This monograph is the r st in Fuzzy Approximation Theory. It contains mostly the author s research work on fuzziness of the last ten years and relies a lot on [10]-[32] and it is a natural outgrowth of them. It belongs to the broader area of Fuzzy Mathematics. Chapters are self-contained and several advanced courses can be taught out of this book. We provide lots of applications but always within the framework of Fuzzy Mathematics. In each chapter is given background and motivations. A c- plete list of references is provided at the end. The topics covered are very diverse. In Chapter 1 we give an extensive basic background on Fuzziness and Fuzzy Real Analysis, as well a complete description of the book. In the following Chapters 2,3 we cover in deep Fuzzy Di?erentiation and Integ- tion Theory, e.g. we present Fuzzy Taylor Formulae. It follows Chapter 4 on Fuzzy Ostrowski Inequalities. Then in Chapters 5, 6 we present results on classical algebraic and trigonometric polynomial Fuzzy Approximation.

This book offers an up-to-date, comprehensive account of determinantal rings and varieties, presenting a multitude of methods used in their study, with tools from combinatorics, algebra, representation theory and geometry. After a concise introduction to Groebner and Sagbi bases, determinantal ideals are studied via the standard monomial theory and the straightening law. This opens the door for representation theoretic methods, such as the Robinson-Schensted-Knuth correspondence, which provide a description of the Groebner bases of determinantal ideals, yielding homological and enumerative theorems on determinantal rings. Sagbi bases then lead to the introduction of toric methods. In positive characteristic, the Frobenius functor is used to study properties of singularities, such as F-regularity and F-rationality. Castelnuovo-Mumford regularity, an important complexity measure in commutative algebra and algebraic geometry, is introduced in the general setting of a Noetherian base ring and then applied to powers and products of ideals. The remainder of the book focuses on algebraic geometry, where general vanishing results for the cohomology of line bundles on flag varieties are presented and used to obtain asymptotic values of the regularity of symbolic powers of determinantal ideals. In characteristic zero, the Borel-Weil-Bott theorem provides sharper results for GL-invariant ideals. The book concludes with a computation of cohomology with support in determinantal ideals and a survey of their free resolutions. Determinants, Groebner Bases and Cohomology provides a unique reference for the theory of determinantal ideals and varieties, as well as an introduction to the beautiful mathematics developed in their study. Accessible to graduate students with basic grounding in commutative algebra and algebraic geometry, it can be used alongside general texts to illustrate the theory with a particularly interesting and important class of varieties.

This monograph shows that, through a recourse to the concepts and methods of abstract algebraic logic, the algebraic theory of regular varieties and the concept of analyticity in formal logic can profitably interact. By extending the technique of Plonka sums from algebras to logical matrices, the authors investigate the different classes of models for logics of variable inclusion and they shed new light into their formal properties. The book opens with the historical origins of logics of variable inclusion and on their philosophical motivations. It includes the basics of the algebraic theory of regular varieties and the construction of Plonka sums over semilattice direct systems of algebra. The core of the book is devoted to an abstract definition of logics of left and right variable inclusion, respectively, and the authors study their semantics using the construction of Plonka sums of matrix models. The authors also cover Paraconsistent Weak Kleene logic and survey its abstract algebraic logical properties. This book is of interest to scholars of formal logic.

The nationwide research project Deduktion', funded by the Deutsche Forschungsgemeinschaft (DFG)' for a period of six years, brought together almost all research groups within Germany engaged in the field of automated reasoning. Intensive cooperation and exchange of ideas led to considerable progress both in the theoretical foundations and in the application of deductive knowledge. This three-volume book covers these original contributions moulded into the state of the art of automated deduction. The three volumes are intended to document and advance a development in the field of automated deduction that can now be observed all over the world. Rather than restricting the interest to purely academic research, the focus now is on the investigation of problems derived from realistic applications. In fact industrial applications are already pursued on a trial basis. In consequence the emphasis of the volumes is not on the presentation of the theoretical foundations of logical deduction as such, as in a handbook; rather the books present the concepts and methods now available in automated deduction in a form which can be easily accessed by scientists working in applications outside of the field of deduction. This reflects the strong conviction that automated deduction is on the verge of being fully included in the evolution of technology. Volume I focuses on basic research in deduction and on the knowledge on which modern deductive systems are based. Volume II presents techniques of implementation and details about system building. Volume III deals with applications of deductive techniques mainly, but not exclusively, to mathematics and the verification of software. Each chapter was read bytwo referees, one an international expert from abroad and the other a knowledgeable participant in the national project. It has been accepted for inclusion on the basis of these review reports. Audience: Researchers and developers in software engineering, formal methods, certification, verification, validation, specification of complex systems and software, expert systems, natural language processing.

This open access book offers a self-contained introduction to the homotopy theory of simplicial and dendroidal sets and spaces. These are essential for the study of categories, operads, and algebraic structure up to coherent homotopy. The dendroidal theory combines the combinatorics of trees with the theory of Quillen model categories. Dendroidal sets are a natural generalization of simplicial sets from the point of view of operads. In this book, the simplicial approach to higher category theory is generalized to a dendroidal approach to higher operad theory. This dendroidal theory of higher operads is carefully developed in this book. The book also provides an original account of the more established simplicial approach to infinity-categories, which is developed in parallel to the dendroidal theory to emphasize the similarities and differences. Simplicial and Dendroidal Homotopy Theory is a complete introduction, carefully written with the beginning researcher in mind and ideally suited for seminars and courses. It can also be used as a standalone introduction to simplicial homotopy theory and to the theory of infinity-categories, or a standalone introduction to the theory of Quillen model categories and Bousfield localization.

The papers in this volume represent a selection of updated talks which were presented in an SDS sponsored International Workshop in Panporovo, Bulgaria, in September 1990. The aim of the text is to bring the reader up to date on research in set-valued analysis and differential inclusions.

This monograph covers some of the most important developments in Ramsey theory from its beginnings in the early 20th century via its many breakthroughs to recent important developments in the early 21st century.

The book first presents a detailed discussion of the roots of Ramsey theory before offering a thorough discussion of the role of parameter sets. It presents several examples of structures that can be interpreted in terms of parameter sets and features the most fundamental Ramsey-type results for parameter sets: Hales-Jewett's theorem and Graham-Rothschild s Ramsey theorem as well as their canonical versions and several applications. Next, the book steps back to the most basic structure, to sets. It reviews classic results as well as recent progress on Ramsey numbers and the asymptotic behavior of classical Ramsey functions. In addition, it presents product versions of Ramsey's theorem, a combinatorial proof of the incompleteness of Peano arithmetic, provides a digression to discrepancy theory and examines extensions of Ramsey's theorem to larger cardinals. The next part of the book features an in-depth treatment of the Ramsey problem for graphs and hypergraphs. It gives an account on the existence of sparse and restricted Ramsey theorem's using sophisticated constructions as well as probabilistic methods. Among others it contains a proof of the induced Graham-Rothschild theorem and the random Ramsey theorem. The book closes with a chapter on one of the recent highlights of Ramsey theory: a combinatorial proof of the density Hales-Jewett theorem.

This book provides graduate students as well as advanced researchers with a solid introduction and reference to the field."

1. Interpolation problems play an important role both in theoretical and applied investigations. This explains the great number of works dedicated to classical and new interpolation problems ([1)-[5], [8), [13)-[16], [26)-[30], [57]). In this book we use a method of operator identities for investigating interpo lation problems. Following the method of operator identities we formulate a general interpolation problem containing the classical interpolation problems (Nevanlinna Pick, Caratheodory, Schur, Humburger, Krein) as particular cases. We write down the abstract form of the Potapov inequality. By solving this inequality we give the description of the set of solutions of the general interpolation problem in the terms of the linear-fractional transformation. Then we apply the obtained general results to a number of classical and new interpolation problems. Some chapters of the book are dedicated to the application of the interpola tion theory results to several other problems (the extension problem, generalized stationary processes, spectral theory, nonlinear integrable equations, functions with operator arguments). 2. Now we shall proceed to a more detailed description of the book contents.

In the beginning of 1983, I came across A. Kaufmann's book "Introduction to the theory of fuzzy sets" (Academic Press, New York, 1975). This was my first acquaintance with the fuzzy set theory. Then I tried to introduce a new component (which determines the degree of non-membership) in the definition of these sets and to study the properties of the new objects so defined. I defined ordinary operations as "n," "U," "+" and ."" over the new sets, but I had began to look more seriously at them since April 1983, when I defined operators analogous to the modal operators of "necessity" and "possibility." The late George Gargov (7 April 1947 - 9 November 1996) is the "god father" of the sets I introduced - in fact, he has invented the name "intu itionistic fuzzy," motivated by the fact that the law of the excluded middle does not hold for them. Presently, intuitionistic fuzzy sets are an object of intensive research by scholars and scientists from over ten countries. This book is the first attempt for a more comprehensive and complete report on the intuitionistic fuzzy set theory and its more relevant applications in a variety of diverse fields. In this sense, it has also a referential character."

The nationwide research project `Deduktion', funded by the `Deutsche Forschungsgemeinschaft (DFG)' for a period of six years, brought together almost all research groups within Germany engaged in the field of automated reasoning. Intensive cooperation and exchange of ideas led to considerable progress both in the theoretical foundations and in the application of deductive knowledge. This three-volume book covers these original contributions moulded into the state of the art of automated deduction. The three volumes are intended to document and advance a development in the field of automated deduction that can now be observed all over the world. Rather than restricting the interest to purely academic research, the focus now is on the investigation of problems derived from realistic applications. In fact industrial applications are already pursued on a trial basis. In consequence the emphasis of the volumes is not on the presentation of the theoretical foundations of logical deduction as such, as in a handbook; rather the books present the concepts and methods now available in automated deduction in a form which can be easily accessed by scientists working in applications outside of the field of deduction. This reflects the strong conviction that automated deduction is on the verge of being fully included in the evolution of technology. Volume I focuses on basic research in deduction and on the knowledge on which modern deductive systems are based. Volume II presents techniques of implementation and details about system building. Volume III deals with applications of deductive techniques mainly, but not exclusively, to mathematics and the verification of software. Each chapter was read by two referees, one an international expert from abroad and the other a knowledgeable participant in the national project. It has been accepted for inclusion on the basis of these review reports. Audience: Researchers and developers in software engineering, formal methods, certification, verification, validation, specification of complex systems and software, expert systems, natural language processing.

Since the late 1980s, a large number of very user-friendly tools for fuzzy control, fuzzy expert systems, and fuzzy data analysis have emerged. This has changed the character of this area and started the area of `fuzzy technology'. The next large step in the development occurred in 1992 when almost independently in Europe, Japan and the USA, the three areas of fuzzy technology, artificial neural nets and genetic algorithms joined forces under the title of `computational intelligence' or `soft computing'. The synergies which were possible between these three areas have been exploited very successfully. Practical Applications of Fuzzy Sets focuses on model and real applications of fuzzy sets, and is structured into four major parts: engineering and natural sciences; medicine; management; and behavioral, cognitive and social sciences. This book will be useful for practitioners of fuzzy technology, scientists and students who are looking for applications of their models and methods, for topics of their theses, and even for venture capitalists who look for attractive possibilities for investments.

Through three editions, Cryptography: Theory and Practice, has been embraced by instructors and students alike. It offers a comprehensive primer for the subject's fundamentals while presenting the most current advances in cryptography. The authors offer comprehensive, in-depth treatment of the methods and protocols that are vital to safeguarding the seemingly infinite and increasing amount of information circulating around the world. Key Features of the Fourth Edition: New chapter on the exciting, emerging new area of post-quantum cryptography (Chapter 9). New high-level, nontechnical overview of the goals and tools of cryptography (Chapter 1). New mathematical appendix that summarizes definitions and main results on number theory and algebra (Appendix A). An expanded treatment of stream ciphers, including common design techniques along with coverage of Trivium. Interesting attacks on cryptosystems, including: padding oracle attack correlation attacks and algebraic attacks on stream ciphers attack on the DUAL-EC random bit generator that makes use of a trapdoor. A treatment of the sponge construction for hash functions and its use in the new SHA-3 hash standard. Methods of key distribution in sensor networks. The basics of visual cryptography, allowing a secure method to split a secret visual message into pieces (shares) that can later be combined to reconstruct the secret. The fundamental techniques cryptocurrencies, as used in Bitcoin and blockchain. The basics of the new methods employed in messaging protocols such as Signal, including deniability and Diffie-Hellman key ratcheting.

This monograph provides the first up-to-date and self-contained presentation of a recently discovered mathematical structure-the Schrodinger-Virasoro algebra. Just as Poincare invariance or conformal (Virasoro) invariance play a key role in understanding, respectively, elementary particles and two-dimensional equilibrium statistical physics, this algebra of non-relativistic conformal symmetries may be expected to apply itself naturally to the study of some models of non-equilibrium statistical physics, or more specifically in the context of recent developments related to the non-relativistic AdS/CFT correspondence.

The study of the structure of this infinite-dimensional Lie algebra touches upon topics as various as statistical physics, vertex algebras, Poisson geometry, integrable systems and supergeometry as well as representation theory, the cohomology of infinite-dimensional Lie algebras, and the spectral theory of Schrodinger operators."

Features Provides a uniquely historical perspective on the mathematical underpinnings of a comprehensive list of games Suitable for a broad audience of differing mathematical levels. Anyone with a passion for games, game theory, and mathematics will enjoy this book, whether they be students, academics, or game enthusiasts Covers a wide selection of topics at a level that can be appreciated on a historical, recreational, and mathematical level.

The axiomatic theory of sets is a vibrant part of pure mathematics, with its own basic notions, fundamental results, and deep open problems. It is also viewed as a foundation of mathematics so that "to make a notion precise" simply means "to define it in set theory." This book gives a solid introduction to "pure set theory" through transfinite recursion and the construction of the cumulative hierarchy of sets, and also attempts to explain how mathematical objects can be faithfully modeled within the universe of sets. In this new edition the author has added solutions to the exercises, and rearranged and reworked the text to improve the presentation.

Fuzzy hardware developments have been a major force driving the applications of fuzzy set theory and fuzzy logic in both science and engineering. This volume provides the reader with a comprehensive up-to-date look at recent works describing new innovative developments of fuzzy hardware. An important research trend is the design of improved fuzzy hardware. There is an increasing interest in both analog and digital implementations of fuzzy controllers in particular and fuzzy systems in general. Specialized analog and digital VLSI implementations of fuzzy systems, in the form of dedicated architectures, aim at the highest implementation efficiency. This particular efficiency is asserted in terms of processing speed and silicon utilization. Processing speed in particular has caught the attention of developers of fuzzy hardware and researchers in the field. The volume includes detailed material on a variety of fuzzy hardware related topics such as: Historical review of fuzzy hardware research Fuzzy hardware based on encoded trapezoids Pulse stream techniques for fuzzy hardware Hardware realization of fuzzy neural networks Design of analog neuro-fuzzy systems in CMOS digital technologies Fuzzy controller synthesis method Automatic design of digital and analog neuro-fuzzy controllers Electronic implementation of complex controllers Silicon compilation of fuzzy hardware systems Digital fuzzy hardware processing Parallel processor architecture for real-time fuzzy applications Fuzzy cellular systems Fuzzy Hardware: Architectures and Applications is a technical reference book for researchers, engineers and scientists interested in fuzzy systems in general and in building fuzzy systems in particular.

At the beginning of the new millennium, fuzzy logic opens a new challenging perspective in information processing. This perspective emerges out of the ideas of the founder of fuzzy logic - Lotfi Zadeh, to develop 'soft' tools for direct computing with human perceptions. The enigmatic nature of human perceptions manifests in their unique capacity to generalize, extract patterns and capture both the essence and the integrity of the events and phenomena in human life. This capacity goes together with an intrinsic imprecision of the perception-based information. According to Zadeh, it is because of the imprecision of the human imprecision that they do not lend themselves to meaning representation through the use of precise methods based on predicate logic. This is the principal reason why existing scientific theories do not have the capability to operate on perception-based information. We are at the eve of the emergence of a theory with such a capability. Its applicative effectiveness has been already demonstrated through the industrial implementation of the soft computing - a powerful intelligent technology centred in fuzzy logic. At the focus of the papers included in this book is the knowledge and experience of the researchers in relation both to the engineering applications of soft computing and to its social and philosophical implications at the dawn of the third millennium. The papers clearly demonstrate that Fuzzy Logic revolutionizes general approaches for solving applied problems and reveals deep connections between them and their solutions.

This second volume of the book series shows R-calculus is a combination of one monotonic tableau proof system and one non-monotonic one. The R-calculus is a Gentzen-type deduction system which is non-monotonic, and is a concrete belief revision operator which is proved to satisfy the AGM postulates and the DP postulates. It discusses the algebraical and logical properties of tableau proof systems and R-calculi in many-valued logics. This book offers a rich blend of theory and practice. It is suitable for students, researchers and practitioners in the field of logic. Also it is very useful for all those who are interested in data, digitization and correctness and consistency of information, in modal logics, non monotonic logics, decidable/undecidable logics, logic programming, description logics, default logics and semantic inheritance networks.

This book furnishes a comprehensive treatment of differential graded Lie algebras, L-infinity algebras, and their use in deformation theory. We believe it is the first textbook devoted to this subject, although the first chapters are also covered in other sources with a different perspective. Deformation theory is an important subject in algebra and algebraic geometry, with an origin that dates back to Kodaira, Spencer, Kuranishi, Gerstenhaber, and Grothendieck. In the last 30 years, a new approach, based on ideas from rational homotopy theory, has made it possible not only to solve long-standing open problems, but also to clarify the general theory and to relate apparently different features. This approach works over a field of characteristic 0, and the central role is played by the notions of differential graded Lie algebra, L-infinity algebra, and Maurer-Cartan equations. The book is written keeping in mind graduate students with a basic knowledge of homological algebra and complex algebraic geometry as utilized, for instance, in the book by K. Kodaira, Complex Manifolds and Deformation of Complex Structures. Although the main applications in this book concern deformation theory of complex manifolds, vector bundles, and holomorphic maps, the underlying algebraic theory also applies to a wider class of deformation problems, and it is a prerequisite for anyone interested in derived deformation theory. Researchers in algebra, algebraic geometry, algebraic topology, deformation theory, and noncommutative geometry are the major targets for the book.

Lectori salutem! The kind reader opens the book that its authors would have liked to read it themselves, but it was not written yet. Then, their only choice was to write this book, to fill a gap in the mathematicalliterature. The idea of convexity has appeared in the human mind since the antiquity and its fertility has led to a huge diversity of notions and of applications. A student intending a thoroughgoing study of convexity has the sensation of swimming into an ocean. It is due to two reasons: the first one is the great number of properties and applications of the classical convexity and second one is the great number of generalisations for various purposes. As a consequence, a tendency of writing huge books guiding the reader in convexity appeared during the last twenty years (for example, the books of P. M. Gruber and J. M. Willis (1993) and R. J. Webster (1994)). Another last years' tendency is to order, from some point of view, as many convexity notions as possible (for example, the book of I. Singer (1997)). These approaches to the domain of convexity follow the previous point of view of axiomatizing it (A. Ghika (1955), W. Prenowitz (1961), D. Voiculescu (1967), V. W. Bryant and R. J. Webster (1969)). Following this last tendency, our book proposes to the reader two classifications of convexity properties for sets, both of them starting from the internal mechanism of defining them.

In this presentation of the Galois correspondence, modern theories of groups and fields are used to study problems, some of which date back to the ancient Greeks. The techniques used to solve these problems, rather than the solutions themselves, are of primary importance. The ancient Greeks were concerned with constructibility problems. For example, they tried to determine if it was possible, using straightedge and compass alone, to perform any of the following tasks? (1) Double an arbitrary cube; in particular, construct a cube with volume twice that of the unit cube. (2) Trisect an arbitrary angle. (3) Square an arbitrary circle; in particular, construct a square with area 1r. (4) Construct a regular polygon with n sides for n > 2. If we define a real number c to be constructible if, and only if, the point (c, 0) can be constructed starting with the points (0,0) and (1,0), then we may show that the set of constructible numbers is a subfield of the field R of real numbers containing the field Q of rational numbers. Such a subfield is called an intermediate field of Rover Q. We may thus gain insight into the constructibility problems by studying intermediate fields of Rover Q. In chapter 4 we will show that (1) through (3) are not possible and we will determine necessary and sufficient conditions that the integer n must satisfy in order that a regular polygon with n sides be constructible.

Geometry for the Artist is based on a course of the same name which started in the 1980s at Maharishi International University. It is aimed both at artists willing to dive deeper into geometry and at mathematicians open to learning about applications of mathematics in art. The book includes topics such as perspective, symmetry, topology, fractals, curves, surfaces, and more. A key part of the book's approach is the analysis of art from a geometric point of view-looking at examples of how artists use each new topic. In addition, exercises encourage students to experiment in their own work with the new ideas presented in each chapter. This book is an exceptional resource for students in a general-education mathematics course or teacher-education geometry course, and since many assignments involve writing about art, this text is ideal for a writing-intensive course. Moreover, this book will be enjoyed by anyone with an interest in connections between mathematics and art. Features Abundant examples of artwork displayed in full color. Suitable as a textbook for a general-education mathematics course or teacher-education geometry course. Designed to be enjoyed by both artists and mathematicians.

This third volume of the book series shows R-calculus is a Gentzen-typed deduction system which is non-monotonic, and is a concrete belief revision operator which is proved to satisfy the AGM postulates and the DP postulates. In this book, R-calculus is taken as Tableau-based/sequent-based/multisequent-based to preserve the satisfiability of the Theory/sequent/multisequent to revise, or sequent-based, to preserve the satisfiability of the sequent to revise. The R-calculi for Post and three-valued logic is given. This book offers a rich blend of theory and practice. It is suitable for students, researchers and practitioners in the field of logic.

This book discusses major theories and applications of fuzzy soft multisets and their generalization which help researchers get all the related information at one place. The primary objective of this book is to help bridge the gap to provide a textbook on the theories in fuzzy soft multisets and their applications in real life. It is targeted to researchers and students working in the field of fuzzy set theory, multiset theory, soft set theory and their applications. Uncertainty, vagueness and the representation of imperfect knowledge have been a problem in many fields of research, including artificial intelligence, network and communication, signal processing, machine learning, computer science, information technology, as well as medical science, economics, environments and engineering. There are many mathematical tools for dealing with uncertainties. They include fuzzy set theory, multiset theory, soft set theory and soft multiset theory.