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).     

Ap和Hp空间的乘子

给出了关于Ａ^p和Ｈ^p空间系数乘子的一些结果     

关于Diophantine 方程xn+2kyn=pz2

设p 为奇素数且对任意的整数m, d, p≠(2^m±1)=/d², 则对任意的素数n > p^8p2, 方程xⁿ+2^kyⁿ=pz², k≥2 没有整数解(x, y, z) 使得x, y, z 两两互素且均不为0.     

1+2x5y+5z11u=2v·11w,xuvw>0,y+z>0的非负整数解

借助计算机,用初等方法给出了指数丢番图方程1+2^x5^y+5^z11^u=2^v·11^w,xuvw>0,y+z>0的全部非负整数解.     

泛系分析：数学.方法论.相对论.辩证思维

基于对广义系统、泛导、泛对称、泛箱原理、泛系相对性提出了新的分析模式与新的相对数学化的简化强化的定义，发展了泛系方法论与泛系相对论的框架与理法，具体内容涉及相对普适的数学化的泛系，200类对偶，对偶转化，泛对称转化，泛系辩证，八畴方法，泛系数学方法，广义量化，逼近转化原则，泛等价定理，供求分析，思维实验，广义灰色系统，等等     

On the subspace L((x ∧ y)m) of Sm(∧2?4)

We take the exterior power ℝ⁴ ∧ ℝ⁴ of the space ℝ⁴, its mth symmetric power V = S ^m (∧²ℝ⁴) = (ℝ⁴ ∧ ℝ⁴) ∨ (ℝ⁴ ∧ ℝ⁴) ∨ ... ∨(ℝ⁴ ∧ ℝ⁴), and put V ₀ = L((x ∧ y)∨ ... ∨(x ∧ y): x, y ∈ ℝ⁴). We find the dimension of V ₀ and an algorithm for distinguishing a basis for V ₀ efficiently. This problem arose in vector tomography for the purpose of reconstructing the solenoidal part of a symmetric tensor field. Original Russian Text Copyright ? 2009 Gubarev V. Yu. The author was supported by the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-344.2008.1). __________ Novosibirsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 50, No. 3, pp. 503–514, May–June, 2009.     

On hypersurfaces of H2×H2

In this paper, we study hypersurfaces of H²× H². We first classify the hypersurfaces with constant principal curvatures and constant product angle functions. Then we classify homogeneous hypersurfaces and isoparametric hypersurfaces, respectively. Finally, we classify the hypersurfaces with at most two distinct constant principal curvatures, as well as those with three distinct constant principal curvatures under some additional conditions.     

Lp( Rn)中的神经网络逼近和系统识别

本文主要研究函数的叠加对Lp(Rn)中的函数,Lp(Rn)上的非线性连续泛函及非线性连续算子的逼近.这些问题与Sigma-Pi型神经网络逼近能力有关.     

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).     

曲面间Gn连续条件

本文在一般情形下研究了曲面间Gⁿ(n阶几何连续)条件,通过分析Gⁿ变换的结构特征给出了一般解、递归线性确定可调函数的方法及各种等价条件;然后对三角域、矩形域Bezier曲面片,给出了一类简单、实用条件。这些条件包含丰富的自由参数,在自由曲面拼接、造型中有着广泛而重要的应用。最后研究了N,M维曲面的q维子流形间Gⁿ条件,使几何连续性问题有了完整的理论基础。     

Embedding of Hpω in the class eL

We obtain the necessary conditions for the embedding H _p ^ω ⊂ e ^L (1≤p∞) with convex modulus of continuity ω in terms of this modulus. In the case p=1 these conditions are also sufficient.     

高维Hausdorff算子在Hp(Rn)上的有界性

本文主要研究以下形式的Hausdorff算子H_Φf(x)=∫_RⁿΦ(u₁….,u_n)f(u₁x₁,…,u_nx_n)du₁…du_n,其中Φ是Rⁿ上的缓增分布.当n≥2,0Φ在H^p(Rⁿ)上有界当且仅当Φ≡0.进一步,当n≥2,n/n+1Φ有合适定义,那么H_Φ在H^p(Rⁿ)上有界当且仅当Φ是常数.这些结果都表明Hausdorff算子H_Φ在H^p(Rⁿ)上的有界性很复杂.此外,我们将H_Φ转化成卷积型算子,得到H_Φ在Lebesgue空间上有界的一些新的结果.     

L2 Extension and Effectiveness of Lp Strong Openness Property

In this note, we present an L² extension approach to the effectiveness result of Lpstrong openness property of multiplier ideal sheaves.     

Balanced colombeau products of the distributions x±-p and x-p

Results on singular products of the distributions x _± ^-p and x ^-p for natural p are derived, when the products are balanced so that their sum exists in the distribution space. These results follow the pattern of a known distributional product published by Jan Mikusiski in 1966. The results are obtained in the Colombeau algebra of generalized functions, which is the most relevant algebraic construction for tackling nonlinear problems of Schwartz distributions.     

Poisson方程的Lp理论

本文对于Poisson方程的Dirichlet问题(第一边值问题)、Neumann问题(第二边值问题)和Robin问题(第三边值问题),给出一个简单而统一的方法来得到解的全局Lp-估计(或称为Calderón-Zygmund估计).这个方法的核心是基本的L²-估计和对于解的二阶偏导数的分布函数的精细分解.     

Extensions of the D(?k2)-triples {k2, k2 ± 1, 4k2 ± 1}

Let n be a nonzero integer. A set of m distinct positive integers is called a D(n)-m-tuple if the product of any two of them increased by n is a perfect square. Let k be a positive integer. In this paper, we show that if {k ², k ² + 1, 4k ² + 1, d} is a D(−k ²)-quadruple, then d = 1, and that if {k ² − 1, k ², 4k ² − 1, d} is a D(k ²)-quadruple, then d = 8k ²(2k ² − 1).     

广义函数x+λ和x-μ的乘积

For any complcx λ≠integral, and any natural number n, the product is calcuated and Hadmard''s finite part of obtained. The product is also obtained when λ+μ is a complex number of any other possible kind, and μ is a complex number ≠ integral. The products are also calculated. Any special example of these kinds of products has not yet been calculated.     

Cpa空间中的非线性逼近

In this paper, we establish the following Jackson and Bernstein inequalities for C_p^a space:(?)where σ=(α+ (1/p)^-1),α > 0,0 < p < ∞. Moreover, these inequalities are used to give characterizations of the approximation spaces resulted from two approximations verpectively by splines with free knots and approximation by rational functions.