Killing spinor initial data sets

A 3+1 decomposition of the twistor and valence-2 Killing spinor equation is made using the space spinor formalism. Conditions on initial data sets for the Einstein vacuum equations are given so that their developments contain solutions to the twistor and/or Killing equations. These lead to the notions of twistor and Killing spinor initial data. These notions are used to obtain a characterisation of initial data sets whose development are of Petrov type N or D.Comment: 31 pages, submitted to J. Geom. Phy

Data sets and data quality in software engineering

OBJECTIVE - to assess the extent and types of techniques used to manage quality within software engineering data sets. We consider this a particularly interesting question in the context of initiatives to promote sharing and secondary analysis of data sets. METHOD - we perform a systematic review of available empirical software engineering studies. RESULTS - only 23 out of the many hundreds of studies assessed, explicitly considered data quality. CONCLUSIONS - first, the community needs to consider the quality and appropriateness of the data set being utilised; not all data sets are equal. Second, we need more research into means of identifying, and ideally repairing, noisy cases. Third, it should become routine to use sensitivity analysis to assess conclusion stability with respect to the assumptions that must be made concerning noise levels

Large isoperimetric surfaces in initial data sets

We study the isoperimetric structure of asymptotically flat Riemannian 3-manifolds (M,g) that are C^0-asymptotic to Schwarzschild of mass m>0. Refining an argument due to H. Bray we obtain an effective volume comparison theorem in Schwarzschild. We use it to show that isoperimetric regions exist in (M, g) for all sufficiently large volumes, and that they are close to centered coordinate spheres. This implies that the volume-preserving stable constant mean curvature spheres constructed by G. Huisken and S.-T. Yau as well as R. Ye as perturbations of large centered coordinate spheres minimize area among all competing surfaces that enclose the same volume. This confirms a conjecture of H. Bray. Our results are consistent with the uniqueness results for volume-preserving stable constant mean curvature surfaces in initial data sets obtained by G. Huisken and S.-T. Yau and strengthened by J. Qing and G. Tian. The additional hypotheses that the surfaces be spherical and far out in the asymptotic region in their results are not necessary in our work.Comment: 29 pages. All comments welcome! This is the final version to appear in J. Differential Geo

Dimensionality reduction of clustered data sets

We present a novel probabilistic latent variable model to perform linear dimensionality reduction on data sets which contain clusters. We prove that the maximum likelihood solution of the model is an unsupervised generalisation of linear discriminant analysis. This provides a completely new approach to one of the most established and widely used classification algorithms. The performance of the model is then demonstrated on a number of real and artificial data sets

Gluing Initial Data Sets for General Relativity

We establish an optimal gluing construction for general relativistic initial data sets. The construction is optimal in two distinct ways. First, it applies to generic initial data sets and the required (generically satisfied) hypotheses are geometrically and physically natural. Secondly, the construction is completely local in the sense that the initial data is left unaltered on the complement of arbitrarily small neighborhoods of the points about which the gluing takes place. Using this construction we establish the existence of cosmological, maximal globally hyperbolic, vacuum space-times with no constant mean curvature spacelike Cauchy surfaces.Comment: Final published version - PRL, 4 page