Curvature-dependent Energies View Full Text


Ontology type: schema:ScholarlyArticle     


Article Info

DATE

2017-04-21

AUTHORS

Emilio Acerbi, Domenico Mucci

ABSTRACT

We report our recent results from [1, 2] on the total curvature of graphs of curves in high codimension Euclidean space. We introduce the corresponding relaxed energy functional and provide an explicit representation formula. In the case of continuous Cartesian curves, i.e., of graphs cu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c_{u}}$$\end{document} of continuous functions u on an interval, the relaxed energy is finite if and only if the curve cu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c_{u}}$$\end{document} has bounded variation and finite total curvature. In this case, moreover, the total curvature does not depend on the Cantor part of the derivative of u. We also deal with the "elastic" case, corresponding to a superlinear dependence on the pointwise curvature. Different phenomena w.r.t. the "plastic" case are observed. A p-curvature functional is well-defined on continuous curves with finite relaxed energy, and the relaxed energy is given by the length plus the p-curvature. We treat the wider class of graphs of one-dimensional BV-functions. More... »

PAGES

41-69

References to SciGraph publications

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s00032-017-0265-x

DOI

http://dx.doi.org/10.1007/s00032-017-0265-x

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1084990180


Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
Incoming Citations Browse incoming citations for this publication using opencitations.net

JSON-LD is the canonical representation for SciGraph data.

TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT

[
  {
    "@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json", 
    "about": [
      {
        "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/01", 
        "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
        "name": "Mathematical Sciences", 
        "type": "DefinedTerm"
      }, 
      {
        "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/0101", 
        "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
        "name": "Pure Mathematics", 
        "type": "DefinedTerm"
      }
    ], 
    "author": [
      {
        "affiliation": {
          "alternateName": "Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Universit\u00e0 degli Studi di Parma, Parco Area delle Scienze, 53/A, 43124, Parma, Italy", 
          "id": "http://www.grid.ac/institutes/grid.10383.39", 
          "name": [
            "Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Universit\u00e0 degli Studi di Parma, Parco Area delle Scienze, 53/A, 43124, Parma, Italy"
          ], 
          "type": "Organization"
        }, 
        "familyName": "Acerbi", 
        "givenName": "Emilio", 
        "id": "sg:person.015135603203.54", 
        "sameAs": [
          "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.015135603203.54"
        ], 
        "type": "Person"
      }, 
      {
        "affiliation": {
          "alternateName": "Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Universit\u00e0 degli Studi di Parma, Parco Area delle Scienze, 53/A, 43124, Parma, Italy", 
          "id": "http://www.grid.ac/institutes/grid.10383.39", 
          "name": [
            "Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Universit\u00e0 degli Studi di Parma, Parco Area delle Scienze, 53/A, 43124, Parma, Italy"
          ], 
          "type": "Organization"
        }, 
        "familyName": "Mucci", 
        "givenName": "Domenico", 
        "id": "sg:person.013336161617.99", 
        "sameAs": [
          "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.013336161617.99"
        ], 
        "type": "Person"
      }
    ], 
    "citation": [
      {
        "id": "sg:pub.10.1007/978-3-7643-8621-4_7", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1085181881", 
          "https://doi.org/10.1007/978-3-7643-8621-4_7"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "sg:pub.10.1007/978-3-662-06218-0", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1029423957", 
          "https://doi.org/10.1007/978-3-662-06218-0"
        ], 
        "type": "CreativeWork"
      }
    ], 
    "datePublished": "2017-04-21", 
    "datePublishedReg": "2017-04-21", 
    "description": "We report our recent results from [1, 2] on the total curvature of graphs of curves in high codimension Euclidean space. We introduce the corresponding relaxed energy functional and provide an explicit representation formula. In the case of continuous Cartesian curves, i.e., of graphs cu\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${c_{u}}$$\\end{document} of continuous functions u on an interval, the relaxed energy is finite if and only if the curve cu\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${c_{u}}$$\\end{document} has bounded variation and finite total curvature. In this case, moreover, the total curvature does not depend on the Cantor part of the derivative of u. We also deal with the \"elastic\" case, corresponding to a superlinear dependence on the pointwise curvature. Different phenomena w.r.t. the \"plastic\" case are observed. A p-curvature functional is well-defined on continuous curves with finite relaxed energy, and the relaxed energy is given by the length plus the p-curvature. We treat the wider class of graphs of one-dimensional BV-functions.", 
    "genre": "article", 
    "id": "sg:pub.10.1007/s00032-017-0265-x", 
    "isAccessibleForFree": false, 
    "isPartOf": [
      {
        "id": "sg:journal.1136491", 
        "issn": [
          "0370-7377", 
          "1424-9286"
        ], 
        "name": "Milan Journal of Mathematics", 
        "publisher": "Springer Nature", 
        "type": "Periodical"
      }, 
      {
        "issueNumber": "1", 
        "type": "PublicationIssue"
      }, 
      {
        "type": "PublicationVolume", 
        "volumeNumber": "85"
      }
    ], 
    "keywords": [
      "total curvature", 
      "relaxed energy", 
      "curvature-dependent energy", 
      "explicit representation formula", 
      "continuous functions u", 
      "finite total curvature", 
      "wide class", 
      "representation formula", 
      "Euclidean space", 
      "p-curvature", 
      "Cantor part", 
      "function u", 
      "continuous curve", 
      "recent results", 
      "graph", 
      "curvature", 
      "pointwise curvature", 
      "functionals", 
      "superlinear dependence", 
      "energy", 
      "Cartesian curve", 
      "formula", 
      "space", 
      "curves", 
      "class", 
      "dependence", 
      "cases", 
      "derivatives", 
      "results", 
      "interval", 
      "length", 
      "variation", 
      "part", 
      "plastic"
    ], 
    "name": "Curvature-dependent Energies", 
    "pagination": "41-69", 
    "productId": [
      {
        "name": "dimensions_id", 
        "type": "PropertyValue", 
        "value": [
          "pub.1084990180"
        ]
      }, 
      {
        "name": "doi", 
        "type": "PropertyValue", 
        "value": [
          "10.1007/s00032-017-0265-x"
        ]
      }
    ], 
    "sameAs": [
      "https://doi.org/10.1007/s00032-017-0265-x", 
      "https://app.dimensions.ai/details/publication/pub.1084990180"
    ], 
    "sdDataset": "articles", 
    "sdDatePublished": "2022-08-04T17:06", 
    "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
    "sdPublisher": {
      "name": "Springer Nature - SN SciGraph project", 
      "type": "Organization"
    }, 
    "sdSource": "s3://com-springernature-scigraph/baseset/20220804/entities/gbq_results/article/article_735.jsonl", 
    "type": "ScholarlyArticle", 
    "url": "https://doi.org/10.1007/s00032-017-0265-x"
  }
]
 

Download the RDF metadata as:  json-ld nt turtle xml License info

HOW TO GET THIS DATA PROGRAMMATICALLY:

JSON-LD is a popular format for linked data which is fully compatible with JSON.

curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/s00032-017-0265-x'

N-Triples is a line-based linked data format ideal for batch operations.

curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/s00032-017-0265-x'

Turtle is a human-readable linked data format.

curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/s00032-017-0265-x'

RDF/XML is a standard XML format for linked data.

curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/s00032-017-0265-x'


 

This table displays all metadata directly associated to this object as RDF triples.

106 TRIPLES      21 PREDICATES      60 URIs      50 LITERALS      6 BLANK NODES

Subject Predicate Object
1 sg:pub.10.1007/s00032-017-0265-x schema:about anzsrc-for:01
2 anzsrc-for:0101
3 schema:author Ncafb4660329244359f880f33e02a19fa
4 schema:citation sg:pub.10.1007/978-3-662-06218-0
5 sg:pub.10.1007/978-3-7643-8621-4_7
6 schema:datePublished 2017-04-21
7 schema:datePublishedReg 2017-04-21
8 schema:description We report our recent results from [1, 2] on the total curvature of graphs of curves in high codimension Euclidean space. We introduce the corresponding relaxed energy functional and provide an explicit representation formula. In the case of continuous Cartesian curves, i.e., of graphs cu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c_{u}}$$\end{document} of continuous functions u on an interval, the relaxed energy is finite if and only if the curve cu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c_{u}}$$\end{document} has bounded variation and finite total curvature. In this case, moreover, the total curvature does not depend on the Cantor part of the derivative of u. We also deal with the "elastic" case, corresponding to a superlinear dependence on the pointwise curvature. Different phenomena w.r.t. the "plastic" case are observed. A p-curvature functional is well-defined on continuous curves with finite relaxed energy, and the relaxed energy is given by the length plus the p-curvature. We treat the wider class of graphs of one-dimensional BV-functions.
9 schema:genre article
10 schema:isAccessibleForFree false
11 schema:isPartOf N13bde8162558442fb5fbc581f8d11241
12 N518cd37a54684ef2b5617bceeaff755b
13 sg:journal.1136491
14 schema:keywords Cantor part
15 Cartesian curve
16 Euclidean space
17 cases
18 class
19 continuous curve
20 continuous functions u
21 curvature
22 curvature-dependent energy
23 curves
24 dependence
25 derivatives
26 energy
27 explicit representation formula
28 finite total curvature
29 formula
30 function u
31 functionals
32 graph
33 interval
34 length
35 p-curvature
36 part
37 plastic
38 pointwise curvature
39 recent results
40 relaxed energy
41 representation formula
42 results
43 space
44 superlinear dependence
45 total curvature
46 variation
47 wide class
48 schema:name Curvature-dependent Energies
49 schema:pagination 41-69
50 schema:productId N08d65c01ad324a7894a0b8275c8a2357
51 N1be1362e2dfc43eb81899b27b490dc87
52 schema:sameAs https://app.dimensions.ai/details/publication/pub.1084990180
53 https://doi.org/10.1007/s00032-017-0265-x
54 schema:sdDatePublished 2022-08-04T17:06
55 schema:sdLicense https://scigraph.springernature.com/explorer/license/
56 schema:sdPublisher N5da957d01b804d3aa65d35c131d473d7
57 schema:url https://doi.org/10.1007/s00032-017-0265-x
58 sgo:license sg:explorer/license/
59 sgo:sdDataset articles
60 rdf:type schema:ScholarlyArticle
61 N08d65c01ad324a7894a0b8275c8a2357 schema:name dimensions_id
62 schema:value pub.1084990180
63 rdf:type schema:PropertyValue
64 N13bde8162558442fb5fbc581f8d11241 schema:volumeNumber 85
65 rdf:type schema:PublicationVolume
66 N1be1362e2dfc43eb81899b27b490dc87 schema:name doi
67 schema:value 10.1007/s00032-017-0265-x
68 rdf:type schema:PropertyValue
69 N518cd37a54684ef2b5617bceeaff755b schema:issueNumber 1
70 rdf:type schema:PublicationIssue
71 N5da957d01b804d3aa65d35c131d473d7 schema:name Springer Nature - SN SciGraph project
72 rdf:type schema:Organization
73 Nc5c9a469ae7f4684b564bde3b80f4beb rdf:first sg:person.013336161617.99
74 rdf:rest rdf:nil
75 Ncafb4660329244359f880f33e02a19fa rdf:first sg:person.015135603203.54
76 rdf:rest Nc5c9a469ae7f4684b564bde3b80f4beb
77 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
78 schema:name Mathematical Sciences
79 rdf:type schema:DefinedTerm
80 anzsrc-for:0101 schema:inDefinedTermSet anzsrc-for:
81 schema:name Pure Mathematics
82 rdf:type schema:DefinedTerm
83 sg:journal.1136491 schema:issn 0370-7377
84 1424-9286
85 schema:name Milan Journal of Mathematics
86 schema:publisher Springer Nature
87 rdf:type schema:Periodical
88 sg:person.013336161617.99 schema:affiliation grid-institutes:grid.10383.39
89 schema:familyName Mucci
90 schema:givenName Domenico
91 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.013336161617.99
92 rdf:type schema:Person
93 sg:person.015135603203.54 schema:affiliation grid-institutes:grid.10383.39
94 schema:familyName Acerbi
95 schema:givenName Emilio
96 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.015135603203.54
97 rdf:type schema:Person
98 sg:pub.10.1007/978-3-662-06218-0 schema:sameAs https://app.dimensions.ai/details/publication/pub.1029423957
99 https://doi.org/10.1007/978-3-662-06218-0
100 rdf:type schema:CreativeWork
101 sg:pub.10.1007/978-3-7643-8621-4_7 schema:sameAs https://app.dimensions.ai/details/publication/pub.1085181881
102 https://doi.org/10.1007/978-3-7643-8621-4_7
103 rdf:type schema:CreativeWork
104 grid-institutes:grid.10383.39 schema:alternateName Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università degli Studi di Parma, Parco Area delle Scienze, 53/A, 43124, Parma, Italy
105 schema:name Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università degli Studi di Parma, Parco Area delle Scienze, 53/A, 43124, Parma, Italy
106 rdf:type schema:Organization
 




Preview window. Press ESC to close (or click here)


...