arbitrary order. Mapping properties are investigated in the Lebesgue spaces. Inversion formulas are proved. As an application, a solution to the boundary value problem for a third order partial differential equation is obtained. PubDate: 2017-09-07 DOI: 10.1007/s10474-017-0748-9
Abstract: Abstract Homogeneous Besov and Triebel–Lizorkin spaces associated with multi-dimensional Laguerre function expansions of Hermite type with index \({\alpha\in [-1/2,\infty)^d\setminus(-1/2,1/2)^d}\) , \({d\geq 1}\) , are defined and investigated. To achieve expected goals Schwartz type spaces on \({\mathbb{R}^d_+}\) are introduced and then tempered type distributions are constructed. Also, ideas from a recent paper of Bui and Duong on Besov and Triebel–Lizorkin spaces associated with Hermite functions expansions are used. This means, in particular, using molecular decomposition and an appropriate form of a Calderón reproducing formula. PubDate: 2017-09-07 DOI: 10.1007/s10474-017-0747-x
Abstract: Abstract We study the individual behaviour of uniform and nonuniform evolutionary processes. In [2] R. Datko gave a necessary and sufficient condition for the uniform exponential stability of an evolutionary process in Banach space. Our aim is to show that for a single vector x and not global, as Datko did in his paper, the trajectory of an evolutionary process on that vector x is exponentially stable. PubDate: 2017-09-07 DOI: 10.1007/s10474-017-0754-y
Abstract: Abstract Let G be a finite group. A subgroup H of G is called an \({\mathcal{H}}\) -subgroup in G if \({N_{G}(H)\cap H^{g} \leq H}\) for all \({g\in G}\) . A subgroup H of G is called weakly c-supplemented in G if G has a subgroup K such that G = HK and \({ H\cap K}\) is an \({\mathcal{H}}\) -subgroup in G. We investigate the structure of finite groups by means of weakly c-supplemented subgroups. Some recent results about supersolvability of finite groups are generalized to a saturated formation containing the class of all supersolvable groups. PubDate: 2017-09-07 DOI: 10.1007/s10474-017-0738-y
Abstract: Abstract We extend holomorphically polyharmonic functions on a real ball to a complex set being the union of rotated balls. We solve a Dirichlet type problem for complex polyharmonic functions with the boundary condition given on the union of rotated spheres. PubDate: 2017-09-07 DOI: 10.1007/s10474-017-0740-4
Abstract: Abstract Let \({\sigma =\{\sigma_{i} i\in I\}}\) be a partition of the set of all primes \({\mathbb{P}}\) and G a finite group. Then \({\sigma(G)=\{\sigma_{i}\ \ \sigma _{i}\cap \pi (G)\ne \emptyset\}}\) . A set \({\mathcal{H}}\) of subgroups of G is said to be a complete Hall \({\sigma }\) -set of G if every member ≠ 1 of \({\mathcal{H}}\) is a Hall \({\sigma _{i}}\) -subgroup of G for some i, and \({\mathcal{H}}\) contains exactly one Hall \({\sigma _{i}}\) -subgroup of G for every \({{\sigma _{i} \in \sigma (G)}}\) . A group is said to be \({\sigma}\) -primary if it is a finite \({\sigma_{i}}\) -group for some i. A subgroup A of G is said to be \({\sigma}\) -semi-per-mutable in G if G possesses a complete Hall \({\sigma}\) -set \({\mathcal{H}}\) such that AH x = H x A for all x∈G and all \({H\in \mathcal{H}}\) such that \({\sigma (H)\cap \sigma (A)=\emptyset}\) ; \({{\sigma}}\) -abnormal in G if L/K L is not \({\sigma}\) -primary whenever \({{A\leq K < L \leq G}}\) . In this paper, we describe finite groups in which every subgroup is either \({{\sigma}}\) -semipermutable or \({{\sigma}}\) -abnormal in G. PubDate: 2017-09-07 DOI: 10.1007/s10474-017-0743-1
Abstract: Abstract We study the classical Dirichlet problem in the disc with the weighted uniform norm for the weight function \({w(x) = v(x)\prod_{j=1}^s \sin(\frac{x-x_j}{2}) ^{\lambda_j}}\) , \({\{\lambda_{j}\}_{j=1}^{s}}\) are positive numbers and v is a strictly positive continuous function on the circle. Remarkably the problem has solution if and only if none of the numbers \({\{\lambda_{j}\}_{j=1}^{s}}\) is natural. PubDate: 2017-09-07 DOI: 10.1007/s10474-017-0749-8
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