The exterior degree d∧(G) of a finite group G has been recently introduced by Rezaei and Niroomand in order to study the probability that two given elements x and y of G commute in the nonabelian exterior square G ∧ G. This notion is related with the probability d(G) that two elements of G commute in the usual sense. Motivated by a paper of Erovenko and Sury of 2008, we compute the exterior deg...

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...

The method of Gröbner bases in commutative polynomial rings introduced by Buchberger (e.g. [1]) is a well-known and very important tool in polynomial ideal theory, for example in solving the ideal membership problem. XIDEAL extends the method to exterior algebras using algorithms from [2] and [3]. There are two main departures from the commutative polynomial case. First, owing to the non-commut...

This article deals with vector valued diierential forms on C 1-manifolds. As a generalization of the exterior product, we introduce an operator that combines Hom(N s (W); Z)-valued forms with Hom(N s (V); W)-valued forms. We discuss the main properties of this operator such as (multi)linearity, associa-tivity and its behavior under pullbacks, push-outs, exterior diierentiation of forms, etc. Fi...

In this paper, working over Z(p) and using algebra perturbation results from [18], p-minimal homological models of twisted tensor products (TTPs) of Cartan’s elementary complexes are obtained. Moreover, making use of the notion of indecomposability of a TTP, we deduce that a homological model of a indecomposable p-minimal TTP of length ` (` ≥ 2) of exterior and divided power algebras is a tenso...

For any Lie algebra g there is a notion of Leibniz cohomology HL(g), which is defined like the classical Lie cohomology, but with the n-th tensor product g⊗n in place of the n-th exterior product Λ g. This Leibniz cohomology is defined on a larger class of algebras : the Leibniz algebras (cf. [L1], [L2]). A Leibniz algebra is a vector space equipped with a product satisfying a variation of the ...

Let EC denote the complexified adjoint Lie-algebra bundle associated to P . For the purposes of this summary, we will assume that E = EC; the more general case will be dealt with in section 7. Let dt : Ω (M ;EC)→Ω (M ;EC) denote the exterior derivative corresponding to At for each t. At t = 0, we wish to calculate the dimension of ker(D0), which gives the number of eigenvalues λα(t) of Dt passi...

Let M̃ be a (2m+ 1)-dimensional almost contact manifold with almost contact structure (φ,ξ,η), that is, a global vector field ξ, a (1,1) tensor field φ, and a 1-form η on M̃ such that φ2X =−X +η(X)ξ, η(ξ) = 1 for any vector field X on M̃. We consider a product manifold M̃×R, whereR denotes a real line. Then a vector field on M̃×R is given by (X , f (d/dt)), where X is a vector field tangent to M̃, t ...

This study describes a “New Automobile Product Development Design Model” (NA-PDDM) using “Dual Corporate Engineering Strategy” for the innovation of automobile product engineering fundamental. Specifically, NA-PDDM contains both “Exterior design strategy” and “Driving performance by “Customer Science Principle” (CSP). Concretely, foundation consists “Automobile Exterior Model with 3 Core Method...