Minor stars in plane graphs with minimum degree five

The weight of a subgraph $H$ in $G$ is the sum of the degrees in $G$ of vertices of $H$. The {\em height} of a subgraph $H$ in $G$ is the maximum degree of vertices of $H$ in $G$. A star in a given graph is minor if its center has degree at most five in the given graph. Lebesgue (1940) gave an approximate description of minor $5$-stars in the class of normal plane maps with minimum degree five. In this paper, we give two descriptions of minor $5$-stars in plane graphs with minimum degree five. By these descriptions, we can extend several results and give some new results on the weight and height for some special plane graphs with minimum degree five.Comment: 11 pages, 3 figure

An Exact Auxiliary Variable Gibbs Sampler for a Class of Diffusions

Stochastic differential equations (SDEs) or diffusions are continuous-valued continuous-time stochastic processes widely used in the applied and mathematical sciences. Simulating paths from these processes is usually an intractable problem, and typically involves time-discretization approximations. We propose an exact Markov chain Monte Carlo sampling algorithm that involves no such time-discretization error. Our sampler is applicable to the problem of prior simulation from an SDE, posterior simulation conditioned on noisy observations, as well as parameter inference given noisy observations. Our work recasts an existing rejection sampling algorithm for a class of diffusions as a latent variable model, and then derives an auxiliary variable Gibbs sampling algorithm that targets the associated joint distribution. At a high level, the resulting algorithm involves two steps: simulating a random grid of times from an inhomogeneous Poisson process, and updating the SDE trajectory conditioned on this grid. Our work allows the vast literature of Monte Carlo sampling algorithms from the Gaussian process literature to be brought to bear to applications involving diffusions. We study our method on synthetic and real datasets, where we demonstrate superior performance over competing methods.Comment: 37 pages, 13 figure

RORS: Enhanced Rule-based OWL Reasoning on Spark

The rule-based OWL reasoning is to compute the deductive closure of an ontology by applying RDF/RDFS and OWL entailment rules. The performance of the rule-based OWL reasoning is often sensitive to the rule execution order. In this paper, we present an approach to enhancing the performance of the rule-based OWL reasoning on Spark based on a locally optimal executable strategy. Firstly, we divide all rules (27 in total) into four main classes, namely, SPO rules (5 rules), type rules (7 rules), sameAs rules (7 rules), and schema rules (8 rules) since, as we investigated, those triples corresponding to the first three classes of rules are overwhelming (e.g., over 99% in the LUBM dataset) in our practical world. Secondly, based on the interdependence among those entailment rules in each class, we pick out an optimal rule executable order of each class and then combine them into a new rule execution order of all rules. Finally, we implement the new rule execution order on Spark in a prototype called RORS. The experimental results show that the running time of RORS is improved by about 30% as compared to Kim & Park's algorithm (2015) using the LUBM200 (27.6 million triples).Comment: 12 page

AJILE Movement Prediction: Multimodal Deep Learning for Natural Human Neural Recordings and Video

Developing useful interfaces between brains and machines is a grand challenge of neuroengineering. An effective interface has the capacity to not only interpret neural signals, but predict the intentions of the human to perform an action in the near future; prediction is made even more challenging outside well-controlled laboratory experiments. This paper describes our approach to detect and to predict natural human arm movements in the future, a key challenge in brain computer interfacing that has never before been attempted. We introduce the novel Annotated Joints in Long-term ECoG (AJILE) dataset; AJILE includes automatically annotated poses of 7 upper body joints for four human subjects over 670 total hours (more than 72 million frames), along with the corresponding simultaneously acquired intracranial neural recordings. The size and scope of AJILE greatly exceeds all previous datasets with movements and electrocorticography (ECoG), making it possible to take a deep learning approach to movement prediction. We propose a multimodal model that combines deep convolutional neural networks (CNN) with long short-term memory (LSTM) blocks, leveraging both ECoG and video modalities. We demonstrate that our models are able to detect movements and predict future movements up to 800 msec before movement initiation. Further, our multimodal movement prediction models exhibit resilience to simulated ablation of input neural signals. We believe a multimodal approach to natural neural decoding that takes context into account is critical in advancing bioelectronic technologies and human neuroscience