## Book of Modules 2013/2014 |
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## Mathematics |

**Credit Weighting: **10

**Teaching Period(s): **Teaching Periods 1 and 2.

**No. of Students: **Min 50, Max 600.

**Pre-requisite(s): **None

**Co-requisite(s): **None

**Teaching Methods: **72 x 1hr(s) Lectures; 20 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Mr Martin Quirke, Department of Mathematics.

**Module Objective: **To introduce the fundamental mathematical techniques of science.

**Module Content: **Revision of trigonometry and polynomial functions. Basic techniques and applications of differentiation, integration, linear regression and ordinary differential equations, emphasizing computational skill in practical examples taken from biology, chemistry, computer science, environmental science, food science and medicine, using real world data.

**Learning Outcomes: **On successful completion of this module, students should be able to:

· Solve scientific problems using differential and integral calculus, algebra, trigonometry, polynomials, exponential and logarithm functions.

· Find derivatives of elementary functions, and apply to graph sketching, maximizing and physical problems. Demonstrate knowledge of derivatives as rates of change in scientific contexts.

· Evaluate indefinite integrals of elementary functions. Apply substitution and integration by parts to evaluate indefinite integrals. Use both definite and indefinite integration in scientific problems.

· Find best fitting curves of various kinds.

· Solve elementary ordinary differential equations and draw planar phase portraits.

**Assessment: **Total Marks 200: End of Year Written Examination 140 marks; Continuous Assessment 60 marks (2 x tests, 30 marks each).

**Compulsory Elements: **End of Year Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **1 x 3 hr(s) paper(s).

**Requirements for Supplemental Examination: **1 x 3 hr(s) paper(s) (which incorporates assessment of both End of Year Written Examination and Continuous Assesment) to be taken in Autumn 2014.

MA1008 Calculus and Linear Algebra for Engineers

**Credit Weighting: **10

**Teaching Period(s): **Teaching Periods 1 and 2.

**No. of Students: **Min 50, Max 250.

**Pre-requisite(s): **None

**Co-requisite(s): **None

**Teaching Methods: **72 x 1hr(s) Lectures; 20 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Dr Stephen Wills, Department of Mathematics.

**Module Objective: **To provide an overview of techniques and applications of one- and multi-variable calculus, and an introduction to differential equations, linear algebra and complex functions.

**Module Content: **Functions, limits, continuity, methods and applications of differentiation and integration, sequences, power series, Taylor expansions. First-order ordinary differential equations and linear 2nd-order ordinary differential equations. Matrices, solutions of simultaneous linear equations, determinants, eigenvalues and eigenvectors. Introduction to multivariable calculus: partial derivatives, tangent planes. Complex numbers, exponential and trigonometric functions.

**Learning Outcomes: **On successful completion of this module, students should be able to:

· Use a broad collection of techniques from calculus of functions of one variable: compute derivatives of standard functions; apply the main techniques of integration; manipulate power series; solve simple ordinary differential equations of first and second order.

· Perform basic calculations in linear algebra: solve systems of linear equations; calculate matrix inverses and solve matrix equations; evaluate determinants; compute eigenvalues and eigenvectors of matrices.

· Perform basic calculations in differential calculus of functions of several variables: compute partial derivatives; find equations of tangent planes; locate and classify critical points of functions of two variables.

· Carry out elementary calculations with complex numbers, interpreting the results geometrically using Argand diagrams.

**Assessment: **Total Marks 200: End of Year Written Examination 160 marks; Continuous Assessment 40 marks (2 x in-class tests; 20 marks each).

**Compulsory Elements: **End of Year Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **None.

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **1 x 3 hr(s) paper(s).

**Requirements for Supplemental Examination: **1 x 3 hr(s) paper(s) to be taken in Autumn 2014. The mark for Continuous Assessment is carried forward.

**Credit Weighting: **15

**Teaching Period(s): **Teaching Periods 1 and 2.

**No. of Students: **Min 5, Max 100.

**Pre-requisite(s): **None

**Co-requisite(s): **None

**Teaching Methods: **120 x 1hr(s) Lectures; 40 x 1hr(s) Workshops.

**Module Co-ordinator: **Dr Thomas Carroll, Department of Mathematics.

**Lecturer(s): ** Dr Thomas Carroll, Department of Mathematics.

**Module Objective: **To provide an introduction to concepts and techniques of higher mathematics.

**Module Content: **Calculus: differentiation and integration of functions of a single variable, applications, approximation techniques. Analysis: real number system, inequalities, completeness, sequences and series, limits. Foundations: sets, proofs, Boolean algebra, relations & functions. Algebra & Number Theory: symmetry groups, primes, groups, rings. Linear Algebra: vectors, dot products, conics & quadrics, matrices, determinants, linear equations.

**Learning Outcomes: **On successful completion of this module, students should be able to:

· recall the basic definitions and theorems of limits, continuity, differentiation and integration;

· solve problems using differentiation, integration and related techniques;

· recall the completeness axiom for the Real Numbers and apply it to solve problems involving infimum and supremum;

· prove and apply basic theorems on sequences and series;

· prove theorems in number theory and algebra using induction;

· recall algebraic concepts such as relations, functions, binary operations, and axiom systems for algebraic groups, rings and fields;

· perform standard matrix-related computations and operations;

· determine whether one vector is a linear combination of others, and deduce the dimension of the kernel and image of a matrix, using Gaussian elimination and examining pivots;

· prove the equivalence of the dozen invertibility criteria of Strang's nutshell using the main theorems of linear algebra.

**Assessment: **Total Marks 300: End of Year Written Examination 250 marks (Paper I 65 marks; Paper II 65 marks; Paper III 55 marks; Paper IV 65 marks); Continuous Assessment 50 marks (in-class tests).

**Compulsory Elements: **End of Year Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **4 x 1½ hr(s) paper(s).

**Requirements for Supplemental Examination: **4 x 1½ hr(s) paper(s) to be taken in Autumn 2014. The mark for Continuous Assessment is carried forward (whether passed or failed).

MA1057 Introduction to Abstract Algebra

**Credit Weighting: **5

**Teaching Period(s): **Teaching Period 1.

**No. of Students: **Min 20, Max 200.

**Pre-requisite(s): **None

**Co-requisite(s): **None

**Teaching Methods: **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Thomas Carroll, Department of Mathematics.

**Lecturer(s): ** Dr Anca Mustata, Department of Mathematics.

**Module Objective: **To provide an introduction to Abstract Algebra, including, the notion of proof, set theory, functions, algebraic systems and the elements of number theory.

**Module Content: **Foundations: sets, proofs, Boolean algebra, relations & functions. Induction. Algebra & Number Theory: symmetry groups, primes, groups, rings.

**Learning Outcomes: **On successful completion of this module, students should be able to:

· Deal with abstraction in algebra;

· Prove theorems in number theory and algebra using mathematical induction and the Well-ordering Principle;

· Work with algebraic concepts such as relations, functions and binary operations;

· Appreciate axiom systems for groups, rings, ordered integral domains and fields.

**Assessment: **Total Marks 100: End of Year Written Examination 75 marks; Continuous Assessment 25 marks (in-class test(s) and/or homework assignment(s), students will be notified of the format and breakdown of marks for continuous assessment at the first lecture.).

**Compulsory Elements: **End of Year Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **None.

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **1 x 1½ hr(s) paper(s).

**Requirements for Supplemental Examination: **1 x 1½ hr(s) paper(s) to be taken in Autumn 2014. The mark for Continuous Assessment is carried forward.

MA1058 Introduction to Linear Algebra

**Credit Weighting: **5

**Teaching Period(s): **Teaching Period 2.

**No. of Students: **Min 20, Max 200.

**Pre-requisite(s): **None

**Co-requisite(s): **None

**Teaching Methods: **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Dr Andrei Mustata, Department of Mathematics.

**Module Objective: **To provide an introduction to Linear Algebra

**Module Content: **Linear Algebra: vectors, dot products, conics & quadrics, matrices, determinants, linear equations.

**Learning Outcomes: **On successful completion of this module, students should be able to:

· Solve systems of linear equations.

· Prove theorems and identities using induction.

· Explain the meanings of the words determinant, eigenvalue, eigenvector, invertibility, kernel, image, and spectrum.

· Find the eigenvectors and eigenvalues of a square matrix.

· Find the determinant and inverse of a 3 x 3 matrix, and solve associated linear equations.

· Deduce the solvability of a system of linear equations, without finding the solutions, via Gaussian elimination.

· Deduce whether one vector is a linear combination of others, and by the same method deduce the dimension of the kernel and image of any matrix, using Gaussian elimination and examining pivots.

· Prove the equivalence of the dozen invertibility criteria of Strang's nutshell using the main theorems of linear algebra.

**Assessment: **Total Marks 100: End of Year Written Examination 80 marks; Continuous Assessment 20 marks (1 x In-class test).

**Compulsory Elements: **End of Year Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **None.

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **1 x 1½ hr(s) paper(s).

**Requirements for Supplemental Examination: **1 x 1½ hr(s) paper(s) to be taken in Autumn 2014. The mark for Continuous Assessment is carried forward.

**Credit Weighting: **5

**Teaching Period(s): **Teaching Period 1.

**No. of Students: **Min 20, Max 200.

**Pre-requisite(s): **None

**Co-requisite(s): **None

**Teaching Methods: **36 x 1hr(s) Lectures; 10 x 1hr(s) Workshops.

**Module Co-ordinator: **Dr Thomas Carroll, Department of Mathematics.

**Lecturer(s): ** Dr Thomas Carroll, Department of Mathematics.

**Module Objective: **To provide an introduction to concepts and techniques of calculus.

**Module Content: **Differentiation and integration of functions of a single variable, applications, approximation techniques.

**Learning Outcomes: **On successful completion of this module, students should be able to:

· Solve problems using differentiation, especially extremal problems;

· Solve problems using integration, especially the computation of areas and volumes;

· Apply the techniques they have learned to compute derivatives and integrals;

· Approximate integrals numerically.

**Assessment: **Total Marks 100: End of Year Written Examination 80 marks; Continuous Assessment 20 marks (1 in-class test).

**Compulsory Elements: **End of Year Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **1 x 1½ hr(s) paper(s).

**Requirements for Supplemental Examination: **1 x 1½ hr(s) paper(s) (which incorporates assessment of both End of Year Written Examination and Continuous Assesment) to be taken in Autumn 2014.

MA1060 Introduction to Analysis

**Credit Weighting: **5

**Teaching Period(s): **Teaching Period 2.

**No. of Students: **Min 20, Max 200.

**Pre-requisite(s): **None

**Co-requisite(s): **None

**Teaching Methods: **36 x 1hr(s) Lectures; 10 x 1hr(s) Workshops.

**Module Co-ordinator: **Dr Thomas Carroll, Department of Mathematics.

**Lecturer(s): ** Dr Thomas Carroll, Department of Mathematics.

**Module Objective: **To provide an introduction to concepts and techniques of mathematical analysis.

**Module Content: **Real number system, complex numbers, inequalities, completeness, sequences and series.

**Learning Outcomes: **On successful completion of this module, students should be able to:

· Recall the basic properties of rational numbers, real numbers and complex numbers;

· Recall the completeness axiom and apply it to solve problems involving infimum and supremum;

· Recall the basic notions of metric spaces;

· Recall and apply the basic definitions and theorems of sequences;

· Recall and apply the basic definitions and theorems of infinite series.

**Assessment: **Total Marks 100: End of Year Written Examination 80 marks; Continuous Assessment 20 marks (1 in-class test).

**Compulsory Elements: **End of Year Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **1 x 1½ hr(s) paper(s).

**Requirements for Supplemental Examination: **1 x 1½ hr(s) paper(s) (which incorporates assessment of both End of Year Written Examination and Continuous Assesment) to be taken in Autumn 2014.

MA1100 Introductory Mathematics for Business I

**Credit Weighting: **5

**Teaching Period(s): **Teaching Period 2.

**No. of Students: **Min 50, Max 400.

**Pre-requisite(s): **None

**Co-requisite(s): **None

**Teaching Methods: **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Martin Kilian, Department of Mathematics.

**Lecturer(s): ** Dr Martin Kilian, Department of Mathematics.

**Module Objective: **To provide an introduction to fundamental quantitative techniques for business.

**Module Content: **Exponential, logarithmic and polynomial functions. Series, annuities and introduction to mathematics of finance. Elementary calculus with business applications.

**Learning Outcomes: **On successful completion of this module, students should be able to:

· Determine present and future values of single payment and annuity financial transactions;

· Compute mortgage payments;

· Interpret graphs of functions;

· Compute equilibrium point for supply and demand functions;

· Determine price elasticities of supply and demand functions.

**Assessment: **Total Marks 100: End of Year Written Examination 80 marks; Continuous Assessment 20 marks (Two In-Class Tests - 10 marks for each).

**Compulsory Elements: **End of Year Written Examination and In-Class Tests.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **1 x 1½ hr(s) paper(s).

**Requirements for Supplemental Examination: **1 x 1½ hr(s) paper(s) to be taken in Autumn 2014. The mark for In-Class Tests is carried forward.

MA1905 Introduction to Mathematical Techniques

**Credit Weighting: **5

**Teaching Period(s): **Teaching Period 2.

**No. of Students: **Min 15, Max 50.

**Pre-requisite(s): **None

**Co-requisite(s): **None

**Teaching Methods: **16 x 1hr(s) Lectures (Evening); 8 x 1hr(s) Tutorials (Evening).

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Staff, Department of Mathematics.

**Module Objective: **To introduce basic mathematical techniques.

**Module Content: **Numbers; Powers; Scientific notation; Algebra; Applied verbal problems; Functions; Co-ordinates; Linear equations; Quadratic equations; Trigonometry.

**Learning Outcomes: **On successful completion of this module, students should be able to:

· Use scientific notation;

· Solve simple applied verbal problems;

· Distinguish between integers, rational and irrational numbers;

· Graph functions in relation to linear and quadratic equations;

· Use basic trigonometry, and the trigonometric functions.

**Assessment: **Total Marks 100: End of Year Written Examination 70 marks; Continuous Assessment 30 marks (3 assignments, 10 marks each).

**Compulsory Elements: **End of Year Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **1 x 1½ hr(s) paper(s).

**Requirements for Supplemental Examination: **1 x 1½ hr(s) paper(s) to be taken in Autumn 2014. The mark for Continuous Assessment is carried forward.

MA2006 Intermediate Calculus for Scientists

**Credit Weighting: **5

**Teaching Period(s): **Teaching Period 2.

**No. of Students: **Min 5, Max 50.

**Pre-requisite(s): **MA1003 or MA1059

**Co-requisite(s): **None

**Teaching Methods: **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Mr Rory Conboye, School of Mathematical Sciences.

**Module Objective: **To study techniques of multivariable calculus.

**Module Content: **Partial differentiation. Double integrals and line integrals.

**Learning Outcomes: **On successful completion of this module, students should be able to:

· sketch the graphs of functions of two variables;

· compute partial derivatives, mixed partial derivatives and higher-order partial derivatives of functions of several variables using various methods including the chain rules;

· find the equations of tangent planes and normal lines to surfaces which are the graphs of functions of several variables;

· compute directional derivatives and maximum/minimum rates of change in terms of the gradient vector;

· solve simple unconstrained and constrained maxima/minima problems, using tests to determine the nature of local extrema of functions of two variables;

· set up line integrals arising in physical and geometrical problems concerned with curvature, mass, and work done by forces, and determine their value;

· calculate Riemann sums of functions of two or more variables, and the values of double and triple integrals of continuous functions by means of iterated integrals and Stokes' theorem of computation.

**Assessment: **Total Marks 100: End of Year Written Examination 75 marks; Continuous Assessment 25 marks (1 x in-class examination).

**Compulsory Elements: **End of Year Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **None.

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **1 x 1½ hr(s) paper(s).

**Requirements for Supplemental Examination: **1 x 1½ hr(s) paper(s) to be taken in Autumn 2014. The mark for Continuous Assessment is carried forward.

**Credit Weighting: **5

**Teaching Period(s): **Teaching Period 1.

**No. of Students: **Min 5, Max 50.

**Pre-requisite(s): **MA1003 or MA1059

**Co-requisite(s): **None

**Teaching Methods: **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Martin Kilian, Department of Mathematics.

**Lecturer(s): ** Dr Martin Kilian, Department of Mathematics.

**Module Objective: **To provide an introduction to techniques of linear algebra.

**Module Content: **Linear algebra. Matrices. Determinants. Linear Equations.

**Learning Outcomes: **On successful completion of this module, students should be able to:

· Manipulate matrices and use them to solve systems of simultaneous equations via Gaussian Elimination, Gauss-Jordan Elimination and the Inverse Matrix Method;

· Calculate determinants of square matrices using minors and cofactors;

· Use Cramer's Rule to solve systems of simultaneous equations;

· Perform calculations involving dot products, norms, the Cauchy-Schwarz inequality and the triangle inequality;

· Identify and provide examples of subspaces of a vector space;

· Manipulate linear combinations of vectors, and determine linear (in)dependence, span, null space, bases and dimension of a vector space;

· Use the Gram-Schmidt orthogonalization process to construct an orthogonal basis of a vector space V from any basis of V;

· Form the characteristic polynomial of a matrix, calculate eigenvalues and eigenvectors of a matrix, and form eigenspaces.

**Assessment: **Total Marks 100: End of Year Written Examination 75 marks; Continuous Assessment 25 marks (1 x in-class examination).

**Compulsory Elements: **End of Year Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **None.

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **1 x 1½ hr(s) paper(s).

**Requirements for Supplemental Examination: **1 x 1½ hr(s) paper(s) to be taken in Autumn 2014.

MA2013 Mathematics for Engineering

**Credit Weighting: **5

**Teaching Period(s): **Teaching Period 1.

**No. of Students: **Min 30, Max 200.

**Pre-requisite(s): **MA1008

**Co-requisite(s): **None

**Teaching Methods: **36 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Mr Liam Floyd, Tyndall Institute.

**Module Objective: ** To provide an introduction to the concepts and techniques of multivariable integral calculus and complex function theory.

**Module Content: **Multivariable integral calculus: line integrals, multiple integrals, Green's theorem, curl and

divergence, surface integrals, Gauss's divergence theorem.

Functions of a complex variable: elementary complex mappings, contour integrals, residue calculus.

**Learning Outcomes: **On successful completion of this module, students should be able to:

· Visualise and work with surfaces and regions in 3-dimensional space;

· Evaluate volume, surface, and line integrals in 2 and higher dimensions, and apply them using Green's theorem and Gauss' theorem;

· Describe the mapping properties of basic functions of a complex variable (powers, fractional, exponential);

· Construct an analytic function from its real part using the Cauchy-Riemann equations;

· Compute residues and apply the Cauchy residue theorem to evaluate certain integrals.

**Assessment: **Total Marks 100: End of Year Written Examination 80 marks; Continuous Assessment 20 marks (2 in-class examinations, 10 marks each).

**Compulsory Elements: **End of Year Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **1 x 1½ hr(s) paper(s).

**Requirements for Supplemental Examination: **1 x 1½ hr(s) paper(s) to be taken in Autumn 2014. The mark for Continuous Assessment is carried forward.

MA2051 Mathematical Analysis I

**Credit Weighting: **5

**Teaching Period(s): **Teaching Period 1.

**No. of Students: **Min 5, Max 75.

**Pre-requisite(s): **MA1059 and MA1060; or MA1055

**Co-requisite(s): **None

**Teaching Methods: **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Stephen Wills, Department of Mathematics.

**Lecturer(s): ** Dr Thomas Carroll, Department of Mathematics.

**Module Objective: **To present elementary classical analysis in a concrete setting, emphasizing specific techniques important to classical analysis and its applications.

**Module Content: **Topology of Euclidean space; continuous and differentiable mappings; Riemann integration

**Learning Outcomes: **On successful completion of this module, students should be able to:

· Formulate basic definitions and results and prove basic results concerning the topology of Euclidean spaces;

· Recall the basic definitions and theorems of limits, continuity and differentiation and their proofs.

· Formulate and prove the Minimum-Maximum Theorem, the Intermediate Value Theorem, the Uniform Continuity Theorem, and some of their applications;

· Give criteria that ensure functions are Riemann-Stieltjes integrable, and establish basic properties of such integrals.

**Assessment: **Total Marks 100: End of Year Written Examination 75 marks; Continuous Assessment 25 marks (In-class test (15 marks), assignments (10 marks)).

**Compulsory Elements: **End of Year Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **1 x 1½ hr(s) paper(s).

**Requirements for Supplemental Examination: **1 x 1½ hr(s) paper(s) to be taken in Autumn 2014. The mark for Continuous Assessment is carried forward.

MA2054 Ordinary Differential Equations

**Credit Weighting: **5

**Teaching Period(s): **Teaching Period 2.

**No. of Students: **Min 5, Max 75.

**Pre-requisite(s): **MA1058, MA1059 and MA1060; or MA1055

**Co-requisite(s): **MA2051

**Teaching Methods: **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Mr David Goulding, School of Mathematical Sciences.

**Module Objective: **To provide an introduction to the theory of ordinary differential equations.

**Module Content: **Uniform convergence. Power series. Existence and uniqueness theory. Picard's theorem. Method of undetermined coefficients, reduction of order, variation of parameters. Theory of linear differential equations.

**Learning Outcomes: **On successful completion of this module, students should be able to:

· Formulate definitions, basic results and their proofs in relation to uniform convergence of sequences and series of functions, the corresponding Cauchy criteria, and some of their applications;

· Formulate and prove basic results about absolute and uniform convergence of power series and determine the radius of convergence.

· Determine existence and uniqueness of solutions of an ordinary differential equation;

· Solve ordinary differential equations by elementary methods such as separation of variables, undetermined coefficients, or variation of parameters;

· Apply reduction of order to differential equations;

· Solve constant-coefficient linear ordinary differential equations of all orders;

· Solve inhomogeneous linear ordinary differential equations, either by Laplace transform or by other elementary methods;

· Apply Picard iteration to approximate solutions of ordinary differential equations.

**Assessment: **Total Marks 100: End of Year Written Examination 75 marks; Continuous Assessment 25 marks (2 x in-class examinations (12.5 marks each).

**Compulsory Elements: **End of Year Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **1 x 1½ hr(s) paper(s).

**Requirements for Supplemental Examination: **1 x 1½ hr(s) paper(s) to be taken in Autumn 2014. The mark for Continuous Assessment is carried forward.

**Credit Weighting: **5

**Teaching Period(s): **Teaching Period 1.

**No. of Students: **Min 5, Max 75.

**Pre-requisite(s): **MA1058, MA1059 and MA1060; or MA1055

**Co-requisite(s): **None

**Teaching Methods: **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Dr Claus Michael Koestler, School of Mathematical Sciences.

**Module Objective: **To provide an introduction to the concepts of the theory of linear algebra.

**Module Content: **Linear equations and matrices; vector spaces; determinants; linear transformations and eigenvalues; norms and inner products; linear operators and normal forms.

**Learning Outcomes: **On successful completion of this module, students should be able to:

· Verify the linearity of mappings on real and complex vector spaces,

· and the linear independence of sets of vectors;

· Evaluate bases, transition matrices and matrices representing linear transformations;

· Compute eigenvalues and eigenvectors of linear operators;

· Construct orthonormal bases for vector spaces;

· Verify properties of projection mappings, adjoint mappings, self-adjoint operators and isometries.

**Assessment: **Total Marks 100: End of Year Written Examination 75 marks; Continuous Assessment 25 marks (2 x in-class examinations, 12.5 marks each).

**Compulsory Elements: **End of Year Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **1 x 1½ hr(s) paper(s).

**Requirements for Supplemental Examination: **1 x 1½ hr(s) paper(s) to be taken in Autumn 2014. The mark for Continuous Assessment is carried forward.

MA2059 Introduction to Discrete Mathematics

**Credit Weighting: **5

**Teaching Period(s): **Teaching Period 2.

**No. of Students: **Min 8, Max 75.

**Pre-requisite(s): **MA1058, MA1059 and MA1060; or MA1055

**Co-requisite(s): **None

**Teaching Methods: **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Dr Anca Mustata, Department of Mathematics.

**Module Objective: **To introduce the student to the techniques of discrete mathematics.

**Module Content: **Basic number-theoretic concepts and techniques such as multiplicative functions and quadratic reciprocity; graph theory including paths and cycles and related theory; permutations, combinations and other methods of combinatorics.

**Learning Outcomes: **On successful completion of this module, students should be able to:

· Use counting methods to solve combinatorial problems;

· Solve recurrence relations;

· Apply the concepts of a graph and a tree to solve mathematical problems, including the use of Kruskal's and Prim's algorithms;

· Determine if graphs are Hamiltonian, Eulerian and semi-Eulerian;

· Use Euclid's algorithm to find the largest common divisor of 2 numbers;

· Solve linear Diophantic equations;

· Prove and apply properties of multiplicative functions such as the Euler phi-function;

· Check when a number is a quadratic residue modulo n.

**Assessment: **Total Marks 100: End of Year Written Examination 75 marks; Continuous Assessment 25 marks (2 in-class examinations (15 marks), assignments (10 marks)).

**Compulsory Elements: **End of Year Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **1 x 1½ hr(s) paper(s).

**Requirements for Supplemental Examination: **1 x 1½ hr(s) paper(s) (which incorporates assessment of both End of Year Written Examination and Continuous Assessment) to be taken in Autumn 2014.

**Credit Weighting: **5

**Teaching Period(s): **Teaching Period 2.

**No. of Students: **Min 5, Max 75.

**Pre-requisite(s): **MA2051, MA2055

**Co-requisite(s): **None

**Teaching Methods: **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Dr Ben McKay, Department of Mathematics.

**Module Objective: **To provide a foundation in multivariable calculus.

**Module Content: **Calculus of several variables, including continuity, differentiability and constrained and unconstrained optimisation. Line, surface and volume integrals.

**Learning Outcomes: **On successful completion of this module, students should be able to:

· Use the definitions to verify continuity and differentiability for simple functions of two (or more) variables;

· Compute partial derivatives, mixed partial derivatives and higher-order partial derivatives using various methods including the chain rules;

· Find the equations of tangent planes and normal lines to surfaces that are the graphs of functions of several variables;

· Compute directional derivatives and maximum/minimum rates of change in terms of the gradient vector;

· Solve simple unconstrained and constrained maxima/minima problems, using tests to determine the nature of local extrema of functions of two variables;

· Set up and calculate line integrals including examples arising in physical and geometrical problems involving curvature, mass, and work done by forces;

· Calculate double and triple integrals of continuous functions, defined over closed, bounded regions, by means of iterated integrals and the fundamental theorem of computation.

**Assessment: **Total Marks 100: End of Year Written Examination 75 marks; Continuous Assessment 25 marks (5 x 0.5hr tests, 5 marks each).

**Compulsory Elements: **End of Year Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **None.

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **1 x 1½ hr(s) paper(s).

**Requirements for Supplemental Examination: **1 x 1½ hr(s) paper(s) to be taken in Autumn 2014. The mark for Continuous Assessment is carried forward.

MA2200 Introductory Mathematics for Business II

**Credit Weighting: **5

**Teaching Period(s): **Teaching Period 1.

**No. of Students: **Min 30, Max 150.

**Pre-requisite(s): **MA1100

**Co-requisite(s): **None

**Teaching Methods: **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Martin Kilian, Department of Mathematics.

**Lecturer(s): ** Dr Andrei Mustata, Department of Mathematics.

**Module Objective: **To complete the introduction to fundamental mathematical techniques for business.

**Module Content: **Least squares approximation and curve fitting. Matrices and simultaneous equations. Further methods and applications of calculus.

**Learning Outcomes: **On successful completion of this module, students should be able to:

· apply integration techniques to business situations such as computing consumer and producer surplus;

· solve elementary first order differential equations, including differential equations describing exponential growth/decay, with emphasis on applying these techniques to calculating revenue, cost, profits, long-term growth predictions etc.;

· formulate and solve a linear programming model from information given in word and table form;

· perform basic matrix calculations, including addition/subtraction, multiplication, transpose, inverses, determinants;

· solve simultaneous equations using Gaussian Elimination, Gauss-Jordan Elimination, Cramer's Rule and the Inverse Matrix Method and use matrices to perform Input/Output analysis;

· form and solve difference equations, given information in word or numerical form and apply this knowledge to income/growth models;

· find curves of best fit to given sets of data and use the least squares approximation method to predict future outcomes.

**Assessment: **Total Marks 100: End of Year Written Examination 80 marks; Continuous Assessment 20 marks (Two in-class tests, each worth 10%).

**Compulsory Elements: **End of Year Written Examination; In-Class Tests.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **1 x 1½ hr(s) paper(s).

**Requirements for Supplemental Examination: **1 x 1½ hr(s) paper(s) to be taken in Autumn 2014. The mark for In-Class Tests is carried forward.

MA3051 Mathematical Analysis II

**Credit Weighting: **5

**Teaching Period(s): **Teaching Period 1.

**No. of Students: **Min 5, Max 75.

**Pre-requisite(s): **MA2051

**Co-requisite(s): **None

**Teaching Methods: **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Dr Stephen Wills, Department of Mathematics.

**Module Objective: **To present elementary linear analysis in a concrete setting, emphasizing specific techniques important to analysis and its applications.

**Module Content: **Normed linear spaces, Banach spaces, bounded linear operators; Hilbert spaces, convexity, orthogonal expansions and their applications, Riesz representation theorem for functionals.

**Learning Outcomes: **On successful completion of this module, students should be able to:

· Verify the axioms of normed vector spaces, Banach spaces and Hilbert spaces in specific examples, applying relevant tests for completeness or the existence of an inner product;

· Compute the norms of operators;

· Use completeness arguments to produce existence proofs for operators with desired properties;

· Compute orthogonal expansions with respect to an orthonormal basis of a Hilbert space, and apply such expansions to solve problems;

· Prove the Riesz Representation Theorem for bounded linear functionals on Hilbert spaces.

**Assessment: **Total Marks 100: End of Year Written Examination 75 marks; Continuous Assessment 25 marks (in-class test(s) and/or assignment(s), students will be notified of the format and breakdown of marks for continuous assessment at the first lecture.).

**Compulsory Elements: **End of Year Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **1 x 1½ hr(s) paper(s).

**Requirements for Supplemental Examination: **1 x 1½ hr(s) paper(s) to be taken in Autumn 2014. The mark for Continuous Assessment is carried forward.

**Credit Weighting: **5

**Teaching Period(s): **Teaching Period 2.

**No. of Students: **Min 5, Max 75.

**Pre-requisite(s): **MA2051

**Co-requisite(s): **None

**Teaching Methods: **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Dr Spyridon Dendrinos, School of Mathematical Sciences.

**Module Objective: **To provide an introduction to the theory of functions of a complex variable.

**Module Content: **Bilinear mappings, complex differentiable functions, power series, complex contour integrals, Cauchy's theorem and integral formula, Taylor's theorem, zeros of analytic functions and Rouche's theorem, maximum modulus principle, singularities and Laurent series, poles and residues, residue calculus.

**Learning Outcomes: **On successful completion of this module, students should be able to:

· Analyse the mapping properties of fundamental functions (bilinear, exponential, trigonometric) of a complex variable;

· Define the derivative of a complex function and derive and apply the Cauchy-Riemann equations;

· Prove and apply Cauchy's integral formula, and Taylor's theorem on the power series expansion of analytic functions;

· Derive Liouville's theorem on entire functions and establish the fundamental theorem of algebra;

· Compute residues and apply Cauchy's theorem to the evaluation of integrals and the summation of series;

· Apply Rouche's theorem on the zeros of analytic functions.

**Assessment: **Total Marks 100: End of Year Written Examination 75 marks; Continuous Assessment 25 marks (2 x in-class examinations, 12.5 marks each).

**Compulsory Elements: **End of Year Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **1 x 1½ hr(s) paper(s).

**Requirements for Supplemental Examination: **1 x 1½ hr(s) paper(s) to be taken in Autumn 2014. The mark for Continuous Assessment is carried forward.

MA3056 Metric Spaces and Topology

**Credit Weighting: **5

**Teaching Period(s): **Teaching Period 1.

**No. of Students: **Min 5, Max 75.

**Pre-requisite(s): **MA2051

**Co-requisite(s): **None

**Teaching Methods: **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Martin Kilian, Department of Mathematics.

**Lecturer(s): ** Dr Martin Kilian, Department of Mathematics.

**Module Objective: **To provide an introduction to the theory and concepts of metric and topological spaces.

**Module Content: **Metric spaces, completeness, topological spaces, compactness, connectedness, product spaces, continuity of functions.

**Learning Outcomes: **On successful completion of this module, students should be able to:

· State the basic concepts of topological and metric spaces.

· Construct examples and counterexamples of topological spaces with certain properties.

· Perform set theoretic computations.

· Formulate the basic properties of continuous functions.

· Apply the theory of various notions of convergence.

**Assessment: **Total Marks 100: End of Year Written Examination 75 marks; Continuous Assessment 25 marks (10 x assignments, 2.5 marks each).

**Compulsory Elements: **End of Year Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **1 x 1½ hr(s) paper(s).

**Requirements for Supplemental Examination: **1 x 1½ hr(s) paper(s) to be taken in Autumn 2014. The mark for Continuous Assessment is carried forward.

MA3060 Topics in Discrete Mathematics

**Credit Weighting: **5

**Teaching Period(s): **Teaching Period 2.

**No. of Students: **Min 8, Max 75.

**Pre-requisite(s): **MA1054, MA1058 or MA1055

**Co-requisite(s): **None

**Teaching Methods: **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Dr Anca Mustata, Department of Mathematics.

**Module Objective: **To develop an understanding of, and the skills required for, aspects of mathematics in a discrete setting.

**Module Content: **The module will include topics chosen from graph theory, combinatorics and number theory, such as graphs and digraphs, Eulerian and Hamiltonian cycles, graph colouring, trees, general counting techniques, the inclusion-exclusion principle, generating functions, recurrence relations, primes and divisibility, congruences, the theorems of Fermat, Euler and Wilson, number theoretic functions, primitive roots, perfect numbers, the Fibonacci sequence, and quadratic reciprocity.

**Learning Outcomes: **On successful completion of this module, students should be able to:

· Use counting methods to solve combinatorial problems;

· Solve recurrence relations;

· Apply the concepts of a graph and a tree to solve mathematical problems, including the use of Kruskal's and Prim's algorithms;

· Determine if graphs are Hamiltonian, Eulerian and semi-Eulerian;

· Use Euclid's algorithm to find the largest common divisor of 2 numbers;

· Solve linear Diophantic equations;

· Prove and apply properties of multiplicative functions such as the Euler phi-function;

· Check when a number is a quadratic residue modulo n.

**Assessment: **Total Marks 100: End of Year Written Examination 75 marks; Continuous Assessment 25 marks (2 in-class examinations (15 marks), homework (10 marks)).

**Compulsory Elements: **End of Year Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **1 x 1½ hr(s) paper(s).

**Requirements for Supplemental Examination: **1 x 1½ hr(s) paper(s) (which incorporates assessment of both End of Year Written Examination and Continuous Assessment) to be taken in Autumn 2014.

MA3062 Introduction to Modern Algebra

**Credit Weighting: **5

**Teaching Period(s): **Teaching Period 1.

**No. of Students: **Min 5, Max 75.

**Pre-requisite(s): **MA1057 or MA1055, MA1058, MA2055

**Co-requisite(s): **None

**Teaching Methods: **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Professor Bernard Hanzon, School of Mathematical Sciences.

**Lecturer(s): ** Dr Andrei Mustata, Department of Mathematics.

**Module Objective: **To provide an introduction to major concepts in Modern Algebra with applications.

**Module Content: **Basic algebraic concepts and techniques such as rings and modules, factorization in integral domains, computational algebra techniques and optimization, groups and group

representations, fields and field extensions, Galois theory.

**Learning Outcomes: **On successful completion of this module, students should be able to:

· Recognize when a ring is a Euclidean domain, a principal ideal domain (PID) or unique factorization domain and perform computations within such rings;

· Compute Groebner bases and apply elimination theory techniques;

· Compute free resolutions of modules; classify modules over PID-s;

· Calculate homology groups of complexes, with applications in algebra, topology and geometry;

· Explain the connection between Lie groups and algebras and use Lie algebras to understand basic properties of the corresponding Lie group;

· Perform irreducible decompositions for given representations by finite groups and some classical groups; classify finite Coxeter groups;

· Solve problems in field extensions of a field F;

· Compute the Galois group of given field extensions;

· Explain the connection between an affine algebraic variety and its ring/field of functions and use these to understand basic properties of given varieties.

**Assessment: **Total Marks 100: End of Year Written Examination 75 marks; Continuous Assessment 25 marks (in-class test(s) and/or homework assignment(s), students will be notified of the format and breakdown of marks for continuous assessment at the beginning of the course).

**Compulsory Elements: **End of Year Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **1 x 1½ hr(s) paper(s).

**Requirements for Supplemental Examination: **1 x 1½ hr(s) paper(s) to be taken in Autumn 2014. The mark for Continuous Assessment is carried forward.

MA3063 Introduction to Differential Geometry

**Credit Weighting: **5

**Teaching Period(s): **Teaching Period 2.

**No. of Students: **Min 5, Max 75.

**Pre-requisite(s): **MA2071

**Co-requisite(s): **MA3051

**Teaching Methods: **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Martin Kilian, Department of Mathematics.

**Lecturer(s): ** Dr Martin Kilian, Department of Mathematics.

**Module Objective: **To provide an introduction to concepts of differential geometry.

**Module Content: **Basic differential geometry concepts and techniques such as differential manifolds and submanifolds, tangent and normal bundles, Riemannian metric, curvatures, geodesics, with emphasis on the cases of curves and surfaces; differentiable maps, geometric aspects of the theory of differential equations, symplectic geometry, homogeneous spaces.

**Learning Outcomes: **On successful completion of this module, students should be able to:

· Explain and manipulate the concepts of differential manifolds, tangent bundle and cotangent bundles, tensor fields, differential forms, differentiable maps, symplectic forms;

· Use the implicit function theorem to pass between parametric and level set descriptions of given manifolds. Use immersions and submersions to describe given manifolds;

· Pass between parametric and level set descriptions of given manifolds;

· Use Riemannian metrics on given manifolds to calculate volumes, the Levi-Civita connection, curvatures, geodesics, with emphasis on surfaces;

· Calculate the induced metric on a submanifold, its second fundamental form, and explain the Gauss equation with particular emphasis on surfaces;

· Calculate the normal and geodesic curvatures of a curve on a surface, and decide whether the curve is a geodesic;

· Calculate critical point indices of a differentiable map on a given manifold, and use this to describe its topological properties;

· Use the concepts of manifold, Lie groups and algebras in the study of differential equations;

· Work with Fuchsian groups and their applications to hyperbolic geometry.

**Assessment: **Total Marks 100: End of Year Written Examination 75 marks; Continuous Assessment 25 marks (in-class test(s) and/or homework assignment(s), students will be notified of the format and breakdown of marks for continuous assessment at the beginning of the course).

**Compulsory Elements: **End of Year Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **1 x 1½ hr(s) paper(s).

**Requirements for Supplemental Examination: **1 x 1½ hr(s) paper(s) to be taken in Autumn 2014. The mark for Continuous Assessment is carried forward.

MA3301 Multivariable Calculus and Optimisation

**Credit Weighting: **5

**Teaching Period(s): **Teaching Period 1.

**No. of Students: **Min 20, Max 100.

**Pre-requisite(s): **MA2200 or equivalent

**Co-requisite(s): **None

**Teaching Methods: **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Martin Kilian, Department of Mathematics.

**Lecturer(s): ** Mr David Goulding, School of Mathematical Sciences.

**Module Objective: **To develop and understanding of techniques required for dealing with multi-parameter quantitative problems in business and management.

**Module Content: **Multivariable functions, partial derivatives, optimisation problems and Language multipliers. Growth rates and applications.

**Learning Outcomes: **On successful completion of this module, students should be able to:

· Calculate first and second order partial derivatives of functions in two or three variables and apply this to problems in relation to economic concepts such as partial elasticity, production and utility;

· Solve unconstrained optimisation problems for functions in two variables and apply this to optimisation problems in economics;

· Solve constrained optimisation problems for functions in two variables and apply this to optimisation problems in economics;

· Apply the Lagrange Multiplier Method to solve constrained optimisation problems in two variables and apply this to optimisation problems in economics;

· Demonstrate understanding of basic mathematical rules for exponential and logarithmic functions and apply these rules to solve problems in economics in relation to relative growth and decay;

· Solve elementary differential equations by the method of separation of variables and apply this to solve problems in economics.

**Assessment: **Total Marks 100: End of Year Written Examination 80 marks; Continuous Assessment 20 marks (In-class Examination).

**Compulsory Elements: **End of Year Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **1 x 1½ hr(s) paper(s) to be taken in Spring 2014.

**Requirements for Supplemental Examination: **1 x 1½ hr(s) paper(s) to be taken in Autumn 2014. The mark for Continuous Assessment is carried forward.

MA3901 Multivariable Calculus and Optimisation

**Credit Weighting: **5

**Teaching Period(s): **Teaching Period 1.

**No. of Students: **Min 20, Max 100.

**Pre-requisite(s): **MA2200 or equivalent

**Co-requisite(s): **None

**Teaching Methods: **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Professor Bernard Hanzon, School of Mathematical Sciences.

**Lecturer(s): ** Mr David Goulding, School of Mathematical Sciences.

**Module Objective: **To develop advanced calculus techniques and then apply them to quantitative problems in business and management.

**Module Content: **Multivariable functions, partial derivatives, optimisation problems and Lagrange multipliers.

**Learning Outcomes: **On successful completion of this module, students should be able to:

· Use the chain rule to compute partial derivatives of functions of several variables;

· Compute equations of tangent planes to two-dimensional surfaces;

· Use partial derivatives to solve problems related to economic concepts such as partial elasticity, production and utility;

· Solve unconstrained optimisation problems for functions of two variables and apply this knowledge to optimisation problems in economics;

· Use the methods of Elimination of Variables and Lagrange Multipliers to solve constrained optimisation problems, including problems from economics.

**Assessment: **Total Marks 100: Continuous Assessment 100 marks (2 x in-class Examinations (1 x 20 marks, 1 x 80 marks)).

**Compulsory Elements: **Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **No End of Year Written Examination.

**Requirements for Supplemental Examination: ** to be taken in Autumn 2014. Marks in passed element(s) of Continuous Assessment are carried forward, Failed element(s) of Continuous Assessment must be repeated (A departmental test will take place during the period set aside by the University for Supplementary examinations; students should consult Department in early August for date.).

**Credit Weighting: **5

**Teaching Period(s): **Teaching Period 1.

**No. of Students: **Min 5, Max 50.

**Pre-requisite(s): **MA3051; MA3056

**Co-requisite(s): **None

**Teaching Methods: **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Dr Stephen Wills, Department of Mathematics.

**Module Objective: **To provide a grounding in modern functional analysis.

**Module Content: **A study of the interaction of topological and algebraic structures, of fundamental importance in contemporary mathematics and its applications. Normed linear spaces, Banach spaces, bounded linear operators; Hilbert spaces, convexity, orthogonal sums, Riesz representation theorem for functionals, adjoint operators, spectrum.

**Learning Outcomes: **On successful completion of this module, students should be able to:

· Verify the axioms of normed vector spaces, Banach spaces and Hilbert spaces in specific examples, applying relevant tests for completeness or the existence of an inner product;

· Compute the norms and spectra of (certain classes of) operators;

· Use completeness arguments to produce existence proofs for operators and linear functionals with desired properties;

· Establish well-known isometric isomorphisms between sequence spaces and their dual spaces;

· Prove the Riesz Representation Theorem for bounded linear functionals on Hilbert spaces.

**Assessment: **Total Marks 100: End of Year Written Examination 75 marks; Continuous Assessment 25 marks (homeworks x 5 (5 marks each)).

**Compulsory Elements: **End of Year Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **1 x 1½ hr(s) paper(s).

**Requirements for Supplemental Examination: **1 x 1½ hr(s) paper(s) to be taken in Autumn 2014. The mark for Continuous Assessment is carried forward.

**Credit Weighting: **5

**Teaching Period(s): **Teaching Periods 1 and 2.

**No. of Students: **Min 1, Max 25.

**Pre-requisite(s): **None

**Co-requisite(s): **None

**Teaching Methods: **12 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Dr Stephen Wills, Department of Mathematics.

**Module Objective: **To develop skills of mathematical investigation and report writing.

**Module Content: **Project on an assigned mathematical topic.

**Learning Outcomes: **On successful completion of this module, students should be able to:

· Write a logical and coherent account of a mathematical topic;

· Construct proofs of mathematical results;

· Assimilate and explain details of selected known theorems and theories;

· Carry out independent library research;

· Give a clear and accurate oral presentation of project results.

**Assessment: **Total Marks 100: Continuous Assessment 100 marks ( 1 x Project).

**Compulsory Elements: **Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Where work is submitted up to and including 7 days late, 10% of the total marks available shall be deducted from the mark achieved. Where work is submitted up to and including 14 days late, 20% of the total marks available shall be deducted from the mark achieved. Work submitted 15 days late or more shall be assigned a mark of zero.

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **No End of Year Written Examination.

**Requirements for Supplemental Examination: **Marks in passed element(s) of Continuous Assessment are carried forward, Failed element(s) of Continuous Assessment must be repeated (resubmit revised project by August 15th, as prescribed by the Department).

MA4058 Measure Theory and Martingales

**Credit Weighting: **5

**Teaching Period(s): **Teaching Period 1.

**No. of Students: **Min 5, Max 50.

**Pre-requisite(s): **MA3051

**Co-requisite(s): **

**Teaching Methods: **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Dr Thomas Carroll, Department of Mathematics.

**Module Objective: **To provide and overview of the theory of measurable sets, integration and Martingales

**Module Content: **Measurable spaces and functions; measures and integrals; integrable functions; conditional expectation; martingales; convergence theorems.

**Learning Outcomes: **On successful completion of this module, students should be able to:

· Identify and formulate the basic concepts and theorems of sigma algebras, measures and abstract measure spaces;

· Discuss the completion and construction of measures including the basics of Carathéodory's Extension Procedure;

· Synthesise techniques that have been developed in the course to solve particular problems;

· Explain the basic concepts and main theorems of Lebesgue and Lesbesgue-Stieltjes integration including the main convergence theorems;

· Solve problems involving Lebesgue and Lesbesgue-Stieltjes integration;

· Explain the basic concepts and theorems of conditional expectation and martingales;

· Evaluate basic martingales.

**Assessment: **Total Marks 100: End of Year Written Examination 75 marks; Continuous Assessment 25 marks (1 x in-class examination).

**Compulsory Elements: **End of Year Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **1 x 1½ hr(s) paper(s).

**Requirements for Supplemental Examination: **1 x 1½ hr(s) paper(s) to be taken in Autumn 2014. The mark for Continuous Assessment is carried forward.

**Credit Weighting: **5

**Teaching Period(s): **Teaching Period 2.

**No. of Students: **Min 6, Max 50.

**Pre-requisite(s): **MA3051, MA3062 MA3063, MA3054

**Co-requisite(s): **None

**Teaching Methods: **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Professor Bernard Hanzon, School of Mathematical Sciences.

**Lecturer(s): ** Dr Ben McKay, Department of Mathematics.

**Module Objective: **To further develop an understanding of, and the skills required for, advanced mathematics.

**Module Content: **The module will include topics from: algebra, analysis and geometry.

**Learning Outcomes: **On successful completion of this module, students should be able to:

· Solve basic problems in an area of current research in mathematics;

· Explain the most important recent results of algebra, analysis, geometry or topology;

· Rigorously prove theorems using geometric intuition, algebraic symbol manipulation, and analytic estimates;

· Use theorems from algebra to give insight in the areas of number theory or complex differential geometry;

· Use the concept of manifold in the study of differential equations.

**Assessment: **Total Marks 100: End of Year Written Examination 75 marks; Continuous Assessment 25 marks (in-class test(s) and/or homework assignment(s), students will be notified of the format and breakdown of marks for continuous assessment at the first lecture).

**Compulsory Elements: **End of Year Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **1 x 1½ hr(s) paper(s).

**Requirements for Supplemental Examination: **1 x 1½ hr(s) paper(s) to be taken in Autumn 2014. The mark for Continuous Assessment is carried forward.

MA4062 Topics in Modern Algebra

**Credit Weighting: **5

**Teaching Period(s): **Teaching Period 1.

**No. of Students: **Min 5, Max 75.

**Pre-requisite(s): **MA1057 or MA1055, MA1058, MA2055

**Co-requisite(s): **None

**Teaching Methods: **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Professor Bernard Hanzon, School of Mathematical Sciences.

**Lecturer(s): ** Dr Andrei Mustata, Department of Mathematics.

**Module Objective: **To further develop major concepts in Modern Algebra with applications.

**Module Content: **A range of algebraic concepts and techniques chosen from rings and modules, factorization in integral domains, computational algebra techniques and optimization, groups and group representations, fields and field extensions, Galois theory, Lie groups and algebra, homology, algebraic varieties.

**Learning Outcomes: **On successful completion of this module, students should be able to:

· Recognize when a ring is a Euclidean domain, a principal ideal domain (PID) or unique factorization domain and perform computations within such rings;

· Compute Groebner bases and apply elimination theory techniques;

· Compute free resolutions of modules; classify modules over PID-s;

· Calculate homology groups of complexes, with applications in algebra, topology and geometry;

· Explain the connection between Lie groups and algebras and use Lie algebras to understand basic properties of the corresponding Lie group;

· Perform irreducible decompositions for given representations by finite groups and some classical groups; classify finite Coxeter groups;

· Solve problems in field extensions of a field F;

· Compute the Galois group of given field extensions;

· Explain the connection between an affine algebraic variety and its ring/field of functions and use these to understand basic properties of given varieties.

**Assessment: **Total Marks 100: End of Year Written Examination 75 marks; Continuous Assessment 25 marks (in-class test(s) and/or homework assignment(s), students will be notified of the format and breakdown of marks for continuous assessment at the beginning of the lecture).

**Compulsory Elements: **End of Year Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **1 x 1½ hr(s) paper(s).

**Requirements for Supplemental Examination: **1 x 1½ hr(s) paper(s) to be taken in Autumn 2014. The mark for Continuous Assessment is carried forward.

MA4063 Topics in Differential Geometry

**Credit Weighting: **5

**Teaching Period(s): **Teaching Period 2.

**No. of Students: **Min 5, Max 75.

**Pre-requisite(s): **MA2071, MA3051

**Co-requisite(s): **

**Teaching Methods: **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Martin Kilian, Department of Mathematics.

**Lecturer(s): ** Dr Martin Kilian, Department of Mathematics.

**Module Objective: **To further develop an understanding of differential geometry.

**Module Content: **Topics to be chosen from the differential geometry of manifolds and submanifolds, tangent and normal bundles, Riemann metric, connections, curvatures, geodesics, with particular emphasis on the cases of curves and surfaces; differential forms, differentiable maps, geometric aspects of the theory of differential equations, Lie groups, symplectic geometry, homogeneous spaces, elliptic and hyperbolic geometries.

**Learning Outcomes: **On successful completion of this module, students should be able to:

· Explain and manipulate the concepts of differential manifolds, tangent bundle and cotangent bundles, tensor fields, differential forms, differentiable maps, symplectic forms.

· Use the implicit function theorem to pass between parametric and level set descriptions of given manifolds. Use immersions and submersions to describe given manifolds.

· Use Riemannian metrics on given manifolds to calculate volumes, the Levi-Civita connection, curvatures, geodesics, with emphasis on surfaces.

· Calculate the induced metric on a submanifold, its second fundamental form, and explain the Gauss equation with particular emphasis on surfaces.

· Calculate the normal and geodesic curvatures of a curve on a surface, and decide whether the curve is a geodesic.

· Calculate critical point indices of a differentiable map on a given manifold, and use this to describe its topological properties.

· Use the concepts of manifold, Lie groups and algebras in the study of differential equations.

· Work with Fuchsian groups and their applications to hyperbolic geometry.

**Assessment: **Total Marks 100: End of Year Written Examination 75 marks; Continuous Assessment 25 marks (in-class test(s) and/or homework assignment(s), students will be notified of the format and breakdown of marks for continuous assessment at the beginning of the course).

**Compulsory Elements: **End of Year Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **1 x 1½ hr(s) paper(s).

**Requirements for Supplemental Examination: **1 x 1½ hr(s) paper(s) to be taken in Autumn 2014. The mark for Continuous Assessment is carried forward.

MA4402 Game Theory and Linear Algebra

**Credit Weighting: **5

**Teaching Period(s): **Teaching Period 2.

**No. of Students: **Min 15, Max 100.

**Pre-requisite(s): **MA2200 or equivalent

**Co-requisite(s): **None

**Teaching Methods: **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Dr Claus Michael Koestler, School of Mathematical Sciences.

**Module Objective: **To provide an introduction to business related applications of linear algebra.

**Module Content: **Applications of linear analysis to game theory, Markov chains, linear programming.

**Learning Outcomes: **On successful completion of this module, students should be able to:

· Solve linear optimization problems subject to linear inequalities;

· Use the simplex method;

· Find optimal strategies and calculate expected values of games;

· Employ Markov matrices in business models.

**Assessment: **Total Marks 100: End of Year Written Examination 80 marks; Continuous Assessment 20 marks (1 x class test).

**Compulsory Elements: **End of Year Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **1 x 1½ hr(s) paper(s).

**Requirements for Supplemental Examination: **1 x 1½ hr(s) paper(s) to be taken in Autumn 2014. The mark for Continuous Assessment is carried forward.

**Credit Weighting: **5

**Teaching Period(s): **Teaching Period 1.

**No. of Students: **Min 15, Max 100.

**Pre-requisite(s): **MA3301; ST1023 or equivalent

**Co-requisite(s): **None

**Teaching Methods: **24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

**Module Co-ordinator: **Dr Ben McKay, Department of Mathematics.

**Lecturer(s): ** Dr Grigory Temnov, School of Mathematical Sciences.

**Module Objective: **To develop facility with quantitative techniques for finance and investment.

**Module Content: **An introduction to the theory of options, the time value of money, rate of return of an investment cash-flow sequence and the arbitrage theorem.

**Learning Outcomes: **On successful completion of this module, students should be able to:

· Calculate probabilities and expectations of events and random variables associated to finite probability spaces and to standard variants of Brownian motion, using conditioning and independence techniques;

· Carry out calculations based on present-value analysis and arbitrage arguments;

· Calculate the price of European call and put options using the multiperiod model;

· Derive and apply the Black-Scholes formula for option pricing;

· Estimate volatility of shares from price history data.

**Assessment: **Total Marks 100: End of Year Written Examination 80 marks; Continuous Assessment 20 marks (One In-Class Test (10 marks) and Homework (10 marks)).

**Compulsory Elements: **End of Year Written Examination; Continuous Assessment.

**Penalties (for late submission of Course/Project Work etc.): **Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

**Pass Standard and any Special Requirements for Passing Module: **40%.

**End of Year Written Examination Profile: **1 x 1½ hr(s) paper(s).

**Requirements for Supplemental Examination: **1 x 1½ hr(s) paper(s) to be taken in Autumn 2014. The mark for Continous Assessment is carried forward.