Algebraic methods in plane geometry

‘Riders’ in geometry are always a pleasure to tackle, and this pleasure is doubled when one finds connections between plane geometry and algebra. This three-part article is about such connections. In Parts 1 and 2, we explore some connections between plane geometry and the algebra of conics and cubics; in Part 1 we give algebraic proofs of results such as Pascal’s Theorem and the Butterfly Theorem, and in Part 2 we study some group theoretic and number theoretic aspects of cubic curves. In Part 3 we look at the role of mappings and transformation groups in plane geometry.     

韦达定理和逆定理在解析几何中的应用

平面解析几何,是用代数方法研究平面几何图形的一个教学分支,它所提出的问题以及问题的结论都是几何形式,而中间的论证和推导基本上是用代数方法。本文通过具体的例子,介绍了韦达定理和逆定理在解析几何中的应用。     

Reflections around the Ramanujan centenary

This article is based on the tape of an extemporaneous talk to a general audience at the Tata Institute of Fundamental Research, Bombay, India, given after the conclusion of the Centenary Conference held there (January 1988). Reproduced fromAtle Selberg, Collected Papers. Vol 1”, pages 695–701, Springer Verlag, 1989.     

Beginnings of group cohomology

Cohomology theory is a powerful mathematical tool. This theory applied to topology is a part of algebraic topology, which associates algebraic invariants to topological spaces. In this article, we give a brief outline of group cohomology. R Sridharan has been an Adjunct Professor at the CMI, Chennai since retiring from TIFR, Bombay in 2000 where he was a senior professor. He is also an INSA senior scientist now. His scholarship permeates to literature (English and Sanskrit) and philosophy as well. Many of the algebraists in the country were his students. Dedicated to my teacher S Eilenberg     

Desargues对偶定理的推广及应用

Desargues对偶定理主要用于证明仿射平面上的共点线,为使Desargues对偶定理能在初等几何中有所应用,将无穷远点还原为直线的平行,并运用其解决欧氏平面上的线共点问题。     

Natural language as a tool for analyzing the proving process: the case of plane geometry proof

In the field of human cognition, language plays a special role that is connected directly to thinking and mental development (e.g., Vygotsky, 1938). Thanks to “verbal thought”, language allows humans to go beyond the limits of immediately perceived information, to form concepts and solve complex problems (Luria, 1975). So, it appears language can be studied as a cognitive process (Chomsky, 1975). In this investigation, I study language as a means for making the cognitive process explicit. In particular, I analyze the role of the verbalization produced by pairs of students solving a plane geometry problem. The basic idea of my research is that, during the resolution process of a plane geometry problem, natural language can play roles beyond that of communication: Natural language can be seen as a tool for supporting students’ cognitive processes (Robotti, 2008), and, at the same time, it can also be seen as a researchers’ tool which allows us to shed light on the evolution of students’ cognitive processes. With regard to language as researchers’ tool, I show how natural language (in our case, students’ verbalization during resolution of a plane geometry problem) can be used by the researcher to make explicit, to study, and to describe the development of the students’ cognitive processes during the resolution process. To this end, I present a model I have developed that allows us to identify, in students’ verbalization, different phases of their cognitive processes.     

George Andrews’ game

This is an expository article showing how Zeck-endorf’s Theorem (every positive integer can be represented in one and only one way as the sum of non-consecutive Fibonacci numbers) can be used to construct a number-guessing game invented by Professor George Andrews. Jerold Mathews is professor (emeritus) of mathematics at Iowa State University, Ames, Iowa, USA, where he served on the faculty from 1961–1995. His interests include research in point set topology, history of mathematics and writing textbooks. He enjoys reading, photography and helping international students learn English.     

Foreign-born academic scientists and engineers: producing more and getting less than their U.S.-born peers?

The Chronicle of Higher Education recently reported that the number of doctoral degrees awarded in the U.S. rose 3.4 percent in 2004, largely because of an increase in foreign students [Smallwood (2005). Doctoral degrees rose 3.4% in 2004, survey finds. The Chronicle of Higher Education, December 9, 2005]. Currently, 20.9 percent [National Science Board (2003). The science and engineering workforce realizing America’s potential. NSB, vol. 3, National Science Foundation] of all science and engineering faculty positions at U.S. universities are held by foreign-born scientists (with even larger percentages in computer science and engineering)—and we can expect higher numbers of foreign-born faculty at U.S. universities in the future. In this paper, we use 2001 Survey of Doctorate Recipients (SDR) data from the National Science Foundation to compare productivity levels, work satisfaction levels and career trajectories of foreign-born scientists and U.S.-born scientists. The results indicate that foreign-born academic scientists and engineers are more productive than their U.S.-born peers in all areas. Yet, average salaries and work satisfaction levels for foreign-born scientists are lower than for U.S.-born scientists. The use of NSF data does not imply NSF endorsement of the research methods or conclusions contained in this report.     

Hilbert‘s Sixteenth Problem and Its Generalization

This paper deals with Hilbert‘s 16th problem and its generalizations.The configurations of all closed branches of an algebraic curve of degreee n are discussed.The maximum number of sheets for an algebraic equation of degree n and the maximum number of limit cycles for a planar algebraic autonomous system are achieved.The author also considers different generalizations and some related problems.     

Pedagogical Content Knowledge and the 2003 Science Teacher Preparation Standards for NCATE Accreditation or State Approval

The 2003 National Science Teachers Association Standards for Science Teacher Preparation (NSTA-SSTP) were developed to provide guidelines and expectations for science teacher preparation programs. This article is the fourth in a special JSTE series on accreditation written to assist science teacher educators in meeting the NSTA-SSTP. In this article, the authors discuss pedagogical content knowledge and how this is expressed in the NSTA-SSTP. Included are competencies and examples needed for a science teacher preparation program to document developing pedagogical content knowledge in preservice science teachers.     

The evolution of compilers

The term compiler was coined in the late forties, by Grace Murray Hopper, a pioneer who rose to the challenges of programming the first computers. The problem of translation from a source language into a target language was viewed as a ‘compilation’ of a sequence of machine language subprograms selected from a library. In this article we briefly trace the evolution of compilers from their beginnings as huge sprawling algorithms in the early fifties, to their current elegant, phase ordered forms. Priti Shankar is with the Department of Computer Science and Automation at the Indian Institute of Science, Bangalore. Her interests are in theoretical computer science and error correcting codes.     

Promoting student collaboration in a detracked, heterogeneous secondary mathematics classroom

Detracking and heterogeneous groupwork are two educational practices that have been shown to have promise for affording all students needed learning opportunities to develop mathematical proficiency. However, teachers face significant pedagogical challenges in organizing productive groupwork in these settings. This study offers an analysis of one teacher’s role in creating a classroom system that supported student collaboration within groups in a detracked, heterogeneous geometry classroom. The analysis focuses on four categories of the teacher’s work that created a set of affordances to support within group collaborative practices and links the teacher’s work with principles of complex systems. An earlier version of this article was presented at the Annual Meeting of the American Educational Research Association, Chicago, 2007 as part of the Tracking and Detracking SIG session Teaching, Learning, and Other Outcomes in Tracked and Detracked Environments.     

Using concept maps to assess change in teachers’ understandings of algebra: a respectful approach

The purpose of this study was to explore teachers’ growth in understanding of algebra using concept maps. The study was set in the context of a five-year National Science Foundation funded teacher retention and renewal professional development project. In this project both beginning and experienced teachers are supported as they increase their understanding about mathematics, their ability to implement effective mathematics practices in their classrooms, and their knowledge of working with English Learners. Results indicate that teachers’ algebraic knowledge structures became more complex and connected as a result of their professional development. In addition, they were able to adapt their knowledge networks to incorporate important aspects of algebra into them. Concept maps are recommended to other leaders of mathematics professional development as a means of assessing change.     

Refined empirical line method to calibrate IKONOS imagery

INTRODUCTION To extract quantitative biophysical parameter such as leaf biomass and leaf chlorophyll concentra- tion from the remotely sensed imagery accurately, the effects of atmospheric scattering and absorption must be removed. Atmospheric effects add to or diminish true ground reflectance, if the atmospheric spectral features are not properly removed. A significant analytical bias could be introduced for data interpre- tation (Ben-Dor and Levin, 2000). Many approaches have been deve…     

Numerical prediction of vortex flow and thermal separation in a subsonic vortex tube

INTRODUCTION Vortex tube is a device for producing hot and cold air when compressed air flows tangentially into the vortex chamber through the inlet nozzles. This causes the vortex and swirl flow movement inside the vortex tube. The air in the middle region of the tube has lower velocity and lower temperature than the inlet air. So the air near the tube wall has higher ve-locity and higher temperature than the inlet air. The cold air in the core region of the tube flows out through the o…     

The prime ordeal

Prime numbers have fascinated mankind through the ages. In fact, one may think that we know all about them. However, this is not so! One does not know the answers to many basic questions on primes. We shall concentrate here mainly on questions and discoveries whose statements are elementary and accessible. Right at the end, we mention a result whose statement is simple but whose proof uses rather sophisticated mathematics. Even here, we do not try to be exhaustive. The subject is too vast for that to be possible. After a long stint (1981–1999) at TIFR, Bombay, B Sury moved to ISI, Bangalore.     

The Adidactic Interaction with the Procedures of Peers in the Transition from Arithmetic to Algebra: A Milieu for the Emergence of New Questions

The purpose of the present article is to give an account of the emergence of knowledge pertaining to the transition from arithmetic to algebra in the course of a debate in a grade 7 classroom. This debate follows two other instances of work: (1) the adidactic interaction between each student and a given problem, (2) the adidactic interaction of each student with the procedures generated by other students during stage 1. The two kinds of processes – adaptation to a milieu and social interactions – play a critical role in the change of “rationality” required for the move from arithmetic to algebra. Both the design of the initial mathematical problem given to the students and the organization of the interactions leading to the debate under study in this article are based on this hypothesis. The research presented in this article is set in a broader work of didactic engineering that aims at studying didactic conditions for making a connection between arithmetic practices and algebraic practices.     

Why Compulsory Science Education Should <Emphasis Type="Italic">Not</Emphasis> Include Philosophy of Science

Like many readers of this journal, I have long been an advocate of having science students introduced to philosophy of science. In particular, influenced by the Philosophy for Children movement founded by Matthew Lipman, I have advocated such an introduction as early as possible and have championed early secondary school as an appropriate place. Further, mainstream science curricula in a number of countries have, for some time now, supported such introductions (albeit of a more limited sort) under the banner of introducing students to the “Nature of Science”. In this paper, I explore a case against such introductions, partly in role as “Devil’s Advocate” and partly exploring genuine qualms that have come to disturb me. Generally speaking, my judgement is that no justification is available in terms of benefit to the individual or to society of sufficient weight to outweigh the loss of freedom of choice involved in such forced learning. One possible exception is a minimalist and intellectually passive “Nature of Science” introduction to some uncontroversial philosophical views about science. An earlier version of this paper was presented to the Seventh International Conference on the History and Philosophy of Science and Science Teaching, University of Winnipeg, Winnipeg and subsequently published in its proceedings (see my 2003). I am grateful to those who engaged in discussion of the paper upon its presentation. I am also grateful to the advice of this journal’s anonymous referees.     

Unpacking Rouché’s Theorem

Abstract

Rouché’s Theorem is a standard topic in undergraduate complex analysis. It is usually covered near the end of the course with applications relating to pure mathematics only (e.g., using it to produce an alternate proof of the Fundamental Theorem of Algebra). The winding number provides a geometric interpretation relating to the conclusion of Rouché’s Theorem, but most undergraduate texts give no geometric insights that lead to an understanding of why Rouché’s Theorem holds. In addition, most texts do not inform students that a stronger version of the theorem exists. In this paper we present a simplified proof of the stronger version, which is a suitable topic for students to pursue as a short project, and provide a geometric argument for the weaker version. Finally, as a project for advanced students, we unpack a standard application of this theorem as used in control systems: the Nyquist stability criterion.     

A geometry in which all triangles are isosceles An introduction to non-archimedean analysis

The purpose of this article is to introduce ‘a new analysis’ to students of mathematics at the undergraduate and postgraduate levels, which in turn introduces a geometry very different from our Euclidean geometry and Riemannian geometry. Some strange things happen: for instance, ‘every triangle becomes isosceles!’.