半参数回归模型非参数分量L1模估计的最优收敛速度

对半参数回归模型，采用分段多项式逼近非参数函数，构造了参数与非参数分量Ｌ１模糊估计，并获得了非参数分量Ｌ１模估计的最优估计收敛速度为Ｏｐ（ｎ＾－ｍ＋ｒ／［２（ｍ＋ｒ）＋１］）。     

正态分布族的参数的经验Bayes估计的收敛速度

对于正态分布族｛Ｎ（μ，σ￣２）：－∞＜μ＜＋∞，σ￣２＞０｝，该文利用密度函数及其偏导数的核估计构造出参数θ＝（μ，σ￣２）的经验Ｂａｙｅｓ（ＥＢ）估计，并在一定条件下证明了θ的ＥＢ估计的收敛速度可任意接近于１．最后给出了一个实例．     

多维参数线性经验Bayes估计一致收敛速度的最优性

证明出任何一个多维参数性经验Ｂａｙｅｓ估计的一致收敛速度不可能超过１，从而说明文［１］中构造的线性经验Ｂａｙｅｓ估计的一致收敛速度１是最优的。     

部分线性模型中估计的收敛速度

考虑回归模型（Ⅰ）：其中（ｘ＿ｉ，ｔ＿ｉ）是固定非随机设计点列，ｘ＿ｉ＝（ｘ＿（ｉｌ），…，ｘ＿（ｉｐ））＇β＝（β＿１，…，β＿ｐ）＇（ｐ＞１），ｇ是定义在［０，１］上的未知函数，β是未知待估参数，０＜ｔ＿ｉ＜１，ｅ＿ｉ是ｉ．ｉ．ｄ．随机误差，且Ｅｅ＿ｉ＝０，Ｅｅ＝σ￣２＜∞。基于ｇ的估计取一类非参数权估计（包括常见的核估计和近邻估计），我们讨论了β的最小二乘估计及ｇ的估计的最优强弱收敛速度。     

关于学生氏U－统计量的渐近性质

设Ｘ１，Ｘ２，…，Ｘｎ（ｎ≥２）为ｉ．ｉ．ｄ随机变量，Ｕｎ为以ｈ（ｘ１，ｘ２）为对称核的Ｕ－统计量，Ｅｈ（Ｘ１，Ｘ２）＝θ，且．设是的ＢｏｏｔｓｔｒａＰ量，施锡铨［１］在关于核ｈ的二阶矩的条件下，证明了：当ｎ→∞时，因此依分布收敛于标准正态变量．本文在关于核ｈ的４阶矩的条件下，讨论了Ｗｎ的分布渐近正态的一致性界限．     

局部凸空间(λ,μ)中的有界集与弱收敛

［１］首次在完备矩阵环Σ（λ）中引入拓扑，使之成为局部凸拓扑代数，并进行了一系列讨论．本文将把这一工作再拓广一步，在（λ，μ）中引入拓扑，使之成为局部凸线性拓扑空间．（λ，μ）的特例（λ，λ）即为Σ（λ）．本文讨论了（λ，μ）中的有界集，正规收敛与弱收敛等价的充要条件及弱收敛与有界且坐标收敛等价的充要条件．从而推广了［２］的结果．作者在［３］，［４］中曾经讨论了特别当λ为Ｅｃｈｅｌｏｎ空间时，（λ，μ）中的有界集和弱收敛．利用本文的结果，可立即得到［３］，［４］中的结果，并得到了进一步的推广．     

泛函型重对数律的收敛速度

泛函型重对数律的收敛速度王启应（南京大学）设｛Ｘ＿ｎ，ｎ≥１｝为ｉ．ｉ．ｄ．随机变量序列为定义在［１，∞）上的实函数。近年来，级数的收敛性问题，引起了众多学者的兴趣。作为一个研究方向，１９６８年，Ｄａｖｌ５￣［１］指出：上述级数的收敛除需要一定矩条件...     

Banach空间中增生算子方程迭代法与非线性收缩半群弱收敛的充要条件

本文在条件（Ｉ）下，证明了增生算子最速下降法和预解式迭代法弱收敛于零点的充要条件，以及非线性收缩半群弱收敛于平衡点的充要条件．所获结果推广了［１］中的基本定理，并与［２］所获得的相应强收敛充要条件对应．     

非同分布随机元和的收敛速度

本文在弱于［１］的矩条件下,对,ｒ＞１讨论了ｐ（１＜ｐ≤２）型空间中非同分布随机元和的收敛速度,同时获得了ｐ型巴拿哈空间的刻划,作为应用,我们给出了随机元随机指标部分和的收敛速度．     

积分算子的n-逼近数

本文研究了积分算子ＴＫ：Ｌｑ［０，１］→Ｌｑ［０，１］，（ｑ≥１）当核 Ｋ（ｓ， ｔ）是 Ｓｏｂｏｌｅｖ空间 Ｗｐｒ（［０， １］２）中元素时ｎ－逼近数 ａｎ（ＴＫ： Ｌｑ→ Ｌｑ）的估计，并把这个估计应用于退化核方法解第二类线性Ｆｒｅｄｈｏｌｍ方程（Ｉ一ＴＫ）ｘ＝ｙ时，Ｂａｄｈｖａｌｏｖ［５］意义下最佳误差的讨论中，所得到的最佳误差之估计当ｑ＝１时，最优化了［１０］的结论．     

不同分布NA列加权和的强极限定理及其在线性模型中的应用

本文讨论了不同分布ＮＡ列Ｓｔｏｕｔ型加权和的完全收敛性和强稳定性，推广并改进了Ｓｔｏｕｔ关于ｉｉｄ列的相应结果，从而将赵林城关于独立误差的方差估计的强收敛速度的理想结果推广到ＮＡ误差的场合。     

截断样本下的强率函数和密度函数核估计的渐进正态性

本文用[1]发展的计数过程去研究截断样本下强率函数核估计的渐进正态性．在弱于[7]和[10]的条件下，得到了更一般的结果．接着我们将这种方法运用到密度函数核估计，在较弱的条件下，得到了截断样本下密度函数核估计的渐进正态性．     

Spectral approximation for compact integral operators by degenerate kernel methods

We establish a new improved error estimate for the solution of the integral equation eigenvalue problem by degenerate kernel methods. In [6] these estimates were proved under the assumption of normality of the original kernel as well as of the approximating degenerate kernel. Now we consider any compact integral operator and a general Banach space situation, in contrast to the Hilbert space setting in [6], This will be done by combining the techniques in [6] with the suitably transformed estimates of [5]. Our results show that degenerate kernel methods have, besides their overall property of furnishing easy approximations to eigenfunctions, for eigenvalues an order of convergence comparable to quadrature methods.     

ESTIMATES OF LINEAR RELATIVE n-WIDTHS IN L~p [0,1]

In this paper we will show that if an approximation process { L n }n ∈ N is shapepreserving relative to the cone of all k-times differentiable functions with non-negative k-th derivative on [0,1],and the operators L n are assumed to be of finite rank n,then the order of convergence of D k L n f to D k f cannot be better than n 2 even for the functions x k,x k+1,x k+2 on any subset of [0,1] with positive measure.Taking into account this fact,we will be able to find some asymptotic estimates of linear relative n-width of sets of differentiable functions in the space L p [0,1],p ∈ N.     

The rate of convergence of Toeplitz based PCG methods for second order nonlinear boundary value problems

Summary. In previous works [21–23] we proposed the use of [5] and band Toeplitz based preconditioners for the solution of 1D and 2D boundary value problems (BVP) by means of the preconditioned conjugate gradient (PCG) methods. As and band Toeplitz linear systems can be solved [4] by using fast sine transforms [8], these methods become especially attractive in a parallel environment of computation. In this paper we extend this technique to the nonlinear, nonsymmetric case and, in addition, we prove some clustering properties for the spectra of the preconditioned matrices showing why these methods exhibit a convergence speed which results to be more than linear. Therefore these methods work much finer than those based on separable preconditioners [18,45], on incomplete LU factorizations [36,13,27], and on circulant preconditioners [9,30,35] since the latter two techniques do not assure a linear rate of convergence. On the other hand, the proposed technique has a wider range of application since it can be naturally used for nonlinear, nonsymmetric problems and for BVP in which the coefficients of the differential operator are not strictly positive and only piecewise smooth. Finally the several numerical experiments performed here and in [22,23] confirm the effectiveness of the theoretical analysis. Received December 19, 1995 / Revised version received September 15, 1997     

Convergence analysis of sectional methods for solving aggregation population balance equations: The fixed pivot technique

In this paper, we introduce the convergence analysis of the fixed pivot technique given by S. Kumar and Ramkrishna (1996) [28] for the nonlinear aggregation population balance equations which are of substantial interest in many areas of science: colloid chemistry, aerosol physics, astrophysics, polymer science, oil recovery dynamics, and mathematical biology. In particular, we investigate the convergence for five different types of uniform and non-uniform meshes which turns out that the fixed pivot technique is second order convergent on a uniform and non-uniform smooth meshes. Moreover, it yields first order convergence on a locally uniform mesh. Finally, the analysis exhibits that the method does not converge on an oscillatory and non-uniform random meshes. Mathematical results of the convergence analysis are also demonstrated numerically.     

THE OVERLAPPING DOMAIN DECOMPOSITION METHODFOR HARMONIC EQUATION OVER EXTERIOR THREE-DIMENSIONAL DOMAIN

1.IntroductionManyscientificandengineeringproblemscallbereducedtoexteriorboundaryvalueproblemsofpartialdifferentialequations.Althoughthenumericalmethodstosolveboundaryvalueproblems,suchasthefiniteelementmethodandthefinitedifferencemethod,areveryeffectiveonboundeddomains,yetweoftenfinditdifficulttousethemtodealwithunboundedproblems.Theboundaryreductionisaforcefulmeanstosolvesomeproblemsoverunboundeddomains.Amongmanyboundaryreductions,thenaturalboundaryreductionfoundedbyK.FengandD.H.Yuhassomed…     

A 2uFunction representation for non-uniform type-2 fuzzy sets: Theory and design

The theoretical and computational complexities involved in non-uniform type-2 fuzzy sets (T2 FSs) are main obstacles to apply these sets to modeling high-order uncertainties. To reduce the complexities, this paper introduces a 2uFunction representation for T2 FSs. This representation captures the ideas from probability theory. By using this representation, any non-uniform T2 FS can be represented by a function of two uniform T2 FSs. In addition, any non-uniform T2 fuzzy logic system (FLS) can be indirectly designed by two uniform T2 FLSs. In particular, a 2uFunction-based trapezoid T2 FLS is designed. Then, it is applied to the problem of forecasting Mackey–Glass time series corrupted by two kinds of noise sources: (1) stationary and (2) non-stationary additive noises. Finally, the performance of the proposed FLS is compared by (1) other types of FLS: T1 FLS and uniform T2 FLS, and (2) other studies: ANFIS [54], IT2FNN-1 [54], T2SFLS [3] and Q-T2FLS [35]. Comparative results show that the proposed design has a low prediction error as well as is suitable for online applications.     

广义U—统计量的指数收敛速度

文「１」研究了一样本Ｕ－统计量的指数收敛速度，本文则研究了广义Ｕ－统计量的指数收敛速度。