The Convex Configurations of "Sei Shonagon Chie no Ita" and Other Dissection Puzzles

The tangram and Sei Shonagon Chie no Ita are popular dissection puzzles consisting of seven pieces. Each puzzle can be formed by identifying edges from sixteen identical right isosceles triangles. It is known that the tangram can form 13 convex polygons. We show that Sei Shonagon Chie no Ita can form 16 convex polygons, propose a new puzzle that can form 19, no 7 piece puzzle can form 20, and 11 pieces are necessary and sufficient to form all 20 polygons formable by 16 identical isosceles right triangles. Finally, we examine the number of convex polygons formable by different quantities of these triangles

A PTAS for Bounded-Capacity Vehicle Routing in Planar Graphs

The Capacitated Vehicle Routing problem is to find a minimum-cost set of tours that collectively cover clients in a graph, such that each tour starts and ends at a specified depot and is subject to a capacity bound on the number of clients it can serve. In this paper, we present a polynomial-time approximation scheme (PTAS) for instances in which the input graph is planar and the capacity is bounded. Previously, only a quasipolynomial-time approximation scheme was known for these instances. To obtain this result, we show how to embed planar graphs into bounded-treewidth graphs while preserving, in expectation, the client-to-client distances up to a small additive error proportional to client distances to the depot

On Wrapping Spheres and Cubes with Rectangular Paper

What is the largest cube or sphere that a given rectangular piece of paper can wrap? This natural problem, which has plagued gift-wrappers everywhere, remains very much unsolved. Here we introduce new upper and lower bounds and consolidate previous results. Though these bounds rarely match, our results significantly reduce the gap

Disentangling the Origin and Heating Mechanism of Supernova Dust: Late-Time Spitzer Spectroscopy of the Type IIn SN 2005ip

This paper presents late-time near-infrared and {\it Spitzer} mid-infrared photometric and spectroscopic observations of warm dust in the Type IIn SN 2005ip in NGC 2906. The spectra show evidence for two dust components with different temperatures. Spanning the peak of the thermal emission, these observations provide strong constraints on the dust mass, temperature, and luminosity, which serve as critical diagnostics for disentangling the origin and heating mechanism of each component. The results suggest the warmer dust has a mass of $\sim 5 \times 10^{-4}~$\msolar, originates from newly formed dust in the ejecta, or possibly the cool, dense shell, and is continuously heated by the circumstellar interaction. By contrast, the cooler component likely originates from a circumstellar shock echo that forms from the heating of a large, pre-existing dust shell $\sim 0.01 - 0.05$~\msolar~by the late-time circumstellar interaction. The progenitor wind velocity derived from the blue edge of the He 1 1.083 \micron~P Cygni profile indicates a progenitor eruption likely formed this dust shell $\sim$100 years prior to the supernova explosion, which is consistent with a Luminous Blue Variable (LBV) progenitor star.Comment: 12 pages, 10 figures, Accepted to Ap

Engineering an Approximation Scheme for Traveling Salesman in Planar Graphs

We present an implementation of a linear-time approximation scheme for the traveling salesman problem on planar graphs with edge weights. We observe that the theoretical algorithm involves constants that are too large for practical use. Our implementation, which is not subject to the theoretical algorithm\u27s guarantee, can quickly find good tours in very large planar graphs

Rapid formation of large dust grains in the luminous supernova SN 2010jl

The origin of dust in galaxies is still a mystery. The majority of the refractory elements are produced in supernova explosions but it is unclear how and where dust grains condense and grow, and how they avoid destruction in the harsh environments of star-forming galaxies. The recent detection of 0.1-0.5 solar masses of dust in nearby supernova remnants suggests in situ dust formation, while other observations reveal very little dust in supernovae the first few years after explosion. Observations of the bright SN 2010jl have been interpreted as pre-existing dust, dust formation or no dust at all. Here we report the rapid (40-240 days) formation of dust in its dense circumstellar medium. The wavelength dependent extinction of this dust reveals the presence of very large (> 1 micron) grains, which are resistant to destructive processes. At later times (500-900 days), the near-IR thermal emission shows an accelerated growth in dust mass, marking the transition of the supernova from a circumstellar- to an ejecta-dominated source of dust. This provides the link between the early and late dust mass evolution in supernovae with dense circumstellar media.Comment: 62 pages, 13 figures, 1 table. Author version of the Letter to Nature, published online July 9 2014 (Nature, 511, 7509, pp. 326-329 (2014)), prior to the final editorial changes to conform to Journal style; includes Methods and Extended Data Figures and the Supplementary Information. See published version http://www.nature.com/nature/journal/v511/n7509/full/nature13558.htm

Constraints on the Progenitor of SN 2010jl and Pre-Existing Hot Dust in its Surrounding Medium

A search for the progenitor of SN~2010jl, an unusually luminous core-collapse supernova of Type~IIn, using pre-explosion {\it Hubble}/WFPC2 and {\it Spitzer}/IRAC images of the region, yielded upper limits on the UV and near-infrared (IR) fluxes from any candidate star. These upper limits constrain the luminosity and effective temperature of the progenitor, the mass of any preexisting dust in its surrounding circumstellar medium (CSM), and dust proximity to the star. A {\it lower} limit on the CSM dust mass is required to hide a luminous progenitor from detection by {\it Hubble}. {\it Upper} limits on the CSM dust mass and constraints on its proximity to the star are set by requiring that the absorbed and reradiated IR emission not exceed the IRAC upper limits. Using the combined extinction-IR emission constraints we present viable $M_d-R_1$ combinations, where $M_d$ and $R_1$ are the CSM dust mass and its inner radius. These depend on the CSM outer radius, dust composition and grain size, and the properties of the progenitor. The results constrain the pre-supernova evolution of the progenitor, and the nature and origin of the observed post-explosion IR emission from SN~2010jl. In particular, an $\eta$~Car-type progenitor will require at least 4~mag of visual extinction to avoid detection by the {\it Hubble}. This can be achieved with dust masses $\gtrsim 10^{-3}$~\msun\ (less than the estimated 0.2-0.5~\msun\ around $\eta$~Car) which must be located at distances of $\gtrsim 10^{16}$~cm from the star to avoid detection by {\it Spitzer}.Comment: Accepted for publication in the ApJ. 14 pages 10 figures. The complete figure set for Figure 10 (24 images) is available in the online journa