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For the first time the atmospheric deposition of trace metals was studied over the entire territory of the Republic of Macedonia. Samples of the terrestrial mosses Hypnum cupressiforme, Camptothecium lutescens, and Homalothecium sericeum were collected in September-October 2002 at 73 sites evenly distributed over the country, and a total of 43 elements were determined by instrumental neutron activation analysis and atomic absorption spectrometry. Principal component factor analysis was used to identify the most polluted areas and characterize different pollution sources. The most important sources of trace metal deposition are ferrous and non-ferrous smelters, oil refineries, fertilizer production plants, and central heating stations. Four areas appear to be particularly exposed to metal pollution: Veles, Skopje, Tetovo, and Kavadarci-Negotino, whereas the predominantly agricultural regions in the south, southwest, and southeast show levels closer to European median values for most elements of mainly pollution origin.

We analyze the Kuramoto model of phase oscillators with natural frequencies distributed according to a unimodal asymmetric function g(omega) . It is obtained that besides a second-, also a first-order phase transition can appear if the distribution of natural frequencies possesses a sufficiently large flat section. It is derived analytically that for the first-order transitions the characteristic exponents describing the order parameter and synchronizing frequency near the critical point are equal to those for the order parameter in the corresponding symmetric case. Stability analysis of the incoherent phase shows that the synchronizing frequency at the onset of synchronization equals the perturbation rotation velocity at the border of stability. The analytic and numerical results are in agreement with numerical simulations.

We consider the Kuramoto model of phase oscillators with natural frequencies distributed according to a unimodal function with the plateau section in the middle representing the maximum and symmetric tails falling off predominantly as |omega-omega0|m, m>0, in the vicinity of the flat region. It is found that the phase transition is of first order as long as there is a finite flat region and that in the vicinity of the critical coupling the following scaling law holds r-rc proportional, variant(K-Kc)2/(2m+3), where r is the order parameter and K is the coupling strength of the interacting oscillators.

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