This page summarises my contributions to Everything2. It’s a bit of a mess; the current approach is chronological. Viewing here rather than on e2 gets you access to beautifully LaTeX-rendered PDF versions, though. Semi-related is this collection of nodes on cryptographic protocol design.
Paraconsistent Logic In the classical logic typically applied to mathematics, a single contradiction is fatal, as it allows anything and everything – including its opposite- to be derived. However, real-world situations such as large software products often feature irresolvable conflicts, from which we would not wish to draw such broad conclusions. Reasoning about such systems motivates the study of paraconsistent logic. The linked article examines the various ways in which the explosion of inconsistency can arise, and thus which rules of inference need to be discarded for such logics.
The abc Conjecture The abc conjecture is deceptively easy to state considering the deep implications it has for number theory. This E2 entry covers a couple of formulations of the conjecture, examines the connection to the ABC theorem for polynomials, and briefly discusses some of the problems it impacts upon.
Volterra’s principle 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.
Linear Programming Summary of the key ideas in Linear programming, from Edexcel A-level module D1 and MA30087: Optimisation methods of Operational Research at the University of Bath. Contents: Example problems (natural language and mathematical); The canonical form- definition, sample constructions, notes on formulation; Theory- feasible, basic and optimal solutions, fundamental theorem of linear programming; Special cases.
Infinity 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.
Hilbert’s Nullstellensatz Discussion of (affine) Varieties, ideals and examples of how they relate. Statement and proof of the Nullstellensatz.
Lyapunov stability 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.
Brachistochrone 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.
Greatest common divisor Integer and polynomial GCD calculations: useful properties, Euclidean techniques,Resultants and the Sylvester matrix, non-Euclidean modular GCD by large/many small primes.
Buchberger’s Algorithm An algorithm for determining a Gröbner basis for a collection of polynomials.
Gröbner Basis A Gröbner basis for a system of polynomials preserves the common roots whilst being simpler relative to an ordering.
Canonical Representation of polynomials Discussion of canonical forms for polynomials- single variable dense and sparse representations; orderings for multivariate polynomials.
The Baire category theorem and cardinality 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.
Internal Rate of Return The internal rate of return (IRR) is a measure that allows for comparisons to be drawn between various investments. In technical terms, the IRR is the discount rate that sets the net present value of the investment to zero. It is of particular use in valuing bonds (the yield to maturity). Article examines IRR through example compound/discount calculations and examination of income streams.
St. Petersburg Paradox A demonstration of how expectation without a notion of utility causes problems in probability theory.
General linear group First introduction to group theory via the properties of the general linear group.
Cholesky factorisation Definition of the Cholesky factorisation, technique for deriving, problem cases and a worked example.
LU factorisation Solving linear equations via LU factorisation- definition, solution method, derivation of LU factorisation (with worked example) and considerations of complexity.
Intermediate Value Theorem Three formulations of the IVT, with proof of equivalence and a non-mathematical explanation of the theorem.
Methods for visualising fluid flow 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.
Register Machine From Turing machines to Register machines- overview, instruction set and simple examples. Simulation of recursive functions by Register machines (with proofs in macro form).
Adjoint Definition and properties of the adjoint. Proof of uniqueness. Notational issues.
Primary decomposition theorem Statement and proof of the primary decomposition theorem for linear operators.
Projection Description of projections and direct-sum decomposition of vector spaces.
Conservative 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.
Vandermonde determinant A special matrix structure with an easy-to-find determinant.
Clausal Form A standard form for first order formulae that consists of a number of clauses (series of atoms in conjunction that imply a disjunction of atoms). Brief overview of clauses (such as the headed horn clause used in Prolog/Cyc); clausal form algorithm; a worked example.
Semantic Tableaux proof method for predicate logic A backward-chaining proof method for predicate logic, motivated by proof by contradiction rather than deduction. Description of the semantic tableau structure and rules, with suggested order of application. Also includes worked examples and a proof of correctness and completeness (with reference to Gödel’s theorems).
Prenex and Skolem normal forms Two standard forms for first order predicate logic- Prenex normal form requires all the quantifiers to be at the front, whilst Skolem form further demands that only universal (forall) quantifiers are used. Gives an algorithm for finding prenex form with worked example; consideration of Skolem functions; Skolem normal form algorithm; a consideration of logical equivalence and the merits of these normal forms.
Language recognition and generation in Prolog A demonstration of weaknesses in Prolog’s goal-matching method in the context of language recognition/generation. Considers decidability, term languages, statement form, arbitrary strings from a restricted alphabet.