Real dimension groups

We show the characterization analogous to dimension groups of partially ordered real vector spaces with interpolation works, but sequential direct limits of simplicial vector spaces only under strong assumptions. We also provide and generalize a proof of a result of Fuchs asserting that the real polynomial algebra with pointwise ordering coming from an interval satisfies Riesz interpolatio

Declarative Specification

Deriving formal specifications from informal requirements is extremely difficult since one has to overcome the conceptual gap between an application domain and the domain of formal specification methods. To reduce this gap we introduce application-specific specification languages, i.e., graphical and textual notations that can be unambiguously mapped to formal specifications in a logic language. We describe a number of realised approaches based on this idea, and evaluate them with respect to their domain specificity vs. generalit

Geographic trends in range sizes explain patterns in bird responses to urbanization in Europe

The probability of occurrence of bird species in towns/cities increases with their range sizes, and Rapoport’s rule states that range sizes increase with latitude. To test the hypothesis that the increasing number of bird species persisting in cities at higher latitudes of Europe is linked to their larger range sizes, we compiled data on bird communities of: a) 41 urban bird atlases; b) 37 city core zones from published sources; c) regions of nine grid cells of the EBCC Atlas of European Breeding Birds around each city. We tested whether the proportion of species from particular regional bird assemblages entering cities (i.e., proportional richness) was related to the geographical  position, mean range size of regional avifaunas, proportion of vegetated areas and city habitat heterogeneity. The mean range sizes of the observed and randomly selected urban avifaunas were contrasted. The proportional richness of urban avifaunas was positively related to the geographic position and mean range size of birds in regional assemblages. The evidence favoured range sizes if considering the European range sizes or latitudinal extents, but was limited for global range sizes. Randomizations tended to show larger range sizes for the real avifaunas than in the randomly selected ones. For urban core zones, the results were less clear-cut with some evidence only in favour of the European range sizes. No role of vegetation or habitat heterogeneity was found. In conclusion, while vegetation availability or heterogeneity did not show any effects, spatial position and range sizes of birds in regional assemblages seemed to influence the proportional richness of cities and their core zones. Factors correlated with spatial position (e.g., climate) might increase the attractivity of particular cities to birds. However, the effects of range sizes indicated that urbanization possibly has more negative impacts on the avifauna in the regions occupied by less widespread species

Set-Theoretic Geology

A ground of the universe V is a transitive proper class W subset V, such that W is a model of ZFC and V is obtained by set forcing over W, so that V = W[G] for some W-generic filter G subset P in W . The model V satisfies the ground axiom GA if there are no such W properly contained in V . The model W is a bedrock of V if W is a ground of V and satisfies the ground axiom. The mantle of V is the intersection of all grounds of V . The generic mantle of V is the intersection of all grounds of all set-forcing extensions of V . The generic HOD, written gHOD, is the intersection of all HODs of all set-forcing extensions. The generic HOD is always a model of ZFC, and the generic mantle is always a model of ZF. Every model of ZFC is the mantle and generic mantle of another model of ZFC. We prove this theorem while also controlling the HOD of the final model, as well as the generic HOD. Iteratively taking the mantle penetrates down through the inner mantles to what we call the outer core, what remains when all outer layers of forcing have been stripped away. Many fundamental questions remain open.Comment: 44 pages; commentary concerning this article can be made at http://jdh.hamkins.org/set-theoreticgeology