关于超实数域R的基数的一些注记

本文证明了在扩大的分析的非标准模型中超实数域~*R,超有理数域~*Q,超自然数集~*N等集合的基数可以大于任何“标准基数”。     

C*-代数上矩阵的数值域及其性质

本定义了C^*-代数上的矩阵，给出了其上矩阵数值域的定义及相关定义，得到了C^ -代数上矩阵的数值域的性质。     

T1^＊和T11／2—型领域空间的遗传性和拓扑性

证明了T1^*和T11／2－型领域空间是可遗传可拓扑不变的。     

试论线性空间V（F,n）的子空间的个数

V（F,n）是数域F上的n维线性空间,Mp是V（F,n)的具有性质P的子空间作成的集合,本给出几种确定的性质P,并证明Mp的基数等于F的基数,即|Mp|=|F|。最后猜想,对于任一种给定的性质P,等式|Mp|=|F|均成立。     

映射空间中几个定理的扩展

本文在映射空间中推广E^*～开拓扑和一致收敛拓扑，引进了E^*～F^*拓扑和紧一致收敛拓扑，并对映射空间的几个定理做了一些扩展．     

关于集合基数的几点注记

讨论了集合关于基数的分解与合成,证明了基数为c的集合可分解为c个互不相交的基数为c的真子集的和集,进而证明了c个基数为c的集合的和集基数为c。     

晶体常数α、β、γ与α^＊、β^＊、γ^＊的几何关系

研究晶体常数α、β、γ与α^*，β^*，γ^*之间的几何关系，给出了它们在球面上的表示。     

浅谈基数的概念

本通过归纳法将自然数定义为集合，再应用函数一一对应的概念对无穷集合的基数进行研究。     

关于阿尔泰语系诸语言中的^*adirGa(-^*atirqa)一语

通过比较分析突厥语族，蒙古语族和满州通古斯语族诸语言和方言材料，对“阿尔泰语系”诸语言中的^*adirGa(-^*atirqa)“儿马”一词进行了语音和词源方面的全面分析和比较研究，并探寻了“阿尔泰语系”诸语言相关语音的对应关系及其发展演变状况，重新构拟了该词的原始形式。     

再论自然数的基数理论

自然数集扩充后，其基数理论起了相应变化，定义与运算法则都需作调整以适应数学教学与应用的需要．     

和为偶数的奇素数对的个数

和为偶数N的奇数对可分为三种情况,第一种是奇合数对(这里把1看做奇合数);第二种是1个是奇合数、1个是奇素数的奇数对;第三种是奇素数对.小于N的奇合数的大约个数可以根据奇合数所含的因数情况来求出,和为N的奇合数对的大约个数也可以根据奇合数对所含的因数情况来求出,小于N的奇合数除两两组成和为N的奇合数对外,其余只能与小于N的奇素数组成和为N的奇数对.求出前两种和为N的奇数对的大约个数,就能求出和为N的奇素数对的大约个数.     

《和为偶数的奇素数对的个数》之续篇

本续篇根据素数定理和有关无穷乘积,再度演化和为偶数的奇素数对的个数的求解公式,得出:和为偶数N的奇素数对的个数大于2N/πln2N,并且举几例比较结果.哥德巴赫猜想应该是和为偶数N的奇素数对的个数为1的一个特例。     

伯努利数与集合的纯偶划分数

引入集合的纯偶划分数,给出了一些它的性质,用纯偶划分数得到了伯努利数的一种表示形式,得到正切数的一种递归表示,指出正切数与二进多项式的一个关系式.     

Fermion number fractionization

Solitons emerge as non-perturbative solutions of non-linear wave equations in classical and quantum theories. These are non-dispersive and localised packets of energy — remarkable properties for solutions of non-linear differential equations. In the presence of such objects, the solutions of Dirac equation lead to the curious phenomenon of ‘fractional fermion number’. Under normal conditions the fermion number takes strictly integral values. In this article, we describe this accidental discovery and its manifestation in polyacetylene chains, which has led to the development of organic conductors. (left) Kumar Rao is a Postdoctoral Fellow at PRL, Ahmedabad. He is interested in particle physics phenomenology as probed in particle colliders and formal aspects of quantum field theory. (right) Narendra Sahu is currently a postdoctoral fellow at Lancaster University, UK. His main research area includes Cosmology and Astroparticle physics. Currently he is working on dark matter and matter anti-matter asymmetry of the universe. (center) P K Panigrahi’s research interests are in the area of quantum computation, solitons in Bose Einstein condensates & nonlinear optical media. He is also deeply interested in science education and derives pleasure from long weekend walks.     

Feeling number: grounding number sense in a sense of quantity

Drawing on results from psychology and from cultural and linguistic studies, we argue for an increased focus on developing quantity sense in school mathematics. We explore the notion of “feeling number”, a phrase that we offer in a twofold sense—resisting tendencies to feel numb-er (more numb) by developing a feeling for numbers and the quantities they represent. First, we distinguish between quantity sense and the relatively vague notion of number sense. Second, we consider the human capacity for quantity sense and place that in the context of related cultural issues, including verbal and symbolic representations of number. Third and more pragmatically, we offer teaching strategies that seem helpful in the development of quantity sense coupled with number sense. Finally, we argue that there is a moral imperative to connect number sense with such a quantity sense that allows students to feel the weight of numbers. It is important that learners develop a feeling for number, which includes a sense of what numbers are and what they can do.     

Littlewood and number theory

J E Littlewood (1885–1977) was a British mathematician well known for his joint work with G H Hardy on Waring’s problem and the development of the circle method. In the first quarter of the 20th century, they created a school of analysis considered the best in the world. Littlewood firmly believed that research should be offset by a certain amount of teaching. In this exposition, we will highlight several notable results obtained by Littlewood in the area of number theory.     

The congruent number problem

In Mathematics, especially number theory, one often comes across problems which arise naturally and are easy to pose, but whose solutions require very sophisticated methods. What is known as ‘The Congruent Number Problem’ is one such. Its statement is very simple and the problem dates back to antiquity, but it was only recently that a breakthrough was made, thanks to current developments in the Arithmetic of elliptic curves, an area of intense research in number theory.