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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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The validity of Shapiro’s cyclic inequality
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by B. A. Troesch PDF
Math. Comp. 53 (1989), 657-664 Request permission

Abstract:

A cyclic sum ${S_N}({\mathbf {x}}) = \sum {{x_i}/({x_{i + 1}} + {x_{i + 2}})}$ is formed with N components of a vector x, where in the sum ${x_{N + 1}} = {x_1}$, ${x_{N + 2}} = {x_2}$, and where all denominators are positive and all numerators are nonnegative. It is known that there exist vectors x for which ${S_N}({\mathbf {x}}) < N/2$ if $N \geq 14$ and even, and if $N \geq 24$. It has been proved that the inequality ${S_N}({\mathbf {x}}) \geq N/2$ holds for $N \leq 13$. Although it has been conjectured repeatedly that the inequality also holds for odd N between 15 and 23, this has apparently never been proved. Here we will confirm that the inequality indeed holds for all odd $N \leq 23$. This settles the question for all N.
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Additional Information
  • © Copyright 1989 American Mathematical Society
  • Journal: Math. Comp. 53 (1989), 657-664
  • MSC: Primary 26D15
  • DOI: https://doi.org/10.1090/S0025-5718-1989-0983563-0
  • MathSciNet review: 983563