Universal entire functions with gap power series
Identifieur interne : 000893 ( Istex/Corpus ); précédent : 000892; suivant : 000894Universal entire functions with gap power series
Auteurs : Wolfgang Luh ; Valeri A. Martirosian ; Jürgen MüllerSource :
- Indagationes Mathematicae [ 0019-3577 ] ; 1998.
Abstract
Let M be the family of all compact sets in C which have connected complement. For K ϵ M we denote by A(K) the set of all functions which are continuous on K and holomorphic in its interior. Suppose that {zn} is any unbounded sequence of complex numbers and let Q be a given sub-sequence of N0. If Q has density Δ(Q) = 1 then there exists a universal entire function ϑ with lacunary power series 1. (1) ϑ(z) = ϵ∞v = 0 ϑvZv, ϑv = 0 for v ∉ Q, which has for all K ϵ M the following properties simultaneously 2. (2) the sequence {ϑ(Z + Zn)} is dense in A(K) 3. (3) the sequence {ϑ (ZZn)} is dense in A(K) if 0 ∉ K. Also a converse result is proved: If ϑ is an entire function of the form (1) which satisfies (3), then Q must have maximal density Δmax(Q) = 1.
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DOI: 10.1016/S0019-3577(98)80032-3
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<record><TEI wicri:istexFullTextTei="biblStruct"><teiHeader><fileDesc><titleStmt><title>Universal entire functions with gap power series</title><author><name sortKey="Luh, Wolfgang" sort="Luh, Wolfgang" uniqKey="Luh W" first="Wolfgang" last="Luh">Wolfgang Luh</name><affiliation><mods:affiliation>Fachbereich IV, Universität Trier, D-54286 Trier, Germany</mods:affiliation></affiliation></author><author><name sortKey="Martirosian, Valeri A" sort="Martirosian, Valeri A" uniqKey="Martirosian V" first="Valeri A." last="Martirosian">Valeri A. Martirosian</name><affiliation><mods:affiliation>Department of Mathematics, Yerevan State University, Alex Manoogian St. 1, Yerevan 375049, Armenia</mods:affiliation></affiliation><affiliation><mods:affiliation>∗ The research work of the second author has been supported by the German Academic Exchange Service (DAAD).</mods:affiliation></affiliation></author><author><name sortKey="Muller, Jurgen" sort="Muller, Jurgen" uniqKey="Muller J" first="Jürgen" last="Müller">Jürgen Müller</name><affiliation><mods:affiliation>Fachbereich IV, Universität Trier, D-54286 Trier, Germany</mods:affiliation></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">ISTEX</idno><idno type="RBID">ISTEX:FF33B736421F5A07D6796054DEFBAAD73A8EFDB3</idno><date when="1998" year="1998">1998</date><idno type="doi">10.1016/S0019-3577(98)80032-3</idno><idno type="url">https://api.istex.fr/document/FF33B736421F5A07D6796054DEFBAAD73A8EFDB3/fulltext/pdf</idno><idno type="wicri:Area/Istex/Corpus">000893</idno><idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Corpus" wicri:corpus="ISTEX">000893</idno></publicationStmt><sourceDesc><biblStruct><analytic><title level="a">Universal entire functions with gap power series</title><author><name sortKey="Luh, Wolfgang" sort="Luh, Wolfgang" uniqKey="Luh W" first="Wolfgang" last="Luh">Wolfgang Luh</name><affiliation><mods:affiliation>Fachbereich IV, Universität Trier, D-54286 Trier, Germany</mods:affiliation></affiliation></author><author><name sortKey="Martirosian, Valeri A" sort="Martirosian, Valeri A" uniqKey="Martirosian V" first="Valeri A." last="Martirosian">Valeri A. Martirosian</name><affiliation><mods:affiliation>Department of Mathematics, Yerevan State University, Alex Manoogian St. 1, Yerevan 375049, Armenia</mods:affiliation></affiliation><affiliation><mods:affiliation>∗ The research work of the second author has been supported by the German Academic Exchange Service (DAAD).</mods:affiliation></affiliation></author><author><name sortKey="Muller, Jurgen" sort="Muller, Jurgen" uniqKey="Muller J" first="Jürgen" last="Müller">Jürgen Müller</name><affiliation><mods:affiliation>Fachbereich IV, Universität Trier, D-54286 Trier, Germany</mods:affiliation></affiliation></author></analytic><monogr></monogr><series><title level="j">Indagationes Mathematicae</title><title level="j" type="abbrev">INDAG</title><idno type="ISSN">0019-3577</idno><imprint><publisher>ELSEVIER</publisher><date type="published" when="1998">1998</date><biblScope unit="volume">9</biblScope><biblScope unit="issue">4</biblScope><biblScope unit="page" from="529">529</biblScope><biblScope unit="page" to="536">536</biblScope></imprint><idno type="ISSN">0019-3577</idno></series><idno type="istex">FF33B736421F5A07D6796054DEFBAAD73A8EFDB3</idno><idno type="DOI">10.1016/S0019-3577(98)80032-3</idno><idno type="PII">S0019-3577(98)80032-3</idno></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0019-3577</idno></seriesStmt></fileDesc><profileDesc><textClass></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">Let M be the family of all compact sets in C which have connected complement. For K ϵ M we denote by A(K) the set of all functions which are continuous on K and holomorphic in its interior. Suppose that {zn} is any unbounded sequence of complex numbers and let Q be a given sub-sequence of N0. If Q has density Δ(Q) = 1 then there exists a universal entire function ϑ with lacunary power series 1. (1) ϑ(z) = ϵ∞v = 0 ϑvZv, ϑv = 0 for v ∉ Q, which has for all K ϵ M the following properties simultaneously 2. (2) the sequence {ϑ(Z + Zn)} is dense in A(K) 3. (3) the sequence {ϑ (ZZn)} is dense in A(K) if 0 ∉ K. Also a converse result is proved: If ϑ is an entire function of the form (1) which satisfies (3), then Q must have maximal density Δmax(Q) = 1.</div></front></TEI><istex><corpusName>elsevier</corpusName><author><json:item><name>Wolfgang Luh</name><affiliations><json:string>Fachbereich IV, Universität Trier, D-54286 Trier, Germany</json:string></affiliations></json:item><json:item><name>Valeri A. Martirosian</name><affiliations><json:string>Department of Mathematics, Yerevan State University, Alex Manoogian St. 1, Yerevan 375049, Armenia</json:string><json:string>∗ The research work of the second author has been supported by the German Academic Exchange Service (DAAD).</json:string></affiliations></json:item><json:item><name>Jürgen Müller</name><affiliations><json:string>Fachbereich IV, Universität Trier, D-54286 Trier, Germany</json:string></affiliations></json:item></author><language><json:string>eng</json:string></language><originalGenre><json:string>Full-length article</json:string></originalGenre><abstract>Let M be the family of all compact sets in C which have connected complement. For K ϵ M we denote by A(K) the set of all functions which are continuous on K and holomorphic in its interior. Suppose that {zn} is any unbounded sequence of complex numbers and let Q be a given sub-sequence of N0. If Q has density Δ(Q) = 1 then there exists a universal entire function ϑ with lacunary power series 1. (1) ϑ(z) = ϵ∞v = 0 ϑvZv, ϑv = 0 for v ∉ Q, which has for all K ϵ M the following properties simultaneously 2. (2) the sequence {ϑ(Z + Zn)} is dense in A(K) 3. (3) the sequence {ϑ (ZZn)} is dense in A(K) if 0 ∉ K. Also a converse result is proved: If ϑ is an entire function of the form (1) which satisfies (3), then Q must have maximal density Δmax(Q) = 1.</abstract><qualityIndicators><score>3.606</score><pdfVersion>1.2</pdfVersion><pdfPageSize>432 x 684 pts</pdfPageSize><refBibsNative>true</refBibsNative><keywordCount>0</keywordCount><abstractCharCount>754</abstractCharCount><pdfWordCount>1746</pdfWordCount><pdfCharCount>10438</pdfCharCount><pdfPageCount>8</pdfPageCount><abstractWordCount>155</abstractWordCount></qualityIndicators><title>Universal entire functions with gap power series</title><pii><json:string>S0019-3577(98)80032-3</json:string></pii><refBibs><json:item><author><json:item><name>N.U. Arakelian</name></json:item><json:item><name>V.A. Martirosian</name></json:item></author><host><volume>235</volume><pages><last>252</last><first>249</first></pages><author></author><title>Dokladi Akad. Nauk USSR</title></host><title>Uniform approximations in the complex plane by gap polynomials</title></json:item><json:item><host><volume>18</volume><pages><last>904</last><first>901</first></pages><author></author><title>Soviet Math. Dokl.</title></host></json:item><json:item><author><json:item><name>L. Bernal-González</name></json:item><json:item><name>A. Montes-Rodriguez</name></json:item></author><host><volume>27</volume><pages><last>56</last><first>47</first></pages><author></author><title>Complex Variables</title></host><title>Universal functions for composition operators</title></json:item><json:item><author><json:item><name>G.D. Birkhoff</name></json:item></author><host><volume>189</volume><pages><last>475</last><first>473</first></pages><author></author><title>C.R. Acad. Sci. Paris</title></host><title>Démonstration d'un théorème élémentaire sur les fonctions entières</title></json:item><json:item><author><json:item><name>M. Dixon</name></json:item><json:item><name>J. Korevaar</name></json:item></author><host><author></author><title>Nederl. Akad. Wetensch. Proc., Ser. A</title></host><serie><author></author><title>Nederl. Akad. Wetensch. Proc., Ser. A</title></serie><title>Approximation by lacunary polynomials</title></json:item><json:item><host><author></author><title>Vorlesungen über Approximation in Komplexen</title></host></json:item><json:item><host><author></author><title>Darstellung und Begründung einiger neuerer Erebnisse der Funktionentheorie</title></host></json:item><json:item><author><json:item><name>W. Luh</name></json:item></author><host><volume>53</volume><pages><last>144</last><first>128</first></pages><author></author><title>J. Approx. Theory</title></host><title>Holomorphic monsters</title></json:item><json:item><author><json:item><name>W. Luh</name></json:item></author><host><volume>31</volume><pages><last>96</last><first>87</first></pages><author></author><title>Complex Variables</title></host><title>Entire functions with various universal properties</title></json:item><json:item><author><json:item><name>V.A. Martirosian</name></json:item></author><host><volume>120</volume><pages><last>472</last><first>451</first></pages><author></author><title>Matem. Sbornik</title></host><title>On the uniform complex approximation by gap polynomials</title></json:item><json:item><host><volume>48</volume><pages><last>462</last><first>445</first></pages><author></author><title>Math. USSR Sbornik</title></host></json:item><json:item><author><json:item><name>S.N. Mergelian</name></json:item></author><host><volume>7</volume><pages><last>122</last><first>31</first></pages><author></author><title>Uspekhi Matem. Nauk</title></host><title>Uniform approximation of functions of a complex variable</title></json:item><json:item><host><author></author><title>Amer. Math. Soc. Transl.</title></host></json:item><json:item><author><json:item><name>J. Müller</name></json:item></author><host><author></author><title>Mitt. Math. Sem. Gieβen</title></host><serie><author></author><title>Mitt. Math. Sem. Gieβen</title></serie><title>Über analytische Fortsetzung mit Matrixverfahren</title></json:item><json:item><author><json:item><name>J. Müller</name></json:item></author><host><author></author><title>Israel Math. Conf. Proc.</title></host><serie><author></author><title>Israel Math. Conf. Proc.</title></serie><title>Approximation with lacunary polynomials in the complex plane</title></json:item><json:item><author><json:item><name>G. Pólya</name></json:item></author><host><volume>29</volume><pages><last>640</last><first>549</first></pages><author></author><title>Math. Z.</title></host><title>Untersuchungen über Lücken und Singularitäten von Potenzreihen (1. Mitteilung)</title></json:item><json:item><host><author></author><title>Real and complex analysis</title></host></json:item><json:item><author><json:item><name>C. Runge</name></json:item></author><host><volume>6</volume><pages><last>244</last><first>228</first></pages><author></author><title>Acta Math.</title></host><title>Zur Theorie der eindeutigen analytischen Funktionen</title></json:item><json:item><author><json:item><name>P. Zappa</name></json:item></author><host><volume>7</volume><pages><last>352</last><first>345</first></pages><author></author><title>Bollettino U.M.I. 2-A</title></host><title>On universal holomorphic functions</title></json:item></refBibs><genre><json:string>research-article</json:string></genre><serie><volume>80</volume><pages><last>194</last><first>176</first></pages><language><json:string>unknown</json:string></language><title>Nederl. Akad. Wetensch. Proc., Ser. 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Korevaar at the meeting of September 29, 1997</note></notesStmt><sourceDesc><biblStruct type="inbook"><analytic><title level="a">Universal entire functions with gap power series</title><author xml:id="author-1"><persName><forename type="first">Wolfgang</forename><surname>Luh</surname></persName><affiliation>Fachbereich IV, Universität Trier, D-54286 Trier, Germany</affiliation></author><author xml:id="author-2"><persName><forename type="first">Valeri A.</forename><surname>Martirosian</surname></persName><affiliation>Department of Mathematics, Yerevan State University, Alex Manoogian St. 1, Yerevan 375049, Armenia</affiliation><affiliation>∗ The research work of the second author has been supported by the German Academic Exchange Service (DAAD).</affiliation></author><author xml:id="author-3"><persName><forename type="first">Jürgen</forename><surname>Müller</surname></persName><affiliation>Fachbereich IV, Universität Trier, D-54286 Trier, Germany</affiliation></author></analytic><monogr><title level="j">Indagationes Mathematicae</title><title level="j" type="abbrev">INDAG</title><idno type="pISSN">0019-3577</idno><idno type="PII">S0019-3577(00)X0015-8</idno><imprint><publisher>ELSEVIER</publisher><date type="published" when="1998"></date><biblScope unit="volume">9</biblScope><biblScope unit="issue">4</biblScope><biblScope unit="page" from="529">529</biblScope><biblScope unit="page" to="536">536</biblScope></imprint></monogr><idno type="istex">FF33B736421F5A07D6796054DEFBAAD73A8EFDB3</idno><idno type="DOI">10.1016/S0019-3577(98)80032-3</idno><idno type="PII">S0019-3577(98)80032-3</idno></biblStruct></sourceDesc></fileDesc><profileDesc><creation><date>1998</date></creation><langUsage><language ident="en">en</language></langUsage><abstract xml:lang="en"><p>Let M be the family of all compact sets in C which have connected complement. For K ϵ M we denote by A(K) the set of all functions which are continuous on K and holomorphic in its interior. Suppose that {zn} is any unbounded sequence of complex numbers and let Q be a given sub-sequence of N0. If Q has density Δ(Q) = 1 then there exists a universal entire function ϑ with lacunary power series 1. (1) ϑ(z) = ϵ∞v = 0 ϑvZv, ϑv = 0 for v ∉ Q, which has for all K ϵ M the following properties simultaneously 2. (2) the sequence {ϑ(Z + Zn)} is dense in A(K) 3. (3) the sequence {ϑ (ZZn)} is dense in A(K) if 0 ∉ K. Also a converse result is proved: If ϑ is an entire function of the form (1) which satisfies (3), then Q must have maximal density Δmax(Q) = 1.</p></abstract></profileDesc><revisionDesc><change when="1998">Published</change></revisionDesc></teiHeader></istex:fulltextTEI><json:item><extension>txt</extension><original>false</original><mimetype>text/plain</mimetype><uri>https://api.istex.fr/document/FF33B736421F5A07D6796054DEFBAAD73A8EFDB3/fulltext/txt</uri></json:item></fulltext><metadata><istex:metadataXml wicri:clean="Elsevier, elements deleted: tail"><istex:xmlDeclaration>version="1.0" encoding="utf-8"</istex:xmlDeclaration><istex:docType PUBLIC="-//ES//DTD journal article DTD version 4.5.2//EN//XML" URI="art452.dtd" name="istex:docType"></istex:docType><istex:document><converted-article version="4.5.2" docsubtype="fla"><item-info><jid>INDAG</jid><aid>98800323</aid><ce:pii>S0019-3577(98)80032-3</ce:pii><ce:doi>10.1016/S0019-3577(98)80032-3</ce:doi><ce:copyright type="unknown" year="1998"></ce:copyright></item-info><head><ce:title>Universal entire functions with gap power series</ce:title><ce:author-group><ce:author><ce:given-name>Wolfgang</ce:given-name><ce:surname>Luh</ce:surname><ce:cross-ref refid="AFF1"><ce:sup>1</ce:sup></ce:cross-ref></ce:author><ce:author><ce:given-name>Valeri A.</ce:given-name><ce:surname>Martirosian</ce:surname><ce:cross-ref refid="AFF2"><ce:sup>2</ce:sup></ce:cross-ref><ce:cross-ref refid="FN1"><ce:sup>∗</ce:sup></ce:cross-ref></ce:author><ce:author><ce:given-name>Jürgen</ce:given-name><ce:surname>Müller</ce:surname><ce:cross-ref refid="AFF1"><ce:sup>1</ce:sup></ce:cross-ref></ce:author><ce:affiliation id="AFF1"><ce:label>a</ce:label><ce:textfn>Fachbereich IV, Universität Trier, D-54286 Trier, Germany</ce:textfn></ce:affiliation><ce:affiliation id="AFF2"><ce:label>b</ce:label><ce:textfn>Department of Mathematics, Yerevan State University, Alex Manoogian St. 1, Yerevan 375049, Armenia</ce:textfn></ce:affiliation><ce:footnote id="FN1"><ce:label>∗</ce:label><ce:note-para>The research work of the second author has been supported by the German Academic Exchange Service (DAAD).</ce:note-para></ce:footnote></ce:author-group><ce:miscellaneous>Communicated by Prof. J. Korevaar at the meeting of September 29, 1997</ce:miscellaneous><ce:abstract><ce:section-title>Abstract</ce:section-title><ce:abstract-sec><ce:simple-para>Let <math altimg="si1.gif"><sc>M</sc></math> be the family of all compact sets in <math altimg="si2.gif"><sc>C</sc></math> which have connected complement. For <ce:italic>K</ce:italic> <ce:italic>ϵ</ce:italic> <ce:italic>M</ce:italic> we denote by <ce:italic>A</ce:italic>(<ce:italic>K</ce:italic>) the set of all functions which are continuous on <ce:italic>K</ce:italic> and holomorphic in its interior.</ce:simple-para><ce:simple-para>Suppose that {z<ce:inf>n</ce:inf>} is any unbounded sequence of complex numbers and let <ce:italic>Q</ce:italic> be a given sub-sequence of <math altimg="si3.gif"><sc>N</sc></math><ce:inf>0</ce:inf>.</ce:simple-para><ce:simple-para>If <ce:italic>Q</ce:italic> has density <ce:italic>Δ</ce:italic>(<ce:italic>Q</ce:italic>) = 1 then there exists a universal entire function ϑ with lacunary power series <ce:list><ce:list-item><ce:label>1.</ce:label><ce:para>(1) <ce:italic>ϑ</ce:italic>(<ce:italic>z</ce:italic>) = <ce:italic>ϵ</ce:italic><ce:sup>∞</ce:sup><ce:inf><ce:italic>v</ce:italic> = 0</ce:inf> <ce:italic>ϑ</ce:italic><ce:inf><ce:italic>v</ce:italic></ce:inf><ce:italic>Z</ce:italic><ce:sup><ce:italic>v</ce:italic></ce:sup>, <ce:italic>ϑ</ce:italic><ce:inf><ce:italic>v</ce:italic></ce:inf> = 0 for <ce:italic>v</ce:italic> ∉ <ce:italic>Q</ce:italic>, which has for all <ce:italic>K</ce:italic> <ce:italic>ϵ</ce:italic> <ce:italic>M</ce:italic> the following properties simultaneously</ce:para></ce:list-item><ce:list-item><ce:label>2.</ce:label><ce:para>(2) the sequence {<ce:italic>ϑ</ce:italic>(<ce:italic>Z</ce:italic> + <ce:italic>Z</ce:italic><ce:inf><ce:italic>n</ce:italic></ce:inf>)} is dense in <ce:italic>A</ce:italic>(<ce:italic>K</ce:italic>)</ce:para></ce:list-item><ce:list-item><ce:label>3.</ce:label><ce:para>(3) the sequence {<ce:italic>ϑ</ce:italic> (<ce:italic>ZZ</ce:italic><ce:inf><ce:italic>n</ce:italic></ce:inf>)} is dense in <ce:italic>A</ce:italic>(<ce:italic>K</ce:italic>) if 0 ∉ <ce:italic>K</ce:italic>.</ce:para></ce:list-item></ce:list></ce:simple-para><ce:simple-para>Also a converse result is proved: If ϑ is an entire function of the form (1) which satisfies (3), then <ce:italic>Q</ce:italic> must have maximal density <ce:italic>Δ</ce:italic><ce:inf><ce:italic>max</ce:italic></ce:inf>(<ce:italic>Q</ce:italic>) = 1.</ce:simple-para></ce:abstract-sec></ce:abstract></head></converted-article></istex:document></istex:metadataXml><mods version="3.6"><titleInfo><title>Universal entire functions with gap power series</title></titleInfo><titleInfo type="alternative" contentType="CDATA"><title>Universal entire functions with gap power series</title></titleInfo><name type="personal"><namePart type="given">Wolfgang</namePart><namePart type="family">Luh</namePart><affiliation>Fachbereich IV, Universität Trier, D-54286 Trier, Germany</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Valeri A.</namePart><namePart type="family">Martirosian</namePart><affiliation>Department of Mathematics, Yerevan State University, Alex Manoogian St. 1, Yerevan 375049, Armenia</affiliation><affiliation>∗ The research work of the second author has been supported by the German Academic Exchange Service (DAAD).</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Jürgen</namePart><namePart type="family">Müller</namePart><affiliation>Fachbereich IV, Universität Trier, D-54286 Trier, Germany</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="research-article" displayLabel="Full-length article"></genre><originInfo><publisher>ELSEVIER</publisher><dateIssued encoding="w3cdtf">1998</dateIssued><copyrightDate encoding="w3cdtf">1998</copyrightDate></originInfo><language><languageTerm type="code" authority="iso639-2b">eng</languageTerm><languageTerm type="code" authority="rfc3066">en</languageTerm></language><physicalDescription><internetMediaType>text/html</internetMediaType></physicalDescription><abstract lang="en">Let M be the family of all compact sets in C which have connected complement. For K ϵ M we denote by A(K) the set of all functions which are continuous on K and holomorphic in its interior. Suppose that {zn} is any unbounded sequence of complex numbers and let Q be a given sub-sequence of N0. If Q has density Δ(Q) = 1 then there exists a universal entire function ϑ with lacunary power series 1. (1) ϑ(z) = ϵ∞v = 0 ϑvZv, ϑv = 0 for v ∉ Q, which has for all K ϵ M the following properties simultaneously 2. (2) the sequence {ϑ(Z + Zn)} is dense in A(K) 3. (3) the sequence {ϑ (ZZn)} is dense in A(K) if 0 ∉ K. Also a converse result is proved: If ϑ is an entire function of the form (1) which satisfies (3), then Q must have maximal density Δmax(Q) = 1.</abstract><note>Communicated by Prof. J. Korevaar at the meeting of September 29, 1997</note><relatedItem type="host"><titleInfo><title>Indagationes Mathematicae</title></titleInfo><titleInfo type="abbreviated"><title>INDAG</title></titleInfo><genre type="journal">journal</genre><originInfo><dateIssued encoding="w3cdtf">19981221</dateIssued></originInfo><identifier type="ISSN">0019-3577</identifier><identifier type="PII">S0019-3577(00)X0015-8</identifier><part><date>19981221</date><detail type="volume"><number>9</number><caption>vol.</caption></detail><detail type="issue"><number>4</number><caption>no.</caption></detail><extent unit="issue pages"><start>477</start><end>636</end></extent><extent unit="pages"><start>529</start><end>536</end></extent></part></relatedItem><identifier type="istex">FF33B736421F5A07D6796054DEFBAAD73A8EFDB3</identifier><identifier type="DOI">10.1016/S0019-3577(98)80032-3</identifier><identifier type="PII">S0019-3577(98)80032-3</identifier><recordInfo><recordContentSource>ELSEVIER</recordContentSource></recordInfo></mods></metadata></istex></record>Pour manipuler ce document sous Unix (Dilib)
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