Newton polygons and local integrability of negative powers of smooth functions in the plane

Let be any smooth real-valued function with . For a sufficiently small neighborhood of the origin, we study the number

It is known that sometimes this number can be expressed in a natural way using the Newton polygon of . We provide necessary and sufficient conditions for this Newton polygon characterization to hold. The behavior of the integral at the supremal is also analyzed.

     

与给定多边形相切的一类含参数三角样条曲线

基于一类与给定多边形相切的三角样条曲线,通过在基函数中引入形状参数λ,在保持原曲线的光滑性及其他基本性质不变的条件下,构造出一类能自由调控曲线形态的含参数三角样条曲线,并结合图例讨论了其相关性质.     

Calculating the Coefficients of the Expansion in Derived Keldysh Chains for an Elliptic Problem with Parameter in the Boundary Condition

Mathematical Notes -     

Asymptotics of Solutions for Nonlocal Elliptic Problems in Plane Angles

In the paper we investigate asymptotics of solutions for nonlocal elliptic problems in plane angles and in ² \ {0}. These problems arise in studying nonlocal problems in bounded domains in the case where support of nonlocal terms intersects with a boundary of a domain. We obtain explicit formulas for the asymptotic coefficients in terms of eigenvectors and associated vectors of both adjoint nonlocal operators acting in spaces of distributions and formally adjoint (with respect to the Green formula) nonlocal transmission problems.     

Directional Newton methods in variables

Directional Newton methods for functions of variables are shown to converge, under standard assumptions, to a solution of . The rate of convergence is quadratic, for near-gradient directions, and directions along components of the gradient of with maximal modulus. These methods are applied to solving systems of equations without inversion of the Jacobian matrix.

     

R~n上一类半线性椭圆方程解的存在唯一性和渐近性质

用上下解方法和位势估计，研究Rn上具有次线性项加超线性项半线性椭圆方程给出了其有界正解的存在性、唯一性和渐近性质，其中为常数，参数．     

基于第二类椭圆积分的椭圆弧长公式变换与应用

以第二类椭圆积分为理论基础,通过推导,将椭圆弧长公式变换为以椭圆离心角、极角等常用角度参数为自变量的第二类椭圆积分的标准形式,建立起椭圆弧长公式与第二类椭圆积分标准形式之间的关系,并分析了椭圆上的弧微分变化规律及椭圆周长与离心率的变化关系.公式反映了椭圆弧长的本质问题即为第二类椭圆积分问题.因此,各类涉及椭圆弧长计算的应用问题,均可化为第二类椭圆的计算问题,应用时直接调用各类编程软件的函数库中的第二类椭圆积分函数,无需复杂编程即可实现椭圆弧长的高精度计算.文章以GPS采用的WGS-84椭球子午线弧长为例进行计算分析,验证了给出的公式及相关分析的正确性及应用价值.     

Hölder Regularity for Nonhomogeneous Elliptic Systems with Nonlinearity Greater than Two

Regularity results for elliptic systems of second order quasilinear PDEs with nonlinear growth of order q > 2 are proved, extending results of [7] and [10]. In particular Hölder regularity of the solutions is obtained if the dimension n is less than or equal to q + 2.     

On the Finite Element Analysis of Problems with Nonlinear Newton Boundary Conditions in Nonpolygonal Domains

The paper is concerned with the study of an elliptic boundary value problem with a nonlinear Newton boundary condition considered in a two-dimensional nonpolygonal domain with a curved boundary. The existence and uniqueness of the solution of the continuous problem is a consequence of the monotone operator theory. The main attention is paid to the effect of the basic finite element variational crimes: approximation of the curved boundary by a polygonal one and the evaluation of integrals by numerical quadratures. With the aid of some important properties of Zlamal's ideal triangulation and interpolation, the convergence of the method is analyzed.     

An Adaptive Subdivision Algorithm for the Identification of the Diffusion Coefficient in Two-dimensional Elliptic Problems

In this work, we consider the identification problem of the diffusion coef-ficient in two-dimensional elliptic equations. For parameterization, we use the zonation method: the diffusion coefficient is assumed to be a piecewise constant space function and unknowns are both the diffusion coefficient values and the geometry of the zones. An algorithm based on geometric principles is developed in order to determine the boundaries between the zones. This algorithm uses the refinement indicators which are easily computed from the gradient of the objective function. The efficiency of the algorithm is proved by testing it in some simple cases with and without noise on the data.      

The Nonlinear Boundary Value Problems of Three Elements for the First Order Quasilinear Elliptic Systems

In this paper, we discuss the nonlinear boundary value problems of three elements with two shifts for the first order quasilinear elliptic systems and the related solvability by using the continuity method.     

A globally convergent inexact Newton method with a new choice for the forcing term

In inexact Newton methods for solving nonlinear systems of equations, an approximation to the step s _k of the Newton’s system J(x _k )s=−F(x _k ) is found. This means that s _k must satisfy a condition like ‖F(x _k )+J(x _k )s _k ‖≤η _k ‖F(x _k )‖ for a forcing term η _k ∈[0,1). Possible choices for η _k have already been presented. In this work, a new choice for η _k is proposed. The method is globalized using a robust backtracking strategy proposed by Birgin et al. (Numerical Algorithms 32:249–260, 2003), and its convergence properties are proved. Several numerical experiments with boundary value problems are presented. The numerical performance of the proposed algorithm is analyzed by the performance profile tool proposed by Dolan and Moré (Mathematical Programming Series A 91:201–213, 2002). The results obtained show a competitive inexact Newton method for solving academic and applied problems in several areas. Supported by FAPESP, CNPq, PRONEX-Optimization.     

带有边界摄动的非线性椭圆型方程奇摄动问题

The nonlinear singularly perturbed problems for elliptic equations with boundary perturbation are considered. Under suitable conditions, by using the theory of differential inequalities the asymptotic behavior of solutions for the boundary value problems is studied.     

一类具有边界摄动的非线性泛函椭圆型方程奇摄动问题

研究了具有边界摄动的非线性泛函椭圆型方程奇摄动边值问题．在适当的条件下．利用伸长变量、微分不等式理论，讨论了问题解的渐近性态和原问题解的存在唯一性。     

具有转向点的椭圆型方程奇摄动边值问题

本文讨论了具有转向点的奇摄动椭动椭圆型方程边值问题,利用多重尺度法和比较定理,确定了边值问题解的渐近性态。     

Globally convergent Jacobian smoothing inexact Newton methods for NCP

A new smoothing algorithm for the solution of nonlinear complementarity problems (NCP) is introduced in this paper. It is based on semismooth equation reformulation of NCP by Fischer–Burmeister function and its related smooth approximation. In each iteration the corresponding linear system is solved only approximately. Since inexact directions are not necessarily descent, a nonmonotone technique is used for globalization procedure. Numerical results are also presented. Research supported by Ministry of Science, Republic of Serbia, grant No. 144006.     

影响阻尼牛顿法收敛性的两个重要参数

对阻尼牛顿算法作了适当的改进,证明了新算法的收敛性.基于新算法,运用计算机代数系统Matlab,研究了迭代次数k,参数对(μ,λ)与初值x0三者间的依赖关系,研究了病态问题在新算法下趋于稳定的渐变(瞬变)过程.数值结果表明:(1)阻尼牛顿迭代中,参数对(μ,λ)与迭代次数k间存在特有的非线性关系;(2)适当的参数对(μ,λ)与阻尼因子α的共同作用能够在迭代中大幅度地降低病态问题的Jacobi阵的条件数,使病态问题逐渐趋于稳定,从而改变原问题的收敛性与收敛速度.     

椭圆曲线上的基的选择与实现

在研究了椭圆曲线y2≡x3+ax+b(modp)上的点的一些性质后,提出了基的计算、验证、选择等算法,同时研究了基的选择、点积效率、安全性三者之间的关系.最后实现了以椭圆曲线基的选择.     

一类次线性椭圆方程组正解的存在唯一性

考虑半线性椭圆方程组■(1)其中A>0,Ω是有界光滑区域.f,g是定义在R_+~2:=[0,∞)×[0,∞)上的实值函数讨论在满足什么条件下此半线性椭圆方程组存在唯一的正解.