A new definition of the fractional Laplacian

It is noted that the standard definition of the fractional Laplacian leads to a hyper-singular convolution integral and is also obscure about how to implement the boundary conditions. This purpose of this note is to introduce a new definition of the fractional Laplacian to overcome these major drawbacks.Comment: This study is carred out with the ongoing project of "mathematical and numerical modelling of medical ultasound wave propagation" sponsored by the Simula Research Laborator

Optimal change-point estimation from indirect observations

We study nonparametric change-point estimation from indirect noisy observations. Focusing on the white noise convolution model, we consider two classes of functions that are smooth apart from the change-point. We establish lower bounds on the minimax risk in estimating the change-point and develop rate optimal estimation procedures. The results demonstrate that the best achievable rates of convergence are determined both by smoothness of the function away from the change-point and by the degree of ill-posedness of the convolution operator. Optimality is obtained by introducing a new technique that involves, as a key element, detection of zero crossings of an estimate of the properly smoothed second derivative of the underlying function.Comment: Published at http://dx.doi.org/10.1214/009053605000000750 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Core reconstruction in pseudopotential calculations

A new method is presented for obtaining all-electron results from a pseudopotential calculation. This is achieved by carrying out a localised calculation in the region of an atomic nucleus using the embedding potential method of Inglesfield [J.Phys. C {\bf 14}, 3795 (1981)]. In this method the core region is \emph{reconstructed}, and none of the simplifying approximations (such as spherical symmetry of the charge density/potential or frozen core electrons) that previous solutions to this problem have required are made. The embedding method requires an accurate real space Green function, and an analysis of the errors introduced in constructing this from a set of numerical eigenstates is given. Results are presented for an all-electron reconstruction of bulk aluminium, for both the charge density and the density of states.Comment: 14 pages, 5 figure

The Zero-Bin and Mode Factorization in Quantum Field Theory

We study a Lagrangian formalism that avoids double counting in effective field theories where distinct fields are used to describe different infrared momentum regions for the same particle. The formalism leads to extra subtractions in certain diagrams and to a new way of thinking about factorization of modes in quantum field theory. In non-relativistic field theories, the subtractions remove unphysical pinch singularities in box type diagrams, and give a derivation of the known pull-up mechanism between soft and ultrasoft fields which is required by the renormalization group evolution. In a field theory for energetic particles, the soft-collinear effective theory (SCET), the subtractions allow the theory to be defined with different infrared and ultraviolet regulators, remove double counting between soft, ultrasoft, and collinear modes, and give results which reproduce the infrared divergences of the full theory. Our analysis shows that convolution divergences in factorization formul\ae occur due to an overlap of momentum regions. We propose a method that avoids this double counting, which helps to resolve a long standing puzzle with singularities in collinear factorization in QCD. The analysis gives evidence for a factorization in rapidity space in exclusive decays.Comment: 92 pages, v4- Journal version. Some improvements to language in sections I, IIA, VI

Full-waveform inversion in three-dimensional PML-truncated elastic media : theory, computations, and field experiments

We are concerned with the high-fidelity subsurface imaging of the soil, which commonly arises in geotechnical site characterization and geophysical explorations. Specifically, we attempt to image the spatial distribution of the Lame parameters in semi-infinite, three-dimensional, arbitrarily heterogeneous formations, using surficial measurements of the soil's response to probing elastic waves. We use the complete waveforms of the medium's response to drive the inverse problem. Specifically, we use a partial-differential-equation (PDE)-constrained optimization approach, directly in the time-domain, to minimize the misfit between the observed response of the medium at select measurement locations, and a computed response corresponding to a trial distribution of the Lame parameters. We discuss strategies that lend algorithmic robustness to the proposed inversion schemes. To limit the computational domain to the size of interest, we employ perfectly-matched-layers (PMLs). The PML is a buffer zone that surrounds the domain of interest, and enforces the decay of outgoing waves. In order to resolve the forward problem, we present a hybrid finite element approach, where a displacement-stress formulation for the PML is coupled to a standard displacement-only formulation for the interior domain, thus leading to a computationally cost-efficient scheme. We discuss several time-integration schemes, including an explicit Runge-Kutta scheme, which is well-suited for large-scale problems on parallel computers. We report numerical results demonstrating stability and efficacy of the forward wave solver, and also provide examples attesting to the successful reconstruction of the two Lame parameters for both smooth and sharp profiles, using synthetic records. We also report the details of two field experiments, whose records we subsequently used to drive the developed inversion algorithms in order to characterize the sites where the field experiments took place. We contrast the full-waveform-based inverted site profile against a profile obtained using the Spectral-Analysis-of-Surface-Waves (SASW) method, in an attempt to compare our methodology against a widely used concurrent inversion approach. We also compare the inverted profiles, at select locations, with the results of independently performed, invasive, Cone Penetrometer Tests (CPTs). Overall, whether exercised by synthetic or by physical data, the full-waveform inversion method we discuss herein appears quite promising for the robust subsurface imaging of near-surface deposits in support of geotechnical site characterization investigations.Civil, Architectural, and Environmental Engineerin

Approximations by Generalized Discrete Singular Operators

Here, we give the approximation properties with rates of generalized discrete versions of Picard, Gauss-Weierstrass, and Poisson-Cauchy singular operators. We cover both the unitary and non-unitary cases of the operators above. We present quantitatively the point-wise and uniform convergences of these operators to the unit operator by involving the higher modulus of smoothness of a uniformly continuous function. We also establish our results with respect to L_p norm, 1≤p\u3c∞. Additionally, we state asymptotic Voronovskaya type expansions for these operators. Moreover, we study the fractional generalized smooth discrete singular operators on the real line regarding their convergence to the unit operator with fractional rates in the uniform norm. Then, we give our results for the operators mentioned above over the real line regarding their simultaneous global smoothness preservation property with respect to L_p norm for 1≤p≤∞, by involving higher order moduli of smoothness. Here we also obtain Jackson type inequalities of simultaneous approximation which are almost sharp, containing neat constants, and they reflect the high order of differentiability of involved function. Next, we cover the approximation properties of on the general complex-valued discrete singular operators over the real line regarding their convergence to the unit operator with rates in the L_p norm for 1≤p≤∞. Finally, we establish the approximation properties of multivariate generalized discrete versions of these operators over R^N,N≥1. We give pointwise, uniform, and L_p convergence of the operators to the unit operator by involving the multivariate higher order modulus of smoothness

La funció conjugada i la transformada de Hilbert

La transformada de Hilbert, tant en el cas periòdic, anomenat també funció conjugada, com per funcions definides a la recta, constitueixen un dels operadors més important en l'anàlisi harmònica. La seva acotació en espais Lp del torus per p>1 implica la convergència en norma p de les sumes parcials de la sèrie de Fourier d'una funció f vers aquesta. En paral.lel al que passa per funcions definides al cercle, tenim resultats anàlegs per la transformada de Hilbert a R que és l'operador integral singular per excel.lència en el cas unidimensional