The qualitative behavior at a vortex point for the Chern-Simon-Higgs equation

In this paper, we study the qualitative behavior at a vortex blow-up point for Chern-Simon-Higgs equation. Roughly speaking, we will establish an energy identity at a each such point, i.e. the local mass is the sum of the bubbles. Moreover, we prove that either there is only one bubble which is a singular bubble or there are more than two bubbles which contains no singular bubble. Meanwhile, we prove that the energies of these bubbles must satisfy a quadratic polynomial which can be used to prove the simple blow-up property when the multiplicity is small. As is well known, for many Liouville type system, Pohozaev type identity is a quadratic polynomial corresponding to energies which can be used directly to compute the local mass at a blow-up point. The difficulty here is that, besides the energy's integration, there is a additional term in the Pohozaev type identity of Chern-Simon-Higgs equation. We need some more detailed and delicated analysis to deal with it

Reverse spatial visual top-k query

With the wide application of mobile Internet techniques an location-based services (LBS), massive multimedia data with geo-tags has been generated and collected. In this paper, we investigate a novel type of spatial query problem, named reverse spatial visual top- $k$ query (RSVQ k ) that aims to retrieve a set of geo-images that have the query as one of the most relevant geo-images in both geographical proximity and visual similarity. Existing approaches for reverse top- $k$ queries are not suitable to address this problem because they cannot effectively process unstructured data, such as image. To this end, firstly we propose the definition of RSVQ k problem and introduce the similarity measurement. A novel hybrid index, named VR 2 -Tree is designed, which is a combination of visual representation of geo-image and R-Tree. Besides, an extension of VR 2 -Tree, called CVR 2 -Tree is introduced and then we discuss the calculation of lower/upper bound, and then propose the optimization technique via CVR 2 -Tree for further pruning. In addition, a search algorithm named RSVQ k algorithm is developed to support the efficient RSVQ k query. Comprehensive experiments are conducted on four geo-image datasets, and the results illustrate that our approach can address the RSVQ k problem effectively and efficiently

Blessing of High-Order Dimensionality: from Non-Convex to Convex Optimization for Sensor Network Localization

This paper investigates the Sensor Network Localization (SNL) problem, which seeks to determine sensor locations based on known anchor locations and partially given anchors-sensors and sensors-sensors distances. Two primary methods for solving the SNL problem are analyzed: the low-dimensional method that directly minimizes a loss function, and the high-dimensional semi-definite relaxation (SDR) method that reformulates the SNL problem as an SDP (semi-definite programming) problem. The paper primarily focuses on the intrinsic non-convexity of the loss function of the low-dimensional method, which is shown in our main theorem. The SDR method, via second-order dimension augmentation, is discussed in the context of its ability to transform non-convex problems into convex ones; while the first-order direct dimension augmentation fails. Additionally, we will show that more edges don't necessarily contribute to the better convexity of the loss function. Moreover, we provide an explanation for the success of the SDR+GD (gradient descent) method which uses the SDR solution as a warm-start of the minimization of the loss function by gradient descent. The paper also explores the parallels among SNL, max-cut, and neural networks in terms of the blessing of high-order dimension augmentation.Comment: 25 pages, 9 figures. References in arxiv. arXiv:1801.06146, arXiv:1810.04805, arXiv preprint arXiv:1906.05474, arXiv preprint arXiv:1801.0614

The C0C^0-convergence at the Neumann boundary for Liouville equations

In this paper, we study the blow-up analysis for a sequence of solutions to the Liouville type equation with exponential Neumann boundary condition. For interior case, i.e. the blow-up point is an interior point, Li \cite{Li} gave a uniform asymptotic estimate. Later, Zhang \cite{Zhang} and Gluck \cite{Gluck} improved Li's estimate in the sense of $C^0$-convergence by using the method of moving planes or classification of solutions of the linearized version of Liouville equation. If the sequence blows up at a boundary point, Bao-Wang-Zhou \cite{Bao-Wang-Zhou} proved a similar asymptotic estimate of Li \cite{Li}. In this paper, we will prove a $C^0$-convergence result in this boundary blow-up process. Our method is different from \cite{Zhang,Gluck}.Comment: 26 page

Investigating the Effectiveness of Road-related Mitigation Measures under Semi-controlled Conditions: A Case Study on Asian Amphibians

Road traffic is the main factor causing the decline in amphibian populations worldwide. The proper design of an amphibian tunnel is one of the most efficient measures to mitigate the negative impacts of road traffic on amphibians. However, no study has investigated the effectiveness of amphibian tunnels under semi-controlled conditions in Asian amphibians. Here, we selected two representative amphibian species, the Chinese brown frog, Rana chensinensis, and the Asiatic toad, Bufo gargarizans, which suffer the most severe road mortality along the roads in Northeast China. We placed experimental arrays of culverts of various sizes (diameters of 1.5, 1, and 0.5 m for circular culverts; side lengths of 1.5, 1, and 0.5 m for box culverts), and substrate type (soil, concrete, and metal) to examine the preferences of both species during the migratory season between May and September in 2016 and 2017. The results revealed that the Chinese brown frog preferred mid- and large-sized culverts as well as soil culverts. We concluded that culverts with a side length ≥ 1 m, lined with soil, and accompanied by a ≥ 0.4 m high guide drift fence and ≤ 45° gradient on the roadside ditch wall would best facilitate road crossings for both species and likely for other amphibian species in Northeast China