A fibrewise coHopf space X over a base B is a sectioned space for which the diagonal map X —> X x _BX may be compressed into X V_BX up to fibrewise pointed homotopy. Such spaces have been investigated by I. M. James in the case where X is a sphere bundle over a sphere. The purpose of this thesis is to demonstrate some of the properties of fibrewise coHopf spaces over more general bases. Particular attention is given to sphere bundles and fibrations with spherical fibre. The fibrewise reduced suspension of a sectioned fibrewise space with closed sec- tion is fibrewise coHopf with associative comultiplication (up to fibrewise pointed homotopy) and a fibrewise inversion. Examples of fibrewise coHopf spaces not of this form are exhibited, and sufficient conditions are given to ensure that a fibrewise coHopf space has the primitive fibrewise pointed homotopy type of a fibrewise re- duced suspension, in terms of the dimension and connectivity of the space, its base and the fibres. It is shown that these conditions may be relaxed if the fibrewise coHopf structure on the space is assumed to be homotopy-associative. An example of a non-associative fibrewise coHopf sphere bundle is given. It is shown that, if q > 1 is odd, a sectioned orientable q-sphere bundle over a finite connected complex is fibrewise coHopf if and only if its fibrewise localisation at the prime 2 is fibrewise coHopf. Moreover, the fibrewise rationalisation of an odd-dimensional sphere bundle over a finite polyhedron whose fibrewise unreduced suspension is fibrewise coHopf is shown to be a trivial fibration. As an application, it is shown that new fibrewise coHopf spherical fibrations may be constructed by mixing. The Thorn space is used to determine the cohomology ring of the total space of a fibrewise coHopf sphere bundle in terms of that of its base, and a generalised Hopf invariant is constructed which vanishes on fibrewise coHopf sphere bundles.

Thesis (Ph. D.)--University of Texas at Austin, 2004. / Supervisor: John Luecke. Vita. Includes bibliographical references. Available also from UMI company.

Proefschrift-Leyden. / Summary in Dutch. Vita. Bibliography: p. 70-72.

Thesis (doctoral)--Bonn, 1998. / Includes bibliographical references (p. 84-86).

Let M be a Riemannian manifold with metric g, and let P be a principal G-bundle over M having connection one-form a. One can define a modified version of the Ricci flow on P by fixing the size of the fiber. These equations are called the Ricci Yang-Mills flow, due to their coupling of the Ricci flow and the Yang-Mills heat flow. In this thesis, we derive the Ricci Yang-Mills flow and show that solutions exist for a short time and are unique. We study obstructions to the long-time existence of the flow and prove a compactness theorem for pointed solutions. We represent the Ricci Yang-Mills flow as a gradient flow and derive monotonicity formulas that can be used to study breather and soliton solutions. Finally, we use maximal regularity theory and ideas of Simonett concerning the asymptotic behavior of abstract quasilinear parabolic partial differential equations to study the stability of the Ricci Yang-Mills flow in dimension 2 at Einstein Yang-Mills metrics. / text

Thesis (Ph. D.)--University of Texas at Austin, 2008. / Vita. Includes bibliographical references.

We will compute relative Gromov--Witten invariants of maximal contact order by applying the virtual localization formula to the moduli space of relative stable maps. In particular, we will enumerate genus 0 stable maps to the Hirzebruch surface 𝔽₁ = ℙ(𝒪_ℙ¹ ⊕ 𝒪_ℙ¹ (1)) relative to the divisor 𝐷 = 𝐵 + 𝐹, where 𝐵 is the base and 𝐹 the fiber of the projective bundle. We will provide an explicit description of the connected components of the fixed locus of the moduli space 𝑀̅₀,𝑛 (𝔽₁ ; 𝐷|𝛽 ; 𝜇) using decorated colored graphs and further determine the weight decomposition of their virtual normal bundles. This thesis contains explicit computations for 𝜇 = (3) and 𝛽 = 3𝐹 + 𝐵), and additionally 𝜇 = (4) and 𝛽 ∈ {4𝐹 + 𝐵, 4𝐹 + 2𝐵}. The same methodology however can be applied to any other ramification pattern 𝜇 and curve class 𝛽.

Natural fibers have the potential to replace glass fibers in fiber-reinforced composite applications. However, the natural fibers’ intrinsic properties cause these issues: 1) the mechanical property variation; 2) moisture uptake by natural fibers and their composites; 3) lack of sound, cost-effective, environmentriendly fiber-matrix compounding processes; 4) incompatibility between natural fibers and polymer matrices; and 5) low heat-resistance of natural fibers and their composites. This dissertation systematically studied the use of kenaf bast fiber bundles, obtained via a mechanical retting method, as a light-weight reinforcement material for fiber-reinforced thermoset polymer composites for automotive applications. Kenaf bast fiber bundle tensile properties were tested, and the effects of locations in the kenaf plant, loading rates, retting methods, and high temperature treatments and their durations on kenaf bast fiber bundle tensile properties were evaluated. A process has been developed for fabricating high fiber loading kenaf bast fiber bundle-reinforced unsaturated polyester composites. The generated composites possessed high elastic moduli and their tensile strengths were close to specification requirements for glass fiber-reinforced sheet molding compounds. Effects of fiber loadings and lengths on resultant composite’s tensile properties were evaluated. Fiber loadings were very important for composite tensile modulus. Both fiber loadings and fiber lengths were important for composite tensile strengths. The distributions of composite tensile, flexural and impact strengths were analyzed. The 2-parameter Weibull model was found to be the most appropriate for describing the composite strength distributions and provided the most conservative design values. Kenaf-reinforced unsaturated polyester composites were also proved to be more cost-effective than glass fiber-reinforced SMCs at high fiber loadings. Kenaf bast fiber bundle-reinforced composite’s water absorption properties were tested. Surface-coating and edge-sealing significantly reduced composite water resistance properties. Encapsulation was a practical method to improve composite water resistance properties. The molding pressure and styrene concentrations on composite and matrix properties were evaluated. Laser and plasma treatment improved fiber-to-matrix adhesion.

Ce mémoire porte sur quelques éléments de la théorie des fibrés symplectiques et leurs usages en étudiant la géométrie hoferienne sur le groupe de difféomorphismes hamiltoniens. En particulier en assumant un certain confort avec les notions de base de la géométrie différentielle et de la topologie algébrique on développe dans le premier chapitre les rudiments nécessaires de la théorie des G-fibrés et, dans la deuxième, tous les faits nécessaires de la topologie symplectique et les difféomorphismes hamiltoniens pour comprendre la théorie de base des fibrés symplectiques, à voir le morphisme de flux et ses liens aux isotopies hamiltoniennes. Le troisième chapitre présente les fondements des fibrés symplectiques se conclu en construisant la forme de couplage dans un langage invariant et en présentant la caractérisation des fibrés symplectiques, dont le groupe de structure réduit au groupe hamiltonien. Le mémoire se termine en présentant quelques applications des fibrés hamiltoniens à la géométrie de Hofer, en particulier une caractérisation de la partie positive de la norme de Hofer d'un lacet hamiltonien en termes du K-aire du fibré au-dessus de la sphère associé et une démonstration de la non-dégénérescence de la norme de Hofer pour des variétés symplectiques fermées. / This thesis presents a reasonably complete account of the elements theory of symplectic and Hamiltonian fibrations. We assume a familiarity and comfort with the basic notions of differential geometry and algebraic topology but little else. Proceeding from this, the first chapter develops the necessary notions from the theory of fiber bundles and G-fiber bundles, while the second chapter develops all the notions and theorems required to understand the later theory of symplectic fibrations. Most notably the second chapter includes a detailed account of the classical relationship between the flux homomorphism and Hamiltonian isotopies. The third chapter is where we develop the theory of symplectic and locally Hamiltonian fiber bundles, and in particular give an invariant construction of the coupling form on a symplectic fibration admitting an extension class. the third chapter ends with a proof of a structure theorem characterizing those symplectic fibrations for which the structure group reduces to the Hamiltonian group. In the final chapter, we present some applications of the theory of Hamiltonian fibrations by the way of characterizing the positive part of the Hofer norm of a Hamiltonian loop as the K-area of its associated Hamiltonian bundle over the sphere, and we finish by giving a proof of the non-degeneracy of the Hofer norm for closed symplectic manifolds.

Made available in DSpace on 2014-06-11T19:26:55Z (GMT). No. of bitstreams: 0 Previous issue date: 2010-02-18Bitstream added on 2014-06-13T20:16:03Z : No. of bitstreams: 1 alves_ff_me_sjrp.pdf: 362207 bytes, checksum: 9a797ca400dea6e139af98c5a9f10378 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Este trablaho apresenta os conceitos de Lyapounov e de espaços próprios e fornece um resultado devido a Oseledets, o qual trata da existência desses expoentes (e, consequentemente, dos espaços próprios) do ponto de vista da teoria da medida. A prova do teorema que nós fornecemos foi dada originalmente por Mañe e posteriormente melhorada por Viana. / This work presents the concepts of Lyapounov exponents and of proper spaces and provides a result due to Oseledets, wich deals with the existence of these exponents (and consequently, of the proper spaces) from a measure-theoretical point of view. The proof of the theorem which we provide was originally given by Mañe later improved by Viana.