交换环上多项式结式的性质

本文主要讨论交换环上多项式结式的一些性质.首先,我们证明了交换环上一种乘积的结式等于结式的乘积的性质,然后,我们证明了交换环上一种和的结式具有的性质,并且给出了交换环上结式为零的一个充分条件.     

库珀教授的数学课

一天,美国斯坦福大学商学院的数学教授库珀让同学们把自己的生日写在小纸片上,然后把所有的小纸片都折起来放在讲台上.他拿出一张5美元的钞票放在讲台上,问:我用5美元打赌,你们中至少有两个人同月同日生.有人敢跟我     

一堂数学课的感悟

《感悟》2011年第5期有一篇文章《斯坦福的趣味数学课》,内容大体如下:美国斯坦福大学商学院的数学教授库珀上一堂有关概率的课,他先让同学们把自己的生日写在小纸片上,然后把所有的小纸片都折起来放在讲台上.他拿出一张5美元的钞票,问:"我用5美元打赌,你们     

分配格上上标准特征向量的求法

本文研究了分配格(L,∧,∨,0,1)上方阵A的上特征向量的性质.利用矩阵的伴随有向图,得到格上方阵A的上标准特征向量的一种实用的新方法.     

库珀教授的数学课

一天,美国斯坦福大学商学院的数学教授库珀让同学们把自己的生日写在小纸片上,然后把所有的小纸片都折起来放在讲台上．他拿出一张5美元的钞票放在讲台上,问：“我用5美元打赌,你们中至少有两个人同月同日生．有人敢跟我赌吗？”     

测量树叶的面积

那天，我们学习了《面积是多少》一课，课上我们通过数格子的办法算出池塘的大小，很有趣!上完课，顾老师布置我们一个任务：回家找一片树叶，并求出面积。     

自行车的奇想和探究（上）

最近,我新买了一辆自行车,意外地发现车轮上有一处红色的油漆印记.在一个阳光明媚的日子里,我骑上这辆自行车,开始放飞一段奇异的梦想：一、试想第一个问题一辆自行车的轮子上有一处红色印记,如果骑车沿笔直的道路行驶,那么从侧面看,这个印记会划出怎样的曲线呢？问题分析首先简化问题：假设这辆自行车的轮子是圆形的,这处红色印记看成一个点，笔直的道路是平坦的．     

射影平面上不动元素的若干性质

射影平面上存在直射变换下的不动元素的充要条件是直射变换式特征方程的系数行列式值为零;平面上的直射变换至少有一不动点与一不动直线,而且不动点和不动直线的数目相同;直射变换的不共线的不变点至多有三个;过一不动点至少有一不动直线;两不动点的连线是不动直线;不动直线上的点不一定都是不动点。     

Q(√6)上的定幺模格的分类

利用推广的邻格方法,完成了Q(√6)上秩4的在一个阿基米德除子上正定,在另一阿基米德除子上负定的所有幺模格种的分类.     

冰川的眼泪

Lucky、贝卡和乐天蚁跟着动物们来到了一处冰川前。大家都围在冰川前议论纷纷。乍看上去．这座冰川很普通．它的外观如同北极荒野上可见的任何一座冰川。但是仔细观察。贝卡他们吓了一跳：一张悲伤痛苦的面孔映在融化的冰壁上，仿佛哭得眼泪流成了河。为此，冰原上、海水里。很多动物发起了呼吁环保的宣传活动。它们认为，大自然在抗议，不能再坐视不理了。     

Parallel Dixon Matrices by Bracket

It is known that the Dixon matrix can be constructed in parallel either by entry or by diagonal. This paper presents another parallel matrix construction, this time by bracket. The parallel by bracket algorithm is the fastest among the three, but not surprisingly it requires the highest number of processors. The method also shows analytically that the Dixon matrix has a total of m(m+1)²(m+2)n(n+1)²(n+2)/36 brackets but only mn(m+1)(n+1)(mn+2m+2n+1)/6 of them are distinct.     

Algebraic structures determined by 3 by 3 matrix geometry

Using a ``3 by 3 matrix trick' we show that multiplication (an algebraic structure) in a *-algebra is determined by the geometry of the *-algebra of the 3 by 3 matrices with entries from , . This is an example of an algebra-geometry duality which, we claim, has applications.

     

Approximation by non-negative algebraic polynomials

Theorem 1 gives an estimate for the approximation of a continuous functionf by polynomials resulting from the convolution off with non-negative algebraic polynomialsp _n . Jackson's theorem can be deduced from it by choosing a particularp _n whose second Chebyshev-Fourier coefficient is sufficiently close to –1.Work supported in part by the Atomic Energy Commission under contract U.S. AEC AT (11-1) 1469, and in part by the National Science Foundation under grant NSF-GJ-812.     

一类插值多项式的导数逼近(英文)

本文给出了以雅可比多项式的零点作为插值节点的一类插值多项式 Bn( f ;x)的导数逼近具有一阶连续导数的函数的收敛阶 .并且指出 limn→∞ Bn′( f;-1 )≠f′( -1 ) .     

On weighted approximation by Bernstein-Durrmeyer operators

In this paper, we consider weighted approximation by Bernstein-Durrmeyer operators in L_p[0, 1] (1≤p≤∞), where the weight function w(x)=x^α(1−x)^β,−1/p<α, β<1-1/p. We obtain the direct and converse theorems. As an important tool we use appropriate K-functionals. Supported by Zhejiang Provincial Science Foundation.     

Holding a regular pyramid by a circle

Among other things, we prove the following (1) If a regular pyramid whose base is a regular polygon of circum-radius 1 has height at least 1, then the pyramid can be held by a circle, while for every ${0 < \varepsilon < 1}$ , there is a regular pyramid with height ${1 - \varepsilon}$ and base polygon of circum-radius 1 that cannot be held by any circle. (2) A regular pyramid of height h whose base is an equilateral triangle of circum-radius 1 can be held by a circle if and only if h > 0.479 . . . (which complements a theorem by Tanoue). (3) A regular pyramid with square base of unit circum-radius can be held by a circle if and only if its height is greater than 0.828 . . ..     

Interpolation by almost quartic splines

In this paper we define a new class of quartic splines of classC ² and solve the problem of (0, 2) interpolation by the elements of this class. We also study the possibility of the solution of the problems of (0, 1) and (1, 2) interpolations by the elements of the same class. Error estimates are obtained in each case.     

Covering boxes by points

Families of boxes in the d-dimensional Euclidean space are considered. A simple proof of an upper bound on the size of a minimum transversal in terms of the space dimension and the independence number of the intersection graph of the family is given. The proof uses an idea by Gyárfás and Lehel. Some examples are constructed, showing that in three instances this upper bound is exact.     

Pseudo-Riemannian geodesic foliations by circles

We investigate under which assumptions an orientable pseudo-Riemannian geodesic foliations by circles is generated by an S ¹-action. We construct examples showing that, contrary to the Riemannian case, it is not always true. However, we prove that such an action always exists when the foliation does not contain lightlike leaves, i.e. a pseudo-Riemannian Wadsley’s Theorem. As an application, we show that every Lorentzian surface all of whose spacelike/timelike geodesics are closed, is finitely covered by ${S^1 \times \mathbb{R}}$ . It follows that every Lorentzian surface contains a nonclosed geodesic.     

Approximating reals by sums of rationals

We consider the question of approximating any real number α by sums of n rational numbers with denominators 1?q₁,q₂,…,qn?N. This leads to inquiries on approximating a real number by rational numbers with a prescribed number of prime factors in the denominator as well as by rational numbers with smooth denominator.