Bachelor's Degree in Modeling and Digital Technology in Materials Science

Second-year mathematics program, taught at Lycée Joffre:

# Linear algebra refresher and Euclidean geometry supplements

# Advanced Probabilities

# Additional analysis of functions of a variable

# Functions of several variables

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The oscillator is an essential concept in physics: matter is often modeled as a collection of oscillators (harmonic or otherwise) interacting with each other and with the external environment. The latter acts on matter via a wave, such as an acoustic or electromagnetic wave. This allows us to lay the theoretical foundations for problems of radiation-matter interaction and thus to construct one of the fundamental tools for the study of matter (in the broad sense): spectroscopy.

Spectroscopy is the basic tool for studying the physical properties of the objects around us, such as molecules, crystals, stars, and galaxies. These properties are deduced either from their spontaneous emission or from their response to external excitation. For example, we measure the absorption, reflection, and transmission properties of applied electromagnetic radiation (visible, infrared, X-rays, neutrons, etc.). The response to this radiation is then a means of discovering the various types of oscillators that make up the medium being studied.

 In short, the study of the physical environments around us requires the use of two fundamental theoretical tools: oscillators and waves, which are precisely the subject of this course.  

The principle adopted here is a step-by-step progression from harmonic oscillators, then coupled oscillators, to waves processed within discrete systems: infinite then finite coupled oscillators with different boundary conditions.

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Complements to linear algebra, reduction theory, and applications 

# Linear algebra supplements, concept of direct sum

# Determinants

# Reduction: diagonalization and applications to the study of discrete and continuous dynamic systems in physics, chemistry, biology, or economics.

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This course continues the training of students in applied mathematics with a focus on numerical methods.

Program (subject to change depending on the year and time available):

# numerical matrix analysis

#digital integration

# numerical solution of differential equations

# Introduction to the qualitative study of nonlinear differential equations

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This module complements and formalizes the concepts of thermodynamics introduced in the Thermodynamics 1 course, exploring several aspects in greater depth: thermodynamic potentials defined using Legendre transformations, thermodynamics of open systems, phase transitions of pure substances and irreversible processes, with forays into the microscopic level to provide an overview of the physical foundations of the theory.

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This course is only taken by students in the Materials Science track of the CPES program. It includes a mathematics section and a chemistry section, focusing on the concept of symmetry groups (of chemical molecules) and their application to determining molecular orbitals through linear combinations of atomic orbitals.

Indicative program:

# introduction to group theory, finite groups, symmetry groups of molecules

# Linear representations of groups, character theory, and applications to molecular orbitals or vibrational modes of a molecule

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The 10th grade computer science course taught at the high school focuses on advanced concepts in algorithms and programming:

# Tree data structures: binary trees, heaps, ABRs, priority queues

# Sort by pile, lower bound on sort

# Graph structure: representations (adjacency matrices, edge lists, neighbor lists)

# Basic algorithms (connectivity, depth-first and breadth-first search, topological sorting)

# Distance calculation (Dijkstra, implementation with heap)

# Introduction to object-oriented programming (encapsulation, late binding, classes, methods, etc.)

# Programming in Python

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Program taught at Lycée Joffre during the second year

# Solid mechanics (statics, dynamics, not torsors)

# Electromagnetism in a vacuum

# Diffraction optics, Fourier optics

# Waves and radiation, wave propagation.

# Kinetics and thermochemistry, introduction to electrochemistry

# Organic chemistry: some reaction mechanisms well suited to quantum modeling with frontier orbitals

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This course complements computer science training by introducing students to the study of information systems and databases (designing processes in an information system and managing relational databases):

# Information systems: introduction to the entity/association model, relational model, process modeling (conceptual process model, organizational process model)

# Databases: creation, manipulation, and querying of relational databases.

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As in the first year, this course consists of supervised projects involving data acquisition, processing, and interpretation: the approach combines numerical methods, mathematical concepts, and modeling.

The topics of study may change from year to year in order to renew students' interest.

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This modern language course focuses on written and oral English, with particular emphasis on oral expression.

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General knowledge (epistemology and philosophy):

# history of ideas

# epistemology of science and digital technology

#argumentation and communication

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General knowledge (epistemology and philosophy):

# history of ideas

# epistemology of science and digital technology

#argumentation and communication

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Introduction to the use of data mining methods:

# Motivations, areas of application

# Data mining tasks (classification, estimation, prediction, etc.)

# Data processing (pre/post-processing)

# Supervised learning (Bayesian method, k-nearest neighbors, neural networks, etc.)

# Unsupervised learning (k-means)

# Evaluation of methods

# Data mining software (R, Scikit, Weka, etc.)

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Second-year mathematics program, taught at Lycée Joffre:

# Linear algebra refresher and Euclidean geometry supplements

# Advanced Probabilities

# Additional analysis of functions of a variable

# Functions of several variables

See the full page

Program taught at Lycée Joffre during the second year

# Solid mechanics (statics, dynamics, not torsors)

# Electromagnetism in a vacuum

# Diffraction optics, Fourier optics

# Waves and radiation, wave propagation.

# Kinetics and thermochemistry, introduction to electrochemistry

# Organic chemistry: some reaction mechanisms well suited to quantum modeling with frontier orbitals

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The first part of this course presents the basics of quantum chemistry for chemists and physical chemists. It begins by reviewing the principles of quantum mechanics and its master equation, the Schrödinger equation. The solution of the Schrödinger equation in simple cases and the concepts of wave functions and quantization are presented and illustrated in simple cases. The hydrogen atom is then studied.

The course also examines approximation methods that can be used to determine the properties of complex systems where Schrödinger's equation cannot be solved directly. The effect of spin on the electronic properties of atoms and molecules will also be discussed.

The second part of this course focuses on the quantum description of molecular properties and reactivity. The qualitative construction of molecular orbitals using symmetry properties will be introduced, and the link between molecular orbital diagrams and chemical bonding will be explained. The link between molecular geometry and electronic structure will be discussed. This course will then focus on Hückel's method, which is used to obtain molecular orbital diagrams of π systems. The classic concepts of conjugation, delocalization, donor or acceptor character, and aromaticity will be studied in this approach. Frontier orbital theory is used to rationalize molecular reactivity (cycloadditions, electrocyclization) and molecular geometries.

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This course continues the training of students in applied mathematics, particularly the use of statistical tools:

# Concepts of populations and random samples.

# Estimators.

Mean square error and bias-variance decomposition.

# Hypothesis testing: null hypothesis, alternative hypothesis. Type I and Type II errors. Power of a test. Test statistic, rejection region. Examples of classical tests.

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This course, similar in spirit to the first-year project, is an introduction to the research process, more advanced than in the first year: it includes immersion in a research laboratory, a bibliographic study, and the completion of a project involving the study of a physical or chemical context, the acquisition of experimental or observational data, and the implementation of a modeling and/or numerical simulation approach.

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This module provides an introduction to the use of computer tools in physics: analyzing a phenomenon, idealizing/modeling it, and then studying it on a computer. Critical interpretation of the results is also part of the module. The examples discussed are chosen in relation to other current topics in the course.

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This course covers the theoretical tools used to describe distributed and conserved quantities, such as density, energy, or charge density.

The first introductory chapter presents the general conservation equation and the vector analysis tools needed to understand it, as well as the various fields in which this equation is used (practically all of physics, but also chemistry, economics, etc.).

The following chapters are mainly devoted to the case of one spatial dimension and one time dimension (1+1 D). Chapter 2 discusses convective transport and presents the characteristic method for solving the corresponding first-order quasi-linear partial differential equation. Diffusive transport is then studied in Chapter 3, with the heat equation and the Fourier solution method. Finally, Chapter 4, devoted to reaction-diffusion equations, considers the case of convective and diffusive transport with the Smoluchowski equation, and the derivation of the famous Einstein equation for diffusion.

The course is accompanied by theoretical and numerical tutorials, covering topics such as road traffic and queue modeling.

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The 10th grade computer science course taught at the high school focuses on advanced concepts in algorithms and programming:

# Tree data structures: binary trees, heaps, ABRs, priority queues

# Sort by pile, lower bound on sort

# Graph structure: representations (adjacency matrices, edge lists, neighbor lists)

# Basic algorithms (connectivity, depth-first and breadth-first search, topological sorting)

# Distance calculation (Dijkstra, implementation with heap)

# Introduction to object-oriented programming (encapsulation, late binding, classes, methods, etc.)

# Programming in Python

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As in the first year, this course consists of supervised projects involving data acquisition, processing, and interpretation: the approach combines numerical methods, mathematical concepts, and modeling.

The topics of study may change from year to year in order to renew students' interest.

See the full page

This modern language course focuses on written and oral English, with particular emphasis on oral expression.

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