Algorithmic computation of knot polynomials of secondary structure elements of proteins

The classification of protein structures is an important and still outstanding problem. The purpose of this paper is threefold. First, we utilize a relation between the Tutte and homfly polynomial to show that the Alexander-Conway polynomial can be algorithmically computed for a given planar graph. Second, as special cases of planar graphs, we use polymer graphs of protein structures. More precisely, we use three building blocks of the three-dimensional protein structure-alpha-helix, antiparallel beta-sheet, and parallel beta-sheet-and calculate, for their corresponding polymer graphs, the Tutte polynomials analytically by providing recurrence equations for all three secondary structure elements. Third, we present numerical results comparing the results from our analytical calculations with the numerical results of our algorithm-not only to test consistency, but also to demonstrate that all assigned polynomials are unique labels of the secondary structure elements. This paves the way for an automatic classification of protein structures.

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.

Les amours de Tristan ([Reprod.]) / [Tristan L'Hermite]

Collection : French books 1601-1700 ; 103.2

Poésies galantes et héroïques de sieur Tristan L'Hermite : contenant ses amours, sa lyre, les plaintes d'Acante, la maison d'Astrée, la belle gueuse, l'aveugle amoureux, les terreurs nocturnes,... ([Reprod.])

Collection : French books 1601-1700 ; 103.4

Les vers héroïques du sieur Tristan L'Hermite ([Reprod.])

Collection : French books 1601-1700 ; 103.5

La Mariane : tragédie ([Reprod.]) / [Tristan L'Hermite]

Collection : French books 1601-1700 ; 103.3

Arrêt du grand conseil qui juge que les brefs de pénitencerie de cour de Rome fulminés par les nonces de Sa Sainteté, quoique confirmés par les brefs de la daterie, n'ont point lieu en France pour le fore extérieur. En conséquence a déclaré la translation d'un hermite de Saint-Augustin dans l'ordre de Saint-Benoît nulle et abusive, et renvoyé ledit hermite dans le couvent de sa profession, à Paris, au faubourg Saint-Germain. Qui maintient un gradué régulier de l'ordre Cluny dans la possession du prieuré de Sainte-Gemme dont il était pourvu comme gradué. Et qui juge nulles tant la permutation faite en fraude des gradués que les collations faites au compermutant, contre les maximes du droit et les termes de l'indult du collateur

[Acte. 1694-09-20. Paris]