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We study the heat kernel expansion of the Laplacian on n-forms defined on a subgraph of a directed complete graph. We derive two expressions for the subgraph heat kernel on 0-forms and compute the coefficients of the expansion. We also obtain the subgraph heat kernel of the Laplacian on 1-forms.

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We study the classes of modules which are generated by a silting module. In the case of either hereditary or perfect rings, it is proved that these are exactly the torsion T such that the regular module has a special T-preenvelope. In particular, every torsion-enveloping class in Mod-R are of the form Gen(T) for a minimal silting module T. For the dual case, we obtain for general rings that the covering torsion-free classes of modules are exactly the classes of the form Cogen(T), where T is a cosilting module.

We study orthogonal polynomials with respect to self-similar measures, focusing on the class of infinite Bernoulli convolutions, which are defined by iterated function systems with overlaps, especially those defined by the Pisot, Garsia, and Salem numbers. By using an algorithm of Mantica, we obtain graphs of the coefficients of the 3-term recursion relation defining the orthogonal polynomials. We use these graphs to predict whether the singular infinite Bernoulli convolutions belong to the Nevai class. Based on our numerical results, we conjecture that all infinite Bernoulli Convolutions with contraction ratios greater than or equal to 1/2 belong to Nevai's class, regardless of the probability weights assigned to the self-similar measures.