We develop a novel approach to tackle the common but challenging problem of conformal inference for missing data in machine learning, focusing on Missing at Random (MAR) data. We propose a new procedure Conformal prediction for Missing data under Multiple Robust Learning (CM–MRL) that combines split conformal calibration with a multiple robust empirical-likelihood (EL) reweighting scheme. The method proceeds via a double calibration by reweighting the complete-case scores by EL so that their distribution matches the full calibration distribution implied by MAR, even when some working models are misspecified. We demonstrate the asymptotic behavior of our estimators through empirical process theory and provide reliable coverage for our prediction intervals, both marginally and conditionally and we further show an interval-length dominance result. We show the effectiveness of the proposed method by several numerical experiments in the presence of missing data.
Let C be a small, connected category with finite hom-sets. We show that if the embedding of a connected subcategory J is both initial and final, then the restriction of any C-module along J preserves the generalized rank-or equivalently, the multiplicity of the "entire" interval modules for C and J. Conversely, we prove that this property characterizes initial and final embeddings when both C and J are posets satisfying certain mild constraints and the embedding is full. For C a poset under these conditions, we describe the minimal full subposet whose embedding is initial or final. This generalizes an observation made by Dey and Lesnick. We also extend a result of Kinser on the generalized rank invariant to small categories. (c) 2026 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).
Large language models have exhibited impressive performance across a broad range of downstream tasks in natural language processing. However, how a language model predicts the next token and generates content is not generally understandable to humans. Furthermore, these models often make errors in prediction and reasoning, known as hallucinations. These errors underscore the urgent need to better understand and interpret the intricate inner workings of language models and how they generate predictive outputs. Motivated by this gap, this paper investigates local explainability and mechanistic interpretability in Transformer-based large language models to foster trust in them. In this regard, our paper aims to make three key contributions. First, we present a review of local explainability and mechanistic interpretability approaches, along with insights from relevant studies in the literature. Furthermore, we describe experimental studies on explainability and reasoning with large language models in two critical domains – healthcare and autonomous driving – and analyze the trust implications of such explanations for explanation receivers. Finally, we summarize the current unaddressed issues in the evolving landscape of LLM explainability and outline opportunities, critical challenges, and future directions for generating human-aligned, trustworthy LLM explanations.
This work investigates the initial value problem (IVP) for the two-parameter family of dispersive wave equations known as the Majda-McLaughlin-Tabak (MMT) model, which arises in the weak turbulence theory of random waves. The MMT model can be viewed as a derivative nonlinear Schrödinger (dNLS) equation where both the nonlinearity and dispersion involve nonlocal fractional derivatives. The purpose of this study is twofold: first, to establish a sharp well-posedness theory for the MMT model; and second, to identify the critical threshold for the derivative in the nonlinearity relative to the dispersive order required to ensure well-posedness. As a by- product, we establish sharp well-posedness for non-local fractional dNLS equations; notably, our results resolve the regularity endpoint left open in https://www.aimsciences.org/article/doi/10.3934/dcdsb.2022039 .
In water-based oil sands extraction, producing cleaner bitumen froth with reduced solids content during primary separation is critical for maintaining high throughput and minimizing downstream solvent loss. This study investigated sodium citrate (Na(3)Cit) as a chemical aid for bitumen froth flotation using synchrotron-based micro-computed tomography (micro-CT) imaging, supported by complementary characterization methods, to elucidate the mechanisms responsible for solids reduction. The addition of Na(3)Cit (150 g/t) reduced the total solids carried to bitumen froth by 18.4 wt% and decreased the solid-to-bitumen (S/B) ratio from 1.15 to 0.97, indicating a significant improvement in froth quality. Micro-CT image segmentation showed that Na(3)Cit selectively influenced sand flotation while leaving the flotation of heavy minerals largely unaffected. In addition, the reduction in solids carryover was primarily associated with decreased water entrainment and reduced true flotation of bitumen-coated sands. The reconstructed X-ray images further suggested enhanced bitumen coalescence accompanied by a notably larger air bubble size and increased total air holdup when Na(3)Cit was used, supporting the hypothesis that Na(3)Cit weakens and softens bitumen-water interfacial film.
