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We examine the difference between several notions of curvature homogeneity and show that the notions introduced by Kowalski and Vanzurova are genuine generalizations of the ordinary notion of k-curvature homogeneity. The homothety group plays an essential role in the analysis. We give a complete classification of homothety homogeneous manifolds which are not homogeneous and which are not VSI and show that such manifolds are cohomogeneity one. We also give a complete description of the local geometry if the homothety character defines a split extension.
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We examine the difference between several notions of curvature homogeneity and show that the notions introduced by Kowalski and Vanv{z}urova are genuine generalizations of the ordinary notion of $k$-curvature homogeneity. The homothety group plays an essential role in the analysis.
k-Curvature homogeneous three-dimensional Walker metrics are described for k=0,1,2. This allows a complete description of locally homogeneous three-dimensional Walker metrics, showing that there exist exactly three isometry classes of such manifolds. As an application one obtains a complete description of all locally homogeneous Lorentzian manifolds with recurrent curvature. Moreover, potential functions are constructed in all the locally homogeneous manifolds resulting in steady gradient Ricci and Cotton solitons.
A Universe with finite age also has a finite causal scale $chi_S$, so the metric can not be homogeneous for $chi>chi_S$, as it is usually assumed. To account for this, we propose a new causal boundary condition, that can be fulfil by fixing the cosmological constant $Lambda$ (a free parameter for gravity). The resulting Universe is inhomogeneous, with possible variation of cosmological parameters on scales $chi simeq chi_S$. The size of $chi_S$ depends on the details of inflation, but regardless of its size, the boundary condition forces $Lambda/8pi G $ to cancel the contribution of a constant vacuum energy $rho_{vac}$ to the measured $rho_Lambda equiv Lambda/8pi G + rho_{vac}$. To reproduce the observed $rho_{Lambda} simeq 2 rho_m$ today with $chi_S rightarrow infty$ we then need a universe filled with evolving dark energy (DE) with pressure $p_{DE}> - rho_{DE}$ and a fine tuned value of $rho_{DE} simeq 2 rho_m$ today. This seems very odd, but there is another solution to this puzzle. We can have a finite value of $chi_S simeq 3 c/H_0$ without the need of DE. This scale corresponds to half the sky at $z sim 1$ and 60deg at $z sim 1000$, which is consistent with the anomalous lack of correlations observed in the CMB.
According to the cosmological principle, galaxy cluster sizes and cluster densities, when averaged over sufficiently large volumes of space, are expected to be constant everywhere, except for a slow variation with look-back time (redshift). Thus, average cluster sizes or correlation lengths provide a means of testing for homogeneity that is almost free of selection biases. Using ~10^6 galaxies from the SDSS DR7 survey, I show that regions of space separated by ~2 Gpc/h have the same average cluster size and density to 5 - 10 percent. I show that the average cluster size, averaged over many galaxies, remains constant to less than 10 percent from small redshifts out to redshifts of 0.25. The evolution of the cluster sizes with increasing redshift gives fair agreement when the same analysis is applied to the Millennium Simulation. However, the MS does not replicate the increase in cluster amplitudes with redshift seen in the SDSS data. This increase is shown to be caused by the changing composition of the SDSS sample with increasing redshifts. There is no evidence to support a model that attributes the SN Ia dimming to our happening to live in a large, nearly spherical void.
Do higher-order network structures aid graph semi-supervised learning? Given a graph and a few labeled vertices, labeling the remaining vertices is a high-impact problem with applications in several tasks, such as recommender systems, fraud detection and protein identification. However, traditional methods rely on edges for spreading labels, which is limited as all edges are not equal. Vertices with stronger connections participate in higher-order structures in graphs, which calls for methods that can leverage these structures in the semi-supervised learning tasks. To this end, we propose Higher-Order Label Spreading (HOLS) to spread labels using higher-order structures. HOLS has strong theoretical guarantees and reduces to standard label spreading in the base case. Via extensive experiments, we show that higher-order label spreading using triangles in addition to edges is up to 4.7% better than label spreading using edges alone. Compared to prior traditional and state-of-the-art methods, the proposed method leads to statistically significant accuracy gains in all-but-one cases, while remaining fast and scalable to large graphs.
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