671 research outputs found
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
A New and Elementary CP^n Dyonic Magnon
We show that the dressing transformation method produces a new type of dyonic
CP^n magnon in terms of which all the other known solutions are either
composites or arise as special limits. In particular, this includes the
embedding of Dorey's dyonic magnon via an RP^3 subspace of CP^n. We also show
how to generate Dorey's dyonic magnon directly in the S^n sigma model via the
dressing method without resorting to the isomorphism with the SU(2) principle
chiral model when n=3. The new dyon is shown to be either a charged dyon or
topological kink of the related symmetric-space sine-Gordon theories associated
to CP^n and in this sense is a direct generalization of the soliton of the
complex sine-Gordon theory.Comment: 21 pages, JHEP3, typos correcte
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Dynamic ham-sandwich cuts in the plane
We design efficient data structures for dynamically maintaining a ham-sandwich cut of two point sets in the plane subject to insertions and deletions of points in either set. A ham-sandwich cut is a line that simultaneously bisects the cardinality of both point sets. For general point sets, our first data structure supports each operation in O(n1/3+Δ) amortized time and O(n4/3+Δ) space. Our second data structure performs faster when each point set decomposes into a small number k of subsets in convex position: it supports insertions and deletions in O(logn) time and ham-sandwich queries in O(klog4n) time. In addition, if each point set has convex peeling depth k , then we can maintain the decomposition automatically using O(klogn) time per insertion and deletion. Alternatively, we can view each convex point set as a convex polygon, and we show how to find a ham-sandwich cut that bisects the total areas or total perimeters of these polygons in O(klog4n) time plus the O((kb)polylog(kb)) time required to approximate the root of a polynomial of degree O(k) up to b bits of precision. We also show how to maintain a partition of the plane by two lines into four regions each containing a quarter of the total point count, area, or perimeter in polylogarithmic time.Engineering and Applied Science
Magnons, their Solitonic Avatars and the Pohlmeyer Reduction
We study the solitons of the symmetric space sine-Gordon theories that arise
once the Pohlmeyer reduction has been imposed on a sigma model with the
symmetric space as target. Under this map the solitons arise as giant magnons
that are relevant to string theory in the context of the AdS/CFT
correspondence. In particular, we consider the cases S^n, CP^n and SU(n) in
some detail. We clarify the construction of the charges carried by the solitons
and also address the possible Lagrangian formulations of the symmetric space
sine-Gordon theories. We show that the dressing, or Backlund, transformation
naturally produces solitons directly in both the sigma model and the symmetric
space sine-Gordon equations without the need to explicitly map from one to the
other. In particular, we obtain a new magnon solution in CP^3. We show that the
dressing method does not produce the more general "dyonic" solutions which
involve non-trivial motion of the collective coordinates carried by the
solitons.Comment: 52 page
The AdS(5)xS(5) Semi-Symmetric Space Sine-Gordon Theory
The generalized symmetric space sine-Gordon theories are a series of
1+1-integrable field theories that are classically equivalent to superstrings
on symmetric space spacetimes F/G. They are formulated in terms of a
semi-symmetric space as a gauged WZW model with fermions and a potential term
to deform it away from the conformal fixed point. We consider in particular the
case of PSU(2,2|4)/Sp(2,2)xSp(4) which corresponds to AdS(5)xS(5). We argue
that the infinite tower of conserved charges of these theories includes an
exotic N=(8,8) supersymmetry that is realized in a mildy non-local way at the
Lagrangian level. The supersymmetry is associated to a double central extension
of the superalgebra psu(2|2)+psu(2|2) and includes a non-trivial R symmetry
algebra corresponding to global gauge transformations, as well as 2-dimensional
spacetime translations. We then explicitly construct soliton solutions and show
that they carry an internal moduli superspace CP(2|1)xCP(2|1) with both bosonic
and Grassmann collective coordinates. We show how to semi-classical quantize
the solitons by writing an effective quantum mechanical system on the moduli
space which takes the form of a co-adjoint orbit of SU(2|2)xSU(2|2). The
spectrum consists of a tower of massive states in the short, or atypical,
symmetric representations, just as the giant magnon states of the string world
sheet theory, although here the tower is truncated.Comment: 39 pages, references adde
Unexposed populations and potential COVID-19 hospitalisations and deaths in European countries as per data up to 21 November 2021.
We estimate the potential remaining COVID-19 hospitalisation and death burdens in 19 European countries by estimating the proportion of each country's population that has acquired immunity to severe disease through infection or vaccination. Our results suggest many European countries could still face high burdens of hospitalisations and deaths, particularly those with lower vaccination coverage, less historical transmission and/or older populations. Continued non-pharmaceutical interventions and efforts to achieve high vaccination coverage are required in these countries to limit severe COVID-19 outcomes
The Cold Peace: Russo-Western Relations as a Mimetic Cold War
In 1989â1991 the geo-ideological contestation between two blocs was swept away, together with the ideology of civil war and its concomitant Cold War played out on the larger stage. Paradoxically, while the domestic sources of Cold War confrontation have been transcended, its external manifestations remain in the form of a âlegacyâ geopolitical contest between the dominant hegemonic power (the United States) and a number of potential rising great powers, of which Russia is one. The post-revolutionary era is thus one of a âcold peaceâ. A cold peace is a mimetic cold war. In other words, while a cold war accepts the logic of conflict in the international system and between certain protagonists in particular, a cold peace reproduces the behavioural patterns of a cold war but suppresses acceptance of the logic of behaviour. A cold peace is accompanied by a singular stress on notions of victimhood for some and undigested and bitter victory for others. The perceived victim status of one set of actors provides the seedbed for renewed conflict, while the âvictoryâ of the others cannot be consolidated in some sort of relatively unchallenged post-conflict order. The âuniversalismâ of the victors is now challenged by Russia's neo-revisionist policy, including not so much the defence of Westphalian notions of sovereignty but the espousal of an international system with room for multiple systems (the Schmittean pluriverse)
The Relativistic Avatars of Giant Magnons and their S-Matrix
The motion of strings on symmetric space target spaces underlies the
integrability of the AdS/CFT correspondence. Although these theories, whose
excitations are giant magnons, are non-relativistic they are classically
equivalent, via the Polhmeyer reduction, to a relativistic integrable field
theory known as a symmetric space sine-Gordon theory. These theories can be
formulated as integrable deformations of gauged WZW models. In this work we
consider the class of symmetric spaces CP^{n+1} and solve the corresponding
generalized sine-Gordon theories at the quantum level by finding the exact
spectrum of topological solitons, or kinks, and their S-matrix. The latter
involves a trignometric solution of the Yang-Baxer equation which exhibits a
quantum group symmetry with a tower of states that is bounded, unlike for
magnons, as a result of the quantum group deformation parameter q being a root
of unity. We test the S-matrix by taking the semi-classical limit and comparing
with the time delays for the scattering of classical solitons. We argue that
the internal CP^{n-1} moduli space of collective coordinates of the solitons in
the classical theory can be interpreted as a q-deformed fuzzy space in the
quantum theory. We analyse the n=1 case separately and provide a further test
of the S-matrix conjecture in this case by calculating the central charge of
the UV CFT using the thermodynamic Bethe Ansatz.Comment: 33 pages, important correction to S-matrix to ensure crossing
symmetr
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SeaWIFS satellite ocean color data from the Southern Ocean
SeaWiFS estimates of surface chlorophyll concentrations are reported for the region of the U.S. JGOFS study in the Southern Ocean (⌠170 °W, 60 °S). Elevated chlorophyll was observed at the Southern Ocean fronts, near the edge of the seasonal ice sheet, and above the Pacific ÂAntarctic Ridge. The elevated chlorophyll levels associated with the Pacific-Antarctic Ridge are surprising since even the crest of the ridge is at depths > 2000 m. This elevated phytoplankton biomass is likely the result of mesoscale physical-biological interactions where the Antarctic Circumpolar Current (ACC) encounters the ridge. Four cruises surveyed this region between October 1997 and March 1998, as part of the U.S. JGOFS. Satellite-derived chlorophyll concentrations were compared with in situ extracted chlorophyll measurements from these cruises. There was good agreement (rÂČ of 0.72, from a linear regression of shipboard vs. satellite chlorophyll), although Sea WiFS underestimated chlorophyll concentrations relative to the ship data
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