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This work proposes a general framework for capturing noise-driven transitions in spatially extended non-equilibrium systems and explains the emergence of coherent patterns beyond the instability onset. The framework relies on stochastic parameterizations to reduce the original equations' complexity while capturing the key effects of unresolved scales. It works for both Gaussian and Levy-type noise. Our parameterizations offer two key advantages. First, they approximate stochastic invariant manifolds when the latter exist. Second, even when such manifolds break down, our formulas can be adapted by a simple optimization of its constitutive parameters. This allows us to handle scenarios with weak time-scale separation where the system has undergone multiple transitions, resulting in large-amplitude solutions not captured by invariant manifold or other time-scale separation methods. The optimized stochastic parameterizations capture how small-scale noise impacts larger scales through the system's nonlinear interactions. This effect is achieved by the very fabric of our parameterizations incorporating non-Markovian coefficients into the reduced equation. Such coefficients account for the noise's past influence using a finite memory length, selected for optimal performance. The specific "memory" function, which determines how this past influence is weighted, depends on the noise's strength and how it interacts with the system's nonlinearities. Remarkably, training our theory-guided reduced models on a single noise path effectively learns the optimal memory length for out-of-sample predictions, including rare events. This success stems from our "hybrid" approach, which combines analytical understanding with data-driven learning. This combination avoids a key limitation of purely data-driven methods: their struggle to generalize to unseen scenarios, also known as the "extrapolation problem."

Maximum distance separable (MDS) and near maximum distance separable (NMDS) codes have been widely used in various fields such as communication systems, data storage, and quantum codes due to their algebraic properties and excellent error-correcting capabilities. This paper focuses on a specific class of linear codes and establishes necessary and sufficient conditions for them to be MDS or NMDS. Additionally, we employ the well-known Schur method to demonstrate that they are non-equivalent to generalized Reed-Solomon codes.

User interactions in recommender systems are inherently complex, often involving behaviors that go beyond simple acceptance or rejection. One particularly common behavior is hesitation, where users deliberate over recommended items, signaling uncertainty. Our large-scale surveys, with 6,644 and 3,864 responses respectively, confirm that hesitation is not only widespread but also has a profound impact on user experiences. When users spend additional time engaging with content they are ultimately uninterested in, this can lead to negative emotions, a phenomenon we term as tolerance. The surveys reveal that such tolerance behaviors often arise after hesitation and can erode trust, satisfaction, and long-term loyalty to the platform. For instance, a click might reflect a need for more information rather than genuine interest, and prolonged exposure to unsuitable content amplifies frustration. This misalignment between user intent and system interpretation introduces noise into recommendation training, resulting in suggestions that increase uncertainty and disengagement. To address these issues, we identified signals indicative of tolerance behavior and analyzed datasets from both e-commerce and short-video platforms. The analysis shows a strong correlation between increased tolerance behavior and decreased user activity. We integrated these insights into the training process of a recommender system for a major short-video platform. Results from four independent online A/B experiments demonstrated significant improvements in user retention, achieved with minimal additional computational costs. These findings underscore the importance of recognizing hesitation as a ubiquitous user behavior and addressing tolerance to enhance satisfaction, build trust, and sustain long-term engagement in recommender systems.

Approximating field variables and data vectors from sparse samples is a key challenge in computational science. Widely used methods such as gappy proper orthogonal decomposition and empirical interpolation rely on linear approximation spaces, limiting their effectiveness for data representing transport-dominated and wave-like dynamics. To address this limitation, we introduce quadratic manifold sparse regression, which trains quadratic manifolds with a sparse greedy method and computes approximations on the manifold through novel nonlinear projections of sparse samples. The nonlinear approximations obtained with quadratic manifold sparse regression achieve orders of magnitude higher accuracies than linear methods on data describing transport-dominated dynamics in numerical experiments.

Copula-based dependence modelling often relies on parametric formulations. This is mathematically convenient but can be statistically inefficient if the parametric families are not suitable for the data and model in focus. To improve the flexibility in modeling dependence, we consider a Bayesian nonparametric mixture model of Archimedean copulas which can capture complex dependence patterns and can be extended to arbitrary dimensions. In particular we use the Poisson-Dirichlet process as mixing distribution over the single parameter of the Archimedean copulas. Properties of the mixture model are studied for the main Archimedenan families and posterior distributions are sampled via their full conditional distributions. Performance of the model is via numerical experiments involving simulated and real data.

Two sequential estimators are proposed for the odds p/(1-p) and log odds log(p/(1-p)) respectively, using independent Bernoulli random variables with parameter p as inputs. The estimators are unbiased, and guarantee that the variance of the estimation error divided by the true value of the odds, or the variance of the estimation error of the log odds, are less than a target value for any p in (0,1). The estimators are close to optimal in the sense of Wolfowitz's bound.

An extremely schematic model of the forces acting an a sailing yacht equipped with a system of foils is here presented and discussed. The role of the foils is to raise the hull from the water in order to reduce the total resistance and then increase the speed. Some CFD simulations are providing the total resistance of the bare hull at some values of speed and displacement, as well as the characteristics (drag and lift coefficients) of the 2D foil sections used for the appendages. A parametric study has been performed for the characterization of a foil of finite dimensions. The equilibrium of the vertical forces and longitudinal moments, as well as a reduced displacement, is obtained by controlling the pitch angle of the foils. The value of the total resistance of the yacht with foils is then compared with the case without foils, evidencing the speed regime where an advantage is obtained, if any.

We consider anisotropic heat flow with extreme anisotropy, as arises in magnetized plasmas for fusion applications. Such problems pose significant challenges in both obtaining an accurate approximation as well in the construction of an efficient solver. In both cases, the underlying difficulty is in forming an accurate approximation of temperature fields that follow the direction of complex, non-grid-aligned magnetic fields. In this work, we construct a highly accurate coarse grid approximation using spectral multiscale basis functions based on local anisotropic normalized Laplacians. We show that the local generalized spectral problems yield local modes that align with magnetic fields, and provide an excellent coarse-grid approximation of the problem. We then utilize this spectral coarse space as an approximation in itself, and as the coarse-grid in a two-level spectral preconditioner. Numerical results are presented for several magnetic field distributions and anisotropy ratios up to $10^{12}$, showing highly accurate results with a large system size reduction, and two-grid preconditioning that converges in $O(1)$ iterations, independent of anisotropy.

We initiate the focused study of constant-cost randomized communication, with emphasis on its connection to graph representations. We observe that constant-cost randomized communication problems are equivalent to hereditary (i.e. closed under taking induced subgraphs) graph classes which admit constant-size adjacency sketches and probabilistic universal graphs (PUGs), which are randomized versions of the well-studied adjacency labeling schemes and induced-universal graphs. This gives a new perspective on long-standing questions about the existence of these objects, including new methods of constructing adjacency labeling schemes. We ask three main questions about constant-cost communication, or equivalently, constant-size PUGs: (1) Are there any natural, non-trivial problems aside from Equality and k-Hamming Distance which have constant-cost communication? We provide a number of new examples, including deciding whether two vertices have path-distance at most k in a planar graph, and showing that constant-size PUGs are preserved by the Cartesian product operation. (2) What structures of a problem explain the existence or non-existence of a constant-cost protocol? We show that in many cases a Greater-Than subproblem is such a structure. (3) Is the Equality problem complete for constant-cost randomized communication? We show that it is not: there are constant-cost problems which do not reduce to Equality.

Due to their inherent capability in semantic alignment of aspects and their context words, attention mechanism and Convolutional Neural Networks (CNNs) are widely applied for aspect-based sentiment classification. However, these models lack a mechanism to account for relevant syntactical constraints and long-range word dependencies, and hence may mistakenly recognize syntactically irrelevant contextual words as clues for judging aspect sentiment. To tackle this problem, we propose to build a Graph Convolutional Network (GCN) over the dependency tree of a sentence to exploit syntactical information and word dependencies. Based on it, a novel aspect-specific sentiment classification framework is raised. Experiments on three benchmarking collections illustrate that our proposed model has comparable effectiveness to a range of state-of-the-art models, and further demonstrate that both syntactical information and long-range word dependencies are properly captured by the graph convolution structure.