Liouville type theorem for a class quasilinear pp-Laplace type equation on the sphere

We use the integral by parts to get a Liouville type theorem for a class quasilinear $p$-Laplace type equation on the sphere, this $p$-Laplace type equation arises from the study of asymptotic behavior near the origin for the semi-linear $p$-Laplace equation on the puncture ball $B_1(o) \subset R^n$. This gives a positive answer to L. V\'{e}ron's question in a paper \cite{Veron92} and his book \cite{Veron} at page 440

The Dirichlet problem of the homogeneous kk-Hessian equation in a punctured domain

In this paper, we consider the Dirichlet problem for the homogeneous $k$-Hessian equation with prescribed asymptotic behavior at $0\in\Omega$ where $\Omega$ is a $(k-1)$-convex bounded domain in the Euclidean space. The prescribed asymptotic behavior at $0$ of the solution is zero if $k>\frac{n}{2}$, it is $\log|x|+O(1)$ if $k=\frac{n}{2}$ and $-|x|^{\frac{2k-n}{n}}+O(1)$ if $k<\frac{n}{2}$. To solve this problem, we consider the Dirichlet problem of the approximating $k$-Hessian equation in $\Omega\setminus \overline{B_r(0)}$ with $r$ small. We firstly construct the subsolution of the approximating $k$-Hessian equation. Then we derive the pointwise $C^{2}$-estimates of the approximating equation based on new gradient and second order estimates established previously by the second author and the third author. In addition, we prove a uniform positive lower bound of the gradient if the domain is starshaped with respect to $0$. As an application, we prove an identity along the level set of the approximating solution and obtain a nearly monotonicity formula. In particular, we get a weighted geometric inequality for smoothly and strictly $(k-1)$-convex starshaped closed hypersurface in $\mathbb R^n$ with $\frac{n}{2}\le k<n$.Comment: 33 pages. arXiv admin note: text overlap with arXiv:2207.1350

Understanding the Political Ideology of Legislators from Social Media Images

In this paper, we seek to understand how politicians use images to express ideological rhetoric through Facebook images posted by members of the U.S. House and Senate. In the era of social media, politics has become saturated with imagery, a potent and emotionally salient form of political rhetoric which has been used by politicians and political organizations to influence public sentiment and voting behavior for well over a century. To date, however, little is known about how images are used as political rhetoric. Using deep learning techniques to automatically predict Republican or Democratic party affiliation solely from the Facebook photographs of the members of the 114th U.S. Congress, we demonstrate that predicted class probabilities from our model function as an accurate proxy of the political ideology of images along a left-right (liberal-conservative) dimension. After controlling for the gender and race of politicians, our method achieves an accuracy of 59.28% from single photographs and 82.35% when aggregating scores from multiple photographs (up to 150) of the same person. To better understand image content distinguishing liberal from conservative images, we also perform in-depth content analyses of the photographs. Our findings suggest that conservatives tend to use more images supporting status quo political institutions and hierarchy maintenance, featuring individuals from dominant social groups, and displaying greater happiness than liberals.Comment: To appear in the Proceedings of International AAAI Conference on Web and Social Media (ICWSM 2020

A Convex Optimization Model and Algorithm for Retinex

Retinex is a theory on simulating and explaining how human visual system perceives colors under different illumination conditions. The main contribution of this paper is to put forward a new convex optimization model for Retinex. Different from existing methods, the main idea is to rewrite a multiplicative form such that the illumination variable and the reflection variable are decoupled in spatial domain. The resulting objective function involves three terms including the Tikhonov regularization of the illumination component, the total variation regularization of the reciprocal of the reflection component, and the data-fitting term among the input image, the illumination component, and the reciprocal of the reflection component. We develop an alternating direction method of multipliers (ADMM) to solve the convex optimization model. Numerical experiments demonstrate the advantages of the proposed model which can decompose an image into the illumination and the reflection components

Search for Quasi-Periodical Oscillations in Precursors of Short and Long Gamma Ray Bursts

The precursors of short and long Gamma Ray Bursts (SGRBs and LGRBs) can serve as probes of their progenitors, as well as shedding light on the physical processes of mergers or core-collapse supernovae. Some models predict the possible existence of Quasi-Periodically Oscillations (QPO) in the precursors of SGRBs. Although many previous studies have performed QPO search in the main emission of SGRBs and LGRBs, so far there was no systematic QPO search in their precursors. In this work, we perform a detailed QPO search in the precursors of SGRBs and LGRBs detected by Fermi/GBM from 2008 to 2019 using the power density spectrum (PDS) in frequency domain and Gaussian processes (GP) in time domain. We do not find any convinced QPO signal with significance above 3 $\sigma$, possibly due to the low fluxes of precursors. Finally, the PDS continuum properties of both the precursors and main emissions are also studied for the first time, and no significant difference is found in the distributions of the PDS slope for precursors and main emissions in both SGRBs and LGRBs.Comment: submitte