AC 2 1 001 Subject Level Semester ID number

Please check the semester your module runs in from the timetable.dundee.ac.uk website as it may have changed.

This module consists of a Calculus and Algebra component. It is part of a series of four modules, Mathematics 1A, 1B, 2A, 2B, which are the core Mathematics modules in years 1 and 2, and provide the foundations in Calculus, Algebra and Geometry for all mathematics modules in higher levels. This module is mandatory for all Level 1 students on Mathematics (including Mathematics combined) degrees. The module is also suitable for students on non-mathematics degrees and recommended for students on physics and computing degrees. If you have questions about this module please contact our Undergraduate Admissions Tutor.

### Prerequisites

Students taking this module should typically have at least a B in Mathematics in Scottish Highers, AS-, or A-Level, or an equivalent qualification.

### Indicative Content

• #### Functions

Number systems (N, Z, Q, R), open and closed intervals, elementary functions, domain, range, composition, inverse. Inequalities. Idea of a limit for functions and for sequences.

• #### Differential Calculus

Derivatives, tangents and rates of change. Simple derivatives by first principles. Treatment of (f + g)′ , (fg)′, (f/g)′, (f ? g)′ and inverse functions. Higher order derivatives. Implicit Differentiation. Revision of index laws and log to base a. Definitions and elementary properties of exp and ln. Solution of equations involving exponential and logarithmic functions. Differentiation of functions involving exponential and logarithmic functions. Logarithmic differentiation. Tangents and Normals to curves. Increasing and decreasing functions. Critical points. Curve sketching (including asymptotes).

• #### Polynomials

Quadratic polynomials. Algebra and geometric representation of complex numbers. Division algorithm, Remainder theorem. Roots of polynomials. Techniques of partial fraction decomposition.

• #### Trigonometry

Definitions and properties of the six trigonometric functions, including formulae for sin(A + B), sin A sin B, sin A + sin B, etc. Solution of trigonometric equations (including a cos(x) + b sin(x) = c).

• #### Series

Series as sequences of partial sums. Summation of series and sigma notation. Convergence of series, geometric series. Examples of finite and infinite series. Binomial theorem.

• #### Conics

Classification, standard forms, parametric representations.

### Delivery and Assessment

The module is delivered in the form of lectures and workshops/tutorials and assessed via coursework (100%) consisting of homeworks, projects and tests.

### Credit Rating

This module is a Scottish Higher Education Level 1 or SCQF level 7 module and is rated as 20 SCOTCAT credits or 10 ECTS credits.

This module consists of a Calculus and Algebra component. It is part of a series of four modules, Mathematics 1A, 1B, 2A, 2B, which are the core Mathematics modules in years 1 and 2, and provide the foundations in Calculus, Algebra and Geometry for all mathematics modules in higher levels. This module is mandatory for all Level 1 students on Mathematics (including Mathematics combined) degrees. The module is also suitable for students on non-mathematics degrees and recommended for students on physics and computing degrees. If you have questions about this module please contact our Undergraduate Admissions Tutor.

### Prerequisites

Students taking this module must have taken the module MA11001, or equivalent.

### Indicative Content

• #### Integral Calculus

Idea of integral, including elementary treatment of the definite integral as a limit using rectangles. Fundamental theorem of calculus. Methods of integration including integration by substitution, by parts and with partial fractions. Relation of integrals with areas. Trapezium and Simpson′s rules for numerical integration.

• #### Differential Equations

First order ordinary differential equations by (a) separation and (b) integrating factor. Second order ordinary differential equations with constant coefficients and simple right- hand sides. (Complex roots included, but no resonance problems.)

• #### Vectors

Vector geometry in R² and R³ vector properties and manipulation. Unit vectors, position vectors, Cartesian coordinates. Scalar product and vector product.

• #### Matrices and linear equations

Matrix properties, addition, multiplication. Inverse matrices, determinants. Linear mappings in R² (rotation, reflection). Systems of linear equations, Gaussian elimination and row operations.

• #### (Further) complex numbers

Polar form, exponential notation. Multiplication, de Moivre′s Theorem, powers and roots.

• #### Lines, planes and spheres

Implicit and parametric equations of lines. Implicit equations of planes. Intersections, distances between points, lines and planes. Equations of spheres, tangent planes. Linear dependence and independence, colinear and coplanar vectors.

### Delivery and Assessment

This module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (50%) and coursework (50%) consisting of homeworks and tests and project work.

### Credit Rating

This module is a Scottish Higher Education Level 1 or SCQF level 7 module and is rated as 20 SCOTCAT credits or 10 ECTS credits.

This module is mandatory for Level 1 students on BSc and MMath Mathematics degrees and is optional for Level 1 students taking the BSc in Mathematics combined with any of Accountancy, Economics, Financial Economics or Psychology. The module is also suitable for students on non-mathematics degrees and recommended for students on physics and computing degrees. If you have questions about this module please contact our Undergraduate Admissions Tutor.

### Prerequisites

Students taking this module should typically have at least a B in Mathematics in Scottish Highers, AS-, or A-Level, or an equivalent qualification.

### Indicative Content

• #### Logic

Propositions, negation, conjunction, disjunction, implication, equivalence. Truth tables

• #### Group Theory

Basic definitions and examples, commutative (Abelian) groups, Cayley tables, order of a group. Permutations and Cycles. Cyclic groups and generators. Subgroups.

• #### Proof

Constructive Proof, Disproof by Counterexample, Proof by Contradiction, Proof by Contrapositive, Proof by Induction.

• #### Number Theory

Integers and Divisibility. Greatest Common Divisor and the Euclidean Algorithm, leading to prime factorisation and the Fundamental Theorem of Arithmetic, and continued fractions. Properties of Primes. Linear Diophantine Equations. Relations, equivalence relations and congruences (modular arithmetic). (Higher degree) Diophantine Equations.

### Delivery and Assessment

The module is delivered in the form of lectures and workshops/tutorials and assessed via coursework (100%) consisting of tests.

### Credit Rating

This module is a Scottish Higher Education Level 1 or SCQF level 7 module and is rated as 20 SCOTCAT credits or 10 ECTS credits.

This module provides an introduction to statistics and probability. This module is mandatory for students taking the BSc or MMath in Mathematics, the BSc or MSci in Mathematics and Physics or the BSc in Mathematics and Accountancy. This module is optional for students taking the BSc in Mathematics combined with any of Economics, Financial Economics or Psychology. If you have questions about this module please contact our Undergraduate Admissions Tutor.

### Prerequisites

Students taking this module must have a C in Mathematics in Scottish Highers, AS-, or A-Level, or an equivalent qualification. Note: You may not take this course if you have previously passed or are currently taking AB12007 or MA22003.

### Indicative Content

• #### Data Analysis

Populations and samples; types of data. Data presentation. Mean; standard deviation. Interpretation of data

• #### Probability

Selection problems. Sample space; events; compound events; complements. Addition rules. Conditional Probability; the multiplication rule; independence. Bayes’ Theorem.

• #### Discrete Random Variables

Probability distribution. Probability mass functions (Uniform, binomial, geometric, Poisson distributions). Joint probability mass functions; covariance and independence. Expected value and variance of sums of random variables.

• #### Continuous Random Variables

Polynomial and negative exponential probability density functions. The Normal distribution and tables. Expected value and variance of continuous random variables. Sums and differences of independent normal random variables. The central limit theorem; Normal approximations. Random samples.

• #### Hypothesis Testing

Hypothesis formulation. Test statistics. p-values. Confidence intervals.

• #### Linear Regression

Least squares. Assessing usefulness of a model. Using a model.

### Delivery and Assessment

This module is delivered in the form of lectures and workshops/computer labs and assessed via an exam (50%) and coursework (50%) consisting of tests and lab reports.

### Credit Rating

This module is a Scottish Higher Education Level 1 or SCQF level 7 module and is rated as 20 SCOTCAT credits or 10 ECTS credits.

This module consists of a Calculus and Algebra component. It is part of a series of four modules, Mathematics 1A, 1B, 2A, 2B, which are the core Mathematics modules in years 1 and 2, and provide the foundations in Calculus, Algebra and Geometry for all mathematics modules in higher levels. This module is mandatory for all Level 2 students on Mathematics (including Mathematics combined) degrees. The module is also suitable for students on non-mathematics degrees and recommended for students on physics and computing degrees. If you have questions about this module please contact our Undergraduate Admissions Tutor or your Advisor of Studies.

### Prerequisites

Students taking this module must have achieved a pass mark in each of the modules MA11001 and MA12001, or equivalents.

### Indicative Content

• #### Differential Equations

Revision of linear differential equations of second order with constant coefficients using undetermined coefficients. General solutions and solutions satisfying initial conditions. Resonance. Equations of higher order.

• #### Hyperbolic Functions

Hyperbolic Functions; Solution of simple equations, inverse functions. Revision of standard methods of definite integration, including hyperbolic substitutions.

• #### Fundamentals of Calculus

Limits, Continuity and Differentiability. Rolle′s Theorem, Mean Value Theorem. Definition and properties of the Riemann integral, Fundamental Theorem of Calculus. L′Hôpital′s Rule and Indeterminate Forms. Infinite and improper integrals. Taylor & Maclaurin series.

• #### Vectors and vector spaces

Definition of a vector space, Rn. Vectors, lines and planes in Rn. Span, linear independence. Basis and dimension. Subspaces.

• #### Inner product

Scalar product, length. Projection. Normal form of hyperplanes in Rn. Orthogonality.

• #### Linear equations and matrices

Systems of linear equations. Gaussian elimination. Matrices and matrix operations, transposes and inverses. Matrix equations. LU factorisation. Determinants.

### Delivery and Assessment

The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (60%) and coursework (40%) consisting of homeworks and tests.

### Credit Rating

This module is a Scottish Higher Education Level 2 or SCQF level 8 module and is rated as 20 SCOTCAT credits or 10 ECTS credits.

The aim of this module is to make the students familiar with a Computer Algebra software package and to use this software to solve a number of problems from the area of Dynamical Systems. This module is mandatory for all Level 2 students on Mathematics (including Mathematics combined) degrees except for those taking the BSc or MSci in Mathematics and Physics, for whom it is optional. If you have questions about this module please contact our Undergraduate Admissions Tutor or your Advisor of Studies.

### Prerequisites

Students taking this module must have achieved a pass mark in each of the modules MA11001 and MA12001, or equivalents.

### Indicative Content

• #### An introduction to Maple

The Maple front end and syntax. Plotting. Integration/differentiation. Differential equations.

• #### An introduction to Dynamical Systems

Vector fields, the gradient field, integrals of motion, fixed points and their classification. Examples of dynamical systems, including mass on a spring, pendulum, Van Der Pol oscillator, non-linear oscillator. Conservation laws for a system of interacting bodies. Orbits in a gravitational field. Non-autonomous systems.

### Delivery and Assessment

The module is delivered in the form of lectures and computer lab workshops. Assessment is entirely computer-based and is via an exam (60%) and coursework (40%) consisting of homeworks and tests.

### Credit Rating

This module is a Scottish Higher Education Level 2 or SCQF level 8 module and is rated as 20 SCOTCAT credits or 10 ECTS credits.

This module provides an introduction for Level 2 students to various topics in Discrete Mathematics. This module is mandatory for Level 2 students taking the BSc or MMath in Mathematics or the BSc or MSci in Mathematical Biology. This module is optional for students taking the BSc in Mathematics combined with any of Accountancy, Economics, Financial Economics or Psychology. If you have questions about this module please contact our Undergraduate Admissions Tutor or your Advisor of Studies.

### Prerequisites

Students taking this module must have achieved a pass mark in each of the modules MA11001 and MA12001, or in EG11003 and EG12003, or equivalents.

### Indicative Content

• #### Sets and Graphs

Sets and subsets: definitions, examples, Set operations, basic identities, power of a set, Cartesian product of sets, relations on sets, Basic graph terminology.

• #### Recurrence relations (Difference Equations)

Definition of a recurrence relation (difference equations), Homogeneous and inhomogeneous difference equations, Nonlinear difference equations: xn+1 = g(xn), Fixed points, linearisation, stability of fixed points. Applications: the Newton and Secant Methods to solve non-linear equations f(x) = 0, Programming: Short introduction to Matlab, Numerical algorithms for difference equations: Newton′s method, Fibonacci sequences, Recursion.

• #### Markov Chains

Definition of Markov chains, probability vectors, and stochastic matrices, Connection between a Markov chain and a second order difference equation, Long time behaviour of a process described by a Markov chain, Random walk as a Markov chain, Absorbing and irreducible Markov chains.

• #### Combinatorics

Permutations and combinations, Binomial coefficients and their properties, Binomial theorem, Principle of inclusion and exclusion. Derangements, Partitions and Stirling numbers, Transpositions and Cycles, Multinomial Theorem, Newton′s Binomial Theorem.

• #### Game Theory

Strategic form games, Dominated Strategies, Nash Equilibrium, Prisoner′s Dilema, Two-person zero-sum games, The minimax Theorem, Extensive form games with perfect information.

### Delivery and Assessment

The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (60%) and coursework (40%) consisting of homeworks and tests.

### Credit Rating

This module is a Scottish Higher Education Level 2 or SCQF level 8 module and is rated as 20 SCOTCAT credits or 10 ECTS credits.

This module consists of a Calculus and Algebra component. It is part of a series of four modules, Mathematics 1A, 1B, 2A, 2B, which are the core Mathematics modules in years 1 and 2, and provide the foundations in Calculus, Algebra and Geometry for all mathematics modules in higher levels. This module is mandatory for all Level 2 students on Mathematics (including Mathematics combined) degrees. The module is also suitable for students on non-mathematics degrees and recommended for students on physics and computing degrees. If you have questions about this module please contact our Undergraduate Admissions Tutor or your Advisor of Studies.

### Prerequisites

Students taking this module must have achieved a pass mark in each of the modules MA11001 and MA12001, and have taken MA21001, or equivalents.

### Indicative Content

• #### Multivariable Calculus

Limits and continuity of functions of two variables. Partial Derivatives. Method of Lagrange Multipliers. Taylor series in two variables. Stationary points for functions of two variables. Double integrals, Jacobian of a change of variables.

• #### Series

Tests for convergence of series of numbers. Convergence of power series, radius of convergence. Application to Taylor and Maclaurin series (mention of Taylor's Theorem).

• #### General vector spaces and subspaces

PnCn and other vector spaces. Span, linear dependence/independence, bases. Reduction to row-ecehelon form, relation to linear independence Intersections, unions and direct sums of subspaces. Range and nullspace of a matrix.

• #### Inner products

Definition of inner products and inner product spaces. Gram-Schmidt orthogonalisation.

• #### Eigenvalues and eigenvectors

Definitions and examples. Complex and repeated eigenvalues, algebraic and geometric multiplicity. Diagonalization of matrices. The Cayley-Hamilton theorem.

• #### Linear mappings

Definitions and matrix representations. Composition of linear mappings. Kernel and image.

### Delivery and Assessment

The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (60%) and coursework (40%).

### Credit Rating

This module is a Scottish Higher Education Level 2 or SCQF level 8 module and is rated as 20 SCOTCAT credits or 10 ECTS credits.

### Prerequisites

Students taking this module must have achieved a pass mark in the module MA12003, or equivalent.

### Indicative Content

• #### Sampling Distributions

Mean and standard deviation of samples, Shape of sampling distributions.

• #### Hypothesis tests

t-tests, Inferences, Confidence intervals, Chi-square tests.

• #### Linear Regression

Least squares, Assessing usefulness of a model, Using a model.

• #### R software package

Appropriate use of computational software to carry out calculations relevant to sampling.

### Delivery and Assessment

The module is delivered in the form of lectures and workshops/computer labs and assessed via an exam (60%) and coursework (40%) consisting of tests and computer lab reports.

### Credit Rating

This module is a Scottish Higher Education Level 2 or SCQF level 8 module and is rated as 20 SCOTCAT credits or 10 ECTS credits.

### Prerequisites

Students taking this module must have achieved a pass mark in each of the modules MA21001 and MA22001, or equivalents.

### Indicative Content

• #### First Order Differential Equations

Separable equations, Linear equations with constant coefficients, Linear equations with variable coefficients, integrating factors, Homogeneous equations, Exact equations and integrating factors.

• #### Second Order Differential Equations

Homogeneous equations with constant coefficients, Fundamental solutions of linear homogeneous equations, Linear independence and the Wronskian (including Abel′s formula), Reduction of order and reduction to the normal form, Nonhomogeneous equations, Method of undetermined coefficients, Initial conditions.

• #### Systems of First Order Linear Equations

Transformation of an nth order equation to a system of n first order equations, Homogeneous linear systems with constant coefficients, Fundamental sets of solutions and fundamental matrices, the Wronskian and Abel′s formula, The exponential of a matrix, Nonhomogeneous linear systems, Variation of parameters, Homogeneous linear systems of two first order equations with constant coefficients, Stability and the phase plane.

• #### Partial Differential Equations and Fourier Series

Fourier series of functions of one variable, Dirichlet′s Conditions, Technique for determining Fourier coefficients (even/odd functions). Gibbs′ phenomena. Introduction to Partial Differential Equations, Technique of separation of variables with application to initial and boundary value problems.

### Delivery and Assessment

The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (80%) and coursework (20%).

### Credit Rating

This module is a Scottish Higher Education Level 3 or SCQF level 9 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

The aim of this module is to provide the Level 3 student with a variety of basic mathematical techniques with which to analyse a wide class of mathematical models arising in science and engineering. This module is mandatory for all Level 3 students on Mathematics (including Mathematics combined) degrees. If you have questions about this module please contact your Advisor of Studies.

### Prerequisites

Students taking this module must have achieved a pass mark in each of the modules MA21001 and MA22001, or equivalents.

### Indicative Content

• Revision of vector products and scalar functions of three variables.

• Orthogonal coordinates.

• Curves in space, parameterization and arc length

• Surfaces in space, parameterization, normal vectors and tangent planes.

• The operators grad, div, curl.

• Line integrals, surface integrals, and volume integrals.

• Divergence Theorem and Stokes Theorem.

• Scalar and vector potentials.

### Delivery and Assessment

The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (80%) and coursework (20%).

### Credit Rating

This module is a Scottish Higher Education Level 3 or SCQF level 9 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

This module provides an in-depth study of Analysis aimed at Level 3 or 4 students in which the concepts are defined precisely and the results are proved rigorously. This module is mandatory for Level 3 students taking a BSc or MMath in Mathematics. This module may be taken in combination with other Level 3 or 4 modules by Level 4 students on Mathematics combined degrees other than those taking the MSci in Mathematics and Physics. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.

### Prerequisites

Students taking this module must have achieved a pass mark in each of the modules MA21001 and MA22001, or equivalents.

### Indicative Content

• #### Normed and Metric Spaces

Supremum, Completeness Axiom. Definitions and properties of normed and metric spaces, convergence of sequences, continuity, closed sets (in terms of limit points)

• #### Connectedness and Completeness

Connected sets: definition in metric spaces; relation to the concept of continuity. Cauchy sequences, completeness and relation to closed sets, Banach's contraction mapping theorem

• #### Compact Sets

General definition with open sets of the notion of compactness, its sequences characterisations on metric spaces, and its connection with closed subsets. "Closed and bounded" characterisation of the compact sets in Rn (i.e., the Heine-Borel Theorem), connection with limit points, Weierstrass Theorem. Connection between continuity and compactness. The concept of uniform continuity and its connection with compactness. Urysohn's Lemma [1 lecture]. Partition of Unity on Rn

• #### Convergence and Equicontinuity

Uniform convergence of sequences of functions. The concept of equicontinuity of a family of functions and Arzela-Ascoli Theorem.

• #### Series

Ratio test, comparison test, Weierstrass M-test. Power series, Taylor series (mention of Taylor's theorem).

### Delivery and Assessment

The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (80%) and coursework (20%).

### Credit Rating

This module is a Scottish Higher Education Level 3 or SCQF level 9 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

This module provides an in-depth study of Operational Research aimed at Level 3 or 4 students. This module is mandatory for Level 3 students taking a BSc or MMath in Mathematics. This module may be taken in combination with other Level 3 or 4 modules by Level 4 students on Mathematics combined degrees other than those taking the MSci in Mathematics and Physics. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.

### Prerequisites

Students taking this module must have achieved a pass mark in each of the modules MA21001 and MA22001, or equivalents.

### Indicative Content

• #### Linear Programming

Structure of Linear Programming problems, Graphical solutions, Simplex method, Duality, Integer linear programming, Branch and Bound method.

• #### Transportation Problems

Transportation and Assignment problems, initial basic feasible solution, Hungarian method.

• #### Network Programming

Graphs, Minimum cost flow, Max-flow/Min-cut theorem, Ford-Fulkerson method.

### Delivery and Assessment

The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (80%) and coursework (20%).

### Credit Rating

This module is a Scottish Higher Education Level 3 or SCQF level 9 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

Matrix algebra is a fundamental and widely used resource for modelling a wide variety of problems in science, technology, industry and commerce. The aim of this course is to use computers to implement algorithms and to solve a number of problems that can be stated in terms of matrix-related equations, and to understand the relevant matrix theory that underpins these algorithms. This module is mandatory for students taking the BSc or MMath in Mathematics and may optionally be taken in combination with other modules by students on any of the Mathematics combined degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.

### Prerequisites

Students taking this module must have achieved a pass mark in each of the modules MA21001 and MA22001, or equivalents.

### Indicative Content

• #### Direct Methods for Solving Linear Systems of Equations

Basic properties of matrices, Gaussian elimination, partial pivoting. LU-factorization. Tridiagonal systems.

• #### Iterative Methods

A general iterative method and convergence, Jacobi method, Gauss-Seidel method, SOR (successive over-relaxation).

• #### Iterative Methods for Solving Eigenvalue Problems

Review of eigenvalue problems, QR factorizations

• #### Using MATLAB to solve problems in linear algebra

Introduction to MATLAB, Application of MATLAB to algorithms for LU factorization, iterative methods and QR factorizations.

### Delivery and Assessment

The module is delivered in the form of lectures and workshops/tutorials and computer labs, and assessed via an exam (70%) and computer homeworks (30%).

### Credit Rating

This module is a Scottish Higher Education Level 3 or SCQF level 9 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

This module introduces the notions of differentiation and integration for functions of a complex variable. It develops the theory with important applications such as evaluation of path integrals via residue calculus, the fundamental theorem of algebra and conformal mappings. This module is mandatory for students taking the BSc or MMath in Mathematics and may optionally be taken in combination with other modules by students on any of the Mathematics combined degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.

### Prerequisites

Students taking this module must have achieved a pass mark in each of the modules MA21001 and MA22001, or equivalents.

### Indicative Content

• Algebraic properties of complex numbers

• Definition of the derivative; Cauchy-Riemann equations

• Power series; radius of convergence

• Logarithmic, exponential and trigonometrical functions; branch points

• Line integrals. The Cauchy integral theorem and integral formula

• The Cauchy formula for derivatives; Taylor series

• Liouville's theorem; fundamental theorem of algebra

• Laurent's theorem; poles and the residue theorem; zeros of analytic functions

• Evaluation of integrals

• Conformal mappings

### Delivery and Assessment

The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (80%) and coursework (20%).

### Credit Rating

This module is a Scottish Higher Education Level 3 or SCQF level 9 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

The basic ideas at the foundations of many physical theories, such as continuum mechanics, fluid dynamics, electromagnetism, thermodynamics, general relativity and gauge theories, are geometrical. This course develops some of the geometrical concepts and tools that are essential for understanding classical and modern physics and engineering. This module is mandatory for students taking the BSc or MMath in Mathematics and may optionally be taken in combination with other modules by students on any of the Mathematics combined degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.

### Prerequisites

Students taking this module must have achieved a pass mark in each of the modules MA21001 and MA22001, or equivalents, and have taken MA31007 or MA32002.

### Indicative Content

• #### Manifolds

Submanifolds in Rn. Implicit Function theorem. Examples

• #### Vector fields on manifolds

The tangent space. Vectors as differential operators. Vector fields and flows.

• #### Covectors and exterior forms

Linear functionals and the dual space. Differential of a function. The pull-back of a covector.

• #### The exterior algebra

The geometric meaning of forms in R^n. Exterior product. Inner product. Exterior differential. Relation to vector analysis.

• #### Integration of forms

Line and surface integrals. Independence of parameterisation. Integrals and pull-backs. Stoke′s Theorem.

• #### Lie-derivative

The Lie-derivative of forms. Relation to equations in physics.

### Delivery and Assessment

The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (80%) and coursework (20%).

### Credit Rating

This module is a Scottish Higher Education Level 3 or SCQF level 9 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

The aim of this course is to introduce you to some biological phenomena and their formulation in terms of mathematical models, which lead to difference equations and ordinary differential equations, and to investigate the solutions of these equations. This module is mandatory for students taking the BSc or MMath in Mathematics or the BSc or MSci in Mathematical Biology, and may optionally be taken in combination with other modules by students on any of the other Mathematics combined degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.

### Prerequisites

Students taking this module must have achieved a pass mark in each of the modules MA21001 and MA22001, or equivalents, and must have taken modules MA31002 and either MA31007 or MA32002.

### Indicative Content

• #### Single Species Population dynamics

Difference equations: graphical analysis, fixed points and linear stability analysis. First order systems of ordinary differential equations: logistic equation, steady states, linearisation, and stability. Harvesting and fisheries.

• #### Interacting Species

Systems of difference equations (host-parasitoid systems). Systems of ordinary differential equation (predator-prey and competition models).

• #### Molecular Dynamics

Biochemical kinetics: Michaelis-Menten kinetics. Metabolic pathways: activation and inhibition.

### Delivery and Assessment

The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (80%) and coursework (20%).

### Credit Rating

This module is a Scottish Higher Education Level 3 or SCQF level 9 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

This module is a personal research project which runs over both semesters. During the first semester there will be classes in computer labs where students will develop skills that will be useful in completing a project. The module consists of a substantial project together with a project report. This module is mandatory for Level 4 students taking the BSc or MMath in Mathematics or the BSc or MSci in Mathematical Biology. If you have questions about this module please contact your Advisor of Studies.

### Prerequisites

Students taking this module must have achieved a pass mark in each of the modules MA31002 and either MA31007 or MA32002, or equivalents.

### Indicative Content

• #### MATLAB

Become familiar with the basic concepts of MATLAB and its use in solving differential equations.

• #### Communicating Mathematics

Learn how to typeset mathematics using LaTeX. Understand how to structure, set out, and typeset a mathematical report.

• #### Presentations

Prepare and deliver presentations on general topics in mathematics, and on your chosen project.

• #### Project

Carry out a substantial project in an area of mathematics and document the work in a project report.

### Delivery and Assessment

The project will require substantial independent work as well as regular meetings with the project supervisor(s), as well as the in-class contact hours in semester 1. Assessment will be based on coursework (100%) consisting of the project report and presentations.

### Credit Rating

This module is a Scottish Higher Education Level 4 or SCQF level 10 module and is rated as 30 SCOTCAT credits or 15 ECTS credits.

Ordinary Differential Equations (ODEs) are an important modelling tool in Science and Engineering. These can rarely be solved exactly and so techniques have been developed to derive approximate solutions that may, in principle, be made as accurate as desired. This module, aimed at Level 4 students, will investigate these techniques. This module is mandatory for Level 4 students taking a BSc or MSci in Mathematical Biology, or an MSci in Mathematics and Physics. This module may be taken in combination with other Level 3 or 4 modules by other Level 4 students. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.

### Prerequisites

Students taking this module must have achieved a pass mark in each of the modules MA31002 and either MA31007 or MA32002, or equivalents.

### Indicative Content

• #### Numerical methods for initial value problems for ODEs

Taylor Series Methods; Linear multi-step methods: one-step methods (Euler, Trapezoidal and Backward Euler methods) and two-step methods; Consistency, zero-stability, weak stability theory and A-stability; Provision of the extra starting values and the potential for instability; Runge-Kutta methods: construction and weak stability theory; Application to systems.

• #### Boundary value problems for ODEs

BVPs for second order ODEs; eigenvalues and eigenfunctions; orthogonality; Green′s functions and maximum principles; Finite difference methods: 2nd order methods; Treatment of boundary conditions; Discrete maximum principles; Convergence.

### Delivery and Assessment

The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (80%) and coursework (20%).

### Credit Rating

This module is a Scottish Higher Education Level 4 or SCQF level 10 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

This module provides an introduction to the Mathematics of Fluids and Plasmas, focusing on Fluid Dynamics. This module is mandatory for students taking the BSc or MMath in Mathematics or the BSc or MSci in Mathematics and Physics, and is optional for students taking any other Mathematics combined degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.

### Prerequisites

Students taking this module must have achieved a pass mark in each of the modules MA31002 and either MA31007 or MA32002, or equivalents.

### Indicative Content

• #### Fundamentals

Fields, flux, potentials. Representation of fields: fieldlines/streamlines, contours, flux surfaces. Gauss' and Stokes' theorems.

• #### Conservation laws

Conservation of mass, conservation of momentum, Euler's Eq., energy equation, equation of state.

• #### Common approximations

Incompressible, irrotational, potential flows, Bernoulli’s the- orem. Laplace’s equation, boundary conditions, uniqueness theorem, separable solutions.

• #### Vorticity

Vorticity and circulation, Kelvin's circulation theorem, vorticity evolution.

• #### The solar wind

Introduction to solar features, Parker's solar wind solution.

• #### Waves

Sound waves, linearisation, dispersion relations, wave properties.

• #### Viscous flow

Stress tensor, viscous stresses, viscosity, energy dissipation, the Reynolds number.

### Delivery and Assessment

The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (80%) and coursework (20%).

### Credit Rating

This module is a Scottish Higher Education Level 4 or SCQF level 10 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

This module aimed at Level 4 students, covers the theory and application of graphs, including both theoretical work and the use of algorithms. This module is mandatory for students taking the BSc or MMath in Mathematics, and is optional for students taking any other Mathematics combined degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.

### Prerequisites

Students taking this module must have achieved a pass mark in each of the modules MA21001 and MA22001, or equivalents.

### Indicative Content

• #### Introduction to Graph Theory

Fundamental definitions of graph theory. Introduction to some special types of graphs. Degree sequences and corresponding graphs.

• #### Connectedness

Sufficient conditions to ensure connectedness. Connectivity and edge connectivity. Tarry's algorithm.

• #### Eulerian and Hamiltonian Graphs

Necessary and sufficient conditions for graphs to be Eulerian. Fleury's algorithm. Necessary conditions for graphs to be Hamiltonian.

• #### Trees

Properties of trees. Spanning trees and labelled spanning trees. Finding minimum weight spanning trees.

• #### Planar and Non-planar Graphs

Necessary conditions for graphs to be Planar. Toroidal graphs. Genus of graphs.

• #### Graph Colourings

Vertex and edge colourings Chromatic polynomials. The 4-colour theorem.

### Delivery and Assessment

The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (80%) and coursework (20%).

### Credit Rating

This module is a Scottish Higher Education Level 4 or SCQF level 10 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

The aim of this module is to introduce Level 4 students to some biological phenomena and their formulation in terms of mathematical models, building on the work in MA32009 (Mathematical Biology I). This module is mandatory for Level 4 students taking the BSc or MSci in Mathematical Biology, and may optionally be taken in combination with other modules by students taking the BSc or MMath in Mathematics or any of the other Mathematics combined degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.

### Prerequisites

Students taking this module must have achieved a pass mark in the module MA32009, or in MA41002, or equivalent.

### Indicative Content

• #### Modelling of Biological Systems using Partial Differential Equations

Derivation of conservation equations. Different models for movement (e.g. diffusion, convection, directed movement). Connection between diffusion and probability.

• #### Linear reaction-diffusion equations

Fundamental solution for linear diffusion equations. Speed of a wave of invasion.

• #### Non-linear reaction-diffusion equations

Travelling wave solutions for monostable equations (e.g. Fisher equation). Travelling wave solutions for bistable equations.

• #### Systems of reaction-diffusion equations

Travelling wave solutions for systems of reaction-diffusion equations. Pattern formations in systems of reaction-diffusion equations. Pattern formations in chemotaxis equations.

• #### Mathematical modelling of infection diseases (SIR)

Derivation of a simple SIR model. Travelling wave solutions for the simple SIR model. Generalisation of the simple SIR model. Stochastic SIR model.

### Delivery and Assessment

The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (80%) and coursework (20%).

### Credit Rating

This module is a Scottish Higher Education Level 4 or SCQF level 10 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

This module gives a broad introduction to PDEs that includes classification into different types, classical solution methods, qualitative properties and, for the majority of problems that cannot be solved exactly, provides techniques for constructing approximate solutions. This module is mandatory for Level 4 students taking the BSc or MSci in Mathematical Biology, and may optionally be taken in combination with other modules by students taking the BSc or MMath in Mathematics or any of the other Mathematics combined degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.

### Prerequisites

Students taking this module must have taken the module MA41003, or equivalent.

### Indicative Content

• #### First and Second Order PDEs

Basic Theory; examples of fundamental solutions Second order linear PDEs; classi- fication, characteristics; dAlemberts solution of the one-dimensional wave equation.

• #### Boundary Value Problems for PDEs

Finite-difference methods for second order problems (Poisson's equation): the treatment of boundary conditions and curved boundaries in two dimensions.

• #### Initial Value Problems for PDEs

Parabolic and Hyperbolic equations: Fundamental solutions. General discussion of basic qualitative properties such as dissipation (energy inequalities) and characteristics. Construction of numerical methods: two-level methods and brief reference to three-level methods (if time permits). Local truncation errors. Stability and choice of norm: Maximum norm, L2 norm via von Neumann's method. Application to hyperbolic systems. The Method of Lines.

### Delivery and Assessment

The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (80%) and coursework (20%).

### Credit Rating

This module is a Scottish Higher Education Level 4 or SCQF level 10 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

This module builds on the foundations established in Mathematics of Fluids and Plasmas I (MA41006). This module may optionally be taken by students on any Mathematics or Mathematics combined degree other than those taking the BSc or MSci in Mathematical Biology. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.

### Prerequisites

Students taking this module must have achieved a pass mark in each of the modules MA31002 and MA32002, or equivalents, and must have taken MA41006.

### Indicative Content

• #### Electromagnetism

Maxwell's equations. Electrostatics. Magnetostatic fields, magnetic effect of currents. Electrodynamics. Waves.

• #### Equations of Magnetohydrodynamics (MHD)

Lorentz force, MHD equations, importance of terms. Diffusion and frozen-in flux. Magnetic field lines and flux tubes.

• #### MHD solutions

Hydrostatic pressure balance, plasma beta. Potential fields. Force-free fields, coronal arcades. Grad-Shafranov equation.

• #### Waves

Linearised MHD equations. Sound waves, Alfven waves, magnetoacoustic waves.

• #### Solar applications

Magnetic reconnection. Magnetic helicity. Dynamo theory. Solar flares, CMEs.

### Delivery and Assessment

The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (80%) and coursework (20%).

### Credit Rating

This module is a Scottish Higher Education Level 4 or SCQF level 10 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

This module is a personal research project which runs over both semesters. The module consists of a substantial project together with a project report. This module is mandatory for Level 5 students taking the MMath in Mathematics or the MSci in Mathematical Biology. If you have questions about this module please contact your Advisor of Studies.

### Prerequisites

Students taking this module must have achieved a pass mark in each of the modules MA31002 and either MA31007 or MA32002, or equivalents.

### Indicative Content

• #### Presentations

Prepare and deliver presentations on your chosen project.

• #### Project

Carry out a substantial project in an area of mathematics and document the work in a project report.

### Delivery and Assessment

The project will require substantial independent work as well as regular meetings with the project supervisor(s). Assessment will be based on coursework (100%) consisting of the project report and presentations.

### Credit Rating

This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 30 SCOTCAT credits or 15 ECTS credits.

This module, aimed at the Level 5 student, takes an advanced look at dynamical systems. The time evolution of many biological, chemical, or physical processes, as well as systems considered in engineering or economics, can be described by difference or differential equations. Dynamical systems theory allows us to study these systems of equations and inver information about the behaviour of the corresponding biological, chemical or physical systems. It addresses questions like the existence and stability of solutions, how the behaviour of solutions changes depending on the system parameters, or determines the existence of strange attractors or chaos in the system.

### Prerequisites

Students taking this module must usually have achieved a pass mark in each of the modules MA31002 and MA32001, or equivalent.

### Indicative Content

• #### One-Dimensional Maps

Definition, Cobweb Plot: Graphical Representation of an Orbit, Stability of Fixed Points, Periodic Points, Chaos: Lyapunov Exponents.

• #### Ordinary Differential Equations

Background, Examples of main Physical and Biological Processes described by Ordinary Differential Equations (ODEs), Existence and uniqueness of solutions of ODEs, Linearised Stability Analysis, Two-dimensional Systems: Hamiltonian and Gradient systems, Periodic solutions: Floquet theory, Poincare Map and Stability of Periodic Orbits, Bifurcation and Chaos.

• #### Partial Differential Equations

Definitions, Background, Well-posedness, Maximum Principles, Spectral Theorem for Laplace Equation, Semigroups for Evolution Equations in Banach Spaces, Nonlinear Evolution Equations: Linearised Stability Analysis for Reaction-Diffusion Equations.

### Delivery and Assessment

The module is delivered in the form of lectures and workshops/presentation classes and assessed via an exam (60%) and coursework (40%).

### Credit Rating

This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

In this module the Level 5 student will learn to write their own code and to apply built-in "black box" solvers in MATLAB and COMSOL to mathematical modelling problems. This module is mandatory for Level 5 students taking the MMath in Mathematics or the MSci in Mathematical Biology. This module may be taken in combination with another at Level 5 by students taking the MSci in Mathematics and Physics. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.

### Prerequisites

Students taking this module must usually have achieved a pass mark in each of the modules MA32005 and MA42003, or equivalent.

### Indicative Content

• #### MATLAB fundamentals

Students will learn basic operations in MATLAB, and implement various finite difference schemes to solve ODEs (primarily initial value problems) originating in celestial mechanics, population dynamics, and cell biomechanics.

• #### MATLAB ODE solvers for initial value problems

Students will learn to use standard built-in solvers with MATLAB, particularly ode45 and ode23s, and possibly dde23. We will apply these solvers to initial value problems (and possibly delay differential equations) stemming from celestial mechanics, cell biomechanics, and population dynamics.

• #### MATLAB random variables, stochastic processes, and SDEs

After a brief introduction to stochastic differential equations (SDEs), students will learn MATLAB solution techniques, with applications to Brownian motion and related physical processes. We will also learn to simulate discrete and continuous stochastic processes, and generate samples from random variables with arbitrary distributions.

• #### MATLAB ODE solvers for boundary value problems

Students will implement a standard "shooting" method to solve a BVP from heat transfer. We will learn to use the standard built-in solvers, particularly bvp4c. We explore alternate solution techniques, such as by formulating the discretised equation as a linear algebraic system, and as the steady state solution to a PDE; these approaches help drive us towards PDE solution methods. The class will apply these solvers to boundary value problems stemming from heat transfer and fluid mechanics.

• #### MATLAB for PDEs

Students will implement explicit finite difference methods in MATLAB, with a focus on reaction-diffusion problems. The overall goal will be to solve coupled reaction-diffusion problems (with heterogeneous coefficients) and cell growth.

• #### Weak formulations for partial differential equations; introduction to FEMs

We repose PDEs using a weak formulation, using the context of function spaces. Using this framework, we develop an understanding of finite element methods (FEMs).

• #### FEMs and COMSOL fundamentals

Students will learn to solve reaction-diffusion equations using the built-in FEMs in COMSOL.

### Delivery and Assessment

Delivery of this module will take a hands-on, interactive approach, where lectures are integrated with guided computer lab time. Assessment will be based on computational coursework (100%).

### Credit Rating

This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

This module, aimed at Level 5 students, gives a non-measure theoretic introduction to stochastic processes, considering the theory and some applications and going on to introduce stochastic differential equations and their solutions. This module may optionally be taken in combination with others by Level 5 students taking the MMath in Mathematics or the MSci in Mathematical Biology or Mathematics and Physics. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.

### Prerequisites

Students taking this module would find it beneficial to have taken each of the modules MA32001 and MA51007, or equivalent.

### Indicative Content

• #### Probability fundamentals

Elementary probability concepts such as random variables, expected value, moment generating and characteristic functions, conditional exceptions, probability inequalities and limit theorems, etc.

• #### The Poisson process

(Homogeneous) Poisson process and related examples such as interarrival and waiting time distributions and conditional distribution of the arrival times. Some practical examples such as the busy period of the M/G/1 queueing system. Introduction to the nonhomogeneous Poisson process.

• #### Markov chains

(Discrete-time) Markov chains and some related examples. Chapman-Kolmogorov equations and classification of states.

• #### Continuous-time Markov chains

Continuous-time Markov chains, birth and death processes, and the Kolmogorov differential equations.

• #### Brownian motion and stochastic differential equations

Basics of Brownian motion, Ito^ integral and Ito^ formula, and then stochastic differential equations (SDEs). A number of commonly used SDEs and their solutions will be discussed.

### Delivery and Assessment

The module is delivered in the form of lectures and workshops/presentation classes and assessed via coursework (100%) consisting of homeworks and a presentation.

### Credit Rating

This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

This is a Level 5 course that offers a robust understanding of the inverse problems theoretical framework and methods suitable for medical and financial applications. The aim is to achieve comprehensive knowledge in the theoretical fundaments and general methodology for inverse problems in various heterogeneous media, including medical applications and finance. This module may optionally be taken in combination with other modules at this level by Level 5 students on the MMath in Mathematics or MSci in Mathematics and Physics degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.

### Prerequisites

Students taking this module must usually have achieved a pass mark in the module MA32001, or equivalent.

### Indicative Content

• #### Examples of Inverse Problems

Examples from medical applications and finance.

• #### Inverse Methodology Preliminary Foundation

Necessary Basic Definitions and Theorems in Measure Theory and Function Spaces

• #### General Regularisation Theory

Tikhonov′s regularization method. Landweber Iteration. The Discrepancy Principle of Morozov. Conjugate gradient method.

• #### Galerkin Methods

Galerkin General formulation. The Least Squares Method. The Dual Least Squares Method.

• #### Inverse Problems in General Heterogeneous Media and Medical Applications

Backward heat conduction problem. Inverse problems in reaction-diffusion equations.

• #### Inverse problems in finance

Formulation of forward model: Black-Scholes and Dupire's Formula. Inverse Problem formulation of market volatility. Reconstruction of time- and price-dependent volatilities.

### Delivery and Assessment

The module is delivered in the form of lectures and assessed via coursework (100%) consisting of tests and homeworks.

### Credit Rating

This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

This module, aimed at the Level 5 student, covers the fundamentals of Measure Theory including Lebesgue Measure and Integration and Lebesgue Spaces as well as their pivotal implications in the modern analysis of partial differential equation and probability theory. This module may optionally be taken in combination with other modules at this level by Level 5 Students on the MMath in Mathematics or MSci in Mathematics and Physics degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.

### Prerequisites

Students taking this module must usually have achieved a pass mark in the module MA32001 or equivalent.

### Indicative Content

• #### Construction of a general measure space

Definition and properties of a σ-algebra. The concept of measures as a non-negative σ-additive real valued set functions defined on a σ-algebra. Measurable sets, σ-finite measurable sets, the notion of algebra of generators of a σ- algebra of parts. Construction of the Lebesgue Measure. Charatheodory Extension Theorem: characterising the unique extension of a finitely additive non-negative real valued function on an algebra of generators to a measure on the generated σ-algebra.

• #### Measurable functions

Definition. Approximation by measurable simple/step functions as well as by continuous functions.

• #### Construction of the Lebesgue Integral

Construction of the integral for measurable indicators (characteristic) functions as well as for measurable simple/step functions. Construction of the integral for a general measurable function. Properties of Lebesgue Integrable functions (additivity, multiplication by scalars, positivity, monotonicity). Definition and basic vectorial properties of the space of Lebesgue Integrable Functions L1.

• #### Limit theorems (concerning point-wise convergence, almost everywhere convergence, and convergence in measure)

Fatou Lemma. Monotone Convergence Theorem. Dominated Convergence Theorem.

### Delivery and Assessment

The module is delivered in the form of lectures and seminars and assessed via an exam (60%) and coursework (40%).

### Credit Rating

This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

This Level 5 module will undertake analysis and computational simulation of mathematical models of cancer growth and treatment. In addition to the mathematical formulation and analysis of such systems which will take place during the formal lectures, through tutorial work students will undertake computational analyses using tools such as MATLAB and COMSOL introduced in MA51004. This module is mandatory for students on the MSci in Mathematical Biology degree and may optionally be taken in combination with other modules by students on the MMath in Mathematics or MSci in Mathematics and Physics. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.

### Prerequisites

Students taking this module must usually have achieved a pass mark in the module MA31002, or equivalent.

### Indicative Content

• #### Overview of Cancer Growth and Spread

The biology of cancer.

• #### Avascular Solid Tumour Growth

ODE models of solid tumour growth Greenspan′s model. Reaction-diffusion moving boundary models.

• #### Tumour-induced Angiogenesis

Continuum PDE models. Hybrid discrete-continuum models.

• #### Cancer Invasion

Continuum PDE models. Hybrid discrete-continuum models.

• #### Immune Response to Cancer

ODE models of the immune response to cancer. PDE models of the immune response to cancer.

### Delivery and Assessment

The module is delivered in the form of lectures and tutorials/computer labs and assessed via exam (100%).

### Credit Rating

This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

In this module we will undertake analysis and computational simulation of mathematical models of ecological and epidemiological systems including predator-prey systems, host-parasitoid systems, plant-herbivore systems, spread and transmission of disease (e.g. AIDS, SARS, measles, rabies). In addition to the mathematical formulation and analysis of such systems which will take place during the formal lectures, through tutorial work students will undertake computational analyses using tools such as MATLAB and COMSOL introduced in MA51004. This module is mandatory for students on the MSci in Mathematical Biology degree and may optionally be taken in combination with other modules by students on the MMath in Mathematics or MSci in Mathematics and Physics. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.

### Prerequisites

Students taking this module must usually have achieved a pass mark in the module MA31002, or equivalent.

### Indicative Content

• #### Non-spatial models

Difference equation models. Ordinary differential equation models. Delay difference/differential equation models.

• #### Spatial models

Integro-difference equation models. Partial differential equation models. Integro-differential equation models.

### Delivery and Assessment

The module is delivered in the form of lectures and tutorials/computer labs and assessed via exam (100%).

### Credit Rating

This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

This module, aimed at Level 5 students, studies mathematical models of cellular physiology in metabolism, genetic networks, and electrophysiology. This module is mandatory for Level 5 students taking the MSci in Mathematical Biology and may optionally be taken in combination with other modules at this level by Level 5 students on the MMath in Mathematics or MSci in Mathematics and Physics degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.

### Prerequisites

Students taking this module must usually have achieved a pass mark in the module MA32009, or MA41002, or equivalent.

### Indicative Content

• #### Biochemical Reactions

Enzyme kinetics. Glycolysis

• #### Genetic Networks

Central Dogma of biology, review of bifurcation theory. Feedback loops and Oscillation. Circadian rhythms. Cell cycle model.

• #### Electrophysiology

The Hodgkin-Huxley model. The FitzHugh-Nagumo equations. Small network dynamics of coupled neurons.

### Delivery and Assessment

The module is delivered in the form of lectures and workshops/presentation classes and assessed via exam (70%) and coursework (30%).

### Credit Rating

This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

Optimisation problems arise from modelling a wide variety of systems in science, technology, industry, business, economics as well as in many other fields. This module, aimed at Level 5 students, covers practical methods of optimisation that are supported by a growing body of mathematical theory. Students are expected to implement the methods and solve problems numerically. This module may optionally be taken in combination with other modules at this level by Level 5 students on the MMath in Mathematics or MSci in Mathematics and Physics degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.

### Prerequisites

Students taking this module must usually have achieved a pass mark in the module MA32005, or equivalent.

### Indicative Content

• #### Introduction

Examples of optimization problems. Mathematical background.

• #### Unconstrained Optimization

Line search and Descent methods. Newton′s method. Conjugate gradient method.

• #### Linear Programming

Simplex method. Slack and artificial variables. Simplex tableau.

• #### Constrained Optimization

Lagrange multipliers. Theory of constrained optimization.

• #### Application in Finance and Energy

Application problems such as factory location problem, oil pipeline problem.

### Delivery and Assessment

The module is delivered in the form of lectures and workshops/presentation classes and assessed via coursework (100%) consisting of homeworks, tests, presentations and project work.

### Credit Rating

This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

This module covers advanced topics in Fluid Dynamics for the Level 5 student who has already undertaken an introductory course. This module may optionally be taken by students on Mathematics or Physics MSc programmes as well as MMath in Mathematics, MSci in Mathematical Biology or MSci Mathematics and Physics.

The course focuses on vortex dynamics and uses this as basis to understand and describe turbulence, one of the most intriguing phenomena of fluid dynamics. It also provides an introduction to non-Newtonian fluids and their fascinating properties. It takes the viewpoint of an applied mathematician/theoretical physicist and derives fundamental properties from the underlying system of equations.

### Prerequisites

Students taking this module must usually have achieved a pass mark in the module MA42007 (Mathematics of Fluids I), or a similar course.

### Indicative Content

• #### Introduction (recap of basic fluid dynamics)

Navier-Stokes equations and their approximations and modifications

• #### Boundary Layers

Nondimensionalisation, Blasius boundary layer

• #### Vortex Dynamics

2D dynamics, point vortices, vortex sheets, dynamics of vortex filaments, effects of viscosity, energy and enstrophy dissipation

• #### Turbulence

Instabilities, scales, description in Fourier space, Kolmogorov′s theory, 3D vs. 2D turbulence

• #### Non-Newtonian Fluids

Types of non-Newtonian fluids, Ostwald - de Waele model, Bingham plastic

### Delivery and Assessment

The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (80%) and coursework (20%) consisting of homeworks.

### Credit Rating

This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

This module, aimed at the Level 5 student, specifically aims to develop students' knowledge, skills and understanding of the fundamentals of function spaces (infinite dimensional spaces) and linear operators and functionals defined on function spaces.

Functional analysis plays an important role in many areas of applied mathematics and is essential for the theory of Partial Differential Equations, Numerical Analysis, Probability theory and Theoretical Physics. Functional analysis originated from classical analysis and is formed by the study of infinite dimensional vector spaces and linear functions defined on these spaces. The theory of functional analysis was developed by some of the most famous mathematicians of the 20th century such as Hilbert, Schmidt, Riesz, Banach and von Neumann. Functional analysis can be characterised as a combination of infinite-dimensional linear algebra and classical analysis. Methods of functional analysis will allow us to analyse the properties of function spaces and to characterise solutions of integral and differential equations, arising in modelling of many biological and physical systems.

### Prerequisites

Students taking this module must usually have achieved a pass mark in MA32001, or equivalent.

### Indicative Content

• #### Function Spaces

• Normed spaces and inner product spaces (Banach and Hilbert spaces)
• Lebesgue integral and Lebesgue spaces
• Orthogonality in Hilbert spaces
• Weak derivatives and Sobolev spaces
• Notion of weak convergence in Function Spaces
• #### Linear Operators and Linear Functionals

• Definition of linear operators and linear functionals
• Definition of a norm of a bounded linear operator
• Riesz and Lax-Milgram theorems
• Definition, main properties and examples of compact operators
• Spectral theory for compact operators
• #### Well-posedness results for Integral and Partial Differential Equations

• Application of Lax-Milgram theorem and theory of compact operators to prove the well-posedness results

### Delivery and Assessment

The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (60%) and coursework (40%) consisting of homeworks and tests.

### Credit Rating

This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

### Prerequisites

Students taking this module must have achieved an average mark of at least 50% (C3) over the Mathematics MSc modules taken in Semesters 1 and 2 and have obtained at least 75 credits in these modules.

### Indicative Content

• #### Project

Carry out a substantial project in an area of mathematics and document the work in a project report.

### Delivery and Assessment

The project will require substantial independent work as well as regular meetings with the project supervisor(s). Assessment will be based on coursework (100%) consisting of the project report.

### Credit Rating

This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 60 SCOTCAT credits or 30 ECTS credits.

### Prerequisites

Students taking this module must have achieved an average mark of at least 50% (C3) over the Mathematics MSc modules taken in Semesters 1 and 2 and have obtained at least 75 credits in these modules.

### Indicative Content

• #### Project

Carry out a substantial project in an area of mathematics and document the work in a project report.

### Delivery and Assessment

The project will require substantial independent work as well as regular meetings with the project supervisor(s). Assessment will be based on coursework (100%) consisting of the project report.

### Credit Rating

This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 60 SCOTCAT credits or 30 ECTS credits.