Examination procedure
Projects with R and oral exam
Examination procedure notes
The projects with R will be designed and developed with the help of a tutor
Prerequisites
Second year. Prerequisite: the first year course, in particular calculus in one and more variables and ordinary differential equations.
Syllabus
Ordinary differential equations
- reminders of the main results seen in the first year (Cauchy problem, Cauchy-Lipschitz theorem, maximal solutions, Gronwall's Lemma, global existence criteria)
- explicit solution methods (equations with separate variables, some linear equations) and qualitative analysis
- asymptotic behavior, stability and instability, oscillations
- detailed study of some examples, from population dynamics (e.g. Lotka-Volterra), virus diffusion models (e.g. SIR), neuronal spike models, tumor growth models.
- projects, also with the help of R software, for particular problems and examples such as differentiation and cell cycle, feedback mechanisms and enzyme-substrate dynamics.
Calculus of Probability and Statistics
- basic elements (events, their probabilities, first rules, conditional probability and independence, Bayes formula and total probabilities)
- examples of discrete and continuous random variables, mean values and their properties
- linear models in statistics (multiple linear regression, principal components method, implementation on data using R software)
- Markov chains (graphs, transition probabilities, state classification, invariant measures) and applications, also to models similar to those of ordinary differential equations.
Fourier series
- Fourier series expansion of regular periodic functions in one variable
- generalizations to less regular functions
- some elements on Fourier series in two variables
- time series analysis with R
Possible advanced topics
- stochastic processes, Brownian motion
- stochastic differential equations (Langevin), Fokker-Planck equations
- graphs and networks
- Bayesian statistics.
Bibliographical references
Teacher's notes.