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Algebra i Analiz, 2013, Volume 25, Issue 4, Pages 139–181 (Mi aa1348)  

This article is cited in 2 scientific papers (total in 2 papers)

Research Papers

Remarks on Hilbert identities, isometric embeddings, and invariant cubature

H. Nozakia, M. Sawab

a Department of Mathematics, Aichi University of Education, Igaya-cho, Kariya-city 448-8542, Japan
b Graduate School of Information Sciences, Nagoya University, Chikusa-ku, Nagoya 464-8601, Japan

Abstract: Victoir (2004) developed a method to construct cubature formulas with various combinatorial objects. Motivated by this, the authors generalize Victoir's method with one more combinatorial object, called regular $t$-wise balanced designs. Many cubature formulas of small indices with few points are provided, which are used to update Shatalov's table (2001) of isometric embeddings in small-dimensional Banach spaces, as well as to improve some classical Hilbert identities. A famous theorem of Bajnok (2007) on Euclidean designs invariant under the Weyl group of Lie type $B$ is extended to all finite irreducible reflection groups. A short proof of the Bajnok theorem is presented in terms of Hilbert identities.

Keywords: cubature formula, Hilbert identity, isometric embedding, Victoir method.

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English version:
St. Petersburg Mathematical Journal, 2014, 25:4, 615–646

Bibliographic databases:

Document Type: Article
Received: 05.04.2012
Language: English

Citation: H. Nozaki, M. Sawa, “Remarks on Hilbert identities, isometric embeddings, and invariant cubature”, Algebra i Analiz, 25:4 (2013), 139–181; St. Petersburg Math. J., 25:4 (2014), 615–646

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. M. Sawa, “On a symmetric representation of Hermitian matrices and its applications to graph theory”, J. Comb. Theory Ser. B, 116 (2016), 484–503  crossref  mathscinet  zmath  isi  scopus
    2. M. Sawa, M. Hirao, “Characterizing D-optimal rotatable designs with finite reflection groups”, Sankhya Ser. A, 79:1 (2017), 101–132  crossref  mathscinet  zmath  isi  scopus
  • Алгебра и анализ St. Petersburg Mathematical Journal
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