Series: Studies in Computational Intelligence, Vol. 6
2005, XIII, 375 p. 75 illus., Hardcover
ISBN: 3-540-26257-1
About this book
"Foundations of Data Mining and Knowledge Discovery"
contains the latest results and new directions in data mining
research. Data mining, which integrates various technologies,
including computational intelligence, database and knowledge
management, machine learning, soft computing, and statistics, is
one of the fastest growing fields in computer science. Although
many data mining techniques have been developed, further
development of the field requires a close examination of its
foundations. This volume presents the results of investigations
into the foundations of the discipline, and represents the state
of the art for much of the current research. This book will prove
extremely valuable and fruitful for data mining researchers, no
matter whether they would like to uncover the fundamental
principles behind data mining, or apply the theories to practical
applications.
Table of contents
ISBN: 0415355281
Pub Date: 12 SEP 2005
Type: Paperback Book
'Nagel and Newman accomplish the wondrous task of clarifying the argumentative outline of Kurt Godel's celebrated logic bomb.' ? The Guardian
In 1931 the mathematical logician Kurt Godel published a revolutionary paper that challenged certain basic assumptions underpinning mathematics and logic. A colleague of physicist Albert Einstein, his theorem proved that mathematics was partly based on propositions not provable within the mathematical system. The importance of Godel's Proof rests upon its radical implications and has echoed throughout many fields, from maths to science to philosophy, computer design, artificial intelligence, even religion and psychology. While others such as Douglas Hofstadter and Roger Penrose have published bestsellers based on Godel?s theorem, this is the first book to present a readable explanation to both scholars and non-specialists alike. A gripping combination of science and accessibility, Godel?s Proof by Nagel and Newman is for both mathematicians and the idly curious, offering those with a taste for logic and philosophy the chance to satisfy their intellectual curiosity.
Contents:
Acknowledgments
Introduction
The Problem of Consistency
Absolute Proofs of Consistency
The Systematic Codification of Formal Logic
An Example of a Successful Absolute Proof of Consistency
The Idea of Mapping and its Use in Mathematics
Godel's Proof
Godel numbering
The arithmetization of meta-mathematics
The heart of Godel?s argument
Concluding Reflections
Appendix: Notes
Brief Bibliography
Index
This is the first English translation of the collected papers of the great German mathematician, Bernhard Riemann
(1826-1866). Riemann surfaces, Riemannian geometry and the Riemann zeta
function are fundamental concepts of modern mathematics. Riemann’s influence
was both broad and deep. In spite of his short life and precarious health,
he provided new and profound insights in many areas of analysis, geometry
and physics.
Highlights of the Collected Papers include:
Riemann’s doctoral thesis on the theory of functions of a complex variable;
The habilitation thesis on the representation of functions by a trigonometric series;
The famous lecture of 1854 on the foundations of geometry;
The beautiful paper on the zeta function and its connection to the distribution of prime numbers;
The great memoir on Abelian functions.
The book includes notes on the individual papers, with an essay on Riemann’s life and work by his friend and colleague Richard Dedekind. The Collected Papers make fruitful reading for anyone interested in the growth of mathematical ideas.
2004. x + 555 pp. ISBN 0-9740427-2-2 (hardback). 0-9740427-3-0 (paperback)
Contents
Hendrik Kloosterman 1900-1968This is a careful and detailed account of the spectral theory of automorphic forms in the upper half plane. The account is restricted to finite index subgroups of the modular group. The tools needed are developed from scratch, including hyperbolic geometry, fundamental regions and the double coset decomposition. Topics developed for use later in the book include special functions, invariant integral operators and Fredholm theory. Non-holomorphic automorphic functions are discussed, beginning from the Fourier expansion. The Eisenstein series are given special attention, including their analytic continuation. The Eisenstein transform is developed and used for the spectral decomposition of square integrable automorphic functions into eigenfunctions of the hyperbolic Laplacian. This is treated both pointwise and in the Hilbert space sense.
In Volume 2, the Kuznetsov formulae will be derived and applied to problems in analytic number theory, along the lines initiated by Deshouillers and Iwaniec in their well-known Inventiones paper of 1982. The present volume is designed to make this important work accessible to non-experts.
From the review in Zentralblatt Math: “... this is a highly welcome introduction to the spectral theory of automorphic functions, starting from very modest prerequisites and leading up to deep results. We may look forward with great excitement to the publication of Vol. 2.” (J. Elstrodt.)
“The pace is leisurely, and the author takes care to include any background material beyond an introductory course in real and complex analysis... This is a helpful introduction to the theory of automorphic forms and may be helpful ... to readers of more advanced texts such as H. Iwaniec, Spectral methods of automorphic forms...” (K. Soundararajan.)
Roger C. Baker, Kloosterman sums and Maass forms, Volume 1. 2003. xiv + 285 pp. Paperback,ISBN 0-9740427-0-6
In 1948 Andre Weil published the proof of the Riemann hypothesis for function fields in one variable over a finite ground field. This was a landmark in both number theory and algebraic geometry. Applications included hitherto unattainable bounds for exponential sums, in particular Kloosterman sums. Weil built on innovative work in the 1920’s and 1930’s of Emil Artin and Helmut Hasse, among others. Later, Grothendieck and Deligne employed profound innovations in algebraic geometry to carry Weil’s work much further.
It came as a surprise to the number theory community when Sergei Stepanov gave elementary proofs of many of Weil’s most significant results in a series of papers published between 1969 and 1974. Stepanov’s method drew inspiration from the work of Axel Thue (1909) in Diophantine approximation. Portraits of Thue, Artin, Hasse and Weil feature on the front cover of this book.
Proofs of Weil’s result in full generality, based on Stepanov’s ideas, were given independently by Wolfgang Schmidt and Enrico Bombieri
in 1973. The present book contains accounts of both methods. Schmidt’s method, which is more elementary, is discussed in Chapters 1 - 6, along with many related matters. This part of the book is, essentially, the content of the first edition, published by Springer in 1976. The remaining chapters, 7 through 9, cover Bombieri’s proof, with necessary material on ‘Valuations and Places’ and ‘The Riemann-Roch theorem’ developed in Chapters 7 and 8.
All chapters are based on the author’s lectures at the University of Colorado. However, the second part existed only in a somewhat rough xeroxed form from 1975 until the present. For the second edition, the whole text has been reset in an attractive typeface, and some inaccuracies have been corrected.
Graduate students will find Wolfgang Schmidt’s book a valuable resource. The necessary tools are developed without the need for substantial prerequisites. The style is leisurely, with many well-chosen examples, and proofs are given in full detail.
Second edition, 2004. ii+333 pp. Paperback, ISBN 0-9740427-1-4