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Modularity (segmentation), homology and heterochrony were essential concepts embraced by Gavin de Beer in his studies of the development and evolution of the vertebrate skull. While his pioneering contributions have stood the test of time, our understanding of the biological processes that underlie each concept has evolved. We assess de Beer's initial training as an experimental embryologist; his switch to comparative and descriptive studies of skulls, jaws and middle ear ossicles; and his later research on the mammalian skull, including his approach to head segmentation. The role of cells of neural crest and mesodermal origin in skull development, and developmental, palaeontological and molecular evidence for the origin of middle ear ossicles in the evolutionary transition from reptiles to mammals are used to illustrate our current understanding of modularity, homology and heterochrony. This article is part of the theme issue ‘The mammalian skull: development, structure and function’. 

Abstract The free multiplicative Brownian motion$$b_{t}$$ $b_t$is the large-Nlimit of the Brownian motion on$$\mathsf {GL}(N;\mathbb {C}),$$ $\mathrm{GL}(N;C),$in the sense of$$*$$ $\ast$-distributions. The natural candidate for the large-Nlimit of the empirical distribution of eigenvalues is thus the Brown measure of$$b_{t}$$ $b_t$. In previous work, the second and third authors showed that this Brown measure is supported in the closure of a region$$\Sigma _{t}$$ ${\Sigma }_t$that appeared in the work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density$$W_{t}$$ $W_t$on$$\overline{\Sigma }_{t},$$ ${\overline{\Sigma }}_t,$which is strictly positive and real analytic on$$\Sigma _{t}$$ ${\Sigma }_t$. This density has a simple form in polar coordinates:$$\begin{aligned} W_{t}(r,\theta )=\frac{1}{r^{2}}w_{t}(\theta ), \end{aligned}$$ $\begin{array}{c}{W}_t(r,\theta )=\frac{1}{r^2}{w}_t\left(\theta \right),\end{array}$where$$w_{t}$$ $w_t$is an analytic function determined by the geometry of the region$$\Sigma _{t}$$ ${\Sigma }_t$. We show also that the spectral measure of free unitary Brownian motion$$u_{t}$$ $u_t$is a “shadow” of the Brown measure of$$b_{t}$$ $b_t$, precisely mirroring the relationship between the circular and semicircular laws. We develop several new methods, based on stochastic differential equations and PDE, to prove these results. 

Graphical perception studies typically measure visualization encoding effectiveness using the error of an “average observer”, leading to canonical rankings of encodings for numerical attributes: e.g., position > area > angle > volume. Yet different people may vary in their ability to read different visualization types, leading to variance in this ranking across individuals not captured by population-level metrics using “average observer” models. One way we can bridge this gap is by recasting classic visual perception tasks as tools for assessing individual performance, in addition to overall visualization performance. In this article we replicate and extend Cleveland and McGill's graphical comparison experiment using Bayesian multilevel regression, using these models to explore individual differences in visualization skill from multiple perspectives. The results from experiments and modeling indicate that some people show patterns of accuracy that credibly deviate from the canonical rankings of visualization effectiveness. We discuss implications of these findings, such as a need for new ways to communicate visualization effectiveness to designers, how patterns in individuals’ responses may show systematic biases and strategies in visualization judgment, and how recasting classic visual perception tasks as tools for assessing individual performance may offer new ways to quantify aspects of visualization literacy. Experiment data, source code, and analysis scripts are available at the following repository: https://osf.io/8ub7t/?view_only=9be4798797404a4397be3c6fc2a68cc0 .