Our Courses and Our Levels
MAN MFSI MMP
(Level Upgrading) (Maths for Scientists and Engineers ) (Maths For les Mathematicians and Physicists)
MAN
MMP
MFSI
ANALYSIS:
logics,
elementary functions,
solving equations and inequalities,
complex numbers.
Goal: Reinforce the understanding of the standard notions of analysis and improve the mastery of calculations involving the classical functions of analysis and the bases of differential and integral calculus.
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LINEAR ALGEBRA AND GEOMETRY:
Goal: Understanding of the fundamental notions of linear algebra in the geometric context that saw them emerge, with the main emphasis on dimensions 2 and 3.
ALGEBRA:
1. Real and complex vector spaces.
2. Linear maps and their matrix representations.
3. Determinants.
4. Eigenvalues and vectors, Jordan form.
5. Spectral theorem.
6.Groups (groups, subgroups, homomorphisms, Lagrange's theorem, cyclic groups, group
symmetric, normal subgroups and quotient groups),
7. Rings (rings and fields, homomorphisms, ideals and quotient rings, Euclidean rings, Gaussian integers,
rings of polynomials),
8.Vector spaces (vector spaces and linear maps on any body, bases and dimension,
rank theorem).
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ANALYSIS:
1. Brief introduction to logic and set theory.
2. Axiomatics of real numbers.
3. Numerical sequences.
4. Continuous functions.
5. Differential calculus.
6. Integral calculus.
7. Elementary functions: logarithm, exponential, trigonometric and hyperbolic functions.
8. Topology of the real line.
9. Numerical series.
10. Metric spaces.
11. Sequences and series of functions.
12. Ordinary differential equations.
13. Multivariate functions (differential calculus).
14. Multiple integrals.
L^p Espaces , Hölder's inequality, Hahn-Banach theorem, topological dual space, reflexive spaces, Baire's theorem, Baire spaces, Banach-Steinhaus theorem, open map theorem, isomorphism theorem of Banach, closed graph theorem, Hilbert spaces, Hilbert bases, Riesz representation theorem, weak topology, weak-* topology, Alaoglu's theorem.
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COMPLEXE ANALYSIS:
1. Complex differentiability: Cauchy-Riemann equations, analytical functions, calculation with series, function exponential,
logarithm.
2. Theory of holomorphic functions: curvilinear integral, Cauchy integral formula, Liouville theorem, extension analytic.
3. Singularities and meromorphic functions: isolated singularities, residue theorem, calculus of integrals, functions meromorphs,
principle of the argument.Singularités et fonctions méromorphes : singularités isolées, théorème des ré- sidus, calcul des
intégrales, fonctions.
4. Complex differentiability: Cauchy-Riemann equations, analytical functions, calculation with series, exponential function,
logarithm. Différentiabilité complexe : équations de Cauchy-Riemann, fonctions analy- tiques, calcul avec des séries, fonction
5. Theory of holomorphic functions: curvilinear integral, Cauchy integral formula, Liouville theorem, extension analytic.
6. Singularities and meromorphic functions: isolated singularities, residue theorem, calculus of integrals, eromorphs functions ,
principle of the argument.Singularités et fonctions méromorphes : singularités isolées, théorème des ré- sidus, calcul des
intégrales, fonctions
7. Fourier series: mean square convergence and simple convergence. Functions with bounded variation. Systems orthogonal.
8. Partial differential equations: wave equation, heat equation, Laplace equation; separation app of variables and Fourier series.
9. Holomorphic functions and harmonic functions.
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REAL ANALYSIS:
1. Functions of several real variables, implicit functions, Lagrange multipliers.
2. Differential forms, exact and closed forms, integrals of differential forms, Green's theorem, Poincaré's lemma,
Stokes theorem.
3. Banach spaces, Lipschitz maps, fixed point theorem.
4. Ordinary differential equations, ODE solving methods, existence and uniqueness of solutions, linear nonlinear ODE systems.
5.Calculus of variations, Euler-Lagrange equations.
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NUMERICAL ANALYSIS:
1. Numercial integration.
2. Interpolation and approximation.
3. Numerical resolution of ordinary differential equations.
4. Numerical linear algebra, method of least squares.
5. Calculation of vectors and eigenvalues.
6. Multivariate nonlinear equations.
ETC.
LOGIC AND SET THEORY:
1. Mathematical reasoning and communication.
2. Set theory.
3. Cardinality.
4. Logic.
5. Equivalence relations and order relations.
6. Numbers: natural and relative, rational, real and complex integers.
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DISCREET MATHEMATICS:
1. Enumeration and enumeration problems.
2. Generating series.
3. Combinatorial techniques.
4. Enumeration of classical objects: permutations, partitions, trees.
5. Théorie des graphes.
ETC.
CONTACT
info@academielab.com
+41 76 548 7678
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