It is common to reduce the dimensionality of data before applying classical multivariate analysis techniques in statistics. Persistent homology, a recent development in computational topology, has been shown to be useful for analyzing high-dimensional (nonlinear) data. In this article, we connect computational topology with the traditional analysis of variance and demonstrate the value of combining these approaches on a three-dimensional orthodontic landmark dataset derived from the maxillary complex. Indeed, combining appropriate techniques of both persistent homology and analysis of variance results in a better understanding of the data’s nonlinear features over and above what could have been achieved by classical means. Supplementary material for this article is available online.

We present and investigate a new class of quantum channels, what we call “universal collective rotation channels,” that includes the class of collective rotation channels as a special case. The fixed point set and noise commutant coincide for a channel in this class. Computing the precise structure of this $C^{*}$ -algebra is a core problem in a particular noiseless subsystem method of quantum error correction. We prove that there is an abundance of noiseless subsystems for every channel in this class and that the Young tableaux combinatorial machine may be used to explicitly compute these subsystems.