arXiv subjects
Publications and source records attributed to Yang Kaiwen.
For solutions to the Minkowski problem of the even dual curvature measures $\widetilde{C}_q$ in $\mathbb{R}^n$, $n\ge 2$, we prove the non-uniqueness for $q>n$ and $n\ge 2$, and the uniqueness for $0<q<n$ and $n=2$. These results are governed by our established logarithmic Brunn-Minkowski inequality for dual quermassintegrals $\widetilde V_q$.
In the 1960s, Grünbaum and Loewner raised problems concerning the minimum possible number of hyperplane sections of a convex body in $R^n$ whose centroids coincide with the centroid of the body itself. By employing the hemispherical transform and Morse theory, we provide a complete solution to these problems.
Technological advancement allows information to be shared in just a single click, which has enabled the rapid spread of false information. This makes automated fact-checking system necessary to ensure the safety and integrity of our online media ecosystem. Previous methods have demonstrated the effectiveness of decomposing the claim into simpler sub-tasks and utilizing LLM-based multi agent system to execute them. However, those models faces two limitations: they often fail to verify every component in the claim and lack of structured framework to logically connect the results of sub-tasks for a final prediction. In this work, we propose a novel automated fact-checking framework called Trification. Our framework begins by generating a comprehensive set of verification actions to ensure complete coverage of the claim. It then structured these actions into a dependency graph to model the logical interaction between actions. Furthermore, the graph can be dynamically modified, allowing the system to adapt its verification strategy. Experimental results on two challenging benchmarks demonstrate that our framework significantly enhances fact-checking accuracy, thereby advancing current state-of-the-art in automated fact-checking system.