Bilinear estimates in Besov spaces generated by the Dirichlet Laplacian

The purpose of this paper is to establish bilinear estimates in Besov spaces generated by the Dirichlet Laplacian on a domain of Euclidian spaces. These estimates are proved by using the gradient estimates for heat semigroup together with the Bony paraproduct formula and the boundedness of spectral multipliers.Comment: 29 page

Besov spaces on open sets

This paper is devoted to giving definitions of Besov spaces on an arbitrary open set of $\mathbb R^n$ via the spectral theorem for the Schr\"odinger operator with the Dirichlet boundary condition. The crucial point is to introduce some test function spaces on $\Omega$. The fundamental properties of Besov spaces are also shown, such as embedding relations and duality, etc. Furthermore, the isomorphism relations are established among the Besov spaces in which regularity of functions is measured by the Dirichlet Laplacian and the Schr\"odinger operators

エコリバブルシティに向けたシビック・エンゲイジメント : 岡山市西川緑道公園のまちづくりを一例に

This article describes the history of civic engagement at Nishigawa Canal Park in Okayama city, Japan. Nishigawa Canal Park was constructed in downtown Okayama from 1974 to 1982, and its total length is 2.4 km. Since 2008, citizens have held events in the park totaling over 70 days per year, with the support of the city’s Nishigawa Performers Project. By analyzing their machizukuri—community development through collaborative actions—we demonstrate the importance of public space in urban development from two points of view: participatory democracy and an eco-livable city. Okayama has rich experiences that show its citizens value a sustainable society. It also faces challenges in urban development carried out through collaboration between public policy makers, political leaders, and citizens

Adsorption of PRG4 on commercially available contact lenses

INTRODUCTION Contact lens wear can cause corneal damage, which destabilizes the tear film that protects the ocular surface and may lead to Dry Eye Syndrome [1]. Contact lens-induced Dry Eye Syndrome causes significant discomfort and affects the quality of life of millions worldwide [2,3]. A possible cause of this discomfort is increased friction between ocular tissues [4]. Proteoglycan 4 (PRG4) is a mucin-like glycoprotein that was originally found in synovial fluid, but has also recently been discovered on the eye. It was shown to function as an effective boundary lubricant for the ocular surfaces [4], specifically between the cornea and eyelid or a contact lens [5]. However, these initial studies were not performed in the presence of tear film proteins, which can accumulate on contact lenses with wear causing discomfort [6]. The objectives of this project were to determine if PRG4 adheres to commercially available contact lenses, and to clarify whether PRG4 is able to maintain its boundary-lubricating ability at a cornea-contact lens biointerface in the presence of tear film proteins. METHODS A western blot was performed to measure adhesion of PRG4 on contact lenses. Samples were prepared by soaking commercial contact lenses Air Optix Aqua (AO), Acuvue Oasys (OAS), Acuvue 2 (Av2) in native bovine PRG4 overnight. The lenses were then rinsed three times in saline to remove excess PRG4, and then heated to 70°C to release PRG4 adhered to the lenses. These samples were loaded onto a gel and immunostained to test for presence of PRG4. The lubricity of commercially available contact lenses OAS and Acuvue TruEye (TE) was analyzed using a custom cornea-contact lens friction test. Lenses were soaked in an artificial tear solution (ATS) or ATS doped with PRG4 (ATS+PRG4), sent from collaborators at the University of Waterloo, to challenge the lenses to a proteinaceous condition. Friction tests measure axial load and torque to calculation friction coefficients for these lubrication conditions. RESULTS Figure 1. Image of a western blot membrane showing PRG4 adhesion of OAS, AO, and Av2 soaked in native PRG4. Western blots showed strong PRG4 adhesion (denoted by the density of the PRG4 bands) on silicone hydrogel contact lenses (OAS and AO), and little adhesion on a conventional hydrogel lens (Av2). AO clearly held onto PRG4 the best (Figure 1). Kinetic friction coefficients were not significantly different for the ATS and ATS+PRG4 conditions. Values of <mkinetic, Neq> were higher for TE than OAS (0.35±0.18 TE; 0.28±0.14 OAS mean±SEM). DISCUSSION AND CONCLUSIONS PRG4 is known to have difficulty sticking to hydrophilic surfaces, such as Av2 conventional hydrogel lenses. Adding silicone to a lens makes it more hydrophobic, allowing PRG4 to better adhere to it [6]. PRG4 had no apparent effect in ATS, possibly due to PRG4 getting bound up in the hydrophobic lipids in ATS before adhering to the lens. Future experiments may examine soaking a contact lens in PRG4 first, before testing in ATS to prevent this, or testing a used contact lens