Some times the lights are shining on me. Other times I can barely see. Lately it occurs to me what a long strange trip its been. Grateful Dead In 1989 when the first edition of the book was completed, my sons David and Greg were 3 and 1, and the cover picture showed the Dow Jones at 2650. The last twenty years have brought many changes but the song remains the same. The title of the book indicates that as we develop the theory, we will focus our attention on examples. Hoping that the book would be a useful reference for people who apply probability in their work, we have tried to emphasize the results that are important for applications, and illustrated their use with roughly 200 examples. Probability is not a spectator sport, so the book contains almost 450 exercises to challenge the reader and to deepen their understanding. The fourth edition has two major changes (in addition to a new publisher): (i) The book has been converted from TeX to LaTeX. The systematic use of labels should eventually eliminate problems with references to other points in the text. In

We introduce the notion of definable compactness and within the context of o-minimal structures prove several topological properties of definably compact spaces. In particular a definable set in an ominimal structure is definably compact (with respect to the subspace topology) if and only if it is closed and bounded. We then apply definable compactness to the study of groups and rings in o-minimal structures. The main result we prove, Theorem 1.2, is that any infinite definable group in an o-minimal structure that is not definably compact contains a definable torsion-free subgroup of dimension one. Using this theorem we give a complete characterization of all rings without zero divisors that are definable in o-minimal structures. The paper concludes with several examples illustrating some limitations on extending Theorem 1.2.

Converting decimal scientific notation into binary floating point is nontrivial, but this conversion can be performed with the best possible accuracy without sacrificing efficiency. 1.

This paper applies the parallel tree contraction techniques developed in Miller and paper [Randomness and Computation, 5, S. Micali, ed., JAI Press, 1989, pp. 47-72] to a number of fundamental graph problems. The paper presents an time and processor, a 0-sided randomized algorithm for testing the isomorphism of trees, and an n) time, n-processor algorithm for maximal isomorphism and for common subexpression elimination. An time, n-processor algorithm for computing the canonical forms of trees and subtrees is given. An Ologn time algorithm for computing the tree of 3-connected components of a graph, an n)time algorithm for computing an explicit planar embedding of a planar graph, and an n)time algorithm for computing a canonical form for a planar graph are also given. All these latter algorithms use only processors on a Parallel Random Access Machine (PRAM) model with concurrent writes and concurrent reads.

To Benoît Mandelbrot, on the occasion of his jubilee. Abstract. We present an overview of a theory of complex dimensions of selfsimilar fractal strings, and compare this theory to the theory of varieties over a finite field from the geometric and the dynamical point of view. Then we combine the several strands to discuss a possible approach to establishing a cohomological interpretation of the complex dimensions. 1.

Let L(x) denote the counting function for Lucas pseudoprimes, and E(x) denote the elliptic pseudoprime counting function. We prove that, for large x, L(x) ≤ x L(x) −1/2 and E(x) ≤ x L(x) −1/3, where L(x) = exp(log xlog log log x / log log x).

Existing black box and other algorithms for explicitly recognising groups of Lie type over GF(q) have asymptotic running times which are polynomial in q, whereas the input size involves only log q. This has represented a serious obstruction to ecient recognition of such groups.

In this paper we shall determine, when 1 = 6, bounds for numbers f(k, I) and F{k, 1) defined as follows: f{k, l)/F(k, I) is defined to be the smallest integer n for which there exists a regular graph/Hamiltonian regular graph of valency k and girth I having n vertices.

We present a non-deterministic polynomial-time algorithm to find a path of length O(log plog log p) between any two vertices of the Cayley graph of SL2(Fp). 1 1 1 0 It is well known that SL2(Fp) is generated by and. It is a much 0 1 1 1 deeper theorem [6] that the Cayley diameter of this group with respect to these generators is O(log p). There are two known proofs. One depends on uniformly bounding the eigenvalues of the Laplacian on L2 0 (X(p)) away from zero [6]. The other uses the circle method to show that any element of SL2(Fp) lifts to an element of SL2(Z) which has a short word representation [7]. Neither method is constructive. A. Lubotzky asked [6] for an efficient algorithm to find short word representations of general elements of SL2(Fp). In this note we give such an algorithm, but for word representations of length O(log p log log p) rather than O(log p). More precisely, we prove Theorem 1: There exist constants c1 and c2 such that for any c3 < 1, there exists c4 such

Abstract. This paper describes the cutting sequences of geodesic flow on the modular surface H/PSL(2, Z) with respect to the standard fundamental domain F = {z = x+iy: −1 1 2 ≤ x ≤