Compressive sensing in Clifford analysis

Compressed sensing is a new paradigm in signal processing which states that for certain matrices sparse representations can be obtained by a simple l1-minimization. In this thesis we explore this paradigm for higher-dimensional signal. In particular three cases are being studied: signals taking values in a bicomplex algebra, quaternionic signals, and complex signals which are representable by a nonlinear Fourier basis, a so-called Takenaka-Malmquist system.

The Legendre Formula in Clifford Analysis

2000 Mathematics Subject Classification: 30A05, 33E05, 30G30, 30G35, 33E20.

Clifford-Hermite polynomials in fractional Clifford analysis

In this paper we generalize radial and standard Clifford-Hermite polynomials to the new framework of fractional Clifford analysis with respect to the Riemann-Liouville derivative in a symbolic way. As main consequence of this approach, one does not require an a priori integration theory. Basic properties such as orthogonality relations, differential equations, and recursion formulas, are proven.

Polinómios de Appell multidimensionais e sua representação matricial

Nesta dissertação é apresentada uma abordagem a polinómios de Appell multidimensionais dando-se especial relevância à estrutura da sua função geradora. Esta estrutura, conjugada com uma escolha adequada de ordenação dos monómios que figuram nos polinómios, confere um carácter unificador à abordagem e possibilita uma representação matricial de polinómios de Appell por meio de matrizes particionadas em blocos. Tais matrizes são construídas a partir de uma matriz de estrutura simples, designada matriz de criação, subdiagonal e cujas entradas não nulas são os sucessivos números naturais. A exponencial desta matriz é a conhecida matriz de Pascal, triangular inferior, onde figuram os números binomiais que fazem parte integrante dos coeficientes dos polinómios de Appell. Finalmente, aplica-se a abordagem apresentada a polinómios de Appell definidos no contexto da Análise de Clifford.

Numerical and combinatorial applications of generalized Appell polynomials

This thesis studies properties and applications of different generalized Appell polynomials in the framework of Clifford analysis. As an example of 3D-quasi-conformal mappings realized by generalized Appell polynomials, an analogue of the complex Joukowski transformation of order two is introduced. The consideration of a Pascal n-simplex with hypercomplex entries allows stressing the combinatorial relevance of hypercomplex Appell polynomials. The concept of totally regular variables and its relation to generalized Appell polynomials leads to the construction of new bases for the space of homogeneous holomorphic polynomials whose elements are all isomorphic to the integer powers of the complex variable. For this reason, such polynomials are called pseudo-complex powers (PCP). Different variants of them are subject of a detailed investigation. Special attention is paid to the numerical aspects of PCP. An efficient algorithm based on complex arithmetic is proposed for their implementation. In this context a brief survey on numerical methods for inverting Vandermonde matrices is presented and a modified algorithm is proposed which illustrates advantages of a special type of PCP. Finally, combinatorial applications of generalized Appell polynomials are emphasized. The explicit expression of the coefficients of a particular type of Appell polynomials and their relation to a Pascal simplex with hypercomplex entries are derived. The comparison of two types of 3D Appell polynomials leads to the detection of new trigonometric summation formulas and combinatorial identities of Riordan-Sofo type characterized by their expression in terms of central binomial coefficients.

Personality analysis of patients complaining of sialorrhoea

BACKGROUND: Sialorrhoea, the symptom of apparent excessive secretion of saliva is a relatively uncommon complaint. Some authors consider that in the absence of clinical findings, then these patients have a psychiatric disorder masquerading as a physical illness. However, there is little evidence in the literature to support this conclusion and a detailed psychological assessment of this population has not previously been reported. METHODS: In total, 18 patients and 18 age- and sex-matched controls were studied. All had a history of a complaint of excess salivation in the absence of any oral mucosal or systemic abnormality. All patients completed an Eysenck Personality Questionnaire. RESULTS: There were no differences in the extroversion of psychoticism scores between the study and control group. However, the result showed significant increases in the neuroticism and Lie Scale score in the patient group. CONCLUSIONS: The overall results of this study indicate that the complaint of sialorrhoea in otherwise healthy individuals does not have an organic basis and suggest that sialorrhoea is associated with high levels of neuroticism and a tendency to dissimulate. © Blackwell Munksgaard 2006. All rights reserved.

Fermionic analysis of Davey-Stewartson dromions

Dromion solutions of the Davey-Stewartson equation are analysed from the point of view of the bilinear formalism. The corresponding τ-functions are expressed in terms of vacuum expectation values of Clifford operators and their group-theoretical content is provided. Explicit computation performed with the help of Wick's theorem allows us to characterize the dromion interaction. © 1990.

Spectrographic Analysis of Montana Limestones

Spectrographic analysis of limestones as a possible method of correlation of geologic formations is an altogether new line of investigation. As far as known the only previous work consists of a few analyses made by Fred Lines in his bachelor thesis work at Montana School of mines in the spring of 1942.