We study the activated random walk model on the one-dimensional ring, in the high-density regime. We develop a toppling procedure that gradually builds an environment that can be used to show that activity will be sustained for a long time. This yields a self-contained and relatively short proof of existence of a slow phase for arbitrarily large sleep rates.

We compute log canonical thresholds of reduced plane curves of degree $d$ at points of multiplicity $d-1$. As a consequence, we describe all possible values of log canonical threshold that are less than $2/(d-1)$ for reduced plane curves of degree $d$. In addition, we compute log canonical thresholds for all reduced plane curves of degree less than 6.

Within the framework of small-amplitude water wave theory, the scattering of obliquely incident monochromatic surface waves by dual thick vertical barriers over an arbitrary bottom topography is analysed. A numerical model based on the finite element method is developed by formulating the governing well-posed mixed boundary value problem over each element within a truncated finite domain. This domain is obtained by limiting the originally infinite domain to a finite distance. Two types of bottom profiles, namely parabolic and rectangular, are considered for the numerical analysis. To ensure the accuracy of the present numerical results, an energy identity relation is derived using Green’s identity and verified numerically. Additionally, for validation purposes, the numerical results are compared with existing results available in the literature. The number of zeros on the reflection and transmission coefficient curves is investigated with respect to the gap between identical and nonidentical barriers. The effects of various physical parameters, including the gap between the vertical barriers, the height of the bottom topography, the thickness and length of the barriers, and the angle of wave incidence on the reflection and transmission coefficients, as well as on the nondimensional horizontal force acting on the front and rear barriers, are examined using the proposed numerical model. This study contributes to understanding of wave–structure interaction and will be useful in addressing similar problems in applied mathematics and fluid mechanics.

Use $L^r$ to $L^s$ bounds to prove sparse form bounds for pseudodifferential operators with Hörmander symbols in $S^m_{\rho,\delta}$ up to, but not including, the sharp end-point in decay $m$. We further develop their technique, obtaining pointwise sparse bounds for rough pseudodifferential operators that are merely measurable in their spatial variables and an alternative proof of their results, which avoids proving geometrically decaying sparse bounds. We also provide sufficient conditions for sparse form bounds to hold and use these to reprove known sparse bounds for pseudodifferential operators with symbols in $S^0_{1,\delta}$ for $\delta \lt 1$.