Vibrations are the oscillation of a mechanical structure. Vibration may be desirable as in the strings of a guitar or in the human vocal cords. More often vibrations are undesirable as for the vibrations of an electrical motor or of an entire car. In both case modelling can inform the designer so that vibration can be precisely obtained or avoided. Although the optimal and cost effective way to minimise the vibration of a structure is by careful engineering early in the design cycle, frequently the engineer must turn to palliative measures to control vibration at a stage in the design when even minor modifications to the structure are prohibitively costly or detrimental to other performance targets. The general aims of this module are to introduce students with little or no previous experience of mechanical vibrations, and with quite different backgrounds, to the basic concepts of vibrational behaviour, to provide a general introduction to vibration modelling, analysis and control and to give students some experience of vibration measurement. This module also promotes the principles which can influence the design process of mechanical structures and it presents a number of commonly adopted techniques for trouble-shooting vibration problems.
Number Theory is the study of integers and their generalisations such as the rationals, algebraic integers or finite fields. The problem more or less defining Number Theory is to find integer solutions to equations, such as the famous Fermat equation x^n + y^n = z^n. In this module we build on the group, ring and number theoretic foundations laid in MATH1001, MATH2003 and MATH3086. We will first prove a structure theorem for the group of units modulo n. We then move on to the famous Gaussian Quadratic Reciprocity Law which yields an algorithm to decide solvability of quadratic equations over finite fields. Using geometric as well as algebraic methods, we will then characterise which integers can be written as the sum of two and four squares, respectively. The former leads us naturally to the study of binary quadratic forms, a central topic of this module. In the final part of this module, we will explore rings of integers in algebraic number fields; they generalise the role the integers play within the rational numbers; the simplest new example is the ring of Gaussian integers, Z[i]. We will investigate to what extent certain central properties of the integers, such as unique prime power factorisation, generalises to these rings. The deviation from unique prime factorisation is measured by the so-called ideal class group, probably the most important invariant of algebraic number fields. It can be seen that it is finite and that its order for quadratic number fields is intimately related to the number of equivalence classes of quadratic forms introduced earlier in the module.
Number Theory is the study of integers and their generalisations such as the rationals, algebraic integers or finite fields. The problem more or less defining Number Theory is to find integer solutions to equations, such as the famous Fermat equation x^n + y^n = z^n. In this module we build on the group, ring and number theoretic foundations laid in MATH1001, MATH2003 and MATH3086. We will first prove a structure theorem for the group of units modulo n. We then move on to the famous Gaussian Quadratic Reciprocity Law which yields an algorithm to decide solvability of quadratic equations over finite fields. Using geometric as well as algebraic methods, we will then characterise which integers can be written as the sum of two and four squares, respectively. The former leads us naturally to the study of binary quadratic forms, a central topic of this module. In the final part of this module, we will explore rings of integers in algebraic number fields; they generalise the role the integers play within the rational numbers; the simplest new example is the ring of Gaussian integers, Z[i]. We will investigate to what extent certain central properties of the integers, such as unique prime power factorisation, generalises to these rings. The deviation from unique prime factorisation is measured by the so-called ideal class group, probably the most important invariant of algebraic number fields. It can be seen that it is finite and that its order for quadratic number fields is intimately related to the number of equivalence classes of quadratic forms introduced earlier in the module. One of the primary domains where number theory finds applications is cryptography. We will study some of the famous cryptosystems where number theory has applications. In particular, Rabin cryptosystem, Goldwasser-Micali cryptosystem, lattice based cryptosystems, elliptic curve cryptography are among those.
This module aims to expand your statistical toolbox by exposing you to a broad set of modelling techniques to employ with data that would not satisfy the assumptions of the mainstream Linear and Generalized Linear models. The first half of the module will follow an explanatory approach, introducing Multilevel (Mixed effects) and Marginal models to understand and deal with the type of correlation found in hierarchical and longitudinal data. The second half of the module will introduce a set of modelling techniques widely used in the predictive approach, such as non-parametric regression, Generalized Additive Models (GAM), Penalized Regression or Classification and Regression Trees (CART).
This module provides a broad introduction to more advanced regression methods such as multilevel models, non-parametric and penalised regression and Generalized Additive models. The module assumes that students are familiar with basic regression techniques such as Linear Regression and Logistic regression.
This course aims to introduce some advanced techniques that hold potential for applications in the future generations of wireless communication systems. Currently, research and development in wireless communications is focused on the sixth generation (6G), which is expected to significantly enhance 5G in both techniques and services. This course will cover several candidate techniques designed to enable 6G wireless systems. The course begins by covering the principles of cooperative communications. Various relay/cooperation protocols are considered and analysed to demonstrate their advantages and challenges. Next, it focuses on non-orthogonal multiple access (NOMA), a technique that allows densely deployed users (or devices) to simultaneously transmit their information. Subsequently, the course addresses the principles of full-duplex communication, exploring the challenges of self-interference and corresponding self-interference cancellation techniques, as well as examining the potential of full-duplex for wireless system design. Then, it introduces integrated sensing and communication (ISAC), providing several examples to explain the principles and illustrate the design trade-offs. A review of the fundamentals of MIMO is then provided, followed by analysing the potential of MIMO for meeting the requirements of future wireless systems. A range of technical options for MIMO transceiver optimisation are discussed. Built on the above theoretical foundation, the course then covers the multi-user MIMO and massive MIMO, with the emphasis on their principles, characteristics, and implementation challenges. Finally, the course covers millimeter wave (mmWave) communications. It begins with an overview of mmWave technology, then characterizes mmWave channels, highlighting key differences from conventional radio frequency (RF) communication channels. The course concludes with an introduction to several advanced techniques for the design and optimization of mmWave systems.
In the last 30 years derivatives have become increasingly important in finance and many different types of derivatives are actively traded on exchanges throughout the world. This module explores the pricing and use of forwards, futures and options with a particular focus on contracts where the underlying asset is a financial asset - for example, a stock index (i.e. stock index futures or stock index options). Students will learn how to price these derivatives using various techniques as well as understand how we can use them for (i) speculation, (ii) hedging strategies and (iii) arbitrage. The nature of the subject makes the module more suitable for students with a solid background in mathematics and familiarity with differential calculus and systems of equations.
We will start from outlining fundamental questions we must answer in order to build up a picture of an astrophysical object, e.g., what is it made of? How luminous? How big? How old? How fast? How heavy? These seemingly simple questions are surprisingly difficult to answer but we will cover the different astrophysical tools used to answer them. We will then move outwards to consider the demography, spatial distribution, and environment of galaxies, in the ‘field’ and in clusters. We will then consider galaxies very distant from us in space and time, discuss galaxy formation and evolution, and have an overview of Active Galaxies, super-massive black holes and their co-evolution with their host galaxies.
This module is designed for students in their third year and aims to introduce the basic concepts and techniques of Galois theory, building on earlier work at level 2. As such, it will provide an introduction to core concepts in rings, fields, polynomials and certain aspects of group theory. Galois theory arose out of attempts to generalize to polynomials of higher degree the well-known formula for the roots of a quadratic polynomial. This turns out to be possible for cubic and quartic polynomials but impossible for polynomials of degree five or more. This impossibility result is one of the main applications of Galois theory. Further applications to be considered are ruler-and-compass constructions; for instance, we determine all natural numbers n for which the regular n-gon can be constructed. Much of this beautiful and fascinating theory was discovered by the French mathematician and revolutionary Évariste Galois, shortly before he was killed in a duel in 1832, aged twenty. It has considerably influenced the development of Algebra and is nowadays a basic tool also in Number Theory and (Algebraic) Geometry. For instance, it features prominently in the famous proof of Fermat's Last Theorem by Andrew Wiles in the 1990s. The main theorem of Galois theory gives a correspondence between the intermediate fields of a finite extension L/K of fields on the one hand and the subgroups of the automorphism group G = Aut (L / K) on the other hand. In particular, this module will introduce the concepts of rings and fields including, for example, the notions of polynomial rings, ideals, quotient rings and homomorphisms, building on material from MATH2046 Algebra and Geometry. Some group theory is also assumed, such as normal subgroups, quotient groups, and familiarity with permutation groups. These topics are all covered in MATH2003 Group Theory, which is also a pre-requisite for this module. On successful completion of the module the students should be able to: • show familiarity with the concepts of ring and field, and their main algebraic properties; • correctly use the terminology and underlying concepts of Galois theory in a problem-solving context; • reproduce the proofs of its main theorems and apply the key ideas in similar arguments; • calculate Galois groups in simple cases and to apply the group-theoretic information to deduce results about fields and polynomials. One of the pre-requisites for MATH3078 and MATH6156
Games design and development is an increasingly important and sophisticated topic, that draws together many of the core aspects of Computer Science and Software Engineering. This course introduces students to the fundamentals of game design, gives them practical experience in developing games within an industry-leading contemporary games framework, and encourages students to consider the wider possibilities of digital entertainment through non-linear narratives and innovative gaming forms.
In the Game Mechanics module, you will delve into the concepts of game mechanics, exploring both mechanics as actions and mechanics as rules. This module combines theoretical understanding with practical application through various hands-on activities.
In the Game Pitching and Presenting module, you will take the work you are undertaking in the Game Project Proposal module and pitch and present your idea to your peers and staff at regular intervals during the various stages of development. The structure of these presentations and pitches will be determined by staff and will follow a pattern that allows you to use this process to inform and support the creation of a strong project proposal. By the end of the module, you will have a series of presentations and materials documenting the process of idea generation and refinement, closely linked to the work within the Games Project Proposal module.
In the Games Project Proposal module, you will work to develop a unique game idea for the module Major Project – Unique Impact Games. The game proposal you develop will start as an individual effort, but you will have the option to form teams around each proposal, encouraged to do so organically. The module will include workshops on how to form a proposal around the concept of a unique impact game, with each workshop building on previous techniques to produce a unique game idea that you will take into production as a solo designer/developer or in your team. The proposed game must be a complete game experience, not just a technical demo, and should have a unique experience and concept that can be conveyed in a finished game. You will be encouraged to think big, lean on your passions, developmental and current needs, and create a project you will make, market, and release in the module Major Project – Unique Impact Games.
In the Game Worlds module, you will embark on a series of creative projects focused on generating and refining ideas for imaginative worlds and settings. This module combines field trips, workshops, and various design and art methodologies to help you create compelling game worlds and associated materials.
This module gives a comprehensive analysis of the modelling of strategic behaviour in modern economics. It will familiarise students with the central concepts in game theory, covering choice in strategic situations under different informational environments. The module provides ample examples and applications for the theories developed. Throughout the module a focus is placed on evaluating outcomes with strategic behaviour with respect to their social desirability and on possible policies to improve upon the outcome. Students will learn to model strategic behaviour using games and evaluate the design of games from a welfare and policy perspective.
This module gives a comprehensive overview of the modelling of strategic behaviour in modern microeconomics, building on the foundations laid in Intermediate Microeconomics. It will familiarise students with the central concepts in game theory, covering choice in strategic situations under different informational environments, including asymmetric information. The module provides ample examples and applications for the theories developed. Throughout the module a focus is placed on evaluating market outcomes with strategic behaviour with respect to their social desirability and on possible policies to improve upon the market outcome. Summarising, students will learn to model strategic behaviour using games, how to best play games and evaluate the design of games.
In the Games as Poems module, you will explore the use of poetry as a lens for game creation. This approach emphasizes a robust understanding of the basic building blocks of games, and ways in which these building blocks can be reconfigured to help players slow down, pay attention, and engage with complex emotions and encounters.
In the Games Design module, you will embark on an individual journey to create a series of documentation, including production planning, design specifications, and core player experience and market analysis, culminating in a video game internal pitching deck. This deck will detail what the game is, provide a story outline or synopsis, game play pillars, game loop details, key art, key prototypes, player experience, and explain why this game is exciting. The presentation of this deck is designed to support a game director in approving projects for further development.
In the Games Design & Culture module, you will delve into a wide array of cultural studies that have shaped and continue to influence game design. By engaging in "thinking through making," you will explore and respond to these cultural and theoretical influences through practical activities.
During the Game Development modules, you will build on a Semester 1 game design pitch to create a prototype that deeply explores your game concept. This interactive prototype will showcase a high level of gameplay and provide a solid proof of concept, addressing complex components such as the game loop and the most challenging aspects of the game design.
