Direct and inverse problems for degenerate differential equations

We are concerned with a general abstract equation that allows to handle various degenerate first and second order differential equations in Banach spaces. We indicate sufficient conditions for existence and uniqueness of a solution. Periodic conditions are assumed to improve previous approaches on the abstract problem to work on \((-\infty ,\infty )\). Related inverse problems are discussed, too. All general results are applied to some systems of partial differential equations. Inverse problems for degenerate evolution integro-differential equations might be described, too.     

Asymptotics for functionals of self-normalized residuals of discretely observed stochastic processes

The purpose of this paper is to derive the stochastic expansion of self-normalized-residual functionals stemming from a class of diffusion type processes observed at high frequency, where total observing period may or may not tend to infinity. The result enables us to construct some explicit statistics for goodness of fit tests, consistent against “presence of a jump component” and “diffusion-coefficient misspecification”; then, the acceptance of the null hypothesis may serve as a collateral evidence for using the correctly specified diffusion type model. Especially, our asymptotic result clarifies how to remove the bias caused by plugging in a diffusion-coefficient estimator.     

Cohen–Macaulay Modules and Holonomic Modules Over Filtered Rings

We study Gorenstein dimension and grade of a module M over a filtered ring whose associated graded ring is a commutative Noetherian ring. An equality or an inequality between these invariants of a filtered module and its associated graded module is the most valuable property for an investigation of filtered rings. We prove an inequality G?dim M ≤ G?dim gr M and an equality grade M = grade gr M, whenever Gorenstein dimension of gr M is finite (Theorems 2.3 and 2.8). We would say that the use of G-dimension adds a new viewpoint for studying filtered rings and modules. We apply these results to a filtered ring with a Cohen–Macaulay or Gorenstein associated graded ring and study a Cohen–Macaulay, perfect, or holonomic module.     

Corrosion and passivation behavior of Mg-Zn-Y-Al alloys prepared by cooling rate-controlled solidification

Highly corrosion-resistant nanocrystalline Mg-Zn-Y-Al multi-phase alloys have been prepared by consolidation of rapidly solidified (RS) ribbons. The relation between corrosion behavior and microstructure evolution of Mg-Zn-Y-Al alloys with a long period stacking ordered phase has been investigated. In order to clarify the influence of rapid solidification on the occurrence of localized corrosion such as filiform corrosion, several Mg_96.75Zn_0.75Y₂Al_0.5 (at.%) alloys with different cooling rates are fabricated by the gravity casting, copper mould injection casting and melt-spinning techniques and their corrosion behavior and microstructures are examined by the salt water immersion test, electrochemical measurements, GDOES, XRD, SEM and TEM. To clarify the effect of aluminium addition on the improvement in corrosion resistance of the alloys, several Mg_97.25−xZn_0.75Y₂Al_x alloys with different aluminium contents are fabricated by consolidating RS ribbons and the formation of corroded films on the Mg-Zn-Y-Al alloys have been investigated. Rapid solidification brings about the grain refinement and an increase in the solid solubility of zinc, yttrium and aluminium into the magnesium matrix, enhancing microstructural and electrochemical homogeneity, which in turn enhanced corrosion resistance. The addition of aluminium to magnesium can modify the structure and chemical composition of surface films and improves the resistance to local breakdown of the films.     

High-quality InAlN/GaN heterostructures grown by metal–organic vapor phase epitaxy

We fabricated high-quality InAlN/GaN heterostructures by metal–organic vapor phase epitaxy (MOVPE). X-ray diffraction measurements revealed that InAlN/GaN heterostructures grown under optimal conditions have flat surfaces and abrupt heterointerfaces. Electron mobility from 1200 to 2000 cm²/V s was obtained at room temperature. To our knowledge, this mobility is the highest ever reported for InAlN/GaN heterostructures. We also investigated the relationship between the Al composition and sheet electron density (N_s) for the first time. N_s increased from 1.0×10¹² to 2.7×10¹³ cm⁻² when the Al composition increased from 0.78 to 0.89.     

Flattened chromatic dispersion in square photonic crystal fibers with low confinement losses

This research presents a simple index-guiding square photonic crystal fibers (SPCFs) that has a silica core surrounded by air hole with two different diameters. It is demonstrated that the designed two-different-size hole-arrayed index-guiding SPCFs has a ultra-flattened chromatic dispersion of 0 ± 0.9 ps/(nm·km) in a wavelength range of 1.34 to 1.61 μm and low confinement loss of less than 10⁻⁷ dB/m in a wavelength range of 1.2 to 1.7 μm. It has also been shown that the proposed SPCFs show reasonable dispersion tolerance.     

The first samarium(II)-mediated aryl radical cyclisation onto an aromatic ring

Intramolecular arylation of aryl radicals was mediated by SmI(2)/HMPA in the presence of i-PrOH to give spirocycles and/or reduced cine-cyclised products, while the reaction in the absence of i-PrOH gave the rearomatised fused rings.