Showing papers in "Open Mathematics in 2006"

Left-symmetric algebras, or pre-Lie algebras in geometry and physics

Abstract: In this survey article we discuss the origin, theory and applications of left-symmetric algebras (LSAs in short) in geometry in physics. Recently Connes, Kreimer and Kontsevich have introduced LSAs in mathematical physics (QFT and renormalization theory), where the name pre-Lie algebras is used quite often. Already Cayley wrote about such algebras more than hundred years ago. Indeed, LSAs arise in many different areas of mathematics and physics. We attempt to give a survey of the fields where LSAs play an important role. Furthermore we study the algebraic theory of LSAs such as structure theory, radical theory, cohomology theory and the classification of simple LSAs. We also discuss applications to faithful Lie algebra representations.

On presentations of Brauer-type monoids

Abstract: We obtain presentations for the Brauer monoid, the partial analogue of the Brauer monoid, and for the greatest factorizable inverse submonoid of the dual symmetric inverse monoid. In all three cases we apply the same approach, based on the realization of all these monoids as Brauer-type monoids.

Rational values of the arccosine function

Abstract: We give a short proof to characterize the cases when arccos(√r), the arccosine of the squareroot of a rational number r ∈ [0, 1], is a rational multiple of π: This happens exactly if r is an integer multiple of 1/4. The proof relies on the well-known recurrence relation for the Chebyshev polynomials of the first kind.

The set of toric minimal log discrepancies

Abstract: We describe the set of minimal log discrepancies of toric log varieties, and study its accumulation points.

Some non-local problems for the parabolic-hyperbolic type equation with non-characteristic line of changing type

Abstract: In this work two non-local problems for the parabolic-hyperbolic type equation with non- characteristic line of changing type are considered. Unique solvability of these problems is proven. The uniqueness of the solution is proven by the method of energy integrals and the existence is proven by the method of integral equations.

Priestley dualities for some lattice-ordered algebraic structures, including MTL, IMTL and MV-algebras

Abstract: In this work we give a duality for many classes of lattice ordered algebras, as Integral Commutative Distributive Residuated Lattices MTL-algebras, IMTL-algebras and MV-algebras (see page 604). These dualities are obtained by restricting the duality given by the second author for DLFI-algebras by means of Priestley spaces with ternary relations (see [2]). We translate the equations that define some known subvarieties of DLFI-algebras to relational conditions in the associated DLFI-space.

Blow-up of regular submanifolds in Heisenberg groups and applications

Abstract: We obtain a blow-up theorem for regular submanifolds in the Heisenberg group, where intrinsic dilations are used. Main consequence of this result is an explicit formula for the density of (p+1)-dimensional spherical Hausdorff measure restricted to a p-dimensional submanifold with respect to the Riemannian surface measure. We explicitly compute this formula in some simple examples and we present a lower semicontinuity result for the spherical Hausdorff measure with respect to the weak convergence of currents. Another application is the proof of an intrinsic coarea formula for vector-valued mappings on the Heisenberg group.

Finite dimensional global attractor for a class of doubly nonlinear parabolic equations

Abstract: Our aim in this paper is to study the long time behavior of a class of doubly nonlinear parabolic equations. In particular, we prove the existence of the global attractor which has, in one and two space dimensions, finite fractal dimension.

On the doubly connected domination number of a graph

Abstract: For a given connected graph G = (V, E), a set \(D \subseteq V(G)\) is a doubly connected dominating set if it is dominating and both 〈D〉 and 〈V (G)-D〉 are connected. The cardinality of the minimum doubly connected dominating set in G is the doubly connected domination number. We investigate several properties of doubly connected dominating sets and give some bounds on the doubly connected domination number.

On hyperbolic virtual polytopes and hyperbolic fans

Abstract: Hyperbolic virtual polytopes arose originally as polytopal versions of counterexamples to the following A.D.Alexandrov's uniqueness conjecture: Let K ⊂ R 3 be a smooth convex body. If for a constant C, at every point of ∂K, we have R1 ≤ C ≤ R2 then K is a ball. (R1 and R2 stand for the principal curvature radii of ∂K.) This paper gives a new (in comparison with the previous construction by Y.Martinez-Maure and by G.Panina) series of counterexamples to the conjecture. In particular, a hyperbolic virtual polytope (and therefore, a hyperbolic herisson) with odd an number of horns is constructed. Moreover, various properties of hyperbolic virtual polytopes and their fans are discussed.

σ-asymptotically lacunary statistical equivalent sequences

Abstract: This paper presents the following definitions which is a natural combination of the definition for asymptotically equivalent, statistically limit, lacunary sequences, and σ-convergence. Let ϑ be a lacunary sequence; Two nonnegative sequences [x] and [y] are S σ,8-asymptotically equivalent of multiple L provided that for every e > 0 $$\mathop {\lim }\limits_r \frac{1}{{h_r }}\left\{ {k \in I_r :\left| {\frac{{x_{\sigma ^k (m)} }}{{y_{\sigma ^k (m)} }} - L} \right| \geqslant \in } \right\} = 0$$ uniformly in m = 1, 2, 3, ..., (denoted by x $$\mathop \sim \limits^{S_{\sigma ,\theta } } $$ y) simply S σ,8-asymptotically equivalent, if L = 1. Using this definition we shall prove S σ,8-asymptotically equivalent analogues of Fridy and Orhan’s theorems in [5] and analogues results of Das and Patel in [1] shall also be presented.

Exact laws for sums of ratios of order statistics from the Pareto distribution

Abstract: Consider independent and identically distributed random variables {Xnk, 1 ≤ k ≤ m, n ≤ 1} from the Pareto distribution. We select two order statistics from each row, Xn(i) ≤ Xn(j), for 1 ≤ i < j ≤ = m. Then we test to see whether or not Laws of Large Numbers with nonzero limits exist for weighted sums of the random variables Rij = Xn(j)/Xn(i).

Local geometry of orbits for an ordinary classical lie supergroup

Abstract: In this paper we identify a real reductive dual pair of Roger Howe with an Ordinary Classical Lie supergroup. In these terms we describe the semisimple orbits of the dual pair in the symplectic space, a slice through a semisimple element of the symplectic space, an analog of a Cartan subalgebra, the corresponding Weyl group and the corresponding Weyl integration formula.

Accelerating the convergence of trigonometric series

Abstract: A nonlinear method of accelerating both the convergence of Fourier series and trigonometric interpolation is investigated. Asymptotic estimates of errors are derived for smooth functions. Numerical results are represented and discussed.

On solutions of third order nonlinear differential equations

Abstract: The aim of our paper is to study oscillatory and asymptotic properties of solutions of nonlinear differential equations of the third order with deviating argument. In particular, we prove a comparison theorem for properties A and B as well as a comparison result on property A between nonlinear equations with and without deviating arguments. Our assumptions on nonlinearity f are related to its behavior only in a neighbourhood of zero and/or of infinity.

On the dynamics of equations with infinite delay

Abstract: We consider a system of ordinary differential equations with infinite delay. We study large time dynamics in the phase space of functions with an exponentially decaying weight. The existence of an exponential attractor is proved under the abstract assumption that the right-hand side is Lipschitz continuous. The dimension of the attractor is explicitly estimated.

Unsaturated solutions for partial difference equations with forcing terms

Abstract: The concept of unsaturated infinite double sequence is introduced by making use of frequency measures. Unsaturated solutions are then studied for a partial difference equation. Conditions for all solutions to be unsaturated are obtained. Since unsaturated solutions are oscillatory, our results yield oscillation criteria.

Ordinary differential equations and their exponentials

Abstract: In the context of Synthetic Differential Geometry, we discuss vector fields/ordinary differential equations as actions; in particular, we exploit function space formation (exponential spaces) in the category of actions.

Cauchy, Ferrers-Jackson and Chebyshev polynomials and identities for the powers of elements of some conjugate recurrence sequences

Abstract: In this paper some decompositions of Cauchy polynomials, Ferrers-Jackson polynomials and polynomials of the form x2n + y2n, n ∈ ℕ, are studied. These decompositions are used to generate the identities for powers of Fibonacci and Lucas numbers as well as for powers of the so called conjugate recurrence sequences. Also, some new identities for Chebyshev polynomials of the first kind are presented here.

An equivalence between varieties of cyclic Post algebras and varieties generated by a finite field

Abstract: In this paper we give a term equivalence between the simple k-cyclic Post algebra of order p, L p,k, and the finite field F(p k) with constants F(p). By using Lagrange polynomials, we give an explicit procedure to obtain an interpretation Φ1 of the variety V(L p,k) generated by L p,k into the variety V(F(p k)) generated by F(p k) and an interpretation Φ2 of V(F(p k)) into V(L p,k) such that Φ2Φ1(B) = B for every B e V(L p,k) and Φ1Φ2(R) = R for every R e V(F(p k)).

Duality triads of higher rank: Further properties and some examples

Abstract: It is shown that duality triads of higher rank are closely related to orthogonal matrix polynomials on the real line. Furthermore, some examples of duality triads of higher rank are discussed. In particular, it is shown that the generalized Stirling numbers of rank r give rise to a duality triad of rank r.

An existence result for an interior electromagnetic casting problem

Abstract: This paper deals with an interior electromagnetic casting (free boundary) problem. We begin by showing that the associated shape optimization problem has a solution which is of class C2. Then, using the shape derivative and the maximum principle, we give a sufficient condition that the minimum obtained solves our problem.

A poset hierarchy

Abstract: This article extends a paper of Abraham and Bonnet which generalised the famous Hausdorff characterisation of the class of scattered linear orders. They gave an inductively defined hierarchy that characterised the class of scattered posets which do not have infinite incomparability antichains (i.e. have the FAC). We define a larger inductive hierarchy κℌ* which characterises the closure of the class of all κ-well-founded linear orders under inversions, lexicographic sums and FAC weakenings. This includes a broader class of “scattered” posets that we call κ-scattered. These posets cannot embed any order such that for every two subsets of size < κ, one being strictly less than the other, there is an element in between. If a linear order has this property and has size κ it is unique and called ℚ(κ). Partial orders such that for every a < b the set {x: a < x < b} has size ≥ κ are called weakly κ-dense, and posets that do not have a weakly κ-dense subset are called strongly κ-scattered. We prove that κℌ* includes all strongly κ-scattered FAC posets and is included in the class of all FAC κ-scattered posets. For κ = ℵ0 the notions of scattered and strongly scattered coincide and our hierarchy is exactly aug(ℌ) from the Abraham-Bonnet theorem.

A note on absolute weighted mean summability factors

Abstract: In this paper we have proved a main theorem concerning the | $$\bar N$$ , p n; δ |k summability methods, which generalizes a result of Bor and Ozarslan [3]

On some new spectral estimates for Schrödinger-like operators

Abstract: We prove the analog of the Cwikel-Lieb-Rozenblum estimate for a wide class of second-order elliptic operators by two different tools: Lieb-Thirring inequalities for Schrodinger operators with matrix-valued potentials and Sobolev inequalities for warped product spaces.

On a generalization of duality triads

Abstract: Some aspects of duality triads introduced recently are discussed In particular, the general solution for the triad polynomials is given Furthermore, a generalization of the notion of duality triad is proposed and some simple properties of these generalized duality triads are derived

Bayoumi Quasi-Differential is different from Fréchet-Differential

Abstract: We prove that the Quasi Differential of Bayoumi of maps between locally bounded F-spaces may not be Frechet-Differential and vice versa. So a new concept has been discovered with rich applications (see [1–6]). Our F-spaces here are not necessarily locally convex

Subsheaves of the cotangent bundle

Abstract: For any smooth projective variety, we study a birational invariant, defined by Campana which depends on the Kodaira dimension of the subsheaves of the cotangent bundle of the variety and its exterior powers.

On differences of two squares

Abstract: The arithmetic function ρ(n) counts the number of ways to write a positive integer n as a difference of two squares. Its average size is described by the Dirichlet summatory function Σn≤xρ(n), and in particular by the error term R(x) in the corresponding asymptotics. This article provides a sharp lower bound as well as two mean-square results for R(x), which illustrates the close connection between ρ(n) and the number-of-divisors function d(n).

Category with a natural cone

Abstract: Generally, in homotopy theory a cylinder object (or, its dual, a path object) is used to define homotopy between morphisms, and a cone object is used to build exact sequences of homotopy groups. Here, an axiomatic theory based on a cone functor is given. Suspension objects are associated to based objects and cofibrations, obtaining homotopy groups referred to an object and relative to a cofibration, respectively. Exact sequences of these groups are built. Algebraic and particular examples are given. We point out that the main results of this paper were already stated in [3], and the purpose of this article is to give full details of the foregoing.