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MTH3100 MTH4100 MTH4101 MTH4102 MTH4103 MTH4104 MTH4105 MTH4106 MTH4107

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MTH3100 Essential Mathematical Skills

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

decompose an integer as a product of prime numbers

calculate the GCD and LCM of a pair of integers

compute quotient and remainder of integer division

simplify arithmetical expressions involving fractions

perform simple estimations

compute quotient and remainder of polynomial division

simplify polynomial and rational expressions

simplify expressions involving square roots

perform algebraic substitutions

solve linear and quadratic equations and inequalities

MTH4100 Calculus I

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

explain and use elements of set theory notation in the context of real line;

solve algebraic equations and inequalities involving the square root and modulus function;

explain the difference between equations and identities;

prove simple identities and inequlities;

recite addition and double-angle formulas for trigonometric functions and use them to express values of trigonometric functions in the surds form;

recognize odd, even, periodic, increasing, decreasing functions;

understand the operation of composition of functions and the concept of functional inverse;

recognize linear, quadratic, power, polynomial, algebraic, rational, trigonometric, exponential, hyperbolic and logarithmic functions and sketch their graphs;

given the graph of f (x) sketch the graph of | f (ax+b)| or a f (|x|)+b;

manipulate piece-wise defined functions;

calculate limits by substitution and by eliminating zero denominators;

calculate limits at infinity of rational functions and rational algebraic;

calculate limits in indeterminate forms by a repeated use of l’Hopital rule, including limits involving (sin x)/x and (1+ 1/x)x;

explain the concepts of rate of change and instantaneous rate of change;

construct derivatives of power, trigonometric, exponential, hyperbolic, logarithmic and inverse trigonometric functions;

recite the basic rules of differentiation and use them to find derivatives of products and quotients;

recite the chain rule and use it to find derivatives of composite functions;

use derivatives to find intervals on which the given function is increasing or decreasing, find maxima and minima of functions;

find tangents and normals to graphs of functions given in explicit, implicit and parametric forms;

estimate change with differentials;

sketch graphs of rational functions including finding asymptotes;

explain the concept of indefinite integral as anti-derivative;

recite standard indefinite integrals and basic rules of indefinite integration;

evaluate integrals by substitution with and without suitable hints;

evaluate integrals of rational functions by partial fractions;

evaluate integrals by a repeated use of integration by parts;

solve separable differential equations and first-order linear differential equations;

explain the concept of definite integral and know the basic properties of definite integrals;

recite the Fundamental Theorem of Calculus and be able to use it for evaluating definite integrals and derivatives of integrals with variable limits of integration;

understand the concept of area of regions with curvilinear boundaries, be able to find area between curves;

calculate volume of solids by slicing and volume of solids of revolutions;

explain the concept of length of a planar curve and be able to find the length of parametric curves;

convert cartesian coordinates in polar coordinates and vice versa;

know the polar equation for lines, circles, circular sectors, annuli, ellipses, parabolas and hyperbolas;

sketch simple polar curves.

MTH4101 Calculus II

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

carry out the operations of addition, multiplication and division on complex numbers, know the meaning of conjugate, real part, imaginary part of complex numbers and find the argument and

modulus of complex numbers.

understand the geometric representation of complex numbers in the Argand diagram, represent geometrically sum, products and quotients of complex numbers.

find complex solutions of quadratic equations with real coefficients.

draw loci and regions in the Argand diagram, e.g. |z−1| = 1, |z−1| < 1, |z| = 2|z−1|, arg(z−a) = p/4, |z| < |z−1|.

know the relation between trigonometric functions and hyperbolic functions

know Euler’s relation eiq = cos q +i sin q and De Moivre’s Theorem, apply this theorem to express (i) sin nq and cos nq in powers of sin q and cos q, and (ii) powers of sin q and cos q in terms of sin and cos of multiple angles

know the nth roots of unity and find n-th roots of complex numbers, e.g. finding the three cubic roots of 8i.

know what is meant for a function of two variables to be continuous, and be able to identify points of discontinuity.

calculate partial derivatives, directional derivatives and estimate the rate of change in a given direction, be able to carry out implicit differentiation.

know the chain rule for functions of two and three variables and be able to use it for finding (i) the rate of change in a function’s values along a curve, and (ii) partial derivatives under transformation of variables.

find gradient vector and directions of maximal, minimal and zero change.

find tangents and normals to level curves, tangent planes and normal lines for surfaces z = f (x,y) and f (x,y, z) = 0.

find local extreme values and classify their type.

use the method of Lagrange multipliers for finding maxima and minima of functions with one constraint.

know the properties of double integrals and reduce double integrals to repeated integrals, be able to reverse the order of integration in repeated integrals.

calculate double integrals over rectangular and simple non-rectangular regions.

find the volume beneath a surface z = f (x,y).

express and evaluate area as a double integral.

for simple coordinate transformations, calculate Jacobians and find the transformed regions.

evaluate double and triple integrals by a given substitution

know the Jacobian of transformation from cartesian to polar coordinates and be able to evaluate double integrals by changing to polar coordinates.

explain the concepts of infinite sequence, converging sequence and diverging to infinity sequence, find the limit of converging sequences

explain the n-th term test for divergence of series, the integral test and the ratio test, and apply these with suitable hints e.g. use the ratio test to determine which of the following series converges and which diverges.

recognize a power series and be able to calculate its radius of convergence.

recognize a geometric series and be able to expand simple algebraic fractions in powers of x.

know Taylor formula with the remainder term in the Lagrange form and apply it to obtain power series and estimate error of approximation by Taylor polynomials

know Taylor series for ex, ln(1+x), sin x, cosx, (1+x)a.

be aware about applications of Taylor series (Euler’s formula for complex numbers, series solution of differential equations, evaluating non-elementary integrals in terms of series, evaluating indeterminate forms of limits)

MTH4102 Differential Equations

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

recognise some standard types of differential equation: separable, first order linear, homogeneous, and know techniques to solve these equations.

integrate first order differential forms, understand the geometrical interpretation of the solutions as a collection of integral curves.

construct sketched direction fields for autonomous/ non-autonomous first order ordinary differential equations.

know the techniques for solving second order linear differential equations.

calculate eigenvalues and eigenvectors of matrices, use matrix calculations to obtain solutions.

understand the concept of phase portrait, know different types of phase portrait. Recognise the stability and instability in a phase portrait.

understand the relevance of stability/instability in a dynamical system e.g. competing populations and mechanics.

recognise nonlinearity in systems. Be able to find the fixed point of simple nonlinear systems and classify its type.

MTH4103 Geometry I

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

convert between vector and cartesian equations of a straight line in R2 or R3.

write equation of a line in R2 or R3 passing through given points, or passing through a point and orthogonal to a line/plane.

use scalar product to calculate length of vector and cosine of angle between two vectors.

form sums of vectors; in R3 form vector product of two vectors.

calculate volume of parallelepiped by determinant.

find all solutions of a set of linear equations in two or three variables, by reduction to echelon form.

multiply two 2×2 or 3×3 matrices over R.

for a 2×2 or 3×3 matrix over R, determine invertibility by determinant, and if invertible, calculate the inverse.

calculate the determinant and (if invertible) the inverse of a 2×2 or 3×3 matrix over R.

calculate characteristic equation, eigenvalues and eigenvectors of a 2×2 or 3×3 matrix over R.

recognise from its matrix a rotation, a reflection, a dilation and a shear in R2 and R3.

compute sin(a +b) and cos(a +b) by taking product of rotation matrices.

MTH4104 Introduction to Algebra

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

recognise and construct a valid proof, and use counterexamples to disprove assertions.

understand the difference between necessary and sufficient conditions.

understand and use the relation between equivalence relations and partitions.

perform the division and Euclidean algorithms on integers and polynomials.

understand the role of abstraction and axiomatisation in mathematics.

know the definitions of group, ring and field, and deduce some consequences of the axioms for these

structures.

MTH4105 Introduction to Mathematical Computing

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

use Maple to solve simple problems in continuous mathematics, in particular to manipulate algebraic expressions, solve equations, differentiate, integrate, take limits, construct series, define new functions.

use Maple to solve simple problems in discrete mathematics, such as perform integer division, study prime numbers, manipulate lists and sets, and perform elementary Boolean logic.

use Maple to plot functions and tabulate data.

write simple programs involving loops, conditional execution and procedures.

MTH4106 Introduction to Statistics

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

plot one and two-dimensional data in an appropriate way and interpret such plots.

calculate summary statistics for a set of data.

understand the concepts of populations and samples and recognise different kinds of variables.

understand the concepts of hypothesis testing: null and alternate hypotheses; type I and type II errors;

p-values; testing at fixed significance levels.

understand concepts of point estimation: sampling distributions; unbiasedness and variance (including

proof of expectation and variance of sample mean and expectation of sample variance); and interval

estimation: confidence intervals.

carry out tests and calculate confidence intervals for one and two sample normal populations with variance

known or unknown.

carry out F-test for equality of two variances.

use Central Limit Theorem to justify normal approximations in calculating probabilities and large

sample tests for population means.

understand the concept of blocking as applied in the paired t-test and when this is appropriate.

carry out chi-squared goodness of fit tests for samples from specified population models.

carry out tests of association for contingency tables and understand different methods of sampling which

lead to such tables.

use an appropriate statistical package to perform these calculations and interpret its output.

MTH4107 Introduction to Probability

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

know and be comfortable with using basic set notation and terminology.

know the definition of and be comfortable with using functions. Understand what it means for functions to be injective, surjective and bijective.

write down the sample space for simple experiments, including sampling with replacement (such as tossing coins or rolling dice), sampling without replacement, and Bernoulli trials with stopping rules.

calculate probabilities in straightforward instances of the above types of experiment.

know the Kolmogorov axioms and make simple deductions from them.

calculate the probability of the complement of an event; and of the union of two disjoint events.

state and use the inclusion-exclusion formula for two events.

define and recognise independent events. Use independence to calculate probabilities.

define conditional probability and calculate it.

know the Theorem of Total Probability and use it in the case of a partition of the sample space into two events.

understand the probability mass function of a discrete random variable.

understand the cumulative distribution function and the probability density function of a continuous random variable, and be able to find each from the other.

find the expectation and variance of discrete and continuous random variables.

know the main properties of Bernoulli, binomial, geometric, Poisson, uniform and exponential random variables.

MTH5100 Algebraic Structures I

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

MTH5102 Calculus III

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

be able to do simple line and surface integrals (e.g. evaluate the integral of F.dr for a given vector field F, with the path given in either parametric or non-parametric form.)

understand three-dimensional cartesian, cylindrical, and spherical polar coordinates geometrically, and be able to express lines, surfaces, and volumes in coordinate or vector notation as appropriate.

be able to do simple manipulations involving gradient, divergence, and curl, and understand their geometrical/physical meaning.

understand Stokes’ theorem and the divergence theorem and be able to do simple problems applying these.

have a basic understanding of the Legendre equation and its solution in Legendre polynomials.

know the important properties of Fourier series and be able to compute coefficients.

understand the variable-separation technique for PDEs and be able to solve simple problems with Laplace’s equation in (at least) 2D Cartesian coordinates.

MTH5103 Complex Variables

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

MTH5104 Convergence and Continuity

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

define the basic concepts underlying continuous mathematics: supremum, limit of a sequence, convergent series, continuous function, derivative.

use criteria for convergence of series and continuity of functions.

follow proofs involving infinite sequences, convergent series and continuous functions.

solve problems relating to convergence of series and continuity of functions.

MTH5105 Differential and Integral Analysis

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

define the derivative and state the properties of the derivative including the chain rule and inverse function rule.

state and use key theorems concerning differentiable functions, such as Rolle's Theorem, the Mean Value Theorem and Taylor's Theorem.

define the Riemann integral, and state its properties.

apply Taylor's Theorem to some well known functions.

distinguish pointwise and uniform convergence.

MTH5106 Dynamics of Physical Systems

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

define momentum, work, kinetic energy, conservative forces and potential energy.

derive the conservation law of energy and use it to obtain qualitative information about motion.

show motion near points of stable equilibrium satisfy SHM and be able to solve this equation.

derive and solve the equation of damped SHM.

derive and solve the equation of forced, damped SHM and understand the phenomenon of resonance.

derive the properties of motion under a central force.

state and apply Newton's sphere theorem.

derive the equation of orbit of planets around the Sun and its solution.

derive Kepler's laws.

explain vector algebra, integration and differentiation of vectors and evaluate the gradient and curl of vectors.

define position vector, velocity and acceleration in both Cartesian and polar coordinates.

solve simple kinematical questions.

state Newton's laws of motion and Newton's law of gravitation.

solve examples of motion under the action of various forms of forces, in one and two dimensions, including resistive forces and force of gravity, leading to classical idea of black holes.

MTH5109 Geometry II: Knots and Surface

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

understand the concepts of knots and Reidemeister moves, and be able to define and compute such measures as writhe, tricolourability, Kauffman bracket and Jones polynomial.

be familiar with the concept of regular parametrised curves and surfaces and their study using tools such as calculus, scalar products and vector products in R3, be able to define and compute such measures as torsion and curvature of curves, unit normal, orientability, fundamental forms, and the geodesic, normal and Gauss curvatures of a surface.

be able to state some of the main theorems and be able to reproduce some of the shorter proofs or parts of proofs. The student should also be able to demonstrate understanding of the main theorems through examples.

MTH5110 Introduction to Numerical Computing

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

identify the causes and sources for numerical errors.

solve mathematical problems by programming.

construct simple routines using Maple.

compare the size of numerical errors of algorithms.

MTH5112 Linear Algebra I

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

solve linear systems and write solutions in vector form.

multiply matrices, including rectangular ones, and calculate the transpose of a matrix where the entries are either scalars or algebraic expressions.

use algebraic equations A(B + C) = A B + A C, (A B)T = BT AT, etc., both with letters for matrices and with examples of matrices whose entries are either scalars or algebraic expressions.

determine whether a given matrix is invertible or not. Calculate the inverse of an invertible matrix.

calculate the determinant of a square matrix (for small matrix dimensions).

determine whether or not a given subset of a vector space is a subspace.

determine whether or not a given vector is in the subspace spanned by a set of vectors.

determine whether given vectors (i) are linear independent, (ii) form a basis for a vector space.

find the coordinates of a vector with respect to a given ordered basis.

calculate the transition matrix corresponding to a change of basis.

calculate the rank of a matrix.

given a mapping from one vector space to another, verify whether it is linear or not.

given a linear mapping from one vector space to another, calculate the matrix of the mapping with respect to given bases.

calculate the scalar product of two vectors and determine whether the vectors are (i) orthogonal, (ii) orthonormal.

find the orthogonal projection of a vector onto a given subspace. Given a vector y and a subspace S, find the vector in S that is closest to y.

determine the set of least-squares solutions of a given linear system.

be able to apply the Gram-Schmidt process.

determine bases for the row and column spaces of a matrix.

calculate eigenvalues and eigenvectors of a square matrix.

given a real square matrix A with distinct eigenvalues, find an invertible matrix P such that P − 1 A P is diagonal.

given a real symmetric matrix A, find an orthogonal matrix Q such that QT A Q is diagonal.

be able to put together a mathematical argument in order to deduce/prove simple facts about vectors, matrices, vectors spaces and linear maps.

MTH5117 Mathematical Writing

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

be able to explain an elementary concept (e.g., vectors, prime numbers), with a minimum level of confidence.

be able to write a summary (title, main points, abstract) of a short document on a level-4 mathematical topic.

be fit for writing a final year project.

be able to write a proof with clarity and style.

MTH5120 Statistical Modelling I

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

take a description of an appropriate problem and express it as a linear model, both in matrix form and as a set of linear equations.

understand the necessary matrix manipulation, and the underlying statistical theory, including proofs.

use the statistical computing package to find both point estimators of parameters and confidence intervals for them

use the statistical computing package to assess the goodness of fit of a model, and to select among several models, all in the context of the normal linear model.

carry out the above calculations by hand in simple cases.

MTH5121 Probability Models

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

explain what is meant by a probability generating function and give suitable examples.

construct the probability mass function from the probability generating function.

construct the expected value and variance from the probability generating function.

construct the probability generating function of the sum of two independent random variables.

compute probabilities and expectations by conditioning, in discrete and continuous settings.

explain what is meant by a random walk, a branching process and a Poisson process.

define and recognise simple random walks.

for a simple random walk on a finite interval, calculate the probability that the random walk will hit one end-point of the interval prior to the other.

calculate the probability that a branching process becomes extinct by generation n.

calculate the probability that a branching process becomes extinct eventually.

explain what is meant by the joint distribution of continuous random variables.

state the main properties of a bivariate normal distribution.

apply simple discrete and continuous probability models in appropriate situations.

state and apply Markov's inequality and Chebyshev's inequality.

state and apply the Central Limit Theorem to the random walk on the integers.

MTH5122 Statistical Methods

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

compute probabilities from the joint probability density function.

explain what is meant by a uniform distribution and by marginal distributions.

state the expectation of a function of random variables.

assess when two random variables are independent.

produce transformations of random variables in simple examples.

describe t- and F-distributions.

apply 1- and 2-sample t tests, F tests and matched pairs t-tests.

explain what is meant by a moment generating function.

describe a Gamma distribution and a chi-squared distribution.

apply a goodness-of-fit test and justify its use.

assess when to use different types of tests in simple situations.

state and apply Chebyshev's inequality.

state and apply the Central Limit Theorem.

state the main properties of a bivariate or multivariate normal distribution.

MTH6100 Actuarial Mathematics

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

valuate cash flows.

use life tables for making predictions (like future lifetime at age x) and for analysing mortality patterns.

valuate life annuities and benefits of life assurance policies.

understand various life insurance products and basic principles of life assurance.

MTH6103 Advanced Statistics Project

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

understand the objectives of the particular project and decide what work you have to do to achieve them.

use and understand new material acquired outside the usual classroom context.

demonstrate independent or critical thought.

manage your time effectively when there are only long-term deadlines.

communicate your work effectively in writing and orally.

MTH6104 Algebraic Structures II

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

state the axioms for groups and rings and prove elementary consequences.

define subgroups, normal subgroups, subrings and ideals, and to test whether a given subset is one of those.

describe standard examples, such as cyclic, symmetric, alternating and dihedral groups, matrix groups in dimension 2 over a field, the ring of integers, the integers modulo n, polynomial rings and matrix rings.

construct quotient rings and quotient groups, and be familiar with their properties.

calculate in quotient groups or quotient rings by using coset representatives.

state, prove and use the three isomorphism theorems and the Correspondence Theorem.

test when a mapping is a group action.

define kernel, orbit, stabilizer, conjugate, centralizer, normalizer, centre.

state, prove and use Cayley’s Theorem, Cauchy’s Theorem, the Orbit-Stabilizer Theorem, the theorem attributed to Burnside, and the theorem that if a group has a subgroup of index n then it has a transitive subgroup of Sn as a quotient.

apply these results to the structure of finite groups of prime-power order.

state and prove Sylow’s three theorems, and apply them to investigate the structure of small finite groups.

define simple groups

explain why alternating groups are simple but the only simple groups of primepower order are cyclic of prime order.

define a composition series and a soluble group.

state, prove and use the Jordan-H¨older Theorem and basic theorems about finite soluble groups.

prove that matrix rings over a field are simple.

define units, associates, irreducibles and highest common factors, a unique factorization domain, a principal ideal domain, and a Noetherian integral domain; know, and be able to prove, the implications between them.

define a Noetherian ring.

state, prove and use the two conditions equivalent to being Noetherian, Hilbert’s Basis Theorem, and other properties of Noetherian rings.

define and describe simple properties of the direct product of groups and the direct sum of rings, and use these to construct examples.

MTH6105 Algorithmic Graph Theory

At the end of this module, students should be able to describe and implement the following algorithms, to be able to estimate their complexity, and to understand the theoretical results on which they are based.

Algorithms to find the components of a graph and the strongly connected components of a digraph.

Algorithms to construct breadth first search and depth first search spanning trees of a connected graph.

The algorithms of Prim and Kruskal to find a minimum weight spanning tree in a connected graph.

Dijkstra's algorithm to find a shortest path spanning tree in a graph or digraph.

Moravéks algorithm to find a longest path spanning tree in an acyclic directed network.

The max flow/min cut algorithm for finding a maximum (s;t)-flow in a network.

Algorithms for finding a maximum matching and a maximum weight matching in a bipartite graph.

Algorithms for finding an Euler trail in a graph or digraph and for solving the Chinese Postman Problem.

MTH6107 Chaos and Fractals

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

understand what is a dynamical system.

compute fixed points and periodic orbits of one-dimensional real maps, and determine the stability of these orbits.

understand what is meant by a diffeomorphism of the real line, and determine the possible patterns of periodic orbits.

describe the main features of the bifurcation diagram for the logistic family of maps, and compute the parameter values of key points in this diagram.

understand conjugacies between maps, and compute them in simple examples.

know what is meant by "chaos"; compute Lyapunov exponents in simple examples.

know what is meant by "fractal"; compute fractal (box-counting) dimension in simple examples.

know how to use iterated function systems to generate fractals.

MTH6108 Coding Theory

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

define error-detecting and error-correcting codes, explain their significance and construct simple examples, such as repetition and parity-check codes.

define the constants A_q(n,d) and calculate small examples.

give (with proof) bounds on the sizes of codes - the Hamming, Singleton and Plotkin bounds.

understand and construct decoding processes, compute error probabilities and state Shannon's Noisy Coding Theorem.

understand the definition and advantages of linear codes.

define, construct and manipulate generator matrices and parity-check matrices.

decode linear codes using Slepian arrays and syndrome decoding.

understand the relationship between a code and a parity-check matrix.

construct Hamming codes, Golay codes and MDS codes, and understand their properties.

MTH6109 Combinatorics

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

calculate the number of ordered and unordered selections with or without repetition from a set.

calculate the number of partitions of a positive integer.

solve 3 term recurrence relations with constant coefficients in specific cases.

derive generating functions for given recurrence relations.

apply the Principle of Inclusion and Exclusion in special cases.

define binomial coefficients and compute their numerical values.

state the Binomial Theorem.

define a permutation and a derangement, and state the number of possible derangements of a set.

define and give the recurrence relations (with initial conditions) for the Fibonacci, Catalan, Bell, and Stirling numbers and their numerical values (without proof).

state the principle of Inclusion and Exclusion.

define systems of distinct representatives (SDRs) and state Hall's theorem.

define Latin squares and mutually orthogonal Latin squares, and state lower and exact bounds for the maximal size of a set of mutually orthogonal Latin squares (without proof).

define Intersecting families and Sperner families and state the bounds on the maximum sizes of these families (without proof).

define a Steiner triple system and state the existence theorems

MTH6110 Communicating and Teaching Mathematics

On successful completion of this module, students will:

have gained substantial experience of working in a challenging and unpredictable working environment;

be able to assess and devise appropriate ways to communicate a difficult principle or concept and will have gained a broad understanding of many of the key aspects of teaching in schools.

Students will also develop a better understanding of and confidence in mathematics. The specific and transferable skills they will have attained include:

communication skills, both one-to-one and with an audience;

understanding the needs of individuals;

interpersonal skills when dealing with colleagues;

staff responsibilities and conduct;

the ability to improvise;

giving (and taking) feedback;

organisational, prioritisation and negotiating skill;

handling difficult and potentially disruptive situations;

public speaking;

team-working;

standard teaching methods;

preparation of lesson plans and teaching materials.

MTH6111 Complex Analysis

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

prove that a complex differentiable function satisfies the Cauchy-Riemann equations.

apply the relationship between conformal maps and harmonic maps to solve problems concerning the latter.

compute complex trigonometric series, logarithms and powers.

differentiate and integrate a power series within its radius of convergence.

prove Cauchy's Theorem for a triangle, deduce it for a convex domain, and state and apply it in more general versions.

prove Cauchy's Integral Formula and apply it to prove Liouville's Theorem and the Fundamental Theorem of Algebra.

state Taylor's Theorem and apply it to compute Taylor series.

compute Laurent expansions for functions with isolated singularities.

classify types of isolated singularities and the behaviour of functions near such singularities.

state Cauchy's Residue Theorem and apply it to evaluate integrals of real functions and sum series of real numbers.

prove that a meromorphic function on the Riemann sphere is a rational function, and that an automorphism of the Riemann sphere is a Moebius transformation.

apply the Maximum Modulus Principle to prove Schwarz's Lemma.

apply Schwarz's Lemma to prove that the automorphisms of the compex upper half-plane are the real Moebius transformations.

construct Riemann surfaces for functions such as square root and logarithm.

prove that there is no holomorphic function function on the complex plane which is doubly-periodic, and that every non-constant elliptic functions has order at least 2.

MTH6115 Cryptography

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

answer questions on cryptography and steganography; plaintext, ciphertext, key.

answer questions on substitution and other traditional ciphers.

answer questions on stream ciphers including Vigenère cipher, one-time pad, shift registers.

answer questions on statistical attack on ciphers; Shannon’s Theorem.

answer questions on public-key cryptography: basic principles including complexity issues; knapsack, RSA and El-Gamal ciphers.

answer questions on digital signatures and authentication; secret sharing.

MTH6116 Design of Experiments

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

given a proposal for an experiment, ask pertinent questions about the aims of the experiment, the treatments and their structure, the plots and any blocks, replication, data recording, costs, resources.

explain the most appropriate designs for experiments on people and animals.

present an experimental plan in a form suitable for the experimenter.

explain why factorial experiments are better than one-factor-at-a-time experiments.

explain what is meant by the interaction between two or more treatment factors, and interpret it to a non-statistician.

decompose the treatment degrees of freedom (both by hand and in Genstat) for complete factorial structures with equal replication and for the structure with two crossed factors plus a control.

for a wide range of orthogonal designs, draw the Hasse diagram for the block factors; calculate degrees of freedom and the null anova table; draw the Hasse diagram for the treatment factors; calculate treatment degrees of freedom; construct a suitable design, either systematically or by using one of the methods below; randomize the design; allocate treatment subspaces to strata and hence calculate the skeleton anova table; analyse data from the designed experiment, both by hand and in Genstat.

construct Latin squares of any size, and a pair of orthogonal Latin squares of side n, where n is a prime number or n = 4 or n = 9.

use these Latin squares to construct fractional factorial designs and lattice designs.

construct factorial designs for experiments in blocks or split plots or fractions, using Latin squares or the method of characters.

MTH6117 Entrepreneurship and Innovation

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

understand entrepreneurship and the enterprise culture.

demonstrate knowledge of the elements required to generate opportunities and to commercialise new ideas.

understand the process involved in protecting and validating ideas.

demonstrate knowledge of the importance of business and financial planning and how to develop a business plan.

identify the sources available to fund ideas.

work through problems as an effective team member.

evaluate and present outcomes through oral presentation.

independently manage learning and the use of a wide range of resources with minimal guidance.

critically appraise progress of independent work.

make informed decisions about career choice and applications for jobs or further study.

MTH6120 Further Topics in Mathematical Finance

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

explain several models appearing in financial mathematics which are either an extension/modification of the geometric Brownian motion or in fact just different from it.

describe how techniques developed in differential equations and probability are applied to analysis of various financial models.

explain how mathematical techniques can be used for for assessing the investment strategies at least in certain relatively simple but still realistic situations.

explain why comprehension of descriptive definitions is essential and, in particular, the ability to translate these ideas to mathematical concepts is paramount.

MTH6121 Introduction to Mathematical Finance

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

explain how summation techniques, differential equations and probability are needed to describe the behaviour of various financial instruments.

explain that even the most basic of financial models requires some deep mixing of techniques from various branches of mathematics.

explain why in financial mathematics, comprehension of descriptive definitions is essential and, in particular, the ability to translate these ideas to mathematical concepts is paramount.

MTH6124 Mathematical Problem Solving

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

MTH6126 Metric Spaces

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

define metric space and associated properties, and recognise these properties in specific examples;

interpret concepts from analysis of a single real variable (convergence, uniform continuity) in the context of metric spaces;

define open and closed sets, and know how they relate to continuity, etc.;

define important concepts such as compactness and completeness, recognise them in concrete examples, and use them to derive conclusions.

MTH6128 Number Theory

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

use continued fractions to develop arbitrarily accurate rational approximations to rational and irrational numbers.

work with Diophantine equations, i.e. polynomial equations with integer solutions.

know what it means to say that an integer is a quadratic reside modulo an odd prime, and calculate whether this relation is true for a given integer and prime.

know some of the famous classical theorems and conjectures in number theory, such as Fermat's Last Theorem and Goldbach's Conjecture, and be aware of some of the tools used to investigate such problems.

MTH6129 Oscillations, Waves and Patterns

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

gain a mastery of the mathematical theory of waves as applied to systems such as strings and fluids.

gain familiarity with simple concepts of nonlinear waves, such as shock formation and solitary waves.

understand the mathematical descriptions of pattern formation via wave growth by instability and in reaction-diffusion systems.

have a grasp of how techniques of calculus, such as Fourier Series, are applied in the theory of waves.

MTH6130 Probability III

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

classify states, calculate probabilities and expectations, and determine the long run behaviour, in a finite Markov chain.

extend this to infinite random walks.

explain the Poisson process and calculate waiting times between rare events.

calculate the equilibrium distributions for birth and death processes and for queuing systems.

model some real situations as stochastic processes.

MTH6132 Relativity

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

explain the principles of special relativity and the key steps leading to the Lorentz transformations.

employ a geometrical approach to special relativistic effects by using Minkowski geometry and spacetime diagrams.

understand and use four–vectors in a variety of different settings relevant to relativistic dynamics and collisions

use the techniques of tensor algebra and tensor calculus in curved (Riemannian) spaces

define the notions of covariant derivatives, connections, parallel transport, geodesics (straight lines) and curvature in curved space.

explain the importance of the metric tensor and the significance and applications of the geodesic equation.

explain the significance of the terms in the Einstein field equations and understand the Newtonian (weak field) limit of the theory.

understand and explain the applications of the General Theory of Relativity, including the Schwarzschild solution, experimental tests of the theory, black holes and gravitational collapse.

MTH6134 Statistical Modelling II

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

carry out an analysis of a completely randomised design: calculate the Analysis of Variance Table, table of means, standard errors of means and standard errors of differences, tests for fixed effects, use contrasts for equi-replicate designs and methods for unplanned comparisons, write down the design matrix for an equi-replicate design, calculate the least squares estimates and justify the ANOVA identity.

analyse a randomised block design: calculate the Analysis of Variance Table, tests for fixed effects and least squares estimates.

carry out an analysis of a factorial design: calculation of the Analysis of Variance Table for a model with two factors, interpretation of the Analysis of Variance Table for a model with three factors, describe the meaning of interaction.

explain when a model for nested factors is used.

explain the interpretation of a least squares estimate as a projection onto a subspace.

interpret Genstat output as seen in practicals and coursework.

MTH6136 Statistical Theory

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

understand the ideas of bias, sufficiency and minimum variance unbiased estimators.

show whether a distribution belongs to an exponential family.

calculate the Cramer-Rao lower bound in a given situation.

derive the likelihood and the maximum likelihood estimator and its large sample distribution, for a given probability model.

derive the method of moments estimator and the least squares estimator, for a given probability model.

obtain confidence intervals using pivots or based on the likelihood.

understand the power function and be able to find it for normal theory problems.

understand simple and composite hypotheses.

state the Neyman-Pearson lemma and apply it to find most powerful and uniformly most powerful tests.

carry out likelihood ratio tests.

MTH6138 Third-year Project

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

study independently towards the understanding of advanced material from a reading list.

independently identify and assimilate background material from a variety of sources.

compose a substantial account of a mathematical topic in an appropriate style, and including the selection and structuring the material.

make short seminar-level presentation of a mathematical topic, including the selection and structuring the material.

MTH6139 Time Series

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

understand the important features of a time plot.

understand a time series model with deterministic trend and seasonality and a stochastic component, and know the methods for eliminating trend and seasonality.

state the definitions of weak and strict stationarity, autocovariance and autocorrelation functions, for stationary time series models.

understand autoregressive (AR), moving average (MA) and ARMA models, and evaluate their properties.

understand the parameter estimation methods for ARMA models.

state the definition of an autoregressive integrated moving average model, evaluate its properties and understand the model-building steps;

interpret Minitab output as seen in practicals and exercise sheets.

MTH6140 Linear Algebra II

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

handle bilinear forms up as far as Gram-Schmidt orthogonalisation, and the simultaneous diagonalisation of two forms over the reals.

reproduce proofs of basic results of linear algebra, such as the Steinitz Exchange Lemma and the formula for multiplying a square matrix by its adjoint.