Polynomial Expansion Research Articles

Coefficient Estimates for New Subclasses of Bi-Univalent Functions with Bounded Boundary Rotation by Using Faber Polynomial Technique

In this article, the authors use the Faber polynomial expansions to find the general coefficient estimates for a few new subclasses of bi-univalent functions with bounded boundary rotation and bounded radius rotation. Some of the results improve the existing coefficient bounds in the literature.

Q-boson model and relations with integrable hierarchies

This work investigates the intricate relationship between the q-boson model, a quantum integrable system, and classical integrable systems such as the Toda and KP hierarchies. Initially, we analyze scalar products of off-shell Bethe states and explore their connections to tau functions of integrable hierarchies. Furthermore, we discuss correlation functions within this formalism, examining their representations in terms of tau functions, as well as their Schur polynomial expansions.

Statistical distribution of static stress resolved onto geometrically-rough faults

The in-situ stress state within fault zones is technically challenging to characterize, requiring the use of indirect methods to estimate. Most work to date has focused on understanding average properties of resolved stress on faults, but fault non-planarity should induce spatial variations in resolved static stress on a single fault. Assuming a particular stochastic model for fault geometry (band-limited fractal) gives an approximate analytic solution for the probability density function (PDF) on fault stress that depends on the mean fault orientation, mean stress ratio, and roughness level. The mean stress is shown to be equal to the planar fault value, while deviations are described by substituting a second-order polynomial expansion of the stress ratio into the inverse distribution on fault slope. The result is an analytical expression for the PDF of shear-to-normal stress ratio on 2-D rough faults in a uniform background stress field. Two end-member distributions exist, one approximately Gaussian when all points on the fault are well away from failure, and one reverse exponential, which occurs when the mean stress ratio approaches the peak. For the range of roughness values expected to apply to crustal faults, stress deviations due to geometry can reach nearly 100% of the background stress level. Consideration of such a distribution of stress on faults suggests that geometric roughness and the resulting stress deviations may play a key role in controlling earthquake behavior.

Construction of some new traveling wave solutions to the space-time fractional modified equal width equation in modern physics

Nonlinear fractional evolution equations are important for determining various complex nonlinear problems that occur in various scientific fields, such as nonlinear optics, molecular biology, quantum mechanics, plasma physics, nonlinear dynamics, water surface waves, elastic media and others. The space-time fractional modified equal width (MEW) equation is investigated in this paper utilizing a variety of solitary wave solutions, with a particular emphasis on their implications for wave propagation characteristics in plasma and optical fibre systems. The fractional-order problem is transformed into an ordinary differential equation using a fractional wave transformation approach. In this article, the polynomial expansion approach and the sardar sub-equation method are successfully used to evaluate the exact solutions of space-time fractional MEW equation. Additionally, in order to graphically represent the physical significance of created solutions, the acquired solutions are shown on contour, 3D and 2D graphs. Based on the results, the employed methods show their efficacy in solving diverse fractional nonlinear evolution equations generated across applied and natural sciences. The findings obtained demonstrate that the two approaches are more effective and suited for resolving various nonlinear fractional differential equations.

FCHG: Fuzzy Cognitive Hypergraph for interpretable fault detection

While deep learning-based fault detection methods for rolling body mechanical components have achieved good results, their application in real-world scenarios is limited, and their reliability is not satisfactory due to the non-interpretability of deep learning. Meanwhile, automated component vibration signals are usually affected by irregularities, which make it challenging to obtain advanced features by conventional feature extraction methods. To solve the above problems, we develop a new fuzzy cognitive graph model that combines the interpretability of fuzzy cognitive graphs with the high-precision categorization of hypergraphs, called fuzzy cognitive hypergraph (FCHG). First, frequency domain analysis is combined with graph analysis to help the network extract high-level features. Second, causal learning is used to construct the FCHG weight matrix to make the FCHG interpretable and enhance the reliability of fault diagnosis. Finally, the FCHG network is built using spectrogram theory and Chebyshev polynomial expansion to capture multiple feature relationships and improve fault diagnosis accuracy. The practicality and effectiveness of the FCHG model for monitoring the health and stability status of rolling body mechanical components are verified on three datasets.

A Preconditioner-Based Data-Driven Polynomial Expansion Method: Application to Compressor Blade With Leading Edge Uncertainty

Abstract In engineering practice, the amount of measured data is often scarce and limited, posing a challenge in uncertainty quantification (UQ) and propagation. Data-driven polynomial chaos (DDPC) is an effective way to tackle this challenge. However, the DDPC method faces problems from the lack of robustness and convergence difficulty. In this paper, a preconditioner-based data-driven polynomial chaos (PDDPC) method is developed to deal with UQ problems with scarce measured data. Two numerical experiments are used to validate the computational robustness, convergence property, and application potential in case of scarce data. Then, the PDDPC is first applied to evaluate the uncertain impacts of real leading edge (LE) errors on the aerodynamic performance of a two-dimensional compressor blade. Results show that the overall performance of compressor blade is degraded and there is a large performance dispersion at off-design incidence conditions. The actual blade performance has a high probability of deviating from the nominal performance. Under the influence of uncertain LE geometry, the probability distributions of the total pressure loss coefficient and static pressure ratio have obvious skewness characteristics. Compared with the PDDPC method, the UQ results obtained by the fitted Gaussian and Beta probability distributions seriously underestimate the performance dispersion of compressor blade. The mechanism analysis illustrates that the large flow variation around the leading edge is the main reason for the overall performance degradation and the fluctuations of the entire flow field.

Sensitivity analysis of turbine stage aerothermal characteristics and blade cooling performance considering combustor swirl and hot spot

This paper investigates the global sensitivity of high-pressure-turbine (HPT) first stage cascades aerothermal characteristics and blade cooling performance under the influence of combustor swirl and hot spot. The combustor outflow conditions impact on turbine working performance and robustness. This paper is focused on the influence of uncertain combustor swirling flow and hot spot, rather than deterministic working conditions. A simplified combustor outflow simulation method is firstly proposed, and the effects of combustor outflow are quantified by combining chaos polynomial expansion with Sobol’ decomposition method. The results show that different rotating directions of the swirl lead to restricted influences on the averaged aerothermal performance of the turbine stage, but significantly affects its robustness. The positive swirl increases the total pressure loss generated by the lower passage vortex of the stator cascade, and the high uncertainty region of the stator cascade total pressure loss profile is the combined products of the residual swirl and the interaction between the passage vortex and mainstream. The total pressure loss of the rotor cascade due to leakage vortex has low sensitivity to the hot spot and swirl strength, while the total pressure loss generated by the passage vortex is affected by hot spot temperature. Temperature distribution affected by the passage vortex is sensitive to swirl strength. Generally, uncertainty variance of stage aerothermal parameters with positive swirl is about 30% larger than that of the negative case, since the positive swirl has stronger interactions with the blades, leading to more significant uncertainty. However, the averaged heat transfer coefficients with positive swirl are 37% less sensitive, compared to the negative swirl case. The results give better understanding on the influence mechanisms of turbine inflow conditions, emphasize the meaning of investigating uncertain characteristics of the turbine cascades. Investigations of this article provide reference for robust turbine blade cooling design.

Conforming Capacitive Load Cells for Conical Pick Cutters.

In underground coal mining, machine operators put themselves at risk when getting close to the machine or cutting face to observe the process. To improve the safety and efficiency of machine operators, a cutting force sensor is proposed. A linear cutting machine is used to cut two separate coal samples cast in concrete with conical pick cutters to simulate mining with a continuous miner. Linear and neural network regression models are fit using 100 random 70:30 test/train splits. The normal force exceeds 60 kN during the rock-cutting tests, and it is averaged using a low pass filter with a 10 Hertz cutoff frequency. The sensor uses measurements of the resonant frequency of capacitive cells in a steel case to determine cutting forces. When used in the rock-cutting experiments, the sensor conforms to the tooling and the stiffness and sensitivity are increased compared to the initial configuration. The sensor is able to track the normal force on the conical picks with a mean absolute error less than 6 kN and an R2 score greater than 0.60 using linear regression. A small neural network with a second-order polynomial expansion is able to improve this to a mean absolute error of less than 4 kN and an R2 score of around 0.80. Filtering measurements before regression fitting is explored. This type of sensor could allow operators to assess tool wear and material type using objective force measurements while maintaining a greater distance from the cutting interface.

Hydrodynamic performance of multi-chamber oscillating water columns in a caisson array

A theoretical model based on the linear potential flow theory and eigenfunction expansion-matching method is developed to analyze the interaction between water waves and multi-chamber oscillating water columns (OWCs) embedded in a caisson array. The semi-analytical solution consists of the diffraction and radiation components of periodic OWCs. The velocity singularity at the tip of the OWC chamber walls is resolved by implementing a Chebyshev polynomial expansion. The unknown expansion coefficients are determined by the continuity equations of velocity and pressure at the interface of subdomains. The semi-analytical solution is verified using the Haskind relation and the conservation law of wave energy flux. It is found that multi-chamber OWCs have higher hydrodynamic efficiency over a broader frequency bandwidth than traditional single- and dual-chamber OWCs. The maximum hydrodynamic efficiency η increases with increasing of chamber number (J), from 0.5 (J = 1) to nearly 1.0 (J = 8 and 12). It is also found that a larger incident wave angle leads to a narrower frequency bandwidth for energy capturing. Increasing the incident wave angle from π/3 to 5π/12 results in a decrease of the first cutoff frequency kc from 2.41 to 2.02, and therefore results in the energy capture bandwidth narrowing accordingly. Both constructive and destructive hydrodynamic interactions between caissons array and OWCs are observed over the tested frequency range. Interestingly, when configuration of the OWC chambers is reversed, the transmission coefficient remains unchanged.

Chebyshev polynomial approach to Loschmidt echo: Application to quench dynamics in two-dimensional quasicrystals.

The understanding of quantum phase transitions in disordered or quasicrystal media is a central issue in condensed matter physics. In this paper we investigate localization properties of the two-dimensional Aubry-André model. We find that the system exhibits self-duality for the transformation between position and momentum spaces at a critical quasiperiodic potential, leading to an energy-independent Anderson transition. Most importantly, we present the implementation of an efficient and accurate algorithm based on the Chebyshev polynomial expansion of the Loschmidt echo, which characterizes the nonequilibrium dynamics of quantum quenched quasiperiodic systems. We analytically prove that the system under quench dynamics displays dynamical quantum phase transitions and further provide numerical verification by computing the polynomial expansion of the Loschmidt echo. Our results may provide insight into the realization of electronic transport in experiments.

Longitudinal failure criterion considering local fiber misplacement for UD composites

A new longitudinal failure criterion for unidirectional fibre reinforced (UD) composites is proposed, incorporating the contribution of local fibre misalignment to fibre failure. The failure function was established based on the quadratic polynomial expansion of stress components in the local coordinate system parallel to the misaligned fibres, with coefficients rationally determined based on the basic strength parameters of UD composites. The local stress components were used to determine the failure status associated with fibre compression, while the tensile load along fibres was assumed to have negligible influence on the development of the fibre misalignment. The predicted failure envelopes agreed well with the experimental data for different stress combinations, particularly in the regimes where the fibre failure was influenced by transverse stresses. This validates the predictive ability of the proposed criteria and highlights the importance of the contribution of matrix-failure-related stresses to the fibre-dominated failure behaviour of fibre reinforced composites.

POLYNOMIAL SOLUTIONS FOR THE KOLMOGOROV–WIENER PREDICTION OF MODELED SMOOTHED HEAVY-TAIL PROCESS

Nowadays telecommunication traffic in systems with data packet transfer is considered as a heavy-tail random process. In a couple of rather simple models traffic is considered to be stationary one. In our recent papers we generated modeled heavy-tail data, which is based on the smoothing of the fractional Gaussian noise. In particular, the applicability if the continuous Kolmogorov–Wiener filter to the prediction of such data was investigated, the corresponding Wiener–Hopf integral equation was solved on the basis of the truncated Walsh function expansion. However, a question occurs – may another truncated orthogonal function expansion be applied to the problem under consideration? So, the corresponding investigation may be an actual question. In our recent papers we investigated theoretical fundamentals of the Kolmogorov–Wiener filter construction for different models, in particular, on the basis of the truncated polynomial expansion method and on the basis of the truncated trigonometric Fourier series expansion method. In this paper we restrict ourselves to the investigation of the applicability of the truncated polynomial expansion method to the problem under consideration, the corresponding method is based on the Сhebyshev polynomials of the first kind. The applicability of another polynomial or trigonometric expansions to the problem under consideration may be discussed in other papers. The aim of the work is to investigate the applicability of the Galerkin method based on the Chebyshev polynomials of the first kind to the Kolmogorov–Wiener prediction of smoothed heavy-tail data. The methodology consists in the solving of the Wiener–Hopf integral equation on the basis of the truncated polynomial expansion method which is based on the Chebyshev polynomials of the first kind. The scientific novelty consists in the proof of the fact that the Galerkin method based on the Chebyshev polynomials of the first kind may be applied to the Kolmogorov–Wiener prediction of smoothed heavy-tail data. The conclusions are as follows. The truncated polynomial expansion method based on the Chebyshev polynomials of the first kind may give reliable results in the framework of the Kolmogorov–Wiener prediction of smoothed heavy-tail data.

On the direct quantization of Maxwell field

In this paper, we apply the generalized integration constants (GCI) method (Belhadi 2023 https://arxiv.org/abs/2303.08236), in field theory to quantize Maxwell and the Klein–Gordon free fields. The study is performed in both position and momentum spaces, to obtain the equal-time Dirac brackets among the fields and their conjugate momenta. The idea is to compute the brackets near the initial instant using the Taylor polynomial expansion, and then deduce directly their expressions at any later time. In the case of the Maxwell field, the interdependence of the field components (constraints) requires the use of the Helmholtz theorem to separate the transversal and longitudinal parts. Our work finishes with the study of the O(3) nonlinear sigma model using the GCI approach.

Investigation of Viscoelastic Guided Wave Properties in Anisotropic Laminated Composites Using a Legendre Orthogonal Polynomials Expansion-Assisted Viscoelastodynamic Model.

This study investigates viscoelastic guided wave properties (e.g., complex-wavenumber-, phase-velocity-, and attenuation-frequency relations) for multiple modes, including different orders of antisymmetric, symmetric, and shear horizontal modes in viscoelastic anisotropic laminated composites. To obtain those frequency-dependent relations, a guided wave characteristic equation is formulated based on a Legendre orthogonal polynomials expansion (LOPE)-assisted viscoelastodynamic model, which fuses the hysteretic viscoelastic model-based wave dynamics and the LOPE-based mode shape approximation. Then, the complex-wavenumber-frequency solutions are obtained by solving the characteristic equation using an improved root-finding algorithm, which leverages coefficient matrix determinant ratios and our proposed local tracking windows. To trace the solutions on the dispersion curves of different wave modes and avoid curve-tracing misalignment in regions with phase-velocity curve crossing, we presented a curve-tracing strategy considering wave attenuation. With the LOPE-assisted viscoelastodynamic model, the effects of material viscosity and fiber orientation on different guided wave modes are investigated for unidirectional carbon-fiber-reinforced composites. The results show that the viscosity in the hysteresis model mainly affects the frequency-dependent attenuation of viscoelastic guided waves, while the fiber orientation influences both the phase-velocity and attenuation curves. We expect the theoretical work in this study to facilitate the development of guided wave-based techniques for the NDT and SHM of viscoelastic anisotropic laminated composites.

Probing New physics with high-redshift quasars: axions and non-standard cosmology

The Hubble diagram of quasars, as candidates to “standardizable” candles, has been used to measure the expansion history of the Universe at late times, up to very high redshifts (z ∼ 7). It has been shown that this history, as inferred from the quasar dataset, deviates at ≳ 3σ level from the concordance (ΛCDM) cosmology model preferred by the cosmic microwave background (CMB) and other datasets. In this article, we investigate whether new physics beyond ΛCDM (BΛCDM) or beyond the Standard Model (BSM) could make the quasar data consistent with the concordance model. We first show that an effective redshift-dependent relation between the quasar UV and X-ray luminosities, complementing previous phenomenological work in the literature, can potentially remedy the discrepancy. Such a redshift dependence can be realized in a BSM model with axion-photon conversion in the intergalactic medium (IGM), although the preferred parameter space is in tension with various other astrophysical constraints on axions, at a level depending on the specific assumptions made regarding the IGM magnetic field. We briefly discuss a variation of the axion model that could evade these astrophysical constraints. On the other hand, we show that models beyond ΛCDM such as one with a varying dark energy equation of state (wCDM) or the phenomenological cosmographic model with a polynomial expansion of the luminosity distance, cannot alleviate the tension. The code for our analysis, based on emcee [1] and corner.py [2], is publicly available at github.com/ChenSun-Phys/high_z_candles.

Energy stability of magnetohydrodynamic flow in channels and ducts

We study the energy stability of pressure-driven laminar magnetohydrodynamic flow in a rectangular duct with a transverse homogeneous magnetic field and electrically insulating walls. For sufficiently strong fields, the laminar velocity distribution has a uniform core and convex Hartmann and Shercliff boundary layers on the walls perpendicular and parallel to the magnetic field. The problem is discretized by a double expansion in Chebyshev polynomials in the cross-stream coordinates. The linear eigenvalue problem for the critical Reynolds number depends on the streamwise wavenumber, Hartmann number and the aspect ratio. We consider the limits of small and large aspect ratios in order to compare with stability models based on one-dimensional base flows. For large aspect ratios, we find good numerical agreement with results based on the quasi-two-dimensional approximation. The lift-up mechanism dominates in the limit of a zero streamwise wavenumber and provides a linear dependence between the critical Reynolds and Hartmann numbers in the duct. As the aspect ratio is reduced away from unity, the duct results converge to Orr's original energy stability result for spanwise uniform perturbations imposed on the plane Poiseuille base flow. We also examine different possible symmetries of eigenmodes as well as the purely hydrodynamic case in the duct geometry.

Cluster cosmology redux: a compact representation for the halo mass function

ABSTRACT Groups and clusters of galaxies imprint coherent, arcminute-scale features across the spectrophotometric sky, especially optical-IR clusters of galaxies, spectral distortions in the cosmic microwave background, and extended sources of X-ray emission. The space–time density of the host dark matter halo population – the halo mass function (HMF) – is a common theoretical basis for modelling such observable features. We explore a compact representation – a dual-quadratic (DQ-HMF) form – that features readily interpretable parameters representing polynomial expansions of the space–time number density surface, first in terms of log-mass, then in redshift. The DQ-HMF form fits Mira-Titan N-body emulator expectations for halo masses $10^{13.7-14.5} \, h^{-1}\, {\rm M}_\odot$ over redshifts, 0.1 < z < 1.5 to within $\sim \! 5~{{\ \rm per\ cent}}$. We provide best-fitting parameters for a Planck 2018 cosmology and demonstrate model self-similarity in the Ωm−$\, \sigma _8$ plane. Convolving with a minimal mass–observable relation (MOR) yields closed-form expressions for counts, mean mass, and mass variance of cluster samples characterized by an observable property. Performing information-matrix forecasts of potential parameter constraints from existing and future surveys, we demonstrate the potential for percent-level constraints on model parameters by an LSST-like optical cluster survey of 300 000 clusters and a richness–mass variance of 0.32. Even better constraints could potentially be achieved by a survey with one-tenth the sample size but with a reduced selection property variance of 0.12. Potential benefits and extensions to the basic MOR parametrization are discussed.

On the orthogonality of Atkin-like polynomials and orthogonal polynomial expansion of generalized Faber polynomials

On the orthogonality of Atkin-like polynomials and orthogonal polynomial expansion of generalized Faber polynomials

Uncertainty Benchmarks for Time-Dependent Transport Problems

Verification solutions for uncertainty quantification (UQ) are presented for time-dependent transport problems where , the scattering ratio, is uncertain. The method of polynomial chaos expansions is employed for quick and accurate calculation of the quantities of interest (QoIs), and uncollided solutions are used to treat part of the uncertainty calculation analytically. We find that approximately six moments in the polynomial expansion are required to represent the solutions to these problems accurately. Additionally, the results show that if the uncertainty interval spans c = 1, which means it is uncertain whether the system is multiplying or not, the confidence interval will grow in time. Finally, since the QoI is a strictly increasing function, the percentile values are known and can be used to verify the accuracy of the expansion. These results can be used to test UQ methods for time-dependent transport problems.

Vibroacoustic analysis of submerged fluid-filled cylindrical shell

This study presents a comprehensive Chebyshev-Fourier spectral method for analyzing the free and forced vibroacoustic characteristics of submerged fluid-filled cylindrical shells in heavy fluids. The vibroacoustic interaction system of submerged cylindrical shells consists of three discrete regions: The Sanders shell theory is applied for the shell structure, whereas the wave equation is used to determine the fluid inside. The external fluid pressure on the shell is described using the Helmholtz boundary integral equation. The governing equations for the shell's motion, the formulation of the external fluid pressure on the fluid-structure-interaction (FSI) surface using first-class Chebyshev polynomial and Fourier series expansions, and the internal sound pressure employing two-dimensional Chebyshev polynomial and Fourier series expansions, are seamlessly integrated with Gauss-Chebyshev-Lobatto sampling techniques. Artificial springs are implemented to replicate the boundary conditions of the cylindrical shell. The precision of the proposed methodology is corroborated by comparisons with existing literature and finite element method (FEM) results. Furthermore, the research examines how the properties of the acoustical medium on the vibroacoustic responses of the cylindrical shell. Results indicate that the presence of air inside the shell slightly influence on its acoustic radiation. However, replacing the internal medium with water significantly alters the shell's original radiation characteristics.