Inviscid Limit for the Free-Boundary problems of MHD Equations with or without Surface Tension

In this paper, we investigate the convergence rates of inviscid limits for the free-boundary problems of the incompressible magnetohydrodynamics (MHD) with or without surface tension in $\mathbb{R}^3$, where the magnetic field is identically constant on the surface and outside of the domain. First, we establish the vorticity, the normal derivatives and the regularity structure of the solutions, and develop a priori co-norm estimates including time derivatives by the vorticity system. Second, we obtain two independent sufficient conditions for the existence of strong vorticity layers: (I) the limit of the difference between the initial MHD vorticity of velocity or magnetic field and that of the ideal MHD equations is nonzero. (II) The cross product of tangential projection on the free surface of the ideal MHD strain tensor of velocity or magnetic field with the normal vector of the free surface is nonzero. Otherwise, the vorticity layer is weak. Third, we prove high order convergence rates of tangential derivatives and the first order normal derivative in standard Sobolev space, where the convergence rates depend on the ideal MHD boundary value.Comment: 87. arXiv admin note: substantial text overlap with arXiv:1608.07276 by other author

Densely tracking sequences of 3D face scans

3D face dense tracking aims to find dense inter-frame correspondences in a sequence of 3D face scans and constitutes a powerful tool for many face analysis tasks, e.g., 3D dynamic facial expression analysis. The majority of the existing methods just fit a 3D face surface or model to a 3D target surface without considering temporal information between frames. In this paper, we propose a novel method for densely tracking sequences of 3D face scans, which ex- tends the non-rigid ICP algorithm by adding a novel specific criterion for temporal information. A novel fitting framework is presented for automatically tracking a full sequence of 3D face scans. The results of experiments carried out on the BU4D-FE database are promising, showing that the proposed algorithm outperforms state-of-the-art algorithms for 3D face dense tracking.Comment: 8 page

A note on the sum of reciprocals

For a fixed positive integer $m$ and any partition $m = m_1 + m_2 + \cdots + m_e$ , there exists a sequence $\{n_{i}\}_{i=1}^{k}$ of positive integers such that $$m=\frac{1}{n_{1}}+\frac{1}{n_{2}}+\cdots+\frac{1}{n_{k}},$$ with the property that partial sums of the series $\{\frac{1}{n_i}\}_{i=1}^{k}$ can only represent the integers with the form $\sum_{i\in I}m_i$, where $I\subset\{1,...,e\}$.Comment: This is a very very preliminary draft, which maybe contains some mistake

Instance-Level Meta Normalization

This paper presents a normalization mechanism called Instance-Level Meta Normalization (ILM~Norm) to address a learning-to-normalize problem. ILM~Norm learns to predict the normalization parameters via both the feature feed-forward and the gradient back-propagation paths. ILM~Norm provides a meta normalization mechanism and has several good properties. It can be easily plugged into existing instance-level normalization schemes such as Instance Normalization, Layer Normalization, or Group Normalization. ILM~Norm normalizes each instance individually and therefore maintains high performance even when small mini-batch is used. The experimental results show that ILM~Norm well adapts to different network architectures and tasks, and it consistently improves the performance of the original models. The code is available at url{https://github.com/Gasoonjia/ILM-Norm

Analytical expressions for greybody factor and dynamic evolution for scalar field in Ho\v{r}ava-Lifshitz black hole

We investigate the propagation and evolution for a massless scalar field in the background of $\lambda=1/2$ Ho\v{r}ava-Lifshitz black hole with the condition of detailed balance. We fortunately obtain an exact solution for the Klein-Gordon equation. Then, we find an analytical expression for the greybody factor which is valid for any frequency; and also exactly show that the perturbation decays without any oscillation. All of these can help us to understand more about the Ho\v{r}ava-Lifshitz gravity.Comment: 13 pages, 4 figures, Accepted for publication in PR

On the Relationship between the Continuum Enhancement and Hard X-ray Emission in a White-Light Flare

We investigate the relationship between the continuum enhancement and the hard X-ray (HXR) emission of a white-light flare on 2002 September 29. By reconstructing the RHESSI HXR images in the impulsive phase, we find two bright conjugate footpoints (FPs) on the two sides of the magnetic neutral line. Using the thick-target model and assuming a low-energy cutoff of 20 keV, the energy fluxes of non-thermal electron beams bombarding FPs A and B are estimated to be 1.0 10^10 and 0.8 10^10 ergs/cm^2/s, respectively. However, the continuum enhancement at the two FPs is not simply proportional to the electron beam flux. The continuum emission at FP B is relatively strong with a maximum enhancement of about 8% and correlates temporally well with the HXR profile; however, that at FP A is less significant with an enhancement of only about 4-5%, regardless of the relatively strong beam flux. By carefully inspecting the Halpha line profiles, we ascribe such a contrast to different atmospheric conditions at the two FPs. The Halpha line profile at FP B exhibits a relatively weak amplitude with a pronounced central reversal, while the profile at FP A is fairly strong without a visible central reversal. This indicates that in the early impulsive phase of the flare, the local atmosphere at FP A has been appreciably heated and the coronal pressure is high enough to prevent most high-energy electrons from penetrating into the deeper atmosphere; while at FP B, the atmosphere has not been fully heated, the electron beam can effectively heat the chromosphere and produce the observed continuum enhancement via the radiative backwarming effect.Comment: 16 pages, 6 figures, accepted for publication in Ap

Crossover transition in the Fluctuation of Internet

Gibrat's law predicts that the standard deviation of the growth rate of a node's degree is constant. On the other hand, the preferential attachment(PA) indicates that such standard deviation decreases with initial degree as a power law of exponent $-0.5$. While both models have been applied to Internet modeling, this inconsistency requires the verification of their validation. Therefore we empirically study the fluctuation of Internet of three different time intervals(daily, monthly and yearly). We find a crossover transition from PA model to Gibrat's law, which has never been reported. Specifically Gibrat-law starts from small degree region and extends gradually with the increase of the observed period. We determine the validated periods for both models and find that the correlation between internal links has large contribution to the emergence of Gibrat law. These findings indicate neither PA nor Gibrat law is applicable to the actual Internet, which requires a more complete model theory.Comment: 5pages,4 figure

SwipeCut: Interactive Segmentation with Diversified Seed Proposals

Interactive image segmentation algorithms rely on the user to provide annotations as the guidance. When the task of interactive segmentation is performed on a small touchscreen device, the requirement of providing precise annotations could be cumbersome to the user. We design an efficient seed proposal method that actively proposes annotation seeds for the user to label. The user only needs to check which ones of the query seeds are inside the region of interest (ROI). We enforce the sparsity and diversity criteria on the selection of the query seeds. At each round of interaction the user is only presented with a small number of informative query seeds that are far apart from each other. As a result, we are able to derive a user friendly interaction mechanism for annotation on small touchscreen devices. The user merely has to swipe through on the ROI-relevant query seeds, which should be easy since those gestures are commonly used on a touchscreen. The performance of our algorithm is evaluated on six publicly available datasets. The evaluation results show that our algorithm achieves high segmentation accuracy, with short response time and less user feedback

On homomorphisms from Ringel-Hall algebras to quantum cluster algebras

In \cite{rupel3},the authors defined algebra homomorphisms from the dual Ringel-Hall algebra of certain hereditary abelian category $\mathcal{A}$ to an appropriate $q$-polynomial algebra. In the case that $\mathcal{A}$ is the representation category of an acyclic quiver, we give an alternative proof by using the cluster multiplication formulas in \cite{DX}. Moreover, if the underlying graph of $Q$ is bipartite and the matrix $B$ associated to the quiver $Q$ is of full rank, we show that the image of the algebra homomorphisms is in the corresponding quantum cluster algebra.Comment: 10 page

Storing and Querying Large-Scale Spatio-Temporal Graphs with High-Throughput Edge Insertions

Real-world graphs often contain spatio-temporal information and evolve over time. Compared with static graphs, spatio-temporal graphs have very different characteristics, presenting more significant challenges in data volume, data velocity, and query processing. In this paper, we describe three representative applications to understand the features of spatio-temporal graphs. Based on the commonalities of the applications, we define a formal spatio-temporal graph model, where a graph consists of location vertices, object vertices, and event edges. Then we discuss a set of design goals to meet the requirements of the applications: (i) supporting up to 10 billion object vertices, 10 million location vertices, and 100 trillion edges in the graph, (ii) supporting up to 1 trillion new edges that are streamed in daily, and (iii) minimizing cross-machine communication for query processing. We propose and evaluate PAST, a framework for efficient PArtitioning and query processing of Spatio-Temporal graphs. Experimental results show that PAST successfully achieves the above goals. It improves query performance by orders of magnitude compared with state-of-the-art solutions, including JanusGraph, Greenplum, Spark and ST-Hadoop