# Level 1 Modules

**MTH1011 Introduction to Algebra and Analysis (Full year)**

*Pre-requisite*: A Level Mathematics Grade A

*Lecturers*: Dr P Siegl, Dr T Huettemann, Dr A Blanco

**Introduction**

The module covers fundamental aspects of basic mathematics, in particular the foundations (logic, proof techniques, concepts of set theory, number systems), introduction to linear algebra (systems of linear equations, vector spaces, bases, dimension) and introduction to analysis (limits, continuity, derivatives). The module builds a solid foundation for further mathematical studies.

**Content**

Elementary logic and set theory, number systems (including integers, rationals, reals and complex numbers), bounds, supremums and infimums, basic combinatorics, functions.

Sequences of real numbers, the notion of convergence of a sequence, completeness, the Bolzano-Weierstrass theorem, limits of series of non-negative reals and convergence tests.

Analytical definition of continuity, limits of functions and derivatives in terms of a limit of a function. Properties of continuous and differentiable functions. L'Hopital's rule, Rolle's theorem, mean-value theorem.

Matrices and systems of simultaneous linear equations, vector spaces, linear dependence, basis, dimension.

**Assessment**

Class tests 15% (x 2)

Final Exam 70%

**MTH1015 Mathematical Reasoning (Sem 1)**

*Pre-requisite*: A Level Mathematics Grade A

*Lecturer*: Dr T Huettemann

**Introduction**

Mathematics is not all about numbers: it is also about convincing other people that the given answer is the correct one. This is done by providing verifiable arguments. The word “verifiable” is key here: this involves both correct logical reasoning and a way of communicating that makes it easy for others to follow the train of thought.

This module will familiarise the student with aspects of formal logic and common mathematical reasoning techniques (known as “proof by induction” and “proof by contradiction”). On this basis, mathematical communication skills will be developed and applied to various group-working tasks in the tutorials, a written project, and an oral presentation. The skills developed here are fundamental, and are used in all areas of mathematics.

**Content**

The notion of mathematical statements and elementary logic. Mathematical symbols and notation. The language of sets. The concept of mathematical proof, and typical examples. Communicating mathematics to others.

**Assessment**

Assignments 40%

Oral Presentations 25%

Report 25%

Tutorial Contribution 10%

** **

**MTH1021 Mathematical Methods 1 (Full year)**

*Pre-requisite*: A Level Mathematics Grade A

*Lecturers*: Dr C Ramsbottom, Dr G De Chiara, Dr D Green

**Introduction**

The module begins with a revision of basic calculus and elementary functions. These fundamental topics form the basis for investigating the properties of functions using derivatives, integrals and power series. Differential equations, complex numbers, vector analysis, matrix manipulation and calculus in several variables are covered in the module. It thus provides many useful mathematical tools for applications in mechanics, statistics, quantum theory etc. Together with the module MTH1011, these methods build a solid foundation for further mathematical studies at higher levels.

**Content**

Review of A-level calculus: elementary functions and their graphs, domains and ranges, trigonometric functions, derivatives and differentials, integration. Maclaurin expansion. Complex numbers and Euler’s formula.

Differential equations (DE); first-order DE: variable separable, linear; second-order linear DE with constant coefficients: homogeneous and inhomogeneous.

Vectors in 3D, definitions and notation, operations on vectors, scalar and vector products, triple products, 2x2 and 3x3 determinants, applications to geometry, equations of a plane and straight line. Rotations and linear transformations in 2D, 2x2 and 3x3 matrices, eigenvectors and eigenvalues.

Newtonian mechanics: kinematics, plane polar coordinates, projectile motion, Newton’s laws, momentum, types of forces, simple pendulum, oscillations (harmonic, forced, damped), planetary motion (universal law of gravity, angular momentum, conic sections, Kepler’s problem).

Curves in 3D (length, curvature, torsion). Functions of several variables, derivatives in 2D and 3D, Taylor expansion, total differential, gradient (nabla operator), stationary points for a function of two variables. Vector functions; div, grad and curl operators and vector operator identities. Line integrals, double integrals, Green's theorem. Surfaces (parametric form, 2nd-degree surfaces). Curvilinear coordinates, spherical and cylindrical coordinates, orthogonal curvilinear coordinates, Lamé coefficients. Volume and surface integrals, Gauss's theorem, Stokes's theorem. Operators div, grad, curl and Laplacian in orthogonal curvilinear coordinates.

**Assessment**

Computer test 10%

Exam 60%

Class Test 15%

Assignment 15%

**MTH1025 Algorithmic Thinking (Sem 2)**

*Pre-requisite*: A Level Mathematics Grade A

*Lecturer*: Dr A Brown

**Introduction**

Modern mathematics makes extensive use of computers to solve problems. While computers are immensely powerful tools, they require clear, step-by-step instructions to perform even simple tasks. In this module you will learn how to break mathematical operations down into these step-by-step instructions (algorithms), and how to implement those operations in computer code. The skills learned in this module are not only key for work you will do later in your degree but are also highly prized by employers.

**Content**

Programming fundamentals (variables, operations, logic, loops); using pseudocode to construct algorithms and plan programs; basic programming skills with Python; using software (LaTeX) to present mathematical content, and to solve mathematical problems (python packages: numpy, matplotlib and sympy).

**Assessment**

Introduction to Python (online course) 20%

Short projects using selected mathematical software 60%

Canvas Quizzes 20%

**SOR1020 Introduction to Probability & Statistics (Full year)**

*Pre-requisite*: This module is intended for students at stage 1 of either an MSci or a BSc Mathematics pathway, and a mathematical knowledge and ability commensurate with this stage is assumed.

*Lecturers*: Dr A Munaro, Dr Z Lin, Dr H Mitchell, Dr F Lamrock

**Introduction**

This module introduces students to the fundamentals of probability and statistics. We start at the very beginning with basic probability before introducing both discrete and continuous random variables, and their properties. Typical standard discrete and continuous distributions such as Binomial, Exponential, Poisson and Normal are discussed with particular emphasis on how these distributions relate to systems in real life. This theory is used to provide a foundation for the formulation of statistical models and to introduce methods of parameter estimation. With this knowledge in particular it is possible to carry out hypothesis tests, both parametric and non-parametric, which are frequently used tools to analyse data in real world scenarios, including in important fields such as epidemiology, finance, pharma and in engineering.

**Content**

Probability: Definitions and laws of probability. Interpretation of probabilities and relationships between probability and statistics. Conditional probability, in particular Bayes Theorem.

Discrete and Continuous Random Variables and Probability Distributions: Key definitions and properties of discrete and continuous random variables and probability distributions. Expected values of discrete and continuous random variables, including properties of expectation and variance operators.

Standard Discrete Distributions: Bernoulli, Geometric, Binomial, Negative Binomial, Hypergeometric and Poisson distributions.

Standard Continuous Random Variables: Uniform, Exponential and Normal distributions to include use of statistical tables, linear combinations of independent normal random variables, central limit theorems and approximations of Binomial and Poisson distributions.

Bivariate Distributions: Key definitions and properties of discrete and continuous bivariate distributions. Properties of independence and expected values; mean, variance, covariance. Correlations coefficients. Means, variances and covariance of linear combinations of random variables.

Statistical Models: Description of "mathematical" modelling. Statistical models. Measurement models, experimental, systematic and random errors; precision and accuracy.

Sampling: Sample surveys. Methods of sampling and probability sampling schemes. Errors in sample surveys. Advantages of sampling. Sampling from infinite populations.

Estimation: Key definitions and properties of estimation. Desirable properties for an estimator and estimation of mean and variance from a single sample and from several samples. Properties of point estimators leading to the method of moments, method of maximum likelihood estimation and method of least squares estimation.

Introduction to Hypothesis Testing: General principles, null and alternative hypotheses, one and two-sided tests, test statistics, critical regions, P-values, significance level, type I and type II errors and power function. Interpretation of results of a significance test, including confidence intervals.

Hypothesis Tests: Parametric tests based upon Normal distribution, t-distribution, F-distribution and chi-squared distribution. Non parametric tests including the Wilcoxon signed-rank test and Mann-Whitney test.

Statistical Quality Control: Process control for systems including Shewhart control charts, upper and lower control limits, upper and lower warning limits. Analysis of patterns and construction of a control chart for attributes and variables.

**Assessment**

Class tests 15% (x2)

Timed Computer Quiz 10%

Exam 60%

**SOR1021 Introduction to Statistical and Operational Research Methods (Sem 2)**

*Pre-requisite*: This module is intended for students at stage 1 of either an MSci or a BSc Mathematics pathway, and a mathematical knowledge and ability commensurate with this stage is assumed.

*Lecturers*: Dr S Moutari

**Introduction**

Introduction to statistical software for applying the following topics in Operational Research and Statistical Methods:

**Content**

Introduction to statistical software for applying the following topics in Operational Research and Statistical Methods:

Linear Programming: Characteristics of linear programming models, general form. Graphical solution. Simplex method: standard form of linear programming problem, conversion procedures, basic feasible solutions. Simplex algorithm: use of artificial variables.

Decision Theory: Characteristics of a decision problem. Decision making under uncertainty: maximax, maximin, generalised maximin (Hurwicz), minimax regret criteria. Decision making under risk: Bayes criterion, value of perfect information. Decision tree; Bayesian decision analysis.

Random Sampling and Simulation: Random sample from a finite population, from a probability distribution. Use of random number tables. General method for drawing a random sample from a discrete distribution. Drawing a random sample from a continuous distribution: inverse transformation method, exponential distribution. Dynamic simulation techniques: application to queueing problems. Computer aspects: random number generators, sampling from normal distributions.

Initial Data Analysis: Scales of measurement. Discrete and continuous variables. Sample mean, variance, standard deviation, percentile for ungrouped data; boxplot. Frequency table for grouped discrete data: relative frequency, cumulative frequency, bar diagram; sample mean, variance, percentile. Frequency table for grouped continuous data: stem-and-leaf plot, histogram, cumulative percentage frequency plot; sample mean, variance, percentile. Linear transformation. Bivariate data; scatter diagram, sample correlation coefficient.

**Assessment**

Timed Exam on Computer 30% (x2)

Timed Exam on Computer 40%