Volterra’s principle resolves a seeming paradox in environmental control- that an attempt to eradicate a pest may increase pest levels, if the intervention also interferes with existing predators. This writeup considers two such examples- the cottony cushion scale insect in the USA, and fishing in the Adriatic Sea – and derives the principle mathematically through consideration of Lotka-Volterra differential equations for predator/prey interaction.

Attempts to give an overview of how mathematicians deal with, or make use of, a notion of infinity. This is done through describing a series of mathematical ‘playgrounds’- groups, the real numbers, extended/hyper reals, polynomials, limits and projective geometry.

Lyapunov (Liapunoff) stability is the standard notion of stability and a vital notion in the study of dynamical systems (including applications such as Mathematical Biology). It appears on the syllabus of many courses at Bath. This writeup covers the defintion (with plain-english interpretation), describes the direct method, and proves the Lyapunov stability theorem. Lyapunov functions and asymptotic stability are briefly mentioned.

Describes the physical interpretation of the brachistochrone, a motivating example of the calculus of variations. Identification of a special case of that method, and its application to the problem.

Application of the Baire category theorem to something more interesting than functional analysis- cardinality. Demonstrates the uncountability of the reals, and the incompleteness of the rationals.

Three formulations of the IVT, with proof of equivalence and a non-mathematical explanation of the theorem.

Techniques for visualising a 4-dimensional spacetime curve: streamlines, particle paths and streaklines. Example calculations for each, and an explanation of what each method demonstrates.

Description of, and theorems on, conservative functions in vector calculus. Proves the equivalence (in simply connected domains) of being conservative, being irrotational, and having a scalar potential.