Approximation of conic section by quartic Bzier curve with endpoints continuity condition

A new method for approximation of conic section by quartic B′ezier curve is presented, based on the quartic B′ezier approximation of circular arcs. Here we give an upper bound of the Hausdorff distance between the conic section and the approximation curve, and show that the error bounds have the approximation order of eight. Furthermore, our method yields quartic G2 continuous spline approximation of conic section when using the subdivision scheme,and the effectiveness of this method is demonstrated by some numerical examples.     

A NEW SMOOTHING APPROXIMATION METHOD FOR SOLVING BOX CONSTRAINED VARIATIONAL INEQUALITIES

In this paper, we first give a smoothing approximation function of nonsmooth system based on box constrained variational inequalities and then present a new smoothing approximation algorithm. Under suitable conditions,we show that the method is globally and superlinearly convergent. A few numerical results are also reported in the paper.     

一类六角系统图的特征多项式及应用

Let Ln be the hexagonal chain graph,Fnbe the hexacyclic system graph and Mn be the M¨obius hexacyclic system graph. Derflinger and Sofer gave the spectra of Ln and Fn by using group theoretical method. Later, Gutman gave the spectra of them using a polynomial result due to Godsil and McKay. In this paper, we give a simple and direct method to determine the characteristic polynomial and spectra of Fn and Ln. By the method, we give the characteristic polynomial and spectrum of Mn that is new. Additionally, the exact values of total π-electron energy and the nullities of Ln, Fn and Mn are obtained, and the bounds for the energy of Ln and Mn are also considered.     

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.     

BEST DIRECT AND CONVERSE RESULTS FOR LAGRANGE-TYPE OPERATORS

The aim of this paper is to give direct and converse theorems for the approximation by using a discretely defined method Ln (see D. H. Mache [10], which is a modification of the Lagrange operator. Furthermore we obtain with a matrix construction technique (see M. D. Ye and D. X. Zhou [11]) a Lagrange-type operator n, for which we get a characterization for Lipschitz functions by the approximation rate of these methods.     

Decomposition and approximation of multivariate functions on the cube

In this paper, we present a decomposition method of multivariate functions. This method shows that any multivariate function f on [0, 1]d is a finite sum of the form ∑jφjψj , where each φj can be extended to a smooth periodic function, each ψj is an algebraic polynomial, and each φjψj is a product of separated variable type and its smoothness is same as f . Since any smooth periodic function can be approximated well by trigonometric polynomials, using our decomposition method, we find that any smooth multivariate function on [0, 1]d can be approximated well by a combination of algebraic polynomials and trigonometric polynomials. Meanwhile, we give a precise estimate of the approximation error.     

Some Remarks on the Convex Feasibility Problem and Best Approximation Problem

In this paper we investigate several solution algorithms for the convex fea- sibility problem(CFP)and the best approximation problem(BAP)respectively.The algorithms analyzed are already known before,but by adequately reformulating the CFP or the BAP we naturally deduce the general projection method for the CFP from well-known steepest decent method for unconstrained optimization and we also give a natural strategy of updating weight parameters.In the linear case we show the connec- tion of the two projection algorithms for the CFP and the BAP respectively.In addition, we establish the convergence of a method for the BAP under milder assumptions in the linear case.We also show by examples a Bauschke's conjecture is only partially correct.     

Convergence of Markov decision processes with constraints and state-action dependent discount factors

This paper is concerned with the convergence of a sequence of discrete-time Markov decision processes(DTMDPs)with constraints,state-action dependent discount factors,and possibly unbounded costs.Using the convex analytic approach under mild conditions,we prove that the optimal values and optimal policies of the original DTMDPs converge to those of the"limit"one.Furthermore,we show that any countablestate DTMDP can be approximated by a sequence of finite-state DTMDPs,which are constructed using the truncation technique.Finally,we illustrate the approximation by solving a controlled queueing system numerically,and give the corresponding error bound of the approximation.     

Explicit Reciprocity Law for Lubin-Tate Formal Groups

In this article, using Fontaine＇s ФГ-module theory, we give a new proof of Coleman＇s explicit reciprocity law, which generalizes that of Artin-Hasse, Iwasawa and Wiles, by giving a complete formula for the norm residue symbol on Lubin-Tate groups. The method used here is different from the classical ones and can be used to study the Iwasawa theory of crystalline representations.     

The Error in the Approximation

In applying any numerical method such as the bisection method to determine a root, it is important to realize that the best we can usually achieve is an approximation ofthe exact root. At each iteration of the method,we obtain a better estimate of the root. Thus it becomes desirable that we be able to estimate how accurate the approximation is at each stage so that we know when to stop the process.With the bisection method,suppose we know that there is a root in some interval [a,b], where a and b are successive integers, say 2 and 3. If we select the midpoint M1 of this interval, then it is obvious that the root R is     

New Approximation Algorithms for Weighted Maximin Dispersion Problem with Box or Ball Constraints

In this paper, we propose new approximation algorithms for a NP-hard problem, i.e., weighted maximin dispersion problem. By using a uniformly distributed random sample method, we first propose a new random approximation algorithm for box constrained or ball constrained weighted maximin dispersion problems and analyze its approximation bound respectively. Moreover, we propose two improved approximation algorithms by combining our technique with an existing binary sample technique for both cases. To the best of our knowledge, they are the best approximation bounds for both box constrained and ball constrained weighted maximin dispersion problems respectively.

     

A Super-Halley Type Approximation in Banach Spaces

The Super-Halley method is one of the best known third-order iteration for solving nonlnear equations. A Newton-like method which is an approximation of this method is studied. Our approach yields a fourth R-order iterative process which is more efficient than its classical predecessor. We establish a Newton-Kantorovich-type convergence theorem using a new system of recurrence relations, and give an explicit expression for the a priori error bound of the iteration.     

用于求根问题的一个基于Thiele连分式的四阶收敛的迭代方法

利用截断的Thiele连分式,本文给出了一个求解非线性单变量方程的单步迭代方法,并证明了所提出的迭代方法具有四阶收敛性.最后,本文通过一些数值例子说明了所提出的方法的有效性和表现.     

A Super-Halley Type Approximation in Banach Spaces

The Super-Halley method is one of the best known third-order iteration for solving nonlnear equations. A Newton-like method which is an approximation of this method is studied. Our approach yields a fourth R-order iterative process which is more efficient than its classical predecessor. We establish a Newton-Kantorovich-type convergence theorem using a new system of recurrence relations, and give an explicit expression for the a priori error bound of the iteration.     

多元Bernstein-Durrmeyer型多项式及其逼近特征

作为Bernstein-Durrmeyer多项式的推广,定义单纯形上的Bernstein-Durrmeyer型多项式.以最佳多项式逼近为度量,给出Bernstein-Durrmeyer型多项式Lp逼近阶的估计,并且以一个逆向不等式的形式建立其Lp逼近的逆定理,从而用最佳多项式逼近刻画该多项式Lp逼近的特征.所获结果包含了多元Bernstein-Durrmeyer多项式的相应结果.     

Nonlinear Chebyshev approximation to set-valued functions

In this paper, we give a characterization of best Chebyshev approximation to set-valued functions from a family of continuous functions with the weak betweeness property. As a consequence, we obtain a characterization of Kolmogorov type for best simultaneous approximation to an infinity set of functions. We introduce the concept of a set-sun and give a characterization of it. In addition, we prove a property of Amir–Ziegler type for a family of real functions and we get a characterization of best simultaneous approximation to two functions     

由斜坡函数激发的神经网络算子逼近

首先,引入一种由斜坡函数激发的神经网络算子,建立了其对连续函数逼近的正、逆定理,给出了其本质逼近阶.其次,引入这种神经网络算子的线性组合以提高逼近阶,并且研究了这种组合的同时逼近问题.最后,利用Steklov函数构造了一种新的神经网络算子,建立了其在L~p[a,b]空间逼近的正、逆定理.     

Nonlinear Approximation with Dictionaries I. Direct Estimates

We study various approximation classes associated with m-term approximation by elements from a (possibly redundant) dictionary in a Banach space. The standard approximation class associated with the best m-term approximation is compared to new classes defined by considering m-term approximation with algorithmic constraints: thresholding and Chebychev approximation classes are studied, respectively. We consider embeddings of the Jackson type (direct estimates) of sparsity spaces into the mentioned approximation classes. General direct estimates are based on the geometry of the Banach space, and we prove that assuming a certain structure of the dictionary is sufficient and (almost) necessary to obtain stronger results. We give examples of classical dictionaries in L^p spaces and modulation spaces where our results recover some known Jackson type estimates, and discuss some new estimates they provide.     

On approximation of functions by algebraic polynomials in Hölder spaces

We study approximation of functions by algebraic polynomials in the Hölder spaces corresponding to the generalized Jacobi translation and the Ditzian–Totik moduli of smoothness. By using modifications of the classical moduli of smoothness, we give improvements of the direct and inverse theorems of approximation and prove the criteria of the precise order of decrease of the best approximation in these spaces. Moreover, we obtain strong converse inequalities for some methods of approximation of functions. As an example, we consider approximation by the Durrmeyer–Bernstein polynomial operators.