Rendezvous of Heterogeneous Mobile Agents in Edge-weighted Networks

We introduce a variant of the deterministic rendezvous problem for a pair of heterogeneous agents operating in an undirected graph, which differ in the time they require to traverse particular edges of the graph. Each agent knows the complete topology of the graph and the initial positions of both agents. The agent also knows its own traversal times for all of the edges of the graph, but is unaware of the corresponding traversal times for the other agent. The goal of the agents is to meet on an edge or a node of the graph. In this scenario, we study the time required by the agents to meet, compared to the meeting time $T_{OPT}$ in the offline scenario in which the agents have complete knowledge about each others speed characteristics. When no additional assumptions are made, we show that rendezvous in our model can be achieved after time $O(n T_{OPT})$ in a $n$-node graph, and that such time is essentially in some cases the best possible. However, we prove that the rendezvous time can be reduced to $\Theta (T_{OPT})$ when the agents are allowed to exchange $\Theta(n)$ bits of information at the start of the rendezvous process. We then show that under some natural assumption about the traversal times of edges, the hardness of the heterogeneous rendezvous problem can be substantially decreased, both in terms of time required for rendezvous without communication, and the communication complexity of achieving rendezvous in time $\Theta (T_{OPT})$

Non-Abelian Tensor Multiplet Equations from Twistor Space

We establish a Penrose-Ward transform yielding a bijection between holomorphic principal 2-bundles over a twistor space and non-Abelian self-dual tensor fields on six-dimensional flat space-time. Extending the twistor space to supertwistor space, we derive sets of manifestly N=(1,0) and N=(2,0) supersymmetric non-Abelian constraint equations containing the tensor multiplet. We also demonstrate how this construction leads to constraint equations for non-Abelian supersymmetric self-dual strings.Comment: v3: 23 pages, revised version published in Commun. Math. Phy

Prolongations of Geometric Overdetermined Systems

We show that a wide class of geometrically defined overdetermined semilinear partial differential equations may be explicitly prolonged to obtain closed systems. As a consequence, in the case of linear equations we extract sharp bounds on the dimension of the solution space.Comment: 22 pages. In the second version, a comparison with the classical theory of prolongations was added. In this third version more details were added concerning our construction and especially the use of Kostant's computation of Lie algebra cohomolog

Harmonic Superspaces in Low Dimensions

Harmonic superspaces for spacetimes of dimension $d\leq 3$ are constructed. Some applications are given.Comment: 16, kcl-th-94-15. Two further references have been added (12 and 13) and a few typographical errors have been correcte

Rendezvous of Distance-aware Mobile Agents in Unknown Graphs

We study the problem of rendezvous of two mobile agents starting at distinct locations in an unknown graph. The agents have distinct labels and walk in synchronous steps. However the graph is unlabelled and the agents have no means of marking the nodes of the graph and cannot communicate with or see each other until they meet at a node. When the graph is very large we want the time to rendezvous to be independent of the graph size and to depend only on the initial distance between the agents and some local parameters such as the degree of the vertices, and the size of the agent's label. It is well known that even for simple graphs of degree $\Delta$, the rendezvous time can be exponential in $\Delta$ in the worst case. In this paper, we introduce a new version of the rendezvous problem where the agents are equipped with a device that measures its distance to the other agent after every step. We show that these \emph{distance-aware} agents are able to rendezvous in any unknown graph, in time polynomial in all the local parameters such the degree of the nodes, the initial distance $D$ and the size of the smaller of the two agent labels $l = \min(l_1, l_2)$. Our algorithm has a time complexity of $O(\Delta(D+\log{l}))$ and we show an almost matching lower bound of $\Omega(\Delta(D+\log{l}/\log{\Delta}))$ on the time complexity of any rendezvous algorithm in our scenario. Further, this lower bound extends existing lower bounds for the general rendezvous problem without distance awareness

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

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

Gathering in Dynamic Rings

The gathering problem requires a set of mobile agents, arbitrarily positioned at different nodes of a network to group within finite time at the same location, not fixed in advanced. The extensive existing literature on this problem shares the same fundamental assumption: the topological structure does not change during the rendezvous or the gathering; this is true also for those investigations that consider faulty nodes. In other words, they only consider static graphs. In this paper we start the investigation of gathering in dynamic graphs, that is networks where the topology changes continuously and at unpredictable locations. We study the feasibility of gathering mobile agents, identical and without explicit communication capabilities, in a dynamic ring of anonymous nodes; the class of dynamics we consider is the classic 1-interval-connectivity. We focus on the impact that factors such as chirality (i.e., a common sense of orientation) and cross detection (i.e., the ability to detect, when traversing an edge, whether some agent is traversing it in the other direction), have on the solvability of the problem. We provide a complete characterization of the classes of initial configurations from which the gathering problem is solvable in presence and in absence of cross detection and of chirality. The feasibility results of the characterization are all constructive: we provide distributed algorithms that allow the agents to gather. In particular, the protocols for gathering with cross detection are time optimal. We also show that cross detection is a powerful computational element. We prove that, without chirality, knowledge of the ring size is strictly more powerful than knowledge of the number of agents; on the other hand, with chirality, knowledge of n can be substituted by knowledge of k, yielding the same classes of feasible initial configurations

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

Modelling 3D hydrodynamics governing island-associated sandbanks in a proposed tidal stream energy site

© 2017 The Authors A 3D numerical modelling study to investigate the existing hydrodynamic regime of the Pentland Firth Inner Sound Channel, Scotland, UK is presented. Hydrodynamics that govern some sensitive sedimentary deposits in the Inner Sound Channel are discussed. A 3D hydrodynamic model Delft3D is set up for Pentland Firth, located between Orkney Islands and mainland Scotland and a full sensitivity analysis of the numerical model is carried out. The current model set up sufficiently captures the existing hydrodynamics during a full spring-neap tidal cycle inside Pentland Firth. Using model results, the 3D structure of the dynamics of the tidal flows in the Inner Sound Channel is investigated. The temporal variability of tidal flows, the residual tidal flows in the channel and local flow interactions with the Island of Stroma are described. It is proved that the tidally dominant flows drive the sediment transport gradient model to explain the principle maintenance mechanisms of two island-associated sandbanks present in the Inner Sound. The present study provides detailed information on the physics of the tidal regime in the Inner Sound and explains the presence of sandbanks in an area of high tidal flows. Due to extremely high tidal flows, Inner Sound is considered as one of the most favourable sites for tidal energy extraction in the UK. The findings of this study will be very useful in assessing the significance of impacts of future tidal energy extraction on natural hydrodynamics and sediment dynamics of the area