Evolution Algebras and their Applications [Paperback]

The author explores evolution algebras, which lie between algebras and dynamical systems. Readers learn the foundations of evolution algebras theory and its applications in non-Mendelian genetics and Markov chains. Theyll also discover evolution algebras connections with other mathematical fields, including graph theory, group theory, stochastic processes, dynamical systems, knot theory, 3-manifolds, and the Ihara-Selberg zeta function.

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.

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. Stochastic 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 identitl#[