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Six mathematical gems from the history of distance geometry.

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Title: Six mathematical gems from the history of distance geometry.
Authors: Liberti, Leo1 liberti@lix.polytechnique.fr, Lavor, Carlile2 clavor@ime.unicamp.br
Source: International Transactions in Operational Research. Sep2016, Vol. 23 Issue 5, p897-920. 24p.
Subjects: Distance geometry, Mathematical formulas, Cauchy transform, Multidimensional scaling, Polyhedra
Abstract: This is a partial account of the fascinating history of distance geometry. We make no claim to completeness, but we do promise a dazzling display of beautiful, elementary mathematics. We prove Heron's formula, Cauchy's theorem on the rigidity of polyhedra, Cayley's generalization of Heron's formula to higher dimensions, Menger's characterization of abstract semimetric spaces, a result of Gödel on metric spaces on the sphere, and Schoenberg's equivalence of distance and positive semidefinite matrices, which is at the basis of multidimensional scaling. [ABSTRACT FROM AUTHOR]
Copyright of International Transactions in Operational Research is the property of Wiley-Blackwell 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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  Data: Six mathematical gems from the history of distance geometry.
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  Data: <searchLink fieldCode="DE" term="%22Distance+geometry%22">Distance geometry</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+formulas%22">Mathematical formulas</searchLink><br /><searchLink fieldCode="DE" term="%22Cauchy+transform%22">Cauchy transform</searchLink><br /><searchLink fieldCode="DE" term="%22Multidimensional+scaling%22">Multidimensional scaling</searchLink><br /><searchLink fieldCode="DE" term="%22Polyhedra%22">Polyhedra</searchLink>
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  Data: This is a partial account of the fascinating history of distance geometry. We make no claim to completeness, but we do promise a dazzling display of beautiful, elementary mathematics. We prove Heron's formula, Cauchy's theorem on the rigidity of polyhedra, Cayley's generalization of Heron's formula to higher dimensions, Menger's characterization of abstract semimetric spaces, a result of Gödel on metric spaces on the sphere, and Schoenberg's equivalence of distance and positive semidefinite matrices, which is at the basis of multidimensional scaling. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of International Transactions in Operational Research is the property of Wiley-Blackwell 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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        Value: 10.1111/itor.12170
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      – Code: eng
        Text: English
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        PageCount: 24
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    Subjects:
      – SubjectFull: Distance geometry
        Type: general
      – SubjectFull: Mathematical formulas
        Type: general
      – SubjectFull: Cauchy transform
        Type: general
      – SubjectFull: Multidimensional scaling
        Type: general
      – SubjectFull: Polyhedra
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              Text: Sep2016
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