例谈概率计算在生物学中的应用

本文通过对生物学中有关概率计算的分析，将典型例题与数学中典型例题进行对比，将生物学中抽象问题转化为数学中同学们熟知的典型概率问题，便于学生更清晰直观地理解、掌握此类问题。     

优化例题教学方法 培养学生解题能力

数学教学离不开解题,数学例题是帮助学生理解、掌握和运用数学概念、定理、公式和法则的数学问题,是数学教师用作示范的具有一定代表性的典型数学问题,例题教学则是数学课堂教学的主要形式．因此,例题教学肩负着激发学生的思维,挖掘学生的创造潜能,提高学生的解题能力的重任,教学方法不可不慎察之．     

跳出题海训练 回归例题教学

多年从事数学教学的实践告诉我们,学生取得好成绩并不是以题海取胜,而是学会在题海中"捞金".教师如何帮助学生从茫茫题海中解脱出来?回归例题教学是实用有效的办法,一个典型例题的剖析既是帮助学生巩固基础知识、形成技能的过程,又是帮助学生提炼和运用数学思想方法以及进行思维训练的过程.笔者在例题教学中作了一些研究,有以下几点肤浅的体会与大家共享.一、掌握依据,夯实基础     

引导探索,培养学生的创新意识

学习数学离不开解题 ,数学思维能力的培养主要是通过解题教学来完成的 .故在解题教学中启动创新思想 ,引导学生大胆探索 ,从而提高学生能力是每个数学教师首要的任务 .1　通过典型例题 ,引导学生推广探索 ,培养抽象概括能力著名物理学家爱因斯坦说 :“提出一个问题 ,比解决一个     

初中数学例题教学方法探讨

数学教学要通过例题和习题使学生获得系统的数学知识.例题的思路分析,解题方法与书写格式都能提高学生的综合能力,使学生在思想上和行为上都受到数学熏陶,对学生的思维及解题行为起着潜移默化的作用,启迪学生掌握各类数学问题的钥匙.数学教学在很大程度上就是数学例题的教学,离开了数学例题,也就无法谈数学教学.     

引申推广例题促进深化理解

高中数学教材中蕴含着许多可以推广的素材,特别是教材中的例题,除了关注例题本身的典型性以外,还应当适度地变化例题的条件,有意识地在分析、解答例题的基础上,对典型例题进行引申推广,充分发挥其以点带面的功能．在对例题引申推广的过程中,既可以挖掘例题所蕴含问题的内涵和外延,又可以激活思维、启迪智慧、拓宽视野、深化理解．笔者结合教学实际,对教材中的一道例题予以引申和推广,以期达到例题增值的目的．     

多法归一:厘清本质,感悟思想--一道例题的教学感悟

数学思想是数学例题教学的内核,让学生经历完整的问题解决与解法交流过程,是帮助学生感悟数学思想的重要途径.数学课上,好的例题教学应关注问题本质,聚焦同质方法,让学生在充分的过程历练中,实现多法归一,使学生充分感悟数学思想的价值所在.     

例题教学中要引导学生进行探究活动

例题教学是数学教学的重要环节．纵观当前的课堂教学，多数是“教师讲、学生听”的模式，严重影响教学质量的提高．我们在例题教学的一些主要环节上，创设情境，有的放矢地引导学生进行探究活动，有效地提高了学生的思维水平，以及独立分析问题、解决问题的能力和灵活驾驭数学问题的能力。     

圆中一道典型例题的变式

典型例题的示范作用显而易见,对典型例题进行深入挖掘产生新的题型、变静态问题为动态问题,增加题目的探究性,也符合新课标的理念.现举例分析如下:     

学科核心素养导向下的高中数学解题教学

高中数学解题教学是培养学生核心素养的重要途径和关键环节之一,基于此,以高中数学典型例题的解题教学为抓手,提出在解题教学中培育学生数学学科核心素养的策略,具体阐述了关于数学建模、直观想象、数学抽象、逻辑推理数学核心素养的教学策略.     

Finite difference calculus and summation of series

The simple question of how much paper is left on my toilet roll is studied from a mathematical modelling perspective. As is typical with applied mathematics, models of increasing complexity are introduced and solved. Solutions produced at each step are compared with the solution from the previous step. This process exposes students to the typical stages of mathematical modelling via an example from everyday life. Two activities are suggested for students to complete, as well as several extensions to stimulate class discussion.     

A Note On Subloop Lattices

Pálfy and Pudlák (Algebra Universalis 11, 22–27, 1980) posed the question: is every finite lattice isomorphic to an interval sublattice of the lattice of subgroups of a finite group? in this paper we will look at examples of lattices that can be realized as subloop lattices but not as subgroup lattices. This is a first step in answering a new question: is every finite lattice isomorphic to an interval sublattice of the subloop lattice of a finite loop?     

Reconstruction of periodic sceneries seen along a random walk

This article was motivated by a question of Kesten. Kesten asked for which random walks periodic sceneries can be reconstructed. Among others, he asked the question for random walks which at each step can move by one or two units to the right. Previously, Howard [C.D. Howard, Distinguishing certain random sceneries on Z$\mathbb{Z}$ via random walks, Statist. Probab. Lett. 34 (2) (1997) 123–132] proved that all periodic sceneries can be reconstructed provided they are observed along the path of a simple random walk.     

A family of nonminimizing abnormal curves

In a four-dimensional sub-Riemannian structure, we study a specific family of abnormal extremals, and we show that they are not length minimizing, answering in the negative to a question that was recently asked. We extend the result to a class of \(4\) -dimensional sub-Riemannian manifolds of step \(5\) .     

How much time do students have to think about teacher questions? An investigation of the quick succession of teacher questions and student responses in the German mathematics classroom

Several studies have shown that the style of the German mathematics classroom at secondary level is mostly based on the so called “fragend-entwickelnde” teaching style which means developing the lesson content by a teacher directed sequence of teacher questions and student responses. In this article we describe a study on the time the students have for thinking about a teacher question in the public classroom interaction. Our investigation is based on a reanalyasis of 22 geometry lessons from grade 8 classes which mainly deal with a challenging proving content. The results show that the average time between a teacher question and a student response is 2.5 seconds. There are no remarkable differences between different phases of the lessons like comparing homework, repetition of content or working on new content. Moreover, for 75% of the teacher questions the first student was called to answer within a three second time interval.     

On the asymptotic joint distribution of an unbounded number of sample extremes

Summary Convergence of the sample maximum to a nondegenerate random variable, as the sample sizen, implies the convergence in distribution of thek largest sample extremes to ak-dimensional random vectorM _k , for all fixedk. If we letk=k(n),k/n0, then a question arises in a natural way: how fast cank grow so that asymptotic probability statements are unaffected when sample extremes are replaced byM _k . We answer this question for two classes of events-the class of all Lebesgue sets inR ^k and the class of events of the form .     

Maximum acyclic and fragmented sets in regular graphs

We show that a typical d‐regular graph G of order n does not contain an induced forest with around vertices, when n ? d ? 1, this bound being best possible because of a result of Frieze and ?uczak [6]. We then deduce an affirmative answer to an open question of Edwards and Farr (see [4]) about fragmentability, which concerns large subgraphs with components of bounded size. An alternative, direct answer to the question is also given. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 149–156, 2008     

SCHOOL OR COLLEGE MATHEMATICS AND WORKPLACE PRACTICE: AN ACTIVITY THEORY PERSPECTIVE

This paper reports on a case study in which we detail how a college mathematics and chemistry student struggles to make sense of the graphical output of an experiment in an industrial chemistry laboratory. The student's attempts to interpret unfamiliar graphical conventions are described and contrasted with those of college mathematics. Our analysis of this draws on activity theory to assist in understanding the position of the student in both the college and the workplace. This highlights the limitations of the experience of the student at college and we question how the mathematics curriculum might be adapted to assist students in making sense of workplace graphical output.     

Normal structure of the unitary group over a local ring

A new approach is used to describe the normal subgroups of unitary groups over the ring of integers of an unramified quadratic extension of a local field of characteristic 2. This method is based on the author's results concerning subgroups of unitary groups normalized by diagonal unitary matrices. By means of this method it has been possible to remove restrictions imposed by other authors in a number of papers pertaining to the same question.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 150–163, 1982.The results in this paper were obtained by the author while he was a post-graduate student under the direction of Z. I. Borevich. The author would like to express his sincere gratitude to Professor Borevich.     

Query complexity,or why is it difficult to separateNP A ∩coNP A fromP A by random oraclesA?

By thequery-time complexity of a relativized algorithm we mean the total length of oracle queries made; thequery-space complexity is the maximum length of the queries made. With respect to these cost measures one can define polynomially time- or space-bounded deterministic, nondeterministic, alternating, etc. Turing machines and the corresponding complexity classes. It turns out that all known relativized separation results operate essentially with this cost measure. Therefore, if certain classes do not separate in the query complexity model, this can be taken as an indication that their relativized separation in the classical cost model will require entirely new principles.A notable unresolved question in relativized complexity theory is the separation of NP^A co NP^A fromP ^A under random oraclesA. We conjecture that the analogues of these classes actually coincide in the query complexity model, thus indicating an answer to the question in the title. As a first step in the direction of establishing the conjecture, we prove the following result, where polynomial bounds refer to query complexity.If two polynomially query-time-bounded nondeterministic oracle Turing machines accept precisely complementary (oracle dependent) languages L^A and {0, 1}^*L^A under every oracle A then there exists a deterministic polynomially query-time-bounded oracle Turing machine that accept L^A. The proof involves a sort of greedy strategy to selecting deterministically, from the large set of prospective queries of the two nondeterministic machines, a small subset that suffices to perform an accepting computation in one of the nondeterministic machines. We describe additional algorithmic strategies that may resolve the same problem when the condition holds for a (1–) fraction of the oracles A, a step that would bring us to a non-uniform version of the conjecture. Thereby we reduce the question to a combinatorial problem on certain pairs of sets of partial functions on finite sets.