No stratification without representation

Sortition is an alternative approach to democracy, in which representatives are not elected but randomly selected from the population. Most electoral democracies fail to accurately represent even a handful of protected groups. By contrast, sortition guarantees that every subset of the population will in expectation fill their fair share of the available positions. This fairness property remains satisfied when the sample is stratified based on known features. Moreover, stratification can greatly reduce the variance in the number of positions filled by any unknown group, as long as this group correlates with the strata. Our main result is that stratification cannot increase this variance by more than a negligible factor, even in the presence of indivisibilities and rounding. When the unknown group is unevenly spread across strata, we give a guarantee on the reduction in variance with respect to uniform sampling. We also contextualize stratification and uniform sampling in the space of fair sampling algorithms. Finally, we apply our insights to an empirical case study.Accepted manuscrip

An ordinal multi-criteria decision-making procedure in the context of uniform qualitative scales

Producción CientíficaIn this contribution, we propose a multi-criteria decision-making procedure that has been devised in a purely ordinal way. Agents evaluate the alternatives regarding several criteria by assigning one or two consecutive terms of a uniform ordered qualitative scale to each alternative in each criterion. Weights assigned to criteria are managed through replications of the corresponding ratings, and alternatives are ranked according to the medians of their ratings after the replications

Polytopality and Cartesian products of graphs

We study the question of polytopality of graphs: when is a given graph the graph of a polytope? We first review the known necessary conditions for a graph to be polytopal, and we provide several families of graphs which satisfy all these conditions, but which nonetheless are not graphs of polytopes. Our main contribution concerns the polytopality of Cartesian products of non-polytopal graphs. On the one hand, we show that products of simple polytopes are the only simple polytopes whose graph is a product. On the other hand, we provide a general method to construct (non-simple) polytopal products whose factors are not polytopal.Comment: 21 pages, 10 figure

Weakly-nonlocal Symplectic Structures, Whitham method, and weakly-nonlocal Symplectic Structures of Hydrodynamic Type

We consider the special type of the field-theoretical Symplectic structures called weakly nonlocal. The structures of this type are in particular very common for the integrable systems like KdV or NLS. We introduce here the special class of the weakly nonlocal Symplectic structures which we call the weakly nonlocal Symplectic structures of Hydrodynamic Type. We investigate then the connection of such structures with the Whitham averaging method and propose the procedure of "averaging" of the weakly nonlocal Symplectic structures. The averaging procedure gives the weakly nonlocal Symplectic Structure of Hydrodynamic Type for the corresponding Whitham system. The procedure gives also the "action variables" corresponding to the wave numbers of $m$-phase solutions of initial system which give the additional conservation laws for the Whitham system.Comment: 64 pages, Late

An update on the Hirsch conjecture

The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to George Dantzig. It states that the graph of a d-dimensional polytope with n facets cannot have diameter greater than n - d. Despite being one of the most fundamental, basic and old problems in polytope theory, what we know is quite scarce. Most notably, no polynomial upper bound is known for the diameters that are conjectured to be linear. In contrast, very few polytopes are known where the bound $n-d$ is attained. This paper collects known results and remarks both on the positive and on the negative side of the conjecture. Some proofs are included, but only those that we hope are accessible to a general mathematical audience without introducing too many technicalities.Comment: 28 pages, 6 figures. Many proofs have been taken out from version 2 and put into the appendix arXiv:0912.423

A reconstruction of the initial conditions of the Universe by optimal mass transportation

Reconstructing the density fluctuations in the early Universe that evolved into the distribution of galaxies we see today is a challenge of modern cosmology [ref.]. An accurate reconstruction would allow us to test cosmological models by simulating the evolution starting from the reconstructed state and comparing it to the observations. Several reconstruction techniques have been proposed [8 refs.], but they all suffer from lack of uniqueness because the velocities of galaxies are usually not known. Here we show that reconstruction can be reduced to a well-determined problem of optimisation, and present a specific algorithm that provides excellent agreement when tested against data from N-body simulations. By applying our algorithm to the new redshift surveys now under way [ref.], we will be able to recover reliably the properties of the primeval fluctuation field of the local Universe and to determine accurately the peculiar velocities (deviations from the Hubble expansion) and the true positions of many more galaxies than is feasible by any other method. A version of the paper with higher-quality figures is available at http://www.obs-nice.fr/etc7/nature.pdfComment: Latex, 4 pages, 3 figure