m次数量幂等矩阵线性组合的可逆性

如果有非零数λ与μ使Pm=λP,Qm=μQ,则称P,Q分别是由λ,μ确定的m次数量幂等矩阵．本文证明了,若有非零数a与b,当μam-1（-1）m-1μbm-1≠0时,使可交换的分别由λ,μ确定的m次数量幂等矩阵P,Q的线性组合aP＋bQ是可逆的,那么对任意非零数u,u,当λμm-1-（-1）m-1μvm-1≠0时,uP＋vQ也是可逆的．本文主要结果和方法的应用,可以推广已有文献的2次、3次幂等矩阵的线性组合可逆的结论．     

一类p-Laplace方程组的正径向解的存在性

本文研究p-Laplace方程组{-(rN-1(φ)(u'))'=λrN-1f(u,v), a＜r＜b,-(rN-1(φ)(v'))'=λrN-1g(u,v), a＜r＜b,u(a)=0=u(b),v(a)=0=v(b)的正解,其中参数λ＞o,(φ)是递增且同伦与R的奇映射,f,g∈[C[0,∞)]2满足适当的条件,讨论了当参数λ很大时正解的存在性.     

三角函数中的一个有用定理

定理 已知f（x）=Acosx＋Bsinx（A,B为常数）,若实数a,b满足f（a）=f（b）=0且a—b≠mπ（m∈Z）,则A=B=0．     

幂等矩阵的组合的零度与秩

本文研究了两个幂等矩阵P与Q的组合Ap Bq-Cpq(a≠0,b≠0)的秩.利用矩阵的核子空间及线性空间的同构的有关性质,得到了:当c＝a b时,Ap Bq-Cpq的秩为一个常数,且等于P-Q的秩;当c≠a b时,Ap Bq-Cpq的秩为一个常数,且等于P Q的秩,推广了J. J. Koliha和V. Rakoeeie[3]的结果.     

Heisenberg群上Folland—Stein定理的注记

该文在G．B．Folland与E．M．Stein研究的算子的基础上．拓展考虑了算子，其中λ＋μ≠0且λ≠α／2n，μ≠—α／2n），证明了：如果，使得有限（其中ψa，b，1（z，t）＝—4（λ＋μ）ab（｜z｜2＋1—it）a-1（｜z｜2＋1＋it）b-1，那么在分布的意义下将有．特别，当λ＝μ＝—1／2时，此结果即原来的Folland－Stein定理．     

Multiple solutions for weighted nonlinear elliptic system involving critical exponents

In this paper,by using the idea of category,we investigate how the shape of the graph of h(x)affects the number of positive solutions to the following weighted nonlinear elliptic system:-div(|x|-2au)-μu|x|2(a+1)=αα+βh(x)|u|α-2|v|βu|x|b2*(a,b)+λK1(x)|u|q-2u,in,-div(|x|-2av)-μv|x|2(a+1)=βα+βh(x)|u|α|v|β-2v|x|b2*(a,b)+σK2(x)|v|q-2v,in,u=v=0,on,where 0∈is a smooth bounded domain in RN(N 3),λ,σ0 are parameters,0μμa(N-2-2a2)2;h(x),K1(x)and K2(x)are positive continuous functions in,1 q2,α,β1 andα+β=2*(a,b)(2*(a,b)2N N-2(1+a-b),is critical Sobolev-Hardy exponent).We prove that the system has at least k nontrivial nonnegative solutions when the pair of the parameters(λ,σ)belongs to a certain subset of R2.     

用矩阵的秩判断两空间直线及直线与平面的位置关系

空间两直线的位置关系可分为两大类:异面、共面;在同一平面内又分为:平行、相交和重合;直线与平面相对位置可分为:平面与直线平行、直线与平面相交、直线在平面上。这里介绍用矩阵的秩来判断两空间直线及直线与平面的位置关系。引理1非齐次线性方程组a11x1+a12x2+…+a1nxn=b1,a21x1+a22x2+…+a2nxn=b2,………………am1x1+am2x2+…+amnxn=bm,有解的充要条件是:系数矩阵A与增广矩阵B的秩RA=RB;且RA=RB=n时有唯一解。引理2设向量βi=(Ai,Bi,Ci)(i=1,2,3),则(β1×β2)·β3=A1B1C1A2B2C2A3B3C3=0三向量共面。定理1设两空间直线L1∶A1…     

S4上仿Blaschke张量的特征值为常数的超曲面

设X：M→S^n＋1州是（n＋1）一维单位球面上不含脐点的超曲面，在S^n＋1的Moebius变换群下浸入x的四个基本不变量是：一个黎曼度量g称为MSbius度量；一个1-形式圣称为Moebius形式；一个对称的（0，2）张量A称为Blaschke张量和一个对称的（0，2）张量B称为MSbius第二基本形式．李海中和王长平研究了满足如下条件的超曲面x：M→S^n＋1：（i）Φ=0；（ii）存在可微函数λ和μ使A＋λg＋μB=0，他们证明了λ和μ都是常数，并且给出了这类超曲面的分类．对称的（0，2）张量A＋λB也是Moebius不变量，称为浸入x的仿Blaschke张量，其中A是常数．因此李海中和王长平也就在Φ=0的条件下给出了A＋λB的特征值全相等的超曲面X：M→s^n＋1州的分类．本文对S^4中满足以下条件的超曲面进行完全分类：（i）Φ=0，（ii）对某一个常数λ，A＋λB具有常数特征值．     

Existence of Nonnegative Solutions for a Class of Systems Involving Fractional(p,q)-Laplacian Operators

The authors study the following Dirichlet problem of a system involving fractional(p, q)-Laplacian operators:{(-△)_p~su=λa(x)|u|+~(p-2)u+λb(x)|u|~(α-2)|u|~βu+μ(x)/αδ|u|~(γ-2)|v|~δu in Ω,(-△)_p~su=λc(x)|v|+~(q-2)v+λb(x)|u|~α|u|~(β-2)v+μ(x)/βγ|u|~γ|v|~(δ-2)v in Ω,u=v=0 on R~N\Ω where λ 0 is a real parameter, ? is a bounded domain in RN, with boundary ?? Lipschitz continuous, s ∈(0, 1), 1 p ≤ q ∞, sq N, while(-?)s pu is the fractional p-Laplacian operator of u and, similarly,(-?)s qv is the fractional q-Laplacian operator of v. Since possibly p = q, the classical definitions of the Nehari manifold for systems and of the Fibering mapping are not suitable. In this paper, the authors modify these definitions to solve the Dirichlet problem above. Then, by virtue of the properties of the first eigenvalueλ_1 for a related system, they prove that there exists a positive solution for the problem when λ λ_1 by the modified definitions. Moreover, the authors obtain the bifurcation property when λ→λ_1~-. Finally, thanks to the Picone identity, a nonexistence result is also obtained when λ≥λ_1.     

具有稳定秩1的置换环

称环R具有稳定秩1,如果对任意的a,b∈R,aR bR=R,则存在Y∈R,使得a by∈U(R).证明了置换环有稳定秩1当且仅当对任意的幂等元e∈R,如果aR b(eR)=R,则存在u,v∈R,使得au b(ev):0且(eR)u (eR)(ev)=eR当且仅当对任意的幂等元e∈R,如果aR b(eR):R,则存在u,t,∈R,使得as b(et)=0当且仅当存在z∈eR,使s=uz,t=vz,从而给出这类置换环新的元素刻画.进一步地,证明了如果R是稳定秩1的置换环,对任意的正则元a∈R,2a总可以表示成两个单位的和.最后对具有降链本原分式的置换环R,证明了对任意的a∈R,2a总可以表示成两个单位的和.     

两个分块矩阵相似性的研究

给出两个分块矩阵相似的两个充分必要条件 .也就是说 ,如果两个方阵 A和 B在 A2 =0和 B2 =0的条件下 ,则两个分块矩阵 A C0 B 和 A 00 B 相似的充分必要条件是 :rank A C0 B =rank(A) +rank(B)和 AC +CB =0 .如果两个方阵 A和 B在 A2 =A和 B2 =B的条件下 ,则两个分块矩阵 A C0 B和 A 00 B 相似的充分必要条件是 :AC +CB =C.     

幂等矩阵的组合的可逆性

本文研究了两个幂等矩阵P与Q的组合aP+bQ-cPQ-dQP-ePQP (其中a,b,c,d,e∈(C),a≠0,b≠0)的可逆性. 利用P-Q的可逆性及幂等矩阵的性质,得到了aP+bQ-cPQ-dQP-ePQP可逆的一些充要条件. 推广了J. J.Koliha 和 V.RakoA(c)eviA(c)[1]及Zuo Kezheng[2]的结论.     

Rank theorems of operators between Banach spaces

Let E and F be Banach spaces, and B(E,F) all of bounded linear operators on E into F. Let T0∈B(E,F) with an outer inverse T#0∈B(F,E). Then a characteristic condition of S=(I+T#0(T-T0))-1T#0 with T∈B(E,F) and ‖T#0(T-T0)‖<1, being a generalized inverse of T, is presented, and hence, a rank theorem of operators on E into F is established (which generalizes the rank theorem of matrices to Banach spaces). Consequently, an improved finite rank theorem and a new rank theorem are deduced. These results will be very useful to nonlinear functional analysis.     

Asymptotic Behavior of Solutions of Some Linear Difference Equations with Oscillatory Decreasing Coefficients

We investigate the asymptotic behavior of solutions of the system x ( n +1)=[ A + B ( n ) V ( n )+ R ( n )] x ( n ), n S n 0 , where A is an invertible m 2 m matrix with real eigenvalues, B ( n )= ~ j =1 r B j e i u j n , u j are real and u j p ~ (1+2 M ) for any M ] Z , B j are constant m 2 m matrices, the matrix V ( n ) satisfies V ( n ) M 0 as n M X , ~ n =0 X Á V ( n +1) m V ( n ) Á < X , ~ n =0 X Á V ( n ) Á 2 < X , and ~ n =0 X Á R ( n ) Á < X . If AV ( n )= V ( n ) A , then we show that the original system is asymptotically equivalent to a system x ( n +1)=[ A + B 0 V ( n )+ R 1 ( n )] x ( n ), where B 0 is a constant matrix and ~ n =0 X Á R 1 ( n ) Á < X . From this, it is possible to deduce the asymptotic behavior of solutions as n M X . We illustrate our method by investigating the asymptotic behavior of solutions of x 1 ( n +2) m 2(cos f 1 ) x 1 ( n +1)+ x 1 ( n )+ a sin n f n g x 2 ( n )=0 x 2 ( n +2) m 2(cos f 2 ) x 2 ( n +1)+ x 2 ( n )+ b sin n f n g x 1 ( n )=0 , where 0< f 1 , f 2 < ~ , 1/2< g h 1, f 1 p f 2 , and 0< f <2 ~ .     

查甫雷金方程的唯一性定理(Ⅰ)

<正>考虑下列混合型议程的唯一性问题 K(y)u_(xx)+u_(yy)=0 (K(0)=0;当y≠0时,■(1) 所考慮的區域D由三條曲綾圍成.其一是雙曲區域(y<0)中由原點引出的特徵线Γ_1,它滿足下面條件     

The Semi-global Isometric Imbedding in R3 of Two Dimensional Riemannian Manifolds with Gaussian Curvature Changing Sign Cleanly

An abstract Riemannian metric ds²= Edu² + 2Fdudv + Gdv² is given in (u, v) ∈ [0, 2&Pi] × [-&delta, &delta] where E, F, G are smooth functions of (u, v) and periodic in u with period 2&Pi. Moneover K|_{v=0} = 0. K_r|_{v=0} ≠ 0. when> K is the Gaussian curvature. We imbed it semiglobally as the graph of a smooth surface x = x(u, v ), y = y(u, v), z = z(u, v) of R³ in the neighborhood of v = 0. In this paper we show that, if [K_rΓ²_{11}]_{v=0}, and three compatibility conditions are satisified, then there exists such an isometric imbedding.     

特征数≠2的非交换主理想整环上线性群的自同构

<正> 华罗庚和J.Dieudonné研究了体上线性群的自同构问题,而华罗庚和Ⅰ.Reiner以及作者则研究了整数环上线性群的自同构问题.因为整数环是一种特殊的环,所以一般体上线性群的自同构的结果不能从整数环上线性群的自同构的结果导     

关于矩阵方程求解的另一点注记

对任意给定的矩阵A∈Pm×n,B∈Pm×s(s≤n),探讨了矩阵方程AX=B有列满秩解,同时BY=A有行满秩解的充分必要条件,并且给出了基于矩阵的等价、齐次方程组的同解、向量组的等价及线性空间语言的推广.