Abstract not available.

For a subset of a cardinal greater than ω1, fatness is strictly stronger than stationarity and strictly weaker than being closed unbounded. For many regular cardinals, being fat is a sufficient condition for having a closed unbounded subset in some generic extension. In this work we characterize fatness for subsets of Pκ(λ). We prove that for many regular cardinals κ and λ, a fat subset of Pκ(λ) obtains a closed unbounded subset in a cardinal-preserving generic extension. Additionally, we work out the conflict produced by two different definitions of fat subset of a cardinal, and introduce a novel (not model-theoretic) proof technique for adding a closed unbounded subset to a fat subset of a cardinal.

No description available.

<p>In this dissertation, we emphasize productiveness not just undecidability since pro- ductiveness implies constructive incompleteness. Analogues of Rice?s Theorem for different classes of languages are investigated, refined and generalized. In particular, several sufficient but general conditions are presented for predicates to be as hard as some widely discussed predicates such as ?= ?? and ?= {0,1}??. These conditions provide several general methods for proving complexity/productiveness results and apply to a large number of simple and natural predicates. As the first step in apply- ing these general methods, we investigate the complexity/productiveness of the pred- icates ?= ??, ?= {0,1}?? and other predicates that can be useful sources of many- one reductions for different classes of languages. Then we use very efficient many- one reductions of these basic source predicates to prove many new non-polynomial complexity lower bounds and productiveness results. Moreover, we study the com- plexity/productiveness of predicates for easily recognizable subsets of instances with important semantic properties. Because of the efficiency of our reductions, intuitively these reductions can preserve many levels of complexity. We apply our general methods to pattern languages [1] and multi-pattern lan- guages [2]. Interrelations between multi-pattern languages (or pattern languages) and standard classes of languages such as context-free languages and regular languages are studied. A way to study the descriptional complexity of standard language descriptors (for examples, context-free grammars and regular expressions) and multi-patterns is illustrated. We apply our general methods to several generalizations of regular ex- pressions. A productiveness result for the predicate ?= {0,1}?? is established for synchronized regular expressions [3]. Because of this, many new productiveness re- sults for synchronized regular expressions follow easily. We also apply our general methods to several classes of Lindenmayer systems [4] and of cellular automata [5]. A way of studying the complexity/productiveness of the 0Lness problem is developed and many new results follow from it. For real time one-way cellular automata, we observe that the predicates ?= ?? and ?= {0,1}?? are both productive. Because vi of this, many more general results are presented. For two-way cellular automata, we prove a strong meta-theorem and give a complete characterization for testing containment of any fixed two-way cellular automaton language. Finally, we generalize our methods and apply them to the theory of functions of real variables. In rings, the equivalence to identically 0 function problem which is an analogue of ?= ?? is studied. We show that the equivalence to identically 0 function problem for some classes of elementary functions is productive for different domains including open and closed bounded intervals of real numbers. Two initial results for real fields are also presented.

<p> Starting in the 1970s, Elliot&rsquo;s classification of AF -algebras and Brown-Douglas-Fillmore&rsquo;s classification of essentially normal operators created an explosion in the use of topological methods in the study of C * -algebras. Kasparov&rsquo;s introduction of KK-theory introduced more advanced machinery. This led to better existence and uniqueness theorems with applications in the classification program. In this thesis, I present such a uniqueness theorem with a proof as presented by Eilers-Dadarlat.</p>

<p>We define a ring R to be right cP-Baer if the right annihilator of a cyclic projective right R-module is generated by an idempotent. We also define a ring R to be right I-extending if each ideal generated by an idempotent is right essential in a direct summand of R. It is shown that the two conditions are equivalent in a semiprime ring. Next we define a right I-prime ring, which generalizes the prime condition. This condition is equivalent to all cyclic projective right R-modules being faithful. For a semiprime ring, we show the existence of a cP-Baer hull. We also provide some results about the p.q.-Baer hull and when it is equal to the cP-Baer hull. Polynomial and formal power series rings are studied with respect to the right cP-Baer condition. In general, a formal power series ring over one indeterminate in which its base ring is right p.q.-Baer ring is not necessarily right p.q.-Baer. However, if the base ring is right cP-Baer then the formal power series ring over one indeterminate is right cP-Baer. The fifth chapter is devoted to matrix extensions of right cP-Baer rings. A characterization of when a 2-by-2 generalized upper triangular matrix ring is right cP-Baer is given. The last major theorem is a decomposition of a cP-Baer ring, satisfying a finiteness condition, into a generalized triangular matrix ring with right I-prime rings down the main diagonal. Examples illustrating and delimiting our results are provided.

Dimension reduction methods have come to the forefront of many applications where the number of covariates, p, far exceed the sample size, N. For example, in survival analysis studies using microarray gene expression data, 10--30K expressions per patient are collected, but only a few hundred patients are available for the study. The focus of this work is on linear dimension reduction methods. Attention is given to the dimension reduction method of Random Projection (RP), in which the original p-dimensional data matrix X is projected onto a k-dimensional subspace using a random matrix Gamma. The motivation of RP is the Johnson-Lindenstrauss (JL) Lemma, which states that a set of N points in p-dimensional Euclidean space can be projected onto a k &ge; 24lnN3e2-2e 3 dimensional Euclidean space such that the pairwise distances between the points are preserved within a factor 1 +/- epsilon. In this work, the JL Lemma is revisited when the random matrix Gamma is defined as standard Gaussian and Achlioptas-typed. An improvement on the lower bound for k is provided by working directly with the distributions of the random distances rather than resorting to the moment generating function technique used in the literature. An improvement on the lower bound for k is also provided when using pairwise L2 distances in the space of the original points and pairwise L 1 distances in the space of the projected points. Another popular dimension reduction method is Partial Least Squares. In this work, a variant of Partial Least Squares is proposed, denoted by Rank-based Modified Partial Least Squares (RMPLS). The weight vectors of RMPLS can be seen to be the solution to an optimization problem. The method is insensitive to outlying values of both the response and the covariates, and takes into account the censoring information in the construction of its weight vectors. Results from simulation and real datasets under the Cox and Accelerated Failure Time (AFT) models indicate that RMPLS outperforms other leading methods for various measures when outliers are present in the response, and is comparable to other methods in the absence of outliers in the response.

<p>The family of algebraic surfaces X defined by the single equation [special characters omitted] over an algebraically closed field <i>k</i> of characteristic zero, where a₁, &hellip;, a_n are distinct, is studied. It is shown that this is a rational surface with a non-rational singularity at the origin. The ideal class group of the surface is computed. The terms of the Chase-Harrison-Rosenberg seven term exact sequence on the open complement of the ramification locus of <i>X</i>&rarr;[special characters omitted] are computed; the Brauer group is also studied in this unramified setting.</p><p> The analysis is extended to the surface <i>X&tilde;</i> obtained by blowing up <i>X</i> at the origin. The interplay between properties of <i>X&tilde;</i> , determined in part by the exceptional curve <i> E</i> lying over the origin, and the properties of <i>X</i> is explored. In particular, the implications that these properties have on the Picard group of the surface <i>X</i> are studied.</p>

<p> The study of factorization in integral domains has a long history. Unique factorization domains, like the integers, have been studied extensively for many years. More recently, mathematicians have turned their attention to generalizations of this such as Dedekind domains or other domains which have weaker factorization properties. Many authors have sought to generalize the notion of factorization in domains. One particular method which has encapsulated many of the generalizations into a single study is that of &tau;-factorization, studied extensively by A. Frazier and D.D. Anderson. </p><p> Another generalization comes in the form of studying factorization in rings with zero-divisors. Factorization gets quite complicated when zero-divisors are present due to the existence of several types of associate relations as well as several choices about what to consider the irreducible elements. </p><p> In this thesis, we investigate several methods for extending the theory of &tau;-factorization into rings with zero-divisors. We investigate several methods including: 1) the approach used by A.G. Ag&caron;arg&uuml;n and D.D. Anderson, S. Chun and S. Valdes-Leon in several papers; 2) the method of U-factorization developed by C.R. Fletcher and extended by M. Axtell, J. Stickles, and N. Baeth and 3) the method of regular factorizations and 4) the method of complete factorizations. </p><p> This thesis synthesizes the work done in the theory of generalized factorization and factorization in rings with zero-divisors. Along the way, we encounter several nice applications of the factorization theory. Using &tau;<i>_z</i>-factorizations, we discover a nice relationship with zero-divisor graphs studied by I. Beck as well as D.D. Anderson, D.F. Anderson, A. Frazier, A. Lauve, and P. Livingston. Using &tau;-U-factorization, we are able to answer many questions that arise when discussing direct products of rings. </p><p> There are several benefits to the regular factorization factorization approach due to the various notions of associate and irreducible coinciding on regular elements greatly simplifying many of the finite factorization property relationships. Complete factorization is a very natural and effective approach taken to studying factorization in rings with zero-divisors. There are several nice results stemming from extending &tau;-factorization in this way. Lastly, an appendix is provided in which several examples of rings satisfying the various finite factorization properties studied throughout the thesis are given.</p>

<p> In 1988, Mayer published a strong form of Vaught's Conjecture for o-minimal theories. She showed Vaught's Conjecture holds, and characterized the number of countable models of an o-minimal theory <i>T</i> if <i> T</i> has fewer than continuum many countable models. Friedman and Stanley have shown that several elementary classes are Borel complete. In this thesis we address the class of countable models of an o-minimal theory <i>T </i> when <i>T</i> has continuum many countable models. Our main result gives a model theoretic dichotomy describing the Borel complexity of isomorphism on the class of countable models of <i>T</i>. The first case is if <i>T</i> has no simple types, isomorphism is Borel on the class of countable models of <i>T</i>. In the second case, <i> T</i> has a simple type over a finite set <i>A</i>, and there is a finite set <i>B</i> containing <i>A</i> such that the class of countable models of the completion of <i>T </i>over <i> B</i> is Borel complete.</p>