• 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 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)$. We show that our newforms are always test vectors for some standard Whittaker functionals, and, in doing so, we give various explicit formulae for newforms. • Cohomology of Line Bundles on Schubert Varieties: The Rank Two Case In this paper we describe vanishing and non-vanishing of cohomology of `most’ line bundles over Schubert subvarieties of flag varieties for rank 2 semisimple algebraic groups. • On the Maximal Dimension of a Completely Entangled Subspace for Finite Level Quantum Systems Let$\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 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 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$\mathcal{A}$. 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 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

• Author Index

• # Proceedings – Mathematical Sciences

Current Issue
Volume 127 | Issue 4
September 2017