The Jackson Inequality for the Best L~2-Approximation of Functions on [0,1] with the Weight x

Let L^2（[0, 1], x） be the space of the real valued, measurable, square summable functions on [0, 1] with weight x, and let n be the subspace of L2（[0, 1], x） defined by a linear combination of Jo（μkX）, where Jo is the Bessel function of order 0 and {μk} is the strictly increasing sequence of all positive zeros of Jo. For f ∈ L^2（[0, 1], x）, let E（f, n） be the error of the best L2（[0, 1], x）, i.e., approximation of f by elements of n. The shift operator off at point x ∈[0, 1] with step t ∈[0, 1] is defined by T（t）f（x）=1/π∫0^π f（√x^2 ＋t^2-2xtcosO）dθ The differences （I- T（t））^r/2f = ∑j=0^∞（-1）^j（j^r/2）T^j（t）f of order r ∈ （0, ∞） and the L^2（[0, 1],x）- modulus of continuity ωr（f,τ） = sup{｜｜（I- T（t））^r/2f｜｜：0≤ t ≤τ] of order r are defined in the standard way, where T^0（t） = I is the identity operator. In this paper, we establish the sharp Jackson inequality between E（f, n） and ωr（f, τ） for some cases of r and τ. More precisely, we will find the smallest constant n（τ, r） which depends only on n, r, and % such that the inequality E（f, n）≤ n（τ, r）ωr（f, τ） is valid.     

τ-可测算子迹的不等式

设m是具有忠实正规半有限迹τ的Hilbert空间上的一个半有限von Neumann代数.隶属于m的—个闭稠定算子x称为τ可测,如果存在常数λ≥0使得τ（e｜x｜（λ,∞））〈∞.将一些很有用的已知的Hilbert空间算子迹的不等式推广到τ-可测算子迹.特别是这些不等式蕴涵了n-元τ-可测算子的Clarkson不等式.同时还给出了τ-可测算子的广义平行四边形法则.     

3-连通、高次和坚韧图周长的估计(Ⅰ)

设G是一个n阶3-连通图,周长为C(G),独立数为,若G是1-坚韧的,且,则G的每一个最长圈是控制圈且;又若G是5/3-坚韧的或,则G是Hamilton图。     

具有奇异系数的发展p-Laplace 方程

该文考虑Rⁿ内在球中和球面系数均有奇异的发展p-Laplac方程的初边值问题的可解性. 在一定条件下证明了轴对称整体解的存在性和不存在性.     

二阶非线性中立型泛函微分方程周期解的存在性

本文利用重合度理论,研究二阶非线性中立型泛函微分方程[x(t)+cx(t- τ)]11+g(t,x(t-σ)=p(t)的周期解的存在性,给出该方程存在周期解的充分性定理.     

一类二阶具偏差变元的微分方程周期解

本文利用重合度理论研究一类二阶具偏差变元的微分方程x＇＇（t）+f（t,x（t）,x（t-τ0（t）））x＇（t）+β（t）g（x（t-τ1（t）））=p（t）的周期解问题,得到了存在周期解的新的结果.     

傅立叶超函数和扩充傅立叶超函数与爱米特热方程

证明了傅立叶超函数和扩充傅立叶超函数可用爱米特热方程的解来表示,且用以表示的解有很良好的性质.     

First Crossing Times of Telegraph Processes with Jumps

Methodology and Computing in Applied Probability - The paper presents exact formulae related to the distribution of the first passage time τx of the jump-telegraph process. In particular, the...     

Un théorème de décomposition pour les polaires génériques d'une courbe plane

In this Note we state a decomposition theorem into bunches of branches (i.e. analytically irreducible germs) for the generic polar curves of a reduced germ of a plane analytic curve, with equation ƒ(x, y) = 0. They are the curves with equation ∂f(x, y)/∂yτ∂f(x, y)/∂x = 0 with generic τ. All the branches of the same bunch have the same contact with each branch of C. A number of the first terms of the Puiseux expansion of each branch of the polar is therefore independent of τ; this number depends only on the bunch to which the branch belongs. This generalizes results of H.J.S. Smith [8], M. Merle [7] (where C is a branch), E. Casas [1], F. Delgado [2]. We show by an example that it is also optimal. We have shown elsewhere that it implies the results of Lê-Michel-Weber [6].     

關於有一無限極限的一個函數的富里哀級數的Borel總和性

<正> 吾人將討論在一點θ的情形並且可以設θ=0.則(1)為此偶函數φ(t)的富里哀級數在t=0由Borel方法求和的公式,吾人知Borel總和法不一定對於一切富里哀級数在其連續點可以求和,但對某一點函數趨於無限大,它的富里哀級數在此點能否     

特殊多面体的笵形和同伦群

<正> 多面体的伦型鉴定问题,已经有过许多拓扑工作者的研究.J.H.C.Whitehead找到 A_n~2多面体的伦型和正则上同调环头(n=2)或正则上同调系统类(n>2)间的一一对应.他和 S.Machane 发见“三型”所对应的代数构造,于是引起(?)     

不动点和一类中立型方程的渐近稳定性

本文用不动点定理研究了一类中立型泛函微分方程[x(t)-P(t)x(t-τ)]′+Q(t)x(t-r(t))=0,t≥t0的零解的渐近稳定性,其中τ∈(0,∞),P,Q∈C([t0,∞),R),r∈C([t0,∞),R+),且当t→∞时t-r(t)→∞.我们讨论了r(t)为常数和不为常数两种情况.所得定理改进和包含了前人已有的结果.     

多连通区域上的Bloch常数

<正> 本文可以看作[1]文的一个应用,在这里将提供一个关于多连通区域的 Bloch 常数的估计的普遍方法.并且特别以于二连通区域的 Bloch 常数进行了具体的估计.     

Approximation of integrable functions by linear methods almost everywhere

It is shown that 2π periodic functions whose (r-1)-th derivatives have bounded variation (r > 0) can be approximated by de La Vallée-Poussin sums σ _n,m (an ?m =m (n) ?An,0 <a<A<1) at almost all points with a rate o(n^?r). For functions belonging to the class Lip (α, L) (0 <α < 1), any natural N, and a positive ?, we have almost everywhere $$|f(x) - \sigma _{n,m} (f;x)| \leqslant c(f,x)n^{ - \alpha } lnn \ldots ln_N^{1 + \varepsilon } n,$$ where \(ln_k x = \underbrace {ln \ldots ln x}_k(k = 1, 2, \ldots )\) . For any triangular method of summation T with bounded coefficients we construct functions belonging to Lip (α, L) (0 < α < 1) and such that almost everywhere, $$\mathop {\overline {\lim } }\limits_{n \to \infty } |f(x) - \tau _n (f;x)|n^a (ln n \ldots ln_N n)^{ - a} = \infty $$ where the τ_n(f; x) are the means of the method T.     

用黎曼求和法(R,1)求和時的基卜斯現象

<正> 假設級數滿足下面兩個條件,即:(甲)在原點的某一鄰域內,對於h(≠0)的一切值級數收斂;     

Periodic solutions of nonlinear delay equations

Existence and uniqueness of 2π-periodic solutions of $\text{d}{}^{\mathrm{j}}\text{x(t)}\text{dt}{}^{\mathrm{j}}\text{+}\text{grad}\text{G(x(t ? τ)) = e(t, x(t), x(t ? τ)) (j = 1, 2)}$, where x(t) is in $\text{R}$ⁿ and e(t, u, v) is a given vector function, 2π-periodic in t, are shown under conditions on the spectrum of the Hessian of G. The equation is studied using a fixed point theorem in the space L²(0, 2π). One feature of this approach is that no relationship between the delay and the period is necessary.     

An explicit example of a noncrossed product division algebra

The paper presents an explicit example of a noncrossed product division algebra of index and exponent 8 over the field ?(s)(t). It is an iterated twisted function field in two variables D(x, σ)(y, τ ) over a quaternion division algebra D which is defined over the number field ?(√3,√?7). The automorphisms σ and τ are computed by solving relative norm equations in extensions of number fields. The example is explicit in the sense that its structure constants are known. Moreover, it is pointed out that the same arguments also yield another example, this time over the field ?((s))((t)), given by an iterated twisted Laurent series ring D((x, σ))((y, τ )) over the same quaternion division algebra D. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Closeness of the τ-Standard Part of a Set

In this paper,the closeness of the τ-standard part of a set is discussed.Some related propositions of the τ-neighborhood system of a set are given.And then some related conclusions of the τ-monad of a set and the τ-standard part of a set are presented.And based on it,the necessary and sufficient conditions of the enlarged model and the saturated model are showed.Finally,some sufficient conditions that the τ-standard part of a set is closed are proved in the enlarged model and the saturated model.     

复合形在欧氏空间中的实现问题Ⅲ

<正> 设 K 是一个有限单形复合形,我们恒可视 K 为一充分高维数 N 的欧氏空间中的欧氏复合形,此时其所定空间将记为(?).在研究 K 是否可实现于某一确定维数 m 的欧氏盘间R~m 中的时候,我们曾引进下面的一些定义(见[1],记号略有不同):     

Semiartinian modules relative to a torsion theory

In the first section we generalize the concept of the socle of a module by replacing simples with τ-simple modules for a hereditary torsion theory τ. The second section is concerned with the τ-Loewy series, and finally these general results are applied in the section 3 to the notions of τ-semiartinian rings and modules.