Taylor Series Research Articles

Investigation of Second Order Taylor Series in the Coarse Operator of Parareal Algorithm for Power System Simulation

Investigation of Second Order Taylor Series in the Coarse Operator of Parareal Algorithm for Power System Simulation

Fekete-Szegö and Hankel inequalities related to the sine function

ABSTRACT The Fekete-Szegö inequality is one of the inequalities for the coefficients which associated with the famous Bieberbach conjecture. Other issues associated with this inequality are determining the Hankel determinant denoted as Hd inequalities which are involved in the study of the singularities and integral coefficients in the Taylor series representations. In this investigation, we discuss such inequalities for certain mappings g for which ( g ′ ( z ) ) α [ 2 z g ′ ( z ) g ( z ) − g ( − z ) ] ( 1 − α ) ≺ 1 + sin ⁡ z , ( 0 ≤ α ≤ 1 ) lies in the image domain starlike about 1 and symmetric about real axis. Particularly, we obtain certain inequality for functions involving Poisson distribution. Furthermore, we also find the second Hankel determinant and other inequalities related to this newly defined class.

Stability of the linearization method applied in tricom-partmental polynomial catenary systems of order (α + β)

This paper aims to identify exchange coefficients of a nonlinear polynomial tri-compartmental catenary system of (α + β) order. This is based on two principal procedures. The first procedure presented is related to the recommended solution consisting of introducing an adequate time t∗ > 0 in a way to be defined. That is to say: wait a moment to allow the exchange to settle in the polynomial (α +β) order nonlinear catenary system after injecting the quantity into the main compartment, then measure this compartment with compartment 2, at this time t∗ > 0. In the second procedure, we apply the Taylor formula to linearize the nonlinear system and identify the exchange coefficients. In the end, we will prove that the linearization method is stable

A quantitative theory for heterogeneous combustion of nonvolatile metal particles in the diffusion-limited regime

The paper presents an analytical theory quantitatively describing the heterogeneous combustion of nonvolatile (metal) particles in the diffusion-limited regime. It is assumed that the particle is suspended in an unconfined, isobaric, quiescent gaseous mixture and the chemisorption of the oxygen takes place evenly on the particle surface. The analytical solution of the particle burn time is derived from the conservation equations of the gas-phase described in a spherical coordinate system with the utilization of constant thermophysical properties, evaluated at a reference film layer. This solution inherently takes the Stefan flow into account. The approximate expression of the time-dependent particle temperature is solved from the conservation of the particle enthalpy by neglecting the higher order terms in the Taylor expansion of the product of the transient particle density and diameter squared. Coupling the solutions for the burn time and time-dependent particle temperature provides quantitative results when initial and boundary conditions are specified. The theory is employed to predict the burn time and temperature of 10–100 μm iron particles, which are then compared with measurements, as the first validation case. The theoretical burn time agrees with the experiments almost perfectly at both low and high oxygen levels. The calculated particle temperature matches the measurements fairly well at relatively low oxygen mole fractions, whereas the theory overpredict the particle peak temperature due to the neglect of evaporation and the possible transition of the combustion regime.Novelty and significance statementFor the first time, we present a comprehensive and quantitative analytical theory elucidating the heterogeneous combustion of nonvolatile (metal) particles in the diffusion-limited regime. This novel theoretical model exhibits a remarkable capacity for quantitative prediction, obviating the need for supplementary information from numerical simulations or experimental data. The derivation process of analytical solutions for burn time and time-dependent particle temperature from conservation equations is elaborated, offering transparency and insight into the model’s foundations. To demonstrate the practical utility of the theory, we apply it to analyze the combustion of iron particles, providing valuable mathematical perspectives on the underlying processes. The model’s predictions for burn time and temperature align closely with experimental results, offering a partial validation of the theory within the realm of its applicable assumptions. This pioneering work contributes a robust and versatile analytical framework, advancing our understanding of diffusion-limited combustion phenomena of nonvolatile particles.

Online measurement of wear depth based on displacement signal of vertical tester

This paper proposes an accurate and applicable online measurement method of wear depth using displacement signal for vertical testers under mixed lubrication. Firstly, displacement signal is revealed to be composed of temperature drift and Brownian motion. Secondly, the first order Taylor expansion of the Brownian motion is revealed to be the unified form of wear models. Further, the online measurement method is established by introducing non-integer order and amplitude coefficient to eliminate systematic randomness. Finally, the method is verified by graphite-metal tribo-pairs on a standard tester and a high speed friction and wear testing bench, relative errors between online and offline measurement results are less than 10 %, extension errors by non-integer order within its application range are less than 2 %.

Efficient numerical modeling of time-fractal tangent hyperbolic fluid flow with heat and mass transfer

This paper introduces a precise method for solving time-fractal parabolic equations specifically designed to represent complex physical phenomena using fractal calculus. The suggested technique attains third-order accuracy by utilizing the fractal Taylor series and implements a concise scheme for spatial discretization. Stability and convergence analyses are offered for scalar and system fractal parabolic equations. The mathematical model analyzes fluid flow over both flat and oscillatory surfaces, considering the effects of heat and mass transfer. This involves transforming the governing equations into dimensionless partial differential equations, which are then solved using the suggested method. From the calculated results, it can be verified that the proposed scheme produces less error than the existing two schemes. This study enhances computational techniques for fractal calculus and introduces novel opportunities for comprehending non-Newtonian fluids in diverse physical contexts.

A data-driven probabilistic evaluation method of hydrogen fuel cell vehicles hosting capacity for integrated hydrogen-electricity network

The growing demand for hydrogen in hydrogen fuel cell vehicles (HFCVs) will present challenges for the safe operation of the integrated distributed hydrogen supply network (DHSN) and power distribution network (PDN). This paper proposes a data-driven probabilistic evaluation method for determining the hosting capacity of hydrogen fuel cell vehicles (HVHC). Firstly, a directional mapping method is proposed to model the maximum safety boundaries of critical electrolyzers. It integrates the thermal-electrical and dynamic operating characteristics while maintaining the full boundaries in dimension reduction to ensure accuracy and efficiency. A probabilistic HVHC evaluation model is developed to consider uncertain factors of high dimensions despite data deficiency, such as HFCV refueling demand and renewable power. The proposed model determines the maximum network tolerance for the HFCV number, constrained by the safety constraints of the PDN integrated with DHSN, on-site and off-site hydrogen supplies coupled by tube trailers. Finally, a cross-term decoupled data-driven polynomial chaos expansion is proposed to efficiently solve the developed probabilistic HVHC model. It is established based on raw and small samples without extracted probability distribution information. The solution approach also combines the cross-terms in expansions using Taylor expansion, making it efficient for high-dimensional problems. Furthermore, the accuracy and scale reduction effect of the solution algorithm are proven based on Wasserstein ambiguity sets. Numerical studies on three systems show that the proposed method has only a 0.01% error in HVHC results and an 8.38% computation time of the Monte Carlo method.

Bond Lengths and Dipole Moments of Diatomic Molecules under Isotropic Pressure with the XP-PCM and GOSTSHYP Models.

While high-pressure chemistry has a well-established history, methods to simulate pressure at the single-molecule level have been somewhat lacking. The current work aims at comparing two static models (XP-PCM and GOSTSHYP) to apply isotropic pressure to single molecules, focusing on the equilibrium bond length and electric dipole moment of diatomic molecules. Numerical challenges arising in the potential energy surface using the XP-PCM method were examined, and a pragmatic approach was followed to mitigate these. The definition of the cavity was scrutinized, and two approaches to retrieve the isotropic character that could potentially be lost when using the standard methodology were suggested. Subsequently, equilibrium bond lengths under pressure were evaluated, showing reasonable agreement between GOSTSHYP and XP-PCM, but some discrepancies persist. A Taylor series analysis introduced elsewhere was then applied to rationalize the observed trends in terms of the bond surface. Finally, the dipole moment was shown to be highly sensitive to the cavity definition, and qualitative agreement necessitates the use of our adapted procedure.

Variational consistent one-point integration with Taylor's expansion-based stabilization in the second-order meshfree Galerkin method for strain gradient elasticity

A generalized variational principle with five independent variables is proposed for strain gradient elasticity, including displacement, strain, strain gradient, stress, and double stress. Based on the principle, a one-point integration scheme is designed for the second order meshfree Galerkin method through nodal smoothed derivatives and their high order derivatives by Taylor's expansion. Since the proposed integration scheme meets the orthogonality conditions, it is variational consistent. The weak form expanded with Taylor's polynomials can be well evaluated by nodal smoothed derivatives and their high order derivatives on one quadrature point. Numerical one- and two-dimensional case studies show that the proposed integration scheme performs better than the standard Gaussian integration method in terms of accuracy, convergence, efficiency, and stability.

A New Perspective for Scientific Modelling: Sparse Reconstruction Based Approach for Learning Time-Space Fractional Differential Equations

Abstract This paper studies a sparse reconstruction based approach to learn time-space fractional differential equations, i.e. to identify parameter values and particularly the order of the fractional derivatives. The approach uses a generalized Taylor series expansion to generate, in every iteration, a feature matrix which is used to learn the fractional orders of both, temporal and spatial derivatives by minimizing the LASSO operator using Differential Evolution algorithm. To verify the robustness of the method, numerical results for time-space fractional diffusion equation, wave equation, and Burgers? equation at different noise levels in the data are presented. Finally, the methodology is applied to a realistic example where underlying fractional differential equation associated to published experimental data obtained from an in-vitro cell culture assay is learned.

Enhancements Self-Scaling Quasi-Newton for Unconstrained Optimization

A self-scaling for the quasi-Newton tecnique is derive by using a second_order Taylor's expansion to achieve optimal computational performance. Following this, new updating formulas for the quasi-Newton method are introduced based on the newly derived self-scaling equation. The numerical results confirm this derivation and suggest that the new method could potentially rival the BFGS method in terms of performance.

Enhancements Self-Scaling Quasi-Newton for Unconstrained Optimization

A self-scaling for the quasi-Newton tecnique is derive by using a second_order Taylor's expansion to achieve optimal computational performance. Following this, new updating formulas for the quasi-Newton method are introduced based on the newly derived self-scaling equation. The numerical results confirm this derivation and suggest that the new method could potentially rival the BFGS method in terms of performance.

Quick and effective approximation of in silico saturation mutagenesis experiments with first-order taylor expansion

To understand the decision process of genomic sequence-to-function models, explainable AI algorithms determine the importance of each nucleotide in a given input sequence to the model's predictions and enable discovery of cis-regulatory motifs for gene regulation. The most commonly applied method is in silico saturation mutagenesis (ISM) because its per-nucleotide importance scores can be intuitively understood as the computational counterpart to invivo saturation mutagenesis experiments. While ISM is highly interpretable, it is computationally challenging to perform for many sequences, and becomes prohibitive as the length of the input sequences and size of the model grows. Here, we use the first-order Taylor approximation to approximate ISM values from the model's gradient, which reduces its computation cost to a single forward pass for an input sequence. We show that the Taylor ISM (TISM) approximation is robust across different model ablations, random initializations, training parameters, and dataset sizes.

Residual Power Series Method for Solving Linear Complex Differential Equations

This work uses the residual power series method (RPSM), primarily based on the general Taylor series formula with a residual error function, to offer solutions to instances of linear complex differential equations (LCDEs) as test problems. After finding approximate solutions to those problems, the current approach's findings have been compared with the exact solutions as shown in the tables and figures, which show the suggested method's reliability, precision, and rapid convergence. So, the importance of this research lies in demonstrating the method's ability to find an approximate solution close to the exact solution for complex differential equations.

RSTC: Residual Swin Transformer Cascade to approximate Taylor expansion for image denoising

Traditional denoising methods establish mathematical models by employing different priors, which can achieve preferable results but they are usually time-consuming and their outputs are not adaptive on regularization parameters. While the success of end-to-end deep learning denoising strategies depends on a large amount of data and lacks a theoretical interpretability. In order to address the above problems, this paper proposes a novel image denoising method, namely Residual Swin Transformer Cascade (RSTC), based on Taylor expansion. The key procedures of our RSTC are specified as follows: Firstly, we discuss the relationship between image denoising model and Taylor expansion, as well as its adjacent derivative parts. Secondly, we use a lightweight deformable convolutional neural network to estimate the basic layer of Taylor expansion and a residual network where swin transformer block is selected as a backbone for pursuing the solution of the derivative layer. Finally, the results of the two networks contribute to the approximation solution of Taylor expansion. In the experiments, we firstly test and discuss the selection of network parameters to verify its effectiveness. Then, we compare it with existing advanced methods in terms of visualization and quantification, and the results show that our method has a powerful generalization ability and performs better than state-of-the-art denoising methods on performance improvement and structure preservation.

Magnetic field interference suppression method based on phase-compensation vector resonance control in near-zero magnetic field environment

The traditional PI controller has limited ability to suppress magnetic field noise with high power spectral density and specific frequencies, leading to degradation in magnetocardiographic imaging quality. A phase-compensated vector resonance controller (PVRC) is proposed to suppress these disturbances. Firstly, by analyzing the coupling effect between the magnetic field and the ferromagnetic boundary of high permeability materials, the magnetic field response is divided into a first-order inertia link and a delay link. Then, the PVRC controller compensates for the phase lag of the disturbance in two stages. Additionally, the discretization method of PVRC is improved by replacing the cosine function with a Taylor series expansion to enhance response speed. The interference suppression effect of PI-PVRC is 27.5% higher than that of PI control.

Dynamic multifield continualization of multilayered lattice-like metamaterials

This work focuses on dynamic continualization of multifield multilayered metamaterials in order to obtain energetically-consistent models able to provide an accurate description of the dispersive behavior of the corresponding discrete system. Continuum models, characterized by constitutive and inertial non-localities, have been identified through a recently proposed enhanced continualization scheme. They are identified by governing equations both of the integro-differential and higher-order gradient-type, whose regularization kernel or pseudo-differential functions accounting for shift operators are formally expanded in Taylor series. The adopted regularization kernel exhibits polar singularities at the edge of the first Brillouin zone, thus assuring the convergence of the frequency spectrum to the one of the Lagrangian system in the entire wave vector domain. The validity of the proposed approach is assessed through the investigation of multilayered discrete lattices with an antitetrachiral topology, where local resonators act as rigid links among the layers. The convergence of dispersion curves of the continuum model to the ones of the Lagrangian model is proved in the whole first Brillouin zone as the adopted continualization order increases, both considering the propagation and the spatial attenuation of Bloch waves inside the metamaterial. A low frequency continualization is also provided, leading to the identification of a first-order medium.

A hybrid model for state of charge estimation of lithium-ion batteries utilizing improved adaptive extended Kalman filter and long short-term memory neural network

With lithium-ion batteries being utilized in all aspects of life, accurately estimating the state of charge (SOC) of a battery has become a key issue in battery management systems. In this paper, an improved hybrid model based on adaptive extended Kalman filter (AEKF) and improved long short-term memory (ILSTM) neural network is proposed. The proposed model is based on a second-order RC equivalent circuit model, and the dynamic forgetting factor recursive least squares (DFFRLS) and AEKF algorithms are utilized to obtain the initial SOC estimates. And the estimation error in the AEKF algorithm due to neglecting the higher order terms of the Taylor expansion equations is compensated by the improvement of the LSTM network. The results under different working conditions indicate that the SOC estimation of the hybrid model has good convergence and high system robustness. The maximum error (MAX) of this algorithm is less than 2.3 %, especially the root mean square error (RMSE) and mean absolute error (MAE) are less than 0.84 % and 0.65 %, respectively.

Closed-Form Method for Unified Far-Field and Near-Field Localization Based on TDOA and FDOA Measurements

When the near-field and far-field information of a target is uncertain, it is necessary to choose a suitable localization method. The modified polar representation (MPR) method integrates the two scenarios and achieves a unified localization with direction of arrival (DOA) estimation in the far field and position estimation in the near field. Previous studies have only proposed solutions for stationary environments and have not considered the motion factor. Therefore, this paper proposes a new unified positioning algorithm using multi-sensor time difference of arrival (TDOA) and frequency difference of arrival (FDOA) measurements without prior target source information. The method represents the position of the target source using MPR and describes the localization problem as a weighted least squares (WLS) problem with two constraints. We first obtain the initial estimates by WLS without considering the constraints and then investigate a two-step error correction method based on the constraints. The first step corrects the initial estimate using the Taylor series expansion technique, and the second step corrects the DOA estimate in the previous step using the direct error compensation technique based on the properties of the second constraint. Simulation experiments show that the method is effective for the unified positioning of moving targets and can achieve the Cramer–Rao lower bound (CRLB).

Finite Element Analysis in The Investigation of Methods to Model AC Resistance in A Coated Conductor

In our electricity dependent and rapidly automated society, the manipulation and usage of electricity to transmit energy is of significant prevalence. In particular, the application of current in high frequency transformers, inductors and motor generators presents a unique efficiency problem: The loss of energy through the spiking of effective resistance intrinsically related to the skin and proximity effects of an AC current. By the usage of Finite Element Analysis (FEA) in the simulative investigation of various variables on AC resistance, this paper seeks to provide insights into how coated conductors might offer advantages over the standard conductor in certain conditions, thus guiding future design and application in high-efficiency electronics. The papers findings reveal that adding a layer of coating with a lower conductivity than the core material could offer reduced energy losses up to the point when the skin depth matches the coating’s thickness. Moreover, the study explored the influence of the coating’s conductivity, suggesting that an increased conductivity of the coating material could further minimize copper losses, albeit with the trade-off of potentially higher costs. The study concludes with a Taylor Series approximation of the relationship between AC resistance and frequency using simulative data.