函数Sp(n2)与Sp(n)的关系

对于任意正整数n,设幂p的原数函数Sp(n)定义为Sp(n)=min{m:pn|m!},其中p为素数.本文目的是运用初等方法研究函数Sp(n2)与Sp(n)之间的关系,并得到一个有趣的恒等式.     

一个关于原数函数Sp(n)的恒等式

对于任意给定的素数p及正整数n,我们定义幂p的原数函数Sp(n)为最小的正整数m使得pn|m!.即就是:Sp(n)=min{m:pn|m!}.本文的主要目的是利用初等方法研究函数Sp(n)的算术性质,并得到一个有趣的恒等式.     

李代数SU(n)和阶化李代数SU(n/1)的不可约表示

本文对李代数SU(n)的生成元,引入了SU(n-1),SU(n-2),…SU(2)的不可约张量.并利用不可约张量的性质,求出了SU(n)的不可约表示,这些不可约表示的表达式比较简单.文中还给出了SU(n)的部分约化系数和Racah系数的代数表达式.在此基础上,求出了阶化李代数SU(n/1)的不可约表示.     

SU(m+n)SU(m)×SU(n) ISOSCALAR FACTORS AND S(f_1+f_2)S(f_1)×S(f_2) OUTER-PRODUCT ISOSCALAR FACTORS

It is proved that the SU(m+n)SU(m)×SU(n) isoscalar factors (ISF) are equal to the S(f_1+f_2) outer-product ISF of the permutation group. Since the latter only depend on the partition labels, the values of the SU(m+n)SU(m)×SU(n) ISF do not depend on m and n explicitely. Consequently for a f(=f_1+f_2)-particle system, by evaluating the S(f) S(f_1)×S(f_2) outer-product ISF we can obtain all (an infinite number) of the SU (m+n) SU(m)×SU(n) ISF (or the f_2-particle coefficients of fractional parentage) for arbitrary m and n at a single stroke, in stead of one m and one n at a time. A simple method, the eigenfunction method, is given for evaluating the SU(m+n) SU(m)×SU(n) single particle ISF, while the many-particle ISF can be calculated in terms of the outer-product reduction coefficients and the transformation coefficients from the Yamanouchi basis to the S(f_1+f_2) S(f_1)×S(f_2) basis.     

循环图C_(2n)(1,(2n+1)/3)的匹配可扩性

称图G是k-偶匹配可扩的,是指G的每一个基数不大于k(1≤k≤(|V(G)|-2)/2)的偶匹配M都可以扩充为G的一个完美匹配.根据循环图的性质研究了图C_(2n)(1,(2n+1)/3)的匹配可扩性,证明了对于任意的n(n≥4),C_(2n)(1,(2n+1)/3)是3-偶匹配可扩的.     

一类具有n~2/2(n为偶数)或n~2+1/2(n为奇数)个细焦点的(E_n)系统

通过对一类特殊的平面n次多项式微分系统进行参数小扰动,得到了一类具有(n~2)/2(n为偶数)或(n~2+1)/2(n为奇数)个2阶细焦点的(E_n)系统.进一步证明了该系统具有(n~2)/2(n为偶数)或(n~2+1)/2(n为奇数)族极限环,得到了S(n)≥(n~2)/2(n为偶数)或S(n)(n~2+1)/2(n为奇数),改进了已有文献的结果.     

秩为2的紧单Lie群的谱理论

<正> 记△为紧 Riemann 对称空间 M 上的 Laplace-Beltraml 算子.△作用在光滑函数空间 C~∞(M)上的谱理论是熟知的,但作用在 P 阶 C~∞外微分形式空间 C~∞((?)~PM),P=1,2,…,dimM 上的谱理论,知道的较少.已有结果为:S.Gallot 与 D.Meyer 及A.Lévy-Bruhl-Laperrière 于1975年解决了 M=S~n 的情形,后者于1977年又解决了M=P~n(C)的情形;随后于1978年 A.Ikeda 与 Y.Taniguci 用不同的方法得到与[2],[3]相同的结果;1981年 C.Tsukamoto 解决了 M 为 SO(n+2)/SO(2)×SO(n)及 Sp(n+1)/Sp(1)×Sp(n)的情形.B.Beers 与 R.Millman 于1977年解决了 M 为SU(2),SU(3),SO(3),SO(4),SO(5)的情形,从而在秩≤2的紧单 Lie 群中,仅有G_2这个情形还没有解决.本文给出一种方法来计算紧半单 Lie 群的谱.应用这个方法我们具体算出了紧单 Lie 群 G_2的所有谱.首先,在§1中我们证明了对一切单连通、连通、紧半单 Lie 群 G 有下面     

Steenod 运算和同伦群(Ⅱ)

<正> [3]中用 Steenrod 运算确定了(n—1)连通空间的一些同伦群的 p 分量群的代数构造,那里的 p 是一些大于2的素数.本文则讨论 p=2 时的情况.G.W.Whitehead 和(?)在1950年分別独立地确定了 n 维球 S~n 的(n+2)维同伦群,张素诚,Hilton,(?)等则确定了(n—1)维连通空间的(n+1)维同伦群.我们很自然有这样一个问题,如何计算(n—1)连通空间的(n+2)维同伦     

Non-naturally reductive Einstein metrics on Sp(n)

We prove that Sp(2k+l)admits at least two non-naturally reductive Einstein metrics which are Ad(Sp(k)×Sp(k)×Sp(l))-invariant ifk<1.It implies that every compact simple Lie group Sp(n)for n≥4 admits at least 2[(n-1)/3]non-naturally reductive left-invariant Einstein metrics.     

四元Heisenberg群上的Twistor-变换与Penrose-积分公式北大核心

利用陪集Sp(2n+4,C)/P的双纤维化,其中P为Sp(2n+4,C)的抛物子群,得到四元Heisenberg群的Twistor-变换,进而得到四元Heisenberg群上切向k-Cauchy-Fueter方程的解:Penrose-积分公式.     

Manifolds homotopy equivalent to P n # P n

We classify, up to homeomorphism, all closed manifolds having the homotopy type of a connected sum of two copies of real projective n-space. Partially supported by a grant from the National Science Foundation.     

On the explicit,purely analytic solution of Von Kármán swirling viscous flow

A new analytic method for highly nonlinear problems, namely the homotopy analysis method, is applied to solve the Von Kármán swirling viscous flow, governed by a set of two fully coupled differential equations with strong nonlinearity. An explicit, purely analytic and uniformly valid solution is given, which agrees well with numerical results.     

A n 4 -polyhedra with free homology

An A _n ^k -polyhedron is a CW-complex which is (n–1)-connected and of dimension (n+k). We compute an algebraic category which is equivalent to the homotopy category of A _n ² -polyhedra with free homology (n3). This computation and a calculation of the -group _n+3(X) (n3) is used to obtain a complete algebraic homotopy invariant for A _n ⁴ -polyhedra with free homology (n3).As an application we compute the group of self-homotopy equivalences Aut(X) for a bouquet X=P².     

Homotopy groups of some homogeneous spaces and some remarks on their sections over spheres

Stable homotopy groups and a few of the unstable homotopy groups of the homogeneous spacesSO(2n)/U(s) andU(2n)/Sp(s) fors, and those ofSO(4n)/Sp(s) fors≤n are computed. The enumeration of homotopy classes of normalized almost complex (n?1) and (n?2) substructures with oriented framed complements are given. Some results about the existence of the equivariant cross sections of these homogeneous spaces over the spheres with respect to certain unitary group actions are proved.     

The log-behavior of $$\root n \of {p(n)}$$ and $$\root n \of {p(n)/n}$$p(n)/nn

Let p(n) denote the partition function and let \(\Delta \) be the difference operator with respect to n. In this paper, we obtain a lower bound for \(\Delta ^2\log \root n-1 \of {p(n-1)/(n-1)}\), leading to a proof of a conjecture of Sun on the log-convexity of \(\{\root n \of {p(n)/n}\}_{n\ge 60}\). Using the same argument, it can be shown that for any real number \(\alpha \), there exists an integer \(n(\alpha )\) such that the sequence \(\{\root n \of {p(n)/n^{\alpha }}\}_{n\ge n(\alpha )}\) is log-convex. Moreover, we show that \(\lim \limits _{n \rightarrow +\infty }n^{\frac{5}{2}}\Delta ^2\log \root n \of {p(n)}=3\pi /\sqrt{24}\). Finally, by finding an upper bound for \(\Delta ^2 \log \root n-1 \of {p(n-1)}\), we establish an inequality on the ratio \(\frac{\root n-1 \of {p(n-1)}}{\root n \of {p(n)}}\).     

关于SSSP（n）和SISP（n）的均值

研究了Smarandache最小平方数列和Smarandache最大平方数列的均值性质,并用初等方法得到了关于这两个数列均值的渐近公式．     

On pseudo-random properties of P (n) and P (n + 1)

We study the pseudo-random properties of the inequalities P (n) &lt; P (n+1) or P (n) &gt; P (n+1) (where P (n) denotes the greatest prime factor of n) for both well-distribution and correlation.     

关于满足I(x,n(x))=n(x)的D-蕴涵的刻画

研究I(x,n(x))=n(x),其中I为由连续三角模T、连续三角余模S和强否定n生成的D-蕴涵,即I(x,y)=S(T(n(x),n(y)),y),给出了满足I(x,n(x))=n(x)的充要条件。     

Fibred sites and stack cohomology

The usual notion of the site associated to a stack is expanded to a definition to a site fibred over a presheaf of categories A on a site . If the presheaf of categories is a presheaf of groupoids G, then the associated homotopy theory is Quillen equivalant to the homotopy theory of simplicial presheaves over BG, and so the homotopy theory for the fibred site is an invariant of the homotopy type of G. Similar homotopy invariance results obtain for presheaves of spectra and presheaves of symmetric spectra on . In particular, stack cohomology can be calculated on the fibred site for any representing presheaf of groupoids within a fixed homotopy type.