Hopf algebras and Markov chains: Two examples and a theory

The operation of squaring (coproduct followed by product) in a combinatorial Hopf algebra is shown to induce a Markov chain in natural bases. Chains constructed in this way include widely studied methods of card shuffling, a natural "rock-breaking" process, and Markov chains on simplicial complexes. Many of these chains can be explictly diagonalized using the primitive elements of the algebra and the combinatorics of the free Lie algebra. For card shuffling, this gives an explicit description of the eigenvectors. For rock-breaking, an explicit description of the quasi-stationary distribution and sharp rates to absorption follow.Comment: 51 pages, 17 figures. (Typographical errors corrected. Further fixes will only appear on the version on Amy Pang's website, the arXiv version will not be updated.

Can scenario-planning support community-based natural resource management? Experiences from three countries in latin america

Community Based Natural Resource Management (CBNRM) is a concept critical to managing socio-ecological systems, but whose implementation needs strengthening. Scenario-planning is one approach that may offer benefits relevant to CBNRM, but whose potential is not yet well understood. We therefore designed, trialled and evaluated a scenario-planning method intended to support CBNRM at three cases, located in Colombia, Mexico and Argentina. Implementing scenario-planning was judged as worthwhile in all three cases, although aspects of it could be challenging to facilitate. The benefits generated were relevant to strengthening CBNRM: encouraging the participation of local people and using their knowledge; enhanced consideration and adaption of future change; and supporting the development of systems thinking. Tracing exactly when and how these benefits arise is challenging, but two elements of the method seemed particularly useful. Firstly, using a systematic approach to discuss how drivers of change may affect local socio-ecological systems helped to foster systems thinking and identify connections between issues. Secondly, explicitly focusing on how to use and respond to scenarios helped identify specific practical activities ('response options') that would support CBNRM despite the pressures of future change. Discussions about response options also highlighted the need for support by other actors (e.g. policy groups): this raises the question of when and how other actors and other sources of knowledge should be involved in scenario-planning, so as to encourage their buy-in to actions identified by the process. We suggest other CBNRM initiatives may benefit from adapting and applying scenario-planning. However, these initiatives should be carefully monitored since further research is required to understand how and when scenario-planning methods may produce benefits, and their strengths and weaknesses versus other methods

New insights into the genetic etiology of Alzheimer's disease and related dementias

Characterization of the genetic landscape of Alzheimer's disease (AD) and related dementias (ADD) provides a unique opportunity for a better understanding of the associated pathophysiological processes. We performed a two-stage genome-wide association study totaling 111,326 clinically diagnosed/'proxy' AD cases and 677,663 controls. We found 75 risk loci, of which 42 were new at the time of analysis. Pathway enrichment analyses confirmed the involvement of amyloid/tau pathways and highlighted microglia implication. Gene prioritization in the new loci identified 31 genes that were suggestive of new genetically associated processes, including the tumor necrosis factor alpha pathway through the linear ubiquitin chain assembly complex. We also built a new genetic risk score associated with the risk of future AD/dementia or progression from mild cognitive impairment to AD/dementia. The improvement in prediction led to a 1.6- to 1.9-fold increase in AD risk from the lowest to the highest decile, in addition to effects of age and the APOE ε4 allele