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PreprintCC BY 4.0arXiv

Dimensionally absolutely stable but non-diagonalizable T-meshes

  • Mao Shi
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Abstract

We construct an infinite family of rectangular T-meshes whose conformality-vector-space dimension is constant over all structurally similar geometric realizations, although the meshes are not diagonalizable. For every integer d≥ 2, the construction has 2(d+2) Tl-edges and intersection graph Kd+2,d+2. Every Tl-edge contains at least d+2 vertices, so no edge is vanishable under the usual vertex-count criterion for bidegree (d,d) and smoothness (d−1,d−1). We prove that the conformality-vector space is one-dimensional throughout the structural similarity class. Its generator is the outer product of two vectors of barycentric interpolation weights, extended by four zero coordinates. Non-diagonalizability follows by examining the last edge in any proposed diagonalizing ordering. Thus, for the geometric structural similarity class specified here, diagonalizability is not a necessary condition for dimensional absolute stability.

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Cite

BibTeXRIS
BibTeX
@misc{shi2026dimensionally,
  title = {{Dimensionally absolutely stable but non-diagonalizable $T$-meshes}},
  author = {Mao Shi},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.05923},
  eprint = {2611.05923},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.05923},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Dimensionally absolutely stable but non-diagonalizable T-meshes
AU  - Mao Shi
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.05923
UR  - https://arxiv.org/abs/2611.05923
AB  - We construct an infinite family of rectangular T-meshes whose conformality-vector-space dimension is constant over all structurally similar geometric realizations, although the meshes are not diagonalizable. For every integer d≥ 2, the construction has 2(d+2) Tl-edges and intersection graph K_d+2,d+2. Every Tl-edge contains at least d+2 vertices, so no edge is vanishable under the usual vertex-count criterion for bidegree (d,d) and smoothness (d−1,d−1). We prove that the conformality-vector space is one-dimensional throughout the structural similarity class. Its generator is the outer product of two vectors of barycentric interpolation weights, extended by four zero coordinates. Non-diagonalizability follows by examining the last edge in any proposed diagonalizing ordering. Thus, for the geometric structural similarity class specified here, diagonalizability is not a necessary condition for dimensional absolute stability.
ER  -