Minimal Degrees Of Unsolvability And The Full Approximation Construction

For the purposes of this monograph, "by a degree" is meant a degree of recursive unsolvability. A degree [script bold]m is said to be minimal if 0 is the unique degree less than [script bold]m. Each of the six chapters of this self-contained monograph is devoted to the proof of an existence theorem for minimal degrees.

Complex interactions of economic, technological, political, and cultural factors have fed the rise of criminal networks worldwide. At the same time, global illegal activities depend on a world of social realities to function. Organized Crime moves beyond traditional concepts of "evil forces" corrupting their host societies, instead analyzing local, national, and international manifestations of organized crime in the situational contexts that aid in its development. The contributors provide up-to-date understanding of various aspects of organized crime, in both classic areas of research (drugs, sex trafficking, labor racketeering) and emerging areas of interest (diamond smuggling, money laundering, eco-crime), in locales as varied as Italy, Quebec, the Sinai, Bulgaria, and the world's tropical rain forests. Topics are explored from a variety of perspectives, including sociology, criminology, political science, and anthropology, giving this book empirical breadth and depth rarely seen in the literature. A sampling of the topics: Symbolic and economic meanings of crime to cultures. The symbiotic relationships between legitimate and criminal activities. Ethical dilemmas of legitimate businesses with criminal clients. Marketing, problem-solving, recruitment: organizational models of criminal enterprises. Innovative law enforcement/administrative strategies for containing and preventing crime in the U.S. and across Europe. Scholars and researchers of organized crime as well as advanced students of criminology will welcome Organized Crime for coverage that is wide-ranging, comparative, and specific enough to match their interests

This volume presents a systematic study of the interaction between local and global degree theory. It introduces the reader to the fascinating combinatorial methods of recursion theory while simultaneously showing how to use these methods to prove global theorems about degrees.

We prove two main results: (1) [lowercase Greek]Omega + 1 is an initial segment of the degrees ([less-than or equal to]≤ a̲ for any r.e. a̲ [not equal to] 0̲ , and (2) given any 0̲ [less than] a̲ [less than] h̲ where a̲ and h̲ are r.e. and h̲ is high, there is a minimal degree m̲ [less than] h̲ such that m̲ [set-theoretic union] a̲ = h̲.

The third in a series of four books presenting the seminal papers from the Caltech-UCLA 'Cabal Seminar'.

This book constitutes the refereed proceedings of the 6th International Conference on Theory and Applications of Models of Computation, TAMC 2009, held in Changsha, China in May 2009. The 39 full papers presented together with 7 invited papers as well as 3 plenary talks were selected from 86 submissions. The papers address the three main themes of the conference which were Computability, Complexity, and Algorithms. The conference aimed to bring together researchers with interests in theoretical computer science, algorithmic mathematics, and applications to the physical sciences.

The chapters of this volume all have their own level of presentation. The topics have been chosen based on the active research interest associated with them. Since the interest in some topics is older than that in others, some presentations contain fundamental definitions and basic results while others relate very little of the elementary theory behind them and aim directly toward an exposition of advanced results. Presentations of the latter sort are in some cases restricted to a short survey of recent results (due to the complexity of the methods and proofs themselves). Hence the variation in level of presentation from chapter to chapter only reflects the conceptual situation itself. One example of this is the collective efforts to develop an acceptable theory of computation on the real numbers. The last two decades has seen at least two new definitions of effective operations on the real numbers.