Small Parameter Research Articles

Space–time structure of weak magnetohydrodynamic turbulence

The two-time energy spectrum of weak magnetohydrodynamic turbulence is found by applying a wave-turbulence closure to the cumulant hierarchy constructed from the dynamical equations. Solutions are facilitated via asymptotic expansions in terms of the small parameter $\varepsilon$ , describing the ratio of time scales corresponding to Alfvénic propagation and nonlinear interactions between counter-propagating Alfvén waves. The strength of nonlinearity at a given spatial scale is further quantified by an integration over all possible delta-correlated modes compliant in a given set of three-wave interactions that are associated with energy flux through the said scale. The wave-turbulence closure for the two-time spectrum uncovers a secularity occurring on a time scale of order $\varepsilon ^{-2}$ , and the asymptotic expansion for the spectrum is reordered in a manner comparable to the one-time case. It is shown that for the regime of stationary turbulence, the two-time energy spectrum exponentially decays on a lagged time scale $(\varepsilon ^2 \gamma _k^s)^{-1}$ in proportion to the strength of the associated three-wave interactions, characterized by nonlinear decorrelation frequency $\gamma _k^s$ . The scaling of the form $k_{\perp } v_0 \chi _0$ exhibited by this frequency is reminiscent of random sweeping by the outer scale with characteristic fluctuation velocity $v_0$ that is modified due to competition with Alfvénic propagation (characterized by $\chi _0$ ) at the said scale. A brief calculation of frequency broadening of the power spectrum due to nonlinear interactions is also presented.

Ni/Bi bilayers: The effect of thickness on the superconducting properties

Nickel/bismuth (Ni/Bi) bilayers have recently attracted attention due to the occurrence of time-reversal symmetry breaking in the superconducting state. Here, we report on the structural, magnetic, and electric characterization of thin film Ni/Bi bilayers with several Bi thicknesses. We observed the formation of a complex layered structure depending on the Bi thickness caused by the inter-diffusion of Bi and Ni which leads to the stabilization of NiBi3 at the Bi/Ni interface. The superconducting transition temperature and the transition width are highly dependent on the Bi thickness and the layer structure. Magnetoelectric transport measurements in perpendicular and parallel magnetic fields were used to investigate the temperature-dependent upper critical field within the framework of the anisotropic Ginzburg–Landau theory and the Werthamer–Helfand–Hohenberg model. For thicker samples, we observed a conventional behavior, similar to that shown by NiBi3 bulk samples, including a small Maki parameter (αM=0), no spin–orbit scattering (λSO=0) and nearly isotropic coherence length (γ=ξ⊥(0)/ξ∥(0)≈1). The values obtained for these properties are close to those characterizing NiBi3 single crystals. On the other hand, in very thin samples, the Maki parameter increases to about αM=2.8. In addition, the coherence length becomes anisotropic (γ=0.32) and spin–orbit scattering (λSO=1.2) must be taken into account. Our results unequivocally show that the properties characterizing the superconducting state in the Ni/Bi are strongly dependent on the sample thickness.

Dynamic theory and experimental testing in ipsilateral offset dual-motor excitation system

This paper addresses unbalanced trajectories of dual-motor vibrating screens by proposing a dynamic theory and experimental testing. First, based on the Lagrange equation, the dynamic model of ipsilateral offset dual-motor excitation system is established. Subsequently, the steady response is worked out by solving the differential equation of motion of the vibrating body. Then, the synchronization condition and synchronization stability criterion of the system are determined employing the small parameter method and the Routh–Hurwitz criterion. Numerical analysis methods are given to reveal the relationship between structural parameters and synchronization characteristics of the vibration system. Moreover, an mechanical–electric coupling model is developed to study dynamic characteristics. The results demonstrate that synchronization condition can be described with capture torque and difference of output torque between motors. The synchronization ability is enhanced by improving installation distance of the motors. Additionally, increasing angle β or decreasing angle θ of the motors is in favor of optimization of centroid trajectory; swing oscillation of the box can be controlled by adjusting the main inclination angle of motors; in such a condition, equilibrium of motion trajectory of system is enhanced. Finally, the validity method for designing vibrating screen is verified through theoretical calculations, simulation analysis, and experiments.

Nonlinear stability of shock-fronted travelling waves in reaction-nonlinear diffusion equations

Reaction-nonlinear diffusion PDEs can be derived as continuum limits of stochastic models for biological and ecological invasion. We numerically investigate the nonlinear stability of shock-fronted travelling waves arising in these RND PDEs, in the presence of a fourth-order spatial derivative multiplied by a small parameter ɛ that models high-order regularization. Once we have verified sectoriality of our linear operator, our task is reduced to checking spectral stability of our family of travelling waves. Motivated by the authors’ recent stability analysis of shock-fronted travelling waves under viscous relaxation, our numerical analysis suggests that near the singular limit, the associated eigenvalue problem for the linearized operator admits a fast–slow decomposition similar to that constructed by Alexander, Gardner, and Jones in the early 90s. In particular, our numerical results suggest a reduction of the complex four-dimensional eigenvalue problem into a real one-dimensional problem defined along the slow manifolds; i.e. slow eigenvalues defined near the tails of the shock-fronted wave for ɛ=0 govern the point spectrum of the linearized operator when 0<ɛ≪1.

Explicit near-symplectic integrators for post-Newtonian Hamiltonian systems

Explicit symplectic integrators are powerful and widely used for Hamiltonian systems. However, once the post-Newtonian (PN) effect is considered to provide more precise modeling for the N-body problem, explicit symplectic methods cannot be constructed due to the nonseparability of the Hamiltonian. Thus, the available symplectic method is either fully implicit or semi-implicit, which decreases the efficiency because of the implicit iteration used during the evolution. In this paper, we aim to explore efficient explicit methods whose performance is mostly like symplectic methods for PN Hamiltonian systems. Taking the small parameter ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document} appearing in PN terms into consideration, we replace the implicit symplectic solver with explicit solvers in the mixed symplectic method to solve the PN term and then derive three explicit methods. It is theoretically shown that the proposed methods are respectively second-order, fourth-order, and pseudo-fourth-order, and that their closeness to the corresponding symplectic methods are O(ε3h3),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {O}}(\\varepsilon ^{3}h^{3}),$$\\end{document}O(ε5h5),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {O}}(\\varepsilon ^{5}h^{5}),$$\\end{document} and O(ε3h3).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {O}}(\\varepsilon ^{3}h^{3}).$$\\end{document} That is, they are explicit near-symplectic methods with the presence of the small parameter ε.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon .$$\\end{document} Numerical experiments with the Hamiltonian problem of spinning compact binaries show that the energy errors and orbital errors of the proposed explicit near-symplectic methods are indistinguishable from the corresponding mixed semi-implicit symplectic methods. The very small magnitude of the difference between the proposed explicit near-symplectic methods and the mixed symplectic methods confirms our theoretical analysis of their closeness to symplecticity. Finally, the much less CPU time consumed by the proposed methods highlights their most important advantage of high efficiency over the mixed symplectic methods.

Bound states of weakly deformed soft waveguides

In this paper we consider the two-dimensional Schrödinger operator with an attractive potential which is a multiple of the characteristic function of an unbounded strip-shaped region, whose thickness is varying and is determined by the function R ∋ x ↦ d + ε f ( x ), where d &gt; 0 is a constant, ε &gt; 0 is a small parameter, and f is a compactly supported continuous function. We prove that if ∫ R f d x &gt; 0, then the respective Schrödinger operator has a unique simple eigenvalue below the threshold of the essential spectrum for all sufficiently small ε &gt; 0 and we obtain the asymptotic expansion of this eigenvalue in the regime ε → 0. An asymptotic expansion of the respective eigenfunction as ε → 0 is also obtained. In the case that ∫ R f d x &lt; 0 we prove that the discrete spectrum is empty for all sufficiently small ε &gt; 0. In the critical case ∫ R f d x = 0, we derive a sufficient condition for the existence of a unique bound state for all sufficiently small ε &gt; 0.

Mechanical circulatory support devices for patients with small anthropometric indicators

Mechanical circulatory support (MCS) devices, designed specifically for patients with small anthropometric parameters, are now emerging. A detailed systematic literature review of existing systems for long-term circulatory support in this patient cohort was conducted. Circulatory support devices and their main technical and biological characteristics were studied in detail. Despite significant scientific and technological progress, there is still no technology for creating an assist pump to support patients with small body surface area (BSA), given the wide range of patient sizes, increased cardiovascular demand due to growth, as well as anatomical and physiological heterogeneity of congenital heart disease.

Hybrid-FHR: a multi-modal AI approach for automated fetal acidosis diagnosis

BackgroundIn clinical medicine, fetal heart rate (FHR) monitoring using cardiotocography (CTG) is one of the most commonly used methods for assessing fetal acidosis. However, as the visual interpretation of CTG depends on the subjective judgment of the clinician, this has led to high inter-observer and intra-observer variability, making it necessary to introduce automated diagnostic techniques.MethodsIn this study, we propose a computer-aided diagnostic algorithm (Hybrid-FHR) for fetal acidosis to assist physicians in making objective decisions and taking timely interventions. Hybrid-FHR uses multi-modal features, including one-dimensional FHR signals and three types of expert features designed based on prior knowledge (morphological time domain, frequency domain, and nonlinear). To extract the spatiotemporal feature representation of one-dimensional FHR signals, we designed a multi-scale squeeze and excitation temporal convolutional network (SE-TCN) backbone model based on dilated causal convolution, which can effectively capture the long-term dependence of FHR signals by expanding the receptive field of each layer’s convolution kernel while maintaining a relatively small parameter size. In addition, we proposed a cross-modal feature fusion (CMFF) method that uses multi-head attention mechanisms to explore the relationships between different modalities, obtaining more informative feature representations and improving diagnostic accuracy.ResultsOur ablation experiments show that the Hybrid-FHR outperforms traditional previous methods, with average accuracy, specificity, sensitivity, precision, and F1 score of 96.8, 97.5, 96, 97.5, and 96.7%, respectively.ConclusionsOur algorithm enables automated CTG analysis, assisting healthcare professionals in the early identification of fetal acidosis and the prompt implementation of interventions.

Detection, Measurement and Classification of Discontinuities of Signals Captured with Noise

In this work, we propose an algorithm for the detection, measurement and classification of discontinuities in signals captured with noise. Our approach is based on the Harten’s subcell-resolution approximation adapted to the presence of noise. This technique has several advantages over other algorithms. The first is that there is a theory that allows us to ensure that discontinuities will be detected as long as we choose a sufficiently small discretization parameter size. The second is that we can consider different types of discretizations such as point values or cell-averages. In this work, we will consider the latter, as it is better adapted to functions with small oscillations, such as those caused by noise, and also allows us to find not only the discontinuities of the function, jumps in functions or edges in images, but also those of the derivative, corners. This also constitutes an advantage over classical procedures that only focus on jumps or edges. We present an application related to heart rate measurements used in sport as a physical indicator. With our algorithm, we are able to identify the different phases of exercise (rest, activation, effort and recovery) based on heart rate measurements. This information can be used to determine the rotation timing of players during a game, identifying when they are in a rest phase. Moreover, over time, we can obtain information to monitor the athlete’s physical progression based on the slope size between zones. Finally, we should mention that regions where heart rate measurements are abnormal indicate a possible cardiac anomaly.

A quasi-static model of a thermoelastic body reinforced by a thin thermoelastic inclusion

The problem of description of quasi-static behavior is studied for a planar thermoelastic body incorporating an inhomogeneity, which geometrically is a strip with a small cross-section. This problem contains a small positive parameter δ describing the thickness of the inhomogeneity, i.e., the size of the cross-section. Relying on the variational formulation of the problem, we investigate the behavior of solutions as δ tends to zero. As the result, by the version of the method of formal asymptotic expansions, we derive a closed limit model in which the inhomogeneity is thin (of zero width). After this, using the Galerkin method and the classical techniques of derivation of energy estimates, we prove existence, uniqueness, and stability of a weak solution to this model.

Asymptotic Representation of Vorticity and Dissipation Energy in the Flux Problem for the Navier–Stokes Equations in Curved Pipes

This study explores the problem of describing viscous fluid motion for Navier–Stokes equations in curved channels, which is important in applications like hemodynamics and pipeline transport. Channel curvature leads to vortex flows and closed vortex zones. Asymptotic models of the flux problem are useful for describing viscous fluid motion in long pipes, thus considering geometric parameters like pipe diameter and characteristic length. This study provides a representation for the vorticity vector and energy dissipation in the flow problem for a curved channel, thereby determining the magnitude of vorticity and energy dissipation depending on the channel’s central line curvature and torsion. The accuracy of the asymptotic formulas are estimated in terms of small parameter powers. Numerical calculations for helical tubes demonstrate the effectiveness of the asymptotic formulas.

Small Darcy number limit of the Navier–Stokes–Darcy system

We study the small Darcy number behavior of the Navier–Stokes–Darcy system with the conservation of mass, Beavers–Joseph–Saffman–Jones condition, and the Lions balance of the normal-force interface boundary conditions imposed on the interface separating the Navier–Stokes flow and Darcy flow. We show that the asymptotic behavior of the coupled system, at small Darcy number, can be captured by two semi-decoupled Darcy number independent sequences: a sequence of (linearized) Navier–Stokes equations, and a sequence of Darcy equations with appropriate initial and boundary data. Approximate solutions to any order of the small parameter (Darcy number) can be constructed via the two sequences. The local in time validity of the asymptotic expansion up to second order is presented. And the global in time convergence is derived under the assumption that the Reynolds number is below a threshold value.

A new treatment for breast cancer using a combination of two drugs: AZD9496 and palbociclib

In this study, we examined a mathematical model of breast cancer (BC) treatment that combines an oral oestrogen receptor inhibitor, AZD9496 with Palbociclib, a selective inhibitor of cyclin- dependent kinases CDK4 and CDK6. Treatment is described by analytical functions that enable us to control the dosage and time interval of the treatment, thus personalising the treatment for each patient. Initially, we investigated the effect of each treatment separately, and finally, we investigated the combination of both treatments. By applying numerical simulations, we confirmed that the combination of AZD9496 with palbociclib was the optimal treatment for BC. The dosage of AZD9496 increased and decreased throughout the treatment period, while the intervals were constant between treatments. Palbociclib changed almost cyclically, whereas the time intervals remained constant. To investigate the mathematical model, we applied the singularly perturbed homotopy analysis method, which is a numerical algorithm. The significant advantage of this method is that the mathematical model does not have to contain a small parameter (as is standard in perturbation theory). However, it is possible to artificially introduce a small parameter into the system of equations, making it possible to study the model using asymptotic methods.

Linearly stable KAM tori for one dimensional forced Kirchhoff equations with refined Töplitz-Lipschitz property

We prove an abstract infinite dimensional KAM theorem, which could be applied to prove the existence and linear stability of small-amplitude quasi-periodic solutions for one dimensional forced Kirchhoff equations with Dirichlet boundary conditionsutt−(1+∫0π|ux|2dx)uxx+Mξu+ϵg(ω¯t,x)=0,u(t,0)=u(t,π)=0, where Mξ is a real Fourier multiplier, g(ω¯t,x)=−g(ω¯t,−x) is real analytic with forced Diophantine frequencies ω¯, ϵ is a small parameter. The proof is based on an improved Kuksin lemma and the off-diagonal decay together with the refined Töplitz-Lipschitz property of the forcing term.

General theory of radiative transfer in a magnetized atmosphere with scattering by electrons

ABSTRACT Here, we consider the general radiative transfer theory in a magnetized atmosphere for any value of parameter $x=\omega _B/\omega \simeq 0.933\times 10^{-8}\lambda (\mu \mathrm{m}) B(\text{G})$, where ωB is the cyclotron frequency of electron rotation and ω is the angular frequency of considered monochromatic radiation. The main term of the radiative transfer equations $\textbf {J}_{\alpha \beta }$ for the Stokes parameters I, V, U, and Q describes the scattering of radiation coming from all directions and distances. All Stokes parameters of the incident radiation mutually transform into each other along their path due to interference and different cross-sections for them. To find this transformation of the Stokes parameters one has to solve the complex system of transfer equations without the sources and term $\textbf {J}_{\alpha \beta }$. This is done in our paper. First, we present the general solution and then give the solution for the case of a homogeneous magnetic field, where the formulas have clear algebraic form. We note that for small parameter x our formulas describe the known Faraday rotation. Our formulas allow us to derive an integral equation for the density of polarized radiation, multiple scattered in a magnetized atmosphere for any values of the parameter x. The obtained correct radiation transfer equation allows us to calculate the Stokes parameters of radiation emerging from an atmosphere, in particular, for the Milne problem.

A novel beam model suitable at the microscale

A novel beam model is proposed to address the limitations of existing microscale beam models by introducing a higher-order displacement field assumption for the modified couple stress theory. The governing equations and boundary conditions are derived using the variational principle. Validation through finite element analysis for a simply supported beam problem demonstrates the beam model successfully captures the width-dependent behavior of microbeams, which existing beam models cannot explain. Additionally, the beam model accurately represents the plane strain behavior as the beam width increases and the classical beam behavior at a small length scale parameter.

Invariants of magnetic lines for Yang-Mills solutions

We construct a new Yang-Mills 3D-solution on the space of negative scalar curvature. We discuss a problem of non-abelian gauge symmetry is broken with the assumption that a scalar curvature of the domain is a negative small parameter. In this case we use the following fact: a geometrical scale related with Vassiliev's discriminant of magnetic lines coincides with a phisical Kolmogorov scale. This gives an estimation of α-effect by the dispersion of the asymptotic ergodic Hopf invariant in the asymptotic limit with a negative scalar curvature parameter.

Prediction method of longitudinal surface settlement caused by double shield tunnelling based on deep learning

The deep learning method faces the challenges of small sample data and high dimensional shield operational parameters in predicting the longitudinal surface settlement caused by shield excavation. In this study, various optimization algorithms were compared, and the slime mould algorithm (SMA) was optimally chosen to optimize the hyperparameters of random forest (RF), and SMA-RF was used for dimensionality reduction and feature contribution analysis. A double-input deep neural network (D-DNN) framework was proposed for the prediction of surface settlement, which considers the influence of twin tunnels and effectively increases the high-fidelity data in the database. The results show that SMA performs best among various optimization algorithms; employing features that have a cumulative contribution value exceeding 90% as input can result in high prediction accuracy; there is significant uncertainty in the feature contribution analysis for small sample data; the reduced shield running parameters show a strong nonlinear relationship with surface settlement; compared with S-DNN, D-DNN takes into account the excavation of twin tunnels and expands the database capacity by more than 1.5 times, with an average increase of 27.85% in the R2 and an average decrease of 53.2% in the MAE.

The Dual Approach to Optimal Control in the Coefficients of Nonlocal Nonlinear Diffusion

We derive the dual variational principle (principle of minimal complementary energy) for the nonlocal nonlinear scalar diffusion problem, which may be viewed as the nonlocal version of the p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p$$\\end{document}-Laplacian operator. We establish existence and uniqueness of solutions (two-point fluxes) as well as their quantitative stability, which holds uniformly with respect to the small parameter (nonlocal horizon) characterizing the nonlocality of the problem. We then focus on the nonlocal analogue of the classical optimal control in the coefficient problem associated with the dual variational principle, which may be interpreted as that of optimally distributing a limited amount of conductivity in order to minimize the complementary energy. We show that this nonlocal optimal control problem Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}-converges to its local counterpart, when the nonlocal horizon vanishes.

Global non-relativistic quasi-neutral limit for a two-fluid Euler-Maxwell system

This paper concerns a two-fluid Euler-Maxwell system for plasmas with two small parameters. We study the non-relativistic quasi-neutral limit for smooth solutions of the system in whole space R3. When the initial data are smooth and sufficiently close to constant equilibrium states, we give a rigorous justification of the limit for all time. The limit system is governed by a compressible Euler system. The proof is based on uniform energy estimates and various dissipative estimates of smooth solutions with respect to the parameters and time. A key step in these estimates is to control the quasi-neutrality of the velocities by using a projection operator.