E-Book Overview
High technology industries are in desperate need for adequate tools to assess the validity of simulations produced by ever faster computers for perennial unstable problems. In order to meet these industrial expectations, applied mathematicians are facing a formidable challenge summarized by these words -- nonlinearity and coupling. This book is unique as it proposes truly original solutions: (1) Using hypercomputation in quadratic algebras, as opposed to the traditional use of linear vector spaces in the 20th century; (2) complementing the classical linear logic by the complex logic which expresses the creative potential of the complex plane.
The book illustrates how qualitative computing has been the driving force behind the evolution of mathematics since Pythagoras presented the first incompleteness result about the irrationality of 2. The celebrated results of Gödel and Turing are but modern versions of the same idea: the classical logic of Aristotle is too limited to capture the dynamics of nonlinear computation. Mathematics provides us with the missing tool, the organic logic, which is aptly tailored to model the dynamics of nonlinearity. This logic will be the core of the "Mathematics for Life" to be developed during this century.
Readership: Graduate students and researchers in applied and pure mathematics.
E-Book Content
Qualitative Computing A Computational Journey into Nonlinearity
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qualitative
Qualitative Computing A Computational Journey into Nonlinearity
Françoise Chatelin Université de Toulouse & CERFACS, France
World Scientific NEW JERSEY
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LONDON
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SINGAPORE
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BEIJING
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SHANGHAI
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TA I P E I
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CHENNAI
Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
QUALITATIVE COMPUTING: A COMPUTATIONAL JOURNEY INTO NONLINEARITY Copyright © 2012 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.
ISBN-13 978-981-4322-92-8 ISBN-10 981-4322-92-X
Printed in Singapore.
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“Man is the measure of all things′′ Protagoras of Abdera (5th century BCE)
qualitative
March 19, 2012
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World Scientific Book - 9in x 6in
qualitative
March 19, 2012
16:22
World Scientific Book - 9in x 6in
The works of six prestigious mathematicians, Cardano, Fermat, Euler, Fourier, Cauchy and Riemann have accompanied me at various stages of my mathematical development. The achievements of these Masters have been a constant source of meditations about Computation. The moonlike influence of my two godfathers in the numerical linear algebra community, the Berkeley Professors Velvel K