It is well acknowledged in image processing domain that the information can be decomposed into different frequency parts and each part has its own merits. However, existing neural networks always ignore the distinctions and straightforwardly feed all the information into neural networks together, treating them equally. In this paper, we propose a novel neural networks framework named J-Net that decomposes images into different frequency bands and then processes them sequentially. Concretely, the images have been decomposed by wavelet transformation and then the wavelet coefficients are fed into neural networks gradually in different depth according to their decomposition levels. An attention module is utilized to facilitate the fusion of neural network features and injected information, yielding significant performance gain. Furthermore, we show how does the information with different frequency impact the accuracy of neural networks. Experiments show that 5.91%, 5.32% and 2.00% accuracy improvements on Caltech 101, Caltech256 and ImageNet, respectively.

Inhomogeneous percolation, for its closer relationship with real-life, can be more useful and reasonable than homogeneous percolation to illustrate the critical phenomena and dynamical behaviour of complex networks. However, due to its intricacy, the theoretical framework of inhomogeneous percolation is far from being complete and many challenging problems are still open. Due to this reasn, in this paper, we investigate inhomogeneous site percolation on Bethe Lattices with two occupation probabilities, and we also extend the result to percolation with m occupation probabilities. The critical behaviour of this inhomogeneous percolation is shown clearly by formulating the percolation probability with given occupation probability, the critical occupation probability, and the average cluster size. Moreover, using the above theory, we discuss in detail the diffusion behaviour of an infectious disease (SARS) and present specific disease-control strategies in consideration of groups with different infection probabilities.

We show that the Hopf elements, the Kervaire classes, and the -family in the stable homotopy groups of spheres are detected by the Hurewicz map from the sphere spectrum to the C 2-fixed points of the Real bordism spectrum. A subset of these families is detected by the C 2-fixed points of Real JohnsonWilson theory E R (n), depending on n. In the proof, we establish an isomorphism between the slice spectral sequence and the C 2-equivariant May spectral sequence of B P R.

In 1960s, Almgren initiated a program to find minimal hypersurfaces in compact manifolds using min-max method. This program was largely advanced by Pitts and Schoen-Simon in 1980s when the manifold has no boundary. In this paper, we finish this program for general compact manifold with nonempty boundary. As a result, we prove the existence of a smooth embedded minimal hypersurface with free boundary in any compact smooth Euclidean domain. An application of our general existence result combined with the work of Marques and Neves shows that for any compact Riemannian manifolds with nonnegative Ricci curvature and convex boundary, there exist infinitely many embedded minimal hypersurfaces with free boundary which are properly embedded.

The classical Minkowski formula is extended to spacelike codimension-two submanifolds in spacetimes which admit "hidden symmetry" from conformal Killing-Yano two-forms. As an application, we obtain an Alexandrov type theorem for spacelike codimension-two submanifolds in a static spherically symmetric spacetime: a codimension-two submanifold with constant normalized null expansion (null mean curvature) must lie in a shear-free (umbilical) null hypersurface. These results are generalized for higher order curvature invariants. In particular, the notion of mixed higher order mean curvature is introduced to highlight the special null geometry of the submanifold. Finally, Alexandrov type theorems are established for spacelike submanifolds with constant mixed higher order mean curvature, which are generalizations of hypersurfaces of constant Weingarten curvature in the Euclidean space.