Related papers: Shape Analysis, Lebesgue Integration and Absolute …
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.…
This book is based on notes compiled over the many years I have been teaching the course "Applied Functional Analysis" in the first year of the Master programme at Delft University of Technology, for students with previous exposure to the…
In this article, we propose a general theory of integration of the Riemann and Lebesgue types with respect to arbitrary measures and functions, connected by a continuous bilinear product, with values in abstract vector spaces endowed with a…
The basic problem of shape complementarity analysis appears fundamental to applications as diverse as mechanical design, assembly automation, robot motion planning, micro- and nano-fabrication, protein-ligand binding, and rational drug…
In this paper we give a detailed measure theoretical analysis of what we call sum-level sets for regular continued fraction expansions. The first main result is to settle a recent conjecture of Fiala and Kleban, which asserts that the…
This text grew out of notes I have used in teaching a one quarter course on integration at the advanced undergraduate level. My intent is to introduce the Lebesgue integral in a quick, and hopefully painless, way and then go on to…
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…
Measure Theory and Integration is exposed with the clear aim to help beginning learners to perfectly master its essence. In opposition of a delivery of the contents in an academic and vertical course, the knowledge is broken into exercises…
The advent of high resolution imaging has made data on surface shape widespread. Methods for the analysis of shape based on landmarks are well established but high resolution data require a functional approach. The starting point is a…
In this article, we introduce the notion of a curved absolute $\mathcal{L}_\infty$-algebra, a structure that behaves like a curved $\mathcal{L}_\infty$-algebra where all infinite sums of operations are well-defined by definition. We develop…
In classical analysis, Lebesgue first proved that $\mathbb{R}$ has the property that each Riemann integrable function from $[a,b]$ into $\mathbb{R}$ is continuous almost everywhere. This property is named as the Lebesgue property. Though…
The prime objective of this paper is to develop the notion of absolute continuity of functions on a more general setting outside $\R$. For this we have considered a topological space which is a measure space as well. We have built axioms…
Given a non-archimedean real closed field with archimedean value group which contains the reals, we establish for the category of semialgebraic sets and functions a full Lebesgue measure and integration theory such that the main results…
Many innovative applications require establishing correspondences among 3D geometric objects. However, the countless possible deformations of smooth surfaces make shape matching a challenging task. Finding an embedding to represent the…
This paper is the second in a series by the author and collaborators devoted to the study of geometric and analytic properties of nonlinear Lebesgue spaces, that is, L^p spaces of mappings taking values in arbitrary metric spaces. The…
We study Lebesgue integration of sums of products of globally subanalytic functions and their logarithms, called constructible functions. Our first theorem states that the class of constructible functions is stable under integration. The…
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…
Multidimensional integration by parts formulas apply under the standard assumption that one of the functions is continuous and the other has bounded Hardy-Krause variation. Motivated by recently developed results in the probabilistic…
We begin the study of completeness of affine connections, especially those on statistical manifolds as well as on affine hypersurfaces. We collect basic facts, prove new theorems and provide examples with remarkable properties.
This article explores the concept of absoluteness in the context of mathematical analysis, focusing specifically on the Riemann integral on $\mathbb{R}^{n}$. In mathematical logic, "absoluteness" refers to the invariance of the truth value…