The Helmholtz equation arises when modeling wave propagation in the frequency domain. The equation is discretized as an indefinite linear system, which is difficult to solve at high wave numbers. In many applications, the solution of the Helmholtz equation is required for a point source. In this case, it is possible to reformulate the equation as two separate equations: one for the travel time ...

We present a formulation of mean-field approximation for layered feed-forward stochastic networks. In this formulation, one can obtain not only estimates of averages for state variables of the networks but also those of intra-layer correlations, the latter of which cannot be obtained by the conventional mean-field approximation. Moreover, this formulation provides a framework to treat “conditio...

A wide variety of different physical systems can be described by a relatively small set of universal equations. For example, small-amplitude nonlinear Schrödinger dark solitons can be described by a Korteweg-de Vries (KdV) equation. Reductive perturbation theory, based on linear boosts and Gallilean transformations, is often employed to establish connections to and between such universal equati...