Projective toric varieties as fine moduli spaces of quiver representations

This paper proves that every projective toric variety is the fine moduli space for stable representations of an appropriate bound quiver. To accomplish this, we study the quiver $Q$ with relations $R$ corresponding to the finite-dimensional algebra $\bigl(\bigoplus_{i=0}^{r} L_i \bigr)$ where $\mathcal{L} := (\mathscr{O}_X,L_1, ...c, L_r)$ is a list of line bundles on a projective toric variety $X$. The quiver $Q$ defines a smooth projective toric variety, called the multilinear series $|\mathcal{L}|$, and a map $X \to |\mathcal{L}|$. We provide necessary and sufficient conditions for the induced map to be a closed embedding. As a consequence, we obtain a new geometric quotient construction of projective toric varieties. Under slightly stronger hypotheses on $\mathcal{L}$, the closed embedding identifies $X$ with the fine moduli space of stable representations for the bound quiver $(Q,R)$.Comment: revised version: improved exposition, corrected typos and other minor change

Multilevel ensemble data assimilation

This thesis aims to investigate and improve the efficiency of ensemble transform methods for data assimilation, using an application of multilevel Monte Carlo. Multilevel Monte Carlo is an interesting framework to estimate statistics of discretized random variables, since it uses a hierarchy of discretizations with a refinement in resolution. This is in contrast to standard Monte Carlo estimators that only use a discretization at a fine resolution. A linear combination of sub-estimators, on different levels of this hierarchy, can provide new statistical estimators to random variables at the finest level of resolution with significantly greater efficiency than a standard Monte Carlo equivalent. Therefore, the extension to computing filtering estimators for data assimilation is a natural, but challenging area of study. These challenges arise due to the fact that correlation must be imparted between ensembles on adjacent levels of resolution and maintained during the assimilation of data. The methodology proposed in this thesis, considers coupling algorithms to establish this correlation. This generates multilevel estimators that significantly reduce the computational expense of propagating ensembles of discretizations through time and space, in between stages of data assimilation. An effective benchmark of this methodology is realised by filtering data into high-dimensional spatio-temporal systems, where a high computational complexity is required to solve the underlying partial differential equations. A novel extension of an ensemble transform localisation framework to finite element approximations within random spatio-temporal systems is proposed, in addition to a multilevel equivalent.Open Acces

Validation of ocean model syntheses against hydrography using a new web application

Results are presented from a new web application called OceanDIVA - Ocean Data Intercomparison and Visualization Application. This tool reads hydrographic profiles and ocean model output and presents the data on either depth levels or isotherms for viewing in Google Earth, or as probability density functions (PDFs) of regional model-data misfits. As part of the CLIVAR Global Synthesis and Observations Panel, an intercomparison of water mass properties of various ocean syntheses has been undertaken using OceanDIVA. Analysis of model-data misfits reveals significant differences between the water mass properties of the syntheses, such as the ability to capture mode water properties

Novel Methods for Measuring the Thermal Diffusivity and the Thermal Conductivity of a Lithium-Ion Battery

Thermal conductivity is a fundamental parameter in every battery pack model. It allows for the calculation of internal temperature gradients which affect cell safety and cell degradation. The accuracy of the measurement for thermal conductivity is directly proportional to the accuracy of any thermal calculation. Currently the battery industry uses archaic methods for measuring this property which have errors up to 50 %. This includes the constituent material approach, the Searle’s bar method, laser/Xeon flash and the transient plane source method. In this paper we detail three novel methods for measuring both the thermal conductivity and the thermal diffusivity to within 5.6 %. These have been specifically designed for bodies like lithium-ion batteries which are encased in a thermally conductive material. The novelty in these methods comes from maintaining a symmetrical thermal boundary condition about the middle of the cell. By using symmetric boundary conditions, the thermal pathway around the cell casing can be significantly reduced, leading to improved measurement accuracy. These novel methods represent the future for thermal characterisation of lithium-ion batteries. Continuing to use flawed measurement methods will only diminish the performance of battery packs and slow the rate of decarbonisation in the transport sector