Related papers: Lebesgue integration. Detailed proofs to be formal…
The $\ell$FEM MATLAB package provides a simple, efficient, and flexible implementation of isoparametric finite elements in bulk domains and on surfaces. The finite element matrix assemblies are based on MATLAB's paged operators and…
Let $(X,d_X,\mu)$ be a metric measure space where $X$ is locally compact and separable and $\mu$ is a Borel regular measure such that $0 <\mu(B(x,r)) <\infty$ for every ball $B(x,r)$ with center $x \in X$ and radius $r>0$. We define…
In this work, we present an efficient numerical implementation of the finite element method for modal analysis that leverages various symmetry operations, including spatial symmetry in point groups and space-time symmetry in…
This paper contains a new elementary proof of the Fundamental Theorem of Calculus for the Lebesgue integral. The hardest part of our proof simply concerns the convergence in ${\rm L}^1$ of a certain sequence of step functions, and we prove…
In this article, we prove the existence of extremal functions in higher-order affine Sobolev inequalities. Proofs rely on concentration-compactness methods in spaces of integer or fractional regularity. The tools we use, available in spaces…
This extended abstract is about an effort to build a formal description of a triangulation algorithm starting with a naive description of the algorithm where triangles, edges, and triangulations are simply given as sets and the most complex…
It has been known that sharp Sobolev embeddings into weak Lebesgue spaces are non-compact but the question of whether the measure of non-compactness of such an embedding equals to its operator norm constituted a well-known open problem. The…
The aim of this work is to certify lower bounds for real-valued multivariate functions, defined by semialgebraic or transcendental expressions. The certificate must be, eventually, formally provable in a proof system such as Coq. The…
In this article we consider the widely used immersed finite element method (IFEM), in both explicit and implicit form, and its relationship to our more recent one-field fictitious domain method (FDM). We review and extend the formulation of…
Numerical and symbolic methods for optimization are used extensively in engineering, industry, and finance. Various methods are used to reduce problems of interest to ones that are amenable to solution by such software. We develop a…
For performance and verification in machine learning, new methods have recently been proposed that optimise learning systems to satisfy formally expressed logical properties. Among these methods, differentiable logics (DLs) are used to…
This paper recalls the principles of the finite-element methods (FEM) theory and declines its application in the EN-MME group, for the numerical modelling and study of particle accelerator equipment. Implicit and explicit methods are…
We introduce a symbolic method for the evaluation of definite integrals containing combinations of various functions, including exponentials, logarithm and products of Bessel functions of different types. The method we develop is naturally…
The safety of automated driving systems must be justified by convincing arguments and supported by compelling evidence to persuade certification agencies, regulatory entities, and the general public to allow the systems on public roads.…
A finite element methodology for large classes of variational boundary value problems is defined which involves discretizing two linear operators: (1) the differential operator defining the spatial boundary value problem; and (2) a Riesz…
Several novel imaging and non-destructive testing technologies are based on reconstructing the spatially dependent coefficient in an elliptic partial differential equation from measurements of its solution(s). In practical applications, the…
We define an integral of real-valued functions with respect to a measure that takes its values in the extended positive cone of a partially ordered vector space $E$. The monotone convergence theorem, Fatou's lemma, and the dominated…
Necessary and sufficient conditions are presented for a fractional Orlicz-Sobolev space on $\rn$ to be continuously embedded into a space of uniformly continuous functions. The optimal modulus of continuity is exhibited whenever these…
This paper deals with the \emph{integral} version of the Dirichlet homogeneous fractional Laplace equation. For this problem weighted and fractional Sobolev a priori estimates are provided in terms of the H\"older regularity of the data. By…
We have applied the Finite Element Method to the self-consistent electronic structure calculations of molecules and solids for the first time. In this approach all the calculations are performed in "real space" and the use of non-uniform…