Related papers: Existence of $\partial$-parameterized Picard-Vessi…
These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new…
The notion of a separable extension is an important concept in Galois theory. Traditionally, this concept is introduced using the minimal polynomial and the formal derivative. In this work, we present an alternative approach to this…
We make explicit certain results around the Galois correspondence in the context of definable automorphism groups, and point out the relation to some recent papers dealing with the Galois theory of algebraic differential equations when the…
Fractional equations have become the model of choice in several applications where heterogeneities at the microstructure result in anomalous diffusive behavior at the macroscale. In this work we introduce a new fractional operator…
The full history recursive multilevel Picard approximation method for semilinear parabolic partial differential equations (PDEs) is the only method which provably overcomes the curse of dimensionality for general time horizons if the…
This is a short survey on the connection between general extension theories and the study of realizations of elliptic operators A on smooth domains in R^n, n > 1. The theory of pseudodifferential boundary problems has turned out to be very…
In this article we study solutions to second order linear difference equations with variable coefficients. Under mild conditions we provide closed form solutions using finite continued fraction representations. The proof of the results are…
An existence result is proved for a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions.…
In this work, we give a variation of parameters formula for nonautonomous linear impulsive differential equations with piecewise constant arguments of generalized type. We cover several cases of differential equations with deviated…
This article is concerned with Galois theory for iterative differential fields (ID-fields) in positive characteristic. More precisely, we consider purely inseparable Picard-Vessiot extensions, because these are the ones having an…
Let $K$ be a number field. We show that, up to allowing a finite set of denominators in the partial quotients, it is possible to define algorithms for $\mathfrak P$-adic continued fractions satisfying the finiteness property on $K$ for…
The aim of these notes is to describe some recent results concerning dispersive estimates for principally normal pseudodifferential operators. The main motivation for this comes from unique continuation problems. Such estimates can be used…
The purpose of this article is to study quasi linear parabolic partial differential equations of second order, posed on a bounded network, satisfying a nonlinear and non dynamical Neumann boundary condition at the vertices. We prove the…
We give some sufficient conditions that ensure oscillations and nonoscillations for nonautonomous impulsive differential equations with piecewise constant arguments of generalized type. We cover several cases of differential equations with…
This paper deals with fundamental properties of Poincar\'e half-maps defined on a straight line for planar linear systems. Concretely, we focus on the analyticity of the Poincar\'e half-maps, their series expansions (Taylor and…
Mixed-dimensional partial differential equations arise in several physical applications, wherein parts of the domain have extreme aspect ratios. In this case, it is often appealing to model these features as lower-dimensional manifolds…
Differential $p$-forms and $q$-vector fields with constant coefficients are studied. Differential $p$-forms of degrees $p=1,2,n-1,n$ with constant coefficients on a smooth $n$-dimensional manifold $M$ are characterized. In the contravariant…
In this paper we describe an algorithm based on the Picard-Vessiot theory that constructs, given any curve invariant under a finite linear algebraic group over the complex numbers, an ordinary linear differential equation whose Schwarz map…
In this article, after recalling and discussing the conventional extremality, local extremality, stationarity and approximate stationarity properties of collections of sets and the corresponding (extended) extremal principle, we focus on…
We propose a theorem that extends the classical Lie approach to the case of fractional partial differential equations (fPDEs) of the Riemann--Liouville type in (1+1) dimensions.