The montane cloud forests of South America are some of the most biodiverse habitats in the world, whilst also being especially vulnerable to climate change and human disturbance. Today much of this ...landscape has been transformed into a mosaic of secondary forest and agricultural fields.This thesis uses palaeoecological proxies (pollen, non-pollen palynomorphs, charcoal, organic content) to interpret ecosystem dynamics during the late Quaternary, unravelling the vegetation history of the landscape and the relationship between people and the montane cloud forest of the eastern Andean flank of Ecuador. Two new sedimentary records are examined from the montane forest adjacent to the Río Cosanga (Vinillos) and in the Quijos Valley (Huila). These sites characterise the natural dynamics of a pre-human arrival montane forest and reveal how vegetation responded during historical changes in local human populations. Non-pollen palynomorphs (NPPs) are employed in a novel approach to analyse a forest cover gradient across these sites. The analysis identifies a distinctive NPP assemblage connected to low forest cover and increased regional burning. Investigation into the late Pleistocene Vinillos sediments show volcanic activity to be the primary landscape-scale driver of ecosystem dynamics prior to human arrival, influencing montane forest populations but having little effect on vegetation composition.
The principal concern of this document is to develop and expose methodology for enumerating idempotents in certain semigroups of diagrams in the sense of 76. These semigroups are known to be ...significant in the representation theory of associated algebras. In particular these algebras are shown in many cases to be semisimple, giving certain idempotents (and in particular those of the monoids of concern) a prominent role in understanding certain features of the representation theory in this situation. The results developed here are mostly theoretical in nature. We propose two viewpoints leading to some combinatorial understanding of the idempotents in the Motzkin (respectively Jones and partial Jones) monoid. In the first instance, we construct a cell complex, whose connected components partition the set of all idempotents into small, manageable chunks that can be analysed uniformly starting from those of particularly low rank. The structure of this complex captures some intricate combinatorics in the semigroup in a fairly simple, uniform way, and reduces our problem to finding and characterising idempotents of particularly low rank. The latter viewpoint takes us closer to pure combinatorics; a family of parameters attached to the elements of the monoids in question. These are examined in the context of ordinary generating functions, counting the elements with various parameter profiles. In particular, important algebraic features of Motzkin pictures, such as degree, rank, idempotency, and membership in the Jones and partial Jones monoids, can be tested against parameter profiles, reducing the problem of understanding all three to that of a parametric underiii iv standing of only the Motzkin monoid. We can then amalgamate these families of techniques into the development of fast linear-space algorithms for counting elements of various parameter profiles by examining certain "convex" elements. In particular, the general problem of enumeration by parameter profile is reduced greatly to enumerating convex elements by parameter profile. As a corollary to this study of convexity, we observe that the sequence of numbers of idempotents (in each semigroup) of some fixed rank-deficiency d = (n − r) is equal (apart from the first couple of values) to some polynomial of degree d; for particularly low rank-deficiency, we calculate these polynomials. Finally, we can show that the problem of understanding these idempotents in this way reduces to the classical open problem in combinatorics of counting meanders, witnessing the fact that significant progress on the former problem would necessitate some development of a better understanding of the latter.
We classify and enumerate the idempotents in several planar diagram monoids: namely, the Motzkin, Jones (a.k.a. Temperley–Lieb) and Kauffman monoids. The classification is in terms of certain vertex- ...and edge-coloured graphs associated to Motzkin diagrams. The enumeration is necessarily algorithmic in nature, and is based on parameters associated to cycle components of these graphs. We compare our algorithms to existing algorithms for enumerating idempotents in arbitrary (regular ⁎-) semigroups, and give several tables of calculated values.
We give a characterisation of the idempotents of the partition monoid, and use this to enumerate the idempotents in the finite partition, Brauer and partial Brauer monoids, giving several formulae ...and recursions for the number of idempotents in each monoid as well as various R-, L- and D-classes. We also apply our results to determine the number of idempotent basis elements in the finite dimensional partition, Brauer and partial Brauer algebras.
We classify and enumerate the idempotents in several planar diagram monoids: namely, the Motzkin, Jones (a.k.a. Temperley-Lieb) and Kauffman monoids. The classification is in terms of certain vertex- ...and edge-coloured graphs associated to Motzkin diagrams. The enumeration is necessarily algorithmic in nature, and is based on parameters associated to cycle components of these graphs. We compare our algorithms to existing algorithms for enumerating idempotents in arbitrary (regular *-) semigroups, and give several tables of calculated values.
We give a characterisation of the idempotents of the partition monoid, and use this to enumerate the idempotents in the finite partition, Brauer and partial Brauer monoids, giving several formulae ...and recursions for the number of idempotents in each monoid as well as various \(\mathscr R\)-, \(\mathscr L\)- and \(\mathscr D\)-classes. We also apply our results to determine the number of idempotent basis elements in the finite dimensional partition, Brauer and partial Brauer algebras.
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