Syllabus

MA260 Arts (honours)

MA286 Real Analysis

• Continuity and differentiability of a function f:R^m->Rⁿ, partial derivatives, directional derivatives, the Chain Rule
• Maxima and minima
• Revision of the main definitions and properties of sequences and series of real numbers
• Lim inf and lim sup, Cauchy's criterion for convergence, Taylor series, power series, Fourier series, uniform convergence, differentiation term by term
• Multiple integrals

Texts

• M. Spiegel. Advanced Calculus. Schaum Outline.
• T.M. Apostol, "Mathematical Analysis" (Addison Wesley)

MA287 Complex Analysis

• Functions of a complex variable: differentiability, the Cauchy-Riemann equations, harmonic conjugates
• Line integrals, logz and e^z
• Cauchy's Integral Theorem, Cauchy's Formula, Cauchy's Inequalities
• The Laurent series of a function, poles, residues
• Contour integration, Rouché's Theorem
• Conformal mappings, Möbius transformations

Texts

• R.V. Churchill & J.W. Brown, "Complex Variables and Applications" (McGraw Hill)
• M.R. Spiegel, "Complex Variables" (Schaum Outline)

MA283 Linear Algebra

• Vector spaces, bases, dimension, linear maps, matrix representation of linear maps
• Matrix algebra, kernels and images, least squares fitting, inner product spaces, the Gram-Schmidt process
• Fourier series, dual spaces, the rank of a matrix, determinants, eigenvalues and eigenvectors, the characteristic polynomial, quadratic forms
• Diagonalisation of a symmetric or Hermitian linear map
• Triangularisation of a linear map
• The Hamilton-Cayley theorem, linear programming

Texts

• T.S. Blyth & E.F. Robertson, "Matrices and Vector Spaces and Linear Algebra" (Chapman & Hall)
• S. Lipschutz, "Linear Algebra" (Schaum Outline)

References

• H. Anton, "Elementary Linear Algebra" (Wiley)
• J.W. Archbold, "Algebra" (Pitman)
• G. Birkhoff & S. MacLane, "A Survey of Modern Algebra" (Macmillan)
• I.N. Herstein, "Topics in Algebra" (Blaisdell)
• S. Lang, "Linear Algebra" (Springer)
• T.A. Whitelaw, "An Introduction to Linear Algebra" (Blackie)

MA284 Discrete Maths

• Enumeration: product rule, sum rule and sieve principle, selections and distributions, the pigeonhole principle
• Graphs, the fundamentals (including various notions of 'path' and 'tree'), plus a study of some of the following topics: colouring problems, bipartite graphs, Hamiltonian graphs, planar graphs and tournaments.
• Algorithms and applications are emphasised throughout.

Texts

• N.L. Biggs, "Discrete Mathematics", (Oxford)

MA227/228 Statistics

• Explanation of statistics through practical examples of its applications.
• Data summarisation and presentation
• Numerical measures of location and spread for both ungrouped and grouped data
• graphical methods including histograms, stem-and-leaf and box plots.
• Probability
• The role of probability theory in modelling random phenomena and in statistical decision making
• Sample spaces and events; some basic probability formulae; conditional probability and independence; Baye's formula; counting techniques; discrete and continuous random variables; hypergeometric and binomial distributions; normal distributions; the distribution of the sample mean when sampling from a normal distribution;.
• The Central Limit Theorem with applications including normal approximations to binomial distributions.
• Statistical Inference
• Concepts of point and interval estimation; concepts in hypothesis testing including Type I and Type II errors and power.
• Confidence intervals and hypothesis tests about a single population mean, a single population proportion, the difference between two population means, a single population variance and the ratio of two population variances; the analysis of enumerative data, including chi-squared goodness-of-fit and contingency table tests
• Correlation and linear regression analysis, including least squares estimation of the parameters of the simple linear regression model, inferences about these parameters and prediction.

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)