Thesis (M.S.)--University of North Carolina Wilmington, 2009. / Title from PDF title page (February 23, 2010)

Thesis--Gothenburg. / Includes bibliographical references (leaves 108-109).

Thesis--University of Wisconsin--Madison. / Typescript. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

No description available.

Assume AD+V=L(R). In the first chapter, let W^1_1 denote the club measure on \omega_1. We analyze the embedding j_{W^1_1}\restr HOD from the point of view of inner model theory. We use our analysis to answer a question of Jackson-Ketchersid about codes for ordinals less than \omega_\omega. In the second chapter, we provide an indiscernibles analysis for models of the form L[T_n,x]. We use our analysis to provide new proofs of the strong partition property on \delta^1_{2n+1}

We extend previous work on classifying o-minimal types, and develop several applications. Marker developed a dichotomy of o-minimal types into "cuts" and "noncuts," with a further dichotomy of cuts being either "uniquely" or "non-uniquely realizable." We use this classification to extend work by van den Dries and Miller on bounding growth rates of definable functions in Chapter 3, and work by Marker on constructing certain "small" extensions in Chapter 4. We further sub-classify "non-uniquely realizable cuts" into three categories in Chapter 2, and we give define the notion of a "decreasing" type in Chapter 5, which is a presentation of a type well-suited for our work. Using this definition, we achieve two results: in Chapter 5.2, we improve a characterization of definable types in o-minimal theories given by Marker and Steinhorn, and in Chapter 6 we answer a question of Speissegger's about extending a continuous function to the boundary of its domain. As well, in Chapter 5.3, we show how every elementary extension can be presented as decreasing.

In this thesis three topics on the model theory of partial differential fields are considered: the generalized Galois theory for partial differential fields, geometric axioms for the theory of partial differentially closed fields, and the existence and properties of the model companion of the theory of partial differential fields with an automorphism. The approach taken here to these subjects is to relativize the algebro geometric notions of prolongation and D-variety to differential notions with respect to a fixed differential structure. It is shown that every differential algebraic group which is not of maximal differential type is definably isomorphic to the sharp points of a relative D-group. Pillay's generalized finite dimensional differential Galois theory is extended to the possibly infinite dimensional partial setting. Logarithmic differential equations on relative D-groups are discussed and the associated differential Galois theory is developed. The notion of generalized strongly normal extension is naturally extended to the partial setting, and a connection betwen these extensions and the Galois extensions associated to logarithmic differential equations is established. A geometric characterization, in the spirit of Pierce-Pillay, for the theory DCF_{0,m+1} (differentially closed fields of characteristic zero with m+1 commuting derivations) is given in terms of the differential algebraic geometry of DCF_{0,m} using relative prolongations. It is shown that this characterization can be rephrased in terms of characteristic sets of prime differential ideals, yielding a first-order geometric axiomatization of DCF_{0,m+1}. Using the machinery of characteristic sets of prime differential ideals it is shown that the theory of partial differential fields with an automorphism has a model companion. Some basic model theoretic properties of this theory are presented: description of its completions, supersimplicity and elimination of imaginaries. Differential-difference modules are introduced and they are used, together with jet spaces, to establish the canonical base property for finite dimensional types, and consequently the Zilber dichotomy for minimal finite dimensional types.

Thesis (Ph. D.)--York University, 1999. Graduate Programme in Mathematics and Statistics. / Typescript. Includes bibliographical references (leaves 87-93). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://wwwlib.umi.com/cr/yorku/fullcit?pNQ43427.

No description available.

In this thesis three topics on the model theory of partial differential fields are considered: the generalized Galois theory for partial differential fields, geometric axioms for the theory of partial differentially closed fields, and the existence and properties of the model companion of the theory of partial differential fields with an automorphism. The approach taken here to these subjects is to relativize the algebro geometric notions of prolongation and D-variety to differential notions with respect to a fixed differential structure. It is shown that every differential algebraic group which is not of maximal differential type is definably isomorphic to the sharp points of a relative D-group. Pillay's generalized finite dimensional differential Galois theory is extended to the possibly infinite dimensional partial setting. Logarithmic differential equations on relative D-groups are discussed and the associated differential Galois theory is developed. The notion of generalized strongly normal extension is naturally extended to the partial setting, and a connection betwen these extensions and the Galois extensions associated to logarithmic differential equations is established. A geometric characterization, in the spirit of Pierce-Pillay, for the theory DCF_{0,m+1} (differentially closed fields of characteristic zero with m+1 commuting derivations) is given in terms of the differential algebraic geometry of DCF_{0,m} using relative prolongations. It is shown that this characterization can be rephrased in terms of characteristic sets of prime differential ideals, yielding a first-order geometric axiomatization of DCF_{0,m+1}. Using the machinery of characteristic sets of prime differential ideals it is shown that the theory of partial differential fields with an automorphism has a model companion. Some basic model theoretic properties of this theory are presented: description of its completions, supersimplicity and elimination of imaginaries. Differential-difference modules are introduced and they are used, together with jet spaces, to establish the canonical base property for finite dimensional types, and consequently the Zilber dichotomy for minimal finite dimensional types.