Second-Year Bachelor's Degree in CPES: Modeling and Numerical Methods in Materials Science

Second-year mathematics curriculum, portion taught at Lycée Joffre:

# Review of Linear Algebra and Additional Concepts in Euclidean Geometry

# Advanced Probability

# Further Analysis of Functions of a Single Variable

# Functions of Several Variables

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The oscillator is a fundamental concept in physics: matter is often modeled as a collection of oscillators (harmonic or otherwise) that interact with one another and with the external environment. The environment acts on the matter through a wave, such as an acoustic or electromagnetic wave. This allows us to lay the theoretical foundations for problems involving radiation-matter interaction and thus to develop one of the fundamental tools for the study of matter (in the broadest sense): spectroscopy.

Spectroscopy is, in fact, the fundamental 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 then provides a way to identify the various types of oscillators that make up the medium under study.

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

The approach taken here is a step-by-step progression, starting with the harmonic oscillator, then moving on to coupled oscillators, and finally to waves analyzed within the framework of discrete systems: infinite and then finite coupled oscillators with different boundary conditions.

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Advanced Topics in Linear Algebra, Reduction Theory, and Applications 

# Additional Concepts in Linear Algebra: The Concept of Direct Sum

# Determinants

# Reduction: Diagonalization and its applications to the study of discrete and continuous dynamical systems in physics, chemistry, biology, and economics.

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This course continues to build students' skills in applied mathematics, with a focus on numerical methods.

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

# Matrix Numerical Analysis

# digital integration

# Numerical Solution of Differential Equations

# Introduction to the Qualitative Analysis of Nonlinear Differential Equations

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This module supplements and formalizes the thermodynamic concepts introduced in the Thermodynamics 1 course unit, exploring several aspects in greater depth: thermodynamic potentials defined using Legendre transformations, the thermodynamics of open systems, phase transitions in 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 unit is taken only 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 a chemical molecule) and its application to the determination of molecular orbitals through the linear combination of atomic orbitals.

Tentative schedule:

# Introduction to Group Theory, Finite Groups, Symmetry Groups of Molecules

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

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

# Tree Data Structures: Binary Trees, Heaps, ABR, Priority Queues

# Sorting into piles, lower bound on the sort

# Graph Structures: Representations (Adjacency Matrices, Edge Lists, Neighbor Lists)

# Basic Algorithms (Connectivity, Breadth-First and Depth-First Search, Topological Sorting)

# Distance Calculation (Dijkstra, Heap Implementation)

# Introduction to Object-Oriented Programming (encapsulation, late binding, classes, methods, etc.)

# Python Programming

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Curriculum taught at Lycée Joffre during the 2nd year

# Solid Mechanics (statics, dynamics—not torsional forces)

# Electromagnetism in a Vacuum

# Diffraction optics, Fourier optics

# Waves and Radiation, Wave Scattering.

# Kinetics and Thermochemistry, Introduction to Electrochemistry

# Organic Chemistry: Some Reaction Mechanisms Well-Suited for Quantum Modeling Using Boundary Orbitals

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

# Information Systems: Introduction to the Entity-Relationship Model, the Relational Model, and Business Process Modeling (Conceptual Business Process Model, Organizational Business Process Model)

# Databases: Creating, Manipulating, and Querying Relational Databases.

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As in the first year, this course consists of supervised projects involving data collection, 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 keep students engaged.

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This modern language course focuses on the written and spoken use of English, with a particular emphasis on speaking.

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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- and 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 curriculum, portion taught at Lycée Joffre:

# Review of Linear Algebra and Additional Concepts in Euclidean Geometry

# Advanced Probability

# Further Analysis of Functions of a Single Variable

# Functions of Several Variables

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Curriculum taught at Lycée Joffre during the 2nd year

# Solid Mechanics (statics, dynamics—not torsional forces)

# Electromagnetism in a Vacuum

# Diffraction optics, Fourier optics

# Waves and Radiation, Wave Scattering.

# Kinetics and Thermochemistry, Introduction to Electrochemistry

# Organic Chemistry: Some Reaction Mechanisms Well-Suited for Quantum Modeling Using Boundary Orbitals

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

The course also examines approximation methods used to determine the properties of complex systems where the Schrödinger 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 connection between molecular orbital diagrams and chemical bonding will be explained. The relationship between molecular geometry and electronic structure will be discussed. This course will then examine the Hückel method, which is used to obtain molecular orbital diagrams for π systems. The classical concepts of conjugation, delocalization, donor and acceptor character, and aromaticity will thus be studied within this framework. Bound orbital theory is used to explain molecular reactivity (cycloadditions, electrocyclization) and molecular geometries.

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

# Concepts of Populations and Random Samples.

# Estimators.

#Mean Square Error and Bias-Variance Decomposition.

# Empirical estimators of the mean and variance. Construction of estimators using the method of moments. Maximum likelihood.
Interval estimation.

# Hypothesis testing: null hypothesis, alternative hypothesis. Type I and Type II errors. Test power. Test statistic, rejection region. Examples of classic tests.

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This course, similar in concept to the first-year project, is an introduction to the research process that goes deeper than the first-year course: it includes an immersion in a research laboratory, a literature review, and the completion of a project involving the study of a physical or chemical context, the collection 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 process of using computer tools in physics: it involves analyzing a phenomenon, idealizing or modeling it, and then studying it on a computer. The critical interpretation of the results is also part of this process. The examples covered are chosen to relate 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 (virtually all of physics, but also chemistry, economics, and more...).

The following chapters are primarily devoted to the case with one spatial dimension and one temporal dimension (1+1 D). Chapter 2 addresses 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, along with the heat equation and the Fourier method. Finally, Chapter 4, devoted to reaction-diffusion equations, considers the case of convective and diffusive transport using the Smoluchowsky equation, and derives the famous Einstein equation for diffusion.

The course includes theoretical and computational tutorials covering topics such as road traffic and queueing models.

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

# Tree Data Structures: Binary Trees, Heaps, ABR, Priority Queues

# Sorting into piles, lower bound on the sort

# Graph Structures: Representations (Adjacency Matrices, Edge Lists, Neighbor Lists)

# Basic Algorithms (Connectivity, Breadth-First and Depth-First Search, Topological Sorting)

# Distance Calculation (Dijkstra, Heap Implementation)

# Introduction to Object-Oriented Programming (encapsulation, late binding, classes, methods, etc.)

# Python Programming

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As in the first year, this course consists of supervised projects involving data collection, 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 keep students engaged.

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This modern language course focuses on the written and spoken use of English, with a particular emphasis on speaking.

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