In this paper, we prove index theorems for integrable metrized line bundles on projective varieties over complete fields and number fields respectively. As applications, we prove a non-archimedean analogue of the Calabi theorem and a rigidity theorem about the preperiodic points of algebraic dynamical systems.

In this paper, we present a staggered discontinuous Galerkin immersed boundary method (SDGIBM) for the numerical approximation of fluid-structure interaction. The immersed boundary method is used to model the fluid-structure interaction, while the fluid flow is governed by incompressible Navier-Stokes equations. One advantage of using Galerkin method over the finite difference method with immersed boundary method is that we can avoid approximations of the Dirac Delta function. Another key ingredient of our method is that our solver for incompressible Navier-Stokes equations combines the advantages of discontinuous Galerkin methods and staggered meshes, and results in many good properties, namely local and global conservations and pointwise divergence-free velocity field by a local postprocessing technique. Furthermore, energy stability is improved by a skew-symmetric discretization of the convection term. We will present numerical results to show the performance of the method.

We propose the backward phase flow method to implement the FourierBrosIagolnitzer (FBI)-transform-based Eulerian Gaussian beam method for solving the Schrdinger equation in the semi-classical regime. The idea of Eulerian Gaussian beams has been first proposed in [12]. In this paper we aim at two crucial computational issues of the Eulerian Gaussian beam method: how to carry out long-time beam propagation and how to compute beam ingredients rapidly in phase space. By virtue of the FBI transform, we address the first issue by introducing the reinitialization strategy into the Eulerian Gaussian beam framework. Essentially we reinitialize beam propagation by applying the FBI transform to wavefields at intermediate time steps when the beams become too wide. To address the second issue, inspired by the original phase flow method, we propose the backward phase flow method which allows us to

In this paper, we will study the (linear) geometric analysis on metric measure spaces. We will establish a local Li-Yau’s estimate for weak solutions of the heat equation and prove a sharp Yau’s gradient gradient for harmonic functions on metric measure spaces, under the Riemannian curvature-dimension condition RCD(K; N).

We count the number$S$($x$) of quadruples $ {\left( {x_{1} ,x_{2} ,x_{3} ,x_{4} } \right)} \in \mathbb{Z}^{4} $ for which $$ p = x^{2}_{1} + x^{2}_{2} + x^{2}_{3} + x^{2}_{4} \leqslant x $$ is a prime number and satisfying the determinant condition:$x$_{1}$x$_{4}−$x$_{2}$x$_{3}= 1. By means of the sieve, one shows easily the upper bound$S$($x$) ≪$x$/log$x$. Under a hypothesis about prime numbers, which is stronger than the Bombieri–Vinogradov theorem but is weaker than the Elliott–Halberstam conjecture, we prove that this order is correct, that is$S$($x$) ≫$x$/log$x$.