The Fréchet distance is a natural measure of similarity between two curves [4]. The Fréchet distance between two curves is often referred to as the “dog-leash distance”. Alt and Godau [4] presented an algorithm to compute the Fréchet distance between two polygonal curves of n and m vertices in O(nm log(nm)) time. There has been a lot of applications using the Fréchet distance to do pattern/curv...

Given a curve f and a surface S, how hard is it to find a simple curve f ′ ⊆ S that is the most similar to f? We introduce and study this simple curve embedding problem for piecewise linear curves and surfaces in R and R, under Hausdorff distance, weak Fréchet distance, and Fréchet distance as similarity measures for curves. Surprisingly, while several variants of the problem turn out to have p...

The Fréchet mean or barycenter generalizes the idea of averaging in spaces where pairwise addition is not well-defined. In general metric spaces, the Fréchet sample mean is not a consistent estimator of the theoretical Fréchet mean. For graph-valued random variables, for instance, the Fréchet sample mean may fail to converge to a unique value. Hence, it becomes necessary to consider the converg...

Computing the Fréchet distance for surfaces is a surprisingly hard problem. It has been shown that it is NP-hard to compute the Fréchet distance between many nice classes of surfaces [God98], [Buc10]. On the other hand, a polynomial time algorithm exists for computing the Fréchet distance between simple polygons [Buc06]. This was the first paper to give an algorithm for computing the Fréchet di...

Let X be a WCG Banach space admitting a Ck-Fréchet smooth norm. Then X admits an equivalent norm which is simultaneously C1-Fréchet smooth, LUR, and a uniform limit of Ck-Fréchet smooth norms. If X = C([0, α]), where α is an ordinal, then the same conclusion holds true with k =∞.

One of our results is a transfer principle between ultrapowers and reduced powers associated with the Fréchet ideal. Although motivated by Elliott classification programme, this result applies to any axiomatizable category. We also show that there exists nonprincipal ultrafilter $\mathcal{U}$ on $\mathbb{N}$ such for every countable (or separable metric) structure $B$ in language quotient map f...

Let F (X) and A(X) be respectively the free topological group and the free Abelian topological group on a Tychonoff space X. For every natural number n we denote by Fn(X) (An(X)) the subset of F (X) (A(X)) consisting of all words of reduced length ≤ n. It is well known that if a space X is not discrete, then neither F (X) nor A(X) is Fréchet-Urysohn, and hence first countable. On the other hand...

We give a unified view of the relation between the SL(2) KdV, the mKdV, and the UrKdV equations through the Fréchet derivatives and their inverses. For this we introduce a new procedure of obtaining the Ur-KdV equation, where we require that it has no nonlocal operators. We extend this method to the SL(3) KdV equation, i.e., Boussinesq(Bsq) equation and obtain the hamiltonian structure of Ur-Bs...