Kantorovich’s theorem for Newton’s method on Lie groups

The convergence criterion of Newton’s method to find the zeros of a map f from a Lie group to its corresponding Lie algebra is established under the assumption that f satisfies the classical Lipschitz condition, and that the radius of convergence ball is also obtained. Furthermore, the radii of the uniqueness balls of the zeros of f are estimated. Owren and Welfert (2000) stated that if the initial point is close sufficiently to a zero of f, then Newton’s method on Lie group converges to the zero; while this paper provides a Kantorovich’s criterion for the convergence of Newton’s method, not requiring the existence of a zero as a priori.     

Kantorovich＇s theorem for Newton＇s method on Lie groups

The convergence criterion of Newton＇s method to find the zeros ofa mapffrom a Lie group to its corresponding Lie algebra is established under the assumption thatfsatisfies the classical Lipschitz condition, and that the radius of convergence ball is also obtained. Furthermore, the radii of the uniqueness balls of the zeros of fare estimated. Owren and Welfert （2000） stated that if the initial point is close sufficiently to a zero off then Newton＇s method on Lie group converges to the zero; while this paper provides a Kantorovich＇s criterion for the convergence of Newton＇s method, not requiring the existence of a zero as a priori.     

Weierstrass’s theorem — Leaving no ‘Stone’ unturned

In this article, we discuss the basic theme of approximating functions by polynomial functions. Although it is exemplified by the classical theorem of Weierstrass, the theme goes much further. Even on the face of it, the advantage of polynomial approximations can be seen from the fact that unlike general continuous functions, it is possible to numerically feed polynomial interpolations of such functions into a computer and the justification that we will be as accurate as we want is provided by the theorems we discuss. In reality, this theme goes deep into subjects like Fourier series and has applications like separability of the space of continuous functions. Marshall Stone’s generalisation to compact Hausdorff spaces is natural and important in mathematics. Applications of the Weierstrass approximation theorem abound in mathematics — to Gaussian quadrature for instance.     

The Archimedes principle and Gauss’s divergence theorem

This article explores the connection between the Archimedes principle in physics and Gauss’s divergence theorem in mathematics.     

A note on the converse of Lagrange’s theorem

In this section of Resonance, we invite readers to pose questions likely to be raised in a classroom situation. We may suggest strategies for dealing with them, or invite responses, or both. ??Classroom?? is equally a forum for raising broader issues and sharing personal experiences and viewpoints on matters related to teaching and learning science.     

Pythagorean theorem from Heron’s formula: Another proof

In this section of Resonance, we invite readers to pose questions likely to be raised in a classroom situation. We may suggest strategies for dealing with them, or invite responses, or both. “Classroom” is equally a forum for raising broader issues and sharing personal experiences and viewpoints on matters related to teaching and learning science.     

Pólya’s one theorem with 100 pages of applications

In 1937, George Pólya wrote what is considered one of the most significant papers in 20th-century mathematics. The article contained one theorem and 100 pages of applications. It introduced a combinatorial method which led to unexpected applications to diverse problems in science like the enumeration of isomers of chemical compounds. Pólya’s theory of enumeration was discussed in detail in [1] by Shriya Anand, a summer student of the author. In what follows, we briefly recall the theory and complement the earlier article by adding some other applications not discussed there.     

Ehrenfest’s theorem and nonclassical states of light 1. Ehrenfest’s theorem in quantum mechanics

When it was first enunciated, Ehrenfest’s Theorem provided a necessary and important link between classical mechanics and quantum mechanics. Today, the content of the theorem is understood to be a natural and immediate consequence of the equation of motion for operators when quantum mechanics is formulated in the Heisenberg picture. Nevertheless, the theorem leads to useful approximations when systems with Hamiltonians of higher than quadratic order in the dynamical variables are considered. In this two-part article, we use it to provide a convenient illustration of the differences between so-called classical and nonclassical states of radiation.     

From Earth to Heaven: Using ‘Newton’s Cannon’ Thought Experiment for Teaching Satellite Physics

Thought Experiments are powerful tools in both scientific thinking and in the teaching of science. In this study, the historical Thought Experiment (TE) ‘Newton’s Cannon’ was used as a tool to teach concepts relating to the motion of satellites to students at upper secondary level. The research instruments were: (a) a teaching-interview designed and implemented according to the Teaching Experiment methodology and (b) an open-ended questionnaire administered to students 2 weeks after the teaching-interview. The sample consisted of forty students divided into eleven groups. The teaching and learning processes which occurred during the teaching-interview were recorded and analyzed. The findings of the present study show that the use of the TE helped students to mentally construct a physical system which has nothing to do with their everyday experience (i.e. they had to imagine themselves as observers in a context in which the whole Earth was visible) and to draw conclusions about phenomena within this system. Specifically, students managed (1) to conclude that if an object is appropriately launched, it may be placed in an orbit around the Earth and to support this conclusion by giving necessary arguments, and (2) to realize that the same laws of physics describe, on the one hand, the motion of the Moon around the Earth (and the motion of other celestial bodies as well) and, on the other hand, the motion of ‘terrestrial’ objects (i.e. objects on the Earth, such as a tennis ball). The main difficulties students met were caused by their idea that there is no gravity in the vacuum (i.e. the area outside of the Earth’s atmosphere) and also by their everyday experience, according to which it is impossible for a projectile to move continuously parallel to the ground.     

An Analysis of UNESCO’s Role In The Development of HigherEducation Policy Around The Globe

In recent years,Globalization enhances the development of education and corporation between higher education organizations.Being part of international organization involved in education policy-making,UNESCO has play an indispensable role in the enhancement of higher education.Present essay introduces the structures and working patterns of UNESCO and its role in accelerating the formulation of higher education policy.Meanwhile,limitations and challenged UNESCO confronted are also provided to give a detailed analysis of the role UNESCO plays in higher education.     

Educational software for improving learning aspects of Newton’s Third Law for student teachers

In this paper, we present the design, development, implementation and evaluation of educational software “Newton-3”, aiming at the learning of Newton’s Third Law by student-teachers who are not Physics majors. We describe the theoretical issues of our teaching approach and the various software tasks that we designed in order to promote students’ understanding. Specifically, the software is designed for the teaching of gravitational and electrostatic interactions between two distant bodies at rest. It is a web-based application and runs on a simple web browser with Macromedia Flash plug-in installed. The development of software and its integration into teaching–learning sequence is based on three main characteristics: the range of contexts in which the concept of force interaction applies, in the specification of the concept, and in an appropriate teaching learning environment (IDRF). We trialled the software on two groups of 8 primary school and 8 pre-school student-teachers, for 3 teaching periods, in the School of Education of our University. The research results indicate that the implementation was effective as the majority of the teacher-students improved their own knowledge concerning the existence and representation of gravitational and electrostatic interactions. An interesting result reveals that student-teachers have difficulty in perceiving the equality of magnitudes of action and reaction forces. This problem seems to be overcome after the teaching of the Inverse square law.     

The Treatment of Cycloidal Pendulum Motion in Newton’s Principia

The discovery of the near isochrony of the simple pendulum offered the possibilityof measuring time intervals more accurately than had been possible before. However,the fact that it was not strictly isochronous for all amplitudes remained a problem. Thecycloidal pendulum provided this strict isochrony and, over a thirty year period from1659 the analysis of the motion of this pendulum was developed. Newtons analysis inhis Principia was both elegant and comprehensive and his argument is illustratedin this paper. It provides insights into the revolutionary nature of Newtons thinkingespecially compared to the Galilean approach to understanding the motion of the simplependulum found in early 18th century textbooks.     

Using Photographs to Probe Students’ Understanding of Physical Concepts: The Case of Newton’s 3rd Law

The starting point of the present research is the following question: since we live in an age that makes increasing use of visual representations of all sorts, is not the visual representation a learner constructs a window into his/her understanding of what is or is not being learned? Following this direction of inquiry, the present preliminary study introduces and evaluates a novel technique for pinpointing learners’ misconceptions, namely, one that has learners create and interpret their own photographs (CIP). 27 high-school students and 26 pre-service teacher trainees were asked to assume the role of textbook designers and create a display—photograph plus attached verbal explanation—which, in their opinion, best depicted Newton’s 3rd law. Subsequent analysis of the participants’ photographs yielded the following six misconception categories: 3rd law not depicted; 3rd law depicts a sequence of events; tendency to introduce irrelevant entities in explanations; the word ‘reaction’ used colloquially; tendency to restrict the application of the third law to dynamic situations; and informal explanations in which the word “force” is absent. The findings indicate that, indeed, the CIP method can be effectively employed to elicit, detect, and investigate learners’ misconceptions. The CIP method joins the growing efforts to utilize the yet relatively untapped potential of visual tools for science education purposes.     

The effect of concept cartoon-embedded worksheets on grade 9 students’ conceptual understanding of Newton’s Laws of Motion

Background: A substantial review study of concept cartoons reports that few studies have indicated their functions. For this reason, the present study illuminates the extent to which concept cartoon-embedded worksheets (through constructivist context) accomplish these functions in conceptual learning.

Purpose: The purpose of the study is to determine the effect of the concept cartoon-embedded worksheets on grade 9 students’ conceptual understanding of Newton’s Laws of Motion.

Sample: Within a quasi-experimental research design with control group, the sample consisted of 102 grade 9 students (aged 15–16 years) enrolled in an Anatolian High School. They were randomly assigned to control (52 students) and experimental groups (50 students).

Design and methods: While the control group followed the existing physics instruction, the experimental one was exposed to the concept cartoon-embedded worksheets. To collect data, two scales (Newton’s Laws of Motion Test –NLMT– and Interview about Instances –IAI) were employed.

Results: The results indicated that the experimental group performed better than did the control group.

Conclusions: It can be deduced that the concept cartoon-embedded worksheets are effective in improving grade 9 students’ conceptual understanding and in replacing their alternative conceptions with scientific ones.     

Newton’s Investigation of the Resistance to Moving Bodies in Continuous Fluids and the Nature of ‘Frontier Science’

Newton’s experiments into the resistance which fluids offer to moving bodies provide some insight into the way he related theory and experiment. His theory demonstrates a way of thought typical of 17th century physics and his experiments are simple enough to be replicated by present day students. Newton’s investigations using pendulums were discussed in a previous paper. In this paper Newton’s theory of the resisted motion of falling bodies is outlined, his investigations into the resistance offered to falling bodies by water and the experiments of others into the resistance offered by air are described and some implications for teaching about the nature of science are discussed.