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Abstract-In the absence of knowledge of the true density function, Bayesian models take the joint density function for a sequence of n random variables to be an average of densities with respect to a prior. We examine the relative entropy distance D,, between the true density and the Bayesian density and show that the asymptotic distance is (d/2Xlogn)+ c, where d is the dimension of the parameter vector. Therefore, the relative entropy rate D,,/n converges to zero at rate (logn)/n. The constant c, which we explicitly identify, depends only on the prior density function and the Fisher information matrix evaluated at the true parameter value. Consequences are given for density estima-tion, universal data compression, composite hypothesis testing, and stock-market portfolio selection. 1.

Given a primitive element g of a finite field GF(q), the discrete logarithm of a nonzero element u GF(q) is that integer k, 1 k q - 1, for which u = g k . The well-known problem of computing discrete logarithms in finite fields has acquired additional importance in recent years due to its applicability in cryptography. Several cryptographic systems would become insecure if an efficient discrete logarithm algorithm were discovered. This paper surveys and analyzes known algorithms in this area, with special attention devoted to algorithms for the fields GF(2 n ). It appears that in order to be safe from attacks using these algorithms, the value of n for which GF(2 n ) is used in a cryptosystem has to be very large and carefully chosen. Due in large part to recent discoveries, discrete logarithms in fields GF(2 n ) are much easier to compute than in fields GF(p) with p prime. Hence the fields GF(2 n ) ought to be avoided in all cryptographic applications. On the other hand, ...

. We prove an asymptotic formula for the probability that, if one chooses a domino tiling of a large Aztec diamond according to the uniform distribution on such tilings, the tiling will contain a domino covering a given pair of adjacent lattice squares. This formula quantifies the effect of the diamond 's boundary conditions on the behavior of typical tilings; in addition, it yields a new proof of the arctic circle theorem of Jockusch, Propp, and Shor. Our approach is to use the saddle point method to estimate certain weighted sums of squares of Krawtchouk polynomials (whose relevance to domino tilings is demonstrated elsewhere), and to combine these estimates with some exponential sum bounds to deduce our final result. This approach generalizes straightforwardly to the case in which the probability distribution on the set of tilings incorporates bias favoring horizontal over vertical tiles or vice versa. We also prove a fairly general large deviation estimate for domino tilings of sim...

Flajolet and Soria established several central limit theorems for the parameter "number of components" in a wide class of combinatorial structures. In this paper, we shall prove a simple theorem which applies to characterize the convergence rates in their central limit theorems. This theorem is also applicable to arithmetical functions. Moreover, asymptotic expressions are derived for moments of integral order. Many examples from different applications are discussed.

This paper develops a unified enumerative and asymptotic theory of directed 2-dimensional lattice paths in half-planes and quarter-planes. The lattice paths are speci ed by a finite set of rules that are both time and space homogeneous, and have a privileged direction of increase. (They are then essentially 1-dimensional objects.) The theory relies on a specific "kernel method" that provides an important decomposition of the algebraic generating functions involved, as well as on a generic study of singularities of an associated algebraic curve. Consequences are precise computable estimates for the number of lattice paths of a given length under various constraints (bridges, excursions, meanders) as well as a characterization of the limit laws associated to several basic parameters of paths.

Abstract. Considered as rest points of ODE on L p, stationary viscous shock waves present a critical case for which standard semigroup methods do not su ce to determine stability. More precisely, there is no spectral gap between stationary modes and essential spectrum of the linearized operator about the wave, a fact which precludes the usual analysis by decomposition into invariant subspaces. For this reason, there have been until recently no results on shock stability from the semigroup perspective except in the scalar or totally compressive case ([Sat], [K.2], resp.), each of which can be reduced to the standard semigroup setting by Sattinger's method of weighted norms. We overcome this di culty in the general case by the introduction of new, pointwise semigroup techniques, generalizing ear-lier work of Howard [H.1], Kapitula [K.1-2], and Zeng [Ze,LZe]. These techniques allow us to do \hard " analysis in PDE within the dynamical systems/semigroup framework: in particular, to obtain sharp, global pointwise bounds on the Green's function of the linearized operator around the wave, su cient for the analysis of linear and nonlinear stability. The method is general, and should nd applications

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Form a random k-SAT formula on n variables by selecting uniformly and independently m = rn clauses out of all 2 possible k-clauses. The Satisfiability Threshold Conjecture asserts that for each k there exists a constant r k such that, as n tends to infinity, the probability that the formula is satisfiable tends to 1 if r < r k and to 0 if r > r k . It has long been known that 2 =k < r k < 2 . We prove that r k > 2 ln 2 d k , where d k ! (1 + ln 2)=2. Our proof also allows a blurry glimpse of the "geometry" of the set of satisfying truth assignments.