The Hodge dual operator ∗ is one of the 3 basic operations on differential forms. (The other 2 are wedge product ∧ and exterior differentiation d.) However most treatments consider only positive-definite inner products, and there are at least 2 standard ways of generalizing this to inner products of arbitrary signature. We outline here a construction of the Hodge dual operator which works for a...

If f f denotes the truncated Lusztig Fourier transform, we show that image by of normalized characteristic function a Coxeter element is alte...

A Stein covering of a complex manifold may be used to realise its analytic cohomology in accordance with the Čech theory. If, however, the Stein covering is parameterised by a smooth manifold rather than just a discrete set, then we construct a cohomology theory in which an exterior derivative replaces the usual combinatorial Čech differential. Our construction is motivated by integral geometry...

Natural analogs of Lie brackets on affine bundles are studied. In particular, a close relation to Lie algebroids and a duality with certain affine analog of Poisson structure is established as well as affine versions of the complete lift and the Cartan exterior calculus.

We consider a cyclic group of order p acting on a module incharacteristic p and show how to reduce the calculation of the symmetric algebra to that of the exterior algebra. Consider a cyclic group of order p acting on a polynomial ring S = k[x1, . . . , xr], where k is a field of characteristic p; this is equivalent to the symmetric algebra S∗(V ) on the module V generated by x1, . . . , xr. We...

A general method for studying boundary value problems for linear and for integrable nonlinear partial differential equations in two dimensions was introduced in [3]. For linear equations in a convex polygon [2,4,5], this method: (a) Expresses the solution q(x,y) in the form of an integral (generalized inverse Fourier transform) in the complex k-plane involving a certain function q̂(k) (generaliz...

<p style='text-indent:20px;'>This paper treats the stationary Stokes problem in exterior domain of <inline-formula><tex-math id="M2">\begin{document}$ {{\mathbb{R}}}^3 $\end{document}</tex-math></inline-formula> with Navier slip boundary condition. The behavior at infinity data and solution are determined by setting id="M3">\begin{document}$ L^p $\end{document}&...

We provide an overview on the application of the exterior calculus of differential forms to the ab initio formulation of lattice field theories, with a focus on irregular or " random " lattices.

The π-exterior derivative d, which is the Finslerian generalization of the (usual) exterior derivative d of Riemannian geometry, is defined. The notion of a d-closed vector field is introduced and investigated. Various characterizations of d-closed vector fields are established. Some results concerning d-closed vector fields in relation to certain special Finsler spaces are obtained. 1