We develop a computational framework for classifying Galois groups of irreducible degree-7 polynomials over~$\mathbb{Q}$, combining explicit resolvent methods with machine learning techniques. A database of over one million normalized projective septics is constructed, each annotated with algebraic invariants~$J_0, \dots, J_4$ derived from binary transvections. For each polynomial, we compute resolvent factorizations to determine its Galois group among the seven transitive subgroups of~$S_7$ identified by Foulkes. Using this dataset, we train a neurosymbolic classifier that integrates invariant-theoretic features with supervised learning, yielding improved accuracy in detecting rare solvable groups compared to coefficient-based models. The resulting database provides a reproducible resource for constructive Galois theory and supports empirical investigations into group distribution under height constraints. The methodology extends to higher-degree cases and illustrates the utility of hybrid symbolic-numeric techniques in computational algebra.

Neural Information Processing Systems http://nips.cc/

In these diverse real-world applications, there is often a degree of uncertainty in the input which has led to the study of stochastic matching models.

In these diverse real-world applications, there is often a degree of uncertainty in the input which has led to the study of stochastic matching models.

We then apply matrix analysis to derive the form of optimal noise covariance. Presented constraint and results are validated by the empirical observations.

Specifically, we extend the normal approximation-based lower bound for Beta distributions by Alfers and Dinges [1984] to Dirichlet distributions.

Existing approaches to posterior collapse often attribute it to the use of neural networks or optimization issues due to variational approximation. In this paper, we consider posterior collapse as a problem of latent variable non-identifiability. We prove that the posterior collapses if and only if the latent variables are non-identifiable in the generative model. This fact implies that posterior collapse is not a phenomenon specific to the use of flexible distributions or approximate inference. Rather, it can occur in classical probabilistic models even with exact inference, which we also demonstrate.

In the context of global sustainability mandates, corporate carbon disclosure has emerged as a critical mechanism for aligning business strategy with environmental responsibility. The Carbon Disclosure Project (CDP) hosts the world's largest longitudinal dataset of climate-related survey responses, combining structured indicators with open-ended narratives, but the heterogeneity and free-form nature of these disclosures present significant analytical challenges for benchmarking, compliance monitoring, and investment screening. This paper proposes a novel decision-support framework that leverages large language models (LLMs) to assess corporate climate disclosure quality at scale. It develops a master rubric that harmonizes narrative scoring across 11 years of CDP data (2010-2020), enabling cross-sector and cross-country benchmarking. By integrating rubric-guided scoring with percentile-based normalization, our method identifies temporal trends, strategic alignment patterns, and inconsistencies in disclosure across industries and regions. Results reveal that sectors such as technology and countries like Germany consistently demonstrate higher rubric alignment, while others exhibit volatility or superficial engagement, offering insights that inform key decision-making processes for investors, regulators, and corporate environmental, social, and governance (ESG) strategists. The proposed LLM-based approach transforms unstructured disclosures into quantifiable, interpretable, comparable, and actionable intelligence, advancing the capabilities of AI-enabled decision support systems (DSSs) in the domain of climate governance.