Title: Persistence of the Conley Index in Combinatorial Dynamical Systems
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. more »« less
Chaidez, Julian; Hutchings, Michael
(, Journal of Computational Dynamics)
null
(Ed.)
We study the combinatorial Reeb flow on the boundary of a four-dimensional convex polytope. We establish a correspondence between "combinatorial Reeb orbits" for a polytope, and ordinary Reeb orbits for a smoothing of the polytope, respecting action and Conley-Zehnder index. One can then use a computer to find all combinatorial Reeb orbits up to a given action and Conley-Zehnder index. We present some results of experiments testing Viterbo's conjecture and related conjectures. In particular, we have found some new examples of polytopes with systolic ratio \begin{document}$ 1 $$\end{document}$.
Ro, Hyunwook; Kino, Motoki; Sohn, Bong Won; Hada, Kazuhiro; Park, Jongho; Nakamura, Masanori; Cui, Yuzhu; Yi, Kunwoo; Chung, Aeree; Hodgson, Jeffrey; et al
(, Astronomy & Astrophysics)
Context. Because of its proximity and the large size of its black hole, M 87 is one of the best targets for studying the launching mechanism of active galactic nucleus jets. Currently, magnetic fields are considered to be an essential factor in the launching and accelerating of the jet. However, current observational estimates of the magnetic field strength of the M 87 jet are limited to the innermost part of the jet (≲100 r s ) or to HST-1 (∼10 5 r s ). No attempt has yet been made to measure the magnetic field strength in between. Aims. We aim to infer the magnetic field strength of the M 87 jet out to a distance of several thousand r s by tracking the distance-dependent changes in the synchrotron spectrum of the jet from high-resolution very long baseline interferometry observations. Methods. In order to obtain high-quality spectral index maps, quasi-simultaneous observations at 22 and 43 GHz were conducted using the KVN and VERA Array (KaVA) and the Very Long Baseline Array (VLBA). We compared the spectral index distributions obtained from the observations with a model and placed limits on the magnetic field strengths as a function of distance. Results. The overall spectral morphology is broadly consistent over the course of these observations. The observed synchrotron spectrum rapidly steepens from α 22 − 43 GHz ∼ −0.7 at ∼2 mas to α 22 − 43 GHz ∼ −2.5 at ∼6 mas. In the KaVA observations, the spectral index remains unchanged until ∼10 mas, but this trend is unclear in the VLBA observations. A spectral index model in which nonthermal electron injections inside the jet decrease with distance can adequately reproduce the observed trend. This suggests the magnetic field strength of the jet at a distance of 2−10 mas (∼900 r s − ∼4500 r s in the deprojected distance) has a range of B = (0.3−1.0 G)( z /2mas) −0.73 . Extrapolating to the Event Horizon Telescope scale yields consistent results, suggesting that the majority of the magnetic flux of the jet near the black hole is preserved out to ∼4500 r s without significant dissipation.
Cholesteric liquid crystals form a right-angle helicoidal structure with the pitch in the submicrometer and micrometer range. Because of the periodic modulation of the refractive index, the structure is capable of Bragg and Raman-Nath diffraction and mirrorless lasing. An attractive feature of cholesterics for optical applications is that the pitch and thus the wavelength of diffraction respond to temperature or chemical composition changes. However, the most desired mode of pitch control, by electromagnetic fields, has so far been elusive. Synthesis of bent-shape flexible dimer molecules resulted in an experimental realization of a new cholesteric state with an oblique helicoidal structure, abbreviated as ChOH. The ChOHstate forms when the material is acted upon by the electric or magnetic field and aligns its axis parallel to the field. The principal advantage of ChOHis that the field changes the pitch but preserves the single-harmonic heliconical structure. As a result, the material shows an extraordinarily broad range of electrically or magnetically tunable robust selective reflection of light, from ultraviolet to visible and infrared, and efficient tunable lasing. The ChOHstructure also responds to molecular reorientation at bounding plates and optical torques. This brief review discusses the recently established features of ChOHelectro-optics and problems to solve.
Tamal K. Dey; Michal Lipinski; Marian Mrozek; Ryan Slechta
(, Leibniz international proceedings in informatics)
Xavier Goaoc; Michael Kerber
(Ed.)
Multivector fields and combinatorial dynamical systems have recently become a subject of interest due to their potential for use in computational methods. In this paper, we develop a method to track an isolated invariant set - a salient feature of a combinatorial dynamical system - across a sequence of multivector fields. This goal is attained by placing the classical notion of the "continuation" of an isolated invariant set in the combinatorial setting. In particular, we give a "Tracking Protocol" that, when given a seed isolated invariant set, finds a canonical continuation of the seed across a sequence of multivector fields. In cases where it is not possible to continue, we show how to use zigzag persistence to track homological features associated with the isolated invariant sets. This construction permits viewing continuation as a special case of persistence.
Abstract We analyze networks of functional correlations between brain regions to identify changes in their structure caused by Attention Deficit Hyperactivity Disorder (adhd). We express the task for finding changes as a network anomaly detection problem on temporal networks. We propose the use of a curvature measure based on the Forman–Ricci curvature, which expresses higher-order correlations among two connected nodes. Our theoretical result on comparing this Forman–Ricci curvature with another well-known notion of network curvature, namely the Ollivier–Ricci curvature, lends further justification to the assertions that these two notions of network curvatures are not well correlated and therefore one of these curvature measures cannot be used as an universal substitute for the other measure. Our experimental results indicate nine critical edges whose curvature differs dramatically in brains ofadhdpatients compared to healthy brains. The importance of these edges is supported by existing neuroscience evidence. We demonstrate that comparative analysis of curvature identifies changes that more traditional approaches, for example analysis of edge weights, would not be able to identify.
Dey, Tamal, Mrozek, Marian, and Slechta, Ryan. Persistence of the Conley Index in Combinatorial Dynamical Systems. Retrieved from https://par.nsf.gov/biblio/10283570. Leibniz international proceedings in informatics .
Dey, Tamal, Mrozek, Marian, & Slechta, Ryan. Persistence of the Conley Index in Combinatorial Dynamical Systems. Leibniz international proceedings in informatics, (). Retrieved from https://par.nsf.gov/biblio/10283570.
Dey, Tamal, Mrozek, Marian, and Slechta, Ryan.
"Persistence of the Conley Index in Combinatorial Dynamical Systems". Leibniz international proceedings in informatics (). Country unknown/Code not available. https://par.nsf.gov/biblio/10283570.
@article{osti_10283570,
place = {Country unknown/Code not available},
title = {Persistence of the Conley Index in Combinatorial Dynamical Systems},
url = {https://par.nsf.gov/biblio/10283570},
abstractNote = {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.},
journal = {Leibniz international proceedings in informatics},
author = {Dey, Tamal and Mrozek, Marian and Slechta, Ryan},
editor = {null}
}
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