Mathematics BSc (Hons)

Many optimisation problems in business and industry can be expressed in the form of a linear programming problem and this is even more apparent with increasing reliance on Artificial Intelligence and Machine Learning.  Businesses and industry use linear programming to determine what to make in order to maximise their profits, Amazon use it to schedule your parcels for delivery and it is also used widely in Game Theory: you can use it to beat your friends at rock-paper-scissors and other games!

In this module, we will study the theoretical background behind the linear programming methods, learn how to express real-world questions as linear programming problems and solve them by hand and using computer programs.  We will also explore some other optimisation methods used in AI and Machine Learning. Topics may include:

• Canonical forms of linear programming problems.
• Theoretical considerations: relevant results from set theory and geometry.
• Integer linear programming.
• Solutions of linear programming models: Simplex and dual simplex methods, Pivot algorithm, and computer-based techniques.
• Degeneracy, cycling, duality.
• Use of a mathematical computer software package, for example, Python, Matlab, Excel etc.
• Application to logistics in transportation and assignment problems, VAM and the Hungarian algorithm.
• Game theory: zero-sum matrix games, multi-phase games.
• Non-linear optimisation techniques in Machine Learning

Stochastic processes serve as essential mathematical models for systems and phenomena exhibiting apparent randomness. Examples encompass diverse scenarios, such as the growth of a bacterial population, fluctuations in electrical current due to thermal noise, or the motion of gas molecules. The applications of stochastic processes span various disciplines, including biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, and financial markets. In order to understand such random behaviour, we introduce and study Markov chains, random walks, Brownian motion and stochastic differential equations. Through these topics, students will not only establish a robust foundation in the principles of stochastic processes but will also gain valuable insights into their diverse applications across numerous domains. The module's goal is to equip students with analytical tools essential for comprehending and modelling complex uncertainties, thereby enhancing their capacity to address real-world challenges in mathematics, statistics, and related fields.

Topics covered include:

• Brief review of Probability Theory via a Measure Theory approach.
• Martingales: Basic definitions, filtrations, stopping times.
• Doob's Martingale inequalities and Convergence Theorem.
• Markov Chains.
• Moment generating functions.
• Characteristic functions.
• Probability generating functions.

Although geometric intuition allows us to understand some simple calculus (differentiation and integration), when things become more complicated we need a more rigorous foundation. Mathematical analysis provides this framework and indispensable to understanding applied techniques such as numerical methods and solving differential equations. By treating the notions of distance and limit formally, this course develops a rigorous theory of convergence, differentiation and integration of real-valued functions.

Topics may include:

• Uniform continuity
• Uniform convergence
• Darboux integral
• Fundamental Theorem of Calculus
• Analysis of functions of more than one variable

Geometry is probably the most accessible branch of mathematics, and can provide an easy route to understanding some of the more complex ideas that mathematics can present. This module intends to introduce students to the major geometrical topics taught at undergraduate level, in a manner that is both accessible and rigorous. The course will consider Euclidean geometry and solid geometry, covering topics such as the equation of lines and planes, inner product, cross product, and triple product. Spherical geometry, featuring concepts including great circles, geodesics, and spherical trigonometry, will be explored for its significant applications in fields like astronomy, navigation and understanding the Earth. Additionally, the study will extend to quadratic curves, quadratic surfaces, and planetary motion.  Through this comprehensive study, students will understand the various geometric concepts, laying a foundation that combines accessibility with the necessary mathematical rigour. 

Topics in Geometry are drawn from the following:

• Geometry of numbers (including the golden ratio and continued fractions).
• Coordinate geometry: lines, planes and hyperplanes.
• Euclidian geometry.
• Solid geometry: vector products, polyhedra.
• Projective geometry: basic concepts & applications.
• Spherical geometry.
• Use of appropriate software to visualise problems (such as Autograph or Matlab, for example).
• Real-life applications of geometry: examples drawn from a wide range of contexts (e.g. cartography, forensic science, navigation, architecture).

This is an experiential learning opportunity that incorporates 35 teaching contact hours/lectures to prepare for the 150 contract hours. L5 students can use all their skills learned to date on an actual real-world (external business) client-driven project, working in teams and producing an artefact. 

You are also expected to undertake around 215 hours of self-study.

This module not only gives them enhanced skills but the opportunity to work for a real client thus giving them a valuable CV and LInkedIn entry as work experience that can contribute to their employability portfolio. 

You will collaborate in teams and produce full client documentation alongside a reflection of their expereince and this all give some much needed contemplation of their skills to date and how to use them. 

This module provides a structured, university-level work placement for 4, 5 or 7 weeks (optional timeframe decided on a 1 2 1 basis) as one continuous block / period with a placement provider (i.e. industry apprioprate sector). It is designed to enhance your professional skills in a real-world job setting.

The placement can either be organised by you or with support from university staff.

All work placements within this module must be university-level; this means:

• Undertaking high-skilled work commensurate with level 5 study (e.g. report writing, attending meetings, delivering presentations, producing spreadsheets, writing content on webpages, social media, marketing services/products etc)
• Physically placed (albeit part of it can be hybrid) within an employer setting in one continuous block / period for 4, 5 or 7 weeks for a minimum of 140-147 hours over the course of the entire work placement

Where applicable, your existing part-time employer can be approached/used as the placement provider, if the high-skilled work.

• criterion above is fulfilled for the full duration of the placement.
• All quality assurances/agreements provided by the University are adhered to, by you and the employer.

The work placement context may not necessarily, reflect your degree discipline per se, but rather, it will give you an enriched experience to enhance your professional skills in a real-world job setting.

The module will provide the opportunity to further develop your language skills, building on your previous learning at advanced level. The second half of the module includes a placement abroad or, alternatively, a project on a sustainability issue in a target language country. The first half of the module will prepare you for placements abroad where appropriate as well as a deeper understanding of sustainability in target language contexts. Students of more than one language may take one language in the first half of the module and spend their time abroad developing a different language. 

The module will provide the opportunity to further develop your language skills, building on your previous learning at intermediate level. The first half of the module includes intensive taught sessions in interactive workshop mode which will prepare you for placements abroad or self-directed language development. The second half of the module includes an placement abroad or, alternatively, a project on a business or tourism issue in a target language country. Students of more than one language may take one language in the first half of the module and spend their time abroad developing a different language. 

The module will provide the opportunity to further develop your language skills, building on your previous learning at beginner level. The first half of the module includes intensive taught sessions in interactive workshop mode which will prepare you for placements abroad or self-directed language development. The second half of the module includes a placement abroad or, alternatively, a project on a cultural issue in a target language country. Students of more than one language may take one language in the first half of the module and spend their time abroad developing a different language. 

• The multiple facets of global citizenship
• Ethical engagement and practice
• The United Nations Sustainable Development Goals
• Cross-cultural issues and sensitivity
• Intercultural communication
• Culture shock
• Cultural adjustment
• Self- assessment of needs: identification of the range of transferable skills, competencies and attitudes employees need and employers expect graduates to possess-with a strong focus on understanding the intercultural competencies (ICC) needed to live and work abroad.
• Critical analysis/evaluation of individual requirements in relation to culture/cultural adjustment/culture shock/visas/medical.
• Critical analysis/evaluation of skills already acquired in relation to key skills related to ICC.
• Devising strategies to improve one’s own prospects of working abroad in the future.
• Devising an action plan to address gaps in transferable skills based on organisational analysis and sector opportunities.

Part A:      

Preparation for Experiential Overseas Learning will take place at the University of Chester during level 5 and will include:  

• The multiple facets of Global citizenship
• Ethical engagement and practice
• Cross-cultural issues and sensitivity
• Intercultural communication

Theories, models and strategies of learning

• Theories and models Intercultural competence
• Theories and models of Integration and Multiculturalism
• Critical thinking skills and models of Reflection

• Experiential learning models
• Self-directed experiential learning

Personal and placement-related skills

• Enhanced independence
• Improved command of multicultural behaviour
• Increased knowledge and confidence in their individual facets of personal identity
• Effective time management and organisational skills
• Project management – working away from University and independent study
• Self-management and personal development
• Team building and team work

Part B:            Overseas

Students will engage in experiential learning activities overseas for at least 150 hours 

Year in Industry placement year for computing students. 

This is a placement module where students acquire a 1 year paid work placement related to their area of study.

Full attendance at the work placement and University-scheduled events is mandatory.

The placement hours stated for this module are only the minimum required hours for this module, it is likely most placements will last for 9 months - 12 months and exceed the minimum 1,180 placements hours.

The traditional academic programme structure is not applicable. The placement content is freely structured and determined by negotiation between the student, placement supervisor and host organisation. It is generally informed by the aims and learning outcomes and by the objective of optimising added value for both the host organisation and the student experience

Preparation for the year abroad will take place in Chester during level 5 and will include:

• Cross-cultural issues and sensitivity
• Host-country orientation, study methods– economic, political and social reality of the country
• Orientation specific to exchange – health, education, gender issues
• The United Nations Sustainable Development Goals
• Practical matters relating to living and studying in the wider world

Theories, models and strategies of learning

• Critical thinking skills, experiential learning and models of reflection

Personal and placement-related transversal skills

• Effective self-motivation and independent resourcefulness
• Effective time management and organisational skills
• Project management – working away from University and independent study
• Self-management and personal development

Whilst abroad:

You will undertake study at one of the University of Chester's partner universities or undertake and approved work placement or virtual placement. If you are a student,  it is expected that you will choose a series of modules at the university abroad which must be agreed by the host institution and the Module Leader. you must supply details of you modules on a learning agreement within 4 weeks of arrival at the host university.

Group theory is the mathematical study of symmetry, a concept which appears throughout mathematics, science and nature.  Physicists use group theory to describe the properties of fundamental particles of matter, Chemists use it to investigate crystal structure and it plays a central role in our every-day lives in cryptography.  In mathematics, groups are found everywhere, in algebraic objects such as rings, field, vector spaces and matrices, in analysis, and in solutions to equations and differential equations.

This module will study groups and their actions on sets and geometric objects.  Highlights include the Jordan-Hölder Theorem, which allows us to break down groups into their fundamental building blocks, called the simple groups; and Sylow's theorems which are of fundamental importance to understanding the structure of finite groups.  Topics covered may include:

• Groups
• Subgroups
• Cosets, normal subgroups, quotient groups
• Homomorphisms and the Isomorphism Theorems
• The Classification of finite abelian groups
• Group actions and the Orbit-Stabiliser Theorem
• The Jordan-Hölder Theorem
• Simple Groups
• Sylow Theory
• Soluble Groups

The project is an opportunity for you to explore deeply an area of mathematics of your choice.  This may be a topic not already covered in the degree, or you may explore in more depth a topic covered before.

You will engage with mathematical literature, seeking out material and learning for yourself, supported by regular meetings with your project supervisor.  You will develop skills of self-discipline and project management, and study the art of mathematical writing and communication.

You will negotiate with tutors the precise title and objectives for the project.

Vector calculus and partial differential equations are indispensable tools in science. These topics are essential to our understanding of electromagnetism and quantum mechanics in physics, and also to the modelling of physical phenomenon such that fluid flow and heat conduction. This course is divided into two halves. The first half will cover topics related to three-dimensional geometry and vector calculus including: vector-valued functions, parametrised curves, line integrals and conservative vector fields, multiple integrals, surface integrals, the theorems of Green and Stokes, and the Divergence Theorem. The second half focusses on the theory of and techniques for solving partial differential equations. Topics covered will include: first order PDEs, classification and techniques for solving linear second order PDEs, Fourier series, the method of separation of variables applied to the heat, wave and Laplace equations, and Fourier transforms.

Driving an acceleration in scientific discovery and giving us new tools that aid our work and daily lives, Artificial Intelligence (AI) and Machine Learning (ML) is changing the world. Underpinning all these intelligent systems are sophisticated mathematical techniques employing machinery from both pure and applied mathematics. By providing a rigorous foundation in the key mathematical concepts and skills essential for understanding and developing modern ML and AI algorithms, this module will equip with students the in-depth knowledge needed to be at the cutting edge of this rapidly developing field.

You will begin by reviewing theory from calculus, linear algebra, optimisation, probability and statistics necessary for the understanding of ML and AI algorithms. You will then go on to learn about specific ML and AI techniques that may include principle component analysis, regression, Gaussian mixture models and support vector machines.

Numerical analysis is the exploration of algorithms that rely on numerical approximation; this is invaluable for the study of many complicated problems, where an exact solution is very difficult, or even impossible to obtain.  Its applications extend across mathematics, physical sciences and engineering, and, in the 21st century, have expanded into life sciences, social sciences, medicine, business, and even the arts.  This module serves as an introduction to the fundamental techniques and methodologies for solving mathematical problems using numerical methods and equips students with the skills needed to analyse, design, and implement numerical algorithms effectively across a broad range of mathematical problems.

The topics include:

• Solving nonlinear equations, finding roots, Newton iteration and related methods.
• Introduction to optimisation, optimisation of functions of several variables, with and without constraints.
• Interpolation and approximations: Lagrange interpolation, Hermite interpolations,
• Numerical integration and differentiation: Trapezoidal method, Simpson method.
• Solutions of ordinary differential equations: Euler method, Runge-Kutta method, multistep methods, stability, convergence.
• Solution techniques for partial differential equations including the heat equation.
• Implementation and programming, e.g., Python, MATLAB.
• Error analysis: developing an understanding of the sources of error in numerical computations and methods for analysing and controlling numerical errors.