Related papers: Measure Construction by Extension in Dependent Typ…
We lay out novel foundations for the computer-aided verification of guaranteed bounds on expected outcomes of imperative probabilistic programs featuring (i) general loops, (ii) continuous distributions, and (iii) conditioning. To handle…
We construct measure which determines a two-variable mean in a very natural way. Using that measure we can extend the mean to infinite sets as well. E.g. we can calculate the geometric mean of any set with positive Lebesgue measure. We also…
We present a new type of integral that is supposed to extend the usability of the Lebesgue integral in certain types of investigations. It is based on the Hausdorff dimension and measure. We examine the basic properties of the integral and…
Lebesgue's dominated convergence theorem is a crucial pillar of modern analysis, but there are certain areas of the subject where this theorem is deficient. Deeper criteria for convergence of integrals are described in this article.
We propose the following way of constructing quantum measure in Regge calculus: the full discrete Regge manifold is made continuous in some direction by tending corresponding dimensions of simplices to zero, then functional integral measure…
Libraries of formalized mathematics use a possibly broad range of different representations for a same mathematical concept. Yet light to major manual input from users remains most often required for obtaining the corresponding variants of…
In the present paper, we study a set that can be treated as a generalised set of subsums for a geometric series. This object was discovered independently in various mathematical aspects. For instance, it is closely related to various…
A universal framework for the joint measurement of multiple localized observables in quantum field theory satisfying spacetime locality and compositionality is still lacking. We present an approach to the problem that is based on the one…
We give a simple and short proof of the classical Lebesgue decomposition theorem of measures via the Riesz orthogonal decomposition theorem of Hilbert spaces. The tools we employ are elementary Hilbert space techniques.
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…
We give a number of formal proofs of theorems from the field of computable analysis. Many of our results specify executable algorithms that work on infinite inputs by means of operating on finite approximations and are proven correct in the…
We introduce the Markov extension, represented schematically as a tower, to the study of dynamical systems with holes. For tower maps with small holes, we prove the existence of conditionally invariant probability measures which are…
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…
The theory of integration over infinite-dimensional spaces is known to encounter serious difficulties. Categorical ideas seem to arise naturally on the path to a remedy. Such an approach was suggested and initiated by Segal in his…
Assurance cases are often required as a means to certify a critical system. Use of formal methods in assurance can improve automation, and overcome problems with ambiguity, faulty reasoning, and inadequate evidentiary support. However,…
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…
We consider nonlinear, or "event-dependent", sampling, i.e. such that the sampling instances {tk} depend on the function being sampled. The use of such sampling in the construction of Lebesgue's integral sums is noted and discussed as…
Recent advances in large language models have demonstrated impressive capabilities in mathematical formalization. However, existing benchmarks focus on logical verification of declarative propositions, often neglecting the task of…
Let $X$ be a complete measure space of finite measure. The Lebesgue transform of an integrable function $f$ on $X$ encodes the collection of all the mean-values of $f$ on all measurable subsets of $X$ of positive measure. In the problem of…
We describe a formalization of higher-order rewriting theory and formally prove that an AFS is strongly normalizing if it can be interpreted in a well-founded domain. To do so, we use Coq, which is a proof assistant based on dependent type…