CENTRAL LIMIT THEOREM AND CONVERGENCE RATES FOR A SUPERCRITICAL BRANCHING PROCESS WITH IMMIGRATION IN A RANDOM ENVIRONMENT

We are interested in the convergence rates of the submartingale Wn=Z_(n)/Π_(n)to its limit W,where(Π_(n))is the usually used norming sequence and(Z_(n))is a supercritical branching process with immigration(Y_(n))in a stationary and ergodic environmentξ.Under suitable conditions,we establish the following central limit theorems and results about the rates of convergence in probability or in law:(i)W-W_(n) with suitable normalization converges to the normal law N(0,1),and similar results also hold for W_(n+k)-W_(n) for each fixed k∈N^(*);(ii)for a branching process with immigration in a finite state random environment,if W_(1) has a finite exponential moment,then so does W,and the decay rate of P(|W-W_(n)|>ε)is supergeometric;(iii)there are normalizing constants an(ξ)(that we calculate explicitly)such that a_(n)(ξ)(W-W_(n))converges in law to a mixture of the Gaussian law.

Convergence Rates for Elliptic Homogenization Problems in Two-dimensional Domain

In this paper, we study the convergence rates of solutions for second order elliptic equations with rapidly oscillating periodic coefficients in two-dimensional domain. We use an extension of the "mixed formulation" approach to obtain the representation formula satisfied by the oscillatory solution and homogenized solution by means of the particularity of solutions for equations in two-dimensional case. Then we utilize this formula in combination with the asymptotic estimates of Green or Neumann functions for operators and uniform regularity estimates of solutions to obtain convergence rates in L^p for solutions as well as gradient error estimates for Dirichlet or Neumann problems respectively.

CONVERGENCE RATES FOR LACUNARY TRIGONOMETRIC INTERPOLATION IN _(2π)~p SPACES

The saturation rate and class of (0,m1,m2, …,mq) trigonometric inter polation operators in . spaces have been determined by Cavaretta and Selvaraj. In this paper, we consider the convergence and saturation problems of these operators in (1≤p≤∞) and obtain complete results.

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.

Error Estimations, Error Computations, and Convergence Rates in FEM for BVPs

This paper presents derivation of a priori error estimates and convergence rates of finite element processes for boundary value problems (BVPs) described by self adjoint, non-self adjoint, and nonlinear differential operators. A posteriori error estimates are discussed in context with local approximations in higher order scalar product spaces. A posteriori error computational framework (without the knowledge of theoretical solution) is presented for all BVPs regardless of the method of approximation employed in constructing the integral form. This enables computations of local errors as well as the global errors in the computed finite element solutions. The two most significant and essential aspects of the research presented in this paper that enable all of the features described above are: 1) ensuring variational consistency of the integral form(s) resulting from the methods of approximation for self adjoint, non-self adjoint, and nonlinear differential operators and 2) choosing local approximations for the elements of a discretization in a subspace of a higher order scalar product space that is minimally conforming, hence ensuring desired global differentiability of the approximations over the discretizations. It is shown that when the theoretical solution of a BVP is analytic, the a priori error estimate (in the asymptotic range, discussed in a later section of the paper) is independent of the method of approximation or the nature of the differential operator provided the resulting integral form is variationally consistent. Thus, the finite element processes utilizing integral forms based on different methods of approximation but resulting in VC integral forms result in the same a priori error estimate and convergence rate. It is shown that a variationally consistent (VC) integral form has best approximation property in some norm, conversely an integral form with best approximation property in some norm is variationally consistent. That is best approximation property of the integral form and the VC of the integral form is equivalent, one cannot exist without the other, hence can be used interchangeably. Dimensional model problems consisting of diffusion equation, convection-diffusion equation, and Burgers equation described by self adjoint, non-self adjoint, and nonlinear differential operators are considered to present extensive numerical studies using Galerkin method with weak form (GM/WF) and least squares process (LSP) to determine computed convergence rates of various error norms and present comparisons with the theoretical convergence rates.

Uniform convergence rates for spot volatility estimation

This study presents the uniform convergence rate for spot volatility estimators based on delta sequences.Kernel and Fourier-based estimators are examples of this type of estimator.We also present the uniform convergence rates for kernel and Fourier-based estimators of spot volatility as applications of the main result.

Higher-order expansions of powered extremes of logarithmic general error distribution

In this paper,Let M_(n)denote the maximum of logarithmic general error distribution with parameter v≥1.Higher-order expansions for distributions of powered extremes M_(n)^(p)are derived under an optimal choice of normalizing constants.It is shown that M_(n)^(p),when v=1,converges to the Frechet extreme value distribution at the rate of 1/n,and if v>1 then M_(n)^(p)converges to the Gumbel extreme value distribution at the rate of(loglogn)^(2)=(log n)^(1-1/v).

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.

New Synchronization Algorithm and Analysis of Its Convergence Rate for Clock Oscillators in Dynamical Network with Time-Delays

New synchronization algorithm and analysis of its convergence rate for clock oscillators in dynamical network with time-delays are presented.A network of nodes equipped with hardware clock oscillators with bounded drift is considered.Firstly,a dynamic synchronization algorithm based on consensus control strategy,namely fast averaging synchronization algorithm (FASA),is presented to find the solutions to the synchronization problem.By FASA,each node computes the logical clock value based on its value of hardware clock and message exchange.The goal is to synchronize all the nodes' logical clocks as closely as possible.Secondly,the convergence rate of FASA is analyzed that proves it is related to the bound by a nondecreasing function of the uncertainty in message delay and network parameters.Then,FASA's convergence rate is proven by means of the robust optimal design.Meanwhile,several practical applications for FASA,especially the application to inverse global positioning system (IGPS) base station network are discussed.Finally,numerical simulation results demonstrate the correctness and efficiency of the proposed FASA.Compared FASA with traditional clock synchronization algorithms (CSAs),the convergence rate of the proposed algorithm converges faster than that of the CSAs evidently.

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.

IMPROVED RATE OF CONVERGENCE FOR THE MODIFIED SZASZ-MIRAKYAN OPERATORS

For some locally bounded function f measurable on the interval we have estimated the rate of convergence of the modified Szasz-Mirakyan operators M f at those points at which one sided limits exist. We have used the Chanturiya’s of variation. Our result improves the estimate of [1], [3], [4], [5], [6] and [9].

ON THE RATE OF POINTWISE CONVERGENCE OF THE DURRMEYER-TYPE OPERATORS

For bounded or some locally bounded functions f measurable on an interval I there is estimated the rate of convergence of the Durrmeyer-type operators Lnf at those points x∈IntI at which the one-sided limits f(x± 0) exist. In the main theorems the Chanturiya’s modulus of variation is used.

Exponential Continuous Non-Parametric Neural Identifier With Predefined Convergence Velocity

This paper addresses the design of an exponential function-based learning law for artificial neural networks(ANNs)with continuous dynamics.The ANN structure is used to obtain a non-parametric model of systems with uncertainties,which are described by a set of nonlinear ordinary differential equations.Two novel adaptive algorithms with predefined exponential convergence rate adjust the weights of the ANN.The first algorithm includes an adaptive gain depending on the identification error which accelerated the convergence of the weights and promotes a faster convergence between the states of the uncertain system and the trajectories of the neural identifier.The second approach uses a time-dependent sigmoidal gain that forces the convergence of the identification error to an invariant set characterized by an ellipsoid.The generalized volume of this ellipsoid depends on the upper bounds of uncertainties,perturbations and modeling errors.The application of the invariant ellipsoid method yields to obtain an algorithm to reduce the volume of the convergence region for the identification error.Both adaptive algorithms are derived from the application of a non-standard exponential dependent function and an associated controlled Lyapunov function.Numerical examples demonstrate the improvements enforced by the algorithms introduced in this study by comparing the convergence settings concerning classical schemes with non-exponential continuous learning methods.The proposed identifiers overcome the results of the classical identifier achieving a faster convergence to an invariant set of smaller dimensions.

The convergence rate and necessary-and-sufficient condition for the consistency of isogeometric collocation method

Although the isogeometric collocation(IGA-C)method has been successfully utilized in practical applications due to its simplicity and efficiency,only a little theoretical results have been established on the numerical analysis of the IGA-C method.In this paper,we deduce the convergence rate of the consistency of the IGA-C method.Moreover,based on the formula of the convergence rate,the necessary and sufficient condition for the consistency of the IGA-C method is developed.These results advance the numerical analysis of the IGA-C method.

CONVERGENCE AND RATE OF APPROXIMATION IN BV_φ FOR A CLASS OF INTEGRAL OPERATORS

We obtain estimates and convergence results with respect to φ-variation in spaces BV φ for a class of linear integral operators whose kernels satisfy a general homogeneity condition. Rates of approximation are also obtained. As applications, we apply our general theory to the case of Mellin convolution operators, to that one of moment operators and finally to a class of operators of fractional order.

Global Convergence of Curve Search Methods for Unconstrained Optimization

In this paper we propose a new family of curve search methods for unconstrained optimization problems, which are based on searching a new iterate along a curve through the current iterate at each iteration, while line search methods are based on finding a new iterate on a line starting from the current iterate at each iteration. The global convergence and linear convergence rate of these curve search methods are investigated under some mild conditions. Numerical results show that some curve search methods are stable and effective in solving some large scale minimization problems.

A RELAXED INERTIAL FACTOR OF THE MODIFIED SUBGRADIENT EXTRAGRADIENT METHOD FOR SOLVING PSEUDO MONOTONE VARIATIONAL INEQUALITIES IN HILBERT SPACES

In this paper,we investigate pseudomonotone and Lipschitz continuous variational inequalities in real Hilbert spaces.For solving this problem,we propose a new method that combines the advantages of the subgradient extragradient method and the projection contraction method.Some very recent papers have considered different inertial algorithms which allowed the inertial factor is chosen in[0;1].The purpose of this work is to continue working in this direction,we propose another inertial subgradient extragradient method that the inertial factor can be chosen in a special case to be 1.Under suitable mild conditions,we establish the weak convergence of the proposed algorithm.Moreover,linear convergence is obtained under strong pseudomonotonicity and Lipschitz continuity assumptions.Finally,some numerical illustrations are given to confirm the theoretical analysis.

Enhancing Iterative Learning Control With Fractional Power Update Law

The P-type update law has been the mainstream technique used in iterative learning control(ILC)systems,which resembles linear feedback control with asymptotical convergence.In recent years,finite-time control strategies such as terminal sliding mode control have been shown to be effective in ramping up convergence speed by introducing fractional power with feedback.In this paper,we show that such mechanism can equally ramp up the learning speed in ILC systems.We first propose a fractional power update rule for ILC of single-input-single-output linear systems.A nonlinear error dynamics is constructed along the iteration axis to illustrate the evolutionary converging process.Using the nonlinear mapping approach,fast convergence towards the limit cycles of tracking errors inherently existing in ILC systems is proven.The limit cycles are shown to be tunable to determine the steady states.Numerical simulations are provided to verify the theoretical results.

THE ZERO LIMIT OF THERMAL DIFFUSIVITY FOR THE 2D DENSITY-DEPENDENT BOUSSINESQ EQUATIONS

This paper is concerned with the asymptotic behavior of solutions to the initial boundary problem of the two-dimensional density-dependent Boussinesq equations.It is shown that the solutions of the Boussinesq equations converge to those of zero thermal diffusivity Boussinesq equations as the thermal diffusivity tends to zero,and the convergence rate is established.In addition,we prove that the boundary-layer thickness is of the valueδ(k)=k^(α)with anyα∈(0,1/4)for a small diffusivity coefficient k>0,and we also find a function to describe the properties of the boundary layer.

Rates of convergence of powered order statistics from general error distribution

Let{Xn:n≥1}be a sequence of independent random variables with common general error distribution GED(v)with shape parameter v>0,and let Mn,r denote the r-th largest order statistics of X1,X2,...,Xn.With different normalizing constants the distributional expansions and the uniform convergence rates of normalized powered order statistics|Mn,r|p are established.An alternative method is presented to estimate the probability of the r-th extremes.Numerical analyses are provided to support the main results.