384 research outputs found
Construction of vacuum initial data by the conformally covariant split system
Using the implicit function theorem, we prove existence of solutions of the
so-called conformally covariant split system on compact 3-dimensional
Riemannian manifolds. They give rise to non-Constant Mean Curvature (non-CMC)
vacuum initial data for the Einstein equations. We investigate the conformally
covariant split system defined on compact manifolds with or without boundaries.
In the former case, the boundary corresponds to an apparent horizon in the
constructed initial data. The case with a cosmological constant is then
considered separately. Finally, to demonstrate the applicability of the
conformal covariant split system in numerical studies, we provide numerical
examples of solutions on manifolds and
On Hawking mass and Bartnik mass of CMC surfaces
Given a constant mean curvature surface that bounds a compact manifold with
nonnegative scalar curvature, we obtain intrinsic conditions on the surface
that guarantee the positivity of its Hawking mass. We also obtain estimates of
the Bartnik mass of such surfaces, without assumptions on the integral of the
squared mean curvature. If the ambient manifold has negative scalar curvature,
our method also applies and yields estimates on the hyperbolic Bartnik mass of
these surfaces.Comment: version accepted by Math. Res. Let
Succinct Representations in Collaborative Filtering: A Case Study using Wavelet Tree on 1,000 Cores
User-Item (U-I) matrix has been used as the dominant data infrastructure of Collaborative Filtering (CF). To reduce space consumption in runtime and storage, caused by data sparsity and growing need to accommodate side information in CF design, one needs to go beyond the UI Matrix. In this paper, we took a case study of Succinct Representations in Collaborative Filtering, rather than using a U-I Matrix. Our key insight is to introduce Succinct Data Structures as a new infrastructure of CF. Towards this, we implemented a User-based K-Nearest-Neighbor CF prototype via Wavelet Tree, by first designing a Accessible Compressed Documents (ACD) to compress U-I data in Wavelet Tree, which is efficient in both storage and runtime. Then, we showed that ACD can be applied to develop an efficient intersection algorithm without decompression, by taking advantage of ACD’s characteristics. We evaluated our design on 1,000 cores of Tianhe-II supercomputer, with one of the largest public data set ml-20m. The results showed that our prototype could achieve 3.7 minutes on average to deliver the results
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