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On the Spectral Gap of a Square Distance Matrix

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Abstract

We consider a square distance matrix which arises from a preconditioned Jacobian matrix for the numerical computation of the Cahn–Hilliard problem. We prove strict negativity of all but one associated eigenvalues. This solves a conjecture in Christieb et al. (J Comput Phys 257:193–215, 2014).

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Notes

  1. Here we used the fact that \(x_i \in [0,1)\). In particular we do not allow the situation \(x_{i_0}=0\), \(x_{j_0}=1\) for some \(i_0\), \(j_0\).

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Acknowledgements

X. Cheng, D. Li and B. Wetton were supported in part by NSERC Discovery grants. This work was supported by a Grant from the Simons Foundation (\(\#359610\), David Shirokoff).

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Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

    Xinyu Cheng, Dong Li & Brian Wetton

  2. Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights, Newark, NJ, 07102, USA

    David Shirokoff

Authors
  1. Xinyu Cheng
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  2. Dong Li
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  3. David Shirokoff
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  4. Brian Wetton
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Corresponding author

Correspondence to Dong Li.

Additional information

Dedicated to Ya.G. Sinai on occasion of his 80th birthday.

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Cheng, X., Li, D., Shirokoff, D. et al. On the Spectral Gap of a Square Distance Matrix. J Stat Phys 166, 1029–1035 (2017). https://doi.org/10.1007/s10955-016-1685-7

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  • DOI: https://doi.org/10.1007/s10955-016-1685-7

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