[{"volume":16,"date_created":"2021-08-08T22:01:28Z","article_type":"original","scopus_import":"1","day":"01","author":[{"last_name":"Graff","full_name":"Graff, Grzegorz","first_name":"Grzegorz"},{"last_name":"Graff","full_name":"Graff, Beata","first_name":"Beata"},{"last_name":"Pilarczyk","id":"3768D56A-F248-11E8-B48F-1D18A9856A87","full_name":"Pilarczyk, Pawel","first_name":"Pawel"},{"last_name":"Jablonski","id":"4483EF78-F248-11E8-B48F-1D18A9856A87","full_name":"Jablonski, Grzegorz","first_name":"Grzegorz","orcid":"0000-0002-3536-9866"},{"first_name":"Dariusz","full_name":"Gąsecki, Dariusz","last_name":"Gąsecki"},{"last_name":"Narkiewicz","full_name":"Narkiewicz, Krzysztof","first_name":"Krzysztof"}],"title":"Persistent homology as a new method of the assessment of heart rate variability","oa_version":"Published Version","file_date_updated":"2021-08-09T09:25:41Z","publication_status":"published","publication_identifier":{"eissn":["19326203"]},"has_accepted_license":"1","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"intvolume":"        16","abstract":[{"lang":"eng","text":"Heart rate variability (hrv) is a physiological phenomenon of the variation in the length of the time interval between consecutive heartbeats. In many cases it could be an indicator of the development of pathological states. The classical approach to the analysis of hrv includes time domain methods and frequency domain methods. However, attempts are still being made to define new and more effective hrv assessment tools. Persistent homology is a novel data analysis tool developed in the recent decades that is rooted at algebraic topology. The Topological Data Analysis (TDA) approach focuses on examining the shape of the data in terms of connectedness and holes, and has recently proved to be very effective in various fields of research. In this paper we propose the use of persistent homology to the hrv analysis. We recall selected topological descriptors used in the literature and we introduce some new topological descriptors that reflect the specificity of hrv, and we discuss their relation to the standard hrv measures. In particular, we show that this novel approach provides a collection of indices that might be at least as useful as the classical parameters in differentiating between series of beat-to-beat intervals (RR-intervals) in healthy subjects and patients suffering from a stroke episode."}],"department":[{"_id":"HeEd"}],"article_number":"e0253851","file":[{"creator":"asandaue","date_updated":"2021-08-09T09:25:41Z","date_created":"2021-08-09T09:25:41Z","file_size":2706919,"file_id":"9832","access_level":"open_access","content_type":"application/pdf","success":1,"file_name":"2021_PLoSONE_Graff.pdf","checksum":"0277aa155d5db1febd2cb384768bba5f","relation":"main_file"}],"month":"07","citation":{"chicago":"Graff, Grzegorz, Beata Graff, Pawel Pilarczyk, Grzegorz Jablonski, Dariusz Gąsecki, and Krzysztof Narkiewicz. “Persistent Homology as a New Method of the Assessment of Heart Rate Variability.” <i>PLoS ONE</i>. Public Library of Science, 2021. <a href=\"https://doi.org/10.1371/journal.pone.0253851\">https://doi.org/10.1371/journal.pone.0253851</a>.","ista":"Graff G, Graff B, Pilarczyk P, Jablonski G, Gąsecki D, Narkiewicz K. 2021. Persistent homology as a new method of the assessment of heart rate variability. PLoS ONE. 16(7), e0253851.","apa":"Graff, G., Graff, B., Pilarczyk, P., Jablonski, G., Gąsecki, D., &#38; Narkiewicz, K. (2021). Persistent homology as a new method of the assessment of heart rate variability. <i>PLoS ONE</i>. Public Library of Science. <a href=\"https://doi.org/10.1371/journal.pone.0253851\">https://doi.org/10.1371/journal.pone.0253851</a>","mla":"Graff, Grzegorz, et al. “Persistent Homology as a New Method of the Assessment of Heart Rate Variability.” <i>PLoS ONE</i>, vol. 16, no. 7, e0253851, Public Library of Science, 2021, doi:<a href=\"https://doi.org/10.1371/journal.pone.0253851\">10.1371/journal.pone.0253851</a>.","ama":"Graff G, Graff B, Pilarczyk P, Jablonski G, Gąsecki D, Narkiewicz K. Persistent homology as a new method of the assessment of heart rate variability. <i>PLoS ONE</i>. 2021;16(7). doi:<a href=\"https://doi.org/10.1371/journal.pone.0253851\">10.1371/journal.pone.0253851</a>","ieee":"G. Graff, B. Graff, P. Pilarczyk, G. Jablonski, D. Gąsecki, and K. Narkiewicz, “Persistent homology as a new method of the assessment of heart rate variability,” <i>PLoS ONE</i>, vol. 16, no. 7. Public Library of Science, 2021.","short":"G. Graff, B. Graff, P. Pilarczyk, G. Jablonski, D. Gąsecki, K. Narkiewicz, PLoS ONE 16 (2021)."},"issue":"7","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","oa":1,"language":[{"iso":"eng"}],"_id":"9821","date_updated":"2023-08-10T14:21:42Z","type":"journal_article","article_processing_charge":"Yes","doi":"10.1371/journal.pone.0253851","publisher":"Public Library of Science","quality_controlled":"1","ddc":["006"],"year":"2021","isi":1,"external_id":{"pmid":["34292957"],"isi":["000678124900050"]},"pmid":1,"date_published":"2021-07-01T00:00:00Z","acknowledgement":"We express our gratitude to the anonymous referees who provided constructive comments that helped us improve the quality of the paper.","status":"public","publication":"PLoS ONE"},{"year":"2016","month":"09","department":[{"_id":"HeEd"}],"publist_id":"6209","publication":"Applied Numerical Mathematics","status":"public","language":[{"iso":"eng"}],"project":[{"_id":"255F06BE-B435-11E9-9278-68D0E5697425","name":"Persistent Homology - Images, Data and Maps","grant_number":"622033","call_identifier":"FP7"}],"citation":{"chicago":"Miyaji, Tomoyuki, Pawel Pilarczyk, Marcio Gameiro, Hiroshi Kokubu, and Konstantin Mischaikow. “A Study of Rigorous ODE Integrators for Multi Scale Set Oriented Computations.” <i>Applied Numerical Mathematics</i>. Elsevier, 2016. <a href=\"https://doi.org/10.1016/j.apnum.2016.04.005\">https://doi.org/10.1016/j.apnum.2016.04.005</a>.","ista":"Miyaji T, Pilarczyk P, Gameiro M, Kokubu H, Mischaikow K. 2016. A study of rigorous ODE integrators for multi scale set oriented computations. Applied Numerical Mathematics. 107, 34–47.","apa":"Miyaji, T., Pilarczyk, P., Gameiro, M., Kokubu, H., &#38; Mischaikow, K. (2016). A study of rigorous ODE integrators for multi scale set oriented computations. <i>Applied Numerical Mathematics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.apnum.2016.04.005\">https://doi.org/10.1016/j.apnum.2016.04.005</a>","mla":"Miyaji, Tomoyuki, et al. “A Study of Rigorous ODE Integrators for Multi Scale Set Oriented Computations.” <i>Applied Numerical Mathematics</i>, vol. 107, Elsevier, 2016, pp. 34–47, doi:<a href=\"https://doi.org/10.1016/j.apnum.2016.04.005\">10.1016/j.apnum.2016.04.005</a>.","ama":"Miyaji T, Pilarczyk P, Gameiro M, Kokubu H, Mischaikow K. A study of rigorous ODE integrators for multi scale set oriented computations. <i>Applied Numerical Mathematics</i>. 2016;107:34-47. doi:<a href=\"https://doi.org/10.1016/j.apnum.2016.04.005\">10.1016/j.apnum.2016.04.005</a>","ieee":"T. Miyaji, P. Pilarczyk, M. Gameiro, H. Kokubu, and K. Mischaikow, “A study of rigorous ODE integrators for multi scale set oriented computations,” <i>Applied Numerical Mathematics</i>, vol. 107. Elsevier, pp. 34–47, 2016.","short":"T. Miyaji, P. Pilarczyk, M. Gameiro, H. Kokubu, K. Mischaikow, Applied Numerical Mathematics 107 (2016) 34–47."},"ec_funded":1,"user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","date_published":"2016-09-01T00:00:00Z","acknowledgement":"MG was partially supported by FAPESP grants 2013/07460-7 and 2010/00875-9, and by CNPq grants 305860/2013-5 and 306453/2009-6, Brazil. The work of HK was partially supported by Grant-in-Aid for Scientific Research (Nos.24654022, 25287029), Ministry of Education, Science, Technology, Culture and Sports, Japan. KM was supported by NSF grants NSF-DMS-0835621, 0915019, 1125174, 1248071, and contracts from AFOSR and DARPA. TM was supported by Grant-in-Aid for JSPS Fellows No. 245312. A part of the research of TM and HK was also supported by JST, CREST.\r\n\r\nResearch conducted by PP has received funding from Fundo Europeu de Desenvolvimento Regional (FEDER) through COMPETE – Programa Operacional Factores de Competitividade (POFC) and from the Portuguese national funds through Fundação para a Ciência e a Tecnologia (FCT) in the framework of the research project FCOMP-01-0124-FEDER-010645 (Ref. FCT PTDC/MAT/098871/2008); from the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme (FP7/2007-2013) under REA grant agreement No. 622033; and from the same sources as HK.\r\n\r\nThe authors express their gratitude to the Department of Mathematics of Kyoto University for making their server available for conducting the computations described in the paper, and to the reviewers for helpful comments that contributed towards increasing the quality of the paper.","scopus_import":1,"day":"01","doi":"10.1016/j.apnum.2016.04.005","author":[{"first_name":"Tomoyuki","full_name":"Miyaji, Tomoyuki","last_name":"Miyaji"},{"last_name":"Pilarczyk","id":"3768D56A-F248-11E8-B48F-1D18A9856A87","full_name":"Pilarczyk, Pawel","first_name":"Pawel"},{"full_name":"Gameiro, Marcio","last_name":"Gameiro","first_name":"Marcio"},{"last_name":"Kokubu","full_name":"Kokubu, Hiroshi","first_name":"Hiroshi"},{"first_name":"Konstantin","last_name":"Mischaikow","full_name":"Mischaikow, Konstantin"}],"publisher":"Elsevier","title":"A study of rigorous ODE integrators for multi scale set oriented computations","oa_version":"None","volume":107,"_id":"1149","date_updated":"2021-01-12T06:48:38Z","date_created":"2018-12-11T11:50:25Z","type":"journal_article","page":"34 - 47","abstract":[{"text":"We study the usefulness of two most prominent publicly available rigorous ODE integrators: one provided by the CAPD group (capd.ii.uj.edu.pl), the other based on the COSY Infinity project (cosyinfinity.org). Both integrators are capable of handling entire sets of initial conditions and provide tight rigorous outer enclosures of the images under a time-T map. We conduct extensive benchmark computations using the well-known Lorenz system, and compare the computation time against the final accuracy achieved. We also discuss the effect of a few technical parameters, such as the order of the numerical integration method, the value of T, and the phase space resolution. We conclude that COSY may provide more precise results due to its ability of avoiding the variable dependency problem. However, the overall cost of computations conducted using CAPD is typically lower, especially when intervals of parameters are involved. Moreover, access to COSY is limited (registration required) and the rigorous ODE integrators are not publicly available, while CAPD is an open source free software project. Therefore, we recommend the latter integrator for this kind of computations. Nevertheless, proper choice of the various integration parameters turns out to be of even greater importance than the choice of the integrator itself. © 2016 IMACS. Published by Elsevier B.V. All rights reserved.","lang":"eng"}],"intvolume":"       107","quality_controlled":"1","publication_status":"published"},{"quality_controlled":"1","page":"140 - 151","date_updated":"2021-01-12T06:49:18Z","_id":"1237","type":"conference","doi":"10.1007/978-3-319-39441-1_13","alternative_title":["LNCS"],"publisher":"Springer","ec_funded":1,"date_published":"2016-06-02T00:00:00Z","acknowledgement":"The research conducted by both authors has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreements no. 291734 (for M. K.) and no. 622033 (for P. P.).","conference":{"location":"Marseille, France","start_date":"2016-06-15","end_date":"2016-06-17","name":"CTIC: Computational Topology in Image Context"},"project":[{"call_identifier":"FP7","name":"International IST Postdoc Fellowship Programme","grant_number":"291734","_id":"25681D80-B435-11E9-9278-68D0E5697425"},{"_id":"255F06BE-B435-11E9-9278-68D0E5697425","name":"Persistent Homology - Images, Data and Maps","grant_number":"622033","call_identifier":"FP7"}],"status":"public","publist_id":"6096","year":"2016","publication_status":"published","abstract":[{"lang":"eng","text":"Bitmap images of arbitrary dimension may be formally perceived as unions of m-dimensional boxes aligned with respect to a rectangular grid in ℝm. Cohomology and homology groups are well known topological invariants of such sets. Cohomological operations, such as the cup product, provide higher-order algebraic topological invariants, especially important for digital images of dimension higher than 3. If such an operation is determined at the level of simplicial chains [see e.g. González-Díaz, Real, Homology, Homotopy Appl, 2003, 83-93], then it is effectively computable. However, decomposing a cubical complex into a simplicial one deleteriously affects the efficiency of such an approach. In order to avoid this overhead, a direct cubical approach was applied in [Pilarczyk, Real, Adv. Comput. Math., 2015, 253-275] for the cup product in cohomology, and implemented in the ChainCon software package [http://www.pawelpilarczyk.com/chaincon/]. We establish a formula for the Steenrod square operations [see Steenrod, Annals of Mathematics. Second Series, 1947, 290-320] directly at the level of cubical chains, and we prove the correctness of this formula. An implementation of this formula is programmed in C++ within the ChainCon software framework. We provide a few examples and discuss the effectiveness of this approach. One specific application follows from the fact that Steenrod squares yield tests for the topological extension problem: Can a given map A → Sd to a sphere Sd be extended to a given super-complex X of A? In particular, the ROB-SAT problem, which is to decide for a given function f: X → ℝm and a value r &gt; 0 whether every g: X → ℝm with ∥g - f ∥∞ ≤ r has a root, reduces to the extension problem."}],"intvolume":"      9667","volume":9667,"date_created":"2018-12-11T11:50:52Z","author":[{"id":"33E21118-F248-11E8-B48F-1D18A9856A87","full_name":"Krcál, Marek","last_name":"Krcál","first_name":"Marek"},{"first_name":"Pawel","last_name":"Pilarczyk","full_name":"Pilarczyk, Pawel","id":"3768D56A-F248-11E8-B48F-1D18A9856A87"}],"scopus_import":1,"day":"02","oa_version":"None","title":"Computation of cubical Steenrod squares","citation":{"apa":"Krcál, M., &#38; Pilarczyk, P. (2016). Computation of cubical Steenrod squares (Vol. 9667, pp. 140–151). Presented at the CTIC: Computational Topology in Image Context, Marseille, France: Springer. <a href=\"https://doi.org/10.1007/978-3-319-39441-1_13\">https://doi.org/10.1007/978-3-319-39441-1_13</a>","mla":"Krcál, Marek, and Pawel Pilarczyk. <i>Computation of Cubical Steenrod Squares</i>. Vol. 9667, Springer, 2016, pp. 140–51, doi:<a href=\"https://doi.org/10.1007/978-3-319-39441-1_13\">10.1007/978-3-319-39441-1_13</a>.","chicago":"Krcál, Marek, and Pawel Pilarczyk. “Computation of Cubical Steenrod Squares,” 9667:140–51. Springer, 2016. <a href=\"https://doi.org/10.1007/978-3-319-39441-1_13\">https://doi.org/10.1007/978-3-319-39441-1_13</a>.","ista":"Krcál M, Pilarczyk P. 2016. Computation of cubical Steenrod squares. CTIC: Computational Topology in Image Context, LNCS, vol. 9667, 140–151.","ieee":"M. Krcál and P. Pilarczyk, “Computation of cubical Steenrod squares,” presented at the CTIC: Computational Topology in Image Context, Marseille, France, 2016, vol. 9667, pp. 140–151.","short":"M. Krcál, P. Pilarczyk, in:, Springer, 2016, pp. 140–151.","ama":"Krcál M, Pilarczyk P. Computation of cubical Steenrod squares. In: Vol 9667. Springer; 2016:140-151. doi:<a href=\"https://doi.org/10.1007/978-3-319-39441-1_13\">10.1007/978-3-319-39441-1_13</a>"},"user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"department":[{"_id":"UlWa"},{"_id":"HeEd"}],"month":"06"},{"page":"1787 - 1801","quality_controlled":"1","main_file_link":[{"url":"https://arxiv.org/abs/1411.7563","open_access":"1"}],"publisher":"American Mathematical Society","doi":"10.1090/proc/12812","article_processing_charge":"No","type":"journal_article","date_updated":"2022-05-24T09:35:58Z","_id":"1252","project":[{"_id":"255F06BE-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","grant_number":"622033","name":"Persistent Homology - Images, Data and Maps"}],"publication":"Proceedings of the American Mathematical Society","status":"public","acknowledgement":"The authors gratefully acknowledge the support of the Lorenz Center which\r\nprovided an opportunity for us to discuss in depth the work of this paper. Research leading to these results has received funding from Fundo Europeu de Desenvolvimento Regional (FEDER) through COMPETE—Programa Operacional Factores de Competitividade (POFC) and from the Portuguese national funds through Funda¸c˜ao para a Ciˆencia e a Tecnologia (FCT) in the framework of the research\r\nproject FCOMP-01-0124-FEDER-010645 (ref. FCT PTDC/MAT/098871/2008),\r\nas well as from the People Programme (Marie Curie Actions) of the European\r\nUnion’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement no. 622033 (supporting PP). The work of the first and third author has\r\nbeen partially supported by NSF grants NSF-DMS-0835621, 0915019, 1125174,\r\n1248071, and contracts from AFOSR and DARPA. The work of the second author\r\nwas supported by Grant-in-Aid for Scientific Research (No. 25287029), Ministry of\r\nEducation, Science, Technology, Culture and Sports, Japan.","date_published":"2016-04-01T00:00:00Z","ec_funded":1,"external_id":{"arxiv":["1411.7563"]},"year":"2016","publist_id":"6075","intvolume":"       144","abstract":[{"text":"We study the homomorphism induced in homology by a closed correspondence between topological spaces, using projections from the graph of the correspondence to its domain and codomain. We provide assumptions under which the homomorphism induced by an outer approximation of a continuous map coincides with the homomorphism induced in homology by the map. In contrast to more classical results we do not require that the projection to the domain have acyclic preimages. Moreover, we show that it is possible to retrieve correct homological information from a correspondence even if some data is missing or perturbed. Finally, we describe an application to combinatorial maps that are either outer approximations of continuous maps or reconstructions of such maps from a finite set of data points.","lang":"eng"}],"publication_status":"published","publication_identifier":{"issn":["1088-6826"]},"title":"Inducing a map on homology from a correspondence","oa_version":"Preprint","author":[{"first_name":"Shaun","last_name":"Harker","full_name":"Harker, Shaun"},{"first_name":"Hiroshi","full_name":"Kokubu, Hiroshi","last_name":"Kokubu"},{"first_name":"Konstantin","full_name":"Mischaikow, Konstantin","last_name":"Mischaikow"},{"last_name":"Pilarczyk","full_name":"Pilarczyk, Pawel","id":"3768D56A-F248-11E8-B48F-1D18A9856A87","first_name":"Pawel"}],"scopus_import":"1","day":"01","article_type":"original","date_created":"2018-12-11T11:50:57Z","volume":144,"language":[{"iso":"eng"}],"oa":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","issue":"4","citation":{"apa":"Harker, S., Kokubu, H., Mischaikow, K., &#38; Pilarczyk, P. (2016). Inducing a map on homology from a correspondence. <i>Proceedings of the American Mathematical Society</i>. American Mathematical Society. <a href=\"https://doi.org/10.1090/proc/12812\">https://doi.org/10.1090/proc/12812</a>","mla":"Harker, Shaun, et al. “Inducing a Map on Homology from a Correspondence.” <i>Proceedings of the American Mathematical Society</i>, vol. 144, no. 4, American Mathematical Society, 2016, pp. 1787–801, doi:<a href=\"https://doi.org/10.1090/proc/12812\">10.1090/proc/12812</a>.","chicago":"Harker, Shaun, Hiroshi Kokubu, Konstantin Mischaikow, and Pawel Pilarczyk. “Inducing a Map on Homology from a Correspondence.” <i>Proceedings of the American Mathematical Society</i>. American Mathematical Society, 2016. <a href=\"https://doi.org/10.1090/proc/12812\">https://doi.org/10.1090/proc/12812</a>.","ista":"Harker S, Kokubu H, Mischaikow K, Pilarczyk P. 2016. Inducing a map on homology from a correspondence. Proceedings of the American Mathematical Society. 144(4), 1787–1801.","ieee":"S. Harker, H. Kokubu, K. Mischaikow, and P. Pilarczyk, “Inducing a map on homology from a correspondence,” <i>Proceedings of the American Mathematical Society</i>, vol. 144, no. 4. American Mathematical Society, pp. 1787–1801, 2016.","short":"S. Harker, H. Kokubu, K. Mischaikow, P. Pilarczyk, Proceedings of the American Mathematical Society 144 (2016) 1787–1801.","ama":"Harker S, Kokubu H, Mischaikow K, Pilarczyk P. Inducing a map on homology from a correspondence. <i>Proceedings of the American Mathematical Society</i>. 2016;144(4):1787-1801. doi:<a href=\"https://doi.org/10.1090/proc/12812\">10.1090/proc/12812</a>"},"month":"04","arxiv":1,"department":[{"_id":"HeEd"}]},{"publication_status":"published","abstract":[{"lang":"eng","text":"We use rigorous numerical techniques to compute a lower bound for the exponent of expansivity outside a neighborhood of the critical point for thousands of intervals of parameter values in the quadratic family. We first compute a radius of the critical neighborhood outside which the map is uniformly expanding. This radius is taken as small as possible, yet large enough for our numerical procedure to succeed in proving that the expansivity exponent outside this neighborhood is positive. Then, for each of the intervals, we compute a lower bound for this expansivity exponent, valid for all the parameters in that interval. We illustrate and study the distribution of the radii and the expansivity exponents. The results of our computations are mathematically rigorous. The source code of the software and the results of the computations are made publicly available at http://www.pawelpilarczyk.com/quadratic/."}],"intvolume":"        25","date_created":"2018-12-11T11:50:58Z","volume":25,"oa_version":"Preprint","title":"Uniform expansivity outside a critical neighborhood in the quadratic family","day":"02","scopus_import":1,"author":[{"first_name":"Ali","full_name":"Golmakani, Ali","last_name":"Golmakani"},{"first_name":"Stefano","full_name":"Luzzatto, Stefano","last_name":"Luzzatto"},{"id":"3768D56A-F248-11E8-B48F-1D18A9856A87","full_name":"Pilarczyk, Pawel","last_name":"Pilarczyk","first_name":"Pawel"}],"user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","citation":{"chicago":"Golmakani, Ali, Stefano Luzzatto, and Pawel Pilarczyk. “Uniform Expansivity Outside a Critical Neighborhood in the Quadratic Family.” <i>Experimental Mathematics</i>. Taylor and Francis, 2016. <a href=\"https://doi.org/10.1080/10586458.2015.1048011\">https://doi.org/10.1080/10586458.2015.1048011</a>.","ista":"Golmakani A, Luzzatto S, Pilarczyk P. 2016. Uniform expansivity outside a critical neighborhood in the quadratic family. Experimental Mathematics. 25(2), 116–124.","apa":"Golmakani, A., Luzzatto, S., &#38; Pilarczyk, P. (2016). Uniform expansivity outside a critical neighborhood in the quadratic family. <i>Experimental Mathematics</i>. Taylor and Francis. <a href=\"https://doi.org/10.1080/10586458.2015.1048011\">https://doi.org/10.1080/10586458.2015.1048011</a>","mla":"Golmakani, Ali, et al. “Uniform Expansivity Outside a Critical Neighborhood in the Quadratic Family.” <i>Experimental Mathematics</i>, vol. 25, no. 2, Taylor and Francis, 2016, pp. 116–24, doi:<a href=\"https://doi.org/10.1080/10586458.2015.1048011\">10.1080/10586458.2015.1048011</a>.","ama":"Golmakani A, Luzzatto S, Pilarczyk P. Uniform expansivity outside a critical neighborhood in the quadratic family. <i>Experimental Mathematics</i>. 2016;25(2):116-124. doi:<a href=\"https://doi.org/10.1080/10586458.2015.1048011\">10.1080/10586458.2015.1048011</a>","ieee":"A. Golmakani, S. Luzzatto, and P. Pilarczyk, “Uniform expansivity outside a critical neighborhood in the quadratic family,” <i>Experimental Mathematics</i>, vol. 25, no. 2. Taylor and Francis, pp. 116–124, 2016.","short":"A. Golmakani, S. Luzzatto, P. Pilarczyk, Experimental Mathematics 25 (2016) 116–124."},"issue":"2","language":[{"iso":"eng"}],"oa":1,"department":[{"_id":"HeEd"}],"month":"04","quality_controlled":"1","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1504.00116"}],"page":"116 - 124","type":"journal_article","_id":"1254","date_updated":"2021-01-12T06:49:25Z","publisher":"Taylor and Francis","doi":"10.1080/10586458.2015.1048011","date_published":"2016-04-02T00:00:00Z","acknowledgement":"AG and PP were partially supported by Abdus Salam International Centre for Theoretical Physics (ICTP). Additionally, AG was supported by BREUDS, and research conducted by PP has received funding from Fundo Europeu de Desenvolvimento Regional (FEDER) through COMPETE—Programa Operacional Factores de Competitividade (POFC) and from the Portuguese national funds through Fundação para a Ciência e a Tecnologia (FCT) in the framework of the research project FCOMP-01-0124-FEDER-010645 (ref. FCT PTDC/MAT/098871/2008); and from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement no. 622033. The  authors  gratefully  acknowledge  the  Department  of\r\nMathematics  of  Kyoto  University  for  providing  access\r\nto  their  server  for  conducting  computations  for  this\r\nproject.","ec_funded":1,"status":"public","publication":"Experimental Mathematics","project":[{"grant_number":"622033","name":"Persistent Homology - Images, Data and Maps","call_identifier":"FP7","_id":"255F06BE-B435-11E9-9278-68D0E5697425"}],"publist_id":"6071","year":"2016"},{"publication_identifier":{"eissn":["1536-0040"]},"publication_status":"published","intvolume":"        14","abstract":[{"lang":"eng","text":"We show that incorporating spatial dispersal of individuals into a simple vaccination epidemic model may give rise to a model that exhibits rich dynamical behavior. Using an SIVS (susceptible-infected-vaccinated-susceptible) model as a basis, we describe the spread of an infectious disease in a population split into two regions. In each subpopulation, both forward and backward bifurcations can occur. This implies that for disconnected regions the two-patch system may admit several steady states. We consider traveling between the regions and investigate the impact of spatial dispersal of individuals on the model dynamics. We establish conditions for the existence of multiple nontrivial steady states in the system, and we study the structure of the equilibria. The mathematical analysis reveals an unusually rich dynamical behavior, not normally found in the simple epidemic models. In addition to the disease-free equilibrium, eight endemic equilibria emerge from backward transcritical and saddle-node bifurcation points, forming an interesting bifurcation diagram. Stability of steady states, their bifurcations, and the global dynamics are investigated with analytical tools, numerical simulations, and rigorous set-oriented numerical computations."}],"date_created":"2018-12-11T11:52:42Z","article_type":"original","volume":14,"oa_version":"Published Version","title":"Rich bifurcation structure in a two patch vaccination model","day":"01","scopus_import":1,"author":[{"last_name":"Knipl","full_name":"Knipl, Diána","first_name":"Diána"},{"last_name":"Pilarczyk","full_name":"Pilarczyk, Pawel","id":"3768D56A-F248-11E8-B48F-1D18A9856A87","first_name":"Pawel"},{"first_name":"Gergely","full_name":"Röst, Gergely","last_name":"Röst"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","citation":{"mla":"Knipl, Diána, et al. “Rich Bifurcation Structure in a Two Patch Vaccination Model.” <i>SIAM Journal on Applied Dynamical Systems</i>, vol. 14, no. 2, Society for Industrial and Applied Mathematics , 2015, pp. 980–1017, doi:<a href=\"https://doi.org/10.1137/140993934\">10.1137/140993934</a>.","apa":"Knipl, D., Pilarczyk, P., &#38; Röst, G. (2015). Rich bifurcation structure in a two patch vaccination model. <i>SIAM Journal on Applied Dynamical Systems</i>. Society for Industrial and Applied Mathematics . <a href=\"https://doi.org/10.1137/140993934\">https://doi.org/10.1137/140993934</a>","ista":"Knipl D, Pilarczyk P, Röst G. 2015. Rich bifurcation structure in a two patch vaccination model. SIAM Journal on Applied Dynamical Systems. 14(2), 980–1017.","chicago":"Knipl, Diána, Pawel Pilarczyk, and Gergely Röst. “Rich Bifurcation Structure in a Two Patch Vaccination Model.” <i>SIAM Journal on Applied Dynamical Systems</i>. Society for Industrial and Applied Mathematics , 2015. <a href=\"https://doi.org/10.1137/140993934\">https://doi.org/10.1137/140993934</a>.","short":"D. Knipl, P. Pilarczyk, G. Röst, SIAM Journal on Applied Dynamical Systems 14 (2015) 980–1017.","ieee":"D. Knipl, P. Pilarczyk, and G. Röst, “Rich bifurcation structure in a two patch vaccination model,” <i>SIAM Journal on Applied Dynamical Systems</i>, vol. 14, no. 2. Society for Industrial and Applied Mathematics , pp. 980–1017, 2015.","ama":"Knipl D, Pilarczyk P, Röst G. Rich bifurcation structure in a two patch vaccination model. <i>SIAM Journal on Applied Dynamical Systems</i>. 2015;14(2):980-1017. doi:<a href=\"https://doi.org/10.1137/140993934\">10.1137/140993934</a>"},"issue":"2","language":[{"iso":"eng"}],"oa":1,"department":[{"_id":"HeEd"}],"month":"01","quality_controlled":"1","main_file_link":[{"open_access":"1","url":"http://discovery.ucl.ac.uk/1473750/1/99393.pdf"}],"ddc":["510"],"page":"980 - 1017","type":"journal_article","_id":"1555","date_updated":"2021-01-12T06:51:34Z","publisher":"Society for Industrial and Applied Mathematics ","article_processing_charge":"No","doi":"10.1137/140993934","acknowledgement":"Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria (pawel.pilarczyk@ist.ac.at). This author’s work was partially supported by the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement 622033, by Fundo Europeu de Desenvolvimento Regional (FEDER) through COMPETE—Programa Operacional Factores de Competitividade (POFC), by the Portuguese national funds through Funda ̧caoparaaCiˆencia e a Tecnologia (FCT) in the framework of the research project FCOMP-01-0124-FEDER-010645 (ref. FCT PTDC/MAT/098871/2008), and by European Research Council through StG 259559 in the framework of the EPIDELAY project.","date_published":"2015-01-01T00:00:00Z","ec_funded":1,"publication":"SIAM Journal on Applied Dynamical Systems","status":"public","project":[{"_id":"255F06BE-B435-11E9-9278-68D0E5697425","name":"Persistent Homology - Images, Data and Maps","grant_number":"622033","call_identifier":"FP7"}],"publist_id":"5616","year":"2015"},{"language":[{"iso":"eng"}],"publication":"Topological Methods in Nonlinear Analysis","status":"public","date_published":"2015-03-01T00:00:00Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","issue":"1","citation":{"chicago":"Graff, Grzegorz, and Pawel Pilarczyk. “An Algorithmic Approach to Estimating the Minimal Number of Periodic Points for Smooth Self-Maps of Simply-Connected Manifolds.” <i>Topological Methods in Nonlinear Analysis</i>. Juliusz Schauder Center for Nonlinear Studies, 2015. <a href=\"https://doi.org/10.12775/TMNA.2015.014\">https://doi.org/10.12775/TMNA.2015.014</a>.","ista":"Graff G, Pilarczyk P. 2015. An algorithmic approach to estimating the minimal number of periodic points for smooth self-maps of simply-connected manifolds. Topological Methods in Nonlinear Analysis. 45(1), 273–286.","apa":"Graff, G., &#38; Pilarczyk, P. (2015). An algorithmic approach to estimating the minimal number of periodic points for smooth self-maps of simply-connected manifolds. <i>Topological Methods in Nonlinear Analysis</i>. Juliusz Schauder Center for Nonlinear Studies. <a href=\"https://doi.org/10.12775/TMNA.2015.014\">https://doi.org/10.12775/TMNA.2015.014</a>","mla":"Graff, Grzegorz, and Pawel Pilarczyk. “An Algorithmic Approach to Estimating the Minimal Number of Periodic Points for Smooth Self-Maps of Simply-Connected Manifolds.” <i>Topological Methods in Nonlinear Analysis</i>, vol. 45, no. 1, Juliusz Schauder Center for Nonlinear Studies, 2015, pp. 273–86, doi:<a href=\"https://doi.org/10.12775/TMNA.2015.014\">10.12775/TMNA.2015.014</a>.","ama":"Graff G, Pilarczyk P. An algorithmic approach to estimating the minimal number of periodic points for smooth self-maps of simply-connected manifolds. <i>Topological Methods in Nonlinear Analysis</i>. 2015;45(1):273-286. doi:<a href=\"https://doi.org/10.12775/TMNA.2015.014\">10.12775/TMNA.2015.014</a>","ieee":"G. Graff and P. Pilarczyk, “An algorithmic approach to estimating the minimal number of periodic points for smooth self-maps of simply-connected manifolds,” <i>Topological Methods in Nonlinear Analysis</i>, vol. 45, no. 1. Juliusz Schauder Center for Nonlinear Studies, pp. 273–286, 2015.","short":"G. Graff, P. Pilarczyk, Topological Methods in Nonlinear Analysis 45 (2015) 273–286."},"month":"03","year":"2015","publist_id":"5608","department":[{"_id":"HeEd"}],"abstract":[{"lang":"eng","text":"For a given self-map $f$ of $M$, a closed smooth connected and simply-connected manifold of dimension $m\\geq 4$, we provide an algorithm for estimating the values of the topological invariant $D^m_r[f]$, which equals the minimal number of $r$-periodic points in the smooth homotopy class of $f$. Our results are based on the combinatorial scheme for computing $D^m_r[f]$ introduced by G. Graff and J. Jezierski [J. Fixed Point Theory Appl. 13 (2013), 63-84]. An open-source implementation of the algorithm programmed in C++ is publicly available at {\\tt http://www.pawelpilarczyk.com/combtop/}."}],"intvolume":"        45","page":"273 - 286","quality_controlled":"1","publication_status":"published","oa_version":"None","publisher":"Juliusz Schauder Center for Nonlinear Studies","title":"An algorithmic approach to estimating the minimal number of periodic points for smooth self-maps of simply-connected manifolds","author":[{"last_name":"Graff","full_name":"Graff, Grzegorz","first_name":"Grzegorz"},{"id":"3768D56A-F248-11E8-B48F-1D18A9856A87","full_name":"Pilarczyk, Pawel","last_name":"Pilarczyk","first_name":"Pawel"}],"doi":"10.12775/TMNA.2015.014","day":"01","scopus_import":1,"type":"journal_article","date_created":"2018-12-11T11:52:44Z","date_updated":"2021-01-12T06:51:37Z","_id":"1563","volume":45},{"doi":"10.1007/s10444-014-9356-1","author":[{"first_name":"Pawel","id":"3768D56A-F248-11E8-B48F-1D18A9856A87","full_name":"Pawel Pilarczyk","last_name":"Pilarczyk"},{"first_name":"Pedro","full_name":"Real, Pedro","last_name":"Real"}],"day":"01","title":"Computation of cubical homology, cohomology, and (co)homological operations via chain contraction","publisher":"Kluwer","date_updated":"2021-01-12T07:59:20Z","_id":"451","volume":41,"type":"journal_article","date_created":"2018-12-11T11:46:33Z","page":"253 - 275","abstract":[{"text":"We introduce algorithms for the computation of homology, cohomology, and related operations on cubical cell complexes, using the technique based on a chain contraction from the original chain complex to a reduced one that represents its homology. This work is based on previous results for simplicial complexes, and uses Serre’s diagonalization for cubical cells. An implementation in C++ of the introduced algorithms is available at http://www.pawelpilarczyk.com/chaincon/ together with some examples. The paper is self-contained as much as possible, and is written at a very elementary level, so that basic knowledge of algebraic topology should be sufficient to follow it.","lang":"eng"}],"intvolume":"        41","main_file_link":[{"open_access":"1","url":"https://idus.us.es/xmlui/handle/11441/38719"}],"quality_controlled":0,"publication_status":"published","year":"2014","month":"01","publist_id":"7371","oa":1,"publication":"Advances in Computational Mathematics","extern":1,"status":"public","issue":"1","citation":{"ista":"Pilarczyk P, Real P. 2014. Computation of cubical homology, cohomology, and (co)homological operations via chain contraction. Advances in Computational Mathematics. 41(1), 253–275.","chicago":"Pilarczyk, Pawel, and Pedro Real. “Computation of Cubical Homology, Cohomology, and (Co)Homological Operations via Chain Contraction.” <i>Advances in Computational Mathematics</i>. Kluwer, 2014. <a href=\"https://doi.org/10.1007/s10444-014-9356-1\">https://doi.org/10.1007/s10444-014-9356-1</a>.","apa":"Pilarczyk, P., &#38; Real, P. (2014). Computation of cubical homology, cohomology, and (co)homological operations via chain contraction. <i>Advances in Computational Mathematics</i>. Kluwer. <a href=\"https://doi.org/10.1007/s10444-014-9356-1\">https://doi.org/10.1007/s10444-014-9356-1</a>","mla":"Pilarczyk, Pawel, and Pedro Real. “Computation of Cubical Homology, Cohomology, and (Co)Homological Operations via Chain Contraction.” <i>Advances in Computational Mathematics</i>, vol. 41, no. 1, Kluwer, 2014, pp. 253–75, doi:<a href=\"https://doi.org/10.1007/s10444-014-9356-1\">10.1007/s10444-014-9356-1</a>.","ama":"Pilarczyk P, Real P. Computation of cubical homology, cohomology, and (co)homological operations via chain contraction. <i>Advances in Computational Mathematics</i>. 2014;41(1):253-275. doi:<a href=\"https://doi.org/10.1007/s10444-014-9356-1\">10.1007/s10444-014-9356-1</a>","ieee":"P. Pilarczyk and P. Real, “Computation of cubical homology, cohomology, and (co)homological operations via chain contraction,” <i>Advances in Computational Mathematics</i>, vol. 41, no. 1. Kluwer, pp. 253–275, 2014.","short":"P. Pilarczyk, P. Real, Advances in Computational Mathematics 41 (2014) 253–275."},"date_published":"2014-01-01T00:00:00Z","acknowledgement":"This research was partially supported from Fundo Europeu de Desenvolvimento Regional (FEDER) through COMPETE – Programa Operacional Factores de Compe titividade (POFC) and from the Portuguese national funds through Fundacaoparaa Ciencia e a Tecnologia (FCT) in the framework of the research project FCOMP-01-0124-FEDER-010645(ref. FCT PTDC/MAT/098871/2008), as well as from the funds distributed through the European Science Foundation (ESF) Research Networking Programme on “Applied and Computational Algebraic Topology” (ACAT). P.Real was additionally supported by the Spanish Ministry of Science and Innovation, project no. MTM2009-12716. "}]