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Aidan Lau

Publications and source records attributed to Aidan Lau.

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Stochastic homogenization of coarse-grained elliptic equations

We prove quenched stochastic homogenization for divergence-form elliptic equations, under the assumption that the coefficients are stationary, ergodic, integrable, and satisfy a coarse-grained ellipticity assumption. The ellipticity assumption requires that the coefficients remain bounded in a negative regularity sense on large scales. As a corollary, we recover a sufficient joint integrability condition on the symmetric and skew-symmetric parts of the coefficient field.

math.AP

Coarse-graining and quantitative stochastic homogenization of parabolic equations in high contrast

We prove quantitative homogenization results for high contrast parabolic equations with random coefficients depending on both space and time. In particular, we prove that under a sufficient decorrelation assumption the homogenization length scale is bounded by $\exp(C\log^2(1+Λ/λ)) + C\sqrtλ$. The proof is based on a parabolic coarse-graining framework which generalizes the results of Armstrong and Kuusi in the elliptic setting.

math.AP

DuoLens: A Framework for Robust Detection of Machine-Generated Multilingual Text and Code

The prevalence of Large Language Models (LLMs) for generating multilingual text and source code has only increased the imperative for machine-generated content detectors to be accurate and efficient across domains. Current detectors, predominantly utilizing zero-shot methods, such as Fast DetectGPT or GPTZero, either incur high computational cost or lack sufficient accuracy, often with a trade-off between the two, leaving room for further improvement. To address these gaps, we propose the fine-tuning of encoder-only Small Language Models (SLMs), in particular, the pre-trained models of RoBERTA and CodeBERTa using specialized datasets on source code and other natural language to prove that for the task of binary classification, SLMs outperform LLMs by a huge margin whilst using a fraction of compute. Our encoders achieve AUROC $= 0.97$ to $0.99$ and macro-F1 $0.89$ to $0.94$ while reducing latency by $8$-$12\times$ and peak VRAM by $3$-$5\times$ at $512$-token inputs. Under cross-generator shifts and adversarial transformations (paraphrase, back-translation; code formatting/renaming), performance retains $\geq 92%$ of clean AUROC. We release training and evaluation scripts with seeds and configs; a reproducibility checklist is also included.

cs.CL

Microscopic Theory of the Magnetic Susceptibility of Insulators

We present a general theory of the magnetic susceptibility of insulators that can be extended to treat spatially varying and finite frequency fields. While there are existing results in the literature for the zero frequency response that appear to be in disagreement with each other, we show that the apparent differences between them vanish with the use of various sum rules, and that our result is in agreement with them. Although our strategy is based on the use of Wannier functions, we show that our result can be written in a ``gauge invariant" form involving Bloch functions. We can write it as the sum of terms that involve the diagonal elements of the Berry connection, and this decomposition is particularly useful in considering the limit of isolated molecules. But these contributions can be repackaged to give a form independent of those diagonal elements, which is thus generally more suitable for numerical computation. We consider an h-BN model to demonstrate the practical considerations in building a model and making calculations within this formalism.

cond-mat.mes-hall