$Open Mathematics$

Open Access Published by De Gruyter Open Access

Volume 9 Issue 4

August 2011

Issue of Open Mathematics

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Contents
Journal Overview

Open Access May 26, 2011

Moment-angle complexes from simplicial posets

Zhi Lü, Taras Panov

Page range: 715-730

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Abstract

We extend the construction of moment-angle complexes to simplicial posets by associating a certain T m-space Z S to an arbitrary simplicial poset S on m vertices. Face rings ℤ[S] of simplicial posets generalise those of simplicial complexes, and give rise to new classes of Gorenstein and Cohen-Macaulay rings. Our primary motivation is to study the face rings ℤ[S] by topological methods. The space Z S has many important topological properties of the original moment-angle complex Z K associated to a simplicial complex K. In particular, we prove that the integral cohomology algebra of Z S is isomorphic to the Tor-algebra of the face ring ℤ[S]. This leads directly to a generalisation of Hochster’s theorem, expressing the algebraic Betti numbers of the ring ℤ[S] in terms of the homology of full subposets in S. Finally, we estimate the total amount of homology of Z S from below by proving the toral rank conjecture for the moment-angle complexes Z S.

Open Access May 26, 2011

Scalar differential invariants of symplectic Monge-Ampère equations

Alessandro Paris, Alexandre Vinogradov

Page range: 731-751

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Abstract

All second order scalar differential invariants of symplectic hyperbolic and elliptic Monge-Ampère equations with respect to symplectomorphisms are explicitly computed. In particular, it is shown that the number of independent second order invariants is equal to 7, in sharp contrast with general Monge-Ampère equations for which this number is equal to 2. We also introduce a series of invariant differential forms and vector fields which allow us to construct numerous scalar differential invariants of higher order. The introduced invariants give a solution of the symplectic equivalence of Monge-Ampère equations. As an example we study equations of the form u xy + f(x, y, u x, u y) = 0 and in particular find a simple linearization criterion.

Open Access May 26, 2011

On the Heegaard genus of contact 3-manifolds

Burak Ozbagci

Page range: 752-756

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Abstract

It is well-known that the Heegaard genus is additive under connected sum of 3-manifolds. We show that the Heegaard genus of contact 3-manifolds is not necessarily additive under contact connected sum. We also prove some basic properties of the contact genus (a.k.a. open book genus [Rubinstein J.H., Comparing open book and Heegaard decompositions of 3-manifolds, Turkish J. Math., 2003, 27(1), 189–196]) of 3-manifolds, and compute this invariant for some 3-manifolds.

Open Access May 26, 2011

Large dimensional sets not containing a given angle

Viktor Harangi

Page range: 757-764

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Abstract

We say that a set in a Euclidean space does not contain an angle α if the angle determined by any three points of the set is not equal to α. The goal of this paper is to construct compact sets of large Hausdorff dimension that do not contain a given angle α ∈ (0,π). We will construct such sets in ℝn of Hausdorff dimension c(α)n with a positive c(α) depending only on α provided that α is different from π/3, π/2 and 2π/3. This improves on an earlier construction (due to several authors) that has dimension c(α) log n. The main result of the paper concerns the case of the angles π/3 and 2π/3. We present self-similar sets in ℝn of Hausdorff dimension $c{{\sqrt[3]{n}} \mathord{\left/ {\vphantom {{\sqrt[3]{n}} {\log n}}} \right. \kern-\nulldelimiterspace} {\log n}}$ with the property that they do not contain the angles π/3 and 2π/3. The constructed sets avoid not only the given angle α but also a small neighbourhood of α.

Open Access May 26, 2011

Quotients of peripherally continuous functions

Jolanta Kosman

Page range: 765-771

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Abstract

We characterize the family of quotients of peripherally continuous functions. Moreover, we study cardinal invariants related to quotients in the case of peripherally continuous functions and the complement of this family.

Open Access May 26, 2011

On the almost monotone convergence of sequences of continuous functions

Zbigniew Grande

Page range: 772-777

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Abstract

A sequence (f n)n of functions f n: X → ℝ almost decreases (increases) to a function f: X → ℝ if it pointwise converges to f and for each point x ∈ X there is a positive integer n(x) such that f n+1(x) ≤ f n (x) (f n+1(x) ≥ f n(x)) for n ≥ n(x). In this article I investigate this convergence in some families of continuous functions.

Open Access May 26, 2011

Real-linear isometries between function algebras

Takeshi Miura

Page range: 778-788

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Abstract

Let A and B be uniformly closed function algebras on locally compact Hausdorff spaces with Choquet boundaries Ch A and ChB, respectively. We prove that if T: A → B is a surjective real-linear isometry, then there exist a continuous function κ: ChB → {z ∈ ℂ: |z| = 1}, a (possibly empty) closed and open subset K of ChB and a homeomorphism φ: ChB → ChA such that T(f) = κ(f ∘φ) on K and $T\left( f \right) = \kappa \overline {fo\phi }$ on ChB \ K for all f ∈ A. Such a representation holds for surjective real-linear isometries between (not necessarily uniformly closed) function algebras.

Open Access May 26, 2011

Kolmogorov-Smirnov isometries and affine automorphisms of spaces of distribution functions

Lajos Molnár

Page range: 789-796

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Abstract

In this paper we describe the structure of surjective isometries of the spaces of all absolutely continuous, singular, or discrete probability distribution functions on R equipped with the Kolmogorov-Smirnov metric. We also study the structure of affine automorphisms of the space of all distribution functions.

Open Access May 26, 2011

Extremely non-complex Banach spaces

Miguel Martín, Javier Merí

Page range: 797-802

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Abstract

A Banach space X is said to be an extremely non-complex space if the norm equality ∥Id +T 2∥ = 1+∥T 2∥ holds for every bounded linear operator T on X. We show that every extremely non-complex Banach space has positive numerical index, it does not have an unconditional basis and that the infimum of diameters of the slices of its unit ball is positive.

Open Access May 26, 2011

Covariant version of the Stinespring type theorem for Hilbert C*-modules

Maria Joiţa

Page range: 803-813

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Abstract

In this paper, we prove a covariant version of the Stinespring theorem for Hilbert C*-modules. Also, we show that there is a bijective correspondence between operator valued completely positive maps, (u′, u)-covariant with respect to the dynamical system (G, η, X) on Hilbert C*-modules and (u′, u)-covariant operator valued completely positive maps on the crossed product G ×η X of X by η.

Open Access May 26, 2011

On a class of nonlocal elliptic operators for compact Lie groups. Uniformization and finiteness theorem

Boris Sternin

Page range: 814-832

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Abstract

We consider a class of nonlocal operators associated with an action of a compact Lie group G on a smooth closed manifold. Ellipticity condition and Fredholm property for elliptic operators are obtained. This class of operators is studied using pseudodifferential uniformization, which reduces the problem to a pseudodifferential operator acting in sections of infinite-dimensional bundles.

Open Access May 26, 2011

On the index of nonlocal elliptic operators for compact Lie groups

Anton Savin

Page range: 833-850

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Abstract

We consider a class of nonlocal operators associated with a compact Lie group G acting on a smooth manifold. A notion of symbol of such operators is introduced and an index formula for elliptic elements is obtained. The symbol in this situation is an element of a noncommutative algebra (crossed product by G) and to obtain an index formula, we define the Chern character for this algebra in the framework of noncommutative geometry.

Open Access May 26, 2011

Nonlinear Leray-Schauder alternatives and application to nonlinear problem arising in the theory of growing cell population

Afif Amar

Page range: 851-865

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Abstract

Motivated by a mathematical model of an age structured proliferating cell population, we state some new variants of Leray-Schauder type fixed point theorems for (ws)-compact operators. Further, we apply our results to establish some new existence and locality principles for nonlinear boundary value problem arising in the theory of growing cell population in L 1-setting. Besides, a topological structure of the set of solutions is provided.

Open Access May 26, 2011

Large time behavior of the solution to an initial-boundary value problem with mixed boundary conditions for a nonlinear integro-differential equation

Temur Jangveladze, Zurab Kiguradze

Page range: 866-873

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Abstract

Large time behavior of the solution to the nonlinear integro-differential equation associated with the penetration of a magnetic field into a substance is studied. Furthermore, the rate of convergence is given. Initial-boundary value problem with mixed boundary conditions is considered.

Open Access May 26, 2011

On a generalized Stokes problem

Václav Mácha

Page range: 874-887

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Abstract

We deal with a generalization of the Stokes system. Instead of the Laplace operator, we consider a general elliptic operator and a pressure gradient with small perturbations. We investigate the existence and uniqueness of a solution as well its regularity properties. Two types of regularity are provided. Aside from the classical Hilbert regularity, we also prove the Hölder regularity for coefficients in VMO space.

Open Access May 26, 2011

The five-variable Volterra system

Janusz Zieliński

Page range: 888-896

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Abstract

We give a description of all polynomial constants of the five-variable Volterra derivation, hence of all polynomial first integrals of its corresponding Volterra system of differential equations. The Volterra system plays a significant role in plasma physics and population biology.

Open Access May 26, 2011

An algorithm for primary decomposition in polynomial rings over the integers

Gerhard Pfister, Afshan Sadiq, Stefan Steidel

Page range: 897-904

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Abstract

We present an algorithm to compute a primary decomposition of an ideal in a polynomial ring over the integers. For this purpose we use algorithms for primary decomposition in polynomial rings over the rationals, resp. over finite fields, and the idea of Shimoyama-Yokoyama, resp. Eisenbud-Hunecke-Vasconcelos, to extract primary ideals from pseudo-primary ideals. A parallelized version of the algorithm is implemented in Singular. Examples and timings are given at the end of the article.

Open Access May 26, 2011

On complements and the factorization problem of Hopf algebras

Sebastian Burciu

Page range: 905-914

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Abstract

Two new results concerning complements in a semisimple Hopf algebra are proved. They extend some well-known results from group theory. The uniqueness of a Krull-Schmidt-Remak type decomposition is proved for semisimple completely reducible Hopf algebras.

Open Access May 26, 2011

Subnormal, permutable, and embedded subgroups in finite groups

James Beidleman, Mathew Ragland

Page range: 915-921

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Abstract

The purpose of this paper is to study the subgroup embedding properties of S-semipermutability, semipermutability, and seminormality. Here we say H is S-semipermutable (resp. semipermutable) in a group Gif H permutes which each Sylow subgroup (resp. subgroup) of G whose order is relatively prime to that of H. We say H is seminormal in a group G if H is normalized by subgroups of G whose order is relatively prime to that of H. In particular, we establish that a seminormal p-subgroup is subnormal. We also establish that the solvable groups in which S-permutability is a transitive relation are precisely the groups in which the subnormal subgroups are all S-semipermutable. Local characterizations of this result are also established.

Open Access May 26, 2011

Modules over group rings of soluble groups with a certain condition of maximality

Olga Dashkova

Page range: 922-928

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Abstract

Let A be an R G-module, where R is an integral domain and G is a soluble group. Suppose that C G(A) = 1 and A/C A(G) is not a noetherian R-module. Let L nnd(G) be the family of all subgroups H of G such that A/C A(H) is not a noetherian R-module. In this paper we study the structure of those G for which L nnd(G) satisfies the maximal condition.

Open Access May 26, 2011

A note on Stone join-semilattices

Shriram Nimbhorkar, Anwari Rahemani

Page range: 929-933

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Abstract

Characterizations for a pseudocomplemented modular join-semilattice with 0 and 1 and its ideal lattice to be a Stone lattice are given.

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.