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.

New method for determination of (E)resveratrol in wine based on microextraction using packed sorbent and ultra-performance liquid chromatography

An ultra-fast and improved analytical methodology based on microextraction by packed sorbent (MEPS) combined with ultra-performance LC (UPLC) was developed and validated for determination of (E)-resveratrol in wines. Important factors affecting the performance of MEPS such as the type of sorbent material (C2, C8, C18, SIL, and M1), number of extraction cycles, and sample volume were studied. The optimal conditions of MEPS extraction were obtained using C8 sorbent and small sample volumes (50–250mL) in one extraction cycle (extract–discard) and in a short time period (about 3 min for the entire sample preparation step). (E)-Resveratrol was eluted by 1 250mL of the mixture containing 95% methanol and 5% water, and the separation was carried out on a highstrength silica HSS T3 analytical column (100 mm 2.1 mm, 1.8mm particle size) using a binary mobile phase composed of aqueous 0.1% formic acid (eluent A) and methanol (eluent B) in the gradient elution mode (10 min of total analysis). The method was fully validated in terms of linearity, detection (LOD) and quantification (LOQ) limits, extraction yield, accuracy, and inter/intra-day precision, using a Madeira wine sample (ET) spiked with (E)-resveratrol at concentration levels ranging from 5 to 60mg/mL. Validation experiments revealed very good recovery rate of 9575.8% RSD, good linearity with r2 values 40.999 within the established concentration range, excellent repeatability (0.52%), and reproducibility (1.67%) values (expressed as RSD), thus demonstrating the robustness and accuracy of the MEPSC8/UPLC-photodiode array (PDA) method. The LOD of the method was 0.21mg/mL, whereas the LOQ was 0.68mg/mL. The validated methodology was applied to 30 commercial wines (24 red wines and six white wines) from different grape varieties, vintages, and regions. On the basis of the analytical validation, the MEPSC8/UPLC-PDA methodology shows to be an improved, sensitive, and ultra-fast approach for determination of (E)-resveratrol in wines with high resolving power within 6 min.