Convergence in LpL^p and its exponential rate for a branching process in a random environment

We consider a supercritical branching process $(Z_n)$ in a random environment $\xi$. Let $W$ be the limit of the normalized population size $W_n=Z_n/E[Z_n|\xi]$. We first show a necessary and sufficient condition for the quenched $L^p$ ($p>1$) convergence of $(W_n)$, which completes the known result for the annealed $L^p$ convergence. We then show that the convergence rate is exponential, and we find the maximal value of $\rho>1$ such that $\rho^n(W-W_n)\rightarrow 0$ in $L^p$, in both quenched and annealed sense. Similar results are also shown for a branching process in a varying environment

Moments, moderate and large deviations for a branching process in a random environment

Let $(Z_{n})$ be a supercritical branching process in a random environment $\xi$, and $W$ be the limit of the normalized population size $Z_{n}/\mathbb{E}[Z_{n}|\xi ]$. We show large and moderate deviation principles for the sequence $\log Z_{n}$ (with appropriate normalization). For the proof, we calculate the critical value for the existence of harmonic moments of $W$, and show an equivalence for all the moments of $Z_{n}$. Central limit theorems on $W-W_n$ and $\log Z_n$ are also established

Weak law of large numbers for some Markov chains along non homogeneous genealogies

We consider a population with non-overlapping generations, whose size goes to infinity. It is described by a discrete genealogy which may be time non-homogeneous and we pay special attention to branching trees in varying environments. A Markov chain models the dynamic of the trait of each individual along this genealogy and may also be time non-homogeneous. Such models are motivated by transmission processes in the cell division, reproduction-dispersion dynamics or sampling problems in evolution. We want to determine the evolution of the distribution of the traits among the population, namely the asymptotic behavior of the proportion of individuals with a given trait. We prove some quenched laws of large numbers which rely on the ergodicity of an auxiliary process, in the same vein as \cite{guy,delmar}. Applications to time inhomogeneous Markov chains lead us to derive a backward (with respect to the environment) law of large numbers and a law of large numbers on the whole population until generation $n$. A central limit is also established in the transient case

Fluorine-Modified Rutaecarpine Exerts Cyclooxygenase-2 Inhibition and Anti-inflammatory Effects in Lungs

Inflammation is the first step that leads to inflammatory cell migration, cytokine release, and myofibroblast formation. Myofibroblasts can deposit excess amounts of extracellular matrix. Cyclooxygenase (COX) inhibitor exhibits strong anti-inflammatory response; however, this is usually achieved with undesirable side effects. In this study, we demonstrated the effects of the fluorine-modified rutaecarpine (RUT), fluoro-2-methoxyrutaecarpine (F-RUT), in inflammatory damage in the lungs. Based on the results, F-RUT retained anti-inflammatory activity both in vitro and in vivo in lungs. Compared to the parent compound, F-RUT showed better COX-2 suppression as a COX-2-selective inhibitor with lower cytotoxicity, and enhanced molecular reactivity and biological activity. F-RUT was also observed to reduce reactive oxygen species (ROS) generation and inflammatory infiltrating neutrophils in lipopolysaccharide (LPS)-stimulated zebrafish and ovalbumin (OVA)/alum-challenged KLF-10-knockout mouse lungs, respectively. Furthermore, F-RUT ameliorated the respiratory function in OVA/alum-challenged BALB/c mice by maintaining the thickness of the blood-air barrier in mouse lungs. Overall, these data suggest that F-RUT may function as an effective therapeutic agent for inflammation-induced lung dysfunction, and a better selection for pharmaceutical purposes than conventionally used anti-inflammatory agents