Zakład Biomatematyki

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O Zakładzie

The Katowice Branch of the Institute of Mathematics was founded in 1966. Jan Mikusiński was the head of the Branch until his retirement in 1984. From 1985 to 1994 the Branch was headed by Piotr Antosik, and then by Ryszard Rudnicki. The following mathematicians worked in the Branch: B. Aniszczyk, P. Antosik, J. Burzyk, T. Dłotko, C. Ferens, P. Hallala, A. Kamiński, W. Kierat, C. Kli¶, S. Krasińska, M. Kuczma, A. Lasota, S. Lewandowska, Z. Lipecki, K. Łoskot, J. Mikusiński, P. Mikusiński, J. Mioduszewski, J. Pochciał, R. Rudnicki, Z. Sadlok, K. Skórnik, W. Smajdor, T. Szarek, Z. Tyc, J. Uryga and P. Uss.

The main line of research has been closely related to Prof. Mikusiński's interests. The dominating topics of investigations are sequential theory of distributions, Mikusiński operational calculus and convergence theory. The main results obtained in this area are: introduction of regular and irregular operations on distributions and local derivatives, functional description of the convergence in the field of Mikusiński operators, axiomatic theory of convergence, diagonal theorems and Paley-Wiener type theorems for regular operators. Moreover, several results concerning applications of operational calculus to differential equations, theory of controllability and special functions have been obtained.

Numerous results obtained by Mikusiński's team are presented in five books written by Mikusiński, Antosik, Sikorski and Boehme. Mikusiński's books have been translated into various languages, for example "Operational Calculus'' was published in Polish, English, Russian, German, Hungarian and Japanese.

In the early nineties a group of scientists connected with Prof. Andrzej Lasota began to work in the Branch. Their main research interests are in probability theory, partial differential equations and biomathematics. The main results obtained are: sufficient conditions for asymptotic stability of Markov operators and semigroups, asymptotic behaviour of solutions of generalized Fokker-Planck equations, constructions of semifractals and global properties of nonlinear models of population dynamics.

Selected publications

Books:

  1. J. Mikusiński, Operational Calculus, Pergamon Press and PWN, 1967; 1983.
  2. J. Mikusiński and T.K. Boehme, Operational Calculus, Volume II, PWN and Pergamon Press, 1987.
  3. P. Antosik, J. Mikusiński and R. Sikorski, Theory of Distributions, The Sequential Approach, Elsevier-PWN, 1973 (Russian edition 1976).
  4. J. Mikusiński, The Bochner Integral, Birkhäuser, 1987; Academic Press, 1978.
  5. P. Antosik and C. Swartz, Matrix Methods in Analysis, Springer, 1985.

Papers:

  1. P. Antosik, On the Mikusiński diagonal theorem, Bull. Acad. Polon. Sci. 20 (1972), 373-377.
  2. J. Burzyk, On convergence in the Mikusiński operational calculus, Studia Math. 75 (1983), 313-333.
  3. J. Burzyk, A Paley-Wiener type theorem for regular operators, Studia Math. 93 (1989), 187-200.
  4. H. Gacki, T. Szarek and S. Wędrychowicz, On existence and stability of solutions of stochastic integral equations with applications to control system, Indian J. Pure Appl. Math. 29 (1998), 175-189.
  5. A. Kamiński, On the Rényi theory of conditional probabilities, Studia Math. 79 (1984), 151-191.
  6. A. Kamiński, D. Kova?ević and S. Pilipović, The equivalence of various definitions of the convolution of ultradistributions, Trudy Mat. Inst. Steklov. 203 (1994) 307-322.
  7. C. Kli¶, An example of a non-complete normed (K) space, Bull. Acad. Polon. Sci. 26 (1978), 415-420.
  8. A. Lasota and J. Myjak, Semifractals, Bull. Polish Acad. Sci. Math. 44 (1996), 5-21.
  9. A. Lasota and J. A. Yorke, When the long time behavior is independent of the initial density, SIAM J. Math. Anal. 27 (1996), 221-240.
  10. K. Łoskot and R. Rudnicki, Limit theorems for stochastically perturbed dynamical systems, J. Appl. Probab. 32 (1995), 459-469.
  11. J. Łuczka and R. Rudnicki, Randomly flashing diffusion: asymptotic properties, J. Statist. Phys. 83 (1996), 1149-1164.
  12. M. C. Mackey and R. Rudnicki, Asymptotic similarity and Malthusian growth in autonomous and nonautonomous populations, J. Math. Anal. Appl. 187 (1994), 548-566.
  13. M. C. Mackey and R. Rudnicki, Global stability in a delayed partial differential equation describing cellular replication, J. Math. Biol. 33 (1994), 89-109.
  14. J. Mikusiński, Sequential theory of the convolution of distributions, Studia Math. 29 (1968), 151-160.
  15. J. Mikusiński, A theorem on vector matrices and its applications in measure theory and functional analysis, Bull. Acad. Polon. Sci. 18 (1970), 151-155.
  16. J. Mikusiński, On full derivatives and on the integral substitution formula, Accad. Naz. Lincei Probl. Atti Sci. Cult. 217 (1975), 377-390.
  17. J. Mikusiński and P. Mikusiński, Quotients de suites et leurs applications dans l'analyse fonctionnelle, C. R. Acad. Sci. Paris Sér. I Math. 293 (1981), 463-464.
  18. K. Pichór and R. Rudnicki, Stability of Markov semigroups and applications to parabolic systems, J. Math. Anal. Appl. 215 (1997), 56-74.
  19. J. Pochciał, Sequential characterizations of metrizability, Czech. Math. J. 41 (1991), 203-215.
  20. R. Rudnicki, Asymptotical stability in L1 of parabolic equations, J. Differential Equations 102 (1993), 391-401.
  21. R. Rudnicki, On asymptotic stability and sweeping for Markov operators, Bull. Polish Acad. Sci. Math. 43 (1995), 245-262.
  22. K. Skórnik, On fractional integrals and derivatives of a class of generalized functions, Soviet Math. Dokl. 22 (1980), 541-543.
  23. K. Skórnik and J. Wloka, m-reduction of ordinary differential equations, Colloq. Math. 78 (1998), 195-212.