Pre-requisite: AMA1020 & PMA1020
Cauchy sequences, especially their characterisation of convergence. Infinite series: further convergence tests (limit comparison, integral test), absolute convergence and conditional convergence, the effects of bracketing and rearrangement, the Cauchy product, key facts about power series (longer proofs omited). Uniform continuity: the two-sequence lemma, preservation of Cauchyness (and the partial converse on bounded domains), equivalence with continuity on closed bounded domains, a gluing lemma, the bounded derivative test. Mean value theorems including that of Cauchy, proof of ''Hôpital's rule, Taylor's theorem with remainder. Riemann integration: definition and study of the main properties, including the fundamental theorem of calculus.
Exam 60% Class Test 15% Projects 15% Continuous Assessment 10%
L. Alcock, How to Think About Analysis (Oxford University Press)
R. Haggarty, Fundamentals of Mathematical Analysis (Addison-Wesley)
A. McCluskey and T.B.M. McMaster, Undergraduate Analysis - a Working Textbook (Oxford University Press)
M. Spivak, Calculus (Cambridge University Press)
Equivalence relations, binary operations, groups, examples and properties, groups of maps, countability, homomorphisms, subgroups, equivalence relations, permutation groups, normal subgroups, quotient groups, structure of finite abelian groups, composition series and solvable groups.
Exam 80% Team Project 20%
PMA2020 Linear Algebra and Complex Variables (1st semester)
Pre-Requisites: PMA1020 & AMA1020
Lecturers: Dr M Mathieu, Dr C Ballance
Basic theory: matrices and Gauss elimination, LU decomposition; abstract vector spaces and subspaces, linear independence, basis, dimension; vector spaces associated to a matrix; general solution to inhomogeneous systems of linear equations; determinants; matrix inversion and computation of determinants; direct sums and direct products; inner products, orthogonality, Cauchy-Schwarz inequality; matrix representation of linear maps, eigenvalues and eigenvectors of matrices and endomorphisms; spectrum, diagonalization, similarity transformations; special matrices (symmetric, hermitian, orthogonal, unitary, normal) and their properties
Additional topics and applications, such as: least-squares approximation, Schur decomposition, orthogonal direct sums, Gram-Schmidt orthogonalization, adjoint maps, Jordan normal form
Basic theory of complex variables: algebra of complex numbers, de Moivre's theorem, integer powers and roots; sequences of complex numbers, convergence, limits; series of complex numbers, convergence and absolute convergence, convergence tests; limits of functions, continuity; complex differentiability, analytic functions; power series; elementary functions: exp, sin, cos, sinh, cosh, log, complex powers; complex integration: paths, length of path, path integrals, fundamental theorem of calculus, Cauchy's theorem, Cauchy's integral formula, Liouville's theorem, Taylor's theorem; Laurent series, Laurent's theorem, classification of isolated singularities, residues, Cauchy's residue theorem; computation of real integrals with complex methods
Additional topics and applications, such as: analytic functions as conformal maps, solutions to Laplace's equation, Moebius transformations, Cauchy-Riemann equations, harmonic functions, maximum modulus principle, Rouche's theorem, isolated zeroes theorem, fundamental theorem of algebra.
One 3-hour written examination 90%, 2 homework assignments 10% .
PMA2021 Graph Theory (1st Semester)
Pre-Requisites: PMA1020 and AMA1020
Lecturers: Dr M Mathieu
- Subgraphs, paths and cycles of a graph. Shortest path problem and Dijkstra's algorithm.
- Bipartite graphs and trees. The connector problem and Kruskal's algorithm. Connectivity.
- Eulerian graphs and tours. The postman problem. Hamiltonian graphs. The travelling salesman problem. Matchings. The marriage problem. Hall's theorem.
- Edge and vertex colourings. Chromatic numbers. Ramsey Theory.
- Graphs on surfaces. Planar graphs. Kuratowski's theorem. Colouring of planar graphs.
- Directed graphs. The one-way traffic problem. Network flows. The max-flow min-cut theorem.
One 3-hour written examination 80%, Coursework 20%.