Related papers: Analyzability in the context of PDEs and applicati…
This paper provides an overview of the necessary and sufficient conditions for guaranteeing the unique solvability of absolute value equations. In addition to discussing the basic form of these equations, we also address several…
These notes form an extended version of a minicourse delivered in Universite de Montreal (June 2002) within the framework of a NATO workshop ``Normal Forms, Bifurcations and Finiteness Problems in Differential Equations''. The focus is on…
We introduce the study of isolated singularities for a semilinear equation involving the fractional Laplacian. In conformal geometry, it is equivalent to the study of singular metrics with constant fractional curvature. Our main ideas are:…
Solvable structures are exploited in order to find families of explicit solutions to evolution PDEs admitting suitable differential constraints. The effectiveness of the method is verified on several explicit examples.
The explicit integrability of second order ordinary differential equations invariant under time-translation and rescaling is investigated. Quadratic systems generated from the linearisable version of this class of equations are analysed to…
We consider elliptic partial differential equations with diffusion coefficients that depend affinely on countably many parameters. We study the summability properties of polynomial expansions of the function mapping parameter values to…
Early developments leading to renormalizable non-Abelian gauge theories for the weak, electromagnetic and strong interactions, are discussed from a personal viewpoint. They drastically improved our view of the role of field theory, symmetry…
For a general formulation of linearised hybrid inverse problems in impedance tomography, the qualitative properties of the solutions are analysed. Using an appropriate scalar pseudo-differential formulation, the problems are shown to permit…
This work deals with singular stochastic PDEs driven by non-translation invariant differential operators. We describe the renormalized equation for a very large class of spacetime dependent renormalization schemes. Our approach bypasses in…
Utilising the notion of measures of non-compactness and Kamke function of order $\alpha$, we address the question of solvability of fractional differential equations in Banach spaces. In particular, we provide sufficient conditions ensuring…
This set of five lectures provides an introduction to regularity structures and their use for the study of singular stochastic partial differential equations. Two appendices provide some additional informations that enter in the main text…
We present a collection of algorithms which utilize dimensional reduction to perform mesh refinement and study possibly singular solutions of time-dependent partial differential equations. The algorithms are inspired by constructions used…
Reynolds' parametricity originally equips types with proof-irrelevant binary propositional relations over the types. But such relations can also be taken proof-relevant or unary, and described either in an indexed or fibred way.…
This paper studies the expressive and computational power of discrete Ordinary Differential Equations (ODEs), a.k.a. (Ordinary) Difference Equations. It presents a new framework using these equations as a central tool for computation and…
This dissertation introduces executable refinement types, which refine structural types by semi-decidable predicates, and establishes their metatheory and accompanying implementation techniques. These results are useful for undecidable type…
Given a closed two dimensional manifold, we prove a general existence result for a class of elliptic PDEs with exponential nonlinearities and negative Dirac deltas on the right-hand side, extending a theory recently obtained for the regular…
Unification in Description Logics has been introduced as a means to detect redundancies in ontologies. We try to extend the known decidability results for unification in the Description Logic $\mathcal{EL}$ to disunification since negative…
We discuss recent developments in the study of semiorthogonal decompositions of algebraic varieties with an emphasis on their behaviour in families. First, we overview new results concerning homological projective duality. Then we introduce…
Partial differential equations often contain unknown functions that are difficult or impossible to measure directly, hampering our ability to derive predictions from the model. Workflows for recovering scalar PDE parameters from data are…
In this review paper, we explain how to apply Renormalization Group ideas to the analysis of the long-time asymptotics of solutions of partial differential equations. We illustrate the method on several examples of nonlinear parabolic…