Rational combinations of Betti diagrams of complete intersections

We investigate decompositions of Betti diagrams over a polynomial ring within the framework of Boij-S\"oderberg theory. That is, given a Betti diagram, we determine if it is possible to decompose it into the Betti diagrams of complete intersections. To do so, we determine the extremal rays of the cone generated by the diagrams of complete intersections and provide a rudimentary algorithm for decomposition.Comment: This research was conducted at the Willamette Mathematics Consortium RE

Helium and Nitrogen Enrichment in Massive Main Sequence Stars: Mechanisms and Implications for the Origin of WNL Stars

The evolutionary paths taken by massive stars with $M \gtrsim 60 \, \mathrm{M}_\odot$ remain substantially uncertain. They begin their lives as main sequence (MS) O-stars. Depending on their masses, rotation rates, and metallicities, they can then encounter a wide range of evolutionary states with an equally broad set of possible surface compositions and spectral classifications. We present a new grid of calculations for the evolution of such stars that covers a broad range in mass, M/M$_\odot = 60$ to $150$, rotation rate, $v \, / \, v_{\rm crit} = 0$ to $0.6$, metallicity, $[\mathrm{Fe}/\mathrm{H}] = -4$ to $0$, and $\alpha$-element enhancement, $[\alpha/\mathrm{Fe}] = 0$ to $0.4$. We show that rotating stars undergo rotationally-induced dredge-up of nucleosynthetic products, mostly He and N, to their surfaces while still on the MS. Non-rotating metal-rich stars also reveal the products of nucleosynthesis on their surfaces because even modest amounts of mass loss expose their "fossil" convective cores: regions that are no longer convective, but which were part of the convective core at an early stage in the star's evolution. Thus surface enhancement of He and N is expected for rotating stars at all metallicities, and for non-rotating stars if they are relatively metal-rich. We calculate a stellar atmosphere for a representative model from our grid, properly accounting for He- and N-enhancement, and show that the resulting spectrum provides a good match to observed WNL stars, strongly suggesting that the physical mechanisms we have identified are the ultimate cause of the WNL phase.Comment: 21 pages, 18 figures, 2 tables, accepted for publication in MNRAS, in pres

Communicating simulation outputs of mesoscale coastal evolution to specialist and non-specialist audiences

Coastal geomorphologists and engineers worldwide are increasingly facing the non-trivial challenge of visualising and communicating mesoscale modelling assumptions, uncertainties and outcomes to both coastal specialists and decision-makers. Visualisation of simulation outcomes is a non-trivial problem because the more abstract scientific visualisation techniques favoured by specialists for data exploration and hypothesis-testing are not always as successful at engaging decision-makers and planners. In this paper, we show how the risk of simulation model outcomes becoming disconnected from more realistic visualisations of model outcomes can be minimised by using the Coastal Modelling Environment (CoastalME). CoastalME is a modelling framework for coastal mesoscale morphological modelling that can achieve close linkages between the scientific model abstractions, in the form of lines, areas and volumes, and the 3D representation of topographic and bathymetric surfaces and shallow sub-surface sediment composition. We propose and illustrate through the study case of Happisburgh (eastern England, UK), a transparent methodology to merge the required variety of data types and formats into a 3D-thickness model that is used to initialise a simulation. We conclude by highlighting some of the barriers to the adoption of the methodology proposed