求轨迹方程问题的一些常用方法

求动点的轨迹方程是解析几何的重要内容之一,也是教学中的一个难点,这部分知识究竟有无规律可循?本文分类介绍几种求轨迹方程的方法。一等式法 1)若问题明确地给出了动点运动过程中所满足的量的关系。那么就把这些量的关系坐标化,列出等式,即得到动点的轨迹方程。这就是所谓“等式法”     

关于陈微分式的一个注记

在陈省身先生关于高斯—波涅公式的著名证明中有一个关键等式，本文对这一等式给出一个稍稍改动的证明，相信更清楚地贴近陈的想法。     

DJKM方程的Bcklund变换及非线性叠加公式

本文考虑了DJKM方程,利用Hirota双线性算子工具,我们给出了其Bcklund变换(BT)并严格证明了相应的非线性叠加公式。     

拟线性椭圆组弱解的正则性和 Pohozaev 恒等式

自1965年文[1]证明了一个积分等式(现称为 Pohozaev 恒等式)以来,发现该等式有多种用途,主要作用之一是证明解的不存在性.近年来,关于这个等式有不少发展.首先是1985年,沈尧天和邓耀华等人合作的工作,最早对重调和和多重调和方程的解建立了这一类积分等式.在1986年 P.Pucci 和 J.Serrin 对一般方程组和高阶方程建立了这一类等式.互相独立的,徐海祥在1987年也对一般方程组建立了这种等     

关于瑕积分的一个极限等式

本文证明如果区间(a,b]上以a为瑕点的收敛的瑕积分∫baf(x)dx中,被积函数f(x)在(a,b]上连续,则成立极限等式∫baf(x)dx=limn→∞∑ni=1f(a+i(b-a)/n)(b-a)/n.利用这一等式可计算一类数列的极限.     

(2+1)维Kadomtsev-Petviashvili方程的留数对称及其相互作用解

运用Painlevé截断展开方法得到(2+1)维Kadomtsev-Petviashvili(KP)方程的非局域留数对称和B?cklund变换.由于非局域对称不能直接对(2+1)维KP方程进行约化求解,因此,需要将非局域对称局域化.然后,利用相容的Riccati展开(CRE)可解的概念证明(2+1)维KP方程的CRE可解性,从而求出(2+1)维KP方程的新的相互作用解.     

也谈一元一次不等式的解集及其数轴表示问题

<正>贵刊(初中版)2009年8月(下)刊发了安徽毛春松老师的《类比方程学好不等式(组)》,文中将一元一次不等式同一元一次方程的解答过程进行了对比,有效帮助同学们掌握解不等式的相关方法.对比解一元一次方程,在解不等式时,"不等式两边同时乘以(或除以)同一个负数时,不等号的开口方向要改变"这一步与解方程不同,最容易出现错误.因此笔者认为很有必要将解不等式步骤中"化系数为1"的这一步凸显出来,如此不仅提醒同学们关注这一步的符号特征从而更加准确地求解,也便于解题之后进行检查.除此之外,本     

（2＋1）维KP方程的三类精确解

研究了（2＋1）维KP方程的孤子解问题．应用Riccati方程映射法，得到了（2＋1）维KP方程的新的显式精确解的结构．根据得到的精确解结构，构造出了该方程的三类精确解．     

恒等式和方程的概念

到現在为止,在教学文献中,關於恒等式和方程的概念的相互關係,还没有確定的观點。在早期的教学文献中,这兩个概念是按照下面的意义而互相比拟的,一个等式倘对於它的兩端所包含的变元(文字)的“所有數值”皆成立,则認为是恒等式,而不是方程。在这个方程定义本身有一个限制条件:方程是“不是对於所有,而只对於某些”未知量的數值成立的等式。在最近的教学文献和教学方法文献中,这种观點遭到了批評。首先,在解方程時,当含有一个未知數(或幾个未知數)的等式对於未知數(或諸未知數)的所有允許值皆成立的情况,是沒有任何合理的理由除外的。其次,在解方程以前,只有在極簡單的情况中,才能立刻說出所提出的方程“在事实上”是方程呢,还是非方程而是恒等式。第三,在解决和研究含参數的方程時,这种观點     

平均反射随机微分方程的传输不等式[英文]

本文研究平均反射随机微分方程的传输不等式的问题. 利用Girsanov定理和鞅表示定理, 针对该方程在连续轨道空间上, 关于一致收敛范数, 获得Talagrand传输不等式. 这推广了吴黎明和张正良(2004)得到的结果.     

An identity involving invariant polynomials of matrix arguments

The purpose of the present paper is to establish an identity involving invariant polynomials of two matrix arguments. This identity is a generalization of a well known identity that gives evaluation of the Gauss hypergeometric function when the argument matrix is identity. Applications of the identity derived in this article are also given.     

A three-term theta function identity and its applications

In this paper, we establish a three-term theta function identity using the complex variable theory of elliptic functions. This simple identity in form turns out to be quite useful and it is a common origin of many important theta function identities. From which the quintuple product identity and one general theta function identity related to the modular equations of the fifth order and many other interesting theta function identities are derived. We also give a new proof of the addition theorem for the Weierstrass elliptic function ℘. An identity involving the products of four theta functions is given and from which one theta function identity by McCullough and Shen is derived. The quintuple product identity is used to derive some Eisenstein series identities found in Ramanujan's lost notebook and our approach is different from that of Berndt and Yee. The proofs are self contained and elementary.     

An AZ-style identity and Bollobás deficiency

The powerful AZ identity is a sharpening of the famous LYM-inequality. More generally, Ahlswede and Zhang discovered a generalization in which the Bollobás inequality for two set families can be lifted to an identity.In this paper, we show another generalization of the AZ identity. The new identity implies an identity which characterizes the deficiency of the Bollobás inequality for an intersecting Sperner family. We also give some consequences relating to Helly families and LYM-style inequalities.     

Convolution identities for tensors

The concept of a convolution identity for tensors is introduced and it is proved that any convolution identity for tensors on a finite-dimensional space follows from a convolution identity equivalent to the classical Cayley-Hamilton identity.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 211–214, 1982.     

有单位元交换环上二阶矩阵的因子分解

对有单位元交换环上矩阵分解问题进行了讨论,给出了有单位元交换环上二阶矩阵可以因式分解的充分必要条件,即单位元交换环上二阶矩阵可以因式分解当且仅当这个矩阵的行列式可以因子分解.     

On identities for coalgebras and hopf algebras

The notion of a polynomial identity for algebras is well-known. In this work the notion of identity is transferred to coalgebras (where one may call it a coidentity). The author studies the basic properties of coidenti-ties, some examples of Hopf algebras with an identity or a coidentity are considered.     

An addition formula for the Jacobian theta function and its applications

In this paper, we prove an addition formula for the Jacobian theta function using the theory of elliptic functions. It turns out to be a fundamental identity in the theory of theta functions and elliptic function, and unifies many important results about theta functions and elliptic functions. From this identity we can derive the Ramanujan cubic theta function identity, Winquist's identity, a theta function identities with five parameters, and many other interesting theta function identities; and all of which are as striking as Winquist's identity. This identity allows us to give a new proof of the addition formula for the Weierstrass sigma function. A new identity about the Ramanujan cubic elliptic function is given. The proofs are self contained and elementary.     

A generalization of Clausen’s identity

The paper gives an extension of Clausen’s identity to the square of any Gauss hypergeometric function. Accordingly, solutions of the related third-order linear differential equation are found in terms of certain bivariate series that can reduce to ₃F₂ series similar to those in Clausen’s identity. The general contiguous variation of Clausen’s identity is found as well. The related Chaundy’s identity is generalized without any restriction on the parameters of the Gauss hypergeometric function. The special case of dihedral Gauss hypergeometric functions is underscored.     

Some Identities in Algebras Obtained by the Cayley-Dickson Process

Polynomial identities in algebras are the central objects of Polynomial Identities Theory. They play an important role in learning of algebras properties. In particular, the Hall identity is fulfilled in the quaternion algebra and does not hold in other non-commutative associative algebras. For this reason, the Hall identity is important for the quaternion algebra. The idea of this work is to generalize the Hall identity to algebras obtained by the Cayley-Dickson process. Starting from the above remarks, in this paper, we prove that the Hall identity is true in all algebras obtained by the Cayley-Dickson process and, in some conditions, the converse of this statement is also true for split quaternion algebras. From Hall identity, we will find some new properties and identities in algebras obtained by the Cayley-Dickson process.     

On Local LYM Identities

The LYMinequality (Lubell, Yamamoto, Meshalkin) is a generalization of Sperner’s theorem for antichains. Kleitman and Harper independently proved that the LYM inequality and the normalized matching property (or local LYM inequality) are equivalent. Many contributions have been proposed to sharpen the LYM inequality. Noticeably, Ahlswede and Zhang lifted the LYM inequality to an identity, called the AZ identity. Thus, one expects that the same sharpening of the local LYM inequality is equivalent to the AZ identity. In this paper, we introduce a local LYM identity which sharpens the local LYM inequality and prove that it is equivalent to the AZ identity. The local LYM identity shows local relationships between components in the AZ identity.