Includes bibliographical references. / It is common to think of our universe according to the "block universe" idea, which says that spacetime consists of many "stacked" 3-surfaces varied as a function of some kind of proper time Ƭ. Standard ideas do not distinguish past and future, but Ellis' "evolving block universe" tries to make a fundamental distinction. One proposal for this proper time is the proper time measured along the timelike Ricci eigenlines, starting from the big bang. The main idea of this work is to investigate the shape of the {Ƭ=constant} surfaces relative to the the null surfaces, and determine what makes these surfaces timelike or spacelike. We use the Lemaître-Tolman metric as our inhomogeneous spacetime model, and we find the necessary and sufficient conditions for these {Ƭ=constant} surfaces to be spacelike or timelike. Furthermore, we indicate whether or not timelike surfaces appear inside black holes and other strong gravity domains, by determining the location of the timelike regions relative to the apparent horizon. Based on this idea, we find that the regions where these surfaces become timelike are often close to the apparent horizons, but always outside them, and in particular timelike regions occur outside black holes. They are always spacelike near the big bang, and at late times (near the crunch or the extreme far future), they are only timelike under special circumstances.

Includes bibliographical references (leaves 143-147).

Includes abstract. / Includes bibliographical references (leaves 82-84). / We study the evolution of scalar cosmological perturbations in the 1+3 Covariant Gauge-Invariant formalism for generic f(R) theories of gravity. Working in the energy frame of the total matter, we give a complete set equations describing the evolution of matter and curvature fluctuations for a multi-fluid cosmological medium. We then specialize to a radiation-dust fluid described by barotropic equations of state. We apply the perturbation equations around a background solution of Rⁿ gravity and look at exact solutions for scales much smaller and much larger than the Hubble radius.

Includes bibliographical references. / We give a slightly more general version of the Rubin-Stark conjecture, but show that in most cases it follows from the standard version. After covering the necessary background, we state the principal Stark conjecture and show that although the conjecture depends on a choice of a set of places and a certain isomorphism of Q[GJ-modules, it is independent of these choices. The conjecture is shown to satisfy certain 'functoriality' properties, and we give proofs of the conjecture in some simple cases. The main body of this dissertation concerns a slightly more general version of the Rubin-Stark conjecture. A number of Galois modules. Connected with the conjecture are defined in chapter 4, and some results on exterior powers and Fitting ideals are stated. In chapter 5 the Rubin-Stark conjecture is stated and we show how its truth is unaffected by lowering the top field, changing a set S of places appropriately, and enlarging moduli. We end by giving proofs of the conjecture in several cases. A number of proofs, which would otherwise have interrupted the flow of the exposition, have been relegated to the appendix, resulting in this dissertation suffering from a bad case of appendicitis.

Includes bibliographical references. / In this thesis we analyse several spatial structures, built from percolation models, by means of an approach used so far in the field of network science. In the first chapter we summarize the major network concepts and characterizations that have been obtained as regards the statistical properties of several data sets or theoretical models, We also give a brief introduction to percolation theory and its applications, adding details in two particular cases where mathematical results are available. In the second chapter we then study one particular application of percolation theory to the modelling of distribution and species abundance at different seales. We mainly focus on the way percolation theory was used to compare two diffcrcnt spatial patterns, particularly the random and the aggrergated distribution.

Includes bibliographical references (leaves 61-68). / The proposal that the physics of quantum critical phase transition in strongly coupled condensed matter systems can be described by a gravitational theory within the frame work of gauge/gravity correspondence is investigated more extensively for s-wave superconductors. We consider a gravitational theory with a black hole solution in anti de Sitter spacetime, coupled to an Abelian-Higgs system in (d + 1)-dimensions. A wide range of negative mass squared for the scalar field that satisfied the Brietenlolmer-Freedman stability bound and the unitarity bound are considered in the probe limit. The dependence of the some of the physical quantities on the scaling dimensions of the dual condensates were thoroughly investigated. We observe that the holographic superconductors can be consistently classified into two, based on the scaling dimensions and the charge of the dual condensates. Holographic superconductors of dimension λ- exhibit features of type II superconductors while those of dimension λ+ show features of type 1. The validity of this classification was confirmed by solving the bulk equations of motion perturbatively near the quantum critical point in order to calculate the superconducting characteristic lengths at a fixed charge q. The results show that there is a critical scaling dimension beyond which a holographic superconductor behave as type I and below this value it is a type II. The properties of holographic superconductors presented in this report are in qualitative agreement with the Ginzburg-Landau theory.

Bibliography: leaves 81-83. / This dissertation looks at some aspects of the representation theory of complex semi-simple Lie groups.

Includes bibliographical references. / While most of Europe is affected by HIV-1 subtype B, sub-Saharan African is dominated by HIV-1 subtype C. Due to costs, most vaccine development is carried out Europe rather than sub-Saharan countries. However since the mechanisms of disease progression in HIV-1 subtype B may be different from those in HIV-1 subtype C, it is interesting to investigate if and how a dendritic cells based vaccine such as the one developed in France and tested on Brazilians (Lu et al, Nature; 2004) can be used on individuals in sub-Saharan Africa. To investigate this, mathematical models and sensitivity analysis techniques are used to understand the mechanisms of disease progression in two HIV-1 subtypes. These models are then extended to explore the ways in which the vaccine could be used to treat these different HIV-1 subtypes. It is found that the level of immune activation plays a large role in determining the mechanism of disease progression and can itself be a means to the development of AIDS. Furthermore, it is also shown that the dendritic cells based vaccine could reduce the viral load but not eliminate the virus resulting in a viral rebound. To maintain a low viral load, vaccination would have to be repeated. Unfortunately, repeated vaccination may lead to the overproduction of proinflamatory cytokines resulting in severe side effects. However this could be avoided by using a carefully planned treatment schedule. We conclude that the dendritic cells based vaccine can be used in individuals in either subtype B or subtype C region as long as the correct treatment schedule is followed.

Bibliography: pages 93-97. / A strategy that is often used in the study of capillary free boundary (FB) problems for viscous incompressible flows is the following: (1) Ignore one of the boundary conditions at the FB and prove that for every chosen position of the FB the resultant problem, here called the auxiliary problem (AP), is well posed. (2) Establish regularity results for the solution of the AP. (3) Using (2) and the remaining boundary condition, determine the position of the FB. We study the existence and uniqueness of the weak solution(s) to the AP, i.e., step (1), under minimal regularity constraints on the data and domain. The analysis is carried out for stationary two-dimensional flows, governed by either the Stokes or Navier-Stokes equations, in the context of four standard examples. A Green's formula is derived which allows the AP to be formulated as a mixed variational problem in which the pressure and normal stress appear as Lagrange multipliers. Existence and uniqueness results are obtained by using the Ladyzhenskaya-Babuska-Brezzi theory for mixed problems. By analogy with step (3), the dependence of the normal stress on the position of the FB is investigated.

Bibliography: leaves 74-76. / For the past three decades, researchers have developed models to price options with non-constant asset price volatility. These models can be divided into deterministic volatility models and stochastic volatility models. Deterministic volatility models assume that volatility is determined by some variables observable in the market. Stochastic volatility models suggest that volatility follows a stochastic process, whose parameters are not directly observable in the market. However, most of these authors have compared the results of their models with the classical Black-Scholes model [6], which assumes that volatility is constant. This dissertation investigates whether there is any model that can completely describe the market. Therefore, instead of com paring the results of the models with that of the Black-Scholes model, we have compared them with the market. For the purpose of this research, the S&P 500 Index option prices extracted from market are used. We investigate and compare for models: the GARCH(l ,I) model, the Constant Elasticity of Variance model, the Hull and White model, and the Heston model. The former two belong to deterministic volatility models and the latter two are stochastic volatility models. We conclude that none of the models under consideration can fully describe the market prices. Moreover, no model dominates the others by producing better results for all options.