q-Orthogonal polynomials and the oscillator quantum group

The oscillator quantum algebra is shown to provide a group-theoretic setting for the q-Laguerre and q-Hermite polynomials.On leave from Laboratoire de Physique Nucléaire, Université de Montréal, Montréal, Canada H3C 3J7.     

Infinite-dimensional geometry and generally covariant quantum closed string-field theory

A version, based on infinite-dimensional geometry, of the generally covariant closed string-field theory is proposed.     

Domination of uniformly continuous semigroups

We prove that a bounded operator on a Banach lattice, satisfying a growth condition, is regular. Also, we prove that the generator of a C ₀-semigroup on such a lattice for which such an operator exists is bounded.     

Maximal inequalities via bracketing with adaptive truncation

The paper provides a recursive interpretation for the technique known as bracketing with adaptive truncation. By way of illustration, a simple bound is derived for the expected value of the supremum of an empirical process, thereby leading to a simpler derivation of a functional central limit theorem due to Ossiander. The recursive method is also abstracted into a framework that consists of only a small number of assumptions about processes and functionals indexed by sets of functions. In particular, the details of the underlying probability model are condensed into a single inequality involving finite sets of functions. A functional central limit theorem of Doukhan, Massart and Rio, for empirical processes defined by absolutely regular sequences, motivates the generalization.     

Fast algorithms for greedy triangulation

We present the first quadratic-time algorithm for the greedy triangulation of a finite planar point set, and the first linear-time algorithm for the greedy triangulation of a convex polygon.     

Stable minimum space partitioning in linear time

In the stable0–1 sorting problem the task is to sort an array ofn elements with two distinct values such that equal elements retain their relative input order. Recently, Munro, Raman and Salowe gave an algorithm which solves this problem inO(n log*n) time and constant extra space. We show that by a modification of their method the stable0–1 sorting is possible inO(n) time andO(1) extra space. Stable three-way partitioning can be reduced to stable0–1 sorting. This immediately yields a stable minimum space quicksort, which sorts multisets in asymptotically optimal time with high probability.     

Page usage in a quadtree index

This paper provides a characterization of the storage needs of a quadtree when used as an index to access large volumes of 2-dimensional data. It is shown that the page occupancy for data in random order approaches 33%. A precise mathematical analysis that involves a modicum of hypergeometric functions and dilogarithms, together with some computer algebra is presented.A brief survey of the analysis of storage usage in tree structures is included. The 33% ratio for quadtrees is to be compared to the figures for binary search trees (50%), tries (69%), and quadtries (46%).The research of this author was done while visiting INRIA, Rocquencourt, France under support from the Ministry of Education of Japanese Government.Work of this author was supported in part by the Basic Research Action of the E.C. under contract No. 3075 (Project ALCOM).     

Differential calculus on the inhomogeneous quantum groups IGL q (n)

We obtain the inhomogeneousq-groups IGL _q (n) via a projection from GL _q (n + 1). The bicovariant differential calculus of IGL _q (n) is constructed, and the corresponding quantum Lie algebra is given explicitly.     

On a necessary but not sufficient condition for a γ-generating pair to be a nehari pair

This paper gives a negative answer to a question due to V.M. Adamjan, D.Z. Arov and M.G. Krein, and (what is the same) gives a counterexample to D.Sarason's conjecture^* concerning exposed points inH ¹.