关于蕴含3Gl可图序列的极值问题

设σ（3C1，n）是具有下述性质的最小正偶数，每个项和至少为σ（3C1，n）的n项可图序列π都有一个实现含有长为3，4，…，l的圈，本文确定了当7≤t≤8且n≥l≥以及当l=9且n≥12时σ（2C1，n）的值。     

关于Erdos-Jacobson-Lehel 问题的门槛

设σ(k,n）表示最小的正整数m，使得对于每个n项正可图序列，当其项和至少为m时，有一个实现含k 1个顶点的团作为其子图。Erdos等人猜想：σ(k,n）＝（k-1)(2n-k) 2.Li等人证明了这个猜想对于k≥5，n≥（^k2））＋3是对的，并且提出如下问题：确定最小的整数N（k），使得这个猜想对于n≥N(k）成立。他们同时指出：当k≥5时，[5k-1/2]≤N(k）≤（^k2) 3.Mubayi猜想：当k≥5时，N(k）＝[5k-1/2]。在本文中，我们证明了N（8）＝20，即Mubayi猜想对于k=8是成立的。     

蕴含K4＋P2-可图序列的刻划

对于给定的图H,若存在可图序列π的一个实现包含H作为子图,则称π为蕴含H-可图的．Gould等人考虑了下述极值问题的变形：确定最小的偶整数σ（H,n）,使得每个满足σ（π）≥σ（H,n）的n项可图序列π=（d1,d2,…,dn）是蕴含H-可图的,其中σ（π）=∑di．本文刻划了蕴含K4＋P2-可图序列,其中K4＋P2是向致的一个顶点添加两条悬挂边后构成的简单图．这一刻划导出σ（K4＋P2,n）的值．     

Precise Rates in the Law of Iterated Logarithm for the Moment of I.I.D. Random Variables

Let{X,Xn;n≥1} be a sequence of i,i.d, random variables, E X = 0, E X^2 = σ^2 〈 ∞.Set Sn=X1＋X2＋…＋Xn,Mn=max k≤n│Sk│,n≥1.Let an=O（1/loglogn）.In this paper,we prove that,for b〉-1,lim ε→0 →^2（b＋1）∑n=1^∞ （loglogn）^b/nlogn n^1/2 E{Mn-σ（ε＋an）√2nloglogn}＋σ2^-b/（b＋1）（2b＋3）E│N│^2b＋3∑k=0^∞ （-1）k/（2k＋1）^2b＋3 holds if and only if EX=0 and EX^2=σ^2〈∞.     

The Threshold for the Erdos, Jacobson and Lehel Conjecture to Be True

Let σ（k, n） be the smallest even integer such that each n-term positive graphic sequence with term sum at least σ（k, n） can be realized by a graph containing a clique of k ＋ 1 vertices. Erdos et al. （Graph Theory, 1991, 439-449） conjectured that σ（k, n） = （k - 1）（2n- k） ＋ 2. Li et al. （Science in China, 1998, 510-520） proved that the conjecture is true for k 〉 5 and n ≥ （k2） ＋ 3, and raised the problem of determining the smallest integer N（k） such that the conjecture holds for n ≥ N（k）. They also determined the values of N（k） for 2 ≤ k ≤ 7, and proved that [5k-1/2] ≤ N（k） ≤ （k2） ＋ 3 for k ≥ 8. In this paper, we determine the exact values of σ（k, n） for n ≥ 2k＋3 and k ≥ 6. Therefore, the problem of determining σ（k, n） is completely solved. In addition, we prove as a corollary that N（k） -= [5k-1/2] for k ≥6.     

σ-相关同伦元素的非平凡性

本文中,通过几何方法证明了σ相关同伦元素在球面稳定同伦群π_mS中是非平凡的,其中m=p~(n+1)q+2p~nq+(s+3)p~2q+(s+3)pq+(s+3)q-8,p≥7是奇素数,n3,0≤sp-3,且q=2(p-1).该σ相关同伦元素在Adams谱序列的E_2-项中由■_s+3■_ng0表示.     

Precise Asymptotics in the Baum-Katz and Davis Laws of Large Numbers of ρ-mixing Sequences

Let {X,Xn;n ≥ 1} be a strictly stationary sequence of ρ-mixing random variables with mean zeros and finite variances. Set Sn =∑k=1^n Xk, Mn=maxk≤n｜Sk｜,n≥1.Suppose limn→∞ESn^2/n=：σ^2〉0 and ∑n^∞=1 ρ^2/d（2^n）〈∞,where d=2 if 1≤r〈2 and d〉r if r≥2.We prove that if E｜X｜^r 〈∞,for 1≤p〈2 and r〉p,then limε→0ε^2（r-p）/2-p ∑∞n=1 n^r/p-2 P{Mn≥εn^1/p}=2p/r-p ∑∞k=1（-1）^k/（2k＋1）^2（r-p）/（2-p）E｜Z｜^2（r-p）/2-p,where Z has a normal distribution with mean 0 and variance σ^2.     

几类B值拟鞅空间上的原子分解

设1〈P≤2,0〈n≤1,X是P一致可光滑空间的Banach空间,则对每个X值拟鞅f=（fn）n≥0∈pHn^σ（X）存在分解fn=∑k∈Zμkαn^k（n≥0）,并且｜｜f｜｜pHα^σ（X）＋｜｜R（f）｜｜α～inf（∑k∈μk^a）^1/a,这里a^k=（an^k）n≥（k∈Z）是一列（1,α,∞;p）拟鞅原子,并且在L^1中收敛,sup k∈z｜｜a^k＊｜｜n〈∞,（μk）k∈Z∈la是非负实数列．对于拟鞅空间pHa^s（X）和qKn（x）成立类似的结果．此外,利用拟鞅原子分解定理,证明了几个拟鞅不等式．     

维林肯-傅里叶级数的（C，α）和

对维林金系统{ψ,n≥1}和0＜α＜ 1定义极大算子σ^α＊f：= sup │σ^αnf│，其中σ^αnf是函数f的（C，α）平均值．证明了算子σ^α＊是（p，p）型（1〈P〈∞）和弱（1，1）型．另外‖σ^α＊f‖1≤C‖f‖H1,，其中H1是Hardy空间．利用上述结果，证明了对任一可积函数f，σ^αnf几乎处处收敛于f．     

关于Erds-Jacobson-Lehel问题的门槛(英文)

设σ(k ,n)表示最小的正整数m ,使得对于每个n项正可图序列 ,当其项和至少为m时 ,有一个实现含k+ 1个顶点的团作为其子图 .Erd s等人猜想 :σ(k ,n) =(k - 1 ) ( 2n-k)+ 2 .Li等人证明了这个猜想对于k≥ 5,n≥ k2 + 3是对的 ,并且提出如下问题 :确定最小的整数N(k) ,使得这个猜想对于n≥N(k)成立 .他们同时指出 :当k≥ 5时 ,5k- 12 ≤N(k)≤ k2 + 3.Mubayi猜想 :当k≥ 5时 ,N(k) =5k - 12 .在本文中 ,我们证明了N( 8) =2 0 ,即Mubayi猜想对于k =8是成立的     

Fibonacci数列的模数列的周期性

对于Fibonacci数列{Fn}以及给定的正整数m,由Fn关于模m的最小非负剩余an,构成一个新的数列{an},称为Fibonacci数列的模数列.本文利用初等数论的知识和数学归纳法,证明了Fibonacci数列的模数列是周期数列,并且是纯周期数列.     

A simple criterion on degree sequences of graphs

A finite sequence of nonnegative integers is called graphic if the terms in the sequence can be realized as the degrees of vertices of a finite simple graph. We present two new characterizations of graphic sequences. The first of these is similar to a result of Havel-Hakimi, and the second equivalent to a result of Erd?s & Gallai, thus providing a short proof of the latter result. We also show how some known results concerning degree sets and degree sequences follow from our results.     

Almost overcomplete and almost overtotal sequences in Banach spaces

We introduce the new concepts of almost overcomplete sequence in a Banach space and almost overtotal sequence in a dual space. We prove that any of such sequences is relatively norm-compact and we present several applications of this fact.     

有限域上二次型序列与几何序列的互相关性

§1Introduction Letpbeaprimenumber,q=pm,andFqdenotethefinitefieldwithqelements.Fo anyn≥1,Trmnm(·)isthetracefunctionfromFqntoFq.LetαbeaprimitiveelementofFq and{α1,α2,...,αn}beabasisofFqnoverFq.Definition1.ForanonlinearfunctionffromFqtoFp,thesequenceS={Si}qn-1i=1withit termdefinedby Si=f(Trmnm(αi))(1iscalledageometricsequence.SuchageometricsequenceShasperioddividingqn-1.Geometricsequences includingm-sequence[1],GMWsequence[2,3],cascadedGMWsequence[4,5]andman others[6],mayhavelar…     

p元扩展序列的线性复杂度

给出了由周期为p~m-1的p元序列导出的周期为p~(em)-1的p元扩展序列的线性复杂度.作为一个实例,计算了扩展Legendre序列的线性复杂度.     

Distinct length modular zero-sum subsequences: A proof of Graham's conjecture

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Let S be a sequence of n nonnegative integers not exceeding n−1 such that S takes at least three distinct values. We show that S has two nonempty zero-sum subsequences with distinct lengths. This proves a conjecture of R.L. Graham. The validity of this conjecture was verified by Erd?s and Szemerédi for all sufficiently large prime n.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=LftJj-E6aQA.     

Graphic sequences with a realization containing intersecting cliques

Let r≥ 1, k≥ 2 and Fm1 ,...,mki;r denote the most general definition of a friendship graph, that is, the graph of Kr+m1 , . . . , Kr+mk meeting in a common r set, where Kr+mi is the complete graph on r + mi vertices. Clearly, | Fm1 ,...,mki;r | = m1+ ··· + mk + r. Let σ(Fm1 ,...,mki;r , n) be the smallest even integer such that every n-term graphic sequence π = (d1, d2, . . . , dn) with term sum σ(π) = d1 + d2 + ··· + dn ≥σ(Fm1 ,...,mki;r,n) has a realization G containing Fm1 ,...,mki;r as a subgraph. In this paper, we determine σ(Fm1 ,...,mki;r,n) for n sufficiently large.     

A pattern sequence approach to Stern’s sequence

Suppose that w∈1{0,1}^∗ and let a_w(n) be the number of occurrences of the word w in the binary expansion of n. Let {s(n)}_n?0 denote the Stern sequence, defined by s(0)=0, s(1)=1, and for n?1, In this note, we show that where denotes the complement of w (obtained by sending 0?1 and 1?0) and [w]₂ denotes the integer specified by the word w∈{0,1}^∗ interpreted in base 2.     

Wythoff''s sequence and -Heap Wythoff''s conjectures

Define a Wythoff's sequence as a sequence of pairs of integers (A_n,B_n) _n>n0 such that there exists a finite set of integers T, A_n=mex( A_i,B_i:i<n T), B_n-A_n=n, and B_n ∩T=. Structural properties and behaviors of Wythoff's sequence are investigated. The main result is that for such a sequence, there always exists an integer α such that when n is large enough, |A_n-nφ-α|1, where , the golden section. The value of α can also be easily determined by a relatively small number of pairs in the sequence. As a corollary, the two conjectures on the N-heap Wythoff's game by Fraenkel [Complexity, appeal and challenges of combinatorial Games, Theoret. Comput. Sci. 313 (2004) 393–415] on the N-heaped Wythoff's game are proved to be equivalent.     

On the boundary sequence of an automatic sequence

In this paper, we prove a conjecture of Chen and Wen that the boundary sequence of an automatic sequence is also automatic. In particular, we study the boundary sequences of the generalized Cantor sequences, and give a complete characterization of the periodic boundary sequences. As an application, for a class of automatic sequences, we prove that their abelian complexities are also automatic.