线性模型参数向量的近似贝叶斯估计

用线性贝叶斯方法去同时估计线性模型中回归系数和误差方差,并在不知道先验分布具体形式的情况下,得到了线性贝叶斯估计的表达式.在均方误差矩阵准则下,证明了其优于最小二乘估计和极大似然估计.与利用MCMC算法得到的贝叶斯估计相比,线性贝叶斯估计具有显式表达式并且更方便使用.对于几种不同的先验分布,数值模拟结果表明线性贝叶斯估...     

错误先验假定下回归系数Bayes估计的小样本性质

本在于错误指定的先验假定获得了回归系数的Ｂａｙｅｓ估计（ＢＥ）,并在均方误差矩阵准则下对其与最小二乘（ＬＳ）估计进行了比较,导出了它们的相对效率的界、讨论了在后验ＰｉｔｍａｎＣｌｏｓｅｎｅｓｓ准则下ＢＥ相对于ＬＳ估计的优良性。     

对积分恒等式的一点补充和有限元的局部校正结果

１．积分恒等式简介其中ｈ＝ｍａｘ｛ｈｅ,ｋｅ｝, 易见用 Ｑ１（ｅ）表示 ｅ上全体双线性函数,ｕＩ∈ Ｑ１（ｅ）为ｕ在 ｅ上的双线性插值．文［１, ２］曾给出了一系列的积分恒等式,列举如下： Ａ：二阶恒等式,用于超收敛估计：对任给ｖ∈Ｑ１（ｅ）,有以及由于Ｉｅ１２的展开式中含有因子б１б２ｖ,将破坏超收敛估计,故对此项关于ｙ分部积分,得展开式其中τ１,τ２分别为ｅ的上下边.（１．３）,（１．４）和（１．６）是作导数超收敛估计的基本展开式．当然我们更有０次项的展开式．在下一节将讨论林氏积分表． 为了进一步…     

与双调和方程相关的连续模和Heinz-Schwarz型不等式

对于给定的两个正整数n ≥ 2和m ≥ 1，假设函数f满足如下条件：（1）在Bⁿ内满足非齐次双调和方程△（△f）=g（g ∈ C（Bⁿ，R^m））；（2）在S^n-1上满足f=ψ₁（ψ₁ ∈ C（S^n-1，R^m）），以及∂f/∂n=ψ₂（ψ₂ ∈ C（S^n-1，R^m）），其中∂/∂n表示内法线方向导数，Bⁿ表示Rⁿ中的单位球以及S^n-1表示Bⁿ的边界.本文主要研究f的连续模和Heinz-Schwarz型不等式.     

错误先验假定下Bayes线性无偏估计的稳健性

本文基于错误的先验假定获得了一般线性模型下可估函数的Bayes线性无偏估计(BLUE), 证明了在均方误差矩阵(MSEM)准则和后验Pitman Closeness (PPC)准则下BLUE相对于最小二乘估计(LSE)的优良性, 并导出了它们的相对效率的界, 从而获得BLUE的稳健性.     

双线性非协调有限元逼近的梯度恢复后验误差估计

给出了二阶椭圆方程的双线性非协调有限元逼近的梯度恢复后验误差估计.该误差估计是在Q_1非协调元上得到的,并给出了误差的上下界.进一步证明该误差估计在拟一致网格上是渐进精确地.证明依赖于clement插值和Helmholtz分解,数值结果验证了理论的正确性.     

关于非协调Qrot1元可计算上界后验误差估计的一个注记

通过数值试验发现Ainsworth建立的非协调Qrot1元可计算上界后误差估计指示子的可靠、有效性差.参照相关文献以及根据Qrot1元的性质,在Ainsworth建立的可计算上界后验误差估计框架下对插值后处理函数的构造和选取分别作了修改和更换,并相应获得可靠且有效的可计算上界后验误差估计,给出了三个不同类型的例子及其实验结果.     

自适应有限元和后验误差估计——等价估计

近年来,自适应有限元计算得到了广泛的研究.后验误差估计是实现自适应计算的基础.70年代末,I. Babuska和 W.Rheinboldt给出了建立等价后验估计的一般原则和方法,在一维情形给出了完整的结果.80年代初,I.Babuska和A.Miller对平面弹性问题正方形双线性元,给出了一种后验估计的方法,并证明了其等价性和渐近准确     

三维区域上Q_1~(rot)元的渐近展开及外推

本文研究了正方体区域上Q₁^rot非协调元渐近展开式.利用林群、吕涛等提出的有限元误差渐近展开法,获得了正方体区域上Q₁^rot非协调元特征值的误差渐近展开式.理论分析和数值实验结果表明三维Q₁^rot非协调元特征值外推公式是有效的,可以把特征值的精度从二阶提高到四阶.     

对流扩散方程迎风有限元的自适应方法

本文对二维发展型对流扩散方程的迎风有限元格式给出了显式后验误差估计,证明了真实误差被后验误差估计器上下界定;并通过误差估计器建立了相应的自适应算法,数值例子表明了方法的有效性.     

Fractal dimensions of fractional integral of continuous functions

In this paper,we mainly explore fractal dimensions of fractional calculus of continuous functions defined on closed intervals.Riemann–Liouville integral of a continuous function f(x) of order v(v0) which is written as D~(-v) f(x) has been proved to still be continuous and bounded.Furthermore,upper box dimension of D~(-v) f(x) is no more than 2 and lower box dimension of D~(-v) f(x) is no less than 1.If f(x) is a Lipshciz function,D~(-v) f(x) also is a Lipshciz function.While f(x) is differentiable on [0,1],D~(-v) f(x) is differentiable on [0,1] too.With definition of upper box dimension and further calculation,we get upper bound of upper box dimension of Riemann–Liouville fractional integral of any continuous functions including fractal functions.If a continuous function f(x) satisfying H?lder condition,upper box dimension of Riemann–Liouville fractional integral of f(x) seems no more than upper box dimension of f(x).Appeal to auxiliary functions,we have proved an important conclusion that upper box dimension of Riemann–Liouville integral of a continuous function satisfying H?lder condition of order v(v0) is strictly less than 2-v.Riemann–Liouville fractional derivative of certain continuous functions have been discussed elementary.Fractional dimensions of Weyl–Marchaud fractional derivative of certain continuous functions have been estimated.     

Justification of a Perturbation Approximation of the Klein–Gordon Equation

Consider the nonlinear wave equation
u_tt − γ ² u_xx + f(u) = 0
with the initial conditions
u ( x ,0) = εφ ( x ), u _t( x ,0) = εψ ( x ),
where f ( u ) is either of the form f ( u )= c ² u −σ u ^{2 s +1}, s =1, 2,…, or an odd smooth function with f '(0)>0 and | f '( u )|≤ C ₀².The initial data φ( x )∈ C ² and ψ( x )∈ C ¹ are odd periodic functions that have the same period. We establish the global existence and uniqueness of the solution u ( x ,  t ; ɛ), and prove its boundedness in x ∈ R and t >0 for all sufficiently small ɛ>0. Furthermore, we show that the error between the solution u ( x ,  t ; ɛ) and the leading term approximation obtained by the multiple scale method is of the order ɛ³ uniformly for x ∈ R and 0≤ t ≤ T /ɛ², as long as ɛ is sufficiently small, T being an arbitrary positive number.     

A Bishop–Phelps–Bollobás Theorem for Asplund Operators

By characterizing Asplund operators through Fréchet differentiability property of convex functions, we show the following Bishop–Phelps–Bollobás theorem: Suppose that X is a Banach space,T : X → C(K) is an Asplund operator with ║T║= 1, and that x_0 ∈ S_X, 0 ε satisfy ║T(x_0)║ 1-ε~2/2.Then there exist x_ε∈ S_X and an Asplund operator S : X → C(K) of norm one so that ║S(x_ε)║ = 1, x_0-x_ε ε and ║T-S║ ε.Making use of this theorem, we further show a dual version of Bishop–Phelps–Bollobás property for a strong Radon–Nikodym operator T : ?_1 → Y of norm one: Suppose that y_0~*∈ S_(Y~*), ε≥ 0 satisfy T~*(y_0~*) 1-ε~2/2. Then there exist y_ε~*∈ S_(Y~*), x_ε∈(±e_n), y_ε∈ S_Y, and a strong Radon–Nikodym operator S : ?_1 → Y of norm one so that (ⅰ)║S(x_ε)║= 1;(ⅱ) S(x_ε) = y_ε;(ⅲ)║T-S║ ε;(ⅳ)║S~*(y_ε~*)║=y_ε~*, y_ε= 1;(ⅴ)║y_0~*-y_ε~*║ ε and (ⅵ)║T~*-S~*║ ε,where(e_n) denotes the standard unit vector basis of ?_1.     

A posteriori error estimate for computing tr(f(A)) by using the Lanczos method

An outstanding problem when computing a function of a matrix, f(A), by using a Krylov method is to accurately estimate errors when convergence is slow. Apart from the case of the exponential function that has been extensively studied in the past, there are no well‐established solutions to the problem. Often, the quantity of interest in applications is not the matrix f(A) itself but rather the matrix–vector products or bilinear forms. When the computation related to f(A) is a building block of a larger problem (e.g., approximately computing its trace), a consequence of the lack of reliable error estimates is that the accuracy of the computed result is unknown. In this paper, we consider the problem of computing tr(f(A)) for a symmetric positive‐definite matrix A by using the Lanczos method and make two contributions: (a) an error estimate for the bilinear form associated with f(A) and (b) an error estimate for the trace of f(A). We demonstrate the practical usefulness of these estimates for large matrices and, in particular, show that the trace error estimate is indicative of the number of accurate digits. As an application, we compute the log determinant of a covariance matrix in Gaussian process analysis and underline the importance of error tolerance as a stopping criterion as a means of bounding the number of Lanczos steps to achieve a desired accuracy.     

Horizontal Growth of Harmonic Functions

We show that positive harmonic functions in the upper halfplane grow at most quadratically in horizontal bands. This bound is sharp in a sense to be specified, which, at least implies that there are examples growing as fast as any power under 2. These results are extended to positive harmonic functions in a half-space of R ^{n +1}, with points represented by ( x , y ), where x ∈R ⁿ , and y ∈R, the sharp maximum rate of growth being now ¦ x ¦ ^{n +1}. The case of Poisson integrals of functions in L_p ( dx /(1+(¦ x ¦)² )^{( n +1)/2}) is also taken up; the bound condition is then O (¦ x ¦^{( n +1)/ p} ).     

Heavy cycles in 2-connected triangle-free weighted graphs

A weighted graph is one in which every edge e is assigned a nonnegative number, called the weight of e. The sum of the weights of the edges incident with a vertex v is called the weighted degree of v, denoted by dw(v). The weight of a cycle is defined as the sum of the weights of its edges. Fujisawa proved that if G is a 2-connected triangle-free weighted graph such that the minimum weighted degree of G is at least d, then G contains a cycle of weight at least 2d. In this paper, we proved that if G is a2-connected triangle-free weighted graph of even size such that dw(u) + dw(v) ≥ 2d holds for any pair of nonadjacent vertices u, v ∈ V(G), then G contains a cycle of weight at least 2d.     

Möbius Homogeneous Hypersurfaces with One Simple Principal Curvature in Sn+1

Let Möb(Sⁿ⁺¹) denote the Möbius transformation group of Sⁿ⁺¹. A hypersurface f:Mⁿ → Sⁿ⁺¹ is called a Möbius homogeneous hypersurface, if there exists a subgroup G ? Möb(Sⁿ⁺¹) such that the orbit G(p)={φ(p)|φ ∈ G}=f(Mⁿ), p ∈ f(Mⁿ). In this paper, we classify the Möbius homogeneous hypersurfaces in Sⁿ⁺¹ with at most one simple principal curvature up to a Möbius transformation.     

Strong skew commutativity preserving maps on rings with involution

Let R be a unital *-ring with the unit I.Assume that R contains a symmetric idempotent P which satisfies ARP = 0 implies A = 0 and AR(I-P) = 0 implies A = 0.In this paper,it is shown that a surjective map Φ:R→R is strong skew commutativity preserving(that is,satisfiesΦ(A)Φ(B)-Φ(B)Φ(A)~w= AB-BA~w for all A,B∈R) if and only if there exist a map f:R→Z_s(R)and an element Z∈Z_s(R) with Z~2=I such that Φ(A)=ZA +f(A) for all A∈R,where Z_s(R) is the symmetric center of R.As applications,the strong skew commutativity preserving maps on unital prime *-rings and von Neumann algebras with no central summands of type I_1 are characterized.     

Multiple Positive Solutions for a Nonlinear Elliptic Equation Involving Hardy–Sobolev–Maz'ya Term

In this paper, we study the existence and nonexistence of multiple positive solutions for the following problem involving Hardy–Sobolev–Maz'ya term:-Δu- λu/|y|2=|u|pt-1u/|y|t+ μf(x), x ∈Ω,where Ω is a bounded domain in RN(N ≥ 2), 0 ∈Ω, x =(y, z) ∈ Rk× RN-kand pt =N +2-2t N-2(0 ≤ t ≤2). For f(x) ∈ C1(Ω)\{0}, we show that there exists a constant μ* 0 such that the problem possessesat least two positive solutions if μ∈(0, μ*) and at least one positive solution if μ = μ*. Furthermore,there are no positive solutions if μ∈(μ*, +∞).     

Tyson型集及Borel函数的图的拟对称极小性

本文将欧氏空间R^d中形如[0，1]×Z的集称为Tyson型集，其中d>1，Z⊂R^d-1.已知当Z是R^d-1中的紧集时，Tyson型集是拟对称极小集.本文改进了这个结果，证明了当Z是R^d-1中的Borel集时，Tyson型集仍是拟对称极小集.作为应用，我们证明了Tyson型集三个形变版本的拟对称极小性，其中一个结果是：设Z是R^d-1中的任一Borel集，h：Z→R¹是Borel函数，满足dim_H（{h≠0}∩Z）=dim_H Z，则h的图G（h）是拟对称极小集，其中h的图G（h）定义为G（h）={（z，y）：z∈Z，y∈[0，h（z）]}.