On the rate of convergence of two generalized Bernstein type operators

In this paper,we introduce the Bézier variant of two new families of generalized Bernstein type operators.We establish a direct approximation by means of the Ditzian-Totik modulus of smoothness and a global approximation theorem in terms of second order modulus of continuity.By means of construction of suitable functions and the method of Bojanic and Cheng,we give the rate of convergence for absolutely continuous functions having a derivative equivalent to a bounded variation function.

The Rate of Convergence of Augmented Lagrangian Method for Minimax Optimization Problems with Equality Constraints

The augmented Lagrangian function and the corresponding augmented Lagrangian method are constructed for solving a class of minimax optimization problems with equality constraints.We prove that,under the linear independence constraint qualification and the second-order sufficiency optimality condition for the lower level problem and the second-order sufficiency optimality condition for the minimax problem,for a given multiplier vectorμ,the rate of convergence of the augmented Lagrangian method is linear with respect to||μu-μ^(*)||and the ratio constant is proportional to 1/c when the ratio|μ-μ^(*)||/c is small enough,where c is the penalty parameter that exceeds a threshold c_(*)>O andμ^(*)is the multiplier corresponding to a local minimizer.Moreover,we prove that the sequence of multiplier vectors generated by the augmented Lagrangian method has at least Q-linear convergence if the sequence of penalty parameters(ck)is bounded and the convergence rate is superlinear if(ck)is increasing to infinity.Finally,we use a direct way to establish the rate of convergence of the augmented Lagrangian method for the minimax problem with a quadratic objective function and linear equality constraints.

SUBGEOMETRIC RATES OF CONVERGENCE OF THE GI/G/1 QUEUEING SYSTEM

The article deals with the waiting time process of the GI/G/1 queueing system.We shall give that the rate of convergence to the stationary distribution and the decay of the stationary tail only depend on the tail of the service distribution,but not on the interarrival distribution.We shall also give explicit criteria for the rate of convergence and decay of stationary tail for three specific types of subgeometric cases（Case 1：the rate function r（n）=exp（sn1/1＋α）,α〉0,s〉0;Case 2：polynomial rate function r（n）=nα,α〉0;Case 3：logarithmic rate function r（n）=logαn,α〉0）.

THE LONG TIME BEHAVIOR OF THE FRACTIONAL ORNSTEIN-UHLENBECK PROCESS WITH LINEAR SELF-REPELLING DRIFT

Let B^(H) be a fractional Brownian motion with Hurst index 1/2≤H<1.In this paper,we consider the equation(called the Ornstein-Uhlenbeck process with a linear self-repelling drift)dX_(t)^(H)=dB_(t)^(H)+σ X_(t)^(H)dt+vdt-θ(∫_(0)^(t)(X_(t)^(H)-X_(s)^(H))ds)dt,whereθ<0,σ,v∈ℝ.The process is an analogue of self-attracting diffusion(Cranston,Le Jan.Math Ann,1995,303:87–93).Our main aim is to study the large time behaviors of the process.We show that the solution X^(H)diverges to infinity as t tends to infinity,and obtain the speed at which the process X^(H)diverges to infinity.

Rate of Convergence of Normal Approximation for Local Median Estimate

Consider the fixed\|design nonparametric median regression model Y \{ni =g(x \{ni )+ε \{ni ,1≤i≤n, where ε \{ni are iid random variables with median zero. In estimating the regression function g(x), local median estimates \{nh (x) are employed, where h is the number of neighbors of x. Under some regularity conditions, the asymptotic normality and rate of convergence of normal approximation are obtained.

ON THE RATE OF CONVERGENCE OF SOLUTIONS FOR A CLASS OF DIFFERENCE EQUATIONS

Consider the following difference equation x(n) - x(n - 1) = p(n)G(x(n), x(n - k(n))) (1)where p(n)≤ M, G(x, .) is non-decreasing, G(., y) is non-increasing and G(x, x) = 0. Under some condition, we prove that every solution of (1) tends to constant and an estimate of the rate of solutions of (1) is given, the main results of paper [4] are extended.

RATE OF CONVERGENCE OF SCHWARZ ALTERNATING METHOD FOR TIME-DEPENDENT CONVECTION-DIFFUSION PROBLEM

Presents a study that examined the application of an overlapping domain decomposition method to the solution of time-dependent convection-diffusion problems. Background on the Schwartz alternating procedure; Application of two kinds of Schwartz alternating procedure to solve the numerical approximation problem; Numerical results.

ON THE RATES OF CONVERGENCE IN THE CENTRAL LIMIT THEOREM FOR TWO－PARAMETER MARTINGALE DIFFERENCES

In this paper we obtain the uniform bounds on the rate of convergence in the central limit theorem (CLT) for a class of two-parameter martingale difference sequences under certain conditions.

ASYMPTOTIC PROPERTIES OF THE MOMENT CONVERGENCE FOR NA SEQUENCES

It is well-known that the complete convergence theorem for i.i.d, random vari- ables has been an active topic since the famous work done by Hsu and Robbins [6]. Chow [4] obtained a moment version of Hsu and Robbins series. However, the series tends to infinity whenever c goes to zero, so it is of interest to investigate the asymptotic behavior of the series as e goes to zero. This note gives some limit theorems of the series generated by moments for NA random variables.

An Adaptive Load Stepping Algorithm for Path-Dependent Problems Based on Estimated Convergence Rates

A new adaptive(automatic)time stepping algorithm,called RCA(Rate of Convergence Algorithm)is presented.The new algorithm was applied in nonlinear finite element analysis of path-dependent problems.The step size is adjusted by monitoring the estimated convergence rate of the nonlinear iterative process.The RCA algorithm is relatively simple to implement,robust and its performance is comparable to,and in some cases better than,the automatic load incrementaion algorithm existent in commercial codes.Discussions about the convergence rate of nonlinear iterative processes,an estimation of the rate and a study of the parameters of the RCA algorithm are presented.To show the capacity of the algorithm to adjust the increment size,detailed discussions based on results for different limit load analyses are presented.The results obtained by RCA algorithm are compared with those by ABAQUS?,one of the most powerful nonlinear FEA(Finite Element Analysis)commercial software,in order to verify the capability of RCA algorithm to adjust the increment size along nonlinear analyses.

MEAN CONVERGENCE OF HERMITE-FEJER TYPE INTERPOLATION ON AN ARBITRARY SYSTEM OF NODES

In this paper sufficient conditions for mean convergence and rate of convergence of Hermite-Fejer type interpolation in the Lp norm on an arbitrary system of nodes are presented.

ON CONVERGENCE OF GENERAL GAMMA TYPE OPERATORS

The present paper deals with the new type of Gamma operators, here we estimate the rate of pointwise convergence of these new Gamma type operators Mn,k for functions of bounded variation, by using some techniques of probability theory.

TRUNCATED HERMITE INTERPOLATION ON AN ARBITRARY SYSTEM OF NODES

The criteria of convergence, including a theorem of Grunwald- type and the rate of convergence in terms of the modulus omega phi (f,t) of Ditzian and Totik for truncated Hermite interpolation on ail arbitrary system of nodes are given.

ASYMPTOTIC APPROXIMATION OF FUNCTIONS AND THEIR DERIVATIVES BY GENERALIZED BASKAKOV-SZAZS-DURRMEYER OPERATORS

We present the complete asymptotic expansion for a generalization of the Baskakov-Szasz-Durrmeyer operators and their derivatives.

The SCUSUM Estimation of Variance Change Point in Linear Process

This paper considers the estimation problem of a variance change-point in linear process.Consistency of a SCUSUM type change-point estimator is proved and its rate of convergence is established.The mean-unknown case is also considered.

On Chlodowsky Variant of Baskakov Type Operators

In the present paper, we deal with Chlodowsky type generalization of the Baskakov operators, special case of these operators includes Chlodowsky type Meyer–K?nig and Zeller operators(see [21]). With the help of Bohman-Korovkin theorem,we obtain some approximation properties for these operators. We give a modification of the operators in the space of differentiable functions and we also present examples of graphs for approximation. Finally, we apply these operators to the solution of the differential equation.

ON APPROXIMATION PROPERTIES OF NON-CONVOLUTION TYPE NONLINEAR INTEGRAL OPERATORS——Dedicated to Professor Paul L. Butzer on his 80^(th) Birthday

In the present paper we state some approximation theorems concerning point- wise convergence and its rate for a class of non-convolution type nonlinear integral opera- tors of the form：Tλ（f;x）=B∫AKλ（t,x,f（t））dr,x∈〈a,b〉λλA.In particular, we obtain the pointwise convergence and its rate at some characteristic points x0 off as （x,λ） → （x0, λ0） in LI 〈A,B 〉, where 〈 a,b 〉 and 〈A,B 〉 are is an arbitrary intervals in R, A is a non-empty set of indices with a topology and X0 an accumulation point of A in this topology. The results of the present paper generalize several ones obtained previously in the papers [191-[23]

Some Approximation Properties of Certain q-Baskakov-Beta Operators

In this paper, we propose the q analogue of modified Baskakov-Beta operators. The Voronovskaja type theorem and some direct results for the above operators are discussed. The rate of convergence and weighted approximation by the operators are studied.

Numerical Solution of Advection Diffusion Equation Using Semi-Discretization Scheme

Numerical diffusion and oscillatory behavior characteristics are averted applying numerical solutions of advection-diffusion equation are themselves immensely sophisticated. In this paper, two numerical methods have been used to solve the advection diffusion equation. We use an explicit finite difference scheme for the advection diffusion equation and semi-discretization on the spatial variable for advection-diffusion equation yields a system of ordinary differential equations solved by Euler’s method. Numerical assessment has been executed with specified initial and boundary conditions, for which the exact solution is known. We compare the solutions of the advection diffusion equation as well as error analysis for both schemes.

Wavelet-Based Density Estimation in Presence of Additive Noise under Various Dependence Structures

We study the following model: . The aim is to estimate the distribution of X when only are observed. In the classical model, the distribution of is assumed to be known, and this is often considered as an important drawback of this simple model. Indeed, in most practical applications, the distribution of the errors cannot be perfectly known. In this paper, the author will construct wavelet estimators and analyze their asymptotic mean integrated squared error for additive noise models under certain dependent conditions, the strong mixing case, the β-mixing case and the ρ-mixing case. Under mild conditions on the family of wavelets, the estimator is shown to be -consistent and fast rates of convergence have been established.