Composition of infinite-dimensional Dirac structures

Mikael Kurula, Arjan van der Schaft, Heiko J. Zwart

Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

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Abstract

In this paper, we define the Dirac structure and give some fundamental tools for its study.We then proceed by defining composition of ``split Dirac structures''. In the finite-dimensional case, composition of two Dirac structures always result in a new Dirac structure, but in the Hilbert space setting this result no longer holds. Thus, the problem of finding necessary and sufficient conditions for the composition of two infinite-dimensional Dirac structures to itself be a Dirac structure arises very naturally. The main result of this paper provides these necessary and sufficient conditions. In addition, we give examples and relate conposition of Dirac structures to the Redheffer star product of unitary operators.
Original languageUndefined
Title of host publicationProceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems
EditorsY Yamamoto
Place of PublicationKyoto
PublisherErasmus University Rotterdam
Pages27-32
Number of pages6
ISBN (Print)not assigned
Publication statusPublished - 28 Jul 2006
Event17th International Symposium on Mathematical Theory of Networks and Systems, MTNS 2006 - Kyoto, Japan
Duration: 24 Jul 200628 Jul 2006
Conference number: 17

Publication series

Name
Numbersupplement

Conference

Conference17th International Symposium on Mathematical Theory of Networks and Systems, MTNS 2006
Abbreviated titleMTNS
CountryJapan
CityKyoto
Period24/07/0628/07/06

Keywords

  • MSC-93C25
  • MSC-46C20
  • IR-66606
  • METIS-237614
  • EWI-8148

Cite this

Kurula, M., van der Schaft, A., & Zwart, H. J. (2006). Composition of infinite-dimensional Dirac structures. In Y. Yamamoto (Ed.), Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems (pp. 27-32). Kyoto: Erasmus University Rotterdam.
Kurula, Mikael ; van der Schaft, Arjan ; Zwart, Heiko J. / Composition of infinite-dimensional Dirac structures. Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems. editor / Y Yamamoto. Kyoto : Erasmus University Rotterdam, 2006. pp. 27-32
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title = "Composition of infinite-dimensional Dirac structures",
abstract = "In this paper, we define the Dirac structure and give some fundamental tools for its study.We then proceed by defining composition of ``split Dirac structures''. In the finite-dimensional case, composition of two Dirac structures always result in a new Dirac structure, but in the Hilbert space setting this result no longer holds. Thus, the problem of finding necessary and sufficient conditions for the composition of two infinite-dimensional Dirac structures to itself be a Dirac structure arises very naturally. The main result of this paper provides these necessary and sufficient conditions. In addition, we give examples and relate conposition of Dirac structures to the Redheffer star product of unitary operators.",
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month = "7",
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isbn = "not assigned",
publisher = "Erasmus University Rotterdam",
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}

Kurula, M, van der Schaft, A & Zwart, HJ 2006, Composition of infinite-dimensional Dirac structures. in Y Yamamoto (ed.), Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems. Erasmus University Rotterdam, Kyoto, pp. 27-32, 17th International Symposium on Mathematical Theory of Networks and Systems, MTNS 2006, Kyoto, Japan, 24/07/06.

Composition of infinite-dimensional Dirac structures. / Kurula, Mikael; van der Schaft, Arjan; Zwart, Heiko J.

Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems. ed. / Y Yamamoto. Kyoto : Erasmus University Rotterdam, 2006. p. 27-32.

Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

TY - GEN

T1 - Composition of infinite-dimensional Dirac structures

AU - Kurula, Mikael

AU - van der Schaft, Arjan

AU - Zwart, Heiko J.

PY - 2006/7/28

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N2 - In this paper, we define the Dirac structure and give some fundamental tools for its study.We then proceed by defining composition of ``split Dirac structures''. In the finite-dimensional case, composition of two Dirac structures always result in a new Dirac structure, but in the Hilbert space setting this result no longer holds. Thus, the problem of finding necessary and sufficient conditions for the composition of two infinite-dimensional Dirac structures to itself be a Dirac structure arises very naturally. The main result of this paper provides these necessary and sufficient conditions. In addition, we give examples and relate conposition of Dirac structures to the Redheffer star product of unitary operators.

AB - In this paper, we define the Dirac structure and give some fundamental tools for its study.We then proceed by defining composition of ``split Dirac structures''. In the finite-dimensional case, composition of two Dirac structures always result in a new Dirac structure, but in the Hilbert space setting this result no longer holds. Thus, the problem of finding necessary and sufficient conditions for the composition of two infinite-dimensional Dirac structures to itself be a Dirac structure arises very naturally. The main result of this paper provides these necessary and sufficient conditions. In addition, we give examples and relate conposition of Dirac structures to the Redheffer star product of unitary operators.

KW - MSC-93C25

KW - MSC-46C20

KW - IR-66606

KW - METIS-237614

KW - EWI-8148

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BT - Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems

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Kurula M, van der Schaft A, Zwart HJ. Composition of infinite-dimensional Dirac structures. In Yamamoto Y, editor, Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems. Kyoto: Erasmus University Rotterdam. 2006. p. 27-32