Lattice Theory: Foundation

Author: George Grätzer

Publisher: Springer Science & Business Media

ISBN: 9783034800181

Category: Mathematics

Page: 614

View: 3378

This book started with Lattice Theory, First Concepts, in 1971. Then came General Lattice Theory, First Edition, in 1978, and the Second Edition twenty years later. Since the publication of the first edition in 1978, General Lattice Theory has become the authoritative introduction to lattice theory for graduate students and the standard reference for researchers. The First Edition set out to introduce and survey lattice theory. Some 12,000 papers have been published in the field since then; so Lattice Theory: Foundation focuses on introducing the field, laying the foundation for special topics and applications. Lattice Theory: Foundation, based on the previous three books, covers the fundamental concepts and results. The main topics are distributivity, congruences, constructions, modularity and semimodularity, varieties, and free products. The chapter on constructions is new, all the other chapters are revised and expanded versions from the earlier volumes. Almost 40 “diamond sections’’, many written by leading specialists in these fields, provide a brief glimpse into special topics beyond the basics. “Lattice theory has come a long way... For those who appreciate lattice theory, or who are curious about its techniques and intriguing internal problems, Professor Grätzer's lucid new book provides a most valuable guide to many recent developments. Even a cursory reading should provide those few who may still believe that lattice theory is superficial or naive, with convincing evidence of its technical depth and sophistication.” Bulletin of the American Mathematical Society “Grätzer’s book General Lattice Theory has become the lattice theorist’s bible.” Mathematical Reviews

General Lattice Theory

Author: George Grätzer,B.A. Davey

Publisher: Springer Science & Business Media

ISBN: 9783764369965

Category: Juvenile Nonfiction

Page: 663

View: 8289

In 20 years, tremendous progress has been made in Lattice Theory. Nevertheless, the change is in the superstructure not in the foundation. Accordingly, I decided to leave the book unchanged and add appendices to record the change. In the first appendix: Retrospective, I briefly review developments from the point of view of this book, specifically, the major results of the last 20 years and solutions of the problems proposed in this book. It is remarkable how many difficult problems have been solved! I was lucky in getting an exceptional group of people to write the other appendices: Brian A. Davey and Hilary A. Priestley on distributive lattices and duality, Friedrich Wehrung on continuous geometries, Marcus Greferath and Stefan E. Schmidt on projective lattice geometries, Peter Jipsen and Henry Rose on varieties, Ralph Freese on free lattices, Bernhard Ganter and Rudolf Wille on formal concept analysis; Thomas Schmidt collaborated with me on congruence lattices. Many of these same people are responsible for the definitive books on the same subjects. I changed very little in the book proper. The diagrams have been redrawn and the book was typeset in ~1EX. To bring the notation up-to-date, I substituted ConL for C(L), IdL for I(L), and so on. Almost 200 mathematicians helped me with this project, from correcting typos to writing long essays on the topics that should go into Retrospective. The last section of Retrospective lists the major contributors. My deeply felt thanks to all of them.

Lattice Theory: Special Topics and Applications

Author: George Grätzer,Friedrich Wehrung

Publisher: Birkhäuser

ISBN: 3319442368

Category: Mathematics

Page: 616

View: 3766

George Grätzer's Lattice Theory: Foundation is his third book on lattice theory (General Lattice Theory, 1978, second edition, 1998). In 2009, Grätzer considered updating the second edition to reflect some exciting and deep developments. He soon realized that to lay the foundation, to survey the contemporary field, to pose research problems, would require more than one volume and more than one person. So Lattice Theory: Foundation provided the foundation. Now we complete this project with Lattice Theory: Special Topics and Applications, in two volumes, written by a distinguished group of experts, to cover some of the vast areas not in Foundation. This second volume is divided into ten chapters contributed by K. Adaricheva, N. Caspard, R. Freese, P. Jipsen, J.B. Nation, N. Reading, H. Rose, L. Santocanale, and F. Wehrung.

Lattice Theory


Author: George Grätzer,Friedrich Wehrung

Publisher: Birkhäuser

ISBN: 9783319482620

Category: Mathematics

Page: N.A

View: 4707

This three-volume-set comprises the complete lattice theory project. Volume 1 of the set, Lattice Theory: Foundation, is the revised and enlarged third edition of General Lattice Theory. It focuses on introducing the field and covers the fundamental concepts and results. The two Special Topics and Applications volumes (volumes 2 and 3 of the set), jointly edited by George Grätzer and Friedrich Wehrung, update the reader on some of the vast areas not in Foundation. Volume 1 is divided into three parts. Part I. Topology and Lattices includes two chapters by Klaus Keimel, Jimmie Lawson and Ales Pultr, Jiri Sichler. Part II. Special Classes of Finite Lattices comprises four chapters by Gabor Czedli, George Grätzer and Joseph P. S. Kung. Part III. Congruence Lattices of Infinite Lattices and Beyond includes four chapters by Friedrich Wehrung and George Grätzer. Volume 2 is divided into ten chapters contributed by K. Adaricheva, N. Caspard, R. Freese, P. Jipsen, J.B. Nation, N. Reading, H. Rose, L. Santocanale, and F. Wehrung.

Lattice Theory

First Concepts and Distributive Lattices

Author: George Gratzer

Publisher: Courier Corporation

ISBN: 048647173X

Category: Mathematics

Page: 212

View: 9242

This outstanding text is written in clear language and enhanced with many exercises, diagrams, and proofs. It discusses historical developments and future directions and provides an extensive bibliography and references. 1971 edition.

Introduction to Lattices and Order

Author: B. A. Davey,H. A. Priestley

Publisher: Cambridge University Press

ISBN: 9780521784511

Category: Mathematics

Page: 298

View: 3939

This new edition of Introduction to Lattices and Order presents a radical reorganization and updating, though its primary aim is unchanged. The explosive development of theoretical computer science in recent years has, in particular, influenced the book's evolution: a fresh treatment of fixpoints testifies to this and Galois connections now feature prominently. An early presentation of concept analysis gives both a concrete foundation for the subsequent theory of complete lattices and a glimpse of a methodology for data analysis that is of commercial value in social science. Classroom experience has led to numerous pedagogical improvements and many new exercises have been added. As before, exposure to elementary abstract algebra and the notation of set theory are the only prerequisites, making the book suitable for advanced undergraduates and beginning graduate students. It will also be a valuable resource for anyone who meets ordered structures.

Undecidable Theories

Author: Alfred Tarski,Andrzej Mostowski,Raphael Mitchel Robinson

Publisher: Elsevier

ISBN: 0444533788

Category: Decidability (Mathematical logic)

Page: 98

View: 6390


Lattice Theory

Author: Garrett Birkhoff

Publisher: American Mathematical Soc.

ISBN: 0821810251

Category: Mathematics

Page: 418

View: 5347

Since its original publication in 1940, this book has been revised and modernized several times, most notably in 1948 (second edition) and in 1967 (third edition). The material is organized into four main parts: general notions and concepts of lattice theory (Chapters I-V), universal algebra (Chapters VI-VII), applications of lattice theory to various areas of mathematics (Chapters VIII-XII), and mathematical structures that can be developed using lattices (Chapters XIII-XVII). At the end of the book there is a list of 166 unsolved problems in lattice theory, many of which still remain open. It is excellent reading, and ... the best place to start when one wishes to explore some portion of lattice theory or to appreciate the general flavor of the field. --Bulletin of the AMS

Probabilistic Lattices

With Applications to Psychology

Author: Louis Narens

Publisher: World Scientific

ISBN: 9814630438

Category: Mathematics

Page: 208

View: 7091

There are many books on lattice theory in the field, but none interfaces with the foundations of probability. This book does. It also develops new probability theories with rigorous foundations for decision theory and applies them to specific well-known problematic examples. There is only one other book that attempts this. It uses quantum probability theory from physics. The new probability theories developed in this book are different; they are not borrowed from physics but are explicitly designed for decision theory. Contents:IntroductionBasic Lattice TheoryPseudo Complemented Distributive LatticesProbability and CoherenceRationality, Heuristics, and Human Judgments of ProbabilityOrthomodular Modeling of Psychological Paradigms Readership: Researchers in probability theory, logic and mathematical psychology. Key Features:Systematic, foundational development of probability theories based on Boolean and non-Boolean algebrasNew theoretical foundations for theories of belief that emphasize the logical structure of event spacesNew methods for modeling the Kahnemann and Tversky heuristics and other psychological paradigms based on intuitionistic logic and a generalization of quantum logicKeywords:Probability on Lattices;Decision Theory;Intuitionistic Logic;Lattice Theory

Mathematical Morphology

Author: Laurent Najman,Hugues Talbot

Publisher: John Wiley & Sons

ISBN: 1118600851

Category: Technology & Engineering

Page: 507

View: 2326

Mathematical Morphology allows for the analysis and processing of geometrical structures using techniques based on the fields of set theory, lattice theory, topology, and random functions. It is the basis of morphological image processing, and finds applications in fields including digital image processing (DSP), as well as areas for graphs, surface meshes, solids, and other spatial structures. This book presents an up-to-date treatment of mathematical morphology, based on the three pillars that made it an important field of theoretical work and practical application: a solid theoretical foundation, a large body of applications and an efficient implementation. The book is divided into five parts and includes 20 chapters. The five parts are structured as follows: Part I sets out the fundamental aspects of the discipline, starting with a general introduction, followed by two more theory-focused chapters, one addressing its mathematical structure and including an updated formalism, which is the result of several decades of work. Part II extends this formalism to some non-deterministic aspects of the theory, in particular detailing links with other disciplines such as stereology, geostatistics and fuzzy logic. Part III addresses the theory of morphological filtering and segmentation, featuring modern connected approaches, from both theoretical and practical aspects. Part IV features practical aspects of mathematical morphology, in particular how to deal with color and multivariate data, links to discrete geometry and topology, and some algorithmic aspects; without which applications would be impossible. Part V showcases all the previously noted fields of work through a sample of interesting, representative and varied applications.

Mathematical Foundations of Public Key Cryptography

Author: Xiaoyun Wang,Guangwu Xu,Mingqiang Wang,Xianmeng Meng

Publisher: CRC Press

ISBN: 1498702244

Category: Computers

Page: 220

View: 2630

In Mathematical Foundations of Public Key Cryptography, the authors integrate the results of more than 20 years of research and teaching experience to help students bridge the gap between math theory and crypto practice. The book provides a theoretical structure of fundamental number theory and algebra knowledge supporting public-key cryptography. Rather than simply combining number theory and modern algebra, this textbook features the interdisciplinary characteristics of cryptography—revealing the integrations of mathematical theories and public-key cryptographic applications. Incorporating the complexity theory of algorithms throughout, it introduces the basic number theoretic and algebraic algorithms and their complexities to provide a preliminary understanding of the applications of mathematical theories in cryptographic algorithms. Supplying a seamless integration of cryptography and mathematics, the book includes coverage of elementary number theory; algebraic structure and attributes of group, ring, and field; cryptography-related computing complexity and basic algorithms, as well as lattice and fundamental methods of lattice cryptanalysis. The text consists of 11 chapters. Basic theory and tools of elementary number theory, such as congruences, primitive roots, residue classes, and continued fractions, are covered in Chapters 1-6. The basic concepts of abstract algebra are introduced in Chapters 7-9, where three basic algebraic structures of groups, rings, and fields and their properties are explained. Chapter 10 is about computational complexities of several related mathematical algorithms, and hard problems such as integer factorization and discrete logarithm. Chapter 11 presents the basics of lattice theory and the lattice basis reduction algorithm—the LLL algorithm and its application in the cryptanalysis of the RSA algorithm. Containing a number of exercises on key algorithms, the book is suitable for use as a textbook for undergraduate students and first-year graduate students in information security programs. It is also an ideal reference book for cryptography professionals looking to master public-key cryptography.

Quark-gluon Plasma

Theoretical Foundations : an Annotated Reprint Collection

Author: Berndt Müller,Johann Rafelski

Publisher: Gulf Professional Publishing

ISBN: 9780444511102

Category: Science

Page: 817

View: 5933

The papers that comprise this collection trace the development of the theoretical understanding of quark-gluon plasma, both in terms of the equation of state and thermal correlation functions, and in terms of its manifestation in high energy nuclear collisions.

Foundations of Crystallography with Computer Applications

Author: Maureen M. Julian

Publisher: CRC Press

ISBN: 1420060767

Category: Science

Page: 368

View: 5816

X-ray crystallography provides a unique opportunity to study the arrangement of atoms in a molecule. This book’s modern computer-graphics centered approach facilitates the extrapolation of these valuable observations. A unified treatment of crystal systems, the book explains how atoms are arranged in crystals using the metric matrix. Featuring two model crystal examples, the text develops theoretical concepts to point and space groups in two dimensions and then extends these ideas to three dimensions. The book interprets the International Tables for Crystallography to bridge the gap between the crystallographic literature and spatial interatomic relationships. Numerous computer-based exercises are integrated throughout the book, with MATLAB® starter programs that help reduce the minutiae of programming.

Formal Concept Analysis

Foundations and Applications

Author: Bernhard Ganter,Gerd Stumme,Rudolf Wille

Publisher: Springer Science & Business Media

ISBN: 3540278915

Category: Computers

Page: 348

View: 7223

Formal concept analysis has been developed as a field of applied mathematics based on the mathematization of concept and concept hierarchy. It thereby allows us to mathematically represent, analyze, and construct conceptual structures. The formal concept analysis approach has been proven successful in a wide range of application fields. This book constitutes a comprehensive and systematic presentation of the state of the art of formal concept analysis and its applications. The first part of the book is devoted to foundational and methodological topics. The contributions in the second part demonstrate how formal concept analysis is successfully used outside of mathematics, in linguistics, text retrieval, association rule mining, data analysis, and economics. The third part presents applications in software engineering.

Dynamical Theory of Crystal Lattices

Author: Max Born,Kun Huang

Publisher: Oxford University Press

ISBN: 0198503695

Category: Science

Page: 420

View: 9932

This is a classic exposition on the dynamics of crystal lattices, co-written by one of the founders of quantum mechanics. It deals with the general statistical mechanics of ideal lattices, and moves on to long lattice waves, thermal, and optical properties of crystals.

Quotient Space Based Problem Solving

A Theoretical Foundation of Granular Computing

Author: Ling Zhang,Bo Zhang

Publisher: Newnes

ISBN: 0124104436

Category: Computers

Page: 396

View: 1067

Quotient Space Based Problem Solving provides an in-depth treatment of hierarchical problem solving, computational complexity, and the principles and applications of multi-granular computing, including inference, information fusing, planning, and heuristic search. Explains the theory of hierarchical problem solving, its computational complexity, and discusses the principle and applications of multi-granular computing Describes a human-like, theoretical framework using quotient space theory, that will be of interest to researchers in artificial intelligence Provides many applications and examples in the engineering and computer science area Includes complete coverage of planning, heuristic search and coverage of strictly mathematical models

Design of Electrical Transmission Lines

Structures and Foundations

Author: Sriram Kalaga,Prasad Yenumula

Publisher: CRC Press

ISBN: 1317627911

Category: Technology & Engineering

Page: 426

View: 7255

This book covers structural and foundation systems used in high-voltage transmission lines, conductors, insulators, hardware and component assembly. In most developing countries, the term “transmission structures” usually means lattice steel towers. The term actually includes a vast range of structural systems and configurations of various materials such as wood, steel, concrete and composites. This book discusses those systems along with associated topics such as structure functions and configurations, load cases for design, analysis techniques, structure and foundation modeling, design deliverables and latest advances in the field. In the foundations section, theories related to direct embedment, drilled shaf ts, spread foundations and anchors are discussed in detail. Featuring worked out design problems for students, the book is aimed at students, practicing engineers, researchers and academics. It contains beneficial information for those involved in the design and maintenance of transmission line structures and foundations. For those in academia, it will be an adequate text-book / design guide for graduate-level courses on the topic. Engineers and managers at utilities and electrical corporations will find the book a useful reference at work.

An Introduction to Mathematical Cryptography

Author: Jeffrey Hoffstein,Jill Pipher,Joseph H. Silverman

Publisher: Springer

ISBN: 1493917110

Category: Mathematics

Page: 538

View: 1696

This self-contained introduction to modern cryptography emphasizes the mathematics behind the theory of public key cryptosystems and digital signature schemes. The book focuses on these key topics while developing the mathematical tools needed for the construction and security analysis of diverse cryptosystems. Only basic linear algebra is required of the reader; techniques from algebra, number theory, and probability are introduced and developed as required. This text provides an ideal introduction for mathematics and computer science students to the mathematical foundations of modern cryptography. The book includes an extensive bibliography and index; supplementary materials are available online. The book covers a variety of topics that are considered central to mathematical cryptography. Key topics include: classical cryptographic constructions, such as Diffie–Hellmann key exchange, discrete logarithm-based cryptosystems, the RSA cryptosystem, and digital signatures; fundamental mathematical tools for cryptography, including primality testing, factorization algorithms, probability theory, information theory, and collision algorithms; an in-depth treatment of important cryptographic innovations, such as elliptic curves, elliptic curve and pairing-based cryptography, lattices, lattice-based cryptography, and the NTRU cryptosystem. The second edition of An Introduction to Mathematical Cryptography includes a significant revision of the material on digital signatures, including an earlier introduction to RSA, Elgamal, and DSA signatures, and new material on lattice-based signatures and rejection sampling. Many sections have been rewritten or expanded for clarity, especially in the chapters on information theory, elliptic curves, and lattices, and the chapter of additional topics has been expanded to include sections on digital cash and homomorphic encryption. Numerous new exercises have been included.

Probability Theory

The Logic of Science

Author: E. T. Jaynes

Publisher: Cambridge University Press

ISBN: 1139435167

Category: Science

Page: N.A

View: 1459

The standard rules of probability can be interpreted as uniquely valid principles in logic. In this book, E. T. Jaynes dispels the imaginary distinction between 'probability theory' and 'statistical inference', leaving a logical unity and simplicity, which provides greater technical power and flexibility in applications. This book goes beyond the conventional mathematics of probability theory, viewing the subject in a wider context. New results are discussed, along with applications of probability theory to a wide variety of problems in physics, mathematics, economics, chemistry and biology. It contains many exercises and problems, and is suitable for use as a textbook on graduate level courses involving data analysis. The material is aimed at readers who are already familiar with applied mathematics at an advanced undergraduate level or higher. The book will be of interest to scientists working in any area where inference from incomplete information is necessary.

Operator Theory in Different Settings and Related Applications

26th IWOTA, Tbilisi, July 2015

Author: Roland Duduchava,Marinus A. Kaashoek,Nikolai Vasilevski,Victor Vinnikov

Publisher: Birkhäuser

ISBN: 3319625276

Category: Mathematics

Page: 309

View: 6140

This book provides a selection of reports and survey articles on the latest research in the area of single and multivariable operator theory and related fields. The latter include singular integral equations, ordinary and partial differential equations, complex analysis, numerical linear algebra, and real algebraic geometry – all of which were among the topics presented at the 26th International Workshop in Operator Theory and its Applications, held in Tbilisi, Georgia, in the summer of 2015. Moreover, the volume includes three special commemorative articles. One of them is dedicated to the memory of Leiba Rodman, another to Murray Marshall, and a third to Boris Khvedelidze, an outstanding Georgian mathematician and one of the founding fathers of the theory of singular integral equations. The book will be of interest to a broad range of mathematicians, from graduate students to researchers, whose primary interests lie in operator theory, complex analysis and applications, as well as specialists in mathematical physics.