First order k-th moment finite element analysis of nonlinear operator equations with stochastic data

We develop and analyze a class of efficient Galerkin approximation methods for uncertainty quantification of nonlinear operator equations. The algorithms are based on sparse Galerkin discretizations of tensorized linearizations at nominal parameters. Specifically, we consider abstract, nonlinear, parametric operator equations J(\alpha ,u)=0 for random input \alpha (\omega ) with almost sure realizations in a neighborhood of a nominal input parameter \alpha _0. Under some structural assumptions on the parameter dependence, we prove existence and uniqueness of a random solution, u(\omega ) = S(\alpha (\omega )). We derive a multilinear, tensorized operator equation for the deterministic computation of k-th order statistical moments of the random solution's fluctuations u(\omega ) - S(\alpha _0). We introduce and analyse sparse tensor Galerkin discretization schemes for the efficient, deterministic computation of the k-th statistical moment equation. We prove a shift theorem for the k-point correlation equation in anisotropic smoothness scales and deduce that sparse tensor Galerkin discretizations of this equation converge in accuracy vs. complexity which equals, up to logarithmic terms, that of the Galerkin discretization of a single instance of the mean field problem. We illustrate the abstract theory for nonstationary diffusion problems in random domains.

The full infinite dimensional moment problem on semi-algebraic sets of generalized functions

We consider a generic basic semi-algebraic subset S of the space of generalized functions, that is a set given by (not necessarily countably many) polynomial constraints. We derive necessary and sufficient conditions for an infinite sequence of generalized functions to be realizable on S, namely to be the moment sequence of a finite measure concentrated on S. Our approach combines the classical results about the moment problem on nuclear spaces with the techniques recently developed to treat the moment problem on basic semi-algebraic sets of Rd. In this way, we determine realizability conditions that can be more easily verified than the well-known Haviland type conditions. Our result completely characterizes the support of the realizing measure in terms of its moments. As concrete examples of semi-algebraic sets of generalized functions, we consider the set of all Radon measures and the set of all the measures having bounded Radon–Nikodym density w.r.t. the Lebesgue measure.

A moment problem for random discrete measures

Let X be a locally compact Polish space. A random measure on X is a probability measure on the space of all (nonnegative) Radon measures on X. Denote by K(X) the cone of all Radon measures η on X which are of the form η =

Coherent states of a particle in a magnetic field and the Stieltjes moment problem

A solution to a version of the Stieltjes moment. problem is presented. Using this solution, we construct a family of coherent states of a charged particle in a uniform magnetic field. We prove that these states form an overcomplete set that is normalized and resolves the unity. By the help of these coherent states we construct the Fock-Bergmann representation related to the particle quantization. This quantization procedure takes into account a circle topology of the classical motion. (C) 2009 Elsevier B.V. All rights reserved.

Sharing the moment : on the Olympic Games as spectacle

Article examining the 'sale' of Olympics drawing on Kant's aesthetics of beauty in the Critique of Judgment.

A modified moment-based image watermarking method robust to cropping attack

Developing a watermarking method that is robust to cropping attack is a challenging task in image watermarking. The moment-based watermarking schemes show good robustness to common signal processing attacks and some geometric attacks but are sensitive to cropping attack. In this paper, we modify the moment-based approach to deal with cropping attack. Firstly, we find the probability density function (pdf) of the pixel value distribution from the original image. Secondly, we reshape and normalize the pdf of the pixel value distribution (PPVD) to form a two dimensional image. Then, the moment invariants are calculated from the PPVD image. Since PPVD is insensitive to cropping, the proposed method is robust to cropping attack. Besides, it also has high robustness against other common attacks. Experimental results demonstrate the effectiveness of the proposed method.

‘Retrieving the Bandung Conference … moment by moment’

This review article discusses two publications: Bandung 1955: Little histories (2010) and Making a world after empire: The Bandung moment and its political afterlives (2010). It concludes that, though thesee two texts offer many prisms through which the conference might be understood, the history of the Bandung Conference remains somewhat fragmented.

Removing the ambiguity of data processing methods: Optimizing the location of peak boundaries for accurate moment calculations

The calculation of the first few moments of elution peaks is necessary to determine: the amount of component in the sample (peak area or zeroth moment), the retention factor (first moment), and the column efficiency (second moment). It is a time consuming and tedious task for the analyst to perform these calculations, thus data analysis is generally completed by data stations associated to modern chromatographs. However, data acquisition software is a black box which provides no information to chromatographers on how their data are treated. These results are too important to be accepted on blind faith. The location of the peak integration boundaries is most important. In this manuscript, we explore the relationships between the size of the integration area, the relative position of the peak maximum within this area, and the accuracy of the calculated moments. We found that relationships between these parameters do exist and that computers can be programmed with relatively simple routines to automatize the extraction of key peak parameters and to select acceptable integration boundaries. It was also found that the most accurate results are obtained when the S/N exceeds 200.