Abstract not found

We describe a very general model of a random graph process whose proportional degree sequence obeys a power law. Such laws have recently been observed in graphs associated with the world wide web.

High order differences of simple number sequences may be analysed asymptotically by means of integral representations, residue calculus, and contour integration. This technique, akin to Mellin transform asymptotics, is put in perspective and illustrated by means of several examples related to combinatorics and the analysis of algorithms like digital tries, digital search trees, quadtrees, and distributed leader election.

This paper presents a chronological overview of the developments in interpolation theory, from the earliest times to the present date. It brings out the connections between the results obtained in different ages, thereby putting the techniques currently used in signal and image processing into historical perspective. A summary of the insights and recommendations that follow from relatively recent theoretical as well as experimental studies concludes the presentation. Keywords—Approximation, convolution-based interpolation, history, image processing, polynomial interpolation, signal processing, splines. “It is an extremely useful thing to have knowledge of the true origins of memorable discoveries, especially those that have been found not by accident but by dint of meditation. It is not so much that thereby history may attribute to each man his own discoveries and others should be encouraged to earn like commendation, as that the art of making discoveries should be extended by considering noteworthy examples of it. ” 1 I.

Continued fractions lie at the heart of a number of classical algorithms like Euclid's greatest common divisor algorithm or the lattice reduction algorithm of Gauss that constitutes a 2-dimensional generalization. This paper surveys the main properties of functional operators, -- transfer operators -- due to Ruelle and Mayer (also following Lévy, Kuzmin, Wirsing, Hensley, and others) that describe precisely the dynamics of the continued fraction transformation. Spectral characteristics of transfer operators are shown to have many consequences, like the normal law for logarithms of continuants associated to the basic continued fraction algorithm and a purely analytic estimation of the average number of steps of the Euclidean algorithm. Transfer operators also lead to a complete analysis of the "Hakmem" algorithm for comparing two rational numbers via partial continued fraction expansions and of the "digital tree" algorithm for completely sorting n real numbers by means of ...

Abstract. The infinite–dimensional unitary group U(∞) is the inductive limit of growing compact unitary groups U(N). In this paper we solve a problem of harmonic analysis on U(∞) stated in [Ol3]. The problem consists in computing spectral decomposition for a remarkable 4–parameter family of characters of U(∞). These characters generate representations which should be viewed as analogs of nonexisting regular representation of U(∞). The spectral decomposition of a character of U(∞) is described by the spectral measure which lives on an infinite–dimensional space Ω of indecomposable characters. The key idea which allows us to solve the problem is to embed Ω into the space of point configurations on the real line without 2 points. This turns the spectral measure into a stochastic point process on the real line. The main result of the paper is a complete description of the processes corresponding to our concrete family of characters. We prove that each of the processes is a determinantal point process. That is, its correlation functions have determinantal form with a certain kernel. Our kernels have a special ‘integrable ’ form and are expressed through the Gauss

In this paper, we shall emphasize the role played by error estimates and annihilation difference operators in the construction of extrapolations processes. It is showed that this approach leads to a unified derivation of many extrapolation algorithms and related devices, to general results about their kernels and that it opens the way to many new algorithms. Their convergence and acceleration properties could also be studied within this framework.

this article we investigate both measures of accuracy. In either case, the accuracy is shown to depend quite heavily on the rule used to bin the data. Various binning rules are discussed in Section 2. The presentation is facilitated through the characterization of a binning rule through a "binning kernel", that has similarities with the usual kernel function. The properties of ~

We prove a new class of relations among multiple zeta values (MZV’s) which contains Ohno’s relation. We also give the formula for the maximal number of independent MZV’s of fixed weight, under our new relations. To derive our formula for MZV’s, we consider the Newton series whose values at non-negative integers are finite multiple harmonic sums. keywords multiple zeta value, the Newton series, Ohno’s relation 1

. We consider orthogonal polynomials pn with respect to an exponential weight function w(x) = exp(\GammaP (x)). The related equations for the recurrence coefficients have been explored by many people, starting essentially with Laguerre [49], in order to study special continued fractions, recurrence relations, and various asymptotic expansions (G. Freud's contribution [28, 56]). Most striking example is n = 2twn + wn (wn+1 + wn + wn\Gamma1 ) for the recurrence coefficients pn+1 = xpn \Gamma wn pn\Gamma1 of the orthogonal polynomials related to the weight w(x) = exp(\Gamma4(tx 2 + x 4 )) (notations of [26, pp.34-- 36]). This example appears in practically all the references below. The connection with discrete Painlev'e equations is described here. 1. Construction of orthogonal polynomials recurrence coefficients. Consider the set fp n g 1 0 of orthonormal polynomials with respect to a weight function w on (a part of) the real line: Z 1 \Gamma1 p n (x)p m (x)w(x) dx = ffi n;m ; n...