Evolution algebras and their applications / Jianjun Paul Tian.

Format:

Book

Author/Creator:

Tian, Jianjun Paul.

Series:

Lecture notes in mathematics (Springer-Verlag) ; 1921.

Lecture notes in mathematics ; 1921

Language:

English

Subjects (All):

Algebra.

Markov processes.

Stochastic processes.

Physical Description:

xi, 125 pages : illustrations ; 24 cm.

Place of Publication:

Berlin : New York : Springer, [2008]

Summary:

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, stochastic 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.

Contents:

2 Motivations 9

2.1 Examples from Biology 9

2.1.1 Asexual propagation 9

2.1.2 Gametic algebras in asexual inheritance 10

2.1.3 The Wright-Fisher model 11

2.2 Examples from Physics 12

2.2.1 Particles moving in a discrete space 12

2.2.2 Flows in a discrete space (networks) 12

2.2.3 Feynman graphs 13

2.3 Examples from Topology 15

2.3.1 Motions of particles in a 3-manifold 15

2.3.2 Random walks on braids with negative probabilities 15

2.4 Examples from Probability Theory 16

2.4.1 Stochastic processes 16

3 Evolution Algebras 17

3.1 Definitions and Basic Properties 17

3.1.1 Departure point 17

3.1.2 Existence of unity elements 22

3.1.4 Ideals of an evolution algebra 24

3.1.5 Quotients of an evolution algebra 25

3.1.6 Occurrence relations 26

3.1.7 Several interesting identities 27

3.2 Evolution Operators and Multiplication Algebras 28

3.2.1 Evolution operators 29

3.2.2 Changes of generator sets (Transformations of natural bases) 30

3.2.3 "Rigidness" of generator sets of an evolution algebra 31

3.2.4 The automorphism group of an evolution algebra 32

3.2.5 The multiplication algebra of an evolution algebra 33

3.2.6 The derived Lie algebra of an evolution algebra 34

3.2.7 The centroid of an evolution algebra 35

3.3 Nonassociative Banach Algebras 36

3.3.1 Definition of a norm over an evolution algebra 37

3.3.2 An evolution algebra as a Banach space 38

3.4 Periodicity and Algebraic Persistency 39

3.4.1 Periodicity of a generator in an evolution algebra 39

3.4.2 Algebraic persistency and algebraic transiency 42

3.5 Hierarchy of an Evolution Algebra 43

3.5.1 Periodicity of a simple evolution algebra 44

3.5.2 Semidirect-sum decomposition of an evolution algebra 45

3.5.3 Hierarchy of an evolution algebra 46

3.5.4 Reducibility of an evolution algebra 49

4 Evolution Algebras and Markov Chains 53

4.1 A Markov Chain and Its Evolution Algebra 53

4.1.1 Markov chains (discrete time) 53

4.1.2 The evolution algebra determined by a Markov chain 54

4.1.3 The Chapman-Kolmogorov equation 56

4.1.4 Concepts related to evolution operators 58

4.1.5 Basic algebraic properties of Markov chains 58

4.2 Algebraic Persistency and Probabilistic Persistency 60

4.2.1 Destination operator of evolution algebra M[subscript X] 60

4.2.2 On the loss of coefficients (probabilities) 64

4.2.3 On the conservation of coefficients (probabilities) 67

4.2.4 Certain interpretations 68

4.2.5 Algebraic periodicity and probabilistic periodicity 69

4.3 Spectrum Theory of Evolution Algebras 69

4.3.1 Invariance of a probability flow 69

4.3.2 Spectrum of a simple evolution algebra 70

4.3.3 Spectrum of an evolution algebra at zeroth level 75

4.4 Hierarchies of General Markov Chains and Beyond 76

4.4.1 Hierarchy of a general Markov chain 76

4.4.2 Structure at the 0th level in a hierarchy 77

4.4.3 1st structure of a hierarchy 80

4.4.4 kth structure of a hierarchy 81

4.4.5 Regular evolution algebras 83

4.4.6 Reduced structure of evolution algebra M[subscript X] 86

4.4.7 Examples and applications 87

5 Evolution Algebras and Non-Mendelian Genetics 91

5.1 History of General Genetic Algebras 91

5.2 Non-Mendelian Genetics and Its Algebraic Formulation 93

5.2.1 Some terms in population genetics 93

5.2.2 Mendelian vs. non-Mendelian genetics 94

5.2.3 Algebraic formulation of non-Mendelian genetics 95

5.3 Algebras of Organelle Population Genetics 96

5.3.1 Heteroplasmy and homoplasmy 96

5.3.2 Coexistence of triplasmy 98

5.4 Algebraic Structures of Asexual Progenies of Phytophthora infestans 100

5.4.1 Basic biology of Phytophthora infestans 101

5.4.2 Algebras of progenies of Phytophthora infestans 102

6 Further Results and Research Topics 109

6.1 Beginning of Evolution Algebras and Graph Theory 109

6.2 Further Research Topics 113

6.2.1 Evolution algebras and graph theory 113

6.2.2 Evolution algebras and group theory, knot theory 114

6.2.3 Evolution algebras and Ihara-Selberg zeta function 115

6.2.4 Continuous evolution algebras 115

6.2.5 Algebraic statistical physics models and applications 115

6.2.6 Evolution algebras and 3-manifolds 116

6.2.7 Evolution algebras and phylogenetic trees, coalescent theory 116

6.3 Background Literature 116.

Notes:

Includes bibliographical references (pages [119]-121) and index.

ISBN:

3540742832

9783540742838

OCLC:

173807298