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Let A be an abelian category and B be the Happel-Reiten-Smalø tilt of A with respect to a torsion pair. We give necessary and sufficient conditions for the existence of a derived equivalence between B and A, which is compatible with the inclusion of B into the derived category of A. In particular, any splitting torsion pair induces a derived equivalence. We prove that for the realization functor of any bounded t-structure, its denseness implies its fullyfaithfulness.

BackgroundAutomatic fundus image processing plays a significant role in computer-assisted retinopathy diagnosis. As retinal vasculature is an important anatomical structure in ophthalmic images, recently, retinal vasculature segmentation has received considerable attention from researchers. A segmentation method usually consists of three steps: preprocessing, segmentation, post-processing. Most of the existing methods emphasize on the segmentation step. In our opinion, the vessels and background can be easily separable when suitable preprocessing exists.MethodsThis paper represents a new matched filter-based vasculature segmentation method for 2-D retinal images. First of all, a raw segmentation is acquired by thresholding the images preprocessed using weighted improved circular gabor filter and multi-directional multi-scale second derivation of Gaussian. After that, the raw segmented image is fine-tuned by a set of novel elongating filters. Finally, we eliminate the speckle like regions and isolated pixels, most of which are non-vessel noises and miss-classified fovea or pathological regions.ResultsThe performance of the proposed method is examined on two popularly used benchmark databases: DRIVE and STARE. The accuracy values are 95.29 and 95.69 %, respectively, without a significant degradation of specificity and sensitivity.Conclusion The performance of the proposed method is significantly better than almost all unsupervised methods, in addition, comparable to most of the existing supervised vasculature segmentation methods.

For a prime number p, let Q p be the field of p-adic numbers. In this paper, we establish the boundedness of a class of p-adic singular integral operators on the p-adic generalized Morrey spaces. We also consider the corresponding boundedness for the commutators generalized by the p-adic singular integral operators and p-adic Lipschitz functions or p-adic generalized Campanato functions.

Artificial neural networks are employed to predict the band structure of the one-dimensional photonic crystal nanobeam, and to inverse-design the geometry structure with on-demand band edges. The data sets generated by 3D finite-difference time-domain based on elliptical-shaped hole nanobeams are used to train the networks and evaluate the networks’ accuracy. Based on the well-trained forward prediction and inverse-design network, an ultrabroad bandgap elliptical hole dielectric mode nanobeam cavity is designed. The bandgap achieves 77.7 THz for the center segment of the structure, and the whole designing process takes only 0.73 s. The approach can also be expanded to fast-design elliptical hole air mode nanobeam cavities. The present work is of significance for further research on the application of artificial neural networks in photonic crystal cavities and other optical devices design.