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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.

17D92 Genetic algebras
92D10 Genetics and epigenetics
Full Text: DOI
[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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