We consider the convergence of the eigenvalues to the support of the equilibrium measure in the β ensemble model under a critical condition. We show a phase transition phenomenon, namely that all eigenvalues will fall in the support of the limiting spectral measure when β > 1, whereas this always fails when β < 1.
In this paper we show the existence of radial positive stationary solutions to the energy critical nonlinear Schrödinger equation on H 3 by reducing the problem to an ODE. We also make an observation that Kenig-Merle's variational argument in  can work even without the existence of a positive stationary solution, based on this, we sketch a possible… (More)
This note records an attempt to prove the bilinear Strichartz estimate on hyperbolic space. The main difficulty is the lack of a con-volution formula for the Fourier transform of a product of functions. We attempt to develop physical-space methods on Euclidean space to prove the Strichartz estimate, with the hope that such methods can also be effective on… (More)
This paper provides an estimate of the sum of a homogeneous polynomial P over the lattice points inside a sphere of radius R. The polynomial P is assumed to be of degree ν and have zero mean over the sphere. It is proved that x∈Z 3 |x|≤R
Based on a survey of peasants who lived in rural community in Wuhan City Hubei Province in China, this paper examines the peasants’ desire for social insurance in rural areas using the multinomial logistic model. The study finds that the peasants’ desire for social insurance in rural areas is remarkably affected by individual and family… (More)