Asymptotics for Two-dimensional Atoms

We prove that the ground state energy of an atom confined to two dimensions with an infinitely heavy nucleus of charge $Z>0$ and $N$ quantum electrons of charge -1 is E(N,Z)=-{1/2}Z^2\ln Z+(E^{\TF}(\lambda)+{1/2}c^{\rm H})Z^2+o(Z^2) when $Z\to \infty$ and $N/Z\to \lambda$, where E^{\TF}(\lambda) is given by a Thomas-Fermi type variational problem and $c^{\rm H}\approx -2.2339$ is an explicit constant. We also show that the radius of a two-dimensional neutral atom is unbounded when $Z\to \infty$, which is contrary to the expected behavior of three-dimensional atoms.Comment: Revised version to appear in Ann. Henri Poincar\'

Amino-functionalized CoFe2O4 magnetic nanoparticles as an efficient heterogeneous catalyst for benzaldehyde cyanosilylation

Diaminosilane-functionalized cobalt spinel ferrite (CoFe2O4) magnetic nanoparticles were synthesized and used as an efficient heterogeneous base catalyst for the cyanosilylation reaction benzaldehyde with trimethylsilyl cyanide. The magnetic nanoparticle catalyst was characterized by X-ray powder diffraction (XRD), transmission electron microscope (TEM), thermogravimetric analysis (TGA), fourier transform infrared (FT-IR), nitrogen physisorption measurements. Quantitative conversion (99%) was achieved under mild conditions. Recovery of catalyst was facilely achieved by magnetic decantation. The supported catalyst could be reused without significant degradation in catalytic activity

Determine the source term of a two-dimensional heat equation

Let $\Omega$ be a two-dimensional heat conduction body. We consider the problem of determining the heat source $F(x,t)=\varphi(t)f(x,y)$ with $\varphi$ be given inexactly and $f$ be unknown. The problem is nonlinear and ill-posed. By a specific form of Fourier transforms, we shall show that the heat source is determined uniquely by the minimum boundary condition and the temperature distribution in $\Omega$ at the initial time $t=0$ and at the final time $t=1$. Using the methods of Tikhonov's regularization and truncated integration, we construct the regularized solutions. Numerical part is given.Comment: 18 page

On the effective quasi-bosonic Hamiltonian of the electron gas: collective excitations and plasmon modes

We consider an effective quasi-bosonic Hamiltonian of the electron gas which emerges naturally from the random phase approximation and describes the collective excitations of the gas. By a rigorous argument, we explain how the plasmon modes can be interpreted as a special class of approximate eigenstates of this model

Determination of the body force of a two-dimensional isotropic elastic body

Let $\Omega$ represent a two$-$dimensional isotropic elastic body. We consider the problem of determining the body force $F$ whose form $\phi(t)(f_1(x),f_2(x))$ with $\phi$ be given inexactly. The problem is nonlinear and ill-posed. Using the Fourier transform, the methods of Tikhonov's regularization and truncated integration, we construct a regularized solution from the data given inexactly and derive the explicitly error estimate. Numerical part is givenComment: 23 page