Local kinematics of K and M giants from Coravel/Hipparcos/Tycho-2 data. Revisiting the concept of superclusters

The availability of the Hipparcos Catalogue triggered many kinematic and dynamical studies of the solar neighbourhood. Nevertheless, those studies generally lacked the third component of the space velocities, i.e., the radial velocities. This work presents the kinematic analysis of 5952 K and 739 M giants in the solar neighbourhood which includes for the first time radial velocity data from a large survey performed with the CORAVEL spectrovelocimeter. It also uses proper motions from the Tycho-2 catalogue, which are expected to be more accurate than the Hipparcos ones. The UV-plane constructed from these data for the stars with precise parallaxes reveals a rich small-scale structure, with several clumps corresponding to the Hercules stream, the Sirius moving group, and the Hyades and Pleiades superclusters. A maximum-likelihood method, based on a bayesian approach, has been applied to the data, in order to make full use of all the available stars and to derive the kinematic properties of these subgroups. Isochrones in the Hertzsprung-Russell diagram reveal a very wide range of ages for stars belonging to these groups. These groups are most probably related to the dynamical perturbation by transient spiral waves rather than to cluster remnants. A possible explanation for the presence of young clusters in the same area of the UV-plane is that they have been put there by the spiral wave associated with their formation, while the kinematics of the older stars of our sample has also been disturbed by the same wave. The term "dynamical stream" for the kinematic groups is thus more appropriate than the traditional term "supercluster" since it involves stars of different ages, not born at the same place nor at the same time.Comment: 22 pages, 16 figures, accepted for publication in A&

The Sketching Complexity of Graph and Hypergraph Counting

Subgraph counting is a fundamental primitive in graph processing, with applications in social network analysis (e.g., estimating the clustering coefficient of a graph), database processing and other areas. The space complexity of subgraph counting has been studied extensively in the literature, but many natural settings are still not well understood. In this paper we revisit the subgraph (and hypergraph) counting problem in the sketching model, where the algorithm's state as it processes a stream of updates to the graph is a linear function of the stream. This model has recently received a lot of attention in the literature, and has become a standard model for solving dynamic graph streaming problems. In this paper we give a tight bound on the sketching complexity of counting the number of occurrences of a small subgraph $H$ in a bounded degree graph $G$ presented as a stream of edge updates. Specifically, we show that the space complexity of the problem is governed by the fractional vertex cover number of the graph $H$. Our subgraph counting algorithm implements a natural vertex sampling approach, with sampling probabilities governed by the vertex cover of $H$. Our main technical contribution lies in a new set of Fourier analytic tools that we develop to analyze multiplayer communication protocols in the simultaneous communication model, allowing us to prove a tight lower bound. We believe that our techniques are likely to find applications in other settings. Besides giving tight bounds for all graphs $H$, both our algorithm and lower bounds extend to the hypergraph setting, albeit with some loss in space complexity

Origami constraints on the initial-conditions arrangement of dark-matter caustics and streams

In a cold-dark-matter universe, cosmological structure formation proceeds in rough analogy to origami folding. Dark matter occupies a three-dimensional 'sheet' of free- fall observers, non-intersecting in six-dimensional velocity-position phase space. At early times, the sheet was flat like an origami sheet, i.e. velocities were essentially zero, but as time passes, the sheet folds up to form cosmic structure. The present paper further illustrates this analogy, and clarifies a Lagrangian definition of caustics and streams: caustics are two-dimensional surfaces in this initial sheet along which it folds, tessellating Lagrangian space into a set of three-dimensional regions, i.e. streams. The main scientific result of the paper is that streams may be colored by only two colors, with no two neighbouring streams (i.e. streams on either side of a caustic surface) colored the same. The two colors correspond to positive and negative parities of local Lagrangian volumes. This is a severe restriction on the connectivity and therefore arrangement of streams in Lagrangian space, since arbitrarily many colors can be necessary to color a general arrangement of three-dimensional regions. This stream two-colorability has consequences from graph theory, which we explain. Then, using N-body simulations, we test how these caustics correspond in Lagrangian space to the boundaries of haloes, filaments and walls. We also test how well outer caustics correspond to a Zel'dovich-approximation prediction.Comment: Clarifications and slight changes to match version accepted to MNRAS. 9 pages, 5 figure