We study the traveling waves of reaction-diffusion equations for a diffusive SIR model.The existence of traveling waves is determined by the basic reproduction number of the corresponding ordinary differential equations and the minimal wave speed.Our proof is based on Schauder fixed point theorem and Laplace transform.

We study the Poisson-Nernst-Planck (PNP) system with an arbitrary number of ion species with arbitrary valences in the absence of fixed charges. Assuming point charges and that the Debye length is small relative to the domain size, we derive an asymptotic formula for the steady-state solution by matching outer and boundary layer solutions. The case of two ionic species has been extensively studied, the uniqueness of the solution has been proved, and an explicit expression for the solution has been obtained. However, the case of three or more ions has received significantly less attention. Previous work has indicated that the solution may be nonunique and that even obtaining numerical solutions is a difficult task since one must solve complicated systems of nonlinear equations. By adopting a methodology that preserves the symmetries of the PNP system, we show that determining the outer solution effectively reduces to solving a single scalar transcendental equation. Due to the simple form of the transcendental equation, it can be solved numerically in a straightforward manner. Our methodology thus provides a standard procedure for solving the PNP system and we illustrate this by solving some practical examples. Despite the fact that for three ions, previous studies have indicated that multiple solutions may exist, we show that all except for one of these solutions are unphysical and thereby prove the existence and uniqueness for the three-ion case.

The proper use of distributed scatterer (DS) can improve both the density and quality of synthetic aperture radar (SAR) interferometry (InSAR) measurements. A critical step in DS interferometry (DSI) is the restoration of a consistent phase series from SAR interferogram stacks. Most state-of-the-art algorithms adopt an approximate likelihood function to calculate the likelihood by replacing the true coherence matrix with its estimation, more specifically, the sample coherence matrix (SCM). However, this approximation has a drawback in that the coherence estimates are greatly biased when the coherence is low. In this study, we derive a new likelihood function without such an approximation. Accordingly, a DSI framework using this function for phase estimation and point selection is provided. In this framework, the new likelihood function serves as a cost function for phase estimation and a quality measure for DS selection. Its performance is investigated by experiments in a simulation study and a real-world case study using Sentinel-1 data over Shenzhen airport in China. The results reveal that the proposed DSI framework outperforms the existing state-of-the-art approaches in different scenarios, in terms of providing a more accurate estimation and improving DS density and coverage.

A local convergence rate is established for a Gauss orthogonal collocation method applied to optimal control problems with control constraints. If the Hamiltonian possesses a strong convexity property, then the theory yields convergence for problems whose optimal state and costate possess two square integrable derivatives. The convergence theory is based on a stability result for the sup-norm change in the solution of a variational inequality relative to a 2-norm perturbation, and on a Sobolev space bound for the error in interpolation at the Gauss quadrature points and the additional point -1. The tightness of the convergence theory is examined using a numerical example.