Intégrales de Selberg complexes et p-adiques et identités de Dyson-Macdonald

Cette thèse fait partie d'un programme de recherche sur la théorie conforme des champs et les représentations de l'algèbre de Lie dollar\frak{sl}_2dollar (réelle, complexe, dollarpdollar-adique, dollarqdollar-déformée). Nous étudions des versions réelle, dollarpdollar-adique et dollarqdollar-déformée d'une intégrale triple apparaissant en physique en connection avec le modèle de Liouville de la théorie conforme des champs. Ces intégrales se trouvent être aussi reliées aux identités de terme constant de Dyson-MacDonald. Et puis, nous donnons une approche différente pour calculer la version comlexe, qui utilise la technique de Bernstein-Reznikov. L'idée principale est d'appliquer des fonctionnelles invariantes à des représentations de séries principales de dollarG=SL(2,\mathbb{C})dollar. Enfin, nous définissons une dollarqdollar-déformation de Jacquet-Langlands de représentations de séries principales de dollarGL_2(\mathbb{R})dollar et nous prouvons l'unicité d'une fonctionnelle triple invariante sur ces objets en utilisant la méthode de H.Y.Loke. Nous trouvons aussi des relations semblables aux équations différentielles de [NSU].This thesis is part of a research program on Conformal field theory and representations of Lie algebra of dollar\frak{sl}_2$(real, complex, dollarpdollar-adic, dollarqdollar-derfomed). We study a real, dollarpdollar-adic and dollarqdollar-deformed versions of a triple integral appeared in physics in connection with the Liouville model of the Conformal field theory. These integrals turn out to be connected with the Dyson-Macdonald constant term identities. We also give another approach to compute the complex case by using Bernstein-Reznikov's technique. The main idea is to apply invariant functionals on principal series representations of dollarG=SL(2,\mathbb C)dollar. Finally, one defines a dollarqdollar-deformation of Jacquet-Langlands principal series representations of dollarGL_2(\mathbb R)$ and prove the uniqueness of an invariant triple functional on them by using method of H.Y.Loke. Alongside, we find out some relations to similar differential equations in [NSU]

Geomechanics in unconventional resource development

To economically produce from very low permeability shale formations, hydraulic fracturing stimulation is typically used to improve their conductivity. This process deforms and breaks the rock, hence requires the geomechanics data and calculation. The development of unconventional reservoirs requires large geomechanical data, and geomechanics has involved in all calculations of the unconventional reservoir projects. Geomechanics has numerous contributions to the development of unconventional reservoirs from reservoir characterization and well construction to hydraulic fracturing and reservoir modeling as well as environmental aspect. This paper reviews and highlights some important aspects of geomechanics on the successful development of unconventional reservoirs as well as outlines the recent development in unconventional reservoir geomechanics. The main objective is to emphasize the importance of geomechanical data and geomechanics and how they are being used in in all aspects of unconventional reservoir projects.Comment: 16 pages, 7 figures, 1 tabl

A Note on Graphs of Linear Rank-Width 1

We prove that a connected graph has linear rank-width 1 if and only if it is a distance-hereditary graph and its split decomposition tree is a path. An immediate consequence is that one can decide in linear time whether a graph has linear rank-width at most 1, and give an obstruction if not. Other immediate consequences are several characterisations of graphs of linear rank-width 1. In particular a connected graph has linear rank-width 1 if and only if it is locally equivalent to a caterpillar if and only if it is a vertex-minor of a path [O-joung Kwon and Sang-il Oum, Graphs of small rank-width are pivot-minors of graphs of small tree-width, arxiv:1203.3606] if and only if it does not contain the co-K_2 graph, the Net graph and the 5-cycle graph as vertex-minors [Isolde Adler, Arthur M. Farley and Andrzej Proskurowski, Obstructions for linear rank-width at most 1, arxiv:1106.2533].Comment: 9 pages, 2 figures. Not to be publishe

Estimating Local Function Complexity via Mixture of Gaussian Processes

Real world data often exhibit inhomogeneity, e.g., the noise level, the sampling distribution or the complexity of the target function may change over the input space. In this paper, we try to isolate local function complexity in a practical, robust way. This is achieved by first estimating the locally optimal kernel bandwidth as a functional relationship. Specifically, we propose Spatially Adaptive Bandwidth Estimation in Regression (SABER), which employs the mixture of experts consisting of multinomial kernel logistic regression as a gate and Gaussian process regression models as experts. Using the locally optimal kernel bandwidths, we deduce an estimate to the local function complexity by drawing parallels to the theory of locally linear smoothing. We demonstrate the usefulness of local function complexity for model interpretation and active learning in quantum chemistry experiments and fluid dynamics simulations.Comment: 19 pages, 16 figure