The idea of keychains, an (n+1)-tuple of non-increasing real numbers in the unit interval always including 1, naturally arises in study of finite fuzzy set theory. They are a useful concept in modeling ideas of uncertainty especially those that arise in Economics, Social Sciences, Statistics and other subjects. In this thesis we define and study some basic properties of keychains with reference to Partially Ordered Sets, Lattices, Chains and Finite Fuzzy Sets. We then examine the role of keychains and their lattice diagrams in representing uncertainties that arise in such problems as in preferential voting patterns, outcomes of competitions and in Economics - Preference Relations.

We study various quotient maps between frames which are defined by stipulating that they satisfy certain conditions on the cozero parts of their domains and codomains. By way of example, we mention that C-quotient and C -quotient maps (as defined by Ball and Walters- Wayland [7]) are typical of the types of homomorphisms we consider in the initial parts of the thesis. To be little more precise, we study uplifting quotient maps, C1- and C2-quotient maps and show that these quotient maps possess some properties akin to those of a C-quotient maps. The study also focuses on R - and G - quotient maps and show, amongst other things, that these quotient maps coincide with the well known C - quotient maps in mildly normal frames. We also study quasi-F frames and give a ring-theoretic characterization that L is quasi-F precisely when the ring RL is quasi-B´ezout. We also show that quasi-F frames are preserved and reflected by dense coz-onto R -quotient maps. We characterize normality and some of its weaker forms in terms of some of these quotient maps. Normality is characterized in terms of uplifting quotient maps, -normally separated frames in terms of C1-quotient maps and mild normality in terms of R - and G -quotient maps. Finally we define cozero complemented frames and show that they are preserved and reflected by dense z#- quotient maps. We end by giving ring-theoretic characterizations of these frames. / Mathematical Science / D. Phil. (Mathematics)

Lattice gauge theory is an important approach to understanding quantum chromodynamics (QCD) due to the large coupling constant in the theory at low energy. In this thesis, we report our study of the topological properties of the gauge fields and we calculate 𝘮_η and 𝘮_η' which are related to the topology of the gauge fields. We also develop two algorithms to speed up the inversion of the Dirac equation which is computationally demanding in lattice QCD calculations. The topology of lattice gauge fields is important but difficult to study because of the large local fluctuations of the gauge fields. In chapter 2, we probe the topological properties of the gauge fields through the measurement of closed quark loops, field strength and low-lying eigenvectors of the Shamir domain wall operator. The closed quark loops suggest the slow evolution of topological modes during the generation of QCD configurations. The chirality of the low-lying eigenvectors is studied and the lattice eigenvectors are compared to the eigenvectors in the continuous theory. The topological charges are calculated from the eigenvectors and the results agree with the topological charges calculated from the smoothed gauge fields. The fermion correlators are also obtained from the eigenvectors. The non-trivial topological properties of QCD gauge fields are important to the mass of the η and η', 𝘮_η and 𝘮_η'. Lattice QCD is an area where 𝘮_{\eta}$ and 𝘮_{\eta'}$ can be calculated by using gauge fields that are sampled over different topological sectors. We calculate 𝘮_η and 𝘮_η' in chapter 3 by including the fermion correlators and the topological charge density correlators. The errors of 𝘮_η and 𝘮_η' are reduced to the percent level and the mixing angle between the octet, singlet states in the SU(3) limit and the physical eigenstates is calculated. An algorithm that reduces communication and increases the usage of the local computational power is developed in chapter 4. The algorithm uses the multisplitting algorithm as a preconditioner in the preconditioned conjugate gradient method. It speeds up the inversion of the Dirac equation during the evolution phase. In chapter 5, we utilize two lattices, called the coarse lattice and the fine lattice, that lie on the renormalization group trajectory and have different lattice spacings. We find that the low-mode space of the coarse lattice corresponds to the low-mode space of the fine lattice. Because of the correspondence, the coarse lattice can be used to solve the low modes of the fine lattice. The coarse lattice is used in the restart algorithm and the preconditioned conjugate gradient algorithm where the latter is called the renormalization group based preconditioned conjugate gradient algorithm (RGPCG). By using the near-null vectors as the filter, RGPCG could reduce the operations of the matrix multiplications on the fine lattice by 33% to 44% for the inversion of Dirac equation. The algorithm works better than the conjugate gradient algorithm when multiple equations are solved.

A cubic lattice with random parameters is reduced to a linear chain by the means of the projection technique. The continued fraction expansion (c.f.e.) approach is herein applied to the density of states. Coefficients of the c.f.e. are obtained numerically by the recursion procedure. Properties of the non-stationary second moments (correlations and dispersions) of their distribution are studied in a connection with the other evidences of transport in a one-dimensional Mori chain. The second moments and the spectral density are computed for the various degrees of disorder in the prototype lattice. The possible directions of the further development are outlined. The physical problem that is addressed in the dissertation is the possibility of the existence of a non-Anderson disorder of a specific type. More precisely, this type of a disorder in the one-dimensional case would result in a positive localization threshold. A specific type of such non-Anderson disorder was obtained by adopting a transformation procedure which assigns to the matrix expressing the physics of the multidimensional crystal a tridiagonal Hamiltonian. This Hamiltonian is then assigned to an equivalent one-dimensional tight-binding model. One of the benefits of this approach is that we are guaranteed to obtain a linear crystal with a positive localization threshold. The reason for this is the existence of a threshold in a prototype sample. The resulting linear model is found to be characterized by a correlated and a nonstationary disorder. The existence of such special disorder is associated with the absence of Anderson localization in specially constructed one-dimensional lattices, when the noise intensity is below the non-zero critical value. This work is an important step towards isolating the general properties of a non-Anderson noise. This gives a basis for understanding of the insulator to metal transition in a linear crystal with a subcritical noise.

In this thesis I present contributions to the development, analysis and application of tensor network state methods for numerically exact quantum dynamics in one and two-dimensional lattice systems. The setting of numerically exact quantum dynamics is introduced in Chapter 2. This includes a discussion of exact diagonalization approaches and massively parallel implementations thereof as well as a brief introduction of tensor network states. In Chapter 3, I perform a detailed analysis of the performance of n-ary tree tensor network states for simulating the dynamics of two-dimensional lattices. This constitutes the first application of this class of tensor network to dynamics in two spatial dimensions, a long-standing challenge, and the method is found to perform on par with existing state-of-the-art approaches. Chapter 4 showcases the efficacy of a novel tensor network format I developed, tailored to electron-phonon coupled problems in their single-electron sector, through an application to the Holstein model. The applicability of the approach to a broad range of parameters of the model allows to reveal the strong influence of the spread of the electron distribution on the initial state of the phonons at the site where the electron is introduced, for which a simple physical picture is offered. I depart from method development in Chapter 5 and analyse the prospects of using tensor network states evolved using the time-dependent variational principle as an approximate approach to determine asymptotic transport properties with a finite, moderate computational effort. The method is shown to not yield the correct asymptotics in a clean, non-integrable system and can thus not be expected to work in generic systems, outside of finely tuned parameter regimes of certain models. Chapters 6 and 7 are concerned with studies of spin transport in long-range interacting systems using tensor network state methods. For the clean case, discussed in Chapter 6, we find that for sufficiently short-ranged interactions, the spreading of the bulk of the excitation is diffusive and thus dominated by the local part of the interaction, while the tail of the excitation decays with a powerlaw that is twice as large as the powerlaw of the interaction. Similarly, in the disordered case, analysed in Chapter 7, we find subdiffusive transport of spin and sub-linear growth of entanglement entropy. This behaviour is in agreement with the behaviour of systems with local interactions at intermediate disorder strength, but provides evidence against the phenomelogical Griffith picture of rare, strongly disordered insulating regions. We generalize the latter to long-ranged interactions and show that it predicts to diffusion, in contrast to the local case where it results in subdiffusive behaviour.

The two neutral kaon states in nature, the 𝘒_𝐿 (long-lived) and 𝘒_s (short-lived) mesons, are the two time-evolution eigenstates of the 𝘒⁰ - 𝘒̅⁰̅ mixing system. The prediction of their mass difference 𝚫m_𝘒 based on the standard model is an important goal of lattice QCD. Non-perturbative formalism has been developed to calculate 𝚫 m_𝘒 and the calculation has been extended from the first exploratory calculation with only connected diagrams to full calculations on near-physical[1] and physical ensembles[2]. In this work, we extend the calculation described in Reference [2] from 59 to 152 configurations and present a new analysis method employed to calculate 𝚫 m_𝘒 with better reduction of statistical error on this larger set of configurations. By using a free-field calculation, we will show that the four-point contractions in our calculation method yields results consistent with the Inami-Lim calculation[3] in the local limit. We also report a series of scaling tests performed on 24³ × 64 and 32³ × 64 lattice ensembles to estimate the size of the finite lattice spacing error in our 𝚫 m_K$ calculation. We will present the 𝚫 m_𝘒 calculation on the ensemble of 64³ × 128 gauge configurations with inverse lattice spacing of 2.36 GeV and physical quark masses obtaining results coming from 2.5 times the Monte Carlo statistics used for the result in [2]. With the new analysis method and estimated finite lattice spacing error, we obtain 𝚫 m_𝘒 = 5.8(0.6)_stat(2.3)_sys × 10¯¹²MeV. Here the first error is statistical and the second is an estimate of largest systematic error due to the finite lattice spacing effects. The new results also imply the validity of the OZI rule for the case of physical kinematics in contrast to the previous calculation of 𝚫 m_𝘒 with unphysical kinematics[1], where contributions from diagrams with disconnected parts are almost half the size of the contributions from fully connected diagrams but with the opposite sign.

Includes bibliographical references. / The thesis is organized as follows. Part I is a general introduction to LGT. The theory is discussed from first principles, so that for the interested reader no previous knowledge is required, although it is assumed that he/she will be familiar with the rudiments of relativistic quantum mechanics. Part II is a review of QCD on the lattice at finite temperature and density. Monte Carlo results and analytical methods are discussed. An attempt has been made to include most relevant data up to the end of 1987, and to update some earlier reviews existing on the subject. To facilitate an understanding of the techniques used in LGT, provision has been made in the form of a separate Chapter on Group Theory and Integration, as well as four Appendices, one of which deals with Grassmann variables and integration.

No description available.

M.S.

In many investigations, an experimenter is interested in comparing the effects of a number of experimental treatments, such as yields of several new agriculture varieties, results of several different production procedures, or effects of various raw materials on a final product. In this type of investigation, an F test of the mean square for treatments is often used to test the hypothesis that all treatment means are homogeneous. Given that this hypothesis can be rejected, the experimenter would like to make decisions about the significances of individual differences among treatment means considered a pair at a time. / Master of Science