PushPush and Push-1 are NP-hard in 2D

We prove that two pushing-blocks puzzles are intractable in 2D. One of our constructions improves an earlier result that established intractability in 3D [OS99] for a puzzle inspired by the game PushPush. The second construction answers a question we raised in [DDO00] for a variant we call Push-1. Both puzzles consist of unit square blocks on an integer lattice; all blocks are movable. An agent may push blocks (but never pull them) in attempting to move between given start and goal positions. In the PushPush version, the agent can only push one block at a time, and moreover when a block is pushed it slides the maximal extent of its free range. In the Push-1 version, the agent can only push one block one square at a time, the minimal extent---one square. Both NP-hardness proofs are by reduction from SAT, and rely on a common construction.Comment: 10 pages, 11 figures. Corrects an error in the conference version: Proc. of the 12th Canadian Conference on Computational Geometry, August 2000, pp. 211-21

EFFICACY OF ELECTRON BEAM IRRADIATION OF PROCESSED PORK PRODUCTS

The research reported on in this paper was conducted as part of a larger project. That project is on-going and is focused on ascertaining if irradiation of processed meats would be effective and economical. It involved the examination, through modeling, of the irradiation of one of many currently produced ready-to-eat (RTE) convenience-oriented, value-added pork products, sliced boneless ham. The results and findings reported in this paper represent the initial estimates of the cost and potential profitability or economic viability of irradiation of processed meats. The results and findings in this paper should be considered preliminary with extension and verification to be reported in a later paper by the authors. The objective of the portion of that project reported on in this paper was to conduct cost analysis of alternative irradiation methods and to ascertain the cost of each of those methods. Three scenarios were considered for cost analysis. The first scenario was the installation of an X-ray irradiator at an existing meat processing plant. The second scenario was the installation of a Cobalt-60 irradiator at an existing meat processing plant. The third scenario assumed that the meat processor contracted for irradiation services from an off-site company providing such service to a number of clients. For purposes of this study it was assumed that irradiation of sliced boneless ham would result in either a $.06/pound reduction in costs from processor to consumer, a$.06/pound increase in willingness to pay [price] or an equivalent combination of reduced costs and increased price. Total cost per pound for the irradiation process applied to sliced boneless ham ranged from $0.008, at the 200 million pound annual throughput rate using Cobalt-60 irradiation, to$0.069 at the 50 million pound annual throughput rate when contracting with an off-site company.Food Consumption/Nutrition/Food Safety,

PushPush is NP-hard in 2D

We prove that a particular pushing-blocks puzzle is intractable in 2D, improving an earlier result that established intractability in 3D [OS99]. The puzzle, inspired by the game *PushPush*, consists of unit square blocks on an integer lattice. An agent may push blocks (but never pull them) in attempting to move between given start and goal positions. In the PushPush version, the agent can only push one block at a time, and moreover, each block, when pushed, slides the maximal extent of its free range. We prove this version is NP-hard in 2D by reduction from SAT.Comment: 18 pages, 13 figures, 1 table. Improves cs.CG/991101

Enumerating Foldings and Unfoldings between Polygons and Polytopes

We pose and answer several questions concerning the number of ways to fold a polygon to a polytope, and how many polytopes can be obtained from one polygon; and the analogous questions for unfolding polytopes to polygons. Our answers are, roughly: exponentially many, or nondenumerably infinite.Comment: 12 pages; 10 figures; 10 references. Revision of version in Proceedings of the Japan Conference on Discrete and Computational Geometry, Tokyo, Nov. 2000, pp. 9-12. See also cs.CG/000701

Examples, Counterexamples, and Enumeration Results for Foldings and Unfoldings between Polygons and Polytopes

We investigate how to make the surface of a convex polyhedron (a polytope) by folding up a polygon and gluing its perimeter shut, and the reverse process of cutting open a polytope and unfolding it to a polygon. We explore basic enumeration questions in both directions: Given a polygon, how many foldings are there? Given a polytope, how many unfoldings are there to simple polygons? Throughout we give special attention to convex polygons, and to regular polygons. We show that every convex polygon folds to an infinite number of distinct polytopes, but that their number of combinatorially distinct gluings is polynomial. There are, however, simple polygons with an exponential number of distinct gluings. In the reverse direction, we show that there are polytopes with an exponential number of distinct cuttings that lead to simple unfoldings. We establish necessary conditions for a polytope to have convex unfoldings, implying, for example, that among the Platonic solids, only the tetrahedron has a convex unfolding. We provide an inventory of the polytopes that may unfold to regular polygons, showing that, for n>6, there is essentially only one class of such polytopes.Comment: 54 pages, 33 figure

Exploring the magnetic properties of the largest single molecule magnets

The giant {Mn₇₀} and {Mn₈₄} wheels are the largest nuclearity single-molecule magnets synthesized to date, and understanding their magnetic properties poses a challenge to theory. Starting from first-principles calculations, we explore the magnetic properties and excitations in these wheels using effective spin Hamiltonians. We find that the unusual geometry of the superexchange pathways leads to weakly coupled {Mn₇} subunits carrying an effective S = 2 spin. The spectrum exhibits a hierarchy of energy scales and massive degeneracies, with the lowest-energy excitations arising from Heisenberg-ring-like excitations of the {Mn₇} subunits around the wheel. We further describe how weak longer-range couplings can select the precise spin ground-state of the Mn wheels out of the nearly degenerate ground-state band

Continuous Blooming of Convex Polyhedra

We construct the first two continuous bloomings of all convex polyhedra. First, the source unfolding can be continuously bloomed. Second, any unfolding of a convex polyhedron can be refined (further cut, by a linear number of cuts) to have a continuous blooming.Comment: 13 pages, 6 figure

Physical Properties of OSIRIS-REx Target Asteroid (101955) 1999 RQ36 derived from Herschel, ESO-VISIR and Spitzer observations

In September 2011, the Herschel Space Observatory performed an observation campaign with the PACS photometer observing the asteroid (101955) 1999 RQ36 in the far infrared. The Herschel observations were analysed, together with ESO VLT-VISIR and Spitzer-IRS data, by means of a thermophysical model in order to derive the physical properties of 1999 RQ36. We find the asteroid has an effective diameter in the range 480 to 511 m, a slightly elongated shape with a semi-major axis ratio of a/b=1.04, a geometric albedo of 0.045 +0.015/-0.012, and a retrograde rotation with a spin vector between -70 and -90 deg ecliptic latitude. The thermal emission at wavelengths below 12 micron -originating in the hot sub-solar region- shows that there may be large variations in roughness on the surface along the equatorial zone of 1999 RQ36, but further measurements are required for final proof. We determine that the asteroid has a disk-averaged thermal inertia of Gamma = 650 Jm-2s-0.5K-1 with a 3-sigma confidence range of 350 to 950 Jm-2s-0.5K-1, equivalent to what is observed for 25143 Itokawa and suggestive that 1999 RQ36 has a similar surface texture and may also be a rubble-pile in nature. The low albedo indicates that 1999 RQ36 very likely contains primitive volatile-rich material, consistent with its spectral type, and that it is an ideal target for the OSIRIS-REx sample return mission.Comment: Accepted for publication in Astronomy & Astrophysics, 9 pages, 7 figure

Thermal Infrared Observations of Asteroid (99942) Apophis with Herschel

The near-Earth asteroid (99942) Apophis is a potentially hazardous asteroid. We obtained far-infrared observations of this asteroid with the Herschel Space Observatory's PACS instrument at 70, 100, and 160 micron. These were taken at two epochs in January and March 2013 during a close Earth encounter. These first thermal measurements of Apophis were taken at similar phase angles before and after opposition. We performed a detailed thermophysical model analysis by using the spin and shape model recently derived from applying a 2-period Fourier series method to a large sample of well-calibrated photometric observations. We find that the tumbling asteroid Apophis has an elongated shape with a mean diameter of 375$^{+14}_{-10}$ m (of an equal volume sphere) and a geometric V-band albedo of 0.30$^{+0.05}_{-0.06}$. We find a thermal inertia in the range 250-800 Jm$^{-2}$s$^{-0.5}$K$^{-1}$ (best solution at 600 Jm$^{-2}$s$^{-0.5}$K$^{-1}$), which can be explained by a mixture of low conductivity fine regolith with larger rocks and boulders of high thermal inertia on the surface. The thermal inertia, and other similarities with (25143) Itokawa indicate that Apophis might also have a rubble-pile structure. If we combine the new size value with the assumption of an Itokawa-like density and porosity we estimate a mass between 4.4 and 6.2 10$^{10}$ kg which is more than 2-3 times larger than previous estimates. We expect that the newly derived properties will influence impact scenario studies and influence the long-term orbit predictions of Apophis.Comment: Accepted for publication in Astronomy & Astrophysics, 21 pages, 8 figures, 2 table