关于向量场旋度的一个附注

若在某单连通区域V∈R3上,向量函数A是某个数量函数u的梯度,则其旋度rotA在所定义的区域上为零.本文证明了其旋度的任意两个分量在所定义的区域V上为零,另一个分量满足在区域V的边界S上为零与其三个分量在所定义的区域V上为零是等价的,解惑教学中学生的问题.     

浅论零向量概念的争议与核心价值

在向量学习中,零向量是一个不容忽视的“特殊向量”.新课标必修四教材对于零向量是这样规定的:长度为0的向量叫做零向量,记作0. 这个定义只规定了零向量的长度,对于零向量的方向没有提及.由于零向量的终点和起点相互重合,所以零向量的方向应理解为“不确定”.“不确定”不等同于“没有”,零向量作为一个向量是必须要有方向的.在处理平行问题时,规定零向量与任一向量都平行.     

关于二阶线性时变系统在第一临界情况下的稳定性

在A（t）为2×2矩阵，而其特征根一个为零，另一个为负数的情况下讨论了系统x=A（t）x零解的稳定性，给出了保证该系统零解稳定的若干充分条件．     

左零因子理想具升链条件之环

Herstein,I.N.(1964)猜想在“左零化子具升链条件”的情况下,一个诣零环必为幂零的(参看[1])。但不久就由Sasiada作出一个非幂零的谐零环而其中的左零化子满足升链条件,这个反例否定了上述猜想。 本文则是把上述条件稍为加强一点而证实了如此的环的诣零单边理想恒为幂零的,自然这样的诣零环就更是幂零的了。     

形式三角矩阵环的零因子图

本文研究了形式三角矩阵环的零因子结构与零因子图的问题.利用零因子的性质及交换环零因子图的有关结论及分类讨论的方法,获得了形式三角矩阵环的零因子图直径为2的充要条件,推广了有限交换环的零因子图的相关结果.     

幂零元生成的子代数是幂零吗?

零代数在什么条件下是幂零的?这是一个重要问题。类似地,我们提出另一问题,对于幂零元生成的子代数在什么条件下是幂零的?一个幂零元生成的子代数显然是幂零的,两个幂零元生成的子代数一般不是诣零的。本文得到一个肯定结果:     

“分母为零”的运用例举

“分母为零”的运用例举３２６５１１江苏如皋市白蒲中学张云飞分母不能为零，为什么？分母为零，分式没意义．这是众所周知的答案，然对于一些具体问题考虑一下分母为零将会怎样，却是很有意义的。１运用“分母为零”。寻求双曲线渐近线的倾斜角、斜率等问题的简洁解法方...     

3-幂零矩阵的相似等价类的计数

主要对定义在一般数域上的3-幂零矩阵的相似等价类的个数问题进行探讨.从中得出n阶3-幂零矩阵秩的范围、n阶3-幂零矩阵的相似等价类的个数的计算公式,以及秩为r的所有n阶3-幂零矩阵的相似等价类的个数的计算公式.     

单调函数判别法的推广

将单调函数的判别法由在开区间内有有限个导数为零,推广到开区间内有可数(列)个导数为零的情形,从而扩大了判别法的应用范围     

在环面同态作用下零维理想的性质

本文以矩阵变换和整理论为工具，研究了在环面同态作用下零维理想的性质，证明了零维理想的环面同态矩阵是满秩的，并且刻画了零维理想在环面作用下的对应关系．     

Contact problems of the theory of elasticity in the presence of wear : PMM vol.40, n 6, 1976, pp. 981–986

The solutions of the contact problems of the theory of elasticity in the presence of wear is given for two cases. In Sect. 1 we consider the problems in which an initially curved beam comes in contact with a half-plane. One of the initial assumptions is that the distance between certain directrices along which the body in contact is sliding and the boundary of the half-plane remains constant. In Sect. 1 the contact between the curved beam and the half-plane is discussed at the assumption that the half-plane is subject to wear. As the result of the wear, the pressure between the beam and the half-plane is gradually reduced. It is naturally assumed that the pressure at the terminal points of the contact area will, in this case, be zero. The conditions characterizing the pressure at these terminal points can be established for various types of contact problems only under certain additional assumptions; this will be discussed below.     

On mathematical modelling of absorption and dispersion of seismic waves for low frequencies

The problem of determining the dispersion and absorption of plane seismic waves by a given seismic quality factor is dealt with in the case that the quality factor tends to a positive limit as the frequency goes to zero. General formulae are derived for the phase velocity and the attenuation coefficient by solving the corresponding Riemann–Hilbert problems for analytic functions in the upper half-plane.     

半平面中有限阶解析函数的因子分解

与经典有限阶整函数的Hadamard因子分解定理和半平面中属于Hardy空间的解析函数的内外函数的因子分解类似,对右半平面中有限阶ρ解析函数f,可以分解为三个解析函数G,eQ和eg的乘积GeQeg,其中G是一个加权Blaschke乘积,Q是一个次数不超过ρ的多项式以及eg是一个加权外函数,log|G|,ReQ和Reg-log|f|在右半平面的边界恒为零．     

Stress intensity factors for three cracks at the interfaces of a graded layer bonding two different materials

We consider two dissimilar elastic half-planes bonded by a nonhomogeneous elastic layer in which there is one crack at the lower interface between the elastic layer and the lower half-plane and two cracks at the upper interface between the elastic layer and the upper half-plane. The stress intensity factors for these three cracks are solved for when tension is applied perpendicular to the interface cracks. The material properties of the bonding layer vary continuously between those of the lower half-plane and those of the upper half-plane. The differences in the crack surface displacements are expanded in a series of functions that are zero outside the cracks. The unknown coefficients in the series are solved by the Schmidt method so as to satisfy the conditions inside the cracks. The stress intensity factors are calculated numerically for selected crack configurations.     

Green functions of a quasi-static problem

In this paper Green functions are constructed in analytic form for a deformable half-plane of a quasi-static problem of thermoelasticity when the heat flow on the boundary x₂=0 of the half-plane is zero. To construct the Green functions, certain integral representations are used whose kernels are known Green functions of the corresponding problems of elasticity theory. The functions constructed make it possible to obtain a wide class of new solutions of boundary-value problems of thermoelasticity, in particular solutions for a piecewise homogeneous half-plane. Bibliography: 6 titles. Translated fromObchyslyuwval’na ta Pryklandna Matematyka, No. 77, 1993, pp. 97–104.     

Dynamics of renormalization group in the lower half-plane of the coupling constants of the fermionic hierarchical model

We consider four-component fermionic (Grassmann-valued) field on the hierarchical lattice. The Gaussian part of the Hamiltonian in the model is invariant under the block-spin renormalization group transformation with given degree of normalization factor (renormalization group parameter). The non-Gaussian part of the Hamiltonian is given by the sum of the selfinteraction forms of the 2-nd and 4-th order. The action of the renormalization group transformation in this model is reduced to the rational map in the plane of coupling constants. We investigate the global dynamics of this map in the case when the coupling constant of the 4-th order form is less than zero (lower half-plane) and the renormalization group parameter belongs to the interval [1, 3/2).     

A cusped prismatic shell-like body under the action of concentrated moments

The elastic equilibrium problem of a cusped prismatic shell-like body, when its projection is a half-plane x ₂ ≥ 0, under the action of a concentrated moment is solved in the explicit form within the framework of the zero approximation of I.Vekua’s hierarchical models of prismatic shells. The thickness of the prismatic shell-like body is proportional to the coordinate x ₂ raised to a non-negative exponent. When the exponent equals to zero, the above solution contains the well-known solution of the classical Carothers’ problem [1] in the case of an elastic half-plane (see also [2], §39).     

Density of Sums of Shifts of a Single Function in Hardy Spaces on the Half-Plane

It is proved that there exists a function defined in the closed upper half-plane for which the sums of its real shifts are dense in all Hardy spaces H_p for 2 ≤ p < ∞, as well as in the space of functions analytic in the upper half-plane, continuous on its closure, and tending to zero at infinity.

     

A cusped prismatic shell-like body under the action of concentrated moments

The elastic equilibrium problem of a cusped prismatic shell-like body, when its projection is a half-plane x ₂ ≥ 0, under the action of a concentrated moment is solved in the explicit form within the framework of the zero approximation of I.Vekua’s hierarchical models of prismatic shells. The thickness of the prismatic shell-like body is proportional to the coordinate x ₂ raised to a non-negative exponent. When the exponent equals to zero, the above solution contains the well-known solution of the classical Carothers’ problem [1] in the case of an elastic half-plane (see also [2], §39).      

右半平面上解析的Laplace-Stieltjes变换对数级

本文研究了零级Laplace-Stieltjes变换的增长性问题.利用对数级和对数下级的定义,获得了这类变换具有对数级的特征,即变换的对数级和对数下级与其系数之间的关系,推广了Dirichlet级数的相关结果.