arXiv subjects

William A. Clark

Publications and source records attributed to William A. Clark.

5 recordsLinked to original sources

arXiv2026

Transfer Operators for Stochastic Hybrid Systems on Manifolds with Guard-Induced Resets

This paper develops a transfer operator framework for stochastic hybrid systems with guard-induced resets, encompassing both the Koopman and Frobenius--Perron operators. Exploiting their duality, we derive a unified formulation in which observables and probability densities evolve under adjoint generators corresponding to the backward and forward Kolmogorov equations. The formulation is developed in a global and intrinsic manner on differentiable manifolds, ensuring consistency with the underlying geometric structure of the state space. In addition, we propose a finite volume computational scheme on manifolds that preserves total probability mass while accurately capturing fluxes across guards and reset-induced transfers. The proposed framework provides a unified and geometrically consistent approach to uncertainty propagation in stochastic hybrid systems, bridging continuous stochastic dynamics and hybrid transitions within a transfer operator perspective.

math.DS↗

arXiv2025

Dynamics and Optimal Control of State-Triggered Affine Hybrid Systems

A study of the dynamics and control for linear and affine hybrid systems subjected to either temporally- or spatially-triggered resets is presented. Hybrid trajectories are capable of degeneracies not found in continuous-time systems namely beating, blocking, and Zeno. These pathologies are commonly avoided by enforcing a lower bound on the time between events. While this constraint is straightforward to implement for temporally-triggered resets, it is impossible to do so for spatially-triggered systems. In particular, linear spatially-triggered hybrid systems always posses trajectories that are beating and blocking while affine systems may also include Zeno trajectories. The hybrid Pontryagin maximum principle is studied in the context of affine hybrid systems. The existence/uniqueness of the induced co-state jump conditions is studied which introduces the notion of strongly and weakly actuated resets. Finally, optimal control in the context of beating and Zeno is discussed. This work concludes with numerical examples.

math.OC↗

arXiv2023

E2PN: Efficient SE(3)-Equivariant Point Network

This paper proposes a convolution structure for learning SE(3)-equivariant features from 3D point clouds. It can be viewed as an equivariant version of kernel point convolutions (KPConv), a widely used convolution form to process point cloud data. Compared with existing equivariant networks, our design is simple, lightweight, fast, and easy to be integrated with existing task-specific point cloud learning pipelines. We achieve these desirable properties by combining group convolutions and quotient representations. Specifically, we discretize SO(3) to finite groups for their simplicity while using SO(2) as the stabilizer subgroup to form spherical quotient feature fields to save computations. We also propose a permutation layer to recover SO(3) features from spherical features to preserve the capacity to distinguish rotations. Experiments show that our method achieves comparable or superior performance in various tasks, including object classification, pose estimation, and keypoint-matching, while consuming much less memory and running faster than existing work. The proposed method can foster the development of equivariant models for real-world applications based on point clouds.

cs.CV↗

arXiv2022

Surprises in a classic boundary-layer problem

We revisit a textbook example of a singularly perturbed nonlinear boundary-value problem. Unexpectedly, it shows a wealth of phenomena that seem to have been overlooked previously, including a pitchfork bifurcation in the number of solutions as one varies the small parameter, and transcendentally small terms in the initial conditions that can be calculated by elementary means. Based on our own classroom experience, we believe this problem could provide an enjoyable workout for students in courses on perturbation methods, applied dynamical systems, or numerical analysis.

math.CA↗

arXiv2019

Interface characteristics in an α+β titanium alloy

The alpha/beta interface in Ti-6Al-2Sn-4Zr-6Mo (Ti-6246) is investigated via centre of symmetry analysis, both as-grown and after 10% cold work. Semi-coherent interface steps are observed at a spacing of 4.5 +/-1.13 atoms in the as-grown condition, in good agreement with theory prediction (4.37 atoms). Lattice accommodation is observed, with elongation along [-1 2 -1 0]alpha and contraction along [1 0 -1 0]alpha . Deformed alpha exhibited larger, less coherent steps with slip bands lying in {110}beta. This indicates dislocation pile-up at the grain boundary, a precursor to globularisation, offering insight into the effect of deformation processing on the interface, which is important for titanium alloy processing route design.

cond-mat.mtrl-sci↗