The random case of Conley's theorem

The well-known Conley's theorem states that the complement of chain recurrent set equals the union of all connecting orbits of the flow $\phi$ on the compact metric space $X$, i.e. $X-\mathcal{CR}(\phi)=\bigcup [B(A)-A]$, where $\mathcal{CR}(\phi)$ denotes the chain recurrent set of $\phi$, $A$ stands for an attractor and $B(A)$ is the basin determined by $A$. In this paper we show that by appropriately selecting the definition of random attractor, in fact we define a random local attractor to be the $\omega$-limit set of some random pre-attractor surrounding it, and by considering appropriate measurability, in fact we also consider the universal $\sigma$-algebra $\mathcal F^u$-measurability besides $\mathcal F$-measurability, we are able to obtain the random case of Conley's theorem.Comment: 15 page

The random case of Conley's theorem: III. Random semiflow case and Morse decomposition

In the first part of this paper, we generalize the results of the author \cite{Liu,Liu2} from the random flow case to the random semiflow case, i.e. we obtain Conley decomposition theorem for infinite dimensional random dynamical systems. In the second part, by introducing the backward orbit for random semiflow, we are able to decompose invariant random compact set (e.g. global random attractor) into random Morse sets and connecting orbits between them, which generalizes the Morse decomposition of invariant sets originated from Conley \cite{Con} to the random semiflow setting and gives the positive answer to an open problem put forward by Caraballo and Langa \cite{CL}.Comment: 21 pages, no figur

Antiviral Activity, Safety, and Tolerability of Multiple Ascending Doses of Elbasvir or Grazoprevir in Participants Infected With Hepatitis C Virus Genotype-1 or -3

Purpose: Elbasvir (MK-8742) and grazoprevir (MK-5172; Merck & Co, Inc, Kenilworth, New Jersey) are hepatitis C virus (HCV)-specific inhibitors of the nonstructural protein 5A phosphoprotein and the nonstructural protein 3/4A protease, respectively. The aims of these studies were to evaluate the antiviral activity and safety of different doses of elbasvir or grazoprevir each administered as monotherapy to participants infected with either HCV genotype (GT) 1 or GT3. Methods: These 2 double-blind, randomized, placebo-controlled, sequential-panel, multiple ascending dose studies were conducted to assess the safety and pharmacodynamics of 5 days of once-daily elbasvir or 7 days of once-daily grazoprevir in adult male participants chronically infected with either HCV GT1 or GT3. Findings: Oral administration of elbasvir or grazoprevir once daily exhibited potent antiviral activity in participants with chronic GT1 or GT3 HCV infections. HCV RNA levels declined rapidly (within 1 day for elbasvir and 2 days for grazoprevir). At 50 mg of elbasvir once daily, the mean maximum reductions in HCV RNA from baseline were 5.21, 4.17, and 3.12 log10 IU/mL for GT1b-, GT1a-, and GT3-infected participants, respectively. At 100 mg of grazoprevir once daily, the mean maximum reductions in HCV RNA from baseline were 4.74 and 2.64 log10 IU/mL for GT1- and GT3-infected participants. Implications: The results in the elbasvir monotherapy study showed that 10 to 50 mg of elbasvir was associated with a rapid decline in HCV viral load; the results in the grazoprevir monotherapy study suggest that doses of 50 mg of grazoprevir and higher are on the maximum response plateau of the dose–response curve for GT1-infected participants. The results of these proof-of-concept studies provided preliminary data for the selection of the dosages of elbasvir and grazoprevir to test in Phase II and III clinical studies. ClinicalTrials.gov identifiers: NCT00998985 (Protocol 5172-004) and NCT01532973 (Protocol 8742-002). © 2018 Elsevier HS Journals, Inc