Asymptotic negative type properties of finite ultrametric spaces

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• 作者：Ian Doust^a ; ^{i.doust@unsw.edu.au" class="auth_mail" title="E-mail the corresponding author} ; Stephen Sá ; nchez^a ; ^{stephen.sanchez@unsw.edu.au" class="auth_mail" title="E-mail the corresponding author} ; Anthony Weston^b ; ^c ; ^{westona@canisius.edu" class="auth_mail" title="E-mail the corresponding author} ; ^{westoar@unisa.ac.za" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Ultrametric space ; (Enhanced) negative type ; Proximity dendrogram
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 February 2017
• 年：2017
• 卷：446
• 期：2
• 页码：1776-1793
• 全文大小：538 K

文摘

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 data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305856&_mathId=si1.gif&_user=111111111&_pii=S0022247X16305856&_rdoc=1&_issn=0022247X&md5=2c619669fd7d5cb26a068936bb1f487c" title="Click to view the MathML source">(X,d)$(X,d)$ be a given finite ultrametric space with minimum non-zero distance α. Then the p  -negative type gap data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305856&_mathId=si2.gif&_user=111111111&_pii=S0022247X16305856&_rdoc=1&_issn=0022247X&md5=a89a72507b75dac00e2e758d997e59ea" title="Click to view the MathML source">Γ_X(p)${\mathrm{\Gamma }}_X(p)$ of data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305856&_mathId=si1.gif&_user=111111111&_pii=S0022247X16305856&_rdoc=1&_issn=0022247X&md5=2c619669fd7d5cb26a068936bb1f487c" title="Click to view the MathML source">(X,d)$(X,d)$ is positive for all data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305856&_mathId=si3.gif&_user=111111111&_pii=S0022247X16305856&_rdoc=1&_issn=0022247X&md5=f930879dd72285404df7e00e42badcc0" title="Click to view the MathML source">p≥0$p\ge 0$. In this paper we compute the value of the limit 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 data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305856&_mathId=si5.gif&_user=111111111&_pii=S0022247X16305856&_rdoc=1&_issn=0022247X&md5=e4808912dc368c5cfb5c361ec1205a75" title="Click to view the MathML source">Γ_X(p)/α^p${\mathrm{\Gamma }}_X(p)/{\alpha }^p$ is a constant independent of p  . The determination of data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305856&_mathId=si321.gif&_user=111111111&_pii=S0022247X16305856&_rdoc=1&_issn=0022247X&md5=0e9ef2006b2509bac73a4ca6738f064e" title="Click to view the MathML source">Γ_X(∞)${\mathrm{\Gamma }}_X(\infty )$ also leads to new, asymptotically sharp, families of enhanced p  -negative type inequalities for data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305856&_mathId=si1.gif&_user=111111111&_pii=S0022247X16305856&_rdoc=1&_issn=0022247X&md5=2c619669fd7d5cb26a068936bb1f487c" title="Click to view the MathML source">(X,d)$(X,d)$. Indeed, suppose that data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305856&_mathId=si66.gif&_user=111111111&_pii=S0022247X16305856&_rdoc=1&_issn=0022247X&md5=35b8bc05e92e30d66e0be2cbe3d617fc" title="Click to view the MathML source">G∈(0,Γ_X(∞))$G\in (0,{\mathrm{\Gamma }}_X(\infty ))$. Then, for all sufficiently large p, the inequality holds for each finite subset data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305856&_mathId=si68.gif&_user=111111111&_pii=S0022247X16305856&_rdoc=1&_issn=0022247X&md5=324e2c53a3652686fa30fcc6056a8f21" title="Click to view the MathML source">{z₁,…,z_n}⊆X$\{z_1,\dots ,z_n\}\subseteq X$, and each scalar n  -tuple data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305856&_mathId=si10.gif&_user=111111111&_pii=S0022247X16305856&_rdoc=1&_issn=0022247X&md5=7f1eb2b2fdb15313d6129848871ba4b5" title="Click to view the MathML source">ζ=(ζ₁,…,ζ_n)∈Rⁿ$\mathit{\zeta }=({\zeta }_1,\dots ,{\zeta }_n)\in {\mathbb{R}}^n$ that satisfies data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305856&_mathId=si11.gif&_user=111111111&_pii=S0022247X16305856&_rdoc=1&_issn=0022247X&md5=1c195f8e4e472e777093e796833384d2" title="Click to view the MathML source">ζ₁+⋯+ζ_n=0${\zeta }_1+\cdots +{\zeta }_n=0$. Notably, these results do not extend to general finite metric spaces.