奇点定性理论中的V函数微分法

<正> 考虑微分方程组 dx/dt=f(x)(1.1)其中x是n维向量,t是时间,(1.1)右端的n维向量、函数组f(x)在原点某邻域內全纯,f(0)=0,原点O是微分方程组(1.1)的奇点. 在文[1]中,一般微分方程组的奇点,分为58类.现在讨论解析系统(1.1).这里的问题是:解析系统(1.1)的奇点,在58种奇点类型中,究竟有哪些奇点类型?对于其中     

关于退化方程的一个始值问题

本文考虑只给出一个始值的退化方程的始值问题其中p是实数。 我们得到如下的结果: (ⅰ)若p≠0,φ(x)在x=0的附近解析,则问题(1),(2)在原点附近存在唯一的解析解。 (ⅱ)若p=0,则问题(1),(2)的解析解不唯一。 (ⅲ)若p=0,φ(x)在x=0附近解析,则问题(1),(2)在原点附近解析解存在的充要条件是 φ~((i+4k))(0)=0,(i=2,3;k=0,1,2,…),     

拟齐次解析亚椭圆偏微分算子的一个必要条件

本文给出了拟齐次线性偏微分算子P（x,D）在原点为解析亚椭圆的一个必要条件：若u∈S,（Rn）;且Pu=0;则存在整函数v;Pv=0,v和u在x=0的某个邻域中重合     

一类重特征方程Goursat问题解的存在性中的离散现象

本文讨论下述Goursat问题■在原点的邻域中解析解的存在性(其中a,b,c为常数).结果如下:(i)若b≠1,3,5,…,则对任何解析函数ψ(x),问题(G_(a,b,c))恒有解析解;(ii)若b=2j+1(j≥0为整数),则问题(G_(a,b,c))有解析解的充要条件为 λφ~((j))(0)+φ~((j+2))(0)=0,若λ≠0, φ~((j+2l))(0)=0,l=1,2,…,若λ=0,其中λ=c-a~2/4,φ(x)=ψ(x)exp(ax/2). 在(i),(ii)两种情形中,均具体给出了解的形式.     

一类具算子系数的Fuchs型方程的H_λ解

<正> 本文讨论的方程的形式如下：■其中B_2是m阶线性偏微分算子,对а/а_t的阶不超过m-2.■方程(0.1)的系数A_k(t,x),B_1(t,x),B_(ij)(t,x)都是(t,x)在I×G R~1×R~n信中的解析函数,I是R~1中的闭区间-1≤t≤1,G是R~n中的有界闭域. A_k(t,x),B_1(t,x),B_(ij)(t,x)对t的展式,有     

LaSalle 定理的推广

LaSalle 在文献[1]中提出一个关于 n 维非自治系统解的正极限集的著名定理.本文利用文献[2]的方法把这个定理推广到更广泛的形式,取消了右端控制项常负的限制.因而在应用上更加方便灵活.设 R~+=[0,+∞),V(t,x):R~+×R~n→R 连续.G 是 R~n 内的任意集合,而(?)是其闭包.对(?)x∈(?),存在 x 的邻域 N_x,使得 V(t,x)对(?)_t≥0及(?)_x∈N_x∩G 是下有界的.     

一道函数题的复数解法

题目：将函数y=f^-1(x)的图象绕原点逆时针旋转2/π后，得到如右图所示的图象，则y=f(x)的解析式可能是。     

Lipschitz域上的帐蓬空间

一、预备知识在[1]中,R.Coifman, Y.Meyer和E.Stein定义了R_+~(n+1)上的帐蓬空间,并讨论它们的一系列性质及应用。本文把帐蓬空间推广到Lipschitz域上。有两种Lip域:有界的与无界的。定义1 设φ:R~(n+1)→R,满足|φ(x)-φ(x′)|≤M|x-x′|,那么区域{(x′,x_n)=(x_1,x_2,…,x_(n-1),x_n): x_n>φ(x′)}是一个无界的Lip域。定义2 设D是一个有界的连通开集。对任意Q∈D,存在以Q为中心的球B,在B内存在一个坐标系x′=(x_1,x_2,…,x_(n-1)),x_n,坐标原点是Q。还存在一个映射φ:R~(n-1)→     

一类直线与抛物线相交的有关问题

<正>已知抛物线y=x²-1与过原点的直线l交于A、B两点,抛物线的顶点为M,判断AM与BM的位置关系,并进行证明.解析由于直线l经过原点,因此可以设直线l的解析式为y=kx(k≠0,k为实数),设A、B两点的横坐标分别是m、n,则m,n是关于x的一元二次方程x2-1与过原点的直线l交于A、B两点,抛物线的顶点为M,判断AM与BM的位置关系,并进行证明.解析由于直线l经过原点,因此可以设直线l的解析式为y=kx(k≠0,k为实数),设A、B两点的横坐标分别是m、n,则m,n是关于x的一元二次方程x²-1=kx的两个实数根,方程x2-1=kx的两个实数根,方程x²-1=kx即是方程x2-1=kx即是方程x²-kx-1=0,因此m+n=k,mn=-1.     

正阻尼摆方程的全局吸引性

本文考虑摆方程d~2x/dt~2+f(t,x)dx/dt+g(t,x)=0,这里f(t,x),g(t,x)∈C~1(R~1/ωZ~1×R~1/ω_1Z~1).我们给出一个充分条件保证上方程在其轨道空间(t,x,x)∈R~1/ωZ~1×R~1/ω_1Z~1×R~1中有唯一的不变环面,且方程的所有解都指数趋于此不变环面.     

Characterization of types of asymptotic discernibility of families of hypotheses

We give a characterization of the types of asymptotic discernibility of families of hypotheses in the case of hypothetical measures that are not, in general, mutually absolutely continuous. The case when the logarithm of the likelihood ratio admits an asymptotic expansion of the type of an expansion with local asymptotic normality is examined in detail. Examples are studied.Translated fromTeoriya Sluchainykh Protsessov, Vol. 15, pp. 64–71, 1987.     

Estimates of stability of characterization of additive types of distributions

An estimate of stability of characterization of distribution types is obtained for the case of additive types. Under some conditions, the estimate has the order ε^1/3L(ε), where L(ε) is a slowly varying function. Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, Russia, 1996, Part I.     

Singularities of Solutions of One Class of Equations of Continuum Mechanics

We analyze one class of families of integral equations and describe the dependence of the singularities of solutions of integral equations on the dimensions of the families of kernels of equations. On the basis of these results, we propose procedures for the construction of approximate solutions for a small parameter.     

Study of the blow-up of solutions of systems of equations of hydrodynamic type

Mathematical Notes - We study the initial boundary-value problem for three-dimensional systems of equations of pseudoparabolic type. The system is similar to the Oskolkov system, but differs from...     

图表示范畴的两个子范畴

引进图表示范畴的两个子范畴,研究它们的同调性质。     

Birational aspects of the geometry of Varieties of Sums of Powers

Varieties of Sums of Powers describe the additive decompositions of a homogeneous polynomial into powers of linear forms. The study of these varieties dates back to Sylvester and Hilbert, but only few of them, for special degrees and number of variables, are concretely identified. In this paper we aim to understand a general birational behavior of VSP. To do this we birationally embed these varieties into Grassmannians and prove the rational connectedness of many VSP in arbitrary degrees and number of variables.     

Estimate of growth of Lebesgue functions of linear methods of summation

Sufficient conditions are obtained in terms of matrix elements for the boundedness of Lebesgue functions of linear methods of summation of expansions in orthogonal systems of polynomial type.Translated from Matematicheskie Zametki, Vol. 6, No. 3, pp. 277–286, September, 1969.     

Emptiness of set of points of lower semicontinuity of Lyapunov exponents

We consider parametric families of differential systems with coefficients that are bounded and continuous on the half-line and uniformly in time continuously depend on a real parameter. For each Lyapunov exponent, we construct a family such that the Lyapunov exponent of its systems treated as a function of the parameter is not a lower semicontinuous function for any value of the parameter.     

Determination of asymptotic solutions of problems of nonlinear theories of thermoviscoelasticity

It is shown that the asymptotic solution of a problem of the nonlinear theory of thermoviscoelasticity, if it exists, can be found directly from the solution of the asymptotic boundary-value problem without completely solving the starting problem.M. V. Lomonosov Moscow State University. Translated from Mekhanika Polimerov, No. 3, pp. 395–400, May–June, 1976.     

一类完全正则半群簇子簇格的性质

本文研究了完全正则半群簇的子簇格［Ｖ＋∩ＰＶ，Ｖ＋∩ＰＶ］的某些格运算性质，我们证明了簇Ｖ＋∩ＰＶ可分解为Ｖ与Ｖ＋∩ＰＶ的并；对任意完全正则半群簇Ｗ，有Ｗ∩（Ｖ∨Ｖ＋∩ＰＶ）＝（Ｗ∩Ｖ）∨（Ｗ∩Ｖ＋∩ＰＶ）．特别地，我们得到了等式Ｖ＋∩ＰＶ＝Ｖ成立的若干条件．