L2-L3 LICENCE MATHEMATIQUES

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Following on from the analysis course in S2, this course deals with the notion of series with terms of any sign. The Riemann integral will be defined and applied to linear and other differential equations. The integration section will be extended to generalized integrals.

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This course introduces probability spaces, the concepts of probability and independence, and defines discrete and density random variables, with an emphasis on modeling.

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This course will cover the notions of symmetric group, determinants and will deal with the reduction of endomorphisms in finite dimension (up to Jordan form) and its applications. This is a first step towards spectral analysis.

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In this course, we introduce an overview of algebraic structures (ring, ideal, body) before tackling K[X] algebra and defining arithmetic on polynomials, drawing parallels with integer arithmetic seen in L1. Calculations on polynomial functions and rational fractions will be covered (explicit factorizations/decompositions).

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This course introduces the basic principles of object-oriented modeling and programming. The supporting languages are UML and Java, with possible elements of Python at the end of the semester.

From a modeling point of view, this course focuses on static view modeling, using class and instance diagrams. These diagrams cover the notions of classes, instances, attributes, operations, associations, interfaces and specialization. Their parallel implementation in Java will give them a concrete application, and show in particular the translation of associations in a programming language that doesn't have them. In Java, particular emphasis will be placed on the notions of class, instance, inheritance, instance variable, class variable and method, visibility and package organization, and static and dynamic binding. Data collections widely used in Java will be presented to translate some of the associations (associative lists and dictionaries). These collections will introduce students to the use of generic classes. Implementation of object-oriented programming concepts with Python may be tackled at the end of the semester, depending on progress.

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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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In this course, we'll look at the particularities of floating-point calculus and then go into detail on the most common elementary numerical methods for solving non-linear equations, interpolating a function and approximating an integral. Students will learn how to implement an algorithm to solve a numerical analysis problem.

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The first-semester course reviews the grammar essential for oral and written communication(tenses and aspect, asking questions, comparisons and superlatives, passive voice) as well as essential general vocabulary(numbers, measurements, shapes); it also includes an introduction to technical vocabulary(basic building materials, plane engine, bike parts, electronic device) through themed lessons and videos in the field of mechanical engineering.

Finally, numerous activities are offered to promote oral expression skills (presentation vocabulary, simulations, role-playing and board games), so that students are able to describe the specific features, functions and uses of a piece of technical equipment of their choice in an oral presentation by groups of two.

S4

Grammatical aspects are limited to a review of modal auxiliaries.

The vocabulary is much more focused on the various elements involved in the design and operation of different types of heat engines, and on emerging technologies(drones, driverless vehicles, 3D-printing).

Students are also expected to produce a CV in English and practice writing emails in a formal style, so as to be prepared for internship or job-seeking situations where fluency in English will either be necessary or an additional skill.

The practice of expression is always the main objective, with an individual oral presentation at the end of the semester of their second-year project in mechanics.

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This course covers the notions of function sequences and series, and the various convergences. Integer and Fourier series will also be developed.

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This course covers an introduction to the topology of R^n, the basics of differential calculus for R^n functions in R and optimization. Parametric curves will also be covered.

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This course is an introduction to bilinear algebra, covering Euclidean and Hermitian spaces. It covers isometries, duality, quadratic forms and endomorphisms.

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This course introduces the main statistical concepts (graphical representation of data, indicators of central tendency and dispersion, relationships between two variables, confidence intervals).

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This course deals with numerical methods applied to linear algebra, and more specifically to matrices. The notions of conditioning, matrix decompositions and iterative methods, and the calculation of eigenvalues will be introduced.

See full page

This course includes presentations on career opportunities, themed lectures and round-table discussions on the various mathematical professions.

For students with a pre-professionalization AED contract, the UE accompanies their activity in the establishment, providing a few elements to enrich their observation and give them perspective. It also prepares them for the written work they will be required to submit.

See full page

See full page

Following on from the analysis course in S2, this course deals with the notion of series with terms of any sign. The Riemann integral will be defined and applied to linear and other differential equations. The integration section will be extended to generalized integrals.

See full page

This course introduces probability spaces, the concepts of probability and independence, and defines discrete and density random variables, with an emphasis on modeling.

See full page

This course will cover the notions of symmetric group, determinants and will deal with the reduction of endomorphisms in finite dimension (up to Jordan form) and its applications. This is a first step towards spectral analysis.

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This unit deals with the study of the mechanics of rigid solids. It is the natural continuation of the unit devoted to the kinematics and statics of rigid solids in L1. In this unit, we'll take a dynamic approach and apply the Fundamental Principle of Dynamics. Writing this principle requires knowledge of the external action torsor, studied in L1, as well as knowledge of the dynamic torsor. The latter can be calculated using the kinetic torsor, which for a rigid solid involves the notion of moment of inertia. The main applications studied in this unit concern rigid solids or simple cases of articulated systems of rigid solids. In addition, we will study the special case of contact and friction actions (Coulomb friction) and the Kinetic Energy Theorem.

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This course is the first step in teaching electromagnetism at university. It covers electrostatics, stationary currents and magnetostatics.

See the syllabus in the "+ info" tab.

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In this course, we introduce an overview of algebraic structures (ring, ideal, body) before tackling K[X] algebra and defining arithmetic on polynomials, drawing parallels with integer arithmetic seen in L1. Calculations on polynomial functions and rational fractions will be covered (explicit factorizations/decompositions).

See full page

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.

See full page

The first-semester course reviews the grammar essential for oral and written communication(tenses and aspect, asking questions, comparisons and superlatives, passive voice) as well as essential general vocabulary(numbers, measurements, shapes); it also includes an introduction to technical vocabulary(basic building materials, plane engine, bike parts, electronic device) through themed lessons and videos in the field of mechanical engineering.

Finally, numerous activities are offered to promote oral expression skills (presentation vocabulary, simulations, role-playing and board games), so that students are able to describe the specific features, functions and uses of a piece of technical equipment of their choice in an oral presentation by groups of two.

S4

Grammatical aspects are limited to a review of modal auxiliaries.

The vocabulary is much more focused on the various elements involved in the design and operation of different types of heat engines, and on emerging technologies(drones, driverless vehicles, 3D-printing).

Students are also expected to produce a CV in English and practice writing emails in a formal style, so as to be prepared for internship or job-seeking situations where fluency in English will either be necessary or an additional skill.

The practice of expression is always the main objective, with an individual oral presentation at the end of the semester of their second-year project in mechanics.

See full page

This course covers the notions of function sequences and series, and the various convergences. Integer and Fourier series will also be developed.

See full page

This course covers an introduction to the topology of R^n, the basics of differential calculus for R^n functions in R and optimization. Parametric curves will also be covered.

See full page

This course is an introduction to bilinear algebra, covering Euclidean and Hermitian spaces. It covers isometries, duality, quadratic forms and endomorphisms.

See full page

This course introduces the main statistical concepts (graphical representation of data, indicators of central tendency and dispersion, relationships between two variables, confidence intervals).

See full page

This course deals with numerical methods applied to linear algebra, and more specifically to matrices. The notions of conditioning, matrix decompositions and iterative methods, and the calculation of eigenvalues will be introduced.

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The first part of this course is designed to consolidate the concepts of magnetostatics and establish the relations between the electromagnetic field at the interface of a plane of charges or currents. We also introduce the expression of Laplace forces (force and moment) acting on volumetric or filiform circuits. The second part is devoted to the properties of fields and potentials in the variable regime. After introducing Faraday's law describing induction phenomena, we establish Maxwell's time-dependent equations. An energetic treatment allows us to define the electric and magnetic energies, as well as the Poynting vector. We apply these concepts to various examples, such as electromechanical conversion or induction heating via eddy currents. A final chapter is devoted to the equations of field and potential propagation, and their application in vacuum-like systems, as well as in perfect conductors and insulators. The notion of skin depth is also introduced.

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This course includes presentations on career opportunities, themed lectures and round-table discussions on the various mathematical professions.

For students with a pre-professionalization AED contract, the UE accompanies their activity in the establishment, providing a few elements to enrich their observation and give them perspective. It also prepares them for the written work they will be required to submit.

See full page

This course introduces the basic principles of object-oriented modeling and programming. The supporting languages are UML and Java, with possible elements of Python at the end of the semester.

From a modeling point of view, this course focuses on static view modeling, using class and instance diagrams. These diagrams cover the notions of classes, instances, attributes, operations, associations, interfaces and specialization. Their parallel implementation in Java will give them a concrete application, and show in particular the translation of associations in a programming language that doesn't have them. In Java, particular emphasis will be placed on the notions of class, instance, inheritance, instance variable, class variable and method, visibility and package organization, and static and dynamic binding. Data collections widely used in Java will be presented to translate some of the associations (associative lists and dictionaries). These collections will introduce students to the use of generic classes. Implementation of object-oriented programming concepts with Python may be tackled at the end of the semester, depending on progress.

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This teaching unit introduces the design of processing in an information system and the management of relational databases. We will cover the following points:
(1) Information systems: Introduction of the entity/association model, Relational model, Processing modeling (Conceptual processing model, Organizational processing model),

(2) Databases: creating, manipulating and querying relational databases.

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Following on from the analysis course in S2, this course deals with the notion of series with terms of any sign. The Riemann integral will be defined and applied to linear and other differential equations. The integration section will be extended to generalized integrals.

See full page

This course introduces probability spaces, the concepts of probability and independence, and defines discrete and density random variables, with an emphasis on modeling.

See full page

This course will cover the notions of symmetric group, determinants and will deal with the reduction of endomorphisms in finite dimension (up to Jordan form) and its applications. This is a first step towards spectral analysis.

See full page

- Formal syntax of propositional logic: symbols, connectors, well-formed formulas, syntactic trees, normal and clausal forms

- Semantics of propositional logic: interpretation, model, truth tables, satisfiability, validity, semantic equivalence, logical consequence

- Modeling: formalizing problems in propositional logic, expressiveness limits of propositional logic

- Formal proof: sequences, inference rules, axioms, theorems, LK system, solution method

- Correctness and completeness of a system with respect to semantics: proof of correctness and completeness of LK and resolution method (reduced to propositional case)

- Curry-Howard correspondence

- Introduction to first-order logic (predicate calculus) without function symbol

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The aim of this course is to describe the main concepts of operating systems, particularly Unix.

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In this course, we'll look at the particularities of floating-point calculus and then go into detail on the most common elementary numerical methods for solving non-linear equations, interpolating a function and approximating an integral. Students will learn how to implement an algorithm to solve a numerical analysis problem.

See full page

This course covers the notions of function sequences and series, and the various convergences. Integer and Fourier series will also be developed.

See full page

See full page

This course covers an introduction to the topology of R^n, the basics of differential calculus for R^n functions in R and optimization. Parametric curves will also be covered.

See full page

Students will be able to model and develop using advanced aspects of object-oriented programming, and will have acquired good programming practices. They will be able to draw UML diagrams expressing the dynamics of interactions in a system, and will consolidate their knowledge of structural modeling.

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This course introduces the main statistical concepts (graphical representation of data, indicators of central tendency and dispersion, relationships between two variables, confidence intervals).

See full page

This course deals with numerical methods applied to linear algebra, and more specifically to matrices. The notions of conditioning, matrix decompositions and iterative methods, and the calculation of eigenvalues will be introduced.

See full page

This course includes presentations on career opportunities, themed lectures and round-table discussions on the various mathematical professions.

For students with a pre-professionalization AED contract, the UE accompanies their activity in the establishment, providing a few elements to enrich their observation and give them perspective. It also prepares them for the written work they will be required to submit.

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This course introduces first-order logic, also known as predicate calculus.

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In the first part: deepen the basic notions of differential calculus seen in L2.

In the second part: introduce the qualitative study of differential equations.

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The aim of this module is to understand the use of formal and numerical calculus both in mathematical experimentation and in helping to solve certain problems in computer science.

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Acquire the basic notions of group and ring theory and illustrate them with examples.

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Acquire the basics of measurement theory and integration.

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Illustrate with a probabilistic point of view the notions seen in the "Measurement - Integration - Fourier" UE, and introduce the tools needed by students who will take the Stochastic Modeling UE in the second semester of L3. 

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The aim of this course is to provide an introduction to the software environments required for modern development. It covers version management, modeling, test development, virtualization, continuous integration and more.

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This module explores some advanced topics in algorithm design and analysis.

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Introduce the basic concepts of topology and their use in the study of functional spaces.

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Deepen the basic notions of group and ring theory covered in the previous semester.

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Learn the basics of mathematical optimization and its applications.

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In the first part: deepen the basic notions of differential calculus seen in L2.

In the second part: introduce the qualitative study of differential equations.

See full page

Acquire the basic notions of group and ring theory and illustrate them with examples.

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Acquire the basics of measurement theory and integration, then use these foundations to introduce the spaces and tools of functional analysis.

See full page

Illustrate with a probabilistic point of view the notions seen in the "Measurement - Integration - Fourier" UE, and introduce the tools needed by students who will take the Stochastic Modeling UE in the second semester of L3. 

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In the first part, the basic concepts of enumeration seen in L1 and L2 are studied in greater depth.

The second part introduces the combinatorial study of graphs.

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• Enable students to discover the teaching profession and learn about the didactics of mathematics.
•  Use practical examples to tackle some of the major themes in mathematics didactics.

• Begin a change of posture on the issue of teaching mathematics and learn to ask questions related to teaching.
• Adopt a reflective stance on issues related to teaching mathematics in secondary schools
• Taking a step back from the mathematical content taught in secondary schools

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This UE will enable :

Manipulate the main results of probability from a practical point of view. Reinforce understanding of random phenomena with numerical illustrations. Introduce students to Monte-Carlo simulation methods for numerically solving integration problems or calculating the probability of complex events. Complete knowledge of the main laws and their properties, with a view to applications to inferential statistics and statistical tests covered in the Master's program.

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Acquire basic notions of numerical methods for differential equations

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This UE will cover the following points:

     - Affine geometry: affine spaces, affine applications, barycenters. Affine geometry of the plane and space, classical theorems.

     - Euclidean affine geometry: Euclidean affine spaces, isometries. Euclidean affine geometry of the plane and space, classical theorems, use of complexes to describe objects in the Euclidean affine plane and isometries.

     - Isometries and groups.

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Revisit undergraduate mathematics content in a cross-disciplinary way.

Problem-solving activity: research and communication.

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Introduce the basic concepts of topology and their use in the study of functional spaces.

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This UE will enable :

Manipulate the main results of probability from a practical point of view. Reinforce understanding of random phenomena with numerical illustrations. Introduce students to Monte-Carlo simulation methods for numerically solving integration problems or calculating the probability of complex events. Complete knowledge of the main laws and their properties, with a view to applications to inferential statistics and statistical tests covered in the Master's program.

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Deepen the basic notions of group and ring theory covered in the previous semester.

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Learn the basics of mathematical optimization and its applications.

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Acquire basic notions of numerical methods for differential equations

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Introduce the basic tools of complex analysis.

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In the first part: deepen the basic notions of differential calculus seen in L2.

In the second part: introduce the qualitative study of differential equations.

See full page

Acquire the basic notions of group and ring theory and illustrate them with examples.

See full page

Acquire the basics of measurement theory and integration, then use these foundations to introduce the spaces and tools of functional analysis.

See full page

Illustrate with a probabilistic point of view the notions seen in the "Measurement - Integration - Fourier" UE, and introduce the tools needed by students who will take the Stochastic Modeling UE in the second semester of L3. 

See full page

See full page

In the first part, the basic concepts of enumeration seen in L1 and L2 are studied in greater depth.

The second part introduces the combinatorial study of graphs.

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• Enable students to discover the teaching profession and learn about the didactics of mathematics.
•  Use practical examples to tackle some of the major themes in mathematics didactics.

• Begin a change of posture on the issue of teaching mathematics and learn to ask questions related to teaching.
• Adopt a reflective stance on issues related to teaching mathematics in secondary schools
• Taking a step back from the mathematical content taught in secondary schools

See full page

See full page

This UE will enable :

Manipulate the main results of probability from a practical point of view. Reinforce understanding of random phenomena with numerical illustrations. Introduce students to Monte-Carlo simulation methods for numerically solving integration problems or calculating the probability of complex events. Complete knowledge of the main laws and their properties, with a view to applications to inferential statistics and statistical tests covered in the Master's program.

See full page

Acquire basic notions of numerical methods for differential equations

See full page

This UE will cover the following points:

     - Affine geometry: affine spaces, affine applications, barycenters. Affine geometry of the plane and space, classical theorems.

     - Euclidean affine geometry: Euclidean affine spaces, isometries. Euclidean affine geometry of the plane and space, classical theorems, use of complexes to describe objects in the Euclidean affine plane and isometries.

     - Isometries and groups.

See full page

Revisit undergraduate mathematics content in a cross-disciplinary way.

Problem-solving activity: research and communication.

See full page

See full page

Introduce the basic concepts of topology and their use in the study of functional spaces.

See full page

See full page

This UE will enable :

Manipulate the main results of probability from a practical point of view. Reinforce understanding of random phenomena with numerical illustrations. Introduce students to Monte-Carlo simulation methods for numerically solving integration problems or calculating the probability of complex events. Complete knowledge of the main laws and their properties, with a view to applications to inferential statistics and statistical tests covered in the Master's program.

See full page

Deepen the basic notions of group and ring theory covered in the previous semester.

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Learn the basics of mathematical optimization and its applications.

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Acquire basic notions of numerical methods for differential equations

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Introduce the basic tools of complex analysis.

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