基于研究型教学的《抽象代数Ⅱ》建设与实践

总结了北航研究生《抽象代数II》课程的建设与实践过程中,探索研究型教学、实践翻转课堂、提高学生学习积极性、主动参与教学过程的一些思路和做法,提倡在教学全过程中鼓励与激发学生探究式学习.     

待定法在"高等代数"教学中的应用

在高等代数的教学中,“特定系数法”思想应用比较普遍.它不仅能够用以分析教材,理解教材中的知识点,还可以帮助我们理解教材中一些章节的知识结构,从而在更深的层次上把握教材.     

《高等代数》课程教学中应注意的几个问题

本文从注重教学理念，改革教学内容和提高教学质量等方面探讨了《高等代数》课程教学中应注意的几个问题。     

《高等代数》中一处证明的简化

通过分析某《高等代数》常用教材中线性空间直和分解定理的证明过程,指出其在叙述上存在的问题,给出该定理的一种简化证明,使读者能更清晰地理解证明过程。     

《高等代数》教学准备的一个尝试

《高等代数》是一门抽象而有用的基础课程.授课教师应当让学生通过有效的学习和训练,深刻理解这门课程的基本概念,熟练掌握一些基本技巧.同时,授课教师应该鼓励学生探索这些基本概念和技巧在其他课程和学术研究中的延伸和应用,从而激发学生自主学习高年级课程甚至从事科研的兴趣.这是《高等代数》教学准备中值得探索的方向.     

在高等代数教学中渗透数学史知识的思考

高等代数是一门内容丰富、思想独特、方法技巧较强的一门基础课程.在授课过程中,适当地渗透数学史知识,有助于学生兴趣的培养,有助于学生理解数学的思想、方法和精神,有助于培养他们抽象思维能力、逻辑推理能力、分析问题和解决问题能力.     

浅谈如何在《高等数学》教学中培养学生的创新能力

创新能力是一个国家科技进步和发展根本，而数学教育在培养学生的创新能力方面扮演着极其重要的角色。本文就如何在《高等数学》教学中培养学生的创新能力作了一些初步的探讨。     

如何在教学中让数学知识生活化

数学源于生活,又广泛应用于生活.数学知识的生活化,就是通过将数学教材中枯糙、脱离学生实际的数学知识还原,取之于学生生活实践并具有一定真实意义的数学问题,以此来沟通"数学与现实生活"的联系     

浅谈如何在《高等数学》教学中提高学生的学习积极性

我们在多年教学实践中,注重通过把握教学重点难点,找准内容突破口;注意数形密切结合,加强知识对比性;加强应用环节教学,增强内容可视性;不断进行教学改革,突出内容联系性;及时进行课堂小结,把握方法规律性;充分应用所学知识,化解学生疑难点;利用寓教于乐方法,吸引学生注意力;恰当应用教学软件,调动学生主动性;介绍数学历史名人,激起学生荣誉感;搞好第二课堂活动,拓宽学生知识面等方法,在教学中提高学生学习的积极性,收到较好的教学效果。     

浅谈如何在《高等数学》教学中提高学生的学习积极性

我们在多年教学实践中．注重通过把握教学重点难点,找准内容突破口;注意数形密切结合．加强知识对比性;加强应用环节教学．增强内容可视性;不断进行教学改革,突出内容联系性;及时进行课堂小结,把握方法规律性;充分应用所学知识．化解学生疑难点;利用寓教于乐方法．吸引学生注意力,恰当应用教学软件,调动学生主动性;介绍数学历史名人．激起学生荣誉感;搞好第二课堂活动．拓宽学生知识面等方法．在教学中提高学生学习的积极性,收到较好的教学效果。     

Vaught's conjecture and the Glimm-Effros property for Polish transformation groups

We extend the original Glimm-Effros theorem for locally compact groups to a class of Polish groups including the nilpotent ones and those with an invariant metric. For this class we thereby obtain the topological Vaught conjecture.

     

The classification problem for torsion-free abelian groups of finite rank

We prove that for each , the classification problem for torsion-free abelian groups of rank is not Borel reducible to that for torsion-free abelian groups of rank .

     

置换群关于变换后基的强生成集的算法

本文给出置换群关于变换后的基的强生成集的一种算法及其理论证明,并讨论了既约强生成集和最小强生成集的问题。     

A Uniformly Quasiconformal Group Not Isomorphic to a Mobius Group

We explicitly give a group not isomorphic to any group of Mobius transformations of the three-dimensional sphere and acting quasiconformally on the three-dimensional sphere.     

Application of group analysis to classification of systems of three second‐order ordinary differential equations

Here, we give a complete group classification of the general case of linear systems of three second‐order ordinary differential equations excluding the case of systems which are studied in the literature. This is given as the initial step in the study of nonlinear systems of three second‐order ordinary differential equations. In addition, the complete group classification of a system of three linear second‐order ordinary differential equations is carried out. Four cases of linear systems of equations are obtained. Copyright © 2015 John Wiley & Sons, Ltd.     

On nice equivalence relations on <Superscript>λ</Superscript>2

Let E be an equivalence relation on the powerset of an uncountable set, which is reasonably definable. We assume that any two subsets with symmetric difference of size exactly 1 are not equivalent. We investigate whether for E there are many pairwise non equivalent sets. I would like to thank Alice Leonhardt for the beautiful typing.This research was supported by The Israel Science Foundation. Publication 724. Mathematics Subject Classification (2000): 03E47, 03E35; 20K20, 20K35     

Counterexample to a problem of Geoghegan-West

Let be a Banach space and its general linear group. Let denote the operator norm and ``' the pointwise convergence topology on . Is the identity map a homotopy equivalence? The answer is negative. One of the possible counterexamples is a well-known James space ---the ``space of counterexamples in Banach spaces theory'.

     

How to generalize known results on equations over groups

Known facts about the solvability of equations over groups are considered from a more general point of view. The theorem about the solvability of unimodular equations over torsion-free groups is generalized. A special case of the generalization is a multivariable version of this theorem. For unimodular equations over torsion-free groups, an analog of Magnus’ Freiheitssatz is proved, which asserts the existence of a solution exhibiting good behavior with respect to the free factors of the initial group.     

An Ergodic Action of the Outer Automorphism Group of a Free Group

For n > 2, the action of the outer automorphism group of the rank n free group F _n on Hom(F _n , SU(2))/SU(2) is ergodic with respect to the Lebesgue measure class. The author gratefully acknowledges support from National Science Foundation grants DMS-0405605 and DMS-0103889. Received: September 2005 Revision: January 2006 Accepted: March 2006