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Conceptual Mathematics
This truly elementary book on categories introduces retracts, graphs, and adjoints to students and scientists.
Conceptual Mathematics
In the last 60 years, the use of the notion of category has led to a remarkable unification and simplification of mathematics. Conceptual Mathematics introduces this tool for the learning, development, and use of mathematics, to beginning students and also to practising mathematical scientists. This book provides a skeleton key that makes explicit some concepts and procedures that are common to all branches of pure and applied mathematics. The treatment does not presuppose knowledge of specific fields, but rather develops, from basic definitions, such elementary categories as discrete dynamical systems and directed graphs; the fundamental ideas are then illuminated by examples in these categories. This second edition provides links with more advanced topics of possible study. In the new appendices and annotated bibliography the reader will find concise introductions to adjoint functors and geometrical structures, as well as sketches of relevant historical developments.
Abstract Algebra and Famous Impossibilities
This textbook develops the abstract algebra necessary to prove the impossibility of four famous mathematical feats: squaring the circle, trisecting the angle, doubling the cube, and solving quintic equations. All the relevant concepts about fields are introduced concretely, with the geometrical questions providing motivation for the algebraic concepts. By focusing on problems that are as easy to approach as they were fiendishly difficult to resolve, the authors provide a uniquely accessible introduction to the power of abstraction. Beginning with a brief account of the history of these fabled problems, the book goes on to present the theory of fields, polynomials, field extensions, and irred...
Language and Equilibrium
A new framework that shows how to derive the meaning of an utterance from first principles by modeling it as a system of interdependent games. In Language and Equilibrium, Prashant Parikh offers a new account of meaning for natural language. He argues that equilibrium, or balance among multiple interacting forces, is a key attribute of language and meaning and shows how to derive the meaning of an utterance from first principles by modeling it as a system of interdependent games. His account results in a novel view of semantics and pragmatics and describes how both may be integrated with syntax. It considers many aspects of meaning—including literal meaning and implicature—and advances a detailed theory of definite descriptions as an application of the framework. Language and Equilibrium is intended for a wide readership in the cognitive sciences, including philosophers, linguists, and artificial intelligence researchers as well as neuroscientists, psychologists, and economists interested in language and communication.
Essential Mathematics for Undergraduates
This textbook covers topics of undergraduate mathematics in abstract algebra, geometry, topology and analysis with the purpose of connecting the underpinning key ideas. It guides STEM students towards developing knowledge and skills to enrich their scientific education. In doing so it avoids the common mechanical approach to problem-solving based on the repetitive application of dry formulas. The presentation preserves the mathematical rigour throughout and still stays accessible to undergraduates. The didactical focus is threaded through the assortment of subjects and reflects in the book’s structure. Part 1 introduces the mathematical language and its rules together with the basic buildi...
A Functorial Model Theory
This book is an introduction to a functorial model theory based on infinitary language categories. The author introduces the properties and foundation of these categories before developing a model theory for functors starting with a countable fragment of an infinitary language. He also presents a new technique for generating generic models with categories by inventing infinite language categories and functorial model theory. In addition, the book covers string models, limit models, and functorial models.
Number Theory
To observe the tenth anniversary of the founding of the Ramanujan Mathematical Society, an international conference on Discrete Mathematics and Number Theory was held in January 1996 in Tiruchirapalli, India. This volume contains proceedings from the number theory component of that conference. Papers are divided into four groups: arithmetic algebraic geometry, automorphic forms, elementary and analytic number theory, and transcendental number theory. This work deals with recent progress in current aspects of number theory and covers a wide variety of topics.
Game Equilibrium Models II
The four volumes of Game Equilibrium Models present applications of non-cooperative game theory. Problems of strategic interaction arising in biology, economics, political science and the social sciences in general are treated in 42 papers on a wide variety of subjects. Internationally known authors with backgrounds in various disciplines have contributed original research. The reader finds innovative modelling combined with advanced methods of analysis. The four volumes are the outcome of a research year at the Center for Interdisciplinary Studies of the University of Bielefeld. The close interaction of an international interdisciplinary group of researchers has produced an unusual collecti...
The Elgar Companion to Recent Economic Methodology
Economic methodology has traditionally been associated with logical positivism in the vein of Milton Friedman, Karl Popper, Imre Lakatos and Thomas Kuhn. However, the emergence and proliferation of new research programs in economics have stimulated many novel developments in economic methodology. This impressive Companion critically examines these advances in methodological thinking, particularly those that are associated with the new research programs which challenge standard economic methodology. Bringing together a collection of leading contributors to this new methodological thinking, the authors explain how it differs from the past and point towards further concerns and future issues. T...
