• Volume 114, Issue 4

November 2004,   pages  299-428

• 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.

• 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.

• 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 onSL2(F). This theory is analogous to the results of Casselman forGL2(F) and Jacquet, Piatetski-Shapiro, and Shalika forGLn(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.

• 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.

• On the maximal dimension of a completely entangled subspace for finite level quantum systems

LetHibe a finite dimensional complex Hilbert space of dimensiondi associated with a finite level quantum system Ai 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 formu1u2 ⊗ ... ⊗uk with ui inHi 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.

• The solutions of then-dimensional Bessel diamond operator and the Fourier-Bessel transform of their convolution

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$$2v_i = 2\alpha _i + 1,\alpha _i &gt; - \tfrac{1}{2}[8],x_i &gt; 0$$,i = 1, 2, ...,nk is a non-negative integer andn is the dimension of ℝn+. 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.

• Some remarks on good sets

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.

• 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.,H1(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.

• Multiple positive solutions to third-order three-point singular semipositone boundary value problem

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 &lt; η &lt; 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.

• Subject Index

• Author Index

• # Proceedings – Mathematical Sciences

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• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019