Semi-local simple connectedness of non-collapsing Ricci limit spaces

Let $X$ be a non-collapsing Ricci limit space and let $x\in X$. We show that for any $\epsilon>0$, there is $r>0$ such that every loop in $B_t(x)$ is contractible in $B_{(1+\epsilon)t}(x)$, where $t\in(0,r]$. In particular, $X$ is semi-locally simply connected.Comment: Slightly modified the proof of Theorem 3.5 to fix a minor error on local covers. Slightly modified the proofs of Lemmas 3.2, 3.6, and 3.8 to fix a minor error on estimating $\rho(t,x)$ by a nearby point. Fixed some typo

Examples of open manifolds with positive Ricci curvature and non-proper Busemann functions

We give the first example of an open manifold with positive Ricci curvature and a non-proper Busemann function at a point. This provides counterexamples to a longtime well-known open question whether the Busemann function at a point of an open manifold with nonnegative Ricci curvature is proper

Examples of Ricci limit spaces with non-integer Hausdorff dimension

We give the first examples of collapsing Ricci limit spaces on which the Hausdorff dimension of the singular set exceeds that of the regular set; moreover, the Hausdorff dimension of these spaces can be non-integers. This answers a question of Cheeger-Colding about collapsing Ricci limit spaces.Comment: Slightly modified the exposition of the introduction. Added some referenc

On-Line Load Balancing with Task Buffer

On-line load balancing is one of the most important problems for applications with resource allocation. It aims to assign tasks to suitable machines and balance the load among all of the machines, where the tasks need to be assigned to a machine upon arrival. In practice, tasks are not always required to be assigned to machines immediately. In this paper, we propose a novel on-line load balancing model with task buffer, where the buffer can temporarily store tasks as many as possible. Three algorithms, namely LPTCP1_α, LPTCP2_α, and LPTCP3_β, are proposed based on the Longest Processing Time (LPT) algorithm and a variety of planarization algorithms. The planarization algorithms are proposed for reducing the difference among each element in a set. Experimental results show that our proposed algorithms can effectively solve the on-line load balancing problem and have good performance in large scale experiments

Surface coverage in wireless sensor networks

Abstract—Coverage is a fundamental problem in Wireless Sensor Networks (WSNs). Existing studies on this topic focus on 2D ideal plane coverage and 3D full space coverage. In many real world applications, the 3D surface of a targeted Field of Interest is complex, however, existing studies do not provide promising results. In this paper, we propose a new coverage model called surface coverage. In surface coverage, the targeted Field of Interest is a surface in 3D space and sensors can be deployed only on the surface. We show that existing 2D plane coverage is merely a special case of surface coverage. Simulations point out that existing sensor deployment schemes for a 2D plane cannot be directly applied to surface coverage cases. In this paper, we target two problems assuming surface coverage to be true. One, under stochastic deployment, how many sensors are needed to reach a certain expected coverage ratio? Two, if sensor deployment can be planned, what is the optimal deployment strategy with guaranteed full coverage with the least number of sensors? We show that the latter problem is NP-complete and propose three approximation algorithms. We further prove that these algorithms have a provable approximation ratio. We also conduct comprehensive simulations to evaluate the performance of the proposed algorithms. I

Attention-based High-order Feature Interactions to Enhance the Recommender System for Web-based Knowledge-Sharing Servic

Providing personalized online learning services has become a hot research topic. Online knowledge-sharing services represents a popular approach to enable learners to use fragmented spare time. User asks and answers questions in the platform, and the platform also recommends relevant questions to users based on their learning interested and context. However, in the big data era, information overload is a challenge, as both online learners and learning resources are embedded in data rich environment. Offering such web services requires an intelligent recommender system to automatically filter out irrelevant information, mine underling user preference, and distil latent information. Such a recommender system needs to be able to mine complex latent information, distinguish differences between users efficiently. In this study, we refine a recommender system of a prior work for web-based knowledge sharing. The system utilizes attention-based mechanisms and involves high-order feature interactions. Our experimental results show that the system outperforms known benchmarks and has great potential to be used for the web-based learning service