Parametric PDEs power modern simulation, design, and digital-twin systems, yet their many-query workloads still hinge on repeatedly solving large finite-element systems. Existing operator-learning approaches accelerate this process but often rely on opaque learned trunks, require extensive labeled data, or break down when boundary and source data vary independently from physical parameters. We introduce RB-DeepONet, a hybrid operator-learning framework that fuses reduced-basis (RB) numerical structure with the branch-trunk architecture of DeepONet. The trunk is fixed to a rigorously constructed RB space generated offline via Greedy selection, granting physical interpretability, stability, and certified error control. The branch network predicts only RB coefficients and is trained label-free using a projected variational residual that targets the RB-Galerkin solution. For problems with independently varying loads or boundary conditions, we develop boundary and source modal encodings that compress exogenous data into low-dimensional coordinates while preserving accuracy. Combined with affine or empirical interpolation decompositions, RB-DeepONet achieves a strict offline-online split: all heavy lifting occurs offline, and online evaluation scales only with the RB dimension rather than the full mesh. We provide convergence guarantees separating RB approximation error from statistical learning error, and numerical experiments show that RB-DeepONet attains accuracy competitive with intrusive RB-Galerkin, POD-DeepONet, and FEONet while using dramatically fewer trainable parameters and achieving significant speedups. This establishes RB-DeepONet as an efficient, stable, and interpretable operator learner for large-scale parametric PDEs.

The quantity of interest in the classical Cramér-Rao theory of unbiased estimation (e.g., the Cramér-Rao lower bound, its exact attainment for exponential families, and asymptotic efficiency of maximum likelihood estimation) is the variance, which represents the instability of an estimator when its value is compared to the value for an independently-sampled data set from the same distribution. In this paper we are interested in a quantity which represents the instability of an estimator when its value is compared to the value for an infinitesimal additive perturbation of the original data set; we refer to this as the "sensitivity" of an estimator. The resulting theory of sensitivity is based on the Wasserstein geometry in the same way that the classical theory of variance is based on the Fisher-Rao (equivalently, Hellinger) geometry, and this insight allows us to determine a collection of results which are analogous to the classical case: a Wasserstein-Cramér-Rao lower bound for the sensitivity of any unbiased estimator, a characterization of models in which there exist unbiased estimators achieving the lower bound exactly, and some concrete results that show that the Wasserstein projection estimator achieves the lower bound asymptotically. We use these results to treat many statistical examples, sometimes revealing new optimality properties for existing estimators and other times revealing entirely new estimators.

Given a graph $G$, its independence sequence is the integral sequence $a_1,a_2,...,a_n$, where $a_i$ is the number of independent sets of vertices of size i. In the late 80's Alavi, Erdos, Malde, Schwenk showed that this sequence need not be unimodal for general graphs, but conjectured that it is always unimodal whenever $G$ is a tree. This conjecture was then naturally generalized to claim that the independence sequence of trees should be log concave, in the sense that $a_i^2$ is always above $a_{i-1}a_{i+1}$. This conjecture stood for many years, until in 2023, Kadrawi, Levit, Yosef, and Mizrachi proved that there were exactly two trees on 26 vertices whose independence sequence was not log concave. In this paper, we use the AI architecture PatternBoost, developed by Charton, Ellenberg, Wagner, and Williamson to train a machine to find counter-examples to the log-concavity conjecture. We will discuss the successes of this approach - finding tens of thousands of new counter-examples to log-concavity with vertex set sizes varying from 27 to 101 - and some of its fascinating failures.

Random permutation set (RPS) is a recently proposed framework designed to represent order-structured uncertain information. Measuring the distance between permutation mass functions is a key research topic in RPS theory (RPST). This paper conducts an in-depth analysis of distances between RPSs from two different perspectives: random finite set (RFS) and transferable belief model (TBM). Adopting the layer-2 belief structure interpretation of RPS, we regard RPST as a refinement of TBM, where the order in the ordered focus set represents qualitative propensity. Starting from the permutation, we introduce a new definition of the cumulative Jaccard index to quantify the similarity between two permutations and further propose a distance measure method for RPSs based on the cumulative Jaccard index matrix. The metric and structural properties of the proposed distance measure are investigated, including the positive definiteness analysis of the cumulative Jaccard index matrix, and a correction scheme is provided. The proposed method has a natural top-weightiness property: inconsistencies between higher-ranked elements tend to result in greater distance values. Two parameters are provided to the decision-maker to adjust the weight and truncation depth. Several numerical examples are used to compare the proposed method with the existing method. The experimental results show that the proposed method not only overcomes the shortcomings of the existing method and is compatible with the Jousselme distance, but also has higher sensitivity and flexibility.

Although conceptually related, variable selection and relative importance (RI) analysis have been treated quite differently in the literature. While RI is typically used for post-hoc model explanation, this paper explores its potential for variable ranking and filter-based selection before model creation. Specifically, we anticipate strong performance from the RI measures because they incorporate both direct and combined effects of predictors, addressing a key limitation of marginal correlation that ignores dependencies among predictors. We implement and evaluate the RI-based variable selection methods using general dominance (GD), comprehensive relative importance (CRI), and a newly proposed, computationally efficient variant termed CRI.Z. We first demonstrate how the RI measures more accurately rank the variables than the marginal correlation, especially when there are suppressed or weak predictors. We then show that predictive models built on these rankings are highly competitive, often outperforming state-of-the-art methods such as the lasso and relaxed lasso. The proposed RI-based methods are particularly effective in challenging cases involving clusters of highly correlated predictors, a setting known to cause failures in many benchmark methods. Although lasso methods have dominated the recent literature on variable selection, our study reveals that the RI-based method is a powerful and competitive alternative. We believe these underutilized tools deserve greater attention in statistics and machine learning communities. The code is available at: https://github.com/tien-endotchang/RI-variable-selection.