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Local existence with mild regularity for the Boltzmann equation
A kinetic description of mutation processes in bacteria
1. | University of Pavia, Department of Mathematics, Via Ferrata 1, 27100 Pavia |
References:
[1] |
W. P. Angerer, An explicit representation of the Luria-Delbrück distribution, J. Math. Biol., 42 (2001), 145-174.
doi: 10.1007/s002850000053. |
[2] |
P. Armitage, The statistical theory of bacterial populations subject to mutation, J. Royal Statist. Soc. B, 14 (1952), 1-40. |
[3] |
M. S. Bartlett, An Introduction to Stochastic Processes With Special Reference to Methods and Applications, Second edition Cambridge University Press, London-New York, 1966. |
[4] |
A. V. Bobylev, The theory of the spatially Uniform Boltzmann equation for Maxwell molecules, Sov. Sci. Review C, 7 (1988), 111-233. |
[5] |
C. Cercignani, The Boltzmann Equation and its Applications, Applied Mathematical Sciences, 67. Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4612-1039-9. |
[6] |
A. Chakraborti, Distributions of money in models of market economy, Int. J. Modern Phys. C, 13 (2002), 1315-1321. |
[7] |
A. Chakraborti and B. K. Chakrabarti, Statistical mechanics of money: Effects of saving propensity, Eur. Phys. J. B, 17 (2000), 167-170. |
[8] |
A. Chatterjee, B. K. Chakrabarti and R. B. Stinchcombe, Master equation for a kinetic model of trading market and its analytic solution, Phys. Rev. E, 72 (2005), 026126. |
[9] |
S. Cordier, L. Pareschi and G. Toscani, On a kinetic model for a simple market economy, J. Stat. Phys., 120 (2005), 253-277.
doi: 10.1007/s10955-005-5456-0. |
[10] |
K. S. Crump and D. G. Hoel, Mathematical models for estimating mutation rates in cell populations, Biometrika, 61 (1974), 237-252. |
[11] |
A. Drăgulescu and V. M. Yakovenko, Statistical mechanics of money, Eur. Phys. Jour. B, 17 (2000), 723-729. |
[12] |
B. Düring, D. Matthes and G. Toscani, Kinetic equations modelling wealth redistribution: A comparison of approaches, Phys. Rev. E, 78 (2008), 056103, 12 pp.
doi: 10.1103/PhysRevE.78.056103. |
[13] |
B. Düring, D. Matthes and G. Toscani, A Boltzmann type approach to the formation of wealth distribution curves, Riv. Mat. Univ. Parma, 1 (2009), 199-261. |
[14] |
E. Gabetta, G. Toscani and B. Wennberg, Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation, J. Statist. Phys., 81 (1995), 901-934.
doi: 10.1007/BF02179298. |
[15] |
B. Hayes, Follow the money, American Scientist, 90 (2002), 400-405. |
[16] |
J. R. Iglesias, S. Gonçalves, S. Pianegonda, J. L. Vega and G. Abramson, Wealth redistribution in our small world, Nonequilibrium statistical mechanics and nonlinear physics (MEDYFINOL '02) (Colonia del Sacramento), Physica A, 327 (2003), 12-17.
doi: 10.1016/S0378-4371(03)00430-8. |
[17] |
M. E. Jones, S. M. Thomas and A. Rogers, Luria-Delbrück fluctuation experiments: Design and analysis, Genetics, 136 (1994), 1209-1216. |
[18] |
E. Kashdan and L. Pareschi, Mean field dynamics and the continuous Luria-Delbrück distribution, Mathematical Biosciences, 240 (2012), 223-230.
doi: 10.1016/j.mbs.2012.08.001. |
[19] |
D. G. Kendall, Stochastic processes and population growth, Journal of the Royal Statistical Society, B, 11 (1949), 230-264. |
[20] |
A. L. Koch, Mutation and growth rates from Luria-Delbrück fluctuation tests, Mutat. Res., 95 (1982), 129-143. |
[21] |
D. E. Lea and C. A. Coulson, The distribution of the numbers of mutants in bacterial populations, J. Genetics, 49 (1949), 264-285. |
[22] |
S. E. Luria and M. Delbrück, Mutations of bacteria from virus sensitivity to virus resistance, Genetics, 28 (1943), 491-511. |
[23] |
W. T. Ma, G. v. H. Sandri and S. Sarkar, Analysis of the Luria and Delbrück distribution using discrete convolution powers, J. Appl. Prob., 29 (1992), 255-267. |
[24] |
B. Mandelbrot, A population birth-and-mutation process, I: Explicit distributions for the number of mutants in an old culture of bacteria, J. Appl. Prob., 11 (1974), 437-444. |
[25] |
D. Matthes and G. Toscani, On steady distributions of kinetic models of conservative economies, J. Stat. Phys., 130 (2008), 1087-1117.
doi: 10.1007/s10955-007-9462-2. |
[26] |
A. G. Pakes, Remarks on the Luria-Delbrück distribution, J. Appl. Prob., 30 (1993), 991-994. |
[27] |
L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic equations & Monte Carlo Methods, Oxford University Press, Oxford 2013. |
[28] |
F. Slanina, Inelastically scattering particles and wealth distribution in an open economy, Phys. Rev. E, 69 (2004), 046102. |
[29] |
F. M. Stewart, D. M. Gordon and B. R. Levin, Fluctuation analysis: The probability distribution of the number of mutants under different conditions, Genetics, 124 (1990), 175-185. |
[30] |
G. Toscani, Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), 481-496. |
[31] |
Q. Zheng, Progress of a half century in the study of the Luria-Delbrück distribution, Math. Biosciences, 162 (1999), 1-32.
doi: 10.1016/S0025-5564(99)00045-0. |
show all references
References:
[1] |
W. P. Angerer, An explicit representation of the Luria-Delbrück distribution, J. Math. Biol., 42 (2001), 145-174.
doi: 10.1007/s002850000053. |
[2] |
P. Armitage, The statistical theory of bacterial populations subject to mutation, J. Royal Statist. Soc. B, 14 (1952), 1-40. |
[3] |
M. S. Bartlett, An Introduction to Stochastic Processes With Special Reference to Methods and Applications, Second edition Cambridge University Press, London-New York, 1966. |
[4] |
A. V. Bobylev, The theory of the spatially Uniform Boltzmann equation for Maxwell molecules, Sov. Sci. Review C, 7 (1988), 111-233. |
[5] |
C. Cercignani, The Boltzmann Equation and its Applications, Applied Mathematical Sciences, 67. Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4612-1039-9. |
[6] |
A. Chakraborti, Distributions of money in models of market economy, Int. J. Modern Phys. C, 13 (2002), 1315-1321. |
[7] |
A. Chakraborti and B. K. Chakrabarti, Statistical mechanics of money: Effects of saving propensity, Eur. Phys. J. B, 17 (2000), 167-170. |
[8] |
A. Chatterjee, B. K. Chakrabarti and R. B. Stinchcombe, Master equation for a kinetic model of trading market and its analytic solution, Phys. Rev. E, 72 (2005), 026126. |
[9] |
S. Cordier, L. Pareschi and G. Toscani, On a kinetic model for a simple market economy, J. Stat. Phys., 120 (2005), 253-277.
doi: 10.1007/s10955-005-5456-0. |
[10] |
K. S. Crump and D. G. Hoel, Mathematical models for estimating mutation rates in cell populations, Biometrika, 61 (1974), 237-252. |
[11] |
A. Drăgulescu and V. M. Yakovenko, Statistical mechanics of money, Eur. Phys. Jour. B, 17 (2000), 723-729. |
[12] |
B. Düring, D. Matthes and G. Toscani, Kinetic equations modelling wealth redistribution: A comparison of approaches, Phys. Rev. E, 78 (2008), 056103, 12 pp.
doi: 10.1103/PhysRevE.78.056103. |
[13] |
B. Düring, D. Matthes and G. Toscani, A Boltzmann type approach to the formation of wealth distribution curves, Riv. Mat. Univ. Parma, 1 (2009), 199-261. |
[14] |
E. Gabetta, G. Toscani and B. Wennberg, Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation, J. Statist. Phys., 81 (1995), 901-934.
doi: 10.1007/BF02179298. |
[15] |
B. Hayes, Follow the money, American Scientist, 90 (2002), 400-405. |
[16] |
J. R. Iglesias, S. Gonçalves, S. Pianegonda, J. L. Vega and G. Abramson, Wealth redistribution in our small world, Nonequilibrium statistical mechanics and nonlinear physics (MEDYFINOL '02) (Colonia del Sacramento), Physica A, 327 (2003), 12-17.
doi: 10.1016/S0378-4371(03)00430-8. |
[17] |
M. E. Jones, S. M. Thomas and A. Rogers, Luria-Delbrück fluctuation experiments: Design and analysis, Genetics, 136 (1994), 1209-1216. |
[18] |
E. Kashdan and L. Pareschi, Mean field dynamics and the continuous Luria-Delbrück distribution, Mathematical Biosciences, 240 (2012), 223-230.
doi: 10.1016/j.mbs.2012.08.001. |
[19] |
D. G. Kendall, Stochastic processes and population growth, Journal of the Royal Statistical Society, B, 11 (1949), 230-264. |
[20] |
A. L. Koch, Mutation and growth rates from Luria-Delbrück fluctuation tests, Mutat. Res., 95 (1982), 129-143. |
[21] |
D. E. Lea and C. A. Coulson, The distribution of the numbers of mutants in bacterial populations, J. Genetics, 49 (1949), 264-285. |
[22] |
S. E. Luria and M. Delbrück, Mutations of bacteria from virus sensitivity to virus resistance, Genetics, 28 (1943), 491-511. |
[23] |
W. T. Ma, G. v. H. Sandri and S. Sarkar, Analysis of the Luria and Delbrück distribution using discrete convolution powers, J. Appl. Prob., 29 (1992), 255-267. |
[24] |
B. Mandelbrot, A population birth-and-mutation process, I: Explicit distributions for the number of mutants in an old culture of bacteria, J. Appl. Prob., 11 (1974), 437-444. |
[25] |
D. Matthes and G. Toscani, On steady distributions of kinetic models of conservative economies, J. Stat. Phys., 130 (2008), 1087-1117.
doi: 10.1007/s10955-007-9462-2. |
[26] |
A. G. Pakes, Remarks on the Luria-Delbrück distribution, J. Appl. Prob., 30 (1993), 991-994. |
[27] |
L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic equations & Monte Carlo Methods, Oxford University Press, Oxford 2013. |
[28] |
F. Slanina, Inelastically scattering particles and wealth distribution in an open economy, Phys. Rev. E, 69 (2004), 046102. |
[29] |
F. M. Stewart, D. M. Gordon and B. R. Levin, Fluctuation analysis: The probability distribution of the number of mutants under different conditions, Genetics, 124 (1990), 175-185. |
[30] |
G. Toscani, Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), 481-496. |
[31] |
Q. Zheng, Progress of a half century in the study of the Luria-Delbrück distribution, Math. Biosciences, 162 (1999), 1-32.
doi: 10.1016/S0025-5564(99)00045-0. |
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