Search

We consider three-dimensional doubly periodic steady water waves with vorticity, under the action of gravity and surface tension; in particular we consider so-called Beltrami flows, for which the velocity field and the vorticity are collinear. We adapt a recent formulation of the corresponding problem for localised waves which involves a generalisation of the classical Dirichlet–Neumann operator. We study this operator in detail, extending some well-known results for the classical Dirichlet–Neumann operator, such as the Taylor expansion in homogeneous powers of the wave profile, the computation of its differential and the asymptotic expansion of its associated symbol. A new formulation of the problem as a single equation for the wave profile is also presented and discussed in a similar vein. As an application of these results we prove existence of doubly periodic gravity-capillary steady waves and construct approximate doubly periodic gravity steady waves. [ABSTRACT FROM AUTHOR]

The displacement function was constructed using Carrera unified formulation (CUF) in combination with Taylor polynomials and the improved Fourier series method (IFSM). This study investigated the vibration characteristics of cylindrical beam made of functionally graded materials (FGM) in a thermal environment. The accuracy of the theoretical model was verified by comparing the calculation results with those obtained using the finite element method. Subsequently, the influential factors were investigated. The variations in material properties with temperature and volume rate index were studied. Additionally, the effects of temperature changes and volume rate index on the thermoelastic vibration of the FGM cylindrical beam were examined. The results indicate that in a uniform temperature field, temperature and volume rate index variations result in changes in the material's physical properties. The modal frequency of the FGM cylindrical beam decreases gradually with increasing temperature and volume rate index. [ABSTRACT FROM AUTHOR]

The complexity of the three-dimensional entry trajectory optimization problem has escalated due to the need to liberalize the angle of attack and bank angle as control variables, thereby enhancing the inherent maneuverability and control capabilities of lifting-body vehicles. The difference-of-convex (DC) properties inherent in the constraints of the problem are exploited in this paper. A DC decomposition approach is utilized to address the nonlinear auxiliary control equations, and the DC relaxation technique is applied to resolve iteration infeasibilities arising from Taylor expansion. The dependence on the initial trajectory is diminished by the implementation of an exact penalty method, thus improving the applicability of the methods. Furthermore, a control variable oscillation suppression mechanism has been constructed to tackle the control variable oscillation issues arising from the relaxation of the angle of attack and bank angle. This mechanism effectively suppresses large jumps in the angle of attack and high-frequency oscillations in the bank angle. Two novel successive DC programming methods are proposed: the successive concave-convex procedure and the successive proximal bundle method, functioning independently of trust-region constraints. Numerical experiments have demonstrated that the two proposed successive DC optimization methods exhibit exceptional performance in accuracy, feasibility, adaptability, and low sensitivity to initial values when applied to solving the three-dimensional entry trajectory optimization problem. [ABSTRACT FROM AUTHOR]

The event‐triggered control of Markov jump systems has attracted more and more interest in field control. However, the problem of how to design a transition probability‐dependent event‐triggered mechanism and controller has not been fully considered. This paper investigates the problem of event‐triggered control for Lipschitz nonlinear Markov jump systems. Through Taylor series expansion, a linear auxiliary system is constructed to obtain the approximate state, whose system matrices are described by the probability‐weighted matrices of nonlinear Markov jump systems. By redefining the measurement error as the difference between the current state and the approximate state, a probability‐dependent event‐triggered mechanism is designed for Markov jump systems. The effectiveness of the developed approach is illustrated by two comparison examples. [ABSTRACT FROM AUTHOR]

We consider the zero-dimensional quartic O(N) vector model and present a complete study of the partition function Z(g, N) and its logarithm, the free energy W(g, N), seen as functions of the coupling g on a Riemann surface. We are, in particular, interested in the study of the transseries expansions of these quantities. The point of this paper is to recover such results using constructive field theory techniques with the aim to use them in the future for a rigorous analysis of resurgence in genuine quantum field theoretical models in higher dimensions. Using constructive field theory techniques, we prove that both Z(g, N) and W(g, N) are Borel summable functions along all the rays in the cut complex plane C π = C \ R - . We recover the transseries expansion of Z(g, N) using the intermediate field representation. We furthermore study the small-N expansions of Z(g, N) and W(g, N). For any g = | g | e ı φ on the sector of the Riemann surface with | φ | < 3 π / 2 , the small-N expansion of Z(g, N) has infinite radius of convergence in N, while the expansion of W(g, N) has a finite radius of convergence in N for g in a subdomain of the same sector. The Taylor coefficients of these expansions, Z n (g) and W n (g) , exhibit analytic properties similar to Z(g, N) and W(g, N) and have transseries expansions. The transseries expansion of Z n (g) is readily accessible: much like Z(g, N), for any n, Z n (g) has a zero- and a one-instanton contribution. The transseries of W n (g) is obtained using Möbius inversion, and summing these transseries yields the transseries expansion of W(g, N). The transseries of W n (g) and W(g, N) are markedly different: while W(g, N) displays contributions from arbitrarily many multi-instantons, W n (g) exhibits contributions of only up to n-instanton sectors. [ABSTRACT FROM AUTHOR]

Let f be a Gaussian random field on R d and let X be the number of critical points of f contained in a compact subset. A long-standing conjecture is that, under mild regularity and non-degeneracy conditions on f, the random variable X has finite moments. So far, this has been established only for moments of order lower than three. In this paper, we prove the conjecture. Precisely, we show that X has finite moment of order p, as soon as, at any given point, the Taylor polynomial of order p of f is non-degenerate. We present a simple and general approach that is not specific to critical points and we provide various applications. In particular, we show the finiteness of moments of the nodal volumes and the number of critical points of a large class of smooth, or holomorphic, Gaussian fields, including the Bargmann-Fock ensemble. [ABSTRACT FROM AUTHOR]

In this work, Korkine and Sonin’s identities are defined on arbitrary time scales. These identities are utilized to establish the Chebyshev and Grüss-type inequalities on time scales. By applying these inequalities, we determine the bounds of the remainders in Montgomery identities that incorporate Taylor’s formula on time scales. Moreover, we derive discrete and quantum inequalities based on these results. [ABSTRACT FROM AUTHOR]

In this paper, the cascade architecture of spline adaptive filtering (CSAF) for nonlinear systems is presented with the normalized version of orthogonal gradient adaptive (NOGA) algorithm. Spline adaptive filtering comprises a sandwich of the first linear adaptive filtering (LAF) and nonlinear adaptive look-up table. In this cascading architecture, SAF is connected to the second LAF. NOGA is considered as the fast convergence applied by stochastic gradient-based approach. Convergence properties of the proposed NOGACSAF algorithm in terms of instantaneous errors can be derived by using Taylor series expansion. Experimental results demonstrate the effectiveness of proposed NOGA-CSAF algorithm using the mean square error scheme. It clearly outperforms the traditional least mean square algorithm on CSAF model in the nonlinear identification system. [ABSTRACT FROM AUTHOR]

This work presents a comprehensive framework for the sensitivity analysis of the Navier–Stokes equations, with an emphasis on the stability estimate of the discretized first‐order sensitivity of the Navier–Stokes equations. The first‐order sensitivity of the Navier–Stokes equations is defined using the polynomial chaos method, and a finite element‐volume numerical scheme for the Navier–Stokes equations is suggested. This numerical method is integrated into the open‐source industrial code TrioCFD developed by the CEA. The finite element‐volume discretization is extended to the first‐order sensitivity Navier–Stokes equations, and the most significant and original point is the discretization of the nonlinear term. A stability estimate for continuous and discrete Navier–Stokes equations is established. Finally, numerical tests are presented to evaluate the polynomial chaos method and to compare it to the Monte Carlo and Taylor expansion methods. [ABSTRACT FROM AUTHOR]

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). [ABSTRACT FROM AUTHOR]

Dielectric elastomer membrane has been applied in a wide variety of engineering fields for their excellent dynamic performances. Functionally graded graphene nanoplatelet-reinforced composite (FG-GNPRC) shows great potential for developing high-performance and multifunctional structures. In this work, the damped nonlinear vibration of the FG-GNPRC dielectric membrane is investigated. The effects of damping and dielectric properties are considered in terms of energy while deriving governing equations. Taylor series expansion and differential quadrature together with direct iterative methods are used to discretize and numerically solve the obtained governing equations. The developed model and numerical solution are verified by comparing present results to existing studies. The influences of functionally graded distribution, damping, stretching ratio, dimensions of the membrane, and the attributes of the electrical field and GNP fillers on the nonlinear vibration of the structure are comprehensively investigated. It is found that compared to a uniform distribution, the membrane with functional distribution of GNP demonstrates exhibits more robust performances. The membrane with a smaller thickness-to-radius ratio is more sensitive to the external electrical field. In addition, the direct current voltage is evidenced to have a more significant effect on the nonlinear vibration when the membrane is subject to a relatively small stretching ratio. The present work suggests that the structural behavior of the FG-GNPRC membrane can be actively adjusted by varying the stretching ratio and the properties of the electrical field and the GNP filler. Effects of damping and dielectric properties are considered for structural behavior in terms of energy. Taylor series expansion and differential quadrature methods are combined to solve governing equations. Two transition regions in frequency ratio are identified due to AC frequency-facilitated polarization and electron tunneling. The effect of the electric field on nonlinear vibration can be regulated by varying the stretching ratio. [ABSTRACT FROM AUTHOR]

The discrete element method (DEM) is a vital numerical approach for analyzing the mechanical behavior of elastoplastic wet sand. However, parameter uncertainty persists within the mapping between constitutive relationships and inherent model parameters. We propose a Parameter calibration neural network based on Attention, Retention, and improved Transformer for Sequential data (PartsNet), which effectively captures the nonlinear mechanical behavior of wet sand and obtains the optimal parameter combination for the Edinburgh elasto-plastic adhesion constitutive model. Variational autoencoder-based principal component ordering is employed by PartsNet to reduce the high-dimensional dynamic response and extract critical parameters along with their weights. Gated recurrent units are combined with a novel sparse multi-head attention mechanism to process sequential data. The fusion information is delivered by residual multilayer perceptron, achieving the association between sequential response and model parameters. The errors in response data generated by calibrated parameters are quantified by PartsNet based on adaptive differentiation and Taylor expansion. Remarkable calibration capabilities are exhibited by PartsNet across six evaluation indicators, surpassing seven other deep learning approaches in the ablation test. The calibration accuracy of PartsNet reaches 91.29%, and MSE loss converges to 0.000934. The validation experiments and regression analysis confirmed the generalization capability of PartsNet in the calibration of wet sand. The improved sparse attention mechanism optimizes multi-head attention, resulting in a convergence speed of 21.25%. PartsNet contributes to modeling and simulating the precise mechanical properties of complex elastoplastic systems and offers valuable insights for diverse engineering applications. [ABSTRACT FROM AUTHOR]

We present an approximation method based on the mixing formula [Hull, J. and White, A., The pricing of options on assets with stochastic volatilities. J. Finance, 1987, 42, 281–300; Romano, M. and Touzi, N., Contingent claims and market completeness in a stochastic volatility model. Math. Finance, 1997, 7, 399–412] for pricing European options in Barndorff-Nielsen and Shephard models. This approximation is based on a Taylor expansion of the option price. It is implemented using a recursive algorithm that allows us to obtain closed form approximations of the option price of any order (subject to technical conditions on the background driving Lévy process). This method can be used for any type of Barndorff-Nielsen and Shephard stochastic volatility model. Explicit results are presented in the case where the stationary distribution of the background driving Lévy process is inverse Gaussian or gamma. In both of these cases, the approximation compares favorably to option prices produced by the characteristic function. In particular, we also perform an error analysis of the approximation, which is partially based on the results of Das and Langrené [Closed-form approximations with respect to the mixing solution for option pricing under stochastic volatility. Stochastics, 2022, 94, 745–788]. We obtain asymptotic results for the error of the $ N{{\rm th}} $ N th order approximation and error bounds when the variance process satisfies an inverse Gaussian Ornstein–Uhlenbeck process or a gamma Ornstein–Uhlenbeck process. [ABSTRACT FROM AUTHOR]

In a recent paper Li et al. (Z Angew Math Phys 73:52, 2022. https://doi.org/10.1007/s00033-022-01681-4) have considered a generalized nonlinear Schrödinger equation which has extensive applications in various fields of physics and engineering. After proving Liouville integrability of this equation, they investigated the phenomenon of the modulational instability for the possible reason of the formation of the rogue waves. By means of the generalized ( 2 , N - 2 )-fold Darboux transformation, authors presented several mixed localized wave solutions, such as breathers, rogue waves and semi-rational solitons for their model equation, and accurately analyzed a number of important physical quantities. It is the aim of this Comment to point out that (i) the baseband modulation instability was developed in a wrong way and (ii) one of the two different types of Taylor series expansions for solution of Lax pair used in that article for building analytical solutions, especially the one obtained with ξ j = Z does not correspond to any solution of the spectral problem (2.1) when using u 0 x , t as the seed solution. Consequently, all mixed localized solutions that involve the mentioned Taylor series are invalid. [ABSTRACT FROM AUTHOR]

AbstractThis article uses Carrera unified formulation (CUF) to analyze the dynamics characteristics of composite shaft with consideration of thermal effects, axial loads, and rotational speeds. The shaft radial displacement is described by Taylor expansion terms, and the axial displacement is described by improved Fourier series. Hamilton’s principle is used to obtain the control equations. The calculation results indicate that: The increase of temperature can reduce composite shaft natural frequency, and the influence is not obvious when the layer angle is small; The effect of axial load on composite shaft natural frequency is roughly linear, but the influence varies under different boundaries, length-radius ratio, layer orders; Due to the gyroscopic effect, the natural frequency of the composite shaft exhibits two states under the action of rotation speed: forward and backward mode; In forward mode, the influence of temperature and axial load on the composite shaft still follows the previous conclusion; In the backward mode, when the rotational speed exceeds the critical speed, the influence of temperature and axial load on the composite shaft shows the opposite conclusion. These findings are of great significance for the practical application of composite shaft in engineering. [ABSTRACT FROM AUTHOR]

In this paper, a class of scalar quartic polynomial delay systems is investigated. We found rich dynamics in this system through numerical simulation, including chaotic attractors, chaotic saddles, and intermittent chaos. Moreover, this chaotic quartic system may serve as an approximation, through Taylor expansion, for a wide class of scalar delay differential equations. Thus, these nonlinear systems may exhibit chaotic behaviors, and the studies in our paper may provide an insight into the emergence of chaos in other time-delay nonlinear systems. We also conduct a detailed theoretical analysis of the system, including the stability of equilibria and Hopf bifurcation analysis based on the theory of normal form and center manifold. Additionally, a numerical analysis is provided to give numerical evidence for the existence of chaos. [ABSTRACT FROM AUTHOR]

This paper proposes a novel approach for broadband acoustic shape sensitivity analysis based on the direct differentiation approach. Since the system matrices of the boundary element method (BEM) for the analysis of acoustic state and acoustic sensitivity have frequency dependence, repeated calculations are needed at different frequencies. This is very time-consuming, especially for sensitivity calculations used in shape optimization design. The Taylor series expansion of the Hankel function is carried out to separate the frequency-dependent and frequency-independent terms in the acoustic shape sensitivity boundary integral equation to construct a frequency-independent system matrix. In addition, due to the formation of asymmetric full-coefficient matrices in acoustic shape sensitivity equations based on the BEM, repeatedly solving system equations is also extremely time-consuming at broadband frequencies for large scale issues. The second-order Arnoldi approach was employed to create a reduced-order model that maintains the key features of the initial full-order model. The strong singular and supersingular integrals within the sensitivity equations can be calculated directly utilizing the singularity elimination technique. Finally, several numerical examples confirm the accuracy and efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Triangular satellite constellations are created by three satellites which are linked through tethers. Such formations serve distributed observation space missions or the creation of high-density satellite communication networks over high-frequency bands. In a tethered multi-satellite system consisting of three-satellites in triangular formation the associated dynamic model exhibits strong nonlinearities. Stabilization and precise positioning of the multi-satellite constellation is a nontrivial task and the solution of the associated nonlinear control problem is an elaborated procedure. In this article a novel nonlinear optimal control method is applied to the above-noted model the tethered multi-satellite system in triangular formation. First, the state-space model of the triangular tethered multi-satellite formation undergoes approximate linearization around a temporary operating point that is recomputed at each time-step of the control method. The linearization relies on Taylor series expansion and on the associated Jacobian matrices. For the linearized state-space model of the tethered satellites a stabilizing optimal (H-infinity) feedback controller is designed. This controller stands for the solution of the nonlinear optimal control problem under model uncertainty and external perturbations. To compute the controller's feedback gains an algebraic Riccati equation is repetitively solved at each iteration of the control algorithm. The stability properties of the control method are proven through Lyapunov analysis. The proposed nonlinear optimal control approach achieves fast and accurate tracking of setpoints under moderate variations of the control inputs and a minimum dispersion of energy when changing the position of the satellites in their triangular tethered formation. [ABSTRACT FROM AUTHOR]

Copyright of Systems Engineering & Electronics is the property of Journal of Systems Engineering & Electronics Editorial Department and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Affected by fault structure and in situ stress, the heterogeneity of the permeability of surrounding rock is universal. Treating it as a fixed value will reduce the prediction accuracy of water inflow and structural head. In view of this problem, considering the coupling effect of ground stress and fault structure, the permeability of surrounding rock is regarded as a spatially discrete type, a plane one-dimensional seepage calculation model in the vertical section is constructed, and the phreatic surface drop curve equation is established. Using Taylor's formula and series expansion theorem, the equation can be reduced to the expression of the falling curve when the permeability of the surrounding rock is homogeneous. Based on Darcy's law and the law of conservation of fluid mass, the calculation formula for tunnel water inflow and external water pressure of the structure was derived and verified through ongoing construction projects. Research shows that the calculation error of water inflow can be reduced from 23.1% to 7.5% when considering the influence of ground stress and fault structure on the permeability of surrounding rock, and the calculation error of water head borne by the supporting structure can be reduced from 43.8% to 30%, which improves the prediction accuracy. Thematic collection: This article is part of the Climate change and resilience in Engineering Geology and Hydrogeology collection available at: https://www.lyellcollection.org/topic/collections/climate-change-and-resilience-in-engineering-geology-and-hydrogeology [ABSTRACT FROM AUTHOR]

This study introduces an innovative model-order reduction (MOR) technique that integrates boundary element and finite element methodologies, streamlining the analysis of wideband vibro-acoustic interactions within aquatic and aerial environments. The external acoustic phenomena are efficiently simulated via the boundary element method (BEM), while the finite element method (FEM) adeptly captures the dynamics of vibrating thin-walled structures. Furthermore, the integration of isogeometric analysis within the finite element/boundary element framework ensures geometric integrity and maintains high-order continuity for Kirchhoff–Love shell models, all without the intermediary step of meshing. Foundational to our reduced-order model is the application of the second-order Arnoldi method coupled with Taylor expansions, effectively eliminating the frequency dependence of system matrices. The proposed technique significantly enhances the computational efficiency of wideband vibro-acoustic coupling analyses, as demonstrated through numerical simulations. [ABSTRACT FROM AUTHOR]

Here we study the roots of the doubly infinite family of Jensen polynomials J PL d , n (x) associated to MacMahon's plane partition function PL (n) . Recently, Ono et al. [Ono et al. in Adv Math 409:108692, 2022] proved that PL (n) is log-concave for all n ≥ 12 , which is equivalent to the polynomials J PL 2 , n (x) having real roots. Moreover, they proved, for each d ≥ 2 , that the J PL d , n (x) have all real roots for sufficiently large n. Here we make their result effective. Namely, if N PL (d) is the minimal integer such that J PL d , n (x) has all real roots for all n ≥ N PL (d) , then we show that N PL (d) ≤ 279928 × d (d - 1) 6 d 3 (22.2) 3 (d - 1) 2 2 d e Γ (2 d 2) (2 π) 2 d + 2 . Moreover, using the ideas that led to the above inequality, we explicitly prove that N PL (3) = 26 , N PL (4) = 46 , N PL (5) = 73 , N PL (6) = 102 and N PL (7) = 136 . [ABSTRACT FROM AUTHOR]

In many real-time kinematic (RTK) positioning applications, reference observations are transmitted over wireless links that can experience frequent interruptions or substantial delays. This results in large differential ages between base and rover observations, which, in turn, leads to a deterioration in positioning performance. To bridge the significant age difference, in this work, we propose a simple and effective scheme for modeling and compensating for such errors. Firstly, the overall differential age error was modeled using truncated Taylor expansion. Then, a time-differenced carrier phase (TDCP)-based observation model was established to estimate the errors with the Kalman framework. Since estimating the receiver's clock error is unnecessary, equivalent transformation and sequential filtering technology were adopted to significantly reduce the computational complexity. Furthermore, a predictor performance monitor was introduced to mitigate the integrity risks that may occur due to model mismatches. The effectiveness of this scheme was validated by static and dynamic field experiments. The static experiment results showed that when the differential age was 60 s, the GPS and BDS satellites' overall root mean square error (RMSE) with the asynchronous RTK (ARTK) prediction method was 2.8 and 5.5 times that of the proposed method, respectively. Moreover, when the differential age was 120 s, these values were 3.3 and 5.4 times that of the proposed method, respectively. The field experiment results showed that when the differential age was 60 s, the integer ambiguity fixed rate and false fixed rate of the ARTK method were 0.90 and 1.63 times that of the proposed method, respectively. Finally, at a 120 s differential age, these values were 0.78 and 4.78 times that of the proposed, respectively. [ABSTRACT FROM AUTHOR]

We propose a new class of high‐order time‐marching schemes with dissipation control and unconditional stability for parabolic equations. High‐order time integrators can deliver the optimal performance of highly accurate and robust spatial discretizations such as isogeometric analysis. The generalized‐α$$ \alpha $$ method delivers unconditional stability and second‐order accuracy in time and controls the numerical dissipation in the discrete spectrum's high‐frequency region. We extend the generalized‐α$$ \alpha $$ methodology to obtain high‐order time marching methods with high accuracy and dissipation control in the discrete high‐frequency range. Furthermore, we maintain the original stability region of the second‐order generalized‐α$$ \alpha $$ method in the new higher‐order methods; we increase the accuracy of the generalized‐α$$ \alpha $$ method while keeping the unconditional stability and user‐control features on the high‐frequency numerical dissipation. The methodology solves k>1,k∈ℕ$$ k>1,k\in \mathbb{N} $$ matrix problems and updates the system unknowns, which correspond to higher‐order terms in Taylor expansions to obtain (3/2k)th$$ \left(3/2k\right)\mathrm{th} $$‐order method for even k$$ k $$ and (3/2k+1/2)th$$ \left(3/2k+1/2\right)\mathrm{th} $$‐order for odd k$$ k $$. A single parameter ρ∞$$ {\rho}^{\infty } $$ controls the high‐frequency dissipation, while the update procedure follows the formulation of the original second‐order method. Additionally, we show that our method is A‐stable, and for ρ∞=0$$ {\rho}^{\infty }=0 $$ we obtain an L‐stable method. Furthermore, we extend this strategy to analyze the accuracy order of a generic method. Lastly, we provide numerical examples that validate our analysis of the method and demonstrate its performance. First, we simulate heat propagation; then, we analyze nonlinear problems, such as the Swift–Hohenberg and Cahn–Hilliard phase‐field models. To conclude, we compare the method to Runge–Kutta techniques in simulating the Lorenz system. [ABSTRACT FROM AUTHOR]

This article presents an innovative approach using the Differential Transform Method (DTM) to analyze the vibration characteristics of cylindrical shells, integrating Taylor's series with Sander's classical theory. It demonstrates DTM's efficiency, accuracy, and potential as an alternative method. The study introduces a novel application of the DTM in exploring the free vibration of cylindrical shells, detailing a technique to address challenges such as normalization, linear solution methodologies, and parameter derivative modifications. A dimensionless parameter analysis evaluates the impact of length, radius, thickness, and modulus of elasticity. Comparative analysis with Hybrid Finite Element Method (FEM) data and validation against existing literature highlights DTM's precision and reliability. In conclusion, DTM offers a robust solution for the eigenvalue problem in coupled differential equations, providing accurate vibration parameters. Additionally, an important relationship between the modulus of elasticity and frequency in the dimensionless state was obtained. [ABSTRACT FROM AUTHOR]

Machine Learning (ML) heavily relies on optimization techniques built upon gradient descent. Numerous gradient-based update methods have been proposed in the scientific literature, particularly in the context of neural networks, and have gained widespread adoption as optimizers in ML software libraries. This paper introduces a novel perspective by framing gradient-based update strategies using the Moreau-Yosida (MY) approximation of the loss function. Leveraging a first-order Taylor expansion, we demonstrate the concrete exploitability of the MY approximation to generalize the model update process. This enables the evaluation and comparison of regularization properties underlying popular optimizers like gradient descent with momentum, ADAGRAD, RMSprop, and ADAM. The MY-based unifying view opens up possibilities for designing new update schemes with customizable regularization properties. To illustrate this potential, we propose a case study that redefines the concept of closeness in the parameter space using network outputs. We present a proof-of-concept experimental procedure, demonstrating the effectiveness of this approach in continual learning scenarios. Specifically, we employ the well-known permuted MNIST dataset, a progressively-permuted MNIST and CIFAR-10 benchmarks, and a non i.i.d. stream. Additionally, we validate the update scheme's efficacy in an offline-learning scenario. By embracing the MY-based unifying view, we pave the way for advancements in optimization techniques for machine learning. [ABSTRACT FROM AUTHOR]

In this paper, we address several scientific and technological challenges with a novel He's non-perturbative approach (NPA), it simplifying processing time compared to traditional methods. The proposed approach transforms nonlinear ordinary differential equations (ODEs) into linear ones, analogous to simple harmonic motion, and producing a new frequency. Studying the periodic solutions leads to enhanced design, performance, reliability, and efficiency across these fields. This new approach is based mainly on the He's frequency formulation (HFF). This method yields highly accurate outcomes, surpassing well-known approximate methodologies, as validated through numerical comparisons in the Mathematical Software (MS). The congruence between numerical solution tests and theoretical predictions further supports our findings. While classical perturbation methods rely on Taylor expansions to simplify restoring forces, the NPA also enables stability analysis. Consequently, for analyzing approximations of highly non-linear oscillators in MS, the NPA serves as a more reliable tool. [ABSTRACT FROM AUTHOR]

When applying the classical multistep schemes for solving differential equations, one often faces the dilemma that smaller time steps are needed with higher-order schemes, making it impractical to use high-order schemes for stiff problems. We construct in this paper a new class of BDF and implicit-explicit schemes for parabolic type equations based on the Taylor expansions at time t n+\beta with \beta > 1 being a tunable parameter. These new schemes, with a suitable \beta, allow larger time steps at higher order for stiff problems than that which is allowed with a usual higherorder scheme. For parabolic type equations, we identify an explicit uniform multiplier for the new second- to fourth-order schemes and conduct rigorously stability and error analysis by using the energy argument. We also present ample numerical examples to validate our findings. [ABSTRACT FROM AUTHOR]

The numerical solution of initial-value problems (IVP) for ordinary differential equations (ODE) is at this time a mature subject, with many high-quality codes freely available. Second-order linear equations without singularities are an especially simple class of problems to solve, even more so if only a single scalar equation such as the Mathieu equation y ′ ′ + (a - 2 q cos 2 x) y = 0 is being considered. Nonetheless, the topic is not yet exhausted, and this paper considers the case of writing an efficient arbitrary-precision code for the solution of such equations. For this purpose, an implicit Hermite–Obreshkov method attains nearly spectral accuracy at a cost only polynomial in the number of bits of accuracy requested. This is interesting for the Mathieu equation in particular because the solutions can be highly oscillatory of variable frequency and be highly ill-conditioned. This paper reports on the details of the prototype Maple implementation of the method and summarizes the approximation theoretic results justifying the choice of a balanced Hermite–Obreshkov method including its backward stability and decent Lebesgue constants. This method may be of especial interest for the solution of so-called D-finite equations, for which Taylor series coefficients up to degree m are available at cost only O(m), instead of the more usual O (m 2) . This paper celebrates the happy occasion of the 90th birthday of John C. Butcher. [ABSTRACT FROM AUTHOR]

This paper develops analytical approximations for volatility index (VIX) option pricing under nonaffine generalized autoregressive conditional heteroskedasticity (GARCH) models as advocated by Christoffersen et al. We obtain the approximation formulae for pricing VIX options and then evaluate their performance with three expansions under four empirically well‐tested models. Our numerical experiments find that the weighted ℒ2 ${{\rm{ {\mathcal L} }}}^{2}$ expansion generated by the fat‐tailed weighting kernel can significantly reduce approximation error over the Gram‐Charlier expansion; the Taylor expansion of conditional moments can lead to divergence for parameters with certain high persistence in the affine GARCH, nonlinear asymmetric GARCH, and Glosten‐Jagannathan‐Runkle GARCH models, while surviving during high persistence in the exponential GARCH. [ABSTRACT FROM AUTHOR]

From both microscopic and macroscopic aspects, we study both discrete and random approximations of the global attractor for a nonlocal lattice dynamical system. First, we establish an abstract Taylor expansion and obtain a global attractor for the continuous-time system. Second, for the discrete-time nonlocal lattice equation via the implicit Euler scheme, we prove the existence of numerical attractors for small time-sizes, and establish the upper semicontinuity of the numerical attractors towards the global attractor as the time-size tends to zero, while the abstract Taylor expansion and error estimation play key roles in the proof. Third, for the random nonlocal lattice equation driven by nonlinear colored noise, we show the existence and upper semicontinuity of random attractors towards the global attractor when the noise-size goes to infinity. Fourth, we establish optimal bounds and several continuities of numerical attractors as well as random attractors under the Hausdorff metric. [ABSTRACT FROM AUTHOR]

This paper proposes algorithms for computing the reduced Gröbner basis of the vanishing ideal of a finite set of points in the frame of ideal interpolation. We also consider the case that the points have multiplicity conditions. First, we introduce the definition of “reverse” reduced team and compute the interpolation monomial basis of a single point ideal interpolation problem; then we translate the interpolation condition functionals into formal power series via Taylor expansion; this will help convert the general ideal interpolation problem to a single point ideal interpolation problem; and finally, the reduced Gröbner basis is read from formal power series by Gaussian elimination. Our algorithm has a polynomial time complexity, and an example is given to illustrate its effectiveness. [ABSTRACT FROM AUTHOR]

In this article, a nonlinear optimal control approach is proposed for the dynamic model of 3-DOF four-cable driven parallel robots (CDPR). To solve the associated nonlinear optimal control problem, the dynamic model of the 3-DOF cable-driven parallel robot undergoes approximate linearization around a temporary operating point that is recomputed at each time-step of the control method. The linearization relies on Taylor series expansion and on the associated Jacobian matrices. For the linearized state-space model of the 3-DOF cable-driven parallel robot a stabilizing optimal (H-infinity) feedback controller is designed. To compute the controller's feedback gains an algebraic Riccati equation is repetitively solved at each iteration of the control algorithm. The stability properties of the control method are proven through Lyapunov analysis. The proposed nonlinear optimal control approach achieves fast and accurate tracking of reference setpoints under moderate variations of the control inputs and a minimum dispersion of energy. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the delayed-advanced Kaldorian business cycle model and consider how time-delay and time-advance affect economic fluctuations. Given the asymptotic stability of the original Kaldor model, we first show that an approximated Kaldor model by a Taylor series expansion can accurately describe the dynamics of the delayed Kaldor model. We also confirm that the delay has a destabilizing effect. When time-delay is replaced with time-advance, we have an advanced Kaldor model. Taking the advanced capital stock formulation, we derive the stability conditions and find that the time-advance has a stabilizing effect. Lastly, we examine these opposite-signed effects in a modified delayed-advanced model. [ABSTRACT FROM AUTHOR]

Summary: In this paper, we lay out a variational framework for correspondence‐based peridynamic (PD) formulations of solid mechanics. Using the framework, we address the numerical instabilities of the original version of correspondence‐based PD by developing a natural stabilization technique that avoids costly bond‐associated approaches and retains the structure of a method with nodal integration. Accuracy, robustness, and efficiency of the proposed naturally stabilized correspondence‐based PD are demonstrated on several computational test cases ranging from linear elastostatics to large deformation elasto‐plasticity. The computational methodology developed is particularly effective for handling materials undergoing nearly‐incompressible deformations. [ABSTRACT FROM AUTHOR]

Objective. Pharmacokinetic parametric images obtained through dynamic fluorescence molecular tomography (DFMT) has ability of capturing dynamic changes in fluorescence concentration, thereby providing three-dimensional metabolic information for applications in biological research and drug development. However, data processing of DFMT is time-consuming, involves a vast amount of data, and the problem itself is ill-posed, which significantly limits the application of pharmacokinetic parametric images reconstruction. In this study, group sparse-based Taylor expansion method is proposed to address these problems. Approach. Firstly, Taylor expansion framework is introduced to reduce time and computational cost. Secondly, group sparsity based on structural prior is introduced to improve reconstruction accuracy. Thirdly, alternating iterative solution based on accelerated gradient descent algorithm is introduced to solve the problem. Main results. Numerical simulation and in vivo experimental results demonstrate that, in comparison to existing methods, the proposed approach significantly enhances reconstruction speed without a degradation of quality, particularly when confronted with background fluorescence interference from other organs. Significance. Our research greatly reduces time and computational cost, providing strong support for real-time monitoring of liver metabolism. [ABSTRACT FROM AUTHOR]

We propose a diffusion approximation method to the continuous-state Markov decision processes that can be utilized to address autonomous navigation and control in unstructured off-road environments. In contrast to most decision-theoretic planning frameworks that assume fully known state transition models, we design a method that eliminates such a strong assumption that is often extremely difficult to engineer in reality. We first take the second-order Taylor expansion of the value function. The Bellman optimality equation is then approximated by a partial differential equation, which only relies on the first and second moments of the transition model. By combining the kernel representation of the value function, we design an efficient policy iteration algorithm whose policy evaluation step can be represented as a linear system of equations characterized by a finite set of supporting states. We first validate the proposed method through extensive simulations in 2 D obstacle avoidance and 2.5 D terrain navigation problems. The results show that the proposed approach leads to a much superior performance over several baselines. We then develop a system that integrates our decision-making framework with onboard perception and conduct real-world experiments in both cluttered indoor and unstructured outdoor environments. The results from the physical systems further demonstrate the applicability of our method in challenging real-world environments. [ABSTRACT FROM AUTHOR]

Future surveys could obtain tighter constraints on the cosmological parameters with the galaxy power spectrum than with the cosmic microwave background. However, the inclusion of multiple overlapping tracers, redshift bins, and more non-linear scales means that generating the necessary ensemble of simulations for model-fitting presents a computational burden. In this work, we combine full-shape fitting of galaxy power spectra, analytical covariance matrix estimates, the massively optimized parameter estimation and data compression (MOPED) method, and the Taylor expansion interpolation of the power spectrum for the first time to constrain the cosmological parameters directly from a state-of-the-art set of galaxy clustering measurements. We find it takes less than a day to compute the analytical covariance while it takes several months to calculate the simulated ones. Combining MOPED with the Taylor expansion interpolation of the power spectrum, we can constrain the cosmological parameters in just a few hours instead of a few days. We also find that even without a priori knowledge of the best-fitting cosmological or galaxy bias parameters, the analytical covariance matrix with the MOPED compression still gives consistent cosmological constraints to within 0.1σ after two iterations. Therefore, the pipeline we have developed here can significantly speed up the analysis for future surveys. [ABSTRACT FROM AUTHOR]

We present a novel 2D oscillator with an asymmetric design and investigate its stable vibration utilizing the Ms-DTM method. Initially, we obtain the equations of motion for the proposed system. Subsequently, by employing Taylor expansion of s i n h z and c o s h z , the derived nonlinear equations are transformed into linear ones, which we solve analytically using the eigenvalues-eigenfunctions technique. Additionally, we solve the obtained nonlinear system using the Ms-DTM method. Lastly, we examine the stability of the nonlinear system by visualizing the closed nature of the phase portrait. [ABSTRACT FROM AUTHOR]

A method without memory as well as a method with memory are developed free of derivatives for solving Banach space valued equations. Their convergence order was established in the scalar case using Taylor expansions and hypotheses on higher order derivatives not appearing on these methods. But this way their applicability is limited. That is why in this paper their ball convergence analysis is provided using only the divided differences of order one that actually appear on these methods. Moreover, we provide computable error distances and uniqueness of the solution results not given before. Our technique is very general, so it can be used to extend the applicability of other methods along the same lines. This way their applicability is expanded. Numerical experiments testing the convergence criteria complete this paper. [ABSTRACT FROM AUTHOR]

We have developed a numerical scheme for the second‐order acoustic wave equation based on the Lie product formula and Taylor‐series expansion. The scheme has been derived from the analytical solution of the wave equation and in the approximation of the time derivative for a wavefield. Through these two equations, we obtained the first‐order differential equation in time, where the time evolution operator of the analytic solution of this differential equation is written as a product of exponential matrices. The new numerical solution using a Lie product formula may be combined with Taylor‐series, Chebyshev, Hermite and Legendre polynomial expansion or any other expansion for the cosine function. We use the proposed scheme combined with the second‐ or fourth‐order Taylor approximations to propagate the wavefields in a recursive procedure, in a stable manner, accurately and efficiently with even larger time steps than the conventional finite‐difference method. Moreover, our numerical scheme has provided results with the same quality as the rapid expansion method but requiring fewer computations of the Laplacian operator per time step. The numerical results have shown that the proposed scheme is efficient and accurate in seismic modelling, reverse time migration and least‐squares reverse time migration. [ABSTRACT FROM AUTHOR]

Gaussian boson sampling (GBS) is considered a candidate problem for demonstrating quantum advantage. We propose an algorithm for the approximate classical simulation of a lossy GBS instance. The algorithm relies on the Taylor series expansion, and increasing the number of terms of the expansion that are used in the calculation yields greater accuracy. The complexity of the algorithm is polynomial in the number of modes given the number of terms is fixed. We describe conditions for the input state squeezing parameter and loss level that provide the best efficiency for this algorithm (by efficient, we mean that the Taylor series converges quickly). In recent experiments that claim to have demonstrated quantum advantage, these conditions are satisfied; thus, this algorithm can be used to classically simulate these experiments. [ABSTRACT FROM AUTHOR]

In this study, a direct position tracking method for non‐circular (NC) signals using distributed passive arrays is proposed. First, we calculate the initial positions of sources using a direct position determination (DPD) approach; next, we transform the tracking into a compensation problem. The offsets of the adjacent time positions are calculated using a first‐order Taylor expansion. The fusion calculation of the noise subspace is performed according to the NC characteristics. Because the proposed method uses the signal information from the previous iteration, it can realize automatic data associations. Compared with traditional DPD and two‐step localization methods, our novel process has lower computational complexity and provides higher accuracy. Moreover, its performance is better than that of the traditional tracking methods. Numerous simulation results support the superiority of our proposed method. [ABSTRACT FROM AUTHOR]

The magnetic levitation (maglev) ball system is a prototypical Single-Input-Single-Output (SISO) system, characterized by its pronounced nonlinearity, rapid response, and open-loop instability. It serves as the basis for many industrial devices. For describing the dynamics of the maglev ball system precisely in the pseudo linear model, the long short-term memory (LSTM) based auto-regressive model with exogenous input variables (LSTM-ARX) is proposed. Firstly, the LSTM network is modified by incorporating the auto-regressive structure with respect to sequence input, allowing it to deduce a locally linearized model without the need for Taylor expansion. Then, the LSTM-ARX model is transformed into a linear parameter varying (LPV) state space model, and upon this foundation, a model predictive controller (MPC) is proposed. Specifically, when deducing the MPC, the deep learning-based model is linearized by fixing its state input at the current state, so that the nonlinear, non-convex optimization problem can be converted to a finite-horizon quadratic programming problem, thereby deriving the explicit form of MPC. To further enhance the efficiency of the controller in real-time control tasks, a predictive functional controller (PFC) is proposed. It employs multiple nonlinear functions to fit the control sequence, thereby reducing the number of decision variables of the on-line optimization problem in MPC. The proposed controller was successfully applied to the real-time control of the maglev ball system. Simulation and real-time control experiments have validated the improvement in transient performance and efficiency of the LSTM-ARX model-based PFC (LSTM-ARX-PFC). • The LSTM-ARX model is proposed, it describes global nonlinearity dynamics and bears pseudo linear structure. • The explicit form of a deep learning based computational efficient MPC is proposed and its stability is proved. • A predictive functional control method is proposed to furtherly facilitate computational efficiency in real-time control. [ABSTRACT FROM AUTHOR]

Industrially, one of the techniques used to enhance the optimum production of petrochemicals is the configuration of process control units such as gas chromatography (G.C. Analyzer) for concentration and thermocouple for temperature in the production system. This research focused on the application of the principles of mass and energy conservation in the development of dynamic or unsteady state models for predicting the behaviour of the system or process involved in the production of ethylene glycol from a reaction involving oxidation of ethylene-to-ethylene oxide which further undergoes hydrolysis to produce ethylene glycol in a continuous stirred tank reactor (CSTR). The high mathematical complexities of the mass and energy balance models of the Process as a result of the reaction kinetic scheme of the nonelementary reaction of the process which integrated both linear and non-linear terms into the models were resolved by the application of the principles of Taylor's series expansion, Laplace Transform, partial fraction and matrix in the development of the dynamic models. Process simulation tool (Simulink) was utilized in the configuration of closed-loop systems with thermocouple and G. C. Analyzer, at the feed point of the reactor for process variables (temperature and concentration) control during the process using proportional, integral and derivative (PID) as controller parameters. The results of the process behaviour showed that manual tuning of different controller parameters experienced a high level of fluctuations and would not attain its stability or steady state even at a processing time above 16 seconds. Whereas, automatic tuning of desired controller gains of 1.361, 1.056 and 0.4109 for KP, KI and KD for temperature and pressure with a tuning time of 143.9 Seconds, requires a maximum time of 8 seconds for the system to attain stability. This study showed that automated tuning of the controller resulted in better performance characteristics for optimum production of ethylene glycol in a non-isotheral continuous stirred tank reactor within the shortest possible time. [ABSTRACT FROM AUTHOR]

Aiming at the problem that there is a big contradiction between accuracy and calculation and cost based on the RSSI positioning algorithm, an accurate and effective cooperative positioning algorithm is proposed in combination with error correction and refinement measures in each stage of positioning. At the ranging stage, the RSSI measurement value is converted to distance by wireless channel modeling and the dynamic acquisition of the power attenuation factor. Then, the ranging correction is carried out by using the known anchor node ranging error information. The Taylor series expansion least-square iterative refinement algorithm is implemented in the position optimization stage, and satisfactory positioning accuracy is obtained. The idea of cooperative positioning is introduced to upgrade the nodes that meet the requirements and are upgraded to anchor nodes and participate in the positioning of other nodes to improve the positioning coverage and positioning accuracy. The experimental results show that the localization effect of this algorithm is close to that of the Taylor series expansion algorithm based on coordinates but far higher than that of the basic least-squares localization algorithm. The positioning accuracy can be improved rapidly with the decrease in the distance measurement error. [ABSTRACT FROM AUTHOR]

The aggregation equation arises naturally in kinetic theory in the study of granular media, and its interpretation as a 2-Wasserstein gradient flow for the nonlocal interaction energy is well-known. Starting from the spatially homogeneous inelastic Boltzmann equation, a formal Taylor expansion reveals a link between this equation and the aggregation equation with an appropriately chosen interaction potential. Inspired by this formal link and the fact that the associated aggregation equation also dissipates the kinetic energy, we present a novel way of interpreting the aggregation equation as a gradient flow, in the sense of curves of maximal slope, of the kinetic energy, rather than the usual interaction energy, with respect to an appropriately constructed transportation metric on the space of probability measures. [ABSTRACT FROM AUTHOR]

Cosmography, as an integral branch of cosmology, strives to characterize the Universe without relying on pre-determined cosmological models. This model-independent approach utilizes Taylor series expansions around the current epoch, providing a direct correlation with cosmological observations and the potential to constrain theoretical models. Various observable quantities in cosmology can be described as different combinations of cosmographic parameters. Furthermore, one can apply cosmography to models with a varying speed of light. In this case, the Hubble parameter can be expressed by the same combination of cosmographic parameters for both the standard model and varying speed-of-light models. However, for the luminosity distance, the two models are represented by different combinations of cosmographic parameters. Hence, luminosity distance might provide a method to constrain the parameters in varying speed-of-light models. [ABSTRACT FROM AUTHOR]

In this paper, a new application of the Taylor series method (TSM) and its properties is used to solve partial differential equation problems with a singular limit value problem. We are particularly interested in applying this method to a certain class of physical problems such as Lane Emden–Fowler partial differential equations. The results obtained show that this analytical technique is very simple and effective compared to other methods. [ABSTRACT FROM AUTHOR]