Genus Ranges of Chord Diagrams

A chord diagram consists of a circle, called the backbone, with line segments, called chords, whose endpoints are attached to distinct points on the circle. The genus of a chord diagram is the genus of the orientable surface obtained by thickening the backbone to an annulus and attaching bands to the inner boundary circle at the ends of each chord. Variations of this construction are considered here, where bands are possibly attached to the outer boundary circle of the annulus. The genus range of a chord diagram is the genus values over all such variations of surfaces thus obtained from a given chord diagram. Genus ranges of chord diagrams for a fixed number of chords are studied. Integer intervals that can, and cannot, be realized as genus ranges are investigated. Computer calculations are presented, and play a key role in discovering and proving the properties of genus ranges.Comment: 12 pages, 8 figure

Unifying Multiple Knowledge Domains Using the ARTMAP Information Fusion System

Sensors working at different times, locations, and scales, and experts with different goals, languages, and situations, may produce apparently inconsistent image labels that are reconciled by their implicit underlying relationships. Even when such relationships are unknown to the user, an ARTMAP information fusion system discovers a hierarchical knowledge structure for a labeled dataset. The present paper addresses the problem of integrating two or more independent knowledge hierarchies based on the same low-level classes. The new system fuses independent domains into a unified knowledge structure, discovering cross-domain rules in this process. The system infers multi-level relationships among groups of output classes, without any supervised labeling of these relationships. In order to self-organize its expert system, ARTMAP information fusion system features distributed code representations that exploit the neural network’s capacity for one-to-many learning. The fusion system software and testbed datasets are available from http://cns.bu.edu/techlabNational Science Foundation (SBE-0354378); National Geospatial-Intelligence Agency (NMA 201-01-1-2016