complexity estimate
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How did Renaissance listeners experience the polyphonic mass ordinary cycle in the soundscape of the church? We hypothesize that the textural differences in complexity between mass movements allowed listeners to track the progress of the service, regardless of intelligibility of the text or sophisticated musical knowledge.  Building on the principles of auditory scene analysis, this article introduces the Auditory Streaming Complexity Estimate, a measure to evaluate the blending or separation of each part in polyphony, resulting in a moment-by-moment tally of how many independent streams or sound objects might be heard. When applied to symbolic scores for a corpus of 216 polyphonic mass ordinary cycles composed between c. 1450 and 1600, we show that the Streaming Complexity Estimate captures information distinct from the number of parts in the score or the distribution of voices active through the piece. While composers did not all follow the same relative complexity strategy for mass ordinary movements, there is a robust hierarchy emergent from the corpus as a whole: a shallow V shape with the Credo as the least complex and the Agnus Dei as the most. The streaming complexity of masses also significantly increased over the years represented in this corpus.

Classification is an important task in time series mining. It is often reported in the literature that nearest neighbor classifiers perform quite well in time series classification, especially if the distance measure properly deals with invariances required by the domain. Complexity invariance was recently introduced, aiming to compensate from a bias towards classes with simple time series representatives in nearest neighbor classification. To this end, a complexity correcting factor based on the ratio of the more complex to the simpler series was proposed. The original formulation uses the length of the rectified time series to estimate its complexity. In this paper the authors investigate an alternative complexity estimate, based on fractal dimension. Results show that this alternative is very competitive with the original proposal, and has a broader application as it does neither depend on the number of points in the series nor on a previous normalization. Furthermore, these results also verify, using a different formulation, the validity of complexity invariance in time series classification.

We apply proof-theoretic techniques of "proof mining" to obtain an effective uniform rate of metastability in the sense of Tao for Baillon's famous nonlinear ergodic theorem in Hilbert space. In fact, we analyze a proof due to Brézis and Browder of Baillon's theorem relative to the use of weak sequential compactness. Using previous results due to the author we show the existence of a bar recursive functional Ω* (using only lowest type bar recursion B0, 1) providing a uniform quantitative version of weak compactness. Primitive recursively in this functional (and hence in T0 + B0, 1) we then construct an explicit bound φ on for the metastable version of Baillon's theorem. From the type level of φ and another result of the author it follows that φ is primitive recursive in the extended sense of Gödel's T. In a subsequent paper also Ω* will be explicitly constructed leading to the refined complexity estimate φ ∈ T4.