#### Filter Results:

#### Publication Year

1987

2012

#### Publication Type

#### Co-author

#### Key Phrase

#### Publication Venue

Learn More

Near a stable fixed point at 0 or ∞, many real-valued dynamical systems follow Benford's law: under iteration of a map T the proportion of values in {x, T (x), T 2 (x),. .. , T n (x)} with mantissa (base b) less than t tends to log b t for all t in [1, b) as n → ∞, for all integer bases b > 1. In particular , the orbits under most power, exponential, and… (More)

- Theodore P Hill
- 2010

are probability measures on the same measurable space (2, Y). Then if all atoms of each Ai have mass a or less, there is a measurable partition Al,..., An of 2 so that pxi(Ai) 2 Vn(a) for all i = 1, ..., n, where Vn(.) is an explicitly given piecewise linear nonincreasing continuous function on [0,1]. Moreover, the bound Vn(a) is attained for all n and all… (More)

- John Elton, Theodore P Hill, Robert P Kertz
- 2010

Suppose fix,... ,fin are nonatomic probability measures on the same measurable space (S, S). Then there exists a measurable partition isi}"=i of 5 such that Pi(Si) > (n + 1 M)'1 for a11 i l,...,n, where M is the total mass of V?=i ßi (tne smallest measure majorizing each m). This inequality is the best possible for the functional M, and sharpens and… (More)

The pointwise limit S of a sequence of Stieltjes transforms (S n) of real Borel probability measures (P n) is itself the Stieltjes transform of a Borel p.m. P if and only if iy S(iy) → −1 as y → ∞, in which case P n converges to P in distribution. Applications are given to several problems in mathematical physics. probability measures. Lévy's classical… (More)

- Theodore P Hill
- 2008

A derivation of Benford's Law or the First-Digit Phenomenon is given assuming only base-invariance of the underlying law. The only base-invariant distributions are shown to be convex combinations of two extremal probabilities, one corresponding to point mass and the other a log-Lebesgue measure. The main tools in the proof are identification of an… (More)

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… (More)

This paper surveys the origin and development of what has come to be known as "prophet inequalities" in optimal stopping theory. Included is a review of all published work to date on these problems, in cl uding extensions and variations, descriptions and examples of the main proof techniques, and a list of a number of basic open problems.

- Theodore P Hill
- 2010

Introduction. Suppose n countries border on a region the ownership of which is in dispute (Fig. 1). Is there a way of partitioning the disputed territory so each country receives a single piece adjacent to itself which it considers at least 1/n the total value of the territory, even though different parts of the territory may be valued differently by… (More)

1. INTRODUCTION. In scientific calculations using digital computers and f1oating-point arithmetic, roundoff errors are inevitable, even with the most elementary of functions. For example, a very simple, hypothetical computer with only one decimal point precision, equipped with the IEEE Stand~d "Unbiase5!" Roundin& approximates the function f(x) = x 2 with a… (More)

- John Elton, Theodore P Hill
- 2008

The distance from the convex hull of the range of an n-dimensional vector-valued measure to the range of that measure is no more than an/2, where a is the largest (one-dimensional) mass of the atoms of the measure. The case a = 0 yields Lyapounov's Convexity Theorem; applications are given to the bisection problem and to the bang-bang principle of optimal… (More)