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Given a smooth minimal surface F: Ω → ℝ3 defined on a simply connected region Ω in the complex plane ℂ, there is a regular SG circle pattern . By the Weierstrass representation of F and the existence theorem of SG circle patterns, there exists an associated SG circle pattern in ℂ with the combinatoric of . Based on the relationship between the circle pattern and the corresponding discrete minimal surface F∊: → ℝ3 defined on the vertex set of the graph of , we show that there exists a family of discrete minimal surface Γ∊: → ℝ3, which converges in C∞(Ω) to the minimal surface F: Ω → ℝ3 as ∊ → 0.
An initial boundary value problem of Riemann type is solved for the nonlinear pseudoparabolic equation with two space variables
The complex function H is measurable on ℂ ×I × ℂ5, with I being an interval of the real line ℝ, Lipschitz continuous with respect to the last five variables, with the Lipschitz constant for the last variable being strictly less than one (ellipticity condition). No smallness assumption is needed in the argument.
Blue photoluminescent emission was observed in pure nanometer-sized γ–Al2O3 powders prepared by the sol-gel process, with aluminum alkoxide as the precursor. The photoluminescent excitation spectrum detected at »em = 422 nm showed four peaks located at 238, 255, 278.5, and 348.5 nm, respectively, the first having the strongest intensity. The photoluminescent emission spectra were made up of a broad band with four peaks located at 404.5, 422, 447, and 484.5 nm. The emission band of 422 nm had the intensity. We suggest that the defect level in the nanometer alumina powder also is the main reason for the appearance of new luminescent emission bands.
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