The homogeneous polynomial solutions for the Grushin operator

In this paper, the author computes the dimension of space of homogeneous Grushin-harmonic functions, and give an orthogonal basis of them. Moreover, the author describes the nodal curves of these homogenous Grushin-harmonic basis. As an application of the orthogonal basis, the author proves a Liouville-type theorem for the Grushin operator, that is the Grushin-harmonic functions are homogeneous polynomials provided that the frequency of such a function is equal to some constant.     

Polarization Maintaining Fibre Loop Mirror for NRZ-to-PRZ Conversion in All-Optical Clock Recovery

We propose a novel configuration for clock extraction by converting the NRZ data into the PRZ data and by employing a polarization-maintaining fibre loop mirror （PMFLM） which is usually used as an optical comb filter. It is found that the PMFLM can simply be constructed by a polarization controller and polarization-maintaining fibre （PMF）. We theoretically analyse the principle of PMFLM for the NRZ-to-PRZ conversion. Experimentally we demonstrate 10 Gbit/s all-optical clock recovery through our proposed setup, It is shown that recovered clock signal with an extinction ratio above 10 dB can be achieved.     

Simulation study on the formation and transition properties of cluster structures in liquid metals during rapid cooling processes

For the first time, a molecular dynamics simulation study has been performed for a liquid metal system consisting of 50000 atoms to deeply investigate the transitions of microstructure configurations dudng the rapid cooling processes. Especially, the cluster-type index method has been adopted to analyze the transforming and evolving processes of clusters and cluster configurations from liquid metal atoms. It has been found that the bigger cluster configurations in the system are formed by means of connecting some small clusters (they are combined by several smaller clusters), and not taken on the multi-shells configuration accumulated with an atom as the center and the surrounding atoms arranged according to some fixed pattern. With the decrease in temperature, the probability of repetitive appearance for clusters increases largely, which reveals that clusters are indeed possessing a certain relative stability and continuity (namely hereditary effect). These results will give us an important enlightenment to understand not only the forming mechanisms and microscopic processes of the short-order sections and disorder sparse sections in amorphous structures but also the freezing processes of liquid metals.``     

Hessian型方程Neumann边值问题的梯度估计

通过构造辅助函数,利用基本对称函数的性质以及函数在极大值点的性质,得到Hessian型方程S_k(D~2u-A(x,u,Du))=B(x,u)的梯度内估计,构造不同的辅助函数,分近边、边界和内部3种情形讨论该方程Neumann边值问题,进而得到全局梯度估计.