Spherical Bessel functions and explicit quadrature formula

An evaluation of the derivative of spherical Bessel functions of order at its zeros is obtained. Consequently, an explicit quadrature formula for entire functions of exponential type is given.

     

Some ramifications of a theorem of Boas and Pollard concerning the completion of a set of functions in

About fifty years ago, R. P. Boas and Harry Pollard proved that an orthonormal system that is completable by the adjunction of a finite number of functions also can be completed by multiplying the elements of the given system by a fixed, bounded, nonnegative measurable function. In subsequent years, several variations and extensions of this theorem have been given by a number of other investigators, and this program is continued here. A mildly surprising corollary of one of the results is that the trigonometric and Walsh systems can be multiplicatively transformed into quasibases for .

     

Boundary Controllability of a Stationary Stokes System with Linear Convection Observed on an Interior Curve

We study the approximate controllability of a stationary Stokes system with linearized convection in a bounded domain of ^N. The control acts on a part of the boundary and the velocity field is observed on an interior curve (N=2) or surface (N=3). We establish the L ²-approximate controllability under certain compatibility conditions and suitable geometrical assumptions on the curve or surface. We build controls of minimal L ²-norm by duality. To compute the control, we propose a numerical method, based on duality techniques, consisting in the minimization of a nonquadratic functional coupled to a Stokes system. It is tested in several situations leading to interesting numerical results.     

An Extension of Itô's Formula for Anticipating Processes

In this paper we introduce a class of square integrable processes, denoted by L^F, defined in the canonical probability space of the Brownian motion, which contains both the adapted processes and the processes in the Sobolev space L^2,2. The processes in the class L^F satisfy that for any time t, they are twice weakly differentiable in the sense of the stochastic calculus of variations in points (r, s) such that r s t. On the other hand, processes belonging to the class L^F are Skorohod integrable, and the indefinite Skorohod integral has properties similar to those of the Ito integral. In particular we prove a change-of-variable formula that extends the classical Itô formula. Those results are generalization of similar properties proved by Nualart and Pardoux⁽⁷⁾ for processes in L^2,2.     

Dirichlet Forms and Dirichlet Operators for Gibbs Measures of Quantum Unbounded Spin Systems: Essential Self-Adjointness and Log-Sobolev Inequality

For each [0, 1] we consider the Dirichlet form and the associated Dirichlet operator for the Gibbs measure of quantum unbounded spin systems interacting via superstable and regular potential. The Gibbs measure is related to the Gibbs state of the system via a (functional) Euclidean integral procedure. The configuration space for the spin systems is given by We formulate Dirichlet forms in the framework of rigged Hilbert spaces which are related to the space . Under appropriate conditions on the potential, we show that the Dirichlet operator is essentially self-adjoint on the domain of smooth cylinder functions. We give sufficient conditions on the potential so that the corresponding Gibbs measure is uniformly log-concave (ULC). This property gives the spectral gap of the Dirichlet operator at the lower end of the spectrum. Furthermore, we prove that under the conditions of (ULC), the unique Gibbs measure satisfies the log-Sobolev inequality (LS). We use an approximate argument used in the study of the same subjects for loop spaces, which in turn is a modification of the method originally developed by S. Albeverio, Yu. G. Kondratiev, and M. Röckner.     

An Extension of Feuerbach's and Luchterhand's Volume Relation

We generalize a result of Feuerbach and Luchterhand from n+2 points on a sphere in Rⁿ to any m n+2 points on the sphere.     

二阶变系数线性微分方程的求解定理

先提出引理,即某函数是二阶变系数线性齐次微分方程的解的充要条件,再给出在已知二阶变系数线性齐次微分方程的某一解的条件下,二阶变系数线性非齐次微分方程的通解公式——即定理1,然后借助引理及定理1提供了几类二阶变系数线性非齐次微分方程通解的积分表达式,从而获得求几类方程通解的统一方法.     

线性的变振动系统动力学方程的一类解法——区间状态转移矩阵逼近算法

依据线性时变微分方程组理论,以时间离散化数值解来逼近相应解析解的方式,在首次放弃将系统在一系列小区间内假定为时不变的前提下,提出了一种建立在线性时变振动系统动力学方程系数矩阵运算基础上的区间状态转移矩阵逼近算法。该算法不仅解题规模小,计算效率高,而且理论依据充足,数值计算可靠,既可用于一般线性时变振动系统的瞬态响应分析,又可用于周期系数线性振动系统的稳态响应求解和动态特性研究,具有很好的适用性。     

有关“二极管特性曲线测定”实验的讨论

普通物理实验教材中,对“二极管特性曲线测定”实验未加讨论直接采用了Id＝Ie（exp（eV/kT）-1）的解析公式,要求由实验结果求出指数e/kT,从而求出电子电荷e,或波尔兹曼常数k,并和理论值比较,考察实验误差．这种要求由于实际结果偏离解析公式较大,因而不尽合理．本文讨论了有关实验结果,并对上述结论作了必要的说明．最后建议对该实验的要求作合理的调整,即主要考察V—A的一般关系．     

熵变与热力学方程关系的一种简便推导

由热力学第二定律的基本微分方程和理想气体状态方程,导出态函数熵的表达式,从而得出常见的过程方程．