R. Ayala

Digital Lighting Functions

In this paper a notion of lighting function is introduced as an axiomatized formalization of the “face membership rules” suggested by Kovalevsky. These functions are defined in the context of the framework for digital topology previously developed by the authors. This enlarged framework provides the (α, β)-connectedness (α, β e {6,18, 26}) defined on ℤ3 within the graph-based approach to digital topology. Furthermore, the Kong-Roscoe (α, β)-surfaces, with (α, β) ≠ (6, 6), (18, 6), are also found as particular cases of a more general notion of digital surface.

Determining the components of the complement of a digital (n-1)-manifold in Zn

The goal of this paper is to determine the components of the complement of digital manifolds in the standard cubical decomposition of Euclidean spaces for arbitrary dimensions. Our main result generalizes the Morgenthaler-Rosenfelds one for (26, 6)-surfaces in ℤ3 [9]. The proof of this generalization is based on a new approach to digital topology sketched in [5] and developed in [2].

Perfect discrete Morse functions on 2-complexes

Highlights? We get conditions under which a 2-complex admits a perfect discrete Morse function. ? We study the topology of comlpexes admitting such kind of functions. ? These results are a first step in the study of these functions on 3-manifolds. This paper is focused on the study of perfect discrete Morse functions on a 2-simplicial complex. These are those discrete Morse functions such that the number of critical i-simplices coincides with the ith Betti number of the complex. In particular, we establish conditions under which a 2-complex admits a perfect discrete Morse function and conversely, we get topological properties necessary for a 2-complex admitting such kind of functions. This approach is more general than the known results in the literature (Lewiner et al., 2003), since our study is not restricted to surfaces. These results can be considered as a first step in the study of perfect discrete Morse functions on 3-manifolds.

Study of DQE dependence with beam quality on GE Essential mammography flat panel

This paper deals with the analysis of the behavior of objective image quality parameters for the new GE Senographe Essential FFDM system, in particular its dependence with beam quality. The detector consists of an indirect conversion a‐Si flat panel coupled to a CsI:Tl scintillator. The system under study has gone through a series of relevant modifications in flat panel with respect to the previous model (GE Senographe DS 2000). These changes in the detector modify its performance and are intended to favor advanced applications like tomosynthesis, which uses harder beam spectra and lower doses per exposure than conventional FFDM. Although our system does not have tomosynthesis implemented, we noticed that most clinical explorations were performed by automatically selecting a harder spectrum than that of typical use in FFDM (Rh/Rh 28–30 kV instead of Mo/Mo 28 kV). Since flat‐panel optimization for tomosynthesis influences the usual FFDM clinical performance, the new detector behavior needed to be investigated. Therefore, the aim of our study is evaluating the dependence of the detector performance for different beam spectra and exposure levels. In this way, we covered the clinical beam quality range (Rh/Rh 28–30 kV) and we extended the study to even harder spectra (Rh/Rh 34 kV). Detector performance is quantified by means of modulation transfer function (MTF), normalized noise power spectrum (NNPS) and detective quantum efficiency (DQE). We found that flat‐panel optimization results in slightly – but statistically significant – higher DQE values as beam quality increases, which is contrary to the expected behavior. This positive correlation between beam quality and DQE is also diametrically opposite to that of the previous model by the same manufacturer. As a direct consequence, usual FFDM takes advantage of the changes in the detector, as less exposure is needed to achieve the same DQE if harder beams are used. PACS number: 87.59.ej

A digital index theorem

This paper is devoted to state and prove a Digital Index Theorem for digital (n - 1)-manifolds in a digital space (Rn, f), where f belongs to a large family of lighting functions on the standard cubical decomposition Rn of the n-dimensional Euclidean space. As an immediate consequence we obtain the corresponding theorems for all (α, β)-surfaces of Kong–Roscoe, with α, β ∈ {6, 18, 26} and (α, β) ≠ (6, 6), (18, 26), (26, 26), as well as for the strong 26-surfaces of Bertrand–Malgouyres.

Properly 3-realizable groups

A finitely presented group G is said to be properly 3-realizable if there exists a compact 2-polyhedron K with π 1 (K) ≅ G and whose universal cover K has the proper homotopy type of a (p.l.) 3-manifold with boundary. In this paper we show that, after taking wedge with a 2-sphere, this property does not depend on the choice of the compact 2-polyhedron K with π 1 (K) ≅ G. We also show that (i) all 0-ended and 2-ended groups are properly 3-realizable, and (ii) the class of properly 3-realizable groups is closed under amalgamated free products (HNN-extensions) over a finite cyclic group (as a step towards proving that ∞-ended groups are properly 3-realizable, assuming 1-ended groups are).

MORSE INEQUALITIES ON CERTAIN INFINITE 2-COMPLEXES

Using the notion of discrete Morse function introduced by R. Forman for finite cw -complexes, we generalize it to the infinite 2-dimensional case in order to get the corresponding version of the well-known discrete Morse inequalities on a non-compact triangulated 2-manifold without boundary and with finite homology. We also extend them for the more general case of a non-compact triangulated 2-pseudo-manifold with a finite number of critical simplices and finite homology.

An Elementary Approach to the Projective Dimension in Proper Homotopy Theory

Abstract This paper presents some basic properties of the category of trees of abelian groups. In particular an elementary proof of the existence of kernels in the category of “finitely generated” trees is included. Also a characterization of projective dimension in the class of “pro-finitely generated” trees is given in terms of the inverse limit functor.

Perfect discrete morse functions on triangulated 3-manifolds

This work is focused on characterizing the existence of a perfect discrete Morse function on a triangulated 3-manifold M, that is, a discrete Morse function satisfying that the numbers of critical simplices coincide with the corresponding Betti numbers. We reduce this problem to the existence of such kind of function on a spine L of M, that is, a 2-subcomplex L such that M−Δ collapses to L, where Δ is a tetrahedron of M. Also, considering the decomposition of every 3-manifold into prime factors, we prove that if every prime factor of M admits a perfect discrete Morse function, then M admits such kind of function.

A Digital Lighting Function for Strong 26-Surfaces

The goal of this paper is to generalize the notion of lighting function given in [3] in order to integrate strong 26-surfaces [5] into our framework for digital topology. In particular, the continuous analogue for strong 26-surfaces introduced in [10] is extended for arbitrary objects.