Given the positions of n sites in a radio network we consider the problem of finding routes between any pair of sites that minimize energy consumption and do not use more than some constant number k of hops. Known exact algorithms for this problem required Ω(n logn) per query pair (p, q). In this paper we relax the exactness requirement and only compute approximate (1+2) solutions which allows ...

Given a real number t > 1, a geometric t-spanner is a geometric graph for a point set in R with straight lines between vertices such that the ratio of the shortest-path distance between every pair of vertices in the graph (with Euclidean edge lengths) to their actual Euclidean distance is at most t. An imprecise point set is modeled by a set R of regions in R. If one chooses a point in each reg...

Given a separated and locally finitely-presented Deligne-Mumford stack X over an algebraic space S, and a locally finitely-presented OX -module F , we prove that the Quot functor Quot(F/X/S) is represented by a separated and locally finitely-presented algebraic space over S. Under additional hypotheses, we prove that the connected components of Quot(F/X/S) are quasi-projective over S.

Instant radiosity methods rely on using a large number of virtual point lights (VPLs) to approximate global illumination. Efficiency considerations require grouping the VPLs into a small number of clusters that are treated as individual lights with respect to each point to be shaded. Two examples of clustering algorithms are Lightcuts [WFA∗05] and LightSlice [OP11]. In this work we use the noti...

[15] provide the notation and terminology for this paper. 1. PRELIMINARIES In this paper N denotes a set with non empty elements. The following three propositions are true: (1) For all sets x, y, z such that x = y and x = z holds {x, y, z} \ {x} = {y, z}. (2) For every non empty non void AMI A over N and for every state s of A and for every object o of A holds s(o) ∈ ObjectKind(o). (3) Let A be...

Lemma 2.2. It suffices to find disjoint L,R ∈ V such that they are (1) Large in Ω(n), or more formally stated, |L||R| ≥ α(n). (2) ∆-separated for ∆ = Ω( 1 logn) Let us define what ∆-separated means so that we can formalize this notion of “farness”. Recall that a distance from a vertex i to a set S is defined as d(i, S) = minj∈S d(i, j) (distance to the closest point in the set) and a distance f...

For any abelian category C satsifying (AB5) over a separated, quasicompact scheme S , we construct a stack of 2-groups GL(C) over the flat site of S. We will give a concrete description of GL(C) when C is the category of quasi-coherent sheaves on a separated, quasi-compact scheme X over S. We will show that the tangent space gl(C) of GL(C) at the origin has a structure as a Lie 2-algebra.