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Probability: Theory and examples
 CAMBRIDGE U PRESS
, 2011
"... 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 t ..."
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Cited by 1347 (16 self)
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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
Definable compactness and definable subgroups of ominimal groups
 J. LONDON MATH. SOC
, 1999
"... We introduce the notion of definable compactness and within the context of ominimal 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 c ..."
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Cited by 55 (2 self)
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We introduce the notion of definable compactness and within the context of ominimal 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 ominimal structures. The main result we prove, Theorem 1.2, is that any infinite definable group in an ominimal structure that is not definably compact contains a definable torsionfree subgroup of dimension one. Using this theorem we give a complete characterization of all rings without zero divisors that are definable in ominimal structures. The paper concludes with several examples illustrating some limitations on extending Theorem 1.2.
Parallel Tree Contraction Part 2: Further Applications
 SIAM JOURNAL ON COMPUTING
, 1991
"... This paper applies the parallel tree contraction techniques developed in Miller and paper [Randomness and Computation, 5, S. Micali, ed., JAI Press, 1989, pp. 4772] to a number of fundamental graph problems. The paper presents an time and processor, a 0sided randomized algorithm for testing the i ..."
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Cited by 34 (3 self)
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This paper applies the parallel tree contraction techniques developed in Miller and paper [Randomness and Computation, 5, S. Micali, ed., JAI Press, 1989, pp. 4772] to a number of fundamental graph problems. The paper presents an time and processor, a 0sided randomized algorithm for testing the isomorphism of trees, and an n) time, nprocessor algorithm for maximal isomorphism and for common subexpression elimination. An time, nprocessor algorithm for computing the canonical forms of trees and subtrees is given. An Ologn time algorithm for computing the tree of 3connected 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.
How to read floating point numbers accurately
 Proceedings of PLDI ’90
, 1990
"... 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. ..."
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Cited by 28 (0 self)
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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.
Vanishing sums of roots of unity
 Proceedings, Bicentennial Congress Wiskundig Genootschap (Vrije Univ
, 1978
"... This paper is a survey of what is known about the magnitude of coefficients appearing in linear relations between roots of unity. The special case of the cyclotomic polynomial is considered in section l; section 2 is devoted to more general relations. Various open problems will be indicated. ..."
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Cited by 21 (0 self)
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This paper is a survey of what is known about the magnitude of coefficients appearing in linear relations between roots of unity. The special case of the cyclotomic polynomial is considered in section l; section 2 is devoted to more general relations. Various open problems will be indicated.
The distribution of Lucas and elliptic pseudoprimes
, 2001
"... 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). ..."
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Cited by 18 (3 self)
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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).
Distribution of eigenvalues for the modular group
 Progress in Math
, 1996
"... The twopoint correlation functions of energy levels for free motion on the modular domain, both with periodic and Dirichlet boundary conditions, are explicitly computed using a generalization of the Hardy– Littlewood method. It is shown that in the limit of small separations they show an uncorrelat ..."
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Cited by 16 (2 self)
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The twopoint correlation functions of energy levels for free motion on the modular domain, both with periodic and Dirichlet boundary conditions, are explicitly computed using a generalization of the Hardy– Littlewood method. It is shown that in the limit of small separations they show an uncorrelated behaviour and agree with the Poisson distribution but they have prominent numbertheoretical oscillations at larger scale. The results agree well with numerical simulations. IPNO/TH 94–43
Constructive recognition of PSL(2, q)
 Trans. Amer. Math. Soc
"... 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. ..."
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Cited by 15 (3 self)
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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.
Rogers–Ramanujan and the Baker–Gammel–Wills (Padé) conjecture
 Annals of Mathematics 157
, 2003
"... In 1961, Baker, Gammel and Wills conjectured that for functions f meromorphic in the unit ball, a subsequence of its diagonal Pade ́ approximants converges uniformly in compact subsets of the ball omitting poles of f. There is also apparently a cruder version of the conjecture due to Pade ́ himself ..."
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Cited by 11 (1 self)
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In 1961, Baker, Gammel and Wills conjectured that for functions f meromorphic in the unit ball, a subsequence of its diagonal Pade ́ approximants converges uniformly in compact subsets of the ball omitting poles of f. There is also apparently a cruder version of the conjecture due to Pade ́ himself, going back to the early twentieth century. We show here that for carefully chosen q on the unit circle, the RogersRamanujan continued fraction 1 + qz
Cutting Sequences for Geodesic Flow on the Modular Surface and Continued Fractions
"... 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 ≤ ..."
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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 ≤