Pith Number

pith:QVTJYM3Q

pith:2026:QVTJYM3QHDSNTF6JKVTSEFOA74

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A multivariable mean equation arising from the spectral geometric mean

Sejong Kim, Vatsalkumar N. Mer

The unique positive definite solution to a proposed nonlinear equation is the spectral geometric mean in the two-variable case.

arxiv:2605.16876 v1 · 2026-05-16 · math.FA

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3 Author claim open · sign in to claim

4 Citations open

5 Replications open

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

In the two-variable case, the unique positive definite solution of this equation is precisely the spectral geometric mean.

C2weakest assumption

The specific nonlinear equation chosen is assumed to be the natural multivariable extension of the two-variable spectral geometric mean definition given in the abstract.

C3one line summary

A nonlinear equation is proposed for the multivariable spectral geometric mean of positive definite operators; it recovers the known two-variable mean but need not have a unique solution in higher dimensions.

References

24 extracted · 24 resolved · 0 Pith anchors

[1] P. C. Alvarez-Esteban, E. del Barrio, J. A. Cuesta-Albertos and C. Matran,A fixed point approach to barycenters in Wasserstein spaces, J. Math. Anal. Appl.441(2016), 744–762 2016

[2] T. Ando and F. Hiai,Log majorization and complementary Golden–Thompson type inequalities, Lin- ear Algebra Appl.197–198(1994), 113–131 1994

[3] Bhatia,Positive Definite Matrices, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, 2007 2007

[4] R. Bhatia and J. Holbrook,Riemannian geometry and matrix geometric means, Linear Algebra Appl. 413(2006), 594–618 2006

[5] R. Bhatia, T. Jain and Y. Lim,On the Bures–Wasserstein distance between positive definite matrices, Expo. Math.37(2019), 165–191 2019

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:03:27.732235Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

85669c337038e4d997c955672215c0ff38c1bb7ff96189cc88e5f8e53a1c4de3

Aliases

arxiv: 2605.16876 · arxiv_version: 2605.16876v1 · doi: 10.48550/arxiv.2605.16876 · pith_short_12: QVTJYM3QHDSN · pith_short_16: QVTJYM3QHDSNTF6J · pith_short_8: QVTJYM3Q

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/QVTJYM3QHDSNTF6JKVTSEFOA74 \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 85669c337038e4d997c955672215c0ff38c1bb7ff96189cc88e5f8e53a1c4de3

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "574868a0f98827e5a3859b15357b8d2eb14380380bf1fcf71fb08c4d0f6e002f",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.FA",
    "submitted_at": "2026-05-16T08:37:32Z",
    "title_canon_sha256": "afc1f1306211a2a9e7ad64424226b7b6f6e034de695972ae7f97202e6024f7e8"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.16876",
    "kind": "arxiv",
    "version": 1
  }
}