$Open Mathematics$

Open Access Published by De Gruyter Open Access

Volume 5 Issue 3

September 2007

Issue of Open Mathematics

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Contents
Journal Overview

Open Access September 1, 2007

Scattering monodromy and the A 1 singularity

Larry Bates, Richard Cushman

Page range: 429-451

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Abstract

We present the notion of scattering monodromy for a two degree of freedom hyperbolic oscillator and apply this idea to determine the Picard-Lefschetz monodromy of the isolated singular point of a quadratic function of two complex variables.

Open Access September 1, 2007

Comparison between different duals in multiobjective fractional programming

Radu Boţ, Robert Chares, Gert Wanka

Page range: 452-469

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Abstract

The present paper is a continuation of [2] where we deal with the duality for a multiobjective fractional optimization problem. The basic idea in [2] consists in attaching an intermediate multiobjective convex optimization problem to the primal fractional problem, using an approach due to Dinkelbach ([6]), for which we construct then a dual problem expressed in terms of the conjugates of the functions involved. The weak, strong and converse duality statements for the intermediate problems allow us to give dual characterizations for the efficient solutions of the initial fractional problem. The aim of this paper is to compare the intermediate dual problem with other similar dual problems known from the literature. We completely establish the inclusion relations between the image sets of the duals as well as between the sets of maximal elements of the image sets.

Open Access May 13, 2007

On the lattice of n-filters of an LM n-algebra

Dumitru Buşneag, Florentina Chirteş

Page range: 470-483

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Abstract

For an n-valued Łukasiewicz-Moisil algebra L (or LM n-algebra for short) we denote by F n(L) the lattice of all n-filters of L. The goal of this paper is to study the lattice F n(L) and to give new characterizations for the meet-irreducible and completely meet-irreducible elements on F n(L).

Open Access September 1, 2007

Distributive implication groupoids

Ivan Chajda, Radomir Halaš

Page range: 484-492

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Abstract

We introduce a concept of implication groupoid which is an essential generalization of the implication reduct of intuitionistic logic, i.e. a Hilbert algebra. We prove several connections among ideals, deductive systems and congruence kernels which even coincide whenever our implication groupoid is distributive.

Open Access September 1, 2007

Harmonic conformal flows on manifolds of constant curvature

Amine Fawaz

Page range: 493-504

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Abstract

We compute the energy of conformal flows on Riemannian manifolds and we prove that conformal flows on manifolds of constant curvature are critical if and only if they are isometric.

Open Access April 28, 2007

Under which conditions is the Jacobi space $$L_{w^{(a,b)} }^p [ - 1,1]$$ subset of $$L_{w^{(\alpha ,\beta )} }^1 [ - 1,1]$$ ?

Michael Felten

Page range: 505-511

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Abstract

Exact conditions for α, β, a, b > −1 and 1 ≤ p ≤ ∞ are determined under which the inclusion property $$L_{w^{(a,b)} }^p [ - 1,1]$$ ⊂ $$L_{w^{(\alpha ,\beta )} }^1 [ - 1,1]$$ is valid. It is shown that the conditions characterize the inclusion property. The paper concludes with some results, in which the inclusion property can be detected in relation with estimates of Jacobi differential operators and with Muckenhoupt’s transplantation theorems and multiplier theorems for Jacobi series.

Open Access September 1, 2007

Holomorphic automorphisms and collective compactness in J*-algebras of operator

José Isidro

Page range: 512-522

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Abstract

Let G be the Banach-Lie group of all holomorphic automorphisms of the open unit ball $$B_\mathfrak{A} $$ in a J*-algebra $$\mathfrak{A}$$ of operators. Let $$\mathfrak{F}$$ be the family of all collectively compact subsets W contained in $$B_\mathfrak{A} $$ . We show that the subgroup F ⊂ G of all those g ∈ G that preserve the family $$\mathfrak{F}$$ is a closed Lie subgroup of G and characterize its Banach-Lie algebra. We make a detailed study of F when $$\mathfrak{A}$$ is a Cartan factor.

Open Access September 1, 2007

Low rank Tucker-type tensor approximation to classical potentials

B. Khoromskij, V. Khoromskaia

Page range: 523-550

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Abstract

This paper investigates best rank-(r 1,..., r d) Tucker tensor approximation of higher-order tensors arising from the discretization of linear operators and functions in ℝd. Super-convergence of the best rank-(r 1,..., r d) Tucker-type decomposition with respect to the relative Frobenius norm is proven. Dimensionality reduction by the two-level Tucker-to-canonical approximation is discussed. Tensor-product representation of basic multi-linear algebra operations is considered, including inner, outer and Hadamard products. Furthermore, we focus on fast convolution of higher-order tensors represented by the Tucker/canonical models. Optimized versions of the orthogonal alternating least-squares (ALS) algorithm is presented taking into account the different formats of input data. We propose and test numerically the mixed CT-model, which is based on the additive splitting of a tensor as a sum of canonical and Tucker-type representations. It allows to stabilize the ALS iteration in the case of “ill-conditioned” tensors. The best rank-(r 1,..., r d) Tucker decomposition is applied to 3D tensors generated by classical potentials, for example $$\tfrac{1}{{\left| {x - y} \right|}}, e^{ - \alpha \left| {x - y} \right|} , \tfrac{{e^{ - \left| {x - y} \right|} }}{{\left| {x - y} \right|}}$$ and $$\tfrac{{erf(|x|)}}{{|x|}}$$ with x, y ∈ ℝd. Numerical results for tri-linear decompositions illustrate exponential convergence in the Tucker rank, and robustness of the orthogonal ALS iteration.

Open Access May 13, 2007

Strengthened Moser’s conjecture, geometry of Grunsky coefficients and Fredholm eigenvalues

Samuel Krushkal

Page range: 551-580

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Abstract

The Grunsky and Teichmüller norms ϰ(f) and k(f) of a holomorphic univalent function f in a finitely connected domain D ∋ ∞ with quasiconformal extension to $$\widehat{\mathbb{C}}$$ are related by ϰ(f) ≤ k(f). In 1985, Jürgen Moser conjectured that any univalent function in the disk Δ* = {z: |z| > 1} can be approximated locally uniformly by functions with ϰ(f) < k(f). This conjecture has been recently proved by R. Kühnau and the author. In this paper, we prove that approximation is possible in a stronger sense, namely, in the norm on the space of Schwarzian derivatives. Applications of this result to Fredholm eigenvalues are given. We also solve the old Kühnau problem on an exact lower bound in the inverse inequality estimating k(f) by ϰ(f), and in the related Ahlfors inequality.

Open Access May 13, 2007

Connections between Romanovski and other polynomials

Hans Weber

Page range: 581-595

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Abstract

A connection between Romanovski polynomials and those polynomials that solve the one-dimensional Schrödinger equation with the trigonometric Rosen-Morse and hyperbolic Scarf potential is established. The map is constructed by reworking the Rodrigues formula in an elementary and natural way. The generating function is summed in closed form from which recursion relations and addition theorems follow. Relations to some classical polynomials are also given.

Open Access May 26, 2007

On hypercentral groups

B. Wehrfritz

Page range: 596-606

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Abstract

Let G be a hypercentral group. Our main result here is that if G/G’ is divisible by finite then G itself is divisible by finite. This extends a recent result of Heng, Duan and Chen [2], who prove in a slightly weaker form the special case where G is also a p-group. If G is torsion-free, then G is actually divisible.

Open Access September 1, 2007

Frequent oscillation in a nonlinear partial difference equation

Jun Yang, Yu Zhang, Sui Cheng

Page range: 607-618

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Abstract

This paper is concerned with a class of nonlinear delay partial difference equations with variable coefficients, which may change sign. By making use of frequency measures, some new oscillatory criteria are established. This is the first time oscillation of these partial difference equations is discussed by employing frequency measures.

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.