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 Acta Mathematica Hungarica   [SJR: 0.53]   [H-I: 29]   [2 followers]  Follow         Hybrid journal (It can contain Open Access articles)    ISSN (Print) 1588-2632 - ISSN (Online) 0236-5294    Published by Springer-Verlag  [2355 journals]
• Finiteness in real real cubic fields
• Authors: Z. Masáková; M. Tinková
Pages: 318 - 333
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: 2017-12-01
DOI: 10.1007/s10474-017-0757-8
Issue No: Vol. 153, No. 2 (2017)

• A note on Jeśmanowicz’ conjecture concerning primitive
Pythagorean triples. II
• Authors: M.-J. Deng; J. Guo
Pages: 436 - 448
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: 2017-12-01
DOI: 10.1007/s10474-017-0751-1
Issue No: Vol. 153, No. 2 (2017)

• Clusters of varieties of completely regular semigroups
• Authors: M. Petrich
Abstract: Abstract The class $${\mathcal{CR}}$$ of completely regular semigroups equipped with the unary operation of inversion forms a variety whose lattice of subvarieties is denoted by $${\mathcal{L(CR)}}$$ . The variety $${\mathcal B}$$ of all bands induces two relations $${\mathbf{B}^{\land}}$$ and $${\mathbf{B}^{\lor} }$$ by meet and join with $${\mathcal B}$$ . Their classes are intervals with lower ends $${\mathcal V_{B^{\land}}}$$ and $${\mathcal V_{B^{\lor} }}$$ , and upper ends $${\mathcal V^{B^{\land}}}$$ and $${\mathcal V^{B^{\lor} } }$$ . These objects induce four operators on $${\mathcal{L(CR)}}$$ . The cluster at a variety $${\mathcal V}$$ is the set of all varieties obtained from $${\mathcal V}$$ by repeated application of these four operators. We identify the cluster at any variety in $${\mathcal{L(CR)}}$$ .
PubDate: 2017-12-21
DOI: 10.1007/s10474-017-0784-5

• Nonlinear *-Lie-type derivations on standard operator algebras
• Authors: W. Lin
Abstract: Abstract Let $$\mathcal{H}$$ be an infinite dimensional complex Hilbert space and $$\mathcal{A}$$ be a standard operator algebra on $$\mathcal{H}$$ which is closed under the adjoint operation. It is shown that each nonlinear *-Lie-type derivation δ on $$\mathcal{A}$$ is a linear *-derivation. Moreover, δ is an inner *-derivation as well.
PubDate: 2017-12-15
DOI: 10.1007/s10474-017-0783-6

• On the stability of a generalization of Jensen functional equation
• Authors: M. Almahalebi
Abstract: Abstract Using the fixed point method, we investigate the stability of a generalization of Jensen functional equation $$\sum_{k=0}^{n-1} f(x+ b_{k}y)=nf(x),$$ where $${n \in \mathbb{N}_{2}}$$ , $${b_{k}=\exp(\frac{2i\pi k}{n})}$$ for $${0\leq k \leq n-1}$$ , in Banach spaces. Also, we prove the hyperstability results of this equation by the fixed point method.
PubDate: 2017-12-15
DOI: 10.1007/s10474-017-0781-8

• Polynomial rings over commutative reduced Hopfian local rings
• Authors: A. M. Dhorajia; H. Mukherjee
Abstract: Abstract We prove that if R is a commutative, reduced, local ring, then R is Hopfian if and only if the ring R[x] is Hopfian. This answers a question of Varadarajan [16], in the case when R is a reduced local ring. We provide examples of non-Noetherian Hopfian commutative domains by proving that the finite dimensional domains are Hopfian. Also, we derive some general results related to Hopfian rings.
PubDate: 2017-12-15
DOI: 10.1007/s10474-017-0782-7

• $${G_\delta}$$ G δ covers of compact spaces
• Authors: S. Spadaro; P. Szeptycki
Abstract: Abstract We solve a long standing question due to Arhangel’skii by constructing a compact space which has a $${G_\delta}$$ cover with no continuum-sized ( $${G_\delta}$$ )-dense subcollection. We also prove that in a countably compact weakly Lindelöf normal space of countable tightness, every $${G_\delta}$$ cover has a $${\mathfrak{c}}$$ -sized subcollection with a $${G_\delta}$$ -dense union and that in a Lindelöf space with a base of multiplicity continuum, every $${G_\delta}$$ cover has a continuum sized subcover. We finally apply our results to obtain a bound on the cardinality of homogeneous spaces which refines De la Vega’s celebrated theorem on the cardinality of homogeneous compacta of countable tightness.
PubDate: 2017-12-15
DOI: 10.1007/s10474-017-0785-4

• Preserved under Sacks forcing again'
• Authors: Y. Y. Zheng
Abstract: Abstract A hierarchy of topological Ramsey spaces $${\mathcal{R}_\alpha}$$ ( $${\alpha < \omega_1}$$ ), generalizing the Ellentuck space, were built by Dobrinen and Todorcevic in order to completely classify certain equivalent classes of ultrafilters Tukey (resp. Rudin–Keisler) below $${\mathcal{U}_\alpha}$$ $${(\alpha < \omega_1)}$$ , where $${\mathcal{U}_\alpha}$$ are ultrafilters constructed by Laflamme satisfying certain partition properties and have complete combinatorics over the Solovay model. We show that Nash–Williams, or Ramsey ultrafilters in these spaces are preserved under countable-support side-by-side Sacks forcing. This is achieved by proving a parametrized theorem for these spaces, and showing that Nash–Williams ultrafilters localizes the theorem. We also show that every Nash–Williams ultrafilter in $${\mathcal{R}_\alpha}$$ is selective.
PubDate: 2017-12-14
DOI: 10.1007/s10474-017-0780-9

• Approximation orders of the unit in the β -dynamical systems
• Authors: C.-Y. Cao; Y.-H. Chen
Abstract: Abstract For any real number β > 1, let S n (β) be the partial sum of the first n items of the β-expansion of 1. It was known that the approximation order of 1 by S n (β) is β −n for Lebesgue almost all β > 1. We consider the size of the set of β > 1 for which 1 can be approximated with the other orders $${\beta^{-\varphi(n)}}$$ , where $${\varphi}$$ is a positive function defined on $${\mathbb N}$$ . More precisely, the size of the sets $$\left\{\beta\in \mathfrak{B}:\limsup_{n\rightarrow\infty}\frac{\log_{\beta}(1-S_n(\beta))}{\varphi(n)}=-1\right\}$$ and $$\left\{\beta\in \mathfrak{B}:\liminf_{n\rightarrow\infty}\frac{\log_{\beta}(1-S_n(\beta))}{\varphi(n)}=-1\right\}$$ are determined, where $${\mathfrak{B}=\{ \beta>1:\beta \text{ is not a simple Parry number}\}}$$ .
PubDate: 2017-12-14
DOI: 10.1007/s10474-017-0776-5

• The imaginary part of the characteristic function
• Authors: S. Norvidas
Abstract: Abstract We consider the conditions under which a continuous function $${\varphi \colon {\mathbb{R}}^n \to \mathbb {R}}$$ is the imaginary part $${\Im f}$$ of the characteristic function f of a probability measure on $${{\mathbb{R}}^n}$$ . A similar problem about such an $${\varphi}$$ that it is the argument of the characteristic function was solved by Ilinskii [Theory Probab. Appl. 20 (1975), 410–415]. In this paper, a characterization of what $${\varphi}$$ might serve as the imaginary part of the characteristic function f is given. As a consequence, we provide an answer to the following question posed by N. G. Ushakov [7]: Is it true that f is never determined by its imaginary part $${\Im f}$$ ' In other words, is it true that for any characteristic function f there exists a characteristic function g such that $${\Im f\equiv \Im g}$$ but $${ f\not\equiv g}$$ ' We prove that the answer to this question is negative. In addition, several examples of characteristic functions which are uniquely determined by their imaginary parts are given.
PubDate: 2017-12-14
DOI: 10.1007/s10474-017-0779-2

• On the generalization of the Salem test
Abstract: Abstract The assertion that the Salem test [5] for the uniform convergence of a trigonometric Fourier series is improvable, is proved. In particular, an example of a continuous function, which does not fulfill the condition of the Salem test but satisfies the condition of the generalized Salem test [10], is constructed. Besides, the theorem which improves Golubov’s [3,4] result for continuous functions of two variables, is given.
PubDate: 2017-12-14
DOI: 10.1007/s10474-017-0774-7

• Improved bounds on the diameter of lattice polytopes
• Authors: A. Deza; L. Pournin
Abstract: Abstract We show that the largest possible diameter $${\delta(d,k)}$$ of a d-dimensional polytope whose vertices have integer coordinates ranging between 0 and k is at most $${kd - \lceil2d/3\rceil-(k-3)}$$ when $${k\geq3}$$ . In addition, we show that $${\delta(4,3)=8}$$ . This substantiates the conjecture whereby $${\delta(d,k)}$$ is at most $${\lfloor(k+1)d/2\rfloor}$$ and is achieved by a Minkowski sum of lattice vectors.
PubDate: 2017-12-14
DOI: 10.1007/s10474-017-0777-4

• Duplication of a module along an ideal
• Authors: E. M. Bouba; N. Mahdou; M. Tamekkante
Abstract: Abstract Let A be a commutative ring and I an ideal of A. The amalgamated duplication of A along I, denoted by $${A \bowtie I}$$ , is the special subring of $${A \times A}$$ defined by $${A \bowtie I } := \pi \times_{\frac{A}{I}} \pi = \{(a, a + i) \mid a \in A, i \in I\}$$ . We are interested in some basic and homological properties of a special kind of $${A \bowtie I}$$ -modules, called the duplication of M along I with M is an A-module, and defined by $${M \bowtie I := \{(m, m') \in M \times M \mid m - m^{\prime} \in IM\}}$$ . The new results generalize some results on amalgamated duplication of a ring along an ideal.
PubDate: 2017-12-14
DOI: 10.1007/s10474-017-0775-6

• Symmetry and measurability
• Authors: D. Fremlin; J. Mala
Abstract: Abstract In the Hewitt–Savage 0-1 law, symmetric measurable sets are considered in a countable product of a σ-algebra Σ with itself. Therefore, it may be of interest to find simple generators for these sets in terms of the generators of Σ. We put the problem into a more general framework of action groups and as an application find simple generators for the finite product case. We also have a partial result in the countable product case.
PubDate: 2017-12-14
DOI: 10.1007/s10474-017-0778-3

• On the Nörlund logarithmic means with respect to Vilenkin system in the
martingale Hardy space H 1
• Authors: L.-E. Persson; G. Tephnadze; P. Wall
Abstract: Abstract We prove and discuss a new divergence result of Nörlund logarithmic means with respect to Vilenkin system in Hardy space H 1.
PubDate: 2017-12-14
DOI: 10.1007/s10474-017-0773-8

• Interpolation by derivatives in $${H^\infty}$$ H ∞
• Authors: F. Tugores; L. Tugores
Abstract: 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

• Erratum to: Computing rotation and self-linking numbers in contact surgery
diagram
• 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

• Upper and lower bounds of large deviations for some dependent sequences
• Authors: X. J. Wang
Abstract: 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

• On general divisor problems involving Hecke eigenvalues
• Authors: D. Wang
Abstract: 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

• Hyperbolic space groups and their supergroups for fundamental simplex
tilings
• Authors: M. Stojanović
Abstract: 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

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