Search results for "Euclidean"

Pieri’s 1900 Point-and-Motion Memoir

2021

This chapter contains an English translation of Mario Pieri’s 1900a memoir, On Elementary Geometry as a Hypothetical Deductive System: Monograph on Point and on Motion.1 By elementary geometry, Pieri meant Euclidean geometry as taught then in elementary courses, except for the theorems dependent on the Euclidean parallel axiom.

Quasisymmetric structures on surfaces

2009

We show that a locally Ahlfors 2-regular and locally linearly locally contractible metric surtace is locally quasisymmetrically equivalent to tne disk. We also discuss an application of this result to the problem of characterizing surfaces embedded in some Euclidean spaces that are locally bi-Lipschitz equivalent to a ball in the plane.

Cartoon filter via adaptive abstraction

2016

We propose a non-parametric methodology to realize abstraction images.The redundant wavelet "a trous" algorithm is applied for details detection.An multi-scale circular median filter is used as a smoothing filter.The proposed algorithm is simple and fast on low-cost entry-level hardware. Abstraction in computer graphics defines a procedure that discriminates the essential information that is worth keeping. Usually details, that correspond to higher frequency components, allow to distinguish otherwise similar images. Vice versa, low frequencies are related to the main information, which are larger structures. Contours themselves may also be identified by high frequencies and separate each pi…

A tour of the theory of absolutely minimizing functions

2004

A detailed analysis of the class of absolutely minimizing functions in Euclidean spaces and the relationship to the infinity Laplace equation

Distance Functions, Clustering Algorithms and Microarray Data Analysis

2010

Distance functions are a fundamental ingredient of classification and clustering procedures, and this holds true also in the particular case of microarray data. In the general data mining and classification literature, functions such as Euclidean distance or Pearson correlation have gained their status of de facto standards thanks to a considerable amount of experimental validation. For microarray data, the issue of which distance function works best has been investigated, but no final conclusion has been reached. The aim of this extended abstract is to shed further light on that issue. Indeed, we present an experimental study, involving several distances, assessing (a) their intrinsic sepa…

A Characterization of Bispecial Sturmian Words

2012

A finite Sturmian word w over the alphabet {a,b} is left special (resp. right special) if aw and bw (resp. wa and wb) are both Sturmian words. A bispecial Sturmian word is a Sturmian word that is both left and right special. We show as a main result that bispecial Sturmian words are exactly the maximal internal factors of Christoffel words, that are words coding the digital approximations of segments in the Euclidean plane. This result is an extension of the known relation between central words and primitive Christoffel words. Our characterization allows us to give an enumerative formula for bispecial Sturmian words. We also investigate the minimal forbidden words for the set of Sturmian wo…

The node-depth encoding

2008

The node-depth encoding has elements from direct and indirect encoding for trees which encodes trees by storing the depth of nodes in a list. Node-depth encoding applies specific search operators that is a typical characteristic for direct encodings. An investigation into the bias of the initialization process and the mutation operators of the node-depth encoding shows that the initialization process has a bias to solutions with small depths and diameters, and a bias towards stars. This investigation, also, shows that the mutation operators are unbiased. The performance of node-depth encoding is investigated for the bounded-diameter minimum spanning tree problem. The results are presented f…

Covering and differentiation

1995

Compactness of a conformal boundary of the Euclidean unit ball

2011

We study conformal metrics d‰ on the Euclidean unit ball B n : We assume that either the density ‰ associated with the metric d‰ satisfies a logarithmic volume growth condition for small balls or that ‰ satisfies a Harnack inequality and a suitable sub-Euclidean volume growth condition. We prove that the ‰-boundary @‰ B n is homeomorphic to S ni1 if and only if @‰ B n is compact. In the planar case, the compactness of @‰ B 2 is further equivalent to local connectivity of the ‰-boundary together with the boundedness of (B 2 ;d‰):

Continuous reformulations and heuristics for the Euclidean travelling salesperson problem

2008

We consider continuous reformulations of the Euclidean travelling salesperson problem (TSP), based on certain clustering problem formulations. These reformulations allow us to apply a generalisation with perturbations of the Weiszfeld algorithm in an attempt to find local approximate solutions to the Euclidean TSP.