Connection between the ideals generated by traces and by supertraces in the superalgebras of observables of Calogero models

If G is a finite Coxeter group, then symplectic reflection algebra H := H_1,η (G) has Lie algebra of inner derivations and can be decomposed under spin: H = H₀ ⊕ H_1/2 ⊕ H₁ ⊕ H_3/2 ⊕ …?We show that if the ideals of all the vectors from the kernel of degenerate bilinear forms B_i(x, y) := sp_i (x · y), where sp_i are (super)traces on H, do exist, then if and only if .     

Pseudo-Kähler metrics on six-dimensional nilpotent Lie algebras

It is well known that any symplectic manifold (M,Ω) has an almost complex structure J which is compatible with Ω. In this paper, we deal with the existence of compatible pairs (J,Ω) on nilpotent Lie algebras of dimension ≤6, J being an integrable almost complex structure. We prove that if such a pair exists, J must satisfy some extra conditions, namely J must be nilpotent in the sense of [Trans. Am. Math. Soc. 352 (2000) 5405]. Associated to any such a compatible pair, there is a pseudo-Kähler metric g which cannot be positive definite unless be abelian. All these metrics are Ricci flat, although many of them are nonflat, and we study the behaviour of its curvature tensor under deformation.     

Lobachevsky triangle altitudes theorem as the Jacobi identity in the Lie algebra of quadratic forms on symplectic plane

An isomorphism between the Lobachevsky and de Sitter’s world geometries with the symplectic geometry and the Lie algebra of binary quadratic forms is used to derive the altitudes concurrence for the Lobachevsky and de Sitter triangles.     

A Maple Package on Symbolic Computation of Hirota Bilinear Form for Nonlinear Equations

An improved algorithm for symbolic computation of Hirota bilinear forms of KdV-type equations with logarithmic transformations is presented. In the algorithm, the general assumption of Hirota bilinear form is successfully reduced based on the property of uniformity in rank. Furthermore, we discard the integral operation in the traditional algorithm. The software package HBFTrans is written in Maple and its running effectiveness is tested by a variety soliton equations.     

A Maple Package on Symbolic Computation of Hirota Bilinear Form for Nonlinear Equations

An improved algorithm for symbolic computation of Hirota bilinear forms of KdV-type equations with logarithmic transformations is presented. In the algorithm, the general assumption of Hirota bilinear form is successfully reduced based on the property of uniformity in rank. Furthermore, we discard the integral operation in the traditional algorithm. The software package HBFTrans is written in Maple and its running effectiveness is tested by a variety soliton equations.     

The extended trace identity and its application

The trace identity is extended to the general loop algebra. The Hamiltonian structures of the integrable systems concerning vector spectral problems and the multi-component integrable hierarchy can be worked out by using the extended trace identity. As its application, we have obtained the Hamiltonian structures of the Yang hierarchy, the Korteweg-de--Vries (KdV) hierarchy, the multi-component Ablowitz--Kaup--Newell--Segur (M-AKNS) hierarchy, the multi-component Ablowitz--Kaup--Newell--Segur Kaup--Newell (M-AKNS--KN) hierarchy and a new multi-component integrable hierarchy separately.     

Elliptic dynamical reflection algebra and partition function of SOS model with reflecting end

In this paper we introduce an elliptic dynamical reflection algebra describing an SOS model with a reflecting end. Using a factorizing Drinfel’d twist, we compute the partition function of this model with domain wall boundary conditions. We show that it can be represented in the form of a single Izergin determinant.     

The coefficients of reflection and refraction on interface of the crystals in the Savart polariscope

According to the boundary conditions of the electromagnetic field, the coefficients of reflection and refraction on the interface of crystal in the Savart polariscope are obtained for ordinary ray (o-ray) and extraordinary ray (e-ray).The coefficients of reflection and refraction are different from an isotropic medium and have more complicated forms than isotropic medium.     

The reflection and scattering of light by a plant leaf

The spectral polarization characteristics of the visible radiation reflected and scattered by a plant leaf were investigated. Measurements of the spectral brightness coefficient, the directed reflection coefficients, and the degree and azimuth of polarization of radiation reflected from a leaf were performed at different angles of incidence of radiation on the leaf and different angles of observation. More than 500 measurements were performed for 36 samples. Maple (Acer platanoides) leaves from different trees were used as objects of investigation. Peculiarities of the mechanism of formation of the flux of radiation reflected from a plant leaf have been revealed. Belarussian State University, 4, F. Skorina Ave., Minsk, 220080, Belarus. Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 66, No. 1, pp. 101–108, January–February, 1999.     

亚皮秒光孤子脉冲在突变界面的反射

本文建立了决定亚皮秒孤子在小突变界面的反射支配方程和边界条件，给出了不同的亚皮秒孤子的界面反射透射特性的数值模拟，讨论了孤子遇到界面后能继续无形变传输的条件。     

全反射对混合型光纤电流传感器检测灵敏度的影响

在利用法拉第(Faraday)效应的混合型光纤传感器中,通常用全反射改变光的传递方向,以增加有效光程,并使结构紧凑。反射将改变光的偏振状态,从而影响传感的灵敏度。本文用琼斯(Jones)矩阵方法推导了有反射时传感头的传光特性,分析了反射对检测灵敏度的影响,获得了与实验一致的结果。     

基于旋转式反射镜的飞行视觉系统设计

在MEMS麦克风贴片工艺中,真空吸嘴吸取MEMS芯片以后需要进行芯片位姿的精确校正。为实现高速高精度贴片功能,提出了一种基于旋转式反射镜的飞行视觉系统,该方案利用齿轮齿条将吸嘴的升降运动转化为反射镜的旋转运动,通过设计齿轮参数,实现反射镜不干涉吸嘴的拾放功能。当反射镜旋转到与水平位置夹角呈45°时,视觉系统可以获得MEMS芯片的静止的清晰图像,CCD相机采集一幅图像后,在贴片头飞行的过程中,控制系统完成MEMS芯片的位置和角度的校正。实验结果表明,该系统可以获取高质量MEMS芯片图像。该系统结构简单紧凑,成像无失真,不限制贴片头的运动轨迹,显著提高了MEMS芯片的贴片质量和贴片效率,可广泛地应用于贴片机系统中。     

Study of reflection and transmission on the moving interface of medium

The laws of refection and transimission on a moving interface between two nonvacuum medium are studied and the new method, which can be used for handling reflection and transimission on the moving interfaces between the vacuum-nonvacuum media and between two nonvacuum medium is suggested. The formula of reflection and transmission coefficients that suitable for both cases are derived. The present paper not only enriches electrodynamical theory, but also laies the foundation of optical waveguide modulators and other similar device.     

Study of reflection and transmission on the moving interface of medium

The laws of refection and transimission on a moving interface between two nonvacuum medium are studied and the new method, which can be used for handling reflection and transimission on the moving interfaces between the vacuum-nonvacuum media and between two nonvacuum medium is suggested. The formula of reflection and transmission coefficients that suitable for both cases are derived. The present paper not only enriches electrodynamical theory, but also laies the foundation of optical waveguide modulators and other similar device.     

基于全反射原理的圆台压平棱镜设计与应用实验

利用全反射原理,设计了特殊的圆台压平棱镜探头,可线性测量角膜的压平面积.本文介绍了圆台压平棱镜的设计方法,给出了测量压平面积的实验装置和测量结果.     

印制线抗电磁干扰模型分析与计算

对双导体传输线的Taylor模型在频域下的非齐次线性微分方程组,采用常数变异法求得其在边界条件下解的表达式,所求得传输线在终端响应的电压和电流值与BLT方程求得的结果相同。同时,还得到了印刷电路板上的平行双线在外电磁场辐射下终端响应电压的解析表达式,并通过解析解分析得到了不产生干扰电压时的电磁波的入射角度和极化方向。     

印制线抗电磁干扰模型分析与计算

对双导体传输线的Taylor模型在频域下的非齐次线性微分方程组,采用常数变异法求得其在边界条件下解的表达式,所求得传输线在终端响应的电压和电流值与BLT方程求得的结果相同。同时,还得到了印刷电路板上的平行双线在外电磁场辐射下终端响应电压的解析表达式,并通过解析解分析得到了不产生干扰电压时的电磁波的入射角度和极化方向。     

The generalized vectorial laws of reflection and refraction applied to the rotation problems in ray optics

This paper deals with the mirror rotation problem and the problem of rotation of refracting surface in ray optics. These two problems of rotation in ray optics have been dealt with on the basis of the generalized vectorial laws of reflection and refraction discovered by the author in 2005. In addition to the development of many interesting physical insights to the aforesaid rotation problems in ray optics, the most remarkable fact that has been discovered in the present study is that the proposition ‘Velocity of light is unattainable’ is not correct. Rather, it is possible to have velocity exceeding the velocity of light - a result not in agreement with the special theory of relativity.