On the cardinality and complexity of the set of codings for self-similar sets with positive Lebesgue measure

Let λ1,…,λn be real numbers in (0,1) and p1,…,pn be points in Rd. Consider the collection of maps fj:Rd→Rd given by fj(x)=λjx+(1−λj)pj. It is a well known result that there exists a unique nonempty compact set Λ⊂Rd satisfying Λ=∪nj=1fj(Λ). Each x∈Λ has at least one coding, that is a sequence (ϵi)∞i=1 ∈{1,…,n}N that satisfies limN→∞fϵ1…fϵN(0)=x. We study the size and complexity of the set of codings of a generic x∈Λ when Λ has positive Lebesgue measure. In particular, we show that under certain natural conditions almost every x∈Λ has a continuum of codings. We also show that almost every x∈Λ has a universal coding. Our work makes no assumptions on the existence of holes in Λ and improves upon existing results when it is assumed Λ contains no holes.

“TEEP: A Course-Driven Assessment Measure”

Conference proceedings paper in Alexander, O. (Ed.) 2007, Proceedings of the 2005 joint BALEAP/SATEFL conference: New Approaches to Materials Development for Language Learning. Bern: Peter Lang.

Anticipating future risk in social-ecological systems using fuzzy cognitive mapping: the case of wildfire in the Chiquitania, Bolivia

Understanding complex social-ecological systems, and anticipating how they may respond to rapid change, requires an approach that incorporates environmental, social, economic, and policy factors, usually in a context of fragmented data availability. We employed fuzzy cognitive mapping (FCM) to integrate these factors in the assessment of future wildfire risk in the Chiquitania region, Bolivia. In this region, dealing with wildfires is becoming increasingly challenging due to reinforcing feedbacks between multiple drivers. We conducted semi-structured interviews and constructed different FCMs in focus groups to understand the regional dynamics of wildfire from diverse perspectives. We used FCM modelling to evaluate possible adaptation scenarios in the context of future drier climatic conditions. Scenarios also considered possible failure to respond in time to the emergent risk. This approach proved of great potential to support decision-making for risk management. It helped identify key forcing variables and generate insights into potential risks and trade-offs of different strategies. All scenarios showed increased wildfire risk in the event of more droughts. The ‘Hands-off’ scenario resulted in amplified impacts driven by intensifying trends, affecting particularly the agricultural production. The ‘Fire management’ scenario, which adopted a bottom-up approach to improve controlled burning, showed less trade-offs between wildfire risk reduction and production compared to the ‘Fire suppression’ scenario. Findings highlighted the importance of considering strategies that involve all actors who use fire, and the need to nest these strategies for a more systemic approach to manage wildfire risk. The FCM model could be used as a decision-support tool and serve as a ‘boundary object’ to facilitate collaboration and integration of different forms of knowledge and perceptions of fire in the region. This approach has also the potential to support decisions in other dynamic frontier landscapes around the world that are facing increased risk of large wildfires.