by Yeung Chung Kuen. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1997. / Includes bibliographical references (leaves 104-105). / Chapter 1 --- Infinitesimal Deformation of Compact Complex Manifolds --- p.3 / Chapter 1.1 --- Differentiable Family --- p.3 / Chapter 1.2 --- Infinitesimal Deformation in Differentiable Family --- p.6 / Chapter 1.3 --- Trivial Differentiable Family --- p.8 / Chapter 1.4 --- Complex Analytic Family --- p.13 / Chapter 1.5 --- Induced Family --- p.19 / Chapter 2 --- Theorem of Existence --- p.22 / Chapter 2.1 --- Introduction --- p.22 / Chapter 2.2 --- "Some Facts on the qth Cohomology Group Hq(M,´ة）" --- p.23 / Chapter 2.3 --- Obstructions to Deformation --- p.24 / Chapter 2.4 --- An Elementary Method for Theorem of Existence --- p.26 / Chapter 2.5 --- Proof of Theorem of Existence --- p.35 / Chapter 3 --- "Comparison between the Number of Moduli m(M) and dim H1 (M,´ة）" --- p.64 / Chapter 3.1 --- Number of Moduli of Compact Complex Manifold --- p.64 / Chapter 3.2 --- Examples --- p.68 / Chapter 4 --- Theorem of Completeness --- p.84 / Chapter 4.1 --- Theorem of Completeness --- p.84 / Chapter 4.2 --- Construction of Formal Power Series of h and g --- p.86 / Chapter 4.3 --- Proof of Convergence --- p.93

by Ip Che-ho. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1997. / Includes bibliographical references (leaf 136). / Covering Page --- p.i / Acknowledgement --- p.ii / Abstract --- p.iii / Table of Content --- p.v / Chapter 1. --- Fundamentals of Complex Analys --- p.is / Chapter 1.1 --- The extended complex plane --- p.1 / Chapter 1.2 --- Stereographic projection --- p.2 / Chapter 1.3 --- Analytic functions --- p.3 / Chapter 1.4 --- Rational functions --- p.5 / Chapter 1.5 --- Mobius transformation --- p.6 / Chapter 2. --- The Topology of the Extended Plane / Chapter 2.1 --- The topology of S2 and C ∞ --- p.9 / Chapter 2.2 --- Smooth map and manifolds --- p.10 / Chapter 2.3 --- Regular points --- p.11 / Chapter 2.4 --- Degree of maps --- p.13 / Chapter 2.5 --- Euler characteristics --- p.14 / Chapter 2.6 --- Covering space --- p.16 / Chapter 2.7 --- Riemann-Hurwritz formula --- p.17 / Chapter 3 --- The Montel Theorem / Chapter 3.1 --- Introduction --- p.21 / Chapter 3.2 --- Normality and Equicontinuous --- p.21 / Chapter 3.3 --- Local boundedness --- p.23 / Chapter 3.4 --- Covering and uniformization --- p.26 / Chapter 3.5 --- Montel's theorem --- p.28 / Chapter 4 --- Fatou Set and Julia Set / Chapter 4.1 --- Iteration of functions --- p.31 / Chapter 4.2 --- Fatou set and Julia set --- p.35 / Chapter 4.3 --- Iteration of Mobius transformtion --- p.39 / Chapter 4.4 --- Fixed points and their classification --- p.44 / Chapter 4.5 --- Periodic points and cycles --- p.45 / Chapter 4.6 --- Critical points --- p.47 / Chapter 4.7 --- Dlustractions of local behaviour of map near periodic points --- p.48 / Chapter 5 --- More about Julia Set / Chapter 5.1 --- Some examples of Julia set --- p.57 / Chapter 5.2 --- Completely invariant set --- p.58 / Chapter 5.3 --- Exceptional set --- p.61 / Chapter 5.4 --- Properties of Julia set --- p.63 / Chapter 5.5 --- Forward and backward convergence of sets --- p.66 / Chapter 6 --- More about Fatou Set / Chapter 6.1 --- Components of Fatou set --- p.97 / Chapter 6.2 --- Simply connected Fatou components --- p.98 / Chapter 6.3 --- Number of components in Fatou set --- p.100 / Chapter 6.4 --- Classification of forward invariant components of the Fatou set --- p.102 / Chapter 6.5 --- Examples illustrating the five possible forward invariant components --- p.104 / Chapter 7 --- Critical Points / Chapter 7.1 --- Introduction --- p.108 / Chapter 7.2 --- Some interesting results --- p.110 / Chapter 7.3 --- The Fatou set of polynomials --- p.114 / Chapter 7.4 --- Quadratic polynomial and Mandelbrot set --- p.116 / Appendix --- p.125 / Reference --- p.136

by Roger Chun Wang Liu. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1996. / Includes bibliographical references (leaves 98-108). / ACKNOWLEDGEMENTS --- p.i / CONTENTS --- p.ii / ABBREVIATIONS --- p.v / LIST OF FIGURES --- p.vi / LIST OF TABLES --- p.vii / Chapter I. --- THE CHEMISTRY OF METALLO-PHTHALOCYANINES AND -NAPHTHALOCYANINES --- p.1 / ABSTRACT --- p.2 / Chapter 1. --- SANDWICH-LIKE BIS(PHTHALOCYANINATO)LANTHANIDE COMPLEXES / Chapter 1.1. --- Introduction --- p.3 / Chapter 1.2. --- Preparation of Substituted Phthalonitriles --- p.8 / Chapter 1.3. --- Condensation of Phthalonitrile --- p.9 / Chapter 1.4. --- Condensation of Substituted Phthalonitriles --- p.10 / Chapter 1.5. --- Spectroscopic and Electrochemical Properties --- p.12 / Chapter 1.6. --- Conclusion --- p.21 / Chapter 2. --- "SUBSTITUTED 2,3-NAPHTHALOCYANINES" / Chapter 2.1. --- Introduction --- p.22 / Chapter 2.2. --- Preparation of Alkyl-Substituted Dicyanonaphthalenes --- p.26 / Chapter 2.3. --- Preparation of Halo-Substituted Dicyanonaphthalenes --- p.30 / Chapter 2.4. --- Condensation of Alkyl-Substituted Dicyanonaphthalenes --- p.30 / Chapter 2.5. --- Condensation of Halo-Substituted Dicyanonaphthalenes --- p.35 / Chapter 2.6. --- Conclusion --- p.38 / Chapter 3. --- EXPERIMENTAL SECTION / Chapter 3.1. --- General Directions --- p.39 / Chapter 3.2. --- Preparation of Substituted Phthalonitriles --- p.40 / Chapter 3.3. --- Condensation of Phthalonitrile --- p.43 / Chapter 3.4. --- Condensation of Substituted Phthalonitriles --- p.44 / Chapter 3.5. --- Preparation of Alkyl-Substituted Dicyanonaphthalenes --- p.46 / Chapter 3.6. --- Preparation of Halo-Substituted Dicyanonaphthalenes --- p.49 / Chapter 3.7. --- Condensation of Alkyl-Substituted Dicyanonaphthalenes --- p.51 / Chapter 3.8. --- Condensation of Halo-Substituted Dicyanonaphthalenes --- p.52 / Chapter II. --- "SYNTHETIC STUDIES OF MIXED AZA-, OXA-, AND THIA-CROWN ETHERS" --- p.55 / ABSTRACT --- p.56 / Chapter 1. --- INTRODUCTION --- p.57 / Chapter 2. --- RESULTS AND DISCUSSION --- p.64 / Chapter 2.1. --- Preparation of Diols and Dithiols --- p.64 / Chapter 2.2. --- Preparation of Ditosylates --- p.66 / Chapter 2.3. --- 1:1 Cyclization -- Preparation of Monoaza- 15-crown-5 --- p.68 / Chapter 2.4. --- Crystal Structure of N-(4-methoxyphenyl) benzomonoaza-15- crown-5 (112) --- p.76 / Chapter 2.5. --- Complexation of Monoaza-15-crown-5 --- p.79 / Chapter 2.6. --- Conclusion --- p.80 / Chapter 3. --- MISCELLANEOUS SYNTHESES --- p.81 / Chapter 3.1. --- Preparation of Tetrabromodibenzo-24-crown-8 --- p.81 / Chapter 3.2. --- Complexation of Tetrabromodibenzo-24-crown-8 --- p.82 / Chapter 4. --- EXPERIMENTAL SECTION --- p.84 / Chapter 4.1. --- General Directions --- p.84 / Chapter 4.2. --- Preparation of Diols and Dithiols --- p.84 / Chapter 4.3. --- Preparation of Ditosylates --- p.88 / Chapter 4.4. --- 1:1 Cyclization -- Preparation of Monoaza- 15-crown-5 --- p.89 / Chapter 4.5. --- Complexation of Monoaza-15-crown-5 --- p.94 / Chapter 4.6. --- Preparation of Tetrabromodibenzo-24-crown-8 --- p.95 / Chapter 4.7. --- Complexation of Tetrabromodibenzo-24-crown-8 --- p.97 / REFERENCES AND NOTES --- p.98 / APPENDIX --- p.109

Or Ming-keung Ben. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (leaves 75-78). / Abstracts in English and Chinese. / Chapter 0 --- Introduction --- p.3 / Chapter 1 --- CR manifolds --- p.6 / Chapter 1.1 --- Introduction to CR manifolds --- p.6 / Chapter 1.2 --- CR functions --- p.11 / Chapter 1.3 --- CR maps and imbedding of CR manifolds --- p.15 / Chapter 1.4 --- Non-degenerate CR structures --- p.19 / Chapter 1.5 --- CR structures by means of differential forms --- p.21 / Chapter 2 --- Segre Family --- p.25 / Chapter 2.1 --- The Segre family associated to a real analytic hyper- surface --- p.25 / Chapter 2.2 --- G-structures on Segre family --- p.30 / Chapter 2.3 --- Local Computations --- p.37 / Chapter 3 --- Projective Structure --- p.41 / Chapter 3.1 --- Construction of the frame bundle over Segre family 。 --- p.41 / Chapter 3.2 --- The associated Cartan Connection --- p.45 / Chapter 3.3 --- Formulation in terms of Projective Connection --- p.54 / Chapter 4 --- Riemann Mapping Theorem --- p.57 / Chapter 4.1 --- Preliminary --- p.57 / Chapter 4.2 --- Generalizations of Poincare's theorem --- p.59 / Chapter 4.3 --- Local G-stucture on the space of hyperplane elements --- p.62 / Chapter 4.4 --- Extension of induced G-structure --- p.66 / Chapter 4.5 --- Proof of Theorem B --- p.70 / Chapter 4.6 --- Domains with continuous boundary --- p.72 / Bibliography --- p.75

a series of complexes of the type MLnX where M=Cu(II), Ni(II), and Cr(III), L=β-diketonate (n=1 for Cu (II) and Ni(II), n=2 for Cr(III)) and X=bridging anion was synthesized in order to study the effect of the bridging group on the magnetic exchange interaction parameter, J.

Many microorganisms have the ability to propel themselves through their fluid environments by periodically actuating their body. The biological fluid environments surrounding these microswimmers are typically complex fluids containing many high-molecular weight protein molecules, which give the fluid non-Newtonian rheological properties. In this thesis, we investigate the effect that one such rheological property, viscoelasticity, has on microswimming. We consider a classical model of a microswimmer, the so-called Taylor's waving sheet and generalise it to arbitrary shapes. We employ the Oldroyd-B model to study its swimming analytically and numerically. We attempt to develop a mechanistic understanding of the swimmer's behaviour in viscoelastic fluids. It has recently been suggested that continuum models of complex biological fluids might not be appropriate for studying the swimming of flagellated microorganisms as the size of biological macromolecules is comparable to the typical width of a microorganism's flagellum. A part of this thesis is devoted to exploring this scenario. We propose an alternative method for modelling complex fluids using a two-fluid depletion region model and we have developed a numerical solver to find the swimming speed and rate of work for the generalised Taylor's waving sheet model swimmer using this alternate depletion region model. This thesis is organised as follows. In the first chapter, we outline a physical mechanism for the slowing down of Taylor's sheet in an Oldroyd-B fluid as the Deborah number increases. We demonstrate how a microswimmer can be designed to avoid this. In the second chapter, we investigate swimming in an Oldroyd-B fluid near a solid boundary and show that, at large amplitudes and low polymer concentrations, the swimming speed of Taylor's sheet increases with De. In the third chapter, we show how the Oldroyd-B model can be adapted using depletion regions. In the final chapter, we investigate optimal swimming in a Newtonian fluid. We show that while the organism's energetics are important, the kinematics of planar-wave microswimmers do not optimise the hydrodynamic 'efficiency' typically used for mathematical optimisation in the literature.

The TAK1 and canonical IKK complexes are the two master protein kinases of the innate immune system that control the production of inflammatory mediators, but the mechanisms by which they are activated in this system are still unclear. In this thesis, I present the research I have carried out to solve these problems. The IKKb component of the canonical IKK complex is required to activate the transcription factors NF-kB and IRF5 and the protein kinase Tpl2, but how IKKβ itself is activated in vivo is still unclear. It was found to require phosphorylation by one or more ‘upstream’ protein kinases in some reports, but by autophosphorylation in others. In the first part of this thesis, I describe my work that has resolved this controversy by demonstrating that the activation of IKKb induced by IL-1 (interleukin-1) or TNF (tumour necrosis factor) in embryonic fibroblasts, or by ligands that activate Toll-like receptors in macrophages, requires two distinct phosphorylation events: first, the TAK1 catalysed phosphorylation of Ser177 and, secondly, the IKKb-catalysed autophosphorylation of Ser181. The phosphorylation of Ser177 by TAK1 is a priming event required for the subsequent autophosphorylation of Ser181, which enables IKKk to phosphorylate exogenous substrates. I also present genetic evidence which indicates that the IL-1-stimulated, LUBAC (linear ubiquitin chain assembly complex)-catalysed formation of Met1-linked/linear ubiquitin (Met1-Ub) chains and their interaction with the NEMO (NF-kB essential modulator) component of the canonical IKK complex permits the TAK1-catalysed priming phosphorylation of IKKb at Ser177 and IKKa at Ser176. These findings may be of general significance for the activation of other protein kinases. The activation of the TAK1 complex by inflammatory stimuli is thought to be triggered by the binding of Lys63-linked ubiquitin chains to the TAB2 or TAB3 components of the TAB1-TAK1-TAB2 and TAB1-TAK1-TAB3 complexes. In the second part of the thesis I tested whether this broadly accepted model was correct by knocking out the genes encoding TAK1 and its regulatory subunits TAB1, TAB2 and TAB3 by CRISPR/Cas9 gene-editing technology, alone and in combination, in an IL-1 receptor expressing human cell line. These genetic studies led me to discover that the IL-1-dependent activation of TAK1 occurs by two different mechanisms. The first, involves the previously described interaction of Lys63-linked ubiquitin chains with TAB2 and TAB3, while the second can take place in the complete absence of TAB2 and TAB3. The second mechanism, which involves activation of the TAB1-TAK1 heterodimer is more transient than the first, but is sufficient for the IL-1-dependent transcription of immediate early genes (A20, IkBa). I show that the activation of the TAB1-TAK1 complex requires the expression of the E3 ubiquitin ligase TRAF6 and the TRAF6-generated formation of Lys63-linked ubiquitin chains, which leads to the phosphorylation of TAK1 at Thr187 and activation. However, neither TAB1 nor TAK1 bind directly to Lys63-linked ubiquitin chains. I identify one novel IL-1-dependent phosphorylation site on TAB1 and two on TAK1 and propose that Lys63-linked ubiquitin chains activate an as yet unidentified protein kinase, which phosphorylates one or more of the novel phosphorylation sites on the TAB1-TAK1 heterodimer inducing a conformational change that permits TAK1 to autophosphorylate Thr187.

In this thesis, we put restrictions on the coefficients of polynomials and give bounds concerning the number of zeros in a specific region. Our results generalize a number of previously known theorems, as well as implying many new corollaries with hypotheses concerning monotonicity of the modulus, real, as well as real and imaginary parts of the coefficients separately. We worked with Enestr\"{o}m-Kakeya type hypotheses, yet we were only concerned with the number of zeros of the polynomial. We considered putting the same type of restrictions on the coefficients of three different types of polynomials: polynomials with a monotonicity``flip" at some index $k$, polynomials split into a monotonicity condition on the even and odd coefficients independently, and ${\cal P}_{n,\mu}$ polynomials that have a gap in between the leading coefficient and the proceeding coefficient, namely the $\mu^{\mbox{th}}$ coefficient.

In this thesis, we explore the effect of restricting the coefficients of polynomials on the bounds for the number of zeros in a given region. The results presented herein build on a body of work, culminating in the generalization of bounds among three classes of polynomials. The hypotheses of monotonicity on each class of polynomials were further subdivided into sections concerning r reversals among the moduli, real parts, and both real and imaginary parts of the coefficients.

Introduction: The nasal septum, a component of the chondrocranium, acts as a growth center that may have a morphogenetic influence on adjacent intramembranous-derived structures of the nasofacial complex. Recent evidence has demonstrated that morphological variation in the nasofacial complex is potentially due to early developmental variation in chondrocranial-derived nasal structures. There are likely both local and systemic factors that affect inter-population nasal variation. If the morphology of the nasal complex is driven, at least in part, by the morphogenetic effects of cartilage during ontogeny, then selection for altered nasal morphology under different climatic conditions is potentially achieved via developmental changes in chondrocranial-derived structures. This suggests that genes influencing the development of cartilage-derived structures may be the targets of climate-mediated selection. The purpose of this study is to further examine the potential influence of variation in chondrocranial-derived structures on gross nasal morphology by utilizing a candidate gene approach to assess phenotype/genotype associations in the nasal complex. Materials and methods: Using cone beam computed tomography scans (CBCT), we collected a series of k=44 landmarks representing different cartilaginous and osseous nasal components from an adult sample (n = 120). A group of 69 loci from 22 genes were selected that have been previously found to have an association to cartilage development or variation in the nasal complex in humans and animal models. Centroid size of coordinate landmark configurations were used to quantify nasal complex size. A principle components analysis was used to quantify nasal complex shape. Phenotypes were characterized using the symmetric component of variation. Subjects were categorized by genotype for each SNP (i.e., AA, AB, BB) analyzed, and significant differences in PC scores were tested using ANOVA. Results: There were no significant associations between nasal complex size and genotype for any of the SNPs analyzed. Phenotype/genotype relationships were assessed for the first four PCs, which accounted for 47.89% of the total variation in the sample. Significant associations between individual PC scores and genotypes were found. Conclusion: Our results indicate that nasal complex variation is associated with a number of genes that have been previously linked to skeletal tissue development and facial morphogenesis.