In practical engineering, pile foundation will be subject to the combined action of vertical and Rayleigh wave, and the existence of vertical dynamic load will cause the second-order effect that will lead to the increase of horizontal displacement. This paper presents a computational model for the effect of vertical load on the lateral response of monopole under Rayleigh wave. The pile cap is equivalent to a rigid block, and the constraint of pile head is regarded as a flexible constraint. By means of operator decomposition theory and variable separation method, the horizontal dynamic response of single-phase soil in uniform free field under Rayleigh wave propagation is obtained. The closed form solution of soil resistance under the combined action of Rayleigh wave and vertical load is obtained by means of operator decomposition theory and variable separation method. Based on Timoshenko beam theory, the dynamic differential equation of pile considering vertical load is established. A comparison with an existing solution is performed to verify the proposed solution. Through numerical examples, the effects of the vertical load, flexible constraints, dimensionless frequency and Poisson’s ratio on the lateral response of monopile are assessed.

Vigilance estimation is a critical task within the field of brain-computer interfaces, extensively applied in monitoring and optimizing user states during human-machine interaction using electroencephalography (EEG). However, most existing vigilance prediction frameworks are prone to spurious correlations stemming from inherent biases in collected data. These biases involve relevant but vigilance-independent information, which may lack robustness when applied to different data distributions, i.e., out-of-distribution (OOD) scenarios. The core idea of this study is to learn constraints that capture causal information from the input based on the assumed underlying data generating process. Leveraging the disentanglement and invariance principles behind the assumptions, we propose a constraint-driven causal representation learning (CCRL) to identify and separate spurious latent variables from biased training data for generalized vigilance estimation. The CCRL training process consists of two phases: self-supervised pretraining and constraint-driven causal information disentanglement. In the first phase, based on the masked autoencoder (MAE) architecture, unlabeled training data are used for reconstructing pretext tasks to capture the comprehensive and intrinsic contextual information from EEG data, which provides a powerful input for downstream disentanglement learning. In the second phase, we propose a novel disentanglement strategy to learn spurious-free latent representations causally related to the vigilance state driven by adversarial and invariance constraints. Comprehensive validation experiments conducted on two well-known public datasets demonstrate the effectiveness and superiority of the proposed framework. In general, this work has promising implications for addressing OOD challenges in vigilance estimation.

In this study, we apply Lie group theory to find solutions to first- and second–ordersecond–order ordinary differential equations. The method is based on identifying a one-parameter Lie group of symmetries admitted by the equations. Additionally, we perform symmetry-based separation of variables using coordinate systems that correspond to the identified symmetries.

We investigate the nonrelativistic quantum dynamics of a spinless particle in a screw-type spacetime endowed with two independent twist controls that interpolate between a pure screw dislocation and a homogeneous twist. From the induced spatial metric, we build the covariant Schrödinger operator, separate variables to obtain a single radial eigenproblem, and include a uniform axial magnetic field and an Aharonov–Bohm (AB) flux by minimal coupling. Analytically, we identify a clean separation between a global, AB-like reindexing set by the screw parameter and a local, curvature-driven mixing generated by the distributed twist. We derive the continuity equation and closed expressions for the azimuthal and axial probability currents, establish practical parameter scalings, and recover limiting benchmarks (AB, Landau, and flat space). Numerically, a finite-difference Sturm–Liouville solver (with core excision near the axis and Langer transform) resolves spectra, wave functions, and currents. The results reveal AB periodicity and reindexing with the screw parameter, Landau fan trends, twist-induced level tilts and avoided crossings, and a geometry-induced near-axis backflow of the axial current with negligible weight in cross-section integrals. The framework maps the geometry and fields directly onto measurable spectral shifts, interferometric phases, and persistent-current signals.

Polyampholyte hydrogels are promising for load-bearing biomedical applications, but the link between composition and compression behavior remains unclear. In this study, we investigate how initial monomer concentration and a neutral comonomer influence swelling and mechanical properties in AMPS–APTAC networks. Terpolymeric AMPS–APTAC–DMAAm hydrogels were prepared with monomer concentrations from 1 to 2 M, MBAAm levels from 1 to 5 mol%, and DMAAm fractions from 0 to 0.16. Swelling was measured in water. Unconfined compression tests at 3 mm·min−1 provided stress–strain curves, Young’s modulus (E), fracture stress (σf), fracture strain (εf), and toughness (W) up to 99% strain. Increasing the monomer concentration produced denser networks, lower swelling, and higher stiffness. For C2M1, E reached 35.4 kPa, σf reached 0.8 MPa, εf was 82%, and W was 65.6 kJ·m−3. Adding DMAAm strengthened the gels through reversible associative interactions. At z = 0.06, σf increased to 4.28 MPa and W to 196.0 kJ·m−3. At z = 0.16, E increased to 103.0 kPa, while σf was 2.34 MPa and W was 191.6 kJ·m−3. Swelling decreased when monomer or crosslinker content increased. These results show that monomer concentration and DMAAm-mediated associations act as separate design variables that can be tuned to optimize stiffness, strength, and toughness in AMPS–APTAC polyampholyte hydrogels.

The thermal therapy mask, as a wearable device, requires precise thermal management to ensure therapeutic efficacy and safety, which necessitates a detailed investigation of its heat conduction behavior under complex conditions. However, the heat convective behavior of an orthotropic thermal therapy mask with an embedded line heat source under practical operational conditions has not yet been rigorously investigated. Therefore, this study addresses this specific problem by abstracting it into a 2D orthotropic transient heat conduction problem with a line heat source under Robin BCs, and derives its analytical solution using the SSM without any assumption of solution form. The SSM first transforms the governing equation into the frequency domain via the Laplace transform technique and reformulates it within the Hamiltonian framework. The original problem is then decomposed into two subproblems, which are solved by the method of separation of variables and the symplectic eigen expansion. The final analytical solution is obtained through superposing the solutions of the subproblems, and its accuracy is validated through comparison with the finite element method. The influence of the heat convection coefficient on the thermal behavior is systematically analyzed, revealing that increasing the heat convection coefficient accelerates the procedure from transient to steady state and results in reduced steady-state temperature. Furthermore, the analysis of orthotropic thermal conductivity reveals a "short-plank effect", where the temperature evolution is limited by the smaller thermal conductivity. This study provides benchmark results for accurate and efficient thermal prediction and may enable an extension to broader applications in flexible electronics such as wearable sensors and displays.

Abstract To address the future demand for higher-precision angular vibration detection and measurement, this paper proposes a multi-channel series magnetohydrodynamics micro angular vibration sensor. The focus lies on analyzing the sensor’s output characteristics and resolving the issue where existing models fail to accurately describe low-frequency behavior. From the perspective of initial prototype development, this paper presents an analytical method for estimating magnetic induction intensity in the sensitive region through theoretical analysis. Additionally, for axially structured magnetohydrodynamics micro angular vibration sensors, an accurate mathematical model for analytical calculations is established using the complex amplitude method and separation of variables. Finite element analysis was employed to verify the sensor’s magnetic field distribution and fluid flow characteristics, while practical tests were conducted to examine its scale factor and phase difference. These two approaches collectively validate the accuracy of the theoretical analysis. Experimental results demonstrate that the proposed analytical calculation method for magnetic induction intensity in the sensitive region offers guiding significance for engineering design. Furthermore, the analytical model of frequency response exhibits a high degree of fitting across the full frequency range, enabling accurate description of the output frequency characteristics of axially structured sensors with broad applicability.

ABSTRACT The flexible deformation characteristics of granular and accumulated embankments result in a relatively weak lateral load transfer capacity, requiring sufficient differences in column–soil settlement to transfer loads from the surrounding soil to the columns. Although large‐diameter columns improve the load transfer ability, not only vertical deformation but also vertical and radial flow occurs inside the column during the consolidation process. To address the consolidation problem of this column–reinforced composite ground, a modified equal strain condition is proposed to consider the difference in settlement between the column and its surrounding soil, and a load transfer‐resistance coefficient is used to represent the lateral load transfer ability. On this basis, the consolidation equations considering the vertical deformation and vertical and radial flows of the columns are established. The analytical consolidation solutions under time‐dependent vertical stress are obtained through the separation of variables method. When the modified equal strain condition is simplified to the equal strain condition, the analytical solution under instantaneous loading in the paper simplifies to the previous corresponding solution. In this case, the concentration region (column or surrounding soil) of the initial pore pressure relies on the vertical permeability coefficient rather than the compressive modulus. Under the modified equal strain condition, the concentration phenomenon of pore pressure occurs later and presents a pseudo Mandel–Cryer effect. Parametric analyses of the consolidation characteristics of composite ground beneath flexible foundations under instantaneous or ramp loading are conducted, providing reference data for design and construction.

There are significant differences in the deformation patterns of different parts of arch dams, and there is a common situation of periodic data loss. To accurately analyze the deformation behavior of arch dams, this paper proposes a safety warning and anomaly diagnosis method for arch dam deformation based on the separation of environmental variable effects in different partitions and a knowledge-driven approach. This method combines various techniques such as an optimized ISODATA clustering method, probabilistic principal component analysis (PPCA), square prediction error (SPE) norm control chart, and contribution chart. By defining data forms and rules, existing engineering specifications and experience are transformed into “knowledge” and applied to the operation and management of arch dams, achieving accurate monitoring of arch dam deformation status and timely diagnosis of outliers. Through monitoring data verification of horizontal displacement in a certain arch dam partition, the results show that this method can accurately identify deformation anomalies in the arch dam and effectively separate the influence of environmental variables and noise interference, providing strong support for the safe operation of the arch dam. Accurate deformation monitoring of arch dams is essential for ensuring structural safety and optimizing operational management. However, conventional early warning indicators and empirical models often fail to capture the spatial heterogeneity of deformation and the complex coupling between environmental variables and structural responses. To overcome these limitations, this study proposes a knowledge-driven safety early warning and anomaly diagnosis model for arch dam deformation, based on spatiotemporal clustering and partitioned environmental variable separation. The method integrates the optimized ISODATA clustering algorithm, probabilistic principal component analysis (PPCA), squared prediction error (SPE) control chart, and contribution chart to establish a comprehensive monitoring framework. The optimized ISODATA identifies deformation zones with similar mechanical behavior, PPCA separates environmental influences such as temperature and reservoir level from structural responses, and the SPE and contribution charts quantify abnormal variations and locate potential risk regions. Application of the proposed method to long-term deformation monitoring data demonstrates that the PPCA-based framework effectively separates environmental effects, improves the interpretability of zoned deformation characteristics, and enhances the accuracy and reliability of anomaly identification compared with conventional approaches. These findings indicate that the proposed knowledge-driven model provides a robust and interpretable framework for precise deformation safety evaluation of arch dams.

In this work, we propose a new physics-informed neural network framework based on the method of separation of variables (SVPINN) to solve the distributed-order time-fractional advection–diffusion equation. We develop a new method for calculating the distributed-order derivative, which enables the fractional integral to be modeled by a network and directly solved by combining automatic differentiation technology. In this way, the approximation of the distributed-order derivative is integrated into the parameter training system of the network, and the data-driven adaptive learning mechanism is used to replace the numerical discretization scheme. In the SVPINN framework, we decompose the kernel function of the Caputo integral into three independent functions using the method of separation of variables, and apply a neural network as a surrogate model for the modified integral and the function related to the time variable. The new physical constraint generated by the modified integral serves as an extra supervised learning task for the network. We systematically evaluated the feasibility of the SVPINN on several numerical experiments and demonstrated its performance.

High morbidity in adults with non-transfusion-dependent thalassemia referred to u.S. specialty centers

This paper studies a mixed PDE containing the second time derivative and a quadratic nonlinearity of the Monge–Ampère type in two spatial variables, which is encountered in geophysical fluid dynamics. The Lie group symmetry analysis of this highly nonlinear PDE is performed for the first time. An invariant point transformation is found that depends on fourteen arbitrary constants and preserves the form of the equation under consideration. One-dimensional symmetry reductions leading to self-similar and some other invariant solutions that described by single ODEs are considered. Using the methods of generalized and functional separation of variables, as well as the principle of structural analogy of solutions, a large number of new non-invariant closed-form solutions are obtained. In general, the extensive list of all exact solutions found includes more than thirty solutions that are expressed in terms of elementary functions. Most of the obtained solutions contain a number of arbitrary constants, and several solutions additionally include two arbitrary functions. Two-dimensional reductions are considered that reduce the original PDE in three independent variables to a single simpler PDE in two independent variables (including linear wave equations, the Laplace equation, the Tricomi equation, and the Guderley equation) or to a system of such PDEs. A number of specific examples demonstrate that the type of the mixed, highly nonlinear PDE under consideration, depending on the choice of its specific solutions, can be either hyperbolic or elliptic. To analyze the equation and construct exact solutions and reductions, in addition to Cartesian coordinates, polar, generalized polar, and special Lorentz coordinates are also used. In conclusion, possible promising directions for further research of the highly nonlinear PDE under consideration and related PDEs are formulated. It should be noted that the described symmetries, transformations, reductions, and solutions can be utilized to determine the error and estimate the limits of applicability of numerical and approximate analytical methods for solving complex problems of mathematical physics with highly nonlinear PDEs.

Abstract This article provides an analytical framework for magnetohydrodynamic (MHD) couple stress fluid (CSF) flow around a partially contaminated liquid drop within a porous medium, utilizing the interfacial slip condition. The Brinkman and Stokes' equations are utilized to investigate the flow dynamics within and surrounding the porous region. The external magnetic field on the fluid flow, which is in the transverse direction, is represented by Lorentz's force. The boundary value problem addresses the continuity of the velocity components and stress distributions. Analytical expressions for stream functions are derived using the variable separable method, drag force on a liquid drop are calculated using the integration method. Specific cases are analyzed, including viscous fluids under both slip and no‐slip conditions, as well as rigid spheres and liquid drops. The findings are consistent with existing literature. Graphical analysis illustrates the relationship among the drag coefficient, couple stress viscosity ratio, and porosity parameter. Couple stress effects enhance rigid sphere drag more than liquid drop drag when the couple stress viscosity ratio increases. As the porosity parameter increases, drag also increases. The inner and outer streamlines get more disturbed as the couple stress viscosity ratio increases, creating more internal streamlines. Furthermore, a comparison investigation discovered that the drag of MHD couple stress fluid flow past a contaminated liquid drop in a porous media is larger than that of MHD viscous flow under the same conditions. This study enables improved control for geothermal and drug delivery applications.

Linking micropollutant mixtures and macroinvertebrate ecological health using AI-based toxicity predictions.

The hydrodynamic interactions of two soft composite spheres in a creeping flow regime are investigated using a semi-analytical framework that combines separation of variables with a boundary collocation scheme. The formulation incorporates non-ideal boundary conditions, including interfacial slip and stress jump, together with complex internal architectures comprising a solid core enveloped by a permeable Brinkman shell. Our analysis quantifies the role of these features and reveals a distinct hierarchy of effects: hydrodynamic confinement due to inter-particle proximity governs the overall drag, while internal structure and interfacial properties modulate its magnitude. High shell permeability can paradoxically increase drag through internal viscous dissipation, whereas an impermeable high-slip core provides a “hydrodynamic cushion” that mitigates resistance. Importantly, these influences diminish in the impermeable limit, where particle behavior is dictated solely by external geometry. Our results highlight that, counterintuitively, a highly permeable shell may amplify drag through enhanced internal dissipation, whereas reduced permeability suppresses this mechanism. Moreover, the overall drag is overwhelmingly dictated by external hydrodynamic confinement, while interfacial slip and stress jump conditions act only as secondary modulators. Limiting-case comparisons and spectral-convergence tests confirm the reliability of the semi-analytical scheme. This work establishes a comprehensive physical framework for predicting particulate transport, clarifying how fluid–structure interactions across multiple scales govern the behavior of soft matter systems.

Love-type wave propagation in a heterogeneous medium between a liquid and a fractured porous half-space: A WKB and variable separation approach

This paper presents a novel extension of trigonometric and hyperbolic functions based on the W α , β , ν -function recently introduced in the literature [Droghei. Properties of the multi-index special function W ( α ¯ , ν ¯ ) ( z ) . Fract Calc Appl Anal. 2023;26(5):2057–2068. doi: 10.1007/s13540-023-00197-6]. The classical factorial is generalized to offer greater flexibility for applications in advanced mathematical fields such as combinatorics, special functions, and complex analysis. Using the W α , β , ν -exponential function, we derive new expressions for generalized trigonometric and hyperbolic functions, including sine, cosine, hyperbolic sine, and hyperbolic cosine. These functions provide exact solutions for non-trivial fractional differential equations with variable coefficients. We discuss some relevant examples of applications to solve linear and nonlinear equations by using operational methods and separation of variables.

The electrostatic potential in certain types of boundary value problems can be found by solving Laplace’s Equation (LE). It is important for students to develop the ability to recognize the utility of LE and apply the method to solve physics problems. To develop students’ problem-solving skills for solving problems that can be solved effectively using Laplace’s equation in an upper-level electricity and magnetism course, we developed and validated a tutorial focused on finding electrostatic potential in a Cartesian coordinate system. The tutorial was implemented across three instructors’ classes, accompanied by scaffolded pretest (after traditional lecture) and posttest (after the tutorial). We also conducted think-aloud interviews with advanced students using both unscaffolded and scaffolded versions of the pretest and posttest. Findings reveal common student difficulties that were included in the tutorial as a guide to help address them. The difference in the performance of students from the pretest after lecture to the posttest after the tutorial was similar on the scaffolded version of the tests (in which the problems posed were broken into sub-problems) for all three instructors’ classes and interviewed students. Equally importantly, interviewed students demonstrated greater differences in scores from the pretest and posttest on the unscaffolded versions in which the problems were not broken into sub-problems, suggesting that the scaffolded version of the tests may have obscured evidence of actual learning from the tutorial. While a scaffolded test is typically intended to guide students through complex reasoning by breaking a problem into sub-problems and offering structured support, it can limit opportunities to demonstrate independent problem-solving and evidence of learning from the tutorial. Additionally, one instructor’s class underperformed relative to others even on the pretest. This instructor had mentioned that the tests and tutorial were not relevant to their current course syllabus and offered a small amount of extra credit for attempting to help education researchers, highlighting how this type of instructor framing of instructional tasks can negatively impact student engagement and performance. Overall, in addition to identifying student difficulties and demonstrating how the tutorial addresses them, this study reveals two unanticipated but critical insights: first, breaking problems into sub-parts can obscure evidence of students’ ability to independently solve problems, and second, instructor framing can significantly influence student engagement and performance.

The diffraction of an electromagnetic TE wave on multilayer diffraction gratings with several two-layer lines in the period was investigated. Such optical structures are employed in spectral beam combining. The diffraction problem was solved by a modified method of variable separation, which requires the solution of two one-dimensional boundary value problems for eigenvalues of second-order differential equations on a segment with piecewise constant coefficients. Each of the boundary value problems for eigenvalues reduces to the calculation of a second-order determinant. The proposed method was applied to solve the problem of the electromagnetic TE wave diffraction on multilayer diffraction gratings with several different configurations, as well as to determine their diffraction efficiency. The numerical results were presented. For modeling the diffraction gratings, materials that are commonly used in thin-film coatings were selected. The method can be further developed to model more complex diffraction gratings with multilayer reflective coatings.

We consider the open XYZ spin chain with boundary fields. We solve the model by the new Separation of Variables approach introduced in [J. M. Maillet and G. Niccoli, J. Stat. Mech.: Theory Exp. 094020 (2019)]. In this framework, the transfer matrix eigenstates are obtained as a particular sub-class of the class of so-called separate states. We consider the problem of computing scalar products of such separate states. As usual, they can be represented as determinants with rows labelled by the inhomogeneity parameters of the model. We notably focus on the special case in which the boundary parameters parametrising the two boundary fields satisfy one constraint, hence enabling for the description of part of the transfer matrix spectrum and eigenstates in terms of some elliptic polynomial QQ-solution of a usual TQTQ-equation. In this case, we show how to transform the aforementioned determinant for the scalar product into some more convenient form for the consideration of the homogeneous and thermodynamic limits: as in the open XXX or XXZ cases, our result can be expressed as some generalisation of the so-called Slavnov determinant.