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Luis Felipe Prieto-Martínez

Publications and source records attributed to Luis Felipe Prieto-Martínez.

16 recordsLinked to original sources
arXiv2026

Shifted Eigenvector Models for Centrality and Occupancy in Urban Networks

This article investigates a family of centrality models for urban networks that incorporate both topological and non-topological factors. Since centrality is inherently recursive, these models can be formulated as fixed-point equations, which we refer to as shifted eigenproblems. Assuming a correlation between node centrality and occupancy, we discuss how experimental data can be used to estimate model parameters via least-squares methods. Furthermore, such data would allow us to infer the intrinsic attraction of each node, as well as the occupancy induced by must-visit points of interest, a task that is conceptually challenging. Once the model parameters are fitted and validated, our framework can be used to assess the impact of urban interventions, such as introducing a must-visit point of interest at a specific node or enhancing its intrinsic attraction. The resulting sensitivity analysis is therefore highly relevant for urban planning decisions. We also provide explicit formulas to facilitate this analysis.

cs.SI↗
arXiv2025

A formula for the Euler characteristic of a poset through the determinant of the order-complement matrix

Given a finite poset $P$, its zeta matrix $\mathbf Z$ encode fundamental incidence-theoretic information about the order structure. In this paper we introduce and study the \emph{order-complement matrix} $\overline{\mathbf Z} = \mathbf J - \mathbf Z$, where $\mathbf J$ is the all-ones matrix. We prove a closed formula for its characteristic polynomial and for its determinant, showing that $\det(\overline{\mathbf Z}) = (-1)^n \tildeχ(P)$, where $n = |P|$ and $\tildeχ(P)$ is the reduced Euler characteristic of $P$. This provides a new, unexpectedly simple linear-algebraic expression for the Euler characteristic of a poset, complementing existing determinant formulas for matrices derived from incidence relations.

math.CO↗
arXiv2025

Roots in the substitution group and in the group of Riordan matrices with ones in the main diagonal

We investigate the existence and uniqueness of iterative roots of order $n$ within the substitution group of formal power series $\mathcal J(Z)$ -- with coefficients in a commutative ring with unity $Z$ -- employing a matrix-based framework grounded in the Riordan group. We analyse the relationship between the substitution group and the Lagrange subgroup -- a group of Riordan matrices -- and explore some classic questions concerning algebraic completeness and uniqueness of root extractions. This approach allows us to obtain various results about the roots in $\mathcal J(Z)$ for different choices of $Z$. Furthermore, the examination of the substitution group facilitates the analysis of roots within the Riordan matrix group.

math.GR↗
arXiv2025

Riordan pattern's quest within simplicial complexes

The aim of this paper is twofold. First, we demonstrate how Riordan matrices can be employed to connect well-known concepts in geometric combinatorics, such as $f$-vectors, $h$-vectors $γ$-vectors, in a similar fashion to the McMullen Correspondence, and the Dehn-Sommerville equations, among others. Second, we investigate the combinatorial properties of the topological join operation, both for simplicial complexes and for Alexandroff spaces. Finally, we explore the Riordan matrices arising from the iteration of this topological operation and analyze their properties.

math.CO↗
arXiv2025

Bi-infinite Riordan matrices: a matricial approach to multiplication and composition of Laurent series

We propose and investigate a bi-infinite matrix approach to the multiplication and composition of formal Laurent series. We generalize the concept of Riordan matrix to this bi-infinite context, obtaining matrices that are not necessarily lower triangular and are determined, not by a pair of formal power series, but by a pair of Laurent series. We extend the First Fundamental Theorem of Riordan Matrices to this setting, as well as the Toeplitz and Lagrange subgroups, that are subgroups of the classical Riordan group. Finally, as an illustrative example, we apply our approach to derive a classical combinatorial identity that cannot be proved using the techniques related to the classical Riordan group, showing that our generalization is not fruitless.

math.GR↗
arXiv2024

Existence and uniqueness of solutions for Leontief's Input-Output Model, graph theory and sensitivity analysis

We provide a complete study of existence and uniqueness (uniqueness up to multiples in the case $\mathbf{d} = \mathbf{0}$) of non-negative and non-trivial solutions $\mathbf{x}$ for the linear system $(\mathbf{I}- \mathbf{A})\mathbf{x} = \mathbf{d}$ with $\mathbf{A} \geq \mathbf{0}, \mathbf{d} \geq 0$ (which, in particular, applies to Leontief's Input Output Model). This study is done in terms of the block triangular form of the matrix $\mathbf{A}$ and is related to a directed graph associated to $\mathbf{A}$. In particular, this study of existence and uniqueness up to multiples of solutions provides a framework that allows us to rigorously perform a sensitivity analysis of the normalized solutions of the linear system $(\mathbf{I} - \mathbf{A})\mathbf{x} = \mathbf{d}$.

math.OC↗
arXiv2023

The concept of center as an equivariant map and a proof of an analogue of the center conjecture for equifacetal simplices

Several authors have remarked the convenience of understanding the different notions of center appearing in Geometry (centroid of a set of points, incenter of a triangle, center of a conic and many others) as functions. The most general way to do so is to define centers as equivariant maps between $G$-spaces. In this paper, we prove that, under certain hypothesis, for any two $G$-spaces $\mathcal A,\mathcal X$, for every $V\in \mathcal A$ and for every point $P\in\mathcal X$ fixed by the symmetry group of $V$, there exists some equivariant map $\mathfrak Z:\mathcal A\to \mathcal X$ such that $\mathfrak Z(V)=P$. As a consequence of this fact, we prove an analogue (for non-neccessarily continuous centers) of the \emph{center conjecture for equifacetal simplices}, proposed by A. L. Edmonds.

math.MG↗
arXiv2022

The relation between symmetries and coincidence and collinearity of polygon centers and centers of multisets of points in the plane

There are several remarkable points, defined for polygons and multisets of points in the plane, called centers (such as the centroid). To make possible their study, there exists a formal definition for the concept of center in both cases. In this paper, the relation between symmetries of polygons and multisets of points in the plane and the coincidence and collinearity of their centers is studied. First, a precise statement for the problem is given. Then, it is proved that, given a polygon or a multiset of points in the plane, a given point in the plane is a center for this object if and only if it belongs to the set of points fixed by its group of symmetries.

math.MG↗
arXiv2022

$n$-gon centers and central lines

In this paper we provide a review of the concept of center of a $n$-gon, generalizing the original idea given by C. Kimberling for triangles. We also generalize the concept of central line for $n$-gons for $n\geq 3$ and establish its basic properties.

math.MG↗
arXiv2021

A solution to two old problems by Menger concerning angle spaces

Around 1930, K. Menger expressed his interest in the concept of abstract angle function. He introduced a general definition of this notion for metric and semi-metric spaces. He also proposed two problems concerning conformal embeddability of spaces endowed with an angle function into Euclidean spaces. These problems received attention in later years but only for some particular cases of metric spaces. In this article we first update the definition of angle function to apply to the larger class of spaces with a notion of betweenness, which seem to us a more natural framework. In this new general setting, we solve the two problems proposed by Menger.

math.MG↗
arXiv2021

Commutators and commutator subgroups of the Riordan group

We calculate the derived series of the Riordan group. To do that, we study a nested sequence of its subgroups, herein denoted by $\mathcal G_k$. By means of this sequence, we first obtain the n-th commutator subgroup of the Associated subgroup. This fact allows us to get some related results about certain groups of formal power series and to complete the proof of our main goal, Theorem 1 in this paper.

math.GR↗
arXiv2020

Geometric continuity of plane curves in terms of Riordan matrices and an application to the F-chordal problem

The first goal of this article is to provide an statement of the conditions for geometric continuity of order k, referred in the bibliography as beta-constraints, in terms of Riordan matrices. The second one is to see this new formulation in action to solve a theoretical cuestion about uniqueness of analytic solution for a general and classical problem in plane geometry: the F-chordal problem.

math.MG↗
arXiv2020

Generalization of Kimberling's concept of triangle center for other polygons

In this article we introduce a general definition of the concept of center of an $n$-gon, for $n\geq 3$, generalizing the idea of C. Kimberling for triangle. We define centers associated to functions instead of to geometrical properties. We discuss the definition of those functions depending on both, the vertices of the polygons or the lengths of sides and diagonals. We explore the problem of characterization of regular polygons in terms of these $n$-gon center functions and we study the relation between our general definition of center of a polygon and other approaches arising from Applied Mathematics.

math.MG↗