We adopt a recent work in Chung, Qian, Uhlmann and Zhao (Inverse Problems, 23 (2007) 309-329) to develop a phase space method for reconstructing pressure wave speed and shear wave speed of an elastic medium from travel time measurements. The method is based on the so-called Stefanov-Uhlmann identity which links two Riemannian metrics with their travel time information. We design a numerical algorithm to solve the resulting inverse problem. The algorithm is a hybrid approach that combines both Lagrangian and Eulerian formulations. In particular the Lagrangian formulation in phase space can take into account multiple arrival times naturally, while the Eulerian formulation for wave speeds allows us to compute the solution in physical space. Numerical examples are shown to validate the method.

In this short note, using Siu-Yau's method [14], we give a new proof that any n-dimensional compact K ahler manifold with positive orthogonal bisectional curvature must be biholomorphic to \mathbb {P}^ n .

G-equations are well-known front propagation models in combustion and are HamiltonJacobi type equations with convex but non-coercive Hamiltonians. Viscous G-equations arise from numerical discretization or modeling dissipative mechanisms. Although viscosity helps to overcome non-coercivity, we prove homogenization of an inviscid G-equation based on approximate correctors and attainability of controlled flow trajectories. We verify the attainability for two-dimensional mean zero incompressible flows, and demonstrate asymptotically and numerically that viscosity reduces the homogenized Hamiltonian in cellular flows. In the case of one-dimensional compressible flows, we found an explicit formula of homogenized Hamiltonians, as well as necessary and sufficient conditions for wave trapping (effective Hamiltonian vanishes identically). Viscosity restores coercivity and wave propagation.

We propose a holographic entanglement entropy prescription for general states and regions in two models of holography beyond AdS/CFT known as flat3/BMSFT and (W)AdS3/WCFT. Flat3/BMSFT is a candidate of holography for asymptotically flat three-dimensional spacetimes, while (W)AdS3/WCFT is relevant in the study of black holes in the real world. In particular, the boundary theories are examples of quantum field theories that feature an infinite dimensional symmetry group but break Lorentz invariance. Our holographic entanglement entropy proposal is given by the area of a swing surface that consists of ropes, which are null geodesics emanating from the entangling surface at the boundary, and a bench, which is a spacelike geodesic connecting the ropes. The proposal is supported by an extension of the Lewkowycz-Maldacena argument, reproduces previous results based on the Rindler method, and satisfies the first law of entanglement entropy.

As a discrete-time quantum walk model on the one-dimensional integer lattice Z, the quantum walk recently constructed byWang and Ye [CaishiWang and Xiaojuan Ye, Quantum walk in terms of quantum Bernoulli noises, Quantum Information Processing 15 (2016), 1897–1908] exhibits quite different features. In this paper, we extend this walk to a higher dimensional case. More precisely, for a general positive integer d ≥ 2, by using quantum Bernoulli noises we introduce a model of discrete-time quantum walk on the d-dimensional integer lattice Z^d, which we call the d-dimensional QBN walk. The d-dimensional QBN walk shares the same coin space with the quantum walk constructed by Wang and Ye, although it is a higher dimensional extension of the latter. Moreover we prove that, for a range of choices of its initial state, the d-dimensional QBN walk has a limit probability distribution of d-dimensional standard Gauss type, which is in sharp contrast with the case of the usual higher dimensional quantum walks. Some other results are also obtained.