商高数的Je?manowícz猜想

利用Bilu、Hanrot 和 Voutier关于本原素因子存在性理论及二次丢番图方程解的表示方面的一些精细结果证明:当a=n+1, b=2n(n+1), c=2n(n+1)+1时, 方程a^x+b^y=c^z仅有正整数解(x,y,z)=(2,2,2).       

关于丢番图方程x3±1=Dy2

对于丢番图方程x³±1=Dy²,D>2,D无平方因子且不能被3或6k+1形的素数整除。本文用初等的方法证明了除开平凡解x=±1,y=0以外,均无其它的整数解。此外,本文指出,用类似的方法还可证明丢番图方程x³±1=3Dy²除开平凡解x=±1,y=0以外,均无其它的整数解,D的条件同前。     

关于丢番图方程(an)x+(bn)y=(cn)z

设n,a,b,c是正整数,gcd(a,b,c)=1,a,b≥3,且丢番图方程a~x+b~y=c~z只有正整数解(x,y,z)=(1,1,1).证明了若(x,y,z)是丢番图方程(an)~x+(bn)~y=(cn)~z的正整数解且(x,y,z)≠(1,1,1),则yzz或xzy.还证明了当(a,b,c)=(3,5,8),(5,8,13),(8,13,21),(13,21,34)时,丢番图方程(an)~x+(bn)~y=(cn)~z只有正整数解(x,y,z)=(1,1,1).     

关于方程ax+by=cz的Terai-Jesmanowicz猜想

设m是正整数,证明了(A)如果b是奇素数,且a=m3-3m,b=3m2-1,c=m2+1,那么丢番图方程ax+by=cz(1)仅有正整数解(x,y,z)=(2,2,3);(B)如果b是奇素数,且a=m|m4-10m2+5|,b=5m4-10m2+1,c=m2+1,那么丢番图方程(1)仅有正整数解(x,y,z)=(2,2,5).     

关于方程ax+by=cz的Terai-Jesmanowicz猜想

设m是正整数,证明了:(A)如果b是奇素数,且a=m3-3m,b=3m2-1,c=m2+1,那么丢番图方程ax+by=cz(1)仅有正整数解(x,y,z)=(2,2,3);(B)如果b是奇素数,且a=m|m4-10m2+5|,b=5m4-10m2+1,c=m2+1,那么丢番图方程(1)仅有正整数解(x,y,z)=(2,2,5).     

关于线性丢番图方程的Frobenius问题

本文考虑线性丢番图方程a_1x_1+…+a_kx_k=b的非负整数解的存在性问题.为解答Frobenius开问题,对于k2,给出整数G(a_1,…,a_k)的表示形式,该整数是使得b≥G a(_1,…,a_k)时,上述丢番图方程总存在非负整数解的最小整数.     

与马猜想有关的一类不定方程

设$p$是奇素数, $b,t,r\in{\rm N}$. 1992 年, 马少麟猜想丢番图方程 $x^2=2^{2b+2}p^{2t}-2^{b+2}p^{t+r}+1$有唯一的正整数解$(x,b,p,t,r)=(49,3,5,1,2)$, 并且证明了这个猜想蕴含McFarland关于乘子为$-1$ 的阿贝尔差集的猜想.在[Ma S L, MaFarland''conjecture on Abelian difference sets with multiplier-1[J]. {\it Designs, Codes and Cryptography,} 1992, 1:321--332.]中, 马少麟证明了: 若$t\geq r$,则丢番图方程$x^2=2^{2b+2}p^{2t}-2^{b+2}p^{t+r}+1$没有正整数解. 本文证明了: 若$a>1$是奇数,$t\geq r$, 那么丢番图方程$x^2=2^{2b+2}a^{2t}-2^{b+2}a^{t+r}+1$的正整数解由$t=r=1, x+a\sqrt{2^{b+2}(2^b-1)}=(2^{b+1}-1+\sqrt{2^{b+2}(2^b-1)})^{n}$给出, 其中$n$为奇数.作者也证明了: 若$p$是奇素数, 则$(x,b,p,t,r)=(7,3,5,1,2)$是丢番图方程$x^4=2^{2b+2}p^{2t}-2^{b+2}p^{t+r}+1$的唯一正整数解.     

含有陡峭势阱和凹凸非线性项的Kirchhoff型问题的多重正解*

在这篇文章中, 作者研究涉及凹凸非线性项的Kirchhoff型问题-(a + b ∫R3|▽u|²dx) Δu + λV (x)u = μf(x)|u|^q?2u + |u|^p?2u, x ∈ R³,u ∈ H¹(R³),其中a,b > 0 是常数, λ, μ > 0 是参数, 1 < q < 2, 4 < p < 6 且 V 是一个非负连续位势. 在f(x) 和 V 的合适条件下,此问题正解的存在性和集中性能够通过Nehari 流形和Ekeland 变分原理得到.     

关于Kenichiro的一个丢番图方程问题

研究了Kenichiro提出的轮换对称形式的丢番图方程,即方程ab bc ca=ya+b+c3的求解问题,利用素数整除的一些性质,证明了该方程仅有平凡解a=b=c以及非平凡解(a,b,c)=(k,k,4k),(k,4k,k),(4k,k,k)(k∈N),从而完全解决了这个方程.     

含有奇异位势和临界指数的拟线性椭圆方程的G-对称解

本文讨论一类奇异拟线性椭圆型方程
-div(|x|^-ap|▽u|^p-2▽u)=μ+h(x)/|x|^(a+1)p|u|^p-2u+k(x)|u|^p-2u/|x|^bq,x∈R^N,
其中1 < p < N, 0 ≤ a < N-p/p, a ≤ b < a + 1, 0 ≤ μ < μ = (N-p/p-a)^p, q=p^*(a, b) = Np/N-(1+a-b)p,h 和k 是R^N上的连续有界函数, 且关于O(N) 的闭子群G满足某些对称性条件. 应用变分方法和Caffarelli-Kohn-Nirenberg 不等式, 在h与k满足适当条件下, 证得了一些G-对称解的存在性和多重性结果.     

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.