A set-theoretical point of view to study algebraic numbers has been introduced. We extend a result of Navarro-Bermudez concerning shift invariant measures in the Cantor space which are topologically equivalent to shift invariant measures which correspond to some algebraic integers. It is known that any transcendental numbers and rational numbers in the unit interval are not binomial. We proved that there are algebraic numbers of degree greater than two so that they are binomial numbers. Algebraic integers of degree 2 are proved not to be binomial numbers. A few compositive relations having to do with algebraic numbers on the unit interval have been studied; for instance, rationally related, integrally related, binomially related, B1-related relations. A formula between binomial numbers and binomial coefficients has been stated. A generalized algebraic equation related to topologically equivalent measures has also been stated.

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Un arrangement central A est un ensemble fini de sous-espaces vectoriels dans un espace vectoriel complexe V de dimension finie. L'espace topologique complémentaire M(A) est l'ensemble des points de V qui n'appartiennent à aucun des sous-espaces de A. Dans ce travail, nous étudions la topologie de l'espace M(A) du point de vue de l'homotopie rationnelle. L'outil clé qui a servi de départ à cette thèse est un modèle rationnel de M(A) qui s'avère relativement simple à manipuler. À l'aide de ce modèle, nous obtenons plusieurs résultats sur la topologie de M(A). Citons par exemple des formules de récursion qui permettent de calculer certains invariants topologiques, dont les nombres de Betti, une preuve du fait que la caractéristique d'Euler de l'espace M(A) est nulle ou encore une description des arrangements (vérifiant une condition technique) dont le complémentaire est un wedge rationnel de sphères. Enfin, les résultats principaux de cet ouvrage sont une caractérisation des arrangements dont le complémentaire a le type d'homotopie d'un produit de sphères, et la preuve du fait que si le complémentaire n'est pas un produit de sphères, alors son algèbre de Lie d'homotopie contient la sous-algèbre de Lie libre à deux générateurs.

Thesis (Ph. D. in Mathematics)--Vanderbilt University, May 2008. / Title from title screen. Includes bibliographical references.

This work constructs and compares different kinds of categorified cohomology of a locally contractible topological space X. Fix a commutative ring k of characteristic 0 and also denote by k the differential graded category with a single object and endomorphisms k. In the Morita model structure k is weakly equivalent to the category of perfect chain complexes over k. We define and compute derived global sections of the constant presheaf k considered as a presheaf of dg-categories with the Morita model structure. If k is a field this is done by showing there exists a suitable local model structure on presheaves of dg-categories and explicitly sheafifying constant presheaves. We call this categorified Cech cohomology Morita cohomology and show that it can be computed as a homotopy limit over a good (hyper)cover of the space X. We then prove a strictification result for dg-categories and deduce that under mild assumptions on X Morita cohomology is equivalent to the category of homotopy locally constant sheaves of k-complexes on X. We also show categorified Cech cohomology is equivalent to a category of ∞-local systems, which can be interpreted as categorified singular cohomology. We define this category in terms of the cotensor action of simplicial sets on the category of dg-categories. We then show ∞-local systems are equivalent to the category of dg-representations of chains on the loop space of X and find an explicit method of computation if X is a CW complex. We conclude with a number of examples.

We investigate quotient-fine nearness frames, showing that they are reflective in the category of strong nearness frames, and that, in those with spatial completion, any near subset is contained in a near grill. We construct two categories, each of which is shown to be equivalent to that of quotient-fine nearness frames. We also consider some subcategories of the category of nearness frames, which are co-hereditary and closed under coproducts. We give due attention to relations between these subcategories. We introduce totally strong nearness frames, whose category we show to be closed under completions. We investigate N-homomorphisms and remote points in the context of totally bounded uniform frames, showing the role played by these uniform N-homomorphisms in the transfer of remote points, and their relationship with C -quotient maps. A further study on grills enables us to establish, among other things, that grills are precisely unions of prime filters. We conclude the thesis by showing that the lattice of all nearnesses on a regular frame is a pseudo-frame, by which we mean a poset pretty much like a frame except for the possible absence of the bottom element. / Mathematical Sciences / Ph.D. (Mathematics)

In this work, an application of the algebraic reconstruction technique to a borehole reconstruction problem is considered. The formulation of the borehole problem gives the attendant electromagnetic wave equations in matrix form. The algebraic reconstruction technique is used to reconstruct a solution. Three sources of errors are identified in the reconstruction process. Suggestions are made which will help minimize or predict the effects of these errors. General limitations of the algebraic reconstruction technique are discussed. The limitations in terms of the borehole problem are explained. Practical limitations for the borehole problem are thus obtained and quantified mathematically. It is found that even in some practical situations, the borehole reconstruction process is impossible.

In this thesis we make several contributions to the theory of moduli spaces of smooth manifolds, especially in dimension two. In Chapter 2 (joint with Soren Galatius) we give a new geometric proof of a generalisation of the Madsen-Weiss theorem, which does not rely on the tangential structure under investigation having homological stability. This allows us to compute the stable homology of moduli spaces of surfaces equipped with many different tangential structures. In Chapter 3 we give a general approach to homological stability problems, especially focused on stability for moduli spaces of surfaces with tangential structure. We give a sufficient condition for a structure to exhibit homological stability, and thus obtain stability ranges for many tangential structures of current interest (orientations, maps to a simply-connected background space, etc.), which match or improve the previously known ranges in all cases. In Chapter 4 we define and study the cobordism category of submanifolds of a fixed background manifold, and extend the work of Galatius-Madsen-Tillmann-Weiss to identify the homotopy type of these categories. We describe several applications of this theory. In Chapter 5 we compute the stable (co)homology of the non-orientable mapping class group, and find a family of geometrically-defined torsion cohomology classes. This is in contrast to the oriented mapping class group, where few are known. In Chapter 6 (joint with Johannes Ebert) we study the divisibility of certain characteristic classes of bundles of unoriented surfaces introduced by Wahl, analogues of the Miller-Morita-Mumford classes for unoriented surfaces. We show them to be indivisible in the free quotient of cohomology.