Group analysis in the SSRS2 catalog

We present an automated method to detect populations of groups in galaxy redshift catalogs. This method uses both analysis of the redshift distribution along lines of sight in fixed cells to detect elementary structures and a friend-of-friend algorithm to merge these elementary structures into physical structures. We apply this method to the SSRS2 galaxy redshift catalog. The groups detected with our method are similar to group catalogs detected with pure friend-of-friend algorithms. They have similar mass distribution, similar abundance versus redshift, similar 2-point correlation function and the same redshift completeness limit, close to 5000 km/s. If instead of SSRS2, we use catalogs of new generation, it would lead to a completeness limit of z$\sim$0.7. We model the luminosity function for nearby galaxy groups by a Schechter function with parameters M*=(-19.99+/-0.36)+5logh and alpha=-1.46 +/- 0.17 to compute the mass to light ratio. The median value of the mass to light ratio is 360 h M/L and we deduce a relation between mass to light ratio and velocity dispersion sigma (M/L=3.79 +/- 0.64)sigma -(294 +/- 570)). The more massive the group, the higher the mass to light ratio, and therefore, the larger the amount of dark matter inside the group. Another explanation is a significant stripping of the gas of the galaxies in massive groups as opposed to low mass groups. This extends to groups of galaxies the mild tendency already detected for rich clusters of galaxies. Finally, we detect a barely significant fundamental plane for these groups but much less narrow than for clusters of galaxies.Comment: 8 pages, 5 figures, accepted in A&A, shortened abstrac

The use of Minimal Spanning Tree to characterize the 2D cluster galaxy distribution

We use the Minimal Spanning Tree to characterize the aggregation level of given sets of points. We test 3 distances based on the histogram of the MST edges to discriminate between the distributions. We calibrate the method by using artificial sets following Poisson, King or NFW distributions. The distance using the mean, the dispersion and the skewness of the histogram of MST edges provides the more efficient results. We apply this distance to a subsample of the ENACS clusters and we show that the bright galaxies are significantly more aggregated than the faint ones. The contamination provided by uniformly distributed field galaxies is neglectible. On the other hand, we show that the presence of clustered groups on the same cluster line of sight masked the variation of the distance with the considered magnitude.Comment: 9 pages, 7 postscript figures, LateX A\{&}A, accepted in A\{&}

Piecewise Extended Chebyshev Spaces: a numerical test for design

Given a number of Extended Chebyshev (EC) spaces on adjacent intervals, all of the same dimension, we join them via convenient connection matrices without increasing the dimension. The global space is called a Piecewise Extended Chebyshev (PEC) Space. In such a space one can count the total number of zeroes of any non-zero element, exactly as in each EC-section-space. When this number is bounded above in the global space the same way as in its section-spaces, we say that it is an Extended Chebyshev Piecewise (ECP) space. A thorough study of ECP-spaces has been developed in the last two decades in relation to blossoms, with a view to design. In particular, extending a classical procedure for EC-spaces, ECP-spaces were recently proved to all be obtained by means of piecewise generalised derivatives. This yields an interesting constructive characterisation of ECP-spaces. Unfortunately, except for low dimensions and for very few adjacent intervals, this characterisation proved to be rather difficult to handle in practice. To try to overcome this difficulty, in the present article we show how to reinterpret the constructive characterisation as a theoretical procedure to determine whether or not a given PEC-space is an ECP-space. This procedure is then translated into a numerical test, whose usefulness is illustrated by relevant examples

Polynomial spaces revisited via weight functions

167-198International audienceExtended Chebyshev spaces are natural generalisations of polynomial spaces due to the same upper bounds on the number of zeroes. In a natural approach, many results of the polynomial framework have been generalised to the larger Chebyshevian framework, concerning Approximation Theory as well as Geometric Design. In the present work, we go the reverse way: considering polynomial spaces as examples of Extended Chebyshev spaces, we apply to them results specifically developed in the Chebyshevian framework. On a closed bounded interval, each Extended Chebyshev space can be defined by means of sequences of generalised derivatives which play the same rôle as the ordinary derivatives for polynomials. We recently achieved an exhaustive description of the infinitely many such sequences. Surprisingly, this issue is closely related to the question of building positive linear operators of the Bernstein type. As Extended Chebyshev spaces, one can thus search for all generalised derivatives which can be associated with polynomials spaces on closed bounded intervals. Though this may a priori seem somewhat nonsensical due to the simplicity of the ordinary derivatives, this actually leads to new interesting results on polynomial and rational Bernstein operators and related results of convergence

Which spline spaces for design?

International audienceWe recently determined the largest class of spaces of sufficient regularity which are suitable for design. How can weconnect different such spaces, possibly with the help of connection matrices, to produce the largest class of splinesusable for design? We present the answer to this question, along with some of the major difficulties encountered toestablish it.We would like to stress that the results we announce are far from being a straightforward generalisationof previous work on piecewise Chebyshevian splines