In this paper we explore some of the geometry that lies behind the real linear, second order, constant coefficient, recurrence relation(1)where a and b are real numbers. Readers will be familiar with the standard method of solving this relation, and, to avoid trivial cases, we shall assume that ab ≠ 0. The auxiliary equation of t2 = at + b of (1) has two (possibly complex) solutionsand the most general solution of (1) is given by

1. (i) when are real and distinct;

2. (ii) when

3. (iii)

This paper focuses on the mathematics behind the repayment of financial debt. The availability of credit, the inevitable accompaniment of which is debt, is an essential component of a monetised economy. Without it most people would not be able to purchase large items like homes, cars and other expensive consumer durables. Businesses would not expand and prosper without access to credit. However, the debt incurred by the availability of credit can also bring huge stress, and indeed distress, from personal and commercial insolvencies to the inhumanities of debt bondage associated with modern-day slavery and human trafficking. The historical origins of debt lie in antiquity and even pre-date the existence of monetised economies [1].

Gerrymandering has affected U. S. politics since at least 1812. A political cartoon that year decried this tactic by then Massachusetts Governor Elbridge Gerry. (Gerrymandering is manipulating the boundaries of districts to benefit a group unfairly.)

This article, like our previous one [1], combines known and new characterisations of parallelograms. Both can be thought of as additions to Martin Josefsson’s series on ‘characterisations of’ and ‘properties of’ various types of quadrilaterals – a series that does not include parallelograms. Josefsson’s publications can be found listed in [2], [3] and [4]. For the importance of characterisations in geometry, see [5].

This Article is a follow-up to a recent Gazette Article about the lost boarding pass problem by Grimmett and Stirzaker [1]. According to their book [2, 1.8.39, p. 10], it seems that they recognised this lovely problem in 2000 or earlier. We quote it with suitable minor changes.

When studying properties of the circumcircle of an integer triangle, it quickly becomes evident that the radius of such a circle (circumradius) need not itself be an integer. When it is not an integer, the circumradius can still be rational but it can also be irrational, as exemplified in the following examples. It is left to the reader to verify that the triangle with sides 10, 24, 26 has circumradius 13, and that the corresponding values for the triangles with sides 13, 14, 15 and 1, 1, 1 are 65/8 and respectively. It is shown in Theorem 1 below that a necessary condition for the circumradius to be an integer is that the area of the triangle is itself an integer (Heronian triangle) but this condition is not in itself sufficient. A simple counterexample is given by the 13, 14, 15 triangle above which has area 84. However, as a consequence of Theorem 1, we can restrict ourselves to consideringHeronian triangles, and relevant properties of such triangles, proved in [1], are given in Theorems 2 and 3 below. We also need to quote some well-known results involving the sums of two squares (see e.g. [2]) and these are listed in Lemma 3. In all that follows, we will use the convention that if T is a triangle with sides a, b, c and z > 0, then zT will denote the triangle, similar to T, with sides za, zb, zc.

What is it that makes us judge two proofs of the same theorem to be the same or different? This is not an idle question: one central aspect of judging mathematics is the novelty of the mathematics presented. This is important everywhere, from the peer-review system, to assigning international prestige, to funding agencies’ grant decisions. It even matters to some extent in examinations, to avoid accusations of collusion. Surprisingly, philosophers of mathematics have not paid the question of novelty much attention. In this Article, we will consider the appealing conjecture that the main ideas that make up the proof, the essence of a proof, can indeed be identified and that very different styles of proofs can share common main ideas. Further, that a particular theorem can be proved using quite different, independent main ideas. As a means of exploring whether this is plausible, we will present a number of novel proofs of the following theorem.

In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph G is a pair G = (V, E), where V and E are the vertex set and the edge set of G, respectively. The order and size of G is the number of vertices and edges of G, respectively. The degree or valency of a vertex u in a graph G (loopless), denoted by deg (u), is the number of edges meeting at u. If, for every vertex ν in G, deg (ν) = k, we say that G is a k-regular graph. The cycle of order n is denoted by Cn and is a connected 2-regular graph. The path graph of order n is denoted by Pn and obtain by deleting an edge of Cn. A tree is an undirected graph in which any two vertices are connected by exactly one path, or equivalently a connected undirected graph without cycle. A leaf (or pendant vertex) of a tree is a vertex of the tree of degree 1. An edge of a graph is said to be pendant if one of its vertices is a pendant vertex. A complete bipartite graph is a graph G with and such that every vertex of the set (part) X is connected to every vertex of the set (part) Y. If , then this graph is denoted by Km,n. The complete bipartite graph K1,n is called the star graph which has n + 1 vertices. The distance between two vertices u and ν of G, denoted by d (u, ν), is defined as the minimum number of edges of the walks between them. The complement of graph G is denoted by and is a graph with the same vertices such that two distinct vertices of are adjacent if, and only if, they are not adjacent in G. For more information on graphs, refer to [1].

The celebrated theorem of Weierstrass, dating from 1885, states that continuous functions can be uniformly approximated by polynomials on any bounded, closed interval. But just how well can we approximate by polynomials of a certain degree? Let us introduce some notation to facilitate the discussion. For an interval I (which will usually be ), denote by C (I) the space of continuous functions on I, and write for (the notation is often used). Uniform convergence of fn to f (on I) equates to the statement that . Denote by the space of polynomials of degree not more than n. This is a linear subspace of C (I) of dimension n + 1. We write