arXiv subjects

Zachary Mullaghy

Publications and source records attributed to Zachary Mullaghy.

4 recordsLinked to original sources

arXiv2025

The Divergence-Free Radiant Transform

This paper presents the rigorous mathematical construction and foundational properties of the Divergence-Free Radiant Transform (DFRT), a spectral transform specifically designed for divergence-free vector fields, with applications in incompressible fluid dynamics and other solenoidal systems. The DFRT basis functions are constructed using a curl-based formulation that ensures the divergence-free condition is satisfied identically. We define the forward and inverse transforms, prove the Parseval identity, and establish the completeness of the basis. The DFRT coefficient space is equipped with an algebraic structure via a spectral coboundary operator, defined using Wigner 3j and 6j symbols to encode angular momentum coupling. This cohomological structure, and its connection to the Geometric Refinement Transform (GRT), is developed in a companion paper using a bigraded cohomology framework. We derive a modal evolution equation for the incompressible Navier-Stokes equations in DFRT coordinates and introduce a persistent regularity class based on cohomological constraints. Finally, we present a variational argument showing that an entropy-maximizing energy distribution leads to exponential decay, offering a new perspective on regularity and singularity prevention in incompressible flows.

math.AP↗

arXiv2025

The Geometric Refinement Transform: A Novel Uncountably Infinite Transform Space

This work introduces a novel and general class of continuous transforms based on hierarchical Voronoi based refinement schemes. The resulting transform space generalizes classical approaches such as wavelets and Radon transforms by incorporating parameters of refinement multiplicity, dispersion, and rotation. We rigorously establish key properties of the transform including completeness, uniqueness, invertibility, closure, and stability using frame bounds over functions of bounded variation and define a natural inner product structure emerging in L2. We identify regions of parameter space that recover known transforms, including multiscale wavelet decompositions and the generalized Radon transform. Applications are discussed across a range of disciplines, with particular emphasis on entropy formulations. Notably, the transform remains well behaved on geometrically complex and even non convex domains, where traditional methods may struggle. Despite the complexity of the underlying geometry, the coefficient spectrum reveals structure, offering insight even in highly irregular settings.

math.NA↗

arXiv2025

Centroidal Voronoi Tessellations as Electrostatic Equilibria: A Generalized Thomson Problem in Convex Domains

We present a variational framework in which Centroidal Voronoi Tessellations (CVTs) arise as local minimizers of a generalized electrostatic energy functional. By modeling interior point distributions in a convex domain as repelling charges balanced against a continuous boundary charge, we show that the resulting equilibrium configurations converge to CVT structures. We prove this by showing that CVTs minimize both the classical centroidal energy and the electrostatic potential, establishing a connection between geometric quantization and potential theory. Finally, we introduce a thermodynamic annealing scheme for global CVT optimization, rooted in Boltzmann statistics and random walk dynamics. By introducing a scheme for varying time steps (faster or slower cooling) we show that the set of minima of the centroid energy functional (and therefore the electrostatic potential) can be recovered. By recovering a set of generator locations corresponding to each minimum we can create a lattice continuation that allows for a customizable framework for individual minimum seeking.

math.NA↗

arXiv2025

Centroidal Voronoi Refinement in the Geometric Refinement Transform: Symmetry, Stability, and Optimal Reconstruction

We extend the Geometric Refinement Transform (GRT) by introducing centroidal Voronoi tessellations (CVTs) into the refinement process, enhancing symmetry, reconstruction accuracy, and numerical stability. By applying Lloyds algorithm at each refinement level, we minimize centroidal energy and generate Voronoi regions that better align with the functions underlying structure. This approach reduces geometric distortion, suppresses reconstruction error, and provides a natural framework for adaptive refinement. We analyze convergence properties, quantify the reduction in reconstruction error using Taylor-based estimates and Lipschitz continuous functions, and propose perturbation strategies to escape symmetry-preserving local minima. The resulting transform offers improved accuracy for applications in medical imaging, signal processing, and physics simulations, while preserving the theoretical completeness and stability guarantees of the original GRT framework.

math.NA↗