Two renewal theorems for general random walks tending to infinity

Summary. Necessary and sufficient conditions for the existence of moments of the first passage time of a random walk S _n into [x, ∞) for fixed x≧ 0, and the last exit time of the walk from (−∞, x], are given under the condition that S _n →∞ a.s. The methods, which are quite different from those applied in the previously studied case of a positive mean for the increments of S _n , are further developed to obtain the “order of magnitude” as x→∞ of the moments of the first passage and last exit times, when these are finite. A number of other conditions of interest in renewal theory are also discussed, and some results for the first time for which the random walk remains above the level x on K consecutive occasions, which has applications in option pricing, are given. Received: 18 September 1995/In revised form: 28 February 1996     

The solubility of argon in aqueous solutions of a complex-ion electrolyte

From measurements of the solubility of argon in aqueous solutions of the complexion electrolyte t-[Co(en)₂NCSCl]Br at 25°C, the Setschenow parameter for this system is found to be +0.33₆. Using the scaled-particle theory of the salt effect with this value ofk ₅ and the Waddington crystal radius of the Br^– ion, 1.98 Å, the radius of the complex cation is calculated to be 3.71 Å. This value is in reasonably good agreement with that estimated from scale models of the t-[Co(en)₂-NCSCl]⁺ cation, about 3.9 Å. It is somewhat larger than the radius calculated from various semiempirical equations relating the molal volume of an ion at infinite dilution to its radius; values of r obtained in this manner range from 3.32 to 3.66 Å.Abstracted, in part, from D.P.'s M.S. thesis, June 1971.     

Concentration dependencies in the luminescence of Li4WO5: U

The uranate ion luminescence in a sequence of (Li_1?xNa_x)₄W_1?yO₅:U_y phosphors was examined at cryogenic temperatures under conditions of high resolution. It was found that as the ratio of U/W or Na/Li was increased, certain features in the sharp-line emission spectra decreased in intensity and other features increased at the same time. Careful examination of several photophysical parameters led to the conclusion that two different uranate sites yielded the observed luminescence features. One of the U(VI) ions occupies a site characterized by the presence of an inversion center, and the other ion occupies a site lacking such symmetry. The lower symmetry site may be considered a defect site, and was found to be an efficient quencher of the higher symmetry site.     

Angular Multiplexing Gratings Characterization in Dichromated Gelatin and Thoughts on Diffractive Optical Memory

We were mainly concerned here with multiple exposure gratings that allow interactions and coupling between gratings. We started the study of interactions and coupling with five gratings recorded in dichromated gelatin (DCG). Having determined experimentally the characteristic parameters of these five gratings, we used a method developed by Oxford University to synthesize the modulation profile. Entering these parameters of each grating into our computing programs (based on the alternative theory of diffraction in modulated media) and by varying the amplitude modulation, the latter was deduced by adjusting the numerical curve to the experimental one. We then extended the study to nine multiplexed gratings. This study allowed us to know the range of the amplitude modulation such that a certain number of angular multiplexing can be done in DCG. Several undesirable effects which we had experimentally observed appeared then during the reconstruction of any one of the multiplexed images, such as cross-talk effect. We noted the good concordance between experimental and theoretical results. We finally illustrated cross-talk effect in a diffractive memory where, if the amplitude modulation was first calculated, the number of multiplexing for an angular range could then be determined and the angular separation achieved to avoid cross-talk.     

Nmr spectral characteristics of N-H protons in purine derivatives

Proton chemical shifts are examined in a series of twelve purine and three 5-azapurine derivatives. The assignment of N-H signals in the xanthine and thioxanthine series, as well as the effects of moisture, concentration and temperature on the N-H and C-H nmr spectral characteristics are discussed.     

Cyclopropanone equivalents from 3-chloropropionic acid 1-piperidino-1-trimethylsilyloxycyclopropane

1-Piperidino-1-trimethylsilyloxycyclopropane and the corresponding 1-hydroxy-1-piperidinocyclopropane are conveniently perpared from the piperidide of 3-chloropropionic acid. Reactions of these cyclopropanone equivalents with various nucleophiles are described.     

Thermochemistry of azide salts and the azide ion using direct minimisation equations to obtain the lattice energy of the univalent azide salts

The opportunity to test a new equation for the computation of the lattice energy and at the same time examine a disparity in the literature data for the enthalpy of formation of the azide ion, $\text{ΔH}{}^{\mathrm{\theta }}{}_{\mathrm{?}}\text{(}\text{N}{}_3{}^{\mathrm{?}}\text{) (}\text{g}\text{)}$ was the motivation for this study. The results confirm our earlier calculation and show the new equation to be reliable. Thermodynamic data produced in the study take values: $\text{ΔH}{}^{\mathrm{\theta }}{}_{\mathrm{?}}\text{(}\text{N}{}_3{}^{\mathrm{?}}\text{)(}\text{g}\text{) = 144}\text{kJ mor}{}^{\mathrm{?1}}$ΔH^θ_hyd(N₃^?) = ?315 KJ mol^?1 or ΔH^θ_hyd(N₃^?) = ?295 KJ mol^?1U_POT(NaN₃) = 732 kJ mol^?1U_POT(KN₃) = 659 kJ mol^?1U_POT(RbN₃) = 637 kJ mol^?1U_POT(CsN₃) = 612 kJ mol^?1U_POT(TIN₃) = 689 kJ mol^?1. The lattice energies of azides whose enthalpies of formation are documented have been calculated as well as the enthalpy of formation of the azide radical.     

Synthesis and characterization of semi- and full-interpenetrating polymer networks of poly(2,6-dimethyl-1,4-phenylene oxide) and polydimethylsiloxane

The synthesis and characterization of pseudo or semi- and full-interpenetrating polymer networks (IPNs) of poly(2,6-dimethyl-1,4-phenylene oxide) and polydimethylsiloxane were performed. We observed that in full IPNs, the elasticity of the IPN samples increased very drastically, as the composition of polydimethylsiloxane increased (i.e. 0–60%) while the tensile strength (TS) and the glass transition temperature (T_g) decreases. The pseudo IPNs appeared to consist of two phases while the full IPNs of lower siloxane content were miscible.