Volume 114, Issue 4
November 2004, pages 299-428
pp 299-308 November 2004 Invited Articles
The congruence subgroup problem
This is a short survey of the progress on the congruence subgroup problem since the sixties when the first major results on the integral unimodular groups appeared. It is aimed at the non-specialists and avoids technical details.
pp 309-318 November 2004 Invited Articles
Random walks in a random environment
Random walks as well as diffusions in random media are considered. Methods are developed that allow one to establish large deviation results for both the ‘quenched’ and the ‘averaged’ case.
pp 319-343 November 2004 Regular Articles
Conductors and newforms for U(1,1)
Let F be a non-Archimedean local field whose residue characteristic is odd. In this paper we develop a theory of newforms forU (1, 1)(F), building on previous work onSL_{2}(F). This theory is analogous to the results of Casselman forGL_{2}(F) and Jacquet, Piatetski-Shapiro, and Shalika forGL_{n}(F). To a representation π ofU(1, 1)(F), we attach an integer c(π) called the conductor of π, which depends only on theL-packet π containing π. A newform is a vector in π which is essentially fixed by a congruence subgroup of level c(π). We show that our newforms are always test vectors for some standard Whittaker functionals, and, in doing so, we give various explicit formulae for newforms.
pp 345-363 November 2004 Regular Articles
Cohomology of line bundles on Schubert varieties: The rank two case
In this paper we describe vanishing and non-vanishing of cohomology of “most” line bundles over Schubert subvarieties of flag varieties for rank 2 semisimple algebraic groups.
pp 365-374 November 2004 Regular Articles
On the maximal dimension of a completely entangled subspace for finite level quantum systems
LetH_{i}be a finite dimensional complex Hilbert space of dimensiond_{i} associated with a finite level quantum system A_{i} for i = 1, 2, ...,k. A subspaceS ⊂$${\mathcal{H}} = {\mathcal{H}}_{A_1 A_2 ...A_k } = {\mathcal{H}}_1 \otimes {\mathcal{H}}_2 \otimes \cdots \otimes {\mathcal{H}}_k $$ is said to becompletely entangled if it has no non-zero product vector of the formu_{1}⊗u_{2} ⊗ ... ⊗u_{k} with u_{i} inH_{i} for each i. Using the methods of elementary linear algebra and the intersection theorem for projective varieties in basic algebraic geometry we prove that$$\mathop {max}\limits_{S \in \varepsilon } dim S = d_1 d_2 ...d_k - (d_1 + \cdots + d_k ) + k - 1$$ where ε is the collection of all completely entangled subspaces.
When$${\mathcal{H}} = {\mathcal{H}}_2 $$ andk = 2 an explicit orthonormal basis of a maximal completely entangled subspace of$${\mathcal{H}}_1 \otimes {\mathcal{H}}_2 $$ is given.
We also introduce a more delicate notion of aperfectly entangled subspace for a multipartite quantum system, construct an example using the theory of stabilizer quantum codes and pose a problem.
pp 375-387 November 2004 Regular Articles
Hüseyin Yildirim M Zeki Sarikaya Sermin öztürk
In this article, the operator$$\diamondsuit _B^k $$ is introduced and named as the Bessel diamond operator iteratedk times and is defined by$$\diamondsuit _B^k = [(B_{x_1 } + B_{x_2 } + \cdots + B_{x_p } )^2 - (B_{x_{p + 1} } + \cdots + B_{x_{p + q} } )^2 ]^k $$$$p + q = n,B_{x_i } = \tfrac{{\partial ^2 }}{{\partial x_i^2 }} + \tfrac{{2v_i }}{{x_i }}\tfrac{\partial }{{\partial x_i }}$$ where$$2v_i = 2\alpha _i + 1,\alpha _i > - \tfrac{1}{2}[8],x_i > 0$$,i = 1, 2, ...,nk is a non-negative integer andn is the dimension of ℝ_{n}^{+}. In this work we study the elementary solution of the Bessel diamond operator and the elementary solution of the operator$$\diamondsuit _B^k $$ is called the Bessel diamond kernel of Riesz. Then, we study the Fourier-Bessel transform of the elementary solution and also the Fourier-Bessel transform of their convolution.
pp 389-397 November 2004 Regular Articles
It is shown that (1) if a good set has finitely many related components, then they are full, (2) loops correspond one-to-one to extreme points of a convex set. Some other properties of good sets are discussed.
pp 399-408 November 2004 Regular Articles
Derivations into duals of ideals of Banach algebras
We introduce two notions of amenability for a Banach algebra A. LetI be a closed two-sided ideal inA, we sayA is I-weakly amenable if the first cohomology group ofA with coefficients in the dual space I* is zero; i.e.,H^{1}(A, I*) = {0}, and,A is ideally amenable ifA isI-weakly amenable for every closed two-sided idealI inA. We relate these concepts to weak amenability of Banach algebras. We also show that ideal amenability is different from amenability and weak amenability. We study theI-weak amenability of a Banach algebraA for some special closed two-sided idealI.
pp 409-422 November 2004 Regular Articles
Multiple positive solutions to third-order three-point singular semipositone boundary value problem
Huimin Yu L Haiyan Yansheng Liu
By using a specially constructed cone and the fixed point index theory, this paper investigates the existence of multiple positive solutions for the third-order threepoint singular semipositone BVP:$$\left\{ \begin{gathered} x'''(t) - \lambda f(t,x) = 0,t \in (0,1); \hfill \\ x(0) = x'(\eta ) = x''(1) = 0, \hfill \\ \end{gathered} \right.$$ where 1/2 < η < 1, the non-linear term ƒ(t, x): (0, 1) × (0, + ∞) → (-∞, + ∞) is continuous and may be singular att = 0,t = 1, andx = 0, also may be negative for some values oft andx, λ is a positive parameter.
pp 423-426 November 2004 Subject Index
pp 427-428 November 2004 Author Index
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