Absorption and scattering by a temporally switched lossy layer: Going beyond the Rozanov bound Article Swipe
YOU?
·
· 2021
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2105.01487
In this paper we study the electromagnetic scattering, absorption, and performance bounds for short time modulated pulses that impinge on a time-varying lossy layer that is sandwiched between vacuum and a perfect electric conductor. The electric characteristics of the layer, namely, the conductivity, permittivity, and permeability are assumed to change abruptly or gradually in time. We demonstrate numerically that a time-varying absorbing layer that undergoes temporal switching of its permittivity and conductance can absorb the power of a modulated, ultra-wideband, as well as a quasi-monochromatic, pulsed wave beyond what is dictated by the time invariant Rozanov bound when integrating over the whole frequency spectrum. We suggest and simulate a practical metamaterial realization that is constructed as a three-dimensional array of resistor loaded dipole. By switching only the dipole's load resistance, desired effective media properties are obtained. Furthermore, we show that Rozanov's bound can be bypassed with abrupt and a more practical gradual, soft, switching thus overcoming some possible causality issue in abrupt switching.
Related Topics
- Type
- preprint
- Language
- en
- Landing Page
- http://arxiv.org/abs/2105.01487
- https://arxiv.org/pdf/2105.01487
- OA Status
- green
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W4307481133
Raw OpenAlex JSON
- OpenAlex ID
-
https://openalex.org/W4307481133Canonical identifier for this work in OpenAlex
- DOI
-
https://doi.org/10.48550/arxiv.2105.01487Digital Object Identifier
- Title
-
Absorption and scattering by a temporally switched lossy layer: Going beyond the Rozanov boundWork title
- Type
-
preprintOpenAlex work type
- Language
-
enPrimary language
- Publication year
-
2021Year of publication
- Publication date
-
2021-05-04Full publication date if available
- Authors
-
Chen Firestein, Amir Shlivinski, Yakir HadadList of authors in order
- Landing page
-
https://arxiv.org/abs/2105.01487Publisher landing page
- PDF URL
-
https://arxiv.org/pdf/2105.01487Direct link to full text PDF
- Open access
-
YesWhether a free full text is available
- OA status
-
greenOpen access status per OpenAlex
- OA URL
-
https://arxiv.org/pdf/2105.01487Direct OA link when available
- Concepts
-
Metamaterial, Permittivity, Scattering, Lossy compression, Perfect conductor, Physics, Materials science, Optics, Computational physics, Optoelectronics, Dielectric, Computer science, Artificial intelligenceTop concepts (fields/topics) attached by OpenAlex
- Cited by
-
0Total citation count in OpenAlex
- Related works (count)
-
10Other works algorithmically related by OpenAlex
Full payload
| id | https://openalex.org/W4307481133 |
|---|---|
| doi | https://doi.org/10.48550/arxiv.2105.01487 |
| ids.doi | https://doi.org/10.48550/arxiv.2105.01487 |
| ids.openalex | https://openalex.org/W4307481133 |
| fwci | |
| type | preprint |
| title | Absorption and scattering by a temporally switched lossy layer: Going beyond the Rozanov bound |
| biblio.issue | |
| biblio.volume | |
| biblio.last_page | |
| biblio.first_page | |
| topics[0].id | https://openalex.org/T10245 |
| topics[0].field.id | https://openalex.org/fields/25 |
| topics[0].field.display_name | Materials Science |
| topics[0].score | 0.9958999752998352 |
| topics[0].domain.id | https://openalex.org/domains/3 |
| topics[0].domain.display_name | Physical Sciences |
| topics[0].subfield.id | https://openalex.org/subfields/2504 |
| topics[0].subfield.display_name | Electronic, Optical and Magnetic Materials |
| topics[0].display_name | Metamaterials and Metasurfaces Applications |
| topics[1].id | https://openalex.org/T11262 |
| topics[1].field.id | https://openalex.org/fields/31 |
| topics[1].field.display_name | Physics and Astronomy |
| topics[1].score | 0.9940999746322632 |
| topics[1].domain.id | https://openalex.org/domains/3 |
| topics[1].domain.display_name | Physical Sciences |
| topics[1].subfield.id | https://openalex.org/subfields/3107 |
| topics[1].subfield.display_name | Atomic and Molecular Physics, and Optics |
| topics[1].display_name | Quantum Mechanics and Non-Hermitian Physics |
| topics[2].id | https://openalex.org/T10069 |
| topics[2].field.id | https://openalex.org/fields/22 |
| topics[2].field.display_name | Engineering |
| topics[2].score | 0.9901000261306763 |
| topics[2].domain.id | https://openalex.org/domains/3 |
| topics[2].domain.display_name | Physical Sciences |
| topics[2].subfield.id | https://openalex.org/subfields/2202 |
| topics[2].subfield.display_name | Aerospace Engineering |
| topics[2].display_name | Antenna Design and Analysis |
| is_xpac | False |
| apc_list | |
| apc_paid | |
| concepts[0].id | https://openalex.org/C110367647 |
| concepts[0].level | 2 |
| concepts[0].score | 0.6121532320976257 |
| concepts[0].wikidata | https://www.wikidata.org/wiki/Q497166 |
| concepts[0].display_name | Metamaterial |
| concepts[1].id | https://openalex.org/C168651791 |
| concepts[1].level | 3 |
| concepts[1].score | 0.5565297603607178 |
| concepts[1].wikidata | https://www.wikidata.org/wiki/Q211569 |
| concepts[1].display_name | Permittivity |
| concepts[2].id | https://openalex.org/C191486275 |
| concepts[2].level | 2 |
| concepts[2].score | 0.540108859539032 |
| concepts[2].wikidata | https://www.wikidata.org/wiki/Q210028 |
| concepts[2].display_name | Scattering |
| concepts[3].id | https://openalex.org/C165021410 |
| concepts[3].level | 2 |
| concepts[3].score | 0.5207217335700989 |
| concepts[3].wikidata | https://www.wikidata.org/wiki/Q55564 |
| concepts[3].display_name | Lossy compression |
| concepts[4].id | https://openalex.org/C16519606 |
| concepts[4].level | 3 |
| concepts[4].score | 0.5206384062767029 |
| concepts[4].wikidata | https://www.wikidata.org/wiki/Q897715 |
| concepts[4].display_name | Perfect conductor |
| concepts[5].id | https://openalex.org/C121332964 |
| concepts[5].level | 0 |
| concepts[5].score | 0.49154195189476013 |
| concepts[5].wikidata | https://www.wikidata.org/wiki/Q413 |
| concepts[5].display_name | Physics |
| concepts[6].id | https://openalex.org/C192562407 |
| concepts[6].level | 0 |
| concepts[6].score | 0.38781213760375977 |
| concepts[6].wikidata | https://www.wikidata.org/wiki/Q228736 |
| concepts[6].display_name | Materials science |
| concepts[7].id | https://openalex.org/C120665830 |
| concepts[7].level | 1 |
| concepts[7].score | 0.35612842440605164 |
| concepts[7].wikidata | https://www.wikidata.org/wiki/Q14620 |
| concepts[7].display_name | Optics |
| concepts[8].id | https://openalex.org/C30475298 |
| concepts[8].level | 1 |
| concepts[8].score | 0.3315032720565796 |
| concepts[8].wikidata | https://www.wikidata.org/wiki/Q909554 |
| concepts[8].display_name | Computational physics |
| concepts[9].id | https://openalex.org/C49040817 |
| concepts[9].level | 1 |
| concepts[9].score | 0.19848382472991943 |
| concepts[9].wikidata | https://www.wikidata.org/wiki/Q193091 |
| concepts[9].display_name | Optoelectronics |
| concepts[10].id | https://openalex.org/C133386390 |
| concepts[10].level | 2 |
| concepts[10].score | 0.16446742415428162 |
| concepts[10].wikidata | https://www.wikidata.org/wiki/Q184996 |
| concepts[10].display_name | Dielectric |
| concepts[11].id | https://openalex.org/C41008148 |
| concepts[11].level | 0 |
| concepts[11].score | 0.13957563042640686 |
| concepts[11].wikidata | https://www.wikidata.org/wiki/Q21198 |
| concepts[11].display_name | Computer science |
| concepts[12].id | https://openalex.org/C154945302 |
| concepts[12].level | 1 |
| concepts[12].score | 0.0 |
| concepts[12].wikidata | https://www.wikidata.org/wiki/Q11660 |
| concepts[12].display_name | Artificial intelligence |
| keywords[0].id | https://openalex.org/keywords/metamaterial |
| keywords[0].score | 0.6121532320976257 |
| keywords[0].display_name | Metamaterial |
| keywords[1].id | https://openalex.org/keywords/permittivity |
| keywords[1].score | 0.5565297603607178 |
| keywords[1].display_name | Permittivity |
| keywords[2].id | https://openalex.org/keywords/scattering |
| keywords[2].score | 0.540108859539032 |
| keywords[2].display_name | Scattering |
| keywords[3].id | https://openalex.org/keywords/lossy-compression |
| keywords[3].score | 0.5207217335700989 |
| keywords[3].display_name | Lossy compression |
| keywords[4].id | https://openalex.org/keywords/perfect-conductor |
| keywords[4].score | 0.5206384062767029 |
| keywords[4].display_name | Perfect conductor |
| keywords[5].id | https://openalex.org/keywords/physics |
| keywords[5].score | 0.49154195189476013 |
| keywords[5].display_name | Physics |
| keywords[6].id | https://openalex.org/keywords/materials-science |
| keywords[6].score | 0.38781213760375977 |
| keywords[6].display_name | Materials science |
| keywords[7].id | https://openalex.org/keywords/optics |
| keywords[7].score | 0.35612842440605164 |
| keywords[7].display_name | Optics |
| keywords[8].id | https://openalex.org/keywords/computational-physics |
| keywords[8].score | 0.3315032720565796 |
| keywords[8].display_name | Computational physics |
| keywords[9].id | https://openalex.org/keywords/optoelectronics |
| keywords[9].score | 0.19848382472991943 |
| keywords[9].display_name | Optoelectronics |
| keywords[10].id | https://openalex.org/keywords/dielectric |
| keywords[10].score | 0.16446742415428162 |
| keywords[10].display_name | Dielectric |
| keywords[11].id | https://openalex.org/keywords/computer-science |
| keywords[11].score | 0.13957563042640686 |
| keywords[11].display_name | Computer science |
| language | en |
| locations[0].id | pmh:oai:arXiv.org:2105.01487 |
| locations[0].is_oa | True |
| locations[0].source.id | https://openalex.org/S4306400194 |
| locations[0].source.issn | |
| locations[0].source.type | repository |
| locations[0].source.is_oa | True |
| locations[0].source.issn_l | |
| locations[0].source.is_core | False |
| locations[0].source.is_in_doaj | False |
| locations[0].source.display_name | arXiv (Cornell University) |
| locations[0].source.host_organization | https://openalex.org/I205783295 |
| locations[0].source.host_organization_name | Cornell University |
| locations[0].source.host_organization_lineage | https://openalex.org/I205783295 |
| locations[0].license | |
| locations[0].pdf_url | https://arxiv.org/pdf/2105.01487 |
| locations[0].version | submittedVersion |
| locations[0].raw_type | text |
| locations[0].license_id | |
| locations[0].is_accepted | False |
| locations[0].is_published | False |
| locations[0].raw_source_name | |
| locations[0].landing_page_url | http://arxiv.org/abs/2105.01487 |
| locations[1].id | pmh:oai:figshare.com:article/21872208 |
| locations[1].is_oa | True |
| locations[1].source.id | https://openalex.org/S4306400572 |
| locations[1].source.issn | |
| locations[1].source.type | repository |
| locations[1].source.is_oa | False |
| locations[1].source.issn_l | |
| locations[1].source.is_core | False |
| locations[1].source.is_in_doaj | False |
| locations[1].source.display_name | OPAL (Open@LaTrobe) (La Trobe University) |
| locations[1].source.host_organization | https://openalex.org/I196829312 |
| locations[1].source.host_organization_name | La Trobe University |
| locations[1].source.host_organization_lineage | https://openalex.org/I196829312 |
| locations[1].license | cc-by |
| locations[1].pdf_url | |
| locations[1].version | submittedVersion |
| locations[1].raw_type | Text |
| locations[1].license_id | https://openalex.org/licenses/cc-by |
| locations[1].is_accepted | False |
| locations[1].is_published | False |
| locations[1].raw_source_name | |
| locations[1].landing_page_url | https://figshare.com/articles/preprint/Absorption_and_scattering_by_a_temporally_switched_lossy_layer_Going_beyond_the_Rozanov_bound/21872208 |
| locations[2].id | doi:10.48550/arxiv.2105.01487 |
| locations[2].is_oa | True |
| locations[2].source.id | https://openalex.org/S4306400194 |
| locations[2].source.issn | |
| locations[2].source.type | repository |
| locations[2].source.is_oa | True |
| locations[2].source.issn_l | |
| locations[2].source.is_core | False |
| locations[2].source.is_in_doaj | False |
| locations[2].source.display_name | arXiv (Cornell University) |
| locations[2].source.host_organization | https://openalex.org/I205783295 |
| locations[2].source.host_organization_name | Cornell University |
| locations[2].source.host_organization_lineage | https://openalex.org/I205783295 |
| locations[2].license | cc-by |
| locations[2].pdf_url | |
| locations[2].version | |
| locations[2].raw_type | article |
| locations[2].license_id | https://openalex.org/licenses/cc-by |
| locations[2].is_accepted | False |
| locations[2].is_published | |
| locations[2].raw_source_name | |
| locations[2].landing_page_url | https://doi.org/10.48550/arxiv.2105.01487 |
| indexed_in | arxiv, datacite |
| authorships[0].author.id | https://openalex.org/A5043982136 |
| authorships[0].author.orcid | https://orcid.org/0000-0002-1991-7308 |
| authorships[0].author.display_name | Chen Firestein |
| authorships[0].author_position | first |
| authorships[0].raw_author_name | Firestein, Chen |
| authorships[0].is_corresponding | False |
| authorships[1].author.id | https://openalex.org/A5005994619 |
| authorships[1].author.orcid | https://orcid.org/0000-0003-4047-2523 |
| authorships[1].author.display_name | Amir Shlivinski |
| authorships[1].author_position | middle |
| authorships[1].raw_author_name | Shlivinski, Amir |
| authorships[1].is_corresponding | False |
| authorships[2].author.id | https://openalex.org/A5027159531 |
| authorships[2].author.orcid | https://orcid.org/0000-0003-4682-9415 |
| authorships[2].author.display_name | Yakir Hadad |
| authorships[2].author_position | last |
| authorships[2].raw_author_name | Hadad, Yakir |
| authorships[2].is_corresponding | False |
| has_content.pdf | False |
| has_content.grobid_xml | False |
| is_paratext | False |
| open_access.is_oa | True |
| open_access.oa_url | https://arxiv.org/pdf/2105.01487 |
| open_access.oa_status | green |
| open_access.any_repository_has_fulltext | False |
| created_date | 2022-11-02T00:00:00 |
| display_name | Absorption and scattering by a temporally switched lossy layer: Going beyond the Rozanov bound |
| has_fulltext | False |
| is_retracted | False |
| updated_date | 2025-11-06T06:51:31.235846 |
| primary_topic.id | https://openalex.org/T10245 |
| primary_topic.field.id | https://openalex.org/fields/25 |
| primary_topic.field.display_name | Materials Science |
| primary_topic.score | 0.9958999752998352 |
| primary_topic.domain.id | https://openalex.org/domains/3 |
| primary_topic.domain.display_name | Physical Sciences |
| primary_topic.subfield.id | https://openalex.org/subfields/2504 |
| primary_topic.subfield.display_name | Electronic, Optical and Magnetic Materials |
| primary_topic.display_name | Metamaterials and Metasurfaces Applications |
| related_works | https://openalex.org/W1995805316, https://openalex.org/W2387503788, https://openalex.org/W1570806970, https://openalex.org/W2174268617, https://openalex.org/W2059658550, https://openalex.org/W2803920876, https://openalex.org/W50067980, https://openalex.org/W2999957348, https://openalex.org/W2100895417, https://openalex.org/W2123492791 |
| cited_by_count | 0 |
| locations_count | 3 |
| best_oa_location.id | pmh:oai:arXiv.org:2105.01487 |
| best_oa_location.is_oa | True |
| best_oa_location.source.id | https://openalex.org/S4306400194 |
| best_oa_location.source.issn | |
| best_oa_location.source.type | repository |
| best_oa_location.source.is_oa | True |
| best_oa_location.source.issn_l | |
| best_oa_location.source.is_core | False |
| best_oa_location.source.is_in_doaj | False |
| best_oa_location.source.display_name | arXiv (Cornell University) |
| best_oa_location.source.host_organization | https://openalex.org/I205783295 |
| best_oa_location.source.host_organization_name | Cornell University |
| best_oa_location.source.host_organization_lineage | https://openalex.org/I205783295 |
| best_oa_location.license | |
| best_oa_location.pdf_url | https://arxiv.org/pdf/2105.01487 |
| best_oa_location.version | submittedVersion |
| best_oa_location.raw_type | text |
| best_oa_location.license_id | |
| best_oa_location.is_accepted | False |
| best_oa_location.is_published | False |
| best_oa_location.raw_source_name | |
| best_oa_location.landing_page_url | http://arxiv.org/abs/2105.01487 |
| primary_location.id | pmh:oai:arXiv.org:2105.01487 |
| primary_location.is_oa | True |
| primary_location.source.id | https://openalex.org/S4306400194 |
| primary_location.source.issn | |
| primary_location.source.type | repository |
| primary_location.source.is_oa | True |
| primary_location.source.issn_l | |
| primary_location.source.is_core | False |
| primary_location.source.is_in_doaj | False |
| primary_location.source.display_name | arXiv (Cornell University) |
| primary_location.source.host_organization | https://openalex.org/I205783295 |
| primary_location.source.host_organization_name | Cornell University |
| primary_location.source.host_organization_lineage | https://openalex.org/I205783295 |
| primary_location.license | |
| primary_location.pdf_url | https://arxiv.org/pdf/2105.01487 |
| primary_location.version | submittedVersion |
| primary_location.raw_type | text |
| primary_location.license_id | |
| primary_location.is_accepted | False |
| primary_location.is_published | False |
| primary_location.raw_source_name | |
| primary_location.landing_page_url | http://arxiv.org/abs/2105.01487 |
| publication_date | 2021-05-04 |
| publication_year | 2021 |
| referenced_works_count | 0 |
| abstract_inverted_index.a | 20, 30, 59, 77, 83, 108, 116, 148 |
| abstract_inverted_index.By | 123 |
| abstract_inverted_index.In | 0 |
| abstract_inverted_index.We | 55, 104 |
| abstract_inverted_index.as | 80, 82, 115 |
| abstract_inverted_index.be | 143 |
| abstract_inverted_index.by | 91 |
| abstract_inverted_index.in | 53, 160 |
| abstract_inverted_index.is | 25, 89, 113 |
| abstract_inverted_index.of | 37, 67, 76, 119 |
| abstract_inverted_index.on | 19 |
| abstract_inverted_index.or | 51 |
| abstract_inverted_index.to | 48 |
| abstract_inverted_index.we | 3, 137 |
| abstract_inverted_index.The | 34 |
| abstract_inverted_index.and | 9, 29, 44, 70, 106, 147 |
| abstract_inverted_index.are | 46, 134 |
| abstract_inverted_index.can | 72, 142 |
| abstract_inverted_index.for | 12 |
| abstract_inverted_index.its | 68 |
| abstract_inverted_index.the | 5, 38, 41, 74, 92, 100, 126 |
| abstract_inverted_index.load | 128 |
| abstract_inverted_index.more | 149 |
| abstract_inverted_index.only | 125 |
| abstract_inverted_index.over | 99 |
| abstract_inverted_index.show | 138 |
| abstract_inverted_index.some | 156 |
| abstract_inverted_index.that | 17, 24, 58, 63, 112, 139 |
| abstract_inverted_index.this | 1 |
| abstract_inverted_index.thus | 154 |
| abstract_inverted_index.time | 14, 93 |
| abstract_inverted_index.wave | 86 |
| abstract_inverted_index.well | 81 |
| abstract_inverted_index.what | 88 |
| abstract_inverted_index.when | 97 |
| abstract_inverted_index.with | 145 |
| abstract_inverted_index.array | 118 |
| abstract_inverted_index.bound | 96, 141 |
| abstract_inverted_index.issue | 159 |
| abstract_inverted_index.layer | 23, 62 |
| abstract_inverted_index.lossy | 22 |
| abstract_inverted_index.media | 132 |
| abstract_inverted_index.paper | 2 |
| abstract_inverted_index.power | 75 |
| abstract_inverted_index.short | 13 |
| abstract_inverted_index.soft, | 152 |
| abstract_inverted_index.study | 4 |
| abstract_inverted_index.time. | 54 |
| abstract_inverted_index.whole | 101 |
| abstract_inverted_index.abrupt | 146, 161 |
| abstract_inverted_index.absorb | 73 |
| abstract_inverted_index.beyond | 87 |
| abstract_inverted_index.bounds | 11 |
| abstract_inverted_index.change | 49 |
| abstract_inverted_index.layer, | 39 |
| abstract_inverted_index.loaded | 121 |
| abstract_inverted_index.pulsed | 85 |
| abstract_inverted_index.pulses | 16 |
| abstract_inverted_index.vacuum | 28 |
| abstract_inverted_index.Rozanov | 95 |
| abstract_inverted_index.assumed | 47 |
| abstract_inverted_index.between | 27 |
| abstract_inverted_index.desired | 130 |
| abstract_inverted_index.dipole. | 122 |
| abstract_inverted_index.impinge | 18 |
| abstract_inverted_index.namely, | 40 |
| abstract_inverted_index.perfect | 31 |
| abstract_inverted_index.suggest | 105 |
| abstract_inverted_index.abruptly | 50 |
| abstract_inverted_index.bypassed | 144 |
| abstract_inverted_index.dictated | 90 |
| abstract_inverted_index.dipole's | 127 |
| abstract_inverted_index.electric | 32, 35 |
| abstract_inverted_index.gradual, | 151 |
| abstract_inverted_index.possible | 157 |
| abstract_inverted_index.resistor | 120 |
| abstract_inverted_index.simulate | 107 |
| abstract_inverted_index.temporal | 65 |
| abstract_inverted_index.Rozanov's | 140 |
| abstract_inverted_index.absorbing | 61 |
| abstract_inverted_index.causality | 158 |
| abstract_inverted_index.effective | 131 |
| abstract_inverted_index.frequency | 102 |
| abstract_inverted_index.gradually | 52 |
| abstract_inverted_index.invariant | 94 |
| abstract_inverted_index.modulated | 15 |
| abstract_inverted_index.obtained. | 135 |
| abstract_inverted_index.practical | 109, 150 |
| abstract_inverted_index.spectrum. | 103 |
| abstract_inverted_index.switching | 66, 124, 153 |
| abstract_inverted_index.undergoes | 64 |
| abstract_inverted_index.conductor. | 33 |
| abstract_inverted_index.modulated, | 78 |
| abstract_inverted_index.overcoming | 155 |
| abstract_inverted_index.properties | 133 |
| abstract_inverted_index.sandwiched | 26 |
| abstract_inverted_index.switching. | 162 |
| abstract_inverted_index.absorption, | 8 |
| abstract_inverted_index.conductance | 71 |
| abstract_inverted_index.constructed | 114 |
| abstract_inverted_index.demonstrate | 56 |
| abstract_inverted_index.integrating | 98 |
| abstract_inverted_index.numerically | 57 |
| abstract_inverted_index.performance | 10 |
| abstract_inverted_index.realization | 111 |
| abstract_inverted_index.resistance, | 129 |
| abstract_inverted_index.scattering, | 7 |
| abstract_inverted_index.Furthermore, | 136 |
| abstract_inverted_index.metamaterial | 110 |
| abstract_inverted_index.permeability | 45 |
| abstract_inverted_index.permittivity | 69 |
| abstract_inverted_index.time-varying | 21, 60 |
| abstract_inverted_index.conductivity, | 42 |
| abstract_inverted_index.permittivity, | 43 |
| abstract_inverted_index.characteristics | 36 |
| abstract_inverted_index.electromagnetic | 6 |
| abstract_inverted_index.ultra-wideband, | 79 |
| abstract_inverted_index.three-dimensional | 117 |
| abstract_inverted_index.quasi-monochromatic, | 84 |
| cited_by_percentile_year | |
| countries_distinct_count | 0 |
| institutions_distinct_count | 3 |
| sustainable_development_goals[0].id | https://metadata.un.org/sdg/7 |
| sustainable_development_goals[0].score | 0.6299999952316284 |
| sustainable_development_goals[0].display_name | Affordable and clean energy |
| citation_normalized_percentile |