The planar fixed-radius near neighbor problem can be stated as follows: Preprocess a set P of N points in the plane so that all points of P lying within some fixed radius r of a new point can be listed effectively. This problem has many practical applications in domains as varied as molecular graphics, statistics, air traffic control or data transmission [3]. Although a great deal of work has b...

We consider countably infinite random geometric graphs, whose vertices are points in R, and edges are added independently with probability p ∈ (0, 1) if the metric distance between the vertices is below a given threshold. Assume that the vertex set is randomly chosen and dense in R. We address the basic question: for what metrics is there a unique isomorphism type for graphs resulting from this...

In this note we show that every n-point ultrametric embeds with constant distortion in l O(logn) p for every ∞ ≥ p ≥ 1. More precisely, we consider a special type of ultrametric with hierarchical structure called a k-hierarchically well-separated tree (k-HST). We show that any k-HST can be embedded with distortion at most 1 + O(1/k) in l O(k2 logn) p . These facts have implications to embedding...

1.1. Normed spaces. Recall that a (real) vector space V is called a normed space if there exists a function ‖ · ‖ : V → R such that (1) ‖f‖ ≥ 0 for all f ∈ V and ‖f‖ = 0 if and only if f = 0. (2) ‖af‖ = |a| ‖f‖ for all f ∈ V and all scalars a. (3) (Triangle inequality) ‖f + g‖ ≤ ‖f ||+ ‖g‖ for all f, g ∈ V . If V is a normed space, then d(f, g) = ‖f−g‖ defines a metric on V . Convergence w.r.t ...

Based on Lloyd iteration, we present a variational method for extracting general quadric surfaces from a 3D mesh surface. This work extends the previous variational methods that extract only planes or special types of quadrics, i.e., spheres and circular cylinders. Instead of using the exact L error metric, we use a new approximate L error metric to make our method more efficient for computing ...

In this paper, we consider the problem of the existence of non-negative weak solution u of A u + u P = O i n ~ u = 0 on Of~ n n + 2 ~ / n 1 having a given closed set S as its singular set. We prove that when < p < _ _ and n 2 n 4 + 2 nv/'n-L--1 S is a closed subset of f2, then there are infinite many positive weak solutions with S as their singular set. Applying this method to the conformal sca...

The integration of marketing and demand with logistics and inventories (supply side of companies) may cause multiple improvements; it can revolutionize the management of the revenue of rental companies, hotels, and airlines. In this paper, we develop a multi-objective pricing-inventory model for a retailer. Maximizing the retailer's profit and the service level are the objectives, and shorta...