Research on the propagation properties of hollow laser beams with three-dimensional trap optical distribution

A new kind of hollow beams - hollow laser beams with three-dimension trap optical distribution - was put forward. With the help of the Collins formula in paraxial optical system, the analytical equation of propagation and transformation of the hollow laser beams was deduced. According to the analytical equation, the propagation properties of the kind of hollow beams that transform in free space were simulated. In the experiment, we obtained the hollow laser beams by means of the combinational optical system of reflecting positive-axis and negative-axis pyramids. The intensity of the vertical loop in different distances was tested, which shows that the analytical equation of propagation and transformation is in agreement with the result.     

An algebraic cubature formula on curvilinear polygons

We implement in Matlab a Gauss-like cubature formula on bivariate domains whose boundary is a piecewise smooth Jordan curve (curvilinear polygons). The key tools are Green’s integral formula, together with the recent software package Chebfun to approximate the boundary curve close to machine precision by piecewise Chebyshev interpolation. Several tests are presented, including some comparisons of this new routine ChebfunGauss with the recent SplineGauss that approximates the boundary by splines.     

贝叶斯公式的几个应用

介绍了贝叶斯公式的一些应用实例及分析,以使在教学中能帮助学生更深入地理解该公式.     

Bounds for increasing multi-state consecutive k-out-of-r-from-n: F system with equal components probabilities

The consecutive k-out-of-r-from-n: F system was generalized to multi-state case. This system consists of n linearly ordered components which are at state below j if and only if at least k_j components out of any r consecutive are in state below j. In this paper we suggest bounds of increasing multi-state consecutive-k-out-of-r-from-n: F system (k₁ ? k₂ ? ? ? k_M) by applying second order Boole–Bonferroni bounds and applying Hunter–Worsley upper bound. Also numerical results are given. The programs in V.B.6 of the algorithms are available upon request from the authors.     

Explicit formulas for Laplace transforms of certain functionals of some time inhomogeneous diffusions

We consider a process given by the SDE , t∈[0,T), with initial condition , where T∈(0,∞], α∈R, (Bt)_t∈[0,T) is a standard Wiener process, b:[0,T)→R?{0} and σ:[0,T)→(0,∞) are continuously differentiable functions. Assuming , t∈[0,T), with some K∈R, we derive an explicit formula for the joint Laplace transform of and for all t∈[0,T) and for all α∈R. Our motivation is that the maximum likelihood estimator (MLE) of α can be expressed in terms of these random variables. As an application, we show that in case of α=K, K≠0,

     

Stochastic calculus for Markov processes associated with non-symmetric Dirichlet forms

Nakao’s stochastic integrals for continuous additive functionals of zero energy are extended from the symmetric Dirichlet forms setting to the non-symmetric Dirichlet forms setting.It? ’s formula in terms of the extended stochastic integrals is obtained.     

构造二元向量值有理插值函数的一种新方法

At present, the methods of constructing vector valued rational interpolation function in rectangular mesh are mainly presented by means of the branched continued fractions. In order to get vector valued rational interpolation function with lower degree and better approximation effect, the paper divides rectangular mesh into pieces by choosing nonnegative integer parameters d1 （0 〈 dl ≤ m） and d2 （0 ≤ d2≤ n）, builds bivariate polynomial vector interpolation for each piece, then combines with them properly. As compared with previous methods, the new method given by this paper is easy to compute and the degree for the interpolants is lower.     

Remarks on asymptotic behaviors of strong solutions to a viscous Boussinesq system

In this paper, we first address the space‐time decay properties for higher‐order derivatives of strong solutions to the Boussinesq system in the usual Sobolev space. The decay rates obtained here are optimal. The proof is based on a parabolic interpolation inequality, bootstrap argument, and some weighted estimates. Secondly, we present a new solution integration formula for the Boussinesq system, which will be employed to establish the existence of strong solutions for small initial data in some scaling invariant function spaces. The smallness conditions are somehow weaker than those presented by Brandolese and Schonbek. We further investigate the asymptotic profiles and decay properties of these strong solutions. Copyright © 2016 John Wiley & Sons, Ltd.