Average customer rating:
- A nice MATH book on this topic
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Finite Model Theory (Springer Monographs in Mathematics)
Heinz-Dieter Ebbinghaus , and
Jörg Flum
Manufacturer: Springer
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ASIN: 3540287876 |
Book Description
The book presents the main results of descriptive complexity theory, that is, the connections between axiomatizability of classes of finite structures and their complexity with respect to time and space bounds. The logics that are important in this context include fixed-point logics, transitive closure logics, and also certain infinitary languages; their model theory is studied in full detail. Other topics include DATALOG languages, quantifiers and oracles, 0-1 laws, and optimization and approximation problems. The book is written in such a way that the respective parts on model theory and descriptive complexity theory may be read independently. This second edition is a thoroughly revised and enlarged version of the original text.
Customer Reviews:
A nice MATH book on this topic.......2000-09-06
This book presents many nice topics in finite model theory. The material on infinitary logic and the finite model properties are nicely presented here and not really available elsewhere. The material on Finite Automata and Logic, and on Descriptive Complexity , is available and better done in other books. This entire field is a cross of computer science and math, and this book is strong on the math end, but not on the computer science angle.
Book Description
One of the problems in economics to which economists have devoted a considerable amount of attention in recent years has been to ensure consistency in the models they employ. Assuming markets to be generally in some state of equilibrium, it must be asked under what circumstances such an equilibrium is possible. The fundamental mathematical tools used to address this concern are fixed point theorems. These outline the conditions under which sets of assumptions have a solution. This book gives the reader access to the mathematical techniques involved and goes on to apply fixed point theorems to proving the existence of equilibria for economics and for cooperative and non-cooperative games. Special emphasis is given to economics and games in cases where the preferences of agents may not be transitive. In addition, the author presents new proofs of old results in order to further clarify the results. He also proposes new results, notably in the last chapter, that refer to the core of a game without transitivity. This book will be useful as a text or reference work for mathematical economists and graduate and advanced undergraduate students.
Customer Reviews:
An excellent survey of more than fixed point theorems........2000-06-02
The author compiles theorems, definitions, and properties on a number of topics. There's some convex analysis, maximization of binary relations, fixed point and selection theorems, and various sufficient conditions for existence of equilibria in games and economies.
A nice feature is that the author explains the equivalence or interconnectedness of theorems from different classes -- comparing the KMM lemma to Brouwer's fixed pt. thm., for instance.
This book is extremely useful for its many variations on common principles. Researchers can benefit from consulting it when they have a problem which does not satisfy the usual criteria -- for instance, if you ever have to ask, "I have a correspondence which satisfies all the conditions for Kakutani's theorem except that it is l.h.c. rather than u.h.c.; am I still able to guarantee a fixed point?"
Grad students in micro/game theory can benefit from the survey of theorems, and familiarizing themselves with the many ways they can get to desired results.
The book is compact and thorough with little exposition. Definitions are very clear, and the author is very good at noting when definitions vary, or the same term has multiple definitions in the literature. Though it presents itself as a mathematical text, its audience is clearly economists -- the conditions given and situations described are clearly ones that economists will recognize, and the applications are economic.
A very nice book for people who already know the applications of such theorems and who need minimal explanation.
the best buy.......2000-03-26
The author, one of the most prominet professors in mathematical economics surveyed fixed point theorems. The theorems are often used to prove the existence of solutions in game theory and economics. If you major in mathematical economics or game theory, you can't avoid to buy this book.
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Diagonalization and Self-Reference (Oxford Logic Guides)
Raymond M. Smullyan
Manufacturer: Oxford University Press, USA
ProductGroup: Book
Binding: Hardcover
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ASIN: 0198534507 |
Book Description
The main purpose of this book is to present a unified treatment of fixed points as they occur in Godel's incompleteness proofs, recursion theory, combinatory logic, semantics, and metamathematics. The book provides a survey of introductory material and a summary of recent research. The first chapters are of an introductory nature and consist mainly of exercises with solutions given to most of them.
Average customer rating:
- Very interesting mathematics
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Unstable Modules over the Steenrod Algebra and Sullivan's Fixed Point Set Conjecture (Chicago Lectures in Mathematics)
Lionel Schwartz
Manufacturer: University Of Chicago Press
ProductGroup: Book
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ASIN: 0226742032 |
Book Description
A comprehensive account of one of the main directions of algebraic topology, this book focuses on the Sullivan conjecture and its generalizations and applications. Lionel Schwartz collects here for the first time some of the most innovative work on the theory of modules over the Steenrod algebra, including ideas on the Segal conjecture, work from the late 1970s by Adams and Wilkerson, and topics in algebraic group representation theory.
This course-tested book provides a valuable reference for algebraic topologists and includes foundational material essential for graduate study.
Customer Reviews:
Very interesting mathematics.......2002-07-21
Fixed points of mappings are of course of enormous interested in the fields of dynamical systems and differential equations. This book however is interested in fixed points from the standpoint of algebraic topology, namely with the study of the homotopy type of the fixed point set of a group action. This motivates the consideration of a homotopy fixed point set, and the author studies specifically the homotopy fixed point set of a finite group acting on a finite complex. From an equivariant point of view, a homotopy fixed point is a set of maps equivariant under the integers modulo 2 (Z/2) from the "antipodal" sphere (i.e the ordinary sphere provided with the antipodal action) to a finite Z/2-CW-complex. A homotopy fixed point is such a map, and ordinary fixed points determine homotopy fixed points. The Sullivan fixed point conjecture asserts that the mapping of ordinary fixed points to homotopy fixed points is a homotopy equivalence, and this conjecture is one of the main topics of the book. The resolution of this conjecture was accomplished by Haynes Miller, with other contributions made by Gunnar Carlsson and J. Lannes. As a point of historical fact, Miller first proved the Sullivan conjecture in the context of pointed maps from the classifying space of a finite group to a finite CW-complex. With the compact-open topology, he showed that this space of maps is weakly contractible. Another line of thought on this topic and considered in this book is that of the fixed point set of a G-space localized at a prime p. The question of whether this fixed point set is weakly homotopy equivalent to the homotopy fixed point set of G acting on the p-localization of a finite CW-complex was solved by Carlsson via a consideration of the Segal conjecture.
The author gives a nice overview of these results, and does so by first considering background material from the theory of unstable modules over the Steenrod algebra. The reader is expected to have a solid background in algebraic topology, particularly in the homotopy theory of CW-complexes, Eilenberg-Maclane spaces, Postnikov systems, the theory of spectral reduced and unreduced cohomology, cohomology operations, and K-theory. The Steenrod algebra has its origins in the consideration of stable Z/2 cohomology operations, where these operations can all be written in terms of Steenrod operations. Consideration of relations among the Steenrod squares result in a family of relations called the Adem relations. This construction can be generalized to a prime p by considering generators other than the Steenrod squares, and dividing out the Adem relations (these are more complex than for the case p = 2). The calculation of the cohomology of Eilenberg-Maclane spaces leads to a characterization of the Steenrod algebra as the algebra of all transformations of mod p cohomology that commute with suspension. Such transformations are called 'stable'.
The mod p cohomology of a space as a module over the Steenrod algebra is unstable, meaning that it is trivial in negative degrees. The author then characterizes the category of unstable modules over the Steenrod algebra (designated U by the author), and shows that is has enough projectives and that it is (locally) Noetherian. That this category has enough injectives is shown using Brown-Gitler technology. This involves the construction of the Brown-Gitler modules, which are related to the Milnor algebra (the dual of the Steenrod algebra, familiar from the elementary theory), and the Carlsson modules. The later are related to Carlsson's work on the Segal conjecture, and their description involves some interesting use of the combinatorics of binary trees. The Lannes functor is introduced as a generalization of this tensor product that still gives an injective category, and its properties are outlined in detail. Modular representation theory is used in the book to study indecomposable reduced U-injectives, and their graded vector space structure is studied using the familiar Poincare series. Then the quotient category of U by its subcategory of nilpotents is studied via a filtration on it, the quotient categories of this filtration being identified with the modular representations of the symmetric groups.
The last part of the book finally gets down to the Sullivan conjecture, beginning with a discussion of the Andre-Quillen cohomology of unstable algebras over the Steenrod algebra. All of the familiar tools from algebraic topology, such as Eilenberg-Moore spectral sequences and the Borel construction are used to prove Miller's version of the Sullivan conjecture and also a generalized version of it.
Book Description
This is the first general introduction to stability of ordinary and functional differential equations by means of fixed point techniques. Suitable for advanced undergraduates and graduate students, it contains an extensive collection of new and classical examples, all worked in detail and presented in an elementary manner. 2006 edition.
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"Granas-Dugundji's book is an encyclopedic survey of the classical fixed point theory of continuous mappings (the work of Poincaré, Brouwer, Lefschetz-Hopf, Leray-Schauder) and all its various modern extensions. This is certainly the most learned book ever likely to be published on this subject."
-Felix Browder, Rutgers University
"The theory of Fixed Points is one of the most powerful tools of modern mathematics. Not only is it used on a daily basis in pure and applied mathematics, but it also serves as a bridge between Analysis and Topology, and provides a very fruitful area of interaction between the two. This book contains a clear, detailed and well-organized presentation of the major results, together with an entertaining set of historical notes and an extensive bibliography describing further developments and applications."
-Haïm Brézis, Université Pierre et Marie Curie
"In this monograph, no effort has been spared, even to the smallest detail, be it mathematical, historical or bibliographical. In particular, the necessary background materials are generously provided for non-specialists. In fact, the book could even serve as an introduction to algebraic topology among others. It is certain that the book will be a standard work on Fixed Point Theory for many years to come."
-Isaac Namioka, University of Washington
This monograph gives a carefully worked out account of the most basic principles and applications of the theory of fixed points. Until now, a treatment of many of the discussed topics has been unavailable in book form. The presentation is self-contained and is accessible to a broad spectrum of readers. The main text is complemented by numerous exercises, detailed comments, and a comprehensive bibliography.
Andrzej Granas studied in Warsaw and then Moscow, where he earned his Ph.D. in 1958 under Lazar Lusternik. Since 1958, he has held various research and teaching posts in Poland, USA, Canada and elsewhere. During the Spring of 1970, he occupied a special chair at the Collège de France. In the early nineties, Dr. Granas founded and edited the journal Topological Methods of Nonlinear Analysis, and since 1992, he has served on the editorial board of the Zentralblatt. He is an Honorary Member of the Gdansk Scientific Society. The first part of this book is based on "Fixed Point Theory I" which was published by PWN, Warsaw in 1982. The second part follows the outline conceived by Andrzej Granas and the late James Dugundji.
Book Description
Many advances have taken place in the field of combinatorial algorithms since Methods of Mathematical Economics first appeared two decades ago. Despite these advances and the development of new computing methods, several basic theories and methods remain important today for understanding mathematical programming and fixed-point theorems. In this easy-to-read classic, readers learn Wolfe's method, which remains useful for quadratic programming, and the Kuhn-Tucker theory, which underlies quadratic programming and most other nonlinear programming methods. In addition, the author presents multiobjective linear programming, which is being applied in environmental engineering and the social sciences. The book presents many useful applications to other branches of mathematics and to economics, and it contains many exercises and examples. The advanced mathematical results are proved clearly and completely.
Average customer rating:
- It can be used as a refresher or as a primer
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Fixed Points (Mathematical World)
Yu. A. Shashkin
Manufacturer: Ameican Mathematical Society
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Intuitive Topology (Mathematical World, Vol 4) (Mathematical World)
ASIN: 082189000X |
Book Description
The theory of fixed points finds its roots in the work of Poincaré, Brouwer, and Sperner and makes extensive use of such topological notions as continuity, compactness, homotopy, and the degree of a mapping. Fixed point theorems have numerous applications in mathematics; most of the theorems ensuring the existence of solutions for differential, integral, operator, or other equations can be reduced to fixed point theorems. In addition, these theorems are used in such areas as mathematical economics and game theory.
This book presents a readable exposition of fixed point theory. The author focuses on the problem of whether a closed interval, square, disk, or sphere has the fixed point property. Another aim of the book is to show how fixed point theory uses combinatorial ideas related to decomposition (triangulation) of figures into distinct parts called faces (simplexes), which adjoin each other in a regular fashion. All necessary background concepts---such as continuity, compactness, degree of a map, and so on---are explained, making the book accessible even to students at the high school level. In addition, the book contains exercises and descriptions of applications. Readers will appreciate this book for its lucid presentation of this fundamental mathematical topic.
Customer Reviews:
It can be used as a refresher or as a primer.......2002-08-31
Many mathematical problems are solved by applying a fixed point theorem, so it is advantageous for all mathematicians to learn a great deal about them. As the title indicates, the purpose of this book is to present the fundamentals of fixed point theorems, and in that capacity, it is well worth reading. The basics of the concept of fixed points are demonstrated in an understandable, yet complete manner. Many diagrams are included as well as over fifty supplemental problems with solutions. Therefore, it can serve as either a supplemental or refresher text in the more detailed points of what is a critical and widely used area of mathematics. I strongly recommend that it be included in all academic mathematics libraries.
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Fixed Point Theory in Probabilistic Metric Spaces (Mathematics and Its Applications)
O. Hadzic , and
E. Pap
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Probabilistic Metric Spaces
ASIN: 1402001290 |
Book Description
Fixed point theory in probabilistic metric spaces can be considered as a part of Probabilistic Analysis, which is a very dynamic area of mathematical research. A primary aim of this monograph is to stimulate interest among scientists and students in this fascinating field. The text is self-contained for a reader with a modest knowledge of the metric fixed point theory.
Several themes run through this book. The first is the theory of triangular norms (t-norms), which is closely related to fixed point theory in probabilistic metric spaces. Its recent development has had a strong influence upon the fixed point theory in probabilistic metric spaces.
In Chapter 1 some basic properties of t-norms are presented and several special classes of t-norms are investigated. Chapter 2 is an overview of some basic definitions and examples from the theory of probabilistic metric spaces. Chapters 3, 4, and 5 deal with some single-valued and multi-valued probabilistic versions of the Banach contraction principle. In Chapter 6, some basic results in locally convex topological vector spaces are used and applied to fixed point theory in vector spaces.
Audience: The book will be of value to graduate students, researchers, and applied mathematicians working in nonlinear analysis and probabilistic metric spaces.
Product Description
This volume is produced from digital images from the Cornell University Library Historical Mathematics Monographs collection.
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