We use potential analysis to study the properties of positive solutions of a discrete Wolff-type equation

$$ \begin{align*} w(i)=W_{\beta,\gamma}(w^q)(i), \quad i \in \mathbb{Z}^n. \end{align*} $$

Here, $n \geq 1$, $\min \{q,\beta \}>0$, $1<\gamma \leq 2$ and $\beta \gamma <n$. Such an equation can be used to study nonlinear problems on graphs appearing in the study of crystal lattices, neural networks and other discrete models. We use the method of regularity lifting to obtain an optimal summability of positive solutions of the equation. From this result, we obtain the decay rate of $w(i)$ when $|i| \to \infty $.

The laboratory generation and diagnosis of uniform near-critical-density (NCD) plasmas play critical roles in various studies and applications, such as fusion science, high energy density physics, astrophysics as well as relativistic electron beam generation. Here we successfully generated the quasistatic NCD plasma sample by heating a low-density tri-cellulose acetate (TCA) foam with the high-power-laser-driven hohlraum radiation. The temperature of the hohlraum is determined to be 20 eV by analyzing the spectra obtained with the transmission grating spectrometer. The single-order diffraction grating was employed to eliminate the high-order disturbance. The temperature of the heated foam is determined to be T = 16.8 ± 1.1 eV by analyzing the high-resolution spectra obtained with a flat-field grating spectrometer. The electron density of the heated foam is about $N_e=4.0\pm 0.3\times {10}^{20}{\text{cm}}^{-3}$ under the reasonable assumption of constant mass density.

In this paper, we are concerned with the non-existence of positive solutions of a Hartree–Poisson system:

\begin{equation*}\left\{\begin{aligned}&-\Delta u=\left(\frac{1}{|x|^{n-2}}\ast v^p\right)v^{p-1},\quad u \gt 0\ \text{in} \ \mathbb{R}^{n},\\&-\Delta v=\left(\frac{1}{|x|^{n-2}}\ast u^q\right)u^{q-1},\quad v \gt 0\ \text{in} \ \mathbb{R}^{n},\end{aligned}\right.\end{equation*}

$n \geq3$

$\min\{p,q\} \gt 1$

This paper is devoted to the American option pricing problem governed by the Black-Scholes equation. The existence of an optimal exercise policy makes the problem a free boundary value problem of a parabolic equation on an unbounded domain. The optimal exercise boundary satisfies a nonlinear Volterra integral equation and is solved by a high-order collocation method based on graded meshes. This free boundary is then deformed to a fixed boundary by the front-fixing transformation. The boundary condition at infinity (due to the fact that the underlying asset's price could be arbitrarily large in theory), is treated by the perfectly matched layer technique. Finally, the resulting initial-boundary value problems for the option price and some of the Greeks on a bounded rectangular space-time domain are solved by a finite element method. In particular, for Delta, one of the Greeks, we propose a discontinuous Galerkin method to treat the discontinuity in its initial condition. Convergence results for these two methods are analyzed and several numerical simulations are provided to verify these theoretical results.

We have studied the growth of GaN on (0001) sapphire and (111) spinel substrates by LP-MOVPE and compared the mosaic structure and cathodoluminescence for the heteroepitaxial films of GaN grown on these substrates.