一道数学竞赛题的另解及推广

针对第67届美国大学生数学竞赛中一道求递推数列极限的试题,运用Stolz定理、Lagrange中值定理和夹逼准则给出一种新解法,并指出原题还可做进一-推广.     

洛必达(L’Hospital)法则在求[1~∞]型极限中的应用

给出了求不定式[1∞]型极限的一个定理,同时通过历年考研试题的应用,说明此定理是有效的、简单的.     

有关二项式定理的高考试题综述

有关二项式定理的高考试题综述２０１５００上海市金山县中学戴丽萍在历年高考中，每年都有涉及二项式定理的试题，本文结合近年来高考试题，根据有关二项式定理给出的不同形式，进行分类并作出解法探讨，以揭示这类问题的玉般解题规律．１求二项展开式中的条件项求二项展...     

圆锥曲线切点弦方程的应用

本文先给出圆锥曲线切点弦所在直线方程,然后证明两个圆锥曲线的一般性质,并利用它们解决一些高考试题和竞赛试题.定理1已知一个圆锥曲线的一般方程为     

一道硕士研究生入学试题的六种证法

费马定理,洛尔定理,拉格朗日中值定理,达布定理,泰勒公式,凸函数,这些内容几乎覆盖了一元函数微分学,应用这些知识,对吉林大学2009年硕士研究生入学考试试题中的一道试题提供六种证法.     

对一道研究生入学试题的四种解答

分别利用洛必达法则,罗尔定理,拉格朗日中值定理等知识,对2007年黑龙江大学硕士研究生入学考试的一道试题给出四种解答,即证明了 f （x）是常函数,如果 f （x）是实数域上可微的周期函数且其导函数在无穷大处存在极限。     

解答三角形构成问题的常用方法

在国内外数学竞赛试题中常常出现一类关于三角形构成问题的试题,这类试题所涉及的知识面较广,解答时需要一定的技巧和方法,通过对有关试题的仔细分析、研究后,现举出若干种方法例说如下。一、定理法所谓定理法就是利用构成三角形的有定理直接解答问题的方法。此方法对许多问题具有较强的实用性。     

从一道硕士研究生考试试题谈起

从2004年哈尔滨工业大学硕士研究生入学考试试题中的一道题目入手,分别利用柯西准则,有限覆盖定理,一致收敛的定义对其给出了三种证明方法.并以此为基础,揭示了几类典型题目之间的内在联系.     

一道2012年北京大学保送生试题的推广及简证

2012年北京大学保送生测试数学部分的第3题为:题目已知f(x)为一元二次函数,且a,f(a),f(f(a)),f(f(f(a)))为正项等比数列.求证:f(a)=a.这是一道构思精巧的试题,涉及到一元二次函数、等比数列甚至复合函数、不动点等概念,很耐人寻味.文[1]利用一元二次函数的性质,给出试题的一种证法.本文将试题推广为下面的定理,并给出一个简证.     

张角定理在解三角形中的妙用

<正>解三角形是高考试题中的必考点,而解三角形的方法多种多样,若仅仅依靠正、余弦定理,有时并不能高效解题．对于一些具有某种特征的问题,除了采用正、余弦定理来解决外,还可以利用张角定理,以此来达到事半功倍的效果．下面通过一道联考题两种解法比较,突显张角定理的“妙”,以及给出应用张角定理解题的相关例子．1解法比较     

A Dirichlet problem for polyharmonic functions

In this article, the Dirichlet problem of polyharmonic functions is considered. As well the explicit expression of the unique solution to the simple Dirichlet problem for polyharmonic functions is obtained by using the decomposition of polyharmonic functions and turning the problem into an equivalent Riemann boundary value problem for polyanalytic functions, as the approach to find the kernel functions of the solution for the general Dirichlet problem is given. Project supported by NNSF of China.     

The Alexander transform of a spirallike function

We show that the Alexander transform of a β-spirallike function is univalent when cosβ?1/2, which settles the problem posed by Robertson. We also solved a problem considered by Y.J. Kim and Merkes.     

带新NCP函数的Lagrangian乘子方法

在经营管理、工程设计、科学研究、军事指挥等方面普遍存在着最优化问题,而实际问题中出现的绝大多数问题都被归纳为非线性规划问题之中。作为带等式、不等式约束的复杂事例,最优化问题的求解向来较为繁琐、困难。适当条件下,非线性互补函数(NCP)可以与约束优化问题相结合,其中NCP函数的无约束极小解对应原约束问题的解及其乘子。本文提出了一类新的NCP函数用于解决等式和不等式约束非线性规划问题,结合新的NCP函数构造了增广Lagrangian函数。在适当假设条件下,证明了增广Lagrangian函数与原问题的解之间的一一对应关系。同时构造了相应算法,并证明了该算法的收敛性和有效性。     

不等式约束优化问题的低阶精确罚函数的光滑化算法

对不等式约束优化问题提出了一个低阶精确罚函数的光滑化算法. 首先给出了光滑罚问题、非光滑罚问题及原问题的目标函数值之间的误差估计,进而在弱的假
设之下证明了光滑罚问题的全局最优解是原问题的近似全局最优解. 最后给出了一个基于光滑罚函数的求解原问题的算法,证明了算法的收敛性,并给出数值算例说明算法的可行性.     

On extensions of partial functions

The problem of extending partial functions is considered from the general viewpoint. Some aspects of this problem are illustrated by examples, which are concerned with typical real-valued partial functions (e.g. semicontinuous, monotone, additive, measurable, possessing the Baire property).     

一类亚解析函数的复合边值问题

本文研究了一类亚解析函数的复合边值问题.利用经典的消去法和对称扩张,得到了边值问题的可解条件和解的表达式.推广了解析函数的相应理论.     

A Continuation Algorithm for Max-Cut Problem

A continuation algorithm for the solution of max-cut problems is proposed in this paper. Unlike the available semi-definite relaxation, a max-cut problem is converted into a continuous nonlinear programming by employing NCP functions, and the resulting nonlinear programming problem is then solved by using the augmented Lagrange penalty function method. The convergence property of the proposed algorithm is studied. Numerical experiments and comparisons with the Geomeans and Williamson randomized algorithm made on some max-cut test problems show that the algorithm generates satisfactory solutions for all the test problems with much less computation costs.     

ON HASEMAN BOUNDARY VALUE PROBLEM FOR A CLASS OF META-ANALYTIC FUNCTIONS

In this article, Haseman boundary value problem for a class of meta-analytic functions is studied. The expression of solution and the condition of solvability for Haseman boundary value problem are obtained by changing the problem discussed into the equivalent Haseman boundary value problem of bi-analytic function. And the expression of solution and the condition of solvability depend on the canonical matrix.     

On Haseman boundary value problem for a class of metaanalytic functions with different factors on the unit circumference

In this article, the Haseman boundary value problem (bvp) is discussed for metaanalytic functions with different factors on the unit circumference. Through a series of appropriate transformations, we obtain general expressions for the solution and the condition of solvability for the problem. Copyright © 2009 John Wiley & Sons, Ltd.     

The modified Green function technique for the exterior Dirichlet problem in linear thermoelasticity

In this work, the modified Green function technique for the exterior Dirichlet problem in linear thermoelasticity is presented. Expressing the solution of the problem as a double‐layer potential of an unknown density, we form the associated boundary integral equation that describes the problem. Exploiting that the discrete spectrum of the irregular values of the associated integral equation is identified with the spectrum of eigenvalues of the corresponding interior homogeneous Neumann problem for the transverse part of the elastic displacement field, we introduce a modification of the fundamental solution of the elastic field. We establish the sufficient conditions that the coefficients of the modification must satisfy to overcome the problem of nonuniqueness for the thermoelastic problem.