{
  "abstract": "We investigate order and spectral properties of the weighted near geometric curve $A\\star_tB=A^{1/2}(B^{1/2}A^{-1}B^{1/2})^tA^{1/2}$ on the cone of positive definite matrices. Our focus is on comparisons governed by the Loewner order, eigenvalue inequalities, and log-majorization. We establish a normalized eigenvalue comparison with the weighted arithmetic mean, give a sufficient condition for the corresponding Loewner inequality, and obtain conditional log-majorization relations. Motivated by the relationship between the weighted geometric and spectral geometric means, we introduce the near spectral geometric curve $A\\widetilde{\\natural}_tB=(A^{-1}\\star B)^tA(A^{-1}\\star B)^t$, where $\\star=\\star_{1/2}$. Setting $D=A^{-1}\\star_1B$, we prove that $A\\widetilde{\\natural}_tB=A\\natural_tD$ and $B\\prec_{\\log}D$, and show that $D=B$ if and only if $A$ and $B$ commute. We further characterize commutativity and near-order monotonicity, prove midpoint isospectrality, and derive affine interpolation and constant-speed identities. Finally, we establish a Lie-Trotter formula and additional log-majorization inequalities. These results connect the two near geometric constructions with the geometric and spectral geometric means and clarify their order and spectral behavior.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.23736",
  "author_total": 2,
  "authors": [
    {
      "name": "Muritala Abiodun Ibrahim"
    },
    {
      "name": "Huajun Huang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bvc1pr6ark87y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.23736",
  "fulltext_url": null,
  "id": "bvc1pr6ark87y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.23736",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "openalex",
      "2604|Mathematical Inequalities and Applications"
    ]
  ],
  "openalex": "W7222961587",
  "pdf_url": "https://arxiv.org/pdf/2610.23736",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bvc1pr6ark87y.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.23736",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.23736"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Order relations for the near geometric and near spectral geometric curves",
  "title_text": "Order relations for the near geometric and near spectral geometric curves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/bvc1pr6ark87y",
  "venue": "arXiv",
  "versions": []
}