We describe two simple obstructions to the existence of Ricci-flat Khler cone metrics on isolated Gorenstein singularities or, equivalently, to the existence of Sasaki-Einstein metrics on the links of these singularities. In particular, this also leads to new obstructions for KhlerEinstein metrics on Fano orbifolds. We present several families of hypersurface singularities that are obstructed, including 3-fold and 4-fold singularities of ADE type that have been studied previously in the physics literature. We show that the AdS/CFT dual of one obstruction is that the Rcharge of a gauge invariant chiral primary operator violates the unitarity bound.

We propose an algorithm for the state feedback pole assignment problem. The algorithm is the first of its kind, making use of the Schur form and minimizing the departure from normality of the closed-loop poles by Newtons method. Eleven illustrative examples, comparing our algorithm with three other existing ones, are given.

In this article, we propose a numerical method based on sparse Gaussian processes (SGPs) to solve nonlinear partial differential equations (PDEs). The SGP algorithm is based on a Gaussian process (GP) method, which approximates the solution of a PDE with the maximum a posteriori probability estimator of a GP conditioned on the PDE evaluated at a finite number of sample points. The main bottleneck of the GP method lies in the inversion of a covariance matrix, whose cost grows cubically with respect to the size of samples. To improve the scalability of the GP method while retaining desirable accuracy, we draw inspiration from SGP approximations, where inducing points are introduced to summarize the information of samples. More precisely, our SGP method uses a Gaussian prior associated with a low-rank kernel generated by inducing points randomly selected from samples. In the SGP method, the size of the matrix to be inverted is proportional to the number of inducing points, which is much less than the size of the samples. The numerical experiments show that the SGP method using less than half of the uniform samples as inducing points achieves comparable accuracy to the GP method using the same number of uniform samples, which significantly reduces the computational cost. We give the existence proof for the approximation to the solution of a PDE and provide rigorous error analysis.

We show that the space of negatively curved metrics of a closed negatively curved Riemannian n-manifold, n  10, is highly nonconnected.

We show that a properly immersed minimal hypersurface in M × R+ equals some M×{c} when M is a complete, recurrent ndimensional Riemannian manifold with bounded curvature. If on the other hand,M is not necessarily recurrent but has nonnegative Ricci curvature with curvature bounded below, the same result holds for any positive entire minimal graph over M.