Jacob Laubacher

In this paper, we investigate properties of a reproducing kernel Hilbert space of a group action. In particular, we introduce an equivalence relation on a compact Hausdorff space$X$ , and consequently establish three equivalent definitions for when two elements are related. We also see how the equivalence classes of$X$correspond to subgroups of the group acting transitively on$X$ , which we aptly refer to as relation stabilizers.

Let$G$be a finite solvable group and let$\Delta(G)$be the character degree graph of$G$ . In this paper, we obtain the metric dimension of certain character degree graphs. Specifically, we calculate the metric dimension for a regular character degree graph, a character degree graph with a diameter of$2$that is not a block, a character degree graph with a diameter of$3$that also has a cut vertex and a character degree graph with Fitting height$2.$We also consider two related parameters, base size and adjacency dimension, and their relation to metric dimension for character degree graphs of solvable groups.

It is well known that the bar resolution can be replaced with any projective resolution of the corresponding algebra when computing the Hochschild (co)homology of that algebra. This is, in fact, a feature of its construction via derived functors. For generalizations and extensions of the Hochschild (co)homology, one uses a bar-like resolution in a simplicial setting in order to accommodate the changing module structures in every dimension. In this note, we present a method in order to replace these bar-like resolutions.