Brzozowski Algorithm Is Generically Super-Polynomial Deterministic Automata

International audienceWe study the number of states of the minimal automaton of the mirror of a rational language recognized by a random deterministic automaton with n states. We prove that, for any d > 0, the probability that this number of states is greater than nd tends to 1 as n tends to infinity. As a consequence, the generic and average complexities of Brzozowski minimization algorithm are super-polynomial for the uniform distribution on deterministic automata

Large-scale study of the NGC 1399 globular cluster system in Fornax

We present a Washington C and Kron-Cousins R photometric study of the globular cluster system of NGC 1399, the central galaxy of the Fornax cluster. A large areal coverage of 1 square degree around NGC 1399 is achieved with three adjoining fields of the MOSAIC II Imager at the CTIO 4-m telescope. Working on such a large field, we can perform the first indicative determination of the total size of the NGC 1399 globular cluster system. The estimated angular extent, measured from the NGC 1399 centre and up to a limiting radius where the areal density of blue globular clusters falls to 30 per cent of the background level, is 45 +/- 5 arcmin, which corresponds to 220 - 275 kpc at the Fornax distance. The bimodal colour distribution of this globular cluster system, as well as the different radial distribution of blue and red clusters, up to these large distances from the parent galaxy, are confirmed. The azimuthal globular cluster distribution exhibits asymmetries that might be understood in terms of tidal stripping of globulars from NGC 1387, a nearby galaxy. The good agreement between the areal density profile of blue clusters and a projected dark-matter NFW density profile is emphasized.Comment: 9 pages, 9 figures. Accepted for publication in A&

2MASS Studies of Differential Reddening Across Three Massive Globular Clusters

J, H, and K_S band data from the Two Micron All-Sky Survey (2MASS) are used to study the effects of differential reddening across the three massive Galactic globular clusters Omega Centauri, NGC 6388, and NGC 6441. Evidence is found that variable extinction may produce false detections of tidal tails around Omega Centauri. We also investigate what appears to be relatively strong differential reddening towards NGC 6388 and NGC 6441, and find that differential extinction may be exaggerating the need for a metallicity spread to explain the width of the red giant branches for these two clusters. Finally, we consider the implications of these results for the connection between unusual, multipopulation globular clusters and the cores of dwarf spheroidal galaxies (dSph).Comment: 40 pages, 14 figures. Accepted for publication in Oct. 2003 A

A Coverage Criterion for Spaced Seeds and its Applications to Support Vector Machine String Kernels and k-Mer Distances

Spaced seeds have been recently shown to not only detect more alignments, but also to give a more accurate measure of phylogenetic distances (Boden et al., 2013, Horwege et al., 2014, Leimeister et al., 2014), and to provide a lower misclassification rate when used with Support Vector Machines (SVMs) (On-odera and Shibuya, 2013), We confirm by independent experiments these two results, and propose in this article to use a coverage criterion (Benson and Mak, 2008, Martin, 2013, Martin and No{\'e}, 2014), to measure the seed efficiency in both cases in order to design better seed patterns. We show first how this coverage criterion can be directly measured by a full automaton-based approach. We then illustrate how this criterion performs when compared with two other criteria frequently used, namely the single-hit and multiple-hit criteria, through correlation coefficients with the correct classification/the true distance. At the end, for alignment-free distances, we propose an extension by adopting the coverage criterion, show how it performs, and indicate how it can be efficiently computed.Comment: http://online.liebertpub.com/doi/abs/10.1089/cmb.2014.017

On the Uniform Random Generation of Non Deterministic Automata Up to Isomorphism

In this paper we address the problem of the uniform random generation of non deterministic automata (NFA) up to isomorphism. First, we show how to use a Monte-Carlo approach to uniformly sample a NFA. Secondly, we show how to use the Metropolis-Hastings Algorithm to uniformly generate NFAs up to isomorphism. Using labeling techniques, we show that in practice it is possible to move into the modified Markov Chain efficiently, allowing the random generation of NFAs up to isomorphism with dozens of states. This general approach is also applied to several interesting subclasses of NFAs (up to isomorphism), such as NFAs having a unique initial states and a bounded output degree. Finally, we prove that for these interesting subclasses of NFAs, moving into the Metropolis Markov chain can be done in polynomial time. Promising experimental results constitute a practical contribution.Comment: Frank Drewes. CIAA 2015, Aug 2015, Umea, Sweden. Springer, 9223, pp.12, 2015, Implementation and Application of Automata - 20th International Conferenc

Subset currents on free groups

We introduce and study the space of \emph{subset currents} on the free group $F_N$. A subset current on $F_N$ is a positive $F_N$-invariant locally finite Borel measure on the space $\mathfrak C_N$ of all closed subsets of $\partial F_N$ consisting of at least two points. While ordinary geodesic currents generalize conjugacy classes of nontrivial group elements, a subset current is a measure-theoretic generalization of the conjugacy class of a nontrivial finitely generated subgroup in $F_N$, and, more generally, in a word-hyperbolic group. The concept of a subset current is related to the notion of an "invariant random subgroup" with respect to some conjugacy-invariant probability measure on the space of closed subgroups of a topological group. If we fix a free basis $A$ of $F_N$, a subset current may also be viewed as an $F_N$-invariant measure on a "branching" analog of the geodesic flow space for $F_N$, whose elements are infinite subtrees (rather than just geodesic lines) of the Cayley graph of $F_N$ with respect to $A$.Comment: updated version; to appear in Geometriae Dedicat