This thesis is comprised of three main parts: The Hermite-Hadamard inequality on discrete time scales, the fractional Hermite-Hadamard inequality, and Karush-Kuhn- Tucker conditions on higher dimensional discrete domains. In the first part of the thesis, Chapters 2 & 3, we define a convex function on a special time scale T where all the time points are not uniformly distributed on a time line. With the use of the substitution rules of integration we prove the Hermite-Hadamard inequality for convex functions defined on T. In the fourth chapter, we introduce fractional order Hermite-Hadamard inequality and characterize convexity in terms of this inequality. In the fifth chapter, we discuss convexity on n{dimensional discrete time scales T = T1 × T2 × ... × Tn where Ti ⊂ R , i = 1; 2,…,n are discrete time scales which are not necessarily periodic. We introduce the discrete analogues of the fundamental concepts of real convex optimization such as convexity of a function, subgradients, and the Karush-Kuhn-Tucker conditions. We close this thesis by two remarks for the future direction of the research in this area.

Flow control can lead to saving millions of dollars in fuel costs each year by making an aircraft more efficient. Synthetic jets, a device for active flow control, operate by introducing small amounts of energy locally to achieve non-local changes in the flow field with large performance gains. These devices consist of a cavity with an oscillating diaphragm that divides it, into active and passive sides. The active side has a small opening where a jet is formed, whereas and the passive side does not directly participate in the fluidic jet.Research has shown that the synthetic jet behavior is dependent on the diaphragm and the cavity design hence, the focus of this work. The performance of the synthetic jet is studied under various factors related to the diaphragm and the cavity geometry. Four diaphragms, manufactured from piezoelectric composites, were selected for this study, Bimorph, Thunder®, Lipca and RFD. The overall factors considered are the driving signals, voltage, frequency, cavity height, orifice size, and passive cavity pressure. Using the average maximum jet velocity as the response variable, these factors are individually studied for each actuator and statistical analysis tools were used to select the relevant factors in the response variable. For all diaphragms, the driving signal was found to be the most important factor, with the sawtooth signal producing significantly higher velocities than the sine signal. Cavity dimensions also proved to be relevant factors when considering the designing of a synthetic jet actuator. The cavities with the smaller orifice produced lower velocities than those with larger orifices and the cavities with smaller volumes followed the same trend. Although there exist a relationship between cavity height and orifice size, the orifice size appears as the dominant factor.Driving frequency of the diaphragm was the only common factor to all diaphragms studied that was not statistically significant having a small effect on jet velocity. However along with waveform, it had a combined effect on jet velocity for all actuators. With the sawtooth signal, the velocity remained constant after a particular low frequency, thus indicating that the synthetic jet cavity could be saturated and the flow choked. No such saturation point was reached with the sine signal, for the frequencies tested. Passive cavity pressure seemed to have a positive effect on the jet velocity up to a particular pressure characteristic of the diaphragm, beyond which the pressure had an adverse effect. For Thunder® and Lipca, the passive cavity pressure that produced a peak was measured at approximately 20 and 18kPa respectively independent of the waveform utilized. For a Bimorph and RFD, this effect was not observed.Linear models for all actuators with the factors found to be statistically significant were developed. These models should lead to further design improvements of synthetic jets.

This thesis examines the relationship between the distribution of structural breaks within a data sample and the estimated parameter of long memory. We use Monte Carlo simulations to generate data from processes with specific values of parameters. Subsequently we join the data with various shifts to mean and examine how the estimates of the parameters vary from their true values. We have discovered that the overestimate of the long memory parameter is higher when the breaks are clustered together. It further increases when the signs of the shifts are positively correlated within the clusters while negative correlation reduces the bias. Our findings enable the improvement of robustness of estimators against the presence structural breaks. Powered by TCPDF (www.tcpdf.org)

Thesis (Ph.D.(Public Health))--University of the Witwatersrand, Faculty of Health Sciences, 2012. / Measuring HIV incidence has proved challenging over the years. A number of serological HIV assays have been proposed, and among these, the BED Capture Enzyme Immunoassay (CEIA) is one of the more widely used. Although the assay performs well among known seroconverting panels, it has been shown to classify some long term infected patients as being recently infected. Information on the performance of the BED assay among low CD4 cell count patients and those on antiretroviral therapy is limited. The risk of onwards transmission of HIV has been reported to be elevated around the seroconversion period compared to the chronic stage of infection. RNA viral load has been reported as the strongest predictor of HIV transmission compared to other HIV markers. Understanding how these markers influence the relationship between the likelihood of being recently infected and the BED assay might help in understanding some of the shortcomings of the BED assay. The main aim of this study was to understand the properties of the BED assay. The performance of the BED assay among advanced HIV disease patients and the influence of ART on BED levels once patients started treatment was investigated. The BED assay and CD4 cell count were used to quantify the risk of in utero and intrapartum transmission to their infants among women believed to have seroconverted during pregnancy. The influence of viral load, haemoglobin and mid-upper arm circumference was investigated on the relationship between the probability of being recently infected and BED ODn levels. Methods Cryopreserved plasma samples from HIV patients on the national antiretroviral treatment (ART) rollout programme at Tygerberg Hospital HIV clinic, South Africa, iv were used to investigate the effect of ART on BED ODn levels once patients commenced treatment. Mixed effect logistic regression models accounting for multiple readings per patient were used. To investigate the risk associated with seroconversion during pregnancy HIV seropositive women who had just given birth were classified into mutually exclusive groups according to their likelihood of having recently seroconverted using BED and CD4 cell count levels. Multinomial logistic regression models adjusting for other factors were used to assess the risk of MTCT in utero and intra-partum infection comparing these groups. To investigate the relationship between BED ODn levels and the probability of being recently infected, BED data from known HIV infected women and women who seroconverted over a 2 year period was used. Fractional polynomial regression models that allow for non-linear functions to be fitted were used, and the influence of viral load, haemoglobin and mid-upper arm circumference was assessed through multi-variable models. Data from the Zimbabwe Vitamin A for Mothers and Babies (ZVITAMBO) project, a double blinded treatment-placebo trial was used for these last two objectives. Results Patients with very low CD4 cell counts were more likely to test false recently infected according to the BED assay than other patients. ART changed BED ODn kinetics among HIV patients on treatment. Over half of advanced disease stage patients were likely to be classified as being recently infected according to the BED assay 2 years into ART treatment. v Women who seemed to have seroconverted during pregnancy had elevated risk of transmitting HIV in-utero compared to chronic HIV patients. BED and CD4 cell count were not predictive of risk of intra-partum infections attributed to seroconversion during pregnancy. The relationship between the probability of being recently infected with HIV and BED ODn levels was described better using Fractional Polynomial regression models than using a linear model in BED ODn or a model in which the BED ODn was categorised. Viral load and haemoglobin were important independent predictors of incident infections. Conclusions If the BED assay is to be used for HIV incidence estimations patients on ART should be accounted for. The BED assay together with other HIV serological markers can be used as prognostic tools to assess the risk of HIV transmission. The risk of in-utero transmission of HIV is higher among women who seroconvert during pregnancy. Repeat HIV testing among pregnant women may help in identifying women who seroconvert during pregnancy, and these women will benefit from Prevention of Mother-to-Child transmission (PMTCT) programmes. It was found that additional markers such as viral load and haemoglobin did not alter the relationship between the probability of having been recently infected and BED ODn.

O método das matrizes operacionais é expandido para o corpo complexo a ordens arbitrárias pela abordagem de Riemann-Liouville e Caputo com ênfase nas séries de Fourier complexas. Elabora-se uma adaptação do formalismo bra-ket de Dirac à linguagem tensorial no espaço de Hilbert de funções com expansões finitas para uso específico na teoria de equações diferenciais e matrizes operacionais, denominado \\Formalismo Coadjunto\". Estende-se o tratamento aos operadores fracionais de Weyl para períodos genéricos a fim de determinar as matrizes operacionais de derivação e integração de ordem arbitrária na base complexa de Fourier. Introduz-se um novo método de resolução de equações diferenciais ordinárias lineares e fracionais não-homogêneas, denominado \\Modelagem Operacional\", que permite a obtenção de soluções de equações de alta ordem com grande precisão sem a necessidade de imposição de condições iniciais ou de contorno. O método apresentado é aperfeiçoado por meio de um novo tipo de expansão, que denominamos \"Séries Associadas de Fourier\", a qual apresenta convergência mais rápida que a série de Fourier original numa restrição de domínio, possibilitando soluções de EDOs e EDFs de alta ordem com maior precis~ao e ampliando a esfera de casos passíveis de resolução. / Operational matrices method is expanded to complex field and arbitrary orders by using the Riemann-Liouville and Caputo approach with emphasis on complex Fourier series. Dirac\'s bra-ket notation is associated to tensor procedures in Hilbert spaces for finite function expansions to be applied specifically to dfferential equations and operational matrices, being called \\Coadjoint Formalism\". This treatment is extended to Weyl fractional operators for generic periods in order to establish the integral and derivative operational matrices of fractional order to complex Fourier basis. A new method to solve linear non-homogeneous ODEs and FDEs, called \\Operational Modelling\"is introduced. It yields high precision solutions on high order dfferential equations without assumption of boundary or initial conditions. The presented method is improved by a new kind of function expansion, called \\Fourier Associated Series\", which yields a faster convergence than original Fourier in a restrict domain, enabling to obtain solutions of high order ODEs and FDEs with excellent precision and broadening the set of solvable equations.

Thesis advisor: Ziqiang Wang / Exotic phases and associated phase transitions in low dimensions have been a fascinating frontier and a driving force in modern condensed matter physics since the 80s. Due to strong correlation effect, they are beyond the description of mean-field theory based on a single-particle picture and Landau's symmetry-breaking theory of phase transitions. These new phases of matter require new physical quantities to characterize them and new languages to describe them. This thesis is devoted to the study on exotic phases of correlated electrons in two spatial dimensions. We present the following efforts in understanding two-dimensional exotic phases: (1) Using Zn vertex algebra, we give a complete classification and characterization of different one-component fractional quantum Hall (FQH) states, including their ground state properties and quasiparticles. (2) In terms of a non-unitary transformation, we obtain the exact form of statistical interactions between composite fermions in the lowest Landau level (LLL) with v=1/(2m), m=1,2... By studying the pairing instability of composite fermions we theoretically explains recently observed FQHE in LLL with v=1/2,1/4. (3) We classify different Z2 spin liquids (SLs) on kagome lattice in Schwinger-fermion representation using projective symmetry group (PSG). We propose one most promising candidate for the numerically discovered SL state in nearest-neighbor Heisenberg model on kagome lattice}. (4) By analyzing different Z2 spin liquids on honeycomb lattice within PSG classification, we find out the nature of the gapped SL phase in honeycomb lattice Hubbard model, labeled sublattice pairing state (SPS) in Schwinger-fermion representation. We also identify the neighboring magnetic phase of SPS as a chiral-antiferromagnetic (CAF) phase and analyze the continuous phase transition between SPS and CAF phase. For the first time we identify a SL called 0-flux state in Schwinger-boson representation with one (SPS) in Schwinger-fermion representation by a duality transformation. (5) We show that when certain non-collinear magnetic order coexists in a singlet nodal superconductor, there will be Majorana bound states in vortex cores/on the edges of the superconductor. This proposal opens a window for discovering Majorana fermions in strongly correlated electrons. (6) Motivated by recent numerical discovery of fractionalized phases in topological flat bands, we construct wavefunctions for spin-polarized fractional Chern insulators (FCI) and time reversal symmetric fractional topological insulators (FTI) by parton approach. We show that lattice symmetries give rise to different FCI/FTI states even with the same filling fraction. For the first time we construct FTI wavefunctions in the absence of spin conservation which preserve all lattice symmetries. The constructed wavefunctions also set up the framework for future variational Monte Carlo simulations. / Thesis (PhD) — Boston College, 2011. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.

The aim of this work is to study rings of algebroid Gorenstein rings. We explore more deeply the symmetry that exists among the sets of values of a fractional ideal and that of its dual and also to express the codimension of a fractional ideal in terms of the maximal points of the value set of the ideal. We apply the formulas we obtained to express the Tjurina number of a complete intersection curve in terms of invariants of its components and the maximal points of the set of values of the Kähler differentials on the curve. / O objetivo desse trabalho é o estudo dos anéis de curvas algebróides de Gorenstein. Expolramos mais aprofundadamente a simetria que existe entre os conjuntos de valores de um ideal fracionário e de seu dual e também expressar a codimensão de um ideal fracionário em função dos pontos maximais de seu conjunto de valores. Aplicamos as fórmulas obtidas para relacionar o número de Tjurina de uma curva de interseção completa com certos invariantes de suas componentes e dos pontos maximais do conjunto de valores das diferenciais de Kähler sobre a curva.

Li, Ming Hin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 42-46). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Inference for Fractional Cointegration --- p.5 / Chapter 2.1 --- Concept of Fractional Cointegration --- p.5 / Chapter 2.1.1 --- Fractional Integration --- p.5 / Chapter 2.1.2 --- Fractional Cointegration --- p.8 / Chapter 2.2 --- Fractional Cointegration Modeling --- p.9 / Chapter 2.2.1 --- Engle-Granger's Methodology --- p.9 / Chapter 2.2.2 --- Johansen's Methodology --- p.10 / Chapter 2.2.2.1 --- Maximum Likelihood Estimators --- p.12 / Chapter 2.2.2.2 --- Cofractional Rank Test --- p.16 / Chapter 3 --- Pairs Trading Strategy --- p.19 / Chapter 3.1 --- Statistical Arbitrage --- p.19 / Chapter 3.2 --- Fractional Cointegration Pairs Trading --- p.20 / Chapter 3.2.1 --- Trading Procedures --- p.22 / Chapter 4 --- Empirical Study --- p.27 / Chapter 4.1 --- Backgrounds --- p.27 / Chapter 4.2 --- Settings --- p.28 / Chapter 4.3 --- Empirical Results --- p.29 / Chapter 5 --- Conclusions and Further Research --- p.39 / Bibliography --- p.42

Ces travaux concernent deux équations paraboliques, dégénérées et non-locales. La première équation est une équation de films minces fractionnaire et la deuxième est une équation des milieux poreux fractionnaire. La présentation des problèmes, les résultats existants dans la littérature, ainsi que le résumé de nos résultats font l'objet de l'introduction. Le deuxième chapitre est consacré à la présentation de la méthode de De Giorgi utilisée pour montrer la régularité Hölder des solutions des équations elliptiques. On présente de plus les résultats utilisant cette approche dans les cas paraboliques local et non-local. Dans le troisième chapitre, on montre l'existence de solutions faibles d'une équation des films minces fractionnaire. C'est une équation parabolique, dégénérée, non-locale d'ordre $alpha+2$ où $0 < alpha < 2$. C'est une généralisation d'une équation étudiée par Imbert et Mellet en 2011 pour $alpha = 1$. Pour construire les solutions, on passe par un problème régularisé. En utilisant les injections de Sobolev, on passe à la limite pour trouver des solutions faibles. Vu la différence des injections de Sobolev, on distingue deux cas $0 <alpha < 1$ et $1 leq alpha < 2$. Dans les deux cas on démontre que la solution est positive si la condition initiale l'est. Le quatrième chapitre concerne une équation des milieux poreux fractionnaire. On montre la régularité Hölder de solutions faibles positives satisfaisant des estimées d'énergie. D'abord, on montre l'existence de solutions faibles qui satisfont des estimées d'énergie. On distingue deux cas $0 <alpha < 1$ et $1 leq alpha < 2$ à cause de problème de divergence. Puis on démontre les lemmes de De Giorgi qui sont des lemmes de réduction de l'oscillation d'en dessus et d'au-dessous. Ces deux lemmes ne suffisent pas pour montrer la régularité Hölder. On a besoin d'améliorer le résultat du lemme de réduction de l'oscillation d'en dessus. Donc, on passe par un lemme des valeurs intermédiaires et on montrer un lemme de réduction de l'oscillation d'en dessus amélioré. Enfin, on montre la régularité Hölder des solutions en utilisant la propriété scaling de ces solutions / In this thesis, we study two degenerate, non-local parabolic equations, a fractional thin film equation and a fractional porous medium equation. The introduction contains a presentation of problems, the previous results in the literature and a brief presentation of our results. In the second chapter, we present a short overview of the De Giorgi method used to prove Hölder regularity of solutions of elliptic equations. Moreover, we present the results using this approach in the local and non-local parabolic cases. In the third chapter we prove existence of weak solutions of a fractional thin film equation. It is a non-local degenerate parabolic equation of order $alpha + 2$ where $0 < alpha < 2$. It is a generalization of an equation studied by Imbert and Mellet in 2011 for $alpha = 1$. To construct these solutions, we consider a regularized problem then we pass to the limit using Sobolev embedding theorem, that's why we distinguish two cases $0 < alpha < 1$ and $1 leq alpha < 2$. We also prove that the solution is positive if the initial condition is so. The fourth chapter is dedicated for a fractional porous medium equation. We prove Hölder regularity of positive weak solutions satisfying energy estimates. First, we prove the existence of weak solutions that satisfy energy estimates. We distiguish two cases $0 < alpha < 1$ and $1 leq alpha < 2$ because of divergence problems. The we prove De Giorgi Lemmas about oscillation reduction from above and from below. This is not suffisant. We need to improve the lemma about oscillation reduction from above. So we pass by an intermediate values lemma and we prove an improved oscillation reduction lemma from above. Finally, we prove Hölder regularity of solutions using the scaling property

Dans cette thèse, nous nous sommes intéressés aux équations différentielles fractionnaires. Nous avons commencé par l'étude d'une équation différentielle fractionnaire en temps. Ensuite, nous avons étudié trois systèmes fractionnaires non linéaires ; le premier avec un Laplacien fractionnaire et les autres avec une dérivée fractionnaire en temps définie au sens de Caputo. Dans le premier chapitre, nous avons établi les propriétés qualitatives de la solution d'une équation différentielle fractionnaire en temps qui modélise l'évolution d'une certaine espèce. Plus précisément, l'existence et l'unicité de la solution globale sont démontrées pour certaines valeurs de la condition initiale. Dans ce cas, nous avons obtenu le comportement asymptotique de la solution en t^α. Sous une autre condition sur la donnée initiale, la solution explose en temps fini. Le profil de la solution et l'estimation du temps d'explosion sont établis et une confirmation numérique de ces résultats est présentée. Les chapitres 4, 5 et 6 sont consacrés à l'étude théorique de trois systèmes fractionnaires : un système de la diffusion anormale qui décrit la propagation d'une épidémie infectieuse de type SIR dans une population confinée, le Brusselator avec une dérivée fractionnaire en temps et un système fractionnaire en temps avec une loi de balance. Pour chaque système, on présente l'existence globale et le comportement asymptotique des solutions. L'existence et l'unicité de la solution locale pour les trois systèmes sont obtenues par le théorème de point fixe de Banach. Cependant, le comportement asymptotique est établi par des techniques différentes : le comportement asymptotique de la solution du premier système est démontré en se basant sur les estimations du semi-groupe et le théorème d'injection de Sobolev. Concernant le Brusselator fractionnaire, la technique utilisée s'appuie sur un argument de feedback. Finalement, un résultat de régularité maximale est utilisé pour l'étude du dernier système. / In this thesis, we are interested in fractional differential equations. We begin by studying a time fractional differential equation. Then we study three fractional nonlinear systems ; the first system contains a fractional Laplacian, while the others contain a time fractional derivative in the sense of Caputo. In the second chapter, we establish the qualitative properties of the solution of a time fractional equation which describes the evolution of certain species. The existence and uniqueness of the global solution are proved for certain values of the initial condition. In this case, the asymptotic behavior of the solution is dominated by t^α. Under another condition, the solution blows-up in a finite time. The solution profile and the blow-up time estimate are established and a numerical confirmation of these results is presented. The chapters 4, 5 and 6 are dedicated to the study of three fractional systems : an anomalous diffusion system which describes the propagation of an infectious disease in a confined population with a SIR type, the time fractional Brusselator and a time fractional reaction-diffusion system with a balance law. The study includes the global existence and the asymptotic behavior. The existence and uniqueness of the local solution for the three systems are obtained by the Banach fixed point theorem. However, the asymptotic behavior is investigated by different techniques. For the first system our results are proved using semi-group estimates and the Sobolev embedding theorem. Concerned the time fractional Brusselator, the used technique is based on an argument of feedback. Finally, a maximal regularity result is used for the last system.