Since the very foundations of geometry, Euclidean lattices have fascinated due to their ability to orderly arrange space in a regular manner, a phenomenon particularly reflected in sphere packings. This mathematical structuring is not merely an abstract curiosity; it carries with it concrete and revolutionary applications, especially in the context of post-quantum cryptography. The theory … Read more
cryptography, euclidean networks, mathematics, network theory, stacking
Matrix analysis: advanced norms and decompositions
Marion Lefevre
The increasing complexity of systems modeled in applied mathematics, physics, and computer science has driven research to deepen the study of matrices beyond simple algebraic calculations. Matrix analysis has now become an essential discipline for understanding, manipulating, and exploiting the properties of matrices in situations where the size and complexity of the data render classical … Read more
advanced mathematics, linear algebra, matrix analysis, matrix decompositions, matrix norms
Monte Carlo methods: simulation and integration
Marion Lefevre
In brief: Understanding the fundamentals of Monte Carlo methods in simulation Monte Carlo methods represent a class of algorithms based on the use of statistics obtained by random repetition of numerical experiments to estimate complex phenomena. This approach stands out for its ability to model situations where variables are subjected to varied and uncertain probabilities. … Read more
integration techniques, monte carlo integration, Monte Carlo methods, monte carlo simulation, numerical simulation
Fractal analysis: dimension and self-similar measurements
In the quest to understand the deep structure of the complex shapes we encounter in nature and dynamic systems, fractal analysis emerges as a fundamental tool. This discipline reveals the extraordinary richness of fractal geometry, allowing for the characterization of the complexity and self-similarity often present in seemingly chaotic objects. From mountain ranges to networks … Read more
Orthogonal polynomials: approximation and quadrature
Orthogonal polynomials are at the heart of many advancements in numerical analysis and functional approximation. These mathematical tools allow for the precise representation of complex functions while controlling the error associated with this approximation. More than just a theoretical concept, they find practical applications in the calculation of numerical integrals, the solving of linear systems, … Read more
Sub-Riemannian Geometry: Metrics and Constrained Geodesics
Subriemannian geometry stands out as a fundamental field in the study of spaces where movements are subject to non-trivial constraints. These spaces, often referred to as Carnot spaces, are characterized by the presence of non-holonomic distributions that limit accessible directions. In this context, classical notions of distance and metrics evolve into adapted concepts, such as … Read more