The existence of the split extension classifier of a crossed module in the category of associative algebras is investigated. According to the equivalence of categories XAss ' Cat-Ass we consider this problem in Cat-Ass. This category is not a category of interest, it satisfies its all axioms except one. The action theory developed in the category of interest is adapted to the new type of catego...

Let Γ denote a Q-polynomial distance-regular graph with diameter D ≥ 3 and standard module V . Recently Ito and Terwilliger introduced four direct sum decompositions of V ; we call these the (μ, ν)–split decompositions of V , where μ, ν ∈ {↓, ↑}. In this paper we show that the (↓, ↓)–split decomposition and the (↑, ↑)–split decomposition are dual with respect to the standard Hermitian form on V...

Code for the Guiding Lines Style Module We show here the python code of the style module that was used the for the “Guiding Lines” layer of Fig. 4 in the paper. The main aspects of this style module are the recursive split used to make strokes start and stop at points of highest curvature and the shader that displaces the vertices of the stroke along its tangent. The functions, predicates, shad...

We consider the Lie algebra of all vector fields on a contact manifold as a module over the Lie subalgebra of contact vector fields. This module is split into a direct sum of two submodules: the contact algebra itself and the space of tangent vector fields. We study the geometric nature of these two modules.

An open problem in robotics is that of using vision to identify a robot’s own body and the world around it. Many models attempt to recover the traditional C-space parameters. Instead, we propose an alternative C-space by deriving generalized coordinates from n images of the robot. We show that the space of such images is bijective to the motion space, so these images lie on a manifold V homeomo...

This paper presents the regularity of a split-quaternionic function and corresponding split- Cauchy-Riemann system split quaternion. The properties an inverse implicit mapping theory for map are investigated. In addition, proposes definition expression biregular in open set C2. obtained results illustrated with some examples.