Numerical Verification Methods for Solutions of the Free Boundary Problem

We propose two methods to enclose the solution of an ordinary free boundary problem. The problem is reformulated as a nonlinear boundary value problem on a fixed interval including an unknown parameter. By appropriately setting a functional space that depends on the finite element approximation, the solution is represented as a fixed point of a compact map. Then, by using the finite element projection with constructive error estimates, a Newton-type verification procedure is derived. In addition, numerical examples confirming the effectiveness of current methods are given.     

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 .     

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.     

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.     

Sensitive HPLC–fluorescence detection of morphine labeled with DIB-Cl in rat brain and blood microdialysates and its application to the preliminarily study of the pharmacokinetic interaction between morphine and diclofenac

A sensitive high-performance liquid chromatography (HPLC)–fluorescence method for determination of morphine (Mor) in rat brain and blood microdialysates was developed using 4-(4,5-diphenyl-1H-imidazol-2-yl)benzoyl chloride (DIB-Cl) as a label. Mor was labeled with DIB-Cl under mild reaction conditions (at room temperature for 10 min). The separation of DIB-Mor was carried out on an octadecylsilica (ODS) column with CH₃CN/0.1 M acetate buffer (pH 5.4) within 14 min. The detection limits of Mor in brain and blood microdialysates at a signal-to-noise ratio of 3 were 0.4 and 0.6 ng mL⁻¹, respectively. The proposed method was successfully applied to the preliminarily study of potential pharmacokinetic interaction between Mor and diclofenac.     

Trehalose and trehalose-based polymers for environmentally benign, biocompatible and bioactive materials

Trehalose is a non-reducing disaccharide that is found in many organisms but not in mammals. This sugar plays important roles in cryptobiosis of selaginella mosses, tardigrades (water bears), and other animals which revive with water from a state of suspended animation induced by desiccation. The interesting properties of trehalose are due to its unique symmetrical low-energy structure, wherein two glucose units are bonded face-to-face by 1-->1-glucoside links. The Hayashibara Co. Ltd., is credited for developing an inexpensive, environmentally benign and industrial-scale process for the enzymatic conversion of alpha-1,4-linked polyhexoses to alpha,alpha-D-trehalose, which made it easy to explore novel food, industrial, and medicinal uses for trehalose and its derivatives. Trehalosechemistry is a relatively new and emerging field, and polymers of trehalose derivatives appear environmentally benign, biocompatible, and biodegradable. The discriminating properties of trehalose are attributed to its structure, symmetry, solubility, kinetic and thermodynamic stability and versatility. While syntheses of trehalose-based polymer networks can be straightforward, syntheses and characterization of well defined linear polymers with tailored properties using trehalose-based monomers is challenging, and typically involves protection and deprotection of hydroxyl groups to attain desired structural, morphological, biological, and physical and chemical properties in the resulting products. In this review, we will overview known literature on trehalose's fascinating involvement in cryptobiology; highlight its applications in many fields; and then discuss methods we used to prepare new trehalose-based monomers and polymers and explain their properties.