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Library Exercises Games Exams Resources Forum News

Biologie – Génétique (3)

From DNA to protein
DNA and Genes
Mendelian genetics

Biologie – Immunologie (1)

The immune system

Biologie – Le vivant (2)

The cell
Cell division: mitosis and meiosis

Biologie – Métabolisme (1)

Photosynthesis

Biologie – Physiologie (2)

Hormonal regulation and homeostasis
The neuron and the nerve message

Biologie – Écologie (1)

Ecosystems

Biologie – Évolution (1)

The evolution of species

Chimie – Chimie supramoléculaire (1)

Supramolecular Self-Assembly: Metal-Organic Cages and Bowls

Chimie – La matière (4)

Atoms, molecules and ions
The periodic table of elements
Chemical bonds
The states of matter and changes of state

Chimie – Organique (2)

Organic chemistry: the main reactions
Organic chemistry: carbon molecules

Chimie – Quantités (3)

The mole and the amount of substance
Gases and the Ideal Gas Law
Solutions and concentration

Chimie – Transformations (7)

Acids, bases and pH
Chemical kinetics and equilibria
Redox reactions
Electrochemistry and Galvanic Cells
Thermochemistry
Chemical reactions and their equations
Acid-base titrations

Cryptographie – Applications (4)

Strong authentication: TOTP and FIDO2
The blockchain: hashing, Merkle trees and proof of work
Steganography: hiding the very existence of the message
Passwords: salting, hashing, and key derivation

Cryptographie – Attaques (2)

Side-channel attacks
Attacks on RSA

Cryptographie – Avancée (1)

Homomorphic encryption

Cryptographie – Clé publique (3)

The Diffie-Hellman key exchange
Elliptic curve cryptography
Public-key cryptography and RSA

Cryptographie – Fondamentaux (6)

Randomness in cryptography
The one-time pad and perfect secrecy
Kerckhoffs's principle and secure design
Secret codes: encryption and decryption
One-way functions: the foundations
Vigenère and the early days of cryptanalysis

Cryptographie – Intégrité (3)

Hash functions and digital signatures
Hash functions: construction and attacks
MAC and HMAC: authenticating messages

Cryptographie – Mathématiques (2)

Modular arithmetic, the foundation of cryptography
Post-quantum cryptography

Cryptographie – Messagerie (2)

End-to-end encryption
Signal: forward secrecy and the double ratchet

Cryptographie – Protocoles (4)

Schnorr identification and the Fiat-Shamir transform
Secure multiparty computation
Shamir's Secret Sharing
Zero-Knowledge Proofs

Cryptographie – Quantique (1)

Quantum key distribution (BB84)

Cryptographie – Réseau (2)

Certificates and the chain of trust
TLS and HTTPS: web security

Cryptographie – Symétrique (3)

AES, the Encryption Standard
Block ciphers and their modes
Modern symmetric encryption: AES and operating modes

Informatique – Algorithmique (6)

Introduction to dynamic programming
Algorithmic Complexity
Binary search
Recursion
Sorting Algorithms
The Basics of Algorithms

Informatique – Architecture (1)

Logic Circuits and Computer Architecture

Informatique – Codage (1)

Error-Correcting Codes

Informatique – Données (1)

Databases and SQL

Informatique – Information (1)

Information Theory and Compression

Informatique – Programmation (4)

Functions
Lists and dictionaries in Python
First steps in Python
Variables, loops and conditions

Informatique – Réseaux (1)

How the Internet works

Informatique – Structures de données (3)

Binary Trees
Graphs and their traversals
Stacks and queues

Informatique – Théorie (1)

Finite automata

Mathématiques (2)

Discovering Algebra
The basics of the calculation

Mathématiques – Algèbre (13)

Expand an expression
Factoring an expression
First-degree inequalities
Literal calculation
The second degree: canonical form
Notable figures
Matrices
Formulating problems as equations
Polynomials and factorisation
Sign of a trinomial
Systems of two equations
First-degree equations
Quadratic equations

Mathématiques – Algèbre linéaire (6)

Linear maps
Determinants
Vector spaces
Linear systems and Gaussian elimination
Invertible matrices and solving systems
Eigenvalues and diagonalisation

Mathématiques – Analyse (31)

Fourier Analysis and Signal Processing
Applications of differentiation
Bolzano-Weierstrass by Bisection
Continuity and the Intermediate Value Theorem
A comparison of growth rates
Calculus assignment: the function ln(1 + x)/x and its associated sequence
Derivatives and variations
Limited developments
Integration by parts
The Area Between Two Curves
Integration by Substitution
The square function
The derivative
The exponential function
The inverse function
The natural logarithm function
The concept of a function
The Average Value of a Function
Integral calculus
The derivative
Proof by induction
Heine's Theorem
Rolle's Theorem and the Mean Value Theorem
The Banach Fixed-Point Theorem
Affine functions
Reference functions
Function limits
Riemann Sums
Primitives of a function
Direction of variation of a function
Differential equations

Mathématiques – Analyse numérique (1)

Numerical Integration

Mathématiques – Arithmétique (14)

Bézout, Gauss and Diophantine equations
Assignment: the Josephus problem
Divisibility, multiples and divisors
The Irrationality of e
Fractions: meaning and representation
Decimal numbers
Whole numbers and the number system
Prime numbers
Relative numbers
The four operations set out
Operations on fractions
GCD and irreducible fractions
Proportionality and percentages
Powers and scientific notation

Mathématiques – Géométrie (12)

Areas and perimeters
Angles and parallelism
Circle, disc and inscribed angles
Solid Geometry
Geometry with reference points in the plane
Pythagoras’s theorem
Thales’ theorem
Rotations and homothecies
Symmetries and translations
Triangles and their properties
Trigonometry of a right-angled triangle
Volumes of common solids

Mathématiques – Logique (2)

Cantor's Diagonal Argument
Logic and Methods of Proof

Mathématiques – Nombres complexes (3)

The d'Alembert–Gauss Theorem
Complex numbers: algebraic form
Modulus, argument and exponential form

Mathématiques – Probabilités (6)

Counting and combinatorics
The binomial distribution
The normal distribution
Conditional probability
Probability on a finite sample space
Random variables, expectation and variance

Mathématiques – Statistiques (2)

Descriptive statistics
Inferential Statistics

Mathématiques – Suites (5)

Digital suites
Limit of a sequence
Direction of variation of a sequence
Arithmetic sequences
Geometric sequences

Mathématiques – Suites et récurrence (1)

The Fibonacci sequence

Mathématiques – Trigonométrie (5)

Cosine and sine of a real number
Sine and cosine functions
The unit circle
Trigonometric formulas
Trigonometric equations

Mathématiques – Vecteurs (6)

Applications of the scalar product
Collinearity and alignment
The scalar product
The vector product
Vectors in the plane
Vectors in space

Physique (2)

Quantities, units and measurement
Motion and forces

Physique – Fluides (4)

Fluid flow
Archimedes’ principle
Bernoulli’s equation
Fluid Pressure and Statics

Physique – Mesure et méthode (6)

Significant figures and uncertainties
Creating and interpreting a graph
Quantities, values and units
The experimental approach
Prefixes and conversions
Scientific notation and orders of magnitude

Physique – Mécanique (22)

Collisions and the conservation of momentum
Conservation of mechanical energy
Describing a movement: the frame of reference
Acceleration
Kinetic energy
Potential gravitational energy
Freefall
Newton’s second law
Universal gravitation
Mechanical power
Momentum
Newton’s third law
Uniform circular motion
Uniform linear motion
The principle of inertia
The work done by a force
Forces: mechanical actions
The motion of projectiles
Oscillators and pendulums
Weight and mass
Satellites and Kepler’s laws
Average and instantaneous speed

Physique – Ondes (2)

The Doppler Effect
Mechanical Waves and Acoustics

Physique – Optique (9)

Dispersion and spectra
Interference and diffraction
The eye and optical instruments
The construction of images
Light: a wave
The propagation of light
The reflection of light
Refraction and Snell–Descartes’ laws
Thin lenses

Physique – Physique nucléaire (1)

Radioactivity and Nuclear Decay

Physique – Physique quantique (10)

The Photoelectric Effect
Quantum tunnelling
Wave-Particle Duality
Energy Quantization and Spectra
The black body and the birth of the quanta
The laser: stimulated emission and population inversion
Heisenberg's uncertainty principle
Spin and the Pauli Principle
Atomic orbitals and quantum numbers
Superposition, entanglement and qubits

Physique – Relativité (1)

Special Relativity

Physique – Thermodynamique (3)

The ideal gas law
Thermodynamics and Heat Engines
Heat Transfer

Physique – Thermodynamique quantique (1)

Autonomizing Quantum Systems and the Work Operator

Physique – Électricité (14)

Charging and discharging a capacitor (RC)
The current intensity
The coil and inductance
Ohm's Law
Electrical voltage
The RL circuit
The electrical circuit
The capacitor
Alternating current
The transistor and amplification
Resistance organisations
Laws of Loops and Knots
Oscillations in an RLC circuit
Electrical power and energy

Physique – Électromagnétisme (7)

Electromagnetic induction
Laplace's force
The Lorentz force
The magnetic field
The electric field
Maxwell’s equations
Electrostatic potential and energy

Physique – Électrostatique (1)

Coulomb's law

272 / 272 courses

Library The Banach Fixed-Point Theorem

The Inner Workings of the Proof

MCQ

Which statements about the Banach fixed-point theorem are correct?

Possible answersThe proof shows that the sequence of iterates is Cauchy, using a geometric sum with ratio

k

Convergence only occurs if the starting point

u_0

is close enough to the fixed pointThe hypothesis

f(X) \subset X

is essential, otherwise the sequence of iterates is not definedUniqueness comes from the inequality

\lvert c - d \rvert \leq k \lvert c - d \rvert

with

k < 1

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