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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Published electronically: November 15, 2000
MathSciNet review: 1814085
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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.


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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: http://dx.doi.org/10.1090/S0002-9939-00-05947-5
PII: S 0002-9939(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