Higher-order Terms Research Articles

A Saturation-Based Unification Algorithm for Higher-Order Rational Patterns

Higher-order unification has been shown to be undecidable [Huet 1973]. Miller discovered the pattern fragment and subsequently showed that higher-order pattern unification is decidable and has most general unifiers [1991]. We extend the algorithm to higher-order rational terms (a.k.a. regular Böhm trees [Huet 1998], a form of cyclic \(\lambda\) -terms) and show that pattern unification on higher-order rational terms is decidable and has most general unifiers. We prove the soundness and completeness of the algorithm.

A New Framework for Understanding Systematic Errors in Cluster Lens Modeling. III. Deflection from Large-Scale Structure

Interpreting and reconstructing distant sources that are gravitationally lensed by galaxy clusters requires accurate and precise lens models. While high-quality data sets have reduced statistical errors in such models, systematic errors remain important. We examine systematic lensing effects caused by density fluctuations due to large-scale structure along the line of sight. We use a multiplane ray-tracing algorithm with the IllustrisTNG 100-3 cosmological simulation of matter distribution and compute the statistical distributions of shear, convergence, and higher-order deflections using two Hubble Frontier Field clusters as examples (A2744 and MACS J0416.1−2403). The cosmic shear distribution is Gaussian in each component, while the cosmic convergence distribution is skewed such that 1 + κ is consistent with a log-normal distribution; the standard deviations for these quantities are at the level of a few to 10%, depending on the redshift of the source. The deflection from higher-order terms beyond convergence and shear has significant scatter: the rms deflection is ∼15″, considerably larger than the image position residuals for current lens models. These results indicate that line-of-sight deflection effects due to large-scale structure can significantly impact lens models and should not be neglected. We present results in forms that can be incorporated into future cluster lens models.

Compositional reservoir simulation with a high-resolution compact stencil adaptive implicit method

The adaptive implicit method (AIM) is the mainstream approach for compositional reservoir simulation. In its standard form, AIM uses single-point upwinding to reconstruct the fluxes across the interfaces of the control volume, and this leads to substantial numerical diffusion and loss of accuracy. Previous efforts to improve the accuracy of AIM focused on using high-order schemes to reconstruct the interfacial fluxes; those schemes introduced additional numerical nonlinearities to the system of equations or compromised the accuracy of the Jacobian by neglecting the high-order terms in its construction. In this work, we describe a high-resolution compact-stencil (HRCS) AIM. The new scheme is applied to compositional reservoir simulation. In addition to the mixed implicit/explicit time discretization of standard AIM, the HRCS AIM scheme uses a mixed time and space discretization. Specifically, we blend low- and high-order fluxes according to a well defined rule that uses a high-order reconstruction in the explicit regions of the domain and a low-order reconstruction in the implicit regions. This strategy ensures that additional nonlinearities introduced by the high-order reconstruction do not impact the Jacobian matrix, thus preserving the algebraic structure of standard AIM. The HRCS AIM method is demonstrated using several compositional problems. The results indicate substantial gains in accuracy with a small additional computational cost compared with standard AIM. Additionally, HRCS AIM is more robust and has a smaller computational cost compared with its full high-resolution counterpart.

Hopf Bifurcation Analysis of a Nose Landing Gear System of Aircraft

The Hopf bifurcation problem of a nose landing gear system is investigated. Taking the nearly smooth landing gear system with high-order nonlinear terms as the research object, the nose landing gear system is simplified to a plane dynamic system considering nonlinear factors by using the center manifold simplification method. The first Lyapunov coefficient of the landing gear system is derived by using the classical bifurcation theory, and the Hopf bifurcation type of the system is determined according to its sign. The bifurcation diagram is obtained by solving the differential equations of motion numerically. It is revealed that there is a stable limit cycle in the neighborhood of the supercritical Hopf bifurcation point. The influence of velocity and other factors on the shimmy oscillations of the landing gear system is analyzed. The dynamical behavior in the neighborhood of the Hopf bifurcation point is explored theoretically and numerically. The research results can provide a theoretical basis for the optimal design of aircraft landing gear systems.

Chained Spatial Beam Adomian Decomposition Model: A Novel Model of Flexible Slender Beams for Large Spatial Deflections

Abstract The main element of compliant mechanisms and continuum robots is flexible slender beams. However, the modeling of beams can be complicated due to the geometric nonlinearity becoming significant at large elastic deflections. This paper presents an explicit nonlinear model called the spatial beam Adomian decomposition model (SBADM) for intermediate spatial deflections of a slender beam with uniform, bisymmetric sections subjected to general end-loading. Specifically, the elongation, bending, torsion, and shear deformations of the beams are modeled based on Timoshenko's assumptions and Cosserat rod theory. Then, the quaternion transformation and Adomian decomposition are used to solve the nonlinear governing differential equations for the beam by truncating the higher-order terms, yielding an explicit expression for spatially deflected beams within intermediate deflection ranges. Simulations demonstrate the accuracy and time-wise efficiency of the SBADM, as well as its advantages over the state-of-the-art. In addition, this paper also introduces a discretization-based scheme called the chained SBADM (CSBADM) for large spatial deflections of flexible beams. Real-world experiments with two different configurations have also been performed to validate the effectiveness of the CSBADM. The results indicate that the CSBADM can accurately calculate the load-displacement relations for large deformed beams.

Generalized XY Models with Arbitrary Number of Phase Transitions

We propose spin models that can display an arbitrary number of phase transitions. The models are based on the standard XY model, which is generalized by including higher-order nematic terms with exponentially increasing order and linearly increasing interaction strength. By employing Monte Carlo simulation we demonstrate that under certain conditions the number of phase transitions in such models is equal to the number of terms in the generalized Hamiltonian and, thus, it can be predetermined by construction. The proposed models produce the desirable number of phase transitions by solely varying the temperature. With decreasing temperature the system passes through a sequence of different phases with gradually decreasing symmetries. The corresponding phase transitions start with a presumably BKT transition that breaks the U(1) symmetry of the paramagnetic phase, and they proceed through a sequence of discrete Z2 symmetry-breaking transitions between different nematic phases down to the lowest-temperature ferromagnetic phase.

Static and free vibration response of FGM plates using higher order shear deformation theory and strain-based finite element formulation

The primary objective and novelty of this work lie in the development of a new four-node quadrilateral finite element based on strain-based high-order shear deformation theory (HSDT). This is the first study to apply this innovative approach to analyze both the static and free vibration behaviors of functionally graded (FG) plates. Another key novelty is the reduction in the number of unknowns to five, unlike other high-order shear deformation theories that employ a larger number of unknowns. This reduction is achieved by applying the condition of zero transverse shear stress at the top and bottom free surfaces of the FG plate and by assuming that the transverse shear strains are sinusoidally distributed through the thickness. The material properties of FG plates are modeled to vary according to a simple power law based on the volume fractions of their constituents. The developed finite element possesses five degrees of freedom (DOFs) per node, resulting from the combination of two strain-based elements: the first is a membrane with two DOFs, and the second is a bending plate with three DOFs. The displacement fields of the proposed element are expressed using high-order terms based on the strain approach, which satisfy compatibility equations and rigid body modes. Furthermore, the concept of the neutral surface is introduced to eliminate membrane-bending coupling. The elementary stiffness and mass matrices are derived using both the total potential energy principle and Hamilton’s principle. The performance and convergence of the proposed element are validated through examples from the literature.

Fermionic Casimir energy in Horava–Lifshitz scenario

In this work, we investigate the violation of Lorentz symmetry through the Casimir effect, one of the most intriguing phenomena in modern physics. The Casimir effect, which represents a macroscopic quantum force between two neutral conducting surfaces, is widely regarded as a triumph of Quantum Field Theory. In this study, we present new results for the Casimir effect, focusing on the contribution of mass associated with fermionic quantum fields confined between two large parallel plates, in the context of Lorentz symmetry violation within the Horava–Lifshitz formalism. To calculate the Casimir energy and pressure, we impose a MIT bag boundary condition on the two plates, compatible with the higher-order derivative term in the modified Dirac equation. Our results reveal a strong influence of Lorentz violation on the Casimir effect. We observe that the Casimir energy is significantly affected, both in intensity and sign, potentially resulting in a repulsive or attractive force between the plates, depending on the critical exponent associated with the Horava–Lifshitz formalism.

General synthetic iterative scheme for non-equilibrium dense gas flows

The recently-developed general synthetic iterative scheme (GSIS), which is tailored for non-equilibrium dilute gas, has been extended to find the steady-state solutions of the non-equilibrium dense gas flows based on the Shakhov-Enskog model, resolving the problems of slow convergence and requirement of ultra-fine grids in near-continuum flows that exist in the conventional iterative scheme. The key ingredient of GSIS is the tight coupling of the mesoscopic kinetic equation and the macroscopic synthetic equations that are exactly derived from the kinetic equation. On the one hand, high-order terms computed from the velocity distribution function provide the higher-order constitutive relations describing the non-equilibrium effects for the macroscopic synthetic equations. On the other hand, the macroscopic quantities obtained from the macroscopic synthetic equations are used to guide the evolution of the velocity distribution function in the kinetic equation. The efficiency and accuracy of GSIS are demonstrated in several test cases, including the shock wave passing through a cylinder and the pressure-driven dense gas flows passing through parallel plates and porous media. The effects of denseness are analyzed in a wide range of gas rarefaction.

High-precision analysis of the critical point in heavy-quark QCD at Nt=6

Binder-cumulant analysis of the critical point in the heavy-quark region of QCD is performed by Monte-Carlo simulations with the hopping-parameter expansion at Nt=6. We extend our previous analysis at Nt=4 to finer lattices and perform high-precision analyses on large spatial volumes up to the aspect ratio LT=Ns/Nt=18. Higher order terms in the hopping-parameter expansion are incorporated effectively up to 14th order. The numerical results show that the violation of the finite-size scaling becomes more prominent on the finer lattice at a given aspect ratio. Published by the American Physical Society 2024

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.

A new nonconvex multi-view subspace clustering via learning a clean low-rank representation tensor

Abstract Recently, low-rank tensor representation has achieved impressive results for multi-view subspace clustering (MSC). The typical MSC methods utilize the tensor nuclear norm as a convex surrogate of the tensor multi-rank to obtain a low-rank representation, which exhibits limited robustness when dealing with noisy and complex data scenarios. In this paper, we introduce an innovative clean low-rank tensor representation approach that combines the idea of tensor robust principal component analysis (TRPCA) with a new nonconvex tensor multi-rank approximation regularization. This integration enhances the robustness of the low-rank representation, resulting in improved performance. Furthermore, to better capture the local geometric features, we employ a higher-order manifold regularization term. To effectively address our new model, we develop an iterative algorithm that has been proven to converge to the desired Karush-Kuhn-Tucker (KKT) point. The numerical experiments on widely used datasets serve to demonstrate the efficacy and effectiveness of our new method.

An insight into the solitonic features of the nonlinear generalized higher-order Schrödinger equation using the solver method

For many nonlinear applications described by the dynamics of nonlinear Schrödinger equation with higher-order terms (HONLSE) such as nonlinear optics, space plasma physics molecular biology, astrophysics, quantum mechanics, superfluid, fluid mechanics, and fiber optics communications, a unique closed-form solution have been obtained using energy equation. In addition, some new solitary solutions HONLSE have been obtained via the unified solver method. The resultant solutions behave as breathers, super solitons, envelope breathers, blow up, localized super waves, periodical super shock, train super solitons, and shock structures. The modulations of Kerr nonlinear, chromatic dispersive, and wave packet drift parameters on the wave characteristics of the obtained solutions have been investigated. It was reported that the model parameters affect the amplitude, steepness, and width of the resultant structures. The provided solution can be used as a box solver for a variety of natural science systems described by distinct nonlinear equations.

Three-dimensional central-moment pseudopotential lattice Boltzmann model with improved discrete additional term

In this work, a three-dimensional central-moment pseudopotential lattice Boltzmann model is developed to simulate a two-phase flow and wetting phenomena. In this model, an improved discrete additional term is proposed to regulate the thermodynamic consistency and surface tension. Different from the discrete additional terms in previous models where only low-order terms are derived at the macroscopic Navier–Stokes equation level, high-order terms are correctly constructed at the mesoscopic lattice Boltzmann equation level in the present improved discrete additional term so that the high-order central moments can be modified in the collision step. With the improved discrete additional term, the simple relationship between the interaction force and the pseudopotential functions is well preserved. On this basis, a simplified wetting boundary scheme is further proposed, which eliminates the complex process for choosing proper characteristic vectors and interpolation. Numerical simulations demonstrate that the proposed model can achieve better performance in thermodynamic consistency, Galilean invariance, numerical stability and computational efficiency, and have great ability to simulate two-phase flow and wetting phenomena on realistic conditions.

Stability of high-order nonlinear Takagi–Sugeno fuzzy impulsive delayed coupled systems

In this study, we consider the stability of high-order nonlinear Takagi–Sugeno fuzzy impulsive delayed coupled systems (TSFICSs). It should be noted that the linear growth condition is removed in the investigation of high-order nonlinear TSFICSs, which weakens the conditions compared with previous studies of impulsive systems. In addition, the existing research methods do not work for high-order nonlinear TSFICSs, so novel impulsive differential inequalities with high-order terms are established to investigate the stabilization of high-order nonlinear TSFICSs and develop the classic impulsive differential inequalities for high-order nonlinear situations. Next, sufficient conditions for the stability are obtained based on the new impulsive differential inequalities together with graph theory and the Lyapunov method. Finally, the theoretical results are applied to modified impulsive delayed coupled Fitzhugh–Nagumo models, and numerical simulations are provided to illustrate the practical value of the derived results.

Through-focus performance and off-axis effects in aspheric monofocal intraocular lenses.

This study aimed to determine the through-focus performance and the effect of misalignment on the optical quality of different aspheric monofocal intraocular lenses (IOLs). To this end, optical quality was assessed in three IOL models with different optic surfaces: standard aberration neutral model and two spherical aberration (SA) correcting, one of which utilizes higher-order aspheric terms. The optical quality was measured by means of the modulation transfer function at 3- and 4.5-mm pupils and under monochromatic and polychromatic light with different corneal SA. The optically derived range of vision and tolerance to misalignment were also tested. The study demonstrated that the type of IOL surface affects the monofocal implant's performance. Although a standard primary-SA correction may improve scotopic image quality, misalignment may diminish this advantage. The higher-order aspheric surface used to correct SA provided an improved performance against decentration and offered a higher optical quality than the aberration-neutral design when tested in a model eye. The latter, however, demonstrated a high tolerance to misalignment, offering a slight extension of the range of vision, potentially resulting from uncorrected optical aberrations.

A combination of XGBoost and neural network in LIBS spectrum processing for precise determination of critical elements in 620 iron ore samples of various origins

China is the worldwide largest importer of iron ores for supplying its iron and steel industries. The necessary inspections of iron ores not only concern total iron content but also involve minor elements, such as silicium, aluminum, magnesium, and calcium. Laser-induced breakdown spectroscopy (LIBS) promises in situ and on-site detection and analysis capabilities, which is required for efficient, reliable, and large-scale impartation inspections. Such opportunity represents at the same time, a challenge for the technique in terms of the quantitative analysis ability. A fundamental issue corresponds to the matrix effect in LIBS analysis due to a large diversity of iron ores in terms of mineralogical and chemical compositions. Demonstrated as effective to deal with the matrix effect in LIBS analyses of geological materials, the machine learning approach needs to be implemented within an optimized data processing pipeline, combining algorithms carefully chosen and tested. In this work, we developed a sequentially connected combination of an extreme gradient boosting (XGBoost) model and a back propagation neural network (BPNN) model, to effectively fit the functional relation between the concentrations and the spectra. The specificity of the approach consists in fitting the linear, low-order nonlinear, and high-order nonlinear terms in a sequential way to substantially improve the model prediction performances. The method has been developed and tested with a significant set of 620 collected iron ore samples of eight production countries and 35 different commercial brands, with a wide range of matrices in terms of mineralogical and chemical compositions. The samples underwent prior characterization using conventional laboratory analysis methods to establish the reference elemental concentrations. The results from this necessary step of laboratory investigation, with root mean square error for prediction (RMSEP) of 0.407 wt%, 0.372 wt%, 0.085 wt%, 0.131 wt%, and 0.114 wt%, respectively for total iron (TFe), SiO2, Al2O3, MgO, and CaO, show the potential for on-site inspections.

A wavelet CNN with appropriate feed-allocation and PSO optimized activations for diabetic retinopathy grading

This work modifies the architecture of conventional CNN with the integration of Multi-resolution Analysis (MRA) in a CNN framework for Diabetic Retinopathy (DR) diagnosis and grading. Here, the HF sub-bands are subjected to optimized activations and are directly fed to the fully connected layers, as it encompasses edge features. Unlike FD-Relu, the proposed function preserves significant negative coefficients, compared to the S-Relu, the proposed third-order S-Relu is optimized such that it sustains the activations in the range suitable for the wavelet coefficients. The coefficients of higher-order terms of the proposed 3rd-order S-Relu are optimized with PSO, fitting the maximum energy of the wavelet sub-bands to ensure High Frequency (HF) edge preservation. The authors re-architecture 3 different CNNs published in the Retinal Image analysis field, with spatial and wavelet inputs with optimized activations. The highest accuracy of 96% is attained with the AlexNet re-architecture, with 35,126 fundus images secured from the Kaggle dataset. As we can infer the proposed re-architecture wavelet CNN outperformed the multiscale shallow CNNs, multiscale attention net, and stacked CNNs with a 6.6, 0.3, 0.7 per cent increase in accuracy. The entire implementation of the wavelet CNN is made available under source code.

Self-resonance during preheating: The case of [formula omitted]-attractor models

In this paper, for the first time, we obtain a new class of solutions for the Hill-type differential equations, which emerge in the preheating self-resonance phase of the expanding Universe. We study, in particular, the class of symmetric and asymmetric scalar field potentials coming from the so-called α-attractor models of the early Universe cosmology. By making a series expansion of the potential and employing perturbative techniques we reformulate the Mukhanov–Sasaki equation, which captures the dynamics of the curvature perturbation in these models, into a Hill equation. This last includes higher-order terms that were never solved in the literature. Namely, those coming from the cubic and quartic contributions of the scalar field potential. Then, we derive the expressions for the Floquet exponents of the Mukhanov–Sasaki variable. Our analytical results are then compared with numerical computations, showing a good agreement and thus making this method valuable for obtaining theoretical predictions with new observational applications in the contexts of Primordial Black Holes and Scalar-Induced Gravitational Waves.

Implicit-Explicit Schemes for Compressible Cahn–Hilliard–Navier–Stokes Equations

The isentropic compressible Cahn–Hilliard–Navier–Stokes equations are a system of fourth-order partial differential equations that model the evolution of some binary fluids under convection. The purpose of this paper is the design of efficient numerical schemes to approximate the solution of initial-boundary value problems with these equations. The efficiency stems from the implicit treatment of the high-order terms in the equations. Our proposal is a second-order linearly implicit-explicit time stepping scheme applied in a method of lines approach, in which the convective terms are treated explicitly and only linear systems have to be solved. Some experiments are performed to assess the validity and efficiency of this proposal.