Dans cette thèse, nous donnons des majorations explicites pour les constantes de Laurent-Stieltjes des séries L de Dirichlet dans deux cas différents. Ces constantes sont les coefficients qui interviennent dans le développement en série de Laurent des séries L de Dirichlet. Cette thèse est composée de trois parties : [A] Dans la première partie, nous donnons, à partir d'une idée due à Matsuoka pour la fonction zêta de Riemann, des majorations explicites de ces coefficients d'ordre élevé lorsque le conducteur du caractère de Dirichlet est fixé. Nous prolongeons la formule de Matsuoka aux fonctions L de Dirichlet et améliorons le résultat de Matsuoka. En utilisant cette majoration, nous déduisons aussi une approximation des fonctions L de Dirichlet au voisinage de z=1 par un polynôme de Taylor relativement court. [B] Dans la deuxième partie de cette thèse, nous donnons une majoration explicite du premier coefficient de Laurent-Stieltjes lorsque le caractère de Dirichlet est un caractère pair qui prend la valeur 1 en 2. Il s'agit là du cas le plus difficile. Ce résultat nous conduit à une amélioration du résultat de Ramaré. Nous en déduisons une majoration explicite pour le nombre des classes pour tout corps quadratique réel et améliorer ainsi un résultat de Le.[C] Dans la troisième partie, nous suivons la méthode de Ramaré pour donner une majoration explicite du premier coefficient lorsque le conducteur du caractère de Dirichlet est divisible par 3, améliorer un résultat de Louboutin. / In this thesis, we give an upper bound for the Laurent- Stieltjes constants for the Dirichlet L- series in two different cases. These constants are the coefficients of the expansion in Laurent series of the Dirichlet L-series. This thesis is divided to three parts: [A] In the first part, we give an explicit upper bound for these constants when the Dirichlet character is fixed and its order goes to infinity, starting from an idea due to Matsuoka for the zeta function. We extend the formula of Matsuoka to the Dirichlet L functions, improving previous results. By using this result, we also deduce an approximation of the Dirichlet L-functions in the neighborhood of z=1 by a short Taylor polynomial. [B] The second part of this thesis deals more specifically with the first Laurent- Stieltjes coefficient. We gave an improvement of the known explicit upper bound due to Ramaré for this quantity in the case when the Dirichlet character is even and takes the value1 at 2 (This is the most difficult case). Thanks to this result, we deduce an upper bound for the class number of any real quadratic field, improving on a result by Le.[C] In the last part, we follow the method of Ramaré for giving an upper bound of the first Laurent Stieltjes coefficient but this time in the case when the conductor of the character is divisible by 3. This result is an improvement on a result of Louboutin.

Thesis (M. A.)--Trinity Evangelical Divinity School, Deerfield, Ill., 1998. / Abstract. Includes bibliographical references (leaves 262-272).

En abordant les nombres spéciaux comme les sommes harmoniques ou les polyzêtassous leur aspect combinatoire, nous introduisons d’abord la définition d’un produitentre mots, dit produit de quasi-mélange q-déformé, une généralisation des produits demélange et de quasi-mélange, ce qui nous permet de construire des structures complètesd’algèbre de Hopf en dualité. En même temps, nous construisons des bases en dualité,contenant des bases de transcendance associées aux mots de Lyndon, et des formules explicitessur lesquelles les sommes harmoniques, les polyzêtas ou les polylogarithmes sontindexés et représentés par la factorisation de la série génératrice noncommutative diagonale.De cette façon, nous déterminons des développements asymptotiques des sommesharmoniques, indexées par ces bases, grâce à leur série génératrice et à la formule d’EulerMaclaurin. Nous établissons également une équation de liaison sur les polyzêtas, quiapparaissent comme les parties finies des développements asymptotiques des sommesharmoniques et des polylogarithmes, reliant entre elles deux structures algébriques. Enidentifiant les coordonnées locales de cette équation, nous trouvons des relations polynomialeshomogènes, en poids, entre les polyzêtas. Pour accompagner cette étude théorique,nous proposons des algorithmes et un package en Maple afin de calculer des bases, lastructure des polyzêtas et des développements asymptotiques des sommes harmoniques. / Approaching special numbers as harmonic sums or polyzetas (multiple zetavalues) in the spirit of combinatorics, we first focus on the study of algebraic structureson words by introducing the definition of a product on words, called q-stuffle product, acommon generalisation of shuffle and quasi-shuffle products, which allows us to completelyconstruct Hopf algebras in duality. Simutaneously, we establish recurrent formulas inorder to compute bases in duality, containing transcendence bases tied to Lyndon wordson which harmonic sums, the polyzetas and polylogarithms are indexed. We use them torepresent the factorization of a diagonal noncommutative generating series. In this respect,we determine asymptotic expansions of harmonic sums thanks to their generatingseries and to Euler Maclaurin formula. We also establish a bridge equation of polyzetas,which appear as fini parts in asymptotic expansions of harmonic sums and of polylogarithms,linking two algebraic structures. Through identification of local coordinates of thisequation, we can deduce homogenous, in weight, polynomial relations among polyzetasindexed on the bases.We also give algorithms and a package in Maple which, in practice,allowed us to find results and examples within this thesis.

A polyhedron P is a subset of a rational vector space V bounded by hyperplanes. If we fix a lattice in V , then we may consider the exponential integral and sum, two meromorphic functions on the dual vector space which serve to generalize the notion of volume of and number of lattice points contained in P, respectively. In 2007, Berline and Vergne constructed an Euler-Maclaurin formula that relates the exponential sum of a given polyhedron to the exponential integral of each face. This formula was "local", meaning that the coefficients in this formula had certain properties independent of the given polyhedron. In this dissertation, the author finds a new construction for this formula which is very different from that of Berline and Vergne. We may 'perturb' any polyhedron by tranlsating its bounding hyperplanes. The author defines a ring of differential operators R(P) on the exponential volume of the perturbed polyhedron. This definition is inspired by methods in the theory of toric varieties, although no knowledge of toric varieties is necessary to understand the construction or the resulting Euler-Maclaurin formula. Each polyhedron corresponds to a toric variety, and there is a dictionary between combinatorial properties of the polyhedron and algebro-geometric properties of this variety. In particular, the equivariant cohomology ring and the group of equivariant algebraic cycles on the corresponding toric variety are equal to a quotient ring and subgroup of R(P), respectively. Given an inner product (or, more generally, a complement map) on V , there is a canonical section of the equivariant cohomology ring into the group of algebraic cycles. One can use the image under this section of a particular differential operator called the Todd class to define the Euler-Maclaurin formula. The author shows that this formula satisfies the same properties which characterize the Berline-Vergne formula.

Universiẗat, Diss., 2003--Jena.

In this thesis, three parts, each with several chapters, are respectively devoted to hydrostatic, viscous and inertial fluids theories and applications. In the hydrostatics part, the classical Maclaurin spheroids theory is generalized, for the first time, to a more realistic multi-layer model, which enables the studies of some gravity problems and direct numerical simulations of flows in fast rotating spheroidal cavities. As an application of the figure theory, the zonal flow in the deep atmosphere of Jupiter is investigated for a better understanding of the Jovian gravity field. High viscosity flows, for example Stokes flows, occur in a lot of processes involving low-speed motions in fluids. Microorganism swimming is such typical a case. A fully three dimensional analytic solution of incompressible Stokes equation is derived in the exterior domain of an arbitrarily translating and rotating prolate spheroid, which models a large family of microorganisms such as cocci bacteria. The solution is then applied to the magnetotactic bacteria swimming problem and good consistency has been found between theoretical predictions and laboratory observations of the moving patterns of such bacteria under magnetic fields. In the analysis of dynamics of planetary fluid systems, which are featured by fast rotation and very small viscosity effects, three dimensional fully nonlinear numerical simulations of Navier-Stokes equations play important roles. A precession driven flow in a rotating channel is studied by the combination of asymptotic analyses and fully numerical simulations. Various results of laminar and turbulent flows are thereby presented. Computational fluid dynamics requires massive computing capability. To make full use of the power of modern high performance computing facilities, a C++ finite-element analysis code is under development based on PETSc platform. The code and data structures will be elaborated, along with the presentations of some preliminary results.

Colin Maclaurin est un mathématicien écossais important dans la vie intellectuelle, sociale et politique de l'Écosse. Pour lui, l'évidence et la certitude des mathématiques justifient leur utilisation dans les autres champs du savoir. Nous montrons donc comment il les utilise en physique, en astronomie, en théologie, etc. De plus, nous montrons comment sa lecture des œuvres de Newton évolue et comment il passe d'un statut de disciple à celui de commentateur, puis à celui de chercheur aux conceptions propres. Enfin, l'étude globale de ses œuvres nous permet de dégager des lignes de forces dans sa production : une volonté de fondation, en particulier en algèbre et dans la méthode des fluxions, un usage fin et privilégié de la géométrie, et une application réciproque de la géométrie et de l'algèbre. Nous analysons de nombreux résultats nouveaux qui résultent de cette approche maclaurinienne des mathématiques.

Consider the D-optimal designs for the dth-degree polynomial regression model with a bounded and positive weight function on a compact interval. As the degree of the model goes to infinity, we show that the D-optimal design converges weakly to the arcsin distribution. If the weight function is equal to 1, we derive the formulae of the values of the D-criterion for five classes of designs including (i) uniform density design; (ii) arcsin density design; (iii) J_{1/2,1/2} density design; (iv) arcsin support design and (v) uniform support design. The comparison of D-efficiencies among these designs are investigated; besides, the asymptotic expansions and limits of their D-efficiencies are also given. It shows that the D-efficiency of the arcsin support design is the highest among the first four designs.