Analyses of ecological network structure have yielded important insights into the functioning of complex ecological systems. However, such analyses almost universally omit non-pairwise interactions, many classes of which are crucial for system structure, function, and resilience. Hypergraphs are mathematical constructs capable of considering such interactions: we discuss their utility for studying ecological networks containing diverse interaction types, and associated challenges and strategies. We demonstrate the approach using a real-world coffee agroecosystem in which resistance to agricultural pests depends upon a large number of TMIIs. A hypergraph representation successfully reflects both the importance of species imposing such effects and the context-dependency of that importance in terms of how it is affected by removal of other species from the system.
Abstract. Alspach conjectured that every connected Cayley graph on a finite Abelian group A is Hamiltondecomposable. Liu has shown that for |A| even, if S = {s 1 , . . . , s k } ⊂ A is an inverse-free strongly minimal generating set of A, then the Cayley graph Cay(A; S ), is decomposable into k Hamilton cycles, where S denotes the inverse-closure of S. Extending these techniques and restricting to the 6-regular case, this article relaxes the constraint of strong minimality on S to require only that S be strongly a-minimal, for some a ∈ S and the index of a be at least four. Strong a-minimality means that 2s / ∈ a for all s ∈ S \ {a, −a}. Some infinite families of open cases for the 6-regular Cayley graphs on even order Abelian groups are resolved. In particular, if |s 1 | ≥ |s 2 | > 2|s 3 |, then Cay(A; {s 1 , s 2 , s 3 } ) is Hamilton-decomposable.
This article argues that epistemological and theoretical issues surrounding time in relation to comparative and international education need further exploration in consideration of the new understandings of time that have emerged (or re-emerged) during the last century. By drawing on an interdisciplinary selection of ideas as well as various scholars' conceptualizations of time that range from antiquity to the present, the article suggests practical ways, supported by sociological and historical examples, to theorize time for application in comparative educational research. Appropriating a more nuanced understanding and abstraction of temporality is necessary and opens different time-worlds for comparison, resulting in knowledge informed and shaped by a time that transcends Newtonian linearity and discreteness. The article is written with the understanding that as technology continues to stimulate globalization and redraw the boundaries of space and time, the importance of reconsidering temporality in regard to educational research is clear.
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