Syllabus

MA380 Science (honours)

 MA341 Metric Spaces

• Metric spaces: examples of metric spaces; convergence in metric spaces; pointwise, uniform and mean convergence; continuity; open and closed sets; closure, interior and boundary; compactness in metric spaces; the Bolzano-Weierstrass theorem
• Completeness: Rⁿ and C[a,b]; contractions; the fixed point theorem; applications to differential equations etc
• Fractal geometry: the space of fractals; the Hausdorff metric; iterated function systems; algorithms for generating fractals; fractal dimension

Texts

• A.N. Kolmogorov & S.V. Fomin, "Real Analysis" (Dover)

MA342 Topology

• Topological spaces: examples; continuity and convergence; subspaces, quotients and product spaces
• Connectedness and path connectedness: components; totally disconnected spaces
• Compactedness and its applications: the Heine-Borel theorem; compactness of subspaces and product spaces; compactness and sequential compactness
• Convergence: the Hausdorff and other separation properties; inadequacy of sequences; nets; filters and ultrafilters

Texts

• S. Lipschutz, "General Topology" (Schaum Outline)

References

• A. Armstrong, "Basic Topology" (Springer)
• S. Barr, "Experiments in Topology" (Murray)
• M. Barnsley, "Fractals Everywhere" (Academic Press)
• N. Bourbaki, "General Topology" (Hermann)
• V. Bryant, "Metric Spaces, Iteration and Applications" (Cambridge)
• E. Copson, "Metric Spaces" (Cambridge)
• K. Falconer, "Fractal Geometry" (Wiley)
• G. Francis, "A Topological Picturebook" (Springer)
• V. Mendelson, "Introduction to Topology" (Alyn & Bacon)
• J. Munkres, "Topology, A First Course" (Prentice Hall)
• I.M. Singer & J.A. Thorpe, "Lecture Notes on Elementary Topology and Geometry" (Springer)
• W.A. Sutherland, "Introduction to Metric and Topological Spaces" (Oxford)
• S. Willard, "General Topology" (Addison Wesley)

MA343 Group Theory (part one)

• Group axioms, cyclic groups, permutation groups.
• Normal subghroups, homomorphisms, isomorphism theorems.
• Permutation groups, isomorphism theorems, linear groups
• Direct and semidirect products
• Finitely generated Abelian groups
• Automorphisms, groups of automorphisms

MA344 Group Theory (part two)

• Group actions, automorphism groups of graphs, application to enumeration.
• Sylow's Theorem, groups of small order
• Simple groups, the Jordan-Hölder Theorem
• Soluble and nilpotent groups
• Semigroups, machines, flip-flop and simple memory machines, flip-flop with reset

References

• J.B. Fraleigh, "A First Course in Abstract Algebra" (Addison Wesley)
• W. Ledermann, "Introduction to the Theory of Finite Groups" (Oliver & Boyd)
• I.D. Macdonald, "The Theory of Groups" (Oxford)
• J.S. Rose, "A Course on Group Theory" (Cambridge)

MA385 Numerical Analysis (Part one)

• Modelling with first order differential equations.
• Euler's method and convergence.
• Higher order one-step methods: Heun's method, modified Euler method, Runge-Kutta method of order 4.
• Practical implementation of the Runge-Kutta method using variable step size.
• Introduction to C programming on the VAX.
• Round-off errors: machine operations, well-conditioned problems, numerical trustworthiness of algorithms, numerical stability of algorithms, error estimation.
• Round-off errors and one-step methods.
• Shooting method for higher order (non-linear) differential equations.
• Iterative methods for finding zeros of non-linear functions: convergence, Aitken's accelerated convergence, quadratic convergence and Newton's method.
• Computer implementation of the shooting method for a non-linear second order boundary value problem.
• Outline of the finite element method for a two-dimensional Dirichlet boundary value problem.

MA378 Numerical Analysis (Part two)

• Numerical quadrature in one variable (and polynomial interpolation): Monte Carlo integration, Newton-Cotes integration and associated errors.
• Gaussian integration (existence, uniqueness and errors of interpolating polynomials, Neville's algorithm, divided differences, Gram-Schmidt process and orthogonal polynomials).
• Numerical quadrature in several variables.
• Systems of linear equations: Gaussian elimination.
• Finer points on the finite element method: completion of a matrix space, dependence of errors on triangulations.

 

J. Stoer & R. Bulirsch. Introduction to Numerical Analysis. Springer.

---

MA387 Statistics

• Introduction to probability theory: probability spaces, properties of probabilities, conditional probability, independence
• Combinatorial analysis and counting
• Random variables: discrete and continuous variables; expectation, variance, covariance, moments
• The Markov, Cauchy-Schwartz-Bunjakowski, and Chebysev's Inequality; correletion coefficients
• Jointly distributed random variables, marginal and conditional densities, iterated expectation
• Distributions of sums, products, and quotients of random variables
• Order statistics, sampling statistics
• Moment generating functions, and characteristic functions, the Uniqueness and Continuity Theorems
• The Weak Law of Large Numbers, the Central Limit Theorem, the normal and Poisson approximations to the binomial density
• Relations between the normal, Chi Squared, Tau, and F densities, and applications to sampling

Texts

• H.J. Larson, "Introduction to Probability" (Addison Wesley)
• Hogg & Craig . "Introduction to Mathematical Statistics" (Pentice Hall)

References

• P. Brémaud, "An Introduction to Probabilistic Modelling" (Springer)
• K.L. Chung, "Elementary Probability Theory with Stochastic Processes" (Springer)
• W. Feller, "An Introduction to Probability Theory and its Applications, Vol I" (Wiley)
• P.G. Hoel, S.C. Port & C.J. Stone, "Introduction to Probability Theory" (Houghton Mifflin)
• R.V. Hogg & A.T. Craig "Introduction to Mathematical Statistics" (Collier Macmillan)

---

Back to 3rd Year Maths Courses.
Back to Higher Diploma In Mathematics.