arXiv subjects

Adam Carter

Publications and source records attributed to Adam Carter.

8 recordsLinked to original sources

arXiv2026

A QUBO Formulation for Nowhere-Zero $k$-Flows

We consider the encoding of graph problems as Quadratic Unconstrained Binary Optimization (QUBO) problems, which are solvable by either quantum or classical annealers. Yet, the class of problems encoded as QUBO problems has not previously included nowhere-zero flows. Nowhere-zero flows are related to Tutte's $5$-flow conjecture and appear in many contexts in graph theory. We provide an encoding of nowhere-zero flows as a QUBO Hamiltonian and prove the correctness of the construction. Our construction yields a Hamiltonian $H_{\mathrm{mod},k}$ whose ground state has zero energy if and only if the graph $G$ has a nowhere-zero $\mathbb Z_k$-flow. By Tutte's equivalence theorem, zero ground energy is equivalent to $\varphi(G)\le k$, and the zero-energy degeneracy is given by the flow polynomial $F(G;k)$. In particular, when the ground-state energy is zero, this is also the ground-state degeneracy. The construction uses one-hot variables to represent the edge flow residues modulo $k$ and auxiliary variables to represent the per-vertex modular quotient. We prove that the correctness of the construction is independent of the choice of orientation, root vertex, and positive penalty weights. We verify the construction on $59$ examples of graphs and values of $k$ that include both yes-instances and no-instances. We exhaustively sweep orientations and root choices on selected robustness instances and test a finite suite of positive penalty weights. The resulting Hamiltonian is implemented using the dimod.BinaryQuadraticModel class, which is compatible with the D-Wave Ocean SDK. Quantum-hardware runs and claims about potential speedup using these devices are left to follow-up work.

quant-ph↗

arXiv2026

The "Intensity" Countoscope: Measuring particle dynamics in real space from microscopy images

Advances in intensity-based microscopy techniques have improved our ability to quantify particle motion at microscopic scales, enabling insight into diffusion and collective dynamics. Building on this foundation, we introduce a novel real-space approach that analyses intensity fluctuations within virtual observation boxes of variable size on microscopy images. By correlating these signals, we uncover distinct temporal regimes in the mean square changes of intensity, $\langle \Delta I^2(t) \rangle$, which are strongly dependent on the box size compared to the particle width. For small boxes or short timescales, $\langle \Delta I^2(t) \rangle$ scales with the mean-square displacement, while for longer timescales and larger boxes, it scales with its square root. We develop a general theoretical framework that captures these regimes and explicitly apply it to a dilute colloidal suspension imaged with confocal microscopy as an experimental model system. This allows for a robust extraction of diffusion coefficients and physical insights into particle dynamics. Our method complements intensity-based and real-space analysis, offering a tool for studying individual and potentially collective behaviour directly from image intensities, even in systems where individual particles cannot be resolved.

cond-mat.soft↗

arXiv2026

Brownian dynamics with soft constraints in soft matter systems

Stiff forces, which bind objects together or otherwise confine motion, are found widely in soft-matter systems -- colloids with short range attractions, ligand-receptor contacts, particles in optical traps, fibres that resist stretching, etc. To assess the long-term effect of these stiff forces on dynamics and structure, it is useful to consider the limit where they are treated as constraints, so the system evolves strictly within allowed configurations. Efforts to derive equations involving both constraints, and the stochastic motion appropriate at the scales of soft matter, began around 50 years ago, yet, we are still lacking a straightforward way to extract the projected equations and apply them in modern formulations of mesoscale dynamics. Here, we address this gap with two key contributions: (1) a practical summary of the constrained Brownian dynamics equations with ``soft'' constraints, i.e. constraints imposed by stiff forces, which is illustrated through several representative examples, taking care to highlight the nontrivial effects of the constraints; and (2) a novel derivation using singular perturbation theory, establishing the validity of these equations over timescales exceeding the relaxation of stiffly constrained degrees of freedom. We further extend our approach to ``soft soft'' constraints, where mobility varies on lengthscales comparable to the restraining forces -- a scenario typical for particles in fluids experiencing hydrodynamic interactions. We hope our results will be useful for soft matter research, as a robust toolkit for studying tethered or confined systems.

cond-mat.soft↗

arXiv2026

The Normalized Difference Layer: A Differentiable Spectral Index Formulation for Deep Learning

Normalized difference indices have been a staple in remote sensing for decades. They stay reliable under lighting changes produce bounded values and connect well to biophysical signals. Even so, they are usually treated as a fixed pre processing step with coefficients set to one, which limits how well they can adapt to a specific learning task. In this study, we introduce the Normalized Difference Layer that is a differentiable neural network module. The proposed method keeps the classical idea but learns the band coefficients from data. We present a complete mathematical framework for integrating this layer into deep learning architectures that uses softplus reparameterization to ensure positive coefficients and bounded denominators. We describe forward and backward pass algorithms enabling end to end training through backpropagation. This approach preserves the key benefits of normalized differences, namely illumination invariance and outputs bounded to $[-1,1]$ while allowing gradient descent to discover task specific band weightings. We extend the method to work with signed inputs, so the layer can be stacked inside larger architectures. Experiments show that models using this layer reach similar classification accuracy to standard multilayer perceptrons while using about 75\% fewer parameters. They also handle multiplicative noise well, at 10\% noise accuracy drops only 0.17\% versus 3.03\% for baseline MLPs. The learned coefficient patterns stay consistent across different depths.

cs.CV↗

arXiv2025

Interpretable Machine Learning-Derived Spectral Indices for Vegetation Monitoring

Spectral indices such as NDVI have driven vegetation monitoring for decades, yet their design remains largely manual and ad hoc. Their usefulness stems not only from their empirical performance, but also from algebraic forms that remain compact and biologically interpretable. However, the space of possible algebraic expressions relating spectral bands is effectively infinite, making systematic search impractical without structural constraints. We introduce the Spectral Feature Polynomial (SFP) framework, a general pipeline that automatically discovers compact, interpretable spectral indices from labeled multispectral imagery. SFP constructs a library of ratio-based spectral features that inherit illumination invariance by construction. It then applies cross-validated feature selection and continuous coefficient optimization to produce a single closed-form equation per task, transparent to domain experts and deployable on any remote sensing platform without requiring standardization statistics. We validate the framework on two agricultural applications. For Kochia (Bassia scoparia) detection in Sentinel-2 imagery near Lucky Lake of Saskatchewan over three growing seasons, the same two-term equation emerged in 44 of 46 independent cross-validation folds, achieving 98.6% mean accuracy, more than 4 percentage points above the best established index under year-held-out evaluation. For wheat plant classification from UAV multispectral imagery, stage-specific indices achieved 99.5%, 97.2%, and 93.5% across three growth stages, compared to 78% or below for the best established index at late season when NIR-based contrasts lose discriminatory power as wheat senesces. In both applications, SFP yielded a single transparent equation that generalized across held-out regions and outperformed established indices.

cs.CV↗

arXiv2025

Collective Hard Core Interactions Leave Multiscale Signatures in Number Fluctuation Spectra

A full understanding of transport in dense, interacting suspensions requires analysis frameworks sensitive to self and collective dynamics across all relevant spatial and temporal scales. Here we introduce a trajectory-free approach to address this problem based on the power spectral density of particle number fluctuations (N-PSD). By combining colloidal experiments and theory we show that the N-PSD naturally probes behaviour across multiple important dynamic regimes and we fully uncover the mechanistic origins of characteristic spectral scalings and timescales. In particular, we demonstrate that while high-frequency scalings link to self-diffusion, low-frequency scalings sensitively capture long-lived correlations and collective dynamics. In this regime, interactions lead to non-trivial spectral signatures, governed by pairwise particle exchange at small length scales and collective rearrangements over large scales. Our findings thus provide important insight into the effect of interactions on microscopic dynamics and fluctuation phenomena and establish a powerful new tool with which to probe dynamics in complex systems.

cond-mat.soft↗

arXiv2025

libMobility: A Python library for hydrodynamics at the Smoluchowski level

Effective hydrodynamic modeling is crucial for accurately predicting fluid-particle interactions in diverse fields such as biophysics and materials science. Developing and implementing hydrodynamic algorithms is challenging due to the complexity of fluid dynamics, necessitating efficient management of large-scale computations and sophisticated boundary conditions. Furthermore, adapting these algorithms for use on massively parallel architectures like GPUs adds an additional layer of complexity. This paper presents the libMobility software library, which offers a suite of CUDA-enabled solvers for simulating hydrodynamic interactions in particulate systems at the Rotne-Prager-Yamakawa (RPY) level. The library facilitates precise simulations of particle displacements influenced by external forces and torques, including both the deterministic and stochastic components. Notable features of libMobility include its ability to handle linear and angular displacements, thermal fluctuations, and various domain geometries effectively. With an interface in Python, libMobility provides comprehensive tools for researchers in computational fluid dynamics and related fields to simulate particle mobility efficiently. This article details the technical architecture, functionality, and wide-ranging applications of libMobility. libMobility is available at https://github.com/stochasticHydroTools/libMobility.

cond-mat.soft↗

arXiv2024

Measuring collective diffusion properties by counting particles in boxes

The collective diffusion coefficient $D_\mathrm{coll}$ is a key quantity for describing the macroscopic transport properties of soft matter systems. However, measuring $D_\mathrm{coll}$ is a fundamental experimental and numerical challenge, as it either relies on nonequilibrium techniques that are hard to interpret or, at equilibrium, on Fourier-based approaches which are fraught with difficulties associated with Fourier transforms. In this work, we investigate the equilibrium diffusive dynamics of a 2D colloidal suspension experimentally and numerically. We use a "Countoscope" technique, which analyses the statistics of particle number counts $N(t)$ in virtual observation boxes of a series of microscopy images at equilibrium, to measure $D_\mathrm{coll}$ for the first time. We validate our results against Fourier-based approaches and establish best practices for measuring $D_\mathrm{coll}$ using fluctuating counts. We show that Fourier techniques yield inaccurate long-range collective measurements because of the non-periodic nature of an experimental image, yet counting exploits this property by using finite observation windows. Finally, we discuss the potential of our method to advance our understanding of collective properties in suspensions, particularly the role of hydrodynamic interactions.

cond-mat.soft↗