Independence, Relative Randomness, and PA Degrees

We study pairs of reals that are mutually Martin-L\"{o}f random with respect to a common, not necessarily computable probability measure. We show that a generalized version of van Lambalgen's Theorem holds for non-computable probability measures, too. We study, for a given real $A$, the \emph{independence spectrum} of $A$, the set of all $B$ so that there exists a probability measure $\mu$ so that $\mu\{A,B\} = 0$ and $(A,B)$ is $\mu\times\mu$-random. We prove that if $A$ is r.e., then no $\Delta^0_2$ set is in the independence spectrum of $A$. We obtain applications of this fact to PA degrees. In particular, we show that if $A$ is r.e.\ and $P$ is of PA degree so that $P \not\geq_{T} A$, then $A \oplus P \geq_{T} 0'$

Assessing the climate vulnerability of the world’s natural and cultural heritage

Climate change is the fastest-growing global threat to the world’s natural and cultural heritage. No systematic approach to assess climate vulnerability of protected areas and their associated communities has existed—until now. The Climate Vulnerability Index (CVI) is scientifically robust, transparent, and repeatable, and has now been applied to various World Heritage properties. The CVI builds upon an established Intergovernmental Panel on Climate Change (IPCC) framework to systematically assess vulnerability through a risk assessment approach that considers the key values of the World Heritage property in question and identifies key climate stressors. The CVI process is then used to assess the climate-related vulnerability of the community (including local residents, domestic visitors, and international tourists) associated with the World Heritage property considering economic, social, and cultural connections. Climate impacts are increasingly adding to a wide range of compounding pressures (e.g., increasing tourism, infrastructure development, changing land use practices) that are affecting places, people, customs, and values. Applications of the CVI to date have led to commitments to integrate outcomes into relevant management plans, and to periodically repeat the process, enabling responsive management to changing future circumstances. The CVI has also demonstrated its potential applicability for protected areas beyond World Heritage properties. The CVI process engages local community members in determining impacts, provides opportunities for identifying adaptation and impact mitigation within the community, and aids broader communication about key climate issues