Small Parameter Research Articles

An integral for self inductance of thin wires

Abstract I propose an elegant method of calculating the self inductance of a thin wire of arbitrary shape. The inductance is broken to into two parts - the inductance of a straight wire of the same length and cross section, and the remainder, the “curve inductance” due to the shape of the axial curve. The curve inductance is finite and can be calculated from a double integral by subtracting the straight wire singularity inside the integrand. Unlike all other methods, this does not involve a small parameter, and can be applied to any curve knowing its arc length parameterization. We use the circular arc and helix as examples and find that the integral is fast and accurate. The limits of the approximation are investigated for touching parallel wires.

Using Delay for Control

This article reviews two techniques that use delay for control: time-delay approaches to control problems (which initially may be free of delays) and the intentional insertion of delays into the feedback. We begin with a now widely used time-delay approach to sampled-data control. In networked control systems with communication constraints, this is the only method that accommodates transmission delays larger than the sampling intervals. We present a predictor-based design that enlarges the maximum allowable delay, which is important for practical implementations. We then discuss methods that use artificial delays via simple Lyapunov functionals that lead to feasible linear matrix inequalities for small delays and simple sampled-data implementations. Finally, we briefly present a new time-delay approach—this time to averaging. Unlike previous results, this approach provides the first quantitative bounds on the small parameter, making averaging-based control (including vibrational and extremum-seeking control) reliable.

Nonlinear Rayleigh wave propagation in a three-layer sandwich structure in dual-phase-lag

Plane, nonlinear Rayleigh wave propagation is investigated in a three-layer sandwich structure of a thermoelastic medium, within the frame of dual-phase-lag theory. The thermal conductivity is taken as a linear function of temperature. This induces nonlinearity in the evolution equations for the heat flux components. A particular solution is found in the form of Poincaré expansion in a small parameter that reflects the fluctuations of temperature about a steady value. This solution is discussed and plots are provided for a special case when both external faces of the structure are traction-free and under prescribed temperature regime. It is noted, in particular, that some quantities of practical interest suffer jumps at the interfaces. Some of the jumps appear only starting from the second order of approximation. The existence of jumps may be favourably used to carry out some measurements of material parameters.

Relationship between rheological parameters and structure formation in high moisture extrusion of plant protein biopolymers

High moisture extrusion (HME) is widely employed to texturize plant-derived protein ingredients, but its development is still much based on trial and error. In this project, it was hypothesized that the rheological properties of plant-based mixtures analyzed when in the molten state and during their cooling can aid in understanding their structure formation. To test this hypothesis, biopolymer formulations containing pea protein isolate (PPI) were examined at different moisture and starch contents, and their rheological properties were analyzed using a closed cavity rheometer (CCR) at temperatures relevant to those applied during HME processing. The results obtained with the CCR were then contrasted with the mechanical properties of HME obtained using a lab scale extruder, measured using oscillatory rheology, large deformation, and dynamic mechanical analysis. Low moisture (55 %) HMEs were stiff and brittle, while high moisture (65 %) HMEs formed more flexible and anisotropic structures. Addition of starch created softer structures. The viscoelastic properties of the biopolymer mixes measured with the CCR were correlated with the mechanical parameters of the final extrudates. Strong correlations were found between small deformation rheological parameters measured in the CCR and the hardness values, while non-linear viscoelastic parameters were correlated with anisotropy indexes. Results demonstrated that the material properties measured at the early cooling stages strongly influence the structural heterogeneity in HMEs. This study highlights the potential to use the viscoelastic properties of the biopolymer mix measured with the CCR to predict their structural features when processed by extrusion.

Solution of the problem with a small parameter in suspension filtration in a porous medium

Relevance. The necessity to calculate the changes in filtration properties during injection of suspensions for subsequent forecasting of technological parameters of wells. This calculation is complicated by the presence of different-scale effects, since the penetration depth of dispersed particles is orders of magnitude smaller than the characteristic dimensions of the formation. These effects have not been studied in detail before. Aim. To analyze the effect of a small parameter on the behavior of flow characteristics using a mathematical model of suspension filtration in a porous medium. Objects. Colmatation coefficient, filtration coefficient, distribution of concentration of dispersed particles in the formation, porosity, mathematical model of deep-bed suspension migration into a porous medium. Methods. Numerical modeling, explicit finite-difference scheme, solution of a problem with a small parameter, introduction of dimensionless parameters, method of characteristics. Results and conclusions. It is shown that the processes of conformance control and colmatation of a porous medium are described within the framework of a unified system of equations of deep-bed suspension migration into a porous medium. It is identified that the concentration of retained particles is a small parameter that allows reducing the complete system of equations of deep-bed suspension migration into a porous medium to a simplified form, in which the solution can be obtained analytically using the method of characteristics. The authors compared the solutions of the complete and simplified systems of equations of deep-bed suspension migration into a porous medium, indicating their compliance with an error of less than 4 %. They obtained the distributions of the concentration of dispersed particles and porosity in the reservoir during colmatation, showing that over time the colmatation front propagates at a constant rate. It is shown that the colmatation coefficient is a small parameter that significantly determines the nature of oil displacement by water, and a decrease in the colmatation coefficient leads to a decrease in the rate of colmatation and the appearance and growth of the size of the stabilized zone near the displacement front.

Broadband spin-dependent anti-reflection in chiral time-varying metamaterials

Time-varying metamaterials have garnered significant attention for their ability to achieve anti-reflection in the time domain. However, current systems face limitations in spin-controlled manipulation, as most studies focus on non-chiral, time-varying metamaterials. Consequently, realizing spin-dependent broadband anti-reflection using time-varying chiral metamaterials remains underexplored. In this work, we propose a time-varying chiral structure composed of four temporal layers, each with distinct impedances and chiral parameters. By carefully adjusting these parameters across the layers, our structure enables broadband anti-reflection for both right- and left-circularly polarized (RCP and LCP) waves under small chiral conditions. Under large chiral parameters, the structure selectively achieves broadband anti-reflection for LCP waves, while consistently reflecting RCP waves across the bandwidth. This unique spin-dependent broadband anti-reflection results from significant phase delays between RCP and LCP waves, a feature not achievable by non-chiral, time-varying multilayer structures. Additionally, the proposed structure allows impedance matching between chiral and non-chiral dielectric spatial-temporal slabs in finite regions under small chiral parameters. These findings offer promising avenues for advanced wave manipulation in chiral metamaterials, with potential applications in broadband absorbers, filters, and quantum information processing systems.

Visualizemi: Visualization, Effect Size, and Replication of Measurement Invariance for Registered Reports.

Latent variable modeling as a lens for psychometric theory is a popular tool for social scientists to examine measurement of constructs. Journals, such as Assessment regularly publish articles supporting measures of latent constructs wherein a measurement model is established. Confirmatory factor analysis can be used to investigate the replicability and generalizability of the measurement model in new samples, while multigroup confirmatory factor analysis is used to examine the measurement model across groups within samples. With the rise of the replication crisis and "psychology's renaissance," interest in divergence in measurement has increased, often focused on small parameter differences within the latent model. This article presents visualizemi, an R package that provides functionality to calculate multigroup models, partial invariance, visualizations for (non)-invariance, effect sizes for models and parameters, and potential replication rates compared with random models. Readers will learn how to interpret the impact and size of the proposed non-invariance in models with a focus on potential replication and how to plan for registered reports.

Synchronization characteristics and stable states of four co-rotating vibrators with balanced slanted elliptical trajectory

In the petroleum drilling and mining screening industries, our goal is to enhance excitation force, integrate circular screening with rectilinear transport to obtain the balanced slanted elliptical trajectory (BSET), and reduce screen mesh clogging. This work investigates the synchronization characteristics and stable states of four co-rotating vibrators. The novelty of this work lies in using a novel vibration model to achieve the BSET in the synchronization state and in further considering the influence of a new skewing angle between two eccentric rotors (ERs), which are few in previous vibration screening field. Initially, the vibration model is extracted from the designed prototype. Then, in accordance with Routh–Hurwitz criterion, the synchronization and stability are theoretically analyzed using small parameters of vibrator velocity and phase differences (PDs). The synchronization characteristics related to mechanical structures are discussed via numerical calculation and experimental verification. Results reveal that the synchronization implementation is adversely affected by the skewing angle, even leading to unordered behaviors. Achieving zero-phase or in-phase synchronization with four vibrators is only feasible with a larger symmetric distance and installation angle. This work can contribute a foundation for designing some new types of large vibration screening and material conveyance machines.

Evaluation of Wellbore Stability by Small Parameter Perturbation Nonlinear Elastoplastic Model

Abstract Wellbore instability is a frequent occurrence during drilling and a popular research topic among petroleum workers. Most of the previous studies simplified the hardening stage and ignored or classified it into the elastic stage. Engineering practice shows that the total stress–strain process of rock can more truly reflect the bearing and deformation characteristics of rock. This study establishes a total stress–strain constitutive model by combining the perturbation index equation with the linear elastic equation. Additionally, the physical model of rock around the well is presented using the total deformation theory for the plastic zone. The stress field of the rock surrounding the well is analyzed based on the physical model of the rock surrounding the well, taking into account the influence of rock strength and geostress state. The relationship between rock strength, geostress state, drilling fluid density, and plastic radius is given. The calculation method of drilling fluid density limit is also given. The practical application demonstrates that the stability of a borehole wall can be evaluated by analyzing the size and direction of the plastic radius, as well as the distribution of radial, tangential, and shear stress for various combinations of stress states and rock strengths. The maximum shear stress is always found in the softening zone, which is also the region where wellbore instability is most likely to happen. Therefore, the area within the softening radius is regarded as an unstable area, and the softening radius can be used to determine the range of wellbore instability.

Medical image segmentation with UNet-based multi-scale context fusion

Histopathological examination holds a crucial role in cancer grading and serves as a significant reference for devising individualized patient treatment plans in clinical practice. Nevertheless, the distinctive features of numerous histopathological image targets frequently contribute to suboptimal segmentation performance. In this paper, we propose a UNet-based multi-scale context fusion algorithm for medical image segmentation, which extracts rich contextual information by extracting semantic information at different encoding stages and assigns different weights to the semantic information at different scales through TBSFF module to improve the learning ability of the network for features. Through multi-scale context fusion and feature selection networks, richer semantic features and detailed information are extracted. The target can be more accurately segmented without significantly increasing the extra overhead. The results demonstrate that our algorithm achieves superior Dice and IoU scores with a relatively small parameter count. Specifically, on the GlaS dataset, the Dice score is 90.56, and IoU is 83.47. For the MoNuSeg dataset, the Dice score is 79.07, and IoU is 65.98.

Analysis of the three‐dimensional elasticity theory problem for a radially inhomogeneous cylinder

AbstractIn this paper an axisymmetric problem of elasticity theory for a radially inhomogeneous cylinder of small thickness is studied. It is assumed that homogeneous mixed boundary conditions are specified on lateral surfaces of the cylinder, and boundary conditions that leave the cylinder in equilibrium are specified on the ends of the cylinder. It is considered that elastic moduli are arbitrary continuous functions of a variable along the radius of the cylinder. The formulated boundary value problem is reduced to a spectral problem containing a small parameter characterizing the thinness of the cylinder walls. To determine its solution, the method of asymptotic integration is used, based on three iteration processes. All solutions of the equilibrium equation are constructed, satisfying the specified homogeneous boundary conditions on lateral surfaces. It is obtained that the solution of the problem consists of a penetrating solution and a solution of the nature of the boundary layer. An analysis of stress‐strain states corresponding to different types of solutions is carried out. Based on the constructed asymptotic solutions, a connection is found between the solutions obtained by the equation of elasticity theory and by the applied theory of shells. It is shown that the penetrating solution determines the internal stress‐strain state of a radially inhomogeneous cylinder. The stress state determined by the penetrating solution is equivalent to the principal vector of stress acting in a cross section perpendicular to the axis of the cylinder.The first terms of asymptotic expansions of the penetrating solution in a small parameter coincide with solutions in the applied theory of shells. New classes of solutions are defined as the character of a boundary layer, which are localized at the ends of the cylinder and decrease exponentially with distance from the ends. These solutions are absent in the applied theories of shells. The first terms of the asymptotic expansion of the solution, having the nature of a boundary layer, are equivalent to the Saint‐Venant edge effect in the theory of heterogeneous plates. Asymptotic formulas for displacement and stresses are constructed, allowing one to calculate the three‐dimensional stress‐strain state of a radially heterogeneous cylinder of small thickness with any predetermined accuracy. Based on the obtained asymptotic expansions, one can estimate the applicability areas of applied theories and construct a refined applied theory for radially heterogeneous cylindrical shells.

Rigid lid limit in shallow water over a flat bottom

AbstractWe perform the so‐called rigid lid limit on different shallow water models such as the abcd Bousssinesq systems or the Green–Naghdi equations. To do so, we consider an appropriate nondimensionalization of these models where two small parameters are involved: the shallowness parameter and a parameter which can be interpreted as a Froude number. When the parameter tends to zero, the surface deformation formally goes to the rest state, hence the name rigid lid limit. We carefully study this limit for different topologies. We also provide rates of convergence with respect to and careful attention is given to the dependence on the shallowness parameter .

Loosely Trapped Surface for Slowly Rotating Black Hole

Abstract We construct the marginal loosely trapped surface (marginal LTS) for the Kerr spacetime with a small Kerr parameter perturbatively, where the LTS condition is saturated. An LTS is a surface that specifies the strong gravity region, which is a generalization of the photon sphere in the Schwarzschild spacetime. It turns out that there are an infinite number of marginal LTSs. At the leading order of the small Kerr parameter, all of the marginal LTSs have the same area. However, one can see that the maximal marginal LTS among them is uniquely determined at the higher order.

Homogenization of a Thermoelastic Bristly Structure Immersed in a Thermofluid

The article considers the mathematical model describing the joint motion of a viscous compressible heat-conducting fluid and a thermoelastic plate with a fine two-level thermoelastic bristly microstructure attached to it. The bristly microstructure consists of a great amount of taller and shorter bristles, which are periodically located on the surface of the plate, and the model under consideration incorporates a small parameter, which is the ratio of the characteristic lengths of the microstructure and the entire plate. Using classical methods in the theory of partial differential equations, we prove that the initial-boundary value problem for the considered model is well-posed. After this, we fulfill the homogenization procedure, i.e., we pass to the limit as the small parameter tends to zero, and, as a result, we derive the effective macroscopic model in which the dynamics of the interaction of the ‘liquid–bristly structure’ is described by equations of two homogeneous thermoviscoelastic layers with memory effects. The homogenization procedure is rigorously justified by means of the Allaire–Briane three-scale convergence method. The developed effective macroscopic model can potentially find application in further mathematical modeling in biotechnology and bionics taking account of heat transfer.

Particulate flow of a viscous fluid driven by a torsionally oscillating disk

Abstract A novel hydrodynamic model is used to study oscillatory flow of a particle-laden fluid driven by a torsionally oscillating disk. The present model is featured by a modified form of Navier-Stokes equations which is conceptually different than existing related models and enjoys relatively concise mathematical formulation. Explicit expressions are derived for the radial, azimuthal and axial velocities using the method of power series expansions of small amplitude parameter, and the derived solutions reduce to Rosenblat's earlier results for a clear fluid driven by a torsionally oscillating disk in the absence of suspended particles. The implications of the present results to dusty gases are discussed in detail with particular interest in the effects of suspended particles on the decay indexes and wavelengths of the induced oscillatory velocity fields. The results obtained in this work can be used to quantify the effects of suspended particles on particulate flow driven by a torsionally oscillating disk.

Control synchronization of the anti-resonance vibration system with two eccentric rotors driven by the motors

The present research works focuses on the control synchronization of anti-resonance vibration system (ARVS) with two eccentric rotors driven by the motors. The small parameters average method and Hamilton’s criterion are employed to analyze the self-synchronous theory of the proposed system from the views of mechanics and mathematics. The research results show that the zero-phase synchronization between two motors is difficult to carry out owing to its own weak coupling property; the ideal dynamic characteristics of the ARVS are easily destroyed by the material quality fluctuation. Therefore, the controllers of the angular speed and phase angle are designed by using the sliding mode control (SMC) algorithm with master-slave control structure. And then Lyapunov theory is used to prove the stability of the controllers. Moreover, some simulation results are given to explore the feasibility and validity of the proposed controllers. The results indicate that the zero-phase synchronization between two motors can be performed by using the proposed control strategy; meanwhile, the ideal dynamic characteristics of the ARVS can be maintained by the proposed controllers with the material quality fluctuation; additionally, a strong robustness of the control system is carried out to reject the structural parametric interferences to the dynamic characteristics. The results can provide a control method for the engineering design of the anti-resonance vibration machine and help anti-resonance vibration machine adapt to more complex working environment.

Gaussian quadrature method with exponential fitting factor for two-parameter singularly perturbed parabolic problem

The parabolic convection-diffusion-reaction problem is examined in this work, where the diffusion and convection terms are multiplied by two small parameters, respectively. The proposed approach is based on a fitted operator finite difference method. The Crank-Nicolson method on uniform mesh is utilized to discretize the time variables in the first step. Two-point Gaussian quadrature rule is used for further discretizing these semi-discrete problems in space, and the second order interpolation of the first derivatives is utilized. The fitting factor’s value, which accounts for abrupt changes in the solution, is calculated using the theory of singular perturbations. The developed scheme is demonstrated to be second-order accurate and uniformly convergent. The proposed method’s applicability is validated by two examples, which yielded more accurate results than some other methods found in the literatures.

Application of blood flow restriction training in lower limb rehabilitation

With the popularity of national fitness campaigns, sports injuries, especially lower limb injuries such as anterior cruciate ligament tears and ankle sprains, are increasing year by year. There are limitations in traditional treatment such as muscle atrophy. In recent years, the blood flow restriction training method (BFRT) which simulates high-intensity exercise with low intensity and limits arterial and venous blood flow to promote muscle growth and strength enhancement, has drawn attention as a novel approach to rehabilitation. This article reviews the application of BFRT in the rehabilitation of lower limb injuries, discusses its effects on muscle function, pain management and quality of life, and evaluates its safety and prospects for clinical application. BFRT reduces mechanical stress, relieves pain, improves endurance and aerobic capacity, and improves cardiovascular function. However, there are discomfort and potential risks, such as muscle injury and thrombosis, which require individualised adjustments and standardised training. BFRT has shown positive result(reducing pain &oedema and improving function)s in rehabilitation of ACL reconstruction (ACLR) and chronic ankle instability (CAI). In addition, BFRT has been shown to aid in the functional recovery of the lower extremity in patients with Achilles tendon injuries and strokes. Despite the positive effects of BFRT, relevant studies have problems such as small sample sizes and non-uniform parameters. BFRT is considered a safe training modality, but contraindications should be examined before clinical application. Future research endeavours should investigate the impact of BFRT in many rehabilitation contexts and enhance tailored rehabilitation strategies. In order to maximize the therapeutic benefit, a rehabilitation training program should be developed and refined with a minimum effective dose based on the quantification of the in vivo physiological response to BFRT. The identification of the optimal prognostic approach will help to advance the standardisation of BFR treatment methods.

A first order in time wave equation modeling nonlinear acoustics

In this paper we focus on a small amplitude approximation of a Navier-Stokes-Fourier system modeling nonlinear acoustics. Omitting all third and higher order terms with respect to certain small parameters, we obtain a first order in time system containing linear and quadratic pressure and velocity terms. Subsequently, the well-posedness of the derived system is shown using the classical method of Galerkin approximation in combination with a fixed point argument. We first prove the well-posedness of a linearized equation using energy estimates and then the well-posedness of the nonlinear system using a Newton-Kantorovich type argument. Based on this, we also obtain global in time well-posedness for small enough data and exponential decay. This is in line with semigroup results for a linear part of the system that we provide as well.

Reply to the Comment on “A New Boson Expansion Theory Utilizing a Norm Operator”

Abstract The comment is based on misunderstandings and, as a result, does not evaluate the norm operator method. We refute the comment. The structure of the norm operator composed of all the excitation modes determines whether the form of a practical boson expansion is a small parameter expansion. KT-3 is chimerical. It should be taken seriously that the small parameter expansions do not hold with truncating the contributions of all phonon excitation modes that are not adopted as boson excitations. The formulation using the boson expansion of the projection operator on to the boson vacuum does not allow the limitation of the phonon excitation number beforehand.