Lorelei Koss

Parametrized dynamics of the Weierstrass elliptic function

We study parametrized dynamics of the Weierstrass elliptic ℘ function by looking at the underlying lattices; that is, we study parametrized families ℘Λ and let Λ vary. Each lattice shape is represented by a point τ in a fundamental period in modular space; for a fixed lattice shape Λ = [1, τ ] we study the parametrized space kΛ. We show that within each shape space there is a wide variety of dynamical behavior, and we conduct a deeper study into certain lattice shapes such as triangular and square. We also use the invariant pair (g2, g3) to parametrize some lattices.

One-sided Lebesgue Bernoulli maps of the sphere of degree n2 and 2n2

We prove that there are families of rational maps of the sphere of degree n2(n=2,3,4,…) and 2n2(n=1,2,3,…) which, with respect to a finite invariant measure equivalent to the surface area measure, are isomorphic to one-sided Bernoulli shifts of maximal entropy. The maps in question were constructed by Boettcher (1903--1904) and independently by Lattes (1919). They were the first examples of maps with Julia set equal to the whole sphere.

Ergodic and Bernoulli properties of analytic maps of complex projective space

We examine the measurable ergodic theory of analytic maps F of complex projective space. We focus on two different classes of maps, Ueda maps of P n , and rational maps of the sphere with parabolic orbifold and Julia set equal to the entire sphere. We construct measures which are invariant, ergodic, weak- or strong-mixing, exact, or automorphically Bernoulli with respect to these maps. We discuss topological pressure and measures of maximal entropy (h μ (F) = h top (F) = log(deg F)). We find analytic maps of P 1 and P 2 which are one-sided Bernoulli of maximal entropy, including examples where the maximal entropy measure lies in the smooth measure class. Further, we prove that for any integer d > 1, there exists a rational map of the sphere which is one-sided Bernoulli of entropy log d with respect to a smooth measure.

Manipulatives for 3-Dimensional Coordinate Systems.

Abstract We describe a set of manipulatives developed to help students learn three-dimensional Cartesian, cylindrical, and spherical coordinate systems.

Elliptic Functions with Critical Orbits Approaching Infinity

We construct examples of elliptic functions, viewed as iterated meromorphic functions from the complex plane to the sphere, with the property that there exist one or more critical points which approach the essential singularity at ∞ under iteration but are not prepoles. We obtain many nonequivalent elliptic functions satisfying this property, including examples with Julia set the whole sphere as well as examples with nonempty Fatou set. These are the first examples of this type known to exist and provide examples to illustrate unusual chaotic measure theoretic behaviour studied by the third author and others.

Differential equations in literature, poetry and film

This paper investigates how differential equations models have been used to study works in literature, poetry and film. We present applications to works by William Shakespeare, Francis Petrarch, Ray Bradbury, Herman Melville, Ridley Scott and others, as well as applications to Greek mythology and the Bible. This paper gives a range of useful examples for teaching, and we discuss how these models have been used in the classroom.

Cantor Julia Sets in a Family of Even Elliptic Functions

In this paper, we investigate topological properties of the Julia set of a family of even elliptic functions on real rectangular lattices. These functions have two real critical points and, depending on the shape of the lattice, one or two non-real critical points. We prove that if 0 is the only real fixed point then the Julia set is Cantor. We show that functions with Cantor Julia sets exist within every equivalence class of real rectangular lattices, and we generally locate the parameters giving rise to these Julia sets in the parameter plane representing real lattices.

Differential equations in music and dance

Abstract This is the second paper in a series that connects ideas from differential equations to relevant and interesting material from the arts and humanities. In this article, we investigate a variety of works in music and dance. We present applications to musical composition, to performance techniques in classical guitar and to choreographing dance.

Writing and Information Literacy in a Cryptology First-Year Seminar

Abstract The author describes a first-year seminar in cryptology with three major assignments which were planned to help students develop information literacy, oral presentation, and writing skills.

Sustainability in a differential equations course: a case study of Easter Island

Easter Island is a fascinating example of resource depletion and population collapse, and its relatively short period of human habitation combined with its isolation lends itself well to investigation by students in a first-semester ordinary differential equations course. This article describes curricular materials for a semester-long case study into environmental and sustainability issues in the history of Easter Island. Using results that appeared in recent journal articles, students investigate the date of arrival of early settlers, the impact they had on natural resources through population growth as well as through the introduction of non-native species, and the effect of European diseases on the population.