Bibliography: pages 140-145. / An interesting problem in universal algebra is the connection between the internal structure of an algebra and the identities which it satisfies. The study of varieties of algebras provides some insight into this problem. Here we are concerned mainly with lattice varieties, about which a wealth of information has been obtained in the last twenty years. We begin with some preliminary results from universal algebra and lattice theory. The next chapter presents some properties of the lattice of all lattice sub-varieties. Here we also discuss the important notion of a splitting pair of varieties and give several characterisations of the associated splitting lattice. The more detailed study of lattice varieties splits naturally into the study of modular lattice varieties and non-modular lattice varieties, dealt with in the second and third chapter respectively. Among the results discussed there are Freese's theorem that the variety of all modular lattices is not generated by its finite members, and several results concerning the question which varieties cover a given variety. The fourth chapter contains a proof of Baker's finite basis theorem and some results about the join of finitely based lattice varieties. Included in the last chapter is a characterisation of the amalgamation classes of certain congruence distributive varieties and the result that there are only three lattice varieties which have the amalgamation property.

Let L be a lattice. For x ∈ L, we say x is a consistent join-irreducible if x V y is a join-irreducible of the lattice [y,1] for all y in L. We say L is consistent if every join-irreducible of L is consistent. In this dissertation, we study the notion of consistent elements in semimodular lattices.

One of the more important concepts in mathematics is the concept of order, that is, the description or comparison of two elements of a set in terms of one preceding or being smaller than or equal to the other. If the elements of a set, as pairs, exhibit certain order-type characteristics, the set is said to be a partially ordered set. The purpose of this paper is to investigate a special class of partially ordered sets, called lattices, and to investigate topologies induced on these lattices by specially defined order related properties called order-convergence and star-convergence.

線性丢番圖方程系統（LDEs）連同其子類--子集和問題--在現實世界有大量且重要的應用。可是當線性番圖方程系統的整數解集被限定在一個有界的多面體中，則此系統屬于NP類問題。同理其子類--子集和問題--也同屬于NP類問題。另一方面，密度（density）接近或等于一的子集和問題已在文獻中被證實爲最難的一類子集和問題，並且現有的針對此最難子集和問題的所有解決方案都不能達到令人滿意的成功率。因此，在這篇論文中，我們旨在提出有效的算法來求解線性番圖方程系統以及其子類問題--密度爲一的子集和問題。 / 我們在此論文中的研究包括：1）基于格理論和LLL算法的性質，采用並改良針對LDEs的格表達式（lattice formulations）；2）提出針對子集和問題的分解（disaggregation）技術；3）創造性地將分解技術與格表達式整合在一起，從而提高求解密度爲一的子集和問題的成功率。 / 數值實驗顯示，我們提出的新整合算法對提高密度爲一的子集和問題的成功率有著顯著的效果。比如，針對維數分別爲20,30和40的密度爲一的子集和問題，對各個維數隨機産生的100個問題，我們的新整合算法均可將成功率提高到100%。同時，針對新整合算法的理論分析顯示，能將短且非0-1的整數解切割掉的分解在達到新整合算法的顯著實驗效果中起到了關鍵作用。 / While systems of linear Diophantine equations (LDEs) with bounded feasible set, including subset sum problem as its special subclass, find wide, and often significant, real-world applications, they unfortunately belong to the NP class in general. Furthermore, the literature has revealed that subset sum problems with their density close to one constitute the hardest subclass of subset sum problems and all the existing solution methods do not perform to a satisfactory level (with low success ratio) even when the problem size is only medium. / We take the challenge in this thesis to investigate lattice formulations for systems of LDEs in which LLL basis reduction algorithm (LLL algorithm) is utilized, propose disaggregation techniques for subset sum problems, and develop a powerful integration of disaggregation techniques with lattice formulations in solving feasible subset sum problems. / More specifically, the contributions in this thesis can be classified into three parts: i) we propose two revised lattice formulations of Aardal et al. (2000) for systems of LDEs to enhance further the computational capability of the LLL algorithm; ii) we study properties related to disaggregation of a single LDE and investigate thoroughly disaggregation schemes based on modular transformations; and iii) we develop a novel version of LLL algorithm by integrating modular disaggregation into the solution process. Promising numerical results have been achieved when applying our newly proposed LLL algorithm in tackling hard subset sum problems with density close to one. For instance, the success ratio can be raised to 100% for 100 randomly generated hard subset sum problems with dimensions 20, 30, and 40, respectively. We carry out theoretical study for possible driving force behind the success of our new algorithm, including dimension reduction of the solution space, information recovering of LDEs, and mechanism in cutting off short non-binary integer solutions when attaching disaggregation with LLL algorithm. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Lu, Bojun. / Thesis (Ph.D.) Chinese University of Hong Kong, 2014. / Includes bibliographical references (leaves 143-153). / Abstracts also in Chinese.

Quantum Chromo - Dynamics ( QCD ) is the part of the Standard Model which describes the interaction of the strong nuclear force with matter. QCD is asymptotically free, so at high energies perturbation expansions in the coupling can be used to calculate expectation values. Away from this limit, however, perturbation expansions in the coupling do not converge. Lattice QCD ( LQCD ) is a non - perturbative approach to calculations in QCD. LQCD first performs a Wick rotation t → - it [subscript E], and then discretises spacetime into a regular lattice with some lattice spacing a. QCD is then expressed in terms of parallel transport operators of the gauge field between grid points, and fermion fields which are defined at the grid points. Operators are evaluated in terms of these quantities, and the lattice spacing is then taken to zero to recover continuum values. We perform computer simulations of Lattice QCD in order to extract a variety of meson observables. In particular, we perform a comprehensive survey of the light and strange meson octets, obtain for the first time exotic meson results consistent with experiment, calculate the charge form - factor of the light and strange pseudoscalar mesons, and determine ( for the first time in Lattice QCD ) all three form - factors of the vector meson. / Thesis (Ph.D.)--University of Adelaide, School of Chemistry and Physics, Discipline of Physics, Centre for the Subatomic Structure of Matter, 2007.

No description available.

We study the four-quark interaction as a first step in understanding the QCD origin of the nuclear force in nature. We simulate QCD on a 20 x 20 x 20 x 32 space-time lattice with the simplifying quenched and static approximations, and with the SU(2) gauge group. Recent four-quark simulations reveal interesting tetrahedral geometry and planar four-quark flux distributions that cannot be explained by existing models. We simulate the flux distribution for the still-unexplored next higher level of geometrical complexity, namely four quarks on the corners of a tetrahedron. In order to complete the simulation within the allotted computing time, we have improved the approach used to simulate the flux distribution. Compared to previous approaches, the new approach nearly eliminates the bottleneck of the computing time, provides more than a 100-time speedup in our project, and also provides a better strategy for improving signal-noise ratio and suppressing signal distortion from the lattice structure. As the result of this improved approach, we have observed the long diagonal flux tube structure, repeated the Helsinki group's 1998 results for the flux distribution of a square geometry, and, for the first time, simulated the flux distribution of a tetrahedron geometry. In this thesis, we also explore some fundamental questions of lattice QCD related to computability theory and complexity theory. / Graduation date: 2001

No description available.

No description available.

No description available.