Classification of asexual diploid organisms by means of strongly isotopic evolution algebras defined over any field

Falcón, Óscar J.; Ganfornina, Raúl Manuel Falcón; Núñez, Juan

doi:10.1016/j.jalgebra.2016.11.018

Cited by 26 publications

(26 citation statements)

References 24 publications

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“…More than ten years have passed since the first papers on this topic appeared in Mathematics literature, and a lot of research effort has been devoted to explore the connections between this abstract object and concepts of other fields. We refer the reader to [4,5,6,7,17] for a survey of properties and results of general evolution algebras; to [2,3,8,13,14] for a connection between evolution algebras and graphs; and to [9,11,12,19,16] for a review of results with relevance in genetics and other applications.…”

Section: Introductionmentioning

confidence: 99%

The connection between evolution algebras, random walks and graphs

2019

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Evolution algebras are a new type of non-associative algebras which are inspired from biological phenomena. A special class of such algebras, called Markov evolution algebras, is strongly related to the theory of discrete time Markov chains. The winning of this relation is that many results coming from Probability Theory may be stated in the context of Abstract Algebra. In this paper we explore the connection between evolution algebras, random walks and graphs. More precisely, we study the relationships between the evolution algebra induced by a random walk on a graph and the evolution algebra determined by the same graph. Given that any Markov chain may be seen as a random walk on a graph we believe that our results may add a new landscape in the study of Markov evolution algebras. ∞ k=1 c ik = 1,for any i, k, then A is called Markov evolution algebra. The name is due to that there is an interesting one-to-one correspondence between A and a discrete time Markov chain (X n ) n≥0 with 2010 Mathematics Subject Classification. 05C25, 17D92, 17D99, 05C81.

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Section: Introductionmentioning

confidence: 99%

The connection between evolution algebras, random walks and graphs

2019

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“…Before to address with the existence of isomorphisms between A RW (G) and A(G) for a given graph G, we start with a more general concept which is the isotopism of algebras introduced by Albert [1] as a generalization of that of isomorphism. This has been recently applied by [9] to study two-dimensional evolution algebras.…”

Section: 1mentioning

confidence: 99%

“…The best general reference of the subject is [14], where the reader can found a review of preliminary definitions and properties, connections with other fields of mathematics, and a list of interesting open problems some of which remain unsolved so far. We refer the reader also to [15] for an update of open problems in the Theory of Evolution Algebras, and to [2]- [9] and references therein for an overview of recent results on this topic. Formally, an evolution algebra is defined as follows.…”

Section: Introductionmentioning

confidence: 99%

On the isomorphisms between evolution algebras of graphs and random walks

Cadavid

Montoya

Rodríguez

2019

Linear and Multilinear Algebra

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Evolution algebras are non-associative algebras inspired from biological phenomena, with applications to or connections with different mathematical fields. There are two natural ways to define an evolution algebra associated to a given graph. While one takes into account only the adjacencies of the graph, the other includes probabilities related to the symmetric random walk on the same graph. In this work we state new properties related to the relation between these algebras, which is one of the open problems in the interplay between evolution algebras and graphs. On the one hand, we show that for any graph both algebras are strongly isotopic. On the other hand, we provide conditions under which these algebras are or are not isomorphic. For the case of finite non-singular graphs we provide a complete description of the problem, while for the case of finite singular graphs we state a conjecture supported by examples and partial results. The case of graphs with an infinite number of vertices is also discussed. As a sideline of our work, we revisit a result existing in the literature about the identification of the automorphism group of an evolution algebra, and we give an improved version of it. ∞ k=1 c ik = 1, for any i, then A is called a Markov evolution algebra. In this case, there is an interesting correspondence between the algebra A and a discrete time Markov chain (X n ) n≥0 with states space {x 1 , x 2 , . . . , x n , . . .} and transition probabilities given by: c ik := P(X n+1 = x k |X n = x i ), 2010 Mathematics Subject Classification. 05C25, 17D92, 17D99, 05C81.

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“…Very recently, in [6] the authors studied the distribution of finite-dimensional evolution algebras over any base field into isotopism 1 classes according to their structure tuples and to the dimension of their annihilators. It is shown the existence of four isotopism classes of two-dimensional evolution algebras, whatever the base field is.…”

Section: Introductionmentioning

confidence: 99%

A class of nilpotent evolution algebras

Omirov

Rozikov²,

Velasco

2019

Communications in Algebra

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Recently, by A. Elduque and A. Labra a new technique and a type of an evolution algebra are introduced. Several nilpotent evolution algebras defined in terms of bilinear forms and symmetric endomorphisms are constructed. The technique then used for the classification of the nilpotent evolution algebras up to dimension five. In this paper we develop this technique for high dimensional evolution algebras. We construct nilpotent evolution algebras of any type. Moreover, we show that, except the cases considered by Elduque and Labra, this construction of nilpotent evolution algebras does not give all possible nilpotent evolution algebras.

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Classification of asexual diploid organisms by means of strongly isotopic evolution algebras defined over any field

Cited by 26 publications

References 24 publications

The connection between evolution algebras, random walks and graphs

The connection between evolution algebras, random walks and graphs

On the isomorphisms between evolution algebras of graphs and random walks

A class of nilpotent evolution algebras

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