Asymptotic negative type properties of finite ultrametric spaces.
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| Title: | Asymptotic negative type properties of finite ultrametric spaces. |
|---|---|
| Authors: | Doust, Ian1 i.doust@unsw.edu.au, Sánchez, Stephen1 stephen.sanchez@unsw.edu.au, Weston, Anthony2,3 westoar@unisa.ac.za |
| Source: | Journal of Mathematical Analysis & Applications. Feb2017, Vol. 446 Issue 2, p1776-1793. 18p. |
| Subjects: | Asymptotic expansions, Metric spaces, Mathematical inequalities, Combinatorics, Mathematical formulas |
| Abstract: | Negative type inequalities arise in the study of embedding properties of metric spaces, but they often reduce to intractable combinatorial problems. In this paper we study more quantitative versions of these inequalities involving the so-called p -negative type gap. In particular, we focus our attention on the class of finite ultrametric spaces which are important in areas such as phylogenetics and data mining. Let ( X , d ) be a given finite ultrametric space with minimum non-zero distance α . Then the p -negative type gap Γ X ( p ) of ( X , d ) is positive for all p ≥ 0 . In this paper we compute the value of the limit Γ X ( ∞ ) : = lim p → ∞ Γ X ( p ) α p . It turns out that this value is positive and it may be given explicitly by an elegant combinatorial formula. This formula allows us to characterize when the ratio Γ X ( p ) / α p is a constant independent of p . The determination of Γ X ( ∞ ) also leads to new, asymptotically sharp, families of enhanced p -negative type inequalities for ( X , d ) . Indeed, suppose that G ∈ ( 0 , Γ X ( ∞ ) ) . Then, for all sufficiently large p , the inequality G ⋅ α p 2 ( ∑ k = 1 n | ζ k | ) 2 + ∑ j , i = 1 n d ( z j , z i ) p ζ j ζ i ≤ 0 holds for each finite subset { z 1 , … , z n } ⊆ X , and each scalar n -tuple ζ = ( ζ 1 , … , ζ n ) ∈ R n that satisfies ζ 1 + ⋯ + ζ n = 0 . Notably, these results do not extend to general finite metric spaces. [ABSTRACT FROM AUTHOR] |
| Copyright of Journal of Mathematical Analysis & Applications is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
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| Items | – Name: Title Label: Title Group: Ti Data: Asymptotic negative type properties of finite ultrametric spaces. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Doust%2C+Ian%22">Doust, Ian</searchLink><relatesTo>1</relatesTo><i> i.doust@unsw.edu.au</i><br /><searchLink fieldCode="AR" term="%22Sánchez%2C+Stephen%22">Sánchez, Stephen</searchLink><relatesTo>1</relatesTo><i> stephen.sanchez@unsw.edu.au</i><br /><searchLink fieldCode="AR" term="%22Weston%2C+Anthony%22">Weston, Anthony</searchLink><relatesTo>2,3</relatesTo><i> westoar@unisa.ac.za</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Journal+of+Mathematical+Analysis+%26+Applications%22">Journal of Mathematical Analysis & Applications</searchLink>. Feb2017, Vol. 446 Issue 2, p1776-1793. 18p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Asymptotic+expansions%22">Asymptotic expansions</searchLink><br /><searchLink fieldCode="DE" term="%22Metric+spaces%22">Metric spaces</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+inequalities%22">Mathematical inequalities</searchLink><br /><searchLink fieldCode="DE" term="%22Combinatorics%22">Combinatorics</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+formulas%22">Mathematical formulas</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: Negative type inequalities arise in the study of embedding properties of metric spaces, but they often reduce to intractable combinatorial problems. In this paper we study more quantitative versions of these inequalities involving the so-called p -negative type gap. In particular, we focus our attention on the class of finite ultrametric spaces which are important in areas such as phylogenetics and data mining. Let ( X , d ) be a given finite ultrametric space with minimum non-zero distance α . Then the p -negative type gap Γ X ( p ) of ( X , d ) is positive for all p ≥ 0 . In this paper we compute the value of the limit Γ X ( ∞ ) : = lim p → ∞ Γ X ( p ) α p . It turns out that this value is positive and it may be given explicitly by an elegant combinatorial formula. This formula allows us to characterize when the ratio Γ X ( p ) / α p is a constant independent of p . The determination of Γ X ( ∞ ) also leads to new, asymptotically sharp, families of enhanced p -negative type inequalities for ( X , d ) . Indeed, suppose that G ∈ ( 0 , Γ X ( ∞ ) ) . Then, for all sufficiently large p , the inequality G ⋅ α p 2 ( ∑ k = 1 n | ζ k | ) 2 + ∑ j , i = 1 n d ( z j , z i ) p ζ j ζ i ≤ 0 holds for each finite subset { z 1 , … , z n } ⊆ X , and each scalar n -tuple ζ = ( ζ 1 , … , ζ n ) ∈ R n that satisfies ζ 1 + ⋯ + ζ n = 0 . Notably, these results do not extend to general finite metric spaces. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Journal of Mathematical Analysis & Applications is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1016/j.jmaa.2016.09.069 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 18 StartPage: 1776 Subjects: – SubjectFull: Asymptotic expansions Type: general – SubjectFull: Metric spaces Type: general – SubjectFull: Mathematical inequalities Type: general – SubjectFull: Combinatorics Type: general – SubjectFull: Mathematical formulas Type: general Titles: – TitleFull: Asymptotic negative type properties of finite ultrametric spaces. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Doust, Ian – PersonEntity: Name: NameFull: Sánchez, Stephen – PersonEntity: Name: NameFull: Weston, Anthony IsPartOfRelationships: – BibEntity: Dates: – D: 15 M: 02 Text: Feb2017 Type: published Y: 2017 Identifiers: – Type: issn-print Value: 0022247X Numbering: – Type: volume Value: 446 – Type: issue Value: 2 Titles: – TitleFull: Journal of Mathematical Analysis & Applications Type: main |
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