Testing isomorphism of graded algebras

We present a new algorithm to decide isomorphism between finite graded algebras. For a broad class of nilpotent Lie algebras, we demonstrate that it runs in time polynomial in the order of the input algebras. We introduce heuristics that often dramatically improve the performance of the algorithm and report on an implementation in Magma

A new species of Rhodomyrtus (Myrtaceae) with brochidodromous venation from north-eastern Queensland, Australia.

Rhodomyrtus verecunda A.J.Ford & Peter G. Wilson from north-east Queensland, Australia, is formally described and illustrated. Notes on habitat, distribution and conservation status are provided, as are details of how it can be distinguished from R. macrocarpa, the only other species of the genus in the region that lacks acrodromous venation. A revised key to the species of Rhodomyrtus in Australia is also presented

Pathwise Conditioning of Gaussian Processes

As Gaussian processes are used to answer increasingly complex questions, analytic solutions become scarcer and scarcer. Monte Carlo methods act as a convenient bridge for connecting intractable mathematical expressions with actionable estimates via sampling. Conventional approaches for simulating Gaussian process posteriors view samples as draws from marginal distributions of process values at finite sets of input locations. This distribution-centric characterization leads to generative strategies that scale cubically in the size of the desired random vector. These methods are prohibitively expensive in cases where we would, ideally, like to draw high-dimensional vectors or even continuous sample paths. In this work, we investigate a different line of reasoning: rather than focusing on distributions, we articulate Gaussian conditionals at the level of random variables. We show how this pathwise interpretation of conditioning gives rise to a general family of approximations that lend themselves to efficiently sampling Gaussian process posteriors. Starting from first principles, we derive these methods and analyze the approximation errors they introduce. We, then, ground these results by exploring the practical implications of pathwise conditioning in various applied settings, such as global optimization and reinforcement learning

Scavenging amphipods of the Tonga Trench: an analysis of community assemblage and population structure

The hadal zone is the common name for the deepest section of the ocean (6,000-11,000 m depth). It encompasses 45 % of the ocean’s depth range, and is mostly represented by oceanic trenches. Trench habitats lack sufficient sampling and the communities within are not well understood. Often, samples are derived from a single depth and thus the population dynamics of trench communities have not been analysed comprehensively. Scavenging amphipods are abundant and diverse taxa in the trench environment, and have been found in every trench sampled to date. They rapidly intercept and consume carrion falls at the deepest trench depths, and act as key prey items to predators in the shallower depths of the hadal zone. There appears to be a relationship of increasing abundance and decreasing diversity of scavenging amphipods with depth. However in the Tonga Trench, sampling of hadal amphipods has been limited, and these patterns remain unclear. The QUELLE (Quest for the Limit of Life) project in 2013 was led by The Japan Agency for Marine-Earth Science and Technology (JAMSTEC). As part of this project, the YOK 13-10 voyage examined scavenging amphipods in the Tonga Trench. The voyage used baited traps to sample depths of ˜6,250 m and ˜10,800 m from October 6 – October 21 in 2013. The main objectives of the present study were to: identify scavenging amphipod assemblages within the Tonga Trench and compare them to other trenches of the South Pacific; analyse the population structure of Hirondellea dubia between depths in the Tonga Trench; and identify a suitable total length proxy for H. dubia. Six species of amphipods were identified from depths of ˜6,250 m and ˜10,800 m in the Tonga Trench. At ˜6,250 m Alicella gigantea, Eurythenes gryllus, H. dubia, Bathycallisoma schellenbergi, an alicellid species, and a gammarid species were recovered. In contrast, H. dubia was the only species recovered from ˜10,800 m. The abundance of amphipods was higher at the ˜10,800 m site while the diversity was much lower. The assemblage of scavenging amphipods in the Tonga Trench was similar to those from past sampling efforts in the same trench. There were also similarities to the assemblages in the adjacent Kermadec Trench, and together these observations support the classification of these two trenches as a single biogeographic province. The assemblages in the Peru-Chile Trench in the South East Pacific were more dissimilar sharing only a few species. The present study provides new Tonga Trench records of the vertical ranges of A. gigantea, E. gryllus, and H. dubia. It also extends the maximum known depth of H. dubia to 10,807 m. This thesis expanded our current knowledge of A. gigantea, by reporting the first instance of this large amphipod in the Tonga Trench, and the second known instance of the species at hadal depths. An analysis of Hirondellea dubia population structure revealed ontogenetic vertical structuring in the Tonga Trench. Juveniles dominated the composition in the shallow end of the H. dubia vertical range, while very few juveniles were found at the deepest site. Juveniles were substantially smaller at ˜6,250 m compared to ˜10,800 m, and this may suggest that juveniles migrate down the trench slope with increasing age. The most likely mechanism for distributing juveniles to the shallower depths is the ascending migration of brooding females. However, this is still not certain as no brooding females were captured. The shallower depth provides a higher quality of food source and the reduced hydrostatic pressure allows for a faster metabolic rate. Thus, this distribution is likely driven by the distribution of food sources throughout the trench in combination with hydrostatic pressure. The dimensions of several established proxies for total length were evaluated for H. dubia. Pereonite 2-7 had the strongest correlation to total length, however it was highly distorted by dorsal curvature. Both the pereonite 2-7 and the pleosome were considered inaccurate due to sexual dimorphism making them inappropriate as proxies. Pereonite 1 was proportionately larger in juvenile lifestages. However, overall pereonite 1 was considered the strongest candidate for a proxy, this is because it was the least influenced by dorsal curvature and was a conspicuous segment that was easy to measure