Every continuous convex function defined on a separable Banach space is Gateaux differentiable on a dense G^ subset of the space E [Mazur]. Suppose we are given a sequence (xn) that Is dense in E. Can we always find a Gateaux differentiable point x such that x = z^=^anxn.for some sequence (an) with infinitely many non-zero terms so that Ση∞=1||anxn|| < co ? According to this paper, such points are called of "simple type," and shown to be dense in E. Mazur's theorem follows directly from the result and Rybakov's theorem (A countably additive vector measure F: E -* X on a cr-field is absolutely continuous with respect to |x*F] for some x* e Xs) can be shown without deep measure theoretic Involvement.

Contracts are legally enforceable agreements between two or more parties. The agreements can contain temporally based conditions, such as actions taken by the contract parties or events that happen, that trigger changes to the state of the contract when the conditions become true. Since the structure of these conditions can be very complex, it can be difficult to write contracts in a natural language in a clear and unambiguous way. A better approach is to have a formal language with a precise semantics to represent contracts. Contracts expressed in such a language have a mathematically precise meaning and can be written, analyzed, and manipulated by software. This thesis presents FCL, a formal language with a precise semantics for writing general contracts that may depend on temporally based conditions. Motivated by carefully selected examples of contracts, we derive a set of desirable requirements that a formal language of contracts should support. Based on the requirements, we clearly de ne the notion of contract and address what it means to fulfill or breach a contract. We present the formal syntax and semantics of FCL. We also successfully formalize different kinds of contracts in FCL and develop a reasoning system for FCL. / Thesis / Doctor of Philosophy (PhD)

Dans cette thèse, nous classifions représentations typiques pour certaines composants Bernstein. Suite aux travaux de Henniart dans le cas de GL_2(F) et Paskunas pour les composants cuspidales, nous classifions représentations typiques pour les composants de niveau zéro pour GL_n(F) pour n> 2, composants de série principale, composants avec Levi sous-groupe de la forme (n, 1) pour n>1 et certains composants avec sous-groupe de Levi de la forme (2,2). Chacun des composants ci-dessus est traité dans un chapitre distinct. La classification utilise la théorie des types développés par Bushnell-Kutzko d'une manière significative. Nous allons donner la classification en termes de types de Bushnell-Kutzko. / In this thesis we classify typical representations for certain non-cuspidal Bernstein components. Following the work of Henniart in the case of GL_2(F) and Paskunas for the cuspidal components, we classify typical representations for of level-zero components for GL_n(F) for n>2, principal series components, components with Levi subgroup of the form (n, 1) for n>1 and certain components with Levi subgroup of the form (2,2). Each of the above component is treated in a separate chapter. The classification uses the theory of types developed by Bushnell-Kutzko in a significant way. We will give the classification in terms of Bushnell-Kutzko types for a given inertial class.