Mathematics

21 pages, 637 KB

Open AccessArticle

SeqRankFL: Sequence-Aware Ranking of LLM-Based Code Representations for Statement-Level Fault Localization

by Dong An, Shihai Wang, Bin Liu and Liandie Zhu

Mathematics 2026, 14(18), 3297; https://doi.org/10.3390/math14183297 - 11 Sep 2026

Software reliability is an important part of reliability assurance for complex engineering systems, and timely fault diagnosis supports safe and continuous operation. After a test failure, statement-level fault localization ranks source-code lines for early inspection. Representation-based approaches can operate without a coverage matrix [...] Read more.

Software reliability is an important part of reliability assurance for complex engineering systems, and timely fault diagnosis supports safe and continuous operation. After a test failure, statement-level fault localization ranks source-code lines for early inspection. Representation-based approaches can operate without a coverage matrix by extracting line-level hidden states from a frozen large language model for code (code LLM), but their downstream readout and training objective are not necessarily aligned with source-window ranking and buggy-version-level evaluation. We propose SeqRankFL, a sequence-aware ranker for LLM-based code representations. It combines a bidirectional long short-term memory network (BiLSTM), which follows code order within a source window, with a Hybrid objective that joins binary cross-entropy (BCE) and the listwise learning-to-rank loss ListNet. On the source-window ranking task using windows of at most 128 physical lines within known buggy files from BugsInPy and Defects4J, relative to a matched LLMAO-style Transformer+BCE reference that shares the same representations, splits, and evaluation protocol, SeqRankFL raises the equally weighted Top-1 from 51.31% to 58.58%, an absolute gain of 7.27 percentage points, with consistent improvements on both datasets. Further controlled analyses show that multi-depth layer mixing and syntax scope mainly improve average ranks, whereas control-flow/data-flow graphs and failure behavior have condition-dependent effects across models, languages, and project partitions. The controlled results identify the listwise objective as the primary driver of the Top-1 improvement, with the BiLSTM readout providing an additional architecture-dependent gain under the ranking-aware objective. Full article

(This article belongs to the Special Issue Mathematical Methods for Reliability Design in Complex Engineering Applications)

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9 pages, 247 KB

Open AccessArticle

On Computing Sums of Double Numerical Series

by Vyacheslav I. Kuzovatov, Alexey A. Kytmanov and Evgeniya K. Myshkina

Mathematics 2026, 14(18), 3296; https://doi.org/10.3390/math14183296 - 10 Sep 2026

Abstract

In this study, we obtain the sums of three families of double numerical series not mentioned in existing handbooks. These sums are expressed by means of known special functions, namely the Bessel function and the modified Bessel function. The technique is based on [...] Read more.

In this study, we obtain the sums of three families of double numerical series not mentioned in existing handbooks. These sums are expressed by means of known special functions, namely the Bessel function and the modified Bessel function. The technique is based on computing the resultant using two different methods. The first method uses the formula for the resultant of an entire function and a polynomial (or an entire function with a finite number of zeros) previously obtained by A. M. Kytmanov and E. K. Myshkina. The second method computes the resultant as the product of values of one function at the roots of another. The proposed approach for computing the sums of double numerical series differs significantly from methods based on residue integrals associated with a system of equations. Full article

(This article belongs to the Section C4: Complex Analysis)

25 pages, 4731 KB

Open AccessFeature PaperArticle

Human–Robot Collaborative Order Picking in Smart Warehouses with Fuzzy Transportation and Processing Time

by Zhiheng Cai, Ziyan Zhao, Yunuo Su and Zijie Yu

Mathematics 2026, 14(18), 3295; https://doi.org/10.3390/math14183295 - 10 Sep 2026

Abstract

Robot mobile fulfillment systems (RMFSs), as human–robot collaborative smart warehouses, transform the traditional person-to-goods order picking mode into a goods-to-person mode. Order picking optimization is a core decision-making challenge in RMFSs to improve the efficiency of the system, which needs to jointly optimize [...] Read more.

Robot mobile fulfillment systems (RMFSs), as human–robot collaborative smart warehouses, transform the traditional person-to-goods order picking mode into a goods-to-person mode. Order picking optimization is a core decision-making challenge in RMFSs to improve the efficiency of the system, which needs to jointly optimize pod selection, robot scheduling, station assignment, and manual picking. Although recent studies have widely investigated integrated operational optimization in RMFSs, most of them rely on deterministic transportation and processing time and ignore uncertainties in practical human–robot collaborative operations. It remains challenging to jointly optimize these coupled decisions under uncertain operation times. To address this challenge, we model the concerned problem with the objective of minimizing fuzzy makespan and design an adaptive large-neighborhood-based variable neighborhood descent algorithm to efficiently solve it. The algorithm adopts three-dimensional coupling encoding and multi-stage heuristic decoding mechanisms. It further integrates a learning-based adaptive destroy operator selection method and a variable neighborhood descent search strategy to enhance its exploration and exploitation abilities. In a large number of systematic experiments, ALVND achieved great performance in solving the concerned problem. The objective function value obtained by it was 5.6–25.1% lower than its competitors, demonstrating its effectiveness in uncertain human–robot collaborative warehouse scenarios. Full article

31 pages, 1674 KB

Open AccessArticle

Dynamic Analysis and Optimal Control Strategy for the Impact of Stem Cell Therapy on Type 1 Diabetes

by Awatif J. Alqarni

Mathematics 2026, 14(18), 3294; https://doi.org/10.3390/math14183294 - 10 Sep 2026

Abstract

Type 1 diabetes (T1D) is a chronic autoimmune disease in which autoreactive effector T cells destroy insulin-producing pancreatic β-cells, leading to impaired insulin production and long-term metabolic complications. This study develops a mathematical model of the interactions among pancreatic β-cells, autoreactive effector T [...] Read more.

Type 1 diabetes (T1D) is a chronic autoimmune disease in which autoreactive effector T cells destroy insulin-producing pancreatic β-cells, leading to impaired insulin production and long-term metabolic complications. This study develops a mathematical model of the interactions among pancreatic β-cells, autoreactive effector T cells, regulatory T cells (Tregs), and stem cell therapy, treating stem cell administration as an immunomodulatory and regenerative strategy that suppresses autoimmune activity and promotes β-cell recovery. The existence, uniqueness, positivity, and boundedness of solutions are established. For the auxiliary subsystem with

S_{E} = 0

and constant treatment input, the disease-free equilibrium and the threshold quantity

R_{0}

are derived, and local asymptotic stability of the DFE is established for

R_{0} < 1

. For the full model with

S_{E} > 0

, the existence of a unique biologically feasible positive equilibrium is established, together with its local asymptotic stability under constant treatment input. An optimal control problem, formulated using Pontryagin’s Maximum Principle, is used to guide therapeutic dosing, and one-year numerical simulations compare two administration protocols: high-dose pulse injections and continuous infusion. Both reduce autoimmune activity and improve β-cell dynamics, with pulse administration producing stronger transient responses and continuous infusion producing smoother treatment-period dynamics, while both protocols approach similar long-term levels. A local sensitivity analysis identifies immune activation, β-cell destruction, and regulatory T-cell activity as the parameters most strongly shaping disease progression and treatment outcomes. By unifying stem cell therapy, stability analysis, sensitivity analysis, and optimal control, and directly comparing pulse and infusion protocols, this framework offers new insight for designing stem cell-based treatments for autoimmune diabetes. Full article

(This article belongs to the Special Issue Dynamic Model and Analysis of Biology and Epidemiology)

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25 pages, 744 KB

Open AccessArticle

LAG-Net: A Deep Unfolding Network for Multi-View Clustering via Learnable Anchor Graph

by Zhiling Cai, Chen Li, Peijun Zhang and Lijin Wang

Mathematics 2026, 14(18), 3293; https://doi.org/10.3390/math14183293 - 10 Sep 2026

Abstract

Multi-view clustering aims to discover consistent cluster structures from heterogeneous features without supervision. Anchor-based methods improve scalability by representing samples through a compact set of anchors, but fixed anchors may be misaligned with the evolving cluster geometry. This mismatch is the main problem [...] Read more.

Multi-view clustering aims to discover consistent cluster structures from heterogeneous features without supervision. Anchor-based methods improve scalability by representing samples through a compact set of anchors, but fixed anchors may be misaligned with the evolving cluster geometry. This mismatch is the main problem addressed here: the shared sample–anchor graph and the global anchor geometry need to be refined together across heterogeneous views, rather than in two disconnected stages. This paper proposes Learnable Anchor Graph Network (LAG-Net), a deep unfolding framework that jointly learns a shared anchor graph, view-specific anchor indicators, and global anchor alignment within a unified model. The global anchor alignment provides geometric guidance for shared anchor graph learning and promotes anchor consistency across heterogeneous views. By unfolding the derived optimization procedure into a trainable network, LAG-Net enables the anchor representations and sample–anchor relationships to be progressively refined. Experiments on six benchmark datasets demonstrate the effectiveness and scalability of the proposed method compared with representative multi-view clustering approaches. Full article

(This article belongs to the Special Issue Causal Inference and Machine Learning: Mathematical Modeling, Analysis and Applications)

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33 pages, 903 KB

Open AccessArticle

A Mathematical Theory of Correct Computation

by Lee Naish, Bernard Pope and Harald Søndergaard

Mathematics 2026, 14(18), 3292; https://doi.org/10.3390/math14183292 - 10 Sep 2026

Abstract

In 1970, Dana Scott proposed his highly influential “mathematical theory of computation” to define the relationship between the text of a program and what the program computes (or denotes)—the “semantics” of the program. Scott used a complete lattice based on the “information ordering”, [...] Read more.

In 1970, Dana Scott proposed his highly influential “mathematical theory of computation” to define the relationship between the text of a program and what the program computes (or denotes)—the “semantics” of the program. Scott used a complete lattice based on the “information ordering”, with the bottom element representing undefined—a program failing to terminate normally, thus producing no information. The top element, however, was unused. Hence most subsequent applications of denotational semantics have used mathematical structures that avoid top elements. We suggest that the information ordering is relevant not only to semanticists but also to working programmers as a basis for determining if a program component or a computation is correct according to their intentions. We also suggest that a return to the use of complete lattices is called for, to broaden formal semantics and allow it to encompass programmer intentions. That is because often those intentions permit more than one runtime behaviour for a given input. In this paper we explore the connections between the information ordering, correctness of computations and programs, and debugging. We present a general theory and describe several instances where the intention for what our logic/functional code computes plus what it actually computes can be described by elements in a complete lattice. For correct code, the information order relates (1) what is intended and what is computed, (2) successive states of a computation, and (3) the left and right sides of program component definitions. For bugs, the information order is violated. The technical results are a reasonably straightforward extension to previous denotational semantics work, but the scheme aligns much better with practical programming and software tools. The theory extends both the theoretical basis and practical flexibility of declarative debugging and reasoning about partial correctness and gives an attractive mathematical framework that encompasses our intentions, our programs and what they compute. Full article

(This article belongs to the Special Issue Analysis of Functional Logic Programming and Its Applications)

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15 pages, 301 KB

Open AccessFeature PaperArticle

Finite and Infinite Weighted Exchangeable Sequences

by Wenpin Tang

Mathematics 2026, 14(18), 3291; https://doi.org/10.3390/math14183291 - 10 Sep 2026

Abstract

Motivated by recent interest in predictive inference under distribution shifts, we study the problem of approximating finite weighted exchangeable sequences by a mixture of finite sequences with independent terms. Two different approaches are developed: a combinatorial approach and a probabilistic approach. Various bounds [...] Read more.

Motivated by recent interest in predictive inference under distribution shifts, we study the problem of approximating finite weighted exchangeable sequences by a mixture of finite sequences with independent terms. Two different approaches are developed: a combinatorial approach and a probabilistic approach. Various bounds are derived in terms of weight functions, extending previous results on finite exchangeable sequences. As a byproduct, we obtain a version of de Finetti’s theorem for infinite weighted exchangeable sequences, recovering an instance of Barber, Candes, Ramdas and Tibshirani 1 (Bernoulli, 30 (2024), pp. 3004–3028) in the Polish space with bounded and continuous weights. Full article

(This article belongs to the Section D1: Probability and Statistics)

26 pages, 2058 KB

Open AccessArticle

Research on High-Precision Prediction of Vertical Hydrodynamic Coefficients for AUV Based on CFD and GA-BP Neural Network

by Dingfeng Yu, Xi Zhang, Xia Yang, Yiyun Peng, Xiong Deng, Yan Luo and Yanyang Wu

Mathematics 2026, 14(18), 3290; https://doi.org/10.3390/math14183290 - 10 Sep 2026

Abstract

Aiming at the engineering pain points of insufficient sample size, inadequate working condition coverage, and poor generalization performance in traditional solution methods of vertical hydrodynamic coefficients for Autonomous Underwater Vehicles (AUVs), this paper constructs a high-precision prediction framework for vertical hydrodynamic coefficients Computational [...] Read more.

Aiming at the engineering pain points of insufficient sample size, inadequate working condition coverage, and poor generalization performance in traditional solution methods of vertical hydrodynamic coefficients for Autonomous Underwater Vehicles (AUVs), this paper constructs a high-precision prediction framework for vertical hydrodynamic coefficients Computational Fluid Dynamics (CFD) numerical simulation and a genetic algorithm-optimized back-propagation (GA-BP) neural network. Firstly, the overset grid technology and User-Defined Function (UDF) are adopted to simulate the pure heave motion of the Planar Motion Mechanism (PMM), completing the unsteady flow field numerical calculation. Secondly, 200 working condition simulation datasets covering different motion amplitudes, heave frequencies and inflow velocities are constructed, and a three-input and two-output BP neural network basic prediction model is established. Subsequently, the basic model is optimized by 5-fold cross-validation and genetic algorithm respectively, and the prediction performances of BP, K-fold-BP and GA-BP models are compared and analyzed. The results show that the determination coefficients of the GA-BP model for the dimensionless vertical hydrodynamic coefficients Z_a and Z_b reach 0.9976 and 0.9975 respectively, and the Mean Absolute Error (MAE) and Root Mean Square Error (RMSE) are significantly lower than those of the other two models, with the optimal prediction accuracy and generalization performance. Finally, 1000 sets of full-working-condition hydrodynamic coefficient prediction are completed based on the GA-BP model, and the optimized dimensionless added mass coefficient and fluid damping coefficient are obtained by fitting. The prediction framework constructed in this paper can provide technical support for the rapid and high-precision acquisition of AUV full-working-condition hydrodynamic coefficients, and provide a reliable parameter basis for dynamic modeling and motion control system design. Full article

25 pages, 3756 KB

Open AccessArticle

CAA-CLIP: Decoupled Pair and Cultural-Term Adaptation for Chinese Ancient Architecture Image–Text Retrieval

by Xiaoyu Zhang, Hao Wang, Wangyu Wu and Qingqing Hu

Mathematics 2026, 14(18), 3289; https://doi.org/10.3390/math14183289 - 10 Sep 2026

Abstract

Image–text retrieval for Chinese ancient architecture requires fine-grained discrimination among visually similar sites and alignment with specialist terms that are sparse in general vision–language corpora. We propose CAA-CLIP, a parameter-efficient framework for exact image–annotation retrieval and annotation-derived cultural-term retrieval. CAA-CLIP adapts five frozen [...] Read more.

Image–text retrieval for Chinese ancient architecture requires fine-grained discrimination among visually similar sites and alignment with specialist terms that are sparse in general vision–language corpora. We propose CAA-CLIP, a parameter-efficient framework for exact image–annotation retrieval and annotation-derived cultural-term retrieval. CAA-CLIP adapts five frozen CLIP-family encoders through separate pair and term branches and fuses row-normalized scores with validation-selected weights. Evaluation uses 581 image–text pairs from 58 sites, five site-grouped outer folds, and three optimization seeds per fold. After seed averaging, CAA-CLIP obtains

29.06 \pm 3.05

% Avg R@1,

63.19 \pm 6.34

% Avg R@5,

78.65 \pm 6.64

% Avg R@10, and

43.82 \pm 6.50

% term mAP. Against matched validation-selected plain fusion, the pair-retrieval differences are 0.43–0.98 points, and the term-mAP difference is 9.05 points. No primary comparison reaches significance after Holm correction over the five folds, although the term difference is positive in every fold. Equal CAA fusion attains 44.39% term mAP, indicating that validation-selected term weights do not improve this endpoint. These findings distinguish expert fusion for exact-pair retrieval from direct term supervision for annotation-defined term retrieval in small, site-structured heritage collections. Full article

(This article belongs to the Special Issue Mathematical Algorithms and Artificial Intelligence Methods in Design: Theory, Computation and Applications)

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34 pages, 2178 KB

Open AccessArticle

Government Regulation, Social Media Auditing, and Consumer Participation in Greenwashing Governance: A Four-Party Evolutionary Game Analysis

by Huili Yan and Zhihua Yan

Mathematics 2026, 14(18), 3288; https://doi.org/10.3390/math14183288 - 10 Sep 2026

Abstract

Greenwashing distorts green markets and impedes sustainable development. This study develops a four-party asymmetric evolutionary game model to examine the strategic interactions among government, companies, social media, and consumers in greenwashing governance. By formulating replicator dynamics and conducting a Jacobian-based local stability analysis, [...] Read more.

Greenwashing distorts green markets and impedes sustainable development. This study develops a four-party asymmetric evolutionary game model to examine the strategic interactions among government, companies, social media, and consumers in greenwashing governance. By formulating replicator dynamics and conducting a Jacobian-based local stability analysis, the research identifies three representative locally asymptotically stable boundary equilibria under specified parameter configurations: such as dual-oversight market withdrawal, consumer-redress-driven governance, and social-media-absent regulation-driven governance. Numerical simulations examine benchmark evolutionary trajectories and selected parameter variations. The results indicate that system outcomes are sensitive to adjustments in complaint compensation, consumer complaint cost, the additional cost of genuine green production, and social media auditing costs. Specifically, high penalties for lenient auditing can induce persistent systemic fluctuations, while strict auditing alone may inadvertently stabilize a market-withdrawal outcome without prompting genuine green behavior. Under the specified parameter conditions of the model, the findings suggest that stable governance is associated with coordinated incentives for genuine production, economically viable consumer redress, and appropriately calibrated auditing policies. Full article

(This article belongs to the Section D2: Operations Research and Fuzzy Decision Making)

29 pages, 33985 KB

Open AccessArticle

SMFLIX: A Unified Body–Head Model for Real-Time Expressive Human Mesh Recovery

by Seokhyeon Heo, Yunseong Oh and Youngwoon Cha

Mathematics 2026, 14(18), 3287; https://doi.org/10.3390/math14183287 - 10 Sep 2026

Abstract

Expressive human mesh recovery reconstructs the body, hands, and face from a single image. Whole-body models such as SMPL-X offer only coarse control over the face, whereas face models such as FLAME are head-only. Combining them requires maintaining two models at inference and [...] Read more.

Expressive human mesh recovery reconstructs the body, hands, and face from a single image. Whole-body models such as SMPL-X offer only coarse control over the face, whereas face models such as FLAME are head-only. Combining them requires maintaining two models at inference and re-annotating training data. Complete eye closure and lip articulation are perceptually important yet difficult to represent, supervise, and evaluate. Existing formulations leave residual gaps or artifacts, and no public dataset, to our knowledge, characterizes eye and lip aperture. We present SMFLIX, a unified body–head model that merges both bases into a single linear block matrix formulation. It eliminates external registration, yields a watertight neck seam, and makes these articulations representable. Parameter compatibility lets us build SMFLIX-Set without full re-annotation, yielding 11M instances with joint whole-body and facial supervision. The dataset includes 536K face-valid test samples with continuous eye and lip aperture ratios, enabling direct evaluation of articulations obscured by mesh-averaged metrics. We train SMFLIX-Net, a one-stage dual-decoder network regressing body and facial parameters separately. Compared with state-of-the-art whole-body, face-only, and combined methods, the network achieves the lowest whole-body, body-part, and aperture errors on SMFLIX-Set. It remains competitive on external benchmarks, runs at 48 FPS on a consumer GPU, and supports a live webcam application. Full article

(This article belongs to the Special Issue Computational Intelligence and Human–Computer Interaction: Modern Methods and Applications, 3rd Edition)

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41 pages, 2391 KB

Open AccessArticle

Inference for a Shared Random-Scale Frailty Birnbaum–Saunders Model Under Progressive Type-II Censoring

by Omar M. Bdair

Mathematics 2026, 14(18), 3286; https://doi.org/10.3390/math14183286 - 10 Sep 2026

Abstract

This paper introduces a shared gamma frailty Birnbaum–Saunders model for clustered lifetime data under progressive Type-II censoring. The frailty term acts on the scale parameter and accounts for unobserved variation among clusters. Likelihood-based and Bayesian formulations are developed, and posterior estimation and prediction [...] Read more.

This paper introduces a shared gamma frailty Birnbaum–Saunders model for clustered lifetime data under progressive Type-II censoring. The frailty term acts on the scale parameter and accounts for unobserved variation among clusters. Likelihood-based and Bayesian formulations are developed, and posterior estimation and prediction are carried out using a Metropolis-within-Gibbs algorithm. A simulation study considers different shape parameters, frailty levels, numbers of clusters, and censoring schemes. The proposed method estimates the shape parameter accurately and gives generally satisfactory results for the frailty variance. Increasing the number of clusters improves estimation, and the frailty model gives lower prediction errors in almost all valid comparisons with the ordinary BS model. A real-data application to kidney catheter infection times illustrates the shared-frailty component of the proposed model. Full article

28 pages, 3507 KB

Open AccessArticle

A Fine-Grained Semantic Steganography Framework with Cross-Modal Drift Regularization

by Khaled Alrawashdeh

Mathematics 2026, 14(18), 3285; https://doi.org/10.3390/math14183285 - 10 Sep 2026

Abstract

Steganography using deep learning can preserve pixel-level image quality while still changing object, attribute, or relational information in captions generated by vision-language models (VLMs). This caption drift creates a detection channel that is not measured by global image-embedding similarity alone. This paper presents [...] Read more.

Steganography using deep learning can preserve pixel-level image quality while still changing object, attribute, or relational information in captions generated by vision-language models (VLMs). This caption drift creates a detection channel that is not measured by global image-embedding similarity alone. This paper presents StegoGuard, a framework that embeds secret payloads while enforcing caption-level semantic consistency. Concept drift is defined as a measurable divergence in object-level, attribute-level, or relational semantics between cover and stego captions and is quantified using CLIP text-embedding cosine distance. The main technical contribution is a cross-modal semantic drift regularization term based on BLIP-2 captions generated for cover and stego images. The framework combines this objective with CLIP-based saliency-guided region selection and a lightweight Vision Transformer encoder-decoder. Saliency-map quality is evaluated against ground-truth segmentation masks, and a deterministic bit-to-patch mapping protocol is provided for reproducibility. Experiments use COCO2017, DIV2K, and BOSSBase. Within the controlled six-baseline protocol, StegoGuard achieved a PSNR of 38.9 dB, an SSIM of 0.976 at 256 bits, detector AUC values of 0.521–0.562, and caption similarity of 0.962 as measured by CLIP text cosine similarity (model-relative, not human-verified). The ablation results show that the drift term reduces the measured concept-shift rates while preserving the reported bit-recovery and image-quality levels. Full article

(This article belongs to the Special Issue Data Hiding, Steganography and Its Application, 2nd Edition)

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39 pages, 435 KB

Open AccessArticle

A Knights and Knaves Game on Paths

by Alexios Karalekas and Athanasios Kehagias

Mathematics 2026, 14(18), 3284; https://doi.org/10.3390/math14183284 - 10 Sep 2026

Abstract

We study a nonzero-sum game in which two players label the N vertices of a path as being either Knights (i.e., truth-tellers) or Knaves (i.e., liars). Each vertex utters the sentence “An even number of my neighbors are Knights” and can be consistent [...] Read more.

We study a nonzero-sum game in which two players label the N vertices of a path as being either Knights (i.e., truth-tellers) or Knaves (i.e., liars). Each vertex utters the sentence “An even number of my neighbors are Knights” and can be consistent (the truth value of its sentence agrees with its type) or inconsistent. The Knight player favors consistent vertices and the Knave player inconsistent ones. After characterizing the pure Nash equilibria of the game, we concentrate on the study of best-response dynamics. It turns out that these are the dynamics of a Rule 90 affine cellular automaton:

x (t + 1) = A_{N} x (t) + d_{N}

, where

A_{N}

is the adjacency matrix of the N-path and the arithmetic is performed in the Galois field

F_{2}

. Using the characteristic polynomial of

A_{N}

, we study the properties of the best response dynamics with special emphasis on its periodic behavior. Full article

45 pages, 2638 KB

Open AccessArticle

Multi-Algorithm Optimization of Explanation Stability in Ensemble Learning for Human Capital Attrition Risk Prediction

by Zihe Qi, Ziwei Chen and Fan Si

Mathematics 2026, 14(18), 3283; https://doi.org/10.3390/math14183283 - 10 Sep 2026

Abstract

Attrition models are tuned for discrimination, but the decision they support is a ranking of retention drivers, and nothing requires that ranking to survive retraining. We treat its reproducibility as an objective: hyperparameter optimization for an ensemble of gradient-boosted and bagged learners becomes [...] Read more.

Attrition models are tuned for discrimination, but the decision they support is a ranking of retention drivers, and nothing requires that ranking to survive retraining. We treat its reproducibility as an objective: hyperparameter optimization for an ensemble of gradient-boosted and bagged learners becomes a bi-objective program maximizing the cost-sensitive area under the precision–recall curve jointly with the expected agreement of SHapley Additive exPlanations (SHAP) global rankings across bootstrap refits. Four results follow, three specializing standard theory. Hoeffding’s U-statistic theory makes the estimator unbiased with variance

4 ζ_{1} / R + O (R^{- 2})

, turning “how many refits” into a variance calculation; an elementary spacing argument makes a ranking’s survival a signal-to-noise rather than a noise-magnitude quantity; SHAP linearity makes the retraining variance in the ensemble attributions a quadratic form, so the mixing weights solve a minimum-variance portfolio. The fourth delimits the third: a ranking is invariant to attribution scale, and this variance is not, so attribution variance is no proxy for explanation stability—minimizing it selects the least stable learner on every benchmark—and stability is optimized directly. The cost of the scheme is given in closed form. Our central empirical finding is a caution. Across five public benchmarks spanning

1.4

–

50.6 %

positives, no test separates the multi-objective solvers from the single-objective ones at a practitioner’s budget, and the objective must be estimated to be optimized: at the refit budget such a search would choose, the estimator’s standard deviation exceeds the spread of true stability across the configurations it ranks, so maximizing the estimate overfits it. Re-estimating each search’s own selection with draws that took no part in selecting it leaves a resolvable stability gain on two of the five benchmarks. Giving the search five and

12.5

times the budget surfaces gains that survive where the practitioner’s budget surfaces none, so part of what a cheap search misses is real; but the gap between what a search advertises and what survives does not close as the budget grows, and on one benchmark widens. Declaring the objective is worth doing; what it pays cannot be read off the search that declared it. The caution applies to any model selection that resamples to estimate its criterion and then maximizes it. Full article

(This article belongs to the Special Issue Trustworthy Machine Learning: Explainability, Robustness and Conformal Prediction)

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27 pages, 957 KB

Open AccessArticle

Density-Conditioned Intensity–Topology Decoupling in Directed Global Stock-Market Volatility Networks: Effective Transfer Entropy and Path Homology

by Xueda Wei, Qiqi Gu and Junda Wu

Mathematics 2026, 14(18), 3282; https://doi.org/10.3390/math14183282 - 10 Sep 2026

Abstract

Weighted directed networks can exhibit stronger aggregate interactions without becoming more integrated or topologically richer. Using daily prices for 32 stock indices over 2006–2026, we construct effective transfer entropy (ETE) networks from range-based variance states and recompute GLMY path homology over a directed [...] Read more.

Weighted directed networks can exhibit stronger aggregate interactions without becoming more integrated or topologically richer. Using daily prices for 32 stock indices over 2006–2026, we construct effective transfer entropy (ETE) networks from range-based variance states and recompute GLMY path homology over a directed edge-density filtration. In 88 overlapping 120-common-date windows, total ETE is associated with a higher density required for global weak connectivity and with smaller integrated first- and second-dimensional Betti curves. These are dependence-aware descriptive associations: reduced-overlap estimates are imprecise, and controls for edge concentration and mean variance reduce the connectivity association to approximately zero. Direct coverage measures show that high-intensity windows can distribute weight broadly while their strongest edges reach nodes unevenly. Exact circular-shift tests yield empty 5% BH-FDR backbones, so individual channels are not treated as established. Null-model, rank-stability, coefficient-field, and discretization checks delimit the fixed-density result. A common absolute-weight contrast yields positive rather than negative TETE–Betti associations, showing that the negative baseline relation is specific to the fixed-density filtration. The results distinguish interaction intensity, dispersion, strong-edge coverage, and directed path-homology organization as separate network dimensions. Full article

(This article belongs to the Special Issue Modeling and Data Analysis of Complex Networks)

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18 pages, 5560 KB

Open AccessArticle

A Disease-Guided Representative Gene Selection Framework for High-Dimensional Gene Expression Analysis

by Cihan Kuzudisli, Bahjat F. Qaqish, Burcu Bakir-Gungor and Malik Yousef

Mathematics 2026, 14(18), 3281; https://doi.org/10.3390/math14183281 - 10 Sep 2026

Abstract

Gene expression datasets provide valuable information for disease classification and biomarker discovery; however, their high dimensionality and limited sample size may limit classification performance and reduce biological interpretability. This study proposes GeDiRep, a prior knowledge-guided framework for identifying compact and informative gene subsets. [...] Read more.

Gene expression datasets provide valuable information for disease classification and biomarker discovery; however, their high dimensionality and limited sample size may limit classification performance and reduce biological interpretability. This study proposes GeDiRep, a prior knowledge-guided framework for identifying compact and informative gene subsets. The method first organizes filtered genes into disease-associated groups using curated gene–disease associations from DisGeNET. Each group is then scored according to its predictive contribution, and representative genes are selected using Random Forest-based feature importance. Representative genes from the top-ranked groups are progressively accumulated, and the resulting gene subsets are used to assess classification performance on the test set. Experiments on eight microarray datasets showed that GeDiRep reduced the average number of selected features from 40.3 to 8.3 compared with G-S-M while improving the average AUC from 0.83 to 0.87. In comparison with traditional feature selection methods using the same number of genes, GeDiRep also achieved competitive AUC values. Biological analyses, including term–gene network, hub gene, and heatmap analyses, supported the functional relevance and stability of several selected genes. Overall, GeDiRep provides a structured and interpretable framework for high-dimensional gene expression analysis by selecting reduced yet discriminative and biologically meaningful gene subsets. Full article

(This article belongs to the Special Issue Current Research in Biostatistics, 2nd Edition)

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35 pages, 12006 KB

Open AccessReview

An Evidence-Based Systematic Literature Review of Deep Reinforcement Learning for Manufacturing Scheduling

by Yi-Kai Su and Chun-Jan Tseng

Mathematics 2026, 14(18), 3280; https://doi.org/10.3390/math14183280 - 10 Sep 2026

Abstract

Deep reinforcement learning (DRL) has become an important approach for manufacturing scheduling because it supports sequential decision-making under complex and changing production conditions. However, existing reviews primarily organize the literature by scheduling problem or learning method, providing less explicit support for tracing how [...] Read more.

Deep reinforcement learning (DRL) has become an important approach for manufacturing scheduling because it supports sequential decision-making under complex and changing production conditions. However, existing reviews primarily organize the literature by scheduling problem or learning method, providing less explicit support for tracing how manufacturing context, Markov Decision Process (MDP) formulation, scheduler architecture, and evaluation choices interact across heterogeneous studies. This study presents an evidence-based systematic literature review of DRL for manufacturing scheduling using a structured methodology for corpus construction, configuration-level coding, evidence traceability, study-quality assessment, and cross-study synthesis. The validated corpus comprises 52 primary studies and 54 independently coded DRL configurations. The evidence is synthesized across manufacturing scheduling characteristics, MDP design, DRL scheduler design, hybrid optimization, and empirical evaluation. The results show that scheduler design is context-dependent and architecturally diverse: manufacturing requirements are associated with differences in state, action, and reward formulation, while DRL schedulers combine different learning algorithms, representation architectures, control structures, and complementary optimization mechanisms. The evidence does not establish universal superiority for individual representations, algorithms, or hybrid architectures because reported outcomes remain strongly conditioned by problem formulation and experimental design. Evaluation evidence further highlights limited generalization, uneven statistical and component-level validation, and a continuing gap between benchmark or simulation studies and live industrial deployment. By linking study-, configuration-, and evidence-level information, this review provides a traceable basis for interpreting methodological relationships, identifying research gaps, and guiding the development and evaluation of DRL-based manufacturing scheduling systems. Full article

(This article belongs to the Topic AI and Data-Driven Advancements in Industry 4.0, 2nd Edition)

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21 pages, 7108 KB

Open AccessArticle

Prediction of Joint Roughness Coefficient Using Metaheuristic-Optimized Random Forest Model

by Shijie Xie, Shanyong Wang, Hang Lin, Tianxing Ma, Gang Ma and Liangxu Shen

Mathematics 2026, 14(18), 3279; https://doi.org/10.3390/math14183279 - 10 Sep 2026

Abstract

Reliable estimation of the joint roughness coefficient (JRC) is important for quantitative characterization of joint roughness and provides an important geometric parameter for subsequent joint shear strength assessment. However, single-parameter empirical models cannot fully capture the combined geometric characteristics of joint profiles. In [...] Read more.

Reliable estimation of the joint roughness coefficient (JRC) is important for quantitative characterization of joint roughness and provides an important geometric parameter for subsequent joint shear strength assessment. However, single-parameter empirical models cannot fully capture the combined geometric characteristics of joint profiles. In this study, a dataset containing 113 joint profiles was compiled. Seven morphological statistical parameters, namely SF, SD_h, i_ave, SD_i, R_max, R_p, and Z₂, were used as model inputs. The Sparrow Search Algorithm (SSA) and Harris Hawks Optimization (HHO) were employed to search for more suitable hyperparameter configurations of Random Forest (RF). Based on this framework, RF, SSA-RF, and HHO-RF models were developed for JRC prediction. HHO-RF model achieved the best test performance, with an R² of 0.935 and an RMSE of 1.152. Its RMSE was 27.64% lower than that of the RF model, and it also outperformed SSA-RF and three single-parameter empirical models. Shapley Additive Explanations (SHAP) analysis identified SD_h, R_max, and SD_i as the most influential inputs, accounting for 62.0% of the total mean absolute SHAP value. This ranking highlights the importance of overall elevation dispersion, maximum local relief, and inclination variability in distinguishing profiles with different roughness levels. By integrating complementary information from multiple morphology descriptors, the proposed HHO-RF model provides an accurate and interpretable approach for rapid JRC estimation from two-dimensional joint profiles. The reported performance applies primarily to two-dimensional joint profiles within the morphological range represented by the present dataset, while broader applicability requires further validation. Full article

(This article belongs to the Special Issue Mathematics Applied in Rock Mechanics and Mining Science)

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