The intersection of two Latin squares of the same order is the set of cells that contain the same entries in both Latin squares. Determining the order of this set can be asked for any type of Latin square and has been solved for most. Generalizing this to Latin squares of different orders leads to a conjecture of Dukes and Mendelsohn, which will be shown to be true. Results on the intersection of Latin squares, idempotent Latin squares, and idempotent symmetric Latin squares are explored. The relationship between the intersection problem for Latin squares and the intersection problem for Steiner triple systems will also be investigated. In addition to new results, past results are included presenting a common and clear notation. The proofs of some new results are able to replace proofs of past results as well as present a straightforward proof structure to new and past results. Two Latin squares of the same order are said to be r-orthogonal if the set of pairs occurring in corresponding cells has size r. Using this notation, two orthogonal Latin squares of order n are n2-orthogonal. The idea of r-orthogonality is generalized to Latin squares of different orders. The set of possible values is established for r and it is shown that this possible set can be obtained for pairs of Latin squares with certain orders.

The relativistic Vlasov-Maxwell system (RVM for short) is a kinetic model that arises in plasma physics and describes the time evolution of an ensemble of charged particles that interact only through their self-induced electromagnetic field. Collisions among the particles are neglected and they are assumed to move at speeds comparable to the speed of light. If the particles are allowed to move in the three dimensional space, then the main open problem concerning this system is to prove (or disprove) that solutions with sufficiently smooth Cauchy data do not develop singularities in finite time. Since the RVM system is essential in the study of dilute hot plasmas, much effort has been directed to the solution of its Cauchy problem. The underlying hyperbolic nature of the Maxwell equations and their nonlinear coupling with the Vlasov equation amount for the challenges imposed by this system. In this thesis, we show that solutions of the RVM system with smooth, compactly supported Cauchy data develop singularities only if the charge density blows-up in finite time. In particular, solutions can not break-down due to shock formations, since in this case scenario the solution would remain bounded while its derivative blows-up. On the other hand, if the transversal component of the displacement current is neglected from the Maxwell equations, then the RVM system reduces to the socalled relativistic Vlasov-Darwin (RVD) system. The latter has useful applications in numeric simulations of collisionless plasma, since the hyperbolic RVM is now reduced to a more tractable elliptic system while preserving a fully coupled magnetic field. As for the RVM system, the main open problem for the RVD system is to prove whether classical solutions with unrestricted Cauchy data exist globally in time. In the second part of this thesis, we show that classical solutions of the RVD system exist provided the Cauchy datum satisfies some suitable smallness assumption. The proof presented here does not require estimates derived from the conservation of the total energy nor those on the transversal component of the electric field. These have been crucial in previous results concerning the RVD system. Instead, we exploit the potential formulation of the model equations. In particular, the Vlasov equation is rewritten in terms of the generalized variables and coupled with the equations satisfied by the scalar and vector Darwin potentials. This allows to use standard estimates for singular integrals and a recursive method to produce the existence of local in time classical solutions. Hence, by means of a bootstrap argument, we show that such solutions can be made global in time provided the Cauchy data is sufficiently small.

This thesis is an account of experiments into the teaching of mathematics to engineering undergraduates which have been conducted over twenty years against a background of changing intake ability, varying output requirements and increasing restrictions on the formal contact time available. The aim has been to improve the efficiency of the teaching-learning process. The main areas of experimentation have been the integration in the syllabus of numerical and analytical methods, the incorporation of case studies into the curriculum and the use of micro-based software to enhance the teaching process. Special attention is paid to courses in Mathematical Engineering and their position in the spectrum of engineering disciplines. A core curriculum in mathematics for undergraduate engineers is proposed and details are provided of its implementation. The roles of case studies and micro-based software are highlighted. The provision of a mathematics learning resource centre is considered a necessary feature of the implementation of the proposed course. Finally, suggestions for further research are made.

Typescript (photocopy) / iv, 160 leaves : ill. ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Civil Engineering, 1979

Thesis (Ph.D.)--University of Western Australia, 2006.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1967. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliography.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1969. / Vita. Typescript. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis (M.S.)--Brigham Young University. Dept. of Mechanical Engineering, 2005. / Includes bibliographical references (p. 83-88).

In this thesis study, three different feature selection methods, LASSO, SLR, and SMLR, were tested and compared using microarray fold change data. Two real datasets were used to first investigate and compare the ability of the algorithms in selecting feature genes on data under two conditions. It was found that SMLR was quite sensitive to its parameter, and was more accurate in selecting differentially expressed genes when compared to SLR and LASSO. In addition, the model coefficients generated by SMLR had a close relationship with the magnitude of fold changes. Also, SMLR's ability in selecting differentially expressed genes with data that had more than two conditions was shown to be successful. The results from simulation experiments agreed with the results from the real dataset experiments. Additionally, it was found that different proportions of differentially expressed genes in the data did not affect the performance of LASSO and SLR, but the number of genes selected by SMLR increased with the proportion of regulated genes. Also, as the number of replicates used to build the model increased, the number of genes selected by SMLR increased. This applied to both correctly and incorrectly selected genes. Furthermore, it was found that SMLR performed the best in identifying future treatment samples.

A permutation array, represented by PA(n, d), is a subset of Sn such that any two distinct elements have a distance of at least d where d is the number of differing positions. We analyze the upper and lower bounds of permutation codes with distance equal to 4. An optimization problem on Young diagrams is used to improve the upper bound for almost all n while the lower bound is improved for small values of n by means of recursive construction methods.