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1
Content available remote On continuous time version of two-phase cell cycle model of Tyrcha
Zwoleński P.
Mathematica Applicanda
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2013
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Vol. 41, No. 1
13--31
EN
We consider a model of two-phase cell cycle in a maturity-structured cellular population, which consists of a system of first order linear partial differential equations (transport equations). The model is based on similar biological assumptions as models of Lasota-Mackey, Tyson-Hannsgen and Tyrcha. We examine behavior of the solutions of the system along characteristics, give conditions for existence of invariant density, and compare results with outcomes of generation.
PL
W artykule rozważamy matematyczny opis dwufazowego modelu cyklu komórkowego w populacjach ze struktura dojrzałości komórkowej. Model, którym się zajmujemy, składa się z układu dwóch równań rózniczkowych cząstkowych (równań transportu) i jest budowany na podobnych biologicznych założeniach co modele Lasoty-Mackeya, Tysona-Hannsgena oraz Tyrchy. Rozwiązania otrzymanego układu równań badamy wzdłuż charakterystyk, podajemy warunki wystarczające na istnienie gęstości niezmienniczej i porównujemy wyniki z wynikami badań nad modelem Tyrchy.
2
Content available remote Generic properties of continuous iterated function systems with place dependent probabilities
Bielaczyc T.
Bulletin of the Polish Academy of Sciences. Mathematics
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2007
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Vol. 55, no 1
81-96
EN
It is shown that for a typical continuous learning system denned on a compact convex subset of R[sup]n the Hausdorff dimension of its invariant measure is equal to zero.
3
Content available remote On residualities in the set of Markov continuous semigroups on C1
Kuna B.
Demonstratio Mathematica
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2006
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Vol. 39, nr 2
439-453
EN
We show that the set of all stochastic strongly continuous semigroups on C1 such that limt-oo |||T(t) - Qx*||| = 0, where Qx* is one-dimensional projection for some state X*, is norm open and dense. Moreover this set forms a norm dense Gb if a state X* is strictly positive.
4
Content available remote The pointwise dimension for invariant measures related with Poisson driven stochastic differential equations
Szarek T.
Bulletin of the Polish Academy of Sciences. Mathematics
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2002
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Vol. 50, no 2
241-250
EN
Sufficient conditions for the asymptotic stability of Poisson driven stochastic differential equations on a separable Banach space are presented. It is also proved that the lower pointwise dimension of a unique invariant measure is greater than or equal to log 2/ log 3.
5
Content available remote Asymptotic behaviour of the iterates of nonnegative operators on Banach spaces with a cone
Socała J.
Bulletin of the Polish Academy of Sciences. Mathematics
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2002
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Vol. 50, no 2
179-187
EN
An asymptotic convergence theorem for nonnegative operators on Banach spaces with a cone is proved. The general result is applied to a class of nonnegative operators on C^1 ([0, 1]).
6
Content available remote Positivity of Schilling functions
Girgensohn R., Morawiec J.
Bulletin of the Polish Academy of Sciences. Mathematics
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2000
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Vol. 48, no 4
407-412
EN
We prove that every non-trivial [L^1]-solution of the Schilling problem is either positive or negative almost everywhere on its support.
7
Content available remote Iterated function systems generated by strict contractions and place-dependent probabilities
Zaharopol R.
Bulletin of the Polish Academy of Sciences. Mathematics
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2000
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Vol. 48, no 4
429-438
EN
Let (w[sub l], w[sub 2],...,w[sub k];p[sub 1],p[sub 2],...p[sub k]) be an iterated function system (IFS for short) with continuous place-dependent probabilities, defined on a metric space (X, d). Assume that every closed ball in X is compact. Our main result is that the IFS has an attractive probability measure whenever the following three conditions are satisfied: (1) w[sub i] : X --> X is a strict contraction for every i = 1,...,k. (2) sum[...]p[sub i](x)p[sub i](y) > 0 for every x, y [belongs to] X. (3) There exists p > 0 in R such that [...] for every x, y [belongs to] X and j = l, 2,...,k. Note that we do not require the p[sub i]'s to be even uniformly continuous. This research was motivated by a question of Barnsley, Demko, Elton, and Geronimo, [1, p. 373], concerning IFS which satisfy only condition (1). We construct a family C of IFS which we use to answer the question. Our main result allows us to distinguish IFS in C which possess attractive probabilities.
8
Content available remote Attractors of multifunetions
Lasota A., Myjak J.
Bulletin of the Polish Academy of Sciences. Mathematics
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2000
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Vol. 48, no 3
319-334
EN
We introduce the notion of a semiattractor and attractor for multifunctions and we show that they have properties similar to semifractals and fractals. Further we show a relationship between the multifunctions and transition functions appearing in the theory of Markov operators.
9
Content available remote Generic properties of continuous iterated function systems
Szarek T.
Bulletin of the Polish Academy of Sciences. Mathematics
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1999
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Vol. 47, no 1
77-89
EN
It is shown that in the space of all nonexpansive continuous IFS's defined on a compact convex subset of R^n the family of asymptotically stable IFS's having a singular stationary distribution is generic.
10
Content available remote An application of the Kantorovich-Rubinstein maximum principle in the stability theory of Markov operators
Gacki H.
Bulletin of the Polish Academy of Sciences. Mathematics
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1998
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Vol. 46, no 2
215-223
EN
We consider the asymptotic behaviour of Markov operators acting on measures, defined on a locally are [sigma]-compact metric space. We prove a new sufficient condition for the asymptotic stability of Markov operators. This condition is applied to stochastically peturbed dynamical systems, and iterated function system.
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