The goal of this thesis is to give an exposition of the following recent result of Freeman, Lennard, Odell, Turett and Randrianantoanina. A Banach space has the Schur property if and only if every weakly compact set is contained in the closed convex hull of a weakly null sequence. This result complements an old result of Grothendieck (now called the Grothendieck Compactness Principle) stating that every norm compact subset of a Banach space is contained in the closed convex hull of a norm null sequence. We include many of the relevant definitions and preliminary results which are required in the proofs of both of these theorems.

O problema de Scarborough-Stone consiste em perguntar se o produto de espaços topológicos sequencialmente compactos precisa ser enumeravelmente compacto. Nesse trabalho estudamos alguns resultados que surgiram tentando resolver tal problema. Começamos com uma resposta negativa em ZFC usando espaços T2 e depois especificamos melhor condições sobre os axiomas de separação envolvendo os espaços do produto. Veremos respostas positivas envolvendo alguns axiomas de separação mais fortes como T6 (usando MA e a negação de CH) e T5 (usando o PFA). Além disso construímos mais respostas negativas usando construções como a Reta de Ostaszewski, espaços de Franklin-Rajagopalan e estruturas envolvendo álgebras Booleanas. / The Scarborough-Stone problem asks if every product of sequentially compact spaces must be a countably compact space. In this work we study some results that have arisen in attempt to solve this problem. We start our results with a negative answer in ZFC using T2 spaces and specify our conditions about the separability axioms of the spaces of the product. We will see positive answers assuming stronger separability axioms like T6 (using MA and the negation of CH) and T5 (using the PFA). We also construct more negative answers using constructions like the Ostaszewski line, Franklin-Rajagopalan spaces and structures involving Boolean algebras.

Assumindo a existência de $\\mathfrak c$ ultrafiltros seletivos dois a dois incomparáveis (segundo a ordem de Rudin-Keisler) provamos que o grupo abeliano livre de cardinalidade $\\mathfrak c$ admite uma topologia de grupo enumeravelmente compacta com uma seqüência não trivial convergente. Sob as mesmas hipóteses, mostramos que um grupo topológico abeliano quase livre de torção $(G, +, \\tau)$ com $|G| = |\\tau| = \\mathfrak c$ admite uma topologia independente de $\\tau$ que o torna um grupo topológico e caracterizamos algebricamente os grupos abelianos de não torção que têm cardinalidade $\\mathfrak c$ e que admitem uma topologia de grupo enumeravelmente compacta (sem seqüências não triviais convergentes). Provamos, ainda, que o grupo abeliano livre de cardinalidade $\\mathfrak c$ admite uma topologia de grupo que torna seu quadrado enumeravelmente compacto e construímos um semigrupo de Wallace cujo quadrado é, também, enumeravelmente compacto. Por fim, assumindo a existência de $2^{\\mathfrak c}$ ultrafiltros seletivos, garantimos que se um grupo abeliano de não torção e cardinalidade $\\mathfrak c$ admite uma topologia de grupo enumeravelmente compacta, então o mesmo admite $2^{\\mathfrak c}$ topologias de grupo enumeravelmente compactas (duas a duas não homeomorfas). / Assuming the existence of $\\mathfrak c$ pairwise incomparable selective ultrafilters (according to the Rudin-Keisler ordering) we prove that the free abelian group of cardinality $\\mathfrak c$ admits a countably compact group topology that contains a non-trivial convergent sequence. Under the same hypothesis, we show that an abelian almost torsion-free topological group $(G, +, \\tau)$ with $|G| = |\\tau| = \\mathfrak c$ admits a group topology independent of $\\tau$ and we algebraically characterize the non-torsion abelian groups of cardinality $\\mathfrak c$ which admit a countably compact group topology (without non-trivial convergent sequences). We also prove that the free abelian group of cardinality $\\mathfrak c$ admits a group topology that makes its square countably compact and we construct a Wallace\'s semigroup whose square is countably compact. Finally, assuming the existence of $2^$ selective ultrafilters, we ensure that if a non-torsion abelian group of cardinality $\\mathfrak c$ admits a countably compact group topology, then it admits $2^$ (pairwise non-homeomorphic) countably compact group topologies.

Sejam X um espaço topológico não necessariamente compacto e T:X->X uma aplicação contínua. Se f:X->R é contínua, daremos condições sobre f que garantam a existência de medidas maximizantes caracterizadas em termos de seu suporte. / Let X be a topological space not necessarily compact, and T:X->X a continuous map. If f:X->R is a continuous function, we seek conditions on f in order to guarantee existence of maximizing measures that are characterized in terms of its support.

Sejam X um espaço topológico não necessariamente compacto e T:X->X uma aplicação contínua. Se f:X->R é contínua, daremos condições sobre f que garantam a existência de medidas maximizantes caracterizadas em termos de seu suporte. / Let X be a topological space not necessarily compact, and T:X->X a continuous map. If f:X->R is a continuous function, we seek conditions on f in order to guarantee existence of maximizing measures that are characterized in terms of its support.

We aim to explain the intuition behind several large cardinal axioms, give characterization theorems for these axioms, and then discuss a few of their properties. As a capstone, we hope to introduce a new large cardinal notion and give a similar characterization theorem of this new notion. Our new notion of near strong compactness was inspired by the similar notion of near supercompactness, due to Jason Schanker.

Water is one of the fundamental components of concrete, not only for its role on the hydration of Portland cement, but also because of the physical functions it develops, which are associated with the main phases of concrete life: fresh state, hardened state and the useful life of the structures. The objective of this PhD Thesis is to study in detail the physical role of water in Portland cement mixtures: the aggregate absorption, the wetting and the fluidization of the granular skeletons that compose the cement pastes. The study covers the mathematical modelling of the mentioned physical functions in a way that it is possible to calculate the volume of water necessary to perform such functions, facilitating the mix-design process. The calculated volume is considered to be the total volume of water needed for production. Moreover, the calculation must take into account the conditions and constraints associated with the production and casting, as well as the technical requirements of the material to be designed. The modelling of the water physical functions allowed the development of a calculation method to quantify the approximate volume of water needed for concrete production. The developed method was used to calculate the volume of water of three different special concretes: a lightweight self-compacting concrete reinforced with fibres, an ultra-high performance concrete reinforced with steel fibres and a concrete with recycled aggregates. What is more, the volume of water for two conventional concretes, with compressive strengths of 25 and 30 MPa, was calculated. Since the calculation was based on granular skeletons for real mixtures, produced in laboratory or/and industrially, the results obtained through the use of the developed method were compared to the experimental results of each concrete. At last, the method was used to quantify the volume of paste necessary for the production of a porous concrete. The results show that the mathematical models used to describe the physical phenomena of absorption, wetting and fluidization fit well to the experimental reproduction of these phenomena. Corrections are needed in some situations due to the ideal boundary conditions adopted during modelling, which facilitate calculation. Anyhow, the errors are corrected through the use of adjusting coefficients. Therefore, the calculation method developed has proven itself effective and applicable in the mix-design of different types of conventional and special concretes, showing the potential to be used for the development of new materials. / El agua es uno de los componentes fundamentales del hormigón, no sólo por ser necesaria a la hidratación del cemento Portland, sino que también por las diferentes funciones físicas que desarrolla, las cuales están asociadas a las principales fases de la vida del hormigón: estado fresco, estado endurecido y vida útil de la estructura. El objetivo de la presente Tesis Doctoral es realizar un estudio detallado de las funciones físicas del agua en las mezclas de cemento Portland: la absorción de esta por los áridos, el mojado y la fluidificación de los conjuntos granulares que componen las pastas de cemento. Dicho estudio se traduce en la modelización matemática de las funciones físicas presentadas, en el sentido de dar una respuesta numérica que facilite el diseño de mezclas, acotando el volumen de agua necesario al desarrollo de las funciones especificadas, siendo éste el volumen de agua total necesario a la producción. Asimismo, el cálculo del referido volumen debe tener en cuenta los condicionantes de producción, puesta en obra, así como los requerimientos técnicos del material que se va diseñar. A través de la modelización de las funciones físicas del agua consideradas, se ha desarrollado un método de cálculo para acotar el volumen de agua total necesario a la producción de hormigones. Se ha utilizado el método desarrollado para el cálculo del volumen de agua de tres hormigones especiales distintos: hormigón ligero autocompactante con fibras, hormigón de ultra-alta resistencia reforzado con fibras de acero y hormigón con áridos reciclados. Asimismo, se ha calculado el volumen de agua para dos hormigones convencionales, de resistencias à compresión 25 y 30 MPa. Se han contrastado los resultados obtenidos por el uso del método desarrollado con los resultados experimentales de cada hormigón, ya que el cálculo se hizo con base en conjuntos granulares de mezclas reales, producidas en laboratorio y/o industrialmente. Por último, se ha utilizado el modelo desarrollado para la cuantificación del volumen de pasta necesario a la producción de un hormigón poroso. Los resultados demuestran que los modelos matemáticos utilizados para describir los fenómenos físicos de absorción, mojado y fluidificación se adecuan bien a la reproducción experimental de dichos fenómenos, en que correcciones son necesarias en algunas situaciones, debido a la adopción de condiciones de contorno ideales en la modelización, que facilitan los cálculos. De cualquier modo, los errores se corrigen a través de coeficientes de ajuste. Así, el método de cálculo desarrollado para acotar el volumen de agua se ha demostrado eficiente en el diseño de diferentes tipos de hormigones convencionales y especiales, pudiendo ser utilizado en el desarrollo de nuevos materiales.

The miscible displacement equations provide the mathematical model for simulating the displacement of a mixture of oil and miscible fluid in underground reservoirs during the Enhance Oil Recovery(EOR) process. In this thesis, I propose a stable numerical scheme combining a mixed finite element method and space-time discontinuous Galerkin method for solving miscible displacement equations under low regularity assumption. Convergence of the discrete solution is investigated using a compactness theorem for functions that are discontinuous in space and time. Numerical experiments illustrate that the rate of convergence is improved by using a high order time stepping method. For petroleum engineers, it is essential to compute finely detailed fluid profiles in order to design efficient recovery procedure thereby increase production in the EOR process. The method I propose takes advantage of both high order time approximation and discontinuous Galerkin method in space and is capable of providing accurate numerical solutions to assist in increasing the production rate of the miscible displacement oil recovery process.

This dissertation is a contribution to the the study of the longtime dynamics of evolutionary equations in unbounded domains. It is of particular interest to prove the existence of global attractors for solutions of such equations. Th this end one need in general some type of asymtotical compactness. In the case the evolutionary PDE is defined on a bounded domain, asymptotical compactness follows from the regularity estimates and the compactnes of Sobolev embeddings and therefore the existence of attractors has been established for most of the disipative equations of mathematocal physics in a bounded domain. The problem is more challenging when the domain is unbounded since the Sobolev embeddings are no longer comapct, so that the usual regularity estimates may not be sufficient.To overcome this obstacle of compactness, A.V. Babin and M.I. Vishik introduced some weighted Sobolev spaces. In their pioneering paper, Proc. Roy. Soc. Edinb.

Recently there have been numerous advances in the development of numerical algorithms to solve conservation laws. Even though the analytical theory (existence-uniqueness) is complete in the case of scalar conservation laws, there are many numerically robust methods for which the question of convergence and error estimates are still open. Usually high order schemes are constructed to be Total Variation Diminishing (TVD) which only guarantees convergence of such schemes to a weak solution. The standard approach in proving convergence to the entropy solution is to try to establish cell entropy inequalities. However, this typically requires additional non-homogeneous limitations on the numerical method, which reduces the modified scheme to first order when the mesh is refined. There are only a few results on the convergence which do not impose such limitations and all of them assume some smoothness on the initial data in addition to L^infinity bound. The Nessyahu-Tadmor (NT) scheme is a typical example of a high order scheme. It is a simple yet robust second order non-oscillatory scheme, which relies on a non-linear piecewise linear reconstruction. A standard reconstruction choice is based on the so-called minmod limiter which gives a maximum principle for the scheme. Unfortunately, this limiter reduces the reconstruction to first order at local extrema. Numerical evidence suggests that this limitation is not necessary. By using MAPR-like limiters, one can allow local nonlinear reconstructions which do not reduce to first order at local extrema. However, use of such limiters requires a new approach when trying to prove a maximum principle for the scheme. It is also well known that the NT scheme does not satisfy the so-called strict cell entropy inequalities, which is the main difficulty in proving convergence to the entropy solution. In this work, the NT scheme with MAPR-like limiters is considered. A maximum principle result for a conservation law with any Lipschitz flux and also with any k-monotone flux is proven. Using this result it is also proven that in the case of strictly convex flux, the NT scheme with a properly selected MAPR-like limiter satisfies an one-sided Lipschitz stability estimate. As a result, convergence to the unique entropy solution when the initial data satisfies the so-called one-sided Lipschitz condition is obtained. Finally, compensated compactness arguments are employed to prove that for any bounded initial data, the NT scheme based on a MAPR-like limiter converges strongly on compact sets to the unique entropy solution of the conservation law with a strictly convex flux.