Evolution Algebras and their Applications / Edition 1

Evolution Algebras and their Applications / Edition 1

by Jianjun Paul Tian
     
 

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 Banach algebras; dynamically, they represent discrete dynamical systems. Evolution algebras have many

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Overview

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 Banach algebras; dynamically, they represent discrete dynamical systems. Evolution algebras have many connections with other mathematical fields including graph theory, group theory, shastic processes, dynamical systems, knot theory, 3-manifolds, and the study of the Ihara-Selberg zeta function. In this volume the foundation of evolution algebra theory and applications in non-Mendelian genetics and Markov chains is developed, with pointers to some further research topics.

Product Details

ISBN-13:
9783540742838
Publisher:
Springer Berlin Heidelberg
Publication date:
10/28/2007
Series:
Lecture Notes in Mathematics Series, #1921
Edition description:
2008
Pages:
133
Product dimensions:
6.63(w) x 9.10(h) x 0.33(d)

Table of Contents

1. Introduction

2. Motivations

2.1. Examples from Biology

2.1.1 Asexual propagation

2.1.2. Gametic algebras in asexual inheritance

2.1.3. The Wright-Fisher model

2.2. Examples from Physics

2.2.1. Particles moving in a discrete space

2.2.2. Flows in a discrete space (networks)

2.2.3. Feynman graphs

2.3. Examples from Topology

2.3.1. Motions of particles in a 3-manifold

2.3.2. Random walks on braids with negative probabilities

2.4. Examples from Probability Theory

2.4.1. Shastic processes

3. Evolution Algebras

3.1. Definitions and Basic Properties

3.1.1. Departure point

3.1.2. Existence of unity elements

3.1.3. Basic definitions

3.1.4. Ideals of an evolution algebra

3.1.5. Quotients of an evolution algebra

3.1.6. Occurrence relations

3.1.7. Several interesting identities

3.2. Evolution Operators and Multiplication Algebras

3.2.1. Evolution operators

3.2.2. Change of generator sets (Transformations of natural bases)

3.2.3. "Rigidness" of generator sets of an evolution algebra

3.2.4. The automorphism group of an evolution algebra

3.2.5. The multiplication algebra of an evolution algebra

3.2.6. The derived Lie algebra of an evolution algebra

3.2.7. The centroid of an evolution algebra

3.3. Non-associative Banach Algebras

3.3.1. Definition of a norm over an evolution algebra

3.3.2. An evolution algebra as a Banach space

3.4. Periodicity and Algebraic Persistency

3.4.1. Periodicity of a generator in an evolution algebra

3.4.2. Algebraic persistency and algebraic transiency

3.5. Hierarchy of an Evolution Algebra

3.5.1. Periodicity of a simple evolution algebra

3.5.2. Semi-direct-sum decomposition of an evolution algebra

3.5.3. Hierarchy of an evolution algebra

3.5.4. Reducibility of an evolution algebra

4. Evolution Algebras and Markov Chains

4.1. Markov Chain and Its Evolution Algebra

4.1.1. Markov chains (discrete time)

4.1.2. The evolution algebra determined by a Markov chain

4.1.3. The Chapman-Kolmogorov equation

4.1.4. Concepts related to evolution operators

4.1.5. Basic algebraic properties of Markov chains

4.2. Algebraic Persistency and Probabilistic Persistency

4.2.1. Destination operator of evolution algebra M(X)

4.2.2. On the loss of coefficients (probabilities)

4.2.3. On the conservation of coefficients (probabilities)

4.2.4. Certain interpretations

4.2.5. Algebraic periodicity and probabilistic periodicity

4.3. Spectrum Theory of Evolution Algebras

4.3.1. Invariance of a probability flow

4.3.2. Spectrum of a simple evolution algebra

4.3.3. Spectrum of an evolution algebra at zero-th level

4.4. Hierarchies of General Markov Chains and Beyond

4.4.1. Hierarchy of a general Markov chain

4.4.2. Structure at the 0-th level in a hierarchy

4.4.3. 1-th structure of a hierarchy

4.4.4. k-th structure of a hierarchy

4.4.5. Regular evolution algebras

4.4.6. Reduced structure of evolution algebra M(X)

4.4.7. Examples and applications

5. Evolution Algebras and Non-Mendelian Genetics

5.1. History of General Genetic Algebras

5.2. Non-Mendelian Genetics and Its Algebraic Formulation

5.2.1. Some terms in population genetics

5.2.2. Mendelian versus non-Mendelian genetics

5.2.3. Algebraic formulation of non-Mendelian genetics

5.3. Algebras of Organelle Population Genetics

5.3.1. Heteroplasmy and homoplasmy

5.3.2. Coexistence of triplasmy

5.4. Algebraic Structures of Asexual Progenies of Phytophthora infestans

5.4.1. Basic biology of Phytophthora infestans

5.4.2. Algebras of progenies of Phytophthora infestans

6. Further Results and Research Topics

6.1. Beginning of Evolution Algebras and Graph Theory

6.2. Further Research Topics

6.2.1. Evolution algebras and graph theory

6.2.2. Evolution algebras and group theory and knot theory

6.2.3. Evolution algebras and Ihara-Selberg zeta function

6.2.4. Continuous evolution algebras

6.2.5. Algebraic statistical physics models and applications

6.2.6. Evolution algebras and 3-manifolds

6.2.7. Evolution algebras and phylogenetic trees, coalescent theory

6.3. Background Literature

References

Index

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