Abstract

This paper presents a symbolic framework for the analysis of discrete elliptic boundary value problems on lattice domains. Discrete symbol classes, a notion of ellipticity, and compatible boundary operators are introduced. By extending the theory of pseudodifferential operators to the discrete setting, elliptic difference operators are formulated via their symbols on the dual torus, and ellipticity is characterized through principal symbol estimates. Using Vasil’ev’s wave factorization method, adapted to discrete symbols, we construct parametrices and establish well-posedness of discrete boundary value problems.

Introduction

Discrete analogues of pseudodifferential operators arise naturally in numerical analysis and discrete physical models. While elliptic boundary value problems in the continuous setting are well understood, their discrete counterparts require specialized analytical tools. Symbolic pseudodifferential calculus provides a natural framework for studying ellipticity, boundary conditions, and regularity, motivating its extension to lattice-based operators.

Methodology

The analysis is carried out on discrete domains Ω_h⊂Zⁿ. Discrete pseudodifferential operators are defined using the discrete Fourier transform and suitable symbol classes. Boundary value problems are formulated by coupling interior difference operators with boundary operators acting on the discrete boundary. Ellipticity is defined through principal symbol estimates, allowing for the construction of parametrices within the discrete symbolic calculus.

Results

The main result shows that elliptic difference operators admit parametrices obtained via wave factorization of their symbols. This approach yields a discrete analogue of the Lopatinski–Shapiro condition and implies Fredholm properties, as well as existence, uniqueness, and regularity of solutions, even for irregular lattice boundaries.

Conclusion

This work establishes a rigorous symbolic theory for discrete elliptic boundary value problems. By adapting Vasil’ev’s wave factorization to the discrete setting, the paper bridges continuous pseudodifferential theory and discrete models, providing a solid analytical foundation for stability analysis and numerical methods for elliptic problems on lattices.

Image restoration is a fundamental problem in image processing aimed at recovering high-quality images from degraded observations corrupted by blur and noise. This task is commonly formulated as a constrained or unconstrained optimization problem consisting of a data fidelity term and a regularization term to preserve important image features such as edges and textures. Due to the large-scale and ill-conditioned nature of image restoration problems, efficient iterative optimization techniques are required.
Conjugate gradient (CG) methods have been widely applied to image restoration because of their low memory requirements and fast convergence properties in solving large sparse linear and nonlinear optimization problems. Several CG variants have been investigated in the literature and successfully integrated into image restoration frameworks. However, the performance of classical CG methods strongly depends on the choice of search direction and step size parameters. In particular, inappropriate line search strategies may lead to slow convergence or instability when applied to nonlinear and nonconvex image restoration problems. To address these limitations, this paper proposes an adaptive hybrid conjugate gradient algorithm that combines the Fletcher–Reeves and Polak–Ribiere schemes through an adaptive weighting strategy. The proposed method exploits the numerical stability of the Fletcher–Reeves approach and the fast convergence behavior of the Polak–Ribiere method, thereby achieving a balanced and robust optimization performance.
Furthermore, the strong Wolfe line search conditions are adopted to guarantee sufficient descent and curvature properties at each iteration, ensuring global convergence of the proposed algorithm. A theoretical convergence analysis is presented under standard assumptions. The experimental results on blurred and noisy images demonstrate that the proposed adaptive hybrid conjugate gradient method outperforms conventional CG algorithms in terms of both convergence speed and restoration quality, as measured by Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity Index (SSIM).

This reseach investigates the dynamics of gradient propagation in complex-valued neural networks (CVNNs) in the presence of planted singularities. Unlike real-valued neural networks, CVNNs operate in the complex domain, enabling the simultaneous representation of magnitude and phase information. This property makes them particularly suitable for applications such as radar signal processing, biomedical imaging, and quantum systems. However, training stability in CVNNs is often compromised by optimization instabilities arising near singular points in parameter space, where gradients may either vanish or diverge.

To address this issue, a rigorous mathematical framework is developed to characterize how the local behavior of activation functions near the origin governs gradient stability. Using Wirtinger calculus and asymptotic analysis of activation derivatives near singularities, the study establishes a trichotomy of gradient regimes: (i) exploding gradients when the derivative diverges, as observed in magnitude-gated activations such as modReLU; (ii) vanishing gradients when the derivative approaches zero, as in zReLU with dead zones; and (iii) bounded and stable gradients when the derivative converges to a finite constant, as in smooth activations such as the Cardioid. The analysis formally demonstrates that while network depth amplifies gradient behavior, it does not alter the underlying regime determined by the activation function.

The theoretical results are supported by extensive experiments on both shallow K-2-K architectures and deeper multi-layer CVNNs trained on synthetic complex-valued datasets. Empirical findings confirm that modReLU frequently leads to gradient explosion, zReLU exhibits prolonged training stagnation, Split ReLU shows irregular convergence due to non-smooth transitions, and the Cardioid activation consistently maintains bounded gradients and stable optimization. These results validate the predictive strength of the proposed theoretical framework and underscore its practical relevance.

Abstract
This paper introduces WPMAXSAT-HNN, a novel neuro-symbolic energy-based framework for optimizing smart contract efficiency and security in blockchain-based financial systems. The framework establishes a precise mathematical mapping from Weighted Partial Maximum Satisfiability (WPMaxSAT) formulations to Hopfield Neural Network (HNN) configurations, enabling simultaneous formal verification and gradient-based optimization. By distinguishing between hard constraints (essential contract semantics) and soft constraints (optimization objectives such as gas cost reduction), the method provides a unified approach for constraint satisfaction and performance enhancement. Experimental validation across 127 production smart contracts, including Automated Market Makers, Lending Protocols, Derivatives, Insurance, Payment Channels, Token Systems, and Governance, demonstrates significant and consistent advantages over existing MAXSAT-HNN and RANMAXSAT-HNN approaches. The proposed framework achieves an average of 18.0% ± 2.9% gas savings while maintaining a 99.0% ± 0.6% satisfaction rate for hard constraints and a 2.04× computational speedup. The model’s efficiency is reflected in the best aggregate Bayesian Information Criterion (-200.5) and the highest F1-Score (0.958 ± 0.014). Performance is particularly strong for Payment Channels (24.8% ± 2.1% gas savings, F1-Score 0.981) and Token Contracts (19.3% ± 2.5% savings, 2.32× speedup). Real-world Ethereum testnet deployments confirm its practical impact, with an average return on investment of 3,744 gas saved per second across major protocols such as Uniswap, Compound, and high-throughput payment networks. The neuro-symbolic architecture effectively integrates exact symbolic reasoning with approximate neural optimization, addressing critical challenges in FinTech: reducing transaction costs, enhancing security through formal verification, and mitigating operational and compliance risks. By combining optimization with provable correctness, WPMAXSAT-HNN offers a foundational technology for the next generation of efficient, secure, and scalable decentralized financial applications.

Deep Survival Machines (DSMs) integrate neural networks with parametric survival distributions, yet their architecture tightly couples representation learning with parameter estimation, limiting interpretability and obscuring the role of latent risk structure in individualized predictions. This tight integration makes it difficult to disentangle how covariate information is transformed into distributional parameters, thereby reducing transparency in model behavior and complicating theoretical understanding of how latent features influence survival dynamics under censoring. This study proposes a Modified Deep Survival Model that explicitly decouples nonlinear feature extraction from probabilistic survival modeling. A multilayer perceptron transforms high-dimensional covariates into compact survival-relevant representations, which are then passed to a New Exponential Power Distribution (NEPD) for parameter estimation and survival function generation. The NEPD accommodates flexible tail behavior, enabling more accurate modeling of heterogeneous time-to-event patterns than conventional parametric forms. The model is trained using censored-likelihood optimization with prior regularization to ensure stable parameter learning. On the SUPPORT benchmark dataset, the Modified DSM achieves a Concordance Index of 88.34%, outperforming the conventional DSM by 1.34%. This improvement demonstrates that separating latent structure learning from distributional estimation enhances both discriminative accuracy and structural transparency. The proposed study offers a robust alternative for clinical risk prediction requiring interpretable, individualized survival distributions

Spreading and cascading processes on complex networks arise in diverse contexts, including infectious disease transmission, infrastructure failures, and information diffusion. A central challenge is to understand when large-scale cascades emerge and how to identify the nodes whose control most effectively mitigates systemic risk. This work addresses these questions through a unified mathematical and computational framework combining epidemic theory, nonlinear dynamics, and data-driven learning.

We first revisit epidemic threshold theory for SIS and SIR dynamics on networks, highlighting the role of structural heterogeneity. Using heterogeneous mean-field arguments and spectral analysis, we show how the epidemic onset is governed by the spectral radius of the adjacency matrix, providing a unifying interpretation of threshold conditions and explaining the heightened vulnerability of highly heterogeneous networks.

To capture realistic cascade behavior beyond static models, we introduce a dynamical network framework in which nonlinear nodal dynamics are coupled through the network topology. The model reproduces subcritical, critical, and supercritical cascade regimes and generates bursty, scale-free activity patterns observed in real systems. Building on this dynamical setting, we propose a forecasting and intervention strategy based on radial basis function (RBF) learning applied to historical system states. The method predicts future cascade growth and quantifies the suppressive impact of virtually deactivating individual nodes, enabling dynamic identification of influential spreaders without requiring explicit knowledge of network topology.

Numerical experiments show that the proposed RBF-based targeting strategy significantly outperforms degree-based and random interventions in suppressing large cascades. Finally, we discuss how graph neural networks can approximate structural risk measures, offering a scalable bridge between network topology and learning-based control in large systems.

Sleep Deprivation (SD) is a growing issue that impairs cognitive, physical, and mental health and increases the risk of accidents and chronic diseases. Conventional detection methods are often intrusive, costly, or impractical for daily use. Aligned with United Nations Sustainable Development Goal 3 (UN SDG 3), this study aims to ensure healthy lives and promote well-being. The development of cost-effective, non-invasive, and accessible tools for SD detection is essential to integrate sleep health into public health approaches. The study developed a Support Vector Machine (SVM) trained by the researchers to classify their mild SD status through voice analysis. The researchers trained the model on an open-access dataset from the Open Science Framework (OSF), which was extracted through Spectro-Temporal Modulation (STM) features. To have a solution, the study evaluated the performance of SVM through STM features. It analyzed its performance across different dimensions of STM features, sessions, and Balanced Accuracy (BAcc) at the population and individual levels. The SVM achieved a training and testing BAcc of 0.8588 and 0.7476, respectively, which indicates sufficient performance and generalization. Statistical analyses are applied to determine the differences between the other trained models in different dimensions of STM: frequency-rate (FR), frequency-scale (FS), and scale-rate (SR). Analysis of Variance (ANOVA), Multivariate Analysis of Variance (MANOVA), and t-tests proved those hypotheses.

The field of artificial intelligence (AI) and computer vision presents significant opportunities to advance physical rehabilitation. However, their application in rehabilitation assessment still remains underexplored. This study addresses the need for accessible and precise real-time evaluation of post-rehabilitation exercises, contributing to the United Nations Sustainable Development Goal 3: Good Health and Well-Being. The proposed system integrates BlazePose, a human pose estimation model, alongside a Long Short-Term Memory (LSTM) model to classify movement correctness and provide real-time feedback. The system was trained on three low back pain rehabilitation exercises from the KinesiothErapy and Rehabilitation for Assisted Ambient Living (KERAAL) dataset. The researchers found that a 13-keypoint skeletal configuration yielded the most optimal performance, achieving an F1-Score of 80% for flank stretch, 78% for torso rotation, and 77% for hiding face. Furthermore, with the same skeletal keypoint configuration and multiple independent training sessions, a Repeated Measures One-Way ANOVA confirmed statistically significant differences (p < .05) in F1-Scores across exercises, indicating varied model performance among the exercise types. The evaluation focused on assessing the robustness and consistency of the model across multiple training runs to ensure reliable performance measurement. The results provide empirical evidence supporting the applicability of deep learning–based pose analysis for automated rehabilitation exercise assessment.

The increasing reliance on machine learning in high-stakes domains such as healthcare, finance, and public policy necessitates models that are not only accurate but also interpretable and causally robust. This study presents a hybrid framework that integrates explainable artificial intelligence (XAI) techniques with causal inference methods to enhance both the transparency and the reliability of machine learning systems.

The proposed framework combines SHAP (Shapley Additive Explanations) for localized interpretability, counterfactual reasoning to support decision justification, and ensemble learning techniques including XGBoost and LightGBM for high-performance prediction. To further enrich the interpretability in complex data scenarios, the model introduces a novel integration of Large Language Models (LLMs) into structured tabular environments through a statistical vector fusion approach.

As a practical demonstration, a synthetic dataset simulating personalized healthcare resource allocation (e.g., ICU prioritization) is constructed based on real-world statistical distributions from OECD and WHO health indicators. Features include clinical variables (e.g., comorbidity index, age, oxygen level) and social determinants (e.g., income level, regional density). The model is evaluated based on both predictive performance (AUC, F1-score) and interpretability metrics (feature attribution stability, causal graph coherence).

Initial results indicate that the hybrid approach not only improves model transparency but also supports ethically aligned and causally coherent decision-making, which is essential for applications where decisions have critical real-life consequences. The framework contributes to the growing literature on explainable and reliable machine learning, offering both theoretical advancements and practical pathways for integration into digital decision support systems.

The application of mathematical theory in engineering is crucial to solve complex challanges nowadays. The insights of recent mathematics development are great momentum to utilize the mathematics into the different fields in education, engineering, and business. In maritime transportation, fuel efficiency plays a vital role in creating efficient operation measures. The concept of optimization theory, along with the least square error method in mathematical theory, has been used to solve this issue. To partly contribute to this topic, a mathematics model has been established based on the least square error theory to accurately estimate the fuel consumption mass of marine diesel engines. In this model, the complexity of uncertain factors impacting the fuel consumption of marine diesel engines is addressed to accurately improve the fuel consumption estimation model. Based on the least square error method, a reduction in fuel consumption estimation error is achieved. A full evaluation of this model will be presented through the performance metric, including the coefficient of determination (R²), the mean absolute error (MAE), and the root mean square error (RMSE). This study is significant in determining the fuel consumption of marine diesel engines nowadays through mathematical theory. In the future, an advanced model will be established from this mathematical model to determine fuel efficiency in maritime transportation.