This is a first course in propositional modal logic, suitable for mathematicians, computer scientists and philosophers. Emphasis is placed on semantic aspects, in the form of labelled transition structures, rather than on proof theory. The book covers all the basic material - propositional languages, semantics and correspondence results, proof systems and completeness results - as well as some topics not usually covered in a modal logic course. It is written from a mathematical standpoint. To help the reader, the material is covered in short chapters, each concentrating on one topic. These are arranged into five parts, each with a common theme. An important feature of the book is the many exercises, and an extensive set of solutions is provided.

This book addresses two-person zero-sum finite games in which the payoffs in any situation are expressed with fuzzy numbers. The purpose of this book is to develop a suite of effective and efficient linear programming models and methods for solving matrix games with payoffs in fuzzy numbers. Divided into six chapters, it discusses the concepts of solutions of matrix games with payoffs of intervals, along with their linear programming models and methods. Furthermore, it is directly relevant to the research field of matrix games under uncertain economic management. The book offers a valuable resource for readers involved in theoretical research and practical applications from a range of different fields including game theory, operational research, management science, fuzzy mathematical programming, fuzzy mathematics, industrial engineering, business and social economics.

This book examines the philosophical conception of abductive reasoning as developed by Charles S. Peirce, the founder of American pragmatism. It explores the historical and systematic connections of Peirce's original ideas and debates about their interpretations. Abduction is understood in a broad sense which covers the discovery and pursuit of hypotheses and inference to the best explanation. The analysis presents fresh insights into this notion of reasoning, which derives from effects to causes or from surprising observations to explanatory theories. The author outlines some logical and AI approaches to abduction as well as studies various kinds of inverse problems in astronomy, physics, medicine, biology, and human sciences to provide examples of retroductions and abductions. The discussion covers also everyday examples with the implication of this notion in detective stories, one of Peirce's own favorite themes. The author uses Bayesian probabilities to argue that explanatory abduction is a method of confirmation. He uses his own account of truth approximation to reformulate abduction as inference which leads to the truthlikeness of its conclusion. This allows a powerful abductive defense of scientific realism. This up-to-date survey and defense of the Peircean view of abduction may very well help researchers, students, and philosophers better understand the logic of truth-seeking.

This book gathers the proceedings of the conference "Cultures of Mathematics and Logic," held in Guangzhou, China. The event was the third in a series of interdisciplinary, international conferences emphasizing the cultural components of philosophy of mathematics and logic. It brought together researchers from many disciplines whose work sheds new light on the diversity of mathematical and logical cultures and practices. In this context, the cultural diversity can be diachronical (different cultures in different historical periods), geographical (different cultures in different regions), or sociological in nature.

Computability and Logic has become a classic because of its accessibility to students without a mathematical background and because it covers not simply the staple topics of an intermediate logic course, such as Godel's incompleteness theorems, but also a large number of optional topics, from Turing's theory of computability to Ramsey's theorem. This 2007 fifth edition has been thoroughly revised by John Burgess. Including a selection of exercises, adjusted for this edition, at the end of each chapter, it offers a simpler treatment of the representability of recursive functions, a traditional stumbling block for students on the way to the Godel incompleteness theorems. This updated edition is also accompanied by a website as well as an instructor's manual.

Introduces the GUHA method of mechanizing hypothesis formation as a data mining tool. Presents examples of data mining with enhanced association rules, histograms, contingency tables and action rules. Provides examples of data mining for exception rules and examples of subgroups discovery. Outlines possibilities of GUHA in business intelligence and big data. Overviews related theoretical results and challenges related to mechanizing hypothesis formation.

This book outlines a vast array of techniques and methods regarding model categories, without focussing on the intricacies of the proofs. Quillen model categories are a fundamental tool for the understanding of homotopy theory. While many introductions to model categories fall back on the same handful of canonical examples, the present book highlights a large, self-contained collection of other examples which appear throughout the literature. In particular, it collects a highly scattered literature into a single volume. The book is aimed at anyone who uses, or is interested in using, model categories to study homotopy theory. It is written in such a way that it can be used as a reference guide for those who are already experts in the field. However, it can also be used as an introduction to the theory for novices.

We humans are collectively driven by a powerful - yet not fully explained - instinct to understand. We would like to see everything established, proven, laid bare. The more important an issue, the more we desire to see it clarified, stripped of all secrets, all shades of gray. What could be more important than to understand the Universe and ourselves as a part of it? To find a window onto our origin and our destiny? This book examines how far our modern cosmological theories - with their sometimes audacious models, such as inflation, cyclic histories, quantum creation, parallel universes - can take us towards answering these questions. Can such theories lead us to ultimate truths, leaving nothing unexplained? Last, but not least, Heller addresses the thorny problem of why and whether we should expect to find theories with all-encompassing explicative power.

This volume presents the main results of the 4th International Conference on Multivariate Approximation, which was held at Witten-Bommerholz, September 24-29, 2000. Nineteen selected, peer-reviewed contributions cover recent topics in constructive approximation on varieties, approximation by solutions of partial differential equations, application of Riesz bases and frames, multiwavelets and subdivision.
Features and Topics:
interpolation and approximation on compact sets, kergin interpolation
error asymptotics
radial basis functions
energy minimizing configurations on the sphere
quadrature and cubature formulae
harmonic functions near a zero
blending functions
frames and approximation of inverse frame operators
The book is an essential resource for researchers and graduates in applied mathematics, computer science and geophysics who are interested in the state-of-the-art developments in multivariate approximation.

Soft computing encompasses various computational methodologies, which, unlike conventional algorithms, are tolerant of imprecision, uncertainty, and partial truth. Soft computing technologies offer adaptability as a characteristic feature and thus permit the tracking of a problem through a changing environment. Besides some recent developments in areas like rough sets and probabilistic networks, fuzzy logic, evolutionary algorithms, and artificial neural networks are core ingredients of soft computing, which are all bio-inspired and can easily be combined synergetically.This book presents a well-balanced integration of fuzzy logic, evolutionary computing, and neural information processing. The three constituents are introduced to the reader systematically and brought together in differentiated combinations step by step. The text was developed from courses given by the authors and offers numerous illustrations as

A compilation of papers presented at the 2001 European Summer Meeting of the Association for Symbolic Logic, Logic Colloquium '01 includes surveys and research articles from some of the world's preeminent logicians. Two long articles are based on tutorials given at the meeting and present accessible expositions of research in two active areas of logic, geometric model theory and descriptive set theory of group actions. The remaining articles cover seperate research topics in many areas of mathematical logic, including applications in Computer Science, Proof Theory, Set Theory, Model Theory, Computability Theory, and aspects of Philosophy. This collection will be of interest not only to specialists in mathematical logic, but also to philosophical logicians, historians of logic, computer scientists, formal linguists and mathematicians in the areas of algebra, abstract analysis and topology. A number of the articles are aimed at non-specialists and serve as good introductions for graduate students.

This book presents the construction and resolution of 50 practical optimization problems and covers an exceptionally wide range, including games-associated problems (Unblock Me, Sudokus), logistical problems, and problems concerning plant distribution, production, operations scheduling, management and resource allocation. The problems are divided into 5 difficulty levels. Problems in the first few levels are focused on learning the model construction methodology, while those in the last level include complex optimization environments. For each problem solution, the specific steps are illustrated, promoting reader comprehension. In addition, all the models are implemented in an optimization library, LINGO, their solutions have been analyzed and their correct construction has been verified. The book also includes a simple guide to implementing models in LINGO in a straightforward manner and in any input data format (text files, spreadsheets or databases). As an ideal companion to the author's previously published work Modelling in Mathematical Programming, the book is intended as a basic tool for students of operations research, and for researchers in any advanced area involving mathematical programming.

The book draws on Prof. Perkowitz's career as successful researcher, teacher, and writer and his broad interests to give him unique insights into how science and scientists connect with general culture and society. The book is especially strong in its coverage of science and art, and science in film. Illustrations from Hollywood films and independent and experimental films increase the book's appeal. The book's mix of varied topics in science and technology, and of short and long pieces written in accessible style, will appeal to general readers.

Professor Atiyah is one of the greatest living mathematicians and is well known throughout the mathematical world. He is a recipient of the Fields Medal, the mathematical equivalent of the Nobel Prize, and is still at the peak of his career. His huge number of published papers, focusing on the areas of algebraic geometry and topology, have here been collected into six volumes, divided thematically for easy reference by individuals interested in a particular subject. From 1977 onwards his interest moved in the direction of gauge theories and the interaction between geometry and physics.

One of the greatest mathematicians in the world, Michael Atiyah has earned numerous honors, including a Fields Medal, the mathematical equivalent of the Nobel Prize. While the focus of his work has been in the areas of algebraic geometry and topology, he has also participated in research with theoretical physicists. For the first time, these volumes bring together Atiyah's collected papers--both monographs and collaborative works-- including those dealing with mathematical education and current topics of research such as K-theory and gauge theory. The volumes are organized thematically. They will be of great interest to research mathematicians, theoretical physicists, and graduate students in these areas.

This book presents a new nominalistic philosophy of mathematics: semantic conventionalism. Its central thesis is that mathematics should be founded on the human ability to create language - and specifically, the ability to institute conventions for the truth conditions of sentences. This philosophical stance leads to an alternative way of practicing mathematics: instead of "building" objects out of sets, a mathematician should introduce new syntactical sentence types, together with their truth conditions, as he or she develops a theory. Semantic conventionalism is justified first through criticism of Cantorian set theory, intuitionism, logicism, and predicativism; then on its own terms; and finally, exemplified by a detailed reconstruction of arithmetic and real analysis. Also included is a simple solution to the liar paradox and the other paradoxes that have traditionally been recognized as semantic. And since it is argued that mathematics is semantics, this solution also applies to Russell's paradox and the other mathematical paradoxes of self-reference. In addition to philosophers who care about the metaphysics and epistemology of mathematics or the paradoxes of self-reference, this book should appeal to mathematicians interested in alternative approaches.

The International Biometric Society (IBS) was formed at the First International Biometric Conference at Woods Hole on September 6, 1947. The History of the International Biometric Society presents a deep dive into the voluminous archival records, with primary focus on IBS's first fifty years. It contains numerous photos and extracts from the archival materials, and features many photos of important leaders who served IBS across the decades. Features: Describes events leading up to and at Woods Hole on September 6, 1947 that led to the formation of IBS Outlines key markers that shaped IBS after the 1947 formation through to the modern day Describes the regional and national group structure, and the formation of regions and national groups Describes events surrounding the key scientific journal of IBS, Biometrics, including the transfer of ownership to IBS, content, editors, policies, management, and importance Describes the other key IBS publications - Biometric Bulletin, Journal of Agricultural Biological and Environmental Statistics, and regional publications Provides details of International Biometric Conferences and key early symposia Describes IBS constitution and by-laws processes, and the evolution of business arrangements Provides a record of international officers, including regional presidents, national group secretaries, journal editors, and the locations of meetings Includes a gallery of international Presidents, and a gallery of Secretaries and Treasurers The History of the International Biometric Society will appeal to anyone interested in the activities of our statistical and biometrical forebearers. The focus is on issues and events that engaged the attention of the officers of IBS. Some of these records are riveting, some entertaining, some intriguing, and some colorful. Some of the issues covered were difficult to handle, but even these often resulted in changes that benefited IBS.

Digital forensics plays a crucial role in identifying, analysing, and presenting cyber threats as evidence in a court of law. Artificial intelligence, particularly machine learning and deep learning, enables automation of the digital investigation process. This book provides an in-depth look at the fundamental and advanced methods in digital forensics. It also discusses how machine learning and deep learning algorithms can be used to detect and investigate cybercrimes. This book demonstrates digital forensics and cyber-investigating techniques with real-world applications. It examines hard disk analytics and style architectures, including Master Boot Record and GUID Partition Table as part of the investigative process. It also covers cyberattack analysis in Windows, Linux, and network systems using virtual machines in real-world scenarios. Digital Forensics in the Era of Artificial Intelligence will be helpful for those interested in digital forensics and using machine learning techniques in the investigation of cyberattacks and the detection of evidence in cybercrimes.

This book is a comprehensive examination of the conception, perception, performance, and composition of time in music across time and culture. It surveys the literature of time in mathematics, philosophy, psychology, music theory, and somatic studies (medicine and disability studies) and looks ahead through original research in performance, composition, psychology, and education. It is the first monograph solely devoted to the theory of construction of musical time since Kramer in 1988, with new insights, mathematical precision, and an expansive global and historical context. The mathematical methods applied for the construction of musical time are totally new. They relate to category theory (projective limits) and the mathematical theory of gestures. These methods and results extend the music theory of time but also apply to the applied performative understanding of making music. In addition, it is the very first approach to a constructive theory of time, deduced from the recent theory of musical gestures and their categories. Making Musical Time is intended for a wide audience of scholars with interest in music. These include mathematicians, music theorists, (ethno)musicologists, music psychologists / educators / therapists, music performers, philosophers of music, audiologists, and acousticians.

Decision Theory An Introduction to Dynamic Programming and Sequential Decisions John Bather University of Sussex, UK Mathematical induction, and its use in solving optimization problems, is a topic of great interest with many applications. It enables us to study multistage decision problems by proceeding backwards in time, using a method called dynamic programming. All the techniques needed to solve the various problems are explained, and the author's fluent style will leave the reader with an avid interest in the subject.
* Tailored to the needs of students of optimization and decision theory
* Written in a lucid style with numerous examples and applications
* Coverage of deterministic models: maximizing utilities, directed networks, shortest paths, critical path analysis, scheduling and convexity
* Coverage of stochastic models: stochastic dynamic programming, optimal stopping problems and other special topics
* Coverage of advanced topics: Markov decision processes, minimizing expected costs, policy improvements and problems with unknown statistical parameters
* Contains exercises at the end of each chapter, with hints in an appendix
Aimed primarily at students of mathematics and statistics, the lucid text will also appeal to engineering and science students and those working in the areas of optimization and operations research.

The starting point for this monograph is the previously unknown connection between the Continuum Hypothesis and the saturation of the non-stationary ideal on 1; and the principle result of this monograph is the identification of a canonical model in which the Continuum Hypothesis is false. This is the first example of such a model and moreover the model can be characterized in terms of maximality principles concerning the universal-existential theory of all sets of countable ordinals. This model is arguably the long sought goal of the study of forcing axioms and iterated forcing but is obtained by completely different methods, for example no theory of iterated forcing whatsoever is required. The construction of the model reveals a powerful technique for obtaining independence results regarding the combinatorics of the continuum, yielding a number of results which have yet to be obtained by any other method. This monograph is directed to researchers and advanced graduate students in Set Theory. The second edition is updated to take into account some of the developments in the decade since the first edition appeared, this includes a revised discussion of -logic and related matters.

This monograph provides a modern introduction to the theory of quantales. First coined by C.J. Mulvey in 1986, quantales have since developed into a significant topic at the crossroads of algebra and logic, of notable interest to theoretical computer science. This book recasts the subject within the powerful framework of categorical algebra, showcasing its versatility through applications to C*- and MV-algebras, fuzzy sets and automata. With exercises and historical remarks at the end of each chapter, this self-contained book provides readers with a valuable source of references and hints for future research. This book will appeal to researchers across mathematics and computer science with an interest in category theory, lattice theory, and many-valued logic.

This unique and contemporary text not only offers an introduction to proofs with a view towards algebra and analysis, a standard fare for a transition course, but also presents practical skills for upper-level mathematics coursework and exposes undergraduate students to the context and culture of contemporary mathematics. The authors implement the practice recommended by the Committee on the Undergraduate Program in Mathematics (CUPM) curriculum guide, that a modern mathematics program should include cognitive goals and offer a broad perspective of the discipline. Part I offers: An introduction to logic and set theory. Proof methods as a vehicle leading to topics useful for analysis, topology, algebra, and probability. Many illustrated examples, often drawing on what students already know, that minimize conversation about "doing proofs." An appendix that provides an annotated rubric with feedback codes for assessing proof writing. Part II presents the context and culture aspects of the transition experience, including: 21st century mathematics, including the current mathematical culture, vocations, and careers. History and philosophical issues in mathematics. Approaching, reading, and learning from journal articles and other primary sources. Mathematical writing and typesetting in LaTeX. Together, these Parts provide a complete introduction to modern mathematics, both in content and practice. Table of Contents Part I - Introduction to Proofs Logic and Sets Arguments and Proofs Functions Properties of the Integers Counting and Combinatorial Arguments Relations Part II - Culture, History, Reading, and Writing Mathematical Culture, Vocation, and Careers History and Philosophy of Mathematics Reading and Researching Mathematics Writing and Presenting Mathematics Appendix A. Rubric for Assessing Proofs Appendix B. Index of Theorems and Definitions from Calculus and Linear Algebra Bibliography Index Biographies Danilo R. Diedrichs is an Associate Professor of Mathematics at Wheaton College in Illinois. Raised and educated in Switzerland, he holds a PhD in applied mathematical and computational sciences from the University of Iowa, as well as a master's degree in civil engineering from the Ecole Polytechnique Federale in Lausanne, Switzerland. His research interests are in dynamical systems modeling applied to biology, ecology, and epidemiology. Stephen Lovett is a Professor of Mathematics at Wheaton College in Illinois. He holds a PhD in representation theory from Northeastern University. His other books include Abstract Algebra: Structures and Applications (2015), Differential Geometry of Curves and Surfaces, with Tom Banchoff (2016), and Differential Geometry of Manifolds (2019).

Chaitin, the inventor of algorithmic information theory, presents in this book the strongest possible version of Goedel's incompleteness theorem, using an information theoretic approach based on the size of computer programs. One half of the book is concerned with studying the halting probability of a universal computer if its program is chosen by tossing a coin. The other half is concerned with encoding the halting probability as an algebraic equation in integers, a so-called exponential diophantine equation.

Linear Logic is a branch of proof theory which provides refined tools for the study of the computational aspects of proofs. These tools include a duality-based categorical semantics, an intrinsic graphical representation of proofs, the introduction of well-behaved non-commutative logical connectives, and the concepts of polarity and focalisation. These various aspects are illustrated here through introductory tutorials as well as more specialised contributions, with a particular emphasis on applications to computer science: denotational semantics, lambda-calculus, logic programming and concurrency theory. The volume is rounded-off by two invited contributions on new topics rooted in recent developments of linear logic. The book derives from a summer school that was the climax of the EU Training and Mobility of Researchers project "Linear Logic in Computer Science." It is an excellent introduction to some of the most active research topics in the area.