arXiv subjects
Publications and source records attributed to Zixuan Qiu.
For the weighted Dirac eigenvalue problem, we show that the two-sided weighted spectrum depends continuously on the weight under continuous deformations within a uniformly elliptic class. Moreover, for differentiable families of weights we obtain a quantitative Lipschitz estimate for the full spectrum in the arsinh--metric, based on a weighted Hellmann--Feynman variational identity.
Wave equations are fundamental to describing a vast array of physical phenomena, yet their simulation in inhomogeneous media poses a computational challenge due to the highly oscillatory nature of the solutions. To overcome the high costs of traditional solvers, we propose the Windowed Fourier Propagator (WFP), a novel neural operator that efficiently learns the solution operator. The WFP's design is rooted in the physical principle of frequency locality, where wave energy scatters primarily to adjacent frequencies. By learning a set of compact, localized propagators, each mapping an input frequency to a small window of outputs, our method avoids the complexity of dense interaction models and achieves computational efficiency. Another key feature is the explicit preservation of superposition, which enables remarkable generalization from simple training data (e.g., plane waves) to arbitrary, complex wave states. We demonstrate that the WFP provides an explainable, efficient and accurate framework for data-driven wave modeling in complex media.
We give a min-max characterization of the weighted Dirac eigenvalues, and show that the weighted eigenvalues and eigenspaces of Dirac operators are continuous with respect to weak $L^p$ convergence of the inverse weight, for any $p>n$. Moreover, we establish a comparison result for such weighted eigenvalue problems when there are no harmonic spinors.
We present INTEGRALBENCH, a focused benchmark designed to evaluate Large Language Model (LLM) performance on definite integral problems. INTEGRALBENCH provides both symbolic and numerical ground truth solutions with manual difficulty annotations. Our evaluation of nine state-of-the-art LLMs reveals significant performance gaps and strong correlations between problem difficulty and model accuracy, establishing baseline metrics for this challenging domain. INTEGRALBENCH aims to advance automated mathematical reasoning by providing a rigorous evaluation framework specifically tailored for definite integral computation.