From natural numbers to numbers and curves in nature - II

In this second part of the article we discuss how simple growth models based on Fibbonachi numbers, golden section, logarithmic spirals, etc. can explain frequently occuring numbers and curves in living objects. Such mathematical modelling techniques are becoming quite popular in the study of pattern formation in nature.     

Triangular numbers

The triangular numbers, which are numbers associated with certain arrays of dots, were known to the ancient Greeks and viewed by them with reverence. Though possessing a simple definition, they are astonishingly rich in properties of various kinds, ranging from simple relationships between them and the square numbers to very complex relationships involving partitions, modular forms, etc. ?? topics which belong to advanced mathematics. They also possess many pretty combinatorial properties. In this expository article we survey a few of these properties.     

Transfinite numbers

In a series of revolutionary articles written during the last quarter of the nineteenth century, the great German mathematician Georg Cantor removed the age-old mistrust of infinity and created an exceptionally beautiful and useful theory of transfinite numbers. This is an introductory article on this topic.     

Complex numbers and plane geometry

The representation of complex numbers as points of the Euclidean plane naturally leads to a two-way interaction between geometry and numbers. The geometry of the plane has a very deep influence in the study complex analytic functions. In this article, we illustrate the other way aspect by a few simple-minded application of complex numbers to give elegant solutions of problems in plane geometry, such as Ptolemy’s Theorem, Euler-line and Nine-point Circle Theorem. Anant R Shastri is a Professor at IIT, Bombay. His research inerests are in algebraic topology and algebraic geometry. He is also keen in math education and music. This article is based on a talk given to an audience consisting mainly students of class IX and X, at Nehru Science Centre under the aegis of Bombay Association for Science Education and Bombay Math. Colloq. on 25th Jan. 2003. An earlier version of this article was published in Bona Mathematica, Vol.14 Nos.1–2, 2003.     

Complementarity, sets and numbers

Niels Bohr's term‘complementarity' has been used by several authors to capture the essential aspects of the cognitive and epistemological development of scientific and mathematical concepts. In this paper we will conceive of complementarity in terms of the dual notions of extension and intension of mathematical terms. A complementarist approach is induced by the impossibility to define mathematical reality independently from cognitive activity itself. R. Thom, in his lecture to the Exeter International Congress on Mathematics Education in 1972,stated ‘‘the real problem which confronts mathematics teaching is not that of rigor,but the problem of the development of‘meaning’, of the ‘existence' of mathematical objects'. Student's insistence on absolute ‘meaning questions’, however,becomes highly counter-productive in some cases and leads to the drying up of all creativity. Mathematics is, first of all,an activity, which, since Cantor and Hilbert, has increasingly liberated itself from metaphysical and ontological agendas. Perhaps more than any other practice,mathematical practice requires acomplementarist approach, if its dynamics and meaning are to be properly understood. The paper has four parts. In the first two parts we present some illustrations of the cognitive implications of complementarity. In the third part, drawing on Boutroux' profound analysis, we try to provide an historical explanation of complementarity in mathematics. In the final part we show how this phenomenon interferes with the endeavor to explain the notion of number. This revised version was published online in July 2006 with corrections to the Cover Date.     

民主党派在解放战争时期的历史作用

抗日战争胜利后,中国历史又进入一个新的阶段———解放战争时期。这一时期的民主党派继续与中国共产党竭诚合作,坚持和平民主,反对内战独裁,积极配合人民解放军作战,并逐步克服自身的弱点,走上新民主主义革命道路,为推翻反动派统治、建立新中国起到了重要作用。     

民主党派在解放战争时期的历史作用

抗日战争胜利后,中国历史又进入一个新的阶段———解放战争时期。这一时期的民主党派继续与中国共产党竭诚合作,坚持和平民主,反对内战独裁,积极配合人民解放军作战,并逐步克服自身的弱点,走上新民主主义革命道路,为推翻反动派统治、建立新中国起到了重要作用。     

数字的警示

素有“万物之灵”之称的人类，凭借自己的聪明才智和勤劳的双手，利用地球上的资源，为自己创造了一个个高度发达的物质文明。高楼大厦随处可见，如网的交通线在地球上延伸，一个个工业区在不同地方勾勒出人类的文明与进步，超市中的商品琳琅满目……当人类在充分享受物质文明的同时，接踵而来的环境危机也在为人类敲响警钟，让人类痛心疾首。让我们一同走进一组组令人触目惊心的数字。     

The influence of a peer-tutoring training model for implementing cooperative groupings with elementary students

This study examined the effects of peer-tutoring training on elementary school student communication and collaboration skills when used in conjunction with cooperative learning. Within six classes (grades 2–6) in an inner-city school, cooperative learning pairs were randomly assigned to two groups (control and training). Multivariate analyses of variance (MANOVA) of quantitative data from a systematic observation instrument used over an entire year showed that, in general, the training group surpassed the control group in both communication and collaborative skills. Students in grades 2–3 showed substantially more improvement than students in grades 4–6; also, students with average or below-average reading levels required more time to acquire these skills than did above-average students. The qualitative data further substantiated these findings while revealing a large variation among teachers in implementing cooperative learning. and is Editor of the Research section of this journal.     

Mayer-Jensen shell model and magic numbers

Many nuclear models have been put forward since 1932. Among them the collective model proposed by Aage Niels Bohr and Ben Roy Mottelson and the nuclear shell model proposed by Maria Goeppert Mayer and Johannes Hans D Jensen are the two most successful models. A number of experimental facts like the existence of magic numbers compiled by Maria Mayer led to the discovery of the nuclear shell model. The addition of a nuclear spin-orbit coupling force to the mean field of the nucleons successfully predicted the nuclear magic numbers and many other properties of nuclei. R Velusamy is the Head of the PG Department of Physics, Ayya Nadar Janaki Ammal College, Sivakasi, Tamilnadu. His current interests are in quantum dots and wave packet dynamics.     

从Fibonacci数的恒等变换到Lucas数的同余式

研究了Fibonacci数的和式∑a b=nUa^mUb^m/a!b!,得出了一些关于Fibonacci数与Lucas数的恒等变换和一些有趣的Lucas数的同余式。     

Are numbers logographs?

Researchers generally assume that logographs such as numerals form a homogeneous set. This paper presents a study whose results challenge that assumption. Using a visual search task, it is shown that native Chinese speakers, who can read and speak English, process strings of Chinese numerals differently from the way they process strings of Western numerals. The different pattern of results found with these two sets of numerals also contrasts with the pattern typically found when native English speakers process Western numerals. An explanation of this result is proposed based on the notion that visual search functions can be the outcome of the combined effect of different basic search procedures.     

关于酉完全数

证明了酉完全数都不是幂数.