Basic counting principal; permutations and combinations. Random phenomena; sample spaces and events; axioms of probability; random selection; rules of probability. Conditional probability; Bayes' rule; stochastic independence. Discrete random variables; expected value and variance; discrete distributions: binomial, Poisson, geometric, hypergeometric and negative-binomial.

Development of the skilled use of vector, differential and integral calculus in the modelling of dynamics of simple physical systems, including the analysis of force fields, motion and modelling assumptions.

Linear systems. Orthogonality: application to curve fitting. Eigenvalues and -vectors: application to systems of differential equations. Singular values: application to image processing. Numerical computations such as LU, QR and SVD factorisation and the computation of eigenvalues and -vectors. Condition numbers: sensitivity of linear systems.

Modelling of a wide variety of applications using ordinary differential equations (DEs). Linear, non-linear, separable and homogeneous DEs as well as systems of DEs. Analytical techniques (including Laplace transforms) as well as numerical methods for solving models. Emphasis on the various steps of the classic modelling process.

Applications of prime factorisation, divisibility, greatest common divisors, the Euler phi function, modular arithmetic, multiplicative inverses, algebraic groups and elementary combinatorics in cryptology (the protection of information) and coding theory (the integrity of information). Introductory graph theory: planarity, colourings, Hamiltonian and Euler graphs.

Numerical stability, and conditioning. Methods for solving non-linear equations; convergence analysis. Interpolation with polynomials and spline functions; error analysis. Numerical differentiation and integration. Numerical methods for solving initial value problems. The use of software like Matlab or Python for numerical calculations.

Modelling of the dynamics of continuous systems; convective and diffusive transport as special cases of the general transport theorem; stress dyadic; energy and heat transport; constitutive equations for fluids; derivation and solution of the Navier-Stokes equation; ideal flow; potential flow; computational simulation of fluid dynamics.

Fourier series, continuous and discrete Fourier transforms, convolution, Laplace transform, Sturm-Liouville theory, orthogonal functions. Applications in signal and image processing, as well as in the solution of ordinary and partial differential equations. Numerical Fourier analysis and the famous FFT (fast Fourier transform).