p-adic Numbers, p-adic Analysis, and Zeta-Functions

Author: Neal Koblitz

Publisher: Springer Science & Business Media

ISBN: 1461211123

Category: Mathematics

Page: 153

View: 4519

The first edition of this work has become the standard introduction to the theory of p-adic numbers at both the advanced undergraduate and beginning graduate level. This second edition includes a deeper treatment of p-adic functions in Ch. 4 to include the Iwasawa logarithm and the p-adic gamma-function, the rearrangement and addition of some exercises, the inclusion of an extensive appendix of answers and hints to the exercises, as well as numerous clarifications.

Lineare Operatoren in Hilberträumen

Teil 1 Grundlagen

Author: Joachim Weidmann

Publisher: Springer-Verlag

ISBN: 9783322800947

Category: Mathematics

Page: 475

View: 1984

Behandelt werden die Grundlagen der Theorie zum Thema Lineare Operatoren in Hilberträumen, wie sie üblicherweise in Standardvorlesungen für Mathematiker und Physiker vorgestellt werden.

P-adic Analysis

A Short Course on Recent Work

Author: Neal Koblitz

Publisher: Cambridge University Press

ISBN: 9780521280600

Category: Mathematics

Page: 163

View: 4082

An introduction to recent work in the theory of numbers and its interrelation with algebraic geometry and analysis.


Author: Reinhold Remmert

Publisher: N.A

ISBN: 9783540127833

Category: Mathematics

Page: 299

View: 8090


p-Adic Analysis and Mathematical Physics

Author: V S Vladimirov,I V Volovich,E I Zelenov

Publisher: World Scientific

ISBN: 9814505765

Category: Science

Page: 340

View: 3084

p-adic numbers play a very important role in modern number theory, algebraic geometry and representation theory. Lately p-adic numbers have attracted a great deal of attention in modern theoretical physics as a promising new approach for describing the non-Archimedean geometry of space-time at small distances. This is the first book to deal with applications of p-adic numbers in theoretical and mathematical physics. It gives an elementary and thoroughly written introduction to p-adic numbers and p-adic analysis with great numbers of examples as well as applications of p-adic numbers in classical mechanics, dynamical systems, quantum mechanics, statistical physics, quantum field theory and string theory. Contents:Analysis on the Field p-Adic Numbers:The Field of p-Adic NumbersAnalytic FunctionsAdditive and Multiplicative CharactersIntegration TheoryThe Gaussian IntegralsGeneralized FunctionsConvolution and the Fourier TransformationHomogeneous Generalized FunctionsPseudo-Differential Operators on the Field of p-Adic Numbers:The Operator D?p-Adic Schrodinger Operatorsp-Adic Quantum Theory:p-Adic Quantum MechanicsSpectral Theory in p-Adic Quantum MechanicsWeyl Systems. Infinite Dimensional Casep-Adic Stringsq-Analysis (Quantum Groups) and p-Adic AnalysisStochastic Processes Over the Field of p-Adic Numbers Readership: Students, postgraduates, mathematical physicists, mathematicians and physicists. keywords:Distribution Theory à la Bruhat;Planck Scale;p-Adic Analysis;Gaussian Integrals;Fourier Theory;Convolution of Generalized Functions;p-Adic Quantum Mechanics;Spectral Theory;Weyl Systems;p-Adic Strings;Quantum Groups;q-Analysis;Stochastic Processes

p-adic Numbers

An Introduction

Author: Fernando Quadros Gouvea

Publisher: Springer Science & Business Media

ISBN: 3642590586

Category: Mathematics

Page: 306

View: 2145

There are numbers of all kinds: rational, real, complex, p-adic. The p-adic numbers are less well known than the others, but they play a fundamental role in number theory and in other parts of mathematics. This elementary introduction offers a broad understanding of p-adic numbers. From the reviews: "It is perhaps the most suitable text for beginners, and I shall definitely recommend it to anyone who asks me what a p-adic number is." --THE MATHEMATICAL GAZETTE

Automorphe Formen

Author: Anton Deitmar

Publisher: Springer-Verlag

ISBN: 3642123902

Category: Mathematics

Page: 252

View: 9924

Das Buch bietet eine Einführung in die Theorie der automorphen Formen. Beginnend bei klassischen Modulformen führt der Autor seine Leser hin zur modernen, darstellungstheoretischen Beschreibung von automorphen Formen und ihren L-Funktionen. Das Hauptgewicht legt er auf den Übergang von der klassischen, elementaren Sichtweise zu der modernen, durch die Darstellungstheorie begründete Herangehensweise. Diese Art der Verbindung von klassischer und moderner Sichtweise war in der Lehrbuchliteratur bisher nicht zu finden.

Lectures on P-adic L-functions

Author: Kenkichi Iwasawa,Kinkichi Iwasawa

Publisher: Princeton University Press

ISBN: 9780691081120

Category: Mathematics

Page: 106

View: 6171

An especially timely work, the book is an introduction to the theory of p-adic L-functions originated by Kubota and Leopoldt in 1964 as p-adic analogues of the classical L-functions of Dirichlet. Professor Iwasawa reviews the classical results on Dirichlet's L-functions and sketches a proof for some of them. Next he defines generalized Bernoulli numbers and discusses some of their fundamental properties. Continuing, he defines p-adic L-functions, proves their existence and uniqueness, and treats p-adic logarithms and p-adic regulators. He proves a formula of Leopoldt for the values of p-adic L-functions at s=1. The formula was announced in 1964, but a proof has never before been published. Finally, he discusses some applications, especially the strong relationship with cyclotomic fields.
Number theory

Introduction to $P$-Adic Analytic Number Theory

Author: M. Ram Murty

Publisher: American Mathematical Soc.

ISBN: 0821847740

Category: Number theory

Page: 149

View: 6882

This book is an elementary introduction to $p$-adic analysis from the number theory perspective. With over 100 exercises included, it will acquaint the non-expert to the basic ideas of the theory and encourage the novice to enter this fertile field of research. The main focus of the book is the study of $p$-adic $L$-functions and their analytic properties. It begins with a basic introduction to Bernoulli numbers and continues with establishing the Kummer congruences. These congruences are then used to construct the $p$-adic analog of the Riemann zeta function and $p$-adic analogs of Dirichlet's $L$-functions. Featured is a chapter on how to apply the theory of Newton polygons to determine Galois groups of polynomials over the rational number field. As motivation for further study, the final chapter introduces Iwasawa theory.


Algebraische Zahlen und Funktionen

Author: Helmut Koch

Publisher: Springer-Verlag

ISBN: 3322803120

Category: Mathematics

Page: 344

View: 7805

Hauptziel des Buches ist die Vermittlung des Grundbestandes der Algebraischen Zahlentheorie einschließlich der Theorie der normalen Erweiterungen bis hin zu einem Ausblick auf die Klassenkörpertheorie. Gleichberechtigt mit algebraischen Zahlen werden auch algebraische Funktionen behandelt. Dies geschieht einerseits um die Analogie zwischen Zahl- und Funktionenkörpern aufzuzeigen, die besonders deutlich im Falle eines endlichen Konstantenkörpers ist. Andererseits erhält man auf diese Weise eine Einführung in die Theorie der "höheren Kongruenzen" als eines wesentlichen Bestandteils der "Arithmetischen Geometrie". Obgleich das Buch hauptsächlich algebraischen Methoden gewidmet ist, findet man in der Einleitung auch einen kurzen Beweis des Primzahlsatzes nach Newman. In den Kapiteln 7 und 8 wird die Theorie der Heckeschen L-Reihen behandelt einschließlich der Verteilung der Primideale algebraischer Zahlkörper in Kegeln.

Ultrametric Calculus

An Introduction to P-Adic Analysis

Author: W. H. Schikhof

Publisher: Cambridge University Press

ISBN: 0521032873

Category: Mathematics

Page: 320

View: 8390

This is an introduction to p-adic analysis which is elementary yet complete and which displays the variety of applications of the subject. Dr Schikhof is able to point out and explain how p-adic and 'real' analysis differ. This approach guarantees the reader quickly becomes acquainted with this equally 'real' analysis and appreciates its relevance. The reader's understanding is enhanced and deepened by the large number of exercises included throughout; these both test the reader's grasp and extend the text in interesting directions. As a consequence, this book will become a standard reference for professionals (especially in p-adic analysis, number theory and algebraic geometry) and will be welcomed as a textbook for advanced students of mathematics familiar with algebra and analysis.

Additive Number Theory: Inverse Problems and the Geometry of Sumsets

Author: Melvyn B. Nathanson

Publisher: Springer Science & Business Media

ISBN: 9780387946559

Category: Mathematics

Page: 296

View: 1881

Many classical problems in additive number theory are direct problems, in which one starts with a set A of natural numbers and an integer H -> 2, and tries to describe the structure of the sumset hA consisting of all sums of h elements of A. By contrast, in an inverse problem, one starts with a sumset hA, and attempts to describe the structure of the underlying set A. In recent years there has been ramrkable progress in the study of inverse problems for finite sets of integers. In particular, there are important and beautiful inverse theorems due to Freiman, Kneser, Plünnecke, Vosper, and others. This volume includes their results, and culminates with an elegant proof by Ruzsa of the deep theorem of Freiman that a finite set of integers with a small sumset must be a large subset of an n-dimensional arithmetic progression.

Calabi-Yau Varieties and Mirror Symmetry

Author: Noriko Yui,James Dominic Lewis

Publisher: American Mathematical Soc.

ISBN: 9780821871430

Category: Mathematics

Page: 367

View: 3340

The idea of mirror symmetry originated in physics, but in recent years, the field of mirror symmetry has exploded onto the mathematical scene. It has inspired many new developments in algebraic and arithmetic geometry, toric geometry, the theory of Riemann surfaces, and infinite-dimensional Lie algebras among others. The developments in physics stimulated the interest of mathematicians in Calabi-Yau varieties. This led to the realization that the time is ripe for mathematicians, armed with many concrete examples and alerted by the mirror symmetry phenomenon, to focus on Calabi-Yau varieties and to test for these special varieties some of the great outstanding conjectures, e.g., the modularity conjecture for Calabi-Yau threefolds defined over the rationals, the Bloch-Beilinson conjectures, regulator maps of higher algebraic cycles, Picard-Fuchs differential equations, GKZ hypergeometric systems, and others. The articles in this volume report on current developments. The papers are divided roughly into two categories: geometric methods and arithmetic methods. One of the significant outcomes of the workshop is that we are finally beginning to understand the mirror symmetry phenomenon from the arithmetic point of view, namely, in terms of zeta-functions and L-series of mirror pairs of Calabi-Yau threefolds. The book is suitable for researchers interested in mirror symmetry and string theory.

Diophantine Analysis

Proceedings at the Number Theory Section of the 1985 Australian Mathematical Society Convention

Author: J. H. Loxton,A. J. van Poorten,Australian Mathematical Society. Number Theory Section

Publisher: Cambridge University Press

ISBN: 9780521339230

Category: Mathematics

Page: 170

View: 9216

These papers were presented at the 1985 Australian Mathematical Society convention. They survey recent work in Diophantine analysis.
Technology & Engineering


Theorie und Anwendungen

Author: Alfred K. Louis,Peter Maaß,Andreas Rieder

Publisher: Springer-Verlag

ISBN: 3322801365

Category: Technology & Engineering

Page: 330

View: 8767

In der 2. Auflage wird u.a. der Vorteil der Wavelet-Transformation gegenüber der gef. Fourier-Transformation deutlich herausgearbeitet. Die Konstruktionsprinzipien orthogonaler und biorthogonaler Wavelets werden durch Beispiele weitergehend erläutert. Zahlreiche Aufgaben erleichtern das Verständnis des Stoffes.

Integration and Probability

Author: Paul Malliavin

Publisher: Springer Science & Business Media

ISBN: 1461242029

Category: Mathematics

Page: 326

View: 1027

An introduction to analysis with the right mix of abstract theories and concrete problems. Starting with general measure theory, the book goes on to treat Borel and Radon measures and introduces the reader to Fourier analysis in Euclidean spaces with a treatment of Sobolev spaces, distributions, and the corresponding Fourier analysis. It continues with a Hilbertian treatment of the basic laws of probability including Doob's martingale convergence theorem and finishes with Malliavin's "stochastic calculus of variations" developed in the context of Gaussian measure spaces. This invaluable contribution gives a taste of the fact that analysis is not a collection of independent theories, but can be treated as a whole.