UA Research Summary No. 12

Recent reports on higher education in the U.S. say it’s in trouble— that it’s too expensive, doesn’t offer enough need-based aid, isn’t educating people for today’s jobs, doesn’t demand enough of instructors or students, and isn’t sufficiently accountable to policymakers and taxpayers.1 Is the University of Alaska (UA)—the state’s only public university —offering a good, affordable education for Alaskans? This paper looks at that question. It first presents the available data on various measures and then summarizes successes and continuing challenges for UA. It ends with a discussion of how UA and the state are addressing higher-education issues and what other steps they might consider.University of Alaska Foundatio

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 dispersionComment: 7 page

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.Comment: 6 pages, no figure

Darboux transformations for differential operators on the superline

We give a full description of Darboux transformations of any order for arbitrary (nondegenerate) differential operators on the superline. We show that every Darboux transformation of such operators factorizes into elementary Darboux transformations of order one. Similar statement holds for operators on the ordinary line

Regularity of Digits and Significant Digits of Random Variables

A random variable X is digit-regular (respectively, significant-digit-regular) if the probability that every block of k given consecutive digits (significant digits) appears in the b-adic expansion of X approaches b &supk; as the block moves to the right, for all integers b > 1 and k ? 1. Necessary and sufficient conditions are established, in terms of convergence of Fourier coefficients, and in terms of convergence in distribution modulo 1, for a random variable to be digit-regular (significant-digit regular), and basic relationships between digit-regularity and various classical classes of probability measures and normal numbers are given. These results provide a theoretical basis for analyses of roundoff errors in numerical algorithms which use floating-point arithmetic, and for detection of fraud in numerical data via using goodness-of-fit of the least significant digits to uniform, complementing recent tests for leading significant digits based on Benford's law.normal numbers, significant digits, Benford's law, digit-regular random variable, significant-digit-regular random variable, law of least significant digits, floating-point numbers, nonleading digits, trailing digits

Rulers in Greek tragedy

This thesis discusses the depiction of rulers in Aeschylus, Sophocles, and Euripides. It aims to demonstrate the complexity and variety both of the ideas underlying the presentation of these rulers (Part One) and of how the playwrights use these ideas to depict these rulers (Part Two). Chapter 1 presents the first detailed study of the features shared between rulers in Herodotus and tragedy. Chapter 2 provides a fresh analysis of how the idea of tyrannis was used in Athenian political and intellectual life, before examining how tragedy fits into this context. Chapter 3 examines how the depiction of tragic rulers draws on ideas current in the Athenian democracy, especially concerning leadership and obedience. Chapter 4 argues that, since many tragic plots revolve around supplication and punishment, the depiction of tragic rulers is greatly affected both by the plays’ plots and by contemporary conceptions of supplication and punishment. After setting out a framework for discussing characters in tragedy (Introduction to Part Two), there follow case studies on five rulers. Chapter 5, on Aeschylus, argues that Eteocles (Seven Against Thebes) can best be understood in light of certain conceptions concerning the behaviour of warriors in epic, and that the presentation of Agamemnon (Agamemnon) draws on ideas about basileia current in the early fifth century. Chapter 6 presents a comparison of Sophocles’ Creon (Antigone) and Oedipus (Oedipus Tyrannus), exploring how Sophocles uses similar underlying ideas to depict them, while nevertheless presenting them differently in moral terms. Chapter 7, on Pentheus in Euripides’ Bacchae, considers how Euripides’ technique in presenting him resembles Sophocles’. The thesis provides many novel observations about fifth-century political thought, about the relationship of tragedy to its political and intellectual context and to other genres (epic, lyric, historiography), and about the tragic playwrights’ technique in presenting characters