关于一个普遍本源公式及其应用

Here presented is a further investigation on a general source formula(GSF) that has been proved capable of deducing more than 30 special formulas for series expansions and summations in the author's recent paper [On a pair of operator series expansions implying a variety of summation formulas.Anal.Theory Appl.,2015,31(3):260–282].It is shown that the pair of series transformation formulas found and utilized by He,Hsu and Shiue [cf.Disc.Math.,2008,308:3427–3440] is also deducible from the GSF as consequences.Thus it is found that the GSF actually implies more than 50 special series expansions and summation formulas.Finally,several expository remarks relating to the(Σ?D) formula class are given in the closing section.     

与中心数相关的一类级数求和公式

This paper provides a pair of summation formulas for a kind of combinatorial series involvingak＋b m as a factor of the summand. The construction of formulas is based on a certain series transformation formula [2, 7, 9] and by making use of the C-numbers [3]. Various consequences and examples including several remarkable classic identities are presented to illustrate some applications of the formulas obtained.     

关于Delta算子的广义Taylor公式的余项估计

本文对哑演算中Ｒｏｔａ提出的至今尚未解决的广义Ｔａｌｏｒ公式的余项估计问题进行了研究。获得了Ｄｅｌｔａ算子关于一般函数的一个不同于Ｌａｇｒａｎｇｅ型的结果,并解决了二阶Ｄｅｌｔａ算子的余项估计问题。     

三类与Riemann Zeta函数有关的级数的求和公式

本文采用组合数学的方法,利用第二类Ｓｔｉｒｌｉｎｇ数和Ｂｅｒｎｏｕｌｌｉ数给出级数∑∞ｋ＝２ｋ＾ｍξ（２ｋ）及∑∞ｋ＝１（２ｋ＋１）＾ｍξ（２ｋ＋１）其中ｍ≥１,ξ（ｘ）＝ξ（ｘ）－１）的求和公式。这些公式表述简洁并有鲜明的规律性。     

KP系统Lax算子及主对称的换位公式

本文讨论的是ＫＰ系统Ｌａｘ算子及主对称的换位公式．通过拓广速降函数空间及对 ＫＰ方程 Ｌａｘ算子的讨论,找到了 Ｌａｘ算子的表示向量;并通过对 Ｌａｘ算子、 Ｌａｘ流、 Ｌａｘ算子表示向量之间联系的讨论,得出了计算 Ｌａｘ算子李括号的表示向量的方法,从而解决了 ＫＰ方程主对称的换位公式问题．最后本文还利用伴随算子给出了从ＫＰ方程任一主对称得到其一个对称的公式．     

关于随机算子方程的随机解

本文研究了随机算子方程（Ａ（ω，ｘ）＝μｘ，（ω，ｘ）∈Ω×Ｄ，μ１）的随机解，得到了若干新的结果，同时，我们推广了著名的Ａｌｔｍａｎ定理．     

Inversion of Trace Formulas for a Sturm-Liouville Operator

This paper revisits the classical problem “Can we hear the density of a string?”, which can be formulated as an inverse spectral problem for a Sturm-Liouville operator. Instead of inverting the map from density to spectral data directly, we propose a novel method to reconstruct the density based on inverting a sequence of trace formulas which bridge the density and its spectral data clearly in terms of a series of nonlinear integral equations. Numerical experiments are presented to verify the validity and effectiveness of the proposed numerical algorithm. The impact of different parameters involved in the algorithm is also discussed.     

Integral Formulas for the r-Mean Curvature Linearized Operator of a Hypersurface

For a normal variation of a hypersurface M ⁿ in a space form Q _c ⁿ⁺¹ by a normal vector field fN, R. Reilly proved:

带对称齐次核的级数算子的范数刻画及其应用

对带对称齐次核K(m,n)的级数算子T:T{an}=Σ_(n=1)~∞ K(m,n)an,{an}∈lω(n),l={{an)| an≥0,Σ_(n=1)~∞ω(n)an<+∞},本文研究了T的范数刻画,并讨论其应用.     

关于一类复合幂级数展开式的收敛性定理

本文研究了一类复合型幂级数展开式,证明了一个收敛性定理并举例说明其应用.在注记中指出了可进一步研究的问题.     

关于Neumann-Bessel级数的线性组合算子

In this paper we construct a new operator Hn,r（N,B） （f; z） by means of the partial sums S（N,S） （f; z） of Neumann-Bessel series. The operator converges uniformly to any fixed continuous function f（z） on the unit circle ｜ z ｜= 1 and has the best approximation order for f（z） on ｜ z ｜= 1.     

On a Class of Analytic Operator Functions and Their Linearizations

We consider an operator function T in a Krein space which can formally be written as (0.1)but the last term on the right of (0.1) is replaced by a relatively form‐compact perturbation of a similar form. We study relations between the operator function T, a selfadjoint operator M in some Krein space, associated with T, and an operator which can be constructed with the help of the operator function –T^–1. The results are applied to a Sturm‐Liouville problem with a coefficient depending rationally on the eigenvalue parameter.     

On a Class of Operator Equations

We assume that E₁ and E₂ are Banach spaces, a:E₁ E₂ is a continuous linear surjective operator, f:E₁ E₂ is a nonlinear completely continuous operator. In this paper, we study existence problems for the equation a(x)=f(x) and estimate the topological dimension dim of the set of solutions.     

一类数项级数和的表示

求出函数f(x)=xk的Fourier系数并将其代人Parseval等式,继而利用第二数学归纳法可证明:数项级数∞∑n=1 1/n2k的和能够表示为π2k/dk的形式.其中对于任意确定的k值.dk以为一常数.证明过程同时给出了求解dk的方法.     

一类强增生算子方程解的Mann迭代收敛问题

设X是实Banach空间,D是X的一个子集,T:X→X是一强增生算子,从而得到带误差的Mann迭代序列的逼近不动点的强收敛问题.     

论一种用到Howard广义斯特灵数的求和公式

It is shown that Howard＇s degenerate weighted Stirling numbers can be used to construct a fruitful summation formula for a class of formal power series involving generalized factorials.This is achieved with the aid of a series-transformation formula due to He,Hsu and Shiue,and several identities involving generalized Stirling numbers and Bell numbers are given to illustrate the application of the formula obtained.     

Z-P-S空间中一类非线性算子方程解的存在性问题

拓扑度理论是研究非线性算子方程解的存在性的有力工具.利用拓扑度的方法,对Z-P-S空间中一类非线性算子方程解的存在性问题进行了研究,得到了若干新的结果.     

Z-P-S空间中一类非线性算子方程解的研究

本文利用Menger PN空间中半闭1-集压缩算子的拓扑度理论研究Z-P-S空间中非线性算子方程Ax=ux解的存在性问题,改进和推广了相关文献中的结果.最后,我们给出主要结果的一个具体应用.     

On the Action of a Linear Operator over Sequences in a Banach Space

In this paper we study the action of a bounded linear operator over different kinds of sequences of a Banach space. Our work is mainly devoted to minimal and M - basic sequences. Plans and García Castellón have characterized the boundedness of a linear operator T by requiring the minimality of any sequence whose image is a minimal sequence (e.g. [P, 1969], [GC, 1990]). We extend these results to other types of sequences like M-basic, basic, strong M-basic, etc., We are also interested on conditions that ensure the minimality of the image of a given minimal sequence. Thus in Corollary 3.7 we characterize semi - Fredholm operators as those which transform every p-minimal sequence into q-minimal. In the last section we deal with M - basis whose image is M - basis or norming M - basis or basis or in general the “best” possible sequence.     

一类非齐次核的Hilbert型级数算子最佳搭配参数的充要条件

选择搭配参数a,b,利用权函数方法,可得核为K(m,n)的级数算子T的不等式:||T((a))||p,β(a,b) ≤ M(a,b)||(a)||p,α(a,b),(a) ={am}一般地,M(a,b)并不是T:lap(a,b)→lβp(a,b)的算子范数,针对非齐次核K(m,n)=G(mλ1/nλ2)(λ1λ2＞0)...