If the point of suspension of a multiple pendulum is suitably oscillated then the pendulum can remain in motion in an upside-down position. Since such pendulums can model flexible materials, this inverted motion is sometimes referred to as an 'Indian rope trick'. Despite the complexity of the governing differential equations, this rope trick can…

In this note, an integrated form of some significant means with two variables is provided, and some chains for mean value inequalities are obtained. At the same time, a concise family of algebraic functions appears, which satisfy Mitrinovic's requirements.

It is reasonably well known that the ratios of consecutive terms of a Fibonacci series converge to the golden ratio. This note presents a simple, complete proof of an interesting generalization of this result to a whole family of 'precious metal ratios'.

The standard reverse-order law for the Moore-Penrose inverse of a matrix product is (AB)[dagger] = B[dagger]A[dagger]. The purpose of this article is to give a set of equivalences of this reverse-order law and other mixed-type reverse-order laws for the Moore-Penrose inverse of matrix products.

Poisson integral formula is revisited. The kernel in the Poisson integral formula can be derived in a series form through the direct BEM free of the concept of image point by using the null-field integral equation in conjunction with the degenerate kernels. The degenerate kernels for the closed-form Green's function and the series form of Poisson…

In this note, the notion of reverse derivation is studied. It is shown that in the class of semiprime rings, this notion coincides with the usual derivation when it maps a semiprime ring into its centre. However, we provide some examples to show that it is not the case in general.

This paper investigates a general identity which expresses an apparently complicated and intriguing sum of fractions as an elegant and straightforward sum of simple terms.

It is very well known that the Cauchy-Schwarz inequality is an important property shared by all inner product spaces and the inner product induces a norm on the space. A proof of the Cauchy-Schwarz inequality for real inner product spaces exists, which does not employ the homogeneous property of the inner product. However, it is shown that a real…

The analysis of tautochrone problems involves the solution of integral equations. The paper shows how a reasonable assumption, based on experience with simple harmonic motion, allows one to greatly simplify such problems. Proposed solutions involve only mathematics available to students from first year calculus.

Within the general framework of a multifactor term structure model, the fundamental partial differential equation (PDE) satisfied by a default-free zero-coupon bond price is derived via a martingale-oriented approach. Using this PDE, a result characterizing a model belonging to an exponential affine class is established using only a system of…

In this paper, the author demonstrates a case study of undergraduate mathematics research in trigonometric sums. Involving undergraduates in research has been a long standing practice in the experimental sciences. However, it has only been recently that undergraduates have been involved in research in mathematics in significant numbers. Obviously,…

Some examples from different areas of mathematics are explored to give a working knowledge of the computer algebra system Maple. Perfect numbers and Mersenne primes, which have fascinated people for a very long time and continue to do so, are studied using Maple and some questions are posed that still await answers.

In this work, the authors present a commonly used example in electrostatics that could be solved exactly in a conventional manner, yet expressed in a compact form, and simultaneously work out special cases using the method of images. Then, by plotting the potentials and electric fields obtained from these two methods, the authors demonstrate that…

A point "E" inside a triangle "ABC" can be coordinatized by the areas of the triangles "EBC," "ECA," and "EAB." These are called the barycentric coordinates of "E." It can also be coordinatized using the six segments into which the cevians through "E" divide the sides of "ABC," or the six angles into which the cevians through "E" divide the angles…

This note takes up the issue of parallel curves while illustrating the utility of "Mathematica" in computations. This work complements results presented earlier. The presented treatment, considering the more general case of parametric curves, provides an analysis of the appearance of cusp singularities, and emphasizes the utility of symbolic…