# School of Mathematical Sciences

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# Learning Outcomes 2011-12

Template

Level 7 Astronomy and Astrophysics: MTH703U-ASTM108 MTH708U-ASTM116 MTH724U-ASTM001 MTH725U-ASTM109 MTH726U-ASTM002 MTH735U-ASTM735 MTH740U-AST740P AST741P

## Template

At the end of this module, students should have:

• knowledge and understanding of:
• A1
• A2
• A3
• intellectual skills enabling them to:
• B1
• B2
• B3
• transferable skills such as:
• C1
• C2
• C3
• practical skills enabling them to:
• D1
• D2
• D3

## MTH3100 Essential Mathematical Skills

At the end of this module, students should have:

• knowledge and understanding of:
• A
• intellectual skills enabling them to:
• decompose an integer as a product of prime numbers;
• calculate the greatest common denominator and lowest common multiple of a pair of integers;
• compute quotient and remainder of integer and polynomial division(s?);
• simplify arithmetical expressions involving fractions, polynomial and rational expressions and expressions involving square roots;
• perform simple estimations and algebraic substitutions;
• solve linear and quadratic equations and inequalities.
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH4100 Calculus I

At the end of this module, students should have:

• knowledge and understanding of:
• the use of elements of set theory notation in the context of real line;
• the difference between equations and identities;
• the operation of composition of functions and the concept of functional inverse;
• the concepts of rate of change and instantaneous rate of change, indefinite integrals as anti-derivative, definite integrals and their basic properties;
• the concept of area of regions with curvilinear boundaries, be able to find area between curves;
• the concept of length of a planar curve and be able to find the length of parametric curves;
• intellectual skills enabling them to:
• solve algebraic equations and inequalities involving the square root and modulus function;
• prove simple identities and inequalities;
• 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;
• recognize linear, quadratic, power, polynomial, algebraic, rational, trigonometric, exponential, hyperbolic and logarithmic functions and sketch their graphs;
• sketch the graph of | f (ax+b)| or a f (|x|)+b when given the graph of f (x) (syllabus?)
• 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;
• 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, and also to 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; (is this a repetition?)
• recite standard indefinite integrals and basic rules of indefinite integration;
• evaluate integrals by substitution with and without suitable hints; and by a repeated use of integration by parts;
• evaluate integrals of rational functions by partial fractions;
• solve separable differential equations and first-order linear differential equations;
• 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;
• calculate volume of solids by slicing and volume of solids of revolutions;
• convert cartesian coordinates in polar coordinates and vice versa;
• state the polar equation for lines, circles, circular sectors, annuli, ellipses, parabolas and hyperbolas;
• sketch simple polar curves.
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH4101 Calculus II

At the end of this module, students should have:

• knowledge and understanding of:
• the relation between trigonometric functions and hyperbolic functions;
• what is meant for a function of two variables to be continuous, and be able to identify points of discontinuity;
• 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);
• intellectual skills enabling them 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|;
• recite Euler’s relation eiq = cos q +i sin q and De Moivre’s Theorem, and 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;
• recite the nth roots of unity and find n-th roots of complex numbers ( e.g. finding the three cubic roots of 8i.)
calculate partial derivatives, directional derivatives and estimate the rate of change in a given direction, and carry out implicit differentiation;
• recite 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, tangents and normals to level curves, tangent planes and normal lines for surfaces z = f (x,y) and f (x,y, z) = 0 and local extreme values and classify their type;
• use the method of Lagrange multipliers for finding maxima and minima of functions with one constraint;
• list 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;
• calculate Jacobians and find the transformed regions for simple coordinate transformations;
• 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 and 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;
• recite 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.
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH4102 Differential Equations

At the end of this module, students should have:

• knowledge and understanding of:
• the techniques for solving second order linear differential equations;
• the concept of phase portrait, and the different types of phase portrait, recognising the stability and instability therein;
• the relevance of stability/instability in a dynamical system (e.g. competing populations and mechanics.);
• intellectual skills enabling them 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.
calculate eigenvalues and eigenvectors of matrices, use matrix calculations to obtain solutions;
• recognise nonlinearity in systems, be able to find the fixed point of simple nonlinear systems and classify its type.
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH4103 Geometry I

At the end of this module, students should have:

• knowledge and understanding of:
• A
• intellectual skills enabling them to:
• convert between vector and cartesian equations of a straight line in R2 or R3;
• write equation of a line in R2or 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.
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH4104 Introduction to Algebra

At the end of this module, students should have:

• knowledge and understanding of:
• the difference between "necessary" and "sufficient" conditions;
• the role of abstraction and axiomatisation in mathematics;
• intellectual skills enabling them to:
• recognise and construct a valid proof, and use counterexamples to disprove assertions;
• use the relation between equivalence relations and partitions;
• perform the division and Euclidean algorithms on integers and polynomials;
• state the definitions of group, ring and field, and deduce some consequences of the axioms for these
structures.
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH4105 Introduction to Mathematical Computing

At the end of this module, students should have:

• knowledge and understanding of:
• A
• intellectual skills enabling them to:
• B
• transferable skills such as:
• C
• practical skills enabling them 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 have:

• knowledge and understanding of:
• the concepts of populations and samples and recognise different kinds of variables;
• the concepts of hypothesis testing (null and alternate hypotheses; type I and type II errors;
p-values; testing at fixed significance levels);
• 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;
• the concept of blocking as applied in the paired t-test and when this is appropriate;
• intellectual skills enabling them to:
• plot one and two-dimensional data in an appropriate way and interpret such plots;
• calculate summary statistics for a set of data;
• 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;
• 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;
• transferable skills such as:
• C
• practical skills enabling them to:
• 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 have:

• knowledge and understanding of:
• basic set notation and terminology;
• injective, surjective and bijective functions;
• the Kolmogorov axioms and how to make simple deductions from them;
• the probability mass function of a discrete random variable;
• the cumulative distribution function and the probability density function of a continuous random variable, and be able to find each from the other;
• intellectual skills enabling them to:
• 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;
• 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 and use independence to calculate probabilities;
• define conditional probability and calculate it;
• use the Theorem of Total Probability in the case of a partition of the sample space into two events;
• find the expectation and variance of discrete and continuous random variables.
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH5100 Algebraic Structures I

At the end of this module, students should have:

• knowledge and understanding of:
• A
• intellectual skills enabling them to:
• B
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH5102 Calculus III

At the end of this module, students should have:

• knowledge and understanding of:
• 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;
• Stokes’ theorem and the divergence theorem and be able to do simple problems applying these;
• the Legendre equation and its solution in Legendre polynomials, to a basic level;
• the important properties of Fourier series and be able to compute coefficients;
• the variable-separation technique for PDEs and be able to solve simple problems with Laplace’s equation in (at least) 2D Cartesian coordinates;
• intellectual skills enabling them 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.);
• "do" (complete?) simple manipulations involving gradient, divergence, and curl, and understand their geometrical/physical meaning.
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH5103 Complex Variables

At the end of this module, students should have:

• knowledge and understanding of:
• A
• intellectual skills enabling them to:
• B
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH5104 Convergence and Continuity

At the end of this module, students should have:

• knowledge and understanding of:
• define the basic concepts underlying continuous mathematics: supremum, limit of a sequence, convergent series, continuous function, derivative;
• intellectual skills enabling them to:
• 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.
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH5105 Differential and Integral Analysis

At the end of this module, students should have:

• knowledge and understanding of:
• the difference between pointwise and uniform convergence;
• intellectual skills enabling them 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.
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH5106 Dynamics of Physical Systems

At the end of this module, students should have:

• knowledge and understanding of:
• momentum, work, kinetic energy, conservative forces and potential energy;
• vector algebra, integration and differentiation of vectors and evaluate the gradient and curl of vectors;
• position vector, velocity and acceleration in both Cartesian and polar coordinates;
• Newton's laws of motion and Newton's law of gravitation;
• intellectual skills enabling them to:
• derive the conservation law of energy and use it to obtain qualitative information about motion;
• show motion near points of stable equilibrium satisfy Simple Harmonic Motion (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;
• solve simple kinematical questions;
• 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.
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH5109 Geometry II: Knots and Surface

At the end of this module, students should have:

• knowledge and understanding of:
• the concepts of knots and Reidemeister moves, and be able to define and compute such measures as writhe, tricolourability, Kauffman bracket and Jones polynomial;
• he 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;
• intellectual skills enabling them to:
• state some of the main theorems and be able to reproduce some of the shorter proofs or parts of proofs, and also be able to demonstrate understanding of the main theorems through examples.
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH5110 Introduction to Numerical Computing

At the end of this module, students should have:

• knowledge and understanding of:
• A
• intellectual skills enabling them to:
• identify the causes and sources for numerical errors;
• solve mathematical problems by programming;
• compare the size of numerical errors of algorithms;
• transferable skills such as:
• C
• practical skills enabling them to:
• construct simple routines using Maple.
• D

## MTH5112 Linear Algebra I

At the end of this module, students should have:

• knowledge and understanding of:
• A
• intellectual skills enabling them 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 = BTAT, 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 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;
• verify whether a mapping from one vector space to another is linear;
• calculate the matrix of a linear mapping from one vector space to another, 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 and, 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;
• apply the Gram-Schmidt process;
• determine bases for the row and column spaces of a matrix;
• calculate eigenvalues and eigenvectors of a square matrix;
• find, when given a real square matrix A with distinct eigenvalues, an invertible matrix P such that P  − 1A P is diagonal;
• find, when given a real symmetric matrix A, an orthogonal matrix Q such that QTA Q is diagonal;
• construct mathematical arguments in order to deduce/prove simple facts about vectors, matrices, vectors spaces and linear maps.
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH5117 Mathematical Writing

At the end of this module, students should have:

• knowledge and understanding of:
• A
• intellectual skills enabling them to:
• B
• transferable skills such as:
• the ability to write a final year project;
• writing a proof with clarity and style;
• practical skills enabling them to:
• explain an elementary concept (e.g., vectors, prime numbers), with a minimum level of confidence;
• write a summary (title, main points, abstract) of a short document on a level-4 mathematical topic.

## MTH5120 Statistical Modelling I

At the end of this module, students should have:

• knowledge and understanding of:
• the necessary matrix manipulation(?), and the underlying statistical theory, including proofs;
• intellectual skills enabling them 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;
• transferable skills such as:
• C
• practical skills enabling them to:
• 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 have:

• knowledge and understanding of:
• what is meant by a probability generating function and give suitable examples;
• what is meant by a random walk, a branching process and a Poisson process;
• what is meant by the joint distribution of continuous random variables;
• intellectual skills enabling them to:
• 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;
• define and recognise simple random walks;
• calculate, for a simple random walk on a finite interval, 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;
• 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.
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH5122 Statistical Methods

At the end of this module, students should have:

• knowledge and understanding of:
• what is meant by a uniform distribution and by marginal distributions
• what is meant by a moment generating function
• when to use different types of tests in simple situations
• intellectual skills enabling them to:
• compute probabilities from the joint probability density function
• 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
• describe a Gamma distribution and a chi-squared distribution
• apply a goodness-of-fit test and justify its use
• state and apply Chebyshev's inequality
• state and apply the Central Limit Theorem
• state the main properties of a bivariate or multivariate normal distribution.
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH6100 Actuarial Mathematics

At the end of this module, students should have:

• knowledge and understanding of:
• various life insurance products and basic principles of life assurance
• intellectual skills enabling them to:
• valuate cash flows
• valuate life annuities and benefits of life assurance policies
• transferable skills such as:
• C
• practical skills enabling them to:
• use life tables for making predictions (like future lifetime at age x) and for analysing mortality patterns

At the end of this module, students should have:

• knowledge and understanding of:
• the objectives of particular projects and decide what work you have to do to achieve them
• intellectual skills enabling them to:
• use and understand new material acquired outside the usual classroom context.
• transferable skills such as:
• independent and critical thinking
• effective time management when there are only long-term deadlines
• effective written and oral communication of your work
• practical skills enabling them to:
• D

## MTH6104 Algebraic Structures II

At the end of this module, students should have:

• knowledge and understanding of:
• the three isomorphism theorems and the Correspondence Theorem;
• composition series and soluble groups;
• the Jordan-Hölder Theorem and basic theorems about finite soluble groups;
• Noetherian rings, the two conditions equivalent to being Noetherian, Hilbert’s Basis Theorem, and other properties of Noetherian rings;
• intellectual skills enabling them 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;
• 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, and 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;
• 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 and describe simple properties of the direct product of groups and the direct sum of rings, and use these to construct examples.
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH6105 Algorithmic Graph Theory

At the end of this module, students should have:

• knowledge and understanding of:
• 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 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;
• intellectual skills enabling them to:
• use the algorithms of Prim and Kruskal to find a minimum weight spanning tree in a connected graph;
• use Dijkstra's algorithm to find a shortest path spanning tree in a graph or digraph;
• use Moravéks algorithm to find a longest path spanning tree in an acyclic directed network.
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH6107 Chaos and Fractals

At the end of this module, students should have:

• knowledge and understanding of:
• the nature of dynamical systems;
• what is meant by a diffeomorphism of the real line, and determine the possible patterns of periodic orbits;
• conjugacies between maps, and compute them in simple examples;
• what is meant by "chaos" and compute Lyapunov exponents in simple examples;
• what is meant by "fractal"; compute fractal (box-counting) dimension in simple examples;
• intellectual skills enabling them to:
• compute fixed points and periodic orbits of one-dimensional real maps, and determine the stability of these 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;
• use iterated function systems to generate fractals.
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH6108 Coding Theory

At the end of this module, students should have:

• knowledge and understanding of:
• the definition and advantages of linear codes;
• the relationship between a code and a parity-check matrix;
• intellectual skills enabling them 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;
• construct decoding processes, compute error probabilities and state Shannon's Noisy Coding Theorem;
• define, construct and manipulate generator matrices and parity-check matrices;
• decode linear codes using Slepian arrays and syndrome decoding;
• construct Hamming codes, Golay codes and MDS codes, and understand their properties.
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH6109 Combinatorics

At the end of this module, students should have:

• knowledge and understanding of:
• the Binomial Theorem;
• the principle of Inclusion and Exclusion;
• intellectual skills enabling them 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;
• 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);
• 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.
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH6110 Communicating and Teaching Mathematics

At the end of this module, students should have a better understanding of and confidence in mathematics, as well as:

• knowledge and understanding of:
• A
• intellectual skills enabling them to:
• B
• transferable skills such as:
• the abilty work in a challenging and unpredictable environment;
• assessing and devising appropriate ways to communicate a difficult principle or concept, with a broad understanding of many of the key aspects of teaching in schools;
• communication, both one-to-one and with an audience;
• understanding the needs of individuals;
• interpersonal skills for dealing with colleagues;
• staff responsibilities and conduct;
• the ability to improvise;
• giving and receiving feedback;
• organisationing, prioritising and negotiating;
• the ability to handle difficult and potentially disruptive situations;
• public speaking;
• team-working;
• standard teaching methods;
• preparation of lesson plans and teaching materials;
• practical skills enabling them to:
• prepare lesson plans and teaching materials.

## MTH6111 Complex Analysis

At the end of this module, students should have:

• knowledge and understanding of:
• the types of isolated singularities and the behaviour of functions near such singularities;
• intellectual skills enabling them 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;
• 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.
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH6115 Cryptography

At the end of this module, students should have:

• knowledge and understanding of:
• cryptography and steganography; plaintext, ciphertext, key;
• substitution and other traditional ciphers;
• stream ciphers including Vigenère cipher, one-time pad, shift registers;
• statistical attack on ciphers; Shannon’s Theorem;
• public-key cryptography: basic principles including complexity issues; knapsack, RSA and El-Gamal ciphers;
• digital signatures and authentication; secret sharing.
• intellectual skills enabling them to:
• B
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH6116 Design of Experiments

At the end of this module, students should have:

• knowledge and understanding of:
• why factorial experiments are better than one-factor-at-a-time experiments;
• the most appropriate designs for experiments on people and animals;
• intellectual skills enabling them to:
• ask pertinent questions, given a proposal for an experiment, about the aims of the experiment, the treatments and their structure, the plots and any blocks, replication, data recording, costs, resources;
• 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;
• draw the Hasse diagram for the block factors of a wide range of orthogonal designs; 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;
• transferable skills such as:
• C
• practical skills enabling them to:
• present an experimental plan in a form suitable for the experimenter.

## MTH6117 Entrepreneurship and Innovation

At the end of this module, students should have:

• knowledge and understanding of:
• entrepreneurship and the enterprise culture;
• the elements required to generate opportunities and to commercialise new ideas;
• the process involved in protecting and validating ideas;
• the importance of business and financial planning;
• intellectual skills enabling them to:
• make informed decisions about career choice and applications for jobs or further study;
• transferable skills such as:
• working through problems as an effective team member;
• critical appraisal of progress of independent work;
• independently managing learning and the use of a wide range of resources with minimal guidance;
• practical skills enabling them to:
• identify the sources available to fund ideas;
• evaluate and present outcomes through oral presentation.

## MTH6120 Further Topics in Mathematical Finance

At the end of this module, students should have:

• knowledge and understanding of:
• models appearing in financial mathematics and their relation to geometric Brownian motion;
• how techniques developed in differential equations and probability are applied to analysis of various financial models;
• how mathematical techniques can be used for for assessing the investment strategies at least in certain relatively simple but still realistic situations;
• why comprehension of descriptive definitions is essential and, in particular, the ability to translate these ideas to mathematical concepts is paramount.
• intellectual skills enabling them to:
• B
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH6121 Introduction to Mathematical Finance

At the end of this module, students should have:

• knowledge and understanding of:
• how summation techniques, differential equations and probability are needed to describe the behaviour of various financial instruments;
• how even the most basic of financial models requires some deep mixing of techniques from various branches of mathematics;
• the importance of comprehension of descriptive definitions in financial mathematics and, in particular, the ability to translate these ideas to mathematical concepts is paramount.
• intellectual skills enabling them to:
• B
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH6124 Mathematical Problem Solving

At the end of this module, students should have:

• knowledge and understanding of:
• A1
• A2
• A3
• intellectual skills enabling them to:
• B1
• B2
• B3
• transferable skills such as:
• C1
• C2
• C3
• practical skills enabling them to:
• D1
• D2
• D3

## MTH6126 Metric Spaces

At the end of this module, students should have:

• knowledge and understanding of:
• A
• intellectual skills enabling them 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.
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH6128 Number Theory

At the end of this module, students should have:

• knowledge and understanding of:
• 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;
• 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;
• intellectual skills enabling them 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.
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH6129 Oscillations, Waves and Patterns

At the end of this module, students should have:

• knowledge and understanding of:
• the mathematical theory of waves as applied to systems such as strings and fluids;
• simple concepts of nonlinear waves, such as shock formation and solitary waves;
• mathematical descriptions of pattern formation via wave growth by instability and in reaction-diffusion systems;
• how techniques of calculus, such as Fourier Series, are applied in the theory of waves.
• intellectual skills enabling them to:
• B
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH6130 Probability III

At the end of this module, students should have:

• knowledge and understanding of:
• A
• intellectual skills enabling them to:
• classify states, calculate probabilities and expectations, and determine the long run behaviour, in a finite Markov chain, and 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.
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH6132 Relativity

At the end of this module, students should have:

• knowledge and understanding of:
• the principles of special relativity and the key steps leading to the Lorentz transformations;
• covariant derivatives, connections, parallel transport, geodesics (straight lines) and curvature in curved space;
• the importance of the metric tensor and the significance and applications of the geodesic equation;
• the significance of the terms in the Einstein field equations and understand the Newtonian (weak field) limit of the theory;
• the applications of the General Theory of Relativity, including the Schwarzschild solution, experimental tests of the theory, black holes and gravitational collapse;
• intellectual skills enabling them to:
• employ a geometrical approach to special relativistic effects by using Minkowski geometry and spacetime diagrams;
• 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.
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH6134 Statistical Modelling II

At the end of this module, students should have:

• knowledge and understanding of:
• when a model for nested factors is used;
• the interpretation of a least squares estimate as a projection onto a subspace;
• Genstat output as seen in practicals and coursework;
• intellectual skills enabling them to:
• analyse 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;
• analyse 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.
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH6136 Statistical Theory

At the end of this module, students should have:

• knowledge and understanding of:
• bias, sufficiency and minimum variance unbiased estimators;
• whether a distribution belongs to an exponential family;
• the power function and be able to find it for normal theory problems;
• simple and composite hypotheses;
• intellectual skills enabling them to:
• 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;
• state the Neyman-Pearson lemma and apply it to find most powerful and uniformly most powerful tests;
• carry out likelihood ratio tests.
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH6138 Third-year Project

At the end of this module, students should have:

• knowledge and understanding of:
• intellectual skills enabling them to:
• make short seminar-level presentation of a mathematical topic, including the selection and structuring the material
• transferable skills such as:
• independent study, assimilating background material from a variety of sources
• practical skills enabling them to:
• compose a substantial account of a mathematical topic in an appropriate style, and including the selection and structuring the material

## MTH6139 Time Series

At the end of this module, students should have:

• knowledge and understanding of:
• the important features of a time plot
• a time series model with deterministic trend and seasonality and a stochastic component, and know the methods for eliminating trend and seasonality
• autoregressive (AR), moving average (MA) and ARMA models, and evaluate their properties
• the parameter estimation methods for ARMA models
• intellectual skills enabling them to:
• state the definitions of weak and strict stationarity, autocovariance and autocorrelation functions, for stationary time series 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
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH6140 Linear Algebra II

At the end of this module, students should have:

• knowledge and understanding of:
• A
• intellectual skills enabling them 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
• transferable skills such as:
• C
• practical skills enabling them to:
• D

## MTH705U-MTHM002 Applied Statistics

At the end of this module, students should have:

• knowledge and understanding of:
• A1
• A2
• A3
• intellectual skills enabling them to:
• B1
• B2
• B3
• transferable skills such as:
• C1
• C2
• C3
• practical skills enabling them to:
• D1
• D2
• D3

## MTH711U-MTHM044 Extremal Combinatorics

At the end of this module, students should have:

• knowledge and understanding of:
• A1
• A2
• A3
• intellectual skills enabling them to:
• B1
• B2
• B3
• transferable skills such as:
• C1
• C2
• C3
• practical skills enabling them to:
• D1
• D2
• D3

the relation between trigonometric functions and hyperbolic functions

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

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)