An essential guide to recognizing bogus numbers and misleading data Numbers are often intimidating, confusing, and even deliberately deceptive-especially when they are really big. The media loves to report on millions, billions, and trillions, but frequently makes basic mistakes or presents such numbers in misleading ways. And misunderstanding numbers can have serious consequences, since they can deceive us in many of our most important decisions, including how to vote, what to buy, and whether to make a financial investment. In this short, accessible, enlightening, and entertaining book, leading computer scientist Brian Kernighan teaches anyone-even diehard math-phobes-how to demystify the numbers that assault us every day. With examples drawn from a rich variety of sources, including journalism, advertising, and politics, Kernighan demonstrates how numbers can mislead and misrepresent. In chapters covering big numbers, units, dimensions, and more, he lays bare everything from deceptive graphs to speciously precise numbers. And he shows how anyone-using a few basic ideas and lots of shortcuts-can easily learn to recognize common mistakes, determine whether numbers are credible, and make their own sensible estimates when needed. Giving you the simple tools you need to avoid being fooled by dubious numbers, Millions, Billions, Zillions is an essential survival guide for a world drowning in big-and often bad-data.

AccordingtoHolzmann 14], protocol speci?cationscomprise ?veelements: the service the protocol provides toits users; the set of messages that are exchanged between protocol entities; the format of each message; the rules governingm- sage exchange (procedures); and the assumptionsabout the environment in which the protocol is intended tooperate. In protocol standards documents, information related to the operatingenvironment isusually writteninformally andmayoccur in several di?erentplaces 37]. This informal speci?cation style canlead to misunderstandings andpossibly incompatible implementations. In contrast, executableformalmodelsrequireprecisespeci?cations oftheoperating environment. Ofparticularsigni?canceisthecommunicationmediumorchannel over which the protocol operates. Channelscan havedi?erent characteristics depending on the physical media (e. g. optical ?bre, copper, cable orunguided media (radio)) they employ. The characteristics also depend on the levelof the protocol inacomputer protocol architecture. Forexample, the link-leveloperates over a singlemedium, whereas the network, transport andapplication levelsmayoperate over a network, or network of networks such as the Internet, which couldemploy several di?erent physical media. Channels (such as satellite links) can be noisy resulting in bit errors in packets. To correct biterrors in packets, many importantprotocols (such the Internet's TransmissionControl Protocol 27]) use CyclicRedundancy Checks (CRCs) 28] to detect errors. On detectingan error, the receiver discards the packet andrelies on the sender to retransmit itforrecovery, known as Au- maticRepeatreQuest(ARQ) 28]. Thisisachievedbythereceiveracknowledging the receipt of good packets, andby the transmitter maintainingatimer. When the timer expires before an acknowledgementhasbeen received, the transmitter retransmits packets that havebeen sent but are as yet notacknowledged. It may also be possibleforpacketsto be lost due to routers in networks discarding packets when congested

Traditional logic as a part of philosophy is one of the oldest scientific disciplines and can be traced back to the Stoics and to Aristotle. Mathematical logic, however, is a relatively young discipline and arose from the endeavors of Peano, Frege, and others to create a logistic foundation for mathematics. It steadily developed during the twentieth century into a broad discipline with several sub-areas and numerous applications in mathematics, informatics, linguistics and philosophy.

This book treats the most important material in a concise and streamlined fashion. The third edition is a thorough and expanded revision of the former. Although the book is intended for use as a graduate text, the first three chapters can easily be read by undergraduates interested in mathematical logic. These initial chapters cover the material for an introductory course on mathematical logic, combined with applications of formalization techniques to set theory. Chapter 3 is partly of descriptive nature, providing a view towards algorithmic decision problems, automated theorem proving, non-standard models including non-standard analysis, and related topics.

The remaining chapters contain basic material on logic programming for logicians and computer scientists, model theory, recursion theory, Godel's Incompleteness Theorems, and applications of mathematical logic. Philosophical and foundational problems of mathematics are discussed throughout the text. Each section of the seven chapters ends with exercises some of which of importance for the text itself. There are hints to most of the exercises in a separate file Solution Hints to the Exercises which is not part of the book but is available from the author's website.

This book constitutes the thoroughly refereed post-conference proceedings of the 19th International Workshop on Recent Trends in Algebraic Development Techniques, WADT 2008, held in Pisa, Italy, on June 13-16, 2008.

The 18 revised full papers presented together with 3 invited talks were carefully reviewed and selected from 33 presentations at the workshop.

The papers focus on the algebraic approaches to the specification and development of systems, and address topics such as formal methods for system development, specification languages and methods, systems and techniques for reasoning about specifications, specification development systems, methods and techniques for concurrent, distributed and mobile systems, and algebraic and co-algebraic foundations.

The core of Volume3 consists of lecture notes for seven sets of lectures Hilbert gave (often in collaboration with Bernays) on the foundations of mathematics between 1917 and 1926. These texts make possible for the first time a detailed reconstruction of the rapid development of Hilbert's foundational thought during this period, and show the increasing dominance of the metamathematical perspective in his logical work: the emergence of modern mathematical logic; the explicit raising of questions of completeness, consistency and decidability for logical systems; the investigation of the relative strengths of various logical calculi; the birth and evolution of proof theory, and the parallel emergence of Hilbert's finitist standpoint. The lecture notes are accompanied by numerous supplementary documents, both published and unpublished, including a complete version of Bernays's "Habilitationschrift" of 1918, the text of the first edition of Hilbert and Ackermann's "Grundzuge der theoretischen Logik" (1928), and several shorter lectures by Hilbert from the later 1920s. These documents, which provide the background to Hilbert and Bernays's monumental "Grundlagen der Mathematik" (1934, 1938), are essential for understanding the development of modern mathematical logic, and for reconstructing the interactions between Hilbert, Bernays, Brouwer, and Weyl in the philosophy of mathematics. "

Edited in collaboration with FoLLI, the Association of Logic, Language and Information, this book constitutes the refereed proceedings of the Second International Workshop on Logic, Rationality, and Interaction, LORI 2009, held in Chongqing, China, in October 2009.

The 24 revised full papers presented together with 8 posters were carefully reviewed and selected from a flood of submissions. The workshops topics include but are not limited to semantic models for knowledge, for belief, and for uncertainty, dynamic logics of knowledge, information flow, and action, logical analysis of the structure of games, belief revision, belief merging, logics for preferences and utilities, logics of intentions, plans, and goals, logics of probability and uncertainty, argument systems and their role in interaction, as well as norms, normative interaction, and normative multiagent systems.

This volume contains the papers presented at SAT 2009: 12th International Conference on Theory and Applications of Satis?ability Testing, held from June 30 to July 3, 2009 in Swansea (UK). The International Conference on Theory and Applications of Satis?ability Testing (SAT) started in 1996 as a series of workshops, and, in parallel with the growthof SAT, developedinto the main eventfor SAT research. This year'sc- ference testi?ed to the strong interest in SAT, regarding theoretical research, - searchonalgorithms, investigationsintoapplications, anddevelopmentofsolvers and software systems. As a core problem of computer science, SAT is central for many research areas, and has deep interactions with many mathematical s- jects. Major impulses for the development of SAT came from concrete practical applications as well as from fundamental theoretical research. This fruitful c- laboration can be seen in virtually all papers of this volume. There were 86 submissions (completed papers within the scope of the c- ference). Each submission was reviewed by at least three, and on average 4. 0 Programme Committee members. The Committee decided to accept 45 papers, consisting of 34 regular and 11 short papers (restricted to 6 pages). A main n- elty was a "shepherding process," where 29% of the papers were accepted only conditionally, and requirements on necessary improvements were formulated by the ProgrammeCommittee and its installment monitored by the "shepherd" for thatpaper(using possibly severalroundsoffeedback).

Thisvolumecontainsthepapersofthe20thInternationalConferenceonRewr- ing Techniques and Applications (RTA 2009), which was held from June 29 to July 1, 2009, in Bras' ?lia, Brazil as part of the 5th International Conference on Rewriting, Deduction, and Programming (RDP 2009) together with the Int- national Conference on Typed Lambda Calculi and Applications (TLCA 2009), the International School on Rewriting (ISR 2009), the 4th Workshop on Logical and Semantic Frameworks with Applications (LSFA 2009), the 10th Inter- tional Workshop on Rule-Based Programming (RULE 2009), the 8th Inter- tional Workshop on Functional and (Constraint) Logic Programming (WFLP 2009), the 9th International Workshop on Reduction Strategies in Rewriting and Programming (WRS 2009), and the annual meeting of the IFIP Working Group 1.6 on term rewriting. RTA is the major forum for the presentation of research on all aspects of rewriting.PreviousRTAconferenceswereheldinDijon(1985),Bordeaux(1987), Chapel Hill (1989), Como (1991), Montreal (1993), Kaiserslautern (1995), R- gers (1996), Sitges (1997), Tsukuba (1998), Trento (1999), Norwich (2000), Utrecht(2001),Copenhagen(2002),Valencia(2003),Aachen(2004),Nara(2005), Seattle (2006), Paris (2007), and Hagenberg (2008).

The symposium "Languages: From Formal to Natural," celebrating the 65th birthday of Nissim Francez, was held on May 24-25, 2009 at the Technion, Haifa. The symposium consisted of two parts, a veri?cation day and a language day, and covered all areas of Nissim's past and present research interests, areas which he has inspiringly in?uenced and to which he has contributed so much. This volume comprises severalpapers presentedat the symposium, as wellas additional articles that were contributed by Nissim's friends and colleagues who were unable to attend the event. We thank the authors for their contributions. Wearealsogratefultothereviewersfor their dedicated and timely work. Nissim Francez was born on January 19, 1944. In 1962 he started his mat- matical education at the Hebrew University. He received a BSc in Mathematics in 1965, and, after four years of military service, started his MSc studies in Computer Science at the Weizmann Institute of Science under the supervision of Amir Pnueli. After completing the MSc program in 1971, Nissim continued his studies toward a PhD, again, at the Weizmann Institute of Science and, again, under the supervisionof Amir Pnueli. Nissim wasawardeda PhDin Computer Science in 1976.

Line graphs have the property that their least eigenvalue is greater than or equal to -2, a property shared by generalized line graphs and a finite number of so-called exceptional graphs. This book deals with all these families of graphs in the context of their spectral properties. The authors discuss the three principal techniques that have been employed, namely 'forbidden subgraphs', 'root systems' and 'star complements'. They bring together the major results in the area, including the recent construction of all the maximal exceptional graphs. Technical descriptions of these graphs are included in the appendices, while the bibliography provides over 250 references. This will be an important resource for all researchers with an interest in algebraic graph theory.

Category theory has experienced a resurgence in popularity recently because of new links with topology and mathematical physics. This book provides a clearly written account of higher order category theory and presents operads and multicategories as a natural language for its study. Tom Leinster has included necessary background material and applications as well as appendices containing some of the more technical proofs that might have disrupted the flow of the text.

The discipline of formal concept analysis (FCA) is concerned with the form- ization of concepts and conceptual thinking. Built on the solid foundation of lattice and order theory, FCA is ?rst and foremost a mathematical discipline. However,its motivation andguiding principles arebasedon strongphilosophical underpinnings. In practice, FCA provides a powerful framework for the qua- tative, formal analysis of data, as demonstrated by numerous applications in diverse areas. Likewise, it emphasizes the aspect of human-centered information processing by employing visualization techniques capable of revealing inherent structure in data in an intuitively graspable way. FCA thereby contributes to structuring and navigating the ever-growing amount of information available in our evolving information society and supports the process of turning data into information and ultimately into knowledge. In response to an expanding FCA community, the International Conference on Formal Concept Analysis (ICFCA) was established to provide an annual opportunity for the exchange of ideas. Previous ICFCA conferences were held in Darmstadt (2003), Sydney (2004), Lens (2005), Dresden (2006), Clermont- Ferrand (2007), as well as Montreal (2008) and are evidence of vivid ongoing interest and activities in FCA theory and applications. ICFCA 2009 took place during May 21-24 at the University of Applied S- ences in Darmstadt. Beyond serving as a host of the very ?rst ICFCA in 2003, Darmstadt can be seen as the birthplace of FCA itself, where this discipline was introduced in the early 1980s and elaborated over the subsequent decades.

This tract presents an exposition of methods for testing sets of special functions for completeness and basis properties, mostly in L2 and L2 spaces. The first chapter contains the theoretical background to the subject, largely in a general Hilbert space setting, and theorems in which the structure of Hilbert space is revealed by properties of its bases are dealt with. Later parts of the book deal with methods: for example, the Vitali criterion, together with its generalisations and applications, is discussed in some detail, and there is an introduction to the theory of stability of bases. The last chapter deals with complete sets as eigenfunctions of differential and a table of a wide variety of bases and complete sets of special functions. Dr Higgins' account will be useful to graduate students of mathematics and professional mathematicians, especially Banach spaces. The emphasis on methods of testing and their applications will also interest scientists and engineers engaged in fields such as the sampling theory of signals in electrical engineering and boundary value problems in mathematical physics.

In the last century, developments in mathematics, philosophy, physics, computer science, economics and linguistics have proven important for the development of logic. There has been an influx of new ideas, concerns, and logical systems reflecting a great variety of reasoning tasks in the sciences. This book embodies the multi-dimensional interplay between logic and science, presenting contributions from the world's leading scholars on new trends and possible developments for research.

The Symposium on Logical Foundations of Computer Science series provides a forum for the fast-growing body of work in the logical foundations of computer science, e.g., those areas of fundamental theoretical logic related to computer science. The LFCS series began with "Logic at Botik," Pereslavl-Zalessky,1989, which was co-organized by Albert R. Meyer (MIT) and Michael Taitslin (Tver). After that, organization passed to Anil Nerode. Currently LFCS is governed by a Steering Committee consisting of Anil Nerode (General Chair), Stephen Cook, Dirk van Dalen, Yuri Matiyasevich, John McCarthy, J. Alan Robinson, Gerald Sacks, and Dana Scott. The 2009 Symposium on Logical Foundations of Computer Science (LFCS 2009) took place in Howard Johnson Plaza Resort, Deer?eld Beach, Florida, USA, during January 3-6. This volume contains the extended abstracts of talks selected by the Program Committee for presentation at LFCS 2009. The scope of the symposium is broad and contains constructive mathematics and type theory; automata and automatic structures; computability and r- domness; logical foundations of programming; logical aspects of computational complexity; logic programmingand constraints;automated deduction and int- active theorem proving; logical methods in protocol and program veri?cation; logical methods in program speci?cation and extraction; domain theory l- ics; logical foundations of database theory; equational logic and term rewriting; lambda andcombinatorycalculi;categoricallogicandtopologicalsemantics;l- ear logic; epistemic and temporal logics; intelligent and multiple agent system logics; logics of proof and justi?cation; nonmonotonic reasoning; logic in game theory and social software; logic of hybrid systems; distributed system logics;

This volume surveys the development of combinatorics since 1930 by presenting in chronological order the fundamental results of the subject proved in over five decades of original papers by: T. van Aardenne-Ehrenfest.- R.L. Brooks.- N.G. de Bruijn.- G.F. Clements.- H.H. Crapo.- R.P. Dilworth.- J. Edmonds.- P. Erd s.- L.R. Ford, Jr.- D.R. Fulkerson.- D. Gale.- L. Geissinger.- I.J. Good.- R.L. Graham.- A.W. Hales.- P. Hall.- P.R. Halmos.- R.I. Jewett.- I. Kaplansky.- P.W. Kasteleyn.- G. Katona.- D.J. Kleitman.- K. Leeb.- B. Lindstr m.- L. Lov sz.- D. Lubell.- C. St. J.A. Nash-Williams.- G. P lya.-R. Rado.- F.P. Ramsey.- G.-C. Rota.- B.L. Rothschild.- H.J. Ryser.- C. Schensted.- M.P. Sch tzenberger.- R.P. Stanley.- G. Szekeres.- W.T. Tutte.- H.E. Vaughan.- H. Whitney.

Over the years, this book has become a standard reference and guide in the set theory community. It provides a comprehensive account of the theory of large cardinals from its beginnings and some of the direct outgrowths leading to the frontiers of contemporary research, with open questions and speculations throughout.

The kernel of this book consists of a series of lectures on in?nitary proof theory which I gave during my time at the Westfalische ] Wilhelms-Universitat ] in Munster ] . It was planned as a successor of Springer Lecture Notes in Mathematics 1407. H- ever, when preparing it, I decided to also include material which has not been treated in SLN 1407. Since the appearance of SLN 1407 many innovations in the area of - dinal analysis have taken place. Just to mention those of them which are addressed in this book: Buchholz simpli?ed local predicativity by the invention of operator controlled derivations (cf. Chapter 9, Chapter 11); Weiermann detected applications of methods of impredicative proof theory to the characterization of the provable recursive functions of predicative theories (cf. Chapter 10); Beckmann improved Gentzen's boundedness theorem (which appears as Stage Theorem (Theorem 6. 6. 1) in this book) to Theorem 6. 6. 9, a theorem which is very satisfying in itself - though its real importance lies in the ordinal analysis of systems, weaker than those treated here. Besides these innovations I also decided to include the analysis of the theory (? -REF) as an example of a subtheory of set theory whose ordinal analysis only 2 0 requires a ?rst step into impredicativity. The ordinal analysis of(? -FXP) of non- 0 1 0 monotone? -de?nable inductive de?nitions in Chapter 13 is an application of the 1 analysis of(? -REF)."

The book offers categorical introductions to order, topology, algebra and sheaf theory, suitable for graduate students, teachers and researchers of pure mathematics. Readers familiar with the very basic notions of category theory will learn about the main tools that are used in modern categorical mathematics but are not readily available in the literature. Hence, in eight rather independent chapters the reader will encounter various ways of how to study ‘spaces’: order-theoretically via their open-set lattices, as objects of a fairly abstract category merely via their interaction with other objects, or via their topoi of set-valued sheaves. Likewise, ‘algebras’ are treated both as models for Lawvere’s algebraic theories and as Eilenberg-Moore algebras for monads, but they appear also as the objects of an abstract category with various levels of ‘exactness’ conditions. The abstract methods are illustrated by applications which, in many cases, lead to results not yet found in more traditional presentations of the various subjects, for instance on the exponentiability of spaces and embeddability of algebras. Suggestions for further studies and research are also given.

This volume is located in a cross-disciplinary ?eld bringing together mat- matics, logic, natural science and philosophy. Re?ection on the e?ectiveness of proof brings out a number of questions that have always been latent in the informal understanding of the subject. What makes a symbolic constr- tion signi?cant? What makes an assumption reasonable? What makes a proof reliable? G] odel, Church and Turing, in di?erent ways, achieve a deep und- standing of the notion of e?ective calculability involved in the nature of proof. Turing's work in particular provides a "precise and unquestionably adequate" de?nition of the general notion of a formal system in terms of a machine with a ?nite number of parts. On the other hand, Eugene Wigner refers to the - reasonable e?ectiveness of mathematics in the natural sciences as a miracle. Where should the boundary be traced between mathematical procedures and physical processes? What is the characteristic use of a proof as a com- tation, as opposed to its use as an experiment? What does natural science tell us about the e?ectiveness of proof? What is the role of mathematical proofs in the discovery and validation of empirical theories? The papers collected in this book are intended to search for some answers, to discuss conceptual and logical issues underlying such questions and, perhaps, to call attention to other relevant questions."

Steven Finch provides 136 essays, each devoted to a mathematical constant or a class of constants, from the well known to the highly exotic. This book is helpful both to readers seeking information about a specific constant, and to readers who desire a panoramic view of all constants coming from a particular field, for example, combinatorial enumeration or geometric optimization. Unsolved problems appear virtually everywhere as well. This work represents a scholarly attempt to bring together all significant mathematical constants in one place.

A satisfactory and coherent theory of orthogonal polynomials in several variables, attached to root systems, and depending on two or more parameters, has developed in recent years. This comprehensive account of the subject provides a unified foundation for the theory to which I.G. Macdonald has been a principal contributor. The first four chapters lead up to Chapter 5 which contains all the main results.

With a never-before published paper by Lord Henry Cavendish, as well as a biography on him, this book offers a fascinating discourse on the rise of scientific attitudes and ways of knowing. A pioneering British physicist in the late 18th and early 19th centuries, Cavendish was widely considered to be the first full-time scientist in the modern sense. Through the lens of this unique thinker and writer, this book is about the birth of modern science.

J. Kung and G.-C. Rota, in their 1984 paper, write: Like the Arabian phoenix rising out of its ashes, the theory of invariants, pronounced dead at the turn of the century, is once again at the forefront of mathematics . The book of Sturmfels is both an easy-to-read textbook for invariant theory and a challenging research monograph that introduces a new approach to the algorithmic side of invariant theory. The Groebner bases method is the main tool by which the central problems in invariant theory become amenable to algorithmic solutions. Students will find the book an easy introduction to this classical and new area of mathematics. Researchers in mathematics, symbolic computation, and computer science will get access to a wealth of research ideas, hints for applications, outlines and details of algorithms, worked out examples, and research problems."

Einstein's equations stem from General Relativity. In the context of Riemannian manifolds, an independent mathematical theory has developed around them. This is the first book which presents an overview of several striking results ensuing from the examination of Einstein 's equations in the context of Riemannian manifolds. Parts of the text can be used as an introduction to modern Riemannian geometry through topics like homogeneous spaces, submersions, or Riemannian functionals.