ε-Regularity theorem and its application to the blow-up solutions of Keller-Segel systems in higher dimensions

Let us consider m(KS) below for all N?2 and general exponents m and q. In particular, the 2-D semi-linear case such as N=2, m=1 and q=2 is included. We establish an ε-regularity theorem for weak solutions. As an application, we give an extension criterion in which coincides with a scaling invariant class of weak solutions associated with m(KS). In addition, the Hausdorff dimension of its singular set is zero if and .     

Comprehensive visualization of detonation-diffraction structures and sizes in unstable and stable mixtures

The geometry and characteristic length of diffraction and re-initiation during a two-dimensional detonation propagation were revealed by visualization. C₂H₄ + 3O₂ (unstable), 2C₂H₂ + 5O₂ + 7Ar (stable) and 2C₂H₂ + 5O₂ + 21Ar (stable) were used as the test mixtures. Experiments were performed over the deviation angle range from 30° to 150° and the initial pressure range from 15.8 to 102.3 kPa. By self-emitting photography, we confirmed that the geometry and the characteristic length of diffraction are not different among test gases, with the exception of the fan-like structure of re-initiation that occurred regardless of whether the mixture was unstable or stable. We conducted a compensative experiment by changing the deviation angle and initial pressure, and summarized the detonation diffraction by shadowgraph. At deviation angles larger than 60°, we measured the distances from the vertex of the channel corner to the point where the transverse detonation wave reflected on the under wall (= wall reflection distance) and confirmed that wall reflection distances are approximately in the range of 10–15 times the cell width, whether the mixture is unstable or stable.     

Structure of the oxidized 4H-SiC(0 0 0 1)-3 × 3 surface

We have determined the structure of the 4H-SiC(0 0 0 1)-3 × 3 surface after exposure to small amounts of molecular oxygen at room temperature using surface X-ray diffraction. The 3 × 3 reconstruction remains until at least an exposure of 10,000 L, but the diffracted intensities change, indicating structural changes. Comparison of the Patterson maps of the clean and oxidized surface shows that the main changes occur at the Si tetramer on top of the 3 × 3 surface. Atomic positions for several models were fitted to the experimental data. A model in which oxygen atoms are inserted into the Si tetramer gives the best fit to the experimental data. The best-fit atomic positions agree well with those obtained using density functional calculations.     

Structural study of Si(1 1 1)-6 × 1-Ag surface using surface X-ray diffraction

The surface structure of Si(1 1 1)-6 × 1-Ag was investigated using surface X-ray diffraction techniques. By analyzing the CTR scattering intensities along 00 rod, the positions of the Ag and reconstructed Si atoms perpendicular to the surface were determined. The results agreed well with the HCC model proposed for a 3 × 1 structure induced by alkali-metals on a Si(1 1 1) substrate. The heights of the surface Ag and Si atoms did not move when the surface structure changed from Si(1 1 1)-√3 × √3-Ag to Si(1 1 1)-6 × 1-Ag by the desorption of the Ag atoms. From the GIXD measurement, the in-plane arrangement of the surface Ag atoms was determined. The results indicate that the Ag atoms move large distances at the phase transition between the 6 × 1 and 3 × 1 structures.     

Cp(*)RuCl-catalyzed formal intermolecular cyclotrimerization of three unsymmetrical alkynes through a boron temporary tether: regioselective four-component coupling synthesis of phthalides

Highly substituted phthalides were efficiently synthesized by sequential Cp(*)RuCl-catalyzed cyclotrimerization of alkynylboronates, propargyl alcohols, and terminal alkynes and palladium(II)-catalyzed carbonylation of the resultant arylboronates. The intermediate arylboronate was isolated and unambiguously characterized by X-ray crystallography. The perfect regioselectivity of the ruthenium-catalyzed formal intermolecular cyclotrimerization was discussed on the basis of the density functional calculations of a boraruthenacycle intermediate.     

One-pot beta-substitution of enones with alkyl groups to beta-alkyl enones

Vinylic hydrogens at the beta-position of enones were effectively substituted with alkyl groups in a one-pot procedure to afford beta-alkyl enones in good to high isolated yields by conjugate addition of higher-order dialkyl cyanocuprates to enones, followed by a reaction with N-tert-butylbenzenesulfinimidoyl chloride at -78 degree C.     

Multicolor digital holography with an achromatic phase shifter

Multiwavelength recording and reconstruction of a three-dimensional object are realized by use of phase-shifting digital holography. Red, green, and blue lines emitted from a white-light He-Cd laser are used for one-step recording of the complex amplitude of the object with a color CCD camera, in which a phase shift is introduced with an achromatic phase shifter based on the geometric phase. Three color images are reconstructed and successfully combined in a computer by use of Fresnel transformation based on a convolution.