An Extensive Adiabatic Invariant for the Klein–Gordon Model in the Thermodynamic Limit View Full Text


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

2015-04

AUTHORS

Antonio Giorgilli, Simone Paleari, Tiziano Penati

ABSTRACT

We construct an extensive adiabatic invariant for a Klein–Gordon chain in the thermodynamic limit. In particular, given a fixed and sufficiently small value of the coupling constant a, the evolution of the adiabatic invariant is controlled up to time scaling as β1/a for any large enough value of the inverse temperature β. The time scale becomes a stretched exponential if the coupling constant is allowed to vanish jointly with the specific energy. The adiabatic invariance is exhibited by showing that the variance along the dynamics, i.e. calculated with respect to time averages, is much smaller than the corresponding variance over the whole phase space, i.e. calculated with the Gibbs measure, for a set of initial data of large measure. All the perturbative constructions and the subsequent estimates are consistent with the extensive nature of the system. More... »

PAGES

897-959

References to SciGraph publications

  • 2001-11. Normal form and exponential stability for some nonlinear string equations in ZEITSCHRIFT FÜR ANGEWANDTE MATHEMATIK UND PHYSIK
  • 2013-07. The Fermi-Pasta-Ulam Problem and Its Underlying Integrable Dynamics in JOURNAL OF STATISTICAL PHYSICS
  • 1899-07. Les méthodes nouvelles de la mécanique céleste in IL NUOVO CIMENTO (1895-1900)
  • 2012-09. Extensive Adiabatic Invariants for Nonlinear Chains in JOURNAL OF STATISTICAL PHYSICS
  • 2011-08. Time-Scales to Equipartition in the Fermi–Pasta–Ulam Problem: Finite-Size Effects and Thermodynamic Limit in JOURNAL OF STATISTICAL PHYSICS
  • 2009-06. The Fermi-Pasta-Ulam Problem: Scaling Laws vs. Initial Conditions in JOURNAL OF STATISTICAL PHYSICS
  • 1986. An invariant torus for nearly integrable Hamiltonian systems with infinitely many degrees of freedom in STOCHASTIC PROCESSES IN CLASSICAL AND QUANTUM SYSTEMS
  • 2007-08. An Averaging Theorem for Hamiltonian Dynamical Systems in the Thermodynamic Limit in JOURNAL OF STATISTICAL PHYSICS
  • 1989-12. Realization of holonomic constraints and freezing of high frequency degrees of freedom in the light of classical perturbation theory. Part II in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1993-05. Exponential stability of states close to resonance in infinite-dimensional Hamiltonian systems in JOURNAL OF STATISTICAL PHYSICS
  • 2012-08. Exponentially Long Stability Times for a Nonlinear Lattice in the Thermodynamic Limit in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1978-04. Formal integrals for an autonomous Hamiltonian system near an equilibrium point in CELESTIAL MECHANICS AND DYNAMICAL ASTRONOMY
  • 2006-06. On Metastability in FPU in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1988-03. A Nekhoroshev-type theorem for Hamiltonian systems with infinitely many degrees of freedom in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • Journal

    TITLE

    Annales Henri Poincaré

    ISSUE

    4

    VOLUME

    16

    Author Affiliations

    Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s00023-014-0335-3

    DOI

    http://dx.doi.org/10.1007/s00023-014-0335-3

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1051777972


    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/2008/0403", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Geology", 
            "type": "DefinedTerm"
          }, 
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/04", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Earth Sciences", 
            "type": "DefinedTerm"
          }
        ], 
        "author": [
          {
            "affiliation": {
              "alternateName": "University of Milan", 
              "id": "https://www.grid.ac/institutes/grid.4708.b", 
              "name": [
                "Dipartimento di Matematica, Universit\u00e0 degli Studi di Milano, Via Saldini, 50, 20133, Milan, Italy"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Giorgilli", 
            "givenName": "Antonio", 
            "id": "sg:person.010532704656.30", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.010532704656.30"
            ], 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "University of Milan", 
              "id": "https://www.grid.ac/institutes/grid.4708.b", 
              "name": [
                "Dipartimento di Matematica, Universit\u00e0 degli Studi di Milano, Via Saldini, 50, 20133, Milan, Italy"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Paleari", 
            "givenName": "Simone", 
            "id": "sg:person.012536754367.53", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.012536754367.53"
            ], 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "University of Milan", 
              "id": "https://www.grid.ac/institutes/grid.4708.b", 
              "name": [
                "Dipartimento di Matematica, Universit\u00e0 degli Studi di Milano, Via Saldini, 50, 20133, Milan, Italy"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Penati", 
            "givenName": "Tiziano", 
            "id": "sg:person.010451767147.39", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.010451767147.39"
            ], 
            "type": "Person"
          }
        ], 
        "citation": [
          {
            "id": "sg:pub.10.1007/s00220-005-1488-1", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1000858754", 
              "https://doi.org/10.1007/s00220-005-1488-1"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01058438", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1003681860", 
              "https://doi.org/10.1007/bf01058438"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1103/physreve.76.022104", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1006323355"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1103/physreve.76.022104", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1006323355"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/pl00001582", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1006510217", 
              "https://doi.org/10.1007/pl00001582"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s10955-007-9332-y", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1012453702", 
              "https://doi.org/10.1007/s10955-007-9332-y"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1073/pnas.47.11.1824", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1016645612"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s10955-012-0568-9", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1016793892", 
              "https://doi.org/10.1007/s10955-012-0568-9"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01218262", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1020365094", 
              "https://doi.org/10.1007/bf01218262"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01218262", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1020365094", 
              "https://doi.org/10.1007/bf01218262"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1016/j.physd.2006.07.017", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1023310655"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s10955-008-9660-6", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1025341457", 
              "https://doi.org/10.1007/s10955-008-9660-6"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01232832", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1028824895", 
              "https://doi.org/10.1007/bf01232832"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01232832", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1028824895", 
              "https://doi.org/10.1007/bf01232832"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf02742713", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1031864213", 
              "https://doi.org/10.1007/bf02742713"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1016/j.physleta.2012.05.006", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1033354892"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1016/s0167-2789(98)00169-9", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1035277359"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/3540171665_69", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1036324358", 
              "https://doi.org/10.1007/3540171665_69"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1016/j.physleta.2003.11.052", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1040830291"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s00220-012-1522-z", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1040875907", 
              "https://doi.org/10.1007/s00220-012-1522-z"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s10955-011-0277-9", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1041003802", 
              "https://doi.org/10.1007/s10955-011-0277-9"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s10955-013-0760-6", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1046718564", 
              "https://doi.org/10.1007/s10955-013-0760-6"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01218157", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1051921197", 
              "https://doi.org/10.1007/bf01218157"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01218157", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1051921197", 
              "https://doi.org/10.1007/bf01218157"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1070/rm1963v018n05abeh004130", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1058193720"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1088/0951-7715/12/6/310", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1059108891"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.2307/121001", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1069397473"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1142/9789812385215_0059", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1088717398"
            ], 
            "type": "CreativeWork"
          }
        ], 
        "datePublished": "2015-04", 
        "datePublishedReg": "2015-04-01", 
        "description": "We construct an extensive adiabatic invariant for a Klein\u2013Gordon chain in the thermodynamic limit. In particular, given a fixed and sufficiently small value of the coupling constant a, the evolution of the adiabatic invariant is controlled up to time scaling as \u03b21/a for any large enough value of the inverse temperature \u03b2. The time scale becomes a stretched exponential if the coupling constant is allowed to vanish jointly with the specific energy. The adiabatic invariance is exhibited by showing that the variance along the dynamics, i.e. calculated with respect to time averages, is much smaller than the corresponding variance over the whole phase space, i.e. calculated with the Gibbs measure, for a set of initial data of large measure. All the perturbative constructions and the subsequent estimates are consistent with the extensive nature of the system.", 
        "genre": "research_article", 
        "id": "sg:pub.10.1007/s00023-014-0335-3", 
        "inLanguage": [
          "en"
        ], 
        "isAccessibleForFree": true, 
        "isPartOf": [
          {
            "id": "sg:journal.1297863", 
            "issn": [
              "1424-0637", 
              "1424-0661"
            ], 
            "name": "Annales Henri Poincar\u00e9", 
            "type": "Periodical"
          }, 
          {
            "issueNumber": "4", 
            "type": "PublicationIssue"
          }, 
          {
            "type": "PublicationVolume", 
            "volumeNumber": "16"
          }
        ], 
        "name": "An Extensive Adiabatic Invariant for the Klein\u2013Gordon Model in the Thermodynamic Limit", 
        "pagination": "897-959", 
        "productId": [
          {
            "name": "readcube_id", 
            "type": "PropertyValue", 
            "value": [
              "f6a414baeae16001dea35022830ad2fc098f081717a969b6adac83d3668834fb"
            ]
          }, 
          {
            "name": "doi", 
            "type": "PropertyValue", 
            "value": [
              "10.1007/s00023-014-0335-3"
            ]
          }, 
          {
            "name": "dimensions_id", 
            "type": "PropertyValue", 
            "value": [
              "pub.1051777972"
            ]
          }
        ], 
        "sameAs": [
          "https://doi.org/10.1007/s00023-014-0335-3", 
          "https://app.dimensions.ai/details/publication/pub.1051777972"
        ], 
        "sdDataset": "articles", 
        "sdDatePublished": "2019-04-10T21:33", 
        "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
        "sdPublisher": {
          "name": "Springer Nature - SN SciGraph project", 
          "type": "Organization"
        }, 
        "sdSource": "s3://com-uberresearch-data-dimensions-target-20181106-alternative/cleanup/v134/2549eaecd7973599484d7c17b260dba0a4ecb94b/merge/v9/a6c9fde33151104705d4d7ff012ea9563521a3ce/jats-lookup/v90/0000000001_0000000264/records_8687_00000496.jsonl", 
        "type": "ScholarlyArticle", 
        "url": "http://link.springer.com/10.1007/s00023-014-0335-3"
      }
    ]
     

    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/s00023-014-0335-3'

    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/s00023-014-0335-3'

    Turtle is a human-readable linked data format.

    curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/s00023-014-0335-3'

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

    curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/s00023-014-0335-3'


     

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

    161 TRIPLES      21 PREDICATES      51 URIs      19 LITERALS      7 BLANK NODES

    Subject Predicate Object
    1 sg:pub.10.1007/s00023-014-0335-3 schema:about anzsrc-for:04
    2 anzsrc-for:0403
    3 schema:author Nd92b5ca9a6a94cef862de07e5e5b997b
    4 schema:citation sg:pub.10.1007/3540171665_69
    5 sg:pub.10.1007/bf01058438
    6 sg:pub.10.1007/bf01218157
    7 sg:pub.10.1007/bf01218262
    8 sg:pub.10.1007/bf01232832
    9 sg:pub.10.1007/bf02742713
    10 sg:pub.10.1007/pl00001582
    11 sg:pub.10.1007/s00220-005-1488-1
    12 sg:pub.10.1007/s00220-012-1522-z
    13 sg:pub.10.1007/s10955-007-9332-y
    14 sg:pub.10.1007/s10955-008-9660-6
    15 sg:pub.10.1007/s10955-011-0277-9
    16 sg:pub.10.1007/s10955-012-0568-9
    17 sg:pub.10.1007/s10955-013-0760-6
    18 https://doi.org/10.1016/j.physd.2006.07.017
    19 https://doi.org/10.1016/j.physleta.2003.11.052
    20 https://doi.org/10.1016/j.physleta.2012.05.006
    21 https://doi.org/10.1016/s0167-2789(98)00169-9
    22 https://doi.org/10.1070/rm1963v018n05abeh004130
    23 https://doi.org/10.1073/pnas.47.11.1824
    24 https://doi.org/10.1088/0951-7715/12/6/310
    25 https://doi.org/10.1103/physreve.76.022104
    26 https://doi.org/10.1142/9789812385215_0059
    27 https://doi.org/10.2307/121001
    28 schema:datePublished 2015-04
    29 schema:datePublishedReg 2015-04-01
    30 schema:description We construct an extensive adiabatic invariant for a Klein–Gordon chain in the thermodynamic limit. In particular, given a fixed and sufficiently small value of the coupling constant a, the evolution of the adiabatic invariant is controlled up to time scaling as β1/a for any large enough value of the inverse temperature β. The time scale becomes a stretched exponential if the coupling constant is allowed to vanish jointly with the specific energy. The adiabatic invariance is exhibited by showing that the variance along the dynamics, i.e. calculated with respect to time averages, is much smaller than the corresponding variance over the whole phase space, i.e. calculated with the Gibbs measure, for a set of initial data of large measure. All the perturbative constructions and the subsequent estimates are consistent with the extensive nature of the system.
    31 schema:genre research_article
    32 schema:inLanguage en
    33 schema:isAccessibleForFree true
    34 schema:isPartOf N59408e0f24a24122a413b4aa8db1dd72
    35 N9b3e5db9aeec4ce38e241e00f8e53c95
    36 sg:journal.1297863
    37 schema:name An Extensive Adiabatic Invariant for the Klein–Gordon Model in the Thermodynamic Limit
    38 schema:pagination 897-959
    39 schema:productId N4ac260229eab44c393bdc9495b8e6f17
    40 N88bcb7fe61a94e94ab825f1665432543
    41 Nda29b6b5491e432cb69ab3e10cedb7b8
    42 schema:sameAs https://app.dimensions.ai/details/publication/pub.1051777972
    43 https://doi.org/10.1007/s00023-014-0335-3
    44 schema:sdDatePublished 2019-04-10T21:33
    45 schema:sdLicense https://scigraph.springernature.com/explorer/license/
    46 schema:sdPublisher N714a8abfc7434f4bb6640cca1b9fa308
    47 schema:url http://link.springer.com/10.1007/s00023-014-0335-3
    48 sgo:license sg:explorer/license/
    49 sgo:sdDataset articles
    50 rdf:type schema:ScholarlyArticle
    51 N4ac260229eab44c393bdc9495b8e6f17 schema:name doi
    52 schema:value 10.1007/s00023-014-0335-3
    53 rdf:type schema:PropertyValue
    54 N59408e0f24a24122a413b4aa8db1dd72 schema:issueNumber 4
    55 rdf:type schema:PublicationIssue
    56 N714a8abfc7434f4bb6640cca1b9fa308 schema:name Springer Nature - SN SciGraph project
    57 rdf:type schema:Organization
    58 N88bcb7fe61a94e94ab825f1665432543 schema:name readcube_id
    59 schema:value f6a414baeae16001dea35022830ad2fc098f081717a969b6adac83d3668834fb
    60 rdf:type schema:PropertyValue
    61 N9b3e5db9aeec4ce38e241e00f8e53c95 schema:volumeNumber 16
    62 rdf:type schema:PublicationVolume
    63 N9ce92dee4aca4b26875440981dbc641d rdf:first sg:person.010451767147.39
    64 rdf:rest rdf:nil
    65 Na91105cf27f547409e6e95d9dc1546b6 rdf:first sg:person.012536754367.53
    66 rdf:rest N9ce92dee4aca4b26875440981dbc641d
    67 Nd92b5ca9a6a94cef862de07e5e5b997b rdf:first sg:person.010532704656.30
    68 rdf:rest Na91105cf27f547409e6e95d9dc1546b6
    69 Nda29b6b5491e432cb69ab3e10cedb7b8 schema:name dimensions_id
    70 schema:value pub.1051777972
    71 rdf:type schema:PropertyValue
    72 anzsrc-for:04 schema:inDefinedTermSet anzsrc-for:
    73 schema:name Earth Sciences
    74 rdf:type schema:DefinedTerm
    75 anzsrc-for:0403 schema:inDefinedTermSet anzsrc-for:
    76 schema:name Geology
    77 rdf:type schema:DefinedTerm
    78 sg:journal.1297863 schema:issn 1424-0637
    79 1424-0661
    80 schema:name Annales Henri Poincaré
    81 rdf:type schema:Periodical
    82 sg:person.010451767147.39 schema:affiliation https://www.grid.ac/institutes/grid.4708.b
    83 schema:familyName Penati
    84 schema:givenName Tiziano
    85 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.010451767147.39
    86 rdf:type schema:Person
    87 sg:person.010532704656.30 schema:affiliation https://www.grid.ac/institutes/grid.4708.b
    88 schema:familyName Giorgilli
    89 schema:givenName Antonio
    90 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.010532704656.30
    91 rdf:type schema:Person
    92 sg:person.012536754367.53 schema:affiliation https://www.grid.ac/institutes/grid.4708.b
    93 schema:familyName Paleari
    94 schema:givenName Simone
    95 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.012536754367.53
    96 rdf:type schema:Person
    97 sg:pub.10.1007/3540171665_69 schema:sameAs https://app.dimensions.ai/details/publication/pub.1036324358
    98 https://doi.org/10.1007/3540171665_69
    99 rdf:type schema:CreativeWork
    100 sg:pub.10.1007/bf01058438 schema:sameAs https://app.dimensions.ai/details/publication/pub.1003681860
    101 https://doi.org/10.1007/bf01058438
    102 rdf:type schema:CreativeWork
    103 sg:pub.10.1007/bf01218157 schema:sameAs https://app.dimensions.ai/details/publication/pub.1051921197
    104 https://doi.org/10.1007/bf01218157
    105 rdf:type schema:CreativeWork
    106 sg:pub.10.1007/bf01218262 schema:sameAs https://app.dimensions.ai/details/publication/pub.1020365094
    107 https://doi.org/10.1007/bf01218262
    108 rdf:type schema:CreativeWork
    109 sg:pub.10.1007/bf01232832 schema:sameAs https://app.dimensions.ai/details/publication/pub.1028824895
    110 https://doi.org/10.1007/bf01232832
    111 rdf:type schema:CreativeWork
    112 sg:pub.10.1007/bf02742713 schema:sameAs https://app.dimensions.ai/details/publication/pub.1031864213
    113 https://doi.org/10.1007/bf02742713
    114 rdf:type schema:CreativeWork
    115 sg:pub.10.1007/pl00001582 schema:sameAs https://app.dimensions.ai/details/publication/pub.1006510217
    116 https://doi.org/10.1007/pl00001582
    117 rdf:type schema:CreativeWork
    118 sg:pub.10.1007/s00220-005-1488-1 schema:sameAs https://app.dimensions.ai/details/publication/pub.1000858754
    119 https://doi.org/10.1007/s00220-005-1488-1
    120 rdf:type schema:CreativeWork
    121 sg:pub.10.1007/s00220-012-1522-z schema:sameAs https://app.dimensions.ai/details/publication/pub.1040875907
    122 https://doi.org/10.1007/s00220-012-1522-z
    123 rdf:type schema:CreativeWork
    124 sg:pub.10.1007/s10955-007-9332-y schema:sameAs https://app.dimensions.ai/details/publication/pub.1012453702
    125 https://doi.org/10.1007/s10955-007-9332-y
    126 rdf:type schema:CreativeWork
    127 sg:pub.10.1007/s10955-008-9660-6 schema:sameAs https://app.dimensions.ai/details/publication/pub.1025341457
    128 https://doi.org/10.1007/s10955-008-9660-6
    129 rdf:type schema:CreativeWork
    130 sg:pub.10.1007/s10955-011-0277-9 schema:sameAs https://app.dimensions.ai/details/publication/pub.1041003802
    131 https://doi.org/10.1007/s10955-011-0277-9
    132 rdf:type schema:CreativeWork
    133 sg:pub.10.1007/s10955-012-0568-9 schema:sameAs https://app.dimensions.ai/details/publication/pub.1016793892
    134 https://doi.org/10.1007/s10955-012-0568-9
    135 rdf:type schema:CreativeWork
    136 sg:pub.10.1007/s10955-013-0760-6 schema:sameAs https://app.dimensions.ai/details/publication/pub.1046718564
    137 https://doi.org/10.1007/s10955-013-0760-6
    138 rdf:type schema:CreativeWork
    139 https://doi.org/10.1016/j.physd.2006.07.017 schema:sameAs https://app.dimensions.ai/details/publication/pub.1023310655
    140 rdf:type schema:CreativeWork
    141 https://doi.org/10.1016/j.physleta.2003.11.052 schema:sameAs https://app.dimensions.ai/details/publication/pub.1040830291
    142 rdf:type schema:CreativeWork
    143 https://doi.org/10.1016/j.physleta.2012.05.006 schema:sameAs https://app.dimensions.ai/details/publication/pub.1033354892
    144 rdf:type schema:CreativeWork
    145 https://doi.org/10.1016/s0167-2789(98)00169-9 schema:sameAs https://app.dimensions.ai/details/publication/pub.1035277359
    146 rdf:type schema:CreativeWork
    147 https://doi.org/10.1070/rm1963v018n05abeh004130 schema:sameAs https://app.dimensions.ai/details/publication/pub.1058193720
    148 rdf:type schema:CreativeWork
    149 https://doi.org/10.1073/pnas.47.11.1824 schema:sameAs https://app.dimensions.ai/details/publication/pub.1016645612
    150 rdf:type schema:CreativeWork
    151 https://doi.org/10.1088/0951-7715/12/6/310 schema:sameAs https://app.dimensions.ai/details/publication/pub.1059108891
    152 rdf:type schema:CreativeWork
    153 https://doi.org/10.1103/physreve.76.022104 schema:sameAs https://app.dimensions.ai/details/publication/pub.1006323355
    154 rdf:type schema:CreativeWork
    155 https://doi.org/10.1142/9789812385215_0059 schema:sameAs https://app.dimensions.ai/details/publication/pub.1088717398
    156 rdf:type schema:CreativeWork
    157 https://doi.org/10.2307/121001 schema:sameAs https://app.dimensions.ai/details/publication/pub.1069397473
    158 rdf:type schema:CreativeWork
    159 https://www.grid.ac/institutes/grid.4708.b schema:alternateName University of Milan
    160 schema:name Dipartimento di Matematica, Università degli Studi di Milano, Via Saldini, 50, 20133, Milan, Italy
    161 rdf:type schema:Organization
     




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


    ...