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Theodore P. Hill

Publications and source records attributed to Theodore P. Hill.

At least 19 recordsLinked to original sources
arXiv2025

A brief survey of Benford's Law in dynamical systems

This article provides a brief overview on a range of basic dynamical systems that conform to the logarithmic distribution of significant digits known as Benford's law. As presented here, most theorems are special cases of known, more general results about dynamical systems whose orbits or trajectories follow this logarithmic law, in one way or another. These results span a wide variety of systems: autonomous and non-autonomous; discrete- and continuous-time; one- and multi-dimensional; deterministic and stochastic. Illustrative examples include familiar systems such as the tent map, Newton's root-finding algorithm, and geometric Brownian motion. The treatise is informal, with the goal of showcasing to the specialists the generality and universal appeal of Benford's law throughout the mathematical field of dynamical systems. References to complete proofs are provided for each known result, while one new theorem is presented in some detail.

math.DS↗
arXiv2024

An Evolutionary Theory for the Variability Hypothesis

An elementary biostatistical theory based on a selectivity-variability principle is proposed to address a question raised by Charles Darwin, namely, how one sex of a sexually dimorphic species might tend to evolve with greater variability than the other sex. Briefly, the theory says that if one sex is relatively selective then from one generation to the next, more variable subpopulations of the opposite sex will generally tend to prevail over those with lesser variability. Moreover, the perhaps less intuitive converse also holds: if a sex is relatively non-selective, then less variable subpopulations of the opposite sex will prevail over those with greater variability. This theory requires certain regularity conditions on the distributions, but makes no assumptions about differences in means between the sexes, nor does it presume that one sex is selective and the other non-selective. Two mathematical models of the selectivity-variability principle are presented: a discrete-time one-step probabilistic model of short-term behavior with an example using normally distributed perceived fitness values; and a continuous-time deterministic model for the long-term asymptotic behavior of the expected sizes of the subpopulations with an example using exponentially distributed fitness levels.

q-bio.PE↗
arXiv2022

A Note on Over- and Under-Representation Among Populations with Normally-Distributed Traits

In every finite mixture of different normal distributions, there will always be exactly one of those distributions that not only is over-represented in the right tail of the mixture, but even completely overwhelms all other subpopulations in the rightmost tails. This property, although not unique to normal distributions, is not shared by other common continuous centrally-symmetric unimodal distributions such as Laplace, nor even by other bell-shaped distributions such as Cauchy (Lorentz) distributions.

math.PR↗
arXiv2021

On the Oval Shapes of Beach Stones

This article introduces a new geophysical theory, in the form of a single simple partial integro-differential equation, to explain how frictional abrasion alone of a stone on a planar beach can lead to the oval shapes observed empirically. The underlying idea in this theory is the intuitive observation that the rate of ablation at a point on the surface of the stone is proportional to the product of the curvature of the stone at that point and how often the stone is likely to be in contact with the beach at that point. Specifically, key roles in this new model are played by both the random wave process and the global (non-local) shape of the stone, i.e., its shape away from the point of contact with the beach. The underlying physical mechanism for this process is the conversion of energy from the wave process into potential energy of the stone. No closed-form or even asymptotic solution is known for the basic equation, even in a 2-dimensional setting, but basic numerical solutions are presented in both the deterministic continuous-time setting using standard curve-shortening algorithms, and a stochastic discrete-time polyhedral-slicing setting using Monte Carlo simulation.

cond-mat.soft↗
arXiv2021

Fundamental Errors in Kane and Mertz's Alleged Debunking of Greater Male Variability in Mathematics Performance

Kane and Mertz's 2012 AMS Notices article "Debunking Myths about Gender and Mathematics Performance" claims to have debunked the greater male variability hypothesis with respect to mathematics abilities. The logical and statistical arguments supporting their claim, however, which are being widely cited in the scientific literature, contain fundamental errors. The methodology is critically flawed, the main logical premise is false, and the article omits reference to numerous published scientific research articles that contradict its findings. Most critically, Kane and Mertz's final conclusion that their data are inconsistent with the greater male variability hypothesis is wrong. The goal of the present note is to correct the scientific record with respect to those claims. Most importantly, by publicizing these errors, the Notices will reduce the chance of similar future errors being repeated.

math.HO↗
arXiv2020

The Mathematics of Benford's Law -- A Primer

This article provides a concise overview of the main mathematical theory of Benford's law in a form accessible to scientists and students who have had first courses in calculus and probability. In particular, one of the main objectives here is to aid researchers who are interested in applying Benford's law, and need to understand general principles clarifying when to expect the appearance of Benford's law in real-life data and when not to expect it. A second main target audience is students of statistics or mathematics, at all levels, who are curious about the mathematics underlying this surprising and robust phenomenon, and may wish to delve more deeply into the subject. This survey of the fundamental principles behind Benford's law includes many basic examples and theorems, but does not include the proofs or the most general statements of the theorems; rather it provides precise references where both may be found.

math.ST↗
arXiv2015

Hubble's Law Implies Benford's Law for Distances to Galaxies

A recent article by Alexopoulos and Leontsinis presented empirical evidence that the first digits of the distances to galaxies are a reasonably good fit to the probabilities predicted by Benford's law, the well known logarithmic statistical distribution of significant digits. The purpose of the present article is to give a theoretical explanation, based on Hubble's law and mathematical properties of Benford's law, why galaxy distances might be expected to follow Benford's law. The new galaxy-distance law derived here, which is robust with respect to change of scale and base, to additive and multiplicative computational or observational errors, and to variability of the Hubble constant in both time and space, predicts that conformity to Benford's law will improve as more data on distances to galaxies becomes available. Conversely, with the logical derivation of this law presented here, the recent empirical observations may be viewed as independent evidence of the validity of Hubble's law.

physics.data-an↗
arXiv2012

Bayesian Posteriors Without Bayes' Theorem

The classical Bayesian posterior arises naturally as the unique solution of several different optimization problems, without the necessity of interpreting data as conditional probabilities and then using Bayes' Theorem. For example, the classical Bayesian posterior is the unique posterior that minimizes the loss of Shannon information in combining the prior and the likelihood distributions. These results, direct corollaries of recent results about conflations of probability distributions, reinforce the use of Bayesian posteriors, and may help partially reconcile some of the differences between classical and Bayesian statistics.

math.ST↗
arXiv2012

Gender Gaps in the Mathematical Sciences: The Creativity Factor

This article presents an overview, and recent history, of studies of gender gaps in the mathematically-intensive sciences. Included are several statistics about gender differences in science, and about public resources aimed at addressing them. We then examine the role that gender differences in creativity play in explaining the recent and current gender differences in the mathematical sciences, and identify several constructive suggestions aimed at improving analytical creativity output in research institutions.

math.HO↗
arXiv2011

Ham Sandwich with Mayo: A Stronger Conclusion to the Classical Ham Sandwich Theorem

The conclusion of the classical ham sandwich theorem of Banach and Steinhaus may be strengthened: there always exists a common bisecting hyperplane that touches each of the sets, that is, intersects the closure of each set. Hence, if the knife is smeared with mayonnaise, a cut can always be made so that it will not only simultaneously bisect each of the ingredients, but it will also spread mayonnaise on each. A discrete analog of this theorem says that n finite nonempty sets in n-dimensional Euclidean space can always be simultaneously bisected by a single hyperplane that contains at least one point in each set. More generally, for n compactly-supported positive finite Borel measures in Euclidean n-space, there is always a hyperplane that bisects each of the measures and intersects the support of each measure. 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.

math.MG↗
arXiv2011

How to Combine Independent Data Sets for the Same Quantity

This paper describes a recent mathematical method called conflation for consolidating data from independent experiments that are designed to measure the same quantity, such as Planck's constant or the mass of the top quark. Conflation is easy to calculate and visualize, and minimizes the maximum loss in Shannon information in consolidating several independent distributions into a single distribution. In order to benefit the experimentalist with a much more transparent presentation than the previous mathematical treatise, the main basic properties of conflation are derived in the special case of normal (Gaussian) data. Included are examples of applications to real data from measurements of the fundamental physical constants and from measurements in high energy physics, and the conflation operation is generalized to weighted conflation for situations when the underlying experiments are not uniformly reliable.

physics.data-an↗
arXiv2010

A Better Definition of the Kilogram

This article reviews several recent proposed redefinitions of the kilogram, and compares them with respect to practical realizations, uncertainties (estimated standard deviations), and educational aspects.

physics.data-an↗
arXiv2010

Fundamental Flaws in Feller's Classical Derivation of Benford's Law

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, Feller gives a sufficient condition ("large spread") for a random variable $X$ to be approximately Benford distributed, that is, for $\log_{10}X$ to be approximately uniformly distributed modulo one. 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 demonstrate that large spread (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

math.PR↗
arXiv2010

Finite-state Markov Chains obey Benford's Law

A sequence of real numbers (x_n) is Benford if the significands, i.e. the fraction parts in the floating-point representation of (x_n) are distributed logarithmically. Similarly, a discrete-time irreducible and aperiodic finite-state Markov chain with probability transition matrix P and limiting matrix P* is Benford if every component of both sequences of matrices (P^n - P*) and (P^{n+1}-P^n) is Benford or eventually zero. Using recent tools that established Benford behavior both for Newton's method and for finite-dimensional linear maps, via the classical theories of uniform distribution modulo 1 and Perron-Frobenius, this paper derives a simple sufficient condition (nonresonant) 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 probabilities are chosen independently and continuously, then the resulting Markov chain is Benford with probability one. Concrete examples illustrate the various cases that arise, and the theory is complemented with several simulations and potential applications.

math.PR↗
arXiv2009

Conflations of Probability Distributions

The conflation of a finite number of probability distributions P_1,..., P_n is a consolidation of those distributions into a single probability distribution Q=Q(P_1,..., P_n), where intuitively Q is the conditional distribution of independent random variables X_1,..., X_n with distributions P_1,..., P_n, respectively, given that X_1= ... =X_n. 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 P_1,..., P_n into a single distribution Q, and also to be the optimal consolidation of the distributions with respect to two minimax likelihood-ratio criteria. When P_1,..., P_n 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.

math.PR↗
arXiv2009

An optimal method to combine results from different experiments

This article describes an optimal method (conflation) to consolidate data from different experiments, and illustrates the advantages of conflation by graphical examples involving gaussian input distributions, and by a concrete numerical example involving the values of lattice spacing of silicon crystals used in determination of the current values of Planck's constant and the Avogadro constant.

physics.data-an↗
arXiv2009

Counterexamples in the theory of fair division

The formal mathematical theory of fair division has a rich history dating back at least to Steinhaus in the 1940's. In recent work in this area, several general classes of errors have appeared along with confusion about the necessity and sufficiency of certain hypotheses. It is the purpose of this article to correct the scientific record and to point out with concrete examples some of the pitfalls that have led to these mistakes. These examples may serve as guideposts for future work.

math.PR↗