1,403 research outputs found
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 \}) 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 , 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
Settling the Reward Hypothesis
The reward hypothesis posits that, "all of what we mean by goals and purposes
can be well thought of as maximization of the expected value of the cumulative
sum of a received scalar signal (reward)." We aim to fully settle this
hypothesis. This will not conclude with a simple affirmation or refutation, but
rather specify completely the implicit requirements on goals and purposes under
which the hypothesis holds
The Birth of a Galaxy - III. Propelling reionisation with the faintest galaxies
Starlight from galaxies plays a pivotal role throughout the process of cosmic
reionisation. We present the statistics of dwarf galaxy properties at z > 7 in
haloes with masses up to 10^9 solar masses, using a cosmological radiation
hydrodynamics simulation that follows their buildup starting with their
Population III progenitors. We find that metal-enriched star formation is not
restricted to atomic cooling ( K) haloes, but can occur
in haloes down to masses ~10^6 solar masses, especially in neutral regions.
Even though these smallest galaxies only host up to 10^4 solar masses of stars,
they provide nearly 30 per cent of the ionising photon budget. We find that the
galaxy luminosity function flattens above M_UV ~ -12 with a number density that
is unchanged at z < 10. The fraction of ionising radiation escaping into the
intergalactic medium is inversely dependent on halo mass, decreasing from 50 to
5 per cent in the mass range . Using our galaxy
statistics in a semi-analytic reionisation model, we find a Thomson scattering
optical depth consistent with the latest Planck results, while still being
consistent with the UV emissivity constraints provided by Ly forest
observations at z = 4-6.Comment: 21 pages, 15 figures, 4 tables. Accepted in MNRA
Hinged Dissections Exist
We prove that any finite collection of polygons of equal area has a common
hinged dissection. That is, for any such collection of polygons there exists a
chain of polygons hinged at vertices that can be folded in the plane
continuously without self-intersection to form any polygon in the collection.
This result settles the open problem about the existence of hinged dissections
between pairs of polygons that goes back implicitly to 1864 and has been
studied extensively in the past ten years. Our result generalizes and indeed
builds upon the result from 1814 that polygons have common dissections (without
hinges). We also extend our common dissection result to edge-hinged dissections
of solid 3D polyhedra that have a common (unhinged) dissection, as determined
by Dehn's 1900 solution to Hilbert's Third Problem. Our proofs are
constructive, giving explicit algorithms in all cases. For a constant number of
planar polygons, both the number of pieces and running time required by our
construction are pseudopolynomial. This bound is the best possible, even for
unhinged dissections. Hinged dissections have possible applications to
reconfigurable robotics, programmable matter, and nanomanufacturing.Comment: 22 pages, 14 figure
CP Violation beyond the Standard Model
In this talk a number of broad issues are raised about the origins of CP
violation and how to test the ideas.Comment: 17 pages, LaTeX, 6 postscript figures. Uses iopart10.clo,
iopart12.clo and iopart.cls. Plenary talk given at the BSM Phenomenology
Workshop, Durham, UK, 6-11 May 2001. To appear in the proceeding
Direct Mediation and Metastable Supersymmetry Breaking for SO(10)
We examine a metastable Macroscopic SO(N) SQCD model of
Intriligator, Seiberg and Shih (ISS). We introduce various baryon and meson
deformations, including multitrace operators and explore embedding an SO(10)
parent of the standard model into two weakly gauged flavour sectors. Direct
fundamental messengers and the symmetric pseudo-modulus messenger mediate SUSY
breaking to the MSSM. Gaugino and sfermion masses are computed and compared for
each deformation type. We also explore reducing the rank of the magnetic quark
matrix of the ISS model and find an additional fundamental messenger.Comment: 43 pages, Latex. Version to appear in JHEP
Who Needs Crossings? Hardness of Plane Graph Rigidity
We exactly settle the complexity of graph realization, graph rigidity, and graph global rigidity as applied to three types of graphs: "globally noncrossing" graphs, which avoid crossings in all of their configurations; matchstick graphs, with unit-length edges and where only noncrossing configurations are considered; and unrestricted graphs (crossings allowed) with unit edge lengths (or in the global rigidity case, edge lengths in {1,2}). We show that all nine of these questions are complete for the class Exists-R, defined by the Existential Theory of the Reals, or its complement Forall-R; in particular, each problem is (co)NP-hard.
One of these nine results - that realization of unit-distance graphs is Exists-R-complete - was shown previously by Schaefer (2013), but the other eight are new. We strengthen several prior results. Matchstick graph realization was known to be NP-hard (Eades & Wormald 1990, or Cabello et al. 2007), but its membership in NP remained open; we show it is complete for the (possibly) larger class Exists-R. Global rigidity of graphs with edge lengths in {1,2} was known to be coNP-hard (Saxe 1979); we show it is Forall-R-complete.
The majority of the paper is devoted to proving an analog of Kempe\u27s Universality Theorem - informally, "there is a linkage to sign your name" - for globally noncrossing linkages. In particular, we show that any polynomial curve phi(x,y)=0 can be traced by a noncrossing linkage, settling an open problem from 2004. More generally, we show that the nontrivial regions in the plane that may be traced by a noncrossing linkage are precisely the compact semialgebraic regions. Thus, no drawing power is lost by restricting to noncrossing linkages. We prove analogous results for matchstick linkages and unit-distance linkages as well
Radio Foregrounds for the 21cm Tomography of the Neutral Intergalactic Medium at High Redshifts
Absorption or emission against the cosmic microwave background radiation
(CMB) may be observed in the redshifted 21cm line if the spin temperature of
the neutral intergalactic medium prior to reionization differs from the CMB
temperature. This so-called 21cm tomography should reveal important information
on the physical state of the intergalactic medium at high redshifts. The
fluctuations in the redshifted 21 cm, due to gas density inhomogeneities at
early times, should be observed at meter wavelengths by the next generation
radio telescopes such as the proposed {\it Square Kilometer Array (SKA)}. Here
we show that the extra-galactic radio sources provide a serious contamination
to the brightness temperature fluctuations expected in the redshifted 21 cm
emission from the IGM at high redshifts. Unless the radio source population
cuts off at flux levels above the planned sensitivity of SKA, its clustering
noise component will dominate the angular fluctuations in the 21 cm signal. The
integrated foreground signal is smooth in frequency space and it should
nonetheless be possible to identify the sharp spectral feature arising from the
non-uniformities in the neutral hydrogen density during the epoch when the
first UV sources reionize the intergalactic medium.Comment: 5 pages emulateapj with 1 figure, accepted to Ap
Continuously Flattening Polyhedra Using Straight Skeletons
We prove that a surprisingly simple algorithm folds the surface of every convex polyhedron, in any dimension, into a flat folding by a continuous motion, while preserving intrinsic distances and avoiding crossings. The flattening respects the straight-skeleton gluing, meaning that points of the polyhedron touched by a common ball inside the polyhedron come into contact in the flat folding, which answers an open question in the book Geometric Folding Algorithms. The primary creases in our folding process can be found in quadratic time, though necessarily, creases must roll continuously, and we show that the full crease pattern can be exponential in size. We show that our method solves the fold-and-cut problem for convex polyhedra in any dimension. As an additional application, we show how a limiting form of our algorithm gives a general design technique for flat origami tessellations, for any spiderweb (planar graph with all-positive equilibrium stress)
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