$Open Mathematics$

Open Access Published by De Gruyter Open Access

Volume 9 Issue 3

June 2011

Issue of Open Mathematics

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Contents
Journal Overview

Open Access March 22, 2011

Threefolds with big and nef anticanonical bundles II

Priska Jahnke, Thomas Peternell, Ivo Radloff

Page range: 449-488

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Abstract

In a follow-up to our paper [Threefolds with big and nef anticanonical bundles I, Math. Ann., 2005, 333(3), 569–631], we classify smooth complex projective threefolds Xwith −K X big and nef but not ample, Picard number γ(X) = 2, and whose anticanonical map is small. We assume also that the Mori contraction of X and of its flop X + are not both birational.

Open Access March 22, 2011

Geography of log models: theory and applications

Vyacheslav Shokurov, Sung Choi

Page range: 489-534

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Abstract

This is an introduction to geography of log models with applications to positive cones of Fano type (FT) varieties and to geometry of minimal models and Mori fibrations.

Open Access March 22, 2011

Hyperholomorphic connections on coherent sheaves and stability

Misha Verbitsky

Page range: 535-557

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Abstract

Let M be a hyperkähler manifold, and F a reflexive sheaf on M. Assume that F (away from its singularities) admits a connection ▿ with a curvature Θ which is invariant under the standard SU(2)-action on 2-forms. If Θ is square-integrable, such sheaf is called hyperholomorphic. Hyperholomorphic sheaves were studied at great length in [21]. Such sheaves are stable and their singular sets are hyperkähler subvarieties in M. In the present paper, we study sheaves admitting a connection with SU(2)-invariant curvature which is not necessary L 2-integrable. We show that such sheaves are polystable.

Open Access March 22, 2011

Integral formulae for a Riemannian manifold with two orthogonal distributions

Vladimir Rovenski

Page range: 558-577

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Abstract

We obtain a series of new integral formulae for a distribution of arbitrary codimension (and its orthogonal complement) given on a closed Riemannian manifold, which start from the formula by Walczak (1990) and generalize ones for foliations by several authors. For foliations on space forms our formulae reduce to the classical type formulae by Brito-Langevin-Rosenberg (1981) and Brito-Naveira (2000). The integral formulae involve the conullity tensor of a distribution, and certain components of the curvature tensor. The formulae also deal with a set of arbitrary functions depending on the scalar invariants of the co-nullity tensor. For a special choice of the functions our formulae involve the Newton transformations of the co-nullity tensor.

Open Access March 22, 2011

Real hypersurfaces in complex two-plane Grassmannians whose normal Jacobi operator is of Codazzi type

Carlos Machado, Juan Dios Pérez, Imsoon Jeong, Young Suh

Page range: 578-582

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Abstract

We prove the non-existence of real hypersurfaces in complex two-plane Grassmannians whose normal Jacobi operator is of Codazzi type.

Open Access March 22, 2011

Monotone weak Lindelöfness

Maddalena Bonanzinga, Filippo Cammaroto, Bruno Pansera

Page range: 583-592

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Abstract

The definition of monotone weak Lindelöfness is similar to monotone versions of other covering properties: X is monotonically weakly Lindelöf if there is an operator r that assigns to every open cover U a family of open sets r(U) so that (1) ∪r(U) is dense in X, (2) r(U) refines U, and (3) r(U) refines r(V) whenever U refines V. Some examples and counterexamples of monotonically weakly Lindelöf spaces are given and some basic properties such as the behavior with respect to products and subspaces are discussed.

Open Access March 22, 2011

A decomposition theorem for compact groups with an application to supercompactness

Wiesław Kubiś, Sławomir Turek

Page range: 593-602

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Abstract

We show that every compact connected group is the limit of a continuous inverse sequence, in the category of compact groups, where each successor bonding map is either an epimorphism with finite kernel or the projection from a product by a simple compact Lie group. As an application, we present a proof of an unpublished result of Charles Mills from 1978: every compact group is supercompact.

Open Access March 22, 2011

On the extent of star countable spaces

Ofelia Alas, Lucia Junqueira, Jan Mill, Vladimir Tkachuk, Richard Wilson

Page range: 603-615

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Abstract

For a topological property P, we say that a space X is star Pif for every open cover Uof the space X there exists Y ⊂ X such that St(Y,U) = X and Y has P. We consider star countable and star Lindelöf spaces establishing, among other things, that there exists first countable pseudocompact spaces which are not star Lindelöf. We also describe some classes of spaces in which star countability is equivalent to countable extent and show that a star countable space with a dense σ-compact subspace can have arbitrary extent. It is proved that for any ω 1-monolithic compact space X, if C p(X)is star countable then it is Lindelöf.

Open Access March 22, 2011

Almost fixed-point-free automorphisms of prime order

Bertram Wehrfritz

Page range: 616-626

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Abstract

Let ϕ be an automorphism of prime order p of the group G with C G(ϕ) finite of order n. We prove the following. If G is soluble of finite rank, then G has a nilpotent characteristic subgroup of finite index and class bounded in terms of p only. If G is a group with finite Hirsch number h, then G has a soluble characteristic subgroup of finite index in G with derived length bounded in terms of p and n only and a soluble characteristic subgroup of finite index in G whose index and derived length are bounded in terms of p, n and h only. Here a group has finite Hirsch number if it is poly (cyclic or locally finite). This is a stronger notion than that used in [Wehrfritz B.A.F., Almost fixed-point-free automorphisms of order 2, Rend. Circ. Mat. Palermo (in press)], where the case p = 2 is discussed.

Open Access March 22, 2011

Haar system on a product of zero-dimensional compact groups

Sergei Lukomskii

Page range: 627-639

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Abstract

In this work, we study the problem of constructing Haar bases on a product of arbitrary compact zero-dimensional Abelian groups. A general scheme for the construction of Haar functions is given for arbitrary dimension. For dimension d=2, we describe all Haar functions.

Open Access March 22, 2011

Iterative algorithms for variational inclusions, mixed equilibrium and fixed point problems with application to optimization problems

Yonghong Yao, Yeol Cho, Yeong-Cheng Liou

Page range: 640-656

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Abstract

In this paper, we introduce an iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of a nonexpansive mapping, and the the set of solutions of a variational inclusion in a real Hilbert space. Furthermore, we prove that the proposed iterative algorithm converges strongly to a common element of the above three sets, which is a solution of a certain optimization problem related to a strongly positive bounded linear operator.

Open Access March 22, 2011

On the basis property of the root functions of differential operators with matrix coefficients

Oktay Veliev

Page range: 657-672

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Abstract

We obtain asymptotic formulas for eigenvalues and eigenfunctions of the operator generated by a system of ordinary differential equations with summable coefficients and periodic or antiperiodic boundary conditions. Then using these asymptotic formulas, we find necessary and sufficient conditions on the coefficients for which the system of eigenfunctions and associated functions of the operator under consideration forms a Riesz basis.

Open Access March 22, 2011

The finite speed of propagation of solutions of the Neumann problem of a degenerate parabolic equation

Jiaqing Pan

Page range: 673-685

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Abstract

In this paper the finite speed of propagation of solutions and the continuous dependence on the nonlinearity of a degenerate parabolic partial differential equation are discussed. Our objective is to derive an explicit expression for the speed of propagation and the large time behavior of the solution and to show that the solution continuously depends on the nonlinearity of the equation.

Open Access March 22, 2011

Blow-up of the solution for higher-order Kirchhoff-type equations with nonlinear dissipation

Qingyong Gao, Fushan Li, Yanguo Wang

Page range: 686-698

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Abstract

In this paper, we consider the nonlinear Kirchhoff-type equation $$ u_{tt} + M(\left\| {D^m u(t)} \right\|_2^2 )( - \Delta )^m u + \left| {u_t } \right|^{q - 2} u_t = \left| {u_t } \right|^{p - 2} u $$ with initial conditions and homogeneous boundary conditions. Under suitable conditions on the initial datum, we prove that the solution blows up in finite time.

Open Access March 22, 2011

Dominating and total dominating partitions in cubic graphs

Justin Southey, Michael Henning

Page range: 699-708

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Abstract

In this paper, we continue the study of domination and total domination in cubic graphs. It is known [Henning M.A., Southey J., A note on graphs with disjoint dominating and total dominating sets, Ars Combin., 2008, 89, 159–162] that every cubic graph has a dominating set and a total dominating set which are disjoint. In this paper we show that every connected cubic graph on nvertices has a total dominating set whose complement contains a dominating set such that the cardinality of the total dominating set is at most (n+2)/2, and this bound is essentially best possible.

Open Access March 22, 2011

A family of L 2-spaces associated to the jumps of a Markov process

Valentin Grecea

Page range: 709-714

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Abstract

Given the (canonical) Markov process associated with a sufficiently general semigroup (P t), we establish a result concerning the uniform completeness of a family of L 2-spaces naturally associated with the jumps of the process. An application of this result is presented.

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.