Exploring the world with mathematics
Abstract
This is an account of my scientific and personal friendship with Prof. Andrzej (Andy) Aleksander Lasota from 1977 until his death 28 December, 2006. It is a tale that fascinates me because of the intertwined links between many people both East and West of several generations, and it illustrates what I feel is the strength and beauty of the personal side of the scientific endeavor.
This contribution is almost identical to the paper “Adventures in Poland: Having fun and doing research with Andrzej Lasota”, Matematyka Stosowana 8 (2007), 5–32. It is in no way to be considered a new contribution, but is rather a record of the second Annual Lecture Commemorating Professor Andrzej Lasota given in Katowice at Uniwersytet Slaski on 16 January, 2009.
References
1. Bohm D., A suggested interpretation of quantum mechanics in terms of “hidden” variables: I, Phys. Rev. 85 (1952), 166–179.
2. Chow S.N., Existence of periodic solutions of autonomous functional differential equations, J. Differential Equations 15 (1974), 350–378.
3. Colijn C., Mackey M.C., A mathematical model of hematopoiesis. I. Periodic chronic myelogenous leukemia, J. Theoret. Biol. 237 (2005), 117–132.
4. Komorník J., Lasota A., Asymptotic decomposition of Markov operators, Bull. Polish Acad. Sci. Math. 35 (1987), 321–327.
5. Lasota A., Ergodic problems in biology, in: Dynamical systems, Vol. 2—Warsaw, Astérisque, No. 50, Soc. Math. France, Paris, 1977, pp. 239–250.
6. Lasota A., Determinism, indeterminism, and mathematics, Found. Sci. 2 (1997), 73–75.
7. Lasota A., Li T.Y., Yorke J.A., Asymptotic periodicity of the iterates of Markov operators, Trans. Amer. Math. Soc. 286 (1984), 751–764.
8. Lasota A., Łoskot K., Mackey M.C., The dynamics of proliferatively coupled cell populations, Acta Biotheor. 39 (1991), 1–14.
9. Lasota A., Mackey M.C., The extinction of slowly evolving dynamical systems, J. Math. Biol. 10 (1980), 333–345.
10. Lasota A., Mackey M.C., Globally asymptotic properties of proliferating cell populations, J. Math. Biol. 19 (1984), 43–62.
11. Lasota A., Mackey M.C., Probabilistic Properties of Deterministic Systems, Cambridge University Press, New York–Cambridge, 1985.
12. Lasota A., Mackey M.C., Noise and statistical periodicity, Phys. D 28 (1987), 143–154.
13. Lasota A., Mackey M.C., Stochastic perturbation of dynamical systems: the weak convergence of measures, J. Math. Anal. Appl. 138 (1989), 232–248.
14. Lasota A., Mackey M.C., Chaos, fractals, and noise, Applied Mathematical Sciences, vol. 97, Springer–Verlag, New York, 1994.
15. Lasota A., Mackey M.C., Cell division and the stability of cellular populations, J. Math. Biol. 38 (1999), 241–261.
16. Lasota A., Mackey M.C., Statistical stability of strongly perturbed dynamical systems, in: Differential equations with applications to biology (Halifax, NS, 1997), Fields Inst. Commun., Vol. 21, Amer. Math. Soc., Providence, RI, 1999, pp. 363–376.
17. Lasota A., Mackey M.C., Tyrcha J., The statistical dynamics of recurrent biological events, J. Math. Biol. 30 (1992), 775–800.
18. Lasota A., Mackey M.C., Ważewska-Czyżewska M., Minimizing therapeutically induced anemia, J. Math. Biol. 13 (1981/82), 149–158.
19. Lasota A., Traple J., Differential equations with dynamical perturbations, J. Differential Equations 63 (1986), 406–417.
20. Losson J., Mackey M.C., Coupled map lattices as models of deterministic and stochastic differential delay equations, Phys. Rev. E 52 (1995), 115–128.
21. Mackey M.C., Time’s Arrow: The Origins of Thermodynamic Behaviour, Springer–Verlag, Berlin–New York–Heidelberg, 1992.
22. Mackey M.C., Glass L., Oscillation and chaos in physiological control systems, Science 197 (1977), 287–289.
23. Mackey M.C., Longtin A., Lasota A., Noise-induced global asymptotic stability, J. Stat. Phys. 60 (1990), 735–751.
24. Mackey M.C., Milton J.G., A deterministic approach to survival statistics, J. Math. Biol. 28 (1990), 33–48.
25. Mackey M.C., Rudnicki R., Global stability in a delayed partial differential equation describing cellular replication, J. Math. Biol. 33 (1994), 89–109.
26. Mackey M.C., Rudnicki R., A new criterion for the global stability of simultaneous cell replication and maturation processes, J. Math. Biol. 38 (1999), 195–219.
27. Mackey M.C., Santavy M., Selepova P., A mitotic oscillator with a strange attractor and distributions of cell cycle times, in: Nonlinear oscillations in biology and chemistry (Salt Lake City, Utah, 1985), Lecture Notes in Biomath., Vol. 66, Springer, Berlin, 1986, pp. 34–45.
28. Mackey M.C., Tyran-Kamińska M., Deterministic Brownian motion: The effects of perturbing a dynamical system by a chaotic semi-dynamical system, Phys. Rep. 422 (2006), 167–222.
29. Mackey M.C., Tyran-Kamińska M., Noise and conditional entropy evolution, Phys. A 365 (2006), 360–382.
30. Mackey M.C., Tyran-Kamińska M., Temporal behavior of the conditional and Gibbs’ entropies, J. Stat. Phys. 124 (2006), 1443–1470.
31. Mackey M.C., Tyran-Kamińska M., Central limit theorem behavior in the skew tent map, Chaos Solitons Fractals 38 (2008), 789–805.
32. Mackey M.C., Tyran-Kamińska M., Central limit theorems for non-invertible measure preserving maps, Colloq. Math. 110 (2008), 167–191.
33. Mackey M.C., Tyran-Kamińska M., Dynamics and density evolution in piecewise deterministic growth processes, Ann. Polon. Math. 94 (2008), 111–129.
34. Murray J.D., Wprowadzenie do biomatematyki (translated by Urszula Foryś and Marek Bodnar), Wydawnictwo Naukowe PWN, Warszawa, 2007.
35. Provatas N., Mackey M.C., Asymptotic periodicity and banded chaos, Phys. D 53 (1991), 295–318.
36. Provatas N., Mackey M.C., Noise-induced asymptotic periodicity in a piecewise linear map, J. Statist. Phys. 63 (1991), 585–612.
37. Rudnicki R., Mackey M.C., Asymptotic similarity and Malthusian growth in autonomous and nonautonomous populations, J. Math. Anal. Appl. 187 (1994), 548–566.
38. Szarski J., Differential Inequalities, Monografie Matematyczne, T. 43, Państwowe Wydawnictwo Naukowe, Warsaw, 1965.
39. Urbaniec J., An interview with Andrzej Lasota: Without the physical world, there would be no mathematics either, Found. Sci. 2 (2002), 183–189.
40. Ważewska-Czyżewska M., Lasota A., Mathematical problems of the dynamics of a system of red blood cells, Mat. Stos. 6 (1976), 23–40.
2. Chow S.N., Existence of periodic solutions of autonomous functional differential equations, J. Differential Equations 15 (1974), 350–378.
3. Colijn C., Mackey M.C., A mathematical model of hematopoiesis. I. Periodic chronic myelogenous leukemia, J. Theoret. Biol. 237 (2005), 117–132.
4. Komorník J., Lasota A., Asymptotic decomposition of Markov operators, Bull. Polish Acad. Sci. Math. 35 (1987), 321–327.
5. Lasota A., Ergodic problems in biology, in: Dynamical systems, Vol. 2—Warsaw, Astérisque, No. 50, Soc. Math. France, Paris, 1977, pp. 239–250.
6. Lasota A., Determinism, indeterminism, and mathematics, Found. Sci. 2 (1997), 73–75.
7. Lasota A., Li T.Y., Yorke J.A., Asymptotic periodicity of the iterates of Markov operators, Trans. Amer. Math. Soc. 286 (1984), 751–764.
8. Lasota A., Łoskot K., Mackey M.C., The dynamics of proliferatively coupled cell populations, Acta Biotheor. 39 (1991), 1–14.
9. Lasota A., Mackey M.C., The extinction of slowly evolving dynamical systems, J. Math. Biol. 10 (1980), 333–345.
10. Lasota A., Mackey M.C., Globally asymptotic properties of proliferating cell populations, J. Math. Biol. 19 (1984), 43–62.
11. Lasota A., Mackey M.C., Probabilistic Properties of Deterministic Systems, Cambridge University Press, New York–Cambridge, 1985.
12. Lasota A., Mackey M.C., Noise and statistical periodicity, Phys. D 28 (1987), 143–154.
13. Lasota A., Mackey M.C., Stochastic perturbation of dynamical systems: the weak convergence of measures, J. Math. Anal. Appl. 138 (1989), 232–248.
14. Lasota A., Mackey M.C., Chaos, fractals, and noise, Applied Mathematical Sciences, vol. 97, Springer–Verlag, New York, 1994.
15. Lasota A., Mackey M.C., Cell division and the stability of cellular populations, J. Math. Biol. 38 (1999), 241–261.
16. Lasota A., Mackey M.C., Statistical stability of strongly perturbed dynamical systems, in: Differential equations with applications to biology (Halifax, NS, 1997), Fields Inst. Commun., Vol. 21, Amer. Math. Soc., Providence, RI, 1999, pp. 363–376.
17. Lasota A., Mackey M.C., Tyrcha J., The statistical dynamics of recurrent biological events, J. Math. Biol. 30 (1992), 775–800.
18. Lasota A., Mackey M.C., Ważewska-Czyżewska M., Minimizing therapeutically induced anemia, J. Math. Biol. 13 (1981/82), 149–158.
19. Lasota A., Traple J., Differential equations with dynamical perturbations, J. Differential Equations 63 (1986), 406–417.
20. Losson J., Mackey M.C., Coupled map lattices as models of deterministic and stochastic differential delay equations, Phys. Rev. E 52 (1995), 115–128.
21. Mackey M.C., Time’s Arrow: The Origins of Thermodynamic Behaviour, Springer–Verlag, Berlin–New York–Heidelberg, 1992.
22. Mackey M.C., Glass L., Oscillation and chaos in physiological control systems, Science 197 (1977), 287–289.
23. Mackey M.C., Longtin A., Lasota A., Noise-induced global asymptotic stability, J. Stat. Phys. 60 (1990), 735–751.
24. Mackey M.C., Milton J.G., A deterministic approach to survival statistics, J. Math. Biol. 28 (1990), 33–48.
25. Mackey M.C., Rudnicki R., Global stability in a delayed partial differential equation describing cellular replication, J. Math. Biol. 33 (1994), 89–109.
26. Mackey M.C., Rudnicki R., A new criterion for the global stability of simultaneous cell replication and maturation processes, J. Math. Biol. 38 (1999), 195–219.
27. Mackey M.C., Santavy M., Selepova P., A mitotic oscillator with a strange attractor and distributions of cell cycle times, in: Nonlinear oscillations in biology and chemistry (Salt Lake City, Utah, 1985), Lecture Notes in Biomath., Vol. 66, Springer, Berlin, 1986, pp. 34–45.
28. Mackey M.C., Tyran-Kamińska M., Deterministic Brownian motion: The effects of perturbing a dynamical system by a chaotic semi-dynamical system, Phys. Rep. 422 (2006), 167–222.
29. Mackey M.C., Tyran-Kamińska M., Noise and conditional entropy evolution, Phys. A 365 (2006), 360–382.
30. Mackey M.C., Tyran-Kamińska M., Temporal behavior of the conditional and Gibbs’ entropies, J. Stat. Phys. 124 (2006), 1443–1470.
31. Mackey M.C., Tyran-Kamińska M., Central limit theorem behavior in the skew tent map, Chaos Solitons Fractals 38 (2008), 789–805.
32. Mackey M.C., Tyran-Kamińska M., Central limit theorems for non-invertible measure preserving maps, Colloq. Math. 110 (2008), 167–191.
33. Mackey M.C., Tyran-Kamińska M., Dynamics and density evolution in piecewise deterministic growth processes, Ann. Polon. Math. 94 (2008), 111–129.
34. Murray J.D., Wprowadzenie do biomatematyki (translated by Urszula Foryś and Marek Bodnar), Wydawnictwo Naukowe PWN, Warszawa, 2007.
35. Provatas N., Mackey M.C., Asymptotic periodicity and banded chaos, Phys. D 53 (1991), 295–318.
36. Provatas N., Mackey M.C., Noise-induced asymptotic periodicity in a piecewise linear map, J. Statist. Phys. 63 (1991), 585–612.
37. Rudnicki R., Mackey M.C., Asymptotic similarity and Malthusian growth in autonomous and nonautonomous populations, J. Math. Anal. Appl. 187 (1994), 548–566.
38. Szarski J., Differential Inequalities, Monografie Matematyczne, T. 43, Państwowe Wydawnictwo Naukowe, Warsaw, 1965.
39. Urbaniec J., An interview with Andrzej Lasota: Without the physical world, there would be no mathematics either, Found. Sci. 2 (2002), 183–189.
40. Ważewska-Czyżewska M., Lasota A., Mathematical problems of the dynamics of a system of red blood cells, Mat. Stos. 6 (1976), 23–40.
MackeyM. C. (2009). Exploring the world with mathematics. Annales Mathematicae Silesianae, 23, 11-42. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14040
Michael C. Mackey
mackey@cnd.mcgill.ca
Departments of Physiology, Physics & Mathematics and Centre for Nonlinear Dynamics, McGill University, Canada Canada
Departments of Physiology, Physics & Mathematics and Centre for Nonlinear Dynamics, McGill University, Canada Canada
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