arXiv subjects

Katarzyna Pichór

Publications and source records attributed to Katarzyna Pichór.

5 recordsLinked to original sources

arXiv2024

Transformation semigroups and their applications

In this chapter we present transformation semigroups and their applications. We begin with Klein's approach to geometry based on invariants of transformation groups. Then we present symmetry groups in chemistry and in classical mechanics. Next we introduce one-parameter semigroups of transformations and their applications in ergodic theory. Our main subject are one-parameter semigroups of operators, in particular stochastic semigroups. We present general results on their existence and long-time behaviour. We also give examples of one-parameter semigroups related to Markov chains, diffusion and processes with jumps. We focus on the applications of semigroups of operators in biology. Among other things, we study models of: DNA evolution; growth of erythrocyte population; gene expression; cell cycle; the movement of bacteria and insects. We also consider models with stochastic noise and different population models.

math.FA↗

arXiv2022

Asymptotic properties of a general model of immune status

We consider a model of dynamics of the immune system. The model is based on three factors: occasional boosting and continuous waning of immunity and a general description of the period between subsequent boosting events. The antibody concentration changes according to a non-Markovian process. The density of the distribution of this concentration satisfies some partial differential equation with an integral boundary condition. We check that this system generates a stochastic semigroup and we study the long-time behaviour of this semigroup. In particular we prove a theorem on its asymptotic stability.

math.PR↗

arXiv2021

Cell cycle length and long-time behaviour of an age-size model

We consider an age-size structured cell population model based on the cell cycle length. The model is described by a first order partial differential equation with initial-boundary conditions. Using the theory of semigroups of positive operators we establish new criteria for an asynchronous exponential growth of solutions to such equations. We discuss the question of exponential size growth of cells. We study in detail a constant size growth model and a model with target size division. We also present versions of the model when the population is heterogeneous.

math.AP↗

arXiv2020

Dynamics of antibody levels: asymptotic properties

We study properties of a piecewise deterministic Markov process modeling the changes in concentration of specific antibodies. The evolution of densities of the process is described by a stochastic semigroup. The long-time behaviour of this semigroup is studied. In particular we prove theorems on its asymptotic stability.

math.PR↗

arXiv2018

Applications of stochastic semigroups to cell cycle models

We consider a generational and continuous-time two-phase model of the cell cycle. The first model is given by a stochastic operator, and the second by a piecewise deterministic Markov process. In the second case we also introduce a stochastic semigroup which describes the evolution of densities of the process. We study long-time behaviour of these models. In particular we prove theorems on asymptotic stability and sweeping. We also show the relations between both models.

math.PR↗