---
_id: '1351'
abstract:
- lang: eng
text: The behaviour of gene regulatory networks (GRNs) is typically analysed using
simulation-based statistical testing-like methods. In this paper, we demonstrate
that we can replace this approach by a formal verification-like method that gives
higher assurance and scalability. We focus on Wagner’s weighted GRN model with
varying weights, which is used in evolutionary biology. In the model, weight parameters
represent the gene interaction strength that may change due to genetic mutations.
For a property of interest, we synthesise the constraints over the parameter space
that represent the set of GRNs satisfying the property. We experimentally show
that our parameter synthesis procedure computes the mutational robustness of GRNs—an
important problem of interest in evolutionary biology—more efficiently than the
classical simulation method. We specify the property in linear temporal logic.
We employ symbolic bounded model checking and SMT solving to compute the space
of GRNs that satisfy the property, which amounts to synthesizing a set of linear
constraints on the weights.
article_processing_charge: No
author:
- first_name: Mirco
full_name: Giacobbe, Mirco
id: 3444EA5E-F248-11E8-B48F-1D18A9856A87
last_name: Giacobbe
orcid: 0000-0001-8180-0904
- first_name: Calin C
full_name: Guet, Calin C
id: 47F8433E-F248-11E8-B48F-1D18A9856A87
last_name: Guet
orcid: 0000-0001-6220-2052
- first_name: Ashutosh
full_name: Gupta, Ashutosh
id: 335E5684-F248-11E8-B48F-1D18A9856A87
last_name: Gupta
- first_name: Thomas A
full_name: Henzinger, Thomas A
id: 40876CD8-F248-11E8-B48F-1D18A9856A87
last_name: Henzinger
orcid: 0000−0002−2985−7724
- first_name: Tiago
full_name: Paixao, Tiago
id: 2C5658E6-F248-11E8-B48F-1D18A9856A87
last_name: Paixao
orcid: 0000-0003-2361-3953
- first_name: Tatjana
full_name: Petrov, Tatjana
id: 3D5811FC-F248-11E8-B48F-1D18A9856A87
last_name: Petrov
orcid: 0000-0002-9041-0905
citation:
ama: Giacobbe M, Guet CC, Gupta A, Henzinger TA, Paixao T, Petrov T. Model checking
the evolution of gene regulatory networks. Acta Informatica. 2017;54(8):765-787.
doi:10.1007/s00236-016-0278-x
apa: Giacobbe, M., Guet, C. C., Gupta, A., Henzinger, T. A., Paixao, T., & Petrov,
T. (2017). Model checking the evolution of gene regulatory networks. Acta Informatica.
Springer. https://doi.org/10.1007/s00236-016-0278-x
chicago: Giacobbe, Mirco, Calin C Guet, Ashutosh Gupta, Thomas A Henzinger, Tiago
Paixao, and Tatjana Petrov. “Model Checking the Evolution of Gene Regulatory Networks.”
Acta Informatica. Springer, 2017. https://doi.org/10.1007/s00236-016-0278-x.
ieee: M. Giacobbe, C. C. Guet, A. Gupta, T. A. Henzinger, T. Paixao, and T. Petrov,
“Model checking the evolution of gene regulatory networks,” Acta Informatica,
vol. 54, no. 8. Springer, pp. 765–787, 2017.
ista: Giacobbe M, Guet CC, Gupta A, Henzinger TA, Paixao T, Petrov T. 2017. Model
checking the evolution of gene regulatory networks. Acta Informatica. 54(8), 765–787.
mla: Giacobbe, Mirco, et al. “Model Checking the Evolution of Gene Regulatory Networks.”
Acta Informatica, vol. 54, no. 8, Springer, 2017, pp. 765–87, doi:10.1007/s00236-016-0278-x.
short: M. Giacobbe, C.C. Guet, A. Gupta, T.A. Henzinger, T. Paixao, T. Petrov, Acta
Informatica 54 (2017) 765–787.
date_created: 2018-12-11T11:51:32Z
date_published: 2017-12-01T00:00:00Z
date_updated: 2023-09-20T11:06:03Z
day: '01'
ddc:
- '006'
- '576'
department:
- _id: ToHe
- _id: CaGu
- _id: NiBa
doi: 10.1007/s00236-016-0278-x
ec_funded: 1
external_id:
isi:
- '000414343200003'
file:
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checksum: 4e661d9135d7f8c342e8e258dee76f3e
content_type: application/pdf
creator: dernst
date_created: 2019-01-17T15:57:29Z
date_updated: 2020-07-14T12:44:46Z
file_id: '5841'
file_name: 2017_ActaInformatica_Giacobbe.pdf
file_size: 755241
relation: main_file
file_date_updated: 2020-07-14T12:44:46Z
has_accepted_license: '1'
intvolume: ' 54'
isi: 1
issue: '8'
language:
- iso: eng
month: '12'
oa: 1
oa_version: Published Version
page: 765 - 787
project:
- _id: 25EE3708-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '267989'
name: Quantitative Reactive Modeling
- _id: 25832EC2-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: S 11407_N23
name: Rigorous Systems Engineering
- _id: 25F42A32-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z211
name: The Wittgenstein Prize
- _id: 25B1EC9E-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '618091'
name: Speed of Adaptation in Population Genetics and Evolutionary Computation
- _id: 25681D80-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '291734'
name: International IST Postdoc Fellowship Programme
- _id: 25B07788-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '250152'
name: Limits to selection in biology and in evolutionary computation
publication: Acta Informatica
publication_identifier:
issn:
- '00015903'
publication_status: published
publisher: Springer
publist_id: '5898'
pubrep_id: '649'
quality_controlled: '1'
related_material:
record:
- id: '1835'
relation: earlier_version
status: public
scopus_import: '1'
status: public
title: Model checking the evolution of gene regulatory networks
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: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 54
year: '2017'
...
---
_id: '1336'
abstract:
- lang: eng
text: Evolutionary algorithms (EAs) form a popular optimisation paradigm inspired
by natural evolution. In recent years the field of evolutionary computation has
developed a rigorous analytical theory to analyse the runtimes of EAs on many
illustrative problems. Here we apply this theory to a simple model of natural
evolution. In the Strong Selection Weak Mutation (SSWM) evolutionary regime the
time between occurrences of new mutations is much longer than the time it takes
for a mutated genotype to take over the population. In this situation, the population
only contains copies of one genotype and evolution can be modelled as a stochastic
process evolving one genotype by means of mutation and selection between the resident
and the mutated genotype. The probability of accepting the mutated genotype then
depends on the change in fitness. We study this process, SSWM, from an algorithmic
perspective, quantifying its expected optimisation time for various parameters
and investigating differences to a similar evolutionary algorithm, the well-known
(1+1) EA. We show that SSWM can have a moderate advantage over the (1+1) EA at
crossing fitness valleys and study an example where SSWM outperforms the (1+1)
EA by taking advantage of information on the fitness gradient.
article_processing_charge: No
author:
- first_name: Tiago
full_name: Paixao, Tiago
id: 2C5658E6-F248-11E8-B48F-1D18A9856A87
last_name: Paixao
orcid: 0000-0003-2361-3953
- first_name: Jorge
full_name: Pérez Heredia, Jorge
last_name: Pérez Heredia
- first_name: Dirk
full_name: Sudholt, Dirk
last_name: Sudholt
- first_name: Barbora
full_name: Trubenova, Barbora
id: 42302D54-F248-11E8-B48F-1D18A9856A87
last_name: Trubenova
orcid: 0000-0002-6873-2967
citation:
ama: Paixao T, Pérez Heredia J, Sudholt D, Trubenova B. Towards a runtime comparison
of natural and artificial evolution. Algorithmica. 2017;78(2):681-713.
doi:10.1007/s00453-016-0212-1
apa: Paixao, T., Pérez Heredia, J., Sudholt, D., & Trubenova, B. (2017). Towards
a runtime comparison of natural and artificial evolution. Algorithmica.
Springer. https://doi.org/10.1007/s00453-016-0212-1
chicago: Paixao, Tiago, Jorge Pérez Heredia, Dirk Sudholt, and Barbora Trubenova.
“Towards a Runtime Comparison of Natural and Artificial Evolution.” Algorithmica.
Springer, 2017. https://doi.org/10.1007/s00453-016-0212-1.
ieee: T. Paixao, J. Pérez Heredia, D. Sudholt, and B. Trubenova, “Towards a runtime
comparison of natural and artificial evolution,” Algorithmica, vol. 78,
no. 2. Springer, pp. 681–713, 2017.
ista: Paixao T, Pérez Heredia J, Sudholt D, Trubenova B. 2017. Towards a runtime
comparison of natural and artificial evolution. Algorithmica. 78(2), 681–713.
mla: Paixao, Tiago, et al. “Towards a Runtime Comparison of Natural and Artificial
Evolution.” Algorithmica, vol. 78, no. 2, Springer, 2017, pp. 681–713,
doi:10.1007/s00453-016-0212-1.
short: T. Paixao, J. Pérez Heredia, D. Sudholt, B. Trubenova, Algorithmica 78 (2017)
681–713.
date_created: 2018-12-11T11:51:27Z
date_published: 2017-06-01T00:00:00Z
date_updated: 2023-09-20T11:14:42Z
day: '01'
ddc:
- '576'
department:
- _id: NiBa
- _id: CaGu
doi: 10.1007/s00453-016-0212-1
ec_funded: 1
external_id:
isi:
- '000400379500013'
file:
- access_level: open_access
checksum: 7873f665a0c598ac747c908f34cb14b9
content_type: application/pdf
creator: system
date_created: 2018-12-12T10:10:19Z
date_updated: 2020-07-14T12:44:44Z
file_id: '4805'
file_name: IST-2016-658-v1+1_s00453-016-0212-1.pdf
file_size: 710206
relation: main_file
file_date_updated: 2020-07-14T12:44:44Z
has_accepted_license: '1'
intvolume: ' 78'
isi: 1
issue: '2'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
page: 681 - 713
project:
- _id: 25B1EC9E-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '618091'
name: Speed of Adaptation in Population Genetics and Evolutionary Computation
publication: Algorithmica
publication_identifier:
issn:
- '01784617'
publication_status: published
publisher: Springer
publist_id: '5931'
pubrep_id: '658'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Towards a runtime comparison of natural and artificial evolution
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: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 78
year: '2017'
...
---
_id: '1084'
abstract:
- lang: eng
text: 'BceRS and PsdRS are paralogous two-component systems in Bacillus subtilis
controlling the response to antimicrobial peptides. In the presence of extracellular
bacitracin and nisin, respectively, the two response regulators (RRs) bind their
target promoters, PbceA or PpsdA, resulting in a strong up-regulation of target
gene expression and ultimately antibiotic resistance. Despite high sequence similarity
between the RRs BceR and PsdR and their known binding sites, no cross-regulation
has been observed between them. We therefore investigated the specificity determinants
of PbceA and PpsdA that ensure the insulation of these two paralogous pathways
at the RR–promoter interface. In vivo and in vitro analyses demonstrate that the
regulatory regions within these two promoters contain three important elements:
in addition to the known (main) binding site, we identified a linker region and
a secondary binding site that are crucial for functionality. Initial binding to
the high-affinity, low-specificity main binding site is a prerequisite for the
subsequent highly specific binding of a second RR dimer to the low-affinity secondary
binding site. In addition to this hierarchical cooperative binding, discrimination
requires a competition of the two RRs for their respective binding site mediated
by only slight differences in binding affinities.'
article_processing_charge: No
author:
- first_name: Chong
full_name: Fang, Chong
last_name: Fang
- first_name: Anna A
full_name: Nagy-Staron, Anna A
id: 3ABC5BA6-F248-11E8-B48F-1D18A9856A87
last_name: Nagy-Staron
orcid: 0000-0002-1391-8377
- first_name: Martin
full_name: Grafe, Martin
last_name: Grafe
- first_name: Ralf
full_name: Heermann, Ralf
last_name: Heermann
- first_name: Kirsten
full_name: Jung, Kirsten
last_name: Jung
- first_name: Susanne
full_name: Gebhard, Susanne
last_name: Gebhard
- first_name: Thorsten
full_name: Mascher, Thorsten
last_name: Mascher
citation:
ama: Fang C, Nagy-Staron AA, Grafe M, et al. Insulation and wiring specificity of
BceR like response regulators and their target promoters in Bacillus subtilis.
Molecular Microbiology. 2017;104(1):16-31. doi:10.1111/mmi.13597
apa: Fang, C., Nagy-Staron, A. A., Grafe, M., Heermann, R., Jung, K., Gebhard, S.,
& Mascher, T. (2017). Insulation and wiring specificity of BceR like response
regulators and their target promoters in Bacillus subtilis. Molecular Microbiology.
Wiley-Blackwell. https://doi.org/10.1111/mmi.13597
chicago: Fang, Chong, Anna A Nagy-Staron, Martin Grafe, Ralf Heermann, Kirsten Jung,
Susanne Gebhard, and Thorsten Mascher. “Insulation and Wiring Specificity of BceR
like Response Regulators and Their Target Promoters in Bacillus Subtilis.” Molecular
Microbiology. Wiley-Blackwell, 2017. https://doi.org/10.1111/mmi.13597.
ieee: C. Fang et al., “Insulation and wiring specificity of BceR like response
regulators and their target promoters in Bacillus subtilis,” Molecular Microbiology,
vol. 104, no. 1. Wiley-Blackwell, pp. 16–31, 2017.
ista: Fang C, Nagy-Staron AA, Grafe M, Heermann R, Jung K, Gebhard S, Mascher T.
2017. Insulation and wiring specificity of BceR like response regulators and their
target promoters in Bacillus subtilis. Molecular Microbiology. 104(1), 16–31.
mla: Fang, Chong, et al. “Insulation and Wiring Specificity of BceR like Response
Regulators and Their Target Promoters in Bacillus Subtilis.” Molecular Microbiology,
vol. 104, no. 1, Wiley-Blackwell, 2017, pp. 16–31, doi:10.1111/mmi.13597.
short: C. Fang, A.A. Nagy-Staron, M. Grafe, R. Heermann, K. Jung, S. Gebhard, T.
Mascher, Molecular Microbiology 104 (2017) 16–31.
date_created: 2018-12-11T11:50:03Z
date_published: 2017-04-01T00:00:00Z
date_updated: 2023-09-20T11:48:43Z
day: '01'
department:
- _id: CaGu
doi: 10.1111/mmi.13597
external_id:
isi:
- '000398059200002'
intvolume: ' 104'
isi: 1
issue: '1'
language:
- iso: eng
month: '04'
oa_version: None
page: 16 - 31
publication: Molecular Microbiology
publication_identifier:
issn:
- ' 0950382X'
publication_status: published
publisher: Wiley-Blackwell
publist_id: '6294'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Insulation and wiring specificity of BceR like response regulators and their
target promoters in Bacillus subtilis
type: journal_article
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 104
year: '2017'
...
---
_id: '954'
abstract:
- lang: eng
text: Understanding the relation between genotype and phenotype remains a major
challenge. The difficulty of predicting individual mutation effects, and particularly
the interactions between them, has prevented the development of a comprehensive
theory that links genotypic changes to their phenotypic effects. We show that
a general thermodynamic framework for gene regulation, based on a biophysical
understanding of protein-DNA binding, accurately predicts the sign of epistasis
in a canonical cis-regulatory element consisting of overlapping RNA polymerase
and repressor binding sites. Sign and magnitude of individual mutation effects
are sufficient to predict the sign of epistasis and its environmental dependence.
Thus, the thermodynamic model offers the correct null prediction for epistasis
between mutations across DNA-binding sites. Our results indicate that a predictive
theory for the effects of cis-regulatory mutations is possible from first principles,
as long as the essential molecular mechanisms and the constraints these impose
on a biological system are accounted for.
article_number: e25192
article_processing_charge: Yes
author:
- first_name: Mato
full_name: Lagator, Mato
id: 345D25EC-F248-11E8-B48F-1D18A9856A87
last_name: Lagator
- first_name: Tiago
full_name: Paixao, Tiago
id: 2C5658E6-F248-11E8-B48F-1D18A9856A87
last_name: Paixao
orcid: 0000-0003-2361-3953
- first_name: Nicholas H
full_name: Barton, Nicholas H
id: 4880FE40-F248-11E8-B48F-1D18A9856A87
last_name: Barton
orcid: 0000-0002-8548-5240
- first_name: Jonathan P
full_name: Bollback, Jonathan P
id: 2C6FA9CC-F248-11E8-B48F-1D18A9856A87
last_name: Bollback
orcid: 0000-0002-4624-4612
- first_name: Calin C
full_name: Guet, Calin C
id: 47F8433E-F248-11E8-B48F-1D18A9856A87
last_name: Guet
orcid: 0000-0001-6220-2052
citation:
ama: Lagator M, Paixao T, Barton NH, Bollback JP, Guet CC. On the mechanistic nature
of epistasis in a canonical cis-regulatory element. eLife. 2017;6. doi:10.7554/eLife.25192
apa: Lagator, M., Paixao, T., Barton, N. H., Bollback, J. P., & Guet, C. C.
(2017). On the mechanistic nature of epistasis in a canonical cis-regulatory element.
ELife. eLife Sciences Publications. https://doi.org/10.7554/eLife.25192
chicago: Lagator, Mato, Tiago Paixao, Nicholas H Barton, Jonathan P Bollback, and
Calin C Guet. “On the Mechanistic Nature of Epistasis in a Canonical Cis-Regulatory
Element.” ELife. eLife Sciences Publications, 2017. https://doi.org/10.7554/eLife.25192.
ieee: M. Lagator, T. Paixao, N. H. Barton, J. P. Bollback, and C. C. Guet, “On the
mechanistic nature of epistasis in a canonical cis-regulatory element,” eLife,
vol. 6. eLife Sciences Publications, 2017.
ista: Lagator M, Paixao T, Barton NH, Bollback JP, Guet CC. 2017. On the mechanistic
nature of epistasis in a canonical cis-regulatory element. eLife. 6, e25192.
mla: Lagator, Mato, et al. “On the Mechanistic Nature of Epistasis in a Canonical
Cis-Regulatory Element.” ELife, vol. 6, e25192, eLife Sciences Publications,
2017, doi:10.7554/eLife.25192.
short: M. Lagator, T. Paixao, N.H. Barton, J.P. Bollback, C.C. Guet, ELife 6 (2017).
date_created: 2018-12-11T11:49:23Z
date_published: 2017-05-18T00:00:00Z
date_updated: 2023-09-22T10:01:17Z
day: '18'
ddc:
- '576'
department:
- _id: CaGu
- _id: NiBa
- _id: JoBo
doi: 10.7554/eLife.25192
ec_funded: 1
external_id:
isi:
- '000404024800001'
file:
- access_level: open_access
checksum: 59cdd4400fb41280122d414fea971546
content_type: application/pdf
creator: system
date_created: 2018-12-12T10:17:49Z
date_updated: 2020-07-14T12:48:16Z
file_id: '5306'
file_name: IST-2017-841-v1+1_elife-25192-v2.pdf
file_size: 2441529
relation: main_file
- access_level: open_access
checksum: b69024880558b858eb8c5d47a92b6377
content_type: application/pdf
creator: system
date_created: 2018-12-12T10:17:50Z
date_updated: 2020-07-14T12:48:16Z
file_id: '5307'
file_name: IST-2017-841-v1+2_elife-25192-figures-v2.pdf
file_size: 3752660
relation: main_file
file_date_updated: 2020-07-14T12:48:16Z
has_accepted_license: '1'
intvolume: ' 6'
isi: 1
language:
- iso: eng
month: '05'
oa: 1
oa_version: Published Version
project:
- _id: 25B1EC9E-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '618091'
name: Speed of Adaptation in Population Genetics and Evolutionary Computation
- _id: 25681D80-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '291734'
name: International IST Postdoc Fellowship Programme
- _id: 2578D616-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '648440'
name: Selective Barriers to Horizontal Gene Transfer
publication: eLife
publication_identifier:
issn:
- 2050084X
publication_status: published
publisher: eLife Sciences Publications
publist_id: '6460'
pubrep_id: '841'
quality_controlled: '1'
scopus_import: '1'
status: public
title: On the mechanistic nature of epistasis in a canonical cis-regulatory element
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: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 6
year: '2017'
...
---
_id: '1007'
abstract:
- lang: eng
text: 'A nonlinear system possesses an invariance with respect to a set of transformations
if its output dynamics remain invariant when transforming the input, and adjusting
the initial condition accordingly. Most research has focused on invariances with
respect to time-independent pointwise transformations like translational-invariance
(u(t) -> u(t) + p, p in R) or scale-invariance (u(t) -> pu(t), p in R>0).
In this article, we introduce the concept of s0-invariances with respect to continuous
input transformations exponentially growing/decaying over time. We show that s0-invariant
systems not only encompass linear time-invariant (LTI) systems with transfer functions
having an irreducible zero at s0 in R, but also that the input/output relationship
of nonlinear s0-invariant systems possesses properties well known from their linear
counterparts. Furthermore, we extend the concept of s0-invariances to second-
and higher-order s0-invariances, corresponding to invariances with respect to
transformations of the time-derivatives of the input, and encompassing LTI systems
with zeros of multiplicity two or higher. Finally, we show that nth-order 0-invariant
systems realize – under mild conditions – nth-order nonlinear differential operators:
when excited by an input of a characteristic functional form, the system’s output
converges to a constant value only depending on the nth (nonlinear) derivative
of the input.'
article_processing_charge: Yes (in subscription journal)
author:
- first_name: Moritz
full_name: Lang, Moritz
id: 29E0800A-F248-11E8-B48F-1D18A9856A87
last_name: Lang
- first_name: Eduardo
full_name: Sontag, Eduardo
last_name: Sontag
citation:
ama: Lang M, Sontag E. Zeros of nonlinear systems with input invariances. Automatica.
2017;81C:46-55. doi:10.1016/j.automatica.2017.03.030
apa: Lang, M., & Sontag, E. (2017). Zeros of nonlinear systems with input invariances.
Automatica. International Federation of Automatic Control. https://doi.org/10.1016/j.automatica.2017.03.030
chicago: Lang, Moritz, and Eduardo Sontag. “Zeros of Nonlinear Systems with Input
Invariances.” Automatica. International Federation of Automatic Control,
2017. https://doi.org/10.1016/j.automatica.2017.03.030.
ieee: M. Lang and E. Sontag, “Zeros of nonlinear systems with input invariances,”
Automatica, vol. 81C. International Federation of Automatic Control, pp.
46–55, 2017.
ista: Lang M, Sontag E. 2017. Zeros of nonlinear systems with input invariances.
Automatica. 81C, 46–55.
mla: Lang, Moritz, and Eduardo Sontag. “Zeros of Nonlinear Systems with Input Invariances.”
Automatica, vol. 81C, International Federation of Automatic Control, 2017,
pp. 46–55, doi:10.1016/j.automatica.2017.03.030.
short: M. Lang, E. Sontag, Automatica 81C (2017) 46–55.
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title: Zeros of nonlinear systems with input invariances
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