This study builds upon results concerning the preservation of fuzzy structures by aggregation operators. We analyze when an aggregation operator preserves the structure of T-subgroup in the last cases remaining unresolved in the characterization of such operators. In order to characterize these operators for different groups, relaxed forms of domination are introduced and their properties and interconnections are studied. We also include a thorough review of the features that an aggregation operator must fulfill to preserve T-subgroups of an arbitrary group.

Our study is centered on the aggregation of fuzzy graphs, looking for conditions under which the aggregation process yields another fuzzy graph. We conduct an in-depth analysis of the preservation of several important properties and structures inherent to fuzzy graphs, like paths, cycles, or bridges. In addition we obtain appropriate criteria for when the aggregation of complete fuzzy graphs is again a complete fuzzy graph.

We study when an aggregation function acting on an n-tuple of T-subgroups preserves the structure of T-subgroup. First we need to consider that there are two known definitions applicable to the aggregation of structures on fuzzy sets. These two notions differ in the domain of the aggregated structure. It is known that for indistinguishability operators, pseudometrics, quasi-pseudometrics among others, the aggregation functions that preserve these structures are the same with both definitions. However this is not the case for quasi-metrics. In this line we study the aggregation of T-subgroups with both definitions and their implications. We see that aggregation functions may preserve the structure of T-subgroups with one definition but not with the other. However, under adequate restrictions, the aggregation functions preserving the structure of T-subgroups are the same with either definition. We also show that the results depend on the structure of the subgroup lattice of the ambient group, the particular T-subgroups being aggregated, or the aggregation function.

In this work, we study when an aggregation operator preserves the structure of T-subgroup of groups whose subgroup lattice is a chain. There are two widely used ways of defining the aggregation of structures in fuzzy logic, previously named on sets and on products. We will focus our attention on the one called aggregation on products. When the lattice of subgroups is not a chain, it is known that the dominance relation between the aggregation operator and the t-norm is crucial. We show that this property is again important for some of the groups in this study. However, for the rest of them, we must define a new property weaker than domination, that will allow us to characterize those operators which preserve T-subgroups.

Copy number changes play an important role in the development of cancer and are commonly associated with changes in gene expression. Persistence curves, such as Betti curves, have been used to detect copy number changes; however, it is known these curves are unstable with respect to small perturbations in the data. We address the stability of lifespan and Betti curves by providing bounds on the distance between persistence curves of Vietoris–Rips filtrations built on data and slightly perturbed data in terms of the bottleneck distance. Next, we perform simulations to compare the predictive ability of Betti curves, lifespan curves (conditionally stable) and stable persistent landscapes to detect copy number aberrations. We use these methods to identify significant chromosome regions associated with the four major molecular subtypes of breast cancer: Luminal A, Luminal B, Basal and HER2 positive. Identified segments are then used as predictor variables to build machine learning models which classify patients as one of the four subtypes. We find that no single persistence curve outperforms the others and instead suggest a complementary approach using a suite of persistence curves. In this study, we identified new cytobands associated with three of the subtypes: 1q21.1-q25.2, 2p23.2-p16.3, 23q26.2-q28 with the Basal subtype, 8p22-p11.1 with Luminal B and 2q12.1-q21.1 and 5p14.3-p12 with Luminal A. These segments are validated by the TCGA BRCA cohort dataset except for those found for Luminal A.

Preservation of structures under aggregation functions is an active area of research with applications in many fields. Among such structures, min-subgroups play an important role, for instance, in mathematical morphology, where they can be used to model translation invariance. Aggregation of min-subgroups has only been studied for binary aggregation functions. However, results concerning preservation of the min-subgroup structure under binary aggregations do not generalize to aggregation functions with arbitrary input size since they are not associative. In this article, we prove that arbitrary self-aggregation functions preserve the min-subgroup structure. Moreover, we show that whenever the aggregation function is strictly increasing on its diagonal, a min-subgroup and its self-aggregation have the same level sets.

The paper studies the aggregation of pairs of T-indistinguishability operators. More concretely we address the question whether A(E,E′) is a T-indistinguishability operator if E,E′ are T-indistinguishability operators. The answer depends on the aggregation function, the t-norm T, and the chosen T-indistinguishability operators. It is well-known that an aggregation function preserves T-transitive relations if and only if it dominates the t-norm T. We show the important role of the minimum t-norm TM in this preservation problem. In particular we develop weaker forms of domination that are used to provide characterizations of TM-indistinguishability preservation under aggregation. We also prove that the existence of a single strictly monotone aggregation that satisfies the indistinguishability operator preservation property guarantees all aggregations to have the same preservation property.

We implement the notion of annihilator into fuzzy subgroups of an abelian group. There are different uses for the term annihilator in algebraic contexts that have been used fuzzy systems. In this paper we refer to another type of annihilator which is essential in classical duality theory and extends the widely applied notion of orthogonal complement in Euclidean spaces. We find that in the natural algebraic duality of a group, a fuzzy subgroup can be recovered after taking the inverse annihilator of its annihilator. We also study the behavior of annihilators with respect to unions and intersections. Some illustrative examples of annihilators of fuzzy subgroups are shown, both with finite and infinite rank.

Numerous problems encountered in computational biology can be formulated as optimization problems. In this context, optimization of drug release characteristics or dosing schedules for anticancer agents has become a prominent area not only for the development of new drugs, but also for established drugs. However, in complex systems, optimization of drug exposure is not a trivial task and cannot be efficiently addressed through trial-error simulation exercises. Finding a solution to those problems is a challenging task which requires more advanced strategies like optimal control theory. In this work, we perform an optimal control analysis on a previously developed computational model for the testosterone effects of triptorelin in prostate cancer patients with the goal of finding optimal drug-release characteristics. We demonstrate how numerical control optimization of non-linear models can be used to find better therapeutic approaches in order to improve the final outcome of the patients.

Polycrystals of thin colloidal deposits, with thickness controlled by spin-coating speed, exhibit axial symmetry with local 4-fold and 6-fold symmetric structures, termed orientationally correlated polycrystals (OCPs). While spin-coating is a very facile technique for producing large-area colloidal deposits, the axial symmetry prevents us from achieving true long-range order. To obtain true long-range order, we break this axial symmetry by introducing a patterned surface topography and thus eliminate the OCP character. We then examine symmetry-independent methods to quantify order in these disordered colloidal deposits. We find that all the information in the bond-orientational order parameters is well captured by persistent homology analysis methods that only use the centers of the particles as input data. It is expected that these methods will prove useful in characterizing other disordered structures.

DNA copy number aberrations (CNAs) play an important role in cancer and can be experimentally detected using microarray comparative genomic hybridization (CGH) techniques. Amplicons, CNAs that extend over large sections of the genome, are difficult to study since they may contain multiple independent and dependent copy number changes. Here, we propose an algorithm to find the CNAs structure within a given amplicon. Our method relies on the observation that co-occurring CNAs can be encoded as 1-dimensional cycles. Applying this method to breast cancer patients known as ERBB2/HER2/NEU amplified we find three regions that can be co-occuring: the first region is in the cytoband 17q12, where the ERBB2 gene is located, the second region expands between 17q21.2 to 17q21.31 and includes the keratin genes, the third one is 17q21.33. We suggest that the first homology group helps uncovering the structure of amplicons.

We use the first Betti number of a complex to analyze the morphological structure of granular samples in mechanical equilibrium. We investigate two-dimensional granular packings after a tapping process by means of both simulations and experiments. States with equal packing fraction obtained with different tapping intensities are distinguished after the introduction of a filtration parameter which determines the particles (nodes in the network) that are joined by an edge. This is accomplished by just using the position of the particles obtained experimentally and no other information about the possible contacts, or magnitude of forces.

We use the first Betti number of a complex to characterize the morphological structure of granular samples in mechanical equilibrium. We analyze two-dimensional granular packings after a tapping process by means of both simulations and experiments. States with equal packing fraction obtained with different tapping intensities are distinguished after the introduction of a filtration parameter which determines the particles (nodes in the network) that are joined by an edge. We first use numerical simulations to characterize the effect of the precision in the particles localization by artificially adding different levels of noise in this magnitude. The outcomes obtained for the simulations are then compared with the experimental results allowing a clear distinction of experimental packings that have the same density. This is accomplished by just using the position of the particles and no other information about the possible contacts, or magnitude of forces.

We present a series of examples of precompact, noncompact, reflexive topological Abelian groups. Some of them are pseudocompact or even countably compact, but we show that there exist precompact non-pseudocompact reflexive groups as well. It is also proved that every pseudocompact Abelian group is a quotient of a reflexive pseudocompact group with respect to a closed reflexive pseudocompact subgroup.

The existence of small order loops of contacts is presented as an intrinsic characteristic of force granular networks. Based on molecular dynamics simulations, it is proposed that the presence of these small order loops — and in particular third order loops of contacts — is important to understand the transition from fluid-like to solid-like behavior of granular packings. In addition, we show a close relationship between the development of third order loops and the small forces of the granular packing in the sense that almost all third order loops allocate a force component smaller than the average.

Semantic memory is the subsystem of human memory that stores knowledge of concepts or meanings, as opposed to life-specific experiences. How humans organize semantic information remains poorly understood. In an effort to better understand this issue, we conducted a verbal fluency experiment on 200 participants with the aim of inferring and representing the conceptual storage structure of the natural category of animals as a network. This was done by formulating a statistical framework for co-occurring concepts that aims to infer significant concept–concept associations and represent them as a graph. The resulting network was analyzed and enriched by means of a missing links recovery criterion based on modularity. Both network models were compared to a thresholded co-occurrence approach. They were evaluated using a random subset of verbal fluency tests and comparing the network outcomes (linked pairs are clustering transitions and disconnected pairs are switching transitions) to the outcomes of two expert human raters. Results show that the network models proposed in this study overcome a thresholded co-occurrence approach, and their outcomes are in high agreement with human evaluations. Finally, the interplay between conceptual structure and retrieval mechanisms is discussed.

Semantic memory is the subsystem of human memory that stores knowledge of concepts or meanings, as opposed to life specific experiences. The organization of concepts within semantic memory can be understood as a semantic network, where the concepts (nodes) are associated (linked) to others depending on perceptions, similarities, etc. Lexical access is the complementary part of this system and allows the retrieval of such organized knowledge. While conceptual information is stored under certain underlying organization (and thus gives rise to a specific topology), it is crucial to have an accurate access to any of the information units, e. g. the concepts, for efficiently retrieving semantic information for real-time need. An example of an information retrieval process occurs in verbal fluency tasks, and it is known to involve two different mechanisms: "clustering", or generating words within a subcategory, and, when a subcategory is exhausted, "switching" to a new subcategory. We extended this approach to random-walking on a network (clustering) in combination to jumping (switching) to any node with certain probability and derived its analytical expression based on Markov chains. Results show that this dual mechanism contributes to optimize the exploration of different network models in terms of the mean first passage time. Additionally, this cognitive inspired dual mechanism opens a new framework to better understand and evaluate exploration, propagation and transport phenomena in other complex systems where switching-like phenomena are feasible.