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''Bernshtein problem''
 
''Bernshtein problem''
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b1201601.png" /> be the simplex in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b1201602.png" /> spanned by the canonical basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b1201603.png" />. Any set of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b1201604.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b1201605.png" />) such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b1201606.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b1201607.png" /> defines a stochastic quadratic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b1201608.png" /> by the formulas
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Let $\Delta ^ { n - 1 }$ be the simplex in ${\bf R} ^ { n }$ spanned by the canonical basis $\{ e _ { i } \} _ { 1 } ^ { n }$. Any set of numbers $p _ { i k , j } \geq 0$ ($1 \leq i , k , j \leq n$) such that $\sum _ { j } p _ { i k,j }  = 1$ and $p _ { i k  ,\, j} = p _ { k i  ,\, j}$ defines a stochastic quadratic mapping $V:\Delta ^ { n - 1 } \rightarrow \Delta ^ { n - 1 }$ by the formulas
  
<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/b120/b120160/b1201609.png" /></td> </tr></table>
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\begin{equation*} x _ { j } ^ { \prime } = \sum _ { i , k } p _ { i k,j } x_i x _ { k } , \quad x _ { i } \geq 0 , \sum _ { i } x _ { i } = 1. \end{equation*}
  
This mapping is called Bernstein (or stationary) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016010.png" />. The Bernstein problem is to explicitly describe all such mappings. This problem was posed by S.N. Bernshtein [[#References|[a1]]] in order to create a mathematical foundation of population genetics. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016011.png" />, this problem has been solved in [[#References|[a2]]]. (For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016012.png" /> the problem is trivial.)
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This mapping is called Bernstein (or stationary) if $V ^ { 2 } = V$. The Bernstein problem is to explicitly describe all such mappings. This problem was posed by S.N. Bernshtein [[#References|[a1]]] in order to create a mathematical foundation of population genetics. For $n = 3$, this problem has been solved in [[#References|[a2]]]. (For $n \leq 2$ the problem is trivial.)
  
 
The classical Mendel mechanism of heredity defines a mapping
 
The classical Mendel mechanism of heredity defines a mapping
  
<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/b120/b120160/b12016013.png" /></td> </tr></table>
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\begin{equation*} x _ { 1 } ^ { \prime } = p ^ { 2 } , x _ { 2 } ^ { \prime } = q ^ { 2 } , x _ { 3 } ^ { \prime } = 2 p q, \end{equation*}
  
<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/b120/b120160/b12016014.png" /></td> </tr></table>
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\begin{equation*} p = x _ { 1 } + \frac { 1 } { 2 } x _ { 3 } , \quad q = x _ { 2 } + \frac { 1 } { 2 } x _ { 3 } \end{equation*}
  
(the Hardy–Weinberg formulas, cf. [[#References|[a5]]]). Biologically, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016016.png" /> are the probabilities of an alternating pair of genes, say <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016018.png" /> respectively, in a population where the individuals may be of genotypes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016021.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016024.png" /> are the probabilities of these genotypes in a generation. If the next generation is formed by random mating, then the probabilities turn into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016027.png" />. As a result, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016029.png" /> and then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016030.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016031.png" />), i.e. the Hardy–Weinberg mapping is stationary. Conversely, if for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016032.png" /> a stationary mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016033.png" /> is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016034.png" /> and all quadratic forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016035.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016036.png" /> is a Hardy–Weinberg mapping (see [[#References|[a2]]], [[#References|[a6]]]). Thus, the only Mendelian heredity is stationary and such that all offsprings for the parental couple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016037.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016038.png" /> (and, in addition, such that all genotypes are present in the next generation).
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(the [[Hardy–Weinberg law]]s, cf. [[#References|[a5]]]). Biologically, $p$ and $q$ are the probabilities of an alternating pair of genes, say $A$ and $a$ respectively, in a population where the individuals may be of genotypes $A A$, $aa$ and $A a$. Then $x_{1} $, $x _ { 2 }$, $x _ { 3 }$ are the probabilities of these genotypes in a generation. If the next generation is formed by random mating, then the probabilities turn into $x _ { 1 } ^ { \prime }$, $x _ { 2 } ^ { \prime }$, $x _ { 3 } ^ { \prime }$. As a result, $p ^ { \prime } = p$, $q ^ { \prime } = q$ and then $x _ { i } ^ { \prime \prime } = x _ { i } ^ { \prime }$ ($1 \leq i \leq 3$), i.e. the Hardy–Weinberg mapping is stationary. Conversely, if for $n = 3$ a stationary mapping $V$ is such that $p _ { 12,3 } = 1$ and all quadratic forms $x _ { j } ^ { \prime } \not\equiv 0$, then $V$ is a Hardy–Weinberg mapping (see [[#References|[a2]]], [[#References|[a6]]]). Thus, the only Mendelian heredity is stationary and such that all offsprings for the parental couple $( A A , a a )$ are $A a$ (and, in addition, such that all genotypes are present in the next generation).
  
For any stochastic quadratic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016039.png" />, the linear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016040.png" /> is called invariant if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016041.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016042.png" /> is called regular if there exists a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016043.png" /> of invariant linear forms such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016044.png" /> for certain constant coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016045.png" />. The Hardy–Weinberg mapping is regular. Another interesting example is the quadrille mapping (see [[#References|[a2]]], [[#References|[a6]]]):
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For any stochastic quadratic mapping $V$, the linear form $f = \sum _ { j } a _ { j} x_j$ is called invariant if $f ^ { \prime } = f$. The mapping $V$ is called regular if there exists a family $\{ f _ { i } \} _ { 1 } ^ { m }$ of invariant linear forms such that $x _ { j } ^ { \prime } = \sum _ { i , k } c _ { i k } f _ { i } f _ { k }$ for certain constant coefficients $c _ { i k }$. The Hardy–Weinberg mapping is regular. Another interesting example is the quadrille mapping (see [[#References|[a2]]], [[#References|[a6]]]):
  
<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/b120/b120160/b12016046.png" /></td> </tr></table>
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\begin{equation*} x _ { 1 } ^ { \prime } = p _ { 1 } q _ { 1 } ,\, x _ { 2 } ^ { \prime } = p _ { 1 } q _ { 2 }, \end{equation*}
  
<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/b120/b120160/b12016047.png" /></td> </tr></table>
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\begin{equation*} x _ { 3 } ^ { \prime } = p _ { 2 } q _ { 1 } , x _ { 4 } ^ { \prime } = p _ { 2 } q _ { 2 } \end{equation*}
  
 
where
 
where
  
<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/b120/b120160/b12016048.png" /></td> </tr></table>
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\begin{equation*} p _ { 1 } = x _ { 1 } + x _ { 2 } , \quad p _ { 2 } = x _ { 3 } + x _ { 4 }, \end{equation*}
  
<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/b120/b120160/b12016049.png" /></td> </tr></table>
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\begin{equation*} q _ { 1 } = x _ { 1 } + x _ { 3 } , \quad q _ { 2 } = x _ { 2 } + x _ { 4 }. \end{equation*}
  
 
For the regular case, the Bernstein problem has been solved in [[#References|[a6]]], [[#References|[a8]]], [[#References|[a9]]]. This is precisely the case when the stationarity is based on a system of genes (see [[#References|[a7]]]). The genes correspond to the extremal rays of the cone of non-negative invariant linear forms. After a normalization, these forms are just the probabilities of the genes.
 
For the regular case, the Bernstein problem has been solved in [[#References|[a6]]], [[#References|[a8]]], [[#References|[a9]]]. This is precisely the case when the stationarity is based on a system of genes (see [[#References|[a7]]]). The genes correspond to the extremal rays of the cone of non-negative invariant linear forms. After a normalization, these forms are just the probabilities of the genes.
  
A standard genetical interpretation also requires <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016050.png" /> to be normal in the sense that:
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A standard genetical interpretation also requires $V$ to be normal in the sense that:
  
1) all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016051.png" />;
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1) all $x _ { i } ^ { \prime } \neq 0$;
  
2) for any pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016053.png" /> the quadratic forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016055.png" /> are not proportional;
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2) for any pair $i$, $k$ the quadratic forms $x _ { i } ^ { \prime }$ and $x _ { k} ^ { \prime }$ are not proportional;
  
3) there is no pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016057.png" /> such that all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016058.png" /> are functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016059.png" /> and of the remaining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016060.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016061.png" /> is normal together with its restrictions to all invariant faces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016062.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016063.png" /> is called ultranormal. All stationary ultranormal mappings are regular [[#References|[a13]]].
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3) there is no pair $i$, $k$ such that all $x _ { j } ^ { \prime }$ are functions of $x _ { i } + x _ {  k }$ and of the remaining $x _ { l }$. If $V$ is normal together with its restrictions to all invariant faces of $\Delta ^ { n - 1 }$, then $V$ is called ultranormal. All stationary ultranormal mappings are regular [[#References|[a13]]].
  
A non-regular stochastic Bernstein mapping appears already for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016064.png" />:
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A non-regular stochastic Bernstein mapping appears already for $n = 3$:
  
<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/b120/b120160/b12016065.png" /></td> </tr></table>
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\begin{equation*} x _ { 1 } ^ { \prime } = x _ { 1 } ( s + v ), \end{equation*}
  
<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/b120/b120160/b12016066.png" /></td> </tr></table>
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\begin{equation*} x _ { 2 } ^ { \prime } = x _ { 3 } ^ { \prime } = \frac { 1 } { 2 } [ ( x _ { 1 } + x _ { 2 } ) s - x _ { 1 } v ], \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016067.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016068.png" />. Here, all invariant linear forms are trivial, i.e. proportional to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016069.png" />.
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where $s = x _ { 1 } + x _ { 2 } + x _ { 3 }$, $v = x_3 - x_2$. Here, all invariant linear forms are trivial, i.e. proportional to $s$.
  
For the non-regular case there are some partial results for the Bernstein problem. In particular, the results cover the low dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016070.png" /> (cf. [[#References|[a3]]], [[#References|[a4]]], [[#References|[a10]]]).
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For the non-regular case there are some partial results for the Bernstein problem. In particular, the results cover the low dimensions $n = 4,5,6$ (cf. [[#References|[a3]]], [[#References|[a4]]], [[#References|[a10]]]).
  
 
In the course of these investigations, Bernstein algebras were introduced as a powerful tool (see [[#References|[a6]]], [[#References|[a8]]] or [[Bernstein algebra|Bernstein algebra]]). The theory of Bernstein algebras was subsequently developed by itself.
 
In the course of these investigations, Bernstein algebras were introduced as a powerful tool (see [[#References|[a6]]], [[#References|[a8]]] or [[Bernstein algebra|Bernstein algebra]]). The theory of Bernstein algebras was subsequently developed by itself.
  
In the algebraic context, the Bernstein problem is to explicitly describe those Bernstein algebras which are stochastic with respect to the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016071.png" />. The latter means that the product of every pair of basis vectors belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016072.png" />.
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In the algebraic context, the Bernstein problem is to explicitly describe those Bernstein algebras which are stochastic with respect to the basis $\{ e _ { i } \} _ { 1 } ^ { n }$. The latter means that the product of every pair of basis vectors belongs to $\Delta ^ { n - 1 }$.
  
The Bernstein algebra corresponding to a regular mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016073.png" /> is regular by definition. This class is the most important from the genetics point of view.
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The Bernstein algebra corresponding to a regular mapping $V$ is regular by definition. This class is the most important from the genetics point of view.
  
Another important tool in the study of the Bernstein problem is a topological structure on the set of essential faces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016074.png" />, the faces such that their intersections with the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120160/b12016075.png" /> are non-empty [[#References|[a11]]].
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Another important tool in the study of the Bernstein problem is a topological structure on the set of essential faces of $\Delta ^ { n - 1 }$, the faces such that their intersections with the image of $V$ are non-empty [[#References|[a11]]].
  
 
See [[#References|[a12]]] for a systematic presentation of the results and methods regarding the Bernstein problem up to the middle of the 1980s.
 
See [[#References|[a12]]] for a systematic presentation of the results and methods regarding the Bernstein problem up to the middle of the 1980s.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S.N. Bernstein, "Mathematical problems in modern biology" ''Science in the Ukraine'' , '''1''' (1922) pp. 14–19 (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> S.N. Bernstein, "Solution of a mathematical problem related to the theory of inheritance" ''Uchen. Zap. Nauch. Issl. Kafedr. Ukrain.'' , '''1''' (1924) pp. 83–115 (In Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> S. Gonzáles, J.C. Gutiérrez, C. Martinez, "The Bernstein problem in dimension 5" ''J. Algebra'' , '''177''' (1995) pp. 676–697</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> J.C. Gutiérrez, "The Bernstein problem in dimension 6" ''J. Algebra'' , '''185''' (1996) pp. 420–439 {{MR|}} {{ZBL|0863.17026}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> G.H. Hardy, "Mendelian proportions in a mixed population" ''Science'' , '''28''' : 706 (1908) pp. 49–50</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> Y.I. Lyubich, "Basic concepts and theorems of evolutionary genetics for free populations" ''Russian Math. Surveys'' , '''26''' : 5 (1971) pp. 51–123</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> Y.I. Lyubich, "Analogues to the Hardy–Weinberg Law" ''Genetics'' , '''9''' : 10 (1973) pp. 139–144 (In Russian)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> Y.I. Lyubich, "Two-level Bernstein populations" ''Math. USSR Sb.'' , '''24''' : 1 (1974) pp. 593–615 {{MR|}} {{ZBL|0317.92017}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> Y.I. Lyubich, "Proper Bernstein populations" ''Probl. Inform. Transmiss.'' , '''Jan.''' (1978) pp. 228–235 {{MR|}} {{ZBL|0398.92025}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> Y.I. Lyubich, "Quasilinear Bernstein populations" ''Teor. Funct. Funct. Anal. Appl.'' , '''26''' (1976) pp. 79–84</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> Y.I. Lyubich, "A topological approach to a problem in mathematical genetics" ''Russian Math. Surveys'' , '''34''' : 6 (1979) pp. 60–66 {{MR|562818}} {{ZBL|0446.92012}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> Y.I. Lyubich, "Mathematical structures in population genetics" , Springer (1992) {{MR|1224676}} {{ZBL|0747.92019}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> Y.I. Lyubich, "A new advance in the Bernstein problem in mathematical genetics" ''Preprint Inst. Math. Sci., SUNY Stony Brook'' , '''9''' (1996) pp. 1–33</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top"> S.N. Bernstein, "Mathematical problems in modern biology" ''Science in the Ukraine'' , '''1''' (1922) pp. 14–19 (In Russian)</td></tr><tr><td valign="top">[a2]</td> <td valign="top"> S.N. Bernstein, "Solution of a mathematical problem related to the theory of inheritance" ''Uchen. Zap. Nauch. Issl. Kafedr. Ukrain.'' , '''1''' (1924) pp. 83–115 (In Russian)</td></tr><tr><td valign="top">[a3]</td> <td valign="top"> S. Gonzáles, J.C. Gutiérrez, C. Martinez, "The Bernstein problem in dimension 5" ''J. Algebra'' , '''177''' (1995) pp. 676–697</td></tr><tr><td valign="top">[a4]</td> <td valign="top"> J.C. Gutiérrez, "The Bernstein problem in dimension 6" ''J. Algebra'' , '''185''' (1996) pp. 420–439 {{MR|}} {{ZBL|0863.17026}} </td></tr><tr><td valign="top">[a5]</td> <td valign="top"> G.H. Hardy, "Mendelian proportions in a mixed population" ''Science'' , '''28''' : 706 (1908) pp. 49–50</td></tr><tr><td valign="top">[a6]</td> <td valign="top"> Y.I. Lyubich, "Basic concepts and theorems of evolutionary genetics for free populations" ''Russian Math. Surveys'' , '''26''' : 5 (1971) pp. 51–123</td></tr><tr><td valign="top">[a7]</td> <td valign="top"> Y.I. Lyubich, "Analogues to the Hardy–Weinberg Law" ''Genetics'' , '''9''' : 10 (1973) pp. 139–144 (In Russian)</td></tr><tr><td valign="top">[a8]</td> <td valign="top"> Y.I. Lyubich, "Two-level Bernstein populations" ''Math. USSR Sb.'' , '''24''' : 1 (1974) pp. 593–615 {{MR|}} {{ZBL|0317.92017}} </td></tr><tr><td valign="top">[a9]</td> <td valign="top"> Y.I. Lyubich, "Proper Bernstein populations" ''Probl. Inform. Transmiss.'' , '''Jan.''' (1978) pp. 228–235 {{MR|}} {{ZBL|0398.92025}} </td></tr><tr><td valign="top">[a10]</td> <td valign="top"> Y.I. Lyubich, "Quasilinear Bernstein populations" ''Teor. Funct. Funct. Anal. Appl.'' , '''26''' (1976) pp. 79–84</td></tr><tr><td valign="top">[a11]</td> <td valign="top"> Y.I. Lyubich, "A topological approach to a problem in mathematical genetics" ''Russian Math. Surveys'' , '''34''' : 6 (1979) pp. 60–66 {{MR|562818}} {{ZBL|0446.92012}} </td></tr><tr><td valign="top">[a12]</td> <td valign="top"> Y.I. Lyubich, "Mathematical structures in population genetics" , Springer (1992) {{MR|1224676}} {{ZBL|0747.92019}} </td></tr><tr><td valign="top">[a13]</td> <td valign="top"> Y.I. Lyubich, "A new advance in the Bernstein problem in mathematical genetics" ''Preprint Inst. Math. Sci., SUNY Stony Brook'' , '''9''' (1996) pp. 1–33</td></tr></table>

Latest revision as of 17:02, 1 July 2020

Bernshtein problem

Let $\Delta ^ { n - 1 }$ be the simplex in ${\bf R} ^ { n }$ spanned by the canonical basis $\{ e _ { i } \} _ { 1 } ^ { n }$. Any set of numbers $p _ { i k , j } \geq 0$ ($1 \leq i , k , j \leq n$) such that $\sum _ { j } p _ { i k,j } = 1$ and $p _ { i k ,\, j} = p _ { k i ,\, j}$ defines a stochastic quadratic mapping $V:\Delta ^ { n - 1 } \rightarrow \Delta ^ { n - 1 }$ by the formulas

\begin{equation*} x _ { j } ^ { \prime } = \sum _ { i , k } p _ { i k,j } x_i x _ { k } , \quad x _ { i } \geq 0 , \sum _ { i } x _ { i } = 1. \end{equation*}

This mapping is called Bernstein (or stationary) if $V ^ { 2 } = V$. The Bernstein problem is to explicitly describe all such mappings. This problem was posed by S.N. Bernshtein [a1] in order to create a mathematical foundation of population genetics. For $n = 3$, this problem has been solved in [a2]. (For $n \leq 2$ the problem is trivial.)

The classical Mendel mechanism of heredity defines a mapping

\begin{equation*} x _ { 1 } ^ { \prime } = p ^ { 2 } , x _ { 2 } ^ { \prime } = q ^ { 2 } , x _ { 3 } ^ { \prime } = 2 p q, \end{equation*}

\begin{equation*} p = x _ { 1 } + \frac { 1 } { 2 } x _ { 3 } , \quad q = x _ { 2 } + \frac { 1 } { 2 } x _ { 3 } \end{equation*}

(the Hardy–Weinberg laws, cf. [a5]). Biologically, $p$ and $q$ are the probabilities of an alternating pair of genes, say $A$ and $a$ respectively, in a population where the individuals may be of genotypes $A A$, $aa$ and $A a$. Then $x_{1} $, $x _ { 2 }$, $x _ { 3 }$ are the probabilities of these genotypes in a generation. If the next generation is formed by random mating, then the probabilities turn into $x _ { 1 } ^ { \prime }$, $x _ { 2 } ^ { \prime }$, $x _ { 3 } ^ { \prime }$. As a result, $p ^ { \prime } = p$, $q ^ { \prime } = q$ and then $x _ { i } ^ { \prime \prime } = x _ { i } ^ { \prime }$ ($1 \leq i \leq 3$), i.e. the Hardy–Weinberg mapping is stationary. Conversely, if for $n = 3$ a stationary mapping $V$ is such that $p _ { 12,3 } = 1$ and all quadratic forms $x _ { j } ^ { \prime } \not\equiv 0$, then $V$ is a Hardy–Weinberg mapping (see [a2], [a6]). Thus, the only Mendelian heredity is stationary and such that all offsprings for the parental couple $( A A , a a )$ are $A a$ (and, in addition, such that all genotypes are present in the next generation).

For any stochastic quadratic mapping $V$, the linear form $f = \sum _ { j } a _ { j} x_j$ is called invariant if $f ^ { \prime } = f$. The mapping $V$ is called regular if there exists a family $\{ f _ { i } \} _ { 1 } ^ { m }$ of invariant linear forms such that $x _ { j } ^ { \prime } = \sum _ { i , k } c _ { i k } f _ { i } f _ { k }$ for certain constant coefficients $c _ { i k }$. The Hardy–Weinberg mapping is regular. Another interesting example is the quadrille mapping (see [a2], [a6]):

\begin{equation*} x _ { 1 } ^ { \prime } = p _ { 1 } q _ { 1 } ,\, x _ { 2 } ^ { \prime } = p _ { 1 } q _ { 2 }, \end{equation*}

\begin{equation*} x _ { 3 } ^ { \prime } = p _ { 2 } q _ { 1 } , x _ { 4 } ^ { \prime } = p _ { 2 } q _ { 2 } \end{equation*}

where

\begin{equation*} p _ { 1 } = x _ { 1 } + x _ { 2 } , \quad p _ { 2 } = x _ { 3 } + x _ { 4 }, \end{equation*}

\begin{equation*} q _ { 1 } = x _ { 1 } + x _ { 3 } , \quad q _ { 2 } = x _ { 2 } + x _ { 4 }. \end{equation*}

For the regular case, the Bernstein problem has been solved in [a6], [a8], [a9]. This is precisely the case when the stationarity is based on a system of genes (see [a7]). The genes correspond to the extremal rays of the cone of non-negative invariant linear forms. After a normalization, these forms are just the probabilities of the genes.

A standard genetical interpretation also requires $V$ to be normal in the sense that:

1) all $x _ { i } ^ { \prime } \neq 0$;

2) for any pair $i$, $k$ the quadratic forms $x _ { i } ^ { \prime }$ and $x _ { k} ^ { \prime }$ are not proportional;

3) there is no pair $i$, $k$ such that all $x _ { j } ^ { \prime }$ are functions of $x _ { i } + x _ { k }$ and of the remaining $x _ { l }$. If $V$ is normal together with its restrictions to all invariant faces of $\Delta ^ { n - 1 }$, then $V$ is called ultranormal. All stationary ultranormal mappings are regular [a13].

A non-regular stochastic Bernstein mapping appears already for $n = 3$:

\begin{equation*} x _ { 1 } ^ { \prime } = x _ { 1 } ( s + v ), \end{equation*}

\begin{equation*} x _ { 2 } ^ { \prime } = x _ { 3 } ^ { \prime } = \frac { 1 } { 2 } [ ( x _ { 1 } + x _ { 2 } ) s - x _ { 1 } v ], \end{equation*}

where $s = x _ { 1 } + x _ { 2 } + x _ { 3 }$, $v = x_3 - x_2$. Here, all invariant linear forms are trivial, i.e. proportional to $s$.

For the non-regular case there are some partial results for the Bernstein problem. In particular, the results cover the low dimensions $n = 4,5,6$ (cf. [a3], [a4], [a10]).

In the course of these investigations, Bernstein algebras were introduced as a powerful tool (see [a6], [a8] or Bernstein algebra). The theory of Bernstein algebras was subsequently developed by itself.

In the algebraic context, the Bernstein problem is to explicitly describe those Bernstein algebras which are stochastic with respect to the basis $\{ e _ { i } \} _ { 1 } ^ { n }$. The latter means that the product of every pair of basis vectors belongs to $\Delta ^ { n - 1 }$.

The Bernstein algebra corresponding to a regular mapping $V$ is regular by definition. This class is the most important from the genetics point of view.

Another important tool in the study of the Bernstein problem is a topological structure on the set of essential faces of $\Delta ^ { n - 1 }$, the faces such that their intersections with the image of $V$ are non-empty [a11].

See [a12] for a systematic presentation of the results and methods regarding the Bernstein problem up to the middle of the 1980s.

References

[a1] S.N. Bernstein, "Mathematical problems in modern biology" Science in the Ukraine , 1 (1922) pp. 14–19 (In Russian)
[a2] S.N. Bernstein, "Solution of a mathematical problem related to the theory of inheritance" Uchen. Zap. Nauch. Issl. Kafedr. Ukrain. , 1 (1924) pp. 83–115 (In Russian)
[a3] S. Gonzáles, J.C. Gutiérrez, C. Martinez, "The Bernstein problem in dimension 5" J. Algebra , 177 (1995) pp. 676–697
[a4] J.C. Gutiérrez, "The Bernstein problem in dimension 6" J. Algebra , 185 (1996) pp. 420–439 Zbl 0863.17026
[a5] G.H. Hardy, "Mendelian proportions in a mixed population" Science , 28 : 706 (1908) pp. 49–50
[a6] Y.I. Lyubich, "Basic concepts and theorems of evolutionary genetics for free populations" Russian Math. Surveys , 26 : 5 (1971) pp. 51–123
[a7] Y.I. Lyubich, "Analogues to the Hardy–Weinberg Law" Genetics , 9 : 10 (1973) pp. 139–144 (In Russian)
[a8] Y.I. Lyubich, "Two-level Bernstein populations" Math. USSR Sb. , 24 : 1 (1974) pp. 593–615 Zbl 0317.92017
[a9] Y.I. Lyubich, "Proper Bernstein populations" Probl. Inform. Transmiss. , Jan. (1978) pp. 228–235 Zbl 0398.92025
[a10] Y.I. Lyubich, "Quasilinear Bernstein populations" Teor. Funct. Funct. Anal. Appl. , 26 (1976) pp. 79–84
[a11] Y.I. Lyubich, "A topological approach to a problem in mathematical genetics" Russian Math. Surveys , 34 : 6 (1979) pp. 60–66 MR562818 Zbl 0446.92012
[a12] Y.I. Lyubich, "Mathematical structures in population genetics" , Springer (1992) MR1224676 Zbl 0747.92019
[a13] Y.I. Lyubich, "A new advance in the Bernstein problem in mathematical genetics" Preprint Inst. Math. Sci., SUNY Stony Brook , 9 (1996) pp. 1–33
How to Cite This Entry:
Bernstein problem in mathematical genetics. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bernstein_problem_in_mathematical_genetics&oldid=24043
This article was adapted from an original article by Yu.I. Lyubich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article