Hamilton–Souplet–Zhang's gradient estimates and Liouville theorems for a nonlinear parabolic equation

In this paper, we study Hamilton–Souplet–Zhang's gradient estimates for positive solutions to the nonlinear parabolic equation

$u_t=\mathrm{\Delta }u+\lambda u^{\alpha }$

on noncompact Riemannian manifolds, where $\lambda ,\alpha$ are two real constants. As an application, we obtain a Liouville-type theorem.     

The mean curvature flow in Minkowski spaces

Studying the geometric flow plays a powerful role in mathematics and physics. In this paper, we introduce the mean curvature flow on Finsler manifolds and give a number of examples of the mean curvature flow. For Minkowski spaces, a special case of Finsler manifolds, we prove the short time existence and uniqueness for solutions of the mean curvature flow and prove that the flow preserves the convexity and mean convexity. We also derive some comparison principles for the mean curvature flow.     

变系数Burgers方程的一种预处理谱方法

本文用预处理Legendre-Galerkin-Chebyshev配置方法来求解二阶变系数Burgers方程,该方法首先对方程进行预处理,然后在空间方向应用Legendre-Galerkin-Chebyshev配置方法,对时间方向采用leap-frog/Crank-Nicolson格式进行离散,这样可使得变系数项和非线性项能够显式计算.我们对该方法进行了误差估计,并通过数值算例验证了算法的有效性和理论分析的正确性.     

Charge-transfer effect at the interface of phthalocyanine-electrode contact studied by scanning tunneling spectroscopy

Scanning tunneling microscopy (STM) and spectroscopy (STS) are used in this work to investigate the charge-transfer effect at the molecule-substrate interface of substituted metal phthalocyanines. STS results revealed that the apparent energy gaps for both fluorinated phthalocyanines and unsubstituted phthalocyanines are essentially the same, which agree with the hybrid density functional calculations. More interestingly, there is a systematic shift of the energy level of valence bands, possibly as the result of charge-transfer effect at the molecule-substrate interface.     

Ab initio electron transport study of carbon and boron-nitrogen nanowires

We report first-principles calculations on the electrical transport properties of two kinds of one-dimensional nanowires: (a) a carbon nanowire (CNW) with alternating single and triple bonds and (b) a boron-nitrogen nanowire (BNNW) with equidistant bonds. We demonstrate the similarity and difference between the carbon nanowire and its boron-nitrogen analogue in the molecular orbital and transport properties, and then explore the potential innovations. The effects of molecular orbitals and nanowire-electrode coupling on the transport properties are analyzed. The cases of the nanowires sandwiched between both nanoscale and bulk electrodes are considered. It suggests that the characteristics of the transmission spectra and the current-voltage characteristics (I-V curves) are determined both by the electrodes and by the molecule as well as their coupling. In particular, the negative differential resistance (NDR) phenomenon is more apparent when the nanowires are positioned between two nanoscale electrodes. The tuning of the transport properties is also probed through the changes of nanowire-electrode separation and the inclusion of a gate voltage. These lead to dramatic variations in the equilibrium conductance, which can be understood from the shift and alignment of the molecular orbital relative to the Fermi level of the electrodes. In the analysis of the effects of nanowire-electrode separation, it shows that the equilibrium conductance has the same variation behavior as that of the projected density of states (PDOS) for CNW, while the localized molecular orbitals of BNNW result in its conductance varies differently from its PDOS. The different molecular orbital characteristics near the Fermi level of these two kinds of nanowires underlie their different transport properties.     

Readily available phosphine-phosphoramidite ligands for highly efficient Rh-catalyzed enantioselective hydrogenations

A new family of air- and moisture-stable phosphine-phosphoramidite ligands (PEAPhos) has been prepared from commercially available and inexpensive (S)-alpha-phenylethylamine through a two- or three-step transformation and successfully applied in the rhodium-catalyzed enantioselective hydrogenations of a variety of substrates, in which up to 99.9% ee was obtained for all of these kinds of substrates.     

A class of binary cyclic codes with five weights

In this paper, the dual code of the binary cyclic code of length 2 n-1 with three zeros α, α t 1 and α t 2 is proven to have five nonzero Hamming weights in the case that n 4 is even and t1 = 2 n/2 + 1, t2 = 2 n-1-2 n/2+1 + 1 or 2 n/2 + 3, where α is a primitive element of the finite field F 2 n . The dual code is a divisible code of level n/2+1, and its weight distribution is also completely determined. When n = 4, the dual code satisfies Ward's bound.     

Random strict convexity and random uniform convexity in random normed modules

Based on the analysis of stratification structure on random normed modules, we first present random strict convexity and random uniform convexity in random normed modules. Then, we establish their respective relations to classical strict and uniform convexity: in the process some known important results concerning strict convexity and uniform convexity of Lebesgue-Bochner function spaces can be obtained as a special case of our results. Further, we also give their important applications to the theory of random conjugate spaces as well as best approximation. Finally, we conclude this paper with some remarks showing that the study of geometry of random normed modules will also motivate the further study of geometry of probabilistic normed spaces.     

A Revisit to the Generalized Radiation Laws of a Kerr Nonlinear Blackbody

Based on the new development of nonextensive statistical mechanics (NSM), i.e., the so-called “optimal Lagrange multipliers” (OLM)approach, we revisit the generalized radiation laws of a Kerr nonlinear blackbody (KNB) and obtain the analytical expressions of generalized Planck radiation law and Stefan-Boltzmann law. In order to illustrate the influence of the parameter q on the two generalized radiation laws, we make a numerical calculation of the two laws under appropriate conditions. Besides, we restate our hypothesis to extend the theory of KNB to the whole Universe.