RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Subscription License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Uspekhi Mat. Nauk: Year: Volume: Issue: Page: Find

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

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.

Full text: PDF file (6353 kB)
References: PDF file   HTML file

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

Bibliographic databases:

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

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} 

• http://mi.mathnet.ru/eng/umn5253
• http://mi.mathnet.ru/eng/umn/v26/i5/p51

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles
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
2. Yu. I. Lyubich, “Two-level Bernsteinian populations”, Math. USSR-Sb., 24:4 (1974), 593–615
3. Yu. I. Lyubich, “Stroenie bernshteinovskikh populyatsii tipa $(2, n-2)$”, UMN, 30:1(181) (1975), 247–248
4. Ivar Heuch, “Genetic algebras for systems with linked loci”, Mathematical Biosciences, 34:1-2 (1977), 35
5. JU. I. LjubiČ, “Algebraic Methods in Evolutionary Genetics”, Biom J, 20:5 (1978), 511
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
7. Yu. I. Lyubich, “A topological approach to a problem in mathematical genetics”, Russian Math. Surveys, 34:6 (1979), 60–66
8. S. M. Lozinskii, “On the hundredth anniversary of the birth of S. N. Bernstein”, Russian Math. Surveys, 38:3 (1983), 163–178
9. Corté Teresa, “Classification of 4-dimensional bernstein algebras∗”, Communications in Algebra, 19:5 (1991), 1429
10. Irvin Roy Hentzel, Luiz Antonio Peresi, “Bernstein algebras given by symmetric bilinear forms”, Linear Algebra and its Applications, 145 (1991), 213
11. Sebastian Walcher, “On Bernstein algebras which are train algebras”, Proc Edin Math Soc, 35:1 (1992), 159
12. Teresa Cortés, “A note on the lattice definability of bernstein algebras”, Linear Algebra and its Applications, 179 (1993), 203
13. S. González, J.C. Gutiérrez, C. Martínez, “On regular bernstein algebras”, Linear Algebra and its Applications, 241-243 (1996), 389
14. Yu. I. Lyubich, “Ultranormal Case of the Bernstein Problem”, Funct. Anal. Appl., 31:1 (1997), 60–62
15. J. Carlos, Gutiérrez Fernández, “Structure of bernstein populations of type (3, n − 3)”, Linear Algebra and its Applications, 269:1-3 (1998), 17
16. F. M. Mukhamedov, “On uniform ergodic theorems for quadratic processes on $C^*$-algebras”, Sb. Math., 191:12 (2000), 1891–1903
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
18. F. M. Mukhamedov, “On the Blum–Hanson theorem for quantum quadratic processes”, Math. Notes, 67:1 (2000), 81–86
19. F. M. Mukhamedov, “Infinite-dimensional quadratic Volterra operators”, Russian Math. Surveys, 55:6 (2000), 1161–1162
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
22. J.Carlos Gutiérrez Fernández, “Solution of the Bernstein Problem in the Non-regular Case”, Journal of Algebra, 223:1 (2000), 109
23. Yuri Lyubich, Valery Kirzhner, Anna Ryndin, “Mathematical Theory of Phenotypical Selection”, Advances in Applied Mathematics, 26:4 (2001), 330
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
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
26. Mukhamedov, F, “On infinite dimensional quadratic Volterra operators”, Journal of Mathematical Analysis and Applications, 310:2 (2005), 533
27. Farruh Mukhamedov, Hasan Akin, Seyit Temir, “On infinite dimensional quadratic Volterra operators”, Journal of Mathematical Analysis and Applications, 310:2 (2005), 533
28. Nadia Boudi, Fouad Zitan, “On Bernstein Algebras Satisfying Chain Conditions”, Comm. in Algebra, 35:8 (2007), 2568
29. Nadia Boudi, Fouad Zitan, “On Bernstein Algebras Satisfying Chain Conditions”, Comm. in Algebra, 35:7 (2007), 2116
30. Reinhard Bürger, “Multilocus selection in subdivided populations I. Convergence properties for weak or strong migration”, J Math Biol, 2008
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
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
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
34. Farrukh Mukhamedov, Hasan Ak{\i}n, Seyit Temir, Abduaziz Abduganiev, “On quantum quadratic operators of and their dynamics”, Journal of Mathematical Analysis and Applications, 376:2 (2011), 641
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
36. N.N. Ganikhodjaev, U.U. Jamilov, R.T. Mukhitdinov, “On Non-Ergodic Transformations onS3”, J. Phys.: Conf. Ser, 435 (2013), 012005
37. Farrukh Mukhamedov, Abduaziz Abduganiev, “On Pure Quasi-Quantum Quadratic Operators of �2(ℂ)”, Open Syst. Inf. Dyn, 20:04 (2013), 1350018
38. Reinhard Bürger, “A survey of migration-selection models in population genetics”, DCDS-B, 19:4 (2014), 883
39. N.N.. GANIKHODJAEV, R.N.. GANIKHODJAEV, U. U. JAMILOV, “Quadratic stochastic operators and zero-sum game dynamics”, Ergod. Th. Dynam. Sys, 2014, 1
40. Ganikhodjaev N., Saburov M., Nawi A.M., “Mutation and Chaos in Nonlinear Models of Heredity”, Sci. World J., 2014, 835069
41. Uygun Jamilov, Manuel Ladra, “Non-Ergodicity of Uniform Quadratic Stochastic Operators”, Qual. Theory Dyn. Syst, 2015
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
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
44. Pirogov S., Rybko A., Kalinina A., Gelfand M., “Recombination Processes and Nonlinear Markov Chains”, J. Comput. Biol., 23:9 (2016), 711–717
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
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
•  Number of views: This page: 1207 Full text: 326 References: 55 First page: 1