An interesting class of purely pandiagonal, i.e. non-magic, whole number (integer) squares of orders (row/column dimension) of the powers of two which are related to Gray codes and square Karnaugh maps has been identified. Treated as matrices these squares possess just two non-zero eigenvalues. The construction of these squares has been automated…

Theon's ladder is an ancient algorithm for calculating rational approximations for the square root of 2. It features two columns of integers (called a ladder), in which the ratio of the two numbers in each row is an approximation to the square root of 2. It is remarkable for its simplicity. This algorithm can easily be generalized to find rational…

This note discusses a simple operator technique based on the differentiation and shifting properties of the Laplace transform to find Laplace transforms for various elementary functions. The method is simpler than known integration techniques to evaluate Laplace transforms.

Parametric differentiation is used to derive the partial fractions decompositions of certain rational functions. Those decompositions enable us to integrate some new combinations of trigonometric functions.

A little known property of determinants is developed in a manner accessible to beginning undergraduates in linear algebra. Using the language of matrix theory, a classical result by Sylvester that describes when two polynomials have a common root is recaptured. Among results concerning the structure of polynomial roots, polynomials with pairs of…

The main purpose in this note is to provide a generalization of the concept of mean, using the algebraic structure of commutative semigroups. Also, the pedagogical aim is to discuss with undergraduate math students the idea of generalizing a concept or a structure as a typical mathematical activity.

This paper explores a geometrical way to sketch graphs of the general quadratic in two variables with Geometer's Sketchpad. To do this, a geometric procedure as described by De Temple is used, bearing in mind that this general quadratic equation (1) represents all the possible conics (conics sections), and the fact that five points (no three of…

Until the early 1970s, ecologists generally assumed that erratic fluctuations observed in natural populations were a product of stochastic noise. It is now known that extremely complex dynamics can arise from basic deterministic processes. This field of study is generally called chaos theory. Here, a computer program, SLAC (Stability, Limits, And…

The UK government's widening participation strategy, and the concomitant development of a mass higher education system, has imposed a variety of pressures on higher education institutions. Not least of these is the changing nature of the student population, and the assumptions that can be made about its skills and knowledge base. It should not be…

This paper explores analytically the connections between three commonly used statistical measures of agreement: sensitivity, specificity, and Cohen's (unweighed) kappa, which are employed often in clinical studies. In particular, we show that if both sensitivity and specificity are below 7/8, then a kappa of 3/4 or greater (which is considered to…

This paper presents an alternative proposal concerning the teaching of mathematics. The present paper can be placed within the broader framework of the teaching of mathematics, but also within the more specific framework of category theory (CT). In other words, new ways will be investigated in which CT can be best developed within the broader…

Latin squares have existed for hundreds of years but it wasn't until rather recently that Latin squares were used in other areas such as statistics, graph theory, coding theory and the generation of random numbers as well as in the design and analysis of experiments. This note describes Latin and diagonal Latin squares, a method of constructing…

A new Newton-like iterative formula for the solution of non-linear equations is proposed. To derive the formula, the convergence criteria of the one-parameter iteration formula, and also the quasilinearization in the derivation of Newton's formula are reviewed. The result is a new formula which eliminates the limitations of other methods. There is…

It is known that any totally skew quantity with (n + 1) indices, each of which ranges over n values, vanishes identically. The aim of this short note is to show that this is equivalent to the simple fact that any (n + 1) vectors in an n-dimensional vector space are linearly dependent.

This note presents a new proof of Gould's famous formula for the Bernoulli numbers.