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It is well-known that every polyhedral cone is finitely generated (i.e. polytopal), and vice versa. Surprisingly, the two notions differ almost always for non-commutative versions of such cones. This was obtained as a byproduct in [3] and later generalized in [8]. In this note we give a direct and constructive proof of the statement. Our proof yields a new and surprising quantitative result: the difference of the two notions can always be seen at the first level of non-commutativity, i.e. for matrices of size 2, independent of dimension and complexity of the initial convex cone. This also answers an open question from [8].

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Preliminary studies on the design of lead-free solders precursors by wet chemistry methods are presented. The main objective is to assess the impact of the way of hydroxide precipitates preparation on the metal elements content of the precipitates. Namely, ternary hydroxide mixtures of three systems: a. Cu(II), Bi(III), Sn(II); b. Cu(II), Bi(III), Sb(III); and c. Cu(II), Bi(III), Zn(II) were prepared, firstly, by single-element precipitation and, secondly, by co-precipitation. Thereafter, all mixtures were reduced by using hydrogen gas. Both, the initial mixtures and the reduced samples were studied by X-ray diffraction, optical and scanning electron microscopes. The chemical compositions of the precipitates were determined experimentally and their dependence on the pH was verified. It was found that alloying occurred during the reduction procedure, but in some cases the reduction was not complete (i.e. oxide phases rest in the samples). This might be a huge obstacle to use such an approach for the preparation of lead-free solders. Moreover, the materials obtained after reductions apparently are bulk alloys, thus, the preparation of small-sized metal particles would be a challenge. Another key feature to be addressed in future studies is the correlation between the chemical compositions of the parent solution and these of the corresponding precipitates