Spectral Gap for Stable Process on Convex Planar Double Symmetric Domains

We study the semigroup of the symmetric α-stable process in bounded domains in R ². We obtain a variational formula for the spectral gap, i.e. the difference between two first eigenvalues of the generator of this semigroup. This variational formula allows us to obtain lower bound estimates of the spectral gap for convex planar domains which are symmetric with respect to both coordinate axes. For rectangles, using “midconcavity” of the first eigenfunction (Bañuelos et al., Potential Anal. 24(3): 205–221, 2006), we obtain sharp upper and lower bound estimates of the spectral gap.     

Albumin adsorption on unmodified and sulfonated polystyrene surfaces, in relation to cell-substratum adhesion

Albumin is commonly applied for blocking the adsorption of other proteins and to prevent the nonspecific adhesion of cells to diverse artificial substrata. Here we address the question of how effective these albumin properties are--by investigating unmodified and sulfonated polystyrene substrata with distinctly different wettabilities. As clearly shown with (125)I-radioisotopic assays, above a concentration of 10-20 μg/mL, the efficiency of bovine serum albumin (BSA) adsorption became markedly higher on the sulfonated surface than on the unmodified one. This study was assisted with the atomic force microscopy. On the unmodified surface, BSA, adsorbed from sufficiently concentrated solutions, formed a monolayer, with occasional intrusions of multilayered patches. Conversely, the arrangement of BSA on the sulfonated surface was chaotic; the height of individual molecules was lower than on the unmodified polystyrene. Importantly, the adhesion study of LNCaP and DU145 cells indicated that both surfaces, subjected to the prior BSA adsorption, did not completely loose their cell-adhesive properties. However, the level of adhesion and the pattern of F-actin organization in adhering cells have shown that cells interacted with unmodified and sulfonated surfaces differently, depending on the arrangement of adsorbed albumin. These results suggest the presence of some bare substratum area accessible for cells after the albumin adsorption to both types of investigated surfaces.     

Muckenhoupt Ap-properties of Distance Functions and Applications to Hardy–Sobolev -type Inequalities

Potential Analysis - Let X be a metric space equipped with a doubling measure. We consider weights w(x) = dist(x,E)?α, where E is a closed set in X and $\alpha \in \mathbb {R}$ . We...