Sparse equidistribution of unipotent orbits in finite-volume quotients of $\text{PSL}(2,\mathbb R)$
Cheng Zheng
In this note, we consider the orbits $\{pu(n^{1+\gamma})|n\in\mathbb N\}$ in $\Gamma\backslash\text{PSL}(2,\mathbb R)$, where $\Gamma$ is a non-uniform lattice in $\text{PSL}(2,\mathbb R)$ and $\{u(t)\}$ is the standard unipotent one-parameter subgroup in $\text{PSL}(2,\mathbb R)$. Under a Diophantine condition on~the initial point $p$, we can prove that the trajectory $\{pu(n^{1+\gamma})|n\in\mathbb N\}$ is equidistributed in $\Gamma\backslash\text{PSL}(2,\mathbb R)$ for small $\gamma>0$, which generalizes a result of Venkatesh [22].
keywords: Sparse equidistribution geodesic excursions into cusps.
Normal and slow growth states of microbial populations in essential resource-based chemostat
Yi Wang Chengmin Zheng
To mimic the striking capability of microbial culture for growth adaptation after the onset of the novel environmental conditions, a modified heterogeneous microbial population model in the chemostat with essential resources is proposed which considers adaptation by spontaneously phenotype-switching between normally growing cells and persister cells having reduced growth rate. A basic reproductive number $R_0$ is introduced so that the population dies out when $R_0<1$, and when $R_0>1$ the population will be asymptotic to a steady state of persister cells, or a steady state of only normal cells, or a steady state corresponding to a heterogeneous population of both normal and persister cells. Our analysis confirms that inherent heterogeneity of bacterial populations is important in adaption to fluctuating environments and in the persistence of bacterial infections.
keywords: Normally growing cells Persister cells Chemostat Essential resources.

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