Related papers: Are there Contact Transformations for Discrete Equ…

The exact and approximate solutions of singular integro-differential equations relating to the problems of interaction of an elastic thin finite or infinite non-homogeneous patch with a plate are considered, provided that the materials of…

We provide explicit formulas for integrating multiplicative forms on local Lie groupoids in terms of infinitesimal data. Combined with our previous work [8], which constructs the local Lie groupoid of a Lie algebroid, these formulas produce…

Let $n \in \mathbb{Z}_{\geq 3}$ be given. We prove Lebesgue-almost everywhere pointwise inversion formulae for the Siegel transforms in the geometry of numbers. These inversion formulae are quite general; for instance, they are valid for…

We define an integer-valued non-degenerate bi-invariant metric (the discriminant metric) on the universal cover of the identity component of the contactomorphism group of any contact manifold. This metric has a very simple geometric…

A contact singularity is a normal singularity $(V,0)$ together with a holomorphic contact form $\eta$ on $V\backslash$ Sing $V$ in a neighbourhood of 0, i.e. $\eta\wedge (d\eta)^r$ has no zero, where dim $V=2r+1$. The main result of this…

In this paper, we investigate the reduction process of a contact Lagrangian system whose Lagrangian is invariant under a group of symmetries. We give explicit coordinate expressions of the resulting reduced differential equations, the…

We show that the classification of simple finite group schemes over an algebraically closed field reduces to the classification of abstract simple finite groups and of simple restricted Lie algebras in positive characteristic. Both these…

We study a quantum Yang-Baxter structure associated with non-ultralocal lattice models. We discuss the canonical structure of a class of integrable quantum mappings, i.e. canonical transformations preserving the basic commutation relations.…

We show that for a big class of contact manifolds the groups of order $\leq n$ invariants (with values in an arbitrary Abelian group) of Legendrian, of transverse and of framed knots are canonically isomorphic. On the other hand for an…

We consider two infinite classes of ordinary difference equations admitting Lax pair representation. Discrete equations in these classes are parameterized by two integers $k\geq 0$ and $s\geq k+1$. We describe the first integrals for these…

In this paper, we present a method to identify integrable complex nonlinear oscillator systems and construct their solutions. For this purpose, we introduce two types of nonlocal transformations which relate specific classes of nonlinear…

This study will explicitly demonstrate by example that an unrestricted infinite and forward recursive hierarchy of differential equations must be identified as an unclosed system of equations, despite the fact that to each unknown function…

A gauge transformation in quantum electrodynamics involves the product of field operators at the same space-time point and hence does not have a well-defined meaning. One way to avoid this difficulty is to generalize the gauge…

We study local Lie algebras of pairs of functions which generate infinitesimal symmetries of almost-cosymplectic-contact structures of odd dimensional manifolds.

Recent years have seen the concept of global symmetry extended to non-invertible (or categorical) symmetries, for which composition of symmetry generators is not necessarily invertible. Such non-invertible symmetries lead to a…

In a way similar to the continuous case formally, we define in different but equivalent manners the difference discrete connection and curvature on discrete vector bundle over the regular lattice as base space. We deal with the difference…

We write down three kinds of scale transformations {\tt i-iii)} on the noncommutative plane. {\tt i)} is the analogue of standard dilations on the plane, {\tt ii)} is a re-scaling of the noncommutative parameter $\theta$, and {\tt iii)} is…

We prove that a K-contact Lie group of dimension five or greater is the central extension of a symplectic Lie group by complexifying the Lie algebra and applying a result from complex contact geometry, namely, that, if the adjoint action of…

Transformation media are at the heart of invisibility devices, perfect lenses and artificial black holes. In this paper, we consider their quantum theory. We show how transformation media map quantum electromagnetism in physical space to…

Starting from square-integrable wave functions on a Lie group, we build an invertible Fourier transform mapping them on wave functions on the dual of the Lie algebra. This is a group-theoretic version of the map from position space to…