A general iterative method for hierarchical variational inequality problems in Hilbert spaces and applications

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Let ?? >?0 and let A be an ??-inverse-strongly monotone mapping of C into H and let B be a maximal monotone operator on H. Let F be a maximal monotone operator on H such that the domain of F is included in C. Let 0?< k?<?1 and let g be a k-contraction of H into itself. Let V be a ${\overline{\gamma}}$ -strongly monotone and L-Lipschitzian continuous operator with ${\overline{\gamma} >0 }$ and L >?0. Take ${\mu, \gamma \in \mathbb R}$ as follows: $${0 < \mu < \frac{2\overline{\gamma}}{L^2}, \quad 0 < \gamma < \frac{\overline{\gamma}-\frac{L^2 \mu}{2}}{k}.}$$ In this paper, under the assumption ${(A+B)^{-1}0 \cap F^{-1}0 \neq \emptyset}$ , we prove a strong convergence theorem for finding a point ${z_0\in (A+B)^{-1}0\cap F^{-1}0}$ which is a unique solution of the hierarchical variational inequality $${\langle (V-\gamma g)z_0, q-z_0 \rangle \geq 0, \quad \forall q\in (A+B)^{-1}0 \cap F^{-1}0.}$$ Using this result, we obtain new and well-known strong convergence theorems in a Hilbert space which are useful in nonlinear analysis and optimization.     

A dendron of subphthalocyanine trefoil

The synthesis of a dendron composed of tetrameric subphthalocyanine (SubPc) is accomplished by substituting the chlorine groups with phenoxy groups at the axial positions of SubPc with SubPc-triol. The present molecular design of the SubPc-triol introduces three phenol groups at the peripheral positions of the SubPc macrocycle as a tritopic template to construct SubPc dendrons. The self-polycondensation of SubPc-triol as a ‘divergent’ synthesis only gave a trace amount of the hyperbranched arrays due to poor solubility of the SubPc-triol. In contrast, a ‘convergent’ synthesis with the terminal SubPc improved the solubility throughout the reaction and a tetrameric SubPc dendron was obtained in moderate isolated yield.     

Measurement of refractive index change induced by dark reaction of photopolymer with digital holographic quantitative phase microscopy

Light-induced refractive index change in photopolymer is quantified by a digital holographic microscope. The refractive index change is induced as the dark reaction which proceeds inside the photopolymer after a writing beam is stopped. Time-lapse phase distribution across the photopolymer is measured by the off-axis digital holographic microscope which enables us to retrieve the 2-D phase map from a single hologram. It is found that the initial phase profile does not coincide with the illumination intensity distribution. This observation suggests that the rate of the refractive index change in dark reaction is not proportional to the illumination intensity in case the exposure energy becomes high.     

In-beam M?ssbauer spectroscopy of 57 Mn implanted into lithium hydride

We measured the temperature dependence of ⁵⁷Fe M?ssbauer spectra obtained after ⁵⁷Mn implantation into polycrystalline LiH with an extremely low implantation dose. Density functional calculations suggested that the Fe atoms were predominantly implanted into both Li and H substitutional sites of the LiH crystal.     

New method to characterize mesoscopic range and very small strain with using multi-wave X-ray diffraction

A new technique to observe mesoscopic-range strain fields (up to several hundreds of nm) is proposed, using modulation of the crystal-truncation-rod (CTR) scattering caused by Bragg reflection. This technique is particularly sensitive to small, long-range strain fields near crystal surfaces and interfaces, which are usually difficult to be discriminated by using Bragg reflection. A simple interpretation can be made for the modulation profile: the technique is physically simple with a few parameters fitted to the data and, independently of any model, is able to determine the total displacement due to mesoscopic strain field for depths up to several hundreds of nanometers. We applied this method to a Si(0 0 1) wafer whose surface is covered with a thermal oxide layer 3 nm thick. On the basis of the expressions we obtained for the modulation profile a least-squares fitting was carried out to give a result that under the oxide layer there exists a total displacement of −0.16 Å. It was also revealed from the visibility of the modulation profile that the total displacement has a static fluctuation of at least ±0.13 Å in the lateral direction. The new method can be used for the correction of the errors of the X-ray standing wave (XSW) method produced by strained layers near crystal surfaces.     

Kinetics of deactivation of Pd/C catalyst repeatedly used in the liquid-phase hydrodechlorination of PCE

The deactivation kinetics of the Pd/C catalyst used in the catalytic hydrodechlorination of PCE in either MeOH or 50/50 water-MeOH mixture with catalyst and solvent reuse was investigated. Empirical power-expressions were used to approximate the experimental data.