Dr Jozef Skokan and Prof Adam Ostoja-Ostaszewski

This course is compulsory on the BSc in Actuarial Science, BSc in Business Mathematics and Statistics, BSc in Mathematics and Economics, BSc in Mathematics with Economics and BSc in Statistics with Finance. This course is available on the BSc in Accounting and Finance, BSc in Econometrics and Mathematical Economics, BSc in Economics, MSc in Econometrics and Mathematical Economics (2 Year Programme) and MSc in Economics (2 Year Programme). This course is available as an outside option to students on other programmes where regulations permit. This course is available with permission to General Course students.

Students should ideally have taken the course Mathematical Methods (MA100) or equivalent, entailing intermediate-level knowledge of calculus and linear algebra, linear independence, eigenvalues, diagonalisation, and proficiency in techniques of differentiation and integration.

This course develops ideas first presented in MA100. It is divided into two halves: calculus and linear algebra. The calculus half explores how integrals may be calculated or transformed by a variety of manipulations, and how they may be applied to the solution of differential equations. This aim is achieved by studying the following topics: Limit calculations. Riemann integral. Multiple integration. Improper integrals. Manipulation of integrals. Laplace transforms. Riemann-Stieltjes integral (permitting application of the Laplace transform to discrete and continuous probability distributions), to a level of detail dependent on time constraints. The linear algebra half covers the following topics: Vector spaces and dimension. Linear transformations, kernel and image. Real inner products. Orthogonal matrices, and the transformations they represent. Complex matrices, diagonalisation, special types of matrix and their properties. Jordan normal form, with applications to the solutions of differential and difference equations. An application to population dynamics. Singular values, and the singular values decomposition. Direct sums, orthogonal projections, least square approximations, Fourier series. Right and left inverses and generalized inverses.

20 hours of lectures and 9 hours of classes in the MT. 20 hours of lectures and 11 hours of classes in the LT. 5 hours of lectures and 1 hour of classes in the ST.

Written answers to set problems will be expected on a weekly basis.

A Ostaszewski, Advanced Mathematical Methods for both halves.

Useful background texts:

(i) for the calculus half: Ken Binmore & Joan Davies, Calculus, Concepts and Methods; M R Spiegel, Laplace Transforms; R A Adams, Calculus.

(ii) for the linear algebra half: Martin Anthony and Michele Harvey, Linear Algebra: Concepts and Methods (Cambridge University Press 2012).

Exam (100%, duration: 3 hours) in the main exam period.