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15 results on '"Erdmann, Grant"'

1. Topological Hierarchical Decompositions

Author

Joyce, Ian Stewart, Erdmann, Grant, Gardner, Kirk P., Kramer, Ryan, and Siegrist, Kyle

Subjects
Computer Science - Computational Geometry, Mathematics - Algebraic Topology
Abstract

Topological data analysis is an emerging field that applies the study of topological invariants to data. Perhaps the simplest of these invariants is the number of connected components or clusters. In this work, we explore a topological framework for cluster analysis and show how it can be used as a basis for explainability in unsupervised data analysis. Our main object of study will be hierarchical data structures referred to as Topological Hierarchical Decompositions (THDs). We give a number of examples of how traditional clustering algorithms can be topologized, and provide preliminary results on the THDs associated with Reeb graphs and the mapper algorithm. In particular, we give a generalized construction of the mapper functor as a pixelization of a cosheaf in order to generalize multiscale mapper.

Published
2023

2. Computation of conserved densities for systems of nonlinear differential-difference equations

Author

Goktas, Unal, Hereman, Willy, and Erdmann, Grant

Subjects
Nonlinear Sciences - Exactly Solvable and Integrable Systems
Abstract

A new method for the computation of conserved densities of nonlinear differential-difference equations is applied to Toda lattices and discretizations of the Korteweg-de Vries and nonlinear Schrodinger equations. The algorithm, which can be implemented in computer algebra languages such as Mathematica, can be used as an indicator of integrability., Comment: submitted to Phys. Lett A, 10 pages, latex

Published
1997
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3. The inverse source problem for wavelet fields

Author

Devaney, Anthony J., Kaiser, Gerald, Marengo Edwin A., Albanese, Richard, and Erdmann, Grant

Subjects
Reliability (Engineering) -- Evaluation, Wavelet transforms -- Evaluation, Beamforming -- Methods, Wave-motion, Theory of -- Evaluation, Mobile communication systems -- Research, Wireless communication systems -- Research, Wireless technology, Business, Computers, Electronics, Electronics and electrical industries
Abstract

The theory of the inverse source problem is employed to compute a class of continuously distributed and compactly supported three-dimensional (volume) sources that radiate the scalar wavelets investigated by Kaiser as well as certain electromagnetic generalizations of these scalar fields. These efforts have shown that the scalar wavelet fields can be radiated by a distributional source (generalized function) supported on a circular disk of radius a or an oblate spheroid surrounding that disk. Our main goal here is to replace this distributional source by a more conventional volume source that radiates the same wavelet field outside its support volume. The equivalent volume sources computed in this paper are supported on (three-dimensional) spherical shells whose outer radius [a.sub.+] > a and inner radius [a.sub.-] < [a.sub.+] are arbitrary. These sources are analytic functions of position within their support volumes for any finite, but arbitrarily large temporal frequency [omega], and possess minimum [L.sup.2] norm among all possible solutions to the inverse source problem with the given support volume constraint. Electromagnetic versions of the wavelet sources and fields are shown to be easily derived from their scalar wave counterparts. Index Terms--Inverse source problem, pulsed beam, reliability, wavelet field.

Published
2008

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