The following reading lists will hopefully be useful for all incoming SAMBa students. They have been put together in order to summarise material that is either needed as a prerequisite for the courses on SAMBa or that will be used within the SAMBa courses.

Copies of all of the books are available in the SAMBa student offices.

## Core Material

This list comprises of areas within Mathematics that are core to core to the SAMBa mandatory units. All students should consider the level of preparation and prior exposure they currently have in these areas before entering SAMBa.

### Numerics and Computation

Students need to have done an introductory course on Numerical Analysis/Matlab. Before entering SAMBa, students should be familiar with the following topics (although some of these will be reviewed in the courses):

• Basic MATLAB Programming for Numerical Analysis.
• Floating point numbers and rounding error.
• Concepts of Convergence and Accuracy: Order of convergence, extrapolation and error estimation.
• Approximation of Functions: Polynomial interpolation, error analysis.
• Integration: Newton-Cotes formulae. Gauss quadrature. Composite formulae. Error analysis.
• ODEs: Euler, Backward Euler, Trapezoidal and explicit Runge-Kutta methods. Stability and convergence.
• Linear Algebra: Gaussian elimination, LU decomposition, pivoting, Matrix norms, conditioning, backward error analysis, iterative refinement.
The list below contains good references for this introductory material:

• Moler, Numerical Computing with Matlab (Link)
• Atkinson, An Introduction to Numerical Analysis , Wiley.
• Süli and Mayers, An Introduction to Numerical Analysis , Cambridge University Press.
• Iserles, A First Course in the Numerical Analysis of Differential Equations , Cambridge University Press.
The above books cover more than is needed. In addtion the list below are the references which are used during the core SAMBa courses:

• Golub and Van Loan, Matrix Computations , Johns Hopkins University Press.
• Demmel, Applied Numerical Linear Algebra , SIAM.
• Trefethen and Bau, Numerical Linear Algebra , SIAM.

### Statistics

Before entering SAMBa, students should be familiar with the following topics in statistics:

• Hypothesis testing.
• Maximum likelihood estimation.
• Properties of multivariate normal random variables.
• The central limit theorem.
• Linear models.
• Generalised linear models.
The SAMBa core unit MA40198 - Applied Statistical Inference is based on the textbook Core Statistics, which is available here. This book is an attempt to cover the minimum that a starting Ph.D. student in statistics should know. It contains a terse review of the prerequisite topics above, as well as the R language for statistical computing, which will be used in computer practicals. A student who is happy with everything in Core Statistics would be in very good shape for the statistical component of SAMBa.

Alternatively, books covering prerequisite material include the following

• Dalgaard, Introductory Statistics with R, Springer Science & Business Media
• Faraway, Linear Models with R, CRC Press
• Faraway, Extending the Linear Models with R, CRC Press
For students with a strong interest in statistics, the following books contain material that is useful for Applied Statistical Inference, and a great deal more besides.

• Davison, Statistical Models, Cambridge University Press.
• Gelman et al., Bayesian Data Analysis, Chapman & Hall/CRC.
• Held and Bove - Applied Statistical Inference, Springer.
Students who find this material challenging are advised to look at lecture notes from courses given by the Department of Mathematical Sciences, which will go at a slower pace to some of the literature above. To obtain these lecture notes, please email M.L.Thomas@bath.ac.uk

This list comprises of other areas within Mathematics that a SAMBa student may follow but are not core to any of the SAMBa mandatory units. Depending on the path through SAMBa that the student envisages, they should consider the level of preparation and prior exposure that they currently have in the following broad areas.

### Probability

Basic combinatorics, independence, random variables, discrete and continuous probability distributions, multivariate Normal distribution, law of large numbers, central limit theorem, conditional probability, conditional expectation, Markov processes, basic measure theory. Reading material includes:

• Ross, A First Course in Probability (9th ed.), Pearson.
• Grimmet and Stirzaker, Probability and Random Processes (3rd ed.), Oxford. (Relevant Chapters: 1 and 6.1-6.11)
• Williams, Probability with Martingales, Cambridge University Press. (For an introduction to measure-theoretic probability. Relevant Chapters: 0-6)
• Bartle, The Elements of Integration and Lebesgue Measure, Wiley (For those with no measure-theoretical background. Relevant Chapters: 1-6.)
For those interested in taking MA50251 - Applied Stochastic Differential Equations in their first year should ensure that they read and are familiar with the following prerequisite material in probability:

### Partial Differential Equations

Solution methods for Laplace’s equation, the heat equation and the wave equation in simple geometries in one, two and three space dimensions. Dirichlet and Neumann boundary conditions. Green’s functions. Fourier series solutions. Fourier and Laplace Transforms. Reading material includes:

• Pinchover and Rubinstein, An Introduction to Partial Differential Equations , Cambridge University Press.
• Strauss, Partial Differential Equations: An Introduction , Wiley

### Dynamical Systems

Linear ODEs, stability and classification of equilibrium points. Stable, unstable and centre manifolds. Codimension-one bifurcations (saddle-node, Hopf) of equilibria. Periodic orbits, Poincaré. maps, bifurcations of periodic orbits. Global bifurcations. Lyapunov exponents and chaos. Reading material includes:

• Glendinning, Stability, Instability and Chaos, Cambridge University Press.
• Strogatz, Nonlinear Dynamics and Chaos, Perseus Books, Cambridge, MA.

• Guckenheimer and Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (later edition), Springer.
• Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer.

### Mathematical Biology

Population dynamics of one or more species in discrete and continuous time, infectious diseases, population genetics, biological motion, molecular and cellular biology, pattern formation in biological systems, and tumour modelling. Reading material includes:

• Britton, Essential Mathematical Biology, Springer.

### Fluid Mechanics and Geophysical Flows

Streamlines and particle paths. Euler and Navier-Stokes equation for incompressible flow. Reynolds number. Streamfunctions. Vorticity. Invisicid flows and Bernoulli’s equation. Potential flow (e.g. around a cylinder) and Stokes flow. Boundary layers. Reading material includes:

• Acheson, An introduction to fluid dynamics, Oxford University Press.