# Undergraduate Catalogue of Courses 2012/2013

# MATHEMATICS

(see also Mathematical Sciences(MX))

*Course Co-ordinator:* Dr A Libman

*Pre-requisite(s):*
SCE H or GCE A level in Mathematics. This course may not be included in a minimum curriculum with EG 1503.

*Note(s):*
The course starts from the beginning of the subject, but it is advantageous to be familiar with the material on Calculus contained in the Scottish Highers syllabus.

Calculus allows for changing situations and complicated averaging processes to be described in precise ways. It was one of the great intellectual achievements of the late 17-th and early 18-th Century. Early applications were made to modeling planetary motion and to calculating tax payable on land. Now the ideas are used in broad areas of mathematics and science and parts of the commercial world. The course begins with an introduction to fundamental mathematical concepts and then develops the basic ideas of the differential calculus of a single variable and explains some of the ways they are applied.

3 one-hour lectures and 1 one-hour tutorial per week; support tutorials to be arranged by the Course Coordinator, as need arises.

1st Attempt: 1 two-hour written examination (70%) and in-course assessment (30%).

Resit: 1 two-hour written examination paper (maximum of (100%) resit and (70%) resit with (30%) in-course assessment).

## Formative Assessment and Feedback Information

Informal assessment of weekly homework through discussions in tutorials.

In-course assignments will normally be marked within one week and feedback provided to students in tutorials. Students will be invited to contact Course Coordinator for feedback on the final examination.

*Course Co-ordinator:* Dr A Gonzales

*Pre-requisite(s):*
SCE H or GCE A level in Mathematics.

The basic course includes a discussion of the following topics: complex numbers and the theory of polynomial equations, vector algebra in two and three dimensions, systems of linear equations and their solution, matrices and determinants.

3 one-hour lectures and 1 one-hour tutorial per week. Support tutorials to be arranged by the Course Coordinator, as need arises.

1st Attempt: 1 two-hour written examination (70%) and in-course assessment (30%).

Resit: 1 two-hour written examination paper (maximum of (100%) resit and (70%) resit with (30%) in-course assessment).

## Formative Assessment and Feedback Information

In-course assignments will normally be marked within one week and feedback provided to students in tutorials. Students will be invited to contact the Course Coordinator for feedback on the final examination.

In-course assessment will be marked and feedback provided to the students.

Support tutorials to be arranged by the Course Coordinator, as need arises.

*Course Co-ordinator:* Dr M Boyle

*Pre-requisite(s):*
S or GCSE or equivalent in Mathematics. This course is not open to students with the equivalent of a Higher in Mathematics at grade B or above.

This is a basic course aimed primarily at helping students achieve greater accuracy, speed and confidence in mathematics. It is suitable both for those who may need mathematics in future study and for students who want to improve their abilities without any intention of studying the subject beyond first year. The course is taught using the interactive computer software CALMAT, enabling students to work in their own way and time but with immediate feedback. Support from staff is available on a daily basis. There is a requirement to attend a single weekly test for continuous assessment. The topics covered include basic arithmetic and algebraic operations, linear and quadratic equations, logarithms and the interpretation of graphs, and an introduction to the calculus.

1 one-hour lecture and 2 one-hour supervised computer classes per week.

1st Attempt: In-course assessment (100%) for students who perform sufficiently well in weekly computerised tests. Any student who fails to achieve by in-course assessment or who wishes to upgrade CAS mark obtained, can take the end of course computerised examination or its resit.

Resit: Computerised examination similar to 1st attempt examination.

## Formative Assessment and Feedback Information

Regular computerised tests discussed during computer practicals.

In-course assessment will be marked and feedback provided to the students.

Informal feedback at computer practicals.

*Course Co-ordinator:* Dr M Boyle

*Pre-requisite(s):*
MA 1007 or equivalent. This course is not open to students with the equivalent of a Higher in Mathematics at Grade B or above.

The course emphasizes accuracy in performing calculations involving trigonometry, exponentials, techniques and application of differentiation and integration, vectors, complex numbers and matrices. The course is taught and examined using the CALMAT computer software.

1 one-hour lecture and 2 one-hour supervised computer classes per week.

1st Attempt: In-course assessment (100%) for students who perform sufficiently well in weekly computerised tests. Any student who fails to achieve by in-course assessment or who wishes to upgrade CAS mark obtained, can take the end of course computerised examination or its resit.

Resit: Computerised examination similar to 1st attempt examination.

## Formative Assessment and Feedback Information

Regular computerised tests discussed during computer practicals.

In-course assessment will be marked and feedback provided to the students.

Informal feedback at computer practicals.

*Course Co-ordinator:* Prof V Gorbunov

*Pre-requisite(s):*
SCE H or GCE A level in Mathematics; MA 1005 (recommended). This course may not be included in a minimum curriculum with EG 1503.

The course is a continuation of Calculus I from the 1st session. It develops the basic ideas concerning the integration of a function of one variable. It introduces Taylor series and determines these series for the most common functions. It also provides a first introduction to differential equations which are fundamental in applications of Mathematics to other sciences.

3 one-hour lectures and 1 one-hour tutorial per week. Support tutorials to be arranged by the Course Coordinator, as need arises.

1st Attempt: 1 two-hour written examination (70%); in-course assessment (30%).

Resit: 1 two-hour written examination paper (maximum of (100%) resit and (70%) resit with (30%) in-course assessment).

## Formative Assessment and Feedback Information

Informal assessment of weekly homework through discussions in tutorials.

In-course assignments will normally be marked within one week and feedback provided to students in tutorials. Students will be invited to contact the Course Coordinator for feedback on the final examination.

*Course Co-ordinator:* Dr D Quinn

*Pre-requisite(s):*
None

- Recurrence.
- Sums.
- Integer functions.
- Elementary number theory.
- Binomial coefficients.

3 hours of lectures and 1 hour tutorial each week.

1st Attempt: 1 two-hour written examination (80%); in-course assessment (20%).

Resit: 1 two-hour written examination paper (maximum of (100%) resit and (80%) resit with (20%) in-course assessment).

## Formative Assessment and Feedback Information

Informal assessment of weekly homework through discussions in tutorials.

In-course assignments will normally be marked within one week and feedback provided to students in tutorials.

*Course Co-ordinator:* Dr R Hepworth

*Pre-requisite(s):*
MA 1006 or, with the permission of the Head of Mathematical Sciences, MA 1507.

This course provides an introduction to algebraic structures and elementary number theory.

The course includes a discussion of:

- Sets (notation, functions, injections, surjections, bijections)
- Countability of the rational numbers and uncountability of the real numbers
- The integers and factorisation
- Prime numbers, Euclidean algorithm, uniqueness of factorisation
- The integers modulo n
- Equivalence relations
- Permutations
- Group axioms
- The symmetric group
- Lagrange's Theorem
- Fermat's Little Theorem
- Definition of commutative ring and of a field with examples, especially polynomial rings
- Vector spaces and linear transformations.

3 one-hour lectures and 1 tutorial per week (to be arranged).

1st Attempt: 1 two-hour written examination (80%); in-course assessment (20%).

Resit: 1 two-hour written examination paper, maximum resit (100%) and resit (80%) with in-course assessment (20%).

## Formative Assessment and Feedback Information

Informal assessment of weekly homework through discussions in tutorials.

In-course assignments will normally be marked within one week and feedback provided to students in tutorials. Students will be invited to contact Course Coordinator for feedback on the final examination.

*Course Co-ordinator:* Dr C Lopez

*Pre-requisite(s):*
MA 1005 or, with the permission of the Head of Mathematical Sciences, both MA 1007 and MA 1507.

- Fundamental properties of real numbers: field operations, order, completeness.
- Sequences and limits: convergence, basic examples, decimal representation of real numbers.
- Functions of one real variable: limits and continuity, elementary functions, basic results on continuous functions.
- Differentiation of functions of one variable: basic definitions and properties, chain rule, first examples of Taylor polynomials and series
- Integration of functions of one variable: basic definitions and properties, the fundamental theorem of calculus, application to arc lengths of plane curves.

3 one-hour lectures and 1 one-hour tutorial per week (to be arranged).

1st Attempt: 1 two-hour written examination (80%); in-course assessment (20%).

Resit: 1 two-hour written examination paper. The CAS mark awarded will be the maximum of 100% resit and 80% resit with 20% in-course assessment.

## Formative Assessment and Feedback Information

Informal assessment of weekly homework through discussions in tutorials.

In-course assignments will normally be marked within one week and feedback provided to students in tutorials. Students will be invited to contact Course Coordinator for feedback on the final examination.

*Course Co-ordinator:* Dr A Libman

*Pre-requisite(s):*
MA 2004, MA 2005

- Sample spaces and the probability function.
- Application of conditional probability and the partition theorem.
- Random variables and distribution functions.
- Expectation and variance.
- Limit theorems and their application.
- Generating functions and their uses.
- Branching processes.
- Markov Chains.

12 week course - 2 one-hour lectures and 1 one-hour tutorial per week (to be arranged).

1st Attempt: 1 two-hour written examination (80%), in-course assessment (20%).

Resit: 1 two-hour written examination paper, maximum resit (100%) and resit (80%) with (20%) in-course assessment (20%).

## Formative Assessment and Feedback Information

Informal assessment of weekly homework through discussions in tutorials.

In-course assignments will normally be marked within one week and feedback provided to students in tutorials. Students will be invited to contact the Course Coordinator for feedback on the final examination.

*Course Co-ordinator:* Dr J Elmer

*Pre-requisite(s):*
MA 1006 and MA 2004 or, with permission of the Head of Mathematical Sciences, MA 1507 and MA 2004.

- Fields.

Solving a linear system over a field; Definition and examples of fields (Q, R, C, Fp ); Elementary row operations, row echelon form, Gaussian algorithm for solving a linear system over a field. - Vector spaces

Definition of a vector space over a field; Examples; subspaces of a vector space, intersection and sum of subspaces; Span, spanning sets; Linear independence; Basis, dimension; Elementary results about bases and dimension; Change of basis matrix. - Linear maps

Definition of a linear map between two K-vector spaces; Kernel, image, injective, surjective linear maps;

Matrix of a linear map; Rank of a matrix; Invertible matrices; Determinants; Change of basis and the matrix of a linear map. - Eigenvalues, eigenvectors and diagonalisation

Linear transformations, eigenvalues and eigenvectors of linear transformations; elementary properties;

Minimal polynomial, characteristic polynomial, Cayley-Hamilton theorem; Triangularisation and diagonalisation. - Inner product spaces, Euclidian/Hermitian spaces

Basic definitions, examples; Constructing an orthogonal basis using Gram-Schmidt; Symmetric matrices, orthogonal transformations and matrices, unitary transformations and matrices.

12 week course - 3 one-hour lectures and 1 one-hour tutorial per week (to be arranged).

1st Attempt: 1 two-hour written examination (80%); in-course assessment (20%).

Resit: 1 two-hour written examination paper, maximum resit (100%) and resit (80%) with in-course assessment (20%).

## Formative Assessment and Feedback Information

Informal assessment of weekly homework through discussions in tutorials.

*Course Co-ordinator:* Dr Jean-Baptiste Gramain

*Pre-requisite(s):*
MA 2005

A) Several variables - Continuity and Partial Differentiation.

- Functions of several variables, graphical surface representations, limits and continuity, partial derivatives, higher order partials, plane tangent, linear approximation, small errors.
- Chain rule, polar coordinates, applications to some elementary PDEs.
- Critical points, second derivative test, some discussion of global global max/min.
- Taylor series and quadratic approximation.

B) Several variables - Multiple Integrals.

- Revision of definite integral as area under curve, approximated by rectangles. Double integral as volume, approximated by rectangular pillars.
- Iteration formulae for rectangles and for more general regions, change of order of integration via double integral.
- Change of variable in double integrals (with emphasis on polars).
- Triple integrals, cylindrical and spherical polar coordinates.
- Applications to volumes, moments and centres of mass.

C) Ordinary differential equations.

- Basic terminology, general solution, integral curves, initial and boundary conditions.
- First order ODEs: linear equations, separable equations, brief treatment of homogeneous and Bernoulli equations, applications (eg. population problems and mixing problems).
- Second order linear ODEs: basic theory for solution of equations with constant coefficients via CF + PI, applications; reduction of order and variation of parameters.

D) Introduction to computing software (4 practical sessions).

- Simple arithmetic, operations, variables, booleans, conditionals, functions, procedures, plotting, functions in two variables, contour plots, parametric plots, basic algebra, differentiation, integration, programming, loops.

12 week course - 3 one-hour lectures per week, 1 one-hour tutorial per week and 4 one-hour practicals over the 12 weeks. To be arranged.

1st Attempt: 1 two-hour written examination (80%); in-course assessment (20%).

Resit: 1 two-hour written examination paper, maximum resit (100%) and resit (80%) with in-course assessment (20%).

## Formative Assessment and Feedback Information

Informal assessment of weekly homework through discussions in tutorials.