The physical models and numerical methodologies of Computational Fluid Dynamics (CFD) are historically linked to the concept continuous medium analysis where continuity, derivation integration defined as limits at a point. first consequence is need extend these notions in multidimensional space by establishing global inertial frame reference order project variables there. In recent decades, eme...

We begin with a short presentation of the basic concepts related to Lie groupoids and Lie algebroids, but the main part of this paper deals with Lie algebroids. A Lie algebroid over a manifold is a vector bundle over that manifold whose properties are very similar to those of a tangent bundle. Its dual bundle has properties very similar to those of a cotangent bundle: in the graded algebra of s...

A discrete (finite-difference) analogue of differential forms is considered, defined on simplicial complexes, including triangulations of continuous manifolds. Various operations are explicitly defined on these forms, including exterior derivative and exterior product. The latter one is non-associative. Instead, as anticipated, it is a part of non-trivial A ∞ structure, involving a chain of pol...

Noncommutative geometric gauge theory is reconstructed based on the superconnection concept. The bosonic action of the Connes-Lott model including the symmetry breaking Higgs sector is obtained by using a new generalized derivative, which consists of the usual 1-form exterior derivative plus an extra element called the matrix derivative, for the curvatures. We first derive the matrix derivative...

The non-commutative differential calculus on the quantum groups SL q (N) is constructed. The quantum external algebra proposed contains the same number of generators as in the classical case. The exterior derivative defined in the constructive way obeys the modified version of the Leibnitz rules.

Using a new local smoothing estimate of the first and third authors, we prove local-in-time Strichartz and smoothing estimates without a loss exterior to a large class of polygonal obstacles with arbitrary boundary conditions and global-in-time Strichartz estimates without a loss exterior to a large class of polygonal obstacles with Dirichlet boundary conditions. In addition, we prove a global-...

The language of modern mechanics is calculus on manifolds, and exterior calculus is an important part of that. It consists of objects like differential forms, general tensors and vector fields on manifolds, and operators that act on these. While the smooth exterior calculus has a long history going back to Cartan, Lie, Grassmann, Hodge, de Rham and many others, the need for a discrete calculus ...

We use the standard notations: differential forms ap, their exterior algebra A, exterior derivative d<p, Hodge's star operator *, coderivative 3f = (_ 1)nP + n 1*d*(P, Laplace-Beltrami operator A = d3 + Md, exterior product (pAg,, inner product (so,*) = fJmpA*4p, and the Dirichlet norm D(sp) = (dep, dp) + (a<p, 5op). For 0forms u the norm reduces to D(u) = (du, du), and Au has the representation

The Exterior-Interior duality expresses a deep connection between the Laplace spectrum in bounded and connected domains in R, and the scattering matrices in the exterior of the domains. Here, this link is extended to the study of the spectrum of the discrete Laplacian on finite graphs. For this purpose, two methods are devised for associating scattering matrices to the graphs. The Exterior -Int...