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On Bernstein algebras which are train algebras. (English) Zbl 0754.17029
The author establishes properties of Bernstein algebras which are train algebras; first, he proves properties of the rank equation; the main result is the following: Let $$\omega(x)$$ be the weight function, $$g_ 0(x)=x$$, $$g_{k+1}(x)={1\over k+1}\sum^ k_{j=0}g_ j(x)g_{k- j}(x)$$ for all $$x\in A$$ and all $$k\geq 0$$. If the Bernstein algebra $$A$$ satisfies a train equation of minimal degree $$r\geq 3$$, then, for all $$x\in A$$ and all $$k\geq r-1$$: $$g_ k(x)-\omega(x)g_{k-1}(x)=0$$.
Then he gives an application to overlapping generations: the behaviour of the population is governed by the differential equation: $$y'=y^ 2-y$$ on the invariant hyperplane $$H$$. The explicit solution for a Bernstein train algebra is given.

##### MSC:
 17D92 Genetic algebras 92D10 Genetics and epigenetics
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##### References:
 [1] DOI: 10.1112/jlms/s2-9.4.613 · Zbl 0365.92025 [2] DOI: 10.1016/0040-5809(73)90024-5 · Zbl 0265.92005 [3] DOI: 10.1214/aoms/1177731642 · Zbl 0063.00333 [4] DOI: 10.1007/BF02414042 · Zbl 0097.14402 [5] DOI: 10.1070/RM1971v026n05ABEH003829 [6] Wörz-Busekros, Algebras in genetics 36 (1980) [7] Walcher, Algebras and differential equations · Zbl 0754.17002 [8] DOI: 10.2307/2372100 · Zbl 0034.02004 [9] DOI: 10.1016/0024-3795(91)90092-B · Zbl 0728.17020
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