[{"doi":"10.1007/s10711-023-00862-3","article_processing_charge":"Yes (via OA deal)","publisher":"Springer Nature","date_updated":"2024-01-11T13:06:32Z","_id":"9651","type":"journal_article","main_file_link":[{"open_access":"1","url":"https://doi.org/10.1007/s10711-023-00862-3"}],"quality_controlled":"1","isi":1,"year":"2023","external_id":{"isi":["001105681500001"],"arxiv":["2102.13046"]},"publication":"Geometriae Dedicata","status":"public","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.","date_published":"2023-11-17T00:00:00Z","author":[{"full_name":"Dymond, Michael","last_name":"Dymond","first_name":"Michael"},{"first_name":"Vojtech","orcid":"0000-0002-2512-8698","id":"21AE5134-9EAC-11EA-BEA2-D7BD3DDC885E","full_name":"Kaluza, Vojtech","last_name":"Kaluza"}],"day":"17","scopus_import":"1","title":"Divergence of separated nets with respect to displacement equivalence","oa_version":"Published Version","article_type":"original","date_created":"2021-07-14T07:01:27Z","abstract":[{"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.","lang":"eng"}],"publication_identifier":{"issn":["0046-5755"],"eissn":["1572-9168"]},"publication_status":"epub_ahead","month":"11","arxiv":1,"department":[{"_id":"UlWa"}],"article_number":"15","oa":1,"language":[{"iso":"eng"}],"citation":{"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>.","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>.","ista":"Dymond M, Kaluza V. 2023. Divergence of separated nets with respect to displacement equivalence. Geometriae Dedicata., 15.","short":"M. Dymond, V. Kaluza, Geometriae Dedicata (2023).","ieee":"M. Dymond and V. Kaluza, “Divergence of separated nets with respect to displacement equivalence,” <i>Geometriae Dedicata</i>. Springer Nature, 2023.","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>"},"user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87"},{"oa_version":"Submitted Version","title":"Highly irregular separated nets","author":[{"first_name":"Michael","full_name":"Dymond, Michael","last_name":"Dymond"},{"id":"21AE5134-9EAC-11EA-BEA2-D7BD3DDC885E","full_name":"Kaluza, Vojtech","last_name":"Kaluza","orcid":"0000-0002-2512-8698","first_name":"Vojtech"}],"scopus_import":"1","day":"01","article_type":"original","date_created":"2021-07-14T07:01:28Z","volume":253,"abstract":[{"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.","lang":"eng"}],"intvolume":"       253","has_accepted_license":"1","publication_identifier":{"eissn":["1565-8511"]},"publication_status":"published","file_date_updated":"2021-07-14T07:41:50Z","arxiv":1,"month":"03","file":[{"checksum":"6fa0a3207dd1d6467c309fd1bcc867d1","relation":"main_file","access_level":"open_access","content_type":"application/pdf","file_name":"separated_nets.pdf","file_id":"9653","creator":"vkaluza","date_updated":"2021-07-14T07:41:50Z","file_size":900422,"date_created":"2021-07-14T07:41:50Z"}],"department":[{"_id":"UlWa"}],"language":[{"iso":"eng"}],"oa":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","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>","ieee":"M. Dymond and V. Kaluza, “Highly irregular separated nets,” <i>Israel Journal of Mathematics</i>, vol. 253. Springer Nature, pp. 501–554, 2023.","short":"M. Dymond, V. Kaluza, Israel Journal of Mathematics 253 (2023) 501–554.","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>.","ista":"Dymond M, Kaluza V. 2023. Highly irregular separated nets. Israel Journal of Mathematics. 253, 501–554.","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>","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>."},"publisher":"Springer Nature","doi":"10.1007/s11856-022-2448-6","article_processing_charge":"No","type":"journal_article","date_updated":"2023-08-14T11:26:34Z","_id":"9652","ddc":["515","516"],"page":"501-554","quality_controlled":"1","external_id":{"arxiv":["1903.05923"],"isi":["000904950300003"]},"isi":1,"year":"2023","keyword":["Lipschitz","bilipschitz","bounded displacement","modulus of continuity","separated net","non-realisable density","Burago--Kleiner construction"],"status":"public","publication":"Israel Journal of Mathematics","date_published":"2023-03-01T00:00:00Z","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_type":"original","date_created":"2021-11-25T13:49:16Z","volume":42,"oa_version":"Preprint","title":"Even maps, the Colin de Verdière number and representations of graphs","author":[{"id":"21AE5134-9EAC-11EA-BEA2-D7BD3DDC885E","full_name":"Kaluza, Vojtech","last_name":"Kaluza","first_name":"Vojtech","orcid":"0000-0002-2512-8698"},{"last_name":"Tancer","id":"38AC689C-F248-11E8-B48F-1D18A9856A87","full_name":"Tancer, Martin","first_name":"Martin","orcid":"0000-0002-1191-6714"}],"scopus_import":"1","day":"01","publication_status":"published","publication_identifier":{"issn":["0209-9683"]},"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."}],"intvolume":"        42","department":[{"_id":"UlWa"}],"month":"12","arxiv":1,"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","citation":{"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>","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>.","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>.","ista":"Kaluza V, Tancer M. 2022. Even maps, the Colin de Verdière number and representations of graphs. Combinatorica. 42, 1317–1345.","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.","short":"V. Kaluza, M. Tancer, Combinatorica 42 (2022) 1317–1345.","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>"},"language":[{"iso":"eng"}],"oa":1,"type":"journal_article","date_updated":"2023-08-02T06:43:27Z","_id":"10335","publisher":"Springer Nature","doi":"10.1007/s00493-021-4443-7","article_processing_charge":"No","quality_controlled":"1","main_file_link":[{"open_access":"1","url":" https://doi.org/10.48550/arXiv.1907.05055"}],"ddc":["514","516"],"page":"1317-1345","external_id":{"arxiv":["1907.05055"],"isi":["000798210100003"]},"isi":1,"year":"2022","date_published":"2022-12-01T00:00:00Z","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.","publication":"Combinatorica","status":"public"}]