Property Testing for Cyclic Groups and Beyond

This paper studies the problem of testing if an input (Gamma,*), where Gamma is a finite set of unknown size and * is a binary operation over Gamma given as an oracle, is close to a specified class of groups. Friedl et al. [Efficient testing of groups, STOC'05] have constructed an efficient tester using poly(log|Gamma|) queries for the class of abelian groups. We focus in this paper on subclasses of abelian groups, and show that these problems are much harder: Omega(|Gamma|^{1/6}) queries are necessary to test if the input is close to a cyclic group, and Omega(|Gamma|^c) queries for some constant c are necessary to test more generally if the input is close to an abelian group generated by k elements, for any fixed integer k>0. We also show that knowledge of the size of the ground set Gamma helps only for k=1, in which case we construct an efficient tester using poly(log|Gamma|) queries; for any other value k>1 the query complexity remains Omega(|Gamma|^c). All our upper and lower bounds hold for both the edit distance and the Hamming distance. These are, to the best of our knowledge, the first nontrivial lower bounds for such group-theoretic problems in the property testing model and, in particular, they imply the first exponential separations between the classical and quantum query complexities of testing closeness to classes of groups.Comment: 15 pages, full version of a paper to appear in the proceedings of COCOON'11. v2: Ref. [14] added and a few modifications to Appendix A don

Temporal and Spatial Data Mining with Second-Order Hidden Models

In the frame of designing a knowledge discovery system, we have developed stochastic models based on high-order hidden Markov models. These models are capable to map sequences of data into a Markov chain in which the transitions between the states depend on the \texttt{n} previous states according to the order of the model. We study the process of achieving information extraction fromspatial and temporal data by means of an unsupervised classification. We use therefore a French national database related to the land use of a region, named Teruti, which describes the land use both in the spatial and temporal domain. Land-use categories (wheat, corn, forest, ...) are logged every year on each site regularly spaced in the region. They constitute a temporal sequence of images in which we look for spatial and temporal dependencies. The temporal segmentation of the data is done by means of a second-order Hidden Markov Model (\hmmd) that appears to have very good capabilities to locate stationary segments, as shown in our previous work in speech recognition. Thespatial classification is performed by defining a fractal scanning ofthe images with the help of a Hilbert-Peano curve that introduces atotal order on the sites, preserving the relation ofneighborhood between the sites. We show that the \hmmd performs aclassification that is meaningful for the agronomists.Spatial and temporal classification may be achieved simultaneously by means of a 2 levels \hmmd that measures the \aposteriori probability to map a temporal sequence of images onto a set of hidden classes

Super-Brownian motion with reflecting historical paths

We consider super-Brownian motion whose historical paths reflect from each other, unlike those of the usual historical super-Brownian motion. We prove tightness for the family of distributions corresponding to a sequence of discrete approximations but we leave the problem of uniqueness of the limit open. We prove a few results about path behavior for processes under any limit distribution. In particular, we show that for any $\gamma>0$, a "typical" increment of a reflecting historical path over a small time interval $\Delta t$ is not greater than $(\Delta t)^{3/4 - \gamma}$.Comment: 2 figure

Homology of spaces of regular loops in the sphere

In this paper we compute the singular homology of the space of immersions of the circle into the $n$-sphere. Equipped with Chas-Sullivan's loop product these homology groups are graded commutative algebras, we also compute these algebras. We enrich Morse spectral sequences for fibrations of free loop spaces together with loop products, this offers some new computational tools for string topology.Comment: 32 pages, 12 figure

Conditioned Brownian trees

We consider a Brownian tree consisting of a collection of one-dimensional Brownian paths started from the origin, whose genealogical structure is given by the Continuum Random Tree (CRT). This Brownian tree may be generated from the Brownian snake driven by a normalized Brownian excursion, and thus yields a convenient representation of the so-called Integrated Super-Brownian Excursion (ISE), which can be viewed as the uniform probability measure on the tree of paths. We discuss different approaches that lead to the definition of the Brownian tree conditioned to stay on the positive half-line. We also establish a Verwaat-like theorem showing that this conditioned Brownian tree can be obtained by re-rooting the unconditioned one at the vertex corresponding to the minimal spatial position. In terms of ISE, this theorem yields the following fact: Conditioning ISE to put no mass on $]-\infty,-\epsilon[$ and letting $\epsilon$ go to 0 is equivalent to shifting the unconditioned ISE to the right so that the left-most point of its support becomes the origin. We derive a number of explicit estimates and formulas for our conditioned Brownian trees. In particular, the probability that ISE puts no mass on $]-\infty,-\epsilon[$ is shown to behave like $2\epsilon^4/21$ when $\epsilon$ goes to 0. Finally, for the conditioned Brownian tree with a fixed height $h$, we obtain a decomposition involving a spine whose distribution is absolutely continuous with respect to that of a nine-dimensional Bessel process on the time interval $[0,h]$, and Poisson processes of subtrees originating from this spine.Comment: 42 page