• 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 for 𝑈(1,1)(𝐹), building on previous work on $SL_2(F)$. This theory is analogous to the results of Casselman for $GL_2(F)$ and Jacquet, Piatetski-Shapiro, and Shalika for $GL_n(F)$. To a representation π of 𝑈(1,1)(𝐹), we attach an integer 𝑐(𝜋) called the conductor of 𝜋, which depends only on the 𝐿-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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      Let $\mathcal{H}_i$ be a finite dimensional complex Hilbert space of dimension $d_i$ associated with a finite level quantum system $A_i$ for $i=1, 2,\ldots,k$. A subspace $S\subset\mathcal{H} = \mathcal{H}_{A_1 A_2\ldots A_k} = \mathcal{H}_1 \otimes \mathcal{H}_2 \otimes\cdots\otimes \mathcal{H}_k$ is said to be completely entangled if it has no non-zero product vector of the form $u_1 \otimes u_2 \otimes\cdots\otimes u_k$ with $u_i$ in $\mathcal{H}_i$ for each 𝑖. Using the methods of elementary linear algebra and the intersection theorem for projective varieties in basic algebraic geometry we prove that

      $$\max\limits_{S\in\mathcal{E}}\dim S=d_1 d_2\ldots d_k-(d_1+\cdots +d_k)+k-1,$$

      where $\mathcal{E}$ is the collection of all completely entangled subspaces.

      When $\mathcal{H}_1 = \mathcal{H}_2$ and $k = 2$ 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 a perfectly entangled subspace for a multipartite quantum system, construct an example using the theory of stabilizer quantum codes and pose a problem.

    • The Solutions of the 𝑛-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 $\Diamond^k_B$ is introduced and named as the Bessel diamond operator iterated 𝑘 times and is defined by

      $$\Diamond^k_B=[(B_{x_1}+B_{x_2}+\cdots +B_{x_p})^2-(B_{x_{p+1}}+\cdots +B_{x_{p+q}})^2]^k,$$

      where $p + q = n, B_{x_i} = \frac{𝜕^2}{𝜕 x^2_i}+\frac{2𝜐_i}{x_i}\frac{𝜕}{𝜕 x_i}$, where $2𝜐_i = 2𝛼_i + 1, 𝛼_i > - \frac{1}{2} [8], x_i > 0, i = 1, 2,\ldots ,n, k$ is a non-negative integer and 𝑛 is the dimension of $\mathbb{R}^+_n$. In this work we study the elementary solution of the Bessel diamond operator and the elementary solution of the operator $\Diamond^k_B$ 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 $\mathcal{A}$. Let 𝐼 be a closed two-sided ideal in $\mathcal{A}$, we say $\mathcal{A}$ is 𝐼-weakly amenable if the first cohomology group of $\mathcal{A}$ with coefficients in the dual space 𝐼* is zero; i.e., $H^1(\mathcal{A},I^*) =\{0\}$, and, $\mathcal{A}$ is ideally amenable if $\mathcal{A}$ is 𝐼-weakly amenable for every closed two-sided ideal 𝐼 in $\mathcal{A}$. 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 the 𝐼-weak amenability of a Banach algebra $\mathcal{A}$ for some special closed two-sided ideal 𝐼.

    • 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 three-point singular semipositone BVP:

      \begin{equation*}\begin{cases}x'"(t)-𝜆 f(t,x)=0, & t\in(0, 1);\\ x(0)=x'(𝜂)=x"(1)=0,\end{cases}\end{equation*}

      where $\frac{1}{2} < 𝜂 < 1$, the non-linear term $f(t,x): (0,1)×(0,=∞)→(-∞ +∞)$ is continuous and may be singular at 𝑡 = 0, 𝑡 = 1, and 𝑥 = 0, also may be negative for some values of 𝑡 and 𝑥, 𝜆 is a positive parameter.

    • Subject Index

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    • Author Index

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