Remark on the Helmholtz decomposition in domains with noncompact boundary

Let \(\varOmega \) be a domain in \(\mathbb {R}^{d+1}\) whose boundary is given as a uniform Lipschitz graph \(x_{d+1}=\eta (x)\) for \(x \in \mathbb {R}^d\) . For such a domain, it is known that the Helmholtz decomposition is not always valid in \(L^p(\varOmega )\) except for the energy space \(L^2 (\varOmega )\) . In this paper we show that the Helmholtz decomposition still holds in certain anisotropic spaces which include vector fields decaying slowly in the \(x_{d+1}\) variable. In particular, these classes include some infinite energy vector fields. For the purpose, we develop a new approach based on a factorization of divergence form elliptic operators whose coefficients are independent of one variable.     

High heat generation ability in AC magnetic field for nano-sized magnetic Y3Fe5O12 powder prepared by bead milling

Nano-sized magnetic Y₃Fe₅O₁₂ ferrite having a high heat generation ability in an AC magnetic field was prepared by bead milling. A commercial powder sample (non-milled sample) of ca. 2.9 μm in particle size did not show any temperature enhancement in the AC magnetic field. The heat generation ability in the AC magnetic field improved with a decrease in the average crystallite size for the bead-milled Y₃Fe₅O₁₂ ferrites. The highest heat ability in the AC magnetic field was for the fine Y₃Fe₅O₁₂ powder with a 15-nm crystallite size (the samples were milled for 4 h using 0.1 mm? beads). The heat generation ability of the excessively milled Y₃Fe₅O₁₂ samples decreased. The main reason for the high heat generation property of the milled samples was ascribed to an increase in the Néel relaxation of the superparamagnetic material. The heat generation ability was not influenced by the concentration of the ferrite powder. For the samples milled for 4 h using 0.1 mm? beads, the heat generation ability (W g⁻¹) was estimated using a 3.58×10⁻⁴ fH² frequency (f/kHz) and the magnetic field (H/kA m⁻¹), which is the highest reported value of superparamagnetic materials.     

New asymmetric vanadium catalyst for highly selective oxidative coupling polymerization

Novel oxovanadium(IV)–bisoxazoline catalysts, such as vanadyl sulfate (VOSO₄)–(R)‐2,2′‐isopropylidenebis(4‐phenyl‐2‐oxazoline) [(R)Phbox], for the asymmetric oxidative coupling polymerization (AOCP) of 2,3‐dihydroxynaphthalene were developed. For example, the AOCP with VOSO₄–(R)Phbox in CH₂Cl₂–MeOH [7/1 (v/v)] at room temperature for 24 h under an O₂ atmosphere, followed by acetylation of the hydroxyl groups, afforded a polymer in a 58% yield with a specific rotation of [α]_D = ?147. The enantioselectivity during the polymerization was estimated to be 80% enantiomeric excess (S), a value much higher than that observed for the polymerizations with the copper(I)‐based catalyst systems and the typical oxovanadium(IV) catalysts reported for the asymmetric oxidative coupling producing the 1,1′‐bi‐2‐naphthol derivatives. © 2005 Wiley Periodicals, Inc., Inc. J Polym Sci Part A: Polym Chem 43: 5872–5878, 2005     

Synthesis and morphology of polyethylene‐block‐poly(methyl methacrylate) through the combination of metallocene catalysis with living radical polymerization

Polyethylene‐block‐poly(methyl methacrylate) (PE‐b‐PMMA) was successfully synthesized through the combination of metallocene catalysis with living radical polymerization. Terminally hydroxylated polyethylene, prepared by ethylene/allyl alcohol copolymerization with a specific zirconium metallocene/methylaluminoxane/triethylaluminum catalyst system, was treated with 2‐bromoisobutyryl bromide to produce terminally esterified polyethylene (PE‐Br). With the resulting PE‐Br as an initiator for transition‐metal‐mediated living radical polymerization, methyl methacrylate polymerization was subsequently performed with CuBr or RuCl₂(PPh₃)₃ as a catalyst. Then, PE‐b‐PMMA block copolymers of different poly(methyl methacrylate) (PMMA) contents were prepared. Transmission electron microscopy of the obtained block copolymers revealed unique morphological features that depended on the content of the PMMA segment. The block copolymer possessing 75 wt % PMMA contained 50–100‐nm spherical polyethylene lamellae uniformly dispersed in the PMMA matrix. Moreover, the PE‐b‐PMMA block copolymers effectively compatibilized homopolyethylene and homo‐PMMA at a nanometer level. © 2003 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 41: 3965–3973, 2003     

An analysis of SEM techniques used to profile chemical etching on the semiconductor Hg1-xCdxTe

Various methods have been used to obtain accurate cross-sectional profiles of Hg_1-xCd_xTe devices after etching. Preliminary measurements from different etching techniques are also presented. Techniques used were scanning electron microscopy with stereo pairs and cleaved cross-sections, topography SEM (multiple detector SEM) and atomic force microscopy.     

Ruthenium-catalyzed cycloaddition of 1,6-diynes with isothiocyanates and carbon disulfide: first transition-metal catalyzed [2 + 2 + 2] cocyclotrimerization involving C=S double bond.

In the presence of 10 mol % Cp*Ru(cod)Cl, 1,6-diynes with a tertiary center at 4-position reacted with various isothiocyanates at their C=S double bond to afford bicyclic (2H)-thiopyranimines in 35-88% yields. The (2H)-thiopyran structure was unequivocally determined by X-ray analysis. The cycloaddition of carbon disulfide with a diyne similarly gave the expected bicyclic dithiopyrone in 50% yield.     

Reconstitution of Biosynthetic Machinery for the Synthesis of the Highly Elaborated Indole Diterpene Penitrem

Penitrem A is one of the most elaborated members of the fungal indole diterpenes. Two separate penitrem gene clusters were identified using genomic and RNA sequencing data, and 13 out of 17 transformations in the penitrem biosynthesis were elucidated by heterologous reconstitution of the relevant genes. These reactions involve 1) a prenylation‐initiated cationic cyclization to install the bicyclo[3.2.0]heptane skeleton (PtmE), 2) a two‐step P450‐catalyzed oxidative processes forming the unique tricyclic penitrem skeleton (PtmK and PtmU), and 3) five sequential oxidative transformations (PtmKULNJ). Importantly, without conventional gene disruption, reconstitution of the biosynthetic machinery provided sufficient data to determine the pathway. It was thus demonstrated that the Aspergillus oryzae reconstitution system is a powerful method for studying the biosynthesis of complex natural products.