We exhibit large classes of examples of noncommutative finitedimensional manifolds which are (non-formal) deformations of classical manifolds. The main result of this paper is a complete description of noncommutative three-dimensional spherical manifolds, a noncommutative version of the sphere S 3 defined by basic K-theoretic equations. We find a 3-parameter family of deformations of the standard 3-sphere S 3 and a corresponding 3-parameter deformation of the 4-dimensional Euclidean space R 4. For generic values of the deformation parameters we show that the obtained algebras of polynomials on the deformed R 4 u are isomorphic to the algebras introduced by Sklyanin in connection with the Yang-Baxter equation. Special values of the deformation parameters do not give rise to Sklyanin algebras and we extract a subclass, the θ-deformations, which we generalize in any dimension and various contexts, and study in some details. Here, and

Digital trees, also known as tries, are a general purpose flexible data structure that implements dictionaries built on sets of words. An analysis is given of three major representations of tries in the form of array-tries, list tries, and bst-tries ("ternary search tries"). The size and the search costs of the corresponding representations are analysed precisely in the average case, while a complete distributional analysis of height of tries is given. The unifying data model used is that of dynamical sources and it encompasses classical models like those of memoryless sources with independent symbols, of finite Markovchains, and of nonuniform densities. The probabilistic behaviour of the main parameters, namely size, path length, or height, appears to be determined by two intrinsic characteristics of the source: the entropy and the probability of letter coincidence. These characteristics are themselves related in a natural way to spectral properties of specific transfer operators of the Ruelle type.

.The Gaussian algorithm for lattice reduction in dimension 2 is analysed under its standard version. It is found that, when applied to random inputs in a continuous model, the complexity is constant on average, the probability distribution decays geometrically, and the dynamics is characterized by a conditional invariant measure. The proofs make use of connections between lattice reduction, continued fractions, continuants, and functional operators. Analysis in the discrete model and detailed numerical data are also presented. Une analyse en moyenne de l'algorithme de Gauss de r'eduction des r'eseaux R'esum'e. L'algorithme de r'eduction des r'eseaux en dimension 2 qui est du `a Gauss est analys'e sous sa forme dite standard. Il est 'etabli ici que, sous un mod`ele continu, sa complexit'e est constante en moyenne et que la distribution de probabilit'es associ'ee decroit g'eom'etriquement tandis que la dynamique est caract'eris'ee par une densit'e conditionnelle invariante. Les preuves f...

Étale groupoids arise naturally as models for leaf spaces of foliations, for orbifolds, and for orbit spaces of discrete group actions. In this paper we introduce a sheaf homology theory for etale groupoids. We prove its invariance under Morita equivalence, as well as Verdier duality between Haefliger cohomology and this homology. We also discuss the relation to the cyclic and Hochschild homologies of Connes' convolution algebra of the groupoid, and derive some spectral sequences which serve as a tool for the computation of these homologies.

To add to the chorus of praise by referring to my own experience would be of little interest, but I am in no way forgetting the facilities for work provided by the Institut des Hautes Études Scientifiques (IHES) for so many years, particularly the constantly renewed opportunities for meetings and exchanges. While there have

A quite general model of source that comes from dynamical systems theory is introduced. Within this model, some important problems about prefixes that intervene in algorithmic information theory contexts are analysed. The main tool is a new object, the generalized Ruelle operator, which can be viewed as a "generating" operator. Its dominant spectral objects are linked with important parameters of the source such as the entropy, and play a central role in all the results. 1 Introduction. In information theory contexts, data items are (infinite) words that are produced by a common mechanism, called a source. Realistic sources are often complex objects. We work here inside a quite general framework of sources related to dynamical systems theory which goes beyond the cases of memoryless and Markov sources. This model can describe nonmarkovian processes, where the dependency on past history is unbounded, and as such, they attain a high level of generality. A probabilistic dynamical source ...

We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored -categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed structure associated with the category of relations does not generalize directly, instead one obtains nuclear ideals. Most tensored -categories have a large class of morphisms which behave as if they were part of a compact closed category, i.e. they allow one to transfer variables between the domain and the codomain. We introduce the notion of nuclear ideals to analyze these classes of morphisms. In compact closed tensored -categories, all morphisms are nuclear, and in the tensored -category of Hilbert spaces, the nuclear morphisms are the Hilbert-Schmidt maps. We also introduce two new examples of tensored -categories, in which integration plays the role of composition. In the first, mor...

We give a general method for computing the cyclic cohomology of crossed products by 'etale groupoids, extending the Feigin-Tsygan-Nistor spectral sequences. In particular we extend the computations performed by Brylinski, Burghelea, Connes, Feigin, Karoubi, Nistor and Tsygan for the convolution algebra C 1 c (G) of an 'etale groupoid, removing the Hausdorffness condition and including the computation of hyperbolic components. Examples like group actions on manifolds and foliations are considered. Keywords: cyclic cohomology, groupoids, crossed products, duality, foliations. Contents 1 Introduction 3 2 Homology and Cohomology of Sheaves on ' Etale Groupoids 4 2.1 ' Etale Groupoids : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2 \Gamma c in the non-Hausdorff case : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.3 Homology and Cohomology of ' Etale Groupoids : : : : : : : : : : : : : : : : : : : : : 8 3 Cyclic Homologies of Sheaves ...

We develop a general framework for the analysis of algorithms of a broad Euclidean type. The average-case 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 algorithm. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory. As a consequence, we obtain precise average-case analyses of algorithms for evaluating the Jacobi symbol of computational number theory fame, thereby solving conjectures of Bach and Shallit. These methods also provide a unifying framework for the analysis of an entire class of gcd-like algorithms together with new results regarding the probable behaviour of their cost functions. 1

Relying on properties of the inductive tensor product, we construct cyclic type homology theories for certain nuclear algebras. In this context, we establish continuity theorems. We compute the periodic cyclic homology of the Schwartz algebra of p-adic GL(n) in terms of compactly supported de Rham cohomology of the tempered dual of GL(n). 1. Complete nuclear locally convex algebras Cyclic type homology groups of an algebra A are computed using chain complexes involving tensor powers of A. When A is a general locally convex algebra, this will involve making a choice of a topological tensor product. A locally convex algebra is a locally convex vector space A over C equipped with a separately continuous multiplication. We shall refer to the projective tensor product\Omega , the injective tensor product\Omega ffl , and the inductive tensor product\Omega i . Let E denote a locally convex space. If E is nuclear then E\Omega F ' E\Omega ffl F . This is the defining property o...