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Uspekhi Mat. Nauk, 1971, Volume 26, Issue 5(161), Pages 51–116 (Mi umn5253)  

This article is cited in 45 scientific papers (total in 46 papers)

Basic concepts and theorems of the evolutionary genetics of free populations

Yu. I. Lyubich

Abstract: It is well known that the principles of biological inheritance, initiated by Mendel in 1865, allow of an exact mathematical formulation. For this reason classical genetics can be regarded as a mathematical discipline.
This article is concerned with the direction in mathematical genetics that stems from the widely known papers of Hardy and Weinberg (1908). It scarcely touches upon purely probabilistic and statistical questions, but uses probabilities (mean values of frequencies) as state coordinates in an “infinitely large” population. Change of state (evolution) occurs under the action of a certain quadratic operator. The paper has two aspects: 1) the structure of free populations; 2) the behaviour of trajectories. The fundamental investigations on these problems were carried out by S. N. Bernstein (1923–1924) and Reiersol (1962). Certain additional results directed towards completing the theory have been found recently by the author and are published here for the first time.
At the beginning of the paper we give a short sketch of the basic notions of classical genetics, in essence simply a minimal glossary. The reader who is familiar with the elements of genetics to the extent, for example, of the popular tract of Auerbach [1] or the appropriate chapters of the textbook by Villee [2], could omit this sketch. For a deeper study of the biological material the books of McKusick [3], Stern [4] and Mayr [5] are recommended.
The elementary mathematical questions of genetics are concerned with certain guiding principles in probability theory (see, for instance, [6]–[8]). The textbooks and monographs [9]–[15] are devoted to mathematical genetics. The sources listed here apply but little to the problems of the present work.
The main results are concentrated in §§ 4, 5, 9, 11. The remaining sections play an auxiliary role.

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English version:
Russian Mathematical Surveys, 1971, 26:5, 51–123

Bibliographic databases:

UDC: 519.9+575.1
MSC: 92D10, 92D15, 92D25, 47N60, 17D92
Received: 18.01.1971

Citation: Yu. I. Lyubich, “Basic concepts and theorems of the evolutionary genetics of free populations”, Uspekhi Mat. Nauk, 26:5(161) (1971), 51–116; Russian Math. Surveys, 26:5 (1971), 51–123

Citation in format AMSBIB
\by Yu.~I.~Lyubich
\paper Basic concepts and theorems of the evolutionary genetics of free populations
\jour Uspekhi Mat. Nauk
\yr 1971
\vol 26
\issue 5(161)
\pages 51--116
\jour Russian Math. Surveys
\yr 1971
\vol 26
\issue 5
\pages 51--123

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    This publication is cited in the following articles:
    1. Yu. I. Lyubich, “Stroenie bernshteinovskikh populyatsii tipa $(n-1,1)$”, UMN, 28:5(173) (1973), 247–248  mathnet  mathscinet  zmath
    2. Yu. I. Lyubich, “Two-level Bernsteinian populations”, Math. USSR-Sb., 24:4 (1974), 593–615  mathnet  crossref  mathscinet  zmath
    3. Yu. I. Lyubich, “Stroenie bernshteinovskikh populyatsii tipa $(2, n-2)$”, UMN, 30:1(181) (1975), 247–248  mathnet  mathscinet  zmath
    4. Ivar Heuch, “Genetic algebras for systems with linked loci”, Mathematical Biosciences, 34:1-2 (1977), 35  crossref
    5. JU. I. LjubiČ, “Algebraic Methods in Evolutionary Genetics”, Biom J, 20:5 (1978), 511  crossref  mathscinet
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    7. Yu. I. Lyubich, “A topological approach to a problem in mathematical genetics”, Russian Math. Surveys, 34:6 (1979), 60–66  mathnet  crossref  mathscinet  zmath
    8. S. M. Lozinskii, “On the hundredth anniversary of the birth of S. N. Bernstein”, Russian Math. Surveys, 38:3 (1983), 163–178  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    9. Corté Teresa, “Classification of 4-dimensional bernstein algebras∗”, Communications in Algebra, 19:5 (1991), 1429  crossref
    10. Irvin Roy Hentzel, Luiz Antonio Peresi, “Bernstein algebras given by symmetric bilinear forms”, Linear Algebra and its Applications, 145 (1991), 213  crossref
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    12. Teresa Cortés, “A note on the lattice definability of bernstein algebras”, Linear Algebra and its Applications, 179 (1993), 203  crossref
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    15. J. Carlos, Gutiérrez Fernández, “Structure of bernstein populations of type (3, n − 3)”, Linear Algebra and its Applications, 269:1-3 (1998), 17  crossref
    16. F. M. Mukhamedov, “On uniform ergodic theorems for quadratic processes on $C^*$-algebras”, Sb. Math., 191:12 (2000), 1891–1903  mathnet  crossref  crossref  mathscinet  zmath  isi
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    18. F. M. Mukhamedov, “On the Blum–Hanson theorem for quantum quadratic processes”, Math. Notes, 67:1 (2000), 81–86  mathnet  crossref  crossref  mathscinet  zmath  isi
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