Some Blow-Up Problems for a Semilinear Parabolic Equation with a Potential

The blow-up rate estimate for the solution to a semilinear parabolic equation $u_t=\Delta u+V(x) |u|^{p-1}u$ in $\Omega \times (0,T)$ with 0-Dirichlet boundary condition is obtained. As an application, it is shown that the asymptotic behavior of blow-up time and blow-up set of the problem with nonnegative initial data u(x,0)=M\vf (x) as $M$ goes to infinity, which have been found in \cite{cer}, are improved under some reasonable and weaker conditions compared with \cite{cer}.Comment: 29 page

Cherry-Picking in Labor Market with Imperfect Information

We study a competitive labor market with imperfect information. In our basic model, the labor market consists of heterogeneous workers and ex ante identical firms who have only imperfect private information about workers' productivities. Firms compete by posting wages in order to cherry-pick more productive workers from the applicant pool. The model predicts many important empirical regularities, including non-degenerated firm size distribution, persistent wage dispersion, and employer size-wage premium. We also consider extensions of the model where firms differ in either productivity or information about worker types, both generating assortative matching with a positive but imperfect correlation of worker and firm types. The main insight of this paper is that identical workers can get different wages depending on productivities of their coworkers in a competitive market with informational frictions. Our model also sheds light on inter-industry wage differential and sorting between industry and worker characteristics.imperfect information, cherry-picking, wage dispersion, size-wage premium, inter-industry wage differential

Glass transitions in two-dimensional suspensions of colloidal ellipsoids

We observed a two-step glass transition in monolayers of colloidal ellipsoids by video microscopy. The glass transition in the rotational degree of freedom was at a lower density than that in the translational degree of freedom. Between the two transitions, ellipsoids formed an orientational glass. Approaching the respective glass transitions, the rotational and translational fastest-moving particles in the supercooled liquid moved cooperatively and formed clusters with power-law size distributions. The mean cluster sizes diverge in power law as approaching the glass transitions. The clusters of translational and rotational fastest-moving ellipsoids formed mainly within pseudo-nematic domains, and around the domain boundaries, respectively

Rainbow gravity corrections to the entropic force

The entropic force attracts a lot of interest for its multifunctional properties. For instance, Einstein's field equation, Newton's law of gravitation and the Friedmann equation can be derived from the entropic force. In this paper, utilizing a new kind of rainbow gravity model that was proposed by Magueijo and Smolin, we explore the quantum gravity corrections to the entropic force. First, we derive the modified thermodynamics of a rainbow black hole via its surface gravity. Then, according to Verlinde's theory, the quantum corrections to the entropic force are obtained. The result shows that the modified entropic force is related not only to the properties of the black hole but also the Planck length $\ell_p$, and the rainbow parameter $\gamma$. Furthermore, based on the rainbow gravity corrected entropic force, the modified Einstein's field equation and the modified Friedmann equation are also derived.Comment: 10 page