Antonio Giraldo

Digitally continuous multivalued functions

We introduce in this paper a notion of continuity in digital spaces which extends the usual notion of digital continuity. Our approach uses multivalued maps. We show how the multivalued approach provides a better framework to define topological notions, like retractions, in a far more realistic way than by using just single-valued digitally continuous functions. In particular, we characterize the deletion of simple points, one of the most important processing operations in digital topology, as a particular kind of retraction.

Computing the Hessenberg matrix associated with a self-similar measure

We introduce in this paper a method to calculate the Hessenberg matrix of a sum of measures from the Hessenberg matrices of the component measures. Our method extends the spectral techniques used by G. Mantica to calculate the Jacobi matrix associated with a sum of measures from the Jacobi matrices of each of the measures. We apply this method to approximate the Hessenberg matrix associated with a self-similar measure and compare it with the result obtained by a former method for self-similar measures which uses a fixed point theorem for moment matrices. Results are given for a series of classical examples of self-similar measures. Finally, we also apply the method introduced in this paper to some examples of sums of (not self-similar) measures obtaining the exact value of the sections of the Hessenberg matrix.

Characterization of the deletion of (26,6)-simple points as multivalued (N,26)-retractions

Abstract In previous papers we have introduced a notion of multivalued continuity in digital spaces which extends the usual notion of digital continuity and allows us to define topological notions, like retractions, in a far more realistic way than by using just single-valued digitally continuous functions. In particular, we have characterized the deletion of simple points in 2-D, one of the most important processing operations in digital topology, as a particular kind of retraction. In this work we extend this result to three-dimensional digital sets, characterizing the deletion of ( 26 , 6 ) -simple points in 3-D as ( N , 26 ) -retractions.

Ronse deletability conditions and ( N , k ) -retractions

Highlights? We consider retractions defined by digitally continuous multivalued maps. ? A special kind of retractions, called ( N , k ) -retractions, is considered. ? Deletion of simple points and many thinning algorithms are ( N , k ) -retractions. ? We characterize thinning algorithms that can be modeled as ( N , k ) -retractions. ? Characterization in terms of Ronse-like conditions for topology preservation. In some recent papers we have introduced a notion of continuity in digital spaces which extends the usual notion of digital continuity. Our approach, which uses multivalued maps, provides a better framework to define topological notions, like retractions, in a far more realistic way than by using just single-valued digitally continuous functions. In particular, we have characterized the deletion of simple points and some well known parallel thinning algorithm as a particular type of retractions, called ( N , k ) -retractions.In this paper we give a full characterization of the thinning algorithms that can be modeled as ( N , k ) -retractions. This class agrees, with minor modifications, with the class of thinning algorithms satisfying Ronse sufficient conditions for preservation of topology.

Thinning Algorithms as Multivalued {\mathcal{N}}-Retractions

In a recent paper we have introduced a notion of continuity in digital spaces which extends the usual notion of digital continuity. Our approach, which uses multivalued maps, provides a better framework to define topological notions, like retractions, in a far more realistic way than by using just single-valued digitally continuous functions. In particular, we characterized the deletion of simple points, one of the most important processing operations in digital topology, as a particular kind of retraction.

The Hessenberg matrix and the Riemann mapping function

Abstract We consider a Jordan arc Γ in the complex plane

Deletion of (26,6)-simple points as multivalued retractions

{\mathbb C}

Starting out at university with team projects

and a regular measure μ whose support is Γ. We denote by D the upper Hessenberg matrix of the multiplication by z operator with respect to the orthonormal polynomial basis associated with μ. We show in this work that, if the Hessenberg matrix D is uniformly asymptotically Toeplitz, then the symbol of the limit operator is the restriction to the unit circle of the Riemann mapping function ϕ(z) which maps conformally the exterior of the unit disk onto the exterior of the support of the measure μ. We use this result to show how to approximate the Riemann mapping function for the support of μ from the entries of the Hessenberg matrix D.

Digital topology java applet

In a recent paper we have introduced a notion of multivalued continuity in digital spaces which extends the usual notion of digital continuity and allows to define topological notions, like retractions, in a far more realistic way than by using just single-valued digitally continuous functions. In particular, we have characterized the deletion of simple points in 2-D, one of the most important processing operations in digital topology, as a particular kind of retraction. In this work we extend some of these results to 3-dimensional digital sets.

Multidisciplinary projects for first year engineering courses

In this work we describe the design and outcomes of a Starting-Out Project, which was conducted at the Universidad Politécnica de Madrids School of Computing before students started their courses, as a welcoming and guidance activity. One goal is to motivate students, while, at the same time, familiarizing them with the institution, its services and associations, and fostering social integration. As an educational objective, the project intends to train and assess students in basic horizontal skills related to teamwork and effective oral communication.