Lattice initial segments of the hyperdegrees

We affirm a conjecture of Sacks [1972] by showing that every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees, $\mathcal{D}_{h}$. In fact, we prove that every sublattice of any hyperarithmetic lattice (and so, in particular, every countable locally finite lattice) is isomorphic to an initial segment of $\mathcal{D}_{h}$. Corollaries include the decidability of the two quantifier theory of $$ and the undecidability of its three quantifier theory. The key tool in the proof is a new lattice representation theorem that provides a notion of forcing for which we can prove a version of the fusion lemma in the hyperarithmetic setting and so the preservation of $\omega _{1}^{CK}$. Somewhat surprisingly, the set theoretic analog of this forcing does not preserve $\omega _{1}$. On the other hand, we construct countable lattices that are not isomorphic to an initial segment of $\mathcal{D}_{h}$

Constructing regional advantage: Platform policies based on related variety and differentiated knowledge bases

The article presents a regional innovation policy model, based on the idea of constructing regional advantage. This policy model brings together concepts like related variety, knowledge bases and policy platforms. Related variety attaches great importance to knowledge spillovers across complementary sectors, possibly in a region. Then, the paper categorises knowledge into ‘analytical’ (science based), ‘synthetic’ (engineering based) and ‘symbolic’ (artistic based) in nature, with different ‘virtual’ and real proximity mixes. Finally, the implications of this are traced for evolving ‘platform policies’ that facilitate economic development within and between regions in action lines appropriate to related variety and differentiated knowledge bases.Related variety; Differentiated knowledge bases; Platform policy, Regional innovation policy

Arithmetic complexity via effective names for random sequences

We investigate enumerability properties for classes of sets which permit recursive, lexicographically increasing approximations, or left-r.e. sets. In addition to pinpointing the complexity of left-r.e. Martin-L\"{o}f, computably, Schnorr, and Kurtz random sets, weakly 1-generics and their complementary classes, we find that there exist characterizations of the third and fourth levels of the arithmetic hierarchy purely in terms of these notions. More generally, there exists an equivalence between arithmetic complexity and existence of numberings for classes of left-r.e. sets with shift-persistent elements. While some classes (such as Martin-L\"{o}f randoms and Kurtz non-randoms) have left-r.e. numberings, there is no canonical, or acceptable, left-r.e. numbering for any class of left-r.e. randoms. Finally, we note some fundamental differences between left-r.e. numberings for sets and reals

Carbon-13 in groundwater from English and Norwegian crystalline rock aquifers: a tool for deducing the origin of alkalinity?

The 13C signature is evaluated for various environmental compartments (vegetation, soils, soil gas, rock and groundwater) for three crystalline rock terrains in England and Norway. The data are used to evaluate the extent to which stable carbon isotopic data can be applied to deduce whether the alkalinity in crystalline bedrock groundwaters has its origin in hydrolysis of carbonate or silicate minerals by CO2. The resolution of this issue has profound implications for the role of weathering of crystalline rocks as a global sink for CO2. In the investigated English terrain (Isles of Scilly), groundwaters are hydrochemically immature and DIC is predominantly in the form of carbonic acid with a soil gas signature. In the Norwegian terrains, the evidence is not conclusive but is consistent with a significant fraction of the groundwater DIC being derived from silicate hydrolysis by CO2. A combined consideration of pH, alkalinity and carbon isotope data, plotted alongside theoretical evolutionary pathways on bivariate diagrams, strongly suggests real evolutionary pathways are likely to be hybrid, potentially involving both open and closed CO2 conditions