Ted Hill Academics

Academic CV

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Research Area I - The Significant Digit Phenomenon or Benford's Law

A century-old empirical observation now called Benford's Law says that the significant digits of many real datasets are logarithmically distributed, rather than uniformly distributed, as might be expected. This research includes discoveries that help explain the ubiquity of Benford data sets. For example, it has now been shown that iterations of many common functions (including all polynomials, power, exponential, and trigonometric functions, as well as compositions thereof), dynamical systems and differential equations, geometric Brownian motion (hence most stock market models), mixtures of data from different sources, and Newton's method, all produce Benford sequences. These publications also include applications to fraud detection and to diagnostic tests for mathematical models, many examples, and open Benford-related problems in dynamical systems, probability, number theory, and differential equations. For American Scientist article on this topic, click here.

For a free searchable online bibliographic database on Benford's Law, see www.benfordonline.net.

Research Area II - Optimal Stopping Theory and Secretary Problems

In many basic processes in science (and in life), there is an element of chance involved, and a crucial problem is deciding when to stop. The process could be debugging large software programs, proofreading a paper, waiting to buy Google stocks, performing medical experiments, looking at houses to buy, gathering laboratory data, or interviewing for a new secretary (or spouse). The typical framework is that a sequence of random variables is being observed, and the objective is to decide when to stop in order to maximize the expected reward. These publications include basic "prophet inequalities" ( comparisons of the expected return of a gambler who has foresight, or inside information, with a gambler who does not' game-theoretic extensions to the classical secretary problem, and determiation of optimal rules and optimal bounds for generalized stopping. For American Scientist article on this topic, click here.

For the French version in Pour la Science, click here.

Research Area III - Fair Division Problems

The general subject of this research is the question of whether an object (such as a cake or piece of land) can be divided among a number of people so that each receives a portion he considers a fair share, according to his own values. (Formally, there are n measures on the same object - a measurable space - and a typical goal is to find a partition of the object into n pieces so that the minimum value of the i-th measure on the i-th piece is as large as possible). These publications include generalizations of Steinhaus'classical "Ham Sandwich Problem", Neyman and Pearson's "Bisection Problem", and Fisher's "Problem of the Nile", determination of optimal bounds and extreme-case measures, generalizations of Lyapounov's Convexity Theorem, and applications to disarmament, dividing inheritances, and lotterized allocations of indivisible goods. For American Scientist article on this topic, click here.

	"Weird" Science and Mathematical Miscellany
	Is the fundamental physical constant called "Avogadro's number" odd or even? For American Scientist article on this topic, click here.
	Is there a natural and purely mathematical definition of the kilogram? For article on this topic, click here.
	How can data from completely different experiments best be consolidated? For articles on this topic, click here and here.
	Can the classical Ham Sandwich Theorem of Steinhaus be strengthened to conclude that the knife will also spread mayonnaise on each ingredient? For article on this topic, click here.
	Raising questions like these is part of my current "recreational" scientific research, and has led to new collaborations, publications and a "Weird Science" award.

Letters to the Editor

Hoisting the Black Flag, Letter to the Editor, Notices of the AMS, January 2010.
Response to Letters to the Editor, American Scientist Online - "Gleaning the Cube", July-August 2007.
Response to Letters to the Editor, American Scientist Online - "Your Number's Up", May-June 2007. (Online link is incorrect.)
Review of The Golden Ratio, Letter to the Editor, Notices of the AMS, Vol. 52, no. 6, 2005.
Response to "Benfords Gesetz", Brief an die Herausgeber, Deutsche Mathematische Verein, Mitteilungen 3/2001, p. 4.