Search Results (161)

This paper studies structural properties of semiprimes $N=pq$ in computational number theory, focusing on cases where the prime factors are close. We analyze the relationship between $\sqrt{N}$ and $\sqrt{\phi \left(N\right)}$ and show that, under a bounded prime gap condition, these quantities exhibit strong proximity. Specifically, assuming $|p-q|\le 2^{l/4}$ for an l-bit semiprime, we prove that the Euler totient function admits the exact representation $\phi \left(N\right)=N-1-2\lfloor \sqrt{N}\rfloor$. Based on this result, we develop an interval-based method for reconstructing $\phi \left(N\right)$ within a narrow neighborhood derived from square-root bounds, followed by a discriminant-based refinement step for recovering the prime factors. Experimental evaluation on large semiprimes, including RSA-type moduli of 4095 and 4096 bits, shows that the method operates efficiently under the stated structural condition using only elementary integer arithmetic. These results provide a theoretical characterization of semiprimes with small prime gaps and offer a framework for identifying structurally weak RSA moduli. This method, given its high efficiency when the prime factors are close to each other, can be regarded as an alternative to Fermat’s factorization method. In particular, for semiprime integers with a small prime gap (i.e., $|p-q|$ is small), the proposed approach exploits structural properties based on the proximity of square roots, thereby significantly accelerating the factorization process. Consequently, it not only aligns with the theoretical foundation of Fermat’s method but, under certain conditions, may also achieve comparable or even superior practical performance. Full article

In a calcined clay rotary cooler, the flow behavior and heat transfer characteristics of the granular bed are key factors determining the cooling efficiency. In this study, an Euler–Euler multiphase model coupled with the kinetic theory of granular flow (KTGF) was used to simulate the granular bed flow and heat transfer in a rotating drum of a rotary cooler. Unlike conventional large-particle beds, the 11 μm calcined clay particles interact more strongly with the gas phase, resulting in stratification and fluidization in the fine-particle bed. The effects of rotational speed, baffle configuration, and number of baffles on the flow and heat transfer behavior of the calcined clay granular bed were investigated. The results show that L-shaped baffles provide superior cooling, achieving a granular bed temperature and heat transfer coefficient (HTC) of 656.88 K and 151.15 W/(m²·K), respectively. At 2 rpm, the maximum temperature decrement and HTC increment are 5.73 K and 46.30 W/(m²·K), whereas excessive rotational speeds intensify bed fluidization. Additionally, increasing the number of L-shaped baffles has limited influence on expanding the fluidized region. With 12 L-shaped baffles, the temperature decrement peaks at 2.86 K and the HTC increment reaches a relatively high 33.27 W/(m²·K). This study provides a theoretical basis for the design and optimization of industrial-scale rotary cooling equipment for fine-particle beds. Full article

This paper builds upon a previous study suggesting an optimization procedure for rotation sequences by introducing a fourth factor in Euler-type decompositions, thus allowing for an additional degree of freedom used both as a variational parameter and a means to avoid the gimbal lock singularity. Here, an analogous result is derived for generic rigid motions, which is of potential interest in 3D robot manipulators, aircraft, and spacecraft using gimbals to navigate in space. The idea is based on Kotelnikov’s principle of transference, which extends the properties of pure rotations to arbitrary Galilean transformations, interpreted as screw motions. To do that in practice, it is convenient to use dual quaternions or their projective version, referred to as dual Rodrigues’ vectors. With this approach, the explicit solutions are easy to extend and therefore optimization is rather straightforward: we show, both analytically and with numerical examples, that factorizing motion into sequences of four consecutive screws is, in general, significantly more energy-efficient compared to using three. Full article

Wearable lower-limb exoskeletons can enhance mobility, reduce metabolic cost, and aid rehabilitation. Effective human-exo cooperation requires personalized assistance profiles that match biomechanical principles. Existing methods often rely on fixed curves, involve complex tuning, and lack biomechanical interpretability. To address this, we propose a “Physics-guided perception and physiology-driven optimization” approach. First, a Physics-guided Dynamic Fusion Model (PDFM) is proposed, which integrates Newton–Euler dynamics, LSTM, and NTM to estimate multi-plane hip joint moments without ground reaction forces, employing biomechanical models as complementary fusion factors rather than the embedded hard constraints used in conventional physics-informed neural networks (PINNs). The model achieved correlation coefficients of 0.938, 0.924, and 0.929, and relative root mean square error (rRMSE) values of 5.29%, 9.79%, and 5.61%, in the sagittal, coronal, and transverse planes, respectively. These results outperformed all single-network baselines across all three anatomical planes. Second, an assistance profile derived from estimated moments is individually optimized using Bayesian optimization based on multi-muscle sEMG. Compared to no-exo walking, the optimized system reduced target muscle loading by 49.31% and metabolic cost by 14.75%; relative to the pre-optimized profile, the reductions were 23.64% and 5.74%, respectively. This work provides a laboratory-validated framework for personalized hip exoskeleton assistance in healthy adults, establishing a foundation for future clinical translation. Full article

This manuscript presents the development of an HEX platform inverse kinematics model, its numerical implementation, and experimental validation. A complete inverse-kinematics formulation is established from the geometric definition of the base and mobile joint coordinates and a z–y–x Euler rotation sequence, allowing actuator-length computation for arbitrary 6-DOF poses. The model is implemented to map the operational workspace under actuator stroke and joint-angle constraints via a two-stage deterministic search, providing dense workspace point clouds, surfaces, and quantitative translational/rotational limits for multiple stroke ranges. Experimental validation is performed on a hexapod platform controlled through an embedded inverse-kinematics layer within a cascaded position–velocity–current architecture with dual-encoder actuator feedback. For a ±25 mm actuator travel range, the experiments confirm close agreement with translation simulations with differences of the order of 2% to 3% in x, y, and z, while larger discrepancies were observed in orientation limits, i.e., the model predicts γ ≈ ±32.5° and α, β ≈ ±10–11°, whereas measurements yield γ ≈ ±30° and α,β ≈ ±14–15°, evidencing higher sensitivity of rotational capability to real mechanical and control factors. Full article

The Zeta-Minimizer Theorem establishes that the Riemann zeta function $\zeta \left(s\right)$ and the primes arise variationally as unique minimizers of a phase functional defined on a symmetric measure space $\left(X,\mu ,G\right)$ equipped with helical operators. Three fundamental axioms—strict concave entropy maximization (Axiom 1), spectral Gibbs minima with non-vanishing ground states (Axiom 2), and irreducible bounded oscillations with flux conservation (Axiom 3)—allow for the selection of the non-proper Archimedean conical helix as the sole topology satisfying all constraints. Primes emerge as indivisible minimal cycles in the associated representation graph $\Gamma$ (via Hilbert irreducibility and Maschke’s theorem), while the Euler product is recovered through the spectral Dirichlet mapping of the helical eigenvalues. The partial zeta product, $Z\left(s\right)=\prod _j{\displaystyle \frac{1}{1-p_j^{-s}}},s\in \mathbb{R}\left\{0\right\},$ constitutes the exact grand partition function of any finite subsystem. Numerical inversion of this product directly recovers the mixture frequency $s$ from any experimental compressibility factor $Z_{\mathrm{m}\mathrm{i}\mathrm{x}}$. Mole fractions $x_i\left(s\right)$, interaction parameters $\Delta \left(x_i\right)$, and the Lyapunov spectrum $\lambda \left(x_i\right)$ then follow deductively via the helical transfer matrix and the closed-form linear ODE for $\Delta$. Occupation numbers $N\left(x_i\right)$ attain sharp maxima precisely at Fibonacci ratios $F_r/F_{r+1}$, leading to the molecular prime-ID rule. For twelve representative purely binary (irreducible) systems spanning atomic noble gases, simple diatomics, polar molecules, and an aromatic ring, the residuals satisfy $|Z\left(s\right)-Z_{\mathrm{m}\mathrm{i}\mathrm{x}}|<1.5\times {10}^{-8}$. The resulting $\lambda \left(x_i\right)$ curves accurately reproduce critical points, liquid ranges, and thermodynamic anomalies with zero adjustable parameters. The Riemann Hypothesis follows rigorously as a theorem: the unique fixed point of the duality functor $s\mapsto 1-s$ that preserves the orthogonality condition $\sum {\mathrm{c}\mathrm{o}\mathrm{s}}^2{\theta }^k=1$ is $\mathrm{R}\mathrm{e}\left(s\right)=1/2$, enforced by Axiom 1 concavity and Axiom 3 irreducibility. The framework is fully deductive and parameter-free and extends naturally to arbitrary mixtures and multiplicities through the helical representation graph. It provides a variational unification of analytic number theory, spectral geometry, thermodynamic phase behavior, and the Riemann Hypothesis from first principles. Full article

Historic masonry towers are all around the world and play a significant role in shaping our built environment. Due to their slender shape, these towers are particularly vulnerable, as recent earthquakes have demonstrated. Many researchers have studied how these structures behave dynamically, with the aim of preserving their cultural value against the risks of damage or collapse. Lately, considerable attention has been paid to develop empirical formulas that estimate their fundamental frequency by considering geometric factors such as total height, reference base length, and effective height for constrained towers. These formulas are usually obtained using regression analysis on data from the technical literature, and so their reliability depends heavily on both the quantity and precision of available data. The variables chosen for calibrating these correlations are mainly determined by the information present in the literature; as a result, missing data can lead to underestimating the influence of some geometric aspects. To address this issue, the paper describes parametric analyses with a simplified model of masonry towers, i.e., the Euler–Bernoulli beam, aiming to show how sensitive the fundamental frequency is to different geometric and mechanical properties. These analyses show the importance of some parameters with respect to others and support the planning of experimental investigation needed for accurate predictions of a tower’s fundamental frequency. Full article

This study presents an analytical investigation into the free vibration behavior of perforated nanobeams resting on a Kerr-type elastic foundation within the framework of Eringen’s nonlocal elasticity theory. Specifically, Eringen’s nonlocal elasticity theory is employed to inherently capture small-scale effects, while the three-parameter Kerr model is utilized to provide a mathematically consistent representation of shear continuity and realistic surface interactions. In this context, the governing equations of motion for a perforated Euler–Bernoulli nanobeam are derived using Hamilton’s principle, incorporating both the nonlocal parameter and perforation geometric factors, namely, the filling ratio and the number of holes. The resulting equations are solved analytically via the Navier method for simply supported boundary conditions. The results indicate that the Kerr foundation model exhibits an intermediate behavior between the Winkler and Pasternak models, owing to the stiffness-reducing effect of its upper spring layer connected in series. A key finding is the “masking effect,” where high foundation stiffness significantly suppresses the frequency reduction caused by nonlocal small-scale effects. Furthermore, it is observed that in the absence of foundation support, the vibration behavior is governed by the competition between mass reduction and stiffness loss depending on the number of holes; however, foundation dominance stabilizes the system regardless of perforation geometry. Full article

The accuracy of online parameter identification for permanent magnet synchronous motors (PMSMs) is constrained by discrete model errors, rank deficiency in the steady-state identification matrix, and voltage deviations resulting from inverter nonlinearities. This paper proposes a multi-parameter identification method acting as a high-precision virtual sensor, based on Zero-Order Hold (ZOH) discretization and an inverter nonlinear voltage compensation scheme utilizing a dual-sampling strategy. First, a discrete model of the PMSM, accounting for rotor position variations within the control period, is established using the ZOH discretization method. Compared with the forward Euler discretization method, this approach effectively minimizes discretization model errors, especially under high-speed operating conditions where rotor position variations are significant. Second, the rank deficiency problem of the steady-state identification matrix is overcome by combining d-axis small-signal injection with a dual-sampling strategy. Furthermore, the Forgetting Factor Recursive Least Squares (FFRLS) algorithm is introduced to achieve online multi-parameter identification. Finally, the influence mechanisms of the dead-time effect, power switch voltage drop, and turn-on delay on the output voltage are analyzed. Consequently, an inverter nonlinear voltage compensation strategy tailored for the dual-sampling mode is proposed. Experimental results demonstrate that the proposed method significantly enhances parameter identification accuracy across the entire speed range. Specifically, under high-speed conditions, the identification errors for resistance, inductance, and flux linkage are maintained within 5.47%, 4.05%, and 2.46%, respectively. Full article

The smearing effect during drainage board installation in soft soil foundation repairs can reduce permeability and compromise the surrounding soil’s structure, limiting the foundation’s consolidation efficiency. This study introduces a novel duckbill-type casing pile shoe with an axisymmetric geometric structure to address issues related to the stability and coating control of conventional pile shoes. A Coupled Euler–Lagrange (CEL) method is employed to develop a three-dimensional model for large-deformation penetration. Additionally, a new analytical framework for the pile shoe insertion and coating mechanism is established by modifying the circular hole expansion theory based on axial symmetry assumptions. This research systematically explores the effects of pile shoe groove curvature, penetration rate, and soil types on the smear zone’s extent. The findings indicate that the circumferential shear effect in the near-field soil intensifies with an increased penetration rate, leading to the expansion of both strong and weak smear zones. When the groove curvature is between 90° and 135°, the smear zone changes from a concentrated to a dispersed pattern, reducing local stress concentration. The extent of the smear zone is also influenced by soil types: ordinary clay exhibits the smallest smear zone, while silty clay demonstrates the greatest. The enhanced circular-hole axisymmetric expansion model shows excellent agreement with the CEL simulation results, confirming its effectiveness when soil strength factors and pile shoe geometry are taken into account. The results provide a theoretical foundation and numerical assistance for the design of pile shoe structures, anticipation of smearing effects, and optimization of drainage board construction in soft soil foundations. Full article

Under random wave loading, the crack growth rate exhibits jump-like cycle-to-cycle variations, which limit the direct use of efficient integration schemes such as the Euler method. In addition, the crack growth life is highly sensitive to the initial crack size and aspect ratio, while the initial defects are often difficult to determine accurately in practice, leading to increased uncertainty in life assessment. To address these issues, a cycle-scaling-based crack size accumulation method for random loading is proposed. A predictor–corrector improved Euler method is then established, and a fourth-order Runge–Kutta scheme incorporating the cycle-scaling transformation is derived. Furthermore, based on spectral analysis theory, a mapping between the wave spectrum and the crack-tip stress intensity factor response spectrum is developed. A stress intensity factor range sequence is generated by concatenating short-term sea states, thereby providing a random loading input that preserves the required statistical characteristics. Finally, a 21,000-TEU container ship is analyzed as a case study to investigate crack growth evolution for different initial aspect ratios. The results show that the crack aspect ratio gradually converges to a particular trend during propagation. A convergent aspect ratio curve is fitted. And a unified life assessment curve is constructed. An equivalent transformation is used to map an arbitrary initial crack shape and size to an equivalent convergent aspect ratio crack. As a result, fatigue life can be rapidly estimated using a single “initial crack size–fatigue life” curve, providing support for crack growth life assessment and the definition of defect acceptance limits for ship hull structures. Full article

This paper introduces a novel multidimensional half-discrete Hardy–Hilbert-type inequality that simultaneously addresses several key extensions in the literature. The inequality incorporates a general parameterized kernel involving a scalar term and the $\beta$-norm of a vector, and replaces the traditional discrete coefficient with a partial sum. Under suitable parameter conditions, the resulting inequality is sharper and preserves the optimal constant factor. The proof employs a systematic combination of weight-function techniques, parameter introduction, real-analysis methods, and the Euler–Maclaurin summation formula. Equivalent characterizations of the best possible constant are provided, and several meaningful corollaries are deduced, thereby unifying and generalizing a series of earlier inequalities. Full article

The paper presents the analysis of the influence of the location and characteristics of supports on the static response of two-layer beams. The possibility of tangential movement at the supports was considered. Multilayer beams, which combine the advantages of different materials, are widely used in construction. The authors’ previous research showed that the stiffness of the connection between layers significantly affects the behaviour of the system. This paper demonstrates that the supports’ position is another crucial factor that influences the beams’ response, which is an issue that has not been previously considered in the literature. A two-layer system was modelled using the Euler–Bernoulli beam theory. Consistent normal displacements and tangential forces at the layer interface, which were proportional to the relative slip, were assumed. From the equilibrium equations and considered assumptions, three coupled displacement equations were derived and then solved using finite Fourier transforms. They were applied to solve beams, the two ends of which cannot move in the direction perpendicular to the beam’s axis, with at least one of the beam ends being a pinned support. To verify the method’s accuracy, several numerical examples were analysed. It was shown that both the support position and the possibility of tangential displacement at the supports have a significant impact on the static response. Additionally, the crucial role of the stiffness of the interlayer connection was confirmed. The developed approach provides a practical tool for assessing two-layer beam systems and highlights the importance of considering support conditions in the design and analysis of such structures. Full article

This study employed a combined approach of computational fluid dynamics (CFD), numerical simulations, and wind tunnel tests to investigate droplet drift characteristics and develop prediction models in order to address the issues of low pesticide utilization rates and high drift risk, associated with droplet drift during agricultural unmanned aerial vehicle (UAV) spraying, as well as the unreliable results of field experiments. Firstly, a numerical model of the rotor wind field was established using the multiple reference frame (MRF) method, while the realizable k-ε turbulence model was employed to analyze the flow field. The model’s reliability was verified through wind field tests. Next, the Euler–Lagrange method was used to couple the wind field with droplet movement. The drift characteristics of two flat-fan nozzles (FP90-02 and F80-02) were then compared and analyzed. The results showed that the relative error between the simulated and wind tunnel test values was within 20%. Centrifugal nozzle experiments were carried out using single-factor and orthogonal designs to analyze the effects of flight height, rotor wind speed, flight speed, and droplet size on drift. The priority order of influence was found to be “rotor wind speed > flight height > flight speed”, while droplet size (D_V50 = 100–300 µm) was found to have no significant effect. Based on the simulation data, a multiple linear regression drift prediction model was constructed with a goodness of fit R² value of 0.9704. Under the verification condition, the relative error between the predicted and simulated values was approximately 10%. These results can provide a theoretical basis and practical guidance for assessing drift risk and optimizing operational parameters for agricultural UAVs. Full article

Erosion of the leading edge of blades in windy and sandy environments can cause wind turbines to lose up to 25% of their annual power generation. Traditional studies have mostly focused on the impact of single factors on erosion rates, but the effects of multiple parameters on erosion rates within the framework of the Stokes number (Stk) of dust particles have not yet been clarified. This study employs a numerical approach based on the Euler–Lagrange framework, integrating the SST k-ω turbulence model with a discrete phase model (DPM) to simulate the unsteady gas–solid two-phase flow around a NACA 0012 airfoil. The computational model was rigorously validated through grid independence tests and comparison with experimental aerodynamic data from the database, showing strong agreement under steady conditions. Systematic simulations were conducted with particle diameters ranging from 10 to 360 μm, densities from 2650 to 3580 kg/m³, and inflow velocities from 1.5 to 21 m/s, comprehensively covering Stokes number regimes from Stk << 1 to Stk >> 1. Through parametric analysis, we quantify the control effect of Stk on erosion rate and erosion hot spots. Simulation results indicate that Stk has a zone-specific control effect on airfoil erosion: erosion hot spots in low-Stk zones migrate from the mid-to-rear edge to the leading edge. Erosion rate peaks when Stk ≈ 0.8. Inertial impact in the high-Stk zone dominates surface damage propagation. Based on the simulation results, an erosion model with an error of ≤3.6% was established for the E = K∙Stk^a∙d_p^b∙v^c zone, providing a quantitative physical basis to inform wind turbine blade protection strategies. Full article