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Integer solutions to interval linear equations and unique measurement


Author: Peter Fishburn
Journal: Proc. Amer. Math. Soc. 129 (2001), 1595-1599
MSC (2000): Primary 05A99, 11D04, 91E45
DOI: https://doi.org/10.1090/S0002-9939-00-05947-5
Published electronically: November 15, 2000
MathSciNet review: 1814085
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Abstract | References | Similar Articles | Additional Information

Abstract:

Every system of $n$ linearly independent homogeneous linear equations in $n+1$ unknowns with coefficients in $\{1,0,-1\}$ has a unique (up to multiplication by $-1$) non-zero solution vector $d= (d_1, d_2, \ldots , d_{n+1} )$ in which the $d_j$'s are integers with no common divisor greater than 1. It is known that, for large $n$, $\vert \sum d_j \vert$ can be arbitrarily greater than $2^n$. We prove that if every equation, written as $\sum_A x_i - \sum_B x_i =0$, is such that $A$ and $B$ are intervals of contiguous indices, then $\vert\sum d_j \vert \le 2^n$. This confirms conjectures of the author and Fred Roberts that arose in the theory of unique finite measurement.


References [Enhancements On Off] (What's this?)

  • 1. P. C. Fishburn, Utility Theory for Decision Making, Wiley, New York, 1970. MR 41:9401
  • 2. P. C. Fishburn, H. M. Marcus-Roberts, and F. S. Roberts, Unique finite difference measurement, SIAM J. Disc. Math., 1 (1988), pp. 334-354. MR 90d:05029
  • 3. P. C. Fishburn and A. M. Odlyzko, Unique subjective probability on finite sets, J. Ramanujan Math. Soc., 4 (1989), pp. 1-23. MR 90k:60062
  • 4. P. C. Fishburn and F. S. Roberts, Uniqueness in finite measurement, in Applications of Combinatorics and Graph Theory in the Biological and Social Sciences, F. S. Roberts, ed., Springer-Verlag, Berlin, 1989, pp. 103-137. MR 90e:92099
  • 5. F. B. Hildebrand, Methods of Applied Mathematics, Prentice-Hall, Englewood Cliffs, NJ, 1952. MR 15:204b
  • 6. D. H. Krantz, R. D. Luce, P. Suppes, and A. Tversky, Foundations of Measurement, Vol. I, Academic Press, New York, 1971. MR 56:17265
  • 7. S. Mac Lane and G. Birkhoff, Algebra, Macmillan, London, 1967. MR 35:5266
  • 8. F. S. Roberts, Measurement Theory, with Applications to Decisionmaking, Utility, and the Social Sciences, Addison-Wesley, Reading, MA, 1979. MR 81b:90003

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Additional Information

Peter Fishburn
Affiliation: AT&T Laboratories, Room C227, 180 Park Avenue, Florham Park, New Jersey 07932
Email: fish@research.att.com

DOI: https://doi.org/10.1090/S0002-9939-00-05947-5
Keywords: Linear equations, integer solutions, measurement theory
Received by editor(s): September 14, 1999
Published electronically: November 15, 2000
Communicated by: Mark J. Ablowitz
Article copyright: © Copyright 2000 American Mathematical Society

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