---
_id: '9821'
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.
acknowledgement: We express our gratitude to the anonymous referees who provided constructive
comments that helped us improve the quality of the paper.
article_number: e0253851
article_processing_charge: Yes
article_type: original
author:
- first_name: Grzegorz
full_name: Graff, Grzegorz
last_name: Graff
- first_name: Beata
full_name: Graff, Beata
last_name: Graff
- first_name: Pawel
full_name: Pilarczyk, Pawel
id: 3768D56A-F248-11E8-B48F-1D18A9856A87
last_name: Pilarczyk
- first_name: Grzegorz
full_name: Jablonski, Grzegorz
id: 4483EF78-F248-11E8-B48F-1D18A9856A87
last_name: Jablonski
orcid: 0000-0002-3536-9866
- first_name: Dariusz
full_name: Gąsecki, Dariusz
last_name: Gąsecki
- first_name: Krzysztof
full_name: Narkiewicz, Krzysztof
last_name: Narkiewicz
citation:
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. PLoS
ONE. 2021;16(7). doi:10.1371/journal.pone.0253851
apa: 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. Public Library of Science. https://doi.org/10.1371/journal.pone.0253851
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.” PLoS ONE. Public Library of Science,
2021. https://doi.org/10.1371/journal.pone.0253851.
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,”
PLoS ONE, vol. 16, no. 7. Public Library of Science, 2021.
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.
mla: Graff, Grzegorz, et al. “Persistent Homology as a New Method of the Assessment
of Heart Rate Variability.” PLoS ONE, vol. 16, no. 7, e0253851, Public
Library of Science, 2021, doi:10.1371/journal.pone.0253851.
short: G. Graff, B. Graff, P. Pilarczyk, G. Jablonski, D. Gąsecki, K. Narkiewicz,
PLoS ONE 16 (2021).
date_created: 2021-08-08T22:01:28Z
date_published: 2021-07-01T00:00:00Z
date_updated: 2023-08-10T14:21:42Z
day: '01'
ddc:
- '006'
department:
- _id: HeEd
doi: 10.1371/journal.pone.0253851
external_id:
isi:
- '000678124900050'
pmid:
- '34292957'
file:
- access_level: open_access
checksum: 0277aa155d5db1febd2cb384768bba5f
content_type: application/pdf
creator: asandaue
date_created: 2021-08-09T09:25:41Z
date_updated: 2021-08-09T09:25:41Z
file_id: '9832'
file_name: 2021_PLoSONE_Graff.pdf
file_size: 2706919
relation: main_file
success: 1
file_date_updated: 2021-08-09T09:25:41Z
has_accepted_license: '1'
intvolume: ' 16'
isi: 1
issue: '7'
language:
- iso: eng
month: '07'
oa: 1
oa_version: Published Version
pmid: 1
publication: PLoS ONE
publication_identifier:
eissn:
- '19326203'
publication_status: published
publisher: Public Library of Science
quality_controlled: '1'
scopus_import: '1'
status: public
title: Persistent homology as a new method of the assessment of heart rate variability
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 16
year: '2021'
...
---
_id: '1149'
abstract:
- lang: eng
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.'
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."
author:
- first_name: Tomoyuki
full_name: Miyaji, Tomoyuki
last_name: Miyaji
- first_name: Pawel
full_name: Pilarczyk, Pawel
id: 3768D56A-F248-11E8-B48F-1D18A9856A87
last_name: Pilarczyk
- first_name: Marcio
full_name: Gameiro, Marcio
last_name: Gameiro
- first_name: Hiroshi
full_name: Kokubu, Hiroshi
last_name: Kokubu
- first_name: Konstantin
full_name: Mischaikow, Konstantin
last_name: Mischaikow
citation:
ama: Miyaji T, Pilarczyk P, Gameiro M, Kokubu H, Mischaikow K. A study of rigorous
ODE integrators for multi scale set oriented computations. Applied Numerical
Mathematics. 2016;107:34-47. doi:10.1016/j.apnum.2016.04.005
apa: 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. Elsevier. https://doi.org/10.1016/j.apnum.2016.04.005
chicago: Miyaji, Tomoyuki, Pawel Pilarczyk, Marcio Gameiro, Hiroshi Kokubu, and
Konstantin Mischaikow. “A Study of Rigorous ODE Integrators for Multi Scale Set
Oriented Computations.” Applied Numerical Mathematics. Elsevier, 2016.
https://doi.org/10.1016/j.apnum.2016.04.005.
ieee: T. Miyaji, P. Pilarczyk, M. Gameiro, H. Kokubu, and K. Mischaikow, “A study
of rigorous ODE integrators for multi scale set oriented computations,” Applied
Numerical Mathematics, vol. 107. Elsevier, pp. 34–47, 2016.
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.
mla: Miyaji, Tomoyuki, et al. “A Study of Rigorous ODE Integrators for Multi Scale
Set Oriented Computations.” Applied Numerical Mathematics, vol. 107, Elsevier,
2016, pp. 34–47, doi:10.1016/j.apnum.2016.04.005.
short: T. Miyaji, P. Pilarczyk, M. Gameiro, H. Kokubu, K. Mischaikow, Applied Numerical
Mathematics 107 (2016) 34–47.
date_created: 2018-12-11T11:50:25Z
date_published: 2016-09-01T00:00:00Z
date_updated: 2021-01-12T06:48:38Z
day: '01'
department:
- _id: HeEd
doi: 10.1016/j.apnum.2016.04.005
ec_funded: 1
intvolume: ' 107'
language:
- iso: eng
month: '09'
oa_version: None
page: 34 - 47
project:
- _id: 255F06BE-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '622033'
name: Persistent Homology - Images, Data and Maps
publication: Applied Numerical Mathematics
publication_status: published
publisher: Elsevier
publist_id: '6209'
quality_controlled: '1'
scopus_import: 1
status: public
title: A study of rigorous ODE integrators for multi scale set oriented computations
type: journal_article
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 107
year: '2016'
...
---
_id: '1237'
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 > 0 whether every g: X → ℝm with ∥g - f ∥∞ ≤ r has a root, reduces to the
extension problem.'
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.).
alternative_title:
- LNCS
author:
- first_name: Marek
full_name: Krcál, Marek
id: 33E21118-F248-11E8-B48F-1D18A9856A87
last_name: Krcál
- first_name: Pawel
full_name: Pilarczyk, Pawel
id: 3768D56A-F248-11E8-B48F-1D18A9856A87
last_name: Pilarczyk
citation:
ama: 'Krcál M, Pilarczyk P. Computation of cubical Steenrod squares. In: Vol 9667.
Springer; 2016:140-151. doi:10.1007/978-3-319-39441-1_13'
apa: 'Krcál, M., & 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. https://doi.org/10.1007/978-3-319-39441-1_13'
chicago: Krcál, Marek, and Pawel Pilarczyk. “Computation of Cubical Steenrod Squares,”
9667:140–51. Springer, 2016. https://doi.org/10.1007/978-3-319-39441-1_13.
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.'
ista: 'Krcál M, Pilarczyk P. 2016. Computation of cubical Steenrod squares. CTIC:
Computational Topology in Image Context, LNCS, vol. 9667, 140–151.'
mla: Krcál, Marek, and Pawel Pilarczyk. Computation of Cubical Steenrod Squares.
Vol. 9667, Springer, 2016, pp. 140–51, doi:10.1007/978-3-319-39441-1_13.
short: M. Krcál, P. Pilarczyk, in:, Springer, 2016, pp. 140–151.
conference:
end_date: 2016-06-17
location: Marseille, France
name: 'CTIC: Computational Topology in Image Context'
start_date: 2016-06-15
date_created: 2018-12-11T11:50:52Z
date_published: 2016-06-02T00:00:00Z
date_updated: 2021-01-12T06:49:18Z
day: '02'
department:
- _id: UlWa
- _id: HeEd
doi: 10.1007/978-3-319-39441-1_13
ec_funded: 1
intvolume: ' 9667'
language:
- iso: eng
month: '06'
oa_version: None
page: 140 - 151
project:
- _id: 25681D80-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '291734'
name: International IST Postdoc Fellowship Programme
- _id: 255F06BE-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '622033'
name: Persistent Homology - Images, Data and Maps
publication_status: published
publisher: Springer
publist_id: '6096'
quality_controlled: '1'
scopus_import: 1
status: public
title: Computation of cubical Steenrod squares
type: conference
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 9667
year: '2016'
...
---
_id: '1252'
abstract:
- lang: eng
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.
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."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Shaun
full_name: Harker, Shaun
last_name: Harker
- first_name: Hiroshi
full_name: Kokubu, Hiroshi
last_name: Kokubu
- first_name: Konstantin
full_name: Mischaikow, Konstantin
last_name: Mischaikow
- first_name: Pawel
full_name: Pilarczyk, Pawel
id: 3768D56A-F248-11E8-B48F-1D18A9856A87
last_name: Pilarczyk
citation:
ama: Harker S, Kokubu H, Mischaikow K, Pilarczyk P. Inducing a map on homology from
a correspondence. Proceedings of the American Mathematical Society. 2016;144(4):1787-1801.
doi:10.1090/proc/12812
apa: Harker, S., Kokubu, H., Mischaikow, K., & Pilarczyk, P. (2016). Inducing
a map on homology from a correspondence. Proceedings of the American Mathematical
Society. American Mathematical Society. https://doi.org/10.1090/proc/12812
chicago: Harker, Shaun, Hiroshi Kokubu, Konstantin Mischaikow, and Pawel Pilarczyk.
“Inducing a Map on Homology from a Correspondence.” Proceedings of the American
Mathematical Society. American Mathematical Society, 2016. https://doi.org/10.1090/proc/12812.
ieee: S. Harker, H. Kokubu, K. Mischaikow, and P. Pilarczyk, “Inducing a map on
homology from a correspondence,” Proceedings of the American Mathematical Society,
vol. 144, no. 4. American Mathematical Society, pp. 1787–1801, 2016.
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.
mla: Harker, Shaun, et al. “Inducing a Map on Homology from a Correspondence.” Proceedings
of the American Mathematical Society, vol. 144, no. 4, American Mathematical
Society, 2016, pp. 1787–801, doi:10.1090/proc/12812.
short: S. Harker, H. Kokubu, K. Mischaikow, P. Pilarczyk, Proceedings of the American
Mathematical Society 144 (2016) 1787–1801.
date_created: 2018-12-11T11:50:57Z
date_published: 2016-04-01T00:00:00Z
date_updated: 2022-05-24T09:35:58Z
day: '01'
department:
- _id: HeEd
doi: 10.1090/proc/12812
ec_funded: 1
external_id:
arxiv:
- '1411.7563'
intvolume: ' 144'
issue: '4'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1411.7563
month: '04'
oa: 1
oa_version: Preprint
page: 1787 - 1801
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
publication_identifier:
issn:
- 1088-6826
publication_status: published
publisher: American Mathematical Society
publist_id: '6075'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Inducing a map on homology from a correspondence
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 144
year: '2016'
...
---
_id: '1254'
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/.
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."
author:
- first_name: Ali
full_name: Golmakani, Ali
last_name: Golmakani
- first_name: Stefano
full_name: Luzzatto, Stefano
last_name: Luzzatto
- first_name: Pawel
full_name: Pilarczyk, Pawel
id: 3768D56A-F248-11E8-B48F-1D18A9856A87
last_name: Pilarczyk
citation:
ama: Golmakani A, Luzzatto S, Pilarczyk P. Uniform expansivity outside a critical
neighborhood in the quadratic family. Experimental Mathematics. 2016;25(2):116-124.
doi:10.1080/10586458.2015.1048011
apa: Golmakani, A., Luzzatto, S., & Pilarczyk, P. (2016). Uniform expansivity
outside a critical neighborhood in the quadratic family. Experimental Mathematics.
Taylor and Francis. https://doi.org/10.1080/10586458.2015.1048011
chicago: Golmakani, Ali, Stefano Luzzatto, and Pawel Pilarczyk. “Uniform Expansivity
Outside a Critical Neighborhood in the Quadratic Family.” Experimental Mathematics.
Taylor and Francis, 2016. https://doi.org/10.1080/10586458.2015.1048011.
ieee: A. Golmakani, S. Luzzatto, and P. Pilarczyk, “Uniform expansivity outside
a critical neighborhood in the quadratic family,” Experimental Mathematics,
vol. 25, no. 2. Taylor and Francis, pp. 116–124, 2016.
ista: Golmakani A, Luzzatto S, Pilarczyk P. 2016. Uniform expansivity outside a
critical neighborhood in the quadratic family. Experimental Mathematics. 25(2),
116–124.
mla: Golmakani, Ali, et al. “Uniform Expansivity Outside a Critical Neighborhood
in the Quadratic Family.” Experimental Mathematics, vol. 25, no. 2, Taylor
and Francis, 2016, pp. 116–24, doi:10.1080/10586458.2015.1048011.
short: A. Golmakani, S. Luzzatto, P. Pilarczyk, Experimental Mathematics 25 (2016)
116–124.
date_created: 2018-12-11T11:50:58Z
date_published: 2016-04-02T00:00:00Z
date_updated: 2021-01-12T06:49:25Z
day: '02'
department:
- _id: HeEd
doi: 10.1080/10586458.2015.1048011
ec_funded: 1
intvolume: ' 25'
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1504.00116
month: '04'
oa: 1
oa_version: Preprint
page: 116 - 124
project:
- _id: 255F06BE-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '622033'
name: Persistent Homology - Images, Data and Maps
publication: Experimental Mathematics
publication_status: published
publisher: Taylor and Francis
publist_id: '6071'
quality_controlled: '1'
scopus_import: 1
status: public
title: Uniform expansivity outside a critical neighborhood in the quadratic family
type: journal_article
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 25
year: '2016'
...
---
_id: '1555'
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.
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.
article_processing_charge: No
article_type: original
author:
- first_name: Diána
full_name: Knipl, Diána
last_name: Knipl
- first_name: Pawel
full_name: Pilarczyk, Pawel
id: 3768D56A-F248-11E8-B48F-1D18A9856A87
last_name: Pilarczyk
- first_name: Gergely
full_name: Röst, Gergely
last_name: Röst
citation:
ama: Knipl D, Pilarczyk P, Röst G. Rich bifurcation structure in a two patch vaccination
model. SIAM Journal on Applied Dynamical Systems. 2015;14(2):980-1017.
doi:10.1137/140993934
apa: Knipl, D., Pilarczyk, P., & Röst, G. (2015). Rich bifurcation structure
in a two patch vaccination model. SIAM Journal on Applied Dynamical Systems.
Society for Industrial and Applied Mathematics . https://doi.org/10.1137/140993934
chicago: Knipl, Diána, Pawel Pilarczyk, and Gergely Röst. “Rich Bifurcation Structure
in a Two Patch Vaccination Model.” SIAM Journal on Applied Dynamical Systems.
Society for Industrial and Applied Mathematics , 2015. https://doi.org/10.1137/140993934.
ieee: D. Knipl, P. Pilarczyk, and G. Röst, “Rich bifurcation structure in a two
patch vaccination model,” SIAM Journal on Applied Dynamical Systems, vol.
14, no. 2. Society for Industrial and Applied Mathematics , pp. 980–1017, 2015.
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.
mla: Knipl, Diána, et al. “Rich Bifurcation Structure in a Two Patch Vaccination
Model.” SIAM Journal on Applied Dynamical Systems, vol. 14, no. 2, Society
for Industrial and Applied Mathematics , 2015, pp. 980–1017, doi:10.1137/140993934.
short: D. Knipl, P. Pilarczyk, G. Röst, SIAM Journal on Applied Dynamical Systems
14 (2015) 980–1017.
date_created: 2018-12-11T11:52:42Z
date_published: 2015-01-01T00:00:00Z
date_updated: 2021-01-12T06:51:34Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1137/140993934
ec_funded: 1
intvolume: ' 14'
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://discovery.ucl.ac.uk/1473750/1/99393.pdf
month: '01'
oa: 1
oa_version: Published Version
page: 980 - 1017
project:
- _id: 255F06BE-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '622033'
name: Persistent Homology - Images, Data and Maps
publication: SIAM Journal on Applied Dynamical Systems
publication_identifier:
eissn:
- 1536-0040
publication_status: published
publisher: 'Society for Industrial and Applied Mathematics '
publist_id: '5616'
quality_controlled: '1'
scopus_import: 1
status: public
title: Rich bifurcation structure in a two patch vaccination model
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 14
year: '2015'
...
---
_id: '1563'
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/}.
author:
- first_name: Grzegorz
full_name: Graff, Grzegorz
last_name: Graff
- first_name: Pawel
full_name: Pilarczyk, Pawel
id: 3768D56A-F248-11E8-B48F-1D18A9856A87
last_name: Pilarczyk
citation:
ama: Graff G, Pilarczyk P. An algorithmic approach to estimating the minimal number
of periodic points for smooth self-maps of simply-connected manifolds. Topological
Methods in Nonlinear Analysis. 2015;45(1):273-286. doi:10.12775/TMNA.2015.014
apa: 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. Juliusz Schauder
Center for Nonlinear Studies. https://doi.org/10.12775/TMNA.2015.014
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.” Topological Methods in Nonlinear Analysis. Juliusz Schauder
Center for Nonlinear Studies, 2015. https://doi.org/10.12775/TMNA.2015.014.
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,”
Topological Methods in Nonlinear Analysis, vol. 45, no. 1. Juliusz Schauder
Center for Nonlinear Studies, pp. 273–286, 2015.
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.
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.” Topological Methods in Nonlinear Analysis, vol. 45, no. 1,
Juliusz Schauder Center for Nonlinear Studies, 2015, pp. 273–86, doi:10.12775/TMNA.2015.014.
short: G. Graff, P. Pilarczyk, Topological Methods in Nonlinear Analysis 45 (2015)
273–286.
date_created: 2018-12-11T11:52:44Z
date_published: 2015-03-01T00:00:00Z
date_updated: 2021-01-12T06:51:37Z
day: '01'
department:
- _id: HeEd
doi: 10.12775/TMNA.2015.014
intvolume: ' 45'
issue: '1'
language:
- iso: eng
month: '03'
oa_version: None
page: 273 - 286
publication: Topological Methods in Nonlinear Analysis
publication_status: published
publisher: Juliusz Schauder Center for Nonlinear Studies
publist_id: '5608'
quality_controlled: '1'
scopus_import: 1
status: public
title: An algorithmic approach to estimating the minimal number of periodic points
for smooth self-maps of simply-connected manifolds
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 45
year: '2015'
...
---
_id: '451'
abstract:
- lang: eng
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.
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. '
author:
- first_name: Pawel
full_name: Pawel Pilarczyk
id: 3768D56A-F248-11E8-B48F-1D18A9856A87
last_name: Pilarczyk
- first_name: Pedro
full_name: Real, Pedro
last_name: Real
citation:
ama: Pilarczyk P, Real P. Computation of cubical homology, cohomology, and (co)homological
operations via chain contraction. Advances in Computational Mathematics.
2014;41(1):253-275. doi:10.1007/s10444-014-9356-1
apa: Pilarczyk, P., & Real, P. (2014). Computation of cubical homology, cohomology,
and (co)homological operations via chain contraction. Advances in Computational
Mathematics. Kluwer. https://doi.org/10.1007/s10444-014-9356-1
chicago: Pilarczyk, Pawel, and Pedro Real. “Computation of Cubical Homology, Cohomology,
and (Co)Homological Operations via Chain Contraction.” Advances in Computational
Mathematics. Kluwer, 2014. https://doi.org/10.1007/s10444-014-9356-1.
ieee: P. Pilarczyk and P. Real, “Computation of cubical homology, cohomology, and
(co)homological operations via chain contraction,” Advances in Computational
Mathematics, vol. 41, no. 1. Kluwer, pp. 253–275, 2014.
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.
mla: Pilarczyk, Pawel, and Pedro Real. “Computation of Cubical Homology, Cohomology,
and (Co)Homological Operations via Chain Contraction.” Advances in Computational
Mathematics, vol. 41, no. 1, Kluwer, 2014, pp. 253–75, doi:10.1007/s10444-014-9356-1.
short: P. Pilarczyk, P. Real, Advances in Computational Mathematics 41 (2014) 253–275.
date_created: 2018-12-11T11:46:33Z
date_published: 2014-01-01T00:00:00Z
date_updated: 2021-01-12T07:59:20Z
day: '01'
doi: 10.1007/s10444-014-9356-1
extern: 1
intvolume: ' 41'
issue: '1'
main_file_link:
- open_access: '1'
url: https://idus.us.es/xmlui/handle/11441/38719
month: '01'
oa: 1
page: 253 - 275
publication: Advances in Computational Mathematics
publication_status: published
publisher: Kluwer
publist_id: '7371'
quality_controlled: 0
status: public
title: Computation of cubical homology, cohomology, and (co)homological operations
via chain contraction
type: journal_article
volume: 41
year: '2014'
...