arXiv subjects

Benjamin McMillan

Publications and source records attributed to Benjamin McMillan.

5 recordsLinked to original sources

arXiv2026

Rheological and Photoelastic Response of Hydrated Soft Granular Particles

Photoelasticity is a qualitative and quantitative optical technique to image internal stress distributions in transparent materials. In the past few decades, discrete photoelastic particles have been used as a proxy for dry granular materials in both static, quasistatic, and dynamic analogue experiments. The technique allows the visualization of force chains, determination of the location and magnitude of contact forces, and outputs a stress tensor for each particle with shear and normal stress components. To date, little to no work has investigated photoelastic suspensions, where photoelastic granular particles are immersed in a fluid medium, despite its relevance in industrial and natural applications. The introduction of a fluid phase yields additional considerations in the rheological and photoelastic behavior of our proxy particles. In this manuscript, we summarize the state-of-the-art in resolving forces in immersed photoelastic granular materials. We introduce characterization techniques to probe changes in rheological and optical properties of hydrated photoelastic particles, and we report considerations for use of photoelastic particles in immersion-based experiments. We intend for this work to provide the leading framework to study the hydrodynamic interactions in 2D systems of photoelastic particles immersed in a fluid medium.

cond-mat.soft↗

arXiv2025

Chern-Weil theory for Haefliger-singular foliations

We give a Chern-Weil map for the Gel'fand-Fuks characteristic classes of Haefliger-singular foliations, those foliations defined by smooth Haefliger structures with dense regular set. Our characteristic map constructs, out of singular geometric structures adapted to singularities, explicit forms representing characteristic classes in de Rham cohomology. The forms are functorial under foliation morphisms. We prove that the theory applies, up to homotopy, to general smooth Haefliger structures: subject only to obvious necessary dimension constraints, every smooth Haefliger structure is homotopic to a Haefliger-singular foliation, and any morphism of Haefliger structures is homotopic to a morphism of Haefliger-singular foliations. As an application, we provide a generalisation to the singular setting of the classical construction of forms representing the Godbillon-Vey invariant.

math.DG↗

arXiv2024

The range of a connection and a Calabi operator for Lorentzian locally symmetric spaces

For semi-Riemannian manifolds of constant sectional curvature, the Calabi operator is a second order linear differential operator that provides local integrability conditions for the range of the Killing operator. In this article, extending earlier results in the Riemannian setting, we identify the Lorentzian locally symmetric spaces on which the Calabi operator is sufficient to identify the range of the Killing operator. Specifically, this is always the case for indecomposable spaces and we identify precisely those products for which it fails. Our method is quite general in that we firstly develop criteria to be in the range of a connection, viewed as a linear differential operator. Then we ascertain how these criteria apply in the case of what we call the Killing connection.

math.DG↗

arXiv2024

A Calabi operator for Riemannian locally symmetric spaces

On a Riemannian manifold of constant curvature, the Calabi operator is a second order linear differential operator that provides local integrability conditions for the range of the Killing operator. We generalise this operator to provide linear second order local integrability conditions on Riemannian locally symmetric spaces, whenever this is possible. Specifically, we show that this generalised operator always works in the irreducible case and we identify precisely those products for which it fails.

math.DG↗

arXiv2024

A characterization of Pfaffian embeddings from (2, 3, 5)- into flat (4, 7)-geometries

Given two smooth manifolds with tangent subbundle distributions, an embedding is Pfaffian if its differential sends the distribution on the source into the distribution on the target. In this paper, we consider the question of existence of Pfaffian embeddings in the specific case where the source is a (2,3,5)-manifold, the target is the 7-dimensional space of isotropic 2-planes in a 6-dimensional symplectic vector space, and the Pfaffian condition is that the derived 3-distribution on the source be mapped into the natural 4-distribution on the target. This is one of the simpler non-trivial cases of the general question on existence of Pfaffian embeddings, but already the answer here requires solution of an interesting differential equation. It turns out that a generic (2,3,5)-manifold does not embed, the first obstruction being the fact that a Pfaffian embeddable (2,3,5)-manifold necessarily has a double root for its Cartan quartic at each point. We determine a complete characterization of embeddable (2,3,5)-manifolds in terms of their associated Cartan geometries, which characterization depends on higher order (non-harmonic) curvature as well.

math.DG↗