A Qubit Algorithm for Simulating the Nonlinear Schroedinger Equation

Recent work in mathematical physics and nonlinear optics has shown that Hamiltonians that are non-Hermitian but still symmetric under parity and time reversal can describe eigenstates of a system with real eigenvalues. Other research has also showed that the nonlinear Schrodinger equation can be generalized to describe PT-symmetric systems, which generates novel solutions not described by its Hermitian equivalent. The Hermitian form of the nonlinear Schroedinger equation can also be extended to describe a particular case of the general PT-symmetric NLS, suggesting a connection between the two. I attempted to generate a unitary operator that will be useful for unitary quantum algorithms describing a coupled set of nonlinear Schroedinger equations and the PT-symmetric version of the NLS

Chow Rings of Vector Space Matroids

The Chow ring of a matroid (or more generally, atomic latice) is an invariant whose importance was demonstrated by Adiprasito, Huh and Katz, who used it to resolve the long-standing Heron-Rota-Welsh conjecture. Here, we make a detailed study of the Chow rings of uniform matroids and of matroids of finite vector spaces. In particular, we express the Hilbert series of such matroids in terms of permutation statistics; in the full rank case, our formula yields the maj-exc $q$-Eulerian polynomials of Shareshian and Wachs. We also provide a formula for the Charney-Davis quantities of such matroids, which can be expressed in terms of either determinants or $q$-secant numbers

Total positivity for matroid Schubert varieties

We define the totally nonnegative matroid Schubert variety $\mathcal Y_V$ of a linear subspace $V \subset \mathbb R^n$. We show that $\mathcal Y_V$ is a regular CW complex homeomorphic to a closed ball, with strata indexed by pairs of acyclic flats of the oriented matroid of $V$. This closely resembles the regularity theorem for totally nonnegative generalized flag varieties. As a corollary, we obtain a regular CW structure on the real matroid Schubert variety of $V$.Comment: Comments welcome

Covid and care: how a ‘stacked’ care system could help places like Hackney

The weakness of local support networks, already cut to the bone, has been cruelly exposed by the pandemic. The LSE’s COVID and Care Research Group looks at the situation in Hackney and explains how an alternative ‘stacked’ care system could help

The Set Splittablity Problem

A collection of sets is called splittable if there is a set S such that for each set B in the collection, the intersection of S and B is half the size of B. Splittability is a generalization of graph colorability, which is an active area of research with numerous applications such as scheduling and matching. We show that the problem of deciding whether a collection is splittable is NP-complete. Nevertheless we characterize splittability for some special collections. Finally we study a further generalization called p-splittability, in which the splitter S is required to contain a given fraction of each set B

Trade-offs and Optimisation of Land-Use for Pastoralism and Carbon in Southeastern Australia

Globally, pressure to ensure future food security is being challenged by competing needs for multiple land-uses in agricultural systems. Rangelands are both a source of greenhouse gas emissions as well as providing opportunities for emissions reduction. Carbon farming is a new land-use option that sequesters carbon in vegetation and soils. National incentive programs in Australia for this option have resulted in significant recent land-use change across Australian rangelands. Beyond the mitigation benefits, the potential for carbon farming income to enhance socio-ecological resilience in rangelands has been identified. However, there are major uncertainties about the impacts of climate change on sequestration rates and trade-offs between land-use for carbon and pastoral production. The AUD$2.45 billion Commonwealth Emissions Reduction Fund has driven recent land-use change and a further AUD$2 billion over the next 10 years, coupled with a fast-growing secondary carbon market is continuing to drive demand for carbon credits. The ability to supply these carbon credits and meet international emissions reduction obligations but limit the trade-offs with pastoral production can be supported through an identification of spatial prioritisation and optimisation at a landscape scale. We use a case study of New South Wales where ~3 million ha of traditional rangeland pastoralism is currently delivering ~27% of the national land sector abatement. Priority areas and optimisation of land-use for carbon farming and production under current and future climates were determined by developing a Carbon Optimisation Model (COM). This high-resolution integrated environmental-economic model provides predictions of spatiotemporal dynamics of land-use options for variations of incentive payment levels and policy settings. Regional downscaling of an ensemble of global circulation models (GCMs) were used to predict the climate impacts on future sequestration rates derived from 3PG forest growth model to quantify carbon supply under future climates. The COM can be used to produce spatial maps to underpin strategic prioritisation abatement activities and allow abatement opportunities to be incorporated into regional NRM planning

Stretched during COVID, Britain’s social infrastructure needs an urgent boost

Besides ‘the economy’ and ‘health’ lies a neglected area of human life during the pandemic: social infrastructures. These vital links, sustained by families and communities, now need to be a priority. The LSE COVID and Care Recovery Group call for both urgent and long term help for voluntary and community groups