This book is for graduate students and researchers, introducing modern foundational research in mathematics, computer science, and philosophy from an interdisciplinary point of view. Its scope includes Predicative Foundations, Constructive Mathematics and Type Theory, Computation in Higher Types, Extraction of Programs from Proofs, and Algorithmic Aspects in Financial Mathematics. By filling the gap between (under-)graduate level textbooks and advanced research papers, the book gives a sch...

This rigorous textbook is intended for a year-long analysis or advanced calculus course for advanced undergraduate or beginning graduate students. Starting with detailed, slow-paced proofs that allow students to acquire facility in reading and writing proofs, it clearly and concisely explains the basics of differentiation and integration of functions of one and several variables, and covers the theorems of Green, Gauss, and Stokes. Minimal prerequisites are assumed, and relevant linear alg...

This valuable book focuses on a collection of powerful methods of analysis that yield deep number-theoretical estimates. Particular attention is given to counting functions of prime numbers and multiplicative arithmetic functions. Both real variable (”elementary”) and complex variable (”analytic”) methods are employed. The reader is assumed to have knowledge of elementary number theory (abstract algebra will also do) and real and complex analysis. Specialized analytic techniques, including...

This text covers topics in algebraic geometry and commutative algebra with a strong perspective toward practical and computational aspects. The first four chapters form the core of the book. A comprehensive chart in the Preface illustrates a variety of ways to proceed with the material once these chapters are covered. In addition to the fundamentals of algebraic geometry—the elimination theorem, the extension theorem, the closure theorem and the Nullstellensatz—this new edition incorporate...

This handbook with exercises reveals in formalisms, hitherto mainly used for hardware and software design and verification, unexpected mathematical beauty. The lambda calculus forms a prototype universal programming language, which in its untyped version is related to Lisp, and was treated in the first author's classic The Lambda Calculus (1984). The formalism has since been extended with types and used in functional programming (Haskell, Clean) and proof assistants (Coq, Isabelle, HOL), u...

This book presents the author's research on automatic learning procedures for categorial grammars of natural languages. The research program spans a number of intertwined disciplines, including syntax, semantics, learnability theory, logic, and computer science. The theoretical framework employed is an extension of categorial grammar that has come to be called multimodal or type-logical grammar. The first part of the book presents an expository summary of how grammatical sentences of any l...

This volume is the first ever collection devoted to the field of proof-theoretic semantics. Contributions address topics including the systematics of introduction and elimination rules and proofs of normalization, the categorial characterization of deductions, the relation between Heyting's and Gentzen's approaches to meaning, knowability paradoxes, proof-theoretic foundations of set theory, Dummett's justification of logical laws, Kreisel's theory of constructions, paradoxical reasoning, ...

Generative Grammar presents a substantial contribution to the field of linguistics in drawing together for the first time the author's most significant work on the theory of generative grammar.The essays collected here display Freidin's role in moving the theory forward in terms of new proposals, and analyse the efforts to understand the evolution and history of the theory by careful investigation of how and why it has changed over the years.

Discriminant equations are an important class of Diophantine equations with close ties to algebraic number theory, Diophantine approximation and Diophantine geometry. This book is the first comprehensive account of discriminant equations and their applications. It brings together many aspects, including effective results over number fields, effective results over finitely generated domains, estimates on the number of solutions, applications to algebraic integers of given discriminant, powe...

This book introduces an important group of logics that have come to be known under the umbrella term 'susbstructural'. Substructural logics have independently led to significant developments in philosophy, computing and linguistics. An Introduction to Substrucural Logics is the first book to systematically survey the new results and the significant impact that this class of logics has had on a wide range of fields.The following topics are covered:* Proof Theory

The book attempts to point out the interconnections between number theory and algebra with a view to making a student understand certain basic concepts in the two areas forming the subject-matter of the book.

This volume presents a multi-dimensional collection of articles highlighting recent developments in commutative algebra. It also includes an extensive bibliography and lists a substantial number of open problems that point to future directions of research in the represented subfields. The contributions cover areas in commutative algebra that have flourished in the last few decades and are not yet well represented in book form. Highlighted topics and research methods include Noetherian and ...