一本别开生面的“数学史”——评《数学纵横》

<正> 《数学纵横》(以下简称《纵横》)是一本装帧秀雅的科普读物,1992年8月由四川教育出版社出版.高隆昌。胡勋玉编著。它不仅是献给大、中数学教师以及一般数学爱好者的一分礼物,即使数学家们也不妨一览。当你读完全书之后,再回到著名的老一辈数学家、中国科学院学部委员柯召教授的提辞上来,就会清楚地     

大概念引领,大单元整合,设计中考备考教学新格局——以函数大单元中考备考教学为例

在初中数学课堂教学中,中考备考教学是学校教学的核心任务.这是在学生已经将初中教材过了一遍之后对知识进行重新整合的过程,也是融入中考指导思想的过程.这一过程就会更加显现出初中数学知识内涵的本质、核心思想或素养,同时构建数学与其他学科知识间的纵横联系,这就是数学大概念.     

挖掘教材潜力,培养学生的思维能力

课堂教学中若能充分挖掘教材中的引言、概念、例题、习题的潜在功能,并恰当地进行演变、引申、拓广,注意纵横渗透、关注教材中隐藏的“二级”结论,则可激发学生的求知欲,启迪学生的思维,达到培养学生的思维能力,提升数学核心素养的数学育人功能.     

欢迎惠稿、订阅《中学数学研究》

《中学数学研究》是由江西师范大学主办面向中学数学教学的普及性学术类月刊.国内统一刊号:CN36-1100/O1,国际标准刊号:ISSN1673-6559.杂志创刊于1980年,一直以来深受广大中学师生、数学教研员及数学爱好者的厚爱.其常设栏目有教学纵横、教例探微、专     

变式教学诱发一题多解

高三数学第一轮复习已经到了“不等式”部分，在教学中重视一题多解，加强各知识之间的纵横联系，能起到举一反三，融汇贯通的作用。     

数学娱乐圈

迷人的数学数学是神奇的,数学是迷人的.记得有位作家这样赞美数学："(数学)是空谷中的幽兰,高寒中的杜鹃,老林中的人参,冰山上的雪莲,绝顶上的灵芝,抽象思维的牡丹."这是多么美的诗句,多么壮丽的画卷.数学是空间关系的浓缩,数学是数量关系的组合,数学是科学发展的桥梁,数学是人类解开愚昧、走向文明的使者,数学是数和图编织的图画,数学是x、y、z合唱出来的高亢的歌.在数学的字里行间充满着魔光幻影,神秘莫测;在数学的字里行间充满着和谐的韵律、抽象的彩虹,让人难以割舍;在数学的字里行间闪烁着人类智慧的火花,璀璨夺目,…     

浅谈一道几何题的多种证法

<正>"一题多解"就是启发和引导从不同角度、不同思路,运用不同的方法和不同的运算过程,解答同一道数学问题,即由多种途径获得同一数学问题的最终结论.一题多解能提高综合运用已学知识解答数学问题的技能技巧,锻炼思维的灵活性,开阔解题思路,能更好地掌握知识间的纵横联系.举一例如下.     

精选改编题目 深挖教学潜能 提高探究能力——初三数学总复习探究式习题课教学案例

习题课是初三数学总复习的主要课型,可灵活地选择经典中考题进行改编和拓展,通过一题多解和一题多变,加强知识间的纵横联系,拓宽解题思路,让学生在探究中巩固所学知识,提高综合运用知识解题的能力和数学素养,本文精选一道中考试题进行改编,给出三种解题思路,然后通过改变结论、改变条件或改变背景,对此题进行变式拓展.     

探析立体几何与相关知识的“交汇性”

在“知识网络的交汇点设计试题”是近几年高考数学命题改革的重要理念和方向．这就要求在数学学习中，要善于有效地把握好知识间的纵横联系和综合应用，打破各章节的界限，对所学内容融会贯通，运用自如，形成有序的网络化的知识体系，以开阔视野，形成能力，全面提高数学素养．本文就立体几何与其它知识的“交汇性”，对这方面问题作如下探析，仅供参考．     

一位中学生读者的来信以及作者的答复

[读者来信] 夏新桥老师: 您好!我是一个数学爱好者,我想您也是吧.您的大名是我在《中学生数学》上找到的.在许多期的趣味数学、数苑纵横等栏目中,我都找到过您发表的一些文章.读了之后,我感到耳目一新,特别是《穿墙术》、《算两次》等等.我想您在数学方面的研究一定很有造诣. 在我的学习过程中碰到了两道难题,历时近两年至今未解决,下面恳请夏老师,为我解     

Double braidings, twists and tangle invariants

A tortile (or ribbon) category defines invariants of ribbon (framed) links and tangles. We observe that these invariants, when restricted to links, string links, and more general tangles which we call turbans, do not actually depend on the braiding of the tortile category. Besides duality, the only pertinent data for such tangles are the double braiding and twist. We introduce the general notions of twine, which is meant to play the rôle of the double braiding (in the absence of a braiding), and the corresponding notion of twist. We show that the category of (ribbon) pure braids is the free category with a twine (a twist). We show that a category with duals and a self-dual twist defines invariants of stringlinks. We introduce the notion of turban category, so that the category of turban tangles is the free turban category. Lastly we give a few examples and a tannaka dictionary for twines and twists.     

Quinn’s Formula and Abelian 3-Cocycles for Quadratic Forms

Algebras and Representation Theory - In pointed braided fusion categories knowing the self-symmetry braiding of simples is theoretically enough to reconstruct the associator and braiding on the...     

Spatial braiding of cylindrical articles

The effective deformative characteristics of spatially reinforced composites made by spatial braiding along the generatrices of a one-sheet hyperboloid are analyzed. The geometrical relationships determining the structure of a unit cell of a braided composite are derived. The effective thermoelastic characteristics are calculated by the method of orientational averaging. The dependences of the bending and torsional stiffnesses of thick-walled cylindrical rods — made by the method suggested and by winding — on the braiding/winding angle are compared. The numerical estimations are given for rods made of carbon (CFRP) and aramid (AFRP) epoxy plastics. Submitted to the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000). Translated from Mekhanika Kompzitnykh Materialov, Vol. 36, No. 3, pp. 341–354, May–June, 2000.     

张量代数上一种带辫子的Poisson结构

对应结合代数的R-冲积构造,考虑了相应半古典极限的Poisson结构构造.进而给出了张量代数上一种带辫子的Poisson结构,该结果推广了Poisson多项式环和双Poisson-Ore扩张.     

Finite GK-dimensional pre-Nichols algebras of super and standard type

We prove that finite GK-dimensional pre-Nichols algebras of super and standard type are quotients of the corresponding distinguished pre-Nichols algebras, except when the braiding matrix is of type super A and the dimension of the braided vector space is three. For these two exceptions we explicitly construct substitutes as braided central extensions of the corresponding pre-Nichols algebras by a polynomial ring in one variable. Via bosonization this gives new examples of finite GK-dimensional Hopf algebras.     

THE GENERALIZED SMASH PRODUCT AND COPRODUCT

91. PreliminariesThroughout the paper all spaces are over a fixed ground field K. If C is a coalgebra,then we always denote the comultiplication and counit by b and E respectively. Set b(c) Z of @ cd for c E C. If M is a right (left) C -comodule, then we use p for the structure mapof M, and set p(m) = ac(o) @ m(1) (p(m) ~ ac(--1) @ m(o)) for m E M. Let H be abialgebra. Denote left H-module category by HM and left H-comodule category by HM.If M and N are in HM, then M @ N is also a l…     

The Weyl groupoid of a Nichols algebra of diagonal type

The theory of Nichols algebras of diagonal type is known to be closely related to that of semi-simple Lie algebras. In this paper the connection between both theories is made closer. For any Nichols algebra of diagonal type invertible transformations are introduced, which remind one of the action of the Weyl group on the root system associated to a semi-simple Lie algebra. They give rise to the definition of a groupoid. As an application an alternative proof of classification results of Rosso, Andruskiewitsch, and Schneider is obtained without using any technical assumptions on the braiding. Mathematics Subject Classification (2000) 17B37, 16W35     

Lifting of Nichols algebras of typeB 2

We compute liftings of the Nichols algebra of a Yetter-Drinfeld module of Cartan typeB ₂ subject to the small restriction that the diagnonal elements of the braiding matrix are primitiventh roots of 1 with oddn≠5. As well, we compute the liftings of a Nichols algebra of Cartan typeA ₂ if the diagonal elements of the braiding matrix are cube roots of 1; this case was not completely covered in previous work of Andruskiewitsch and Schneider. We study the problem of when the liftings of a given Nichols algebra are quasi-isomorphic. The Appendix (with I. Rutherford) contains a generalization of the quantum binomial formula. This formula was used in the computation of liftings of typeB ₂ but is also of interest independent of these results. With an appendix “A generalization of theq-binomial theorem” with Ian Rutherford, Department of Mathematics and Computer Science, Mount Allison University, Sackville, N.B., Canada E4L 1E6. Research partially supported by NSERC. A visit to University of Bucharest in 1999 was partially supported by CNCSIS, Grant C12. She would like to thank the department for their warm hospitality. Received partial support from CNCSIS, Grants C12 and 199. Thanks to Mount Allison University for their hospitality in June 1999.     

Pseudosymmetric braidings, twines and twisted algebras

A laycle is the categorical analogue of a lazy cocycle. Twines (introduced by Bruguières) and strong twines (as introduced by the authors) are laycles satisfying some extra conditions. If c is a braiding, the double braiding c² is always a twine; we prove that it is a strong twine if and only if c satisfies a sort of modified braid relation (we call such cpseudosymmetric, as any symmetric braiding satisfies this relation). It is known that the category of Yetter-Drinfeld modules over a Hopf algebra H is symmetric if and only if H is trivial; we prove that the Yetter-Drinfeld category _HYD^H over a Hopf algebra H is pseudosymmetric if and only if H is commutative and cocommutative. We introduce as well the Hopf algebraic counterpart of pseudosymmetric braidings under the name pseudotriangular structures and prove that all quasitriangular structures on the 2ⁿ⁺¹-dimensional pointed Hopf algebras E(n) are pseudotriangular. We observe that a laycle on a monoidal category induces a so-called pseudotwistor on every algebra in the category, and we obtain some general results (and give some examples) concerning pseudotwistors, inspired by the properties of laycles and twines.     

Yang–Baxter operators in symmetric categories

We introduce non-degenerate solutions of the Yang–Baxter equation in the setting of symmetric monoidal categories. Our theory includes non-degenerate set-theoretical solutions as basic examples. However, infinite families of non-degenerate solutions (that are not of set-theoretical type) appear. As in the classical theory of Etingof, Schedler, and Soloviev, non-degenerate solutions are classified in terms of invertible 1-cocycles. Braces and matched pairs of cocommutative Hopf algebras (or braiding operators) are also generalized to the context of symmetric monoidal categories and turn out to be equivalent to invertible 1-cocycles.