Faces of platonic solids in all dimensions

This paper considers Platonic solids/polytopes in the real Euclidean space R^n of dimension 3 <= n < infinity. The Platonic solids/polytopes are described together with their faces of dimensions 0 <= d <= n-1. Dual pairs of Platonic polytopes are considered in parallel. The underlying finite Coxeter groups are those of simple Lie algebras of types An, Bn, Cn, F4 and of non-crystallographic Coxeter groups H3, H4. Our method consists in recursively decorating the appropriate Coxeter-Dynkin diagram. Each recursion step provides the essential information about faces of a specific dimension. If, at each recursion step, all of the faces are in the same Coxeter group orbit, i.e. are identical, the solid is called Platonic.Comment: 11 pages, 1 figur

Dominant weight multiplicities in hybrid characters of Bn, Cn, F4, G2

The characters of irreducible finite dimensional representations of compact simple Lie group G are invariant with respect to the action of the Weyl group W(G) of G. The defining property of the new character-like functions ("hybrid characters") is the fact that W(G) acts differently on the character term corresponding to the long roots than on those corresponding to the short roots. Therefore the hybrid characters are defined for the simple Lie groups with two different lengths of their roots. Dominant weight multiplicities for the hybrid characters are determined. The formulas for "hybrid dimensions" are also found for all cases as the zero degree term in power expansion of the "hybrid characters".Comment: 15 page

Testing a hatchback type passenger car fire

DOI nepřidělenThe paper presents comments on statistics, referring to passenger cars fi res in Poland. The emphasis is placed on arsons. Experiments on fi res started by the “Molotov cocktails” are presented. The paper includes description of a fi re test conducted on a hatchback car with a LPG system. The photographs include the burning gas outfl ow, witnessed after triggering the safety valves in LPG system. Hazards are pointed out, including those concerning the driver, passengers, rescuers and others. The hazards are connected with a leak of fuel or a gas stream and hazardous explosions of car parts. A description of temperature taken on the car body surface is provided. An infrared camera has been used

Probiotics for preventing healthcare-associated diarrhea in children: A meta-analysis of randomized controlled trials

AbstractAimTo systematically update evidence on the efficacy of using probiotics for the prevention of healthcare-associated diarrhea in children.MethodsMEDLINE, EMBASE, The Cochrane Library, Health Source: Nursing/Academic Edition, two clinical trials and reference lists were searched in June 2013, for randomized controlled trials (RCTs) performed in children aged 1 month to 18 years that compared the effects of the administration of probiotics with placebo or no intervention. The primary outcome measure was the incidence of healthcare-associated diarrhea.ResultsSix RCTs involving 1343 children met the inclusion criteria. Administration of Lactobacillus rhamnosus GG (LGG) compared with placebo reduced the risk of healthcare-associated diarrhea (2 RCTs, n=823, RR 0.37; 95% CI 0.23–0.59), reduced the risk of rotavirus gastroenteritis (3 RCTs, n=1043, RR 0.49, 95% CI 0.28–0.86), but did not reduce the risk of asymptomatic rotavirus infection (2 RCTs, n=301, RR 1.39, 95% CI 0.74–2.62). Administration of Bifidobacterium bifidum & Streptococcus thermophilus compared with placebo reduced the risk of healthcare-associated diarrhea (1 RCT, n=55, RR 0.22, 95% CI 0.05–0.96), rotavirus gastroenteritis (1 RCT, n=55, RR 0.27, 95% CI 0.08–0.87), and rotavirus asymptomatic infection (1 RCT, n=55, RR 0.27, 95% CI 0.08–0.87). Administration of two other probiotics (i.e., Lactobacillus reuteri DSM 17938 and Lactobacillus delbrueckii H2B20) was ineffective.ConclusionIn hospitalized children, the administration of LGG, compared with placebo, reduced the incidence of healthcare-associated diarrhea, including rotavirus diarrhea. Evidence on the effects of other probiotics, whether positive or negative, is limited

TWODIMENSIONAL HYBRIDS WITH MIXED BOUNDARY VALUE PROBLEMS

Boundary value problems are considered on a simplex F in the real Euclidean space R2. The recent discovery of new families of special functions, orthogonal on F, makes it possible to consider not only the Dirichlet or Neumann boundary value problems on F, but also the mixed boundary value problem which is a mixture of Dirichlet and Neumann type, ie. on some parts of the boundary of F a Dirichlet condition is fulfilled and on the other Neumann’s works

Four types of special functions of G_2 and their discretization

Properties of four infinite families of special functions of two real variables, based on the compact simple Lie group G2, are compared and described. Two of the four families (called here C- and S-functions) are well known, whereas the other two (S^L- and S^S-functions) are not found elsewhere in the literature. It is shown explicitly that all four families have similar properties. In particular, they are orthogonal when integrated over a finite region F of the Euclidean space, and they are discretely orthogonal when their values, sampled at the lattice points F_M \subset F, are added up with a weight function appropriate for each family. Products of ten types among the four families of functions, namely CC, CS, SS, SS^L, CS^S, SS^L, SS^S, S^SS^S, S^LS^S and S^LS^L, are completely decomposable into the finite sum of the functions. Uncommon arithmetic properties of the functions are pointed out and questions about numerous other properties are brought forward.Comment: 18 pages, 4 figures, 4 table