Evolution Algebras Research Articles

On real chains of evolution algebras

In this paper, we define a chain of -dimensional evolution algebras corresponding to a permutation of numbers. We show that a chain of evolution algebras (CEA) corresponding to a permutation is non-trivial if and only if the permutation has a fixed point. We show that a CEA is a chain of nilpotent algebras (independently on time) if it is trivial. We construct a wide class of chains of three-dimensional EAs and a class of symmetric -dimensional CEAs. A construction of arbitrary dimensional CEAs is given. Moreover, for a chain of three-dimensional EAs, we study the behaviour of the baric property, the behaviour of the set of absolute nilpotent elements and dynamics of the set of idempotent elements depending on the time.

CLASSIFICATION DYNAMICS OF TWO-DIMENSIONAL CHAINS OF EVOLUTION ALGEBRAS

Recently by Casas et al. a notion of chain of evolution algebras (CEAs) is introduced. This chain is a dynamical system the state of which at each given time is an evolution algebra. It is known 25 distinct classes of chains of two-dimensional evolution algebras. In our previous paper we gave the classification of two-dimensional real evolution algebras. This classification contains seven (pairwise non-isomorphic) such algebras. For each known CEA we study its dynamics to be an element of a given class.

The multiplicative spectrum and the uniqueness of the complete norm topology

We define the spectrum of an element a in a non-associative algebra A according to a classical notion of invertibility (a is invertible if the multiplication operators La and Ra are bijective). Around this notion of spectrum, we develop a basic theoretical support for a non-associative spectral theory. Thus we prove some classical theorems of automatic continuity free of the requirement of associativity. In particular, we show the uniqueness of the complete norm topology of m-semisimple algebras, obtaining as a corollary of this result a well-known theorem of Barry E. Johnson (1967). The celebrated result of C.E. Rickart (1960) about the continuity of dense-range homomorphisms is also studied in the non-associative framework. Finally, because non-associative algebras are very suitable models in genetics, we provide here a hint of how to apply this approach in that context, by showing that every homomorphism from a complete normed algebra onto a particular type of evolution algebra is automatically continuous.

Service net algebra based on logic Petri nets

Web service process reuse can help us efficiently to construct a new service or service process by using the existing service processes generated by service composition. Since service requests are characteristic of multiformity, the existing service processes can hardly be reused unless they are modified and transformed to some specific scenarios. In this paper, a service process is modeled as a service net (SN) using logic Petri nets. Inspired by the relational algebra, service net algebra (SNA) is proposed to provide a formal foundation for structural transformations of SNs including composition and decomposition. It is constructed on the basis of logic Petri nets and consists of structure algebra, evolution algebra and synthesis algebra. Some algebra operators of SNA are defined, and their corresponding operational rules are presented. The properties and soundness preservation of SNA are analyzed. A framework for service process reuse is also given based on SNA.

Dynamics of two-dimensional evolution algebras

Recently in [2] a notion of chain of evolution algebras is introduced. This chain is a dynamical system the state of which at each given time is an evolution algebra. The sequence of matrices of the structural constants for this chain of evolution algebras satisfies the Chapman-Kolmogorov equation. In this paper we construct 25 distinct examples of chains of two-dimensional evolution algebras. For all of these 25 chains we study the behavior of the baric property, the behavior of the set of absolute nilpotent elements and dynamics of the set of idempotent elements depending on the time.

FINITELY GENERATED NIL BUT NOT NILPOTENT EVOLUTION ALGEBRAS

To use evolution algebras to model population dynamics that both allow extinction and introduction of certain gametes in finite generations, nilpotency must be built into the algebraic structures of these algebras with the entire algebras not to be nilpotent if the populations are assumed to evolve for a long period of time. To adequately address this need, evolution algebras over rings with nilpotent elements must be considered instead of evolution algebras over fields. This paper develops some criteria, which are computational in nature, about the nilpotency of these algebras, and shows how to construct finitely generated evolution algebras which are nil but not nilpotent.

The Convergence Mechanism and Strategies for Non-Elitist Genetic Programming

Genetic programming is an evolutionary algorithm that proposed to solve the automatic computer program design problem by J.R.Koza in the 1990s. It has good universality and intelligence, and has been widely applied in the field of computer engineering. But genetic programming is essentially a stochastic optimization algorithm, lack theoretic basis on the convergence of algorithm, which limit the scope of its application in some extent. The convergence mechanism of non-elitist genetic programming was studied in this paper. A recursive estimation of the probability of population contains satisfactory solution with the evolution algebra was established by the analysis of operators characteristic parameters, then a sufficient condition of population converge in probability was derived from this estimation, and thereby some operational convergence strategies for many common evolution modes were provided.

On nilpotent index and dibaricity of evolution algebras

An evolution algebra corresponds to a quadratic matrix A of structural constants. It is known the equivalence between nil, right nilpotent evolution algebras and evolution algebras which are defined by upper triangular matrices A. We establish a criterion for an n-dimensional nilpotent evolution algebra to be with maximal nilpotent index 2n-1+1. We give the classification of finite-dimensional complex evolution algebras with maximal nilpotent index. Moreover, for any s=1,…,n-1 we construct a wide class of n-dimensional evolution algebras with nilpotent index 2n-s+1. We show that nilpotent evolution algebras are not dibaric and establish a criterion for two-dimensional real evolution algebras to be dibaric.

Mathematical tools for the future: Graph Theory and graphicable algebras

This study constitutes the continuation of innovative research in Discrete Mathematics introduced in earlier papers on algebras in general, regarding the use of graphs to study the particular case of graphicable algebras, which form a subset of evolution algebras. Evolution algebras are particularly interesting since they are intrinsically linked with other mathematical fields, such as group theory, stochastics processes, and dynamical systems, for instance. Our advances in this study are obtained by setting a natural correspondence between evolution algebras and direct graphs, in order to translate the general concepts of graphicable algebras: subalgebra, ideal, centralizer, normalizer…to the language of graphs. These translations will enable advances in the application of these algebras to various branches of Mathematics.

Evolution algebra of a bisexual population

We introduce an (evolution) algebra identifying the coefficients of inheritance of a bisexual population as the structure constants of the algebra. The basic properties of the algebra are studied. We prove that this algebra is commutative (and hence flexible), not associative and not necessarily power-associative. We show that the evolution algebra of the bisexual population is not a baric algebra, but a dibaric algebra and hence its square is baric. Moreover, we show that the algebra is a Banach algebra. The set of all derivations of the evolution algebra is described. We find necessary conditions for a state of the population to be a fixed point or a zero point of the evolution operator which corresponds to the evolution algebra. We also establish upper estimate of the limit points set for trajectories of the evolution operator. Using the necessary conditions we give a detailed analysis of a special case of the evolution algebra (of bisexual population in which type “1” of females and males have preference). For such a special case we describe the full set of idempotent elements and the full set of absolute nilpotent elements.

Particle Swarm Optimization Based on Multiobjective Optimization

PSO will population each individual as the search space without a volume and quality of particle. These particles in the search space at a certain speed flight, the speed according to its own flight experience and the entire population of flight experience dynamic adjustment. We describe the standard PSO, multi-objective optimization and MOPSO. The main focus of this thesis is several PSO algorithms which are introduced in detail and studied. MOPSO algorithm introduced adaptive grid mechanism of the external population, not only to groups of particle on variation, but also to the value scope of the particles and variation, and the variation scale and population evolution algebra in proportion.

Evolution algebras generated by Gibbs measures

In this article we study algebraic structures of function spaces defined by graphs and state spaces equipped with Gibbs measures by associating evolution algebras. We give a constructive description of associating evolution algebras to the function spaces (cell spaces) defined by graphs and state spaces and Gibbs measure µ. For finite graphs we find some evolution subalgebras and other useful properties of the algebras. We obtain a structure theorem for evolution algebras when graphs are finite and connected. We prove that for a fixed finite graph, the function spaces has a unique algebraic structure since all evolution algebras are isomorphic to each other for whichever Gibbs measures assigned. When graphs are infinite graphs then our construction allows a natural introduction of thermodynamics in studying of several systems of biology, physics and mathematics by theory of evolution algebras.

A chain of evolution algebras

We introduce a notion of chain of evolution algebras. The sequence of matrices of the structural constants for this chain of evolution algebras satisfies an analogue of Chapman–Kolmogorov equation. We give several examples (time homogenous, time non-homogenous, periodic, etc.) of such chains. For a periodic chain of evolution algebras we construct a continuum set of non-isomorphic evolution algebras and show that the corresponding discrete time chain of evolution algebras is dense in the set. We obtain a criteria for an evolution algebra to be baric and give a concept of a property transition. For several chains of evolution algebras we describe the behavior of the baric property depending on the time. For a chain of evolution algebras given by the matrix of a two-state evolution we define a baric property controller function and under some conditions on this controller we prove that the chain is not baric almost surely (with respect to Lebesgue measure). We also construct examples of the almost surely baric chains of evolution algebras. We show that there are chains of evolution algebras such that if it has a unique (resp. infinitely many) absolute nilpotent element at a fixed time, then it has unique (resp. infinitely many) absolute nilpotent element any time; also there are chains of evolution algebras which have not such property. For an example of two dimensional chain of evolution algebras we give the full set of idempotent elements and show that for some values of parameters the number of idempotent elements does not depend on time, but for other values of parameters there is a critical time t c such that the chain has only two idempotent elements if time t ⩾ t c and it has four idempotent elements if time t < t c .

Algebraic model of non-Mendelian inheritance

Evolution algebra theory is used to study non-Mendelian inheritance, particularly organelle heredity and population genetics of Phytophthora infectans. We not only can explain a puzzling feature of establishment of homoplasmy from heteroplasmic cell population and the coexistence of mitochondrial triplasmy, but also can predict all mechanisms to form the homoplasmy of cell populations, which are hypothetical mechanisms in current mitochondrial disease research. The algebras also provide a way to easily find different genetically dynamic patterns from the complexity of the progenies of Phytophthora infectans which cause the late blight of potatoes and tomatoes. Certain suggestions to pathologists are made as well.

SOME PROPERTIES OF EVOLUTION ALGEBRAS

The paper is devoted to the study of finite dimensional complex evolution algebras. The class of evolution algebras isomorphic to evolution algebras with Jordan form matrices is described. For finite dimensional complex evolution algebras the criterium of nilpotency is established in terms of the properties of corresponding matrices. Moreover, it is proved that for nilpotent <TEX>$n$</TEX>-dimensional complex evolution algebras the possible maximal nilpotency index is <TEX>$1+2^{n-1}$</TEX>.

The derivations of some evolution algebras

In this work, we investigate the derivations of n-dimensional complex evolution algebras, depending on the rank of the appropriate matrices. For evolution algebra with non-singular matrices we prove that the space of derivations is zero. The spaces of derivations for evolution algebras with matrices of rank n − 1 are described.

On Homogeneous Weighted Algebras

We extend the concept of the “join” to the case of infinitely many weighted algebras. We study the problem of its uniqueness (up to weighted isomorphism) which gives rise to a natural notion of homogeneous weighted algebras. We show that several classes of weighted algebras coming from genetics are homogeneous and that homogeneity is preserved by duplication. Finally, we examine some well-known weighted algebras satisfying identities, as Bernstein, train, and evolution algebras.