关于两点Birkhoff插值的一致收敛性

设给定函数F(t),它在[0,1]上有各阶导数,作级数: sum from j=0 to ∞[F~(2j)(0)f_(2j 1)(t) F~(2j)(1)g_(2j 1)(t)],0≤t≤1, (1)其中f_(2j 1)(t)与g_(2j 1)(t)为[1]中定义的2n 1次多项式。[2]中给出了下述定理: 定理A.已给函数F(t),0≤t≤1。若偶阶导数序列{F~(2j)(t)}在[0,1]上一致有界,即存在M>0,使得     

奇异(n-1,1)共轭边值问题的多重正解

利用锥映射不动点指数定理证明了非线性(n-1,1)共轭边值问题u(n)+a(t)[f(u)+m2u]=0,u(j)(0)=u(1)=0,0≤j≤n-2至少存在两个正解.本文允许a(t)在[0,1]两端点处具有奇性,并允许a(t)在[0,1]某些子区间上恒为零.     

矩形上的Bernstein多项式的逼近阶

1己1.当上.J二二J设j(动是定义在汇0,l]区间上的函数,与它相联系的Bernstein多项式为“·(‘;小一‘t0j(分:(x)(1 .1)这儿人们熟知(例如见〔1]) 当f任C[0,1]时,了:闭:一(冷‘(卜劝卜‘·,用B,(f)来逼近函数f有如下结果:一l厂一刀。(厂)l!_一。(。(。一去))当,任e‘ro,1]时,i一了一刀二(了)11一。(n一告二(了,,,一专)) 当f任CZ[0,1]时,}!f一B,(f){}。=O(n一‘)1981年,H.H.Gonska(见〔2〕,〔3了)证明了(1 .2)(1 .3)(1 .4)24第五卷 }1了一刀。(j)一l一镇3 .2502(f;,一香)n一朴是二阶的平滑模。(l .5)包含了(l .2)一(l.刃的结果。(1 .5)这…     

方程x~((n))+f_1(t)x~((n-1))+f_2(t)x~((n-2))+…+f_(n-1)(t)■+f_n(t)x=0解的稳定性

考虑方程z~((n))+f_1(t)x~((n-1))+f_2(t)x~((n-2))+……+f_(n-1)(t)x+f_n(t)=0 (1)在现有文献中,对方程(1)的研究几乎能集中在二阶,对于 n>2的情形,很少见到。本文应用变换技巧以及.直接方法,特别利用文[9]的推广了的方法研究了 n 为任意正整数的方程(1)的平凡解的稳定性,得到了其平凡解全局一致渐近稳定性的充分条件。特别当 n=2时所得的结果包含了文献[1-8]的有关结果。有些结果就非文[8]的条件所能得到,如本文定理4、5及其推论1。     

方程dy/dx=(q_(00)+q_(10)x+q_(01)y+q_(20)x~2+q_(11)xy+q_(02)y~2)/p_(00)+p_(10)x+p_(01)y+p_(20)x~2+p_(11)xy+p_(02)y~2)所定义的积分曲线的定性研究(Ⅱ):极限圈的唯一性

<正> 作为[1]的定理1的特例,我們知道方程dy/dt=x,dx/dt=-y+mxy+ny~2(mn≠0)(1)沒有周期解.此方程有一个指标为+1的初等奇点(0,0)和一个鞍点(0,1/n).(0,0)的稳定性由mn的符号决定,当mn>0时为不稳定,mn<0时为稳定.今后不妨設m<0(否則将x,t改号),n<0(否則将y,t改号),于是由旋轉向量場的理論[2],知道当d<0而絕对值足够小时方程     

Bernstein-Durrmeyer-Bezier算子在L_P[0,1]上的逼近性质

对[0,1]上的L—可积函数ф及α>0定义下列B—D—B算子;本文研究了M_(na)(ф,x)当α>0时,在L_P(0,1](1≤p<+∞)的一致逼近;当α≥1时在L_P[O,1]及L~1_P[0,1]逼近度的量化估计。作者在文[4]中定义了B—D—B算子:其中f_(nk)(X)称为Bézeief基函数文[4]研究的是B—D—B称子在C[0,1]空间中的逼近性质,本文继续[4]的工作,专研究这个算子在L_P[0,1](1≤P<+∞)的逼近性质,证明了M_(na)(ф X)当α>0时在L_P[0,1]中为一致逼近,并得到了当α≥1时在L_P[0,1]及L~1_P[0,1]中逼近度的量化估计。     

Bernstein—Durrmeger—Bézier算子的逼近定理

对[0,1]上的 L—可积函数φ及α>0定义下列 B-D-B 算子:■其中■■且规定 f_((n,n)+1)(x)=0.f_(nk)(x)为 Bézier 基函数。本文研究了 M_(na)(φ;x)在 C[0,1]的一致逼近,在 C[0,1],C~1[0,1]逼近度的量化估计及 C~2[0,1]中当0<α<1情形下的 Vonorovskya 型渐近等式。     

方程(dy)/(dx)=(q_(00)+q_(10)x+q(10)y+q_(20)x~2+q_(11)xy+q_(02)y~2)/(p_(00)+p_(10)x+p(10)y+p_(20)x~2+p_(11)xy+p_(02)y~2)所定义的积分曲线的定性研究(Ⅲ):Ⅰ类方程的极限环的个数

<正> 按照文[1]的分类,我们研究其中的I类方程,它是最一般形式可化为dx/dy=-y+dx+lx~2+xy+ny~2=P(x,y),dy/dt=x=Q(x,y).当 d=0时文[1]已证明此方程不存在极限环,这时有限远奇点 O(0,0)为焦点,l+n>0时为稳定,l+n<0时为不稳定,当 n≠0 时还有另一奇点 N(0,1/n),为鞍点.为确定起见,以下均假定 l+n>0(l+n=0 时以原点为中心,由旋转向量场的理论可知加上 dx 项以后不产生极限环故不必讨论,l+n<0 时则将 y,t 改号即可化为 l+n>0的情况).由旋转向量场理论可知 d<0 而|d|甚小时在原点 O 附近产生不稳定极     

二阶非线性泛函微分方程的渐近性与振动性

本文讨论如下类型二阶泛函微分方程(E) [r(t)x′(t)]′ sum from 1 to m(f_i(t,x(g_(il)(t)),…,x(g_(in)(t))))=0解的渐近性与振动性。 假设i)r(t)对t≥a连续且为正; ii)对一切t≥a,有g_(ij)(t)≥t(或≤t),连续,且当t→∞时,趋于 ∞,i=1,2,…,m,j=1,2,…,n;     

半参数回归模型中误差方差估计之分布的非一致性收敛速度

Engle等人将气候条件对电力需求关系归结成半参数回归模型其中{e_j,1≤j≤n}是iid.的随机误差,均值为0,方差σ~2>0,{(X_j;T_j),1≤j≤n}是R~p×[0,1]上的随机设计点列且与{e_j,1≤j≤n}相互独立,{T_j,1≤j≤n}iid.,β是p维未知回归参数,g(t)是定义在[0,1]上的未知回归函数。     

Qualitative Investigation of the Singular Cauchy Problem \sum\limits_{k = 1}^n {(a_{k1} t + a_{k2} x)(x')^k = b_1 t + b_2 x + f(t,x,x'),x(0) = 0}

We prove the existence of continuously differentiable solutions $x:(0,\rho ] \to {\mathbb{R}}$ with required asymptotic properties as t → +0 and determine the number of these solutions.     

On a family of diophantine triples {k, A 2 k + 2A, (A + 1)2 k + 2 (A + 1)} with two parameters

Let A and k be positive integers. We study the Diophantine quadruples $$ \{ k,A^2 k + 2A,(A + 1)^2 k + 2(A + 1),d\} $$ . We prove that if d is a positive integer such that the product of any two distinct elements of the set increased by 1 is a perfect square, then $$ \begin{gathered} d = (4A^4 + 8A^3 + 4A^2 )k^3 + (16A^3 + 24A^2 + 8A)k^2 \hfill \\ + (20A^2 + 20A + 4)k + (8A + 4) \hfill \\ \end{gathered} $$ when 3 ≦ A ≦ 10. This extends a theorem obtained by Dujella [7] for A = 1, and also, a classical theorem of Baker and Davenport [2] for A = k = 1.     

On the regularity of the locally integrable solutions of the functional equations $$\sum\limits_{i = 1}^k {a_i \left( {x,t} \right)f\left( {x + \varphi _i \left( t \right)} \right) = b\left( {x,t} \right)} $$

Aequationes mathematicae -     

Iterative algorithm for solving a class of general Sylvester-conjugate matrix equation \sum_{i = 1}^{s} A_{i}V + \sum_{j = 1}^{t} B_{j}W = \sum_{l = 1}^{m} E_{l}\overline{V}F_{l} + C

This paper is concerned with iterative solution to general Sylvester-conjugate matrix equation of the form $\sum_{i = 1}^{s} A_{i}V + \sum_{j = 1}^{t} B_{j}W = \sum_{l = 1}^{m} E_{l}\overline{V}F_{l} + C$ . An iterative algorithm is established to solve this matrix equation. When this matrix equation is consistent, for any initial matrices, the solutions can be obtained within finite iterative steps in the absence of round off errors. Some lemmas and theorems are stated and proved where the iterative solutions are obtained. Finally, a numerical example is given to verify the effectiveness of the proposed algorithm.     

Une formule asymptotique pour\sum\limits_{n \leqq x} {a_z (n)d(n + 1)}

Sans résumé     

The stability of RDDE\ddot x(t) + p_1 (t)\dot x(t) + q_1 (t)x(t) + p_2 (t)\dot x(t - r(t)) + q_2 (t)x(t - r(t)) = O

In this paper, the author considered the stability of zero solution of linear RDDE $$\begin{gathered} \ddot x(t) + p_1 (t)\dot x(t) + q_1 (t)x(t) + p_2 (t)\dot x(t - r(t)) + q_2 (t)x(t - r(t)) = O, \hfill \\ \ddot x(t) + p_1 (t)\dot x(t) + q_1 (t)x(t) + p_2 (t)\dot x(t - r(t)) = O \hfill \\ \end{gathered} $$ using Liapunov-Razumikhin functional and transformations and obtained some sufficient conditions for the stability of Eqs.(1) and (2). These results are suitable both for boundedp _i (t),q _i (t) andr(t).i=1,2.