---
_id: '9651'
abstract:
- lang: eng
  text: We introduce a hierachy of equivalence relations on the set of separated nets
    of a given Euclidean space, indexed by concave increasing functions ϕ:(0,∞)→(0,∞).
    Two separated nets are called ϕ-displacement equivalent if, roughly speaking,
    there is a bijection between them which, for large radii R, displaces points of
    norm at most R by something of order at most ϕ(R). We show that the spectrum of
    ϕ-displacement equivalence spans from the established notion of bounded displacement
    equivalence, which corresponds to bounded ϕ, to the indiscrete equivalence relation,
    coresponding to ϕ(R)∈Ω(R), in which all separated nets are equivalent. In between
    the two ends of this spectrum, the notions of ϕ-displacement equivalence are shown
    to be pairwise distinct with respect to the asymptotic classes of ϕ(R) for R→∞.
    We further undertake a comparison of our notion of ϕ-displacement equivalence
    with previously studied relations on separated nets. Particular attention is given
    to the interaction of the notions of ϕ-displacement equivalence with that of bilipschitz
    equivalence.
acknowledgement: 'Open access funding provided by Institute of Science and Technology
  (IST Austria). This work was started while both authors were employed at the University
  of Innsbruck and enjoyed the full support of Austrian Science Fund (FWF): P 30902-N35.
  It was continued when the first named author was employed at University of Leipzig
  and the second named author was employed at Institute of Science and Technology
  of Austria, where he was supported by an IST Fellowship.'
article_number: '15'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Michael
  full_name: Dymond, Michael
  last_name: Dymond
- first_name: Vojtech
  full_name: Kaluza, Vojtech
  id: 21AE5134-9EAC-11EA-BEA2-D7BD3DDC885E
  last_name: Kaluza
  orcid: 0000-0002-2512-8698
citation:
  ama: Dymond M, Kaluza V. Divergence of separated nets with respect to displacement
    equivalence. <i>Geometriae Dedicata</i>. 2023. doi:<a href="https://doi.org/10.1007/s10711-023-00862-3">10.1007/s10711-023-00862-3</a>
  apa: Dymond, M., &#38; Kaluza, V. (2023). Divergence of separated nets with respect
    to displacement equivalence. <i>Geometriae Dedicata</i>. Springer Nature. <a href="https://doi.org/10.1007/s10711-023-00862-3">https://doi.org/10.1007/s10711-023-00862-3</a>
  chicago: Dymond, Michael, and Vojtech Kaluza. “Divergence of Separated Nets with
    Respect to Displacement Equivalence.” <i>Geometriae Dedicata</i>. Springer Nature,
    2023. <a href="https://doi.org/10.1007/s10711-023-00862-3">https://doi.org/10.1007/s10711-023-00862-3</a>.
  ieee: M. Dymond and V. Kaluza, “Divergence of separated nets with respect to displacement
    equivalence,” <i>Geometriae Dedicata</i>. Springer Nature, 2023.
  ista: Dymond M, Kaluza V. 2023. Divergence of separated nets with respect to displacement
    equivalence. Geometriae Dedicata., 15.
  mla: Dymond, Michael, and Vojtech Kaluza. “Divergence of Separated Nets with Respect
    to Displacement Equivalence.” <i>Geometriae Dedicata</i>, 15, Springer Nature,
    2023, doi:<a href="https://doi.org/10.1007/s10711-023-00862-3">10.1007/s10711-023-00862-3</a>.
  short: M. Dymond, V. Kaluza, Geometriae Dedicata (2023).
date_created: 2021-07-14T07:01:27Z
date_published: 2023-11-17T00:00:00Z
date_updated: 2024-01-11T13:06:32Z
day: '17'
department:
- _id: UlWa
doi: 10.1007/s10711-023-00862-3
external_id:
  arxiv:
  - '2102.13046'
  isi:
  - '001105681500001'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.1007/s10711-023-00862-3
month: '11'
oa: 1
oa_version: Published Version
publication: Geometriae Dedicata
publication_identifier:
  eissn:
  - 1572-9168
  issn:
  - 0046-5755
publication_status: epub_ahead
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Divergence of separated nets with respect to displacement equivalence
type: journal_article
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
year: '2023'
...
---
_id: '9652'
abstract:
- lang: eng
  text: In 1998 Burago and Kleiner and (independently) McMullen gave examples of separated
    nets in Euclidean space which are non-bilipschitz equivalent to the integer lattice.
    We study weaker notions of equivalence of separated nets and demonstrate that
    such notions also give rise to distinct equivalence classes. Put differently,
    we find occurrences of particularly strong divergence of separated nets from the
    integer lattice. Our approach generalises that of Burago and Kleiner and McMullen
    which takes place largely in a continuous setting. Existence of irregular separated
    nets is verified via the existence of non-realisable density functions ρ:[0,1]d→(0,∞).
    In the present work we obtain stronger types of non-realisable densities.
acknowledgement: 'This work was done while both authors were employed at the University
  of Innsbruck and enjoyed the full support of Austrian Science Fund (FWF): P 30902-N35.'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Michael
  full_name: Dymond, Michael
  last_name: Dymond
- first_name: Vojtech
  full_name: Kaluza, Vojtech
  id: 21AE5134-9EAC-11EA-BEA2-D7BD3DDC885E
  last_name: Kaluza
  orcid: 0000-0002-2512-8698
citation:
  ama: Dymond M, Kaluza V. Highly irregular separated nets. <i>Israel Journal of Mathematics</i>.
    2023;253:501-554. doi:<a href="https://doi.org/10.1007/s11856-022-2448-6">10.1007/s11856-022-2448-6</a>
  apa: Dymond, M., &#38; Kaluza, V. (2023). Highly irregular separated nets. <i>Israel
    Journal of Mathematics</i>. Springer Nature. <a href="https://doi.org/10.1007/s11856-022-2448-6">https://doi.org/10.1007/s11856-022-2448-6</a>
  chicago: Dymond, Michael, and Vojtech Kaluza. “Highly Irregular Separated Nets.”
    <i>Israel Journal of Mathematics</i>. Springer Nature, 2023. <a href="https://doi.org/10.1007/s11856-022-2448-6">https://doi.org/10.1007/s11856-022-2448-6</a>.
  ieee: M. Dymond and V. Kaluza, “Highly irregular separated nets,” <i>Israel Journal
    of Mathematics</i>, vol. 253. Springer Nature, pp. 501–554, 2023.
  ista: Dymond M, Kaluza V. 2023. Highly irregular separated nets. Israel Journal
    of Mathematics. 253, 501–554.
  mla: Dymond, Michael, and Vojtech Kaluza. “Highly Irregular Separated Nets.” <i>Israel
    Journal of Mathematics</i>, vol. 253, Springer Nature, 2023, pp. 501–54, doi:<a
    href="https://doi.org/10.1007/s11856-022-2448-6">10.1007/s11856-022-2448-6</a>.
  short: M. Dymond, V. Kaluza, Israel Journal of Mathematics 253 (2023) 501–554.
date_created: 2021-07-14T07:01:28Z
date_published: 2023-03-01T00:00:00Z
date_updated: 2023-08-14T11:26:34Z
day: '01'
ddc:
- '515'
- '516'
department:
- _id: UlWa
doi: 10.1007/s11856-022-2448-6
external_id:
  arxiv:
  - '1903.05923'
  isi:
  - '000904950300003'
file:
- access_level: open_access
  checksum: 6fa0a3207dd1d6467c309fd1bcc867d1
  content_type: application/pdf
  creator: vkaluza
  date_created: 2021-07-14T07:41:50Z
  date_updated: 2021-07-14T07:41:50Z
  file_id: '9653'
  file_name: separated_nets.pdf
  file_size: 900422
  relation: main_file
file_date_updated: 2021-07-14T07:41:50Z
has_accepted_license: '1'
intvolume: '       253'
isi: 1
keyword:
- Lipschitz
- bilipschitz
- bounded displacement
- modulus of continuity
- separated net
- non-realisable density
- Burago--Kleiner construction
language:
- iso: eng
month: '03'
oa: 1
oa_version: Submitted Version
page: 501-554
publication: Israel Journal of Mathematics
publication_identifier:
  eissn:
  - 1565-8511
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Highly irregular separated nets
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 253
year: '2023'
...
---
_id: '10335'
abstract:
- lang: eng
  text: "Van der Holst and Pendavingh introduced a graph parameter σ, which coincides
    with the more famous Colin de Verdière graph parameter μ for small values. However,
    the definition of a is much more geometric/topological directly reflecting embeddability
    properties of the graph. They proved μ(G) ≤ σ(G) + 2 and conjectured σ(G) ≤ σ(G)
    for any graph G. We confirm this conjecture. As far as we know, this is the first
    topological upper bound on σ(G) which is, in general, tight.\r\nEquality between
    μ and σ does not hold in general as van der Holst and Pendavingh showed that there
    is a graph G with μ(G) ≤ 18 and σ(G) ≥ 20. We show that the gap appears at much
    smaller values, namely, we exhibit a graph H for which μ(H) ≥ 7 and σ(H) ≥ 8.
    We also prove that, in general, the gap can be large: The incidence graphs Hq
    of finite projective planes of order q satisfy μ(Hq) ∈ O(q3/2) and σ(Hq) ≥ q2."
acknowledgement: 'V. K. gratefully acknowledges the support of Austrian Science Fund
  (FWF): P 30902-N35. This work was done mostly while he was employed at the University
  of Innsbruck. During the early stage of this research, V. K. was partially supported
  by Charles University project GAUK 926416. M. T. is supported by the grant no. 19-04113Y
  of the Czech Science Foundation(GA ˇCR) and partially supported by Charles University
  project UNCE/SCI/004.'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Vojtech
  full_name: Kaluza, Vojtech
  id: 21AE5134-9EAC-11EA-BEA2-D7BD3DDC885E
  last_name: Kaluza
  orcid: 0000-0002-2512-8698
- first_name: Martin
  full_name: Tancer, Martin
  id: 38AC689C-F248-11E8-B48F-1D18A9856A87
  last_name: Tancer
  orcid: 0000-0002-1191-6714
citation:
  ama: Kaluza V, Tancer M. Even maps, the Colin de Verdière number and representations
    of graphs. <i>Combinatorica</i>. 2022;42:1317-1345. doi:<a href="https://doi.org/10.1007/s00493-021-4443-7">10.1007/s00493-021-4443-7</a>
  apa: Kaluza, V., &#38; Tancer, M. (2022). Even maps, the Colin de Verdière number
    and representations of graphs. <i>Combinatorica</i>. Springer Nature. <a href="https://doi.org/10.1007/s00493-021-4443-7">https://doi.org/10.1007/s00493-021-4443-7</a>
  chicago: Kaluza, Vojtech, and Martin Tancer. “Even Maps, the Colin de Verdière Number
    and Representations of Graphs.” <i>Combinatorica</i>. Springer Nature, 2022. <a
    href="https://doi.org/10.1007/s00493-021-4443-7">https://doi.org/10.1007/s00493-021-4443-7</a>.
  ieee: V. Kaluza and M. Tancer, “Even maps, the Colin de Verdière number and representations
    of graphs,” <i>Combinatorica</i>, vol. 42. Springer Nature, pp. 1317–1345, 2022.
  ista: Kaluza V, Tancer M. 2022. Even maps, the Colin de Verdière number and representations
    of graphs. Combinatorica. 42, 1317–1345.
  mla: Kaluza, Vojtech, and Martin Tancer. “Even Maps, the Colin de Verdière Number
    and Representations of Graphs.” <i>Combinatorica</i>, vol. 42, Springer Nature,
    2022, pp. 1317–45, doi:<a href="https://doi.org/10.1007/s00493-021-4443-7">10.1007/s00493-021-4443-7</a>.
  short: V. Kaluza, M. Tancer, Combinatorica 42 (2022) 1317–1345.
date_created: 2021-11-25T13:49:16Z
date_published: 2022-12-01T00:00:00Z
date_updated: 2023-08-02T06:43:27Z
day: '01'
ddc:
- '514'
- '516'
department:
- _id: UlWa
doi: 10.1007/s00493-021-4443-7
external_id:
  arxiv:
  - '1907.05055'
  isi:
  - '000798210100003'
intvolume: '        42'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: ' https://doi.org/10.48550/arXiv.1907.05055'
month: '12'
oa: 1
oa_version: Preprint
page: 1317-1345
publication: Combinatorica
publication_identifier:
  issn:
  - 0209-9683
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Even maps, the Colin de Verdière number and representations of graphs
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 42
year: '2022'
...