We introduce a modified version of the Hybrid Physics-Aware Sparse Neural Network (Hy-PAS), designed to tackle both ordinary and partial differential equations (PDEs). The approach blends classical numerical reasoning with modern deep learning, offering a sparse and interpretable framework that respects the underlying physics. Rather than treating PDE solutions as purely data driven, Hy-PAS reinterprets traditional mesh-free representations through a neural network perspective. In doing so, it bridges the gap between dense neural formulations such as Physics-Informed Neural Networks (PINNs) and established mesh-free numerical schemes. What makes Hy-PAS distinctive is that its parameters correspond directly to physical quantities like node locations, kernel widths, and basis coefficients. This connection allows the model to represent mesh adaptivity naturally and to handle steep gradients or discontinuities with improved stability and accuracy. Hy-PAS is sparse, which means it needs a lot fewer trainable parameters than fully linked networks, which are often used to approximate PDEs. We also show that classical representations, including Fourier and wavelet expansions, emerge as special cases of the proposed architecture, situating Hy-PAS within a broader family of physics-structured neural operators. Extensive numerical experiments with elliptic, parabolic, hyperbolic, and nonlinear PDEs, as well as benchmarks in fluid dynamics, demonstrate the accuracy, robustness, and computational efficiency of Hy-PAS. The framework lays out a mathematically sound way to create neural solvers for scientific computing that are easy to understand and can handle large amounts of data.

Introduction:
Attention mechanisms are a core component of modern artificial intelligence, especially in Transformer-based architectures. However, widely used formulations such as scaled dot-product attention are primarily heuristic and lack a clear energy-based interpretation. This limits theoretical understanding of their stability, robustness, and optimization behavior.

Methods:
In this work, we propose a novel attention mechanism inspired by the quantum harmonic oscillator (QHO). Query–key interactions are modeled as energy states within a bounded harmonic potential, where similarity scores are mapped to energy values using a Hamiltonian-based formulation. Instead of conventional softmax normalization, attention weights are computed through an exponential energy decay function motivated by quantum principles. We further analyze the proposed formulation to establish properties such as bounded gradients, Lipschitz continuity, and improved conditioning of the optimization landscape.

Results:
The proposed QHO-based attention is integrated into Transformer architectures and evaluated on standard classification and sequence modeling tasks. Experimental results show that it achieves performance comparable to conventional attention mechanisms while providing improved training stability and reduced sensitivity to initialization. Empirical analysis also indicates more controlled gradient behavior and enhanced robustness under noisy inputs and adversarial perturbations.

Conclusions:
This work introduces a physically interpretable and mathematically grounded alternative to traditional attention mechanisms. By framing attention through an energy-based perspective, it strengthens the theoretical foundation of neural architectures and offers a promising direction for building more stable, robust, and interpretable deep learning models.

Repeating decimal fractions, both simple and compound, have long posed significant computational challenges in mathematics and education. Traditional methods, which convert these decimals into fractions for algebraic operations, are often time-consuming, error-prone, and complex, particularly when dealing with long, multiple, or nested repeating cycles. This study presents a novel direct method for performing addition, subtraction, and division on repeating decimals without requiring conversion to fractions. By aligning and synchronizing periodic sequences, the method enhances computational speed, maintains high accuracy, and minimizes human error. The approach was systematically developed through detailed analysis of decimal structures, construction of original problems, and rigorous verification across a wide range of cases, including simple cycles, composite cycles, and multiple simultaneous repeating sequences. Comparative evaluation with conventional fraction-based methods demonstrates that the proposed approach significantly reduces computational complexity while achieving identical results. Preliminary observations further indicate promising potential for extending the method to multiplication of repeating decimals. The simplicity, efficiency, and broad applicability of this method make it particularly suitable for educational contexts, theoretical research, and practical applications in engineering, computer science, cryptography, and other related fields. Overall, this study provides a clear, systematic, and reliable framework for handling repeating decimal computations, offering both pedagogical and computational advantages over traditional fraction-based techniques, and opening opportunities for further development and software implementation.

The accurate modeling of spray deposition dynamics is essential for assessing the efficiency and environmental consequences of UAV-based precision agriculture. Traditional fluid mechanics and physical dispersion models, while valuable, often struggle to capture the complexity of real-world spraying conditions influenced by canopy heterogeneity, wind variation, and nozzle type. This study proposes an AI-augmented mathematical framework that integrates partial differential equation (PDE)-based spray dispersion models with statistical learning methods to improve prediction accuracy. Random forests and gradient boosting algorithms are employed to calibrate and refine predictions from field-collected UAV spray data, while principal component analysis and multivariate regression identify the dominant factors affecting deposition efficiency, including droplet size, nozzle angle, wind speed, and canopy density. Experimental validation in orchard conditions demonstrates that the hybrid PDE–AI model achieves superior accuracy over traditional physics-only approaches, thus reducing the prediction error by more than 20%. Beyond operational optimization, the framework also quantifies reductions in pesticide use, water footprint, and CO₂ emissions, thereby linking mathematical modeling with sustainability objectives. The results confirm that combining mathematical analysis with AI not only improves predictive capability but also supports decision-making for environmentally responsible precision spraying practices. This research highlights the potential of hybrid mathematical–statistical approaches to advance sustainable agriculture.

The scalability of incompressible Navier–Stokes solvers on massive parallel clusters is fundamentally constrained by the pressure projection step, where the global elliptic coupling of the standard Pressure Poisson Equation (PPE) necessitates expensive all-to-all communication and creates a severe latency bottleneck. To overcome this barrier, we introduce Locality-Certified Screened Projection (LCSP), a novel framework enabling a fully parallel, communication-free pressure solve. By relaxing the strict incompressibility constraint into a penalized form ∇· uⁿ⁺¹ + ηψ = 0, we transform the PPE into a screened Helmholtz problem (-Δ + κ²) ψ = ƒ. This operator exhibits intrinsic locality characterized by the exponential Yukawa decay of its Green's function. Leveraging this property, we implement a single-pass Overlap-Restrict assembly strategy: the computational domain is partitioned into overlapping tiles where local problems are solved entirely independently, and solutions are then restricted to the core without any inter-subdomain trace exchange. Our rigorous error analysis demonstrates that artifacts from artificial tile boundaries decay exponentially with the overlap width, allowing the mass conservation error to be explicitly controlled via the screening parameter κ. Extensive numerical benchmarks confirm that LCSP successfully decouples the global dependency, reduces the peak memory footprint to Ο(|tile|), and achieves optimal linear weak scaling for large-scale flow simulations. Ultimately, LCSP establishes a mathematically grounded trade-off between exact incompressibility and parallel efficiency, providing a robust, highly scalable solution for high-fidelity CFD simulations on next-generation heterogeneous supercomputers.

In this study, a fractional-order mathematical model is proposed to describe the transmission dynamics of Monkeypox using the Caputo fractional derivative, which effectively captures memory effects and long-range temporal dependencies inherent in disease spread processes. The population is divided into epidemiologically relevant compartments, and the qualitative behaviour of the resulting fractional system is rigorously investigated. Fundamental properties such as positivity and boundedness of solutions are established, ensuring the biological feasibility and well-posedness of the model. The basic reproduction number, R₀​, is derived using the next-generation matrix approach and is identified as a key threshold parameter that governs the persistence or extinction of the disease within the population. Local stability analysis reveals that the disease-free equilibrium is locally asymptotically stable when R₀<1, whereas for R₀>1, the system admits a unique endemic equilibrium that is also locally asymptotically stable. To reduce the disease burden while balancing the costs associated with intervention strategies, an optimal control problem is formulated by incorporating time-dependent controls representing vaccination of susceptible individuals, isolation of infectious cases, and treatment of infected individuals. The necessary conditions for optimality are derived using Pontryagin’s maximum principle, leading to a coupled system of state and adjoint equations. Numerical simulations based on the Adams–Bashforth–Moulton predictor–corrector method are performed to validate the analytical findings and demonstrate the effectiveness of the proposed control strategies in significantly reducing Monkeypox transmission.

Cervical cancer, one of the deadliest diseases, is becoming a serious public health issue. In particular, developing countries face a significantly higher burden due to a lack of diagnostic and treatment facilities. In 2022, cervical cancer accounted for over 350,000 deaths and 660,000 new cases, making it the fourth most prevalent disease among women globally. Almost 94% of all deaths were reported in nations with low and middle incomes. Chronic infection with high-risk strains of the human papillomavirus (HPV) is the primary cause of this disease. The research developed a cervical cancer epidemic model to analyze the transmission dynamics of human papillomavirus (HPV) infection and the progression of cervical cancer. The existence, uniqueness, non-negativity, and boundedness of the model's solution are demonstrated. Two types of equilibrium points, disease-free equilibrium and endemic equilibrium, are calculated. The stability of both equilibrium points is also discussed. In order to identify the parameters that significantly contribute to the spread and persistence of the disease in the population, sensitivity indices are determined. Numerical simulation is performed through the Levenberg-Marquardt algorithm in conjunction with artificial neural networks (LM-ANNs), which support our analytical findings. The LM-ANNs' effectiveness, precision, reliability, validity, efficacy, accuracy, and convergence rate are evaluated through time series analysis, regression analysis, examination of the training state, evaluation of error histograms, convergence analysis, and the calculation of both absolute errors and statistical metrics. The numerical results are supported by analytical results.

Introduction:

Fractional calculus is a branch of mathematical analysis that studies the possibility of extending the order of the differentiation and integration operators to a noninteger order. We focus on the stability of the implicit boundary value problem for Caputo fractional differential equations of variable order.

where is a continuous function, , and is the caputo fractional derivative of variable order.

Ulam Hyers stability: Assuming

be a partition of the interval and let be a PWCF with respect to , i.e., where are constants with (). Using (H1) the BVP(1) becomes

Theorem 1:

The (1) is (UH) stable if there exists such that for any , and for every solution of the following inequality , there exists a solution of (1) with

Theorem 2: Assume that (H1) is satisfied and

(H2)

and the inequality holds, then (1) is (UH) stable.

Conclusion:

This study is a valuable contribution to the expanding field of fractional calculus, in which we skillfully proved the stability.

This communication presents a foundational framework for semantic projections in Natural Language Processing (NLP), focusing on word embeddings and projection-based semantic indices. We formalize two complementary perspectives: (i) geometric embeddings, where terms are represented in $\mathbb{R}^d$ and semantic proximity is measured through a metric; and (ii) set-based representations, where meaning is modeled in a finite measure space and projections arise as normalized overlap ratios.

Under Lipschitz regularity assumptions, we show that projection estimators admit explicit error bounds, ensuring stability and consistency across representations. Building on these foundations, we address two practical stability questions in NLP pipelines: \emph{projection coherence} (consistency of projections across sources) and \emph{universe transfer} (estimation of projections when the semantic universe changes). To quantify and control instability, we exploit Lipschitz regularity assumptions and metric-based estimators, including McShane–Whitney-type extensions.

We introduce computable metric-based estimators and prove that they preserve Lipschitz constants, yielding controlled transfer errors. In particular, we show that any Lipschitz semantic projection defined on a universe $U_0$ can be extended to a larger universe $U_1 \supset U_0$ by using the McShane--Whitney formulas and interpolating, preserving the Lipschitz constant and enabling stable semantic transfer.

Empirically, we validate the framework on multi-source semantic projections (DOAJ, Scholar, Google, and Arxiv), showing that coherence can be quantified using correlation and clustering, and that universe transfer can be evaluated through RMSE-type errors. These results demonstrate that modeling choices (data source, universe design, and estimator selection,) have measurable effects on semantic conclusions, providing both theoretical guarantees and practical diagnostics for robust NLP pipelines. This work extends our previous study published in \emph{Axioms} (14(5), 389).

The Monte Carlo simulation is a numerical computing algorithm that computes its outcome through repeated random sampling. It is a technique used to model the different types of uncertainty that impact the value of the investment or underlying instrument under consideration. This study used the Monte Carlo method on a GPU to investigate and assess the Arithmetic Asian Option. By utilizing the GPU's built-in parallelization capability, the study was able to accelerate pricing options more effectively and decisively when compared to the CPU implementation. It also evaluated the amount of power consumed by the GPU and optimized the power for greater efficiency. The effectiveness of the method was examined by utilizing the various paths that were constructed using a randomized function to test this assertion. Ten paths (samples) with ranges between 100,000 and 1,000,000 were taken into consideration. The GPU was utilized to enhance performance in terms of speedup, computation time, power consumption, and time complexity. The GPU's power utilization is then further optimized to improve performance. A detailed experiment using GPU and quad-core multiprocessor systems was conducted. The optimized accelerator for the GPU was coded using the CUDA programming language, libraries, and Application Programming Interface (API). Experimental results using different simulated scenarios demonstrated that the GPU was found to be more efficient in every scenario, both in terms of speed and power consumption. Additionally, the optimized GPU-accelerated optimizer results also showed increased speed and optimally reduced power consumption compared to the quad-core CPU counterpart. This approach will be beneficial in mathematical financial computing and stock market price forecasting.