Conformal Maps to Multiply-Slit Domains and Applications

By exploiting conformal maps to vertically slit regions in the complex plane, a recently developed rational spectral method [Tee and Trefethen, 2006] is able to solve PDEs with interior layer-like behaviour using significantly fewer collocation points than traditional spectral methods. The conformal maps are chosen to 'enlarge the region of analyticity' in the solution: an idea which can be extended to other numerical methods based upon global polynomial interpolation. Here we show how such maps can be rapidly computed in both periodic and nonperiodic geometries, and apply them to some challenging differential equations

Flat Foldings of Plane Graphs with Prescribed Angles and Edge Lengths

When can a plane graph with prescribed edge lengths and prescribed angles (from among $\{0,180^\circ, 360^\circ$\}) be folded flat to lie in an infinitesimally thin line, without crossings? This problem generalizes the classic theory of single-vertex flat origami with prescribed mountain-valley assignment, which corresponds to the case of a cycle graph. We characterize such flat-foldable plane graphs by two obviously necessary but also sufficient conditions, proving a conjecture made in 2001: the angles at each vertex should sum to $360^\circ$, and every face of the graph must itself be flat foldable. This characterization leads to a linear-time algorithm for testing flat foldability of plane graphs with prescribed edge lengths and angles, and a polynomial-time algorithm for counting the number of distinct folded states.Comment: 21 pages, 10 figure

The difficulty of folding self-folding origami

Why is it difficult to refold a previously folded sheet of paper? We show that even crease patterns with only one designed folding motion inevitably contain an exponential number of `distractor' folding branches accessible from a bifurcation at the flat state. Consequently, refolding a sheet requires finding the ground state in a glassy energy landscape with an exponential number of other attractors of higher energy, much like in models of protein folding (Levinthal's paradox) and other NP-hard satisfiability (SAT) problems. As in these problems, we find that refolding a sheet requires actuation at multiple carefully chosen creases. We show that seeding successful folding in this way can be understood in terms of sub-patterns that fold when cut out (`folding islands'). Besides providing guidelines for the placement of active hinges in origami applications, our results point to fundamental limits on the programmability of energy landscapes in sheets.Comment: 8 pages, 5 figure

All Sky Survey Mission Observing Scenario Strategy

This paper develops a general observing strategy for missions performing all-sky surveys, where a single spacecraft maps the celestial sphere subject to realistic constraints. The strategy is flexible such that targeted observations and variable coverage requirements can be achieved. This paper focuses on missions operating in Low Earth Orbit, where the thermal and stray-light constraints due to the Sun, Earth, and Moon result in interacting and dynamic constraints. The approach is applicable to broader mission classes, such as those that operate in different orbits or that survey the Earth. First, the instrument and spacecraft configuration is optimized to enable visibility of the targeted observations throughout the year. Second, a constraint-based high-level strategy is presented for scheduling throughout the year subject to a simplified subset of the constraints. Third, a heuristic-based scheduling algorithm is developed to assign the all-sky observations over short planning horizons. The constraint-based approach guarantees solution feasibility. The approach is applied to the proposed SPHEREx mission, which includes coverage of the North and South Celestial Poles, Galactic plane, and a uniform coverage all-sky survey, and the ability to achieve science requirements demonstrated and visualized. Visualizations demonstrate the how the all-sky survey achieves its objectives

Repeated minimizers of pp-frame energies

For a collection of $N$ unit vectors $\mathbf{X}=\{x_i\}_{i=1}^N$, define the $p$-frame energy of $\mathbf{X}$ as the quantity $\sum_{i\neq j} |\langle x_i,x_j \rangle|^p$. In this paper, we connect the problem of minimizing this value to another optimization problem, so giving new lower bounds for such energies. In particular, for $p<2$, we prove that this energy is at least $2(N-d) p^{-\frac p 2} (2-p)^{\frac {p-2} 2}$ which is sharp for $d\leq N\leq 2d$ and $p=1$. We prove that for $1\leq m<d$, a repeated orthonormal basis construction of $N=d+m$ vectors minimizes the energy over an interval, $p\in[1,p_m]$, and demonstrate an analogous result for all $N$ in the case $d=2$. Finally, in connection, we give conjectures on these and other energies

Effect of Computer Keyboard Slope on Wrist Position and Forearm Electromyography of Typists Without Musculoskeletal Disorders

Positioning a computer keyboard with a downward slope reduces wrist extension needed to use the keyboard and has been shown to decrease pressure in the carpal tunnel. However, whether a downward slope of the keyboard reduces electromyographic (EMG) activity of the forearm muscles, in particular the wrist extensors, is not known. Subjects and Methods. Sixteen experienced typists participated in this study and typed on a conventional keyboard that was placed on slopes. Electromyographic activity of the extensor carpi ulnaris (ECU), flexor carpi ulnaris (FCU), and flexor carpi radialis (FCR) muscles was measured with surface electrodes, while the extension and ulnar deviation angles of the right and left wrists were measured with electrogoniometers. Results. Wrist extension angle decreased from approximately 12 degrees of extension while typing on a keyboard with a 7.5-degree slope to 3 degrees of flexion with the keyboard at a slope of –15 degrees. Although the differences were in the range of 1% to 3% of maximum voluntary contraction (MVC), amplitude probability distribution function (APDF) of root-mean-square EMG data points from the ECU, FCU, and FCR muscles varied across keyboard slopes. Discussion and Conclusion. Wrist extension decreased as the keyboard slope decreased. Furthermore, a slight decrease in percentage of MVC of the ECU muscle was noted as the keyboard slope decreased. Based on biomechanical modeling and published work on carpal tunnel pressure, both of these findings appear to be positive with respect to comfort and fatigue, but the exact consequences of these findings on the reduction or prevention of injuries have yet to be determined. The results may aid physical therapists and ergonomists in their evaluations of computer keyboard workstations and in making recommendations for interventions with regard to keyboard slope angle. [Simoneau GG, Marklin RW, Berman JE. Effect of computer keyboard slope on wrist position and forearm electromyography of typists without musculoskeletal disorders. Phys Ther. 2003;83:816–830.

Suppression of backscattered diffraction from sub-wavelength ‘moth-eye’ arrays

The eyes and wings of some species of moth are covered with arrays of nanoscale features that dramatically reduce reflection of light. There have been multiple examples where this approach has been adapted for use in antireflection and antiglare technologies with the fabrication of artificial moth-eye surfaces. In this work, the suppression of iridescence caused by the diffraction of light from such artificial regular moth-eye arrays at high angles of incidence is achieved with the use of a new tiled domain design, inspired by the arrangement of features on natural moth-eye surfaces. This bio-mimetic pillar architecture contains high optical rotational symmetry and can achieve high levels of diffraction order power reduction. For example, a tiled design fabricated in silicon and consisting of domains with 9 different orientations of the traditional hexagonal array exhibited a ~96% reduction in the intensity of the ?1 diffraction order. It is suggested natural moth-eye surfaces have evolved a tiled domain structure as it confers efficient antireflection whilst avoiding problems with high angle diffraction. This combination of antireflection and stealth properties increases chances of survival by reducing the risk of the insect being spotted by a predator. Furthermore, the tiled domain design could lead to more effective artificial moth-eye arrays for antiglare and stealth applications

Validation and application of a computational model for wrist and hand movements using surface markers

A kinematic model is presented based on surface marker placement generating wrist, metacarpal arch, fingers and thumb movements. Standard calculations are used throughout the model and then applied to the specified marker placement. A static trial involving eight unimpaired participants was carried out to assess inter-rater reliability. The standard deviations across the data were comparable to manual goniometers. In addition, a test retest trial of ten unimpaired participants is also reported to illustrate the variability of movement at the wrist joint, metacarpal arch, and index finger as an example of model output when repeating the same task many times. Light and heavyweight versions of the tasks are assessed and characteristics of individual movement strategies presented. The participant trial showed moderate correlation in radial/ulnar deviation of the wrist ( = 0 65), and strong correlation in both metacarpal arch joints ( = 075 and = 085), the MCP ( = 079), and PIP ( = 087) joints of the index finger. The results indicate that individuals use repeated strategies of movement when lifting light and heavyweight versions of the same object, but showed no obvious repeated pattern of movement across the population