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...free course is an introduction to group theory, one of the three main branches of pure mathematics. Section 1 looks at the set of symmetries of a two-dimensional figure which are then viewed as functions. Section 2 introduces an algebraic notation for recording symmetries and calculating composites and inverses of symmetries. Section 3 introduces definitions and looks at...

...free course consolidates and builds on group theory studied at OU level 2 or equivalent. Section 1 describes how to construct a group called the direct product of two given groups, and then describes certain conditions under which a group can be regarded as the direct product of its subgroups. Section 2 describes the key properties of the structure of cyclic groups....

...free course concerns the calculus of variations. Section 1 introduces some key ingredients by solving a seemingly simple problem – finding the shortest distance between two points in a plane. The section also introduces the notions of a functional and of a stationary path. Section 2 describes basic problems that can be formulated in terms of functionals. Section 3 looks...

...free course, matrices are used as a concise way of representing systems of linear equations which occur frequently in mathematics. Section 1 looks at simultaneous linear equations in two and three unknowns and then generalises the ideas to systems of linear equations. Section 2 develops a strategy for solving systems of linear equations. Section 3 looks at the algebra of...

...free course, Introduction to number theory, is a branch of mathematics concerned with the properties of integers. Section 1 introduces Euclid’s algorithm, which is used to find the HCF of two integers, and the idea of congruences, mathematical statements used to compare remainders when two integers are each divided by another integer. Section 2 introduces modular...

...free course examines the basic kinematics of two-dimensional fluid flows. Section 1 introduces the differential equations for pathlines and streamlines. Section 2 introduces a scalar field, called the stream function, which for an incompressible fluid provides an alternative method of modelling the flow and finding the streamlines. Sections 2 and 3 derive the stream...

...free course examines the formulation and solution of small linear programming problems. Section 1 deals with the formulation of linear programming models, describing how mathematical models of suitable real-world problems can be constructed. Section 2 looks at graphical representations of two-dimensional models, considers some theoretical implications and examines the...

...free course, Mosquito resistance to insecticides, further demonstrates how traits that are advantageous in one environment can have effects that are disadvantageous in other environments, a situation known as a trade-off...Learn about how allele frequency can change rapidly in a population in response to selective pressure by considering how evolutionary change can occur...