L2 - General Math

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This course will approach, in the continuity of the analysis course of S2, the notion of series with terms of any sign. The Riemann integral will be defined and applied to treat differential equations, especially linear ones. The integration part will be extended to generalized integrals.

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This course will introduce probability spaces, the notions of probability and independence and will define 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. It is a first step towards spectral analysis.

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

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The teaching unit presents the basic principles of modeling and programming by objects. The supporting languages are UML and Java, with possibly elements of Python at the end of the semester

From a modeling point of view, the teaching unit focuses on the modeling of static views, with class and instance diagrams. Through these diagrams, the notions of classes, instances, attributes, operations, associations, interfaces and specialization will be seen. Their parallel implementation in Java will allow to give them a concrete application and to show in particular the translation of associations in a programming language which does not have them. In Java, the focus will be on the notions of class, instance, inheritance, instance variable, class variable and method, visibility and organization in packages, and static and dynamic bindings. Data collections widely used in Java will be presented to translate some of the associations (lists and associative dictionaries). These collections will introduce students to the use of generic classes. The implementation of the concepts of object-oriented programming with Python may be addressed at the end of the semester depending on the progress.

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This module completes and formalizes the notions of thermodynamics introduced by the EU Thermodynamics 1, by deepening several aspects: thermodynamic potentials defined from Legendre transformations, thermodynamics of open systems, phase transitions of the pure body and irreversible processes, with incursions at the microscopic level in order to give an overview of the physical foundations of the theory.

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In this course, the particularities of floating-point calculus will be discussed, followed by details of the usual elementary numerical methods for solving nonlinear equations, interpolating a function and approximating an integral. The student 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 will cover the concepts of sequences and series of functions and the various convergences. The integer and Fourier series will also be developed.

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In this course, an introduction to the topology of R^n, the basic notions of differential calculus of R^n functions in R and optimization will be covered. Parametric curves will also be covered.

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This course is an introduction to bilinear algebra and will cover Euclidean and Hermitian spaces. It will deal with isometries, duality, quadratic forms and endomorphisms.

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Introduction in this course of the main statistical concepts (graphic representation of data, indicators of central tendency and dispersion, relationship between two variables, confidence intervals).

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This course will cover numerical methods applied to linear algebra and more particularly to matrices. The notions of conditioning, matrix decompositions and iterative methods, and eigenvalue computation will be introduced.

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This course will allow students to discover the various careers in mathematics through presentations of job opportunities, thematic conferences and round tables.

For students who benefit from a pre-professionalization contract, the UE accompanies their activity in the institution, by bringing some elements intended to enrich their observation and to give them some distance. It is also a question of preparing the written work that they will have to submit.

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