Moduli of Continuity for Viscosity Solutions

In this paper, we investigate the moduli of continuity for viscosity solutions of a wide class of nonsingular quasilinear evolution equations and also for the level set mean curvature flow, which is an example of singular degenerate equations. We prove that the modulus of continuity is a viscosity subsolution of some one dimensional equation. This work extends B. Andrews' recent result on moduli of continuity for smooth spatially periodic solutions.Comment: 8 page

Jet magnetically accelerated from disk-corona around a rotating black hole

A jet acceleration model for extracting energy from disk-corona surrounding a rotating black hole is proposed. In the disk-corona scenario, we obtain the ratio of the power dissipated in the corona to the total for such disk-corona system by solving the disk dynamics equations. The analytical expression of the jet power is derived based on the electronic circuit theory of the magnetosphere. It is shown that jet power increases with the increasing black hole (BH) spin, and concentrates in the inner region of the disk-corona. In addition, we use a sample consisting of 37 radio loud quasars to explore their jet production mechanism, and show that our jet formation mechanism can simulate almost all sources with high power jet, that fail to be explained by the Blandford-Znajek (BZ) process

Sharp lower bound for the first eigenvalue of the Weighted pp-Laplacian II

Combined with our previous work \cite{LW19eigenvalue}, we prove sharp lower bound estimates for the first nonzero eigenvalue of the weighted $p$-Laplacian with $1< p< \infty$ on a compact Bakry-\'Emery manifold $(M^n,g,f)$, without boundary or with a convex boundary and Neumann boundary condition, satisfying $\text{Ric}+\nabla^2 f \geq \kappa \, g$ for some $\kappa \in \mathbb{R}$.Comment: Final version, to appear Mathematical Research Letter

Moduli of Continuity for Viscosity Solutions on Manifolds

We establish the estimates of modulus of continuity for viscosity solutions of nonlinear evolution equations on manifolds, extending previous work of B. Andrews and J. Clutterbuck for regular solutions on manifolds \cite{AC3} and the first author's recent work for viscosity solutions in Euclidean spaces \cite{me1}.Comment: 16 page

Nonparametric hypersurfaces moving by powers of Gauss curvature

We study asymptotic behavior of nonparametric hypersurfaces moving by $\alpha$ powers of Gauss curvature $\alpha > 1/n$. Our work generalizes the results of V. Oliker [Oli91] for $\alpha= 1$.Comment: 7 pages. Any comments are welcom

Parabolic frequency monotonicity on compact manifolds

This work is devoted to the study of parabolic frequency for solutions of the heat equation on Riemannian manifolds. We show that the parabolic frequency functional is almost increasing on compact manifolds with nonnegative sectional curvature, which generalizes a monotonicity result proved by C. Poon and by L. Ni. The proof is based on a generalization of R. Hamilton's matrix Harnack inequality for small time. As applications, we obtain a unique continuation result. Monotonicity of a new quantity under two-dimensional Ricci flow, closely related to the parabolic frequency functional, is derived as well.Comment: 17 page

Oriented diameter and rainbow connection number of a graph

The oriented diameter of a bridgeless graph $G$ is $\min\{diam(H)\ | H\ is\ an orientation\ of\ G\}$. A path in an edge-colored graph$G$, where adjacent edges may have the same color, is called rainbow if no two edges of the path are colored the same. The rainbow connection number$rc(G)$of$G$is the smallest integer$k$for which there exists a$k$-edge-coloring of$G$such that every two distinct vertices of$G$are connected by a rainbow path. In this paper, we obtain upper bounds for the oriented diameter and the rainbow connection number of a graph in terms of$rad(G)$and$\eta(G)$, where$rad(G)$is the radius of$G$and$\eta(G)$is the smallest integer number such that every edge of$G$is contained in a cycle of length at most$\eta(G)$. We also obtain constant bounds of the oriented diameter and the rainbow connection number for a (bipartite) graph$G$in terms of the minimum degree of$G$.Comment: 16 page

Four-dimentional Gradient Shrinking Solitons with Positive Isotropic Curvature

We show that a four-dimensional complete gradient shrinking Ricci soliton with positive isotropic curvature is either a quotient of S^4 or a quotient of S^3 cross R. This gives a clean classification result removing the earlier additional assumptions in [13] by Wallach and the second author.Comment: 9 pages, Any comments are welcom

Two-Stream Multi-Task Network for Fashion Recognition

In this paper, we present a two-stream multi-task network for fashion recognition. This task is challenging as fashion clothing always contain multiple attributes, which need to be predicted simultaneously for real-time industrial systems. To handle these challenges, we formulate fashion recognition into a multi-task learning problem, including landmark detection, category and attribute classifications, and solve it with the proposed deep convolutional neural network. We design two knowledge sharing strategies which enable information transfer between tasks and improve the overall performance. The proposed model achieves state-of-the-art results on large-scale fashion dataset comparing to the existing methods, which demonstrates its great effectiveness and superiority for fashion recognition.Comment: Accepted by ICIP 201

DeepRebirth: Accelerating Deep Neural Network Execution on Mobile Devices

Deploying deep neural networks on mobile devices is a challenging task. Current model compression methods such as matrix decomposition effectively reduce the deployed model size, but still cannot satisfy real-time processing requirement. This paper first discovers that the major obstacle is the excessive execution time of non-tensor layers such as pooling and normalization without tensor-like trainable parameters. This motivates us to design a novel acceleration framework: DeepRebirth through "slimming" existing consecutive and parallel non-tensor and tensor layers. The layer slimming is executed at different substructures: (a) streamline slimming by merging the consecutive non-tensor and tensor layer vertically; (b) branch slimming by merging non-tensor and tensor branches horizontally. The proposed optimization operations significantly accelerate the model execution and also greatly reduce the run-time memory cost since the slimmed model architecture contains less hidden layers. To maximally avoid accuracy loss, the parameters in new generated layers are learned with layer-wise fine-tuning based on both theoretical analysis and empirical verification. As observed in the experiment, DeepRebirth achieves more than 3x speed-up and 2.5x run-time memory saving on GoogLeNet with only 0.4% drop of top-5 accuracy on ImageNet. Furthermore, by combining with other model compression techniques, DeepRebirth offers an average of 65ms inference time on the CPU of Samsung Galaxy S6 with 86.5% top-5 accuracy, 14% faster than SqueezeNet which only has a top-5 accuracy of 80.5%.Comment: AAAI 201