Tianhao Dong | Searcharxiv

arXiv2026

Uniqueness for the spatially homogeneous Boltzmann equation in critical Sobolev spaces

We study the spatially homogeneous Boltzmann equation without angular cutoff for very soft potentials satisfying the inverse power law relation $\gamma+4s=1$. Our main results establish the existence, uniqueness, stability and regularization estimates of solutions in the critical Sobolev space $ H^{-(\gamma + 2s + \frac{3}{2})} $ with a logarithmic correction. Combined with the recently established monotonicity of the Fisher information, the solutions extend globally in time. Our primary tools are energy estimates based on a simultaneous dyadic localization in the phase and frequency variables, together with sharp commutator estimates between the collision operator and the localization operators.

math.AP↗

arXiv2026

Language-Agnostic Visual Embeddings for Cross-Script Handwriting Retrieval

Handwritten word retrieval is vital for digital archives but remains challenging due to large handwriting variability and cross-lingual semantic gaps. While large vision-language models offer potential solutions, their prohibitive computational costs hinder practical edge deployment. To address this, we propose a lightweight asymmetric dual-encoder framework that learns unified, style-invariant visual embeddings. By jointly optimizing instance-level alignment and class-level semantic consistency, our approach anchors visual embeddings to language-agnostic semantic prototypes, enforcing invariance across scripts and writing styles. Experiments show that our method outperforms 28 baselines and achieves state-of-the-art accuracy on within-language retrieval benchmarks. We further conduct explicit cross-lingual retrieval, where the query language differs from the target language, to validate the effectiveness of the learned cross-lingual representations. Achieving strong performance with only a fraction of the parameters required by existing models, our framework enables accurate and resource-efficient cross-script handwriting retrieval.

cs.CV↗