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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 797 (10 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
Functional Equations And Distribution Functions
, 1994
"... . We consider the functional equation f(t) = 1 b b\Gamma1 X =0 f i t \Gamma fi a j for all t 2 IR; (F) where 0 ! a ! 1, b 2 IN n f1g and \Gamma1 = fi 0 fi 1 : : : fi b\Gamma1 = 1 are given parameters, f : IR ! IR is the unknown. We show that there is a unique bounded function f which s ..."
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Cited by 8 (3 self)
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. We consider the functional equation f(t) = 1 b b\Gamma1 X =0 f i t \Gamma fi a j for all t 2 IR; (F) where 0 ! a ! 1, b 2 IN n f1g and \Gamma1 = fi 0 fi 1 : : : fi b\Gamma1 = 1 are given parameters, f : IR ! IR is the unknown. We show that there is a unique bounded function f which solves (F) and satisfies f(t) = 0 for t ! \Gamma1=(1 \Gamma a), f(t) = 1 for t ? 1=(1 \Gamma a). This solution can be interpreted as the distribution function of a certain random series. It is known to be either singular or absolutely continuous, but the problem for which parameters it is absolutely continuous is largely open. We collect some previously established partial answers and generalize them. We also point out an interesting connection to the socalled Schilling equation. Dedicated to Prof. J'anos Acz'el on the occasion of his 70th birthday. 1 Introduction: A random series and its distribution function Let a 2 (0; 1), b 2 IN n f1g and \Gamma1 = fi 0 fi 1 : : : fi b\Gamma1 = 1. By...
Some studies on arithmetical chaos in classical and quantum mechanics
 Internat. J. Modern Phys
, 1993
"... Several aspects of classical and quantum mechanics applied to a class of strongly chaotic systems are studied. The latter consists of single particles moving without external forces on surfaces of constant negative Gaussian curvature whose corresponding fundamental groups are supplied with an arithm ..."
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Several aspects of classical and quantum mechanics applied to a class of strongly chaotic systems are studied. The latter consists of single particles moving without external forces on surfaces of constant negative Gaussian curvature whose corresponding fundamental groups are supplied with an arithmetic structure. It is shown that the arithmetical features of the considered systems lead to exceptional properties of the corresponding spectra of lengths of closed geodesics (periodic orbits). The most significant one is an exponential growth of degeneracies in these geodesic length spectra. Furthermore, the arithmetical systems are distinguished by a structure that appears as a generalization of geometric symmetries. These pseudosymmetries occur in the quantization of the classical arithmetic systems as Hecke operators, which form an infinite algebra of selfadjoint operators commuting with the Hamiltonian. The statistical properties of quantum energies in the arithmetical systems have previously been identified as exceptional. They do not fit into the general scheme of random matrix theory. It is shown with the help of a simplified model for the spectral form factor
Continued fractions from Euclid to the present day
, 2000
"... this paper to indicate how continued fractions are relevant to ..."
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this paper to indicate how continued fractions are relevant to
ON THE STRUCTURE OF THE STABLE NORM OF PERIODIC METRICS
"... Abstract. We study the differentiability of the stable norm ‖· ‖ associated with a Z n periodic metric on R n.Extending one of the main results of [Ba2], we prove that if p ∈ R n and the coordinates of p are linearly independent over Q, then there is a linear 2plane V containing p such that the res ..."
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Abstract. We study the differentiability of the stable norm ‖· ‖ associated with a Z n periodic metric on R n.Extending one of the main results of [Ba2], we prove that if p ∈ R n and the coordinates of p are linearly independent over Q, then there is a linear 2plane V containing p such that the restriction of ‖· ‖ to V is differentiable at p.We construct examples where ‖· ‖ it is not differentiable at a point with coordinates linearly independent over Q.
RamanujanFourier series, the WienerKhintchine formula and the distribution of prime pairs
, 1999
"... The WienerKhintchine formula plays a central role in statistical mechanics. It is shown here that the problem of prime pairs is related to autocorrelation and hence to a WienerKhintchine formula. "Experimental" evidence is given for this. c # 1999 Elsevier Science B.V. All rights reserved. PA ..."
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Cited by 4 (2 self)
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The WienerKhintchine formula plays a central role in statistical mechanics. It is shown here that the problem of prime pairs is related to autocorrelation and hence to a WienerKhintchine formula. "Experimental" evidence is given for this. c # 1999 Elsevier Science B.V. All rights reserved. PACS: 05.40+j; 02.30.Nw; 02.10.Lh Keywords: Twin primes; RamanujanFourier series; WienerKhintchine formula 1. Introduction " The WienerKhintchine theorem states a relationship between two important characteristics of a random process: the power spectrum of the process and the correlation function of the process" [1]. One of the outstanding problems in number theory is the problem of prime pairs which asks how primes of the form p and p+h (where h is an even integer) are distributed. One immediately notes that this is a problem of #nding correlation between primes. We make two key observations. First of all there is an arithmetical function (a function de#ned on integers) which traps the...
Some heuristics and results for small cycles of the discrete logarithm
 Mathematics of Computation
"... Abstract. Brizolis asked the question: does every prime p have a pair (g, h) such that h is a fixed point for the discrete logarithm with base g? The first author previously extended this question to ask about not only fixed points but also twocycles, and gave heuristics (building on work of Zhang, ..."
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Cited by 4 (3 self)
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Abstract. Brizolis asked the question: does every prime p have a pair (g, h) such that h is a fixed point for the discrete logarithm with base g? The first author previously extended this question to ask about not only fixed points but also twocycles, and gave heuristics (building on work of Zhang, Cobeli, Zaharescu, Campbell, and Pomerance) for estimating the number of such pairs given certain conditions on g and h. In this paper we extend these heuristics and prove results for some of them, building again on the aforementioned work. We also make some new conjectures and prove some average versions of the results. 1. Introduction and Statement
Algorithmic information theory,” in Encyclopedia of Statistical Sciences
 G. J. Chaitin
, 1982
"... ensemble notion; it is a measure of the degree of ignorance concerning which possibility holds in an ensemble with a given a priori probability distribution* n� H(p1,...,pn) ≡ − pk log2 pk. In algorithmic information theory the primary concept is that of the information content of an individual obje ..."
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ensemble notion; it is a measure of the degree of ignorance concerning which possibility holds in an ensemble with a given a priori probability distribution* n� H(p1,...,pn) ≡ − pk log2 pk. In algorithmic information theory the primary concept is that of the information content of an individual object, which is a measure of how difficult it is to specify or describe how to construct or calculate that object. This notion is also known as informationtheoretic complexity. For introductory expositions, see refs. 1, 4, and 6. For the necessary background on computability theory and mathematical logic, see refs. 3, 7, and 8. For a more technical survey of algorithmic information theory and a more complete bibliography, see ref. 2. See also ref. 5. The original formulation of the concept of algorithmic information is independently due to R. J. Solomonoff [22], A. N. Kolmogorov * [19], and G. J. Chaitin [10]. The information content I(x) of a binary string x is defined to be the size in bits (binary digits) of the smallest program for a canonical universal computer U to calculate x. (That the 1 k=1
The Low Activity Phase of Some Dirichlet Series
 J. Math. Phys
, 1996
"... We show that a rigorous statistical mechanics description of some Dirichlet series is possible. Using the abstract polymer model language of statistical mechanics and the polymer expansion theory we characterize the low activity phase by the suitable exponential decay of the truncated correlation fu ..."
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We show that a rigorous statistical mechanics description of some Dirichlet series is possible. Using the abstract polymer model language of statistical mechanics and the polymer expansion theory we characterize the low activity phase by the suitable exponential decay of the truncated correlation functions. 1 Introduction The idea to relate number theory and equilibrium statistical mechanics or, more precisely, zeta functions and partition functions, is now already quite old. One motivation for pursuing this idea lies in the probabilistic aspects of the prime number distribution. Statistical mechanics as an intrinsically probabilistic theory is hoped to be an appropriate language for these phenomena. The book [15] by Kac nicely presents this kind of probabilistic reasoning. More concretely, the formulation of the famous LeeYang theorem was influenced by a paper [23] by P'olya on the Riemann zeta function. In that paper P'olya took the asymptotics of the Fourier transformed zeta funct...
Random Harmonic Series
"... The harmonic series is the first nontrivial divergent series we encounter. We learn that, although the individual terms 1/j converge to zero, together they accumulate so that their sum is infinite: ..."
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The harmonic series is the first nontrivial divergent series we encounter. We learn that, although the individual terms 1/j converge to zero, together they accumulate so that their sum is infinite: