arXiv subjects

Pablo Padilla-Longoria

Publications and source records attributed to Pablo Padilla-Longoria.

6 recordsLinked to original sources

arXiv2026

On a Stochastic PDE Model for Epigenetic Dynamics

We propose a stochastic model to investigate epigenetic mutations, i.e., modifications of the genetic information that control gene expression patterns in a cell but do not alter the DNA sequence. Epigenetic mutations are related to environmental fluctuations, which leads us to consider (additive) noise as the driving element for such mutations (noise-induced transitions in Waddington's epigenetic landscape). We focus on two applications: firstly, molecular biochemistry of cancer immunology involving macrophages' epigenetic modifications, where we show the relevance of random perturbations in the tumor microenvironment, and secondly, cell fate determination and mutation of the flower Arabidopsis thaliana. Due to the complexities of cancer biology for the first case, we present the details in [1] since our principal objective here is to validate our system as an appropriate epigenetic model for more general biological applications, with emphasis on mathematical oncology and developmental biology; for such results, we rely on the theory of Stochastic PDE, theory of large deviations, and ergodic theory. Moreover, since epigenetic mutations are reversible, a fact currently exploited to develop so-called epi-drugs to treat diseases such as cancer, we also investigate an optimal control problem for our system to study the reversal of epigenetic mutations; our control problem is also relevant for studying epigenetic stabilizers and transcription factors in immunotherapies for cancer [1].

math.AP↗

arXiv2025

Biological network dynamics: Poincaré-Lindstedt series and the effect of delays

This paper focuses on the Hopf bifurcation in an activator-inhibitor system without diffusion which can be modeled as a delay differential equation. The main result of this paper is the existence of the Poincaré-Lindstedt series to all orders for the bifurcating periodic solutions. The model has a non-linearity which is non-polynomial, and yet this allows us to exploit the use of Fourier-Taylor series to develop order-by-order calculations that lead to linear recurrence equations for the coefficients of the Poincaré-Lindstedt series. As applications, we implement the computation of the coefficients of these series for any finite order, and use a pseudo-arclength continuation to compute branches of periodic solutions.

math.DS↗

arXiv2024

Analysis and Visualization of Musical Structure using Networks

In this article, a framework for defining and analysing a family of graphs or networks from symbolic music information is discussed. Such graphs concern different types of elements, such as pitches, chords and rhythms, and the relations among them, and are built from quantitative or categorical data contained in digital music scores. They are helpful in visualizing musical features at once, thus leading to a computational tool for understanding the general structural elements of a music fragment. Data obtained from a digital score undergoes different analytical procedures from graph and network theory, such as computing their centrality measures and entropy, and detecting their communities. We analyze pieces of music coming from different styles, and compare some of our results with conclusions from traditional music analysis techniques.

cs.SI↗

arXiv2023

A Probabilistic Approach to the Existence of Solutions to Semilinear Elliptic Equations

We study a semilinear elliptic equation with a pure power nonlinearity with exponent $p>1$, and provide sufficient conditions for the existence of positive solutions. These conditions involve expected exit times from the domain, $D$, where a solution is defined, and expected occupation times in suitable subdomains of $D$. They provide an alternative new approach to the geometric or topological sufficient conditions given in the literature for exponents close to the critical Sobolev exponent. Moreover, unlike standard results, in our probabilistic approach no \emph{a priori} upper bound restriction is imposed on $p$, which might be supercritical. The proof is based on a fixed point argument using a probabilistic representation formula. We also prove a multiplicity result and discuss possible extensions to the existence of sign changing solutions. Finally, we conjecture that necessary conditions for the existence of solutions might be obtained using a similar probabilistic approach. This motivates a series of natural questions related to the characterisation of topological and geometrical properties of a domain in probabilistic terms.

math.AP↗

arXiv2022

A Framework for Topological Music Analysis (TMA)

In the present article we describe and discuss a framework for applying different topological data analysis (TDA) techniques to a music fragment given as a score in traditional Western notation. We first consider different sets of points in Euclidean spaces of different dimensions that correspond to musical events in the score, and obtain their persistent homology features. Then we introduce two families of simplicial complexes that can be associated with chord sequences, and leverage homology to compute their salient features. Finally, we show the results of applying the described methods to the analysis and stylistic comparison of fragments from three Brandenburg Concertos by J.S. Bach and two Graffiti by Mexican composer Armando Luna.

math.AT↗

arXiv2022

Rhythm and form in music: a complex systems approach

There has been an everlasting discussion around the concept of form in music. This work is motivated by such debate by using a complex systems framework in which we study the form as an emergent property of rhythm. Such a framework corresponds with the traditional notion of musical form and allows us to generalize this concept to more general shapes and structures in music. We develop the three following metrics of the rhythmic complexity of a musical piece and its parts: 1) the rhythmic heterogeneity, based on the permutation entropy, where high values indicate a wide variety of rhythmic patterns; 2) the syncopation, based on the distribution of on-beat onsets, where high values indicate a high proportion of off-the-beat notes; and 3) the component extractor, based on the communities of a visibility graph of the rhythmic figures over time, where we identify structural components that constitute the piece at a (to be explained) perceptual level. With the same parameters, our metrics are comparable within a piece or between pieces.

eess.AS↗