In this paper, we provide a general framework for counting geometric structures in pseudo-random graphs. As applications, our theorems recover and improve several results on the finite field analog of questions originally raised in the continuous setting. The results present interactions between discrete geometry, geometric measure theory, and graph theory.

The theta graph ${\Theta _{\ell ,t}}$ consists of two vertices joined by t vertex-disjoint paths, each of length $\ell $ . For fixed odd $\ell $ and large t, we show that the largest graph not containing ${\Theta _{\ell ,t}}$ has at most ${c_\ell }{t^{1 - 1/\ell }}{n^{1 + 1/\ell }}$ edges and that this is tight apart from the value of ${c_\ell }$ .

The classical Kővári–Sós–Turán theorem states that if G is an n-vertex graph with no copy of Ks,t as a subgraph, then the number of edges in G is at most O(n2−1/s). We prove that if one forbids Ks,t as an induced subgraph, and also forbids any fixed graph H as a (not necessarily induced) subgraph, the same asymptotic upper bound still holds, with different constant factors. This introduces a non-trivial angle from which to generalize Turán theory to induced forbidden subgraphs, which this paper explores. Along the way, we derive a non-trivial upper bound on the number of cliques of fixed order in a Kr-free graph with no induced copy of Ks,t. This result is an induced analogue of a recent theorem of Alon and Shikhelman and is of independent interest.

Like the Messianic Banquet, the Messianic Torah is a concept frequently encountered among New Testament scholars. It seems to be presupposed as a category well known to first century Judaism and, therefore, to be a yardstick against which Jesus’ ‘legislative’ activity could and can be measured. An examination of this concept begins with the tension in Jewish thought between the eternity of the Mosaic Law and the various strategies designed to supplement its inadequacy to deal with contemporary situations. Given that the unique circumstances of the Messianic Age involve a further complication, the relevant texts are reviewed for their contributions to the problem. The Old Testament writers look to a deepening of observance of the Mosaic Law in the eschatological times. The Pseudepigrapha offer little more apart from a hint of Messianic legislation in Psalms of Solomon 17. Even the gospels do not show us a Jesus who fits readily into the model of a Messianic legislator. In fact St John probably pictures him as the embodiment of the Torah, the Word made flesh. The conclusion is that, whatever the later rabbinic teaching on the subject, the idea of the Messianic Torah is a scholarly construct as far as the New Testament is concerned.

An integral part of decommissioning the Nuclear Decommissioning Agency's (NDA) Dounreay site is the management of the solid low level radioactive waste (LLW). The United Kingdom Atomic Energy Authority (UKAEA) has developed and progressed a technical and stakeholder programme that has enabled it to submit a robust Planning Application to Highland Council (HC) for New LLW Disposal Facilities at Dounreay and to submit substantive preliminary safety and environmental cases to the Nuclear Installations Inspectorate (NII) and the Scottish Environment Protection Agency (SEPA). To UKAEA’s knowledge this is the most advanced project in the UK for new LLW disposal facilities. Experience has been gained in progressing the Best Practicable Environmental Option (BPEO) Study, working with regulators in unfamiliar areas, and undertaking groundbreaking stakeholder consultation. Key lessons learnt are that stakeholders should be engaged in dialogue on the project as early as possible, documentation must be high quality and tailored to its audience, and internationally respected and credible consultants must be involved.

Nutrigenomics is the study of how constituents of the diet interact with genes, and their products, to alter phenotype and, conversely, how genes and their products metabolise these constituents into nutrients, antinutrients, and bioactive compounds. Results from molecular and genetic epidemiological studies indicate that dietary unbalance can alter gene–nutrient interactions in ways that increase the risk of developing chronic disease. The interplay of human genetic variation and environmental factors will make identifying causative genes and nutrients a formidable, but not intractable, challenge. We provide specific recommendations for how to best meet this challenge and discuss the need for new methodologies and the use of comprehensive analyses of nutrient–genotype interactions involving large and diverse populations. The objective of the present paper is to stimulate discourse and collaboration among nutrigenomic researchers and stakeholders, a process that will lead to an increase in global health and wellness by reducing health disparities in developed and developing countries.