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We introduce harmonic persistent homology spaces for filtrations of finite simplicial complexes. As a result we can associate concrete subspaces of cycles to each bar of the barcode of the filtration. We prove stability of the harmonic persistent homology subspaces, as well as the subspaces associated to the bars of the barcodes, under small perturbations of functions defining them. We relate the notion of ``essential simplices'' introduced in an earlier work to identify simplices which play a significant role in the birth of a bar, with that of harmonic persistent homology. We prove that the harmonic representatives of simple bars maximizes the ``relative essential content'' amongst all representatives of the bar, where the relative essential content is the weight a particular cycle puts on the set of essential simplices. \footnote{An extended abstract of the paper appeared in the Proceedings of the IEEE Symposium on the Foundations of Computer Science, 2021.}
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Bar code reader for the THz region
We demonstrate a bar code sensing system for the THz region using leaky parallel plate waveguide and an off-axis parabolic mirror. The bars of the bar code are made from metal with air as gaps between them. We use up to 6 bars in the barcode system which can store up to 64 bits. Because the system employs coherent detection, we can further increase the bit density by adding Teflon strips to the barcode, encoding information in both amplitude and phase delay. These bar codes can be manufactured easily and inexpensively, offering a versatile alternative to RFID tags.
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- Award ID(s):
- 1923782
- PAR ID:
- 10249813
- Publisher / Repository:
- Optical Society of America
- Date Published:
- Journal Name:
- Optics Express
- Volume:
- 29
- Issue:
- 13
- ISSN:
- 1094-4087; OPEXFF
- Format(s):
- Medium: X Size: Article No. 20240
- Size(s):
- Article No. 20240
- Sponsoring Org:
- National Science Foundation
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