This thesis uses the powerful mathematics of wavelet packet signal processing to efficiently extract features from sampled acoustic spectra for the purpose of discriminating between different classes of sounds. An algorithm called dictionary projection pursuit (DPP) is developed which is a fast approximate version of the projection pursuit (PP) algorithm [P.J. Huber Projection Pursuit, Annals of Statistics, 13 ( 2) 435–525, 1985]. When used with a wavelet packet or cosine packet dictionary, this algorithm is significantly faster than the PP algorithm with relatively little degradation in performance provided that the multivariate vectors are samples of an underlying continuous waveform or image. The DPP algorithm is applied to the problem of approximating the Karhunen-Loève transform (KLT) in high dimensional spaces and simulations are performed to compare this algorithm to Wickerhauser's approximate KLT algorithm [M.V. Wickerhauser. Adapted Wavelet Analysis from Theory to Software, A.K. Peters Ltd, 1994]. Both algorithms perform very well relative to the eigenanalysis form of the KLT algorithm at a small fraction of the computational cost. The DPP algorithm is then applied to the problem of finding discriminant features in acoustic spectra for sound recognition tasks; extensive simulations are performed to compare this algorithm to previously developed dictionary methods for discrimination such as Saito and Coifman's local discriminant bases [N. Saito and R. Coifman. Local Discriminant Bases and their Applications. Journal of Mathematical Imaging and Vision, 5 (4) 337–358, 1995] and Buckheit and Donoho's discriminant pursuit [J. Buckheit and D. Donoho. Improved Linear Discrimination Using Time-Frequency Dictionaries. Proceedings of SPIE Wavelet Applications in Signal and Image Processing III Vol 2569, 540–551, July, 1995]. It is found that each feature extraction algorithm performs well under different conditions, but the DPP algorithm is the most flexible and consistent performer. / Graduate

M.Sc.

No description available.

We consider the spectrum of the Schrödinger operator on an octagonal lattice using the Floquet-Bloch transform of the Laplacian. We will first consider the spectrum of the Laplacian in detail and prove various properties thereof, including spectral-band limits and locations of singularities. In addition, we will prove that Schrödinger operators with 1-1 periodic potentials can open at most two gaps in the spectrum precisely at energies $pm1$, and that a third gap can open at 0 for 2-2 periodic potentials. We describe in detail the structure of these operators for higher periods, and motivate our expectations of their spectra. / Master of Science / In quantum physics, we would like the capability to model environments, such as magnetic fields, that interact with electrons or other quantum entities. The fields of graph theory and functional analysis within mathematics provide tools which relate well-understood mathematical concepts to these physical interactions. In this work, we use these tools to describe these environments using previously employed techniques in new ways.

The spectral theory of the Laplace operator has long been studied in connection with physics. It appears in the wave equation, the heat equation, Schroedinger's equation and in the expression of quantum effects such as the Casimir force. The Casimir effect can be studied in terms of spectral invariants computed entirely from the spectrum of the Laplace operator. It is these spectral invariants and their computation that are the object of study in the present work. The objective of this thesis is to present a computational framework for the spectral zeta function $\zeta(s)$ and its derivative on a Euclidean domain in $\mathbb{R}^2$, with rigorous theoretical error bounds when this domain is polygonal. To obtain error bounds that remain practical in applications an improvement to existing heat trace estimates is necessary. Our main result is an original estimate and proof of a heat trace estimate for polygons that improves the one of van den Berg and Srisatkunarajah, using finite propagation speed of the corresponding wave kernel. We then use this heat trace estimate to obtain a rigorous error bound for $\zeta(s)$ computations. We will provide numerous examples of our computational framework being used to calculate $\zeta(s)$ for a variety of situations involving a polygonal domain, including examples involving cutouts and extrusions that are interesting in applications. Our second result is the development a new eigenvalue solver for a planar polygonal domain using a partition of unity decomposition technique. Its advantages include multiple precision and ease of use, as well as reduced complexity compared to Finite Elemement Method. While we hoped that it would be able to contend with existing packages in terms of speed, our implementation was many times slower than MPSPack when dealing with the same problem (obtaining the first 5 digits of the principal eigenvalue of the regular unit hexagon). Finally, we present a collection of numerical examples where we compute the spectral determinant and Casimir energy of various polygonal domains. We also use our numerical tools to investigate extremal properties of these spectral invariants. For example, we consider a square with a small square cut out of the interior, which is allowed to rotate freely about its center.

Dissertation (PhD)--University of Stellenbosch, 2002 / ENGLISH ABSTRACT: Singular spectrum analysis (SSA) originated in the field of Physics. The technique is non-parametric by nature and inter alia finds application in atmospheric sciences, signal processing and recently in financial markets. The technique can handle a very broad class of time series that can contain combinations of complex periodicities, polynomial or exponential trend. Forecasting techniques are reviewed in this study, and a new coordinate free joint-horizon k-period-ahead forecasting formulation is derived. The study also considers model selection in SSA, from which it become apparent that forward validation results in more stable model selection. The roots of SSA are outlined and distributional assumptions of signal senes are considered ab initio. Pitfalls that arise in the multivariate statistical theory are identified. Different approaches of recurrent one-period-ahead forecasting are then reviewed. The forecasting approaches are all supplied in algorithmic form to ensure effortless adaptation to computer programs. Theoretical considerations, underlying the forecasting algorithms, are also considered. A new coordinate free joint-horizon kperiod- ahead forecasting formulation is derived and also adapted for the multichannel SSA case. Different model selection techniques are then considered. The use of scree-diagrams, phase space portraits, percentage variation explained by eigenvectors, cross and forward validation are considered in detail. The non-parametric nature of SSA essentially results in the use of non-parametric model selection techniques. Finally, the study also considers a commercial software package that is available and compares it with Fortran code, which was developed as part of the study. / AFRIKAANSE OPSOMMING: Singulier spektraalanalise (SSA) het sy oorsprong in die Fisika. Die tegniek is nieparametries van aard en vind toepassing in velde soos atmosferiese wetenskappe, seinprossesering en onlangs in finansiële markte. Die tegniek kan 'n wye verskeidenheid tydreekse hanteer wat kombinasies van komplekse periodisiteite, polinomiese- en eksponensiële tendense insluit. Vooruitskattingstegnieke word ook in hierdie studie beskou, en 'n nuwe koërdinaatvrye gesamentlike horison k-periodevooruitskattingformulering word afgelei. Die studie beskou ook model seleksie in SSA, waaruit duidelik blyk dat voorwaartse validasie meer stabiele model seleksie tot gevolg het. Die agtergrond van SSA word ab initio geskets en verdelingsaannames van seinreekse beskou. Probleemgevalle wat voorkom in die meervoudige statistiese teorie word duidelik geïdentifiseer. Verskeie tegnieke van herhalende toepassing van een-periode-vooruitskatting word daarna beskou. Die benaderings tot vooruitskatting word in algororitmiese formaat verskaf wat die aanpassing na rekenaarprogrammering vergemaklik. Teoretiese vraagstukke, onderliggend aan die vooruitskattings-algortimes, word ook beskou. 'n Nuwe koërdinaatvrye gesamentlike horison k-periode-vooruitskattingsformulering word afgelei en aangepas vir die multikanaal SSA geval. Verskillende model seleksie tegnieke is ook beskou. Die gebruik van "scree"- diagramme, fase ruimte diagramme, persentasie variasie verklaar deur eievektore, kruis- en voorwaartse validasie word ook aangespreek. Die nie-parametriese aard van SSA noop die gebruik van nie-parametriese model seleksie tegnieke. Die studie vergelyk laastens 'n kommersiële sagtewarepakket met die Fortran bronkode wat as deel van hierdie studie ontwikkel is.

Thesis (PhD)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: See full text. / AFRIKAANSE OPSOMMING: Sien volteks.

Thesis (Ph.D.)--University of the Witwatersrand, Faculty of Science, School of Mathematics, 2017. / ForwardandinversespectralproblemsconcerningSturm-Liouvilleoperatorswithoutdiscontinuitieshavebeenstudiedextensively. Bycomparison,therehasbeenlimitedworktacklingthecase where the eigenfunctions have discontinuities at interior points, a case which appears naturally in physical applications. We refer to such discontinuity conditions as transmission conditions. We consider Sturm-Liouville problems with transmission conditions rationally dependent on the spectral parameter. We show that our problem admits geometrically double eigenvalues, necessitating a new analysis. We develop the forward theory associated with this problem and also consider a related inverse problem. In particular, we prove a uniqueness result analogous to that of H. Hochstadt on the determination of the potential from two sequences of eigenvalues. In addition, we consider the problem of extending Sturm’s oscillation theorem, regarding the number of zeroes of eigenfunctions, from the classical setting to discontinuous problems with general constant coefficient transmission conditions / GR2018

A Thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg for the Degree of Doctor of Philosophy. Johannesburg, 1979

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1982. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE / Includes bibliographical references. / by Ilan Vardi. / Ph.D.