5,261 research outputs found
Cluster Algebras of Grassmannians are Locally Acyclic
Considered as commutative algebras, cluster algebras can be very unpleasant
objects. However, the first author introduced a condition known as "local
acyclicity" which implies that cluster algebras behave reasonably. One of the
earliest and most fundamental examples of a cluster algebra is the homogenous
coordinate ring of the Grassmannian. We show that the Grassmannian is locally
acyclic. Morally, we are in fact showing the stronger result that all positroid
varieties are locally acyclic. However, it has not been shown that all
positroid varieties have cluster structure, so what we actually prove is that
certain cluster varieties associated to Postnikov's alternating strand diagrams
are locally acylic. Moreover, we actually establish a slightly stronger
property than local acyclicity, which we term the Louise property, that is
designed to facilitate proofs involving the Mayer-Vietores sequence.Comment: 14 pages, 8 figures, minor edits from previous version, added
references to recent work of LeCler
The twist for positroid varieties
The purpose of this document is to connect two maps related to certain graphs embedded in the disc. The first is Postnikovâs boundary measurement map, which combines partition functions of matchings in the graph into a map from an algebraic torus to an open positroid variety in a Grassmannian. The second is a rational map from the open positroid variety to an algebraic torus, given by certain PlĂŒcker coordinates which are expected to be a cluster in a cluster structure.This paper clarifies the relationship between these two maps, which has been ambiguous since they were introduced by Postnikov in 2001. The missing ingredient supplied by this paper is a twist automorphism of the open positroid variety, which takes the target of the boundary measurement map to the domain of the (conjectural) cluster. Among other applications, this provides an inverse to the boundary measurement map, as well as Laurent formulas for twists of PlĂŒcker coordinates.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/139990/1/plms12056.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/139990/2/plms12056_am.pd
The twist for positroids
International audienceThere are two reasonable ways to put a cluster structure on a positroid variety. In one, the initial seed is a set of Plu Ìcker coordinates. In the other, the initial seed consists of certain monomials in the edge weights of a plabic graph. We will describe an automorphism of the positroid variety, the twist, which takes one to the other. For the big positroid cell, this was already done by Marsh and Scott; we generalize their results to all positroid varieties. This also provides an inversion of the boundary measurement map which is more general than Talaska's, in that it works for all reduced plabic graphs rather than just Le-diagrams. This is the analogue for positroid varieties of the twist map of Berenstein, Fomin and Zelevinsky for double Bruhat cells. Our construction involved the combinatorics of dimer configurations on bipartite planar graphs
Nanoscale assembly processes revealed in the nacroprismatic transition zone of Pinna nobilis mollusc shells
Intricate biomineralization processes in molluscs engineer hierarchical
structures with meso-, nano-, and atomic architectures that give the final
composite material exceptional mechanical strength and optical iridescence on
the macroscale. This multiscale biological assembly inspires new synthetic
routes to complex materials. Our investigation of the prism-nacre interface
reveals nanoscale details governing the onset of nacre formation using
high-resolution scanning transmission electron microscopy. A wedge polishing
technique provides unprecedented, large-area specimens required to span the
entire interface. Within this region, we find a transition from nanofibrillar
aggregation to irregular early-nacre layers, to well-ordered mature nacre
suggesting the assembly process is driven by aggregation of nanoparticles
(~50-80 nm) within an organic matrix that arrange in fiber-like polycrystalline
configurations. The particle number increases successively and, when critical
packing is reached, they merge into early-nacre platelets. These results give
new insights into nacre formation and particle-accretion mechanisms that may be
common to many calcareous biominerals.Comment: 5 Figure
Two-Dimensional Hydrodynamics of Pre-Core Collapse: Oxygen Shell Burning
By direct hydrodynamic simulation, using the Piecewise Parabolic Method (PPM)
code PROMETHEUS, we study the properties of a convective oxygen burning shell
in a SN 1987A progenitor star prior to collapse. The convection is too
heterogeneous and dynamic to be well approximated by one-dimensional
diffusion-like algorithms which have previously been used for this epoch.
Qualitatively new phenomena are seen.
The simulations are two-dimensional, with good resolution in radius and
angle, and use a large (90-degree) slice centered at the equator. The
microphysics and the initial model were carefully treated. Many of the
qualitative features of previous multi-dimensional simulations of convection
are seen, including large kinetic and acoustic energy fluxes, which are not
accounted for by mixing length theory. Small but significant amounts of
carbon-12 are mixed non-uniformly into the oxygen burning convection zone,
resulting in hot spots of nuclear energy production which are more than an
order of magnitude more energetic than the oxygen flame itself. Density
perturbations (up to 8%) occur at the `edges' of the convective zone and are
the result of gravity waves generated by interaction of penetrating flows into
the stable region. Perturbations of temperature and electron fraction at the
base of the convective zone are of sufficient magnitude to create angular
inhomogeneities in explosive nucleosynthesis products, and need to be included
in quantitative estimates of yields. Combined with the plume-like velocity
structure arising from convection, the perturbations will contribute to the
mixing of nickel-56 throughout supernovae envelopes. Runs of different
resolution, and angular extent, were performed to test the robustness of theseComment: For mpeg movies of these simulations, see
http://www.astrophysics.arizona.edu/movies.html Submitted to the
Astrophysical Journa
Strain Relaxation in Core-Shell Pt-Co Catalyst Nanoparticles
Surface strain plays a key role in enhancing the activity of Pt-alloy
nanoparticle oxygen reduction catalysts. However, the details of strain effects
in real fuel cell catalysts are not well-understood, in part due to a lack of
strain characterization techniques that are suitable for complex supported
nanoparticle catalysts. This work investigates these effects using strain
mapping with nanobeam electron diffraction and a continuum elastic model of
strain in simple core-shell particles. We find that surface strain is relaxed
both by lattice defects at the core-shell interface and by relaxation across
particle shells caused by Poisson expansion in the spherical geometry. The
continuum elastic model finds that in the absence of lattice dislocations,
geometric relaxation results in a surface strain that scales with the average
composition of the particle, regardless of the shell thickness. We investigate
the impact of these strain effects on catalytic activity for a series of Pt-Co
catalysts treated to vary their shell thickness and core-shell lattice
mismatch. For catalysts with the thinnest shells, the activity is consistent
with an Arrhenius dependence on the surface strain expected for coherent strain
in dislocation-free particles, while catalysts with thicker shells showed
greater activity losses indicating strain relaxation caused by dislocations as
well.Comment: 23 pages,7 figures, includes appendi
Towards Gravitational Wave Signals from Realistic Core Collapse Supernova Models
We have computed the gravitational wave signal from supernova core collapse
using the presently most realistic input physics available. We start from
state-of-the-art progenitor models of rotating and non-rotating massive stars,
and simulate the dynamics of their core collapse by integrating the equations
of axisymmetric hydrodynamics together with the Boltzmann equation for the
neutrino transport including an elaborate description of neutrino interactions,
and a realistic equation of state. We compute the quadrupole wave amplitudes,
the Fourier wave spectra, the amount of energy radiated in form of
gravitational waves, and the S/N ratios for the LIGO and the tuned Advanced
LIGO interferometers resulting both from non-radial mass motion and anisotropic
neutrino emission. The simulations demonstrate that the dominant contribution
to the gravitational wave signal is produced by neutrino-driven convection
behind the supernova shock. For stellar cores rotating at the extreme of
current stellar evolution predictions, the core-bounce signal is detectable
with advanced LIGO up to a distance of 5kpc, whereas the signal from post-shock
convection is observable up to a distance of about 100kpc. If the core is
non-rotating its gravitational wave emission can be measured up to a distance
of 15kpc, while the signal from the Ledoux convection in the deleptonizing,
nascent neutron star can be detected up to a distance of 10kpc. Both kinds of
signals are generically produced by convection in any core collapse supernova.Comment: 9 pages, 13 figures, Latex, submitted to ApJ, error in ps-file fixed;
figures in full resolution are available upon reques
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