A new Nystrom method for solving boundary value problems on the real axis

Production control optimization in process industries is often challenged by complex physicochemical processes that are difficult to model mathematically. As a model-free approach, Reinforcement Learning (RL) offers a promising solution. However, online action exploration through trial-and-error risks compromising equipment safety and efficiency, while offline training suffers from sampling bias due to limited and imbalanced datasets, particularly the scarcity of faulty operation data. To address these issues, this study proposes a world model-driven operational framework that integrates conditional diffusion with offline RL. By leveraging the distribution approximation capability of diffusion models, we introduce a conditional trajectory generation mechanism constrained by operational parameters and historical state transitions. This allows the diffusion model to produce near-realistic state trajectories and reward signals, constructing an interactive virtual state–action–reward space. We further employ autoregressive generation of imagined trajectories to support RL agent training. During world model training, a spatiotemporal Transformer architecture is incorporated to capture dependencies along state–action trajectories. For offline agent training, a Twin-Delayed Deep Deterministic policy gradient-based RL model regularized by behavior cloning is adopted. Experiments on a tobacco leaf-processing line demonstrate that the proposed conditional diffusion-based offline RL method accurately constructs a virtual sample space with a mean squared error of 1.27e−4, significantly reducing policy acquisition costs. The resulting RL-driven parameter adjustment achieves an approximately 12% improvement in the product qualification rate compared to other state-of-the-art offline RL algorithms. Our algorithm implementation and evaluation dataset can be found here: https://github.com/sizizuo0076/WM-PIO-ORL . • A diffusion-based world model is proposed to guide offline decision agent training. • A spatiotemporal Transformer is used for noise prediction in the diffusion model. • Reinforcement learning with behavior cloning is designed for continuous control. • The proposed framework is validated on a real tobacco shredding production line. • The validation shows a 17.2% quality improvement for process production control.

We present an algorithm for efficient evaluation of Boys functions F0,…,Fkmax tailored to modern computing architectures, in particular graphical processing units, where maximum throughput is high and data movement is costly. The method combines rational minimax approximations with upward and downward recurrence relations. The non-negative real axis is partitioned into three regions, [0, ∞⟩ = A ∪ B ∪ C, where regions A and B are treated using rational minimax approximations and region C by an asymptotic approximation. This formulation avoids lookup tables and irregular memory access, making it well-suited for hardware with high maximum throughput and low latency. The rational minimax coefficients are generated using the rational Remez algorithm. For a target maximum absolute error of ɛtol = 5 × 10-14, the corresponding approximation regions and coefficients for Boys functions F0, …, F32 are provided in AppendixD.

Extrachromosomal DNA (ecDNA) plays a key role in cancer pathology. EcDNAs mediate high oncogene amplification and expression and worse patient outcomes. Accurately determining the structure of these circular molecules is essential for understanding their function, yet reconstructing ecDNA cycles from sequencing data remains challenging. We introduce Cycle-Extractor (CE) for reconstruction. CE accepts a breakpoint graph derived from either short or long read sequencing data as input and extracts a cycle with the maximum length-weighted-copy-number. CE utilizes a mixed-integer linear program (MILP) and a separate traversal procedure, enabling fast optimization and compatibility with free solvers. We evaluated CE against CoRAL (long-read-based quadratic optimization), Decoil (long-reads), and AmpliconArchitect (AA for short reads) on both simulated data and real cancer cell lines. On simulated ecDNA, CE achieves performance comparable to CoRAL across three accuracy metrics and consistently outperforms AA and Decoil. On cancer cell lines, CE produces longer and heavier cycles than AA, and achieves performance similar to CoRAL. Moreover, CE is, on average, 40× faster than CoRAL. These results demonstrate that CE accurately reconstructs ecDNA from both short- and long-read sequencing data, while long-read inputs allow CE to recover more complete and higher-confidence ecDNA structures. CE improved the prediction of many ecDNA structures. On a PC3 ecDNA containing MYC , CE uses ONT data to reconstruct a substantially larger and higher-copy sequence (4.2 Mbp) compared to the short-read-derived reconstruction (690 Kbp). CRISPR-CATCH experiments confirm the presence of a large ecDNA molecule, validating the long-read-based CE reconstruction.

A bstract We obtain infinitely many boundary operators in the Brownian loop soup in the subcritical phase by analyzing the conformal block expansion of the two-point function that computes the probability of having two marked points on the upper half-plane being separated by Brownian loops. The resulting boundary operators are primary operators in a 2D CFT with central charge c ≤ 1 and have conformal dimensions that are non-negative integers. By comparing the above-mentioned conformal block expansion with probabilities in the Brownian loop soup, we provide a physical interpretation of the boundary operators of even dimensions as operators that insert multiple outer boundaries of Brownian loops at points on the real axis.

Abstract The family of normal distributions on $$\mathbb {R}$$ R is closed under scaling and translation. Another example of an exponential family on $$\mathbb {R}$$ R with this property is the family of exponential-polynomial distributions. We prove that, conversely, an exponential family with such property is essentially the family of exponential-polynomial distributions.

Given a perfect Polish space \(X\), a compact subset \(K\subset X\) and a countable ordinal \(\alpha<\omega_1\), we show that there exists a compact subset \(\widehat K\subset X\) such that \[ \widehat K^{(\alpha)} = K \] where \(\widehat K^{(\alpha)}\) denotes the \(\alpha\)-th Cantor–Bendixson derivative of \(\widehat K\). In other words, every compact subset of a perfect Polish space admits an \(\alpha\)-primitive with respect to the Cantor–Bendixson derivative. This extends to perfect Polish spaces a result previously known for countable compact subsets of the real line. The proof proceeds in three steps: first, we construct primitives for singletons; then, for countable compact subsets; and finally, for arbitrary compact subsets, using separability of Polish spaces.

Abstract While General Fractional Calculus has successfully expanded the scope of memory operators beyond power-laws, standard formulations remain predominantly restricted to the half-line via Riemann-Liouville or Caputo definitions. This constraint artificially truncates the system's history, limiting the thermodynamic consistency required for modeling processes on unbounded domains. To overcome these barriers, we construct the Weighted Weyl-Sonine Framework, a generalized formalism that extends non-local theory to the entire real line without history truncation.Unlike recent algebraic approaches based on conjugation for finite intervals, we develop a rigorous harmonic analysis framework. Our central contribution is the Generalized Spectral Mapping Theorem, which establishes the Weighted Fourier Transform as a unitary diagonalization map for these operators. This result allows us to rigorously classify and solve distinct physical regimes under a single algebraic structure. We explicitly derive exact solutions for diffusive relaxation (governed by Complete Bernstein Functions), inertial wave propagation (exhibiting oscillatory dynamics), and retarded aging (via distributed order), proving that our framework unifies the description of anomalous transport and wave mechanics in complex, time-deformed media.

There is an elegant style of mathematical argument by which one proves an existence claim not by exhibiting any particular instance, but rather by proving that a randomly chosen instance from a suitable class exhibits the property with nonzero probability.

For a metric continuum X , for a Whitney map μ for C ( X ) , and for p ∈ X , we define the midpoint hyperspace M i d μ ( p , X ) as the set containing { p } and all arcs in X whose midpoint is p with respect to μ . We show geometric models of M i d μ ( p , X ) , in the case when X is the arc, the simple closed curve, the simple triod, compactifications of the ray [ 0 , ∞ ) with remainder an arc, compactifications of the real line with remainder an arc, and indecomposable arc continua. Moreover, we present characterizations of dendrites and the arc in terms of this hyperspace.

Let W 1 , … , W n be non-negative random variables. We consider an undirected random graph model on the node set { 1 , … , n } , where two nodes i < j are adjacent if W i < W j . In our setting, the W i ‘s are independent but not necessarily identically distributed, resulting in a model that generalizes the classical random permutation graphs. The model exhibits a certain dependence among the edges. Moreover, when nodes have physical interpretations—such as points on the real line R with node i located at position x = i —the model gains spatial structure and becomes, in particular, distance-dependent. We derive theoretical results on degree distributions, the number of isolated vertices, and the number of close neighbors. Simulation-based observations are also provided for the average clustering and the global efficiency.

Abstract A classical problem in data-driven model order reduction (MOR) of linear time-invariant (LTI) systems is the preservation of structural properties of the underlying large-scale dynamics. When dealing with MOR based on transfer function measurements, one relevant problem is how to force the reduced-order model (ROM) to inherit the asymptotic stability of the reference system, i.e., to enforce the poles of the ROM transfer function to have strictly negative real part. In this work, we tackle the more general problem of placing such poles in arbitrary linear matrix inequality (LMI) regions of the complex plane, which include a rich class of convex sets symmetric with respect to the real axis. LTI systems with poles constrained to this kind of regions are called $$\mathcal {D}$$ D -stable. Combining well-established results from control theory and recent developments in asymptotically stable rational approximation algorithms, we show that the problem can be solved efficiently via standard convex optimization routines. Several numerical testbenches of engineering interest confirm the effectiveness of the proposed methodology in practical applications.

Wafer map defect pattern recognition is a key task for ensuring yield in integrated circuit manufacturing. However, in real production lines it commonly suffers from scarce labeled data, long-tailed class distributions, and limited feature representations, which cause existing deep learning models to degrade in performance, particularly for minority defect classes and complex defect morphologies. To address these challenges, we propose a semi-supervised classification method for wafer maps based on a multimodal Wasserstein autoencoder (MM-WAE). The framework constructs three parallel feature branches in the spatial, frequency, and texture domains, using a multi-head attention mechanism and gating mechanism for adaptive multimodal fusion. This allows defect patterns to be comprehensively characterized by macroscopic geometric distributions, spectral periodic structures, and microscopic texture details. The Wasserstein autoencoder is introduced, with the latent space distribution regularized by a maximum mean discrepancy (MMD) loss using an inverse multiquadratic kernel. Additionally, an inverse class-frequency weighted cross-entropy loss and a modality consistency loss between the encoder and classifier jointly optimize the reconstruction and classification paths while leveraging large amounts of unlabeled wafer maps for semi-supervised learning. Experimental results show that MM-WAE mitigates performance limitations caused by insufficient labels and class imbalance, significantly improving the accuracy and robustness of wafer defect classification, with promising potential for industrial application and further development.

Artificial intelligence (AI) and computer vision (CV) offer promising avenues for automated quality control in food manufacturing, yet many prior works in that sector focused on agricultural primary production tasks. This study evaluates object detection for in-line quality monitoring on a real production line for ready-to-eat chicken-type products. Overhead cameras captured images at four processing steps: forming, coating, frying, and cooking. For each step, we labeled 2000 images containing multiple products with multiple classes of quality deviations. Separate YOLOv12x models (default and hyperparameter-tuned) were trained per step and evaluated using mAP50-95, F1-curves, and confusion matrices. Step-specific models, i.e., models applicable solely for a specific processing step, achieved similar peak mAP50-95 (0.50-0.60), and hyperparameter tuning did not yield any major gains despite high computational cost. Performance was strongly tied to class frequency: common classes achieved high F1-Scores, whereas rare classes were often misclassified. To mitigate imbalance and improve robustness, we trained a single model on a combined dataset spanning all steps, which attained a higher peak mAP50-95 of 0.7331 ± 0.0040 and produced more balanced F1-curves, albeit with some loss of step-specific strengths, such as detection of certain deviations specific to that step. The results indicate that out-of-the-box detectors can add practical value to industrial CV-enhanced quality control in food processing, and that further improvements will primarily come from targeted data collection for minority classes, instance-centric datasets, higher-resolution or multi-scale training, and methods that address class imbalance.

This paper studies a generalized heavenly equation, which includes the standard heavenly equation as a special case. The work starts from a vector nonlinear Riemann-Hilbert problem on the real axis. Next, the large-time asymptotic behavior for the solutions of the Cauchy problem is derived by solving the corresponding inverse Riemann-Hilbert problem. Then, by parameterizing the Riemann-Hilbert data, a class of implicit solutions is constructed.

We introduce a natural topology on powers of a space that is inspired by the Vietoris topology on compact subsets. We then place this topology in context with other product topologies; specifically, we compare this topology with the Tychonoff product, the box product, and Bell's uniform box topology. We identify a variety of topological properties for the specific case when the ground space is discrete. When the ground space is the Euclidean real line, we show that the resulting power is not Lindelöf, and hence, not Menger. This shows that, unlike the the Vietoris topology on unordered compact subsets, covering properties of the ground space need not transfer to the Vietoris power.

This paper focuses on the essential role of the quasi-conformal curvature tensor in developing the theoretical foundation and applications of quasi-contactmetric geometry. The quasi-conformal curvature tensor of NC10-manifolds using the space associated with G structures (AGS-space) has been investigated. By deriving explicit expressions for the tensor components, the necessary and sufficient conditionsfor considering the quasi-conformal NC10-manifold as flat quasi-conformal have beendetermined. Noteworthy, it indicated that any flat quasi-conformal NC10-manifold is locally isometric to the product of the Kähler manifold and the real line, thus connecting abstract tensor properties to classical geometric structures. Moreover, theconditions for the NC10-manifold that yield the η-Einstein structure are established. New analogues of Gray’s identities for the quasi-conformal curvature tensor havebeen introduced for NC10-manifold.

Homeomorphism theorem for sums of translates on the real axis

We prove two results on approximation of Borel measures in the Kantorovich metric. The first result provides the rate of convergence in this metric of normalized surface measures on $ n $ -dimensional spheres to the standard Gaussian measure on the countable power of the real line, restricted to an arbitrary continuously embedded separable Banach space of full measure. The second result, for an arbitrary weakly convergent sequence of probability Borel measures on a separable Fréchet space, gives a sufficient condition for convergence in the Kantorovich metric generated by the norm of a compactly embedded separable Banach space.

The zeros of exponential sums with complex coefficients and distinct real-valued exponents, also known as exponential polynomials, are among the most intensively studied of all entire functions. Nevertheless, the zeros of exponential sums possess intriguing properties that are yet to be discovered. A prototype worthy of study is the three-term exponential sum in which the zeros are contained in one or at most two disjoint vertical strips in the complex plane. In other words, the real parts of the roots are confined to one or two closed and bounded intervals of the real axis: [a, b] and [c, d]. Here, it is shown that the real parts of the roots are dense in the intervals [a, b] and [c, d] if and only if one key parameter is irrational. This result is an important consequence of Weyl’s equidistribution theorem.