Compatible schemes localize degrees of freedom according to the physical nature of the underlying fields and operate a clear distinction between topological laws and closure relations. For elliptic problems, the cornerstone in the scheme design is the discrete Hodge operator linking gradients to fluxes by means of a dual mesh, while a structure-preserving discretization is employed for the grad...

We give a relatively simple algorithm to do Hodge decomposition on the variational bicomplex based on the imbedding of the total differentiation operator in a Heisenberg algebra. The method is illustrated with several examples taken from the literature. It has been implemented by the authors in the computer algebra system Form [Ver91].

It is shown that in the first order gauge theories under some general assumptions gauge conditions can play the role of new local symmetry generators, while the original constraints become gauge fixing terms. It is possible to associate with this new symmetry a second BRST charge and its anticommutator with the original BRST charge is the Hodge operator of the corresponding cohomology complex.

We give a pedagogical introduction into an old, but unfortunately not very well-known formulation of GR in terms of self-dual two-forms due to Plebanski. Our presentation is rather explicit in that we show how the familiar textbook solutions: Schwarzschild, Volkoff-Oppenheimer, as well as those describing the Newtonian limit, graviton and homogeneous isotropic Universe can be obtained within th...

It is shown that in the first order gauge theories under some general assumptions gauge conditions can play the role of new local symmetry generators, while the original constraints become gauge fixing terms. It is possible to associate with this new symmetry a second BRST charge and its anticommutator with the original BRST charge is the Hodge operator of the corresponding cohomology complex.

We study the continuum limit for Dirac–Hodge operators defined on n dimensional square lattice $$h\mathbb {Z}^n$$ as h goes to 0. This result extends a first order discrete differential operator known convergence of Schrödinger their continuous counterpart. To be able define such analog, we start by defining an alternative framework higher–dimensional calculus. believe that this framework, gene...

We first present the natural definitions of horizontal differential, divergence (as an adjoint operator), and a $p$-harmonic form on Finsler manifold. Next, we prove Hodge-type theorem for manifold in sense that $p$-form is harmonic if only Laplacian vanishes. This viewpoint provides new appropriate definition vector fields geometry. approach leads to Bochner-Yano type classification based Ricc...

Abstract We study a p -adic Maass–Shimura operator in the context of Mumford curves defined by [15]. prove that this arises from splitting Hodge filtration, thus answering question also relation with generalized Heegner cycles, spirit [1, 4, 19, 28].

We introduce a Hodge operator in a framework of noncommutative geometry. The complete integrability of 2-dimensional classical harmonic maps into groups (σ-models or principal chiral models) is then extended to a class of ‘noncommutative’ harmonic maps into matrix algebras.