Group-Like Structure Underlying the Unit Ball in Real Inner Product Spaces

Abstraction of the relativistic velocity addition law and of the Thomas rotation of the special theory of relativity yields a means of endowing the unit ball in any real inner product space with a group- like structure, in which the standard associative- commutative laws are relaxed by means of the Thomas rotation. The resulting group- like object is called a complete weakly associative- commutative groupoid. Any complete WACG can be extended to a group analogous to the Lorentz group of the special theory of relativity.     

A study of the intersystem crossing reaction induced in gaseous sulfur dioxide molecules by collisions with nitrogen and cyclohexane at 27°C

The quantum yields of the sulfur dioxide triplet (³SO₂)-sensitized phosphorescence of biacetyl (Φ_sens) were determined in experiments with N₂–SO₂–Ac₂ and c-C₆H₁₂–SO₂–Ac₂ mixtures excited at 2875 Å at 27°C. The fraction of the biacetyl triplets which reacts homogeneously by radiative or nonradiative decay reactions was determined in a series of runs at constant [SO₂]/[M] and [SO₂]/[Ac₂] ratios but at varied total pressure. A kinetic treatment of the Φ_sens results and singlet sulfur dioxide (¹SO₂) quenching rate constant data gave the following new kinetic estimates: ¹SO₂ + M → (SO₂–M) (1b) ¹SO₂ + M → ³SO₂ + M (2b); for ¹SO₂–N₂ collisions, k_2b/(k_1b + k_2b) = 0.033 ± 0.008; for ¹SO₂–c-C₆H₁₂ collisions, k_2b/(k_1b ± k_2b) = 0.073 ± 0.024; previous studies have shown this ratio to be 0.095 ± 0.005 for ¹SO₂–SO₂ collisions. It was concluded that the inter-system crossing ratio in ¹SO₂ induced by collision is relatively insensitive to the nature of the collision partner M. However, the individual rate constants for the collision-induced spin inversion of ¹SO₂ (k_2b) and the total ¹SO₂-quenching constants (k_1b + k_2b) are quite sensitive to the nature of M: k_2b/k_2a varies from 0.10 ± 0.03 for M = N₂ to 1.11 ± 0.37 for M = c-C₆H₁₂, and (k_1b + k_2b)/(k_1a + k_2a) varies from 0.29 for M = N₂ to 1.44 for M = c-C₆H₁₂; k_1a and k_1b are the rate constants for the reactions ¹SO₂ - SO₂ → (2SO₂) (1a) and ¹SO₂ + SO₂ → ³SO₂ + SO₂ (2a), respectively.     

Space and time-dependent effects in optical bistability

We present numerical studies of optical bistability which admit variations in both space and time in the governing c-number equations. We justify mean field theory for both the steady state and time dependent regimes for low enough cavity transmissivity T: but for T ? 0.2 mean field theory is scarcely applicable. Higher harmonic fields are important at low T (high reflexion). As expected the steady state hysterisis behaviour at low T is considerably modified in the time dependent regime. Transistor action at high T ～ 0.9 is described. In general the numerical results show good qualitative agreement with the experiment of Gibbs et al.     

Dynamics of a Local Perturbation in the XY Model. I—Approach to Equilibrium

The magnetisation and spin correlation functions of the X–Y model are analysed exactly when the system is subjected to a local perturbation of the magnetic field in the z-direction. When such a perturbation is removed, these quantities approach their equilibrium values asymptotically as (time)^?1.     

Metalloid derivatives for synthesis: Monoderivatization of symmetric diols

The monoderivatization of symmetric diols by condensation of their stannoxanes with activated halides is described.     

Half-factorial-domains

LetR be a commutative domain with 1. We termR an HFD (Half-Factorial-Domain) provided the equality Π _i=1 ⁿ χ_i=Π{_f=1/^m}y _f impliesm=n, whenever thex’s and they’s are non-zero, non-unit and irreducible elements ofR. The purpose of this note is to study HFD’s, in particular, Krull domains that are HFD’s, and to provide examples of HFD’s, that contradict a conjecture of Narkiewicz.     

The NMR spectra and conformations of cyclic compounds: IX—conformational studies of bicyclo(3,1,0)hexane derivatives by 13C NMR

The ¹³C spectra of α-thujene ( 1 ), isothujone ( 2 ), (?)isothujol ( 3 ), (+)neoisothujol ( 4 ), sabinol ( 5 ), dihydroumbellulone ( 6 ) and umbellulone ( 7 ) and the alcohol acetates are recorded and assigned. The C-6 chemical shift may be used in conjunction with the steric shift mechanism as a conformational probe in these molecules. The results obtained indicate that isothujol, neoisothujol, isothujone and dihydroumbellulone adopt boat-like conformations whilst sabinol has a much flatter conformation. Conjugation of the isolated double bond of α-thujene and the carbonyl group of dihydroumbellulone with the cyclopropane ring has virtually no effect on the ¹³C shifts, but those of umbellulone itself are anomalous, indicating conjugation of the α,β unsaturated ketone system with the cyclopropyl ring.     

Mode-pulling,mode-splitting and pulsing in a high gain He-Xe laser

High-gain, single-mode, helium-xenon lasers operating at 3.51 μm have been found to emit light in pulses rather than in a continuous wave. Two different pulsing regimes have been found, one correlated with oscillation in the vicinity of the Lamb dip and the other apparently due to mode splitting. Both instabilities require high gain and relatively low homogeneous broadening to be observed.     

The conformation of 4-phenylcyclohexanone in solution by 1H- and 13C-Lanthanide induced shift (L.I.S.) analysis : Evidence in favour of phenyl over t-butyl as a conformational lock

The degree of pucker of the cyclohexanone ring in 4-phenylcyclohexanone has been refined by L.I.S. analysis. The results indicate puckering intermediate between that in cyclohexanone and 4-t-butylcyclohexanone, suggesting that phenyl may be preferable to t-butyl as a locking group in conformational analysis. The Yb (fod)3 shift reagent, although binding primarily at the carbonyl oxygen atom, also complexes weakly with the phenyl group.