Discrete groups of slow subgroup growth

It is known that the subgroup growth of finitely generated linear groups is either polynomial or at least $n^{\frac{{\log n}}{{\log \log n}}} $ . In this paper we prove the existence of a finitely generated group whose subgroup growth is of type $n^{\frac{{\log n}}{{(\log \log n)^2 }}} $ . This is the slowest non-polynomial subgroup growth obtained so far for finitely generated groups. The subgroup growth typen ^logn is also realized. The proofs involve analysis of the subgroup structure of finite alternating groups and finite simple groups in general. For example, we show there is an absolute constantc such that, ifT is any finite simple group, thenT has at mostn ^{c logn} subgroups of indexn.     

A characterization of reflexive Banach spaces

A Banach space is not reflexive if and only if there exist a closed separable subspace of and a convex closed subset of with empty interior which contains translates of all compact sets in . If, moreover, is separable, then it is possible to put .

     

Synthesis and structure of a dithioether-substituted butene-2-olide-4-ylidene-4 cobalt carbonyl complex

The complex (1) has been prepared and its structure determined. Complex1 is the first stable thioether derivative of a metal carbonyl containing only cobalt as metal. It crystallizes in the monoclinic P2₁/c space group, witha = 13.975( 4 ),b = 11.363( 3 ),c = 14.686(2)A, = 112.25(2)°,V = 2158.5(7)A³.Z = 4. The solution using direct methods and the anisotropic refinement of all nonhydrogen atoms led toR = 0.039,wR = 0.041 for 3296 reflections with F> 4.0(F) Analogous but less stable complexes are formed with 1,6-diphenyl-2,5-dithiahexane and the crown thioether 1,4,7-trithiacyclonane, the latter functioning either as a bidenlate or a tridentate ligand.     

Experimental determination of the flame expansion factor

The flame expansion factor (ratio of volumes of flame gases before and after burning) has been determined by mercury vapour introduction. The method is based on the determination of atomic concentration ratio in the flame and the flameless gas jet using the same sample introduction rate. A good agreement has been found between the expansion factors determined by the proposed method and those calculated by the gas law.     

Properties study of anhydrite(II) prepared from waste gypsums

Anhydrites(II)- CaSO₄ were prepared from two waste gypsums (PRECHEZA Perov, FOSFA Potorná). The samples of anhydrites(II) with sulphate activators were tested on spillage, beginning and the end of setting time, bending and pressure strength. Both sodium and potassium sulphates accelerated setting of anhydrite, more impressively that prepared from Potorná gypsum. The addition of Na₂SO₄ influenced the strength of this anhydrite very favourably. On the contrary K₂SO₄ influenced favourably the strength of Perov anhydrite. Results confirm the necessity to evaluate properties of such materials with respect to their origin.