$Open Mathematics$

Open Access Published by De Gruyter Open Access

Volume 8 Issue 2

April 2010

Issue of Open Mathematics

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Contents
Journal Overview

Open Access April 14, 2010

Large continuum, oracles

Saharon Shelah

Page range: 213-234

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Abstract

Our main theorem is about iterated forcing for making the continuum larger than ℵ2. We present a generalization of [2] which deal with oracles for random, (also for other cases and generalities), by replacing ℵ1,ℵ2 by λ, λ + (starting with λ = λ <λ > ℵ1). Well, we demand absolute c.c.c. So we get, e.g. the continuum is λ + but we can get cov(meagre) = λ and we give some applications. As in non-Cohen oracles [2], it is a “partial” countable support iteration but it is c.c.c.

Open Access April 14, 2010

From factorizations of noncommutative polynomials to combinatorial topology

Vladimir Retakh

Page range: 235-243

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Abstract

This is an extended version of a talk given by the author at the conference “Algebra and Topology in Interaction” on the occasion of the 70th Anniversary of D.B. Fuchs at UC Davis in September 2009. It is a brief survey of an area originated around 1995 by I. Gelfand and the speaker.

Open Access April 14, 2010

Realizability and automatic realizability of Galois groups of order 32

Helen Grundman, Tara Smith

Page range: 244-260

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Abstract

This article provides necessary and sufficient conditions for each group of order 32 to be realizable as a Galois group over an arbitrary field. These conditions, given in terms of the number of square classes of the field and the triviality of specific elements in related Brauer groups, are used to derive a variety of automatic realizability results.

Open Access April 14, 2010

Infinite dimensional linear groups with many G — invariant subspaces

Leonid Kurdachenko, Alexey Sadovnichenko, Igor Subbotin

Page range: 261-265

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Abstract

Let F be a field, A be a vector space over F, GL(F, A) be the group of all automorphisms of the vector space A. A subspace B of A is called nearly G-invariant, if dimF(BFG/B) is finite. A subspace B is called almost G-invariant, if dim F(B/Core G(B)) is finite. In the current article, we study linear groups G such that every subspace of A is either nearly G-invariant or almost G-invariant in the case when G is a soluble p-group where p = char F.

Open Access April 14, 2010

On locally finite minimal non-solvable groups

Ahmet Arıkan, Sezgin Sezer, Howard Smith

Page range: 266-273

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Abstract

In the present work we consider infinite locally finite minimal non-solvable groups, and give certain characterizations. We also define generalizations of the centralizer to establish a result relevant to infinite locally finite minimal non-solvable groups.

Open Access April 14, 2010

Separable K-linear categories

Andrei Chiteș, Costel Chiteș

Page range: 274-281

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Abstract

We define and investigate separable K-linear categories. We show that such a category C is locally finite and that every left C-module is projective. We apply our main results to characterize separable linear categories that are spanned by groupoids or delta categories.

Open Access April 14, 2010

The groups of points on abelian varieties over finite fields

Sergey Rybakov

Page range: 282-288

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Abstract

Let A be an abelian variety with commutative endomorphism algebra over a finite field k. The k-isogeny class of A is uniquely determined by a Weil polynomial f A without multiple roots. We give a classification of the groups of k-rational points on varieties from this class in terms of Newton polygons of f A(1 − t).

Open Access April 14, 2010

On the singularities of multiple L-functions

Alexandru Zaharescu, Mohammad Zaki

Page range: 289-298

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Abstract

We investigate the singularities of a class of multiple L-functions considered by Akiyama and Ishikawa [2].

Open Access April 14, 2010

The isomorphism relation between tree-automatic Structures

Olivier Finkel, Stevo Todorčević

Page range: 299-313

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Abstract

An ω-tree-automatic structure is a relational structure whose domain and relations are accepted by Muller or Rabin tree automata. We investigate in this paper the isomorphism problem for ω-tree-automatic structures. We prove first that the isomorphism relation for ω-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n ≥ 2) is not determined by the axiomatic system ZFC. Then we prove that the isomorphism problem for ω-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n ≥ 2) is neither a Σ21-set nor a Π21-set.

Open Access April 14, 2010

Continuous tree-like scales

James Cummings

Page range: 314-318

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Abstract

Answering a question raised by Luis Pereira, we show that a continuous tree-like scale can exist above a supercompact cardinal. We also show that the existence of a continuous tree-like scale at ℵω is consistent with Martin’s Maximum.

Open Access April 14, 2010

Altitude of wheels and wheel-like graphs

Tomasz Dzido, Hanna Furmańczyk

Page range: 319-326

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Abstract

An edge-ordering of a graph G=(V, E) is a one-to-one mapping f:E(G)→{1, 2, ..., |E(G)|}. A path of length k in G is called a (k, f)-ascent if f increases along the successive edges forming the path. The altitude α(G) of G is the greatest integer k such that for all edge-orderings f, G has a (k, f)-ascent. In our paper we give exact values of α(G) for all helms and wheels. Furthermore, we use our result to obtain altitude for graphs that are subgraphs of helms.

Open Access April 14, 2010

Stable bundles on hypercomplex surfaces

Ruxandra Moraru, Misha Verbitsky

Page range: 327-337

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Abstract

A hypercomplex manifold is a manifold equipped with three complex structures I, J, K satisfying the quaternionic relations. Let M be a 4-dimensional compact smooth manifold equipped with a hypercomplex structure, and E be a vector bundle on M. We show that the moduli space of anti-self-dual connections on E is also hypercomplex, and admits a strong HKT metric. We also study manifolds with (4,4)-supersymmetry, that is, Riemannian manifolds equipped with a pair of strong HKT-structures that have opposite torsion. In the language of Hitchin’s and Gualtieri’s generalized complex geometry, (4,4)-manifolds are called “generalized hyperkähler manifolds”. We show that the moduli space of anti-self-dual connections on M is a (4,4)-manifold if M is equipped with a (4,4)-structure.

Open Access April 14, 2010

Topological K-equivalence of analytic function-germs

Sérgio Alvarez, Lev Birbrair, João Costa, Alexandre Fernandes

Page range: 338-345

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Abstract

We study the topological K-equivalence of function-germs (ℝn, 0) → (ℝ, 0). We present some special classes of piece-wise linear functions and prove that they are normal forms for equivalence classes with respect to topological K-equivalence for definable functions-germs. For the case n = 2 we present polynomial models for analytic function-germs.

Open Access April 14, 2010

Daugavet centers and direct sums of Banach spaces

Tetiana Bosenko

Page range: 346-356

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Abstract

A linear continuous nonzero operator G: X → Y is a Daugavet center if every rank-1 operator T: X → Y satisfies ||G + T|| = ||G|| + ||T||. We study the case when either X or Y is a sum X 1⊕F X 2 of two Banach spaces X 1 and X 2 by some two-dimensional Banach space F. We completely describe the class of those F such that for some spaces X 1 and X 2 there exists a Daugavet center acting from X 1⊕F X 2, and the class of those F such that for some pair of spaces X 1 and X 2 there is a Daugavet center acting into X 1⊕F X 2. We also present several examples of such Daugavet centers.

Open Access April 14, 2010

Non-self mappings in modular spaces and common fixed point theorems

Abdolrahman Razani, Valdimir Rakočević, Zahraa Goodarzi

Page range: 357-366

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Abstract

The aim of this paper, is to introduce the convex structure (specially, Takahashi convex structure) on modular spaces. Moreover, we are interested in proving some common fixed point theorems for non-self mappings in modular space.

Open Access April 14, 2010

Multivalued fractals in b-metric spaces

Monica Boriceanu, Marius Bota, Adrian Petruşel

Page range: 367-377

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Abstract

Fractals and multivalued fractals play an important role in biology, quantum mechanics, computer graphics, dynamical systems, astronomy and astrophysics, geophysics, etc. Especially, there are important consequences of the iterated function (or multifunction) systems theory in several topics of applied sciences. It is known that examples of fractals and multivalued fractals are coming from fixed point theory for single-valued and multivalued operators, via the so-called fractal and multi-fractal operators. On the other hand, the most common setting for the study of fractals and multi-fractals is the case of operators on complete or compact metric spaces. The purpose of this paper is to extend the study of fractal operator theory for multivalued operators on complete b-metric spaces.

Open Access April 14, 2010

Basis properties of a fourth order differential operator with spectral parameter in the boundary condition

Ziyatkhan Aliyev

Page range: 378-388

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Abstract

We consider a fourth order eigenvalue problem containing a spectral parameter both in the equation and in the boundary condition. The oscillation properties of eigenfunctions are studied and asymptotic formulae for eigenvalues and eigenfunctions are deduced. The basis properties in L p(0; l); p ∈ (1;∞); of the system of eigenfunctions are investigated.

Open Access April 14, 2010

On an approximation processes in the space of analytical functions

Akif Gadjiev, Arash Ghorbanalizadeh

Page range: 389-398

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Abstract

In this paper we obtain various approximation theorems by means of k-positive linear operators defined on the space of all analytic functions on a bounded domain of the complex plane.

Open Access April 14, 2010

On q-Szász-Durrmeyer operators

Nazim Mahmudov, Havva Kaffaoğlu

Page range: 399-409

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Abstract

In the present paper, we introduce the q-Szász-Durrmeyer operators and justify a local approximation result for continuous functions in terms of moduli of continuity. We also discuss a Voronovskaya type result for the q-Szász-Durrmeyer operators.

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.