Asymptotic Methods applied to an American Option under a CEV Process

We consider an American put option under the Constant Elasticity of Variance (CEV) process. This corresponds to a free boundary problem for a partial differential equation (PDE). We show that this free boundary satisfies a nonlinear integral equation, and analyze it in the limit of small $\rho$ = $2r/ \sigma^2$, where $r$ is the interest rate and $\sigma$ is the volatility. We find that the free boundary behaves differently for five ranges of time to expiry. We then analyze option price $P(S,t)$, as a function of the asset price $S$ and time to expiry $t$. We obtain the asymptotic expansion of $P$ as $\rho \rightarrow 0$, first via an integral equation formulation, and then using the PDE satisfied by $P$, and analyzing it by perturbation theory and matched asymptotic expansions

Facile Fabrication of Highly Efficient g‑C₃N₄/Ag₂O Heterostructured Photocatalysts with Enhanced Visible-Light Photocatalytic Activity

Highly efficient visible-light-driven g-C₃N₄/Ag₂O heterostructured photocatalysts were prepared by a simple liquid phase synthesis method at room temperature. The composition, structure, morphology, and optical absorption properties of the as-prepared g-C₃N₄/Ag₂O composites were characterized by XRD, FTIR, XPS, TEM, and UV–vis DRS, respectively. We found interestingly that the photogenerated charge carriers separations of the as-prepared g-C₃N₄/Ag₂O composites were closely related to the mass ratio of g-C₃N₄ and Ag₂O. When the mass ratio of g-C₃N₄ and Ag₂O reached 1:4, the as-prepared composite exhibited the highest photocatalytic activity, which was almost 11 and 1.2 times as high as that of individual g-C₃N₄ and Ag₂O, respectively. The enhancement of photocatalytic activity could be attributed to the synergetic effects between g-C₃N₄ and Ag₂O as well as the improved dispersibility and the decreased particle size of Ag₂O. Moreover, the as-prepared composites showed excellent stability toward the photodegradation of methyl orange (MO). Finally, a possible photocatalytic and charge separation mechanism was proposed

Media 3: Electrically actuated liquid iris

Originally published in Optics Letters on 01 March 2015 (ol-40-5-831

Media 1: Electrically actuated liquid iris

Originally published in Optics Letters on 01 March 2015 (ol-40-5-831

Media 2: Electrically actuated liquid iris

Originally published in Optics Letters on 01 March 2015 (ol-40-5-831

Media 2: A polarization converter array using a twisted-azimuthal liquid crystal in cylindrical polymer cavities

Originally published in Optics Express on 01 July 2013 (oe-21-13-16222

Additional file 1 of The serum lipid profiles in immune thrombocytopenia: Mendelian randomization analysis and a retrospective study

Supplementary Material

Media 1: A polarization converter array using a twisted-azimuthal liquid crystal in cylindrical polymer cavities

Originally published in Optics Express on 01 July 2013 (oe-21-13-16222

Media 1: An adaptive liquid lens with a reciprocating movement in a cylindrical hole

Originally published in Optics Express on 15 December 2014 (oe-22-25-31041

Dielectric Liquid Microlens Array with Tunable Focal Length Based on Microdroplet Array Created via Dip-Coating Method

A dielectric liquid microlens array (LMA) with a tunable focal length was fabricated by using a microdroplet array generated through the dip-coating method. The process began with treating the octadecyltrichlorosilane (OTS) layer with selective UV/O3 irradiation for 20 min to establish a hydrophilic–hydrophobic patterning surface. The substrate was subsequently immersed in glycerol and then withdrawn at a constant rate to create a microdroplet array. Upon filling the cell with matching oil (SL5267) and placing it within a square array of a 200 μm diameter glycerol microdroplet array, the LMA was produced. The focal length ranged from approximately −0.96 to −0.3 mm within a voltage range of 0 to 60 Vrms. The glycerol microdroplets, characterized by their shapes, sizes, curvatures, and filling factors, can be precisely controlled by designing an OTS patterning or adjusting the dip-coating speed. This approach offers a rapid and high-throughput method for preparation. Our approach to fabricating tunable LMA offers several advantages, including simplicity of fabrication, uniform structural properties, cost-effectiveness, polarization independence, and excellent optical performance. These focus-tunable LMAs hold significant potential for applications in image processing, 3D displays, medical endoscopy, and military technologies