STRONG CODING THEOREM AND ASYMPTOTIC ERROR EXPONENT OF ARBITRARILY VARYING SOURCE

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An extension of the method of least squares

The standard method of least squares assumes that there is a linear relationship relating two variables, only one of which has an error in it. A discussion of the validity of this assumption leads to the desirability of a technique which applies to the more general situation where both variables are subject to error. This includes relating the relevant statistical distance to be minimized to the Euclidean distance. The corresponding statistical formulae are derived, and simple comparisons are then made for various ranges of the statistical parameters.     

关于离散形式结点域定理证明的几点注记

本文指出了文[1]的主要定理(定理4与定理6)证明中的错误,并对其进行了修正,给出了正确的证明.     

Nonlinear half-dirichlet problems for first order elliptic equations in the unit ball of Rm(m≥3)

On the basis of an a priori estimate for solutions of a linear half-Dirichlet problem in the unit ball first order equations with the Di-rac operator as principal part for Clifford algebravalued functions an existence and uniqueness result is proved for a related nonlinear problemAn approximation procedure is used and an error estimate given.     

Corrigendum to “Renormalised solutions in thermo-visco-plasticity for a Norton–Hoff type model. Part I: The truncated case” [Nonlinear Anal. RWA 28 (2016) 140–152]

We correct an error in the original paper [Renormalised solutions in thermo-visco-plasticity for a Norton–Hoff type model. Part I: The truncated case, Nonlinear Anal. RWA 28 (2016) 140–152]. The proof of Theorem 3.2 is wrong and therefore we present here correct proof based on the same methods.     

Radial kernels and their reproducing kernel Hilbert spaces

We describe how to use Schoenberg’s theorem for a radial kernel combined with existing bounds on the approximation error functions for Gaussian kernels to obtain a bound on the approximation error function for the radial kernel. The result is applied to the exponential kernel and Student’s kernel. To establish these results we develop a general theory regarding mixtures of kernels. We analyze the reproducing kernel Hilbert space (RKHS) of the mixture in terms of the RKHS’s of the mixture components and prove a type of Jensen inequality between the approximation error function for the mixture and the approximation error functions of the mixture components.     

Solving Continuous-Time Linear Programming Problems Based on the Piecewise Continuous Functions

The numerical method is proposed in this article to solve a general class of continuous-time linear programming problems in which the functions appeared in the coefficients of this problem are assumed to be piecewise continuous. In order to make sure that all the subintervals of time interval will not contain the discontinuities, a different methodology for not equally partitioning the time interval is proposed. The main issue of this article is to obtain an analytic formula of error upper bound. In this article, we shall propose two kinds of computational procedure to evaluate the error upper bounds. One needs to solve the dual problem of the discretized linear programming problem, and another one does not need to solve the dual problem. Finally, we present a numerical example to demonstrate the usefulness of the numerical method.     

Numerical verification of solutions for nonlinear hyperbolic equations

We consider a numerical method to verify the solutions for nonlinear hyperbolic problems with guaranteed error bounds in the one-space dimensional case. We present verification procedures and show some numerical examples.     

Hypothesis testing for arbitrarily varying source

Hypothesis testing for arbitrarily varying source (AVS), which is to decide between the two hypotheses for the varying behavior of the distribution of AVS, is considered in this paper. We determine the best asymptotic exponent of the second kind of error probability when the first kind of error probability is fixed. This result generalizes the well-known lemma of Stein in statistics. As a corollary, Strassen's coding theorem for AVS is obtained.Supported by the Young Teacher Foundation of Chinese Educational Ministry and by the National Natural Science Foundation of China.     

Almost optimal convergence of the point vortex method for vortex sheets using numerical filtering

Standard numerical methods for the Birkhoff-Rott equation for a vortex sheet are unstable due to the amplification of roundoff error by the Kelvin-Helmholtz instability. A nonlinear filtering method was used by Krasny to eliminate this spurious growth of round-off error and accurately compute the Birkhoff-Rott solution essentially up to the time it becomes singular. In this paper convergence is proved for the discretized Birkhoff-Rott equation with Krasny filtering and simulated roundoff error. The convergence is proved for a time almost up to the singularity time of the continuous solution. The proof is in an analytic function class and uses a discrete form of the abstract Cauchy-Kowalewski theorem. In order for the proof to work almost up to the singularity time, the linear and nonlinear parts of the equation, as well as the effects of Krasny filtering, are precisely estimated. The technique of proof applies directly to other ill-posed problems such as Rayleigh-Taylor unstable interfaces in incompressible, inviscid, and irrotational fluids, as well as to Saffman-Taylor unstable interfaces in Hele-Shaw cells.

     

Szász–Mirakjan type operators providing a better error estimation

In this work, giving a modification of the well-known Szasz–Mirakjan operators, we prove that the error estimation of our operators is better than that of the classical Szasz–Mirakjan operators. Furthermore, we obtain a Voronovskaya type theorem for these modified operators.     

Detection of C2‐singularities using uniform spline approximations

For the detection of C²‐singularities, we present lower estimates for the error in Schoenberg variation‐diminishing spline approximation with equidistant knots in terms of the classical second‐order modulus of smoothness. To this end, we investigate the behaviour of the iterates of the Schoenberg operator. In addition, we show an upper bound of the second‐order derivative of these iterative approximations. Finally, we provide an example of how to detect singularities based on the decay rate of the approximation error.     

CONVERGENCE OF AN ITERATION METHOD WITHOUT DERIVATIVE

In this paper, we give a convergence theorem and error estimates for an iteration method under new Kantorovitch-Ostrowski type condition using the information of higher derivatives at initial points. Compared with the corresponding study in [3],the convergence determination is established under one global condition, instead of two , on the function.     

Approximation Scheme for Solutions of BSDEs with Two Reflecting Barriers

Abstract

In this work, we interest to the simulation of solution for the BSDEs with two reflecting barriers. Specially, we present some properties of the solution, give a representation theorem and suggest a backward discretization scheme. After, we study the L ² induced error.     

On Shige Peng’s central limit theorem

We give error estimates in Peng’s central limit theorem for not necessarily nondegenerate case. The exposition uses the language of the classical probability theory instead of the language of the theory of sublinear expectations. We only consider the one-dimensional case. The higher dimensional extension is left to the interested reader.     

对称矩阵方程Q+BKS+(BKS)~T+BKRK~TB~T=0的向后误差分析

本文研究了对称矩阵方程Q+BKS+(BKS)~T+BKRK~TB~T=0近似解的最佳向后误差,得到了向后误差量的上界与下界,并用一个简单的数值例子来说明所得结论     

Note on the integral mean value method for Fredholm integral equations of the second kind

We study the paper of Avazzadeh et al. [Z. Avazzadeh, M. Heydari, G.B., Loghmani, Numerical solution of Fedholm integral equations of the second kind by using integral mean value theorem, Appl. Math. Model. 35 (2011) 2374–2383] with the integral mean value method for Fredholm integral equations of the second kind. The objective of the note is threefold. First, we point out a basic error in the paper. Second, we find that the given numerical examples are only related to the special cases of Fredholm integral equations of the second kind with the degenerate kernels, which can be solved simply. Third, due to the basic error, our observations reveal that generally the suggested method should not be considered for a Fredholm integral equation of the second kind.     

THE ERROR ESTIMATES OF HALLEY''''S METHOD

In this paper we give an almost sharp error estimate of Halley's iteration for the majorizing sequence. Compared with the corresponding results in [6,14], it is far better. Meanwhile,the convergence theorem is established .for Halley's iteration in Banach spaces.     

A robust Kantorovich’s theorem on the inexact Newton method with relative residual error tolerance

We prove that under semi-local assumptions, the inexact Newton method with a fixed   relative residual error tolerance converges Q$Q$-linearly to a zero of the nonlinear operator under consideration. Using this result we show that the Newton method for minimizing a self-concordant function or to find a zero of an analytic function can be implemented with a fixed relative residual error tolerance.     

再生核空间二元函数展开

For convenience, we use D to denote ［0,1］×［0,1］. PutH2={u(x,y)|u∈L2(D) is a real and totally continuous function withux, uy, 2ux2, 2uy2, 2uxy, 3uxy2, 3ux2y, 4ux2y2∈L2(D)} For any u and v∈H2, we define an inner product〈u,v〉=Duv+2uxvx+2uyvy+42uxy2vxy+2ux22vx2+2uy22vy2+23ux2y3vx2y+23uxy23vxy2+4ux2y24vx2y2dxdy.(1)Selecting a norm as ‖·‖=〈·,·〉1/2, we find that, as the proof in ［1］, H2 is a reproducing kernel space (see ［2］), and (1) defines indeed an inner product on the …