The most eye-catching effect of digitalization on the law of enforcement jurisdiction is the fading into irrelevance of territoriality. Insofar as the “physical” location of digital data—on a server—may be entirely fortuitous and may in fact not be known by the territorial state, it appears unreasonable for that state to invoke its territorial sovereignty as a shield against another state’s claims over such data. To prevent a jurisdictional free-for-all, however, it is key that the exercise of extraterritorial enforcement jurisdiction in cyberspace becomes subject to a stringent test weighting all relevant connections and interests in concrete cases. Introducing such a weighting test means that extraterritorial enforcement jurisdiction is no longer governed by binary rules (allowed and not allowed), but becomes a matter of degree, requiring a granular, contextual assessment. It remains the case that such a flexible attitude towards extraterritorial enforcement jurisdiction is not universally shared, and that relevant state practice and expert opinion in favor of the “un-territoriality of data” has a particular Western slant.

Theorems of hyperarithmetic analysis (THAs) occupy an unusual neighborhood in the realms of reverse mathematics and recursion-theoretic complexity. They lie above all the fixed (recursive) iterations of the Turing jump but below ATR $_{0}$ (and so $\Pi _{1}^{1}$ -CA $_{0}$ or the hyperjump). There is a long history of proof-theoretic principles which are THAs. Until the papers reported on in this communication, there was only one mathematical example. Barnes, Goh, and Shore [1] analyze an array of ubiquity theorems in graph theory descended from Halin’s [9] work on rays in graphs. They seem to be typical applications of ACA $_{0}$ but are actually THAs. These results answer Question 30 of Montalbán’s Open Questions in Reverse Mathematics [19] and supply several other natural principles of different and unusual levels of complexity.

This work led in [25] to a new neighborhood of the reverse mathematical zoo: almost theorems of hyperarithmetic analysis (ATHAs). When combined with ACA $_{0}$ they are THAs but on their own are very weak. Denizens both mathematical and logical are provided. Generalizations of several conservativity classes ( $\Pi _{1}^{1}$ , r- $\Pi _{1}^{1}$ , and Tanaka) are defined and these ATHAs as well as many other principles are shown to be conservative over RCA $_{0}$ in all these senses and weak in other recursion-theoretic ways as well. These results answer a question raised by Hirschfeldt and reported in [19] by providing a long list of pairs of principles one of which is very weak over RCA $_{0}$ but over ACA $_{0}$ is equivalent to the other which may be strong (THA) or very strong going up a standard hierarchy and at the end being stronger than full second-order arithmetic.

There is no consensus on the question of whether or not free-living protist species have biogeographies, with the strongest disagreement coming from advocates of the hypothesis that the extraordinary abundance of protists drives their ubiquitous dispersal. If the probability of a species being ubiquitous is a function of its absolute global abundance, then the species that are least likely to be ubiquitous are those with relatively small global populations, i.e. the largest species. Among the free-living ciliated protozoa, a prime candidate for such an organism must be the large (~1200 μm long), unmistakable, fragile, non-encysting karyorelictid Loxodes rex. This ciliate was known only from fresh waters in tropical Africa and it was long considered to be a rare example of an endemic ciliate. Here it is reported that Loxodes rex is thriving in a pond in Thailand. The status of other alleged endemic ciliate species is discussed.

Given a finite lexicon L of relational symbols and equality, one may view the collection of all L-structures on the set of natural numbers ω as a space in several different ways. We consider it as: (i) the space of outcomes of certain infinite two-person games; (ii) a compact metric space; and (iii) a probability measure space. For each of these viewpoints, we can give a notion of relative ubiquity, or largeness, for invariant sets of structures on ω. For example, in every sense of relative ubiquity considered here, the set of dense linear orderings on ω is ubiquitous in the set of linear orderings on ω.