Taylor Series Research Articles

Classical correspondence beyond the Ehrenfest time for open quantum systems with general Lindbladians

Quantum and classical systems evolving under the same formal Hamiltonian H may exhibit dramatically different behavior after the Ehrenfest timescale tE∼log(ħ-1), even as ħ→0. Coupling the system to a Markovian environment results in a Lindblad equation for the quantum evolution. Its classical counterpart is given by the Fokker–Planck equation on phase space, which describes Hamiltonian flow with friction and diffusive noise. The quantum and classical evolutions may be compared via the Wigner-Weyl representation. Due to decoherence, they are conjectured to match closely for times far beyond the Ehrenfest timescale as ħ→0. We prove a version of this correspondence, bounding the error between the quantum and classical evolutions for any sufficiently regular Hamiltonian H(x, p) and Lindblad functions Lk(x,p). The error is small when the strength of the diffusion D associated to the Lindblad functions satisfies D≫ħ4/3, in particular allowing vanishing noise in the classical limit. Our method uses a time-dependent semiclassical mixture of variably squeezed Gaussian states. The states evolve according to a local harmonic approximation to the Lindblad dynamics constructed from a second-order Taylor expansion of the Lindbladian. Both the exact quantum trajectory and its classical counterpart can be expressed as perturbations of this semiclassical mixture, with the errors bounded using Duhamel’s principle. We present heuristic arguments suggesting the 4/3 exponent is optimal and defines a boundary in the sense that asymptotically weaker diffusion permits a breakdown of quantum-classical correspondence at the Ehrenfest timescale. Our presentation aims to be comprehensive and accessible to both mathematicians and physicists. In a shorter companion paper, we treat the special case of Hamiltonians that decompose into kinetic and potential energy with linear Lindblad operators, with explicit bounds that can be applied directly to physical systems.

Generalizing the Classical Remainder Theorem: A Reflection-Based Methodological Strategy

The framework of this paper is the presentation of a case study in which university students are required to extend a particular problem of division of polynomials in one variable over the field of real numbers (as generalizing action) clearly influenced by prior strategies (as reflection generalization). Specifically, the objective of this paper is to present a methodology for generalizing the classical Remainder Theorem to the case in which the divisor is a product of binomials (x−a1)n1(x−a2)n2⋯(x−ak)nk, where a1,a2,⋯,ak∈R and n1,n2,⋯,nk∈N. A first approach to this issue is the Taylor expansion of the dividend P(x) at a point a, which clearly shows the quotient and the remainder of the division of P(x) by (x−a)k, where the degree of P(x), say n, must be greater than or equal to k. The methodology used in this paper is the proof by induction which allows to obtain recurrence relations different from those obtained by other scholars dealing with the generalization of the classical Remainder Theorem.

A reduced-order boundary element method for two-dimensional acoustic scattering

This study presents a novel method for wideband acoustic analysis using the Boundary Element Method (BEM), addressing significant computational challenges. Traditional BEM requires repetitive computations across different frequencies due to the frequency-dependent system matrix, resulting in high computational costs. To overcome this, the Hankel function is expanded into a Taylor series, enabling the separation of frequency-dependent and frequency-independent components in the boundary integral equations. This results in a frequency-independent system matrix, improving computational efficiency. Additionally, the method addresses the issue of full-rank, asymmetric coefficient matrices in BEM, which complicate the solution of system equations over wide frequency ranges, particularly for large-scale problems. A Reduced-Order Model (ROM) is developed using the Second-Order Arnoldi (SOAR) method, which retains the key characteristics of the original Full-Order Model (FOM). The singularity elimination technique is employed to directly compute the strong singular and super-singular integrals in the acoustic equations. Numerical examples demonstrate the accuracy and efficiency of the proposed approach, showing its potential for large-scale applications in noise control and acoustic design, where fast and precise analysis is crucial.

Mixed QCD-EW corrections to W-pair production at electron-positron colliders

The discrepancy between the CDF measurement and the Standard Model theoretical prediction for the W-boson mass underscores the importance of conducting high-precision studies on the W boson, which is one of the predominant objectives of proposed future e+e− colliders. We investigate in detail the production of W-boson pairs at e+e− colliders, and compute the next-to-next-to-leading order mixed QCD-EW corrections to both the integrated cross section and various kinematic distributions. By employing the method of differential equations, we analytically calculate the two-loop master integrals for the mixed QCD-EW virtual corrections to e+e− → W+W−. Utilizing the Magnus transformation, we derive a set of canonical master integrals for each integral family. This canonical basis satisfies a system of differential equations in which the dependence on the dimensional regulator is linearly factorized from the kinematics. We then express all these canonical master integrals as Taylor series in ϵ up to ϵ4, with coefficients articulated in terms of Goncharov polylogarithms up to weight four. Upon applying our analytic expressions of these master integrals to the phenomenological analysis of W-pair production, we observe that the O(ααs) corrections are significantly impactful in the α(0) scheme, particularly in certain phase-space regions. However, these mixed QCD-EW corrections can be heavily suppressed by adopting the Gμ scheme.

Optimal coordination of directional overcurrent relays: A fast and precise quadratically constrained quadratic programming solution methodology

AbstractNowadays, power system protection is increasingly important because of the growing number of customers and the pressing need for timely fault resolution and relay operations. This paper addresses the non‐linear nature of the objective function in the optimal coordination of directional overcurrent relays (DOCRs) by employing a quadratic Taylor series expansion around an operating point, converting the problem into a quadratically constrained quadratic programming problem, ensuring a global optimal solution with increased computational efficiency. Additionally, the quadratic constraints are converted into second‐order cone constraints for compatibility with the CPLEX solver. Using the least square method, the operating point values are determined and further fine‐tuned using iterations with the DOCR operation times. The IEEE 3‐bus, IEEE 8‐bus, and IEEE 14‐bus test systems are used to test the method, which shows higher improvement rates in reducing DOCR operation times and enhancing cooperation than conventional and metaheuristic methods. The simulation results verify the numerical superiority of the method in optimizing the protection system's efficiency while obtaining rapid and accurate solutions. The proposed method was tested on IEEE 3‐bus, 8‐bus, and 14‐bus systems, optimizing relay operating times to 0.87, 2.96, and 7.05 s, respectively, demonstrating the method's efficiency over conventional approaches.

Uncertainty and worst-case expectation analysis for multiphysics QPR simulations

Abstract Quadrupole resonators (QPRs) serve to characterize superconducting samples. Like any cavity, during operation, they will deviate from the design geometry for various reasons. Those deviations can be static, stemming from manufacturing variations reflected in the manufacturing tolerances, or dynamic, such as electromagnetic radiation pressure (Lorentz detuning) or microphonics. As a result, a QPR's measurement accuracy and general operation can be severely limited. In particular, during operation, it became evident that the third operating mode of typical QPRs is mainly affected. In this work, by solving the underlying multiphysics problem with random input parameters, we predict the predominant sources of significant measurement bias in surface resistance. On the one hand, we employ the stochastic collocation method compound with the polynomial chaos expansion (PC-SCM) to quantify uncertainties in the physical model governed by a coupled electro-stress-heat (E-S-H) problem. On the other hand, we explore the perturbation analysis to calculate the mean-worst-scenario bound of the merit functions due to the first-order truncation of the Taylor expansion around mean parameter values. The developed method allows us to study the effect of a small nonlinear deformation on the performance of the QPR. Finally, we discuss the simulation results and their implication for the operational conditions of the QPRs.

Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine

Abstract In this article, the authors introduce Qi’s normalized remainder of the Maclaurin series expansion of Qi’s normalized remainder for the cosine function. By virtue of a monotonicity rule for the quotient of two series and with the aid of an increasing monotonicity of a sequence involving the quotient of two consecutive non-zero Bernoulli numbers, they prove the logarithmic convexity of Qi’s normalized remainder. In view of a higher order derivative formula for the quotient of two functions, they expand the logarithm of Qi’s normalized remainder into a Maclaurin series whose coefficients are expressed in terms of determinants of a class of specific Hessenberg matrices. In light of a monotonicity rule for the quotient of two series, they present the monotonicity of the ratio between two normalized remainders. Finally, the authors connect two of their main results with the generalized hypergeometric functions.

Generalized Taylor Series and Peano Kernel Theorem

ABSTRACTAs in the polynomial case, non‐polynomial divided differences can be viewed as a discrete analog of derivatives. This link between non‐polynomial divided differences and derivatives is defined by a generalization of the derivative operator. In this study, we obtain a generalization of Taylor series using the link between non‐polynomial divided differences and derivatives, and state generalized Taylor theorem. With the definition of a definite integral, the relation between the non‐polynomial divided difference and non‐polynomial B‐spline functions is given in terms of integration. Also, we derive a general form of the Peano kernel theorem based on a generalized Taylor expansion with the integral remainder. As in the polynomial case, it is shown that the non‐polynomial B‐splines are in fact the Peano kernels of non‐polynomial divided differences.MSC2020 Classification: 65D05, 65D07

Thermal vibration analysis of FGM beams using Carrera unified formulation

Through the extension of Carrera unified formulation (CUF), the vibration equation of functionally graded material (FGM) beams in different temperature distributions is established, and the free vibration characteristics are analyzed. The material properties of FGM beams are defined by four-parameter power-law distribution and Voigt mixture law. The different temperature distributions are considered as functions of thickness. Utilizing CUF, the displacement function is constructed by Taylor polynomials and Fourier series. The vibration equation is solved by state-space method. The convergence and accuracy of the proposed method are verified by comparing with the simulation results of finite element software. The variation of material properties under different temperature distributions, as well as the influence of different factors on the vibration characteristics of FGM beams are studied.

Nonlinear Optimal and Multi-Loop Flatness-Basedcontrol of Omnidirectional 3-Wheel Mobile Robots

In this article, the control problem for omnidirectional 3-wheel autonomous mobile robots is solved with the use of (i) a nonlinear optimal control method (ii) a flatness-based control approach which is implemented in successive loops. To apply method (i) that is nonlinear optimal control, the dynamic model of the omnidirectional 3-wheel autonomous mobile robots undergoes approximate linearization at each sampling instant with the use of first-order Taylor series expansion and through the computation of the associated Jacobian matrix. The linearization point is defined by the present value of the system's state vector and by the last sampled value of the control inputs vector. To compute the feedback gains of the optimal controller an algebraic Riccati equation is repetitively solved at each time-step of the control algorithm. The global stability properties of the nonlinear optimal control method are proven through Lyapunov analysis. To implement control method (ii), that is flatness-based control in successive loops, the state-space model of the omnidirectional 3-wheel autonomous mobile robot is separated into chained subsystems, which are connected in cascading loops. Each one of these subsystems can be viewed independently as a differentially flat system and control about it can be performed with inversion of its dynamics as in the case of input-output linearized flat systems. The state variables of the preceding (i-th) subsystem become virtual control inputs for the subsequent (i+1-th) subsystem. In turn, exogenous control inputs are applied to the last subsystem. The whole control method is implemented in successive loops and its global stability properties are also proven through Lyapunov stability analysis. The proposed method achieves trajectory tracking and autonomous navigation for the omnidirectional 3-wheel autonomous mobile robots without the need of diffeomorphisms and complicated state-space model transformations.

HOPE: High-Order Polynomial Expansion of Black-Box Neural Networks.

Despite their remarkable performance, deep neural networks remain mostly "black boxes", suggesting inexplicability and hindering their wide applications in fields requiring making rational decisions. Here we introduce HOPE (High-order Polynomial Expansion), a method for expanding a network into a high-order Taylor polynomial on a reference input. Specifically, we derive the high-order derivative rule for composite functions and extend the rule to neural networks to obtain their high-order derivatives quickly and accurately. From these derivatives, we can then derive the Taylor polynomial of the neural network, which provides an explicit expression of the network's local interpretations. We combine the Taylor polynomials obtained under different reference inputs to obtain the global interpretation of the neural network. Numerical analysis confirms the high accuracy, low computational complexity, and good convergence of the proposed method. Moreover, we demonstrate HOPE's wide applications built on deep learning, including function discovery, fast inference, and feature selection. We compared HOPE with other XAI methods and demonstrated our advantages.

Constrain the Jerk Parameters with DESI 2024 Data

Abstract The deceleration coefficient q and the jerk coefficient j obtained by the Taylor expansion of the scale factor a(t) play an important role in the study of cosmology. The current value of these coefficients for a cosmological model reflects the transition time between the phases dominated by dark energy and matter and can be used to determine if and how much the universe is decelerating. Thus, these coefficient values offer a way of constraining a particular cosmology model. Research based on this scenario was completed by Orlando Luongo and Marco Muccino. However, some approaches in this method should be tested prudently because some conditions such as dd L dz > 0 and dH dz > 0 may not be guaranteed. In this study, we used the MAPAge model to reconstruct the jerk parameters (q 0 and j 0) with DESI 2024 data. Using the MAPAge model ensures particular physical circumstances are satisfied in the approach of determining the jerk parameters. Compared to the previous method, which used the Taylor expansion series q 0, j 0, and s 0 as model-independent parameters, we obtained more physical and slightly different results for the jerk parameters. Our results suggest that the DESI 2024 BAO data set favours different jerk parameters compared to the jerk parameters in the standard ΛCDM model.

A High-Order Shifted Interface Method for Lagrangian Shock Hydrodynamics

We present a new method for two-material Lagrangian hydrodynamics, which combines the Shifted Interface Method (SIM) with a high-order Finite Element Method. Our approach relies on an exact (or sharp) material interface representation, that is, it uses the precise location of the material interface. The interface is represented by the zero level-set of a continuous high-order finite element function that moves with the material velocity. This strategy allows to evolve curved material interfaces inside curved elements. By reformulating the original interface problem over a surrogate (approximate) interface, located in proximity of the true interface, the SIM avoids cut cells and the associated problematic issues regarding implementation, numerical stability, and matrix conditioning. Accuracy is maintained by modifying the original interface conditions using Taylor expansions. We demonstrate the performance of the proposed algorithms on established numerical benchmarks in one, two and three dimensions.

On the Growth Orders and Types of Biregular Functions

One of the main aims of Clifford analysis is to study the growth properties of regular functions. Biregular functions are a well-known generalization of regular functions. In this paper, the growth orders and types of biregular functions are studied. First, generalized growth orders and types of biregular functions are defined in the context of Clifford analysis. Then, using the methods of Wiman and Valiron, generalized Lindelöf–Pringsheim theorems are proved, which show the relationship between growth orders, growth types, and Taylor series. These connections allow us to calculate the growth order and determine the type of biregular functions.

More accurate representation of interaction at the fluid–structure interface with an improved smoothed field gradient method

Flexible structures are wind-sensitive with a significant fluid–structure interaction (FSI). The FSI analysis, however, often has poor numerical stability and low convergence efficiency due to drastic changes of the physical fields induced by computation errors in local regions of the fluid–structure interface. This paper aims at addressing these problems with the proposal of a new method to smooth the gradient of the pressure field at the fluid–structure interface for an efficient convergence in the FSI analysis. The smoothed gradient theory is modified by introducing weight coefficients. The field of fluid pressure in each smoothing domain with large numerical fluctuations at the interface is then gradient smoothed with the proposed method and the modified field is obtained from the linear Taylor series expansion. The convergence of fluid and structure solvers for the proposed method is ensured within the commercial software FLUENT and ANSYS adopted. The proposed method is validated with experimental results from the literature. It is also numerically validated with a thin plate in viscous flow with different site categories and average wind velocities through comparison of results from conventional methods. The proposed method is found valid and accurate in the FSI analysis. It is relatively independent of a wide range of parameters with satisfactory robustness and notable improvement in the convergence of the FSI analysis.

A gradient-enhanced univariate dimension reduction method for uncertainty propagation

The univariate dimension reduction (UDR) method stands as a way to estimate the statistical moments of the output that is effective in a large class of uncertainty quantification (UQ) problems. UDR's fundamental strategy is to approximate the original function using univariate functions so that the UQ cost only scales linearly with the dimension of the problem. Nonetheless, UDR's effectiveness can diminish when uncertain inputs have high variance, particularly when assessing the output's second and higher-order statistical moments. This paper proposes a new method, gradient-enhanced univariate dimension reduction (GUDR), that enhances the accuracy of UDR by incorporating univariate gradient function terms into the UDR approximation function. Theoretical results indicate that the GUDR approximation is expected to be one order more accurate than UDR in approximating the original function, and it is expected to generate more accurate results in computing the output's second and higher-order statistical moments. Our proposed method uses a computational graph transformation strategy to efficiently evaluate the GUDR approximation function on tensor-grid quadrature inputs, and use the tensor-grid input-output data to compute the statistical moments of the output. With an efficient automatic differentiation method to compute the gradients, our method preserves UDR's linear scaling of computation time with problem dimension. Numerical results show that the GUDR is more accurate than UDR in estimating the standard deviation of the output and has a performance comparable to the method of moments using a third-order Taylor series expansion.

Sharp Second-Order Hankel Determinants Bounds for Alpha-Convex Functions Connected with Modified Sigmoid Functions

The study of the Hankel determinant generated by the Maclaurin series of holomorphic functions belonging to particular classes of normalized univalent functions is one of the most significant problems in geometric function theory. Our goal in this study is first to define a family of alpha-convex functions associated with modified sigmoid functions and then to investigate sharp bounds of initial coefficients, Fekete-Szegö inequality, and second-order Hankel determinants. Moreover, we also examine the logarithmic and inverse coefficients of functions within a defined family regarding recent issues. All of the estimations that were found are sharp.

An analytical model for estimating aerodynamic damping of wind turbines in the shutdown state

As wind farms are constantly being constructed, the risk of tower failure for wind turbines increases significantly under strong winds. Compared with the extensively concerned wind-induced behaviors during the operating state, those ones during the shutdown state attract little attention but may lead to serious problems of damages or even collapse. To clearly grasp the aero-structure interaction in the shutdown state, this paper develops an analytical model for estimating aerodynamic damping of wind turbines. In this method, an analytical expression of aerodynamic damping coupling matrix is derived via the combination of multibody dynamics and first-order Taylor expansion. This matrix is further quantified as the ratio of modal aerodynamic damping with the aid of state-space equation and complex eigenvalue analysis. This treatment can facilitate the straightforward application of efficient calculation methods, such as frequency domain analysis and uncoupled analysis. More importantly, the developed model is able to simultaneously consider multiple realistic factors, such as blade flexibility, tower top rotation, yaw error, wind shear, and pitch angle. This model may have the high calculation efficiency and accuracy, as well as strong applicability for estimating the aerodynamic damping. Numerical examples based on a typical 5 MW wind turbine are employed to validate the effectiveness of the developed model. Experimental analyses demonstrate that this model outperforms the existing formula and presents a high consistency with OpenFAST in the estimation of aerodynamic damping. Meanwhile, the influence of multiple realistic factors is quantitatively analyzed, which even makes the estimation error exceed 70%.

Temporal asynchrony analysis for dynamic operation of hydraulic-thermal-electricity multiple energy networks based on holomorphic embedding method

Analyzing the operational states of multiple energy networks (MEN) in multi-energy systems is crucial for ensuring system stability. The dynamic operational characteristics of different energy flows pose challenges for computational analysis. Traditional steady-state methods are inadequate for addressing the dynamics of MEN, especially when dealing with temporal discrepancies between hydraulic and thermal flows in thermal networks (TN) and the heterogeneity between TN and electrical networks. Therefore, this paper proposes a novel holomorphic embedding method (HEM) based on multi-stage decomposition method. The developed HEM constructs a time coefficient matrix and utilize inner-outer loop recursion to handle the time lag between thermal flow and hydraulic flow in the TN. Additionally, we reconstruct a holomorphic matrix, integrating hydraulic flow to bridge thermal and electric power flows, thereby improving the operational heterogeneity among different networks. Real-case simulations show that when the Taylor expansion order in HEM is equal to 4, the proposed method achieves a mere 1% discrepancy from actual operational data, enhancing computational efficiency by 60% compared to the Newton-Raphson method. Moreover, in this real-case, the TN exhibits a maximum delay response time of 180s compared to electrical networks. Exploiting this delay time effectively increases renewable energy generation within multi-energy systems by 961.58kW per day.

A high-order no image point sharp interface immersed boundary method for compressible flows

A high-order no-image point sharp interface immersed boundary method for compressible flow is presented. The method comprises a stable high-order compact scheme and a ghost point value determination method. By regulating dissipation, the stability of the compact scheme for either Dirichlet or Neumann boundary conditions is validated by the von Neumann method in one dimension. With regard to the use of ghost points, mirror points or Lagrange points are no longer employed. The boundary conditions at the intersection of arbitrary geometries and Cartesian grids are imposed on the basis function of Taylor polynomial interpolation, along with weighted least squares error minimization, in order to determine the values of the ghost points. Third-order accuracy is maintained for both subsonic and supersonic inviscid flow. Numerical simulations of several two-dimensional benchmark problems are carried out to provide evidence about the convergence order of the method.