Conformal Einstein equations and Cartan conformal connection

Necessary and sufficient conditions for a space-time to be conformal to an Einstein space-time are interpreted in terms of curvature restrictions for the corresponding Cartan conformal connection

Differential Calculi on Some Quantum Prehomogeneous Vector Spaces

This paper is devoted to study of differential calculi over quadratic algebras, which arise in the theory of quantum bounded symmetric domains. We prove that in the quantum case dimensions of the homogeneous components of the graded vector spaces of k-forms are the same as in the classical case. This result is well-known for quantum matrices. The quadratic algebras, which we consider in the present paper, are q-analogues of the polynomial algebras on prehomogeneous vector spaces of commutative parabolic type. This enables us to prove that the de Rham complex is isomorphic to the dual of a quantum analogue of the generalized Bernstein-Gelfand-Gelfand resolution.Comment: LaTeX2e, 51 pages; changed conten

L∞L_\infty-Algebras, the BV Formalism, and Classical Fields

We summarise some of our recent works on $L_\infty$-algebras and quasi-groups with regard to higher principal bundles and their applications in twistor theory and gauge theory. In particular, after a lightning review of $L_\infty$-algebras, we discuss their Maurer-Cartan theory and explain that any classical field theory admitting an action can be reformulated in this context with the help of the Batalin-Vilkovisky formalism. As examples, we explore higher Chern-Simons theory and Yang-Mills theory. We also explain how these ideas can be combined with those of twistor theory to formulate maximally superconformal gauge theories in four and six dimensions by means of $L_\infty$-quasi-isomorphisms, and we propose a twistor space action.Comment: 19 pages, Contribution to Proceedings of LMS/EPSRC Durham Symposium Higher Structures in M-Theory, August 201

Foresight—a generative pretrained transformer for modelling of patient timelines using electronic health records: a retrospective modelling study

Background: An electronic health record (EHR) holds detailed longitudinal information about a patient's health status and general clinical history, a large portion of which is stored as unstructured, free text. Existing approaches to model a patient's trajectory focus mostly on structured data and a subset of single-domain outcomes. This study aims to evaluate the effectiveness of Foresight, a generative transformer in temporal modelling of patient data, integrating both free text and structured formats, to predict a diverse array of future medical outcomes, such as disorders, substances (eg, to do with medicines, allergies, or poisonings), procedures, and findings (eg, relating to observations, judgements, or assessments). / Methods: Foresight is a novel transformer-based pipeline that uses named entity recognition and linking tools to convert EHR document text into structured, coded concepts, followed by providing probabilistic forecasts for future medical events, such as disorders, substances, procedures, and findings. The Foresight pipeline has four main components: (1) CogStack (data retrieval and preprocessing); (2) the Medical Concept Annotation Toolkit (structuring of the free-text information from EHRs); (3) Foresight Core (deep-learning model for biomedical concept modelling); and (4) the Foresight web application. We processed the entire free-text portion from three different hospital datasets (King's College Hospital [KCH], South London and Maudsley [SLaM], and the US Medical Information Mart for Intensive Care III [MIMIC-III]), resulting in information from 811 336 patients and covering both physical and mental health institutions. We measured the performance of models using custom metrics derived from precision and recall. / Findings: Foresight achieved a precision@10 (ie, of 10 forecasted candidates, at least one is correct) of 0·68 (SD 0·0027) for the KCH dataset, 0·76 (0·0032) for the SLaM dataset, and 0·88 (0·0018) for the MIMIC-III dataset, for forecasting the next new disorder in a patient timeline. Foresight also achieved a precision@10 value of 0·80 (0·0013) for the KCH dataset, 0·81 (0·0026) for the SLaM dataset, and 0·91 (0·0011) for the MIMIC-III dataset, for forecasting the next new biomedical concept. In addition, Foresight was validated on 34 synthetic patient timelines by five clinicians and achieved a relevancy of 33 (97% [95% CI 91–100]) of 34 for the top forecasted candidate disorder. As a generative model, Foresight can forecast follow-on biomedical concepts for as many steps as required. / Interpretation: Foresight is a general-purpose model for biomedical concept modelling that can be used for real-world risk forecasting, virtual trials, and clinical research to study the progression of disorders, to simulate interventions and counterfactuals, and for educational purposes. / Funding: National Health Service Artificial Intelligence Laboratory, National Institute for Health and Care Research Biomedical Research Centre, and Health Data Research UK

Is there a Jordan geometry underlying quantum physics?

There have been several propositions for a geometric and essentially non-linear formulation of quantum mechanics. From a purely mathematical point of view, the point of view of Jordan algebra theory might give new strength to such approaches: there is a ``Jordan geometry'' belonging to the Jordan part of the algebra of observables, in the same way as Lie groups belong to the Lie part. Both the Lie geometry and the Jordan geometry are well-adapted to describe certain features of quantum theory. We concentrate here on the mathematical description of the Jordan geometry and raise some questions concerning possible relations with foundational issues of quantum theory.Comment: 30 page

The kernel of the edth operators on higher-genus spacelike two-surfaces

The dimension of the kernels of the edth and edth-prime operators on closed, orientable spacelike 2-surfaces with arbitrary genus is calculated, and some of its mathematical and physical consequences are discussed.Comment: 12 page

3-dimensional Cauchy-Riemann structures and 2nd order ordinary differential equations

The equivalence problem for second order ODEs given modulo point transformations is solved in full analogy with the equivalence problem of nondegenerate 3-dimensional CR structures. This approach enables an analog of the Feffereman metrics to be defined. The conformal class of these (split signature) metrics is well defined by each point equivalence class of second order ODEs. Its conformal curvature is interpreted in terms of the basic point invariants of the corresponding class of ODEs

The influence of waves on morphodynamic impacts of energy extraction at a tidal stream turbine site in the Pentland Firth

Extraction of energy from tidal streams has the potential to impact on the morphodynamics of areas such as sub-tidal sandbanks via alteration of hydrodynamics. Marine sediment transport is forced by both wave and tidal currents. Past work on tidal stream turbine impacts has largely ignored the contribution of waves. Here, a fully coupled hydrodynamic, spectral wave and sediment transport model is used to assess the importance of including waves in simulations of turbine impact on seabed morphodynamics. Assessment of this is important due to the additional expense of including waves in simulations. Focus is given to a sandbank in the Inner Sound of the Pentland Firth. It is found that inclusion of wave action alters hydrodynamics, although extent of alteration is dependant of wave direction. Magnitude of sediment transport is increased when waves are included in the simulations and this has implications for morphological and volumetric changes. Volumetric changes are substantially increased when wave action is included: the impact of including waves is greater than the impact of including tidal stream turbines. Therefore it is recommended that at tidal turbine array sites exposed to large swell or wind-seas, waves should be considered for inclusion in simulations of physical impact

Large scale three-dimensional modelling for wave and tidal energy resource and environmental impact : methodologies for quantifying acceptable thresholds for sustainable exploitation

We describe a modelling project to estimate the potential effects of wave & tidal stream renewables on the marine environment. • Realistic generic devices to be used by those without access to the technical details available to developers are described. • Results show largely local sea bed effects at the level of the currently proposed renewables developments in our study area. • Large scale 3D modelling is critical to quantify the direct, indirect and cumulative effects of renewable energy extraction. • This is critical to comply with planning & environmental impact assessment regulations and achieve Good Environmental Status