相关论文: Lebesgue integration. Detailed proofs to be formal…
To obtain the highest confidence on the correction of numerical simulation programs for the resolution of Partial Differential Equations (PDEs), one has to formalize the mathematical notions and results that allow to establish the soundness…
Integration, just as much as differentiation, is a fundamental calculus tool that is widely used in many scientific domains. Formalizing the mathematical concept of integration and the associated results in a formal proof assistant helps in…
Formalization of mathematics is a major topic, that includes in particular numerical analysis, towards proofs of scientific computing programs. The present study is about the finite element method, a popular method to numerically solve…
Lebesgue integration is a well-known mathematical tool, used for instance in probability theory, real analysis, and numerical mathematics. Thus its formalization in a proof assistant is to be designed to fit different goals and projects.…
We report on an original formalization of measure and integration theory in the Coq proof assistant. We build the Lebesgue measure following a standard construction that had not yet been formalized in proof assistants based on dependent…
To obtain the highest confidence on the correction of numerical simulation programs implementing the finite element method, one has to formalize the mathematical notions and results that allow to establish the soundness of the method. The…
Formalization of real analysis offers a chance to rebuild traditional proofs of important theorems as unambiguous theories that can be interactively explored. This paper provides a comprehensive overview of the Lebesgue Differentiation…
The Bochner integral is a generalization of the Lebesgue integral, for functions taking their values in a Banach space. Therefore, both its mathematical definition and its formalization in the Coq proof assistant are more challenging as we…
Advancements in modern science have led to an increased prevalence of functional data, which are usually viewed as elements of the space of square-integrable functions $L^2$. Core methods in functional data analysis, such as functional…
We extend in this article the classical imbedding theorems for fractional Lebesgue-Sobolev's spaces into the so-called Grand Lebesgue spaces, with sharp constant evaluation.
As shape analysis of the form presented in Srivastava and Klassen's textbook 'Functional and Shape Data Analysis' is intricately related to Lebesgue integration and absolute continuity, it is advantageous to have a good grasp of the latter…
Given a finite Borel measure $\mu$ on R n and basic semi-algebraic sets $\Omega$\_i $\subset$ R n , i = 1,. .. , p, we provide a systematic numerical scheme to approximate as closely as desired $\mu$(\cup\_i $\Omega$\_i), when all moments…
We develop the theory of mixed finite elements in terms of special inverse systems of complexes of differential forms, defined over cellular complexes. Inclusion of cells corresponds to pullback of forms. The theory covers for instance…
Spatial numerical integration is essential for finite element analysis. Currently, numerical integration schemes, mostly based on Gauss quadrature, are widely used. Herein, we present an alternative semi-analytical approach for mass matrix…
The stability, robustness, accuracy, and efficiency of space-time finite element methods crucially depend on the choice of approximation spaces for test and trial functions. This is especially true for high-order, mixed finite element…
The main purpose of this article is to facilitate the implementation of space-time finite element methods in four-dimensional space. In order to develop a finite element method in this setting, it is necessary to create a numerical…
This paper describes a formalization of discrete real closed fields in the Coq proof assistant. This abstract structure captures for instance the theory of real algebraic numbers, a decidable subset of real numbers with good algorithmic…
We present three projects concerned with applications of proof assistants in the area of programming language theory and mathematics. The first project is about a certified compilation technique for a domain-specific programming language…
Functional integrals are central to modern theories ranging from quantum mechanics and statistical thermodynamics to biology, chemistry, and finance. In this work we present a new method for calculating functional integrals based on a…
The present paper is devoted to a theory of profile decomposition for bounded sequences in \emph{homogeneous} Sobolev spaces, and it enables us to analyze the lack of compactness of bounded sequences. For every bounded sequence in…