The present paper opens a new branch in the theory of variational problems with branching extremals, the investigation of one-dimensional minimal fillings of finite pseudo-metric spaces. On the one hand, this problem is a one-dimensional version of a generalization of Gromov's minimal fillings problem to the case of stratified manifolds (the filling in our case is a weighted graph). On the other hand, this problem is interesting in itself and also can be considered as a generalization of another classical problem, namely, the Steiner problem on the construction of a shortest network joining a given set of terminals. Besides the statement of the problem, we discuss several properties of the minimal fillings, describe minimal fillings of additive spaces, and state several conjectures. We also include some announcements concerning the very recent results obtained in our group, including a formula calculating the weight of the minimal filling for an arbitrary finite pseudo-metric space and the concept of pseudo-additive space which generalizes the classical concept of additive space. We hope that the theory of one-dimensional minimal fillings refreshes the interest in the Steiner problem and gives an opportunity to solve several long standing problems, such as the calculation of the Steiner ratio, in particular the verification of the Gilbert-Pollack conjecture on the Steiner ratio of the Euclidean plane.Bibliography: 33 items.Recent results of our group, see [10], [11], [12], and Sections 8 and 9, show that sometimes it is usefull to allow the weight function of a filling to take negative values. The corresponding objects defined in [10] are referred as generalized fillings. It turns out, that among minimal generalized fillings of an arbitrary pseudo-metric space there exists a minimal filling, see Theorem 8.1. Therefore one can calculate the weight of minimal generalized filling instead of ordinary minimal filling. Investigating the generalized fillings, A. Eremin has found out that Conjectures 7.1 and 7.3 are not valid, but Formula ( ‡) can be turned to a true formula, see Theorem 8.2, by means of the generalized fillings and the concept of a multi-tour introduced by Eremin in [11]. A multi-tour of multiplicity k around a tree G joining M can be defined as a set of boundary paths forming an Euler tour in the 2k-plication of G (that is, the multigraph obtained from G by taking 2k copies of each its edge). The half-perimeter of a multi-tour is defined as the sum of the corresponding kn distances divided by 2k. Eremin proved that even Formula ( †) becomes true providing its left-hand part means the weight of minimal parametric generalized filling, and the maximum in the right-hand part is taken over all multi-tours around G.Section 9 is devoted to the additive spaces that appear in many applications. Recall that a pseudo-metric space M = (M, ρ) is additive, if there exists a weighted tree G = (G, ω) joining M , such that d ω = ρ. In this case the tree G is said to be generating. A well-known description of additive s...

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Spherical Si solar cells were fabricated based on polycrystalline Si spheres with a diameter of 1 mm produced by a dropping method. To decrease the cooling rates of Si spheres by decreasing the convection heat transfer to ambient, the Si spheres were dropped in a free-fall tower at a pressure of 0.2 atm. The conversion efficiency of low-pressure spherical Si solar cells was higher than that of normal-pressure spherical Si solar cells. Both Si spheres were polycrystals that consisting of crystal grains of about 200 mm. The distribution between electric active defects and the crystal quality were characterized by electron beam induced current (EBIC) measurements and transmission electron microscopy (TEM). By EBIC measurements, the lowpressure spherical Si solar cells were clearly observed to have the lower recombination sinks than the normal-pressure spherical Si solar cells. By TEM, the dislocation density in the low-pressure spherical Si solar cells was observed to be more reduced than that in the normal-pressure spherical Si solar cells. The dislocation density in the low-pressure Si spheres decreased because the reduction in the stress generated in the crystal grain.

The aim of this note is to clear some background information and references to readers interested in understanding the current status of the Gilbert-Pollak Conjecture, in particular, to show that A.O. Ivanov and A.A. Tuzhilin were the first who understood the nature of the real gaps in Du-Hwang proof, what has reflected in their publications starting from 2002.

It is proved that the Gromov-Hausdorff metric on the space of compact metric spaces considered up to an isometry is strictly intrinsic, i.e., the corresponding metric space is geodesic. In other words, each two points of this space (each two compact metric spaces) can be connected by a geodesic. For finite metric spaces a geodesic is constructed explicitly.