2nd ed. 2002. X, 254 p. Softcover
3-540-43871-8
This introduction to Probability Theory can be used, at the beginning graduate level, for a one-semester course on Probability Theory or for self-direction without benefit of a formal course; the measure theory needed is developed in the text. It will also be useful for students and teachers in related areas such as Finance Theory (Economics), Electrical Engineering, and Operations Research. The text covers the essentials in a directed and lean way with 28 short chapters. Assuming of readers only an undergraduate background in mathematics, it brings them from a starting knowledge of the subject to a knowledge of the basics of Martingale Theory. After learning Probability Theory from this text, the interested student will be ready to continue with the study of more advanced topics, such as Brownian Motion and Ito Calculus, or Statistical Inference. The second edition contains some additions to the text and to the references and some parts are completely rewritten.
Keywords: Probability, measure theory, central limit theorem, martingales . MSC ( 2000 ): 60-01, 60E05, 60E10, 60G42
"(The book is) a lean and largely self
contained
introduction to the modern theory of probability,
aimed at
advanced undergraduate or beginning graduate
students. The 28
short chapters belie the book's genesis as
polished lecture
notes; the exposition is sleek and rigorous
and each chapter ends
with a supporting collection of mainly routine
exercises. ... The
authors make it clear what luggage is required
for this
exhilarating trek,... a good knowledge of
advanced calculus, some
linear algebra, and some "mathematical
sophistication".
With this understood, the itinary is immaculately
paced and
planned with just the right balances of technical
ascents and
pauses to admire the scenery. Within the
constraints of a slim
volume, it is hard to imagine how th authors
could have don a
more effective or more attractive job."
The Mathematical
Gazette, Vol. 84, No 500, 2000 "The
authors provide the
shortest path through the twenty-eight chapter
headings. The
topis are treated in a mathematically and
pedagogically
digestible way. The writing is concise and
crisp: the average
chapter length is about eight pages. ...
Numerous exercises add
to the value of the text as a teaching tool.
In conclusion, this
is an excellent text for the intended audience."
Short Book Reviews, Vol. 21, No. 2, 2001
Contents: 1. Introduction 2. Axioms of Probability 3. Conditional Probability and Independence 4. Probabilites on a Countable Space 5. Random Variables on a Countable Space 6. Construction of a Probability Measure 7. Construction of a Probability Measure on R 8. Random Variables 9. Integration with Respect to a Probability Measure 10. Independent Random Variables 11. Probability Distributions on R 12. Probability Distributions on Rn 13. Characteristic Functions 14. Properties of Characteristic Functions 15. Sums of Independent Random Variables 16. Gaussian Random Variables (The Normal and the Multivariate Normal Distributions) 17. Convergence of Random Variables 18. Weak Convergence 19. Weak Convergence and Characteristic Functions 20. The Laws of Large Numbers 21. The Central Limit Theorem 22. L2 and Hilbert Spaces 23. Conditional Expectation 24. Martingales 25. Supermartingales and Submartingales 26. Martingale Inequalities 27. Martingales Convergence Theorems 28. The Radon-Nikodym Theorem
Series: Universitext.
2002. X, 535 p. Hardcover
3-540-43714-2
This book gives the complete classification of Moufang polygons, starting from first principles. In particular, it may serve as an introduction to the various important algebraic concepts which arise in this classification including alternative division rings, quadratic Jordan division algebras of degree three, pseudo-quadratic forms, BN-pairs and norm splittings of quadratic forms. This book also contains a new proof of the classification of irreducible spherical buildings of rank at least three based on the observation that all the irreducible rank two residues of such a building are Moufang polygons. In an appendix, the connection between spherical buildings and algebraic groups is recalled and used to describe an alternative existence proof for certain Moufang polygons.
Keywords: Buildings, generalized polygons, incidence geometry, graph theory, combinatorics .
Contents: Preliminary Results.- Nine Families of Moufang Polygons.- The Classification of Moufang Polygons.- More Results on Moufang Polygons.- Moufang Polygons and Spherical Buildings.- Bibliography.- Index of Notation.- Index
Series: Springer Monographs in Mathematics.
2002. XII, 223 p. Hardcover
3-540-43796-7
Homology 3-sphere is a closed 3-dimensional manifold whose homology equals that of the 3-sphere. These objects may look rather special but they have played an outstanding role in geometric topology for the past fifty years. The book gives a systematic exposition of diverse ideas and methods in the area, from algebraic topology of manifolds to invariants arising from quantum field theories. The main topics covered are constructions and classification of homology 3-spheres, Rokhlin invariant, Casson invariant and its extensions, including invariants of Walker and Lescop, Herald and Lin invariants of knots, and equivariant Casson invariants, Floer homology and gauge-theoretical invariants of homology cobordism. Many of the topics covered in the book appear in monograph form for the first time. The book gives a rather broad overview of ideas and methods and provides a comprehensive bibliography. It will be appealing to both graduate students and researchers in mathematics and theoretical physics.
Keywords: Invariants of knots and 3--manifolds, Rokhlin invariant, Casson invariant, Floer homology, homology cobordism
Series: Encyclopaedia of Mathematical Sciences. VOL. 140
2002. Approx. 620 p. 200 illus. Hardcover
3-540-43908-0
Integrability, chaos and patterns are three of the most important concepts in nonlinear dynamics. These are covered in this book from fundamentals to recent developments. The book presents a self-contained treatment of the subject to suit the needs of students, teachers and researchers in physics, mathematics, engineering and applied sciences who wish to gain a broad knowledge of nonlinear dynamics. It describes fundamental concepts, theoretical procedures, experimental and numerical techniques and technological applications of nonlinear dynamics. Numerous examples and problems are included to facilitate the understanding of the concepts and procedures described. In addition to 16 chapters of main material, the book contains 10 appendices which present in-depth mathematical formulations involved in the analysis of various nonlinear systems.
Keywords: Nonlinear dynamics, Chaos, Solitons, Integrable systems, Spatiotemporal patterns
Contents: What is Nonlinearity?- Linear and Nonlinear Oscillators.- Qualitative Features.- Bifurcations and Onset of Chaos in Dissipative Systems.- Chaos in Dissipative Nonlinear Oscillators and Criteria for Chaos.- Chaos in Nonlinear Electronic Circuits.- Chaos in Conservative Systems.- Characterization of Regular and Chaotic Motions.- Further Developments in Chaotic Dynamics.- Finite Dimensional Integrable Nonlinear Dynamical Systems.- Linear and Nonlinear Dispersive Waves.- Korteweg-De Vries Equation and Solitons.- Basic Soliton Theory of KdV Equation.- Other Ubiquitous Soliton Equations.- Spatio-Temporal Patterns.- Nonlinear Dynamics: From Theory to Technology.- Appendices.
Series: Advanced Texts in Physics.
2002. Approx. 255 pp. Hardcover
0-387-95519-4
This book covers functional analysis and its applications to continuum mechanics. The mathematical material is treated in a non-abstract manner and is fully illuminated by the underlying mechanical ideas. The presentation is concise but complete, and is intended for specialists in continuum mechanics who wish to understand the mathematical underpinnings of the discipline. Graduate students and researchers in mathematics, physics, and engineering will find this book useful. Exercises and examples are included throughout with detailed solutions provided in the appendix.
Contents: Metric Spaces.- Elements of theory of operators.- Elements of nonlinear functional analysis.- Hints for selected problems.
Series: Springer Monographs in Mathematics.
2002. VIII, 536 p. Hardcover
3-540-43581-6
This book opens a novel dimension in the 50 year history of mathematical theories of "information" since the birth of Shannon theory. First of all, it introduces, in place of the traditional notion of entropy and mutual information, the completely new and highly unconventional approach of "information-spectrum" as a basic but powerful tool for constructing the general theory of information. Reconstructing step-by-step all the essential major topics in information theory from the viewpoint of such an "information-spectrum", this comprehensive work provides an accessible introduction to the new type of mathematical theory of information that focuses mainly on general nonstationary and /or nonergodic sources and channels, in clear contrast with the traditional theories of information. This book is a new non-traditional theoretical reference for communication professionals and statisticians specializing in information theory.
Keywords: information-spectrum, information theory, source coding, channel coding, decoder, encoder MSC ( 2000 ): 60xx ; 62xx ; 94xx
Contents: I Source Coding.- II Random Number Generation.- III Channel Coding.- IV Hypothesis Testing.- V Rate-Distortion Theory.- VI Identification Code and Channel Resolvability.- VII Multi-Terminal Information Theory.
Series: Applications of Mathematics. VOL. 50
3rd ed. 2002. Approx. 450 pp. 60 figs. Hardcover
0-387-95534-8
Derived from a course given at the University
of Maryland for
advanced graduate students, this book deals
with some of the
latest developments in our attempts to construct
a unified theory
of the fundamental interactions of nature.
Among the topics
covered are spontaneous symmetry breaking,
grand unified
theories, supersymmetry, and supergravity.
the book starts with a
quick review of elementary particle theory
and continues with a
discussion of composite quarks, leptons,
Higgs bosons, and CP
violation; it concludes with consideration
of supersymmetric
unification schemes, in which bosons and
leptons are considered
in some sense equivalent. The third edition
will be completely
revised and brought up to date, particularly
by including
discussions of the many experimental developments
in recent years.
Contents: Important Basic Concepts in Particle Physics.- Spontaneous Symmetry Breaking.- The SU(2) Asub LU X U(1) Model.- CP-Violation: Weak and Strong.- Grand Unification and the SU(5) Model.- Left-Right Symmetric Models of Weak Interactions and Massive Neutrinos.- SO(10) Grand Unification.- Technicolor and Compositeness.- Global Supersymmetry.- Field Theories with Global Supersymmetry.- Broken Supersymmetry and Application to Particle Physics.- Minimal Supersymmetric Standard Model.- Supersymmetric Grand Unification.- Local Supersymmetry (N=1).- Application of Supergravity (N=1) to Particle Physics.- Beyond N=1 Supergravity.- Superstrings and Quark-Lepton Physics.- Index.
Series: Graduate Texts in Contemporary Physics.
2002. Approx. 470 pp. 14 figs. Hardcover
0-387-95464-3
The largest part of this book is devoted to normal forms, divided into semisimple theory, applied when the linear part is diagonalizable, and the general theory, applied when the linear part is the sum of the semisimple and nilpotent matrices. One of the objectives of this book is to develop all of the necessary theory 'from scratch' in just the form that is needed for the application to normal forms, with as little unnecessary terminology as possible. The intended audience is Ph.D. students and researchers in applied mathematics, theoretical physics, and advanced engineering, though in principle it could be read by anyone with a sufficient background in linear algebra and differential equations.
Contents: Preface.- 1. Two Examples.- 2. The splitting problem for linear operators.- 3. Linear Normal Forms.- 4. Nonlinear Normal Forms.- 5. Geometrical Structures in Normal Forms.- 6. Selected Topics in Local Bifurcation Theory.- Appendix A: Rings.- Appendix B: Modules.- Appendix C: Format 2b: Generated Recursive (Hori).- Appendix D: Format 2c: Generated Recursive (Deprit).- Appendix E: On Some Algorithms in Linear Algebra.- Bibliography.- Index.
Series: Springer Monographs in Mathematics.