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1. Persistence of the Conley Index in Combinatorial Dynamical Systems

   Dey, Tamal; Mrozek, Marian; Slechta, Ryan (January 2020, Leibniz international proceedings in informatics)

   null (Ed.)

   A combinatorial framework for dynamical systems provides an avenue for connecting classical dynamics with data-oriented, algorithmic methods. Combinatorial vector fields introduced by Forman [R. Forman, 1998; R. Forman, 1998] and their recent generalization to multivector fields [Mrozek, 2017] have provided a starting point for building such a connection. In this work, we strengthen this relationship by placing the Conley index in the persistent homology setting. Conley indices are homological features associated with so-called isolated invariant sets, so a change in the Conley index is a response to perturbation in an underlying multivector field. We show how one can use zigzag persistence to summarize changes to the Conley index, and we develop techniques to capture such changes in the presence of noise. We conclude by developing an algorithm to "track" features in a changing multivector field. 

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2. Persistent spectral graph

   https://doi.org/10.1002/cnm.3376  

   Wang, Rui; Nguyen, Duc Duy; Wei, Guo‐Wei (August 2020, International Journal for Numerical Methods in Biomedical Engineering)

   Abstract Persistent homology is constrained to purely topological persistence, while multiscale graphs account only for geometric information. This work introduces persistent spectral theory to create a unified low‐dimensional multiscale paradigm for revealing topological persistence and extracting geometric shapes from high‐dimensional datasets. For a point‐cloud dataset, a filtration procedure is used to generate a sequence of chain complexes and associated families of simplicial complexes and chains, from which we construct persistent combinatorial Laplacian matrices. We show that a full set of topological persistence can be completely recovered from the harmonic persistent spectra, that is, the spectra that have zero eigenvalues, of the persistent combinatorial Laplacian matrices. However, non‐harmonic spectra of the Laplacian matrices induced by the filtration offer another powerful tool for data analysis, modeling, and prediction. In this work, fullerene stability is predicted by using both harmonic spectra and non‐harmonic persistent spectra, while the latter spectra are successfully devised to analyze the structure of fullerenes and model protein flexibility, which cannot be straightforwardly extracted from the current persistent homology. The proposed method is found to provide excellent predictions of the protein B‐factors for which current popular biophysical models break down. 

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3. Advancing an interdisciplinary framework to study seed dispersal ecology

   https://doi.org/10.1093/aobpla/plz048  

   Beckman, Noelle G; Aslan, Clare E; Rogers, Haldre S; Kogan, Oleg; Bronstein, Judith L; Bullock, James M; Hartig, Florian; HilleRisLambers, Janneke; Zhou, Ying; Zurell, Damaris; et al (April 2020, AoB PLANTS)

   McConkey, Kim (Ed.)

   Abstract Although dispersal is generally viewed as a crucial determinant for the fitness of any organism, our understanding of its role in the persistence and spread of plant populations remains incomplete. Generalizing and predicting dispersal processes are challenging due to context dependence of seed dispersal, environmental heterogeneity and interdependent processes occurring over multiple spatial and temporal scales. Current population models often use simple phenomenological descriptions of dispersal processes, limiting their ability to examine the role of population persistence and spread, especially under global change. To move seed dispersal ecology forward, we need to evaluate the impact of any single seed dispersal event within the full spatial and temporal context of a plant’s life history and environmental variability that ultimately influences a population’s ability to persist and spread. In this perspective, we provide guidance on integrating empirical and theoretical approaches that account for the context dependency of seed dispersal to improve our ability to generalize and predict the consequences of dispersal, and its anthropogenic alteration, across systems. We synthesize suitable theoretical frameworks for this work and discuss concepts, approaches and available data from diverse subdisciplines to help operationalize concepts, highlight recent breakthroughs across research areas and discuss ongoing challenges and open questions. We address knowledge gaps in the movement ecology of seeds and the integration of dispersal and demography that could benefit from such a synthesis. With an interdisciplinary perspective, we will be able to better understand how global change will impact seed dispersal processes, and potential cascading effects on plant population persistence, spread and biodiversity. 

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4. Numerical Continuation on a Graphical Processing Unit for Kinematic Synthesis

   https://doi.org/10.1115/1.4047240  

   Glabe, Jeffrey; McCarthy, J. Michael (December 2020, Journal of Computing and Information Science in Engineering)

   Abstract This paper presents an implementation of a homotopy path tracking algorithm for polynomial numerical continuation on a graphical processing unit (GPU). The goal of this algorithm is to track homotopy curves from known roots to the unknown roots of a target polynomial system. The path tracker solves a set of ordinary differential equations to predict the next step and uses a Newton root finder to correct the prediction so the path stays on the homotopy solution curves. In order to benefit from the computational performance of a GPU, we organize the procedure so it is executed as a single instruction set, which means the path tracker has a fixed step size and the corrector has a fixed number iterations. This trade-off between accuracy and GPU computation speed is useful in numerical kinematic synthesis where a large number of solutions must be generated to find a few effective designs. In this paper, we show that our implementation of GPU-based numerical continuation yields 85 effective designs in 63 s, while an existing numerical continuation algorithm yields 455 effective designs in 2 h running on eight threads of a workstation. 

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5. Covering action on Conley index theory

   https://doi.org/10.1017/etds.2022.13  

   LIMA, D. V.; R. DA SILVEIRA, M.; VIEIRA, E. R. (January 2022, Ergodic Theory and Dynamical Systems)

   Abstract In this paper we apply Conley index theory in a covering space of an invariant set S , possibly not isolated, in order to describe the dynamics in S . More specifically, we consider the action of the covering translation group in order to define a topological separation of S which distinguishes the connections between the Morse sets within a Morse decomposition of S . The theory developed herein generalizes the classical connection matrix theory, since one obtains enriched information on the connection maps for non-isolated invariant sets, as well as for isolated invariant sets. Moreover, in the case of an infinite cyclic covering induced by a circle-valued Morse function, one proves that the Novikov differential of f is a particular case of the p -connection matrix defined herein. 

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