Parameterization of non-linear manifolds

C. W. Gear

Home/Archive/arXiv

arXiv · 1208.5246

Parameterization of non-linear manifolds

C. W. Gear

View original source ↗Read source document ↗Export metadata ↓

Abstract

In this report we consider the parameterization of low-dimensional manifolds that are specified (approximately) by a set of points very close to the manifold in the original high-dimensional space. Our objective is to obtain a parameterization that is (1-1) and non singular (in the sense that the Jacobian of the map between the manifold and the parameter space is bounded and non singular).

Explore related subjects

physics.comp-ph cs.NA math.NA

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

C. W. Gear. 2013-04-08. Parameterization of non-linear manifolds. https://arxiv.org/abs/1208.5246

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Mechanical resilience of ultra-low-density racing-shoe foams

High-performance racing shoes rely on ultra-low-density elastomeric foams that undergo large, repeated deformations during running. Yet little is known about how their mechanical properties vary throughout the shoe or change with repeated use. Here, we characterize the midsole foam in an elite-level racing shoe from the heel, midfoot, and toe of a new shoe and a shoe worn for 300 miles. Microscopy reveals a characteristic pore length scale of 128+/-18 um and supports an approximately isotropic continuum description. We then quantify the mechanical response under tension, compression, and shear. Remarkably, despite 300 miles of real-world use, the foam retains its mechanical response across all three loading modes and shoe regions. Energy return remains largely unchanged, with values of 85-93% in tension and compression and 64-71% in shear. At the same time, we observe strong regional variations, with tensile and compressive stiffnesses 38-51% lower in the toe than in the heel. The foam also exhibits a pronounced tension-compression asymmetry in Poisson's ratio. Together, these findings reveal a spatially structured and mode-dependent mechanical response that remains largely preserved after 300 miles of use. This mechanical resilience may extend the functional lifetime of racing shoes, with implications for runners, replacement recommendations, and sustainability.

physics.comp-ph↗

arXiv2026

Fundamental Bounds on the Polarizability of Macroscopic Scatterers

Polarizability predicts how an object responds to an incident electromagnetic field, the interactions between small particles, or the optical forces exerted upon them. Polarizability is responsible for the effective-medium properties of artificial materials or metasurfaces. Despite significant progress in all these areas, it is unclear what the limits of the strength of such interactions are or, more specifically, what the upper bounds on the polarizability of a given spatial region that an unknown and designed particle would occupy are. This work connects the electromagnetic field description via an integral equation with a dual formulation of quadratic programming to derive fundamental bounds on components of all four polarizability tensors or on their specific combinations. In particular, the work establishes an intuitive visualization of what strong polarizability means and how strong it can be. The developed fundamental bound also answers which materials and domains are best for the given demands on polarizability. These findings establish a versatile platform that can accommodate a wide range of demands on polarizable bodies, providing an absolute measure of their performance against which the results of human-powered or automated design procedures can be compared.

physics.comp-ph↗

arXiv2026

Perspective on Magnetic Nanoparticle Modeling: Interactions, Timescales and Regimes

The response of magnetic nanoparticles (MNPs) to applied magnetic fields underpins a broad range of biomedical and technological applications. In this Perspective, we review the principal modeling approaches for describing MNP dynamics across different physical regimes, ranging from coarse-grained macrospin descriptions to spatially resolved micromagnetic simulations. Selecting an appropriate model depends on the relevant energy scales and timescales, including those associated with magnetic anisotropy. We compare the assumptions, computational requirements, and regimes of applicability of the fixed-point-dipole, effective-field, thermal Stoner-Wohlfarth, diffusion-jump, coupled Landau-Lifshitz-Gilbert, egg, and micromagnetic models. Particular attention is given to coupling magnetization dynamics with translational and rotational particle motion, hydrodynamic interactions, and long-range dipolar interactions. By relating the relevant physical regimes to the resolution and computational cost of each approach, we provide practical guidance for model selection and outline challenges for predictive multiscale simulations of interacting MNP systems.

physics.comp-ph↗