The Riemann-Hilbert method applied to the theory of orthogonal polynomials

Doutoramento em Matemática

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.

Monogenic polynomials of four variables with binomial expansion

In the recent past one of the main concern of research in the field of Hypercomplex Function Theory in Clifford Algebras was the development of a variety of new tools for a deeper understanding about its true elementary roots in the Function Theory of one Complex Variable. Therefore the study of the space of monogenic (Clifford holomorphic) functions by its stratification via homogeneous monogenic polynomials is a useful tool. In this paper we consider the structure of those polynomials of four real variables with binomial expansion. This allows a complete characterization of sequences of 4D generalized monogenic Appell polynomials by three different types of polynomials. A particularly important case is that of monogenic polynomials which are simply isomorphic to the integer powers of one complex variable and therefore also called pseudo-complex powers.

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.

Left ventricle functional analysis from coronary CT angiography

Coronary CT angiography is widely used in clinical practice for the assessment of coronary artery disease. Several studies have shown that the same exam can also be used to assess left ventricle (LV) function. LV function is usually evaluated using just the data from end-systolic and end-diastolic phases even though coronary CT angiography (CTA) provides data concerning multiple cardiac phases, along the cardiac cycle. This unused wealth of data, mostly due to its complexity and the lack of proper tools, has still to be explored in order to assess if further insight is possible regarding regional LV functional analysis. Furthermore, different parameters can be computed to characterize LV function and while some are well known by clinicians others still need to be evaluated concerning their value in clinical scenarios. The work presented in this thesis covers two steps towards extended use of CTA data: LV segmentation and functional analysis. A new semi-automatic segmentation method is presented to obtain LV data for all cardiac phases available in a CTA exam and a 3D editing tool was designed to allow users to fine tune the segmentations. Regarding segmentation evaluation, a methodology is proposed in order to help choose the similarity metrics to be used to compare segmentations. This methodology allows the detection of redundant measures that can be discarded. The evaluation was performed with the help of three experienced radiographers yielding low intraand inter-observer variability. In order to allow exploring the segmented data, several parameters characterizing global and regional LV function are computed for the available cardiac phases. The data thus obtained is shown using a set of visualizations allowing synchronized visual exploration. The main purpose is to provide means for clinicians to explore the data and gather insight over their meaning, as well as their correlation with each other and with diagnosis outcomes. Finally, an interactive method is proposed to help clinicians assess myocardial perfusion by providing automatic assignment of lesions, detected by clinicians, to a myocardial segment. This new approach has obtained positive feedback from clinicians and is not only an improvement over their current assessment method but also an important first step towards systematic validation of automatic myocardial perfusion assessment measures.

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.