Name of Degree Programme:

BACHELOR OF SCIENCE HONOURS DEGREE IN APPLIED MATHEMATICS AND COMPUTATIONAL SCIENCE

Actual Credit Load:

528

Real number system, intervals, inequalities and their solutions, absolute value, functions, Limits and continuity of single variable functions, Differentiation: rules of differentiation, differentiation from first principles, L’Hopital’s Rule, Rolle’s Theorem, Mean Value Theorem of Differential Calculus. The module also covers: Integration – indefinite and definite integrals, integration techniques; substitution method, integration by parts, tabular integration, trigonometric substitutions, reduction formulae, Mean Value Theorem of Integral Calculus.

Vectors: Basic properties and operations. Real and complex vectors in n dimensions Linear independence, bases for  and ,the scalar product of two vectors, vector product, geometrical representation of real vectors. Complex numbers: geometric representation, algebra. De Moivre’s theorem polynomials and roots of polynomial equations. Matrices and determinants: algebra of matrices, inverses, definition and manipulation of determinants, solutions of simultaneous linear equations, applications to geometry and vectors. Differential equations: separable, homogeneous, exact, integrating factors, linear equation with constant coefficients. Application: Use of practical big data in linear equations about business problems, tax problems, economic planner’s models, and input-output matrix for economic producing transportation problems, and interpret big data analytic problems. Explore real data using eigenvalues, and eigenvectors to discover information from the big data. Applications in structural engineering, control theory problem, vibration analysis problem, electric circuits problem, and advanced dynamic problem and others.

HMTCS131 Introduction to Machine Learning

Introduction to modelling and algorithmic techniques for machines to learn concepts from data extracting meaningful patterns from random samples of large data sets. Generalization: underfitting, overfitting, cross-validation. Tasks: classification, regression, clustering. Optimization-based learning: loss minimization. regularization. Statistical learning: maximum likelihood, Bayesian learning. Algorithms: nearest neighbour, (generalized) linear regression, mixtures of Gaussians, Gaussian processes, kernel methods, support vector machines, deep learning, sequence learning, ensemble techniques. Large scale learning: distributed learning and stream learning. Common paradigms: neural networks, kernel methods and support-vector machines, boosting, nearest neighbour classifiers Applications to data mining, human computer interaction.

HMTCS132 Web Database Systems

An Introduction to Information Systems. Information gathering and requirements specification. An introduction to the main software design cycle phases: understanding the difference between requirements specification, analysis and design. An introduction to systems development using a structured methodology. Basic systems development with Data driven approach. The Unified Modelling Language (UML). ER modelling concepts, introduction to databases and SQL. Object oriented approach to systems development.
Basic concepts of objects and OOAD. SQL and database software.

 HMAT139 Mathematical Discourse and Structures

 Logic, set theory, relations, functions, algebraic structures and graph theory. Sets:  formulae,  propositions, Boolean algebra and its applications. Logic, mathematical reasoning and proof:examples taken from various areas of mathematics. Relations: binary, n-ary, reflexive, symmetric, transitive, equivalence relations and classes, partitions, order relations, inverse relations.  Functions:  one to one, onto, inverse functions. Operations: binary, n-ary closed, associative, distributive.  Structures: sets with one or two binary operations: permutations, symmetry groups, modular Arithmetic

HMAT140 Mathematical Computing and Simulation

 Introduction to ideas of mathematical modelling: testing and validating a model. Review of essential concepts in statistics. Deriving and solving models with deterministic and probabilistic demand. Models with Poisson arrival, steady-state analysis, cost analysis.  Application to simple problems of Operational Research including business and physical systems (e.g. queueing, inventory control).
  Using software such as spreadsheets for calculation and data presentation.  Introduction to computer programming for mathematical work: routines, functions, control structures for selection and repetition. Designing a computer program, testing and debugging. Mathematical logic and truth tables.  Introduction to mathematical software: MATLAB, Python, Mathematica, Minitab and R. Any other suitable mathematical software will be included. Introduction to software capable of performing simulation. Case Study: Group work to use Operational Research techniques to model and solve realistic business and analyse the output in a professional manner.

HMTCS141 Introduction to Group Theory

Group theory (An introduction): Axioms and examples of groups and subgroups. Vector Spaces (A more abstract and subspaces over arbitrary fields) Linear dependence and independence.  Basic and Dimension.  Linear conformation. Rings and Fields:  Axioms and examples of rings and fields. Application of group theory (Atomic and molecular spectroscopy,  molecular orbital theory, computer graphics, medical image analysis, cryptography and public key algorithms)

HMAT133 Linear Algebra II

Vector spaces, linear dependence and independence, bases and dimension. Inner product spaces. Basic definitions with examples, the notion of norm and distance, the Cauchy-Schwarz inequality, the Gram-Schmidt orthogonalisation process, orthonormal basis. Linear transformations. Operations on linear operators, algebra of operators. Eigenvectors, eigenvalues, orthogonality of eigenvectors, geometric and algebraic multiplicity of eigenvalues. Applications of diagonalisation of matrices, quadratic and bilinear forms, Jordan Normal form of a matrix, solutions of systems of differential equations. The Cayley Hamilton Theorem and its applications. Revision of the basic techniques for solutions of first and second order differential equations.

HSTA139 Probability and Statistics

Counting rules used in Probability: summation notation, product notation, random experiments, and basic principles of counting. Sets and events: operations with sets and subsets, De Morgan’s rules, sample spaces and events. Probability: concept of probability, conditionally probability, law of total probability, properties of probability measure. Random variables: random variables and probability, discrete and continuous random variables. Special probability distributions: Expectation and moments: Chebyshev’s inequality. Generating functions: probability generating function and its properties, moment generating function and its properties, Normal approximation, Laws of large numbers. Joint probability distributions: discrete and continuous random vectors, independence, conditional distributions, covariance and correlation. Bayesian inference with known priors, probability intervals, conjugate priors, Bayesian inference with unknown priors, Frequentist significance tests and confidence intervals, Resampling methods: bootstrapping, Linear regression. Application: Classifying data sets into categories that describe the shape of the data distribution. Graphical visualization ranging from simple graphics such as histogram, boxplot, and scatterplot to advanced graphics such as PCA projection plots, trellis plots, maps, etc. Students must use available software packages to explore big data using graphics to discover information from the real data.

HMTCS133 Web Programming                                                                                                                  

Review of HTML/CSS and HTML Forms. Client / Server relationship – HTTP . Client-side scripting and CSS frameworks. Relational databases: creating, updating and normalisation in MySQL, Server-side PHP PDO programming. Website security including basic data encryption. Understanding of Content Management Systems. Accessibility and user-testing. Review of JavaScript. Introduction to Node JS and server-side JS Intro to Mongo DB and NoSQL ExpressJS , VueJS. Client / Server relationship – HTTP Accessibility and user-testing.  Ethical, Legal and Social Issues.

HMAT211 Ordinary Differential Equations

Modelling with first order ODEs in population dynamics, and second order ODEs (mass-spring systems, RLC circuits). Methods of undetermined coefficients, reduction of order and method of variation of parameters. Existence and uniqueness of solutions, revision of continuous functions and Lipschitz conditions. Series solutions of ODEs, solutions near ordinary and singular points. Systems of linear first order ODEs, solution and stability. Differential equations of special functions. Laplace transforms and inverse laplace transforms, applications to solving IVPs, Heaviside and Dirac functions.

 Application: Use the practical big data for differential equations such as simple chemical conversion problems, growth of population problems, price of commodities models, Newton’s law of cooling problems, and many physical problems.

HMTCS231 Information Security and Visualisation                                                                                    

Fundamental principles of information security and concepts. Security risk assessment and controls for different systems and organizations. Algorithms and procedures covering cryptography and its applications. Principles of secure design. Approaches to cyber security. Fundamental concepts in information visualisation.  Good design practices in information visualisation.  Different types of information visualisation and the options for using them.  Interactive tools for information visualisation and dashboards.  Visual analytics for identifying trends and patterns in datasets.  Practical experience of data exploration.

HMTCS232 Advanced Programming                                                                                                      

Programming with threads; Using reflection; Creating software components; Test-driven development (e.g. with JUnit); Version control and build tools (e.g. Git); Creating Generic classes; Design patterns; Building libraries. Lambda expressions

HMAT241 Analysis 1

Revision of elementary functions including hyperbolic functions and their inverses. Limits and continuity of functions. Techniques of differentiation. Higher order derivatives. Applications to curve sketching and extreme value problems. Taylor series. L’Hopital’s rule. Methods of integration. The fundamental theorem of calculus. Applications of the integral. Complex exponentials. De Moivre’s theorem. Historical development of the real number system: Countability, cardinal numbers, existence of transcendental numbers. The real numbers as a complete, ordered field. Supremum axiom, Archimedean property, principle of monotone bounded convergence, nested interval theorem, Bolzano-Weierstrass theorem, convergence of Cauchy sequences. Limits and continuity of real functions. Boundedness theorem, intermediate value theorem, interval theorem, application to fixed –point theorem, Uniform continuity. Differentiability, Local extrema, Rolle’s theorem, mean – value theorem, L’Hospital’s rule, Leibnitz’s theorem. Taylor’s theorem. Applications to finding roots, curve sketching, classification of local extrema, and approximation by polynomials. The Riemann integral, Integrability, properties of the Riemann integral, the mean-value theorem for integrals, the fundamental theorem of calculus.

HMAT223 Algebra

Sets and relations, mapping, binary operation including closure of sets.  Equivalence relations. Identities, inverse and zero divisors. Group theory (An introduction):  Axioms and examples of groups.  Subgroups. Homomorphisms and isomorphisms between groups.  Automorphisms of a group. Vector Spaces (A more abstract and subspaces over arbitrary fields) Linear dependence and independence.  Basic and Dimension.  Linear conformation. Rings and Fields:  Axioms and examples of rings and fields.

HMAT248 Numerical Methods

Introduction to simple numerical methods for solving problems in Mathematics and Science; Computer arithmetic and rounding errors; Root location in one- and higher-dimension (Newton-Raphson, Bisection), Numerical Linear Algebra (Gaussian elimination, LU decomposition, Gauss-Seidel method, Eigenvalue and Eigenvector iterative methods). Polynomial interpolation and splines; Solution of linear algebraic equations, scaled partial pivoting; Solutions to Ordinary Differential Equations including Initial Value Problems (Euler’s method, Runge-Kutta methods) and Boundary Value Problems (tridiagonal system and Thomas algorithm for tridiagonal matrices), Solutions to Partial Differential Equations (finite-difference methods). Examples from applications (e.g. engineering, economics, internet search engines, bio-sciences).

HMAT210 Calculus II

Theorems on differentiation, higher order derivatives and Leizbnitz’s formula. The mean value theorems. Rolle’s theorem, the mean value theorem, the generalised mean value theorem. Taylor’s theorem. Applications to maxima and minima, curve sketching,

HMTCS231 Introduction to Partial Differential Equations and Fourier Series

Partial differential equations of mathematical physics and introduction to their study. Classification of second order partial differential equations in two independent variables. Derivation of the wave, Laplace and Poisson equations, method of separation of variables and Laplace transform techniques. Orthogonal sets of functions in an inner product space. Introduction to Hilbert spaces. Fourier sine and Fourier Cosine series. Discussion of convergence theorem. Integration and differentiation of Fourier series. Application of Fourier series to boundary value problems. Fourier series in two variables. The Fourier transform and its inverse. The convolution theorem. Applications. Bessel functions J(x). The zeros of J(X). Orthogonal sets of Bessel functions. Fourier Bessel series. Applications of the theory to the solution of partial differential equations will be stressed throughout.

HMAT224 Complex Analysis/Variables

Analytic Functions. Cauchy-Riemann Equations. Conformal Mappings. Line integrals. Cauchy’s Integral Theorem and Formula. Power series. Taylor series, Laurent series. Zeros and singularities. The Residue Theorem. Evaluation of real integrals.

HMAT214 Vector Calculus

Brief review of line: surface and volume integrals and applications to work done, flax through surfaces etc. Derivatives of Vector Functions. Directional Derivatives. Gradient of Scalar Fields. Divergence and Curl of Vector Fields. Constrained extremal problems. Line and surface integrals. Green’s Theorem, Gauss’ Divergence Theorem, Stokes’ Theorem and their applications. Orthogonal Curvilinear Coordinates. The Laplacian in polar, cylindrical and spherical coordinates.

HMTCS270 Mini Research Project

The mini research project involves the specification, design and implementation of a piece of software or hardware or the use of existing software to develop some computational mathematical application. Work may be drawn from one of several proposed research themes and is guided by an academic supervisor. The mini research project aims to develop research, analytic and report writing skills.

HMAT424 Optimization Techniques

Formulations of combinatorial optimization problems, greedy algorithms, dynamic programming, branch-and-bound, cutting plane algorithms, decomposition techniques in integer programming, approximation algorithms. Nonlinear Optimization A course on the fundamentals of nonlinear optimization, including both the mathematical and the computational aspects. Necessary and sufficient optimality conditions for unconstrained and constrained problems. Convexity and its applications. Computational techniques and their analysis. Deterministic OR Models: An applications-oriented approach illustrating how various mathematical models and methods of optimization can be used to solve problems arising in business, industry, and science. Computational optimization methodologies underlying portfolio problems in finance. Computational linear algebra, determining derivatives, quadratic, and nonlinear optimization. The efficient frontier problem. Applications:  Use and interpret practical big data in business/finance optimization, economic optimization and other economic producing problems, and interpret big data analytic problems.  

HMAT433 Partial Differential Equations

Second Order PDE’s and Method of Separation of Variables derivation of the wave and diffusion equations, solutions of second order equations, initial and boundary conditions, solution of the wave equation, Laplace’s equation, wave equation and the heat flow equation. Parabolic Partial Differential Equations- equation for time-dependent heat flow, explicit method, Crank-Nicolson method, the Theta method, parabolic equations in Two or Three dimensions. Hyperbolic Partial Differential Equations, wave equation. Introduction to solution of PDEs using finite differences and finite elements. Solution procedures: elliptic equations — Green functions, conformal mapping; hyperbolic equations — generalized d’Alembert solution, spherical means, method of descent; transform methods — Fourier, multiple Fourier, Laplace, Hankel (for all three types of partial differential equations); Duhamel’s method for inhomogeneous hyperbolic and parabolic equations.

HMTCS440 Mathematical Finance

Theory of interest rates: simple interest, compound interest, nominal rates of interest, accumulation factors, force of interest, present values, Stoodley’s formula for the force of interest, present values of cash flows. Basic compound interest functions: the equation of value and yield of a transaction,  annuities certain, present values and accumulations, deferred annuities, continuously payable annuities, the general loan schedule, the loan schedule for a level annuity. Nominal rates of interest: annuities payable , annuities payable , present values and accumulations, annuities  payable at intervals of time r, where r>1, the loan schedule for a pannuity. Discounted cash flow: Net cash   flows, Net present values and yields, comparison of two investment projects, different interest rates for lending and borrowing, effects of inflation, the yield of a fund, measurement of investment performance. Capital redemption policies: Introduction to premium calculations, policy values, policy values when premiums are payable p, surrender values, paid up policy values and policy alterations, variations in interest rates, Stoodley’s logistic model for the force of interest, reinvestment rates.Application: Use of practical big data in the financial sector.

HMTCS441 Mathematics Computing for Life Sciences

Population modelling – predator-prey models, dynamic modelling, simulation, analysis of behaviours. Lotka-Volterra equations and other population models. Modelling the spread of disease.  The mathematics of DNA. The topology of viruses. Evolutionary concepts: kin selection, group selection. Mathematics of evolution: game theory, Hawk-dove models, evolutionary steady states.  The Prisoner’s Dilemma as a model for the evolution of altruism. Tit for tat and other strategies for the Iterated Prisoner’s Dilemma. The Tragedy of the Commons. Modelling co-operative behaviour. Cycles of behaviour patterns. Punishment and reward. Trust and reputation. “Game Theory can save the world” (Nowak and Highfield). Climate change as a case study.

HMAT442 Real Analysis

Sets. Algebra of sets, Borel sets. Topology of real line. Outer measure on R, its properties. Measurable sets and Lebesgue measure. Properties of Lebesgue measure. The sigma algebra of measurable sets. The sigma algebra of Borel sets. Example of a non-measureable set. Measurable functions, equivalent definitions, properties. Convergence almost everywhere, convergence in measure. Lebesgue integral. Properties. Classes of integrable functions. Comparison with Riemann integral. Bounded convergence theorem. Lebesgue theorem. Fatou’s lemma. Monotone convergence theorem. Lebesgue convergence theorem. Absolutely continuous functions. Product measures. The Fubini theorem. Completion of product measures. Distribution functions. Integrating on product spaces. Introduction to linear vector spaces. Normed spaces. spaces, L∞space. The Holder and Minkowski inequalities. Convergence in LP. Comparison with other kinds of convergence. Measure spaces.

HMTCS442 Numerical Computations for Financial Modelling

Introduction to the mathematical models used in finance and economics with emphasis on pricing derivative instruments. Financial markets and instruments; elements from basic probability theory; interest rates and present value analysis; normal distribution of stock returns; option pricing; arbitrage pricing theory; the multiperiod binomial model; the Black-Scholes option pricing formula; proof of the Black-Scholes option pricing formula and applications; trading and hedging of options; Delta hedging; utility functions and portfolio theory; elementary stochastic calculus; Ito’s Lemma; the Black-Scholes equation and its conversion to the heat equation.

HMTCS443 Coding and Cryptography

Introduction to coding theory. Parity checking, error detection and correction, code bounds. Examples of simple codes such as Hamming codes and Hadamard codes. Linear codes, polynomial codes and cyclic codes, including historical and modern examples. Modern applications such as Reed-Solomon codes. Introduction to cryptography. Historical ciphers, including substitution ciphers, stream ciphers (e.g. linear shift register) and block ciphers (e.g. DES/AES). Public-key cryptography, including the mathematical theory behind the ciphers (e.g. RSA, elliptic curves). Privacy, authentication, integrity and non-repudiation. Digital signatures and PKI infrastructures. Future developments, including quantum computing and quantum cryptography.

HMTCS444 Database Management and Administration

Exploring the database architecture; Creating the database; Managing the database instance; Managing database storage structures; Managing transactional processing and locking mechanism; Database security; Monitoring the database and using the advisors; Backup and recovery concepts; investigating, reporting, and resolving problems.

HMTCS446 Applications of Graph Theory

Introduction to graphs and digraphs, walks, paths, trails and cycles. Definitions and characterization of classes of special graphs. Distance and connectedness measures.  Eulerian and Hamiltonian graphs. Connectivity – Whitney’s theorem and Menger’s theorem. Planarity – Kuratowski’s theorem and Euler’s theorem. Colourability – 5-colour theorem, 4-colour theorem. Algorithms – Greedy algorithm, shortest path algorithms, Fleury’s algorithm. NP completeness. Applications in computing, engineering, finance, mathematical biology, social networks.

HMTCS447 Human Computer Interaction and Design

Design Process, Prototyping, Prototyping Tools, User Research, Research Methods, Evaluation of HCI solutions, Usability testing, Cognitive Psychology, Information Processing, HCI methodologies, standards and guidelines, Multi-Modal Interactions, Multi-Sensory Interactions, Tangible User Interfaces,  Brain Computer Interfaces,  Natural User Interfaces, Dialog Systems, Metaphors, Conceptual Models, Relevant legal, social, ethical and professional issues.

HMTCS410 Stochastic Processes for Applied Mathematics

Random variables, expectations, conditional probabilities, conditional expectations, convergence of a sequence of random variables, limit theorems, minimum mean square error estimation, the orthogonality principle, random process, discrete-time and continuous-time Markov chains and applications, forward and backward equation, invariant distribution, Gaussian process and Brownian motion, expectation maximization algorithm, linear discrete stochastic equations, linear innovation sequences, Kalman filter, various applications.

HMAT436 Mathematical Programming

Introduction to mathematical programming problems: Linear programming problem formulation; simplex method; Chmens and two phase techniques; sensitivity analysis; duality in LP; Dual simplex method; transportation and assignment methods; integer programming; dynamic programming; quadratics and separable programming; K T conditions for optimality.

HMAT444 Mathematical Modelling

This module introduces mathematical modelling in a variety of contexts including using Newton’s laws of motion, Newton’s law of gravitation, population models, exponential growth, density dependent growth, and predator-prey models. Deriving differential equations from big data; dimensional analysis; discrete models and difference equations: steady states and their stability; continuous models and ordinary differential equations: steady states and their stability; the slope fields and phase lines;  applications of Linear Algebra (in lower dimensions): systems of linear ordinary differential equations; linear phase plane analysis and stability; electrical networks; vector algebra, vector geometry, vector equations, coordinate systems and vector differentiation; application in mechanics: Newton’s laws for a single particle in 3-D; conserved quantities; angular velocity, angular momentum, moment of a force; harmonic motion.

HCSCI132 Principles of Programming Languages

Refer to Department of Computer Science and Computer Engineering. 

CS131 Communications Skills

Refer to Communication Skills Centre.

GS231 Gender Studies

Refer to the Gender Institute.

Refer to Entrepreneurship Department.