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Haldane linearisation done right: Solving the nonlinear recombination equation the easy way
1. | Technische Fakultät, Universität Bielefeld, Postfach 100131, 33501 Bielefeld |
2. | Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld |
References:
[1] |
M. Aigner, Combinatorial Theory, reprint, Springer, Berlin, 1997.
doi: 10.1007/978-3-642-59101-3. |
[2] |
H. Amann, Gewöhnliche Differentialgleichungen, 2nd ed., de Gryuter, Berlin, 1995. |
[3] |
E. Baake, Deterministic and stochastic aspects of single-crossover recombination, in: Proceedings of the International Congress of Mathematicians, Hyderabad, India, 2010, Vol. VI, ed. J. Bhatia, Hindustan Book Agency, New Delhi (2010), 3037-3053; arXiv:1101.2081. |
[4] |
E. Baake, M. Baake and M. Salamat, The general recombination equation in continuous time and its solution, Discr. Cont. Dynam. Syst. A, 36 (2016), 63-95, arXiv:1409.1378.
doi: 10.3934/dcds.2016.36.63. |
[5] |
M. Baake, Recombination semigroups on measure spaces}, Monatsh. Math., 146 (2005), 267-278 and 150 (2007), 83-84 (Addendum); arXiv:math.CA/0506099.
doi: 10.1007/s00605-005-0326-z. |
[6] |
M. Baake and E. Baake, An exactly solved model for mutation, recombination and selection, Can. J. Math., 55 (2003), 3-41 and 60 (2008), 264-265 (Erratum); arXiv:math.CA/0210422.
doi: 10.4153/CJM-2003-001-0. |
[7] |
M. Baake and E. Shamsara, The recombination equation for interval partitions, preprint, arXiv:1508.04985. |
[8] |
J. H. Bennett, On the theory of random mating, Ann. Human Gen., 18 (1954), 311-317. |
[9] |
R. Bürger, The Mathematical Theory of Selection, Recombination and Mutation, Wiley, Chichester, 2000. |
[10] |
F. B. Christiansen, Population Genetics of Multiple Loci, Wiley, Chichester (1999). |
[11] |
K. J. Dawson, The decay of linkage disequilibrium under random union of gametes: How to calculate Bennett's principal components, Theor. Popul. Biol., 58 (2000), 1-20.
doi: 10.1006/tpbi.2000.1471. |
[12] |
K. J. Dawson, The evolution of a population under recombination: How to linearise the dynamics, Lin. Alg. Appl., 348 (2002), 115-137.
doi: 10.1016/S0024-3795(01)00586-9. |
[13] |
H. Geiringer, On the probability theory of linkage in Mendelian heredity, Ann. Math. Stat., 15 (1944), 25-57.
doi: 10.1214/aoms/1177731313. |
[14] |
H. S. Jennings, The numerical results of diverse systems of breeding, with respect to two pairs of characters, linked or independent, with special relation to the effects of linkage. Genetics, 2 (1917), 97-154. |
[15] |
Y. I. Lyubich, Mathematical Structures in Population Genetics, Springer, Berlin, 1992.
doi: 10.1007/978-3-642-76211-6. |
[16] |
S. Martínez, A probabilistic analysis of a discrete-time evolution in recombination, preprint, arXiv:1603.07201, and dto., part II: On partitions, preprint, arXiv:1604.05124. |
[17] |
D. McHale and G. A. Ringwood, Haldane linearisation of baric algebras, J. London Math. Soc., 28 (1983), 17-26.
doi: 10.1112/jlms/s2-28.1.17. |
[18] |
T. Nagylaki, J. Hofbauer and P. Brunovski, Convergence of multilocus systems under weak epistasis or weak selection, J. Math. Biol., 38 (1999), 103-133.
doi: 10.1007/s002850050143. |
[19] |
J. R. Norris, Markov Chains, Cambridge University Press, Cambridge, 1998. |
[20] |
R. B. Robbins, Some applications of mathematics to breeding problems III. Genetics, 3 (1918), 375-389. |
[21] |
U. von Wangenheim, E. Baake and M. Baake, Single-crossover recombination in discrete time, J. Math. Biol., 60 (2010), 727-760; arXiv:0906.1678.
doi: 10.1007/s00285-009-0277-4. |
show all references
References:
[1] |
M. Aigner, Combinatorial Theory, reprint, Springer, Berlin, 1997.
doi: 10.1007/978-3-642-59101-3. |
[2] |
H. Amann, Gewöhnliche Differentialgleichungen, 2nd ed., de Gryuter, Berlin, 1995. |
[3] |
E. Baake, Deterministic and stochastic aspects of single-crossover recombination, in: Proceedings of the International Congress of Mathematicians, Hyderabad, India, 2010, Vol. VI, ed. J. Bhatia, Hindustan Book Agency, New Delhi (2010), 3037-3053; arXiv:1101.2081. |
[4] |
E. Baake, M. Baake and M. Salamat, The general recombination equation in continuous time and its solution, Discr. Cont. Dynam. Syst. A, 36 (2016), 63-95, arXiv:1409.1378.
doi: 10.3934/dcds.2016.36.63. |
[5] |
M. Baake, Recombination semigroups on measure spaces}, Monatsh. Math., 146 (2005), 267-278 and 150 (2007), 83-84 (Addendum); arXiv:math.CA/0506099.
doi: 10.1007/s00605-005-0326-z. |
[6] |
M. Baake and E. Baake, An exactly solved model for mutation, recombination and selection, Can. J. Math., 55 (2003), 3-41 and 60 (2008), 264-265 (Erratum); arXiv:math.CA/0210422.
doi: 10.4153/CJM-2003-001-0. |
[7] |
M. Baake and E. Shamsara, The recombination equation for interval partitions, preprint, arXiv:1508.04985. |
[8] |
J. H. Bennett, On the theory of random mating, Ann. Human Gen., 18 (1954), 311-317. |
[9] |
R. Bürger, The Mathematical Theory of Selection, Recombination and Mutation, Wiley, Chichester, 2000. |
[10] |
F. B. Christiansen, Population Genetics of Multiple Loci, Wiley, Chichester (1999). |
[11] |
K. J. Dawson, The decay of linkage disequilibrium under random union of gametes: How to calculate Bennett's principal components, Theor. Popul. Biol., 58 (2000), 1-20.
doi: 10.1006/tpbi.2000.1471. |
[12] |
K. J. Dawson, The evolution of a population under recombination: How to linearise the dynamics, Lin. Alg. Appl., 348 (2002), 115-137.
doi: 10.1016/S0024-3795(01)00586-9. |
[13] |
H. Geiringer, On the probability theory of linkage in Mendelian heredity, Ann. Math. Stat., 15 (1944), 25-57.
doi: 10.1214/aoms/1177731313. |
[14] |
H. S. Jennings, The numerical results of diverse systems of breeding, with respect to two pairs of characters, linked or independent, with special relation to the effects of linkage. Genetics, 2 (1917), 97-154. |
[15] |
Y. I. Lyubich, Mathematical Structures in Population Genetics, Springer, Berlin, 1992.
doi: 10.1007/978-3-642-76211-6. |
[16] |
S. Martínez, A probabilistic analysis of a discrete-time evolution in recombination, preprint, arXiv:1603.07201, and dto., part II: On partitions, preprint, arXiv:1604.05124. |
[17] |
D. McHale and G. A. Ringwood, Haldane linearisation of baric algebras, J. London Math. Soc., 28 (1983), 17-26.
doi: 10.1112/jlms/s2-28.1.17. |
[18] |
T. Nagylaki, J. Hofbauer and P. Brunovski, Convergence of multilocus systems under weak epistasis or weak selection, J. Math. Biol., 38 (1999), 103-133.
doi: 10.1007/s002850050143. |
[19] |
J. R. Norris, Markov Chains, Cambridge University Press, Cambridge, 1998. |
[20] |
R. B. Robbins, Some applications of mathematics to breeding problems III. Genetics, 3 (1918), 375-389. |
[21] |
U. von Wangenheim, E. Baake and M. Baake, Single-crossover recombination in discrete time, J. Math. Biol., 60 (2010), 727-760; arXiv:0906.1678.
doi: 10.1007/s00285-009-0277-4. |
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