We introduce an invariant, called mean rank, for any module{\mathcal{M}}of the integral group ring of a discrete amenable group Γ, as an analogue of the rank of an abelian group. It is shown that the mean dimension of the induced Γ-action on the Pontryagin dual of{\mathcal{M}}, the mean rank of{\mathcal{M}}, and the von Neumann–Lück rank of{\mathcal{M}}all coincide. As applications, we establish an addition formula for mean dimension of algebraic actions, prove the analogue of the Pontryagin–Schnirelmann theorem for algebraic actions, and show that for elementary amenable groups with an upper bound on the orders of finite subgroups, algebraic actions with zero mean dimension are inverse limits of finite entropy actions.
The topological pressure for any sub-additive potentials of a countable discrete amenable group action and any given open cover is defined. A local variational principle for the topological pressure is established.
The rapid emergence of bacterial coinfection caused by cytosolic bacteria has become a huge threat to public health worldwide. Past efforts have been devoted to discover the broad-spectrum antibiotics, while the emergence of antibiotic resistance encourages the development of antibacterial agents. In essence, bacterial virulence is a factor in antibiotic tolerance. However, the discovery and development of new antibacterial drugs and special antitoxin drugs is much more difficult in the antibiotic resistance era. Herein, we hypothesize that antitoxin hemolytic activity can serve as a screening principle to select antibacterial drugs to combat coinfection from natural products. Being the most abundant natural drug of plant origins, flavonoids were selected to assess the ability of antibacterial coinfections in this paper. Firstly, we note that four flavonoids, namely, baicalin, catechin, kaempferol, and quercetin, have previously exhibited antibacterial abilities. Then, we found that baicalin, kaempferol, and quercetin have better inhibitions of hemolytic activity of Hla than catechin. In addition, kaempferol and quercetin, have therapeutic effectivity for the coinfections of Staphylococcus aureus and Pseudomonas aeruginosa in vitro and in vivo. Finally, our results indicated that kaempferol and quercetin therapied the bacterial coinfection by inhibiting S. aureus α-hemolysin (Hla) and reduced the host inflammatory response. These results suggest that antitoxins may play a promising role as a potential target for screening flavonoids to combat bacterial coinfection.
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