The Shape of Space, Third Edition maintains the standard of excellence set by the previous editions. This lighthearted textbook covers the basic geometry and topology of two- and three-dimensional spaces-stretching students' minds as they learn to visualize new possibilities for the shape of our universe. Written by a master expositor, leading researcher in the field, and MacArthur Fellow, its informal exposition and engaging exercises appeal to an exceptionally broad audience, from liberal arts students to math undergraduate and graduate students looking for a clear intuitive understanding to supplement more formal texts, and even to laypeople seeking an entertaining self-study book to expand their understanding of space. Features of the Third Edition: Full-color figures throughout "Picture proofs" have replaced algebraic proofs Simpler handles-and-crosscaps approach to surfaces Updated discussion of cosmological applications Intuitive examples missing from many college and graduate school curricula About the Author: Jeffrey R. Weeks is a freelance geometer living in Canton, New York. With support from the U.S. National Science Foundation, the MacArthur Foundation and several science museums, his work spans pure mathematics, applications in cosmology and-closest to his heart-exposition for the general public.

This text deals with three basic techniques for constructing models of Zermelo-Fraenkel set theory: relative constructibility, Cohen's forcing, and Scott-Solovay's method of Boolean valued models. Our main concern will be the development of a unified theory that encompasses these techniques in one comprehensive framework. Consequently we will focus on certain funda mental and intrinsic relations between these methods of model construction. Extensive applications will not be treated here. This text is a continuation of our book, "I ntroduction to Axiomatic Set Theory," Springer-Verlag, 1971; indeed the two texts were originally planned as a single volume. The content of this volume is essentially that of a course taught by the first author at the University of Illinois in the spring of 1969. From the first author's lectures, a first draft was prepared by Klaus Gloede with the assistance of Donald Pelletier and the second author. This draft was then rcvised by the first author assisted by Hisao Tanaka. The introductory material was prepared by the second author who was also responsible for the general style of exposition throughout the text. We have inc1uded in the introductory material al1 the results from Boolean algebra and topology that we need. When notation from our first volume is introduced, it is accompanied with a deflnition, usually in a footnote. Consequently a reader who is familiar with elementary set theory will find this text quite self-contained.

This book grew out of lectures. It is intended as an introduction to classical two-valued predicate logic. The restriction to classical logic is not meant to imply that this logic is intrinsically better than other, non-classical logics; however, classical logic is a good introduction to logic because of its simplicity, and a good basis for applications because it is the foundation of classical mathematics, and thus of the exact sciences which are based on it. The book is meant primarily for mathematics students who are already acquainted with some of the fundamental concepts of mathematics, such as that of a group. It should help the reader to see for himself the advantages of a formalisation. The step from the everyday language to a formalised language, which usually creates difficulties, is dis cussed and practised thoroughly. The analysis of the way in which basic mathematical structures are approached in mathematics leads in a natural way to the semantic notion of consequence. One of the substantial achievements of modern logic has been to show that the notion of consequence can be replaced by a provably equivalent notion of derivability which is defined by means of a calculus. Today we know of many calculi which have this property."

This book contains fundamental concepts on discrete mathematical structures in an easy to understand style so that the reader can grasp the contents and explanation easily. The concepts of discrete mathematical structures have application to computer science, engineering and information technology including in coding techniques, switching circuits, pointers and linked allocation, error corrections, as well as in data networking, Chemistry, Biology and many other scientific areas. The book is for undergraduate and graduate levels learners and educators associated with various courses and progammes in Mathematics, Computer Science, Engineering and Information Technology. The book should serve as a text and reference guide to many undergraduate and graduate programmes offered by many institutions including colleges and universities. Readers will find solved examples and end of chapter exercises to enhance reader comprehension. Features Offers comprehensive coverage of basic ideas of Logic, Mathematical Induction, Graph Theory, Algebraic Structures and Lattices and Boolean Algebra Provides end of chapter solved examples and practice problems Delivers materials on valid arguments and rules of inference with illustrations Focuses on algebraic structures to enable the reader to work with discrete structures

The maths needed to succeed in AS and A Level Psychology is harder now than ever before. Suitable for all awarding bodies, this practical handbook covers all of the maths skills needed for the AS and A Level Psychology specifications. Worked examples, practice questions, 'remember points' and 'stretch yourself' questions give students the key knowledge and then the opportunity to practise and build confidence.

Reflecting many of the recent advances and trends in this area, Discrete Structures with Contemporary Applications covers the core topics in discrete structures as well as an assortment of novel applications-oriented topics. The applications described include simulations, genetic algorithms, network flows, probabilistic primality tests, public key cryptography, and coding theory. A modern and comprehensive introduction to discrete structures With clear definitions and theorems and carefully explained proofs, this classroom-tested text presents an accessible yet rigorous treatment of the material. Numerous worked-out examples illustrate key points while figures and tables help students grasp the more subtle and difficult concepts. "Exercises for the Reader" are interspersed throughout the text, with complete solutions included in an appendix. In addition to these, each section ends with extensive, carefully crafted exercise sets ranging from routine to nontrivial; answers can be found in another appendix. Most sections also contain computer exercises that guide students through the process of writing their own programs on any computing platform. Accommodates various levels of computer implementation Although the book highly encourages the use of computing platforms, it can be used without computers. The author explains algorithms in ordinary English and, when appropriate, in a natural and easy-to-understand pseudo code that can be readily translated into any computer language. A supporting website provides an extensive set of sample programs.

Using basic category theory, this Element describes all the central concepts and proves the main theorems of theoretical computer science. Category theory, which works with functions, processes, and structures, is uniquely qualified to present the fundamental results of theoretical computer science. In this Element, readers will meet some of the deepest ideas and theorems of modern computers and mathematics, such as Turing machines, unsolvable problems, the P=NP question, Kurt Goedel's incompleteness theorem, intractable problems, cryptographic protocols, Alan Turing's Halting problem, and much more. The concepts come alive with many examples and exercises.

Yearning for the Impossible: The Surprising Truth of Mathematics, Second Edition explores the history of mathematics from the perspective of the creative tension between common sense and the "impossible" as the author follows the discovery or invention of new concepts that have marked mathematical progress. The author puts these creations into a broader context involving related "impossibilities" from art, literature, philosophy, and physics. This new edition contains many new exercises and commentaries, clearly discussing a wide range of challenging subjects.

This book is the first book-length treatment of hybrid logic and its proof-theory. Hybrid logic is an extension of ordinary modal logic which allows explicit reference to individual points in a model. The extra expressive power is useful for many applications, for example, when reasoning about time one often wants to formulate a series of statements about what happens at specific times. There is little consensus about proof-theory for ordinary modal logic. Many modal-logical proof systems lack important properties and the relationships between proof systems for different modal logics are often unclear. The present book demonstrates that hybrid-logical proof-theory remedies this lack of uniformity in ordinary modal-logical proof systems. It considers a spectrum of different versions of hybrid logic (propositional, first-order, international first-order, and intuitionist) and of different types of proof-systems for hybrid-logic (natural deduction, Gentzen, tableaux, and axiom systems). All these systems can be motivated independently, but the fact that the systems can be given in a uniform way shows that hybrid logic and hybrid-logical proof theory is a natural enterprise.

Der Band 1A beginnt mit einem Vorwort zur Gesamtedition. Den Hauptteil des Bandes bilden Hausdorffs Arbeiten uber geordnete Mengen aus den Jahren 1901-1909. Diese haben der Entwicklung der Mengenlehre nachhaltige Impulse verliehen. Sie enthalten zahlreiche fur die Untersuchung geordneter Mengen grundlegende neue Begriffe sowie tiefliegendere Resultate. Alle diese Arbeiten sind sorgfaltig kommentiert. Die Kommentare zeigen, dass einige von Hausdorff's Ideen und Resultaten fur die moderne Grundlagenforschung hochaktuell sind.

Ferner enthalt der Band Hausdorff's kritische Besprechung von Russells "The Principles of Mathematics," aus dem Nachlass seine Vorlesung "Mengenlehre" von 1901 (eine der ersten Vorlesungen uber dieses Gebiet uberhaupt) sowie einen Essay "Hausdorff als akademischer Lehrer."

This book collects chapters which discuss interdisciplinary solutions to complex problems by using different approaches in order to save money, time and resources. The book presents the results on the recent advancements in artificial intelligence, computational intelligence, decision-making problems, emerging problems and practical achievements in the broad knowledge management field. q-ROFS is one of the hot topics for all the researchers, industrialists as well as academicians. This book is of interest to professionals and researchers working in the field of decision making and computational intelligence, as well as postgraduate and undergraduate students studying applications of fuzzy sets. The book helps solve different kinds of the decision-making problems such as medical diagnosis, pattern recognition, construction problems and technology selection under the uncertain fuzzy environment. Containing 19 chapters, the book begins by giving a topology of the q-ROFSs and their applications. It then progresses in a logical fashion, dedicating a chapter to each approach, including the generalized information measures for q-ROFSs, implementation of q-ROFSs to medical diagnosis, inventory model, multi-attribute decision-making and approaches to real-life industrial problems such as green campus transportation, social responsibility evaluation pattern and extensions of the q-ROFSs.

Real Scientists Don't Wear Ties links science to general and popular culture and everyday life in an easy-to-understand style. When a gifted writer of science selects his best pieces published in the world's most reputable periodicals such as Nature, Discover, and MIT Technology Review, we get an eminently readable collection of his varied work in book form. That it covers all-time relevant topics like quantum physics, gravitational waves, genetic engineering, space exploration, and artificial intelligence is an added delight. Prof. Perkowitz also discusses how science can be found in medical practice, cooking, soccer, and art, and also science and science fiction in the media. On the lighter side, he reports on his efforts to teach a computer to understand poetry, explains why scientists resist dressing up, and shows that unlike many people, scientists actually enjoy math.