A spectral radius theorem for matrix seminorms

AbstractA matrix seminorm ∥·∥ is called supspectral if it satisfies the condition that the spectral radius of a square matrix A is lim sup ∥An∥1/n as n→∞. This property is shown to be equivalent to each of two conditions on ∥·∥, one characterizing behavior on idempotent A, and the other characterizing behavior on non-nilpotent A. Examples of supspectral seminorms are provided

The internal representation of pitch sequences in tonal music

A model for the internal representation of pitch sequences in tonal music is advanced. This model assumes that pitch sequences are retained as hierarchical networks. At each level of the hierarchy, elements are organized as structural units in accordance with laws of figural goodness, such as proximity and good continuation. Further, elements that are present at each hierarchical level are elaborated by further elements so as to form structural units at the next-lower level, until the lowest level is reached. Processing advantages of the system are discussed. It may generally be stated that we tend to encode and retain information in the form of hierarchies when given the opportunity to do so. For example, programs of behavior tend to be retained as hierarchies (Miller, Galanter, & Pribram, 1960) and goals in problem solving as hierarchies of subgoal