Smooth center-stable/unstable manifolds and foliations of stochastic evolution equations with non-dense domain

The current paper is devoted to the asymptotic behavior of a class of stochastic PDE. More precisely, with the help of the theory of integrated semigroups and a crucial estimate of the random Stieltjes convolution, we study the existence and smoothness of center-unstable invariant manifolds and center-stable foliations for a class of stochastic PDE with non-dense domain through the Lyapunov-Perron method. Finally, we give two examples about a stochastic age-structured model and a stochastic parabolic equation to illustrate our results.Comment: 46 page

Second Quantized Reduced Bloch Equations and the Exact Solutions for Pairing Hamiltonian

In this article, we present a set of hierarchy Bloch equations for the reduced density operators in either canonical or grand canonical ensembles in the occupation number representation. They provide a convenient tool for studying the equilibrium quantum statistical mechanics for some model systems. As an example of their applications, we solve the equations for the model system with a pairing Hamiltonian. With the aid of its symplectic group symmetry, we obtain the statistical reduced density matrices with different orders. As a special instance for the solutions, we also get the reduced density matrices of the ground state for a superconductor

Robust estimation of bacterial cell count from optical density

Optical density (OD) is widely used to estimate the density of cells in liquid culture, but cannot be compared between instruments without a standardized calibration protocol and is challenging to relate to actual cell count. We address this with an interlaboratory study comparing three simple, low-cost, and highly accessible OD calibration protocols across 244 laboratories, applied to eight strains of constitutive GFP-expressing E. coli. Based on our results, we recommend calibrating OD to estimated cell count using serial dilution of silica microspheres, which produces highly precise calibration (95.5% of residuals <1.2-fold), is easily assessed for quality control, also assesses instrument effective linear range, and can be combined with fluorescence calibration to obtain units of Molecules of Equivalent Fluorescein (MEFL) per cell, allowing direct comparison and data fusion with flow cytometry measurements: in our study, fluorescence per cell measurements showed only a 1.07-fold mean difference between plate reader and flow cytometry data

Lactobacillus plantarum modulate gut microbiota and intestinal immunity in cyclophosphamide-treated mice model

Gut microbiota (GM) contributes to the production of immune-regulatory molecules and cytokines. However, our understanding regarding intricate relationship between Lactobacillus plantarum and GM on regulation of immune function remained limited. To investigate the effect of Lactobacillus plantarum on an immunosuppressed mouse model, we employed cyclophosphamide treatment and conducted various analysis including H&E (hematoxylineosin staining), immunohistochemistry, 16S rRNA gene sequencing, and RT-PCR. Our results demonstrated that the administration of Lactobacillus plantarum had significant immunoenhancing effects in the immune-suppressed mice, as evidenced by the restoration of functional expression of specific immune markers in the spleen and an increase in the number of goblet cells in intestine (P < 0.05). Microbial taxonomic analysis revealed alterations in the gut microbiota composition, characterized by a decrease in the richness of Firmicutes and an increase in the proportion of Verrucomicrobia and Actinobacteria following cyclophosphamide treatment. Furthermore, cyclophosphamide treatment significantly suppressed the mRNA expression of inflammatory cytokines (P < 0.05), which were subsequently restored after administration of Lactobacillus plantarum. These observations provide valuable insights into the complex interplay between probiotics, gut microbiota, and immune system functioning.ISSN:0753-3322ISSN:1950-600