This thesis investigates the set stabilization problem for systems with Lie group symmetry. Initially, we examine left-invariant systems on Lie groups where the target set is a left or right coset of a closed subgroup. We broaden the scope to systems defined on smooth manifolds that are invariant under a Lie group action. Inspired by the solution of this problem for linear time-invariant systems, we show its equivalence to an equilibrium stabilization problem for a suitable quotient control system. We provide necessary and sufficient conditions for the existence of the quotient control system and analyze various properties of such a system. This theory is applied to the formation stabilization of three kinematic unicycles, the path stabilization of a particle in a gravitational field, and the conversion and temperature control of a continuously stirred tank reactor.

This thesis investigates the set stabilization problem for systems with Lie group symmetry. Initially, we examine left-invariant systems on Lie groups where the target set is a left or right coset of a closed subgroup. We broaden the scope to systems defined on smooth manifolds that are invariant under a Lie group action. Inspired by the solution of this problem for linear time-invariant systems, we show its equivalence to an equilibrium stabilization problem for a suitable quotient control system. We provide necessary and sufficient conditions for the existence of the quotient control system and analyze various properties of such a system. This theory is applied to the formation stabilization of three kinematic unicycles, the path stabilization of a particle in a gravitational field, and the conversion and temperature control of a continuously stirred tank reactor.

Different aspects of specific models in supersymmetry as well as constraints on decaying dark matter are analysed in this thesis. In chapter 1 we give a general introduction to supersymmetry, and briefly discuss some of the concepts that are used throughout the thesis. In chapter 2 we present a version of Gauge Mediated Supersymmetry Breaking which preserves an $R$-symmetry---the gauginos are Dirac particles, the $A$-terms are zero, and there are four Higgs doublets. This offers an alternative way for gauginos to acquire mass in the supersymmetry-breaking models of Intriligator, Seiberg, and Shih \cite{Intriligator:2006dd} . Additionally, we investigate the possibility of using $R$-symmetric gauge mediation to realise the spectrum and large sfermion mixing of the model of Kribs, Poppitz, and Weiner \cite{Kribs:2007ac}. In chapter 3 we investigate the Higgs sector of the $R$-symmetric model presented in chapter 2. Furthermore, a scan of the parameter space and sample spectra are provided. Other attributes like the tuning of the model are discussed. In chapter 4 we present a complete analysis of the cosmological constraints on decaying dark matter. In order to do this, we have updated and extended previous analyses to include Lyman-$\alpha$ forest, large scale structure, and weak lensing observations. Astrophysical constraints are not considered in this thesis. The bounds on the lifetime of decaying dark matter are dominated by either the late-time integrated Sachs-Wolfe effect for the scenario with weak reionization, or CMB polarisation observations when there is significant reionization. For the respective scenarios, the lifetimes for decaying dark matter are $\Gamma^{-1} \gtrsim 100$ Gyr and $ (f \Gamma) ^{-1} \gtrsim 5.3 \times 10^8$ Gyr (at 95.4\% confidence level), where the phenomenological parameter $f$ is the fraction of the decay energy deposited in baryonic gas. This allows us to constrain particle physics models with dark matter candidates through investigation of dark matter decays into Standard Model particles via effective operators. For decaying dark matter of $\sim 100$ GeV mass, we found that the size of the coupling constant in the effective dimension-4 operators responsible for dark matter decay has to generically be $ \lesssim 10^{-22}$.

Different aspects of specific models in supersymmetry as well as constraints on decaying dark matter are analysed in this thesis. In chapter 1 we give a general introduction to supersymmetry, and briefly discuss some of the concepts that are used throughout the thesis. In chapter 2 we present a version of Gauge Mediated Supersymmetry Breaking which preserves an $R$-symmetry---the gauginos are Dirac particles, the $A$-terms are zero, and there are four Higgs doublets. This offers an alternative way for gauginos to acquire mass in the supersymmetry-breaking models of Intriligator, Seiberg, and Shih \cite{Intriligator:2006dd} . Additionally, we investigate the possibility of using $R$-symmetric gauge mediation to realise the spectrum and large sfermion mixing of the model of Kribs, Poppitz, and Weiner \cite{Kribs:2007ac}. In chapter 3 we investigate the Higgs sector of the $R$-symmetric model presented in chapter 2. Furthermore, a scan of the parameter space and sample spectra are provided. Other attributes like the tuning of the model are discussed. In chapter 4 we present a complete analysis of the cosmological constraints on decaying dark matter. In order to do this, we have updated and extended previous analyses to include Lyman-$\alpha$ forest, large scale structure, and weak lensing observations. Astrophysical constraints are not considered in this thesis. The bounds on the lifetime of decaying dark matter are dominated by either the late-time integrated Sachs-Wolfe effect for the scenario with weak reionization, or CMB polarisation observations when there is significant reionization. For the respective scenarios, the lifetimes for decaying dark matter are $\Gamma^{-1} \gtrsim 100$ Gyr and $ (f \Gamma) ^{-1} \gtrsim 5.3 \times 10^8$ Gyr (at 95.4\% confidence level), where the phenomenological parameter $f$ is the fraction of the decay energy deposited in baryonic gas. This allows us to constrain particle physics models with dark matter candidates through investigation of dark matter decays into Standard Model particles via effective operators. For decaying dark matter of $\sim 100$ GeV mass, we found that the size of the coupling constant in the effective dimension-4 operators responsible for dark matter decay has to generically be $ \lesssim 10^{-22}$.

The main topic of this thesis is nonrigid image registration for medical applications. We start with an overview and classification of existing registration techniques. We develop a general nonrigid image registration algorithm. It uses spline functions to describe the deformation and uses multi-scale strategy to search for the optimal transformation. Then we present a new registration operator that is transitive and symmetric. We investigate the theoretical implication of these properties and apply this operator to the registration of sequences of MR cardiac images. In the second part of the thesis, two methods, one 2D and one 3D, for validation of nonrigid image registration algorithms are proposed and compared to a manual validation strategy. Both methods provide pairs of deformed images as well as corresponding true displacement fields with known accuracy. Nonrigid registration algorithms can be run on the pairs of images and their outputs can be compared to the true displacement fields that were generated manually by five observers. While these phantom validation studies do not provide physically correct deformations, they are certainly a useful way to test the algorithm's ability to recover various deformation patterns.

Results of spin density functional theory (SDFT) calculations were used to construct and check features of a generally applicable semi-quantitative approach to understanding magnetic coupling in gadolinium-containing molecules, clusters, and solids. Using fragments based on structures of metal-rich lanthanide compounds, we have investigated molecular and low-dimensional extended structures, and have shown that open-d-shell clusters facilitate strong ferromagnetic coupling whereas closed-d-shell systems prefer antiferromagnetic coupling. The qualitative features can be interpreted using a perturbative molecular orbital (PMO) model that focuses the influence of the 4f 7- d exchange interaction on the d-based molecular orbitals. The f-d exchange interaction, mediated by spin polarization of both filled and partially-filled metal-metal bonding orbitals, is described for the model system Gd3I6(OPH3)12 n+ using basic perturbation methods. This approach is successful for predicting the magnetic ground state for Gd2Cl3, a semiconducting system for which calculations predict antiferromagnetic ordering of the 4f 7 moments in a pattern consistent with published neutron diffraction data. An attempt to account for the calculated magnetic energies of spin patterns using an Ising model was unsuccessful, indicating that the Ising model is inappropriate. Instead, the d-electron mediated f-f exchange interaction was interpreted using our basic perturbation theory approach. Computed density of states and spin polarization information was used to support the perturbation-theoretic analysis. This method has also been successful evaluating the ground state for Gd[Gd6FeI12]. Using the model [Gd6CoI12](OPH3)6, which has three unpaired electrons in the HOMO, the 4f moments prefer spin alignment with the unpaired electrons in the system and the ferromagnetic 4f 7 spin arrangement is the ground state. We have extended our analysis of R6X12 clusters to include nonmetal interstitial atoms, the bioctahedral cluster compounds Gd10Cl17C4 and Gd10I16C4, and Gd5(O)(OPri)5. Finally, we have shown that we can successfully predict the ground state magnetic structures of several metallic and semiconducting Gd-containing compounds, Gd2Cl3, GdB2C2,alpha-Gd2S3, Gd5Si4, and Gd5Ge4, using semi-empirical calculations which closely simulates the exchange effects exerted by the 4f electrons. In a more speculative vein, ideas concerning the incorporation of anisotropic rare-earth metal atoms to the cluster framework are touched upon.

The symmetry energy is the energy difference between symmetric nuclear matter and pure neutron matter at a given density. Around normal nuclear density, i.e. p/p0 =1, and temperature, i.e. T = 0, the symmetry energy is approximately 23.5 MeV/nucleon for finite nuclear matter and 30 MeV/nucleon for infinite nuclear matter, but at other densities, the symmetry energies are very poorly understood. Since the symmetry energy is very important in understanding many aspects of heavy ion reactions, structure, and nuclear astrophysics, many different models have been developed and some predications of the density dependence of symmetry energy have been made. Intermediate energy heavy ion collisions provide a unique tool to probe the nuclear equation of state. The initial compression and the thermal shock in Fermi- Energy heavy ion collisions lead naturally to the production of nucleonic matter at varying temperatures and densities which are interesting in this context. Since the light particle emission during this stage witnesses each stage of the reaction, it carries essential information on the early dynamics and on the degree of equilibration at each stage of the reaction. The kinematic features and yields of emitted light particles and clusters in the invairant velocity frame have been exploited to probe the nature of the intermediate system and information on the Equation Of State (EOS) with emphasis on the properties of the low density participant matter produced in such collisions. In order to pursue this effort and broaden the density range over which the symmetry energies are experimentally determined we have now carried out a series of experiments in which the reactions of 112Sn and 124Sn with projectiles, ranging from 4He,10B, 20Ne, 40Ar to 64Zn, all at the same energy per nucleon, 47 Mev/u, were performed. In this series of experiments different collision systems should lead to different average densities. By careful comparisons of the yields, spectra and angular distributions observed for particle emission from these different systems we attempted to cleanly separate early emission resulting from nucleon-nucleon collisions from that resulting from evaporation from the thermalized system and obtain a much cleaner picture of the dynamic evolution of the hotter systems. The Albergo Model has been used to calculate the density and temperature, symmetry free energies with the isoscaling technique for systems with different N/Z ratios. Those are compared with Roepke Model results. Also other models like VEOS, Lattimer, and Shen-Toki have been added to calculate the alpha mass fraction in order to understand the properties of low density matter further.

Thesis (Ph.D.)--University of Wollongong, 2009. / Typescript. Includes bibliographical references: p. 267-287.

A novel holographic model of chiral symmetry breaking has been proposed by Kuperstein and Sonnenschein by embedding non-supersymmetric probe D7 and anti-D7 branes in the Klebanov-Witten background. We study the dynamics of the probe flavours in this model in the presence of finite temperature and a constant electromagnetic field. In keeping with the weakly coupled field theory intuition, we find the magnetic field promotes spontaneous breaking of chiral symmetry whereas the electric field restores it. The former effect is universally known as the ``magnetic catalysis" in chiral symmetry breaking. In the presence of an electric field such a condensation is inhibited and a current flows. Thus we are faced with a steady-state situation rather than a system in equilibrium. We conjecture a definition of thermodynamic free energy for this steady-state phase and using this proposal we study the detailed phase structure when both electric and magnetic fields are present in two representative configurations: mutually perpendicular and parallel. / text

We consider the pair of a smooth complex projective variety together with an anti-canonical simple normal crossing divisor (we call it "log Calabi- Yau"). Standard examples are toric varieties together with their toric boundaries (we call them "toric pairs"). We provide a numerical criterion for a general log Calabi-Yau to be toric by an inequality between its dimension, Picard number and the number of boundary components. The problem originates in birational geometry and our proof is constructive, motivated by mirror symmetry. / text