The nearly analytic discrete (NAD) method is a kind of finite difference method with advantages of high accuracy and stability. Previous studies have investigated the NAD method for simulating wave propagation in the time-domain. This study applies the NAD method to solving three-dimensional (3D) acoustic wave equations in the frequency-domain. This forward modeling approach is then used as the “engine” for implementing 3D frequency-domain full waveform inversion (FWI). In the numerical modeling experiments, synthetic examples are first given to show the superiority of the NAD method in forward modeling compared with traditional finite difference methods. Synthetic 3D frequency-domain FWI experiments are then carried out to examine the effectiveness of the proposed methods. The inversion results show that the NAD method is more suitable than traditional methods, in terms of computational cost and stability, for 3D frequency-domain FWI, and represents an effective approach for inversion of subsurface model structures.

Two-dimensional localized edge modes in optical honeycomb lattices are found and analyzed analytically and computationally. Weak nonlinearity and transverse modulation are found to introduce self-phase modulation in the phase and create internal nonlinear interactions as the electromagnetic field propagates through the lattice. Even with relatively strong nonlinearity localization and persistence of modes along the edge are found.

No abstract uploaded!

No abstract uploaded!

We introduce an irrational factor of order <i>k</i> defined by I k ( n ) = i = 1 l p i i , where I k ( n ) = i = 1 l p i i is the factorization of <i>n</i> and $${\beta_{i} = \left\{\begin{array}{ll}\alpha_i, \quad \quad {\rm if} \quad \alpha_i &lt; k \\ \frac{1}{\alpha_i},\quad \quad {\rm if} \quad \alpha_i \geqq k \end{array}\right.}$$. It turns out that the function I k ( n ) = i = 1 l p i i well approximates the characteristic function of <i>k</i>-free integers. We also derive asymptotic formulas for I k ( n ) = i = 1 l p i i and I k ( n ) = i = 1 l p i i .