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We examine the long-term behavior of nonintegrable, energy-conserved, 1D systems of macroscopic grains interacting via a contact-only generalized Hertz potential and held between stationary walls. Existing dynamical studies showed the absence of energy equipartitioning in such systems, hence their long-term dynamics was described as quasi-equilibrium. Here, we show that these systems do in fact reach thermal equilibrium at sufficiently long times, as indicated by the calculated heat capacity. This phase is described by equilibrium statistical mechanics, opening up the possibility that the machinery of nonequilibrium statistical mechanics may be used to understand the behavior of these systems away from equilibrium.

Shortest path plays an important role in the study of complex networks. But in real transportation systems, choosing the shortest path may not be the best way for the drivers. Based on the traffic equilibrium theory, we generalize the concept of shortest path. Flux distribution is also investigated by using the generalized concept on various types of complex networks. We find that the flux differs little in all the edges of lattice while in small-world and scale-free networks, the flux distribution follows a power law, and in the random network, the flux distribution has an exponential tail. We consider lattice may be the optimal topology in design a transportation network.

In this paper, we present two epidemic models with a nonlinear incidence and transfer from infectious to recovery. For epidemic models, the basic reproductive number is calculated. A dynamic system based on threshold, using LaSalle’s invariance principle and Lyapunov function, is structured completely by the basic reproductive number. By studying the SIR and SIRS models under the nonlinear condition, the general validity of the method is verified.

The interaction between iron(II) tetrasulfophthalocyanine ([Fe^IITSPc]⁴⁺) and histamine results in the oxidation of the central metal by oxygen in the former, with the formation of a complex denoted as [(His)Fe^IIITSPc]³⁻ (where His = histamine). The rate constant for the formation of the complex is k_f = 2.41 × 10⁻²dm³.mol⁻¹.s⁻¹ and an equilibrium constant of 6.3 dm³.mol^-1 was obtained. The oxidation state of the central metal of [Fe^IITSPc]⁴⁻ before and after the coordination of histamine is confirmed by spectroelectrochemistry. Further electrochemical oxidation of this [(His)Fe^IIITsPc]³⁻ derivative results in a metal-based process proposed to involve an Fe^IV phthalocyanine species.