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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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Lower bounds for the disk packing constant
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by David W. Boyd PDF
Math. Comp. 24 (1970), 697-704 Request permission

Abstract:

An osculatory packing of a disk, $U$, is an infinite sequence of disjoint disks, $\{ {U_n}\}$, contained in $U$, chosen so that, for $n \geqq 2$, ${U_n}$ has the largest possible radius, ${r_n}$, of all disks fitting in $U\backslash ({U_1} \cup \cdots \cup {U_{n - 1}})$. The exponent of the packing, $S$, is the least upper bound of numbers, $t$, such that $\sum {r_n^t} = \infty$. Here, we present a number of methods for obtaining lower bounds for $S$, based on obtaining weighted averages of the curvatures of the ${U_n}$. We are able to prove that $S > 1.28467$. We use a number of well-known results from the analytic theory of matrices.
References
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Additional Information
  • © Copyright 1970 American Mathematical Society
  • Journal: Math. Comp. 24 (1970), 697-704
  • MSC: Primary 52.45; Secondary 40.00
  • DOI: https://doi.org/10.1090/S0025-5718-1970-0278193-8
  • MathSciNet review: 0278193