Regularity from pp-harmonic potentials to ∞\infty-harmonic potentials in convex rings

The exploration of shape metamorphism, surface reconstruction, and image interpolation raises fundamental inquiries concerning the $C^1$ and higher-order regularity of $\infty$-harmonic potentials -- a specialized category of $\infty$-harmonic functions. Additionally, it prompts questions regarding their corresponding approximations using $p$-harmonic potentials. It is worth noting that establishing $C^1$ and higher-order regularity for $\infty$-harmonic functions remains a central concern within the realm of $\infty$-Laplace equations and $L^\infty$-variational problems. In this study, we investigate the regularity properties from $p$-harmonic potentials to $\infty$-harmonic potentials within arbitrary convex ring domains $\Omega=\Omega_0\backslash \overline \Omega_1$ in $\mathbb R^n$. Here $\Omega_0$ is a bounded convex domain in $\mathbb R^n$ and $\overline\Omega_1\subset \Omega_0$ is a compact convex set. We prove the interior $C^1$ and some Sobolev regularity for $\infty$-harmonic potentials.Comment: 46 Page

Jacobian determinants for (nonlinear) gradient of planar ∞\infty-harmonic functions and applications

In dimension 2, we introduce a distributional Jacobian determinant $\det DV_\beta(Dv)$ for the nonlinear complex gradient $(x_1,x_2)\mapsto |Dv|^\beta(v_{x_1},-v_{x_2})$ for any $\beta>-1$, whenever $v\in W^{1,2 }_{\text{loc}}$ and $\beta |Dv|^{1+\beta}\in W^{1,2}_{\text{loc}}$. Then for any planar $\infty$-harmonic function $u$, we show that such distributional Jacobian determinant is a nonnegative Radon measure with some quantitative local lower and upper bounds. We also give the following two applications. (i) Applying this result with $\beta=0$, we develop an approach to build up a Liouville theorem, which improves that of Savin [33]. Precisely, if $u$ is $\infty$-harmonic functions in whole ${\mathbb R}^2$ with $$\liminf_{R\to\infty}\inf_{c\in\mathbb R}\frac1 {R^3}\int_{B(0,R)}|u(x)-c|\,dx<\infty,$$ then $u=b+a\cdot x$ for some $b\in{\mathbb R}$ and $a\in{\mathbb R}^2$. (ii) Denoting by $u_p$ the $p$-harmonic function having the same nonconstant boundary condition as $u$, we show that $\det DV_\beta(Du_p) \to \det DV_\beta(Du)$ as $p\to\infty$ in the weak-$\star$ sense in the space of Radon measure. Recall that $V_\beta(Du_p)$ is always quasiregular mappings, but $V_\beta(Du)$ is not in general.Comment: 31 pages, some minor changes, submitte

Shadow of topologically charged rotating braneworld black hole

In this paper, we discuss optical properties of the topologically charged rotating black hole. We study the horizon, the photon region, the shadow of the black hole and other observables. The results show that in addition to the black hole spin parameter $a$, the other two parameters, tidal charge $\beta$ and electric charge $q$, are also found to affect the horizon, the photon region and the black hole shadow. In a certain range, with the increase of the three parameters, the horizon distance, shape of the photon region and the black hole shadow will all shrink. Moreover, with the increase of these three parameters, the distortion parameter $\delta_{s}$ gradually increases, while the peak of the black hole energy emission rate decreases

(E)-3-(1-Phenyl­ethyl­idene)indolin-2-one

In the title mol­ecule, C16H13NO, the indoline-2-one ring system is nearly planar [maximum atomic deviation = 0.082 (2) Å] and is oriented at a dihedral angle of 66.60 (12)° with respect to the phenyl ring. In the crystal, inter­molecular N—H⋯O hydrogen bonds link the mol­ecules into supra­molecular dimers

AoM: Detecting Aspect-oriented Information for Multimodal Aspect-Based Sentiment Analysis

Multimodal aspect-based sentiment analysis (MABSA) aims to extract aspects from text-image pairs and recognize their sentiments. Existing methods make great efforts to align the whole image to corresponding aspects. However, different regions of the image may relate to different aspects in the same sentence, and coarsely establishing image-aspect alignment will introduce noise to aspect-based sentiment analysis (i.e., visual noise). Besides, the sentiment of a specific aspect can also be interfered by descriptions of other aspects (i.e., textual noise). Considering the aforementioned noises, this paper proposes an Aspect-oriented Method (AoM) to detect aspect-relevant semantic and sentiment information. Specifically, an aspect-aware attention module is designed to simultaneously select textual tokens and image blocks that are semantically related to the aspects. To accurately aggregate sentiment information, we explicitly introduce sentiment embedding into AoM, and use a graph convolutional network to model the vision-text and text-text interaction. Extensive experiments demonstrate the superiority of AoM to existing methods. The source code is publicly released at https://github.com/SilyRab/AoM.Comment: Findings of ACL 202