Skip to main content
SpringerLink
Log in

Alexander Duality and Moments in Reliability Modelling

  • Published:
Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract.

There are strong connections between coherent systems in reliability for systems which have components with a finite number of states and certain algebraic structures. A special case is binary systems where there are two states: fail and not fail. The connection relies on an order property in the system and a way of coding states \({{\alpha=(\alpha_1, \ldots, \alpha_d)}}\) with monomials \({{x^\alpha=(x_1^{{\alpha_1}}, \ldots, x_d^{{\alpha_d}})}}\). The algebraic entities are the Scarf complex and the associated Alexander duality. The failure ‘‘event’’ can be studied using these ideas and identities and bounds derived when the algebra is combined with probability distributions on the states. The x α coding aids the use of moments μα=E(X α) with respect to the underlying distribution.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bayer, D., Peeva, I., Sturmfels, B.: Monomial resolutions. Math. Res. Lett. 5, 31–46 (1998)

    MathSciNet  MATH  Google Scholar 

  2. Dohmen, K.: An improvement of the inclusion-exclusion principle. Arch. Math. 72(4), 298–303 (1999a)

    Article  MATH  Google Scholar 

  3. Dohmen, K.: Improved inclusion-exclusion identities and Inequalities based on a particular class of abstract tubes. Electr. J. Probab. 4(5), 12 (electronic), (1999b)

    MATH  Google Scholar 

  4. Dohmen, K.: Improved Inclusion-Exclusion Identities and Bonferroni Inequalities with Applications to Reliability Analysis of Coherent Systems. Habilitationsschrift, University of Humboldt, Berlin, 2000

  5. Fröberg, R.: An introduction to Gröbner Bases. New York: Wiley, 1997

  6. Giglio, B., Naiman, D., Wynn, H. P.: Abstract tube theory and Gröbner bases for improved inclusion-exclusion reliability bounds. In Laboratoire Statistique Mathématique et ses Applications, editor, MMR’2000 – Second International Conference on Mathematical Methods in Reliability, Bordeaux, volume 1, pp. 451–454, Bordeaux, 2000. Univérsité Victor Segalen, Bordeaux 2

  7. Giglio, B.: Naiman, D. Q., Wynn, H. P.: Gröbner bases, abstract tubes, and inclusion-exclusion reliability bounds. IEEE Transactions on Reliability, 51(3), 358–366 (2002)

    Google Scholar 

  8. Giglio, B., Wynn, H. P.: Monomial Ideals and the Scarf Complex for Coherent Systems in Reliability Theory. (To appear in Ann. Statis. 2004)

  9. Miller, E.: The Alexander duality functors and local duality with monomial support. J. Algebra 231, 180–234 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Miller, E.: Resolutions of monomial ideals. PhD thesis, University of California at Berkeley, 2000

  11. Miller, E., Perkinson, D.: Eight Lectures on monomial ideals. To appear in Queen’s Papers in Pure and Applied Mathematics

  12. Naiman, D., Wynn, H. P.: Improved inclusion-exclusion inequalities for simplex and orthant arrangements. J. Inequal. Pure. App. Math. 2, 16 (2001)

    MATH  Google Scholar 

  13. Pistone G., Riccomagno, E. M., Wynn, H. P.: Algebraic Statistics. Chapman and Hall, CRC, 2001

  14. Shier, D. R.: Network Reliability and Algebraic Structures, Clarendon Press. Oxford Science Publications, 1991

Download references

Author information

Authors and Affiliations

  1. Financial Services Authority, 25, The North Colonnade, London, E14 5HS

    Beatrice Giglio

  2. Department of Statistics, London School of Economics, Houghton Street, London, WC2A 2AE, UK

    Henry P. Wynn

Authors
  1. Beatrice Giglio
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Henry P. Wynn
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Beatrice Giglio.

Additional information

Keywords: Reliability, Monomial ideals, Scarf complex, Alexander dual, Hilbert series, Moments

Rights and permissions

Reprints and permissions

About this article

Cite this article

Giglio, B., Wynn, H. Alexander Duality and Moments in Reliability Modelling. AAECC 14, 153–174 (2003). https://doi.org/10.1007/s00200-002-0115-z

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00200-002-0115-z

Keywords

Search

Navigation