Normal form of equivariant maps in infinite dimensions

Local normal form theorems for smooth equivariant maps between infinite-dimensional manifolds are established. These normal form results are new even in finite dimensions. The proof is inspired by the Lyapunov–Schmidt reduction for dynamical systems and by the Kuranishi method for moduli spaces. It uses a slice theorem for Fréchet manifolds as the main technical tool. As a consequence, the abstract moduli space obtained by factorizing a level set of the equivariant map with respect to the group action carries the structure of a Kuranishi space, i.e., such moduli spaces are locally modeled on the quotient by a compact group of the zero set of a smooth map. The general results are applied to the moduli space of anti-self-dual instantons, the Seiberg–Witten moduli space and the moduli space of pseudoholomorphic curves.

Deep Learning for Detection of Colonic Polyps from Computed Tomography Colonoscopy Images Combined with Colonoscopy

The objective of this study was to investigate the diagnosis of colonic polyps (CP) through the computed tomography (CT) images combined with colonoscopy based on Fourier central slice theorem algorithm. In this study, 86 patients with CP admitted to hospital were selected as research objects. CT imaging and colonoscopy were applied to diagnose the patients based on the algorithm of Fourier central slice theorem. The results showed that the diagnostic detection rates of CP and colon cancer (CC) were 88.2% and 94.2%, respectively. The occurrence site of CP was the sigmoid and ascending colon. 38 patients were positive for serosal invasion of CP while 42 patients were negative for serosal invasion of CP, and there were no statistical differences ( P > 0.05 ). The lesion positions of remaining 6 cases were hard to find and could not be detected accurately. Besides, the diagnostic accuracy of preoperative and postoperative stages III and IV was all 100.00%. The combination of CT imaging and colonoscopy was employed to diagnose CP, which was found to be able to accurately locate the lesions, to effectively evaluate the tumor stage before and after surgery, and to have a good diagnostic efficacy in detecting tumor serosal layer.

Short survey on the existence of slices for the space of Riemannian metrics

We review the well-known slice theorem of Ebin for the action of the diffeomorphism group on the space of Riemannian metrics of a closed manifold. We present advances in the study of the spaces of Riemannian metrics, and produce a more concise proof for the existence of slices.