arXiv subjects

Chong Shen

Publications and source records attributed to Chong Shen.

22 records · Page 2Linked to original sources

arXiv2020

$ω$-Rudin spaces, well-filtered determined spaces and first-countable spaces

We investigate some versions of $d$-space, well-filtered space and Rudin space concerning various countability properties. The main results include: (i) if the sobrification of a $T_0$ space $X$ is first-countable, then $X$ is an $ω$-Rudin space; (ii) every $ω$-well-filtered space is sober if its sobrification is first-countable; (iii) if a $T_0$ space is second-countable or first-countable and with a countable underlying set, then it is a $ω$-Rudin space; (iv) every first-countable $T_0$ space is well-filtered determined; (v) every irreducible closed subset in a first-countable $ω$-well-filtered space is countably-directed; (vi) every first-countable $ω$-well-filtered $ω^\ast$-$d$-space is sober.

math.GN↗

arXiv2019

First countability, $ω$-well-filtered spaces and reflections

We first introduce and study two new classes of subsets in $T_0$ spaces - $ω$-Rudin sets and $ω$-well-filtered determined sets lying between the class of all closures of countable directed subsets and that of irreducible closed subsets, and two new types of spaces - $ω$-$d$ spaces and $ω$-well-filtered spaces. We prove that an $ω$-well-filtered $T_0$ space is locally compact iff it is core compact. One immediate corollary is that every core compact well-filtered space is sober, answering Jia-Jung problem with a new method. We also prove that all irreducible closed subsets in a first countable $ω$-well-filtered $T_0$ space are directed. Therefore, a first countable $T_0$ space $X$ is sober iff $X$ is well-filtered iff $X$ is an $ω$-well-filtered $d$-space. Using $ω$-well-filtered determined sets, we present a direct construction of the $ω$-well-filtered reflections of $T_0$ spaces, and show that products of $ω$-well-filtered spaces are $ω$-well-filtered.

math.GN↗

arXiv2019

On $T_0$ spaces determined by well-filtered spaces

We first introduce and study two new classes of subsets in $T_0$ spaces - Rudin sets and $\wdd$ sets lying between the class of all closures of directed subsets and that of irreducible closed subsets. Using such subsets, we define three new types of topological spaces - $\mathsf{DC}$ spaces, Rudin spaces and $\wdd$ spaces. The class of Rudin spaces lie between the class of $\wdd$ spaces and that of $\dc$ spaces, while the class of $\dc$ spaces lies between the class of Rudin spaces and that of sober spaces. Using Rudin sets and $\wdd$ sets, we formulate and prove a number of new characterizations of well-filtered spaces and sober spaces. For a $T_0$ space $X$, it is proved that $X$ is sober if{}f $X$ is a well-filtered Rudin space if{}f $X$ is a well-filtered $\mathsf{WD}$ space. We also prove that every locally compact $T_0$ space is a Rudin space, and every core compact $T_0$ space is a $\wdd$ space. One immediate corollary is that every core compact well-filtered space is sober, giving a positive answer to Jia-Jung problem. Using $\wdd$ sets, we present a more directed construction of the well-filtered reflections of $T_0$ spaces, and prove that the products of any collection of well-filtered spaces are well-filtered. Our study also leads to a number of problems, whose answering will deepen our understanding of the related spaces and structures.

math.GN↗

arXiv2019

Spatial and Temporal fluctuation of an ABP in an optical trap

A colloidal suspension of active Brownian particles (ABPs) driven by controllable forces into directed or persistent motions can serve as a model for understanding the biological systems. Experiments and numerical simulations are established to investigate the motions of an ABP, a single, induced-charge electrophoretic (ICEP) metallic Janus particle, confined in a quadratic potential well. On the one hand, 1-D position histograms of the trapped active particle, behaving differently from that of a Boltzmann distribution, reveal a splitting from a single peak of the ABP positional distribution to a bimodal distribution. Decoupling the thermal and non-thermal contributions from the overall histogram is non-trivial. However, the two contributions can be examined by convoluting numerically generated thermal and non-thermal contributions into a full histogram. On the other hand, temporal fluctuations analyzed by the power spectral density (PSD), reveal two unique frequencies characterizing the stiffness of the trap and the rotational diffusion of the particle, respectively. Connections between the spatial and temporal fluctuations are obtained by the separate analysis of the temporal and spatial fluctuations of an ABP trapped in a quadratic potential well. This study reveals how thermal and nonthermal fluctuations play against each other in a confined environment.

cond-mat.soft↗