The key to achieving efficient separation of C2H2/CO2/C2H4 is to prepare materials with both high adsorption capacity and high selectivity. Flexible two-dimensional metal-organic frameworks (2D MOFs) exhibit excellent performance in difficult separation applications due to their high structural toughness within the layers and strong stimulus responsiveness between layers, but their controllable preparation is a challenge. Herein, under the guidance of crystal engineering, we develop a framework dimensionality reduction strategy (3D → 2D) for tailoring of pillaring ligands to directionally construct flexible 2D MOFs, based on a 3D MOF (UPC-191) constructed by terpyridine pillaring ligands and organic carboxylic acid ligands. The interlayer and intralayer pore environments are synergistically optimized by co-modification of semi-pillared ligands and carboxylic acid ligands as well as metal ion regulation. Remarkably, the optimal structure of difluoro-functionalized MOF (UPC-197) exhibits temperature-responsive framework flexibility with a unique "gate-opening effect" for C2H2 and reverse adsorption behavior for CO2/C2H4 at 195 K, achieving efficient separation of C2H2/C2H4 and CO2/C2H4 as well as C2H2/CO2/C2H4, which has been confirmed by adsorption isotherms, experimental breakthrough curves, and density functional theory calculations. This work provides a guidance for the precise construction of flexible 2D MOFs for the challenging separation of multi-component low-carbon hydrocarbons.

Abstract Steel is a pillar industry of the national economy, in which scrap, as an important secondary raw material for steel production, is of great significance for its rapid identification in order to realize accurate classification for recycling and reuse. In this paper, Laser-Induced Breakdown Spectroscopy (LIBS) combined with the Restricted Boltzmann machine-Backpropagation algorithm (RBM-BP) is used for the rapid identification of 13 steel samples. Based on the collected spectral data, spectral preprocessing was performed using the discrete wavelet transform (DWT) to eliminate redundant information such as spectral interference and background noise. In particular, the number of DWT decomposition layers was 10, the wavelet function was selected as db2, and the calibration RMSEC was 0.99%. The preprocessed data were subjected to downscaling and feature extraction using Restricted Boltzmann machine (RBM) and Principal Component Analysis (PCA), respectively, and then the back propagation algorithm (BP) was used to classify and model the steel samples and compare the performance of the two models, RBM-BP and PCA-BP. The results show that the classification accuracy of the RBM-BP model is up to 99.88%, and the dimensionality reduction time is 16.74 s, which is much lower than the 78.73 s of the PCA-BP model. The measured results show that LIBS combined with the RBM-BP algorithm can realize the fast and accurate classification of steel, and this technology has great potential in the accurate classification of scrap steel for recycling and reuse, which can provide important support for the sustainable development of the steel industry and the construction of a resource-saving and environment-friendly society.

ABSTRACT The key to achieving efficient separation of C 2 H 2 /CO 2 /C 2 H 4 is to prepare materials with both high adsorption capacity and high selectivity. Flexible two‐dimensional metal‐organic frameworks (2D MOFs) exhibit excellent performance in difficult separation applications due to their high structural toughness within the layers and strong stimulus responsiveness between layers, but their controllable preparation is a challenge. Herein, under the guidance of crystal engineering, we develop a framework dimensionality reduction strategy (3D → 2D) for tailoring of pillaring ligands to directionally construct flexible 2D MOFs, based on a 3D MOF (UPC‐191) constructed by terpyridine pillaring ligands and organic carboxylic acid ligands. The interlayer and intralayer pore environments are synergistically optimized by co‐modification of semi‐pillared ligands and carboxylic acid ligands as well as metal ion regulation. Remarkably, the optimal structure of difluoro‐functionalized MOF (UPC‐197) exhibits temperature‐responsive framework flexibility with a unique “gate‐opening effect” for C 2 H 2 and reverse adsorption behavior for CO 2 /C 2 H 4 at 195 K, achieving efficient separation of C 2 H 2 /C 2 H 4 and CO 2 /C 2 H 4 as well as C 2 H 2 /CO 2 /C 2 H 4 , which has been confirmed by adsorption isotherms, experimental breakthrough curves, and density functional theory calculations. This work provides a guidance for the precise construction of flexible 2D MOFs for the challenging separation of multi‐component low‐carbon hydrocarbons.

This study maps European countries’ resilience to cognitive warfare by developing a cross-national composite measure. The framework integrates three pillars: information ecology, institutional-digital capacity, and socioeconomic context—drawing on a systemic perspective linking social structures to societal functions. Publicly available secondary indicators are compiled from online sources for EU (European Union) and EEA (European Economics Area) states. The dataset is examined through descriptive analysis, association testing, multivariate modelling, dimensionality reduction to derive a composite resilience score, and unsupervised clustering to produce a country typology. Indicators capture governance effectiveness, e-government maturity, public-sector AI (Artificial Intelligence) readiness, digital connectivity and infrastructure, media freedom and broader media-ecosystem quality, academic freedom, and socioeconomic vulnerabilities such as youth labour market exclusion. Results show that resilience aligns most strongly with institutional capacity and governance performance; a healthy ecology acts as a reinforcing layer. Digital infrastructure appears necessary but insufficient without capable, credible institutions and coherent public policy. Socioeconomic vulnerabilities tend to erode resilience and heighten susceptibility to hostile cognitive influence. The study concludes that policy efforts should prioritise governance integrity and effectiveness, end-to-end digital government, responsible public-sector AI capability, and safeguards for media and academic autonomy, alongside measures that improve youth inclusion.

This article provides a review of the gauge-theoretic approach to Khovanov homology, framed in terms of a generalisation of Witten's original proposal. Concretely, the physical arguments underlying Witten's insights suggest that there is a one-parameter family of Haydys-Witten instanton Floer homology groups $HF_{\theta}\bigl(W^4\bigr)$ for four-manifolds. At the heart of the proposal is a systematic investigation of the dimensional reductions of the Haydys-Witten equations. It is shown that on the five-dimensional cylinder $M^5=\mathbb{R}_s\times W^4$ with nowhere-vanishing vector field $v=\cos\theta \partial_s+\sin\theta w$, the Haydys-Witten equations provide flow equations for the $\theta$-Kapustin-Witten equations on $W^4$. Similar reductions to lower dimensions include the twisted extended Bogomolny equations on three-manifolds and the twisted octonionic Nahm equations on one-manifolds, whose solutions provide natural boundary conditions along the boundary and corners of $W^4$. These reductions determine the indicial roots of the Haydys-Witten and $\theta$-Kapustin-Witten equations with twisted Nahm-pole boundary conditions, which are required to establish elliptic regularity. Motivated by these insights, the groups $HF_{\theta}\bigl(W^4\bigr)$ are defined in analogy with Yang-Mills instanton Floer theory: solutions of the $\theta$-Kapustin-Witten equations on $W^4$ modulo Haydys-Witten instantons on the cylinder $\mathbb{R}_s\times W^4$ interpolating between them. The relation to knot invariants observed by Witten arises when the four-manifold is the geometric blow-up $W^4=\bigl[X^3\times\mathbb{R}^+,K\bigr]$ along a knot $K\subset X^3\times{0}$ in its three-dimensional boundary. This yields a precise restatement of Witten's conjecture as the equality between $HF^\bullet_{\pi/2}\bigl(\bigl[S^3\times\mathbb{R}^+,K\bigr]\bigr)$ and Khovanov homology $\mathrm{Kh}^\bullet(K)$.

BackgroundCancer-associated fibroblasts (CAFs) play critical roles in tumor progression and immunosuppression; however, their contribution to the functional classification and personalized treatment of gastric cancer remains poorly defined. This study aimed to identify effective therapeutic targets to facilitate individualized treatment strategies for patients with gastric cancer.MethodsSingle-cell and bulk transcriptomic analyses were integrated to characterize gastric cancer fibroblasts. “Seurat”, “Slingshot”, and “CellChat” were used for dimensionality reduction, trajectory inference, and cell–cell communication analyses, respectively. Key metastasis-associated fibroblast modules were identified using High-dimensional weighted gene co-expression network analysis (hdWGCNA) to construct a prognostic model, which was further evaluated for immune infiltration, therapeutic response, and mutational features. The expression and function of the core gene tripeptidyl peptidase 1 (TPP1) were validated through immunoblotting, PCR, and functional assays.ResultsEight fibroblast subpopulations associated with gastric cancer metastasis exhibited distinct differentiation trajectories and transcriptional heterogeneity. Prognostic analysis indicated that metastasis-associated fibroblasts correlated with poor clinical outcomes. The high-risk subgroup showed marked immunosuppression, resistance to immunotherapy, and reduced mutational burden, with tumor progression–related pathways significantly enriched in this group. In vitro experiments further confirmed that TPP1 knockdown suppressed gastric cancer cell metastasis, invasion, and clonogenic capacity while inducing apoptosis.ConclusionThis study characterized the heterogeneity of gastric cancer–associated fibroblasts using single-cell transcriptomic analysis and established a prognostic model based on metastasis-related fibroblast markers. The model demonstrated strong predictive performance for patient prognosis, immune landscape, and immunotherapy response. Furthermore, the findings highlighted the pivotal role of TPP1 in gastric cancer progression and its potential as a therapeutic target.

This study aims to evaluate the impact of four SNPs of human 8-oxoguanine DNA glycosylase 1 (hOGG1) gene, and its interaction with smoking and alcohol drinking on the risk of nasopharyngeal carcinoma (NPC). Hardy-Weinberg equilibrium (HWE) and the relationship between four SNPs of the hOGG1 gene and the risk of NPC were tested using the SNPStats online software ( https://www.snpstats.net/start.htm ). Generalized multifactor dimensionality reduction (GMDR) was utilized to screen the optimal interaction combinations among four hOGG1 gene SNPs, smoking, and alcohol drinking. We found that rs1052133- Cys allele was associated with increased NPC risk, ORs (95% CI) were 1.42 (1.12-1.85) for Ser/Cys genotype, 1.95 (1.51-2.48) for Cys/Cys genotype, 1.57 (1.18-2.05) for Ser/Cys or Cys/Cys genotype, compared to Ser/Ser genotype. We also found that rs159153- T allele was associated with increased NPC risk, ORs (95% CI) were 1.35 (1.06-1.73) for CT genotype, 2.18 (1.62-2.85) for TT genotype, 1.45 (1.09-1.90) for CT or TT genotype, compared to CC genotype. However, we did not find any statistically significant impact of the minor alleles of rs3219008 and rs293795 on the risk of NPC. GMDR model found a significant gene-environment interaction combination (two-locus model with P = 0.018) between rs1052133 and smoking. Compared to never smokers with rs1052133 Ser/Ser genotype, ever or currently smokers with rs1052133- Ser/Cys or Cys/Cys genotype have the highest NPC risk, OR (95% CI) = 3.18 (1.94-4.42). The rs1052133 and rs159153 minor alleles, the interaction between rs1052133 and smoking, were all associated with increased NPC risk.

ABSTRACTWhen developing clinical prediction models, it can be challenging to balance between global models that are valid for all patients and personalized models tailored to individuals or potentially unknown subgroups. To aid such decisions, we propose a diagnostic tool for contrasting global regression models and patient‐specific (local) regression models. The core utility of this tool is to identify where and for whom a global model may be inadequate. We focus on regression models and specifically suggest a localized regression approach that identifies regions in the predictor space where patients are not well represented by the global model. As localization becomes challenging when dealing with many predictors, we propose modeling in a dimension‐reduced latent representation obtained from an autoencoder. Using such a neural network architecture for dimension reduction enables learning a latent representation simultaneously optimized for both good data reconstruction and for revealing local outcome‐related associations suitable for robust localized regression. We illustrate the proposed approach with a clinical study involving patients with chronic obstructive pulmonary disease. Our findings indicate that the global model is adequate for most patients but that indeed specific subgroups benefit from personalized models. We also demonstrate how to map these subgroup models back to the original predictors, providing insight into why the global model falls short for these groups. Thus, the principal application and diagnostic yield of our tool is the identification and characterization of patients or subgroups whose outcome associations deviate from the global model.

We study the design of efficient approximation algorithms for the ℓ-center clustering and minimum-diameter ℓ-clustering problems in high-dimensional Euclidean and Hamming spaces. Our main tool is randomized dimension reduction. First, we present a general method of reducing the dependency of the running time of a hypothetical algorithm for the ℓ-center problem in a high-dimensional Euclidean space on the dimension. Utilizing this method in part, we provide (2+ϵ)-approximation algorithms for the ℓ-center clustering and minimum-diameter ℓ-clustering problems in Euclidean and Hamming spaces that are substantially faster than the known 2-approximation algorithms when both ℓ and the dimension are super-logarithmic. Next, we apply the general method to the recent fast approximation algorithms with higher approximation guarantees for the ℓ-center clustering problem in a high-dimensional Euclidean space. Finally, we provide a speed-up of the known O(1)-approximation method for the generalization of the ℓ-center clustering problem that allows z outliers (i.e., z input points may be ignored when computing the maximum distance from an input point to a center) in high-dimensional Euclidean and Hamming spaces.

Sliced inverse regression (SIR), which includes linear discriminant analysis (LDA) as a special case, is a popular and powerful dimension reduction tool. In this article, we extend SIR to address the challenges of decentralized data, prioritizing privacy and communication efficiency. Our approach, termed as federated sliced inverse regression (FSIR), facilitates distributed computing of the sufficient dimension reduction subspace among multiple clients, solely sharing local estimates to protect sensitive datasets from exposure. To guard against potential adversary attacks, FSIR employs diverse perturbation strategies, including a novel vectorized Gaussian mechanism that guarantees ( ε , δ ) -differential privacy at a low cost of statistical accuracy. Additionally, FSIR achieves a tight composition of various privacy mechanisms by adopting a hypothesis testing perspective on differential privacy. It also incorporates a collaborative feature screening procedure, enabling effective handling of high-dimensional client data with varying feature sets. Theoretical properties of FSIR are established for both low-dimensional and high-dimensional settings, supported by extensive numerical experiments and real data analysis. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Network data are commonly collected in a variety of applications, representing either directly measured or statistically inferred connections between subjects or features of interest. In an increasing number of domains, these networks are collected over time, such as repeated interactions between users of a social media platform, or across multiple subjects, such as in multi-subject neuroimaging studies. When analyzing multiple large networks, dimensionality reduction techniques are often used to embed networks in a more tractable low-dimensional space. To this end, we develop a framework for principal components analysis (PCA) on collections of networks via a specialized tensor decomposition, termed Semi-Symmetric Tensor PCA or SST-PCA, and analyze it theoretically. Notably, we show that SST-PCA achieves the same accuracy as classical matrix PCA, with error proportional to the square root of the number of vertices and not the number of edges as might be expected. Our framework inherits many of the strengths of classical PCA and is suitable for a wide range of unsupervised learning tasks, including identifying principal networks, isolating changepoints and outliers, and for characterizing the “variability network” of the most varying edges. Finally, we demonstrate the effectiveness of SST-PCA in simulation and on an example from empirical social studies. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Accurate endpoint control in basic oxygen furnace (BOF) steelmaking is essential for reducing production costs and improving steel quality. To overcome the limited mechanism support and poor transparency of purely data-driven models, this study proposes a physics-aware and interpretable framework for cumulative decarburization prediction based on real industrial data. Historical multi-heat data from the same converter were integrated, and an averaged full-spectrum cross-correlation method was used to estimate and correct the transport delay of off-gas signals, thereby constructing a heat-wise large-sample dataset. Key elemental features with clear physical significance were then extracted from high-dimensional flame spectra by incorporating their underlying radiation mechanisms. On this basis, a Stacking-based ensemble model was developed for cumulative decarburization prediction, and SHAP was introduced to interpret the model decision logic. Results show that the proposed framework outperforms conventional single models and purely data-driven dimensionality reduction methods. SHAP analysis further indicates that model decisions are mainly dominated by four core elemental spectral features, namely Fe, C, O, and Mn. Overall, the proposed method combines predictive performance, physical constraints, and interpretability, and provides a new solution for auxiliary soft sensing and decision support in BOF endpoint control.

Upper extremity (UE) pain and pathology are prevalent among manual wheelchair users (MWUs) due to repetitive loading demands, highlighting the need for tools to identify high-risk tasks and inform injury prevention. This study investigated the feasibility of classifying activity intensity for wheelchair-related tasks using wearable sensors and supervised machine learning. Twenty-four MWUs with chronic spinal cord injury completed a standardized mobility course and simulated activities of daily living while UE electromyography (EMG) and inertial measurement unit (IMU) data were collected. Signals segmented into 3, 5, and 10 s windows, and time- and frequency-domain features were extracted and labeled as low, moderate, or high intensity. Multiple classification algorithms were evaluated using subject-dependent and subject-independent cross-validation, and dimensionality reduction was explored to assess class separability. Subject-dependent analyses demonstrated performance above chance but below 75% accuracy, with decision tree models demonstrating superior performance, particularly when trained on data segmented into 5 s windows. IMU features outperformed EMG features, but combining signal types enhanced performance. Subject-independent analyses revealed similar overall accuracy across signal types, but decreased high-intensity classification for EMG data, indicating subject dependency. Findings support the potential of wearable sensor-based machine learning with population-specific findings for activity intensity classification in MWUs, while highlighting challenges related to inter-subject variability for injury risk prediction.

Low-dimensional coherent patterns underlie the behavior of many complex dynamical systems. We propose a novel dynamic mode decomposition (DMD) framework to discover reduced-order models of complex physics from spatiotemporal data. This algorithm combines time-delay embedding and reduced-order spectral projections to transform the initial dynamics into a latent space where DMD can efficiently resolve their evolution in space and time. Here, the efficiency and accuracy of this approach, which we call time-augmented, space-contracted DMD (TASC-DMD), is demonstrated in several benchmark tests, and is then utilized to compactly characterize human brain deformation. Sparse identification of nonlinear dynamics (SINDy) is then employed to discover a parsimonious model for brain deformation in TASC coordinates. This integrated algorithm (TASC-SINDy) is trained on 4D strain data from in vivo tagged magnetic resonance imaging (tagged MRI) in 36 human subjects from a cohort of 45 subjects. A unique and sparse set of governing equations describing the temporal dynamics of the brain was discovered using only three generic modes. The TASC-SINDy model achieved exceptional dimensionality reduction and accurately predicted dynamic strain fields for the nine test subjects not used for training. This data-driven approach can systematically unravel dynamics and improve predictions in many complex physical systems.

Metastable states and the conformational transitions in between them are key to understanding the dynamical behavior and function of large-scale molecular systems. By combining basic dimensionality reduction techniques with a state-of-the-art approximation of the Koopman operator associated with molecular dynamics simulations (MD), we show that these states and transitions can be analyzed very efficiently based on MD simulation data. To construct the Koopman approximation, we employ a kernel-based method and solve the associated matrix equations using random Fourier features, leading to accurate solutions while maintaining low computational effort. On a benchmark set of fast-folding proteins, we demonstrate that key properties such as transition timescales, free energies, secondary structure elements, and hydrogen bonding patterns can be computed with remarkable robustness across hyperparameter regimes.

Protocol for generating high-quality, 45-color spectral flow cytometry data for unsupervised clustering to investigate aging in human PBMCs.

We present a compact and generative machine-learning framework that predicts oil contamination based on microbial community compositions from experimental samples. Our method combines dimensionality reduction with data augmentation and generative modeling to address high-dimensional, non-linear, and sparse microbial data. To reduce the 503-dimensional bacterial composition dataset, we compared three dimensionality reduction techniques: feature importance from random forest, principal component analysis (PCA), and t-distributed stochastic neighbor embedding (t-SNE). Feature importance outperformed PCA and t-SNE, improving predictive performance and identifying microbial species most strongly correlated with oil contamination. To mitigate data scarcity, we augmented the training data using an augmented data neural network (ADNN) with noise injection. Samples generated by a variational autoencoder (VAE) were used as controlled perturbations to probe model robustness during stress testing. Using the top 3-10 bacterial features, our model achieved an R² value of up to 0.99 in both training and stress testing for predicting oil contamination from microbial data. In a bottle-level hold-out evaluation (22 splits at an 80/20 bottle ratio), performance on held-out bottles was lower and variable (mean test R² = -0.150), indicating limited generalization within this cohort. These results should be interpreted as a feasibility demonstration requiring validation on larger independent datasets.

Background and purpose The aim is to study the role and function of junior athletes’ emotional intelligence in shaping productive strategies, including self-control, accepting responsibility, problem-solving planning, positive reappraisal, and social support. Material and methods The sample (n = 120) consisted of randomly selected junior male (n = 55; 45.83%) and female (n = 65; 54.17%) athletes with different qualifications, who were educated and trained at the academies of professional clubs and sports schools for children and youth, belonged to the country’s Olympic Reserve, had experience in participating in All-Ukrainian and international sports competitions, and were prize-winners in European and World championships. The respondents’ ages ranged from 15 to 19 years (М = 16.45; SD = ±2.97). Results It was substantiated that emotional intelligence, in shaping productive coping strategies, is a foundation for respondents’ emotional self-regulation. The profile of junior athletes’ emotional self-regulation was created. Psychological correlations between the studied parameters were established. The only inverse correlation between “accepting responsibility” and “self-motivation” was explained by respondents’ creative manifestations, which partially contradicts the coping strategy “accepting responsibility”. There was a caveat that empathy and social support are the most autonomous parameters that can pose a latent danger if their values are excessively high or low. The profile of emotional self-regulation was designed to control changes in these parameters. Based on the grouping variables of emotional intelligence, differences were found in two productive coping strategies: “self-control” – for the grouping variable “managing emotions” (t = -2.183; p = 0.031; d = -0.401) and “accepting responsibility” – for the grouping variable “self-motivation” (t = 2.382; p = 0.019; d = 0.436). It was substantiated that the parameters of emotional intelligence, “managing emotions” and “self-motivation”, are the most sensitive to shaping productive coping strategies. It was emphasised that the three factors resulting from dimensionality reduction, “self-motivational regulation”, “positive construction”, and “conscious control”, relevantly reflected the psychological content features of emotional self-regulation. Conclusions It was substantiated that research into junior athletes’ emotional intelligence in shaping productive coping strategies is a confirmatory empirical study, which allowed for establishing correlations, identifying significant advantages in productive coping strategies of self-control, accepting responsibility, problem-solving planning, positive reappraisal, and social support based on the grouping variables of emotional intelligence. It was summarised that the established scientific facts complement the theoretical knowledge of the formation of emotional self-regulation, the development of the emotional and volitional sphere, and the construction of worldviews in junior sports representatives.

In the context of the rapid advancement of financial technology, the issue of credit card default has become increasingly salient, emerging as one of the crucial risks that financial institutions are eagerly addressing. Traditional credit card default risk prediction models predominantly rely on statistical correlations for feature selection. This approach not only makes it challenging to uncover the genuine causal relationships between variables but also leads to limitations in prediction accuracy and interpretability. To overcome these limitations, this paper presents a novel credit card default risk prediction model that integrates causal feature screening, interaction feature construction, and interpretability enhancement. Initially, by leveraging the information value (IV) and eXtreme gradient boosting (XGBoost), we perform initial feature dimensionality reduction. Subsequently, we introduce the Peter Clark algorithm (PC) augmented with perturbation enhancement and bootstrap sampling to identify a stable set of causal features. Building on this foundation, we proceed to construct higher-order interaction features to bolster the model’s nonlinear modeling capacity. These causal features and their interaction counterparts are then fed into a variety of mainstream machine learning models for training and evaluation purposes. Furthermore, on the basis of the causal feature set identified via the PC algorithm, we construct a causal path diagram. We also incorporate the causal forest double machine learning (causal forest DML) method to estimate the causal effects of features. Additionally, we design a counterfactual explanation mechanism to aid in analyzing the direction and magnitude of the impact of variable interventions on default probability. Empirical tests conducted using four typical credit datasets reveal the following findings: (1) the introduction of causal features generally enhances the model’s performance in terms of the F1 score, area under the curve (AUC), and geometric mean (G-mean). This improvement is especially pronounced in models that are highly reliant on feature quality, such as logistic regression (LR). (2) Causal features offer significant advantages in terms of model interpretability, stability, and compliance, thereby presenting a new research paradigm for credit risk prevention and control in high-risk financial scenarios.

Abstract We present theoretical and practical results on the order theory of lattices of functions, focusing on Galois connections that abstract (sets of) functions – a topic known as higher-order abstract interpretation . We are motivated by the challenge of inferring closed-form bounds on functions which are defined recursively, i.e. as the fixed point of an operator or, equivalently, as the solution to a functional equation. This has multiple applications in program analysis (e.g. cost analysis, loop acceleration, declarative language analysis) and in hybrid systems governed by differential equations. Our main contribution is a new family of constraint-based abstract domains for abstracting numerical functions, $\mathfrak {B}$ B -bound domains , which abstract a function $f$ f by a conjunction of bounds from a preselected set of boundary functions. They allow inferring highly non-linear numerical invariants , which classical numerical abstract domains struggle with. We uncover a convexity property in the constraint space that simplifies, and, in some cases, fully automates , transfer function design. We also introduce domain abstraction , a functor that lifts arbitrary mappings in value space to Galois connections in function space. This supports abstraction from symbolic to numerical functions (i.e. size abstraction ), and enables dimensionality reduction of equations. We base our constructions of transfer functions on a simple operator language , starting with sequences , and extending to more general functions , including multivariate, piecewise, and non-discrete domains.