On the state space geometry of the Kuramoto-Sivashinsky flow in a periodic domain

The continuous and discrete symmetries of the Kuramoto-Sivashinsky system restricted to a spatially periodic domain play a prominent role in shaping the invariant sets of its chaotic dynamics. The continuous spatial translation symmetry leads to relative equilibrium (traveling wave) and relative periodic orbit (modulated traveling wave) solutions. The discrete symmetries lead to existence of equilibrium and periodic orbit solutions, induce decomposition of state space into invariant subspaces, and enforce certain structurally stable heteroclinic connections between equilibria. We show, on the example of a particular small-cell Kuramoto-Sivashinsky system, how the geometry of its dynamical state space is organized by a rigid `cage' built by heteroclinic connections between equilibria, and demonstrate the preponderance of unstable relative periodic orbits and their likely role as the skeleton underpinning spatiotemporal turbulence in systems with continuous symmetries. We also offer novel visualizations of the high-dimensional Kuramoto-Sivashinsky state space flow through projections onto low-dimensional, PDE representation independent, dynamically invariant intrinsic coordinate frames, as well as in terms of the physical, symmetry invariant energy transfer rates.Comment: 31 pages, 17 figures; added references, corrected typos. Due to file size restrictions some figures in this preprint are of low quality. A high quality copy may be obtained from http://www.cns.gatech.edu/~predrag/papers/preprints.html#rp

3 case studies: a hybrid educational strategy for ART/SCI collaborations

In this paper we report on a transdisciplinary university course designed to bring together fine art/visual communication design and computer science students for the creation and implementation of collaborative visual/audio projects that draw upon the specialized knowledge of both these disciplines. While an overview of the syllabus and the teaching methodologies is undertaken in the introduction, the focus of the paper concentrates upon an in-depth discussion and analysis of 3 specific projects that were developed by 3 distinct teams of students comprised of one artist/designer and one engineer each

Symmetry Detection of Rational Space Curves from their Curvature and Torsion

We present a novel, deterministic, and efficient method to detect whether a given rational space curve is symmetric. By using well-known differential invariants of space curves, namely the curvature and torsion, the method is significantly faster, simpler, and more general than an earlier method addressing a similar problem. To support this claim, we present an analysis of the arithmetic complexity of the algorithm and timings from an implementation in Sage.Comment: 25 page

Fostering collaborative knowledge construction with visualization tools

This study investigates to what extent collaborative knowledge construction can be fostered by providing students with visualization tools as structural support. Thirty-two students of Educational Psychology took part in the study. The students were subdivided into dyads and asked to solve a case problem of their learning domain under one of two conditions: 1) with content-specific visualization 2) with content-unspecific visualization. Results show that by being provided with a content-specific visualization tool, both the process and the outcome of the cooperative effort improved. More specifically, dyads under that condition referred to more adequate concepts, risked more conflicts, and were more successful in integrating prior knowledge into the collaborative solution. Moreover, those learning partners had a more similar individual learning outcome