In this paper a rigorous proof of the mean field limit for a pedestrian flow model in two dimensions is given by using a probabilistic method. The model under investigation is an interacting particle system coupled to the eikonal equation on the microscopic scale. For stochastic initial data, it is proved that the solution of the N -particle pedestrian flow system with properly chosen cut-off converges in the probability sense to the solution of the characteristics of the non-cut-off Vlasov equation. Furthermore, the result on propagation of chaos is also deduced in terms of bounded Lipschitz distance.
Carbon dots (CDs), as the most important type of carbon materials, have been widely used in many fields because of their unique fluorescence characteristics and excellent properties of biocompatibility. In previous studies, the fluorescence of CDs was mainly concentrated in the blue and green, whereas the red fluorescence was relatively less. Herein, we prepared efficient red-emitting CDs from 1,4-diaminonaphthalene using solvothermal methods. We discussed the effects of different solvothermal solvents on CDs. The results show that CDs prepared with octane and acetone as reaction media have the best fluorescence properties. The CDs dispersed in different organic solvents exhibited tunable emission across a wide spectrum from 427 nm to 679 nm. We further demonstrated the application of red light-emitting diode (LED) optoelectronics and fluorescence detection of Fe3+ in aqueous solution.
In this study, amino hyperbranched polymers (HBP)-grafted polyacrylonitrile (PAN) fiber was prepared through an amidation reaction in an autoclave. The prepared PAN-G-HBP fiber can complex Ag+ through amino groups of amino HBP, and in a hot steaming condition, Ag+ can be converted to Ag0 through the reducibility of HBP. PAN-G-HBP and Ag nanoparticles (NPs)-coated fibers were then characterized through FTIR, UV–VIS DRS, FE-SEM, EDS, XPS and antibacterial measurement. FTIR results confirmed HBP was grafted on the surface of PAN fiber. FE-SEM showed that after grafting with HBP, the average diameter of PAN fibers was amplified. EDS, XPS, and UV–VIS DRS method indicated that under hot steaming condition and with the reducibility of HBP, Ag NPs uniform coating on the PAN-G-HBP. Ag NPs-coated fibers exhibits excellent antibacterial property against Escherichia coli and Staphylococcus aureus. Even under 20 times home washing conditions, the antibacterial reduction of Ag NPs-coated PAN fiber can achieved more than 98.94%.
The functional lipid components found in breast milk, vernix caseosa, and meconium are Branched‐chain Fatty Acids (BCFA). The goal of this study was to establish the existence of BCFA in vernix and meconium in infants born at different gestational ages. TLC plates and GC‐MS were examined for the lipids in vernix caseosa and meconium. The results indicated that there were nine BCFA in vernix caseosa, including iso‐12:0, anteiso‐13:0, iso‐14:0, iso‐15:0, anteiso‐15:0, iso‐16:0, anteiso‐17:0, iso‐18:0, and iso‐20:0. Five BCFA (iso‐12:0, anteiso‐13:0, iso‐14:0, iso‐15:0, and anteiso‐15:0) were not contained in the meconium, suggesting that some of the BCFA may be digested and consumed by infants. In the vernix caseosa, the content of BCFA in triacylglycerol (TAG) and free fatty acid (FFA) was 15.59% and 11.82%, respectively. The vernix caseosa's wax ester fraction contained the highest content of BCFA, reaching up to 16.81%. The carbon chain length of fatty acids (FA) ranged from 12 to 24 in the vernix caseosa and 14 to 22 in meconium samples. The gestational age was likely to affect BCFA concentrations, with the vernix caseosa and meconium BCFA content being significantly higher in full‐term infants than in preterm infants (p < .001). Further research is required into the relationship between BCFA and gut microbiotas.
In this paper, we consider the mean field limit and non-relativistic limit of the relativistic Vlasov–Maxwell particle system to the Vlasov–Poisson equation. With the relativistic Vlasov–Maxwell particle system being a starting point, we carry out the estimates (with respect to N and c) between the characteristic equation of both the Vlasov–Maxwell particle model and the Vlasov–Poisson equation, where the probabilistic method is exploited. In the last step, we take both a large N limit and a non-relativistic limit (meaning c tending to infinity) to close the argument. Deriving such a mean-field limit for interactions with Coulomb singularity is a difficult task; therefore, we introduce an N-dependent mollifier to smoothen the interactions. Note that the length scale of our mollifier is N−α for an α that can be chosen arbitrarily close to but smaller than 1/3, which improves previous results in the literature on deriving Vlasov–Maxwell in the non-relativistic setting.
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