Richard Varro | Université de Montpellier (original) (raw)

Papers by Richard Varro

arXiv (Cornell University), 2015

We introduce the gonosomal algebra. Gonosomal algebra extend the evolution algebra of the bisexua... more We introduce the gonosomal algebra. Gonosomal algebra extend the evolution algebra of the bisexual population (EABP) defined by Ladra and Rozikov. We show that gonosomal algebras can represent algebraically a wide variety of sex determination systems observed in bisexual populations. We illustrate this by about twenty genetic examples, most of these examples cannot be represented by an EABP. We give seven algebraic constructions of gonosomal algebras, each is illustrated by genetic examples. We show that unlike the EABP gonosomal algebras are not dibaric. We approach the existence of dibaric function and idempotent in gonosomal algebras.

HAL (Le Centre pour la Communication Scientifique Directe), 2011

arXiv (Cornell University), Nov 16, 2019

We are interested in the evolution operators defined on commutative and nonassociative algebras w... more We are interested in the evolution operators defined on commutative and nonassociative algebras when the scalar field is of characteristic 2. We distinguish four types : nilpotent, quasi-constant, ultimately periodic and plenary train operators. They are studied and classified for non baric and for baric algebras.

WORLD SCIENTIFIC eBooks, Apr 22, 2020

In this paper we consider discrete-time dynamical systems generated by gonosomal evolution operat... more In this paper we consider discrete-time dynamical systems generated by gonosomal evolution operators of sex linked inheritance. Mainly we study dynamical systems of a hemophilia, which biologically is a group of hereditary genetic disorders that impair the body's ability to control blood clotting or coagulation, which is used to stop bleeding when a blood vessel is broken. We give an algebraic model of the biological system corresponding to the hemophilia. The evolution of such system is studied by a nonlinear (quadratic) gonosomal operator. In a general setting, this operator is considered as a mapping from R n , n ≥ 2 to itself. In particular, for a gonosomal operator at n = 4 we explicitly give all (two) fixed points. Then limit points of the trajectories of the corresponding dynamical system are studied. Moreover we consider a normalized version of the gonosomal operator. In the case n = 4, for the normalized gonosomal operator we show uniqueness of fixed point and study limit points of the dynamical system.

arXiv (Cornell University), Nov 16, 2019

We are interested in the evolution operators defined on commutative and nonassociative algebras w... more We are interested in the evolution operators defined on commutative and nonassociative algebras when the scalar field is of characteristic 2. We distinguish four types : nilpotent, quasi-constant, ultimately periodic and plenary train operators. They are studied and classified for non baric and for baric algebras.

MONTPELLIER-BU Sciences (341722106) / SudocSudocFranceF

HAL (Le Centre pour la Communication Scientifique Directe), 2011

arXiv (Cornell University), Jan 5, 2020

In this paper, we initiate the study of a discrete-time dynamical system modelling a trophic netw... more In this paper, we initiate the study of a discrete-time dynamical system modelling a trophic network connecting the three types of plankton (phytoplankton, zooplankton, mixoplankton) and bacteria. The nonlinear operator V associated with this dynamical system is of type 4-Volterra quadratic stochastic operator (QSO) with twelve parameters. We give conditions on the parameters under which this operator maps the five-dimensional standard simplex to itself and we find its fixed points. Moreover, we study the limit points of trajectories for this operator. For each situations we give some biological interpretations.

CRC Press eBooks, May 20, 2019

We show that if A2 is a train algebra of rank r, then the duplicate of A is train of rank r + 1. ... more We show that if A2 is a train algebra of rank r, then the duplicate of A is train of rank r + 1. Also, if A” is a Bernstein algebra of order n and period 11. the commutative duplicate of A is Bernstein of order )L + 1 and period p.

Communications in Algebra, 2016

We study the ideal of polynomial identities of a single indeterminate satis ed by all backcrossin... more We study the ideal of polynomial identities of a single indeterminate satis ed by all backcrossing algebras. For this we distinguish two categories according to whether or not these algebras satisfy an identity for the plenary powers. For each category, we give the generators for the vector space of identities, a condition for any object belonging to one of these two categories verify a given identity, a necessary and su cient condition that a polynomial is an identity and we study the existence of an idempotent element. We give a method which brings the search of identities sati ed by the backcrossing algebras to the solution of linear systems and we illustrate this method by constructing generators of homogeneous and non homogeneous identities of degrees less than 8.

arXiv (Cornell University), Mar 22, 2015

We introduce the gonosomal algebra. Gonosomal algebra extend the evolution algebra of the bisexua... more We introduce the gonosomal algebra. Gonosomal algebra extend the evolution algebra of the bisexual population (EABP) defined by Ladra and Rozikov. We show that gonosomal algebras can represent algebraically a wide variety of sex determination systems observed in bisexual populations. We illustrate this by about twenty genetic examples, most of these examples cannot be represented by an EABP. We give seven algebraic constructions of gonosomal algebras, each is illustrated by genetic examples. We show that unlike the EABP gonosomal algebras are not dibaric. We approach the existence of dibaric function and idempotent in gonosomal algebras.

Communications in Algebra, 2020

Abstract We are interested in the evolution operators defined on commutative and non-associative ... more Abstract We are interested in the evolution operators defined on commutative and non-associative algebras when the characteristic of the scalar field is 2. We distinguish four types: nilpotent, quasi-constant, ultimately periodic, and plenary train operators. They are studied and classified for non-baric and for baric algebras.

Http Www Theses Fr, 1992

Dans le but de modeliser des populations dont la composition genetique suit une succession cycliq... more Dans le but de modeliser des populations dont la composition genetique suit une succession cyclique d'etats d'equilibre, nous presentons une generalisation de la notion d'algebre de Bernstein d'ordre n en introduisant la notion de periodicite. Nous nous sommes interesses aux conditions d'existence d'idempotents generalises et a leurs proprietes, a la structure vectorielle qui apparait lors de la decomposition de Peierce et a des problemes de transport de structures dans la dupliquee de ces algebres. Enfin nous etudions les algebres qui modelisent les populations d'organismes soumises seulement a la mutation des genes et nous etablissons la condition necessaire et suffisante pour que ces populations atteignent a la premiere generation une situation d'equilibre periodique

Proceedings of the Edinburgh Mathematical Society

Proceedings of the Edinburgh Mathematical Society, 1992

Nous Présentons ici l'étude, dans l'optique exhibée dans [7], d'une catégorie d'a... more Nous Présentons ici l'étude, dans l'optique exhibée dans [7], d'une catégorie d'algèbres à idempotents, commutatives, non associatives, définies par des relations. On y trouve une structure qui n'est en général ni pondérée ni de Jordan et où, cependant, ces conditions sont équivalentes.Following a previous paper, we study here a class of commutative and non associative algebras with idempotents and defined by relations. We find a structure which is not, in general, baric or Jordan but where these notions are equivalent.

A l’aide des alg`ebres de mutation on contruit des structures v´erifiant des identit´ es ω-polynom... more A l’aide des alg`ebres de mutation on contruit des structures v´erifiant des identit´ es ω-polynomiales arbitraires. Avec cette optique, concernant les trainalg`ebres d’ordre n, on ´etablit des r´esultats relatifs `a l’existence d’idempotents, en particulier quand ces alg`ebres ont la valeur 1 2 comme train-racine.

arXiv (Cornell University), 2015

We introduce the gonosomal algebra. Gonosomal algebra extend the evolution algebra of the bisexua... more We introduce the gonosomal algebra. Gonosomal algebra extend the evolution algebra of the bisexual population (EABP) defined by Ladra and Rozikov. We show that gonosomal algebras can represent algebraically a wide variety of sex determination systems observed in bisexual populations. We illustrate this by about twenty genetic examples, most of these examples cannot be represented by an EABP. We give seven algebraic constructions of gonosomal algebras, each is illustrated by genetic examples. We show that unlike the EABP gonosomal algebras are not dibaric. We approach the existence of dibaric function and idempotent in gonosomal algebras.

HAL (Le Centre pour la Communication Scientifique Directe), 2011

arXiv (Cornell University), Nov 16, 2019

We are interested in the evolution operators defined on commutative and nonassociative algebras w... more We are interested in the evolution operators defined on commutative and nonassociative algebras when the scalar field is of characteristic 2. We distinguish four types : nilpotent, quasi-constant, ultimately periodic and plenary train operators. They are studied and classified for non baric and for baric algebras.

WORLD SCIENTIFIC eBooks, Apr 22, 2020

In this paper we consider discrete-time dynamical systems generated by gonosomal evolution operat... more In this paper we consider discrete-time dynamical systems generated by gonosomal evolution operators of sex linked inheritance. Mainly we study dynamical systems of a hemophilia, which biologically is a group of hereditary genetic disorders that impair the body's ability to control blood clotting or coagulation, which is used to stop bleeding when a blood vessel is broken. We give an algebraic model of the biological system corresponding to the hemophilia. The evolution of such system is studied by a nonlinear (quadratic) gonosomal operator. In a general setting, this operator is considered as a mapping from R n , n ≥ 2 to itself. In particular, for a gonosomal operator at n = 4 we explicitly give all (two) fixed points. Then limit points of the trajectories of the corresponding dynamical system are studied. Moreover we consider a normalized version of the gonosomal operator. In the case n = 4, for the normalized gonosomal operator we show uniqueness of fixed point and study limit points of the dynamical system.

arXiv (Cornell University), Nov 16, 2019

We are interested in the evolution operators defined on commutative and nonassociative algebras w... more We are interested in the evolution operators defined on commutative and nonassociative algebras when the scalar field is of characteristic 2. We distinguish four types : nilpotent, quasi-constant, ultimately periodic and plenary train operators. They are studied and classified for non baric and for baric algebras.

MONTPELLIER-BU Sciences (341722106) / SudocSudocFranceF

HAL (Le Centre pour la Communication Scientifique Directe), 2011

arXiv (Cornell University), Jan 5, 2020

In this paper, we initiate the study of a discrete-time dynamical system modelling a trophic netw... more In this paper, we initiate the study of a discrete-time dynamical system modelling a trophic network connecting the three types of plankton (phytoplankton, zooplankton, mixoplankton) and bacteria. The nonlinear operator V associated with this dynamical system is of type 4-Volterra quadratic stochastic operator (QSO) with twelve parameters. We give conditions on the parameters under which this operator maps the five-dimensional standard simplex to itself and we find its fixed points. Moreover, we study the limit points of trajectories for this operator. For each situations we give some biological interpretations.

CRC Press eBooks, May 20, 2019

We show that if A2 is a train algebra of rank r, then the duplicate of A is train of rank r + 1. ... more We show that if A2 is a train algebra of rank r, then the duplicate of A is train of rank r + 1. Also, if A” is a Bernstein algebra of order n and period 11. the commutative duplicate of A is Bernstein of order )L + 1 and period p.

Communications in Algebra, 2016

We study the ideal of polynomial identities of a single indeterminate satis ed by all backcrossin... more We study the ideal of polynomial identities of a single indeterminate satis ed by all backcrossing algebras. For this we distinguish two categories according to whether or not these algebras satisfy an identity for the plenary powers. For each category, we give the generators for the vector space of identities, a condition for any object belonging to one of these two categories verify a given identity, a necessary and su cient condition that a polynomial is an identity and we study the existence of an idempotent element. We give a method which brings the search of identities sati ed by the backcrossing algebras to the solution of linear systems and we illustrate this method by constructing generators of homogeneous and non homogeneous identities of degrees less than 8.

arXiv (Cornell University), Mar 22, 2015

We introduce the gonosomal algebra. Gonosomal algebra extend the evolution algebra of the bisexua... more We introduce the gonosomal algebra. Gonosomal algebra extend the evolution algebra of the bisexual population (EABP) defined by Ladra and Rozikov. We show that gonosomal algebras can represent algebraically a wide variety of sex determination systems observed in bisexual populations. We illustrate this by about twenty genetic examples, most of these examples cannot be represented by an EABP. We give seven algebraic constructions of gonosomal algebras, each is illustrated by genetic examples. We show that unlike the EABP gonosomal algebras are not dibaric. We approach the existence of dibaric function and idempotent in gonosomal algebras.

Communications in Algebra, 2020

Abstract We are interested in the evolution operators defined on commutative and non-associative ... more Abstract We are interested in the evolution operators defined on commutative and non-associative algebras when the characteristic of the scalar field is 2. We distinguish four types: nilpotent, quasi-constant, ultimately periodic, and plenary train operators. They are studied and classified for non-baric and for baric algebras.

Http Www Theses Fr, 1992

Dans le but de modeliser des populations dont la composition genetique suit une succession cycliq... more Dans le but de modeliser des populations dont la composition genetique suit une succession cyclique d'etats d'equilibre, nous presentons une generalisation de la notion d'algebre de Bernstein d'ordre n en introduisant la notion de periodicite. Nous nous sommes interesses aux conditions d'existence d'idempotents generalises et a leurs proprietes, a la structure vectorielle qui apparait lors de la decomposition de Peierce et a des problemes de transport de structures dans la dupliquee de ces algebres. Enfin nous etudions les algebres qui modelisent les populations d'organismes soumises seulement a la mutation des genes et nous etablissons la condition necessaire et suffisante pour que ces populations atteignent a la premiere generation une situation d'equilibre periodique

Proceedings of the Edinburgh Mathematical Society

Proceedings of the Edinburgh Mathematical Society, 1992

Nous Présentons ici l'étude, dans l'optique exhibée dans [7], d'une catégorie d'a... more Nous Présentons ici l'étude, dans l'optique exhibée dans [7], d'une catégorie d'algèbres à idempotents, commutatives, non associatives, définies par des relations. On y trouve une structure qui n'est en général ni pondérée ni de Jordan et où, cependant, ces conditions sont équivalentes.Following a previous paper, we study here a class of commutative and non associative algebras with idempotents and defined by relations. We find a structure which is not, in general, baric or Jordan but where these notions are equivalent.

A l’aide des alg`ebres de mutation on contruit des structures v´erifiant des identit´ es ω-polynom... more A l’aide des alg`ebres de mutation on contruit des structures v´erifiant des identit´ es ω-polynomiales arbitraires. Avec cette optique, concernant les trainalg`ebres d’ordre n, on ´etablit des r´esultats relatifs `a l’existence d’idempotents, en particulier quand ces alg`ebres ont la valeur 1 2 comme train-racine.