A special project which can be given to students of ordinary differential equations is described in detail. Students create new differential equations by changing the dependent variable in the familiar linear first-order equation (dv/dx)+p(x)v=q(x) by means of a substitution v=f(y). The student then creates a table of the new equations and…

In this short note, a mathematical proposition on a functional equation for f(xy)=xf(y) + yf(x)for x,y [does not equal] 0, which is encountered in calculus, is generalized step by step. These steps involve continuity, differentiability, a functional equation, an ordinary differential linear equation of the first order, and relationships between…

The mean and variance of the number of turning points in random permutations of two kinds of distinct elements are evaluated. These results are applied to a Wald-Wolfowitz run test.

In this note 4 x 4 most-perfect pandiagonal magic squares are considered in which rows, columns and the two main, along with the broken, diagonals add up to the same sum. It is shown that the Moore-Penrose inverse of these squares has the same magic property.

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.

The sixth-order wave equation governing the propagation of one-dimensional acoustic waves in a viscous, heat conducting gaseous medium subject to relaxation effects has been considered. It has been reduced to a system of lower order equations corresponding to the finite speeds occurring in the equation, following a method due to Whitham. The lower…

A semigroup G is a group if it has a left identity and every element has a left inverse. The purpose of this note is to weaken this condition further in two different ways. A semigroup G with an identity e is a group if every element x in G has an inverse. It is well known that this statement can be weakened. A semigroup G with a left identity e…

Teachers of first-year college mathematics and engineering courses must often spend considerable time reviewing material originally taught in high school. Instead of this being a mere exercise in repetition, this article suggests that such a review can enrich and revitalize by unifying some of the subjects that need to be re-taught. In the example…

Using MAPLE enables students to consider many examples which would be very tedious to work out by hand. This applies to graph plotting as well as to algebraic manipulation. The challenge is to use these observations to develop the students' understanding of mathematical concepts. In this note an interesting relationship arising from inverse…

A family of integrals over infinite intervals appears frequently in the statistical literature on non-parametric regression. This note presents two simple transform methods, based on beta function and Parseval identity, to explicitly evaluate these integrals. These methods will benefit researchers and practitioners working in non-parametric…

This article describes a general instructional strategy designed to help students in the learning process from textbooks and to furnish opportunities for practice in critical reading. Students participate in cooperative learning by breaking the class up into small groups--the Study Teams--and providing them with worksheets and reading organizers,…

This note offers a derivation of the Euler-Maclaurin formula that is simple and elementary. In addition, the paper shows that the derivation provides Euler-Maclaurin formulas for a variety of functionals other than the trapezoid rule.

The following theorem is proved: let f : X --> Y and suppose points are closed in Y (that is, singleton sets are closed in Y ). Then, f is continous if and only if f[1] preserves separated sets. We also give a counter example in the event points are not closed in Y.

A simple and novel method for evaluating the partial fraction expansion of proper rational functions is presented. The technique involves simultaneous determination of the partial fraction coefficients associated with each of the multiple poles in the expansion in turn. Only synthetic division is required, which makes the process very suitable for…

In this article quadrilaterals with concurrent maltitudes are characterized. A generalization of the maltitudes is given, and a larger family of quadrilaterals for which an analogous property of concurrency holds is determined and studied.