• Volume 114, Issue 4

      November 2004,   pages  299-428

    • The congruence subgroup problem

      M S Raghunathan

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      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

      S R S Varadhan

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      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)

      Joshua Lansky A Raghuram

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      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

      K Paramasamy

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      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

      K R Parthasarathy

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      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

      Hüseyin Yildirim M Zeki Sarikaya Sermin öztürk

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      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 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

      K Gowri Navada

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      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

      M E Gorgi T Yazdanpanah

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      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

      Huimin Yu L Haiyan Yansheng Liu

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      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.

    • Subject Index

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    • Author Index

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