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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2024 MCQ for Mathematics of Computation is 1.78.

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On Shapiro’s cyclic inequality for $N=13$
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by B. A. Troesch PDF
Math. Comp. 45 (1985), 199-207 Request permission

Abstract:

A cyclic sum $S({\mathbf {x}}) = \Sigma \;{x_i}/({x_{i + 1}} + {x_{i + 2}})$ is formed with the N components of a vector x, where ${x_{N + 1}} = {x_1}$, ${x_{N + 2}} = {x_2}$, and where all denominators are positive and all numerators are nonnegative. It is known that there exist vectors x for which $S({\mathbf {x}}) < N/2$ if $N \geqslant 14$ and even, and if $N \geqslant 25$. It has been proved that the inequality $S({\mathbf {x}}) \geqslant N/2$ holds for $N \leqslant 12$. Although it has been conjectured repeatedly that the inequality also holds for odd N between 13 and 23. this has apparently not yet been proved. Here we will confirm that the inequality indeed holds for $N = 13$.
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Additional Information
  • © Copyright 1985 American Mathematical Society
  • Journal: Math. Comp. 45 (1985), 199-207
  • MSC: Primary 26D15; Secondary 05A20
  • DOI: https://doi.org/10.1090/S0025-5718-1985-0790653-0
  • MathSciNet review: 790653