一道立几例题的应用价值

一道立几例题的应用价值甘大旺（武汉东西湖吴一中４３００４０）高中课本《立体几何》第一章倒数第二个例题用浓黑字体排：如果两个平面互相垂直，那么经过第一个平面内的一点垂直于第二个平面的直线，在第一个平面内．这个例题的结论既是两个平面垂直的一个性质命题，也...     

例谈平面向量数量积的取值范围

<正>平面向量是高中数学的核心知识,两个向量的数量积是平面向量最重要、最活跃的内容,它的应用十分广泛,也是高考重点考查的内容.许多同学对于求解平面向量数量积的取值范围的问题有时感觉困难.本文结合一道例题来谈谈此类问题的解题思路和方法.例题如图1,△ABC是边长为     

一堂"跑题"的数学课

1在教学中碰到问题 在上数学归纳法的复习课时,笔者分析讲解了如下一道例题. 例平面内有n个两两相交的圆,且任意3个圆不经过同一点,求平面被n个圆分割成的区域数.     

一道平面向量例题的拓广探究及其应用

教材资源是达成核心素养的知识载体和能力沃土,研究教材例题,揭示价值内核,有利于夯实基础知识,培养高阶思维,发展关键能力,本文对一道基于平面向量基本定理的教材例题进行拓广探究,并介绍相关结论的迁移应用.     

一道课本例题的潜在价值

高级中学课本《平面解析几何》全一册 (必修 )第66页例 2如下 :已知一曲线是与两个定点 O(0 ,0 )、A(3,0 )距离的比为 12 的点的轨迹 ,求这个曲线的方程 ,并画出曲线 .这一例题 ,如能充分展示其深刻背景 ,纵横拓展 ,便可成为挖掘例题潜在智能价值 ,优化数学解题教学的典型材料     

例谈平面向量数量积的取值范围

平面向量是高中数学的核心知识,两个向量的数量积是平面向量最重要、最活跃的内容,它的应用十分广泛,也是高考重点考查的内容.许多同学对于求解平面向量数量积的取值范围的问题有时感觉困难.本文结合一道例题来谈谈此类问题的解题思路和方法.     

形状优化的全解析敏度分析

在形状优化设计中,建立了边界元的全解析敏度分析技术,并将该技术与通用的形状优化设计算法相结合,对二维平面应力下的弹性体进行形状优化。在优化该文的例题时,用加权求和法处理该例题的多目标问题,最后获得满意的结果。     

对一节课中例题设置的商榷及建议

笔者最近教授上教版高中数学第八章第三节“平面向量的分解定理”一课时,发现例题的设置及解法会影响本节课目标的达成.尽管教参与考试手册中都提到了理解平面向量分解定理这一目标,但学生普遍感到困惑:不学习这节课的内容,这3道例题也可以很好地完成,那为什么还要学习分解定理?这3道例题存在的意义何在?     

数学竞赛中的“三点共线定理”

从课本例题出发,深入探究了平面向量“三点共线定理”,运用该定理求解了两道北京市中学生数学竞赛题,并推广了三角形面积比例的一类问题.     

发挥例题的中心作用上好复习课

在复习课中,发挥课本例题的中心作用,很有好处。利用课本上的例题“一题多变”,把某一内容的基本知识串起来,下面谈一谈我个人上复习课一点的体会。一以例题为中心串通“形异质同”的题例1 已知四边形ABCD是空间四边形(四个顶点不在同一平面内的四边形),E、H分别是AB、AD的中点,F、G分别是BC、DC的三等分     

格蕴涵代数中的扩张滤子

在格蕴涵代数中提出了扩张滤子的概念,讨论了扩张滤子与滤子,扩张滤子与素滤子,扩张滤子与滤子的根,扩张滤子与准素滤子,扩张滤子与最大滤子之间的关系.得到了扩张滤子的一些性质.最后,证明了在格H蕴涵代数中,扩张滤子与扩张滤子的根相等.     

pth Power Lagrangian Method for Integer Programming

When does there exist an optimal generating Lagrangian multiplier vector (that generates an optimal solution of an integer programming problem in a Lagrangian relaxation formulation), and in cases of nonexistence, can we produce the existence in some other equivalent representation space? Under what conditions does there exist an optimal primal-dual pair in integer programming? This paper considers both questions. A theoretical characterization of the perturbation function in integer programming yields a new insight on the existence of an optimal generating Lagrangian multiplier vector, the existence of an optimal primal-dual pair, and the duality gap. The proposed pth power Lagrangian method convexifies the perturbation function and guarantees the existence of an optimal generating Lagrangian multiplier vector. A condition for the existence of an optimal primal-dual pair is given for the Lagrangian relaxation method to be successful in identifying an optimal solution of the primal problem via the maximization of the Lagrangian dual. The existence of an optimal primal-dual pair is assured for cases with a single Lagrangian constraint, while adopting the pth power Lagrangian method. This paper then shows that an integer programming problem with multiple constraints can be always converted into an equivalent form with a single surrogate constraint. Therefore, success of a dual search is guaranteed for a general class of finite integer programming problems with a prominent feature of a one-dimensional dual search.     

Carleman estimates for second-order hyperbolic systems in anisotropic cases and applications. Part II: an inverse source problem

In this paper, we consider Carleman-type estimate and consider an inverse problem for second order hyperbolic systems in an anisotropic case. In the previous Part I paper, we established a Carleman-type estimate for hyperbolic systems in which the coefficient matrices satisfy suitable conditions. We apply a Carleman estimate in the previous Part I paper to an inverse source problem for second-order hyperbolic systems in an anisotropic case and prove an estimate of the Hölder type.     

Probing interactions in exploratory teaching: a case study

This study investigates an exploratory teaching style used in an undergraduate geometry course to help students identify an ellipse. We attempt to probe beneath the surface of exploration to understand how the actions of teachers can contribute to developing students’ competence in justifying an ellipse. We analyse the complex interactions between student, content, and teacher, and discuss explicit pedagogical strategies that help students develop a higher level of geometric reasoning. The findings indicate that students engaged in guided explorations by the teacher and in group discussions with peers were able to identify an ellipse and justify their reasoning.     

B-Construction and C-Construction

B-construction is a way of obtaining a graded algebra from the triple consisting of an additive category, an object, and an autoequivalence, while C-construction is a way of obtaining an algebra (without unity) from the pair consisting of an additive category and a set of objects. In this article, we study and compare three important classes of algebras in noncommutative algebraic geometry and representation theory of finite dimensional algebras, namely, quantum polynomial algebras, preprojetive algebras and trivial extensions, via these constructions.     

Donaldson's theorem,Heegaard Floer homology,and knots with unknotting number one

We establish an obstruction to unknotting an alternating knot by a single crossing change. The obstruction is lattice-theoretic in nature, and combines Donaldson's diagonalization theorem with an obstruction developed by Ozsváth and Szabó using Heegaard Floer homology. As an application, we enumerate the alternating 3-braid knots with unknotting number one, and show that each has an unknotting crossing in its standard alternating diagram.     

Bounded approximation properties in non‐archimedean Banach spaces

Some non‐archimedean bounded approximation properties are introduced and studied in this paper. As an application, an affirmative answer is given, for non‐spherically complete base fields, to the following problem, posed in 13 , p. 95: Does there exist an absolutely convex edged set B in a non‐archimedean locally convex space such that its closure $\overline{B}Some non‐archimedean bounded approximation properties are introduced and studied in this paper. As an application, an affirmative answer is given, for non‐spherically complete base fields, to the following problem, posed in 13 , p. 95: Does there exist an absolutely convex edged set B in a non‐archimedean locally convex space such that its closure $\overline{B}$ is not edged?     

Distributions on almost contact manifolds

It is known that any hypersurface in an almost complex space admits an almost contact manifold [11, 14]. In this article we show that a hyperplane in an almost contact manifold has an almost complex structure. Along with this result, we explain how to determine when an almost contact structure induces a contact structure, followed by examples of a manifold with a closed G₂-structure.     

用最少的虚工序构建等效多阶段工序网络

运用网络计划可以直观地表示项目管理中的诸多疑难问题, 便于分析和求解. 但是它也存在明显的缺点, 如, (1) 工序网络的有向无回路性表明很多时候适合运用动态规划法, 但它在通常情况下的无阶段性使得该方法无法直接应用; (2) 任意构建的工序网络容易表现得错综复杂, 不利于研究; (3) 用最少的虚工序表示双代号网络是NP-难问题, 因此对一个工序系统可能构建出多个差别迥异的工序网络, 有碍于进度计划管理研究, 等等. 如果能将工序网络构建成等效的多阶段网络, 各工序分别表示在相应的阶段中, 无疑有助于上述问题的解决. 构建等效多阶段工序网络需要添加虚工序. 通过添加最少的虚工序将工序网络构建成等效多阶段网络, 从而有助于建立更合理的工序网络表示法.     

L-Fuzzy拟序集的理想完备化

建立[4]中定义的L-fuzzy拟序集的一种理想完备化,即证明了L-fuzzy拟序集上的所有L-fuzzy理想组成的集合,赋以合适的程度映射构成L-fuzzydomain,任意从L-fuzzy拟序集到L-fuzzydomain的L-fuzzy单调映射都可以扩张成为一个L-fuzzyScott连续映射。