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<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  相似文献

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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.  相似文献

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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.  相似文献

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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.  相似文献

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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.  相似文献

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M. Kochetov 《代数通讯》2013,41(3):1211-1221
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.  相似文献

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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.  相似文献

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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.  相似文献

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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.  相似文献

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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.  相似文献