Hodge algebra structures on certain rings of invariants and applications

Let B be a commutative ring with identity, m, n, and r be positive integers such that r ≤ min{m, n}, a ₁, …, a _r (resp. b ₁, … b _r ) be integers such that 1 ≤ a ₁< … < a _r ≤ m (resp. 1 ≤ b ₁ < … < b _r < n) and U (resp. V) be the most general m × r (resp. r × n) matrix such that s-minors of first a _s ? 1 rows (resp. b _s ? 1 columns) of U (resp. V) are all zero for s = 1, …, r. We investigate the B-algebra C generated by all the entries of UV and all the r-minors of U and V. We introduce a Hodge algebra structure, to which the discrete Hodge algebra associate is Cohen Macaulay, on C and prove that C is Cohen-Macaulay if so is B. Using this Hodge algebra structure, we show that C is the ring of absolute invariants of a certain group action, compute the divisor class group and the canonical class of C, and give a criterion of Gorenstein property of C in terms of a ₁ ,…, a _r and b ₁…, b _r .     

Shapes of lipid monolayer domains: Solutions using elliptic functions

Solid lipid monolayer domains surrounded by a fluid phase at an air-water interface exhibit complex shapes. These intriguing shapes can be understood in terms of a competition between line tension and long-range dipole-dipole interaction. The dipolar energy has recently been relevant to a negative line tension and a positive curvature energy at the boundary, and a corresponding shape equation was derived by the variation of the approximated domain energy (Phys. Rev. Lett. 93, 206101 (2004)). Here we further incorporate surface pressure into the shape equation and show that the equation can be analytically solved: the curvature of the domain boundary is exactly obtained as an elliptic function of arc-length. We find that a circular domain can grow into bean-and peach-like domains with pressure, i.e., dipping and cuspidal transitions of circle by compression. The comparison with the experimental observation shows nice agreement.     

Unusual Attractive Au–π Interactions in Small Diacetylene‐Modified Gold Clusters

It is well known that alkynes act as π‐acids in the formation of complexes with metals. We found unprecedented attractive Au–π interactions in diacetylene‐modified [core+exo]‐type [Au₈]⁴⁺ clusters. The 4‐phenyl‐1,3‐butadiynyl‐modified cluster has unusually short Au–C_α distances in the crystal structure, revealing the presence of attractive interactions between the coordinating C≡C moieties and the neighboring bitetrahedral Au₆ core, which is further supported by IR and NMR spectra. Such weak interactions are not found in mono‐acetylene‐modified clusters, which indicates that they are specific for diacetylenic ligands. The attractive Au–π interactions are likely associated with the low energy of the π* orbital in the diacetylenic moieties, into which the valence electrons of the gold core may be back donated. The [Au₈]⁴⁺ clusters show clear red‐shifts of >10 nm with respect to the corresponding mono‐acetylenic clusters in UV/Vis absorption bands, which indicates substantial electronic perturbation effects of the Au–π interactions.     

Non‐Thermoresponsive Decanano‐sized Domains in Thermoresponsive Hydrogel Microspheres Revealed by Temperature‐Controlled High‐Speed Atomic Force Microscopy

Despite the tremendous efforts devoted to the structural analysis of hydrogel microspheres (microgels), many details of their structures remain unclear. Reported in this study is that thermoresponsive poly(N‐isopropyl acrylamide) (pNIPAm)‐based microgels exhibit not only the widely accepted core–shell structures, but also inhomogeneous decanano‐sized non‐thermoresponsive spherical domains within their dense cores, which was revealed by temperature‐controlled high‐speed atomic force microscopy (TC‐HS‐AFM). Based on a series of experiments, it is concluded that the non‐thermoresponsive domains are characteristic for pNIPAm microgels synthesized by precipitation polymerization, and plausible structures for microgels prepared by other polymerization techniques are proposed.     

Mixed layer formation of copper overlayers on Ni(1 1 0) surface

Copper overlayer formation on the Ni(1 1 0) surface was studied by scanning tunneling microscopy (STM) in an ultrahigh vacuum. Atom-resolved STM images showed that initially deposited Cu is replaced with surface Ni atoms forming atom-size depressions on the Ni(1 1 0) terraces and a Ni-rich quasi-one-dimensional island along the direction. Further Cu deposition yields a mosaic structure on the islands, indicating Cu/Ni mixed layer formation. From the quantitative measurement of the Cu/Ni ratio on the substrate and the islands, impinging Cu will be replaced with surface Ni whereas expelled Ni and directly impinging Cu to the island form the mixed island. The number of Cu atoms in the islands, however, more than the directly impinging Cu, indicate significant Cu/Ni replacement at the periphery of the island.     

Copper(II)–acid catalyzed cyclopropanation of 1,3-dienamides by electrophilic activation of α-aryl diazoesters

Copper(II)–acid catalyzed cyclopropanation of electron-rich alkenes, such as 1,3-dienamides, with α-aryl diazoesters are described. The reaction could be performed without rare metal catalysts, excess substrate, or the need for the slow addition of the diazoesters.     

Hybrid Planar Wiggler with a Period of 2 cm for Higher Harmonic FEL Lasing

A hybrid planar wiggler with a period of 20 mm has been studied as the simplest one which gives the strong field including some higher harmonic components by selecting proper sizes of the ferromagnetic and the permanent magnet. Small gap length of the wiggler and small width of permendur satisfy these conditions to a certain degree.Gain analysis of FEL suggests that for high wiggler field of K>1 1.6, higher harmonic gains are improved primarily due to its strong field, and for low wiggler field of K< 1 1.6, they are mainly due to the modification of the wiggler field distribution.     

Structural analysis of Na2OGa2O3SiO2 glasses by the X-ray diffraction method

The structure of Na₂OGa₂O₃SiO₂ glasses of four different compositions containing up to 45 mol. % Ga₂O₃ has been determined by the X-ray diffraction method. The radial distribution function D(r) obtained indicates that Ga³⁺ ions are in fourfold coordination and GaO₄ tetrahedra are formed in these glasses. The reduced intensity function S·i(S) and radial distribution function D(r) which were calculated based on a structural model for A-25, 0.25(Ga₂O₃)·0.75(Na₂O·2SiO₂)/3 glass agreed well with the observed ones.     

Rheological study on rapid recovery of hydrogel based on oligomeric electrolyte

The hydrogel consisting of an oligomeric electrolyte, poly[pyridinium-1,4-diyl-iminocarbonyl-1,4-phenylenemethylene chloride] ( 1-Cl) underwent self-healing at temperatures lower than its gelation temperature after destruction of the gel network in a shear flow. The self-healing mechanism was investigated by rheological measurements on three different kinds of gels including a low-molecular weight organogelator and a polymeric hydrogelator. Although all of the three gels exhibited thermo-reversible hysteresis loops in the shear moduli, only 1-Cl hydrogel recovered its mechanical properties after vigorous agitation. It is conjectured that the self-healing is due to formation of network structure via a chlorine ion mediated hydrogen bond for which the activation energy is on the order of 10 kJ/mol.