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.

In this article, we introduce a formal definition for integral arithmagons. Informally, an integral arithmagon is a polygonal figure with integer labeled vertices and edges in which, under a binary operation, adjacent vertices equal the included edge. By considering the group of automorphisms for the associated graph, we count the number of integral arithmagons whose exterior sum or product equ...

From algebraic K-theory, we show that there exists a spectral sequence that has real cohomology of SL(n)(Z) as its E(1)-terms and converges to the tensor product of a polynomial algebra and an exterior algebra. On the basis of this spectral sequence, we discovered several families of real unstable cohomology classes of SL(n)(Z).

and Applied Analysis 3 We denote by ∧l ∧l R the space of l-covectors in R and the direct sum ∧ R n ⊕ l 0 ∧l R 2.1 is a graded algebra with respect to the wedge product ∧. We will make use of the exterior derivative: d : C∞ ( Ω,∧l ) −→ C∞ ( Ω,∧l 1 ) 2.2 and its formal adjoint operator d∗ −1 nl 1 ∗ d∗ : C∞ ( Ω,∧l 1 ) −→ C∞ ( Ω,∧l ) , 2.3 known as the Hodge codifferential, where the symbol ∗ denot...

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

We construct projections from HΛk(Ω), the space of differential k forms on Ω which belong to L2(Ω) and whose exterior derivative also belongs to L2(Ω), to finite dimensional subspaces of HΛk(Ω) consisting of piecewise polynomial differential forms defined on a simplicial mesh of Ω. Thus, their definition requires less smoothness than assumed for the definition of the canonical interpolants base...

The fundamental quantities of classical dynamics, such as position, velocity and acceleration, are expressed in terms of the Cartan tetrad, and the first Cartan structure equation is used to develop classical dynamics. The velocity is defined as the covariant exterior derivative of the position, and the acceleration as the exterior covariant derivative of the velocity. The resulting terms are a...