arXiv subjects

Sadashige Ishida

Publications and source records attributed to Sadashige Ishida.

7 recordsLinked to original sources

arXiv2026

Ricci curvature for fluid models on the torus via Zeitlin's quantization

Ricci curvature measures the average stability of geodesics under transverse perturbations, but how it should be defined in infinite dimensions is often unclear. This paper proposes a definition of Ricci curvature on the space of Hamiltonian diffeomorphisms on the two-dimensional flat torus $\mathrm{HDiff}(\mathbb{T}^2)$, the state space for ideal fluids. Our definition is based on Zeitlin's model, which approximates $\mathrm{HDiff}(\mathbb{T}^2)$ by finite-dimensional Lie groups $\mathrm{SU}(N)$. We derive a formula for the Ricci curvature tensor on $\mathrm{SU}(N)$ and provide numerical evidence for its convergence in the large-$N$ limit to our conjectured finite value. Additionally, we explore potential applications for hydrodynamics through the Lyapunov stability of gravest wave modes and Arnold's tradewind estimates for long-term weather predictability. Our framework extends to a wide range of settings. We demonstrate this by introducing Ricci curvature on the state spaces of fluids on rectangular domains, the Lagrangian averaged Euler equation induced by the $H^1$-Sobolev metric, and the quasi-geostrophic equation incorporating the Coriolis effect.

math.DG↗

arXiv2026

L'Hopital rules for complex-valued functions in higher dimensions

In calculus, l'Hopital's rule provides a simple way to evaluate the limits of quotient functions when both the numerator and denominator vanish. But what happens when we move beyond real functions on a real interval? In this article, we study when the quotient of two complex-valued functions in higher dimension can be defined continuously at the points where both functions vanish. Surprisingly, the answer is far subtler than in the real-valued setting. We provide a complete characterization for the continuity of the quotient function. We also point out why extending this result to smoother quotients remains an intriguing challenge.

math.CV↗

arXiv2025

Implicit representations of codimension-2 submanifolds and their prequantum structure

This paper explores the geometry of the space of codimension-2 submanifolds. We implicitly represent these submanifolds by a class of complex-valued functions. We show that the space of all these implicit representations admits a prequantum bundle structure over the space of submanifolds, equipped with the well-known Marsden-Weinstein symplectic structure. This bundle allows a new geometric interpretation of the Marsden-Weinstein structure as the curvature of a connection form, which measures the average of volumes swept by the deformation of the S^1-family of hypersurfaces, defined as the phase level sets of the complex function implicitly representing a submanifold.

math.SG↗

arXiv2025

Quantitative convergence of a discretization of dynamic optimal transport using the dual formulation

We present a discretization of the dynamic optimal transport problem for which we can obtain the convergence rate for the value of the transport cost to its continuous value when the temporal and spatial stepsize vanish. This convergence result does not require any regularity assumption on the measures, though experiments suggest that the rate is not sharp. Via an analysis of the duality gap we also obtain the convergence rates for the gradient of the optimal potentials and the velocity field under mild regularity assumptions. To obtain such rates, we discretize the dual formulation of the dynamic optimal transport problem and use the mature literature related to the error due to discretizing the Hamilton-Jacobi equation.

math.NA↗

arXiv2024

Symplectic structures on the space of space curves

We present symplectic structures on the shape space of unparameterized space curves that generalize the classical Marsden-Weinstein structure. Our method integrates the Liouville 1-form of the Marsden-Weinstein structure with Riemannian structures that have been introduced in mathematical shape analysis. We also derive Hamiltonian vector fields for several classical Hamiltonian functions with respect to these new symplectic structures.

math.SG↗

arXiv2023

Area formula for spherical polygons via prequantization

We present a formula for the signed area of a spherical polygon via prequantization. In contrast to the traditional formula based on the Gauss-Bonnet theorem that requires measuring angles, the new formula mimics Green's theorem and is applicable to a wider range of degenerate spherical curves and polygons.

math.DG↗

arXiv2022

Hidden Degrees of Freedom in Implicit Vortex Filaments

This paper presents a new representation of curve dynamics, with applications to vortex filaments in fluid dynamics. Instead of representing these filaments with explicit curve geometry and Lagrangian equations of motion, we represent curves implicitly with a new co-dimensional 2 level set description. Our implicit representation admits several redundant mathematical degrees of freedom in both the configuration and the dynamics of the curves, which can be tailored specifically to improve numerical robustness, in contrast to naive approaches for implicit curve dynamics that suffer from overwhelming numerical stability problems. Furthermore, we note how these hidden degrees of freedom perfectly map to a Clebsch representation in fluid dynamics. Motivated by these observations, we introduce untwisted level set functions and non-swirling dynamics which successfully regularize sources of numerical instability, particularly in the twisting modes around curve filaments. A consequence is a novel simulation method which produces stable dynamics for large numbers of interacting vortex filaments and effortlessly handles topological changes and re-connection events.

cs.GR↗