# Calculating different topological indices of total graph of ℤ n

Authors: M. Nikmehr, L. Heidarzadeh and N. Soleimani

A recently published paper [6] considered the total graph of commutative ring R. In this paper, we compute Wiener, hyper-Wiener, reverse Wiener, Randić, Zagreb, ABC and GA indices of zero-divisor graph.

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# A classifier for simple space curve singularities

Authors: Faira Janjua and Gerhard Pfister

The classification of Bruce and Gaffney respectively Gibson and Hobbs for simple plane curve singularities respectively simple space curve singularities is characterized in terms of invariants. This is the basis for the implementation of a classifier in the computer algebra system singular.

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# Convergence of logarithmic means of multiple Walsh-Fourier series

Authors: György Gát and Ushangi Goginava

The maximal Orlicz spaces such that the mixed logarithmic means of multiple Walsh-Fourier series for the functions from these spaces converge in measure and in norm are found.

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# Erdös-Ko-Rado theorem in some linear groups and some projective special linear group

Authors: Milad Ahanjideh and Neda Ahanjideh

Let V be the 2-dimensional column vector space over a finite field

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q$$ \end{document}
(where q is necessarily a power of a prime number) and let ℙq be the projective line over
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q$$ \end{document}
. In this paper, it is shown that GL 2(q), for q ≠ 3, and SL 2(q) acting on V − {0} have the strict EKR property and GL 2(3) has the EKR property, but it does not have the strict EKR property. Also, we show that GL n(q) acting on
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left( {\mathbb{F}_q } \right)^n - \left\{ 0 \right\}$$ \end{document}
has the EKR property and the derangement graph of PSL 2(q) acting on ℙq, where q ≡ −1 (mod 4), has a clique of size q + 1.

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# Generalized ideal transforms

Authors: Tran Nam and Nguyen Tri

We study basic properties of the generalized ideal transforms D I (M, N) and the set of associated primes of the modules R i D I (M, N).

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# The influence of weakly \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{H}$$ \end{document}-subgroups on the structure of finite groups

Authors: M. Asaad, M. Al-Shomrani and A. Heliel

Let G be a finite group. A subgroup H of G is called an

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{H}$$ \end{document}
-subgroup in G if N G(H) ∩ H gH for all gG. A subgroup H of G is called a weakly
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{H}$$ \end{document}
-subgroup in G if there exists a normal subgroup K of G such that G = HK and HK is an
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{H}$$ \end{document}
-subgroup in G. In this article, we investigate the structure of a group G in which every subgroup with order p m of a Sylow p-subgroup P of G is a weakly
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{H}$$ \end{document}
-subgroup in G, where m is a fixed positive integer. Our results improve and extend the main results of Skiba [13], Jaraden and Skiba [11], Guo and Wei [8], Tong-Veit [15] and Li et al. [12].

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# A new generalization of the logarithmic series distribution

Author: A. Mishra

A new generalization of the logarithmic series distribution has been obtained as a limiting case of the zero-truncated Mishra’s [10] generalized negative binomial distribution (GNBD). This distribution has an advantage over the Mishra’s [9] quasi logarithmic series distribution (QLSD) as its moments appear in compact forms unlike the QLSD. This makes the estimation of parameters easier by the method of moments. The first four moments of this distribution have been obtained and the distribution has been fitted to some well known data-sets to test its goodness of fit.

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# A note on a Turán-type extremal question on 3-graphs

Author: Vajk Széecsi

An upper bound is given on the size of a k-fan-free 3-graph, and an infinite family reaching this bound is also described.

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# On a second order coercive dirichlet problem with a non-differentiable action functional

Author: Marek Galewski

Applying certain convexity arguments we investigate the existence of a classical solution for a Dirichlet problem for which the Euler action functional is not necessarily differentiable in the sense of Gâteaux.

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# On semifields of order q 4 with center \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q$$ \end{document}, admitting a Klein 4-group of automorphisms

Authors: Mashhour Bani-Ata and Ra’ed Al-Nouty

The aim of this paper is to investigate the semifields of order q 4 over a finite field of order q, q an odd prime power, admitting a Klein 4-group of automorphisms.

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# On the maximal operators of Riesz logarithmic means of Vilenkin-Fourier series

The main aim of this paper is to investigate (H p, L p) and (H p, L p,∞) type inequalities for maximal operators of Riesz logarithmic means of one-dimensional Vilenkin—Fourier series.

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# Riesz decomposition for super-polyharmonic functions in the punctured unit ball

Authors: Toshihide Futamura, Yoshihiro Mizuta and Takao Ohno

We consider a Riesz decomposition theorem for super-polyharmonic functions satisfying certain growth condition on surface integrals in the punctured unit ball. We give a condition that super-polyharmonic functions u have the bound

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$u\left( x \right) = O\left( {\mathcal{R}_2 \left( x \right)} \right),$$ \end{document}
where
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{R}_2$$ \end{document}
denotes the fundamental solution for −Δu in ℝn.

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# On Devaney chaotic generalized shift dynamical systems

Authors: Fatemah Shirazi, Javad Sarkooh and Bahman Taherkhani

In the following text we prove that in a generalized shift dynamical system (X Г, σ φ) for infinite countable Г and discrete X with at least two elements the following statements are equivalent:

1. the dynamical system (X Г, σ φ) is chaotic in the sense of Devaney
2. the dynamical system (X Г, σ φ) is topologically transitive
3. the map φ: Г → Г is one to one without any periodic point.
Also for infinite countable Г and finite discrete X with at least two elements (X Г, σ φ) is exact Devaney chaotic, if and only if φ: Г → Г is one to one and φ: Г → Г has niether periodic points nor φ-backwarding infinite sequences.

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# C-coherent rings, C-semihereditary rings and C-regular rings

Author: Zhanmin Zhu

Let C be a class of some finitely presented left R-modules. A left R-module M is called C-injective, if ExtR 1(C, M) = 0 for each CC. A right R-module M is called C-flat, if Tor1 R(M, C) = 0 for each CC. A ring R is called C-coherent, if every CC is 2-presented. A ring R is called C-semihereditary, if whenever 0 → KPC → 0 is exact, where CC and P is finitely generated projective and K is finitely generated, then K is also projective. A ring R is called C-regular, if whenever P/KC, where P is finitely generated projective and K is finitely generated, then K is a direct summand of P. Using the concepts of C-injectivity and C-flatness of modules, we present some characterizations of C-coherent rings, C-semihereditary rings, and C-regular rings.

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# Bernstein-type operators which preserve exactly two test functions

Authors: Ovidiu Pop, Dan Bǎrbosu and Petru Braica

A general class of linear and positive operators dened by nite sum is constructed. Some of their approximation properties, including a convergence theorem and a Voronovskaja-type theorem are established. Next, the operators of the considered class which preserve exactly two test functions from the set {e 0, e 1, e 2} are determined. It is proved that the test functions e 0 and e 1 are preserved only by the Bernstein operators, the test functions e 0 and e 2 only by the King operators while the test functions e 1 and e 2 only by the operators recently introduced by P. I. Braica, O. T. Pop and A. D. Indrea in [4].

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# The beta generalized half-normal geometric distribution

Authors: Thiago Ramires, Edwin Ortega, Gauss Cordeiro and Gholamhoss Hamedani

The beta generalized half-normal distribution is commonly used to model lifetimes. We propose a new wider distribution called the beta generalized half-normal geometric distribution, whose failure rate function can be decreasing, increasing or upside-down bathtub. Its density function can be expressed as a linear combination of beta generalzed half-normal density functions. We derive quantile function, moments and generating unction. We characterize the proposed distribution using a simple relationship between wo truncated moments. The method of maximum likelihood is adapted to estimate the model parameters and its potentiality is illustrated with an application to a real fatigue data set. Further, we propose a new extended regression model based on the logarithm of the new distribution. This regression model can be very useful for the analysis of real data and provide more realistic fits than other special regression models.

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Authors: Zoltán Sebestyén and Zsigmond Tarcsay

The purpose of this paper is to revise von Neumann’s characterizations of selfadjoint operators among symmetric ones. In fact, we do not assume that the underlying Hilbert space is complex, nor that the corresponding operator is densely defined, moreover, that it is closed. Following Arens, we employ algebraic arguments instead of the geometric approach of von Neumann using the Cayley transform.

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# Comment on the paper of J. Mills: “certain congruences on orthodox semigroups”

Author: Roman Gigoń

In the paper we give some remarks on the article of Janet Mills. In particular, the proof of Lemma 1.2 (in her work) is incorrect, and so the proof of Theorem 3.5 is not valid, too. Using different methods we show the mentioned theorem. Moreover, we find a new equivalent condition to the statements in Theorem 3.5. In particular, an explicit definition of a new class of orthodox semigroups is introduced.

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# Empirical results on distance of two-dimensional samples

Author: Csaba Noszály

The distance of two-dimensional samples is studied. The distance of two samples is based on the optimal matching method. Simulation results are obtained when the samples are drawn from normal and uniform distributions.

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# On app skew generalized power series rings

Authors: A. Majidinya and A. Moussavi

By [12], a ring R is left APP if R has the property that “the left annihilator of a principal ideal is pure as a left ideal”. Equivalently, R is a left APP-ring if R modulo the left annihilator of any principal left ideal is flat. Let R be a ring, (S, ≦) a strictly totally ordered commutative monoid and ω: S → End(R) a monoid homomorphism. Following [16], we show that, when R is a (S, ω)-weakly rigid and (S, ω)-Armendariz ring, then the skew generalized power series ring R[[S , ω]] is right APP if and only if r R(A) is S-indexed left s-unital for every S-indexed generated right ideal A of R. We also show that when R is a (S, ω)-strongly Armendariz ring and ω(S) ⫅ Aut(R), then the ring R[[S , ω]] is left APP if and only if R(∑a As S s(a)) is S-indexed right s-unital, for any S-indexed subset A of R. In particular, when R is Armendariz relative to S, then R[[S ]] is right APP if and only if r R(A) is S-indexed left s-unital, for any S-indexed generated right ideal A of R.

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# On the distribution of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\sqrt p$$ \end{document} modulo one involving primes of special type

Author: Yingchun Cai

Let P r denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper we show that the inequality

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left\{ {\sqrt p } \right\} < p^{ - \tfrac{1} {{15.5}}}$$ \end{document}
has infinitely many solutions in primes p such that p + 2 = P 4.

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# Paths on the doubly covered region of a covering of the plane by unit discs

Given a covering of the plane by closed unit discs

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{F}$$ \end{document}
and two points A and B in the region doubly covered by
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{F}$$ \end{document}
, what is the length of the shortest path connecting them that stays within the doubly covered region? This is a problem of G. Fejes-Tóth and he conjectured that if the distance between A and B is d, then the length of this path is at most
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\sqrt {2d} + O(1)$$ \end{document}
. In this paper we give a bound of 2.78d + O(1).

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# Shortest path avoiding balls

Author: Gábor Tóth

It is shown that in a packing of open circular discs with radii not exceeding 1, any two points lying outside the circles at distance d from one another can be connected by a path traveling outside the circles and having length at most

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\tfrac{4} {\pi }d + O\left( {\sqrt d } \right)$$ \end{document}
. Given a packing of open balls with bounded radii in E n and two points outside the balls at distance d from one another, the length of the shortest path connecting the two points and avoiding the balls is d + O(d/n) as d and n approaches infinity.

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# Approximation by complex modified Szász-Mirakjan operators

Authors: Nursel Çetin and Nurhayat İspir

In this study, we investigate approximation properties and obtain Voronovskaja type results for complex modified Szász-Mirakjan operators. Also, we estimate the exact orders of approximation in compact disks and prove that the complex modified Szász-Mirakjan operators attached to an analytic function preserve the univalence, starlikeness, convexity and spirallikeness in the unit disk.

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# Linearization of the products of the generalized Lauricella polynomials and the multivariate Laguerre polynomials via their integral representations

Authors: Shuoh-Jung Liu, Shy-Der Lin, Han-Chun Lu and H. Srivastava

In this paper, the authors investigate the linearization problems associated with two families of generalized Lauricella polynomials of the first and second kinds. By means of their multiple integral representations, it is shown how one can linearize the product of two different members of each of these two families of the generalized Lauricella polynomials. Upon suitable specialization of the main results presented in this paper, the corresponding integral representations are deduced for such familiar classes of multivariable hypergeometric polynomials as (for example) the Lauricella polynomials F A (r) in r variables, the Appell polynomials F 2 in two variables and the multivariable Laguerre polynomials. Each of these integral representations, which are derived as special cases of the main results in this paper, may also be viewed as a linearization relationship for the product of two different members of the associated family of multivariable hypergeometric polynomials.

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# On absolutely nonmeasurable homomorphisms of commutative groups

Author: A. Kharazishvili

We give a characterization of all those commutative groups which admit at least one absolutely nonmeasurable homomorphism into the real line (or into the one-dimensional torus). These are exactly those commutative groups (G, +) for which the quotient group G/G 0 is uncountable, where G 0 denotes the torsion subgroup of G.

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# On the extensibility of D(−1)-triples {1, b, c} in the ring \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{Z}\left[ {\sqrt { - t} } \right]$$ \end{document}, t > 0

Author: Ivan Soldo

Let b = 2, 5, 10 or 17 and t > 0. We study the existence of D(−1)-quadruples of the form {1, b, c, d} in the ring \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{Z}\left[ {\sqrt { - t} } \right]$$ \end{document}. We prove that if {1, b, c} is a D(−1)-triple in \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{Z}\left[ {\sqrt { - t} } \right]$$ \end{document}, then c is an integer. As a consequence of this result, we show that for t ∉ {1, 4, 9, 16} there does not exist a subset of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{Z}\left[ {\sqrt { - t} } \right]$$ \end{document} of the form {1, b, c, d} with the property that the product of any two of its distinct elements diminished by 1 is a square of an element in \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{Z}\left[ {\sqrt { - t} } \right]$$ \end{document}.

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# On the minimum cardinality of a planar point set containing two disjoint convex polygons

Authors: Liping Wu and Wanbing Lu

Let N(k, l) be the smallest positive integer such that any set of N(k, l) points in the plane, no three collinear, contains both a convex k-gon and a convex l-gon with disjoint convex hulls. In this paper, we prove that N(3, 4) = 7, N(4, 4) = 9, N(3, 5) = 10 and N(4, 5) = 11.

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# The complements in an affine group

Author: Pál Hegedűs

In this paper we analyse the natural permutation module of an affine permutation group. For this the regular module of an elementary Abelian p-group is described in detail. We consider the inequivalent permutation modules coming from nonconjugate complements. We prove their strong structural similarity well exceeding the fact that they have equal Brauer characters.

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# Diminution of the convergence classes of divergent permutations

Author: Roman Wituła

The purpose of this paper is to investigate the relations of incomparability between so called convergence classes of the permutations of ℕ. The convergence class of any permutation p of ℕ, denoted by Σ(p), is defined to be the family of all real series Σa n such that both Σa n and Σa p(n) are convergent. A permutation p of ℕ is called a divergent permutation if there exists a conditionally convergent real series Σa n such that the p-rearranged series Σa p(n) is divergent.It is proved that for every divergent permutation p of ℕ there exists a family \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathfrak{F}$ \end{document}(p) of divergent permutations of ℕ such that card \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathfrak{F}$ \end{document}(p) = \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathfrak{c}$ \end{document} and for every q\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathfrak{F}$ \end{document}(p) the family Σ(q) is a proper subset of Σ(p) and, furthermore, Σ(q 1)\Σ(q 2) ≠ ∅ and Σ(q 2)\Σ(q 1) ≠ ∅ whenever q 1; q 2\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathfrak{F}$ \end{document}(p) are different. Permutations q 1, q 2 of ℕ satisfying the above relations are called the incomparable permutations.This result, like many other results of the paper, is given in more general context resulting from the more subtle discussion on the subfamilies of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathfrak{P}$ \end{document} and concepts of incomparability of the families of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathfrak{P}$ \end{document}.

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# Inclusion ideals associated to uniformly increasing hypergraphs

Authors: I. Anwar, S. Ahmad, A. Inam and A. Haider

In this paper, we introduce inclusion ideals I(H) associated to a special class of non uniform hypergraphs H(gC; ɛ; d), namely, the uniformly increasing hypergraphs. We discuss some algebraic properties of the inclusion ideals. In particular, we give an upper bound of the Castelnouvo-Mumford regularity of the special dual ideal I [*](H).

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# On the csáki-vincze transformation

Author: Hatem Hajri

Csáki and Vincze have defined in 1961 a discrete transformation T which applies to simple random walks and is measure preserving. In this paper, we are interested in ergodic and asymptotic properties of T. We prove that T is exact: ∩k≧1 σ(T k(S)) is trivial for each simple random walk S and give a precise description of the lost information at each step k. We then show that, in a suitable scaling limit, all iterations of T “converge” to the corresponding iterations of the continuous Lévy transform of Brownian motion.

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# On the Móri-Székely conjectures for the Borel-Cantelli lemma

Authors: Chunrong Feng and Liangpan Li

The purpose of this note is to show by constructing counterexamples that two conjectures of Móri and Székely for the Borel-Cantelli lemma are false.

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# On the variety of strict pseudosemilattices

Authors: K. Auinger and L. Oliveira

A new model, in terms of finite bipartite graphs, of the free pseudosemilattice is presented. This will then be used to obtain several results about the variety SPS of all strict pseudosemilattices: (i) an identity basis for SPS is found, (ii) SPS is shown to be inherently non-finitely based, (iii) SPS is shown to have no irredundant identity basis, and (iv) SPS is shown to have no covers and to be ∩-prime in the lattice of all varieties of pseudosemilattices. Some applications to e-varieties of locally inverse semigroups are also derived.

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# On the volume product of planar polar convex bodies — Lower estimates with stability

Authors: K. Böröczky, E. Makai, M. Meyer and S. Reisner

Let K ⊂ ℝ2 be an o-symmetric convex body, and K* its polar body. Then we have |K| · |K*| ≧ 8, with equality if and only if K is a parallelogram. (|·| denotes volume). If K ⊂ ℝ2 is a convex body, with o ∈ int K, then |K| · |K*| ≧ 27/4, with equality if and only if K is a triangle and o is its centroid. If K ⊂ ℝ2 is a convex body, then we have |K| · |[(KK)/2)]*| ≧ 6, with equality if and only if K is a triangle. These theorems are due to Mahler and Reisner, Mahler and Meyer, and to Eggleston, respectively. We show an analogous theorem: if K has n-fold rotational symmetry about o, then |K| · |K*| ≧ n 2 sin2(π/n), with equality if and only if K is a regular n-gon of centre o. We will also give stability variants of these four inequalities, both for the body, and for the centre of polarity. For this we use the Banach-Mazur distance (from parallelograms, or triangles), or its analogue with similar copies rather than affine transforms (from regular n-gons), respectively. The stability variants are sharp, up to constant factors. We extend the inequality |K| · |K*| ≧ n 2 sin2(π/n) to bodies with o ∈ int K, which contain, and are contained in, two regular n-gons, the vertices of the contained n-gon being incident to the sides of the containing n-gon. Our key lemma is a stability estimate for the area product of two sectors of convex bodies polar to each other. To several of our statements we give several proofs; in particular, we give a new proof for the theorem of Mahler-Reisner.

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# Symplectic Kloosterman sums

Author: Árpád Tóth

We give optimal bounds for Kloosterman sums that arise in the estimation of Fourier coefficients of Siegel modular forms of genus 2.

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# Characterization of multidimensional A-strong convergence

Author: Mehmet Ünver

Recently Khan and Orhan have proved that an ordinary (single) sequence is A-strongly convergent if and only if it is A-statistically convergent and A-uniformly integrable. In this paper we consider the similar problem for multidimensional sequences when A is a multivariable-to-single matrix. We also study the same question when A is a multivariable-to-multivariable matrix.

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# Characterization of the convergence of weighted averages in a more general setting

Authors: Ferenc Móricz and Ulrich Stadtmüller

Let ν be a positive Borel measure on ℝ̄+:= [0;∞) and let p: ℝ̄+ → ℝ̄+ be a weight function which is locally integrable with respect to ν. We assume that \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $P(t): = \int\limits_0^t {p(u)d\nu (u) \to \infty } andP(t - 0)/P(t) \to 1ast \to \infty .$ \end{document} Let f: ℝ̄+ → ℂ be a locally integrable function with respect to p dν, and define its weighted averages by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\sigma _t (f;pd\nu ): = \frac{1}{{P(t)}}\int\limits_0^t {f(u)p(u)d\nu (u)}$ \end{document} for large enough t, where P(t) > 0. We prove necessary and sufficient conditions under which the finite limit \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\sigma _t (f;pd\nu ) \to Last \to \infty$ \end{document} exists. This characterization is a unified extension of the results in [5], and it may find application in Probability Theory and Stochastic Processes.

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# Filling words in fundamental groups of Riemann surfaces

Author: C. Zhang

The purpose of this article is to utilize some exiting words in the fundamental group of a Riemann surface to acquire new words that are represented by filling closed geodesics.

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# A non representable infinite dimensional quasi-polyadic equality algebra with a representable cylindric reduct

Authors: Hajnal Andréka, István Németi and Tarek Ahmed

We construct an infinite dimensional quasi-polyadic equality algebra \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathfrak{A}$ \end{document} such that its cylindric reduct is representable, while \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathfrak{A}$ \end{document} itself is not representable.

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# On an open problem by Nasr-Isfahani on strict inner amenability

Authors: Mohammad Ghanei and Mehdi Nemati

For two locally compact groups G and H, we show that if L 1(G) is strictly inner amenable, then L 1(G × H) is strictly inner amenable. We then apply this result to show that there is a large class of locally compact groups G such that L 1(G) is strictly inner amenable, but G is not even inner amenable.

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# Some properties of local homology and local cohomology modules

Author: Tran Nam

We study some properties of representable or I-stable local homology modules H i I (M) where M is a linearly compact module. By duality, we get some properties of good or at local cohomology modules H I i (M) of A. Grothendieck.

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# The sub-bifractional Brownian motion

Authors: Charles El-Nouty and Jean-Lin Journé

The sub-bifractional Brownian motion, which is a quasi-helix in the sense of Kahane, is presented. The upper classes of some of its increments are characterized by an integral test.

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# Weak convergence of random walks, conditioned to stay away from small sets

Authors: Zsolt Pajor-Gyulai and Domokos Szász

Let {X n}n∈ℕ be a sequence of i.i.d. random variables in ℤd. Let S k = X 1 + … + X k and Y n(t) be the continuous process on [0, 1] for which Y n(k/n) = S k/n 1/2 for k = 1, … n and which is linearly interpolated elsewhere. The paper gives a generalization of results of ([2]) on the weak limit laws of Y n(t) conditioned to stay away from some small sets. In particular, it is shown that the diffusive limit of the random walk meander on ℤd: d ≧ 2 is the Brownian motion.

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# Characterizations of the Amoroso distribution

Author: G. Hamedani

Characterizations of the Amoroso distribution based on a simple relationship between two truncated moments are presented. A remark regarding the characterization of certain special cases of the Amoroso distribution based on hazard function is given. We will also point out that a sub-family of the Amoroso family is a member of the generalized Pearson system.

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# Fundamental group of Desargues configuration spaces

Authors: Barbu Berceanu and Saima Parveen

We compute the fundamental group of various spaces of Desargues configurations in complex projective spaces: planar and non-planar configurations, with a fixed center and also with an arbitrary center.

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# Joint distributions of circular runs of various lengths based on multi-colour Pólya urn model

Author: Sonali Bhattacharya

In this paper, we have used Eryilmaz’s (2008) multi-colour Pólya urn model to obtain joint distributions of runs of t-types of exact lengths (k 1, k 2, …, k t), at least lengths (k 1, k 2, …, k t), non-overlapping runs of lengths (k 1, k 2, … k t) and overlapping runs of lengths (k 1, k 2, … k t) when counting of runs is done in a circular setup. We have also derived joint distributions of longest runs of various types under similar conditions. Distributions of runs have found applications in fields of reliability of consecutive-k-out-of n: F system, consecutive k-out-of-r-from n: F system, start-up demonstration test, molecular biology, radar detection, time sharing systems and quality control. The literature is profound in discussion of marginal distribution and joint distribution of runs of various types under linear and circular setup using techniques like urn model with balls of two or more colours, probability generating function and compounding discrete distribution with suitable beta functions. Through this paper for first time effort been made to discuss joint distributions of runs of various lengths and types using Multi-colour urn model.

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# The largest family of subsets satisfying sequential-evaluation convergence

Authors: Aihong Chen and Ronglu Li

Suppose X is a locally convex space, Y is a topological vector space and λ(X)βY is the β-dual of some X valued sequence space λ(X). When λ(X) is c 0(X) or l (X), we have found the largest M ⊂ 2λ(X) for which (A j) ∈ λ(X)βY if and only if Σ j=1 A j(x j) converges uniformly with respect to (x j) in any MM. Also, a remark is given when λ(X) is l p(X) for 0 < p < + ∞.

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# New characterizations of p-soluble and p-supersoluble finite groups

Authors: Wenbin Guo and Alexander Skiba

Let G be a finite group and H a subgroup of G. H is said to be S-quasinormal in G if HP = PH for all Sylow subgroups P of G. Let H sG be the subgroup of H generated by all those subgroups of H which are S-quasinormal in G and H sG the intersection of all S-quasinormal subgroups of G containing H. The symbol |G|p denotes the order of a Sylow p-subgroup of G. We prove the followingTheorem A. Let G be a finite group and p a prime dividing |G|. Then G is p-supersoluble if and only if for every cyclic subgroup H of (G) of prime order or order 4 (if p = 2), has a normal subgroup T such that H sḠ and HT=H sḠT.Theorem B. A soluble finite group G is p-supersoluble if and only if for every 2-maximal subgroup E of G such that O p′ (G) ≦ E and |G: E| is not a power of p, G has an S-quasinormal subgroup T with cyclic Sylow p-subgroups such that E sG = ET and |ET|p = |E sGT|p.Theorem C. A finite group G is p-soluble if for every 2-maximal subgroup E of G such that O p (G) ≦ E and |G: E| is not a power of p, G has an S-quasinormal subgroup T such that E sG = ET and |ET p = |E sGT|p.

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# Nilpotent elements and McCoy rings

Authors: Liang Zhao, Xiaosheng Zhu and Qinqin Gu

We introduce the concept of nil-McCoy rings to study the structure of the set of nilpotent elements in McCoy rings. This notion extends the concepts of McCoy rings and nil-Armendariz rings. It is proved that every semicommutative ring is nil-McCoy. We shall give an example to show that nil-McCoy rings need not be semicommutative. Moreover, we show that nil-McCoy rings need not be right linearly McCoy. More examples of nil-McCoy rings are given by various extensions. On the other hand, the properties of α-McCoy rings by considering the polynomials in the skew polynomial ring R[x; α] in place of the ring R[x] are also investigated. For a monomorphism α of a ring R, it is shown that if R is weak α-rigid and α-reversible then R is α-McCoy.

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# On invariant convex subsets in algebras defined on a locally compact group G

Author: Ali Ghaffari

Suppose that A is either the Banach algebra L 1(G) of a locally compact group G, or measure algebra M(G), or other algebras (usually larger than L 1(G) and M(G)) such as the second dual, L 1(G)**, of L 1(G) with an Arens product, or LUC(G)* with an Arenstype product. The left translation invariant closed convex subsets of A are studied. Finally, we obtain necessary and sufficient conditions for LUC(G)* to have 1-dimensional left ideals.

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# Parameterized affine codes

Authors: Hiram López, Eliseo Sarmiento, Maria Pinto and Rafael Villarreal

Let K be a finite field and let X* be an affine algebraic toric set parameterized by monomials. We give an algebraic method, using Gröbner bases, to compute the length and the dimension of C X* (d), the parameterized affine code of degree d on the set X*. If Y is the projective closure of X*, it is shown that C X* (d) has the same basic parameters that C Y (d), the parameterized projective code on the set Y. If X* is an affine torus, we compute the basic parameters of C X* (d). We show how to compute the vanishing ideals of X* and Y.

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# Varieties generated by 2-testable monoids

Author: Edmond Lee

The smallest monoid containing a 2-testable semigroup is defined to be a 2-testable monoid. The well-known Brandt monoid B 2 1 of order six is an example of a 2-testable monoid. The finite basis problem for 2-testable monoids was recently addressed and solved. The present article continues with the investigation by describing all monoid varieties generated by 2-testable monoids. It is shown that there are 28 such varieties, all of which are finitely generated and precisely 19 of which are finitely based. As a comparison, the sub-variety lattice of the monoid variety generated by the monoid B 2 1 is examined. This lattice has infinite width, satisfies neither the ascending chain condition nor the descending chain condition, and contains non-finitely generated varieties.

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# Wavelet approximation and Fourier widths of classes of periodic functions of several variables. II

Author: Д. Бaзaрхaноь

## Реэюме

Получены точные в смысле порядка оценки для поперечников Фурье классов типа Никольского-Бесова

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$B_{pq}^{sm} (\mathbb{T}^k )$$ \end{document}
и Лиэоркина-Трибеля
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$L_{pq}^{sm} (\mathbb{T}^k )$$ \end{document}
в метрике
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$L_r (\mathbb{T}^k )$$ \end{document}
для ряда соотношений между параметрами s, p, q, r (эдесъ s ∈ (0, ∞)n, 1 ≤ p, r, q, ≤ ∞, 1 ≤ nk, m = (m 1, …, m n) ∈ ℕn: m 1 + … + m n = k).

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# Об одном многомерном варианте неравенства типа Гильберта

Authors: Mario Krnić and Predrag Vuković

## Abstract

The main objective of this paper is a study of some new multidimensional Hilbert type inequalities with a general homogeneous kernel. We derive a pair of equivalent inequalities, and also establish the conditions under which the constant factors included in the obtained inequalities are the best possible. Some applications in particular settings are also considered.

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# Равномерное приближение интегралов Пуассона функций иэ класса H ω суммами Валле Пуссена

Authors: A. Serdyuk and Ie. Ovsii

## Abstract

We obtain asymptotic equalities for least upper bounds of deviations in the uniform metric of de la Vallée Poussin sums on the sets C β q H ω of Poisson integrals of functions from the class H ω generated by convex upwards moduli of continuity ω(t) which satisfy the condition ω(t)/t → ∞ as t → 0. As an implication, a solution of the Kolmogorov-Nikol’skii problem for de la Vallée Poussin sums on the sets of Poisson integrals of functions belonging to Lipschitz classes H α, 0 < α < 1, is obtained.

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# Cubes of integral vectors in dimension four

Authors: Emil Kiss and Péter Kutas

A system of m nonzero vectors in ℤn is called an m-icube if they are pairwise orthogonal and have the same length. The paper describes m-icubes in ℤ4 for 2 ≦ m ≦ 4 using Hurwitz integral quaternions, counts the number of them with given edge length, and proves that unlimited extension is possible in ℤ4.

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# Deformation theory of Fuchsian equations and logarithmic connections

Author: Szilárd Szabó

Motivated by a remark and a question of Nicholas Katz, we characterize the tangent space of the space of Fuchsian equations with given generic exponents inside the corresponding moduli space of logarithmic connections: we construct a weight 1 Hodge structure on the tangent space of the moduli of logarithmic connections such that deformations of Fuchsian equations correspond to the (1, 0)-part.

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# Devaney chaos and Li-Yorke sensitivity for product systems

Authors: Xinxing Wu and Peiyong Zhu

This paper mainly discusses how Devaney chaos and Li-Yorke sensitivity carry over to product systems. First, two results on the periodic points of product systems are obtained. By using them, the following two results are Proved: (1) A finite product system is mixing and Devaney chaotic if and only if each factor system is mixing and Devaney chaotic. (2) An infinite product map Π i=1 f i is mixing and Devaney chaotic if and only if each factor map f i is mixing and Devaney chaotic and sup {min P(f i): i ∈ ℕ} < + ∞, where P(f i) is the set of all periods of f i. Besides, we obtain that the product system is Li-Yorke sensitive (sensitive) if and only if there exists a factor system that is Li-Yorke sensitive (sensitive).

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# Double uniform density and corresponding convergence of double sequences

Authors: Pratulananda Das and Ekrem Savas

We introduce the concept of double uniform density of subsets of ℕ×ℕ and the study the corresponding convergence (namely, I u-convergence) of double sequences. Further we solve an inequality related to the I u-limit superior of bounded double sequences in line of [5].

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# A fourier analytic approach to the problem of mutually unbiased bases

Author: Máté Matolcsi
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