$Open Mathematics$

Open Access Published by De Gruyter Open Access

Volume 9 Issue 5

October 2011

Issue of Open Mathematics

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Contents
Journal Overview

Open Access July 26, 2011

Editorial Material

Ludmil Katzarkov, Tony Pantev, Yuri Tschinkel, Yuri Zarhin

Page range: 935-935

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Open Access July 26, 2011

Unramified cohomology of alternating groups

Fedor Bogomolov, Tihomir Petrov

Page range: 936-948

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Abstract

We prove vanishing results for the unramified stable cohomology of alternating groups.

Open Access July 26, 2011

3-dimensional sundials

Enrico Carlini, Maria Catalisano, Anthony Geramita

Page range: 949-971

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Abstract

R. Hartshorne and A. Hirschowitz proved that a generic collection of lines on ℙn, n≥3, has bipolynomial Hilbert function. We extend this result to a specialization of the collection of generic lines, by considering a union of lines and 3-dimensional sundials (i.e., a union of schemes obtained by degenerating pairs of skew lines).

Open Access July 26, 2011

Hori-Vafa mirror models for complete intersections in weighted projective spaces and weak Landau-Ginzburg models

Victor Przyjalkowski

Page range: 972-977

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Abstract

We prove that Hori-Vafa mirror models for smooth Fano complete intersections in weighted projective spaces admit an interpretation as Laurent polynomials.

Open Access July 26, 2011

The Lindelöf number of C p(X)×C p(X) for strongly zero-dimensional X

Oleg Okunev

Page range: 978-983

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Abstract

We prove that if X is a strongly zero-dimensional space, then for every locally compact second-countable space M, C p(X, M) is a continuous image of a closed subspace of C p(X). It follows in particular, that for strongly zero-dimensional spaces X, the Lindelöf number of C p(X)×C p(X) coincides with the Lindelöf number of C p(X). We also prove that l(C p(X n)κ) ≤ l(C p(X)κ) whenever κ is an infinite cardinal and X is a strongly zero-dimensional union of at most κcompact subspaces.

Open Access July 26, 2011

Short separating geodesics for multiply connected domains

Mark Comerford

Page range: 984-996

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Abstract

We consider the following questions: given a hyperbolic plane domain and a separation of its complement into two disjoint closed sets each of which contains at least two points, what is the shortest closed hyperbolic geodesic which separates these sets and is it a simple closed curve? We show that a shortest geodesic always exists although in general it may not be simple. However, one can also always find a shortest simple curve and we call such a geodesic a meridian of the domain. We prove that, although they are not in general uniquely defined, if one of the sets of the separation of the complement is connected, then they are unique and are also the shortest possible geodesics which separate the complement in this fashion.

Open Access July 26, 2011

The arithmetic of distributions in free probability theory

Gennadii Chistyakov, Friedrich Götze

Page range: 997-1050

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Abstract

We give an analytical approach to the definition of additive and multiplicative free convolutions which is based on the theory of Nevanlinna and Schur functions. We consider the set of probability distributions as a semigroup M equipped with the operation of free convolution and prove a Khintchine type theorem for the factorization of elements of this semigroup. An element of M contains either indecomposable (“prime”) factors or it belongs to a class, say I 0, of distributions without indecomposable factors. In contrast to the classical convolution semigroup, in the free additive and multiplicative convolution semigroups the class I 0 consists of units (i.e. Dirac measures) only. Furthermore we show that the set of indecomposable elements is dense in M.

Open Access July 26, 2011

Multiplicative isometries on the Smirnov class

Osamu Hatori, Yasuo Iida

Page range: 1051-1056

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Abstract

We show that T is a surjective multiplicative (but not necessarily linear) isometry from the Smirnov class on the open unit disk, the ball, or the polydisk onto itself, if and only if there exists a holomorphic automorphism Φ such that T(f)=f ○ Φ for every class element f or T(f) = $\overline {f^\circ \bar \varphi } $ for every class element f, where the automorphism Φ is a unitary transformation in the case of the ball and Φ(z 1, ..., z n) = $(\lambda _1 z_{i_1 } ,...,\lambda _n z_{i_n } )$ for |λ j| = 1, 1 ≤ j ≤ n, and (i 1; ..., i n)is some permutation of the integers from 1through n in the case of the n-dimensional polydisk.

Open Access July 26, 2011

A category Ψ-density topology

Władysław Wilczyński, Wojciech Wojdowski

Page range: 1057-1066

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Abstract

Ψ-density point of a Lebesgue measurable set was introduced by Taylor in [Taylor S.J., On strengthening the Lebesgue Density Theorem, Fund. Math., 1958, 46, 305–315] and [Taylor S.J., An alternative form of Egoroff’s theorem, Fund. Math., 1960, 48, 169–174] as an answer to a problem posed by Ulam. We present a category analogue of the notion and of the Ψ-density topology. We define a category analogue of the Ψ-density point of the set A at a point x as the Ψ-density point at x of the regular open representation of A.

Open Access July 26, 2011

A solution of an open problem concerning Lagrangian mean-type mappings

Dorota Głazowska

Page range: 1067-1073

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Abstract

The problem of invariance of the geometric mean in the class of Lagrangian means was considered in [Głazowska D., Matkowski J., An invariance of geometric mean with respect to Lagrangian means, J. Math. Anal. Appl., 2007, 331(2), 1187–1199], where some necessary conditions for the generators of Lagrangian means have been established. The question if all necessary conditions are also sufficient remained open. In this paper we solve this problem.

Open Access July 26, 2011

Galois theory and Lubin-Tate cochains on classifying spaces

Andrew Baker, Birgit Richter

Page range: 1074-1087

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Abstract

We consider brave new cochain extensions F(BG +,R) → F(EG +,R), where R is either a Lubin-Tate spectrum E n or the related 2-periodic Morava K-theory K n, and G is a finite group. When R is an Eilenberg-Mac Lane spectrum, in some good cases such an extension is a G-Galois extension in the sense of John Rognes, but not always faithful. We prove that for E n and K n these extensions are always faithful in the K n local category. However, for a cyclic p-group $C_{p^r } $, the cochain extension $F(BC_{p^r + } ,E_n ) \to F(EC_{p^r + } ,E_n )$ is not a Galois extension because it ramifies. As a consequence, it follows that the E n-theory Eilenberg-Moore spectral sequence for G and BGdoes not always converge to its expected target.

Open Access July 26, 2011

Natural dualities between abelian categories

Flaviu Pop

Page range: 1088-1099

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Abstract

In this paper we consider a pair of right adjoint contravariant functors between abelian categories and describe a family of dualities induced by them.

Open Access July 26, 2011

Conditions for strong Morita equivalence of partially ordered semigroups

Lauri Tart

Page range: 1100-1113

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Abstract

We investigate when a partially ordered semigroup (with various types of local units) is strongly Morita equivalent to a posemigroup from a given class of partially ordered semigroups. Necessary and sufficient conditions for such equivalence are obtained for a series of well-known classes of posemigroups. A number of sufficient conditions for several classes of naturally ordered posemigroups are also provided.

Open Access July 26, 2011

Rectangular table negotiation problem revisited

Dalibor Froncek, Michael Kubesa

Page range: 1114-1120

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Abstract

We solve the last missing case of a “two delegation negotiation” version of the Oberwolfach problem, which can be stated as follows. Suppose we have two negotiating delegations with n=mk members each and we have a seating arrangement such that every day the negotiators sit at m tables with k people of the same delegation at one side of each table. Every person can effectively communicate just with three nearest persons across the table. Our goal is to guarantee that over the course of several days, every member of each delegation can communicate with every member of the other delegation exactly once. We denote by H(k, 3) the graph describing the communication at one table and by mH(k, 3) the graph consisting of m disjoint copies of H(k, 3). We completely characterize all complete bipartite graphs K n,n that can be factorized into factors isomorphic to G =mH(k, 3) for k ≡ 2 (mod 4) by showing that the necessary conditions n=mk and m ≡ 0 mod(3k−2)/4 are also sufficient. This results complement previous characterizations for k ≡ 0, 1, 3 (mod 4) to settle the problem in full.

Open Access July 26, 2011

Pattern avoiding partitions and Motzkin left factors

Toufik Mansour, Mark Shattuck

Page range: 1121-1134

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Abstract

Let L n, n ≥ 1, denote the sequence which counts the number of paths from the origin to the line x = n − 1 using (1, 1), (1, −1), and (1, 0) steps that never dip below the x-axis (called Motzkin left factors). The numbers L n count, among other things, certain restricted subsets of permutations and Catalan paths. In this paper, we provide new combinatorial interpretations for these numbers in terms of finite set partitions. In particular, we identify four classes of the partitions of size n, all of which have cardinality L n and each avoiding a set of two classical patterns of length four. We obtain a further generalization in one of the cases by considering a pair of statistics on the partition class. In a couple of cases, to show the result, we make use of the kernel method to solve a functional equation arising after a certain parameter has been introduced.

Open Access July 26, 2011

A trichotomy result for non-autonomous rational difference equations

Frank Palladino, Michael Radin

Page range: 1135-1142

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Abstract

We study non-autonomous rational difference equations. Under the assumption of a periodic non-autonomous parameter, we show that a well known trichotomy result in the autonomous case is preserved in a certain sense which is made precise in the body of the text. In addition we discuss some questions regarding whether periodicity preserves or destroys boundedness.

Open Access July 26, 2011

Approximate weak invariance for semilinear differential inclusions in Banach spaces

Alina Lazu, Victor Postolache

Page range: 1143-1155

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Abstract

In this paper we give a criterion for a given set K in Banach space to be approximately weakly invariant with respect to the differential inclusion x′(t) ∈ Ax(t) + F(x(t)), where A generates a C 0-semigroup and F is a given multi-function, using the concept of a tangent set to another set. As an application, we establish the relation between approximate solutions to the considered differential inclusion and solutions to the relaxed one, i.e., x′(t) ∈ Ax(t) + $\overline {co} $ F(x(t)), without any Lipschitz conditions on the multi-function F.

Open Access July 26, 2011

Positivity conditions and bounds for Green’s functions for higher order two-point BVP

Michael Gil’

Page range: 1156-1163

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Abstract

We consider a linear nonautonomous higher order ordinary differential equation and establish the positivity conditions and two-sided bounds for Green’s function for the two-point boundary value problem. Applications of the obtained results to nonlinear equations are also discussed.

Open Access July 26, 2011

On some finite difference schemes for solution of hyperbolic heat conduction problems

Raimondas Čiegis, Aleksas Mirinavičius

Page range: 1164-1170

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Abstract

We consider the accuracy of two finite difference schemes proposed recently in [Roy S., Vasudeva Murthy A.S., Kudenatti R.B., A numerical method for the hyperbolic-heat conduction equation based on multiple scale technique, Appl. Numer. Math., 2009, 59(6), 1419–1430], and [Mickens R.E., Jordan P.M., A positivity-preserving nonstandard finite difference scheme for the damped wave equation, Numer. Methods Partial Differential Equations, 2004, 20(5), 639–649] to solve an initial-boundary value problem for hyperbolic heat transfer equation. New stability and approximation error estimates are proved and it is noted that some statements given in the above papers should be modified and improved. Finally, two robust finite difference schemes are proposed, that can be used for both, the hyperbolic and parabolic heat transfer equations. Results of numerical experiments are presented.

Open Access July 26, 2011

Interval algorithm for absolute value equations

Aixiang Wang, Haijun Wang, Yongkun Deng

Page range: 1171-1184

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Abstract

We investigate the absolute value equations Ax−|x| = b. Based on ɛ-inflation, an interval verification method is proposed. Theoretic analysis and numerical results show that the new proposed method is effective.

Open Access July 26, 2011

Algebraic axiomatization of tense intuitionistic logic

Ivan Chajda

Page range: 1185-1191

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Abstract

We introduce two unary operators G and H on a relatively pseudocomplemented lattice which form an algebraic axiomatization of the tense quantifiers “it is always going to be the case that” and “it has always been the case that”. Their axiomatization is an extended version for the classical logic and it is in accordance with these operators on many-valued Łukasiewicz logic. Finally, we get a general construction of these tense operators on complete relatively pseudocomplemented lattice which is a power lattice via the so-called frame.

About this journal

Open Mathematics is a fully peer-reviewed open access journal that publishes original and relevant works on topics across the full span of mathematics. The journal's primary goal is to maintain the high quality of its published content and to provide free, instant, and continued access to all content worldwide.

AIMS AND SCOPE
Open Mathematics aims to promote knowledge directly relevant to all fields of mathematics. Subject areas suitable for publication include, but are not limited to, the following:

Algebra, Algebraic Geometry, Category Theory, Complex Analysis, Control Theory and Optimization, Differential Equations, Differential Geometry, Discrete Mathematics, Dynamical Systems and Ergodic Theory, Functional Analysis, Geometry, Mathematical Logic and Foundations, Mathematical Physics, Number Theory, Numerical Analysis, Operator Theory, Probability Theory and Statistics, Real Analysis, and Topology.

The journal accepts submissions of research articles, reviews, and communications. Submission of a manuscript implies that the work described is not copyrighted, published or submitted elsewhere, except in abstract form. The corresponding author should ensure that all authors approve the manuscript before its submission.