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''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 [[#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

\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 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 $V$, 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]]]):

\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 [[#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 $V$ 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 [[#References|[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. [[#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 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 [[#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.

====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>