Related papers: Report on the absolute differential equations I
Symmetry groups allow to transform solutions of differential equations continuously into other solutions. This property can be used for the observability analysis of infinite-dimensional systems with input and output. In this contribution,…
We present a new scheme of defining invariant observables for general relativistic systems. The scheme is based on the introduction of an observer which endowes the construction with a straightforward physical interpretation. The…
Conventional finite-difference schemes for solving partial differential equations are based on approximating derivatives by finite-differences. In this work, an alternative theory is proposed which view finite-difference schemes as…
The quest for complete observables in general relativity has been a longstanding open problem. We employ methods from descriptive set theory to show that no complete observable on rich enough collections of spacetimes is Borel definable. In…
These lecture notes provide an introduction to the theory and application of symmetry methods for ordinary differential equations, building on minimal prerequisites. Their primary purpose is to enable a quick and self-contained approach for…
We investigate a large class of infinitesimal, but fully nonlinear in the field, transformations of the Galileon and search for extended symmetries. The transformations involve powers of the coordinates $x$ and the field $\pi$ up to any…
We study deterministic and quantum dynamics from a constructive "finite" point of view, since the introduction of a continuum, or other actual infinities in physics poses serious conceptual and technical difficulties, without any need for…
We study substitutions on countably infinite alphabet (without compactification) as Borel dynamical systems. We construct stationary and non-stationary generalized Bratteli-Vershik models for a class of such substitutions, known as left…
We completely characterize all nonlinear partial differential equations leaving a given finite-dimensional vector space of analytic functions invariant. Existence of an invariant subspace leads to a re duction of the associated dynamical…
We study the projective systems in both continuous and discrete settings. These systems are linearizable by construction and thus, obviously, integrable. We show that in the continuous case it is possible to eliminate all variables but one…
We discuss a Lie algebraic and differential geometry construction of solutions to some multidimensional nonlinear integrable systems describing diagonal metrics on Riemannian manifolds, in particular those of zero and constant curvature.…
We consider a system of differential equations and obtain its solutions with exponential asymptotics and analyticity with respect to the spectral parameter. Solutions of such type have importance in studying spectral properties of…
A class of two-dimensional globally scale-invariant, but not conformally invariant, theories is obtained. These systems are identified in the process of discussing global and local scaling properties of models related by duality…
An extension of the Dirac procedure for the quantization of constrained systems is necessary to address certain issues that are left open in Dirac's original proposal. These issues play an important role especially in the context of…
These notes provide an opportunity to discover the beauty of Bourbaki set theory, and I hope that they will facilitate the task to those who find it difficult to read this book, one of the most critical elements of the mathematics of…
This paper addresses questions regarding controllability for `generic parameter' dynamical systems, i.e. the question whether a dynamical system is `structurally controllable'. Unlike conventional methods that deal with structural…
Can a large system be fully characterized using its subsystems via inductive reasoning? Is it possible to completely reduce the behavior of a complex system to the behavior of its simplest "atoms"? In the following paper we answer these…
We continue the study of symmetries in the Lagrangian formalism of arbitrary order with the help of the so-called Anderson-Duchamp-Krupka equations. For the case of second-order equations and arbitrary vector fields we are able to establish…
We review the status of a certain (infinite) class of four-dimensional generally covariant theories propagating two degrees of freedom that are formulated without any direct mention of the metric. General relativity itself (in its Plebanski…
In this note we present some ideas on when Lie symmetries, both point and generalized, can depend on arbitrary functions. We show on a few examples, both in partial differential and partial difference equations when this happens. Moreover…