Extendible elements of the alternating groups

An element σ of A_n, the Alternating group of degree n, is extendible in S_n, the Symmetric group of degree n, if there exists a subgroup H of S_n but not A_n whose intersection with A_n is the cyclic group generated by σ. A simple number-theoretic criterion, in terms of the cycle-decomposition, for an element of A_n to be extendible in S_n is given here.     

Why is space three-dimensional and how may groups be seen?

First we propose a model of visual perception essentially based on the Keldysh-Chernavsky-Sossinsky three-channel theorem, from which three-dimensionality of space follows. Second, we associate with a system of subgroups H ₁, ..., H_s of a given group G a geometric object, called a group crystal, in order to visualize G. How this notion works is illustrated via the Burnside problem.     

The use of fluorescence in determining formation constants of complexes : Part II. Complexes which fluoresce

Four different methods for the calculation of the formation constants of fluorescing complexes are discussed. The methods were used to determine constants for the complexes of aluminum(III), gallium(III) and indium(III) with 8-quinolinol-5-sulfonate anion.     

Synthesis ofS-(+)-hydroprene

A novel path toS-(+)-hydroprene (1) starting from the technical gradeS-(+)-dihydromyrcene (2, e.e. 50%) is proposed. The latter was selectively transformed intoS-3,7-dimethyloctanal (5) in three steps including hydroalumination. The reactions of5 with allyl- or methallylmagnesium chloride followed, respectively, either by oxygenation in the presence of PdCl₂/CuCl or by ozonolysis, affordS,E-6,10-dimethyl-3-undecen-2-one (7) which was treated with ethoxyethynylmagnesium bromide to give the title juvenile hormone analogue in 23% overall yield.Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No 1, pp. 110–112, January, 1993.     

Determination of the valence distribution of vanadium in polyolefin catalysts prepared from vanadium trichloride

An electrochemical method has been developed for determining the relative amounts of V(II) and V(III) in polyolefin catalysts prepared using aluminum alkyls and vanadium trichloride. The catalyst, usually obtained as a slurry in n-heptane, is titrated potentiometrically with ceric sulfate using a platinum indicator electrode. Under proper experimental conditions, 2 breaks are obtained, corresponding to the reactions V(II) →V(III) + e and V(III) →V(IV) + e. Interferences from other components of the catalyst are negligible. The method is useful for the analysis of both laboratory catalysts and grab samples taken from pilot plant reactors.     

Vibrational and high pressure conformational studies of 1,1,2-trichloroethane

The infrared spectra of 1,1,2-trichloroethane were recorded from 4000 to 50 cm^?1 in the vapour and liquid states. Additional spectra above 200 cm^?1 of the low temperature crystal and of two crystalline solids, prepared by compressing the sample in a diamond anvil cell, were obtained. Also, infrared spectra of the liquid and of the compound dissolved in CS₂ were recorded at increased pressures. Raman spectral data of the liquid (including polarization measurements) and of the low temperature crystal were obtained.The fundamental frequencies for each of the two conformers, C₁ and C_s, were assigned and the results checked by normal coordinate analysis. The same diagonal and off-diagonal force constants were employed for the two conformers and the force fields were derived by means of a least squares refinement, including data for chloroethane, 1,1-dichloroethane and various deuterated species. A standard deviation of 3 % was obtained in the final fit.From the changes in relative intensities for infrared bands belonging to the C₁ and C_s conformers with increasing pressure, the volume differences (ΔV¯between the conformers were determined in CS₂ solution and in the pure liquid.     

Spencer cohomologies and symmetry groups

We propose the construction of a spectral sequence converging to Spencer cohomologies. By using symmetry groups of differential equations systems, we manage to unify computations by reduction to the invariant systems over a homogeneous space. The conditions of coincidence of Spencer cohomologies with the cohomologies of an invariant Spencer complex we obtain from the arithmetic of a -characteristic manifold with respect to fundamental weights of the homogeneous space.     

Growth functions for semi-regular tilings of the hyperbolic plane

This paper studies the growth function, with respect to the generating set of edge identifications, of a surface group with fundamental domainD in the hyperbolic plane ann-gon whose angles alternate between /p and /q. The possibilities ofn,p andq for which a torsion-free surface group can have such a fundamental polygon are classified, and the growth functions are computed. Conditions are given for which the denominator of the growth function is a product of cyclotomic polynomials and a Salem polynomial.This work was supported in part by NSF Research Grants.     

Shock fluctuations in the asymmetric simple exclusion process

Summary We consider the one dimensional nearest neighbors asymmetric simple exclusion process with ratesq andp for left and right jumps respectively;q<p. Ferrari et al. (1991) have shown that if the initial measure isv _, , a product measure with densities and to the left and right of the origin respectively, <, then there exists a (microscopic) shock for the system. A shock is a random positionX _t such that the system as seen from this position at timet has asymptotic product distributions with densities and to the left and right of the origin respectively, uniformly int. We compute the diffusion coefficient of the shockD=lim _t t ^–1(E(X _t )²–(EX _t )²) and findD=(p–q)(–)^–1((1–)+(1–)) as conjectured by Spohn (1991). We show that in the scale the position ofX _t is determined by the initial distribution of particles in a region of length proportional tot. We prove that the distribution of the process at the average position of the shock converges to a fair mixture of the product measures with densities and . This is the so called dynamical phase transition. Under shock initial conditions we show how the density fluctuation fields depend on the initial configuration.