关于奇完全数和孤立数的几个命题

设E(a,b,m)=1/m(a~(2~n)+b~(2n)),这里a,b,m,,n是正整数适合gcd(a,b)=1,ab,m是a~(2~n)+b~(2n)的因数,且当2+ab时,m≡2(mod 4),当2|ab时,m≡1(mod2).运用初等方法证明了:i)当nlog_2log_2log_2a时,E(a,b,m)都不是奇完全数;ii)当nmax{7,logloga}或nmax{5,3 logloga}时,E(a,1,m)都是孤立数.从而改进了相关文献中的结果.     

RN中一类带Hardy-Sobolev临界指数项的Kirchhoff型问题的正解*

研究如下一类带Hardy-Sobolev临界指数的Kirchhoff型问题: 其中N ≥ 3, a, λ ≥ 0, b, m > 0, 0 ≤ s < 2, h ∈ L^2*/2*-1 (R^N )为非零非负函数.当λ = 0时, 利用分析技巧获得了问题可解性结果. 当λ > 0时, 利用变分方法获得该问题正解的存在性.特别地, 当0 < m < 2-s/N-2时, 获得了问题至少存在两个正解.     

单形中的一类不等式

<正> 在ΔABC 中,设 BC=a,CA=b,AB=c,又 x_a,x_b,x_c 分别为三边 a,b,c 上的任一点至该边所对的顶点之间的线段,若ΔABC 的面积为Δ,则对于正实数 α,u,v(u≤v)有∑a~α(-ux_a~α+vx_b~α+vx_c~α)≥3(2v-u)2~αΔ~α.(1)当 x_a,x_b,x_c 分别为正ΔABC 的边 a,b,c 上的高时等号成立.在(1)中,特别地取 x 为ΔABC 的高线,内角平分线及中线时,则有∑a~α(-uh_a~α+vh_b~α+vh_c~α)≥3(2v-u)2~αΔ~α,(2)∑a~α(-ut_a~α+vt_b~α+vt_c~α)≥3(2v-u)2~αΔ~α,(3)∑a~α(-um_a~α+vm_b~α+vm_c~α)≥3(2v-u)2~αΔ~α.(4)当且仅当ΔABC 为正三角形时等号成立.若有两个ΔABC 和ΔA′B′C′的面积分别为Δ和Δ′,h_a,h_b,h_c 及 h_(a′),h_(b′),h_(c′)分别为     

Robin型二阶m 点边值问题正解的存在性

设 a∈C［0,1］, b∈C(［0,1］,(-∞, 0)). 设\-1(t)为线性边值问题  u″+a(t)u′+b(t)u=0， u′(0)=0,\ u(1)=1  的唯一正解. 该文研究非线性二阶常微分方程m 点边值问题  u″+a(t)u′+b(t)u+h(t) f(u)=0，\= u′(0)=0, u(1)-∑[DD(]m-2[]i=1[DD)]α\-i u(ξ\-i)=d  正解的存在性. 其中 d 为参数, ξ\-i∈(0,1), α\-i∈(0,∞) 为满足 ∑[DD(]m-2[]i=1[DD)]α\-i\-1(ξ\-i)<1的常数, i∈{1,\:，m-2}. 在适当的条件下证得: 存在正常数 d\+*, 使 当0d\+*时无正解.     

不等式1/a+1/b≥4/(a+b)的应用

由不等式a2+b2≥2ab,我们可以得到(a+b)2≥4ab.当a,b∈R+时,容易得到a+b/ab≥4a+b,即1a+1b≥4a+b.这又是一个非常有用的基本不等式,下面我们用这个不等式来处理几个问题,看看它的威力。     

无理函数y=（a1x+b1）～（1/2）+（a2x+b2）～（1/2）的最值求法

无理函数y=（a₁x+b₁）^1/2+（a₂x+b₂）^1/2（a1,a2,b1,b2均不为0）（1）的最值问题,是代数中较为典型的一类最值问题之一.当a1a2≥0时,函数（1）为单调函数,求出定义域后利用单调性很容易确定最大值和最小值.但当a1a2<0时,函数（1）最值的求解具有一定的难度.其实,当a1a2<0时,无理函数（1）可改写成如下形式:y=a（x-b）^1/2+c（d-x）^1/2（a,c>0,b,d≠0）（2）当b≤d时,函数才有意义.当b=d时,函数值域为单点集{0}.本文考虑b相似文献   

关于群中乘积元素的阶

讨论群中两个元素a,b的阶不相等时其乘积ab的阶的一类计算问题.设ㄧaㄧ=m,ㄧ bㄧ=n,若(m,n)=1,且存在k∈N使a=bk,则有ㄧabㄧ=mn/d1d2,其中d1=(m,k+1),d2=(n,k+1).若m≠n,ab=ba,且(m,n)ㄧm/(m,n),或(m,n)ㄧn/(m,n),则有ㄧabㄧ=[m,n].     

勾股数问题的推广——空间勾股数

我们知道m>n,m、n都是正整数时,m2-n2、2mn、m2+n2为一组勾股数,当k为正整数时,用k乘以上各数,也可以得出另一组勾股数:k(m2-n2)、2kmn、k(m2+n2).如图1,若设过长方体一个顶点的三条棱长分别为a、b、c,长方体对角线的长为d.则a2+b2+c2=d2.下面我们就探索a、b、c、d都为正整数的构造方法,暂称这四     

奇异摄动非线性边值问题解的渐近性质

其中a相似文献   

一类服务员强制休假的M/M/1排队模型的稳态解

当一次能接受服务的最大顾客数为2时研究服务员强制休假的M/M/1排队模型的主算子的特征值并证明当顾客的到达率λ,服务员的服务率μ和服务员的休假率b满足λ(μ+b)μb时,0不是该主算子的特征值.由此说明该模型不存在稳态解。因此,部分回答一个公开问题.     

RECENT PROGRESS ON SPHERICAL HARMONIC APPROXIMATION MADE BY BNU RESEARCH GROUP ——In memory of Professor Sun Yongsheng

As early as in 1990, Professor Sun Yongsheng, suggested his students at Beijing Normal University to consider research problems on the unit sphere. Under his guidance and encouragement his students started the research on spherical harmonic analysis and approximation. In this paper, we incompletely introduce the main achievements in this area obtained by our group and relative researchers during recent 5 years （2001-2005）. The main topics are： convergence of Cesaro summability, a.e. and strong summability of Fourier-Laplace series; smoothness and K-functionals; Kolmogorov and linear widths.     

Instructions for Authors

<正>Submission Authors must use LaTeX for typewriting,and visit our website www.actamath.com to submit your paper.Our address is Editorial Office of Acta Mathematica Sinica,Academy of Mathematics and Systems Science,Chinese Academy of Sciences,Beijing 100190,P.R.China.     

THE 8~(th) INTERNATIONAL CONGRESS ON INDUSTRIAL AND APPLIED MATHEMATICS

正August 10-14,2015Beijin,China The International Congress on Industrial and Applied Mathematics(ICIAM)is the premier international congress in the field of applied mathematics held every four years under the auspices of the International Council for Industrial and Applied Mathematics.From August 10 to 14,2015,mathematicians,scientists     

Boundedness for commutators of multipliers on Hardy type spaces

In this paper, we study the commutators generalized by multipliers and a BMO function. Under some assumptions, we establish its boundedness properties from certain atomic Hardy space Hb^p（R^n） into the Lebesgue space L^p with p 〈 1.     

A unified approach to certain problems of best local approximation

In this paper we study best local quasi-rational approximation and best local approximation from finite dimensional subspaces of vectorial functions of several variables. Our approach extends and unifies several problems concerning best local multi-point approximation in different norms.     

Towards Excellence in Science Chinese Academy of Sciences

<正>May 26,2014,Beijing Science is a human enterprise in the pursuit of knowledge.The scientific revolution that occurred in the 17th Century initiated the advances of modern science.The scientific knowledge system created by     

THE 8~(th) INTERNATIONAL CONGRESS ON INDUSTRIAL AND APPLIED MATHEMATICS

<正>August 10-14,2015Beijing,ChinaThe International Congress on Industrial and Applied Mathematics(ICIAM)is the premier international congress in the field of applied mathematics held every four years under the auspices of the International Council for Industrial and Applied Mathematics.From August 10 to 14,2015,mathematicians,scientists     

Bounds for the zeros of polynomials

Let P(z)=∑↓j=0↑n ajx^j be a polynomial of degree n. In this paper we prove a more general result which interalia improves upon the bounds of a class of polynomials. We also prove a result which includes some extensions and generalizations of Enestrǒm-Kakeya theorem.