No abstract uploaded!

Let$M$be a connected, noncompact, complete Riemannian manifold, consider the operator$L=Δ+∇V$for some$V∈C$^{2}(M) with exp[$V$] integrable with respect to the Riemannian volume element. This paper studies the existence of the spectral gap of$L$. As a consequence of the main result, let ϱ be the distance function from a point o, then the spectral gap exists provided lim_{ϱ→∞}sup$L$_{ϱ<0}while the spectral gap does not exist if o is a pole and lim_{ϱ→∞}inf$L$_{ϱ≥0}. Moreover, the elliptic operators on$R$^{$d$}are also studied.

In this paper, we consider the time-dependent Maxwell’s equations when Cole–Cole dispersive medium is involved. The Cole–Cole model contains a fractional time derivative term, which couples with the standard Maxwell’s equations in free space and creates some challenges in developing and analyzing time-domain finite element methods for solving this model as mentioned in our earlier work [J. Li, J. Sci. Comput., 47 (2001), pp. 1–26]. By adopting some techniques developed for the fractional diffusion equations [V.J. Ervin, N. Heuer, and J.P. Roop, SIAM J. Numer. Anal., 45 (2007), pp. 572–591], [Y. Lin and C. Xu, J. Comput. Phys., 225 (2007), pp. 1533–1552], [F. Liu, P. Zhuang, V. Anh, I. Turner, and K. Burrage, Appl. Math. Comput., 191 (2007), pp. 12–20], we propose two fully discrete mixed finite element schemes for the Cole–Cole model. Numerical stability and optimal error estimates are proved for both schemes. The proposed algorithms are implemented and detailed numerical results are provided to justify our theoretical analysis.

The universal Teichmu¨ller space is an infinitely dimensional generalization of the classical Teichmu¨ller space of Riemann surfaces. It carries a natural Hilbert structure, on which one can define a natural Riemannian metric, the Weil-Petersson metric. In this paper we investigate the Weil-Petersson Riemannian curvature operator ˜ Q of the universal Teichmu¨ller space with the Hilbert structure, and prove the following: (i) ˜ Q is non-positive definite. (ii) ˜ Q is a bounded operator. (iii) ˜ Q is not compact; the set of the spectra of ˜ Q is not discrete. As an application, we show that neither the Quaternionic hyperbolic space nor the Cayley plane can be totally geodesically immersed in the universal Teichmu¨ller space endowed with the Weil-Petersson metric.

In this paper, we study the existence of local and global saddle points for nonlinear second-order cone programming problems. The existence of local saddle points is developed by using the second-order sufficient conditions, in which a sigma-term is added to reflect the curvature of second-order cone. Furthermore, by dealing with the perturbation of the primal problem, we establish the existence of global saddle points, which can be applicable for the case of multiple optimal solutions. The close relationship between global saddle points and exact penalty representations are discussed as well.