Related papers: Braiding and exponentiating noncommutative vector …
A very first step to develop non-commutative algebraic geometry is the arithmetic of polynomials in non-commuting variables over a commutative field, that is, the study of elements in free associative algebras. This investigation is…
We explicitly construct Green functions for a field in an arbitrary representation of gauge group propagating in noncommutative instanton backgrounds based on the ADHM construction. The propagators for spinor and vector fields can be…
We discuss the noncommutative generalizations of polynomial algebras which after appropriate completions can be used as coordinate algebras in various noncommutative settings, (noncommutative differential geometry, noncommutative algebraic…
The Rankin--Cohen brackets provide a basic example of ``non-elementary" differential symmetry breaking operators. They can be interpreted as bi-differential operators remarkable for reflecting the structure of fusion rules for holomorphic…
We study, theoretically and experimentally, a 1-parameter family of transformations and their limiting vector field on the space of plane polygons. These transformations are discrete analogs of completely integrable transformation on closed…
Learning useful representations is a key ingredient to the success of modern machine learning. Currently, representation learning mostly relies on embedding data into Euclidean space. However, recent work has shown that data in some domains…
The analogue of the Poisson bracket for the De Donder-Weyl (DW) Hamiltonian formulation of field theory is proposed. We start from the Hamilton- Poincar\'{e}-Cartan (HPC) form of the multidimensional variational calculus and define the…
We initiate the systematic study of modular representations of symmetric groups that arise via the braiding in (symmetric) tensor categories over fields of positive characteristic. We determine what representations appear for certain…
We review and extend a technique for recovering a smooth function from its averages over a wide class of curves in a general region of Euclidean space. The method is based on complexification of the underlying vector fields defining the…
This paper gives a classification of the topology of vector fields which are nowhere tangent to the fibers of a Seifert fibering.
Thermodynamic equivalence between classical many-body system and some auxiliary nonlinear auxiliary field is proved. Connection between Hamiltonians of the many-body system and the auxiliary field is derived.
We classify fields having finitely many finite non-commutative (not necessarily central) division algebras over them. In the process, we introduce the notion of anti-closure of a field and also make comments on fields having a linear…
A rigorous mathematical proof is given of a class of vector identities that provide a way to separate an arbitrary vector field (over a linear space) into the sum of a radial (i.e., pointing toward the radial unit vector) vector field,…
Exponential actions defined by vector configurations provide a universal framework for several constructions of holomorphic dynamics, non-Kaehler complex geometry, toric geometry and topology. These include leaf spaces of holomorphic…
We consider shadowing properties for vector fields corresponding to different type of reparametrisations. We give an example of a vector field which has the oriented shadowing properties, but does not have the standard shadowing property.
The aim of the present paper is to investigate intrinsically the notion of a concircular $\pi$-vector field in Finsler geometry. This generalizes the concept of a concircular vector field in Riemannian geometry and the concept of a…
In the framework of Abstract Differential Geometry, we show that to a given principal sheaf and a representation of its stuctural sheaf in $A^n$, where A is a sheaf of associative, commutative, unital algebras (over R or C), we associate a…
This paper shows that various relevant dynamical systems can be described as vector fields associated to smooth functions via a bracket that defines what we call a Leibniz structure. We show that gradient flows, some dissipative systems,…
Using the recently developed formalism of braided noncommutative field theory, we construct an explicit example of braided electrodynamics, that is, a noncommutative $U(1)$ gauge theory coupled to a Dirac fermion. We construct the braided…
A general approach is proposed to constructing covariant Poisson brackets in the space of histories of a classical field-theoretical model. The approach is based on the concept of Lagrange anchor, which was originally developed as a tool…