Quantifying the risk carried by an aggregate position Sd := ∑d i=1Xi comprising many risk factors Xi is fundamental to both insurance and financial risk management. Fréchet inequalities quantify the worst-case risk carried by the aggregate position given distributional information concerning its composing factors but without assuming independence. This marginal factor modeling of the aggregate ...

Assuming Martin’s Axiom, we provide an example of two Fréchet-Urysohn 〈α4〉-spaces, whose product is a non-Fréchet-Urysohn 〈α4〉-space. This gives a consistent negative answer to a question raised by T. Nogura.

A Fréchet space, X , with a sequence of generating seminorms {‖·‖k} ∞ k=1 is called tame in case there exists an increasing function σ : N → N, such that for every continuous linear operator T from X into itself, there exists an N0 and C > 0 such that ‖T (x)‖n ≤ C‖x‖σ(n), ∀x ∈ X and n ≥ N0. This property does not depend upon the choice of fundamental system of semi-norms for X and is a property...

We describe two (1 + ε)-approximation algorithms for computing the Fréchet distance between two homeomorphic piecewise linear surfaces R and S of genus zero and total complexity n, with Fréchet distance δ. 1. A 2 (( n+ Area(R)+Area(S) (εδ)2 )2) time algorithm if R and S are composed of fat triangles (triangles with angles larger than a constant). 2. An O(D/(εδ)2) · n + 2O(D/(εδ)) time algorithm...

We study the decision tree complexity of the discrete Fréchet distance (decision version) under the L1 and L∞ metrics over R. While algorithms for the Euclidean (L2) discrete Fréchet distance were studied extensively, the problem in other metrics such as L1 and L∞ seems to be much less investigated. For the L1 discrete Fréchet distance in R we present a 2d-linear decision tree with depth O(n lo...

In this paper we consider Bernstein’s Lethargy Theorem (BLT) in the context of Fréchet spaces. Let X be an infinite-dimensional Fréchet space and let V = {Vn} be a nested sequence of subspaces of X such that Vn ⊆ Vn+1 for any n ∈ N and X = ⋃∞ n=1 Vn. Let en be a decreasing sequence of positive numbers tending to 0. Under an additional natural condition on sup{dist(x, Vn)}, we prove that there e...

Suppose a two-member team is required to navigate through dangerous territory (a plane with polygonal obstacles) along two preselected polygonal paths. The team could consist of police officers, soldiers, or even robots on the surface of Mars. Team members are allowed to coordinate their movements, and it is imperative to keep the team close together during the mission so that if an emergency o...

Measuring the similarity of two polygonal curves is a fundamental computational task. Among alternatives, the Fréchet distance is one of the most well studied similarity measures. Informally, the Fréchet distance is described as the minimum leash length required for a man on one of the curves to walk a dog on the other curve continuously from the starting to the ending points. In this paper we ...