We consider the problem of detecting whether or not, in a given sensor network, there is a cluster of sensors which exhibit an "unusual behavior." Formally, suppose we are given a set of nodes and attach a random variable to each node. We observe a realization of this process and want to decide between the following two hypotheses: under the null, the variables are i.i.d. standard normal; under the alternative, there is a cluster of variables that are i.i.d. normal with positive mean and unit variance, while the rest are i.i.d. standard normal. We also address surveillance settings where each sensor in the network collects information over time. The resulting model is similar, now with a time series attached to each node. We again observe the process over time and want to decide between the null, where all the variables are i.i.d. standard normal, and the alternative, where there is an emerging cluster of i.i.d. normal variables with positive mean and unit variance. The growth models used to represent the emerging cluster are quite general and, in particular, include cellular automata used in modeling epidemics. In both settings, we consider classes of clusters that are quite general, for which we obtain a lower bound on their respective minimax detection rate and show that some form of scan statistic, by far the most popular method in practice, achieves that same rate to within a logarithmic factor. Our results are not limited to the normal location model, but generalize to any one-parameter exponential family when the anomalous clusters are large enough.Comment: Published in at http://dx.doi.org/10.1214/10-AOS839 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Abstract. We study the enumeration complexity of the natural extension of acyclic conjunctive queries with disequalities. In this language, a number of NP-complete problems can be expressed. We first improve a previous result of Papadimitriou and Yannakakis by proving that such queries can be computed in time c.|M|.|ϕ(M)| where M is the structure, ϕ(M) is the result set of the query and c is a simple exponential in the size of the formula ϕ. A consequence of our method is that, in the general case, tuples of such queries can be enumerated with a linear delay between two tuples. We then introduce a large subclass of acyclic formulas called CCQ = and prove that the tuples of a CCQ = query can be enumerated with a linear time precomputation and a constant delay between consecutive solutions. Moreover, under the hypothesis that the multiplication of two n × n boolean matrices cannot be done in time O(n 2 ), this leads to the following dichotomy for acyclic queries: either such a query is in CCQ = or it cannot be enumerated with linear precomputation and constant delay. Furthermore we prove that testing whether an acyclic formula is in CCQ = can be performed in polynomial time. Finally, the notion of free-connex treewidth of a structure is defined. We show that for each query of free-connex treewidth bounded by some constant k, enumeration of results can be done with O(|M| k+1 ) precomputation steps and constant delay.

A central problem motivated by Diophantine approximation is to determine the size properties of subsets of R d (d ∈ N) of the formwhere · denotes an arbitrary norm, I a denumerable set, (x i , r i ) i∈I a family of elements of R d ×(0, ∞) and ϕ a nonnegative nondecreasing function defined on [0, ∞). We show that if F Id , where Id denotes the identity function, has full Lebesgue measure in a given nonempty open subset V of R d , the set F ϕ belongs to a class G h (V ) of sets with large intersection in V with respect to a given gauge function h. We establish that this class is closed under countable intersections and that each of its members has infinite Hausdorff g-measure for every gauge function g which increases faster than h near zero. In particular, this yields a sufficient condition on a gauge function g such that a given countable intersection of sets of the form F ϕ has infinite Hausdorff g-measure. In addition, we supply several applications of our results to Diophantine approximation. For any nonincreasing sequence ψ of positive real numbers converging to zero, we investigate the size and large intersection properties of the sets of all points that are ψ-approximable by rationals, by rationals with restricted numerator and denominator and by real algebraic numbers. This enables us to refine the analogs of Jarník's theorem for these sets. We also study the approximation of zero by values of integer polynomials and deduce several new results concerning Mahler's and Koksma's classifications of real transcendental numbers.