Beginning with a short biography of Kurt Godel, "Godel's Theorem in Focus" provides the reader with a clear guide to the mechanics of Godel's proof in a format intelligible to the non-mathematician. The book moves on to explanations of the mechanics of Godel's proof and its significance for mathematical logic and the philosophy of mathematics. In the final section, S. G. Shanker presents a major new critique of Godel's theorem.

This book deals with the determinants of linear operators in Euclidean, Hilbert and Banach spaces. Determinants of operators give us an important tool for solving linear equations and invertibility conditions for linear operators, enable us to describe the spectra, to evaluate the multiplicities of eigenvalues, etc. We derive upper and lower bounds, and perturbation results for determinants, and discuss applications of our theoretical results to spectrum perturbations, matrix equations, two parameter eigenvalue problems, as well as to differential, difference and functional-differential equations.

This book brings together the scattered literature associated with the seemingly unrelated regression equations (SURE) model used by econometricians and others. It focuses on the theoretical statistical results associated with the SURE model.

Fuzzy knowledge and fuzzy systems affect our lives today as systems enter the world of commerce. Fuzzy systems are incorporated in domestic appliances (washing machine, air conditioning, microwave, telephone) and in transport systems (a pilotless helicopter has recently completed a test flight). Future applications are expected to have dramatic implications for the demand for labor, among other things. It was with such thoughts in mind that this first international survey of future applications of fuzzy logic has been undertaken. The results are likely to be predictive for a decade beyond the millenium. The predictive element is combined with a bibliography which serves as an historical anchor as well as being both extensive and extremely useful. Analysis and Evaluation of Fuzzy Systems is thus a milestone in the development of fuzzy logic and applications of three representative subsystems: Fuzzy Control, Fuzzy Pattern Recognition and Fuzzy Communications.

This book includes articles on denotational semanitcs, recursion theoretic aspects of computer science, model theory and algebra, automath and automated reasoning, stability theory, topoi and mathematics, and topoi and logic. It is intended for mathematical logicians and computer scientists.

This book explores the limits of our knowledge. The author shows how uncertainty and indefiniteness not only define the borders confining our understanding, but how they feed into the process of discovery and help to push back these borders. Starting with physics the author collects examples from economics, neurophysiology, history, ecology and philosophy. The first part shows how information helps to reduce indefiniteness. Understanding rests on our ability to find the right context, in which we localize a problem as a point in a network of connections. New elements must be combined with the old parts of the existing complex knowledge system, in order to profit maximally from the information. An attempt is made to quantify the value of information by its ability to reduce indefiniteness. The second part explains how to handle indefiniteness with methods from fuzzy logic, decision theory, hermeneutics and semiotics. It is not sufficient that the new element appears in an experiment, one also has to find a theoretical reason for its existence. Indefiniteness becomes an engine of science, which gives rise to new ideas.

First published in 1990, this book consists of a detailed exposition of results of the theory of "interpretation" developed by G. Kreisel - the relative impenetrability of which gives the elucidation contained here great value for anyone seeking to understand his work. It contains more complex versions of the information obtained by Kreisel for number theory and clustering around the no-counter-example interpretation, for number-theorectic forumulae provide in ramified analysis. It also proves the omega-consistency of ramified analysis. The author also presents proofs of Schutte's cut-elimination theorems which are based on his consistency proofs and essentially contain them - these went further than any published work up to that point, helping to squeeze the maximum amount of information from these proofs.

Published with the aid of a grant from the National Endowment for the Humanities. Contains the only complete English-language text of The Concept of Truth in Formalized Languages. Tarski made extensive corrections and revisions of the original translations for this edition, along with new historical remarks. It includes a new preface and a new analytical index for use by philosophers and linguists as well as by historians of mathematics and philosophy.

Das Buch behandelt Matrizengleichungen und -funktionen sowie die computergerechte Darstellung und Losung der Bewegungsgleichungen von Schwingungssystemen mit endlich vielen Freiheitsgraden und fuhrt in die Grundlagen der Naherungsmethoden von Rayleigh und Ritz ein. Das Eigenwertproblem wird, anders als sonst ublich, von einem allgemeinen Standpunkt aus betrachtet. Dadurch gewinnt die Darstellung an Verstandlichkeit und an Anwendungsbreite. Das Buch ist sowohl fur Studierende als auch fur Physiker und Ingenieure in der Praxis geschrieben.

The Only Undergraduate Textbook to Teach Both Classical and Virtual Knot Theory An Invitation to Knot Theory: Virtual and Classical gives advanced undergraduate students a gentle introduction to the field of virtual knot theory and mathematical research. It provides the foundation for students to research knot theory and read journal articles on their own. Each chapter includes numerous examples, problems, projects, and suggested readings from research papers. The proofs are written as simply as possible using combinatorial approaches, equivalence classes, and linear algebra. The text begins with an introduction to virtual knots and counted invariants. It then covers the normalized f-polynomial (Jones polynomial) and other skein invariants before discussing algebraic invariants, such as the quandle and biquandle. The book concludes with two applications of virtual knots: textiles and quantum computation.

Category theory is unmatched in its ability to organize and layer abstractions and to find commonalities between structures of all sorts. No longer the exclusive preserve of pure mathematicians, it is now proving itself to be a powerful tool in science, informatics, and industry. By facilitating communication between communities and building rigorous bridges between disparate worlds, applied category theory has the potential to be a major organizing force. This book offers a self-contained tour of applied category theory. Each chapter follows a single thread motivated by a real-world application and discussed with category-theoretic tools. We see data migration as an adjoint functor, electrical circuits in terms of monoidal categories and operads, and collaborative design via enriched profunctors. All the relevant category theory, from simple to sophisticated, is introduced in an accessible way with many examples and exercises, making this an ideal guide even for those without experience of university-level mathematics.

The present study is an extension of the topic introduced in Dr. Hailperin's Sentential Probability Logic, where the usual true-false semantics for logic is replaced with one based more on probability, and where values ranging from 0 to 1 are subject to probability axioms. Moreover, as the word "sentential" in the title of that work indicates, the language there under consideration was limited to sentences constructed from atomic (not inner logical components) sentences, by use of sentential connectives ("no," "and," "or," etc.) but not including quantifiers ("for all," "there is"). An initial introduction presents an overview of the book. In chapter one, Halperin presents a summary of results from his earlier book, some of which extends into this work. It also contains a novel treatment of the problem of combining evidence: how does one combine two items of interest for a conclusion-each of which separately impart a probability for the conclusion-so as to have a probability for the conclusion based on taking both of the two items of interest as evidence? Chapter two enlarges the Probability Logic from the first chapter in two respects: the language now includes quantifiers ("for all," and "there is") whose variables range over atomic sentences, not entities as with standard quantifier logic. (Hence its designation: ontological neutral logic.) A set of axioms for this logic is presented. A new sentential notion-the suppositional-in essence due to Thomas Bayes, is adjoined to this logic that later becomes the basis for creating a conditional probability logic. Chapter three opens with a set of four postulates for probability on ontologically neutral quantifier language. Many properties are derived and a fundamental theorem is proved, namely, for any probability model (assignment of probability values to all atomic sentences of the language) there will be a unique extension of the probability values to all closed sentences of the language.

This volume, which presents the cumulation of the authors' research in the field, deals with Lidstone, Hermite, Abel-Gontscharoff, Birkhoff, piecewise Hermite and Lidstone, spline and Lidstone-spline interpolating problems. Explicit representations of the interpolating polynomials and associated error functions are given, as well as explicit error inequalities in various norms. Numerical illustrations are provided of the importance and sharpness of the various results obtained. Also demonstrated are the significance of these results in the theory of ordinary differential equations such as maximum principles, boundary value problems, oscillation theory, disconjugacy and disfocality. The book should be useful for mathematicians, numerical analysts, computer scientists and engineers.

This book presents a philosophy of science, based on panenmentalism: an original modal metaphysics, which is realist about individual pure (non-actual) possibilities and rejects the notion of possible worlds. The book systematically constructs a new and novel way of understanding and explaining scientific progress, discoveries, and creativity. It demonstrates that a metaphysics of individual pure possibilities is indispensable for explaining and understanding mathematics and natural sciences. It examines the nature of individual pure possibilities, actualities, mind-dependent and mind-independent possibilities, as well as mathematical entities. It discusses in detail the singularity of each human being as a psychical possibility. It analyses striking scientific discoveries, and illustrates by means of examples of the usefulness and vitality of individual pure possibilities in the sciences.

Fuzzy social choice theory is useful for modeling the uncertainty and imprecision prevalent in social life yet it has been scarcely applied and studied in the social sciences. Filling this gap, Application of Fuzzy Logic to Social Choice Theory provides a comprehensive study of fuzzy social choice theory. The book explains the concept of a fuzzy maximal subset of a set of alternatives, fuzzy choice functions, the factorization of a fuzzy preference relation into the "union" (conorm) of a strict fuzzy relation and an indifference operator, fuzzy non-Arrowian results, fuzzy versions of Arrow's theorem, and Black's median voter theorem for fuzzy preferences. It examines how unambiguous and exact choices are generated by fuzzy preferences and whether exact choices induced by fuzzy preferences satisfy certain plausible rationality relations. The authors also extend known Arrowian results involving fuzzy set theory to results involving intuitionistic fuzzy sets as well as the Gibbard-Satterthwaite theorem to the case of fuzzy weak preference relations. The final chapter discusses Georgescu's degree of similarity of two fuzzy choice functions.

Aimed at graduate students and research logicians and mathematicians, this much-awaited text covers over forty years of work on relative classification theory for non-standard models of arithmetic. With graded exercises at the end of each chapter, the book covers basic isomorphism invariants: families of types realized in a model, lattices of elementary substructures and automorphism groups. Many results involve applications of the powerful technique of minimal types due to Haim Gaifman, and some of the results are classical but have never been published in a book form before.

This volume, first published in 2000, presents a classical approach to the foundations and development of the geometry of vector fields, describing vector fields in three-dimensional Euclidean space, triply-orthogonal systems and applications in mechanics. Topics covered include Pfaffian forms, systems in n-dimensional space, and foliations and their Godbillion-Vey invariant. There is much interest in the study of geometrical objects in n-dimensional Euclidean space and this volume provides a useful and comprehensive presentation.

This text explores the many transformations that the mathematical proof has undergone from its inception to its versatile, present-day use, considering the advent of high-speed computing machines. Though there are many truths to be discovered in this book, by the end it is clear that there is no formalized approach or standard method of discovery to date. Most of the proofs are discussed in detail with figures and equations accompanying them, allowing both the professional mathematician and those less familiar with mathematics to derive the same joy from reading this book.

This book argues for a view in which processes of dialogue and interaction are taken to be foundational to reasoning, logic, and meaning. This is both a continuation, and a substantial modification, of an inferentialist approach to logic. As such, the book not only provides a critical introduction to the inferentialist view, but it also provides an argument that this shift in perspective has deep and foundational consequences for how we understand the nature of logic and its relationship with meaning and reasoning. This has been upheld by several technical results, including, for example a novel approach to logical paradox and logical revision, and an account of the internal justification of logical rules. The book shows that inferentialism is greatly strengthened, such that it can answer the most stringent criticisms of the view. This leads to a view of logic that emphasizes the dynamics of reasoning, provides a novel account of the justification and normativity of logical rules, thus leading to a new, attractive approach to the foundations of logic. The book addresses readers interested in philosophy of language, philosophical and mathematical logic, theories of reasoning, and also those who actively engage in current debates involving, for example, logical revision, and the relationship between logic and reasoning, from advanced undergraduates, to professional philosophers, mathematicians, and linguists.

Convergence of Blockchain, AI and IoT: A Digital Platform discusses the convergence of three powerful technologies that play into the digital revolution and blur the lines between biological, digital, and physical objects. This book covers novel algorithms, solutions for addressing issues in applications, security, authentication, and privacy. Discusses innovative technological upgradation and significant challenges in the current era Gives an overview of clinical scientific research that enables smart diagnosis through artificial intelligence Provides an insight into how disruptive technology enabled with the self-running devices and protection mechanism is involved in an augmented reality with blockchain mechanism Talks about neural science being capable of enhancing deep brain waves to predict an overall improvement in human thoughts and behaviours Covers the digital currency mechanism in detail Enhances the knowledge of the readers about smart contract and ledger mechanism with artificial intelligence and blockchain mechanism Targeted audiences range from those interested in the technical revolution of blockchain, big data and the Internet of Things, to research scholars and the professional market.

Introduction to Mathematical Modeling and Chaotic Dynamics focuses on mathematical models in natural systems, particularly ecological systems. Most of the models presented are solved using MATLAB (R). The book first covers the necessary mathematical preliminaries, including testing of stability. It then describes the modeling of systems from natural science, focusing on one- and two-dimensional continuous and discrete time models. Moving on to chaotic dynamics, the authors discuss ways to study chaos, types of chaos, and methods for detecting chaos. They also explore chaotic dynamics in single and multiple species systems. The text concludes with a brief discussion on models of mechanical systems and electronic circuits. Suitable for advanced undergraduate and graduate students, this book provides a practical understanding of how the models are used in current natural science and engineering applications. Along with a variety of exercises and solved examples, the text presents all the fundamental concepts and mathematical skills needed to build models and perform analyses.

This volume brings together a selection of Solomon Feferman's most important recent writings, covering the relation between logic and mathematics, proof theory, objectivity and intensionality in mathematics, and key issues in the work of Gödel, Hilbert, and Turing.

This book digs deeper and shows not only that quantum gravity is more than just a physical theory-describing physical aspects-but also that, in fact, it covers "it all."

Descriptive complexity theory establishes a connection between the computational complexity of algorithmic problems (the computational resources required to solve the problems) and their descriptive complexity (the language resources required to describe the problems). This groundbreaking book approaches descriptive complexity from the angle of modern structural graph theory, specifically graph minor theory. It develops a 'definable structure theory' concerned with the logical definability of graph theoretic concepts such as tree decompositions and embeddings. The first part starts with an introduction to the background, from logic, complexity, and graph theory, and develops the theory up to first applications in descriptive complexity theory and graph isomorphism testing. It may serve as the basis for a graduate-level course. The second part is more advanced and mainly devoted to the proof of a single, previously unpublished theorem: properties of graphs with excluded minors are decidable in polynomial time if, and only if, they are definable in fixed-point logic with counting.

In topology the three basic concepts of metrics, topologies and uniformities have been treated so far as separate entities by means of different methods and terminology. This work treats all three concepts as a special case of the concept of approach spaces. This theory provides an answer to natural questions in the interplay between topological and metric spaces by introducing a well suited supercategory of TOP and MET. The theory makes it possible to equip initial structures of metricizable topological spaces with a canonical structure, preserving the numerical information of the metrics. It provides a solid basis for approximation theory, turning ad hoc notions into canonical concepts, and it unifies topological and metric notions. The book explains the richness of approach structures in detail; it provides a comprehensive explanation of the categorical set-up, develops the basic theory and provides many examples, displaying links with various areas of mathematics such as approximation theory, probability theory, analysis and hyperspace theory. This book is intended for lecturers, researchers and graduate students in the following areas: topology, categorical theory, category th