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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 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*}
where $n \geq3$ and $\min\{p,q\} \gt 1$. We prove that the system has no positive solution under a Serrin-type condition. In addition, the system has no positive radial classical solution in a Sobolev-type subcritical case. In addition, the system has no positive solution with some integrability in this Sobolev-type subcritical case. Finally, the relation between a Liouville theorem and the estimate of boundary blowing-up rates is given.
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.
To determine the prevalence of high weight at different characteristics, understand the perceptions and behaviours towards high body weight, and determine potential influencing factors of body weight misperception among high-weight adults in Jilin Province.
Design
A cross-sectional survey with complex sampling design was conducted. We described the prevalence and perception of high body weight.
Setting
Northeast China in 2012.
Subjects
Adults (n 20 552) aged 18–79 years.
Results
Of overweight individuals, 37·4 % considered themselves as ‘normal weight’, 4·8 % reported themselves as being ‘very thin’ and only 53·1 % were aware of their own weight being ‘overweight’. About 1·8 % of both male and female obese individuals perceived themselves as ‘very thin’. Only 29·1 % of obese people thought of themselves as ‘too fat’. Nearly 30·0 % of centrally obese men and women perceived that their waist circumference was about right and they were of ‘normal weight’; 5·7 % of the centrally obese even perceived themselves as being ‘very thin’. Only 51·8 and 12·5 % of centrally obese individuals reported themselves to be ‘overweight’ or ‘too fat’. Body weight misperception was more common in rural residents (OR; 95 % CI: 1·340; 1·191, 1·509). The prevalence of body weight misperception increased with age (middle age: 1·826; 1·605, 2·078; old people: 3·101; 2·648, 3·632) and declined with increased education level (junior middle school: 0·628; 0·545, 0·723; senior middle school: 0·498; 0·426, 0·583; undergraduate and above: 0·395; 0·320, 0·487).
Conclusions
Body weight misperception was common among adults from Jilin Province.
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.
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