OptimalL ∞-error estimates for piecewise linear finite elements inR n for 1≦n≦3

Summary In this paper, for two second order elliptic boundary value problems, uniform convergence error estimates are derived using piecewise linear (conforming and non-conforming) finite elements. The estimates are valid for polyhedral convex regions R ⁿ with 1n3. The proof of the estimates is mainly based on Sobolev's inequality, by which the uniform estimates are reduced to estimates in the mean. In this way, it is possible to obtain the desired uniformO(h ²)-estimate in the plane as a special case, thus confirming theoretically Natterer's numerical results.

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Zusammenfassung In dieser Arbeit werden für zwei elliptische Randwertprobleme zweiter Ordnung Fehlerabschätzungen in der Maximumnorm hergeleitet, wenn die finiten Elemente stückweise linear (konform oder nichtkonform) sind. Die Fehlerabschätzungen gelten für konvexe polyederförmige Gebiete R ⁿ mit 1n3. Der Beweis für die Fehlerabschätzungen beruht hauptsächlich auf der Sobolewschen Ungleichung, mit deren Hilfe die gleichmäßigen Abschätzungen auf Abschätzungen im Mittel zurückgeführt werden. Auf diese Weise erhält man für den Spezialfall der Ebene die gewünschte gleichmäßigeO(h ²)-Abschätzung, womit Natterer's numerische Ergebnisse bestätigt werden.

     

О мультипликативном дополнении некоторы х неполных ортонормированных с истем до базисов вL p , 1≦p<∞

The problem of the possibility of multiplicative completion to basis for the sequence of functions which arise after the deletion of a finite collection of elements from a complete orthonormal system of functions is solved.For some complete orthonormal systems (for instance the Walsh system, the trigonometric system etc.) there exists no such measurable function.On the other hand, if any finite collection of functions is deleted from the Haar system, there exists a bounded function such that after multiplying the remaining set of functions by this function the system obtained will be a basis in all spacesL ^p , 1p<.For the Haar system certain infinite deletions are also possible.

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Investigation of the Wilson remainders in the interval 3≦p<50,000

The Wilson remainders for all primes less than 50,000 have been computed and tabulated. The distribution of the remainders divided by the corresponding primes has also been examined.     

Sur les Suites Presque Echangeables DansL q , 1≦q<2

Let 1≦q<p<2. We construct a bounded sequence (X _n ) _n∈N inL _q which defines a typeσ onL _q , such that:

1. (X _n ) _n∈N is equivalent to the unit vector basis ofl _p .
2. The l-conic classK ₁(σ) generated byσ is not relatively compact for the topology of uniform convergence on bounded sets ofL _q .
3. (X _n ) _n∈N has no almost exchangeable subsequence after any change of density.

This sequence does not verify the two natural conditions inL _q -spaces that ensure the existence of an almost symmetric subsequence.     

Strong laws for the k-th order statistic when k≦c log2 n

Summary Under general regularity assumptions, we characterize the upper and lower almost sure classes of U _{k, n} , where U _{1, n} ...U _{n, n} are the order statistics of an i.i.d. sample of size n from the uniform distribution on (0, 1), and where k=k _n is a non-decreasing integer sequence such that 1k =O(log₂ n) as n .     

次b凸函数和次b凸规划

研究一种称为次b 凸函数的广义凸函数, 并介绍了次b 凸集的概念. 分别在一般情形及可微情形下讨论了次b 凸函数的相关性质, 得到了次b 凸函数成为拟凸函数及伪凸函数的充分条件. 最后, 在次b 凸函数的条件下给出了无约束及带不等式约束规划的最优性条件.     

Decrease orders of theL p -moduli of continuity (0<p≦∞)

k — , 0<p, _k(p) — k- L _p ¹ ; >0, - , (0, ), (t)0 (t0), (t) ,t ^–(t) . _k(p), , 0<p1 _{k(p) (k– 1)p+1}, 1<p — _k , .     

比较a/b与(a 2b)/(a b)的大小

设a,b∈R~ 且a>b,讨论并比较a/b与a 2b/a b的大小。原题中含有两个变元a、b,为使问题化繁为简,引进新的变元t,且令t=a/b,显然有t>l,则(a 2b)/(a b)=(t 2)/(t 1)>1,这样,要讨论的两个分式的大小比较就转化为讨论t与(t 2)/(t 1)的大小比较了。 1°设t=t 2/t 1,则有t~2 t=t 2t~2=2,由     

不等式|a|-|b|≤|a±b|≤|a|+|b|的补注

“|ａ| -|ｂ|≤ |ａ±ｂ|≤ |ａ| + |ｂ|”是高中数学中的一个重要不等式定理 ,它是处理含有绝对值问题的一个重要工具 .课本主要介绍它在证明不等式中的应用 ,而其它方面很少涉及 ,且何时取等号也未指明 ,但在高考中却多次考查到 .为此本文加以补充并例谈其应用 .一、定理的补注1.等号成立的条件|ａ +ｂ| =|ａ| + |ｂ| ａｂ≥ 0 ;|ａ -ｂ| =|ａ| + |ｂ| ａｂ≤ 0 ;|ａ| -|ｂ| =|ａ +ｂ| (ａ +ｂ)ｂ≤ 0 ;|ａ| -|ｂ| =|ａ -ｂ| (ａ -ｂ)ｂ≥ 0 .2 .字母ａ ,ｂ的范围其实ａ ,ｂ不仅在实数中成立 ,且在复数集中也成立 .同时右边不等式…     

用不等式|a|-|b|≤|a b|≤|a| |b|解题

（事实上，也成立，证明略）在高中《代数》下册不等式一章给出，教材主要研究了它在绝对值不等式证明中的运用．而在其它方面的运用几乎没有涉及．有关资料一般也未作探讨，为了强化《考试说明》中提及的这一重要基础，下面就其运用举例说明．1证明不等式例1用数学归纳法证明：（1985年全国高考上海试题）证明（1）n=1时，不等式显然成立．（2）假设成立，那么n＝k十1时，＜Silk。OSSI＋ICOSk。i。l＜ksinxl＋Dsinx【＝（k＋1）Isinx【．不等式成立．综合（1）、（2）有lsinn6I。nlsin8I．NZ已知a、b、c是实数，函数（1996年全国（3…     

Using ax=b

叼:少飞二参 We have known a variety of applieations of multiplieation: areas of reetangles;arrays;rate faetors;expansions;eontraetions. Any of these applieations ean lead to anequation of the form ax~b.Hereis an example involving areas of reetan- gle     

Solving ax=b

Here is an equation 3w=6.How tosolve it? Suppose there is a balance scale. Three boxes with unknown weight w per box are on the left side of the scale.They balance the 6 one-kilogram weights on the right side.Theis situation pictures 3w=6.     

Über die Differentialungleichung 0<α≦rt−S 2≦\<∞

Ohne Zusammenfassung Leon Lichtenstein zum GedächtnisDiese Arbeit wurde teilweise von einem Kontrakt des Office of Naval Research in Stanford University unterstützt.     

1.|a b|>|a| |b|

有众所周知}a bl(la! !b{,在这里我们却}a bl》ial }b}.诸肴如下证明:*为一且二_八亘万_压二画三二一’一’I口 bl可(a 吞)1心(a b)(a b) ︼一人U. 一t,一,一.办 一.︸︺叭一办一 一‘U一之一al一一叭U李,一b 叼a︸ 一a_旦口十石 。 b不),雌 !b}}a b}篇-_I,口。、乓乏‘石不十石落j, (l) (2)得{a} }b}}a b} a-下-r十a十O:}a bl》}a! b。 b}bl...‘l,‘ ‘1.|a b|>|a| |b|@曹存富$宁波市北仓区教研室~~…     

ab≥0■|a+b|=|a|+|b|

初中代数介绍有理数(后来为实数)加法时,法则分两部分。第一是符号法则,第二是绝对值法则。关于后者,最后可归纳成: ab≥0|a+b|=|a|+|b|。可逆的箭头,表示可逆的法则。例1 已知同号两数a、b的绝对值为2和5,求a+b。解:ab>0得 |a+b|=|a|+|b|=2+5=7。所以有 a+b=±7。以上是法则的正用,以下看法则的逆用,     

不等式|a|-|b|≤|a+b|≤|a|+|b|取等号的充要条件

高中数学课本证阴了不等式!。}一}川《1。一卜日《卜!理l划,但没有指出何时才能取得‘“”号,本文指出其‘二’号成立的充要条件,并给出应用. 定理1.不等式la+bl《卜l+lb}(。,b〔R)取‘二”号的充要条件是。b>0. 证明:卜一:一6!二扣}+!乙}令今!。十bl忍二·(卜卜!b!)s令冷,十2}。}lb!十b’二。:+:卜b!+bZ令今:b》0. 用数学归纳法可证明推论:不等式l::十。:十…+口。I《!。:卜卜:卜…+}。。】取‘=”号的充要条件是所有a。a;》o(‘,j=l,2,…,。) 定理2.不等式}口{一}b】《】a+‘}(a,b, ‘刃)取“~’号能充妥系件是口二b=。或一l《b/:‘〔. …     

不等式“|a|-|b|≤|a+b|≤|a|+|b|”应用例析

不等式“|a|- |b|≤ |a b|≤ |a| |b|[1] ”(以下简称 [1])是高中数学的一个重要知识点 ,考试大纲说明中对此有明确要求 :会应用不等式“|a|- |b|≤ |a b|≤ |a| |b|”.事实上 ,如何有效地、灵活地应用不等式 [1]证明有关综合性代数推理题是高中数学的难点 .以下简述知识要点