Eighty-eight strains of microorganisms were isolated from soils collected in northern and southern Chile, and 10 fungi which showed the highest solubilizing action upon the iron in granodiorite were then selected. These fungi were incubated with the following iron-containing minerals: augite, hornblende, biotite, magnetite, hematite, and the igneous rock granodiorite. The solubility of iron in these minerals depended on their nature, crystalline structure, the concentration of metabolic products, or all three. Complex formation could be the mechanism involved, as a strong cation-exchange resin was not able to extract Fe from culture solutions. This conclusion is also confirmed by the Rp values obtained by thin-layer chromatography of iron-containing culture solutions. Certain groups of microorganisms can directly or indirectly transform rocks and minerals in quantities large enough to influence their geological distribution (11, 14). These transformations include enzymatic oxidations and reductions and the formation of chelates and complexes with proteins, amino acids, organic acids, etc. (4, 11). The action of some organic products of microbial metabolism on minerals has been reported (4, 5). It was demonstrated that microorganisms transform crystalline biotite (2), mica to vermiculite (15), and certain rocks to an amorphous state (12). The present work examines the solubilization of iron minerals of different composition and crystalline forms by soil microorganisms. Complexing of iron by metabolic products of Aspergillus niger was also studied. MATERALS AND METHODS Eighty-eight strains of microorganisms were isolated from soils collected in northern and southern Chile. The soils were diluted in water and cultivated on glucose agar and Sabouraud glucose agar. Ten fungi were selected which, when incubated in the presence of 100 mg of granodiorite, solubilized the highest amounts of iron. For incubation, a sterile basal medium was used which had the following composition, per liter: (NH4)2504, 0.5 g; KH2PO,, 0.5 g; and sucrose, 50 g; it was dissolved in distilled water and adjusted to pH 7.0 with a diluted NaOH solution. The selected fungi were numbered from 1 to 10, corresponding to the following genera: Penicilliwn (1 and 10), Mucor (2), Aspergillus (3, 4, and 5), Cephalosporium (6, 7, and 8) and Fusarium (9). They were maintained in Sabouraud agar medium.

A g-circulant matrix of order n is defined as a matrix of order n where each row is a right cyclic shift in g−places to the preceding row. Using number theory, certain nonnegative g-circulant real matrices are constructed. In particular, it is shown that spectra with sufficient conditions so that it can be the spectrum of a real g-circulant matrix is not a spectrum with sufficient conditions so that it can be the spectrum of a real circulant matrix of the same order. The obtained results are applied to Nonnegative Inverse Eigenvalue Problem to construct nonnegative, g-circulant matrices with given appropriated spectrum. Moreover, nonnegative g-circulant matrices by blocks are also studied and in this case, their orders can be a multiple of a prime number.

A list Λ = { λ 1 , λ 2 , … , λ n } \Lambda =\left\{{\lambda }_{1},{\lambda }_{2},\ldots ,{\lambda }_{n}\right\} of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix and is said to be universally realizable (UR), if it is realizable for each possible Jordan canonical form allowed by Λ \Lambda . In 1981, Minc proved that if Λ \Lambda is diagonalizably positively realizable, then Λ \Lambda is UR [Proc. Amer. Math. Society 83 (1981), 665–669]. The question whether this result holds for nonnegative realizations was open for almost 40 years. Recently, two extensions of Mins’s result have been obtained by Soto et al. [Spec. Matrices 6 (2018), 301–309], [Linear Algebra Appl. 587 (2020), 302–313]. In this work, we exploit these extensions to generate new universal realizability criteria. Moreover, we also prove that under certain conditions, the union of two lists UR is also UR, and for certain criteria, if Λ \Lambda is UR, then for t ≥ 0 t\ge 0 , Λ t = { λ 1 + t , λ 2 ± t , λ 3 , … , λ n } {\Lambda }_{t}=\left\{{\lambda }_{1}+t,{\lambda }_{2}\pm t,{\lambda }_{3},\ldots ,{\lambda }_{n}\right\} is also UR.

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