arXiv subjects
Publications and source records attributed to Andrew Tausz.
In this paper, we review a method for computing and parameterizing the set of homotopy classes of chain maps between two chain complexes. This is then applied to finding topologically meaningful maps between simplicial complexes, which in the context of topological data analysis, can be viewed as an extension of conventional unsupervised learning methods to simplicial complexes.
The theory of zigzag persistence is a substantial extension of persistent homology, and its development has enabled the investigation of several unexplored avenues in the area of topological data analysis. In this paper, we discuss three applications of zigzag persistence: topological bootstrapping, parameter thresholding, and the comparison of witness complexes.
We discuss some properties of conditional expectation operators, and use these facts to prove an interesting counterexample regarding sufficient statistics. In particular, we show that there exists sufficient random variables X and Y, such that (X, Y) are jointly not sufficient. We follow the work of Burkholder and Chow, presenting complete proofs and supplying missing steps.
We propose and experimentally demonstrate a single-mode fiber length and dispersion measurement system based on a novel frequency-shifted asymmetric Sagnac interferometer incorporating an acousto-optic modulator (AOM). By sweeping the driving frequency of the AOM, which is asymmetrically placed in the Sagnac loop, the optical length of the fiber can be determined by measuring the corresponding variation in the phase delay between the two counter-propagating light beams. Combined with a high-resolution data processing algorithm, this system yields a dynamic range from a few centimeters to 60km (limited only by our availability of long fibers) with a resolution about 1ppm for long fibers.