Studies in non-Gaussian analysis

This thesis presents general methods in non-Gaussian analysis in infinite dimensional spaces. As main applications we study Poisson and compound Poisson spaces. Given a probability measure μ on a co-nuclear space, we develop an abstract theory based on the generalized Appell systems which are bi-orthogonal. We study its properties as well as the generated Gelfand triples. As an example we consider the important case of Poisson measures. The product and Wick calculus are developed on this context. We provide formulas for the change of the generalized Appell system under a transformation of the measure. The L² structure for the Poisson measure, compound Poisson and Gamma measures are elaborated. We exhibit the chaos decomposition using the Fock isomorphism. We obtain the representation of the creation, annihilation operators. We construct two types of differential geometry on the configuration space over a differentiable manifold. These two geometries are related through the Dirichlet forms for Poisson measures as well as for its perturbations. Finally, we construct the internal geometry on the compound configurations space. In particular, the intrinsic gradient, the divergence and the Laplace-Beltrami operator. As a result, we may define the Dirichlet forms which are associated to a diffusion process. Consequently, we obtain the representation of the Lie algebra of vector fields with compact support. All these results extends directly for the marked Poisson spaces.

Enzymatic inhibitory activity of hydroxycinnamates (HCs): in silico studies

Diabetes is a worldwide health issue that has been expanding mainly in developed countries. It is characterized by abnormal levels of blood sugar due to several factors. The most common are resistance to insulin and the production of defective insulin which exerts little or no effect. Its most common symptoms include tissue damage to several systems due to elevated levels of blood sugar. One of the key enzymes in hydrocarbon metabolism is α-glucosidase (EC 3.2.1.20). It catalyzes the breakdown of complex carbohydrates into their respective monomers (glucose) which allows them to be absorbed. In this work, caffeoyl quinic acids and their metabolites were analyzed as potential inhibitors for α-glucosidase. The search for the best inhibitor was conducted using molecular docking. The affinity of each compound was compared to the inhibitor present in the crystal structure of the protein. As no inhibitor with a similar affinity was´found, a new approach was used, in situ drug design. It was not possible to achieve an inhibitor capable of competing with the one present in the crystal structure of the enzyme, which is also its current commercial inhibitor. It is possible to draw some conclusions as to which functional groups interact best with certain residues of the active site. This work was divided into three main sections. The first section, Diabetes, serves as an introduction to what is Diabetes, its symptoms and/or side effects and how caffeoyl quinic acids could be used as a treatment. The second section, Caffeoylquinic acids and their metabolites as inhibitors for Alfa-glucosidase, corresponds to the search through molecular docking of caffeoyl quinic acids as inhibitors for α-glucosidase and what was possible to draw from this search. The last section, In situ design of an inhibitor for α-glucosidase (EC 3.2.1.20), corresponds to the in situ drug design study and what it achieved. The representation of each of the molecules used as a ligand can be found in the Annexes.