We consider codimension-2 ribbon knottings of circles and 2-spheres. We find that if a given ribbon knot has two ribbon disks, those disks are related by ambient isotopy together with a finite number of local modifications to be described. This allows a complete set of moves to be developed for the representation of ribbon 2-knots by abstract or planar graphs. Similar results hold for classical...

An open ribbon is a square with one side called the seam. A closed ribbon is a cylinder with one boundary component called the seam. We sew an open (resp. closed) ribbon onto a graph by identifying the seam with an open (resp. closed) walk in the graph. A ribbon complex is a graph with a nite number of ribbons sewn on. We investigate when a ribbon complex embeds in 3-dimensional Euclidean space...

Andruskiewitsch and Schneider classify a large class of pointed Hopf algebras with abelian coradical. The quantum double of each such Hopf algebra is investigated. The quantum doubles of a family of Hopf algebras from the above classification are ribbon Hopf algebras. Introduction Quasitriangular Hopf algebras have an universal R-matrix which is a solution of the Yang-Baxter equation and their ...

BACKGROUND AND PURPOSE Large admission DWI infarct volume (>70 mL) is an established marker for poor clinical outcome in acute stroke. Outcome is more variable in patients with small infarcts (<70 mL). Percentage insula ribbon infarct correlates with infarct growth. We hypothesized that percentage insula ribbon infarct can help identify patients with stroke likely to have poor clinical outcome,...

Abstract Nature and technology often adopt structures that can be described as tubular helical assemblies. However, the role mechanisms of these remain elusive. In this paper, we study mechanical response under compression extension a assembly composed 8 Kirchhoff rods, arranged in pairs with opposite chirality connected by pin joints, both analytically numerically. We first focus on find that,...

The mixing of slurry in an anaerobic digester (AD) is one many key parameters, which have a significant effect on specific biogas yield (BY) and volatile solid (VS) removal rate. determination the optimum regime very complex as it depends large number internal external factors such microbial community, rheology slurry, impeller design, intensity, intervals. novelty this study investigation lab-...