Pre-requisite: AMA1020 & PMA1020
Sequences & Series: Supremum and infimum, Cauchy sequences, convergent sequences. The Bolzano-Weierstrass theorem. Infinite series, convergence tests.
Limits & Continuity: Limit of a function at a point. Continuity. Intermediate value theorem. Bounds of a continuous function on a bounded closed interval.
Differentation: Definition of derivative. Basic results on the derivative. Rolle's theorem. Mean value theorems. L'Hôpital's rule. Taylor's theorem. Local maxima and minima.
Riemann Integration: Definition of the Riemann integral and study of its main properties. Differentiation of the indefinite Riemann integral.
Exam 60% Class test 15% Projects 15% Continuous Assessment 10%
R. G. Bartle and D. R. Sherbert, Introduction to Real Analysis (Wiley & Sons)
R. Haggarty, Fundamentals of Mathematical Analysis (Addison-Wesley)
K. E. Hirst, Numbers, Sequences and Series (Arnold)
P. E. Kopp, Analysis (Arnold)
K. G. Binmore, Mathematical Analysis (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 T Todorov, Dr S Shkarin
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; special 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%.