Search Results (500)

The System for Teaching and Assessment using a Computer Algebra Kernel (STACK) is an open source, computer algebra-based online assessment system for teaching and learning mathematics at university. Although the popularity is increasing around the world, its problem generation needs a complex procedure such as algebraic scripting, dynamic randomization, and grading logic, which poses a substantial workload. In this paper, we propose a mathematics problem generation tool using Generative AI for STACK. It adopts a Retrieval-Augmented Generation (RAG) framework to guide the AI to produce pedagogically aligned problems across Depth of Knowledge (DoK) levels, while a Computer Algebra System (CAS) validates mathematical precision. The output is rendered into an XML template and is imported into the STACK system. For evaluation, we measured the success rate of generating 90 problem files for STACK by the proposal and compared the completion time with their manual generation. Learning Object Review Instrument (LORI) was also evaluated for user satisfactions. The results showed that the success rate was 79% while the time was reduced by 35.71%. Furthermore, the LORI evaluations demonstrated a feasibility score of 82.1%, confirming the potential to mitigate teacher workload. Full article

Accurate harmonic estimation at the per-cycle timescale is increasingly required in modern power-quality (PQ) monitoring, where fast-varying distortion sources demand high temporal resolution. However, when harmonic phasors are estimated from a single cycle using windowed discrete Fourier transform techniques, off-nominal fundamental frequency introduces spectral interference between harmonics, leading to systematic amplitude and phase errors that conventional correction methods cannot remove. This paper presents a lightweight, non-iterative harmonic estimation module designed to operate on fixed-rate, one-cycle data streams. The method leverages a frequency estimate provided by an external tracker to explicitly model the spectral interference induced by windowing under off-nominal conditions. By formulating this effect as a linear mixing process, the proposed approach applies an algebraic inversion to recover unbiased harmonic phasors without requiring adaptive resampling, variable window lengths, or modifications to the acquisition system. The module is designed as a plugin component compatible with existing PQ processing chains and shared sampled-value architectures. Experimental validation across frequency sweeps, Monte Carlo noise trials, and dynamic streaming scenarios demonstrates machine-precision accuracy in ideal conditions and noise-limited performance in realistic settings. Compared to iterative alternatives, the proposed solution achieves equivalent accuracy with a $484\times$ reduction in computation time. A sensitivity analysis further quantifies the relationship between frequency-tracking accuracy and harmonic estimation error, providing practical guidelines for system integration. These results show that accurate, real-time harmonic estimation can be achieved from single-cycle data using fixed-rate acquisition, enabling improved monitoring and protection capabilities in modern power systems. Full article

With the development of the socio-economic landscape and the increasing demand for urban power supply, user expectations for power supply reliability have risen significantly. To address this challenge, dual-petal distribution networks, characterized by multiple tie-line structures and inter-regional load transfer paths, have significantly enhanced fault recovery capability and are gradually replacing traditional radial configurations as a key form of modern distribution systems. However, their multi-regional coupling characteristics introduce complex issues such as dynamic changes in load transfer paths and islanded operation, resulting in significant limitations in the accuracy and adaptability of existing reliability assessment methods. To this end, this paper proposes an analytical method for calculating reliability indices of dual-petal distribution networks, considering the characteristics of load transfer zones. First, typical operation modes of dual-petal distribution networks are extracted, and a time-sequential component reliability analysis model is established. Second, a load transfer zone matrix is constructed based on the impact of distribution network faults on load nodes across different regions. Third, based on the fault ride-through capability of distributed generation (DG), a load restoration strategy considering load transfer zone characteristics is formulated, and the DG Island Recovery Matrix (DGIRM) is derived. Finally, by performing algebraic operations among various matrices and reliability parameter vectors, an explicit analytical calculation of reliability indices for dual-petal distribution networks with different DG configurations is achieved. The effectiveness of the proposed method is validated using a typical dual-petal network. The results demonstrate that the proposed method offers high computational efficiency and accuracy, effectively quantifying the impact of DG on the power supply reliability of dual-petal distribution networks, and providing theoretical and methodological support for the reliability assessment and planning of complex distribution systems. Full article

This paper presents a novel optimal control-based beamforming framework for phased antenna arrays, targeting advanced wireless communication and radar applications, including 5G systems. Unlike conventional beamforming techniques, such as Fourier-based methods and adaptive algorithms (e.g., LMS and RLS), the proposed approach formulates the beam synthesis problem as a discrete-time optimal control problem. The antenna array is modeled using a state-space representation, and a quadratic cost function is introduced to jointly minimize the deviation from a desired radiation pattern and the excitation power. The optimal excitation weights are derived using the Linear Quadratic Regulator (LQR) framework by solving the discrete-time algebraic Riccati equation. This formulation enables an effective trade-off between sidelobe suppression, main lobe accuracy, and power efficiency. Simulation results demonstrate that the proposed method achieves a well-focused main beam, significantly reduced sidelobe levels, and improved directivity compared to conventional approaches. Furthermore, the framework offers robustness and computational efficiency, making it a promising candidate for future FPGA and embedded implementations. Overall, the proposed optimal control-based beamforming approach provides a flexible, robust, and computationally efficient solution for next-generation antenna systems in 5G, beyond-5G (B5G), and radar applications. Full article

Initial value problems of the third order featuring explicit dependence on velocity, denoted as $y^{‴}=f(x,y,y^{\prime })$, emerge regularly across applications such as electromechanical networks, structural mechanics, and robotic trajectory control. Despite their practical prevalence, these differential equations remain insufficiently addressed by standard numerical integration techniques. Orthodox Runge–Kutta–Nyström (RKN) schemes are fundamentally formulated for differential equations lacking the first derivative, specifically $y^{‴}=f(x,y)$. Due to this algorithmic constraint, researchers frequently resort to computationally demanding first-order system reductions or rely upon standard Runge–Kutta methods. The present study resolves this methodological gap by defining an explicit s-stage integration architecture that natively incorporates the first derivative within the internal stage evaluations. Such structural modifications require the deployment of a supplementary coefficient matrix, denoted as D, to formulate the corresponding order theory. The complete set of algebraic order conditions is systematically established up to the seventh order, accompanied by a generic mathematical framework for generating schemes of arbitrary order. Based on this analytical foundation, an embedded $6(4)$ method is constructed. This specific pair achieves strict error tolerances utilizing merely six function evaluations per integration step, representing a substantial operational reduction compared to the eight computations strictly required by equivalent Runge–Kutta pairs. Direct numerical integration of the native third-order system prevents the dimensionality increase from reducing to first-order systems. Performance validation of the numerical solver involves two representative physical benchmarks: a coupled robotic appendage subjected to platform excitation and an electromechanical actuator array regulated by transient control inputs. Both dynamical systems exhibit severe velocity-dependent dissipation mechanisms and nonlinear external forcing. Quantitative numerical evaluations confirm that the constructed 6(4) pair yields higher precision and demands less computational expenditure than prevailing RK and RKN integrators. The analytical and empirical findings establish that derivative-capable Nyström integration algorithms furnish mathematically rigorous and computationally efficient numerical solutions for velocity-coupled third-order dynamics. Full article

Edge artificial intelligence (Edge AI) requires real-time inference under stringent constraints on computation, memory, energy, and connectivity. Although training can be offloaded to servers, efficient, high-capacity inference and rapid on-device adaptation remain central challenges. Cloud-based inference offers substantial computational power but depends on connectivity, latency, privacy, and reliability conditions that edge deployments cannot always guarantee. Classical model-compression methods—including quantization, pruning, distillation, and neural architecture search—have extended the feasibility of on-device inference, yet they leave largely unchanged the fundamental cost of the linear-algebraic, sampling, and optimization primitives that dominate modern deep learning. Quantum computing has therefore been proposed as a complementary accelerator for selected AI workloads, with theoretical advantages in linear systems, singular value decomposition, sampling, kernel evaluation, and optimization. This review surveys the emerging field of quantum-accelerated AI for edge systems under a hybrid architectural premise: edge devices remain classical, while quantum processors operate as remote, cloud, MEC, or near-edge accelerators. We synthesize advances across quantum learning models, hybrid optimization methods, hardware and deployment architectures, and quantum-inspired approaches suitable for constrained devices. We also assess the practical barriers that currently separate asymptotic quantum advantage from deployable edge intelligence, including data loading, measurement overhead, noise, latency, and benchmarking gaps. Finally, we outline a staged research roadmap from near-term hybrid workflows to fault-tolerant and integrated quantum-edge architectures. Full article

Multibody dynamics (MBD) simulations involving hundreds to thousands of bodies give rise to large-scale, sparse, and structurally indefinite linear systems. Traditional direct solvers incur prohibitive memory and computational costs, while iterative methods suffer from slow convergence due to severe ill-conditioning. This paper proposes HPI-MBD, a hybrid preconditioned iterative framework. It combines an algebraic multigrid (AMG) for global error smoothing with a block Jacobi preconditioner tailored to the kinematic constraint graph. The framework exploits graph topology to construct a block-diagonal Schur complement approximation, incorporates Tikhonov regularisation for redundant constraints, and maintains $\mathcal{O}\left(n\right)$ work per iteration, where n is the number of degrees of freedom. A rigorous spectral analysis supports the problem-size independent convergence of the Minimal Residual (MINRES) solver. Evaluated on five benchmark systems with ${10}^4$ to ${10}^6$ degrees of freedom, the HPI-MBD achieves speedups up to $12.7\times$ and memory reductions up to 68% against MA57, with comparable gains against PARDISO. All solutions maintain relative residuals below ${10}^{-6}$. Comparisons against ILU(0)-preconditioned Generalised Minimal Residual (GMRES), Finite Element Tearing and Interconnecting method (FETI-1), and a block-Jacobi-only variant confirm the essential role of AMG. The framework exhibits near-linear scalability and strong parallel efficiency on up to 32 processors, along with robust performance under redundant constraints and varying time step sizes. These results position HPI-MBD as a scalable, memory-efficient alternative for real-time simulation in virtual prototyping, robotics, and biomechanics. Full article

We develop a matrix-theoretic framework for the natural embedding of the exceptional Lie algebra ${\mathfrak{g}}_2=\mathrm{Der}\left(\mathbb{O}\right)$ in $\mathfrak{so}\left(8\right)$, use it to make constructive a recent existence result on octonionic triality, and derive geometric applications for moduli spaces of principal bundles. Specifically, the derivation condition for $\widehat{D}\in \mathfrak{so}\left(7\right)$ is reformulated as a homogeneous linear system in the 21 entries of $\widehat{D}$, whose solution space is identified with ${\mathfrak{g}}_2=ker\mathsf{\Psi }$, where $\mathsf{\Psi }:\mathfrak{so}\left(7\right)\to {\Lambda }^3{\left({\mathbb{R}}^7\right)}^{*}$ is the Lie derivative with respect to the associative 3-form $\phi$ on $\mathrm{Im}\left(\mathbb{O}\right)$. It is proved that $\mathrm{rank}\mathsf{\Psi }=7$, and an algorithm is given for computing an orthonormal basis of ${\mathfrak{g}}_2$. The image $\mathsf{\Psi }\left({\widehat{A}}_{\sigma }\right)$ of the triality generator is computed for all triples, yielding six nonzero components and squared norm 12. As geometric applications, the map $\mathsf{\Psi }$ is globalized to a morphism of adjoint bundles, giving an intrinsic characterization of the $G_2$-reductible locus in $\mathcal{M}\left(\mathrm{SO}\right(7\left)\right)$. The orthogonal decomposition of $\mathfrak{so}\left(8\right)$ globalizes to an explicit splitting of the adjoint bundle of any $\mathrm{SO}\left(8\right)$-principal bundle admitting a $G_2$-reduction. Finally, $\mathcal{M}\left(G_2\right)$ is identified as a connected component of the triality fixed-point locus in $\mathcal{M}\left(\mathrm{Spin}\right(8\left)\right)$, with an explicit description of the tangent and normal spaces in terms of the Lie-algebraic decomposition. Full article

Heat transport in heterogeneous materials can deviate markedly from classical Fourier behavior when microstructural disorder, trapping effects, nonlinear mobility, and long-range temporal correlations interact across multiple spatial and temporal scales. These mechanisms may produce delayed relaxation, persistent thermal footprints, front deformation, and non-classical spreading patterns that are not adequately represented by conventional integer-order diffusion models. In this study, a modeling and simulation framework is developed for anomalous heat transport in heterogeneous media using a two-dimensional time-fractional porous medium equation. The model combines a Caputo fractional time derivative, which represents thermal memory, with nonlinear degenerate porous-medium diffusion, spatially heterogeneous conductivity, localized volumetric heating, and Robin-type convective boundary exchange. A conservative fully discrete numerical scheme is constructed using flux-based finite differences for the heterogeneous nonlinear diffusion operator and an $L1$ approximation for the Caputo derivative. The nonlinear algebraic system at each time level is solved using an under-relaxed Picard frozen-coefficient iteration with non-negativity enforcement and sparse direct solution of the resulting linear systems. The numerical implementation is verified through a manufactured-solution convergence study, and additional analyses are performed to examine computational cost, Picard iteration behavior, coefficient-regularization sensitivity, strong-source effects, heterogeneous conductivity structures, and long-time thermal-footprint persistence. The results show that heterogeneous conductivity mainly redirects heat through preferential pathways and enlarges the spatial footprint while producing negligible changes in global heat content. Stronger fractional memory, represented by smaller fractional order, increases the persistence and spatial reach of moderate heating, whereas larger porous-medium exponents confine heat near the source and preserve higher local peaks. Source amplitude increases the thermal burden and footprint monotonically over the tested range, including strong forcing, without producing an abrupt localization-spreading transition. Boundary exchange remains secondary in the short-time interior-heating regime considered. These findings demonstrate that the proposed two-dimensional time-fractional porous medium framework provides a verified and physically interpretable model for non-Fourier heat transport in heterogeneous materials, where local intensity, global heat retention, and spatial thermal exposure must be assessed jointly. Full article

The simulation of modern wireless communication circuits remains challenging because of the coexistence of nonlinear behavior, heterogeneous subsystems, and widely separated time scales. This review presents a structured overview of advanced numerical methods for solving multitime partial differential–algebraic equations (MPDAEs) arising in circuit-level modeling of RF and microwave systems. Compared with previous survey papers, the main contribution of this work is to organize the literature according to the underlying numerical strategy, distinguishing purely time-domain, hybrid time–frequency, multidimensional frequency-domain, and circuit-block partitioning approaches. The reviewed methods show that multitime formulations can deliver substantial computational gains over conventional simulation techniques, particularly for multirate and multiscale circuits. Time-domain techniques are generally more robust for strongly nonlinear regimes, whereas frequency-domain and hybrid methods are often more efficient when the waveform can be represented with a limited number of harmonics. Circuit-block partitioning further improves efficiency by exploiting active and latent variables, but the computational complexity of MPDAE methods increases rapidly with the number of time scales, and their applicability becomes more limited for aperiodic or highly general multirate excitations. Overall, this review highlights both the strengths and the practical limitations of current MPDAE-based numerical approaches and identifies open challenges for future research. Full article

Accurate prediction of contaminant transport and self-purification processes in rivers remains challenging because pollutant dispersion, biochemical reactions, and hydrodynamic conditions interact across multiple spatial scales. This study aims to develop and compare mathematical models for soluble contaminant transport and biodegradable organic matter removal in channels and rivers. Unsteady advection–diffusion–reaction equations were formulated for one-dimensional (1D), two-dimensional (2D), and three-dimensional (3D) transport scenarios and solved through numerical techniques based on the transformation of partial differential equations into systems of ordinary differential or algebraic equations. In parallel, the classical Streeter–Phelps model and an extended formulation incorporating turbulent diffusion were implemented to evaluate organic load degradation and oxygen deficit dynamics. Simulations were performed using a Matlab R2019a-based computational framework under representative hydraulic and reaction conditions obtained from literature data and empirical correlations. The results showed that, under specific conditions, the 3D model reproduced trends comparable to those predicted by the 2D model, while the latter approached the behavior of the 1D formulation. The Streeter–Phelps model predicted an organic load removal efficiency of 97.74%, a purification index of 1.9564, a critical time of 18.43 h, and a critical distance of 6.93 km. These findings provide a useful framework for river water-quality assessment and support future applications involving complex hydrodynamic and pollutant-loading scenarios. Full article

Autonomous mobile robots are used in optimizing warehouse logistics, yet achieving precise positioning during docking maneuvers and autonomous planning remains a technical challenge. This study presents a custom vision-based control system designed for an autonomous omnidirectional wheeled robot. The proposed methodology acquires visual feedback using a stereo camera integrated within the Robot Operating System framework. Two visual feedback control laws are formulated and rigorously evaluated: a Classic Position-Based Visual Servoing algorithm, which minimizes pose error using a quaternion-based approach, and a second solution that utilizes Dual Lie Algebra to compute the 3D visual sensor’s velocities, ensuring convergence towards the desired point-feature configuration. Experimental validation reveals that while both methods achieve docking, the dual pose-free approach enables more robust, effortless movement of the robot platform than Classic Position-Based Visual Servoing. Consequently, these findings indicate that integrating depth-based feature recovery with advanced algebraic strategies offers a stable control strategy for automated industrial scenarios. Full article

Topological data analysis (TDA) has evolved into a flexible and robust paradigm for obtaining qualitative, geometry-inspired insights from high-dimensional, noisy, and complex data. Grounded in algebraic topology, geometry, statistics, and machine learning (ML), TDA provides multiscale descriptions through persistent homology, Mapper (a graph-based method that summarizes the shape of high-dimensional data), and related topological signatures that are often inaccessible to standard linear and metric methods. In recent years, and especially during 2024–2025, TDA has expanded rapidly across science, engineering, biomedical research, and socio-economic studies, while also being integrated with modern learning paradigms such as deep learning (DL) and graph learning. This survey summarizes recent developments in TDA using a carefully selected set of articles, with emphasis on 2024–2025. We first present the mathematical and computational foundations of TDA, covering simplicial complexes, filtrations, persistent homology, the Mapper algorithm, and computational advances such as data simplification, stability, and efficiency. We then review applications in time series and dynamical systems, biomedical imaging and precision medicine, engineering and physical sciences, finance and risk analysis, DL and interpretability, and security and critical infrastructure systems. Throughout, we highlight how TDA can extract informative features, function as a model component, and provide a conceptual lens for studying complex systems. However, the survey also emphasizes recurrent failure patterns: TDA performance is highly sensitive to filtration, embedding, and vectorization choices; aggressive simplification can dilute or remove informative topological signals; and integration into standard ML workflows still lacks uniform validation and reporting protocols. We conclude by outlining key challenges—including scalability, statistical foundations, interpretability, and compatibility with rapidly evolving artificial intelligence (AI) paradigms—and by identifying directions for future research. The survey also provides a unifying design perspective for TDA systems, highlighting methodological trade-offs and emerging research directions for integrating topology with modern ML. Full article

Conventional transient stability assessment in multi-machine power systems relies predominantly on fixed thresholds, which exhibit limited adaptability to varying operating conditions and fail to provide a unified analytical framework for rotor angle and voltage stability. To address these challenges, this paper proposes an adaptive threshold warning method based on geometric algebra. A multi-dimensional unified state vector incorporating generator rotor angles, speeds, electromagnetic powers and bus voltage magnitudes and phases is constructed to map system dynamics onto a high-dimensional geometric trajectory. The second- and third-order wedge products of this trajectory are computed to quantify disturbance severity and volumetric expansion preceding instability. An adaptive threshold mechanism is established utilizing sliding window robust statistics (Median Absolute Deviation) to track the trajectory’s instantaneous dimension in real time. Validation on the IEEE 39-bus system demonstrates that the proposed method issues a warning at t = 4.90 s, achieving a detection advance of 0.30 s relative to the conventional 30° rotor angle separation threshold. The method exhibits strong noise robustness with only 40 ms warning delay under 20 dB SNR conditions, and effectively captures rotor angle–voltage coupling characteristics. The geometric algebra framework offers a unified assessment tool with distinct advantages in computational speed, adaptivity, and interpretability. Full article

This paper introduces and investigates a novel two-parameter sequence, termed the biparametric Appell extension (B-App-Ex) and denoted by ${\mathcal{B}}_n(x;\lambda ,\alpha )$. Standard classical Appell sequences often lack sufficient structural parameters, which can limit their operational flexibility in certain advanced spectral schemes. To address this limitation, we construct an enhanced operational framework by integrating a binomial structural kernel ${(1+w)}^{\lambda }$ with a linear exponential scaling $e^{\alpha xw}$ entirely within the Appell class. We provide a rigorous logical deduction of the fundamental properties of this sequence, including its explicit power series representation, a characteristic three-term recurrence relation, and a governing second-order differential equation (DEq.). A significant contribution of this work is the establishment of analytically exact connection and inverse connection formulas between the B-App-Ex basis and various classical orthogonal polynomial (COP) families. Numerical verification via a collocation-based projection framework demonstrates that these algebraic kernels achieve near-machine epsilon precision (≈${10}^{-15}$), remaining stable even for high-order approximations. Furthermore, by isolating the dilation factor $\alpha$, we establish an $O\left(N\right)$ computational complexity that offers a reduction in latency by approximately two orders of magnitude compared to classical matrix-based transformations. The results demonstrate that the proposed biparametric (Bip.) extension offers a versatile and highly optimized analytical template for modeling complex dynamic systems where structural shifting and spatial scaling must be tuned simultaneously. Full article