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We generalize an approach to deriving diagnosability results of various interconnection networks in terms of the popular $g$-good-neighbor and $g$-extra fault-tolerant models, as well as mainstream diagnostic models such as the PMC and the MM* models. As demonstrative examples, we show how to follow this constructive, and effective, process to derive the $g$-extra diagnosabilities of the hypercube, the $(n, k)$-star, and the arrangement graph. These results agree with those achieved individually, without duplicating structure independent technical details. Some of them come with a larger applicable range than those already known, and the result for the arrangement graph in terms of the MM* model is new., Comment: Preliminary versions of some results of this paper were announced (without proofs) at 2019 International Conference on Modeling, Simulation, Optimization and Algorithm (ICMSOA 2019), November 9-10, 2019, Sanya, China. J. Phys: Conf. Ser. 1409 (2019) 012024

Dolomite crystals in partially dolomitized limestone from the Platteville Formation are both compositionally and microstructurally heterogeneous. A single dolomite crystal usually contains three phases: the host Ca-rich dolomite [Ca1.14Mg0.86(CO3)2], an Fe-bearing dolomite [Ca1.06Mg0.80Fe0.14(CO3)2], and calcite inclusions. These three phases show similar orientations. The Ca-rich dolomite exhibits modulated microstructures with wavelength ranging from 7 to 30 nm. The modulated microstructures are not evident in Fe-bearing dolomite. Modulations in the Ca-rich dolomite have three predominant orientation ranges in the studied sample: from (205) to (104), from (001) to (-101), and ~(110), which are consistent with previous studies. Bright-field (BF) and high-angle annular dark-field (HAADF) images confirm that these modulations are due to chemical variation rather than strain or diffraction contrast. The Ca-rich lamellae are Mg-rich calcite with compositions ranging from Ca0.85Mg0.15CO3 to Ca0.70Mg0.30CO3. The observed results indicate that these Ca-rich exsolution lamellae formed during diagenesis. In this study, three kinds of "c"-reflections, which are weak spots in the halfway position between the principal reflections along the (104)*, (112)*, and (110)* directions, have been found in the diffraction patterns of some Ca-rich dolomite. Mg-Ca ordering in x-y planes was not observed directly in Z-contrast images. FFT patterns from the Z-contrast images do not show "c"-reflections. STEM images confirm that the "c"-reflections could result from multiple diffraction between the host dolomite and twinned Mg-calcite nano-lamellae under TEM imaging and diffraction modes.

Nano-precipitates with tripled periodicity along the c-axis are observed in a Ca-rich dolomite sample from Proterozoic carbonate rocks with "molar tooth" structure. This observation is consistent with previous description of d reflections. High-angle annular dark-field STEM imaging (or Z-contrast imaging) that avoids dynamic diffraction as seen in electron diffraction and high-resolution TEM imaging modes, confirms that d reflections correspond to nanoscale precipitates aligned parallel to (001) of the host dolomite. The lamellae precipitates have a cation ordering sequence of Ca-Ca-Mg-Ca-Ca- Mg along the c direction resulting in a chemical composition of Ca0.67Mg0.33CO3. This superstructure is attributed to the extra or d reflections, thus is referred to as the d superstructure in this study. The structure can be simply described as interstratified calcite/dolomite. The crystal structure of the d superstructure calculated from density functional theory (DFT) has a space group of P31c and has a and c unit-cell parameters of 4.879 and 16.260 {\AA}, respectively, values between those of dolomite and calcite. The detailed structural characteristics and parameters obtained from ab initio calculations are also reported in this paper. The method of combining Z-contrast imaging and ab initio calculations can be used for solving structures of other nano-precipitates and nano-phases.

Dehydration of water from surface Mg2+ is most likely the rate-limiting step in the dolomite growth at low temperature. Here, we investigate the role of polysaccharide in this step using classical molecular dynamics (MD) calculations. Free energy (potential of mean force, PMF) calculations have been performed for water molecules leaving the first two hydration layers above the dolomite (104) surface under the following three conditions: without catalyst, with monosaccharide (mannose) and with oligosaccharide (three units of mannose). MD simulations reveal that there is no obvious effect of monosaccharide in lowering the dehydration barrier for surface Mg2+. However, we found that there are metastable configurations of oligosaccharide, which can decrease the dehydration barrier of surface Mg2+ by about 0.7-1.1 kcal/mol. In these configurations, the molecule lies relatively flat on the surface and forms a bridge shape. The hydrophobic space near the surface created by the non-polar -CH groups of the oligosaccharide in the bridge conformation is the reason for the observed reduction of dehydration barrier.

The key kinetic barrier to dolomite formation is related to the surface Mg2+-H2O complex, which hinders binding of surface Mg2+ ions to the CO3 2- ions in solution. It has been proposed that this reaction can be catalyzed by dissolved hydrogen sulfide. To characterize the role of dissolved hydrogen sulfide in the dehydration of surface Mg 2+ ions, ab initio simulations based on density functional theory (DFT) were carried out to study the thermodynamics of competitive adsorption of hydrogen sulfide and water on dolomite (104) surfaces from solution. We find that water is thermodynamically more stable on the surface with the difference in adsorption energy of -13.6 kJ/mol (in vacuum) and -12.8 kJ/mol (in aqueous solution). However, aqueous hydrogen sulfide adsorbed on the surface increases the Mg2+-H2O distances on surrounding surface sites. Two possible mechanisms were proposed for the catalytic effects of adsorbed hydrogen sulfide on the anhydrous Ca-Mg-carbonate crystallization at low temperature.