Multivariate Approaches to Classification in Extragalactic Astronomy

Clustering objects into synthetic groups is a natural activity of any science. Astrophysics is not an exception and is now facing a deluge of data. For galaxies, the one-century old Hubble classification and the Hubble tuning fork are still largely in use, together with numerous mono-or bivariate classifications most often made by eye. However, a classification must be driven by the data, and sophisticated multivariate statistical tools are used more and more often. In this paper we review these different approaches in order to situate them in the general context of unsupervised and supervised learning. We insist on the astrophysical outcomes of these studies to show that multivariate analyses provide an obvious path toward a renewal of our classification of galaxies and are invaluable tools to investigate the physics and evolution of galaxies.Comment: Open Access paper. http://www.frontiersin.org/milky\_way\_and\_galaxies/10.3389/fspas.2015.00003/abstract\>. \<10.3389/fspas.2015.00003 \&g

Collaboration in Social Networks

The very notion of social network implies that linked individuals interact repeatedly with each other. This allows them not only to learn successful strategies and adapt to them, but also to condition their own behavior on the behavior of others, in a strategic forward looking manner. Game theory of repeated games shows that these circumstances are conducive to the emergence of collaboration in simple games of two players. We investigate the extension of this concept to the case where players are engaged in a local contribution game and show that rationality and credibility of threats identify a class of Nash equilibria -- that we call "collaborative equilibria" -- that have a precise interpretation in terms of sub-graphs of the social network. For large network games, the number of such equilibria is exponentially large in the number of players. When incentives to defect are small, equilibria are supported by local structures whereas when incentives exceed a threshold they acquire a non-local nature, which requires a "critical mass" of more than a given fraction of the players to collaborate. Therefore, when incentives are high, an individual deviation typically causes the collapse of collaboration across the whole system. At the same time, higher incentives to defect typically support equilibria with a higher density of collaborators. The resulting picture conforms with several results in sociology and in the experimental literature on game theory, such as the prevalence of collaboration in denser groups and in the structural hubs of sparse networks

Concepts of Classification and Taxonomy. Phylogenetic Classification

Phylogenetic approaches to classification have been heavily developed in biology by bioinformaticians. But these techniques have applications in other fields, in particular in linguistics. Their main characteristics is to search for relationships between the objects or species in study, instead of grouping them by similarity. They are thus rather well suited for any kind of evolutionary objects. For nearly fifteen years, astrocladistics has explored the use of Maximum Parsimony (or cladistics) for astronomical objects like galaxies or globular clusters. In this lesson we will learn how it works. 1 Why phylogenetic tools in astrophysics? 1.1 History of classification The need for classifying living organisms is very ancient, and the first classification system can be dated back to the Greeks. The goal was very practical since it was intended to distinguish between eatable and toxic aliments, or kind and dangerous animals. Simple resemblance was used and has been used for centuries. Basically, until the XVIIIth century, every naturalist chose his own criterion to build a classification. At the end, hundreds of classifications were available, most often incompatible to each other. The criteria for this traditional way of classifying is the subjective appearance of the living organisms. During the XVIIIth a revolution occurred. Scientists like Adanson and Linn{\'e} devised new ways of classifying the objects and naming the classes. Adanson realised that all the observable traits should be used, giving birth to the mutivariate clustering and classification activity (Adanson, 1763). Linn{\'e} based his binomial nomenclature on neutral names unrelated whatsoever to any property of the classes. We can realise the success of these two ideas more than two centuries and a half later

Coined quantum walks on percolation graphs

Quantum walks, both discrete (coined) and continuous time, form the basis of several quantum algorithms and have been used to model processes such as transport in spin chains and quantum chemistry. The enhanced spreading and mixing properties of quantum walks compared with their classical counterparts have been well-studied on regular structures and also shown to be sensitive to defects and imperfections in the lattice. As a simple example of a disordered system, we consider percolation lattices, in which edges or sites are randomly missing, interrupting the progress of the quantum walk. We use numerical simulation to study the properties of coined quantum walks on these percolation lattices in one and two dimensions. In one dimension (the line) we introduce a simple notion of quantum tunneling and determine how this affects the properties of the quantum walk as it spreads. On two-dimensional percolation lattices, we show how the spreading rate varies from linear in the number of steps down to zero, as the percolation probability decreases to the critical point. This provides an example of fractional scaling in quantum walk dynamics.Comment: 25 pages, 14 figures; v2 expanded and improved presentation after referee comments, added extra figur

Solving the Canonical Representation and Star System Problems for Proper Circular-Arc Graphs in Log-Space

We present a logspace algorithm that constructs a canonical intersection model for a given proper circular-arc graph, where `canonical' means that models of isomorphic graphs are equal. This implies that the recognition and the isomorphism problems for this class of graphs are solvable in logspace. For a broader class of concave-round graphs, that still possess (not necessarily proper) circular-arc models, we show that those can also be constructed canonically in logspace. As a building block for these results, we show how to compute canonical models of circular-arc hypergraphs in logspace, which are also known as matrices with the circular-ones property. Finally, we consider the search version of the Star System Problem that consists in reconstructing a graph from its closed neighborhood hypergraph. We solve it in logspace for the classes of proper circular-arc, concave-round, and co-convex graphs.Comment: 19 pages, 3 figures, major revisio

On the Minimum Ropelength of Knots and Links

The ropelength of a knot is the quotient of its length and its thickness, the radius of the largest embedded normal tube around the knot. We prove existence and regularity for ropelength minimizers in any knot or link type; these are $C^{1,1}$ curves, but need not be smoother. We improve the lower bound for the ropelength of a nontrivial knot, and establish new ropelength bounds for small knots and links, including some which are sharp.Comment: 29 pages, 14 figures; New version has minor additions and corrections; new section on asymptotic growth of ropelength; several new reference