《数学分析》教学过程中容易被忽视的环节的教学探讨

国内大部分高等院校数学专业使用华东师大版的《数学分析》作为教材.在教学实践中,发现教材有一些重要的环节没有处理好,容易被教师忽略.本文从凹凸函数的连续性、微积分学基本定理、牛顿-莱布尼茨公式和定积分分部积分法、泰勒级数的和函数、多元函数的极值与最值问题、多元函数的可微性充分条件、有界图形面积为零的充要条件等多方面指出了教材的一些瑕疵,分析了存在的问题,并根据教学实践给出了教学建议.     

一本教材中的一道错误习题

指出北航版工科研究生教材《矩阵论引论》中一道有关满秩分解定理应用的习题存在错误,并进行纠正.此题经过适当修改,还可得到一道构造性的习题.     

关于函数最值的一个注记

最值是刻画函数形态的一个重要指标.本文指出《数学分析》教材(华东师范大学数学系编,第三版)函数最值例题证明过程中的一个不当之处并给出两种完善的证明方法.     

高中数学教材三角函数内容比较研究——以北师大版与人教A版为例

从结构系统和内容系统两方面比较北师大版与人教A版教材三角函数的内容,研究结果表明,两版教材都注重发展学生的数学学科核心素养,将数学文化与数学知识相融合,注重与实际生活的联系.同时二者又有各自的编写特点.北师大版教材注重学习过程的启发性和引导性;人教A版教材突出了单元整体的任务意识,有利于学生从整体上掌握知识.     

高中数学教材与课程标准的一致性分析——以人教A版和苏教版必修一“不等式”为例

通过SEC一致性分析方法,人教A版教材(2019年版)和苏教版教材(2020年版)必修一“不等式”部分与2017年版课程标准总体上一致性显著.在认知水平上,两版教材与课程标准之间仍存在一定差异,在理解水平和探究水平的侧重高于课程标准,在识记水平和应用水平的侧重低于课程标准.建议教师整合题目资源,充分利用习题素材;建议教材编写者重视高水平认知,及时更新习题配置.     

向量组关系与其构造的齐次线性方程组的解关系探讨

本文定义了一个向量组所能构造的两个齐次线性方程组:分量型和向量型齐次线性方程组.随后得出:向量组之间的线性表示和等价性与它们对应的分量型齐次线性方程组的解之间关系不大,但与它们对应的向量型齐次线性方程组的解之间的关系密切,可以相互刻划.在此基础上指出了教材《线性代数》(第二版,同济大学数学教研室编)中有关部分的安排及叙述的不妥之处.     

新课程理念下一元二次不等式及其解法的教学设计

一元二次不等式及其解法是高中数学教学中比较稳定的内容,但从大纲版教材到新课标教材,这部分内容位置变了,教材相关内容的安排也有相当大的变化.除了沪版新教材还是沿用旧大纲版教材,将这块内容还放在集合交集的模型外,     

一道不等式证明题证法的探究及启示

正北师大版高中数学选修4-5《不等式选讲》第22页习题1-4题5是:用求商比较法证明:当a2,b2时,a+bab.这是教材讲授不等式证明后的一道习题,此题虽然难度不大,但是如果我们不满足于用求商比较法给出证明,那么这题就可能成为一道思维训练的好题、妙题,而且能为巩固我们     

HPM视角下三版高中数学教材中数列内容的对比研究

为了解数学教材中数学史分布情况,促进数学史融入数学课堂.研究以人教版、苏教版和北师大版高中数学教材为对象,从章前页、内容编排、呈现方式、数学史四个方面对数列内容进行对比分析.基于此,提出几点建议:统筹规划教材中数学史的“数量”及“质量”;基于HPM视角下设计教学,促进数学教学的多元化;数学史融入要符合学生的认知发展,合理设计教材内容.     

回归课本好处多深入探究更精彩

1 问题提出 　　在高三复习排练组合二项式定理时,笔者为学生出了一道习题: 　　求证:C1n+2C2n+3C3n+...+nCnn=n*2n-1. 　　此题是全日制普通高级中学课程标准实验教科书(人教社B版)选修2-3P38的一道习题,且教材中已经给出证明的提示,选题目的是引导学生回归教材,复习通性通法,注重知识的前后联系,培养学生的探究精神和创新意识.以下是这节课的课堂教学实录,敬请批评指正.……     

Corrigendum to the paper “Ovoidal packings of PG(3,q) for even q”

We point out that the proof of Theorem 3.3 of Bagchi and Sastry (2013) contains a serious flaw. Accordingly, this theorem needs to be modified. In consequence, we also have to retract Corollary 3.4, Corollary 3.6 and Theorem 3.8 of Bagchi and Sastry (2013).     

A note on the strong authenticated key exchange with auxiliary inputs

Recently, Chen et al. proposed a framework for authenticated key exchange (AKE) protocols (referred to as CMYSG scheme) in Designs, Codes and Cryptography (available at http://link.springer.com/article/10.1007/s10623-016-0295-3). It is claimed that the proposed AKE protocol is secure in a new leakage-resilient eCK model w.r.t. auxiliary inputs (AI-LR-eCK). The main tool used for the generic construction is the smooth projective hash function (SPHF). In this note, we revisit the CMYSG scheme and point out a subtle flaw in the original security proof. Precisely, we show that the AI-LR-eCK security of the proposed construction cannot be successfully reduced to a pseudo-random SPHF and thus the CMYSG scheme is not secure as claimed. To restore the security proof, we replace the underlying typical SPHF with a 2-smooth SPHF, and show that such a replacement combined with a \(\pi \hbox {PRF}\) suffices to overcome the subtle flaw.     

Cryptanalysis of a cryptosystem using multiple one-dimensional chaotic maps

Recently, a new chaotic cryptosystem using external 128-bit key and multiple chaotic maps has been proposed. In this paper, a fundamental flaw of this cryptosystem is pointed out and a known plaintext attack is presented. Furthermore, a remedial modification is suggested, which avoids the flaw while keeping all the merits of the original cryptosystem.     

关于Darboux定理

本文细致地讨论了 Darboux定理及有关导函数的几种情形 ,并指出了文献 [1 ]中的一点不足之处 .     

A Revision of the Trapezoidal Branch-and-Bound Algorithm for Linear Sum-of-Ratios Problems

In this paper, we point out a theoretical flaw in Kuno [(2002)Journal of Global Optimization 22, 155–174] which deals with the linear sum-of-ratios problem, and show that the proposed branch-and-bound algorithm works correctly despite the flaw. We also note a relationship between a single ratio and the overestimator used in the bounding operation, and develop a procedure for tightening the upper bound on the optimal value. The procedure is not expensive, but the revised algorithms incorporating it improve significantly in efficiency. This is confirmed by numerical comparisons between the original and revised algorithms. The author was partially supported by the Grand-in-Aid for Scientific Research (C)(2) 15560048 from the Japan Society for the Promotion of Science.     

The projective translation equation and rational plane flows. II. Corrections and additions

In this second part of the work, we correct the flaw which was left in the proof of the main Theorem in the first part. This affects only a small part of the text in this first part and two consecutive papers. Yet, some additional arguments are needed to claim the validity of the classification results. With these new results, algebraic and rational flows can be much more easily and transparently classified. It also turns out that the notion of an algebraic projective flow is a very natural one. For example, we give an inductive (on dimension) method to build algebraic projective flows with rational vector fields, and ask whether these account for all such flows. Further, we expand on results concerning rational flows in dimension 2. Previously we found all such flows symmetric with respect to a linear involution \(i_{0}(x,y)=(y,x)\). Here we find all rational flows symmetric with respect to a non-linear 1-homogeneous involution \(i(x,y)=(\frac{y^2}{x},y)\). We also find all solenoidal rational flows. Up to linear conjugation, there appears to be exactly two non-trivial examples.     

On three-dimensional fibrous flaws in unidirectional fibre reinforced elastic composites

This paper examines the problem of the stress concentration at a flaw in a unidirectional fibre reinforced composite. The geometry of the flaw corresponds to a spheroidal region in which the reinforcing fibres exhibit continuity across the flaw surface. The composite containing the flaw is subjected to a uniaxial stress field which acts along the fibre direction. An exact solution is devoloped for the stress concentration factor at the reinforced flaw boundary. Typical numerical results presented in the paper illustrate the manner in which the stress concentration at the bridged flaw is influenced by the fibre volume fraction, the fibre-matrix modular ratio and the flaw geometry.     

Electrostrictive stresses near crack-like flaws

Slit cracks in purely dielectric material systems do not perturb any applied uniform electric field. Furthermore, when the dielectric is unconstrained and does not support any conducting plates or mechanical loads, there are no additional mechanical stresses generated in the material upon introduction of the crack. This situation applies to both electrostrictive and piezoelectric materials. However, flaws which have finite thickness such as thin elliptical or ellipsoidal voids will cause severe inhomogeneous concentration of the electric field. In turn, this can generate substantial mechanical stress from electrostrictive or piezoelectric sources. The effect of an elliptical through flaw in an infinite isotropic body is considered. It is found that, in the case of thin ellipses, the near flaw tip mechanical stresses approximate the singular stresses near a slit crack with an equivalent stress intensity factor. In that sense, the flaw may be considered as a slit crack and treated in terms of linear elastic fracture mechanics. However, except for impermeable and conducting flaws, the value of the equivalent stress intensity factor depends on the aspect ratio of the flaw. As the aspect ratio of the flaw diminishes, the magnitude of the equivalent stress intensity factor falls and disappears in the limit of a slit crack. The results are used to show that a flaw-like crack in a material with a very high dielectric constant can be treated by fracture mechanics as an impermeable slit crack when the flaw aspect ratio is an order of magnitude greater than the ratio of dielectric permittivities (flaw value divided by the value for the surrounding material).     

A note on the additive data envelopment analysis model

This paper considers the objective function of the Additive (Pareto–Koopmans) Data Envelopment Analysis model and finds a flaw which is induced by the manner in which units invariance is sought. This flaw is rectified, in the spirit of the original proposal, but the consequence is a non-linear objective function for the efficiency calculations. However, the resulting non-linear programming problems solve readily with existing NLP software.     

关于《偏序集上的一种拓扑排序》一文的几点意见

就文献《偏序集上的一种拓扑排序》一义提出了几点看法，探讨了文献中给出的祖先数算法、支配排序算法中的问题，并就其中的dominate函数、函数的时间复杂度的计算以及文献中给出的定理2的正确性进行了分析和论证，并指出了文献中所举例子中存在的差错．最后，对拓扑序列的合理性做了简单的讨论．