四类取整数列通项性质的探究与应用

函数f（x）：[x]（x∈R）表示不超过实数x的最大整数,称为取整函数．定义在正整数集上的某些类型取整数列的通项是否具有一些特有的性质呢？利用这些类型取整数列的通项性质可以表示哪些类型数列的通项公式呢？笔者发现在近年来的高考和竞赛试题中经常可以找到此类数列问题的影子!本文将展开四类取整数列通项性质的探究,以实例说明其应用．     

Fibonacci数列倒数的无穷和

利用初等方法以及取整函数的性质研究了Fibonacci数列三次倒数的求和问题,获得了该和式倒数取整后的确切值,也就是给出了一个包含Fibonacci数列有趣的恒等式.     

一些取整数列求和公式及应用

一些取整数列求和公式及应用秦宗慈（镇江高专数理系２１２００３）由取整函数［Ｘ］产生的数列可称作取整数列．关于取整数列的求和，我们已经知道等［１］．本文将证明另外一些求和公式，其中一切字母表示自然数．在公式④中，令ｓ＝２，ｍ＝ｌ便得①；令ｓ＝２，ｍ＝２...     

两个数列的N—互补性

两个数列的Ｎ—互补性华强（上海市杨浦区教育学院）我们先考察下列两对整数数列｛。。｝和｛６。｝．它们的通项依次为：玉。ｎ一３ｎ，＼一ｎ干卜二７二１，。ＥＮ．这里ｈｉ表示不超过Ｘ的最大整数吓同）２ａ。一ｎ＋【Ｉｎ叶ｂ。一。十【ｅ＊ｎＥＮ；这里Ｉｎｘ为自然...     

递推数列与函数“不动点”

数列综合题是高考数学中的热点和难点之一，特别是已知递推关系但又难求通项的数列综合题，充分运用函数的相关性质是解决这类问题的着手点和关键．与递推关系对应的函数的“不动点”决定着递推数列的增减情况，这里我们以例题的形式说明函数“不动点”与递推数列之间的关系，以及怎样利用函数“不动点”来分析、解决与递推数列有关的综合题，以期对同学们有所帮助．     

周期数列的几种形式

数列是一种特殊的函数，所以数列中也存在周期的问题，且以各种形式给出的周期函数问题在数列中同样成立．     

运用图像性质妙解数列问题

数列是一种特殊的函数，借助函数的图像解决数列问题，可使问题变得形象、直观，易于求解．现举例说明如下：     

数列

数列是一种特殊的函数，它不仅是高中数学的重要内容之一，而且是初等数学和高等数学的重要衔接点．本单元以函数方法为基础，以等差数列和等比数列这两个基本数列为载体，研究和探索数列的通项公式、数列的求和以及数列和其它知识的综合应用．     

与高斯函数有关的高考压轴题

早年,数学王子高斯发现并定义了取整函数,即设x∈R,用[x]表示不超过x的最大整数,并用{x}表示x的非负纯小数,则y=[x]称为高斯函数,也叫取整函数.高斯函数[x]的定义域是R,值域为Z,它的图象是不连续的水平线段.高斯函数在数论中也有非常重要的作用,在各种数学竞赛和高考中经常出现含有取整函数的问题,高考中多以信息题的形式出现在压轴题的     

“破解”数列综合问题

数列是特殊的函数．《课程标准》对数列内容的处理更加突出了函数思想、数学模型思想以及离散与连续的关系，要求从函数的观点、模型的观点、连续与离散的角度认识数列．     

环Z/(p~e)上本原序列压缩映射的新结果

令Z/(pe)表示整数剩余类环,其中p为素数且e 2为正整数.令f(x)表示Z/(pe)上的n次本原多项式,G′(f(x),pe)表示Z/(pe)上所有由f(x)生成的本原序列构成的集合.设序列a∈G′(f(x),pe),它有唯一的p进制展开a=a0+a1p+···+ae-1pe-1.令φ(x0,x1,...,xe-1)=g(xe-1)+μ(x0,x1,...,xe-2)表示由Fe p到Fp的一个e变元多项式.那么,φ可以诱导出一个从G′(f(x),pe)到F∞p的压缩映射.在p为奇素数且f(x)为强本原多项式的条件下,人们已经证明该压缩映射是保熵的.而本文证明该压缩映射在f(x)为本原多项式的条件下仍然是保熵的.当deg(g(x))2时,我们还要求deg(g(x))为奇数,或者g(x)=xk+∑k-2i=0cixi.     

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…     

A thorough analysis of the discrepancy of shifted Hammersley and van der Corput point sets

We study the star discrepancy of Hammersley nets and van der Corput sequences which are important examples of low-dimensional quasi-Monte Carlo point sets. By a so-called digital shift, the quality of distribution of these point sets can be improved. In this paper, we advance and extend existing bounds on digitally shifted Hammersley and van der Corput point sets and establish criteria for the choice of digital shifts leading to optimal results. Our investigations are partly based on a close analysis of certain sums of distances to the nearest integer. Mathematics Subject Classi cation (2000) 11K38; 11K09     

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.     

On ratio block sequences with extreme distribution function

Distribution functions of ratio block sequences formed from sequences of positive integers are investigated in the paper. We characterize the case when the set of all distribution functions of a ratio block sequence contains c ₀, the greatest possible distribution function. Presented results complete some previously published results. Supported by grant MSM 6198898701, and VEGA no. 1/4006/07.     

Log-convex and Stieltjes moment sequences

We show that Stieltjes moment sequences are infinitely log-convex, which parallels a famous result that (finite) Pólya frequency sequences are infinitely log-concave. We introduce the concept of q-Stieltjes moment sequences of polynomials and show that many well-known polynomials in combinatorics are such sequences. We provide a criterion for linear transformations and convolutions preserving Stieltjes moment sequences. Many well-known combinatorial sequences are shown to be Stieltjes moment sequences in a unified approach and therefore infinitely log-convex, which in particular settles a conjecture of Chen and Xia about the infinite log-convexity of the Schröder numbers. We also list some interesting problems and conjectures about the log-convexity and the Stieltjes moment property of the (generalized) Apéry numbers.     

迭代数列收敛性与特征函数符号之间的关系及应用

利用特征函数的符号研究迭代数列的收敛性,得到根据特征函数的符号判断一类迭代数列收敛性并求其极限的几个定理,并举例说明定理的一些应用.     

Partitions and Compositions Defined by Inequalities

We consider sequences of integers (₁,..., _k) defined by a system of linear inequalities _{i j>i}a_ijj with integer coefficients. We show that when the constraints are strong enough to guarantee that all _i are nonnegative, the generating function for the integer solutions of weight n has a finite product form , where the b_i are positive integers that can be computed from the coefficients of the inequalities. The results are proved bijectively and are used to give several examples of interesting identities for integer partitions and compositions. The method can be adapted to accommodate equalities along with inequalities and can be used to obtain multivariate forms of the generating function. We show how to extend the technique to obtain the generating function when the coefficients a_i,i+1 are allowed to be rational, generalizing the case of lecture hall partitions. Our initial results were conjectured thanks to the Omega package (G.E. Andrews, P. Paule, and A. Riese, European J. Comb. 22(7) (2001), 887–904).Research supported by NSA grants MDA 904-00-1-0059 and MDA 904-01-0-0083.     

输电系统的部分可观Petri网故障诊断器设计

采用部分可观Petri网的故障诊断方法来解决变电站输电系统中不可观事件和不可观运行状态的故障诊断问题.首先,将系统可观测序列分解为长度为1的基础观测序列,应用线性不等式矩阵计算与基础观测序列相符的点火序列集;然后,基于整数线性规划问题,利用向前向后函数拓宽诊断区间,同时应用参数K限定故障诊断序列长度,通过分析系统可观事件和系统部分可观状态,给出故障诊断结果.最后,构造变电站输电系统的部分可观Petri网模型,应用提出的故障诊断算法对输电系统进行诊断,诊断结果准确给出了故障发生与否及故障发生位置.算法适用于在线故障诊断,计算复杂性线性相关于观序列长度.     

混合误差半参数回归模型估计的相合性

研究了误差为ψ混合和ψ混合序列的半参数回归模型,综合最小二乘法和非参数权函数估计方法,分别定义了待估参数β和未知函数夕的估计量βm,n和9m,n(χ).利用混合序列的矩不等式及凸函数的性质,在较弱的条件下证明了这些估计量的强相合性与矩相合性,这些结果推广了已有的相应的研究结果.