某些组合问题(Ⅰ)

In this note and its sequels we will present various unsolved problems mainly occured in our researches of combinatorial analysis. 1. Problems concerning the maximal degree of combinatorial identities Combinatorial identities are sometimes called binomial coefficient summat     

关于Euler-Maclaurin求和公式的一种余项表示

Let f(t) be a functiin defined on [a,b] and Let f^(r-1)(t)(r≥2) be of bounded variation there. Then for any x∈[0,l] we have (?).It is shown that the expression given above has various consequences including those mentioned in [1], [2],[3], respectively. Also the result included in this note is a refined estimate fcr the remainder term of an asymptotic expansion of a kind of oscillatory integral (See (13) - (16)).     

关于∫f(x,{Nx})dx的渐近展开及其应用

本文在最广条件下找到了大参数积分integral from n=0 to 1 f(x,{Nx})dx的一对渐近展开公式,其中{Nx}表示非负实数Nx的分数部分。这些公式对振荡函数的积分计算及线积分逼近重积分方法都有应用。 [2]曾就大参数N为正整数的情形,得出了所论积分的渐近展开式。但就N为非整数的情形,原来的论证方法不再适用。考虑到在实际问题中,例如当f(x,y)对变元y具有     

一类含有调和数的组合和

本文给出了含有调和数的一类组合和的求和公式．该公式有二熟知特例．     

Two Wider Classes of Combinatorial Identities

Here introduced are two general classes of combinatorial identities that may be defined via inverse relations due to Gould and Hsu(cf.Duke Math J. 40(1973), 885 891). Classes(∑) and(∑~*). Any identity is said to belong to the class(∑) if it can be embedded in either of the forms(reciprocal relations)     

关于含有Stirling公式的双边不等式

本文证明了有关ｎ！的一个便于应用的双边不等式,它对一切自然数都成立,且当ｎ变大时,上界不式能给出误差界为Ｏ（ｎ＾－３）的Ｓｔｉｒｌｉｎｇ渐近公式,从一定意义上说,文中的上界不等式具有最优形式,因为其中的常数０．５已作了最佳选择,文末还给出了关于Ｃａｔａｌａｎ数的一个双边不等式。     

论G?del不完备性定理

The object of this expository paper is to disclose the real essential idea involved in G?del's imcompleteness theorem through a detailed analysis of the proof presented in Arbib's booklet. In §1 very extensive definitions have been given of the potential infinite and the real infinite, respectively. In §2 the general concept concerning types of infinite extension processes as well as the principle of relatively complete exhaustion have been expounded. In§3 a deeper analysis has been made of the well-known proof of G?del's imcompleteness theorem, thus leading to the conclusion that G?del's theorem is actually implied by the most general principle that any infinite process of lower type cannot exhaust and exhibit the whole content of the corresponding process with relatively higher type, where the words "exhaust" and "exhibit" may be given precise meanings in certain special cases including the case for G?del's arithmetic logic.     

论非标准时空连续统模型及其对Zeno悖论分析的应用

1960年美国数理逻辑学家A.Robinson运用现代数理逻辑方法成功地构造出包含无限小和无限大等非标准数在内的数域~*R,并在~*R上建立了非标准分析理论。1980年,本文作者之一及其合作者曾引进了一个非康托自然数序列模型,其中包含着,v为无限自然数},(ω) 为ω的银河(Galaxy)。N及被解释为生成观点下的无限性总体。这一模型的特点是包含这个有限与无限之间即与(ω)之间的潜变段或飞跃段。中的元素v是既非有限又非无限的潜变无限大。飞跃段的