形如sun from i=1 to n(a_ib_(n-i+1))序列极限的几个命题

<正> 在高等数学中关于极限的讨论,一般是既普遍又复杂的一个问题。我们在此就sum from i=1 to n a_1b_(n-i+1)形式的极限进行讨论。定理1(Mertens)设sum from a_n t 和sum from b_n 两无穷级数收敛且至少有一个绝对收敛,又C_n=a_1b_n+a_2b_(n-1)+…+     

ON THE EXISTENCE OF INFINITELY MANY CRITICAL POINTS OF THE EVEN FUNCTIONAL integral from n=Q to (F(x,u,Du)) integral from n=■Q to (G(x,u))

In this paper we deal with the existence of infinitely many critical points of the even functional I(u)=integral from n=Q to (F(x,u,Du)) integral from n=(?)Q to (G(x,u)), u∈W~(1,p)(Ω),where G(x, u)=integral from n=o to u (g(x,t)dt), under the weak structure conditions on F(x, u, q) by the Mountain Pass Lemma.     

关于(sum from i=1 to n a_ib_i)~2下限值的探讨

对a、b两组实数a_i,b_i(i=1,2…,n),切贝雪夫不等式给出sum from(a_ib_i)(本文略去求和上、下限)上下限: 若a_i,b_i同序,有sum from(a_ib_i)≥1/n(sum from(a_i))(sum from(b_i));若a_i,b_i逆序,有sum from(a_ib_i)≤1/n(sum from(a_i))(sum from(b_i)),柯西不等式给出了(sum from(a_ib_i))~2的上限值     

不等式〔∑from i=1 to n (a_i~3)〕〔∑from i=1 to n (b_i~3)〕〔∑from i=1 to n (c_i~3)〕≥〔∑ from i=1 to n (a_ib_ic_i)〕~3的恒等式构造

文[1]为证明2001年第42届IMO第2题而通过独特的思路给出了一个恒等式:设实数ai,bi∈R,A3=n∑i=1ai3,B3=n∑i=1bi3,且AB≠0,则有恒等式n∑i=1ai3 2/3n∑i=1bi3 1/3=n∑i=1ai2bi 13A2Bn∑i=12aiA biBaiA-biB2(1)根据恒等式(1),我们自然会考虑更一般形式的3×N维形式的不等式n∑i=1ai3n∑i=1bi3n∑i=1ci3≥n∑i=1aibici3(2)通过对(2)的研究,本文通过构造方法给出了式(2)的一个新的恒等式.定理设实数ai,bi∈R,A=3∑ni=1ai3,B=3∑ni=1bi3,C=3∑ni=1ci3,且ABC≠0,则有恒等式3(n∑i=1ai3)(n∑i=1bi3)n∑i=1ci3=n∑i=1aibici ABC6Ω(3)其…     

关于完全t部图K(n1,n2,…,nt)的色唯一性

设P（G，λ）是图G的色多项式，如果对任意使P（G，λ）=P（H，λ）的图H都与G同构，则称G是色唯一图。这里通过比较图的特征子图的个数，讨论了由Koh和Teo在文献[1]中提出的问题（若｜ni-nj｜≤2，1≤i，j≤t且min{n1，n2，…，nt}充分大，K（n1，n2，…，nt）是否为色唯一图？）。证明了，若｜ni—nj｜≤2且t↑∑↑i=1 ni〉t^2/2＋t√t-1，则K（n1，n2，…，nt）是色唯一图；若αi=0或k，t↑∑↑i=1 n＋αi〉t^2k^2/8＋｜tk｜/2√t-1，则K（n＋α1，n＋α2，…，n＋αt）是色唯一图。其条件比文献[4]中的条件较好一些。     

幂级数sun from n=0 to ∞((2n)!/(n!)~2(1/2)~(2n))的收敛性讨论

利用stirling公式和阿拉伯判别法可证级数sun from n=0 to ∞((2n)!/(n!)~2(1/2)~(2n))发散,但其相应的交错级数条件收敛.     

图^nUi=1 Fmi,t和^nUi=1 Hmi,t的优美性

本文构造了两类非连通图^nUi=1 Fmi,t和^nUi=1 Hmi,t并证明了这两类图是优美的，且也是交错的。     

三路树P(m,n,t)是边幻图的证明

文 [1 ]中猜测每一棵树是边幻图 .本文证明了三路树 P( m,n,t) ,当 ( i) n,t为偶数且相等 ;( ii)t=n+1 ;( iii) n为奇数且 t=n+2时为边幻图 .     

sum from n=1 to +∞(1/n~(2m))的初等解法

本文用初等的方法研究sum from n=1 to(1/n~(2m))(m∈N)的求和问题。这个问题最先由Euler[8]解决。文献[1][6]给出了另两种求解方法。特别地,对于m=1的情形,即sum from n=1 to ∞(1/n~2)=((π~2)/6),已有许多不同的证明方法,可见文献[2][3][4][5]以及那里的参考文献。本文的想法,主要受文献[5][6]的启发而来的。     

完全t部图 K(n1,n2,…,nt)的色唯一性

本文使用比较两个色等价图的色划分数的方法,得出了完全t部图的色等价图类仍为完全t部图的一般形式数值条件,进一步得出了K(n1,n2,n3)和K(n1,n2,n3,n4)为色唯一图的一般形式数值条件.     

SU(n,1)/S(U(n)×U(1))上的中心极限定理

Let G=SU(n,1),K=S(U(n)×u(1)),and for l∈Z,let{η)1∈Z be a one-Dimensional K-type and let E1 be the line bundle over G/K associated to η.In this work we obtain a central limit theorem for the space El.     

有关循环图C(n;{1,k})的独立数的一些结果

令G=(V(G),E(G))是一个简单有限无向图.如果V(G)的子集S中任意两个顶点均不相邻,则S是图G的一个独立集.顶点独立集大小的最大值,称为图G的独立数,记作α(G).本文研究了循环图C(n;{1,k})的独立数问题,并给出了当k=2,3,4,5时的准确值.     

Young tableaux and crystal base for U_q(osp(1|2n))

Let Uq(osp(1|2n)) be the quantized enveloping superalgebra corresponding to the Lie superalgebra osp(1|2n). In terms of semistandard Young tableaux satisfying some additional conditions, a realization of crystal graph of finite-dimensional irreducible modules of Uq(osp(1|2n)) is given. Also, the generalized LittlewoodRichardson rule for tensor product of crystal graphs is established.     

The stability of RDDE\ddot x(t) + p_1 (t)\dot x(t) + q_1 (t)x(t) + p_2 (t)\dot x(t - r(t)) + q_2 (t)x(t - r(t)) = O

In this paper, the author considered the stability of zero solution of linear RDDE $$\begin{gathered} \ddot x(t) + p_1 (t)\dot x(t) + q_1 (t)x(t) + p_2 (t)\dot x(t - r(t)) + q_2 (t)x(t - r(t)) = O, \hfill \\ \ddot x(t) + p_1 (t)\dot x(t) + q_1 (t)x(t) + p_2 (t)\dot x(t - r(t)) = O \hfill \\ \end{gathered} $$ using Liapunov-Razumikhin functional and transformations and obtained some sufficient conditions for the stability of Eqs.(1) and (2). These results are suitable both for boundedp _i (t),q _i (t) andr(t).i=1,2.     

Periodic solutions of the differential delay equation\dot x(t) = - f(x(t - 1))

For an odd function 1/2(x) defined only on a finite interval, this paper deals with the existence of periodic solutions and the number of simple periodic solutions of the differential delay equation (DDE) . By use of the method of qualitative analysis combined with the constructing of special solutions a series of interesting results are obtained on these problems.Supported by the National Natural Science Foundation of China.     

多项式(∑ki=0t~i)~n系数的性质

本文讨论了多项式∑ｋｉ＝０ｔｉｎ的系数的一些性质，得出了一些有趣的结果     

The Factorization of Adjoint Polynomials of E^G（i）-class Graphs and Chromatically Equivalence Analysis

Let Sn be the star with n vertices,and let G be any connected graph with p vertices.We denote by EG(i)rp (r-1) the graph obtained from Sr and rG by coinciding the i-th vertex of G with the vertex of degree r-1 of Sr,while the i-th vertex of each component of (r-1)G be adjacented to r-1 vertices of degree 1 of Sr,respectively.By applying the properties of adjoint polynomials,We prove that factorization theorem of adjoint polynomials of kinds of graphs EG(i)rp (r-1)U(r-1)K1(1≤i≤p).Furthermore,we obtain structure characteristics of chromatically equivalent graphs of their complements.     

Iterative algorithm for solving a class of general Sylvester-conjugate matrix equation \sum_{i = 1}^{s} A_{i}V + \sum_{j = 1}^{t} B_{j}W = \sum_{l = 1}^{m} E_{l}\overline{V}F_{l} + C

This paper is concerned with iterative solution to general Sylvester-conjugate matrix equation of the form $\sum_{i = 1}^{s} A_{i}V + \sum_{j = 1}^{t} B_{j}W = \sum_{l = 1}^{m} E_{l}\overline{V}F_{l} + C$ . An iterative algorithm is established to solve this matrix equation. When this matrix equation is consistent, for any initial matrices, the solutions can be obtained within finite iterative steps in the absence of round off errors. Some lemmas and theorems are stated and proved where the iterative solutions are obtained. Finally, a numerical example is given to verify the effectiveness of the proposed algorithm.     

一类裂变图的L(_(d,d,1)~(0,1,2))-标号数上界

将图的标号问题由每个顶点需要一个标号的情况推广到每个顶点需要多个标号的情况,给出裂变图的概念以及赋权图的L(_(d,d,1)~(0,1,2))-标号的概念,给出R-单位球图对应裂变图的L(_(d,d,1)~(0,1,2))-标号数的一个上界。