Geometry and nonlinear evolution equations

We briefly review the nonlinear dynamics of diverse physical systems which can be described in terms of moving curves and surfaces. The interesting connections that exist between the underlying differential geometry of these systems and the corresponding nonlinear partial differential equations are highlighted by considering classic examples such as the motion of a vortex filament in a fluid and the dynamics of a spin chain. The association of the dynamics of a non-stretching curve with a hierarchy of completely integrable soliton-supporting equations is discussed. The application of the surface embeddability approach is shown to be useful in obtaining such connections as well as exact solutions of some nonlinear systems such as the Belavin-Polyakov equation and the inhomogeneous Heisenberg chain.     

Mathematical methods for a class of mixed boundary-value problems of planar pentagonal quasicrystal and some solutions

The mathematical theory of elasticity for planar pentagonal quasicrystals is developed and some analytic solutions for a class of mixed boundary-value problems (corresponding to a Griffith crack) of the theory are offered. An alternate procedure and a direct integral approach are proposed. Some analytical solutions are constructed and the stress and displacement fields of a Griffith crack in the quasicrystals are determined. A basis for further studying the mechanical behavior of the material related to planar defects is provided. Project supported by the Foundation of State Education Commission of China for Doctorate Station.     

Higher-dimensional cosmological model with variable gravitational constant and bulk viscosity in Lyra geometry

We have studied five-dimensional homogeneous cosmological models with variableG and bulk viscosity in Lyra geometry. Exact solutions for the field equations have been obtained and physical properties of the models are discussed. It has been observed that the results of new models are well within the observational limit.     

GEOM: A new tool for molecular modelling based on distance geometry calculations with NMR data

Summary GEOM is a new graphics tool which allows the use of distance geometry to compute linear and cyclic structures typically arising in drug design situations. Modified amino acids or monomeric organic entities can be easily constructed in an interactive way and deposited in the library of the distance geometry program together with geometric information required for structure calculation in dihedral angle space. In addition, GEOM is able to produce all files needed to calculate a structure based on NMR data (NOE and J-coupling constraints) and it permits the graphic analysis and comparison of computed structures. The application of GEOM is demonstrated in three examples: modelling of cyclosporin A structures with and without a limited set of H-bond constraints and modelling of a cyclic hexapeptide with a full NMR data set.     

Determination of Optical Waveguide Refractive-Index Profiles with the Inverse Analytic Transfer Matrix Method

In this paper, we present an inverse analytic transfer matrix (IATM) method to predict the refractive index profiles from the measurement of mode indices based on the analytic transfer matrix (ATM) method, which can exactly determine the modal characteristics of a planar optical waveguide with arbitrary index profiles. IATM method has been proved to be universally reliable not only in slowly changing profiles but also the profiles changing rapidly in which inverse WKB method would find difficult through numerical analysis for several typical refractive index profiles.     

An approximation algorithm for cutting out convex polygons

We provide an O(logn)-approximation algorithm for the following problem. Given a convex n-gon P, drawn on a convex piece of paper, cut P out of the piece of paper in the cheapest possible way. No polynomial-time approximation algorithm was known for this problem posed in 1985.     

Geometric Formulation of Gauge Theory of Gravity

Differential geometric formulation of quantum gauge theory of gravity is studied in this paper. The quantumgauge theory of gravity is formulated completely in the framework of traditional quantum field theory. In order to studythe relationship between quantum gauge theory of gravity and traditional quantum gravity which is formulated in curvedspace, it is important to set up the geometry picture of quantum gauge theory of gravity. The correspondence betweenquantum gauge theory of gravity and differential geometry is discussed and the geometry picture of quantum gaugetheory of gravity is studied.     

Normals in a Minkowski Plane

In this paper we will establish bounds on the average number of normals through a point in a convex body in a Minkowski plane for certain classes of convex bodies. Also, a related Euler relation is discussed.     

粘弹性半空间上刚体的垂直振动

根据弹性力学混合边值条件，建立了粘弹性半空间上刚体垂直振动的对偶积分方程，并用正交多项式化积分方程为线性代数方程组，并提出可以用围线积分和解析开拓原理把方程组系数的无穷积分化为有穷积分。计算结果与实测资料进行了比较，说明本方法是正确的。