Mathematical logic Books

Book SynopsisThis book provides the first comprehensive treatment of Benford's law, the surprising logarithmic distribution of significant digits discovered in the late nineteenth century. Establishing the mathematical and statistical principles that underpin this intriguing phenomenon, the text combines up-to-date theoretical results with overviews of the law'Trade Review"This is a marvelous and excellent introduction."--Adhemar Bultheel, European Mathematical Society Bulletin "A must-read for novices and experts alike. It can be used for a graduate-level topics course or as a reference text for researchers in the field. The exposition is outstanding, with hundreds of carefully chosen examples, figures and diagrams to illustrate the theory. For those who are up for a challenge, the book contains several open problems as well. An Introduction to Benford's Law will surely be the go-to text on the subject for years to come."--Pieter C. Allaart, Mathematical ReviewsTable of ContentsPreface vii 1 Introduction 1 1.1 History 3 1.2 Empirical evidence 4 1.3 Early explanations 6 1.4 Mathematical framework 7 2 Significant Digits and the Significand 11 2.1 Significant digits 11 2.2 The significand 12 2.3 The significand sigma-algebra 14 3 The Benford Property 22 3.1 Benford sequences 23 3.2 Benford functions 28 3.3 Benford distributions and random variables 29 4 The Uniform Distribution and Benford's Law 43 4.1 Uniform distribution characterization of Benford's law 43 4.2 Uniform distribution of sequences and functions 46 4.3 Uniform distribution of random variables 54 5 Scale-, Base-, and Sum-Invariance 63 5.1 The scale-invariance property 63 5.2 The base-invariance property 74 5.3 The sum-invariance property 80 6 Real-valued Deterministic Processes 90 6.1 Iteration of functions 90 6.2 Sequences with polynomial growth 93 6.3 Sequences with exponential growth 97 6.4 Sequences with super-exponential growth 101 6.5 An application to Newton's method 111 6.6 Time-varying systems 116 6.7 Chaotic systems: Two examples 124 6.8 Differential equations 127 7 Multi-dimensional Linear Processes 135 7.1 Linear processes, observables, and difference equations 135 7.2 Nonnegative matrices 139 7.3 General matrices 145 7.4 An application to Markov chains 162 7.5 Linear difference equations 165 7.6 Linear differential equations 170 8 Real-valued Random Processes 180 8.1 Convergence of random variables to Benford's law 180 8.2 Powers, products, and sums of random variables 182 8.3 Mixtures of distributions 202 8.4 Random maps 213 9 Finitely Additive Probability and Benford's Law 216 9.1 Finitely additive probabilities 217 9.2 Finitely additive Benford probabilities 219 10 Applications of Benford's Law 223 10.1 Fraud detection 224 10.2 Detection of natural phenomena 225 10.3 Diagnostics and design 226 10.4 Computations and Computer Science 228 10.5 Pedagogical tool 230 List of Symbols 231 Bibliography 234 Index 245

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Book SynopsisA sweeping, in-depth history of NSA, whose famous “cult of silence” has left the agency shrouded in mystery for decades   The National Security Agency was born out of the legendary codebreaking programs of World War II that cracked the famed Enigma machine and other German and Japanese codes, thereby turning the tide of Allied victory. In the postwar years, as the United States developed a new enemy in the Soviet Union, our intelligence community found itself targeting not soldiers on the battlefield, but suspected spies, foreign leaders, and even American citizens. Throughout the second half of the twentieth century, NSA played a vital, often fraught and controversial role in the major events of the Cold War, from the Korean War to the Cuban Missile Crisis to Vietnam and beyond. In Code Warriors, Stephen Budiansky—a longtime expert in cryptology—tells the fascinating story of how NSA came to be, from its roots in World War II

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Book SynopsisAdvances in Modal Logic is a unique forum for presenting the latest results and new directions of research in modal logic. The topics dealt with are of interdisciplinary interest and range from mathematical, computational, and philosophical problems to applications in knowledge representation and formal linguistics.Volume 3 presents substantial advances in the relational model theory and the algorithmic treatment of modal logics. It contains invited and contributed papers from the third conference on “Advances in Modal Logic”, held at the University of Leipzig (Germany) in October 2000. It includes papers on dynamic logic, description logic, hybrid logic, epistemic logic, combinations of modal logics, tense logic, action logic, provability logic, and modal predicate logic.Table of ContentsFrom description to hybrid logics, and back, C. Areces and M. de Rijke; homophonic theory of truth for tense logic, Torben Brauner; weak necessity on weak Kleene matrices, F. Correia; bimodal logics for reasoning about continuous dynamics, J.M. Davoren and R.P. Gore; from bisimulation quantifiers to classifying toposes, S. Ghilardi and M. Zawadowski; normal products of modal logics, Y. Hasimoto; a tableau algorithm for the clique guarded fragment, C. Hirsch and S. Tobies; the complexity of reasoning with Boolean modal logics, C. Lutz and U. Sattler; outline of a logic of action, K. Segerberg; belief, names, and modes of presentation, R. Ye and M. Fitting. (Part contents)

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Book SynopsisThis volume is devoted to the main areas of mathematical logic and applications to computer science. There are articles on weakly o-minimal theories, algorithmic complexity of relations, models within the computable model theory, hierarchies of randomness tests, computable numberings, and complexity problems of minimal unsatisfiable formulas. The problems of characterization of the deduction-detachment theorem, Δ1-induction, completeness of Leśniewski's systems, and reduction calculus for the satisfiability problem are also discussed.The coverage includes the answer to Kanovei's question about the upper bound for the complexity of equivalence relations by convergence at infinity for continuous functions. The volume also gives some applications to computer science such as solving the problems of inductive interference of languages from the full collection of positive examples and some negative data, the effects of random negative data, methods of formal specification and verification on the basis of model theory and multiple-valued logics, interval fuzzy algebraic systems, the problems of information exchange among agents on the base topological structures, and the predictions provided by inductive theories.Table of ContentsAnother Characterization of the Deduction-Detachment Theorem (S V Babyonyshev); On Behavior of 2-Formulas in Weakly o-Minimal Theories (B S Baizhanov & B Sh Kulpeshov); Arithmetic Turing Degrees and Categorical Theories of Computable Models (E Fokina); Negative Data in Learning Languages (S Jain & E Kinber); Effective Cardinals in the Nonstandard Universe (V Kanovei & M Reeken); Model-Theoretic Methods of Analysis of Computer Arithmetic (S P Kovalyov); The Functional Completeness of Le niewski's Systems (F Lepage); Hierarchies of Randomness Tests (J Reimann & F Stephan); Intransitive Linear Temporal Logic Based on Integer Numbers, Decidability, Admissible Logical Consecutions (V V Rybakov); The Logic of Prediction (E Vityaev); Conceptual Semantic Systems Theory and Applications (K E Wolff); Complexity Results on Minimal Unsatisfiable Formulas (X Zhao); and other papers.

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Book SynopsisThis volume is a collection of written versions of the talks given at the Workshop on Computational Prospects of Infinity, held at the Institute for Mathematical Sciences from 18 June to 15 August 2005. It consists of contributions from many of the leading experts in recursion theory (computability theory) and set theory. Topics covered include the structure theory of various notions of degrees of unsolvability, algorithmic randomness, reverse mathematics, forcing, large cardinals and inner model theory, and many others.Table of ContentsPrompt Simplicity, Array Computability and Cupping (R Downey et al.); A Simpler Short Extenders Forcing-Gap 3 (M Gitik); The Strength of Some Combinatorial Principles Related to Ramsey's Theorem for Pairs (D R Hirschfeldt et al.); Absoluteness for Universally Baire Sets and the Uncountable II (I Farah et al.); Modaic Definability of Ordinals (I Neeman); Eliminating Concepts (A Nies); Diamonds on P (M Shioya); Rigidity and Biinterpretability in the Hyperdegrees (R A Shore); Some Fundamentals Issues Concerning Degrees or Unsolvability (S G Simpson); A tt Version of the Poner-Robinson Theorem (W H Woodin); and other papers.

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Book SynopsisThis book analyzes the generation of the arrow-categories of a given category, which is a foundational and distinguishable Category Theory phenomena, in analogy to the foundational role of sets in the traditional set-based Mathematics, for defi nition of natural numbers as well. This inductive transformation of a category into the infinite hierarchy of the arrowcategories is extended to the functors and natural transformations. The author considers invariant categorial properties (the symmetries) under such inductive transformations. The book focuses in particular on Global symmetry (invariance of adjunctions) and Internal symmetries between arrows and objects in a category (in analogy to Field Theories like Quantum Mechanics and General Relativity). The second part of the book is dedicated to more advanced applications of Internal symmetry to Computer Science: for Intuitionistic Logic, Untyped Lambda Calculus with Fixpoint Operators, Labeled Transition Systems in Process Algebras and Modal logics as well as Data Integration Theory.

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Book SynopsisCellular automata are a class of spatially and temporally discrete mathematical systems characterized by local interaction and synchronous dynamical evolution. Introduced by the mathematician John von Neumann in the 1950s as simple models of biological self-reproduction, they are prototypical models for complex systems and processes consisting of a large number of simple, homogeneous, locally interacting components. Cellular automata have been the focus of great attention over the years because of their ability to generate a rich spectrum of very complex patterns of behavior out of sets of relatively simple underlying rules. Moreover, they appear to capture many essential features of complex self-organizing cooperative behavior observed in real systems.This book provides a summary of the basic properties of cellular automata, and explores in depth many important cellular-automata-related research areas, including artificial life, chaos, emergence, fractals, nonlinear dynamics, and self-organization. It also presents a broad review of the speculative proposition that cellular automata may eventually prove to be theoretical harbingers of a fundamentally new information-based, discrete physics. Designed to be accessible at the junior/senior undergraduate level and above, the book will be of interest to all students, researchers, and professionals wanting to learn about order, chaos, and the emergence of complexity. It contains an extensive bibliography and provides a listing of cellular automata resources available on the World Wide Web.Table of Contents* Introduction: Preliminary Musings * Formalism * Phenomenological Studies of Generic CA * Dynamical Systems Theory Approach * Analytic Approach * Cellular Automata and Language Theory * Probabilistic CA * Generalized Models * CA Models of Fluid Dynamics * Neural Networks * Artificial-Life * Is Nature, Underneath It All, a CA?

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Book SynopsisThis fresh approach blends theory and connections with other parts of mathematics so that readers can understand the place of set theory within the wider context. The author, an experienced instructor, includes examples and exercises designed for undergraduate students. It is also a useful refresher for more advanced readers.Table of ContentsNote to the instructor; Acknowledgments; 1. Preliminaries; 2. ZFC; 3. Order; 4. Cardinality; 5. Trees; 6. Dense linear orderings; 7. Filters and ideals; Appendix. Summary of exercises on Boolean algebra; Index.

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Book SynopsisAn Introduction to Metalogic is a uniquely accessible introduction to the metatheory of first-order predicate logic. No background knowledge of logic is presupposed, as the book is entirely self-contained and clearly defines all of the technical terms it employs. Yaqub begins with an introduction to predicate logic and ends with detailed outlines of the proofs of the incompleteness, undecidability, and indefinability theorems, covering many related topics in between.Trade Review“This is an excellent text in intermediate logic that should be warmly received by students. The well-paced exposition is remarkably clear. The choice of material, including remarks on the philosophical significance of the principal theorems and welcome sections on Second-Order logic, seems just right. And the ample exercises at the end of each chapter, which range from easy to difficult, not only test the reader’s understanding, but move the subject forward. Highly recommended.” — David Keyt, Research Professor, University of Arizona; Professor Emeritus, University of Washington“An Introduction to Metalogic provides extensive coverage of the usual meta-theoretic results concerning the standard logical systems. It covers the completeness theorems, compactness, the Löwenheim-Skolem theorems, and the incompleteness of arithmetic. There is also a good sketch of more advanced topics, giving the enthusiastic reader and student incentive to keep studying the field. As with Professor Yaqub’s earlier logic text, An Introduction to Logical Theory, the presentation is clear and straightforward.” — Stewart Shapiro, Ohio State University“Yaqub has produced a pedagogically sensitive, yet completely rigorous tour of the introductory metalogical terrain. His textbook will be welcomed by instructors in philosophy, mathematics, computer science and linguistics who endeavor to offer their students something more substantial than standard introductory logic.” — Kai Wehmeier, University of California, IrvineTable of Contents Introduction Chapter One: First-Order Predicate Logic 1. The Syntax of PL 2. The Semantics of PL 3. Logical Concepts in PL 4. PL Proof Theory 5. Exercises Chapter Two: Resources of the Metatheory 1. Linguistic and Logical Resources 2. Arithmetical Resources 3. Set-Theoretic Resources 4. An Economical Version of PL 5. Exercises Chapter Three: The Soundness and Completeness Theorems 1. The Soundness Theorem 2. The Completeness Theorem 3. The Compactness Theorem 4. PL Interpretations and PL Sets 5. The Löwenheim-Skolem Theorem 6. Exercises Chapter Four: Computability 1. Effective Procedures and Computable Functions 2. Turing Computability 3. The Halting Problem 4. Partial Recursive Functions 5. Exercises Chapter Five: The Incompleteness Theorems 1. Peano Arithmetic 2. Representability in Peano Arithmetic 3. The Arithmetization of the Metatheory 4. Diagonalization and the First Incompleteness Theorem 5. Consequences of Diagonalization and Incompleteness 6. The Incompleteness of Second-Order Predicate Logic 7. Gödel's Second Incompleteness Theorem 8. Exercises

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Book SynopsisThis book covers a broad range of up-to-date issues in non-classical logic that are of interest not only to philosophical and mathematical logicians but also to computer scientists and researchers in artificial intelligence. The problems addressed range from methodological issues in paraconsistent and deontic logic to the revision theory of truth and infinite Turing machines. The book identifies a number of important current trends in contemporary non-classical logic. Among them are dialogical and substructural logic, the classification of concepts of negation, truthmaker theory, and mathematical and foundational aspects of modal and temporal logic.Table of ContentsFine-Grained Theories of Time (P Blackburn); Revision Sequences and Computers with an Infinite Amount of Time (B Lowe); On Frege's Nightmare: A Combination of Intuitionistic, Free and Paraconsistent Logics (S Rahman); Truthmakers, Entailment and Necessity (S Read); Global Definability in Basic Modal Logic (M de Rijke & H Sturm); Ackermann's Implication for Typefree Logic (K Robering); Why Dialogical Logic? (H Ruckert); Semantics for Constructive Negations (Y Shramko); Recent Trends in Paraconsistent Logic (M Urchs); Obligations, Authorities, and History Dependence (H Wansing).

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Book Synopsis''An invaluable companion for anyone who wants a deep understanding of what's under the hood of often inscrutable machines'' Melanie Mitchell A rich, narrative explanation of the mathematics that has brought us machine learning and the ongoing explosion of artificial intelligenceMachine-learning systems are making life-altering decisions for us: approving mortgage loans, determining whether a tumour is cancerous, or deciding whether someone gets bail. They now influence discoveries in chemistry, biology and physics - the study of genomes, extra-solar planets, even the intricacies of quantum systems.We are living through a revolution in artificial intelligence that is not slowing down. This major shift is based on simple mathematics, some of which goes back centuries: linear algebra and calculus, the stuff of eighteenth-century mathematics. Indeed by the mid-1850s, a lot of the groundwork was all done. It took the development of computer science and the kindling of 1990s computer chips designed for video games to ignite the explosion of AI that we see all around us today. In this enlightening book, Anil Ananthaswamy explains the fundamental maths behind AI, which suggests that the basics of natural and artificial intelligence might follow the same mathematical rules.As Ananthaswamy resonantly concludes, to make the most of our most wondrous technologies we need to understand their profound limitations - the clues lie in the maths that makes AI possible.

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Book SynopsisWhat is the necessary raison d'etre that puzzles serve? Why did they emerge at the same time in history as myth, magic and the occult arts? And why can't we put them down? This book presents an exploration of the philosophical implications of the puzzle instinct.Trade Review"Danesi, a professor of semiotics and anthropology (Univ. of Toronto), explores why puzzles, having arisen in earliest human history at the same time as mystery cults, are an intrinsic part of human life. Will Shortz, crossword puzzle editor of the New York Times, has suggested enigmatology as the study of the relationship between puzzles and culture. This book, which explores the puzzle genres that have survived over the years, is a contribution to that rubric. After first asking the question Why puzzles? (and developing several possible answers, among which is that they provide comic relief from unanswerable larger questions), Danesi devotes chapters to each of several types of puzzle. These include language puzzles (e.g., riddles and anagrams); pictures (e.g., optical illusions and mazes); logic (e.g., deductions and paradoxes); numbers (e.g., mathematical recreations); and games (e.g., chess). A final chapter synopsizes the discussion. A detailed list of references is included, as are solutions to the specific puzzles posed. The book is well written, has no mathematical prerequisites, and is quite suitable for a general audience as well as lower- and upper-division undergraduates." —D. Robbins, Trinity College (CT), Choice, December 2002Table of ContentsPreface 1. Why Puzzles? 2. Puzzling Language: Riddles, Anagrams, and Other Verbal Perplexities 3. Puzzling Pictures: Optical Illusions, Mazes, and Other Visual Mind-Bogglers 4. Puzzling Logic: Deductions, Paradoxes, and Other Forms of Mind Play 5. Puzzling Numbers: Magic Squares, Cryptarithms, and Other Mathematical Recreations 6. Puzzling Games: Chess, Checkers, and Other Games 7. The Puzzle of Life Solutions Bibliography and General Reading List Index

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Book SynopsisSelected Logic Papers, long out of print and now reissued with eight additional essays, includes much of the author's important work on mathematical logic and the philosophy of mathematics from the past sixty years.Trade Review[Quine] is at once the most elegant expounder of systematic logic in the older, pre-Gödelian style of Frege and Russell, the most distinguished American recruit to logical empiricism, probably the contemporary American philosopher most admired in the profession, and an original philosophical thinker of the first rank… This is an amazing feat of condensation with something solid to say in its brief scope about every major topic of interest in modern formal logic. * New York Review of Books *What [Quine] is expert in is, of course, logic… What [this book offers] is a view of the expert at work. Selected Logic Papers shows him actually doing logic… Logic is not a guide to life, but then Quine has never maintained that it was. It is a powerful adjunct to empirical inquiry, whose proper use requires prior discipline; its virtue lies in the fact that if we supply it with truth, it will never yield falsehood. Few have shown the manner of its use with more authority. * Partisan Review *This book is of continuing, not just historical interest. Quine is the greatest American philosopher of the twentieth century. His work in logic is inseparable from his work in other parts of philosophy. -- George Boolos, Massachusetts Institute of TechnologyTable of ContentsWhitehead and the Rise of Modern Logic (1941); Logic, Symbolic (1954); A Method of Generating Part of Arithmetic Without Use of Intuitive Logic (1934); Definition of Substitution (1936); Concatenation as a Basis for Arithmetic (1946); Set-theoretic Foundations for Logic (1936); Logic Based on Inclusion and Abstraction (1937); On Ordered Pairs and Relations (1945-46); On w-Inconsistency and a So-called Axiom of Infinity (1952); Element and Number (1941); On an Application of Tarski's Theory of Truth (1952); On Frege's Way Out (1954); Completeness of the Propositional Calculus (1937); On Cores and Prime Implicants of Truth Functions (1958); Two Theorems about Truth Functions (1951); On Boolean Functions (1949); On the Logic of Quantification (1945); A Proof Procedure for Quantification Theory (1954); Interpretations of Sets of Conditions (1953); Church's Theorem on the Decision Problem (1954); Quantification and the Empty Domain (1953); Reduction to a Dyadic Predicate (1953); Variables Explained Away (1960); Truth, Paradox, and Godel's Theorem (1992); Immanence and Validity (1991); MacHale on Boole (1985); Peirce's Logic (1989); Peano as Logician (1982); Free Logic, Description, and Virtual Classes (1994); The Inception of "New Foundations" (1987); Pythagorean Triples and Fermat's Last Theorem (1992).

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Book SynopsisSurveys the assortment of methods put forth for fixing Frege's system, in an attempt to determine just how much of mathematics can be reconstructed in each. This work considers every proposed fix, each with its distinctive philosophical advantages and drawbacks.Trade ReviewCo-Winner of the 2007 Shoenfield Prize, Association for Symbolic Logic "Fixing Frege fills a serious gap in the Frege's literature (always increasing but perhaps with an excessive attention paid to semantics and the philosophy of language) and should remain for a long time a necessary reference for scholars in the field."--Ignacio Angelelli, Review of Modern LogicTable of ContentsAcknowledgments ix CHAPTER 1: Frege, Russell, and After 1 CHAPTER 2: Predicative Theories 86 CHAPTER 3: Impredicative Theories 146 Tables 215 Notes 227 References 241 Index 249

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Book SynopsisWhat did mathematicians rely on for their work before then? And how did mathematical notations evolve into what we know today? This book explains the history behind the development of our mathematical notation system.Trade Review"Mazur (Euclid in the Rainforest) gives readers the fascinating history behind the mathematical symbols we use, and completely take for granted, every day. Mathematical notation turns numbers into sentences--or, to the uninitiated, a mysterious and impenetrable code. Mazur says the story of math symbols begins some 3,700 years ago, in ancient Babylon, where merchants incised tallies of goods on cuneiform tablets, along with the first place holder--a blank space. Many early cultures used letters for both numbers and an alphabet, but convenient objects like rods, fingers, and abacus beads, also proved popular. Mazur shows how our 'modern' system began in India, picking up the numeral 'zero' on its way to Europe, where it came into common use in the 16th century, thanks to travelers and merchants as well as mathematicians like Fibonacci. Signs for addition, subtraction, roots, and equivalence followed, but only became standardized through the influence of scientists and mathematicians like Rene Descartes and Gottfried Leibniz. Mazur's lively and accessible writing makes what could otherwise be a dry, arcane history as entertaining as it is informative."--Publishers Weekly "[A] fascinating narrative... This is a nuanced, intelligently framed chronicle packed with nuggets--such as the fact that Hindus, not Arabs, introduced Arabic numerals. In a word: enlightening."--George Szpiro, Nature "Mazur begins by illustrating how the ancient Incas and Mayans managed to write specific, huge numbers. Then, for more than 200 pages, he traces the history of division signs, square roots, pi, exponents, graph axes and other symbols in the context of cognition, communication, and analysis."--Washington Post "Mazur delivers a solid exposition of an element of mathematics that is fundamental to its history."--Library Journal "Mazur treats only a subset of F. Cajori's monumental A History of Mathematical Notation (Dover, 1993 first edition 1922) and there is overlap with many other mathematical history books, but Mazur adds new findings and insights and it is so much more entertaining ... and these features make it an interesting addition to the existing literature for anybody with only a slight interest in mathematics or its history."--European Mathematical Society "Symbols like '+' and '=' are so ingrained that it's hard to conceive of math without them. But a new book, Enlightening Symbols: A Short History of Mathematical Notation and its Hidden Power, offers a surprising reminder: Until the early 16th century, math contained no symbols at all."--Kevin Hartnett, Boston Globe "Enlightening Symbols retraces the winding road that has led to the way we now teach, study, and conceive mathematics... Thanks to Mazur's playful approach to the subject, Enlightening Symbols offers an enjoyable read."--Gaia Donati, Science "If you enjoy reading about history, languages and science, then you'll enjoy this book... The best part is the writing is compelling enough that you don't have to be a mathematician to enjoy this informative book."--Guardian.com's GrrlScientist blog "[I]nformative, highly readable and scholarly."--Brian Rotman, Literary Review "[T]his insightful account of the historical development of a highly characteristic feature of the mathematical enterprise also represents a valuable contribution to our understanding of the nature of mathematics."--Eduard Glas, Mathematical Reviews Clippings "Joseph Mazur's beautiful book Enlightening Symbols tells the story of human civilization through the development of mathematical notation. Surprises abound... The book is visually exquisite, great care having been taken with illustrations and figures. Mazur's discussion of the emergence of particular symbols affords the reader an overview of the often difficult primary literature."--Donal O'Shea, Sarasota Herald-Tribune "At whatever depth one chooses to read it, Enlightening Symbols has something for everyone. It is entertaining and eclectic, and Mazur's personal and easy style helps connect us with those who led the long and winding search for the best ways to quantify and analyze our world. Their success has liberated us from 'the shackles of our physical impressions of space'--and of the particular and the concrete--'enabling imagination to wander far beyond the tangible world we live in, and into the marvels of generality.'"--Robyn Arianrhod, Notices of the Notices of the American Mathematical Society "Mazur introduces the reader to major characters, weaves in relevant aspects of wider culture and gives a feel for the breadth of mathematical history. It is a useful book for both student and interested layperson alike."--Mark McCartney, London Mathematical Society "[T]his is a good book. It is well written by an experienced author and is full of interesting facts about how the symbols used in mathematics have arisen. It would certainly interest anyone who studies the history of mathematics."--Phil Dyke, Leonardo "Mazur is a master story teller."--John Stillwell, Bulletin of the American Mathematical SocietyTable of ContentsIntroduction ix Definitions xxi Note on the Illustrations xxiii Part 1 Numerals 1 1. Curious Beginnings 3 2. Certain Ancient Number Systems 10 3. Silk and Royal Roads 26 4. The Indian Gift 35 5. Arrival in Europe 51 6. The Arab Gift 60 7. Liber Abbaci 64 8. Refuting Origins 73 Part 2 Algebra 81 9. Sans Symbols 85 10. Diophantus's Arithmetica 93 11. The Great Art 109 12. Symbol Infancy 116 13. The Timid Symbol 127 14. Hierarchies of Dignity 133 15. Vowels and Consonants 141 16. The Explosion 150 17. A Catalogue of Symbols 160 18. The Symbol Master 165 19. The Last of the Magicians 169 Part 3 The Power of Symbols 177 20. Rendezvous in the Mind 179 21. The Good Symbol 189 22. Invisible Gorillas 192 23. Mental Pictures 210 24. Conclusion 216 Appendix A Leibniz's Notation 221 Appendix B Newton's Fluxion of xn 223 Appendix C Experiment 224 Appendix D Visualizing Complex Numbers 228 Appendix E Quaternions 230 Acknowledgments 233 Notes 235 Index 269

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Book SynopsisHow can reposting on Twitter kill a movie's opening weekend? How can you use mathematics to find your celebrity look-alike? What is Homer Simpson's method for disproving Fermat's Last Theorem? This title deals with these questions.Trade ReviewOne of Choice's Outstanding Academic Titles for 2014 "[Math Bytes] manages to fulfill its aim of providing a 'tasty byte of math and computing' while following a clear, concise, and to-the-point format... [P]rovides interesting insights to some of the complex, highly sophisticated algorithms that lie behind some of our most widely used technological resources."--Rachael Skyner, Science "The author's humanistic, sincere enjoyment of communicating his selections permeates and I think defines the book... The choice of topics and exposition details make the book entertainingly relevant."--Alexander Bogomolny, Cut the Knot "Math Bytes is fun and full of puzzles and brainteasers in a mashup of mathematics, social media, and pop culture referencing Beyonce, Google, and Twitter among others... This reviewer is simply a fool for math puzzles (though some readers might stop at simply a fool). Math Bytes scratches that itch quite nicely."--Robert Schaefer, New York Journal of Books "Printed on glossy paper and filled with colourful photographs, illustrations and diagrams on nearly every page, this book discusses mathematical ideas that can help us make sense of the digital world. The author's offbeat viewpoint comes from a truly unusual background; he studied mime with the legendary Marcel Marceau whilst pursuing his doctorate in applied maths."--Devorah Bennu, GrrlScientist, The Guardian "Tim Chartier, a professor of mathematics at Davidson College, has put together a delightful book of recreational mathematics. His presentation of a large array of topics is accompanied by excellent graphics, many in color... In sum, a fun book."--Philip J. Davis, SIAM News "The mathematical topics are interesting and informative; if you teach at the college level, you are likely to find some interesting fodder for your courses here. You also may, as I did, learn some new things yourself... This book, which is filled with photos, drawings, and anecdotes, is a treasure trove of amusing mathematical vignettes. It makes for very pleasant summer reading."--Mark Hunacek, MAA Reviews "The reader is constantly challenged to think about it or answer certain questions and to solve some problems (some solutions are provided at the end). Most of all, it's such a lovely little booklet that does not give you the time to get bored with. The average chapter length, including the many illustrations, is only 10 pages. Just enough to catch your interest and get bitten by the mathematics."--A. Bultheel, European Mathematical Society "For readers who love math, computing and puzzles, Math Bytes will be a welcome gift."--George Erdosh, San Francisco Book Review "[L]ively and entertaining book... All of the applications are presented in an accessible and engaging way, enabling beginners and advanced readers alike to learn and explore at their own pace--a bit and a byte at a time."--Zentralblatt MATH "A mathematical Pandora's box released is perhaps the best way to describe this book. It overflows with ideas, flitting from one fascinating topic to the next, often without an apparent connection. The range is impressive... Everyone should find a 'math byte' of interest in this book, and perhaps end up finding other topics of interest as well."--Choice "I definitely learnt a few new things and was entertained as I did so. Furthermore, anything that spreads the message that mathematics is both interesting and useful has to be a good thing, so I wish this book success."--Rob Ashmore, Mathematics Today "A fun collection of mathematical applications that has something for everyone, even mathphobes."--Anne Quinn, National Council of Teachers of MathematicsTable of ContentsPreface ix 1Your First Byte 1 2Deceiving Arithmetic 5 3Two by Two 11 4Infinite Detail 21 5Plot the Course 32 6Doodling into a Labyrinth 42 7Obama-cize Yourself 54 8Painting with M&Ms 61 9Distorting Reality 73 10A Pretty Mathematical Face 86 11March MATHness 98 12Ranking a Googol of Bits 105 13A Byte to Go 124 14Up to the Challenge 125 Bibliography 131 Index 133 Image Credits 135

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Book SynopsisJohn Mather's seminal works in Hamiltonian dynamics represent some of the most important contributions to our understanding of the complex balance between stable and unstable motions in classical mechanics. His novel approach--known as Aubry-Mather theory--singles out the existence of special orbits and invariant measures of the system, which posseTable of ContentsPreface vii 1 Tonelli Lagrangians and Hamiltonians on Compact Manifolds 1 1.1 Lagrangian Point of View 1 1.2 Hamiltonian Point of View 4 2 From KAM Theory to Aubry-Mather Theory 8 2.1 Action-Minimizing Properties of Measures and Orbits on KAM Tori 8 3 Action-Minimizing Invariant Measures for Tonelli Lagrangians 18 3.1 Action-Minimizing Measures and Mather Sets 18 3.2 Mather Measures and Rotation Vectors 24 3.3 Mather's a-and B-Functions 28 3.4 The Symplectic Invariance of Mather Sets 35 3.5 An Example: The Simple Pendulum (Part I) 39 3.6 Holonomic Measures and Generic Properties of Tonelli Lagrangians 45 4 Action-Minimizing Curves for Tonelli Lagrangians 48 4.1 Global Action-Minimizing Curves: Aubry and Mane Sets 48 4.2 Some Topological and Symplectic Properties of the Aubry and Mane Sets 66 4.3 An Example: The Simple Pendulum (Part II) 68 4.4 Mather's Approach: Peierls' Barrier 71 5 The Hamilton-Jacobi Equation and Weak KAM Theory 76 5.1 Weak Solutions and Subsolutions of Hamilton-Jacobi and Fathi's Weak KAM theory 76 5.2 Regularity of Critical Subsolutions 85 5.3 Non-Wandering Points of the Mane Set 87 Appendices A On the Existence of Invariant Lagrangian Graphs 89 A.1 Symplectic Geometry of the Phase Space 89 A.2 Existence and Nonexistence of Invariant Lagrangian Graphs 91 B Schwartzman Asymptotic Cycle and Dynamics 97 B.1 Schwartzman Asymptotic Cycle 97 B.2 Dynamical Properties 99 Bibliography 107 Index 113

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Book SynopsisTable of Contents*Frontmatter, pg. i*Contents, pg. vii*Chapter 1. Introduction, pg. 1*Chapter 2. Auxiliary Results, pg. 29*Chapter 3. Reduction to Restriction Estimates near the Principal Root Jet, pg. 50*Chapter 4. Restriction for Surfaces with Linear Height below 2, pg. 57*Chapter 5. Improved Estimates by Means of Airy-Type Analysis, pg. 75*Chapter 6. The Case When hlin(PHI) => 2: Preparatory Results, pg. 105*Chapter 7. How to Go beyond the Case hlin(PHI) => 5, pg. 131*Chapter 8. The Remaining Cases Where m = 2 and B = 3 or B = 4, pg. 181*Chapter 9. Proofs of Propositions 1.7 and 1.17, pg. 244*Bibliography, pg. 251*Index, pg. 257

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Book SynopsisTrade Review"Mazur (Euclid in the Rainforest) gives readers the fascinating history behind the mathematical symbols we use, and completely take for granted, every day. Mathematical notation turns numbers into sentences--or, to the uninitiated, a mysterious and impenetrable code. Mazur says the story of math symbols begins some 3,700 years ago, in ancient Babylon, where merchants incised tallies of goods on cuneiform tablets, along with the first place holder--a blank space. Many early cultures used letters for both numbers and an alphabet, but convenient objects like rods, fingers, and abacus beads, also proved popular. Mazur shows how our 'modern' system began in India, picking up the numeral 'zero' on its way to Europe, where it came into common use in the 16th century, thanks to travelers and merchants as well as mathematicians like Fibonacci. Signs for addition, subtraction, roots, and equivalence followed, but only became standardized through the influence of scientists and mathematicians like Rene Descartes and Gottfried Leibniz. Mazur's lively and accessible writing makes what could otherwise be a dry, arcane history as entertaining as it is informative."--Publishers Weekly "[A] fascinating narrative... This is a nuanced, intelligently framed chronicle packed with nuggets--such as the fact that Hindus, not Arabs, introduced Arabic numerals. In a word: enlightening."--George Szpiro, Nature "Mazur begins by illustrating how the ancient Incas and Mayans managed to write specific, huge numbers. Then, for more than 200 pages, he traces the history of division signs, square roots, pi, exponents, graph axes and other symbols in the context of cognition, communication, and analysis."--Washington Post "Mazur delivers a solid exposition of an element of mathematics that is fundamental to its history."--Library Journal "Mazur treats only a subset of F. Cajori's monumental A History of Mathematical Notation (Dover, 1993 first edition 1922) and there is overlap with many other mathematical history books, but Mazur adds new findings and insights and it is so much more entertaining ... and these features make it an interesting addition to the existing literature for anybody with only a slight interest in mathematics or its history."--European Mathematical Society "Symbols like '+' and '=' are so ingrained that it's hard to conceive of math without them. But a new book, Enlightening Symbols: A Short History of Mathematical Notation and its Hidden Power, offers a surprising reminder: Until the early 16th century, math contained no symbols at all."--Kevin Hartnett, Boston Globe "Enlightening Symbols retraces the winding road that has led to the way we now teach, study, and conceive mathematics... Thanks to Mazur's playful approach to the subject, Enlightening Symbols offers an enjoyable read."--Gaia Donati, Science "If you enjoy reading about history, languages and science, then you'll enjoy this book... The best part is the writing is compelling enough that you don't have to be a mathematician to enjoy this informative book."--Guardian.com's GrrlScientist blog "[I]nformative, highly readable and scholarly."--Brian Rotman, Literary Review "[T]his insightful account of the historical development of a highly characteristic feature of the mathematical enterprise also represents a valuable contribution to our understanding of the nature of mathematics."--Eduard Glas, Mathematical Reviews Clippings "Joseph Mazur's beautiful book Enlightening Symbols tells the story of human civilization through the development of mathematical notation. Surprises abound... The book is visually exquisite, great care having been taken with illustrations and figures. Mazur's discussion of the emergence of particular symbols affords the reader an overview of the often difficult primary literature."--Donal O'Shea, Sarasota Herald-Tribune "At whatever depth one chooses to read it, Enlightening Symbols has something for everyone. It is entertaining and eclectic, and Mazur's personal and easy style helps connect us with those who led the long and winding search for the best ways to quantify and analyze our world. Their success has liberated us from 'the shackles of our physical impressions of space'--and of the particular and the concrete--'enabling imagination to wander far beyond the tangible world we live in, and into the marvels of generality.'"--Robyn Arianrhod, Notices of the Notices of the American Mathematical Society "Mazur introduces the reader to major characters, weaves in relevant aspects of wider culture and gives a feel for the breadth of mathematical history. It is a useful book for both student and interested layperson alike."--Mark McCartney, London Mathematical Society "[T]his is a good book. It is well written by an experienced author and is full of interesting facts about how the symbols used in mathematics have arisen. It would certainly interest anyone who studies the history of mathematics."--Phil Dyke, Leonardo "Mazur is a master story teller."--John Stillwell, Bulletin of the American Mathematical SocietyTable of ContentsIntroduction ix Definitions xxi Note on the Illustrations xxiii Part 1 Numerals 1 1. Curious Beginnings 3 2. Certain Ancient Number Systems 10 3. Silk and Royal Roads 26 4. The Indian Gift 35 5. Arrival in Europe 51 6. The Arab Gift 60 7. Liber Abbaci 64 8. Refuting Origins 73 Part 2 Algebra 81 9. Sans Symbols 85 10. Diophantus's Arithmetica 93 11. The Great Art 109 12. Symbol Infancy 116 13. The Timid Symbol 127 14. Hierarchies of Dignity 133 15. Vowels and Consonants 141 16. The Explosion 150 17. A Catalogue of Symbols 160 18. The Symbol Master 165 19. The Last of the Magicians 169 Part 3 The Power of Symbols 177 20. Rendezvous in the Mind 179 21. The Good Symbol 189 22. Invisible Gorillas 192 23. Mental Pictures 210 24. Conclusion 216 Appendix A Leibniz's Notation 221 Appendix B Newton's Fluxion of xn 223 Appendix C Experiment 224 Appendix D Visualizing Complex Numbers 228 Appendix E Quaternions 230 Acknowledgments 233 Notes 235 Index 269

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Book SynopsisThis book develops arithmetic without the induction principle, working in theories that are interpretable in Raphael Robinson's theory Q. Certain inductive formulas, the bounded ones, are interpretable in Q. A mathematically strong, but logically very weak, predicative arithmetic is constructed. Originally published in 1986. The Princeton LegacyTable of Contents*FrontMatter, pg. i*Acknowledgments, pg. v*Table of Contents, pg. vii*Chapter 1. The impredicativity of induction, pg. 1*Chapter 2. Logical terminology, pg. 3*Chapter 3. The axioms of arithmetic, pg. 8*Chapter 4. Order, pg. 10*Chapter 5. Induction by relativization, pg. 12*Chapter 6. Interpretability in Robinson's theory, pg. 16*Chapter 7. Bounded induction, pg. 23*Chapter 8. The bounded least number principle, pg. 29*Chapter 9. The euclidean algorithm, pg. 32*Chapter 10. Encoding, pg. 36*Chapter 11. Bounded separation and minimum, pg. 43*Chapter 12. Sets and functions, pg. 46*Chapter 13. Exponential functions, pg. 51*Chapter 14. Exponentiation, pg. 54*Chapter 15. A stronger relativization scheme, pg. 60*Chapter 16. Bounds on exponential functions, pg. 64*Chapter 17. Bounded replacement, pg. 70*Chapter 18. An impassable barrier, pg. 73*Chapter 19. Sequences, pg. 82*Chapter 20. Cardinality, pg. 90*Chapter 21. Existence of sets, pg. 95*Chapter 22. Semibounded Replacement, pg. 98*Chapter 23. Formulas, pg. 101*Chapter 24. Proofs, pg. 111*Chapter 25. Derived rules of inference, pg. 115*Chapter 26. Special constants, pg. 134*Chapter 27. Extensions by definition, pg. 136*Chapter 28. Interpretations, pg. 152*Chapter 29. The arithmetization of arithmetic, pg. 157*Chapter 30. The consistency theorem, pg. 162*Chapter 31. Is exponentiation total?, pg. 173*Chapter 32. A modified Hilbert program, pg. 178*Bibliography, pg. 181*General index, pg. 183*Index of defining axioms, pg. 186

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Book SynopsisThe book develops the technical machinery to study the minimal model of the Largest Suslin Axiom, and it is a contribution to the Gödel's Program, a central program in the philosophy of mathematics. It will be indispensable for graduate students and researchers in mathematics and philosophy of mathematics.

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Book SynopsisIs college worth the cost? Should I worry about arsenic in my rice? Can we recycle pollution? Real questions of personal finance, public health, and social policy require sober, data-driven analyses. This unique text provides students with the tools of quantitative reasoning to answer such questions. The text models how to clarify the question, recognize and avoid bias, isolate relevant factors, gather data, and construct numerical analyses for interpretation. Themes and techniques are repeated across chapters, with a progression in mathematical sophistication over the course of the book, which helps the student get comfortable with the process of thinking in numbers. This textbook includes references to source materials and suggested further reading, making it user-friendly for motivated undergraduate students. The many detailed problems and worked solutions in the text and extensive appendices help the reader learn mathematical areas such as algebra, functions, graphs, and probability. End-of-chapter problem material provides practice for students, and suggested projects are provided with each chapter. A solutions manual is available online for instructors.Trade Review'Quantitative Reasoning: Thinking in Numbers is as engaging as it is informative. Zaslow takes intriguing questions, such as 'What impact do undocumented immigrants have on the US economy?' and 'Why is it dark at night?', and skillfully communicates the disciplinary knowledge, data, mathematical tools, and logical reasoning skills needed to approach these questions.' Allison K. Henrich, Seattle University'With chapter titles such as 'Is College Worth the Cost?', 'How Many People Died in the Civil War?', and 'Why Is It Dark at Night?', the author draws the reader in by means of topics that are relevant and also pique one's curiosity. The writing is very user-friendly, inviting readers to think of and write down possible answers to the posed questions, before getting into the actual mathematics. I very much appreciate the way in which perhaps nervous students will be made to feel that they can think through the questions at hand, and come to well-reasoned answers without getting bogged down in unnecessary complexities. Students using this book will improve both their quantitative reasoning skills and their self-confidence in their ability to solve real-life questions involving numerical data.' Graeme Bird, Harvard University'A number of our students enter university without the quantitative skills needed to succeed in further STEM studies, and Zaslow's pre-college summer course - and text - in quantitative reasoning was so effective in addressing this common obstacle at Northwestern that we made the course a permanent part of the curriculum open to all students.' Mary Finn, Northwestern University'Any late high school/early college math text that begins with the question of whether college is worth the cost should be required reading. What a powerful and enduring illustration for every instructor. And the rest of the text follows in kind. Ostensibly, Zaslow has written a text for mathematics students. In reality, he's provided a research-based, easily applicable framework for analyzing any rich problem. It's the best form of professional development and growth I've seen in years. Instructors, read it, teach it, understand it, and then apply the principles to your own favorite problems and those of your students.' Dave Meyers, Co-founder and CEO of Teachers Connect'I have really enjoyed reading this book and I am sure others will find the same. Whilst the whole structure is designed for a specific course, nevertheless the approach is helpful and useful.' Peter Hall, The Mathematical GazetteTable of Contents1. Is college worth the cost?; 2. How many people died in the Civil War?; 3. How much will this car cost?; 4. Should we worry about arsenic in rice?; 5. What is the economic impact of the undocumented?; 6. Should I buy health insurance?; 7. Can we recycle pollution?; 8. Why is it dark at night?; 9. Where do the stars go in the daytime?; 10. Should I take this drug for my headache?; Appendix 1. Numeracy; Appendix 2. Arithmetic; Appendix 3. Algebra; Appendix 4. Geometry; Appendix 5. Units and scientific notation; Appendix 6. Functions; Appendix 7. Probability; Appendix 8. Statistics; Appendix 9. Estimation.

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Book SynopsisCategory theory reveals commonalities between structures of all sorts. This self-contained tour of applied category theory shows its potential in science, engineering, and beyond. Each chapter discusses a real-world application using category-theoretic tools, all of which are introduced in an accessible way with many examples and exercises.Trade Review'Category theory was always applied, but traditionally within pure mathematics. Now it is being used to clarify and synthesize a broad range of topics outside mathematics: from computer science to linguistics, from quantum theory to chemistry, and beyond. Charmingly informal yet crystal clear, Fong and Spivak's book does a wonderful job of demonstrating the power of category theory to beginners – even beginners without much background in pure mathematics.' John Baez, University of California, Riverside'The authors quite rightly describe category theory as a tool for thinking. So if your work requires thinking, this book is for you.' Bartosz Milewski, author of Category Theory for Programmers'This book provides a fantastic introduction to how category is not just abstract nonsense but can be applied to real-world engineering problems, pedagogical while still broad, and fun. A must read for all those entering the exciting emerging field of applied category theory by two key players of this community.' Bob Coecke, University of Oxford'An invitation to Applied Category Theory: Seven Sketches in Compositionality provides a grand tour of the fascinating emergent field of applied category theory that centers examples and use cases before gently introducing the accompanying abstract notions. Fong and Spivak should be congratulated for providing this accessible broad viewpoint to illustrate what category theory is all about vis-à-vis the real world.' Emily Riehl, The Johns Hopkins University'An Invitation to Applied Category Theory is clearly and entertainingly written, and provides a great entry into the world of applied category theory. It is chock full of concrete examples and illustrated with clear diagrams … Fong and Spivak will whet your appetite for learning about categories and how they - and the categorical way of thinking - can be applied in and beyond mathematics. And they will give you the means to do that in a self-contained text.' David Jaz Myers, MAA Reviews'Fong and Spivak's book is highly recommendable for anyone with even a passing interest in category theory in general. And it is mandatory reading for scholars aiming to apply category theory to real world problems.' Fernando A. Tohme, MathSciNet'The presentation is highly visual, employing graphs (nodes and edges), directed graphs, and hypergraphs. In addition, exercises intersperse each presentation, and the solutions to many of the exercises are included. Finally, the chapters include concluding summaries, with suggestions for further study. The book contains scores of references. In short, an excellent self-study resource for those interested in learning about applications of category theory to real-world problems.' J. T. Saccoman, Choice'… highly recommended.' Berthold Stoge, IUCr Journals CRYSTALLOGRAPHY JOURNALS ONLINETable of ContentsPreface; 1. Generative effects: orders and Galois connections; 2. Resource theories: monoidal preorders and enrichment; 3. Databases: categories, functors, and universal constructions; 4. Collaborative design: profunctors, categorification, and monoidal categories; 5. Signal flow graphs: props, presentations, and proofs; 6. Electric circuits: hypergraph categories and operads; 7. Logic of behavior: sheaves, toposes, and internal languages; Appendix. Exercise solutions; References; Index.

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Book Synopsisthis section gained proofs of the Schroeder–Bernstein theorem and the Trichotomy Law for Sets, and lost most of the material about finite and countable sets, which has now been moved to a new section devoted to those two types of sets.Trade Review“This is a well-written book, based on very sound pedagogical ideas. It would be an excellent choice as a textbook for a ‘transition’ course.” (Margret Höft, zbMATH 1012.00013, 2021)“The contents of the book is organized in three parts … . this is a nice book, which also this reviewer has used with profit in his teaching of beginner students. It is written in a highly pedagogical style and based upon valuable didactical ideas.” (R. Steinbauer, Monatshefte für Mathematik, Vol. 174, 2014)“Books in this category are meant to teach mathematical topics and techniques that will become valuable in more advanced courses. This book meets these criteria. … This book is well suited as a textbook for a transitional course between calculus and more theoretical courses. I also recommend it for academic libraries.” (Edgar R. Chavez, ACM Computing Reviews, February, 2012)“This is an improved edition of a good book that can serve in the undergraduate curriculum as a bridge between computationally oriented courses like calculus and more abstract courses like algebra.” (Teun Koetsier, Zentralblatt MATH, Vol. 1230, 2012)Table of ContentsPreface to the Second Edition Preface to the First Edition To the Student To the Instructor Part I. Proofs 1. Informal Logic 2. Strategies for Proofs Part II. Fundamentals 3. Sets 4. Functions 5. Relations 6. Finite and Infinite Sets Part III. Extras 7. Selected Topics 8. Explorations Appendix: Properties of Numbers Bibliography Index

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Book SynopsisCircular analyses of philosophical, linguistic, or computational phenomena have been attacked on the assumption that they conflict with mathematical rigour. Barwise and Moss have undertaken to prove this assumption false. This volume is concerned with extending the modelling capabilities of set theory to provide a uniform treatment of circular phenomena. As a means of guiding the reader through the concrete examples of the theory, the authors have included many exercises and solutions: these exercises range in difficulty and ultimately stimulate the reader to come up with new results. Vicious Circles is intended for use by researchers who want to use hypersets; although some experience in mathematics is necessary, the book is accessible to people with widely differing backgrounds and interests.Trade Review' ... a book to learn from.' L'Enseignement MathématiqueTable of ContentsPart I. Background: 1. Introduction; 2. Background on set theory; Part II. Vicious Circles: 3. Circularity in computer science; 4. Circularity in philosophy; 5. Circularity and paradox; Part III. Basic Theory: 6. The solution dilemma; 7. Bisimulation; Part IV. Elementary applications: 8. Graphs; 9. Modal logic; 10. Streams; 11. Games; 12. Modeling the semantic paradoxes; Part V. Further Theory: 13. Greatest fixed points; 14. Uniform operators; 15. Corecursion; Part VI. Further Applications: 16. Some applications; 17. Modeling partial information; 18. Circularity and the notion of set; 19. Conclusions and future directions.

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Book SynopsisDonald Knuth's influence in computer science ranges from the invention of literate programming to the development of the TeX programming language. One of the foremost figures in the field of mathematical sciences, his papers are widely referenced and stand as milestones of development over a wide range of topics. This volume assembles more than three dozen of Professor Knuth's pioneering contributions to discrete mathematics. It includes a variety of topics in combinatorial mathematics (finite geometries, graph theory, enumeration, partitions, tableaux, matroids, codes); discrete algebra (finite fields, groupoids, closure operators, inequalities, convolutions, Pfaffians); and concrete mathematics (recurrence relations, special numbers and notations, identities, discrete probability). Of particular interest are two fundamental papers in which the evolution of random graphs is studied by means of generating functions.Table of Contents1. Discussion of Mr. Riordan's paper 'Abel identities and inverse relations'; 2. Duality in addition chains; 3. Combinatorial analysis and computers; 4. Tables of finite fields; 5. Finite semifields and projective planes; 6. A class of projective planes; 7. Construction of a random sequence; 8. Oriented subtrees of an arc digraph; 9. Another enumeration of trees; 10. Notes on central groupoids; 11. Permutations, matrices, and generalized Young tableaux; 12. A note on solid partitions; 13. Subspaces, subsets, and partitions; 14. Enumeration of plane partitions; 15. Complements and transitive closures; 16. Permutations with nonnegative partial sums; 17. Wheels within wheels; 18. The asymptotic number of geometries; 19. Random matroids; 20. Identities from partition involutions; 21. Huffman's algorithm via algebra; 22. A permanent inequality; 23. Efficient balanced codes; 24. The power of a prime that divides a generalized binomial coefficient; 25. The first cycles in an evolving graph; 26. The birth of the giant component; 27. Polynomials involving the floor function; 28. The sandwich theorem; 29. Aztec diamonds, checkerboard graphs, and spanning trees.

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Book SynopsisInverse problems arise in many disciplines and hold great importance to practical applications. However, sound new methods are needed to solve these problems. Over the past few years, Japanese and Korean mathematicians have obtained a number of very interesting and unique results in inverse problems.Inverse Problems and Related Topics compiles papers authored by some of the top researchers in Korea and Japan. It presents a number of original and useful results and offers a unique opportunity to explore the current trends of research in inverse problems in these countries. Highlighting the existence and active work of several Japanese and Korean groups, it also serves as a guide to those seeking future scientific exchange with researchers in these countries.Trade Review"The aim of this book is to fill the gap between high-school mathematics and mathematics taught at university…the reader is shown what it means to prove something rigourously…This book is easy to read for anyone with a high-school mathematics background." - European Mathematical Society NewsletterTable of ContentsA Finite Difference Model for Calderón's Boundary Inverse Problem. Inverse Problems for Equations with Memory. Parameter Estimation of Elastic Media. The Probe Method and its Applications. Recent Progress in the Inverse Conductivity Problem with Single Measurement. A Moment Method on Inverse Problems for the Heat Equation. Some Remarks on Free Boundaries of Recirculation Euler Flows with Constant Vorticity. Algorithms for the Identification of Spatially Varying/Invariant Stiffness and Dampings in Flexible Beams. Numerical Solutions of the Cauchy Problem in Potential and Elastostatics. Inverse Source Problems in the Helmholtz Equations. A Numerical Method for a Magnetostatic Inverse Problem using the Edge Element. Exact Controllability Method and Multidimensional Linear Inverse Problems. Impedance Computed Tomo-Electrocardiography. An Inverse Problem for Free Channel Scattering. Surface Impedance Tensor and Boundary Value Problem. Aysmptotics for the Spectral and Weyl Functions of the Operator-Value Sturm-Liouville Problem. Exact Controllability Method and Multidimensional Linear Inverse Problems

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Book SynopsisMild Cognitive Impairment (MCI) has been identified as an important clinical transition between normal aging and the early stages of Alzheimer's disease (AD). Since treatments for AD are most likely to be most effective early in the course of the disease, MCI has become a topic of great importance and has been investigated in different populations of interest in many countries. This book brings together these differing perspectives on MCI for the first time. This volume provides a comprehensive resource for clinicians, researchers, and students involved in the study, diagnosis, treatment, and rehabilitation of people with MCI. Clinical investigators initially defined mild cognitive impairment (MCI) as a transitional condition between normal aging and the early stages of Alzheimer’s disease (AD). Because the prevalence of AD increases with age and very large numbers of older adults are affected worldwide, these clinicians saw a pressing need to identify AD as early as possible. It is at this very early stage in the disease course that treatments to slow the progress and control symptoms are likely to be most effective.Since the first introduction of MCI, research interest has grown exponentially, and the utility of the concept has been investigated from a variety of perspectives in different populations of interest (e.g., clinical samples, volunteers, population-based screening) in many different countries. Much variability in findings has resulted. Although it has been acknowledged that the differences observed between samples may be ‘legitimate variations’, there has been no attempt to understand what it is we have learned about MCI (i.e., common features and differences) from each of these perspectives.This book brings together information about MCI in different populations from around the world. Mild Cognitive Impairment will be an important resource for any clinician, researcher, or student involved in the study, detection, treatment, and rehabilitation of people with MCI.Trade Review"This valuable volume brings the kind of broad perspective to mild cognitive impairment that has long been needed. Rather than basing conclusions on a single sample or framework, the editors have pulled together articles from leading research groups around the world. This is the kind of comprehensive approach that is needed for developing systematic and valid definitions of MCI and identifying better tools that make it possible to differentiate between benign memory changes in later life and the early signs of pathological processes." - Steven H. Zarit, Department of Human Development and Family Studies, The Pennsylvania State University"This volume provides the most comprehensive overview of mild cognitive impairment currently available. The conceptual and methodological challenges for studying MCI are tackled with rigor, and the complexities of defining the syndrome are not underestimated. This book is certain to become a classic text for those studying or researching cognitive agin, MCI and dementia, and for clinicians seeking an authoritative reference on the clinical manifestations of MCI." - Kaarin J. Anstey, Centre for Mental Health Research, Australian National University"The editors of this book have done a great job. The description of the issues is laid out in a well-written introduction, making the descriptions of the research papers very accessible, even to the less well-informed reader. The conclusion likewise pulled together the various strands, including defining what still needs to be done to further refine the concept of MCI." - Graham A. Jackson, Laverndale Hospital, Scotland. In Dementia, August, 2008"This valuable volume brings the kind of broad perspective to mild cognitive impairment that has long been needed. Rather than basing conclusions on a single sample or framework, the editors have pulled together articles from leading research groups around the world. This is the kind of comprehensive approach that is needed for developing systematic and valid definitions of MCI and identifying better tools that make it possible to differentiate between benign memory changes in later life and the early signs of pathological processes." - Steven H. Zarit, Department of Human Development and Family Studies, The Pennsylvania State University"This volume provides the most comprehensive overview of mild cognitive impairment currently available. The conceptual and methodological challenges for studying MCI are tackled with rigor, and the complexities of defining the syndrome are not underestimated. This book is certain to become a classic text for those studying or researching cognitive agin, MCI and dementia, and for clinicians seeking an authoritative reference on the clinical manifestations of MCI." - Kaarin J. Anstey, Centre for Mental Health Research, Australian National University"The editors of this book have done a great job. The description of the issues is laid out in a well-written introduction, making the descriptions of the research papers very accessible, even to the less well-informed reader. The conclusion likewise pulled together the various strands, including defining what still needs to be done to further refine the concept of MCI." - Graham A. Jackson, Laverndale Hospital, Scotland. In Dementia, August, 2008Table of ContentsPart 1. Introduction. H. Tuokko, I. McDowell, An Overview of Mild Cognitive Impairment. Part 2. General Population Research on MCI. C. Fabrigoule, P. Barberger-Gateau, J.-F. Dartigues, The PAQUID Study. K. Palmer, L. Bäckman, B.J. Small, L. Fratiglioni, Cognitive Impairment in Elderly Persons without Dementia: Findings from the Kungsholmen Project. J. Fleming, F.E. Matthews, M. Chatfield, C. Brayne, Population Levels of Mild Cognitive Impairment in England and Wales. A. Collie, P. Maruff, D.G. Darby, C. Masters, J. Currie, The Melbourne Aging Study. Part 3. Specific Samples. R. Wilson, N.T. Aggarwal, D.A. Bennett, Mild Cognitive Impairment in the Religious Orders Study. G. Smith, M. Machulda, K. Kantarci, A Perspective from the Mayo Clinic. M.C. Tierney, Prediction of Probable Alzheimer's Disease: The Sunnybrook Memory Study. H. Wolf, H.-J. Gertz, Studies in the Leipzig Memory Clinic: Contribution to the Concept of Mild Cognitive Impairment. Part 4. Interventions. H. Chertkow, Emerging Pharmacological Therapies for Mild Cognitive Impairment. B. Woods, L. Clare, Cognition-based Therapies and Mild Cognitive Impairment. K. Peters, G. Winocur, Combined Therapies in Mild Cognitive Impairment. Part 5. Summary and Future Directions. H. Tuokko, D.F. Hultsch, The Future of Mild Cognitive Impairment.

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Book Synopsis

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