Chebyshev Action on Finite Fields

Given a polynomial f and a finite field F one can construct a directed graph where the vertices are the values in the finite field, and emanating from each vertex is an edge joining the vertex to its image under f. When f is a Chebyshev polynomial of prime degree, the graphs display an unusual degree of symmetry. In this paper we provide a complete description of these graphs, and also provide some examples of how these graphs can be used to determine the decomposition of primes in certain field extensions

Discriminants of Chebyshev Radical Extensions

Let t be any integer and fix an odd prime ell. Let Phi(x) = T_ell^n(x)-t denote the n-fold composition of the Chebyshev polynomial of degree ell shifted by t. If this polynomial is irreducible, let K = bbq(theta), where theta is a root of Phi. A theorem of Dedekind's gives a condition on t for which K is monogenic. For other values of t, we apply the Montes algorithm to obtain a formula for the discriminant of K and to compute basis elements for the ring of integers O_K.Comment: This update contains proofs for the conjectures appearing in a earlier version of this paper. This article draws heavily from arXiv:0906.262

Aqueduct Metadata Document, Aqueduct Global Maps 2.0

This document describes the specific characteristics of the indicator data and calculations for the Aqueduct Water Risk Atlas Global Maps. Complete guidelines and processes for data collection, calculations, and mapping techniques are described fully in the Aqueduct Water Risk Framework. The Aqueduct Water Risk Atlas makes use of a Water Risk Framework, that includes 12 global indicators grouped into three categories of risk and one overall score

Aqueduct Country and River Basin Rankings: A Weighted Aggregation of Spatially Distinct Hydrological Indicators

More and more countries around the world face high levels of water stress, but measuring and communicating that stress consistently is challenging. This paper ranks countries and river basins worldwide based on their exposure water-related risks. Specifically, it provides national and basin-level scores derived from more localized water-risk scores from the Aqueduct Water Risk Atlas. Rankings are available for 181 countries, the world's 100-largest river basins by area, and the planet's 100-most populous river basins for five different measures of water supply and demand

Night Sky of the Flint Hills

While enjoying the Flint Hills during the day, it is the vast openness that first strikes the observer. There is literally nothing for miles and miles. But when the sun goes down, one experiences something that is awe-inspiring