Gibbs states of the Hopfield model with extensively many patterns

We consider the Hopfield model withM(N)=N patterns, whereN is the number of neurons. We show that if is sufficiently small and the temperature sufficiently low, then there exist disjoint Gibbs states for each of the stored patterns, almost surely with respect to the distribution of the random patterns. This solves a provlem left open in previous work. The key new ingredient is a self-averaging result on the free energy functional. This result has considerable additional interest and some consequences are discussed. A similar result for the free energy of the Sherrington-Kirkpatrick model is also given.     

Energy relaxation in aϕ 4 with long range interactions

We investigate the influence of long range interactions on the relaxation behaviour of a lattice model with an on-site potential of ⁴-type and infinite range harmonic interactions. For finite number of particlesN, it is shown that the autocorrelation functions <E _n(t)E _n > of the fluctuations of the one-particle energiesE _n(t) decays exponentially. The corresponding relaxation time is proportional toN and is given by (T, N) =N₀(T). The temperature dependent time scale ₀ can explicitly be related to the dynamics of a one-particle correlator of the noninteracting system. The results are derived using Mori-Zwanzig projection formalism. The corresponding memory kernel is calculated within a mode coupling approximation and by a perturbative approach. Both results agree in leading order in 1/N. It is speculated that any interaction of range generates a timescale .     

Symmetric and asymmetric fission of charged sodium clusters

A deformed-jellium model is used to calculate the fission barrier height of Na ₂₄ ²⁺ → Na ₂₁ ⁺ +Na ₃ ⁺ and Na ₄₂ ²⁺ → 2Na ₂₁ ⁺ within the Kohn—Sham Density Functional Theory (i.e., including shell effects). Although the shape of the barrier obviously depends on the parametrization of the fission path, we have found that the barrier maximum corresponds to a configuration in which the emerging fragments are already formed and rather well apart. The two examples chosen serve to illustrate the influence of the electronic shell effects.     

Spectrophotometric determination of zinc in copper-base alloys with TAN

TAN reacts with zinc(II) forming a red complex with composition 1:2 Zn(II)-TAN and absorption maximum at 582 nm. Zinc can be determined with this reagent in the presence of Triton X-100, in the pH range 6.20-8.00 with a molar absorptivity of 4.5×10⁴ l/mol/cm Beer's Law was obeyed up to least 1.55 g/ml. Copper interference was eliminated with a mixture of thiosulfate and ascorbic acid and nickel separated by precipitation with dimethylglyoxime. The proposed method was used for zinc determination in several copper-base alloys and the results of analysis in comparison with certified values indicated that the procedure was accurate and precise. A derivative procedure is also proposed, allowing zinc determination with high sensitivity (5-400 ng/ml).