Abstract. The ‘classical ’ random graph models, in particular G(n, p), are ‘homogeneous’, in the sense that the degrees (for example) tend to be concentrated around a typical value. Many graphs arising in the real world do not have this property, having, for example, power-law degree distributions. Thus there has been a lot of recent interest in defining and studying ‘inhomogeneous ’ random graph models. One of the most studied properties of these new models is their ‘robustness’, or, equivalently, the ‘phase transition ’ as an edge density parameter is varied. For G(n, p), p = c/n, the phase transition at c = 1 has been a central topic in the study of random graphs for well over 40 years. Many of the new inhomogenous models are rather complicated; although there are exceptions, in most cases precise questions such as determining exactly the critical point of the phase transition are approachable only when there is independence between the edges. Fortunately, some models studied have this already, and others can be approximated by models with

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. We study relations between quantum dynamics and spectral properties, concentrating on spectral decompositions which arise from decomposing measures with respect to dimensional Hausdorff measures. 1. Introduction Let H be a separable Hilbert space, H : H ! H a self adjoint operator, and / 2 H (with k/k = 1). The spectral measure ¯/ of / (and H ) is uniquely defined by [24]: h/ ; f(H)/i = Z oe(H) f(x) d¯/ (x) ; (1:1) for any measurable (Borel) function f . The time evolution of the state / , in the Schrodinger picture of quantum mechanics, is given by /(t) = e \GammaiHt / : (1:2) The relations between various properties of the spectral measure ¯/ (with an emphasis on "fractal" properties) and the nature of the time evolution have been the subject of several recent papers [7,13,15--18,20,22,33,36,39]. Our purpose in this paper is twofold: First, we use a theory, due to Rogers and Taylor [28,29], of decomposing singular continuous measures with respect to Hausdorff measures to i...

We compute the divisor of Selberg's zeta function for convex co-compact, torsion-free discrete groups ; acting on a real hyperbolic space of dimension n +1. The divisor is determined by the eigenvalues and scattering poles of the Laplacian on X = ;nH n+1 together with the Euler characteristic of X compactified to a manifold with boundary.Ifn is even, the singularities of the zeta funciton associated to the Euler characteristic of X are identified using work of Bunke and Olbrich.

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DSF-xx-01 hep-th/yymmnnn preliminary version

Abstract. Determinants of invertible pseudo-differential operators (PDOs) close to positive self-adjoint ones are defined through the zeta-function regularization. We define a multiplicative anomaly as the ratio det(AB)/(det(A)det(B)) considered as a function on pairs of elliptic PDOs. We obtained an explicit formula for the multiplicative anomaly in terms of symbols of operators. For a certain natural class of PDOs on odd-dimensional manifolds generalizing the class of elliptic differential operators, the multiplicative anomaly is identically 1. For elliptic PDOs from this class a holomorphic determinant and a determinant for zero orders PDOs are introduced. Using various algebraic, analytic, and topological tools we study local and global properties of the multiplicative anomaly and of the determinant Lie group closely related with it. The Lie algebra for the determinant Lie group has a description in terms of symbols only. Our main discovery is that there is a quadratic non-linearity hidden in the definition of determinants of PDOs through zeta-functions. The natural explanation of this non-linearity follows from complex-analytic properties of a new trace functional TR on PDOs of non-integer orders. Using TR we easily reproduce known facts about noncommutative residues of PDOs and obtain several new results. In particular, we describe a structure of derivatives of zetafunctions at zero as of functions on logarithms of elliptic PDOs. We propose several definitions extending zeta-regularized determinants to general elliptic PDOs. For elliptic PDOs of nonzero complex orders we introduce a canonical determinant in its natural domain of definition. Contents

Several random particle systems approaches were recently suggested to solve numerically non linear filtering problems. The present analysis is concerned with genetic-type interacting particle systems. Our aim is to study the fluctuations on path space of such particle approximating models. Keywords : Central Limit, Interacting random processes, Filtering, Stochastic approximation. code A.M.S : 60F05, 60G35, 93E11, 62L20. 1 Introduction 1.1 Background and motivations The Non Linear Filtering problem consists in recursively computing the conditional distributions of a non linear signal given its noisy observations. This problem has been extensively studied in the literature and, with the notable exception of the linear-Gaussian situation or wider classes of models (B`enes filters [2]) optimal filters have no finitely recursive solution (ChaleyatMaurel /Michel [7]). Although Kalman filtering ([26],[29]) is a popular tool in handling estimation problems its optimality heavily depends on...

Given an arbitrary distribution on a countable set S consider the number of independent samples required until the first repeated value is seen. Exact and asymptotic formulae are derived for the distribution of this time and of the times until subsequent repeats. Asymptotic properties of the repeat times are derived by embedding in a Poisson process. In particular, necessary and sufficient conditions for convergence are given and the possible limits explicitly described. Under the same conditions the finite dimensional distributions of the repeat times converge to the arrival times of suitably modified Poisson processes, and random trees derived from the sequence of independent Research supported in part by N.S.F. Grants DMS 92-24857, 94-04345, 92-24868 and 97-03691 trials converge in distribution to an inhomogeneous continuum random tree. 1 Introduction Recall the classical birthday problem: given that each day of the year is equally likely as a possible birthday, and that birth...

We discuss resonances for Schrödinger operators in whole- and half-line problems. One of our goals is to connect the Fredholm determinant approach of Froese to the Fourier transform approach of Zworski. Another is to prove a result on the number of antibound states namely, in a half-line problem there are an odd number of antibound states between any two bound states.