Characterization of $$H^1$$ H 1 Sobolev Spaces by Square Functions of Marcinkiewicz Type

We establish characterization of \(H^1\) Sobolev spaces by certain square functions of Marcinkiewicz type. The square functions are related to the Lusin area integrals. Also, in the one dimensional case, the non-periodic version of the function of Marcinkiewicz is applied to characterize weighted \(H^1\) Sobolev spaces.

     

Polymerization of 5-Substituted Cyclooctenes with Tungsten and Molybdenum Catalysts

Polymerizations of cyclooctene, 5-methyl, 5-chloro-, and 5-methoxycyclooctenes were studied. Cyclooctene (CO) and 5-methylcyclooctene (MCO) provided high polymers in 80% yield with the use of WCl₆/AlEti._B Cl_u5 or WCl₆/AlEtCl₂ catalyst. 5-Chlorocyclooctene gave oligomer in 50% yield with WCl₆/AlEt₂Cl catalyst. Neither polymer nor oligomer was produced from 5-methoxycyclooctene. These polymers were found to be produced through a ring-opening mechanism. The ratio of cis to trans structure in poly(CO) and poly(MCO) was determined by measurements of the decoupled ′H-NMR spectrum. Poly(CO) containing more than 50% trans structure was a crystalline solid at room temperature, while the polymer containing 30% of trans structure did not crystallize at room temperature. Poly(MCO) was amorphous, regardless of the content of trans structure. Poly(CO) and poly(MCO) obtained with MoCU/AlEtaCl or MoCU/AlEtCb catalyst contained no carbon-carbon double bond, and a vinyl polymerization mechanism was expected for this system.     

Conditions for Shake-Gel Formation: The Relationship between the Size of Poly(Ethylene Oxide) and the Distance between Silica Particles

Colloidal silica suspensions are widely used in many fields, including environmental restoration, oil drilling, and food and medical industries. To control the rheological property of suspensions, poly(ethylene oxide) (PEO) polymers are often used. Under specific conditions, the silica-PEO suspension can create a phenomenon called a shake-gel. Previous works discussed the conditions necessary to form a shake-gel and suggested that the bridging effect of the polymer is one of the important mechanisms for shake-gel formation. However, we noted that the influence of PEO size compared to the separation distance between silica particles regarding shake-gel formation has not been systematically investigated, while the PEO size should be larger than the particle–particle separation distance for polymer bridging in order to form gels. Thus, we conducted a series of experiments to examine the effects of the radius of gyration of the PEO and the distance between the silica particles by controlling the PEO molecular weight and the silica concentration. Our results elucidated that the radius of gyration of the PEO should be 2.5 times larger than the distance between the silica surfaces in order to promote the formation of a shake-gel. This result supports the hypothesis that the bridging effect is the main cause of shake-gel formation, which can help us to understand the conditions necessary for shake-gel preparation.     

Wastes Treatment: Hydrothermal electrolytic oxidation of wastewater

Abstract

Electrochemical evolution of hydrogen gas, oxygen gas and chlorine at the electrodes is the usual reaction in conventional electrolysis of aqueous salt solution. However, here we demonstrate that the electrolysis governing reaction in hydrothermal solutions is different from the electrolysis performed at atmospheric pressure and temperatures up to 100°C. Experimental electrolytic reaction of aqueous salt solution carried out inside a sealed 300-mL batch autoclave showed that; accumulation of hydrogen gas, oxygen gas and chlorine is highly suppressed under hydrothermal (250°C and 7 MPa) conditions. We have also observed that, when organics are present in the aqueous salt solution being hydrothermally electrolyzed, an effective oxidation of organics is accomplished. Furthermore, for hydrothermal electrolytic oxidation (HEO) with oxygen gas added, experimentally observed TOC removal profile demonstrates apparent anodic oxidation electrical current efficiency of almost 200% for highconcentrated acetic acid solutions.     

Frequency dependence of H2O2 generation from distilled water

The frequency dependence of H2O2 generation from H2O by a sonochemical reaction was detected experimentally. The results are in good agreement with previous experimental results, which indicate that in sonochemical reactions, frequencies higher than 90 kHz are more effective than frequencies of several tens of kilohertz. The phonon concept of acoustic waves makes it clear that energy depends on frequency, i.e. on the condition of equal phonon density; higher frequency means higher energy. The concentration and accumulation of acoustic energy will be performed through a bubble surface. From the analogy of photoelectric effects, the frequency dependence of the sonochemical reaction was discussed using the phonon concept.     

Electrons and holes in HgTe and Hg0.82Cd0.18Te with controlled deviations from stoichiometry

The deviations from the stoichiometric composition of HgTe and Hg_0.82Cd_0.18Te crystals have been controlled by heat treatment under Hg vapor pressure. The magnetic field dependence of the Hall coefficient always shows the presence of two different sets of electrons and one set of holes. A low mobility electron is shown to belong to the conduction band. Vapor pressure dependence of hole concentration in HgTe shows that the concentration of nonstoichiometric defects decreases with increasing Hg vapor pressure, but the hole concentration is always higher than the electron concentration. In the case of Hg_0.82Cd_0.18Te, the electron concentration exceeds the hole concentration at high Hg vapor pressures. The dependence of the conduction band electron mobility in HgTe upon carrier density shows that the scattering by holes and impurities is predominant. In Hg_0.82Cd_0.18Te, however, optical phonon scattering is dominant when the deviation from stoichiometry is small and the effect of residual impurities can be neglected, and scattering by holes is dominant when the hole concentration is over $\text{10}{}^{17}\text{cm}{}^3$.     

Transverse mode control by manipulating gain distribution in a Yb:YAG ceramic thin disk

We demonstrated that transverse mode can be controlled by manipulating gain distribution in a Yb:YAG ceramic thin disk. Several elongated higher order transverse modes of a Hermite-Gaussian beam were observed when a straight-line-shaped gain region was created by using a diode laser output from an optical fiber. An orthogonally crossing pair of straight-line-shaped gain regions created with an additional optical fiber output resulted in the generation of a doughnut-shaped Laguerre-Gaussian beam. This implies that easy and quick switching of Hermite-Gaussian and Laguerre-Gaussian beams is possible without mechanical handling.