Finiteness of graded local cohomology modules

Let R=?_n∈N0R_n be a Noetherian homogeneous ring with local base ring (R₀,m₀) and irrelevant ideal R₊, let M be a finitely generated graded R-module. In this paper we show that is Artinian and is Artinian for each i in the case where R₊ is principal. Moreover, for the case where , we prove that, for each i∈N₀, is Artinian if and only if is Artinian. We also prove that is Artinian, where and c is the cohomological dimension of M with respect to R₊. Finally we present some examples which show that and need not be Artinian.     

Classification of quasifinite modules over the Lie algebras of Weyl type

For a nondegenerate additive subgroup Γ of the n-dimensional vector space over an algebraically closed field of characteristic zero, there is an associative algebra and a Lie algebra of Weyl type spanned by all differential operators uD₁^m₁?D_n^m_n for (the group algebra), and m₁,…,m_n?0, where D₁,…,D_n are degree operators. In this paper, it is proved that an irreducible quasifinite -module is either a highest or lowest weight module or else a module of the intermediate series; furthermore, a classification of uniformly bounded -modules is completely given. It is also proved that an irreducible quasifinite -module is a module of the intermediate series and a complete classification of quasifinite -modules is also given, if Γ is not isomorphic to .     

Realizable classes of tetrahedral extensions

Let k be a number field and O_k its ring of integers. Let Γ be the alternating group A₄. Let be a maximal O_k-order in k[Γ] containing O_k[Γ] and its class group. We denote by the set of realizable classes, that is the set of classes such that there exists a Galois extension N/k at most tamely ramified, with Galois group isomorphic to Γ, for which the class of is equal to c, where O_N is the ring of integers of N. In this article we determine and we prove that it is a subgroup of provided that k and the 3rd cyclotomic field of are linearly disjoint, and the class number of k is odd.     

On a conjecture of Erd?s, Graham and Spencer, II

It is conjectured by Erd?s, Graham and Spencer that if 1≤a₁≤a₂≤?≤a_s are integers with , then this sum can be decomposed into n parts so that all partial sums are ≤1. This is not true for as shown by a₁=?=a_n−2=1, . In 1997 Sandor proved that Erd?s-Graham-Spencer conjecture is true for . Recently, Chen proved that the conjecture is true for . In this paper, we prove that Erd?s-Graham-Spencer conjecture is true for .     

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 .     

A new majorization between functions, polynomials, and operator inequalities

Let P₊ be the set of all non-negative operator monotone functions defined on [0,∞), and put . Then and . For a function and a strictly increasing function h we write if is operator monotone. If and and if and , then . We will apply this result to polynomials and operator inequalities. Let and be non-increasing sequences, and put for t≧a₁ and for t≧b₁. Then v₊?u₊ if m≦n and : in particular, for a sequence of orthonormal polynomials, (p_n-1)₊?(p_n)₊. Suppose 0<r,p and s=0 or 1≦s≦1+p/r. Then 0≦A≦B implies for 0<α≦r/(p+r).     

Nilpotent orbits and some small unitary representations of indefinite orthogonal groups

For 2?m?l/2, let G be a simply connected Lie group with as Lie algebra, let be the complexification of the usual Cartan decomposition, let K be the analytic subgroup with Lie algebra , and let be the universal enveloping algebra of . This work examines the unitarity and K spectrum of representations in the “analytic continuation” of discrete series of G, relating these properties to orbits in the nilpotent radical of a certain parabolic subalgebra of .The roots with respect to the usual compact Cartan subalgebra are all ±e_i±e_j with 1?i<j?l. In the usual positive system of roots, the simple root e_m−e_m+1 is noncompact and the other simple roots are compact. Let be the parabolic subalgebra of for which e_m−e_m+1 contributes to and the other simple roots contribute to , let L be the analytic subgroup of G with Lie algebra , let , let be the sum of the roots contributing to , and let be the parabolic subalgebra opposite to .The members of are nilpotent members of . The group acts on with finitely many orbits, and the topological closure of each orbit is an irreducible algebraic variety. If Y is one of these varieties, let R(Y) be the dual coordinate ring of Y; this is a quotient of the algebra of symmetric tensors on that carries a fully reducible representation of .For , let . Then λ_s defines a one-dimensional module . Extend this to a module by having act by 0, and define . Let be the unique irreducible quotient of . The representations under study are and , where and Π_S is the Sth derived Bernstein functor.For s>2l−2, it is known that π_s=π_s′ and that π_s′ is in the discrete series. Enright, Parthsarathy, Wallach, and Wolf showed for m?s?2l−2 that π_s=π_s′ and that π_s′ is still unitary. The present paper shows that π_s′ is unitary for 0?s?m−1 even though π_s≠π_s′, and it relates the K spectrum of the representations π_s′ to the representation of on a suitable R(Y) with Y depending on s. Use of a branching formula of D. E. Littlewood allows one to obtain an explicit multiplicity formula for each K type in π_s′; the variety Y is indispensable in the proof. The chief tools involved are an idea of B. Gross and Wallach, a geometric interpretation of Littlewood's theorem, and some estimates of norms.It is shown further that the natural invariant Hermitian form on π_s′ does not make π_s′ unitary for s<0 and that the K spectrum of π_s′ in these cases is not related in the above way to the representation of on any R(Y).A final section of the paper treats in similar fashion the simply connected Lie group with Lie algebra , 2?m?l/2.     

Estimation of location parameters for spherically symmetric distributions

Estimation of the location parameters of a p×1 random vector with a spherically symmetric distribution is considered under quadratic loss. The conditions of Brandwein and Strawderman [Ann. Statist. 19(1991) 1639-1650] under which estimators of the form dominate are (i) where -h is superharmonic, (ii) is nonincreasing in R, where has a uniform distribution in the sphere centered at with a radius R, and (iii) . In this paper, we not only drop their condition (ii) to show the dominance of over but also obtain a new bound for a which is sometimes better than that obtained by Brandwein and Strawderman. Specifically, the new bound of a is 0<a<[μ₁/(p²μ_-1)][1-(p-1)μ₁/(pμ_-1μ₂)]^-1 with for i=-1,1,2. The generalization to concave loss functions is also considered. Additionally, we investigate estimators of the location parameters when the scale is unknown and the observation contains a residual vector.     

Distance irredundance and connected domination numbers of a graph

Let k be a positive integer and G be a connected graph. This paper considers the relations among four graph theoretical parameters: the k-domination number γ_k(G), the connected k-domination number ; the k-independent domination number and the k-irredundance number ir_k(G). The authors prove that if an ir_k-set X is a k-independent set of G, then , and that for k?2, if ir_k(G)=1, if ir_k(G) is odd, and if ir_k(G) is even, which generalize some known results.     

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, ).     

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.     

Generalization of transitive fraternal augmentations for directed graphs and its applications

Let be a directed graph. A transitive fraternal augmentation of is a directed graph with the same vertex set, including all the arcs of and such that for any vertices x,y,z,

1.

if and then or (fraternity);

2.

if and then (transitivity).

In this paper, we explore some generalization of the transitive fraternal augmentations for directed graphs and its applications. In particular, we show that the 2-coloring number col₂(G)≤O(∇₁(G)∇₀(G)²), where ∇_k(G) (k≥0) denotes the greatest reduced average density with depth k of a graph G; we give a constructive proof that ∇_k(G) bounds the distance (k+1)-coloring number col_k+1(G) with a function f(∇_k(G)). On the other hand, ∇_k(G)≤(col_2k+1(G))^2k+1. We also show that an inductive generalization of transitive fraternal augmentations can be used to study nonrepetitive colorings of graphs.     

Inverse spectral theory for symmetric operators with several gaps: scalar-type Weyl functions

Let S be the orthogonal sum of infinitely many pairwise unitarily equivalent symmetric operators with non-zero deficiency indices. Let J be an open subset of R. If there exists a self-adjoint extension S₀ of S such that J is contained in the resolvent set of S₀ and the associated Weyl function of the pair {S,S₀} is monotone with respect to J, then for any self-adjoint operator R there exists a self-adjoint extension such that the spectral parts and R_J are unitarily equivalent. It is shown that for any extension of S the absolutely continuous spectrum of S₀ is contained in that one of . Moreover, for a wide class of extensions the absolutely continuous parts of and S are even unitarily equivalent.     

Toeplitz operators associated to commuting row contractions

Let H be a complex Hilbert space and let {Tn}_n?1 be a sequence of commuting bounded operators on H such that . Let denote the space of all operators X in B(H) for which and suppose that . We will show that there exists a triple {K,Γ,{Un}_n?1} where K is a Hilbert space, Γ:K→H is a bounded operator and {Un}_n?1⊂B(K) is a sequence of commuting normal operators with such that TnΓ=ΓUn for n?1, and for which the mapping Y?ΓYΓ^∗ is a complete isometry from the commutant of {Un}_n?1 onto the space . Moreover we show that the inverse of this mapping can be extended to a ^∗-homomorphism