Thesis (Sc. D.)--Massachusetts Institute of Technology and Woods Hole Oceanographic Institution, 1987. / "January 1987." Bibliography: p. 193-200.

Zhang, Hai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 106-109). / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- Preliminaries --- p.13 / Chapter 2.1 --- Maxwell equations --- p.13 / Chapter 2.2 --- Reflection principle --- p.15 / Chapter 3 --- Scattering by General Polyhedral Obstacle --- p.19 / Chapter 3.1 --- Direct problem --- p.19 / Chapter 3.2 --- Inverse problem and statement of main results --- p.21 / Chapter 3.3 --- Proof of the main results --- p.22 / Chapter 3.3.1 --- Preliminaries --- p.23 / Chapter 3.3.2 --- Properties of perfect planes --- p.24 / Chapter 3.3.3 --- Proofs --- p.33 / Chapter 4 --- Scattering by Bi-periodic Polyhedral Grating (I) --- p.35 / Chapter 4.1 --- Direct problem --- p.36 / Chapter 4.2 --- Inverse problem and statement of main results --- p.38 / Chapter 4.3 --- Preliminaries --- p.39 / Chapter 4.4 --- Classification of unidentifiable periodic structures --- p.41 / Chapter 4.4.1 --- Observations and auxiliary tools --- p.41 / Chapter 4.4.2 --- First class of unidentifiable gratings --- p.45 / Chapter 4.4.3 --- Preparation for finding other classes of unidentifiable gratings --- p.47 / Chapter 4.4.4 --- A simple transformation --- p.52 / Chapter 4.4.5 --- Second class of unidentifiable gratings --- p.53 / Chapter 4.4.6 --- Third class of unidentifiable gratings --- p.58 / Chapter 4.4.7 --- Excluding the case with L --- p.61 / Chapter 4.4.8 --- Summary on all unidentifiable gratings --- p.65 / Chapter 4.5 --- Proof of Main results --- p.65 / Chapter 5 --- Scattering by Bi-periodic Polyhedral Grating (II) --- p.69 / Chapter 5.1 --- Preliminaries --- p.70 / Chapter 5.2 --- Classification of unidentifiable periodic structures --- p.72 / Chapter 5.2.1 --- First class of unidentifiable gratings --- p.72 / Chapter 5.2.2 --- Preparation for finding other classes of unidentifiable gratings --- p.73 / Chapter 5.2.3 --- Studying of the case L --- p.76 / Chapter 5.2.4 --- Study of the case with L --- p.89 / Chapter 5.2.5 --- Study of the case with L --- p.95 / Chapter 5.2.6 --- Summary on all unidentifiable gratings --- p.104 / Chapter 5.3 --- Unique determination of bi-periodic polyhedral grating --- p.104 / Bibliography --- p.106

Thesis (Ph.D.)--University of Delaware, 2007. / Principal faculty advisor: Daniel S. Weile, Dept. of Electrical and Computer Engineering. Includes bibliographical references.

Inverse problems arise in numerous fields of science and engineering where one tries to find out the desired information of an unknown object or the cause of an observed effect. They are of fundamental importance in many areas including radar and sonar applications, nondestructive testing, image processing, medical imaging, remote sensing, geophysics and astronomy among others. This study is concerned with three issues in scattering theory, fractional operators, as well as some of their inverse problems. The first topic is scattering problems for electromagnetic waves governed by Maxwell equations. It will be proved in the current study that an inhomogeneous EM medium with a corner on its support always scatters by assuming certain regularity and admissible conditions. This result implies that one cannot achieve invisibility for such materials. In order to verify the result, an integral of solutions to certain interior transmission problem is to be analyzed, and complex geometry optics solutions to corresponding Maxwell equations with higher order estimate for the residual will be constructed. The second problem involves the linearized elastic or seismic wave scattering described by the Lamei system. We will consider the elastic or seismic body wave which is composed of two different type of sub-waves, that is, the compressional or primary (P-) and the shear or secondary (S-) waves. We shall prove that the P- and the S-components of the total wave can be completely decoupled under certain geometric and boundary conditions. This is a surprising finding since it is known that the P- and the S-components of the elastic or seismic body wave are coupled in general. Results for decoupling around local boundary pieces, for boundary value problems, and for scattering problems are to be established. This decoupling property will be further applied to derive uniqueness and stability for the associated inverse problem of identifying polyhedral elastic obstacles by an optimal number of scattering measurements. Lastly, we consider a type of fractional (and nonlocal) elliptic operators and the associated Calderoin problem. The well-posedness for a kind of forward problems concerning the fractional operator will be established. As a consequence, the corresponding Dirichlet to Neumann map with certain mapping property is to be defined. As for the inverse problem, it will be shown that a potential can be uniquely identified by local Cauchy data of the associated nonlocal operator, in dimensions larger than or equal to two.

No description available.

The completeness of eigenfunctions for linearized equations is critical for many applications, such as the study of stability of solitary waves. In this thesis, we work with the Nonlinear Schr{\"o}dinger (NLS) equation, associated with the Zakharov-Shabat spectral problem. Firstly, we construct a complete set of eigenfunctions for the spectral problem. Our method involves working with an adjoint spectral problem and deriving completeness and orthogonality relations between eigenfunctions and adjoint eigenfunctions. Furthermore, we prove completeness of squared eigenfunctions, which are used to represent solutions of the linearized NLS equation. For this, we find relations between the variation of potential and the variation of scattering data. Moreover, we show the connection between the squared eigenfunctions of the Zakharov-Shabat spectral problem and solutions of the linearized NLS equation. / Thesis / Master of Science (MSc)

Thanks to the advancement of technological progress in networking and signal processing, we can access a large amount of musical content. In order for users to search among these vast catalogs, they need to have access to music-related information beyond the pure digital music file. Manual annotation of music is too expensive, therefore automated annotation would be of great use. A meaningful description of the musical pieces requires the incorporation of information about the instruments present in them. In this work, we present an approach for musical instrument recognition using the scattering transform, which is a transformation that gives a translation invariant representation, that is stable to deformations and preserves high frequency information for classication. We study recognition in both singleinstrument and multiple-instrument contexts. We compare the performance of models using the scattering transform to those using other standard features. We also examine the impact of the amount of training data. The experiments carried out do not show a clear superior performance of either feature representation. Still, the scattering transform is worth taking into account when choosing a way to extract features if we want to be able to characterize non-stationary signal structures. / Tack vare den tekniska utvecklingen i nätverk och signalbehandling kan vi få tillgång till en stor mängd musikaliskt innehåll. For att användare ska söka bland dessa stora kataloger måste de ha tillgång till musikrelaterad information utöver den rena digitala musikfilen. Eftersom den manuella annotationsprocessen skulle vara för dyr måste den automatiseras. En meningsfull beskrivning av musikstyckena kräver införlivande av information om instrumenten som finns i dem. I det här arbetet presenterar vi en metod for igenkänning av musikinstrument med hjälp av den scattering transform, som är en transformation som ger en översattnings-invariant representation, som är stabil för deformationer och bevarar högfrekvensinformation för klassicering. Vi studerar igenkännande i både enskilda instrument- och flera instrumentförhållanden. Vi jämför modellerna med den scattering transforms prestanda med de som använder andra standardfunktioner. Vi undersöker också effekterna av mangden traningsdata. Experimenten som utförs visar inte en tydlig överlagsen prestanda for någon av representationsföreställningarna jämfört med den andra. Fortfarande är den scattering transform värd att ta hänsyn till när man väljer ett sätt att extrahera funktioner om vi vill kunna karakterisera icke-stationära signalstrukturer.

We are concerned with the quantum inverse scattering problem. The corresponding Marchenko integral equation is solved by using the collocation method together with piece-wise polynomials, namely, Hermite splines. The scarcity of experimental data and the lack of phase information necessitate the generation of the input reflection coefficient by choosing a specific profile and then applying our method to reconstruct it. Various aspects of the single and coupled channels inverse problem and details about the numerical techniques employed are discussed. We proceed to apply our approach to synthetic seismic reflection data. The transformation of the classical one-dimensional wave equation for elastic displacement into a Schr¨odinger-like equation is presented. As an application of our method, we consider the synthetic reflection travel-time data for a layered substrate from which we recover the seismic impedance of the medium. We also apply our approach to experimental seismic reflection data collected from a deep water location in the North sea. The reflectivity sequence and the relevant seismic wavelet are extracted from the seismic reflection data by applying the statistical estimation procedure known as Markov Chain Monte Carlo method to the problem of blind deconvolution. In order to implement the Marchenko inversion method, the pure spike trains have been replaced by amplitudes having a narrow bell-shaped form to facilitate the numerical solution of the Marchenko integral equation from which the underlying seismic impedance profile of the medium is obtained. / Physics / D.Phil.(Physics)

In this thesis, we establish an analog of the Paley-Wiener Theorem for the ZS-AKNS scattering transform on a set of real potentials. We also demonstrate one application of our techniques to the study of an inverse spectral problem for a half-line Miura potential Schroedinger equation.

The Manakov system appears in the physics of optical fibers, as well as in quantum mechanics, as multi-component versions of the Nonlinear Schr\"odinger and the Gross-Pitaevskii equations.Although the Manakov system is completely integrable its solutions are far from being explicit in most cases. However, the Inverse Scattering Transform (IST) can be exploited to obtain asymptotic information about solutions.This thesis will describe the IST of the Manakov system, and its asymptotic behavior at short times. I will compare the focusing and defocusing behavior, numerically and analytically, for squared barrier initial potentials. Finally, I will show that the continuous spectrum gives the dominant contribution at short-times.