<p> A number of questions are examined concerning many body correlations in connection with electron gas at metallic densities (2 < = rs <= 5.7) and the annihilation of a positron in simple metals, by means of a technique involving the two particle correlation Green's function. Estimates are made of low temperature contributions to angular correlation data, which describe the momentum distribution of annihilating electron positron pairs, in the form of smearing at the sharp cutoff corresponding to the Fermi momentum from electron- and positron- photon interactions, and in the form of broad tails beyond the cutoff resulting from the high-momentum components introduced into the electron wave function by the presence of a periodic crystal lattice. </p> <p> Phonon effects are introduced into the perturbation expansion of the two particle Green's function describing an electron positron pair. A calculation of the lowest order phonon contribution seems to indicate that such effects do not explain the smearing at the Fermi momentum. </p> <p> A Green's function calculation of the first-order enhancement of the lattice tails, due to the positron-electron correlation, is made by introducing particle-lattice interactions explicitly in a model based on a simple metal such as sodium. It considers a weak potential and treats as zero the lattice components corresponding to other than nearest-neighbours points in reciprocal lattice space. The enhancement for rs =4, which is almost a constant, is very similar to that for the main part of angular correlation data. This indicates that, for simple metals at least, angular correlation data can be interpreted directly from a free-particle model. </p> <p> Short-range correlations among opposite-spin electrons are examined by field-theoretic techniques as a step to obtaining a fundamental understanding of the correlations among electrons at metallic densities. A calculation of the p.d.f. for opposite-spin electrons is positive over a wide range of metallic densities and seems to account for short-range correlations of the Coulomb hole through the multiple scattering of particle-particle ladders. </p> / Thesis / Doctor of Philosophy (PhD)

The present work presents a number of contributions to the areas of numerical integration, singular integrals, and boundary element methods. The first contribution is an elemental distortion technique, based on the Duffy transformation, used to improve efficiency for the numerical integration of near hypersingular integrals. Results show that this method can reduce quadrature expense by up to 75 percent over the standard Duffy transformation. The second contribution is an improvement to integration of weakly singular integrals by using regularization to smooth weakly singular integrals. Errors show that the method may reduce errors by several orders of magnitude for the same quadrature order. The final work investigated the use of regularization applied to hypersingular integrals in the context of the boundary element method in three dimensions. This work showed that by using the simple solutions technique, the BEM is reduced to a weakly singular form which directly supports numerical integration. Results support that the method is more efficient than the state-of-the-art.

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京都大学 / 新制・課程博士 / 博士(理学) / 甲第23675号 / 理博第4765号 / 新制||理||1683(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 坂上 貴之, 教授 泉 正己, 教授 國府 寛司 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Quantum walks have been utilized by many quantum algorithms which provide improved performance over their classical counterparts. Quantum search algorithms, the quantum analogues of spatial search algorithms, have been studied on a wide variety of structures. We study quantum walks and searches on the Cayley graphs of finitely-generated free groups. Return properties are analyzed via Green’s functions, and quantum searches are examined. Additionally, the stopping times and success rates of quantum searches on random networks are experimentally estimated. / Mathematics

To follow up on the unexpectedly-good performance of coupled-cluster models with approx- imate inclusion of 3-body clusters [J. Chem. Phys. 151, 064102 (2019)] we performed a more complete assessment of the 3CC method [J. Chem. Phys. 125, 204105 (2006)] for accurate computational thermochemistry in the standard HEAT framework. New spin- integrated implementation of the 3CC method applicable to closed- and open-shell systems utilizes a new automated toolchain for derivation, optimization, and evaluation of operator algebra in many-body electronic structure. We found that with a double-zeta basis set the 3CC correlation energies and their atomization energy contributions are almost always more accurate (with respect to the CCSDTQ reference) than the CCSDT model as well as the standard CCSD(T) model. The mean errors in { 3CC, CCSDT, and CCSD(T) } electronic (per valence electron) and atomization energies were {23, 69, 125} μEh/e and {0.39, 1.92, 2.57} kJ/mol, respectively. The significant and systematic reduction of the error by the 3CC method and its lower cost than CCSDT suggests it as a viable candidate for post-CCSD(T) thermochemistry application. / Doctor of Philosophy / Stepping into the information age, the computing power has rapidly grown over the last half century. Solving chemical problems on computers has improved lives by reducing the cost and time of researching critical technologies. Scientific research is evolving and experimental finding are now supported with a computational model. Doing chemistry on computers requires quantum simulations, which is essentially solving the Schr ̈odinger equation on a computer that simulates a wave function for all the electrons in a system. Different models are built based on how these inter electronic interactions are treated. To predict results with accuracy on par with the experimental findings requires using higher-order wave functions methods.These are computationally expensive and often not practical. The lower-order methods that are easy to implement can be found in all quantum chemistry software packages. On the other hand, the higher-order methods are laborious and error prone to implement manually due to the sheer complexity of theory. Debugging such implementations often requires a lot of effort with the uncertainty in returns. To solve this problem, we implemented a second-quantization toolkit (SeQuant version 2.0) that derives many-body methods, specifically the general-order coupled cluster (CC) model. The CC model is systematically improvable and accurate. One such CC model, the CCSD(T), has been called the gold standard in quantum chemistry. For compactness, these equations are usually derived in their spin-orbital form. The evaluation and storage cost of these methods is reduced up to four-fold by transforming the spin-orbital expressions to a spin-traced form. In this work, the spin-tracing algorithms are described in detail. The general-order coupled cluster approach is used to derive the internally corrected approximate coupled cluster methods. These methods improve the accuracy of a model at a reduced cost. For small molecules, it was observed that the spin-traced evaluation was over three times faster than spin-orbital coupled cluster. To further reduce the cost of calculations, we added explicit correlation to our CC models. These methods improved the quality of our results with a modest increase in the computational cost.

A simulation model which incorporates the statistical- and numerical-based Lattice Green Function Local Load Sharing Model and a Fracture Mechanics-based Residual Strength Model has been developed. The model simulates creep rupture by imposing a fixed load of constant stress on the composite over the simulation duration. Simulation of the fatigue of glass fiber-reinforced composites is achieved by replacing the constant stress parameter in the model with a sinusoidal wave function. Results from the creep rupture model using fused silica fiber parameters, compare well with S-2 glass/epoxy systems. Results using Mandell's postulate that fatigue failure in glass fiber-reinforced polymeric composites is a fiber-dominated mechanism, with a characteristic slope of 10 %UTS/decade are consistent with available experimental data. The slopes of fatigue curves for simulated composites for three frequencies namely: 2, 5 and 10 Hz are within 12-14 %UTS/decade compared with that of 10.6-13.0%UTS/decade for unidirectionl glass reinforced composites (epoxy and vinyl ester) obtained from Demers' [40] data. / Master of Science

In this work, we investigate the spectrum of the non-relativistic Krönig-Penney Hamiltonian H_α= -d²/dx² +αΣ_m∈Zδ(-(2m+1)π) perturbed by a short-range potential λW and the spectrum of its relativistic counterpart obtained by replacing the Schrödinger Hamiltonian H_α with its relativistic analogue H̅_α. The interesting feature of both spectra is that they have gaps and that bound states may occur in such gaps as a consequence of the presence of the short-range potential representing the impurity. Such bound states, often called "impurity states" in the solid state physics literature. are important with regard to the conductivity properties of solids We show the existence of such bound states of H_α + λW in each sufficiently remote gap of its essential spectrum if the integral of W is different from zero and the 1 + 𝛅-moment of W is finite for some 𝛅 > 0. Furthermore, if the potential has a constant sign we prove that there is only one bound state in each sufficiently remote gap. We shall see that in the relativistic case one may have more than one bound state in each remote gap under the same assumptions on W. Nevertheless, we shall see that such additional bound states cannot appear in the range of energies of solid state physics. / Ph. D.

Thesis (PhD) -- Stellenbosch University, 2014. / ENGLISH ABSTRACT: This work considers the efficient numerical analysis of large, aperiodic finite antenna arrays. A Method of Moments (MoM) based domain decomposition technique called the Domain Green's Function Method (DGFM) is formulated to address a wide range of array problems in a memory and runtime efficient manner. The DGFM is a perturbation approach that builds on work initially conducted by Skrivervik and Mosig for disjoint arrays on multi-layered substrates, a detailed review of which will be provided in this thesis. Novel extensions considered for the DGFM are as follows: a formulation on a higher block matrix factorisation level that allows for the treatment of a wider range of applications, and is essentially independent of the elemental basis functions used for the MoM matrix formulation of the problem. As an example of this, both conventional Rao-Wilton-Glisson elements and also hierarchical higher order basis functions were used to model large array structures. Acceleration techniques have been developed for calculating the impedance matrix for large arrays including one based on using the Adaptive Cross Approximation (ACA) algorithm. Accuracy improvements that extend the initial perturbation assumption on which the method is based have also been formulated. Finally, the DGFM is applied to array geometries in complex environments, such as that in the presence of finite ground planes, by using the Numerical Green's Function (NGF) method in the hybrid NGF-DGFM formulation. In addition to the above, the DGFM is combined with the existing domain decomposition method, viz., the Characteristic Basis Function Method (CBFM), to be used for the analysis of very large arrays consisting of sub-array tiles, such as the Low-Frequency Array (LOFAR) for radio astronomy. Finally, interesting numerical applications for the DGFM are presented, in particular their usefulness for the electromagnetic analysis of large, aperiodic sparse arrays. For this part, the accuracy improvements of the DGFM are used to calculate quantities such as embedded element patterns, which is a major extension from its original formulation. The DGFM has been integrated as part of an efficient array analysis tool in the commercial computational electromagnetics software package, FEKO. / AFRIKAANSE OPSOMMING: In hierdie werkstuk word die doeltre ende analise van eindige, aperiodiese antenna samestellings behandel. Eindige gebied benaderings wat op die Moment Metode (MoM) berus, word as vetrekpunt gebruik. `n Tegniek genaamd die Gebied Green's Funksie Metode (GGFM) word voorgestel en is geskik vir die analise van `n verskeidenheid van ontkoppelde samestellings. Die e ektiewe gebruik van rekenaargeheue en looptyd is onderliggend in die implementasie daarvan. Die GGFM is 'n perturbasie metode wat op die oorspronklike werk van Skrivervik en Mosig berus. Laasgenoemde is hoofsaaklik ontwikkel vir die analise van ontkoppelde antenna samestellings op multilaag di elektrikums. `n Deeglike oorsig van voorafgaande word in die tesis verskaf. In hierdie tesis is die bogenoemde werk op `n unieke wyse uitgebrei: `n ho er blok matriks vlak formulering is ontwikkel wat dit moontlik maak vir die analise van `n verskeidenheid strukture en wat onafhanklik is van die onderliggende basis funksies. Beide lae-vlak Rao-Wilton-Glisson (RWG) basis funksies, asook ho er orde hierargiese basis funksies word gebruik vir die modellering van groot antenna samestellings. Die oorspronklike perturbasie aanname is uitgebrei deur akkuraatheidsverbeteringe vir die tegniek voor te stel. Die Aanpasbare Kruis Benaderings (AKB) tegniek is onder andere gebruik om spoed verbeteringe vir die GGFM te bewerkstellig. Die GGFM is verder uitgebrei vir die analise van antenna samestellings in `n komplekse omgewing, bv. `n antenna samestelling bo `n eindige grondplaat. Die Numeriese Green's Funksie (NGF) metode is hiervoor ingespan en die hibriede NGF-GGFM is ontwikkel. Die GGFM is verder met die Karakteristieke Basis Funksie Metode (KBFM) gekombineer. Die analise van groot skikkings wat bestaan uit sub-skikkings, soos die wat tans by die \Low- Frequency Array (LOFAR) " vir radio astronomie in Nederland gebruik word, kan hiermee gedoen word. In die werkstuk word die GGFM ook toegepas op `n reeks interessante numeriese voorbeelde, veral die toepaslike EM analise van groot aperiodiese samestellings. Die akkuraatheidsverbeteringe vir die GGFM maak die berekening van elementpatrone vir skikkings moontlik. Die GGFM is by the sagteware pakket FEKO geintegreer.

Thesis (MScIng)--University of Stellenbosch, 2003. / ENGLISH ABSTRACT: A complete mixed-potential integral equation formulation for the analysis of arbitrarily shaped scatterers in a planarly layered medium is presented. The integral equation is able to solve for simultaneous electric and magnetic surface currents using a Method of Moments (MoM) procedure. The MoM formulation which was developed uses vector-valued basis functions defined over a triangular mesh and are used to model electric currents on conducting scatterers and magnetic currents on slotline structures. The Green’s functions employed in the analysis were developed for a stratified medium using a Sommerfeld plane wave formulation. The scheme used for filling the method of moments matrix was designed to simultaneously solve multiple problems that are stacked and separated by an infinite conducting ground plane. The filling algorithm also efficiently packs partially symmetric matrices, which are present when solving problems that support a combination of electric and magnetic currents. Several examples are presented to illustrate and validate the analysis method. Numerical predictions of the scattering parameters (both magnitude and phase) show good correspondence with results from literature and measured data. / AFRIKAANSE OPSOMMING: ’n Volledige gemengde potensiaal integraalvergelyking formulering vir die analise van stralers van arbitrˆere vorm binne gelaagde strukture word aangebied. Die integraalvergelyking kan gelyktydige elektriese en magnetiese oppervlakstrome oplos deur die Metode van Momente (MoM) te gebruik. Die MoM formulering gebruik vektor basis funksies wat oor ’n driehoekige diskretisering gedefinieer word om elektriese strome op geleidende stralers en magnetiese strome op gleuflyn strukture te modelleer. Die Green’s funksies wat in die analise gebruik word, is ontwikkel vir gelaagde media deur gebruik te maak van Sommerfeld se platvlakgolf formulering. Die metode wat gebruik word om the moment matriks te vul, is ontwerp om meervoudige gestapelde probleme wat deur oneindig geleidende grondvlakke geskei word, gelyktydig op te los. Gedeeltelik simmetriese matrikse word ook effektief gevul. Hierdie matrikse kom voor wanneer probleme ’n kombinasie van elektriese en magnetiese strome ondersteun. Verskeie voorbeelde word gebruik om die analise metode te verifieer. Numeriese voorspellings van strooiparameters (beide grootte en hoek) vergelyk baie goed met resultate en gemete data wat in die literatuur gevind is. iv