arXiv subjects

Miguel Lopez

Publications and source records attributed to Miguel Lopez.

7 recordsLinked to original sources

arXiv2026

Asymmetrically Weighted Dowker Persistence and Applications in Dynamical Systems

By their nature it is difficult to differentiate chaotic dynamical systems through measurement. In recent years, work has begun on using methods of Topological Data Analysis (TDA) to qualitatively type dynamical data by approximating the topology of the underlying attracting set. This comes with the additional challenges of high dimensionality incurring computational complexity along with the lack of directional information encoded in the approximated topology. Due to the latter fact, standard methods of TDA for this high dimensional dynamical data do not differentiate between periodic cycles and non-periodic cycles in the attractor. We present a framework to address both of these challenges. We begin by binning the dynamical data, and capturing the sequential information in the form of a coarse-grained weighted and directed network. We then calculate the persistent Dowker homology of the asymmetric network, encoding spatial and temporal information. Analytically, we highlight the differences in periodic and non-periodic cycles by providing a full characterization of their one-dimensional Dowker persistences. We prove how the homologies of graph wedge sums can be described in terms of the wedge component homologies. Finally, we generalize our characterization to cactus graphs with arbitrary edge weights and orientations. Our analytical results give insight into how our method captures temporal information in its asymmetry, producing a persistence framework robust to noise and sensitive to dynamical structure.

math.AT↗

arXiv2025

Addressing Bias in LLMs: Strategies and Application to Fair AI-based Recruitment

The use of language technologies in high-stake settings is increasing in recent years, mostly motivated by the success of Large Language Models (LLMs). However, despite the great performance of LLMs, they are are susceptible to ethical concerns, such as demographic biases, accountability, or privacy. This work seeks to analyze the capacity of Transformers-based systems to learn demographic biases present in the data, using a case study on AI-based automated recruitment. We propose a privacy-enhancing framework to reduce gender information from the learning pipeline as a way to mitigate biased behaviors in the final tools. Our experiments analyze the influence of data biases on systems built on two different LLMs, and how the proposed framework effectively prevents trained systems from reproducing the bias in the data.

cs.AI↗

arXiv2025

Clearing Sections of Lattice Liability Networks

Modern financial networks involve complex obligations that transcend simple monetary debts: multiple currencies, prioritized claims, supply chain dependencies, and more. We present a mathematical framework that unifies and extends these scenarios by recasting the classical Eisenberg-Noe model of financial clearing in terms of lattice liability networks. Each node in the network carries a complete lattice of possible states, while edges encode nominal liabilities. Our framework generalizes the scalar-valued clearing vectors of the classical model to lattice-valued clearing sections, preserving the elegant fixed-point structure while dramatically expanding its descriptive power. Our main theorem establishes that such networks possess clearing sections that themselves form a complete lattice under the product order. This structure theorem enables tractable analysis of equilibria in diverse domains, including multi-currency financial systems, decentralized finance with automated market makers, supply chains with resource transformation, and permission networks with complex authorization structures. We further extend our framework to chain-complete lattices for term structure models and multivalued mappings for complex negotiation systems. Our results demonstrate how lattice theory provides a natural language for understanding complex network dynamics across multiple domains, creating a unified mathematical foundation for analyzing systemic risk, resource allocation, and network stability.

q-fin.MF↗

arXiv2025

Categorical Diffusion of Weighted Lattices

We introduce a categorical formalization of diffusion processes for network-structured data, motivated by applications in data science and information dynamics. At the heart of our construction is the Lawvere Laplacian, an endofunctor on a product category indexed by a graph and enriched in a quantale. This framework enables the systematic study of diffusion processes on network sheaves taking values in categories enriched in quantales, analogous to classical diffusion operators on metric spaces and vector spaces. Our main theoretical contribution extends Tarski's fixed point theorem to the quantale-enriched categorical setting, establishing that both prefix and suffix points of the Lawvere Laplacian form complete quantale-enriched categories. We develop a discrete-time distributed algorithm - harmonic flow - that provably converges to these fixed points, providing a constructive method for computing fuzzy global sections. This computational approach bridges sheaf-theoretic and dynamical perspectives on network diffusion, with applications ranging from discrete event systems to preference dynamics and path-finding problems.

math.CT↗

arXiv2024

Lattice-Valued Bottleneck Duality

This note reformulates certain classical combinatorial duality theorems in the context of order lattices. For source-target networks, we generalize bottleneck path-cut and flow-cut duality results to edges with capacities in a distributive lattice. For posets, we generalize a bottleneck version of Dilworth's theorem, again weighted in a distributive lattice. These results are applicable to a wide array of non-numerical network flow problems, as shown. All results, proofs, and applications were created in collaboration with AI language models. An appendix documents their role and impact.

math.OC↗

arXiv2024

Equivariant Cosheaves and Finite Group Representations in Graphic Statics

This work extends the theory of reciprocal diagrams in graphic statics to frameworks that are invariant under finite group actions by utilizing the homology and representation theory of cellular cosheaves, recent tools from applied algebraic topology. By introducing the structure of an equivariant cellular cosheaf, we prove that pairs of self-stresses and reciprocal diagrams of symmetric frameworks are classified by the irreducible representations of the underlying group. We further derive the symmetry-aligned Euler characteristics of a finite dimensional equivariant chain complex, which for the force cosheaf yields a new formulation of the symmetry-adapted Maxwell counting rule for detecting symmetric self-stresses and kinematic degrees of freedom in frameworks. A freely available program is used to implement the relevant cosheaf homologies and illustrate the theory with examples.

math.AT↗

arXiv2018

Combinatorics of $k$-Farey graphs

With an eye towards studying curve systems on low-complexity surfaces, we introduce and analyze the $k$-Farey graphs $\mathcal{F}_k$ and $\mathcal{F}_{\leqslant k}$, two natural variants of the Farey graph in which we relax the edge condition to indicate intersection number $=k$ or $\le k$, respectively. The former, $\mathcal{F}_k$, is disconnected when $k>1$. In fact, we find that the number of connected components is infinite if and only if $k$ is not a prime power. Moreover, we find that each component of $\mathcal{F}_k$ is an infinite-valence tree whenever $k$ is even, and $\mathrm{Aut}(\mathcal{F}_k)$ is uncountable for $k>1$. As for $\mathcal{F}_{\leqslant k}$, Agol obtained an upper bound of $1+\min\{p:p\text{ is a prime}>k\}$ for both chromatic and clique numbers, and observed that this is an equality when $k$ is either one or two less than a prime. We add to this list the values of $k$ that are three less than a prime equivalent to $11\ (\mathrm{mod}\ 12)$, and we show computer-assisted computations of many values of $k$ for which equality fails.

math.GT↗