# 2. Teaching

If you have any questions regarding my lectures,
feel free to stop by the Math Support Center (Coventry Uni.)
every Friday between 2pm and 3pm.
or email me at thierry dot platini at coventry dot ac dot uk

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I recently started making recordings (see the link which says videos).
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Lecture notes, module guide, tutorial sheets with solutions and reading material can be found on the Moodle web page

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### Complex Variables and Linear Mathematics – 322MSE

This module concentrates on two aspects of linear mathematics. The first is calculus for functions of a complex variable and, in particular, integration, which is a linear operation. The second considers linear integral operators and solution of integral equations. Applications will include the evaluation of real integrals and the solution of linear ordinary differential equations in terms of integral operators and Greens functions.

Indicative content
Integral operators and Laplace transforms.
Inverse Laplace transforms and conjugacy problems
Convolution and the Convolution Theorem for Laplace transforms
Variation of parameters technique, and Greens functions
Wronskians and their applications
Green functions for boundary value problems
Voltera integral equations and integro-differential equations solved by convolution. Integral equations of the Fredholm type
Fredholm integral equations with degenerate kernels
Links to differential equations and extension of the Fredholm alternative theorem.

Video – Laplace Transform and Operator of Translation

Video – Initial and Final Value Theorem

Video – Volterra and Fredholm Operators and Adjoint Operators

Adjoint of Volterra Operators

Adjoint of Fredholm Operators

Volterra Integral Equations

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### Practical Applications of Mathematics – 323MSE

#### – Statistical Description of Stochastic Processes : Application to BioPhysics of Gene Expression

In nature, systems are governed by well-known ’deterministic’ physical laws. However the microscopic details are usually unreachable and a full description of the system is impossible. To face such situations, our best approximation is to assume that the interactions between the microscopic elements occur according to some probabilistic rules. It follows that the natural reformulation of complex problems takes the form of stochastic processes (meaning that they are randomly occurring). In this module, we will focus our attention on some of the simplest stochastic processes. For each process, we will see how to give a complete description of the system using the Master Equation.
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### Abstract Algebra – 224MSE

This module provides an introduction to abstract algebra (including some number theory), providing a first serious exposure to axiomatically defined mathematical structures and showing how they can be applied to combinatorial problems.

Indicative Content
GroupsDefinitions and basic properties.
Number TheoryDivisibility, greatest common divisor, Euclid ’s algorithm, Euler phi-function, fundamental theorem of arithmetic. Congruence arithmetic, Euler’s Theorem, Fermat’s Little Theorem. Applications to combinatorial problems and cryptography.
RingsRings, fields, ring of polynomials over a field. Division of polynomials, Euclid ’s algorithm, factorization, irreducible polynomials. …

Video – The number we are dealing with

Video – Solving Diophantine Equations

Video – $\sqrt{2}$ is no rational

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