A new summation method for power series with rational coefficients

We show that an asymptotic summation method, recently proposed by the authors, can be conveniently applied to slowly convergent power series whose coefficients are rational functions of the summation index. Several numerical examples are presented.

     

Orbital and intrinsic angular momentum of single photons and entangled pairs of photons generated by parametric down-conversion

For systems that exhibit a second-order phase transition with a spontaneously broken continuous O(N) symmetry at low temperatures, we give a criterion for judging at which temperature T(K) long-range directional fluctuations of the order field destroy the order when approaching the critical temperature from below. The temperature T(K) lies always significantly below the famous Ginzburg temperature T(G) at which size fluctuations of finite range become important.     

Screened indicators in anhydrous acetic acid medium: Chromatic evaluation of their transitions

Three efficient indicators consisting of Malachite Green and a screening dye for use in anhydrous acetic acid media have been designed by use of complementary tristimulus data. The transition quality can be evaluated by plotting the relative greyness of the indicator against free perchloric acid concentration. Several bases have been successfully determined titrimetrically with use of these indicators.     

Ion pairing in solutions of sodium tetraphenylborate

Conductance data are presented for sodium tetraphenylborate (NaBPh₄) in three solvent systems: water-dimethyl sulfoxide, water-acetone, and acetone-dimethyl sulfoxide. These and other data from the literature were first analyzed by the three-parameter equation Λ^′=Λ₀+Aτ²+Bτ³, where $$\Lambda \prime = [\Lambda (obs.) + \beta _0 c^{1/2} (1 + 0.5\tau ln 2\tau )]/[1 - \alpha _0 c^{1/2} + (\tau ^2 /3) ln 2\tau ]$$ Here, β₀ is the Onsager electrophoresis coefficient, α_oc^1/2 the limiting relaxation term, τ=e²κ/2DkT, κ^?1 is the Debye-Hückel distance, andA andB are empirical constants. The fact that three parameters are necessary to reproduce the data shows that ion pairing is not negligible for NaBPh₄ solutions despite the fact that the conductance curve lies above the limiting tangent even in ethanol (D=24.30). The quantity Λ_o is the limit of Λ^′ at zero concentration; the sum Aτ²+Bτ³ represents the part of the observed conductance from which the pairing constantK _Λ and the distance parameterR are derived. Conductance experiments should therefore be designed so that this sum is as large as possible, subject to the restriction that the highest concentration should not exceed 10^?7D³ (where the effects of short-range three-ion interactions are no longer negligible). Pairing constants for NaBPh₄ in the various solvents range from 4.6 in water to 70 in a mixture of Me₂SO and Me₂CO with dielectric constantD=23.80. The dependence ofE _s (the difference in free energy between solvent-separated pairs and contact pairs) on solvent composition is different for aprotic solvents and for mixtures containing water; for the latter, theE _s/kT vs. l/D plot shows a maximum while for the former the curve is monotone descending. The Walden produces (Λ_oη=limiting conductance times viscosity) plotted againstD ^?1 follow the same pattern.     

9-Imino derivatives of dihydroxyanthraquinones as analytical reagents

The synthesis, characteristics, properties and reactions with metallic ions of the reagents isticin-9-imine, alizarin-9-imine and 3-sulphoalizarin-9-imine have been studied.     

Radical addition of nitrones to acrylates mediated by SmI2: asymmetric synthesis of gamma-amino acids employing carbohydrate-based chiral auxiliaries

Alkyl nitrones possessing N-substituted sugars as chiral auxiliaries were found to effectively undergo an SmI(2)-mediated radical addition to n-butyl acrylate affording gamma-amino acid derivatives with high diastereomeric control.