Related papers: 2nd order PDEs: geometric and functional considera…
We compute invariants for the two-variable M\"obius transformation. In particular we are interested in partial differential equations in two dependent and two independent variables that are kept invariant under this transformation.
New approaches to the study of stability of solutions of Set Differential Equations (SDEs) based on convex geometry and the theory of mixed volumes were proposed. The stability of the forms of program solutions of linear SDEs with a stable…
We present conservativeness criteria for sub-Markovian semigroups generated by divergence type operators with specified infinitesimally invariant measures. The conservativeness criteria in this article are derived by $L^1$-uniqueness and…
Foundation models, such as large language models, have demonstrated success in addressing various language and image processing tasks. In this work, we introduce a multi-modal foundation model for scientific problems, named PROSE-PDE. Our…
For quasi-linear elliptic equations we detect relevant properties which remain invariant under the action of a suitable class of diffeomorphisms. This yields a connection between existence theories for equations with degenerate and…
We introduce new aspects in conformal geometry of some very natural second-order differential operators. These operators are termed shift operators. In the flat space, they are intertwining operators which are closely related to symmetry…
The geometrical theory of partial differential equations in the absolute sense, without any additional structures, is developed. In particular the symmetries need not preserve the hierarchy of independent and dependent variables. The order…
Differential equations (DEs) are commonly used to describe dynamic systems evolving in one (ordinary differential equations or ODEs) or in more than one dimensions (partial differential equations or PDEs). In real data applications the…
A new formulation of boundary value problems in gradient elasticity is presented in this work. The main outcome is the construction of partial differential systems of second order, which are typically equivalent with the well known fourth…
Using a generalization of complexes, called 2-complexes, this paper defines and analyzes new Sobolev spaces of matrix fields and their interrelationships within a commuting diagram. These spaces have very weak second-order derivatives. An…
We explore the existence of a class of generalised Laplace maps for third order partial differential operators of the form…
We first strictly expressed the basic notions and research methods of abstract operators, which systematically expounded the main results of abstract operator theory. By combining abstract operators with the Laplace transform, we can easily…
For a second-order linear differential equation with two irregular singular points of rank three, multiple Laplace-type contour integral solutions are considered. An explicit formula in terms of the Stokes multipliers is derived for the…
For discretisations of hyperbolic conservation laws, mimicking properties of operators or solutions at the continuous (differential equation) level discretely has resulted in several successful methods. While well-posedness for nonlinear…
We define a class of discrete operators acting on infinite, finite or periodic sequences mimicking the standard properties of pseudo-differential operators. In particular we can define the notion of order and regularity, and we recover the…
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…
In this article we study abstract and embedded invariants of reduced curve germs via topological techniques. One of the most important numerical analytic invariants of an abstract curve is its delta invariant. Our primary goal is to develop…
This note provides a short guide to dimensional analysis in Lorentzian and general relativity and in differential geometry. It tries to revive Dorgelo and Schouten's notion of 'intrinsic' or 'absolute' dimension of a tensorial quantity. The…
In this paper, we initiate the study of a new interrelation between linear ordinary differential operators and complex dynamics which we discuss in details in the simplest case of operators of order $1$. Namely, assuming that such an…
Local superlinear convergence of the semismooth Newton method usually necessitates assumptions on the uniform invertibility of the utilized, generalized Jacobian matrices, such as, e.g., BD- or CD-regularity. For certain composite-type…