Jacobi Elliptic Functions and the Complete Solution to the Bead on the Hoop Problem

Jacobi elliptic functions are flexible functions that appear in a variety of problems in physics and engineering. We introduce and describe important features of these functions and present a physical example from classical mechanics where they appear: a bead on a spinning hoop. We determine the complete analytical solution for the motion of a bead on the driven hoop for arbitrary initial conditions and parameter values.Comment: Accepted for publication in American Journal of Physics. 9 pages, 6 figure

Necessary Conditions for the Generic Global Rigidity of Frameworks on Surfaces

A result due in its various parts to Hendrickson, Connelly, and Jackson and Jord\'an, provides a purely combinatorial characterisation of global rigidity for generic bar-joint frameworks in $\mathbb{R}^2$. The analogous conditions are known to be insufficient to characterise generic global rigidity in higher dimensions. Recently Laman-type characterisations of rigidity have been obtained for generic frameworks in $\mathbb{R}^3$ when the vertices are constrained to lie on various surfaces, such as the cylinder and the cone. In this paper we obtain analogues of Hendrickson's necessary conditions for the global rigidity of generic frameworks on the cylinder, cone and ellipsoid.Comment: 13 page

Break it Down for Me: A Study in Automated Lyric Annotation

Comprehending lyrics, as found in songs and poems, can pose a challenge to human and machine readers alike. This motivates the need for systems that can understand the ambiguity and jargon found in such creative texts, and provide commentary to aid readers in reaching the correct interpretation. We introduce the task of automated lyric annotation (ALA). Like text simplification, a goal of ALA is to rephrase the original text in a more easily understandable manner. However, in ALA the system must often include additional information to clarify niche terminology and abstract concepts. To stimulate research on this task, we release a large collection of crowdsourced annotations for song lyrics. We analyze the performance of translation and retrieval models on this task, measuring performance with both automated and human evaluation. We find that each model captures a unique type of information important to the task.Comment: To appear in Proceedings of EMNLP 201

Is what you see what you get? representations, metaphors and tools in mathematics didactics

This paper is exploratory in character. The aim is to investigate ways in which it is possible to use the theoretical concepts of representations, tools and metaphors to try to understand what learners of mathematics ‘see’ during classroom interactions (in their widest sense) and what they might get from such interactions. Through an analysis of a brief classroom episode, the suggestion is made that what learners see may not be the same as what they get. From each of several theoretical perspectives utilised in this paper, what learners ‘get’ appears to be something extra. According to our analysis, this something ‘extra’ is likely to depend on the form of technology being used and the representations and metaphors that are available to both teacher and learner

Cascading Proximity Effects in Rotating Magnetizations

We demonstrate two effects that occur in all diffusive superconducting-magnetic heterostructures with rotating magnetization: the reappearance of singlet $|0,0>$ correlations deep in the magnetic material and a cascade of $s=1$, $m=0, \pm 1$ components (in the two spin-$1/2$ basis $|s,m>$). We do so by examining the order parameter and Josephson current through a multilayer with five mutually perpendicular ferromagnets. The properties of the middle layer determine whether the current is due to $m=0$ or $m\neq 0$ contributions. We conclude that so-called long- and short-range components are present across a proximity system with rotating magnetization.Comment: 5 pages, 4 figure