Thesis (Ph. D.)--University of Washington, 2002. / Vita. Includes bibliographical references (leaves 51-52).

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Isomorphisms of automorphism groups of reduced torsion abelian p-groups have recently been classified by W. Liebert [L1] and [L2] for p ≠ 2. The primary objective of this study is to investigate the isomorphisms of automorphism groups of reduced mixed modules M and N of torsion-free ranks < ∞ over a complete discrete valuation ring with totally projective torsion submodules t(M) and t(N) respectively. For modules over ℤ(p), p ≠ 2, we show that if AutM and AutN are isomorphic and the quotient modules M/t(M) and N /t(N) are divisible, then M ≃ N.

Uniform design has found successful applications in manufacturing, system engineering, pharmaceutics and natural sciences since it appeared in 1980's. Recently, research related to uniform design is emerging. Discussions are mainly focusing on the construction and the theoretical properties of uniform design. On one hand, new construction methods can help researchers to search for uniform designs in more efficient and effective ways. On the other hand, since uniformity has been accepted as an essential criterion for comparing fractional factorial designs, it is interesting to explore its relationship with other criteria, such as aberration, orthogonality, confounding, etc. The first goal of this thesis is to propose new uniform design construction methods and recommend designs with good uniformity. A novel stochastic heuristic technique, the adjusted threshold accepting algorithm, is proposed for searching uniform designs. This algorithm has successfully generated a number of uniform designs, which outperforms the existing uniform design tables in the website https://uic.edu.hk/~isci/UniformDesign/UD%20Tables.html. In addition, designs with good uniformity are recommended for screening either qualitative or quantitative factors via a comprehensive study of symmetric orthogonal designs with 27 runs, 3 levels and 13 factors. These designs are also outstanding under other traditional criteria. The second goal of this thesis is to give an in-depth study of the uniformity property of fractional factorial designs. Close connections between different criteria and lower bounds of the average uniformity have been revealed, which can be used as benchmarks for selecting the best designs. Moreover, we find non-isomorphic designs have different combinatorial and geometric properties in their projected and level permutated designs. Two new non-isomorphic detection methods are proposed and utilized for classifying fractional factorial designs. The new methods take advantages over the existing ones in terms of computation efficiency and classification capability. Finally, the relationship between uniformity and isomorphism of fractional factorial designs has been discussed in detail. We find isomorphic designs may have different geometric structure and propose a new isomorphic identification method. This method significantly reduces the computational complexity of the procedure. A new uniformity criterion, the uniformity pattern, is proposed to evaluate the overall uniformity performance of an isomorphic design set.

Uniform design has found successful applications in manufacturing, system engineering, pharmaceutics and natural sciences since it appeared in 1980's. Recently, research related to uniform design is emerging. Discussions are mainly focusing on the construction and the theoretical properties of uniform design. On one hand, new construction methods can help researchers to search for uniform designs in more efficient and effective ways. On the other hand, since uniformity has been accepted as an essential criterion for comparing fractional factorial designs, it is interesting to explore its relationship with other criteria, such as aberration, orthogonality, confounding, etc. The first goal of this thesis is to propose new uniform design construction methods and recommend designs with good uniformity. A novel stochastic heuristic technique, the adjusted threshold accepting algorithm, is proposed for searching uniform designs. This algorithm has successfully generated a number of uniform designs, which outperforms the existing uniform design tables in the website https://uic.edu.hk/~isci/UniformDesign/UD%20Tables.html. In addition, designs with good uniformity are recommended for screening either qualitative or quantitative factors via a comprehensive study of symmetric orthogonal designs with 27 runs, 3 levels and 13 factors. These designs are also outstanding under other traditional criteria. The second goal of this thesis is to give an in-depth study of the uniformity property of fractional factorial designs. Close connections between different criteria and lower bounds of the average uniformity have been revealed, which can be used as benchmarks for selecting the best designs. Moreover, we find non-isomorphic designs have different combinatorial and geometric properties in their projected and level permutated designs. Two new non-isomorphic detection methods are proposed and utilized for classifying fractional factorial designs. The new methods take advantages over the existing ones in terms of computation efficiency and classification capability. Finally, the relationship between uniformity and isomorphism of fractional factorial designs has been discussed in detail. We find isomorphic designs may have different geometric structure and propose a new isomorphic identification method. This method significantly reduces the computational complexity of the procedure. A new uniformity criterion, the uniformity pattern, is proposed to evaluate the overall uniformity performance of an isomorphic design set.

Thesis (M.S.)--University of North Texas, Dec., 2009. / Title from title page display. Includes bibliographical references.

Thesis (Ph. D.)--Ohio State University, 2005. / Title from first page of PDF file. Includes bibliographical references (p. 299-303).

This paper examines idam's conjuecture and some of its generalizations. In terms of Adam's conjecture, we prove Alspach and Parson's results f or Zpq and ZP2. More generally, we prove Babai's characterization of the CI-property, Palfy's characterization of CI-groups, and Brand's result for Zpr for polynomial isomorphism's. We also prove for the first time a characterization of the CI-property for 1 SG, and prove that Zn is a CI-Pn-group where Pn is the group of permutation polynomials on Z,, and n is square free.

Factorial design has played a prominent role in the field of experimental design because of its richness in both theory and application. It explores the factorial effects by allowing the arrangement of efficient and economic experimentation, among which orthogonal design, uniform design and some other factorial designs have been widely used in various scientific investigations. The main contribution of this thesis shows the recent advances in the classification and selection of orthogonal designs. Design isomorphism is essential to the classification, selection and construction of designs. It also covers various popular design criteria as necessary conditions, such connection has led to a rapid growth of research on the novel approaches for either detecting the non-isomorphism or identifying the isomorphism. But further classification of non-isomorphic designs has received little attention, and hence remains an open question. It motivates us to propose the degree of isomorphism, as a more general view of isomorphism, for classifying non-isomorphic subclasses in orthogonal designs, and develop the column-wise identification framework accordingly. Selecting designs in sequential experiments is another concern. As a well-recognized strategy for improving the initial design, fold-over techniques have been widely applied to construct combined designs with better property in a certain sense. While each fold-over method has been comprehensively studied, there is no discussion on the comparison of them. It is the motivation behind our survey on the existing fold-over methods in view of statistical performance and computational complexity. The thesis involves five chapters and it is organized as follows. In the beginning chapter, the underlying statistical models in factorial design are demonstrated. In particular, we introduce orthogonal design and uniform design associated with commonly-used criteria of aberration and uniformity. In Chapter 2, the motivation and previous work of design isomorphism are reviewed. It attempts to explain the evolution of strategies from identification methods to detection methods, especially when the superior efficiency of the latter has been gradually appreciated by the statistical community. In Chapter 3, the concepts including the degree of isomorphism and pairwise distance are proposed. It allows us to establish the hierarchical clustering of non-isomorphic orthogonal designs. By applying the average linkage method, we present a new classification of L 27 (3 13 ) with six different clusters. In Chapter 4, an efficient algorithm for measuring the degree of isomorphism is developed, and we further extend it to a general framework to accommodate different issues in design isomorphism, including the detection of non-isomorphic designs, identification of isomorphic designs and the determination of non-isomorphic subclass for unclassified designs. In Chapter 5, a comprehensive survey of the existing fold-over techniques is presented. It starts with the background of these methods, and then explores the connection between the initial designs and their combined designs in a general framework. The dictionary cross-entropy loss is introduced to simplify a class of criteria that follows the dictionary ordering from pattern into scalar, it allows the statistical performance to be compared in a more straightforward way with visualization