Skip to main content Accessibility help
×
Home

Integral p-adic Normal Matrices Satisfying the Incidence Equation

  • J. K. Goldhaber (a1)

Extract

The problem of arranging v elements into v sets in such a way that every set contains exactly k distinct elements and that every pair of sets has exactly λ = k(k — l)/(v — 1) elements in common, where 0 < » < k < v, is equivalent to finding a normal integral v by v matrix A such that AT A = B, where B is the v by v matrix having k in every position on the main diagonal and λ in all other positions (10). Utilizing the fact that for the existence of a λ, k, v design it is necessary that I (the v by v identity matrix) represent B rationally, (2) and (3) have proved the non-existence of certain λ, k, v designs. Neither of the proofs utilize the fact that it is necessary that A be normal. However, Albert (1) for the projective plane case and Hall and Ryser (5) for the general design proved that if there exists a rational A such that ATA = B then there exists a normal rational matrix satisfying the same equation. Thus the requirement of normality does not exclude any λ, k, v which were not previously excluded.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Integral p-adic Normal Matrices Satisfying the Incidence Equation
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Integral p-adic Normal Matrices Satisfying the Incidence Equation
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Integral p-adic Normal Matrices Satisfying the Incidence Equation
      Available formats
      ×

Copyright

References

Hide All
1. Albert, A.A., Rational normal matrices satisfying the incidence equation, Proc. Amer. Math. Soc, 4 (1953), 554–9.
2. Bruck, R.H. and Ryser, H.J., The nonexistence of certain finite projective planes, Can. J. Math., 1 (1949), 8893.
3. Chowla, S. and Ryser, H. J., Combinational problems, Can. J. Math., 2 (1950), 93–9.
4. Eichler, M., Quadratische formen und orthogonale gruppen (Berlin, 1952).
5. Hall, Marshall and Ryser, H.J., Normal completions of incidence matrices, Amer. J. Math., 76 (1954), 581–9.
6. Jones, B.W., A canonical quadratic form for the ring of 2-adic integers, Duke Math. J., 11 (1944), 715–27.
7. Jones, B.W., The arithmetic theory of quadratic forms, Carus Math. Monographs, 10 (1950).
8. Magnus, W., Ueber die Anzahl der in einem Geschlecht enthaltenen Klassen von positiv definiten quadratischen Formen, Math. Ann. 114 (1937), 465-75.
9. Mayer, A., Zurich naturf. Ges., 36 (1891), 241.
10. Ryser, H.J., Matrices with integer elements in combinational investigations, Amer. J. Math., 74 (1952), 769–73.
11. Siegel, C.L., Equivalence of quadratic forms, Amer. J. Math. 68 (1941), 658-80.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Integral p-adic Normal Matrices Satisfying the Incidence Equation

  • J. K. Goldhaber (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed