Some Geometry of the p-adic Rationals

Given a prime p, we introduce the p-adic absolute value on the rational numbers. We define triangles and angles using this absolute value and investigate their behavior in p-adic geometry and how it differs from Euclidean geometry. Finally, we consider the existence of other polygons using the new absolute value

Nineteenth-century craftsmen and tradesmen in Hertfordshire: a spatial, economic and social study of self-sufficiency

This is an exploration of the validity of the generally accepted belief that rural self-sufficiency declined in Victorian England. Three phases of rural self-sufficiency are identified which follow a chronological sequence. The first is household independence, many basic goods and services are produced within the household: there are a small number of specialised craftsmen and tradesmen within the village, but few, if any, shops. This is followed by decline in household independence, but maintenance of village self-sufficiency with the presence of at least basic specialised crafts and trades and some shops. The final stage is erosion of village self-sufficiency: loss of essential services, leading to hierarchical dependence on services provided by towns. The main data sources used are the unpublished census enumerators' books and the trade directories. Their validity as primary sources is assessed in the context of their suitability for macro- and micro-level studies of rural crafts and trades. Rural service provision over time is explored, initially, at the county level, then focusing on two agricultural areas containing 31 parishes. The status of craftsmen and tradesmen within their local communities is examined in two self-sufficient villages, St. Paul's Walden and Much Hadham. The county-wide study showed that a significant proportion of Hertfordshire villages retained their self sufficiency in basic crafts and trades at least until the beginning of the twentieth century. Self-sufficiency was the predominant system of rural service provision in villages in the Buntingford area from the mid-nineteenth century until at least the 1890s. In contrast, the predominant system of rural service provision in villages in the Stevenage area changed from self sufficiency at mid-century to hierarchical dependence on central villages and neighbouring towns by the 1890s. The study of individual craftsmen and tradesmen revealed evidence of declining economic status and downward social mobility among craftsmen and tradesmen in both the parishes examined in detail

What do new findings about social interaction in autistic adults mean for neurodevelopmental research?

Rachael Davis – ORCID: 0000-0002-3887-6003 https://orcid.org/0000-0002-3887-6003Deficit-based accounts of social and communication abilities continue to dominate autism research. However, emerging findings suggest that this view may be overly simplistic and discount the two-way nature of interaction. Here we discuss the reconceptualization of social cognition to consider such difficulties as examples of bidirectional, multifaceted misattunement between autistic and nonautistic individuals. Aligned with progressive theoretical frameworks, emerging empirical research indicates that mismatches in communication styles can contribute to autistic social difficulties and the important role that nonautistic difficulties play. We highlight two areas of future research with the aim of providing empirical support for the views that the autistic community has proposed over the past 2 decades. We discuss the impact of such a paradigm shift on a number of levels, including how bridging the gap between different interaction styles can reduce stigma and increase understanding. Adopting such a framework will provide radical opportunities for transformative societal changes and education around inclusion.https://doi.org/10.1177/174569162095801016pubpub