Nonlinear Form Research Articles

Sensitivity of the Ramsey’s Regression Specification Error Term Test on the Degree of Nonlinearity of the Functional Form

The present paper aims to demonstrate that the Ramsey’s Regression Specification Error Term Test (RESET) is very sensitive to the degree of nonlinearity between the variables of the under-specification functional form. This widely used test, for testing the functional specification of a model, is based on the notion that if nonlinear combinations of the explanatory variables have any power in explaining the predictor, the model is mis-specified and the data generating mechanism might be approximated by a nonlinear functional form. Using Monte Carlo techniques, we find that the power of the Ramsey’s RESET test is highly influenced and related with the degree of nonlinearity between the dependent and the independent variables of the under-specification functional form.

Retrieval of Optical Solitons with Anti-Cubic Nonlinearity

Purpose: In this article, two main subjects are discussed. First, the nonlinear Schrödinger equation (NLSE) with an anti-cubic (AC) nonlinearity equation is examined, which has a great working area, importance and popularity among the study areas of soliton behavior in optical fibers, by using the enhanced modified extended tanh expansion method and a wide range of optical soliton solutions is obtained. Second, the effects of AC parameters on soliton behavior are examined for each obtained soliton type. Methodology: In order to apply the method, the non-linear ordinary differential equation form (NLODE) of the investigated NLSE-AC is obtained by applying the defined wave transformation. Then, with the help of the proposed algorithm for the NLODE form, polynomial form, an algebraic equation system is obtained by setting the coefficients of this form to zero, and the solution of this system is also obtained. After determining the suitable solution set, the optical soliton solution of the investigated problem is obtained with the help of the serial form of the proposed method, a Riccati solution and wave transform. After checking that the solution satisfies the investigated problem, 3D and 2D graphics are obtained for the special parameter values and the necessary comments are made in the relevant sections. Findings: With the proposed method, many optical soliton solutions, such as topological, anti-peaked, combined peaked-bright, combined anti-peaked dark, singular, combined singular-anti peaked, periodic singular, composite kink anti-peaked, kink, periodic and periodic, with different amplitudes are obtained, and 3D and 2D representations have been made. Then, the effect of AC parameters on the soliton behavior in each case has been successfully studied. It has been shown that AC parameters have a significant effect on the soliton behavior, and this effect changes depending on the soliton shape and the parameters. Moreover, providing and maintaining the delicate balance between the soliton shape and the parameters and the interaction of the parameters with each other involves great difficulties. Originality: Although some soliton types of the NLSE-AC equation have been presented for the first time in this study, there is no study in the literature showing the effect of AC parameters on soliton behavior, especially for the NLSE-AC equation.

Inflation expectations and nonlinearities in the Phillips curve

SummaryThis paper examines the presence of nonlinearities in the Phillips curve. We allow for a flexible form of nonlinearity and estimate a threshold regression model with the number and location of thresholds determined directly from the data. Over the estimation period starting in the late 1960s, we document that the linear model cannot be rejected if we properly control for inflation expectations. More precisely, not controlling for consumer expectations may lead the econometrician to overestimate the degree of nonlinearity. Our results hold with aggregate data, regional data, and controlling for cost‐push shocks directly or using instrumental variables.

Critical Review and Discussion of the Nonlinear Form of Radke–Prausnitz Model in Adsorption Solid–Liquid Phases

The Radke–Prausnitz model, a typical three-parameter model belonging to the Langmuir–Freundlich-type equation, is increasingly being used to correlate the equilibrium data of various adsorption processes. However, very few published works provide the correct information on (1) the nonlinear form of this model, (2) the units of its parameters, and (3) its original citation. This work provides a critical analysis of the Radke–Prausnitz model with special emphasis on promoting its proper application in adsorption research. The correct expression and citation of the models used in adsorption studies are provided. The correct units of its parameters aRP, bRP, and β are L/kg, (mol/kg)/(mol/L)β, and unitless when the solution phase concentration Ce is given in mol/L and the adsorbed phase concentration qe is defined in mol/kg. The Radke–Prausnitz model can be reduced to the Henry, Freundlich, Langmuir, or Redlich–Peterson model in some special cases. The Radke—Prausnitz model is valid when the exponent β is in the range of 0≤β≤1. A new adsorption equilibrium constant of this model is introduced. The feasibility of using the new equilibrium constant to calculate the thermodynamic parameters (ΔG°, ΔH°, and ΔS°) of adsorption processes is discussed herein.

On the investigation of chiral solitons via modified new Kudryashov method

Purpose: This study includes the examination of the cases where the [Formula: see text]-dimensional chiral nonlinear Schrödinger equation also has Bohm potential. This review is not to obtain different soliton solutions for both cases but to obtain a certain type of soliton and to observe the effect of the problem parameters. By using the modified new Kudryashovs scheme. This observation also includes how the soliton behavior is effective by comparing the [Formula: see text]-dimensional chiral nonlinear Schrödinger equation (C-NLSE) and [Formula: see text]-dimensional chiral nonlinear Schrödinger equation with Bohm potential (C-NLSE-BP), especially by examining the Bohm potential parameter. Methodology: In order to apply the proposed analytical method which is the modified Kudryasovs scheme (m-NKM), as in many studies, the nonlinear partial ordinary differential equation (NLPDE) is first converted into nonlinear ordinary differential equation form (NLODE) by using wave transform. Then, in order to determine which degree the solution function to be proposed will be, the balancing constant is calculated. The next step is to determine the unknown parameters of the problem by applying the m-NKM on NODE, obtaining solution sets, and combine the solution of the Riccati equation, which is the basis of the method, with the proposed solution function and wave transform. Obtaining the optical solution by providing the main NLPDE is the next step that follows this stage. The final stage is the graphical analysis and interpretation of the parameter effect for both problems using the obtained solution function. Findings: The examination of the case with and without Bohm potential was carried out on the behavior of the bright soliton shape, which is one of the basic soliton shapes that many effects can be clearly studied on it. It was successfully shown that both equation parameters and the Bohm potential had a significant effect on the soliton behavior, graphical presentations were made and interpreted in detail. Originality: Such an examination has not been studied before in the literature for the investigated equations.

Hamiltonian formalism for nonlinear Schrödinger equations

We study the Hamiltonian formalism for second and fourth order nonlinear Schrödinger equations. In the case of the second order equation, we consider cubic and logarithmic nonlinearities. Since the Lagrangians generating these nonlinear equations are degenerate, we follow the Dirac–Bergmann formalism to construct their corresponding Hamiltonians. In order to obtain consistent equations of motion, the Dirac–Bergmann formalism imposes some set of constraints that contribute to the total Hamiltonian along with their Lagrange multipliers. The order of the Lagrangian degeneracy determines the number of primary constraints. If a constraint is not a constant of motion, a secondary constraint is introduced to force the consistency condition. We show that for second order and fourth order nonlinear Schrödinger equations we only have primary constraints, and the form of nonlinearity or the order of derivatives does not change the constraint dynamics of the system. However, we observe that introducing new fields to treat higher derivatives in the Lagrangians of these equations changes the constraint dynamics, and secondary constraints are needed to construct a consistent set of Hamilton equations.

Critical brain wave dynamics of neuronal avalanches.

Analytical expressions for scaling of brain wave spectra derived from the general non-linear wave Hamiltonian form show excellent agreement with experimental "neuronal avalanche" data. The theory of the weakly evanescent non-linear brain wave dynamics reveals the underlying collective processes hidden behind the phenomenological statistical description of the neuronal avalanches and connects together the whole range of brain activity states, from oscillatory wave-like modes, to neuronal avalanches, to incoherent spiking, showing that the neuronal avalanches are just the manifestation of the different non-linear side of wave processes abundant in cortical tissue. In a more broad way these results show that a system of wave modes interacting through all possible combinations of the third order non-linear terms described by a general wave Hamiltonian necessarily produces anharmonic wave modes with temporal and spatial scaling properties that follow scale free power laws. To the best of our knowledge this has never been reported in the physical literature and may be applicable to many physical systems that involve wave processes and not just to neuronal avalanches.

A new approach to model a system with both friction and geometric nonlinearity

This paper presents a new reduced-order model that can capture the nonlinear dynamics of a structure containing both friction and geometric nonlinearity, and proposes a procedure to derive the parameters of the model from quasi-static simulations. The Single-mode Implicit Condensation and Expansion (SICE) method was recently shown to be capable of capturing the resonant behavior of a geometrically nonlinear structure, and the modal Iwan model has proven effective at capturing the hysteretic behavior of a mode that exhibits nonlinearity due to friction. This paper combines the two into a new model, referred to here as the Iwan model with Geometric Nonlinearity (or the IGNL model). It consists of an Iwan model with an additional spring-slider unit, where the slider has infinite strength and the spring consists of polynomial terms that define the SICE-ROM. A procedure is also proposed to use Quasi-Static Modal Analysis (QSMA) to estimate the parameters of the IGNL model from a set of nonlinear force–displacement curves. Existing literature shows how QSMA can be used to characterize the force–displacement behavior of a nonlinear mode where the nonlinearity is due to friction or bending–stretching coupling, but not both simultaneously. This paper proposes certain adjustments to the QSMA procedure so that the two forms of nonlinearity can be isolated. Two case studies are presented to test the efficacy of the IGNL model, one consisting of a simple spring–mass model and the second, a finite element model containing both contact and geometric nonlinearity. The proposed method is shown to be significantly faster than performing dynamic simulations to estimate the amplitude-dependent frequency and damping behavior.

Nonlinear function-on-scalar regression via functional universal approximation.

We consider general nonlinear function-on-scalar (FOS) regression models, where the functional response depends on multiple scalar predictors in a general unknown nonlinear form. Existing methods either assume specific model forms (e.g., additive models) or directly estimate the nonlinear function in a space with dimension equal to the number of scalar predictors, which can only be applied to models with a few scalar predictors. To overcome these shortcomings, motivated by the classic universal approximation theorem used in neural networks, we develop a functional universal approximation theorem which can be used to approximate general nonlinear FOS maps and can be easily adopted into the framework of functional data analysis. With this theorem and utilizing smoothness regularity, we develop a novel method to fit the general nonlinear FOS regression model and make predictions. Our new method does not make any specific assumption on the model forms, and it avoids the direct estimation of nonlinear functions in a space with dimension equal to the number of scalar predictors. By estimating a sequence of bivariate functions, our method can be applied to models with a relatively large number of scalar predictors. The good performance of the proposed method is demonstrated by empirical studies on various simulated and realdatasets.

Anisotropy-assisted bias-free spin Hall nano-oscillator

Ferromagnet/nonmagnet (FM/NM) bilayer-based spin Hall nano-oscillators (SHNOs)—a sub-class of spintronic oscillator devices—have promising potential toward realizing low-power physical reservoir computing systems because of their inherent nonlinearity and miniature form factor. However, most of the studies on SHNOs indicate that an external biasing magnetic field is necessary for their operation, creating a bottleneck for their practical implementation in designing small and compact RC hardware. In this report, using micromagnetic simulation, we demonstrate biasing field-free operation of a FM/NM bilayer-based SHNO by exploiting the magnetic anisotropy. Our results reveal that the magnetic anisotropy in the FM layer provides active control over the DC tunability of auto-oscillation frequency and the threshold value of current needed for sustained auto-oscillations. We show that the increase in uniaxial anisotropy substantially modifies the spatial profile of auto-oscillation and eventually leads to the reduction in the threshold current for auto-oscillation, which could be utilized to design low-power computing hardware using SHNO devices.

ON THE NONLINEARITY OF DEFORMATION OF REINFORCED CONCRETE FLEXURAL ELEMENTS

It is shown that in most calculations of reinforced concrete elements, the descending branch of the concrete deformation diagram should not be taken into account. This is due to the fact that in the calculation it is impossible to exceed the value of the stress in concrete, the value of the design resistance. It is shown that the "moment-curvature" dependence has a linear character both before the formation of cracks and after the formation of cracks. At the same time, calculations using software systems are greatly simplified with almost no loss of accuracy. The use of nonlinear methods for determining the stress-strain state in design leads to significant difficulties. In addition, these methods do not add accuracy to the calculation, but lead to ill-conditioned design schemes. The ascending branch of the concrete diagram can be approximated by a 63 straight line, which greatly simplifies the calculations without a noticeabledecrease in accuracy. The use of a non-linear form of the concrete diagram complicates the calculations, especially when used in software systems. This is because the roots of any equation greater than the first degree have two or more values for the same value of moment or stress. Therefore, it is difficult to choose the desired root in the software package. The use of a linear function always has a unique solution, while the accuracy decreases quite slightly. The statement about the linear relationship "moment-curvature" is also proved by the approach that is adopted in the US standards. If we pass from the formula for determining the moment of inertia of an element with cracks to the “moment-curvature” dependence, then we can see that this dependence is linear. The article shows how to use linear dependencies in the calculations of beam elements and slabs. The proposed approach to the design of reinforced concrete slabs using the finite element method involves changing the rigidity of the finite element depending on the presence or absence of cracks. In this case, the stiffness can have only two values, which are approximated by linear dependencies

Some new soliton solutions to the (3 + 1)-dimensional generalized KdV-ZK equation via enhanced modified extended tanh-expansion approach

In this article, The (3 + 1)-dimensional generalized Korteweg-de-Vries-Zakharov-Kuznetsov equation (gKdV-ZKe) which explains the influence of the magnetic field on the weak nonlinear ion-acoustic waves investigated in the field of plasma conjured up including both cold and hot electrons. GKdV-zk techniques solutions are obtained using the improved modified extended tanh expansion method, which is one of the most efficient algebraic methods for obtaining accurate solution to nonlinear partial differential equations. We aim to show how the analyzed model’s parameter impact soliton behavior by choosing different bright and single soliton forms and by developing various analytical optical soliton solutions for the explored equation. MethodologyIn order to apply the suggested method, we used a complex wave transform to derive the nonlinear ordinary differential form of the analyzed equation. Then, using the method, we were able to obtain the polynomial form, leading to a set of linear equations. The conclusion of solving the linear equations problem, the outcomes of the analyzed model, and the suggested strategy are all included in different solution sets. After choosing the appropriate set from these sets, using the solution functions, and utilizing the wave transformation provided by the approach, we were able to arrive at the optical soliton solutions by providing the central equation. FindingThe proposed method has successfully produced a number of soliton solutions and several analytical optical solutions to such model. The research shows that the parameters of the model may have a variety of effects on the behavior of solitons, categories based on the soliton type. The findings we get in this article can be used to research and compare numerical and experimental data with analytical solving problems in plasma physics. OriginalityThis study differs from others in that it assessed the impact that parameters of the model have on the actions of solitons, despite the fact that the proposed technique was applied for the first time on the topic under investigation and numerous soliton types were created. This study focuses on the influence of model parameters on solitons behavior.

Optical solitons in birefringent fibers with the generalized coupled space–time fractional non-linear Schrödinger equations

The nonlinear Schrödinger (NLS) equation is an ideal model for describing optical soliton transmission. This paper first introduces an integer-order generalized coupled NLS equation describing optical solitons in birefringence fibers. Secondly, the semi-inverse and fractional variational method is used to extend the integer‐order model to the space–time fractional order. Moreover, various nonlinear forms of fractional NLS equations are discussed, including the Kerr, power, parabolic, dual-power, and log law. The exact soliton solutions, such as bright, dark, and singular solitons, are given. Finally, the behavior of the solution is shown by three-dimensional figures with different fractional orders, which reveals the propagation characteristics of optical solitons in birefringence fibers described by the generalized coupled space–time fractional NLS equation.

Green-Lazarsfeld index of square-free monomial ideals and their powers

Let K be a field and I be a square-free monomial ideal in the polynomial ring K[x1,…,xn]. The Green-Lazarsfeld index, index(I), counts the number of steps to reach to a syzygy minimally generated by a nonlinear form in a graded minimal free resolution of I. In this paper, we study this invariant for I and its powers from a combinatorial point of view. We characterize all square-free monomial ideals I generated in degree 3 such that index(I)>1. Utilizing this result, we also characterize all square-free monomial ideals generated in degree 3 such that index(I)>1 and index(I2)=1. In case n≤5, it is shown that index(Ik)>1 for all k if I is any square-free monomial ideal with index(I)>1.

Computational Mathematics: Solving Dual Fully Fuzzy Nonlinear Matrix Equations Numerically using Broyden’s Method

Fuzzy numbers have many applications in various mathematical models in both linear and nonlinear forms. In the form of a nonlinear system, fuzzy nonlinear equations can be constructed in the form of matrix equations. Unfortunately, the matrix equations used to solve the problem of dual fully fuzzy nonlinear systems are still relatively few found in the publication of research results. This article attempts to solve a dual fully fuzzy nonlinear equation system involving triangular fuzzy numbers using Broyden’s method. This article provides the pseudocode algorithm and the implementation of the algorithm into the MATLAB program for the iteration process to be carried out quickly and easily. The performance of the given algorithm is the fastest in finding system solutions and provides a minimum error value.

Three-dimensional solid particle self-assembly in thermovibrational flow: The case with unidirectional temperature gradient and concurrent vibrations

As a follow-up to earlier work [Crewdson and Lappa, “Spatial and temporal evolution of three-dimensional thermovibrational convection in a cubic cavity with various thermal boundary conditions,” Phys. Fluids 34, 014108 (2022)], where the main focus was on the modes of convection in a three-dimensional cubic enclosure filled with a Pr = 7 liquid undergoing vibrations in a direction “parallel” to the imposed temperature gradient, the present study considers the modes of particle clustering, which occur when solid spheres, with density ratio ξ = 1.85 or 0.3 and Stokes number (St) between 0.5 and 3.5 × 10−5, are added to the fluid. Starting from a uniform distribution of solid particles and fluid in quiescent conditions, the governing equations for the involved phases are numerically solved in their complete, time-dependent, and non-linear form for a representative vibrational Rayleigh number (8.34 × 104), angular frequency Ω = 50, and non-dimensional acceleration amplitude (γ) spanning the interval 0.4 × 107 ≤ γ ≤ 3.4 × 107. It is shown that, while for relatively high values of St and/or γ, only degenerate states are obtained, where all particles collapse on planar structures, for intermediate values of such parameters, interesting (heretofore unseen) patterns are enabled. The hallmark of these phenomena is an endless squeezing and expansion of the particle formations along the direction of the temperature gradient. As confirmed by the numerical simulations, the underlying formation mechanisms rely on the combined action of the body force acting on particles due to their different densities with respect to the host fluid and the additional drag that is produced when the carrier thermovibrational flow enters a specific stage, known as “convective burst,” where the magnitude of the fluid velocity increases dramatically.

Deformation analysis of buried subsea pipeline under different pressure-reverse​ fault displacement loading paths

The combined action of subsea seismic faults and pressure loads can cause pipelines to reach the limit state. Consequently, various failure modes, such as global buckling and system collapse, can occur. In this study, the deformation of a buried subsea pipeline under different pressure-reverse fault displacement loading paths is investigated using the nonlinear vector form intrinsic finite element (VFIFE). To quantify the damage to the pipeline and identify its failure modes under accidental reverse fault displacement, the loading path ‘fault displacement → internal/external pressure’ was analysed first. The post-earthquake resumption and reconditioning of unpressurised and pressurised pipelines, respectively, are evaluated based on the performance criteria in terms of strain and deformation. The corresponding loading paths are simplified as ‘ambient hydrostatic pressure → fault displacement → operation pressure’ and ‘operation pressure → fault displacement → external over pressure’. Moreover, the residual bearing capacity, weak position, and possible failure behaviour of the pipeline are comprehensively analysed for all subsequent external overpressures. The results show that the loading paths have a significant effect on structural responses, particularly on the elasto-plastic deformation. The two yield bends of a deformed pipeline under fault displacement are dangerous regions. Their inadequate relative residual strength leads to several successive local collapses and determines the occurrence and direction of buckling propagation during the subsequent external overpressure. The previous internal pressure has a significant influence on the residual capacity and failure modes of the pipeline, particularly the strain hardening effects with the influence of large seismic faults. The critical points of the first local collapse have specific characteristics, and the minimum residual resistance to external over pressure occurs when the maximum flattening parameter is 15% under initial fault displacement. The results of this study can guide the development of seismic design methodologies and engineering practice for buried subsea pipelines.

Higher-Order Dispersive and Nonlinearity Modulations on the Propagating Optical Solitary Breather and Super Huge Waves

The nonlinearity form of the Schrödinger equation (NLSE) gives a sterling account for energy and solitary transmission properties in modern communications with optical-fiber energ- reinforcement actions. The solitary representation during fiber transmissions was regulated by NLSE coefficients such as nonlinear Kerr, evolutions, and dispersions, which controlled the energy changes through the model. Sometimes, the energy values predicted from the NLSEs computations may diverge due to variations in the amplitude and width caused by scattering, dispersive, and dissipative features of fiber materials. Higher-order nonlinear Schrödinger equations (HONLSEs) should be explored to alleviate these implications in energy and wave features. The unified solver approach is employed in this work to evaluate the HONLSEs. Steepness, HO dispersions, and nonlinearity self-frequency influences have been taken into consideration. The energy and solitary features were altered by higher-order actions. The unified solver approach is employed in this work to reform the HONLSE solutions and its energy properties. The steepness, HO dispersions, and nonlinearity self-frequency influences have been taken into consideration. The energy and soliton features in the investigated model were altered by the higher-order impacts. Furthermore, the new HONLSE solutions explain a wide range of important complex phenomena in wave energy and its applications.

Optimal Control of Dynamical Systems using Calculus of Variations

This paper explores the application of calculus of variations techniques for optimizing the control of complex dynamical systems. Determining control strategies that minimize cost metrics and satisfy constraints is critical across engineering disciplines like robotics, aerospace, and process operations. Classical optimal control methods, such as Pontryagin's Maximum Principle, transform an optimal control problem into a calculus of variations framework amenable to analytical and numerical optimization. We present a unified framework for applying these techniques, enabling dynamical systems defined by ordinary and partial differential equations to be optimized by conversion into a nonlinear programming form. Analytical approaches provide theoretical guarantees on control performance while numerical methods, like direct collocation and transcription, enable large-scale optimal control problems to be efficiently solved. The connections between dynamical systems, calculus of variations, and modern numerical optimization methods establish a holistic methodology for control engineers and applied mathematicians to design optimal controllers for physical systems. Case studies on real-time trajectory optimization, adaptive path planning, and dynamic process operations demonstrate the efficacy of the proposed optimal control framework across robotics, aerospace, power systems, and other applications.

Optimal Control of Dynamical Systems using Calculus of Variations

This paper explores the application of calculus of variations techniques for optimizing the control of complex dynamical systems. Determining control strategies that minimize cost metrics and satisfy constraints is critical across engineering disciplines like robotics, aerospace, and process operations. Classical optimal control methods, such as Pontryagin's Maximum Principle, transform an optimal control problem into a calculus of variations framework amenable to analytical and numerical optimization. We present a unified framework for applying these techniques, enabling dynamical systems defined by ordinary and partial differential equations to be optimized by conversion into a nonlinear programming form. Analytical approaches provide theoretical guarantees on control performance while numerical methods, like direct collocation and transcription, enable large-scale optimal control problems to be efficiently solved. The connections between dynamical systems, calculus of variations, and modern numerical optimization methods establish a holistic methodology for control engineers and applied mathematicians to design optimal controllers for physical systems. Case studies on real-time trajectory optimization, adaptive path planning, and dynamic process operations demonstrate the efficacy of the proposed optimal control framework across robotics, aerospace, power systems, and other applications.