Integration is an important topic discussed in first year calculus. Many subsequent topics, such as solving first-order or second-order differential equations, requires the evaluation of at least one integral, and sometimes several.

The most challenging material in integration is arguably the techniques of integration. Unfortunately, due to time constraints, only basic techniques are commonly taught, while in reality, many integrals having special forms are solvable only by using more sophisticated techniques.

This problem was posed in the 1930s by Paul Erdős [1]. It is the simplest case of a still unsolved conjecture.

If k2 + 1 squares with no interior point in common are drawn inside a unit square then the maximum possible sum of their side-lengths is k [2].

We shall use the notation S(n) to denote the maximum possible sum of the side-lengths for n squares drawn with no interior point in common inside a unit square. The main aim of this article will be to develop an approach to the study of the function S which will give surprisingly simple proofs of a number of known results. This approach will then be used to prove a new result about the asymptotic behaviour of S.

During the 19th century various criticisms of Euclid's geometry emerged and alternative axiom systems were constructed. That of David Hilbert ([1], 1899) paid particular attention to the independence of the axioms, and it is his insights which have shaped many of the further developments during the 20th century.

We can, from his insights, define an affine plane as a set of points, with distinguished subsets called lines such that

Axiom 1: Given two distinct points, there is a unique line containing them both.

Axiom 2: Given a line L and a point, p, not contained in L, there is a unique line containing p which does not intersect L.

Axiom 3: There exist at least three points, not belonging to the same line.

TantrixTM tiles are black hexagons imprinted with three coloured paths [1] joining pairs of edges. There are three different kinds of path. One is a straight line going from an edge to the opposite edge, one a circular arc joining adjacent edges and one an arc of larger radius joining alternate edges (or two apart). Tiles can be rotated but, since they are opaque, they cannot be turned over. A careful enumeration would indicate that, identifying tiles under rotation but not under reflection, there are 16 such tiles. However, the two tiles consisting of three straight lines (meeting at the centre of the hexagon) are not part of the set, so actually there are only 14 different tiles. The game is played by matching tiles to connect paths of the same colour; the goal is to create loops or long paths of a single colour This easy to learn yet hard to master game has inspired research on strategy (e.g. [2]) and complexity (e.g. [3]).

In the performance of digital communication systems over fading channels, the error analysis is typically modelled using a Gaussian probability distribution. One function central to the analysis is what engineers routinely refer to as the (Gaussian) Q-function and is defined by

1

This is the canonical representation used for the function. In this paper a number of derivations of an important alternative representation for the Q-function known as Craig's formula will be given.

In 1879, the Irish astronomer Robert Stawell Ball published a slim book entitled simply Mechanics [1]. This book appeared as part of the series of ‘London Science Class-Books’, published by Longmans, Green & Co. These books were intended as elementary science texts for use in schools, and, as a consequence, their mathematical content was quite basic — even for those books on supposedly mathematical topics. In this article, I will look at Ball's handling of his subject, and compare his book to its distant ancestor: Newton's Principia.