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.     

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…     

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.     

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.     

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.     

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.     

Banach and Statistical Cores of Bounded Sequences

In this paper, we are mainly concerned with characterizing matrices that map every bounded sequence into one whose Banach core is a subset of the statistical core of the original sequence.     

A construction of binary Golay sequence pairs from odd‐length Barker sequences

Binary Golay sequence pairs exist for lengths 2, 10 and 26 and, by Turyn's product construction, for all lengths of the form 2^a10^b26^c where a, b, c are non‐negative integers. Computer search has shown that all inequivalent binary Golay sequence pairs of length less than 100 can be constructed from five “seed” pairs, of length 2, 10, 10, 20 and 26. We give the first complete explanation of the origin of the length 26 binary Golay seed pair, involving a Barker sequence of length 13 and a related Barker sequence of length 11. This is the special case m=1 of a general construction for a length 16m+10 binary Golay pair from a related pair of Barker sequences of length 8m+5 and 8m+3, for integer m≥0. In the case m=0, we obtain an alternative explanation of the origin of one of the length 10 binary Golay seed pairs. The construction cannot produce binary Golay sequence pairs for m>1, having length greater than 26, because there are no Barker sequences of odd length greater than 13. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 478–491, 2009     

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.     

Random sequences of binary digits in which missing values can almost certainly be restored

Infinite sequences of random (nearly independent) binary digits can be generated having (almost certainly) a remarkable property. Observation of any nonzero fraction such as sequence makes it possible to calculate the values of all the unobserved digits.     

广义的k阶Fibonacci-Jacobsthal序列及其性质

定义了一类广义的k阶Fibonacci-Jacobsthal序列,并给出了第四个初值条件.借助矩阵的方法得到了Jacobsthal序列与Jacobsthal-Lucas序列的关系,广义k阶Fibonacci-Jacobsthal序列与Jacobsthal序列,Fibonacci序列的关系,同时给出了k阶Fibonacc...     

前馈序列与反馈序列的关系

本文引入移位寄存器序列的向量值表示讨论了线性移位寄存器的前馈序列与反馈序列同时为ｍ序列时的关系,得到了它们在向量值表示下的一个关系式。     

一类随机变量序列部分和的强大数定律

设{Xn,n≥1}是随机变量序列.文[4]在二阶矩限制下,获得了任意随机变量序列的Hajek-Renyi型不等式,并给出了随机变量序列的强大数定律.本文利用胡舒合等获得的强大数定律,给出了随机变量序列的一些几乎必然收敛性,并给出了结果在PA,NA和两两NQD序列场合下的应用.     

An Example of a Positive Semidefinite Double Sequence Which is Not a Moment Sequence

The first explicit example of a positive semidefinite double sequence which is not a moment sequence was given by Friedrich. We present an example with a simpler definition and more moderate growth as (m, n) .     

关于序列的不等式及其应用

本文证明了一些关于新的序列的不等式.作为定理的应用,引进了一类新的迭 代序列并证明了它的收敛性.进一步地,证明了它与著名的Ishikawa迭代序列等价.     

A short constructive proof of the Erd?s-Gallai characterization of graphic lists

Erd?s and Gallai proved that a nonincreasing list (d₁,…,d_n) of nonnegative integers is the list of degrees of a graph (with no loops or multi-edges) if and only if the sum is even and the list satisfies for 1≤k≤n. We give a short constructive proof of the characterization.     

图的度序列

图的度序列是图论研究中一个重要的课题．至今已发表了４００余篇文章．本文概述这一课题的某些进展，其中包括了可图序列的判准、蕴含Ｐ可图序列和强迫Ｐ可图序列的一些主要结论，同时列出了一些有待进一步研究的问题．     

G-frames and g-frame sequences in Hilbert spaces

In this paper, we first determine the relations among the best bounds A and B of the g-frame, the g-frame operator S and the pre-frame operator Q and give a necessary and sufficient condition for a g-frame with bounds A and B in a complex Hilbert space. We also introduce the definition of a g-frame sequence and obtain a necessary and sufficient condition for a g-frame sequence with bounds A and B in a complex Hilbert space. Lastly, we consider the stability of a g-frame sequence for a complex Hilbert space under perturbation.