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The admissible derivations assigned to w are denoted by ΔW (w) ⊆ Δ and we call the cone w ΔW (w) the Janet cone of w with respect to W . This construction ensures disjointness of cones but not necessarily that the union of cones equals W Δ . For the Janet completion a finite set W ⊃ W is successively created by adding any w ˜ = ∂i wj ∈ w∈W w ΔW (w) to W , where wj ∈ W and ∂i ∈ Δ \ ΔW (wj ). This leads to the disjoint Janet decomposition W Δ= w ΔW (w) w∈W that separates a Δ-closed set W into finitely many cones w ΔW (w) after finitely many steps.

Algorithm: 1: x ← ld(p) 2: while exists q= ∈ (ST )= and i1 , . .

Moreover, the matrix formalism and symbolic formulae for some matrix operations lead us to more efficient computational procedures for the distributed and parallel computational process. Additional computational flows are connected with the next two procedures. The first of them is devoted to visualization of all necessary information including auxiliary procedures (for example, analysis of images using differential geometry methods) and space-charge force computing. Here we can use the methods proposed in our previous publications (see, for example, [16,17,18]).

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