This thesis presents a brief synopsis of the derivations of the BCS and Eliashberg equations. An analytic formula for the critical temperature $T_c$ in Eliashberg theory is derived, which contains a sum of iterative integral corrections. These iterative integral corrections are the result of an iterative expression for the gap quotient $\Delta(\iw, T)/\Delta(0,T)$, which is derived. At the critical temperature this expression contains no reference to the critical temperature itself due to the gap approaching zero in this limit, $\lim_{T \rightarrow T_c} \Delta(\iw, T) = 0$. This enables explicit calculation of the critical temperature through the aforementioned iterative expression.\\ \\The behaviour of the iterative expression and its corrections are explored numerically with a toy spectral function $\sF$. Through these numerical experiments, this formula is found to be consistent with, though not equal to the successful McMillan formula for the coupling parameter $\lambda$ in the range $0.3 \leq \lambda \leq 1.5$. Below this value, the McMillan formula is found to approach zero critical temperature $T_c$ more rapidly, raising the future question of which of the two expressions is most successful in predicting the critical temperature $T_c$ in this range. \\ \\ For a toy spectral function with a single mode, the zeroth order correction of the iterative expression for the critical temperature $T_c$ is found to be adequate for most practical purposes due to the magnitude of measurement errors in real life measurements of model parameters. / Detta examensarbete går igenom en kort derivation av BCS och Eliashberg ekvationerna. En analytisk formel för den kritiska temperaturen $T_c$ i Eliashbergteori ges. Denna formel innehåller en summa av iterativa integraler som resulterar från ett uttryckt för energigapets kvot. Vid den kritiska temperaturen så kan man explicit lösa ut denna och på så sätt få ett analytiskt uttryck. Den uttrycket för den kritiska temperaturen utforskas numeriskt med en leksaks-spektralfunktion. Genom dessa numeriska experiment visas det hur det iterativa uttrycket sammanstämmer med McMillans formel för kopplingsparametern $0.3 < \lambda < 1.5$, även om dem ej är lika. Under detta intervall så närmar sig McMillans uttryck noll snabbare, vilket höjer frågan vilken utav dem två uttrycken som fungerar bäst i denna gräns.  För en leksaks-spektralfunktion med ett läge så räcker den nollte korrektionen i det iterativa uttrycket för att få godtagbara resultat, med bakgrund av dem relativt stora mätfelen för riktiga parametrar.

Thesis (Ph.D. (Physics)) -- University of Limpopo, 2017 / Density functional theory (DFT) study is employed in order to investigate the surfaces of, β-MnO2, β-TiO2 and β-VO2 (β-MO2) which act as catalysts in Li/Na-air batteries. Adsorption and co-adsorption of metal (Li/Na) and oxygen on (110) β-MO2 surface is investigated, which is important in the discharging and charging of Li/Na– air batteries. Due of the size of the supercell, and assuming that oxygen atoms occupy bulk-like positions around the surface metal atoms, only five values of (gamma) Γ are possible if constraint to a maximum of 1 monolayer (ML) of adatoms or vacancies: Γ= 0 surface is the stoichiometric surface, Γ= 1, 2 are the partially and totally oxidised surfaces, and Γ=-1, -2 are the partially and totally reduced surfaces. The manganyl, titanyl and vanadyl terminated surface is not the only surface that can be formed with Γ= +2. Oxygen can be adsorbed also as peroxo species (O2)2-, with less electron transfer from the surface vanadium atoms to the adatoms than in the case of manganyl and titanyl formation. The redox properties of the (110) surfaces are investigated by calculating the relative surface free energies of the non-stoichiometric compositions as a function of oxygen chemical potential. Increasing the temperature and lowering the pressure (i.e. more reducing conditions) we find the stoichiometric surface reduces first partially and then entirely at higher temperatures. The lithium orientation between two bridging oxygen and in-plane oxygen (bbi) orientation is much more stable for the three metal oxides, thus lithium generally prefers to adsorb where it will be triply coordinated to two bridging oxygens and one in-plane oxygen atom. However, sodium prefers to orientate itself on the bridging oxygen on the surface, but a triple coordination on sodium is also favourable. Oxygen adsorption on Li/MO2 was simulated and it was found that in all ii the metal oxides (MnO2, TiO2 and VO2) the most stable orientation is the dissociated composition where there is an oxygen atom on the “bulk-like” positions on top of each of the M cations. The surface lithium peroxide for MO2 simulated produces clusters with oxygen - oxygen bond lengths that are comparable to the calculated bulk and monomer discharge products reported in literature. Adsorption of oxygen on Na/MO2 was investigated and it was observed that the catalysts used encourage formation of the discharge product reported in literature, i.e. NaO2. The surface NaO2 appears to have comparable bond lengths to the calculated bulk and monomer NaO2. / National Research Foundation, South African Research Chair Initiative of the Department of Science Technology and Department of Energy storage Programme

Landau-Ginzburg mirror symmetry takes place in the context of affine singularities in CN. Given such a singularity defined by a quasihomogeneous polynomial W and an appropriate group of symmetries G, one can construct the FJRW theory (see [3]). This construction fills the role of the A-model in a mirror symmetry proposal of Berglund and H ubsch [1]. The conjecture is that the A-model of W and G should match the B-model of a dual singularity and dual group (which we denote by WT and GT). The B-model construction is based on the Milnor ring, or local algebra, of the singularity. We verify this conjecture for a wide class of singularities on the level of Frobenius algebras, generalizing work of Krawitz [10]. We also review the relevant parts of the constructions.

<p>Over the past two decades, open systems that are described by a non-Hermitian Hamiltonian have become a subject of intense research. These systems encompass classical wave systems with balanced gain and loss, semi-classical models with mode selective losses, and lossy quantum systems. The rapidly growing research on these systems has mainly focused on the wide range of novel functionalities they demonstrate. In this thesis, I intend to present some intriguing properties of a class of open systems which possess parity (P) and time-reversal (T) symmetry with a theoretical background, accompanied by the experimental platform these are realized on. These systems show distinct regions of broken and unbroken symmetries separated by a special phase boundary in the parameter space. This separating boundary is called the PT-breaking threshold or the PT transition threshold.</p> <p>We investigate non-Hermitian systems in two settings: tight binding lattice models, and electrical circuits, with the help of theoretical and numerical techniques. </p> <p><br></p> <p>With lattice models, we explore the PT-symmetry breaking threshold in discrete realizations of systems with balanced gain and loss which is determined by the effective coupling between the gain and loss sites. In one-dimensional chains, this threshold is maximum when the two sites are closest to each other or the farthest. We investigate the fate of this threshold in the presence of parallel, strongly coupled, Hermitian (neutral) chains, and find that it is increased by a factor proportional to the number of neutral chains. These results provide a surprising way to engineer the PT threshold in experimentally accessible samples.</p> <p>In another example, we investigate the PT-threshold for a one-dimensional, finite Kitaev chain—a prototype for a p-wave superconductor— in the presence of a single pair of gain and loss potentials as a function of the superconducting order parameter, onsite potential, and the distance between the gain and loss sites. In addition to a robust, non-local</p> <p>threshold, we find a rich phase diagram for the threshold that can be qualitatively understood in terms of the band-structure of the Hermitian Kitaev model.</p> <p>Finally, with electrical circuits, we propose a protocol to study the properties of a PT-symmetric system in a single LC oscillator circuit which is contrary to the notion that these systems require a pair of spatially separated balanced gain and loss elements. With a dynamically tunable LC oscillator with synthetically constructed circuit elements, we demonstrate static and</p> <p>Floquet PT breaking transitions by tracking the energy of the circuit. Distinct from traditional mechanisms to implement gain and loss, our protocol enables parity-time symmetry in a minimal classical system.</p>

The problem of systematically determining groups of canonical transformations that leave invariant Hamilton's equations of motion for bound Keplerian system in the plane is formulated and examined in detail. Previously unknown invariance groups are found, and a procedure for finding further ones is outlined. The methods used are directly generalizable to other systems in spaces of arbitrary dimension.

No description available.

In the first part of this thesis we study the generalization of the recent algebraic approach to classical field theory by proposing a more general setting based on the manifold of smooth sections of a non-trivial fiber bundle. Central is the notion of observables/functionals over such sections, i.e. appropriate smooth functions on them. The kinematic will be further specified by means of Peierls brackets, which in turn are defined via the causal propagators of linearized field equations. In the second part we implement deformation quantization of the algebras obtained previously in the simpler setting of scalar field theory. Wick powers and time ordered products for quantum field theories in curved spacetimes are defined by giving a set of axioms which, when implemented, defines uniquely, up to some classifiable ambiguities, the aforementioned quantities. Those ambiguities are known to be tightly restrained by locality, covariance and other regularity conditions. One of the additional constraints used was to require continuous and analytic dependence on the metric and coupling parameters. It was recently shown that this rather strong requirement could be weakened, in the case of Wick powers, to the so-called parametrized microlocal spectrum condition. We therefore show the existence of Wick powers satisfying the above condition and extend this axiom to time ordered products, while reestablishing the usual uniqueness and existence results in light of the new constraint.

This problem has been selected from the mathematical theory of elasticity. We consider a rectangular plate of thickness h, length a, and width b. The plate is subjected to compressive forces. These forces act in the neutral plane and give the plate a tendency to buckle. However, this problem differs from other plate problems in that it is assumed that there are two intermediate supports located on the edges of the plate parallel to the compressive forces.

L'information quantique est un nouveau domaine de la physique, qui consiste à employer les systèmes quantiques dans le calcul et la transmission de l'information. Cette thèse est consacrée à l'étude de certains aspects théoriques de l'information quantique. Les ordinateurs quantiques utilisent les lois de la mécanique quantique pour exécuter des calculs d'une manière bien plus efficace que les ordinateurs existants. Les ordinateurs quantiques envisageables dans la pratique seraient influencés par des perturbations diverses. Parmi ces sources de perturbations, les interactions résiduelles statiques (indépendantes du temps) à l'intérieur de l'ordinateur sont connues pour être les plus dangereuses dans le sens où elles peuvent s'ajouter de façon cohérente, tandis que les autres perturbations ont la forme d'un bruit aléatoire avec une moyenne égale à zéro. Nous étudions, dans les cas de deux calculs quantiques très différents, l'efficacité des ordinateurs quantiques en présence d'imperfections statiques. Nous trouvons le domaine des paramètres dans lequel l'ordinateur quantique est robuste en présence des imperfections. Une des raisons fondamentales de l'efficacité extraordinaire de l'ordinateur quantique et de l'existence d'autres applications de l'information quantique est l'effet de l'intrication quantique. L'intrication consiste dans l'impossibilité de considérer un état pur générique d'un système quantique composé comme le simple produit des états purs de ses sous-systèmes. Dans cette thèse nous étudions certaines propriétés importantes d'une certaine mesure quantitative d'intrication largement utilisée. Nous considérons également l'entropie informationnelle moyenne des états quantiques, puis nous trouvons une expression explicite pour cette quantité et étudions ses propriétés les plus importantes.

Nous proposons dans cette recherche une solution nouvelle quant à l'existence et à la nature de la singularité initiale d'espace-temps. Dans le contexte de la supergravité N=2 et de la théorie topologique des champs, nous considérons que la singularité initiale d'espace-temps correspond à un instanton gravitationnel singulier de taille zéro caractérisé par une configuration Riemanienne de la métrique (++++) en dimension D = 4. Associée à un état topologique correspondant à l'échelle zéro de l'espace-temps, la singularité initiale n'est pas ici considérée en termes de divergences des champs physiques, mais peut être résolue dans la cadre de la théorie topologique des champs. Nous obtenons ce résultat à partir de l'observation physique selon laquelle le pré espace-temps doit être considéré en équilibre thermique à l'échelle de Planck. En conséquences, nous suggérons de manière naturelle qu'à l'échelle de Planck l'espace-temps à l'équilibre doit être soumis à la condition KMS. Dans ce contexte, l'état KMS dans lequel se trouve le pré espace-temps à l'échelle de Planck pourrait être interprété comme le résultat d'une unification entre " état physique" (métrique lorentzienne +++-) et "état topologique" (métrique riemanienne ++++). Ceci correspond à la phase d'oscillation quantique de la signature de la métrique déjà mise en évidence dans des travaux antérieurs. Nous suggérons alors que "la singularité d'échelle zéro" doit être comprise en termes d'invariants topologiques, en particulier le premier invariant de Donaldson. En conséquences, nous proposons ici un nouvel invariant topologique, asssocié à l'échelle 0 et de la forme Z = TR (-1)s, que nous appelons "invariant de singularité". Enfin, dans ce contexte, nous proposons la conjecture selon laquelle le problème de l'interaction inertielle pourrait être expliqué en termes d'amplitude topologique liée à l'instanton gravitationnel singulier caractérisant, dans notre approche, l'échelle zéro de l'espace-temps.