We offer you lively and modern programmes of study centred on the application of mathematics, presented in a stimulating environment, by experts in the field.
Overview
Mathematics is essential:
 to the study of science  few scientific developments are possible without underlying mathematical theories
 essential in the construction of models of the economy, which influence the decisions of business and government
 to the development of methods of secure communication  used by us all for banking, internet transactions, mobile phone conversations
We believe that undergraduates are best served by studying in a Division that is active in research to the highest level, and we are fortunate in having an excellent international reputation for our research work. Our courses are taught by those who have great enthusiasm for their subject, a dedication to it, and an appreciation of the needs of students.
As well as offering degrees in Mathematics and Mathematical Biology, we offer a wide range of Joint Honours Degrees, where Mathematics can be combined with a variety of other subjects, for example, Accountancy, Economics, English, Physics, and Psychology.
The core Mathematics programmes (BSc Mathematics and MMath) are accredited by the Institute of Mathematics and its Applications
We are a relatively small Division and are able to operate with an excellent staff/student ratio. One advantage of this is that we can get to know each student personally, and so can offer a friendly and supportive learning experience. In addition, our StudentStaff Committee meets regularly to discuss matters of importance to our students.
Computing power is used extensively in modern mathematics. Our students have access to a dedicated computer pool 24 hours a day 7 days a week. Students use computers to assist their studies at all levels, using specialist mathematics software, writing up projects and obtaining information from the internet.
Furthermore, you can join DuMaS (Dundee Mathematical Society), our student society, which regularly organises social events such as the annual ball, bowling or cinema nights, careers events and BBQs.
Entry Requirements
The following are the minimum, uptodate entry requirements.
Courses starting 2018  

Qualification  Level 1 Entry  Advanced Entry to Level 2 
SQA Higher/Advanced Higher  BBBB (minimum)  AABB (typical) at Higher including mathematics at B  AB at Advanced Higher including mathematics at A, plus AB at Higher in different subjects 
GCE ALevel  BSc: BCC (minimum)  BBB (typical) including ALevel mathematics at B MMath: ABB at ALevel including mathematics at B 
BBC  ABB at ALevel including ALevel mathematics at A 
Irish Leaving Certificate (ILC)  H2H2H3H3 at Higher Level including mathematics at H3  Level 2 entry is not possible with this qualification 
International Baccalaureate (IB) Diploma  BSc: 30 points at Higher Level grades 5, 5, 5 to include mathematics MMath: 32 points at Higher Level grades 6, 5, 5 to include mathematics. A combination of IB Certificate plus other qualifications, such as ALevels, Advanced Placement Tests or the International Baccalaureate Careerrelated Programme (IBCP), will also be considered. 
34 points at Higher Level grades 6, 6, 5 to include mathematics at grade 6 
Graduate Entry  
BTEC  A relevant BTEC Level 3 Extended Diploma with DDM  A relevant BTEC Level 3 Extended Diploma with DDD. 
SQA Higher National (HNC/HND)  A relevant HNC with B in the Graded Unit including Mathematics for Engineering 1  A relevant HNC with A in the Graded Unit including Mathematics for Engineering 2 and 120 SCQF points. A relevant HND with BB in the Graded Units including Mathematics for Engineering 2 
Scottish Baccalaureate  Pass with BC at Advanced Higher in Mathematics and a Science/Engineering subject  Distinction with AB at Advanced Higher in Mathematics and a Science/Engineering subject 
SWAP Access  Relevant science subjects with ABB grades including Mathematics and Physics Units at SCQF Level 6  Level 2 entry is not possible with this qualification 
Advanced Diploma  Grade B with ASLA Levels at AB in Mathematics and a Science/Engineering subject  Grade B with ASLA Level at AA in Mathematics and a Science/Engineering subject 
Welsh Baccalaureate  Pass with A level at AB in Mathematics and a Science/Engineering subject  Pass with A level at AA in Mathematics and a Science/Engineering subject 
European Baccalaureate  70% overall with 7 in Mathematics  75% overall with 7.5 in Mathematics 
Other Qualifications  
Notes 
EU and International qualifications
English Language Requirement
For non EU students
IELTS Overall  6.0 

Listening  5.5 
Reading  5.5 
Writing  6.0 
Speaking  5.5 
Equivalent grades from other test providers
English Language Programmes
We offer PreSessional and Foundation Programme(s) throughout the year. These are designed to prepare you for university study in the UK when you have not yet met the language requirements for direct entry onto a degree programme.
Teaching Excellence Framework (TEF)
The University of Dundee has been given a Gold award – the highest possible rating – in the 2017 Teaching Excellence Framework (TEF).
Teaching & Assessment
How you will be taught
You will learn by traditional methods such as lectures, tutorials, and workshops as well as via computer assisted learning. From Level 2 onwards we teach the use of professional mathematical software packages in order to allow you to explore mathematics far beyond the limits of traditional teaching. This also prepares you for the way in which Mathematicians work in finance, industry and research.
Our excellent staff student ratio allows for close contact between students and lecturers and provides a lively learning environment. For example, we run "Maths Base'', a dropin facility in the Division, which provides another source of help with your studies over and above what you can expect from your lectures and tutorials.
How you will be assessed
Assessment for most modules involves a final examination and takes into account varying amounts of continuous assessment, including assignments and class tests.
What you will study
MMath Honours Degree
An MMath honours degree normally takes five years, full time, you study levels 15, as described below.
BSc or MA Honours Degree
An honours degree normally takes four years, full time, you study levels 14, as described below.
Advanced Entry Honours Degree
It is possible to study for most of our honours degrees in one year less if you have the required grades and subjects as listed in the Entry Requirements section. You study levels 24 for the BSc/MA or 25 for the MMath. There are definite advantages to considering this route as the time needed to study is reduced by one year which enables you to start working and earning earlier.
Typical Degree Programme
Level 1
At Level 1 students taking a degree fully within Mathematics spend two thirds of their time devoted to Mathematics and one third to another subject of their choice. Joint degrees typically involve a 5050 split between Mathematics and the other subject. Mathematics modules:
About the module
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 nonmathematics 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 ALevel, 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.
About the module
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 nonmathematics 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
Calculus Component

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.)
Algebra Component

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.
About the module
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 nonmathematics 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 ALevel, 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.
About the module
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 ALevel, 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. pvalues. 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.
Level 2
At Level 2 students taking a degree fully within Mathematics take either 4 or 5 modules in Mathematics, with the remaining 1 or 2 modules being taken in a subject of their choice. Joint degrees typically involve a 5050 split between Mathematics and the other subject.
About the module
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 nonmathematics 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
Calculus Component

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.
Algebra Component

Vectors and vector spaces
Definition of a vector space, R^{n}. Vectors, lines and planes in R^{n}. Span, linear independence. Basis and dimension. Subspaces.

Inner product
Scalar product, length. Projection. Normal form of hyperplanes in R^{n}. 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.
About the module
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, nonlinear oscillator. Conservation laws for a system of interacting bodies. Orbits in a gravitational field. Nonautonomous systems.
Delivery and Assessment
The module is delivered in the form of lectures and computer lab workshops. Assessment is entirely computerbased 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.
About the module
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: x_{n+1} = g(x_{n}), Fixed points, linearisation, stability of fixed points. Applications: the Newton and Secant Methods to solve nonlinear 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, Twoperson zerosum 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.
About the module
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 nonmathematics 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
Calculus Component

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).
Algebra Component

General vector spaces and subspaces
P_{n}, C^{n} and other vector spaces. Span, linear dependence/independence, bases. Reduction to rowecehelon 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. GramSchmidt orthogonalisation.

Eigenvalues and eigenvectors
Definitions and examples. Complex and repeated eigenvalues, algebraic and geometric multiplicity. Diagonalization of matrices. The CayleyHamilton 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.
About the module
This module provides the Level 2 student with further understanding of statistics and probability. This module is optional for students taking the BSc or MMath in Mathematics. 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 the module MA12003, or equivalent.
Indicative Content

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

Hypothesis tests
ttests, Inferences, Confidence intervals, Chisquare 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.
Level 3
Students taking a degree fully within Mathematics typically take modules in the following topics. Joint Honours students typically take half of these modules.
About the module
This module provides an indepth study of Differential Equations aimed at Level 3 students. 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

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.
About the module
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.
About the module
This module provides an indepth 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 R^{n} (i.e., the HeineBorel 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 R^{n}

Convergence and Equicontinuity
Uniform convergence of sequences of functions. The concept of equicontinuity of a family of functions and ArzelaAscoli Theorem.

Series
Ratio test, comparison test, Weierstrass Mtest. 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.
About the module
This module provides an indepth 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, Maxflow/Mincut theorem, FordFulkerson 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.
About the module
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 matrixrelated 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. LUfactorization. Tridiagonal systems.

Iterative Methods
A general iterative method and convergence, Jacobi method, GaussSeidel method, SOR (successive overrelaxation).

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.
About the module
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; CauchyRiemann 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.
About the module
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 R^{n}. 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 pullback 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 pullbacks. Stoke′s Theorem.

Liederivative
The Liederivative 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.
About the module
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 (hostparasitoid systems). Systems of ordinary differential equation (predatorprey and competition models).

Molecular Dynamics
Biochemical kinetics: MichaelisMenten 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.
Level 4
Students taking a degree fully within Mathematics complete a personal project, and typically select six modules from the following list. Joint Honours students typically take four of these modules, or two plus a project.
About the module
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 multistep methods: onestep methods (Euler, Trapezoidal and Backward Euler methods) and twostep methods; Consistency, zerostability, weak stability theory and Astability; Provision of the extra starting values and the potential for instability; RungeKutta 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.
About the module
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.

Turbulence and Chaos
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.
About the module
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 Nonplanar Graphs
Necessary conditions for graphs to be Planar. Toroidal graphs. Genus of graphs.

Graph Colourings
Vertex and edge colourings Chromatic polynomials. The 4colour 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.
About the module
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/MA41002 (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 reactiondiffusion equations
Fundamental solution for linear diffusion equations. Speed of a wave of invasion.

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

Systems of reactiondiffusion equations
Travelling wave solutions for systems of reactiondiffusion equations. Pattern formations in systems of reactiondiffusion 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.
About the module
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 onedimensional wave equation.

Boundary Value Problems for PDEs
Finitedifference 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: twolevel methods and brief reference to threelevel 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.
About the module
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.

Introduction to properties of plasmas, especially on the Sun

Equations of Magnetohydrodynamics (MHD)
Lorentz force, MHD equations, importance of terms. Diffusion and frozenin flux. Magnetic field lines and flux tubes.

MHD solutions
Hydrostatic pressure balance, plasma beta. Potential fields. Forcefree fields, coronal arcades. GradShafranov 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.
About the module
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.
This module may optionally be taken by students on the MMath in Mathematics, or the MSci in Mathematical Biology or Mathematics and Physics degrees. If you have questions about this module, please contact your Advisor of Studies.
Prerequisites
Students taking this module must usually have achieved a pass mark in each of the modules MA31002 and MA32001, or equivalent.
Indicative Content

OneDimensional 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, Twodimensional Systems: Hamiltonian and Gradient systems, Periodic solutions: Floquet theory, Poincare Map and Stability of Periodic Orbits, Bifurcation and Chaos.

Partial Differential Equations
Definitions, Background, Wellposedness, Maximum Principles, Spectral Theorem for Laplace Equation, Semigroups for Evolution Equations in Banach Spaces, Nonlinear Evolution Equations: Linearised Stability Analysis for ReactionDiffusion 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.
About the module
In this module the Level 5 student will learn to write their own code and to apply builtin "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 builtin 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 builtin 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 reactiondiffusion problems. The overall goal will be to solve coupled reactiondiffusion 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 reactiondiffusion equations using the builtin FEMs in COMSOL.
Delivery and Assessment
Delivery of this module will take a handson, 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.
About the module
This module, aimed at Level 5 students, gives a nonmeasure 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
(Discretetime) Markov chains and some related examples. ChapmanKolmogorov equations and classification of states.

Continuoustime Markov chains
Continuoustime 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.
About the module
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.

The Truncated Singular Value Decomposition Method

Stable inversion via the Mollification Method

Inverse Problems in General Heterogeneous Media and Medical Applications
Backward heat conduction problem. Inverse problems in reactiondiffusion equations.

Inverse problems in finance
Formulation of forward model: BlackScholes and Dupire's Formula. Inverse Problem formulation of market volatility. Reconstruction of time and pricedependent 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.
About the module
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 σalgebra. The concept of measures as a nonnegative σadditive real valued set functions defined on a σalgebra. Measurable sets, σfinite 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 nonnegative real valued function on an algebra of generators to a measure on the generated σalgebra.

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 L^{1}.

Limit theorems (concerning pointwise convergence, almost everywhere convergence, and convergence in measure)
Fatou Lemma. Monotone Convergence Theorem. Dominated Convergence Theorem.

Product Measures and the Fubini Theorem

Absolutely continuous measures and the LebesgueRadonNikodym Theorem

Definition and properties of general Lebesgue Spaces L^{p} and their embedding relations
If time permits then the following will also be covered:

The Sobolev Spaces H^{1} and their implications for the analysis of elliptic equations partial differential equations

Connection with probability theory, including random variables, cumulative distribution functions and martingales
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.
About the module
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.
About the module
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 infinitedimensional 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.
This module may optionally be taken by students on the MMath in Mathematics, or the MSci in Mathematical Biology or 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 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 LaxMilgram theorems
 Definition, main properties and examples of compact operators
 Spectral theory for compact operators

Wellposedness results for Integral and Partial Differential Equations
 Application of LaxMilgram theorem and theory of compact operators to prove the wellposedness 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.
Level 5
This year is followed by the MMath/MSci Honours students. Students undertake a yearlong personal project, and typically select six of the following modules:
About the module
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.
This module may optionally be taken by students on the MMath in Mathematics, or the MSci in Mathematical Biology or Mathematics and Physics degrees. If you have questions about this module, please contact your Advisor of Studies.
Prerequisites
Students taking this module must usually have achieved a pass mark in each of the modules MA31002 and MA32001, or equivalent.
Indicative Content

OneDimensional 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, Twodimensional Systems: Hamiltonian and Gradient systems, Periodic solutions: Floquet theory, Poincare Map and Stability of Periodic Orbits, Bifurcation and Chaos.

Partial Differential Equations
Definitions, Background, Wellposedness, Maximum Principles, Spectral Theorem for Laplace Equation, Semigroups for Evolution Equations in Banach Spaces, Nonlinear Evolution Equations: Linearised Stability Analysis for ReactionDiffusion 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.
About the module
In this module the Level 5 student will learn to write their own code and to apply builtin "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 builtin 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 builtin 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 reactiondiffusion problems. The overall goal will be to solve coupled reactiondiffusion 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 reactiondiffusion equations using the builtin FEMs in COMSOL.
Delivery and Assessment
Delivery of this module will take a handson, 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.
About the module
This module, aimed at Level 5 students, gives a nonmeasure 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
(Discretetime) Markov chains and some related examples. ChapmanKolmogorov equations and classification of states.

Continuoustime Markov chains
Continuoustime 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.
About the module
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.

The Truncated Singular Value Decomposition Method

Stable inversion via the Mollification Method

Inverse Problems in General Heterogeneous Media and Medical Applications
Backward heat conduction problem. Inverse problems in reactiondiffusion equations.

Inverse problems in finance
Formulation of forward model: BlackScholes and Dupire's Formula. Inverse Problem formulation of market volatility. Reconstruction of time and pricedependent 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.
About the module
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 σalgebra. The concept of measures as a nonnegative σadditive real valued set functions defined on a σalgebra. Measurable sets, σfinite 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 nonnegative real valued function on an algebra of generators to a measure on the generated σalgebra.

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 L^{1}.

Limit theorems (concerning pointwise convergence, almost everywhere convergence, and convergence in measure)
Fatou Lemma. Monotone Convergence Theorem. Dominated Convergence Theorem.

Product Measures and the Fubini Theorem

Absolutely continuous measures and the LebesgueRadonNikodym Theorem

Definition and properties of general Lebesgue Spaces L^{p} and their embedding relations
If time permits then the following will also be covered:

The Sobolev Spaces H^{1} and their implications for the analysis of elliptic equations partial differential equations

Connection with probability theory, including random variables, cumulative distribution functions and martingales
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.
About the module
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. Reactiondiffusion moving boundary models.

Tumourinduced Angiogenesis
Continuum PDE models. Hybrid discretecontinuum models.

Cancer Invasion
Continuum PDE models. Hybrid discretecontinuum models.

Immune Response to Cancer
ODE models of the immune response to cancer. PDE models of the immune response to cancer.

Chemotherapy and Radiotherapy Treatment
Chemotherapy drug scheduling models. The linearquadratic radiobiological model.
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.
About the module
In this module we will undertake analysis and computational simulation of mathematical models of ecological and epidemiological systems including predatorprey systems, hostparasitoid systems, plantherbivore 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

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

Spatial models
Integrodifference equation models. Partial differential equation models. Integrodifferential 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.
About the module
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 HodgkinHuxley model. The FitzHughNagumo 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.
About the module
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.
About the module
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 nonNewtonian 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.
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 MA42007 (Mathematics of Fluids I), or a similar course.
Indicative Content

Introduction (recap of basic fluid dynamics)
NavierStokes 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

NonNewtonian Fluids
Types of nonNewtonian 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.
About the module
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 infinitedimensional 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.
This module may optionally be taken by students on the MMath in Mathematics, or the MSci in Mathematical Biology or 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 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 LaxMilgram theorems
 Definition, main properties and examples of compact operators
 Spectral theory for compact operators

Wellposedness results for Integral and Partial Differential Equations
 Application of LaxMilgram theorem and theory of compact operators to prove the wellposedness 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.
Careers
Mathematics is central to the sciences, and to the development of a prosperous, modern society. The demand for people with mathematical qualifications is considerable, and a degree in mathematics is a highly marketable asset.
Mathematics graduates are consistently amongst those attracting the highest graduate salaries and can choose from an everwidening range of careers in research, industry, science, engineering, commerce, finance and education.
Many of our graduates enter the financial sector following career paths in accountancy, banking, the stock market and insurance.
Even if you do not take your mathematics any further than university, employers know that mathematics graduates are intelligent, logical problem solvers. With this training behind you, the career options become almost limitless.
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Fees for students starting 201819
Fee category  Fees for students starting 201819 

Scottish and EU students  £1,820 per year of study (for Sept 2017 entry). Fees for September 2018 will be confirmed by the Scottish Government in early 2018. 
Rest of UK students  BSc  £9,250 per year, for a maximum of 3 years, even if you are studying a four year degree. MMath/MSci  £9,250 per year, for a maximum of 4 years, even if you are studying a five year degree. See our scholarships for rest of UK applicants. 
Overseas students (nonEU)  £19,950 per year of study. See our scholarships for international applicants. 
Scottish and EU students can apply to the Students Award Agency for Scotland (SAAS) to have tuition fees paid by the Scottish Government.
Rest of the UK students can apply for financial assistance, including a loan to cover the full cost of the tuition fees, from the Student Loans Company.
Tuition fees for Overseas (nonEU) students are guaranteed not to increase by more than 3% per year, for the length of your course.
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