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This paper is devoted to the derivation of an a posteriori residual-based error estimator for the A-φ magnetodynamic harmonic formulation of the Maxwell system. The weak continuous and discrete formulations are established, and the well-posedness of both of them is addressed. Some useful analytical tools are derived. Among them, an ad hoc Helmholtz decomposition is proven, which allows to pertinently split the error. Consequently, an a posteriori error estimator is obtained, which is proven to be reliable and locally efficient. Finally, numerical tests confirm the theoretical results.

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We have developed synthetic techniques involving coprecipitation of iron(III) and manganese(II) ions from aqueous solutions using NaOH to create single-crystal, stoichiometric MnFe2O4 particles. The particle diameter could be varied between 5 and 25 nm by varying the metallic-ion to hydroxide-ion concentration ratio. Magnetic measurements revealed that the saturation magnetization decreased, the ferrimagnetic to paramagnetic transition broadened, and the apparent transition temperature increased, all with decreasing particle size.

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