Evolution Algebras and their Applications
Behind genetics and Markov chains, there is an intrinsic algebraic structure. It is defined as a type of new algebra: as evolution algebra. This concept lies between algebras and dynamical systems. Algebraically, evolution algebras are non-associative Ban
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1921
Jianjun Paul Tian
Evolution Algebras and their Applications
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Author Jianjun Paul Tian Mathematical Biosciences Institute The Ohio State University 231 West 18th Avenue Columbus, OH 43210-1292 USA Contact after August 2007 Mathematics Department College of William and Mary P. O. Box 8795 Williamsburg VA 23187-8795 USA e-mail: [email protected]
Library of Congress Control Number: 2007933498 Mathematics Subject Classification (2000): 08C92, 17D92, 60J10, 92B05, 05C62, 16G99 ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 ISBN 978-3-540-74283-8 Springer Berlin Heidelberg New York DOI 10.1007/978-3-540-74284-5 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2008 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting by the author and SPi using a Springer LATEX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper
SPIN: 12109265
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To my parents Bi-Yuan Tian and Yu-Mei Liu My father, the only person I know who can operate two abaci using his left and right hand simultaneously in his business.
Preface
In this book, we introduce a new type of algebra, which we call evolution algebras. These are algebras in which the multiplication tables are of a special type. They are motivated by evolution laws of genetics. We view alleles (or organelles or cells, etc,) as generators of algebras. Therefore we define the 0 if i = j. However, multiplication of two “alleles” Gi and Gj by Gi · Gj = Gi · Gi is viewed as “self-reproduction,” so that Gi · Gi = j pij Gj , where the summation is taken over all generators Gj . Thus, reproduction in genetics is represented by multiplication in algebra. It seems obvious that this type of algebra is nonassociative, but commutative. When the pij s form Markovian transition probabilities, the properties of algebras are associated with properties of Markov chains. Markov chains allow us to develop an algebra theory at deeper hierarchical levels than standard algebras. After we introduce several new algebraic concepts, particularly algebraic persistency, algebraic transiency, algebraic periodicity, and their relative versions, we establish hierarchical structures for ev
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