L2 - Double Computer Science Mathematics

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 course presents 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: creation, manipulation and interrogation of relational databases.

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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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- 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: formalization of problems in propositional logic, expressiveness limit of propositional logic

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

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

- Correspondence from Curry-Howard

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

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The objective of this course is to describe the main concepts of operating systems and in particular of Unix.

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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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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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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 create UML diagrams expressing the dynamics of interactions in a system and will consolidate their knowledge of structural modeling.

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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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