# Mathematics and Computer Science with a Year in Industry BSc (Joint Honours)

- Course length: 4 years
- UCAS code: GG16
- Year of entry: 2018
- Typical offer: A-level : ABB / IB : 33 / BTEC : D*DD

## Honours Select

×This programme offers Honours Select combinations.

## Honours Select 100

×This programme is available through Honours Select as a Single Honours (100%).

## Honours Select 75

×This programme is available through Honours Select as a Major (75%).

## Honours Select 50

×This programme is available through Honours Select as a Joint Honours (50%).

## Honours Select 25

×This programme is available through Honours Select as a Minor (25%).

## Study abroad

×This programme offers study abroad opportunities.

## Year in China

×This programme offers the opportunity to spend a Year in China.

## Accredited

×This programme is accredited.

### Module details

### Programme Year One

Year 1 of the programme has been designed as an even split between subjects related to Computing and Mathematics. You will take

- COMP101 Introduction to Programming

or

- COMP105 Programming Language Paradigms (depending on your prior programming experience)

all of

- COMP107 Graduates for the Digital Society
- COMP108 Algorithmic Foundations
- COMP122 Object-Oriented Programming
- MATH101 Calculus I
- MATH102 Calculus II
- MATH103 Introduction to Linear Algebra

and one of

- MATH122 Dynamic Modelling
- MATH142 Numbers, Groups and Codes
- MATH162 Introduction to Statistics

#### Year One Compulsory Modules

##### Introduction to Programming (COMP101)

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**0:100 **Aims**- To introduce the concepts and principles of problem solving using computational thinking.
- To identify and employ algorithms in the solution of identified problems.
- To develop sound principles in designing programming solutions to identified problems using appropriate data structures.
- To introduce the concepts of implementing solutions in a high level programming language.

**Learning Outcomes**Identify the principles and practice of using high-level programming constructs to solve a problem

Use relevant data structures to solve problems

Produce documentation in support of a programmed solution

Use a suitable Integrated Development Environment to carry out Implementation, interpretation/compilation, testing and execution.

Identify appropriate design approaches to formulate a solution to a program

Design and apply effective test cases

Develop debugging skills to correct a program

Specific learning outcomes are listed above.

General learning outcomes:

- An understanding of the principles and practice of analysis and design in the construction of robust, maintainable programs which satisfy their requirements;
- A competence to design, write, compile, test and execute straightforward programs using a high-level language;
- An appreciate of the principles of procedural programming;
- An awareness of the need for a professional approach to design and the importance of good documentation to the finished programs.

##### Programming Language Paradigms (COMP105)

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**25:75 **Aims**- To introduce the functional programming paradigm, and to compare and contrastit with the imperative programming paradigm.
- To explore the common techniques that are employed to solve problems in a functional way.

**Learning Outcomes**Describe the imperative and functional programming paradigms including the differences between them.

Apply recursion to solve algorithmic tasks.

Apply common functional programming idioms such as map, filter, reduce and fold.

Write programs using a functional programming language.

##### Graduates for the Digital Society (COMP107)

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**0:100 **Aims**To provide the students with a wide-ranging understanding of the discipline of computing, and to introduce students to concepts of professional ethics as well as social and legal aspects of computing.

To equip the students with the communication, time and project management, and employability skills required for a computing professional.

To allow the students to gain an understanding of the importance of appropriate and efficient database design strategies, at the conceptual and logical level, and how to communicate them effectively to stakeholders

To provide the students with practical experience of database programming, including data manipulation and query in SQL.**Learning Outcomes**Identify and appraise professional, ethical, legal and social issues related to the work of a professional within the IT industry with particular regard to the BCS Codes of Conduct and Practice.

Recognise employability and entrepreneurship skills that prepare students to undertake paid work experience during the course of their degree or independently

Identify, describe and discuss economic, historical, organisational, research, and social aspects of computing as a discipline and computing in practice;

Identify and apply principles of database conceptual design using ER and UML design methodologies

Recognise database logical design principles, and issues related to database physical design;

Use SQL as a data definition and manipulation language, and as a language for querying database

##### Data Structures and Algorithms (COMP108)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**60:40 **Aims**1. To introduce the notation, terminology, and techniques underpinning the study of algorithms.

2. To introduce basic data structures and associated algorithms.

3. To introduce standard algorithmic design paradigms and efficient use of data structures employed in the development of efficient algorithmic solutions.**Learning Outcomes**Be able to describe the principles of and apply a variety of data structures and their associated algorithms; Be able to describe standard algorithms, apply a given pseudo code algorithm in order to solve a given problem, and carry out simple asymptotic analyses of algorithms;

Be able to describe and apply different algorithm design principles and distinguish the differences between these principles;

Be able to choose and justify the use of appropriate data structures to enable efficient implementation of algorithms;

##### Object-oriented Programming (COMP122)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**0:100 **Aims**- To develop understanding of object-oriented software methodology, in theory and practice.
- To further develop sound principles in software design and software development.
- To understand basic concepts of software testing principles and software version control systems.

**Learning Outcomes**Describe object hierarchy structure and how to design such a hierarchy of related classes.

Describe the concept of object polymorphism in theory and demonstrate this concept in practice.

Design and code iterators for collection-based data management.

Design simple unit tests using appropriate software tools.

Demonstrate concepts of event-driven programming and be able to design simple GUI to demonstrate this understanding.

Identify and describe the task and issues involved in the process of developing interactive products for people, and the techniques used to perform these tasks.

- To develop understanding of object-oriented software methodology, in theory and practice.
##### Calculus I (MATH101)

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**80:20 **Aims**1. To introduce the basic ideas of differential and integral calculus, to develop the basic skills required to work with them and to apply these skills to a range of problems.

2. To introduce some of the fundamental concepts and techniques of real analysis, including limits and continuity.

3. To introduce the notions of sequences and series and of their convergence.

**Learning Outcomes**differentiate and integrate a wide range of functions;

sketch graphs and solve problems involving optimisation and mensuration

understand the notions of sequence and series and apply a range of tests to determine if a series is convergent

##### Calculus II (MATH102)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**80:20 **Aims**· To discuss local behaviour of functions using Taylor’s theorem.

· To introduce multivariable calculus including partial differentiation, gradient, extremum values and double integrals.

**Learning Outcomes**use Taylor series to obtain local approximations to functions;

obtain partial derivaties and use them in several applications such as, error analysis, stationary points change of variables

evaluate double integrals using Cartesian and Polar Co-ordinates

##### Introduction to Linear Algebra (MATH103)

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**80:20 **Aims**- To develop techniques of complex numbers and linear algebra, including equation solving, matrix arithmetic and the computation of eigenvalues and eigenvectors.
- To develop geometrical intuition in 2 and 3 dimensions.
- To introduce students to the concept of subspace in a concrete situation.
- To provide a foundation for the study of linear problems both within mathematics and in other subjects.

**Learning Outcomes**manipulate complex numbers and solve simple equations involving them

solve arbitrary systems of linear equations

understand and use matrix arithmetic, including the computation of matrix inverses

compute and use determinants

understand and use vector methods in the geometry of 2 and 3 dimensions

calculate eigenvalues and eigenvectors and, if time permits, apply these calculations to the geometry of conics and quadrics

##### Newtonian Mechanics (MATH122)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**80:20 **Aims**To provide a basic understanding of the principles of Classical Mechanics and their application to simple dynamical systems.

Learning Outcomes:

After completing the module students should be able to analyse real world problems

involving:

- the motions of bodies under simple force systems

- conservation laws for momentum and energy

- rigid body dynamics using centre of mass,

angular momentum and moments of inertia**Learning Outcomes**

After completing the module students should be able to analyse

real-world problems involving:the motions of bodies under simple force systems

conservation laws for momentum and energy

rigid body dynamics using centre of mass, angular momentum and moments

oscillation, vibration, resonance

##### Numbers, Groups and Codes (MATH142)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**90:10 **Aims**· To provide an introduction to rigorous reasoning in axiomatic systems exemplified by the framework of group theory.

· To give an appreciation of the utility and power of group theory as the study of symmetries.

· To introduce public-key cryptosystems as used in the transmission of confidential data, and also error-correcting codes used to ensure that transmission of data is accurate. Both of these ideas are illustrations of the application of algebraic techniques.

**Learning Outcomes** Use the division algorithm to construct the greatest common divisor of a pair of positive integers;

Solve linear congruences and find the inverse of an integer modulo a given integer;

Code and decode messages using the public-key method

Manipulate permutations with confidence

Decide when a given set is a group under a specified operation and give formal axiomatic proofs;

Understand the concept of a subgroup and use Lagrange''s theorem;

Understand the concept of a group action, an orbit and a stabiliser subgroup

Understand the concept of a group homomorphism and be able to show (in simple cases) that two groups are isomorphic;

### Programme Year Two

In Year 2 you continue with a mix of modules related to Computer Science and Mathematics but also have the opportunity to specialise in certain subject areas of your choice.

In Semester 1 you take two of the following modules:

- COMP111 Introduction to Artificial Intelligence
- COMP201 Software Engineering I
- COMP207 Database Development

and two of

- MATH201 Ordinary Differential Equations
- MATH225 Vector Calculus with Applications in Fluid Mechanics
- MATH227 Math Models: Micro-economics & Population Dynamics
- MATH241 Metric Spaces and Calculus
- MATH243 Complex Functions
- MATH244 Linear Algebra and Geometry
- MATH261 Introduction to Methods of Operational Research
- MATH268 Operational Research: Probabilistic Models

In Semester 2 you take

- COMP202 Complexity of algorithms

one of

- COMP124 Computer Systems
- COMP218 Decision, Computation and Language

and two of

- MATH206 Group Project Module
- MATH224 Introduction to the Methods of Applied Mathematics
- MATH228 Classical Mechanics
- MATH247 Commutative Algebra
- MATH248 Geometry of Curves
- MATH262 Financial Mathematics II
- MATH263 Statistical Theory and Methods I
- MATH264 Statistical Theory and Methods II
- MATH2661 Numerical Methods

#### Year Two Compulsory Modules

##### Complexity of Algorithms (COMP202)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**80:20 **Aims**- To demonstrate how the study of algorithmics has been applied in a number of different domains.
- To introduce formal concepts of measures of complexity and algorithms analysis.
- To introduce fundamental methods in data structures and algorithms design.
- To make students aware of computationally hard problems and possible ways of coping with them.

**Learning Outcomes**At the conclusion of the module students should have an appreciation of the diversity of computational fields to which algorithmics has made significant contributions.

At the conclusion of the module students should have fluency in using basic data structures (queues, stacks, trees, graphs, etc) in conjunction with classical algorithmic problems (searching, sorting, graph algorithms, security issues) and be aware of basic number theory applications, etc.

At the conclusion of the module students should be familiar with formal theories providing evidence that many important computational problems are inherently intractable, e.g., NP-completeness.

#### Year Two Optional Modules

##### Introduction to Artificial Intelligence (COMP111)

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**80:20 **Aims**1. To provide an introduction to AI through studying search problems, reasoning under uncertainty, knowledge representation, planning, and learning in intelligent systems.

2. To equip the students with an awareness of the main applications of AI and the history, philosophy, and ethics of AI.**Learning Outcomes**Students should be able to identify and describe the characteristics of intelligent agents and the environments that they can inhabit.

Students should be able to identify, contrast and apply to simple examples the basic search techniques that have been developed for problem-solving in AI.

Students should be able to apply to simple examples the basic notions of probability theory that have been applied to reasoning under uncertainty in AI.

Students should be able to identify and describe logical agents and the role of knowledge bases and logical inference in AI.

Students should be able to identify and describe some approaches to learning in AI and apply these to simple examples.

##### Computer Systems (COMP124)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**80:20 **Aims**1. Tointroduce how computers function at the instruction operation level.

2. Tointroduce the relationships between the instruction operation level and boththe higher (software) and lower (hardware) levels

3. Tointroduce students to the structure and functionality of modern operating systems.

4. Toexplain how the principal components of computer-based systems perform theirfunctions and how they interact with each other.

**Learning Outcomes**Describe the structure and operation of computer hardware at the register transfer level.

Implement/reason about simple algorithms at the level of machine code.

Describe the overall structure and functionality of a modern operating system and its interactions with computer hardware and user processes.

Construct/reason about programs that involve the management of concurrent processes.

Explain at a simple level the operation and organisation of compilers.

##### Software Engineering I (COMP201)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**80:20 **Aims**The module is intended to develop an understanding of the problems associated with the development of significant computing systems (that is, systems that are too large to be designed and developed by a single person,and are designed to be used by many users) and to appreciate the techniques and tools necessary to develop such systems efficiently, in a cost-effective manner.

**Learning Outcomes** At the end of the module, the student is expected to

- realise the problems in designing and building significant computer systems;

- understand the need to design systems that fully meet the requirements of the intended users including functional and non functional elements

- appreciate the need to ensure that the implementation of a design is adequately tested to ensure that the completed system meets the specifications

- be fully aware of the principles and practice of an O-O approach to the design and development of computer systems

- be able to apply these principles in practice.

##### Database Development (COMP207)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**80:20 **Aims**- To introduce students to the problems arising from concurrency in databases, information security considerations and how they are solved
- To introduce students to the problems arising from the integration of heterogeneous sources of information and the use of semi-structured data;
- To introduce students to non-relational databases and the economic factors involved in their selection
- To introduce students to techniques for analyzing large amounts of data, the security issues and commercial factors involved with them

**Learning Outcomes**At the end of this module the student will be able to identify and apply the principles underpinning transaction management within DBMS and the main security issues involved in securing transaction;

Demonstrate an understanding of advanced SQL topics;

Illustrate the issues related to Web technologies as a semi-structured data representation formalism;

Identify the principles underlying object relational models and the economic factors in their uptake and development;

Interpret the main concepts and security aspects in data warehousing, and the concepts of data mining and commercial considerations involved in adopting the paradigm.

##### Decision, Computation and Language (COMP218)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**80:20 **Aims**- To introduce formal concepts of automata, grammars and languages.
To introduce ideas of computability and decidability.

To illustrate the importance of automata, formal language theory and general models of computation in Computer Science and Artificial Intelligence.

**Learning Outcomes**Be familiar with the relationships between language as an object recognised by an automaton and as a set generated by a formal grammar. Be able to apply standard translations between non-deterministic and deterministic finite automata.

Be familiar with the distinct types of formal grammar (e.g. Chomsky hierarchy) and the concept of normal forms for grammars.

Be aware of the limitations (with respect to expressive power) of different automata and grammar forms.

Understand the distinction between recursive and recursively enumerable languages.- To introduce formal concepts of automata, grammars and languages.
##### Ordinary Differential Equations (MATH201)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**90:10 **Aims**To familiarize students with basic ideas and fundamental techniques to solve ordinary differential equations.

To illustrate the breadth of applications of ODEs and fundamental importance of related concepts.

**Learning Outcomes**After completing the module students should be:

- familiar with elementary techniques for the solution of ODE''s, and the idea of reducing a complex ODE to a simpler one;

- familiar with basic properties of ODE, including main features of initial value problems and boundary value problems, such as existence and uniqueness of solutions;

- well versed in the solution of linear ODE systems (homogeneous and non-homogeneous) with constant coefficients matrix;

- aware of a range of applications of ODE.

##### Vector Calculus With Applications in Fluid Mechanics (MATH225)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**85:15 **Aims**To provide an understanding of the various vector integrals, the operators div, grad and curl and the relations between them.

To give an appreciation of the many applications of vector calculus to physical situations.

To provide an introduction to the subjects of fluid mechanics and electromagnetism.

**Learning Outcomes**After completing the module students should be able to:

- Work confidently with different coordinate systems.

- Evaluate line, surface and volume integrals.

- Appreciate the need for the operators div, grad and curl together with the associated theorems of Gauss and Stokes.

- Recognise the many physical situations that involve the use of vector calculus.

- Apply mathematical modelling methodology to formulate and solve simple problems in electromagnetism and inviscid fluid flow.

All learning outcomes are assessed by both examination and course work.

##### Mathematical Models: Microeconomics and Population Dynamics (MATH227)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**90:10 **Aims**1. To provide an understanding of the techniques used in constructing, analysing, evaluating and interpreting mathematical models.

2. To do this in the context of two non-physical applications, namely microeconomics and population dynamics.

3. To use and develop mathematical skills introduced in Year 1 - particularly the calculus of functions of several variables and elementary differential equations.

**Learning Outcomes**After completing the module students should be able to:

- Use techniques from several variable calculus in tackling problems in microeconomics.

- Use techniques from elementary differential equations in tackling problems in population dynamics.

- Apply mathematical modelling methodology in these subject areas.

All learning outcomes are assessed by both examination and course work.

##### Metric Spaces and Calculus (MATH241)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**90:10 **Aims**To introduce the basic elements of the theories of metric spaces and calculus of several variables.

**Learning Outcomes**After completing the module students should:

Be familiar with a range of examples of metric spaces.

Have developed their understanding of the notions of convergence and continuity.

Understand the contraction mapping theorem and appreciate some of its applications.

Be familiar with the concept of the derivative of a vector valued function of several variables as a linear map.

Understand the inverse function and implicit function theorems and appreciate their importance.

Have developed their appreciation of the role of proof and rigour in mathematics.

##### Complex Functions (MATH243)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**80:20 **Aims**To introduce the student to a surprising, very beautiful theory having intimate connections with other areas of mathematics and physical sciences, for instance ordinary and partial differential equations and potential theory.

**Learning Outcomes**After completing this module students should:

- appreciate the central role of complex numbers in mathematics;

- be familiar with all the classical holomorphic functions;

- be able to compute Taylor and Laurent series of such functions;

- understand the content and relevance of the various Cauchy formulae and theorems;

- be familiar with the reduction of real definite integrals to contour integrals;

- be competent at computing contour integrals.

##### Linear Algebra and Geometry (MATH244)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**90:10 **Aims**To introduce general concepts of linear algebra and its applications in geometry and other areas of mathematics.

**Learning Outcomes**After completing the module students should be able to:

• appreciate the geometric meaning of linear algebraic ideas,

• appreciate the concept of an abstract vector space and how it is used in different mathematical

situations,

• apply a change of coordinates to simplify a linear map,

• manipulate matrix groups (in particular Gln, On and Son),

• understand bilinear forms from a geometric point of view.

##### Introduction to Methods of Operational Research (MATH261)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**90:10 **Aims**- Appreciate the operational research approach.
- Be able to apply standard methods to a wide range of real-world problems as well as applications in other areas of mathematics.
- Appreciate the advantages and disadvantages of particular methods.
- Be able to derive methods and modify them to model real-world problems.
- Understand and be able to derive and apply the methods of sensitivity analysis.

**Learning Outcomes**Appreciate the operational research approach.Be able to apply standard methods to a wide range of real-world problems as well asapplications in other areas of mathematics.

Appreciate the advantages and disadvantages of particular methods.

Be able to derive methods and modify them to model real-world problems.Understand and be able to derive and apply the methods of sensitivity analysis. Appreciate the importance of sensitivity analysis.

##### Operational Research: Probabilistic Models (MATH268)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**90:10 **Aims**To introduce a range of models and techniques for solving under uncertainty in Business, Industry, and Finance.

**Learning Outcomes**The ability to understand and describe mathematically real-life optimization problems.

Understanding the basic methods of dynamical decision making.

Understanding the basics of forecasting and simulation.

The ability to analyse elementary queueing systems.

##### Introduction to the Methods of Applied Mathematics (MATH224)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**90:10 **Aims**To provide a grounding in elementary approaches to solution of some of the standard partial differential equations encountered in the applications of mathematics.

To introduce some of the basic tools (Fourier Series) used in the solution of differential equations and other applications of mathematics.

**Learning Outcomes**After completing the module students should:

- be fluent in the solution of basic ordinary differential equations, including systems of first order equations;

- be familiar with the concept of Fourier series and their potential application to the solution of both ordinary and partial differential equations;

- be familiar with the concept of Laplace transforms and their potential application to the solution of both ordinary and partial differential equations;

- be able to solve simple first order partial differential equations;

- be able to solve the basic boundary value problems for second order linear partial differential equations using the method of separation of variables.

##### Classical Mechanics (MATH228)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**90:10 **Aims**To provide a basic understanding of the principles of Classical Mechanics and their application to simple dynamical systems.

**Learning Outcomes**the motions of bodies under simple force systems, including calculations of the orbits of satellites, comets and planetary motions

motion relative to a rotating frame, Coriolis and centripetal forces, motion under gravitry over the Earth''s surface

rigid body dynamics using centre of mass, angular momentum and moments of inertia

##### Commutative Algebra (MATH247)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**90:10 **Aims**To give an introduction to abstract commutative algebra and show how it both arises naturally, and is a useful tool, in number theory.

**Learning Outcomes**After completing the module students should be able to:

• Work confidently with the basic tools of algebra (sets, maps, binary operations and equivalence relations).

• Recognise abelian groups, different kinds of rings (integral, Euclidean, principal ideal and unique factorisation domains) and fields.

• Find greatest common divisors using the Euclidean algorithm in Euclidean domains.

• Apply commutative algebra to solve simple number-theoretic problems.

##### Geometry of Curves (MATH248)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**90:10 **Aims**To introduce geometric ideas and develop the basic skills in handling them.

To study the line, circle, ellipse, hyperbola, parabola, cubics and many other curves.

To study theoretical aspects of parametric, algebraic and projective curves.

To study and sketch curves using an appropriate computer package.

**Learning Outcomes**After completing this module students should be able to:

- use a computer package to study curves and their evolution in both parametric and algebraic forms.

-determine and work with tangents, inflexions, curvature, cusps, nodes, length and other features.

-calculate envelopes and evolutes.

- solve the position and shape of some algebraic curves including conics.

The first learning outcome is assessed by coursework, the others by both coursework and examination.

##### Financial Mathematics II (MATH262)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**90:10 **Aims** to provide an understanding of the basic financial mathematics theory used in the study process of actuarial/financial interest,

to provide an introduction to financial methods and derivative pricing financial instruments ,

to understand some financial models with applications to financial/insurance industry,

to prepare the students adequately and to develop their skills in order to be ready to sit for the CT1 & CT8 subject of the Institute of Actuaries (covers 20% of CT1 and material of CT8).

**Learning Outcomes**After completing the module students should:

(a) Understand the assumptions of CAMP, explain the no riskless lending or borrowing and other lending and borrowing assumptions, be able to use the formulas of CAMP, be able to derive the capital market line and security market line,

(b) Describe the Arbitrage Theory Model (APT) and explain its assumptions, perform estimating and testing in APT,

(c) Be able to explain the terms long/short position, spot/delivery/forward price, understand the use of future contracts, describe what a call/put option (European/American) is and be able to makes graphs and explain their payouts, describe the hedging for reducing the exposure to risk, be able to explain arbitrage, understand the mechanism of short sales,

(d) Explain/describe what arbitrage is, and also the risk neutral probability measure, explain/describe and be able to use (perform calculation) the binomial tree for European and American style options,

(e) Understand the probabilistic interpretation and the basic concept of the random walk of asset pricing,

(f) Understand the concepts of replication, hedging, and delta hedging in continuous time,

(g) Be able to use Ito''s formula, derive/use the the Black‐Scholes formula, price contingent claims (in particular European/American style options and forward contracts), be able to explain the properties of the Black‐Scholes formula, be able to use the Normal distribution function in numerical examples of pricing,

(h) Understand the role of Greeks , describe intuitively what Delta, Theta, Gamma is, and be able to calculate them in numerical examples.

##### Statistical Theory and Methods I (MATH263)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**85:15 **Aims**To introduce statistical methods with a strong emphasis on applying standard statistical techniques appropriately and with clear interpretation. The emphasis is on applications.

**Learning Outcomes**After completing the module students should have a conceptual and practical understanding of a range of commonly applied statistical procedures. They should have also developed some familiarity with the statistical package MINITAB.

##### Statistical Theory and Methods II (MATH264)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**90:10 **Aims**To introduce statistical distribution theory which forms the basis for all applications of statistics, and for further statistical theory.

**Learning Outcomes**After completing the module students should understand basic probability calculus. They should be familiar with a range of techniques for solving real life problems of the probabilistic nature.

### Programme Year Three

Year 3 of the programme is taken up with a placement in a professional software industry environment.

#### Year Three Compulsory Modules

##### Industrial Placement Y3 (COMP299)

**Level**2 **Credit level**120 **Semester**Whole Session **Exam:Coursework weighting**0:100 **Aims**The aim of this module is to provide students with working experience within a commercial or industrial environment and to gain an understanding of the various operational aspects of a company, of its products and working practices.

**Learning Outcomes**At the end of the module students should have an appreciation of the working practices of the organisation hosting the placement and show an ability to make a contribution towards its daily operaton.

Students should gain practical experience of software design and development, and coding practices within industrial/commercial environments;

Preparation of reports and other relevant documentation

Working practices within industrial/commercial environments

Experience with management of time and resources

### Programme Year Four

You choose 4 out of

- COMP219 Artificial Intelligence
- COMP304 Knowledge Representation and Reasoning
- COMP305 Biocomputation
- COMP309 Efficient Sequential Algorithms
- COMP310 Multi-Agent Systems
- COMP313 Formal Methods
- COMP315 Technologies for E-Commerce
- COMP319 Software Engineering II
- COMP323 Introduction to Computational Game Theory
- COMP326 Computational Game Theory and Mechanism Design
- COMP331 Optimization
- COMP3911 Final Year First Semester 15 Credit Project
- COMP3921 Final Year Second Semester 15 Credit Project

as well as 4 of

- MATH322 Chaos and Dynamical Systems
- MATH323 Further Methods of Applied Mathematics
- MATH324 Cartesian Tensors and Mathematical Models of Solids and Viscous Fluids
- MATH325 Quantum Mechanics
- MATH343 Group Theory
- MATH344 Combinatorics
- MATH351 Analysis and Number Theory
- MATH362 Applied Probability
- MATH363 Linear Statistical Models
- MATH367 Networks in Theory and Practice.

#### Year Four Optional Modules

##### Artificial Intelligence (COMP219)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**80:20 **Aims**To provide a grounding in the AI programming language Prolog.

**Learning Outcomes**At the end of this module, students should be able to:

identify, contrast and apply to simple examples the major search techniques that have been developed for problem-solving in AI;

distinguish the characteristics, and advantages and disadvantages, of the major knowledge representation paradigms that have been used in AI, such as production rules, semantic networks, propositional logic and first-order logic;

solve simple knowledge-based problems using the AI representations studied;

identify or describe approaches used to solve planning problems in AI and apply these to simple examples;

identify or describe the major approaches to learning in AI and apply these to simple examples;

identify or describe some of the major applications of AI;

understand and write Prolog code to solve simple knowledge-based problems.

##### Knowledge Representation and Reasoning (COMP304)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**75:24 **Aims**- To introduce Knowledge Representation as a research area.
To give a complete and critical understanding of the notion of representation languages and logics.

To study description logic and their use;

To study epistemic logic and its use

To study methods for reasoning under uncertainty

**Learning Outcomes**be able to explain and discuss the need for formal approaches to knowledge representation in artificial intelligence, and in particular the value of logic as such an approach; be able to demonstrate knowledge of the basics of propositional logic

be able to determine the truth/satisfiability of modal formula;

be able to perform modal logic model checking on simple examples

be able to perform inference tasks in description logic

be able to model problems concenring agents'' knowledge using epistemic logic;

be able to indicate how updates and other epistemic actions determine changes on epistemic models;

have sufficient knowledge to build "interpreted systems" from a specification, and to verify the "knowledge" properties of such systems;

be familiar with the axioms of a logic for knowledge of multiple agents;

be able to model simple problems involving uncertainty, using probability and decision theory;

be able to demonstrate knowledge of the basics of probability and decision theory, and their use in addressing problems in knowledge representation.

- To introduce Knowledge Representation as a research area.
##### Biocomputation (COMP305)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**80:20 **Aims**- To introduce students to some of the established work in the field of neural computation.
- To highlight some contemporary issues within the domain of neural computation with regard to biologically-motivated computing particularly in relation to multidisciplinary research.
- To equip students with a broad overview of the field of evolutionary computation, placing it in a historical and scientific context.
- To emphasise the need to keep up-to-date in developing areas of science and technology and provide some skills necessary to achieve this.
To enable students to make reasoned decisions about the engineering of evolutionary ("selectionist") systems.

**Learning Outcomes**Account for biological and historical developments neural computation Describe the nature and operation of MLP and SOM networks and when they are used

Assess the appropriate applications and limitations of ANNs

Apply their knowledge to some emerging research issues in the field

Understand how selectionist systems work in general terms and with respect to specific examples

Apply the general principles of selectionist systems to the solution of a number of real world problems

Understand the advantages and limitations of selectionist approaches and have a considered view on how such systems could be designed

##### Efficient Sequential Algorithms (COMP309)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**80:20 **Aims**- To learn some advanced topics in the design and analysis of efficient sequential algorithms, and a few key results related to the study of their complexity.

**Learning Outcomes**At the conclusion of the module students should have an understanding of the role of algorithmics within Computer Science;

have expanded their knowledge of computational complexity theory;

be aware of current research-level concerns in the field of algorithm design.

##### Multi-agent Systems (COMP310)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**- To introduce the student to the concept of an agent and multi-agent systems, and the main applications for which they are appropriate;
- To introduce the main issues surrounding the design of intelligent agents;
- To introduce the main issues surrounding the design of a multi-agent society.
- To introduce a contemporary platform for implementing agents and multi-agent systems.

**Learning Outcomes**- Understand the notion of an agent, how agents are distinct from other software paradigms (eg objects) and understand the characteristics of applications that lend themselves to an agent-oriented solution;
- Understand the key issues associated with constructing agents capable of intelligent autonomous action, and the main approaches taken to developing such agents;
- Understand the key issues in designing societies of agents that can effectively cooperate in order to solve problems, including an understanding of the key types of multi-agent interactions possible in such systems
- Understand the main application areas of agent-based solutions, and be able to develop a meaningful agent-based system using a contemporary agent development platform.

##### Formal Methods (COMP313)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**As more complex computational systems are used within critical applications, it is becoming essential that these systems are formally specified. Such specifications are used to give a precise and unambiguous description of the required system. While this is clearly important in criticial systems such as industrial process management and air/spacecraft control, it is also becoming essential when applications involving E-commerce and mobile code are developed. In addition, as computational systems become more complex in general, formal specification can allow us to define the key characteristics of systems in a clear way and so help the development process.

Formal specifications provide the basis for verification of properties of systems. While there are a number of ways in which this can be achieved, the model-checking approach is a practical and popular way to verify the temporal properties of finite-state systems. Indeed, such temporal verification is widely used within the design of critical parts of integrated circuits, has recently been used to verify parts of the control mechanism for one of NASA’s space probes, and is now beginning to be used to verify general Java programs.

**Learning Outcomes**Upon completing this module, a student will:

- understand the principles of standard formal methods, such as Z;
- understand the basic notions of temporal logic and its use in relation to reactive systems;
- understand the use of model checking techniques in the verification of reactive systems;
- be aware of some of the current research issues related to formal methods.

##### Technologies for E-commerce (COMP315)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**- To introduce the environment in which e-commerce takes place, the main technologies for supporting e-commerce, and how these technologies fit together;
- To introduce security as a major issue in secure e-commerce, and to provide an overview of security issues;
- To introduce encryption as a means of ensuring security, and to describe how secure encryption can be delivered;
- To introduce issues relating to privacy; and
- To introduce auction protocols and negotiation mechanisms as emerging e-commerce technologies

**Learning Outcomes**Upon completing this module, a student will:

- understand the main technologies behind e-commerce systems and how these technologies interact;
- understand the security issues which relate to e-commerce;
- understand how encryption can be provided and how it can be used to ensure secure commercial transactions;
- understand implementation aspects of e-commerce and cryptographic systems;
- have an appreciation of privacy issues; and
- understand auction protocols and interaction mechanisms.

##### Software Engineering II (COMP319)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**The overall aim of this module is to introduce students to a range of advanced, near-research level topics in contemporary software engineering. The actual choice of topics will depend upon the interests of the lecturer and the topics current in the software engineering research literature at that time. The course will introduce issues from a problem (user-driven) perspective and a technology-driven perspective – where users have new categories of software problems that they need to be solved, and where technology producers create technologies that present new opportunities for software products. It will be expected that students will read articles in the software engineering research literature, and will discuss these articles in a seminar-style forum.

**Learning Outcomes**Understand the key problems driving research and development in contemporary software engineering (eg the need to develop software for embedded systems). Be conversant with approaches to these problems, as well as their advantages, disadvantages, and future research directions.

Understand the key technological drivers behind contemporary software engineering research (eg the increased use of the Internet leading to the need to engineer systems on and for the web).

Be able to present, analyse, and give a reasoned critique of articles in the software engineering research literature. ble to read and understand articles in the research literature of software engineering.

Be able to read and understand articles in the research literature of software engineering.

##### Introduction to Computational Game Theory (COMP323)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**80:20 **Aims**- to introduce the student to the notion of a game, its solutions concepts, and other basic notions and tools of game theory, and the main applications for which they are appropriate, including electricity trading markets;
to formalize the notion of strategic thinking and rational choice by using the tools of game theory, and to provide insights into using game theory in modeling applications;

to draw the connections between game theory, computer science, and economics, especially emphasizing the computational issues;

to introduce contemporary topics in the intersection of game theory, computer science, and economics;

**Learning Outcomes**Given a real world situation a student should be able to identify its key strategic aspects and based on these be able to connect them to appropriate game theoretic concepts;

A student will understand the key connections and interactions between game theory, computer science and economics;

A student will understand the impact of game theory on its contemporary applications, and be able to identify the key such application areas;

- to introduce the student to the notion of a game, its solutions concepts, and other basic notions and tools of game theory, and the main applications for which they are appropriate, including electricity trading markets;
##### Computational Game Theory and Mechanism Design (COMP326)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**80:20 **Aims**- To provide an understanding of the inefficiency arising from uncontrolled, decentralized resource allocation.
- To provide a foundation for modelling various mechanism design problems together with their algorithmic aspects.
- To provide the tools and paradigms for the design and analysis of efficient algorithms/mechanisms that are robust in environments that involve interactions of selfish agents.
- To review the links and interconnections between algorithms and computational issues with selfish agents.

**Learning Outcomes**Have a systematic understanding of current problems and important concepts in the field of computational game theory.

Ability to quantify the inefficiency of equilibria.

The ability to formulate mechanism design models or network games for the purpose of modeling particular applications.

The ability to use, describe and explain appropriate algorithmic paradigms and techniques in context of a particular game-theoretic or mechanism design problem.

##### Optimisation (COMP331)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**75:25 **Aims**- To provide a foundation for modelling various continuous and discrete optimisation problems.
- To provide the tools and paradigms for the design and analysis of algorithms for continuous and discrete optimisation problems. Apply these tools to real-world problems.
- To review the links and interconnections between optimisation and computational complexity theory.
- To provide an in-depth, systematic and critical understanding of selected significant topics at the intersection of optimisation, algorithms and (to a lesser extent) complexity theory, together with the related research issues.

**Learning Outcomes**A conceptual understanding of current problems and techniques in the field of optimisation.

The ability to formulate optimisation models for the purpose of modelling particular applications.The ability to use appropriate algorithmic paradigms and techniques in context of a particular optimisation model. ##### Final Year First Semester 15 Credit Project (COMP391)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**0:100 **Aims**- To give students the opportunity to work in a guided but independent fashion to explore an individual problem in depth, making practical use of principles, techniques and methodologies acquired elsewhere in the course.
- To give experience of carrying out a sustained piece of individual work and in producing a dissertation.
- To enhance communication skills, both oral and written.

**Learning Outcomes**After completing the module students should be able:

- To specify a problem, and produce a plan to address the problem
- To manage their time effectively so as to carry out their plan
- To locate and make use of information relevant to their project
- To design a solution to their problem
- To implement and test their solution
- To evaluate in a critical fashion the work they have done, and to place it in the context of related work
- To prepare and deliver a formal presentation
- To structure and write a dissertation describing their project

##### Final Year Second Semester 15 Credit Project (COMP392)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**0:100 **Aims**- To give students the opportunity to work in a guided but independent fashion to explore an individual problem in depth, making practical use of principles, techniques and methodologies acquired elsewhere in the course.
- To give experience of carrying out a sustained piece of individual work and in producing a dissertation.
- To enhance communication skills, both oral and written.

**Learning Outcomes**After completing the module students should be able:

- To specify a problem, and produce a plan to address the problem
- To manage their time effectively so as to carry out their plan
- To locate and make use of information relevant to their project
- To design a solution to their problem
- To implement and test their solution
- To evaluate in a critical fashion the work they have done, and to place it in the context of related work
- To prepare and deliver a formal presentation
- To structure and write a dissertation describing their project

##### Further Methods of Applied Mathematics (MATH323)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**To give an insight into some specific methods for solving important types of ordinary differential equations.

To provide a basic understanding of the Calculus of Variations and to illustrate the techniques using simple examples in a variety of areas in mathematics and physics.

To build on the students'' existing knowledge of partial differential equations of first and second order.

**Learning Outcomes**After completing the module students should be able to:

- use the method of "Variation of Arbitrary Parameters" to find the solutions of some inhomogeneous ordinary differential equations.

- solve simple integral extremal problems including cases with constraints;

- classify a system of simultaneous 1st-order linear partial differential equations, and to find the Riemann invariants and general or specific solutions in appropriate cases;

- classify 2nd-order linear partial differential equations and, in appropriate cases, find general or specific solutions. [This might involve a practical understanding of a variety of mathematics tools; e.g. conformal mapping and Fourier transforms.]

##### Cartesian Tensors and Mathematical Models of Solids and VIscous Fluids (MATH324)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**To provide an introduction to the mathematical theory of viscous fluid flows and solid elastic materials. Cartesian tensors are first introduced. This is followed by modelling of the mechanics of continuous media. The module includes particular examples of the flow of a viscous fluid as well as a variety of problems of linear elasticity.

**Learning Outcomes**After completing the module, students should be able to understand and actively use the basic concepts of continuum mechanics such as stress, deformation and constitutive relations, and apply mathematical methods for analysis of problems involving the flow of viscous fluid or behaviour of solid elastic materials.

##### Quantum Mechanics (MATH325)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**The development of Quantum Mechanics, requiring as it did revolutionary changes in our understanding of the nature of reality, was arguably the greatest conceptual achievement of all time. The aim of the module is to lead the student to an understanding of the way that relatively simple mathemactics (in modern terms) led Bohr, Einstein, Heisenberg and others to a radical change and improvement in our understanding of the microscopic world.

**Learning Outcomes**After completing the module students should be able to solve Schrodinger''s equation for simple systems, and have some intuitive understanding of the significance of quantum mechanics for both elementary systems and the behaviour of matter.

##### Relativity (MATH326)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**To impart

(i) a firm grasp of the physical principles behind Special and General Relativity and their main consequences;

(ii) technical competence in the mathematical framework of the subjects - Lorentz transformation, coordinate transformations and geodesics in Riemann space;

(iii) knowledge of some of the classical tests of General Relativity - perihelion shift, gravitational deflection of light;

(iv) basic concepts of black holes and (if time) relativistic cosmology.

**Learning Outcomes**After completing this module students should

(i) understand why space-time forms a non-Euclidean four-dimensional manifold;

(ii) be proficient at calculations involving Lorentz transformations, energy-momentum conservation, and the Christoffel symbols.

(iii) understand the arguments leading to the Einstein''s field equations and how Newton''s law of gravity arises as a limiting case.

(iv) be able to calculate the trajectories of bodies in a Schwarzschild space-time.

##### Mathematical Economics (MATH331)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**· To explore, from a game-theoretic point of view, models which have been used to understand phenomena in which conflict and cooperation occur.

· To see the relevance of the theory not only to parlour games but also to situations involving human relationships, economic bargaining (between trade union and employer, etc), threats, formation of coalitions, war, etc..

· To treat fully a number of specific games including the famous examples of "The Prisoners'' Dilemma" and "The Battle of the Sexes".

· To treat in detail two-person zero-sum and non-zero-sum games.

· To give a brief review of n-person games.

· In microeconomics, to look at exchanges in the absence of money, i.e. bartering, in which two individuals or two groups are involved. To see how the Prisoner''s Dilemma arises in the context of public goods.

**Learning Outcomes**After completing the module students should:

· Have further extended their appreciation of the role of mathematics in modelling in Economics and the Social Sciences.

· Be able to formulate, in game-theoretic terms, situations of conflict and cooperation.

· Be able to solve mathematically a variety of standard problems in the theory of games.

· To understand the relevance of such solutions in real situations.

##### Population Dynamics (MATH332)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**- To provide a theoretical basis for the understanding of population ecology

- To explore the classical models of population dynamics

- To learn basic techniques of qualitative analysis of mathematical models

**Learning Outcomes** The ability to relate the predictions of the mathematical models to experimental results obtained in the field.

The ability to recognise the limitations of mathematical modelling in understanding the mechanics of complex biological systems.

The ability to use analytical and graphical methods to investigate population growth and the stability of equilibrium states for continuous-time and discrete-time models of ecological systems.

##### Number Theory (MATH342)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**To give an account of elementary number theory with use of certain algebraic methods and to apply the concepts to problem solving.

**Learning Outcomes**After completing this module students should be able to understand and solve a wide range of problems about the integers, and have a better understanding of the properties of prime numbers.

##### Group Theory (MATH343)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**90:10 **Aims**To introduce the basic techniques of finite group theory with the objective of explaining the ideas needed to solve classification results.

**Learning Outcomes**Understanding of abstract algebraic systems (groups) by concrete, explicit realisations (permutations, matrices, Mobius transformations).

The ability to understand and explain classification results to users of group theory.

The understanding of connections of the subject with other areas of Mathematics.

To have a general understanding of the origins and history of the subject.

##### Combinatorics (MATH344)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**90:10 **Aims**To provide an introduction to the problems and methods of Combinatorics, particularly to those areas of the subject with the widest applications such as pairings problems, the inclusion-exclusion principle, recurrence relations, partitions and the elementary theory of symmetric functions.

**Learning Outcomes**After completing the module students should be able to:

- understand of the type of problem to which the methods of Combinatorics apply, and model these problems;

- solve counting and arrangement problems;

- solve general recurrence relations using the generating function method;

- appreciate the elementary theory of partitions and its application to the study of symmetric functions.

##### Differential Geometry (MATH349)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**85:15 **Aims**This module is designed to provide an introduction to the methods of differential geometry, applied in concrete situations to the study of curves and surfaces in euclidean 3-space. While forming a self-contained whole, it will also provide a basis for further study of differential geometry, including Riemannian geometry and applications to science and engineering.

**Learning Outcomes**1. Knowledge and understanding

After the module, students should have a basic understanding of

a) invariants used to describe the shape of explicitly given curves and surfaces,

b) special curves on surfaces,

c) the difference between extrinsically defined properties and those which depend only on the surface metric,

d) understanding the passage from local to global properties exemplified by the Gauss-Bonnet Theorem.

2. Intellectual abilities

After the module, students should be able to

a) use differential calculus to discover geometric properties of explicitly given curves and surface,

b) understand the role played by special curves on surfaces.

3. Subject-based practical skills

Students should learn to

a) compute invariants of curves and surfaces,

b) interpret the invariants of curves and surfaces as indicators of their geometrical properties.

4. General transferable skills

Students will improve their ability to

a) think logically about abstract concepts,

b) combine theory with examples in a meaningful way.

##### Analysis and Number Theory (MATH351)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**To provide an introduction to uniform distribution modulo one as a meeting ground for other branches of pure mathematics such as topology, analysis and number theory.

To give an idea of how these subjects work together in this context to shape modern analysis by focusing on one particular rich context.

**Learning Outcomes**After completing the module students should be able to:

understand completions and irrationality;

understand diophantine approximation and its relation to uniform distribution;

appreciate that analysis has a complex unity and to have a feel for basic computations in analysis.

calculate rational approximations to real and p adic numbers and to put this to use in number theoretic situations.

calculate approximations to functions from families of simpler functions e.g. continuous functions by polynomials and step functions by trigonometric sums.

work with basic tools from analysis, like Fourier series and continuous functions to prove distributional properties of sequences of numbers.

##### Theory of Statistical Inference (MATH361)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**90:10 **Aims**To introduce some of the concepts and principles which provide theoretical underpinning for the various statistical methods, and, thus, to consolidate the theory behind the other second year and third year statistics options.

**Learning Outcomes**After completing the module students should have a good understanding of the classical approach to, and especially the likelihood methods for, statistical inference.

The students should also gain an appreciation of the blossoming area of Bayesian approach to inference

##### Applied Probability (MATH362)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**To give examples of empirical phenomena for which stochastic processes provide suitable mathematical models. To provide an introduction to the methods of probabilistic model building for ‘‘dynamic" events occuring over time. To familiarise students with an important area of probability modelling.

**Learning Outcomes**1. Knowledge and Understanding

After the module, students should have a basic understanding of:

(a) some basic models in discrete and continuous time Markov chains such as random walk and Poisson processes

(b) important subjects like transition matrix, equilibrium distribution, limiting behaviour etc. of Markov chain

(c) special properties of the simple finite state discrete time Markov chain and Poisson processes, and perform calculations using these.

2. Intellectual Abilities

After the module, students should be able to:

(a) formulate appropriate situations as probability models: random processes

(b) demonstrate knowledge of standard models

(c) demonstrate understanding of the theory underpinning simple dynamical systems

3. General Transferable Skills

(a) numeracy through manipulation and interpretation of datasets

(b) communication through presentation of written work and preparation of diagrams

(c) problem solving through tasks set in tutorials

(d) time management in the completion of practicals and the submission of assessed work

(e) choosing, applying and interpreting results of probability techniques for a range of different problems.

##### Linear Statistical Models (MATH363)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**· to understand how regression methods for continuous data extend to include multiple continuous and categorical predictors, and categorical response variables.

· to provide an understanding of how this class of models forms the basis for the analysis of experimental and also observational studies.

· to understand generalized linear models.

· to develop familiarity with the computer package SPSS.

**Learning Outcomes**After completing the module students should be able to:

understand the rationale and assumptions of linear regression and analysis of variance.

· understand the rationale and assumptions of generalized linear models.

· recognise the correct analysis for a given experiment.

· carry out and interpret linear regressions and analyses of variance, and derive appropriate theoretical results.

· carry out and interpret analyses involving generalised linear models and derive appropriate theoretical results.

· perform linear regression, analysis of variance and generalised linear model analysis using the SPSS computer package.

##### Medical Statistics (MATH364)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**The aims of this module are to:

- demonstrate the purpose of medical statistics and the role it plays in the control of disease and promotion of health
- explore different epidemiological concepts and study designs
- apply statistical methods learnt in other programmes, and some new concepts, to medical problems and practical epidemiological research
- enable further study of the theory of medical statistics by using this module as a base.

**Learning Outcomes**identify the types of problems encountered in medical statistics

demonstrate the advantages and disadvantages of different epidemiological study designs

apply appropriate statistical methods to problems arising in epidemiology and interpret results

explain and apply statistical techniques used in survival analysis

critically evaluate statistical issues in the design and analysis of clinical trials

discuss statistical issues related to systematic review and apply appropriate methods of meta-analysisapply Bayesian methods to simple medical problems.

##### Mathematical Risk Theory (MATH366)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims** to provide an understanding of the mathematical risk theory used in the study process of actuarial interest,

to provide an introduction to mathematical methods for managing the risk in insurance and finance (calculation of risk measures/quantities),

to develop skills of calculating the ruin probability and the total claim amount distribution in some non‐life actuarial risk models with applications to insurance industry,

to prepare the students adequately and to develop their skills in order to be ready to sit for the exams of CT6 subject of the Institute of Actuaries (MATH366 covers 50% of CT6 in much more depth).

**Learning Outcomes**After completing the module students should be able to:

(a) Define the loss/risk function and explain intuitively the meaning of it, describe and determine optimal strategies of game theory, apply the decision criteria''s, be able to decide a model due to certain model selection criterion, describe and perform calculations with Minimax and Bayes rules.

(b) Understand the concept (and the mathematical assumptions) of the sums of independent random variables, derive the distribution function and the moment generating function of finite sums of independent random variables,

(c) Define and explain the compound Poisson risk model, the compound binomial risk model, the compound geometric risk model and be able to derive the distribution function, the probability function, the mean, the variance, the moment generating function and the probability generating function for exponential/mixture of exponential severities and gamma (Erlang) severities, be able to calculate the distribution of sums of independent compound Poisson random variables.

(d) Understand the use of convolutions and compute the distribution function and the probability function of the compound risk model for aggregate claims using convolutions and recursion relationships ,

(e) Define the stop‐loss reinsurance and calculate the (mean) stop‐loss premium for exponential and mixtures of exponential severities, be able to compare the original premium and the stoploss premium in numerical examples,

(f) Understand and be able to use Panjer''s equation when the number of claims belongs to the

R(a, b, 0) class of distributions, use the Panjer''s recursion in order to derive/evaluate the probability function for the total aggregate claims,(g) Explain intuitively the individual risk model, be able to calculate the expected losses (as well as the variance) of group life/non‐life insurance policies when the benefits of the each person of the group are assumed to have deterministic variables,

(h) Derive a compound Poisson approximations for a group of insurance policies (individual risk model as approximation),

(i) Understand/describe the classical surplus process ruin model and calculate probabilities of the number of the risks appearing in a specific time period, under the assumption of the Poisson process,

(j) Derive the moment generating function of the classical compound Poisson surplus process, calculate and explain the importance of the adjustment coefficient, also be able to make use of Lundberg''s inequality for exponential and mixtures of exponential claim severities,

(k) Derive the analytic solutions for the probability of ruin, psi(u), by solving the corresponding integro‐differential equation for exponential and mixtures of exponential claim amount severities,

(l) Define the discrete time surplus process and be able to calculate the infinite ruin probability, psi(u,t) in numerical examples (using convolutions),

(m) Derive Lundberg''s equation and explain the importance of the adjustment coefficient under the consideration of reinsurance schemes,

(n) Understand the concept of delayed claims and the need for reserving, present claim data as a triangle (most commonly used method), be able to fill in the lower triangle by comparing present data with past (experience) data,

(o) Explain the difference and adjust the chain ladder method, when inflation is considered,

(p) Describe the average cost per claim method and project ultimate claims, calculate the required reserve (by using the claims of the data table),

(q) Use loss ratios to estimate the eventual loss and hence outstanding claims,

(r) Describe the Bornjuetter‐Ferguson method (be able to understand the combination of the estimated loss ratios with a projection method), use the aforementioned method to calculate the revised ultimate losses (by making use of the credibility factor).

##### Networks in Theory and Practice (MATH367)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**To develop an appreciation of network models for real world problems.

To describe optimisation methods to solve them.

To study a range of classical problems and techniques related to network models.

**Learning Outcomes**After completing the module students should

. be able to model problems in terms of networks.

· be able to apply effectively a range of exact and heuristic optimisation techniques.

##### Projects in Mathematics (MATH399)

**Level**3 **Credit level**15 **Semester**Whole Session **Exam:Coursework weighting**0:100 **Aims**a) To study in depth an area of pure mathematics and report on it; or

b) To construct and study mathematical models of a chosen problem; or

c) To demonstrate a critical understanding and historical appreciation of some branch of mathematics by means of directed reading and preparation of a report.; or

d) To study in depth a particular problem in statistics, probability or operational research.

**Learning Outcomes**a) (Pure Maths option) - After completing the report with suitable guidance the student should have

· gained a greater understanding of the chosen mathematical topic

· gained experience in applying his/her mathematical skills

· had experience in consulting relevant literature

· learned how to construct a written project report

· had experience in making an oral presentation

b) (Applied Mathematics) - After completing the project with suitable guidance the students should have:

- learned strategies for simple model building

- gained experience in choosing and using appropriate mathematics

- understood the nature of approximations used

- made critical appraisal of results

- had experience in consulting related relevant literature

- learned how to construct a written project report

- had experience in making an oral presentation.

c) (Applied Maths/Theoretical Physics) - After researching and preparing the mathematical essay the student should have:

· gained a greater understanding of the chosen mathematical topic

· gained an appreciation of the historical context

· learned how to abstract mathematical concepts and explain them

· had experience in consulting related relevant literature

· learned how to construct a written project report

· had experience in making an oral presentation.

d) (Statistics, Probability and Operational Research) -

After completing the project the student should have:

· gained an in-depth understanding of the chosen topic

· had experience in consulting relevant literature

· learned how to construct a written project report;

· had experience in making an oral presentation.

e) Mathematics in Society Projects. Only available to G1X3 students

Students interested in doing such a project should see Dr A Pratoussevitch and Dr T Eckl initially.

The programme detail and modules listed are illustrative only and subject to change.

#### Teaching and Learning

Teaching is by a mix of formal lectures, small group tutorials and supervised laboratory-based practical sessions. Students also undertake individual and group projects. Key problem solving skills and employability skills, like presentation and teamwork skills, are developed throughout the programme.