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。     

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”     

(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.     

（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．     

非线性波方程新的显式精确解

利用Darboux和一个可化为标准Bernoulli方程的4阶常微分方程,统一地处理了三个著名方程KdV方程,Kadomtsev-Petviashvili(KP)方程和Hirota-Satsuma(HS)方程的求解问题.给出了这些方程一批新的具有更为丰富形式的精确解,其中包括孤波解和行波解.     

一个KP型方程的新型Darboux变换

KP型方程是物理上有重要意义的1+1维和2+1维的几个非线性发展方程的统一和推广.基于KP型方程的Lax对的Painlevé展开,给出了KP型方程的一个新型Darboux变换,并给出证明.然后适当选取原方程的平凡解,利用新型Darboux变换求出方程新的精确解.进而,利用图形展示了所得解的性质.     

例谈构造法解竞赛题

“构造法”是最富活力的数学转化方法之一 .恰当地运用这一方法解题 ,能收到以简驭繁、化难为易、事半功倍之效 ,且有助于发展创造性思维品质和探索创新能力 .本文以各类竞赛题为例 ,对常用的构造法予以说明 .一构造方程例 1　若ａｂ≠ 1,且有 5ａ2 + 2 0 0 1ａ + 9=0及 9ｂ2 + 2 0 0 1ｂ + 5 =0 ,则 ａｂ的值是 (　　 ) .　 (Ａ) 95 (Ｂ) 59(Ｃ) -2 0 0 15 (Ｄ) -52 0 0 1(2 0 0 1年全国初中数学联赛题 )分析　抓住题设两等式的结构特征 ,对其中一等式稍加变形 ,即可利用方程根的定义构造一个一元二次方程 ,再由韦达定理迅速获解 .解　∵　…     

一些非线性演化方程族的新对称

众所周知,具有无穷多对称是孤立子方程的一大特色,近年来国内外许多学者致力于寻找一些孤立子方程(显含时间变量t)的新对称,在1980年,P.J.Olver发现了Kdv方程具有新的对称.与此同时H.H.Chen等用不同的方法找到了诸如KdV,MKdv,NLS,KP等几个著名方程的新对称.其后B.Fuchssteiner通过构造BO,KP等方程的mas-tersymmetries导出了这些方程的新对称.最近李翊神和朱国城从KdV方程族入手,得到了该方程的一串新对称,并指出了新的对称与谱可变演化方程之间的关系,同时还证明了所求得的两串对称(新的和旧的刘称)构成了无穷维李代数,进而他们又把相应的结果推广     

相似变换及三维柱KP方程的衰减解（英文）

本文首先导出三维柱KP方程与三维常系数KP方程解之间的相似变换及三维柱KP方程的系数所满足的约束条件;借助于该相似变换及三维常系数KP方程的解,得到三维柱KP方程的解;最后,讨论四个特殊的三维柱KP方程,特别地,得到这些三维柱KP方程的衰减解.     

求解具有单连续变量背包问题的精确算法

具有单连续变量背包问题(KPC)是标准0-1背包问题(0-1KP)的一个新颖扩展形式,由于其中的背包载重不再固定不变,而是由一个连续变量进行连续调整,因此KPC是一个比0-1KP更难求解的背包问题.首先提出了一个带有实函数的变载重背包问题(0-1KP(Σ,f)),基于动态规划法给出了求解它的一般方法;然后,利用放缩法将KPC中的连续变量离散化,在建立KPC的一个新数学模型的基础上,将它转化成为0-1KP(Σ,f)的一个特例,利用0-1KP(Σ,f)的求解方法给出了KPC的一个简单且易于实现的精确算法.     

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.     

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.     

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.     

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.     

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.     

Obtaining the dividends-penalty identities by interpretation

The dividends-penalty identity is a relation between three functions: the discounted penalty function without dividends, the discounted penalty function if a barrier dividend strategy is applied, and the expected discounted dividends until ruin. The classical model of risk theory is modified in that the deterministic premiums are replaced by a compound Poisson process with exponential jumps. In this model, the dividends-penalty identity is new and can be derived by interpretation. Then the dividends-penalty identity in the classical model is obtained as a limit.     

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.