This dissertation examines the topological properties of simplicial complexes that arise from two distinct combinatorial objects. In 2003, A. Björner and M. de Longueville proved that the neighborhood complex of the stable Kneser graph SGn,k is homotopy equivalent to a k-sphere. Further, for n = 2 they showed that the neighborhood complex deformation retracts to a subcomplex isomorphic to the associahedron. They went on to ask whether or not, for all n and k, the neighborhood complex of SGn,k contains as a deformation retract the boundary complex of a simplicial polytope. Part one of this dissertation provides a positive answer to this question in the case k = 2. In this case it is also shown that, after partially subdividing the neighborhood complex, the resulting complex deformation retracts onto a subcomplex arising as a polyhedral boundary sphere that is invariant under the action induced by the automorphism group of SGn,2. Part two of this dissertation studies simplicial complexes that arise from non-attacking rook placements on a subclass of Ferrers boards that have ai rows of length i where ai > 0 and i ≤ n for some positive integer n. In particular, enumerative properties of their facets, homotopy type, and homology are investigated.

The Kaapvaal Craton was the scene of two major magmatic events around 2.0 Ga, namely the Bushveld Complex and the Phalaborwa Complex. Both complexes have been the subject of numerous palaeomagnetic studies during and prior to the 1980s. Despite these studies, systematic inconstancies for emplacement ages, in particular for the Bushveld Complex, have been found between the palaeomagnetic findings and well constrained ages. The greatest concern with the Bushveld Complex results are the large spread in pole positions previously determined for the different zones. This has been interpreted in prior studies to indicate that the Bushveld Complex was emplaced and cooled below the Curie temperature of magnetite over a time span of 50 my. The results obtained previously for the Phalaborwa Complex appear to be out of position (~16°) with respect to those for the Bushveld Complex. This is of concern because new geochronological data show that the Rustenburg Layered Suite of the Bushveld Complex was emplaced approximately 1 my after the Phalaborwa Complex. These inconsistencies have prompted the current re-investigation of the palaeomagnetic results for both the Bushveld Complex and the Phalaborwa Complex. New palaeomagnetic data collected from all zones of the Rustenburg Layered Suite from the Eastern, Northern and Western Lobes of the Bushveld Complex, yielded palaeomagnetic poles that eliminated the spread in the apparent polar wander path. This observation is in agreement with precise age data, constraining the time period of emplacement of the complex to ~ 6 my. Resulting beddingcorrected high blocking components from all zones produced better groupings, thereby supporting a primary magnetic signature and indicating that the complex was intruded in a near-horizontal position. Dual polarities identified within each zone of the complex and positive reversal tests have identified one of the oldest known reversals of the Earth’s magnetic field. iii Palaeomagnetic data from the Phalaborwa Complex have produced a pole position that is in close proximity to those obtained from the coeval Bushveld Complex. In an attempt to achieve a better understanding of tectonic events occurring in the Kaapvaal Craton a number of dual polarity dykes within the Bushvled and Phalaborwa Complexes were palaeomagnetic analysed. Results revealed that the acquired pole positions are in agreement with ~1.9 Ga dykes, indicating the possibility that the dykes occurring in both complexes are part of the same magmatic event. Palaeopoles generated during this study were used in refining the Kaapvaal Craton apparent polar wander path around 2.0 Ga, and in conjunction with other welldefined 2.0 Ga poles for the Kaapvaal Craton, a robust cratonic pole was produced that was used in Precambrian palaeographic reconstructions with emphasis on the postulated Vaalbara continent and the Columbia supercontinent. Palaeomagnetic reconstruction derived in this study has cast doubt on the existence of the Vaalbara continent at 2.0 Ga. Although, some support is given to the existence of the Columbia supercontinent at the same period.

This dissertation concerns modeling two aspects of dynamics of complex networks: (1) response dynamics and (2) growth and formation. A particularly challenging class of networks are ones in which both nodes and links are evolving over time – the most prominent example is a financial network. In the first part of the dissertation we present a model for the response dynamics in networks near a metastable point. We start with a Landau-Ginzburg approach and show that the most general lowest order Lagrangians for dynamical weighted networks can be used to derive conditions for stability under external shocks. Using a closely related model, which is easier to solve numerically, we propose a powerful and intuitive set of equations for response dynamics of financial networks. We find the stability conditions of the model and find two phases: “calm” phase , in which changes are sub-exponential and where the system moves to a new, close-by equilibrium; “frantic” phase, where changes are exponential, with negative blows resulting in crashes and positive ones leading to formation of "bubbles". We empirically verify these claims by analyzing data from Eurozone crisis of 2009-2012 and stock markets. We show that the model correctly identifies the time-line of the Eurozone crisis, and in the stock market data it correctly reproduces the auto-correlations and phases observed in the data. The second half of the dissertation addresses the following question: Do networks that form due to local interactions (local in real space, or in an abstract parameter space) have characteristics different from networks formed of random or non-local interactions? Using interacting fields obeying Fokker-Planck equations we show that many network characteristics such as degree distribution, degree-degree correlation and clustering can either be derived analytically or there are analytical bounds on their behaviour. In particular, we derive recursive equations for all powers of the ensemble average of the adjacency matrix. We analyze a few real world networks and show that some networks that seem to form from local interactions indeed have characteristics almost identical to simulations based on our model, in contrast with many other networks.

The transition metal complexes cyclopentadienylrhenium tricarbonyl [CpRe(CO)3, Cp = cyclopentadienyl] and (isopropylcyclopentadienyl)rhenium tricarbonyl [(i-PrCp)Re(CO)3, i-Pr = isopropyl] were photolysed in alkanes at low temperature and the resulting alkane complexes, of the general formula Cp'Re(CO)2(alkane) (Cp' = Cp or (i-PrCp)), were studied using NMR spectroscopy. Characteristic proton chemical shifts (δ) and couplings (3JHH) were observed for alkane complexes of several linear, branched and cyclic alkanes of up to eight carbons. Alkanes with chemically distinct methyl (CH3) and/or methylene (CH2) units were observed alternatively binding through each unit to rhenium. No bound methine unit was observed. Large C-H coupling constants (1JCH) were observed for protons of several bound CH3 and CH2 units, indicating the bound C-H is intact. These species are, thus, alkane sigma (σ) complexes, wherein the alkane has an agostic (M-H-C, 3 centre 2 electron) interaction with the rhenium centre. The CH3 binding mode of (i-PrCp)Re(CO)2(1-pentane) was elucidated; sequential deuteration in the bound CH3 revealed an equilibrium isotope effect (EIE) in the remaining proton/s, confirming that only one C-H has an agostic interaction with rhenium at any instant . NMR parameters δ(1H) (-8.22), δ(13C) ( 42.4) and 1JCH (85 Hz) for the complexed C-H reveal it is unequivocally intact and yet strongly interacting with the rhenium centre, hallmarks for the agostic interaction. Intramolecular exchange was identified between pentane complex isomers Cp'Re(CO)2(1-pentane), Cp'Re(CO)2(2-pentane) and Cp'Re(CO)2(3-pentane). Equilibrium constants were determined, revealing a preference for CH2 binding over CH3. The inequivalent hydrogens found in methylene groups of cyclohexane at low temperature permitted simultaneous observation of axial and equatorial C-H protons of a bound CH2 in CpRe(CO)2(cyclohexane); an EIE, upon deuteration, indicated rapid exchange between complexed C-H bonds in the bound CH2 unit. The rhenium centre was found to prefer complexation of the axial C-H bond, over the equatorial, with K ~2.9. Intermolecular exchange of alkane ligands with free solvent was directly observed, in the competitive complexation of the [CpRe(CO)2] fragment to different alkanes in binary mixtures. The preference cyclohexane > cyclopentane > pentane > isobutane was established and equilibrium constants determined. The kinetics were followed by NMR and modelled, revealing rate constants; decay rates were also determined.

The classical Lu Qi-keng Conjecture asks whether the Bergman kernel function for every domain is zero free. The answer is no, and several counterexamples exist in the literature. However, the more general Lu Qi-keng Problem, that of determining which domains in Cn have vanishing kernels, remains a difficult open problem in several complex variables. A challenge in studying the Lu Qi-keng Problem is that concrete formulas for kernels are generally difficult or impossible to compute. Our primary focus is on developing methods of computing concrete formulas in order to study the Lu Qi-keng Problem. The kernel for the annulus was historically the first counterexample to the Lu Qi-keng Conjecture. We locate the zeros of the kernel for the annulus more precisely than previous authors. We develop a theory giving a formula for the weighted kernel on a general planar domain with weight the modulus squared of a meromorphic function. A consequence of this theory is a technique for computing explicit, closed-form formulas for such kernels where the weight is associated to a meromorphic kernel with a finite number of zeros on the domain. For kernels associated to meromorphic functions with an arbitrary number of zeros on the domain, we obtain a weighted version of the classical Ramadanov's Theorem which says that for a sequence of nested bounded domains exhausting a limiting domain, the sequence of associated kernels converges to the kernel associated to the limiting domain. The relationship between the zeros of the weighted kernels and the zeros of the corresponding unweighted kernels is investigated, and since these weighted kernels are related to unweighted kernels in C^2, this investigation contributes to the study of the Lu Qi-keng Problem. This theory provides a much easier technique for computing certain weighted kernels than classical techniques and provides a unifying explanation of many previously known kernel formulas. We also present and explore a generalization of the Lu Qi-keng Problem.

No description available.

The transition metal complexes cyclopentadienylrhenium tricarbonyl [CpRe(CO)3, Cp = cyclopentadienyl] and (isopropylcyclopentadienyl)rhenium tricarbonyl [(i-PrCp)Re(CO)3, i-Pr = isopropyl] were photolysed in alkanes at low temperature and the resulting alkane complexes, of the general formula Cp'Re(CO)2(alkane) (Cp' = Cp or (i-PrCp)), were studied using NMR spectroscopy. Characteristic proton chemical shifts (δ) and couplings (3JHH) were observed for alkane complexes of several linear, branched and cyclic alkanes of up to eight carbons. Alkanes with chemically distinct methyl (CH3) and/or methylene (CH2) units were observed alternatively binding through each unit to rhenium. No bound methine unit was observed. Large C-H coupling constants (1JCH) were observed for protons of several bound CH3 and CH2 units, indicating the bound C-H is intact. These species are, thus, alkane sigma (σ) complexes, wherein the alkane has an agostic (M-H-C, 3 centre 2 electron) interaction with the rhenium centre. The CH3 binding mode of (i-PrCp)Re(CO)2(1-pentane) was elucidated; sequential deuteration in the bound CH3 revealed an equilibrium isotope effect (EIE) in the remaining proton/s, confirming that only one C-H has an agostic interaction with rhenium at any instant . NMR parameters δ(1H) (-8.22), δ(13C) ( 42.4) and 1JCH (85 Hz) for the complexed C-H reveal it is unequivocally intact and yet strongly interacting with the rhenium centre, hallmarks for the agostic interaction. Intramolecular exchange was identified between pentane complex isomers Cp'Re(CO)2(1-pentane), Cp'Re(CO)2(2-pentane) and Cp'Re(CO)2(3-pentane). Equilibrium constants were determined, revealing a preference for CH2 binding over CH3. The inequivalent hydrogens found in methylene groups of cyclohexane at low temperature permitted simultaneous observation of axial and equatorial C-H protons of a bound CH2 in CpRe(CO)2(cyclohexane); an EIE, upon deuteration, indicated rapid exchange between complexed C-H bonds in the bound CH2 unit. The rhenium centre was found to prefer complexation of the axial C-H bond, over the equatorial, with K ~2.9. Intermolecular exchange of alkane ligands with free solvent was directly observed, in the competitive complexation of the [CpRe(CO)2] fragment to different alkanes in binary mixtures. The preference cyclohexane > cyclopentane > pentane > isobutane was established and equilibrium constants determined. The kinetics were followed by NMR and modelled, revealing rate constants; decay rates were also determined.

Thesis (Ph. D.)--University of Hong Kong, 1990.

No description available.

No description available.