Removal and Contraction Operations in nnD Generalized Maps for Efficient Homology Computation

In this paper, we show that contraction operations preserve the homology of $n$D generalized maps, under some conditions. Removal and contraction operations are used to propose an efficient algorithm that compute homology generators of $n$D generalized maps. Its principle consists in simplifying a generalized map as much as possible by using removal and contraction operations. We obtain a generalized map having the same homology than the initial one, while the number of cells decreased significantly. Keywords: $n$D Generalized Maps; Cellular Homology; Homology Generators; Contraction and Removal Operations.Comment: Research repor

3D Well-composed Polyhedral Complexes

A binary three-dimensional (3D) image $I$ is well-composed if the boundary surface of its continuous analog is a 2D manifold. Since 3D images are not often well-composed, there are several voxel-based methods ("repairing" algorithms) for turning them into well-composed ones but these methods either do not guarantee the topological equivalence between the original image and its corresponding well-composed one or involve sub-sampling the whole image. In this paper, we present a method to locally "repair" the cubical complex $Q(I)$ (embedded in $\mathbb{R}^3$) associated to $I$ to obtain a polyhedral complex $P(I)$ homotopy equivalent to $Q(I)$ such that the boundary of every connected component of $P(I)$ is a 2D manifold. The reparation is performed via a new codification system for $P(I)$ under the form of a 3D grayscale image that allows an efficient access to cells and their faces

Topological signature for periodic motion recognition

In this paper, we present an algorithm that computes the topological signature for a given periodic motion sequence. Such signature consists of a vector obtained by persistent homology which captures the topological and geometric changes of the object that models the motion. Two topological signatures are compared simply by the angle between the corresponding vectors. With respect to gait recognition, we have tested our method using only the lowest fourth part of the body's silhouette. In this way, the impact of variations in the upper part of the body, which are very frequent in real scenarios, decreases considerably. We have also tested our method using other periodic motions such as running or jumping. Finally, we formally prove that our method is robust to small perturbations in the input data and does not depend on the number of periods contained in the periodic motion sequence.Comment: arXiv admin note: substantial text overlap with arXiv:1707.0698

MONOGRAFÍA MOTIVACIÓN

MONOGRAFÍ

Geometric Objects and Cohomology Operations

Cohomology operations (including the cohomology ring) of a geometric object are finer algebraic invariants than the homology of it. In the literature, there exist various algorithms for computing the homology groups of simplicial complexes but concerning the algorithmic treatment of cohomology operations, very little is known. In this paper, we establish a version of the incremental algorithm for computing homology which saves algebraic information, allowing us the computation of the cup product and the effective evaluation of the primary and secondary cohomology operations on the cohomology of a finite simplicial complex. We study the computational complexity of these processes and a program in Mathematica for cohomology computations is presented.Comment: International Conference Computer Algebra and Scientific Computing CASC'02. Big Yalta, Crimea (Ucrania). Septiembre 200

Límites y alcances de la interpretación constitucional de la Suplencia de la Queja en Amparo Penal y los Derechos Humanos a partir de la Reforma Constitucional y Legal de 2011

El Juicio de Amparo en México con motivo de la reforma constitucional de 2011 y legal se ha transformado sustancialmente, principalmente su procedencia, ya que actualmente se ejerce contra normas generales, actos y omisiones de autoridad y excepcionalmente contra particulares que violen los derechos humanos consagrados en la Constitución y en los Tratados Internacionales en el que el Estado mexicano se aparte, procedimiento que es regulado por el sistema de Control Constitucional y Convencional; se puede tramitar por dos vías, la primera de manera indirecta ante Jueces de Distrito en competencia ordinaria, concurrente y auxiliar y ante los tribunales Unitarios de Circuito, y la segunda de manera directa ante los Tribunales Colegiados de Circuito, procedimientos en los que se deben observar una serie de principios sobre los cuales se regula su substanciación, algunos han subsistido en lo general, pero otros se han transformado, lo que ha impactado en sus excepciones, como es la denominada Suplencia de la Queja, sobre todo en materia penal, puesto que éste no será aplicado de manera total e indiscriminadamente, puesto que tiene alcances y límites, los que serán analizados para conocer su aplicabilidad y eficacia