一阶导数形式不变性

针对数学教材中由导数基本公式到复合函数求导法则过渡形式不一致现象,提出一阶导数的一致形式、修订复合函数求导法则、给出导数基本公式的一般形式.     

A Nonlinear Sandwich Shell Theory Accounting for Transverse Core Compressibility

The present study is concerned with an advanced theory of sandwich shells with transversely compressible core. The model is based on the standard Kirchhoff‐Love hypothesis for the face sheets and a third‐order displacement expansion for the core. Consistent equations of motion and boundary conditions are derived by means of Hamilton's variational principle. The model is applied to a postbuckling analysis of cylindrical shells under axial compression.     

用张量导数法求横观各向同性弹性材料的弹性张量的一般形式

用对张量函数求导的方法导出了横观各向同性材料和各向同性材料的弹性张量的一般形式与应力-应变关系式.从推导过程可更清楚地看出为什么横观各向同性材料和各向同性材料分别有五个和两个独立的弹性常数,即材料有几个独立的弹性常数是由其应变能函数的形式所决定的.     

一阶线性微分方程组的一种解法

利用矩阵知识给出了一阶线性微分方程组的一种用公式表达的解法,其优点在于一方面可以避免繁琐的复矩阵运算以及求复特征向量的运算,另一方面可以简化求解过程.     

A Shell Model for Optimal Mixing

What is the maximum mixing efficiency of an incompressible flow? To address this question we introduce a shell model—a reduced model mimicking the kinematics of advection and diffusion—to study the evolution of an initially inhomogeneous tracer concentration carried by a given incompressible fluid on a periodic spatial domain. We pose the mixing task as an optimization problem: Find the divergence-free velocity field (the control variable) that produces a well-mixed tracer concentration field (the state variable). We consider two alternative objectives: local-in-time optimization (maximize the instantaneous mixing rate) and global-in-time optimization (maximize mixing at a prescribed final time). Throughout, we use a shell-model analog of the \(H^{-1}\) mix-norm to measure mixing. In addition, lower bounds on the mix-norm are obtained and rule out perfect mixing in finite time in particular cases.     

Structure of a Linear Array of Uniform Vortices

The shape and properties of an infinite steady linear array of uniform vortices are calculated. A nonlinear singular integrodifferential equation is obtained for the shapes, which is solved numerically by Newton's method and Euler continuation to give a one parameter family of shapes as size over separation is varied. The kinematic properties and energy of the array are obtained. It is found that there exists an array of maximum area, for given separation, which also possesses minimum energy in accordance with a general argument of Kelvin. A simple model based on elliptical vortices is constructed, which reproduces the qualitative kinematic properties and is quantitatively quite accurate. Continuation of the numerical solution past the array of maximum area leads to a limit of finite, lens shaped, touching vortices. This array is also shown to be limit of a finite amplitude bifurcation of a vortex sheet of finite thickness. The stability of the array to two dimensional subharmonic and superharmonic disturbances is considered. General arguments, based on ideas of Kelvin, are given to show that the array is stable to superharmonic disturbances if the area is less than the maximum and otherwise unstable, and that it is always unstable to subharmonic disturbances, of which the pairing instability is a special case. It is verified by direct calculation in an Appendix that hollow vortices, whose shapes can be determined analytically in closed form, are unstable to the pairing instability whatever their size. Some speculations are made about the possible relevance of the results to the observed properties of organized structures in the turbulent mixing layer.     

Buckling of a Complete Spherical Shell Under Uniform External Pressure

Axisymmetric buckling of a complete spherical shell under uniform external pressure is treated by the spectral method. Among other results the lower critical pressures of 7% and 5% of classical linear buckling pressures are computed for radius-thickness ratios of 100 and 200 respectively. Both symmetric (non-physical) and non-symmetric buckling are computed. The non-symmetric buckled shapes at the two lower critical pressures mentioned are virtually identical.     

非对称自变量线性各向同性张量函数的表示

本文给出了非对称自变量的线性各向同性标量值和张量值张量函数表示定理的数学证明.     

Inflation-Extension and Eversion of a Tube of Incompressible Isotropic Elastic Material

Adopting a form of strain-energy function for highly deformableincompressible elastic materials recently proposed by Ogden,we study in this paper two deformations of a circular cylindricaltube. In the first deformation the tube is inflated by increasingits internal radius by an assigned factor, and in the secondthe tube is turned inside out. In each case it is assumed thatthe tube is free of longitudinal constraint to the extent thatthe normal tractions on the end-faces have zero resultant. Inflationand eversion are therefore accompanied by axial shortening orelongation. Detailed numerical results are presented and comparedwith the limited experimental data to hand. Consideration isalso given to the existence and uniqueness of deformations whichpreserve the shape of the tube and comply with the assumed boundaryconditions. Restrictions are obtained on the material constantsappearing in the strain-energy function which ensure the existenceof precisely one such solution to the problem of combined inflationand extension, and sufficient conditions for the existence ofat least one solution to the eversion problem are also found.These requirements agree with criteria for "physically realisticresponse" which have been established elsewhere.     

一类二阶线性时变系统的稳定性

本文讨论了一类二阶线性时变系统在临界情况下的稳定性，给出了保证该系统零解稳定的充分条件，这一结果将拓宽控制论中二维线性时变控制系统的研究范围。     

对流扩散方程特征线三角元法的一致估计

利用三角形线性元的积分恒等式,给出了二维非定常对流占优扩散方程的特征线有限元解和真解的一致最优估计,并利用插值后处理算子,得到了有限元解梯度的一致超收敛估计,即只与初值和右端项有关,而与ε无关.     

协变量区间删失时线性模型的参数估计

本文研究了当协变量为区间数据时的线性模型，通过构造区间数据变量的条件均值，得到了回归参数的估计，当协变量的分布已知时，证明了估计的无偏性与强相合性．时协变量的分布未知的情形也作了讨论．文中还作了若干模拟计算，从模拟的结果不难发现，利用本文提出的方法所获得的估计简便且具有较高的精度．     

Uniform Asymptoti Integration of a Family of Linear Differential Systems

Consider a family of linear differential systems y′ =A(x, z)y depending on a parameter z Assume, as in Levinson's Theorem, that A = Λ + R with Λ diagonal and R integrable. We discuss the problem of asymptotically solving this systemwith uniform (in z) control on the error terms.Questions of this type occur in the spectral theory of differential operators.     

Shrinkage Estimators under Spherical Symmetry for the General Linear Model

This paper is primarily concerned with extending the results of Brandwein and Strawderman in the usual canonical setting of a general linear model when sampling from a spherically symmetric distribution. When the location parameter belongs to a proper linear subspace of the sampling space, we give an unbiased estimator of the difference of the risks between the least squares estimator φ₀ and a general shrinkage estimator φ = φ₀ − X − φ₀ ² · g φ₀. We obtain a general condition of domination for φ over φ₀ which is weaker than that of Brandwein and Strawderman. We do not need any superharmonicity condition on g. Our results are valid for general quadratic loss.     

可压缩反应气体模型的一致爆破率

研究一类带有非线性非局部源项的可压缩反应气体模型方程的Dirichlet初始边界值问题．在一些假设条件下,证明了爆破率在有界域的所有紧子集中是一致的,并且证明了爆破率只与非线性非局部源的指数有关,而与扩散项的指数无关．     

修正的Durrmeyer-Bernstein算子线性组合的一致逼近

讨论了修正的 Durrmeyer- Bernstein算子线性组合的一致逼近问题 ,给出了正定理、逆定理和特征刻划 .     

求解不确定型二层线性规划模型的悲观决策方法

基于多参数线性规划理论，将不确定型二层线性规划问题转化为多个关于不确定参数的线性规划问题。利用不确定型决策方法中的悲观准则．从最不利的结果中选择最有利的结果，从而得到不确定型二层线性规划的最优解。数值实例的仿真结果表明，所提出的悲观决策方法对解决诸如不确定供应链的规划与运作等问题不失为一种有效的决策支持工具。