Weighted estimates for the multilinear maximal function on the upper half‐spaces

For a general dyadic grid, we give a Calderón–Zygmund type decomposition, which is the principle fact about the multilinear maximal function on the upper half‐spaces. Using the decomposition, we study the boundedness of . We obtain a natural extension to the multilinear setting of Muckenhoupt's weak‐type characterization. We also partially obtain characterizations of Muckenhoupt's strong‐type inequalities with one weight. Assuming the reverse Hölder's condition, we get a multilinear analogue of Sawyer's two weight theorem. Moreover, we also get Hytönen–Pérez type weighted estimates.     

Optimal strategy of vaccination and treatment in an SIRS model with Markovian switching

Medical treatment and vaccination decisions are often sequential and uncertain. Markov decision process is an appropriate means to model and handle such stochastic dynamic decisions. This paper studies the near‐optimality of a stochastic SIRS epidemic model that incorporates vaccination and saturated treatment with regime switching. The stochastic model takes white noises and color noise into account. We first prove some priori estimates of the susceptible, infected, and recovered populations. Moreover, we establish some sufficient and necessary conditions of the near‐optimality by Pontryagin stochastic maximum principle. Our results show that the two kinds of environmental noises have great impacts on the infectious diseases. Finally, we illustrate our conclusions through numerical simulations.     

Synthesis,Crystal Structure,and Selected Properties of [Au(S2CNH2)2]SCN: A Precursor for Gold Macro-Needles Consisting of Gold Nanoparticles Glued by Graphitic Carbon Nitride

A new preparation route is developed for the synthesis of needle-like crystals of [Au(S₂CNH₂)₂]SCN, which avoids disproportionation of the Au^I salt used as a starting material. In the crystal structure, the two crystallographically independent Au^III centers are in a square-planar environment of two S₂CNH₂ ligands. The Hirshfeld surface analysis reveals the presence of noncovalent intermolecular S⋅⋅⋅S interactions, which are essential for the spatial arrangement of the molecules. Density functional theory (DFT) calculations including dispersion and damping corrections result in a unit cell volume very close to the value determined experimentally. Thermal decomposition in an inert atmosphere generates black needles with lengths of up to 500 μm. X-ray powder diffraction and pair distribution function analyses demonstrate that the needles are composed of nanosized crystals with a volume-weighted average domain size of 20(1) nm. According to results of X-ray photoemission experiments, the black needles are covered by a nitrogen-rich carbon nitride with composition near (CN)₂N. ¹³C solid-state NMR investigations indicate that two different carbon species are present, with signals corresponding well to heptazine units as in melon and triazine units as in poly(triazin imide) type compounds. Scanning transmission electron microscopy tomography evidences that the needles are composed of slightly elongated nanoparticles.     

Dynamics of water in real space and time

Owing to marked advances in instrumentation in X-ray and neutron scattering the time-dependent pair correlation function, the Van Hove function, can now be determined by inelastic X-ray and neutron scattering measurements. The local dynamics of water in real space and time is visualised by this approach. We discuss how the dynamic properties, such as viscosity and diffusion, can be elucidated through the Van Hove function of water.     

Statistics of reaction progress variable and mixture fraction gradients of a pulverised coal jet flame using Direct Numerical Simulation data

The statistical behaviour of the variances, covariance and gradients of the reaction progress variable (c), and the mixture fraction (ξ) have been analysed in a pulverised coal jet flame using a three-dimensional carrier phase direct numerical simulation (DNS) dataset. It has been observed that the Favre-probability density functions (PDFs) of c and ξ can be parametrised by the standard β function. Furthermore, the log-normal distribution has been found to accurately represent |?c| and |?ξ|. It is also found that ?c and ?ξ remain aligned throughout the flame brush. Finally the joint PDF of |?c| and |?ξ| has been compared with the product of the PDF of |?c| and PDF of |?ξ| extracted from carrier phase DNS, and it has been found that |?c| and |?ξ| are not statistically independent in the case investigated.The bivariate log-normal distributions with and without correlation have also been considered, and the former is found to be in better agreement with the carrier phase DNS data.     

Smarandache函数在两数列上的下界估计

设S(n)是Smarandache函数,其中n是一正整数.讨论Smarandache函数S(n)在数列F((2k),1)=F(n,1)=n2n+1(n=2k)与数列G(2n,1)=(2n)2n+1上的下界估计.基于初等方法证明了:当偶数n≥6时,有S(F((2k),1))=S(F(n,1))≥6×2n+1;当n≥4时,有S(G(2n,1))≥6×2n+1.