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Books in Set theory

Irreducible Tensorial Sets
1st Edition
U. Fano + 1 more
H. S. W. Massey + 1 more
English
Irreducible Tensorial Sets discusses mathematical methods originating from the theory of coupling and recoupling of angular momenta in atomic physics that constitute an extension of vector and tensor algebra. The book presents a unified treatment with a compact system of notations from different approaches, such as group theory, algebra, and quantum mechanical transformation theory. It discusses irreducible tensorial sets that cover different sets of quantities such as tensor components and states of atomic systems. It also explains quantum mechanical applications, coupling and recoupling of atomic and nuclear states, the Wigner-Eckart theorem, and the products of tensorial sets of operators. The text shows how to calculate the interaction energy between atomic systems couple with one another with a constant total angular momentum. The book also explains the correlations which are functions of the Euler angles between the frame of reference in which a radiation is observed and a frame of reference attached to the orienting radiation or field. It then cites sample problems related to the angular distribution of radiations. The book will prove useful for physicists, for mathematicians, or for readers with some knowledge in theoretical physics, particularly on theory of groups and quantum mechanics.
Readings in Fuzzy Sets for Intelligent Systems
1st Edition
Didier J. Dubois + 2 more
English
Readings in Fuzzy Sets for Intelligent Systems is a collection of readings that explore the main facets of fuzzy sets and possibility theory and their use in intelligent systems. Basic notions in fuzzy set theory are discussed, along with fuzzy control and approximate reasoning. Uncertainty and informativeness, information processing, and membership, cognition, neural networks, and learning are also considered. Comprised of eight chapters, this book begins with a historical background on fuzzy sets and possibility theory, citing some forerunners who discussed ideas or formal definitions very close to the basic notions introduced by Lotfi Zadeh (1978). The reader is then introduced to fundamental concepts in fuzzy set theory, including symmetric summation and the setting of fuzzy logic; uncertainty and informativeness; and fuzzy control. Subsequent chapters deal with approximate reasoning; information processing; decision and management sciences; and membership, cognition, neural networks, and learning. Numerical methods for fuzzy clustering are described, and adaptive inference in fuzzy knowledge networks is analyzed. This monograph will be of interest to both students and practitioners in the fields of computer science, information science, applied mathematics, and artificial intelligence.
Essentials of Elementary School Mathematics
1st Edition
Max D. Larsen + 1 more
English
Essentials of Elementary School Mathematics is an introductory text on the essentials of mathematics taught in elementary schools. It presents a systematic development of the mathematics of arithmetic. A primary objective is to give students a background sufficient to understand and answer at an appropriate level the various quite penetrating questions asked by young students. Some examples and exercises are concerned primarily with pedagogical aspects of arithmetic. Comprised of 14 chapters, this book begins with an overview of the language of mathematics, focusing on concepts such as the conjunction (and); negation (not); disjunction (or); and conditional (if...then...). The discussion then turns to the theory of sets; the concept of binary operations; and recognition and identification of properties of various relations. The next section deals with the number systems of arithmetic: whole numbers, integers, rational numbers, and real numbers. Number theory and clock arithmetic are also examined, along with counting techniques and probability. The final section is devoted to motion geometry and analytic geometry. This monograph should be of interest to students and teachers of mathematicians at the elementary level.
Nonmeasurable Sets and Functions
1st Edition
Volume 195
Alexander Kharazishvili
English
The book is devoted to various constructions of sets which are nonmeasurable with respect to invariant (more generally, quasi-invariant) measures. Our starting point is the classical Vitali theorem stating the existence of subsets of the real line which are not measurable in the Lebesgue sense. This theorem stimulated the development of the following interesting topics in mathematics:1. Paradoxical decompositions of sets in finite-dimensional Euclidean spaces;2. The theory of non-real-valued-meas... cardinals;3. The theory of invariant (quasi-invariant)ext... of invariant (quasi-invariant) measures.These topics are under consideration in the book. The role of nonmeasurable sets (functions) in point set theory and real analysis is underlined and various classes of such sets (functions) are investigated . Among them there are: Vitali sets, Bernstein sets, Sierpinski sets, nontrivial solutions of the Cauchy functional equation, absolutely nonmeasurable sets in uncountable groups, absolutely nonmeasurable additive functions, thick uniform subsets of the plane, small nonmeasurable sets, absolutely negligible sets, etc. The importance of properties of nonmeasurable sets for various aspects of the measure extension problem is shown. It is also demonstrated that there are close relationships between the existence of nonmeasurable sets and some deep questions of axiomatic set theory, infinite combinatorics, set-theoretical topology, general theory of commutative groups. Many open attractive problems are formulated concerning nonmeasurable sets and functions.
The Infinite-Dimensional Topology of Function Spaces
1st Edition
Volume 64
J. van Mill
English
In this book we study function spaces of low Borel complexity.Technique... from general topology, infinite-dimensional topology, functional analysis and descriptive set theoryare primarily used for the study of these spaces. The mix ofmethods from several disciplines makes the subjectparticularly interesting. Among other things, a complete and self-contained proof of the Dobrowolski-Marcisze... Theorem that all function spaces of low Borel complexity are topologically homeomorphic, is presented.In order to understand what is going on, a solid background ininfinite-dimension... topology is needed. And for that a fair amount of knowledge of dimension theory as well as ANR theory is needed. The necessary material was partially covered in our previous book `Infinite-dimensiona... topology, prerequisites and introduction'. A selection of what was done there can be found here as well, but completely revised and at many places expanded with recent results. A `scenic' route has been chosen towards theDobrowolski-Marci... Theorem, linking theresults needed for its proof to interesting recent research developments in dimension theory and infinite-dimensional topology.The first five chapters of this book are intended as a text forgraduate courses in topology. For a course in dimension theory, Chapters 2 and 3 and part of Chapter 1 should be covered. For a course in infinite-dimensional topology, Chapters 1, 4 and 5. In Chapter 6, which deals with function spaces, recent research results are discussed. It could also be used for a graduate course in topology but its flavor is more that of a research monograph than of a textbook; it is thereforemore suitable as a text for a research seminar. The bookconsequently has the character of both textbook and a research monograph. In Chapters 1 through 5, unless statedotherwise, all spaces under discussion are separable andmetrizable. In Chapter 6 results for more general classes of spaces are presented.In Appendix A for easy reference and some basic facts that are important in the book have been collected. The book is not intended as a basis for a course in topology; its purpose is to collect knowledge about general topology.The exercises in the book serve three purposes: 1) to test the reader's understanding of the material 2) to supply proofs of statements that are used in the text, but are not proven there3) to provide additional information not covered by the text.Solutions to selected exercises have been included in Appendix B.These exercises are important or difficult.
Theory of Relations
1st Edition
Volume 145
R. Fraisse
English
Relation theory originates with Hausdorff (Mengenlehre 1914) and Sierpinski (Nombres transfinis, 1928) with the study of order types, specially among chains = total orders = linear orders. One of its first important problems was partially solved by Dushnik, Miller 1940 who, starting from the chain of reals, obtained an infinite strictly decreasing sequence of chains (of continuum power) with respect to embeddability. In 1948 I conjectured that every strictly decreasing sequence of denumerable chains is finite. This was affirmatively proved by Laver (1968), in the more general case of denumerable unions of scattered chains (ie: which do not embed the chain Q of rationals), by using the barrier and the better orderin gof Nash-Williams (1965 to 68).Another important problem is the extension to posets of classical properties of chains. For instance one easily sees that a chain A is scattered if the chain of inclusion of its initial intervals is itself scattered (6.1.4). Let us again define a scattered poset A by the non-embedding of Q in A. We say that A is finitely free if every antichain restriction of A is finite (antichain = set of mutually incomparable elements of the base). In 1969 Bonnet and Pouzet proved that a poset A is finitely free and scattered iff the ordering of inclusion of initial intervals of A is scattered. In 1981 Pouzet proved the equivalence with the a priori stronger condition that A is topologically scattered: (see 6.7.4; a more general result is due to Mislove 1984); ie: every non-empty set of initial intervals contains an isolated elements for the simple convergence topology.In chapter 9 we begin the general theory of relations, with the notions of local isomorphism, free interpretability and free operator (9.1 to 9.3), which is the relationist version of a free logical formula. This is generalized by the back-and-forth notions in 10.10: the (k,p)-operator is the relationist version of the elementary formula (first order formula with equality).Chapter 12 connects relation theory with permutations: theorem of the increasing number of orbits (Livingstone, Wagner in 12.4). Also in this chapter homogeneity is introduced, then more deeply studied in the Appendix written by Norbert Saucer.Chapter 13 connects relation theory with finite permutation groups; the main notions and results are due to Frasnay. Also mention the extension to relations of adjacent elements, by Hodges, Lachlan, Shelah who by this mean give an exact calculus of the reduction threshold.The book covers almost all present knowledge in Relation Theory, from origins (Hausdorff 1914, Sierpinski 1928) to classical results (Frasnay 1965, Laver 1968, Pouzet 1981) until recent important publications (Abraham, Bonnet 1999).All results are exposed in axiomatic set theory. This allows us, for each statement, to specify if it is proved only from ZF axioms of choice, the continuum hypothesis or only the ultrafilter axiom or the axiom of dependent choice, for instance.
Hausdorff Gaps and Limits
1st Edition
Volume 132
R. Frankiewicz + 1 more
English
Gaps and limits are two phenomena occuring in the Boolean algebra P(&ohgr;)/fin. Both were discovered by F. Hausdorff in the mid 1930's. This book aims to show how they can be used in solving several kinds of mathematical problems and to convince the reader that they are of interest in themselves. The forcing technique, which is not commonly known, is used widely in the text. A short explanation of the forcing method is given in Chapter 11. Exercises, both easy and more difficult, are given throughout the book.
Rings, Fields and Groups
2nd Edition
Reg Allenby
English
'Rings, Fields and Groups' gives a stimulating and unusual introduction to the results, methods and ideas now commonly studied on abstract algebra courses at undergraduate level. The author provides a mixture of informal and formal material which help to stimulate the enthusiasm of the student, whilst still providing the essential theoretical concepts necessary for serious study.Retaining the highly readable style of its predecessor, this second edition has also been thoroughly revised to include a new chapter on Galois theory plus hints and solutions to many of the 800 exercises featured.
Almost Free Modules
Set-Theoretic Methods
1st Edition
Volume 46
P.C. Eklof + 1 more
English
This is an extended treatment of the set-theoretic techniques which have transformed the study of abelian group and module theory over the last 15 years. Part of the book is new work which does not appear elsewhere in any form. In addition, a large body of material which has appeared previously (in scattered and sometimes inaccessible journal articles) has been extensively reworked and in many cases given new and improved proofs. The set theory required is carefully developed with algebraists in mind, and the independence results are derived from explicitly stated axioms. The book contains exercises and a guide to the literature and is suitable for use in graduate courses or seminars, as well as being of interest to researchers in algebra and logic.
Descriptive Set Theory
1st Edition
Volume 100
Y.N. Moschovakis
English
Now available in paperback, this monograph is a self-contained exposition of the main results and methods of descriptive set theory. It develops all the necessary background material from logic and recursion theory, and treats both classical descriptive set theory and the effective theory developed by logicians.