On tori without conjugate points

Summary In this paper we consider Riemannian metrics without conjugate points on an n-torus. Recent work of J. Heber established that the gradient vector fields of Busemann functions on the universal cover of such a manifold induce a natural foliation (akin to the weak stable foliation for a Riemannian manifold with negative sectional curvature) on the unit tangent bundle. The main result in the paper is that the metric is flat if this foliation is Lipschitz. We also prove that this foliation is Lipschitz if and only if the metric has bounded asymptotes. This confirms a conjecture of E. Hopf in this case.Oblatum 22-IX-1993 & 25-IV-1994Supported in part by NSF grant #DMS90-01707 and #DMS85-05550 while at MSRISupported by an NSF Postdoctoral Fellowship     

Graph generators

We consider the construction of highly symmetrical vertex transitive graphs. Some such graphs represent the degenerate rearrangements in which a molecule or an ion is formed by breaking and making bonds so that the final and the initial skeleton is identical. The approach is closely related to Cayley's graphs for selected groups. We restrict the choice of generators to symmetric matrices. Successive multiplications of such matrices generate other permutation matrices of the same dimension, each new matrix representing a new vertex for a transitive graph under the construction. In particular we restrict our discussion to matrices of dimension 3 and 4 and proceed to construct systematically all transitive graphs using 4 × 4 symmetric matrices as generators.     

Homo-and copolymerization of vinylcyclohexane with α-olefins in the presence of heterogeneous and homogeneous catalytic systems

The polymerization and copolymerization of vinylcyclohexane with α-olefins in the presence of several heterogeneous and homogeneous catalytic systems were studied. It was shown that, with respect to activity in the polymerization of vinylcyclohexane, the tested catalysts can be arranged in the following order: α-TiCl₃ < titanium-magnesium catalyst < metallocene catalyst. Poly(vinylcyclohexane) prepared with heterogeneous catalytic systems is a solid semicrystalline polymer. The properties of polymers synthesized with homogeneous systems differ substantially depending on the type of the metallocene used. In the presence of metallocenes with a C ₂ symmetry, crystalline powderlike products arise, while in the case of metallocenes with C ₁ and C _s symmetries, polymerization yields amorphous viscous products. Molecular-mass distributions of poly(vinylcyclohexane) samples prepared using both heterogeneous titanium-magnesium catalysts and homogeneous metallocene complexes show a bimodal pattern, indicating the heterogeneity of active centers of these catalysts. Upon introduction of a comonomer (ethylene, propylene, and 1-hexene) into the reaction mixture, the activity of all studied catalytic systems increases. When Me₂C(3-Me-Cp)(Flu)ZrCl₂ and rac-Me₂SiInd₂ZrCl₂ are used as catalysts, the degree of crystallinity of the copolymers grows owing to the presence of ethylene or propylene units in poly(vinylcyclohexane) chains.     

The structure of copolymers of vinyl cyclohexane with styrene

Copolymerization of vinyl cyclohexane (monomer-1) with styrene was investigated in the presence of the stereospecific complex catalyst TiCl₃ + Al(iso-C₄H₉)₃. Monomer reactivity ratios were r₁ = 0·177 ± 0·051 and r₂ = 2·117 ± 0·370. The monomer unit distributions in the copolymers were estimated by comparison of the i.r.-spectra of copolymers and the isotactic homopolymers using absorption bands at 565 and 1084 cm^?1 which correspond to the vibrations of styrene blocks containing ? 5 styrene units and the band at 985 cm^?1 characterizing polystyrene crystallinity. The data indicate the tendency towards alternation in the copolymerization. Analysis of the experimental and literature data led to the conclusion that distribution of the units in copolymers of vinyl cyclohexane with α-olefins is determined by the nature of the α-olefin. The following activity series is proposed for α-olefins in their copolymerization with vinyl cyclohexane in the presence of catalytic systems based on titanium salts and organo-aluminium compounds: propylene >; 4-methylpentene-1 >; styrene >; 3-methylbutene-1 ～ vinyl cyclohexane.     

Probing the Methyl Torsional Barriers of the E and Z Isomers of Butadienyl Acetate by Microwave Spectroscopy

The Fourier transform microwave spectra of the E and Z isomers of butadienyl acetate were measured in the frequency range from 2 to 26.5 GHz under molecular‐jet conditions. The most stable conformer of each isomer, in which all heavy atoms are located in a symmetry plane, was identified after analyzing the spectrum by comparison with the results from quantum‐chemical calculations. The barriers to internal rotation of the acetyl methyl group were found to be 149.1822(20) and 150.2128(48) cm^?1 for the E and Z isomers, respectively, which are similar to that of vinyl acetate. A comparison between two theoretical approaches treating internal rotation, the rho axis method and combined axis method, was also performed. The influence of the alkyl R chain on the methyl torsional barriers in CH₃ ‐COOR acetates was explored.     

Annealing‐induced increase of permeability and amorphous‐phase mobility in an ethylene–vinylacetate copolymer

The 50% increased permeability after annealing of semicrystalline poly(ethylene/vinylacetate) containing 3 mol % vinylacetate is linked to increased mobility in the amorphous phase, as identified by line‐narrowing of ¹H wideline nuclear magnetic resonance (NMR) spectra and by reduced cross‐polarization efficiency in ¹³C NMR. Other morphological parameters, such as crystallinity, measured as 30 to 35% by differential scanning calorimetry (DSC) and NMR, are hardly changed by annealing. Small‐angle X‐ray scattering and NMR studies, using spin diffusion as well as T_1ρ and T₁ relaxation, detected only a small increase in crystallite thickness. The annealing‐induced enhancement in segmental mobility in the amorphous regions corresponds to a temperature shift of about 10 K, from which an increase of the motional rate by a factor of 2 is estimated, and which can account for the enhancement in the permeability. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 2774–2780, 2001     

Influence of a wearer's voice on noise dosimeter measurements

In recent years, interest in personal noise exposure has expanded beyond a workplace safety measure to become an effective means of investigating physiological effects of the acoustic environment on an individual. This work investigates the effects of the wearer's voice as a possible dominant sound source on body-mounted noise dosimeters and develops methods to improve the application of dosimeter measurements in medium-level noise environments. Subjects experienced a controlled set of acoustic conditions while wearing a dosimeter. In each condition, sound pressure levels were recorded with and without the subject speaking controlled phrases. Three experimental variables were considered-room type, noise type, and noise level. All three variables had a statistically significant effect upon the contribution of speech to a dosimeter measurement; for example, noise level was shown to cause a change in speech contribution by as much as 5.5 dB between sequential levels. Based upon the analysis, a method of predicting the decibel contribution of a wearer's voice was developed. The results of this study can be used to estimate the effect of a wearer's voice on dosimeter measurements in medium-level noise environments.