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。     

图Kn和Kn,n的7—匹配设计和简单设计

本文给出图Ｋｎ和Ｋｎ，ｎ的７－匹配设计的存在性和由两个简单的（ｎ，ｋ，λ）－设计（ｉ＝１，２）构造简单的（ｎ，ｋ，λ１＋λ２）设计的条件。     

不等式研究成果集锦(17)

20 2　设 xi >0 ,i =1,2 ,… ,n,n≥ 2 ,∑ni= 1xi =1,记 Ek(x) =Ek(x1 ,x2 ,… ,xn) =∑1≤ i1 <… 0 )时 ,有Ek(1x1 - m,… ,1xn - m)≥ Ckn(n - m) k.(续铁权 .2 0 0 1,1)2 0 3　设 Ai >0 ,λk>0　 (i =1,2 ,… ,n;k = 1,2 ,… ,n) ,∑ni=1Ai ≤π,n∈ N.(1)若 0≤λ≤ 1,有C2n(1-λ21 λ2 ) 2 (λπ) 2 ≤ (n - 1 cosλπ) .∑nk= 1cos2 λAk - cosλπ(∑ni=1cosλAi) 2 ≤ C2n(λπ) 2 ,等号同时成立当且仅当λ=0 .(2 )若 0≤λ≤ 1,有4λ2 C2ncos2 λ2 π≤ (n - 1 cosλ…     

最优(m,n,4,1)光正交签名码的进一步结果

为支持高速多址网络中二维图像的传输,Kitayama首次提出码分多址并行图像传输系统的概念.作为码分多址并行图像传输系统的首选光地址码,光正交签名码(OOSPC)是一族具有良好相关性的Hamming重量为k的m×n(0,1)-矩阵.用Θ(m,n,k,λ)表示所有参数为(m,n,k,λ)的OOSPC中码字容量可能的最大值,则称码字容量为Θ(m,n,k,λ)的(m,n,k,λ)-OOSPC是最优的.本文将针对满足下列条件之一的正整数m和n:(1)mn≡8,16(mod 24),gcd(m,n,2)=2,且mn≡16(mod 32)和gcd(m,n,4)=2不同时成立,其中m和n的所有奇素因子均模6余1;(2)mn≡0(mod 24)且gcd(m,n,6)=2,证明Θ(m,n,4,1)=|mn-1/12|,即构造码字容量为|mn-1/12|的最优(m,n,4,1)-OOSPC.     

具有色多项式∑n＋2／k[k n＋2—k]（λ）k＋1的图

本文讨论了色项式为∑/k≤n 2(n 2)/k[k n 2-k](λ）k 1的图的结构,给出了具有这种色多项式的全部色等价图。     

f与f~(k)具有一个CM公共值的亚纯函数

得到如下结果：设f（z）为非常数亚纯函数，f与f(k)以1为CM公共值，如果（r，f） （r）＜λT（r，f），k=1，0＜λ＜;或3（r，f） （r，） 3（r，）＜λT（r，f），k≥2，0＜λ＜；或（r,） 3（r,）＜λT（r,f），k≥3，0＜则-C，其中C为某一非零常数．     

从一道习题谈ξ(2n)的求法

一般的高等数学教科书或习题集在Fourier级数这一章都安排有类似以下的例题或习题:求x2/4-π|x|/2 π2/6在[-π,π]上的Fourier级数展开式,并计算∑∞n=11/n2的值.它的答案是x24-π2|x| π26=∑∞n=11n2cosnx,-π≤x≤π.(1)　在上式中令x=0得∑∞n=11n2=π26.仔细观察(1)式的右边会发现如果对它积分2次,再令x=0就会出现和式∑∞n=11/n4.一般地对(1)式右边不断积分重复2k-2次,再令x=0就会出现和式∑∞n=11/n2k.这就启示我们也许可以通过上述方式来求级数∑∞n=11/n2k的值.下面我们就来实现它.为符号简单起见,记ξ(2k)=∑∞n=11n2k,k≥1.把(…     

运用S_(m n)公式解高考题

设数列｛an}的前n项和为Sn则Sm＋n＝Sn＋（am＋1＋…＋an＋n）．（1）若数列如｛an}是公差为d的等差数列，则Sm＋n＝Sm＋Sn＋mnd（1）特别地，sn＋1=a1＋Sn＋nd．推论等差数列的前n项和为A，次n项和为B，后n项和为C，则（2）若数列｛an｝是公比为q的等比数列，则am＋1＋…＋am＋n特别地，Sn 1=a1＋qSn(2)推论对等比数列有SS＋Sg。一战（SZ。＋Ss。）．在处理某些等差（或等比）数列的“和”问题时，运用上述公式可简捷求解．例1已知k。）是等比数列，若。1＋。2＋a。218，a；＋a3＋a。—一人且入一al＋a。＋…＋a。，那么tims＂的值…     

偶数阶调和算子△~(2m)的特征值界的估计

设Ω■R~m,m≥2,是边界充分光滑的有界区域,若正数ι满足下列三条件之一:1.l=2~k;2.l=2_0~(k k_1-1) 2~k_0-1,3.l=2~k_0 k_1-2~k,k=0,1,2,…,k_0,k_1=1,2,…,△~(2l)u-λu=0,x∈Ω,则问题u=аu/аn=…=а~(2l-1)u/аn~(2l-1)=0,x∈аΩ,（n是αΩ的单位外法向）的第n 1个特征值λ_(n 1)有下述隐式和显式表示的界     

关于k√n的不等式猜想的再研讨

文[1]给出了一个关于k√n的不等式猜想，文[2]指出该猜想的右侧不等式，即对于正整数n，k〉1，不等式k√n〈kn＋（k-1）/k＋1k√n-k（n-1）＋（k-1）/k＋1k√n-1在k=2时不成立，当k〉2时成立．本文研究了该猜想的左侧不等式，对于正整数n，k〉1，不等式     

Graph Designs for all Graphs with Six Vertices and Eight Edges

Graph designs for all graphs with six vertices and eight edges are discussed. The existence of these graph designs are completely solved except in two possible cases of order 32.     

The Asymptotic Existence of Resolvable Group Divisible Designs

A group divisible design (GDD) is a triple which satisfies the following properties: (1) is a partition of X into subsets called groups; (2) is a collection of subsets of X, called blocks, such that a group and a block contain at most one element in common; and (3) every pair of elements from distinct groups occurs in a constant number λ blocks. This parameter λ is usually called the index. A k‐GDD of type is a GDD with block size k, index , and u groups of size g. A GDD is resolvable if the blocks can be partitioned into classes such that each point occurs in precisely one block of each class. We denote such a design as an RGDD. For fixed integers and , we show that the necessary conditions for the existence of a k‐RGDD of type are sufficient for all . As a corollary of this result and the existence of large resolvable graph decompositions, we establish the asymptotic existence of resolvable graph GDDs, G‐RGDDs, whenever the necessary conditions for the existence of ‐RGDs are met. We also show that, with a few easy modifications, the techniques extend to general index. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 21: 112–126, 2013     

A survey on the existence of G‐Designs

A G‐design of order n is a decomposition of the complete graph on n vertices into edge‐disjoint subgraphs isomorphic to G. We survey the current state of knowledge on the existence problem for G‐designs. This includes references to all the necessary designs and constructions, as well as a few new designs. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 373–410, 2008     

Finiteness Results in Triangle‐Free Quasi‐Symmetric Designs

Triangle‐free quasi‐symmetric 2‐ designs with intersection numbers ; and are investigated. Possibility of triangle‐free quasi‐symmetric designs with or is ruled out. It is also shown that, for a fixed x and a fixed ratio , there are only finitely many triangle‐free quasi‐symmetric designs. © 2012 Wiley Periodicals, Inc. J Combin Designs 00: 1‐6, 2012     

λKv的最大K2,3填充设计和最小K2,3覆盖设计

对于一个有限简单图G,λK_v的G-设计(G-填充,G-覆盖),记为(v,G,λ)-GD((v,G,λ)-PD,(v,G,λ)-CD),是一个(X,B),其中X是K_v的顶点集,B是K_v的子图族,每个子图(称为区组)均同构于G,且K_v中任一边都恰好(最多,至少)出现在B的λ个区组中.一个填充(覆盖)设计称为是最大(最小)的,如果没有其它的这种填充(覆盖)设计具有更多(更少)的区组.本文对于λ>1确定了(v,K_2,3,λ)-GD的存在谱,并对任意λ构造了λK_v的最大K_2,3-填充设计和最小K_2,3-覆盖设计.     

On GMW Designs and Cyclic Hadamard Designs

We generalise results of Jackson concerning cyclic Hadamard designs admitting SL(2,2ⁿ) as a point transitive automorphism group. The generalisation concerns the designs of Gordon, Mills and Welch and we characterise these as designs admitting GM(m,qⁿ) acting in a certain way. We also generalise a construction given by Maschietti, using hyperovals, of cyclic Hadamard designs, and characterise these amongst the designs of Gordon, Mills and Welch.     

λKv的最大K2,3填充设计和最小K2,3覆盖设计

对于一个有限简单图G,λKv的G-设计(G-填充,G-覆盖),记为(v,G,λ)-GD((v,G,λ)-PD,(v,G,λ)-CD),是一个(X,B),其中X是Kb的顶点集,B是Kv的子图族,每个子图(称为区组)均同构于G,且Kv中任一边都恰好(最多,至少)出现在B的λ个区组中.一个填充(覆盖)设计称为是最大(最小)的,如果没有其它的这种填充(覆盖)设计具有更多(更少)的区组.本文对于λ>1确定了(v,K2,3,λ)-GD的存在谱,并对任意λ构造了λKv的最大K2,3-填充设计和最小K2,3-覆盖设计.     

Optimal (2, n) visual cryptographic schemes

In (2,n) visual cryptographic schemes, a secret image(text or picture) is encrypted into n shares, which are distributed among n participants. The image cannot be decoded from any single share but any two participants can together decode it visually, without using any complex decoding mechanism. In this paper, we introduce three meaningful optimality criteria for evaluating different schemes and show that some classes of combinatorial designs, such as BIB designs, PBIB designs and regular graph designs, can yield a large number of black and white (2,n) schemes that are optimal with respect to all these criteria. For a practically useful range of n, we also obtain optimal schemes with the smallest possible pixel expansion.     

任意n个完备二分图的并图的k－优美性和算术性

证明了对于正整数k,n,si,ti（si,ti≥2,i=1,2,…,n）,图n/U/i=1,Ksi,ti是k-优美图;对于正整数k,d（d≥2）,k≠0（roodd）及n,si,ti（si,ti≥2,i=1,2,…,n）,图n/U/i=1,Ksi,ti是（k,d）-算术图,前一结论推广了文[6]的相应结果。