by Ki-sing Ng. / Thesis (M.Phil.)--Chinese University of Hong Kong. / Bibliography: leaves 30.

We study a random homogenization problem concerning the flow of a viscous fluid through a permeable membrane with a highly oscillatory geometry and nonlinear boundary condition on it. Along an interface we consider a periodic distribution of small permeable obstacles with a random geometry. Leak boundary conditions of threshold type are considered on the obstacle part of the membrane: the normal velocity of the fluid is zero until the jump of the normal component of the stress acting on it reaches a certain limit, and then the fluid may pass freely. The problem is studied first in the deterministic case, and then in the random case, for which assumptions on the randomness of the solid obstacles are needed in order to obtain a limiting behaviour. The description of the obstacles is given in terms of a random set-valued variable defined on a probability space and a dynamical system acting on it. Effective boundary conditions for the fluid are derived, and these depend on the relative size of the obstacles. We establish two major cases, in one of them we obtain an effective permeability across the membrane and in the critical case a slip boundary condition of Navier type. If the dynamical system is assumed to be ergodic, the limiting behaviour of the fluid is deterministic. The approach is based on the Mosco convergence, which also allows us to pass from the stationary case to the time dependent case via the convergence of the associated semigroups.

L’objectif de réduction des écarts de niveaux de vie entre les pays membres de l’Union Monétaire Européenne nécessite la convergence des productivités entre les économies. Notre contribution est de comparer la dynamique sur longue période des Productivités Globales des Facteurs (PGF) des pays de la zone euro à celles d’un ensemble de pays dénommé "autres OCDE". Cet objectif général est subordonné à deux autres plus spécifiques : 1) Analyser les gains de productivité par une approche paramétrique des frontières de production afin d’examiner la sigma-convergence des PGF des deux zones. 2) Etudier les évolutions des inefficacités productives sous-jacentes aux dynamiques de la PGF en s’appuyant sur un modèle d’activités non paramétrique. Nous étudions les évolutions de 11 pays de la zone euro et de 11 pays « autres OCDE entre 1965 et 2015. Nos résultats montrent que la zone euro connait un meilleur taux de croissance tendanciel de la PGF que celui des autres pays de l’OCDE ; bien que les niveaux de productivité de ceux-ci soient plus élevés. Par ailleurs, les variations des inefficacités productives attestent l’existence d’un rattrapage technologique au sein de chaque zone. Ce phénomène se fait non pas vers la frontière mais en dessous de celle-ci. De plus, la réduction des inefficacités structurelles entre le milieu des années 1970 et le début des années 2000 témoignent d’une homogénéisation des mixes d’output/inputs entre les pays membres. La convergence conditionnelle aux variables monétaires budgétaires et réelles conduit à des résultats différents suivant qu’il s’agit du processus de rattrapage technique ou de celui de la convergence structurelle. Toutefois au début des années 2000, un mouvement de divergence s’installe entre les pays de l’Euroland et s’accentue à partir de 2008-2009. / The objective of reducing differences in living standards among the Eurozone countries requires the convergence of productivity levels between economies. Our contribution is to compare the long-run dynamics of the Total Factor Productivity (TFP) among the Eurozone countries to that of another group, referred to as "other OECD countries". This general objective is divided into two more specific research subjects: 1) To analyze the productivity gains based on a parametric activity framework of production set to investigate the absolute Sigma-convergence of the TFPs in the two zones. 2) To examine the evolution of the productive inefficiencies underlying the dynamics of the TFP, based on a nonparametric activity model. We observed the evolutions of 11 Eurozone countries, and 11 other OECD countries over the period 1965-2015. Our empirical results show that the level of productivity still seems to be higher in other OECD countries even the growth rate of productivity is greater in the Eurozone. This indicates a catch- up phenomenon between two groups and a sigma-convergence is observed before the end of 1990s. However, the catching up efficiency processes do not move the countries toward the frontier but make them converge below the frontier. Additionally, a significant decrease in structural inefficiencies is noticed between the mid-1970s and the early 2000s which indicates the homogenization of output / input mixes for each zone. The convergence process is conditional to real and monetary variables which may differ according to the technical catch-up or the structural efficiency dynamics. However, it seems that these dynamics have been interrupted since the creation of the Euro and have been even turned to a movement of divergence since the subprime crisis in 2008.

This thesis deals with the convergence project the IASB and FASB on the example of rules for leases. The first chapter introduces the general problems. The next chapters are introduced the IASB and the FASB, its objectives, structures, activities. The thesis focuses on the convergence project. The next two chapters, discusses the current leases arrangements in IFRS and U.S. GAAP, including its comparison. The main part analyses the draft of a new joint standard also discusses its possible impact and response to this proposal.

This work is about convergence to equilibrium problems for equations coming from kinetic theory. The bulk of the work is about Hypocoercivity. Hypocoercivity is the phenomenon when a semigroup shows exponentially relaxation towards equilibrium without the corresponding coercivity (dissipativity) inequality on the Dirichlet form in the natural space, i.e. a lack of contractivity. In this work we look at showing hypocoercivity in weak measure distances, and using probabilistic techniques. First we review the history of convergence to equilibrium for kinetic equations, particularly for spatially inhomogeneous kinetic theory (Boltzmann and Fokker-Planck equations) which motivates hypocoercivity. We also review the existing work on showing hypocoercivity using probabilistic techniques. We then present three different ways of showing hypocoercivity using stochastic tools. First we study the kinetic Fokker-Planck equation on the torus. We give two different coupling strategies to show convergence in Wasserstein distance, $W_2$. The first relies on explicitly solving the stochastic differential equation. In the second we couple the driving Brownian motions of two solutions with different initial data, in a well chosen way, to show convergence. Next we look at a classical tool to show convergence to equilibrium for Markov processes, Harris's theorem. We use this to show quantitative convergence to equilibrium for three Markov jump processes coming from kinetic theory: the linear relaxation/BGK equation, the linear Boltzmann equation, and a jump process which is similar to the kinetic Fokker-Planck equation. We show convergence to equilibrium for these equations in total variation or weighted total variation norms. Lastly, we revisit a version of Harris's theorem in Wasserstein distance due to Hairer and Mattingly and use this to show quantitative hypocoercivity for the kinetic Fokker-Planck equation with a confining potential via Malliavin calculus. We also look at showing hypocoercivity in relative entropy. In his seminal work work on hypocoercivity Villani obtained results on hypocoercivity in relative entropy for the kinetic Fokker-Planck equation. We review this and subsequent work on hypocoercivity in relative entropy which is restricted to diffusions. We show entropic hypocoercivity for the linear relaxation Boltzmann equation on the torus which is a non-local collision equation. Here we can work around issues arising from the fact that the equation is not in the H\"{o}rmander sum of squares form used by Villani, by carefully modulating the entropy with hydrodynamical quantities. We also briefly review the work of others to show a similar result for a close to quadratic confining potential and then show hypocoercivity for the linear Boltzmann equation with close to quadratic confining potential using similar techniques. We also look at convergence to equilibrium for Kac's model coupled to a non-equilibrium thermostat. Here the equation is directly coercive rather than hypocoercive. We show existence and uniqueness of a steady state for this model. We then show that the solution will converge exponentially fast towards this steady state both in the GTW metric (a weak measure distance based on Fourier transforms) and in $W_2$. We study how these metrics behave with the dimension of the state space in order to get rates of convergence for the first marginal which are uniform in the number of particles.

In this thesis I have connected different evolutionary studies of Lioalemus lizards. In Chapter 1, I followed an integrative approach to delimit species in the Liolaemus walkeri complex. Using mitochondrial markers, morphological data, bioclimatic information and methods appropriate for each data type, we found that the name L. walkeri was covering three new lineages. Three new species were described and one of them (L. chavin) is now categorized as Near Threatened in the IUCN red list. In Chapter 2, I change the subject from species boundaries to the study of viviparity and placentation. In this paper we employed scanning electron and confocal microscopy to compare the placental ultra-structure and pattern of blood vessels in two Liolaemus species. One of the most remarkable traits found is the complete reduction of the eggshell in both placentae, a possible adaptation to improve gas exchange in the hypoxic environments of the high Andes. In chapter 3, I returned to the issue of species delimitation and employed two integrative approaches: a hypothetical deductive framework and a model-based procedure. I applied both approaches in lowland and highland Liolaemus species of the montanus group. I found that in only one case (of four) an unnamed lowland lineage ("Nazca") was delimited concordantly by both procedures. In Chapter 4, I focus on a study of convergent evolution of desert phenotype in Liolaemus species and Ctenoblepharys adspersa. I performed a Bayesian time calibrated and maximum likelihood tree based on 55 taxa and seven molecular markers. We employed quantitative and categorical traits based on 400 specimens and non-metric multidimensional scaling to obtain new quantitative variables. I used three phylogenetic comparative methods to identify and measure the strength of convergence. My results found a strong case of convergent traits in C. adspersa, L. lentus, L. manueli, L. poconchilensis and L. stolzmanni that are probably related to predator avoidance in the Peruvian-Atacama and Monte deserts. In addition, my time calibrated tree resolves the origin of these traits first in C. adspersa at about 80 million years (My) and later independently in Liolaemus species at about 25 My suggesting the present of evolutionary constraints.

The purpose of this thesis is to explore methodology concerning Markov Chain Monte Carlo (MCMC), a powerful technique in the Bayesian framework, on binary variables. The primary application of interest in this thesis is applying this methodology to phase haplotypes, a type of categorical variable. Haplotypes are the combination of variants present in an individual’s genome. Phasing refers to estimating the true haplotype. By considering only biallelic and heterozygous variants, the haplotype can be expressed as a vector of binary variables. Accounting for differences in haplotypes is essential for the study of associations between genotype and disease. MCMC is an extremely popular class of statistical methods for simulating autocorrelated draws from target distributions, including posterior distributions in Bayesian analysis. Techniques for sampling categorical variables in MCMC have been developed in a variety of disparate settings. Samplers include Gibbs, Metropolis-Hastings, and exact Hamiltonian based samplers. A review of these techniques is presented and their relevance to the genetic model discussed. An important consideration in using simulated MCMC draws for inference is that they have converged to the distribution of interest. Since the distribution is typically of a non-standard form, convergence cannot generally be proven and, instead, is assessed with convergence diagnostics. The convergence diagnostics developed so far focus on continuous variables and may be inappropriate for binary variables or categorical variables in general. Two convergence diagnostics are proposed that are tailor-made for categorical variables by modeling the data using categorical time series models. Performance of the convergence diagnostics is evaluated under various simulations. The methodology developed in the thesis is applied to estimate haplotypes. There are two main challenges involved in accounting for haplotype differences. One is estimating the true combination of genetic variants on a single chromosome, known as haplotype phasing. The other is the phenomenon of allele-specific expression (ASE) in which haplotypes can be expressed non-equally. No existing method addresses these two intrinsically linked challenges together. Rather, current strategies rely on known haplotypes or family trio data, i.e. having data on subject of interest and their parents. A novel method is presented, named IDP-ASE, which is capable of phasing haplotypes and quantifying ASE using only RNA-seq data. This model leverages the strengths of both Second Generation Sequencing (SGS) data and Third Generation Sequencing (TGS) data. The long read length of TGS data facilitates phasing, while the accuracy and depth of SGS data facilitates estimation of ASE. Moreover, IDP-ASE is capable of estimating ASE at both the gene and isoform level.

In the last 15 years it has been widely debated whether or not the United States should adopt IFRS. Convergence efforts in the 21st century have limited the distinctions between U.S. GAAP and IFRS, but significant differences still exist. This paper takes an in-depth look at the most important remaining differences between U.S. GAAP and IFRS, and examines both sides of the argument on adoption. Finally, I conclude that the U.S. should continue to use and refine its own standards.

Burgers’ Equation ut + cuux = νuxx is a nonlinear partial differential equation which arises in models of traffic and fluid flow. It is perhaps the simplest equation describing waves under the influence of diffusion. We consider the large time behavior of solutions with exponentially localized initial conditions, analyzing the rate of convergence to a known self similar single-hump solution. We use the Cole-Hopf Transformation to convert the problem into a heat equation problem with exponentially localized initial conditions. The solution to this problem converges to a Gaussian. We then find an optimal Gaussian approximation which is accurate to order t−2. Transforming back to Burgers’ Equation yields a solution accurate to order t−2.

The boundary layer problems in periodic homogenization arise naturally from the quantitative analysis of convergence rates. Formally they are second-order linear elliptic systems with periodically oscillating coefficient matrix, subject to periodically oscillating Dirichelt or Neumann boundary data. In this dissertation, for either Dirichlet problem or Neumann problem, we establish the homogenization results and obtain the nearly sharp convergence rates, provided the domain is strictly convex. Also, we show that the homogenized boundary data is in W1,p for any p ∈ (1,∞), which implies the Cα-Hölder continuity for any α ∈ (0,1).