Michael Wijaya | Searcharxiv

arXiv2026

The distribution of $k$-free ideals in ray class groups

In this paper, we extend the classical problem of studying the distribution of $k$-free integers in arithmetic progressions to the setting of arbitrary number fields. Using the language of ray class groups, we establish asymptotic formulas, together with error terms, for the number of $k$-free ideals of bounded norm lying in a given ray class. In particular, our results show that $k$-free ideals are equidistributed among ray classes. We also obtain improved error estimates in the cases of ideal class groups and narrow class groups by using sharper ideal counting asymptotics due to Landau. Our results recover the classical formulas of Gegenbauer and Cohen--Robinson over $\mathbb{Q}$ and extend previous work of Benkowski, Nymann, and Sittinger to the setting of ray class groups. We also present explicit computational examples that illustrate the asymptotic formulas and the equidistribution of $k$-free ideals among ray classes.

math.NT↗

arXiv2025

SafeMed-R1: Adversarial Reinforcement Learning for Generalizable and Robust Medical Reasoning in Vision-Language Models

Vision--Language Models (VLMs) show significant promise for Medical Visual Question Answering (VQA), yet their deployment in clinical settings is hindered by severe vulnerability to adversarial attacks. Standard adversarial training, while effective for simpler tasks, often degrades both generalization performance and the quality of generated clinical reasoning. We introduce SafeMed-R1, a hybrid defense framework that ensures robust performance while preserving high-quality, interpretable medical reasoning. SafeMed-R1 employs a two-stage approach: at training time, we integrate Adversarial Training with Group Relative Policy Optimization (AT-GRPO) to explicitly robustify the reasoning process against worst-case perturbations; at inference time, we augment the model with Randomized Smoothing to provide certified $L_2$-norm robustness guarantees. We evaluate SafeMed-R1 on the OmniMedVQA benchmark across eight medical imaging modalities comprising over 88,000 samples. Our experiments reveal that standard fine-tuned VLMs, despite achieving 95\% accuracy on clean inputs, collapse to approximately 25\% under PGD attacks. In contrast, SafeMed-R1 maintains 84.45\% accuracy under the same adversarial conditions, representing a 59 percentage point improvement in robustness. Furthermore, we demonstrate that models trained with explicit chain-of-thought reasoning exhibit superior adversarial robustness compared to instruction-only variants, suggesting a synergy between interpretability and security in medical AI systems.

cs.AI↗