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Title:           Plane Curves with Proportional Affine and Euclidean Curvatures
Subject:         Qualitative Theory of Dynamical Systems, https://doi.org/10.1007/s12346-026-01475-3
Keywords:        Euclidean curvature; Affine curvature; Ordinary differential equation
Author:          Ronaldo A. Garcia 
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      Title: Plane Curves with Proportional Affine and Euclidean Curvatures
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        Title: Abstract
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       Item: 
        Title: 1 Introduction
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       Item: 
        Title: 2 Euclidean and Affine Curvatures
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         Item: 
          Title: 2.1 Euclidean Curvature
          Destination: subsection.2.1
         Item: 
          Title: 2.2 Affine Curvature
          Destination: subsection.2.2
         Item: 
          Title: 2.3 More About Both Curvatures
          Destination: subsection.2.3
       Item: 
        Title: 3 Second Order Differential Equations
        Destination: section.3
        Children: 
         Item: 
          Title: 3.1 Using Euclidean Arc Length Parametrization
          Destination: subsection.3.1
         Item: 
          Title: 3.2 Using the Support Function Parametrization
          Destination: subsection.3.2
       Item: 
        Title: 4 Curves with Proportional Euclidean and Affine Curvatures
        Destination: section.4
       Item: 
        Title: 5 Conclusion
        Destination: section.5
       Item: 
        Title: Acknowledgements
        Destination: ackcount.1
       Item: 
        Title: References
        Destination: myseccnt.1
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    Keywords: Euclidean curvature; Affine curvature; Ordinary differential equation
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                           <pdfaProperty:valueType>URI</pdfaProperty:valueType>
                        </rdf:li>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>external</pdfaProperty:category>
                           <pdfaProperty:description>The Digital Object Identifier for the article. The DOI may also be used as the dc:identifier. If used as a dc:identifier, the URI form should be captured, and the bare identifier should also be captured using prism:doi. If an alternate unique identifier is used as the required dc:identifier, then the DOI should be specified as a bare identifier within prism:doi only. If the URL associated with a DOI is to be specified, then prism:url may be used in conjunction with prism:doi in order to provide the service endpoint (i.e. the URL). </pdfaProperty:description>
                           <pdfaProperty:name>doi</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
                        </rdf:li>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>external</pdfaProperty:category>
                           <pdfaProperty:description>ISSN for an electronic version of the issue in which the resource occurs. Permits publishers to include a second ISSN, identifying an electronic version of the issue in which the resource occurs (therefore e(lectronic)Issn. If used, prism:eIssn MUST contain the ISSN of the electronic version.</pdfaProperty:description>
                           <pdfaProperty:name>issn</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
                        </rdf:li>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>Volume number</pdfaProperty:description>
                           <pdfaProperty:name>volume</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
                        </rdf:li>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>Issue number</pdfaProperty:description>
                           <pdfaProperty:name>number</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
                        </rdf:li>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>Starting page</pdfaProperty:description>
                           <pdfaProperty:name>startingPage</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
                        </rdf:li>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>Ending page</pdfaProperty:description>
                           <pdfaProperty:name>endingPage</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
                        </rdf:li>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>external</pdfaProperty:category>
                           <pdfaProperty:description>The aggregation type specifies the unit of aggregation for a content collection. Comment PRISM recommends that the PRISM Aggregation Type Controlled Vocabulary be used to provide values for this element. Note: PRISM recommends against the use of the #other value currently allowed in this controlled vocabulary. In lieu of using #other please reach out to the PRISM group at info@prismstandard.org to request addition of your term to the Aggregation Type Controlled Vocabulary. </pdfaProperty:description>
                           <pdfaProperty:name>aggregationType</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
                        </rdf:li>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>external</pdfaProperty:category>
                           <pdfaProperty:description>Title of the magazine, or other publication, in which a resource was/will be published. Typically this will be used to provide the name of the magazine an article appeared in as metadata for the article, along with information such as the article title, the publisher, volume, number, and cover date. Note: Publication name can be used to differentiate between a print magazine and the online version if the names are different such as “magazine” and “magazine.com.”</pdfaProperty:description>
                           <pdfaProperty:name>publicationName</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
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                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>external</pdfaProperty:category>
                           <pdfaProperty:description>Copyright</pdfaProperty:description>
                           <pdfaProperty:name>copyright</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
                        </rdf:li>
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                  <pdfaSchema:prefix>pdf</pdfaSchema:prefix>
                  <pdfaSchema:schema>Adobe PDF Schema</pdfaSchema:schema>
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                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>A name object indicating whether the document has been modified to include trapping information</pdfaProperty:description>
                           <pdfaProperty:name>Trapped</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
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                  <pdfaSchema:schema>XMP Media Management Schema</pdfaSchema:schema>
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                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>UUID based identifier for specific incarnation of a document</pdfaProperty:description>
                           <pdfaProperty:name>InstanceID</pdfaProperty:name>
                           <pdfaProperty:valueType>URI</pdfaProperty:valueType>
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                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>The common identifier for all versions and renditions of a document.</pdfaProperty:description>
                           <pdfaProperty:name>DocumentID</pdfaProperty:name>
                           <pdfaProperty:valueType>URI</pdfaProperty:valueType>
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                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>The common identifier for all versions and renditions of a document.</pdfaProperty:description>
                           <pdfaProperty:name>OriginalDocumentID</pdfaProperty:name>
                           <pdfaProperty:valueType>URI</pdfaProperty:valueType>
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                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>A reference to the original document from which this one is derived. It is a minimal reference; missing components can be assumed to be unchanged. For example, a new version might only need to specify the instance ID and version number of the previous version, or a rendition might only need to specify the instance ID and rendition class of the original.</pdfaProperty:description>
                           <pdfaProperty:name>DerivedFrom</pdfaProperty:name>
                           <pdfaProperty:valueType>ResourceRef</pdfaProperty:valueType>
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                        <rdf:li rdf:parseType="Resource">
                           <pdfaType:description>Identifies a portion of a document. This can be a position at which the document has been changed since the most recent event history (stEvt:changed). For a resource within an xmpMM:Ingredients list, the ResourceRef uses this type to identify both the portion of the containing document that refers to the resource, and the portion of the referenced resource that is referenced.</pdfaType:description>
                           <pdfaType:namespaceURI>http://ns.adobe.com/xap/1.0/sType/Part#</pdfaType:namespaceURI>
                           <pdfaType:prefix>stPart</pdfaType:prefix>
                           <pdfaType:type>Part</pdfaType:type>
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                  <pdfaSchema:namespaceURI>http://www.aiim.org/pdfa/ns/id/</pdfaSchema:namespaceURI>
                  <pdfaSchema:prefix>pdfaid</pdfaSchema:prefix>
                  <pdfaSchema:schema>PDF/A ID Schema</pdfaSchema:schema>
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                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>Part of PDF/A standard</pdfaProperty:description>
                           <pdfaProperty:name>part</pdfaProperty:name>
                           <pdfaProperty:valueType>Integer</pdfaProperty:valueType>
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                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>Amendment of PDF/A standard</pdfaProperty:description>
                           <pdfaProperty:name>amd</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
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                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>Conformance level of PDF/A standard</pdfaProperty:description>
                           <pdfaProperty:name>conformance</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
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                     <rdf:Seq>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>XMP08 Spec: An ordered array of plate names that are needed to print the document (including any in contained documents).</pdfaProperty:description>
                           <pdfaProperty:name>PlateNames</pdfaProperty:name>
                           <pdfaProperty:valueType>Seq Text</pdfaProperty:valueType>
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                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>external</pdfaProperty:category>
                           <pdfaProperty:description>Values for Journal Article Version are one of the following:&#xA;AO = Author’s Original&#xA;SMUR = Submitted Manuscript Under Review&#xA;AM = Accepted Manuscript&#xA;P = Proof&#xA;VoR = Version of Record&#xA;CVoR = Corrected Version of Record&#xA;EVoR = Enhanced Version of Record</pdfaProperty:description>
                           <pdfaProperty:name>journal_article_version</pdfaProperty:name>
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