Topological Chaos in a Three-Dimensional Spherical Fluid Vortex

In chaotic deterministic systems, seemingly stochastic behavior is generated by relatively simple, though hidden, organizing rules and structures. Prominent among the tools used to characterize this complexity in 1D and 2D systems are techniques which exploit the topology of dynamically invariant structures. However, the path to extending many such topological techniques to three dimensions is filled with roadblocks that prevent their application to a wider variety of physical systems. Here, we overcome these roadblocks and successfully analyze a realistic model of 3D fluid advection, by extending the homotopic lobe dynamics (HLD) technique, previously developed for 2D area-preserving dynamics, to 3D volume-preserving dynamics. We start with numerically-generated finite-time chaotic-scattering data for particles entrained in a spherical fluid vortex, and use this data to build a symbolic representation of the dynamics. We then use this symbolic representation to explain and predict the self-similar fractal structure of the scattering data, to compute bounds on the topological entropy, a fundamental measure of mixing, and to discover two different mixing mechanisms, which stretch 2D material surfaces and 1D material curves in distinct ways.Comment: 14 pages, 11 figure

A K Nearest Classifier design

This paper presents a multi-classifier system design controlled by the topology of the learning data. Our work also introduces a training algorithm for an incremental self-organizing map (SOM). This SOM is used to distribute classification tasks to a set of classifiers. Thus, the useful classifiers are activated when new data arrives. Comparative results are given for synthetic problems, for an image segmentation problem from the UCI repository and for a handwritten digit recognition problem

Visualizing engineering design data using a modified two-level self-organizing map clustering approach

Engineers tasked with designing large and complex systems are continually in need of decision-making aids able to sift through enormous amounts of data produced through simulation and experimentation. Understanding these systems often requires visualizing multidimensional design data. Visual cues such as size, color, and symbols are often used to denote specific variables (dimensions) as well as characteristics of the data. However, these cues are unable to effectively convey information attributed to a system containing more than three dimensions. Two general techniques can be employed to reduce the complexity of information presented to an engineer: dimension reduction, and individual variable comparison. Each approach can provide a comprehensible visualization of the resulting design space, which is vital for an engineer to decide upon an appropriate optimization algorithm. Visualization techniques, such as self-organizing maps (SOMs), offer powerful methods able to surmount the difficulties of reducing the complexity of n-dimensional data by producing simple to understand visual representations that quickly highlight trends to support decision-making. The SOM can be extended by providing relevant output information in the form of contextual labels. Furthermore, these contextual labels can be leveraged to visualize a set of output maps containing statistical evaluations of each node residing within a trained SOM. These maps give a designer a visual context to the data set’s natural topology by highlighting the nodal performance amongst the maps. A drawback to using SOMs is the clustering of promising points with predominately less desirable data. Similar data groupings can be revealed from the trained output maps using visualization techniques such as the SOM, but these are not inherently cluster analysis methods. Cluster analysis is an approach able to assimilate similar data objects into “natural groups” from an otherwise unknown prior knowledge of a data set. Engineering data composed of design alternatives with associated variable parameters often contain data objects with unknown classification labels. Consequently, identifying the correct classifications can be difficult and costly. This thesis applies a cluster analysis technique to SOMs to segment a high-dimensional dataset into “meta-clusters”. Furthermore, the thesis will describe the algorithm created to establish these meta-clusters through the development of several computational metrics involving intra and inter cluster densities. The results from this work show the presented algorithm’s ability to narrow a large-complex system’s plethora of design alternatives into a few overarching set of design groups containing similar principal characteristics, which saves the time a designer would otherwise spend analyzing numerous design alternatives

Topological barriers for locally homeomorphic quasiregular mappings in 3-space

We construct a new type of locally homeomorphic quasiregular mappings in the 3-sphere and discuss their relation to the M.A.Lavrentiev problem, the Zorich map with an essential singularity at infinity, the Fatou's problem and a quasiregular analogue of domains of holomorphy in complex analysis. The construction of such mappings comes from our construction of non-trivial compact 4-dimensional cobordisms $M$ with symmetric boundary components and whose interiors have complete 4-dimensional real hyperbolic structures. Such locally homeomorphic quasiregular mappings are defined in the 3-sphere $S^3$ as mappings equivariant with the standard conformal action of uniform hyperbolic 3-lattices $\Gamma$ in the unit 3-ball and its complement in $S^3$ and with its discrete representation $G=\rho(\Gamma)$ in the group of isometries of $H^4$. Here $G$ is the fundamental group of our non-trivial hyperbolic 4-cobordism $M=(H^4\cup\Omega(G))/G$ and the kernel of the homomorphism $\rho \!:\! \Gamma\rightarrow G$ is a free group $F_3$ on three generators.Comment: 21 pages, 8 figures; Corrected typo