TAILIEUCHUNG - Evolution Algebras and their Applications

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 multiplication of two “alleles” Gi and Gj by Gi · Gj = 0 if i = j. However, Gi ·Gi is viewed as “self-reproduction,” so that Gi ·Gi = j pijGj, where the summation is taken over all generators Gj . Thus, reproduction in genetics is represented by multiplication in algebra | Lecture Notes in Mathematics 1921 Editors . Morel Cachan F. Takens Groningen B. Teissier Paris Jianjun Paul Tian Evolution Algebras and their Applications Aspringer 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 jtian@ 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 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-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 41 SPi 5432 .

• Sinh học
• Stochastic processes
• Basic definitions
• interesting identities
• evolution algebra
• automorphism group
• Algebraic Persistency
• Queueing system
• nite number
• shop assistants
• a forming of queues
• Queueing systems
• without queues
• Some general types
• stochastic processes Download free ebooks at bookboon
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• Probability Examples c 9 – Stochastic Processes 2 © 2009 Leif Mejlbro & Ventus Publishing ApS ISBN 978 87 7681 525 7 Download free ebooks at bookboon
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• Stochastic Processes 2 Contents Contents Introduction 1 1
• 1 1
• 2 1
• 3 1
• 4 1
• 5 1
• 6 1
• 7 2 3 4 5 Theoretical background The Poisson process Birth and death processes Queueing theory in general Queueing system of innitely many shop assistants Queueing system of a nite number of shop assistants
• and with forming of queues Queueing systems with a nite number of shop assistants and without queues Some general types of stochastic processes The Poisson process Birth and death processes Queueing theory Other types of stochastic processes Index 5 6 6 8 11 11 12 15 17 19 37 52 117 126 The next step for top performing gradua
• Probability random variables and stochastic processes
• Probability random variables
• Probability variables
• Discrete time stochastic processes
• Markov chains
• Continuous time stochastic processes
• Itô’s lemma
• Stochastic integral
• Preliminaries
• Lecture Introduction to Stochastic Processes
• Introduction to Stochastic Processes
• Disjoint time intervals
• The stochastic process
• Homogeneous Poisson proces
• Markov chain
• Discrete parameter stochastic process
• Time homogeneous Markov chain
• Lecture Probability Theory
• Probability Theory
• Lý thuyết xác suất
• Stationary Stochastic Processes
• Systems with Stochastic Inputs
• Homogeneous poisson processes
• Holds trivially
• Probability density function
• Order statistics property
• Conditional distribution
• Theorem superposition
• Probability generating function
• Poisson process
• stochastic calculus
• mathematics of finance
• Process Limits
• Random variables
• Invariant measur
• Unique stationary distribution
• Symmetric transition probability matrix
• SLLN for Markov chains
• The long run proportion
• Dissection principle
• Limiting distribution
• Aperiodic Markov chain
• Stationary distribution
• Homogeneous Poisson process
• Discrete Markov chains
• Positive real numbers
• Heuristic justification
• Rigorous justification
• Positive random variables
• Exponential distribution
• Poissonian increments
• Inhomogeneous Poisson process
• Discrete parameter Markov chain
• Countable state space
• Mathematical description
• Transition probabilities
• Simple random walk
• On off process
• Random walks on groups
• Gambler's ruin problem
• Chapman Kolmogorov equation
• Decomposition of state space
• Deterministically monotone Markov chain
• Communication classes
• The dissection principle
• Non empty closed subsets
• Recurrent Markov chain
• E valued random variables
• Power series
• Abel's limit theorem
• Proof of the theorem