Ghent University

One of the most important principles of J.-Y. Girard's Π 2 1 -logic is induction on dilators. In particular, Girard used this principle to construct his famous functor Λ. He claimed that the totality of Λ is equivalent to the set existence axiom of Π 1 1 -comprehension from reverse mathematics. While Girard provided a plausible description of a proof around 1980, it seems that the very technical details have not been worked out to this day. A few years ago, a loosely related approach led to an equivalence between Π 1 1 -comprehension and a certain Bachmann-Howard principle. The present paper closes the circle. We relate the Bachmann-Howard principle to induction on dilators. This allows us to show that Π 1 1 -comprehension is equivalent to the totality of a functor J due to P. Päppinghaus, which can be seen as a streamlined version of Λ.

Sub-visible particle analysis using flow imaging microscopy combined with deep learning has proven effective in identifying particle types, enabling the distinction of harmless components such as silicone oil from protein particles. However, the scarcity of available data and severe imbalance between particle types within datasets remain substantial hurdles when applying multi-class classifiers to such problems, often forcing researchers to rely on less effective methods. The aforementioned issue is particularly challenging for particle types that appear unintentionally and in lower numbers, such as silicone oil and air bubbles, as opposed to protein particles, where obtaining large numbers of images through controlled settings is comparatively straightforward. In this work, we develop a state-of-the-art diffusion model to address data imbalance by generating high-fidelity images that can augment training datasets, enabling the effective training of multi-class deep neural networks. We validate this approach by demonstrating that the generated samples closely resemble real particle images in terms of visual quality and structure. To assess the effectiveness of using diffusion-generated images in training datasets, we conduct large-scale experiments on a validation dataset comprising 500,000 protein particle images and demonstrate that this approach improves classification performance with no observable downside. Finally, to promote open research and reproducibility, we publicly release both our diffusion models and the trained multi-class deep neural network classifiers, along with a straightforward interface for easy integration into future studies, at https://github.com/utkuozbulak/svp-generative-ai.