The Hilbert–Poincaré Series for Some Algebras of Invariants of Cyclic Groups

Let be a linear representation of a finite group over a field of characteristic 0. Further, let R be the corresponding algebra of invariants, and let P (t) be its Hilbert–Poincaré series. Then the series P (t) represents a rational function (t)/(t). If R is a complete intersection, then (t) is a product of cyclotomic polynomials. Here we prove the inverse statement for the case where is an almost regular (in particular, regular) representation of a cyclic group. This yields an answer to a question of R. Stanley in this very special case. Bibliography: 3 titles.     

Subgroups of a finite group whose algebra of invariants is a complete intersection

Let G be a finite subgroup of GL(V), where V is a finite-dimensional vector space over the field K and char KG. We show that if the algebra of invariants K(V)^G of the symmetric algebra of V is a complete intersection then K(V)^H is also a complete intersection for all subgroups H of G such that H={ Gv (v)=v for all v V^H}.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 116, pp. 63–67, 1982.     

35Cl NQR Studies of the Geometry of Aromatic Imines

The preferred conformation of solidphase R4benzylidene3,4dichloroaniline molecules has been established by nuclear quadrupole resonance (NQR) combined with quantum chemical calculations. The conformational effect of substitutes reported previously is proved. The rotation angle of the plane of the aniline ring with respect to the plane of the azomethine bond is given.     

Fatty acid composition of the main phospholipids of the Antarctic krill Euphausia superba

The Antarctic krillEuphausia superba Dana contains about 5% of its natural weight of extractable lipids, more than half of which are in the form of phospholipids — phosphatidylcholine (33–36% of the sum of the lipids), phosphatidylethanolamine (15–17%), lysophosphatidylcholine (3–4%), and others (2–3%) — while among the phosphorus-free components triacylglycerols predominate (32–35%). In the first two phospholipids the dominating fatty acid residue is the arachidonic acid residue (more than 40% of the sum of the acyl residues) and the amounts of residues of eicosapentaenoic acid (C_{20 : 5w3}) are about 13 and 28%, respectively. A simple procedure for isolating the phosphatidylcholine and phosphatidylethanolamine is proposed. For the analysis of the fatty acid a new method was used, consisting of the HPLC of their (7-diethylaminocoumarin-4-yl)hydrazides, with fluorometric detection.Scientific Research Laboratory of Biologically Active Substances of Hydrobionts, USSR Ministry of Health, Moscow. K. A. Timiryazev Moscow Agricultural Academy. Translated from Khimiya Prirodnykh Soedinenii, No. 2, pp. 181–187, March–April, 1990.     

Quantum chemical study of nickel(II) complexes with cyclic diimine ligands on the base of bis[3,3′-iminopropyl]methylamine

A quantum chemical study of spatial and electronic structures of molecules in the frame complexes, bis[3,3′(RR′)-ketiminepropyl]methylamine nickel dichlorides, where R = H, CH₃, and R′ = H, CH₃, has been performed by DFT(B3LYP/LANL2DZ) method. The molecules of these complexes were found to be in a triplet state. Energy stability of the endo form of the complex molecules was shown. In the molecule of bis[3,3′-aldiminopropyl]methylamine nickel dichloride (R = R′ = H), a considerable strengthening of the bond Ni-N(amine) takes place when passing from the diamagnetic into paramagnetic state, and all bonds Ni-N become equivalent with respect to interatomic distance values. The topology analysis of the electron density for the complexes with R = R′ =H and R = R′ = CH₃ was carried out. It is stated that all Ni-N bonds are covalent in the molecules of paramagnetic complexes.     

35Cl NQR and structural-dynamic properties of crystalline CCl3C(O)OC6H2Cl3-2,4,6

The temperature dependences of the resonance frequency and quadrupole spin-lattice relaxation time of the chlorine-35 nuclei in crystalline CCl₃C(O)OC₆H₂Cl₃-2,4,6 were studied. Owing to the presence of resonant nuclei in various fragments of the molecules, the character of dynamics of these molecular fragments and the entire molecule was established. It is shown that thermal librations of molecules are quasiharmonic over the whole temperature range from 77 K to the melting point of the crystal. The reorientational motion of the CCl₃ group bonded to the three-coordinated carbon atom was revealed. This motion causes an exponential increase in the spin-lattice relaxation time of the chlorine nuclei of this group and subsequent damping of NQR signals (chlorine-35 resonance signals of the aryl radical were observed before melting of the sample). The activation energy of the reorientational motion is found; its value, which is 27.3 kJ/mole, is considered in comparison with the activation energies obtained by the Cl NQR method in solids for CCl₃ reorientations in similar molecular structures.     

On fractal nature of the structure of nanoporous carbon obtained from carbides

The curves describing small-angle x-ray scattering at npor-C nanoporous carbon samples obtained from polycrystalline α-SiC, TiC, and Mo₂C and a 6H-SiC single crystal have been analyzed. An algorithm is developed for taking into account the corrections to experimental curves for the intensity of the primary beam transmitted through the sample and the height of the inlet slit in these measurements. Two systems of nanoclusters observed in the npor-C structure differ in the type of stacking of structural elements: small-scale mass fractals of a dimension 1<D ₂<3 and a size L ₂=50–90 Å, which depend on the type of the initial carbide, and large-scale nanoclusters having a size L ₁>550 Å. In most samples, large-scale nanoclusters can be regarded as objects with a fractal surface and a dimension 2<D ₁<13, which also depends on the type of the initial carbide. Large-scale nanoclusters in npor-C obtained from Mo₂C prove to be mass fractals with a dimension D ₁>2. Peculiarities of the structure formation of nanoporous carbon obtained from various carbides are discussed.     

Covering numbers for Chevalley groups

LetG be a quasisimple Chevalley group. We give an upper bound for the covering number cn(G) which is linear in the rank ofG, i.e. we give a constantd such that for every noncentral conjugacy classC ofG we haveC ^rd =G, wherer=rankG. Research supported in part by NSERC Canada Grant A7251. Research supported in part by the Hermann Minkowski-Minerva Center for Geometry at Tel Aviv University.