arXiv subjects

William Green

Publications and source records attributed to William Green.

7 recordsLinked to original sources

arXiv2026

A Systematic Evaluation of Molecular Mixture Behavior Prediction

Machine learning for molecular property prediction has focused largely on pure compounds, even though many practical applications depend on mixtures with intermolecular interactions. Recent work has expanded the availability of mixture datasets, but evaluation still focuses mainly on absolute accuracy. However, absolute errors in mixtures conflate pure-component contributions with deviations from ideal mixing. We propose an evaluation framework that decomposes mixture-property error into pure-compound and interaction (non-ideal) components. The framework combines leakage-aware split protocols, ideal-mixture baselines, and excess-property metrics. To support reproducible benchmarking, we curate seven matched pure and mixture physicochemical property datasets. Across multiple mixture-property tasks and model families, we find that strong absolute accuracy can mask poor recovery of non-ideal mixture behavior, and that performance drops substantially under strict molecule splits. These results identify transfer to unseen molecules as a central challenge in molecular mixture machine learning and motivate evaluation beyond absolute accuracy alone.

cs.LG↗

arXiv2025

Dispersive estimates for fractional order Schr\"odinger operators

We prove dispersive bounds for fractional Schr\"odinger operators on $\mathbb R^n$ of the form $H=(-\Delta)^{\alpha}+V$ with $V$ a real-valued, decaying potential and $\alpha \notin\mathbb N$. We derive pointwise bounds on the resolvent operators for all $0<\alpha<\frac{n}{2}$, a quantitative limiting absorption principle for $\frac12<\alpha<\frac{n}{2}$, and establish global dispersive estimates in dimension $n\geq 2$ for the range $\frac{n+1}{4}\leq \alpha <\frac{n}2$.

math.AP↗

arXiv2025

The $L^p$-continuity of wave operators for fractional order Schr\"odinger operators

We consider fractional Schr\"odinger operators $H=(-\Delta)^\alpha+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>2\alpha$, $\alpha>1$. We show that the wave operators extend to bounded operators on $L^p(\mathbb R^n)$ for all $1\leq p\leq\infty$ under conditions on the potential that depend on $n$ and $\alpha$ analogously to the case when $\alpha\in \mathbb N$. As a consequence, we deduce a family of dispersive and Strichartz estimates for the perturbed fractional Schr\"odinger operator.

math.AP↗

arXiv2024

Generalizable, Fast, and Accurate DeepQSPR with fastprop

Quantitative Structure Property Relationship studies aim to define a mapping between molecular structure and arbitrary quantities of interest. This was historically accomplished via the development of descriptors which requires significant domain expertise and struggles to generalize. Thus the field has morphed into Molecular Property Prediction and been given over to learned representations which are highly generalizable. The paper introduces fastprop, a DeepQSPR framework which uses a cogent set of molecular level descriptors to meet and exceed the performance of learned representations on diverse datasets in dramatically less time. fastprop is freely available on github at github.com/JacksonBurns/fastprop.

cs.LG↗

arXiv2021

Diversity of symptom phenotypes in SARS-CoV-2 community infections observed in multiple large datasets

Through the use of cutting-edge unsupervised classification techniques from statistics and machine learning, we characterise symptom phenotypes among symptomatic SARS-CoV-2 PCR-positive community cases. We first analyse each dataset in isolation and across age bands, before using methods that allow us to compare multiple datasets. While we observe separation due to the total number of symptoms experienced by cases, we also see a separation of symptoms into gastrointestinal, respiratory and other types, and different symptom co-occurrence patterns at the extremes of age. In this way, we are able to demonstrate the deep structure of symptoms of COVID-19 without usual biases due to study design. This is expected to have implications for the identification and management of community SARS-CoV-2 cases and could be further applied to symptom-based management of other diseases and syndromes.

stat.AP↗

arXiv2021

The $L^p$-continuity of wave operators for higher order Schr\"odinger operators

We consider the higher order Schr\"odinger operator $H=(-\Delta)^m+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>2m$, $m\in \mathbb N$, $m>1$. When $n$ is odd, we prove that the wave operators extend to bounded operators on $L^p(\mathbb R^n)$ for all $1\leq p\leq\infty$ under $n$ and $m$ dependent conditions on the potential analogous to the case when $m=1$. Further, if $V$ is small in certain norms, that depend $n$ and $m$, the wave operators are bounded on the same range for even $n$. We further show that if the smallness assumption is removed in even dimensions the wave operators remain bounded in the range $1<p<\infty$.

math.AP↗

arXiv2017

Order out of Randomness : Self-Organization Processes in Astrophysics

Self-organization is a property of dissipative nonlinear processes that are governed by an internal driver and a positive feedback mechanism, which creates regular geometric and/or temporal patterns and decreases the entropy, in contrast to random processes. Here we investigate for the first time a comprehensive number of 16 self-organization processes that operate in planetary physics, solar physics, stellar physics, galactic physics, and cosmology. Self-organizing systems create spontaneous {\sl order out of chaos}, during the evolution from an initially disordered system to an ordered stationary system, via quasi-periodic limit-cycle dynamics, harmonic mechanical resonances, or gyromagnetic resonances. The internal driver can be gravity, rotation, thermal pressure, or acceleration of nonthermal particles, while the positive feedback mechanism is often an instability, such as the magneto-rotational instability, the Rayleigh-Bénard convection instability, turbulence, vortex attraction, magnetic reconnection, plasma condensation, or loss-cone instability. Physical models of astrophysical self-organization processes involve hydrodynamic, MHD, and N-body formulations of Lotka-Volterra equation systems.

astro-ph.SR↗