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

Thesis (Ph. D.)--University of Wisconsin--Madison, 1970. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 1997. / Includes bibliographical references. Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2005. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file viewed on (November 14, 2006) Vita. Includes bibliographical references.

Thesis (Ph. D.)--University of Texas at Austin, 2007. / Vita. Includes bibliographical references.

Thesis (Ph.D.)--Hong Kong University of Science and Technology, 2007. / Includes bibliographical references (leaves 90-92). Also available in electronic version.

This thesis extends the study of the notion of termination equivalence of abstract structures first proposed by Kfoury. The connection with abstract data types (ADTs) is made by demonstrating that many kinds of equivalence between ADT implementations are in fact instances of termination equivalence between their underlying algebras. The results in the thesis extend the original work in two directions. The first is to consider how the termination equivalence of structures is dependent upon the choice of programming formalism. The termination equivalences for all of the common classes of programs and for some new classes of non-computable schemes are studied, and their relative strengths are established. The other direction is a study of a congruence property of equivalences relative to the join or addition datatype building operation. We decide which of the termination equivalences are congruences for all structures and for all computable structures, and for those equivalences which are not, we characterise those congruences closest to them (both stronger and weaker). These programmes of work involved the use of constructions and properties of structures relating to program termination which are of interest in themselves. These are examined and are used to prove some general results about the relative strengths of termination equivalences.

The overall goal of this work is to combine the complementary contributions of algebra and coalgebra to specification, in order to provide a formal framework for the specification of state-based, dynamical systems. Algebraic specification methods benefit from the availability of inductive techniques for defining and reasoning about structures that involve computation; coalgebraic specification methods complement algebraic ones both in their objectives and in their means of achieving them, by employing coinductive techniques for defining and reasoning about structures that involve observation. State-based, dynamical systems comprise a computational aspect, concerned with the construction of (new) system states, and an observational aspect, concerned with the observation of (existing) system states, with the two aspects overlapping on features concerned with the evolution of system states. Existing formalisms for the specification of such systems typically exploit the overlap between computational and observational features to employ either algebraic or coalgebraic techniques for specification and reasoning. However, such a choice limits the expressiveness of these formalisms w.r.t. either observational or computational features. Furthermore, the accounts given by such approaches to the concepts of indistinguishability by observations and respectively of reachability under computations are somewhat artificial, due to the failure to distinguish between computational and observational features. The approach taken here is to clearly separate the two categories of features (by shifting the features concerned with the evolution of system states to the computational component), and to use algebra and respectively coalgebra in formalising them. In particular, such an approach yields a coalgebraically-defined notion of indistinguishability by observations, and an algebraically-defined notion of reachability under computations. The relationship between computing new states and observing the resulting states is specified by suitably lifting the coalgebraic structure of the semantic domains induced by the observational component to computations over these semantic domains. Such an approach automatically results in a compatibility between computational and observational features, with the observational indistinguishability of states being preserved by computations, and with the reachability of states under computations being preserved by observations. Correctness properties of system behaviour are formalised using equational sentences. This is a standard technique in algebraic specification. A similar technique is used here for coalgebraic specification, with the resulting notion of sentence capturing system invariants quantified over state spaces. Moreover, a sound and complete calculus for reasoning about the specified behaviours is formulated in a concrete setting obtained by syntactically dualising the setting of many-sorted algebra. Equational sentences are then used to formalise the equivalence of computations as well as various system invariants, with the associated notions of satisfaction abstracting away observationally indistinguishable and respectively unreachable states, and with the associated proof techniques employing coinduction and respectively induction. Suitably instantiating the resulting approach yields a formalism for the specification and verification of objects.

We construct a semi-orthogonal decomposition for the equivariant derived category of a hypersurface associated to the sum of two potentials. More specifically, if $f,g$ are two homogeneous poynomials of degree $d$ defining smooth Calabi-Yau or general type hypersurfaces in $\mathbb{P}^n$, we construct a semi-orthogonal decomposition of $D[V(f\oplus g)/\mu_d]$. Moreover, every component of the semi-orthogonal decomposition is explicitly given by Fourier-Mukai functors.

M.Sc.