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Average BitComplexity of Euclidean Algorithms
 Proceedings ICALP’00, Lecture Notes Comp. Science 1853, 373–387
, 2000
"... We obtain new results regarding the precise average bitcomplexity of five algorithms of a broad Euclidean type. We develop a general framework for analysis of algorithms, where the averagecase complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set ..."
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Cited by 17 (6 self)
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We obtain new results regarding the precise average bitcomplexity of five algorithms of a broad Euclidean type. We develop a general framework for analysis of algorithms, where the averagecase complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithms. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory and provide a unifying framework for the analysis of an entire class of gcdlike algorithms. Keywords: Averagecase Analysis of algorithms, BitComplexity, Euclidean Algorithms, Dynamical Systems, Ruelle operators, Generating Functions, Dirichlet Series, Tauberian Theorems. 1 Introduction Motivations. Euclid's algorithm was analysed first in the worst case in 1733 by de Lagny, then in the averagecase around 1969 independently by Heilbronn [12] and Dixon [6], and finally in distribution by Hensley [13] who proved in 1994 that the Eu...
Quantum statistical mechanics over function fields
"... It has become increasingly evident, starting from the seminal paper of Bost and Connes [3] and continuing with several more recent developments ([8], [10], [12], [13], [22], [24]), that there is a rich interplay between quantum statistical mechanics and arithmetic. In the case of number fields, the ..."
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Cited by 16 (9 self)
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It has become increasingly evident, starting from the seminal paper of Bost and Connes [3] and continuing with several more recent developments ([8], [10], [12], [13], [22], [24]), that there is a rich interplay between quantum statistical mechanics and arithmetic. In the case of number fields, the symmetries and equilibrium states of the Bost–Connes system are closely
Limiting modular symbols and the Lyapunov spectrum
 J. Number Theory
"... This paper consists of variations upon a theme, that of limiting modular symbols introduced in [16]. We show that, on level sets of the Lyapunov exponent for the shift map T of the continued fraction expansion, the limiting modular symbol can be computed as a Birkhoff average. We show that the limit ..."
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Cited by 15 (10 self)
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This paper consists of variations upon a theme, that of limiting modular symbols introduced in [16]. We show that, on level sets of the Lyapunov exponent for the shift map T of the continued fraction expansion, the limiting modular symbol can be computed as a Birkhoff average. We show that the limiting modular symbols vanish
Digits and Continuants in Euclidean Algorithms. Ergodic versus Tauberian Theorems
, 2000
"... We obtain new results regarding the precise average case analysis of the main quantities that intervene in algorithms of a broad Euclidean type. We develop a general framework for the analysis of such algorithms, where the averagecase complexity of an algorithm is related to the analytic behaviou ..."
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Cited by 14 (5 self)
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We obtain new results regarding the precise average case analysis of the main quantities that intervene in algorithms of a broad Euclidean type. We develop a general framework for the analysis of such algorithms, where the averagecase complexity of an algorithm is related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithms. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory and provide a unifying framework for the analysis of the main parameters digits and continuants that intervene in an entire class of gcdlike algorithms. We operate a general transfer from the continuous case (Continued Fraction Algorithms) to the discrete case (Euclidean Algorithms), where Ergodic Theorems are replaced by Tauberian Theorems.
Intermittency and regularized Fredholm determinants
 Invent. Math
, 1999
"... We consider realanalytic maps of the interval I = [0,1] which are expanding everywhere except for a neutral fixed point at 0. We show that on a certain function space the spectrum of the associated PerronFrobenius operator M has a decomposition sp(M) = σc ∪ σp where σc = [0,1] is the continuous s ..."
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Cited by 13 (1 self)
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We consider realanalytic maps of the interval I = [0,1] which are expanding everywhere except for a neutral fixed point at 0. We show that on a certain function space the spectrum of the associated PerronFrobenius operator M has a decomposition sp(M) = σc ∪ σp where σc = [0,1] is the continuous spectrum of M and σp is the pure point spectrum with no points of accumulation outside 0 and 1. We construct a regularized Fredholm determinant d(λ) which has a holomorphic extension to λ ∈ C − σc and can be analytically continued from each side of σc to an open neighborhood of σc − {0,1} (on different Riemann sheets). In C − σc the zeroset of d(λ) is in onetoone correspondence with the point spectrum of M. Through the conformal transformation λ(z) = 1 4z (1 + z)2 the function d ◦ λ(z) extends to a holomorphic function in a domain which contains the unit disc. Shorttitle: Intermittency and Regularized Fredholm Determinants. 1 Assumptions and statement of results.
Calculating Hausdorff Dimension Of Julia Sets And Kleinian Limit Sets
 Amer. J. Math
"... We present a new algorithm for efficiently computing the Hausdorff dimension of sets X invariant under conformal expanding dynamical systems. By locating the periodic points of period up to N , we construct approximations s N which converge to dim(X) superexponentially fast in N . This method can b ..."
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Cited by 12 (1 self)
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We present a new algorithm for efficiently computing the Hausdorff dimension of sets X invariant under conformal expanding dynamical systems. By locating the periodic points of period up to N , we construct approximations s N which converge to dim(X) superexponentially fast in N . This method can be used to give rigorous estimates for important examples, including hyperbolic Julia sets and limit sets of Schottky and quasifuchsian groups.
Computing The Dimension Of Dynamically Defined Sets I: E2 and Bounded . . .
"... We present a powerful approach to computing the Hausdorff dimension of certain conformally selfsimilar sets. We illustrate this method for the dimension dim H (E 2 ) of the set E 2 , consisting of those real numbers whose continued fraction expansions contain only the digits 1 or 2. A very striking ..."
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Cited by 12 (2 self)
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We present a powerful approach to computing the Hausdorff dimension of certain conformally selfsimilar sets. We illustrate this method for the dimension dim H (E 2 ) of the set E 2 , consisting of those real numbers whose continued fraction expansions contain only the digits 1 or 2. A very striking feature of this method is that the successive approximations converge to dim(E 2 ) at a superexponential rate.
New Approach to Arakelov Geometry
"... The principal aim of this work is to provide an alternative algebraic framework for Arakelov geometry, and to demonstrate its usefulness by presenting several simple applications. This framework, called theory of generalized ..."
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Cited by 11 (0 self)
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The principal aim of this work is to provide an alternative algebraic framework for Arakelov geometry, and to demonstrate its usefulness by presenting several simple applications. This framework, called theory of generalized
Hopf algebras in noncommutative geometry
 in Geometrical and Topological Methods in Quantum Field Theory
, 2003
"... We give an introductory survey to the use of Hopf algebras in several problems of noncommutative geometry. The main example, the Hopf algebra of rooted trees, is a graded, connected Hopf algebra arising from a universal construction. We show its relation to the algebra of transverse differential ope ..."
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We give an introductory survey to the use of Hopf algebras in several problems of noncommutative geometry. The main example, the Hopf algebra of rooted trees, is a graded, connected Hopf algebra arising from a universal construction. We show its relation to the algebra of transverse differential operators introduced by Connes and Moscovici in order to compute a local index formula in cyclic cohomology, and to the several Hopf algebras defined by Connes and Kreimer to simplify the combinatorics of perturbative renormalization. We explain how characteristic classes for a Hopf module algebra can be obtained from the cyclic cohomology of the Hopf algebra which acts on it. Finally, we discuss the theory of noncommutative spherical manifolds and show how they arise as homogeneous spaces of certain compact quantum groups.
Size and Path length of Patricia Tries: Dynamical Sources Context.
, 2001
"... Digital trees, also known as tries, and Patricia tries are flexible data structures that occur in a variety of computer and communication algorithms including dynamic hashing, partial match retrieval, searching and sorting, conflict resolution algorithms for broadcast communication, data compression ..."
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Cited by 10 (1 self)
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Digital trees, also known as tries, and Patricia tries are flexible data structures that occur in a variety of computer and communication algorithms including dynamic hashing, partial match retrieval, searching and sorting, conflict resolution algorithms for broadcast communication, data compression, and so forth. We consider here tries and Patricia tries built from $n$ words emitted by a probabilistic dynamical source. Such sources encompass classical and many more models of sources as memoryless sources and finite Markov chains. The probabilistic behavior of the main parameters, namely the size and path length, appears to be determined by some intrinsic characteristics of the source, namely the entropy and two other constants, themselves related in a natural way to spectral properties of specific transfer operators of Ruelle type. Keywords: Averagecase Analysis of datastructures, Information Theory, Trie, Mellin analysis, Dynamical systems, Ruelle operator, Functional Analysis.