Analytic aspects of evolution algebras

Mellon, Pauline; Velasco, María V.

doi:10.1215/17358787-2018-0018

Cited by 10 publications

(9 citation statements)

References 15 publications

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“…They were introduced in 2008 by Tian [1] to enlighten the study of non-Mendelian genetics. Since then, a large literature has flourished on this topic (see for instance [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]) motivated by the fact that these algebras have connections with group theory, Markov processes, theory of knots, systems and graph theory. For instance, in [2], the theory of evolution algebras was related to that of pulse processes on weighted digraphs and applications were provided by reviewing and enlightening a report of the National Science Foundation about air pollution achieved by the Rand Corporation.…”

Section: Introductionmentioning

confidence: 99%

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Determining When an Algebra Is an Evolution Algebra

2020

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Evolution algebras are non-associative algebras that describe non-Mendelian hereditary processes and have connections with many other areas. In this paper, we obtain necessary and sufficient conditions for a given algebra A to be an evolution algebra. We prove that the problem is equivalent to the so-called SDC problem, that is, the simultaneous diagonalisation via congruence of a given set of matrices. More precisely we show that an n-dimensional algebra A is an evolution algebra if and only if a certain set of n symmetric n×n matrices {M1,…,Mn} describing the product of A are SDC. We apply this characterisation to show that while certain classical genetic algebras (representing Mendelian and auto-tetraploid inheritance) are not themselves evolution algebras, arbitrarily small perturbations of these are evolution algebras. This is intringuing, as evolution algebras model asexual reproduction, unlike the classical ones.

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Section: Introductionmentioning

confidence: 99%

“…To date, most literature on evolution algebras is on finite-dimensional ones. However, in [12] it is shown that every infinite-dimensional Banach evolution algebra is the direct sum of a finite-dimensional evolution algebra and a zero-product algebra.…”

Section: Introductionmentioning

confidence: 99%

Determining When an Algebra Is an Evolution Algebra

2020

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“…The analytic study of evolution algebras, carried on in [9] and focused on the ergodic behaviour of evolution operators of finite-dimensional evolution algebras, can be understood as a generalization of previously mentioned results in the case of primitive nonnegative matrices. Definition 2.11: The evolution operator L : E → E of an evolution algebra (w.r.t.…”

Section: Proposition 28: Markov Evolution Algebras Are Not Nil (Nor Nilpotent)mentioning

confidence: 99%

“…In this section, we address the elementary properties of this class of evolution algebras, such as the (non)existence of nilpotent or idempotent elements. We also study the behaviour of their evolution operator, and show that results in [9] can be seen as a generalization for the ergodic theorems for Markov evolution algebras having primitive structure matrices [10].…”

Section: Introductionmentioning

confidence: 99%

Markov evolution algebras

Paniello

2021

Linear and Multilinear Algebra

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We study Markov evolution algebras, that is, evolution algebras having Markov structure matrices. We first consider the discrete-time case, and delve into their algebraic structure for later application to continuous-time Markov evolution algebras that arise defined by standard stochastic semigroups. The study of embeddable Markov evolution algebras, that is, of Markov evolution algebras with structure matrices existing within standard stochastic semigroups is then equivalent to the embedding problem for Markov matrices.

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“…These results confirm and increase the ones by Mellon and Velasco, who recently proved that every evolution algebra is a normed algebra, for a determined norm defined in terms of a fixed natural basis. They gave a necessary and sufficient condition for a normed evolution algebra to be Banach and showed that every non-degenerate Banach evolution algebra is finite-dimensional [14]. Note that, in general, the completion of a normed evolution algebra is not an evolution algebra.…”

Section: Certain Conclusionmentioning

confidence: 99%