Application of polynomial algebras to non-linear equation solvers

Ponència presentada en: Mathematical Modelling in Engineering & Human Behaviour 2022 (MME&HB 2022). July 13th-15th, 202

Tools to detect structures in dynamical systems using Jet Transport

This paper is devoted to the development of some dynamical indicators that allow the determination of regions and structures that separate different dynamic regimes in autonomous and non-autonomous dynamical systems. The underlying idea is closely related to the Lagrangian coherent structures concept introduced by Haller. In the present paper, instead of using the Cauchy-Green tensor, that determines the domains where the flow associated to a differential equation is expanding in the normal direction, the Jet Transport methodology is used. This is a semi-numerical tool, that has as basic ingredients a polynomial algebra package and a numerical integration method, allowing, at each integration step, the propagation under a flow of a neighbourhood U instead of a single initial condition. The output of the procedure is a polynomial in several variables that represents the image of U up to a selected order, containing high order terms of the variational equations. Using these high order representation, the places where the normal direction expands can be easily detected, in a similar manner as the procedures for calculating the Lagrangian coherent structures do. In order to illustrate the methodology, first the results obtained in the determination of the separatrices of the simple and the periodically perturbed pendulum are given. Later, the applications to the circular restricted three body problem are considered, where the aim is the detection of invariant manifolds of libration point orbits, as well as in the non-autonomous vector field defined by the elliptic restricted three body problem.Peer ReviewedPostprint (author's final draft

Tools to detect structures in dynamical systems using Jet Transport

This paper is devoted to the development of some dynamical indicators that allow the determination of regions and structures that separate different dynamic regimes in autonomous and non-autonomous dynamical systems. The underlying idea is closely related to the Lagrangian Coherent Structures concept introduced by Haller. In the present paper, instead of using the Cauchy-Green tensor, that determines the domains where the flow associated to a differential equation is expanding in the normal direction, the Jet Transport methodology is used. This is a semi-numerical tool, that has as basic ingredients a polynomial algebra package and a numerical integration method, allowing, at each integration step, the propagation under a flow of a neighbourhood $U$ instead of a single initial condition. The output of the procedure is a polynomial in several variables that represents the image of $U$ up to a selected order, containing high order terms of the variational equations. Using these high order representation, the places where the normal direction expands can be easily detected, in a similar manner as the procedures for calculating the Lagrangian Coherent Structures do. In order to illustrate the methodology, first the results obtained in the determination of the separatrices of the simple and the periodically perturbed pendulum are given. Later, the applications to the circular restricted three body problem are considered, where the aim is the detection of invariant manifolds of libration point orbits, as well as in the non-autonomous vector field defined by the elliptic restricted three body problem

A new subdivision algorithm for the flow propagation using polynomial algebras

The Jet Transport method has emerged as a powerful tool for the numerical integration of ordinary differential equations; it uses polynomial expansions to approximate the flow map associated to the differential equation in the neighbourhood of a reference solution. One of the main drawbacks of the method is that the region of accuracy becomes smaller along the integration. In this paper we introduce a procedure to determine a ball covering the set of given initial conditions that keeps the accuracy of the integration within a selected threshold. The paper gives detailed explanations of the algorithm illustrated with some examples of applicability, as well as a comparison with a previous existing method for the same purposePeer ReviewedPostprint (author's final draft

Dispersal-enhanced resilience in two-patch metapopulations: origin's instability type matters

Many populations of animals or plants, exhibit a metapopulation structure with close, spatially-separated subpopulations. The field of metapopulation theory has made significant advancements since the influential Levins model. Various modeling approaches have provided valuable insights to theoretical Ecology. Despite extensive research on metapopulation models, there are still challenging questions that are difficult to answer from ecological metapopulational data or multi-patch models. Low-dimension mathematical models offer a promising avenue to address these questions, especially for global dynamics which have been scarcely investigated. In this study, we investigate a two-patch metapopulation model with logistic growth and diffusion between patches. By using analytical and numerical methods, we thoroughly analyze the impact of diffusion on the dynamics of the metapopulation. We identify the equilibrium points and assess their local and global stability. Furthermore, we analytically derive the optimal diffusion rate that leads to the highest metapopulation values. Our findings demonstrate that increased diffusion plays a crucial role in the preservation of both subpopulations and the full metapopulation, especially under the presence of stochastic perturbations. Specifically, at low diffusion values, the origin is a repeller, causing orbits starting around it to travel closely parallel to the axes. This configuration makes the metapopulation less resilient and thus more susceptible to local and global extinctions. However, as diffusion increases, the repeller transitions to a saddle point, and orbits starting near the origin rapidly converge to the unstable manifold of the saddle. This phenomenon reduces the likelihood of stochastic extinctions and the metapopulation becomes more resilient due to these changes in the vector field of the phase space.Comment: submitted to International Journal of Bifurcation and Chao

Family of fourth-order optimal classes for solving multiple-root nonlinear equations

[EN] We present a new iterative procedure for solving nonlinear equations with multiple roots with high efficiency. Starting from the arithmetic mean of Newton's and Chebysev's methods, we generate a two-step scheme using weight functions, resulting in a family of iterative methods that satisfies the Kung and Traub conjecture, yielding an optimal family for different choices of weight function. We have performed an in-depth analysis of the stability of the family members, in order to select those members with the highest stability for application in solving mathematical chemistry problems. We show the good characteristics of the selected methods by applying them on four relevant chemical problems.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. This research was partially supported by Grant PGC2018-095896-B-C22, funded by MCIN/AEI/10.13039/5011000113033 by "ERDF A way of making Europe", European Union; and by the internal research project ADMIREN of Universidad Internacional de La Rioja (UNIR).Chicharro, FI.; Garrido-Saez, N.; Jerezano, JH.; Pérez-Palau, D. (2023). Family of fourth-order optimal classes for solving multiple-root nonlinear equations. Journal of Mathematical Chemistry. 61(4):736-760. https://doi.org/10.1007/s10910-022-01429-573676061

Semi-automatic 3D reconstruction of atheroma plaques from intravascular ultrasound images using an ad-hoc algorithm

The occurrence of atheroma plaques in the arteries can eventually obstruct them, leading to diseases such as atherosclerosis, which can cause, among others, a myocardial infarction or a stroke. As a consequence, it is necessary to shorten the time spent in locating and reconstructing the atheroma plaque that can be developed in an artery. This localization is usually conducted manually from the contours located on the cross-sectional radiographs of the artery and then reconstructed by creating the volumes using different techniques. This paper presents a 3-D reconstruction of the atheroma plaque by applying an image processing algorithm ad-hoc developed in order to obtain the boundaries of the atheroma, from a set of intravascular ultrasound images. The advantage of the approach developed in this paper is that it can be implemented in common medical procedures, as an important complementary decision-support tool. By reconstructing the atheroma instead of the artery, this work provides a different approach to improve its location and treatment. Results presented herein can be implemented in machine-learning-based algorithms, able to predict the growth and extent of incipient atheroma plaques, which ultimately contribute to an early detection of this pathology

Investigar educa en ciudadanía

Este documento recoge los resultados de la segunda parte del proyecto “Educación para el desarrollo basada en investigación y evaluación formativa”, mediante el cual se pretendía incrementar el corpus teórico de la educación para el desarrollo en los ámbitos de didáctica y evaluación.Está publicación ha sido realizada con el apoyo financiero de la Agencia Española de Cooperación Internacional para el Desarrollo (AECID)

A crowdsourcing database for the copy-number variation of the spanish population

Background: Despite being a very common type of genetic variation, the distribution of copy-number variations (CNVs) in the population is still poorly understood. The knowledge of the genetic variability, especially at the level of the local population, is a critical factor for distinguishing pathogenic from non-pathogenic variation in the discovery of new disease variants. Results: Here, we present the SPAnish Copy Number Alterations Collaborative Server (SPACNACS), which currently contains copy number variation profiles obtained from more than 400 genomes and exomes of unrelated Spanish individuals. By means of a collaborative crowdsourcing effort whole genome and whole exome sequencing data, produced by local genomic projects and for other purposes, is continuously collected. Once checked both, the Spanish ancestry and the lack of kinship with other individuals in the SPACNACS, the CNVs are inferred for these sequences and they are used to populate the database. A web interface allows querying the database with different filters that include ICD10 upper categories. This allows discarding samples from the disease under study and obtaining pseudo-control CNV profiles from the local population. We also show here additional studies on the local impact of CNVs in some phenotypes and on pharmacogenomic variants. SPACNACS can be accessed at: http://csvs.clinbioinfosspa.es/spacnacs/. Conclusion: SPACNACS facilitates disease gene discovery by providing detailed information of the local variability of the population and exemplifies how to reuse genomic data produced for other purposes to build a local reference database.This work is supported by Grants PID2020-117979RB-I00 from the Spanish Ministry of Science and Innovation; by the Institute of Health Carlos III (project IMPaCT-Data, exp. IMP/00019, IMP/00009 and PI20/01305), co-funded by the European Union, European Regional Development Fund (ERDF, “A way to make Europe”)

A crowdsourcing database for the copy-number variation of the Spanish population

Background: Despite being a very common type of genetic variation, the distribution of copy-number variations (CNVs) in the population is still poorly understood. The knowledge of the genetic variability, especially at the level of the local population, is a critical factor for distinguishing pathogenic from non-pathogenic variation in the discovery of new disease variants. Results: Here, we present the SPAnish Copy Number Alterations Collaborative Server (SPACNACS), which currently contains copy number variation profiles obtained from more than 400 genomes and exomes of unrelated Spanish individuals. By means of a collaborative crowdsourcing effort whole genome and whole exome sequencing data, produced by local genomic projects and for other purposes, is continuously collected. Once checked both, the Spanish ancestry and the lack of kinship with other individuals in the SPACNACS, the CNVs are inferred for these sequences and they are used to populate the database. A web interface allows querying the database with different filters that include ICD10 upper categories. This allows discarding samples from the disease under study and obtaining pseudo-control CNV profiles from the local population. We also show here additional studies on the local impact of CNVs in some phenotypes and on pharmacogenomic variants. SPACNACS can be accessed at: http://csvs.clinbioinfosspa.es/spacnacs/. Conclusion: SPACNACS facilitates disease gene discovery by providing detailed information of the local variability of the population and exemplifies how to reuse genomic data produced for other purposes to build a local reference database