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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
\Bibitem{Lyu71}
\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
\mathnet{http://mi.mathnet.ru/umn5253}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=446581}
\zmath{https://zbmath.org/?q=an:0276.92021}
\transl
\jour Russian Math. Surveys
\yr 1971
\vol 26
\issue 5
\pages 51--123
\crossref{https://doi.org/10.1070/RM1971v026n05ABEH003829}


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    Erratum

    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
    6. M. I. Zakharevich, “On the behaviour of trajectories and the ergodic hypothesis for quadratic mappings of a simplex”, Russian Math. Surveys, 33:6 (1978), 265–266  mathnet  crossref  mathscinet  zmath
    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
    11. Sebastian Walcher, “On Bernstein algebras which are train algebras”, Proc Edin Math Soc, 35:1 (1992), 159  crossref  mathscinet  zmath  isi
    12. Teresa Cortés, “A note on the lattice definability of bernstein algebras”, Linear Algebra and its Applications, 179 (1993), 203  crossref
    13. S. González, J.C. Gutiérrez, C. Martínez, “On regular bernstein algebras”, Linear Algebra and its Applications, 241-243 (1996), 389  crossref
    14. Yu. I. Lyubich, “Ultranormal Case of the Bernstein Problem”, Funct. Anal. Appl., 31:1 (1997), 60–62  mathnet  crossref  crossref  mathscinet  zmath  isi
    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
    17. N. N. Ganikhodzhaev, F. M. Mukhamedov, “Ergodic properties of discrete quadratic stochastic processes defined on von Neumann algebras”, Izv. Math., 64:5 (2000), 873–890  mathnet  crossref  crossref  mathscinet  zmath  isi
    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
    19. F. M. Mukhamedov, “Infinite-dimensional quadratic Volterra operators”, Russian Math. Surveys, 55:6 (2000), 1161–1162  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    20. Mukhamedov, FM, “On decomposition quantum quadratic stochastic processes”, Doklady Akademii Nauk, 371:2 (2000), 167  mathnet  mathscinet  zmath  isi
    21. Kevin J. Dawson, “The Decay of Linkage Disequilibrium under Random Union of Gametes: How to Calculate Bennett's Principal Components”, Theoretical Population Biology, 58:1 (2000), 1  crossref
    22. J.Carlos Gutiérrez Fernández, “Solution of the Bernstein Problem in the Non-regular Case”, Journal of Algebra, 223:1 (2000), 109  crossref
    23. Yuri Lyubich, Valery Kirzhner, Anna Ryndin, “Mathematical Theory of Phenotypical Selection”, Advances in Applied Mathematics, 26:4 (2001), 330  crossref
    24. Kevin J. Dawson, “The evolution of a population under recombination: how to linearise the dynamics”, Linear Algebra and its Applications, 348:1-3 (2002), 115  crossref
    25. F. M. Mukhamedov, “On expansion of quantum quadratic stochastic processes into fibrewise Markov processes defined on von Neumann algebras”, Izv. Math., 68:5 (2004), 1009–1024  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    26. Mukhamedov, F, “On infinite dimensional quadratic Volterra operators”, Journal of Mathematical Analysis and Applications, 310:2 (2005), 533  mathscinet  zmath  adsnasa  isi
    27. Farruh Mukhamedov, Hasan Akin, Seyit Temir, “On infinite dimensional quadratic Volterra operators”, Journal of Mathematical Analysis and Applications, 310:2 (2005), 533  crossref
    28. Nadia Boudi, Fouad Zitan, “On Bernstein Algebras Satisfying Chain Conditions”, Comm. in Algebra, 35:8 (2007), 2568  crossref
    29. Nadia Boudi, Fouad Zitan, “On Bernstein Algebras Satisfying Chain Conditions”, Comm. in Algebra, 35:7 (2007), 2116  crossref
    30. Reinhard Bürger, “Multilocus selection in subdivided populations I. Convergence properties for weak or strong migration”, J Math Biol, 2008  crossref  isi
    31. Murray R. Bremner, Yunfeng Piao, Sheldon W. Richards, “Polynomial Identities for Bernstein Algebras of Simple Mendelian Inheritance”, Communications in Algebra, 37:10 (2009), 3438  crossref  elib
    32. U. A. Rozikov, N. B. Shamsiddinov, “On Non-Volterra Quadratic Stochastic Operators Generated by a Product Measure”, Stochastic Analysis and Applications, 27:2 (2009), 353  crossref
    33. N. Ganikhodja, J.I. Daoud, M. Usmanova, “Linear and Nonlinear Models of Heredity for Blood Groups and Rhesus Factor”, J Applied Sci, 10:16 (2010), 1748  crossref
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    35. Ganikhodzhaev R., Mukhamedov F., Rozikov U., “Quadratic Stochastic Operators and Processes: Results and Open Problems”, Infin Dimens Anal Quantum Probab Relat Top, 14:2 (2011), 279–335  crossref  isi
    36. N.N. Ganikhodjaev, U.U. Jamilov, R.T. Mukhitdinov, “On Non-Ergodic Transformations onS3”, J. Phys.: Conf. Ser, 435 (2013), 012005  crossref
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    38. Reinhard Bürger, “A survey of migration-selection models in population genetics”, DCDS-B, 19:4 (2014), 883  crossref
    39. N.N.. GANIKHODJAEV, R.N.. GANIKHODJAEV, U. U. JAMILOV, “Quadratic stochastic operators and zero-sum game dynamics”, Ergod. Th. Dynam. Sys, 2014, 1  crossref
    40. Ganikhodjaev N., Saburov M., Nawi A.M., “Mutation and Chaos in Nonlinear Models of Heredity”, Sci. World J., 2014, 835069  crossref  isi
    41. Uygun Jamilov, Manuel Ladra, “Non-Ergodicity of Uniform Quadratic Stochastic Operators”, Qual. Theory Dyn. Syst, 2015  crossref
    42. Ganikhodjaev N. Hamzah Nur Zatul Akmar, “on Gaussian Nonlinear Transformations”, 22Nd National Symposium on Mathematical Sciences (Sksm22), AIP Conference Proceedings, 1682, ed. Mohamed I. How L. Mui A. Bin W., Amer Inst Physics, 2015, 040009  crossref  isi
    43. Ganikhodjaev N. Hamzah Nur Zatul Akmar, “on Volterra Quadratic Stochastic Operators With Continual State Space”, International Conference on Mathematics, Engineering and Industrial Applications 2014 (Icomeia 2014), AIP Conference Proceedings, 1660, ed. Ramli M. Junoh A. Roslan N. Masnan M. Kharuddin M., Amer Inst Physics, 2015, 050025  crossref  isi
    44. Pirogov S., Rybko A., Kalinina A., Gelfand M., “Recombination Processes and Nonlinear Markov Chains”, J. Comput. Biol., 23:9 (2016), 711–717  crossref  mathscinet  isi  elib  scopus
    45. Ganikhodjaev N., Hamzah Nur Zatul Akmar, “On (3,3)-Gaussian Quadratic Stochastic Operators”, 37Th International Conference on Quantum Probability and Related Topics (Qp37), Journal of Physics Conference Series, 819, eds. Accardi L., Mukhamedov F., Hee P., IOP Publishing Ltd, 2017, UNSP 012007  crossref  isi
    46. Hamzah Nur Zatul Akmar Ganikhodjaev N., “On Non-Ergodic Gaussian Quadratic Stochastic Operators”, AIP Conference Proceedings, 1974, ed. Mohamad D. Akbarally A. Maidinsah H. Jaffar M. Mohamed M. Sharif S. Rahman W., Amer Inst Physics, 2018, UNSP 030021  crossref  mathscinet  isi  scopus
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