函数系数线性自回归模型的样条估计

基于多项式样条全局光滑方法,建立函数系数线性自回归模型中系数函数的样条估计.在适当条件下,证明了系数函数多项式样条估计的相合性,并给出了它们的收敛速度.模拟例子验证了理论结果的正确性.     

插值多项式样条的存在唯一性

对给定的区间[a,b]上的分划。 △_N:a=x_0相似文献   

三奇次散乱点多项式自然样条插值

为解决较为复杂的三变量散乱数据插值问题,提出了一种三元多项式自然样条插值方法.在使得对一种带自然边界条件的目标泛函极小的情况下,用Hilbert空间样条函数方法,构造出了插值问题的解,并可表为一个分块三元三奇次多项式.其表示形式简单,且系数可由系数矩阵对称的线性代数方程组确定.     

二次样条的一种推广

在样条函数的讨论中,除了通常的多项式样条,T-样条等外,[1,2,3]分别讨论了更为一般的样条,本文考虑二次样条的一种推广,二次多项式样条是满足一定光滑性条件的分段二次多项式.设Δ:0=x_0相似文献   

自然数方幂和的通项公式

用初等方法证明sum from i=1 to n i2k+1为n2(n+1)2与n(n+1)的(k-1)次有理多项式的乘积,sum from i=1 to ni2k为n(n+1)(2n+1)与n(n+1)的(k-1)次有理多项式的乘积,提出关于上述公式系数符号的一个猜想.     

样条指函数

提出了指性表示的概念,证明了函数系1,x,x2,…,xn,x-x1n+,x-x2n+,…,x-xNn+为n次样条指函数集合Sn(x1,x2,…,xN)的指基.给出了应用样条指函数求最大似然估计量的例子.     

关于用矩阵多项式表示矩阵函数的存在性与唯一性问题

利用Banach 空间的极限理论证明用矩阵多项式表示矩阵函数(收敛的方阵幂级数)的存在性与唯一性.     

二元三次样条空间S_3~(1,2)(△_(mn)~(2))的样条拟插值

本文首先利用由两组具有局部最小支集的样条所组成的基函数,构造非均匀2型三角剖分上二元三次样条空间S1,23(△(2)mn)的若干样条拟插值算子.这些变差缩减算子由样条函数B1ij支集上5个网格点或中心和样条函数B2ij支集上5个网格点处函数值定义.这些样条拟插值算子具有较好的逼近性,甚至算子Vmn(f)能保持近最优的三次多项式性.然后利用连续模,分析样条拟插值算子Vmn(f)一致逼近于充分光滑的实函数.最后推导误差估计.     

Herstein定理的推广

1955年Herstein将著名的Jacobson定理推广为:定理A.如果对R中任意x,y,存在可依赖于x,y的整系数多项式p(t),使[x-x2p(x),y]=0,则R是交换的.本文利用多项式的系数和定义了n元多项式,f(x1,x2…,xn)的Fk性质,并以此证明了一个环的交换性定理,当n=1时,即得到定理A.     

LP[0,1](1＜p＜∞)空间正系数多项式倒数逼近的一个推广

本文推广了LP[0,1](1＜p＜∞)空间函数的正系数多项式的倒数逼近的结论,即证明了:设f(x)∈LP[0,1],1＜p＜∞,且在(0,1)内严格1次变号,则存在一点x0∈(0,1)及一个n次多项式Pn(x)∈∏n(+)使得‖f(x)-x-x0/Pn(x)‖LP[0,1]≤Cpω(f,n-1/2)LP[0,1],其中∏n(+)为次数不超过n的正系数多项式的全体.     

Odd multiperfect numbers of abundancy 4

Euler's structure theorem for any odd perfect number is extended to odd multiperfect numbers of abundancy power of 2. In addition, conditions are found for classes of odd numbers not to be 4-perfect: some types of cube, some numbers divisible by 9 as the maximum power of 3, and numbers where 2 is the maximum even prime power.     

用Paley图计算对角Ramsey数下界的新方法

本文研究了对角Paley数的下界问题.利用一个新发现的Paley图的自同构,给出了计算Paley图团数的一个新方法,获得了2个对角Rasey数的新下界:R(20,20)≥18877,R(21,21)≥25949.     

正相协列乘积和的重对数律与强大数律

证明了强平稳正相协列乘积和的重对数律与不同分布正相协列乘积和的强大数律,指出了部分和服从强大数律但乘积和未必服从强大数律这一事实,并讨论了定理2中一个条件的必要性．     

All square chiliagonal numbers

A square chiliagonal number is a number which is simultaneously a chiliagonal number and a perfect square (just as the well-known square triangular number is both triangular and square). In this work, we determine which of the chiliagonal numbers are perfect squares and provide the indices of the corresponding chiliagonal numbers and square numbers. The study revealed that the determination of square chiliagonal numbers naturally leads to a generalized Pell equation x² ? Dy² = N with D = 1996 and N = 996², and has six fundamental solutions out of which only three yielded integer values for use as indices of chiliagonal numbers. The crossing/independent recurrence relations satisfied by each class of indices of the corresponding chiliagonal numbers and square numbers are obtained. Finally, the generating functions serve as a clothesline to hang up the indices of the corresponding chiliagonal numbers and square numbers for easy display and this was used to obtain the first few sequence of square chiliagonal numbers.     

K_m~-□P_n的交叉数

Klesc等人先后确定了K_m~-□P_n(4≤m≤6)的交叉数,本文利用构造法确定了K_m-2K_2(4≤m≤12,m≠10,12)的交叉数.在此基础上,可进一步确定K_m~-□P_n(4≤m≤9,m≠8)的交叉数.相比而言,我们所采用的方法更具一般性.     

A note on some constants related to the zeta-function and their relationship with the Gregory coefficients

In this article, new series for the first and second Stieltjes constants (also known as generalized Euler’s constant), as well as for some closely related constants are obtained. These series contain rational terms only and involve the so-called Gregory coefficients, which are also known as the (reciprocal) logarithmic numbers, the Cauchy numbers of the first kind and the Bernoulli numbers of the second kind. In addition, two interesting series with rational terms for Euler’s constant \(\gamma \) and the constant \(\ln 2\pi \) are given, and yet another generalization of Euler’s constant is proposed and various formulas for the calculation of these constants are obtained. Finally, we mention in the paper that almost all the constants considered in this work admit simple representations via the Ramanujan summation.     

由轮图导出的某些平面图类的星色数

图的星色数是通常色数概念的推广.本文求出了几类由轮图导出的平面图的星色数.前两类是由3-或5-轮图经细分等构造出的,其星色数分别为2+2/(2n+1),2+3/(3n+1)和2+3/(3n-1).第三类平面图是由n-轮图经过Hajos构造得到的,其星色数为3+1/n.本类图的星色数结果推广了已有结论.     

Some small ramsey numbers

In previous work, the Ramsey numbers have been evaluated for all pairs of graphs with at most four points. In the present note, Ramsey numbers are tabulated for pairs F₁, F₂ of graphs where F₁ has at most four points and F₂ has exactly five points. Exact results are listed for almost all of these pairs.     

Romanoff定理的定量形式

1934年,Romanoff证明了能表成2的方幂与一个素数之和形式的正整数在正整数集合中有正的比例．最近,本文作者证明了对充分大的x,能表成2的方幂与一个素数之和形式的正整数在不超过x的正整数中至少有0．0868x个．本文证明了:设 x≥5,则在不超过x的正整数中,能表成2的方幂与一个素数之和的数的个数不少于 0．005x,即给出了Romanoff定理的定量形式．     

Periodicity of Balancing Numbers

The balancing numbers originally introduced by Behera and Panda [2] as solutions of a Diophantine equation on triangular numbers possess many interesting properties. Many of these properties are comparable to certain properties of Fibonacci numbers, while some others are more interesting. Wall [14] studied the periodicity of Fibonacci numbers modulo arbitrary natural numbers. The periodicity of balancing numbers modulo primes and modulo terms of certain sequences exhibits beautiful results, again, some of them are identical with corresponding results of Fibonacci numbers, while some others are more fascinating. An important observation concerning the periodicity of balancing numbers is that, the period of this sequence coincides with the modulus of congruence if the modulus is any power of 2. There are three known primes for which the period of the sequence of balancing numbers modulo each prime is equal to the period modulo its square, while for the Fibonacci sequence, till date no such prime is available.