Licence 2 CPES Modeling and Numerics in Material Sciences

Second year mathematics program, part taught at Lycée Joffre :

# Reminders of linear algebra and complements of Euclidean geometry

# Advanced probability

# Analysis of functions of one variable

# Functions of several variables

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The oscillator is an essential concept in physics: matter is often modeled by a collection of oscillators (harmonic or not) 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 lays the theoretical foundations for the problems of radiation-matter inter-action, and thus for the construction of one of the fundamental tools for the study of matter (in the broadest sense): spectroscopy.

Spectroscopy is the basic tool for studying the physical properties of the objects that surround 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 making up the medium under study.

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

The principle adopted here is a step-by-step progression from the harmonic oscillator, then coupled oscillators, to waves treated as discrete systems: infinite then finite coupled oscillators with different edge conditions.

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Linear Algebra Complements, Reduction Theory, and Applications 

# Complements of linear algebra, the notion of direct sum

# Determinants

# Reduction: diagonalization and applications to the study of discrete and continuous dynamical 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 (may be modified depending on the year and the time available):

# Matrix Numerical Analysis

# Digital Integration

# numerical solution of diffential equations

# Introduction to the qualitative study of nonlinear differential equations

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This module completes and formalizes the notions of thermodynamics introduced in EU Thermodynamics 1, by exploring several aspects in greater depth: thermodynamic potentials defined on the basis of Legendre transformations, thermodynamics of open systems, pure-body phase transitions and irreversible processes, with incursions at the microscopic level to provide an insight into the physical foundations of the theory.

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This course is followed only by students of the Material Sciences track of the CPES. It includes a mathematical part and a chemical part, around the notion of symmetry group (of a chemical molecule) and its application to the determination of molecular orbitals by linear combination of atomic orbitals.

Indicative programme:

# Introduction to group theory, finite groups, moldecule symmetry groups

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

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The 2nd year computer science course taught at the Lycée is focused on advanced concepts in algorithmics and programming:

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

# Sort by heap, lower bound on sorting

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

# Basic algorithms (connectedness, depth and width paths, topological sorting)

# Distance calculation (Dijkstra, heap implementation)

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

# Programming in Python

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Program taught at the Lycée Joffre during the 2nd year

# Solid mechanics (static, dynamic, not torsors)

# Electromagnetism in a vacuum

# Diffraction optics, Fourier optics

# Waves and radiation, diffusion of waves.

# Kinetics and thermochemistry, intro to electrochemistry

# Organic chemistry: some reaction mechanisms that lend themselves well to quantum modeling with boundary orbitals

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This course completes the training in computer science by opening up to the study of information systems and databases (design of processing in an information system and the management of relational databases):

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

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

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As in the first year,; This teaching consists of tutored projects with data acquisition, processing and interpretation: the approach combines numerical methods, mathematical ideas and modelling.

The themes of study may evolve from year to year, in order to renew the interests of the students.

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This modern language teaching is focused on the written and oral practice of English, with a particular emphasis on oral expression.

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General culture (epistemology and philosophy):

# history of ideas

# Epistemology of science and digital technology

#argumentatition and communication

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General culture (epistemology and philosophy):

# history of ideas

# Epistemology of science and digital technology

#argumentatition 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.)

# Processing on the data (pre/post-processing)

# Supervised learning (Bayes 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, part taught at Lycée Joffre :

# Reminders of linear algebra and complements of Euclidean geometry

# Advanced probability

# Analysis of functions of one variable

# Functions of several variables

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Program taught at the Lycée Joffre during the 2nd year

# Solid mechanics (static, dynamic, not torsors)

# Electromagnetism in a vacuum

# Diffraction optics, Fourier optics

# Waves and radiation, diffusion of waves.

# Kinetics and thermochemistry, intro to electrochemistry

# Organic chemistry: some reaction mechanisms that lend themselves well to quantum modeling with boundary orbitals

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The first part of this course presents the basics of quantum chemistry for chemists and physical chemists. He began by taking up the principles of quantum mechanics and his master equation, the Schrödinger equation. The solution of the Schrödinger equation in simple cases and the notions of wave functions and quantization are presented and illustrated in simple cases. The hydrogen atom is then studied.

The teaching also focuses on approximation methods that make it possible 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 link between molecular orbital diagram and chemical bonding is made. The link between molecular geometry and electronic structure will be discussed. This course will then focus on the Huckel method which allows to obtain diagrams of molecular orbitals of π systems. The classical notions of conjugation, delocalization, donor or acceptor character and aromaticity will be studied in this approach. The theory of boundary orbitals is used to rationalize molecular reactivity (cycloaddition, electrocyclization) and molecular geometries.

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

# Notions of populations and random samples.

# Estimators.

#Erreur mean square and bias-variance decomposition.

# Empirical estimators of mean and variance. Construction of estimators by the method of moments. Maximum plausibility.
Interval estimation.

# Hypothesis tests: null hypothesis, alternative hypothesis. Errors of the first and second kinds. Power of a test. Test Statistic, Rejection Region. Examples of classic 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 an immersion in a research laboratory, a bibliographic study, and the realization 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 is an introduction to the use of computer tools in physics: it involves analyzing a phenomenon, idealizing/modeling it, then studying it on a computer. Critical interpretation of results is also included. Examples are chosen in relation to other current subjects in the course.

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This course discusses the theoretical tools for describing distributed and conserved quantities, such as a density, an energy or a charge density.

The first introductory chapter presents the general conservation equation, and the vector analysis tools necessary to understand it, as well as the different fields where this equation comes into play (practically all physics but also chemistry, economics, etc.).

The following chapters are mainly devoted to the case of a dimension of space and a dimension of time (1+1 D). In Chapter 2, we discuss convective transport, and we present the method of characteristics to solve the corresponding first-order quasilinear 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 Smoluchowsky equation, and the derivation of the famous Einstein equation for diffusion.

The course is accompanied by theoretical and numerical tutorials, for example on road traffic and queue modelling.

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The 2nd year computer science course taught at the Lycée is focused on advanced concepts in algorithmics and programming:

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

# Sort by heap, lower bound on sorting

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

# Basic algorithms (connectedness, depth and width paths, topological sorting)

# Distance calculation (Dijkstra, heap implementation)

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

# Programming in Python

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As in the first year,; This teaching consists of tutored projects with data acquisition, processing and interpretation: the approach combines numerical methods, mathematical ideas and modelling.

The themes of study may evolve from year to year, in order to renew the interests of the students.

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This modern language teaching is focused on the written and oral practice of English, with a particular emphasis on oral expression.

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