On Bernstein algebras which are train algebras

Walcher, Sebastian

doi:10.1017/s0013091500005411

Cited by 14 publications

(13 citation statements)

References 9 publications

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“…A random sampling of current references might include Wörz-Busekros [37]; Walcher [33]; Guzzo [18], [19]; Costa and Guzzo [7], [8]; Hentzel, Peresi, and Holgate [21]; Peresi [31]; Cortés [6]; Martinez [28]; Burgueño, Neuberg, and Suazo [5]; González and Martinez [16]; and González, Martinez, and Vicente [17]. This list is by no means complete, but it does provide a starting point for some of the current work being done in the field.…”

Section: Resultsmentioning

confidence: 99%

Algebraic structure of genetic inheritance

Reed

1997

Bull. Amer. Math. Soc.

113

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Abstract. In this paper we will explore the nonassociative algebraic structure that naturally occurs as genetic information gets passed down through the generations. While modern understanding of genetic inheritance initiated with the theories of Charles Darwin, it was the Augustinian monk Gregor Mendel who began to uncover the mathematical nature of the subject. In fact, the symbolism Mendel used to describe his first results (e.g., see his 1866 paper Experiments in Plant-Hybridization [30]) is quite algebraically suggestive. Seventy four years later, I.M.H. Etherington introduced the formal language of abstract algebra to the study of genetics in his series of seminal papers [9], [10], [11]. In this paper we will discuss the concepts of genetics that suggest the underlying algebraic structure of inheritance, and we will give a brief overview of the algebras which arise in genetics and some of their basic properties and relationships. With the popularity of biologically motivated mathematics continuing to rise, we offer this survey article as another example of the breadth of mathematics that has biological significance. The most comprehensive reference for the mathematical research done in this area (through 1980) is .

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Section: Resultsmentioning

confidence: 99%

Algebraic structure of genetic inheritance

Reed

1997

Bull. Amer. Math. Soc.

113

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“…We first summarize some facts which were proved in [2] and [5]: Now let lK denote the field of real or complex numbers and A a finite dimensional commutative lK-algebra with left multiplication L (x). Then A is called a genetic algebra if there is a nontrivial homomorphism co : A --+ IK and the coefficients of the characteristic polynomial of any transformation f (L (xl) ..... L (xs)) only depend on co(x1) ..... co(Xs) for any polynomial f in s noncommuting indeterminates; the second requirement is equivalent to A (in the real case A ® C) permitting coordinates such that L (x) is lower triangular for all x C A and strictly lower triangular for all x E Ker co (see [6] for details).…”

Section: Continuous Time Models and Genetic Algebrasmentioning

confidence: 99%

Long-time behavior of continuous time models in genetic algebras

Gradl

1994

J. Math. Biol.

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“…Now let G(y, t) = ~k>~o tkgk(Y) be the Taylor series expansion about t = 0. From [7] it is known that the gk satisfy the recursion (k + l)g, +, (y) = Dg,(y)y 2 for all k~>0. (In particular go(Y)=Y, gl(Y)=y2, gz(Y)=y3 and g3(Y)= }(2y4+yZy2).)…”

Section: Continuous Time Modelsmentioning

confidence: 99%

“…In [7] the general solution of the differential equation in question and its long time behavior were determined for Bernstein algebras satisfying a train equation.…”

Section: Bernstein Algebrasmentioning

confidence: 99%

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On continuous time models in genetic and Bernstein algebras

Gradl¹,

Walcher²

1992

J. Math. Biol.

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We discuss the long-time behavior of Andreoli's differential equation for genetic algebras and for Bernstein algebras and show convergence to an equilibrium in both cases. For a class of Bernstein algebras this equilibrium is determined explicitly. Continuous time modelsAndreoli [1] introduced the differential equation ~ : X 2 --X, x ( 0 ) = y in the standard simplex of a genetic algebra with genetic realization to model the time dependence of the genotype frequencies of a population in the limiting case of continuously overlapping generations. While Andreoli restricted his attention to three-dimensional systems, Heuch [5] later considered genetic algebras of arbitrary (finite) dimension and showed that this equation can be solved by elementary functions; see also W6rz-Busekros [9].The long-time behavior of the solutions seems to have been unknown so far in the general case, although it was known for special classes of algebras [9]. We show in this note that for any genetic algebra with genetic realization every solution in the standard simplex tends to a stationary point for t--* oo.Using different methods, we show that the same holds for Bernstein algebras (which are not necessarily genetic).In the remainder of this section we derive some preliminary results about the differential equation 2 = x 2 -x in a real or complex commutative algebra A. The first one is proved in [7]:Lemma 0.1 Let G(y, t) resp. S(y, t) be the solution of 2 = x 2 resp. 2 = x 2 -x with x(O) = y. Then S(y, t) = e -" G(y, 1 -e -9 , wherever both sides are defined. Now let G(y, t) = ~k>~o tkgk(Y) be the Taylor series expansion about t = 0. From [7] it is known that the gk satisfy the recursion

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On Bernstein algebras which are train algebras

Abstract: The class of non-associative algebras over a field of characteristic zero named in the title is studied using a result of Ouattara [9]. As an application, the differential equation for overlapping generations in the timecontinuous model is solved.

Cited by 14 publications

References 9 publications

Algebraic structure of genetic inheritance

Algebraic structure of genetic inheritance

Long-time behavior of continuous time models in genetic algebras

On continuous time models in genetic and Bernstein algebras

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