Hilbert schemes, polygraphs and the Macdonald positivity conjecture

We study the isospectral Hilbert scheme , defined as the reduced fiber product of with the Hilbert scheme of points in the plane , over the symmetric power . By a theorem of Fogarty, is smooth. We prove that is normal, Cohen-Macaulay and Gorenstein, and hence flat over . We derive two important consequences.

(1) We prove the strong form of the conjecture of Garsia and the author, giving a representation-theoretic interpretation of the Kostka-Macdonald coefficients . This establishes the Macdonald positivity conjecture, namely that .

(2) We show that the Hilbert scheme is isomorphic to the -Hilbert scheme of Nakamura, in such a way that is identified with the universal family over . From this point of view, describes the fiber of a character sheaf at a torus-fixed point of corresponding to .

The proofs rely on a study of certain subspace arrangements , called polygraphs, whose coordinate rings carry geometric information about . The key result is that is a free module over the polynomial ring in one set of coordinates on . This is proven by an intricate inductive argument based on elementary commutative algebra.

     

Derived Lengths of Solvable Groups Having Five Irreducible Character Degrees I

Let G be a solvable group with five character degrees. We show that the derived length of G is at most 5. This verifies that the Taketa inequality, dl(G)|cd(G)|, is valid for solvable groups with at most five character degrees.     

Picosecond X-ray diffraction probed transient structural changes in organic solids

In this Letter, we report on the experimental characterization of the geometry of short-lived electronically excited states in organic solids by time-resolved x-ray diffraction. Here, the structure factor of the organic crystal is measured as a function of time. Since this technique gives complete structural information, it is a very useful tool for learning more about atom motions on the excited-state energy surface-"beyond" the broad band typical of conventional spectroscopy. Although we used molecular crystals rather than free molecules, the compounds show detectable transient structural changes on the ps to ns time scale in our study.     

Hydrothermal synthesis of [Mn(HTPO)(BIPY)] (H3TPO = tris-(4-carboxylphenyl)phosphine oxide): a crosslinked coordination polymer

The hydrothermal synthesis and crystal structure of the title compound is reported. The structure consists of 1-D looped chains with the HTPO ligand binding through the phosphoryl and two carboxylate oxygens to manganese atoms. The carboxylic acid group is hydrogen bonded to adjacent chains forming a 2-D hydrogen bonded network. The crystals are triclinic and the space group isP-1 with a = 11.6767(8) Å, b = 10.8680(8) Å and c = 11.4232(87rpar; Å = 92.301(2)°, = 106.932(2)° and = 103.2390(10)° V = 1341.02(16) ų; Z = 2.     

Synthesis of [Mn(O2CCH2CH2CO2)(2,2′-bipyridyl) (H2O)2]n·H2O

The title compound exists as a 1D coordination polymer consisting of succinate anions, manganese cations and 2,2-bipyridyl molecules. Twelve water molecules are present in the unit cell and are part of a hydrogen bonding system. Some of these waters are binding to manganese cations while others are only participating in hydrogen bonding. The structure has a monoclinic space group P2₁/c with a = 8.2750(3) Å, b = 11.5104(3) Å and c = 16.9113(6) Å, = 95.2727(16); V = 1603.96(9) ų and Z = 4.     

Plasmonic enhancing nanoantennas for photodetection

This paper discusses the use of plasmonic nanostructured systems as nanoantennas for photodetection. Even though semiconductors and their heterostructures have many useful properties and widely used in photodetection, their electron density is very small compared to that of metals and, therefore, they have low absorption cross sections. The idea of using metal nanostructured antennas is to combine the high optical responses of metals with the functional electric properties of semiconductors.     

Approximate importance sampling Monte Carlo for data assimilation

Importance sampling Monte Carlo offers powerful approaches to approximating Bayesian updating in sequential problems. Specific classes of such approaches are known as particle filters. These procedures rely on the simulation of samples or ensembles of the unknown quantities and the calculation of associated weights for the ensemble members. As time evolves and/or when applied in high-dimensional settings, such as those of interest in many data assimilation problems, these weights typically display undesirable features. The key difficulty involves a collapse toward approximate distributions concentrating virtually all of their probability on an implausibly few ensemble members.

After reviewing ensembling, Monte Carlo, importance sampling and particle filters, we present some approximations intended to moderate the problem of collapsing weights. The motivations for these suggestions are combinations of (i) the idea that key dynamical behavior in many systems actually takes place on a low dimensional manifold, and (ii) notions of statistical dimension reduction. We illustrate our suggestions in a problem of inference for ocean surface winds and atmospheric pressure. Real observational data are used.     

Power-Law Inflation in Spacetimes Without Symmetry

We consider models of accelerated cosmological expansion described by the Einstein equations coupled to a nonlinear scalar field with a suitable exponential potential. We show that homogeneous and isotropic solutions are stable under small nonlinear perturbations without any symmetry assumptions. Our proof is based on results on the nonlinear stability of de Sitter spacetime and Kaluza-Klein reduction techniques.     

A high-throughput search for direct methanol fuel cell anode electrocatalysts of type PtxBiyPbz

We used a high-throughput method to screen for direct methanol fuel cell anode electrocatalysts in the Pt-Bi-Pb system. Previous studies showed that PtBi and PtPb (both NiAs structure type) were active electrocatalysts for the oxidation of formic acid, but only PtPb was active in oxidizing methanol. We synthesized thin films with continuous composition spreads of the three elements by magnetron sputtering at deposition temperatures from ambient to 510 °C. A fluorescence method was then used to identify compositions that were active toward methanol oxidation. Only films deposited between temperatures of 160 and 400 °C showed electrocatalytic activity. The areas that were active for methanol oxidation showed predominantly the NiAs structure type according to XRD, with optimal activity for compositions near PtBi_0.01Pb_0.53.     

Measuring particle size-dependent physicochemical structure in airborne single walled carbon nanotube agglomerates

As-produced single-walled carbon nanotube (SWCNT) material is a complex matrix of carbon nanotubes, bundles of nanotubes (nanoropes), non-tubular carbon and metal catalyst nanoparticles. The pulmonary toxicity of material released during manufacture and handling will depend on the partitioning and arrangement of these components within airborne particles. To probe the physicochemical structure of airborne SWCNT aggregates, a new technique was developed and applied to aerosolized as-produced material. Differential Mobility Analysis-classified aggregates were analyzed using an Aerosol Particle Mass Monitor, and a structural parameter Γ (proportional to the square of particle mobility diameter, divided by APM voltage) derived. Using information on the constituent components of the SWCNT, modal values of Γ were estimated for specific particle compositions and structures, and compared against measured values. Measured modal values of Γ for 150 nm mobility diameter aggregates suggested they were primarily composed of non-tubular carbon from one batch of material, and thin nanoropes from a second batch of material – these findings were confirmed using Transmission Electron Microscopy. Measured modal values of Γ for 31 nm mobility diameter aggregates indicated that they were comprised predominantly of thin carbon nanoropes with associated nanometer-diameter metal catalyst particles; there was no indication that either catalyst particles or non-tubular carbon particles were being preferentially released into the air. These results indicate that the physicochemistry of aerosol particles released while handling as-produced SWCNT may vary significantly by particle size and production batch, and that evaluations of potential health hazards need to account for this. Disclaimer: The mention of any company or product does not constitute an endorsement by the Centers for Disease Control and Prevention. The findings and conclusions in this paper are those of the authors and do not necessarily represent the views of the National Institute for Occupational Safety and Health.