Istnienie gęstości niezmienniczych dla semiukładów dynamicznych ze skokami

In this dissertation we are going to analyze the problem of existence of invariant densities for piecewise deterministic Markow processes (PDMP) called semiflows with jumps. This class of processes is numerously applied in biological processes modeling. In particular, the existence of invariant densities is crucial for studying these models. PDMP is a continuous-time process {X( t ) }ₜ ≥0 for which there exists an increasing sequence of so called jump times (tₙ). Between two consecutive jumps the process is deterministic. We give a precise definition of such semiflow in Chapter 1. We study PDMP using the theory of substochastic semigroups { P (t)}ₜ ≥0 on L¹ space of functions integrable with respect to a fixed measure m. In Chapter 2 we define many necessary notions, such as substochastic semigroups and invariant densities. We also quote results which allow us to give sufficient conditions for the existence and uniqueness of invariant densities for Markow operators. Moreover, we cite theorems on asymptotic stability of substochastic semigroups. In Chapter 3 we analyze the problem of existence of invariant densities for substochastic semigroups. We obtain the existence of a so called minimal substochastic semigroup { P( t ) }ₜ ≥0 for a process {X( t ) }ₜ ≥0 and for a unique Markow operator K on L¹ which satisfies: if the distribution of the random variable X (0) has a density f , i.e., P r (X (0) ϵ B ) = ʃ B f (x)m(dx) for all measurable subsets B of the space of states of the process, then X (t₁) has a density K f . Relationships between invariant densities for the operator K and invariant densities for the minimal semigroup { P (t)}ₜ ≥0 are the main topic of this Chapter. Here the most important results are obtained in Theorems 3.5 and 3.12 and also in Corollary 3.14 following from these theorems. In Chapter 4 we study the problem of existence of invariant densities for semiflows with jumps. One of the main result is Theorem 4.2 which gives sufficient conditions for the existence of a unique invariant density for a piecewise deterministic Markow process. We do not have to assume that the process is non-explosive and we look for absolutely continuous invariant measures. Additionally, we obtain asymptotic stability of the semigroup {P(t)}ₜ ≥0 (Theorem 4.6), i.e. the fact th a t the density of X( t ) converges to the invariant density in L¹ irrespective of the density of X (0). In Section 4.2 we provide sufficient conditions for the existence of invariant densities and asymptotic stability of a semigroup { P (t)}ₜ ≥0 in a form which makes the application easier, i.e., in terms of local characteristics of semiflows with jumps. In the final section of Chapter 4 we show how dynamical systems with random switching can be studied with our methods. In the last two Chapters we present how to apply our results in analyzing biological models. These examples are a two dimensional model of gene expression with bursting and a fragmentation process. Therefore, our framework can be used to analyze biological processes described by PDMPs, e.g. for gene expression with bursting, for dynamics with switching or for fragmentation process

One sessional hypnotherapy of nicotine addiction and smoking cessation

Najbardziej rozpowszechnionym nałogiem na świecie jest palenie papierosów. Przeprowadzone badania miały na celu określenie skuteczności hipnoterapii w leczeniu nałogu nikotynowego. Przebadano 60 osób (30 kobiet i 30 mężczyzn) w wieku od 20 do 76 lat. Trzydziestu badanych poddano jednosesyjnej hipnozie, pozostałe trzydzieści osób stanowiło grupę kontrolną nie poddawaną żadnym oddziaływaniom. W badaniach wykorzystano własne tłumaczenie indukcji hipnotycznej Herberta Spiegla (1970) oraz autorski kwestionariusz dotyczący historii palenia oraz motywacji do zrywania z nałogiem. Uzyskane wyniki nie potwierdziły hipotezy. Osoby poddane hipnozie nie rzucały palenia częściej niż przedstawiciele grupy kontrolnej. Zaobserwowano spadek średniej ilości wypalanych papierosów jeden dzień po przeprowadzeniu hipnozy dlatego podejmując badania w przyszłości warto zwrócić uwagę na zwiększenie ilości sesji.The most common addiction in the world is smoking cigarettes. The study aimed to determine the effectiveness of hypnotherapy in the treatment of nicotine addiction. For the study purposes, 60 patients were examined (30 women and 30 men) aged 20 to 76 years. Thirty respondents took part in one session of hypnosis, the other thirty people attended as a control group not subjected to any impacts. The author used for the study a translation of Herbert Spiegel’s hypnotic induction (1970) and prepared a questionnaire on smoking history and motivation to eliminate addiction. The results have not confirmed the hypothesis. People who took part in hypnosis session did not quite smoking more often than representatives of the control group. A decrease in the average number of smoked cigarettes was observed but only one day after the hypnosis therefore in the future it would be worth noting the increase in the number of sessions during the examination