arXiv subjects

Luis Martínez

Publications and source records attributed to Luis Martínez.

At least 19 recordsLinked to original sources

arXiv2026

A Framework for Reputation Aware Uninorm-driven Consensus Algorithms for Blockchain Networks

The operation of blockchain is governed by consensus algorithms (CA). Several consensus mechanisms require significant computational power, while others necessitate high amounts of stakes to select the participant to validate and verify the transactions in the block, leading to centralisation of power and participant exclusion. This paper proposes a novel methodology to address these issues in reputation-based consensus algorithms by studying the reputation behaviour of the validator using intuitionistic fuzzy sets (IFSs) and uninorm aggregation operations (UAOs). Our approach uses IFSs to express the "reputation" because the reputation values in a consensus algorithm eventually imply uncertainty, and IFSs facilitate the representation of a lack of precise knowledge about reputation. Moreover, this methodology utilises uninorm aggregation operations to monitor reputation over time and reinforces the importance of negative and positive reputation. Consequently, this solution allows validators to rectify past failures in subsequent verification processes and foster an equitable consensus algorithm design. The proposed framework maintains linear computational complexity and does not introduce additional communication overhead beyond the underlying consensus protocol. Supported by experimental results, our methodology demonstrates improved performance and evaluation, promising advancements in blockchain network fairness and inclusivity.

cs.DC↗

arXiv2026

Wythoff-Fibonacci Sequences and a Perturbed Greedy Almost-involution

We introduce the lower and upper Wythoff-Fibonacci sequences, obtained from the classical Wythoff sequences by a Fibonacci correction. Specifically, if we put $$ε(j)=\begin{cases}(-1)^k, & \text{if }j=F_k\text{ for some }k\\ 0, & \text{in other case}\end{cases},$$ where $F_k$ is the $k$-th Fibonacci number, then we define the general terms of the lower and upper Wythoff-Fibonacci sequences by $$LWF(n)=\begin{cases} 1, & \text{if }n=1,\\ 3, & \text{if }n=2,\\ a(n)+ε(n), & \text{if }n\geq 3.\end{cases}$$ and $$UWF(n)=\begin{cases} 2, & \text{if }n=1,\\ b(n)+ε(n), & \text{if }n\geq 2,\end{cases}$$ respectively. We show that these sequences partition the set of natural numbers and use them to give an explicit formula for a sequence $q^{\star}_j$, defined from a greedy construction studied by the first author and his coauthors in a previous paper, but with the additional condition that $q^{\star}_1=3$, instead of being defined by the greedy rule. This sequence is a permutation of the set of non-negative integers and has the property that every integer appears exactly once in the sequence of differences $q^{\star}_j-j$. We prove that $q^{\star}_{q^{\star}_j}=j\ \forall j\geq 5$, so that $q^{\star}_j$ is an almost-involution. We also give another greedy algorithm generating $q^{\star}_j$.

math.CO↗

arXiv2026

Almost Orthogonal Arrays: Search Three Ways

Orthogonal arrays play a fundamental role in many applications. However, constructing orthogonal arrays with the required parameters for an application usually is extremely difficult and, sometimes, even impossible. Hence there is an increasing need for a relaxation of orthogonal arrays to allow a wider flexibility. The latter has lead to various types of arrays under the name of ``nearly-orthogonal arrays'', and less often ``almost orthogonal arrays''. In this paper, we explore how to find almost orthogonal arrays three ways: using integer programming, local search meta-heuristics and algebraic methods. We compare all our search results with the ones existing in the literature, and we show that they are competitive, improving some of the existing arrays for many non-orthogonality measures. All our found almost orthogonal arrays are available at a public repository.

math.CO↗

arXiv2025

Least Absolute Deviation Utility for Trapezoidal Fuzzy Preference Relations

Preference relations (PRs) are widely used to model expert judgments because they allow for eliciting the decision-makers' opinions from pairwise comparisons. Traditionally, PRs have been elicited using real numbers. However, in real-world decision-makers usually feel more comfortable using linguistic expressions closer to natural language. In this context, our purpose is to extend the classical idea of PR into the environment of Trapezoidal Fuzzy Numbers (TrFNs) by addressing several drawbacks in current research. Existing fuzzy extensions for Fuzzy Preference Relations (FPRs) and Multiplicative Preference Relations (MPRs) assume that the notion of neutrality must be modeled by a crisp real number, which fails to capture the subjective and diverse ways in which decision-makers may perceive indifference. Moreover, current research lacks a theoretical framework that unifies both FPRs and MPRs for fuzzy numbers and simultaneously generalizes the involved classical notions (e.g., reciprocity, consistency) from the perspective of fuzzy arithmetic. Therefore, we introduce the notion of neutral TrFN, a fuzzy number that can model indifference on a certain fuzzy scale. This provides a flexible representation of neutrality, leading to a more realistic extension of PRs into the fuzzy environment. Building upon this, we propose the Trapezoidal FPR (TrFPR), analyze its properties and how reciprocity and consistency are extended using fuzzy arithmetic. Furthermore, we introduce the Least Absolute Deviation (LAD) utility vector associated with a consistent TrFPR and develop an optimization model to derive it from an inconsistent TrFPR to compute priority values. These ideas are also extended to MPRs in the fuzzy domain. Finally, the applicability of the proposed LAD utility method is demonstrated in evaluating land development projects, where comparison with fuzzy AHP illustrates its superiority.

math.GM↗

arXiv2025

Mutual Consensus and its Application in Minimum Cost Consensus Models

This paper introduces the concept of {mutual consensus} as a novel non-compensatory consensus measure that accounts for the maximum disparity among opinions to ensure robust consensus evaluation. Incorporating this concept, several new Minimum Cost Consensus (MCC) models are proposed, and their properties are analyzed. To show their applicability, these mutual consensus-based MCC models are then considered in the context of the {OWA-MCC} model, which employs Ordered Weighted Averaging (OWA) operators for preference aggregation. Concretely, we include a linearized formulation under symmetry conditions as well as examples of the non-convexity of the feasible region in the general case. Finally, mutual consensus is utilized to obtain approximate solutions for the OWA-MCC model, demonstrating its practical effectiveness and advancing the theoretical and applied dimensions of consensus modeling in group decision-making.

math.OC↗

arXiv2025

Enhancing Explainability and Reliable Decision-Making in Particle Swarm Optimization through Communication Topologies

Swarm intelligence effectively optimizes complex systems across fields like engineering and healthcare, yet algorithm solutions often suffer from low reliability due to unclear configurations and hyperparameters. This study analyzes Particle Swarm Optimization (PSO), focusing on how different communication topologies Ring, Star, and Von Neumann affect convergence and search behaviors. Using an adapted IOHxplainer , an explainable benchmarking tool, we investigate how these topologies influence information flow, diversity, and convergence speed, clarifying the balance between exploration and exploitation. Through visualization and statistical analysis, the research enhances interpretability of PSO's decisions and provides practical guidelines for choosing suitable topologies for specific optimization tasks. Ultimately, this contributes to making swarm based optimization more transparent, robust, and trustworthy.

cs.LG↗

arXiv2025

Fibonacci-like partitions and their associated piecewise-defined permutations

In this paper we introduce a family of partitions of the set of natural numbers, Fibonacci-like partitions. In particular, we introduce a Fibonacci-like partition in a number of parts corresponding to the Fibonacci numbers, the standard Fibonacci-like partitions of the first kind. That partition refines the well-known partition in two parts associated to the Wythoff sequences. In a similar way, we introduce another family of Fibonacci-like partitions of the natural numbers, with an initial segment removed, in a number of parts which also follows the Fibonacci sequence, the standard Fibonacci-like partitions of the second kind. We study some piecewise-defined permutations over Fibonacci-like partitions, which give both torsion-free and torsion elements in the symmetric group over the set of natural numbers. We give an application of Fibonacci-like partitions to the study of a sequence analyzed by A. Shapovalov and by B.J. Venkatachala. We give also a generalization of the mentioned sequence, involving techniques that are not related to Fibbonaci numbers nor Beatty sequences, obtaining an infinite family of sequences that we conjecture that produce an infinite orthogonal array in which the number of symbols, trials and factors is countable infinite, and having an automorphism group isomorphic to Z acting regularly on the symbol set.

math.CO↗

arXiv2025

Building Interval Type-2 Fuzzy Membership Function: A Deck of Cards based Co-constructive Approach

Since its inception, Fuzzy Set has been widely used to handle uncertainty and imprecision in decision-making. However, conventional fuzzy sets, often referred to as type-1 fuzzy sets (T1FSs) have limitations in capturing higher levels of uncertainty, particularly when decision-makers (DMs) express hesitation or ambiguity in membership degree. To address this, Interval Type-2 Fuzzy Sets (IT2FSs) have been introduced by incorporating uncertainty in membership degree allocation, which enhanced flexibility in modelling subjective judgments. Despite their advantages, existing IT2FS construction methods often lack active involvement from DMs and that limits the interpretability and effectiveness of decision models. This study proposes a socio-technical co-constructive approach for developing IT2FS models of linguistic terms by facilitating the active involvement of DMs in preference elicitation and its application in multicriteria decision-making (MCDM) problems. Our methodology is structured in two phases. The first phase involves an interactive process between the DM and the decision analyst, in which a modified version of Deck-of-Cards (DoC) method is proposed to construct T1FS membership functions on a ratio scale. We then extend this method to incorporate ambiguity in subjective judgment and that resulted in an IT2FS model that better captures uncertainty in DM's linguistic assessments. The second phase formalizes the constructed IT2FS model for application in MCDM by defining an appropriate mathematical representation of such information, aggregation rules, and an admissible ordering principle. The proposed framework enhances the reliability and effectiveness of fuzzy decision-making not only by accurately representing DM's personalized semantics of linguistic information.

cs.AI↗

arXiv2024

Fuzzychain: An Equitable Consensus Mechanism for Blockchain Networks

Blockchain technology has become a trusted method for establishing secure and transparent transactions through a distributed, encrypted network. The operation of blockchain is governed by consensus algorithms, among which Proof of Stake (PoS) is popular yet has its drawbacks, notably the potential for centralising power in nodes with larger stakes or higher rewards. Fuzzychain, our proposed solution, introduces the use of fuzzy sets to define stake semantics, promoting decentralised and distributed processing control. This system selects validators based on their degree of membership to the stake fuzzy sets rather than just the size of their stakes. As a pioneer proposal in applying fuzzy sets to blockchain, Fuzzychain aims to rectify PoS's limitations. Our results indicate that Fuzzychain not only matches PoS in functionality but also ensures a fairer distribution of stakes among validators, leading to more inclusive validator selection and a better-distributed network.

cs.CR↗

arXiv2024

Generalized tableaux over arbitrary digraphs and their associated differential equations

We revisit the concepts of acyclic orderings and number of acyclic orderings of acyclic digraphs in terms of dispositions and counters for arbitrary multidigraphs. We prove that when we add a sequence of nested directed paths to a directed graph there is a unique polynomial such that the generatrix function of the family of counters is the product of the polynomial and the exponential function. We give an application, by considering a kind of digraphs arranged in rows introduced by the authors in a previous paper, called dispositional digraphs, in the particular case in which the digraph has two rows, to obtain new families of linear differential equations of small order whose coefficients are polynomials of small degree which admit polynomial solutions. In particular, we obtain a new differential equation associated to Catalan numbers, and the corresponding associated polynomials, which are solution of this differential equation; we term them Catalan differencial equation and Catalan polynomials, respectively. We prove that the Catalan polynomials obtained when we connect the directed path to the second vertex of the lower row of the digraph are orthogonal polynomials for an appropriate weight function. We characterize the digraphs that maximize the counter of connected dispositional digraphs and we find a new differential equation associated to these digraphs. We introduce also dispositions and counters in any multidigraph with non-strict inequalities in the dispositions, and we find new differential equations associated to some of them.

math.CO↗

arXiv2024

Twisted products: Enveloping actions and equivariant absolute neighborhood extensors

The classical notion of twisted product is studied in the context of partial actions, in particular, we show that the globalization of a partial action is a twisted product. In addition, we establish conditions for the metrizability of twisted products, and some homotopy and categorical properties are proved. Furthermore, sufficient conditions for the enveloping space to be an equivariant absolute neighborhood extensor are also studied.

math.GN↗

arXiv2023

A note on adjunction spaces and G-ANE's

We provide a different proof of the equivariant version of the Borsuk-Whitehead-Hanner Theorem in the category of proper G-spaces which are metrizable by a G-invariant metric.

math.GN↗

arXiv2023

Creating Synthetic Datasets for Collaborative Filtering Recommender Systems using Generative Adversarial Networks

Research and education in machine learning needs diverse, representative, and open datasets that contain sufficient samples to handle the necessary training, validation, and testing tasks. Currently, the Recommender Systems area includes a large number of subfields in which accuracy and beyond accuracy quality measures are continuously improved. To feed this research variety, it is necessary and convenient to reinforce the existing datasets with synthetic ones. This paper proposes a Generative Adversarial Network (GAN)-based method to generate collaborative filtering datasets in a parameterized way, by selecting their preferred number of users, items, samples, and stochastic variability. This parameterization cannot be made using regular GANs. Our GAN model is fed with dense, short, and continuous embedding representations of items and users, instead of sparse, large, and discrete vectors, to make an accurate and quick learning, compared to the traditional approach based on large and sparse input vectors. The proposed architecture includes a DeepMF model to extract the dense user and item embeddings, as well as a clustering process to convert from the dense GAN generated samples to the discrete and sparse ones, necessary to create each required synthetic dataset. The results of three different source datasets show adequate distributions and expected quality values and evolutions on the generated datasets compared to the source ones. Synthetic datasets and source codes are available to researchers.

cs.IR↗

arXiv2023

Partial actions of groups on profinite spaces

We show that for a partial action $η$ with closed domain of a compact group $G$ on a profinite space $X$ the space of orbits $X/\!\sim_G$ is profinite, this leads to the fact that when $G$ is profinite the enveloping space $X_G$ is also profinite. Moreover, we provide conditions for the induced quotient map $π_G: X\to X/\!\sim_G$ of $η$ to have a continuous section. Relations between continuous sections of $π_G$ and continuous sections of the quotient map induced by the enveloping action of $η$ are also considered. At the end of this work we prove that the category of actions on profinite spaces with countable number of clopen sets is reflective in the category of actions of compact Hausdorff spaces having countable number of clopen sets.

math.GN↗

arXiv2021

A topological correspondence between partial actions of groups and inverse semigroup actions

We present some generalizations of the well-known correspondence, found by R. Exel, between partial actions of a group $G$ on a set $X$ and semigroup homomorphism of $S(G)$ on the semigroup $I(X)$ of partial bijections of $X,$ being $S(G)$ an inverse monoid introduced by Exel. We show that any unital premorphism $θ:G\to S$, where $S$ is an inverse monoid, can be extended to a semigroup homomorphism $θ^*:T\to S$ for any inverse semigroup $T$ with $S(G)\subseteq T\subseteq P^*(G)\times G,$ being $P^*(G)$ the semigroup of non-empty subset of $G$, and such that $E(S)$ satisfies some lattice theoretical condition. We also consider a topological version of this result. We present a minimal Hausdorff inverse semigroup topology on $Γ(X)$, the inverse semigroup of partial homeomorphism between open subsets of a locally compact Hausdorff space $X$.

math.GN↗

arXiv2021

Relative Torsion Classes, relative tilting and relative silting modules

Let $Λ$ be an Artin algebra. In 2014, T. Adachi, O. Iyama and I. Reiten proved that the torsion funtorially finite classes in $\mathrm{mod}\,(Λ)$ can be described by the $τ$-tilting theory. The aim of this paper is to introduce the notion of $F$-torsion class in $\mathrm{mod}\,(Λ)$, where $F$ is an additive subfunctor of $\mathrm{Ext}^1_Λ,$ and to characterize when these clases are preenveloping and $F$-preenveloping. In order to do that, we introduce the notion of $F$-presilting $Λ$-module. The latter is both a generalization of $τ$-rigid and $F$-tilting in $\mathrm{mod}(Λ).$

math.RT↗

arXiv2020

On the structure of nearly epsilon and epsilon-strongly graded rings

In this work we study the classes of epsilon and nearly epsilon-strongly graded rings by a group $G$. In particular, we extend Dade's theorem to the realm of nearly epsilon-strongly graded rings. Moreover, we introduce the category SIM$S-$gr of symmetrically graded modules and use it to present a new characterization of strongly graded rings. A functorial approach is used to obtain a characterization of epsilon-strongly graded rings. Finally, we determine conditions for which an epsilon-strongly graded ring can be written as a direct sum of strongly graded rings and a trivially graded ring.

math.RA↗

arXiv2019

The GC-content of a family of cyclic codes with applications to DNA-codes

Given a prime power $q$ and a positive integer $r>1$ we say that a cyclic code of length $n$, $C\subseteq F_{q^r}^n$, is Galois supplemented if for any non-trivial element $σ$ in the Galois group of the extension $ F_{q^r}/ F_q$, $C+C^σ= F_{q^r}^n$, where $C^σ=\{(x_1^σ,\dots,x_n^σ)\mid (x_1,\dots,x_n)\in C\}$. This family includes the quadratic-residue (QR) codes over $ F_{q^2}$. Some important properties QR-codes are then extended to Galois supplemented codes and a new one is also considered, which is actually the motivation for the introduction of this family of codes: in a Galois supplemented code we can explicitly count the number of words that have a fixed number of coordinates in $ F_q$. In connection with DNA-codes the number of coordinates of a word in $ F_4^n$ that lie in $ F_2$ is sometimes referred to as the $GC$-content of the word and codes over $ F_4$ all of whose words have the same $GC$-content have a particular interest. Therefore our results have some direct applications in this direction.

cs.IT↗