Related papers: Formalizing Schwartz functions and tempered distri…
We consider a broad family of test function spaces and their dual (distribution) space. The family includes Gelfand-Shilov spaces, a family of test function spaces introduced by S. Pilipovic. We deduce different characterizations of such…
We study a new method - called Schrodingerisation introduced in [Jin, Liu, Yu, arXiv: 2212.13969] - for solving general linear partial differential equations with quantum simulation. This method converts linear partial differential…
This dissertation is an exposition of Kontsevich's proof of the formality theorem and the classification of deformation quantisation on a Poisson manifold. We begin with an account of the physical background and introduce the Weyl-Moyal…
Evaluation of the angular distribution function of particles scattered in an amorphous medium is improved by deforming the integration path in the Fourier integral representation into the complex plane. That allows us to present the…
Let G be a solvable Lie group endowed with right Haar measure. We define and study a dense Frechet *-subalgebra S of L1(G), consisting of smooth functions rapidly-decreasing at infinity on G. When G is nilpotent, we recover the classical…
Fractional calculus is a generalization of classical theories of integration and differentiation to arbitrary order (i.e., real or complex numbers). In the last two decades, this new mathematical modeling approach has been widely used to…
The paper is devoted to frame expansions in Fr\'echet spaces. First we review some results which concern series expansions in general Fr\'echet spaces via Fr\'echet and General Fr\'echet frames. Then we present some new results on series…
We report on a formalization of the change of variables formula in integrals, in the mathlib library for Lean. Our version of this theorem is extremely general, and builds on developments in linear algebra, analysis, measure theory and…
We present an extension to the $\mathtt{mathlib}$ library of the Lean theorem prover formalizing the foundations of computability theory. We use primitive recursive functions and partial recursive functions as the main objects of study, and…
We give an extension of the theory of relaxation of variational integrals in classical Sobolev spaces to the setting of metric Sobolev spaces. More precisely, we establish a general framework to deal with the problem of finding an integral…
We introduce a new category called Quasi-Nash, unifying Nash manifolds and algebraic varieties. We define Schwartz functions, tempered functions and tempered distributions in this category. We show that properties that hold on affine…
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…
The theory of commutative monads on cartesian closed categories provides a framework where aspects of the theory of distributions and other extensive quantities can be formulated and some results proved. We make explicit a link between our…
This paper considers a new version of fractional Sobolev spaces $\widetilde{\mathcal{W}}_{\mathcal{U}}^{\beta,p}(\mathbb{C}^{n})$ defined using the concept of tempered ultradistributions with respect to the spaces of ultradifferentiable…
In analogy to the classical isomorphism between $\mathcal{L}(\mathcal{S}(\mathbb{R}^{n}) ,\mathcal{S}^{\prime}(\mathbb{R}^{m}) ) $ and $\mathcal{S}^{\prime}(\mathbb{R}^{n+m}) $, we show that a large class of moderate linear mappings acting…
The nonlinear Fourier transform (NFT), a powerful tool in soliton theory and exactly solvable models, is a method for solving integrable partial differential equations governing wave propagation in certain nonlinear media. The NFT…
In this work, a general definition of convolution between two arbitrary four dimensional Lorentz invariant (fdLi) Tempered Ultradistributions is given, in both: Minkowskian and Euclidean Space (Spherically symmetric tempered…
Lie algebras are an important class of algebras which arise throughout mathematics and physics. We report on the formalisation of Lie algebras in Lean's Mathlib library. Although basic knowledge of Lie theory will benefit the reader, none…
Building on Olander's work on algebraic spaces, we prove Orlov's representability theorem relating fully faithful functors and Fourier--Mukai transforms between the bounded derived category of coherent sheaves to the case of smooth, proper,…
Formalizing mathematical proofs using computerized verification languages like Lean 4 has the potential to significantly impact the field of mathematics, it offers prominent capabilities for advancing mathematical reasoning. However,…