The present book deals with coalition games in which expected pay-offs are only vaguely known. In fact, this idea about vagueness of expectations ap pears to be adequate to real situations in which the coalitional bargaining anticipates a proper realization of the game with a strategic behaviour of players. The vagueness being present in the expectations of profits is mod elled by means of the theory of fuzzy set and fuzzy quantities. The fuzziness of decision-making and strategic behaviour attracts the attention of mathematicians and its particular aspects are discussed in sev eral works. One can mention in this respect in particular the book "Fuzzy and Multiobjective Games for Conflict Resolution" by Ichiro Nishizaki and Masatoshi Sakawa (referred below as 43]) which has recently appeared in the series Studies in Fuzziness and Soft Computing published by Physica-Verlag in which the present book is also apperaing. That book, together with the one you carry in your hands, form in a certain sense a complementary pair. They present detailed views on two main aspects forming the core of game theory: strategic (mostly 2-person) games, and coalitional (or cooperative) games. As a pair they offer quite a wide overview of fuzzy set theoretical approaches to game theoretical models of human behaviour."

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 capabilities of modern technology are rapidly increasing, spurred on to a large extent by the tremendous advances in communications and computing. Automated vehicles and global wireless connections are some examples of these advances. In order to take advantage of such enhanced capabilities, our need to model and manipulate our knowledge of the geophysical world, using compatible representations, is also rapidly increasing. In response to this one fundamental issue of great concern in modern geographical research is how to most effectively capture the physical world around us in systems like geographical information systems (GIS). Making this task even more challenging is the fact that uncertainty plays a pervasive role in the representation, analysis and use of geospatial information. The types of uncertainty that appear in geospatial information systems are not the just simple randomness of observation, as in weather data, but are manifested in many other forms including imprecision, incompleteness and granularization. Describing the uncertainty of the boundaries of deserts and mountains clearly require different tools than those provided by probability theory. The multiplicity of modalities of uncertainty appearing in GIS requires a variety of formalisms to model these uncertainties. In light of this it is natural that fuzzy set theory has become a topic of intensive interest in many areas of geographical research and applications This volume, Fuzzy Modeling with Spatial Information for Geographic Problems, provides many stimulating examples of advances in geographical research based on approaches using fuzzy sets and related technologies.

1 2 Harald Atmanspacher and Hans Primas 1 Institute for Frontier Areas of Psychology, Freiburg, Germany, [email protected] 2 ETH Zurich, Switzerland, [email protected] Thenotionofrealityisofsupremesigni?canceforourunderstandingofnature, the world around us, and ourselves. As the history of philosophy shows, it has been under permanent discussion at all times. Traditional discourse about - ality covers the full range from basic metaphysical foundations to operational approaches concerning human kinds of gathering and utilizing knowledge, broadly speaking epistemic approaches. However, no period in time has ex- rienced a number of moves changing and, particularly, restraining traditional concepts of reality that is comparable to the 20th century. Early in the 20th century, quite an in?uential move of such a kind was due to the so-called Copenhagen interpretation of quantum mechanics, laid out essentially by Bohr, Heisenberg, and Pauli in the mid 1920s. Bohr's dictum, quoted by Petersen (1963, p.12), was that "it is wrong to think that the task of physics is to ?nd out how nature is. Physics concerns what we can say about nature." Although this standpoint was not left unopposed - Einstein, Schr] odinger, and others were convinced that it is the task of science to ?nd out about nature itself - epistemic, operational attitudes have set the fashion for many discussions in the philosophy of physics (and of science in general) until today."

This book reflects the progress made in the forty years since the appearance of Abraham Robinson 's revolutionary book Nonstandard Analysis in the foundations of mathematics and logic, number theory, statistics and probability, in ordinary, partial and stochastic differential equations and in education. The contributions are clear and essentially self-contained.

Extensively researched, this book traces the life and work of Abraham De Moivre as well as the state of probability and statistics in eighteenth-century Britain. It is the first extensive biography of De Moivre and is based on recently discovered material and translations, including some of De Moivre's letters. The book begins with discussions on De Moivre's early life in France and his initial work in pure mathematics with some excursions into celestial mechanics. It then describes his fundamental contributions to probability theory and applications, including those in finance and actuarial science. The author explores how De Moivre's wide network of personal and professional connections often motivated his research. The book also covers De Moivre's contemporaries and his impact on the field. Written in a clear, approachable style, this biography will appeal to historians and practitioners of the art of probability and statistics in a wide range of applications, including finance and actuarial science.

Categories and sheaves, which emerged in the middle of the last century as an enrichment for the concepts of sets and functions, appear almost everywhere in mathematics nowadays.

This book covers categories, homological algebra and sheaves in a systematic and exhaustive manner starting from scratch, and continues with full proofs to an exposition of the most recent results in the literature, and sometimes beyond.

The authors present the general theory of categories and functors, emphasising inductive and projective limits, tensor categories, representable functors, ind-objects and localization. Then they study homological algebra including additive, abelian, triangulated categories and also unbounded derived categories using transfinite induction and accessible objects. Finally, sheaf theory as well as twisted sheaves and stacks appear in the framework of Grothendieck topologies.

Parameterized complexity theory is a recent branch of computational complexity theory that provides a framework for a refined analysis of hard algorithmic problems. The central notion of the theory, fixed-parameter tractability, has led to the development of various new algorithmic techniques and a whole new theory of intractability.

This book is a state-of-the-art introduction to both algorithmic techniques for fixed-parameter tractability and the structural theory of parameterized complexity classes, and it presents detailed proofs of recent advanced results that have not appeared in book form before. Several chapters are each devoted to intractability, algorithmic techniques for designing fixed-parameter tractable algorithms, and bounded fixed-parameter tractability and subexponential time complexity. The treatment is comprehensive, and the reader is supported with exercises, notes, a detailed index, and some background on complexity theory and logic.

The book will be of interest to computer scientists, mathematicians and graduate students engaged with algorithms and problem complexity.

These notes are devoted to the study of some classical problems in the Geometry of Banach spaces. The novelty lies in the fact that their solution relies heavily on techniques coming from Descriptive Set Theory. Thecentralthemeisuniversalityproblems.Inparticular, thetextprovides an exposition of the methods developed recently in order to treat questions of the following type: (Q) LetC be a class of separable Banach spaces such that every space X in the classC has a certain property, say property (P). When can we ?nd a separable Banach space Y which has property (P) and contains an isomorphic copy of every member ofC? We will consider quite classical properties of Banach spaces, such as "- ing re?exive," "having separable dual," "not containing an isomorphic copy of c," "being non-universal," etc. 0 It turns out that a positive answer to problem (Q), for any of the above mentioned properties, is possible if (and essentially only if) the classC is "simple." The "simplicity" ofC is measured in set theoretic terms. Precisely, if the classC is analytic in a natural "coding" of separable Banach spaces, then we can indeed ?nd a separable space Y which is universal for the class C and satis?es the requirements imposed above.

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.

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)."

An Introduction to Mathematical Proofs presents fundamental material on logic, proof methods, set theory, number theory, relations, functions, cardinality, and the real number system. The text uses a methodical, detailed, and highly structured approach to proof techniques and related topics. No prerequisites are needed beyond high-school algebra. New material is presented in small chunks that are easy for beginners to digest. The author offers a friendly style without sacrificing mathematical rigor. Ideas are developed through motivating examples, precise definitions, carefully stated theorems, clear proofs, and a continual review of preceding topics. Features Study aids including section summaries and over 1100 exercises Careful coverage of individual proof-writing skills Proof annotations and structural outlines clarify tricky steps in proofs Thorough treatment of multiple quantifiers and their role in proofs Unified explanation of recursive definitions and induction proofs, with applications to greatest common divisors and prime factorizations About the Author: Nicholas A. Loehr is an associate professor of mathematics at Virginia Technical University. He has taught at College of William and Mary, United States Naval Academy, and University of Pennsylvania. He has won many teaching awards at three different schools. He has published over 50 journal articles. He also authored three other books for CRC Press, including Combinatorics, Second Edition, and Advanced Linear Algebra.

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.

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 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.

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

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).

The theory of sets, described in the preface to this book as 'Georg Cantor's magnificent theory' was first developed in the 1870s, and was recognised as one of the most important new branches of mathematical science. W. H. Young and his wife Grace Chisholm Young wrote this book, published in 1906, as a 'simple presentation'; but they warn that it is effectively a work in progress: the writing 'has necessarily involved attempts to extend the frontier of existing knowledge, and to fill in gaps which broke the connexion between isolated parts of the subject.' The Young's were a dynamic force in mathematical research: William had been Grace's tutor at Girton College; she was subsequently the first woman to be awarded a Ph. D by the University of G ttingen. Cantor himself said of the book: 'It is a pleasure for me to see with what diligence, skill and success you have worked.'

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 study of higher dimensional categories has mostly been developed in the globular form of 2-categories, n-categories, omega-categories and their weak versions. Here we study a different form: double categories, n-tuple categories and multiple categories, with their weak and lax versions.We want to show the advantages of this form for the theory of adjunctions and limits. Furthermore, this form is much simpler in higher dimension, starting with dimension three where weak 3-categories (also called tricategories) are already quite complicated, much more than weak or lax triple categories.This book can be used as a textbook for graduate and postgraduate studies, and as a basis for research. Notions are presented in a 'concrete' way, with examples and exercises; the latter are endowed with a solution or hints. Part I, devoted to double categories, starts at basic category theory and is kept at a relatively simple level. Part II, on multiple categories, can be used independently by a reader acquainted with 2-dimensional categories.

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 book, suitable for interested post-16 school pupils or undergraduates looking for a supplement to their course text, develops our modern view of space-time and its implications in the theories of gravity and cosmology. While aspects of this topic are inevitably abstract, the book seeks to ground thinking in observational and experimental evidence where possible. In addition, some of Einstein's philosophical thoughts are explored and contrasted with our modern views. Written in an accessible yet rigorous style, Jonathan Allday, a highly accomplished writer, brings his trademark clarity and engagement to these fascinating subjects, which underpin so much of modern physics. Features: Restricted use of advanced mathematics, making the book suitable for post-16 students and undergraduates Contains discussions of key modern developments in quantum gravity, and the latest developments in the field, including results from the Laser Interferometer Gravitational-Wave Observatory (LIGO) Accompanied by appendices on the CRC Press website featuring detailed mathematical arguments for key derivations

Elements of Mathematics takes readers on a fascinating tour that begins in elementary mathematics--but, as John Stillwell shows, this subject is not as elementary or straightforward as one might think. Not all topics that are part of today's elementary mathematics were always considered as such, and great mathematical advances and discoveries had to occur in order for certain subjects to become "elementary." Stillwell examines elementary mathematics from a distinctive twenty-first-century viewpoint and describes not only the beauty and scope of the discipline, but also its limits. From Gaussian integers to propositional logic, Stillwell delves into arithmetic, computation, algebra, geometry, calculus, combinatorics, probability, and logic. He discusses how each area ties into more advanced topics to build mathematics as a whole. Through a rich collection of basic principles, vivid examples, and interesting problems, Stillwell demonstrates that elementary mathematics becomes advanced with the intervention of infinity. Infinity has been observed throughout mathematical history, but the recent development of "reverse mathematics" confirms that infinity is essential for proving well-known theorems, and helps to determine the nature, contours, and borders of elementary mathematics. Elements of Mathematics gives readers, from high school students to professional mathematicians, the highlights of elementary mathematics and glimpses of the parts of math beyond its boundaries.

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.

This monograph gives an introductory treatment of the most important iterative methods for constructing fixed points of nonlinear contractive type mappings. For each iterative method considered, it summarizes the most significant contributions in the area by presenting some of the most relevant convergence theorems. It also presents applications to the solution of nonlinear operator equations as well as the appropriate error analysis of the main iterative methods.