Regularity of degenerate Hessian equations

We show a second order a priori estimate for solutions to the complex k-Hessian equation on a compact Kähler manifold provided the (k−1)st root of the right hand side is C1,1. This improves an estimate of Hou-Ma-Wu (Math Res Lett 17:547-561, 2010). An example is provided to show that the exponent is sharp

Carleman estimates for elliptic operators with jumps at an interface: Anisotropic case and sharp geometric conditions

We consider a second-order selfadjoint elliptic operator with an anisotropic diffusion matrix having a jump across a smooth hypersurface. We prove the existence of a weight-function such that a Carleman estimate holds true. We moreover prove that the conditions imposed on the weight function are necessary.Comment: 43 page

Introducing Structure to Expedite Quantum Search

We present a novel quantum algorithm for solving the unstructured search problem with one marked element. Our algorithm allows generating quantum circuits that use asymptotically fewer additional quantum gates than the famous Grover's algorithm and may be successfully executed on NISQ devices. We prove that our algorithm is optimal in the total number of elementary gates up to a multiplicative constant. As many NP-hard problems are not in fact unstructured, we also describe the \emph{partial uncompute} technique which exploits the oracle structure and allows a significant reduction in the number of elementary gates required to find the solution. Combining these results allows us to use asymptotically smaller number of elementary gates than the Grover's algorithm in various applications, keeping the number of queries to the oracle essentially the same. We show how the results can be applied to solve hard combinatorial problems, for example Unique k-SAT. Additionally, we show how to asymptotically reduce the number of elementary gates required to solve the unstructured search problem with multiple marked elements.Comment: 22 pages, 7 figure