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The modes of occurrence of elements in coal are important not only because they can provide insights into the sources of mineral matter in coal but also because they are vital in determining the behavior of their environmental and human health impacts. Besides a number of physical and chemical analyses for determining the modes of occurrence in coal, some statistical methods have been commonly adopted to investigate elements in coal. Among many statistical methods, the hierarchy clustering algorithm is the most common method for deducing modes of occurrence of elements in coal. However, different hierarchical clustering algorithms with a number of similarity measures sometimes result in different modes of occurrence of elements in coal, and subsequently in some cases, such results could be confusing. Therefore, which algorithm is more effective in determining the modes of occurrence in coal deserves to be investigated. In this paper, the data sets of coals from the Adaohai coal mine in Inner Mongolia, China, are used for this performance evaluation. From the analytical results with the average linkage hierarchical clustering algorithm on Adaohai coal samples, many instructive and surprising insights can be concluded. For example, selenium, Be, and Tl do not appear to be in agreement with geochemical principles, that is, substituting for P, associated with rare earth elements, and occurring in Fe-sulfides, respectively. In conclusion, the average linkage hierarchical clustering algorithm with correlation similarity is much better in the analysis of the geological processes than the previous statistical method used in Adaohai coal samples, that is, centroid linkage hierarchical clustering algorithm with Pearson correlation similarity.

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A class of nonlinear fractional multipoint boundary value problems at resonance is considered in this article. The existence results are obtained by the method of the coincidence degree theory of Mawhin. An example is given to illustrate the results.