函数在一点可导的等价形式

通过对一道考研试题的推广,得到函数在某点的可导的一个等价形式,即若函数 f （x）在 x ＝0处连续,且 f （0）＝0,limx f （x）- af （bx）→0 x＝ K ,其中0＜｜ ab｜≠1,0＜｜ b｜≤1,且 f （x）在 x ＝0处满足Lipschitz条件,则有 f′（0）＝ K 1＋ ab 。     

涉及双中值点存在性的一道习题的另证

有别于一般文献所使用的构造辅助函数方法,针对在[0,1]上连续,在（0,1）上可导,且满足f （0）＝0,f （1）＝1的函数 f （x）,先用反证法可证,存在 a ,b ∈（0,1）,使得 f′（a）＜1＜ f′（b）,进而利用导函数的介值性可证,存在ξ,η∈（0,1）,使得 f′（ξ） f′（η）＝1（ξ≠η）。     

再谈柯西中值定理

将柯西中值定理改叙并证明之：如果f(x)和F(x)在闭区间[a，b]上连续，在(a，b)内可导，且F（a)≠F(b),则在（a,b)内至少存在一点ξ,使f′（ξ）=f(b)-f(a)/F(b)-F(a) F′（ξ），进一步地，若F′（ξ）≠0,则有f(b)-f(a)/F(b)-F(a)=f′（ξ)/F′（ξ）。     

一类四阶两点边值问题的可解性

本文考虑如下四阶完全拟线性两点过值问题：y（4）＝f（x，y，y′，y″，y），y（0）＝y″（0）＝y′（1）＝y（1）＝0，给出了不限制f增长的可解性条件。     

河南省暨重庆市(1998)高中数学竞赛试题及解答

一、选择题：（本题满分30分，每小题5分）1．在等差数列｛an｝中，若a4＋a6＋a8＋a10＋a12＝120，则2a9－a10的值为（C）2．已知椭圆的两个焦点分别为F1，F2，过右焦点F2作倾角为的弦AB，则△ABF1的面积为（B）．5．一半径为a的半球内切于顶角为90°的圆锥，半球的底面在圆锥的底面内，则V半球：V圆锥等于（A）6．函数f（x）是定义在R上的奇函数，且f（2）＝0，对任意x∈R，都有f（x＋4）＝f（x）＋f（2）成立．则f（1998＝（D）．即f（x）是以4为周期的函数．∴f（1998）＝f（499×4＋2）＝f（2）＝0．二、填空题（本题满分30分，…     

例谈极值问题转化的策略

函数y=f（x）在点x=a的函数值f（a）比它在点x=a附近其它点的函数值都大（都小）,f′（a）=0,而且在点x=a附近的左侧f′（x）〈0（f′（x）〉0）,右侧f′（x）〉0（f′（x）〈0）,就把点a叫函数y=f（x）的极小值（极大值）点,f（a）叫函数y=f（x）的极小值（极大值）.可见极值点a处一定有f′（a）=0,但是f′（a）=0的点a不一定为极值点.处理极值问题除了课本上常见的列表定义判断外,     

求不定积分中某些常见错误简析

计算不定积分时，同学们往往只拘泥于会积出原函数，而忽视了一些细节问题，导致不易发现的错误结果。下面举例说明之。一、不定积分与原函数概念常见错误例亚若F’（x）＝f（x），则称F（x）是f（x）的原函数，F（x）＋C是f（x）的不定积分。析F’（x）＝f（x），未注明x所属区间。。。。jsdx。－。。。。，。。。－，。。。。。。，。“JH。x－JZ。x若理解为前一原函数族中某一特殊原函数与后一原函数族中任一原函数之差，则一般应等于一非0常数！若后者正巧取得是前一特殊原函数，那么其差才为0。二、换元积分中易出现的错误析今x—t…     

一个二阶三点边值问题的可解性

本文考虑如下这位问题：x″＝f（t，x，x′）（0＜t＜1），x（0）＝0，x（ξ）＝x（1），其中ξ∈（0，1）是给定的．利用基于度理论的一定不动点定理，得到了以上过值问题有解的某些充分条件     

函数的右导数与导函数的右极限的关系

通过例子和定理讨论了函数f（x）在点x0处的右导数f′＋（x0）与导函数当x→x0^＋时的右极限f′（x0^＋）=limx→x0^＋f′（x）之间的关系．     

拉格朗日定理的一个推广

利用条件极值判别法和连续函数的介值性定理，通过构造辅助函数获得拉格郎日定理的一个推广，即若 f（x）在（a ，b）内2n次可导（n≥2，n∈Z），f （2n）（ξ）≠0，f （3）（ξ）＝ f （4）（ξ）＝…＝ f （2n-1）（ξ）＝0（a＜ξ＜ b），则存在 a1，b1∈（a ，b），使得 f（b1）- f （a1）＝ f′（ξ）（b1- a1）。     

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.     

CONTENTS OF VOLUME 30

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Instructions for contributions

正Applied Mathematics-A Journal of Chinese Universities,Series B(Appl.Math.J.Chinese Univ.,Ser.B)is a comprehensive applied mathematics journal jointly sponsored by Zhejiang University,China Society for Industrial and Applied Mathematics,and Springer-Verlag.It is a quarterly journal with     

Journal of Mathematical Research with Applications Guide for Authors

正Journal overview:Journal of Mathematical Research with Applications(JMRA),formerly Journal of Mathematical Research and Exposition(JMRE)created in 1981,one of the transactions of China Society for Industrial and Applied Mathematics,is a home for original research papers of the highest quality in all areas of mathematics with applications.The target audience comprises:pure and applied mathematicians,graduate students in broad fields of sciences and technology,scientists and engineers interested in mathematics.     

Staircase transportation problems with superadditive rewards and cumulative capacities

A cumulative-capacitated transportation problem is studied. The supply nodes and demand nodes are each chains. Shipments from a supply node to a demand node are possible only if the pair lies in a sublattice, or equivalently, in a staircase disjoint union of rectangles, of the product of the two chains. There are (lattice) superadditive upper bounds on the cumulative flows in all leading subrectangles of each rectangle. It is shown that there is a greatest cumulative flow formed by the natural generalization of the South-West Corner Rule that respects cumulative-flow capacities; it has maximum reward when the rewards are (lattice) superadditive; it is integer if the supplies, demands and capacities are integer; and it can be calculated myopically in linear time. The result is specialized to earlier work of Hoeffding (1940), Fréchet (1951), Lorentz (1953), Hoffman (1963) and Barnes and Hoffman (1985). Applications are given to extreme constrained bivariate distributions, optimal distribution with limited one-way product substitution and, generalizing results of Derman and Klein (1958), optimal sales with age-dependent rewards and capacities.To our friend, Philip Wolfe, with admiration and affection, on the occasion of his 65th birthday.Research was supported respectively by the IBM T.J. Watson and IBM Almaden Research Centers and is a minor revision of the IBM Research Report [6].