We address the formulation of a scale-space theory for discrete signals. In one dimension it is possible to characterize the smoothing transformations completely and an exhaustive treatment is given, answering the following two main questions: 1. Which linear transformations remove structure in the sense that the number of local extrema (or zero-crossings) in the output signal does not exceed the number of local extrema (or zero-crossings) in the original signal? 2. How should one create a multi-resolution family of representations with the property that a signal at a coarser level of scale never contains more structure than a signal at a finer level of scale? We propose that there is only one reasonable way to define a scale-space for 1D discrete signals comprising a continuous scale parameter, namely by (discrete) convolution with the family of kernels T (n; t) = e I n (t), where I n are the modified Bessel functions of integer order. Similar arguments applied in the continuous case uniquely lead to the Gaussian kernel.

this paper we extend the results of [3, 4] to the whole variety G0 . We will try to make the point that the natural framework for the study of G0 is provided by the decomposition of G into the disjoint union of double Bruhat cells G = BuB " B \Gamma vB \Gamma ; here B and B \Gamma are two opposite Borel subgroups in G, and u and v belong to the Weyl group W of G. We believe these double cells to be a very interesting object of study in its own right. The term "cells" might be misleading: in fact, the topology of G is in general quite nontrivial. (In some special cases, the "real part" of G was studied in [20, 21]. V. Deodhar [9] studied the intersections BuB " B \Gamma vB whose properties are very different from those of G .) We study a family of birational parametrizations of G , one for each reduced expression i of the element (u; v) in the Coxeter group W \Theta W . Every such parametrization can be thought of as a system of local coordinates in G call these coordinates the factorization parameters associated to i. They are obtained by expressing a generic element x 2 G as an element of the maximal torus H = B " B \Gamma multiplied by the product of elements of various one-parameter subgroups in G associated with simple roots and their negatives; the reduced expression i prescribes the order of factors in this product. The main technical result of this paper (Theorem 1.9) is an explicit formula for these factorization parameters as rational functions on the double Bruhat cell G . Theorem 1.9 is formulated in terms of a special family of regular functions \Delta fl ;ffi on the group G. These functions are suitably normalized matrix coefficients corresponding to pairs of extremal weights (fl; ffi ) in some fundamental representation of G. Again, we b...

We analyze when it is possible to compute the singular values and singular vectors of a matrix with high relative accuracy. This means that each computed singular value is guaranteed to have some correct digits, even if the singular values have widely varying magnitudes. This is in contrast to the absolute accuracy provided by conventional backward stable algorithms, whichin general only guarantee correct digits in the singular values with large enough magnitudes. It is of interest to compute the tiniest singular values with several correct digits, because in some cases, such as #nite element problems and quantum mechanics, it is the smallest singular values that havephysical meaning, and should be determined accurately by the data. Many recent papers have identi#ed special classes of matrices where high relative accuracy is possible, since it is not possible in general. The perturbation theory and algorithms for these matrix classes have been quite di#erent, motivating us to seek a co...

Networks of linear units are the simplest kind of networks, where the basic questions related to learning, generalization, and self-organisation can sometimes be answered analytically. We survey most of the known results on linear networks, including: (1) back-propagation learning and the structure of the error function landscape; (2) the temporal evolution of generalization; (3) unsupervised learning algorithms and their properties. The connections to classical statistical ideas, such as principal component analysis (PCA), are emphasized as well as several simple but challenging open questions. A few new results are also spread across the paper, including an analysis of the effect of noise on back-propagation networks and a unified view of all unsupervised algorithms. Keywords--- linear networks, supervised and unsupervised learning, Hebbian learning, principal components, generalization, local minima, self-organisation I. Introduction This paper addresses the problems of supervise...

We study the a.s. exponential stability of the optimal filter w.r.t. its initial conditions. A bound is provided on the exponential rate (equivalently, on the memory length of the filter) for a general setting both in discrete and in continuous time, in terms of Birkhoff's contraction coefficient. Criteria for exponential stability and explicit bounds on the rate are given in the specific cases of a diffusion process on a compact manifold, and discrete time Markov chains on both continuous and discrete-countable state spaces. R'esum'e Nous 'etudions la stabilit'e du filtre optimal par raport `a ses conditions initiales. Le taux de d'ecroissance exponentielle est calcul'e dans un cadre g'en'eral, pour temps discret et temps continu, en terme du coefficient de contraction de Birkhoff. Des crit`eres de stabilit'e exponentielle et des bornes explicites sur le taux sont calcul'ees pour les cas particuliers d'une diffusion sur une vari'ete compacte, ainsi que pour des chaines de Markov sur ...

A matrix is totally positive (resp. totally nonnegative) if all its minors are positive (resp. nonnegative) real numbers. The first systematic study of these classes of matrices was undertaken in the 1930s by F. R. Gantmacher and M. G. Krein [20, 21, 22], who established their remarkable spectral properties (in particular,

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.

This paper is intended to serve as a postscript to the fundamental 1966 paper by Curry and Schoenberg on B-splines. It is also intended to promote the point of view that B-splines are truly basic splines: B-splines express the essentially local, but not completely local, character of splines; certain facts about splines take on their most striking form when put into B-spline terms, and many theorems about splines are most easily proved with the aid of B-splines; the computational determination of a specific spline from some information about it is usually facilitated when B-splines are used in its construction.

The work of Harper and subsequent authors has shown that nite sequences (a 0;;an) arising from combinatorial problems are often such that the polynomial A(z): = P n k=0 akz k has only real zeros. Basic examples include rows from the arrays of binomial coe cients, Stir-ling numbers of the rst and second kinds, and Eulerian numbers. Assuming the ak are non-negative, A(1)> 0 and that A(z) is not constant, it is known that A(z) has only real zeros i the normalized sequence (a 0=A(1);;an=A(1)) is the probability distribution of the Research supported in part by N.S.F. Grant MCS9404345 1 number of successes in n independent trials for some sequence of suc-cess probabilities. Such sequences (a 0;;an) are also known to be characterized by total positivity of the in nite matrix (ai,j) indexed by non-negative integers i and j. This papers reviews inequalities and approximations for such sequences, called Polya frequency sequences which follow from their probabilistic representation. In combinatorial examples these inequalities yield a number of improvements of known estimates.

This paper studies collusion in repeated Bertrand oligopoly when stochastic demand levels for the product of each firm are their private information and are positively correlated. It derives general sufficient conditions for efficient collusion through communication and a simple grim-trigger strategy. This analysis is then applied to a model where the demand signal has multiple random components which respond differently to price deviations. In this model, it is shown that the above sufficient conditions hold if idiosyncratic noise terms are sufficiently small.