Geometric Integration Methods that Preserve Lyapunov Functions

We consider ordinary differential equations (ODEs) with a known Lyapunov function V. To ensure that a numerical integrator reflects the correct dynamical behaviour of the system, the numerical integrator should have V as a discrete Lyapunov function. Only second-order geometric integrators of this type are known for arbitrary Lyapunov functions. In this paper we describe projection-based methods of arbitrary order that preserve any given Lyapunov function. AMS subject classification (2000) 65L05, 65L06, 65L20, 65P40     

普物实验教学应科学引导学生进行知能整合

以“电池电动势和内阻的测定”为例,简述了在普物实验教学中,科学引导学生进行知能整合的过程,使科学知识与相关知识的应用实现了最佳的整合,提高了学生设计实验、实验测量、科学评定实验过程及测量结果的能力。     

A potential integration method for Birkhoffian system

This paper is intended to apply the potential integration method to the differential equations of the Birkhoffian system. The method is that, for a given Birkhoffian system, its differential equations are first rewritten as 2n first-order differential equations. Secondly, the corresponding partial differential equations are obtained by potential integration method and the solution is expressed as a complete integral. Finally, the integral of the system is obtained.     

三维轴对称折叠腔CO2激光器的位相锁定

在一新型三维轴对称腔CO2激光器的基础上提出了一种对此激光器位相锁定的方法。通过在输出镜后放置一具有反射和整形作用的凹凸镜，让作为输出注入光束的谐振腔优先起振。此腔输出的光束到达凹凸镜时被其反射。利用重新定义光斑半径的方法，证明此光束能够覆盖其他放电管输出的光束，根据几何光学，此光束能反射进入各个放电管。然后利用光子简并度的概念证明了反射光束能够控制腔内的本征模式，并且数值分析了注入锁定后腔内模式匹配。在反射注入锁定的情况下，给出了数值计算的光强输出及其模拟，结果证明经位相锁定后，激光器输出光束的质量有了很大的提高。     

Small Ball Probabilities of Fractional Brownian Sheets via Fractional Integration Operators

We investigate the small ball problem for d-dimensional fractional Brownian sheets by functional analytic methods. For this reason we show that integration operators of Riemann–Liouville and Weyl type are very close in the sense of their approximation properties, i.e., the Kolmogorov and entropy numbers of their difference tend to zero exponentially. This allows us to carry over properties of the Weyl operator to the Riemann–Liouville one, leading to sharp small ball estimates for some fractional Brownian sheets. In particular, we extend Talagrand's estimate for the 2-dimensional Brownian sheet to the fractional case. When passing from dimension 1 to dimension d2, we use a quite general estimate for the Kolmogorov numbers of the tensor products of linear operators.     

Exact Gaussian expansions of Slater-type atomic orbitals

Three exact Slater-type function (STO) integral transforms are presented. The STO-NG basis set can then be developed using either only 1s Gaussian functions, the same Gaussian exponents for each shell, or using the first Gaussian of each symmetry. The use of any of these three alternatives depends only on appropriate numerical integration techniques.     

Theoretical Investigation of Transparent (3–5 μm) and High Reflective (1.3 μm) Filter by Sol-Gel Process

When designing a cut-off interference filter, extra layers are needed to add on each side of periodic symmetrical multilayer to match the incident medium and the substrate. In this paper, we use sol-gel film with adjustable refractive index as the match layer. The influence of the refractive index and thickness of the match layer on the film system is discussed in detail. As an example, a special filter which is transparent (>80%) at the wavelength 3–5 m and high reflective (>90%) at the wavelength 1.3 m is designed. Finally a series of optimized film parameters are presented. The whole design process is a good reference to the other multilayer special filter made by sol-gel process.