Strong Positivity in Right-Invariant Order on Braid Groups and Quasipositivity

Dehornoy constructed a right invariant order on the braid group B _n uniquely defined by the condition 1{\text{ if }}\beta _0 ,\beta _1$$ " align="middle" border="0"> are words in . A braid is called strongly positive if 1$$ " align="middle" border="0"> for any . In the present paper it is proved that the braid is strongly positive if the word does not contain . We also provide a geometric proof of the result by Burckel and Laver that the standard generators of a braid group are strongly positive. Finally, we discuss relations between the right invariant order and quasipositivity.     

Endomorphisms of matrix semigroups over division rings

Let be an arbitrary division ring and M _n ( ) the multiplicative semigroup of all n × n matrices over . We describe the general form of endomorphisms of M _n ( ). Supported in part by a grant from the Ministry of Science of Slovenia.     

The Number of Compatible Totally Bounded Quasi-Uniformities

We prove that a topological space that admits a nontransitive totally bounded quasi-uniformity, admits at least nontransitive totally bounded quasi-uniformities. Furthermore we show that each infinite T ₂-space admits at least transitive totally bounded quasi-uniformities. In fact, each supersober space having a discrete subspace of infinite cardinality admits at least transitive totally bounded quasi-uniformities.     

Numerical bifurcation and stability analysis for steady-states of reaction diffusion equations

Solutions of a semilinear elliptic boundary value problem, (with bounded below) can be put into a one-to-one correspondence with zeros of a function . Often d is small. The function is called the bifurcation function. It can also be shown that the eigenvalues of the matrix characterize the stability properties of the solutions of the elliptic problem as rest points of . A finite element method that can be used for computing B and B _c has recently been proposed. An overview of these results and the finite element method is given. This revised version was published online in June 2006 with corrections to the Cover Date.     

Levi Classes Generated by Nilpotent Groups

Let be a class of all groups G for which the normal closure (x) ^G of every element x belongs to a class . is a Levi class generated by . Let and ₀ be classes of finitely generated nilpotent groups and of torsion-free, finitely generated, nilpotent groups, respectively. We prove that and , and so and . It is shown that quasivarieties and are closed under free products, and that each contains at most one maximal proper subquasivariety. It is also proved that is closed under free products if so is .     

Denseness of holomorphic functions attaining their numerical radii

For two complex Banach spaces X and Y, (B _X; Y) will denote the space of bounded and continuous functions from B _X to Y that are holomorphic on the open unit ball. The numerical radius of an element h in (B _X; X) is the supremum of the set

Complexities of finite families of polynomials,Weyl systems,and constructions in combinatorial number theory

We introduce two notions of complexity of a system of polynomials p ₁,..., p _r ∈ ℤ[n] and apply them to characterize the limits of the expressions of the form where T is a skew-product transformation of a torus and are measurable sets. The dynamical results obtained allow us to construct subsets of integers with specific combinatorial properties related to the polynomial Szemerédi theorem. Bergelson and Leibman were supported by NSF grants DMS-0345350 and DMS-0600042.     

Biorthogonal bases and multiple interpolation in weighted Hardy spaces

Let {z_k=x_k+iy_k} be a sequence on upper half plane and {s_i} be the number of appearence of z_k in {z₁,z₂,...,z_k}. Suppose sup s_i<+∞. Let ω(x) be a weight belonging to A^∞ and . We Consider the weighted Hardy space and operator T_p mapping f(z)∈H _+w ^p into a sequence defined by , 0<p≤+∞, j=1,2,.... Then T_p(H _+w ^p )=l^p if and only if {z_k} is uniformly separated. Besides the effective solution for interpolation is obtained. Supported by National Science Foundation of China and Shanghai Youth Science Foundation     

Quantum duality in quantum deformations

In accordance with the quantum duality principle, the twisted algebra is equivalent to the quantum group and has two preferred bases: one inherited from the universal enveloping algebra and the other generated by coordinate functions of the dual Lie group . We show howthe transformation can be explicitly obtained for any simple Lie algebra and a factorable chain of extended Jordanian twists. In the algebra , we introduce a natural vector grading , compatible with the adjoint representation of the algebra. Passing to the dual-group coordinates allows essentially simplifying the costructure of the deformed Hopf algebra , considered as a quantum group . The transformation can be used to construct new solutions of the twist equations. We construct a parameterized family of extended Jordanian deformations and study it in terms of ; we find new realizations of the parabolic twist. Dedicated to the birthday of my teacher, Yurii Novozhilov __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 1, pp. 112–125, July, 2006.     

On posets with isomorphic interval posets

Let be a partially ordered set, Int the system of all (nonempty) intervals of partially ordered by the set-theoretical inclusion . We are interested in partially ordered sets with Int isomorphic to Int . We are going to show that they correspond to couples of binary relations on A satisfying some conditions. If is a directed partially ordered set, the only with Int isomorphic to Int are corresponding to direct decompositions of ( denotes the dual of . The present results include those presented in the paper [11] by V. Slavík. Systems of intervals, particularly of lattices, have been investigated by many authors, cf. [1]–[11].     

Energy transfer in the Kuramoto-Sivashinskii equation on a multidimensional torus with Riemannian metric

In this paper, we consider the Kuramoto-Sivashinskii equation on the multidimensional torus with a Riemannian metric: where . For this equation the theorem on energy transfer holds. It is formulated in the following way. Let be the Fourier expansion of the function u. Denote by P _N and P _N ^⊥ the operators of rejection of the “leading” and, respectively, “lowest” terms of the Fourier expansion (harmonics), i.e., . For any ρ > 0,N ∈ ℕ, s ≥ n/2+5, and λ ∈ (0, 1), there exists R such that for any function. ϕ ∈ lying outside the ball in the ball , there exists an instant of time t ∈ (0, 1) such that g _KS ^t ϕ=ψ and . Here, R is a constant depending on the metric (g), n _s is the sth Sobolev norm, and is the C ¹-norm. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 24, Dynamical Systems and Optimization, 2005.     

Construction of conjugate quadrature filters with specified zeros

Let ℂ denote the complex numbers and denote the ring of complex-valued Laurent polynomial functions on ℂ\{0}. Furthermore, we denote by the subsets of Laurent polynomials whose restriction to the unit circle is real, nonnegative, respectively. We prove that for any two Laurent polynomials , which have no common zeros in ℂ\{0} there exists a pair of Laurent polynomials satisfying the equation Q ₁ P ₁ + Q ₂ P ₂ = 1. We provide some information about the minimal length Laurent polynomials Q ₁ and Q ₂ with these properties and describe an algorithm to compute them. We apply this result to design a conjugate quadrature filter whose zeros contain an arbitrary finite subset Λ⊂ℂ\{0} with the property that for every implies and . This revised version was published online in June 2006 with corrections to the Cover Date.     

On the Durrmeyer-type operators for bivariate functions

For measurable functions f of two real variables there are considered the Boolean sums of parametric extensions of certain univariate Durmeyer-type operators and . The weighted mixed moduli of continuity of are estimated and the degrees of approximation of f by in some weighted norms are investigated.     

Metrics in the sphere of a C*-module

Given a unital C*-algebra and a right C*-module over , we consider the problem of finding short smooth curves in the sphere = {x ∈ : 〈x, x〉 = 1}. Curves in are measured considering the Finsler metric which consists of the norm of at each tangent space of . The initial value problem is solved, for the case when is a von Neumann algebra and is selfdual: for any element x ₀ ∈ and any tangent vector ν at x ₀, there exists a curve γ(t) = e ^tZ (x ₀), Z ∈ , Z* = −Z and ∥Z∥ ≤ π, such that γ(0) = x ₀ and (0) = ν, which is minimizing along its path for t ∈ [0, 1]. The existence of such Z is linked to the extension problem of selfadjoint operators. Such minimal curves need not be unique. Also we consider the boundary value problem: given x ₀, x ₁ ∈ , find a curve of minimal length which joins them. We give several partial answers to this question. For instance, let us denote by f ₀ the selfadjoint projection I − x ₀ ⊗ x ₀, if the algebra f ₀ f ₀ is finite dimensional, then there exists a curve γ joining x ₀ and x ₁, which is minimizing along its path.      

Precise asymptotics in the Baum-Katz and davis law of large numbers for positively associated sequences

§ 1　 Introduction and resultsL et { X,Xi;i≥ 1} be a sequence of i.i.d.random variables,and set Sn= ni=1 Xi,n≥1.Hsu and Robbins[1 ] introduced the conceptof complete convergence.They together withErdos[2 ] proved n≥ 1 P(|Sn|≥εn) <∞ ,ε>0 (1)if and only if EX=0 and EX2 <∞ .L ater,Spitzer[3] proved n≥ 11n P(|Sn|≥εn) <∞ ,ε>0if and only if EX =0 and E|X|<∞ .More generally,it was shown by Baum and Katz[4 ]that,for 0 0 (…     

Imprimitive Q-polynomial Association Schemes

It is well known that imprimitive P-polynomial association schemes with are either bipartite or antipodal, i.e., intersection numbers satisfy either for all for all . In this paper, we show that imprimitive -polynomial association schemes with are either dual bipartite or dual antipodal, i.e., dual intersection numbers satisfy either .     

Generalized (anti) Yetter-Drinfeld modules as components of a braided T-category

If H is a Hopf algebra with bijective antipode and α, β ∈ Aut _Hopf (H), we introduce a category , generalizing both Yetter-Drinfeld modules and anti-Yetter-Drinfeld modules. We construct a braided T-category having all the categories as components, which, if H is finite dimensional, coincides with the representations of a certain quasitriangular T-coalgebra DT(H) that we construct. We also prove that if (α, β) admits a so-called pair in involution, then is isomorphic to the category of usual Yetter-Drinfeld modules . Research partially supported by the programme CERES of the Romanian Ministry of Education and Research, contract no. 4-147/2004.     

Equational Theories for Classes of Finite Semigroups

It is proved that there exists an infinite sequence of finitely based semigroup varieties such that, for all i, an equational theory for and for the class of all finite semigroups in is undecidable while an equational theory for and for the class of all finite semigroups in is decidable. An infinite sequence of finitely based semigroup varieties is constructed so that, for all i, an equational theory for and for the class of all finite semigroups in is decidable whicle an equational theory for and for the class of all finite semigroups in is not.     

Isometries and direct decompositions of pseudo MV-algebras

In the paper isometries in pseudo MV-algebras are investigated. It is shown that for every isometry f in a pseudo MV-algebra = (A, ⊕, ⁻, ^∼, 0, 1) there exists an internal direct decomposition of with commutative such that and for each x ∈ A. On the other hand, if is an internal direct decomposition of a pseudo MV-algebra = (A, ⊕, ⁻, ^∼, 0, 1) with commutative, then the mapping g: A → A defined by is an isometry in and .      

K-Theory for the Banach Algebra of Operators on James's Quasi-reflexive Banach Spaces

We prove that the K-groups of the Banach algebra of bounded, linear operators on the pth James space , where 1 < p < , are given by and . Moreover, for each Banach space and each non-zero, closed ideal contained in the ideal of inessential operators, we show that and . This enables us to calculate the K-groups of for each Banach space which is a direct sum of finitely many James spaces and -spaces.