Higher-order Terms Research Articles

The marginality principle revisited: Should "higher-order" terms always be accompanied by "lower-order" terms in regression analyses?

The marginality principle guides analysts to avoid omitting lower-order terms from models in which higher-order terms are included as covariates. Lower-order terms are viewed as "marginal" to higher-order terms. We consider how this principle applies to three cases: regression models that may include the ratio of two measured variables; polynomial transformations of a measured variable; and factorial arrangements of defined interventions. For each case, we show that which terms or transformations are considered to be lower-order, and therefore marginal, depends on the scale of measurement, which is frequently arbitrary. Understanding the implications of this point leads to an intuitive understanding of the curse of dimensionality. We conclude that the marginality principle may be useful to analysts in some specific cases but caution against invoking it as a context-freerecipe.

On soliton solutions for higher-order nonlinear Schrödinger equation with cubic-quintic-septic law

The topic of dispersive optical solitons is one of the essential areas of telecommunications engineering. This paper presents the nonlinear Schrödinger equation (NLSE) with the cubic-quintic-septic (CQS) law which includes higher-order dispersion terms. This equation examines the dynamics of dispersive optical solitons propagation in nonlinear optical fibers. The presented model is significant in mathematics and theoretical physics sciences because long-distance optical fibers are used in optical communication utilized for transoceanic and intercontinental data transmission. First, the considered equation is transformed into a nonlinear ordinary differential equation (NLODE) using the traveling wave transformation. Then, two powerful techniques, Sinh-Gordon expansion approach and the auxiliary equation method (AEM), based on the homogeneous balance rule, are operated to produce analytical solutions for the NLSE with CQS law. Thus, bright, dark, singular, and periodic-singular soliton solutions are acquired. We demonstrate the three-dimensional (3D) and two-dimensional (2D) representations to interpret the physical behavior of some derived solutions. Besides, we investigate the effects of some parameters in the model with graphical illustrations. Therefore, it can be noted that the obtained soliton solutions have a significant role in soliton dynamics.

Charged spherically symmetric black holes in scalar-tensor Gauss–Bonnet gravity

We derive a novel class of four-dimensional black hole (BH) solutions in Gauss–Bonnet (GB) gravity coupled with a scalar field in presence of Maxwell electrodynamics. In order to derive such solutions, we assume the ansatz for metric potentials. Due to the choice of the ansatz of the metric, the Reissner Nordström gauge potential cannot be recovered because of the presence of higher-order terms which are not allowed to be vanishing. Moreover, the scalar field is not allowed to vanish. If it vanishes, a function of the solution results undefined. Furthermore, it is possible to show that the electric field is of higher-order in the monopole expansion: this fact explicitly comes from the contribution of the scalar field. Therefore, we can conclude that the GB scalar field acts as non-linear electrodynamics creating monopoles, quadrupoles, etc in the metric potentials. We compute the invariants associated with the BHs and show that, when compared to Schwarzschild or Reissner–Nordström space-times, they have a soft singularity. Also, it is possible to demonstrate that these BHs give rise to three horizons in AdS space-time and two horizons in dS space-time. Finally, thermodynamic quantities can be derived and we show that the solution can be stable or unstable depending on a critical value of the temperature.

Adaptive fuzzy fixed-time tracking control for high-order nonlinear delayed systems with mismatched disturbances

The fixed-time tracking control issue is investigated for a class of high-order nonlinear delayed systems subject to mismatched disturbances in this paper. Wherein, the fuzzy logic system is applied in dealing with the nonlinear functions and the power integrator technique is appropriately introduced to address the high-order terms in the considered system. Different from the conventional methods of using the Lyapunov-Krasovskii functional approach to cope with the delayed terms in the previous work, the adaptive backstepping method is utilized in this paper. The main purpose of the addressed control problem is to design a suitable adaptive fuzzy fixed-time controller such that the closed-loop system is stable and guaranteed to be bounded in a fixed-time interval. Subsequently, the specific control law is deduced through the inductive method, and the upper bound on the settling time for the fixed-time control is given. Finally, the simulation examples are given to illustrate the feasibility and effectiveness of the proposed method.

The inverse Rytov series for diffuse optical tomography

The Rytov approximation is known in near-infrared spectroscopy including diffuse optical tomography. In diffuse optical tomography, the Rytov approximation often gives better reconstructed images than the Born approximation. Although related inverse problems are nonlinear, the Rytov approximation is almost always accompanied by the linearization of nonlinear inverse problems. In this paper, we will develop nonlinear reconstruction with the inverse Rytov series. By this, linearization is not necessary and higher order terms in the Rytov series can be used for reconstruction. The convergence and stability are discussed. We find that the inverse Rytov series has a recursive structure similar to the inverse Born series.

Modulated wave patterns brought by higher-order dispersion and cubic–quintic nonlinearity in monoatomic chains with anharmonic potential

This work investigates the modulation instability and wave patterns in the extended nonlinear Schrödinger equation with higher-order dispersion and cubic–quintic nonlinearity. We use the one-dimensional monoatomic chain with anharmonic potential to derive the discrete model. From the multiple scale method combined with a quasidiscreteness approximation, we derive the cubic–quintic nonlinear schrödinger equation, and thereafter an expression of the modulation instability gain is obtained by using a linearized expression. One notices that the modulation instability growth is sensitive to the higher-order dispersion and nonlinearities terms. A split-step Fourier method is used to assess the analytical predictions. A long evolution of the continuous wave is shown to lead to the formation of the bright soliton, and Akhemediev breathers also emerge to manifest the modulation instability. We have also demonstrated that the excitation wave number generates the train of waves to confirm the fact that the continuous wave can grow exponentially with any value of the latter. We mention equally that the model of the extended cubic–quintic nonlinearity with complex envelope has opened new features of the modulated wave patterns in monoatomic chains.

Quantum droplets in one-dimensional Bose-Bose mixtures: beyond the Lee-Huang-Yang description

The static and dynamic properties of self-bound quantum droplets in a one-dimensional Bose-Bose mixture are discussed in the spirit of the Hartree–Fock-Bogoliubov theory. This latter enables us to provide beyond the Lee-Huang-Yang (LHY) quantum corrections to the equation of state at both zero and finite temperatures. In the uniform case our results for the ground-state energy and the critical temperature are confirmed through comparison with Quantum Monte-Carlo simulation and with available theoretical results. The density profiles are supported by numerical simulations of the generalized Gross-Pitaevskii equation which selfconsistently includes higher-order terms originating from the normal and anomalous fluctuations under the local density approximation. We show that the density exhibits a dip near its center in the flat-top plateau region for large interspecies interactions. We exemplify the impact of the beyond LHY corrections on the spatiotemporal evolution of the self-bound droplet in the presence of excitation induced by periodic density modulation. It is found that higher-order corrections may lead to the formation of a train of small droplets. We then extend our study for the case of inhomogeneous droplets in quasi one-dimensional Bose mixtures.

Event-Triggered Synchronization for Delayed Quaternion-Valued Inertial Fuzzy Neural Networks via Nonreduced Order Approach

This paper mainly studies the synchronization problem for a class of Takagi-Sugeno (T-S) fuzzy quaternion-valued inertial neural networks with time-varying delay through event-triggered control (ETC) scheme. Firstly, a class of quaternion-valued inertial fuzzy neural networks (QVIFNNs) model with time-varying delay is proposed. To avoid the increase of the number of state variables caused by reduced-order method, non-reduced order approach is utilized fully, and a fuzzy exponential gain event-triggered controller is designed so as to occupy the less communication resources. Then, by establishing a novel Lyapunov functional well handling the high-order term, a new synchronization condition is derived under static event-triggered scheme. And then, to further reduce the number of triggers, a dynamic event-triggered condition is developed and the corresponding sufficient criterion is given. Meanwhile, it is proved that the lower bound of event intervals are non-zero positive and the Zeno phenomenons do not exist via rigorous mathematical derivation. Finally, a numerical example is offered to show the effectiveness of the proposed method. A application example for the considered QVIFNNs with time-varying delay is given.

On a simplified multi-layered plate model of growth: Asymptotic analyses and numerical implementation

In this paper, a simplified multi-layered plate model is proposed, which can be used to study the growth-induced deformations of multi-layered hyperelastic plates. First, we introduce a full-form multi-layered plate theory of growth, which has a relatively complicated plate equation system. To simplify the plate equations, systematic asymptotic analyses are conducted with some specified magnitudes of growth functions and surface tractions. It is found that the terms containing a high-order stress tensor always have higher asymptotic orders. By dropping these higher-order terms from the original plate equations, a simplified multi-layered plate model can be derived. Despite its simplicity, this new plate model is valid in a wide range of growth and loading conditions. To be prepared for practical applications, the associated weak forms of the simplified plate equations are derived and implemented in a finite element software. Some typical examples are further studied to show the efficiency of the plate model. By virtue of the plate model, the growth deformations of multi-layered plate samples are simulated, which show very good consistency with the results of 3D volume model. Especially, the distributions of residual stresses on the interfaces of different layers can be predicted accurately, which has an important meaning for the design of soft devices with multi-layered plate structures.

Single UHD image dehazing via Interpretable Pyramid Network

Currently, most dehazing-based deep approaches are developed as an end-to-end manner to reconstruct a degraded image in an unintelligible fashion. For these dehazing models, the absence of ‘Credible Modeling’ to guide the network design is a barrier to be applied commercially in the open-world. To address this problem, we introduce the Taylor approximation principle as the soul and materialize this principle with the help of the Laplacian Pyramid. Specifically, we assume that the N paths of Laplacian pyramid model correspond to the N terms (sub-functions) in Taylor’s theorem. We attempt to use bottom paths and top paths to reconstruct the low-/high-frequency information of the clear image, respectively. Further, we develop a T-Unet module that focuses on regularizing the feature maps generated at bottom paths and design an attention sharing weight K to help approximate the Taylor high-order terms. Extensive experimental results demonstrate that our approach can run a 4K image on a single GPU with 24G RAM in real-time (80fps) and have unparalleled interpretability. The url of our code at https://github.com/zzr-idam/Interpretable-Pyramid-Network.

TaBE: Decoupling spatial and spectral processing with Taylor’s unfolding method in the beamspace domain for multi-channel speech enhancement

In recent years, significant advancements have been made in neural beamforming, leveraging spectral and spatial cues to enhance their performance in multi-channel speech enhancement. When the frame-wise processing mechanism is required, there exists a trade-off between performance and algorithmic delay for existing all-neural beamformers. However, from the perspective of multi-source information fusion, the network is often encapsulated into a black box to entangle and fuse the spatial and spectral features into a non-linear feature space, which hinders our understanding of how they work collaboratively for target speech extraction. In this regard, this paper proposes to decouple the spatial and spectral domain processing inspired by Taylor’s approximation theory. Specifically, we reformulate the time-variant beamforming defined in the spatial domain into the adaptive weighting and mixing of different beam components in the beamspace domain. This reformulation enables us to model the recovery of target speech as a weighted sum operation in the beamspace domain, where each beam component is associated with an introduced unknown term for residual interference cancellation. By virtue of Taylor’s series expansion, the recovery process can be decomposed into the superimposition of the 0th-order non-derivative and high-order derivative terms, where the former acts as spatial filtering in the spatial domain, and the latter serves as a residual interference canceller in the spectral domain. We conduct extensive experiments on the spatialized LibriSpeech and L3DAS Challenge datasets. Experimental results show that, compared with existing advanced approaches, the proposed method not only achieves competitive performance in terms of multiple objective metrics but also provides feasible guidance in multi-channel speech enhancement pipeline design.

Crack branching at low tip speeds: spilling the T

Using the criterion that a crack will extend along the direction of maximum circumferential stress, this paper demonstrates the influence of the coupling between the crack-parallel T-stress and the tip speed on the directional (in)stability of dynamics cracks in brittle materials, i.e., branching, turning, and limiting velocities. The proposed (in)stability criterion evolves within the theory of dynamic fracture: we build on the work of Ramulu and Kobayashi (1983) by introducing a reference distance ahead of the crack-tip to incorporate the contribution of the higher-order terms in the asymptotic solution of the elastic crack-tip fields. The theoretical aspect is first explored, a methodology to numerically (and experimentally) advocate the instability—as a co-action of T-stress and a fast-running crack—is then proposed and validated on Borden et al. (2012)’s branching benchmark. An experimental setup combining Ultra-High-Speed High-Resolution imaging with advanced Digital Image Correlation algorithms and a novel crack-branching inertial impact test enables for never-seen-before quantification of the rich dynamical behaviour of the fracture. This permits the experimental validation of the developed crack (in)stability criterion.

Time-Asymptotic Expansion with Pointwise Remainder Estimates for 1D Viscous Compressible Flow

We construct a time-asymptotic expansion with pointwise remainder estimates for solutions to 1D compressible Navier–Stokes equations. The leading-order term is the well-known diffusion wave and the higher-order terms are a newly introduced family of waves which we call higher-order diffusion waves. In particular, these provide an accurate description of the power-law asymptotics of the solution around the origin x=0, where the diffusion wave decays exponentially. The expansion is valid locally and also globally in the L^p({mathbb {R}})-norm for all 1le ple infty . The proof is based on pointwise estimates of Green’s function.

Detecting bifurcations in a fractional-order neural network with nonidentical delays via Cramer’s rule

This article is dedicated to reseaching the bifurcations of a fractional-order neural network (FONN) with nonidentical self-connection and comunication delays. In accordance with eigenvalue analysis, we apply Cramer’s rule to ingeniously calculate the specific value of the bifurcation point of an equation set with quartic transcendence term. It is noteworthy that the method proposed in this article is more concise than the existing methods for solving higher-order transcendental terms, and has a certain degree of generalization, which can be applied to the case involving n degree transcendental terms. Furthermore, it detects that the devised FONN can ameliorate dramatically the stability attributions in comparison with its integer-order counterpart. This article ultimately provides two experimental fruits for bifurcation caused by different delays to underpin the correctness of the developed methodology.

On the polymatroid Tutte polynomial

The Tutte polynomial is a well-studied invariant of matroids. The polymatroid Tutte polynomial TP(x,y), introduced by Bernardi, Kálmán, and Postnikov, is an extension of the classical Tutte polynomial from matroids to polymatroids P. In this paper, we first prove that TP(x,t) and TP(t,y) are interpolating for any fixed real number t≥1, and then we study the coefficients of high-order terms in TP(x,1) and TP(1,y). These results generalize results on the interior and exterior polynomials of hypergraphs.

New formulations for the mean wall-normal velocity and Reynolds shear stress in a turbulent boundary layer under zero pressure gradient

In this study, a novel analytical solution is derived rigorously for the mean wall-normal velocity in a turbulent boundary layer (TBL) under zero pressure gradient (ZPG), without relying on any a priori assumptions regarding the mean streamwise velocity. By neglecting the higher-order terms in the exact solution, an approximate formulation for the mean wall-normal velocity is obtained. The accuracy of this approximation is then validated through a comparison with data from direct numerical simulations (DNS). Drawing upon the distinct behaviours of force balance in the inner and outer regions of the ZPG TBL, simplified approximate mean momentum equations are derived by decomposing the Reynolds shear stress into inner and outer parts. The inner and outer Reynolds shear stresses are then obtained through integration of the corresponding approximate mean momentum equations. The outer Reynolds shear stress is revealed to exhibit a strong connection to the mean wall-normal velocity. Specifically, it is observed that $R_{uv-{out}}/u^2_\tau \approx 1 - UV/(U_e V_e)$ , where $u_\tau$ represents the friction velocity, and $U_e$ and $V_e$ denote the mean streamwise and wall-normal velocities at the boundary layer edge, respectively. Finally, an approximate formulation for the Reynolds shear stress across the entire TBL is developed by synthesizing the inner and outer parts. Extensive validation against both DNS and experimental data, spanning a wide range of Reynolds numbers, demonstrates the excellent agreement achieved.

Review of Wind-Induced Effects Estimation through Nonlinear Analysis of Tall Buildings, High-Rise Structures, Flexible Bridges and Transmission Lines

The nonlinear effects exhibited by structures under the action of wind loads have gradually stepped into the vision of wind-resistant researchers. By summarizing the prominent wind-induced nonlinear problems of four types of wind-sensitive structures, namely tall buildings, high-rise structures, flexible bridges, and transmission lines, the occurrence mechanism of their nonlinear effects is revealed, providing cutting-edge research progress in theoretical studies, experimental methods and vibration control. Aerodynamic admittance provides insights into the aerodynamic nonlinearity (AN) between the wind pressure spectrum and wind speed spectrum of tall building surfaces. The equivalent nonlinear equation method is used to solve nonlinear vibration equations with generalized van-der-Pol-type aerodynamic damping terms. The elastic–plastic finite element method and multiscale modeling method are widely employed to analyze the effects of geometric nonlinearity (GN) and material nonlinearity (MN) at local nodes on the wind-induced response of latticed tall structures. The AN in blunt sections of bridges arises from the amplitude dependence of the aerodynamic derivative and the higher-order term of the self-excited force. Volterra series aerodynamic models are more suitable for the nonlinear aerodynamic modeling of bridges than the polynomial models studied more in the past. The improved Lindstedt–Poincare perturbation method, which considers the strong GN in the response of ice-covered transmission lines, offers high accuracy. The complex numerical calculations and nonlinear analyses involved in wind-induced nonlinear effects continue to consume significant computational resources and time, especially for complex wind field conditions and flexible and variable structural forms. It is necessary to further develop analytical, modeling and identification tools to facilitate the modeling of nonlinear features in the future.

Modeling fracture in brittle materials by higher-order phase field method using C1 non-Sibsonian interpolants

PurposeThis study implements the fourth-order phase field method (PFM) for modeling fracture in brittle materials. The weak form of the fourth-order PFM requires C1 basis functions for the crack evolution scalar field in a finite element framework. To address this, non-Sibsonian type shape functions that are nonpolynomial types based on distance measures, are used in the context of natural neighbor shape functions. The capability and efficiency of this method are studied for modeling cracks.Design/methodology/approachThe weak form of the fourth-order PFM is derived from two governing equations for finite element modeling. C0 non-Sibsonian shape functions are derived using distance measures on a generalized quad element. Then these shape functions are degree elevated with Bernstein-Bezier (BB) patch to get higher-order continuity (C1) in the shape function. The quad element is divided into several background triangular elements to apply the Gauss-quadrature rule for numerical integration. Both fourth-order and second-order PFMs are implemented in a finite element framework. The efficiency of the interpolation function is studied in terms of convergence and accuracy for capturing crack topology in the fourth-order PFM.FindingsIt is observed that fourth-order PFM has higher accuracy and convergence than second-order PFM using non-Sibsonian type interpolants. The former predicts higher failure loads and failure displacements compared to the second-order model due to the addition of higher-order terms in the energy equation. The fracture pattern is realistic when only the tensile part of the strain energy is taken for fracture evolution. The fracture pattern is also observed in the compressive region when both tensile and compressive energy for crack evolution are taken into account, which is unrealistic. Length scale has a certain specific effect on the failure load of the specimen.Originality/valueFourth-order PFM is implemented using C1 non-Sibsonian type of shape functions. The derivation and implementation are carried out for both the second-order and fourth-order PFM. The length scale effect on both models is shown. The better accuracy and convergence rate of the fourth-order PFM over second-order PFM are studied using the current approach. The critical difference between the isotropic phase field and the hybrid phase field approach is also presented to showcase the importance of strain energy decomposition in PFM.

Dynamic asymptotic homogenization for wave propagation in magneto-electro-elastic laminated composite periodic structure

In the present work, anti-plane wave propagation with oblique incidence in a magneto-electro-elastic multi-laminated composite periodic structure with perfect contact is studied. A general asymptotic homogenization method is applied to analyze dispersion phenomena and dynamic process. On assuming the solution having a single-frequency dependency, the higher-order terms for the displacement, electric potential and magnetic potential in asymptotic expansions are studied. Higher order local problems up to a finite order and a generalization of the local problem of infinite order are derived through rigorous calculations satisfying the necessary and sufficient condition for the existence of 1-periodic solutions. The effective equation of motion also known as “Good” Boussinesq equation has been achieved directly which is one of the highlight of the present work. Moreover, closed-form expression of dispersion equation and closed-form solutions of first and second local problems have been obtained. The analytical results have been validated for particular elastic case. Graphical illustrations have been done for tri-laminated magneto-electro-elastic composite periodic structure and the effect of size of unit cell, angle of incidence of wave, volume fraction of the composite and magneto-electric properties on the dispersion curve and also slowness curves have been examined. The numerical results have been validated with previous work and compared with results obtained by other approximation method. The present dynamic homogenization method has been proven to provide a more dispersive system and better approximation.

Event-Based Fuzzy Adaptive Consensus Tracking for Stochastic High-Order Nonlinear Multiagent Networks With Specified-Time Convergence

In this paper, a specified-time event-triggered fuzzy adaptive control algorithm is developed to solve the consensus tracking control problem for stochastic high-order nonlinear multi-agent networks, which is intrinsically challenging due to the existence of stochasticity and high-order (positive odd integers greater than one) terms. More precisely, a novel specified-time performance function (STPF) is incorporated into time-varying high-order tan-type barrier Lyapunov function to guarantee that the tracking errors remain under time-varying constraints within specified time. Combining fuzzy logic systems with the adding on power integrator technique, an adaptive approximation policy is introduced to handle the system uncertainties. Moreover, a new switching threshold event-triggered mechanism is devised to determine the control signals updating instants, which reduces the transmission and computation burden, while resizing the triggering threshold in real-time. The Zeno phenomenon is excluded by guaranteeing that the triggering intervals is lower bounded by a positive constant. Two simulation examples are provided to demonstrate the effectiveness of the designed algorithm.