酉辛群上的调和分析Ⅲ——Fourier级数的Abel求和

Abelian Summability may define as the limit case of Cesaro suminability whera α→∞, The kernel of Abelian mean, the so-called Poisson kernel of USp(2n) is.     

A Formula for Khovanov Typ e Link Homology of Pretzel Knots

Khovanov type homology is a generalization of Khovanov homology. The main result of this paper is to give a recursive formula for Khovanov type homology of pretzel knots P (?n,?m, m). The computations reveal that the rank of the homology of pretzel knots is an invariant of n. The proof is based on a “shortcut” and two lemmas that recursively reduce the computational complexity of Khovanov type homology.     

一类Ginzburg-Landau型泛函的径向极小元的渐近性

The behavior of radial minimizers for a Ginzburg-Landau type functional is considered. The weak convergence of minimizers in W1,n is improved to the strong convergence in W1,n. Some estimates of the rate of the convergence for the module of minimizers are presented.     

The Solution of Scholz's Problem

Def. The natural n is called a satisfiable number of a(in first order logic)formula α if α has a model of cardinality n. Def.The set M is called a finite spectrum if M is the set of all the satis-fiable numbers of a certain formula of first order logic.     

Invariant Mean—Value Property and M—Harmonicity in the Unit Ball of R^n

In 1993,Ahern,Flores and Rudin showed that,if f is integrable over the unit ball BC^n of C^n and satisfies∫BC^nfoφdv=f（φ（0)) for every φ∈Aut(BC^n),then f is M-harmonic if and only if n≤11.The present paper is about an analogous question in the context of the unit ball Bn of R^n as well as in the weighted setting.     

对称正交对称矩阵反问题的最小二乘解

Let P ∈ Rn×n be a symmetric orthogonal matrix. A∈Rn×n is called a symmetric orthogonal symmetric matrix if AT = A and (PA) T = PA. The set of all n × n symmetric orthogonal symmetric matrices is denoted by SRnxnp. This paper discusses the following problems: Problem I. Given X,B∈ Rn×m, find A ∈SRn×np such that||AX - B|| = min Problem II. Given A∈ Rn×n, find A∈SL such thatwhere ||·|| is the Frobenius norm, and SL is the solution set of Problem I.The general form of SL is given. The solvability conditions for the inverseproblem AX = B in SRn×nP are obtained. The expression of the solution toProblem II is presented.     

Minimum congestion spanning trees in bipartite and random graphs

The first problem considered in this article reads: is it possible to find upper estimates for the spanning tree congestion in bipartite graphs, which are better than those for general graphs? It is proved that there exists a bipartite version of the known graph with spanning tree congestion of order n 3 2 , where n is the number of vertices. The second problem is to estimate spanning tree congestion of random graphs. It is proved that the standard model of random graphs cannot be used to find graphs whose spanning tree congestion has order greater than n 3 2 .     

A branching particle system approximation for nonlinear stochastic filtering

The optimal filter π = {π t,t ∈ [0,T ]} of a stochastic signal is approximated by a sequence {π n t } of measure-valued processes defined by branching particle systems in a random environment(given by the observation process).The location and weight of each particle are governed by stochastic differential equations driven by the observation process,which is common for all particles,as well as by an individual Brownian motion,which applies to this specific particle only.The branching mechanism of each particle depends on the observation process and the path of this particle itself during its short lifetime δ = n 2α,where n is the number of initial particles and α is a fixed parameter to be optimized.As n →∞,we prove the convergence of π n t to π t uniformly for t ∈ [0,T ].Compared with the available results in the literature,the main contribution of this article is that the approximation is free of any stochastic integral which makes the numerical implementation readily available.     

FOURIER—CHEBYSHEV COEFFICIENTS AND GAUSS—TURAN QUADRATURE WITH CHEBYSHEV WEIGHT

The main purpose of this paper is to derive an explicit expression for Fourier-Chebyshev coefficient,Akn(f)=2/π∫^1-1f(x)Tkn(x)dx/√-1-x^2,k,n∈No,which is intiated by L.Gori and C.A.Micchelli.     

ON THE EQUATION□Φ=|▽Φ|~2 IN FOUR SPACE DIMENSIONS

This paper considers the following Cauchy problem for semilinear wave equations in n space dimensionswhere A is the wave operator, F is quadratic in (?) with (?) = ( ).The minimal value of s is determined such that the above Cauchy problem is locally well-posed in H8. It turns out that for the general equation s must satisfyThis is due to Ponce and Sideris (when n = 3) and Tataru (when n≥5). The purpose of this paper is to supplement with a proof in the case n = 2,4.     

一个包含Z（n）和D（n）函数的方程及其它的正整数解

对于任意正整数n,著名的伪Smarandache函数Z（n）定义为最小的正整数m使得n｜m（m＋1）／2．而数论函数D（n）定义为最小的正整数m使得n｜d（1）d（2）d（3）…d（m）,其中d（n）为Dirichlet除数函数．本文的主要目的是利用初等方法研究一类包含伪Smarandache函数Z（n）和数论函数D（n）的方程2^z（n）=D（n）的可解性,并获得了该方程的所有正整数解．     

Smarandache可求积因数对问题

对任意正整数n,设d(n)表示n的Dirichlet除数函数,即就是n的所有不同正因数的个数.Smarandache可求积因数对问题是:求所有正整数对m及n使得d(m)+d(n)=d(mn).主要目的是利用初等方法以及除数函数的性质研究这一问题,并给予彻底解决.具体地说也就是证明了正整数对m及n满足方程d(m)+d(n)=d(mn)当且仅当(m,n)=(pq~α,q)或者(m,n)=(p,p~αq),其中p及q为不同的素数,α为非负整数.     

关于伪Smarandache函数的一个问题

对任意正整n,著名的伪Smarandache函数Z(n)定义为最小的正整数m使得n|m(m 1)/2.本文的主要目的是利用初等方法研究Kenichiro Kashihara提出的"求所有正整数n使得伪smarandache函数Z(n)为n的原根"这一问题,并得到彻底解决.即就是证明了Z(n)为n的原根当且仅当n=2,3,4.     

On Factorisations and Generators in Transformation Semigroups

It is shown that the classical decomposition of permutations into disjoint cycles can be extended to more general mappings by means of path-cycles, and an algorithm is given to obtain the decomposition. The device is used to obtain information about generating sets for the semigroup of all singular selfmaps of $X_{n} = \{1, 2, \dots, n\}$. Let $T_{n,r} = S_{n}\cup K_{n,r}$, where $S_{n}$ is the symmetric group and $K_{n,r}$ is the set of maps $\alpha\,:\, X_{n} \to X_{n}$ such that $|im(\alpha)| \le r$. The smallest number of elements of $K_{n,r}$ which, together with $S_{n}$, generate $T_{n,r}$ is $p_{r}(n)$, the number of partitions of $n$ with $r$ terms.     

双对称非负定阵一类逆特征值问题的最小二乘解

１．引言 逆特征值问题在工程中有广泛的应用,其研究已有一些很好的结果［１－５］．最近,文［６］还研究了双对称矩阵逆特征值问题,即研究了如下两个问题： 问题Ａ．已知Ｘ∈Ｒｎｘｍ,Ａ＝ｄｉａｇ（λ１…,λｍ）,求Ａ∈ＢＳＲｎｘｎ使 ＡＸ＝ＸＡ,其中 Ｒｎｘｍ表示全体 ｎ ｘ ｍ实矩阵集合, ＢＳＲｎｘｎ表示全体 ｎ ｘ ｎ双对称阵集合． 问题Ｂ．已知Ａ＊ＥＲｎｘｎ,求Ａ∈ＳＥ使 ｜｜Ａ＊－Ａ｜｜＝ ｉｎｆ ｜｜Ａ＊－Ａ｜｜ ＡＦＳＥ其中 ＳＥ是问题 Ａ的解集合,｜｜． ｜｜表示 Ｆｒｏｂｅｎｉｕｓ范数． 在实际问题中, …     

关于奇算术图

本文讨论了$n$个$m$长圈有一个公共结点图$C^n_m$, $n$个$m$长圈与$t$长路有一个公共结点图$C^n_m\cdot P_t$, $n$个$m$阶完全图有一个公共结点图$K^n_m$和星形图的同胚图的奇算术性问题.给出了完全图,完全二部图和圈是奇算术的充要条件.     

相伴于随机自相似分形的随机测度的强测度的收敛性

构造了随机自相似分形及其上的记忆函数,并得出了有关结论,在此基础上,我们可以定义一个随机概率测度dΦn(τ)=Kn(τ)dτ,Φn(τ)弱收敛于Φ,进一步可得到强测度序列Ψn(·)=EΦn(·),则{Ψn}弱收敛于Ψ=EΦ.     

立方幂补数除数函数的均值

设 n是正整数 ,S(n)是 n的立方幂补数 ,τ(n)表示 n的除数函数 .本文的主要目的是探讨∑n xτ(S(n) )n 和 ∑n xτ(S(n) ) 的渐近性质 ,得到了两个渐近公式 ,进一步解决 F.Smarandache教授提出的第2 8个问题 .     

关于Smarandache对偶函数

定义Smarandache对偶函数S*(n)为最大的正整数m使得m!|n.定义另一种双阶乘函数S**(n)为最大的正整数2m-1使得(2m-1)!!|n,其中2 n;且当2|n时,为最大的正整数2m使得(2m)!!|n.本文的主要目的是利用初等方法研究一个包含S**(n)的无穷级数的收敛性,并给出一个有趣的恒等式.     

立方幂补数倒数的均值

设 n为任意正整数 ,S( n) 表示 n的立方幂补数 .本文的主要目的是研究 ∑n x1S( n) 和 ∑n xnS( n) 的渐近性质 ,并用初等方法得到了两个渐近公式 ,进一步解决 F.Smarandache教授在文献 [1 ]中提出的第 2 8个问题 .