This book presents a large amount of material, both classic and recent (on occasion, unpublished) about the relations of Algebra and Topology. It therefore belongs to the area called Topological Algebra. More specifically, the objects of the study are subtle and sometimes unexpected phenomena that occur when the continuity meets and properly feeds an algebraic operation. Such a combination gives rise to many classic structures, including topological groups and semigroups, paratopological groups, etc. Special emphasis is given to tracing the influence of compactness and its generalizations on the properties of an algebraic operation, causing on occasion the automatic continuity of the operation. The main scope of the book, however, is outside of the locally compact structures, thus distinguishing the monograph from a series of more traditional textbooks.The book is unique in that it presents very important material, dispersed in hundreds of research articles, not covered by any monograph in existence. The reader is gently introduced to an amazing world at the interface of Algebra, Topology, and Set Theory. He/she will find that the way to the frontier of the knowledge is quite short -- almost every section of the book contains several intriguing open problems whose solutions can contribute significantly to the area.

TOPOLOGICAL GROUPS AND RELATED STRUCTURES (Atlantis Studies in Mathematics)

Science & Math

Arhangel'skii, Alexander

Tkachenko, Mikhail

Mathematics

Atlantis Press / World Scientific

Continued Fractions consists of two volumes -- Volume 1: Convergence Theory; and Volume 2: Representation of Functions (tentative title), which is expected in 2011. Volume 1 is dedicated to the convergence and computation of continued fractions, while Volume 2 will treat representations of meromorphic functions by continued fractions. Taken together, the two volumes will present the basic continued fractions theory without requiring too much previous knowledge; some basic knowledge of complex functions will suffice. Both new and advanced graduate students of continued fractions shall get a comprehensive understanding of how these infinite structures work in a number of applications, and why they work so well. A varied buffet of possible applications to whet the appetite is presented first, before the more basic but modernized theory is given.This new edition is the result of an increasing interest in computing special functions by means of continued fractions. The methods described in detail are, in many cases, very simple, yet reliable and efficient.

CONTINUED FRACTIONS - VOL 1: CONVERGENCE THEORY (2ND EDITION) (Atlantis Studies in Mathematics for Engineering and Science)

Waadeland, Haakon

Lorentzen, Lisa

This book focuses on systematic and intelligent information processing theories and methods based on linguistic values. The main contents include topics such as the 2-tuple model of linguistic values, 2-tuple linguistic aggregation operator hedge algebras, complete hedge algebras and linguistic value reasoning, reasoning with linguistic quantifiers, linguistic reasoning based on a random set, the fuzzy number model of linguistic values, linguistic value aggregation based on the fuzzy number model, extraction of linguistic proposition from a database and handling of linguistic information on the Internet, linguistic truth value lattice implication algebras, linguistic truth value resolution automatic reasoning, linguistic truth value reasoning based on lattice implication algebra, and related applications in decision-making and forecast.

LINGUISTIC VALUES BASED INTELLIGENT INFORMATION PROCESSING: THEORY, METHODS AND APPLICATIONS (Atlantis Computational Intelligence Systems)

Computers & Technology

Liu, Jun

Ruan, Da

Xu, Yang

Pei, Zheng

Computer Science

A mathematical theory is introduced in this book to unify a large class of nonlinear partial differential equation (PDE) models for better understanding and analysis of the physical and biological phenomena they represent. The so-called mean field approximation approach is adopted to describe the macroscopic phenomena from certain microscopic principles for this unified mathematical formulation. Two key ingredients for this approach are the notions of "duality" according to the PDE weak solutions and "hierarchy" for revealing the details of the otherwise hidden secrets, such as physical mystery hidden between particle density and field concentration, quantized blow up biological mechanism sealed in chemotaxis systems, as well as multi-scale mathematical explanations of the Smoluchowski-Poisson model in non-equilibrium thermodynamics, two-dimensional turbulence theory, self-dual gauge theory, and so forth. This book shows how and why many different nonlinear problems are inter-connected in terms of the properties of duality and scaling, and the way to analyze them mathematically.

MEAN FIELD THEORIES AND DUAL VARIATION: A MATHEMATICAL PROFILE EMERGED IN THE NONLINEAR HIERARCHY (Atlantis Studies in Mathematics for Engineering and Science)

Suzuki, Takashi

This book highlights various topics on measure theory and vividly demonstrates that the different questions of this theory are closely connected with the central measure extension problem. Several important aspects of the measure extension problem are considered separately: set-theoretical, topological and algebraic. Also, various combinations (e.g., algebraic-topological) of these aspects are discussed by stressing their specific features. Several new methods are presented for solving the above mentioned problem in concrete situations. In particular, the following new results are obtained: the measure extension problem is completely solved for invariant or quasi-invariant measures on solvable uncountable groups; non-separable extensions of invariant measures are constructed by using their ergodic components; absolutely non-measurable additive functionals are constructed for certain classes of measures; the structure of algebraic sums of measure zero sets is investigated.The material presented in this book is essentially self-contained and is oriented towards a wide audience of mathematicians (including postgraduate students). New results and facts given in the book are based on (or closely connected with) traditional topics of set theory, measure theory and general topology such as: infinite combinatorics, Martin's Axiom and the Continuum Hypothesis, Luzin and Sierpinski sets, universal measure zero sets, theorems on the existence of measurable selectors, regularity properties of Borel measures on metric spaces, and so on. Essential information on these topics is also included in the text (primarily, in the form of Appendixes or Exercises), which enables potential readers to understand the proofs and follow the constructions in full details. This not only allows the book to be used as a monograph but also as a course of lectures for students whose interests lie in set theory, real analysis, measure theory and general topology.

TOPICS IN MEASURE THEORY AND REAL ANALYSIS (Atlantis Studies in Mathematics)

Kharazishvili, Alexander B.

In this book, an attempt is made to provide a hybrid grand unified theory to understand the universe, both in its micro/quantum aspects as well as macro/galactic aspects. The book describes a truly hybrid theory as it encompasses both the modern and ancient theories of the universe, together with its functioning at all levels of human comprehension. One of its authors, Dr Escultura, was nominated in 2005 for a Nobel Prize for his flux theory of gravitation. From then on this theory has been improved, clarified and is now known as the hybrid grand unified theory. This book deals with this new bold theory, unifying mathematics and physics, and answering some open fundamental questions and paradoxes in these disciplines. The book also describes what the ancient scientists knew about these matters.

HYBRID GRAND UNIFIED THEORY, THE (Atlantis Studies in Mathematics for Engineering and Science)

Lakshmikantham, Vangipuram

Escultura, E

Leela, Srinivasa G

Conventional on-chip communication design mostly use ad-hoc approaches that fail to meet the challenges posed by the next-generation MultiCore Systems on-chip (MCSoC) designs. These major challenges include wiring delay, predictability, diverse interconnection architectures, and power dissipation. A Network-on-Chip (NoC) paradigm is emerging as the solution for the problems of interconnecting dozens of cores into a single system on-chip. However, there are many problems associated with the design of such systems. These problems arise from non-scalable global wire delays, failure to achieve global synchronization, and difficulties associated with non-scalable bus-based functional interconnects.The book consists of three parts, with each part being subdivided into four chapters. The first part deals with design and methodology issues. The architectures used in conventional methods of MCSoCs design and custom multiprocessor architectures are not flexible enough to meet the requirements of different application domains and not scalable enough to meet different computation needs and different complexities of various applications. Several chapters of the first part will emphasize on the design techniques and methodologies.The second part covers the most critical part of MCSoCs design -- the interconnections. One approach to addressing the design methodologies is to adopt the so-called reusability feature to boost design productivity. In the past years, the primitive design units evolved from transistors to gates, finite state machines, and processor cores. The network-on-chip paradigm offers this attractive property for the future and will be able to close the productivity gap.The last part of this book delves into MCSoCs validations and optimizations. A more qualitative approach of system validation is based on the use of formal techniques for hardware design. The main advantage of formal methods is the possibility to prove the validity of essential design requirements. As formal languages have a mathematical foundation, it is possible to formally extract and verify these desired properties of the complete abstract state space. Online testing techniques for identifying faults that can lead to system failure are also surveyed. Emphasis is given to analytical redundancy-based techniques that have been developed for fault detection and isolation in the automatic control area.

MULTICORE SYSTEMS ON-CHIP: PRACTICAL SOFTWARE/HARDWARE DESIGN (Atlantis Ambient and Pervasive Intelligence)

Engineering & Transportation

Abderazek, Ben Abadallh

Engineering

The book "Enaction, Embodiment, Evolutionary Robotics" proposes how a particular kind of simulation model, i.e. Evolutionary Robotics simulations, can help to solve several problems in Cognitive Science. Examples discussed in the book ranges from motor control, neuroscientific theory, social contingency and time perception. It is argued that methodological minimalism can be a merit, not a shortcoming, even when studying something as complex as the human mind. The book concludes by proposing a new minimalist interdisciplinary framework for the study of perception, combining simulation modeling, experimental methods and accounts of subjective experience. This book endorses an enactive and constructivist view on the human mind, in opposition to the traditional information-processing view. Furthermore, the book discusses and presents the enactive approach, clarifies the assets of this view and how it differs from other proposed alternatives to the computationalist paradigm in AI and Cognitive Science, crucial and missing in the ongoing "embodied turn". The book also presents new experimental results on a number of topics and points out connections between them. Finally, the book proposes a novel framework for the study of perception that combines a number of methods, including computational modeling, in a previously unseen and promising way.

ENACTION, EMBODIMENT, EVOLUTIONARY ROBOTICS: SIMULATION MODELS FOR A POST-COGNITIVIST SCIENCE OF MIND (Atlantis Thinking Machines)

Rohde, Marieke

This book contains many important new results on the theory of wavelet transform, such as its relations with Hilbert transform and other fractional integral operators, convolution for wavelet transform. Wavelet transforms on certain distribution spaces, on spaces of type S and type W, and on generalized Sobolev space have been studied. Asymptotic expansions of the wavelet transform when translation or dilation parameter is large/small have been obtained. So far, these results are not published in book form. The results — well-illustrated by means of specific examples and relevant figures — will find applications in approximation theory, signal processing and in the study of partial differential equations. An overview of the various topics covered in the book facilitates easy reference. Research workers, interested in these topics, will find many open problems being treated in this book.

WAVELET TRANSFORM, THE (Atlantis Studies in Mathematics for Engineering and Science)

Pathak, RAM Shankar

In recent years, there has been a growing interest in the need for designing intelligent systems to address complex decision systems. One of the most challenging issues for the intelligent system is to effectively handle real-world uncertainties that cannot be eliminated. These uncertainties include various types of information that are incomplete, imprecise, fragmentary, not fully reliable, vague, contradictory, deficient, and overloading. The uncertainties result in a lack of the full and precise knowledge of the decision system, including the determining and selection of evaluation criteria, alternatives, weights, assignment scores, and the final integrated decision result. Computational intelligent techniques (including fuzzy logic, neural networks, and genetic algorithms etc.), which are complimentary to the existing traditional techniques, have shown great potential to solve these demanding, real-world decision problems that exist in uncertain and unpredictable environments. These technologies have formed the foundation for intelligent systems.

COMPUTATIONAL INTELLIGENCE IN COMPLEX DECISION SYSTEMS (Atlantis Computational Intelligence Systems)

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Books by Atlantis Press / World Scientific