This note presents and discusses a class of real integrals involving a hyperbolic function by means of complex integration. An integral representation is obtained which appears in several fields of physics, including statistical mechanics, condensed matter and quantum optics and more in the so called dispersion relations.

Whether or not there is an interaction between two factors in their effects on a dependent variable is often a central question. This paper proposes a general mechanism by which an interaction may arise: (a) the two factors are the same thing--or, at least, have a dimension in common--in the sense that it is meaningful to add (or subtract) them;…

This article discusses the process of integrating the use a software system into a conventional course in Abstract Algebra. It is a description of how and why a software system was used. It uses a particular system, Groups32, as an example.

This paper analyses a lecture by an excellent teaching award winner professor of mathematics, given to high school mathematics teachers. The analysis is based upon two sources: (i) the lecture plan, as expressed in a series of 29 transparencies, prepared by the lecturer in advance; (ii) the actual implementation of the lecture, as transcribed from…

This paper describes students' solutions to a commonly taught and not commonly taught inequality. The findings showed students' difficulties. Participants implicitly and explicitly exhibited two intuitive beliefs: inequalities must result in inequalities and solving inequalities and equations are the same process. Following the analysis of…

The origin of the function concept is usually traced to Galileo's work on motion. We argue that specific proto-function concepts appeared in the work of Tartaglia a century before the publication of Galileo's Two New Sciences. The study of Tartaglia's ideas can be used in the classroom as a historical introduction to various function concepts,…

Scenarios in which a teacher provides the student with an initial description or demonstration of some procedure to be learned are common in many science and technology fields. Structuring such initial presentations to assist rather than inhibit the student's representation of the target procedure is a real pedagogical concern. Production systems…

An analysis is made of the role of Fibonacci numbers in some quadratic Diophantine equations. A general solution is obtained for finding factors in sums of Fibonacci numbers. Interpretation of the results is facilitated by the use of a modular ring which also permits extension of the analysis.

The present paper describes an educational experiment dealing with teaching the theory of formal languages and automata as well as their application concepts. It presents a practical application of an educational experiment and initial results based on comparative instruction of two samples of students (n = 56). The application concept should…

The traditional approach to expressing cumulants in terms of moments is by expansion of the cumulant generating function which is represented as an embedded power series of the moments. The moments are then obtained in terms of cumulants through successive reverse substitutions. In this note we demonstrate how cumulant-moment relations are…

This note discusses a possible method for teaching an example of problem solving to intermediate grade students. Under the guidance of the instructor, students may see the links between the 'guess' method and the arithmetic operation 'strings'. This may help enhance their logical thinking ability as well as arithmetic operation skills. For…

Categorical data sequences occur in many applications such as forecasting, data mining and bioinformatics. In this note, we present higher-order Markov chain models for modelling categorical data sequences with an efficient algorithm for solving the model parameters. The algorithm can be implemented easily in a Microsoft EXCEL worksheet. We give a…

The following note contains the following sections: Introduction; The Kronecker Product and Some Formulae; and Proof of Formula. It discusses in detail the Kronecker product and how it relates to differential systems.

This paper describes an interactive modelling, simulation, animation, and real-time control (MoSART) environment for a class of 'cart-pendulum' electromechanical systems that may be used to enhance learning within differential equations and linear algebra classes. The environment is useful for conveying fundamental mathematical/systems concepts…

The topic 'relative velocity' has recently been introduced into the Cambridge Ordinary Level Additional Mathematics syllabus under the application of Vectors. In this note, the results of relative velocity and the 'reduction to rest' technique of teaching relative velocity are derived mathematically from vector algebra, in the hope of providing…