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

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

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 Postnikov’s 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 Plücker 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 Plücker 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 ̈cker 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

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

Mapping local optical densities of states in silicon photonic structures with nanoscale electron spectroscopy

Relativistic electrons in a structured medium generate radiative losses such as Cherenkov and transition radiation that act as a virtual light source, coupling to the photonic densities of states. The effect is most pronounced when the imaginary part of the dielectric function is zero, a regime where in a non-retarded treatment no loss or coupling can occur. Maps of the resultant energy losses as a sub-5nm electron probe scans across finite waveguide structures reveal spatial distributions of optical modes in a spectral domain ranging from near-infrared to far ultraviolet.Comment: 18 pages, 4 figure