Kallenberg, O., University of Auburn, AL, USA

Foundations of Modern Probability, 2nd ed.

2002. Approx. 650 pp. Hardcover
0-387-95313-2

From the reviews of the first editions: "... Kallenberg's present book would have to qualify as the assimilation of probability par excellence. It is a great edifice of material, clearly and ingeniously presented, without any non-mathematical distractions. Readers wishing to venture into it may do so with confidence that they are in very capable hands." F.B. Knight, Mathematical Reviews
"...Indeed the monograph has the potential to become a (possibly even "the") major reference book on large parts of probability theory for the next decade or more." M. Scheutzow, Zentralblatt
"The theory of probability has grown exponentially during the second half of the twentieth century and the idea of writing a single volume that could serve as a general reference for much of the modern theory seems almost foolhardy. Yet this is precisely what Professor Kallenberg has attempted in the volume under review and he has accomplished it brilliantly...It is astonishing that a single volume of just over five hundred pages could contain so much material presented with complete rigor and still be at least formally self-contained..." R.K. Getoor, Metrika
This new edition contains four new chapters as well as numerous improvements throughout the text.
Olav Kallenberg was educated in Sweden, where he received his Ph.D. in 1972 from Chalmers University. After teaching for many years at Swedish universities, he moved in 1985 to the U.S., where he is currently a Professor of Mathematics at Auburn University. He is known for his book "Random Measures" (4th edition, 1986) and for numerous research papers in all areas of probability. In 1977, he was the second recipient ever of the prestigious Rollo Davidson Prize from Cambridge University. In 1991-94, he served as the Editor-in-Chief of "Probability Theory and Related Fields."


From the reviews of the first editions: "... Kallenberg's present book would have to qualify as the assimilation of probability par excellence. It is a great edifice of material, clearly and ingeniously presented, without any non-mathematical distractions. Readers wishing to venture into it may do so with confidence that they are in very capable hands." F.B. Knight, Mathematical Reviews
"...Indeed the monograph has the potential to become a (possibly even "the") major reference book on large parts of probability theory for the next decade or more." M. Scheutzow, Zentralblatt
"The theory of probability has grown exponentially during the second half of the twentieth century and the idea of writing a single volume that could serve as a general reference for much of the modern theory seems almost foolhardy. Yet this is precisely what Professor Kallenberg has attempted in the volume under review and he has accomplished it brilliantly...It is astonishing that a single volume of just over five hundred pages could contain so much material presented with complete rigor and still be at least formally self-contained..." R.K. Getoor, Metrika


Contents: Measure Theory-Basic Notions * Measure Theory-Key Results.- Processes, Distributions, and Independence.- Random Sequences, Series, and Averages.- Characteristic Functions and Classical Limit Theorems.- Conditioning and Disintegration.- Martingales and Optional Times.- Markov Processes and Discrete-Time Chains.- Random Walks and Renewal Theory.- Stationary Processes and Ergodic Theory.- Special Notions of Symmetry and Invariance.- Poisson and Pure Jump- Type Markov Processes.- Gaussian Processes and Brownian Motion.- Skorohod Embedding and Invariance Principles.- Independent Increments and Infinite Divisibility.- Convergence of Random Processes, Measures, and Sets.- Stochastic Integrals and Quadratic Variation.- Continuous Martingales and Brownian Motion.- Feller Processes and Semigroups.- Ergodic Properties of Markov Processes.- Stochastic Differential Equations and Martingale Problems.- Local Time, Excursions, and Additive Functionals.- One-Dimensional SDEs and Diffusions.- Connections with PDEs and Potential Theory.- Predictability, Compensation, and Excessive Functions.- Semimartingales and General Stochastic Integration.- Large Deviations.- Appendix 1: Advanced.

Series: Probability and its Applications.

Lang, S., Yale University, New Haven, CT, USA

Short Calculus

The Original Edition of "A First Course in Calculus"

Reprint of the 1st ed. Addison-Wesley, 1964.
2001. Approx. 260 pp. 30 figs. Softcover
0-387-95327-2

This is a reprint of "A First Course in Calculus," which has gone through five editions since the early sixties. It covers all the topics traditionally taught in the first-year calculus sequence in a brief and elementary fashion. As sociological and educational conditions have evolved in various ways over the past four decades, it has been found worthwhile to make the original edition available again. The audience consists of those taking the first calculus course, in high school or college. The approach is the one which was successful decades ago, involving clarity, and adjusted to a time when the students'background was not as substantial as it might be. We are now back to those times, so its time to start over again. There are no epsilon-deltas, but this does not imply that the book is not rigorous. Lang learned this attitude from Emil Artin, around 1950.

Keywords: Calculus

"...Lang's present book is a source of interesting ideas and brilliant techniques."
Acta Scientarium Mathematicarum

"... It is an admirable straightforward introduction to calculus."
Mathematika

Contents: Numbers and Functions.- Graphs and Curves.- The Derivative.- Sine and Cosine.- The Mean Value Theorem.- Sketching Curves.- Inverse Functions.- Exponents and Logarithms.- Integration.- Properties of the Integral.- Techniques of Integration.- Some Substantial Exercises.- Applications of Integration.- Taylor's Formula.- Series.- Appendix 1. Epsilon and Delta.- Appendix 2. Physics and Mathematics.- Answers.- Index.

Series: Undergraduate Texts in Mathematics

Stillwell, J., Monash University, Clayton, VIC, Australia

Mathematics and its History, 2nd ed

 2002. Approx. 520 pp. 177 figs. Hardcover
0-387-95336-1

From the reviews of the first edition: "There are many books on the history of mathematics in which mathematics is subordinated to history. This is a book in which history is definitely subordinated to mathematics. It can be described as a collection of critical historical essays dealing with a large variety of mathematical disciplines and issues, and intended for a broad audience...we know of no book on mathematics and its history that covers half as much nonstandard material. Even when dealing with standard material, Stillwell manages to dramatize it and to make it worth rethinking. In short, his book is a splendid addition to the genre of works that build royal roads to mathematical culture for the many." Mathematical Intelligencer
"The discussion is at a deep enough level that I suspect most trained mathematicians will find much that they do not know, as well as good intuitive explanations of familiar facts. The careful exposition, lightness of touch, and the absence of technicalities should make the book accessible to most senior undergraduates." American Mathematical Monthly
"...The book is a treasure, which deserves wide adoption as a text and much consultation by historians and mathematicians alike." Physis - Revista Internazionale di Storia della Scienza
"A beautiful little book, certain to be treasured by several generations of mathematics lovers, by students and teachers so enlightened as to think of mathematics not as a forest of technical details but as the beautiful coherent creation of a richly diverse population of extraordinary people...His writing is so luminous as to engage the interest of utter novices, yet so dense with particulars as to stimulate the imagination of professionals." Book News, Inc.
This second edition includes new chapters on Chinese and Indian number theory, on hypercomplex numbers, and on algebraic number theory. Many more exercises have been added, as well as commentary to the exercises expalining how they relate to the preceding section, and how they foreshadow later topics. The index has been given added structure to make searching easier, the references have been redone, and hundreds of minor improvements have been made throughout the text.

Keywords: History of Mathematics

From the reviews of the first edition:
"There are many books on the history of mathematics in which mathematics is subordinated to history. This is a book in which history is definitely subordinated to mathematics. It can be described as a collection of critical historical essays dealing with a large variety of mathematical disciplines and issues, and intended for a broad audience...we know of no book on mathematics and its history that covers half as much nonstandard material. Even when dealing with standard material, Stillwell manages to dramatize it and to make it worth rethinking. In short, his book is a splendid addition to the genre of works that build royal roads to mathematical culture for the many."
Mathematical Intelligencer
"The discussion is at a deep enough level that I suspect most trained mathematicians will find much that they do not know, as well as good intuitive explanations of familiar facts. The careful exposition, lightness of touch, and the absence of technicalities should make the book accessible to most senior undergraduates." American Mathematical Monthly
"...The book is a treasure, which deserves wide adoption as a text and much consultation by historians and mathematicians alike." Physis - Revista Internazionale di Storia della Scienza
"A beautiful little book, certain to be treasured by several generations of mathematics lovers, by students and teachers so enlightened as to think of mathematics not as a forest of technical details but as the beautiful coherent creation of a richly diverse population of extraordinary people...His writing is so luminous as to engage the interest of utter novices, yet so dense with particulars as to stimulate the imagination of professionals." Book News, Inc.


Contents: The Theorem of Pythagoras.- Greek Geometry.- Greek Number Theory.- Infinity in Greek Mathematics.- Number Theory in Asia.- Polynomial Equations.- Analytic Geometry.- Projective Geometry.- Calculus.- Infinite Series.- The Revival of Number Theory.- Elliptic Functions.- Mechanics.- Complex Numbers in Algebra.- Complex Numbers and Curves.- Complex Numbers and Functions.- Differential Geometry.- Noneuclidean Geometry.- Group Theory.- Hypercomplex Numbers.- Algebraic Number Theory.- Topology.- Sets, Logic, and Computation.- Bibliography.- Index.

Series: Undergraduate Texts in Mathematics.

Aubert, G., University of Nice-Sophia Antipolis, Nice, France
Kornprobst, P., INRIA, Sophia Antipolis, France

Mathematical Problems in Image Processing
Partial Differential Equations and the Calculus of Variations

2001. Approx. 315 pp. 93 figs. Hardcover
0-387-95326-4

Partial differential equations and variational methods were introduced into image processing about 15 years ago, and intensive research has been carried out since then. The main goal of this work is to present the variety of image analysis applications and the precise mathematics involved. It is intended for two audiences. The first is the mathematical community, to show the contribution of mathematics to this domain and to highlight some unresolved theoretical questions. The second is the computer vision community, to present a clear, self-contained, and global overview of the mathematics involved in image processing problems.
The book is divided into five main parts. Chapter 1 is a detailed overview. Chapter 2 describes and illustrates most of the mathematical notions found throughout the work. Chapters 3 and 4 examine how PDEs and variational methods can be successfully applied in image restoration and segmentation processes. Chapter 5, which is more applied, describes some challenging computer vision problems, such as sequence analysis or classification. This book will be useful to researchers and graduate students in mathematics and computer vision.

Keywords: Image Processing, Partial Differential Equations, PDE, Calculus of Variations, PDEs in image processing, variational methods in image processing, image analysis

Contents: Introduction.- Mathematical Preliminaries.- Image Restoration.- The Segmentation Problem.- Other Challenging Applications.- Appendices.

Series: Applied Mathematical Sciences. VOL. 147

Chaitin, G.J., IBM Research Division, Hawthorne, NY, USA

Conversations with a Mathematician
Math, Art, Science and the Limits of Reason

2002. VII, 158 pp. Hardcover
1-85233-549-1

G. J. Chaitin is at the IBM Thomas J. Watson Research Center in New York. He has shown that God plays dice not only in quantum mechanics, but even in the foundations of mathematics, where Chaitin discovered mathematical facts that are true for no reason, that are true by accident. This book collects his most wide-ranging and non-technical lectures and interviews, and it will be of interest to anyone concerned with the philosophy of mathematics, with the similarities and differences between physics and mathematics, or with the creative process and mathematics as an art.
"Chaitin has put a scratch on the rock of eternity."
Jacob T. Schwartz, Courant Institute, New York University, USA
"(Chaitin is) one of the great ideas men of mathematics and computer science."
Marcus Chown, author of The Magic Furnace, in NEW SCIENTIST
"Finding the right formalization is a large component of the art of doing great mathematics."
John Casti, author of Mathematical Mountaintops, on Godel, Turing and Chaitin in NATURE
"What mathematicians over the centuries - from the ancients, through Pascal, Fermat, Bernoulli, and de Moivre, to Kolmogorov and Chaitin - have discovered, is that it ArandomnessU is a profoundly rich concept."
Jerrold W. Grossman in the MATHEMATICAL INTELLIGENCER

Contents: A Century of Controversy over the foundations of mathematics.- How to be a mathematician.- The creative life: science vs art.- Algorithmic information theory and the foundations of mathematics.- Randomness in arithmetic.- The reason for my life.- Undecidability and randomness in pure mathematics.- Math, science and fantasy.- Sensual mathematics.- Final thoughts.