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Title: Mixing properties of colourings of the ℤ d lattice
Abstract We study and classify proper q -colourings of the ℤ d lattice, identifying three regimes where different combinatorial behaviour holds. (1) When $q\le d+1$ , there exist frozen colourings, that is, proper q -colourings of ℤ d which cannot be modified on any finite subset. (2) We prove a strong list-colouring property which implies that, when $q\ge d+2$ , any proper q -colouring of the boundary of a box of side length $n \ge d+2$ can be extended to a proper q -colouring of the entire box. (3) When $q\geq 2d+1$ , the latter holds for any $n \ge 1$ . Consequently, we classify the space of proper q -colourings of the ℤ d lattice by their mixing properties.  more » « less
Award ID(s):
1855464
NSF-PAR ID:
10338572
Author(s) / Creator(s):
; ; ; ;
Date Published:
Journal Name:
Combinatorics, Probability and Computing
Volume:
30
Issue:
3
ISSN:
0963-5483
Page Range / eLocation ID:
360 to 373
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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We used a variety of techniques such as the file locking mechanism, multithreading, circular buffers, real-time event decoding, and signal-decision plotting to realize the system. A video demonstrating the system is available at: https://www.isip.piconepress.com/projects/nsf_pfi_tt/resources/videos/realtime_eeg_analysis/v2.5.1/video_2.5.1.mp4. The final conference submission will include a more detailed analysis of the online performance of each module. ACKNOWLEDGMENTS Research reported in this publication was most recently supported by the National Science Foundation Partnership for Innovation award number IIP-1827565 and the Pennsylvania Commonwealth Universal Research Enhancement Program (PA CURE). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the official views of any of these organizations. REFERENCES [1] A. Craik, Y. He, and J. L. 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