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In this thesis we investigate the existence and stability of periodic solutions of Hamiltonian systems with a discrete symmetry. The global existence of periodic motions can be proven using the classical techniques of the calculus of variations; our particular interest is in how the stability type of the solutions thus obtained can be determined analytically using solely the variational problem and the symmetries of the system -- we make no use of numerical or perturbation techniques. Instead, we use a method introduced in [41] in the context of a special case of the three-body problem. Using techniques from symplectic geometry, and specifically the Maslov index for curves of Lagrangian subspaces along the minimizing trajectories, we verify conditions which preclude the existence of eigenvalues of the monodromy matrix on the unit circle. We study the applicability of this method in two specific cases. Firstly, we consider another special case from celestial mechanics: the hip-hop solutions of the 2N-body problem. This is a family of Z_2-symmetric, periodic orbits which arise as collision-free minimizers of the Lagrangian action on a space of symmetric loops [14, 53]. Following a symplectic reduction, it is shown that the hip-hop solutions are brake orbits which are generically hyperbolic on the reduced energy-momentum surface. Secondly we consider a class of natural Hamiltonian systems of two degrees of freedom with a homogeneous potential function. The associated action functional is unbounded above and below on the function space of symmetric curves, but saddle points can be located by minimization subject to a certain natural constraint of a type first considered by Nehari [37, 38]. Using the direct method of the calculus of variations, we prove the existence of symmetric solutions of both prescribed period and prescribed energy. In the latter case, we employ a variational principle of van Groesen [55] based upon a modification of the Jacobi functional, which has not been widely used in the literature. We then demonstrate that the (constrained) minimizers are again hyperbolic brake orbits; this is the first time the method has been applied to solutions which are not globally minimizing. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2013-09-25 10:47:53.257

This thesis presents electronic structure calculations on problems related to the bonding in inorganic coordination compounds and clusters. A wide range of molecules with the ability to exist in different structural forms or electronic states has been selected and density functional theory is systematically applied in order to gain detailed insight into their characteristics and reactivity at the electronic level. First, we address the question of redox non-innocent behaviour of bipyridine in a series of 1st row transition metal complexes. Complexes of the type [M(2,2'-bipyridine)(mes)₂]⁰ (M = Cr, Mn, Fe, Co, Ni; mes = 2,4,6-Me₃C6H₂) and their one-electron reduced forms have been explored. The results clearly show that the anions are best described as complexes of the monoanionic bipyridine radical (S_bpy = 1/2), giving a rationale for the observed structural changes within the ligand. Likewise, we have identified dianionic bipyridine in both the complexes [Zn2(4,4'-bpy)(mes)₄]²⁻ and [Fe(2,2'-bpy)₂]²⁻. In no case have we found evidence for significant metal-to-ligand backbonding. The subject of redox-noninnocence is further revisited in a comparative study of the two complexes [M(o-Clpap)₃] (M = Cr, Mo; o-Clpap = 2-[(2-chloro-phenyl)azo]-pyridine), and their associated electron transfer series. The results indicate that all electron transfer processes are primarily ligand-based, although in the case of the Mo analogue these are coupled to substantial electron density changes at the metal. The ability of pap to form radical anions finds a contrasting case in the di- nuclear Rh complex [Rh₂(μ-p-Clpap)₂ (cod)Cl₂], where the two ligand bridges act as acceptors of strong dπ∗ backbonding from a formally Rh^–I centre. We then direct our attention to the endohedral Zintl clusters [Fe@Ge₁₀]³⁻ and [Mn@Pb₁₂]³⁻, which reveal peculiar topologies. We have probed the electronic factors that influence their geometric preferences, and propose a model based on the shift of electron density from the endo- hedral metal to the cage to account for the observed geometries. Subsequently, we reassess the electronic structure of the xenophilic clusters Mn₂(thf)₄(Fe(CO)₄)₂ and [Mn(Mn(thf)₂)₃(Mn(CO)₄)₃]^–. We conclude that these are best viewed as exchange coupled Mn^II centres bridged by closed- shell carbonylate fragments. In the closing chapter the reduction of NO₂^– to NO by the complex [Cu(tct)(NO₂)]⁺ (tct = cis,cis-1,3,5-tris(cinnamylideneamino)cyclohexane) is studied, a process that mimics the enzyme-catalysed reaction. Two viable pathways for the reaction have been traced and key inter-mediates identified. Both direct release of NO or via decomposition of a Cu-NO complex are kinetically and thermodynamically feasible.

<p>The conventional mechanism of fermion mass generation in the Standard Model involves Spontaneous Symmetry Breaking (SSB). In this thesis, we study an alternate mechanism for the generation of fermion masses that does not require SSB, in the context of lattice field theories. Being inherently strongly coupled, this mechanism requires a non-perturbative approach like the lattice approach.</p><p>In order to explore this mechanism, we study a simple lattice model with a four-fermion interaction that has massless fermions at weak couplings and massive fermions at strong couplings, but without any spontaneous symmetry breaking. Prior work on this type of mass generation mechanism in 4D, was done long ago using either mean-field theory or Monte-Carlo calculations on small lattices. In this thesis, we have developed a new computational approach that enables us to perform large scale quantum Monte-Carlo calculations to study the phase structure of this theory. In 4D, our results confirm prior results, but differ in some quantitative details of the phase diagram. In contrast, in 3D, we discover a new second order critical point using calculations on lattices up to size $ 60^3$. Such large scale calculations are unprecedented. The presence of the critical point implies the existence of an alternate mechanism of fermion mass generation without any SSB, that could be of interest in continuum quantum field theory.</p> / Dissertation

This thesis is concerned with graphical Gaussian models with equality constraints on the concentration or partial correlation matrix introduced by Højsgaard and Lauritzen (2008) as RCON and RCOR models. The models can be represented by vertex and edge coloured graphs G = (V,ε), where parameters associated with equally coloured vertices or edges are restricted to being identical. In the first part of this thesis we study the problem of estimability of a non-zero model mean μ if the covariance structure Σ is restricted to satisfy the constraints of an RCON or RCOR model but is otherwise unknown. Exploiting results in Kruskal (1968), we obtain a characterisation of suitable linear spaces Ω such that if Σ is restricted as above, the maximum likelihood estimator μ(with circumflex) and the least squares estimator μ* of μ coincide for μ ∈ Ω, thus allowing μ and Σ to be estimated independently. For the special case of Ω being specified by equality relations among the entries of μ according to a partition M of the model variables V, our characterisation translates into a necessary and sufficient regularity condition on M and (V,ε). In the second part we address model selection of RCON and RCOR models. Due to the large number of models, we study the structure of four model classes lying strictly within the sets of RCON and RCOR models, each of which is defined by desirable statistical properties corresponding to colouring regularity conditions. Two of these appear in Højsgaard and Lauritzen (2008), while the other two arise from the regularity condition ensuring equality of estimators μ(with circumflex) = μ* we find in the first part. We show each of the colouring classes to form complete lattices, which qualifies the corresponding model spaces for an Edwards-Havránek model selection procedure (Edwards and Havránek, 1987). We develop a coresponding algorithm for one of the model classes and give an algorithm for a systematic search in accordance with the Edwards-Havránek principles for a second class. Both are applied to data sets previously analysed in the literature, with very encouraging performances.

In this paper we consider a radially symmetric nonlinear Dirichlet problem in a ball, where the nonlinearity is "superlinear" and "superlinear with jumping."

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Symmetry in biological systems is the occurrence of an event on both sides of the system. The term bilateralism was introduced to represent this phenomenon, and it was defined as the conditional co-occurrence of two events given that at least at one of them has occurred. This phenomenon is highly associated with the prevalence of each of the events. Two parameters were developed to evaluate the presence of this phenomenon, testing whether events co-occur with higher probability than would be expected by chance. Nonparametric confidence intervals were constructed using the bootstrap percentile method. These non parametric confidence intervals were used in testing the null hypothesis of no bilateralism.A simulation study was performed to examine the properties of the two bilateralism parameters' estimates. The size and power of the tests of bilateralism were examined under a variety of sample sizes and prevalences of the two events. The simulation study showed that both parameter estimates have similar properties, and the tests have similar size and power. The power of the test was affected by the prevalence of either event, the differences in the prevalences, the sample size and by number of events that occur simultaneously. The methodology of testing for bilateralism was applied on data from the Pain in Sickle Cell Epidemiology Study (PiSCES). This study collected up to 6 months worth of daily diaries about pain and medical utilization from patients with sickle cell disease. Each diary recorded the body site and side where pain was experienced over the past 24 hours. The sample consists of 119 subjects who completed at least 50 daily pain diaries (reference). Information about the subjects age, gender and sickle cell genotype were also available. Nine body sites (5 upper peripheral, and 4 lower peripheral site) were analyzed to test for bilateralism. Bilateralism was tested for each subject and each site separately. The associations of prevalence of bilateralism on each site, and percentages of sites that hurt bilaterally with age, gender and genotype where studied.The results show a high prevalence of bilateral pain among sickle cell patients at all sites. Age gender and genotype were associated with higher prevalence in bilateral pain in some, but not all sites. The percentage of sites that have bilateral pain is also associated with the number of sites that have pain.