In this paper, we analyse results of the 1 5 th International Henryk Wieniawski Violin Competition by comparing the properties of its results network to the properties of generic networks of votings. Suppose that a competition Comp is given. In this competition, k contestants are rated by n jurors. Suppose that the Borda count is used as a voting method, i.e. every juror gives k points for the best contestant, k−1 points for the second best contestant, and so on. In particular, the worts contestant gets 1 point. For such a competition, we create a weighted network N ( Comp ) in the following way. The node set of N ( Comp ) corresponds to jurors and the link set of N ( Comp ) consists of all links { J s , J t : s ≠ t } . For link l st connecting nodes J s and J t , we assign the weight w ( l st ) = w st , where w st = LF 2 ( α s α t − 1 ) . Here LF 2 is a Lehmer norm on the permutation group S k , whereas α s and α t denotes the votes of jurors J s and J t , respectively. In particular, for i = 1 , 2 , . . . , k , α s ( i ) is the number of points given to the i-th contestant by juror J s . The similar holds for juror J t . Note that α s and α t can be considered as elements of S k . Suppose now that the probability measure P is given on space V of all possible votings of a single juror, i.e. on space S k . Suppose that every juror votes independently according to P. We repeat such a voting process 100 times and for every j = 1 , 2 , . . . , 1 0 0 , we create a network N j in the way described above. In this paper, we compare some statistical properties of networks N j , for probability measures P being the convex combinations of two Dirac probability measures and a uniform probability measure, to the properties of network of jurors’ votings in the 2016 Wieniawski Competition.

In this paper we prove that the Pontriagin sphere and the Pontriagin nonorientable surface occurs as the Gromov boundary of a 7-systolic group acting geometrically on 7-systolic normal pseudomanifold of dimension 3.MSC: 20F67, 20F65, 57P99, 57M60

We introduce two new classes of compacta, called trees of manifolds with boundary and boundary trees of manifolds with boundary. We establish their basic properties. Introduction.In this paper we describe two classes of metric compacta. Each space in each of the two classes is uniquely determined by a fixed, countable (finite or infinite) family M of compact topological manifolds with boundary, all having the same dimension n. The spaces in the first class, denoted by X (M), are called regular trees of manifolds with boundary, while those in the second class, denoted by X bd (M), are called regular boundary trees of manifolds with boundary. The spaces in each class are typically "wild", e.g. they are not ANRs and, as we show in this paper, their topological dimension is equal to n − 1. To show that the corresponding spaces are uniquely determined by M and to calculate their dimension, we derive a few not very well known properties of topological manifolds with boundary.Trees of manifolds with boundary are analogues of spaces which are called trees of closed manifolds. In the orientable case they were examined by W. Jakobsche [J1], [J2], who described them in terms of inverse limits of certain inverse systems of closed oriented manifolds. Every regular tree of closed manifolds depends on a closed oriented n-manifold L and a family M of closed oriented n-manifolds, and we denote it, following W. Jakobsche, by X(L, M). Similar constructions were considered earlier in different contexts by L. S. Pontryagin [P], R. F. Williams [W] and F. D. Ancel and L. C. Siebenmann [AS]. Jakobsche's construction was extended to the nonorientable case by P. R. Stallings [St].Trees of closed manifolds occur as boundaries of nonpositively curved groups. H. Fisher [F] claimed that the boundary of a right-angled Coxeter group, whose nerve is a flag PL-triangulation of a closed oriented manifold N ,