Search Results (495)

We introduce a geometric framework in which classical transforms are represented as coordinate charts on a symbolic manifold. The construction defines symbolic curvature (κ), strain (τ), compressibility (σ), and the ratio Γ = κ/τ, which together provide a diagnostic coordinate system for comparing representational stability across chart transitions. Within this setting, transforms such as Fourier, Laplace, wavelet, Jordan, and polynomial projection can be treated as charts connected by transition maps that preserve Γ on specified domains. We also introduce a symmetric positive-definite metric tensor Gab to quantify displacement in the invariant coordinates and to formalize minimal-effort paths (geodesics) under modeling assumptions stated in the text. The resulting framework provides a reproducible screening method for evaluating transform stability, diagnosing closure failure, and comparing transform behavior under a shared set of invariants. Full article

This Editorial introduces the Symmetry Special Issue “Theory and Applications of Special Functions II” and summarizes the nine contributions collected therein. The papers span the analytic continuation of multivariate hypergeometric functions; stability theory for differential equations via integral transforms; numerical schemes for multi-space fractional partial differential equations based on nonstandard finite differences and orthogonal polynomials; applications of the Lambert W function to viscoelastic creep modeling; algebraic constructions of new Hermite-type polynomial families via the monomiality principle; higher-level generalizations of poly-Cauchy numbers; Bell-polynomial expansions for Laplace transforms of higher-order nested functions; and two complementary studies on the physical implementation and algebraic description of Gaussian quantum states. Beyond the contributions of the Special Issue, we highlight methodological connections—continued fractions and complex analysis, transform techniques, special polynomials, and combinatorial sequences—and emphasize the unifying role of symmetry across mathematical structures and applications. Full article

Interferometers are essential tools for quality control of optical surfaces. While interferometric techniques like phase-shifting interferometry offer high accuracy, they involve complex setups, require stringent calibration, and are sensitive to phase shift errors, noise, and surface inhomogeneities. In this research, we introduce an alternative algorithm that integrates Moving Average and Fast Fourier Transform (MAFFT) techniques with Polynomial Fitting. The proposed method achieves results comparable to a Zygo interferometer under standard conditions, with an error margin under 2%. It also maintains measurement stability in noisy environments and in the presence of significant local inhomogeneities, operating in real-time to enable wavefront measurements at 30 Hz. We have validated the algorithm through simulations assessing noise-induced errors and through experimental comparisons with a Zygo interferometer. Full article

This paper presents a novel high-order numerical method. The proposed scheme utilizes polynomial generating functions to achieve p order accuracy in time for the Grünwald–Letnikov fractional derivatives, while maintaining second-order spatial accuracy. By incorporating a short-memory principle, the method remains computationally efficient for long-time simulations. The authors rigorously analyze the stability of equilibrium points for the fractional vegetation–water model and perform a weakly nonlinear analysis to derive amplitude equations. Convergence analysis confirms the scheme’s consistency, stability, and convergence. Numerical simulations demonstrate the method’s effectiveness in exploring how different fractional derivative orders influence system dynamics and pattern formation, providing a robust tool for studying complex fractional systems in theoretical ecology. Full article

Pumped-storage power stations (PSPSs) are vital for grid stability, yet pump-turbines (PTs) operating in the S-shaped region often induce severe hydraulic instability. To reveal the mechanism of these self-excited oscillations, this study establishes a nonlinear mathematical model based on rigid water column theory and a cubic polynomial approximation of the PT’s nonlinear characteristics. Both analytical derivations and numerical simulations were conducted. Analytical results indicate that, in the absence of surge tanks, self-excited oscillations occur when the PT’s negative hydraulic impedance modulus exceeds the pipeline impedance. With a single surge tank, the system behaves analogously to the Van der Pol oscillator, exhibiting oscillations that converge to a stable limit cycle governed by system parameters. Numerical simulations for a dual-surge-tank system further reveal that, due to initial negative damping, the PT transitions to alternative stable equilibria. Crucially, the transition direction is governed by the polarity of the initial disturbance: negative perturbations lead to the regular turbine region, while positive ones lead to the reverse pump region. Additionally, pipe friction causes the steady-state discharge to deviate slightly from the theoretical static value, with deviations remaining below 2.96%. This work provides a theoretical basis for stability prediction in PSPSs. Full article

Rainfall infiltration is one of the main factors inducing slope instability, while the spatial heterogeneity and uncertainty of soil parameters have profound impacts on slope response characteristics and stability evolution. Traditional deterministic analysis methods struggle to reveal the dynamic risk evolution process of the system under heavy rainfall. Therefore, this paper proposes an uncertainty analysis framework combining Karhunen–Loève Expansion (KLE) random field theory, Polynomial Chaos Kriging (PCK) surrogate modeling, and Monte Carlo simulation to efficiently quantify the probabilistic characteristics and spatial risks of rainfall-induced slope instability. First, for key strength parameters such as cohesion and internal friction angle, a two-dimensional random field with spatial correlation is constructed to realistically depict the regional variability of soil mechanical properties. Second, a PCK surrogate model optimized by the LARS algorithm is developed to achieve high-precision replacement of finite element calculation results. Then, large-scale Monte Carlo simulations are conducted based on the surrogate model to obtain the probability distribution characteristics of slope safety factors and potential instability areas at different times. The research results show that the slope enters the most unstable stage during the middle of rainfall (36–54 h), with severe system response fluctuations and highly concentrated instability risks. Deterministic analysis generally overestimates slope safety and ignores extreme responses in tail samples. The proposed method can effectively identify the multi-source uncertainty effects of slope systems, providing theoretical support and technical pathways for risk early warning, zoning design, and protection optimization of slope engineering during rainfall periods. Full article

Satellite Clock Bias (SCB) is a major source of error in Precise Point Positioning (PPP). The real-time service products from the International GNSS Service (IGS) are susceptible to network interruptions. Such disruptions can compromise product availability and, consequently, degrade positioning accuracy. We introduce the CNN-LSTM-Attention model to address this challenge. The model enhances a Long Short-Term Memory (LSTM) network by integrating Convolutional Neural Networks (CNNs) and an Attention mechanism. The proposed model can efficiently extract data features and balance the weight allocation in the Attention mechanism, thereby improving both the accuracy and stability of predictions. Across various forecasting horizons (1, 2, 4, and 6 h), the CNN-LSTM-Attention model demonstrates prediction accuracy improvements of (76.95%, 66.84%, 65.92%, 84.33%, and 43.87%), (72.59%, 65.61%, 74.60%, 82.98%, and 51.13%), (70.45%, 68.52%, 81.63%, 88.44%, and 60.49%), and (70.26%, 70.51%, 84.28%, 93.66%, and 66.76%), respectively, across the five benchmark models: Linear Polynomial (LP), Quadratic Polynomial (QP), Autoregressive Integrated Moving Average (ARIMA), Backpropagation Neural Network (BP), and LSTM models. Furthermore, in dynamic PPP experiments utilizing IGS tracking stations, the model predictions achieve positioning accuracy comparable to that of post-processed products. This proves that the proposed model demonstrates superior accuracy and stability for predicting SCB, while also satisfying the demands of positioning applications. Full article

Amid fossil-fuel depletion and worsening environmental impacts, battery electric vehicles (BEVs) are pivotal to the energy transition. Energy management in BEVs relies on accurate motor efficiency maps, yet real-time onboard control demands models that balance fidelity with computational cost. To address map inaccuracy under real driving and the high runtime cost of 2-D interpolation, we propose a driving-cycle-aware, physically interpretable quadratic polynomial-surface framework. We extract priority operating regions on the speed–torque plane from typical driving cycles and model electrical power $P_e$ as a function of motor speed $n$ and mechanical power $P_m$. A nested model family (M3–M6) and three fitting strategies—global, local, and region-weighted—are assessed using $R^2$, RMSE, a computational complexity index (CCI), and an Integrated Criterion for accuracy–complexity and stability (ICS). Simulations on the Worldwide Harmonized Light Vehicles Test Cycle, the China Light-Duty Vehicle Test Cycle, and the Urban Dynamometer Driving Schedule show that region-weighted fitting consistently achieves the best or near-best ICS; relative to Global fitting, mean ICS decreases by 49.0%, 46.4%, and 90.6%, with the smallest variance. Regarding model order, the four-term M4 $\left(+P_m^2\right)$ offers the best accuracy–complexity trade-off. Finally, the region-weighted fitting M4 $\left(+P_m^2\right)$ polynomial model was integrated into the vehicle-level economic speed planning model based on the dynamic programming algorithm. In simulations covering a 27 km driving distance, this model reduced computational time by approximately 87% compared to a linear interpolation method based on a two-dimensional lookup table, while achieving an energy consumption deviation of about 0.01% relative to the lookup table approach. Results demonstrate that the proposed model significantly alleviates computational burden while maintaining high energy consumption prediction accuracy, thereby providing robust support for real-time in-vehicle applications in whole-vehicle energy management. Full article

The description of the Earth’s gravitational field, governed by the fundamental potential equation (the Laplace equation), is conventionally expressed using spherical harmonics, yet the Cartesian formulation, using a Taylor series representation, offers significant algebraic advantages. This paper proposes a novel Direct Cartesian Method for generating spherical basis functions and coefficients directly within the Cartesian coordinate system, utilising the partial derivatives of the inverse distance ($1/R$) function. The present study investigates the structural correspondence between the Cartesian form of spherical basis functions and the high-order partial derivatives of $1/R$. The study reveals that spherical basis functions can be categorised into four distinct groups based on the parity of the degree n and order m. It is demonstrated that each spherical basis function is equivalent to a weighted summation of the partial derivatives of the inverse distance ($1/R$) with respect to Cartesian coordinates. Specifically, the basis functions are combined with those derivatives that share the same order of Z-differentiation and possess matching parities in their orders of differentiation with respect to X and Y. In order to facilitate the practical calculation of these high-degree derivatives, a recursive numerical algorithm has been developed. The method generates the polynomial coefficients for the numerator of the $1/R$ derivatives. A pivotal innovation is the implementation of a step-wise normalization scheme within the recursive relations. The integration of the recursive ratios of global normalization factors (including full Schmidt normalization) into each step of the algorithm effectively neutralises factorial growth, rendering the process immune to numerical overflow. The validity and numerical stability of the proposed method are demonstrated through a detailed step-by-step derivation of a sectorial basis function ($n=8,m=2$). Full article

Bairstow’s method employs synthetic division to express a polynomial $p\left(x\right)$ of degree n in the form $p\left(x\right)=q\left(x\right)(x^2+Bx+C)+R(B,C)x+S(B,C)$, where $q\left(x\right)$ is the quotient polynomial of degree $n-2$, and $R(B,C)$, $S(B,C)$ are the remainder coefficients that depend nonlinearly on the quadratic parameters B and C. The original algorithm proposed by Bairstow uses Newton–Raphson method to solve $R(B,C)=S(B,C)=0$; it requires initial guesses within very narrow attraction basins for ill-conditioned polynomials and fails at singular Jacobian matrices. To address these issues, we reformulate Bairstow’s method as a constrained optimization problem that maximizes $C^2$ as the objective function subject to the constraints $R(B,C)=S(B,C)=0$. While modern, highly optimized, non-linear solvers (available in commercial software like MATLAB) have largely superseded classical iterative polynomial rootfinding techniques, our reformulated Bairstow approach offers distinct advantages for selective root extraction and application-specific constraints. Specifically, the optimization formulation enables the extraction of specific roots of interest rather than computing all roots simultaneously, naturally accommodates additional constraints for application-specific factorization x (such as discriminant conditions for real versus complex root extraction). The $C^2$ objective automatically selects the quadratic factor with the largest root magnitude, enhancing numerical stability during deflation. Numerical experiments validate the approach on polynomials with degree bigger than 10 including cases with simple real roots, multiple roots, mixed real and complex roots, and Chebyshev polynomials, achieving machine precision accuracy with robust handling of the discriminant constraint. Full article

This paper presents an integrated approach to the structural decomposition of concurrent control systems using exact transversal hypergraphs (XT-hypergraphs). The proposed method combines formal properties of XT-hypergraphs with invariant-based Petri net analysis to enable automatic partitioning of complex, concurrent specifications into deterministic and independent components. The approach focuses on preserving behavioral correctness while minimizing inter-component dependencies and computational complexity. By exploiting the uniqueness of minimal transversal covers, reducibility, and structural stability of XT-hypergraphs, the method achieves a deterministic decomposition process with polynomial-delay generation of exact transversals. The research provides practical insights into the construction, reduction, and classification of XT structures, together with quality metrics evaluating decomposition efficiency and structural compactness. The developed methodology is validated on representative real-world control and embedded systems, showing its applicability in deterministic modeling, analysis, and implementation of concurrent architectures. Future work includes the integration of XT-hypergraph algorithms with adaptive decomposition and verification frameworks to enhance scalability and automation in modern system design and integration with currently popular AI and machine learning methods. Full article

Electromagnetic pumps are developed for industrial, medical and scientific applications, moving electrically conductive liquids and molten solder in electronics manufacturing using electromagnetism instead of mechanical parts. This study presents a comprehensive thermal analysis of an electromagnetic diaphragm pump, focusing on the influence of operating current, permanent magnet switching speed, and cooling conditions on pumping performance. The pump utilizes a flexible diaphragm embedded with a permanent neodymium magnet, which interacts with time-varying magnetic fields generated by electromagnets to drive fluid motion. Temperature monitoring is conducted using a waterproof DS18B20 sensor and an uncooled FLIR A325sc infrared camera, allowing accurate mapping of thermal distribution across the pump surface. Experimental results demonstrate that higher current and increased magnet switching speed lead to faster temperature rise, impacting the volume of fluid pumped. Incorporation of an automatic cooling fan effectively reduces coil temperature and stabilizes pump performance. Polynomial regression models describe the relationship between temperature, pumped liquid volume, and magnet switching speed, providing information to optimize pump operation and cooling strategies. The novel relationship between temperature and the volume of the pumped liquid is considered as a fourth-degree polynomial. In particular the model describes a quantitative evaluation of the effect of heating on pumping efficiency. An initial increase in temperature correlates with a higher pumped volume, but excessive heating leads to efficiency saturation or even decline. Indeed, mathematical dependencies are crucial in mechanical pump engineering for analyzing physical phenomena; this is achieved by using a mathematical equation to define how different physical variables are related to each other, enabling engineers to calculate performance and optimize the pump design. Full article

This study develops a rigorous analytic framework for solving the Cauchy problem of polyharmonic equations in ${\mathbb{R}}^{\mathrm{n}}$, highlighting the crucial role of symmetry in the structure, stability, and solvability of solutions. Polyharmonic equations, as higher-order extensions of Laplace and biharmonic equations, frequently arise in elasticity, potential theory, and mathematical physics, yet their Cauchy problems are inherently ill-posed. Using hyperspherical harmonics and homogeneous harmonic polynomials, whose orthogonality reflects the underlying rotational and reflectional symmetries, the study constructs explicit, uniformly convergent series solutions. Through analytic continuation of integral representations, necessary and sufficient solvability criteria are established, ensuring convergence of all derivatives on compact domains. Furthermore, newly derived Green-type identities provide a systematic method to reconstruct boundary information and enforce stability constraints. This approach not only generalizes classical Laplace and biharmonic results to higher-order polyharmonic equations but also demonstrates how symmetry governs boundary data admissibility, convergence, and analytic structure, offering both theoretical insights and practical tools for elasticity, inverse problems, and mathematical physics. Full article

New two-step simultaneous iterative techniques are proposed for solving polynomial equations with multiple roots of unknown multiplicity. The developed schemes achieve a local convergence order of ten and address key limitations of existing solvers, namely their dependence on prior multiplicity information and their reduced efficiency when dealing with clustered or repeated roots. Root multiplicities are adaptively estimated within the iterative process, avoiding additional function evaluations beyond those required for parallel updates. The robustness and stability of the proposed methods are assessed using both random and distant initial guesses and validated on benchmark polynomials as well as nonlinear models from biomedical engineering. The numerical results show notable improvements in residual error, iteration count, CPU time, memory usage, and overall convergence rate compared with established classical techniques. These findings demonstrate that the proposed schemes provide reliable, high-order, and computationally efficient tools for solving challenging nonlinear problems in science and engineering. Full article

It has been shown for the first time that in the case of a pressure flow of a Newtonian fluid in a circular pipeline, the influence of forces of rheological origin, ion electrostatic and Van der Waals nature on the radius of the undeformed flow core is described by a third-degree polynomial with respect to the thickness of the layer, where the suspension structure is destroyed and its shear flow occurs. In this polynomial, the contributions of rheological forces and the influence of the hydraulic size of the solid-phase particles in the suspension enter as linear terms; ionic electrostatic and Van der Waals forces enter as quadratic and constant terms, respectively. For conditions typical of water–coal fuel, we demonstrate that the hydraulic (size) term is several orders of magnitude smaller than the leading terms and may be neglected, and that the quadratic term is negligible compared with the constant (free) term, so that the limiting value of the undeformed core radius is obtained as the real root of a cubic equation containing cubic, linear and constant terms. At DLVO equilibrium, the constant term vanishes, and the limiting relative core radius reduces to the rheological–hydraulic expression; away from equilibrium, the constant term becomes positive or negative, thereby altering the admissible interval of the relative core radius. Using Cardan’s method, we show analytically that (i) when the cubic discriminant is positive, a single real root exists and physically admissible solutions occur only for a negative constant term; (ii) when the discriminant is negative, three real roots exist and the maximum relative radius at which the suspension structure is preserved shifts above or below the rheological-only radius depending on the sign of the constant term. Numerical evaluation of the proposed lyophobicity model for proportionality coefficients k₁ in the range 1–10 yields a lyophobicity function varying approximately from 0.67 to 1.06, confirming the modest but non-negligible role of interparticle interaction energy in modifying the undeformed core size under water–coal fuel conditions. These results quantify the competing roles of rheology and interparticle forces in determining the stability and extent of the undeformed core in pipeline transport of structured suspensions. Full article