Perspex machine IX: transreal analysis

We introduce transreal analysis as a generalisation of real analysis. We find that the generalisation of the real exponential and logarithmic functions is well defined for all transreal numbers. Hence, we derive well defined values of all transreal powers of all non-negative transreal numbers. In particular, we find a well defined value for zero to the power of zero. We also note that the computation of products via the transreal logarithm is identical to the transreal product, as expected. We then generalise all of the common, real, trigonometric functions to transreal functions and show that transreal (sin x)/x is well defined everywhere. This raises the possibility that transreal analysis is total, in other words, that every function and every limit is everywhere well defined. If so, transreal analysis should be an adequate mathematical basis for analysing the perspex machine - a theoretical, super-Turing machine that operates on a total geometry. We go on to dispel all of the standard counter "proofs" that purport to show that division by zero is impossible. This is done simply by carrying the proof through in transreal arithmetic or transreal analysis. We find that either the supposed counter proof has no content or else that it supports the contention that division by zero is possible. The supposed counter proofs rely on extending the standard systems in arbitrary and inconsistent ways and then showing, tautologously, that the chosen extensions are not consistent. This shows only that the chosen extensions are inconsistent and does not bear on the question of whether division by zero is logically possible. By contrast, transreal arithmetic is total and consistent so it defeats any possible "straw man" argument. Finally, we show how to arrange that a function has finite or else unmeasurable (nullity) values, but no infinite values. This arithmetical arrangement might prove useful in mathematical physics because it outlaws naked singularities in all equations.

Characterising the use of urban space: a geochemical case study from Calleva Atrebatum (Silchester, Hampshire, UK) Insula IX during the late first/early second century AD

The geochemical analysis of soil samples from the Roman town of Calleva Atrebatum (Silchester, Hampshire, UK) has been undertaken in order to enhance our understanding of urban occupation during the late first/early second century AD. Samples taken from a variety of occupation deposits within several, contemporary timber buildings, including associated hearths, have been analysed using laboratory-based x-ray fluorescence for a suite of elements (Cu, Zn, Pb, Sr, P and Ca). The patterns of elemental enrichment seen across the site have allowed us to compare and contrast the buildings that were occupied during this time in an attempt to distinguish different uses, such as between domestic and work-space. Two of the buildings stand out as having high concentrations of elements which suggest that they were dirtier work spaces, whilst other buildings appear to be have lower chemical loadings suggesting they were cleaner.