The Projective Line Over the Finite Quotient Ring GF(2)[xx]/<x3−x>< x^{3} - x> and Quantum Entanglement I. Theoretical Background

The paper deals with the projective line over the finite factor ring $R\_{\clubsuit} \equiv$ GF(2)[$x$]/$$. The line is endowed with 18 points, spanning the neighbourhoods of three pairwise distant points. As $R\_{\clubsuit}$ is not a local ring, the neighbour (or parallel) relation is not an equivalence relation so that the sets of neighbour points to two distant points overlap. There are nine neighbour points to any point of the line, forming three disjoint families under the reduction modulo either of two maximal ideals of the ring. Two of the families contain four points each and they swap their roles when switching from one ideal to the other; the points of the one family merge with (the image of) the point in question, while the points of the other family go in pairs into the remaining two points of the associated ordinary projective line of order two. The single point of the remaining family is sent to the reference point under both the mappings and its existence stems from a non-trivial character of the Jacobson radical, ${\cal J}\_{\clubsuit}$, of the ring. The factor ring $\widetilde{R}\_{\clubsuit} \equiv R\_{\clubsuit}/ {\cal J}\_{\clubsuit}$ is isomorphic to GF(2) $\otimes$ GF(2). The projective line over $\widetilde{R}\_{\clubsuit}$ features nine points, each of them being surrounded by four neighbour and the same number of distant points, and any two distant points share two neighbours. These remarkable ring geometries are surmised to be of relevance for modelling entangled qubit states, to be discussed in detail in Part II of the paper.Comment: 8 pages, 2 figure

Terrace reconstruction and long profile projection: a case study from the Solent river system near Southampton, England

River terrace sequences are important frameworks for archaeological evidence and as such it is important to produce robust correlations between what are often fragmentary remnants of ancient terraces. This paper examines both conceptual and practical issues related to such correlations, using a case study from the eastern part of the former Solent River system near Southampton, England. In this region two recent terrace schemes have been constructed using different data to describe the terrace deposits: one based mainly on terrace surfaces; the other on gravel thicknesses, often not recording the terrace surface itself. The utility of each of these types of data in terrace correlation is discussed in relation to the complexity of the record, the probability of post-depositional alteration of surface sediments and comparison of straight-line projections with modern river long profiles. Correlation using age estimates is also discussed, in relation to optically-stimulated luminescence dating of sand lenses within terrace gravels in this region during the PASHCC project. It is concluded that the need for replication at single sites means that this approach has limited use for correlative purposes, although dating of sediments is important for understanding wider landscape evolution and patterns of human occupation