基于无限度量的一元粗糙函数及其数学分析性质

一元粗糙函数及其数学分析性质具有意义,但当前研究主要局限于有限度量.基于无限度量研究一元粗糙函数及其数学分析性质.将度量从有限集扩展到无限集,讨论粗糙函数分类;基于无限度量研究粗糙函数的粗糙连续、粗糙极限、粗糙导数.采用无限度量,推进了一元粗糙函数及其分析性质.     

一个有趣的二元函数

在一元函数中,“可导”和“可微”是等价的,“可微”的重要性似乎不大明显.在二元函数中情形就不一样了.“两个偏导数都存在”不能保证可微,甚至不能保证连续.也不能保证有极限;由可微却可以得到上述所有其它性质.还可以保证有方向导数.所以,对于二元函数,“可微”和“有两个偏导数”不等价,“可微”有重要的作用.     

本质有穷的全纯函数及其导数的Borel例外值

本文对于复平面上的全纯函数,推广通常的增长级为p阶增长级,引进本质有穷的概念,进而研究本质有穷的全纯函数及其导数的Borel例外值的存在性.将通常意义下有限级全纯函数的Hadamard因子分解定理推广到本质有穷的全纯函数上来,在此基础上,将熟知的Borel例外值定理推广到本质有穷全纯函数的情形.然后,将Milloux等人关于全纯函数及其导数的Picard值的存在性定理推广为本质有穷的全纯函数及其导数的Borel例外值的存在性定理.     

Mathematica数学软件的图形功能在微积分中的应用问题

讨论如何正确使用Math.的图形功能,对微积分中二元函数在一点极限不存在的情形及二元函数在点不连续但偏导数存在的情形给出几何解释。     

关于导数极限定理的几个应用及其推广

本文首先利用导数极限定理,总结了一些导函数分析性质,并举例说明各种类型间断点在导函数中出现的可能性,最后将导函数极限定理推广到二元函数情形.     

纵向数据变系数模型中的减元估计法

纵向数据变系数模型常应用于传染病学、生物医学和环境科学等领域. 本文提出了一种称为减元估计法的方法来估计模型中的未知函数和它们的导数. 减元估计法既适用于系数函数具有相同光滑度的情形, 也适用于系数函数具有不同光滑度的情形; 既适用于变量不依赖于时间的情形, 也适用于变量依赖于时间的情形. 给出了一般条件下估计量的局部渐近偏差、方差和渐近正态性, 并且渐近性结果显示: 当系数函数具有不同的光滑度时, 减元估计量的渐近方差比现有方法得到的估计量的渐近方差要少. 本文还通过 Monte Carlo 模拟研究了估计量的有限样本性质.     

聚焦导数定义

导数是高中数学的新增内容,为高中数学 注入了新的活力,利用导数可从更深的角度来 研究函数的性质.本文聚焦导数定义,对平时 遇到的有关导数概念的题目分类分析,以期加 深对导数概念的理解和应用. f(x)在点x0处的导数:     

单调函数判别法的推广

将单调函数的判别法由在开区间内有有限个导数为零,推广到开区间内有可数(列)个导数为零的情形,从而扩大了判别法的应用范围     

四阶抛物偏微分方程的H1-Galerkin混合元方法及数值模拟

到目前为止, H¹-Galerkin 混合有限元方法研究的问题仅局限于二阶发展方程. 然而对于高阶发展方程, 特别是重要的四阶发展方程问题的研究却没有出现. 本文首次提出四阶发展方程的H¹-Galerkin 混合有限元方法, 为了给出理论分析的需要, 我们考虑四阶抛物型发展方程. 通过引进三个适当的中间辅助变量, 形成四个一阶方程组成的方程组系统, 提出四阶抛物型方程的H¹-Galerkin 混合有限元方法. 得到了一维情形下的半离散和全离散格式的最优收敛阶误差估计和多维情形的半离散格式误差估计, 并采用迭代方法证明了全离散格式的稳定性. 最后, 通过数值例子验证了提出算法的可行性. 在一维情况下我们能够同时得到未知纯量函数、一阶导数、负二阶导数和负三阶导数的最优逼近解, 这一点是以往混合元方法所不能得到的.     

具有Caputo导数的分数阶退化脉冲微分系统的有限时间稳定性

本文通过构建新的Lyapunov泛函,并利用Caputo导数的相关性质以及广义的Gronwall不等式研究了同时带有扰动和脉冲因素的分数阶退化线性系统在Caputo导数意义下的有限时间稳定性问题.在此基础上给出了在没有扰动的情形下分数阶退化脉冲微分系统的有限时间稳定性的判据,所获得的结果推广了相关文献的结论.最后针对不同的情况给出具体数值例子验证了定理条件的有效性.     

On Discrete Limits of Sequences of Approximately Continuous Functions and tae-Continuous Functions

We show that the classes of all discrete limits of sequences of ap- proximately continuous functions, of all discrete limits of sequences of derivatives and of all discrete limits of sequences of Baire 1 functions are the same. We describe also the discrete limits of sequences of quasicontinuous functions, and of sequences of almost everywhere continuous functions, and we present anec- essary condition which must be satisfied by the discrete limits of sequences of T_ae -continuous functions.     

Constructions of strict Lyapunov functions for discrete time and hybrid time-varying systems

We provide explicit closed form expressions for strict Lyapunov functions for time-varying discrete time systems. Our Lyapunov functions are expressed in terms of known nonstrict Lyapunov functions for the dynamics and finite sums of persistency of excitation parameters. This provides a discrete time analog of our previous continuous time Lyapunov function constructions. We also construct explicit strict Lyapunov functions for systems satisfying nonstrict discrete time analogs of the conditions from Matrosov’s Theorem. We use our methods to build strict Lyapunov functions for time-varying hybrid systems that contain mixtures of continuous and discrete time evolutions.     

Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition

An essential feature of nonstandard finite difference schemes for differential equations is the precise manner in which the discretization of derivatives is made. We demonstrate, for differential equations modeling systems where the solutions satisfy a positivity condition, that procedures can be formulated to calculate the so‐called denominator functions that appear in the discrete derivatives. These procedures are applied to a number of both ordinary and partial model differential equations to illustrate their use. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

A product formula for orthogonal polynomials associated with infinite distance-transitive graphs

The infinite, locally finite distance-transitive graphs form an extension of homogeneous trees and are described by two discrete parameters. The associated orthogonal polynomials may be regarded as spherical functions of certain Gelfand pairs or as characters of some polynomial hypergroups; they are certain Bernstein polynomials and admit a discrete nonnegative product formula. In this paper we use the graph-theoretic origin of these polynomials to derive the existence of positive dual continuous product and transfer formulas. The dual product formulas will be computed explicitly.     

HOW TO PROVE THE DISCRETE RELIABILITY FOR NONCONFORMING FINITE ELEMENT METHODS

Optimal convergence rates of adaptive finite element methods are well understood in terms of the axioms of adaptivity.One key ingredient is the discrete reliability of a residualbased a posteriori error estimator,which controls the error of two discrete finite element solutions based on two nested triangulations.In the error analysis of nonconforming finite element methods,like the Crouzeix-Raviart or Morley finite element schemes,the difference of the piecewise derivatives of discontinuous approximations to the distributional gradients of global Sobolev functions plays a dominant role and is the object of this paper.The nonconforming interpolation operator,which comes natural with the definition of the aforementioned nonconforming finite element in the sense of Ciarlet,allows for stability and approximation properties that enable direct proofs of the reliability for the residual that monitors the equilibrium condition.The novel approach of this paper is the suggestion of a right-inverse of this interpolation operator in conforming piecewise polynomials to design a nonconforming approximation of a given coarse-grid approximation on a refined triangulation.The results of this paper allow for simple proofs of the discrete reliability in any space dimension and multiply connected domains on general shape-regular triangulations beyond newest-vertex bisection of simplices.Particular attention is on optimal constants in some standard discrete estimates listed in the appendices.     

Backward Euler Scheme,Singular Hölder Norms,and Maximal Regularity for Parabolic Difference Equations

We show finite difference analogues of maximal regularity results for discretizations of abstract linear parabolic problems. The involved spaces are discrete versions of spaces of Hölder continuous functions, which can be singular in 0. The main tools are real interpolation and Da Prato–Grisvard's theory of the sum of linear operators.     

On a high order numerical method for functions with singularities

By splitting a given singular function into a relatively smooth part and a specially structured singular part, it is shown how the traditional Fourier method can be modified to give numerical methods of high order for calculating derivatives and integrals. Singular functions with various types of singularities of importance in applications are considered. Relations between the discrete and the continuous Fourier series for the singular functions are established. Of particular interest are piecewise smooth functions, for which various important applications are indicated, and for which numerous numerical results are presented.

     

Green's functions for elliptic and parabolic equations with random coefficients II

This paper is concerned with linear parabolic partial differential equations in divergence form and their discrete analogues. It is assumed that the coefficients of the equation are stationary random variables, random in both space and time. The Green's functions for the equations are then random variables. Regularity properties for expectation values of Green's functions are obtained. In particular, it is shown that the expectation value is a continuously differentiable function in the space variable whose derivatives are bounded by the corresponding derivatives of the Green's function for the heat equation. Similar results are obtained for the related finite difference equations. This paper generalises results of a previous paper which considered the case when the coefficients are constant in time but random in space.

     

Les derivees partielles des fonctions pseudo-booleennes generalisees

This paper is devoted to a study of differential calculus for generalised pseudo-Boolean functions with finite domain and antidomain P with a ring structure which may be infinite. A completely new definition is given of partial derivatives of generalised pseudo-Boolean functions $\text{?f}{}_{\mathrm{q}}\text{?x}{}_{\mathrm{i}}\text{=f(x}{}_1\text{,…,x}{}_{\mathrm{i?1}}\text{,a,x}{}_{\mathrm{i+1}}\text{,…,x}{}_{\mathrm{n}}\text{)?f(x}{}_1\text{,…,x}{}_{\mathrm{n}}\text{)}$. Further are given some properties of these partial derivatives and a new Taylor expansion.M. Davio, J.P. Deschamps, A. Thayse, Belgian mathematicians, have studied the differential calculus for discrete functions with finite domain and antidomain which is a totally ordered lattice 0<1<?<m?1 with a structure of a ring of integers modulo m. If P is finite, then generalised pseudo-Boolean functions are discrete functions, but the partial derivatives of generalised pseudo-Boolean functions are different from the partial derivatives of discrete functions which were studied by the Belgian mathematicians.