Related papers: Twisted symmetries of differential equations
I give a short review of the theory of twisted symmetries of differential equations, emphasizing geometrical aspects. Some open problems are also mentioned.
After the introduction of $\lambda$-symmetries by Muriel and Romero, several other types of so called "twisted symmetries" have been considered in the literature (their name refers to the fact they are defined through a deformation of the…
We consider the theory of \emph{twisted symmetries} of differential equations, in particular $\lambda$ and $\mu$-symmetries, and discuss their geometrical content. We focus on their interpretation in terms of gauge transformations on the…
Symmetry properties are at the basis of integrability. In recent years, it appeared that so called "twisted symmetries" are as effective as standard symmetries in many respects (integrating ODEs, finding special solutions to PDEs). Here we…
Following the usual definition of $\lambda$-symmetries of differential equations, we introduce the analogous concept for difference equations and apply it to some examples.
We extend two of the methods previously introduced to find discrete symmetries of differential equations to the case of difference and differential-difference equations. As an example of the application of the methods, we construct the…
We discuss twisted cohomology, not just for ordinary cohomology but also for $K$-theory and other exceptional cohomology theories, and discuss several of the applications of these in mathematical physics. Our list of applications is by no…
We discuss some recent advances concerning the symmetry of stochastic differential equations, and in particular the interrelations between these and the integrability -- complete or partial -- of the equations.
Twisted symmetries, widely studied in the last decade, proved to be as effective as standard ones in the analysis and reduction of nonlinear equations. We explain this effectiveness in terms of a Lie-Frobenius reduction; this requires to…
A twisted ring is a ring endowed with a family of endomorphisms satisfying certain relations. One may then consider the notions of twisted module and twisted differential module. We study them and show that, under some general hypothesis,…
The discrete heat equation is worked out in order to illustrate the search of symmetries of difference equations. It is paid an special attention to the Lie structure of these symmetries, as well as to their dependence on the derivative…
We examine some kinds of discrete symmetries which are dynamically preserved, using the (generalized) Gowdy models of the first kind.
I will sketchily illustrate how the theory of symmetry helps in determining solutions of (deterministic) differential equations, both ODEs and PDEs, staying within the classical theory. I will then present a quick discussion of some more…
We construct explicitly the symmetries of the isospectral deformations as twists of Lie algebras and demonstrate that they are isometries of the deformed spectral triples.
Deligne cohomology can be viewed as a differential refinement of integral cohomology, hence captures both topological and geometric information. On the other hand, it can be viewed as the simplest nontrivial version of a differential…
A novel approach to an old symmetry problem is developed. A new proof is given for the following symmetry problem, studied earlier.
This article is an expository account of the theory of twisted commutative algebras, which simply put, can be thought of as a theory for handling commutative algebras with large groups of linear symmetries. Examples include the coordinate…
Starting with a novel definition of divided differences, this essay derives and discusses the basic properties of, and facts about, (univariate) divided differences.
Over the past years a lot of progress has been made in the understanding of single spin asymmetries in hard scattering processes. We briefly review this subject, covering the non-vanishing of time-reversal odd parton distributions,…
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…