Exact evaluation of the multidimensional Franck–Condon integrals based on the contour integral method

Adopting a different method from the Sharp–Rosenstock generating function method and the Doktorov coherent state method, a closed form expression for the Franck–Condon integrals between arbitrary multidimensional harmonic oscillators is derived exactly under the Duschinsky mixing effects on the base of the contour integral form expression of Hermite polynomials. In addition, an analytical expression for the calculation of the four-mode Franck–Condon factors is presented.     

Phase diagram of two-dimensional binary Yukawa mixtures

We have used Ramakrishnan–Yussouff (RY) density functional theory (DFT) to explore the topology of the phase diagram of two-component charge stabilised colloidal suspensions confined to a two-dimensional plane. The particles of the system interact via purely repulsive soft core Yukawa potential. Pair correlation functions (PCFs) used as input informations in DFT were calculated by solving both the hypernetted chain (HNC) and Percus–Yevick (PY) integral equation theories. To test the relative performance of the HNC and PY theories in the context of phase transitions, we have also studied the corresponding one-component systems. We found that RY DFT with HNC PCFs does not stabilise solid in both the one- and two-component cases, whereas the PY theory does. By considering the freezing into the substitutionally disordered triangular solid, we found that the temperature-composition phase diagrams of the binary mixture are narrow spindles whose thickness depends on the symmetry of the mixture components and the value of the screening constant of the Yukawa potential. Although the phase diagram obtained by RY DFT with structural inputs calculated by the PY theory is found to be shifted to higher temperature region in the temperature-composition plane, however, it captures qualitatively all the essential features of the phase diagram. Our results are in principle verifiable through computer simulations and experiments.     

用一种分数阶算法研究非马尔可夫过程中阻尼与涨落的竞争机制

基于分数阶朗之万方程和随机行走理论, 建立了一种用于研究非马尔可夫系统中随机变量随时间演化的数值模拟算法, 称之为分数阶随机行走模拟法. 进一步运用此算法分别数值研究了无阻尼有涨落、 有阻尼无涨落和阻尼与涨落兼备三种情况下, 受欠扩散分数阶朗之万方程约束的随机变量随时间的演化行为. 结果显示阻尼和涨落存在竞争关系: 高斯型涨落的影响会随着时间的增长被"抹平", 从而凸显阻尼使系统趋于平衡的作用; 而长尾型涨落则由于包含"小概率大贡献"事件, 使得长时间演化之后系统变量仍以一定概率出现突然变化. 关键词： 非马尔可夫 欠扩散 阻尼与涨落 分数阶朗之万方程     

一种分数阶非线性系统稳定性判定定理的问题及分析

分数阶非线性系统稳定性理论的研究对于分数阶混沌系统同步控制的应用具有重要价值,将分数阶非线性系统稳定性判断转化为相应整数阶非线性系统稳定性判断的探讨很有意义.通过实例表明了:对于时变系数矩阵,如果整数阶系统稳定,其对应的阶次小于1的分数阶系统也稳定的判定定理是错误的,并分析了问题产生的原因.     

关于谐振子第一积分的研究

提出一维谐振子基本积分的概念, 并利用它们构造其他积分. 将上述概念和方法推广到多维谐振子, 利用直接构造法证明不同类型的二维谐振子都存在三个与时间无关的独立的第一积分, n维谐振子系统存在2n-1个与时间无关的独立的第一积分. 讨论了分振动频率比为有理数和无理数的二维非对称谐振子的特征. 关键词： 谐振子 第一积分 基本积分 不含时积分     

数值积分求解均匀带电圆环平面上的电场分布

本文用数值积分的方法计算椭圆积分，求出在均匀带电圆环平面上的场强与电势．     

分数布朗运动和反常扩散

本文评述了分数布朗运动和反常扩散现象及描写它们的几种数学方式。报告了我们在弹道扩散的产生条件、起源和长时间效应方面的工作。     

One-body and Two-body Fractional Parentage Coefficients for Spinor Bose-Einstein Condensation

A very effective tool, namely, the analytical expression of the fractional parentage coefficients (FPC), is introduced in this paper to deal with the total spin states of N-body spinor bosonic systems, where N is supposed to be large and the spin of each boson is one. In particular, the analytical forms of the one-body and two-body FPC for the total spin states with {N} and {N−1,1} permutation symmetries have been derived. These coefficients facilitate greatly the calculation of related matrix elements, and they can be used even in the case of N→∞. They appear as a powerful tool for the establishment of an improved theory of spinor Bose-Einstein condensation, where the eigenstates have the total spin S and its Z-component being both conserved.     

On the positive almost periodic type solutions for some nonlinear delay integral equations

In this paper, we discuss the existence of positive almost periodic type solutions for some nonlinear delay integral equations, by constructing a new fixed point theorem in the cone. Some known results are extended.     

Integral Representations in Linearly Convex Polyhedra

We introduce the notion of mixed Levian (Levi determinant) of a system of k Hermitian and k linear forms. In terms of these Levians we obtain an integral formula for holomorphic functions in linearly convex polyhedra.Original Russian Text Copyright © 2005 Krivokolesko V. P. and Tsikh A. K.The first author was supported by a grant of the President of the Russian Federation and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-1212.2003.1); the second author was supported by the Ministry for Education of the Russian Federation (St. Petersburg, Grant 02-1-138).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 579–593, May–June, 2005.