Schwarz-Pick estimates for bounded holomorphic functions in the unit ball of ℂ n

We give a Schwarz-Pick estimate for bounded holomorphic functions on unit ball in Cn, and generalize some early work of Schwarz-Pick estimates for bounded holomorphic functions on unit disk in C.     

Convergence of Solutions of Discrete Reflected Backward SDE＇s and Simulations

The objective of this paper is to introduce elementary discrete reflected backward equations and to give a simple method to discretize in time a （continuous） reflected backward equation. A presentation of numerical simulations is also described.     

Convergence analysis for the Cauchy problem of Laplace’s equation by a regularized method of fundamental solutions

In this paper, we consider the Cauchy problem of Laplace’s equation in the neighborhood of a circle. The method of fundamental solutions (MFS) combined with the discrete Tikhonov regularization is applied to obtain a regularized solution from noisy Cauchy data. Under the suitable choices of a regularization parameter and an a priori assumption to the Cauchy data, we obtain a convergence result for the regularized solution. Numerical experiments are presented to show the effectiveness of the proposed method.     

A second order nonconforming rectangular finite element method for approximating Maxwell’s equations

The main objective of this paper is to present a new rectangular nonconforming finite element scheme with the second order convergence behavior for approximation of Maxwell’s equations.Then the corresponding optimal error estimates are derived.The difficulty in construction of this finite element scheme is how to choose a compatible pair of degrees of freedom and shape function space so as to make the consistency error due to the nonconformity of the element being of order O(h 3 ) ,properly one order higher than that of its interpolation error O(h 2 ) in the broken energy norm,where h is the subdivision parameter tending to zero.     

The Superlinear Convergence Analysis of a Nonmonotone BFGS Algorithm on Convex Objective Functions

We prove the superlinear convergence of a nonmonotone BFGS algorithm on convex objective functions under suitable conditions.     

Convergence analysis of Oja’s iteration for solving online PCA with nonzero-mean samples

Principal component analysis(PCA)is one of the most popular multivariate data analysis techniques for dimension reduction and data mining,and is widely used in many fields ranging from industry and biology to finance and social development.When working on big data,it is of great necessity to consider the online version of PCA,in which only a small subset of samples could be stored.To handle the online PCA problem,Oja(1982)presented the stochastic power method under the assumption of zero-mean samples,and there have been lots of theoretical analysis and modified versions of this method in recent years.However,a common circumstance where the samples have nonzero mean is seldom studied.In this paper,we derive the convergence rate of a nonzero-mean version of Oja’s algorithm with diminishing stepsizes.In the analysis,we succeed in handling the dependency between each iteration,which is caused by the updated mean term for data centering.Furthermore,we verify the theoretical results by several numerical tests on both artificial and real datasets.Our work offers a way to deal with the top-1 online PCA when the mean of the given data is unknown.     

Convergence and error estimate of cascade algorithms with infinitely supported masks in L p (? s )

The cascade algorithm plays an important role in computer graphics and wavelet analysis.In this paper,we first investigate the convergence of cascade algorithms associated with a polynomially decaying mask and a general dilation matrix in L p (R s) (1 p ∞) spaces,and then we give an error estimate of the cascade algorithms associated with truncated masks.It is proved that under some appropriate conditions if the cascade algorithm associated with a polynomially decaying mask converges in the L p-norm,then the cascade algorithms associated with the truncated masks also converge in the L p-norm.Moreover,the error between the two resulting limit functions is estimated in terms of the masks.     

Convergence of the gradient projection method and Newton’s method as applied to optimization problems constrained by intersection of a spherical surface and a convex closed set

The gradient projection method and Newton’s method are generalized to the case of nonconvex constraint sets representing the set-theoretic intersection of a spherical surface with a convex closed set. Necessary extremum conditions are examined, and the convergence of the methods is analyzed.     

An application of Chen’s theorem to convergence of Fourier multiplier transforms

Let V be a star shaped region. In 2006, Colzani, Meaney and Prestini proved that if function f satisfies some condition, then the multiplier transform with the characteristic function of tV as the multiplier tends to f almost everywhere, when t goes to∞. In this paper we use a Theorem established by K.K.Chen to show that if we change their multiplier, then the condition on f can be weakened.     

Almost Everywhere Convergence of Sequences of Cesàro and Riesz Means of Integrable Functions with Respect to the Multidimensional Walsh System

The aim of this paper is to prove the a.e.convergence of sequences of the Cesa`ro and Riesz means of the Walsh–Fourier series of d variable integrable functions.That is,let a=(a1,...,ad):N→Nd(d∈P)such that aj(n+1)≥δsupk≤n aj(k)(j=1,...,d,n∈N)for someδ0 and a1(+∞)=···=ad(+∞)=+∞.Then,for each integrable function f∈L1(Id),we have the a.e.relation for the Ces`aro means limn→∞σαa(n)f=f and for the Riesz means limn→∞σα,γa(n)f=f for any 0αj≤1≤γj(j=1,...,d).A straightforward consequence of our result is the so-called cone restricted a.e.convergence of the multidimensional Ces`aro and Riesz means of integrable functions,which was proved earlier by Weisz.     

Convergence analysis of the corrected Uzawa algorithmfor symmetric saddle point problems简

For the large sparse saddle point problems, Pan and Li recently proposed in [H. K. Pan, W. Li, Math. Numer. Sinica, 2009, 31（3）： 231-242] a corrected Uzawa algorithm based on a nonlinear Uzawa algorithm with two nonlinear approximate inverses, and gave the detailed convergence analysis. In this paper, we focus on the convergence analysis of this corrected Uzawa algorithm, some inaccuracies in [H. K. Pan, W. Li, Math. Numer. Sinica, 2009, 31（3）： 231-242] are pointed out, and a corrected convergence theorem is presented. A special case of this modified Uzawa algorithm is also discussed.     

Convergence Theorems of Mann and Ishikawa Iterative Processes with Errors for Multivalued Φ－strongly Accretive Mapping

Let X be a real Banach space and A : X→ 2^X a bounded uniformly continuous Φ-strongly accretive multivalued mapping. For any f∈ X, Mann and Ishikawa iterative processes with errors converge strongly to the unique solution of Ax(←∈) f. The conclusion in this paper weakens the stronger conditions about errors in Chidume and Moore‘s theorem (J.Math.Anal .Appl.,245(2000),142-160).     

Modeling and prediction of children’s growth data via functional principal component analysis

We use the functional principal component analysis (FPCA) to model and predict the weight growth in children. In particular, we examine how the approach can help discern growth patterns of underweight children relative to their normal counterparts, and whether a commonly used transformation to normality plays any constructive roles in a predictive model based on the FPCA. Our work supplements the conditional growth charts developed by Wei and He (2006) by constructing a predictive growth model based on a small number of principal components scores on individual’s past. This work was supported by National Natural Science Foundation of China (Grant No. 10828102), a Changjiang Visiting Professorship, the Training Fund of Northeast Normal University’s Scientific Innovation Project (Grant No. NENU-STC07002) and the National Institutes of Health Grant of USA (Grant No. R01GM080503-01A1).     

Optimal error estimates and modified energy conservation identities of the ADI-FDTD scheme on staggered grids for 3D Maxwell’s equations

This paper is concerned with the optimal error estimates and energy conservation properties of the alternating direction implicit finite-difference time-domain(ADI-FDTD) method which is a popular scheme for solving the 3D Maxwell’s equations.Precisely,for the case with a perfectly electric conducting(PEC) boundary condition we establish the optimal second-order error estimates in both space and time in the discrete H 1-norm for the ADI-FDTD scheme,and prove the approximate divergence preserving property that if the divergence of the initial electric and magnetic fields are zero,then the discrete L 2-norm of the discrete divergence of the ADI-FDTD solution is approximately zero with the second-order accuracy in both space and time.The key ingredient is two new discrete modified energy norms which are second-order in time perturbations of two new energy conservation laws for the Maxwell’s equations introduced in this paper.Furthermore,we prove that,in addition to two known discrete modified energy identities which are second-order in time perturbations of two known energy conservation laws,the ADI-FDTD scheme also satisfies two new discrete modified energy identities which are second-order in time perturbations of the two new energy conservation laws.This means that the ADI-FDTD scheme is unconditionally stable under the four discrete modified energy norms.Experimental results which confirm the theoretical results are presented.     

Dunkl’s theory and best approximation by entire functions of exponential type in L 2-metric with power weight

In this paper, we study the sharp Jackson inequality for the best approximation of f ∈L2,κ(Rd) by a subspace E2κ(σ)(SE2κ(σ)), which is a subspace of entire functions of exponential type(spherical exponential type) at most σ. Here L2,κ(Rd) denotes the space of all d-variate functions f endowed with the L2-norm with the weight v2κ(x) =ξ∈R, which is defined by a positive+|(ξ, x)|κ(ξ)subsystem R+ of a finite root system RRdand a function κ(ξ) : R → R+ invariant under the reflection group G(R) generated by R. In the case G(R) = Zd2, we get some exact results. Moreover,the deviation of best approximation by the subspace E2κ(σ)(SE2κ(σ)) of some class of the smooth functions in the space L2,κ(Rd) is obtained.