Computer Science and Mathematics

Abstract: This article reviews the linear solvers available in OpenFOAM and assesses their impact on the convergence behaviour of the SIMPLE algorithm. The discretisation of transport equations in CFD results in large and sparse linear systems, for which the choice of linear solver strongly influences the computational time. Although the solver does not change the final discrete solution, the difference in speed and robustness between the solvers can be more than an order of magnitude. A brief overview is given of how the velocity and pressure fields are decoupled in OpenFOAM, followed by a detailed review of the main linear solver families, including direct methods, basic iterative methods, multigrid methods and Krylov subspace methods, with attention to their practical strengths and weaknesses. The performance of the most advanced solvers is evaluated on a full-scale non-reacting kiln case consisting of 2.3 million cells. The pressure-corrector equation is identified as the main bottleneck in the SIMPLE algorithm. The Conjugate Gradient (CG) solver with the Generalised Geometric–Algebraic MultiGrid (GAMG) preconditioner is found to be the fastest and most stable method, achieving speed-ups of up to a factor of 7 compared to the slower advanced methods. Using GAMG as preconditioner also improves the robustness of the Bi-CGStab method.

Abstract: This paper studies fixed-point iteration methods within the contraction-mapping framework and develops a center-based iterative algorithm constrained by Mahalanobis-distance rejection. Building on Banach’s fixed-point theorem, we relate contractivity to the Jacobian norm and derive estimators for the order of convergence, showing the method exhibits linear convergence under the stated conditions. The algorithm modifies standard k-means by using conditional expectations for parameter updates and discarding low-probability tail points of Gaussian components via a p-quantile criterion of the chi-squared Mahalanobis distribution. The procedure is analyzed in both continuous (marginal Gaussian) and discrete noisy settings, with numerical experiments that quantify convergence behavior and robustness to outliers. A spherical clustering validity index is introduced to optimize the p-quantile for pattern detection. Applications to industrial-scene, RGB segmented images demonstrate effective detection of spherical patterns in highly noisy structures, illustrating the method’s practical potential for robust Gaussian mixture estimation and pattern recognition.

Abstract: We rigorously disprove the existence of global smooth solutions to the three‑dimensional incompressible Navier–Stokes equations in a periodic rectangular cuboid domain $\Omega$ subject to the prescribed smooth body force. The smooth initial data of the flow field is derived from a two-dimensional stationary exact solution. The analysis is grounded in Sobolev space regularity, the decomposition of velocity into a time‑averaged mean flow and a disturbance flow, the local vanishing of sum of the viscous term and the body force, and the Energy–Velocity Monotonicity Principle (EVMP). When the Reynolds number exceeds the critical value for turbulent transition, the nonlinear convective term dominates over the viscous term and external force term, nonlinear interactions amplify disturbances, leading to local cancellation of the sum of the mean flow viscous term and the body force with the disturbance viscous term at a finite critical time $t^*>0$ and interior point $\boldsymbol{x}^*\in\Omega$. This cancellation leads to the local mechanical energy gradient along the streamline being zero when the time derivativer is zero, which by EVMP requires $|\boldsymbol{u}(\boldsymbol{x}^*,t^*)|=0$, contradicting the existing non‑vanishing velocity. The contradiction generates a finite‑time regularity singularity, under which the velocity gradient $L^\infty$‑norm diverges. This violates the Sobolev embedding condition required for global smoothness of solutions. This study resolves the problem statement (D) in Fefferman (2006).

Abstract: We present GLYBATOMAQ™, a rank-centric and quantum-geometric framework for GLIPR1-focused in silico screening. The framework treats docking as a fixed-protocol comparative oracle and places the main methodological emphasis on auditable rank movement, positive-semidefinite operator geometry, DFT-derived electronic descriptors, QMC-style uncertainty auditing, and MQWalk topology validation. To make the quantum-geometric contribution explicit for quantum-geometry reporting, we introduce Bures/Fubini-Study-style distance controls, local metric and curvature penalties, quantum Fisher information-inspired sensitivity diagnostics, Berry-type gauge-consistency checks, and a candidate-level assembly certificate. Quantum-geometry reporting elements define the reporting schema, connect each mathematical object to a computational decision, and show how DFT, QMC, MQWalk, curvature, and diagnostic penalties are fused without claiming experimental affinity, efficacy, or biomolecular quantum transport. The output is a reproducible leaderboard and audit bundle for GLIPR1-oriented computational hypotheses: rank shifts are accepted only when supported by electronic descriptors, uncertainty-aware energetic evidence, operator-overlap topology, and chemistry-safe HMC/HSX feasibility constraints.

Abstract: This paper presents new optimal eighth-order families with weight functions for solving nonlinear systems, obtained as a generalization of the first optimal eighth-order CTT8 method introduced by Cordero, Torregrosa and Triguero-Navarro. The proposed schemes are constructed by combining a Newton-type predictor with high-order correction steps whose weight functions are suitably chosen to preserve optimal convergence while keeping a low computational cost. To the best of our knowledge, this work introduces the first family of optimal eighth-order methods for nonlinear systems, in the sense of the Cordero Torregrosa conjecture, developed through a weight-function technique. A complete local convergence analysis is carried out under standard smoothness assumptions, proving eighth-order convergence for nondegenerate solutions. The computational efficiency of the proposed methods is also studied and compared with several existing high-order iterative schemes. Numerical experiments on nonlinear systems of different dimensions confirm the theoretical order of convergence and show the robustness of the new families. In addition, a Fredholm integral equation is solved, followed by a semilinear elliptic Dirichlet problem, further illustrating the reliability and computational performance of the proposed weight-function-based methods.

Abstract: In this paper, we investigate the higher-order partial differentiation formulas for the new extended Srivastava hypergeometric functions ^KĤ_A, ^KĤ_B, and ^KĤ_C with respect to the parameters κ, ω, and τ. By employing their triple series representations together with classical identities of the Pochhammer symbol and the beta function, we derive explicit formulas for the r-th order partial derivatives. The obtained results show that repeated differentiation preserves the structural form of these functions, up to multiplicative Pochhammer factors and systematic shifts in the parameters. This reveals an inherent invariance property of the considered extended hypergeometric family under differential operators. The presented formulas provide a unified and consistent framework for all three classes of ^KĤ-functions and may be useful in the study of fractional differential equations, integral transforms, and related applications in mathematical physics and applied analysis.

Abstract: This paper presents an efficient numerical framework for solving the Merton jump–diffusion PIDE in European option pricing. To handle the nonlocal integral term caused by asset price jumps, we employ an IMEX scheme that preserves the tridiagonal structure of the linear system. A nonuniform spatial grid and fast Gaussian quadrature with spline interpolation are used to enhance accuracy and computational speed. We prove the unconditional stability and convergence of the IMEX–Crank–Nicolson scheme. Numerical experiments confirm the method’s effectiveness and superiority over traditional approaches in handling jump-diffusion dynamics. Furthermore, the proposed method significantly reduces computational cost while maintaining high precision, making it suitable for real-time applications. The results demonstrate robust performance across various market conditions and jump parameters.

Abstract: FEFLOW is used to analyze seepage flow through a tailings storage facility constructed by on-dam cycloning. Partial saturation of tailings beach material is accounted for by solving Richards’ transient flow equation throughout facility staged construction, and seepage analysis of idealized 1D and 2D staged construction processes are used to benchmark FEFLOW simulation results. These results include design intended phreatic surface level, drain flows, and water balance of the tailings storage facility. Transient seepage analysis is also used to examine how as-built rise in the facility’s dam crest phreatic surface levels may be controlled by both hydraulic conductivity gradient of the tailings beaches and hydraulic conductivity of the dam downstream shells.

Abstract: We establish a complete asymptotic expansion for Quantum Neural Network Operators (QNNOs) approximating arbitrary quantum channels, providing a non-commutative analogue of the classical Voronovskaya theorem. Within a rigorous functional analytic framework, we introduce quantum Sobolev and Hölder spaces C^m,γ(H) based on Fréchet differentiability in the Liouville representation, and we measure approximation errors using the diamond norm. Our main result, the Quantum Voronovskaya–Damasclin Theorem, reveals a multiscale decomposition of the error into three distinct contributions: integer-order terms involving Fréchet derivatives and even kernel moments, fractional corrections governed by Marchaud fractional derivatives that capture Hölder regularity of order γ, and intrinsically non-commutative commutator terms that vanish in classical settings. The remainder is sharply bounded by O(n−^(m+γ)(log n)^3m/2) with an explicit constant depending on m, γ, and the Hilbert space dimension d. As applications, we derive a quantum central limit theorem for QNNO fluctuations, construct optimal interpolation geodesics between quantum channels using Kubo–Ando means, and develop a quantum Richardson extrapolation method that reveals fundamental acceleration limits imposed by fractional smoothness. Our results establish a rigorous bridge between classical approximation theory, fractional calculus, and quantum machine learning, providing a powerful tool for the design and analysis of quantum neural networks in finite-dimensional settings.

Abstract: This work investigates the exact boundary controllability of the heat equation posed in a spherical domain, using the method of moments. Starting from the spectral decomposition of the radial Laplacian in the weighted space \( L^2_{r^2}(0,R) \), we derive a sequence of moment equations whose solvability is established via biorthogonal sequences, following the Fattorini–Russell theory. The resulting control function is expressed as a series expansion whose convergence in \( L^2(0,t_f) \) is rigorously proved for any initial temperature distribution in \( L^2_{r^2}(0,R) \) and any final time tf exceeding a minimal threshold. The numerical implementation employs linear finite elements in space and the implicit Euler scheme in time. The accuracy of the solver is rigorously verified through the Method of Manufactured Solutions (MMS), confirming optimal convergence rates: second-order in space and first-order in time. Numerical experiments show that the computed control drives the temperature to zero with a residual norm of 1.96×10⁻⁴, consistent with the spatial discretization error. A comparative analysis with two alternative approaches—the Hilbert Uniqueness Method (HUM) and gradient-based optimization—demonstrates that the proposed moment-based strategy achieves an 18.9% reduction in total control energy relative to HUM, making it particularly attractive for energy-constrained thermal control applications.

Abstract: Permutation flow shop scheduling is an important production planning problem handled in different contexts. Just-in-time measures have been significant in the optimization of real problems and one is specifically addressed here: the total earliness and tardiness of jobs. The most used approach in the literature to mathematically express this measure is to sum them up using unit weights thus obtainning a mono-objective function. In this paper it is shown that this is a simplification of a problem that is inherently multi-objective, highlighting how a more comprehensive approach can better support decision-making. A bi-objective mathematical optimization model and tools capable of analyzing the mono-objective solution within the multi-objective perspective are proposed. A computational study to analyze the benefits and difficulties of the solution using the bi-objective approach is presented. The results show that for large scale instances in which the tardiness factor is small, the conflict between the objectives of minimizing the total earliness and minimizing the total tardiness of jobs increases significantly. Therefore, the multi-objective approach has a greater potential to support decision-makers. Furthermore, we show that the choice of the solution method must be carefully considered, since the Pareto frontier associated with most instances has many non-supported points.

Abstract: This paper investigates bifurcation dynamics in a fractional-order extension of the classical Susceptible–Latent–Breaking–Out model for computer virus propagation. The proposed framework incorporates two distinct transmission-related time delays and employs Caputo fractional derivatives of incommensurate orders, with the delays associated with infection rate and latent period selected as the primary bifurcation parameters. Due to the combined influence of multiple delays and incommensurate fractional exponents, the resulting system exhibits a complexity that goes beyond most existing models in the literature. By linearizing the model around its endemic equilibrium and analyzing the associated characteristic roots, we characterize how the system’s qualitative behavior depends on the magnitudes of the time delays, and establish explicit sufficient conditions for bifurcation to occur. In particular, the endemic equilibrium remains asymptotically stable as long as each delay stays below a certain critical value; once any delay exceeds its threshold, the system undergoes a Hopf bifurcation, leading to sustained periodic oscillations in virus prevalence. Numerical simulations are provided to support the analytical results, and they show strong agreement between predicted and observed system responses. These findings enhance theoretical insight into bifurcation mechanisms in fractional-order delay models of epidemic dynamics on networks, and may offer useful guidance for designing containment strategies in large-scale interconnected systems.

Abstract: This study investigates the regularity of the three-dimensional (3D) incompressible Navier-Stokes equations (NSE) for plane Couette flow, a canonical shear-driven flow model with a well-defined laminar-turbulent transition threshold. Employing Sobolev space theory and the Energy-Velocity Monotonicity Principle (EVMP), we rigorously prove that no global smooth solutions exist as the Reynolds number exceeds the critical value \( Re_{cr} \). Prior studies have revealed that a zero velocity gradient on the velocity profile is the necessary and sufficient condition for turbulence generation in 3D plane Couette flow, yet they lack mathematical theoretical proof from the perspective of partial differential equation framework. This study fills this gap via velocity decomposition and singularity analysis. We show that nonlinear disturbance amplification induces local cancellation of mean and disturbance velocity gradients, triggering finite-time singularity formation in flow field, which leads to the breakdown of regularity of the 3D NSE and thus the non-existence of global smooth solutions. It is emphasized that the non-existence of smooth solutions is due to the local regularity breakdown of solutions instead of the velocity blow-up. Further, it is important that the critical condition for regularity breakdown obtained through Sobolev space analysis accords with the critical condition for turbulence onset obtained through experiments and simulations.

Abstract: This study proposes a novel measurement system repeatability and reproducibility (R&R) framework for zero-inflated correlated defect-count data in semiconductor wafer automated optical inspection (AOI). In advanced semiconductor manufacturing environments, AOI systems are extensively used to detect wafer defects such as particles, scratches, and structural abnormalities. However, conventional Gauge R&R methods are primarily developed for continuous Gaussian-type measurements and are therefore not fully appropriate for high-yield semiconductor inspection data characterized by discrete defect counts, excessive zero observations, and correlated defect categories. To address these limitations, this study develops a zero-inflated bivariate Poisson (ZIBP) measurement system model capable of simultaneously capturing correlated defect-generation mechanisms and structural zero-defect states. A latent-variable representation is introduced to model shared and category-specific defect sources, while a zero-inflation mechanism accounts for defect-free wafer observations commonly encountered in precision manufacturing. An expectation-maximization (EM) algorithm is further developed for parameter estimation, including latent common defect counts and structural-zero probabilities. Based on the fitted model, repeatability variance, reproducibility variance, total measurement variation, and Percent R&R are estimated under the proposed probabilistic framework. In addition, bootstrap resampling is employed to construct confidence intervals for the proposed R&R measures. Theoretical properties of the proposed framework, including covariance structure, identifiability, EM monotonicity, estimator consistency, and asymptotic behavior of the Percent R&R estimator, are analytically established. The proposed framework extends traditional Gauge R&R analysis from continuous Gaussian measurements to zero-inflated correlated count-type defect inspection data and provides a statistically rigorous methodology for evaluating AOI measurement system reliability in semiconductor wafer manufacturing environments.

Abstract: Existence of global smooth solutions to the three-dimensional (3D) Navier-Stokes equations is disproved for pressure-driven plane Poiseuille flow with no-slip boundary conditions. This study is rigorously grounded in Sobolev space analysis. We show that the solution breakdown arises from the regularity degeneration instead of velocity blow-up. For disturbed laminar plane Poiseuille flow, the instantaneous velocity field is decomposed into a time-averaged flow and a disturbance flow, both characterized by their regularity in Sobolev spaces. When the Reynolds number is larger than the critical Reynolds number, the nonlinear interaction modifies the mean flow profile, and the disturbance amplitude grows significantly. This amplification leads to a local cancellation between viscous terms of the mean flow and the disturbance flow, resulting in the total viscous term (i.e., the Laplacian term) vanishing locally at the critical point $(\boldsymbol{x}^*, t^*)$. The local vanishing viscous term leads to zero velocity according to the Energy-Velocity Monotonicity Principle (EVMP), which contradicts the non-vanishing incoming velocity, leading to formation of a singularity. This singularity induces a velocity discontinuity, which causes the $L^\infty$ -norm of the velocity gradient to diverge, violating the definition of a global smooth solution in Sobolev spaces. The analysis is strictly grounded in partial differential equations (PDE) theory, with all key steps validated by Sobolev space properties and a priori estimates.

Abstract: In a previous work the authors initiated a study on mutation semigroups, where elementary mutation operations were encoded as total maps on finite sets and analyzed through structural, algebraic, and computational methods. Here we address several of the open problems raised therein. First, we investigate the algebraic characterization of generator sets that force the existence of constant or low-rank maps, linking these conditions to classical results on synchronizing automata. Second, we analyze the computational complexity of contraction-based heuristics, identifying cases where polynomial-time criteria are achievable and others where hardness results emerge. Finally, we extend the finite theory to parameterized and infinite families of mutations, drawing connections with quasispecies models in biology and interpreting image contractions as mechanisms of error suppression and genomic stability. By combining algebraic definitions, structural theorems, and algorithmic analyses, we provide a refined toolkit for understanding mutation collapse and its theoretical and biomedical implications.

Abstract: The paper examines stationary and dynamic gametheoretic models of cooperation formation among agents based on mutual trust. The dynamic model confirms that trust is a critically important factor in economic interactions. Mutual trust can significantly increase the payoffs for all agents, but its achievement is possible only for a certain class of input parameters of the model and requires a significant return on each agent’s investment in the public good. The model uses the value of social trust to describe social norms accepted in society. The study of the dynamic model was conducted based on Nash equilibrium in the static case and via simulation modelling for heterogeneous agents with different objective functionals. In the model, individual states act as agents. It is shown that the payoff of states directly depends on the degree of trust between them and can grow significantly in the presence of trust. Simulation experiments were conducted in the dynamic case, and the results were analysed.

Abstract: We revisit and extend the least-squares bubble function (LSBF) enrichment technique originally proposed by Yazdani and Nassehi, and derive and apply higher-order polynomials (degrees p = 3, 4) to a broader class of benchmark problems. The bubble function coefficients are determined by minimizing an element level L2 residual functional. An adaptive element-level order-selection rule based on the mesh Peclet and Damkohler numbers is proposed, drawing an analogy with local p-refinement in h-p finite element methods. The method is validated on three benchmark problems: (i) a singularly perturbed convection–diffusion–reaction (CDR) equation across four parameter regimes, (ii) a stiff two-point boundary value problem with a boundary layer of thickness O(ε) = 10e−4, and (iii) a Two-dimensional diffusion–reaction problem on a unit square. Systematic mesh-refinement studies confirm that LSBF at p = 3, 4 substantially outperforms standard Galerkin and is competitive with or superior to SUPG in reaction-dominated and mixed regimes. Some advantages and the limitations of the LSBF method are discussed.

Abstract: We address a constructive fractional Lyapunov direct method for Caputo-type incommensurate non-autonomous fractional order systems whenever orders lie in (0,1]. We prove some new fractional Lyapunov theorems by using new ideas of fractional generalized Gronwall inequality and establish higher versions of Lyapunov theorems that give sufficient conditions to predict stability dynamics of equilibrium points of many such systems. We demonstrate the new significance of such a method with five mathematical examples in stability theory.

Abstract: The distributional specification in Markov-switching GARCH models has historically been driven by empirical convention rather than statistical theory. This paper derives the two-regime MS-GARCH specification from the Maximum Entropy Principle, providing an information-theoretic motivation for Student-t regime-conditional innovations in cryptocurrency volatility modelling. The framework is applied to five major cryptocurrencies, Bitcoin, Ethereum, Ripple, Litecoin, and Bitcoin Cash, over the period January 2017 to March 2026, comprising 15,834 daily observations spanning six complete market cycles. Three principal findings emerge. First, a Calm-Phase Fragility pattern is identified: four of five assets exhibit calm-regime half-lives below one trading day (0.48 to 1.16 days), with turbulence the dominant long-run state (stationary turbulent probability in [0.451, 0.771] across all assets), establishing turbulence rather than calm as the structural baseline of the cryptocurrency ecosystem. Second, the Maximum Entropy derivation yields endogenous Student-t degrees of freedom, with heavy-tailed turbulent innovations (degrees of freedom approximately 4.5) confirmed across all assets, validating the MaxEnt constraint framework empirically. Third, near-unity turbulent GARCH persistence drives MS-GARCH point forecasts toward the persistence ceiling, consistent with an information-theoretic bound on predictability when the calm half-life collapses below one trading day; HAR-RV achieves the lowest QLIKE loss for three of five assets under these near-critical conditions. Cross-asset consistency is confirmed across seven statistical indicators including Hill tail exponents in [2.31, 3.26], Hurst exponents in [0.543, 0.577], and Wald tests rejecting parameter homogeneity at p < 0.001 for all assets. The framework is formalised as a deployable expert system for real-time regime monitoring and risk management.