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Around 1900, S.N. Bernstein (cf. [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]]) worked on an important problem concerning the laws of formal genetics. This problem is known today as the Bernstein problem. Following Yu.I. Lyubich (cf. [[#References|[a10]]]), this problem can be expressed as follows. The state of a population in a given generation is described by a vector in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b1103401.png" /> whose coordinates satisfy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b1103402.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b1103403.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b1103404.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b1103405.png" /> of all states is a simplex in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b1103406.png" /> and the vertices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b1103407.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b1103408.png" />) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b1103409.png" /> are the different types of individuals in the population. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034010.png" /> is the [[Probability|probability]] that an individual <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034011.png" /> appears in the next generation from parents of types <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034013.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034014.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034015.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034016.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034017.png" />). In absence of selection and under random hypothesis, the state of the population in the next generation can be written, in terms of coordinates, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034018.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034019.png" />). These relations define a quadratic operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034020.png" /> called the evolutionary quadratic operator. The Bernstein stationarity principle says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034021.png" /> and the Bernstein problem aims at describing all quadratic operators satisfying this principle. Bernstein solved his problem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034022.png" /> and much progress was achieved recently (cf. [[#References|[a6]]], [[#References|[a8]]]) in this direction. The Bernstein problem can be translated in terms of algebra structure. In fact, over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034023.png" /> an algebra structure can be defined via the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034024.png" /> by
+
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034025.png" /></td> </tr></table>
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for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034026.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034027.png" /> is the mapping defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034028.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034029.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034030.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034031.png" />. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034032.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034033.png" />. Of course, to define this multiplication over the whole space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034034.png" /> starting from the simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034035.png" />, one has to make convenient extensions of this multiplication by bilinearity. Now, in general, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034036.png" /> is a (commutative) [[Field|field]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034037.png" /> is a commutative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034038.png" />-algebra, then a weighted algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034039.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034040.png" /> is said to be a Bernstein algebra if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034041.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034042.png" /> (cf. [[#References|[a2]]]). In recent years (1990s), the theory of Bernstein algebras has been substantially improved. V.M. Abraham (cf. [[#References|[a1]]]) suggests the construction of a generalized Bernstein algebra. In this perspective, for an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034043.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034044.png" /> is a weighted algebra, the plenary powers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034045.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034046.png" /> are defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034048.png" /> for all integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034049.png" />. The plenary powers can be interpreted by saying that they represent random mating between discrete non-overlapping generations. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034050.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034052.png" />th order Bernstein algebra if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034053.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034054.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034055.png" /> is the smallest such integer (cf. [[#References|[a11]]]). Second-order Bernstein algebras are simply called Bernstein algebras and first-order Bernstein algebras are also called gametic diploid algebras. The interpretation of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034056.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034057.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034058.png" />) is that equilibrium in the population is reached after exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110340/b11034059.png" /> generations of intermixing. For genetic properties of Bernstein algebras, see [[#References|[a7]]] and [[#References|[a12]]].
+
Around 1900, S.N. Bernstein (cf. [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]]) worked on an important problem concerning the laws of formal genetics. This problem is known today as the Bernstein problem. Following Yu.I. Lyubich (cf. [[#References|[a10]]]), this problem can be expressed as follows. The state of a population in a given generation is described by a vector in  $  \mathbf R  ^ {n} $
 +
whose coordinates satisfy  $  x _ {i} \geq  0 $(
 +
$  i = 1 \dots n $)
 +
and  $  \sum _ {i = 1 }  ^ {n} x _ {i} = 1 $.  
 +
The set  $  S $
 +
of all states is a simplex in  $  \mathbf R  ^ {n} $
 +
and the vertices  $  e _ {i} $(
 +
$  i = 1 \dots n $)
 +
of  $  S $
 +
are the different types of individuals in the population. If  $  \gamma _ {ijk }  $
 +
is the [[Probability|probability]] that an individual  $  e _ {k} $
 +
appears in the next generation from parents of types  $  e _ {i} $
 +
and  $  e _ {j} $,  
 +
then $  \sum _ {k = 1 }  ^ {n} \gamma _ {ijk }  = 1 $(
 +
$  i,j = 1 \dots n $)
 +
and  $  \gamma _ {ijk }  = \gamma _ {jik }  $(
 +
$  i,j,k = 1 \dots n $).  
 +
In absence of selection and under random hypothesis, the state of the population in the next generation can be written, in terms of coordinates, as  $  x _ {k}  ^  \prime  = \sum _ {i,j = 1 }  ^ {n} \gamma _ {ijk }  x _ {i} x _ {j} $(
 +
$  k = 1 \dots n $).
 +
These relations define a quadratic operator  $  V : S \rightarrow S $
 +
called the evolutionary quadratic operator. The Bernstein stationarity principle says that  $  V  ^ {2} = V $
 +
and the Bernstein problem aims at describing all quadratic operators satisfying this principle. Bernstein solved his problem for $  n = 3 $
 +
and much progress was achieved recently (cf. [[#References|[a6]]], [[#References|[a8]]]) in this direction. The Bernstein problem can be translated in terms of algebra structure. In fact, over  $  \mathbf R  ^ {n} $
 +
an algebra structure can be defined via the operator  $  V $
 +
by
 +
 
 +
$$
 +
xy = {
 +
\frac{1}{2}
 +
} ( V ( x + y ) - V ( x ) - V ( y ) )
 +
$$
 +
 
 +
for all $  x, y \in \mathbf R  ^ {n} $,
 +
and if  $  \omega : {\mathbf R  ^ {n} } \rightarrow \mathbf R $
 +
is the mapping defined by  $  x = ( x _ {1} \dots x _ {n} ) \mapsto \sum _ {i = 1 }  ^ {n} x _ {i} $,
 +
then  $  V  ^ {2} = V $
 +
if and only if  $  ( x  ^ {2} )  ^ {2} = \omega ( x )  ^ {2} x  ^ {2} $
 +
for all  $  x \in \mathbf R  ^ {n} $.  
 +
Moreover,  $  \omega ( xy ) = \omega ( x ) \omega ( y ) $
 +
for all  $  x, y \in \mathbf R  ^ {n} $.  
 +
Of course, to define this multiplication over the whole space $  \mathbf R  ^ {n} $
 +
starting from the simplex $  S $,  
 +
one has to make convenient extensions of this multiplication by bilinearity. Now, in general, if $  K $
 +
is a (commutative) [[Field|field]] and $  A $
 +
is a commutative $  K $-
 +
algebra, then a weighted algebra $  ( A, \omega ) $
 +
over $  K $
 +
is said to be a Bernstein algebra if $  ( x  ^ {2} )  ^ {2} = \omega ( x )  ^ {2} x  ^ {2} $
 +
for all $  x \in A $(
 +
cf. [[#References|[a2]]]). In recent years (1990s), the theory of Bernstein algebras has been substantially improved. V.M. Abraham (cf. [[#References|[a1]]]) suggests the construction of a generalized Bernstein algebra. In this perspective, for an element $  x \in A $,  
 +
where $  ( A, \omega ) $
 +
is a weighted algebra, the plenary powers $  x ^ {[ m ] } $
 +
of $  x $
 +
are defined by $  x ^ {[ 1 ] } = x $
 +
and $  x ^ {[ m -1 ] } x ^ {[ m -1 ] } = x ^ {[ m ] } $
 +
for all integer $  m \geq  2 $.  
 +
The plenary powers can be interpreted by saying that they represent random mating between discrete non-overlapping generations. $  ( A, \omega ) $
 +
is called an $  n $
 +
th order Bernstein algebra if $  x ^ {[ n + 2 ] } = \omega ( x ) ^ {2  ^ {n} } x ^ {[ n + 1 ] } $
 +
for all $  x \in A $,  
 +
where $  n \geq  1 $
 +
is the smallest such integer (cf. [[#References|[a11]]]). Second-order Bernstein algebras are simply called Bernstein algebras and first-order Bernstein algebras are also called gametic diploid algebras. The interpretation of the equation $  x ^ {[ n + 2 ] } = x ^ {[ n + 1 ] } $(
 +
$  x \in A $
 +
such that $  \omega ( x ) = 1 $)  
 +
is that equilibrium in the population is reached after exactly $  n $
 +
generations of intermixing. For genetic properties of Bernstein algebras, see [[#References|[a7]]] and [[#References|[a12]]].
  
 
See also [[Genetic algebra|Genetic algebra]]; [[Baric algebra|Baric algebra]].
 
See also [[Genetic algebra|Genetic algebra]]; [[Baric algebra|Baric algebra]].

Revision as of 10:58, 29 May 2020


Around 1900, S.N. Bernstein (cf. [a3], [a4], [a5]) worked on an important problem concerning the laws of formal genetics. This problem is known today as the Bernstein problem. Following Yu.I. Lyubich (cf. [a10]), this problem can be expressed as follows. The state of a population in a given generation is described by a vector in $ \mathbf R ^ {n} $ whose coordinates satisfy $ x _ {i} \geq 0 $( $ i = 1 \dots n $) and $ \sum _ {i = 1 } ^ {n} x _ {i} = 1 $. The set $ S $ of all states is a simplex in $ \mathbf R ^ {n} $ and the vertices $ e _ {i} $( $ i = 1 \dots n $) of $ S $ are the different types of individuals in the population. If $ \gamma _ {ijk } $ is the probability that an individual $ e _ {k} $ appears in the next generation from parents of types $ e _ {i} $ and $ e _ {j} $, then $ \sum _ {k = 1 } ^ {n} \gamma _ {ijk } = 1 $( $ i,j = 1 \dots n $) and $ \gamma _ {ijk } = \gamma _ {jik } $( $ i,j,k = 1 \dots n $). In absence of selection and under random hypothesis, the state of the population in the next generation can be written, in terms of coordinates, as $ x _ {k} ^ \prime = \sum _ {i,j = 1 } ^ {n} \gamma _ {ijk } x _ {i} x _ {j} $( $ k = 1 \dots n $). These relations define a quadratic operator $ V : S \rightarrow S $ called the evolutionary quadratic operator. The Bernstein stationarity principle says that $ V ^ {2} = V $ and the Bernstein problem aims at describing all quadratic operators satisfying this principle. Bernstein solved his problem for $ n = 3 $ and much progress was achieved recently (cf. [a6], [a8]) in this direction. The Bernstein problem can be translated in terms of algebra structure. In fact, over $ \mathbf R ^ {n} $ an algebra structure can be defined via the operator $ V $ by

$$ xy = { \frac{1}{2} } ( V ( x + y ) - V ( x ) - V ( y ) ) $$

for all $ x, y \in \mathbf R ^ {n} $, and if $ \omega : {\mathbf R ^ {n} } \rightarrow \mathbf R $ is the mapping defined by $ x = ( x _ {1} \dots x _ {n} ) \mapsto \sum _ {i = 1 } ^ {n} x _ {i} $, then $ V ^ {2} = V $ if and only if $ ( x ^ {2} ) ^ {2} = \omega ( x ) ^ {2} x ^ {2} $ for all $ x \in \mathbf R ^ {n} $. Moreover, $ \omega ( xy ) = \omega ( x ) \omega ( y ) $ for all $ x, y \in \mathbf R ^ {n} $. Of course, to define this multiplication over the whole space $ \mathbf R ^ {n} $ starting from the simplex $ S $, one has to make convenient extensions of this multiplication by bilinearity. Now, in general, if $ K $ is a (commutative) field and $ A $ is a commutative $ K $- algebra, then a weighted algebra $ ( A, \omega ) $ over $ K $ is said to be a Bernstein algebra if $ ( x ^ {2} ) ^ {2} = \omega ( x ) ^ {2} x ^ {2} $ for all $ x \in A $( cf. [a2]). In recent years (1990s), the theory of Bernstein algebras has been substantially improved. V.M. Abraham (cf. [a1]) suggests the construction of a generalized Bernstein algebra. In this perspective, for an element $ x \in A $, where $ ( A, \omega ) $ is a weighted algebra, the plenary powers $ x ^ {[ m ] } $ of $ x $ are defined by $ x ^ {[ 1 ] } = x $ and $ x ^ {[ m -1 ] } x ^ {[ m -1 ] } = x ^ {[ m ] } $ for all integer $ m \geq 2 $. The plenary powers can be interpreted by saying that they represent random mating between discrete non-overlapping generations. $ ( A, \omega ) $ is called an $ n $ th order Bernstein algebra if $ x ^ {[ n + 2 ] } = \omega ( x ) ^ {2 ^ {n} } x ^ {[ n + 1 ] } $ for all $ x \in A $, where $ n \geq 1 $ is the smallest such integer (cf. [a11]). Second-order Bernstein algebras are simply called Bernstein algebras and first-order Bernstein algebras are also called gametic diploid algebras. The interpretation of the equation $ x ^ {[ n + 2 ] } = x ^ {[ n + 1 ] } $( $ x \in A $ such that $ \omega ( x ) = 1 $) is that equilibrium in the population is reached after exactly $ n $ generations of intermixing. For genetic properties of Bernstein algebras, see [a7] and [a12].

See also Genetic algebra; Baric algebra.

References

[a1] V.M. Abraham, "Linearising quadratic transformations in genetic algebras" Thesis, Univ. London (1975)
[a2] M.T. Alcalde, C. Burgueno, A. Labra, A. Micali, "Sur les algèbres de Bernstein" Proc. London Math. Soc. (3) , 58 (1989) pp. 51–68
[a3] S.N. Bernstein, "Principe de stationarité et généralisation de la loi de Mendel" C.R. Acad. Sci. Paris , 177 (1923) pp. 528–531
[a4] S.N. Bernstein, "Démonstration mathématique de la loi d'hérédité de Mendel" C.R. Acad. Sci. Paris , 177 (1923) pp. 581–584
[a5] S.N. Bernstein, "Solution of a mathematical problem connected with the theory of heredity" Ann. Math. Stat. , 13 (1942) pp. 53–61
[a6] S. González, J.C. Gutiérrez, C. Martínez, "The Bernstein problem in dimension " J. Algebra , 177 (1995) pp. 676–697
[a7] A.N. Griskhov, "On the genetic property of Bernstein algebras" Soviet Math. Dokl. , 35 (1987) pp. 489–492 (In Russian)
[a8] J.C. Gutiérrez, "The Bernstein problem for type " J. Algebra , 181 (1996) pp. 613–627
[a9] P. Holgate, "Genetic algebras satisfying Bernstein's stationarity principle" J. London Math. Soc. (2) , 9 (1975) pp. 613–623
[a10] Yu.I. Lyubich, "Mathematical structures in population genetics" Biomathematics , 22 (1992)
[a11] C. Mallol, A. Micali, M. Ouattara, "Sur les algèbres de Bernstein IV" Linear Alg. & Its Appl. , 158 (1991) pp. 1–26
[a12] A. Micali, M. Ouattara, "Structure des algèbres de Bernstein" Linear Alg. & Its Appl. , 218 (1995) pp. 77–88
How to Cite This Entry:
Bernstein algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bernstein_algebra&oldid=46025
This article was adapted from an original article by A. Micali (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article