arXiv subjects
Publications and source records attributed to Theodore P. Hill.
Without additional hypotheses, Proposition 7.1 in Brams and Taylor's book "Fair Division" (Cambridge University Press, 1996) is false, as are several related Pareto-optimality theorems of Brams, Jones and Klamler in their 2006 cake-cutting paper.
This article contains counterexamples to theorems and claims in Brams, Jones and Klamler's article "Better Ways to Cut a Cake" in the December 2006 Notices of the American Mathematical Society.
Fixing the value of Avogadro's constant, the number of atoms in 12 grams of carbon-12, at exactly 84446886^3 would imply that one gram is the mass of exactly 18x14074481^3 carbon-12 atoms. This new definition of the gram, and thereby also the kilogram, is precise, elegant and unchanging in time, unlike the current 118-year-old artifact kilogram in Paris and the proposed experimental definitions of the kilogram using man-made silicon spheres or the watt balance apparatus.
An exact value for Avodagro's number, namely NA* = (84446888)^3, is proposed. The number 84446888 represents the side length of a cube of atoms whose volume is closest, among all integral side lengths, to the current official NIST value of Avogadro's number. This value NA* is nearly dead center of the estimated range for the value of Avogadro's number, and is within the official standard level of uncertainty. Adoption of this value as the fixed value for NA would eliminate the current time-dependent definition of Avogadro's number, which depends on the definition of kilogram via an unstable physical artifact. It would also eliminate the need for the kilogram artifact altogether, since then, by definition, a kilogram would be exactly 1000/12 the mass of NA* atoms of carbon-12.