A Power Geometry and Hypergeometric Functions
Abstract
This paper is a corrected English translation of an original Japanese work introducing "power geometry," a research programme centred on power transformations, deformations and dualities of functions in the sense of optimisation theory. Its two main motivations are the origin of probability distributions (via maximum-likelihood principles, rational-function parameters and ODEs linked to information geometry, the Pearson system and Schwarzian equations) and the construction of matrix dynamical systems that compute eigenvalues, singular values and principal or minor subspaces. Building on Brockett's double-bracket flows, the author extends these systems to rectangular matrices, flag manifolds and difference-of-convex potentials, and shows how a continuous power parameter can interpolate between largest- and smallest-eigenvalue problems. Dual potentials, topological distinctions between principal and minor subspace flows, and links via matrix Schwarzian--Riccati equations are examined, with a view toward design principles for classical and quantum information-processing algorithms.
Related
- Sharp power-mean comparisons for Gauss hypergeometric functions
- Natural coordinates for constrained correlation functions: Partial autocorrelations and the geometry of positive power spectra
- The second type of power mean bounds for the hypergeometric mean
- The first type of power mean bounds for the Gaussian hypergeometric function
- Extensions of several famous combinatorial identities via hypergeometric functions
BibTeX
@misc{yoshizawa2026power,
title = {{A Power Geometry and Hypergeometric Functions}},
author = {Shintaro Yoshizawa},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05474},
eprint = {2609.05474},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05474},
}
RIS
TY - UNPB
TI - A Power Geometry and Hypergeometric Functions
AU - Shintaro Yoshizawa
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.05474
UR - https://arxiv.org/abs/2609.05474
AB - This paper is a corrected English translation of an original Japanese work introducing "power geometry," a research programme centred on power transformations, deformations and dualities of functions in the sense of optimisation theory. Its two main motivations are the origin of probability distributions (via maximum-likelihood principles, rational-function parameters and ODEs linked to information geometry, the Pearson system and Schwarzian equations) and the construction of matrix dynamical systems that compute eigenvalues, singular values and principal or minor subspaces. Building on Brockett's double-bracket flows, the author extends these systems to rectangular matrices, flag manifolds and difference-of-convex potentials, and shows how a continuous power parameter can interpolate between largest- and smallest-eigenvalue problems. Dual potentials, topological distinctions between principal and minor subspace flows, and links via matrix Schwarzian--Riccati equations are examined, with a view toward design principles for classical and quantum information-processing algorithms.
ER -