A characterization ofL n (q) by the normalizers' orders of their sylow subgroups

LetG be a finite group. If for every primer, whereR ₁ Syl _r G andR ₂ Syl _r (L _n (q)), thenG L _n (q).     

Critical exponents,hyperscaling, and universal amplitude ratios for two- and three-dimensional self-avoiding walks

We make a high-precision Monte Carlo study of two- and three-dimensional self-avoiding walks (SAWs) of length up to 80,000 steps, using the pivot algorithm and the Karp-Luby algorithm. We study the critical exponentsv and 2 ₄ – as well as several universal amplitude ratios; in particular, we make an extremely sensitive test of the hyperscaling relationdv = 2 ₄ –. In two dimensions, we confirm the predicted exponentv=3/4 and the hyperscaling relation; we estimate the universal ratios <R _g ² >/<R _e ² >=0.14026±0.00007, <R _m ² >/<R _e ² >=0.43961±0.00034, and ^*=0.66296±0.00043 (68% confidence limits). In three dimensions, we estimatev=0.5877±0.0006 with a correctionto-scaling exponent ₁=0.56±0.03 (subjective 68% confidence limits). This value forv agrees excellently with the field-theoretic renormalization-group prediction, but there is some discrepancy for ₁. Earlier Monte Carlo estimates ofv, which were 0.592, are now seen to be biased by corrections to scaling. We estimate the universal ratios <R _g ² >/<R _e ² >=0.1599±0.0002 and ^*=0.2471±0.0003; since ^*>0, hyperscaling holds. The approach to ^* is from above, contrary to the prediction of the two-parameter renormalization-group theory. We critically reexamine this theory, and explain where the error lies. In an appendix, we prove rigorously (modulo some standard scaling assumptions) the hyperscaling relationdv = 2 ₄ – for two-dimensional SAWs.     

Titantetrafluorid – Eine überraschend einfache Kolumnarstruktur

Titanium Tetrafluoride – a Surprising Simple Column Structure For the first time single crystals of TiF₄ have been obtained by solvothermal decomposition of (O₂)₂Ti₇F₃₀ in anhydrous HF. The colourless, transparent needles crystallize orthorhombic in spacegroup Pnma–D (No. 62) with a = 2 281.1, b = 384.8, c = 956.8 pm, Z = 12. The new type of structure is dominated by isolated columns of corner-linked TiF₆-octahedra.     

Molecular dynamics of cyclic and linear poly(dimethylsiloxanes)

Dielectric spectroscopy (10^–1 Hz to 10⁷ Hz) has been employed to study the molecular dynamics of a series of cyclic and linear polydimethylsiloxanes (PDMS) of various molecular weights ranging from 300 to 10 000 g/mol in the temperature range above the glass transition (from 130 K to 190 K). The observed -relaxation depends strongly on both molecular weight and structure of the samples. For linear PDMS oligomers, the -relaxation shifts towards lower temperatures with decreasing molecular weight in good accordance with the Fox-Flory-model. Cyclic PDMS reveals a qualitatively different molecular weight dependence: for a given temperature the -relaxation time increases with decreasing ring length, but has a maximum for small oligomers (degree of polymerizationn6). The shape of relaxation curves and, with it, the relaxation time distribution is independent from length and architecture of the chains The observed experimental findings are in qualitative agreement with dynamic Monte-Carlo simulations.Dedicated to Prof. E.W. Fischer on the occasion of his 65th birthday Fast macht' das WLF ihn krank, jetzt raucht er wieder, Gott sei Dank! (frei nach Wilhelm Busch)     

Kerr constants and dipole moments of aminobenzoic acids in dioxane

Aminobenzoic acids in dioxane have been investigated by dipole moment and Kerr effect methods.m-Aminobenzoic acid exists in a solution mainly (60 %) in thesyn-form. Inp-aminobenzoic acid, conjugation flattens the pyramidal configuration of the nitrogen atom, which is even more flattened ino-aminobenzoic acid owing to an intramolecular hydrogen bond.Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 5, pp. 859–861, May, 1994.     

Alkylation of cyclododecanone with α,ω-dibromoalkanes under conditions of phase-transfer catalysis

Alkylation of cyclododecanone with ,-dibromoalkanes Br(CH) _n Br (n=3, 4, 5) under conditions of phase-transfer catalysis in the presence of KOH results in the formation of either bicyclic ketones forn=3 and 5 or a mixture of bicyclic and spirocyclic ketones forn=4.Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 8, pp. 1436–1438, August, 1994.     

On the continuity of midpoint hull convex set-valued functions

Summary The concept of hull convexity (midpoint hull convexity) for set-valued functions in vector spaces is examined. This concept, introduced by A. V. Fiacco and J. Kyparisis (Journal of Optimization Theory and Applications,43 (1986), 95–126), is weaker than one of convexity (midpoint convexity).The main result is a sufficient condition for a midpoint hull convex set-valued function to be continuous. This theorem improves a result obtained by K. Nikodem (Bulletin of the Polish Academy of Sciences, Mathematics,34 (1986), 393–399).     

Symmetric balanced ternary designs with ϱ1 = 1 or 2

Summary We prove the following two non-existence theorems for symmetric balanced ternary designs. If ₁ = 1 and 0 (mod 4) then eitherV = + 1 or 4₂ – + 1 is a square and (4₂ – + 1) divides ² – 1. If ₁ = 2 thenV = ((m + 1)/2) ² + 2,K = (m ² + 7)/4 and = ((m – 1)/2)² + 1 wherem 3 (mod 4). An example belonging to the latter series withV = 18 is constructed.     

Ein Beitrag über Ba3Pt4HgO11: Das erste Erdkalimetall-Oxoplatinat(II,V)/Oxomercurat

A Contribution on Ba₃Pt₄HgO₁₁: The First Alkaline-Earth Oxoplatinate(II,V)/Oxomercurate Single crystals of Ba₃Pt₄HgO₁₁ were prepared by oxygen high pressure technique (4 200–3 600 bar) and investigated by X-ray methods. It crystallizes with hexagonal symmetry, space group D? P6 2c, a = 6.021, c = 17.374 Å, Z = 2. Ba₃Pt₄HgO₁₁ represents a new structure type, showing structural relationships to Ba₂Hg₃Pd₇O₁₄ and to the precious metal 6L-perovskites. The Hg²⁺ ions show dumb-bell like coordination, Pt²⁺ a square-planar surrounding and Pt⁵⁺ face shared double octahedra.     

Kinetics and mechanisms of halide ion catalysis of CO substitution reactions in Os3(CO)12 and Ru3(CO)12 metal carbonyl clusters

The results of kinetic studies on ligand substitution in [M₃(CO)₁₁X]^– complexes (M = Ru, Os; X = Cl, Br, I) are summarized. The [Os₃(CO)₁₁X]^– complexes react with PPh₃ under mild conditions to initially yield monosubstituted products [Os₃(CO)₁₀(PPh₃)X]^–. The rate of CO substitution obeys a first-order equation with respect to the concentration of the complex and does not depend on the ligand concentration. The rates of the reactions decrease in the order Cl > Br > I withH values increasing from 15 to 18 kcal mol^–1 and S values varying from –19 to –13 cal mol^–1 K^–1. The enhanced reactivities of these complexes as well as the low activation energies and negative activation entropies are discussed in terms of the effects of -X bridge formation on the transition state of the reaction. Reactions of PPN[Ru₃(CO)_11–x (Cl)] (PPN is the bis(triphenylphosphine)iminium cation;x=0, 1) and PPN[Ru₃(CO)₉(₃-I)] with alkynes are also reported. The reactivities of alkynes follow the order BuCCH PhCCH EtCCEt PhCCPh. The higher rates of the reactions of monosubstituted acetylenes compared with those of their disubstituted analogs are explained by agostic interaction between the metal atom and the C-H bond in the reaction transition state and by steric effects. The results obtained attest that the reaction with alkynes occursvia intermediates containing halide bridges and that ₃-halide complexes are more reactive than ₂-halide complexes.Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 9, pp. 1540–1545, September, 1994.This work was supported by a Presidential Grant from Northwestern University. One of the authors (F. Basolo) wishes to thank Academician M. E. Vol'pin for the invitation to participate in the Workshop The Modern Problems of Organometallic Chemistry (INEOS-94) and Academician O. M. Nefedov for the invitation to publish a review in theRussian Chemical Bulletin.