CIP - Convergence, Integration and Probability
Ref: 1SL1000
Description
The objective is to have students master the mathematical concepts and formalisms of the modelling of complex systems, through
• the analysis of integro-differential models ; the techniques of approximation and their convergence;
• the construction of transforms to solve problems;
• the deep knowledge of measure theory, as the basis of the modern framework of probability, necessary to describe fluctuating phenomena, and of data processing.
This theoretical framework will allow to tackle the techniques of signal processing and the statistical study of data in common core courses. In second year, students will be able to refine these bases by studying stochastic processes and nonlinear phenomena in advanced courses.
• the analysis of integro-differential models ; the techniques of approximation and their convergence;
• the construction of transforms to solve problems;
• the deep knowledge of measure theory, as the basis of the modern framework of probability, necessary to describe fluctuating phenomena, and of data processing.
This theoretical framework will allow to tackle the techniques of signal processing and the statistical study of data in common core courses. In second year, students will be able to refine these bases by studying stochastic processes and nonlinear phenomena in advanced courses.
Quarter number
SG1 and ST2
Prerequisites (in ters of CS courses)
None
Syllabus
Topology
Metric spaces
Normed vector spaces
Hilbert spaces
Measured spaces
Construction of the integral with respect to a measure
L^p spaces
Interchanging limits and integrals
Probability, random variables
Product measures, probability on R^N, independence
Convolution, Fourier transform and characteristic functions
Gaussian vectors
Convergence of sequences of random variables
Conditional expectation
Random walks
Class components (lecture, labs,etc.)
Several tracks are proposed within this course, depending on choices and selection of students.
Students will be assigned in one of the following 3 tracks:
Track 1 : classroom setting in French
Lectures and tutorials in classroom in French (several groups of different level, limited enrollment)
• Lectures: 11 slots of 1,5 hrs (16,5 HPE / 25 HEE) in classroom
• Tutorials: 10 slots of 1,5 hrs (15 HPE / 22,5 HEE) in classroom
• Autonomy : In addition to the usual work between each session, students will have 4 homeworks to prepare, which corresponds to the content of 1,5hrs of course material and 3hrs of tutorials, spaced between the classroom sessions (0 HPE / 9,5 HEE).
Some students will be assigned in moderate track with reinforcement (selection from the Direction des Etudes):
• 3 groups of 25 students in French and 1 group of 25 students in English,
• 10 supplementary slots of 1,5 hrs for reinforcement.
Track 2 : blended in French
Lectures are online, prerecorded videos in French (choice of the language by students). Tutorials happen in classroom in the same language as the lectures. The course content is transmitted via prerecorded video sequences. Interaction with the teachers takes place during the scheduled tutorials and through digital means. This track allows students to adapt their learning pace, especially when the concepts are more difficult to grasp.
• Lectures, 12 slots of 1,5 hrs (1,5 HPE / 28,5 HEE): the first lecture in class, then online videos;
• Tutorials: 11 slots of 1,5 hrs (16,5 HPE / 28,5 HEE): in class with a quiz at the beginning of each session.
Track 3 : classroom setting in English
Lectures and tutorials in English (choice of the language by students, limited enrollment).
• Lectures: 11 slots of 1,5 hrs (16,5 HPE / 25 HEE) in classroom
• Tutorials: 10 slots of 1,5 hrs (15 HPE / 22,5 HEE) in classroom
• Autonomy : In addition to the usual work between each session, students will have 4 homeworks to prepare, which corresponds to the content of 1,5hrs of course material and 3hrs of tutorials, spaced between the classroom sessions (0 HPE / 9,5 HEE).
Some students will be assigned in moderate track with reinforcement (selection from the Direction des Etudes):
• 3 groups of 25 students in French and 1 group of 25 students in English,
• 10 supplementary slots of 1,5 hrs for reinforcement.
Math Libre 1:
In place of the traditional CIP class, a small group of selected students will follow a special class directed by Erick Herbin. The advanced topics and pedagogical approach of this class will be detailed by Erick Herbin in the beginning of September.
Important: Students assigned to groups where the teaching takes place in a classroom setting will make an honorary commitment to be present at all sessions.
Resources
• Teaching staff (instructor(s) names):
Julien Bect, Philippe Bouafia, John Cagnol, Ioane Muni Toke, Alexandre Richard - lectures in French
Antonio Silveti-Falls - lectures in EnglishErick Herbin - special classes Math Libre 1 in French
• Maximum enrollment: 21 tutorial groups of between 25 and 60 students, depending on level groups
• Software, number of licenses required: None
• Equipment-specific classrooms (specify the department and room capacity): 0
• Software, number of licenses required: None
• Equipment-specific classrooms (specify the department and room capacity): 0
Course support, bibliography
Printed Lecture Notes (in English) and an extensive bibliography (books, electronic documents, exercise book)