A Theorem for Numerical Verification on Local Uniqueness of Solutions to Fixed-Point Equations

We give a theoretical result with respect to numerical verification of existence and local uniqueness of solutions to fixed-point equations which are supposed to have Fréchet differentiable operators. The theorem is based on Banach's fixed-point theorem and gives sufficient conditions in order that a given set of functions includes a unique solution to the fixed-point equation. The conditions are formulated to apply readily to numerical verification methods.

We already derived such a theorem in [11 N. Yamamoto ( 1998 ). A numerical verification method for solutions of boundary value problems with local uniqueness by Banach's fixed-point theorem . SIAM J. Numer. Anal. 35 : 2004 – 2013 .[Crossref] , [Google Scholar]], which is suitable to Nakao's methods on numerical verification for PDEs. The present theorem has a more general form and one may apply it to many kinds of differential equations and integral equations which can be transformed into fixed-point equations.     

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 .     

Measurement of the vector A y and tensor A yy , A xx , A xz analyzing powers for the dd → 3H p reaction at 200 MeV

The data on the vector A _y and tensor A _yy , A _xx , A _xz analyzing powers for the d → ³H p reaction have been obtained at the energy of the initial deuteron of 200MeV in the angular range of 0–95 degrees in the c.m.s. The calculations performed within one-nucleon exchange model with the use of the standard three-nucleon bound state wave functions fail to reproduce the data on the tensor analyzing powers.     

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.     

New alkylamide from the stems of Zanthoxylum nitidum

A new alkylamide, named (2E,6E,8E)-N-(2-methylpropyl)-10-oxo-2,6,8-decatrienamide (1), together with 22 known compounds (2–23), were isolated from the stems of Zanthoxylum nitidum. Their structures were elucidated by spectroscopic methods, including 1D and 2D NMR spectroscopy. The isolated compounds exhibited slightly antioxidant activities through DPPH and ABTS radical scavenging assays but showed no antibacterial activity against Streptococcus mutans ATCC2517, a dental caries causing bacteria.     

On the Smith normal form of a skew‐symmetric d‐optimal design of order

We show that the Smith normal form of a skew‐symmetric D ‐optimal design of order is determined by its order. Furthermore, we show that the Smith normal form of such a design can be written explicitly in terms of the order , thereby proving a recent conjecture of Armario. We apply our result to show that certain D ‐optimal designs of order are not equivalent to any skew‐symmetric D ‐optimal design. We also provide a correction to a result in the literature on the Smith normal form of D ‐optimal designs.     

A new type of amino amide organocatalyzed enantioselective crossed aldol reaction of ketones with aromatic aldehydes

A new type of amino amide organocatalysts was designed and synthesized from commercially available amino acids in easy steps. Their catalytic activities were examined in enantioselective crossed aldol reaction of various acyclic and cyclic ketones with aromatic aldehydes to afford the corresponding chiral anti-aldol adducts with good to excellent chemical yields, diastereoselectivities and enantioselectivities (up to 99%, up to syn:anti?=?1:99, up to 97% ee).     

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.     

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.