本文继续[Ⅰ]的讨论,建立了亚正定阵的行列式理论。给出了许多新的结果:例如广义Minkowski不等式、广义凸性不等式等等,对某些种类的亚正定阵,还将关于任意阵的行列式的Hadamard不等式作了改进。最后,将Open-heim关于正定阵的Hadamavd乘积的著名结果推广到亚正定阵上。

1.Consider an analytic function (s) defined by a Dirichlet series whose axis of convergence is σ = 0, where 0 ≤λ_n< + ∞ and lim (log n/λ_n) = 0. We define the Ritt order and proximate order of f(s) in σ > 0 and have obtained some relations between the growth of f(s) and the coefficients, which extend some of G. Valiron's results.2. Let (Ω, P) be a probability space. Consider a random Dirichlet series whose abscissa of convergence is σ(ω), where a_n(ω) are random variables in (Ω,P) and λ_n satisfy the same conditions as in 1(n = 0,1,2,…). To σ(ω) we extend L. Arnold's results on the radius of a.s. convergence of random Taylor series and have solved some of his problems.In the case σ(ω) = 0 a.s. the results mentioned in 1 is applied to studying the a.s. growth of f(s, ω) in σ> 0. As a special case we find that if |a_n(ω)| are independent and have the same non-degenerate distribution function F(x) and if the radius of a.s. convergence of g(s, ω)is 1, then the a.s. growth of g(s, ω |z|<1 ean be determined by the convergence or divergence of(logx)~k dF(x) (k≥1).3. Let the probability space in 2 be such that Ω = [0, 1], is composed of all Lebesgue measurable sets E on Ω and P[E] is the Lebesgue measure of E. Let a_n(ω)= b_nε_n(ω) or b_nγ_n(ω), where {ε_n(ω)} or {γ_n(ω)}is a Rademaeher or Steinhaus sequenee and b_n are constants such that the abscissa of convergence of f(s, ω) is zero. Then, under certain conditions every point on σ = 0 is a Pieard or Borel (R) point of f(s, ω) a.S.

<正> §1 引言设 P_1,P_2,…,P_N 是正 N 边形 S_N 的顶点,所有线段 P_iP_j 之长的平方和∑r_(ij)~2记为 N_1;所有三角形△P_iP_jP_k 的面积平方和∑△_(ijk)~2记为 N_2.对 N=3,4,…进行计算表明,总有

关于矩阵的迹,[5,6]推广了Bellman不等式,在A_1,A_2,…,A_m为n阶两两可换的正定Hermite矩阵的条件下,证明了本文中的不等式(2).本文对半正定Hermite矩阵乘积的迹证明了一个新的更强的不等式(1).从而不等式(1)和(2)成立的条件只要求A_1,A_2,…,A_m是半正定的Hermite矩阵.

本文给出了S-闭空间的另一等价定义,建立了刻划S-闭空间特征的定理1,在应用上比[1]中的特征定理2方便得多.以此为依据我们得出了S-闭空间的若干性质,包括:(1)为使Hausdorff空间x是S-闭空间,必须且只须X是极不连通的H-闭空间.(2)为使正则空间是S-闭空间,必须且只须X是极不连通的紧空间.(3)为使拓扑空间X是S-闭空间,必须且只须X的半正则化是S-闭空间.(4)满足第一可数公理的S-闭的Hausdorff空间是有限的.此外,我们指出了[2]的主要结论的证明是错误的.

<正> 本文是作者工作[1]的继续. 关于增算子的不动点定理,在数学的许多领域,特别是在非线性微分方程和非线性积分方程中,有着广泛的应用(见[2][3]L4][5][6]).设E是Banach空间,P是E中的锥,D=[u_o,ν_o]是E中的序区间,A:D→E是增算子,满足u_o≤Au_o,Au_o≤ν_o.关于增算子的三个有代表性的结论是:

<正> 关于丢番图方程x~4-Dy~2=1,D>0且不是平方数,(1)有过一系列工作,其主要结果如下:Nagell 证明了 D≡3(mod 8)是素数,(1)无正整数解.Ljunggren 证明了(1)最多只有两组正整数解.Cohn 证明了 D 使得 x~2-Dy~2=-4有解 x≡y≡1(mod 2),则(1)除开有限个D 的值外,仅有整数解 x=1.

本文对取值于 Banach 空间的Φ-混合序列得到了一矩不等式,作为推论,对独立情形获得了比 Marcinkeiwicz-Zygmud 更为实用的矩不等式,作为应用,讨论了实值 (?)-混合序列的完全收敛性,取值于 p-型空间独立和的完全收敛性以及实值 (?)-混合序列的几乎处处不变原理等问题,获得了理想的结果.

In this paper, we use partial order theory to study mixed monotone operators,and get existence and uniqueness of fixed point without assuming the operator to be compact or continuous. And apply the new results to Hammerstain integral equations on RN.

<正> 全文中我们用 ∑_A,∑_B表示n维欧氏空间E~n中的单形;其顶点分别为a_1,a_2,…,a_(n+1)和b_1,b_2,…,b_(n+1);其稜长分别为 a_(ij)=|a_ia_j|和b_(ij)=|b_ib_j|;其体积分别为V(A),V(B). 令∑_A,∑_B的顶点集{a_i},{b_i}的Cayley-Menger阵分别为n+2阶方阵:

<正> 广义逆矩阵理论最近分别被推广到域上与有限域上.本文将讨论p-除环上矩阵的广义逆.我们要用到下面的 引理1 设A是除环△上矩阵的r×r可逆子阵,则

在[4]中,S.V.Du和V.Lakshmikantham利用紧型条件证明了必存在单调序列{v_n}和{w_n},一致收敛于Banach空间常微分方程初值问题u′=f(t,u),u(0)=u_0的最大解和最小解。在本文中我们证明了,如果Banach空间是弱序列完备的,则[4]中的紧型条件(这是[4]中的一个主要条件)是可以删掉的。我们还对Banach空间周期边值问题证明了类似的结果。

本文中,首先给出了一阶线性偏差变元微分方程所有解振动的充分必要条件,然后应用这个结果研究了具正负系数的中立型微分方程的振动性.

In this paper we consider the existence, uniqueness, stability, unstability of periodic solutions to a class of differential equations with deviating aguments. New results are obtained.

In this paper, we firstly discuss the existence, uniqueness and non-symmetric iterative approximation of solution for a class of systems of mixed monotone operator equations, and obtain some new fixed point theorems of mixed monotone operators, increasing operators and decreasing operators which have no continuous and compact conditions. Secondly, we study the fixed point of mixed monotone operators which have a-concave and -a-convex conditions while the operators also have no continuous and compact conditions, and get a new result. Finally, we apply the new results presented in this paper to Hammerstein integral equations on RN (see [1-12]).

本文利用 Fourier 级数理论研究二阶常系数线性中立型方程的周期解问题.获得了保证周期解存在、唯一的充分必要条件以及一些简便的充分条件.

Given a positive integer n, the definition of the famous Smarandache function is: S(n) = min{m : m∈N, n|m!}. The main purpose of this paper is to study the value distribution of the Smarandache function by using the elementary method, and give some interesting asymptotic formulae.

本文是在[1]文的基础上,证明自共轭四元数矩阵 A 的行列式‖A‖的展开定理,而当 A 为实对称矩阵或复 Hermitian 矩阵时,‖A‖的展开式即与通常的行列式|A|的展开式一致.并由此进一步得出 A 的特征多项式 f(λ)就是 A 的特征矩阵的行展开式,从而得到 f(λ)的直接计算法,且由此又得到正定与半正定自共轭矩阵的另一等价命题,完善了[2]中(?)4的结果.还有一些关于实、复正定与半正定矩阵的重要定理,也可应用展开定理把它们加以推广.

In this paper, Hardy-Hilbert's integral inequalities are all-sidely generalized, and many fine and new inequalities are obtained.

In this paper, we discuss the following nonlinear neutral delay equation: [x(t) + cx(t - τ)]" + g(t,x(t - σ)) - p(t), by using the coincidence degree theory. A sufficient condition for the existence of periodic solution of the equation is given.

<正>多元齿(Splinc)的实际背景是明显的。除了飞机外形设计及数学船体放样外,在有限元法的模型函数构造等方面都有直接的意义。由于多元齿的存在性与对区域的剖分关系极大(这是和一元齿本质不同的),因此必须搞清它们的关系。本文正是顺着这个思路展开的。本文建立了在给定剖分下多元齿存在的充要条件,并且给出了多元非乘积型截断多项式,多元齿的最一般(非乘积型)的表达式以及多元非乘积型齿插值存在的充要条件。

In this paper, we first establish a basic theorem on zeros of general transcendental functions. Based on the basic theorem, combining the global Hopf bifurcation theorem for general functional differential equations, which was established by Wu J. H. al using degree theory methods, we investigate the stability and global Hopf bifurcation for a lossless transmission line network which is described by a one delay differential-difference equation of neutral type.

<正> 工程设计中的运动系统常常是多维自由度的,例如飞机是六维的运动.实际工作者常把高维运动分解为低维运动,例如将飞机的运动分解为纵向运动和横向运动.研究分解后的低维问题的稳定性与原来高维问题的稳定性之间的关系,1960年秦元勋曾提出稳定性理论中的分解问题,并得到由分解系统的稳定性保证原来系统的稳定性的若干充分条件.在工程设计中,这个参数的稳定区越大,则设计的自由度越大.因此,有必要进一步精估这一区域.1965年 Bailey 提出向量李雅普诺夫函数的概念,这种概念适合于用来研究分解问题.本文第一部分即利用这种方法来研究秦元勋所提出的分解问题,放宽了原得到的充分条件,在某些情况下,例如 n=2实际得到了充要条件,即最佳的范围.

In this paper we study the relative position and number of limit cycles of the quadratic differential system. dx/dt=-y+lx~2+mxy+ny~2,dy/dt=x(1+ax+by)(1)for which O(0,0) is a fine focus. We prove that limit cycles of system (1) Can only be distributed around one of the two focus, provided that system (1) apart from these focus, has a third singuler point.On the other hand system (1) has only two singular points: a fine focus O(0,0) and a rough focus N(0, 1), then (1) may have limit cycles around the two singular points simultaneously. It is important to note that if we choose suitably the Coefficents in system (1), and add a sufficiently small term x to the right hand side of the first equation, there may exist three limit cycles around O(0, 0) and one limit cycle around N(0, 1). Then system (1) has at least four limit cycles.

本文在紧性条件下证明了若干新的非连续的增算子的不动点定理.如果不假定增算子 A 满足连续性条件和紧性条件,我们证明了算子 A 有广义不动点.

<正> 设 K 为一体,F 为其中心.如果 K 上的 n 阶矩阵 A 的特征矩阵λI—A 在 K[λ]上等价于

<正> 在邮局搞綫性規划时,发現了下述問題:“一个投递員每次上班,要走遍他負責送信的段,然后回到邮局.問应該怎样走才能使所走的路程最短.” 这个問題可以归結为 “在平面土給出一个連通的綫性图,要求将这个綫性图从某一点开始一笔画出(允許重复),并且最后仍回到起点,問怎样画才能使重复路线最短.”

In this paper, sufficient conditions are obtained for the oscillation of all solutions of the homogeneous Neumann, Dirichlet and Robin's boundary value problems associated with the hyperbolic system for i = 1, 2, ???, m, (x, t) ? ft x (0, oo), $1 C Rn, where A denotes the Laplacian in E".

In this paper, the decreasing operator with convexity or concavity is studied,and existence, uniqueness and iteration of the fixed point on decreasing operator areobtained. Finaly, some applications to various equations are also done.

本文中,作者研究了具有离散参量的非齐次马尔可夫链函数的强大数定律,并且给出了直接加于链和过程样本函数上的充分条件(条件Ⅰ—Ⅲ).对于链是齐次且函数与参数无关的特殊情形,它概括了各种有关的已知结果.

考虑具有正负系数的中立型时滞微分方程d/dt[x(t)-C(t)x(t-r)]+P(t)x(t-r)-Q(t)x(t-δ)=0(1)我们获得了(1)的所有解振动的“sharp”条件,即条件在系数C(t),P(t)及Q(t)为常数时是充分必要的.作为其推论也大大地推广并改进了文[2—5,7—9]的相应定理.

<正> 则它称为是对称的.以下恒设 l(y)为对称的微分算式.令(?)表示 L~2[0,∞)内由 l(y)所生成的最小算子,(?)为其定义域.假定(?)的亏指数为(m,m),则知 m 必满足[(n+1)/2]≤m≤n.当 m=n 时,l(y)称为是极限圆型的(简称圆型).根据 Hilbert 空间内无界算子的一般理论,在亏指数为等值的情况

<正> 近十多年来计算机应用与数控技术的发展大大推动了插值计算方法和理论的研究.一般的样条函数,特别是常用的三次样条函数,方法简单易行,具有一系列良好的性质,对于不出现近于垂直切线的所谓“小挠度”函数的插值和逼近效果很好,但不能直接处理有近于垂直切线的“大挠度”曲线或多值曲线,否则会破坏计算的稳定,甚至得出荒谬的结果.为了解决这个问题,国外有人提出了一些方法,如常用的“参数样条”或矢量样条

<正> 设 g 是一秩为 l 的复单李代数,h 是其 Cartan 子代数,∑,Π={β_1,…,β_l}是 g 的根系和基础根系.熟知其 Weyl 群 W(g)是由反射

Let (F(S),ρ) be the logic metric space of two-valued propositional logic.It is proved that a logic theory G is fully divergent iff D(G) is dense in (F(S),ρ),and G is consistent iff D(G) contains no interior point.Moreover,it is proved that (F(S),ρ) is zero-dimensional and possesses a property similar to the Key Fan's property for describing connectedness of topological spaces.

在本文中,我们研究了具有强迫项的n阶非线性时滞微分方程的振动性与渐近性.建立了某些充分条件,在这些条件下,如果 n 为偶数,方程的所有解为振动的;而 n 为奇数,方程的所有解或者振动,或者解本身及它的 1 至 n-1 阶导数都单调趋于零.

关于有限超可解的研究已有不少结果,本文在苏向盈研究有限群的半正规子群的基础上又得到了一系列新的结果(文中定理2—9),本文还对[1]中定理6的证明进行了改进并得到了重要推论,改进的方法在本文得到了充分应用.文中所说群均为有限群,使用的符号是规范的.

<正> §1.引言 将预给棱长的单形嵌入于n维欧氏空间E~n的问题是距离几何的经典问题之一.这一问题首先由Frechet提出而被K.Menger圆满解决.Menger所找到的构成n维单形的充分必要条件是,预给的(2n+1)条棱长满足一组具有行列式形式的不等式.作者

This paper discusses the oscillation of second order nonlinear ordinary differential equations and delay differential equations. Some new oscillation criteria for the equations are obtained.

Auerbach 与 Banach 曾证明,当0<σ<τ≤1时,在满足σ阶 Lipschitz 条件的函数中,存在函数 f(x)使关系式(?)处处成立.本文将推广这个定理,并从而得到如下的推论:设φ(x)是定义在[0,1]上的增函数,(?)φ(x)=0,如果φ(x)是比 x 较低阶的无穷小,则在连续模ω_f(δ)≤φ(δ)的函数 f(x)所组成的类中,存在处处不可微的函数.