Authors:Z. Masáková; M. Tinková Pages: 318 - 333 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: 2017-12-01 DOI: 10.1007/s10474-017-0757-8 Issue No:Vol. 153, No. 2 (2017)

Authors:H. F. de Lima; A. M. S. Oliveira; M. S. Santos; M. A. L. Velásquez Pages: 334 - 349 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: 2017-12-01 DOI: 10.1007/s10474-017-0731-5 Issue No:Vol. 153, No. 2 (2017)

Authors:D. Głazowska; P. Leonetti; J. Matkowski; S. Tringali Pages: 350 - 355 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: 2017-12-01 DOI: 10.1007/s10474-017-0734-2 Issue No:Vol. 153, No. 2 (2017)

Authors:F. Weisz Pages: 356 - 381 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: 2017-12-01 DOI: 10.1007/s10474-017-0737-z Issue No:Vol. 153, No. 2 (2017)

Authors:T. Komatsu; P. Yuan Pages: 382 - 400 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: 2017-12-01 DOI: 10.1007/s10474-017-0744-0 Issue No:Vol. 153, No. 2 (2017)

Authors:Z. D. Mitrović; S. Radenović Pages: 401 - 407 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: 2017-12-01 DOI: 10.1007/s10474-017-0750-2 Issue No:Vol. 153, No. 2 (2017)

Authors:Z. Hao L. Li Pages: 417 - 429 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: 2017-12-01 DOI: 10.1007/s10474-017-0741-3 Issue No:Vol. 153, No. 2 (2017)

Authors:P. Kórus Pages: 430 - 435 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: 2017-12-01 DOI: 10.1007/s10474-017-0735-1 Issue No:Vol. 153, No. 2 (2017)

Authors:M.-J. Deng; J. Guo Pages: 436 - 448 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: 2017-12-01 DOI: 10.1007/s10474-017-0751-1 Issue No:Vol. 153, No. 2 (2017)

Authors:Cs. Nagy Pages: 449 - 489 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: 2017-12-01 DOI: 10.1007/s10474-017-0755-x Issue No:Vol. 153, No. 2 (2017)

Authors:X.-W. Ma Pages: 524 - 536 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: 2017-12-01 DOI: 10.1007/s10474-017-0746-y Issue No:Vol. 153, No. 2 (2017)

Authors:S. De Marchi; A. Kroó Abstract: We investigate Marcinkiewicz–Zygmund type inequalities for multivariate polynomials on various compact domains in \({\mathbb{R}^d}\) . These inequalities provide a basic tool for the discretization of the L p norm and are widely used in the study of the convergence properties of Fourier series, interpolation processes and orthogonal expansions. Recently Marcinkiewicz–Zygmund type inequalities were verified for univariate polynomials for the general class of doubling weights, and for multivariate polynomials on the ball and sphere with doubling weights. The main goal of the present paper is to extend these considerations to more general multidimensional domains, which in particular include polytopes, cones, spherical sectors, toruses, etc. Our approach will rely on application of various polynomial inequalities, such as Bernstein–Markov, Schur and Videnskii type estimates, and also using symmetry and rotation in order to generate results on new domains. PubDate: 2017-10-20 DOI: 10.1007/s10474-017-0769-4

Authors:C. L. Samuels Abstract: For an algebraic number α, the metric Mahler measure \({m_1(\alpha)}\) was first studied by Dubickas and Smyth [4] and was later generalized to the t-metric Mahler measure \({m_t(\alpha)}\) by the author [16]. The definition of \({m_t(\alpha)}\) involves taking an infimum over a certain collection N-tuples of points in \(\overline{\mathbb{Q}}\) , and from previous work of Jankauskas and the author, the infimum in the definition of \({m_t(\alpha)}\) is attained by rational points when \({\alpha\in \mathbb{Q}}\) . As a consequence of our main theorem in this article, we obtain an analog of this result when \({\mathbb{Q}}\) is replaced with any imaginary quadratic number field of class number equal to 1. Further, we study examples of other number fields to which our methods may be applied, and we establish various partial results in those cases. PubDate: 2017-10-20 DOI: 10.1007/s10474-017-0770-y

Authors:F. Tugores; L. Tugores Abstract: We pose and solve two interpolation problems for bounded analytic functions in the unit disc. In the first, we require interpolation by some derivative and prove that the corresponding interpolating sequences are the uniformly separated ones. In the second, we interpolate by the function or some derivative on each of two disjoint subsequences whose union is the given sequence, proving that it is possible if and only if both subsequences are uniformly separated. PubDate: 2017-10-20 DOI: 10.1007/s10474-017-0772-9

Authors:S. Durst; M. Kegel Abstract: There was a minor mistake in the formula for computing the Poincarédual of the Euler class of the contact structure in Theorem 5.1(1). PubDate: 2017-10-09 DOI: 10.1007/s10474-017-0759-6

Authors:X. J. Wang Abstract: Let \({\{X_{n}, n\geq1\}}\) be a sequence of random variables with \({S_n=\sum_{i=1}^nX_i}\) and \({M_n=\max \{X_1,X_2,\ldots, X_n\}}\) . Under some suitable conditions, we establish the upper bound of large deviations for \({S_n}\) and \({M_n}\) based on some dependent sequences including acceptable random variables, widely acceptable random variables and a class of random variables that satisfies the Marcinkiewicz–Zygmund type inequality and Rosenthal type inequality. In addition, the lower bound of large deviations for some dependent sequences is also obtained. The results obtained in the paper generalize and improve some corresponding ones for independent random variables and negatively associated random variables. PubDate: 2017-10-09 DOI: 10.1007/s10474-017-0764-9

Authors:D. Wang Abstract: We study two general divisor problems related to Hecke eigenvalues of classical Maass cusp forms. We give the relevant estimation and improve previous results. PubDate: 2017-10-09 DOI: 10.1007/s10474-017-0763-x

Authors:M. Stojanović Abstract: We investigate and partially classify supergroups of some hyperbolic space groups. Fundamental domains of the considered groups are truncated tetrahedra (trunc-simplices) obtained by truncating tetrahedra which belong to the family F2, by the previous notation given in [9] and recited here in figures. PubDate: 2017-10-09 DOI: 10.1007/s10474-017-0761-z

Authors:W. H. Sun; J. C. Wu; X. Zhang Abstract: Monotone normality in generalized topological spaces is introduced. The characterizations and several preservation theorems of μ-monotonically normal spaces are given and the “monotone variant” of Urysohn’s lemma for μ-monotonically normal spaces is established. PubDate: 2017-10-09 DOI: 10.1007/s10474-017-0762-y

Authors:V. M. Phung Abstract: We construct new Hermite interpolation schemes in \({\mathbb{R}^2}\) , and Lagrange and Hermite interpolation schemes on the unit sphere in \({\mathbb{R}^3}\) . We show that the Hermite projectors in \({\mathbb{R}^2}\) are the limits of Lagrange projectors at Bos configurations distributed on lines and circles in \({\mathbb{R}^2}\) . Using this result we prove that the Lagrange projectors on the unit sphere converge to the Hermite projectors when the interpolation points coalesce. PubDate: 2017-10-09 DOI: 10.1007/s10474-017-0760-0