Inaug.--diss.G̲̲öttingen. / Lebenslauf. Includes bibliographical references.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1964. / Vita. Typescript. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis--University of Wisconsin--Madison. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves [94-95]).

Thesis (Ph. D.)--University of Wisconsin--Madison, 1984. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 91-94).

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

Thesis (Ph. D.)--University of Wisconsin--Madison, 1984. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 82-83).

Thesis--University of Wisconsin--Madison. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 99-101).

Thesis (M.S.)--University of North Carolina at Greensboro, 2007. / Title from PDF title page screen. Advisor: Theresa Vaughan; submitted to the Dept. of Mathematics and Statistics. Includes bibliographical references (p. 56).

This thesis presents the results of an investigation into the use of puzzle-based combinatorial structures for erasure correction purposes. The research encompasses two main combinatorial structures: the well-known number placement puzzle Sudoku and a novel three component construction designed specifically with puzzle-based erasure correction in mind. The thesis describes the construction of outline erasure correction schemes incorporating each of the two structures. The research identifies that both of the structures contain a number of smaller sub-structures, the removal of which results in a grid with more than one potential solution - a detrimental property for erasure correction purposes. Extensive investigation into the properties of these sub-structures is carried out for each of the two outline erasure correction schemes, and results are determined that indicate that, although the schemes are theoretically feasible, the prevalence of sub-structures results in practically infeasible schemes. The thesis presents detailed classifications for the different cases of sub-structures observed in each of the outline erasure correction schemes. The anticipated similarities in the sub-structures of Sudoku and sub-structures of Latin Squares, an established area of combinatorial research, are observed and investigated, the proportion of Sudoku puzzles free of small sub-structures is calculated and a simulation comparing the recovery rates of small sub-structure free Sudoku and standard Sudoku is carried out. The analysis of sub-structures for the second erasure correction scheme involves detailed classification of a variety of small sub-structures; the thesis also derives probabilistic lower bounds for the expected numbers of case-specific sub-structures within the puzzle structure, indicating that specific types of sub-structure hinder recovery to such an extent that the scheme is infeasible for practical erasure correction. The consequences of complex cell inter-relationships and wider issues with puzzle-based erasure correction, beyond the structures investigated in the thesis are also discussed, concluding that while there are suggestions in the literature that Sudoku and other puzzle-based combinatorial structures may be useful for erasure correction, the work of this thesis suggests that this is not the case.

Three concepts of ordinal numbers are examined with a view to their intuitiveriess and existence in two principle systems of axiomatic set theory. The first is based on equivalence classes of the similarity relation between well-ordered sets. Two alternatives are suggested in later chapters for overcoming the problems arizing from this definition. Next, ordinal numbers are defined as certain representatives of these equivalence classes,, and one of several such possible definitions is taken for proving the fundamental properties of these ordinals. Finally, a generalization of Peano's axioms provides us with a method of defining ordinal numbers which are the ultimate result of abstractions. / Science, Faculty of / Mathematics, Department of / Graduate