Quantifications des algèbres de Hopf d'arbres plans décorés et lien avec les groupes quantiques

We introduce a functor from the category of braided spaces into the category of braided Hopf algebras which associates to a braided space V a braided Hopf algebra of planar rooted trees . We show that the Nichols algebra of V is a subquotient of . We construct a Hopf pairing between and , generalising one of the results of [Bull. Sci. Math. 126 (2002) 193-239]. When the braiding of c is given by c(v_i⊗v_j)=q_i,jv_j⊗v_i, we obtain a quantification of the Hopf algebras introduced in [Bull. Sci. Math. 126 (2002) 193-239; 126 (2002) 249-288]. When q_i,j=q^a_i,j, with q an indeterminate and (a_i,j)_i,j the Cartan matrix of a semi-simple Lie algebra , then is a subquotient of . In this case, we construct the crossed product of with a torus and then the Drinfel'd quantum double of this Hopf algebra. We show that is a subquotient of .     

Superinjective simplicial maps of complexes of curves and injective homomorphisms of subgroups of mapping class groups

Let S be a closed, connected, orientable surface of genus at least 3, be the complex of curves on S and be the extended mapping class group of S. We prove that a simplicial map, , preserves nondisjointness (i.e. if α and β are two vertices in and i(α,β)≠0, then i(λ(α),λ(β))≠0) iff it is induced by a homeomorphism of S. As a corollary, we prove that if K is a finite index subgroup of and is an injective homomorphism, then f is induced by a homeomorphism of S and f has a unique extension to an automorphism of .     

Multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations

In this paper we study the existence and qualitative property of standing wave solutions for the nonlinear Schrödinger equation with E being a critical frequency in the sense that . We show that if the zero set of W−E has several isolated connected components Z_i(i=1,…,m) such that the interior of Z_i is not empty and ∂Z_i is smooth, then for ?>0 small there exists, for any integer k,1?k?m, a standing wave solution which is trapped in a neighborhood of , where is any given subset of . Moreover the amplitude of the standing wave is of the level . This extends the result of Byeon and Wang (Arch. Rational Mech. Anal. 165 (2002) 295) and is in striking contrast with the non-critical frequency case , which has been studied extensively in the past 20 years.     

Geometric degree of finite extensions of mappings from a smooth variety

Let f:V→W be a finite polynomial mapping of algebraic subsets V,W of and , respectively, with n≤m. It is known that f can be extended to a finite polynomial mapping . Moreover, it is known that, if V,W are smooth of dimension k,4k+2≤n=m, and f is dominated on every component (without vertical components) then there exists a finite polynomial extension such that , where means the number of points in the generic fiber of h. In this note we improve this result. Namely we show that there exists a finite polynomial extension such that .     

Multivariate stress-strength reliability model and its evaluation for coherent structures

Consider a system which has n independent components (or subsystems) each consisting of m dependent elements. Let , i=1,2,…,n denote the random strength vector of the ith component, where denotes the random strength of the jth element of the ith component. The elements of the components are subjected to a common random stress over time. In this paper, we setup a multivariate stress-strength model based on the conditional ordering between s and and evaluate the reliability of coherent structures in this setup.     

Progressive Type II censored order statistics for multivariate observations

For a sequence of independent and identically distributed random vectors , i=1,2,…,n, we consider the conditional ordering of these random vectors with respect to the magnitudes of , where N is a p-variate continuous function defined on the support set of X₁ and satisfying certain regularity conditions. We also consider the Progressive Type II right censoring for multivariate observations using conditional ordering. The need for the conditional ordering of random vectors exists for example, in reliability analysis when a system has n independent components each consisting of p arbitrarily dependent and parallel connected elements. Let the vector of life lengths for the ith component of the system be , where denotes the life length of the jth element of the ith component. Then the first failure in the system occurs at time , and for this case . In this paper we introduce the conditionally ordered and Progressive Type II right-censored conditionally ordered statistics for multivariate observations and to study their distributional properties.     

Quasi sure analysis of local times of anticipating smooth semimartingales

Let be the anticipating smooth semimartingale and be its generalized local time. In this paper, we give some estimates about the quasi sure property of Xt and its quadratic variation process t〈X〉. We also study the fractional smoothness of and prove that the quadratic variation process of can be constructed as the quasi sure limit of the form , where is a sequence of subdivisions of [a,b], , i=0,1,…,n².     

Weak and strong convergence theorems for finite families of asymptotically nonexpansive mappings in Banach spaces

Let E be a real uniformly convex Banach space whose dual space E^∗ satisfies the Kadec-Klee property, K be a closed convex nonempty subset of E. Let be asymptotically nonexpansive mappings of K into E with sequences (respectively) satisfying kin→1 as n→∞, i=1,2,…,m, and . For arbitrary ?∈(0,1), let be a sequence in [?,1−?], for each i∈{1,2,…,m} (respectively). Let {xn} be a sequence generated for m?2 by

     

On the Spaltenstein correspondence

In the article [Spa1], N. Spaltenstein has established a bijection between the irreducible components of _χ, the space of full flags fixed by a nilpotent element χ ? M(n, k), where k is an algebraically closed field, and the standard tableaux associated to the Young diagram of χ. In this present work we determine, when χ is of hook type, for each irreducible component X of _χ, the unique Schubert cell _X of the full flag manifold = (V) (where V is vector space of dimension n over k), such that X ∩ _X is a dense subspace in X. This result will allow us to optimize the computation of _χ and when k = is the complex field, to see that the graph resolution of the partition (2, 1, …, 1) of n is related to the Dynkin diagram of sl(n, ).     

A fractional Helly theorem for convex lattice sets

A set of the form , where is convex and denotes the integer lattice, is called a convex lattice set. It is known that the Helly number of d-dimensional convex lattice sets is 2^d. We prove that the fractional Helly number is only d+1: For every d and every α∈(0,1] there exists β>0 such that whenever F₁,…,F_n are convex lattice sets in such that for at least index sets I⊆{1,2,…,n} of size d+1, then there exists a (lattice) point common to at least βn of the F_i. This implies a (p,d+1)-theorem for every p?d+1; that is, if is a finite family of convex lattice sets in such that among every p sets of , some d+1 intersect, then has a transversal of size bounded by a function of d and p.