ISBN: 048645083X
Clear and simple, this introduction to the theory of sets employs
the discoveries of Cantor, Russell, Weierstrass, Zermelo,
Bernstein, Dedekind, and other mathematicians. It analyzes
concepts and principles, offering numerous examples. Starting
with the rudiments of set theory, the text advances to arbitrary
sets and their cardinal numbers, ordered sets and their order
types, and well-ordered sets and their ordinal numbers. With its
emphasis on fundamentals, the presentation is easily
comprehensible to students acquainted with college-level algebra.
1950 ed.
Table of Contents for Theory of Sets
INTRODUCTION
CHAPTER I. THE RUDIMENTS OF SET THEORY
1. A First Classification of Sets
2. Three Remarkable Examples of Enumerable Sets
3. "Subset, Sum, and Intersection of Sets; in Particular, of
Enumerable Sets"
4. An Example of a Nonenumerable Set
CHAPTER II. ARBITRARY SETS AND THEIR CARDINAL NUMBERS
1. Extensions of the Number Concept
2. Equivalence of Sets
3. Cardinal Numbers
4. Introductory Remarks Concerning the Scale of Cardinal Numbers
5. F. Bernstein's Equivalence-Theorem
6. The Sum of Two Cardinal Numbers
7. The Product of Two Cardinal Numbers
8. The Sum of Arbitrarily Many Cardinal Numbers
9. The Product of Arbitrarily Many Cardinal Numbers
10. The Power
11. Some Examples of the Evaluation of Powers
CHAPTER III. ORDERED SETS AND THEIR ORDER TYPES
1. Definition of Ordered Set
2. Similarity and Order Type
3. The Sum of Order Types
4. The Product of Two Order Types
5. Power of Type Classes
6. Dense Sets
7. Continuous Sets
CHAPTER IV. WELL-ORDERED SETS AND THEIR ORDINAL NUMBERS
1. Definition of Well-ordering and of Ordinal Number
2. "Addition of Arbitrarily Many, and Multiplication of Two,
Ordinal Numbers"
3. Subsets and Similarity Mappings of Well-ordered Sets
4. The Comparison of Ordinal Numbers
5. Sequences of Ordinal Numbers
6. Operating with Ordinal Numbers
7. "The Sequence of Ordinal Numbers, and Transfinite
Induction"
8. The Product of Arbitrarily Many Ordinal Numbers
9. Powers of Ordinal Numbers
10. Polynomials in Ordinal Numbers
11. The Well-ordering Theorem
12. An Application of the Well-ordering Theorem
13. The Well-ordering of Cardinal Numbers
14. Further Rules of Operation for Cardinal Numbers. Order Type
of Number Classes
15. Ordinal Numbers and Sets of Points
CONCLUDING REMARKS
BIBLIOGRAPHY
KEY TO SYMBOLS
INDEX
ISBN: 0486449807
This evenhanded treatment addresses the decades-old dispute among
probability theorists, asserting that both statistical and
inductive probabilities may be treated as sentence-theoretic
measurements, and that the latter qualify as estimates of the
former. Beginning with a survey of the essentials of sentence
theory and of set theory, the author examines statistical
probabilities, showing that statistical probabilities may be
passed on to sentences, and thereby qualify as truth-values. An
exploration of inductive probabilities follows, demonstrating
their reinterpretation as estimates of truth-values. Each chapter
is preceded by a summary of its contents. Illustrations and
footnotes elucidate definitions, theorems, and technicalities.
1962 ed.
Table of Contents for Statistical and Inductive Probabilities
1. The Languages L
2. Statistical Probabilities: Part One
3. Statistical Probabilities: Part Two
4. Inductive Probabilities
List of Symbols
Bibliographical References
Index of Authors
Index of Matters
ISBN: 0486450821
This stimulating introduction employs the language of point set
topology to define and discuss topological groups. It examines
set-theoretic topology and its applications in function spaces as
well as homotopy and the fundamental group. This new theoretical
knowledge is applied to concrete problems, such as the
calculation of the fundamental group of the circle and a proof of
the fundamental theorem of algebra. The abstract development
concludes with the classification of topological groups by
equivalence under local isomorphism. Well-chosen exercises and
problems serve as reinforcements. 1967 ed. 99 illustrations.
Table of Contents for Topology: An Introduction with Application
to Topological Groups
Introduction
1. Sets and Functions
2. Groups
3. Metric Spaces
4. Topologies
5. Topological Groups
6. Compactness and Connectedness
7. Function Spaces
8. The Fundamental Group
9. The Fundamental Group of the Circle
10. Locally Isomorphic Groups
Indexes
ISBN: 0486449785
This concise, elementary treatment illustrates the ways in which
an atomic-molecular perspective yields new insights and powers
operative in the realms of macroscopic thermodynamics. Starting
with an analysis of some very simple microcanonical ensembles, it
proceeds to the Boltzmann distribution law and a systematic
exploration of the proper formulation, evaluation, and
application of partition functions. Encompassing virtually all of
the forms of statistical mechanics customary to undergraduate
physical chemistry books, this brief text requires prior
acquaintance with only the rudiments of the calculus and a few of
the simplest propositions of classical thermodynamics. 1974 ed.
Table of Contents for Elements of Statistical Thermodynamics:
Second Edition
1. The Statistical Viewpoint
2. The Partition Function
3. Evaluation of Partition Functions
4. Applications
Problems
ISBN: 0486449769
Just how random is a card shuffle or a throw of the dice? Is
bluffing a valid poker strategy? How can you tell if a puzzle is
unsolvable? How large a role does luck play in games like golf
and soccer? This book examines each of these issues and many
others, along with the general principles behind such classic
puzzles as peg solitaire and Rubik's cube, showing how simple
mathematical analysis can throw unexpected light on games of
every type--games of chance, games of skill, games of chance and
skill, and automatic games. Lucid, instructive, and full of
surprises, it will fascinate mathematicians and gamesters alike.
1989 ed.
Table of Contents for The Mathematics of Games
1. Introduction
2. The luck of the deal
3. The luck of the die
4. To err is human
5. If A beats B, and B beats C...
6. Bluff and double bluff
7. The analysis of puzzles
8. Sauce for the gander
9. The measure of a game
10. When the counting has to stop
11. Round and round in circles
Further reading
Index
*
ISBN: 0486450309
This text for undergraduates was designed as a short introductory
course to give students the tools of vector algebra and calculus,
as well as a brief glimpse into these subjects' manifold
applications. The applications are developed to the extent that
the uses of the potential function, both scalar and vector, are
fully illustrated. Moreover, the basic postulates of vector
analysis are brought to the foreground, placing their logical
structure in sharp relief. The text concludes with a brief
introduction to abstract vector spaces, together with the ideas
of linear dependence, basis, and dimension. The exposition of
these abstract concepts is kept simple and clear. Numerous
figures appear throughout the text. 1957 ed. 86 figures.
Table of Contents for Vector Analysis
1. Vector Algebra
2. Line Vectors
3. Vector Functions of One Variable
4. Differential Invariants
5. Integral Theories
6. Dynamics
7. Fluid Mechanics
8. Electrodynamics
9. Vector Spaces
Appendix
ISBN: 0486449688
A world-renowned mathematician takes a lighthearted look at the
philosophy, form, and recreations of mathematics in this
fascinating book. Nathan A. Court discusses and illustrates the
charm of mathematics as well as the science's vast practical
utility and its vital significance to our cultural history. In
lucid and witty language, he explains how mathematicians think,
what they do, and the role of math in everyday life. Dr. Court
introduces the might and plight of reasoning, the relationship
between mathematics and genius, the lure of the infinite, and the
appeal of pure art in mathematics. He also addresses some notable
perplexities, including famous problems in recreational
mathematics (and their solutions). 1958 ed.
Table of Contents for Mathematics in Fun and in Earnest
1. Mathematics and Philosophy
2. Some Sociologic Aspects of Mathematics
3. The Lure of the Infinite
4. Mathesis the Beautiful
5. Mathematics and the Mathematician
6. Mathematical Asides
7. Mathematics as Recreation
ISBN: 0486449939
Self-contained and suitable for undergraduate students, this text
offers a working knowledge of calculus and statistics. A
coordinated study that develops the interrelationships between
calculus, probability, and statistics, it surveys random
variables and graphs, the derivative, applications of the
derivative, sequences and series, and integration. Additional
topics include the integral and continuous variates, some basic
discrete distributions as well as other important distributions,
hypothesis testing, functions of several variables, and
regression and correlation. The text concludes with an appendix,
answers to selected exercises, a general index, and an index of
symbols. 1970 ed. 201 figures. 36 tables.
Table of Contents for Calculus and Statistics
1. The Basic Concepts of Function and Probability
2. Some Specific Probabilities
3. Random Variables. Graphs.
4. The Derivative
5. Applications of the Derivative
6. Sequences and Series
7. Integration
8. The Integral and Continuous Variates
9. Some Basic Discrete Distributions
10. Other Important Distributions
11. Hypothesis Testing
12. Functions of Several Variables
13. Regression and Correlation
Appendix
Answers to Selected Exercises
General Index
Index of Symbols
ISBN: 0486449726
Suitable for advanced undergraduates and graduate students, this
text explores aspects of matrix theory that are most useful in
developing and appraising computational methods for solving
systems of linear equations and for finding characteristic roots.
An introductory chapter covers the Lanczos algorithm, orthogonal
polynomials, and determinantal identities. Succeeding chapters
examine norms, bounds and convergence; localization theorems and
other inequalities; and methods of solving systems of linear
equations. The final chapters illustrate mathematical principles
underlying linear equations and their interrelationships,
including methods of successive approximation, direct methods of
inversion, normalization and reduction of the matrix, and proper
values and vectors. 1964 ed.
Table of Contents for The Theory of Matrices in Numerical
Analysis
1. Some Basic Identities and Inequalities
2. Norms, Bounds and Convergence
3. Localization Theorems and Other Inequalities
4. The Solution of Linear Systems: Methods of Successive
Approximation
5. Direct Methods of Inversion
6. Proper Values and Vectors: Normalization and Reduction of the
Matrix
7. Proper Values and Vectors: Successive Approximation
Bibliography
Index
ISBN: 0486450333
Intended for advanced undergraduates and graduate students, this
concise text focuses on the convergence of real series.
Definitions of the terms and summaries of those results in
analysis that are of special importance in the theory of series
are specified at the outset. Series whose terms are complex and
real infinite products are examined only as illustrations of the
main theme, and infinite integrals appear solely in connection
with the integral test for convergence. Topics include functions
and limits, real sequences and series, series of non-negative
terms, general series, series of functions, the multiplication of
series, infinite products, and double series. Prerequisites
include a familiarity with the principles of elementary analysis.
1959 ed.
Table of Contents for Infinite Series 1. Functions and Limits
2. Some Properties of Particular Functions
3. Real Sequences and Series
4. Series of Non-negative Terms
5. General Series
6. Series of Functions
7. The Multiplication of Series
8. Infinite Products
9. Double Series
Index