Number Worlds: Visual and Experimental Access to Elementary Number Theory Concepts

Recent research demonstrates that many issues related to the structure of natural numbers and the relationship among numbers are not well grasped by students. In this article, we describe a computer-based learning environment called Number Worlds that was designed to support the exploration of elementary number theory concepts by making the essential relationships and patterns more accessible to learners. Based on our research with pre-service elementary school teachers, we show how both the visual representations embedded in the microworld, and the possibilities afforded for experimentation affect learners' understanding and appreciation of basic concepts in elementary number theory. We also discuss the aesthetic and affective dimensions of the research participants' engagement with the learning environment. This revised version was published online in July 2006 with corrections to the Cover Date.     

Developing Student Understanding: Contextualizing Calculus Concepts

This study looked at the practice of one high school teacher who provided students with concrete examples from their physics class to give them a contextually rich environment in which to explore the abstractions of calculus. Students discovered connections between the physics concepts of position, velocity, and acceleration and the calculus concepts of function, derivative, and antiderivative. The qualitative study sought to describe several critical aspects of understanding: students' ability to explain concepts and procedures, to apply concepts in a physics context, and to explore their own learning. It included 32 seniors at a large, urban, comprehensive, religious school in a midwestern stale. Samples of student work and reflections were collected by the teacher, as well as by students in individual portfolios. The teacher kept a reflective journal. This study suggests that making connections between calculus and physics can yield deep understandings of semantic as well as procedural knowledge.     

从“距离”的角度理解有关“实数”的一些概念和定理

从＂距离＂的角度来描述和理解数学分析中有关实数的一些概念和定理,并作出相关的证明,特别给出实数完备性的一个新的描述.     

The Adaptable Chromatic Number and the Chromatic Number

We prove that the adaptable chromatic number of a graph is at least asymptotic to the square root of the chromatic number. This is best possible.     

Developing Number Sense and Basic Computational Skills in Students with Special Needs

This article presents an investigation of the effectiveness of procedures undertaken to develop number sense and basic computational skills in learning disabled students. Twelve students in a K-1 classroom who had been identified as learning disabled (LD) were presented tasks which required them to subitize (i.e., recognize the number of objects in a set without actually counting them). Consistent with other such studies in special education, a qualitative research methodology was employed, involving a case study of an intact group of LD students. Also consistent with many such studies in special education, observational rather than quantitative data were collected. At the end of a four-week period, all students were consistently successful in recognizing and matching the numbers 0 through 5 and adding sums to five as determined on a teacher-administered test. Increased time-on-task and pupil independence also are reported. Suggestions for further instructional research related to other arithmetic skills are presented.     

Euler数，Bernoulli数及有序Bell数的渐近估计

利用特殊函数的发生函数为亚纯函数的特点,给出Euler数、Bernoulli数、有序Bell数等几个组合数的带有余项的渐进估计．     

The Number of Cylindrical Shells

Given a set P of n points in three dimensions, a cylindrical shell (or zone cylinder) is formed by two circular cylinders with the same axis such that all points of P are between the two cylinders. We prove that the number of cylindrical shells enclosing P passing through combinatorially different subsets of P has size (n ³) and O(n ⁴) (the previously known bound was O(n ⁵)). As a consequence, the minimum enclosing shell can be found in O(n ⁴) time.     

The p-Bondage Number of Trees

Let p be a positive integer and G = (V, E) be a simple graph. A p-dominating set of G is a subset D  í  V{D\,{\subseteq}\, V} such that every vertex not in D has at least p neighbors in D. The p-domination number of G is the minimum cardinality of a p-dominating set of G. The p-bondage number of a graph G with (ΔG) ≥ p is the minimum cardinality among all sets of edges B í E{B\subseteq E} for which γ _p (G − B) > γ _p (G). For any integer p ≥ 2 and tree T with (ΔT) ≥ p, this paper shows that 1 ≤  b _p (T) ≤ (ΔT) − p + 1, and characterizes all trees achieving the equalities.     

网络割的计数

指出了文献[1],[2]所出现的欠妥处,给出了网络割的计数.     

The Number of Hierarchical Orderings

An ordered set-partition (or preferential arrangement) of n labeled elements represents a single “hierarchy” these are enumerated by the ordered Bell numbers. In this note we determine the number of “hierarchical orderings” or “societies”, where the n elements are first partitioned into m ≤ n subsets and a hierarchy is specified for each subset. We also consider the unlabeled case, where the ordered Bell numbers are replaced by the composition numbers. If there is only a single hierarchy, we show that the average rank of an element is asymptotic to n/(4 log 2) in the labeled case and to n/4 in the unlabeled case. This revised version was published online in September 2006 with corrections to the Cover Date.     

模糊概念的EI代数分解

应用 AFS结构 [6] 上的 EI代数[6] 分析模糊概念的数学结构 ,证明有限集上任意一个模糊概念都是一类极其简单的模糊概念的 EI代数分解。     

几何分析中的基本概念与原理

本文是根据作者在中国科学院“中德几何分析伙伴小组”成立仪式上的演讲整理而成，主要目的在于通过回顾几何分析中的基本概念及基本定理，了解现阶段的研究情况和将来的研究线路，主要描述黎曼几何与量子场论中的思想，特别特别是极化物理量与几何量产生的变分问题。     

The Developing Organization

Journal of the Operational Research Society -     

联系数在梯形模糊数多属性决策中的应用

属性权重和属性值都是梯形模糊数的多属性决策问题是一种带有不确定性的决策,需要作不确定性分析后才能得出结论.为此把梯形模糊数表示的属性值和属性权重先用其特征函数"均值+偏差"联系数(特征联系数)表示,再利用联系数运算法则作"加权求和"运算,不仅获得与其它决策方法相同的结果,而且借助联系数中i的不同取值考察决策对象的排序变化,方法简便易行,且具有较强适用性.     

The Number 71.428571 in Hindu Cosmology

Zimmer, in his 1962 book, Myths and Symbols in Indian Art and Civilization, speculates that the number 14, used to divide a kalpa into manvantaras, was chosen in an attempt to reconcile a mahayuga chronology with a tradition of flood heroes. Each manvantara was divided by 71 and an unspecified fraction to produce the mahayuga. The purpose of this research was to find this fraction and discover the reason for its use. The repeating cyclic fraction, 1/7, provides a plausible explanation for use of the number 71.428571 (71 and 3/7) in cosmology; for 22/7 as a value forπ and for concepts of time and circle division related to breaths per minute. The discovery of cyclic decimal fractions by Vedic scholars likely contributed to cyclic theories of cosmology and specifically to the concept of infinity in mathematics.     

Simple Concepts

To talk about simple concepts presupposes that the notion of concept has been aptly explicated. I argue that a most adequate explication should abandon the set-theoretical paradigm and use a procedural approach. Such a procedural approach is offered by Tichý´s Transparent Intensional Logic (TIL). Some main notions and principles of TIL are briefly presented, and as a result, concepts are explicated as a kind of abstract procedure. Then it can be shown that simplicity, as applied to concepts, is well definable as a property relative to conceptual systems, each of which is determined by a finite set of simple (‘primitive’) concepts. Refinement as a method of replacing simple concepts by compound concepts is defined.