Theodore P. Hillhttps://works.bepress.com/tphill/Recent works by Theodore P. Hillen-usCopyright (c) 2019 All rights reserved.Mon, 01 Aug 2011 07:00:00 +00003600Criticisms of the proposed “new SI”https://works.bepress.com/tphill/77/Mon, 01 Aug 2011 07:00:00 +0000https://works.bepress.com/tphill/77/ArticlesA Stronger Conclusion to the Classical Ham Sandwich Theoremhttps://works.bepress.com/tphill/70/The conclusion of the classical ham sandwich theorem for bounded Borel sets may be strengthened, without additional hypotheses – there always exists a common bisecting hyperplane that touches each of the sets, that is, that intersects the closure of each set. In the discrete setting, where the sets are finite (and the measures are counting measures), there always exists a bisecting hyperplane that contains at least one point in each of the sets. Both these results follow from the main theorem of this note, which says that for n compactly supported positive finite Borel measures in Rn, there is always an (n − 1)-dimensional hyperplane that bisects each of the measures and intersects the support of each measure. Thus, for example, at any given instant of time, there is one planet, one moon and one asteroid in our solar system and a single plane touching all three that exactly bisects the total planetary mass, the total lunar mass, and the total asteroidal mass of the solar system. In contrast to the bisection conclusion of the classical ham sandwich theorem, this bisection-and-intersection conclusion does not carry over to unbounded sets of finite measure.John H. Elton et al.Fri, 01 Jul 2011 07:00:00 +0000https://works.bepress.com/tphill/70/ArticlesFinite-state Markov Chains Obey Benford’s Lawhttps://works.bepress.com/tphill/80/<p>A sequence of real numbers (xn) is Benford if the significands, i.e., the fraction parts in the floating-point representation of (xn), are distributed logarithmically. Similarly, a discrete-time irreducible and aperiodic finite-state Markov chain with transition probability matrix P and limiting matrix P* is Benford if every component of both sequences of matrices (Pn−P*) and (Pn+1−Pn) is Benford or eventually zero. Using recent tools that established Benford behavior for finite-dimensional linear maps, via the classical theories of uniform distribution modulo 1 and Perron–Frobenius, this paper derives a simple sufficient condition (“nonresonance”) guaranteeing that P, or the Markov chain associated with it, is Benford. This result in turn is used to show that almost all Markov chains are Benford, in the sense that if the transition probability matrix is chosen in an absolutely continuous manner, then the resulting Markov chain is Benford with probability one. Concrete examples illustrate the various cases that arise, and the theory is complemented with simulations and potential applications.</p>
Arno Berger et al.Fri, 01 Jul 2011 00:00:00 +0000https://works.bepress.com/tphill/80/ArticlesConflations of Probability Distributionshttps://works.bepress.com/tphill/79/<p>The conflation of a finite number of probability distributions P1,...,Pn is a consolidation of those distributions into a single probability distribution Q = Q(P1,...,Pn), where intuitively Q is the conditional distribution of independent random variables X1,...,Xn with distributions P1,...,Pn, respectively, given that X1 = ···= Xn. Thus, in large classes of distributions the conflation is the distribution determined by the normalized product of the probability density or probability mass functions. Q is shown to be the unique probability distribution that minimizes the loss of Shannon information in consolidating the combined information from P1,...,Pn into a single distribution Q, and also to be the optimal consolidation of the distributions with respect to two minimax likelihood-ratio criteria. In that sense, conflation may be viewed as an optimal method for combining the results from several different independent experiments. When P1,...,Pn are Gaussian, Q is Gaussian with mean the classical weighted-mean-squares reciprocal of variances. A version of the classical convolution theorem holds for conflations of a large class of a.c. measures.</p>
Theodore P. HillWed, 01 Jun 2011 00:00:00 +0000https://works.bepress.com/tphill/79/ArticlesTowards a Better Definition of the Kilogramhttps://works.bepress.com/tphill/72/It is widely accepted that improvement of the current International System of Units (SI) is necessary, and that central to this problem is redefinition of the kilogram. This paper compares the relative advantages of two main proposals for a modern scientific definition of the kilogram: an ‘electronic kilogram’ based on a fixed value of Planck’s constant, and an ‘atomic kilogram’ based on a fixed value for Avogadro’s number. A concrete and straightforward atomic definition of the kilogram is proposed. This definition is argued to be more experimentally neutral than the electronic kilogram, more realizable by school and university laboratories than the electronic kilogram, and more readily comprehensible than the electronic kilogram.Tue, 01 Mar 2011 08:00:00 +0000https://works.bepress.com/tphill/72/ArticlesBenford’s Law Strikes Back: No Simple Explanation in Sight for Mathematical Gemhttps://works.bepress.com/tphill/74/Arno Berger et al.Tue, 01 Mar 2011 00:00:00 +0000https://works.bepress.com/tphill/74/ArticlesA Basic Theory of Benford’s Lawhttps://works.bepress.com/tphill/78/Drawing from a large, diverse body of work, this survey presents a comprehensive and unified introduction to the mathematics underlying the prevalent logarithmic distribution of significant digits and significands, often referred to as Benford’s Law (BL) or, in a special case, as the First Digit Law. The invariance properties that characterize BL are developed in detail. Special attention is given to the emergence of BL in a wide variety of deterministic and random processes. Though mainly expository in nature, the article also provides strengthened versions of, and simplified proofs for, many key results in the literature. Numerous intriguing problems for future research arise naturally.Arno Berger et al.Sat, 01 Jan 2011 08:00:00 +0000https://works.bepress.com/tphill/78/ArticlesCutting Cakes Carefullyhttps://works.bepress.com/tphill/73/Theodore P Hill et al.Wed, 01 Sep 2010 07:00:00 +0000https://works.bepress.com/tphill/73/ArticlesFundamental Flaws in Feller’s Classical Derivation of Benford’s Lawhttps://works.bepress.com/tphill/75/Feller’s classic text An Introduction to Probability Theory and its Applications contains a derivation of the well known significant-digit law called Benford’s law. More specifically, Fellergives a sufficient condition (“large spread”) for a random variable X to be approximately Benford distributed, that is, for log10X to be approximately uniformly distributed moduloone. This note shows that the large-spread derivation, which continues to be widely cited and used, contains serious basic errors. Concrete examples and a new inequality clearly demonstratethat larges pread (or large spread on a logarithmic scale) does not imply that a random variable is approximately Benford distributed, for any reasonable definition of “spread” or measure of dispersion.Fri, 14 May 2010 07:00:00 +0000https://works.bepress.com/tphill/75/ArticlesObituary for Lester Eli Dubins, 1921-2010https://works.bepress.com/tphill/76/David Gilat et al.Sat, 01 May 2010 07:00:00 +0000https://works.bepress.com/tphill/76/Articles