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This thesis is a model-theoretic study of exponential differential equations in the context of differential algebra. I define the theory of a set of differential equations and give an axiomatization for the theory of the exponential differential equations of split semiabelian varieties. In particular, this includes the theory of the equations satisfied by the usual complex exponential function and the Weierstrass p-functions. The theory consists of a description of the algebraic structure on the solution sets together with necessary and sufficient conditions for a system of equations to have solutions. These conditions are stated in terms of a dimension theory; their necessity generalizes Ax’s differential field version of Schanuel’s conjecture and their sufficiency generalizes recent work of Crampin. They are shown to apply to the solving of systems of equations in holomorphic functions away from singularities, as well as in the abstract setting. The theory can also be obtained by means of a Hrushovski-style amalgamation construction, and I give a category-theoretic account of the method. Restricting to the usual exponential differential equation, I show that a “blurring” of Zilber’s pseudo-exponentiation satisfies the same theory. I conjecture that this theory also holds for a suitable blurring of the complex exponential maps and partially resolve the question, proving the necessity but not the sufficiency of the aforementioned conditions. As an algebraic application, I prove a weak form of Zilber’s conjecture on intersections with subgroups (known as CIT) for semiabelian varieties. This in turn is used to show that the necessary and sufficient conditions are expressible in the appropriate first order language.

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This work advances the use of nonclassical logics for developing qualitative models of real-world systems. Abstract mathematics is "qualitative" inasmuch as it relegates numerical considerations to the background and focuses explicitly on topological, algebraic, logical, or other types of conceptual forms. Mathematical logic, the present topic, serves to explicate alternative modes of reasoning for use in general research design and in model construction. The central thesis is that the theory of formal logical systems, and particularly, of logical systems based on nonclassical modes of reasoning, offers important new techniques for developing qualitative models of real-world systems. This thesis is supported in three major parts. Part I develops a semantically complete axiomatization of L. A. Zadeh's theory of approximate reasoning. This mode of reasoning is based on the conception of a "fuzzy set," by which means it yields a realistic representation of the "vagueness" ordinarily inherent in natural languages, such as English. All axiomatizations of this mode of reasoning to date have been deficient in that their linguistic structures are adequate for expressing only the simplest fuzzy linguistic ideas. The axiomatization developed herein goes beyond these limitations in a two-leveled formal system, which, at the inner level, is a multivalent logic that accommodates fuzzy assertions, and at the outer level, is a bivalent formalization of segments of the metalanguage. This system is adequate for expressing most of the basic fuzzy linguistic ideas, including: linguistic terms, hedges, and connectives; semantic equivalence and entailment; possibilistic reasoning; and linguistic truth. The final chapter of Part I applies the theory of approximate reasoning to a class of structural models for use in forecasting. The result is a direct mathematical link between the imprecision in a model and the uncertainty which that imprecision contributes to the model's forecasted events. Part II studies the systems of logical "form" which have beeen developed by G. Spencer-Brown and F. J. Varela. Spencer-Brown's "laws of form" is here shown to be essentially isomorphic with the axiomatized propositional calculus, and Varela's "calculus for self-reference" is shown to be isomorphically translatable into a system which axiomatizes a three-valued logic developed by S. C. Kleene. No semantically complete axiomatization of Kleene's logic has heretofore been known. Following on Kleene's original interpretation of his logic in the theory of partial recursion, this leads to a proof that Varela's concept of logical "autonomy" is exactly isomorphic with the notion of a "totally undecidable" partial recursive set. In turn, this suggests using Kleene-Varela type systems as formal tools for representing "mechanically unknowable" or empirically unverifiable system properties. Part III is an essay on the theoretical basis and methodological framework for implementing nonstandard logics in the modeling exercise. The evolution of mathematical logic is considered from the standpoint of its providing the opportunity to "select" alternative modes of reasoning. These general theoretical considerations serve to motivate the methodological ones, which begin by addressing the discussions of P. Suppes and M. Bunge regarding the role of formal systems in providing "the semantics of science." Bunge's work extends that of Suppes and is herein extended in turn to a study of the manner in which formal systems (both classical and nonclassical) can be implemented for mediation between the observer and the observed, i.e., for modeling. Whether real-world systems in fact obey the laws of one logic versus another must remain moot, but models based on alternative modes of reasoning to satisfy Bunge's criteria for empirical testability, and therefore do provide viable systems perspectives and methods of research.

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A Hamilton cycle is a cycle which passes through every vertex of a graph. A Hamilton cycle decomposition of a k-regular graph is defined as the partition of the edge set into Hamilton cycles if k is even, or a partition into Hamilton cycles and a 1-factor, if k is odd. Consequently, for 2-regular or 3-regular graphs, finding a Hamilton cycle decompositon is equilvalent to finding a Hamilton cycle. Two classes of graphs are studies in this thesis and both have significant symmetry. The first class of graphs is the 6-regular circulant graphs. These are a king of Cayley graph. Given a finite group A and a subset S ⊆ A, the Cayley Graph Cay(A,S) is the simple graph with vertex set A and edge set {{a, as}|a ∈ A, s ∈ S}. If the group A is cyclic then the graph is called a circulant graph. This thesis proves two results on 6-regular circulant graphs: 1. There is a Hamilton cycle decomposition of every 6-regular circulant graph Cay(Z[subscript n],S) in which S has an element of order n; 2. There is a Hamilton cycle decomposition of every connected 6-regular circulant graph of odd order. The second class of graphs examined in this thesis is a futher generalization of the Generalized Petersen graphs. The Petersen graph is well known as a highly symmetrical graph which does not contain a Hamilton cycle. In 1983 Alspach completely determined which Generalized Petersen graphs contain Hamilton cycles. In this thesis we define a larger class of graphs which includes the Generalized Petersen graphs as a special case. We call this larger class spoked Cayley graphs. We determine which spoked Cayley graphs on Abelian groups are Hamiltonian. As a corollary, we determine which are 1-factorable.