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131	Bounded sets in topological groups Chis, Cristina 09 February 2010 (has links) A boundedness structure (bornology) on a topological space is an ideal of subsets containing all singletons, that is, closed under taking subsets and unions of finitely many elements. In this paper we deal with the structure of the whole family of bounded subsets rather than the specific properties of them by means of certain functions that we define on a metrizable topological group. Our motivation is twofold: on the one hand, we obtain useful information about the structural features of certain remarkable classes of bounded systems, cofinality, local properties, etc. For example, we estimate the cofinality of these boundedness notions. In the second part of the paper, we apply duality methods in order to obtain estimations of the size of a local base for an important class of groups. This translation, which has been widely exhibited in the Pontryagin-van Kampen duality theory of locally compact abelian groups, is often very relevant and has been extended by many authors to more general classes of topological groups. In this work we follow basically the pattern and terminology given by Vilenkin in 1998. tightness Pontryagin-van Kampen duality theory duality theory invariant cardinals bornology boundedness bounded set general topology topological spaces harmonic analysis topological groups character weight cofinality local base 51 512 515.1 517
132	Cohomologies on sympletic quotients of locally Euclidean Frolicher spaces Tshilombo, Mukinayi Hermenegilde 08 1900 (has links) This thesis deals with cohomologies on the symplectic quotient of a Frölicher space which is locally diffeomorphic to a Euclidean Frölicher subspace of Rn of constant dimension equal to n. The symplectic reduction under consideration in this thesis is an extension of the Marsden-Weinstein quotient (also called, the reduced space) well-known from the finite-dimensional smooth manifold case. That is, starting with a proper and free action of a Frölicher-Lie-group on a locally Euclidean Frölicher space of finite constant dimension, we study the smooth structure and the topology induced on a small subspace of the orbit space. It is on this topological space that we will construct selected cohomologies such as : sheaf cohomology, Alexander-Spanier cohomology, singular cohomology, ~Cech cohomology and de Rham cohomology. Some natural questions that will be investigated are for instance: the impact of the symplectic structure on these di erent cohomologies; the cohomology that will give a good description of the topology on the objects of category of Frölicher spaces; the extension of the de Rham cohomology theorem in order to establish an isomorphism between the five cohomologies. Beside the algebraic, topological and geometric study of these new objects, the thesis contains a modern formalism of Hamiltonian mechanics on the reduced space under symplectic and Poisson structures. / Mathematical Sciences / D. Phil. (Mathematics) Constant dimension Locally Euclidean Frolicher spaces Symplectic structure Symplectic geometry Symplectic quotient or reduced space Exterior algebra Ringed Frolicher space Smooth Gelfand representation Sheaf cohomology Alexander-Spanier cohomology Singular cohomology Cech cohomology and de Rham cohomology Vector fields and mechanics 514.224 Homology theory Symplectic manifolds Topological spaces Sheaf theory Geometry, Differential Hamiltonian graph theory Algebraic spaces
133	Cohomologies on sympletic quotients of locally Euclidean Frolicher spaces Tshilombo, Mukinayi Hermenegilde 08 1900 (has links) This thesis deals with cohomologies on the symplectic quotient of a Frölicher space which is locally diffeomorphic to a Euclidean Frölicher subspace of Rn of constant dimension equal to n. The symplectic reduction under consideration in this thesis is an extension of the Marsden-Weinstein quotient (also called, the reduced space) well-known from the finite-dimensional smooth manifold case. That is, starting with a proper and free action of a Frölicher-Lie-group on a locally Euclidean Frölicher space of finite constant dimension, we study the smooth structure and the topology induced on a small subspace of the orbit space. It is on this topological space that we will construct selected cohomologies such as : sheaf cohomology, Alexander-Spanier cohomology, singular cohomology, ~Cech cohomology and de Rham cohomology. Some natural questions that will be investigated are for instance: the impact of the symplectic structure on these di erent cohomologies; the cohomology that will give a good description of the topology on the objects of category of Frölicher spaces; the extension of the de Rham cohomology theorem in order to establish an isomorphism between the five cohomologies. Beside the algebraic, topological and geometric study of these new objects, the thesis contains a modern formalism of Hamiltonian mechanics on the reduced space under symplectic and Poisson structures. / Mathematical Sciences / D. Phil. (Mathematics) Constant dimension Locally Euclidean Frolicher spaces Symplectic structure Symplectic geometry Symplectic quotient or reduced space Exterior algebra Ringed Frolicher space Smooth Gelfand representation Sheaf cohomology Alexander-Spanier cohomology Singular cohomology Cech cohomology and de Rham cohomology Vector fields and mechanics 514.224 Homology theory Symplectic manifolds Topological spaces Sheaf theory Geometry, Differential Hamiltonian graph theory Algebraic spaces

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