The Transfer Operator Approach to Selberg’s Zeta Function and Modular and Maass Wave Forms For P S L (2, ℤ)
Identifieur interne : 000387 ( Istex/Corpus ); précédent : 000386; suivant : 000388The Transfer Operator Approach to Selberg’s Zeta Function and Modular and Maass Wave Forms For P S L (2, ℤ)
Auteurs : Cheng-Hung Chang ; Dieter H. MayerSource :
- The IMA Volumes in Mathematics and its Applications [ 0940-6573 ] ; 1999.
Abstract
Abstract: In this paper we discuss the transfer operator approach to Selberg’s zeta function for P S L(2, ℤ). Since this function can be expressed as the Fredholm determinant det(1 - L β ) of the transfer operator L β , β ∈ C for the geodesic flow on the modular surface, the zeros and poles of the Selberg function are closely related to those β-values, where L β has an eigenvalue λ = 1 respectively where L β has poles. It turns out that the corresponding eigenfunctions of L β for eigenvalues λ = 1 are closely related to both holomorphic and non-holomorphic modular forms respectively the Maass wave forms. Therefore these eigenfunctions, which by definition of L β are holomorphic functions, are by themselves interesting quantities for the group P S L (2, ℤ): indeed special cases are the period polynomials and functions of the Manin-Eichler and Shimura theory of periods for this group. Another special example of such an eigenfunction is the well known density of Gauss’s measure for the continued fraction expansion. The transfer operator approach hence in a surprising way combines several aspects of the theory of modular and Maass wave forms for the modular group, which up to now were not directly related.
Url:
DOI: 10.1007/978-1-4612-1544-8_3
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<record><TEI wicri:istexFullTextTei="biblStruct"><teiHeader><fileDesc><titleStmt><title xml:lang="en">The Transfer Operator Approach to Selberg’s Zeta Function and Modular and Maass Wave Forms For P S L (2, ℤ)</title><author><name sortKey="Chang, Cheng Hung" sort="Chang, Cheng Hung" uniqKey="Chang C" first="Cheng-Hung" last="Chang">Cheng-Hung Chang</name><affiliation><mods:affiliation>Institut für Theoretische Physik C, TU Clausthal, Arnold-Sommerfeld-Str. 6, 38678, Clausthal-Zellerfeld, Germany</mods:affiliation></affiliation></author><author><name sortKey="Mayer, Dieter H" sort="Mayer, Dieter H" uniqKey="Mayer D" first="Dieter H." last="Mayer">Dieter H. Mayer</name><affiliation><mods:affiliation>Institut für Theoretische Physik C, TU Clausthal, Arnold-Sommerfeld-Str. 6, 38678, Clausthal-Zellerfeld, Germany</mods:affiliation></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">ISTEX</idno><idno type="RBID">ISTEX:1273B34EC314C75A1F671BC13B8218FFBE44FE97</idno><date when="1999" year="1999">1999</date><idno type="doi">10.1007/978-1-4612-1544-8_3</idno><idno type="url">https://api.istex.fr/document/1273B34EC314C75A1F671BC13B8218FFBE44FE97/fulltext/pdf</idno><idno type="wicri:Area/Istex/Corpus">000387</idno><idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Corpus" wicri:corpus="ISTEX">000387</idno></publicationStmt><sourceDesc><biblStruct><analytic><title level="a" type="main" xml:lang="en">The Transfer Operator Approach to Selberg’s Zeta Function and Modular and Maass Wave Forms For P S L (2, ℤ)</title><author><name sortKey="Chang, Cheng Hung" sort="Chang, Cheng Hung" uniqKey="Chang C" first="Cheng-Hung" last="Chang">Cheng-Hung Chang</name><affiliation><mods:affiliation>Institut für Theoretische Physik C, TU Clausthal, Arnold-Sommerfeld-Str. 6, 38678, Clausthal-Zellerfeld, Germany</mods:affiliation></affiliation></author><author><name sortKey="Mayer, Dieter H" sort="Mayer, Dieter H" uniqKey="Mayer D" first="Dieter H." last="Mayer">Dieter H. Mayer</name><affiliation><mods:affiliation>Institut für Theoretische Physik C, TU Clausthal, Arnold-Sommerfeld-Str. 6, 38678, Clausthal-Zellerfeld, Germany</mods:affiliation></affiliation></author></analytic><monogr></monogr><series><title level="s">The IMA Volumes in Mathematics and its Applications</title><imprint><date>1999</date></imprint><idno type="ISSN">0940-6573</idno><idno type="ISSN">0940-6573</idno></series></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0940-6573</idno></seriesStmt></fileDesc><profileDesc><textClass></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">Abstract: In this paper we discuss the transfer operator approach to Selberg’s zeta function for P S L(2, ℤ). Since this function can be expressed as the Fredholm determinant det(1 - L β ) of the transfer operator L β , β ∈ C for the geodesic flow on the modular surface, the zeros and poles of the Selberg function are closely related to those β-values, where L β has an eigenvalue λ = 1 respectively where L β has poles. It turns out that the corresponding eigenfunctions of L β for eigenvalues λ = 1 are closely related to both holomorphic and non-holomorphic modular forms respectively the Maass wave forms. Therefore these eigenfunctions, which by definition of L β are holomorphic functions, are by themselves interesting quantities for the group P S L (2, ℤ): indeed special cases are the period polynomials and functions of the Manin-Eichler and Shimura theory of periods for this group. Another special example of such an eigenfunction is the well known density of Gauss’s measure for the continued fraction expansion. The transfer operator approach hence in a surprising way combines several aspects of the theory of modular and Maass wave forms for the modular group, which up to now were not directly related.</div></front></TEI><istex><corpusName>springer-ebooks</corpusName><author><json:item><name>Cheng-Hung Chang</name><affiliations><json:string>Institut für Theoretische Physik C, TU Clausthal, Arnold-Sommerfeld-Str. 6, 38678, Clausthal-Zellerfeld, Germany</json:string></affiliations></json:item><json:item><name>Dieter H. Mayer</name><affiliations><json:string>Institut für Theoretische Physik C, TU Clausthal, Arnold-Sommerfeld-Str. 6, 38678, Clausthal-Zellerfeld, Germany</json:string></affiliations></json:item></author><arkIstex>ark:/67375/HCB-49CL5BP1-8</arkIstex><language><json:string>eng</json:string></language><originalGenre><json:string>OriginalPaper</json:string></originalGenre><abstract>Abstract: In this paper we discuss the transfer operator approach to Selberg’s zeta function for P S L(2, ℤ). Since this function can be expressed as the Fredholm determinant det(1 - L β ) of the transfer operator L β , β ∈ C for the geodesic flow on the modular surface, the zeros and poles of the Selberg function are closely related to those β-values, where L β has an eigenvalue λ = 1 respectively where L β has poles. It turns out that the corresponding eigenfunctions of L β for eigenvalues λ = 1 are closely related to both holomorphic and non-holomorphic modular forms respectively the Maass wave forms. Therefore these eigenfunctions, which by definition of L β are holomorphic functions, are by themselves interesting quantities for the group P S L (2, ℤ): indeed special cases are the period polynomials and functions of the Manin-Eichler and Shimura theory of periods for this group. Another special example of such an eigenfunction is the well known density of Gauss’s measure for the continued fraction expansion. 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Since this function can be expressed as the Fredholm determinant det(1 - L β ) of the transfer operator L β , β ∈ C for the geodesic flow on the modular surface, the zeros and poles of the Selberg function are closely related to those β-values, where L β has an eigenvalue λ = 1 respectively where L β has poles. It turns out that the corresponding eigenfunctions of L β for eigenvalues λ = 1 are closely related to both holomorphic and non-holomorphic modular forms respectively the Maass wave forms. Therefore these eigenfunctions, which by definition of L β are holomorphic functions, are by themselves interesting quantities for the group P S L (2, ℤ): indeed special cases are the period polynomials and functions of the Manin-Eichler and Shimura theory of periods for this group. Another special example of such an eigenfunction is the well known density of Gauss’s measure for the continued fraction expansion. The transfer operator approach hence in a surprising way combines several aspects of the theory of modular and Maass wave forms for the modular group, which up to now were not directly related.</p></abstract><textClass><keywords scheme="Book-Subject-Collection"><list><label>SUCO11649</label><item><term>Mathematics and Statistics</term></item></list></keywords></textClass><textClass><keywords scheme="Book-Subject-Group"><list><label>SCM</label><label>SCM25001</label><label>SCM29010</label><item><term>Mathematics</term></item><item><term>Number Theory</term></item><item><term>Combinatorics</term></item></list></keywords></textClass></profileDesc><revisionDesc><change when="1999">Published</change><change xml:id="refBibs-istex" who="#ISTEX-API" when="2017-11-28">References added</change></revisionDesc></teiHeader></istex:fulltextTEI><json:item><extension>txt</extension><original>false</original><mimetype>text/plain</mimetype><uri>https://api.istex.fr/document/1273B34EC314C75A1F671BC13B8218FFBE44FE97/fulltext/txt</uri></json:item></fulltext><metadata><istex:metadataXml wicri:clean="corpus springer-ebooks not found" wicri:toSee="no header"><istex:xmlDeclaration>version="1.0" encoding="UTF-8"</istex:xmlDeclaration><istex:docType PUBLIC="-//Springer-Verlag//DTD A++ V2.4//EN" URI="http://devel.springer.de/A++/V2.4/DTD/A++V2.4.dtd" name="istex:docType"></istex:docType><istex:document><Publisher><PublisherInfo><PublisherName>Springer New York</PublisherName><PublisherLocation>New York, NY</PublisherLocation><PublisherImprintName>Springer</PublisherImprintName></PublisherInfo><Series><SeriesInfo SeriesType="Series" TocLevels="0"><SeriesID>811</SeriesID><SeriesPrintISSN>0940-6573</SeriesPrintISSN><SeriesTitle Language="En">The IMA Volumes in Mathematics and its Applications</SeriesTitle></SeriesInfo><SeriesHeader><EditorGroup><Editor AffiliationIDS="Aff1"><EditorName DisplayOrder="Western"><GivenName>Willard</GivenName><FamilyName>Miller</FamilyName><Suffix>Jr.</Suffix></EditorName></Editor><Affiliation ID="Aff1"><OrgDivision>Institute for Mathematics and its Applications</OrgDivision><OrgName>University of Minnesota</OrgName><OrgAddress><City>Minneapolis</City><State>MN</State><Postcode>55455</Postcode><Country>USA</Country></OrgAddress></Affiliation></EditorGroup></SeriesHeader><Book Language="En"><BookInfo BookProductType="Monograph" ContainsESM="No" Language="En" MediaType="eBook" NumberingStyle="Unnumbered" OutputMedium="All" TocLevels="0"><BookID>978-1-4612-1544-8</BookID><BookTitle>Emerging Applications of Number Theory</BookTitle><BookVolumeNumber>109</BookVolumeNumber><BookSequenceNumber>110</BookSequenceNumber><BookDOI>10.1007/978-1-4612-1544-8</BookDOI><BookTitleID>61436</BookTitleID><BookPrintISBN>978-1-4612-7186-4</BookPrintISBN><BookElectronicISBN>978-1-4612-1544-8</BookElectronicISBN><BookChapterCount>28</BookChapterCount><BookCopyright><CopyrightHolderName>Springer-Verlag New York, Inc.</CopyrightHolderName><CopyrightYear>1999</CopyrightYear></BookCopyright><BookSubjectGroup><BookSubject Code="SCM" Type="Primary">Mathematics</BookSubject><BookSubject Code="SCM25001" Priority="1" Type="Secondary">Number Theory</BookSubject><BookSubject Code="SCM29010" Priority="2" Type="Secondary">Combinatorics</BookSubject><SubjectCollection Code="SUCO11649">Mathematics and Statistics</SubjectCollection></BookSubjectGroup><BookContext><SeriesID>811</SeriesID></BookContext></BookInfo><BookHeader><EditorGroup><Editor AffiliationIDS="Aff2"><EditorName DisplayOrder="Western"><GivenName>Dennis</GivenName><GivenName>A.</GivenName><FamilyName>Hejhal</FamilyName></EditorName></Editor><Editor AffiliationIDS="Aff3"><EditorName DisplayOrder="Western"><GivenName>Joel</GivenName><FamilyName>Friedman</FamilyName></EditorName></Editor><Editor AffiliationIDS="Aff4"><EditorName DisplayOrder="Western"><GivenName>Martin</GivenName><GivenName>C.</GivenName><FamilyName>Gutzwiller</FamilyName></EditorName></Editor><Editor AffiliationIDS="Aff5"><EditorName DisplayOrder="Western"><GivenName>Andrew</GivenName><GivenName>M.</GivenName><FamilyName>Odlyzko</FamilyName></EditorName></Editor><Affiliation ID="Aff2"><OrgDivision>School of Mathematics</OrgDivision><OrgName>University of Minnesota</OrgName><OrgAddress><City>Minneapolis</City><State>MN</State><Postcode>55455</Postcode><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff3"><OrgDivision>Department of Mathematics</OrgDivision><OrgName>University of British Columbia</OrgName><OrgAddress><City>Vancouver</City><State>British Columbia</State><Postcode>V6T 1Z2</Postcode><Country>Canada</Country></OrgAddress></Affiliation><Affiliation ID="Aff4"><OrgName>IBM Thomas J. Watson Research Center</OrgName><OrgAddress><City>Yorktown Heights</City><State>NY</State><Postcode>10598</Postcode><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff5"><OrgName>AT&T Labs—Research</OrgName><OrgAddress><City>Florham Park</City><State>NJ</State><Postcode>07932-0971</Postcode><Country>USA</Country></OrgAddress></Affiliation></EditorGroup></BookHeader><Chapter ID="Chap3" Language="En"><ChapterInfo ChapterType="OriginalPaper" ContainsESM="No" Language="En" NumberingStyle="Unnumbered" TocLevels="0"><ChapterID>3</ChapterID><ChapterDOI>10.1007/978-1-4612-1544-8_3</ChapterDOI><ChapterSequenceNumber>3</ChapterSequenceNumber><ChapterTitle Language="En">The Transfer Operator Approach to Selberg’s Zeta Function and Modular and Maass Wave Forms For <Emphasis Type="Italic">P S L</Emphasis> (2, ℤ)</ChapterTitle><ChapterFirstPage>73</ChapterFirstPage><ChapterLastPage>141</ChapterLastPage><ChapterCopyright><CopyrightHolderName>Springer Science+Business Media New York</CopyrightHolderName><CopyrightYear>1999</CopyrightYear></ChapterCopyright><ChapterHistory><RegistrationDate><Year>2011</Year><Month>8</Month><Day>2</Day></RegistrationDate></ChapterHistory><ChapterGrants Type="Regular"><MetadataGrant Grant="OpenAccess"></MetadataGrant><AbstractGrant Grant="OpenAccess"></AbstractGrant><BodyPDFGrant Grant="Restricted"></BodyPDFGrant><BodyHTMLGrant Grant="Restricted"></BodyHTMLGrant><BibliographyGrant Grant="Restricted"></BibliographyGrant><ESMGrant Grant="Restricted"></ESMGrant></ChapterGrants><ChapterContext><SeriesID>811</SeriesID><BookID>978-1-4612-1544-8</BookID><BookTitle>Emerging Applications of Number Theory</BookTitle></ChapterContext></ChapterInfo><ChapterHeader><AuthorGroup><Author AffiliationIDS="Aff6"><AuthorName DisplayOrder="Western"><GivenName>Cheng-Hung</GivenName><FamilyName>Chang</FamilyName></AuthorName></Author><Author AffiliationIDS="Aff6"><AuthorName DisplayOrder="Western"><GivenName>Dieter</GivenName><GivenName>H.</GivenName><FamilyName>Mayer</FamilyName></AuthorName></Author><Affiliation ID="Aff6"><OrgName>Institut für Theoretische Physik C, TU Clausthal</OrgName><OrgAddress><Street>Arnold-Sommerfeld-Str. 6</Street><Postcode>38678</Postcode><City>Clausthal-Zellerfeld</City><Country>Germany</Country></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1" Language="En" OutputMedium="All"><Heading>Abstract</Heading><Para>In this paper we discuss the transfer operator approach to Selberg’s zeta function for <Emphasis Type="Italic">P S L</Emphasis>(2, ℤ). Since this function can be expressed as the Fredholm determinant <Emphasis Type="Italic">det</Emphasis>(1 - <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">β</Emphasis></Subscript>) of the transfer operator <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">β</Emphasis></Subscript>, <Emphasis Type="Italic">β</Emphasis> ∈ <Emphasis Type="Italic">C</Emphasis> for the geodesic flow on the modular surface, the zeros and poles of the Selberg function are closely related to those <Emphasis Type="Italic">β-</Emphasis>values, where <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">β</Emphasis></Subscript> has an eigenvalue λ = 1 respectively where <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">β</Emphasis></Subscript> has poles. It turns out that the corresponding eigenfunctions of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">β</Emphasis></Subscript> for eigenvalues λ = 1 are closely related to both holomorphic and non-holomorphic modular forms respectively the Maass wave forms. Therefore these eigenfunctions, which by definition of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">β</Emphasis></Subscript> are holomorphic functions, are by themselves interesting quantities for the group <Emphasis Type="Italic">P S L</Emphasis> (2, ℤ): indeed special cases are the period polynomials and functions of the Manin-Eichler and Shimura theory of periods for this group. Another special example of such an eigenfunction is the well known density of Gauss’s measure for the continued fraction expansion. The transfer operator approach hence in a surprising way combines several aspects of the theory of modular and Maass wave forms for the modular group, which up to now were not directly related.</Para></Abstract><KeywordGroup Language="En"><Heading>Key words</Heading><Keyword>quantum chaos</Keyword><Keyword>Selberg zeta function</Keyword><Keyword>dynamical zeta function</Keyword><Keyword>transfer operator</Keyword><Keyword>functional equation</Keyword><Keyword>modular forms</Keyword><Keyword>Maass wave forms</Keyword><Keyword>period polynomial</Keyword><Keyword>period function</Keyword></KeywordGroup><ArticleNote Type="Misc"><SimplePara>The work is supported by DFG Schwerpunktprogramm “Ergodentheorie, Analysis und effiziente Simulation dynamischer Systeme”.</SimplePara></ArticleNote></ChapterHeader><NoBody></NoBody></Chapter></Book></Series></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>The Transfer Operator Approach to Selberg’s Zeta Function and Modular and Maass Wave Forms For P S L (2, ℤ)</title></titleInfo><titleInfo type="alternative" contentType="CDATA" lang="en"><title>The Transfer Operator Approach to Selberg’s Zeta Function and Modular and Maass Wave Forms For P S L (2, ℤ)</title></titleInfo><name type="personal"><namePart type="given">Cheng-Hung</namePart><namePart type="family">Chang</namePart><affiliation>Institut für Theoretische Physik C, TU Clausthal, Arnold-Sommerfeld-Str. 6, 38678, Clausthal-Zellerfeld, Germany</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Dieter</namePart><namePart type="given">H.</namePart><namePart type="family">Mayer</namePart><affiliation>Institut für Theoretische Physik C, TU Clausthal, Arnold-Sommerfeld-Str. 6, 38678, Clausthal-Zellerfeld, Germany</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre displayLabel="OriginalPaper" authority="ISTEX" authorityURI="https://content-type.data.istex.fr" type="research-article" valueURI="https://content-type.data.istex.fr/ark:/67375/XTP-1JC4F85T-7">research-article</genre><originInfo><publisher>Springer New York</publisher><place><placeTerm type="text">New York, NY</placeTerm></place><dateIssued encoding="w3cdtf">1999</dateIssued><dateIssued encoding="w3cdtf">1999</dateIssued><copyrightDate encoding="w3cdtf">1999</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><abstract lang="en">Abstract: In this paper we discuss the transfer operator approach to Selberg’s zeta function for P S L(2, ℤ). Since this function can be expressed as the Fredholm determinant det(1 - L β ) of the transfer operator L β , β ∈ C for the geodesic flow on the modular surface, the zeros and poles of the Selberg function are closely related to those β-values, where L β has an eigenvalue λ = 1 respectively where L β has poles. It turns out that the corresponding eigenfunctions of L β for eigenvalues λ = 1 are closely related to both holomorphic and non-holomorphic modular forms respectively the Maass wave forms. Therefore these eigenfunctions, which by definition of L β are holomorphic functions, are by themselves interesting quantities for the group P S L (2, ℤ): indeed special cases are the period polynomials and functions of the Manin-Eichler and Shimura theory of periods for this group. Another special example of such an eigenfunction is the well known density of Gauss’s measure for the continued fraction expansion. The transfer operator approach hence in a surprising way combines several aspects of the theory of modular and Maass wave forms for the modular group, which up to now were not directly related.</abstract><relatedItem type="host"><titleInfo><title>Emerging Applications of Number Theory</title></titleInfo><name type="personal"><namePart type="given">Dennis</namePart><namePart type="given">A.</namePart><namePart type="family">Hejhal</namePart><affiliation>School of Mathematics, University of Minnesota, 55455, Minneapolis, MN, USA</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Joel</namePart><namePart type="family">Friedman</namePart><affiliation>Department of Mathematics, University of British Columbia, V6T 1Z2, Vancouver, British Columbia, Canada</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Martin</namePart><namePart type="given">C.</namePart><namePart type="family">Gutzwiller</namePart><affiliation>IBM Thomas J. Watson Research Center, 10598, Yorktown Heights, NY, USA</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Andrew</namePart><namePart type="given">M.</namePart><namePart type="family">Odlyzko</namePart><affiliation>AT&T Labs—Research, 07932-0971, Florham Park, NJ, USA</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><genre type="book-series" displayLabel="Monograph" authority="ISTEX" authorityURI="https://publication-type.data.istex.fr" valueURI="https://publication-type.data.istex.fr/ark:/67375/JMC-0G6R5W5T-Z">book-series</genre><originInfo><publisher>Springer</publisher><copyrightDate encoding="w3cdtf">1999</copyrightDate><issuance>monographic</issuance></originInfo><subject><genre>Book-Subject-Collection</genre><topic authority="SpringerSubjectCodes" authorityURI="SUCO11649">Mathematics and Statistics</topic></subject><subject><genre>Book-Subject-Group</genre><topic authority="SpringerSubjectCodes" authorityURI="SCM">Mathematics</topic><topic authority="SpringerSubjectCodes" authorityURI="SCM25001">Number Theory</topic><topic authority="SpringerSubjectCodes" authorityURI="SCM29010">Combinatorics</topic></subject><identifier type="DOI">10.1007/978-1-4612-1544-8</identifier><identifier type="ISBN">978-1-4612-7186-4</identifier><identifier type="eISBN">978-1-4612-1544-8</identifier><identifier type="ISSN">0940-6573</identifier><identifier type="BookTitleID">61436</identifier><identifier type="BookID">978-1-4612-1544-8</identifier><identifier type="BookChapterCount">28</identifier><identifier type="BookVolumeNumber">109</identifier><identifier type="BookSequenceNumber">110</identifier><part><date>1999</date><detail type="volume"><number>109</number><caption>vol.</caption></detail><extent unit="pages"><start>73</start><end>141</end></extent></part><recordInfo><recordOrigin>Springer-Verlag New York, Inc., 1999</recordOrigin></recordInfo></relatedItem><relatedItem type="series"><titleInfo><title>The IMA Volumes in Mathematics and its Applications</title></titleInfo><name type="personal"><namePart type="given">Willard</namePart><namePart type="family">Miller Jr.</namePart><affiliation>Institute for Mathematics and its Applications, University of Minnesota, 55455, Minneapolis, MN, USA</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><originInfo><publisher>Springer</publisher><copyrightDate encoding="w3cdtf">1999</copyrightDate><issuance>serial</issuance></originInfo><identifier type="ISSN">0940-6573</identifier><identifier type="SeriesID">811</identifier><recordInfo><recordOrigin>Springer-Verlag New York, Inc., 1999</recordOrigin></recordInfo></relatedItem><identifier type="istex">1273B34EC314C75A1F671BC13B8218FFBE44FE97</identifier><identifier type="ark">ark:/67375/HCB-49CL5BP1-8</identifier><identifier type="DOI">10.1007/978-1-4612-1544-8_3</identifier><identifier type="ChapterID">3</identifier><identifier type="ChapterID">Chap3</identifier><accessCondition type="use and reproduction" contentType="copyright">Springer-Verlag New York, Inc., 1999</accessCondition><recordInfo><recordContentSource authority="ISTEX" authorityURI="https://loaded-corpus.data.istex.fr" valueURI="https://loaded-corpus.data.istex.fr/ark:/67375/XBH-3XSW68JL-F">springer</recordContentSource><recordOrigin>Springer Science+Business Media New York, 1999</recordOrigin></recordInfo></mods><json:item><extension>json</extension><original>false</original><mimetype>application/json</mimetype><uri>https://api.istex.fr/document/1273B34EC314C75A1F671BC13B8218FFBE44FE97/metadata/json</uri></json:item></metadata><annexes><json:item><extension>txt</extension><original>true</original><mimetype>text/plain</mimetype><uri>https://api.istex.fr/document/1273B34EC314C75A1F671BC13B8218FFBE44FE97/annexes/txt</uri></json:item></annexes></istex></record>Pour manipuler ce document sous Unix (Dilib)
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