arXiv subjects
Publications and source records attributed to Ruixuan Zhu.
Let $n\ge3$. We prove a priori interior gradient estimate for the Lagrangian mean curvature equation \[ \sum_{i=1}^n\arctan\lambda_i(D^2u)=\theta(x) \] under the critical and supercritical phase condition $\theta\ge (n-2)\pi/2$, assuming only that the Lipschitz norm of the prescribed phase is bounded.
We establish interior $C^{2,\alpha}$ regularity for admissible solutions of the quadratic Hessian equation with positive $C^\alpha$ right-hand side on the full positive branch. The main ingredients are a quantitative large-trace propagation argument and an adaptive Dirichlet comparison method.
We consider convex admissible viscosity solutions of $\sigma_2(D^2u)=f>0$ in an open subset of $\mathbb R^n$, where $n\ge2$, $0<\alpha<1$, and $f\in C_{\mathrm{loc}}^{0,\alpha}$. We prove that every such solution belongs to $C_{\mathrm{loc}}^{2,\alpha}$. For solutions in $B_2$, we also prove a uniform $C^{2,\alpha}(B_{1/4})$ estimate under an $L^\infty$ bound for $u$, a positive lower bound for $f$, and a $C^{0,\alpha}$ bound for $f$.
Spoilers in movie reviews are important on platforms like IMDb and Rotten Tomatoes, offering benefits and drawbacks. They can guide some viewers' choices but also affect those who prefer no plot details in advance, making effective spoiler detection essential. Existing spoiler detection methods mainly analyze review text, often overlooking the impact of movie genres and user bias, limiting their effectiveness. To address this, we analyze movie review data, finding genre-specific variations in spoiler rates and identifying that certain users are more likely to post spoilers. Based on these findings, we introduce a new spoiler detection framework called GUSD (The code is available at https://github.com/AI-explorer-123/GUSD) (Genre-aware and User-specific Spoiler Detection), which incorporates genre-specific data and user behavior bias. User bias is calculated through dynamic graph modeling of review history. Additionally, the R2GFormer module combines RetGAT (Retentive Graph Attention Network) for graph information and GenreFormer for genre-specific aggregation. The GMoE (Genre-Aware Mixture of Experts) model further assigns reviews to specialized experts based on genre. Extensive testing on benchmark datasets shows that GUSD achieves state-of-the-art results. This approach advances spoiler detection by addressing genre and user-specific patterns, enhancing user experience on movie review platforms.
We prove the existence and regularity of convex solutions to the first initial-boundary value problem for the parabolic Monge-Ampère equationn $$ \left\{\begin{eqnarray} &&-u_t+\det D^2u= ψ(x,t) \quad\quad\ \text{ in } Q_T,\newline &&u=ϕ\quad\text{ on }\partial_pQ_T, \end{eqnarray}\right. $$ where $ψ,ϕ$ are given functions, $Q_T=Ω\times(0,T]$, $\partial_p Q_T$ is the parabolic boundary of $Q_T$, and $Ω\subset\mathbb{R}^n$ is a uniformly convex domain. Our approach can also be used to prove similar results for the $γ$-Gauss curvature flow with any $0<γ\le 1$.
We study the first derivative estimates for solutions to Monge-Ampère equations in terms of modulus of continuity. As a result, we establish the optimal global log-Lipschitz continuity for the gradient of solutions to the Monge-Ampère equation with natural boundary conditions.