Global asymptotic stability in a class of Putnam-type equations

In this paper, we initiate the study of a class of Putnam-type equation of the form x(n-1) = A(1)x(n) + A(2)x(n-1) + A(3)x(n-2)x(n-3) + A(4)/B(1)x(n)x(n-1) + B(2)x(n-2) + B(3)x(n-3) + B-4 n = 0, 1, 2,..., where A(1), A(2), A(3), A(4), B-1, B-2, B-3, B-4 are positive constants with A(1) + A(2) + A(3) + A(4) = B-1 + B-2 + B-3 + B-4, x(-3), x(-2), x(-1), x(0) are positive numbers. A sufficient condition is given for the global asymptotic stability of the equilibrium point c = 1 of such equations. (c) 2005 Elsevier Ltd. All rights reserved.

Ball milling as an effective route for the preparation of doped bornite: synthesis, stability and thermoelectric properties

Bornite, Cu5FeS4, is a naturally-occuring mineral with an ultralow thermal conductivity and potential for thermoelectric power generation. We describe here a new, easy and scalable route to synthesise bornite, together with the thermoelectric behaviour of manganese-substituted derivatives, Cu5Fe1-xMnxS4 (0 ≤ x ≤ 0.10). The electrical and thermal transport properties of Cu5Fe1-xMnxS4 (0 ≤ x ≤ 0.10), which are p-type semiconductors, were measured from room temperature to 573 K. The stability of bornite was investigated by thermogravimetric analysis under inert and oxidising atmospheres. Repeated measurements of the electrical transport properties confirm that bornite is stable up to 580 K under an inert atmosphere, while heating to 890 K results in rapid degradation. Ball milling leads to a substantial improvement in the thermoelectric figure of merit of unsusbtituted bornite (ZT = 0.55 at 543 K), when compared to bornite prepared by conventional high-temperature synthesis (ZT < 0.3 at 543 K). Manganese-substituted samples have a ZT comparable to that of unsubstituted bornite.