$Open Mathematics$

Open Access Published by De Gruyter Open Access

Volume 3 Issue 1

March 2005

Issue of Open Mathematics

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Open Access March 1, 2005

Some alternating sums of Lucas numbers

Zvonko Čerin

Page range: 1-13

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Abstract

We consider alternating sums of squares of odd and even terms of the Lucas sequence and alternating sums of their products. These alternating sums have nice representations as products of appropriate Fibonacci and Lucas numbers.

Open Access March 1, 2005

On two theorems for flat, affine group schemes over a discrete valuation ring

Adrian Vasiu

Page range: 14-25

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Abstract

We include short and elementary proofs of two theorems that characterize reductive group schemes over a discrete valuation ring, in a slightly more general context.

Open Access March 1, 2005

Quaternionic geometry of matroids

Tamás Hausel

Page range: 26-38

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Abstract

Building on a recent paper [8], here we argue that the combinatorics of matroids are intimately related to the geometry and topology of toric hyperkähler varieties. We show that just like toric varieties occupy a central role in Stanley’s proof for the necessity of McMullen’s conjecture (or g-inequalities) about the classification of face vectors of simplicial polytopes, the topology of toric hyperkähler varieties leads to new restrictions on face vectors of matroid complexes. Namely in this paper we will give two proofs that the injectivity part of the Hard Lefschetz theorem survives for toric hyperkähler varieties. We explain how this implies the g-inequalities for rationally representable matroids. We show how the geometrical intuition in the first proof, coupled with results of Chari [3], leads to a proof of the g-inequalities for general matroid complexes, which is a recent result of Swartz [20]. The geometrical idea in the second proof will show that a pure O-sequence should satisfy the g-inequalities, thus showing that our result is in fact a consequence of a long-standing conjecture of Stanley.

Open Access March 1, 2005

An existence result for a quadrature surface free boundary problem

Mohammed Barkatou, Diaraf Seck, Idrissa Ly

Page range: 39-57

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Abstract

The aim of this paper is to present two different approachs in order to obtain an existence result to the so-called quadrature surface free boundary problem. The first one requires the shape derivative calculus while the second one depends strongly on the compatibility condition of the Neumann problem. A necessary and sufficient condition of existences is given in the radial case.

Open Access March 1, 2005

On the homotopy type of (n-1)-connected (3n+1)-dimensional free chain Lie algebra

Mahmoud Benkhalifa, Nabilah Abughzalah

Page range: 58-75

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Abstract

Let R be a subring ring of Q. We reserve the symbol p for the least prime which is not a unit in R; if R ⊒Q, then p=∞. Denote by DGL nnp, n≥1, the category of (n-1)-connected np-dimensional differential graded free Lie algebras over R. In [1] D. Anick has shown that there is a reasonable concept of homotopy in the category DGL nnp. In this work we intend to answer the following two questions: Given an object (L(V), ϖ) in DGL n3n+2 and denote by S(L(V), ϖ) the class of objects homotopy equivalent to (L(V), ϖ). How we can characterize a free dgl to belong to S(L(V), ϖ)? Fix an object (L(V), ϖ) in DGL n3n+2. How many homotopy equivalence classes of objects (L(W), δ) in DGL n3n+2 such that H * (W, d′)≊H * (V, d) are there? Note that DGL n3n+2 is a subcategory of DGL nnp when p>3. Our tool to address this problem is the exact sequence of Whitehead associated with a free dgl.

Open Access March 1, 2005

Bolzano’s intermediate-value theorem for quasi-holomorphic maps

Aboubakr Bayoumi

Page range: 76-82

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Abstract

We extend Bolzano’s intermediate-value theorem to quasi-holomorphic maps of the space of continuous linear functionals from l p into the scalar field, (0< p<1). This space is isomorphic to l ∞.

Open Access March 1, 2005

Congruences, ideals and annihilators in standard QBCC-algebras

Radomír Halaš, Luboš Plojhar

Page range: 83-97

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Abstract

We characterize congruence lattices of standard QBCC-algebras and their connection with the congruence lattices of congruence kernels.

Open Access March 1, 2005

Oscillations of linear integro-differential equations

Rudolf Olach, Helena Šamajová

Page range: 98-104

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Abstract

Sufficient conditions which guarantee that certain linear integro-differential equation cannot have a positive solution are established.

Open Access March 1, 2005

Localization of LM n-algebras

Florentina Chirte§

Page range: 105-124

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Abstract

The aim of this paper is to define the localization LM n-algebra of an LM n—algebra L with respect to a topology F on L; in Section 5 we prove that the maximal LM n-algebra of fractions (defined in [3]) and the LM n-algebra of fractions relative to an Λ—closed system (defined in Section 2) are LM n-algebras of localization.

Open Access March 1, 2005

Left-sided quasi-invertible bimodules over Nakayama algebras

Zygmunt Pogorzały

Page range: 125-142

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Abstract

Bimodules over triangular Nakayama algebras that give stable equivalences of Morita type are studied here. As a consequence one obtains that every stable equivalence of Morita type between triangular Nakayama algebras is a Morita equivalence.

Open Access March 1, 2005

On the two-point boundary-value problem for the Riccati matrix differential equation

V. Laptinsky, I. Makovetsky

Page range: 143-154

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Abstract

Constructive sufficient conditions for univalent resolvability of a two-point boundary value problem for nonlinear Riccati equation are obtained. An illustrative example is 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.