A friend of mine [1] mentioned a problem to me, which he was told had an interesting solution involving an unexpected square root. I have not seen this problem described elsewhere, so I have carried out my own analysis, which I will present here. In fact the solution involves not only square roots, but also higher roots … and a logarithm.

Many contend that U.S. state parties are increasingly polarized and nationalized, meaning that they have adopted divergent positions matching their national counterparts’ positions. Such trends reflect a transformation of America's historically decentralized party system. Yet, the precise timing of these related trends—as well as the mechanisms underpinning them—remain unclear. We assess these dynamics using a novel data set of 1,783 state party platforms between 1918 and 2017. Applying tools from automated and manual content analysis, we document a dramatic divergence in the topics emphasized by Democrats and Republicans starting in the mid-1990s, just as congressional speech became polarized. During this period, cross-state differences in each party's agenda decreased and regional/sectoral issues became less prominent, suggesting tight connections between polarization, nationalization, and state agendas. We also find that innovative phrases increasingly debut in state (not national) platforms. Overall, the evidence undercuts claims of top-down polarization emanating from national party leaders in Washington, DC. Polarization at the state and federal levels coincided with the development of an integrated network of activists spanning multiple levels of the polity.

We present the data and initial results from the first pilot survey of the Evolutionary Map of the Universe (EMU), observed at 944 MHz with the Australian Square Kilometre Array Pathfinder (ASKAP) telescope. The survey covers $270 \,\mathrm{deg}^2$ of an area covered by the Dark Energy Survey, reaching a depth of 25–30 $\mu\mathrm{Jy\ beam}^{-1}$ rms at a spatial resolution of $\sim$ 11–18 arcsec, resulting in a catalogue of $\sim$ 220 000 sources, of which $\sim$ 180 000 are single-component sources. Here we present the catalogue of single-component sources, together with (where available) optical and infrared cross-identifications, classifications, and redshifts. This survey explores a new region of parameter space compared to previous surveys. Specifically, the EMU Pilot Survey has a high density of sources, and also a high sensitivity to low surface brightness emission. These properties result in the detection of types of sources that were rarely seen in or absent from previous surveys. We present some of these new results here.

The two major approaches to studying macroevolution in deep time are the fossil record and reconstructed relationships among extant taxa from molecular data. Results based on one approach sometimes conflict with those based on the other, with inconsistencies often attributed to inherent flaws of one (or the other) data source. Any contradiction between the molecular and fossil records represents a failure of our ability to understand the imperfections of our data, as both are limited reflections of the same evolutionary history. We therefore need to develop conceptual and mathematical models that jointly explain our observations in both records. Fortunately, the different limitations of each record provide an opportunity to test or calibrate the other, and new methodological developments leverage both records simultaneously. However, we must reckon with the distinct relationships between sampling and time in the fossil record and molecular phylogenies. These differences impact our recognition of baselines and the analytical incorporation of age estimate uncertainty.