{"title":"Representations of the twisted SU(2) quantum group and some q-hypergeometric orthogonal polynomials","authors":"T.H. Koornwinder","doi":"10.1016/S1385-7258(89)80020-4","DOIUrl":"10.1016/S1385-7258(89)80020-4","url":null,"abstract":"<div><p>The matrix elements of the irreducible unitary representations of the twisted <em>SU</em>(2) quantum group are computed explicitly. It is shown that they can be identified with two different classes of <em>p</em>-hypergeometric orthogonal polynomials: with the little <em>q</em>-Jacobi polynomials and with certain <em>q</em>-analogues of Krawtchouk polynomials. The orthogonality relations for these polynomials correspond to Schur type orthogonality relations in the first case and to the unitarity conditions for the representations in the second case. The paper also contains a new proof of Woronowicz' classification of the unitary irreducible representations of this quantum group. It avoids infinitesimal methods. Symmetries of the matrix elements of the irreducible unitary representations are proved without using the explicit expressions.</p></div>","PeriodicalId":100664,"journal":{"name":"Indagationes Mathematicae (Proceedings)","volume":"92 1","pages":"Pages 97-117"},"PeriodicalIF":0.0,"publicationDate":"1989-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S1385-7258(89)80020-4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56486212","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the non-rationality of the Fano variety of ℙ5, which contains three planes two by two meeting in one point","authors":"Elisabetta Ambrogio , Daniela Romagnoli","doi":"10.1016/S1385-7258(89)80012-5","DOIUrl":"10.1016/S1385-7258(89)80012-5","url":null,"abstract":"<div><p>We prove that the Fano variety <em>W</em> of degree 6 in ℙ<sup>5</sup>, complete intersection of a smooth quadric hypersurface with a smooth cubic hypersurface of ℙ<sup>5</sup>, is not rational if it contains 3 planes two by two meeting at least in one point.</p></div>","PeriodicalId":100664,"journal":{"name":"Indagationes Mathematicae (Proceedings)","volume":"92 1","pages":"Pages 15-19"},"PeriodicalIF":0.0,"publicationDate":"1989-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S1385-7258(89)80012-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"97010689","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A characterization of Riesz spaces which are Riesz isomorphic to C(X) for some completely regular space X","authors":"Hong-Yun Xiong","doi":"10.1016/S1385-7258(89)80019-8","DOIUrl":"10.1016/S1385-7258(89)80019-8","url":null,"abstract":"<div><p>Let <em>E</em> be an Archimedean Riesz space possessing a weak unit <em>e</em> and let <em>Ω</em> be the collection of all Riesz homomorphisms <em>ø</em> from <em>E</em> onto ℝ such that <em>ø</em>(<em>e</em>)=1. The Gelfand mapping <em>G</em> :<em>x</em>→<em>x</em>^ on <em>E</em> is defined by <em>x</em>^(<em>ø</em>) = <em>ø</em>(<em>x</em>) for all <em>ø</em>∈Ω. We endow Ω with the topology induced by <em>E</em> (i.e., the weakest topology such that each <em>x</em>^ is continuous on <em>Ω</em>). The principal ideal in <em>E</em> generated by <em>e</em> is denoted by <em>I<sub>d</sub></em>(<em>e</em>). The main theorem in this paper says that the following statements (A) and (B) are equivalent.</p><ul><li><span>(A)</span><span><p>There exists a completely regular space <em>X</em> such that <em>E</em> is Riesz isomorphic to the space <em>C</em>(<em>X</em>) of all real continuous functions on <em>X</em>.</p></span></li><li><span>(B)</span><span><p>The following conditions for the Riesz space <em>E</em> hold: (1) <em>E</em> is Archimedean and has a weak unit <em>e</em>; (2) <em>Ω</em> separates the points of <em>E</em>; (3) <em>E</em> is uniformly complete; (4) <em>G</em>(<em>I<sub>d</sub></em>(<em>e</em>)) is norm dense in the space <em>C<sub>b</sub></em>(<em>Ω</em>) of all real bounded continuous functions on <em>Ω</em>; (5) <em>E</em> is 2-universally complete with carrier space <em>Ω</em>.</p></span></li></ul><p>Some other conditions are mentioned and an example is given to show that condition (5) is necessary for (B) ⇒(A).</p></div>","PeriodicalId":100664,"journal":{"name":"Indagationes Mathematicae (Proceedings)","volume":"92 1","pages":"Pages 87-95"},"PeriodicalIF":0.0,"publicationDate":"1989-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S1385-7258(89)80019-8","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"108379557","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Polynomial congruences over incomplete residue systems, modulo k","authors":"J.H.H. Chalk","doi":"10.1016/S1385-7258(89)80016-2","DOIUrl":"10.1016/S1385-7258(89)80016-2","url":null,"abstract":"<div><p>This is a sequel to the previous article (Proceedings Kon. Ned. Akad. van Wetensch., <strong>A 83</strong> (4), (1980), 367–374, see MR, 82d: 10053, 10G10.) on the Mordell and Tietäväinen inequalities for the distribution of zeros of polynomial congruences in incomplete residue systems <em>modulo k</em>. Here the emphasis is on composite k and applies to a general class of polynomials satisfying mild conditions of non-degeneracy for each prime <em>p</em>∣<em>k</em>.</p></div>","PeriodicalId":100664,"journal":{"name":"Indagationes Mathematicae (Proceedings)","volume":"92 1","pages":"Pages 49-62"},"PeriodicalIF":0.0,"publicationDate":"1989-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S1385-7258(89)80016-2","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"96873186","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Derative of analytic elements on infraconnected clopen sets","authors":"Alain Escassut","doi":"10.1016/S1385-7258(89)80017-4","DOIUrl":"10.1016/S1385-7258(89)80017-4","url":null,"abstract":"","PeriodicalId":100664,"journal":{"name":"Indagationes Mathematicae (Proceedings)","volume":"92 1","pages":"Pages 63-70"},"PeriodicalIF":0.0,"publicationDate":"1989-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S1385-7258(89)80017-4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"112164102","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A characterization of dual Banach lattices","authors":"V. Caselles","doi":"10.1016/S1385-7258(89)80015-0","DOIUrl":"10.1016/S1385-7258(89)80015-0","url":null,"abstract":"<div><p>In this paper we give a characterization of dual Banach lattices. In fact, we prove that a Banach function space <em>E</em> on a separable measure space which has the Fatou property is a dual Banach lattice if and only if all positive operators from <em>L</em><sup>1</sup>(0,1) into <em>E</em> are abstract kernel operators, hence extending the fact, proved by M. Talagrand, that separable Banach lattices with the Radon-Nikodym property are dual Banach lattices.</p></div>","PeriodicalId":100664,"journal":{"name":"Indagationes Mathematicae (Proceedings)","volume":"92 1","pages":"Pages 35-47"},"PeriodicalIF":0.0,"publicationDate":"1989-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S1385-7258(89)80015-0","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"106846702","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}