Factorization and criticality in the anisotropic XY chain via correlations

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Detection of small buried objects: asymptotic factorization and MUSIC

In this work, we consider a simple model problem for the electromagnetic exploration of small perfectly conducting objects buried within the lower halfspace of an unbounded two–layered background medium. In possible applications, such as, e.g., humanitarian demining, the two layers would correspond to air and soil. Moving a set of electric devices parallel to the surface of ground to generate a time–harmonic field, the induced field is measured within the same devices. The goal is to retrieve information about buried scatterers from these data. In mathematical terms, we are concerned with the analysis and numerical solution of the inverse scattering problem to reconstruct the number and the positions of a collection of finitely many small perfectly conducting scatterers buried within the lower halfspace of an unbounded two–layered background medium from near field measurements of time–harmonic electromagnetic waves. For this purpose, we first study the corresponding direct scattering problem in detail and derive an asymptotic expansion of the scattered field as the size of the scatterers tends to zero. Then, we use this expansion to justify a noniterative MUSIC–type reconstruction method for the solution of the inverse scattering problem. We propose a numerical implementation of this reconstruction method and provide a series of numerical experiments.

The factorization method for real elliptic problems

The Factorization Method localizes inclusions inside a body from measurements on its surface. Without a priori knowing the physical parameters inside the inclusions, the points belonging to them can be characterized using the range of an auxiliary operator. The method relies on a range characterization that relates the range of the auxiliary operator to the measurements and is only known for very particular applications. In this work we develop a general framework for the method by considering symmetric and coercive operators between abstract Hilbert spaces. We show that the important range characterization holds if the difference between the inclusions and the background medium satisfies a coerciveness condition which can immediately be translated into a condition on the coefficients of a given real elliptic problem. We demonstrate how several known applications of the Factorization Method are covered by our general results and deduce the range characterization for a new example in linear elasticity.

Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem

In various imaging problems the task is to use the Cauchy data of the solutions to an elliptic boundary value problem to reconstruct the coefficients of the corresponding partial differential equation. Often the examined object has known background properties but is contaminated by inhomogeneities that cause perturbations of the coefficient functions. The factorization method of Kirsch provides a tool for locating such inclusions. In this paper, the factorization technique is studied in the framework of coercive elliptic partial differential equations of the divergence type: Earlier it has been demonstrated that the factorization algorithm can reconstruct the support of a strictly positive (or negative) definite perturbation of the leading order coefficient, or if that remains unperturbed, the support of a strictly positive (or negative) perturbation of the zeroth order coefficient. In this work we show that these two types of inhomogeneities can, in fact, be located simultaneously. Unlike in the earlier articles on the factorization method, our inclusions may have disconnected complements and we also weaken some other a priori assumptions of the method. Our theoretical findings are complemented by two-dimensional numerical experiments that are presented in the framework of the diffusion approximation of optical tomography.

Factorization method and irregular inclusions in electrical impedance tomography

In electrical impedance tomography, one tries to recover the conductivity inside a physical body from boundary measurements of current and voltage. In many practically important situations, the investigated object has known background conductivity but it is contaminated by inhomogeneities. The factorization method of Andreas Kirsch provides a tool for locating such inclusions. Earlier, it has been shown that under suitable regularity conditions positive (or negative) inhomogeneities can be characterized by the factorization technique if the conductivity or one of its higher normal derivatives jumps on the boundaries of the inclusions. In this work, we use a monotonicity argument to generalize these results: We show that the factorization method provides a characterization of an open inclusion (modulo its boundary) if each point inside the inhomogeneity has an open neighbourhood where the perturbation of the conductivity is strictly positive (or negative) definite. In particular, we do not assume any regularity of the inclusion boundary or set any conditions on the behaviour of the perturbed conductivity at the inclusion boundary. Our theoretical findings are verified by two-dimensional numerical experiments.

Factorization and non-local 1/m b corrections in the decay B X s gamma

In this thesis, a systematic analysis of the bar B to X_sgamma photon spectrum in the endpoint region is presented. The endpoint region refers to a kinematic configuration of the final state, in which the photon has a large energy m_b-2E_gamma = O(Lambda_QCD), while the jet has a large energy but small invariant mass. Using methods of soft-collinear effective theory and heavy-quark effective theory, it is shown that the spectrum can be factorized into hard, jet, and soft functions, each encoding the dynamics at a certain scale. The relevant scales in the endpoint region are the heavy-quark mass m_b, the hadronic energy scale Lambda_QCD and an intermediate scale sqrt{Lambda_QCD m_b} associated with the invariant mass of the jet. It is found that the factorization formula contains two different types of contributions, distinguishable by the space-time structure of the underlying diagrams. On the one hand, there are the direct photon contributions which correspond to diagrams with the photon emitted directly from the weak vertex. The resolved photon contributions on the other hand arise at O(1/m_b) whenever the photon couples to light partons. In this work, these contributions will be explicitly defined in terms of convolutions of jet functions with subleading shape functions. While the direct photon contributions can be expressed in terms of a local operator product expansion, when the photon spectrum is integrated over a range larger than the endpoint region, the resolved photon contributions always remain non-local. Thus, they are responsible for a non-perturbative uncertainty on the partonic predictions. In this thesis, the effect of these uncertainties is estimated in two different phenomenological contexts. First, the hadronic uncertainties in the bar B to X_sgamma branching fraction, defined with a cut E_gamma > 1.6 GeV are discussed. It is found, that the resolved photon contributions give rise to an irreducible theory uncertainty of approximately 5 %. As a second application of the formalism, the influence of the long-distance effects on the direct CP asymmetry will be considered. It will be shown that these effects are dominant in the Standard Model and that a range of -0.6 < A_CP^SM < 2.8 % is possible for the asymmetry, if resolved photon contributions are taken into account.

Machine Learning applicato al Web Semantico: Statistical Relational Learning vs Tensor Factorization

Obiettivo della tesi è analizzare e testare i principali approcci di Machine Learning applicabili in contesti semantici, partendo da algoritmi di Statistical Relational Learning, quali Relational Probability Trees, Relational Bayesian Classifiers e Relational Dependency Networks, per poi passare ad approcci basati su fattorizzazione tensori, in particolare CANDECOMP/PARAFAC, Tucker e RESCAL.