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Metric Coordinate Systems

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Craig Calcaterra, Axel Boldt, Michael Green, David Bleecker

Dynamical Systems · Submitted 24 Jun 2002

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Nagumo's Invariance Theorem is invoked to prove the analogue of the classical Cauchy-Lipschitz Theorem for vector fields on a locally compact coordinatized space.

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arXiv IDmath/0206253 (version 1)
Pith IDFANVKESV, Pith's permanent record of this paper and its review
CitationAxel Boldt, Craig Calcaterra, David Bleecker, Michael Green. "Metric Coordinate Systems." arXiv:math/0206253 (2002). https://pith.science/paper/FANVKESV
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,080 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.06 · human participants 0.02 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.21 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

Coordinate systems are defined on general metric spaces with the purpose of generalizing vector fields on a manifold. Conversion formulae are available between metric and Cartesian coordinates on a Hilbert space. Nagumo's Invariance Theorem is invoked to prove the analogue of the classical Cauchy-Lipschitz Theorem for vector fields on a locally compact coordinatized space. A metric space version of Nagumo's Theorem is one consequence. Examples are given throughout.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Coordinate systems are defined on general metric spaces with the purpose of generalizing vector fields on a manifold. contribution 0.53 0.36 0.13 asserted 1.00
s1 Conversion formulae are available between metric and Cartesian coordinates on a Hilbert space. contribution 0.44 0.44 0.07 asserted 1.00
s2 Nagumo's Invariance Theorem is invoked to prove the analogue of the classical Cauchy-Lipschitz Theorem for vector fields on a locally compact coordinatized space. method 0.52 0.79 0.38 asserted 1.00
s3 A metric space version of Nagumo's Theorem is one consequence. result 0.90 0.79 0.15 asserted 1.00
s4 Examples are given throughout. contribution 0.52 0.79 0.03 asserted 0.99

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 1.00.

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  • figures and tables
  • references and citing sentences
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  • P abstract
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  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

Coordinate systems are defined on general metric spaces with the purpose of generalizing vector fields on a manifold. Conversion formulae are available between metric and Cartesian coordinates on a Hilbert space. Nagumo's Invariance Theorem is invoked to prove the analogue of the classical Cauchy-Lipschitz Theorem for vector fields on a locally compact coordinatized space. A metric space version of Nagumo's Theorem is one consequence. Examples are given throughout.

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