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313
Nuclear and Trace Ideals in Tensored *Categories
, 1998
"... 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 ..."
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Cited by 28 (10 self)
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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 HilbertSchmidt maps. We also introduce two new examples of tensored categories, in which integration plays the role of composition. In the first, mor...
Diffraction of Random Tilings: Some Rigorous Results
 J. STAT. PHYS
, 1999
"... The diffraction of stochastic point sets, both Bernoulli and Markov, and of random tilings with crystallographic symmetries is investigated in rigorous terms. In particular, we derive the diffraction spectrum of 1D random tilings, of stochastic product tilings built from cuboids, and of planar rando ..."
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Cited by 26 (16 self)
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The diffraction of stochastic point sets, both Bernoulli and Markov, and of random tilings with crystallographic symmetries is investigated in rigorous terms. In particular, we derive the diffraction spectrum of 1D random tilings, of stochastic product tilings built from cuboids, and of planar random tilings based on solvable dimer models, augmented by a brief outline of the diraction from the classical 2D Ising lattice gas. We also give a summary of the measure theoretic approach to mathematical diraction theory which underlies the unique decomposition of the diffraction spectrum into its pure point, singular continuous and absolutely continuous parts.
Diffractive point sets with entropy
"... Dedicated to HansUde Nissen on the occasion of his 65th birthday After a brief historical survey, the paper introduces the notion of entropic model sets (cut and project sets), and, more generally, the notion of diffractive point sets with entropy. Such sets may be thought of as generalizations of ..."
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Cited by 24 (16 self)
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Dedicated to HansUde Nissen on the occasion of his 65th birthday After a brief historical survey, the paper introduces the notion of entropic model sets (cut and project sets), and, more generally, the notion of diffractive point sets with entropy. Such sets may be thought of as generalizations of lattice gases. We show that taking the site occupation of a model set stochastically results, with probabilistic certainty, in welldefined diffractive properties augmented by a constant diffuse background. We discuss both the case of independent, but identically distributed (i.i.d.) random variables and that of independent, but different (i.e., site dependent) random variables. Several examples are shown.
The Intrinsic Structure of Optic Flow Incorporating Measurement Duality
 International Journal of Computer Vision
, 1997
"... The purpose of this report 1 is to define optic flow for scalar and density images without using a priori knowledge other than its defining conservation principle, and to incorporate measurement duality, notably the scalespace paradigm. It is argued that the design of optic flow based applicati ..."
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Cited by 20 (13 self)
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The purpose of this report 1 is to define optic flow for scalar and density images without using a priori knowledge other than its defining conservation principle, and to incorporate measurement duality, notably the scalespace paradigm. It is argued that the design of optic flow based applications may benefit from a manifest separation between factual image structure on the one hand, and goalspecific details and hypotheses about image flow formation on the other. The approach is based on a physical symmetry principle known as gauge invariance. Dataindependent models can be incorporated by means of admissible gauge conditions, each of which may single out a distinct solution, but all of which must be compatible with the evidence supported by the image data. The theory is illustrated by examples and verified by simulations, and performance is compared to several techniques reported in the literature. 1 Introduction The conventional "spacetime" representation of a movie as...
Várilly, “Dixmier traces on noncompact isospectral deformations
 J. Funct. Anal
"... We extend the isospectral deformations of Connes, Landi and DuboisViolette to the case of Riemannian spin manifolds carrying a proper action of the noncompact abelian group R l. Under deformation by a torus action, a standard formula relates Dixmier traces of measurable operators to integrals of fu ..."
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Cited by 18 (8 self)
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We extend the isospectral deformations of Connes, Landi and DuboisViolette to the case of Riemannian spin manifolds carrying a proper action of the noncompact abelian group R l. Under deformation by a torus action, a standard formula relates Dixmier traces of measurable operators to integrals of functions on the manifold. We show that this relation persists for actions of R l, under mild restrictions on the geometry of the manifold which guarantee the Dixmier traceability of those operators.
Semiparametrically efficient rankbased inference for shape I: Optimal rankbased tests for sphericity
 Ann. Statist
, 2006
"... A class of Restimators based on the concepts of multivariate signed ranks and the optimal rankbased tests developed in Hallin and Paindaveine [Ann. Statist. 34 (2006)] is proposed for the estimation of the shape matrix of an elliptical distribution. These Restimators are rootn consistent under a ..."
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Cited by 16 (12 self)
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A class of Restimators based on the concepts of multivariate signed ranks and the optimal rankbased tests developed in Hallin and Paindaveine [Ann. Statist. 34 (2006)] is proposed for the estimation of the shape matrix of an elliptical distribution. These Restimators are rootn consistent under any radial density g, without any moment assumptions, and semiparametrically efficient at some prespecified density f. When based on normal scores, they are uniformly more efficient than the traditional normaltheory estimator based on empirical covariance matrices (the asymptotic normality of which, moreover, requires finite moments of order four), irrespective of the actual underlying elliptical density. They rely on an original rankbased version of Le Cam’s onestep methodology which avoids the unpleasant nonparametric estimation of crossinformation quantities that is generally required in the context of Restimation. Although they are not strictly affineequivariant, they are shown to be equivariant in a weak asymptotic sense. Simulations confirm their feasibility and excellent finitesample performances. 1. Introduction. 1.1. Rankbased inference for elliptical families. An elliptical density over Rk is determined by a location center θ ∈ Rk, a scale parameter σ ∈ R + 0, a realvalued positive definite symmetric k × k matrix V = (Vij) with V11 = 1,
Some nonlinear SPDE’s that are second order in time
 ELECTRONIC J. OF PROBABILITY, VOL
, 2003
"... We extend Walsh’s theory of martingale measures in order to deal with hyperbolic stochastic partial differential equations that are second order in time, such as the wave equation and the beam equation, and driven by spatially homogeneous Gaussian noise. For such equations, the fundamental solution ..."
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Cited by 15 (3 self)
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We extend Walsh’s theory of martingale measures in order to deal with hyperbolic stochastic partial differential equations that are second order in time, such as the wave equation and the beam equation, and driven by spatially homogeneous Gaussian noise. For such equations, the fundamental solution can be a distribution in the sense of Schwartz, which appears as an integrand in the reformulation of the s.p.d.e. as a stochastic integral equation. Our approach provides an alternative to the Hilbert space integrals of HilbertSchmidt operators. We give several examples, including the beam equation and the wave equation, with nonlinear multiplicative noise terms.
Spaces of Distributions and Interpolation by Translates of a Basis Function
 Numer. Math
, 1997
"... Interpolation with translates of a basis function is a common process in approximation theory. One starts with a single function (the basis function) and a set of interpolation points. The most elementary form of the interpolant then consists of a linear combination of all translates by interpolatio ..."
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Cited by 15 (1 self)
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Interpolation with translates of a basis function is a common process in approximation theory. One starts with a single function (the basis function) and a set of interpolation points. The most elementary form of the interpolant then consists of a linear combination of all translates by interpolation points of the basis function. Frequently, low degree polynomials are added to the interpolant. One of the significant features of this type of interpolant is that it is often the solution of a variational problem. To make this work, one needs an appropriate space in which to carry out the variational arguments. In this paper we concentrate on developing such spaces for a wide class of basis functions. We also show how the theory leads to efficient ways of calculating the interpolant and to error estimates. 1 Introduction Radial basis function interpolation is now a wellestablished method for performing interpolation to data specified at points in IR n . The form of the interpolant used...
An introduction to total variation for image analysis
 in Theoretical Foundations and Numerical Methods for Sparse Recovery, De Gruyter
, 2010
"... These notes address various theoretical and practical topics related to Total Variationbased image reconstruction. They focuse first on some theoretical results on functions which minimize the total variation, and in a second part, describe a few standard and less standard algorithms to minimize th ..."
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Cited by 15 (2 self)
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These notes address various theoretical and practical topics related to Total Variationbased image reconstruction. They focuse first on some theoretical results on functions which minimize the total variation, and in a second part, describe a few standard and less standard algorithms to minimize the total variation in a finitedifferences setting, with a series of applications from simple denoising to stereo, or deconvolution issues, and even more exotic uses like the minimization of minimal partition problems.