无穷小量在1~∞型极限中的应用

通过无穷小量讨论1∞型极限是否存在,并且介绍一种用无穷小量的等价代换快速求解1∞型极限的方法     

无穷小量在1∞型极限中的应用

通过无穷小量讨论1^∞型极限是否存在,并且介绍一种用无穷小量的等价代换快速求解1^∞型极限的方法。     

极限lim x→＋∞f（x）=0的一个充分条件的探讨

从无穷积分∫＋∞ a f（x）dx收敛与无穷远极限lim x→＋∞f（x）=0之间的关系展开论述,研究在广义积分∫＋∞ a f（x）dx收敛的前提下,无穷远极限lim x→＋∞f（x）=0的一个充分条件．在此基础上,适当减弱条件得到该条件的推广形式,为更好的解决无穷远极限lim x→＋∞f（x）=0的问题提供更一般的方法．     

再论无穷多个无穷小量的乘积

对文[6]提出的质疑给出回答,表明由于不同的无穷小量趋近于0的速度有快有慢,因此无穷多个无穷小量的乘积∏∞k=1{x_n~(k)}∞n=1,有可能不是无穷小量(其中对每个正整数k,{x_n~(k)}_(n=1)~∞表示极限为0的数列),而验证∏∞k=1{x_n~(k)}∞n=1是否是无穷多个无穷小量的乘积,只需验证对每个正整数k,当n→+∞时,{x_n~(k))_(n=1)~∞是否趋近于0,而无需考虑函数列{{x_n~(k)}_(n=1)~∞}_(k=1)~∞的极限limk→∞x_n~(k)是不是无穷小量.进而,对无穷多个无穷小量的乘积是无穷小量或不是无穷小量给出了一些充分条件,     

无穷限反常积分收敛时被积函数的极限状态

根据无穷限反常积分∫a^＋∞f（x）dx收敛的柯西准则和定积分的性质,讨论被积函数f（x）当x→∞时。的极限状态,并得出当无穷限反常积分∫a^＋∞f（x）dx收敛且f（x）在[a,＋∞）上连续,或者无穷限反常积分∫a^＋∞f（x）dx绝对收敛时,存在数列{xn}∩[a,＋∞]且xn→＋∞（n→∞）,使limn→∞xnf（xn）=0.     

等价代换在极限计算中的应用

在计算极限时，如能正确使用等价代换，会使问题大为简化，但是学生在使用这种方法时经常出现这样或那样的错误，针对这种情况，本文重点介绍等价代换在极限计算中的应用。先介绍几个有关概念。若lima=0（∞），limβ=0（∞），且（C为任何实数或无穷大），则称α与β是该过程下可以比较的无穷小（大）。特别地，若limα=0（∞），limβ=0（∞），且（α为常数，且a／0，1），则称α与β是该过程下的同阶无穷小（大）。若lima—0（co），limP二0（co），且tim舌21，则称a与卢是该过程下的等价无穷小—“““““－””一”’“““““””…     

∞^0型极限为1的充分条件

综合使用洛必达法则及等价无穷小代换的方法,可得到关于一般抽象函数的∞^0型极限为1的两个充分条件,最后借助实例展示其应用便捷性．     

关于利用等价无穷小代换来求极限的探讨

一、利用等价无穷小代换来求极限的一些容易证明的定理定理1设无穷小量f（x）～（x），且limf（x）·g（x）存在，则这里，（1）无穷小量f（x）～（x），表示f（x）与（x）是当x→x0或x→∞时的等价无穷小；（2）limf（x）表示limf（x）或limf（x）．下同。定理2设无穷小量f（x）～（x），且存在，则由这二个定理可知，一般在乘或除的情况下是可用等价无穷小代换来求极限的。此外在幂指函数求极限中，也常利用等价无穷小代换，这有下面二个定理，这里只证后一个定理。定理3设八x）＞0，无穷小量g（x）～～（x），且tim八x）”“’存在，则定…     

利用Taylor公式选择恰当的等价无穷小

在求函数极限的过程中,利用等价无穷小代换常常会使计算简单,但对形如的极限,若无穷小量有时权限并不相等.这里的关键是与的选取不当．本文着重讨论这种情况下利用Taylor公式选取适当的等价无穷小量作代换保持权限不变．为叙述方便,我们只讨论的情况．同时假定下文中所涉及的都是当时的连续且具有任意阶导数的函数无穷小量,以后不再交代．引理1若引理2若则,这里不全为0引理3若a在处的Taylor展开式（带皮亚诺余项）为（按前面假设,常数项为0）：证明显然,从而实际上,通常我们所用的等价无穷小都是取Tayfor展式的第一个非零项．如…     

不定式定值法的几点注记

1.等价无穷小代换问题在求0/0型不定式的极限过程中,有时为了方便运算,而进行等价无穷小代换,但当0/0型不定式的分子(或分母)是两个无穷小量相加减时,应如何代换?我们分4种情况来归纳这个问题.1.0/0型不定式的分子(或分母)是两个无穷小量a_1、a_2相加,并且当lim（a_1/a_2）≠-1时,可分别用各自的等价无穷小代换,特别是相加的两项中一项比另一项高阶时,可以删去高阶项(其结果都相当于把分子(或分母)作为整体进行了等价无穷小代换).     

Qualitative Analysis of Bifurcating Solutions in the Lengyel-Epstein Model

One of the most fundamental problems in theoretical biology is to explain the mechanisms by which patterns and forms are created in the＇living world. In his seminal paper ＂The Chemical Basis of Morphogenesis＂, Turing showed that a system of coupled reaction-diffusion equations can be used to describe patterns and forms in biological systems. However, the first experimental evidence to the Turing patterns was observed by De Kepper and her associates（1990） on the CIMA reaction in an open unstirred reactor, almost 40 years after Turing＇s prediction. Lengyel and Epstein characterized this famous experiment using a system of reaction-diffusion equations. The Lengyel-Epstein model is in the form as follows     

(0,1 ;0)-INTERPOLATION ON INFINITE INTERVAL (-∞, +∞)

In this paper, we study the explicit representation and convergence of (0, 1; 0)-interpolation on infinite interval, which means to determine a polynomial of degree ≤ 3n - 2 when the function values are prescribed at two set of points namely the zeros of Hn(x) and H′n(x) and the first derivatives at the zeros of H′n(x).     

RECENT PROGRESS ON SPHERICAL HARMONIC APPROXIMATION MADE BY BNU RESEARCH GROUP ——In memory of Professor Sun Yongsheng

As early as in 1990, Professor Sun Yongsheng, suggested his students at Beijing Normal University to consider research problems on the unit sphere. Under his guidance and encouragement his students started the research on spherical harmonic analysis and approximation. In this paper, we incompletely introduce the main achievements in this area obtained by our group and relative researchers during recent 5 years （2001-2005）. The main topics are： convergence of Cesaro summability, a.e. and strong summability of Fourier-Laplace series; smoothness and K-functionals; Kolmogorov and linear widths.     

On a class of self-similar processes with stationary increments in higher order Wiener chaoses

We study a class of self-similar processes with stationary increments belonging to higher order Wiener chaoses which are similar to Hermite processes. We obtain an almost sure wavelet-like expansion of these processes. This allows us to compute the pointwise and local Hölder regularity of sample paths and to analyse their behaviour at infinity. We also provide some results on the Hausdorff dimension of the range and graphs of multidimensional anisotropic self-similar processes with stationary increments defined by multiple Wiener–Itô integrals.     

A REPRESENTATION FORMULA RELATED TO SCHR(O)DINGER OPERATORS

Schr(o)dinger operator is a central subject in the mathematical study of quantum mechanics.Consider the Schrodinger operator H = -△ V on R, where △ = d2/dx2 and the potential function V is real valued. In Fourier analysis, it is well-known that a square integrable function admits an expansion with exponentials as eigenfunctions of -△. A natural conjecture is that an L2 function admits a similar expansion in terms of "eigenfunctions" of H, a perturbation of the Laplacian (see [7], Ch. Ⅺ and the notes), under certain condition on V.     

On a class of Riemann surfaces

It is considered the class of Riemann surfaces with dimT1 = 0, where T1 is a subclass of exact harmonic forms which is one of the factors in the orthogonal decomposition of the spaceΩH of harmonic forms of the surface, namely The surfaces in the class OHD and the class of planar surfaces satisfy dimT1 = 0. A.Pfluger posed the question whether there might exist other surfaces outside those two classes. Here it is shown that in the case of finite genus g, we should look for a surface S with dimT1 = 0 among the surfaces of the form Sg\K , where Sg is a closed surface of genus g and K a compact set of positive harmonic measure with perfect components and very irregular boundary.     

Journal of the Operations Research Society of China Special issue on Operations Research in Healthcare Management

正Guest Editors:Hong Chen,Shanghai Jiao Tong University,Shanghai,China Guohua Wan,Shanghai Jiao Tong University,Shanghai,China David Yao,Columbia University,New York,USA Scope:Healthcare delivery worldwide has been fraught with high cost,low efficiency and poor quality of patient care service.For the field of operations research(OR),healthcare offers some of the biggest challenges as well as best opportunities in     

ANALYSIS IN THEORY AND APPLICATIONS Vol. 30, 2014 CONTENTS

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An alternative mathematical algorithm for the photo- and videokeratoscope

Due to the resolution of current laser technology, the accuracy of corneal topography as measured by the videokeratoscope is no longer adequate to provide precise enough data for refractive surgery or for the fitting of customized contact lenses. We present an algorithm for recovering corneal topography that makes use of modern differential geometric techniques and numerical descent in Sobolev spaces. We believe this algorithm may be used with the photo- and videokeratoscope to increase the accuracy of the recovered corneal topography.