The covariogram of a measurable set A ⊂ R is the function gA which to each y ∈ R associates the Lebesgue measure of A ∩ (y + A). This paper proves two formulas. The first equates the directional derivatives at the origin of gA to the directional variations of A. The second equates the average directional derivative at the origin of gA to the perimeter of A. These formulas, previously known with...

We study maximal m-rigid objects in the m-cluster category C H associated with a finite dimensional hereditary algebra H with n nonisomorphic simple modules. We show that all maximal m-rigid objects in these categories have exactly n nonisomorphic indecomposable summands, and that any almost complete m-rigid object in C H has exactly m + 1 nonisomorphic complements. We also show that the maxima...

We provide a new constant factor approximation algorithm for the (connected) distance-r dominating set problem on graph classes of bounded expansion. Classes of bounded expansion include many familiar classes of sparse graphs such as planar graphs and graphs with excluded (topological) minors, and notably, these classes form the most general subgraph closed classes of graphs for which a sequent...

In the class of Carnot groups we study fine properties of sets of finite perimeter. Improving a recent result by Ambrosio-Kleiner-Le Donne, we show that the perimeter measure is local, i.e., that given any pair of sets of finite perimeter their perimeter measures coincide on the intersection of their essential boundaries. This solves a question left open in [4]. As a consequence we prove a gene...

Here A :D(A) ⊂H →H is a self-adjoint operator, K = {x ∈H :g(x) ≤ 1}, where g :H → R is convex and of class C∞, NK(x) is the normal cone to K at x and W (t) is a cylindrical Wiener process in H (see Hypothesis 1.1 for more precise assumptions). Obviously the expression in (1.1) is formal and its precise meaning should be defined. When H is finite-dimensional a solution to (1.1) is a pair of cont...

Let Z andX be Banach spaces, U ⊂ Z be an open convex set and f : U → X be a mapping. We say that f is a d.c. mapping if there exists a continuous convex function h on U such that y ◦ f + h is a continuous convex function for each y ∈ Y , ‖y∗‖ = 1. We say that f : U → X is locally d.c. if for each x ∈ U there exists open convex U ′ such that x ∈ U ′ ⊂ U and f |U ′ is d.c. This notion of d.c. map...

Dv = B′(ũ)∇uLN +B′(ũ)Dcu+ (B(u)−B(u))νuH Ju , where ∇u is the absolutely continuous part of Du, Du is the Cantor part of Du and Ju is the jump set of u (for the definition of these and other relevant quantities, see Sect.2). A delicate issue about this formula concerns the meaning of the first two terms on the right hand side. In fact, in order to understand why they are well defined, one has t...

This paper describes the Loquendo – Politecnico di Torino system evaluated on the 2006 NIST speaker recognition evaluation dataset. This system was among the best participants in this evaluation. It combines the results of two independent GMM systems: a Phonetic GMM and a classical GMM. Both systems rely on an intersession variation compensation approach, performed in the feature domain. It all...

As well known, the Moreau-Rockafellar-Robinson internal point qualification condition is sufficient to ensure that the infimal convolution of the conjugates of two extended-real-valued convex lower semi-continuous functions defined on a locally convex space is exact, and that the sub-differential of the sum of these functions is the sum of their sub-differentials. This note is devoted to provin...

In this era, automatic Web video categorization has become an important multimedia task for organizing and retrieving the plentiful videos on the Web. Due to unbounded variation in both content and quality of Web videos and deficiency in precisely labeled training data, Web video categorization remains a challenging task. In this paper, a novel three-stage framework is proposed for Web video cl...